LMFDB - Embedded newform 3456.2.i.l.2305.5

Newspace parameters

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.i (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
Defining polynomial:	$ x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} - 216 x^{3} + 243 x^{2} - 486 x + 729 $ x^12 - 2x^11 + 3x^10 - 8x^9 + 22x^8 - 42x^7 + 51x^6 - 126x^5 + 198x^4 - 216x^3 + 243x^2 - 486*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2305.5
Root			$0.952418 - 1.44669i$ of defining polynomial
Character	$\chi$	$=$	3456.2305
Dual form			3456.2.i.l.1153.5

$q$-expansion

$f(q)$	$=$	$q+(1.24278 - 2.15256i) q^{5} +(0.909142 + 1.57468i) q^{7} +O(q^{10})$	$q+(1.24278 - 2.15256i) q^{5} +(0.909142 + 1.57468i) q^{7} +(-0.598407 - 1.03647i) q^{11} +(-2.83342 + 4.90762i) q^{13} +5.30021 q^{17} -4.55980 q^{19} +(2.01328 - 3.48711i) q^{23} +(-0.589008 - 1.02019i) q^{25} +(3.01513 + 5.22236i) q^{29} +(2.81647 - 4.87827i) q^{31} +4.51946 q^{35} +5.18127 q^{37} +(-4.57620 + 7.92621i) q^{41} +(3.99129 + 6.91313i) q^{43} +(-1.39470 - 2.41570i) q^{47} +(1.84692 - 3.19896i) q^{49} +1.54470 q^{53} -2.97475 q^{55} +(1.85725 - 3.21686i) q^{59} +(4.01513 + 6.95441i) q^{61} +(7.04263 + 12.1982i) q^{65} +(-6.91372 + 11.9749i) q^{67} +11.1794 q^{71} +12.3969 q^{73} +(1.08807 - 1.88460i) q^{77} +(4.36480 + 7.56006i) q^{79} +(-8.89267 - 15.4025i) q^{83} +(6.58700 - 11.4090i) q^{85} +0.455297 q^{89} -10.3039 q^{91} +(-5.66683 + 9.81524i) q^{95} +(-1.01640 - 1.76045i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{5} + 6 q^{7}+O(q^{10}) $ 12 * q + 2 * q^5 + 6 * q^7	$ 12 q + 2 q^{5} + 6 q^{7} - 4 q^{11} + 10 q^{13} - 4 q^{17} + 4 q^{19} - 8 q^{23} - 14 q^{25} + 2 q^{29} + 8 q^{31} - 8 q^{35} + 2 q^{41} - 2 q^{43} + 14 q^{47} - 18 q^{49} - 24 q^{53} - 16 q^{55} - 6 q^{59} + 14 q^{61} + 8 q^{65} + 4 q^{67} + 28 q^{71} + 60 q^{73} - 2 q^{77} + 16 q^{79} - 24 q^{83} + 16 q^{85} + 48 q^{89} - 52 q^{91} + 20 q^{95} - 14 q^{97}+O(q^{100}) $ 12 * q + 2 * q^5 + 6 * q^7 - 4 * q^11 + 10 * q^13 - 4 * q^17 + 4 * q^19 - 8 * q^23 - 14 * q^25 + 2 * q^29 + 8 * q^31 - 8 * q^35 + 2 * q^41 - 2 * q^43 + 14 * q^47 - 18 * q^49 - 24 * q^53 - 16 * q^55 - 6 * q^59 + 14 * q^61 + 8 * q^65 + 4 * q^67 + 28 * q^71 + 60 * q^73 - 2 * q^77 + 16 * q^79 - 24 * q^83 + 16 * q^85 + 48 * q^89 - 52 * q^91 + 20 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	1.24278	−	2.15256i	0.555788	−	0.962654i	−0.442053	−	0.896989i	$-0.645750\pi$
$5$	1.24278	−	2.15256i	0.555788	−	0.962654i	0.997842	−	0.0656650i	$-0.0209169\pi$
$6$		0			0
$7$	0.909142	+	1.57468i	0.343623	+	0.595173i	0.985103	−	0.171967i	$-0.0550123\pi$
$7$	0.909142	+	1.57468i	0.343623	+	0.595173i	−0.641479	+	0.767140i	$0.721679\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.598407	−	1.03647i	−0.180426	−	0.312508i	0.761599	−	0.648048i	$-0.224414\pi$
$11$	−0.598407	−	1.03647i	−0.180426	−	0.312508i	−0.942026	+	0.335540i	$0.891081\pi$
$12$		0			0
$13$	−2.83342	+	4.90762i	−0.785848	+	1.36113i	0.142643	+	0.989774i	$0.454440\pi$
$13$	−2.83342	+	4.90762i	−0.785848	+	1.36113i	−0.928491	+	0.371355i	$0.878893\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.30021			1.28549			0.642745	−	0.766080i	$-0.277795\pi$
$17$	5.30021			1.28549			0.642745	+	0.766080i	$0.277795\pi$
$18$		0			0
$19$	−4.55980			−1.04609			−0.523045	−	0.852305i	$-0.675204\pi$
$19$	−4.55980			−1.04609			−0.523045	+	0.852305i	$0.675204\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.01328	−	3.48711i	0.419798	−	0.727112i	−0.576120	−	0.817365i	$-0.695434\pi$
$23$	2.01328	−	3.48711i	0.419798	−	0.727112i	0.995919	+	0.0902526i	$0.0287674\pi$
$24$		0			0
$25$	−0.589008	−	1.02019i	−0.117802	−	0.204038i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.01513	+	5.22236i	0.559896	+	0.969768i	0.997505	+	0.0706027i	$0.0224923\pi$
$29$	3.01513	+	5.22236i	0.559896	+	0.969768i	−0.437609	+	0.899166i	$0.644174\pi$
$30$		0			0
$31$	2.81647	−	4.87827i	0.505853	−	0.876163i	−0.494124	−	0.869391i	$-0.664511\pi$
$31$	2.81647	−	4.87827i	0.505853	−	0.876163i	0.999977	−	0.00677135i	$-0.00215540\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.51946			0.763928
$36$		0			0
$37$	5.18127			0.851796			0.425898	−	0.904771i	$-0.359958\pi$
$37$	5.18127			0.851796			0.425898	+	0.904771i	$0.359958\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.57620	+	7.92621i	−0.714682	+	1.23787i	0.248400	+	0.968658i	$0.420095\pi$
$41$	−4.57620	+	7.92621i	−0.714682	+	1.23787i	−0.963082	+	0.269208i	$0.913238\pi$
$42$		0			0
$43$	3.99129	+	6.91313i	0.608667	+	1.05424i	0.991460	+	0.130408i	$0.0416287\pi$
$43$	3.99129	+	6.91313i	0.608667	+	1.05424i	−0.382794	+	0.923834i	$0.625038\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.39470	−	2.41570i	−0.203438	−	0.352366i	0.746196	−	0.665727i	$-0.231878\pi$
$47$	−1.39470	−	2.41570i	−0.203438	−	0.352366i	−0.949634	+	0.313361i	$0.898545\pi$
$48$		0			0
$49$	1.84692	−	3.19896i	0.263846	−	0.456994i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.54470			0.212181			0.106091	−	0.994356i	$-0.466167\pi$
$53$	1.54470			0.212181			0.106091	+	0.994356i	$0.466167\pi$
$54$		0			0
$55$	−2.97475			−0.401116
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.85725	−	3.21686i	0.241794	−	0.418799i	−0.719431	−	0.694563i	$-0.755598\pi$
$59$	1.85725	−	3.21686i	0.241794	−	0.418799i	0.961225	+	0.275764i	$0.0889309\pi$
$60$		0			0
$61$	4.01513	+	6.95441i	0.514085	+	0.890421i	0.999866	+	0.0163411i	$0.00520175\pi$
$61$	4.01513	+	6.95441i	0.514085	+	0.890421i	−0.485781	+	0.874080i	$0.661465\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.04263	+	12.1982i	0.873531	+	1.51300i
$66$		0			0
$67$	−6.91372	+	11.9749i	−0.844645	+	1.46297i	0.0412836	+	0.999147i	$0.486855\pi$
$67$	−6.91372	+	11.9749i	−0.844645	+	1.46297i	−0.885929	+	0.463821i	$0.846478\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.1794			1.32675			0.663376	−	0.748287i	$-0.269123\pi$
$71$	11.1794			1.32675			0.663376	+	0.748287i	$0.269123\pi$
$72$		0			0
$73$	12.3969			1.45095			0.725473	−	0.688251i	$-0.241621\pi$
$73$	12.3969			1.45095			0.725473	+	0.688251i	$0.241621\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.08807	−	1.88460i	0.123998	−	0.214770i
$78$		0			0
$79$	4.36480	+	7.56006i	0.491079	+	0.850573i	0.999947	−	0.0102710i	$-0.00326940\pi$
$79$	4.36480	+	7.56006i	0.491079	+	0.850573i	−0.508869	+	0.860844i	$0.669936\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.89267	−	15.4025i	−0.976097	−	1.69065i	−0.676267	−	0.736657i	$-0.736403\pi$
$83$	−8.89267	−	15.4025i	−0.976097	−	1.69065i	−0.299830	−	0.953993i	$-0.596930\pi$
$84$		0			0
$85$	6.58700	−	11.4090i	0.714461	−	1.23748i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.455297			0.0482614			0.0241307	−	0.999709i	$-0.492318\pi$
$89$	0.455297			0.0482614			0.0241307	+	0.999709i	$0.492318\pi$
$90$		0			0
$91$	−10.3039			−1.08014
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.66683	+	9.81524i	−0.581405	+	1.00702i
$96$		0			0
$97$	−1.01640	−	1.76045i	−0.103199	−	0.178747i	0.809802	−	0.586704i	$-0.199575\pi$
$97$	−1.01640	−	1.76045i	−0.103199	−	0.178747i	−0.913001	+	0.407957i	$0.866241\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.06107	+	5.30192i	0.304587	+	0.527561i	0.977169	−	0.212462i	$-0.0681481\pi$
$101$	3.06107	+	5.30192i	0.304587	+	0.527561i	−0.672582	+	0.740023i	$0.734815\pi$
$102$		0			0
$103$	3.09086	−	5.35352i	0.304551	−	0.527498i	−0.672610	−	0.739997i	$-0.734827\pi$
$103$	3.09086	−	5.35352i	0.304551	−	0.527498i	0.977161	+	0.212499i	$0.0681602\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.2193			−1.56798			−0.783990	−	0.620774i	$-0.786818\pi$
$107$	−16.2193			−1.56798			−0.783990	+	0.620774i	$0.786818\pi$
$108$		0			0
$109$	2.08460			0.199669			0.0998344	−	0.995004i	$-0.468169\pi$
$109$	2.08460			0.199669			0.0998344	+	0.995004i	$0.468169\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.45933	+	2.52763i	−0.137282	+	0.237779i	−0.926467	−	0.376376i	$-0.877170\pi$
$113$	−1.45933	+	2.52763i	−0.137282	+	0.237779i	0.789185	+	0.614156i	$0.210503\pi$
$114$		0			0
$115$	−5.00414	−	8.66742i	−0.466638	−	0.808241i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.81865	+	8.34614i	0.441725	+	0.765090i
$120$		0			0
$121$	4.78382	−	8.28582i	0.434893	−	0.753256i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.49978			0.849686
$126$		0			0
$127$	15.0618			1.33652			0.668261	−	0.743926i	$-0.267039\pi$
$127$	15.0618			1.33652			0.668261	+	0.743926i	$0.267039\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.89311	+	3.27896i	−0.165402	+	0.286484i	−0.936798	−	0.349871i	$-0.886225\pi$
$131$	−1.89311	+	3.27896i	−0.165402	+	0.286484i	0.771396	+	0.636355i	$0.219559\pi$
$132$		0			0
$133$	−4.14551	−	7.18023i	−0.359461	−	0.622605i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.39448	−	11.0756i	−0.546317	−	0.946250i	−0.998523	−	0.0543357i	$-0.982696\pi$
$137$	−6.39448	−	11.0756i	−0.546317	−	0.946250i	0.452205	−	0.891914i	$-0.350637\pi$
$138$		0			0
$139$	4.46539	−	7.73428i	0.378749	−	0.656013i	−0.612131	−	0.790756i	$-0.709688\pi$
$139$	4.46539	−	7.73428i	0.378749	−	0.656013i	0.990881	+	0.134743i	$0.0430209\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.78214			0.567151
$144$		0			0
$145$	14.9886			1.24473
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.65170	−	8.05698i	0.381082	−	0.660054i	−0.610135	−	0.792297i	$-0.708885\pi$
$149$	4.65170	−	8.05698i	0.381082	−	0.660054i	0.991217	+	0.132244i	$0.0422182\pi$
$150$		0			0
$151$	7.83527	+	13.5711i	0.637625	+	1.10440i	0.985953	+	0.167026i	$0.0534164\pi$
$151$	7.83527	+	13.5711i	0.637625	+	1.10440i	−0.348328	+	0.937373i	$0.613250\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.00051	−	12.1252i	−0.562294	−	0.973922i
$156$		0			0
$157$	9.75491	−	16.8960i	0.778526	−	1.34845i	−0.154265	−	0.988030i	$-0.549301\pi$
$157$	9.75491	−	16.8960i	0.778526	−	1.34845i	0.932791	−	0.360418i	$-0.117366\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.32144			0.577010
$162$		0			0
$163$	−5.02888			−0.393892			−0.196946	−	0.980414i	$-0.563102\pi$
$163$	−5.02888			−0.393892			−0.196946	+	0.980414i	$0.563102\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.65066	+	13.2513i	−0.592026	+	1.02542i	0.401933	+	0.915669i	$0.368338\pi$
$167$	−7.65066	+	13.2513i	−0.592026	+	1.02542i	−0.993959	+	0.109750i	$0.964995\pi$
$168$		0			0
$169$	−9.55650	−	16.5523i	−0.735115	−	1.27326i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.30192	+	9.18320i	0.403098	+	0.698186i	0.994098	−	0.108486i	$-0.0346001\pi$
$173$	5.30192	+	9.18320i	0.403098	+	0.698186i	−0.591000	+	0.806671i	$0.701267\pi$
$174$		0			0
$175$	1.07098	−	1.85500i	0.0809588	−	0.140225i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.27314			0.693107			0.346553	−	0.938030i	$-0.387352\pi$
$179$	9.27314			0.693107			0.346553	+	0.938030i	$0.387352\pi$
$180$		0			0
$181$	−2.32975			−0.173169			−0.0865845	−	0.996245i	$-0.527595\pi$
$181$	−2.32975			−0.173169			−0.0865845	+	0.996245i	$0.527595\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.43918	−	11.1530i	0.473418	−	0.819985i
$186$		0			0
$187$	−3.17169	−	5.49352i	−0.231937	−	0.401726i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.54208	+	14.7953i	0.618083	+	1.07055i	0.989835	+	0.142219i	$0.0454238\pi$
$191$	8.54208	+	14.7953i	0.618083	+	1.07055i	−0.371752	+	0.928332i	$0.621243\pi$
$192$		0			0
$193$	−12.1360	+	21.0202i	−0.873568	+	1.51306i	−0.0152882	+	0.999883i	$0.504867\pi$
$193$	−12.1360	+	21.0202i	−0.873568	+	1.51306i	−0.858280	+	0.513182i	$0.828467\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.6971			−1.11837			−0.559186	−	0.829042i	$-0.688886\pi$
$197$	−15.6971			−1.11837			−0.559186	+	0.829042i	$0.688886\pi$
$198$		0			0
$199$	−14.4764			−1.02620			−0.513101	−	0.858328i	$-0.671503\pi$
$199$	−14.4764			−1.02620			−0.513101	+	0.858328i	$0.671503\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.48237	+	9.49574i	−0.384787	+	0.666470i
$204$		0			0
$205$	11.3744	+	19.7011i	0.794424	+	1.37598i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.72862	+	4.72610i	0.188742	+	0.326911i
$210$		0			0
$211$	3.21103	−	5.56167i	0.221056	−	0.382881i	−0.734073	−	0.679071i	$-0.762383\pi$
$211$	3.21103	−	5.56167i	0.221056	−	0.382881i	0.955129	+	0.296190i	$0.0957162\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.8412			1.35316
$216$		0			0
$217$	10.2423			0.695291
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−15.0177	+	26.0114i	−1.01020	+	1.74972i
$222$		0			0
$223$	−7.90683	−	13.6950i	−0.529481	−	0.917087i	−0.999409	−	0.0343825i	$-0.989054\pi$
$223$	−7.90683	−	13.6950i	−0.529481	−	0.917087i	0.469928	−	0.882705i	$-0.344280\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.00928	+	5.21223i	0.199733	+	0.345948i	0.948442	−	0.316951i	$-0.102659\pi$
$227$	3.00928	+	5.21223i	0.199733	+	0.345948i	−0.748709	+	0.662899i	$0.769326\pi$
$228$		0			0
$229$	13.8177	−	23.9329i	0.913098	−	1.58153i	0.103436	−	0.994636i	$-0.467016\pi$
$229$	13.8177	−	23.9329i	0.913098	−	1.58153i	0.809662	−	0.586896i	$-0.199650\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.1858			0.798321			0.399161	−	0.916881i	$-0.369302\pi$
$233$	12.1858			0.798321			0.399161	+	0.916881i	$0.369302\pi$
$234$		0			0
$235$	−6.93324			−0.452275
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.82794	−	13.5584i	0.506347	−	0.877019i	−0.493626	−	0.869674i	$-0.664329\pi$
$239$	7.82794	−	13.5584i	0.506347	−	0.877019i	0.999973	−	0.00734451i	$-0.00233785\pi$
$240$		0			0
$241$	9.39281	+	16.2688i	0.605044	+	1.04797i	0.992045	+	0.125887i	$0.0401778\pi$
$241$	9.39281	+	16.2688i	0.605044	+	1.04797i	−0.387001	+	0.922079i	$0.626489\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.59064	−	7.95121i	−0.293285	−	0.507984i
$246$		0			0
$247$	12.9198	−	22.3778i	0.822068	−	1.42386i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.1684			−0.704942			−0.352471	−	0.935823i	$-0.614658\pi$
$251$	−11.1684			−0.704942			−0.352471	+	0.935823i	$0.614658\pi$
$252$		0			0
$253$	−4.81905			−0.302971
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.19941	+	5.54154i	−0.199574	+	0.345672i	−0.948390	−	0.317106i	$-0.897289\pi$
$257$	−3.19941	+	5.54154i	−0.199574	+	0.345672i	0.748817	+	0.662777i	$0.230622\pi$
$258$		0			0
$259$	4.71051	+	8.15885i	0.292697	+	0.506966i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.71632	−	11.6330i	−0.414146	−	0.717322i	0.581192	−	0.813766i	$-0.302586\pi$
$263$	−6.71632	−	11.6330i	−0.414146	−	0.717322i	−0.995338	+	0.0964440i	$0.969253\pi$
$264$		0			0
$265$	1.91973	−	3.32507i	0.117928	−	0.204257i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	26.6295			1.62363			0.811814	−	0.583916i	$-0.198480\pi$
$269$	26.6295			1.62363			0.811814	+	0.583916i	$0.198480\pi$
$270$		0			0
$271$	14.9630			0.908936			0.454468	−	0.890763i	$-0.349829\pi$
$271$	14.9630			0.908936			0.454468	+	0.890763i	$0.349829\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.704933	+	1.22098i	−0.0425091	+	0.0736278i
$276$		0			0
$277$	8.36861	+	14.4949i	0.502821	+	0.870912i	0.999995	+	0.00326057i	$0.00103787\pi$
$277$	8.36861	+	14.4949i	0.502821	+	0.870912i	−0.497174	+	0.867651i	$0.665629\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.80547	+	4.85922i	0.167361	+	0.289877i	0.937491	−	0.348009i	$-0.113142\pi$
$281$	2.80547	+	4.85922i	0.167361	+	0.289877i	−0.770131	+	0.637886i	$0.779809\pi$
$282$		0			0
$283$	6.41074	−	11.1037i	0.381079	−	0.660048i	−0.610138	−	0.792295i	$-0.708886\pi$
$283$	6.41074	−	11.1037i	0.381079	−	0.660048i	0.991217	+	0.132247i	$0.0422194\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.6417			−0.982326
$288$		0			0
$289$	11.0923			0.652487
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.24047	+	12.5409i	−0.422993	+	0.732645i	−0.996231	−	0.0867441i	$-0.972354\pi$
$293$	−7.24047	+	12.5409i	−0.422993	+	0.732645i	0.573238	+	0.819389i	$0.305687\pi$
$294$		0			0
$295$	−4.61632	−	7.99570i	−0.268772	−	0.465528i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.4089	+	19.7609i	0.659796	+	1.14280i
$300$		0			0
$301$	−7.25731	+	12.5700i	−0.418304	+	0.724524i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	19.9597			1.14289
$306$		0			0
$307$	−19.4320			−1.10905			−0.554523	−	0.832169i	$-0.687099\pi$
$307$	−19.4320			−1.10905			−0.554523	+	0.832169i	$0.687099\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.47547	+	6.01968i	−0.197076	+	0.341345i	−0.947579	−	0.319522i	$-0.896478\pi$
$311$	−3.47547	+	6.01968i	−0.197076	+	0.341345i	0.750503	+	0.660867i	$0.229811\pi$
$312$		0			0
$313$	−2.19252	−	3.79756i	−0.123929	−	0.214651i	0.797385	−	0.603471i	$-0.206216\pi$
$313$	−2.19252	−	3.79756i	−0.123929	−	0.214651i	−0.921314	+	0.388820i	$0.872883\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.6742	+	27.1485i	0.880350	+	1.52481i	0.850952	+	0.525244i	$0.176026\pi$
$317$	15.6742	+	27.1485i	0.880350	+	1.52481i	0.0293983	+	0.999568i	$0.490641\pi$
$318$		0			0
$319$	3.60855	−	6.25020i	0.202040	−	0.349944i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.1679			−1.34474
$324$		0			0
$325$	6.67562			0.370297
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.53597	−	4.39243i	0.139812	−	0.242162i
$330$		0			0
$331$	11.1515	+	19.3150i	0.612943	+	1.06165i	0.990742	+	0.135760i	$0.0433477\pi$
$331$	11.1515	+	19.3150i	0.612943	+	1.06165i	−0.377799	+	0.925888i	$0.623319\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	17.1845	+	29.7644i	0.938888	+	1.62620i
$336$		0			0
$337$	−10.4077	+	18.0266i	−0.566943	+	0.981974i	0.429923	+	0.902866i	$0.358541\pi$
$337$	−10.4077	+	18.0266i	−0.566943	+	0.981974i	−0.996866	+	0.0791086i	$0.974793\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.74158			−0.365077
$342$		0			0
$343$	19.4444			1.04990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.4912	−	25.0995i	0.777929	−	1.34741i	−0.155204	−	0.987882i	$-0.549604\pi$
$347$	14.4912	−	25.0995i	0.777929	−	1.34741i	0.933133	−	0.359530i	$-0.117063\pi$
$348$		0			0
$349$	−7.60709	−	13.1759i	−0.407198	−	0.705288i	0.587377	−	0.809314i	$-0.300161\pi$
$349$	−7.60709	−	13.1759i	−0.407198	−	0.705288i	−0.994575	+	0.104026i	$0.966827\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.78818	+	11.7575i	0.361298	+	0.625787i	0.988175	−	0.153332i	$-0.0490003\pi$
$353$	6.78818	+	11.7575i	0.361298	+	0.625787i	−0.626877	+	0.779119i	$0.715667\pi$
$354$		0			0
$355$	13.8935	−	24.0643i	0.737393	−	1.27720i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−31.6623			−1.67107			−0.835536	−	0.549436i	$-0.814842\pi$
$359$	−31.6623			−1.67107			−0.835536	+	0.549436i	$0.814842\pi$
$360$		0			0
$361$	1.79179			0.0943046
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.4066	−	26.6850i	0.806419	−	1.39676i
$366$		0			0
$367$	4.90625	+	8.49788i	0.256104	+	0.443585i	0.965195	−	0.261532i	$-0.0842276\pi$
$367$	4.90625	+	8.49788i	0.256104	+	0.443585i	−0.709091	+	0.705117i	$0.750894\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.40435	+	2.43241i	0.0729105	+	0.126285i
$372$		0			0
$373$	4.98487	−	8.63404i	0.258107	−	0.447054i	−0.707628	−	0.706585i	$-0.750235\pi$
$373$	4.98487	−	8.63404i	0.258107	−	0.447054i	0.965735	+	0.259531i	$0.0835681\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−34.1725			−1.75997
$378$		0			0
$379$	−30.2351			−1.55307			−0.776537	−	0.630072i	$-0.783026\pi$
$379$	−30.2351			−1.55307			−0.776537	+	0.630072i	$0.783026\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.86835	−	17.0925i	0.504249	−	0.873386i	−0.495739	−	0.868472i	$-0.665103\pi$
$383$	9.86835	−	17.0925i	0.504249	−	0.873386i	0.999988	−	0.00491371i	$-0.00156409\pi$
$384$		0			0
$385$	−2.70447	−	4.68429i	−0.137833	−	0.238733i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.60577	−	6.24537i	−0.182820	−	0.316653i	0.760020	−	0.649900i	$-0.225189\pi$
$389$	−3.60577	−	6.24537i	−0.182820	−	0.316653i	−0.942840	+	0.333247i	$0.891856\pi$
$390$		0			0
$391$	10.6708	−	18.4824i	0.539647	−	0.934696i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	21.6980			1.09174
$396$		0			0
$397$	−20.8930			−1.04859			−0.524296	−	0.851536i	$-0.675671\pi$
$397$	−20.8930			−1.04859			−0.524296	+	0.851536i	$0.675671\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.59174	−	2.75697i	0.0794877	−	0.137677i	−0.823541	−	0.567256i	$-0.808005\pi$
$401$	1.59174	−	2.75697i	0.0794877	−	0.137677i	0.903029	+	0.429580i	$0.141338\pi$
$402$		0			0
$403$	15.9605	+	27.6443i	0.795047	+	1.37706i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.10051	−	5.37024i	−0.153687	−	0.266193i
$408$		0			0
$409$	11.5046	−	19.9265i	0.568865	−	0.985302i	−0.427814	−	0.903867i	$-0.640716\pi$
$409$	11.5046	−	19.9265i	0.568865	−	0.985302i	0.996679	−	0.0814356i	$-0.0259505\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.75403			0.332344
$414$		0			0
$415$	−44.2065			−2.17001
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.36640	−	2.36667i	0.0667528	−	0.115619i	−0.830717	−	0.556694i	$-0.812069\pi$
$419$	1.36640	−	2.36667i	0.0667528	−	0.115619i	0.897470	+	0.441075i	$0.145403\pi$
$420$		0			0
$421$	−2.98079	−	5.16288i	−0.145275	−	0.251624i	0.784201	−	0.620507i	$-0.213073\pi$
$421$	−2.98079	−	5.16288i	−0.145275	−	0.251624i	−0.929475	+	0.368884i	$0.879740\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.12187	−	5.40724i	−0.151433	−	0.262289i
$426$		0			0
$427$	−7.30065	+	12.6451i	−0.353303	+	0.611939i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.5125			−0.988051			−0.494026	−	0.869447i	$-0.664475\pi$
$431$	−20.5125			−0.988051			−0.494026	+	0.869447i	$0.664475\pi$
$432$		0			0
$433$	−41.5464			−1.99659			−0.998295	−	0.0583639i	$-0.981412\pi$
$433$	−41.5464			−1.99659			−0.998295	+	0.0583639i	$0.981412\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.18017	+	15.9005i	−0.439147	+	0.760625i
$438$		0			0
$439$	15.0834	+	26.1253i	0.719894	+	1.24689i	0.961042	+	0.276404i	$0.0891428\pi$
$439$	15.0834	+	26.1253i	0.719894	+	1.24689i	−0.241148	+	0.970488i	$0.577524\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.67554	+	16.7585i	0.459699	+	0.796222i	0.998945	−	0.0459267i	$-0.0146241\pi$
$443$	9.67554	+	16.7585i	0.459699	+	0.796222i	−0.539246	+	0.842148i	$0.681291\pi$
$444$		0			0
$445$	0.565834	−	0.980054i	0.0268231	−	0.0464590i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.4220			−0.633424			−0.316712	−	0.948522i	$-0.602579\pi$
$449$	−13.4220			−0.633424			−0.316712	+	0.948522i	$0.602579\pi$
$450$		0			0
$451$	10.9537			0.515790
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.8055	+	22.1798i	−0.600331	+	1.03980i
$456$		0			0
$457$	−3.06037	−	5.30072i	−0.143158	−	0.247957i	0.785526	−	0.618828i	$-0.212392\pi$
$457$	−3.06037	−	5.30072i	−0.143158	−	0.247957i	−0.928684	+	0.370871i	$0.879059\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5.80084	−	10.0473i	−0.270172	−	0.467952i	0.698734	−	0.715382i	$-0.253747\pi$
$461$	−5.80084	−	10.0473i	−0.270172	−	0.467952i	−0.968906	+	0.247430i	$0.920414\pi$
$462$		0			0
$463$	4.42830	−	7.67005i	0.205801	−	0.356457i	−0.744587	−	0.667526i	$-0.767353\pi$
$463$	4.42830	−	7.67005i	0.205801	−	0.356457i	0.950388	+	0.311068i	$0.100687\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.20231			0.333283			0.166642	−	0.986018i	$-0.446708\pi$
$467$	7.20231			0.333283			0.166642	+	0.986018i	$0.446708\pi$
$468$		0			0
$469$	−25.1422			−1.16096
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.77684	−	8.27372i	0.219639	−	0.380426i
$474$		0			0
$475$	2.68576	+	4.65187i	0.123231	+	0.213443i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.2521	−	21.2213i	−0.559815	−	0.969628i	−0.997511	−	0.0705051i	$-0.977539\pi$
$479$	−12.2521	−	21.2213i	−0.559815	−	0.969628i	0.437696	−	0.899123i	$-0.355794\pi$
$480$		0			0
$481$	−14.6807	+	25.4277i	−0.669382	+	1.15940i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.05263			−0.229428
$486$		0			0
$487$	−37.7200			−1.70926			−0.854629	−	0.519239i	$-0.826215\pi$
$487$	−37.7200			−1.70926			−0.854629	+	0.519239i	$0.826215\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.203651	−	0.352734i	0.00919063	−	0.0159186i	−0.861394	−	0.507938i	$-0.830408\pi$
$491$	0.203651	−	0.352734i	0.00919063	−	0.0159186i	0.870584	+	0.492020i	$0.163741\pi$
$492$		0			0
$493$	15.9808	+	27.6796i	0.719741	+	1.24663i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.1637	+	17.6040i	0.455903	+	0.789647i
$498$		0			0
$499$	−3.73644	+	6.47171i	−0.167266	+	0.289714i	−0.937458	−	0.348099i	$-0.886827\pi$
$499$	−3.73644	+	6.47171i	−0.167266	+	0.289714i	0.770192	+	0.637813i	$0.220161\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	34.5169			1.53903			0.769517	−	0.638626i	$-0.220497\pi$
$503$	34.5169			1.53903			0.769517	+	0.638626i	$0.220497\pi$
$504$		0			0
$505$	15.2169			0.677145
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.41136	+	2.44454i	−0.0625572	+	0.108352i	−0.895608	−	0.444845i	$-0.853259\pi$
$509$	−1.41136	+	2.44454i	−0.0625572	+	0.108352i	0.833051	+	0.553197i	$0.186592\pi$
$510$		0			0
$511$	11.2705	+	19.5211i	0.498579	+	0.863564i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.68252	−	13.3065i	−0.338532	−	0.586355i
$516$		0			0
$517$	−1.66920	+	2.89114i	−0.0734114	+	0.127152i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.11300			−0.399248			−0.199624	−	0.979873i	$-0.563972\pi$
$521$	−9.11300			−0.399248			−0.199624	+	0.979873i	$0.563972\pi$
$522$		0			0
$523$	1.96313			0.0858418			0.0429209	−	0.999078i	$-0.486334\pi$
$523$	1.96313			0.0858418			0.0429209	+	0.999078i	$0.486334\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	14.9279	−	25.8559i	0.650269	−	1.12630i
$528$		0			0
$529$	3.39339	+	5.87752i	0.147539	+	0.255544i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−25.9325	−	44.9165i	−1.12326	−	1.94555i
$534$		0			0
$535$	−20.1570	+	34.9130i	−0.871465	+	1.50942i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.42084			−0.190419
$540$		0			0
$541$	3.46053			0.148780			0.0743899	−	0.997229i	$-0.476299\pi$
$541$	3.46053			0.148780			0.0743899	+	0.997229i	$0.476299\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.59070	−	4.48723i	0.110974	−	0.192212i
$546$		0			0
$547$	−4.80884	−	8.32915i	−0.205611	−	0.356129i	0.744716	−	0.667381i	$-0.232585\pi$
$547$	−4.80884	−	8.32915i	−0.205611	−	0.356129i	−0.950327	+	0.311252i	$0.899252\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−13.7484	−	23.8129i	−0.585702	−	1.01447i
$552$		0			0
$553$	−7.93645	+	13.7463i	−0.337492	+	0.584554i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.47602			−0.401512			−0.200756	−	0.979641i	$-0.564340\pi$
$557$	−9.47602			−0.401512			−0.200756	+	0.979641i	$0.564340\pi$
$558$		0			0
$559$	−45.2360			−1.91328
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.8982	+	36.1967i	−0.880753	+	1.52551i	−0.0302481	+	0.999542i	$0.509630\pi$
$563$	−20.8982	+	36.1967i	−0.880753	+	1.52551i	−0.850505	+	0.525967i	$0.823704\pi$
$564$		0			0
$565$	3.62725	+	6.28257i	0.152599	+	0.264310i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.74041	−	16.8709i	−0.408339	−	0.707265i	0.586364	−	0.810047i	$-0.300559\pi$
$569$	−9.74041	−	16.8709i	−0.408339	−	0.707265i	−0.994704	+	0.102783i	$0.967225\pi$
$570$		0			0
$571$	9.27352	−	16.0622i	0.388085	−	0.672182i	−0.604107	−	0.796903i	$-0.706470\pi$
$571$	9.27352	−	16.0622i	0.388085	−	0.672182i	0.992192	+	0.124721i	$0.0398035\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.74336			−0.197812
$576$		0			0
$577$	28.5458			1.18838			0.594188	−	0.804326i	$-0.297473\pi$
$577$	28.5458			1.18838			0.594188	+	0.804326i	$0.297473\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.1694	−	28.0062i	0.670820	−	1.16189i
$582$		0			0
$583$	−0.924361	−	1.60104i	−0.0382831	−	0.0663083i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.49840	−	2.59531i	−0.0618457	−	0.107120i	0.833445	−	0.552603i	$-0.186365\pi$
$587$	−1.49840	−	2.59531i	−0.0618457	−	0.107120i	−0.895290	+	0.445483i	$0.853032\pi$
$588$		0			0
$589$	−12.8425	+	22.2439i	−0.529168	+	0.916545i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.7528			−1.01648			−0.508238	−	0.861216i	$-0.669703\pi$
$593$	−24.7528			−1.01648			−0.508238	+	0.861216i	$0.669703\pi$
$594$		0			0
$595$	23.9541			0.982022
$596$		0			0
$597$		0			0
$598$		0			0
$599$	19.2808	−	33.3954i	0.787794	−	1.36450i	−0.139522	−	0.990219i	$-0.544557\pi$
$599$	19.2808	−	33.3954i	0.787794	−	1.36450i	0.927316	−	0.374280i	$-0.122110\pi$
$600$		0			0
$601$	−15.4398	−	26.7426i	−0.629804	−	1.09085i	−0.987591	−	0.157050i	$-0.949802\pi$
$601$	−15.4398	−	26.7426i	−0.629804	−	1.09085i	0.357786	−	0.933803i	$-0.383532\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.8905	−	20.5949i	−0.483417	−	0.837302i
$606$		0			0
$607$	−14.1298	+	24.4736i	−0.573512	+	0.993352i	0.422689	+	0.906275i	$0.361086\pi$
$607$	−14.1298	+	24.4736i	−0.573512	+	0.993352i	−0.996202	+	0.0870777i	$0.972247\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.8071			0.639487
$612$		0			0
$613$	−28.6419			−1.15684			−0.578419	−	0.815740i	$-0.696330\pi$
$613$	−28.6419			−1.15684			−0.578419	+	0.815740i	$0.696330\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.80372	+	3.12413i	−0.0726150	+	0.125773i	−0.900047	−	0.435794i	$-0.856468\pi$
$617$	−1.80372	+	3.12413i	−0.0726150	+	0.125773i	0.827432	+	0.561566i	$0.189801\pi$
$618$		0			0
$619$	4.76132	+	8.24686i	0.191374	+	0.331469i	0.945706	−	0.325024i	$-0.105372\pi$
$619$	4.76132	+	8.24686i	0.191374	+	0.331469i	−0.754332	+	0.656493i	$0.772039\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.413930	+	0.716947i	0.0165837	+	0.0287239i
$624$		0			0
$625$	14.7512	−	25.5498i	0.590047	−	1.02199i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	27.4619			1.09498
$630$		0			0
$631$	30.5885			1.21771			0.608855	−	0.793281i	$-0.291629\pi$
$631$	30.5885			1.21771			0.608855	+	0.793281i	$0.291629\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.7186	−	32.4215i	0.742824	−	1.28661i
$636$		0			0
$637$	10.4662	+	18.1280i	0.414686	+	0.718257i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.09007	−	7.08420i	−0.161548	−	0.279809i	0.773876	−	0.633337i	$-0.218315\pi$
$641$	−4.09007	−	7.08420i	−0.161548	−	0.279809i	−0.935424	+	0.353528i	$0.884982\pi$
$642$		0			0
$643$	13.0611	−	22.6225i	0.515079	−	0.892144i	−0.484768	−	0.874643i	$-0.661096\pi$
$643$	13.0611	−	22.6225i	0.515079	−	0.892144i	0.999847	−	0.0175005i	$-0.00557087\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−22.0170			−0.865577			−0.432789	−	0.901495i	$-0.642470\pi$
$647$	−22.0170			−0.865577			−0.432789	+	0.901495i	$0.642470\pi$
$648$		0			0
$649$	−4.44557			−0.174504
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.14611	+	1.98512i	−0.0448508	+	0.0776839i	−0.887579	−	0.460655i	$-0.847615\pi$
$653$	−1.14611	+	1.98512i	−0.0448508	+	0.0776839i	0.842729	+	0.538339i	$0.180948\pi$
$654$		0			0
$655$	4.70544	+	8.15006i	0.183857	+	0.318449i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.29926	−	5.71449i	−0.128521	−	0.222605i	0.794583	−	0.607156i	$-0.207690\pi$
$659$	−3.29926	−	5.71449i	−0.128521	−	0.222605i	−0.923104	+	0.384551i	$0.874356\pi$
$660$		0			0
$661$	2.88305	−	4.99359i	0.112138	−	0.194228i	−0.804494	−	0.593960i	$-0.797564\pi$
$661$	2.88305	−	4.99359i	0.112138	−	0.194228i	0.916632	+	0.399732i	$0.130897\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.6078			−0.799137
$666$		0			0
$667$	24.2813			0.940174
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.80537	−	8.32314i	0.185509	−	0.321311i
$672$		0			0
$673$	−22.6226	−	39.1835i	−0.872038	−	1.51041i	−0.859885	−	0.510488i	$-0.829465\pi$
$673$	−22.6226	−	39.1835i	−0.872038	−	1.51041i	−0.0121528	−	0.999926i	$-0.503868\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.70089	−	6.41013i	−0.142237	−	0.246361i	0.786102	−	0.618097i	$-0.212096\pi$
$677$	−3.70089	−	6.41013i	−0.142237	−	0.246361i	−0.928339	+	0.371736i	$0.878763\pi$
$678$		0			0
$679$	1.84810	−	3.20100i	0.0709235	−	0.122843i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.88608			−0.378281			−0.189140	−	0.981950i	$-0.560570\pi$
$683$	−9.88608			−0.378281			−0.189140	+	0.981950i	$0.560570\pi$
$684$		0			0
$685$	−31.7878			−1.21455
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.37679	+	7.58082i	−0.166742	+	0.288806i
$690$		0			0
$691$	−4.43367	−	7.67934i	−0.168665	−	0.292136i	0.769286	−	0.638905i	$-0.220612\pi$
$691$	−4.43367	−	7.67934i	−0.168665	−	0.292136i	−0.937951	+	0.346769i	$0.887279\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−11.0990	−	19.2240i	−0.421009	−	0.729209i
$696$		0			0
$697$	−24.2548	+	42.0106i	−0.918717	+	1.59126i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.9889			−1.47259			−0.736296	−	0.676660i	$-0.763427\pi$
$701$	−38.9889			−1.47259			−0.736296	+	0.676660i	$0.763427\pi$
$702$		0			0
$703$	−23.6256			−0.891055
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.56589	+	9.64040i	−0.209327	+	0.362564i
$708$		0			0
$709$	−5.12709	−	8.88039i	−0.192552	−	0.333510i	0.753543	−	0.657398i	$-0.228343\pi$
$709$	−5.12709	−	8.88039i	−0.192552	−	0.333510i	−0.946095	+	0.323888i	$0.895010\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−11.3407	−	19.6427i	−0.424712	−	0.735623i
$714$		0			0
$715$	8.42872	−	14.5990i	0.315216	−	0.545970i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16.8003			0.626545			0.313272	−	0.949663i	$-0.398575\pi$
$719$	16.8003			0.626545			0.313272	+	0.949663i	$0.398575\pi$
$720$		0			0
$721$	11.2401			0.418604
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.55187	−	6.15203i	0.131913	−	0.228481i
$726$		0			0
$727$	9.98309	+	17.2912i	0.370252	+	0.641296i	0.989604	−	0.143818i	$-0.0459379\pi$
$727$	9.98309	+	17.2912i	0.370252	+	0.641296i	−0.619352	+	0.785113i	$0.712605\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21.1547	+	36.6410i	0.782436	+	1.35522i
$732$		0			0
$733$	−2.98307	+	5.16683i	−0.110182	+	0.190841i	−0.915844	−	0.401535i	$-0.868477\pi$
$733$	−2.98307	+	5.16683i	−0.110182	+	0.190841i	0.805661	+	0.592376i	$0.201810\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	16.5489			0.609585
$738$		0			0
$739$	0.673830			0.0247872			0.0123936	−	0.999923i	$-0.496055\pi$
$739$	0.673830			0.0247872			0.0123936	+	0.999923i	$0.496055\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.4737	+	26.8013i	−0.567676	+	0.983244i	0.429119	+	0.903248i	$0.358824\pi$
$743$	−15.4737	+	26.8013i	−0.567676	+	0.983244i	−0.996795	+	0.0799963i	$0.974509\pi$
$744$		0			0
$745$	−11.5621	−	20.0261i	−0.423602	−	0.733700i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−14.7457	−	25.5402i	−0.538794	−	0.933219i
$750$		0			0
$751$	0.0124745	−	0.0216064i	0.000455200	−	0.000788429i	−0.865798	−	0.500394i	$-0.833188\pi$
$751$	0.0124745	−	0.0216064i	0.000455200	−	0.000788429i	0.866253	+	0.499606i	$0.166522\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	38.9501			1.41754
$756$		0			0
$757$	−2.61883			−0.0951829			−0.0475914	−	0.998867i	$-0.515155\pi$
$757$	−2.61883			−0.0951829			−0.0475914	+	0.998867i	$0.515155\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.19437	+	7.26486i	−0.152046	+	0.263351i	−0.931979	−	0.362511i	$-0.881919\pi$
$761$	−4.19437	+	7.26486i	−0.152046	+	0.263351i	0.779934	+	0.625862i	$0.215253\pi$
$762$		0			0
$763$	1.89520	+	3.28258i	0.0686109	+	0.118837i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.5247	+	18.2294i	0.380027	+	0.658225i
$768$		0			0
$769$	−1.00513	+	1.74094i	−0.0362460	+	0.0627800i	−0.883579	−	0.468281i	$-0.844873\pi$
$769$	−1.00513	+	1.74094i	−0.0362460	+	0.0627800i	0.847333	+	0.531061i	$0.178207\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−10.2064			−0.367099			−0.183550	−	0.983010i	$-0.558759\pi$
$773$	−10.2064			−0.367099			−0.183550	+	0.983010i	$0.558759\pi$
$774$		0			0
$775$	−6.63569			−0.238361
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.8666	−	36.1419i	0.747622	−	1.29492i
$780$		0			0
$781$	−6.68983	−	11.5871i	−0.239381	−	0.414620i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24.2464	−	41.9960i	−0.865392	−	1.49890i
$786$		0			0
$787$	13.6213	−	23.5928i	0.485547	−	0.840991i	−0.514315	−	0.857601i	$-0.671954\pi$
$787$	13.6213	−	23.5928i	0.485547	−	0.840991i	0.999862	+	0.0166097i	$0.00528727\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5.30694			−0.188693
$792$		0			0
$793$	−45.5062			−1.61597
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.73924	+	11.6727i	−0.238716	+	0.413468i	−0.960346	−	0.278810i	$-0.910060\pi$
$797$	−6.73924	+	11.6727i	−0.238716	+	0.413468i	0.721630	+	0.692279i	$0.243393\pi$
$798$		0			0
$799$	−7.39223	−	12.8037i	−0.261518	−	0.452963i
$800$		0			0
$801$		0			0
$802$		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:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.24278 - 2.15256i) q^{5} +(0.909142 + 1.57468i) q^{7} +O(q^{10})\)	\(q+(1.24278 - 2.15256i) q^{5} +(0.909142 + 1.57468i) q^{7} +(-0.598407 - 1.03647i) q^{11} +(-2.83342 + 4.90762i) q^{13} +5.30021 q^{17} -4.55980 q^{19} +(2.01328 - 3.48711i) q^{23} +(-0.589008 - 1.02019i) q^{25} +(3.01513 + 5.22236i) q^{29} +(2.81647 - 4.87827i) q^{31} +4.51946 q^{35} +5.18127 q^{37} +(-4.57620 + 7.92621i) q^{41} +(3.99129 + 6.91313i) q^{43} +(-1.39470 - 2.41570i) q^{47} +(1.84692 - 3.19896i) q^{49} +1.54470 q^{53} -2.97475 q^{55} +(1.85725 - 3.21686i) q^{59} +(4.01513 + 6.95441i) q^{61} +(7.04263 + 12.1982i) q^{65} +(-6.91372 + 11.9749i) q^{67} +11.1794 q^{71} +12.3969 q^{73} +(1.08807 - 1.88460i) q^{77} +(4.36480 + 7.56006i) q^{79} +(-8.89267 - 15.4025i) q^{83} +(6.58700 - 11.4090i) q^{85} +0.455297 q^{89} -10.3039 q^{91} +(-5.66683 + 9.81524i) q^{95} +(-1.01640 - 1.76045i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{5} + 6 q^{7}+O(q^{10}) \) 12 * q + 2 * q^5 + 6 * q^7	\( 12 q + 2 q^{5} + 6 q^{7} - 4 q^{11} + 10 q^{13} - 4 q^{17} + 4 q^{19} - 8 q^{23} - 14 q^{25} + 2 q^{29} + 8 q^{31} - 8 q^{35} + 2 q^{41} - 2 q^{43} + 14 q^{47} - 18 q^{49} - 24 q^{53} - 16 q^{55} - 6 q^{59} + 14 q^{61} + 8 q^{65} + 4 q^{67} + 28 q^{71} + 60 q^{73} - 2 q^{77} + 16 q^{79} - 24 q^{83} + 16 q^{85} + 48 q^{89} - 52 q^{91} + 20 q^{95} - 14 q^{97}+O(q^{100}) \) 12 * q + 2 * q^5 + 6 * q^7 - 4 * q^11 + 10 * q^13 - 4 * q^17 + 4 * q^19 - 8 * q^23 - 14 * q^25 + 2 * q^29 + 8 * q^31 - 8 * q^35 + 2 * q^41 - 2 * q^43 + 14 * q^47 - 18 * q^49 - 24 * q^53 - 16 * q^55 - 6 * q^59 + 14 * q^61 + 8 * q^65 + 4 * q^67 + 28 * q^71 + 60 * q^73 - 2 * q^77 + 16 * q^79 - 24 * q^83 + 16 * q^85 + 48 * q^89 - 52 * q^91 + 20 * q^95 - 14 * q^97

Introduction

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