LMFDB - Embedded newform 768.3.h.g.641.14

Newspace parameters

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.h (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$x^{16} - 4 x^{15} + 14 x^{14} - 28 x^{13} + 50 x^{12} - 104 x^{11} - 66 x^{10} + 640 x^{9} + 555 x^{8} - 7060 x^{7} + 17714 x^{6} - 25496 x^{5} + 24840 x^{4} - 17932 x^{3} + 11724 x^{2} - 7056 x + 2401$
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{34}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.14
Root			$1.57980 - 0.654376i$ of defining polynomial
Character	$\chi$	$=$	768.641
Dual form			768.3.h.g.641.13

$q$-expansion

$f(q)$	$=$	$q+(2.55118 + 1.57844i) q^{3} -1.31534 q^{5} +10.2329 q^{7} +(4.01705 + 8.05378i) q^{9} +O(q^{10})$	\(q+(2.55118 + 1.57844i) q^{3} -1.31534 q^{5} +10.2329 q^{7} +(4.01705 + 8.05378i) q^{9} +16.6620 q^{11} -18.7454i q^{13} +(-3.35566 - 2.07618i) q^{15} -4.38114i q^{17} -11.5544i q^{19} +(26.1059 + 16.1520i) q^{21} -16.7490i q^{23} -23.2699 q^{25} +(-2.46419 + 26.8873i) q^{27} +12.5498 q^{29} -20.3167 q^{31} +(42.5079 + 26.3001i) q^{33} -13.4597 q^{35} -18.5778i q^{37} +(29.5885 - 47.8230i) q^{39} +78.6737i q^{41} +36.4860i q^{43} +(-5.28377 - 10.5934i) q^{45} -19.9175i q^{47} +55.7118 q^{49} +(6.91537 - 11.1771i) q^{51} +81.3064 q^{53} -21.9162 q^{55} +(18.2380 - 29.4775i) q^{57} +29.9477 q^{59} -72.0687i q^{61} +(41.1060 + 82.4133i) q^{63} +24.6565i q^{65} -56.3520i q^{67} +(26.4374 - 42.7298i) q^{69} +136.465i q^{71} +80.8141 q^{73} +(-59.3657 - 36.7301i) q^{75} +170.501 q^{77} -86.0317 q^{79} +(-48.7266 + 64.7048i) q^{81} -80.4263 q^{83} +5.76268i q^{85} +(32.0168 + 19.8091i) q^{87} +131.830i q^{89} -191.820i q^{91} +(-51.8315 - 32.0687i) q^{93} +15.1980i q^{95} -20.4375 q^{97} +(66.9323 + 134.192i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16q - 16q^{7} + O(q^{10}) $	$ 16q - 16q^{7} + 32q^{15} + 80q^{25} - 112q^{31} + 16q^{33} + 208q^{39} + 144q^{49} - 384q^{55} + 80q^{57} + 528q^{63} + 160q^{73} - 816q^{79} + 144q^{81} + 736q^{87} + 192q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.55118	+	1.57844i	0.850394	+	0.526147i
$3$	2.55118	+	1.57844i	0.850394	+	0.526147i
$4$		0			0
$5$	−1.31534			−0.263067			−0.131534	−	0.991312i	$-0.541990\pi$
$5$	−1.31534			−0.263067			−0.131534	+	0.991312i	$0.541990\pi$
$6$		0			0
$7$	10.2329			1.46184			0.730920	−	0.682463i	$-0.239091\pi$
$7$	10.2329			1.46184			0.730920	+	0.682463i	$0.239091\pi$
$8$		0			0
$9$	4.01705	+	8.05378i	0.446339	+	0.894864i
$10$		0			0
$11$	16.6620			1.51473			0.757366	−	0.652991i	$-0.226486\pi$
$11$	16.6620			1.51473			0.757366	+	0.652991i	$0.226486\pi$
$12$		0			0
$13$		−	18.7454i		−	1.44196i	−0.692958	−	0.720978i	$-0.743693\pi$
$13$		−	18.7454i		−	1.44196i	0.692958	−	0.720978i	$-0.256307\pi$
$14$		0			0
$15$	−3.35566	−	2.07618i	−0.223711	−	0.138412i
$16$		0			0
$17$		−	4.38114i		−	0.257714i	−0.991663	−	0.128857i	$-0.958869\pi$
$17$		−	4.38114i		−	0.257714i	0.991663	−	0.128857i	$-0.0411308\pi$
$18$		0			0
$19$		−	11.5544i		−	0.608129i	−0.952652	−	0.304064i	$-0.901656\pi$
$19$		−	11.5544i		−	0.608129i	0.952652	−	0.304064i	$-0.0983438\pi$
$20$		0			0
$21$	26.1059	+	16.1520i	1.24314	+	0.769143i
$22$		0			0
$23$		−	16.7490i		−	0.728219i	−0.931356	−	0.364109i	$-0.881373\pi$
$23$		−	16.7490i		−	0.728219i	0.931356	−	0.364109i	$-0.118627\pi$
$24$		0			0
$25$	−23.2699			−0.930796
$26$		0			0
$27$	−2.46419	+	26.8873i	−0.0912663	+	0.995827i
$28$		0			0
$29$	12.5498			0.432752			0.216376	−	0.976310i	$-0.430576\pi$
$29$	12.5498			0.432752			0.216376	+	0.976310i	$0.430576\pi$
$30$		0			0
$31$	−20.3167			−0.655377			−0.327688	−	0.944786i	$-0.606270\pi$
$31$	−20.3167			−0.655377			−0.327688	+	0.944786i	$0.606270\pi$
$32$		0			0
$33$	42.5079	+	26.3001i	1.28812	+	0.796971i
$34$		0			0
$35$	−13.4597			−0.384562
$36$		0			0
$37$		−	18.5778i		−	0.502103i	−0.967974	−	0.251052i	$-0.919224\pi$
$37$		−	18.5778i		−	0.502103i	0.967974	−	0.251052i	$-0.0807764\pi$
$38$		0			0
$39$	29.5885	−	47.8230i	0.758681	−	1.22623i
$40$		0			0
$41$			78.6737i			1.91887i	0.281930	+	0.959435i	$0.409025\pi$
$41$			78.6737i			1.91887i	−0.281930	+	0.959435i	$0.590975\pi$
$42$		0			0
$43$			36.4860i			0.848511i	0.905543	+	0.424255i	$0.139464\pi$
$43$			36.4860i			0.848511i	−0.905543	+	0.424255i	$0.860536\pi$
$44$		0			0
$45$	−5.28377	−	10.5934i	−0.117417	−	0.235410i
$46$		0			0
$47$		−	19.9175i		−	0.423777i	−0.977294	−	0.211888i	$-0.932039\pi$
$47$		−	19.9175i		−	0.423777i	0.977294	−	0.211888i	$-0.0679613\pi$
$48$		0			0
$49$	55.7118			1.13698
$50$		0			0
$51$	6.91537	−	11.1771i	0.135596	−	0.219158i
$52$		0			0
$53$	81.3064			1.53408			0.767041	−	0.641598i	$-0.221728\pi$
$53$	81.3064			1.53408			0.767041	+	0.641598i	$0.221728\pi$
$54$		0			0
$55$	−21.9162			−0.398476
$56$		0			0
$57$	18.2380	−	29.4775i	0.319965	−	0.517149i
$58$		0			0
$59$	29.9477			0.507589			0.253794	−	0.967258i	$-0.418321\pi$
$59$	29.9477			0.507589			0.253794	+	0.967258i	$0.418321\pi$
$60$		0			0
$61$		−	72.0687i		−	1.18145i	−0.806872	−	0.590727i	$-0.798841\pi$
$61$		−	72.0687i		−	1.18145i	0.806872	−	0.590727i	$-0.201159\pi$
$62$		0			0
$63$	41.1060	+	82.4133i	0.652476	+	1.30815i
$64$		0			0
$65$			24.6565i			0.379331i
$66$		0			0
$67$		−	56.3520i		−	0.841074i	−0.907275	−	0.420537i	$-0.861842\pi$
$67$		−	56.3520i		−	0.841074i	0.907275	−	0.420537i	$-0.138158\pi$
$68$		0			0
$69$	26.4374	−	42.7298i	0.383150	−	0.619273i
$70$		0			0
$71$			136.465i			1.92204i	0.276479	+	0.961020i	$0.410832\pi$
$71$			136.465i			1.92204i	−0.276479	+	0.961020i	$0.589168\pi$
$72$		0			0
$73$	80.8141			1.10704			0.553521	−	0.832835i	$-0.313284\pi$
$73$	80.8141			1.10704			0.553521	+	0.832835i	$0.313284\pi$
$74$		0			0
$75$	−59.3657	−	36.7301i	−0.791543	−	0.489735i
$76$		0			0
$77$	170.501			2.21429
$78$		0			0
$79$	−86.0317			−1.08901			−0.544504	−	0.838758i	$-0.683282\pi$
$79$	−86.0317			−1.08901			−0.544504	+	0.838758i	$0.683282\pi$
$80$		0			0
$81$	−48.7266	+	64.7048i	−0.601563	+	0.798825i
$82$		0			0
$83$	−80.4263			−0.968992			−0.484496	−	0.874793i	$-0.660997\pi$
$83$	−80.4263			−0.968992			−0.484496	+	0.874793i	$0.660997\pi$
$84$		0			0
$85$			5.76268i			0.0677962i
$86$		0			0
$87$	32.0168	+	19.8091i	0.368009	+	0.227691i
$88$		0			0
$89$			131.830i			1.48124i	0.671923	+	0.740621i	$0.265468\pi$
$89$			131.830i			1.48124i	−0.671923	+	0.740621i	$0.734532\pi$
$90$		0			0
$91$		−	191.820i		−	2.10791i
$92$		0			0
$93$	−51.8315	−	32.0687i	−0.557328	−	0.344824i
$94$		0			0
$95$			15.1980i			0.159979i
$96$		0			0
$97$	−20.4375			−0.210696			−0.105348	−	0.994435i	$-0.533596\pi$
$97$	−20.4375			−0.210696			−0.105348	+	0.994435i	$0.533596\pi$
$98$		0			0
$99$	66.9323	+	134.192i	0.676083	+	1.35548i
$100$		0			0
$101$	−143.374			−1.41954			−0.709772	−	0.704431i	$-0.751202\pi$
$101$	−143.374			−1.41954			−0.709772	+	0.704431i	$0.751202\pi$
$102$		0			0
$103$	−113.748			−1.10435			−0.552173	−	0.833730i	$-0.686201\pi$
$103$	−113.748			−1.10435			−0.552173	+	0.833730i	$0.686201\pi$
$104$		0			0
$105$	−34.3381	−	21.2453i	−0.327029	−	0.202336i
$106$		0			0
$107$	−6.26024			−0.0585069			−0.0292535	−	0.999572i	$-0.509313\pi$
$107$	−6.26024			−0.0585069			−0.0292535	+	0.999572i	$0.509313\pi$
$108$		0			0
$109$			147.657i			1.35465i	0.735682	+	0.677327i	$0.236862\pi$
$109$			147.657i			1.35465i	−0.735682	+	0.677327i	$0.763138\pi$
$110$		0			0
$111$	29.3240	−	47.3954i	0.264180	−	0.426986i
$112$		0			0
$113$			51.3905i			0.454783i	0.973803	+	0.227392i	$0.0730197\pi$
$113$			51.3905i			0.454783i	−0.973803	+	0.227392i	$0.926980\pi$
$114$		0			0
$115$			22.0306i			0.191571i
$116$		0			0
$117$	150.971	−	75.3013i	1.29035	−	0.643601i
$118$		0			0
$119$		−	44.8317i		−	0.376737i
$120$		0			0
$121$	156.624			1.29441
$122$		0			0
$123$	−124.182	+	200.711i	−1.00961	+	1.63179i
$124$		0			0
$125$	63.4912			0.507929
$126$		0			0
$127$	145.579			1.14629			0.573147	−	0.819452i	$-0.305722\pi$
$127$	145.579			1.14629			0.573147	+	0.819452i	$0.305722\pi$
$128$		0			0
$129$	−57.5909	+	93.0823i	−0.446441	+	0.721568i
$130$		0			0
$131$	−45.3993			−0.346560			−0.173280	−	0.984873i	$-0.555437\pi$
$131$	−45.3993			−0.346560			−0.173280	+	0.984873i	$0.555437\pi$
$132$		0			0
$133$		−	118.235i		−	0.888987i
$134$		0			0
$135$	3.24124	−	35.3659i	0.0240092	−	0.261969i
$136$		0			0
$137$		−	179.157i		−	1.30771i	−0.756618	−	0.653857i	$-0.773150\pi$
$137$		−	179.157i		−	1.30771i	0.756618	−	0.653857i	$-0.226850\pi$
$138$		0			0
$139$		−	50.8841i		−	0.366073i	−0.983106	−	0.183036i	$-0.941407\pi$
$139$		−	50.8841i		−	0.366073i	0.983106	−	0.183036i	$-0.0585926\pi$
$140$		0			0
$141$	31.4386	−	50.8131i	0.222969	−	0.360377i
$142$		0			0
$143$		−	312.337i		−	2.18418i
$144$		0			0
$145$	−16.5072			−0.113843
$146$		0			0
$147$	142.131	+	87.9378i	0.966877	+	0.598216i
$148$		0			0
$149$	−76.0410			−0.510342			−0.255171	−	0.966896i	$-0.582132\pi$
$149$	−76.0410			−0.510342			−0.255171	+	0.966896i	$0.582132\pi$
$150$		0			0
$151$	−179.918			−1.19151			−0.595754	−	0.803167i	$-0.703147\pi$
$151$	−179.918			−1.19151			−0.595754	+	0.803167i	$0.703147\pi$
$152$		0			0
$153$	35.2847	−	17.5993i	0.230619	−	0.115028i
$154$		0			0
$155$	26.7233			0.172408
$156$		0			0
$157$			67.2180i			0.428140i	0.976818	+	0.214070i	$0.0686720\pi$
$157$			67.2180i			0.428140i	−0.976818	+	0.214070i	$0.931328\pi$
$158$		0			0
$159$	207.427	+	128.337i	1.30457	+	0.807153i
$160$		0			0
$161$		−	171.391i		−	1.06454i
$162$		0			0
$163$		−	102.723i		−	0.630204i	−0.949058	−	0.315102i	$-0.897961\pi$
$163$		−	102.723i		−	0.630204i	0.949058	−	0.315102i	$-0.102039\pi$
$164$		0			0
$165$	−55.9122	−	34.5934i	−0.338862	−	0.209657i
$166$		0			0
$167$			80.7935i			0.483793i	0.970302	+	0.241897i	$0.0777695\pi$
$167$			80.7935i			0.483793i	−0.970302	+	0.241897i	$0.922231\pi$
$168$		0			0
$169$	−182.391			−1.07924
$170$		0			0
$171$	93.0569	−	46.4148i	0.544192	−	0.271431i
$172$		0			0
$173$	−169.185			−0.977950			−0.488975	−	0.872298i	$-0.662629\pi$
$173$	−169.185			−0.977950			−0.488975	+	0.872298i	$0.662629\pi$
$174$		0			0
$175$	−238.118			−1.36067
$176$		0			0
$177$	76.4021	+	47.2707i	0.431650	+	0.267066i
$178$		0			0
$179$	170.982			0.955205			0.477602	−	0.878576i	$-0.341506\pi$
$179$	170.982			0.955205			0.477602	+	0.878576i	$0.341506\pi$
$180$		0			0
$181$			39.6292i			0.218946i	0.993990	+	0.109473i	$0.0349163\pi$
$181$			39.6292i			0.218946i	−0.993990	+	0.109473i	$0.965084\pi$
$182$		0			0
$183$	113.756	−	183.860i	0.621618	−	1.00470i
$184$		0			0
$185$			24.4361i			0.132087i
$186$		0			0
$187$		−	72.9988i		−	0.390368i
$188$		0			0
$189$	−25.2158	+	275.135i	−0.133417	+	1.45574i
$190$		0			0
$191$			239.917i			1.25611i	0.778170	+	0.628054i	$0.216148\pi$
$191$			239.917i			1.25611i	−0.778170	+	0.628054i	$0.783852\pi$
$192$		0			0
$193$	−13.4972			−0.0699337			−0.0349669	−	0.999388i	$-0.511133\pi$
$193$	−13.4972			−0.0699337			−0.0349669	+	0.999388i	$0.511133\pi$
$194$		0			0
$195$	−38.9189	+	62.9033i	−0.199584	+	0.322581i
$196$		0			0
$197$	−16.6040			−0.0842843			−0.0421422	−	0.999112i	$-0.513418\pi$
$197$	−16.6040			−0.0842843			−0.0421422	+	0.999112i	$0.513418\pi$
$198$		0			0
$199$	143.674			0.721979			0.360990	−	0.932570i	$-0.382439\pi$
$199$	143.674			0.721979			0.360990	+	0.932570i	$0.382439\pi$
$200$		0			0
$201$	88.9483	−	143.764i	0.442529	−	0.715244i
$202$		0			0
$203$	128.421			0.632614
$204$		0			0
$205$		−	103.482i		−	0.504792i
$206$		0			0
$207$	134.893	−	67.2817i	0.651657	−	0.325032i
$208$		0			0
$209$		−	192.521i		−	0.921151i
$210$		0			0
$211$		−	83.6553i		−	0.396471i	−0.980154	−	0.198235i	$-0.936479\pi$
$211$		−	83.6553i		−	0.396471i	0.980154	−	0.198235i	$-0.0635210\pi$
$212$		0			0
$213$	−215.402	+	348.146i	−1.01128	+	1.63449i
$214$		0			0
$215$		−	47.9913i		−	0.223215i
$216$		0			0
$217$	−207.898			−0.958056
$218$		0			0
$219$	206.171	+	127.560i	0.941422	+	0.582467i
$220$		0			0
$221$	−82.1263			−0.371612
$222$		0			0
$223$	−146.898			−0.658734			−0.329367	−	0.944202i	$-0.606835\pi$
$223$	−146.898			−0.658734			−0.329367	+	0.944202i	$0.606835\pi$
$224$		0			0
$225$	−93.4763	−	187.410i	−0.415450	−	0.832936i
$226$		0			0
$227$	28.5250			0.125661			0.0628304	−	0.998024i	$-0.479987\pi$
$227$	28.5250			0.125661			0.0628304	+	0.998024i	$0.479987\pi$
$228$		0			0
$229$			345.118i			1.50707i	0.657410	+	0.753533i	$0.271652\pi$
$229$			345.118i			1.50707i	−0.657410	+	0.753533i	$0.728348\pi$
$230$		0			0
$231$	434.978	+	269.125i	1.88302	+	1.16504i
$232$		0			0
$233$			105.272i			0.451809i	0.974149	+	0.225905i	$0.0725338\pi$
$233$			105.272i			0.451809i	−0.974149	+	0.225905i	$0.927466\pi$
$234$		0			0
$235$			26.1982i			0.111482i
$236$		0			0
$237$	−219.482	−	135.796i	−0.926086	−	0.572978i
$238$		0			0
$239$		−	280.456i		−	1.17346i	−0.809783	−	0.586729i	$-0.800415\pi$
$239$		−	280.456i		−	1.17346i	0.809783	−	0.586729i	$-0.199585\pi$
$240$		0			0
$241$	−369.833			−1.53458			−0.767289	−	0.641302i	$-0.778395\pi$
$241$	−369.833			−1.53458			−0.767289	+	0.641302i	$0.778395\pi$
$242$		0			0
$243$	−226.443	+	88.1616i	−0.931865	+	0.362805i
$244$		0			0
$245$	−73.2798			−0.299101
$246$		0			0
$247$	−216.593			−0.876894
$248$		0			0
$249$	−205.182	−	126.948i	−0.824025	−	0.509832i
$250$		0			0
$251$	−27.4959			−0.109545			−0.0547727	−	0.998499i	$-0.517443\pi$
$251$	−27.4959			−0.109545			−0.0547727	+	0.998499i	$0.517443\pi$
$252$		0			0
$253$		−	279.073i		−	1.10306i
$254$		0			0
$255$	−9.09604	+	14.7016i	−0.0356708	+	0.0576534i
$256$		0			0
$257$		−	4.92447i		−	0.0191614i	−0.999954	−	0.00958068i	$-0.996950\pi$
$257$		−	4.92447i		−	0.0191614i	0.999954	−	0.00958068i	$-0.00304967\pi$
$258$		0			0
$259$		−	190.105i		−	0.733995i
$260$		0			0
$261$	50.4132	+	101.073i	0.193154	+	0.387254i
$262$		0			0
$263$		−	263.575i		−	1.00219i	−0.865394	−	0.501093i	$-0.832932\pi$
$263$		−	263.575i		−	1.00219i	0.865394	−	0.501093i	$-0.167068\pi$
$264$		0			0
$265$	−106.945			−0.403567
$266$		0			0
$267$	−208.087	+	336.323i	−0.779351	+	1.25964i
$268$		0			0
$269$	−236.023			−0.877409			−0.438704	−	0.898631i	$-0.644562\pi$
$269$	−236.023			−0.877409			−0.438704	+	0.898631i	$0.644562\pi$
$270$		0			0
$271$	−433.288			−1.59885			−0.799424	−	0.600767i	$-0.794862\pi$
$271$	−433.288			−1.59885			−0.799424	+	0.600767i	$0.794862\pi$
$272$		0			0
$273$	302.776	−	489.367i	1.10907	−	1.79255i
$274$		0			0
$275$	−387.724			−1.40991
$276$		0			0
$277$		−	244.067i		−	0.881107i	−0.897726	−	0.440553i	$-0.854782\pi$
$277$		−	244.067i		−	0.881107i	0.897726	−	0.440553i	$-0.145218\pi$
$278$		0			0
$279$	−81.6131	−	163.626i	−0.292520	−	0.586473i
$280$		0			0
$281$		−	431.627i		−	1.53604i	−0.640426	−	0.768020i	$-0.721242\pi$
$281$		−	431.627i		−	1.53604i	0.640426	−	0.768020i	$-0.278758\pi$
$282$		0			0
$283$		−	35.1843i		−	0.124326i	−0.998066	−	0.0621631i	$-0.980200\pi$
$283$		−	35.1843i		−	0.124326i	0.998066	−	0.0621631i	$-0.0197999\pi$
$284$		0			0
$285$	−23.9891	+	38.7728i	−0.0841723	+	0.136045i
$286$		0			0
$287$			805.058i			2.80508i
$288$		0			0
$289$	269.806			0.933583
$290$		0			0
$291$	−52.1397	−	32.2594i	−0.179174	−	0.110857i
$292$		0			0
$293$	214.900			0.733446			0.366723	−	0.930330i	$-0.380480\pi$
$293$	214.900			0.733446			0.366723	+	0.930330i	$0.380480\pi$
$294$		0			0
$295$	−39.3914			−0.133530
$296$		0			0
$297$	−41.0585	+	447.998i	−0.138244	+	1.50841i
$298$		0			0
$299$	−313.968			−1.05006
$300$		0			0
$301$			373.356i			1.24039i
$302$		0			0
$303$	−365.773	−	226.307i	−1.20717	−	0.746889i
$304$		0			0
$305$			94.7946i			0.310802i
$306$		0			0
$307$		−	425.639i		−	1.38645i	−0.720723	−	0.693223i	$-0.756190\pi$
$307$		−	425.639i		−	1.38645i	0.720723	−	0.693223i	$-0.243810\pi$
$308$		0			0
$309$	−290.191	−	179.544i	−0.939128	−	0.581048i
$310$		0			0
$311$			7.67121i			0.0246663i	0.999924	+	0.0123331i	$0.00392586\pi$
$311$			7.67121i			0.0246663i	−0.999924	+	0.0123331i	$0.996074\pi$
$312$		0			0
$313$	−313.959			−1.00306			−0.501532	−	0.865139i	$-0.667230\pi$
$313$	−313.959			−1.00306			−0.501532	+	0.865139i	$0.667230\pi$
$314$		0			0
$315$	−54.0682	−	108.401i	−0.171645	−	0.344131i
$316$		0			0
$317$	−361.930			−1.14174			−0.570868	−	0.821042i	$-0.693393\pi$
$317$	−361.930			−1.14174			−0.570868	+	0.821042i	$0.693393\pi$
$318$		0			0
$319$	209.105			0.655503
$320$		0			0
$321$	−15.9710	−	9.88142i	−0.0497539	−	0.0307832i
$322$		0			0
$323$	−50.6216			−0.156723
$324$		0			0
$325$			436.204i			1.34217i
$326$		0			0
$327$	−233.068	+	376.701i	−0.712748	+	1.15199i
$328$		0			0
$329$		−	203.813i		−	0.619493i
$330$		0			0
$331$		−	283.000i		−	0.854984i	−0.904019	−	0.427492i	$-0.859397\pi$
$331$		−	283.000i		−	0.854984i	0.904019	−	0.427492i	$-0.140603\pi$
$332$		0			0
$333$	149.622	−	74.6280i	0.449314	−	0.224108i
$334$		0			0
$335$			74.1218i			0.221259i
$336$		0			0
$337$	242.847			0.720614			0.360307	−	0.932834i	$-0.382672\pi$
$337$	242.847			0.720614			0.360307	+	0.932834i	$0.382672\pi$
$338$		0			0
$339$	−81.1168	+	131.106i	−0.239283	+	0.386745i
$340$		0			0
$341$	−338.517			−0.992720
$342$		0			0
$343$	68.6810			0.200236
$344$		0			0
$345$	−34.7740	+	56.2041i	−0.100794	+	0.162910i
$346$		0			0
$347$	−35.8055			−0.103186			−0.0515929	−	0.998668i	$-0.516430\pi$
$347$	−35.8055			−0.103186			−0.0515929	+	0.998668i	$0.516430\pi$
$348$		0			0
$349$		−	119.019i		−	0.341030i	−0.985355	−	0.170515i	$-0.945457\pi$
$349$		−	119.019i		−	0.341030i	0.985355	−	0.170515i	$-0.0545431\pi$
$350$		0			0
$351$	504.014	+	46.1923i	1.43594	+	0.131602i
$352$		0			0
$353$			158.225i			0.448230i	0.974563	+	0.224115i	$0.0719492\pi$
$353$			158.225i			0.448230i	−0.974563	+	0.224115i	$0.928051\pi$
$354$		0			0
$355$		−	179.497i		−	0.505626i
$356$		0			0
$357$	70.7641	−	114.374i	0.198219	−	0.320375i
$358$		0			0
$359$			127.881i			0.356214i	0.984011	+	0.178107i	$0.0569974\pi$
$359$			127.881i			0.356214i	−0.984011	+	0.178107i	$0.943003\pi$
$360$		0			0
$361$	227.495			0.630180
$362$		0			0
$363$	399.576	+	247.221i	1.10076	+	0.681051i
$364$		0			0
$365$	−106.298			−0.291227
$366$		0			0
$367$	92.6498			0.252452			0.126226	−	0.992002i	$-0.459714\pi$
$367$	92.6498			0.252452			0.126226	+	0.992002i	$0.459714\pi$
$368$		0			0
$369$	−633.620	+	316.036i	−1.71713	+	0.856466i
$370$		0			0
$371$	831.998			2.24258
$372$		0			0
$373$			9.37918i			0.0251453i	0.999921	+	0.0125726i	$0.00400210\pi$
$373$			9.37918i			0.0251453i	−0.999921	+	0.0125726i	$0.995998\pi$
$374$		0			0
$375$	161.977	+	100.217i	0.431940	+	0.267245i
$376$		0			0
$377$		−	235.251i		−	0.624009i
$378$		0			0
$379$			353.377i			0.932392i	0.884681	+	0.466196i	$0.154376\pi$
$379$			353.377i			0.932392i	−0.884681	+	0.466196i	$0.845624\pi$
$380$		0			0
$381$	371.400	+	229.789i	0.974802	+	0.603120i
$382$		0			0
$383$			106.620i			0.278382i	0.990266	+	0.139191i	$0.0444502\pi$
$383$			106.620i			0.278382i	−0.990266	+	0.139191i	$0.955550\pi$
$384$		0			0
$385$	−224.266			−0.582509
$386$		0			0
$387$	−293.850	+	146.566i	−0.759302	+	0.378723i
$388$		0			0
$389$	659.144			1.69446			0.847229	−	0.531227i	$-0.178269\pi$
$389$	659.144			1.69446			0.847229	+	0.531227i	$0.178269\pi$
$390$		0			0
$391$	−73.3799			−0.187672
$392$		0			0
$393$	−115.822	−	71.6601i	−0.294712	−	0.182341i
$394$		0			0
$395$	113.161			0.286483
$396$		0			0
$397$			523.170i			1.31781i	0.752227	+	0.658904i	$0.228980\pi$
$397$			523.170i			1.31781i	−0.752227	+	0.658904i	$0.771020\pi$
$398$		0			0
$399$	186.627	−	301.639i	0.467738	−	0.755989i
$400$		0			0
$401$		−	255.566i		−	0.637323i	−0.947869	−	0.318661i	$-0.896767\pi$
$401$		−	255.566i		−	0.637323i	0.947869	−	0.318661i	$-0.103233\pi$
$402$		0			0
$403$			380.845i			0.945024i
$404$		0			0
$405$	64.0919	−	85.1086i	0.158252	−	0.210145i
$406$		0			0
$407$		−	309.545i		−	0.760552i
$408$		0			0
$409$	630.598			1.54180			0.770902	−	0.636954i	$-0.219806\pi$
$409$	630.598			1.54180			0.770902	+	0.636954i	$0.219806\pi$
$410$		0			0
$411$	282.788	−	457.061i	0.688050	−	1.11207i
$412$		0			0
$413$	306.452			0.742014
$414$		0			0
$415$	105.788			0.254910
$416$		0			0
$417$	80.3175	−	129.815i	0.192608	−	0.311306i
$418$		0			0
$419$	475.721			1.13537			0.567686	−	0.823245i	$-0.307839\pi$
$419$	475.721			1.13537			0.567686	+	0.823245i	$0.307839\pi$
$420$		0			0
$421$		−	239.614i		−	0.569155i	−0.958653	−	0.284578i	$-0.908147\pi$
$421$		−	239.614i		−	0.569155i	0.958653	−	0.284578i	$-0.0918534\pi$
$422$		0			0
$423$	160.411	−	80.0096i	0.379222	−	0.189148i
$424$		0			0
$425$			101.949i			0.239879i
$426$		0			0
$427$		−	737.470i		−	1.72710i
$428$		0			0
$429$	493.006	−	796.828i	1.14920	−	1.85741i
$430$		0			0
$431$		−	618.539i		−	1.43513i	−0.696494	−	0.717563i	$-0.745258\pi$
$431$		−	618.539i		−	1.43513i	0.696494	−	0.717563i	$-0.254742\pi$
$432$		0			0
$433$	211.763			0.489061			0.244530	−	0.969642i	$-0.421366\pi$
$433$	211.763			0.489061			0.244530	+	0.969642i	$0.421366\pi$
$434$		0			0
$435$	−42.1129	−	26.0557i	−0.0968112	−	0.0598981i
$436$		0			0
$437$	−193.526			−0.442851
$438$		0			0
$439$	−56.6884			−0.129131			−0.0645654	−	0.997913i	$-0.520566\pi$
$439$	−56.6884			−0.129131			−0.0645654	+	0.997913i	$0.520566\pi$
$440$		0			0
$441$	223.797	+	448.690i	0.507476	+	1.01744i
$442$		0			0
$443$	−687.493			−1.55190			−0.775952	−	0.630792i	$-0.782730\pi$
$443$	−687.493			−1.55190			−0.775952	+	0.630792i	$0.782730\pi$
$444$		0			0
$445$		−	173.401i		−	0.389666i
$446$		0			0
$447$	−193.994	−	120.026i	−0.433992	−	0.268515i
$448$		0			0
$449$			490.455i			1.09233i	0.837679	+	0.546163i	$0.183912\pi$
$449$			490.455i			1.09233i	−0.837679	+	0.546163i	$0.816088\pi$
$450$		0			0
$451$			1310.86i			2.90657i
$452$		0			0
$453$	−459.002	−	283.989i	−1.01325	−	0.626908i
$454$		0			0
$455$			252.307i			0.554522i
$456$		0			0
$457$	537.122			1.17532			0.587661	−	0.809107i	$-0.300049\pi$
$457$	537.122			1.17532			0.587661	+	0.809107i	$0.300049\pi$
$458$		0			0
$459$	117.797	+	10.7960i	0.256639	+	0.0235206i
$460$		0			0
$461$	−129.931			−0.281847			−0.140923	−	0.990021i	$-0.545007\pi$
$461$	−129.931			−0.281847			−0.140923	+	0.990021i	$0.545007\pi$
$462$		0			0
$463$	−262.112			−0.566118			−0.283059	−	0.959103i	$-0.591349\pi$
$463$	−262.112			−0.566118			−0.283059	+	0.959103i	$0.591349\pi$
$464$		0			0
$465$	68.1759	+	42.1811i	0.146615	+	0.0907121i
$466$		0			0
$467$	−418.978			−0.897169			−0.448585	−	0.893740i	$-0.648072\pi$
$467$	−418.978			−0.897169			−0.448585	+	0.893740i	$0.648072\pi$
$468$		0			0
$469$		−	576.643i		−	1.22952i
$470$		0			0
$471$	−106.100	+	171.485i	−0.225264	+	0.364087i
$472$		0			0
$473$			607.931i			1.28527i
$474$		0			0
$475$			268.871i			0.566043i
$476$		0			0
$477$	326.612	+	654.823i	0.684720	+	1.37280i
$478$		0			0
$479$		−	306.699i		−	0.640290i	−0.947369	−	0.320145i	$-0.896268\pi$
$479$		−	306.699i		−	0.640290i	0.947369	−	0.320145i	$-0.103732\pi$
$480$		0			0
$481$	−348.249			−0.724011
$482$		0			0
$483$	270.530	−	437.249i	0.560104	−	0.905278i
$484$		0			0
$485$	26.8822			0.0554272
$486$		0			0
$487$	−387.411			−0.795506			−0.397753	−	0.917493i	$-0.630210\pi$
$487$	−387.411			−0.795506			−0.397753	+	0.917493i	$0.630210\pi$
$488$		0			0
$489$	162.143	−	262.066i	0.331580	−	0.535922i
$490$		0			0
$491$	−13.1250			−0.0267311			−0.0133655	−	0.999911i	$-0.504255\pi$
$491$	−13.1250			−0.0267311			−0.0133655	+	0.999911i	$0.504255\pi$
$492$		0			0
$493$		−	54.9824i		−	0.111526i
$494$		0			0
$495$	−88.0385	−	176.508i	−0.177855	−	0.356582i
$496$		0			0
$497$			1396.43i			2.80971i
$498$		0			0
$499$		−	239.526i		−	0.480012i	−0.970771	−	0.240006i	$-0.922851\pi$
$499$		−	239.526i		−	0.480012i	0.970771	−	0.240006i	$-0.0771494\pi$
$500$		0			0
$501$	−127.528	+	206.119i	−0.254546	+	0.411415i
$502$		0			0
$503$		−	595.462i		−	1.18382i	−0.806004	−	0.591911i	$-0.798374\pi$
$503$		−	595.462i		−	1.18382i	0.806004	−	0.591911i	$-0.201626\pi$
$504$		0			0
$505$	188.585			0.373436
$506$		0			0
$507$	−465.312	−	287.893i	−0.917775	−	0.567837i
$508$		0			0
$509$	600.556			1.17987			0.589937	−	0.807449i	$-0.299153\pi$
$509$	600.556			1.17987			0.589937	+	0.807449i	$0.299153\pi$
$510$		0			0
$511$	826.961			1.61832
$512$		0			0
$513$	310.668	+	28.4724i	0.605591	+	0.0555017i
$514$		0			0
$515$	149.616			0.290517
$516$		0			0
$517$		−	331.866i		−	0.641908i
$518$		0			0
$519$	−431.622	−	267.049i	−0.831642	−	0.514545i
$520$		0			0
$521$		−	394.528i		−	0.757251i	−0.925550	−	0.378626i	$-0.876397\pi$
$521$		−	394.528i		−	0.757251i	0.925550	−	0.378626i	$-0.123603\pi$
$522$		0			0
$523$			639.213i			1.22221i	0.791551	+	0.611103i	$0.209274\pi$
$523$			639.213i			1.22221i	−0.791551	+	0.611103i	$0.790726\pi$
$524$		0			0
$525$	−607.482	−	375.855i	−1.15711	−	0.715914i
$526$		0			0
$527$			89.0102i			0.168900i
$528$		0			0
$529$	248.470			0.469697
$530$		0			0
$531$	120.302	+	241.192i	0.226557	+	0.454223i
$532$		0			0
$533$	1474.77			2.76692
$534$		0			0
$535$	8.23433			0.0153913
$536$		0			0
$537$	436.205	+	269.884i	0.812300	+	0.502578i
$538$		0			0
$539$	928.273			1.72221
$540$		0			0
$541$		−	606.241i		−	1.12059i	−0.828292	−	0.560296i	$-0.810687\pi$
$541$		−	606.241i		−	1.12059i	0.828292	−	0.560296i	$-0.189313\pi$
$542$		0			0
$543$	−62.5524	+	101.101i	−0.115198	+	0.186190i
$544$		0			0
$545$		−	194.219i		−	0.356365i
$546$		0			0
$547$		−	409.425i		−	0.748492i	−0.927330	−	0.374246i	$-0.877902\pi$
$547$		−	409.425i		−	0.748492i	0.927330	−	0.374246i	$-0.122098\pi$
$548$		0			0
$549$	580.425	−	289.503i	1.05724	−	0.527329i
$550$		0			0
$551$		−	145.006i		−	0.263169i
$552$		0			0
$553$	−880.352			−1.59196
$554$		0			0
$555$	−38.5709	+	62.3409i	−0.0694972	+	0.112326i
$556$		0			0
$557$	994.040			1.78463			0.892316	−	0.451411i	$-0.149079\pi$
$557$	994.040			1.78463			0.892316	+	0.451411i	$0.149079\pi$
$558$		0			0
$559$	683.945			1.22351
$560$		0			0
$561$	115.224	−	186.233i	0.205391	−	0.331966i
$562$		0			0
$563$	29.5954			0.0525673			0.0262837	−	0.999655i	$-0.491633\pi$
$563$	29.5954			0.0525673			0.0262837	+	0.999655i	$0.491633\pi$
$564$		0			0
$565$		−	67.5958i		−	0.119639i
$566$		0			0
$567$	−498.614	+	662.117i	−0.879389	+	1.16775i
$568$		0			0
$569$			355.324i			0.624471i	0.950005	+	0.312235i	$0.101078\pi$
$569$			355.324i			0.624471i	−0.950005	+	0.312235i	$0.898922\pi$
$570$		0			0
$571$			133.482i			0.233769i	0.993145	+	0.116885i	$0.0372908\pi$
$571$			133.482i			0.233769i	−0.993145	+	0.116885i	$0.962709\pi$
$572$		0			0
$573$	−378.694	+	612.071i	−0.660898	+	1.06819i
$574$		0			0
$575$			389.748i			0.677823i
$576$		0			0
$577$	−228.735			−0.396422			−0.198211	−	0.980159i	$-0.563513\pi$
$577$	−228.735			−0.396422			−0.198211	+	0.980159i	$0.563513\pi$
$578$		0			0
$579$	−34.4338	−	21.3046i	−0.0594712	−	0.0367954i
$580$		0			0
$581$	−822.993			−1.41651
$582$		0			0
$583$	1354.73			2.32372
$584$		0			0
$585$	−198.578	+	99.0465i	−0.339450	+	0.169310i
$586$		0			0
$587$	−633.349			−1.07896			−0.539480	−	0.841998i	$-0.681379\pi$
$587$	−633.349			−1.07896			−0.539480	+	0.841998i	$0.681379\pi$
$588$		0			0
$589$			234.748i			0.398553i
$590$		0			0
$591$	−42.3598	−	26.2084i	−0.0716748	−	0.0443459i
$592$		0			0
$593$		−	189.510i		−	0.319578i	−0.987151	−	0.159789i	$-0.948919\pi$
$593$		−	189.510i		−	0.319578i	0.987151	−	0.159789i	$-0.0510813\pi$
$594$		0			0
$595$			58.9688i			0.0991071i
$596$		0			0
$597$	366.538	+	226.781i	0.613967	+	0.379867i
$598$		0			0
$599$			340.082i			0.567750i	0.958861	+	0.283875i	$0.0916200\pi$
$599$			340.082i			0.567750i	−0.958861	+	0.283875i	$0.908380\pi$
$600$		0			0
$601$	−528.406			−0.879211			−0.439605	−	0.898191i	$-0.644882\pi$
$601$	−528.406			−0.879211			−0.439605	+	0.898191i	$0.644882\pi$
$602$		0			0
$603$	453.846	−	226.369i	0.752647	−	0.375404i
$604$		0			0
$605$	−206.013			−0.340517
$606$		0			0
$607$	−659.649			−1.08674			−0.543368	−	0.839495i	$-0.682851\pi$
$607$	−659.649			−1.08674			−0.543368	+	0.839495i	$0.682851\pi$
$608$		0			0
$609$	327.624	+	202.704i	0.537971	+	0.332848i
$610$		0			0
$611$	−373.362			−0.611067
$612$		0			0
$613$			67.7073i			0.110452i	0.998474	+	0.0552262i	$0.0175880\pi$
$613$			67.7073i			0.110452i	−0.998474	+	0.0552262i	$0.982412\pi$
$614$		0			0
$615$	163.341	−	264.002i	0.265595	−	0.429272i
$616$		0			0
$617$			807.698i			1.30907i	0.756031	+	0.654536i	$0.227136\pi$
$617$			807.698i			1.30907i	−0.756031	+	0.654536i	$0.772864\pi$
$618$		0			0
$619$			122.858i			0.198479i	0.995064	+	0.0992393i	$0.0316409\pi$
$619$			122.858i			0.198479i	−0.995064	+	0.0992393i	$0.968359\pi$
$620$		0			0
$621$	450.337	+	41.2728i	0.725180	+	0.0664619i
$622$		0			0
$623$			1349.01i			2.16534i
$624$		0			0
$625$	498.235			0.797176
$626$		0			0
$627$	303.882	−	491.155i	0.484661	−	0.783341i
$628$		0			0
$629$	−81.3921			−0.129399
$630$		0			0
$631$	141.286			0.223908			0.111954	−	0.993713i	$-0.464289\pi$
$631$	141.286			0.223908			0.111954	+	0.993713i	$0.464289\pi$
$632$		0			0
$633$	132.045	−	213.420i	0.208602	−	0.337156i
$634$		0			0
$635$	−191.486			−0.301553
$636$		0			0
$637$		−	1044.34i		−	1.63947i
$638$		0			0
$639$	−1099.06	+	548.186i	−1.71996	+	0.857881i
$640$		0			0
$641$			240.601i			0.375352i	0.982231	+	0.187676i	$0.0600955\pi$
$641$			240.601i			0.375352i	−0.982231	+	0.187676i	$0.939904\pi$
$642$		0			0
$643$		−	774.975i		−	1.20525i	−0.798025	−	0.602624i	$-0.794122\pi$
$643$		−	774.975i		−	1.20525i	0.798025	−	0.602624i	$-0.205878\pi$
$644$		0			0
$645$	75.7515	−	122.435i	0.117444	−	0.189821i
$646$		0			0
$647$			823.719i			1.27314i	0.771220	+	0.636568i	$0.219647\pi$
$647$			823.719i			1.27314i	−0.771220	+	0.636568i	$0.780353\pi$
$648$		0			0
$649$	498.991			0.768861
$650$		0			0
$651$	−530.386	−	328.155i	−0.814724	−	0.504078i
$652$		0			0
$653$	−142.733			−0.218581			−0.109290	−	0.994010i	$-0.534858\pi$
$653$	−142.733			−0.218581			−0.109290	+	0.994010i	$0.534858\pi$
$654$		0			0
$655$	59.7154			0.0911685
$656$		0			0
$657$	324.634	+	650.859i	0.494116	+	0.990652i
$658$		0			0
$659$	−485.418			−0.736598			−0.368299	−	0.929707i	$-0.620060\pi$
$659$	−485.418			−0.736598			−0.368299	+	0.929707i	$0.620060\pi$
$660$		0			0
$661$		−	89.9502i		−	0.136082i	−0.997683	−	0.0680410i	$-0.978325\pi$
$661$		−	89.9502i		−	0.136082i	0.997683	−	0.0680410i	$-0.0216749\pi$
$662$		0			0
$663$	−209.519	−	129.632i	−0.316017	−	0.195523i
$664$		0			0
$665$			155.519i			0.233863i
$666$		0			0
$667$		−	210.197i		−	0.315138i
$668$		0			0
$669$	−374.763	−	231.869i	−0.560184	−	0.346591i
$670$		0			0
$671$		−	1200.81i		−	1.78958i
$672$		0			0
$673$	779.599			1.15839			0.579197	−	0.815188i	$-0.303366\pi$
$673$	779.599			1.15839			0.579197	+	0.815188i	$0.303366\pi$
$674$		0			0
$675$	57.3415	−	625.665i	0.0849503	−	0.926911i
$676$		0			0
$677$	207.151			0.305984			0.152992	−	0.988227i	$-0.451109\pi$
$677$	207.151			0.305984			0.152992	+	0.988227i	$0.451109\pi$
$678$		0			0
$679$	−209.134			−0.308003
$680$		0			0
$681$	72.7724	+	45.0250i	0.106861	+	0.0661160i
$682$		0			0
$683$	680.228			0.995941			0.497970	−	0.867194i	$-0.334079\pi$
$683$	680.228			0.995941			0.497970	+	0.867194i	$0.334079\pi$
$684$		0			0
$685$			235.652i			0.344017i
$686$		0			0
$687$	−544.748	+	880.458i	−0.792938	+	1.28160i
$688$		0			0
$689$		−	1524.12i		−	2.21208i
$690$		0			0
$691$			480.570i			0.695470i	0.937593	+	0.347735i	$0.113049\pi$
$691$			480.570i			0.695470i	−0.937593	+	0.347735i	$0.886951\pi$
$692$		0			0
$693$	684.910	+	1373.17i	0.988326	+	1.98149i
$694$		0			0
$695$			66.9297i			0.0963018i
$696$		0			0
$697$	344.680			0.494520
$698$		0			0
$699$	−166.165	+	268.567i	−0.237718	+	0.384216i
$700$		0			0
$701$	−854.903			−1.21955			−0.609774	−	0.792575i	$-0.708740\pi$
$701$	−854.903			−1.21955			−0.609774	+	0.792575i	$0.708740\pi$
$702$		0			0
$703$	−214.656			−0.305343
$704$		0			0
$705$	−41.3523	+	66.8364i	−0.0586558	+	0.0948034i
$706$		0			0
$707$	−1467.13			−2.07515
$708$		0			0
$709$			990.069i			1.39643i	0.715888	+	0.698215i	$0.246022\pi$
$709$			990.069i			1.39643i	−0.715888	+	0.698215i	$0.753978\pi$
$710$		0			0
$711$	−345.593	−	692.880i	−0.486067	−	0.974514i
$712$		0			0
$713$			340.285i			0.477258i
$714$		0			0
$715$			410.828i			0.574585i
$716$		0			0
$717$	442.684	−	715.495i	0.617411	−	0.997901i
$718$		0			0
$719$		−	502.244i		−	0.698531i	−0.937024	−	0.349266i	$-0.886431\pi$
$719$		−	502.244i		−	0.698531i	0.937024	−	0.349266i	$-0.113569\pi$
$720$		0			0
$721$	−1163.97			−1.61438
$722$		0			0
$723$	−943.511	−	583.760i	−1.30499	−	0.807413i
$724$		0			0
$725$	−292.032			−0.402803
$726$		0			0
$727$	718.280			0.988005			0.494002	−	0.869461i	$-0.335533\pi$
$727$	718.280			0.988005			0.494002	+	0.869461i	$0.335533\pi$
$728$		0			0
$729$	−716.856	−	132.511i	−0.983341	−	0.181771i
$730$		0			0
$731$	159.850			0.218673
$732$		0			0
$733$		−	1362.31i		−	1.85854i	−0.369397	−	0.929272i	$-0.620436\pi$
$733$		−	1362.31i		−	1.85854i	0.369397	−	0.929272i	$-0.379564\pi$
$734$		0			0
$735$	−186.950	−	115.668i	−0.254354	−	0.157371i
$736$		0			0
$737$		−	938.939i		−	1.27400i
$738$		0			0
$739$			1057.01i			1.43033i	0.698957	+	0.715164i	$0.253648\pi$
$739$			1057.01i			1.43033i	−0.698957	+	0.715164i	$0.746352\pi$
$740$		0			0
$741$	−552.568	−	341.879i	−0.745705	−	0.461375i
$742$		0			0
$743$		−	88.0799i		−	0.118546i	−0.998242	−	0.0592731i	$-0.981122\pi$
$743$		−	88.0799i		−	0.118546i	0.998242	−	0.0592731i	$-0.0188783\pi$
$744$		0			0
$745$	100.019			0.134254
$746$		0			0
$747$	−323.076	−	647.736i	−0.432499	−	0.867116i
$748$		0			0
$749$	−64.0603			−0.0855278
$750$		0			0
$751$	11.4389			0.0152315			0.00761576	−	0.999971i	$-0.497576\pi$
$751$	11.4389			0.0152315			0.00761576	+	0.999971i	$0.497576\pi$
$752$		0			0
$753$	−70.1470	−	43.4006i	−0.0931566	−	0.0576369i
$754$		0			0
$755$	236.652			0.313447
$756$		0			0
$757$		−	38.6736i		−	0.0510879i	−0.999674	−	0.0255440i	$-0.991868\pi$
$757$		−	38.6736i		−	0.0510879i	0.999674	−	0.0255440i	$-0.00813178\pi$
$758$		0			0
$759$	440.501	−	711.966i	0.580370	−	0.938032i
$760$		0			0
$761$		−	814.341i		−	1.07009i	−0.844822	−	0.535047i	$-0.820294\pi$
$761$		−	814.341i		−	1.07009i	0.844822	−	0.535047i	$-0.179706\pi$
$762$		0			0
$763$			1510.96i			1.98029i
$764$		0			0
$765$	−46.4113	+	23.1489i	−0.0606684	+	0.0302601i
$766$		0			0
$767$		−	561.383i		−	0.731921i
$768$		0			0
$769$	490.085			0.637302			0.318651	−	0.947872i	$-0.396770\pi$
$769$	490.085			0.637302			0.318651	+	0.947872i	$0.396770\pi$
$770$		0			0
$771$	7.77298	−	12.5632i	0.0100817	−	0.0162947i
$772$		0			0
$773$	−747.603			−0.967145			−0.483572	−	0.875304i	$-0.660661\pi$
$773$	−747.603			−0.967145			−0.483572	+	0.875304i	$0.660661\pi$
$774$		0			0
$775$	472.767			0.610022
$776$		0			0
$777$	300.069	−	484.991i	0.386189	−	0.624184i
$778$		0			0
$779$	909.030			1.16692
$780$		0			0
$781$			2273.78i			2.91137i
$782$		0			0
$783$	−30.9251	+	337.430i	−0.0394957	+	0.430946i
$784$		0			0
$785$		−	88.4143i		−	0.112630i
$786$		0			0
$787$			975.671i			1.23973i	0.784707	+	0.619867i	$0.212814\pi$
$787$			975.671i			1.23973i	−0.784707	+	0.619867i	$0.787186\pi$
$788$		0			0
$789$	416.037	−	672.427i	0.527297	−	0.852252i
$790$		0			0
$791$			525.873i			0.664820i
$792$		0			0
$793$	−1350.96			−1.70360
$794$		0			0
$795$	−272.837	−	168.807i	−0.343191	−	0.212336i
$796$		0			0
$797$	417.874			0.524309			0.262154	−	0.965026i	$-0.415567\pi$
$797$	417.874			0.524309			0.262154	+	0.965026i	$0.415567\pi$
$798$		0			0
$799$	−87.2613			−0.109213
$800$		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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.55118 + 1.57844i) q^{3} -1.31534 q^{5} +10.2329 q^{7} +(4.01705 + 8.05378i) q^{9} +O(q^{10})\)	\(q+(2.55118 + 1.57844i) q^{3} -1.31534 q^{5} +10.2329 q^{7} +(4.01705 + 8.05378i) q^{9} +16.6620 q^{11} -18.7454i q^{13} +(-3.35566 - 2.07618i) q^{15} -4.38114i q^{17} -11.5544i q^{19} +(26.1059 + 16.1520i) q^{21} -16.7490i q^{23} -23.2699 q^{25} +(-2.46419 + 26.8873i) q^{27} +12.5498 q^{29} -20.3167 q^{31} +(42.5079 + 26.3001i) q^{33} -13.4597 q^{35} -18.5778i q^{37} +(29.5885 - 47.8230i) q^{39} +78.6737i q^{41} +36.4860i q^{43} +(-5.28377 - 10.5934i) q^{45} -19.9175i q^{47} +55.7118 q^{49} +(6.91537 - 11.1771i) q^{51} +81.3064 q^{53} -21.9162 q^{55} +(18.2380 - 29.4775i) q^{57} +29.9477 q^{59} -72.0687i q^{61} +(41.1060 + 82.4133i) q^{63} +24.6565i q^{65} -56.3520i q^{67} +(26.4374 - 42.7298i) q^{69} +136.465i q^{71} +80.8141 q^{73} +(-59.3657 - 36.7301i) q^{75} +170.501 q^{77} -86.0317 q^{79} +(-48.7266 + 64.7048i) q^{81} -80.4263 q^{83} +5.76268i q^{85} +(32.0168 + 19.8091i) q^{87} +131.830i q^{89} -191.820i q^{91} +(-51.8315 - 32.0687i) q^{93} +15.1980i q^{95} -20.4375 q^{97} +(66.9323 + 134.192i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q - 16q^{7} + O(q^{10}) \)	\( 16q - 16q^{7} + 32q^{15} + 80q^{25} - 112q^{31} + 16q^{33} + 208q^{39} + 144q^{49} - 384q^{55} + 80q^{57} + 528q^{63} + 160q^{73} - 816q^{79} + 144q^{81} + 736q^{87} + 192q^{97} + O(q^{100}) \)

Introduction

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