LMFDB - Embedded newform 1078.2.c.b.1077.10

Newspace parameters

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$ x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1 $ x^16 - 6x^15 + 18x^14 - 36x^13 + 34x^12 + 18x^11 - 72x^10 + 132x^9 - 93x^8 - 102x^7 + 144x^6 - 432x^5 + 502x^4 + 288x^3 + 72x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1077.10
Root			$2.24352 + 0.601150i$ of defining polynomial
Character	$\chi$	$=$	1078.1077
Dual form			1078.2.c.b.1077.7

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{2} -2.59518i q^{3} -1.00000 q^{4} -3.53900i q^{5} +2.59518 q^{6} -1.00000i q^{8} -3.73495 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.59518i q^{3} -1.00000 q^{4} -3.53900i q^{5} +2.59518 q^{6} -1.00000i q^{8} -3.73495 q^{9} +3.53900 q^{10} +(-2.29457 - 2.39477i) q^{11} +2.59518i q^{12} -5.01680 q^{13} -9.18432 q^{15} +1.00000 q^{16} +3.89362 q^{17} -3.73495i q^{18} +4.65637 q^{19} +3.53900i q^{20} +(2.39477 - 2.29457i) q^{22} -1.55962 q^{23} -2.59518 q^{24} -7.52450 q^{25} -5.01680i q^{26} +1.90732i q^{27} +0.100205i q^{29} -9.18432i q^{30} +0.279729i q^{31} +1.00000i q^{32} +(-6.21487 + 5.95482i) q^{33} +3.89362i q^{34} +3.73495 q^{36} +0.705430 q^{37} +4.65637i q^{38} +13.0195i q^{39} -3.53900 q^{40} +2.94809 q^{41} +9.03556i q^{43} +(2.29457 + 2.39477i) q^{44} +13.2180i q^{45} -1.55962i q^{46} +6.56949i q^{47} -2.59518i q^{48} -7.52450i q^{50} -10.1046i q^{51} +5.01680 q^{52} +5.54058 q^{53} -1.90732 q^{54} +(-8.47510 + 8.12048i) q^{55} -12.0841i q^{57} -0.100205 q^{58} -14.5344i q^{59} +9.18432 q^{60} -11.1272 q^{61} -0.279729 q^{62} -1.00000 q^{64} +17.7544i q^{65} +(-5.95482 - 6.21487i) q^{66} +3.98392 q^{67} -3.89362 q^{68} +4.04750i q^{69} -8.45381 q^{71} +3.73495i q^{72} -3.89771 q^{73} +0.705430i q^{74} +19.5274i q^{75} -4.65637 q^{76} -13.0195 q^{78} +7.69982i q^{79} -3.53900i q^{80} -6.25501 q^{81} +2.94809i q^{82} +2.87320 q^{83} -13.7795i q^{85} -9.03556 q^{86} +0.260049 q^{87} +(-2.39477 + 2.29457i) q^{88} -5.94902i q^{89} -13.2180 q^{90} +1.55962 q^{92} +0.725946 q^{93} -6.56949 q^{94} -16.4789i q^{95} +2.59518 q^{96} -16.7587i q^{97} +(8.57010 + 8.94436i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} - 32 q^{9}+O(q^{10}) $ 16 * q - 16 * q^4 - 32 * q^9	$ 16 q - 16 q^{4} - 32 q^{9} - 16 q^{11} - 8 q^{15} + 16 q^{16} - 8 q^{22} - 32 q^{23} + 32 q^{36} + 32 q^{37} + 16 q^{44} + 56 q^{53} + 24 q^{58} + 8 q^{60} - 16 q^{64} - 24 q^{67} + 8 q^{71} - 16 q^{78} + 16 q^{81} - 40 q^{86} + 8 q^{88} + 32 q^{92} + 88 q^{93} + 56 q^{99}+O(q^{100}) $ 16 * q - 16 * q^4 - 32 * q^9 - 16 * q^11 - 8 * q^15 + 16 * q^16 - 8 * q^22 - 32 * q^23 + 32 * q^36 + 32 * q^37 + 16 * q^44 + 56 * q^53 + 24 * q^58 + 8 * q^60 - 16 * q^64 - 24 * q^67 + 8 * q^71 - 16 * q^78 + 16 * q^81 - 40 * q^86 + 8 * q^88 + 32 * q^92 + 88 * q^93 + 56 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$.

$n$	$199$	$981$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	2.59518i		−	1.49833i	−0.662385	−	0.749163i	$-0.730456\pi$
$3$		−	2.59518i		−	1.49833i	0.662385	−	0.749163i	$-0.269544\pi$
$4$	−1.00000			−0.500000
$5$		−	3.53900i		−	1.58269i	−0.611372	−	0.791344i	$-0.709382\pi$
$5$		−	3.53900i		−	1.58269i	0.611372	−	0.791344i	$-0.290618\pi$
$6$	2.59518			1.05948
$7$		0			0
$7$		0			0
$8$		−	1.00000i		−	0.353553i
$9$	−3.73495			−1.24498
$10$	3.53900			1.11913
$11$	−2.29457	−	2.39477i	−0.691839	−	0.722052i
$11$	−2.29457	−	2.39477i	−0.691839	−	0.722052i
$12$			2.59518i			0.749163i
$13$	−5.01680			−1.39141			−0.695704	−	0.718328i	$-0.744908\pi$
$13$	−5.01680			−1.39141			−0.695704	+	0.718328i	$0.744908\pi$
$14$		0			0
$15$	−9.18432			−2.37138
$16$	1.00000			0.250000
$17$	3.89362			0.944342			0.472171	−	0.881507i	$-0.343471\pi$
$17$	3.89362			0.944342			0.472171	+	0.881507i	$0.343471\pi$
$18$		−	3.73495i		−	0.880335i
$19$	4.65637			1.06825			0.534123	−	0.845407i	$-0.320642\pi$
$19$	4.65637			1.06825			0.534123	+	0.845407i	$0.320642\pi$
$20$			3.53900i			0.791344i
$21$		0			0
$22$	2.39477	−	2.29457i	0.510568	−	0.489204i
$23$	−1.55962			−0.325204			−0.162602	−	0.986692i	$-0.551989\pi$
$23$	−1.55962			−0.325204			−0.162602	+	0.986692i	$0.551989\pi$
$24$	−2.59518			−0.529738
$25$	−7.52450			−1.50490
$26$		−	5.01680i		−	0.983875i
$27$			1.90732i			0.367064i
$28$		0			0
$29$			0.100205i			0.0186075i	0.999957	+	0.00930376i	$0.00296152\pi$
$29$			0.100205i			0.0186075i	−0.999957	+	0.00930376i	$0.997038\pi$
$30$		−	9.18432i		−	1.67682i
$31$			0.279729i			0.0502408i	0.999684	+	0.0251204i	$0.00799691\pi$
$31$			0.279729i			0.0502408i	−0.999684	+	0.0251204i	$0.992003\pi$
$32$			1.00000i			0.176777i
$33$	−6.21487	+	5.95482i	−1.08187	+	1.03660i
$34$			3.89362i			0.667750i
$35$		0			0
$36$	3.73495			0.622491
$37$	0.705430			0.115972			0.0579860	−	0.998317i	$-0.481532\pi$
$37$	0.705430			0.115972			0.0579860	+	0.998317i	$0.481532\pi$
$38$			4.65637i			0.755364i
$39$			13.0195i			2.08478i
$40$	−3.53900			−0.559564
$41$	2.94809			0.460415			0.230207	−	0.973142i	$-0.426060\pi$
$41$	2.94809			0.460415			0.230207	+	0.973142i	$0.426060\pi$
$42$		0			0
$43$			9.03556i			1.37791i	0.724804	+	0.688955i	$0.241930\pi$
$43$			9.03556i			1.37791i	−0.724804	+	0.688955i	$0.758070\pi$
$44$	2.29457	+	2.39477i	0.345919	+	0.361026i
$45$			13.2180i			1.97042i
$46$		−	1.55962i		−	0.229954i
$47$			6.56949i			0.958259i	0.877744	+	0.479129i	$0.159048\pi$
$47$			6.56949i			0.958259i	−0.877744	+	0.479129i	$0.840952\pi$
$48$		−	2.59518i		−	0.374582i
$49$		0			0
$50$		−	7.52450i		−	1.06412i
$51$		−	10.1046i		−	1.41493i
$52$	5.01680			0.695704
$53$	5.54058			0.761057			0.380529	−	0.924769i	$-0.375742\pi$
$53$	5.54058			0.761057			0.380529	+	0.924769i	$0.375742\pi$
$54$	−1.90732			−0.259553
$55$	−8.47510	+	8.12048i	−1.14278	+	1.09496i
$56$		0			0
$57$		−	12.0841i		−	1.60058i
$58$	−0.100205			−0.0131575
$59$		−	14.5344i		−	1.89222i	−0.323853	−	0.946108i	$-0.604978\pi$
$59$		−	14.5344i		−	1.89222i	0.323853	−	0.946108i	$-0.395022\pi$
$60$	9.18432			1.18569
$61$	−11.1272			−1.42469			−0.712346	−	0.701828i	$-0.752367\pi$
$61$	−11.1272			−1.42469			−0.712346	+	0.701828i	$0.752367\pi$
$62$	−0.279729			−0.0355256
$63$		0			0
$64$	−1.00000			−0.125000
$65$			17.7544i			2.20217i
$66$	−5.95482	−	6.21487i	−0.732987	−	0.764997i
$67$	3.98392			0.486713			0.243356	−	0.969937i	$-0.421752\pi$
$67$	3.98392			0.486713			0.243356	+	0.969937i	$0.421752\pi$
$68$	−3.89362			−0.472171
$69$			4.04750i			0.487262i
$70$		0			0
$71$	−8.45381			−1.00328			−0.501641	−	0.865076i	$-0.667270\pi$
$71$	−8.45381			−1.00328			−0.501641	+	0.865076i	$0.667270\pi$
$72$			3.73495i			0.440168i
$73$	−3.89771			−0.456192			−0.228096	−	0.973639i	$-0.573250\pi$
$73$	−3.89771			−0.456192			−0.228096	+	0.973639i	$0.573250\pi$
$74$			0.705430i			0.0820045i
$75$			19.5274i			2.25483i
$76$	−4.65637			−0.534123
$77$		0			0
$78$	−13.0195			−1.47417
$79$			7.69982i			0.866297i	0.901322	+	0.433149i	$0.142598\pi$
$79$			7.69982i			0.866297i	−0.901322	+	0.433149i	$0.857402\pi$
$80$		−	3.53900i		−	0.395672i
$81$	−6.25501			−0.695001
$82$			2.94809i			0.325562i
$83$	2.87320			0.315375			0.157687	−	0.987489i	$-0.449596\pi$
$83$	2.87320			0.315375			0.157687	+	0.987489i	$0.449596\pi$
$84$		0			0
$85$		−	13.7795i		−	1.49460i
$86$	−9.03556			−0.974330
$87$	0.260049			0.0278801
$88$	−2.39477	+	2.29457i	−0.255284	+	0.244602i
$89$		−	5.94902i		−	0.630595i	−0.948993	−	0.315297i	$-0.897896\pi$
$89$		−	5.94902i		−	0.630595i	0.948993	−	0.315297i	$-0.102104\pi$
$90$	−13.2180			−1.39330
$91$		0			0
$92$	1.55962			0.162602
$93$	0.725946			0.0752771
$94$	−6.56949			−0.677591
$95$		−	16.4789i		−	1.69070i
$96$	2.59518			0.264869
$97$		−	16.7587i		−	1.70159i	−0.525502	−	0.850793i	$-0.676122\pi$
$97$		−	16.7587i		−	1.70159i	0.525502	−	0.850793i	$-0.323878\pi$
$98$		0			0
$99$	8.57010	+	8.94436i	0.861327	+	0.898942i
$100$	7.52450			0.752450
$101$	−1.40290			−0.139594			−0.0697971	−	0.997561i	$-0.522235\pi$
$101$	−1.40290			−0.139594			−0.0697971	+	0.997561i	$0.522235\pi$
$102$	10.1046			1.00051
$103$			2.06291i			0.203264i	0.994822	+	0.101632i	$0.0324064\pi$
$103$			2.06291i			0.203264i	−0.994822	+	0.101632i	$0.967594\pi$
$104$			5.01680i			0.491937i
$105$		0			0
$106$			5.54058i			0.538149i
$107$		−	12.2655i		−	1.18575i	−0.805295	−	0.592875i	$-0.797993\pi$
$107$		−	12.2655i		−	1.18575i	0.805295	−	0.592875i	$-0.202007\pi$
$108$		−	1.90732i		−	0.183532i
$109$		−	6.72491i		−	0.644129i	−0.946718	−	0.322065i	$-0.895623\pi$
$109$		−	6.72491i		−	0.644129i	0.946718	−	0.322065i	$-0.104377\pi$
$110$	−8.12048	−	8.47510i	−0.774257	−	0.808069i
$111$		−	1.83072i		−	0.173764i
$112$		0			0
$113$	−11.9900			−1.12792			−0.563960	−	0.825802i	$-0.690723\pi$
$113$	−11.9900			−1.12792			−0.563960	+	0.825802i	$0.690723\pi$
$114$	12.0841			1.13178
$115$			5.51950i			0.514696i
$116$		−	0.100205i		−	0.00930376i
$117$	18.7375			1.73228
$118$	14.5344			1.33800
$119$		0			0
$120$			9.18432i			0.838410i
$121$	−0.469894	+	10.9900i	−0.0427176	+	0.999087i
$122$		−	11.1272i		−	1.00741i
$123$		−	7.65083i		−	0.689852i
$124$		−	0.279729i		−	0.0251204i
$125$			8.93419i			0.799098i
$126$		0			0
$127$		−	13.2199i		−	1.17308i	−0.809922	−	0.586538i	$-0.800491\pi$
$127$		−	13.2199i		−	1.17308i	0.809922	−	0.586538i	$-0.199509\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	23.4489			2.06456
$130$	−17.7544			−1.55717
$131$	16.4986			1.44149			0.720746	−	0.693200i	$-0.243800\pi$
$131$	16.4986			1.44149			0.720746	+	0.693200i	$0.243800\pi$
$132$	6.21487	−	5.95482i	0.540935	−	0.518300i
$133$		0			0
$134$			3.98392i			0.344158i
$135$	6.74999			0.580947
$136$		−	3.89362i		−	0.333875i
$137$	6.64374			0.567613			0.283807	−	0.958882i	$-0.408403\pi$
$137$	6.64374			0.567613			0.283807	+	0.958882i	$0.408403\pi$
$138$	−4.04750			−0.344546
$139$	−14.3533			−1.21743			−0.608716	−	0.793388i	$-0.708315\pi$
$139$	−14.3533			−1.21743			−0.608716	+	0.793388i	$0.708315\pi$
$140$		0			0
$141$	17.0490			1.43578
$142$		−	8.45381i		−	0.709428i
$143$	11.5114	+	12.0141i	0.962631	+	1.00467i
$144$	−3.73495			−0.311246
$145$	0.354624			0.0294499
$146$		−	3.89771i		−	0.322577i
$147$		0			0
$148$	−0.705430			−0.0579860
$149$		−	2.66587i		−	0.218397i	−0.994020	−	0.109198i	$-0.965172\pi$
$149$		−	2.66587i		−	0.218397i	0.994020	−	0.109198i	$-0.0348284\pi$
$150$	−19.5274			−1.59441
$151$			10.3903i			0.845554i	0.906234	+	0.422777i	$0.138945\pi$
$151$			10.3903i			0.845554i	−0.906234	+	0.422777i	$0.861055\pi$
$152$		−	4.65637i		−	0.377682i
$153$	−14.5425			−1.17569
$154$		0			0
$155$	0.989959			0.0795154
$156$		−	13.0195i		−	1.04239i
$157$		−	8.37473i		−	0.668376i	−0.942506	−	0.334188i	$-0.891538\pi$
$157$		−	8.37473i		−	0.668376i	0.942506	−	0.334188i	$-0.108462\pi$
$158$	−7.69982			−0.612565
$159$		−	14.3788i		−	1.14031i
$160$	3.53900			0.279782
$161$		0			0
$162$		−	6.25501i		−	0.491440i
$163$	3.06508			0.240075			0.120038	−	0.992769i	$-0.461698\pi$
$163$	3.06508			0.240075			0.120038	+	0.992769i	$0.461698\pi$
$164$	−2.94809			−0.230207
$165$	21.0741	+	21.9944i	1.64061	+	1.71226i
$166$			2.87320i			0.223003i
$167$	10.0731			0.779482			0.389741	−	0.920925i	$-0.372565\pi$
$167$	10.0731			0.779482			0.389741	+	0.920925i	$0.372565\pi$
$168$		0			0
$169$	12.1682			0.936018
$170$	13.7795			1.05684
$171$	−17.3913			−1.32995
$172$		−	9.03556i		−	0.688955i
$173$	0.0766026			0.00582399			0.00291200	−	0.999996i	$-0.499073\pi$
$173$	0.0766026			0.00582399			0.00291200	+	0.999996i	$0.499073\pi$
$174$			0.260049i			0.0197142i
$175$		0			0
$176$	−2.29457	−	2.39477i	−0.172960	−	0.180513i
$177$	−37.7193			−2.83516
$178$	5.94902			0.445898
$179$	−8.97491			−0.670817			−0.335408	−	0.942073i	$-0.608874\pi$
$179$	−8.97491			−0.670817			−0.335408	+	0.942073i	$0.608874\pi$
$180$		−	13.2180i		−	0.985209i
$181$		−	15.1450i		−	1.12572i	−0.826554	−	0.562858i	$-0.809702\pi$
$181$		−	15.1450i		−	1.12572i	0.826554	−	0.562858i	$-0.190298\pi$
$182$		0			0
$183$			28.8771i			2.13465i
$184$			1.55962i			0.114977i
$185$		−	2.49651i		−	0.183547i
$186$			0.725946i			0.0532289i
$187$	−8.93419	−	9.32434i	−0.653332	−	0.681864i
$188$		−	6.56949i		−	0.479129i
$189$		0			0
$190$	16.4789			1.19550
$191$	6.00000			0.434145			0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000			0.434145			0.217072	+	0.976156i	$0.430349\pi$
$192$			2.59518i			0.187291i
$193$		−	14.8281i		−	1.06735i	−0.845690	−	0.533674i	$-0.820811\pi$
$193$		−	14.8281i		−	1.06735i	0.845690	−	0.533674i	$-0.179189\pi$
$194$	16.7587			1.20320
$195$	46.0759			3.29956
$196$		0			0
$197$			14.4745i			1.03126i	0.856811	+	0.515631i	$0.172443\pi$
$197$			14.4745i			1.03126i	−0.856811	+	0.515631i	$0.827557\pi$
$198$	−8.94436	+	8.57010i	−0.635648	+	0.609050i
$199$			11.2259i			0.795780i	0.917433	+	0.397890i	$0.130258\pi$
$199$			11.2259i			0.795780i	−0.917433	+	0.397890i	$0.869742\pi$
$200$			7.52450i			0.532062i
$201$		−	10.3390i		−	0.729255i
$202$		−	1.40290i		−	0.0987080i
$203$		0			0
$204$			10.1046i			0.707466i
$205$		−	10.4333i		−	0.728693i
$206$	−2.06291			−0.143729
$207$	5.82511			0.404873
$208$	−5.01680			−0.347852
$209$	−10.6844	−	11.1510i	−0.739054	−	0.771329i
$210$		0			0
$211$		−	6.44038i		−	0.443374i	−0.975118	−	0.221687i	$-0.928844\pi$
$211$		−	6.44038i		−	0.443374i	0.975118	−	0.221687i	$-0.0711563\pi$
$212$	−5.54058			−0.380529
$213$			21.9391i			1.50324i
$214$	12.2655			0.838451
$215$	31.9768			2.18080
$216$	1.90732			0.129777
$217$		0			0
$218$	6.72491			0.455468
$219$			10.1152i			0.683525i
$220$	8.47510	−	8.12048i	0.571391	−	0.547482i
$221$	−19.5335			−1.31397
$222$	1.83072			0.122870
$223$		−	4.01793i		−	0.269061i	−0.990909	−	0.134530i	$-0.957047\pi$
$223$		−	4.01793i		−	0.269061i	0.990909	−	0.134530i	$-0.0429526\pi$
$224$		0			0
$225$	28.1036			1.87357
$226$		−	11.9900i		−	0.797560i
$227$	12.7293			0.844874			0.422437	−	0.906392i	$-0.361175\pi$
$227$	12.7293			0.844874			0.422437	+	0.906392i	$0.361175\pi$
$228$			12.0841i			0.800290i
$229$		−	6.01345i		−	0.397380i	−0.980062	−	0.198690i	$-0.936331\pi$
$229$		−	6.01345i		−	0.397380i	0.980062	−	0.198690i	$-0.0636687\pi$
$230$	−5.51950			−0.363945
$231$		0			0
$232$	0.100205			0.00657875
$233$		−	6.46694i		−	0.423663i	−0.977306	−	0.211832i	$-0.932057\pi$
$233$		−	6.46694i		−	0.423663i	0.977306	−	0.211832i	$-0.0679428\pi$
$234$			18.7375i			1.22491i
$235$	23.2494			1.51662
$236$			14.5344i			0.946108i
$237$	19.9824			1.29800
$238$		0			0
$239$			15.5775i			1.00762i	0.863813	+	0.503812i	$0.168070\pi$
$239$			15.5775i			1.00762i	−0.863813	+	0.503812i	$0.831930\pi$
$240$	−9.18432			−0.592846
$241$	−23.4762			−1.51224			−0.756119	−	0.654434i	$-0.772907\pi$
$241$	−23.4762			−1.51224			−0.756119	+	0.654434i	$0.772907\pi$
$242$	−10.9900	−	0.469894i	−0.706461	−	0.0302059i
$243$			21.9548i			1.40840i
$244$	11.1272			0.712346
$245$		0			0
$246$	7.65083			0.487799
$247$	−23.3601			−1.48637
$248$	0.279729			0.0177628
$249$		−	7.45646i		−	0.472534i
$250$	−8.93419			−0.565048
$251$			18.2818i			1.15394i	0.816766	+	0.576968i	$0.195764\pi$
$251$			18.2818i			1.15394i	−0.816766	+	0.576968i	$0.804236\pi$
$252$		0			0
$253$	3.57867	+	3.73495i	0.224989	+	0.234814i
$254$	13.2199			0.829489
$255$	−35.7603			−2.23940
$256$	1.00000			0.0625000
$257$		−	9.61966i		−	0.600058i	−0.953930	−	0.300029i	$-0.903004\pi$
$257$		−	9.61966i		−	0.600058i	0.953930	−	0.300029i	$-0.0969964\pi$
$258$			23.4489i			1.45986i
$259$		0			0
$260$		−	17.7544i		−	1.10108i
$261$		−	0.374259i		−	0.0231660i
$262$			16.4986i			1.01929i
$263$			0.578666i			0.0356820i	0.999841	+	0.0178410i	$0.00567928\pi$
$263$			0.578666i			0.0356820i	−0.999841	+	0.0178410i	$0.994321\pi$
$264$	5.95482	+	6.21487i	0.366494	+	0.382499i
$265$		−	19.6081i		−	1.20452i
$266$		0			0
$267$	−15.4388			−0.944837
$268$	−3.98392			−0.243356
$269$		−	22.0896i		−	1.34683i	−0.739266	−	0.673413i	$-0.764827\pi$
$269$		−	22.0896i		−	1.34683i	0.739266	−	0.673413i	$-0.235173\pi$
$270$			6.74999i			0.410791i
$271$	4.26222			0.258911			0.129456	−	0.991585i	$-0.458677\pi$
$271$	4.26222			0.258911			0.129456	+	0.991585i	$0.458677\pi$
$272$	3.89362			0.236085
$273$		0			0
$274$			6.64374i			0.401363i
$275$	17.2655	+	18.0195i	1.04115	+	1.08662i
$276$		−	4.04750i		−	0.243631i
$277$		−	18.3653i		−	1.10346i	−0.834022	−	0.551731i	$-0.813968\pi$
$277$		−	18.3653i		−	1.10346i	0.834022	−	0.551731i	$-0.186032\pi$
$278$		−	14.3533i		−	0.860855i
$279$		−	1.04477i		−	0.0625489i
$280$		0			0
$281$			3.72890i			0.222448i	0.993795	+	0.111224i	$0.0354771\pi$
$281$			3.72890i			0.222448i	−0.993795	+	0.111224i	$0.964523\pi$
$282$			17.0490i			1.01525i
$283$	12.7315			0.756811			0.378405	−	0.925640i	$-0.376473\pi$
$283$	12.7315			0.756811			0.378405	+	0.925640i	$0.376473\pi$
$284$	8.45381			0.501641
$285$	−42.7657			−2.53322
$286$	−12.0141	+	11.5114i	−0.710408	+	0.680683i
$287$		0			0
$288$		−	3.73495i		−	0.220084i
$289$	−1.83972			−0.108219
$290$			0.354624i			0.0208242i
$291$	−43.4917			−2.54953
$292$	3.89771			0.228096
$293$	−26.2251			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$293$	−26.2251			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$294$		0			0
$295$	−51.4371			−2.99478
$296$		−	0.705430i		−	0.0410023i
$297$	4.56760	−	4.37648i	0.265039	−	0.253949i
$298$	2.66587			0.154430
$299$	7.82431			0.452492
$300$		−	19.5274i		−	1.12742i
$301$		0			0
$302$	−10.3903			−0.597897
$303$			3.64079i			0.209158i
$304$	4.65637			0.267061
$305$			39.3791i			2.25484i
$306$		−	14.5425i		−	0.831337i
$307$	25.5942			1.46074			0.730369	−	0.683053i	$-0.239348\pi$
$307$	25.5942			1.46074			0.730369	+	0.683053i	$0.239348\pi$
$308$		0			0
$309$	5.35360			0.304556
$310$			0.989959i			0.0562259i
$311$			1.90970i			0.108289i	0.998533	+	0.0541446i	$0.0172432\pi$
$311$			1.90970i			0.108289i	−0.998533	+	0.0541446i	$0.982757\pi$
$312$	13.0195			0.737083
$313$		−	30.4503i		−	1.72115i	−0.509323	−	0.860576i	$-0.670104\pi$
$313$		−	30.4503i		−	1.72115i	0.509323	−	0.860576i	$-0.329896\pi$
$314$	8.37473			0.472613
$315$		0			0
$316$		−	7.69982i		−	0.433149i
$317$	34.3942			1.93177			0.965885	−	0.258973i	$-0.0833841\pi$
$317$	34.3942			1.93177			0.965885	+	0.258973i	$0.0833841\pi$
$318$	14.3788			0.806323
$319$	0.239967	−	0.229926i	0.0134356	−	0.0128734i
$320$			3.53900i			0.197836i
$321$	−31.8311			−1.77664
$322$		0			0
$323$	18.1302			1.00879
$324$	6.25501			0.347501
$325$	37.7489			2.09393
$326$			3.06508i			0.169759i
$327$	−17.4523			−0.965116
$328$		−	2.94809i		−	0.162781i
$329$		0			0
$330$	−21.9944	+	21.0741i	−1.21075	+	1.16009i
$331$	15.5128			0.852663			0.426332	−	0.904567i	$-0.359806\pi$
$331$	15.5128			0.852663			0.426332	+	0.904567i	$0.359806\pi$
$332$	−2.87320			−0.157687
$333$	−2.63474			−0.144383
$334$			10.0731i			0.551177i
$335$		−	14.0991i		−	0.770314i
$336$		0			0
$337$		−	7.70882i		−	0.419926i	−0.977709	−	0.209963i	$-0.932666\pi$
$337$		−	7.70882i		−	0.419926i	0.977709	−	0.209963i	$-0.0673344\pi$
$338$			12.1682i			0.661865i
$339$			31.1161i			1.68999i
$340$			13.7795i			0.747299i
$341$	0.669887	−	0.641857i	0.0362764	−	0.0347585i
$342$		−	17.3913i		−	0.940415i
$343$		0			0
$344$	9.03556			0.487165
$345$	14.3241			0.771183
$346$			0.0766026i			0.00411818i
$347$			21.7413i			1.16713i	0.812065	+	0.583567i	$0.198343\pi$
$347$			21.7413i			1.16713i	−0.812065	+	0.583567i	$0.801657\pi$
$348$	−0.260049			−0.0139401
$349$	13.4565			0.720307			0.360154	−	0.932893i	$-0.382724\pi$
$349$	13.4565			0.720307			0.360154	+	0.932893i	$0.382724\pi$
$350$		0			0
$351$		−	9.56862i		−	0.510735i
$352$	2.39477	−	2.29457i	0.127642	−	0.122301i
$353$		−	6.70769i		−	0.357014i	−0.983939	−	0.178507i	$-0.942873\pi$
$353$		−	6.70769i		−	0.357014i	0.983939	−	0.178507i	$-0.0571267\pi$
$354$		−	37.7193i		−	2.00476i
$355$			29.9180i			1.58788i
$356$			5.94902i			0.315297i
$357$		0			0
$358$		−	8.97491i		−	0.474339i
$359$			31.0463i			1.63856i	0.573391	+	0.819282i	$0.305627\pi$
$359$			31.0463i			1.63856i	−0.573391	+	0.819282i	$0.694373\pi$
$360$	13.2180			0.696648
$361$	2.68183			0.141149
$362$	15.1450			0.796002
$363$	28.5209	+	1.21946i	1.49696	+	0.0640050i
$364$		0			0
$365$			13.7940i			0.722010i
$366$	−28.8771			−1.50943
$367$		−	29.6492i		−	1.54768i	−0.633384	−	0.773838i	$-0.718335\pi$
$367$		−	29.6492i		−	1.54768i	0.633384	−	0.773838i	$-0.281665\pi$
$368$	−1.55962			−0.0813010
$369$	−11.0110			−0.573208
$370$	2.49651			0.129788
$371$		0			0
$372$	−0.725946			−0.0376385
$373$		−	8.78055i		−	0.454640i	−0.973820	−	0.227320i	$-0.927004\pi$
$373$		−	8.78055i		−	0.454640i	0.973820	−	0.227320i	$-0.0729963\pi$
$374$	9.32434	−	8.93419i	0.482150	−	0.461976i
$375$	23.1858			1.19731
$376$	6.56949			0.338796
$377$		−	0.502706i		−	0.0258907i
$378$		0			0
$379$	30.8135			1.58278			0.791391	−	0.611311i	$-0.209358\pi$
$379$	30.8135			1.58278			0.791391	+	0.611311i	$0.209358\pi$
$380$			16.4789i			0.845349i
$381$	−34.3080			−1.75765
$382$			6.00000i			0.306987i
$383$			5.29482i			0.270553i	0.990808	+	0.135276i	$0.0431922\pi$
$383$			5.29482i			0.270553i	−0.990808	+	0.135276i	$0.956808\pi$
$384$	−2.59518			−0.132435
$385$		0			0
$386$	14.8281			0.754729
$387$		−	33.7473i		−	1.71547i
$388$			16.7587i			0.850793i
$389$	0.860237			0.0436158			0.0218079	−	0.999762i	$-0.493058\pi$
$389$	0.860237			0.0436158			0.0218079	+	0.999762i	$0.493058\pi$
$390$			46.0759i			2.33314i
$391$	−6.07258			−0.307104
$392$		0			0
$393$		−	42.8168i		−	2.15982i
$394$	−14.4745			−0.729213
$395$	27.2496			1.37108
$396$	−8.57010	−	8.94436i	−0.430664	−	0.449471i
$397$			12.3473i			0.619695i	0.950786	+	0.309847i	$0.100278\pi$
$397$			12.3473i			0.619695i	−0.950786	+	0.309847i	$0.899722\pi$
$398$	−11.2259			−0.562702
$399$		0			0
$400$	−7.52450			−0.376225
$401$	33.7252			1.68416			0.842078	−	0.539355i	$-0.181332\pi$
$401$	33.7252			1.68416			0.842078	+	0.539355i	$0.181332\pi$
$402$	10.3390			0.515661
$403$		−	1.40334i		−	0.0699054i
$404$	1.40290			0.0697971
$405$			22.1365i			1.09997i
$406$		0			0
$407$	−1.61866	−	1.68935i	−0.0802339	−	0.0837377i
$408$	−10.1046			−0.500254
$409$	−38.3692			−1.89723			−0.948616	−	0.316428i	$-0.897516\pi$
$409$	−38.3692			−1.89723			−0.948616	+	0.316428i	$0.897516\pi$
$410$	10.4333			0.515264
$411$		−	17.2417i		−	0.850470i
$412$		−	2.06291i		−	0.101632i
$413$		0			0
$414$			5.82511i			0.286289i
$415$		−	10.1682i		−	0.499139i
$416$		−	5.01680i		−	0.245969i
$417$			37.2494i			1.82411i
$418$	11.1510	−	10.6844i	0.545412	−	0.522590i
$419$			33.5023i			1.63670i	0.574723	+	0.818348i	$0.305110\pi$
$419$			33.5023i			1.63670i	−0.574723	+	0.818348i	$0.694890\pi$
$420$		0			0
$421$	−23.0155			−1.12171			−0.560853	−	0.827915i	$-0.689527\pi$
$421$	−23.0155			−1.12171			−0.560853	+	0.827915i	$0.689527\pi$
$422$	6.44038			0.313513
$423$		−	24.5367i		−	1.19302i
$424$		−	5.54058i		−	0.269074i
$425$	−29.2975			−1.42114
$426$	−21.9391			−1.06295
$427$		0			0
$428$			12.2655i			0.592875i
$429$	31.1787	−	29.8741i	1.50532	−	1.44234i
$430$			31.9768i			1.54206i
$431$		−	26.8381i		−	1.29275i	−0.763021	−	0.646373i	$-0.776285\pi$
$431$		−	26.8381i		−	1.29275i	0.763021	−	0.646373i	$-0.223715\pi$
$432$			1.90732i			0.0917659i
$433$			30.6912i			1.47493i	0.675387	+	0.737463i	$0.263976\pi$
$433$			30.6912i			1.47493i	−0.675387	+	0.737463i	$0.736024\pi$
$434$		0			0
$435$		−	0.920312i		−	0.0441256i
$436$			6.72491i			0.322065i
$437$	−7.26219			−0.347398
$438$	−10.1152			−0.483325
$439$	−16.8452			−0.803976			−0.401988	−	0.915645i	$-0.631681\pi$
$439$	−16.8452			−0.803976			−0.401988	+	0.915645i	$0.631681\pi$
$440$	8.12048	+	8.47510i	0.387129	+	0.404035i
$441$		0			0
$442$		−	19.5335i		−	0.929114i
$443$	−0.790590			−0.0375620			−0.0187810	−	0.999824i	$-0.505979\pi$
$443$	−0.790590			−0.0375620			−0.0187810	+	0.999824i	$0.505979\pi$
$444$			1.83072i			0.0868819i
$445$	−21.0536			−0.998035
$446$	4.01793			0.190255
$447$	−6.91841			−0.327230
$448$		0			0
$449$	15.9734			0.753833			0.376917	−	0.926247i	$-0.376984\pi$
$449$	15.9734			0.753833			0.376917	+	0.926247i	$0.376984\pi$
$450$			28.1036i			1.32482i
$451$	−6.76461	−	7.06002i	−0.318533	−	0.332443i
$452$	11.9900			0.563960
$453$	26.9648			1.26692
$454$			12.7293i			0.597416i
$455$		0			0
$456$	−12.0841			−0.565891
$457$		−	34.9331i		−	1.63410i	−0.576564	−	0.817052i	$-0.695607\pi$
$457$		−	34.9331i		−	1.63410i	0.576564	−	0.817052i	$-0.304393\pi$
$458$	6.01345			0.280990
$459$			7.42637i			0.346633i
$460$		−	5.51950i		−	0.257348i
$461$	20.8346			0.970364			0.485182	−	0.874413i	$-0.338753\pi$
$461$	20.8346			0.970364			0.485182	+	0.874413i	$0.338753\pi$
$462$		0			0
$463$	13.1724			0.612172			0.306086	−	0.952004i	$-0.400981\pi$
$463$	13.1724			0.612172			0.306086	+	0.952004i	$0.400981\pi$
$464$			0.100205i			0.00465188i
$465$		−	2.56912i		−	0.119140i
$466$	6.46694			0.299575
$467$		−	16.8184i		−	0.778262i	−0.921182	−	0.389131i	$-0.872775\pi$
$467$		−	16.8184i		−	0.778262i	0.921182	−	0.389131i	$-0.127225\pi$
$468$	−18.7375			−0.866140
$469$		0			0
$470$			23.2494i			1.07241i
$471$	−21.7339			−1.00145
$472$	−14.5344			−0.668999
$473$	21.6381	−	20.7327i	0.994922	−	0.953292i
$474$			19.9824i			0.917822i
$475$	−35.0369			−1.60760
$476$		0			0
$477$	−20.6938			−0.947503
$478$	−15.5775			−0.712498
$479$	−22.8582			−1.04442			−0.522208	−	0.852818i	$-0.674891\pi$
$479$	−22.8582			−1.04442			−0.522208	+	0.852818i	$0.674891\pi$
$480$		−	9.18432i		−	0.419205i
$481$	−3.53900			−0.161364
$482$		−	23.4762i		−	1.06931i
$483$		0			0
$484$	0.469894	−	10.9900i	0.0213588	−	0.499544i
$485$	−59.3089			−2.69308
$486$	−21.9548			−0.995891
$487$	31.2628			1.41665			0.708327	−	0.705884i	$-0.249450\pi$
$487$	31.2628			1.41665			0.708327	+	0.705884i	$0.249450\pi$
$488$			11.1272i			0.503705i
$489$		−	7.95442i		−	0.359711i
$490$		0			0
$491$		−	22.4979i		−	1.01532i	−0.861558	−	0.507659i	$-0.830511\pi$
$491$		−	22.4979i		−	1.01532i	0.861558	−	0.507659i	$-0.169489\pi$
$492$			7.65083i			0.344926i
$493$			0.390159i			0.0175719i
$494$		−	23.3601i		−	1.05102i
$495$	31.6541	−	30.3295i	1.42274	−	1.36321i
$496$			0.279729i			0.0125602i
$497$		0			0
$498$	7.45646			0.334132
$499$	−31.9097			−1.42847			−0.714236	−	0.699905i	$-0.753226\pi$
$499$	−31.9097			−1.42847			−0.714236	+	0.699905i	$0.753226\pi$
$500$		−	8.93419i		−	0.399549i
$501$		−	26.1415i		−	1.16792i
$502$	−18.2818			−0.815957
$503$	−30.2687			−1.34962			−0.674808	−	0.737994i	$-0.735773\pi$
$503$	−30.2687			−1.34962			−0.674808	+	0.737994i	$0.735773\pi$
$504$		0			0
$505$			4.96487i			0.220934i
$506$	−3.73495	+	3.57867i	−0.166039	+	0.159091i
$507$		−	31.5787i		−	1.40246i
$508$			13.2199i			0.586538i
$509$		−	23.0542i		−	1.02186i	−0.859622	−	0.510930i	$-0.829301\pi$
$509$		−	23.0542i		−	1.02186i	0.859622	−	0.510930i	$-0.170699\pi$
$510$		−	35.7603i		−	1.58349i
$511$		0			0
$512$			1.00000i			0.0441942i
$513$			8.88119i			0.392114i
$514$	9.61966			0.424305
$515$	7.30061			0.321703
$516$	−23.4489			−1.03228
$517$	15.7325	−	15.0742i	0.691912	−	0.662961i
$518$		0			0
$519$		−	0.198797i		−	0.00872624i
$520$	17.7544			0.778583
$521$			7.98303i			0.349743i	0.984591	+	0.174872i	$0.0559510\pi$
$521$			7.98303i			0.349743i	−0.984591	+	0.174872i	$0.944049\pi$
$522$	0.374259			0.0163809
$523$	5.58280			0.244119			0.122059	−	0.992523i	$-0.461050\pi$
$523$	5.58280			0.244119			0.122059	+	0.992523i	$0.461050\pi$
$524$	−16.4986			−0.720746
$525$		0			0
$526$	−0.578666			−0.0252310
$527$			1.08916i			0.0474444i
$528$	−6.21487	+	5.95482i	−0.270467	+	0.259150i
$529$	−20.5676			−0.894242
$530$	19.6081			0.851721
$531$			54.2851i			2.35577i
$532$		0			0
$533$	−14.7900			−0.640625
$534$		−	15.4388i		−	0.668101i
$535$	−43.4075			−1.87667
$536$		−	3.98392i		−	0.172079i
$537$			23.2915i			1.00510i
$538$	22.0896			0.952350
$539$		0			0
$540$	−6.74999			−0.290473
$541$		−	13.4022i		−	0.576204i	−0.957600	−	0.288102i	$-0.906976\pi$
$541$		−	13.4022i		−	0.576204i	0.957600	−	0.288102i	$-0.0930242\pi$
$542$			4.26222i			0.183078i
$543$	−39.3039			−1.68669
$544$			3.89362i			0.166938i
$545$	−23.7994			−1.01946
$546$		0			0
$547$			24.5161i			1.04823i	0.851647	+	0.524116i	$0.175604\pi$
$547$			24.5161i			1.04823i	−0.851647	+	0.524116i	$0.824396\pi$
$548$	−6.64374			−0.283807
$549$	41.5595			1.77372
$550$	−18.0195	+	17.2655i	−0.768353	+	0.736203i
$551$			0.466590i			0.0198774i
$552$	4.04750			0.172273
$553$		0			0
$554$	18.3653			0.780265
$555$	−6.47890			−0.275014
$556$	14.3533			0.608716
$557$			3.61157i			0.153027i	0.997069	+	0.0765137i	$0.0243789\pi$
$557$			3.61157i			0.153027i	−0.997069	+	0.0765137i	$0.975621\pi$
$558$	1.04477			0.0442287
$559$		−	45.3296i		−	1.91724i
$560$		0			0
$561$	−24.1983	+	23.1858i	−1.02165	+	0.978905i
$562$	−3.72890			−0.157294
$563$	5.85803			0.246887			0.123443	−	0.992352i	$-0.460606\pi$
$563$	5.85803			0.246887			0.123443	+	0.992352i	$0.460606\pi$
$564$	−17.0490			−0.717892
$565$			42.4324i			1.78515i
$566$			12.7315i			0.535146i
$567$		0			0
$568$			8.45381i			0.354714i
$569$			34.6147i			1.45112i	0.688158	+	0.725561i	$0.258420\pi$
$569$			34.6147i			1.45112i	−0.688158	+	0.725561i	$0.741580\pi$
$570$		−	42.7657i		−	1.79126i
$571$		−	7.96636i		−	0.333382i	−0.986009	−	0.166691i	$-0.946692\pi$
$571$		−	7.96636i		−	0.333382i	0.986009	−	0.166691i	$-0.0533082\pi$
$572$	−11.5114	−	12.0141i	−0.481315	−	0.502335i
$573$		−	15.5711i		−	0.650491i
$574$		0			0
$575$	11.7354			0.489399
$576$	3.73495			0.155623
$577$		−	18.3764i		−	0.765018i	−0.923952	−	0.382509i	$-0.875060\pi$
$577$		−	18.3764i		−	0.765018i	0.923952	−	0.382509i	$-0.124940\pi$
$578$		−	1.83972i		−	0.0765223i
$579$	−38.4815			−1.59924
$580$	−0.354624			−0.0147249
$581$		0			0
$582$		−	43.4917i		−	1.80279i
$583$	−12.7133	−	13.2684i	−0.526529	−	0.549523i
$584$			3.89771i			0.161288i
$585$		−	66.3118i		−	2.74166i
$586$		−	26.2251i		−	1.08335i
$587$		−	17.2267i		−	0.711021i	−0.934672	−	0.355511i	$-0.884307\pi$
$587$		−	17.2267i		−	0.711021i	0.934672	−	0.355511i	$-0.115693\pi$
$588$		0			0
$589$			1.30252i			0.0536695i
$590$		−	51.4371i		−	2.11763i
$591$	37.5638			1.54517
$592$	0.705430			0.0289930
$593$	41.2427			1.69363			0.846817	−	0.531884i	$-0.178516\pi$
$593$	41.2427			1.69363			0.846817	+	0.531884i	$0.178516\pi$
$594$	4.37648	+	4.56760i	0.179569	+	0.187411i
$595$		0			0
$596$			2.66587i			0.109198i
$597$	29.1331			1.19234
$598$			7.82431i			0.319960i
$599$	9.95440			0.406726			0.203363	−	0.979103i	$-0.434813\pi$
$599$	9.95440			0.406726			0.203363	+	0.979103i	$0.434813\pi$
$600$	19.5274			0.797203
$601$	−4.76767			−0.194477			−0.0972386	−	0.995261i	$-0.531001\pi$
$601$	−4.76767			−0.194477			−0.0972386	+	0.995261i	$0.531001\pi$
$602$		0			0
$603$	−14.8797			−0.605949
$604$		−	10.3903i		−	0.422777i
$605$	38.8934	+	1.66295i	1.58124	+	0.0676087i
$606$	−3.64079			−0.147897
$607$	−4.55345			−0.184819			−0.0924094	−	0.995721i	$-0.529457\pi$
$607$	−4.55345			−0.184819			−0.0924094	+	0.995721i	$0.529457\pi$
$608$			4.65637i			0.188841i
$609$		0			0
$610$	−39.3791			−1.59441
$611$		−	32.9578i		−	1.33333i
$612$	14.5425			0.587844
$613$			1.44894i			0.0585223i	0.999572	+	0.0292611i	$0.00931544\pi$
$613$			1.44894i			0.0585223i	−0.999572	+	0.0292611i	$0.990685\pi$
$614$			25.5942i			1.03290i
$615$	−27.0762			−1.09182
$616$		0			0
$617$	−2.96752			−0.119468			−0.0597340	−	0.998214i	$-0.519025\pi$
$617$	−2.96752			−0.119468			−0.0597340	+	0.998214i	$0.519025\pi$
$618$			5.35360i			0.215354i
$619$			11.5475i			0.464133i	0.972700	+	0.232067i	$0.0745488\pi$
$619$			11.5475i			0.464133i	−0.972700	+	0.232067i	$0.925451\pi$
$620$	−0.989959			−0.0397577
$621$		−	2.97470i		−	0.119371i
$622$	−1.90970			−0.0765720
$623$		0			0
$624$			13.0195i			0.521196i
$625$	−6.00443			−0.240177
$626$	30.4503			1.21704
$627$	−28.9387	+	27.7279i	−1.15570	+	1.10734i
$628$			8.37473i			0.334188i
$629$	2.74668			0.109517
$630$		0			0
$631$	−27.5215			−1.09562			−0.547808	−	0.836604i	$-0.684537\pi$
$631$	−27.5215			−1.09562			−0.547808	+	0.836604i	$0.684537\pi$
$632$	7.69982			0.306282
$633$	−16.7139			−0.664319
$634$			34.3942i			1.36597i
$635$	−46.7851			−1.85661
$636$			14.3788i			0.570156i
$637$		0			0
$638$	0.229926	+	0.239967i	0.00910288	+	0.00950040i
$639$	31.5745			1.24907
$640$	−3.53900			−0.139891
$641$	33.6547			1.32928			0.664639	−	0.747164i	$-0.268585\pi$
$641$	33.6547			1.32928			0.664639	+	0.747164i	$0.268585\pi$
$642$		−	31.8311i		−	1.25627i
$643$			5.95261i			0.234748i	0.993088	+	0.117374i	$0.0374476\pi$
$643$			5.95261i			0.234748i	−0.993088	+	0.117374i	$0.962552\pi$
$644$		0			0
$645$		−	82.9855i		−	3.26755i
$646$			18.1302i			0.713321i
$647$		−	11.7737i		−	0.462873i	−0.972850	−	0.231437i	$-0.925657\pi$
$647$		−	11.7737i		−	0.462873i	0.972850	−	0.231437i	$-0.0743426\pi$
$648$			6.25501i			0.245720i
$649$	−34.8066	+	33.3502i	−1.36628	+	1.30911i
$650$			37.7489i			1.48063i
$651$		0			0
$652$	−3.06508			−0.120038
$653$	28.8670			1.12965			0.564827	−	0.825209i	$-0.308943\pi$
$653$	28.8670			1.12965			0.564827	+	0.825209i	$0.308943\pi$
$654$		−	17.4523i		−	0.682440i
$655$		−	58.3886i		−	2.28143i
$656$	2.94809			0.115104
$657$	14.5577			0.567952
$658$		0			0
$659$		−	44.6812i		−	1.74053i	−0.492581	−	0.870266i	$-0.663947\pi$
$659$		−	44.6812i		−	1.74053i	0.492581	−	0.870266i	$-0.336053\pi$
$660$	−21.0741	−	21.9944i	−0.820307	−	0.856130i
$661$			39.6341i			1.54159i	0.637084	+	0.770794i	$0.280140\pi$
$661$			39.6341i			1.54159i	−0.637084	+	0.770794i	$0.719860\pi$
$662$			15.5128i			0.602924i
$663$			50.6929i			1.96875i
$664$		−	2.87320i		−	0.111502i
$665$		0			0
$666$		−	2.63474i		−	0.102094i
$667$		−	0.156281i		−	0.00605124i
$668$	−10.0731			−0.389741
$669$	−10.4272			−0.403141
$670$	14.0991			0.544694
$671$	25.5321	+	26.6471i	0.985657	+	1.02870i
$672$		0			0
$673$			2.71886i			0.104804i	0.998626	+	0.0524022i	$0.0166878\pi$
$673$			2.71886i			0.104804i	−0.998626	+	0.0524022i	$0.983312\pi$
$674$	7.70882			0.296933
$675$		−	14.3516i		−	0.552394i
$676$	−12.1682			−0.468009
$677$	−6.05945			−0.232884			−0.116442	−	0.993198i	$-0.537149\pi$
$677$	−6.05945			−0.232884			−0.116442	+	0.993198i	$0.537149\pi$
$678$	−31.1161			−1.19501
$679$		0			0
$680$	−13.7795			−0.528420
$681$		−	33.0348i		−	1.26590i
$682$	0.641857	+	0.669887i	0.0245780	+	0.0256513i
$683$	−6.03409			−0.230888			−0.115444	−	0.993314i	$-0.536829\pi$
$683$	−6.03409			−0.230888			−0.115444	+	0.993314i	$0.536829\pi$
$684$	17.3913			0.664973
$685$		−	23.5122i		−	0.898354i
$686$		0			0
$687$	−15.6060			−0.595405
$688$			9.03556i			0.344478i
$689$	−27.7960			−1.05894
$690$			14.3241i			0.545309i
$691$		−	19.8053i		−	0.753427i	−0.926330	−	0.376714i	$-0.877054\pi$
$691$		−	19.8053i		−	0.753427i	0.926330	−	0.376714i	$-0.122946\pi$
$692$	−0.0766026			−0.00291200
$693$		0			0
$694$	−21.7413			−0.825288
$695$			50.7963i			1.92681i
$696$		−	0.260049i		−	0.00985712i
$697$	11.4788			0.434789
$698$			13.4565i			0.509334i
$699$	−16.7828			−0.634786
$700$		0			0
$701$			26.0155i			0.982591i	0.870993	+	0.491296i	$0.163476\pi$
$701$			26.0155i			0.982591i	−0.870993	+	0.491296i	$0.836524\pi$
$702$	9.56862			0.361144
$703$	3.28475			0.123887
$704$	2.29457	+	2.39477i	0.0864799	+	0.0902565i
$705$		−	60.3363i		−	2.27240i
$706$	6.70769			0.252447
$707$		0			0
$708$	37.7193			1.41758
$709$	−24.8917			−0.934826			−0.467413	−	0.884039i	$-0.654814\pi$
$709$	−24.8917			−0.934826			−0.467413	+	0.884039i	$0.654814\pi$
$710$	−29.9180			−1.12280
$711$		−	28.7584i		−	1.07853i
$712$	−5.94902			−0.222949
$713$		−	0.436271i		−	0.0163385i
$714$		0			0
$715$	42.5178	−	40.7388i	1.59008	−	1.52354i
$716$	8.97491			0.335408
$717$	40.4263			1.50975
$718$	−31.0463			−1.15864
$719$		−	13.2970i		−	0.495895i	−0.968773	−	0.247948i	$-0.920244\pi$
$719$		−	13.2970i		−	0.495895i	0.968773	−	0.247948i	$-0.0797561\pi$
$720$			13.2180i			0.492604i
$721$		0			0
$722$			2.68183i			0.0998073i
$723$			60.9250i			2.26583i
$724$			15.1450i			0.562858i
$725$		−	0.753989i		−	0.0280025i
$726$	−1.21946	+	28.5209i	−0.0452583	+	1.05851i
$727$			5.67083i			0.210319i	0.994455	+	0.105160i	$0.0335354\pi$
$727$			5.67083i			0.210319i	−0.994455	+	0.105160i	$0.966465\pi$
$728$		0			0
$729$	38.2116			1.41525
$730$	−13.7940			−0.510538
$731$			35.1810i			1.30122i
$732$		−	28.8771i		−	1.06733i
$733$	41.3759			1.52825			0.764127	−	0.645066i	$-0.223170\pi$
$733$	41.3759			1.52825			0.764127	+	0.645066i	$0.223170\pi$
$734$	29.6492			1.09437
$735$		0			0
$736$		−	1.55962i		−	0.0574885i
$737$	−9.14137	−	9.54058i	−0.336727	−	0.351432i
$738$		−	11.0110i		−	0.405320i
$739$			41.5403i			1.52808i	0.645167	+	0.764042i	$0.276788\pi$
$739$			41.5403i			1.52808i	−0.645167	+	0.764042i	$0.723212\pi$
$740$			2.49651i			0.0917736i
$741$			60.6236i			2.22706i
$742$		0			0
$743$			39.7404i			1.45793i	0.684549	+	0.728967i	$0.259999\pi$
$743$			39.7404i			1.45793i	−0.684549	+	0.728967i	$0.740001\pi$
$744$		−	0.725946i		−	0.0266145i
$745$	−9.43451			−0.345654
$746$	8.78055			0.321479
$747$	−10.7312			−0.392636
$748$	8.93419	+	9.32434i	0.326666	+	0.340932i
$749$		0			0
$750$			23.1858i			0.846626i
$751$	5.36526			0.195781			0.0978905	−	0.995197i	$-0.468791\pi$
$751$	5.36526			0.195781			0.0978905	+	0.995197i	$0.468791\pi$
$752$			6.56949i			0.239565i
$753$	47.4445			1.72897
$754$	0.502706			0.0183075
$755$	36.7714			1.33825
$756$		0			0
$757$	25.8962			0.941215			0.470607	−	0.882343i	$-0.344035\pi$
$757$	25.8962			0.941215			0.470607	+	0.882343i	$0.344035\pi$
$758$			30.8135i			1.11920i
$759$	9.69285	−	9.28727i	0.351828	−	0.337107i
$760$	−16.4789			−0.597752
$761$	−15.2206			−0.551747			−0.275873	−	0.961194i	$-0.588967\pi$
$761$	−15.2206			−0.551747			−0.275873	+	0.961194i	$0.588967\pi$
$762$		−	34.3080i		−	1.24285i
$763$		0			0
$764$	−6.00000			−0.217072
$765$			51.4657i			1.86075i
$766$	−5.29482			−0.191310
$767$			72.9160i			2.63284i
$768$		−	2.59518i		−	0.0936454i
$769$	−22.5146			−0.811898			−0.405949	−	0.913896i	$-0.633059\pi$
$769$	−22.5146			−0.811898			−0.405949	+	0.913896i	$0.633059\pi$
$770$		0			0
$771$	−24.9647			−0.899083
$772$			14.8281i			0.533674i
$773$		−	28.0333i		−	1.00829i	−0.863619	−	0.504144i	$-0.831808\pi$
$773$		−	28.0333i		−	1.00829i	0.863619	−	0.504144i	$-0.168192\pi$
$774$	33.7473			1.21302
$775$		−	2.10482i		−	0.0756073i
$776$	−16.7587			−0.601601
$777$		0			0
$778$			0.860237i			0.0308410i
$779$	13.7274			0.491836
$780$	−46.0759			−1.64978
$781$	19.3979	+	20.2450i	0.694110	+	0.724422i
$782$		−	6.07258i		−	0.217155i
$783$	−0.191122			−0.00683014
$784$		0			0
$785$	−29.6381			−1.05783
$786$	42.8168			1.52723
$787$	−12.7432			−0.454246			−0.227123	−	0.973866i	$-0.572932\pi$
$787$	−12.7432			−0.454246			−0.227123	+	0.973866i	$0.572932\pi$
$788$		−	14.4745i		−	0.515631i
$789$	1.50174			0.0534634
$790$			27.2496i			0.969499i
$791$		0			0
$792$	8.94436	−	8.57010i	0.317824	−	0.304525i

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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.59518i q^{3} -1.00000 q^{4} -3.53900i q^{5} +2.59518 q^{6} -1.00000i q^{8} -3.73495 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.59518i q^{3} -1.00000 q^{4} -3.53900i q^{5} +2.59518 q^{6} -1.00000i q^{8} -3.73495 q^{9} +3.53900 q^{10} +(-2.29457 - 2.39477i) q^{11} +2.59518i q^{12} -5.01680 q^{13} -9.18432 q^{15} +1.00000 q^{16} +3.89362 q^{17} -3.73495i q^{18} +4.65637 q^{19} +3.53900i q^{20} +(2.39477 - 2.29457i) q^{22} -1.55962 q^{23} -2.59518 q^{24} -7.52450 q^{25} -5.01680i q^{26} +1.90732i q^{27} +0.100205i q^{29} -9.18432i q^{30} +0.279729i q^{31} +1.00000i q^{32} +(-6.21487 + 5.95482i) q^{33} +3.89362i q^{34} +3.73495 q^{36} +0.705430 q^{37} +4.65637i q^{38} +13.0195i q^{39} -3.53900 q^{40} +2.94809 q^{41} +9.03556i q^{43} +(2.29457 + 2.39477i) q^{44} +13.2180i q^{45} -1.55962i q^{46} +6.56949i q^{47} -2.59518i q^{48} -7.52450i q^{50} -10.1046i q^{51} +5.01680 q^{52} +5.54058 q^{53} -1.90732 q^{54} +(-8.47510 + 8.12048i) q^{55} -12.0841i q^{57} -0.100205 q^{58} -14.5344i q^{59} +9.18432 q^{60} -11.1272 q^{61} -0.279729 q^{62} -1.00000 q^{64} +17.7544i q^{65} +(-5.95482 - 6.21487i) q^{66} +3.98392 q^{67} -3.89362 q^{68} +4.04750i q^{69} -8.45381 q^{71} +3.73495i q^{72} -3.89771 q^{73} +0.705430i q^{74} +19.5274i q^{75} -4.65637 q^{76} -13.0195 q^{78} +7.69982i q^{79} -3.53900i q^{80} -6.25501 q^{81} +2.94809i q^{82} +2.87320 q^{83} -13.7795i q^{85} -9.03556 q^{86} +0.260049 q^{87} +(-2.39477 + 2.29457i) q^{88} -5.94902i q^{89} -13.2180 q^{90} +1.55962 q^{92} +0.725946 q^{93} -6.56949 q^{94} -16.4789i q^{95} +2.59518 q^{96} -16.7587i q^{97} +(8.57010 + 8.94436i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} - 32 q^{9}+O(q^{10}) \) 16 * q - 16 * q^4 - 32 * q^9	\( 16 q - 16 q^{4} - 32 q^{9} - 16 q^{11} - 8 q^{15} + 16 q^{16} - 8 q^{22} - 32 q^{23} + 32 q^{36} + 32 q^{37} + 16 q^{44} + 56 q^{53} + 24 q^{58} + 8 q^{60} - 16 q^{64} - 24 q^{67} + 8 q^{71} - 16 q^{78} + 16 q^{81} - 40 q^{86} + 8 q^{88} + 32 q^{92} + 88 q^{93} + 56 q^{99}+O(q^{100}) \) 16 * q - 16 * q^4 - 32 * q^9 - 16 * q^11 - 8 * q^15 + 16 * q^16 - 8 * q^22 - 32 * q^23 + 32 * q^36 + 32 * q^37 + 16 * q^44 + 56 * q^53 + 24 * q^58 + 8 * q^60 - 16 * q^64 - 24 * q^67 + 8 * q^71 - 16 * q^78 + 16 * q^81 - 40 * q^86 + 8 * q^88 + 32 * q^92 + 88 * q^93 + 56 * q^99

Introduction

L-functions

Modular forms

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