LMFDB - Embedded newform 177.7.c.a.58.8

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$40.7195728007$
Analytic rank:	$0$
Dimension:	$60$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.8
Character	$\chi$	$=$	177.58
Dual form			177.7.c.a.58.53

$q$-expansion

$f(q)$	$=$	$q-13.2493i q^{2} -15.5885 q^{3} -111.544 q^{4} -224.166 q^{5} +206.536i q^{6} +460.347 q^{7} +629.930i q^{8} +243.000 q^{9} +O(q^{10})$	\(q-13.2493i q^{2} -15.5885 q^{3} -111.544 q^{4} -224.166 q^{5} +206.536i q^{6} +460.347 q^{7} +629.930i q^{8} +243.000 q^{9} +2970.05i q^{10} -1566.88i q^{11} +1738.80 q^{12} +3966.82i q^{13} -6099.29i q^{14} +3494.40 q^{15} +1207.30 q^{16} +6700.13 q^{17} -3219.58i q^{18} +8706.20 q^{19} +25004.5 q^{20} -7176.11 q^{21} -20760.0 q^{22} -7021.62i q^{23} -9819.63i q^{24} +34625.4 q^{25} +52557.7 q^{26} -3788.00 q^{27} -51349.1 q^{28} -38321.7 q^{29} -46298.5i q^{30} +35296.6i q^{31} +24319.6i q^{32} +24425.2i q^{33} -88772.1i q^{34} -103194. q^{35} -27105.3 q^{36} -59583.7i q^{37} -115351. i q^{38} -61836.6i q^{39} -141209. i q^{40} -7065.43 q^{41} +95078.5i q^{42} -9877.97i q^{43} +174776. i q^{44} -54472.4 q^{45} -93031.6 q^{46} +87041.4i q^{47} -18820.0 q^{48} +94270.7 q^{49} -458763. i q^{50} -104445. q^{51} -442477. i q^{52} -2972.73 q^{53} +50188.3i q^{54} +351241. i q^{55} +289987. i q^{56} -135716. q^{57} +507736. i q^{58} +(193592. - 68575.8i) q^{59} -389781. q^{60} +75860.0i q^{61} +467655. q^{62} +111864. q^{63} +399485. q^{64} -889227. i q^{65} +323617. q^{66} -305808. i q^{67} -747361. q^{68} +109456. i q^{69} +1.36725e6i q^{70} +351118. q^{71} +153073. i q^{72} -146464. i q^{73} -789443. q^{74} -539757. q^{75} -971128. q^{76} -721307. i q^{77} -819293. q^{78} +188502. q^{79} -270636. q^{80} +59049.0 q^{81} +93612.1i q^{82} -371074. i q^{83} +800454. q^{84} -1.50194e6 q^{85} -130876. q^{86} +597376. q^{87} +987022. q^{88} +609230. i q^{89} +721722. i q^{90} +1.82612e6i q^{91} +783222. i q^{92} -550219. i q^{93} +1.15324e6 q^{94} -1.95164e6 q^{95} -379105. i q^{96} -998439. i q^{97} -1.24902e6i q^{98} -380751. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) $	$ 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	13.2493i		−	1.65616i	−0.560607	−	0.828082i	$-0.689432\pi$
$2$		−	13.2493i		−	1.65616i	0.560607	−	0.828082i	$-0.310568\pi$
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	−111.544			−1.74288
$5$	−224.166			−1.79333			−0.896664	−	0.442711i	$-0.854017\pi$
$5$	−224.166			−1.79333			−0.896664	+	0.442711i	$0.854017\pi$
$6$			206.536i			0.956187i
$7$	460.347			1.34212			0.671060	−	0.741403i	$-0.265839\pi$
$7$	460.347			1.34212			0.671060	+	0.741403i	$0.265839\pi$
$8$			629.930i			1.23033i
$9$	243.000			0.333333
$10$			2970.05i			2.97005i
$11$		−	1566.88i		−	1.17722i	−0.808418	−	0.588609i	$-0.799676\pi$
$11$		−	1566.88i		−	1.17722i	0.808418	−	0.588609i	$-0.200324\pi$
$12$	1738.80			1.00625
$13$			3966.82i			1.80556i	0.430099	+	0.902782i	$0.358479\pi$
$13$			3966.82i			1.80556i	−0.430099	+	0.902782i	$0.641521\pi$
$14$		−	6099.29i		−	2.22277i
$15$	3494.40			1.03538
$16$	1207.30			0.294751
$17$	6700.13			1.36375			0.681877	−	0.731466i	$-0.261164\pi$
$17$	6700.13			1.36375			0.681877	+	0.731466i	$0.261164\pi$
$18$		−	3219.58i		−	0.552055i
$19$	8706.20			1.26931			0.634656	−	0.772795i	$-0.281142\pi$
$19$	8706.20			1.26931			0.634656	+	0.772795i	$0.281142\pi$
$20$	25004.5			3.12556
$21$	−7176.11			−0.774874
$22$	−20760.0			−1.94967
$23$		−	7021.62i		−	0.577104i	−0.957464	−	0.288552i	$-0.906826\pi$
$23$		−	7021.62i		−	0.577104i	0.957464	−	0.288552i	$-0.0931737\pi$
$24$		−	9819.63i		−	0.710332i
$25$	34625.4			2.21603
$26$	52557.7			2.99031
$27$	−3788.00			−0.192450
$28$	−51349.1			−2.33916
$29$	−38321.7			−1.57127			−0.785634	−	0.618691i	$-0.787663\pi$
$29$	−38321.7			−1.57127			−0.785634	+	0.618691i	$0.787663\pi$
$30$		−	46298.5i		−	1.71476i
$31$			35296.6i			1.18481i	0.805642	+	0.592403i	$0.201821\pi$
$31$			35296.6i			1.18481i	−0.805642	+	0.592403i	$0.798179\pi$
$32$			24319.6i			0.742175i
$33$			24425.2i			0.679667i
$34$		−	88772.1i		−	2.25860i
$35$	−103194.			−2.40686
$36$	−27105.3			−0.580960
$37$		−	59583.7i		−	1.17631i	−0.808748	−	0.588156i	$-0.799854\pi$
$37$		−	59583.7i		−	1.17631i	0.808748	−	0.588156i	$-0.200146\pi$
$38$		−	115351.i		−	2.10219i
$39$		−	61836.6i		−	1.04244i
$40$		−	141209.i		−	2.20639i
$41$	−7065.43			−0.102515			−0.0512575	−	0.998685i	$-0.516323\pi$
$41$	−7065.43			−0.102515			−0.0512575	+	0.998685i	$0.516323\pi$
$42$			95078.5i			1.28332i
$43$		−	9877.97i		−	0.124240i	−0.998069	−	0.0621201i	$-0.980214\pi$
$43$		−	9877.97i		−	0.124240i	0.998069	−	0.0621201i	$-0.0197862\pi$
$44$			174776.i			2.05175i
$45$	−54472.4			−0.597776
$46$	−93031.6			−0.955778
$47$			87041.4i			0.838363i	0.907902	+	0.419181i	$0.137683\pi$
$47$			87041.4i			0.838363i	−0.907902	+	0.419181i	$0.862317\pi$
$48$	−18820.0			−0.170175
$49$	94270.7			0.801288
$50$		−	458763.i		−	3.67011i
$51$	−104445.			−0.787364
$52$		−	442477.i		−	3.14688i
$53$	−2972.73			−0.0199677			−0.00998384	−	0.999950i	$-0.503178\pi$
$53$	−2972.73			−0.0199677			−0.00998384	+	0.999950i	$0.503178\pi$
$54$			50188.3i			0.318729i
$55$			351241.i			2.11114i
$56$			289987.i			1.65125i
$57$	−135716.			−0.732837
$58$			507736.i			2.60228i
$59$	193592.	−	68575.8i	0.942609	−	0.333899i
$59$	193592.	−	68575.8i	0.942609	−	0.333899i
$60$	−389781.			−1.80454
$61$			75860.0i			0.334213i	0.985939	+	0.167107i	$0.0534424\pi$
$61$			75860.0i			0.334213i	−0.985939	+	0.167107i	$0.946558\pi$
$62$	467655.			1.96223
$63$	111864.			0.447374
$64$	399485.			1.52392
$65$		−	889227.i		−	3.23797i
$66$	323617.			1.12564
$67$		−	305808.i		−	1.01678i	−0.861128	−	0.508388i	$-0.830242\pi$
$67$		−	305808.i		−	1.01678i	0.861128	−	0.508388i	$-0.169758\pi$
$68$	−747361.			−2.37686
$69$			109456.i			0.333191i
$70$			1.36725e6i			3.98616i
$71$	351118.			0.981021			0.490510	−	0.871435i	$-0.336810\pi$
$71$	351118.			0.981021			0.490510	+	0.871435i	$0.336810\pi$
$72$			153073.i			0.410111i
$73$		−	146464.i		−	0.376499i	−0.982121	−	0.188249i	$-0.939719\pi$
$73$		−	146464.i		−	0.376499i	0.982121	−	0.188249i	$-0.0602813\pi$
$74$	−789443.			−1.94817
$75$	−539757.			−1.27942
$76$	−971128.			−2.21226
$77$		−	721307.i		−	1.57997i
$78$	−819293.			−1.72646
$79$	188502.			0.382326			0.191163	−	0.981558i	$-0.438774\pi$
$79$	188502.			0.382326			0.191163	+	0.981558i	$0.438774\pi$
$80$	−270636.			−0.528586
$81$	59049.0			0.111111
$82$			93612.1i			0.169782i
$83$		−	371074.i		−	0.648972i	−0.945890	−	0.324486i	$-0.894809\pi$
$83$		−	371074.i		−	0.648972i	0.945890	−	0.324486i	$-0.105191\pi$
$84$	800454.			1.35051
$85$	−1.50194e6			−2.44566
$86$	−130876.			−0.205762
$87$	597376.			0.907172
$88$	987022.			1.44837
$89$			609230.i			0.864194i	0.901827	+	0.432097i	$0.142226\pi$
$89$			609230.i			0.864194i	−0.901827	+	0.432097i	$0.857774\pi$
$90$			721722.i			0.990016i
$91$			1.82612e6i			2.42328i
$92$			783222.i			1.00582i
$93$		−	550219.i		−	0.684048i
$94$	1.15324e6			1.38847
$95$	−1.95164e6			−2.27629
$96$		−	379105.i		−	0.428495i
$97$		−	998439.i		−	1.09397i	−0.837142	−	0.546986i	$-0.815775\pi$
$97$		−	998439.i		−	1.09397i	0.837142	−	0.546986i	$-0.184225\pi$
$98$		−	1.24902e6i		−	1.32706i
$99$		−	380751.i		−	0.392406i
$100$	−3.86227e6			−3.86227
$101$			1.19215e6i			1.15709i	0.815651	+	0.578544i	$0.196379\pi$
$101$			1.19215e6i			1.15709i	−0.815651	+	0.578544i	$0.803621\pi$
$102$			1.38382e6i			1.30400i
$103$			9419.64i			0.00862031i	0.999991	+	0.00431015i	$0.00137197\pi$
$103$			9419.64i			0.00862031i	−0.999991	+	0.00431015i	$0.998628\pi$
$104$	−2.49882e6			−2.22144
$105$	1.60864e6			1.38960
$106$			39386.6i			0.0330698i
$107$	−413246.			−0.337332			−0.168666	−	0.985673i	$-0.553946\pi$
$107$	−413246.			−0.337332			−0.168666	+	0.985673i	$0.553946\pi$
$108$	422529.			0.335417
$109$		−	1.77849e6i		−	1.37332i	−0.726978	−	0.686661i	$-0.759076\pi$
$109$		−	1.77849e6i		−	1.37332i	0.726978	−	0.686661i	$-0.240924\pi$
$110$	4.65370e6			3.49639
$111$			928818.i			0.679144i
$112$	555778.			0.395592
$113$		−	2.68112e6i		−	1.85815i	−0.369888	−	0.929076i	$-0.620604\pi$
$113$		−	2.68112e6i		−	1.85815i	0.369888	−	0.929076i	$-0.379396\pi$
$114$			1.79815e6i			1.21370i
$115$			1.57401e6i			1.03494i
$116$	4.27456e6			2.73853
$117$			963938.i			0.601854i
$118$	−908582.	−	2.56496e6i	−0.552991	−	1.56112i
$119$	3.08439e6			1.83032
$120$			2.20123e6i			1.27386i
$121$	−683541.			−0.385841
$122$	1.00509e6			0.553512
$123$	110139.			0.0591870
$124$		−	3.93713e6i		−	2.06498i
$125$	−4.25926e6			−2.18074
$126$		−	1.48213e6i		−	0.740924i
$127$	3.22453e6			1.57418			0.787092	−	0.616835i	$-0.211586\pi$
$127$	3.22453e6			1.57418			0.787092	+	0.616835i	$0.211586\pi$
$128$		−	3.73645e6i		−	1.78168i
$129$			153982.i			0.0717301i
$130$	−1.17816e7			−5.36261
$131$		−	2.07176e6i		−	0.921562i	−0.887514	−	0.460781i	$-0.847569\pi$
$131$		−	2.07176e6i		−	0.921562i	0.887514	−	0.460781i	$-0.152431\pi$
$132$		−	2.72449e6i		−	1.18458i
$133$	4.00788e6			1.70357
$134$	−4.05175e6			−1.68395
$135$	849140.			0.345126
$136$			4.22061e6i			1.67787i
$137$	−280882.			−0.109235			−0.0546176	−	0.998507i	$-0.517394\pi$
$137$	−280882.			−0.109235			−0.0546176	+	0.998507i	$0.517394\pi$
$138$	1.45022e6			0.551819
$139$	−2.20910e6			−0.822568			−0.411284	−	0.911507i	$-0.634920\pi$
$139$	−2.20910e6			−0.822568			−0.411284	+	0.911507i	$0.634920\pi$
$140$	1.15107e7			4.19488
$141$		−	1.35684e6i		−	0.484029i
$142$		−	4.65207e6i		−	1.62473i
$143$	6.21552e6			2.12554
$144$	293374.			0.0982504
$145$	8.59042e6			2.81780
$146$	−1.94055e6			−0.623544
$147$	−1.46954e6			−0.462624
$148$			6.64622e6i			2.05017i
$149$		−	1.69318e6i		−	0.511851i	−0.966697	−	0.255925i	$-0.917620\pi$
$149$		−	1.69318e6i		−	0.511851i	0.966697	−	0.255925i	$-0.0823801\pi$
$150$			7.15141e6i			2.11894i
$151$		−	958302.i		−	0.278337i	−0.990269	−	0.139169i	$-0.955557\pi$
$151$		−	958302.i		−	0.278337i	0.990269	−	0.139169i	$-0.0444430\pi$
$152$			5.48430e6i			1.56167i
$153$	1.62813e6			0.454585
$154$	−9.55683e6			−2.61669
$155$		−	7.91230e6i		−	2.12475i
$156$			6.89753e6i			1.81685i
$157$			3.40122e6i			0.878894i	0.898269	+	0.439447i	$0.144826\pi$
$157$			3.40122e6i			0.878894i	−0.898269	+	0.439447i	$0.855174\pi$
$158$		−	2.49752e6i		−	0.633195i
$159$	46340.3			0.0115283
$160$		−	5.45163e6i		−	1.33096i
$161$		−	3.23238e6i		−	0.774543i
$162$		−	782359.i		−	0.184018i
$163$	−182298.			−0.0420938			−0.0210469	−	0.999778i	$-0.506700\pi$
$163$	−182298.			−0.0420938			−0.0210469	+	0.999778i	$0.506700\pi$
$164$	788109.			0.178671
$165$		−	5.47530e6i		−	1.21887i
$166$	−4.91648e6			−1.07481
$167$	4.78402e6			1.02717			0.513587	−	0.858038i	$-0.328316\pi$
$167$	4.78402e6			1.02717			0.513587	+	0.858038i	$0.328316\pi$
$168$		−	4.52044e6i		−	0.953352i
$169$	−1.09089e7			−2.26006
$170$			1.98997e7i			4.05042i
$171$	2.11561e6			0.423104
$172$			1.10183e6i			0.216536i
$173$		−	8.66796e6i		−	1.67409i	−0.547135	−	0.837044i	$-0.684282\pi$
$173$		−	8.66796e6i		−	1.67409i	0.547135	−	0.837044i	$-0.315718\pi$
$174$		−	7.91482e6i		−	1.50243i
$175$	1.59397e7			2.97418
$176$		−	1.89169e6i		−	0.346986i
$177$	−3.01780e6	+	1.06899e6i	−0.544216	+	0.192777i
$178$	8.07188e6			1.43125
$179$			5.24760e6i			0.914960i	0.889220	+	0.457480i	$0.151248\pi$
$179$			5.24760e6i			0.914960i	−0.889220	+	0.457480i	$0.848752\pi$
$180$	6.07608e6			1.04185
$181$	3.23180e6			0.545015			0.272507	−	0.962154i	$-0.412147\pi$
$181$	3.23180e6			0.545015			0.272507	+	0.962154i	$0.412147\pi$
$182$	2.41948e7			4.01336
$183$		−	1.18254e6i		−	0.192958i
$184$	4.42313e6			0.710029
$185$			1.33566e7i			2.10951i
$186$	−7.29003e6			−1.13290
$187$		−	1.04983e7i		−	1.60544i
$188$		−	9.70897e6i		−	1.46117i
$189$	−1.74379e6			−0.258291
$190$			2.58578e7i			3.76991i
$191$			1.90730e6i			0.273728i	0.990590	+	0.136864i	$0.0437023\pi$
$191$			1.90730e6i			0.273728i	−0.990590	+	0.136864i	$0.956298\pi$
$192$	−6.22736e6			−0.879833
$193$	−1.15832e7			−1.61122			−0.805612	−	0.592443i	$-0.798163\pi$
$193$	−1.15832e7			−1.61122			−0.805612	+	0.592443i	$0.798163\pi$
$194$	−1.32286e7			−1.81180
$195$			1.38617e7i			1.86944i
$196$	−1.05154e7			−1.39655
$197$	−2.50665e6			−0.327864			−0.163932	−	0.986472i	$-0.552418\pi$
$197$	−2.50665e6			−0.327864			−0.163932	+	0.986472i	$0.552418\pi$
$198$	−5.04469e6			−0.649888
$199$	1.30088e6			0.165074			0.0825371	−	0.996588i	$-0.473698\pi$
$199$	1.30088e6			0.165074			0.0825371	+	0.996588i	$0.473698\pi$
$200$			2.18116e7i			2.72645i
$201$			4.76708e6i			0.587036i
$202$	1.57952e7			1.91633
$203$	−1.76413e7			−2.10883
$204$	1.16502e7			1.37228
$205$	1.58383e6			0.183843
$206$	124804.			0.0142766
$207$		−	1.70625e6i		−	0.192368i
$208$			4.78915e6i			0.532192i
$209$		−	1.36415e7i		−	1.49426i
$210$		−	2.13134e7i		−	2.30141i
$211$		−	5.44641e6i		−	0.579779i	−0.957060	−	0.289890i	$-0.906381\pi$
$211$		−	5.44641e6i		−	0.579779i	0.957060	−	0.289890i	$-0.0936186\pi$
$212$	331591.			0.0348013
$213$	−5.47339e6			−0.566393
$214$			5.47523e6i			0.558677i
$215$			2.21431e6i			0.222804i
$216$		−	2.38617e6i		−	0.236777i
$217$			1.62487e7i			1.59015i
$218$	−2.35638e7			−2.27445
$219$			2.28316e6i			0.217372i
$220$		−	3.91789e7i		−	3.67946i
$221$			2.65782e7i			2.46235i
$222$	1.23062e7			1.12477
$223$	1.43801e7			1.29672			0.648360	−	0.761334i	$-0.275455\pi$
$223$	1.43801e7			1.29672			0.648360	+	0.761334i	$0.275455\pi$
$224$			1.11955e7i			0.996089i
$225$	8.41398e6			0.738676
$226$	−3.55230e7			−3.07741
$227$			1.47692e7i			1.26264i	0.775521	+	0.631321i	$0.217487\pi$
$227$			1.47692e7i			1.26264i	−0.775521	+	0.631321i	$0.782513\pi$
$228$	1.51384e7			1.27725
$229$		−	160746.i		−	0.0133855i	−0.999978	−	0.00669275i	$-0.997870\pi$
$229$		−	160746.i		−	0.0133855i	0.999978	−	0.00669275i	$-0.00213038\pi$
$230$	2.08545e7			1.71402
$231$			1.12441e7i			0.912195i
$232$		−	2.41400e7i		−	1.93318i
$233$		−	1.32519e7i		−	1.04764i	−0.851829	−	0.523819i	$-0.824507\pi$
$233$		−	1.32519e7i		−	1.04764i	0.851829	−	0.523819i	$-0.175493\pi$
$234$	1.27715e7			0.996770
$235$		−	1.95117e7i		−	1.50346i
$236$	−2.15941e7	+	7.64924e6i	−1.64285	+	0.581946i
$237$	−2.93845e6			−0.220736
$238$		−	4.08660e7i		−	3.03132i
$239$	1.99713e7			1.46289			0.731446	−	0.681899i	$-0.238846\pi$
$239$	1.99713e7			1.46289			0.731446	+	0.681899i	$0.238846\pi$
$240$	4.21880e6			0.305179
$241$	3.58286e6			0.255964			0.127982	−	0.991777i	$-0.459150\pi$
$241$	3.58286e6			0.255964			0.127982	+	0.991777i	$0.459150\pi$
$242$			9.05644e6i			0.639016i
$243$	−920483.			−0.0641500
$244$		−	8.46175e6i		−	0.582493i
$245$	−2.11323e7			−1.43697
$246$		−	1.45927e6i		−	0.0980234i
$247$			3.45360e7i			2.29182i
$248$	−2.22344e7			−1.45770
$249$			5.78447e6i			0.374684i
$250$			5.64322e7i			3.61166i
$251$	2.20964e7			1.39734			0.698668	−	0.715446i	$-0.253776\pi$
$251$	2.20964e7			1.39734			0.698668	+	0.715446i	$0.253776\pi$
$252$	−1.24778e7			−0.779719
$253$	−1.10020e7			−0.679376
$254$		−	4.27228e7i		−	2.60711i
$255$	2.34130e7			1.41200
$256$	−2.39384e7			−1.42684
$257$	−1.57000e7			−0.924911			−0.462456	−	0.886642i	$-0.653032\pi$
$257$	−1.57000e7			−0.924911			−0.462456	+	0.886642i	$0.653032\pi$
$258$	2.04016e6			0.118797
$259$		−	2.74292e7i		−	1.57875i
$260$			9.91882e7i			5.64339i
$261$	−9.31216e6			−0.523756
$262$	−2.74493e7			−1.52626
$263$	9.82007e6			0.539818			0.269909	−	0.962886i	$-0.413006\pi$
$263$	9.82007e6			0.539818			0.269909	+	0.962886i	$0.413006\pi$
$264$	−1.53862e7			−0.836216
$265$	666385.			0.0358086
$266$		−	5.31016e7i		−	2.82139i
$267$		−	9.49696e6i		−	0.498943i
$268$			3.41112e7i			1.77212i
$269$		−	8.38467e6i		−	0.430754i	−0.976531	−	0.215377i	$-0.930902\pi$
$269$		−	8.38467e6i		−	0.430754i	0.976531	−	0.215377i	$-0.0690980\pi$
$270$		−	1.12505e7i		−	0.571586i
$271$	−4.37105e6			−0.219623			−0.109812	−	0.993952i	$-0.535025\pi$
$271$	−4.37105e6			−0.219623			−0.109812	+	0.993952i	$0.535025\pi$
$272$	8.08907e6			0.401968
$273$		−	2.84663e7i		−	1.39908i
$274$			3.72150e6i			0.180911i
$275$		−	5.42538e7i		−	2.60875i
$276$		−	1.22092e7i		−	0.580712i
$277$	2.03885e7			0.959280			0.479640	−	0.877465i	$-0.340767\pi$
$277$	2.03885e7			0.959280			0.479640	+	0.877465i	$0.340767\pi$
$278$			2.92691e7i			1.36231i
$279$			8.57707e6i			0.394936i
$280$		−	6.50052e7i		−	2.96124i
$281$	1.58550e7			0.714575			0.357288	−	0.933994i	$-0.383702\pi$
$281$	1.58550e7			0.714575			0.357288	+	0.933994i	$0.383702\pi$
$282$	−1.79772e7			−0.801632
$283$		−	2.80181e7i		−	1.23618i	−0.786109	−	0.618088i	$-0.787908\pi$
$283$		−	2.80181e7i		−	1.23618i	0.786109	−	0.618088i	$-0.212092\pi$
$284$	−3.91652e7			−1.70980
$285$	3.04230e7			1.31422
$286$		−	8.23514e7i		−	3.52024i
$287$	−3.25255e6			−0.137587
$288$			5.90966e6i			0.247392i
$289$	2.07541e7			0.859827
$290$		−	1.13817e8i		−	4.66674i
$291$			1.55641e7i			0.631605i
$292$			1.63373e7i			0.656192i
$293$	−3.79082e7			−1.50706			−0.753530	−	0.657414i	$-0.771650\pi$
$293$	−3.79082e7			−1.50706			−0.753530	+	0.657414i	$0.771650\pi$
$294$			1.94703e7i			0.766181i
$295$	−4.33968e7	+	1.53724e7i	−1.69041	+	0.598790i
$296$	3.75336e7			1.44725
$297$			5.93532e6i			0.226556i
$298$	−2.24334e7			−0.847709
$299$	2.78535e7			1.04200
$300$	6.02069e7			2.22988
$301$		−	4.54730e6i		−	0.166745i
$302$	−1.26968e7			−0.460972
$303$		−	1.85838e7i		−	0.668046i
$304$	1.05110e7			0.374131
$305$		−	1.70052e7i		−	0.599354i
$306$		−	2.15716e7i		−	0.752867i
$307$	−7.94326e6			−0.274526			−0.137263	−	0.990535i	$-0.543831\pi$
$307$	−7.94326e6			−0.274526			−0.137263	+	0.990535i	$0.543831\pi$
$308$			8.04578e7i			2.75369i
$309$		−	146838.i		−	0.00497694i
$310$	−1.04832e8			−3.51893
$311$	1.84910e6			0.0614722			0.0307361	−	0.999528i	$-0.490215\pi$
$311$	1.84910e6			0.0614722			0.0307361	+	0.999528i	$0.490215\pi$
$312$	3.89527e7			1.28255
$313$		−	1.39026e7i		−	0.453382i	−0.973967	−	0.226691i	$-0.927209\pi$
$313$		−	1.39026e7i		−	0.453382i	0.973967	−	0.226691i	$-0.0727907\pi$
$314$	4.50639e7			1.45559
$315$	−2.50762e7			−0.802288
$316$	−2.10263e7			−0.666349
$317$	5.90330e7			1.85318			0.926589	−	0.376077i	$-0.122727\pi$
$317$	5.90330e7			1.85318			0.926589	+	0.376077i	$0.122727\pi$
$318$		−	613977.i		−	0.0190928i
$319$			6.00453e7i			1.84972i
$320$	−8.95511e7			−2.73288
$321$	6.44187e6			0.194759
$322$	−4.28269e7			−1.28277
$323$	5.83327e7			1.73103
$324$	−6.58658e6			−0.193653
$325$			1.37353e8i			4.00118i
$326$			2.41532e6i			0.0697142i
$327$			2.77239e7i			0.792888i
$328$		−	4.45073e6i		−	0.126127i
$329$			4.00693e7i			1.12518i
$330$	−7.25440e7			−2.01864
$331$	3.69169e7			1.01798			0.508992	−	0.860771i	$-0.330018\pi$
$331$	3.69169e7			1.01798			0.508992	+	0.860771i	$0.330018\pi$
$332$			4.13912e7i			1.13108i
$333$		−	1.44788e7i		−	0.392104i
$334$		−	6.33850e7i		−	1.70117i
$335$			6.85519e7i			1.82341i
$336$	−8.66372e6			−0.228395
$337$		−	3.10257e7i		−	0.810647i	−0.914173	−	0.405324i	$-0.867159\pi$
$337$		−	3.10257e7i		−	0.810647i	0.914173	−	0.405324i	$-0.132841\pi$
$338$			1.44535e8i			3.74303i
$339$			4.17946e7i			1.07281i
$340$	1.67533e8			4.26249
$341$	5.53054e7			1.39477
$342$		−	2.80304e7i		−	0.700729i
$343$	−1.07621e7			−0.266695
$344$	6.22243e6			0.152857
$345$		−	2.45364e7i		−	0.597521i
$346$	−1.14844e8			−2.77257
$347$			7.35778e7i			1.76100i	0.474049	+	0.880498i	$0.342792\pi$
$347$			7.35778e7i			1.76100i	−0.474049	+	0.880498i	$0.657208\pi$
$348$	−6.66339e7			−1.58109
$349$		−	3.00755e7i		−	0.707516i	−0.935337	−	0.353758i	$-0.884904\pi$
$349$		−	3.00755e7i		−	0.707516i	0.935337	−	0.353758i	$-0.115096\pi$
$350$		−	2.11191e8i		−	4.92573i
$351$		−	1.50263e7i		−	0.347481i
$352$	3.81058e7			0.873702
$353$			7.81394e7i			1.77642i	0.459437	+	0.888210i	$0.348051\pi$
$353$			7.81394e7i			1.77642i	−0.459437	+	0.888210i	$0.651949\pi$
$354$	1.41634e7	+	3.99838e7i	0.319270	+	0.901310i
$355$	−7.87088e7			−1.75929
$356$		−	6.79562e7i		−	1.50619i
$357$	−4.80808e7			−1.05674
$358$	6.95271e7			1.51532
$359$	−2.92202e7			−0.631538			−0.315769	−	0.948836i	$-0.602263\pi$
$359$	−2.92202e7			−0.631538			−0.315769	+	0.948836i	$0.602263\pi$
$360$		−	3.43138e7i		−	0.735463i
$361$	2.87521e7			0.611150
$362$		−	4.28191e7i		−	0.902634i
$363$	1.06553e7			0.222765
$364$		−	2.03693e8i		−	4.22349i
$365$			3.28324e7i			0.675186i
$366$	−1.56679e7			−0.319570
$367$			7.33923e7i			1.48475i	0.669987	+	0.742373i	$0.266300\pi$
$367$			7.33923e7i			1.48475i	−0.669987	+	0.742373i	$0.733700\pi$
$368$		−	8.47720e6i		−	0.170102i
$369$	−1.71690e6			−0.0341716
$370$	1.76966e8			3.49370
$371$	−1.36849e6			−0.0267990
$372$			6.13738e7i			1.19221i
$373$	4.77308e7			0.919754			0.459877	−	0.887983i	$-0.347893\pi$
$373$	4.77308e7			0.919754			0.459877	+	0.887983i	$0.347893\pi$
$374$	−1.39095e8			−2.65887
$375$	6.63952e7			1.25905
$376$	−5.48299e7			−1.03146
$377$		−	1.52015e8i		−	2.83702i
$378$			2.31041e7i			0.427773i
$379$	−2.31283e7			−0.424841			−0.212421	−	0.977178i	$-0.568135\pi$
$379$	−2.31283e7			−0.424841			−0.212421	+	0.977178i	$0.568135\pi$
$380$	2.17694e8			3.96730
$381$	−5.02655e7			−0.908856
$382$	2.52704e7			0.453338
$383$	−7.21705e7			−1.28459			−0.642293	−	0.766459i	$-0.722017\pi$
$383$	−7.21705e7			−1.28459			−0.642293	+	0.766459i	$0.722017\pi$
$384$			5.82455e7i			1.02865i
$385$			1.61693e8i			2.83340i
$386$			1.53469e8i			2.66845i
$387$		−	2.40035e6i		−	0.0414134i
$388$			1.11370e8i			1.90666i
$389$	6.16152e7			1.04674			0.523370	−	0.852105i	$-0.324674\pi$
$389$	6.16152e7			1.04674			0.523370	+	0.852105i	$0.324674\pi$
$390$	1.83658e8			3.09610
$391$		−	4.70457e7i		−	0.787028i
$392$			5.93839e7i			0.985850i
$393$			3.22955e7i			0.532064i
$394$			3.32113e7i			0.542997i
$395$	−4.22557e7			−0.685637
$396$			4.24706e7i			0.683916i
$397$			2.31569e7i			0.370091i	0.982730	+	0.185046i	$0.0592433\pi$
$397$			2.31569e7i			0.370091i	−0.982730	+	0.185046i	$0.940757\pi$
$398$		−	1.72358e7i		−	0.273390i
$399$	−6.24766e7			−0.983556
$400$	4.18033e7			0.653177
$401$		−	7.13538e7i		−	1.10658i	−0.832988	−	0.553291i	$-0.813372\pi$
$401$		−	7.13538e7i		−	1.10658i	0.832988	−	0.553291i	$-0.186628\pi$
$402$	6.31606e7			0.972228
$403$	−1.40015e8			−2.13924
$404$		−	1.32978e8i		−	2.01667i
$405$	−1.32368e7			−0.199259
$406$			2.33735e8i			3.49257i
$407$	−9.33603e7			−1.38477
$408$		−	6.57928e7i		−	0.968719i
$409$		−	2.42489e7i		−	0.354423i	−0.984173	−	0.177212i	$-0.943292\pi$
$409$		−	2.42489e7i		−	0.354423i	0.984173	−	0.177212i	$-0.0567077\pi$
$410$		−	2.09847e7i		−	0.304474i
$411$	4.37852e6			0.0630670
$412$		−	1.05071e6i		−	0.0150242i
$413$	8.91196e7	−	3.15687e7i	1.26509	−	0.448133i
$414$	−2.26067e7			−0.318593
$415$			8.31822e7i			1.16382i
$416$	−9.64715e7			−1.34004
$417$	3.44365e7			0.474910
$418$	−1.80741e8			−2.47473
$419$		−	1.33284e8i		−	1.81191i	−0.423377	−	0.905954i	$-0.639155\pi$
$419$		−	1.33284e8i		−	1.81191i	0.423377	−	0.905954i	$-0.360845\pi$
$420$	−1.79435e8			−2.42191
$421$			9.90160e6i			0.132696i	0.997797	+	0.0663482i	$0.0211348\pi$
$421$			9.90160e6i			0.132696i	−0.997797	+	0.0663482i	$0.978865\pi$
$422$	−7.21612e7			−0.960210
$423$			2.11511e7i			0.279454i
$424$		−	1.87261e6i		−	0.0245669i
$425$	2.31995e8			3.02212
$426$			7.25187e7i			0.938039i
$427$			3.49220e7i			0.448554i
$428$	4.60952e7			0.587929
$429$	−9.68904e7			−1.22718
$430$	2.93380e7			0.368999
$431$		−	1.02767e8i		−	1.28357i	−0.766884	−	0.641786i	$-0.778194\pi$
$431$		−	1.02767e8i		−	1.28357i	0.766884	−	0.641786i	$-0.221806\pi$
$432$	−4.57325e6			−0.0567249
$433$	6.29630e7			0.775571			0.387786	−	0.921750i	$-0.373240\pi$
$433$	6.29630e7			0.775571			0.387786	+	0.921750i	$0.373240\pi$
$434$	2.15284e8			2.63356
$435$	−1.33911e8			−1.62686
$436$			1.98381e8i			2.39354i
$437$		−	6.11316e7i		−	0.732524i
$438$	3.02502e7			0.360003
$439$	5.99040e7			0.708047			0.354023	−	0.935237i	$-0.384813\pi$
$439$	5.99040e7			0.708047			0.354023	+	0.935237i	$0.384813\pi$
$440$	−2.21257e8			−2.59740
$441$	2.29078e7			0.267096
$442$	3.52143e8			4.07805
$443$		−	2.24838e6i		−	0.0258617i	−0.999916	−	0.0129309i	$-0.995884\pi$
$443$		−	2.24838e6i		−	0.0258617i	0.999916	−	0.0129309i	$-0.00411614\pi$
$444$		−	1.03604e8i		−	1.18367i
$445$		−	1.36569e8i		−	1.54978i
$446$		−	1.90526e8i		−	2.14758i
$447$			2.63940e7i			0.295517i
$448$	1.83902e8			2.04528
$449$	1.02985e7			0.113772			0.0568861	−	0.998381i	$-0.481883\pi$
$449$	1.02985e7			0.113772			0.0568861	+	0.998381i	$0.481883\pi$
$450$		−	1.11480e8i		−	1.22337i
$451$			1.10707e7i			0.120682i
$452$			2.99064e8i			3.23854i
$453$			1.49384e7i			0.160698i
$454$	1.95682e8			2.09114
$455$		−	4.09353e8i		−	4.34574i
$456$		−	8.54917e7i		−	0.901633i
$457$			1.32085e8i			1.38390i	0.721945	+	0.691950i	$0.243248\pi$
$457$			1.32085e8i			1.38390i	−0.721945	+	0.691950i	$0.756752\pi$
$458$	−2.12978e6			−0.0221686
$459$	−2.53800e7			−0.262455
$460$		−	1.75572e8i		−	1.80377i
$461$	−1.68560e8			−1.72049			−0.860245	−	0.509881i	$-0.829689\pi$
$461$	−1.68560e8			−1.72049			−0.860245	+	0.509881i	$0.829689\pi$
$462$	1.48976e8			1.51074
$463$		−	5.21499e7i		−	0.525425i	−0.964874	−	0.262713i	$-0.915383\pi$
$463$		−	5.21499e7i		−	0.525425i	0.964874	−	0.262713i	$-0.0846171\pi$
$464$	−4.62658e7			−0.463133
$465$			1.23340e8i			1.22672i
$466$	−1.75579e8			−1.73506
$467$		−	8.72860e7i		−	0.857026i	−0.903536	−	0.428513i	$-0.859038\pi$
$467$		−	8.72860e7i		−	0.857026i	0.903536	−	0.428513i	$-0.140962\pi$
$468$		−	1.07522e8i		−	1.04896i
$469$		−	1.40778e8i		−	1.36464i
$470$	−2.58517e8			−2.48998
$471$		−	5.30198e7i		−	0.507430i
$472$	4.31979e7	+	1.21949e8i	0.410806	+	1.15972i
$473$	−1.54776e7			−0.146258
$474$			3.89325e7i			0.365575i
$475$	3.01456e8			2.81283
$476$	−3.44046e8			−3.19003
$477$	−722373.			−0.00665589
$478$		−	2.64606e8i		−	2.42279i
$479$	−1.18080e8			−1.07441			−0.537204	−	0.843452i	$-0.680519\pi$
$479$	−1.18080e8			−1.07441			−0.537204	+	0.843452i	$0.680519\pi$
$480$			8.49825e7i			0.768433i
$481$	2.36358e8			2.12390
$482$		−	4.74704e7i		−	0.423918i
$483$			5.03879e7i			0.447182i
$484$	7.62451e7			0.672474
$485$			2.23816e8i			1.96185i
$486$			1.21958e7i			0.106243i
$487$	−1.31294e6			−0.0113673			−0.00568365	−	0.999984i	$-0.501809\pi$
$487$	−1.31294e6			−0.0113673			−0.00568365	+	0.999984i	$0.501809\pi$
$488$	−4.77865e7			−0.411193
$489$	2.84174e6			0.0243029
$490$			2.79989e8i			2.37986i
$491$	−9.85100e7			−0.832216			−0.416108	−	0.909315i	$-0.636606\pi$
$491$	−9.85100e7			−0.832216			−0.416108	+	0.909315i	$0.636606\pi$
$492$	−1.22854e7			−0.103156
$493$	−2.56760e8			−2.14282
$494$	4.57578e8			3.79563
$495$			8.53515e7i			0.703713i
$496$			4.26136e7i			0.349223i
$497$	1.61636e8			1.31665
$498$	7.66403e7			0.620539
$499$	−8.98227e7			−0.722911			−0.361455	−	0.932389i	$-0.617720\pi$
$499$	−8.98227e7			−0.722911			−0.361455	+	0.932389i	$0.617720\pi$
$500$	4.75096e8			3.80077
$501$	−7.45755e7			−0.593039
$502$		−	2.92763e8i		−	2.31422i
$503$		−	2.20691e7i		−	0.173412i	−0.996234	−	0.0867062i	$-0.972366\pi$
$503$		−	2.20691e7i		−	0.173412i	0.996234	−	0.0867062i	$-0.0276341\pi$
$504$			7.04667e7i			0.550418i
$505$		−	2.67240e8i		−	2.07504i
$506$			1.45769e8i			1.12516i
$507$	1.70052e8			1.30485
$508$	−3.59678e8			−2.74361
$509$			1.05035e8i			0.796488i	0.917280	+	0.398244i	$0.130380\pi$
$509$			1.05035e8i			0.796488i	−0.917280	+	0.398244i	$0.869620\pi$
$510$		−	3.10206e8i		−	2.33851i
$511$		−	6.74245e7i		−	0.505307i
$512$			7.80340e7i			0.581399i
$513$	−3.29791e7			−0.244279
$514$			2.08014e8i			1.53181i
$515$		−	2.11156e6i		−	0.0154590i
$516$		−	1.71759e7i		−	0.125017i
$517$	1.36383e8			0.986936
$518$	−3.63418e8			−2.61467
$519$			1.35120e8i			0.966536i
$520$	5.60151e8			3.98377
$521$	1.73155e8			1.22440			0.612198	−	0.790704i	$-0.290285\pi$
$521$	1.73155e8			1.22440			0.612198	+	0.790704i	$0.290285\pi$
$522$			1.23380e8i			0.867426i
$523$	−1.42119e7			−0.0993456			−0.0496728	−	0.998766i	$-0.515818\pi$
$523$	−1.42119e7			−0.0993456			−0.0496728	+	0.998766i	$0.515818\pi$
$524$			2.31093e8i			1.60617i
$525$	−2.48476e8			−1.71714
$526$		−	1.30109e8i		−	0.894027i
$527$			2.36492e8i			1.61579i
$528$			2.94885e7i			0.200333i
$529$	9.87328e7			0.666952
$530$		−	8.82914e6i		−	0.0593050i
$531$	4.70429e7	−	1.66639e7i	0.314203	−	0.111300i
$532$	−4.47056e8			−2.96912
$533$		−	2.80273e7i		−	0.185097i
$534$	−1.25828e8			−0.826331
$535$	9.26358e7			0.604947
$536$	1.92638e8			1.25097
$537$		−	8.18020e7i		−	0.528252i
$538$	−1.11091e8			−0.713399
$539$		−	1.47711e8i		−	0.943290i
$540$	−9.47168e7			−0.601514
$541$		−	2.54744e8i		−	1.60884i	−0.594062	−	0.804420i	$-0.702476\pi$
$541$		−	2.54744e8i		−	1.60884i	0.594062	−	0.804420i	$-0.297524\pi$
$542$			5.79135e7i			0.363732i
$543$	−5.03787e7			−0.314664
$544$			1.62944e8i			1.01215i
$545$			3.98678e8i			2.46282i
$546$	−3.77159e8			−2.31711
$547$	−7.57754e7			−0.462985			−0.231492	−	0.972837i	$-0.574361\pi$
$547$	−7.57754e7			−0.462985			−0.231492	+	0.972837i	$0.574361\pi$
$548$	3.13308e7			0.190384
$549$			1.84340e7i			0.111404i
$550$	−7.18826e8			−4.32051
$551$	−3.33636e8			−1.99443
$552$	−6.89497e7			−0.409935
$553$	8.67763e7			0.513128
$554$		−	2.70133e8i		−	1.58873i
$555$		−	2.08210e8i		−	1.21793i
$556$	2.46413e8			1.43364
$557$	2.79672e8			1.61839			0.809197	−	0.587538i	$-0.199903\pi$
$557$	2.79672e8			1.61839			0.809197	+	0.587538i	$0.199903\pi$
$558$	1.13640e8			0.654078
$559$	3.91841e7			0.224324
$560$	−1.24587e8			−0.709426
$561$			1.63652e8i			0.926899i
$562$		−	2.10068e8i		−	1.18345i
$563$		−	2.66407e8i		−	1.49287i	−0.665460	−	0.746433i	$-0.731765\pi$
$563$		−	2.66407e8i		−	1.49287i	0.665460	−	0.746433i	$-0.268235\pi$
$564$			1.51348e8i			0.843605i
$565$			6.01017e8i			3.33228i
$566$	−3.71221e8			−2.04731
$567$	2.71831e7			0.149125
$568$			2.21180e8i			1.20698i
$569$		−	7.39562e7i		−	0.401456i	−0.979647	−	0.200728i	$-0.935669\pi$
$569$		−	7.39562e7i		−	0.401456i	0.979647	−	0.200728i	$-0.0643307\pi$
$570$		−	4.03084e8i		−	2.17656i
$571$		−	2.66103e8i		−	1.42936i	−0.699451	−	0.714680i	$-0.746572\pi$
$571$		−	2.66103e8i		−	1.42936i	0.699451	−	0.714680i	$-0.253428\pi$
$572$	−6.93306e8			−3.70456
$573$		−	2.97319e7i		−	0.158037i
$574$			4.30941e7i			0.227867i
$575$		−	2.43127e8i		−	1.27888i
$576$	9.70749e7			0.507972
$577$	−1.28656e8			−0.669733			−0.334866	−	0.942266i	$-0.608691\pi$
$577$	−1.28656e8			−0.669733			−0.334866	+	0.942266i	$0.608691\pi$
$578$		−	2.74978e8i		−	1.42401i
$579$	1.80564e8			0.930241
$580$	−9.58212e8			−4.91109
$581$		−	1.70823e8i		−	0.870999i
$582$	2.06214e8			1.04604
$583$			4.65790e6i			0.0235063i
$584$	9.22623e7			0.463218
$585$		−	2.16082e8i		−	1.07932i
$586$			5.02258e8i			2.49594i
$587$			1.86641e8i			0.922769i	0.887200	+	0.461384i	$0.152647\pi$
$587$			1.86641e8i			0.922769i	−0.887200	+	0.461384i	$0.847353\pi$
$588$	1.63918e8			0.806298
$589$			3.07299e8i			1.50389i
$590$	2.03673e8	+	5.74978e8i	0.991695	+	2.79959i
$591$	3.90747e7			0.189293
$592$		−	7.19354e7i		−	0.346719i
$593$	4.19768e7			0.201300			0.100650	−	0.994922i	$-0.467908\pi$
$593$	4.19768e7			0.201300			0.100650	+	0.994922i	$0.467908\pi$
$594$	7.86389e7			0.375213
$595$	−6.91415e8			−3.28237
$596$			1.88864e8i			0.892094i
$597$	−2.02788e7			−0.0953056
$598$		−	3.69040e8i		−	1.72572i
$599$	−5.23574e7			−0.243611			−0.121806	−	0.992554i	$-0.538868\pi$
$599$	−5.23574e7			−0.243611			−0.121806	+	0.992554i	$0.538868\pi$
$600$		−	3.40009e8i		−	1.57412i
$601$		−	1.99557e8i		−	0.919269i	−0.888108	−	0.459635i	$-0.847980\pi$
$601$		−	1.99557e8i		−	0.919269i	0.888108	−	0.459635i	$-0.152020\pi$
$602$	−6.02486e7			−0.276158
$603$		−	7.43115e7i		−	0.338925i
$604$			1.06893e8i			0.485109i
$605$	1.53227e8			0.691940
$606$	−2.46222e8			−1.10639
$607$	−1.17781e6			−0.00526633			−0.00263317	−	0.999997i	$-0.500838\pi$
$607$	−1.17781e6			−0.00526633			−0.00263317	+	0.999997i	$0.500838\pi$
$608$			2.11731e8i			0.942051i
$609$	2.75000e8			1.21753
$610$	−2.25308e8			−0.992628
$611$	−3.45278e8			−1.51372
$612$	−1.81609e8			−0.792287
$613$		−	7.99912e7i		−	0.347265i	−0.984811	−	0.173632i	$-0.944450\pi$
$613$		−	7.99912e7i		−	0.347265i	0.984811	−	0.173632i	$-0.0555505\pi$
$614$			1.05243e8i			0.454660i
$615$	−2.46895e7			−0.106142
$616$	4.54373e8			1.94388
$617$	8.34368e7			0.355224			0.177612	−	0.984101i	$-0.443163\pi$
$617$	8.34368e7			0.355224			0.177612	+	0.984101i	$0.443163\pi$
$618$	−1.94550e6			−0.00824262
$619$	1.44257e8			0.608225			0.304113	−	0.952636i	$-0.401640\pi$
$619$	1.44257e8			0.608225			0.304113	+	0.952636i	$0.401640\pi$
$620$			8.82572e8i			3.70318i
$621$			2.65979e7i			0.111064i
$622$		−	2.44993e7i		−	0.101808i
$623$			2.80457e8i			1.15985i
$624$		−	7.46554e7i		−	0.307261i
$625$	4.13758e8			1.69475
$626$	−1.84200e8			−0.750874
$627$			2.12651e8i			0.862709i
$628$		−	3.79387e8i		−	1.53181i
$629$		−	3.99218e8i		−	1.60420i
$630$			3.32243e8i			1.32872i
$631$	2.82421e8			1.12411			0.562055	−	0.827100i	$-0.310011\pi$
$631$	2.82421e8			1.12411			0.562055	+	0.827100i	$0.310011\pi$
$632$			1.18743e8i			0.470388i
$633$			8.49011e7i			0.334736i
$634$		−	7.82147e8i		−	3.06917i
$635$	−7.22831e8			−2.82303
$636$	−5.16899e6			−0.0200925
$637$			3.73955e8i			1.44678i
$638$	7.95559e8			3.06345
$639$	8.53217e7			0.327007
$640$			8.37586e8i			3.19514i
$641$	1.25319e8			0.475819			0.237910	−	0.971287i	$-0.423538\pi$
$641$	1.25319e8			0.475819			0.237910	+	0.971287i	$0.423538\pi$
$642$		−	8.53503e7i		−	0.322552i
$643$	4.06213e8			1.52799			0.763996	−	0.645221i	$-0.223235\pi$
$643$	4.06213e8			1.52799			0.763996	+	0.645221i	$0.223235\pi$
$644$			3.60554e8i			1.34993i
$645$		−	3.45176e7i		−	0.128636i
$646$		−	7.72868e8i		−	2.86687i
$647$	−5.97655e7			−0.220667			−0.110334	−	0.993895i	$-0.535192\pi$
$647$	−5.97655e7			−0.220667			−0.110334	+	0.993895i	$0.535192\pi$
$648$			3.71967e7i			0.136704i
$649$	−1.07450e8	−	3.03335e8i	−0.393071	−	1.10966i
$650$	1.81983e9			6.62661
$651$		−	2.53292e8i		−	0.918075i
$652$	2.03343e7			0.0733644
$653$	2.07427e8			0.744947			0.372474	−	0.928043i	$-0.378510\pi$
$653$	2.07427e8			0.744947			0.372474	+	0.928043i	$0.378510\pi$
$654$	3.67323e8			1.31315
$655$			4.64418e8i			1.65266i
$656$	−8.53010e6			−0.0302164
$657$		−	3.55909e7i		−	0.125500i
$658$	5.30890e8			1.86349
$659$		−	1.87199e8i		−	0.654106i	−0.945006	−	0.327053i	$-0.893944\pi$
$659$		−	1.87199e8i		−	0.654106i	0.945006	−	0.327053i	$-0.106056\pi$
$660$			6.10739e8i			2.12434i
$661$	−2.21184e7			−0.0765860			−0.0382930	−	0.999267i	$-0.512192\pi$
$661$	−2.21184e7			−0.0765860			−0.0382930	+	0.999267i	$0.512192\pi$
$662$		−	4.89123e8i		−	1.68595i
$663$		−	4.14313e8i		−	1.42164i
$664$	2.33751e8			0.798451
$665$	−8.98431e8			−3.05506
$666$	−1.91835e8			−0.649388
$667$			2.69080e8i			0.906784i
$668$	−5.33631e8			−1.79024
$669$	−2.24163e8			−0.748662
$670$	9.08266e8			3.01987
$671$	1.18863e8			0.393441
$672$		−	1.74520e8i		−	0.575092i
$673$		−	8.70516e7i		−	0.285582i	−0.989753	−	0.142791i	$-0.954392\pi$
$673$		−	8.70516e7i		−	0.285582i	0.989753	−	0.142791i	$-0.0456077\pi$
$674$	−4.11069e8			−1.34256
$675$	−1.31161e8			−0.426475
$676$	1.21682e9			3.93901
$677$	−8.24674e7			−0.265776			−0.132888	−	0.991131i	$-0.542425\pi$
$677$	−8.24674e7			−0.265776			−0.132888	+	0.991131i	$0.542425\pi$
$678$	5.53749e8			1.77674
$679$		−	4.59629e8i		−	1.46824i
$680$		−	9.46118e8i		−	3.00897i
$681$		−	2.30230e8i		−	0.728987i
$682$		−	7.32758e8i		−	2.30998i
$683$		−	1.62347e8i		−	0.509546i	−0.967001	−	0.254773i	$-0.917999\pi$
$683$		−	1.62347e8i		−	0.509546i	0.967001	−	0.254773i	$-0.0820007\pi$
$684$	−2.35984e8			−0.737419
$685$	6.29643e7			0.195895
$686$			1.42591e8i			0.441691i
$687$			2.50579e6i			0.00772812i
$688$		−	1.19257e7i		−	0.0366200i
$689$		−	1.17923e7i		−	0.0360529i
$690$	−3.25090e8			−0.989593
$691$			3.43304e8i			1.04051i	0.854012	+	0.520253i	$0.174162\pi$
$691$			3.43304e8i			1.04051i	−0.854012	+	0.520253i	$0.825838\pi$
$692$			9.66862e8i			2.91774i
$693$		−	1.75278e8i		−	0.526656i
$694$	9.74856e8			2.91650
$695$	4.95206e8			1.47513
$696$			3.76305e8i			1.11612i
$697$	−4.73393e7			−0.139805
$698$	−3.98480e8			−1.17176
$699$			2.06577e8i			0.604855i
$700$	−1.77799e9			−5.18364
$701$		−	3.99694e8i		−	1.16031i	−0.814506	−	0.580155i	$-0.802992\pi$
$701$		−	3.99694e8i		−	1.16031i	0.814506	−	0.580155i	$-0.197008\pi$
$702$	−1.99088e8			−0.575485
$703$		−	5.18748e8i		−	1.49311i
$704$		−	6.25944e8i		−	1.79398i
$705$			3.04158e8i			0.868023i
$706$	1.03529e9			2.94204
$707$			5.48803e8i			1.55295i
$708$	3.36619e8	−	1.19240e8i	0.948502	−	0.335986i
$709$	−7.80876e7			−0.219101			−0.109550	−	0.993981i	$-0.534941\pi$
$709$	−7.80876e7			−0.219101			−0.109550	+	0.993981i	$0.534941\pi$
$710$			1.04284e9i			2.91368i
$711$	4.58059e7			0.127442
$712$	−3.83772e8			−1.06325
$713$	2.47839e8			0.683756
$714$			6.37038e8i			1.75013i
$715$	−1.39331e9			−3.81179
$716$		−	5.85340e8i		−	1.59466i
$717$	−3.11322e8			−0.844601
$718$			3.87148e8i			1.04593i
$719$			1.51010e8i			0.406273i	0.979150	+	0.203137i	$0.0651136\pi$
$719$			1.51010e8i			0.406273i	−0.979150	+	0.203137i	$0.934886\pi$
$720$	−6.57645e7			−0.176195
$721$			4.33631e6i			0.0115695i
$722$		−	3.80946e8i		−	1.01217i
$723$	−5.58512e7			−0.147781
$724$	−3.60488e8			−0.949895
$725$	−1.32690e9			−3.48198
$726$		−	1.41176e8i		−	0.368936i
$727$	5.97563e8			1.55518			0.777589	−	0.628773i	$-0.216442\pi$
$727$	5.97563e8			1.55518			0.777589	+	0.628773i	$0.216442\pi$
$728$	−1.15033e9			−2.98144
$729$	1.43489e7			0.0370370
$730$	4.35006e8			1.11822
$731$		−	6.61837e7i		−	0.169433i
$732$			1.31906e8i			0.336303i
$733$	4.51581e8			1.14663			0.573315	−	0.819335i	$-0.305657\pi$
$733$	4.51581e8			1.14663			0.573315	+	0.819335i	$0.305657\pi$
$734$	9.72397e8			2.45898
$735$	3.29420e8			0.829637
$736$	1.70763e8			0.428312
$737$	−4.79164e8			−1.19697
$738$			2.27477e7i			0.0565939i
$739$			4.61246e8i			1.14288i	0.820645	+	0.571439i	$0.193614\pi$
$739$			4.61246e8i			1.14288i	−0.820645	+	0.571439i	$0.806386\pi$
$740$		−	1.48986e9i		−	3.67663i
$741$		−	5.38362e8i		−	1.32318i
$742$			1.81315e7i			0.0443836i
$743$	3.52308e8			0.858927			0.429464	−	0.903084i	$-0.358703\pi$
$743$	3.52308e8			0.858927			0.429464	+	0.903084i	$0.358703\pi$
$744$	3.46599e8			0.841606
$745$			3.79553e8i			0.917917i
$746$		−	6.32400e8i		−	1.52326i
$747$		−	9.01710e7i		−	0.216324i
$748$			1.17102e9i			2.79808i
$749$	−1.90237e8			−0.452740
$750$		−	8.79691e8i		−	2.08519i
$751$			1.58247e8i			0.373609i	0.982397	+	0.186804i	$0.0598130\pi$
$751$			1.58247e8i			0.373609i	−0.982397	+	0.186804i	$0.940187\pi$
$752$			1.05085e8i			0.247108i
$753$	−3.44449e8			−0.806753
$754$	−2.01410e9			−4.69858
$755$			2.14819e8i			0.499150i
$756$	1.94510e8			0.450171
$757$	−2.71720e8			−0.626375			−0.313187	−	0.949691i	$-0.601397\pi$
$757$	−2.71720e8			−0.626375			−0.313187	+	0.949691i	$0.601397\pi$
$758$			3.06434e8i			0.703607i
$759$	1.71504e8			0.392238
$760$		−	1.22939e9i		−	2.80059i
$761$	−2.64372e8			−0.599876			−0.299938	−	0.953959i	$-0.596966\pi$
$761$	−2.64372e8			−0.599876			−0.299938	+	0.953959i	$0.596966\pi$
$762$			6.65983e8i			1.50521i
$763$		−	8.18724e8i		−	1.84316i
$764$		−	2.12749e8i		−	0.477075i
$765$	−3.64972e8			−0.815220
$766$			9.56209e8i			2.12749i
$767$	2.72028e8	+	7.67945e8i	0.602875	+	1.70194i
$768$	3.73162e8			0.823785
$769$		−	6.69624e8i		−	1.47249i	−0.676716	−	0.736244i	$-0.736597\pi$
$769$		−	6.69624e8i		−	1.47249i	0.676716	−	0.736244i	$-0.263403\pi$
$770$	2.14232e9			4.69258
$771$	2.44739e8			0.533998
$772$	1.29204e9			2.80817
$773$		−	6.33270e8i		−	1.37104i	−0.728053	−	0.685521i	$-0.759575\pi$
$773$		−	6.33270e8i		−	1.37104i	0.728053	−	0.685521i	$-0.240425\pi$
$774$	−3.18029e7			−0.0685874
$775$			1.22216e9i			2.62557i
$776$	6.28946e8			1.34595
$777$			4.27579e8i			0.911493i
$778$		−	8.16359e8i		−	1.73357i
$779$	−6.15131e7			−0.130123
$780$		−	1.54619e9i		−	3.25821i
$781$		−	5.50159e8i		−	1.15487i
$782$	−6.23324e8			−1.30345
$783$	1.45162e8			0.302391
$784$	1.13813e8			0.236181
$785$		−	7.62439e8i		−	1.57615i

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-13.2493i q^{2} -15.5885 q^{3} -111.544 q^{4} -224.166 q^{5} +206.536i q^{6} +460.347 q^{7} +629.930i q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-13.2493i q^{2} -15.5885 q^{3} -111.544 q^{4} -224.166 q^{5} +206.536i q^{6} +460.347 q^{7} +629.930i q^{8} +243.000 q^{9} +2970.05i q^{10} -1566.88i q^{11} +1738.80 q^{12} +3966.82i q^{13} -6099.29i q^{14} +3494.40 q^{15} +1207.30 q^{16} +6700.13 q^{17} -3219.58i q^{18} +8706.20 q^{19} +25004.5 q^{20} -7176.11 q^{21} -20760.0 q^{22} -7021.62i q^{23} -9819.63i q^{24} +34625.4 q^{25} +52557.7 q^{26} -3788.00 q^{27} -51349.1 q^{28} -38321.7 q^{29} -46298.5i q^{30} +35296.6i q^{31} +24319.6i q^{32} +24425.2i q^{33} -88772.1i q^{34} -103194. q^{35} -27105.3 q^{36} -59583.7i q^{37} -115351. i q^{38} -61836.6i q^{39} -141209. i q^{40} -7065.43 q^{41} +95078.5i q^{42} -9877.97i q^{43} +174776. i q^{44} -54472.4 q^{45} -93031.6 q^{46} +87041.4i q^{47} -18820.0 q^{48} +94270.7 q^{49} -458763. i q^{50} -104445. q^{51} -442477. i q^{52} -2972.73 q^{53} +50188.3i q^{54} +351241. i q^{55} +289987. i q^{56} -135716. q^{57} +507736. i q^{58} +(193592. - 68575.8i) q^{59} -389781. q^{60} +75860.0i q^{61} +467655. q^{62} +111864. q^{63} +399485. q^{64} -889227. i q^{65} +323617. q^{66} -305808. i q^{67} -747361. q^{68} +109456. i q^{69} +1.36725e6i q^{70} +351118. q^{71} +153073. i q^{72} -146464. i q^{73} -789443. q^{74} -539757. q^{75} -971128. q^{76} -721307. i q^{77} -819293. q^{78} +188502. q^{79} -270636. q^{80} +59049.0 q^{81} +93612.1i q^{82} -371074. i q^{83} +800454. q^{84} -1.50194e6 q^{85} -130876. q^{86} +597376. q^{87} +987022. q^{88} +609230. i q^{89} +721722. i q^{90} +1.82612e6i q^{91} +783222. i q^{92} -550219. i q^{93} +1.15324e6 q^{94} -1.95164e6 q^{95} -379105. i q^{96} -998439. i q^{97} -1.24902e6i q^{98} -380751. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) \)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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