LMFDB - Embedded newform 177.7.c.a.58.9

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			58.9
Character	$\chi$	$=$	177.58
Dual form			177.7.c.a.58.52

$q$-expansion

$f(q)$	$=$	$q-12.5406i q^{2} -15.5885 q^{3} -93.2673 q^{4} +48.4743 q^{5} +195.489i q^{6} -222.016 q^{7} +367.030i q^{8} +243.000 q^{9} +O(q^{10})$	\(q-12.5406i q^{2} -15.5885 q^{3} -93.2673 q^{4} +48.4743 q^{5} +195.489i q^{6} -222.016 q^{7} +367.030i q^{8} +243.000 q^{9} -607.898i q^{10} -2577.25i q^{11} +1453.89 q^{12} -4161.02i q^{13} +2784.22i q^{14} -755.640 q^{15} -1366.32 q^{16} +5170.04 q^{17} -3047.37i q^{18} +1327.36 q^{19} -4521.07 q^{20} +3460.89 q^{21} -32320.3 q^{22} -3347.39i q^{23} -5721.43i q^{24} -13275.2 q^{25} -52181.7 q^{26} -3788.00 q^{27} +20706.8 q^{28} +16077.2 q^{29} +9476.20i q^{30} +28530.9i q^{31} +40624.4i q^{32} +40175.4i q^{33} -64835.5i q^{34} -10762.1 q^{35} -22663.9 q^{36} -52681.0i q^{37} -16646.0i q^{38} +64863.8i q^{39} +17791.5i q^{40} -4948.00 q^{41} -43401.7i q^{42} -21051.9i q^{43} +240373. i q^{44} +11779.3 q^{45} -41978.3 q^{46} +59914.8i q^{47} +21298.9 q^{48} -68357.8 q^{49} +166480. i q^{50} -80593.0 q^{51} +388087. i q^{52} -105642. q^{53} +47503.8i q^{54} -124931. i q^{55} -81486.5i q^{56} -20691.5 q^{57} -201618. i q^{58} +(-45352.4 + 200309. i) q^{59} +70476.5 q^{60} +152913. i q^{61} +357795. q^{62} -53949.9 q^{63} +422011. q^{64} -201703. i q^{65} +503824. q^{66} +115751. i q^{67} -482195. q^{68} +52180.6i q^{69} +134963. i q^{70} +181869. q^{71} +89188.2i q^{72} -648928. i q^{73} -660653. q^{74} +206940. q^{75} -123799. q^{76} +572192. i q^{77} +813433. q^{78} +576666. q^{79} -66231.6 q^{80} +59049.0 q^{81} +62051.0i q^{82} -316797. i q^{83} -322788. q^{84} +250614. q^{85} -264004. q^{86} -250618. q^{87} +945928. q^{88} +208482. i q^{89} -147719. i q^{90} +923813. i q^{91} +312201. i q^{92} -444753. i q^{93} +751369. q^{94} +64343.0 q^{95} -633272. i q^{96} -180262. i q^{97} +857250. i q^{98} -626272. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	12.5406i		−	1.56758i	−0.621027	−	0.783789i	$-0.713284\pi$
$2$		−	12.5406i		−	1.56758i	0.621027	−	0.783789i	$-0.286716\pi$
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	−93.2673			−1.45730
$5$	48.4743			0.387795			0.193897	−	0.981022i	$-0.437887\pi$
$5$	48.4743			0.387795			0.193897	+	0.981022i	$0.437887\pi$
$6$			195.489i			0.905042i
$7$	−222.016			−0.647278			−0.323639	−	0.946181i	$-0.604906\pi$
$7$	−222.016			−0.647278			−0.323639	+	0.946181i	$0.604906\pi$
$8$			367.030i			0.716855i
$9$	243.000			0.333333
$10$		−	607.898i		−	0.607898i
$11$		−	2577.25i		−	1.93633i	−0.250319	−	0.968163i	$-0.580535\pi$
$11$		−	2577.25i		−	1.93633i	0.250319	−	0.968163i	$-0.419465\pi$
$12$	1453.89			0.841373
$13$		−	4161.02i		−	1.89395i	−0.321302	−	0.946977i	$-0.604120\pi$
$13$		−	4161.02i		−	1.89395i	0.321302	−	0.946977i	$-0.395880\pi$
$14$			2784.22i			1.01466i
$15$	−755.640			−0.223893
$16$	−1366.32			−0.333575
$17$	5170.04			1.05232			0.526159	−	0.850386i	$-0.323632\pi$
$17$	5170.04			1.05232			0.526159	+	0.850386i	$0.323632\pi$
$18$		−	3047.37i		−	0.522526i
$19$	1327.36			0.193521			0.0967606	−	0.995308i	$-0.469152\pi$
$19$	1327.36			0.193521			0.0967606	+	0.995308i	$0.469152\pi$
$20$	−4521.07			−0.565134
$21$	3460.89			0.373706
$22$	−32320.3			−3.03534
$23$		−	3347.39i		−	0.275120i	−0.990493	−	0.137560i	$-0.956074\pi$
$23$		−	3347.39i		−	0.275120i	0.990493	−	0.137560i	$-0.0439260\pi$
$24$		−	5721.43i		−	0.413876i
$25$	−13275.2			−0.849615
$26$	−52181.7			−2.96892
$27$	−3788.00			−0.192450
$28$	20706.8			0.943278
$29$	16077.2			0.659198			0.329599	−	0.944121i	$-0.393086\pi$
$29$	16077.2			0.659198			0.329599	+	0.944121i	$0.393086\pi$
$30$			9476.20i			0.350970i
$31$			28530.9i			0.957702i	0.877896	+	0.478851i	$0.158947\pi$
$31$			28530.9i			0.957702i	−0.877896	+	0.478851i	$0.841053\pi$
$32$			40624.4i			1.23976i
$33$			40175.4i			1.11794i
$34$		−	64835.5i		−	1.64959i
$35$	−10762.1			−0.251011
$36$	−22663.9			−0.485767
$37$		−	52681.0i		−	1.04004i	−0.854155	−	0.520019i	$-0.825925\pi$
$37$		−	52681.0i		−	1.04004i	0.854155	−	0.520019i	$-0.174075\pi$
$38$		−	16646.0i		−	0.303360i
$39$			64863.8i			1.09347i
$40$			17791.5i			0.277992i
$41$	−4948.00			−0.0717923			−0.0358961	−	0.999356i	$-0.511429\pi$
$41$	−4948.00			−0.0717923			−0.0358961	+	0.999356i	$0.511429\pi$
$42$		−	43401.7i		−	0.585813i
$43$		−	21051.9i		−	0.264781i	−0.991198	−	0.132390i	$-0.957735\pi$
$43$		−	21051.9i		−	0.264781i	0.991198	−	0.132390i	$-0.0422653\pi$
$44$			240373.i			2.82181i
$45$	11779.3			0.129265
$46$	−41978.3			−0.431272
$47$			59914.8i			0.577086i	0.957467	+	0.288543i	$0.0931708\pi$
$47$			59914.8i			0.577086i	−0.957467	+	0.288543i	$0.906829\pi$
$48$	21298.9			0.192590
$49$	−68357.8			−0.581032
$50$			166480.i			1.33184i
$51$	−80593.0			−0.607556
$52$			388087.i			2.76006i
$53$	−105642.			−0.709594			−0.354797	−	0.934943i	$-0.615450\pi$
$53$	−105642.			−0.709594			−0.354797	+	0.934943i	$0.615450\pi$
$54$			47503.8i			0.301681i
$55$		−	124931.i		−	0.750897i
$56$		−	81486.5i		−	0.464004i
$57$	−20691.5			−0.111730
$58$		−	201618.i		−	1.03334i
$59$	−45352.4	+	200309.i	−0.220823	+	0.975314i
$59$	−45352.4	+	200309.i	−0.220823	+	0.975314i
$60$	70476.5			0.326280
$61$			152913.i			0.673683i	0.941561	+	0.336841i	$0.109359\pi$
$61$			152913.i			0.673683i	−0.941561	+	0.336841i	$0.890641\pi$
$62$	357795.			1.50127
$63$	−53949.9			−0.215759
$64$	422011.			1.60985
$65$		−	201703.i		−	0.734465i
$66$	503824.			1.75246
$67$			115751.i			0.384859i	0.981311	+	0.192430i	$0.0616368\pi$
$67$			115751.i			0.384859i	−0.981311	+	0.192430i	$0.938363\pi$
$68$	−482195.			−1.53354
$69$			52180.6i			0.158841i
$70$			134963.i			0.393479i
$71$	181869.			0.508141			0.254070	−	0.967186i	$-0.418231\pi$
$71$	181869.			0.508141			0.254070	+	0.967186i	$0.418231\pi$
$72$			89188.2i			0.238952i
$73$		−	648928.i		−	1.66812i	−0.551671	−	0.834062i	$-0.686010\pi$
$73$		−	648928.i		−	1.66812i	0.551671	−	0.834062i	$-0.313990\pi$
$74$	−660653.			−1.63034
$75$	206940.			0.490526
$76$	−123799.			−0.282019
$77$			572192.i			1.25334i
$78$	813433.			1.71411
$79$	576666.			1.16962			0.584808	−	0.811172i	$-0.301170\pi$
$79$	576666.			1.16962			0.584808	+	0.811172i	$0.301170\pi$
$80$	−66231.6			−0.129359
$81$	59049.0			0.111111
$82$			62051.0i			0.112540i
$83$		−	316797.i		−	0.554047i	−0.960863	−	0.277024i	$-0.910652\pi$
$83$		−	316797.i		−	0.554047i	0.960863	−	0.277024i	$-0.0893480\pi$
$84$	−322788.			−0.544602
$85$	250614.			0.408083
$86$	−264004.			−0.415065
$87$	−250618.			−0.380588
$88$	945928.			1.38807
$89$			208482.i			0.295732i	0.989007	+	0.147866i	$0.0472405\pi$
$89$			208482.i			0.295732i	−0.989007	+	0.147866i	$0.952760\pi$
$90$		−	147719.i		−	0.202633i
$91$			923813.i			1.22591i
$92$			312201.i			0.400933i
$93$		−	444753.i		−	0.552929i
$94$	751369.			0.904627
$95$	64343.0			0.0750465
$96$		−	633272.i		−	0.715776i
$97$		−	180262.i		−	0.197510i	−0.995112	−	0.0987548i	$-0.968514\pi$
$97$		−	180262.i		−	0.197510i	0.995112	−	0.0987548i	$-0.0314860\pi$
$98$			857250.i			0.910813i
$99$		−	626272.i		−	0.645442i
$100$	1.23815e6			1.23815
$101$		−	1.54557e6i		−	1.50012i	−0.661371	−	0.750059i	$-0.730025\pi$
$101$		−	1.54557e6i		−	1.50012i	0.661371	−	0.750059i	$-0.269975\pi$
$102$			1.01069e6i			0.952392i
$103$			1.64324e6i			1.50379i	0.659281	+	0.751897i	$0.270861\pi$
$103$			1.64324e6i			1.50379i	−0.659281	+	0.751897i	$0.729139\pi$
$104$	1.52722e6			1.35769
$105$	167764.			0.144921
$106$			1.32482e6i			1.11234i
$107$	−925802.			−0.755730			−0.377865	−	0.925861i	$-0.623342\pi$
$107$	−925802.			−0.755730			−0.377865	+	0.925861i	$0.623342\pi$
$108$	353296.			0.280458
$109$			1.70317e6i			1.31516i	0.753384	+	0.657581i	$0.228420\pi$
$109$			1.70317e6i			1.31516i	−0.753384	+	0.657581i	$0.771580\pi$
$110$	−1.56671e6			−1.17709
$111$			821216.i			0.600466i
$112$	303346.			0.215916
$113$			1.24607e6i			0.863589i	0.901972	+	0.431794i	$0.142119\pi$
$113$			1.24607e6i			0.863589i	−0.901972	+	0.431794i	$0.857881\pi$
$114$			259485.i			0.175145i
$115$		−	162262.i		−	0.106690i
$116$	−1.49947e6			−0.960650
$117$		−	1.01113e6i		−	0.631318i
$118$	2.51200e6	+	568747.i	1.52888	+	0.346157i
$119$	−1.14783e6			−0.681142
$120$		−	277342.i		−	0.160499i
$121$	−4.87066e6			−2.74936
$122$	1.91763e6			1.05605
$123$	77131.6			0.0414493
$124$		−	2.66100e6i		−	1.39566i
$125$	−1.40092e6			−0.717271
$126$			676566.i			0.338219i
$127$	−1.76313e6			−0.860741			−0.430371	−	0.902652i	$-0.641617\pi$
$127$	−1.76313e6			−0.860741			−0.430371	+	0.902652i	$0.641617\pi$
$128$		−	2.69232e6i		−	1.28380i
$129$			328167.i			0.152871i
$130$	−2.52948e6			−1.15133
$131$		−	1.05099e6i		−	0.467505i	−0.972296	−	0.233752i	$-0.924899\pi$
$131$		−	1.05099e6i		−	0.467505i	0.972296	−	0.233752i	$-0.0751005\pi$
$132$		−	3.74705e6i		−	1.62917i
$133$	−294696.			−0.125262
$134$	1.45160e6			0.603297
$135$	−183621.			−0.0746311
$136$			1.89756e6i			0.754359i
$137$	−2.10794e6			−0.819778			−0.409889	−	0.912135i	$-0.634433\pi$
$137$	−2.10794e6			−0.819778			−0.409889	+	0.912135i	$0.634433\pi$
$138$	654377.			0.248995
$139$	3.12213e6			1.16253			0.581267	−	0.813713i	$-0.302557\pi$
$139$	3.12213e6			1.16253			0.581267	+	0.813713i	$0.302557\pi$
$140$	1.00375e6			0.365798
$141$		−	933979.i		−	0.333181i
$142$		−	2.28075e6i		−	0.796550i
$143$	−1.07240e7			−3.66731
$144$	−332017.			−0.111192
$145$	779331.			0.255633
$146$	−8.13797e6			−2.61491
$147$	1.06559e6			0.335459
$148$			4.91341e6i			1.51565i
$149$		−	4.06210e6i		−	1.22798i	−0.789313	−	0.613990i	$-0.789563\pi$
$149$		−	4.06210e6i		−	1.22798i	0.789313	−	0.613990i	$-0.210437\pi$
$150$		−	2.59516e6i		−	0.768937i
$151$			3.77394e6i			1.09613i	0.836434	+	0.548067i	$0.184636\pi$
$151$			3.77394e6i			1.09613i	−0.836434	+	0.548067i	$0.815364\pi$
$152$			487181.i			0.138727i
$153$	1.25632e6			0.350773
$154$	7.17564e6			1.96471
$155$			1.38302e6i			0.371392i
$156$		−	6.04967e6i		−	1.59352i
$157$		−	4.32923e6i		−	1.11870i	−0.828933	−	0.559348i	$-0.811052\pi$
$157$		−	4.32923e6i		−	1.11870i	0.828933	−	0.559348i	$-0.188948\pi$
$158$		−	7.23176e6i		−	1.83346i
$159$	1.64680e6			0.409684
$160$			1.96924e6i			0.480772i
$161$			743174.i			0.178079i
$162$		−	740511.i		−	0.174175i
$163$	8.17608e6			1.88791			0.943957	−	0.330068i	$-0.107072\pi$
$163$	8.17608e6			1.88791			0.943957	+	0.330068i	$0.107072\pi$
$164$	461486.			0.104623
$165$			1.94747e6i			0.433531i
$166$	−3.97283e6			−0.868512
$167$	2.13038e6			0.457411			0.228706	−	0.973496i	$-0.426551\pi$
$167$	2.13038e6			0.457411			0.228706	+	0.973496i	$0.426551\pi$
$168$			1.27025e6i			0.267893i
$169$	−1.24872e7			−2.58706
$170$		−	3.14286e6i		−	0.639703i
$171$	322549.			0.0645071
$172$			1.96346e6i			0.385866i
$173$			8.44401e6i			1.63084i	0.578872	+	0.815419i	$0.303493\pi$
$173$			8.44401e6i			1.63084i	−0.578872	+	0.815419i	$0.696507\pi$
$174$			3.14291e6i			0.596602i
$175$	2.94732e6			0.549937
$176$			3.52136e6i			0.645910i
$177$	706974.	−	3.12251e6i	0.127492	−	0.563098i
$178$	2.61449e6			0.463583
$179$		−	4.54211e6i		−	0.791952i	−0.918261	−	0.395976i	$-0.870406\pi$
$179$		−	4.54211e6i		−	0.791952i	0.918261	−	0.395976i	$-0.129594\pi$
$180$	−1.09862e6			−0.188378
$181$	3.26935e6			0.551349			0.275674	−	0.961251i	$-0.411099\pi$
$181$	3.26935e6			0.551349			0.275674	+	0.961251i	$0.411099\pi$
$182$	1.15852e7			1.92172
$183$		−	2.38368e6i		−	0.388951i
$184$	1.22859e6			0.197221
$185$		−	2.55368e6i		−	0.403321i
$186$	−5.57748e6			−0.866760
$187$		−	1.33245e7i		−	2.03763i
$188$		−	5.58809e6i		−	0.840988i
$189$	840996.			0.124569
$190$		−	806902.i		−	0.117641i
$191$			5.28174e6i			0.758014i	0.925394	+	0.379007i	$0.123734\pi$
$191$			5.28174e6i			0.758014i	−0.925394	+	0.379007i	$0.876266\pi$
$192$	−6.57850e6			−0.929444
$193$	−2.45643e6			−0.341690			−0.170845	−	0.985298i	$-0.554650\pi$
$193$	−2.45643e6			−0.341690			−0.170845	+	0.985298i	$0.554650\pi$
$194$	−2.26059e6			−0.309612
$195$			3.14423e6i			0.424044i
$196$	6.37554e6			0.846738
$197$	8.21269e6			1.07420			0.537102	−	0.843517i	$-0.319519\pi$
$197$	8.21269e6			1.07420			0.537102	+	0.843517i	$0.319519\pi$
$198$	−7.85384e6			−1.01178
$199$	3.79081e6			0.481030			0.240515	−	0.970645i	$-0.422684\pi$
$199$	3.79081e6			0.481030			0.240515	+	0.970645i	$0.422684\pi$
$200$		−	4.87241e6i		−	0.609051i
$201$		−	1.80439e6i		−	0.222199i
$202$	−1.93824e7			−2.35155
$203$	−3.56939e6			−0.426684
$204$	7.51668e6			0.885392
$205$	−239851.			−0.0278407
$206$	2.06072e7			2.35731
$207$		−	813415.i		−	0.0917067i
$208$			5.68530e6i			0.631776i
$209$		−	3.42095e6i		−	0.374720i
$210$		−	2.10387e6i		−	0.227175i
$211$			1.02197e7i			1.08791i	0.839116	+	0.543953i	$0.183073\pi$
$211$			1.02197e7i			1.08791i	−0.839116	+	0.543953i	$0.816927\pi$
$212$	9.85296e6			1.03409
$213$	−2.83506e6			−0.293375
$214$			1.16101e7i			1.18467i
$215$		−	1.02048e6i		−	0.102681i
$216$		−	1.39031e6i		−	0.137959i
$217$		−	6.33432e6i		−	0.619899i
$218$	2.13588e7			2.06162
$219$			1.01158e7i			0.963091i
$220$			1.16519e7i			1.09428i
$221$		−	2.15126e7i		−	1.99304i
$222$	1.02986e7			0.941277
$223$	−1.74440e7			−1.57301			−0.786507	−	0.617582i	$-0.788112\pi$
$223$	−1.74440e7			−1.57301			−0.786507	+	0.617582i	$0.788112\pi$
$224$		−	9.01929e6i		−	0.802469i
$225$	−3.22588e6			−0.283205
$226$	1.56265e7			1.35374
$227$		−	1.44622e7i		−	1.23640i	−0.786022	−	0.618199i	$-0.787863\pi$
$227$		−	1.44622e7i		−	1.23640i	0.786022	−	0.618199i	$-0.212137\pi$
$228$	1.92984e6			0.162824
$229$		−	1.71115e7i		−	1.42489i	−0.701727	−	0.712446i	$-0.747588\pi$
$229$		−	1.71115e7i		−	1.42489i	0.701727	−	0.712446i	$-0.252412\pi$
$230$	−2.03487e6			−0.167245
$231$		−	8.91958e6i		−	0.723617i
$232$			5.90080e6i			0.472549i
$233$		−	2.25364e6i		−	0.178163i	−0.996024	−	0.0890815i	$-0.971607\pi$
$233$		−	2.25364e6i		−	0.178163i	0.996024	−	0.0890815i	$-0.0283932\pi$
$234$	−1.26802e7			−0.989640
$235$			2.90433e6i			0.223791i
$236$	4.22989e6	−	1.86823e7i	0.321805	−	1.42133i
$237$	−8.98934e6			−0.675278
$238$			1.43945e7i			1.06774i
$239$	4.46792e6			0.327274			0.163637	−	0.986521i	$-0.447677\pi$
$239$	4.46792e6			0.327274			0.163637	+	0.986521i	$0.447677\pi$
$240$	1.03245e6			0.0746853
$241$	2.48112e7			1.77255			0.886273	−	0.463163i	$-0.153286\pi$
$241$	2.48112e7			1.77255			0.886273	+	0.463163i	$0.153286\pi$
$242$			6.10811e7i			4.30984i
$243$	−920483.			−0.0641500
$244$		−	1.42618e7i		−	0.981758i
$245$	−3.31360e6			−0.225321
$246$		−	967279.i		−	0.0649750i
$247$		−	5.52318e6i		−	0.366520i
$248$	−1.04717e7			−0.686533
$249$			4.93838e6i			0.319879i
$250$			1.75684e7i			1.12438i
$251$	−1.31287e7			−0.830232			−0.415116	−	0.909769i	$-0.636259\pi$
$251$	−1.31287e7			−0.830232			−0.415116	+	0.909769i	$0.636259\pi$
$252$	5.03176e6			0.314426
$253$	−8.62705e6			−0.532722
$254$			2.21107e7i			1.34928i
$255$	−3.90669e6			−0.235607
$256$	−6.75465e6			−0.402608
$257$	−3.85412e6			−0.227052			−0.113526	−	0.993535i	$-0.536215\pi$
$257$	−3.85412e6			−0.227052			−0.113526	+	0.993535i	$0.536215\pi$
$258$	4.11542e6			0.239638
$259$			1.16960e7i			0.673193i
$260$			1.88122e7i			1.07034i
$261$	3.90675e6			0.219733
$262$	−1.31801e7			−0.732850
$263$	−1.01716e7			−0.559141			−0.279570	−	0.960125i	$-0.590192\pi$
$263$	−1.01716e7			−0.559141			−0.279570	+	0.960125i	$0.590192\pi$
$264$	−1.47456e7			−0.801400
$265$	−5.12094e6			−0.275177
$266$			3.69567e6i			0.196358i
$267$		−	3.24991e6i		−	0.170741i
$268$		−	1.07958e7i		−	0.560856i
$269$		−	2.34107e7i		−	1.20270i	−0.798985	−	0.601350i	$-0.794630\pi$
$269$		−	2.34107e7i		−	1.20270i	0.798985	−	0.601350i	$-0.205370\pi$
$270$			2.30272e6i			0.116990i
$271$	2.23212e7			1.12152			0.560762	−	0.827977i	$-0.310508\pi$
$271$	2.23212e7			1.12152			0.560762	+	0.827977i	$0.310508\pi$
$272$	−7.06395e6			−0.351027
$273$		−	1.44008e7i		−	0.707782i
$274$			2.64349e7i			1.28507i
$275$			3.42136e7i			1.64513i
$276$		−	4.86674e6i		−	0.231479i
$277$	−2.40338e7			−1.13079			−0.565397	−	0.824819i	$-0.691277\pi$
$277$	−2.40338e7			−1.13079			−0.565397	+	0.824819i	$0.691277\pi$
$278$		−	3.91534e7i		−	1.82236i
$279$			6.93301e6i			0.319234i
$280$		−	3.95001e6i		−	0.179938i
$281$	7.96445e6			0.358953			0.179476	−	0.983762i	$-0.442560\pi$
$281$	7.96445e6			0.358953			0.179476	+	0.983762i	$0.442560\pi$
$282$	−1.17127e7			−0.522287
$283$		−	1.63519e7i		−	0.721456i	−0.932671	−	0.360728i	$-0.882528\pi$
$283$		−	1.63519e7i		−	0.721456i	0.932671	−	0.360728i	$-0.117472\pi$
$284$	−1.69624e7			−0.740514
$285$	−1.00301e6			−0.0433281
$286$			1.34485e8i			5.74880i
$287$	1.09854e6			0.0464695
$288$			9.87174e6i			0.413253i
$289$	2.59175e6			0.107374
$290$		−	9.77329e6i		−	0.400725i
$291$			2.81000e6i			0.114032i
$292$			6.05238e7i			2.43096i
$293$	2.31684e7			0.921071			0.460536	−	0.887641i	$-0.347657\pi$
$293$	2.31684e7			0.921071			0.460536	+	0.887641i	$0.347657\pi$
$294$		−	1.33632e7i		−	0.525858i
$295$	−2.19843e6	+	9.70985e6i	−0.0856339	+	0.378222i
$296$	1.93355e7			0.745556
$297$			9.76261e6i			0.372646i
$298$	−5.09413e7			−1.92496
$299$	−1.39285e7			−0.521065
$300$	−1.93008e7			−0.714843
$301$			4.67387e6i			0.171387i
$302$	4.73275e7			1.71828
$303$			2.40931e7i			0.866093i
$304$	−1.81361e6			−0.0645539
$305$			7.41236e6i			0.261251i
$306$		−	1.57550e7i		−	0.549864i
$307$	3.50205e7			1.21034			0.605170	−	0.796096i	$-0.293105\pi$
$307$	3.50205e7			1.21034			0.605170	+	0.796096i	$0.293105\pi$
$308$		−	5.33667e7i		−	1.82649i
$309$		−	2.56155e7i		−	0.868216i
$310$	1.73439e7			0.582185
$311$	−2.49635e7			−0.829898			−0.414949	−	0.909845i	$-0.636201\pi$
$311$	−2.49635e7			−0.829898			−0.414949	+	0.909845i	$0.636201\pi$
$312$	−2.38069e7			−0.783863
$313$			9.97787e6i			0.325390i	0.986676	+	0.162695i	$0.0520187\pi$
$313$			9.97787e6i			0.325390i	−0.986676	+	0.162695i	$0.947981\pi$
$314$	−5.42913e7			−1.75364
$315$	−2.61519e6			−0.0836703
$316$	−5.37841e7			−1.70448
$317$	4.14321e7			1.30065			0.650324	−	0.759657i	$-0.274633\pi$
$317$	4.14321e7			1.30065			0.650324	+	0.759657i	$0.274633\pi$
$318$		−	2.06519e7i		−	0.642212i
$319$		−	4.14349e7i		−	1.27642i
$320$	2.04567e7			0.624289
$321$	1.44318e7			0.436321
$322$	9.31986e6			0.279153
$323$	6.86252e6			0.203646
$324$	−5.50734e6			−0.161922
$325$			5.52385e7i			1.60913i
$326$		−	1.02533e8i		−	2.95945i
$327$		−	2.65498e7i		−	0.759309i
$328$		−	1.81606e6i		−	0.0514646i
$329$		−	1.33021e7i		−	0.373535i
$330$	2.44225e7			0.679593
$331$	−1.94238e7			−0.535611			−0.267805	−	0.963473i	$-0.586298\pi$
$331$	−1.94238e7			−0.535611			−0.267805	+	0.963473i	$0.586298\pi$
$332$			2.95468e7i			0.807413i
$333$		−	1.28015e7i		−	0.346679i
$334$		−	2.67162e7i		−	0.717028i
$335$			5.61098e6i			0.149246i
$336$	−4.72870e6			−0.124659
$337$		−	7.24837e7i		−	1.89387i	−0.321422	−	0.946936i	$-0.604161\pi$
$337$		−	7.24837e7i		−	1.89387i	0.321422	−	0.946936i	$-0.395839\pi$
$338$			1.56598e8i			4.05542i
$339$		−	1.94243e7i		−	0.498593i
$340$	−2.33741e7			−0.594700
$341$	7.35313e7			1.85442
$342$		−	4.04497e6i		−	0.101120i
$343$	4.12965e7			1.02337
$344$	7.72669e6			0.189809
$345$			2.52942e6i			0.0615976i
$346$	1.05893e8			2.55646
$347$			2.29379e7i			0.548991i	0.961588	+	0.274496i	$0.0885109\pi$
$347$			2.29379e7i			0.548991i	−0.961588	+	0.274496i	$0.911489\pi$
$348$	2.33745e7			0.554631
$349$		−	3.75880e7i		−	0.884247i	−0.896954	−	0.442123i	$-0.854225\pi$
$349$		−	3.75880e7i		−	0.884247i	0.896954	−	0.442123i	$-0.145775\pi$
$350$		−	3.69612e7i		−	0.862069i
$351$			1.57619e7i			0.364492i
$352$	1.04699e8			2.40058
$353$			2.25945e7i			0.513663i	0.966456	+	0.256831i	$0.0826785\pi$
$353$			2.25945e7i			0.513663i	−0.966456	+	0.256831i	$0.917321\pi$
$354$	−3.91582e7	−	8.86589e6i	−0.882700	−	0.199854i
$355$	8.81598e6			0.197054
$356$		−	1.94445e7i		−	0.430971i
$357$	1.78929e7			0.393258
$358$	−5.69609e7			−1.24145
$359$	−6.84724e7			−1.47990			−0.739950	−	0.672662i	$-0.765151\pi$
$359$	−6.84724e7			−1.47990			−0.739950	+	0.672662i	$0.765151\pi$
$360$			4.32334e6i			0.0926642i
$361$	−4.52840e7			−0.962550
$362$		−	4.09997e7i		−	0.864282i
$363$	7.59261e7			1.58734
$364$		−	8.61615e7i		−	1.78653i
$365$		−	3.14564e7i		−	0.646889i
$366$	−2.98928e7			−0.609711
$367$		−	6.78171e7i		−	1.37196i	−0.727621	−	0.685980i	$-0.759374\pi$
$367$		−	6.78171e7i		−	1.37196i	0.727621	−	0.685980i	$-0.240626\pi$
$368$			4.57361e6i			0.0917732i
$369$	−1.20236e6			−0.0239308
$370$	−3.20247e7			−0.632237
$371$	2.34543e7			0.459304
$372$			4.14809e7i			0.805784i
$373$	3.72116e6			0.0717053			0.0358527	−	0.999357i	$-0.488585\pi$
$373$	3.72116e6			0.0717053			0.0358527	+	0.999357i	$0.488585\pi$
$374$	−1.67097e8			−3.19415
$375$	2.18382e7			0.414117
$376$	−2.19905e7			−0.413687
$377$		−	6.68974e7i		−	1.24849i
$378$		−	1.05466e7i		−	0.195271i
$379$	5.20364e7			0.955850			0.477925	−	0.878401i	$-0.341389\pi$
$379$	5.20364e7			0.955850			0.477925	+	0.878401i	$0.341389\pi$
$380$	−6.00110e6			−0.109365
$381$	2.74844e7			0.496949
$382$	6.62363e7			1.18825
$383$	−3.35438e7			−0.597058			−0.298529	−	0.954401i	$-0.596496\pi$
$383$	−3.35438e7			−0.597058			−0.298529	+	0.954401i	$0.596496\pi$
$384$			4.19691e7i			0.741201i
$385$			2.77366e7i			0.486039i
$386$			3.08052e7i			0.535626i
$387$		−	5.11562e6i		−	0.0882603i
$388$			1.68125e7i			0.287831i
$389$	−7.19227e7			−1.22185			−0.610924	−	0.791690i	$-0.709202\pi$
$389$	−7.19227e7			−1.22185			−0.610924	+	0.791690i	$0.709202\pi$
$390$	3.94306e7			0.664722
$391$		−	1.73061e7i		−	0.289514i
$392$		−	2.50893e7i		−	0.416515i
$393$			1.63834e7i			0.269914i
$394$		−	1.02992e8i		−	1.68390i
$395$	2.79535e7			0.453571
$396$			5.84107e7i			0.940604i
$397$			2.23068e7i			0.356505i	0.983985	+	0.178252i	$0.0570443\pi$
$397$			2.23068e7i			0.356505i	−0.983985	+	0.178252i	$0.942956\pi$
$398$		−	4.75391e7i		−	0.754052i
$399$	4.59386e6			0.0723200
$400$	1.81383e7			0.283410
$401$			1.21946e8i			1.89118i	0.325357	+	0.945591i	$0.394516\pi$
$401$			1.21946e8i			1.89118i	−0.325357	+	0.945591i	$0.605484\pi$
$402$	−2.26281e7			−0.348314
$403$	1.18718e8			1.81384
$404$			1.44151e8i			2.18612i
$405$	2.86236e6			0.0430883
$406$			4.47624e7i			0.668861i
$407$	−1.35772e8			−2.01385
$408$		−	2.95800e7i		−	0.435530i
$409$			1.06170e8i			1.55178i	0.630867	+	0.775891i	$0.282699\pi$
$409$			1.06170e8i			1.55178i	−0.630867	+	0.775891i	$0.717301\pi$
$410$			3.00788e6i			0.0436424i
$411$	3.28595e7			0.473299
$412$		−	1.53260e8i		−	2.19148i
$413$	1.00690e7	−	4.44718e7i	0.142934	−	0.631299i
$414$	−1.02007e7			−0.143757
$415$		−	1.53565e7i		−	0.214857i
$416$	1.69039e8			2.34805
$417$	−4.86691e7			−0.671190
$418$	−4.29008e7			−0.587404
$419$			1.13196e7i			0.153883i	0.997036	+	0.0769415i	$0.0245155\pi$
$419$			1.13196e7i			0.153883i	−0.997036	+	0.0769415i	$0.975485\pi$
$420$	−1.56469e7			−0.211194
$421$		−	8.51436e7i		−	1.14105i	−0.821279	−	0.570527i	$-0.806739\pi$
$421$		−	8.51436e7i		−	1.14105i	0.821279	−	0.570527i	$-0.193261\pi$
$422$	1.28161e8			1.70538
$423$			1.45593e7i			0.192362i
$424$		−	3.87738e7i		−	0.508676i
$425$	−6.86335e7			−0.894066
$426$			3.55534e7i			0.459888i
$427$		−	3.39492e7i		−	0.436060i
$428$	8.63470e7			1.10133
$429$	1.67170e8			2.11732
$430$	−1.27974e7			−0.160960
$431$		−	6.60582e7i		−	0.825078i	−0.910940	−	0.412539i	$-0.864642\pi$
$431$		−	6.60582e7i		−	0.825078i	0.910940	−	0.412539i	$-0.135358\pi$
$432$	5.17563e6			0.0641966
$433$	5.92256e7			0.729534			0.364767	−	0.931099i	$-0.381149\pi$
$433$	5.92256e7			0.729534			0.364767	+	0.931099i	$0.381149\pi$
$434$	−7.94363e7			−0.971740
$435$	−1.21486e7			−0.147590
$436$		−	1.58850e8i		−	1.91659i
$437$		−	4.44319e6i		−	0.0532416i
$438$	1.26858e8			1.50972
$439$	1.45660e7			0.172165			0.0860826	−	0.996288i	$-0.472565\pi$
$439$	1.45660e7			0.172165			0.0860826	+	0.996288i	$0.472565\pi$
$440$	4.58532e7			0.538284
$441$	−1.66109e7			−0.193677
$442$	−2.69782e8			−3.12425
$443$			9.32610e7i			1.07273i	0.843987	+	0.536363i	$0.180202\pi$
$443$			9.32610e7i			1.07273i	−0.843987	+	0.536363i	$0.819798\pi$
$444$		−	7.65925e7i		−	0.875059i
$445$			1.01060e7i			0.114683i
$446$			2.18759e8i			2.46582i
$447$			6.33218e7i			0.708975i
$448$	−9.36933e7			−1.04202
$449$	−1.71584e8			−1.89556			−0.947778	−	0.318930i	$-0.896676\pi$
$449$	−1.71584e8			−1.89556			−0.947778	+	0.318930i	$0.896676\pi$
$450$			4.04546e7i			0.443946i
$451$			1.27522e7i			0.139013i
$452$		−	1.16217e8i		−	1.25851i
$453$		−	5.88299e7i		−	0.632854i
$454$	−1.81366e8			−1.93815
$455$			4.47812e7i			0.475403i
$456$		−	7.59441e6i		−	0.0800939i
$457$		−	1.46508e8i		−	1.53502i	−0.641037	−	0.767510i	$-0.721496\pi$
$457$		−	1.46508e8i		−	1.53502i	0.641037	−	0.767510i	$-0.278504\pi$
$458$	−2.14589e8			−2.23363
$459$	−1.95841e7			−0.202519
$460$			1.51338e7i			0.155480i
$461$	−1.66731e8			−1.70182			−0.850912	−	0.525308i	$-0.823950\pi$
$461$	−1.66731e8			−1.70182			−0.850912	+	0.525308i	$0.823950\pi$
$462$	−1.11857e8			−1.13433
$463$		−	1.07211e8i		−	1.08018i	−0.841608	−	0.540089i	$-0.818391\pi$
$463$		−	1.07211e8i		−	1.08018i	0.841608	−	0.540089i	$-0.181609\pi$
$464$	−2.19666e7			−0.219892
$465$		−	2.15591e7i		−	0.214423i
$466$	−2.82621e7			−0.279285
$467$		−	1.05249e8i		−	1.03340i	−0.856168	−	0.516698i	$-0.827161\pi$
$467$		−	1.05249e8i		−	1.03340i	0.856168	−	0.516698i	$-0.172839\pi$
$468$			9.43050e7i			0.920020i
$469$		−	2.56987e7i		−	0.249111i
$470$	3.64221e7			0.350810
$471$			6.74861e7i			0.645879i
$472$	−7.35193e7	−	1.66457e7i	−0.699158	−	0.158298i
$473$	−5.42561e7			−0.512702
$474$			1.12732e8i			1.05855i
$475$	−1.76211e7			−0.164419
$476$	1.07055e8			0.992629
$477$	−2.56711e7			−0.236531
$478$		−	5.60305e7i		−	0.513028i
$479$	1.23350e8			1.12237			0.561183	−	0.827692i	$-0.310347\pi$
$479$	1.23350e8			1.12237			0.561183	+	0.827692i	$0.310347\pi$
$480$		−	3.06975e7i		−	0.277574i
$481$	−2.19207e8			−1.96978
$482$		−	3.11149e8i		−	2.77860i
$483$		−	1.15849e7i		−	0.102814i
$484$	4.54273e8			4.00665
$485$		−	8.73807e6i		−	0.0765932i
$486$			1.15434e7i			0.100560i
$487$	−1.98572e7			−0.171922			−0.0859611	−	0.996298i	$-0.527396\pi$
$487$	−1.98572e7			−0.171922			−0.0859611	+	0.996298i	$0.527396\pi$
$488$	−5.61237e7			−0.482933
$489$	−1.27452e8			−1.08999
$490$			4.15546e7i			0.353208i
$491$	−1.31211e8			−1.10848			−0.554238	−	0.832358i	$-0.686990\pi$
$491$	−1.31211e8			−1.10848			−0.554238	+	0.832358i	$0.686990\pi$
$492$	−7.19385e6			−0.0604041
$493$	8.31197e7			0.693686
$494$	−6.92641e7			−0.574549
$495$		−	3.03581e7i		−	0.250299i
$496$		−	3.89824e7i		−	0.319465i
$497$	−4.03779e7			−0.328908
$498$	6.19303e7			0.501436
$499$	−1.87364e8			−1.50794			−0.753972	−	0.656907i	$-0.771865\pi$
$499$	−1.87364e8			−1.50794			−0.753972	+	0.656907i	$0.771865\pi$
$500$	1.30660e8			1.04528
$501$	−3.32093e7			−0.264086
$502$			1.64642e8i			1.30145i
$503$			1.92336e8i			1.51132i	0.654964	+	0.755660i	$0.272684\pi$
$503$			1.92336e8i			1.51132i	−0.654964	+	0.755660i	$0.727316\pi$
$504$		−	1.98012e7i		−	0.154668i
$505$		−	7.49206e7i		−	0.581738i
$506$			1.08189e8i			0.835084i
$507$	1.94657e8			1.49364
$508$	1.64442e8			1.25436
$509$		−	9.10904e6i		−	0.0690748i	−0.999403	−	0.0345374i	$-0.989004\pi$
$509$		−	9.10904e6i		−	0.0690748i	0.999403	−	0.0345374i	$-0.0109958\pi$
$510$			4.89923e7i			0.369333i
$511$			1.44073e8i			1.07974i
$512$		−	8.76009e7i		−	0.652678i
$513$	−5.02804e6			−0.0372432
$514$			4.83330e7i			0.355922i
$515$			7.96548e7i			0.583163i
$516$		−	3.06073e7i		−	0.222780i
$517$	1.54415e8			1.11743
$518$	1.46676e8			1.05528
$519$		−	1.31629e8i		−	0.941564i
$520$	7.40308e7			0.526505
$521$	−4.65924e7			−0.329459			−0.164730	−	0.986339i	$-0.552675\pi$
$521$	−4.65924e7			−0.329459			−0.164730	+	0.986339i	$0.552675\pi$
$522$		−	4.89931e7i		−	0.344448i
$523$	1.13804e8			0.795522			0.397761	−	0.917489i	$-0.369787\pi$
$523$	1.13804e8			0.795522			0.397761	+	0.917489i	$0.369787\pi$
$524$			9.80233e7i			0.681295i
$525$	−4.59441e7			−0.317506
$526$			1.27558e8i			0.876497i
$527$			1.47506e8i			1.00781i
$528$		−	5.48926e7i		−	0.372917i
$529$	1.36831e8			0.924309
$530$			6.42197e7i			0.431361i
$531$	−1.10206e7	+	4.86751e7i	−0.0736076	+	0.325105i
$532$	2.74855e7			0.182544
$533$			2.05887e7i			0.135971i
$534$	−4.07559e7			−0.267650
$535$	−4.48776e7			−0.293068
$536$	−4.24842e7			−0.275888
$537$			7.08045e7i			0.457233i
$538$	−2.93585e8			−1.88533
$539$			1.76175e8i			1.12507i
$540$	1.71258e7			0.108760
$541$		−	2.89674e7i		−	0.182944i	−0.995808	−	0.0914718i	$-0.970843\pi$
$541$		−	2.89674e7i		−	0.182944i	0.995808	−	0.0914718i	$-0.0291571\pi$
$542$		−	2.79921e8i		−	1.75808i
$543$	−5.09642e7			−0.318321
$544$			2.10030e8i			1.30462i
$545$			8.25602e7i			0.510013i
$546$	−1.80595e8			−1.10950
$547$	−2.65654e6			−0.0162313			−0.00811567	−	0.999967i	$-0.502583\pi$
$547$	−2.65654e6			−0.0162313			−0.00811567	+	0.999967i	$0.502583\pi$
$548$	1.96602e8			1.19466
$549$			3.71579e7i			0.224561i
$550$	4.29060e8			2.57887
$551$	2.13402e7			0.127569
$552$	−1.91518e7			−0.113866
$553$	−1.28029e8			−0.757066
$554$			3.01399e8i			1.77261i
$555$			3.98079e7i			0.232857i
$556$	−2.91192e8			−1.69416
$557$	2.85251e8			1.65068			0.825339	−	0.564638i	$-0.190984\pi$
$557$	2.85251e8			1.65068			0.825339	+	0.564638i	$0.190984\pi$
$558$	8.69442e7			0.500424
$559$	−8.75975e7			−0.501483
$560$	1.47045e7			0.0837310
$561$			2.07708e8i			1.17643i
$562$		−	9.98792e7i		−	0.562686i
$563$			4.31220e7i			0.241643i	0.992674	+	0.120821i	$0.0385528\pi$
$563$			4.31220e7i			0.241643i	−0.992674	+	0.120821i	$0.961447\pi$
$564$			8.71097e7i			0.485544i
$565$			6.04024e7i			0.334895i
$566$	−2.05063e8			−1.13094
$567$	−1.31098e7			−0.0719197
$568$			6.67514e7i			0.364263i
$569$		−	5.01143e7i		−	0.272035i	−0.990706	−	0.136017i	$-0.956570\pi$
$569$		−	5.01143e7i		−	0.272035i	0.990706	−	0.136017i	$-0.0434303\pi$
$570$			1.25784e7i			0.0679202i
$571$			1.37021e8i			0.736004i	0.929825	+	0.368002i	$0.119958\pi$
$571$			1.37021e8i			0.736004i	−0.929825	+	0.368002i	$0.880042\pi$
$572$	1.00020e9			5.34438
$573$		−	8.23342e7i		−	0.437639i
$574$		−	1.37763e7i		−	0.0728446i
$575$			4.44373e7i			0.233746i
$576$	1.02549e8			0.536615
$577$	1.66358e8			0.865996			0.432998	−	0.901395i	$-0.357456\pi$
$577$	1.66358e8			0.865996			0.432998	+	0.901395i	$0.357456\pi$
$578$		−	3.25021e7i		−	0.168317i
$579$	3.82919e7			0.197275
$580$	−7.26860e7			−0.372535
$581$			7.03340e7i			0.358622i
$582$	3.52392e7			0.178754
$583$			2.72266e8i			1.37401i
$584$	2.38176e8			1.19580
$585$		−	4.90137e7i		−	0.244822i
$586$		−	2.90546e8i		−	1.44385i
$587$		−	2.97718e8i		−	1.47194i	−0.677012	−	0.735972i	$-0.736725\pi$
$587$		−	2.97718e8i		−	1.47194i	0.677012	−	0.735972i	$-0.263275\pi$
$588$	−9.93849e7			−0.488864
$589$			3.78708e7i			0.185336i
$590$	1.21768e8	+	2.75696e7i	0.592892	+	0.134238i
$591$	−1.28023e8			−0.620192
$592$			7.19793e7i			0.346931i
$593$	2.48438e8			1.19139			0.595694	−	0.803211i	$-0.296877\pi$
$593$	2.48438e8			1.19139			0.595694	+	0.803211i	$0.296877\pi$
$594$	1.22429e8			0.584152
$595$	−5.56404e7			−0.264143
$596$			3.78861e8i			1.78954i
$597$	−5.90928e7			−0.277723
$598$			1.74672e8i			0.816810i
$599$	−9.40947e7			−0.437809			−0.218904	−	0.975746i	$-0.570248\pi$
$599$	−9.40947e7			−0.437809			−0.218904	+	0.975746i	$0.570248\pi$
$600$			7.59533e7i			0.351636i
$601$		−	1.55857e8i		−	0.717965i	−0.933344	−	0.358982i	$-0.883124\pi$
$601$		−	1.55857e8i		−	0.717965i	0.933344	−	0.358982i	$-0.116876\pi$
$602$	5.86133e7			0.268662
$603$			2.81276e7i			0.128286i
$604$		−	3.51985e8i		−	1.59740i
$605$	−2.36102e8			−1.06619
$606$	3.02142e8			1.35767
$607$	−4.24758e7			−0.189922			−0.0949611	−	0.995481i	$-0.530273\pi$
$607$	−4.24758e7			−0.189922			−0.0949611	+	0.995481i	$0.530273\pi$
$608$			5.39234e7i			0.239920i
$609$	5.56414e7			0.246346
$610$	9.29557e7			0.409531
$611$	2.49306e8			1.09297
$612$	−1.17173e8			−0.511181
$613$		−	3.73187e8i		−	1.62011i	−0.586353	−	0.810056i	$-0.699437\pi$
$613$		−	3.73187e8i		−	1.62011i	0.586353	−	0.810056i	$-0.300563\pi$
$614$		−	4.39179e8i		−	1.89730i
$615$	3.73890e6			0.0160738
$616$	−2.10011e8			−0.898463
$617$	−1.67464e8			−0.712961			−0.356480	−	0.934303i	$-0.616023\pi$
$617$	−1.67464e8			−0.712961			−0.356480	+	0.934303i	$0.616023\pi$
$618$	−3.21235e8			−1.36100
$619$	−2.22010e8			−0.936053			−0.468027	−	0.883714i	$-0.655035\pi$
$619$	−2.22010e8			−0.936053			−0.468027	+	0.883714i	$0.655035\pi$
$620$		−	1.28990e8i		−	0.541229i
$621$			1.26799e7i			0.0529469i
$622$			3.13058e8i			1.30093i
$623$		−	4.62864e7i		−	0.191421i
$624$		−	8.86250e7i		−	0.364756i
$625$	1.39517e8			0.571461
$626$	1.25129e8			0.510075
$627$			5.33273e7i			0.216345i
$628$			4.03776e8i			1.63028i
$629$		−	2.72363e8i		−	1.09445i
$630$			3.27961e7i			0.131160i
$631$	−1.76655e8			−0.703132			−0.351566	−	0.936163i	$-0.614351\pi$
$631$	−1.76655e8			−0.703132			−0.351566	+	0.936163i	$0.614351\pi$
$632$			2.11654e8i			0.838445i
$633$		−	1.59309e8i		−	0.628102i
$634$		−	5.19585e8i		−	2.03887i
$635$	−8.54664e7			−0.333791
$636$	−1.53592e8			−0.597033
$637$			2.84438e8i			1.10045i
$638$	−5.19620e8			−2.00089
$639$	4.41942e7			0.169380
$640$		−	1.30508e8i		−	0.497850i
$641$	−3.17034e8			−1.20374			−0.601869	−	0.798595i	$-0.705577\pi$
$641$	−3.17034e8			−1.20374			−0.601869	+	0.798595i	$0.705577\pi$
$642$		−	1.80984e8i		−	0.683967i
$643$	−2.99097e8			−1.12507			−0.562535	−	0.826773i	$-0.690174\pi$
$643$	−2.99097e8			−1.12507			−0.562535	+	0.826773i	$0.690174\pi$
$644$		−	6.93138e7i		−	0.259515i
$645$			1.59077e7i			0.0592827i
$646$		−	8.60603e7i		−	0.319231i
$647$	1.74573e7			0.0644561			0.0322281	−	0.999481i	$-0.489740\pi$
$647$	1.74573e7			0.0644561			0.0322281	+	0.999481i	$0.489740\pi$
$648$			2.16727e7i			0.0796505i
$649$	5.16247e8	+	1.16884e8i	1.88853	+	0.427585i
$650$	6.92725e8			2.52244
$651$			9.87423e7i			0.357899i
$652$	−7.62560e8			−2.75126
$653$	4.06062e8			1.45832			0.729160	−	0.684344i	$-0.239911\pi$
$653$	4.06062e8			1.45832			0.729160	+	0.684344i	$0.239911\pi$
$654$	−3.32951e8			−1.19028
$655$		−	5.09462e7i		−	0.181296i
$656$	6.76056e6			0.0239481
$657$		−	1.57690e8i		−	0.556041i
$658$	−1.66816e8			−0.585545
$659$		−	1.24543e8i		−	0.435174i	−0.976041	−	0.217587i	$-0.930181\pi$
$659$		−	1.24543e8i		−	0.435174i	0.976041	−	0.217587i	$-0.0698185\pi$
$660$		−	1.81636e8i		−	0.631785i
$661$	−2.93091e8			−1.01484			−0.507421	−	0.861698i	$-0.669401\pi$
$661$	−2.93091e8			−1.01484			−0.507421	+	0.861698i	$0.669401\pi$
$662$			2.43586e8i			0.839611i
$663$			3.35349e8i			1.15068i
$664$	1.16274e8			0.397171
$665$	−1.42852e7			−0.0485759
$666$	−1.60539e8			−0.543446
$667$		−	5.38165e7i		−	0.181359i
$668$	−1.98694e8			−0.666586
$669$	2.71926e8			0.908180
$670$	7.03652e7			0.233955
$671$	3.94096e8			1.30447
$672$			1.40597e8i			0.463306i
$673$			5.89657e7i			0.193443i	0.995311	+	0.0967217i	$0.0308357\pi$
$673$			5.89657e7i			0.193443i	−0.995311	+	0.0967217i	$0.969164\pi$
$674$	−9.08991e8			−2.96879
$675$	5.02865e7			0.163509
$676$	1.16465e9			3.77013
$677$	−2.84900e8			−0.918178			−0.459089	−	0.888390i	$-0.651824\pi$
$677$	−2.84900e8			−0.918178			−0.459089	+	0.888390i	$0.651824\pi$
$678$	−2.43593e8			−0.781584
$679$			4.00210e7i			0.127844i
$680$			9.19829e7i			0.292537i
$681$			2.25444e8i			0.713835i
$682$		−	9.22128e8i		−	2.90695i
$683$			7.07388e7i			0.222022i	0.993819	+	0.111011i	$0.0354089\pi$
$683$			7.07388e7i			0.222022i	−0.993819	+	0.111011i	$0.964591\pi$
$684$	−3.00833e7			−0.0940062
$685$	−1.02181e8			−0.317905
$686$		−	5.17884e8i		−	1.60421i
$687$			2.66742e8i			0.822661i
$688$			2.87638e7i			0.0883243i
$689$			4.39579e8i			1.34394i
$690$	3.17205e7			0.0965590
$691$		−	1.82554e8i		−	0.553295i	−0.960971	−	0.276648i	$-0.910776\pi$
$691$		−	1.82554e8i		−	0.553295i	0.960971	−	0.276648i	$-0.0892235\pi$
$692$		−	7.87550e8i		−	2.37662i
$693$			1.39043e8i			0.417780i
$694$	2.87656e8			0.860587
$695$	1.51343e8			0.450825
$696$		−	9.19844e7i		−	0.272826i
$697$	−2.55813e7			−0.0755483
$698$	−4.71377e8			−1.38613
$699$			3.51308e7i			0.102863i
$700$	−2.74888e8			−0.801424
$701$			1.76642e8i			0.512790i	0.966572	+	0.256395i	$0.0825348\pi$
$701$			1.76642e8i			0.512790i	−0.966572	+	0.256395i	$0.917465\pi$
$702$	1.97664e8			0.571369
$703$		−	6.99268e7i		−	0.201269i
$704$		−	1.08763e9i		−	3.11719i
$705$		−	4.52740e7i		−	0.129206i
$706$	2.83349e8			0.805207
$707$			3.43142e8i			0.970993i
$708$	−6.59375e7	+	2.91228e8i	−0.185794	+	0.820603i
$709$	1.13091e8			0.317314			0.158657	−	0.987334i	$-0.449284\pi$
$709$	1.13091e8			0.317314			0.158657	+	0.987334i	$0.449284\pi$
$710$		−	1.10558e8i		−	0.308898i
$711$	1.40130e8			0.389872
$712$	−7.65191e7			−0.211997
$713$	9.55039e7			0.263483
$714$		−	2.24389e8i		−	0.616462i
$715$	−5.19838e8			−1.42216
$716$			4.23630e8i			1.15411i
$717$	−6.96480e7			−0.188952
$718$			8.58687e8i			2.31986i
$719$			3.86770e8i			1.04056i	0.853997	+	0.520278i	$0.174172\pi$
$719$			3.86770e8i			1.04056i	−0.853997	+	0.520278i	$0.825828\pi$
$720$	−1.60943e7			−0.0431196
$721$		−	3.64825e8i		−	0.973372i
$722$			5.67889e8i			1.50887i
$723$	−3.86769e8			−1.02338
$724$	−3.04924e8			−0.803481
$725$	−2.13428e8			−0.560065
$726$		−	9.52161e8i		−	2.48829i
$727$	−4.75413e7			−0.123728			−0.0618640	−	0.998085i	$-0.519705\pi$
$727$	−4.75413e7			−0.123728			−0.0618640	+	0.998085i	$0.519705\pi$
$728$	−3.39067e8			−0.878802
$729$	1.43489e7			0.0370370
$730$	−3.94483e8			−1.01405
$731$		−	1.08839e8i		−	0.278634i
$732$			2.22319e8i			0.566818i
$733$	−4.83665e8			−1.22810			−0.614049	−	0.789268i	$-0.710460\pi$
$733$	−4.83665e8			−1.22810			−0.614049	+	0.789268i	$0.710460\pi$
$734$	−8.50469e8			−2.15065
$735$	5.16539e7			0.130089
$736$	1.35986e8			0.341083
$737$	2.98321e8			0.745214
$738$			1.50784e7i			0.0375133i
$739$		−	1.08119e8i		−	0.267897i	−0.990988	−	0.133949i	$-0.957234\pi$
$739$		−	1.08119e8i		−	0.267897i	0.990988	−	0.133949i	$-0.0427657\pi$
$740$			2.38174e8i			0.587760i
$741$			8.60978e7i			0.211611i
$742$		−	2.94131e8i		−	0.719995i
$743$	4.95229e8			1.20737			0.603684	−	0.797224i	$-0.293699\pi$
$743$	4.95229e8			1.20737			0.603684	+	0.797224i	$0.293699\pi$
$744$	1.63237e8			0.396370
$745$		−	1.96908e8i		−	0.476205i
$746$		−	4.66656e7i		−	0.112404i
$747$		−	7.69817e7i		−	0.184682i
$748$			1.24274e9i			2.96944i
$749$	2.05543e8			0.489167
$750$		−	2.73864e8i		−	0.649160i
$751$			6.79194e8i			1.60352i	0.597647	+	0.801759i	$0.296102\pi$
$751$			6.79194e8i			1.60352i	−0.597647	+	0.801759i	$0.703898\pi$
$752$		−	8.18630e7i		−	0.192501i
$753$	2.04656e8			0.479335
$754$	−8.38935e8			−1.95711
$755$			1.82939e8i			0.425075i
$756$	−7.84374e7			−0.181534
$757$	2.68360e8			0.618628			0.309314	−	0.950960i	$-0.399901\pi$
$757$	2.68360e8			0.618628			0.309314	+	0.950960i	$0.399901\pi$
$758$		−	6.52569e8i		−	1.49837i
$759$	1.34482e8			0.307567
$760$			2.36158e7i			0.0537975i
$761$	−3.16771e8			−0.718772			−0.359386	−	0.933189i	$-0.617014\pi$
$761$	−3.16771e8			−0.718772			−0.359386	+	0.933189i	$0.617014\pi$
$762$		−	3.44672e8i		−	0.779007i
$763$		−	3.78132e8i		−	0.851275i
$764$		−	4.92614e8i		−	1.10465i
$765$	6.08993e7			0.136028
$766$			4.20661e8i			0.935935i
$767$	8.33489e8	+	1.88712e8i	1.84720	+	0.418228i
$768$	1.05295e8			0.232446
$769$		−	3.68040e8i		−	0.809311i	−0.914469	−	0.404656i	$-0.867391\pi$
$769$		−	3.68040e8i		−	0.809311i	0.914469	−	0.404656i	$-0.132609\pi$
$770$	3.47834e8			0.761904
$771$	6.00797e7			0.131089
$772$	2.29104e8			0.497945
$773$		−	2.02647e8i		−	0.438734i	−0.975642	−	0.219367i	$-0.929601\pi$
$773$		−	2.02647e8i		−	0.438734i	0.975642	−	0.219367i	$-0.0703992\pi$
$774$	−6.41531e7			−0.138355
$775$		−	3.78754e8i		−	0.813678i
$776$	6.61614e7			0.141586
$777$		−	1.82323e8i		−	0.388668i
$778$			9.01955e8i			1.91534i
$779$	−6.56778e6			−0.0138933
$780$		−	2.93254e8i		−	0.617959i
$781$		−	4.68722e8i		−	0.983926i
$782$	−2.17030e8			−0.453836
$783$	−6.09003e7			−0.126863
$784$	9.33989e7			0.193818
$785$		−	2.09857e8i		−	0.433824i

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)

\(f(q)\)	\(=\)	\(q-12.5406i q^{2} -15.5885 q^{3} -93.2673 q^{4} +48.4743 q^{5} +195.489i q^{6} -222.016 q^{7} +367.030i q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-12.5406i q^{2} -15.5885 q^{3} -93.2673 q^{4} +48.4743 q^{5} +195.489i q^{6} -222.016 q^{7} +367.030i q^{8} +243.000 q^{9} -607.898i q^{10} -2577.25i q^{11} +1453.89 q^{12} -4161.02i q^{13} +2784.22i q^{14} -755.640 q^{15} -1366.32 q^{16} +5170.04 q^{17} -3047.37i q^{18} +1327.36 q^{19} -4521.07 q^{20} +3460.89 q^{21} -32320.3 q^{22} -3347.39i q^{23} -5721.43i q^{24} -13275.2 q^{25} -52181.7 q^{26} -3788.00 q^{27} +20706.8 q^{28} +16077.2 q^{29} +9476.20i q^{30} +28530.9i q^{31} +40624.4i q^{32} +40175.4i q^{33} -64835.5i q^{34} -10762.1 q^{35} -22663.9 q^{36} -52681.0i q^{37} -16646.0i q^{38} +64863.8i q^{39} +17791.5i q^{40} -4948.00 q^{41} -43401.7i q^{42} -21051.9i q^{43} +240373. i q^{44} +11779.3 q^{45} -41978.3 q^{46} +59914.8i q^{47} +21298.9 q^{48} -68357.8 q^{49} +166480. i q^{50} -80593.0 q^{51} +388087. i q^{52} -105642. q^{53} +47503.8i q^{54} -124931. i q^{55} -81486.5i q^{56} -20691.5 q^{57} -201618. i q^{58} +(-45352.4 + 200309. i) q^{59} +70476.5 q^{60} +152913. i q^{61} +357795. q^{62} -53949.9 q^{63} +422011. q^{64} -201703. i q^{65} +503824. q^{66} +115751. i q^{67} -482195. q^{68} +52180.6i q^{69} +134963. i q^{70} +181869. q^{71} +89188.2i q^{72} -648928. i q^{73} -660653. q^{74} +206940. q^{75} -123799. q^{76} +572192. i q^{77} +813433. q^{78} +576666. q^{79} -66231.6 q^{80} +59049.0 q^{81} +62051.0i q^{82} -316797. i q^{83} -322788. q^{84} +250614. q^{85} -264004. q^{86} -250618. q^{87} +945928. q^{88} +208482. i q^{89} -147719. i q^{90} +923813. i q^{91} +312201. i q^{92} -444753. i q^{93} +751369. q^{94} +64343.0 q^{95} -633272. i q^{96} -180262. i q^{97} +857250. i q^{98} -626272. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} + O(q^{10}) \)	\( 60q - 1920q^{4} - 408q^{7} + 14580q^{9} - 1944q^{12} - 4536q^{15} + 56616q^{16} + 8480q^{17} + 11376q^{19} + 40796q^{20} - 8232q^{22} + 197940q^{25} + 147252q^{26} + 71640q^{28} + 63456q^{29} - 364432q^{35} - 466560q^{36} + 99632q^{41} - 470316q^{46} + 171072q^{48} + 1737420q^{49} + 60912q^{51} + 92240q^{53} + 186624q^{57} + 917264q^{59} + 1063368q^{60} - 115768q^{62} - 99144q^{63} - 1107444q^{64} + 1172232q^{66} - 4247232q^{68} + 1498048q^{71} + 1161448q^{74} - 1477440q^{75} - 1045320q^{76} - 1060452q^{78} - 90600q^{79} + 77096q^{80} + 3542940q^{81} - 2225880q^{84} - 693408q^{85} - 1567768q^{86} + 1821528q^{87} + 62892q^{88} + 5268696q^{94} + 296128q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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