LMFDB - Embedded newform 1550.3.d.a.1549.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1550,3,Mod(1549,1550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1550, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1550.1549"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1550 = 2 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1550.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$42.2344409758$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.9265217536.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 20x^{5} + 144x^{4} - 332x^{3} + 696x^{2} - 868x + 961 $ x^8 - 4x^7 - 20x^5 + 144x^4 - 332x^3 + 696x^2 - 868x + 961
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 62)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1549.7
Root			$-2.35291 + 2.38882i$ of defining polynomial
Character	$\chi$	$=$	1550.1549
Dual form			1550.3.d.a.1549.3

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} +3.34239 q^{3} -2.00000 q^{4} +4.72685i q^{6} -1.58579i q^{7} -2.82843i q^{8} +2.17157 q^{9} -2.76893i q^{11} -6.68478 q^{12} -14.7540 q^{13} +2.24264 q^{14} +4.00000 q^{16} +17.5230 q^{17} +3.07107i q^{18} +12.2132 q^{19} -5.30032i q^{21} +3.91585 q^{22} +22.2498 q^{23} -9.45371i q^{24} -20.8653i q^{26} -22.8233 q^{27} +3.17157i q^{28} -14.7540i q^{29} +(21.5858 - 22.2498i) q^{31} +5.65685i q^{32} -9.25483i q^{33} +24.7812i q^{34} -4.34315 q^{36} +8.30678 q^{37} +17.2721i q^{38} -49.3137 q^{39} +5.34315 q^{41} +7.49578 q^{42} +60.0646 q^{43} +5.53785i q^{44} +31.4660i q^{46} +39.7401i q^{47} +13.3696 q^{48} +46.4853 q^{49} +58.5685 q^{51} +29.5080 q^{52} +92.9151 q^{53} -32.2770i q^{54} -4.48528 q^{56} +40.8213 q^{57} +20.8653 q^{58} -13.7868 q^{59} -69.8543i q^{61} +(31.4660 + 30.5269i) q^{62} -3.44365i q^{63} -8.00000 q^{64} +13.0883 q^{66} -109.054i q^{67} -35.0459 q^{68} +74.3675 q^{69} +2.75736 q^{71} -6.14214i q^{72} -74.0077 q^{73} +11.7476i q^{74} -24.4264 q^{76} -4.39093 q^{77} -69.7401i q^{78} +102.942i q^{79} -95.8284 q^{81} +7.55635i q^{82} +110.872 q^{83} +10.6006i q^{84} +84.9442i q^{86} -49.3137i q^{87} -7.83171 q^{88} +50.4718i q^{89} +23.3967i q^{91} -44.4996 q^{92} +(72.1481 - 74.3675i) q^{93} -56.2010 q^{94} +18.9074i q^{96} -113.711i q^{97} +65.7401i q^{98} -6.01293i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 40 q^{9} - 16 q^{14} + 32 q^{16} - 72 q^{19} + 184 q^{31} - 80 q^{36} - 304 q^{39} + 88 q^{41} + 304 q^{49} + 16 q^{51} + 32 q^{56} - 280 q^{59} - 64 q^{64} + 512 q^{66} - 16 q^{69} + 56 q^{71}+ \cdots - 608 q^{94}+O(q^{100}) $ 8 * q - 16 * q^4 + 40 * q^9 - 16 * q^14 + 32 * q^16 - 72 * q^19 + 184 * q^31 - 80 * q^36 - 304 * q^39 + 88 * q^41 + 304 * q^49 + 16 * q^51 + 32 * q^56 - 280 * q^59 - 64 * q^64 + 512 * q^66 - 16 * q^69 + 56 * q^71 + 144 * q^76 - 744 * q^81 - 608 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1550\mathbb{Z}\right)^\times$.

$n$	$251$	$1427$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$	3.34239			1.11413			0.557065	−	0.830469i	$-0.311927\pi$
$3$	3.34239			1.11413			0.557065	+	0.830469i	$0.311927\pi$
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$			4.72685i			0.787809i
$7$		−	1.58579i		−	0.226541i	−0.993564	−	0.113270i	$-0.963867\pi$
$7$		−	1.58579i		−	0.226541i	0.993564	−	0.113270i	$-0.0361327\pi$
$8$		−	2.82843i		−	0.353553i
$9$	2.17157			0.241286
$10$		0			0
$11$		−	2.76893i		−	0.251721i	−0.992048	−	0.125860i	$-0.959831\pi$
$11$		−	2.76893i		−	0.251721i	0.992048	−	0.125860i	$-0.0401691\pi$
$12$	−6.68478			−0.557065
$13$	−14.7540			−1.13492			−0.567462	−	0.823399i	$-0.692075\pi$
$13$	−14.7540			−1.13492			−0.567462	+	0.823399i	$0.692075\pi$
$14$	2.24264			0.160189
$15$		0			0
$16$	4.00000			0.250000
$17$	17.5230			1.03076			0.515381	−	0.856961i	$-0.327650\pi$
$17$	17.5230			1.03076			0.515381	+	0.856961i	$0.327650\pi$
$18$			3.07107i			0.170615i
$19$	12.2132			0.642800			0.321400	−	0.946943i	$-0.395847\pi$
$19$	12.2132			0.642800			0.321400	+	0.946943i	$0.395847\pi$
$20$		0			0
$21$		−	5.30032i		−	0.252396i
$22$	3.91585			0.177993
$23$	22.2498			0.967383			0.483691	−	0.875239i	$-0.339296\pi$
$23$	22.2498			0.967383			0.483691	+	0.875239i	$0.339296\pi$
$24$		−	9.45371i		−	0.393904i
$25$		0			0
$26$		−	20.8653i		−	0.802513i
$27$	−22.8233			−0.845306
$28$			3.17157i			0.113270i
$29$		−	14.7540i		−	0.508759i	−0.967104	−	0.254380i	$-0.918129\pi$
$29$		−	14.7540i		−	0.508759i	0.967104	−	0.254380i	$-0.0818713\pi$
$30$		0			0
$31$	21.5858	−	22.2498i	0.696316	−	0.717736i
$31$	21.5858	−	22.2498i	0.696316	−	0.717736i
$32$			5.65685i			0.176777i
$33$		−	9.25483i		−	0.280450i
$34$			24.7812i			0.728859i
$35$		0			0
$36$	−4.34315			−0.120643
$37$	8.30678			0.224508			0.112254	−	0.993680i	$-0.464193\pi$
$37$	8.30678			0.224508			0.112254	+	0.993680i	$0.464193\pi$
$38$			17.2721i			0.454528i
$39$	−49.3137			−1.26445
$40$		0			0
$41$	5.34315			0.130321			0.0651603	−	0.997875i	$-0.479244\pi$
$41$	5.34315			0.130321			0.0651603	+	0.997875i	$0.479244\pi$
$42$	7.49578			0.178471
$43$	60.0646			1.39685			0.698426	−	0.715682i	$-0.253884\pi$
$43$	60.0646			1.39685			0.698426	+	0.715682i	$0.253884\pi$
$44$			5.53785i			0.125860i
$45$		0			0
$46$			31.4660i			0.684043i
$47$			39.7401i			0.845534i	0.906238	+	0.422767i	$0.138941\pi$
$47$			39.7401i			0.845534i	−0.906238	+	0.422767i	$0.861059\pi$
$48$	13.3696			0.278533
$49$	46.4853			0.948679
$50$		0			0
$51$	58.5685			1.14840
$52$	29.5080			0.567462
$53$	92.9151			1.75311			0.876557	−	0.481298i	$-0.159834\pi$
$53$	92.9151			1.75311			0.876557	+	0.481298i	$0.159834\pi$
$54$		−	32.2770i		−	0.597722i
$55$		0			0
$56$	−4.48528			−0.0800943
$57$	40.8213			0.716163
$58$	20.8653			0.359747
$59$	−13.7868			−0.233675			−0.116837	−	0.993151i	$-0.537276\pi$
$59$	−13.7868			−0.233675			−0.116837	+	0.993151i	$0.537276\pi$
$60$		0			0
$61$		−	69.8543i		−	1.14515i	−0.819852	−	0.572576i	$-0.805944\pi$
$61$		−	69.8543i		−	1.14515i	0.819852	−	0.572576i	$-0.194056\pi$
$62$	31.4660	+	30.5269i	0.507516	+	0.492370i
$63$		−	3.44365i		−	0.0546611i
$64$	−8.00000			−0.125000
$65$		0			0
$66$	13.0883			0.198308
$67$		−	109.054i		−	1.62767i	−0.581097	−	0.813835i	$-0.697376\pi$
$67$		−	109.054i		−	1.62767i	0.581097	−	0.813835i	$-0.302624\pi$
$68$	−35.0459			−0.515381
$69$	74.3675			1.07779
$70$		0			0
$71$	2.75736			0.0388360			0.0194180	−	0.999811i	$-0.493819\pi$
$71$	2.75736			0.0388360			0.0194180	+	0.999811i	$0.493819\pi$
$72$		−	6.14214i		−	0.0853074i
$73$	−74.0077			−1.01380			−0.506902	−	0.862004i	$-0.669209\pi$
$73$	−74.0077			−1.01380			−0.506902	+	0.862004i	$0.669209\pi$
$74$			11.7476i			0.158751i
$75$		0			0
$76$	−24.4264			−0.321400
$77$	−4.39093			−0.0570250
$78$		−	69.7401i		−	0.894104i
$79$			102.942i			1.30307i	0.758620	+	0.651533i	$0.225874\pi$
$79$			102.942i			1.30307i	−0.758620	+	0.651533i	$0.774126\pi$
$80$		0			0
$81$	−95.8284			−1.18307
$82$			7.55635i			0.0921506i
$83$	110.872			1.33581			0.667906	−	0.744246i	$-0.267191\pi$
$83$	110.872			1.33581			0.667906	+	0.744246i	$0.267191\pi$
$84$			10.6006i			0.126198i
$85$		0			0
$86$			84.9442i			0.987723i
$87$		−	49.3137i		−	0.566824i
$88$	−7.83171			−0.0889967
$89$			50.4718i			0.567099i	0.958958	+	0.283549i	$0.0915120\pi$
$89$			50.4718i			0.567099i	−0.958958	+	0.283549i	$0.908488\pi$
$90$		0			0
$91$			23.3967i			0.257107i
$92$	−44.4996			−0.483691
$93$	72.1481	−	74.3675i	0.775786	−	0.799651i
$94$	−56.2010			−0.597883
$95$		0			0
$96$			18.9074i			0.196952i
$97$		−	113.711i		−	1.17228i	−0.810212	−	0.586138i	$-0.800648\pi$
$97$		−	113.711i		−	1.17228i	0.810212	−	0.586138i	$-0.199352\pi$
$98$			65.7401i			0.670818i
$99$		−	6.01293i		−	0.0607366i
$100$		0			0
$101$	172.255			1.70549			0.852747	−	0.522325i	$-0.174935\pi$
$101$	172.255			1.70549			0.852747	+	0.522325i	$0.174935\pi$
$102$			82.8284i			0.812043i
$103$			3.75231i			0.0364302i	0.999834	+	0.0182151i	$0.00579836\pi$
$103$			3.75231i			0.0364302i	−0.999834	+	0.0182151i	$0.994202\pi$
$104$			41.7307i			0.401257i
$105$		0			0
$106$			131.402i			1.23964i
$107$			4.81623i			0.0450115i	0.999747	+	0.0225058i	$0.00716441\pi$
$107$			4.81623i			0.0450115i	−0.999747	+	0.0225058i	$0.992836\pi$
$108$	45.6465			0.422653
$109$	−80.6812			−0.740195			−0.370097	−	0.928993i	$-0.620676\pi$
$109$	−80.6812			−0.740195			−0.370097	+	0.928993i	$0.620676\pi$
$110$		0			0
$111$	27.7645			0.250131
$112$		−	6.34315i		−	0.0566352i
$113$		−	175.853i		−	1.55622i	−0.628128	−	0.778110i	$-0.716179\pi$
$113$		−	175.853i		−	1.55622i	0.628128	−	0.778110i	$-0.283821\pi$
$114$			57.7300i			0.506404i
$115$		0			0
$116$			29.5080i			0.254380i
$117$	−32.0394			−0.273841
$118$		−	19.4975i		−	0.165233i
$119$		−	27.7877i		−	0.233510i
$120$		0			0
$121$	113.333			0.936637
$122$	98.7889			0.809745
$123$	17.8589			0.145194
$124$	−43.1716	+	44.4996i	−0.348158	+	0.358868i
$125$		0			0
$126$	4.87006			0.0386513
$127$	−105.810			−0.833146			−0.416573	−	0.909102i	$-0.636769\pi$
$127$	−105.810			−0.833146			−0.416573	+	0.909102i	$0.636769\pi$
$128$		−	11.3137i		−	0.0883883i
$129$	200.759			1.55627
$130$		0			0
$131$	−169.848			−1.29655			−0.648274	−	0.761407i	$-0.724509\pi$
$131$	−169.848			−1.29655			−0.648274	+	0.761407i	$0.724509\pi$
$132$			18.5097i			0.140225i
$133$		−	19.3675i		−	0.145621i
$134$	154.225			1.15094
$135$		0			0
$136$		−	49.5624i		−	0.364429i
$137$	−1.85954			−0.0135733			−0.00678663	−	0.999977i	$-0.502160\pi$
$137$	−1.85954			−0.0135733			−0.00678663	+	0.999977i	$0.502160\pi$
$138$			105.172i			0.762113i
$139$			24.4453i			0.175865i	0.996126	+	0.0879326i	$0.0280260\pi$
$139$			24.4453i			0.175865i	−0.996126	+	0.0879326i	$0.971974\pi$
$140$		0			0
$141$			132.827i			0.942035i
$142$			3.89949i			0.0274612i
$143$			40.8528i			0.285684i
$144$	8.68629			0.0603215
$145$		0			0
$146$		−	104.663i		−	0.716867i
$147$	155.372			1.05695
$148$	−16.6136			−0.112254
$149$	62.0244			0.416271			0.208136	−	0.978100i	$-0.433260\pi$
$149$	62.0244			0.416271			0.208136	+	0.978100i	$0.433260\pi$
$150$		0			0
$151$		−	28.2627i		−	0.187170i	−0.995611	−	0.0935852i	$-0.970167\pi$
$151$		−	28.2627i		−	0.187170i	0.995611	−	0.0935852i	$-0.0298328\pi$
$152$		−	34.5442i		−	0.227264i
$153$	38.0524			0.248708
$154$		−	6.20971i		−	0.0403228i
$155$		0			0
$156$	98.6274			0.632227
$157$			179.569i			1.14375i	0.820341	+	0.571874i	$0.193784\pi$
$157$			179.569i			1.14375i	−0.820341	+	0.571874i	$0.806216\pi$
$158$	−145.582			−0.921407
$159$	310.558			1.95320
$160$		0			0
$161$		−	35.2834i		−	0.219152i
$162$		−	135.522i		−	0.836555i
$163$		−	263.718i		−	1.61790i	−0.587877	−	0.808950i	$-0.700036\pi$
$163$		−	263.718i		−	1.61790i	0.587877	−	0.808950i	$-0.299964\pi$
$164$	−10.6863			−0.0651603
$165$		0			0
$166$			156.797i			0.944561i
$167$	125.192			0.749653			0.374826	−	0.927095i	$-0.377702\pi$
$167$	125.192			0.749653			0.374826	+	0.927095i	$0.377702\pi$
$168$	−14.9916			−0.0892355
$169$	48.6812			0.288055
$170$		0			0
$171$	26.5219			0.155099
$172$	−120.129			−0.698426
$173$		−	134.711i		−	0.778674i	−0.921095	−	0.389337i	$-0.872704\pi$
$173$		−	134.711i		−	0.778674i	0.921095	−	0.389337i	$-0.127296\pi$
$174$	69.7401			0.400805
$175$		0			0
$176$		−	11.0757i		−	0.0629302i
$177$	−46.0809			−0.260344
$178$	−71.3779			−0.400999
$179$			316.183i			1.76639i	0.469008	+	0.883194i	$0.344611\pi$
$179$			316.183i			1.76639i	−0.469008	+	0.883194i	$0.655389\pi$
$180$		0			0
$181$			329.217i			1.81888i	0.415837	+	0.909439i	$0.363489\pi$
$181$			329.217i			1.81888i	−0.415837	+	0.909439i	$0.636511\pi$
$182$	−33.0880			−0.181802
$183$		−	233.480i		−	1.27585i
$184$		−	62.9320i		−	0.342021i
$185$		0			0
$186$	105.172	+	102.033i	0.565439	+	0.548564i
$187$		−	48.5198i		−	0.259464i
$188$		−	79.4802i		−	0.422767i
$189$			36.1928i			0.191496i
$190$		0			0
$191$	−250.213			−1.31002			−0.655008	−	0.755622i	$-0.727335\pi$
$191$	−250.213			−1.31002			−0.655008	+	0.755622i	$0.727335\pi$
$192$	−26.7391			−0.139266
$193$			70.3970i			0.364751i	0.983229	+	0.182376i	$0.0583787\pi$
$193$			70.3970i			0.364751i	−0.983229	+	0.182376i	$0.941621\pi$
$194$	160.811			0.828924
$195$		0			0
$196$	−92.9706			−0.474340
$197$	−321.623			−1.63260			−0.816302	−	0.577626i	$-0.803979\pi$
$197$	−321.623			−1.63260			−0.816302	+	0.577626i	$0.803979\pi$
$198$	8.50356			0.0429473
$199$		−	107.235i		−	0.538868i	−0.963019	−	0.269434i	$-0.913163\pi$
$199$		−	107.235i		−	0.538868i	0.963019	−	0.269434i	$-0.0868366\pi$
$200$		0			0
$201$		−	364.500i		−	1.81344i
$202$			243.605i			1.20597i
$203$	−23.3967			−0.115255
$204$	−117.137			−0.574201
$205$		0			0
$206$	−5.30657			−0.0257600
$207$	48.3171			0.233416
$208$	−59.0161			−0.283731
$209$		−	33.8175i		−	0.161806i
$210$		0			0
$211$	−55.0172			−0.260745			−0.130373	−	0.991465i	$-0.541617\pi$
$211$	−55.0172			−0.260745			−0.130373	+	0.991465i	$0.541617\pi$
$212$	−185.830			−0.876557
$213$	9.21617			0.0432684
$214$	−6.81118			−0.0318280
$215$		0			0
$216$			64.5540i			0.298861i
$217$	−35.2834	−	34.2304i	−0.162596	−	0.157744i
$218$		−	114.101i		−	0.523397i
$219$	−247.362			−1.12951
$220$		0			0
$221$	−258.534			−1.16984
$222$			39.2649i			0.176869i
$223$	209.522			0.939561			0.469780	−	0.882783i	$-0.344333\pi$
$223$	209.522			0.939561			0.469780	+	0.882783i	$0.344333\pi$
$224$	8.97056			0.0400472
$225$		0			0
$226$	248.693			1.10041
$227$		−	173.515i		−	0.764382i	−0.924083	−	0.382191i	$-0.875170\pi$
$227$		−	173.515i		−	0.764382i	0.924083	−	0.382191i	$-0.124830\pi$
$228$	−81.6426			−0.358082
$229$		−	38.4867i		−	0.168064i	−0.996463	−	0.0840321i	$-0.973220\pi$
$229$		−	38.4867i		−	0.168064i	0.996463	−	0.0840321i	$-0.0267798\pi$
$230$		0			0
$231$	−14.6762			−0.0635333
$232$	−41.7307			−0.179874
$233$		−	136.019i		−	0.583774i	−0.956453	−	0.291887i	$-0.905717\pi$
$233$		−	136.019i		−	0.583774i	0.956453	−	0.291887i	$-0.0942831\pi$
$234$		−	45.3106i		−	0.193635i
$235$		0			0
$236$	27.5736			0.116837
$237$			344.073i			1.45179i
$238$	39.2977			0.165116
$239$			396.876i			1.66057i	0.557340	+	0.830284i	$0.311822\pi$
$239$			396.876i			1.66057i	−0.557340	+	0.830284i	$0.688178\pi$
$240$		0			0
$241$		−	152.406i		−	0.632391i	−0.948694	−	0.316196i	$-0.897594\pi$
$241$		−	152.406i		−	0.632391i	0.948694	−	0.316196i	$-0.102406\pi$
$242$			160.277i			0.662302i
$243$	−114.887			−0.472784
$244$			139.709i			0.572576i
$245$		0			0
$246$			25.2563i			0.102668i
$247$	−180.194			−0.729530
$248$	−62.9320	−	61.0538i	−0.253758	−	0.246185i
$249$	370.579			1.48827
$250$		0			0
$251$			148.589i			0.591987i	0.955190	+	0.295994i	$0.0956507\pi$
$251$			148.589i			0.591987i	−0.955190	+	0.295994i	$0.904349\pi$
$252$			6.88730i			0.0273306i
$253$		−	61.6081i		−	0.243510i
$254$		−	149.637i		−	0.589123i
$255$		0			0
$256$	16.0000			0.0625000
$257$		−	383.818i		−	1.49346i	−0.665129	−	0.746728i	$-0.731624\pi$
$257$		−	383.818i		−	1.49346i	0.665129	−	0.746728i	$-0.268376\pi$
$258$			283.917i			1.10045i
$259$		−	13.1728i		−	0.0508602i
$260$		0			0
$261$		−	32.0394i		−	0.122756i
$262$		−	240.201i		−	0.916798i
$263$	194.710			0.740344			0.370172	−	0.928963i	$-0.379299\pi$
$263$	194.710			0.740344			0.370172	+	0.928963i	$0.379299\pi$
$264$	−26.1766			−0.0991539
$265$		0			0
$266$	27.3898			0.102969
$267$			168.696i			0.631822i
$268$			218.108i			0.813835i
$269$			180.686i			0.671695i	0.941916	+	0.335847i	$0.109023\pi$
$269$			180.686i			0.671695i	−0.941916	+	0.335847i	$0.890977\pi$
$270$		0			0
$271$		−	48.9889i		−	0.180771i	−0.995907	−	0.0903855i	$-0.971190\pi$
$271$		−	48.9889i		−	0.180771i	0.995907	−	0.0903855i	$-0.0288099\pi$
$272$	70.0918			0.257690
$273$			78.2010i			0.286451i
$274$		−	2.62978i		−	0.00959774i
$275$		0			0
$276$	−148.735			−0.538895
$277$	359.397			1.29746			0.648731	−	0.761018i	$-0.275300\pi$
$277$	359.397			1.29746			0.648731	+	0.761018i	$0.275300\pi$
$278$	−34.5708			−0.124356
$279$	46.8751	−	48.3171i	0.168011	−	0.173179i
$280$		0			0
$281$	203.333			0.723605			0.361803	−	0.932255i	$-0.382161\pi$
$281$	203.333			0.723605			0.361803	+	0.932255i	$0.382161\pi$
$282$	−187.846			−0.666120
$283$			347.789i			1.22894i	0.788942	+	0.614468i	$0.210629\pi$
$283$			347.789i			1.22894i	−0.788942	+	0.614468i	$0.789371\pi$
$284$	−5.51472			−0.0194180
$285$		0			0
$286$	−57.7746			−0.202009
$287$		−	8.47309i		−	0.0295230i
$288$			12.2843i			0.0426537i
$289$	18.0538			0.0624700
$290$		0			0
$291$		−	380.065i		−	1.30607i
$292$	148.015			0.506902
$293$		−	73.4071i		−	0.250536i	−0.992123	−	0.125268i	$-0.960021\pi$
$293$		−	73.4071i		−	0.250536i	0.992123	−	0.125268i	$-0.0399791\pi$
$294$			219.729i			0.747378i
$295$		0			0
$296$		−	23.4951i		−	0.0793754i
$297$			63.1960i			0.212781i
$298$			87.7157i			0.294348i
$299$	−328.274			−1.09791
$300$		0			0
$301$		−	95.2497i		−	0.316444i
$302$	39.9695			0.132349
$303$	575.743			1.90014
$304$	48.8528			0.160700
$305$		0			0
$306$			53.8142i			0.175863i
$307$			340.532i			1.10922i	0.832109	+	0.554612i	$0.187133\pi$
$307$			340.532i			1.10922i	−0.832109	+	0.554612i	$0.812867\pi$
$308$	8.78185			0.0285125
$309$			12.5417i			0.0405880i
$310$		0			0
$311$	−318.252			−1.02332			−0.511659	−	0.859189i	$-0.670969\pi$
$311$	−318.252			−1.02332			−0.511659	+	0.859189i	$0.670969\pi$
$312$			139.480i			0.447052i
$313$	−380.123			−1.21445			−0.607225	−	0.794530i	$-0.707718\pi$
$313$	−380.123			−1.21445			−0.607225	+	0.794530i	$0.707718\pi$
$314$	−253.948			−0.808752
$315$		0			0
$316$		−	205.884i		−	0.651533i
$317$			384.470i			1.21284i	0.795145	+	0.606420i	$0.207395\pi$
$317$			384.470i			1.21284i	−0.795145	+	0.606420i	$0.792605\pi$
$318$			439.196i			1.38112i
$319$	−40.8528			−0.128065
$320$		0			0
$321$			16.0977i			0.0501487i
$322$	49.8983			0.154964
$323$	214.011			0.662574
$324$	191.657			0.591534
$325$		0			0
$326$	372.953			1.14403
$327$	−269.668			−0.824673
$328$		−	15.1127i		−	0.0460753i
$329$	63.0193			0.191548
$330$		0			0
$331$			409.000i			1.23565i	0.786316	+	0.617825i	$0.211986\pi$
$331$			409.000i			1.23565i	−0.786316	+	0.617825i	$0.788014\pi$
$332$	−221.745			−0.667906
$333$	18.0388			0.0541705
$334$			177.048i			0.530085i
$335$		0			0
$336$		−	21.2013i		−	0.0630990i
$337$	113.404			0.336510			0.168255	−	0.985744i	$-0.446187\pi$
$337$	113.404			0.336510			0.168255	+	0.985744i	$0.446187\pi$
$338$			68.8457i			0.203685i
$339$		−	587.769i		−	1.73383i
$340$		0			0
$341$	−61.6081	−	59.7695i	−0.180669	−	0.175277i
$342$			37.5076i			0.109671i
$343$		−	151.419i		−	0.441456i
$344$		−	169.888i		−	0.493862i
$345$		0			0
$346$	190.510			0.550606
$347$	143.329			0.413053			0.206526	−	0.978441i	$-0.433784\pi$
$347$	143.329			0.413053			0.206526	+	0.978441i	$0.433784\pi$
$348$			98.6274i			0.283412i
$349$	−52.9361			−0.151679			−0.0758396	−	0.997120i	$-0.524164\pi$
$349$	−52.9361			−0.151679			−0.0758396	+	0.997120i	$0.524164\pi$
$350$		0			0
$351$	336.735			0.959359
$352$	15.6634			0.0444983
$353$	598.746			1.69616			0.848082	−	0.529865i	$-0.177757\pi$
$353$	598.746			1.69616			0.848082	+	0.529865i	$0.177757\pi$
$354$		−	65.1682i		−	0.184091i
$355$		0			0
$356$		−	100.944i		−	0.283549i
$357$		−	92.8772i		−	0.260160i
$358$	−447.151			−1.24902
$359$	−370.806			−1.03289			−0.516443	−	0.856322i	$-0.672744\pi$
$359$	−370.806			−1.03289			−0.516443	+	0.856322i	$0.672744\pi$
$360$		0			0
$361$	−211.838			−0.586808
$362$	−465.583			−1.28614
$363$	378.803			1.04354
$364$		−	46.7935i		−	0.128553i
$365$		0			0
$366$	330.191			0.902161
$367$	489.217			1.33302			0.666509	−	0.745497i	$-0.267788\pi$
$367$	489.217			1.33302			0.666509	+	0.745497i	$0.267788\pi$
$368$	88.9992			0.241846
$369$	11.6030			0.0314445
$370$		0			0
$371$		−	147.343i		−	0.397152i
$372$	−144.296	+	148.735i	−0.387893	+	0.399825i
$373$			423.461i			1.13528i	0.823276	+	0.567642i	$0.192144\pi$
$373$			423.461i			1.13528i	−0.823276	+	0.567642i	$0.807856\pi$
$374$	68.6173			0.183469
$375$		0			0
$376$	112.402			0.298942
$377$			217.681i			0.577404i
$378$	−51.1844			−0.135408
$379$	−208.711			−0.550688			−0.275344	−	0.961346i	$-0.588792\pi$
$379$	−208.711			−0.550688			−0.275344	+	0.961346i	$0.588792\pi$
$380$		0			0
$381$	−353.657			−0.928233
$382$		−	353.855i		−	0.926322i
$383$	−716.500			−1.87076			−0.935379	−	0.353648i	$-0.884941\pi$
$383$	−716.500			−1.87076			−0.935379	+	0.353648i	$0.884941\pi$
$384$		−	37.8148i		−	0.0984761i
$385$		0			0
$386$	−99.5563			−0.257918
$387$	130.435			0.337041
$388$			227.421i			0.586138i
$389$			70.1733i			0.180394i	0.995924	+	0.0901971i	$0.0287497\pi$
$389$			70.1733i			0.180394i	−0.995924	+	0.0901971i	$0.971250\pi$
$390$		0			0
$391$	389.882			0.997141
$392$		−	131.480i		−	0.335409i
$393$	−567.698			−1.44452
$394$		−	454.843i		−	1.15442i
$395$		0			0
$396$			12.0259i			0.0303683i
$397$		−	484.696i		−	1.22090i	−0.792057	−	0.610448i	$-0.790990\pi$
$397$		−	484.696i		−	1.22090i	0.792057	−	0.610448i	$-0.209010\pi$
$398$	151.653			0.381037
$399$		−	64.7339i		−	0.162240i
$400$		0			0
$401$		−	424.819i		−	1.05940i	−0.848185	−	0.529700i	$-0.822304\pi$
$401$		−	424.819i		−	1.05940i	0.848185	−	0.529700i	$-0.177696\pi$
$402$	515.481			1.28229
$403$	−318.477	+	328.274i	−0.790266	+	0.814576i
$404$	−344.510			−0.852747
$405$		0			0
$406$		−	33.0880i		−	0.0814975i
$407$		−	23.0009i		−	0.0565132i
$408$		−	165.657i		−	0.406022i
$409$			644.352i			1.57543i	0.616038	+	0.787716i	$0.288737\pi$
$409$			644.352i			1.57543i	−0.616038	+	0.787716i	$0.711263\pi$
$410$		0			0
$411$	−6.21530			−0.0151224
$412$		−	7.50462i		−	0.0182151i
$413$			21.8629i			0.0529368i
$414$			68.3307i			0.165050i
$415$		0			0
$416$		−	83.4614i		−	0.200628i
$417$			81.7056i			0.195937i
$418$	47.8251			0.114414
$419$	522.037			1.24591			0.622955	−	0.782257i	$-0.285932\pi$
$419$	522.037			1.24591			0.622955	+	0.782257i	$0.285932\pi$
$420$		0			0
$421$	−584.127			−1.38748			−0.693738	−	0.720228i	$-0.744037\pi$
$421$	−584.127			−1.38748			−0.693738	+	0.720228i	$0.744037\pi$
$422$		−	77.8061i		−	0.184375i
$423$			86.2986i			0.204016i
$424$		−	262.804i		−	0.619820i
$425$		0			0
$426$			13.0336i			0.0305954i
$427$	−110.774			−0.259424
$428$		−	9.63247i		−	0.0225058i
$429$			136.546i			0.318289i
$430$		0			0
$431$	−401.172			−0.930793			−0.465396	−	0.885102i	$-0.654088\pi$
$431$	−401.172			−0.930793			−0.465396	+	0.885102i	$0.654088\pi$
$432$	−91.2931			−0.211327
$433$	47.5876			0.109902			0.0549510	−	0.998489i	$-0.482500\pi$
$433$	47.5876			0.109902			0.0549510	+	0.998489i	$0.482500\pi$
$434$	48.4092	−	49.8983i	0.111542	−	0.114973i
$435$		0			0
$436$	161.362			0.370097
$437$	271.741			0.621834
$438$		−	349.823i		−	0.798684i
$439$	−189.434			−0.431512			−0.215756	−	0.976447i	$-0.569221\pi$
$439$	−189.434			−0.431512			−0.215756	+	0.976447i	$0.569221\pi$
$440$		0			0
$441$	100.946			0.228903
$442$		−	365.622i		−	0.827200i
$443$			145.115i			0.327573i	0.986496	+	0.163786i	$0.0523708\pi$
$443$			145.115i			0.327573i	−0.986496	+	0.163786i	$0.947629\pi$
$444$	−55.5290			−0.125065
$445$		0			0
$446$			296.309i			0.664370i
$447$	207.310			0.463780
$448$			12.6863i			0.0283176i
$449$		−	749.924i		−	1.67021i	−0.550091	−	0.835105i	$-0.685407\pi$
$449$		−	749.924i		−	1.67021i	0.550091	−	0.835105i	$-0.314593\pi$
$450$		0			0
$451$		−	14.7948i		−	0.0328044i
$452$			351.706i			0.778110i
$453$		−	94.4651i		−	0.208532i
$454$	245.387			0.540500
$455$		0			0
$456$		−	115.460i		−	0.253202i
$457$	−689.761			−1.50932			−0.754662	−	0.656114i	$-0.772199\pi$
$457$	−689.761			−1.50932			−0.754662	+	0.656114i	$0.772199\pi$
$458$	54.4284			0.118839
$459$	−399.931			−0.871309
$460$		0			0
$461$		−	193.465i		−	0.419664i	−0.977737	−	0.209832i	$-0.932708\pi$
$461$		−	193.465i		−	0.419664i	0.977737	−	0.209832i	$-0.0672917\pi$
$462$		−	20.7553i		−	0.0449248i
$463$	−174.951			−0.377864			−0.188932	−	0.981990i	$-0.560503\pi$
$463$	−174.951			−0.377864			−0.188932	+	0.981990i	$0.560503\pi$
$464$		−	59.0161i		−	0.127190i
$465$		0			0
$466$	192.360			0.412791
$467$		−	778.144i		−	1.66626i	−0.553076	−	0.833131i	$-0.686546\pi$
$467$		−	778.144i		−	1.66626i	0.553076	−	0.833131i	$-0.313454\pi$
$468$	64.0789			0.136921
$469$	−172.936			−0.368734
$470$		0			0
$471$			600.188i			1.27428i
$472$			38.9949i			0.0826164i
$473$		−	166.315i		−	0.351616i
$474$	−486.593			−1.02657
$475$		0			0
$476$			55.5753i			0.116755i
$477$	201.772			0.423002
$478$	−561.267			−1.17420
$479$	−670.335			−1.39945			−0.699724	−	0.714414i	$-0.746693\pi$
$479$	−670.335			−1.39945			−0.699724	+	0.714414i	$0.746693\pi$
$480$		0			0
$481$	−122.558			−0.254799
$482$	215.535			0.447168
$483$		−	117.931i		−	0.244164i
$484$	−226.666			−0.468318
$485$		0			0
$486$		−	162.474i		−	0.334309i
$487$	570.402			1.17126			0.585628	−	0.810580i	$-0.300848\pi$
$487$	570.402			1.17126			0.585628	+	0.810580i	$0.300848\pi$
$488$	−197.578			−0.404872
$489$		−	881.448i		−	1.80255i
$490$		0			0
$491$		−	410.147i		−	0.835330i	−0.908601	−	0.417665i	$-0.862849\pi$
$491$		−	410.147i		−	0.835330i	0.908601	−	0.417665i	$-0.137151\pi$
$492$	−35.7178			−0.0725971
$493$		−	258.534i		−	0.524410i
$494$		−	254.833i		−	0.515856i
$495$		0			0
$496$	86.3431	−	88.9992i	0.174079	−	0.179434i
$497$		−	4.37258i		−	0.00879795i
$498$			524.077i			1.05236i
$499$		−	204.181i		−	0.409180i	−0.978848	−	0.204590i	$-0.934414\pi$
$499$		−	204.181i		−	0.409180i	0.978848	−	0.204590i	$-0.0655862\pi$
$500$		0			0
$501$	418.441			0.835211
$502$	−210.136			−0.418598
$503$		−	391.101i		−	0.777536i	−0.921336	−	0.388768i	$-0.872901\pi$
$503$		−	391.101i		−	0.777536i	0.921336	−	0.388768i	$-0.127099\pi$
$504$	−9.74012			−0.0193256
$505$		0			0
$506$	87.1270			0.172188
$507$	162.712			0.320930
$508$	211.619			0.416573
$509$		−	218.066i		−	0.428421i	−0.976788	−	0.214211i	$-0.931282\pi$
$509$		−	218.066i		−	0.428421i	0.976788	−	0.214211i	$-0.0687179\pi$
$510$		0			0
$511$			117.360i			0.229668i
$512$			22.6274i			0.0441942i
$513$	−278.745			−0.543363
$514$	542.801			1.05603
$515$		0			0
$516$	−401.519			−0.778137
$517$	110.037			0.212838
$518$	18.6291			0.0359636
$519$		−	450.256i		−	0.867545i
$520$		0			0
$521$	64.7006			0.124185			0.0620927	−	0.998070i	$-0.480223\pi$
$521$	64.7006			0.124185			0.0620927	+	0.998070i	$0.480223\pi$
$522$	45.3106			0.0868019
$523$	949.222			1.81496			0.907478	−	0.420100i	$-0.138005\pi$
$523$	949.222			1.81496			0.907478	+	0.420100i	$0.138005\pi$
$524$	339.696			0.648274
$525$		0			0
$526$			275.362i			0.523502i
$527$	378.247	−	389.882i	0.717736	−	0.739815i
$528$		−	37.0193i		−	0.0701124i
$529$	−33.9462			−0.0641705
$530$		0			0
$531$	−29.9390			−0.0563824
$532$			38.7351i			0.0728103i
$533$	−78.8329			−0.147904
$534$	−238.573			−0.446765
$535$		0			0
$536$	−308.451			−0.575468
$537$			1056.81i			1.96799i
$538$	−255.528			−0.474960
$539$		−	128.714i		−	0.238802i
$540$		0			0
$541$	655.337			1.21134			0.605672	−	0.795714i	$-0.292904\pi$
$541$	655.337			1.21134			0.605672	+	0.795714i	$0.292904\pi$
$542$	69.2808			0.127824
$543$			1100.37i			2.02647i
$544$			99.1248i			0.182215i
$545$		0			0
$546$	−110.593			−0.202551
$547$		−	661.947i		−	1.21014i	−0.796172	−	0.605071i	$-0.793145\pi$
$547$		−	661.947i		−	1.21014i	0.796172	−	0.605071i	$-0.206855\pi$
$548$	3.71907			0.00678663
$549$		−	151.694i		−	0.276309i
$550$		0			0
$551$		−	180.194i		−	0.327031i
$552$		−	210.343i		−	0.381056i
$553$	163.244			0.295198
$554$			508.264i			0.917444i
$555$		0			0
$556$		−	48.8905i		−	0.0879326i
$557$	−46.9902			−0.0843631			−0.0421815	−	0.999110i	$-0.513431\pi$
$557$	−46.9902			−0.0843631			−0.0421815	+	0.999110i	$0.513431\pi$
$558$	68.3307	+	66.2914i	0.122456	+	0.118802i
$559$	−886.195			−1.58532
$560$		0			0
$561$		−	162.172i		−	0.289077i
$562$			287.556i			0.511666i
$563$			486.556i			0.864221i	0.901821	+	0.432110i	$0.142231\pi$
$563$			486.556i			0.864221i	−0.901821	+	0.432110i	$0.857769\pi$
$564$		−	265.654i		−	0.471018i
$565$		0			0
$566$	−491.848			−0.868989
$567$			151.963i			0.268013i
$568$		−	7.79899i		−	0.0137306i
$569$		−	435.542i		−	0.765452i	−0.923862	−	0.382726i	$-0.874985\pi$
$569$		−	435.542i		−	0.765452i	0.923862	−	0.382726i	$-0.125015\pi$
$570$		0			0
$571$		−	19.8744i		−	0.0348064i	−0.999849	−	0.0174032i	$-0.994460\pi$
$571$		−	19.8744i		−	0.0348064i	0.999849	−	0.0174032i	$-0.00553989\pi$
$572$		−	81.7056i		−	0.142842i
$573$	−836.310			−1.45953
$574$	11.9828			0.0208759
$575$		0			0
$576$	−17.3726			−0.0301607
$577$			193.622i			0.335567i	0.985824	+	0.167784i	$0.0536610\pi$
$577$			193.622i			0.335567i	−0.985824	+	0.167784i	$0.946339\pi$
$578$			25.5320i			0.0441729i
$579$			235.294i			0.406380i
$580$		0			0
$581$		−	175.820i		−	0.302616i
$582$	537.494			0.923529
$583$		−	257.275i		−	0.441295i
$584$			209.325i			0.358434i
$585$		0			0
$586$	103.813			0.177156
$587$	−212.488			−0.361989			−0.180995	−	0.983484i	$-0.557932\pi$
$587$	−212.488			−0.361989			−0.180995	+	0.983484i	$0.557932\pi$
$588$	−310.744			−0.528476
$589$	263.632	−	271.741i	0.447592	−	0.461361i
$590$		0			0
$591$	−1074.99			−1.81893
$592$	33.2271			0.0561269
$593$		−	1018.91i		−	1.71822i	−0.511788	−	0.859112i	$-0.671017\pi$
$593$		−	1018.91i		−	1.71822i	0.511788	−	0.859112i	$-0.328983\pi$
$594$	−89.3726			−0.150459
$595$		0			0
$596$	−124.049			−0.208136
$597$		−	358.420i		−	0.600369i
$598$		−	464.250i		−	0.776337i
$599$	−177.090			−0.295643			−0.147822	−	0.989014i	$-0.547226\pi$
$599$	−177.090			−0.295643			−0.147822	+	0.989014i	$0.547226\pi$
$600$		0			0
$601$			640.354i			1.06548i	0.846278	+	0.532741i	$0.178838\pi$
$601$			640.354i			1.06548i	−0.846278	+	0.532741i	$0.821162\pi$
$602$	134.703			0.223760
$603$		−	236.818i		−	0.392734i
$604$			56.5255i			0.0935852i
$605$		0			0
$606$			814.223i			1.34360i
$607$		−	870.583i		−	1.43424i	−0.696950	−	0.717119i	$-0.745460\pi$
$607$		−	870.583i		−	1.43424i	0.696950	−	0.717119i	$-0.254540\pi$
$608$			69.0883i			0.113632i
$609$	−78.2010			−0.128409
$610$		0			0
$611$		−	586.327i		−	0.959618i
$612$	−76.1047			−0.124354
$613$	−812.069			−1.32475			−0.662373	−	0.749175i	$-0.730450\pi$
$613$	−812.069			−1.32475			−0.662373	+	0.749175i	$0.730450\pi$
$614$	−481.585			−0.784340
$615$		0			0
$616$			12.4194i			0.0201614i
$617$		−	674.975i		−	1.09396i	−0.837145	−	0.546981i	$-0.815777\pi$
$617$		−	674.975i		−	1.09396i	0.837145	−	0.546981i	$-0.184223\pi$
$618$	−17.7366			−0.0287000
$619$		−	145.820i		−	0.235573i	−0.993039	−	0.117787i	$-0.962420\pi$
$619$		−	145.820i		−	0.235573i	0.993039	−	0.117787i	$-0.0375799\pi$
$620$		0			0
$621$	−507.813			−0.817735
$622$		−	450.076i		−	0.723595i
$623$	80.0375			0.128471
$624$	−197.255			−0.316114
$625$		0			0
$626$		−	537.575i		−	0.858746i
$627$		−	113.031i		−	0.180273i
$628$		−	359.137i		−	0.571874i
$629$	145.559			0.231414
$630$		0			0
$631$			688.302i			1.09081i	0.838172	+	0.545406i	$0.183624\pi$
$631$			688.302i			1.09081i	−0.838172	+	0.545406i	$0.816376\pi$
$632$	291.165			0.460704
$633$	−183.889			−0.290504
$634$	−543.723			−0.857607
$635$		0			0
$636$	−621.117			−0.976599
$637$	−685.845			−1.07668
$638$		−	57.7746i		−	0.0905558i
$639$	5.98781			0.00937059
$640$		0			0
$641$			315.020i			0.491450i	0.969340	+	0.245725i	$0.0790260\pi$
$641$			315.020i			0.491450i	−0.969340	+	0.245725i	$0.920974\pi$
$642$	−22.7656			−0.0354605
$643$	−1101.41			−1.71293			−0.856465	−	0.516204i	$-0.827344\pi$
$643$	−1101.41			−1.71293			−0.856465	+	0.516204i	$0.827344\pi$
$644$			70.5669i			0.109576i
$645$		0			0
$646$			302.658i			0.468510i
$647$	−42.7961			−0.0661454			−0.0330727	−	0.999453i	$-0.510529\pi$
$647$	−42.7961			−0.0661454			−0.0330727	+	0.999453i	$0.510529\pi$
$648$			271.044i			0.418277i
$649$			38.1746i			0.0588207i
$650$		0			0
$651$	−117.931	−	114.412i	−0.181154	−	0.175747i
$652$			527.436i			0.808950i
$653$			1084.35i			1.66056i	0.557345	+	0.830281i	$0.311820\pi$
$653$			1084.35i			1.66056i	−0.557345	+	0.830281i	$0.688180\pi$
$654$		−	381.368i		−	0.583132i
$655$		0			0
$656$	21.3726			0.0325802
$657$	−160.713			−0.244616
$658$			89.1228i			0.135445i
$659$	−56.4445			−0.0856518			−0.0428259	−	0.999083i	$-0.513636\pi$
$659$	−56.4445			−0.0856518			−0.0428259	+	0.999083i	$0.513636\pi$
$660$		0			0
$661$	594.200			0.898941			0.449471	−	0.893295i	$-0.351613\pi$
$661$	594.200			0.898941			0.449471	+	0.893295i	$0.351613\pi$
$662$	−578.413			−0.873736
$663$	−864.122			−1.30335
$664$		−	313.594i		−	0.472281i
$665$		0			0
$666$			25.5107i			0.0383043i
$667$		−	328.274i		−	0.492165i
$668$	−250.384			−0.374826
$669$	700.304			1.04679
$670$		0			0
$671$	−193.421			−0.288258
$672$	29.9831			0.0446177
$673$	420.707			0.625122			0.312561	−	0.949898i	$-0.398813\pi$
$673$	420.707			0.625122			0.312561	+	0.949898i	$0.398813\pi$
$674$			160.377i			0.237948i
$675$		0			0
$676$	−97.3625			−0.144027
$677$	−895.449			−1.32267			−0.661336	−	0.750090i	$-0.730010\pi$
$677$	−895.449			−1.32267			−0.661336	+	0.750090i	$0.730010\pi$
$678$	831.231			1.22600
$679$	−180.321			−0.265568
$680$		0			0
$681$		−	579.954i		−	0.851621i
$682$	84.5268	−	87.1270i	0.123940	−	0.127752i
$683$		−	518.934i		−	0.759786i	−0.925030	−	0.379893i	$-0.875961\pi$
$683$		−	518.934i		−	0.759786i	0.925030	−	0.379893i	$-0.124039\pi$
$684$	−53.0437			−0.0775493
$685$		0			0
$686$	214.139			0.312156
$687$		−	128.638i		−	0.187245i
$688$	240.259			0.349213
$689$	−1370.87			−1.98965
$690$		0			0
$691$	677.247			0.980097			0.490048	−	0.871695i	$-0.336979\pi$
$691$	677.247			0.980097			0.490048	+	0.871695i	$0.336979\pi$
$692$			269.421i			0.389337i
$693$	−9.53522			−0.0137593
$694$			202.698i			0.292072i
$695$		0			0
$696$	−139.480			−0.200403
$697$	93.6277			0.134330
$698$		−	74.8629i		−	0.107253i
$699$		−	454.630i		−	0.650400i
$700$		0			0
$701$	−333.382			−0.475580			−0.237790	−	0.971317i	$-0.576423\pi$
$701$	−333.382			−0.475580			−0.237790	+	0.971317i	$0.576423\pi$
$702$			476.215i			0.678369i
$703$	101.452			0.144314
$704$			22.1514i			0.0314651i
$705$		0			0
$706$			846.755i			1.19937i
$707$		−	273.159i		−	0.386364i
$708$	92.1617			0.130172
$709$			1028.87i			1.45115i	0.688143	+	0.725575i	$0.258426\pi$
$709$			1028.87i			1.45115i	−0.688143	+	0.725575i	$0.741574\pi$
$710$		0			0
$711$			223.547i			0.314412i
$712$	142.756			0.200500
$713$	480.280	−	495.054i	0.673604	−	0.694325i
$714$	131.348			0.183961
$715$		0			0
$716$		−	632.367i		−	0.883194i
$717$			1326.51i			1.85009i
$718$		−	524.399i		−	0.730361i
$719$			59.1145i			0.0822176i	0.999155	+	0.0411088i	$0.0130890\pi$
$719$			59.1145i			0.0822176i	−0.999155	+	0.0411088i	$0.986911\pi$
$720$		0			0
$721$	5.95036			0.00825293
$722$		−	299.584i		−	0.414936i
$723$		−	509.401i		−	0.704566i
$724$		−	658.434i		−	0.909439i
$725$		0			0
$726$			535.709i			0.737891i
$727$			622.217i			0.855870i	0.903810	+	0.427935i	$0.140759\pi$
$727$			622.217i			0.855870i	−0.903810	+	0.427935i	$0.859241\pi$
$728$	66.1760			0.0909010
$729$	478.460			0.656324
$730$		0			0
$731$	1052.51			1.43982
$732$			466.960i			0.637924i
$733$			995.965i			1.35875i	0.733791	+	0.679376i	$0.237749\pi$
$733$			995.965i			1.35875i	−0.733791	+	0.679376i	$0.762251\pi$
$734$			691.858i			0.942586i
$735$		0			0
$736$			125.864i			0.171011i
$737$	−301.962			−0.409718
$738$			16.4092i			0.0222346i
$739$		−	217.418i		−	0.294206i	−0.989121	−	0.147103i	$-0.953005\pi$
$739$		−	217.418i		−	0.294206i	0.989121	−	0.147103i	$-0.0469949\pi$
$740$		0			0
$741$	−602.278			−0.812791
$742$	208.375			0.280829
$743$	1045.54			1.40719			0.703596	−	0.710600i	$-0.251576\pi$
$743$	1045.54			1.40719			0.703596	+	0.710600i	$0.251576\pi$
$744$	−210.343	−	204.066i	−0.282719	−	0.274282i
$745$		0			0
$746$	−598.864			−0.802767
$747$	240.767			0.322312
$748$			97.0395i			0.129732i
$749$	7.63752			0.0101970
$750$		0			0
$751$	−617.458			−0.822181			−0.411091	−	0.911595i	$-0.634852\pi$
$751$	−617.458			−0.822181			−0.411091	+	0.911595i	$0.634852\pi$
$752$			158.960i			0.211384i
$753$			496.642i			0.659551i
$754$	−307.848			−0.408286
$755$		0			0
$756$		−	72.3857i		−	0.0957482i
$757$	−703.565			−0.929412			−0.464706	−	0.885465i	$-0.653840\pi$
$757$	−703.565			−0.929412			−0.464706	+	0.885465i	$0.653840\pi$
$758$		−	295.161i		−	0.389395i
$759$		−	205.918i		−	0.271302i
$760$		0			0
$761$			702.343i			0.922922i	0.887161	+	0.461461i	$0.152674\pi$
$761$			702.343i			0.922922i	−0.887161	+	0.461461i	$0.847326\pi$
$762$		−	500.146i		−	0.656360i
$763$			127.943i			0.167684i
$764$	500.426			0.655008
$765$		0			0
$766$		−	1013.28i		−	1.32283i
$767$	203.411			0.265203
$768$	53.4782			0.0696331
$769$	686.734			0.893022			0.446511	−	0.894778i	$-0.352666\pi$
$769$	686.734			0.893022			0.446511	+	0.894778i	$0.352666\pi$
$770$		0			0
$771$		−	1282.87i		−	1.66390i
$772$		−	140.794i		−	0.182376i
$773$	−850.908			−1.10079			−0.550393	−	0.834905i	$-0.685522\pi$
$773$	−850.908			−1.10079			−0.550393	+	0.834905i	$0.685522\pi$
$774$			184.463i			0.238324i
$775$		0			0
$776$	−321.622			−0.414462
$777$		−	44.0286i		−	0.0566648i
$778$	−99.2401			−0.127558
$779$	65.2569			0.0837701
$780$		0			0
$781$		−	7.63493i		−	0.00977583i
$782$			551.377i			0.705085i
$783$			336.735i			0.430058i
$784$	185.941			0.237170
$785$		0			0
$786$		−	802.846i		−	1.02143i
$787$	−303.486			−0.385623			−0.192812	−	0.981236i	$-0.561761\pi$
$787$	−303.486			−0.385623			−0.192812	+	0.981236i	$0.561761\pi$
$788$	643.246			0.816302
$789$	650.798			0.824839
$790$		0			0
$791$	−278.865			−0.352547
$792$	−17.0071			−0.0214736
$793$			1030.63i			1.29966i
$794$	685.463			0.863304
$795$		0			0
$796$			214.470i			0.269434i
$797$	728.804			0.914434			0.457217	−	0.889355i	$-0.348846\pi$
$797$	728.804			0.914434			0.457217	+	0.889355i	$0.348846\pi$
$798$	91.5475			0.114721
$799$			696.364i			0.871545i
$800$		0			0
$801$			109.603i			0.136833i
$802$	600.785			0.749109
$803$			204.922i			0.255195i
$804$			729.001i			0.906718i
$805$		0			0
$806$	−464.250	−	450.395i	−0.575992	−	0.558802i
$807$			603.923i			0.748355i
$808$		−	487.210i		−	0.602983i
$809$		−	698.264i		−	0.863120i	−0.902084	−	0.431560i	$-0.857963\pi$
$809$		−	698.264i		−	0.863120i	0.902084	−	0.431560i	$-0.142037\pi$
$810$		0			0
$811$	796.317			0.981895			0.490948	−	0.871189i	$-0.336651\pi$
$811$	796.317			0.981895			0.490948	+	0.871189i	$0.336651\pi$
$812$	46.7935			0.0576274
$813$		−	163.740i		−	0.201402i
$814$	32.5281			0.0399609
$815$		0			0
$816$	234.274			0.287101
$817$	733.582			0.897897
$818$	−911.251			−1.11400
$819$			50.8077i			0.0620363i
$820$		0			0
$821$		−	1071.39i		−	1.30498i	−0.757796	−	0.652491i	$-0.773724\pi$
$821$		−	1071.39i		−	1.30498i	0.757796	−	0.652491i	$-0.226276\pi$
$822$		−	8.78976i		−	0.0106931i
$823$	−671.508			−0.815928			−0.407964	−	0.912998i	$-0.633761\pi$
$823$	−671.508			−0.815928			−0.407964	+	0.912998i	$0.633761\pi$
$824$	10.6131			0.0128800
$825$		0			0
$826$	−30.9188			−0.0374320
$827$	−1029.64			−1.24503			−0.622513	−	0.782610i	$-0.713888\pi$
$827$	−1029.64			−1.24503			−0.622513	+	0.782610i	$0.713888\pi$
$828$	−96.6341			−0.116708
$829$			633.432i			0.764092i	0.924143	+	0.382046i	$0.124780\pi$
$829$			633.432i			0.764092i	−0.924143	+	0.382046i	$0.875220\pi$
$830$		0			0
$831$	1201.24			1.44554
$832$	118.032			0.141866
$833$	814.559			0.977862
$834$	−115.549			−0.138548
$835$		0			0
$836$			67.6349i			0.0809030i
$837$	−492.658	+	507.813i	−0.588600	+	0.606706i
$838$			738.271i			0.880992i
$839$	491.720			0.586079			0.293039	−	0.956100i	$-0.405333\pi$
$839$	491.720			0.586079			0.293039	+	0.956100i	$0.405333\pi$
$840$		0			0
$841$	623.319			0.741164
$842$		−	826.080i		−	0.981093i
$843$	679.618			0.806190
$844$	110.034			0.130373
$845$		0			0
$846$	−122.045			−0.144261
$847$		−	179.722i		−	0.212187i
$848$	371.660			0.438279
$849$			1162.45i			1.36919i
$850$		0			0
$851$	184.824			0.217185
$852$	−18.4323			−0.0216342
$853$		−	663.523i		−	0.777870i	−0.921265	−	0.388935i	$-0.872843\pi$
$853$		−	663.523i		−	0.777870i	0.921265	−	0.388935i	$-0.127157\pi$
$854$		−	156.658i		−	0.183440i
$855$		0			0
$856$	13.6224			0.0159140
$857$			852.250i			0.994457i	0.867620	+	0.497229i	$0.165649\pi$
$857$			852.250i			0.994457i	−0.867620	+	0.497229i	$0.834351\pi$
$858$	−193.105			−0.225064
$859$			168.429i			0.196076i	0.995183	+	0.0980381i	$0.0312567\pi$
$859$			168.429i			0.196076i	−0.995183	+	0.0980381i	$0.968743\pi$
$860$		0			0
$861$		−	28.3204i		−	0.0328924i
$862$		−	567.342i		−	0.658170i
$863$	905.754			1.04954			0.524771	−	0.851244i	$-0.324151\pi$
$863$	905.754			1.04954			0.524771	+	0.851244i	$0.324151\pi$
$864$		−	129.108i		−	0.149430i
$865$		0			0
$866$			67.2990i			0.0777125i
$867$	60.3429			0.0695997
$868$	70.5669	+	68.4609i	0.0812982	+	0.0788720i
$869$	285.040			0.328009
$870$		0			0
$871$			1608.98i			1.84728i
$872$			228.201i			0.261698i
$873$		−	246.931i		−	0.282853i
$874$			384.300i			0.439703i
$875$		0			0
$876$	494.725			0.564755
$877$		−	839.258i		−	0.956965i	−0.878097	−	0.478482i	$-0.841187\pi$
$877$		−	839.258i		−	0.956965i	0.878097	−	0.478482i	$-0.158813\pi$
$878$		−	267.899i		−	0.305125i
$879$		−	245.355i		−	0.279130i
$880$		0			0
$881$		−	1139.46i		−	1.29337i	−0.762757	−	0.646686i	$-0.776155\pi$
$881$		−	1139.46i		−	1.29337i	0.762757	−	0.646686i	$-0.223845\pi$
$882$			142.759i			0.161859i
$883$	342.234			0.387581			0.193790	−	0.981043i	$-0.437922\pi$
$883$	342.234			0.387581			0.193790	+	0.981043i	$0.437922\pi$
$884$	517.068			0.584919
$885$		0			0
$886$	−205.223			−0.231629
$887$			1128.08i			1.27179i	0.771775	+	0.635896i	$0.219369\pi$
$887$			1128.08i			1.27179i	−0.771775	+	0.635896i	$0.780631\pi$
$888$		−	78.5299i		−	0.0884345i
$889$			167.791i			0.188742i
$890$		0			0
$891$			265.342i			0.297802i
$892$	−419.044			−0.469780
$893$			485.354i			0.543510i
$894$			293.180i			0.327942i
$895$		0			0
$896$	−17.9411			−0.0200236
$897$	−1097.22			−1.22321
$898$	1060.55			1.18102
$899$	−328.274	−	318.477i	−0.365155	−	0.354257i
$900$		0			0
$901$	1628.15			1.80704
$902$	20.9230			0.0231962
$903$		−	318.362i		−	0.352560i
$904$	−497.387			−0.550207
$905$		0			0
$906$	133.594			0.147455
$907$		−	508.301i		−	0.560420i	−0.959939	−	0.280210i	$-0.909596\pi$
$907$		−	508.301i		−	0.560420i	0.959939	−	0.280210i	$-0.0904040\pi$
$908$			347.029i			0.382191i
$909$	374.064			0.411511
$910$		0			0
$911$			752.710i			0.826246i	0.910675	+	0.413123i	$0.135562\pi$
$911$			752.710i			0.826246i	−0.910675	+	0.413123i	$0.864438\pi$
$912$	163.285			0.179041
$913$		−	306.997i		−	0.336251i
$914$		−	975.469i		−	1.06725i
$915$		0			0
$916$			76.9734i			0.0840321i
$917$			269.342i			0.293721i
$918$		−	565.588i		−	0.616109i
$919$	910.396			0.990638			0.495319	−	0.868711i	$-0.335051\pi$
$919$	910.396			0.990638			0.495319	+	0.868711i	$0.335051\pi$
$920$		0			0
$921$			1138.19i			1.23582i
$922$	273.601			0.296747
$923$	−40.6821			−0.0440760
$924$	29.3524			0.0317666
$925$		0			0
$926$		−	247.418i		−	0.267190i
$927$			8.14841i			0.00879009i
$928$	83.4614			0.0899368
$929$			961.821i			1.03533i	0.855583	+	0.517665i	$0.173199\pi$
$929$			961.821i			1.03533i	−0.855583	+	0.517665i	$0.826801\pi$
$930$		0			0
$931$	567.734			0.609811
$932$			272.039i			0.291887i
$933$	−1063.72			−1.14011
$934$	1100.46			1.17822
$935$		0			0
$936$			90.6212i			0.0968175i
$937$		−	515.024i		−	0.549652i	−0.961494	−	0.274826i	$-0.911380\pi$
$937$		−	515.024i		−	0.549652i	0.961494	−	0.274826i	$-0.0886202\pi$
$938$		−	244.569i		−	0.260734i
$939$	−1270.52			−1.35306
$940$		0			0
$941$			736.670i			0.782858i	0.920208	+	0.391429i	$0.128019\pi$
$941$			736.670i			0.782858i	−0.920208	+	0.391429i	$0.871981\pi$
$942$	−848.794			−0.901055
$943$	118.884			0.126070
$944$	−55.1472			−0.0584186
$945$		0			0
$946$	235.204			0.248630
$947$	348.443			0.367945			0.183972	−	0.982931i	$-0.441104\pi$
$947$	348.443			0.367945			0.183972	+	0.982931i	$0.441104\pi$
$948$		−	688.146i		−	0.725893i
$949$	1091.91			1.15059
$950$		0			0
$951$			1285.05i			1.35126i
$952$	−78.5954			−0.0825582
$953$	−617.497			−0.647951			−0.323976	−	0.946065i	$-0.605020\pi$
$953$	−617.497			−0.647951			−0.323976	+	0.946065i	$0.605020\pi$
$954$			285.348i			0.299107i
$955$		0			0
$956$		−	793.752i		−	0.830284i
$957$	−136.546			−0.142681
$958$		−	947.997i		−	0.989558i
$959$			2.94883i			0.00307490i
$960$		0			0
$961$	−29.1076	−	960.559i	−0.0302889	−	0.999541i
$962$		−	173.324i		−	0.180170i
$963$			10.4588i			0.0108606i
$964$			304.812i			0.316196i
$965$		0			0
$966$	166.780			0.172650
$967$	1058.96			1.09510			0.547551	−	0.836772i	$-0.315560\pi$
$967$	1058.96			1.09510			0.547551	+	0.836772i	$0.315560\pi$
$968$		−	320.554i		−	0.331151i
$969$	715.310			0.738194
$970$		0			0
$971$	−1112.85			−1.14609			−0.573043	−	0.819526i	$-0.694237\pi$
$971$	−1112.85			−1.14609			−0.573043	+	0.819526i	$0.694237\pi$
$972$	229.773			0.236392
$973$	38.7650			0.0398407
$974$			806.670i			0.828203i
$975$		0			0
$976$		−	279.417i		−	0.286288i
$977$			1859.00i			1.90277i	0.308008	+	0.951384i	$0.400338\pi$
$977$			1859.00i			1.90277i	−0.308008	+	0.951384i	$0.599662\pi$
$978$	1246.56			1.27460
$979$	139.753			0.142750
$980$		0			0
$981$	−175.205			−0.178599
$982$	580.035			0.590667
$983$	922.826			0.938785			0.469393	−	0.882990i	$-0.344473\pi$
$983$	922.826			0.938785			0.469393	+	0.882990i	$0.344473\pi$
$984$		−	50.5125i		−	0.0513339i
$985$		0			0
$986$	365.622			0.370814
$987$	210.635			0.213410
$988$	360.388			0.364765
$989$	1336.43			1.35129
$990$		0			0
$991$		−	1536.86i		−	1.55082i	−0.631461	−	0.775408i	$-0.717544\pi$
$991$		−	1536.86i		−	1.55082i	0.631461	−	0.775408i	$-0.282456\pi$
$992$	125.864	+	122.108i	0.126879	+	0.123092i
$993$			1367.04i			1.37667i
$994$	6.18377			0.00622109
$995$		0			0
$996$	−741.157			−0.744134
$997$			1334.88i			1.33890i	0.742858	+	0.669449i	$0.233470\pi$
$997$			1334.88i			1.33890i	−0.742858	+	0.669449i	$0.766530\pi$
$998$	288.756			0.289334
$999$	−189.588			−0.189778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1550.3.d.a.1549.7		8
5.2	odd	4		1550.3.c.a.1301.1		4
5.3	odd	4		62.3.b.a.61.4	yes	4
5.4	even	2	inner	1550.3.d.a.1549.2		8
15.8	even	4		558.3.d.a.433.2		4
20.3	even	4		496.3.e.d.433.2		4
31.30	odd	2	inner	1550.3.d.a.1549.6		8
155.92	even	4		1550.3.c.a.1301.2		4
155.123	even	4		62.3.b.a.61.3	✓	4
155.154	odd	2	inner	1550.3.d.a.1549.3		8
465.278	odd	4		558.3.d.a.433.1		4
620.123	odd	4		496.3.e.d.433.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.3.b.a.61.3	✓	4	155.123	even	4
62.3.b.a.61.4	yes	4	5.3	odd	4
496.3.e.d.433.2		4	20.3	even	4
496.3.e.d.433.3		4	620.123	odd	4
558.3.d.a.433.1		4	465.278	odd	4
558.3.d.a.433.2		4	15.8	even	4
1550.3.c.a.1301.1		4	5.2	odd	4
1550.3.c.a.1301.2		4	155.92	even	4
1550.3.d.a.1549.2		8	5.4	even	2	inner
1550.3.d.a.1549.3		8	155.154	odd	2	inner
1550.3.d.a.1549.6		8	31.30	odd	2	inner
1550.3.d.a.1549.7		8	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1550 = 2 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1550.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +3.34239 q^{3} -2.00000 q^{4} +4.72685i q^{6} -1.58579i q^{7} -2.82843i q^{8} +2.17157 q^{9} -2.76893i q^{11} -6.68478 q^{12} -14.7540 q^{13} +2.24264 q^{14} +4.00000 q^{16} +17.5230 q^{17} +3.07107i q^{18} +12.2132 q^{19} -5.30032i q^{21} +3.91585 q^{22} +22.2498 q^{23} -9.45371i q^{24} -20.8653i q^{26} -22.8233 q^{27} +3.17157i q^{28} -14.7540i q^{29} +(21.5858 - 22.2498i) q^{31} +5.65685i q^{32} -9.25483i q^{33} +24.7812i q^{34} -4.34315 q^{36} +8.30678 q^{37} +17.2721i q^{38} -49.3137 q^{39} +5.34315 q^{41} +7.49578 q^{42} +60.0646 q^{43} +5.53785i q^{44} +31.4660i q^{46} +39.7401i q^{47} +13.3696 q^{48} +46.4853 q^{49} +58.5685 q^{51} +29.5080 q^{52} +92.9151 q^{53} -32.2770i q^{54} -4.48528 q^{56} +40.8213 q^{57} +20.8653 q^{58} -13.7868 q^{59} -69.8543i q^{61} +(31.4660 + 30.5269i) q^{62} -3.44365i q^{63} -8.00000 q^{64} +13.0883 q^{66} -109.054i q^{67} -35.0459 q^{68} +74.3675 q^{69} +2.75736 q^{71} -6.14214i q^{72} -74.0077 q^{73} +11.7476i q^{74} -24.4264 q^{76} -4.39093 q^{77} -69.7401i q^{78} +102.942i q^{79} -95.8284 q^{81} +7.55635i q^{82} +110.872 q^{83} +10.6006i q^{84} +84.9442i q^{86} -49.3137i q^{87} -7.83171 q^{88} +50.4718i q^{89} +23.3967i q^{91} -44.4996 q^{92} +(72.1481 - 74.3675i) q^{93} -56.2010 q^{94} +18.9074i q^{96} -113.711i q^{97} +65.7401i q^{98} -6.01293i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 40 q^{9} - 16 q^{14} + 32 q^{16} - 72 q^{19} + 184 q^{31} - 80 q^{36} - 304 q^{39} + 88 q^{41} + 304 q^{49} + 16 q^{51} + 32 q^{56} - 280 q^{59} - 64 q^{64} + 512 q^{66} - 16 q^{69} + 56 q^{71}+ \cdots - 608 q^{94}+O(q^{100}) \) 8 * q - 16 * q^4 + 40 * q^9 - 16 * q^14 + 32 * q^16 - 72 * q^19 + 184 * q^31 - 80 * q^36 - 304 * q^39 + 88 * q^41 + 304 * q^49 + 16 * q^51 + 32 * q^56 - 280 * q^59 - 64 * q^64 + 512 * q^66 - 16 * q^69 + 56 * q^71 + 144 * q^76 - 744 * q^81 - 608 * q^94

Introduction

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