LMFDB - Embedded newform 405.5.d.a.404.43

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,5,Mod(404,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.404"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [44] 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:	$41.8648350490$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.43
Character	$\chi$	$=$	405.404
Dual form			405.5.d.a.404.44

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+7.50729 q^{2} +40.3595 q^{4} +(20.1247 - 14.8322i) q^{5} -69.8446i q^{7} +182.874 q^{8} +(151.082 - 111.350i) q^{10} -52.1241i q^{11} +120.420i q^{13} -524.344i q^{14} +727.134 q^{16} -80.3611 q^{17} -254.262 q^{19} +(812.223 - 598.621i) q^{20} -391.311i q^{22} -250.541 q^{23} +(185.009 - 596.990i) q^{25} +904.028i q^{26} -2818.89i q^{28} +289.055i q^{29} +22.1320 q^{31} +2532.83 q^{32} -603.294 q^{34} +(-1035.95 - 1405.60i) q^{35} +2475.27i q^{37} -1908.82 q^{38} +(3680.28 - 2712.43i) q^{40} -486.356i q^{41} -99.8171i q^{43} -2103.70i q^{44} -1880.89 q^{46} +3478.61 q^{47} -2477.26 q^{49} +(1388.92 - 4481.78i) q^{50} +4860.09i q^{52} +2919.00 q^{53} +(-773.118 - 1048.98i) q^{55} -12772.7i q^{56} +2170.02i q^{58} +1751.56i q^{59} +5582.26 q^{61} +166.151 q^{62} +7380.58 q^{64} +(1786.10 + 2423.42i) q^{65} +3843.98i q^{67} -3243.33 q^{68} +(-7777.19 - 10552.3i) q^{70} -988.642i q^{71} +6017.38i q^{73} +18582.6i q^{74} -10261.9 q^{76} -3640.59 q^{77} +63.2910 q^{79} +(14633.4 - 10785.0i) q^{80} -3651.22i q^{82} +1455.65 q^{83} +(-1617.25 + 1191.94i) q^{85} -749.356i q^{86} -9532.13i q^{88} +12837.4i q^{89} +8410.68 q^{91} -10111.7 q^{92} +26114.9 q^{94} +(-5116.95 + 3771.28i) q^{95} -7321.26i q^{97} -18597.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) $ 44 * q + 324 * q^4 + 28 * q^10 + 2116 * q^16 - 8 * q^19 + 296 * q^25 + 2224 * q^31 + 872 * q^34 + 1700 * q^40 - 5668 * q^46 - 10792 * q^49 - 3072 * q^55 - 5564 * q^61 + 8348 * q^64 - 9564 * q^70 + 3552 * q^76 - 15800 * q^79 - 13532 * q^85 + 1560 * q^91 + 37652 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$	7.50729			1.87682			0.938412	−	0.345519i	$-0.112297\pi$
$2$	7.50729			1.87682			0.938412	+	0.345519i	$0.112297\pi$
$3$		0			0
$3$		0			0
$4$	40.3595			2.52247
$5$	20.1247	−	14.8322i	0.804989	−	0.593290i
$5$	20.1247	−	14.8322i	0.804989	−	0.593290i
$6$		0			0
$7$		−	69.8446i		−	1.42540i	−0.701469	−	0.712700i	$-0.747472\pi$
$7$		−	69.8446i		−	1.42540i	0.701469	−	0.712700i	$-0.252528\pi$
$8$	182.874			2.85740
$9$		0			0
$10$	151.082	−	111.350i	1.51082	−	1.11350i
$11$		−	52.1241i		−	0.430778i	−0.976528	−	0.215389i	$-0.930898\pi$
$11$		−	52.1241i		−	0.430778i	0.976528	−	0.215389i	$-0.0691020\pi$
$12$		0			0
$13$			120.420i			0.712545i	0.934382	+	0.356272i	$0.115952\pi$
$13$			120.420i			0.712545i	−0.934382	+	0.356272i	$0.884048\pi$
$14$		−	524.344i		−	2.67522i
$15$		0			0
$16$	727.134			2.84037
$17$	−80.3611			−0.278066			−0.139033	−	0.990288i	$-0.544399\pi$
$17$	−80.3611			−0.278066			−0.139033	+	0.990288i	$0.544399\pi$
$18$		0			0
$19$	−254.262			−0.704327			−0.352163	−	0.935939i	$-0.614554\pi$
$19$	−254.262			−0.704327			−0.352163	+	0.935939i	$0.614554\pi$
$20$	812.223	−	598.621i	2.03056	−	1.49655i
$21$		0			0
$22$		−	391.311i		−	0.808494i
$23$	−250.541			−0.473613			−0.236806	−	0.971557i	$-0.576101\pi$
$23$	−250.541			−0.473613			−0.236806	+	0.971557i	$0.576101\pi$
$24$		0			0
$25$	185.009	−	596.990i	0.296015	−	0.955183i
$26$			904.028i			1.33732i
$27$		0			0
$28$		−	2818.89i		−	3.59552i
$29$			289.055i			0.343704i	0.985123	+	0.171852i	$0.0549750\pi$
$29$			289.055i			0.343704i	−0.985123	+	0.171852i	$0.945025\pi$
$30$		0			0
$31$	22.1320			0.0230301			0.0115151	−	0.999934i	$-0.496335\pi$
$31$	22.1320			0.0230301			0.0115151	+	0.999934i	$0.496335\pi$
$32$	2532.83			2.47347
$33$		0			0
$34$	−603.294			−0.521881
$35$	−1035.95	−	1405.60i	−0.845675	−	1.14743i
$36$		0			0
$37$			2475.27i			1.80809i	0.427440	+	0.904044i	$0.359415\pi$
$37$			2475.27i			1.80809i	−0.427440	+	0.904044i	$0.640585\pi$
$38$	−1908.82			−1.32190
$39$		0			0
$40$	3680.28	−	2712.43i	2.30018	−	1.69527i
$41$		−	486.356i		−	0.289325i	−0.989481	−	0.144663i	$-0.953790\pi$
$41$		−	486.356i		−	0.289325i	0.989481	−	0.144663i	$-0.0462097\pi$
$42$		0			0
$43$		−	99.8171i		−	0.0539843i	−0.999636	−	0.0269922i	$-0.991407\pi$
$43$		−	99.8171i		−	0.0539843i	0.999636	−	0.0269922i	$-0.00859292\pi$
$44$		−	2103.70i		−	1.08662i
$45$		0			0
$46$	−1880.89			−0.888888
$47$	3478.61			1.57474			0.787372	−	0.616478i	$-0.211441\pi$
$47$	3478.61			1.57474			0.787372	+	0.616478i	$0.211441\pi$
$48$		0			0
$49$	−2477.26			−1.03176
$50$	1388.92	−	4481.78i	0.555567	−	1.79271i
$51$		0			0
$52$			4860.09i			1.79737i
$53$	2919.00			1.03916			0.519581	−	0.854421i	$-0.326088\pi$
$53$	2919.00			1.03916			0.519581	+	0.854421i	$0.326088\pi$
$54$		0			0
$55$	−773.118	−	1048.98i	−0.255576	−	0.346772i
$56$		−	12772.7i		−	4.07294i
$57$		0			0
$58$			2170.02i			0.645071i
$59$			1751.56i			0.503178i	0.967834	+	0.251589i	$0.0809531\pi$
$59$			1751.56i			0.503178i	−0.967834	+	0.251589i	$0.919047\pi$
$60$		0			0
$61$	5582.26			1.50021			0.750103	−	0.661321i	$-0.230004\pi$
$61$	5582.26			1.50021			0.750103	+	0.661321i	$0.230004\pi$
$62$	166.151			0.0432235
$63$		0			0
$64$	7380.58			1.80190
$65$	1786.10	+	2423.42i	0.422745	+	0.573591i
$66$		0			0
$67$			3843.98i			0.856310i	0.903705	+	0.428155i	$0.140836\pi$
$67$			3843.98i			0.856310i	−0.903705	+	0.428155i	$0.859164\pi$
$68$	−3243.33			−0.701412
$69$		0			0
$70$	−7777.19	−	10552.3i	−1.58718	−	2.15352i
$71$		−	988.642i		−	0.196120i	−0.995180	−	0.0980601i	$-0.968736\pi$
$71$		−	988.642i		−	0.196120i	0.995180	−	0.0980601i	$-0.0312637\pi$
$72$		0			0
$73$			6017.38i			1.12918i	0.825373	+	0.564588i	$0.190965\pi$
$73$			6017.38i			1.12918i	−0.825373	+	0.564588i	$0.809035\pi$
$74$			18582.6i			3.39346i
$75$		0			0
$76$	−10261.9			−1.77664
$77$	−3640.59			−0.614031
$78$		0			0
$79$	63.2910			0.0101412			0.00507058	−	0.999987i	$-0.498386\pi$
$79$	63.2910			0.0101412			0.00507058	+	0.999987i	$0.498386\pi$
$80$	14633.4	−	10785.0i	2.28647	−	1.68516i
$81$		0			0
$82$		−	3651.22i		−	0.543013i
$83$	1455.65			0.211300			0.105650	−	0.994403i	$-0.466308\pi$
$83$	1455.65			0.211300			0.105650	+	0.994403i	$0.466308\pi$
$84$		0			0
$85$	−1617.25	+	1191.94i	−0.223840	+	0.164974i
$86$		−	749.356i		−	0.101319i
$87$		0			0
$88$		−	9532.13i		−	1.23091i
$89$			12837.4i			1.62068i	0.585963	+	0.810338i	$0.300717\pi$
$89$			12837.4i			1.62068i	−0.585963	+	0.810338i	$0.699283\pi$
$90$		0			0
$91$	8410.68			1.01566
$92$	−10111.7			−1.19467
$93$		0			0
$94$	26114.9			2.95552
$95$	−5116.95	+	3771.28i	−0.566975	+	0.417870i
$96$		0			0
$97$		−	7321.26i		−	0.778113i	−0.921214	−	0.389056i	$-0.872801\pi$
$97$		−	7321.26i		−	0.778113i	0.921214	−	0.389056i	$-0.127199\pi$
$98$	−18597.5			−1.93644
$99$		0			0
$100$	7466.87	−	24094.2i	0.746687	−	2.40942i
$101$		−	3825.42i		−	0.375005i	−0.982264	−	0.187502i	$-0.939961\pi$
$101$		−	3825.42i		−	0.375005i	0.982264	−	0.187502i	$-0.0600393\pi$
$102$		0			0
$103$		−	18268.8i		−	1.72201i	−0.508593	−	0.861007i	$-0.669834\pi$
$103$		−	18268.8i		−	1.72201i	0.508593	−	0.861007i	$-0.330166\pi$
$104$			22021.6i			2.03602i
$105$		0			0
$106$	21913.8			1.95032
$107$	−7204.92			−0.629306			−0.314653	−	0.949207i	$-0.601888\pi$
$107$	−7204.92			−0.629306			−0.314653	+	0.949207i	$0.601888\pi$
$108$		0			0
$109$	405.823			0.0341573			0.0170786	−	0.999854i	$-0.494563\pi$
$109$	405.823			0.0341573			0.0170786	+	0.999854i	$0.494563\pi$
$110$	−5804.02	−	7875.03i	−0.479671	−	0.650829i
$111$		0			0
$112$		−	50786.4i		−	4.04866i
$113$	−8915.91			−0.698246			−0.349123	−	0.937077i	$-0.613521\pi$
$113$	−8915.91			−0.698246			−0.349123	+	0.937077i	$0.613521\pi$
$114$		0			0
$115$	−5042.07	+	3716.09i	−0.381253	+	0.280990i
$116$			11666.1i			0.866980i
$117$		0			0
$118$			13149.5i			0.944376i
$119$			5612.79i			0.396355i
$120$		0			0
$121$	11924.1			0.814430
$122$	41907.7			2.81562
$123$		0			0
$124$	893.234			0.0580927
$125$	−5131.43	−	14758.4i	−0.328412	−	0.944535i
$126$		0			0
$127$		−	20437.2i		−	1.26711i	−0.773698	−	0.633554i	$-0.781595\pi$
$127$		−	20437.2i		−	1.26711i	0.773698	−	0.633554i	$-0.218405\pi$
$128$	14882.8			0.908375
$129$		0			0
$130$	13408.8	+	18193.3i	0.793418	+	1.07653i
$131$			17157.1i			0.999772i	0.866091	+	0.499886i	$0.166625\pi$
$131$			17157.1i			0.999772i	−0.866091	+	0.499886i	$0.833375\pi$
$132$		0			0
$133$			17758.8i			1.00395i
$134$			28857.9i			1.60714i
$135$		0			0
$136$	−14695.9			−0.794546
$137$	−26301.1			−1.40130			−0.700652	−	0.713503i	$-0.747107\pi$
$137$	−26301.1			−1.40130			−0.700652	+	0.713503i	$0.747107\pi$
$138$		0			0
$139$	−22957.3			−1.18821			−0.594103	−	0.804389i	$-0.702493\pi$
$139$	−22957.3			−1.18821			−0.594103	+	0.804389i	$0.702493\pi$
$140$	−41810.4	−	56729.4i	−2.13319	−	2.89436i
$141$		0			0
$142$		−	7422.02i		−	0.368083i
$143$	6276.79			0.306949
$144$		0			0
$145$	4287.33	+	5817.15i	0.203916	+	0.276678i
$146$			45174.2i			2.11926i
$147$		0			0
$148$			99900.6i			4.56084i
$149$			30202.5i			1.36041i	0.733022	+	0.680205i	$0.238109\pi$
$149$			30202.5i			1.36041i	−0.733022	+	0.680205i	$0.761891\pi$
$150$		0			0
$151$	−22357.2			−0.980537			−0.490269	−	0.871571i	$-0.663101\pi$
$151$	−22357.2			−0.980537			−0.490269	+	0.871571i	$0.663101\pi$
$152$	−46497.8			−2.01254
$153$		0			0
$154$	−27331.0			−1.15243
$155$	445.399	−	328.266i	0.0185390	−	0.0136635i
$156$		0			0
$157$		−	27628.0i		−	1.12086i	−0.828203	−	0.560428i	$-0.810637\pi$
$157$		−	27628.0i		−	1.12086i	0.828203	−	0.560428i	$-0.189363\pi$
$158$	475.144			0.0190332
$159$		0			0
$160$	50972.6	−	37567.6i	1.99112	−	1.46748i
$161$			17498.9i			0.675087i
$162$		0			0
$163$			674.702i			0.0253943i	0.999919	+	0.0126972i	$0.00404174\pi$
$163$			674.702i			0.0253943i	−0.999919	+	0.0126972i	$0.995958\pi$
$164$		−	19629.1i		−	0.729814i
$165$		0			0
$166$	10928.0			0.396573
$167$	−40517.2			−1.45280			−0.726401	−	0.687271i	$-0.758808\pi$
$167$	−40517.2			−1.45280			−0.726401	+	0.687271i	$0.758808\pi$
$168$		0			0
$169$	14060.0			0.492280
$170$	−12141.1	+	8948.21i	−0.420109	+	0.309627i
$171$		0			0
$172$		−	4028.56i		−	0.136174i
$173$	−26552.3			−0.887176			−0.443588	−	0.896231i	$-0.646295\pi$
$173$	−26552.3			−0.887176			−0.443588	+	0.896231i	$0.646295\pi$
$174$		0			0
$175$	−41696.5	−	12921.9i	−1.36152	−	0.421939i
$176$		−	37901.3i		−	1.22357i
$177$		0			0
$178$			96373.9i			3.04172i
$179$			42490.3i			1.32612i	0.748565	+	0.663062i	$0.230743\pi$
$179$			42490.3i			1.32612i	−0.748565	+	0.663062i	$0.769257\pi$
$180$		0			0
$181$	46294.0			1.41308			0.706542	−	0.707671i	$-0.250254\pi$
$181$	46294.0			1.41308			0.706542	+	0.707671i	$0.250254\pi$
$182$	63141.5			1.90622
$183$		0			0
$184$	−45817.4			−1.35330
$185$	36713.8	+	49814.2i	1.07272	+	1.45549i
$186$		0			0
$187$			4188.75i			0.119785i
$188$	140395.			3.97224
$189$		0			0
$190$	−38414.5	+	28312.1i	−1.06411	+	0.784268i
$191$		−	4961.70i		−	0.136008i	−0.997685	−	0.0680038i	$-0.978337\pi$
$191$		−	4961.70i		−	0.136008i	0.997685	−	0.0680038i	$-0.0216630\pi$
$192$		0			0
$193$			44113.4i			1.18428i	0.805834	+	0.592142i	$0.201717\pi$
$193$			44113.4i			1.18428i	−0.805834	+	0.592142i	$0.798283\pi$
$194$		−	54962.9i		−	1.46038i
$195$		0			0
$196$	−99981.0			−2.60259
$197$	−16721.9			−0.430876			−0.215438	−	0.976518i	$-0.569118\pi$
$197$	−16721.9			−0.430876			−0.215438	+	0.976518i	$0.569118\pi$
$198$		0			0
$199$	−56999.6			−1.43935			−0.719673	−	0.694313i	$-0.755708\pi$
$199$	−56999.6			−1.43935			−0.719673	+	0.694313i	$0.755708\pi$
$200$	33833.3	−	109174.i	0.845832	−	2.72934i
$201$		0			0
$202$		−	28718.6i		−	0.703818i
$203$	20188.9			0.489915
$204$		0			0
$205$	−7213.75	−	9787.78i	−0.171654	−	0.232904i
$206$		−	137150.i		−	3.23192i
$207$		0			0
$208$			87561.5i			2.02389i
$209$			13253.2i			0.303409i
$210$		0			0
$211$	31351.7			0.704201			0.352101	−	0.935962i	$-0.385468\pi$
$211$	31351.7			0.704201			0.352101	+	0.935962i	$0.385468\pi$
$212$	117809.			2.62125
$213$		0			0
$214$	−54089.5			−1.18110
$215$	−1480.51	−	2008.79i	−0.0320284	−	0.0434568i
$216$		0			0
$217$		−	1545.80i		−	0.0328271i
$218$	3046.63			0.0641072
$219$		0			0
$220$	−31202.6	−	42336.4i	−0.644682	−	0.874720i
$221$		−	9677.09i		−	0.198135i
$222$		0			0
$223$		−	47772.0i		−	0.960646i	−0.877092	−	0.480323i	$-0.840519\pi$
$223$		−	47772.0i		−	0.960646i	0.877092	−	0.480323i	$-0.159481\pi$
$224$		−	176905.i		−	3.52568i
$225$		0			0
$226$	−66934.3			−1.31049
$227$	−18624.5			−0.361437			−0.180718	−	0.983535i	$-0.557842\pi$
$227$	−18624.5			−0.361437			−0.180718	+	0.983535i	$0.557842\pi$
$228$		0			0
$229$	94004.1			1.79257			0.896284	−	0.443480i	$-0.146256\pi$
$229$	94004.1			1.79257			0.896284	+	0.443480i	$0.146256\pi$
$230$	−37852.3	+	27897.8i	−0.715545	+	0.527368i
$231$		0			0
$232$			52860.5i			0.982098i
$233$	−72051.8			−1.32719			−0.663595	−	0.748092i	$-0.730970\pi$
$233$	−72051.8			−1.32719			−0.663595	+	0.748092i	$0.730970\pi$
$234$		0			0
$235$	70006.1	−	51595.6i	1.26765	−	0.934280i
$236$			70692.1i			1.26925i
$237$		0			0
$238$			42136.8i			0.743889i
$239$		−	53627.3i		−	0.938837i	−0.882976	−	0.469418i	$-0.844464\pi$
$239$		−	53627.3i		−	0.938837i	0.882976	−	0.469418i	$-0.155536\pi$
$240$		0			0
$241$	−6325.23			−0.108904			−0.0544518	−	0.998516i	$-0.517341\pi$
$241$	−6325.23			−0.108904			−0.0544518	+	0.998516i	$0.517341\pi$
$242$	89517.5			1.52854
$243$		0			0
$244$	225297.			3.78422
$245$	−49854.2	+	36743.4i	−0.830558	+	0.612134i
$246$		0			0
$247$		−	30618.2i		−	0.501864i
$248$	4047.35			0.0658063
$249$		0			0
$250$	−38523.2	−	110795.i	−0.616371	−	1.77272i
$251$			11578.9i			0.183790i	0.995769	+	0.0918948i	$0.0292924\pi$
$251$			11578.9i			0.183790i	−0.995769	+	0.0918948i	$0.970708\pi$
$252$		0			0
$253$			13059.2i			0.204022i
$254$		−	153428.i		−	2.37814i
$255$		0			0
$256$	−6359.55			−0.0970390
$257$	−38296.6			−0.579821			−0.289911	−	0.957054i	$-0.593626\pi$
$257$	−38296.6			−0.579821			−0.289911	+	0.957054i	$0.593626\pi$
$258$		0			0
$259$	172884.			2.57725
$260$	72086.0	+	97807.9i	1.06636	+	1.44686i
$261$		0			0
$262$			128803.i			1.87639i
$263$	125852.			1.81949			0.909745	−	0.415167i	$-0.136277\pi$
$263$	125852.			1.81949			0.909745	+	0.415167i	$0.136277\pi$
$264$		0			0
$265$	58744.2	−	43295.4i	0.836513	−	0.616524i
$266$			133321.i			1.88423i
$267$		0			0
$268$			155141.i			2.16001i
$269$			5288.09i			0.0730792i	0.999332	+	0.0365396i	$0.0116335\pi$
$269$			5288.09i			0.0730792i	−0.999332	+	0.0365396i	$0.988366\pi$
$270$		0			0
$271$	4773.83			0.0650022			0.0325011	−	0.999472i	$-0.489653\pi$
$271$	4773.83			0.0650022			0.0325011	+	0.999472i	$0.489653\pi$
$272$	−58433.3			−0.789810
$273$		0			0
$274$	−197450.			−2.63000
$275$	−31117.6	−	9643.45i	−0.411472	−	0.127517i
$276$		0			0
$277$		−	61566.1i		−	0.802383i	−0.915994	−	0.401192i	$-0.868596\pi$
$277$		−	61566.1i		−	0.802383i	0.915994	−	0.401192i	$-0.131404\pi$
$278$	−172347.			−2.23005
$279$		0			0
$280$	−189448.	−	257048.i	−2.41643	−	3.27867i
$281$			125433.i			1.58855i	0.607560	+	0.794274i	$0.292149\pi$
$281$			125433.i			1.58855i	−0.607560	+	0.794274i	$0.707851\pi$
$282$		0			0
$283$			15937.2i			0.198993i	0.995038	+	0.0994967i	$0.0317233\pi$
$283$			15937.2i			0.198993i	−0.995038	+	0.0994967i	$0.968277\pi$
$284$		−	39901.0i		−	0.494706i
$285$		0			0
$286$	47121.7			0.576088
$287$	−33969.3			−0.412404
$288$		0			0
$289$	−77063.1			−0.922679
$290$	32186.2	+	43671.0i	0.382714	+	0.519275i
$291$		0			0
$292$			242858.i			2.84831i
$293$	−107300.			−1.24987			−0.624936	−	0.780676i	$-0.714875\pi$
$293$	−107300.			−1.24987			−0.624936	+	0.780676i	$0.714875\pi$
$294$		0			0
$295$	25979.6	+	35249.7i	0.298530	+	0.405053i
$296$			452662.i			5.16643i
$297$		0			0
$298$			226739.i			2.55325i
$299$		−	30170.2i		−	0.337470i
$300$		0			0
$301$	−6971.68			−0.0769492
$302$	−167842.			−1.84029
$303$		0			0
$304$	−184883.			−2.00055
$305$	112342.	−	82797.5i	1.20765	−	0.890056i
$306$		0			0
$307$		−	72255.8i		−	0.766647i	−0.923614	−	0.383324i	$-0.874779\pi$
$307$		−	72255.8i		−	0.766647i	0.923614	−	0.383324i	$-0.125221\pi$
$308$	−146932.			−1.54887
$309$		0			0
$310$	3343.74	−	2464.39i	0.0347944	−	0.0256440i
$311$			17225.3i			0.178093i	0.996027	+	0.0890465i	$0.0283820\pi$
$311$			17225.3i			0.178093i	−0.996027	+	0.0890465i	$0.971618\pi$
$312$		0			0
$313$		−	131715.i		−	1.34446i	−0.740344	−	0.672228i	$-0.765338\pi$
$313$		−	131715.i		−	1.34446i	0.740344	−	0.672228i	$-0.234662\pi$
$314$		−	207411.i		−	2.10365i
$315$		0			0
$316$	2554.39			0.0255807
$317$	−120318.			−1.19732			−0.598661	−	0.801003i	$-0.704300\pi$
$317$	−120318.			−1.19732			−0.598661	+	0.801003i	$0.704300\pi$
$318$		0			0
$319$	15066.7			0.148060
$320$	148532.	−	109471.i	1.45051	−	1.06905i
$321$		0			0
$322$			131370.i			1.26702i
$323$	20432.8			0.195849
$324$		0			0
$325$	71889.5	+	22278.8i	0.680611	+	0.210924i
$326$			5065.18i			0.0476606i
$327$		0			0
$328$		−	88941.7i		−	0.826718i
$329$		−	242962.i		−	2.24464i
$330$		0			0
$331$	−50097.6			−0.457258			−0.228629	−	0.973514i	$-0.573424\pi$
$331$	−50097.6			−0.457258			−0.228629	+	0.973514i	$0.573424\pi$
$332$	58749.0			0.532997
$333$		0			0
$334$	−304174.			−2.72665
$335$	57014.8	+	77359.0i	0.508040	+	0.689320i
$336$		0			0
$337$			49723.7i			0.437828i	0.975744	+	0.218914i	$0.0702514\pi$
$337$			49723.7i			0.437828i	−0.975744	+	0.218914i	$0.929749\pi$
$338$	105553.			0.923923
$339$		0			0
$340$	−65271.1	+	48105.9i	−0.564629	+	0.416141i
$341$		−	1153.61i		−	0.00992087i
$342$		0			0
$343$			5326.54i			0.0452749i
$344$		−	18253.9i		−	0.154255i
$345$		0			0
$346$	−199336.			−1.66507
$347$	174658.			1.45054			0.725268	−	0.688467i	$-0.241716\pi$
$347$	174658.			1.45054			0.725268	+	0.688467i	$0.241716\pi$
$348$		0			0
$349$	−71605.8			−0.587892			−0.293946	−	0.955822i	$-0.594969\pi$
$349$	−71605.8			−0.587892			−0.293946	+	0.955822i	$0.594969\pi$
$350$	−313028.	−	97008.4i	−2.55533	−	0.791905i
$351$		0			0
$352$		−	132022.i		−	1.06552i
$353$	73880.0			0.592895			0.296447	−	0.955049i	$-0.404198\pi$
$353$	73880.0			0.592895			0.296447	+	0.955049i	$0.404198\pi$
$354$		0			0
$355$	−14663.8	−	19896.1i	−0.116356	−	0.157875i
$356$			518109.i			4.08810i
$357$		0			0
$358$			318987.i			2.48890i
$359$			192955.i			1.49715i	0.663048	+	0.748577i	$0.269263\pi$
$359$			192955.i			1.49715i	−0.663048	+	0.748577i	$0.730737\pi$
$360$		0			0
$361$	−65671.8			−0.503924
$362$	347543.			2.65211
$363$		0			0
$364$	339451.			2.56197
$365$	89251.2	+	121098.i	0.669928	+	0.908974i
$366$		0			0
$367$		−	50798.4i		−	0.377153i	−0.982059	−	0.188577i	$-0.939613\pi$
$367$		−	50798.4i		−	0.377153i	0.982059	−	0.188577i	$-0.0603873\pi$
$368$	−182177.			−1.34524
$369$		0			0
$370$	275621.	+	373970.i	2.01330	+	2.73170i
$371$		−	203877.i		−	1.48122i
$372$		0			0
$373$			69503.2i			0.499559i	0.968303	+	0.249780i	$0.0803581\pi$
$373$			69503.2i			0.499559i	−0.968303	+	0.249780i	$0.919642\pi$
$374$			31446.2i			0.224815i
$375$		0			0
$376$	636146.			4.49967
$377$	−34808.0			−0.244904
$378$		0			0
$379$	−81228.0			−0.565493			−0.282747	−	0.959195i	$-0.591246\pi$
$379$	−81228.0			−0.565493			−0.282747	+	0.959195i	$0.591246\pi$
$380$	−206517.	+	152207.i	−1.43018	+	1.05406i
$381$		0			0
$382$		−	37248.9i		−	0.255262i
$383$	95986.8			0.654356			0.327178	−	0.944963i	$-0.393902\pi$
$383$	95986.8			0.654356			0.327178	+	0.944963i	$0.393902\pi$
$384$		0			0
$385$	−73265.8	+	53998.1i	−0.494288	+	0.364298i
$386$			331172.i			2.22269i
$387$		0			0
$388$		−	295482.i		−	1.96276i
$389$			33206.9i			0.219447i	0.993962	+	0.109723i	$0.0349965\pi$
$389$			33206.9i			0.219447i	−0.993962	+	0.109723i	$0.965003\pi$
$390$		0			0
$391$	20133.8			0.131696
$392$	−453026.			−2.94816
$393$		0			0
$394$	−125536.			−0.808678
$395$	1273.71	−	938.747i	0.00816352	−	0.00601664i
$396$		0			0
$397$		−	180216.i		−	1.14344i	−0.820449	−	0.571720i	$-0.806276\pi$
$397$		−	180216.i		−	1.14344i	0.820449	−	0.571720i	$-0.193724\pi$
$398$	−427912.			−2.70140
$399$		0			0
$400$	134527.	−	434092.i	0.840791	−	2.71307i
$401$		−	137745.i		−	0.856615i	−0.903633	−	0.428307i	$-0.859110\pi$
$401$		−	137745.i		−	0.856615i	0.903633	−	0.428307i	$-0.140890\pi$
$402$		0			0
$403$			2665.13i			0.0164100i
$404$		−	154392.i		−	0.945937i
$405$		0			0
$406$	151564.			0.919483
$407$	129021.			0.778884
$408$		0			0
$409$	34418.4			0.205752			0.102876	−	0.994694i	$-0.467195\pi$
$409$	34418.4			0.205752			0.102876	+	0.994694i	$0.467195\pi$
$410$	−54155.7	−	73479.8i	−0.322164	−	0.437119i
$411$		0			0
$412$		−	737321.i		−	4.34372i
$413$	122337.			0.717229
$414$		0			0
$415$	29294.5	−	21590.5i	0.170094	−	0.125362i
$416$			305004.i			1.76246i
$417$		0			0
$418$			99495.6i			0.569444i
$419$			150513.i			0.857325i	0.903465	+	0.428662i	$0.141015\pi$
$419$			150513.i			0.857325i	−0.903465	+	0.428662i	$0.858985\pi$
$420$		0			0
$421$	−109126.			−0.615693			−0.307847	−	0.951436i	$-0.599608\pi$
$421$	−109126.			−0.615693			−0.307847	+	0.951436i	$0.599608\pi$
$422$	235367.			1.32166
$423$		0			0
$424$	533809.			2.96930
$425$	−14867.5	+	47974.8i	−0.0823117	+	0.265604i
$426$		0			0
$427$		−	389891.i		−	2.13839i
$428$	−290787.			−1.58740
$429$		0			0
$430$	−11114.6	−	15080.6i	−0.0601116	−	0.0815607i
$431$		−	344386.i		−	1.85392i	−0.375159	−	0.926960i	$-0.622412\pi$
$431$		−	344386.i		−	1.85392i	0.375159	−	0.926960i	$-0.377588\pi$
$432$		0			0
$433$		−	42720.4i		−	0.227855i	−0.993489	−	0.113928i	$-0.963657\pi$
$433$		−	42720.4i		−	0.227855i	0.993489	−	0.113928i	$-0.0363432\pi$
$434$		−	11604.7i		−	0.0616107i
$435$		0			0
$436$	16378.8			0.0861606
$437$	63703.1			0.333578
$438$		0			0
$439$	199882.			1.03716			0.518579	−	0.855029i	$-0.326461\pi$
$439$	199882.			1.03716			0.518579	+	0.855029i	$0.326461\pi$
$440$	−141383.	−	191831.i	−0.730283	−	0.990865i
$441$		0			0
$442$		−	72648.7i		−	0.371863i
$443$	−100200.			−0.510578			−0.255289	−	0.966865i	$-0.582171\pi$
$443$	−100200.			−0.510578			−0.255289	+	0.966865i	$0.582171\pi$
$444$		0			0
$445$	190407.	+	258349.i	0.961530	+	1.30463i
$446$		−	358638.i		−	1.80296i
$447$		0			0
$448$		−	515493.i		−	2.56843i
$449$			26310.5i			0.130508i	0.997869	+	0.0652539i	$0.0207857\pi$
$449$			26310.5i			0.130508i	−0.997869	+	0.0652539i	$0.979214\pi$
$450$		0			0
$451$	−25350.9			−0.124635
$452$	−359841.			−1.76130
$453$		0			0
$454$	−139819.			−0.678353
$455$	169263.	−	124749.i	0.817595	−	0.602581i
$456$		0			0
$457$		−	66055.8i		−	0.316285i	−0.987416	−	0.158143i	$-0.949449\pi$
$457$		−	66055.8i		−	0.316285i	0.987416	−	0.158143i	$-0.0505505\pi$
$458$	705716.			3.36433
$459$		0			0
$460$	−203495.	+	149979.i	−0.961698	+	0.708787i
$461$		−	77216.9i		−	0.363338i	−0.983360	−	0.181669i	$-0.941850\pi$
$461$		−	77216.9i		−	0.363338i	0.983360	−	0.181669i	$-0.0581499\pi$
$462$		0			0
$463$		−	162833.i		−	0.759593i	−0.925070	−	0.379797i	$-0.875994\pi$
$463$		−	162833.i		−	0.759593i	0.925070	−	0.379797i	$-0.124006\pi$
$464$			210182.i			0.976245i
$465$		0			0
$466$	−540914.			−2.49090
$467$	115736.			0.530681			0.265340	−	0.964155i	$-0.414516\pi$
$467$	115736.			0.530681			0.265340	+	0.964155i	$0.414516\pi$
$468$		0			0
$469$	268481.			1.22058
$470$	525556.	−	387343.i	2.37916	−	1.75348i
$471$		0			0
$472$			320315.i			1.43778i
$473$	−5202.88			−0.0232553
$474$		0			0
$475$	−47040.8	+	151792.i	−0.208491	+	0.672761i
$476$			226529.i			0.999793i
$477$		0			0
$478$		−	402596.i		−	1.76203i
$479$		−	429811.i		−	1.87330i	−0.350271	−	0.936648i	$-0.613911\pi$
$479$		−	429811.i		−	1.87330i	0.350271	−	0.936648i	$-0.386089\pi$
$480$		0			0
$481$	−298072.			−1.28834
$482$	−47485.3			−0.204393
$483$		0			0
$484$	481249.			2.05437
$485$	−108591.	−	147338.i	−0.461646	−	0.626372i
$486$		0			0
$487$		−	116726.i		−	0.492166i	−0.969249	−	0.246083i	$-0.920856\pi$
$487$		−	116726.i		−	0.492166i	0.969249	−	0.246083i	$-0.0791435\pi$
$488$	1.02085e6			4.28669
$489$		0			0
$490$	−374270.	+	275843.i	−1.55881	+	1.14887i
$491$			106115.i			0.440163i	0.975481	+	0.220081i	$0.0706323\pi$
$491$			106115.i			0.440163i	−0.975481	+	0.220081i	$0.929368\pi$
$492$		0			0
$493$		−	23228.8i		−	0.0955723i
$494$		−	229860.i		−	0.941911i
$495$		0			0
$496$	16092.9			0.0654140
$497$	−69051.2			−0.279549
$498$		0			0
$499$	111205.			0.446603			0.223302	−	0.974749i	$-0.428316\pi$
$499$	111205.			0.446603			0.223302	+	0.974749i	$0.428316\pi$
$500$	−207102.	−	595639.i	−0.828408	−	2.38256i
$501$		0			0
$502$			86926.4i			0.344941i
$503$	−48033.5			−0.189849			−0.0949246	−	0.995484i	$-0.530261\pi$
$503$	−48033.5			−0.189849			−0.0949246	+	0.995484i	$0.530261\pi$
$504$		0			0
$505$	−56739.6	−	76985.6i	−0.222487	−	0.301875i
$506$			98039.6i			0.382913i
$507$		0			0
$508$		−	824834.i		−	3.19624i
$509$		−	150934.i		−	0.582575i	−0.956636	−	0.291287i	$-0.905916\pi$
$509$		−	150934.i		−	0.582575i	0.956636	−	0.291287i	$-0.0940836\pi$
$510$		0			0
$511$	420281.			1.60953
$512$	−285868.			−1.09050
$513$		0			0
$514$	−287504.			−1.08822
$515$	−270968.	−	367656.i	−1.02165	−	1.38620i
$516$		0			0
$517$		−	181320.i		−	0.678365i
$518$	1.29789e6			4.83704
$519$		0			0
$520$	326630.	+	443180.i	1.20795	+	1.63898i
$521$		−	398056.i		−	1.46645i	−0.679984	−	0.733227i	$-0.738013\pi$
$521$		−	398056.i		−	1.46645i	0.679984	−	0.733227i	$-0.261987\pi$
$522$		0			0
$523$			303039.i			1.10789i	0.832554	+	0.553943i	$0.186878\pi$
$523$			303039.i			1.10789i	−0.832554	+	0.553943i	$0.813122\pi$
$524$			692450.i			2.52189i
$525$		0			0
$526$	944810.			3.41486
$527$	−1778.55			−0.00640390
$528$		0			0
$529$	−217070.			−0.775691
$530$	441010.	−	325031.i	1.56999	−	1.15711i
$531$		0			0
$532$			716736.i			2.53242i
$533$	58567.0			0.206157
$534$		0			0
$535$	−144997.	+	106865.i	−0.506584	+	0.373361i
$536$			702962.i			2.44682i
$537$		0			0
$538$			39699.2i			0.137157i
$539$			129125.i			0.444461i
$540$		0			0
$541$	298904.			1.02126			0.510632	−	0.859800i	$-0.329412\pi$
$541$	298904.			1.02126			0.510632	+	0.859800i	$0.329412\pi$
$542$	35838.5			0.121998
$543$		0			0
$544$	−203541.			−0.687788
$545$	8167.07	−	6019.26i	0.0274962	−	0.0202652i
$546$		0			0
$547$			132213.i			0.441875i	0.975288	+	0.220938i	$0.0709118\pi$
$547$			132213.i			0.441875i	−0.975288	+	0.220938i	$0.929088\pi$
$548$	−1.06150e6			−3.53474
$549$		0			0
$550$	−233609.	−	72396.2i	−0.772260	−	0.239326i
$551$		−	73495.6i		−	0.242080i
$552$		0			0
$553$		−	4420.53i		−	0.0144552i
$554$		−	462195.i		−	1.50593i
$555$		0			0
$556$	−926545.			−2.99721
$557$	347933.			1.12146			0.560731	−	0.827998i	$-0.310520\pi$
$557$	347933.			1.12146			0.560731	+	0.827998i	$0.310520\pi$
$558$		0			0
$559$	12020.0			0.0384662
$560$	−753276.	−	1.02206e6i	−2.40203	−	3.25913i
$561$		0			0
$562$			941665.i			2.98142i
$563$	−514119.			−1.62199			−0.810993	−	0.585055i	$-0.801073\pi$
$563$	−514119.			−1.62199			−0.810993	+	0.585055i	$0.801073\pi$
$564$		0			0
$565$	−179430.	+	132243.i	−0.562081	+	0.414262i
$566$			119645.i			0.373476i
$567$		0			0
$568$		−	180796.i		−	0.560394i
$569$			456112.i			1.40879i	0.709807	+	0.704396i	$0.248782\pi$
$569$			456112.i			1.40879i	−0.709807	+	0.704396i	$0.751218\pi$
$570$		0			0
$571$	262872.			0.806256			0.403128	−	0.915144i	$-0.367923\pi$
$571$	262872.			0.806256			0.403128	+	0.915144i	$0.367923\pi$
$572$	253328.			0.774267
$573$		0			0
$574$	−255018.			−0.774010
$575$	−46352.4	+	149570.i	−0.140196	+	0.452387i
$576$		0			0
$577$		−	297110.i		−	0.892414i	−0.894930	−	0.446207i	$-0.852775\pi$
$577$		−	297110.i		−	0.892414i	0.894930	−	0.446207i	$-0.147225\pi$
$578$	−578535.			−1.73171
$579$		0			0
$580$	173034.	+	234777.i	0.514371	+	0.697910i
$581$		−	101669.i		−	0.301187i
$582$		0			0
$583$		−	152151.i		−	0.447648i
$584$			1.10042e6i			3.22651i
$585$		0			0
$586$	−805534.			−2.34579
$587$	306666.			0.889999			0.444999	−	0.895531i	$-0.353204\pi$
$587$	306666.			0.889999			0.444999	+	0.895531i	$0.353204\pi$
$588$		0			0
$589$	−5627.31			−0.0162207
$590$	195036.	+	264630.i	0.560289	+	0.760212i
$591$		0			0
$592$			1.79985e6i			5.13563i
$593$	84728.9			0.240947			0.120474	−	0.992717i	$-0.461559\pi$
$593$	84728.9			0.240947			0.120474	+	0.992717i	$0.461559\pi$
$594$		0			0
$595$	83250.2	+	112956.i	0.235153	+	0.319062i
$596$			1.21895e6i			3.43159i
$597$		0			0
$598$		−	226496.i		−	0.633372i
$599$			554273.i			1.54479i	0.635140	+	0.772397i	$0.280942\pi$
$599$			554273.i			1.54479i	−0.635140	+	0.772397i	$0.719058\pi$
$600$		0			0
$601$	5420.18			0.0150060			0.00750299	−	0.999972i	$-0.497612\pi$
$601$	5420.18			0.0150060			0.00750299	+	0.999972i	$0.497612\pi$
$602$	−52338.4			−0.144420
$603$		0			0
$604$	−902325.			−2.47337
$605$	239969.	−	176861.i	0.655607	−	0.483193i
$606$		0			0
$607$			111817.i			0.303481i	0.988420	+	0.151740i	$0.0484878\pi$
$607$			111817.i			0.303481i	−0.988420	+	0.151740i	$0.951512\pi$
$608$	−644003.			−1.74213
$609$		0			0
$610$	843381.	−	621585.i	2.26654	−	1.67048i
$611$			418894.i			1.12208i
$612$		0			0
$613$			631873.i			1.68155i	0.541387	+	0.840774i	$0.317900\pi$
$613$			631873.i			1.68155i	−0.541387	+	0.840774i	$0.682100\pi$
$614$		−	542445.i		−	1.43886i
$615$		0			0
$616$	−665767.			−1.75453
$617$	168643.			0.442994			0.221497	−	0.975161i	$-0.428906\pi$
$617$	168643.			0.442994			0.221497	+	0.975161i	$0.428906\pi$
$618$		0			0
$619$	202821.			0.529336			0.264668	−	0.964340i	$-0.414738\pi$
$619$	202821.			0.529336			0.264668	+	0.964340i	$0.414738\pi$
$620$	17976.1	−	13248.7i	0.0467640	−	0.0344658i
$621$		0			0
$622$			129316.i			0.334249i
$623$	896621.			2.31011
$624$		0			0
$625$	−322168.	−	220897.i	−0.824751	−	0.565497i
$626$		−	988823.i		−	2.52331i
$627$		0			0
$628$		−	1.11505e6i		−	2.82732i
$629$		−	198916.i		−	0.502768i
$630$		0			0
$631$	539826.			1.35580			0.677900	−	0.735154i	$-0.262890\pi$
$631$	539826.			1.35580			0.677900	+	0.735154i	$0.262890\pi$
$632$	11574.2			0.0289773
$633$		0			0
$634$	−903260.			−2.24716
$635$	−303129.	−	411293.i	−0.751763	−	1.02001i
$636$		0			0
$637$		−	298312.i		−	0.735177i
$638$	113110.			0.277882
$639$		0			0
$640$	299513.	−	220746.i	0.731232	−	0.538930i
$641$			23293.7i			0.0566920i	0.999598	+	0.0283460i	$0.00902402\pi$
$641$			23293.7i			0.0566920i	−0.999598	+	0.0283460i	$0.990976\pi$
$642$		0			0
$643$		−	100081.i		−	0.242063i	−0.992649	−	0.121032i	$-0.961380\pi$
$643$		−	100081.i		−	0.242063i	0.992649	−	0.121032i	$-0.0386203\pi$
$644$			706248.i			1.70288i
$645$		0			0
$646$	153395.			0.367575
$647$	−315310.			−0.753232			−0.376616	−	0.926370i	$-0.622912\pi$
$647$	−315310.			−0.753232			−0.376616	+	0.926370i	$0.622912\pi$
$648$		0			0
$649$	91298.7			0.216758
$650$	539696.	+	167254.i	1.27739	+	0.395866i
$651$		0			0
$652$			27230.6i			0.0640563i
$653$	−156142.			−0.366178			−0.183089	−	0.983096i	$-0.558610\pi$
$653$	−156142.			−0.366178			−0.183089	+	0.983096i	$0.558610\pi$
$654$		0			0
$655$	254478.	+	345282.i	0.593154	+	0.804805i
$656$		−	353646.i		−	0.821791i
$657$		0			0
$658$		−	1.82399e6i		−	4.21279i
$659$			258381.i			0.594963i	0.954728	+	0.297482i	$0.0961467\pi$
$659$			258381.i			0.594963i	−0.954728	+	0.297482i	$0.903853\pi$
$660$		0			0
$661$	311759.			0.713535			0.356768	−	0.934193i	$-0.383879\pi$
$661$	311759.			0.713535			0.356768	+	0.934193i	$0.383879\pi$
$662$	−376098.			−0.858192
$663$		0			0
$664$	266199.			0.603768
$665$	263403.	+	357391.i	0.595631	+	0.808166i
$666$		0			0
$667$		−	72420.1i		−	0.162782i
$668$	−1.63525e6			−3.66464
$669$		0			0
$670$	428027.	+	580756.i	0.953501	+	1.29373i
$671$		−	290971.i		−	0.646256i
$672$		0			0
$673$		−	648563.i		−	1.43193i	−0.698136	−	0.715965i	$-0.745987\pi$
$673$		−	648563.i		−	1.43193i	0.698136	−	0.715965i	$-0.254013\pi$
$674$			373290.i			0.821726i
$675$		0			0
$676$	567455.			1.24176
$677$	−249966.			−0.545386			−0.272693	−	0.962101i	$-0.587914\pi$
$677$	−249966.			−0.545386			−0.272693	+	0.962101i	$0.587914\pi$
$678$		0			0
$679$	−511350.			−1.10912
$680$	−295751.	+	217974.i	−0.639601	+	0.471396i
$681$		0			0
$682$		−	8660.48i		−	0.0186197i
$683$	−100280.			−0.214968			−0.107484	−	0.994207i	$-0.534279\pi$
$683$	−100280.			−0.214968			−0.107484	+	0.994207i	$0.534279\pi$
$684$		0			0
$685$	−529302.	+	390104.i	−1.12803	+	0.831379i
$686$			39987.9i			0.0849729i
$687$		0			0
$688$		−	72580.4i		−	0.153335i
$689$			351507.i			0.740449i
$690$		0			0
$691$	504589.			1.05677			0.528387	−	0.849004i	$-0.322797\pi$
$691$	504589.			1.05677			0.528387	+	0.849004i	$0.322797\pi$
$692$	−1.07164e6			−2.23787
$693$		0			0
$694$	1.31121e6			2.72240
$695$	−462010.	+	340508.i	−0.956492	+	0.704950i
$696$		0			0
$697$			39084.1i			0.0804516i
$698$	−537566.			−1.10337
$699$		0			0
$700$	−1.68285e6	−	521520.i	−3.43438	−	1.06433i
$701$		−	309177.i		−	0.629175i	−0.949228	−	0.314588i	$-0.898134\pi$
$701$		−	309177.i		−	0.629175i	0.949228	−	0.314588i	$-0.101866\pi$
$702$		0			0
$703$		−	629367.i		−	1.27348i
$704$		−	384706.i		−	0.776218i
$705$		0			0
$706$	554639.			1.11276
$707$	−267185.			−0.534532
$708$		0			0
$709$	−691554.			−1.37573			−0.687866	−	0.725838i	$-0.741452\pi$
$709$	−691554.			−1.37573			−0.687866	+	0.725838i	$0.741452\pi$
$710$	−110085.	−	149366.i	−0.218380	−	0.296303i
$711$		0			0
$712$			2.34762e6i			4.63092i
$713$	−5544.97			−0.0109074
$714$		0			0
$715$	126319.	−	93098.9i	0.247090	−	0.182109i
$716$			1.71489e6i			3.34510i
$717$		0			0
$718$			1.44857e6i			2.80989i
$719$		−	903805.i		−	1.74830i	−0.485652	−	0.874152i	$-0.661418\pi$
$719$		−	903805.i		−	1.74830i	0.485652	−	0.874152i	$-0.338582\pi$
$720$		0			0
$721$	−1.27598e6			−2.45456
$722$	−493018.			−0.945776
$723$		0			0
$724$	1.86840e6			3.56446
$725$	172563.	+	53477.8i	0.328300	+	0.101741i
$726$		0			0
$727$		−	567672.i		−	1.07406i	−0.843563	−	0.537030i	$-0.819546\pi$
$727$		−	567672.i		−	1.07406i	0.843563	−	0.537030i	$-0.180454\pi$
$728$	1.53809e6			2.90215
$729$		0			0
$730$	670035.	+	909119.i	1.25734	+	1.70598i
$731$			8021.41i			0.0150112i
$732$		0			0
$733$			580304.i			1.08006i	0.841646	+	0.540030i	$0.181587\pi$
$733$			580304.i			1.08006i	−0.841646	+	0.540030i	$0.818413\pi$
$734$		−	381358.i		−	0.707850i
$735$		0			0
$736$	−634579.			−1.17147
$737$	200364.			0.368880
$738$		0			0
$739$	−257218.			−0.470991			−0.235495	−	0.971875i	$-0.575671\pi$
$739$	−257218.			−0.470991			−0.235495	+	0.971875i	$0.575671\pi$
$740$	1.48175e6	+	2.01047e6i	2.70590	+	3.67143i
$741$		0			0
$742$		−	1.53056e6i		−	2.77999i
$743$	−24500.2			−0.0443805			−0.0221902	−	0.999754i	$-0.507064\pi$
$743$	−24500.2			−0.0443805			−0.0221902	+	0.999754i	$0.507064\pi$
$744$		0			0
$745$	447970.	+	607816.i	0.807117	+	1.09512i
$746$			521781.i			0.937584i
$747$		0			0
$748$			169056.i			0.302153i
$749$			503225.i			0.897012i
$750$		0			0
$751$	−831719.			−1.47468			−0.737339	−	0.675523i	$-0.763918\pi$
$751$	−831719.			−1.47468			−0.737339	+	0.675523i	$0.763918\pi$
$752$	2.52942e6			4.47285
$753$		0			0
$754$	−261314.			−0.459642
$755$	−449933.	+	331608.i	−0.789322	+	0.581743i
$756$		0			0
$757$			181974.i			0.317553i	0.987315	+	0.158777i	$0.0507550\pi$
$757$			181974.i			0.317553i	−0.987315	+	0.158777i	$0.949245\pi$
$758$	−609803.			−1.06133
$759$		0			0
$760$	−935756.	+	689667.i	−1.62008	+	1.19402i
$761$			558208.i			0.963889i	0.876202	+	0.481944i	$0.160069\pi$
$761$			558208.i			0.963889i	−0.876202	+	0.481944i	$0.839931\pi$
$762$		0			0
$763$		−	28344.5i		−	0.0486877i
$764$		−	200251.i		−	0.343075i
$765$		0			0
$766$	720601.			1.22811
$767$	−210923.			−0.358537
$768$		0			0
$769$	−144332.			−0.244068			−0.122034	−	0.992526i	$-0.538942\pi$
$769$	−144332.			−0.244068			−0.122034	+	0.992526i	$0.538942\pi$
$770$	−550028.	+	405379.i	−0.927691	+	0.683723i
$771$		0			0
$772$			1.78039e6i			2.98731i
$773$	−629040.			−1.05274			−0.526368	−	0.850257i	$-0.676447\pi$
$773$	−629040.			−1.05274			−0.526368	+	0.850257i	$0.676447\pi$
$774$		0			0
$775$	4094.61	−	13212.5i	0.00681726	−	0.0219980i
$776$		−	1.33887e6i		−	2.22338i
$777$		0			0
$778$			249294.i			0.411863i
$779$			123662.i			0.203780i
$780$		0			0
$781$	−51532.1			−0.0844842
$782$	151150.			0.247170
$783$		0			0
$784$	−1.80130e6			−2.93059
$785$	−409785.	−	556005.i	−0.664992	−	0.902277i
$786$		0			0
$787$		−	1.12674e6i		−	1.81917i	−0.415519	−	0.909585i	$-0.636400\pi$
$787$		−	1.12674e6i		−	1.81917i	0.415519	−	0.909585i	$-0.363600\pi$
$788$	−674886.			−1.08687
$789$		0			0
$790$	9562.14	−	7047.45i	0.0153215	−	0.0112922i
$791$			622728.i			0.995280i
$792$		0			0
$793$			672216.i			1.06896i
$794$		−	1.35294e6i		−	2.14603i
$795$		0			0
$796$	−2.30047e6			−3.63070
$797$	260076.			0.409433			0.204717	−	0.978821i	$-0.434373\pi$
$797$	260076.			0.409433			0.204717	+	0.978821i	$0.434373\pi$
$798$		0			0
$799$	−279545.			−0.437883
$800$	468598.	−	1.51208e6i	0.732184	−	2.36262i
$801$		0			0
$802$		−	1.03409e6i		−	1.60771i
$803$	313651.			0.486424
$804$		0			0
$805$	259549.	+	352161.i	0.400522	+	0.543438i
$806$			20007.9i			0.0307987i
$807$		0			0
$808$		−	699569.i		−	1.07154i
$809$			278756.i			0.425920i	0.977061	+	0.212960i	$0.0683103\pi$
$809$			278756.i			0.425920i	−0.977061	+	0.212960i	$0.931690\pi$
$810$		0			0
$811$	384901.			0.585204			0.292602	−	0.956234i	$-0.405479\pi$
$811$	384901.			0.585204			0.292602	+	0.956234i	$0.405479\pi$
$812$	814813.			1.23579
$813$		0			0
$814$	968602.			1.46183
$815$	10007.3	+	13578.2i	0.0150662	+	0.0204421i
$816$		0			0
$817$			25379.7i			0.0380226i
$818$	258389.			0.386161
$819$		0			0
$820$	−291143.	−	395030.i	−0.432991	−	0.587492i
$821$		−	345314.i		−	0.512304i	−0.966637	−	0.256152i	$-0.917545\pi$
$821$		−	345314.i		−	0.512304i	0.966637	−	0.256152i	$-0.0824548\pi$
$822$		0			0
$823$			1.14509e6i			1.69060i	0.534296	+	0.845298i	$0.320577\pi$
$823$			1.14509e6i			1.69060i	−0.534296	+	0.845298i	$0.679423\pi$
$824$		−	3.34089e6i		−	4.92048i
$825$		0			0
$826$	918421.			1.34611
$827$	−62165.6			−0.0908948			−0.0454474	−	0.998967i	$-0.514471\pi$
$827$	−62165.6			−0.0908948			−0.0454474	+	0.998967i	$0.514471\pi$
$828$		0			0
$829$	710786.			1.03426			0.517130	−	0.855907i	$-0.327000\pi$
$829$	710786.			1.03426			0.517130	+	0.855907i	$0.327000\pi$
$830$	219922.	−	162086.i	0.319237	−	0.235282i
$831$		0			0
$832$			888769.i			1.28393i
$833$	199076.			0.286898
$834$		0			0
$835$	−815397.	+	600961.i	−1.16949	+	0.861932i
$836$			534892.i			0.765338i
$837$		0			0
$838$			1.12994e6i			1.60905i
$839$			4499.46i			0.00639199i	0.999995	+	0.00319600i	$0.00101732\pi$
$839$			4499.46i			0.00639199i	−0.999995	+	0.00319600i	$0.998983\pi$
$840$		0			0
$841$	623728.			0.881868
$842$	−819241.			−1.15555
$843$		0			0
$844$	1.26534e6			1.77632
$845$	282954.	−	208542.i	0.396280	−	0.292065i
$846$		0			0
$847$		−	832832.i		−	1.16089i
$848$	2.12251e6			2.95160
$849$		0			0
$850$	−111615.	+	360161.i	−0.154484	+	0.498492i
$851$		−	620157.i		−	0.856333i
$852$		0			0
$853$			642383.i			0.882869i	0.897294	+	0.441434i	$0.145530\pi$
$853$			642383.i			0.882869i	−0.897294	+	0.441434i	$0.854470\pi$
$854$		−	2.92702e6i		−	4.01338i
$855$		0			0
$856$	−1.31759e6			−1.79818
$857$	458409.			0.624154			0.312077	−	0.950057i	$-0.398975\pi$
$857$	458409.			0.624154			0.312077	+	0.950057i	$0.398975\pi$
$858$		0			0
$859$	251745.			0.341173			0.170587	−	0.985343i	$-0.445434\pi$
$859$	251745.			0.341173			0.170587	+	0.985343i	$0.445434\pi$
$860$	−59752.6	−	81073.7i	−0.0807904	−	0.109618i
$861$		0			0
$862$		−	2.58541e6i		−	3.47948i
$863$	−167993.			−0.225564			−0.112782	−	0.993620i	$-0.535976\pi$
$863$	−167993.			−0.225564			−0.112782	+	0.993620i	$0.535976\pi$
$864$		0			0
$865$	−534357.	+	393830.i	−0.714167	+	0.526352i
$866$		−	320714.i		−	0.427644i
$867$		0			0
$868$		−	62387.5i		−	0.0828053i
$869$		−	3298.99i		−	0.00436859i
$870$		0			0
$871$	−462892.			−0.610159
$872$	74214.2			0.0976010
$873$		0			0
$874$	478238.			0.626067
$875$	−1.03079e6	+	358403.i	−1.34634	+	0.468118i
$876$		0			0
$877$			122287.i			0.158995i	0.996835	+	0.0794973i	$0.0253315\pi$
$877$			122287.i			0.158995i	−0.996835	+	0.0794973i	$0.974668\pi$
$878$	1.50058e6			1.94656
$879$		0			0
$880$	−562161.	−	762752.i	−0.725931	−	0.984959i
$881$			1.16563e6i			1.50179i	0.660420	+	0.750897i	$0.270378\pi$
$881$			1.16563e6i			1.50179i	−0.660420	+	0.750897i	$0.729622\pi$
$882$		0			0
$883$		−	417334.i		−	0.535257i	−0.963522	−	0.267629i	$-0.913760\pi$
$883$		−	417334.i		−	0.535257i	0.963522	−	0.267629i	$-0.0862400\pi$
$884$		−	390562.i		−	0.499788i
$885$		0			0
$886$	−752235.			−0.958265
$887$	324198.			0.412062			0.206031	−	0.978545i	$-0.433945\pi$
$887$	324198.			0.412062			0.206031	+	0.978545i	$0.433945\pi$
$888$		0			0
$889$	−1.42743e6			−1.80614
$890$	1.42944e6	+	1.93950e6i	1.80462	+	2.44855i
$891$		0			0
$892$		−	1.92805e6i		−	2.42320i
$893$	−884478.			−1.10913
$894$		0			0
$895$	630227.	+	855106.i	0.786775	+	1.06751i
$896$		−	1.03948e6i		−	1.29480i
$897$		0			0
$898$			197521.i			0.244940i
$899$			6397.34i			0.00791554i
$900$		0			0
$901$	−234574.			−0.288956
$902$	−190317.			−0.233918
$903$		0			0
$904$	−1.63048e6			−1.99517
$905$	931655.	−	686644.i	1.13752	−	0.838368i
$906$		0			0
$907$		−	311144.i		−	0.378223i	−0.981956	−	0.189111i	$-0.939439\pi$
$907$		−	311144.i		−	0.378223i	0.981956	−	0.189111i	$-0.0605607\pi$
$908$	−751674.			−0.911712
$909$		0			0
$910$	1.27070e6	−	936530.i	1.53448	−	1.13094i
$911$			974766.i			1.17453i	0.809395	+	0.587264i	$0.199795\pi$
$911$			974766.i			1.17453i	−0.809395	+	0.587264i	$0.800205\pi$
$912$		0			0
$913$		−	75874.3i		−	0.0910234i
$914$		−	495900.i		−	0.593611i
$915$		0			0
$916$	3.79395e6			4.52169
$917$	1.19833e6			1.42507
$918$		0			0
$919$	917855.			1.08678			0.543392	−	0.839479i	$-0.317140\pi$
$919$	917855.			1.08678			0.543392	+	0.839479i	$0.317140\pi$
$920$	−922062.	+	679574.i	−1.08939	+	0.802900i
$921$		0			0
$922$		−	579690.i		−	0.681921i
$923$	119052.			0.139744
$924$		0			0
$925$	1.47771e6	+	457948.i	1.72705	+	0.535220i
$926$		−	1.22244e6i		−	1.42562i
$927$		0			0
$928$			732127.i			0.850141i
$929$		−	276565.i		−	0.320454i	−0.987080	−	0.160227i	$-0.948777\pi$
$929$		−	276565.i		−	0.320454i	0.987080	−	0.160227i	$-0.0512226\pi$
$930$		0			0
$931$	629874.			0.726698
$932$	−2.90797e6			−3.34779
$933$		0			0
$934$	868861.			0.995994
$935$	62128.6	+	84297.5i	0.0710671	+	0.0964254i
$936$		0			0
$937$		−	1.51274e6i		−	1.72300i	−0.507755	−	0.861502i	$-0.669524\pi$
$937$		−	1.51274e6i		−	1.72300i	0.507755	−	0.861502i	$-0.330476\pi$
$938$	2.01556e6			2.29082
$939$		0			0
$940$	2.82541e6	−	2.08237e6i	3.19761	−	2.35669i
$941$			540106.i			0.609958i	0.952359	+	0.304979i	$0.0986494\pi$
$941$			540106.i			0.609958i	−0.952359	+	0.304979i	$0.901351\pi$
$942$		0			0
$943$			121852.i			0.137028i
$944$			1.27362e6i			1.42921i
$945$		0			0
$946$	−39059.5			−0.0436460
$947$	−1.10857e6			−1.23613			−0.618066	−	0.786126i	$-0.712084\pi$
$947$	−1.10857e6			−1.23613			−0.618066	+	0.786126i	$0.712084\pi$
$948$		0			0
$949$	−724613.			−0.804588
$950$	−353149.	+	1.13955e6i	−0.391301	+	1.26265i
$951$		0			0
$952$			1.02643e6i			1.13255i
$953$	−1.08856e6			−1.19857			−0.599287	−	0.800534i	$-0.704549\pi$
$953$	−1.08856e6			−1.19857			−0.599287	+	0.800534i	$0.704549\pi$
$954$		0			0
$955$	−73593.1	−	99852.8i	−0.0806920	−	0.109485i
$956$		−	2.16437e6i		−	2.36818i
$957$		0			0
$958$		−	3.22672e6i		−	3.51585i
$959$			1.83699e6i			1.99742i
$960$		0			0
$961$	−923031.			−0.999470
$962$	−2.23772e6			−2.41799
$963$		0			0
$964$	−255283.			−0.274706
$965$	654300.	+	887769.i	0.702623	+	0.953335i
$966$		0			0
$967$		−	818736.i		−	0.875570i	−0.899080	−	0.437785i	$-0.855763\pi$
$967$		−	818736.i		−	0.875570i	0.899080	−	0.437785i	$-0.144237\pi$
$968$	2.18060e6			2.32715
$969$		0			0
$970$	−815222.	−	1.10611e6i	−0.866428	−	1.17559i
$971$			914500.i			0.969941i	0.874530	+	0.484971i	$0.161170\pi$
$971$			914500.i			0.969941i	−0.874530	+	0.484971i	$0.838830\pi$
$972$		0			0
$973$			1.60344e6i			1.69367i
$974$		−	876300.i		−	0.923708i
$975$		0			0
$976$	4.05906e6			4.26114
$977$	333226.			0.349100			0.174550	−	0.984648i	$-0.444153\pi$
$977$	333226.			0.349100			0.174550	+	0.984648i	$0.444153\pi$
$978$		0			0
$979$	669137.			0.698152
$980$	−2.01209e6	+	1.48294e6i	−2.09505	+	1.54409i
$981$		0			0
$982$			796636.i			0.826108i
$983$	−109944.			−0.113780			−0.0568900	−	0.998380i	$-0.518118\pi$
$983$	−109944.			−0.113780			−0.0568900	+	0.998380i	$0.518118\pi$
$984$		0			0
$985$	−336523.	+	248023.i	−0.346851	+	0.255634i
$986$		−	174385.i		−	0.179372i
$987$		0			0
$988$		−	1.23574e6i		−	1.26594i
$989$			25008.3i			0.0255677i
$990$		0			0
$991$	−1.49460e6			−1.52187			−0.760933	−	0.648830i	$-0.775259\pi$
$991$	−1.49460e6			−1.52187			−0.760933	+	0.648830i	$0.775259\pi$
$992$	56056.6			0.0569643
$993$		0			0
$994$	−518388.			−0.524665
$995$	−1.14710e6	+	845431.i	−1.15866	+	0.853949i
$996$		0			0
$997$			1.28231e6i			1.29004i	0.764168	+	0.645018i	$0.223150\pi$
$997$			1.28231e6i			1.29004i	−0.764168	+	0.645018i	$0.776850\pi$
$998$	834846.			0.838196
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.d.a.404.43		44
3.2	odd	2	inner	405.5.d.a.404.1		44
5.4	even	2	inner	405.5.d.a.404.2		44
9.2	odd	6		135.5.h.a.44.22		44
9.4	even	3		135.5.h.a.89.1		44
9.5	odd	6		45.5.h.a.29.22	yes	44
9.7	even	3		45.5.h.a.14.1	✓	44
15.14	odd	2	inner	405.5.d.a.404.44		44
45.4	even	6		135.5.h.a.89.22		44
45.14	odd	6		45.5.h.a.29.1	yes	44
45.29	odd	6		135.5.h.a.44.1		44
45.34	even	6		45.5.h.a.14.22	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.h.a.14.1	✓	44	9.7	even	3
45.5.h.a.14.22	yes	44	45.34	even	6
45.5.h.a.29.1	yes	44	45.14	odd	6
45.5.h.a.29.22	yes	44	9.5	odd	6
135.5.h.a.44.1		44	45.29	odd	6
135.5.h.a.44.22		44	9.2	odd	6
135.5.h.a.89.1		44	9.4	even	3
135.5.h.a.89.22		44	45.4	even	6
405.5.d.a.404.1		44	3.2	odd	2	inner
405.5.d.a.404.2		44	5.4	even	2	inner
405.5.d.a.404.43		44	1.1	even	1	trivial
405.5.d.a.404.44		44	15.14	odd	2	inner

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.50729 q^{2} +40.3595 q^{4} +(20.1247 - 14.8322i) q^{5} -69.8446i q^{7} +182.874 q^{8} +(151.082 - 111.350i) q^{10} -52.1241i q^{11} +120.420i q^{13} -524.344i q^{14} +727.134 q^{16} -80.3611 q^{17} -254.262 q^{19} +(812.223 - 598.621i) q^{20} -391.311i q^{22} -250.541 q^{23} +(185.009 - 596.990i) q^{25} +904.028i q^{26} -2818.89i q^{28} +289.055i q^{29} +22.1320 q^{31} +2532.83 q^{32} -603.294 q^{34} +(-1035.95 - 1405.60i) q^{35} +2475.27i q^{37} -1908.82 q^{38} +(3680.28 - 2712.43i) q^{40} -486.356i q^{41} -99.8171i q^{43} -2103.70i q^{44} -1880.89 q^{46} +3478.61 q^{47} -2477.26 q^{49} +(1388.92 - 4481.78i) q^{50} +4860.09i q^{52} +2919.00 q^{53} +(-773.118 - 1048.98i) q^{55} -12772.7i q^{56} +2170.02i q^{58} +1751.56i q^{59} +5582.26 q^{61} +166.151 q^{62} +7380.58 q^{64} +(1786.10 + 2423.42i) q^{65} +3843.98i q^{67} -3243.33 q^{68} +(-7777.19 - 10552.3i) q^{70} -988.642i q^{71} +6017.38i q^{73} +18582.6i q^{74} -10261.9 q^{76} -3640.59 q^{77} +63.2910 q^{79} +(14633.4 - 10785.0i) q^{80} -3651.22i q^{82} +1455.65 q^{83} +(-1617.25 + 1191.94i) q^{85} -749.356i q^{86} -9532.13i q^{88} +12837.4i q^{89} +8410.68 q^{91} -10111.7 q^{92} +26114.9 q^{94} +(-5116.95 + 3771.28i) q^{95} -7321.26i q^{97} -18597.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) 44 * q + 324 * q^4 + 28 * q^10 + 2116 * q^16 - 8 * q^19 + 296 * q^25 + 2224 * q^31 + 872 * q^34 + 1700 * q^40 - 5668 * q^46 - 10792 * q^49 - 3072 * q^55 - 5564 * q^61 + 8348 * q^64 - 9564 * q^70 + 3552 * q^76 - 15800 * q^79 - 13532 * q^85 + 1560 * q^91 + 37652 * q^94

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