LMFDB - Embedded newform 405.5.d.a.404.7

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			404.7
Character	$\chi$	$=$	405.404
Dual form			405.5.d.a.404.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.74722 q^{2} +17.0305 q^{4} +(-17.1405 - 18.1989i) q^{5} -47.8220i q^{7} -5.92246 q^{8} +(98.5104 + 104.593i) q^{10} -84.5766i q^{11} +223.417i q^{13} +274.843i q^{14} -238.450 q^{16} -434.998 q^{17} +378.461 q^{19} +(-291.912 - 309.937i) q^{20} +486.080i q^{22} -653.651 q^{23} +(-37.4035 + 623.880i) q^{25} -1284.03i q^{26} -814.433i q^{28} -496.753i q^{29} -302.995 q^{31} +1465.18 q^{32} +2500.03 q^{34} +(-870.311 + 819.695i) q^{35} +55.6914i q^{37} -2175.09 q^{38} +(101.514 + 107.783i) q^{40} -475.050i q^{41} -816.458i q^{43} -1440.38i q^{44} +3756.67 q^{46} -870.464 q^{47} +114.054 q^{49} +(214.966 - 3585.57i) q^{50} +3804.90i q^{52} -2339.28 q^{53} +(-1539.21 + 1449.69i) q^{55} +283.224i q^{56} +2854.95i q^{58} +1177.42i q^{59} -6913.03 q^{61} +1741.38 q^{62} -4605.53 q^{64} +(4065.96 - 3829.49i) q^{65} -3875.51i q^{67} -7408.23 q^{68} +(5001.86 - 4710.97i) q^{70} -5822.30i q^{71} +6443.82i q^{73} -320.071i q^{74} +6445.37 q^{76} -4044.62 q^{77} +4893.43 q^{79} +(4087.17 + 4339.54i) q^{80} +2730.21i q^{82} +6514.72 q^{83} +(7456.10 + 7916.50i) q^{85} +4692.36i q^{86} +500.901i q^{88} +13898.7i q^{89} +10684.3 q^{91} -11132.0 q^{92} +5002.75 q^{94} +(-6487.02 - 6887.59i) q^{95} +13488.8i q^{97} -655.493 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.74722			−1.43680			−0.718402	−	0.695628i	$-0.755126\pi$
$2$	−5.74722			−1.43680			−0.718402	+	0.695628i	$0.755126\pi$
$3$		0			0
$3$		0			0
$4$	17.0305			1.06441
$5$	−17.1405	−	18.1989i	−0.685622	−	0.727958i
$5$	−17.1405	−	18.1989i	−0.685622	−	0.727958i
$6$		0			0
$7$		−	47.8220i		−	0.975960i	−0.872855	−	0.487980i	$-0.837734\pi$
$7$		−	47.8220i		−	0.975960i	0.872855	−	0.487980i	$-0.162266\pi$
$8$	−5.92246			−0.0925384
$9$		0			0
$10$	98.5104	+	104.593i	0.985104	+	1.04593i
$11$		−	84.5766i		−	0.698980i	−0.936940	−	0.349490i	$-0.886355\pi$
$11$		−	84.5766i		−	0.698980i	0.936940	−	0.349490i	$-0.113645\pi$
$12$		0			0
$13$			223.417i			1.32200i	0.750388	+	0.660998i	$0.229867\pi$
$13$			223.417i			1.32200i	−0.750388	+	0.660998i	$0.770133\pi$
$14$			274.843i			1.40226i
$15$		0			0
$16$	−238.450			−0.931446
$17$	−434.998			−1.50518			−0.752591	−	0.658488i	$-0.771196\pi$
$17$	−434.998			−1.50518			−0.752591	+	0.658488i	$0.771196\pi$
$18$		0			0
$19$	378.461			1.04837			0.524184	−	0.851605i	$-0.324371\pi$
$19$	378.461			1.04837			0.524184	+	0.851605i	$0.324371\pi$
$20$	−291.912	−	309.937i	−0.729780	−	0.774843i
$21$		0			0
$22$			486.080i			1.00430i
$23$	−653.651			−1.23564			−0.617818	−	0.786322i	$-0.711983\pi$
$23$	−653.651			−1.23564			−0.617818	+	0.786322i	$0.711983\pi$
$24$		0			0
$25$	−37.4035	+	623.880i	−0.0598457	+	0.998208i
$26$		−	1284.03i		−	1.89945i
$27$		0			0
$28$		−	814.433i		−	1.03882i
$29$		−	496.753i		−	0.590670i	−0.955394	−	0.295335i	$-0.904569\pi$
$29$		−	496.753i		−	0.590670i	0.955394	−	0.295335i	$-0.0954312\pi$
$30$		0			0
$31$	−302.995			−0.315292			−0.157646	−	0.987496i	$-0.550390\pi$
$31$	−302.995			−0.315292			−0.157646	+	0.987496i	$0.550390\pi$
$32$	1465.18			1.43084
$33$		0			0
$34$	2500.03			2.16265
$35$	−870.311	+	819.695i	−0.710458	+	0.669139i
$36$		0			0
$37$			55.6914i			0.0406804i	0.999793	+	0.0203402i	$0.00647493\pi$
$37$			55.6914i			0.0406804i	−0.999793	+	0.0203402i	$0.993525\pi$
$38$	−2175.09			−1.50630
$39$		0			0
$40$	101.514	+	107.783i	0.0634464	+	0.0673641i
$41$		−	475.050i		−	0.282600i	−0.989967	−	0.141300i	$-0.954872\pi$
$41$		−	475.050i		−	0.282600i	0.989967	−	0.141300i	$-0.0451281\pi$
$42$		0			0
$43$		−	816.458i		−	0.441567i	−0.975323	−	0.220784i	$-0.929139\pi$
$43$		−	816.458i		−	0.441567i	0.975323	−	0.220784i	$-0.0708615\pi$
$44$		−	1440.38i		−	0.743999i
$45$		0			0
$46$	3756.67			1.77537
$47$	−870.464			−0.394053			−0.197027	−	0.980398i	$-0.563129\pi$
$47$	−870.464			−0.394053			−0.197027	+	0.980398i	$0.563129\pi$
$48$		0			0
$49$	114.054			0.0475027
$50$	214.966	−	3585.57i	0.0859865	−	1.43423i
$51$		0			0
$52$			3804.90i			1.40714i
$53$	−2339.28			−0.832780			−0.416390	−	0.909186i	$-0.636705\pi$
$53$	−2339.28			−0.832780			−0.416390	+	0.909186i	$0.636705\pi$
$54$		0			0
$55$	−1539.21	+	1449.69i	−0.508828	+	0.479236i
$56$			283.224i			0.0903138i
$57$		0			0
$58$			2854.95i			0.848677i
$59$			1177.42i			0.338242i	0.985595	+	0.169121i	$0.0540929\pi$
$59$			1177.42i			0.338242i	−0.985595	+	0.169121i	$0.945907\pi$
$60$		0			0
$61$	−6913.03			−1.85784			−0.928921	−	0.370277i	$-0.879263\pi$
$61$	−6913.03			−1.85784			−0.928921	+	0.370277i	$0.879263\pi$
$62$	1741.38			0.453013
$63$		0			0
$64$	−4605.53			−1.12440
$65$	4065.96	−	3829.49i	0.962357	−	0.906389i
$66$		0			0
$67$		−	3875.51i		−	0.863334i	−0.902033	−	0.431667i	$-0.857926\pi$
$67$		−	3875.51i		−	0.863334i	0.902033	−	0.431667i	$-0.142074\pi$
$68$	−7408.23			−1.60213
$69$		0			0
$70$	5001.86	−	4710.97i	1.02079	−	0.961422i
$71$		−	5822.30i		−	1.15499i	−0.816394	−	0.577495i	$-0.804030\pi$
$71$		−	5822.30i		−	1.15499i	0.816394	−	0.577495i	$-0.195970\pi$
$72$		0			0
$73$			6443.82i			1.20920i	0.796530	+	0.604599i	$0.206667\pi$
$73$			6443.82i			1.20920i	−0.796530	+	0.604599i	$0.793333\pi$
$74$		−	320.071i		−	0.0584497i
$75$		0			0
$76$	6445.37			1.11589
$77$	−4044.62			−0.682176
$78$		0			0
$79$	4893.43			0.784078			0.392039	−	0.919949i	$-0.371770\pi$
$79$	4893.43			0.784078			0.392039	+	0.919949i	$0.371770\pi$
$80$	4087.17	+	4339.54i	0.638620	+	0.678054i
$81$		0			0
$82$			2730.21i			0.406040i
$83$	6514.72			0.945670			0.472835	−	0.881151i	$-0.343231\pi$
$83$	6514.72			0.945670			0.472835	+	0.881151i	$0.343231\pi$
$84$		0			0
$85$	7456.10	+	7916.50i	1.03199	+	1.09571i
$86$			4692.36i			0.634446i
$87$		0			0
$88$			500.901i			0.0646825i
$89$			13898.7i			1.75467i	0.479881	+	0.877333i	$0.340680\pi$
$89$			13898.7i			1.75467i	−0.479881	+	0.877333i	$0.659320\pi$
$90$		0			0
$91$	10684.3			1.29021
$92$	−11132.0			−1.31522
$93$		0			0
$94$	5002.75			0.566178
$95$	−6487.02	−	6887.59i	−0.718783	−	0.763167i
$96$		0			0
$97$			13488.8i			1.43361i	0.697274	+	0.716805i	$0.254396\pi$
$97$			13488.8i			1.43361i	−0.697274	+	0.716805i	$0.745604\pi$
$98$	−655.493			−0.0682521
$99$		0			0
$100$	−637.001	+	10625.0i	−0.0637001	+	1.06250i
$101$			18214.1i			1.78552i	0.450533	+	0.892760i	$0.351234\pi$
$101$			18214.1i			1.78552i	−0.450533	+	0.892760i	$0.648766\pi$
$102$		0			0
$103$		−	12938.7i		−	1.21960i	−0.792555	−	0.609800i	$-0.791250\pi$
$103$		−	12938.7i		−	1.21960i	0.792555	−	0.609800i	$-0.208750\pi$
$104$		−	1323.18i		−	0.122335i
$105$		0			0
$106$	13444.3			1.19654
$107$	−1372.35			−0.119867			−0.0599333	−	0.998202i	$-0.519089\pi$
$107$	−1372.35			−0.119867			−0.0599333	+	0.998202i	$0.519089\pi$
$108$		0			0
$109$	8324.99			0.700698			0.350349	−	0.936619i	$-0.386063\pi$
$109$	8324.99			0.700698			0.350349	+	0.936619i	$0.386063\pi$
$110$	8846.15	−	8331.68i	0.731086	−	0.688568i
$111$		0			0
$112$			11403.2i			0.909054i
$113$	20812.7			1.62994			0.814972	−	0.579501i	$-0.196752\pi$
$113$	20812.7			1.62994			0.814972	+	0.579501i	$0.196752\pi$
$114$		0			0
$115$	11203.9	+	11895.8i	0.847178	+	0.899490i
$116$		−	8459.96i		−	0.628712i
$117$		0			0
$118$		−	6766.89i		−	0.485987i
$119$			20802.5i			1.46900i
$120$		0			0
$121$	7487.80			0.511427
$122$	39730.7			2.66936
$123$		0			0
$124$	−5160.16			−0.335598
$125$	11995.1	−	10012.9i	0.767685	−	0.640828i
$126$		0			0
$127$		−	3442.90i		−	0.213460i	−0.994288	−	0.106730i	$-0.965962\pi$
$127$		−	3442.90i		−	0.213460i	0.994288	−	0.106730i	$-0.0340380\pi$
$128$	3026.01			0.184693
$129$		0			0
$130$	−23367.9	+	22008.9i	−1.38272	+	1.30230i
$131$			5512.51i			0.321223i	0.987018	+	0.160611i	$0.0513466\pi$
$131$			5512.51i			0.321223i	−0.987018	+	0.160611i	$0.948653\pi$
$132$		0			0
$133$		−	18098.8i		−	1.02316i
$134$			22273.4i			1.24044i
$135$		0			0
$136$	2576.26			0.139287
$137$	4334.15			0.230921			0.115460	−	0.993312i	$-0.463166\pi$
$137$	4334.15			0.230921			0.115460	+	0.993312i	$0.463166\pi$
$138$		0			0
$139$	3374.05			0.174631			0.0873157	−	0.996181i	$-0.472171\pi$
$139$	3374.05			0.174631			0.0873157	+	0.996181i	$0.472171\pi$
$140$	−14821.8	+	13959.8i	−0.756215	+	0.712236i
$141$		0			0
$142$			33462.0i			1.65949i
$143$	18895.9			0.924048
$144$		0			0
$145$	−9040.39	+	8514.62i	−0.429983	+	0.404976i
$146$		−	37034.0i		−	1.73738i
$147$		0			0
$148$			948.452i			0.0433004i
$149$			33535.9i			1.51056i	0.655403	+	0.755279i	$0.272499\pi$
$149$			33535.9i			1.51056i	−0.655403	+	0.755279i	$0.727501\pi$
$150$		0			0
$151$	1034.59			0.0453746			0.0226873	−	0.999743i	$-0.492778\pi$
$151$	1034.59			0.0453746			0.0226873	+	0.999743i	$0.492778\pi$
$152$	−2241.42			−0.0970143
$153$		0			0
$154$	23245.3			0.980154
$155$	5193.51	+	5514.20i	0.216171	+	0.229519i
$156$		0			0
$157$			4545.86i			0.184424i	0.995739	+	0.0922119i	$0.0293937\pi$
$157$			4545.86i			0.184424i	−0.995739	+	0.0922119i	$0.970606\pi$
$158$	−28123.6			−1.12657
$159$		0			0
$160$	−25114.1	−	26664.8i	−0.981018	−	1.04159i
$161$			31258.9i			1.20593i
$162$		0			0
$163$		−	35857.6i		−	1.34960i	−0.737999	−	0.674802i	$-0.764229\pi$
$163$		−	35857.6i		−	1.34960i	0.737999	−	0.674802i	$-0.235771\pi$
$164$		−	8090.33i		−	0.300801i
$165$		0			0
$166$	−37441.5			−1.35874
$167$	22217.6			0.796644			0.398322	−	0.917246i	$-0.369593\pi$
$167$	22217.6			0.796644			0.398322	+	0.917246i	$0.369593\pi$
$168$		0			0
$169$	−21354.2			−0.747671
$170$	−42851.8	−	45497.9i	−1.48276	−	1.57432i
$171$		0			0
$172$		−	13904.7i		−	0.470007i
$173$	35192.2			1.17586			0.587928	−	0.808913i	$-0.299944\pi$
$173$	35192.2			1.17586			0.587928	+	0.808913i	$0.299944\pi$
$174$		0			0
$175$	29835.2	+	1788.71i	0.974210	+	0.0584069i
$176$			20167.3i			0.651062i
$177$		0			0
$178$		−	79878.9i		−	2.52111i
$179$		−	44189.5i		−	1.37915i	−0.724213	−	0.689577i	$-0.757797\pi$
$179$		−	44189.5i		−	1.37915i	0.724213	−	0.689577i	$-0.242203\pi$
$180$		0			0
$181$	−41327.3			−1.26148			−0.630740	−	0.775994i	$-0.717249\pi$
$181$	−41327.3			−1.26148			−0.630740	+	0.775994i	$0.717249\pi$
$182$	−61404.8			−1.85378
$183$		0			0
$184$	3871.22			0.114344
$185$	1013.53	−	954.581i	0.0296136	−	0.0278913i
$186$		0			0
$187$			36790.6i			1.05209i
$188$	−14824.4			−0.419433
$189$		0			0
$190$	37282.3	+	39584.4i	1.03275	+	1.09652i
$191$			23923.1i			0.655768i	0.944718	+	0.327884i	$0.106335\pi$
$191$			23923.1i			0.655768i	−0.944718	+	0.327884i	$0.893665\pi$
$192$		0			0
$193$			2349.32i			0.0630708i	0.999503	+	0.0315354i	$0.0100397\pi$
$193$			2349.32i			0.0630708i	−0.999503	+	0.0315354i	$0.989960\pi$
$194$		−	77523.2i		−	2.05982i
$195$		0			0
$196$	1942.40			0.0505622
$197$	9314.30			0.240004			0.120002	−	0.992774i	$-0.461710\pi$
$197$	9314.30			0.240004			0.120002	+	0.992774i	$0.461710\pi$
$198$		0			0
$199$	−6189.82			−0.156305			−0.0781524	−	0.996941i	$-0.524902\pi$
$199$	−6189.82			−0.156305			−0.0781524	+	0.996941i	$0.524902\pi$
$200$	221.521	−	3694.90i	0.00553802	−	0.0923726i
$201$		0			0
$202$		−	104680.i		−	2.56544i
$203$	−23755.8			−0.576470
$204$		0			0
$205$	−8645.41	+	8142.61i	−0.205721	+	0.193756i
$206$			74361.7i			1.75233i
$207$		0			0
$208$		−	53273.9i		−	1.23137i
$209$		−	32008.9i		−	0.732788i
$210$		0			0
$211$	1433.62			0.0322010			0.0161005	−	0.999870i	$-0.494875\pi$
$211$	1433.62			0.0322010			0.0161005	+	0.999870i	$0.494875\pi$
$212$	−39839.1			−0.886416
$213$		0			0
$214$	7887.21			0.172225
$215$	−14858.7	+	13994.5i	−0.321442	+	0.302748i
$216$		0			0
$217$			14489.9i			0.307712i
$218$	−47845.5			−1.00677
$219$		0			0
$220$	−26213.4	+	24688.9i	−0.541600	+	0.510102i
$221$		−	97186.0i		−	1.98984i
$222$		0			0
$223$		−	372.434i		−	0.00748926i	−0.999993	−	0.00374463i	$-0.998808\pi$
$223$		−	372.434i		−	0.00748926i	0.999993	−	0.00374463i	$-0.00119196\pi$
$224$		−	70068.1i		−	1.39645i
$225$		0			0
$226$	−119615.			−2.34191
$227$	−30233.4			−0.586726			−0.293363	−	0.956001i	$-0.594774\pi$
$227$	−30233.4			−0.586726			−0.293363	+	0.956001i	$0.594774\pi$
$228$		0			0
$229$	−44173.2			−0.842341			−0.421171	−	0.906981i	$-0.638381\pi$
$229$	−44173.2			−0.842341			−0.421171	+	0.906981i	$0.638381\pi$
$230$	−64391.4	−	68367.5i	−1.21723	−	1.29239i
$231$		0			0
$232$			2942.00i			0.0546597i
$233$	22978.1			0.423255			0.211628	−	0.977350i	$-0.432124\pi$
$233$	22978.1			0.423255			0.211628	+	0.977350i	$0.432124\pi$
$234$		0			0
$235$	14920.2	+	15841.5i	0.270172	+	0.286854i
$236$			20052.0i			0.360027i
$237$		0			0
$238$		−	119556.i		−	2.11066i
$239$			68002.9i			1.19051i	0.803538	+	0.595253i	$0.202948\pi$
$239$			68002.9i			1.19051i	−0.803538	+	0.595253i	$0.797052\pi$
$240$		0			0
$241$	69463.8			1.19598			0.597991	−	0.801503i	$-0.295966\pi$
$241$	69463.8			1.19598			0.597991	+	0.801503i	$0.295966\pi$
$242$	−43034.0			−0.734820
$243$		0			0
$244$	−117732.			−1.97750
$245$	−1954.95	−	2075.66i	−0.0325689	−	0.0345800i
$246$		0			0
$247$			84554.6i			1.38594i
$248$	1794.48			0.0291766
$249$		0			0
$250$	−68938.3	+	57546.5i	−1.10301	+	0.920744i
$251$			44839.6i			0.711728i	0.934538	+	0.355864i	$0.115813\pi$
$251$			44839.6i			0.711728i	−0.934538	+	0.355864i	$0.884187\pi$
$252$		0			0
$253$			55283.6i			0.863684i
$254$			19787.1i			0.306700i
$255$		0			0
$256$	56297.3			0.859029
$257$	512.345			0.00775704			0.00387852	−	0.999992i	$-0.498765\pi$
$257$	512.345			0.00775704			0.00387852	+	0.999992i	$0.498765\pi$
$258$		0			0
$259$	2663.28			0.0397024
$260$	69245.3	−	65218.1i	1.02434	−	0.964765i
$261$		0			0
$262$		−	31681.6i		−	0.461534i
$263$	46915.1			0.678268			0.339134	−	0.940738i	$-0.389866\pi$
$263$	46915.1			0.678268			0.339134	+	0.940738i	$0.389866\pi$
$264$		0			0
$265$	40096.5	+	42572.4i	0.570972	+	0.606229i
$266$			104017.i			1.47009i
$267$		0			0
$268$		−	66001.8i		−	0.918937i
$269$		−	18619.3i		−	0.257311i	−0.991689	−	0.128655i	$-0.958934\pi$
$269$		−	18619.3i		−	0.257311i	0.991689	−	0.128655i	$-0.0410661\pi$
$270$		0			0
$271$	119151.			1.62240			0.811201	−	0.584767i	$-0.198814\pi$
$271$	119151.			1.62240			0.811201	+	0.584767i	$0.198814\pi$
$272$	103725.			1.40200
$273$		0			0
$274$	−24909.3			−0.331788
$275$	52765.6	+	3163.46i	0.697727	+	0.0418309i
$276$		0			0
$277$		−	27803.4i		−	0.362359i	−0.983450	−	0.181179i	$-0.942009\pi$
$277$		−	27803.4i		−	0.362359i	0.983450	−	0.181179i	$-0.0579914\pi$
$278$	−19391.4			−0.250911
$279$		0			0
$280$	5154.38	−	4854.61i	0.0657446	−	0.0619211i
$281$			5480.66i			0.0694097i	0.999398	+	0.0347049i	$0.0110491\pi$
$281$			5480.66i			0.0694097i	−0.999398	+	0.0347049i	$0.988951\pi$
$282$		0			0
$283$		−	120114.i		−	1.49975i	−0.661577	−	0.749877i	$-0.730113\pi$
$283$		−	120114.i		−	1.49975i	0.661577	−	0.749877i	$-0.269887\pi$
$284$		−	99156.7i		−	1.22938i
$285$		0			0
$286$	−108599.			−1.32768
$287$	−22717.8			−0.275806
$288$		0			0
$289$	105702.			1.26558
$290$	51957.1	−	48935.4i	0.617801	−	0.581871i
$291$		0			0
$292$			109741.i			1.28708i
$293$	20263.0			0.236031			0.118016	−	0.993012i	$-0.462347\pi$
$293$	20263.0			0.236031			0.118016	+	0.993012i	$0.462347\pi$
$294$		0			0
$295$	21427.8	−	20181.6i	0.246226	−	0.231906i
$296$		−	329.830i		−	0.00376450i
$297$		0			0
$298$		−	192738.i		−	2.17038i
$299$		−	146037.i		−	1.63350i
$300$		0			0
$301$	−39044.7			−0.430952
$302$	−5945.99			−0.0651944
$303$		0			0
$304$	−90244.0			−0.976498
$305$	118493.	+	125810.i	1.27378	+	1.35243i
$306$		0			0
$307$			132432.i			1.40512i	0.711623	+	0.702562i	$0.247961\pi$
$307$			132432.i			1.40512i	−0.711623	+	0.702562i	$0.752039\pi$
$308$	−68881.9			−0.726113
$309$		0			0
$310$	−29848.2	−	31691.3i	−0.310595	−	0.329774i
$311$			136455.i			1.41081i	0.708804	+	0.705406i	$0.249235\pi$
$311$			136455.i			1.41081i	−0.708804	+	0.705406i	$0.750765\pi$
$312$		0			0
$313$		−	109120.i		−	1.11382i	−0.830574	−	0.556909i	$-0.811987\pi$
$313$		−	109120.i		−	1.11382i	0.830574	−	0.556909i	$-0.188013\pi$
$314$		−	26126.0i		−	0.264981i
$315$		0			0
$316$	83337.5			0.834577
$317$	−42839.0			−0.426305			−0.213153	−	0.977019i	$-0.568373\pi$
$317$	−42839.0			−0.426305			−0.213153	+	0.977019i	$0.568373\pi$
$318$		0			0
$319$	−42013.7			−0.412867
$320$	78941.2	+	83815.8i	0.770910	+	0.818513i
$321$		0			0
$322$		−	179652.i		−	1.73269i
$323$	−164630.			−1.57798
$324$		0			0
$325$	−139385.	−	8356.59i	−1.31963	−	0.0791157i
$326$			206082.i			1.93912i
$327$		0			0
$328$			2813.46i			0.0261513i
$329$			41627.4i			0.384580i
$330$		0			0
$331$	−109570.			−1.00008			−0.500039	−	0.866003i	$-0.666681\pi$
$331$	−109570.			−1.00008			−0.500039	+	0.866003i	$0.666681\pi$
$332$	110949.			1.00658
$333$		0			0
$334$	−127689.			−1.14462
$335$	−70530.1	+	66428.3i	−0.628471	+	0.591920i
$336$		0			0
$337$			9207.78i			0.0810765i	0.999178	+	0.0405383i	$0.0129073\pi$
$337$			9207.78i			0.0810765i	−0.999178	+	0.0405383i	$0.987093\pi$
$338$	122727.			1.07426
$339$		0			0
$340$	126981.	+	134822.i	1.09845	+	1.16628i
$341$			25626.3i			0.220383i
$342$		0			0
$343$		−	120275.i		−	1.02232i
$344$			4835.44i			0.0408619i
$345$		0			0
$346$	−202257.			−1.68947
$347$	−22555.7			−0.187326			−0.0936628	−	0.995604i	$-0.529858\pi$
$347$	−22555.7			−0.187326			−0.0936628	+	0.995604i	$0.529858\pi$
$348$		0			0
$349$	203804.			1.67326			0.836629	−	0.547770i	$-0.184523\pi$
$349$	203804.			1.67326			0.836629	+	0.547770i	$0.184523\pi$
$350$	−171469.	−	10280.1i	−1.39975	−	0.0839193i
$351$		0			0
$352$		−	123920.i		−	1.00013i
$353$	−64610.3			−0.518504			−0.259252	−	0.965810i	$-0.583476\pi$
$353$	−64610.3			−0.518504			−0.259252	+	0.965810i	$0.583476\pi$
$354$		0			0
$355$	−105960.	+	99797.4i	−0.840784	+	0.791886i
$356$			236702.i			1.86768i
$357$		0			0
$358$			253966.i			1.98157i
$359$		−	39569.8i		−	0.307026i	−0.988147	−	0.153513i	$-0.950941\pi$
$359$		−	39569.8i		−	0.307026i	0.988147	−	0.153513i	$-0.0490587\pi$
$360$		0			0
$361$	12911.4			0.0990741
$362$	237517.			1.81250
$363$		0			0
$364$	181958.			1.37331
$365$	117271.	−	110451.i	0.880246	−	0.829053i
$366$		0			0
$367$			194407.i			1.44338i	0.692218	+	0.721688i	$0.256634\pi$
$367$			194407.i			1.44338i	−0.692218	+	0.721688i	$0.743366\pi$
$368$	155863.			1.15093
$369$		0			0
$370$	−5824.95	+	5486.18i	−0.0425489	+	0.0400744i
$371$			111869.i			0.812760i
$372$		0			0
$373$			107367.i			0.771709i	0.922560	+	0.385854i	$0.126093\pi$
$373$			107367.i			0.771709i	−0.922560	+	0.385854i	$0.873907\pi$
$374$		−	211444.i		−	1.51165i
$375$		0			0
$376$	5155.29			0.0364651
$377$	110983.			0.780863
$378$		0			0
$379$	231550.			1.61200			0.806002	−	0.591913i	$-0.201627\pi$
$379$	231550.			1.61200			0.806002	+	0.591913i	$0.201627\pi$
$380$	−110477.	−	117299.i	−0.765077	−	0.812320i
$381$		0			0
$382$		−	137491.i		−	0.942210i
$383$	−46562.9			−0.317426			−0.158713	−	0.987325i	$-0.550734\pi$
$383$	−46562.9			−0.317426			−0.158713	+	0.987325i	$0.550734\pi$
$384$		0			0
$385$	69327.1	+	73607.9i	0.467715	+	0.496596i
$386$		−	13502.1i		−	0.0906204i
$387$		0			0
$388$			229721.i			1.52594i
$389$		−	15134.3i		−	0.100015i	−0.998749	−	0.0500074i	$-0.984075\pi$
$389$		−	15134.3i		−	0.100015i	0.998749	−	0.0500074i	$-0.0159245\pi$
$390$		0			0
$391$	284337.			1.85986
$392$	−675.481			−0.00439583
$393$		0			0
$394$	−53531.3			−0.344838
$395$	−83876.1	−	89055.3i	−0.537581	−	0.570776i
$396$		0			0
$397$			49119.2i			0.311652i	0.987784	+	0.155826i	$0.0498040\pi$
$397$			49119.2i			0.311652i	−0.987784	+	0.155826i	$0.950196\pi$
$398$	35574.3			0.224579
$399$		0			0
$400$	8918.88	−	148764.i	0.0557430	−	0.929777i
$401$			134026.i			0.833490i	0.909023	+	0.416745i	$0.136829\pi$
$401$			134026.i			0.833490i	−0.909023	+	0.416745i	$0.863171\pi$
$402$		0			0
$403$		−	67694.4i		−	0.416814i
$404$			310195.i			1.90052i
$405$		0			0
$406$	136529.			0.828274
$407$	4710.19			0.0284348
$408$		0			0
$409$	−54219.7			−0.324124			−0.162062	−	0.986781i	$-0.551814\pi$
$409$	−54219.7			−0.324124			−0.162062	+	0.986781i	$0.551814\pi$
$410$	49687.0	−	46797.3i	0.295580	−	0.278390i
$411$		0			0
$412$		−	220353.i		−	1.29815i
$413$	56306.6			0.330110
$414$		0			0
$415$	−111666.	−	118561.i	−0.648372	−	0.688408i
$416$			327347.i			1.89157i
$417$		0			0
$418$			183962.i			1.05287i
$419$		−	176655.i		−	1.00623i	−0.864219	−	0.503115i	$-0.832187\pi$
$419$		−	176655.i		−	1.00623i	0.864219	−	0.503115i	$-0.167813\pi$
$420$		0			0
$421$	143504.			0.809657			0.404829	−	0.914393i	$-0.367331\pi$
$421$	143504.			0.809657			0.404829	+	0.914393i	$0.367331\pi$
$422$	−8239.33			−0.0462665
$423$		0			0
$424$	13854.3			0.0770642
$425$	16270.5	−	271386.i	0.0900787	−	1.50248i
$426$		0			0
$427$			330595.i			1.81318i
$428$	−23371.8			−0.127587
$429$		0			0
$430$	85396.0	−	80429.6i	0.461850	−	0.434990i
$431$		−	96119.3i		−	0.517435i	−0.965953	−	0.258718i	$-0.916700\pi$
$431$		−	96119.3i		−	0.517435i	0.965953	−	0.258718i	$-0.0832999\pi$
$432$		0			0
$433$			313068.i			1.66979i	0.550406	+	0.834897i	$0.314473\pi$
$433$			313068.i			1.66979i	−0.550406	+	0.834897i	$0.685527\pi$
$434$		−	83276.3i		−	0.442122i
$435$		0			0
$436$	141779.			0.745827
$437$	−247381.			−1.29540
$438$		0			0
$439$	−104331.			−0.541356			−0.270678	−	0.962670i	$-0.587248\pi$
$439$	−104331.			−0.541356			−0.270678	+	0.962670i	$0.587248\pi$
$440$	9115.88	−	8585.72i	0.0470862	−	0.0443477i
$441$		0			0
$442$			558549.i			2.85902i
$443$	−30727.1			−0.156572			−0.0782860	−	0.996931i	$-0.524945\pi$
$443$	−30727.1			−0.156572			−0.0782860	+	0.996931i	$0.524945\pi$
$444$		0			0
$445$	252942.	−	238232.i	1.27732	−	1.20304i
$446$			2140.46i			0.0107606i
$447$		0			0
$448$			220246.i			1.09737i
$449$		−	47332.1i		−	0.234781i	−0.993086	−	0.117391i	$-0.962547\pi$
$449$		−	47332.1i		−	0.234781i	0.993086	−	0.117391i	$-0.0374529\pi$
$450$		0			0
$451$	−40178.1			−0.197531
$452$	354451.			1.73492
$453$		0			0
$454$	173758.			0.843010
$455$	−183134.	−	194442.i	−0.884599	−	0.939222i
$456$		0			0
$457$		−	117727.i		−	0.563694i	−0.959459	−	0.281847i	$-0.909053\pi$
$457$		−	117727.i		−	0.563694i	0.959459	−	0.281847i	$-0.0909470\pi$
$458$	253873.			1.21028
$459$		0			0
$460$	190808.	+	202591.i	0.901741	+	0.957423i
$461$		−	43945.2i		−	0.206781i	−0.994641	−	0.103390i	$-0.967031\pi$
$461$		−	43945.2i		−	0.206781i	0.994641	−	0.103390i	$-0.0329691\pi$
$462$		0			0
$463$			37895.2i			0.176776i	0.996086	+	0.0883878i	$0.0281715\pi$
$463$			37895.2i			0.176776i	−0.996086	+	0.0883878i	$0.971829\pi$
$464$			118451.i			0.550177i
$465$		0			0
$466$	−132060.			−0.608135
$467$	−229722.			−1.05334			−0.526671	−	0.850069i	$-0.676560\pi$
$467$	−229722.			−1.05334			−0.526671	+	0.850069i	$0.676560\pi$
$468$		0			0
$469$	−185335.			−0.842579
$470$	−85749.8	−	91044.7i	−0.388184	−	0.412154i
$471$		0			0
$472$		−	6973.22i		−	0.0313004i
$473$	−69053.2			−0.308647
$474$		0			0
$475$	−14155.8	+	236114.i	−0.0627402	+	1.04649i
$476$			354276.i			1.56361i
$477$		0			0
$478$		−	390827.i		−	1.71052i
$479$		−	49199.3i		−	0.214431i	−0.994236	−	0.107216i	$-0.965806\pi$
$479$		−	49199.3i		−	0.214431i	0.994236	−	0.107216i	$-0.0341935\pi$
$480$		0			0
$481$	−12442.4			−0.0537792
$482$	−399224.			−1.71839
$483$		0			0
$484$	127521.			0.544365
$485$	245483.	−	231206.i	1.04361	−	0.982914i
$486$		0			0
$487$		−	154002.i		−	0.649335i	−0.945828	−	0.324668i	$-0.894748\pi$
$487$		−	154002.i		−	0.649335i	0.945828	−	0.324668i	$-0.105252\pi$
$488$	40942.2			0.171922
$489$		0			0
$490$	11235.5	+	11929.3i	0.0467951	+	0.0496847i
$491$		−	168740.i		−	0.699931i	−0.936763	−	0.349965i	$-0.886193\pi$
$491$		−	168740.i		−	0.699931i	0.936763	−	0.349965i	$-0.113807\pi$
$492$		0			0
$493$			216087.i			0.889066i
$494$		−	485954.i		−	1.99132i
$495$		0			0
$496$	72249.3			0.293677
$497$	−278434.			−1.12722
$498$		0			0
$499$	174509.			0.700838			0.350419	−	0.936593i	$-0.386039\pi$
$499$	174509.			0.700838			0.350419	+	0.936593i	$0.386039\pi$
$500$	204282.	−	170525.i	0.817128	−	0.682101i
$501$		0			0
$502$		−	257703.i		−	1.02261i
$503$	−384923.			−1.52138			−0.760691	−	0.649115i	$-0.775140\pi$
$503$	−384923.			−1.52138			−0.760691	+	0.649115i	$0.775140\pi$
$504$		0			0
$505$	331477.	−	312199.i	1.29978	−	1.22419i
$506$		−	317727.i		−	1.24095i
$507$		0			0
$508$		−	58634.2i		−	0.227208i
$509$			133845.i			0.516613i	0.966063	+	0.258306i	$0.0831644\pi$
$509$			133845.i			0.516613i	−0.966063	+	0.258306i	$0.916836\pi$
$510$		0			0
$511$	308156.			1.18013
$512$	−371969.			−1.41895
$513$		0			0
$514$	−2944.55			−0.0111453
$515$	−235471.	+	221777.i	−0.887818	+	0.836184i
$516$		0			0
$517$			73620.9i			0.275436i
$518$	−15306.4			−0.0570446
$519$		0			0
$520$	−24080.5	+	22680.0i	−0.0890550	+	0.0838758i
$521$			128028.i			0.471662i	0.971794	+	0.235831i	$0.0757812\pi$
$521$			128028.i			0.471662i	−0.971794	+	0.235831i	$0.924219\pi$
$522$		0			0
$523$			396160.i			1.44833i	0.689627	+	0.724165i	$0.257775\pi$
$523$			396160.i			1.44833i	−0.689627	+	0.724165i	$0.742225\pi$
$524$			93880.7i			0.341911i
$525$		0			0
$526$	−269631.			−0.974538
$527$	131802.			0.474572
$528$		0			0
$529$	147419.			0.526794
$530$	−230443.	−	244673.i	−0.820375	−	0.871032i
$531$		0			0
$532$		−	308231.i		−	1.08906i
$533$	106134.			0.373595
$534$		0			0
$535$	23522.9	+	24975.4i	0.0821831	+	0.0872578i
$536$			22952.5i			0.0798916i
$537$		0			0
$538$			107009.i			0.369705i
$539$		−	9646.31i		−	0.0332035i
$540$		0			0
$541$	−411433.			−1.40574			−0.702870	−	0.711318i	$-0.748098\pi$
$541$	−411433.			−1.40574			−0.702870	+	0.711318i	$0.748098\pi$
$542$	−684786.			−2.33108
$543$		0			0
$544$	−637352.			−2.15368
$545$	−142695.	−	151506.i	−0.480414	−	0.510079i
$546$		0			0
$547$			259611.i			0.867658i	0.900995	+	0.433829i	$0.142838\pi$
$547$			259611.i			0.867658i	−0.900995	+	0.433829i	$0.857162\pi$
$548$	73812.8			0.245794
$549$		0			0
$550$	−303255.	−	18181.1i	−1.00250	−	0.0601028i
$551$		−	188002.i		−	0.619239i
$552$		0			0
$553$		−	234014.i		−	0.765229i
$554$			159792.i			0.520638i
$555$		0			0
$556$	57461.8			0.185879
$557$	261444.			0.842691			0.421345	−	0.906900i	$-0.361558\pi$
$557$	261444.			0.842691			0.421345	+	0.906900i	$0.361558\pi$
$558$		0			0
$559$	182411.			0.583750
$560$	207526.	−	195457.i	0.661753	−	0.623267i
$561$		0			0
$562$		−	31498.5i		−	0.0997282i
$563$	370219.			1.16800			0.583999	−	0.811755i	$-0.301487\pi$
$563$	370219.			1.16800			0.583999	+	0.811755i	$0.301487\pi$
$564$		0			0
$565$	−356742.	−	378770.i	−1.11752	−	1.18653i
$566$			690320.i			2.15485i
$567$		0			0
$568$			34482.3i			0.106881i
$569$			511072.i			1.57855i	0.614042	+	0.789273i	$0.289542\pi$
$569$			511072.i			1.57855i	−0.614042	+	0.789273i	$0.710458\pi$
$570$		0			0
$571$	18725.8			0.0574338			0.0287169	−	0.999588i	$-0.490858\pi$
$571$	18725.8			0.0574338			0.0287169	+	0.999588i	$0.490858\pi$
$572$	321806.			0.983562
$573$		0			0
$574$	130564.			0.396279
$575$	24448.9	−	407800.i	0.0739474	−	1.23342i
$576$		0			0
$577$		−	165587.i		−	0.497364i	−0.968585	−	0.248682i	$-0.920003\pi$
$577$		−	165587.i		−	0.497364i	0.968585	−	0.248682i	$-0.0799975\pi$
$578$	−607493.			−1.81838
$579$		0			0
$580$	−153962.	+	145008.i	−0.457676	+	0.431059i
$581$		−	311547.i		−	0.922936i
$582$		0			0
$583$			197848.i			0.582097i
$584$		−	38163.3i		−	0.111897i
$585$		0			0
$586$	−116456.			−0.339131
$587$	559279.			1.62313			0.811563	−	0.584265i	$-0.198617\pi$
$587$	559279.			1.62313			0.811563	+	0.584265i	$0.198617\pi$
$588$		0			0
$589$	−114672.			−0.330542
$590$	−123150.	+	115988.i	−0.353778	+	0.333203i
$591$		0			0
$592$		−	13279.6i		−	0.0378916i
$593$	−398735.			−1.13390			−0.566950	−	0.823752i	$-0.691877\pi$
$593$	−398735.			−1.13390			−0.566950	+	0.823752i	$0.691877\pi$
$594$		0			0
$595$	378583.	−	356566.i	1.06937	−	1.00718i
$596$			571133.i			1.60785i
$597$		0			0
$598$			839305.i			2.34702i
$599$			234704.i			0.654135i	0.945001	+	0.327067i	$0.106060\pi$
$599$			234704.i			0.654135i	−0.945001	+	0.327067i	$0.893940\pi$
$600$		0			0
$601$	−231821.			−0.641805			−0.320903	−	0.947112i	$-0.603986\pi$
$601$	−231821.			−0.641805			−0.320903	+	0.947112i	$0.603986\pi$
$602$	224398.			0.619193
$603$		0			0
$604$	17619.5			0.0482970
$605$	−128345.	−	136270.i	−0.350645	−	0.372297i
$606$		0			0
$607$		−	530761.i		−	1.44053i	−0.693700	−	0.720264i	$-0.744021\pi$
$607$		−	530761.i		−	1.44053i	0.693700	−	0.720264i	$-0.255979\pi$
$608$	554515.			1.50005
$609$		0			0
$610$	−681006.	−	723057.i	−1.83017	−	1.94318i
$611$		−	194477.i		−	0.520937i
$612$		0			0
$613$			688057.i			1.83106i	0.402247	+	0.915531i	$0.368229\pi$
$613$			688057.i			1.83106i	−0.402247	+	0.915531i	$0.631771\pi$
$614$		−	761112.i		−	2.01889i
$615$		0			0
$616$	23954.1			0.0631275
$617$	−141738.			−0.372321			−0.186160	−	0.982519i	$-0.559604\pi$
$617$	−141738.			−0.372321			−0.186160	+	0.982519i	$0.559604\pi$
$618$		0			0
$619$	−552312.			−1.44146			−0.720731	−	0.693215i	$-0.756193\pi$
$619$	−552312.			−1.44146			−0.720731	+	0.693215i	$0.756193\pi$
$620$	88448.0	+	93909.5i	0.230094	+	0.244302i
$621$		0			0
$622$		−	784237.i		−	2.02706i
$623$	664665.			1.71248
$624$		0			0
$625$	−387827.	−	46670.6i	−0.992837	−	0.119477i
$626$			627134.i			1.60034i
$627$		0			0
$628$			77418.2i			0.196302i
$629$		−	24225.6i		−	0.0612314i
$630$		0			0
$631$	231361.			0.581073			0.290537	−	0.956864i	$-0.406166\pi$
$631$	231361.			0.581073			0.290537	+	0.956864i	$0.406166\pi$
$632$	−28981.1			−0.0725574
$633$		0			0
$634$	246205.			0.612517
$635$	−62657.1	+	59013.1i	−0.155390	+	0.146353i
$636$		0			0
$637$			25481.6i			0.0627984i
$638$	241462.			0.593208
$639$		0			0
$640$	−51867.5	−	55070.2i	−0.126630	−	0.134449i
$641$		−	450526.i		−	1.09649i	−0.836318	−	0.548244i	$-0.815297\pi$
$641$		−	450526.i		−	1.09649i	0.836318	−	0.548244i	$-0.184703\pi$
$642$		0			0
$643$			306023.i			0.740170i	0.928998	+	0.370085i	$0.120671\pi$
$643$			306023.i			0.740170i	−0.928998	+	0.370085i	$0.879329\pi$
$644$			532355.i			1.28360i
$645$		0			0
$646$	946162.			2.26725
$647$	200004.			0.477782			0.238891	−	0.971046i	$-0.423216\pi$
$647$	200004.			0.477782			0.238891	+	0.971046i	$0.423216\pi$
$648$		0			0
$649$	99582.2			0.236424
$650$	801078.	+	48027.1i	1.89604	+	0.113674i
$651$		0			0
$652$		−	610673.i		−	1.43653i
$653$	−515618.			−1.20921			−0.604604	−	0.796526i	$-0.706669\pi$
$653$	−515618.			−1.20921			−0.604604	+	0.796526i	$0.706669\pi$
$654$		0			0
$655$	100322.	−	94487.3i	0.233837	−	0.220237i
$656$			113276.i			0.263226i
$657$		0			0
$658$		−	239241.i		−	0.552567i
$659$			348173.i			0.801722i	0.916139	+	0.400861i	$0.131289\pi$
$659$			348173.i			0.801722i	−0.916139	+	0.400861i	$0.868711\pi$
$660$		0			0
$661$	−24648.3			−0.0564136			−0.0282068	−	0.999602i	$-0.508980\pi$
$661$	−24648.3			−0.0564136			−0.0282068	+	0.999602i	$0.508980\pi$
$662$	629720.			1.43692
$663$		0			0
$664$	−38583.2			−0.0875108
$665$	−329378.	+	310222.i	−0.744821	+	0.701504i
$666$		0			0
$667$			324703.i			0.729852i
$668$	378377.			0.847952
$669$		0			0
$670$	405352.	−	381778.i	0.902989	−	0.850474i
$671$			584681.i			1.29860i
$672$		0			0
$673$		−	317993.i		−	0.702082i	−0.936360	−	0.351041i	$-0.885828\pi$
$673$		−	317993.i		−	0.702082i	0.936360	−	0.351041i	$-0.114172\pi$
$674$		−	52919.1i		−	0.116491i
$675$		0			0
$676$	−363673.			−0.795825
$677$	−131911.			−0.287809			−0.143904	−	0.989592i	$-0.545966\pi$
$677$	−131911.			−0.287809			−0.143904	+	0.989592i	$0.545966\pi$
$678$		0			0
$679$	645063.			1.39915
$680$	−44158.4	−	46885.2i	−0.0954984	−	0.101395i
$681$		0			0
$682$		−	147280.i		−	0.316647i
$683$	539667.			1.15687			0.578435	−	0.815729i	$-0.303664\pi$
$683$	539667.			1.15687			0.578435	+	0.815729i	$0.303664\pi$
$684$		0			0
$685$	−74289.8	−	78877.1i	−0.158324	−	0.168101i
$686$			691246.i			1.46887i
$687$		0			0
$688$			194685.i			0.411296i
$689$		−	522635.i		−	1.10093i
$690$		0			0
$691$	−389711.			−0.816181			−0.408091	−	0.912941i	$-0.633805\pi$
$691$	−389711.			−0.816181			−0.408091	+	0.912941i	$0.633805\pi$
$692$	599341.			1.25159
$693$		0			0
$694$	129632.			0.269150
$695$	−57833.1	−	61404.2i	−0.119731	−	0.127124i
$696$		0			0
$697$			206646.i			0.425364i
$698$	−1.17131e6			−2.40414
$699$		0			0
$700$	508108.	+	30462.7i	1.03696	+	0.0621687i
$701$			831525.i			1.69215i	0.533063	+	0.846076i	$0.321041\pi$
$701$			831525.i			1.69215i	−0.533063	+	0.846076i	$0.678959\pi$
$702$		0			0
$703$			21077.0i			0.0426480i
$704$			389520.i			0.785931i
$705$		0			0
$706$	371329.			0.744989
$707$	871035.			1.74260
$708$		0			0
$709$	41638.5			0.0828329			0.0414164	−	0.999142i	$-0.486813\pi$
$709$	41638.5			0.0828329			0.0414164	+	0.999142i	$0.486813\pi$
$710$	608974.	−	573557.i	1.20804	−	1.13778i
$711$		0			0
$712$		−	82314.6i		−	0.162374i
$713$	198053.			0.389586
$714$		0			0
$715$	−323885.	−	343885.i	−0.633548	−	0.672668i
$716$		−	752568.i		−	1.46798i
$717$		0			0
$718$			227416.i			0.441136i
$719$			800950.i			1.54934i	0.632364	+	0.774671i	$0.282085\pi$
$719$			800950.i			1.54934i	−0.632364	+	0.774671i	$0.717915\pi$
$720$		0			0
$721$	−618757.			−1.19028
$722$	−74204.8			−0.142350
$723$		0			0
$724$	−703825.			−1.34273
$725$	309914.	+	18580.3i	0.589611	+	0.0353490i
$726$		0			0
$727$			834328.i			1.57858i	0.614018	+	0.789292i	$0.289552\pi$
$727$			834328.i			1.57858i	−0.614018	+	0.789292i	$0.710448\pi$
$728$	−63277.1			−0.119394
$729$		0			0
$730$	−673980.	+	634783.i	−1.26474	+	1.19119i
$731$			355157.i			0.664640i
$732$		0			0
$733$		−	415308.i		−	0.772970i	−0.922296	−	0.386485i	$-0.873689\pi$
$733$		−	415308.i		−	0.772970i	0.922296	−	0.386485i	$-0.126311\pi$
$734$		−	1.11730e6i		−	2.07385i
$735$		0			0
$736$	−957719.			−1.76800
$737$	−327777.			−0.603453
$738$		0			0
$739$	−862015.			−1.57843			−0.789216	−	0.614115i	$-0.789513\pi$
$739$	−862015.			−1.57843			−0.789216	+	0.614115i	$0.789513\pi$
$740$	17260.8	−	16257.0i	0.0315209	−	0.0296877i
$741$		0			0
$742$		−	642936.i		−	1.16778i
$743$	−304217.			−0.551068			−0.275534	−	0.961291i	$-0.588855\pi$
$743$	−304217.			−0.551068			−0.275534	+	0.961291i	$0.588855\pi$
$744$		0			0
$745$	610318.	−	574824.i	1.09962	−	1.03567i
$746$		−	617062.i		−	1.10879i
$747$		0			0
$748$			626563.i			1.11985i
$749$			65628.7i			0.116985i
$750$		0			0
$751$	−955276.			−1.69375			−0.846874	−	0.531793i	$-0.821518\pi$
$751$	−955276.			−1.69375			−0.846874	+	0.531793i	$0.821518\pi$
$752$	207562.			0.367040
$753$		0			0
$754$	−637845.			−1.12195
$755$	−17733.4	−	18828.4i	−0.0311098	−	0.0330308i
$756$		0			0
$757$			233252.i			0.407037i	0.979071	+	0.203518i	$0.0652377\pi$
$757$			233252.i			0.407037i	−0.979071	+	0.203518i	$0.934762\pi$
$758$	−1.33077e6			−2.31613
$759$		0			0
$760$	38419.1	+	40791.4i	0.0665151	+	0.0706223i
$761$			357583.i			0.617458i	0.951150	+	0.308729i	$0.0999036\pi$
$761$			357583.i			0.617458i	−0.951150	+	0.308729i	$0.900096\pi$
$762$		0			0
$763$		−	398118.i		−	0.683853i
$764$			407422.i			0.698003i
$765$		0			0
$766$	267607.			0.456079
$767$	−263056.			−0.447154
$768$		0			0
$769$	−208275.			−0.352196			−0.176098	−	0.984373i	$-0.556348\pi$
$769$	−208275.			−0.352196			−0.176098	+	0.984373i	$0.556348\pi$
$770$	−398438.	−	423041.i	−0.672015	−	0.713511i
$771$		0			0
$772$			40010.1i			0.0671329i
$773$	−328500.			−0.549764			−0.274882	−	0.961478i	$-0.588639\pi$
$773$	−328500.			−0.549764			−0.274882	+	0.961478i	$0.588639\pi$
$774$		0			0
$775$	11333.1	−	189033.i	0.0188688	−	0.314727i
$776$		−	79887.1i		−	0.132664i
$777$		0			0
$778$			86980.4i			0.143702i
$779$		−	179788.i		−	0.296268i
$780$		0			0
$781$	−492430.			−0.807315
$782$	−1.63414e6			−2.67225
$783$		0			0
$784$	−27196.2			−0.0442462
$785$	82729.9	−	77918.5i	0.134253	−	0.126445i
$786$		0			0
$787$			212791.i			0.343561i	0.985135	+	0.171781i	$0.0549520\pi$
$787$			212791.i			0.343561i	−0.985135	+	0.171781i	$0.945048\pi$
$788$	158627.			0.255461
$789$		0			0
$790$	482054.	+	511820.i	0.772399	+	0.820093i
$791$		−	995308.i		−	1.59076i
$792$		0			0
$793$		−	1.54449e6i		−	2.45606i
$794$		−	282299.i		−	0.447783i
$795$		0			0
$796$	−105416.			−0.166372
$797$	151634.			0.238715			0.119358	−	0.992851i	$-0.461916\pi$
$797$	151634.			0.238715			0.119358	+	0.992851i	$0.461916\pi$
$798$		0			0
$799$	378650.			0.593123
$800$	−54803.1	+	914099.i	−0.0856298	+	1.42828i
$801$		0			0
$802$		−	770276.i		−	1.19756i
$803$	544996.			0.845206
$804$		0			0
$805$	568879.	−	535795.i	0.877866	−	0.826812i
$806$			389054.i			0.598880i
$807$		0			0
$808$		−	107872.i		−	0.165229i
$809$			66493.9i			0.101598i	0.998709	+	0.0507989i	$0.0161768\pi$
$809$			66493.9i			0.101598i	−0.998709	+	0.0507989i	$0.983823\pi$
$810$		0			0
$811$	1.27019e6			1.93120			0.965602	−	0.260025i	$-0.0837307\pi$
$811$	1.27019e6			1.93120			0.965602	+	0.260025i	$0.0837307\pi$
$812$	−404572.			−0.613598
$813$		0			0
$814$	−27070.5			−0.0408552
$815$	−652571.	+	614619.i	−0.982455	+	0.925318i
$816$		0			0
$817$		−	308997.i		−	0.462925i
$818$	311612.			0.465702
$819$		0			0
$820$	−147236.	+	138673.i	−0.218970	+	0.206235i
$821$			479423.i			0.711267i	0.934625	+	0.355634i	$0.115735\pi$
$821$			479423.i			0.711267i	−0.934625	+	0.355634i	$0.884265\pi$
$822$		0			0
$823$		−	145421.i		−	0.214698i	−0.994221	−	0.107349i	$-0.965764\pi$
$823$		−	145421.i		−	0.214698i	0.994221	−	0.107349i	$-0.0342362\pi$
$824$			76629.1i			0.112860i
$825$		0			0
$826$	−323606.			−0.474304
$827$	893602.			1.30657			0.653286	−	0.757111i	$-0.273390\pi$
$827$	893602.			1.30657			0.653286	+	0.757111i	$0.273390\pi$
$828$		0			0
$829$	−659734.			−0.959975			−0.479988	−	0.877275i	$-0.659359\pi$
$829$	−659734.			−0.959975			−0.479988	+	0.877275i	$0.659359\pi$
$830$	641768.	+	681396.i	0.931584	+	0.989108i
$831$		0			0
$832$		−	1.02895e6i		−	1.48645i
$833$	−49613.3			−0.0715003
$834$		0			0
$835$	−380822.	−	404337.i	−0.546196	−	0.579923i
$836$		−	545128.i		−	0.779984i
$837$		0			0
$838$			1.01527e6i			1.44576i
$839$		−	920528.i		−	1.30771i	−0.756618	−	0.653857i	$-0.773150\pi$
$839$		−	920528.i		−	1.30771i	0.756618	−	0.653857i	$-0.226850\pi$
$840$		0			0
$841$	460517.			0.651109
$842$	−824751.			−1.16332
$843$		0			0
$844$	24415.3			0.0342749
$845$	366023.	+	388625.i	0.512620	+	0.544273i
$846$		0			0
$847$		−	358082.i		−	0.499132i
$848$	557802.			0.775690
$849$		0			0
$850$	−93509.8	+	1.55972e6i	−0.129425	+	2.15878i
$851$		−	36402.8i		−	0.0502661i
$852$		0			0
$853$		−	724893.i		−	0.996268i	−0.867100	−	0.498134i	$-0.834019\pi$
$853$		−	724893.i		−	0.996268i	0.867100	−	0.498134i	$-0.165981\pi$
$854$		−	1.90000e6i		−	2.60518i
$855$		0			0
$856$	8127.70			0.0110923
$857$	−984027.			−1.33982			−0.669908	−	0.742444i	$-0.733667\pi$
$857$	−984027.			−1.33982			−0.669908	+	0.742444i	$0.733667\pi$
$858$		0			0
$859$	731552.			0.991422			0.495711	−	0.868487i	$-0.334907\pi$
$859$	731552.			0.991422			0.495711	+	0.868487i	$0.334907\pi$
$860$	−253051.	+	238334.i	−0.342145	+	0.322247i
$861$		0			0
$862$			552418.i			0.743453i
$863$	−498062.			−0.668747			−0.334374	−	0.942441i	$-0.608525\pi$
$863$	−498062.			−0.668747			−0.334374	+	0.942441i	$0.608525\pi$
$864$		0			0
$865$	−603213.	−	640461.i	−0.806193	−	0.855974i
$866$		−	1.79927e6i		−	2.39917i
$867$		0			0
$868$			246769.i			0.327531i
$869$		−	413870.i		−	0.548055i
$870$		0			0
$871$	865854.			1.14132
$872$	−49304.4			−0.0648415
$873$		0			0
$874$	1.42175e6			1.86124
$875$	−478839.	−	573629.i	−0.625422	−	0.749229i
$876$		0			0
$877$			1.36837e6i			1.77911i	0.456827	+	0.889556i	$0.348986\pi$
$877$			1.36837e6i			1.77911i	−0.456827	+	0.889556i	$0.651014\pi$
$878$	599611.			0.777822
$879$		0			0
$880$	367024.	−	345679.i	0.473946	−	0.446383i
$881$		−	1.19744e6i		−	1.54277i	−0.636368	−	0.771386i	$-0.719564\pi$
$881$		−	1.19744e6i		−	1.54277i	0.636368	−	0.771386i	$-0.280436\pi$
$882$		0			0
$883$			461117.i			0.591412i	0.955279	+	0.295706i	$0.0955548\pi$
$883$			461117.i			0.591412i	−0.955279	+	0.295706i	$0.904445\pi$
$884$		−	1.65513e6i		−	2.11800i
$885$		0			0
$886$	176595.			0.224963
$887$	−854483.			−1.08607			−0.543033	−	0.839711i	$-0.682724\pi$
$887$	−854483.			−1.08607			−0.543033	+	0.839711i	$0.682724\pi$
$888$		0			0
$889$	−164646.			−0.208328
$890$	−1.45371e6	+	1.36917e6i	−1.83526	+	1.72853i
$891$		0			0
$892$		−	6342.73i		−	0.00797162i
$893$	−329436.			−0.413113
$894$		0			0
$895$	−804202.	+	757431.i	−1.00397	+	0.945578i
$896$		−	144710.i		−	0.180253i
$897$		0			0
$898$			272028.i			0.337334i
$899$			150514.i			0.186233i
$900$		0			0
$901$	1.01758e6			1.25349
$902$	230912.			0.283814
$903$		0			0
$904$	−123263.			−0.150832
$905$	708373.	+	752114.i	0.864898	+	0.918304i
$906$		0			0
$907$			465527.i			0.565888i	0.959136	+	0.282944i	$0.0913111\pi$
$907$			465527.i			0.565888i	−0.959136	+	0.282944i	$0.908689\pi$
$908$	−514890.			−0.624514
$909$		0			0
$910$	1.05251e6	+	1.11750e6i	1.27099	+	1.34948i
$911$			688448.i			0.829535i	0.909927	+	0.414767i	$0.136137\pi$
$911$			688448.i			0.829535i	−0.909927	+	0.414767i	$0.863863\pi$
$912$		0			0
$913$		−	550993.i		−	0.661005i
$914$			676602.i			0.809917i
$915$		0			0
$916$	−752291.			−0.896593
$917$	263619.			0.313501
$918$		0			0
$919$	−692392.			−0.819825			−0.409912	−	0.912125i	$-0.634441\pi$
$919$	−692392.			−0.819825			−0.409912	+	0.912125i	$0.634441\pi$
$920$	−66354.8	−	70452.2i	−0.0783965	−	0.0832374i
$921$		0			0
$922$			252563.i			0.297103i
$923$	1.30080e6			1.52689
$924$		0			0
$925$	−34744.8	−	2083.06i	−0.0406075	−	0.00243454i
$926$		−	217792.i		−	0.253992i
$927$		0			0
$928$		−	727835.i		−	0.845157i
$929$			497050.i			0.575928i	0.957641	+	0.287964i	$0.0929784\pi$
$929$			497050.i			0.575928i	−0.957641	+	0.287964i	$0.907022\pi$
$930$		0			0
$931$	43165.0			0.0498003
$932$	391328.			0.450515
$933$		0			0
$934$	1.32026e6			1.51345
$935$	669551.	−	630612.i	0.765879	−	0.721338i
$936$		0			0
$937$			761906.i			0.867805i	0.900960	+	0.433903i	$0.142864\pi$
$937$			761906.i			0.867805i	−0.900960	+	0.433903i	$0.857136\pi$
$938$	1.06516e6			1.21062
$939$		0			0
$940$	254099.	+	269789.i	0.287572	+	0.305329i
$941$			257395.i			0.290684i	0.989381	+	0.145342i	$0.0464282\pi$
$941$			257395.i			0.290684i	−0.989381	+	0.145342i	$0.953572\pi$
$942$		0			0
$943$			310517.i			0.349190i
$944$		−	280756.i		−	0.315054i
$945$		0			0
$946$	396864.			0.443465
$947$	−463023.			−0.516300			−0.258150	−	0.966105i	$-0.583113\pi$
$947$	−463023.			−0.516300			−0.258150	+	0.966105i	$0.583113\pi$
$948$		0			0
$949$	−1.43966e6			−1.59855
$950$	81356.2	−	1.35700e6i	0.0901454	−	1.50360i
$951$		0			0
$952$		−	123202.i		−	0.135939i
$953$	323955.			0.356696			0.178348	−	0.983967i	$-0.442925\pi$
$953$	323955.			0.356696			0.178348	+	0.983967i	$0.442925\pi$
$954$		0			0
$955$	435375.	−	410054.i	0.477371	−	0.449609i
$956$			1.15812e6i			1.26718i
$957$		0			0
$958$			282759.i			0.308096i
$959$		−	207268.i		−	0.225370i
$960$		0			0
$961$	−831715.			−0.900591
$962$	71509.3			0.0772702
$963$		0			0
$964$	1.18300e6			1.27301
$965$	42755.2	−	40268.7i	0.0459129	−	0.0432427i
$966$		0			0
$967$			590965.i			0.631988i	0.948761	+	0.315994i	$0.102338\pi$
$967$			590965.i			0.631988i	−0.948761	+	0.315994i	$0.897662\pi$
$968$	−44346.2			−0.0473266
$969$		0			0
$970$	−1.41084e6	+	1.32879e6i	−1.49946	+	1.41225i
$971$			1.14727e6i			1.21682i	0.793622	+	0.608412i	$0.208193\pi$
$971$			1.14727e6i			1.21682i	−0.793622	+	0.608412i	$0.791807\pi$
$972$		0			0
$973$		−	161354.i		−	0.170433i
$974$			885084.i			0.932967i
$975$		0			0
$976$	1.64841e6			1.73048
$977$	858573.			0.899473			0.449737	−	0.893161i	$-0.351518\pi$
$977$	858573.			0.899473			0.449737	+	0.893161i	$0.351518\pi$
$978$		0			0
$979$	1.17551e6			1.22648
$980$	−33293.7	−	35349.6i	−0.0346665	−	0.0368071i
$981$		0			0
$982$			969785.i			1.00566i
$983$	−1.16064e6			−1.20113			−0.600566	−	0.799575i	$-0.705058\pi$
$983$	−1.16064e6			−1.20113			−0.600566	+	0.799575i	$0.705058\pi$
$984$		0			0
$985$	−159652.	−	169511.i	−0.164552	−	0.174713i
$986$		−	1.24190e6i		−	1.27741i
$987$		0			0
$988$			1.44001e6i			1.47520i
$989$			533679.i			0.545616i
$990$		0			0
$991$	−1.59778e6			−1.62693			−0.813467	−	0.581611i	$-0.802422\pi$
$991$	−1.59778e6			−1.62693			−0.813467	+	0.581611i	$0.802422\pi$
$992$	−443944.			−0.451133
$993$		0			0
$994$	1.60022e6			1.61960
$995$	106097.	+	112648.i	0.107166	+	0.113783i
$996$		0			0
$997$			1.30341e6i			1.31127i	0.755080	+	0.655633i	$0.227598\pi$
$997$			1.30341e6i			1.31127i	−0.755080	+	0.655633i	$0.772402\pi$
$998$	−1.00294e6			−1.00697
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.d.a.404.7		44
3.2	odd	2	inner	405.5.d.a.404.37		44
5.4	even	2	inner	405.5.d.a.404.38		44
9.2	odd	6		135.5.h.a.44.4		44
9.4	even	3		135.5.h.a.89.19		44
9.5	odd	6		45.5.h.a.29.4	yes	44
9.7	even	3		45.5.h.a.14.19	yes	44
15.14	odd	2	inner	405.5.d.a.404.8		44
45.4	even	6		135.5.h.a.89.4		44
45.14	odd	6		45.5.h.a.29.19	yes	44
45.29	odd	6		135.5.h.a.44.19		44
45.34	even	6		45.5.h.a.14.4	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.h.a.14.4	✓	44	45.34	even	6
45.5.h.a.14.19	yes	44	9.7	even	3
45.5.h.a.29.4	yes	44	9.5	odd	6
45.5.h.a.29.19	yes	44	45.14	odd	6
135.5.h.a.44.4		44	9.2	odd	6
135.5.h.a.44.19		44	45.29	odd	6
135.5.h.a.89.4		44	45.4	even	6
135.5.h.a.89.19		44	9.4	even	3
405.5.d.a.404.7		44	1.1	even	1	trivial
405.5.d.a.404.8		44	15.14	odd	2	inner
405.5.d.a.404.37		44	3.2	odd	2	inner
405.5.d.a.404.38		44	5.4	even	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)

\(f(q)\)	\(=\)	\(q-5.74722 q^{2} +17.0305 q^{4} +(-17.1405 - 18.1989i) q^{5} -47.8220i q^{7} -5.92246 q^{8} +(98.5104 + 104.593i) q^{10} -84.5766i q^{11} +223.417i q^{13} +274.843i q^{14} -238.450 q^{16} -434.998 q^{17} +378.461 q^{19} +(-291.912 - 309.937i) q^{20} +486.080i q^{22} -653.651 q^{23} +(-37.4035 + 623.880i) q^{25} -1284.03i q^{26} -814.433i q^{28} -496.753i q^{29} -302.995 q^{31} +1465.18 q^{32} +2500.03 q^{34} +(-870.311 + 819.695i) q^{35} +55.6914i q^{37} -2175.09 q^{38} +(101.514 + 107.783i) q^{40} -475.050i q^{41} -816.458i q^{43} -1440.38i q^{44} +3756.67 q^{46} -870.464 q^{47} +114.054 q^{49} +(214.966 - 3585.57i) q^{50} +3804.90i q^{52} -2339.28 q^{53} +(-1539.21 + 1449.69i) q^{55} +283.224i q^{56} +2854.95i q^{58} +1177.42i q^{59} -6913.03 q^{61} +1741.38 q^{62} -4605.53 q^{64} +(4065.96 - 3829.49i) q^{65} -3875.51i q^{67} -7408.23 q^{68} +(5001.86 - 4710.97i) q^{70} -5822.30i q^{71} +6443.82i q^{73} -320.071i q^{74} +6445.37 q^{76} -4044.62 q^{77} +4893.43 q^{79} +(4087.17 + 4339.54i) q^{80} +2730.21i q^{82} +6514.72 q^{83} +(7456.10 + 7916.50i) q^{85} +4692.36i q^{86} +500.901i q^{88} +13898.7i q^{89} +10684.3 q^{91} -11132.0 q^{92} +5002.75 q^{94} +(-6487.02 - 6887.59i) q^{95} +13488.8i q^{97} -655.493 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) 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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