LMFDB - Embedded newform 405.5.d.a.404.9

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			404.9
Character	$\chi$	$=$	405.404
Dual form			405.5.d.a.404.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.23381 q^{2} +11.3928 q^{4} +(3.38042 + 24.7704i) q^{5} -57.8140i q^{7} +24.1135 q^{8} +(-17.6925 - 129.644i) q^{10} -169.160i q^{11} -114.462i q^{13} +302.588i q^{14} -308.489 q^{16} -209.860 q^{17} +26.9887 q^{19} +(38.5123 + 282.203i) q^{20} +885.349i q^{22} +917.283 q^{23} +(-602.145 + 167.469i) q^{25} +599.071i q^{26} -658.661i q^{28} +356.673i q^{29} -1258.54 q^{31} +1228.76 q^{32} +1098.36 q^{34} +(1432.08 - 195.436i) q^{35} -368.530i q^{37} -141.254 q^{38} +(81.5137 + 597.300i) q^{40} -1837.31i q^{41} +320.590i q^{43} -1927.19i q^{44} -4800.88 q^{46} +2448.58 q^{47} -941.461 q^{49} +(3151.51 - 876.500i) q^{50} -1304.04i q^{52} -1898.83 q^{53} +(4190.15 - 571.831i) q^{55} -1394.10i q^{56} -1866.76i q^{58} +4859.73i q^{59} +947.679 q^{61} +6586.95 q^{62} -1495.26 q^{64} +(2835.27 - 386.930i) q^{65} -2168.49i q^{67} -2390.88 q^{68} +(-7495.21 + 1022.87i) q^{70} +9078.87i q^{71} +984.836i q^{73} +1928.81i q^{74} +307.475 q^{76} -9779.80 q^{77} -10251.0 q^{79} +(-1042.82 - 7641.40i) q^{80} +9616.15i q^{82} -4780.77 q^{83} +(-709.415 - 5198.31i) q^{85} -1677.91i q^{86} -4079.02i q^{88} +858.461i q^{89} -6617.50 q^{91} +10450.4 q^{92} -12815.4 q^{94} +(91.2332 + 668.521i) q^{95} -4265.61i q^{97} +4927.43 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.23381			−1.30845			−0.654226	−	0.756299i	$-0.727006\pi$
$2$	−5.23381			−1.30845			−0.654226	+	0.756299i	$0.727006\pi$
$3$		0			0
$3$		0			0
$4$	11.3928			0.712047
$5$	3.38042	+	24.7704i	0.135217	+	0.990816i
$5$	3.38042	+	24.7704i	0.135217	+	0.990816i
$6$		0			0
$7$		−	57.8140i		−	1.17988i	−0.807448	−	0.589939i	$-0.799152\pi$
$7$		−	57.8140i		−	1.17988i	0.807448	−	0.589939i	$-0.200848\pi$
$8$	24.1135			0.376773
$9$		0			0
$10$	−17.6925	−	129.644i	−0.176925	−	1.29644i
$11$		−	169.160i		−	1.39801i	−0.715115	−	0.699007i	$-0.753626\pi$
$11$		−	169.160i		−	1.39801i	0.715115	−	0.699007i	$-0.246374\pi$
$12$		0			0
$13$		−	114.462i		−	0.677289i	−0.940914	−	0.338644i	$-0.890032\pi$
$13$		−	114.462i		−	0.677289i	0.940914	−	0.338644i	$-0.109968\pi$
$14$			302.588i			1.54381i
$15$		0			0
$16$	−308.489			−1.20504
$17$	−209.860			−0.726158			−0.363079	−	0.931758i	$-0.618274\pi$
$17$	−209.860			−0.726158			−0.363079	+	0.931758i	$0.618274\pi$
$18$		0			0
$19$	26.9887			0.0747609			0.0373805	−	0.999301i	$-0.488099\pi$
$19$	26.9887			0.0747609			0.0373805	+	0.999301i	$0.488099\pi$
$20$	38.5123	+	282.203i	0.0962809	+	0.705508i
$21$		0			0
$22$			885.349i			1.82923i
$23$	917.283			1.73399			0.866997	−	0.498313i	$-0.166047\pi$
$23$	917.283			1.73399			0.866997	+	0.498313i	$0.166047\pi$
$24$		0			0
$25$	−602.145	+	167.469i	−0.963433	+	0.267950i
$26$			599.071i			0.886200i
$27$		0			0
$28$		−	658.661i		−	0.840129i
$29$			356.673i			0.424106i	0.977258	+	0.212053i	$0.0680150\pi$
$29$			356.673i			0.424106i	−0.977258	+	0.212053i	$0.931985\pi$
$30$		0			0
$31$	−1258.54			−1.30961			−0.654807	−	0.755796i	$-0.727250\pi$
$31$	−1258.54			−1.30961			−0.654807	+	0.755796i	$0.727250\pi$
$32$	1228.76			1.19996
$33$		0			0
$34$	1098.36			0.950143
$35$	1432.08	−	195.436i	1.16904	−	0.159540i
$36$		0			0
$37$		−	368.530i		−	0.269196i	−0.990900	−	0.134598i	$-0.957026\pi$
$37$		−	368.530i		−	0.269196i	0.990900	−	0.134598i	$-0.0429744\pi$
$38$	−141.254			−0.0978211
$39$		0			0
$40$	81.5137	+	597.300i	0.0509461	+	0.373312i
$41$		−	1837.31i		−	1.09299i	−0.837463	−	0.546494i	$-0.815962\pi$
$41$		−	1837.31i		−	1.09299i	0.837463	−	0.546494i	$-0.184038\pi$
$42$		0			0
$43$			320.590i			0.173386i	0.996235	+	0.0866929i	$0.0276299\pi$
$43$			320.590i			0.173386i	−0.996235	+	0.0866929i	$0.972370\pi$
$44$		−	1927.19i		−	0.995451i
$45$		0			0
$46$	−4800.88			−2.26885
$47$	2448.58			1.10846			0.554229	−	0.832364i	$-0.313013\pi$
$47$	2448.58			1.10846			0.554229	+	0.832364i	$0.313013\pi$
$48$		0			0
$49$	−941.461			−0.392112
$50$	3151.51	−	876.500i	1.26061	−	0.350600i
$51$		0			0
$52$		−	1304.04i		−	0.482262i
$53$	−1898.83			−0.675981			−0.337991	−	0.941149i	$-0.609747\pi$
$53$	−1898.83			−0.675981			−0.337991	+	0.941149i	$0.609747\pi$
$54$		0			0
$55$	4190.15	−	571.831i	1.38517	−	0.189035i
$56$		−	1394.10i		−	0.444546i
$57$		0			0
$58$		−	1866.76i		−	0.554923i
$59$			4859.73i			1.39607i	0.716062	+	0.698037i	$0.245943\pi$
$59$			4859.73i			1.39607i	−0.716062	+	0.698037i	$0.754057\pi$
$60$		0			0
$61$	947.679			0.254684			0.127342	−	0.991859i	$-0.459355\pi$
$61$	947.679			0.254684			0.127342	+	0.991859i	$0.459355\pi$
$62$	6586.95			1.71357
$63$		0			0
$64$	−1495.26			−0.365053
$65$	2835.27	−	386.930i	0.671069	−	0.0915810i
$66$		0			0
$67$		−	2168.49i		−	0.483068i	−0.970392	−	0.241534i	$-0.922349\pi$
$67$		−	2168.49i		−	0.483068i	0.970392	−	0.241534i	$-0.0776505\pi$
$68$	−2390.88			−0.517058
$69$		0			0
$70$	−7495.21	+	1022.87i	−1.52964	+	0.208750i
$71$			9078.87i			1.80101i	0.434850	+	0.900503i	$0.356801\pi$
$71$			9078.87i			1.80101i	−0.434850	+	0.900503i	$0.643199\pi$
$72$		0			0
$73$			984.836i			0.184807i	0.995722	+	0.0924035i	$0.0294550\pi$
$73$			984.836i			0.184807i	−0.995722	+	0.0924035i	$0.970545\pi$
$74$			1928.81i			0.352230i
$75$		0			0
$76$	307.475			0.0532333
$77$	−9779.80			−1.64948
$78$		0			0
$79$	−10251.0			−1.64252			−0.821260	−	0.570555i	$-0.806728\pi$
$79$	−10251.0			−1.64252			−0.821260	+	0.570555i	$0.806728\pi$
$80$	−1042.82	−	7641.40i	−0.162941	−	1.19397i
$81$		0			0
$82$			9616.15i			1.43012i
$83$	−4780.77			−0.693972			−0.346986	−	0.937870i	$-0.612795\pi$
$83$	−4780.77			−0.693972			−0.346986	+	0.937870i	$0.612795\pi$
$84$		0			0
$85$	−709.415	−	5198.31i	−0.0981889	−	0.719489i
$86$		−	1677.91i		−	0.226867i
$87$		0			0
$88$		−	4079.02i		−	0.526733i
$89$			858.461i			0.108378i	0.998531	+	0.0541889i	$0.0172573\pi$
$89$			858.461i			0.108378i	−0.998531	+	0.0541889i	$0.982743\pi$
$90$		0			0
$91$	−6617.50			−0.799118
$92$	10450.4			1.23469
$93$		0			0
$94$	−12815.4			−1.45036
$95$	91.2332	+	668.521i	0.0101089	+	0.0740743i
$96$		0			0
$97$		−	4265.61i		−	0.453354i	−0.973970	−	0.226677i	$-0.927214\pi$
$97$		−	4265.61i		−	0.453354i	0.973970	−	0.226677i	$-0.0727862\pi$
$98$	4927.43			0.513060
$99$		0			0
$100$	−6860.09	+	1907.93i	−0.686009	+	0.190793i
$101$		−	4285.04i		−	0.420061i	−0.977695	−	0.210030i	$-0.932644\pi$
$101$		−	4285.04i		−	0.420061i	0.977695	−	0.210030i	$-0.0673563\pi$
$102$		0			0
$103$		−	10473.2i		−	0.987196i	−0.869690	−	0.493598i	$-0.835681\pi$
$103$		−	10473.2i		−	0.987196i	0.869690	−	0.493598i	$-0.164319\pi$
$104$		−	2760.07i		−	0.255184i
$105$		0			0
$106$	9938.12			0.884489
$107$	−17871.6			−1.56097			−0.780487	−	0.625171i	$-0.785029\pi$
$107$	−17871.6			−1.56097			−0.780487	+	0.625171i	$0.785029\pi$
$108$		0			0
$109$	−14838.6			−1.24894			−0.624469	−	0.781049i	$-0.714685\pi$
$109$	−14838.6			−1.24894			−0.624469	+	0.781049i	$0.714685\pi$
$110$	−21930.4	+	2992.86i	−1.81243	+	0.247343i
$111$		0			0
$112$			17835.0i			1.42180i
$113$	−9378.47			−0.734472			−0.367236	−	0.930128i	$-0.619696\pi$
$113$	−9378.47			−0.734472			−0.367236	+	0.930128i	$0.619696\pi$
$114$		0			0
$115$	3100.81	+	22721.5i	0.234466	+	1.71807i
$116$			4063.49i			0.301984i
$117$		0			0
$118$		−	25434.9i		−	1.82670i
$119$			12132.8i			0.856778i
$120$		0			0
$121$	−13974.0			−0.954441
$122$	−4959.97			−0.333242
$123$		0			0
$124$	−14338.2			−0.932506
$125$	−6183.78	−	14349.3i	−0.395762	−	0.918353i
$126$		0			0
$127$			3644.02i			0.225929i	0.993599	+	0.112965i	$0.0360347\pi$
$127$			3644.02i			0.225929i	−0.993599	+	0.112965i	$0.963965\pi$
$128$	−11834.2			−0.722305
$129$		0			0
$130$	−14839.2	+	2025.12i	−0.878061	+	0.119829i
$131$		−	5365.86i		−	0.312678i	−0.987703	−	0.156339i	$-0.950031\pi$
$131$		−	5365.86i		−	0.312678i	0.987703	−	0.156339i	$-0.0499692\pi$
$132$		0			0
$133$		−	1560.32i		−	0.0882087i
$134$			11349.5i			0.632071i
$135$		0			0
$136$	−5060.44			−0.273596
$137$	−2444.59			−0.130246			−0.0651230	−	0.997877i	$-0.520744\pi$
$137$	−2444.59			−0.130246			−0.0651230	+	0.997877i	$0.520744\pi$
$138$		0			0
$139$	15008.0			0.776772			0.388386	−	0.921497i	$-0.373033\pi$
$139$	15008.0			0.776772			0.388386	+	0.921497i	$0.373033\pi$
$140$	16315.3	−	2226.55i	0.832413	−	0.113600i
$141$		0			0
$142$		−	47517.1i		−	2.35653i
$143$	−19362.3			−0.946859
$144$		0			0
$145$	−8834.94	+	1205.71i	−0.420211	+	0.0573464i
$146$		−	5154.44i		−	0.241811i
$147$		0			0
$148$		−	4198.57i		−	0.191680i
$149$		−	12623.5i		−	0.568599i	−0.958736	−	0.284299i	$-0.908239\pi$
$149$		−	12623.5i		−	0.568599i	0.958736	−	0.284299i	$-0.0917610\pi$
$150$		0			0
$151$	−23282.4			−1.02111			−0.510556	−	0.859845i	$-0.670560\pi$
$151$	−23282.4			−1.02111			−0.510556	+	0.859845i	$0.670560\pi$
$152$	650.790			0.0281679
$153$		0			0
$154$	51185.6			2.15827
$155$	−4254.40	−	31174.5i	−0.177082	−	1.29759i
$156$		0			0
$157$			17090.1i			0.693340i	0.937987	+	0.346670i	$0.112688\pi$
$157$			17090.1i			0.693340i	−0.937987	+	0.346670i	$0.887312\pi$
$158$	53651.6			2.14916
$159$		0			0
$160$	4153.73	+	30436.8i	0.162255	+	1.18894i
$161$		−	53031.8i		−	2.04590i
$162$		0			0
$163$		−	26012.6i		−	0.979059i	−0.871987	−	0.489530i	$-0.837169\pi$
$163$		−	26012.6i		−	0.979059i	0.871987	−	0.489530i	$-0.162831\pi$
$164$		−	20932.1i		−	0.778259i
$165$		0			0
$166$	25021.7			0.908029
$167$	−45355.8			−1.62630			−0.813148	−	0.582057i	$-0.802248\pi$
$167$	−45355.8			−1.62630			−0.813148	+	0.582057i	$0.802248\pi$
$168$		0			0
$169$	15459.5			0.541280
$170$	3712.94	+	27206.9i	0.128475	+	0.941417i
$171$		0			0
$172$			3652.41i			0.123459i
$173$	−5696.27			−0.190326			−0.0951630	−	0.995462i	$-0.530337\pi$
$173$	−5696.27			−0.190326			−0.0951630	+	0.995462i	$0.530337\pi$
$174$		0			0
$175$	9682.05	+	34812.4i	0.316149	+	1.13673i
$176$			52183.9i			1.68466i
$177$		0			0
$178$		−	4493.02i		−	0.141807i
$179$			28582.7i			0.892066i	0.895017	+	0.446033i	$0.147164\pi$
$179$			28582.7i			0.892066i	−0.895017	+	0.446033i	$0.852836\pi$
$180$		0			0
$181$	34257.3			1.04567			0.522837	−	0.852433i	$-0.324874\pi$
$181$	34257.3			1.04567			0.522837	+	0.852433i	$0.324874\pi$
$182$	34634.7			1.04561
$183$		0			0
$184$	22118.9			0.653322
$185$	9128.63	−	1245.79i	0.266724	−	0.0363999i
$186$		0			0
$187$			35499.8i			1.01518i
$188$	27896.1			0.789274
$189$		0			0
$190$	−477.497	−	3498.91i	−0.0132271	−	0.0969227i
$191$			71200.0i			1.95170i	0.218443	+	0.975850i	$0.429902\pi$
$191$			71200.0i			1.95170i	−0.218443	+	0.975850i	$0.570098\pi$
$192$		0			0
$193$		−	61488.8i		−	1.65075i	−0.564584	−	0.825375i	$-0.690964\pi$
$193$		−	61488.8i		−	1.65075i	0.564584	−	0.825375i	$-0.309036\pi$
$194$			22325.4i			0.593192i
$195$		0			0
$196$	−10725.8			−0.279202
$197$	14233.7			0.366763			0.183382	−	0.983042i	$-0.441296\pi$
$197$	14233.7			0.366763			0.183382	+	0.983042i	$0.441296\pi$
$198$		0			0
$199$	−58493.1			−1.47706			−0.738531	−	0.674220i	$-0.764480\pi$
$199$	−58493.1			−1.47706			−0.738531	+	0.674220i	$0.764480\pi$
$200$	−14519.8	+	4038.25i	−0.362995	+	0.100956i
$201$		0			0
$202$			22427.1i			0.549629i
$203$	20620.7			0.500393
$204$		0			0
$205$	45511.0	−	6210.90i	1.08295	−	0.147791i
$206$			54814.5i			1.29170i
$207$		0			0
$208$			35310.2i			0.816158i
$209$		−	4565.39i		−	0.104517i
$210$		0			0
$211$	−33332.5			−0.748691			−0.374346	−	0.927289i	$-0.622133\pi$
$211$	−33332.5			−0.748691			−0.374346	+	0.927289i	$0.622133\pi$
$212$	−21632.9			−0.481331
$213$		0			0
$214$	93536.5			2.04246
$215$	−7941.15	+	1083.73i	−0.171793	+	0.0234447i
$216$		0			0
$217$			72761.2i			1.54518i
$218$	77662.6			1.63418
$219$		0			0
$220$	47737.3	−	6514.73i	0.986309	−	0.134602i
$221$			24020.9i			0.491819i
$222$		0			0
$223$		−	46467.3i		−	0.934411i	−0.884149	−	0.467205i	$-0.845261\pi$
$223$		−	46467.3i		−	0.934411i	0.884149	−	0.467205i	$-0.154739\pi$
$224$		−	71039.5i		−	1.41581i
$225$		0			0
$226$	49085.1			0.961021
$227$	25943.3			0.503470			0.251735	−	0.967796i	$-0.418999\pi$
$227$	25943.3			0.503470			0.251735	+	0.967796i	$0.418999\pi$
$228$		0			0
$229$	−24196.8			−0.461410			−0.230705	−	0.973024i	$-0.574103\pi$
$229$	−24196.8			−0.461410			−0.230705	+	0.973024i	$0.574103\pi$
$230$	−16229.0	−	118920.i	−0.306787	−	2.24801i
$231$		0			0
$232$			8600.62i			0.159792i
$233$	59802.9			1.10157			0.550783	−	0.834648i	$-0.314329\pi$
$233$	59802.9			1.10157			0.550783	+	0.834648i	$0.314329\pi$
$234$		0			0
$235$	8277.25	+	60652.4i	0.149882	+	1.09828i
$236$			55365.7i			0.994070i
$237$		0			0
$238$		−	63500.9i		−	1.12105i
$239$		−	74336.0i		−	1.30138i	−0.759345	−	0.650689i	$-0.774480\pi$
$239$		−	74336.0i		−	1.30138i	0.759345	−	0.650689i	$-0.225520\pi$
$240$		0			0
$241$	−55615.2			−0.957546			−0.478773	−	0.877939i	$-0.658918\pi$
$241$	−55615.2			−0.957546			−0.478773	+	0.877939i	$0.658918\pi$
$242$	73137.1			1.24884
$243$		0			0
$244$	10796.7			0.181347
$245$	−3182.54	−	23320.4i	−0.0530202	−	0.388511i
$246$		0			0
$247$		−	3089.17i		−	0.0506347i
$248$	−30347.7			−0.493427
$249$		0			0
$250$	32364.7	+	75101.3i	0.517836	+	1.20162i
$251$			31731.5i			0.503666i	0.967771	+	0.251833i	$0.0810334\pi$
$251$			31731.5i			0.503666i	−0.967771	+	0.251833i	$0.918967\pi$
$252$		0			0
$253$		−	155167.i		−	2.42415i
$254$		−	19072.1i		−	0.295618i
$255$		0			0
$256$	85862.3			1.31015
$257$	43521.4			0.658927			0.329463	−	0.944168i	$-0.393132\pi$
$257$	43521.4			0.658927			0.329463	+	0.944168i	$0.393132\pi$
$258$		0			0
$259$	−21306.2			−0.317619
$260$	32301.5	−	4408.19i	0.477832	−	0.0652100i
$261$		0			0
$262$			28083.9i			0.409124i
$263$	−40447.3			−0.584760			−0.292380	−	0.956302i	$-0.594447\pi$
$263$	−40447.3			−0.584760			−0.292380	+	0.956302i	$0.594447\pi$
$264$		0			0
$265$	−6418.86	−	47034.8i	−0.0914042	−	0.669773i
$266$			8166.44i			0.115417i
$267$		0			0
$268$		−	24705.1i		−	0.343967i
$269$		−	16975.9i		−	0.234600i	−0.993097	−	0.117300i	$-0.962576\pi$
$269$		−	16975.9i		−	0.234600i	0.993097	−	0.117300i	$-0.0374239\pi$
$270$		0			0
$271$	79819.5			1.08685			0.543426	−	0.839457i	$-0.317127\pi$
$271$	79819.5			1.08685			0.543426	+	0.839457i	$0.317127\pi$
$272$	64739.4			0.875046
$273$		0			0
$274$	12794.5			0.170421
$275$	28329.0	+	101859.i	0.374598	+	1.34689i
$276$		0			0
$277$		−	88616.1i		−	1.15492i	−0.816418	−	0.577461i	$-0.804043\pi$
$277$		−	88616.1i		−	1.15492i	0.816418	−	0.577461i	$-0.195957\pi$
$278$	−78549.1			−1.01637
$279$		0			0
$280$	34532.3	−	4712.64i	0.440463	−	0.0601101i
$281$		−	73330.5i		−	0.928693i	−0.885654	−	0.464346i	$-0.846289\pi$
$281$		−	73330.5i		−	0.928693i	0.885654	−	0.464346i	$-0.153711\pi$
$282$		0			0
$283$			58841.7i			0.734704i	0.930082	+	0.367352i	$0.119736\pi$
$283$			58841.7i			0.734704i	−0.930082	+	0.367352i	$0.880264\pi$
$284$			103433.i			1.28240i
$285$		0			0
$286$	101339.			1.23892
$287$	−106223.			−1.28959
$288$		0			0
$289$	−39479.9			−0.472695
$290$	46240.4	−	6310.44i	0.549826	−	0.0750350i
$291$		0			0
$292$			11220.0i			0.131591i
$293$	−101266.			−1.17959			−0.589793	−	0.807554i	$-0.700791\pi$
$293$	−101266.			−1.17959			−0.589793	+	0.807554i	$0.700791\pi$
$294$		0			0
$295$	−120378.	+	16428.0i	−1.38325	+	0.188773i
$296$		−	8886.52i		−	0.101426i
$297$		0			0
$298$			66068.8i			0.743984i
$299$		−	104994.i		−	1.17442i
$300$		0			0
$301$	18534.6			0.204574
$302$	121855.			1.33608
$303$		0			0
$304$	−8325.72			−0.0900896
$305$	3203.56	+	23474.4i	0.0344376	+	0.252345i
$306$		0			0
$307$		−	43494.6i		−	0.461487i	−0.973015	−	0.230743i	$-0.925884\pi$
$307$		−	43494.6i		−	0.461487i	0.973015	−	0.230743i	$-0.0741158\pi$
$308$	−111419.			−1.17451
$309$		0			0
$310$	22266.7	+	163161.i	0.231703	+	1.69783i
$311$			75876.3i			0.784486i	0.919862	+	0.392243i	$0.128301\pi$
$311$			75876.3i			0.784486i	−0.919862	+	0.392243i	$0.871699\pi$
$312$		0			0
$313$		−	67453.4i		−	0.688518i	−0.938875	−	0.344259i	$-0.888130\pi$
$313$		−	67453.4i		−	0.688518i	0.938875	−	0.344259i	$-0.111870\pi$
$314$		−	89446.5i		−	0.907202i
$315$		0			0
$316$	−116787.			−1.16955
$317$	−181151.			−1.80269			−0.901345	−	0.433102i	$-0.857419\pi$
$317$	−181151.			−1.80269			−0.901345	+	0.433102i	$0.857419\pi$
$318$		0			0
$319$	60334.7			0.592906
$320$	−5054.61	−	37038.1i	−0.0493614	−	0.361701i
$321$		0			0
$322$			277558.i			2.67696i
$323$	−5663.83			−0.0542882
$324$		0			0
$325$	19168.8	+	68922.7i	0.181480	+	0.652522i
$326$			136145.i			1.28105i
$327$		0			0
$328$		−	44304.0i		−	0.411808i
$329$		−	141562.i		−	1.30784i
$330$		0			0
$331$	156119.			1.42495			0.712475	−	0.701698i	$-0.247574\pi$
$331$	156119.			1.42495			0.712475	+	0.701698i	$0.247574\pi$
$332$	−54466.2			−0.494141
$333$		0			0
$334$	237383.			2.12793
$335$	53714.4	−	7330.42i	0.478631	−	0.0653190i
$336$		0			0
$337$			139621.i			1.22939i	0.788765	+	0.614695i	$0.210721\pi$
$337$			139621.i			1.22939i	−0.788765	+	0.614695i	$0.789279\pi$
$338$	−80912.0			−0.708239
$339$		0			0
$340$	−8082.19	−	59223.0i	−0.0699151	−	0.512310i
$341$			212894.i			1.83086i
$342$		0			0
$343$		−	84381.8i		−	0.717234i
$344$			7730.54i			0.0653270i
$345$		0			0
$346$	29813.2			0.249032
$347$	608.529			0.00505385			0.00252693	−	0.999997i	$-0.499196\pi$
$347$	608.529			0.00505385			0.00252693	+	0.999997i	$0.499196\pi$
$348$		0			0
$349$	−62762.8			−0.515290			−0.257645	−	0.966240i	$-0.582946\pi$
$349$	−62762.8			−0.515290			−0.257645	+	0.966240i	$0.582946\pi$
$350$	−50674.0	−	182202.i	−0.413665	−	1.48736i
$351$		0			0
$352$		−	207856.i		−	1.67756i
$353$	−108966.			−0.874460			−0.437230	−	0.899350i	$-0.644040\pi$
$353$	−108966.			−0.874460			−0.437230	+	0.899350i	$0.644040\pi$
$354$		0			0
$355$	−224887.	+	30690.4i	−1.78447	+	0.243527i
$356$			9780.24i			0.0771702i
$357$		0			0
$358$		−	149596.i		−	1.16723i
$359$			11414.0i			0.0885621i	0.999019	+	0.0442811i	$0.0140997\pi$
$359$			11414.0i			0.0885621i	−0.999019	+	0.0442811i	$0.985900\pi$
$360$		0			0
$361$	−129593.			−0.994411
$362$	−179296.			−1.36821
$363$		0			0
$364$	−75391.5			−0.569010
$365$	−24394.8	+	3329.17i	−0.183110	+	0.0249890i
$366$		0			0
$367$			129731.i			0.963191i	0.876394	+	0.481595i	$0.159942\pi$
$367$			129731.i			0.963191i	−0.876394	+	0.481595i	$0.840058\pi$
$368$	−282972.			−2.08953
$369$		0			0
$370$	−47777.5	+	6520.21i	−0.348996	+	0.0476275i
$371$			109779.i			0.797575i
$372$		0			0
$373$			56741.2i			0.407832i	0.978988	+	0.203916i	$0.0653669\pi$
$373$			56741.2i			0.407832i	−0.978988	+	0.203916i	$0.934633\pi$
$374$		−	185799.i		−	1.32831i
$375$		0			0
$376$	59043.8			0.417637
$377$	40825.5			0.287242
$378$		0			0
$379$	−98981.2			−0.689087			−0.344544	−	0.938770i	$-0.611966\pi$
$379$	−98981.2			−0.689087			−0.344544	+	0.938770i	$0.611966\pi$
$380$	1039.40	+	7616.29i	0.00719804	+	0.0527444i
$381$		0			0
$382$		−	372647.i		−	2.55371i
$383$	181737.			1.23893			0.619464	−	0.785025i	$-0.287350\pi$
$383$	181737.			1.23893			0.619464	+	0.785025i	$0.287350\pi$
$384$		0			0
$385$	−33059.9	−	242249.i	−0.223038	−	1.63434i
$386$			321821.i			2.15993i
$387$		0			0
$388$		−	48597.0i		−	0.322810i
$389$			163698.i			1.08179i	0.841090	+	0.540895i	$0.181914\pi$
$389$			163698.i			1.08179i	−0.841090	+	0.540895i	$0.818086\pi$
$390$		0			0
$391$	−192501.			−1.25915
$392$	−22701.9			−0.147737
$393$		0			0
$394$	−74496.5			−0.479892
$395$	−34652.6	−	253920.i	−0.222097	−	1.62743i
$396$		0			0
$397$			170885.i			1.08424i	0.840302	+	0.542118i	$0.182378\pi$
$397$			170885.i			1.08424i	−0.840302	+	0.542118i	$0.817622\pi$
$398$	306142.			1.93266
$399$		0			0
$400$	185755.	−	51662.4i	1.16097	−	0.322890i
$401$			63937.9i			0.397622i	0.980038	+	0.198811i	$0.0637079\pi$
$401$			63937.9i			0.397622i	−0.980038	+	0.198811i	$0.936292\pi$
$402$		0			0
$403$			144055.i			0.886987i
$404$		−	48818.4i		−	0.299103i
$405$		0			0
$406$	−107925.			−0.654741
$407$	−62340.3			−0.376340
$408$		0			0
$409$	45111.5			0.269675			0.134838	−	0.990868i	$-0.456949\pi$
$409$	45111.5			0.269675			0.134838	+	0.990868i	$0.456949\pi$
$410$	−238196.	+	32506.7i	−1.41699	+	0.193377i
$411$		0			0
$412$		−	119318.i		−	0.702930i
$413$	280961.			1.64720
$414$		0			0
$415$	−16161.0	−	118422.i	−0.0938368	−	0.687599i
$416$		−	140646.i		−	0.812719i
$417$		0			0
$418$			23894.4i			0.136755i
$419$			124503.i			0.709171i	0.935024	+	0.354585i	$0.115378\pi$
$419$			124503.i			0.709171i	−0.935024	+	0.354585i	$0.884622\pi$
$420$		0			0
$421$	−182499.			−1.02967			−0.514834	−	0.857290i	$-0.672146\pi$
$421$	−182499.			−1.02967			−0.514834	+	0.857290i	$0.672146\pi$
$422$	174456.			0.979627
$423$		0			0
$424$	−45787.4			−0.254691
$425$	126366.	−	35145.0i	0.699604	−	0.194574i
$426$		0			0
$427$		−	54789.1i		−	0.300496i
$428$	−203607.			−1.11149
$429$		0			0
$430$	41562.5	−	5672.04i	0.224783	−	0.0306763i
$431$		−	52320.0i		−	0.281652i	−0.990034	−	0.140826i	$-0.955024\pi$
$431$		−	52320.0i		−	0.281652i	0.990034	−	0.140826i	$-0.0449758\pi$
$432$		0			0
$433$			342116.i			1.82473i	0.409383	+	0.912363i	$0.365744\pi$
$433$			342116.i			1.82473i	−0.409383	+	0.912363i	$0.634256\pi$
$434$		−	380818.i		−	2.02180i
$435$		0			0
$436$	−169053.			−0.889303
$437$	24756.3			0.129635
$438$		0			0
$439$	−260933.			−1.35394			−0.676970	−	0.736011i	$-0.736707\pi$
$439$	−260933.			−1.35394			−0.676970	+	0.736011i	$0.736707\pi$
$440$	101039.	−	13788.8i	0.521896	−	0.0712233i
$441$		0			0
$442$		−	125721.i		−	0.643521i
$443$	186462.			0.950128			0.475064	−	0.879951i	$-0.342425\pi$
$443$	186462.			0.950128			0.475064	+	0.879951i	$0.342425\pi$
$444$		0			0
$445$	−21264.4	+	2901.96i	−0.107383	+	0.0146545i
$446$			243201.i			1.22263i
$447$		0			0
$448$			86446.9i			0.430718i
$449$		−	65910.0i		−	0.326933i	−0.986549	−	0.163466i	$-0.947732\pi$
$449$		−	65910.0i		−	0.326933i	0.986549	−	0.163466i	$-0.0522675\pi$
$450$		0			0
$451$	−310799.			−1.52801
$452$	−106847.			−0.522978
$453$		0			0
$454$	−135782.			−0.658766
$455$	−22370.0	−	163918.i	−0.108054	−	0.791779i
$456$		0			0
$457$			165886.i			0.794288i	0.917756	+	0.397144i	$0.129999\pi$
$457$			165886.i			0.794288i	−0.917756	+	0.397144i	$0.870001\pi$
$458$	126641.			0.603733
$459$		0			0
$460$	35326.7	+	258860.i	0.166950	+	1.22335i
$461$		−	25241.4i		−	0.118771i	−0.998235	−	0.0593857i	$-0.981086\pi$
$461$		−	25241.4i		−	0.118771i	0.998235	−	0.0593857i	$-0.0189142\pi$
$462$		0			0
$463$		−	102699.i		−	0.479076i	−0.970887	−	0.239538i	$-0.923004\pi$
$463$		−	102699.i		−	0.479076i	0.970887	−	0.239538i	$-0.0769959\pi$
$464$		−	110030.i		−	0.511063i
$465$		0			0
$466$	−312997.			−1.44135
$467$	−88039.3			−0.403685			−0.201843	−	0.979418i	$-0.564693\pi$
$467$	−88039.3			−0.403685			−0.201843	+	0.979418i	$0.564693\pi$
$468$		0			0
$469$	−125369.			−0.569961
$470$	−43321.6	−	317443.i	−0.196114	−	1.43704i
$471$		0			0
$472$			117185.i			0.526002i
$473$	54230.9			0.242396
$474$		0			0
$475$	−16251.1	+	4519.77i	−0.0720271	+	0.0200322i
$476$			138226.i			0.610066i
$477$		0			0
$478$			389060.i			1.70279i
$479$			279176.i			1.21677i	0.793644	+	0.608383i	$0.208181\pi$
$479$			279176.i			1.21677i	−0.793644	+	0.608383i	$0.791819\pi$
$480$		0			0
$481$	−42182.6			−0.182324
$482$	291079.			1.25290
$483$		0			0
$484$	−159202.			−0.679607
$485$	105661.	−	14419.6i	0.449191	−	0.0613012i
$486$		0			0
$487$		−	250438.i		−	1.05595i	−0.849261	−	0.527974i	$-0.822952\pi$
$487$		−	250438.i		−	1.05595i	0.849261	−	0.527974i	$-0.177048\pi$
$488$	22851.8			0.0959579
$489$		0			0
$490$	16656.8	+	122054.i	0.0693744	+	0.508348i
$491$			80046.8i			0.332033i	0.986123	+	0.166016i	$0.0530905\pi$
$491$			80046.8i			0.332033i	−0.986123	+	0.166016i	$0.946910\pi$
$492$		0			0
$493$		−	74851.3i		−	0.307968i
$494$			16168.1i			0.0662531i
$495$		0			0
$496$	388246.			1.57813
$497$	524886.			2.12497
$498$		0			0
$499$	−42860.6			−0.172130			−0.0860652	−	0.996290i	$-0.527429\pi$
$499$	−42860.6			−0.172130			−0.0860652	+	0.996290i	$0.527429\pi$
$500$	−70450.3	−	163478.i	−0.281801	−	0.653911i
$501$		0			0
$502$		−	166076.i		−	0.659023i
$503$	189896.			0.750549			0.375274	−	0.926914i	$-0.377548\pi$
$503$	189896.			0.750549			0.375274	+	0.926914i	$0.377548\pi$
$504$		0			0
$505$	106142.	−	14485.3i	0.416203	−	0.0567993i
$506$			812116.i			3.17188i
$507$		0			0
$508$			41515.4i			0.160872i
$509$			20906.5i			0.0806948i	0.999186	+	0.0403474i	$0.0128465\pi$
$509$			20906.5i			0.0806948i	−0.999186	+	0.0403474i	$0.987154\pi$
$510$		0			0
$511$	56937.3			0.218050
$512$	−260039.			−0.991970
$513$		0			0
$514$	−227783.			−0.862174
$515$	259424.	−	35403.7i	0.978130	−	0.133486i
$516$		0			0
$517$		−	414201.i		−	1.54964i
$518$	111512.			0.415589
$519$		0			0
$520$	68368.0	−	9330.21i	0.252840	−	0.0345052i
$521$		−	54650.3i		−	0.201334i	−0.994920	−	0.100667i	$-0.967902\pi$
$521$		−	54650.3i		−	0.201334i	0.994920	−	0.100667i	$-0.0320977\pi$
$522$		0			0
$523$		−	543509.i		−	1.98703i	−0.113723	−	0.993513i	$-0.536278\pi$
$523$		−	543509.i		−	1.98703i	0.113723	−	0.993513i	$-0.463722\pi$
$524$		−	61132.0i		−	0.222641i
$525$		0			0
$526$	211693.			0.765130
$527$	264116.			0.950986
$528$		0			0
$529$	561567.			2.00674
$530$	33595.1	+	246171.i	0.119598	+	0.876366i
$531$		0			0
$532$		−	17776.4i		−	0.0628088i
$533$	−210302.			−0.740269
$534$		0			0
$535$	−60413.6	−	442687.i	−0.211070	−	1.54664i
$536$		−	52289.8i		−	0.182007i
$537$		0			0
$538$			88848.6i			0.306963i
$539$			159257.i			0.548178i
$540$		0			0
$541$	−17290.1			−0.0590748			−0.0295374	−	0.999564i	$-0.509403\pi$
$541$	−17290.1			−0.0590748			−0.0295374	+	0.999564i	$0.509403\pi$
$542$	−417760.			−1.42209
$543$		0			0
$544$	−257867.			−0.871360
$545$	−50160.9	−	367559.i	−0.168878	−	1.23747i
$546$		0			0
$547$		−	81452.9i		−	0.272227i	−0.990693	−	0.136114i	$-0.956539\pi$
$547$		−	81452.9i		−	0.272227i	0.990693	−	0.136114i	$-0.0434612\pi$
$548$	−27850.6			−0.0927412
$549$		0			0
$550$	−148268.	−	533109.i	−0.490144	−	1.76234i
$551$			9626.14i			0.0317066i
$552$		0			0
$553$			592649.i			1.93797i
$554$			463800.i			1.51116i
$555$		0			0
$556$	170983.			0.553098
$557$	449197.			1.44786			0.723930	−	0.689873i	$-0.242334\pi$
$557$	449197.			1.44786			0.723930	+	0.689873i	$0.242334\pi$
$558$		0			0
$559$	36695.4			0.117432
$560$	−441780.	+	60289.9i	−1.40874	+	0.192251i
$561$		0			0
$562$			383798.i			1.21515i
$563$	245018.			0.773003			0.386501	−	0.922289i	$-0.373683\pi$
$563$	245018.			0.773003			0.386501	+	0.922289i	$0.373683\pi$
$564$		0			0
$565$	−31703.2	−	232308.i	−0.0993130	−	0.727726i
$566$		−	307966.i		−	0.961325i
$567$		0			0
$568$			218923.i			0.678570i
$569$		−	107375.i		−	0.331648i	−0.986155	−	0.165824i	$-0.946972\pi$
$569$		−	107375.i		−	0.331648i	0.986155	−	0.165824i	$-0.0530284\pi$
$570$		0			0
$571$	307679.			0.943681			0.471841	−	0.881684i	$-0.343590\pi$
$571$	307679.			0.943681			0.471841	+	0.881684i	$0.343590\pi$
$572$	−220590.			−0.674208
$573$		0			0
$574$	555948.			1.68737
$575$	−552338.	+	153616.i	−1.67059	+	0.464624i
$576$		0			0
$577$			310091.i			0.931404i	0.884942	+	0.465702i	$0.154198\pi$
$577$			310091.i			0.931404i	−0.884942	+	0.465702i	$0.845802\pi$
$578$	206630.			0.618499
$579$		0			0
$580$	−100654.	+	13736.3i	−0.299210	+	0.0408333i
$581$			276396.i			0.818802i
$582$		0			0
$583$			321206.i			0.945031i
$584$			23747.8i			0.0696302i
$585$		0			0
$586$	530008.			1.54343
$587$	67084.6			0.194691			0.0973456	−	0.995251i	$-0.468965\pi$
$587$	67084.6			0.194691			0.0973456	+	0.995251i	$0.468965\pi$
$588$		0			0
$589$	−33966.3			−0.0979079
$590$	630033.	−	85980.8i	1.80992	−	0.247000i
$591$		0			0
$592$			113687.i			0.324391i
$593$	−82855.9			−0.235621			−0.117810	−	0.993036i	$-0.537588\pi$
$593$	−82855.9			−0.235621			−0.117810	+	0.993036i	$0.537588\pi$
$594$		0			0
$595$	−300535.	+	41014.1i	−0.848909	+	0.115851i
$596$		−	143816.i		−	0.404869i
$597$		0			0
$598$			549518.i			1.53667i
$599$			12789.1i			0.0356439i	0.999841	+	0.0178220i	$0.00567321\pi$
$599$			12789.1i			0.0356439i	−0.999841	+	0.0178220i	$0.994327\pi$
$600$		0			0
$601$	−521356.			−1.44340			−0.721698	−	0.692209i	$-0.756638\pi$
$601$	−521356.			−1.44340			−0.721698	+	0.692209i	$0.756638\pi$
$602$	−97006.6			−0.267675
$603$		0			0
$604$	−265250.			−0.727080
$605$	−47237.9	−	346141.i	−0.129057	−	0.945675i
$606$		0			0
$607$			144599.i			0.392452i	0.980559	+	0.196226i	$0.0628687\pi$
$607$			144599.i			0.392452i	−0.980559	+	0.196226i	$0.937131\pi$
$608$	33162.6			0.0897100
$609$		0			0
$610$	−16766.8	−	122860.i	−0.0450599	−	0.330181i
$611$		−	280269.i		−	0.750746i
$612$		0			0
$613$		−	39623.7i		−	0.105447i	−0.998609	−	0.0527235i	$-0.983210\pi$
$613$		−	39623.7i		−	0.105447i	0.998609	−	0.0527235i	$-0.0167902\pi$
$614$			227643.i			0.603833i
$615$		0			0
$616$	−235825.			−0.621481
$617$	−162914.			−0.427946			−0.213973	−	0.976840i	$-0.568640\pi$
$617$	−162914.			−0.427946			−0.213973	+	0.976840i	$0.568640\pi$
$618$		0			0
$619$	−279974.			−0.730695			−0.365348	−	0.930871i	$-0.619050\pi$
$619$	−279974.			−0.730695			−0.365348	+	0.930871i	$0.619050\pi$
$620$	−48469.3	−	355163.i	−0.126091	−	0.923942i
$621$		0			0
$622$		−	397122.i		−	1.02646i
$623$	49631.1			0.127873
$624$		0			0
$625$	334533.	−	201681.i	0.856405	−	0.516304i
$626$			353038.i			0.900893i
$627$		0			0
$628$			194704.i			0.493691i
$629$			77339.5i			0.195479i
$630$		0			0
$631$	521595.			1.31001			0.655005	−	0.755624i	$-0.272666\pi$
$631$	521595.			1.31001			0.655005	+	0.755624i	$0.272666\pi$
$632$	−247186.			−0.618856
$633$		0			0
$634$	948107.			2.35873
$635$	−90263.7	+	12318.3i	−0.223854	+	0.0305495i
$636$		0			0
$637$			107761.i			0.265573i
$638$	−315780.			−0.775789
$639$		0			0
$640$	−40004.8	−	293139.i	−0.0976679	−	0.715671i
$641$		−	80795.4i		−	0.196639i	−0.995155	−	0.0983197i	$-0.968653\pi$
$641$		−	80795.4i		−	0.196639i	0.995155	−	0.0983197i	$-0.0313468\pi$
$642$		0			0
$643$		−	188924.i		−	0.456946i	−0.973550	−	0.228473i	$-0.926627\pi$
$643$		−	188924.i		−	0.456946i	0.973550	−	0.228473i	$-0.0733733\pi$
$644$		−	604178.i		−	1.45678i
$645$		0			0
$646$	29643.4			0.0710335
$647$	−315560.			−0.753829			−0.376915	−	0.926248i	$-0.623015\pi$
$647$	−315560.			−0.753829			−0.376915	+	0.926248i	$0.623015\pi$
$648$		0			0
$649$	822070.			1.95173
$650$	−100326.	−	360728.i	−0.237458	−	0.853794i
$651$		0			0
$652$		−	296355.i		−	0.697136i
$653$	−505252.			−1.18490			−0.592450	−	0.805607i	$-0.701839\pi$
$653$	−505252.			−1.18490			−0.592450	+	0.805607i	$0.701839\pi$
$654$		0			0
$655$	132915.	−	18138.9i	0.309806	−	0.0422794i
$656$			566792.i			1.31709i
$657$		0			0
$658$			740911.i			1.71125i
$659$		−	771267.i		−	1.77596i	−0.459878	−	0.887982i	$-0.652107\pi$
$659$		−	771267.i		−	1.77596i	0.459878	−	0.887982i	$-0.347893\pi$
$660$		0			0
$661$	−56127.3			−0.128461			−0.0642304	−	0.997935i	$-0.520459\pi$
$661$	−56127.3			−0.128461			−0.0642304	+	0.997935i	$0.520459\pi$
$662$	−817096.			−1.86448
$663$		0			0
$664$	−115281.			−0.261470
$665$	38649.9	−	5274.56i	0.0873986	−	0.0119273i
$666$		0			0
$667$			327170.i			0.735398i
$668$	−516727.			−1.15800
$669$		0			0
$670$	−281131.	+	38366.0i	−0.626266	+	0.0854668i
$671$		−	160309.i		−	0.356051i
$672$		0			0
$673$			528543.i			1.16694i	0.812133	+	0.583472i	$0.198306\pi$
$673$			528543.i			1.16694i	−0.812133	+	0.583472i	$0.801694\pi$
$674$		−	730747.i		−	1.60860i
$675$		0			0
$676$	176126.			0.385417
$677$	453465.			0.989388			0.494694	−	0.869067i	$-0.335280\pi$
$677$	453465.			0.989388			0.494694	+	0.869067i	$0.335280\pi$
$678$		0			0
$679$	−246612.			−0.534903
$680$	−17106.4	−	125349.i	−0.0369949	−	0.271084i
$681$		0			0
$682$		−	1.11425e6i		−	2.39559i
$683$	−210879.			−0.452055			−0.226027	−	0.974121i	$-0.572574\pi$
$683$	−210879.			−0.452055			−0.226027	+	0.974121i	$0.572574\pi$
$684$		0			0
$685$	−8263.74	−	60553.4i	−0.0176115	−	0.129050i
$686$			441638.i			0.938466i
$687$		0			0
$688$		−	98898.7i		−	0.208936i
$689$			217344.i			0.457835i
$690$		0			0
$691$	−24578.0			−0.0514742			−0.0257371	−	0.999669i	$-0.508193\pi$
$691$	−24578.0			−0.0514742			−0.0257371	+	0.999669i	$0.508193\pi$
$692$	−64896.2			−0.135521
$693$		0			0
$694$	−3184.93			−0.00661273
$695$	50733.5	+	371755.i	0.105033	+	0.769638i
$696$		0			0
$697$			385578.i			0.793682i
$698$	328488.			0.674232
$699$		0			0
$700$	110305.	+	396610.i	0.225113	+	0.809407i
$701$			349841.i			0.711925i	0.934500	+	0.355963i	$0.115847\pi$
$701$			349841.i			0.711925i	−0.934500	+	0.355963i	$0.884153\pi$
$702$		0			0
$703$		−	9946.13i		−	0.0201254i
$704$			252937.i			0.510349i
$705$		0			0
$706$	570305.			1.14419
$707$	−247735.			−0.495620
$708$		0			0
$709$	188543.			0.375076			0.187538	−	0.982257i	$-0.439949\pi$
$709$	188543.			0.375076			0.187538	+	0.982257i	$0.439949\pi$
$710$	1.17702e6	−	160628.i	2.33489	−	0.318643i
$711$		0			0
$712$			20700.5i			0.0408338i
$713$	−1.15444e6			−2.27086
$714$		0			0
$715$	−65452.9	−	479612.i	−0.128031	−	0.938163i
$716$			325635.i			0.635193i
$717$		0			0
$718$		−	59738.6i		−	0.115879i
$719$			570498.i			1.10356i	0.833989	+	0.551781i	$0.186051\pi$
$719$			570498.i			1.10356i	−0.833989	+	0.551781i	$0.813949\pi$
$720$		0			0
$721$	−605496.			−1.16477
$722$	678263.			1.30114
$723$		0			0
$724$	390285.			0.744569
$725$	−59731.7	−	214769.i	−0.113639	−	0.408598i
$726$		0			0
$727$		−	357336.i		−	0.676095i	−0.941129	−	0.338047i	$-0.890234\pi$
$727$		−	357336.i		−	0.676095i	0.941129	−	0.338047i	$-0.109766\pi$
$728$	−159571.			−0.301086
$729$		0			0
$730$	127678.	−	17424.2i	0.239590	−	0.0326970i
$731$		−	67279.0i		−	0.125905i
$732$		0			0
$733$		−	358614.i		−	0.667451i	−0.942670	−	0.333725i	$-0.891694\pi$
$733$		−	358614.i		−	0.667451i	0.942670	−	0.333725i	$-0.108306\pi$
$734$		−	678988.i		−	1.26029i
$735$		0			0
$736$	1.12712e6			2.08072
$737$	−366821.			−0.675335
$738$		0			0
$739$	−121677.			−0.222802			−0.111401	−	0.993776i	$-0.535534\pi$
$739$	−121677.			−0.222802			−0.111401	+	0.993776i	$0.535534\pi$
$740$	104000.	−	14192.9i	0.189920	−	0.0259185i
$741$		0			0
$742$		−	574563.i		−	1.04359i
$743$	−672849.			−1.21882			−0.609411	−	0.792854i	$-0.708594\pi$
$743$	−672849.			−1.21882			−0.609411	+	0.792854i	$0.708594\pi$
$744$		0			0
$745$	312688.	−	42672.7i	0.563377	−	0.0768842i
$746$		−	296973.i		−	0.533628i
$747$		0			0
$748$			404440.i			0.722855i
$749$			1.03323e6i			1.84176i
$750$		0			0
$751$	151460.			0.268546			0.134273	−	0.990944i	$-0.457130\pi$
$751$	151460.			0.268546			0.134273	+	0.990944i	$0.457130\pi$
$752$	−755362.			−1.33573
$753$		0			0
$754$	−213673.			−0.375843
$755$	−78704.3	−	576714.i	−0.138072	−	1.01173i
$756$		0			0
$757$			667098.i			1.16412i	0.813146	+	0.582060i	$0.197753\pi$
$757$			667098.i			1.16412i	−0.813146	+	0.582060i	$0.802247\pi$
$758$	518048.			0.901638
$759$		0			0
$760$	2199.95	+	16120.3i	0.00380877	+	0.0279092i
$761$		−	108034.i		−	0.186548i	−0.995640	−	0.0932742i	$-0.970267\pi$
$761$		−	108034.i		−	0.186548i	0.995640	−	0.0932742i	$-0.0297333\pi$
$762$		0			0
$763$			857881.i			1.47360i
$764$			811163.i			1.38970i
$765$		0			0
$766$	−951177.			−1.62108
$767$	556254.			0.945545
$768$		0			0
$769$	−682312.			−1.15380			−0.576900	−	0.816815i	$-0.695738\pi$
$769$	−682312.			−1.15380			−0.576900	+	0.816815i	$0.695738\pi$
$770$	173029.	+	1.26789e6i	0.291835	+	2.13845i
$771$		0			0
$772$		−	700527.i		−	1.17541i
$773$	−70204.3			−0.117491			−0.0587455	−	0.998273i	$-0.518710\pi$
$773$	−70204.3			−0.117491			−0.0587455	+	0.998273i	$0.518710\pi$
$774$		0			0
$775$	757823.	−	210766.i	1.26172	−	0.350911i
$776$		−	102859.i		−	0.170811i
$777$		0			0
$778$		−	856762.i		−	1.41547i
$779$		−	49586.7i		−	0.0817128i
$780$		0			0
$781$	1.53578e6			2.51783
$782$	1.00751e6			1.64754
$783$		0			0
$784$	290431.			0.472509
$785$	−423330.	+	57771.9i	−0.686972	+	0.0937514i
$786$		0			0
$787$			1.17856e6i			1.90285i	0.307888	+	0.951423i	$0.400378\pi$
$787$			1.17856e6i			1.90285i	−0.307888	+	0.951423i	$0.599622\pi$
$788$	162161.			0.261153
$789$		0			0
$790$	181365.	+	1.32897e6i	0.290603	+	2.12942i
$791$			542207.i			0.866587i
$792$		0			0
$793$		−	108473.i		−	0.172495i
$794$		−	894382.i		−	1.41867i
$795$		0			0
$796$	−666398.			−1.05174
$797$	518290.			0.815936			0.407968	−	0.912996i	$-0.366237\pi$
$797$	518290.			0.815936			0.407968	+	0.912996i	$0.366237\pi$
$798$		0			0
$799$	−513859.			−0.804915
$800$	−739891.	+	205779.i	−1.15608	+	0.321530i
$801$		0			0
$802$		−	334639.i		−	0.520269i
$803$	166594.			0.258363
$804$		0			0
$805$	1.31362e6	−	179270.i	2.02711	−	0.276641i
$806$		−	753954.i		−	1.16058i
$807$		0			0
$808$		−	103327.i		−	0.158267i
$809$		−	1.25627e6i		−	1.91948i	−0.280881	−	0.959742i	$-0.590627\pi$
$809$		−	1.25627e6i		−	1.91948i	0.280881	−	0.959742i	$-0.409373\pi$
$810$		0			0
$811$	−28967.0			−0.0440415			−0.0220207	−	0.999758i	$-0.507010\pi$
$811$	−28967.0			−0.0440415			−0.0220207	+	0.999758i	$0.507010\pi$
$812$	234927.			0.356304
$813$		0			0
$814$	326277.			0.492423
$815$	644343.	−	87933.7i	0.970067	−	0.132385i
$816$		0			0
$817$			8652.31i			0.0129625i
$818$	−236105.			−0.352857
$819$		0			0
$820$	518496.	−	70759.3i	0.771112	−	0.105234i
$821$			368265.i			0.546354i	0.961964	+	0.273177i	$0.0880744\pi$
$821$			368265.i			0.546354i	−0.961964	+	0.273177i	$0.911926\pi$
$822$		0			0
$823$			632492.i			0.933803i	0.884309	+	0.466902i	$0.154630\pi$
$823$			632492.i			0.933803i	−0.884309	+	0.466902i	$0.845370\pi$
$824$		−	252544.i		−	0.371949i
$825$		0			0
$826$	−1.47049e6			−2.15528
$827$	290970.			0.425439			0.212719	−	0.977113i	$-0.431768\pi$
$827$	290970.			0.425439			0.212719	+	0.977113i	$0.431768\pi$
$828$		0			0
$829$	893630.			1.30032			0.650158	−	0.759799i	$-0.274703\pi$
$829$	893630.			1.30032			0.650158	+	0.759799i	$0.274703\pi$
$830$	84583.8	+	619796.i	0.122781	+	0.899690i
$831$		0			0
$832$			171150.i			0.247247i
$833$	197575.			0.284735
$834$		0			0
$835$	−153322.	−	1.12348e6i	−0.219903	−	1.61136i
$836$		−	52012.4i		−	0.0744208i
$837$		0			0
$838$		−	651624.i		−	0.927916i
$839$		−	139605.i		−	0.198325i	−0.995071	−	0.0991625i	$-0.968384\pi$
$839$		−	139605.i		−	0.198325i	0.995071	−	0.0991625i	$-0.0316164\pi$
$840$		0			0
$841$	580065.			0.820134
$842$	955166.			1.34727
$843$		0			0
$844$	−379749.			−0.533104
$845$	52259.6	+	382938.i	0.0731902	+	0.536309i
$846$		0			0
$847$			807891.i			1.12612i
$848$	585769.			0.814582
$849$		0			0
$850$	−661375.	+	183942.i	−0.915399	+	0.254591i
$851$		−	338046.i		−	0.466785i
$852$		0			0
$853$		−	810721.i		−	1.11423i	−0.830437	−	0.557113i	$-0.811909\pi$
$853$		−	810721.i		−	1.11423i	0.830437	−	0.557113i	$-0.188091\pi$
$854$			286756.i			0.393185i
$855$		0			0
$856$	−430946.			−0.588133
$857$	−830019.			−1.13012			−0.565062	−	0.825048i	$-0.691148\pi$
$857$	−830019.			−1.13012			−0.565062	+	0.825048i	$0.691148\pi$
$858$		0			0
$859$	305567.			0.414115			0.207057	−	0.978329i	$-0.433611\pi$
$859$	305567.			0.414115			0.207057	+	0.978329i	$0.433611\pi$
$860$	−90471.6	+	12346.7i	−0.122325	+	0.0166937i
$861$		0			0
$862$			273833.i			0.368528i
$863$	1.37893e6			1.85149			0.925746	−	0.378147i	$-0.123439\pi$
$863$	1.37893e6			1.85149			0.925746	+	0.378147i	$0.123439\pi$
$864$		0			0
$865$	−19255.8	−	141099.i	−0.0257353	−	0.188578i
$866$		−	1.79057e6i		−	2.38757i
$867$		0			0
$868$			828950.i			1.10024i
$869$			1.73405e6i			2.29626i
$870$		0			0
$871$	−248210.			−0.327177
$872$	−357811.			−0.470566
$873$		0			0
$874$	−129570.			−0.169621
$875$	−829589.	+	357509.i	−1.08354	+	0.466951i
$876$		0			0
$877$		−	714623.i		−	0.929133i	−0.885538	−	0.464566i	$-0.846210\pi$
$877$		−	714623.i		−	0.929133i	0.885538	−	0.464566i	$-0.153790\pi$
$878$	1.36567e6			1.77157
$879$		0			0
$880$	−1.29262e6	+	176404.i	−1.66918	+	0.227794i
$881$			390992.i			0.503751i	0.967760	+	0.251875i	$0.0810473\pi$
$881$			390992.i			0.503751i	−0.967760	+	0.251875i	$0.918953\pi$
$882$		0			0
$883$		−	1.05235e6i		−	1.34971i	−0.737950	−	0.674855i	$-0.764206\pi$
$883$		−	1.05235e6i		−	1.34971i	0.737950	−	0.674855i	$-0.235794\pi$
$884$			273664.i			0.350198i
$885$		0			0
$886$	−975904.			−1.24320
$887$	−865730.			−1.10036			−0.550181	−	0.835046i	$-0.685441\pi$
$887$	−865730.			−1.10036			−0.550181	+	0.835046i	$0.685441\pi$
$888$		0			0
$889$	210675.			0.266569
$890$	111294.	−	15188.3i	0.140505	−	0.0191748i
$891$		0			0
$892$		−	529391.i		−	0.665344i
$893$	66084.0			0.0828693
$894$		0			0
$895$	−708004.	+	96621.6i	−0.883873	+	0.120622i
$896$			684185.i			0.852231i
$897$		0			0
$898$			344960.i			0.427776i
$899$		−	448887.i		−	0.555415i
$900$		0			0
$901$	398488.			0.490869
$902$	1.62666e6			1.99933
$903$		0			0
$904$	−226147.			−0.276729
$905$	115804.	+	848567.i	0.141393	+	1.03607i
$906$		0			0
$907$			1.08101e6i			1.31405i	0.753867	+	0.657027i	$0.228186\pi$
$907$			1.08101e6i			1.31405i	−0.753867	+	0.657027i	$0.771814\pi$
$908$	295566.			0.358494
$909$		0			0
$910$	117080.	+	857916.i	0.141384	+	1.03601i
$911$		−	251847.i		−	0.303459i	−0.988422	−	0.151729i	$-0.951516\pi$
$911$		−	251847.i		−	0.303459i	0.988422	−	0.151729i	$-0.0484843\pi$
$912$		0			0
$913$			808714.i			0.970182i
$914$		−	868217.i		−	1.03929i
$915$		0			0
$916$	−275668.			−0.328546
$917$	−310222.			−0.368922
$918$		0			0
$919$	−1.32687e6			−1.57108			−0.785541	−	0.618809i	$-0.787615\pi$
$919$	−1.32687e6			−1.57108			−0.785541	+	0.618809i	$0.787615\pi$
$920$	74771.1	+	547893.i	0.0883402	+	0.647322i
$921$		0			0
$922$			132109.i			0.155407i
$923$	1.03918e6			1.21980
$924$		0			0
$925$	61717.3	+	221909.i	0.0721312	+	0.259353i
$926$			537507.i			0.626847i
$927$		0			0
$928$			438265.i			0.508910i
$929$		−	705682.i		−	0.817669i	−0.912609	−	0.408834i	$-0.865935\pi$
$929$		−	705682.i		−	0.817669i	0.912609	−	0.408834i	$-0.134065\pi$
$930$		0			0
$931$	−25408.8			−0.0293146
$932$	681320.			0.784367
$933$		0			0
$934$	460781.			0.528203
$935$	−879343.	+	120004.i	−1.00585	+	0.137269i
$936$		0			0
$937$		−	1.20291e6i		−	1.37011i	−0.728493	−	0.685053i	$-0.759779\pi$
$937$		−	1.20291e6i		−	1.37011i	0.728493	−	0.685053i	$-0.240221\pi$
$938$	656159.			0.745767
$939$		0			0
$940$	94300.7	+	690998.i	0.106723	+	0.782025i
$941$		−	1.60965e6i		−	1.81782i	−0.416987	−	0.908912i	$-0.636914\pi$
$941$		−	1.60965e6i		−	1.81782i	0.416987	−	0.908912i	$-0.363086\pi$
$942$		0			0
$943$		−	1.68534e6i		−	1.89524i
$944$		−	1.49918e6i		−	1.68232i
$945$		0			0
$946$	−283834.			−0.317163
$947$	−1.16938e6			−1.30394			−0.651968	−	0.758246i	$-0.726056\pi$
$947$	−1.16938e6			−1.30394			−0.651968	+	0.758246i	$0.726056\pi$
$948$		0			0
$949$	112726.			0.125168
$950$	85055.2	−	23655.6i	0.0942440	−	0.0262112i
$951$		0			0
$952$			292564.i			0.322810i
$953$	−92982.8			−0.102380			−0.0511902	−	0.998689i	$-0.516301\pi$
$953$	−92982.8			−0.102380			−0.0511902	+	0.998689i	$0.516301\pi$
$954$		0			0
$955$	−1.76365e6	+	240686.i	−1.93378	+	0.263903i
$956$		−	846891.i		−	0.926642i
$957$		0			0
$958$		−	1.46115e6i		−	1.59208i
$959$			141331.i			0.153674i
$960$		0			0
$961$	660399.			0.715088
$962$	220776.			0.238562
$963$		0			0
$964$	−633611.			−0.681818
$965$	1.52310e6	−	207858.i	1.63559	−	0.223210i
$966$		0			0
$967$			427556.i			0.457235i	0.973516	+	0.228618i	$0.0734206\pi$
$967$			427556.i			0.457235i	−0.973516	+	0.228618i	$0.926579\pi$
$968$	−336961.			−0.359607
$969$		0			0
$970$	−553009.	+	75469.3i	−0.587744	+	0.0802097i
$971$		−	967416.i		−	1.02606i	−0.858369	−	0.513032i	$-0.828522\pi$
$971$		−	967416.i		−	1.02606i	0.858369	−	0.513032i	$-0.171478\pi$
$972$		0			0
$973$		−	867674.i		−	0.916496i
$974$			1.31074e6i			1.38166i
$975$		0			0
$976$	−292349.			−0.306903
$977$	534429.			0.559888			0.279944	−	0.960016i	$-0.409684\pi$
$977$	534429.			0.559888			0.279944	+	0.960016i	$0.409684\pi$
$978$		0			0
$979$	145217.			0.151514
$980$	−36257.9	−	265683.i	−0.0377529	−	0.276638i
$981$		0			0
$982$		−	418950.i		−	0.434449i
$983$	−811794.			−0.840115			−0.420057	−	0.907498i	$-0.637990\pi$
$983$	−811794.			−0.840115			−0.420057	+	0.907498i	$0.637990\pi$
$984$		0			0
$985$	48116.0	+	352575.i	0.0495926	+	0.363395i
$986$			391757.i			0.402961i
$987$		0			0
$988$		−	35194.2i		−	0.0360543i
$989$			294072.i			0.300650i
$990$		0			0
$991$	−25780.3			−0.0262507			−0.0131254	−	0.999914i	$-0.504178\pi$
$991$	−25780.3			−0.0262507			−0.0131254	+	0.999914i	$0.504178\pi$
$992$	−1.54644e6			−1.57148
$993$		0			0
$994$	−2.74715e6			−2.78042
$995$	−197732.	−	1.44890e6i	−0.199724	−	1.46350i
$996$		0			0
$997$		−	803814.i		−	0.808659i	−0.914613	−	0.404329i	$-0.867505\pi$
$997$		−	803814.i		−	0.808659i	0.914613	−	0.404329i	$-0.132495\pi$
$998$	224324.			0.225224
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.d.a.404.9		44
3.2	odd	2	inner	405.5.d.a.404.35		44
5.4	even	2	inner	405.5.d.a.404.36		44
9.2	odd	6		135.5.h.a.44.5		44
9.4	even	3		135.5.h.a.89.18		44
9.5	odd	6		45.5.h.a.29.5	yes	44
9.7	even	3		45.5.h.a.14.18	yes	44
15.14	odd	2	inner	405.5.d.a.404.10		44
45.4	even	6		135.5.h.a.89.5		44
45.14	odd	6		45.5.h.a.29.18	yes	44
45.29	odd	6		135.5.h.a.44.18		44
45.34	even	6		45.5.h.a.14.5	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.h.a.14.5	✓	44	45.34	even	6
45.5.h.a.14.18	yes	44	9.7	even	3
45.5.h.a.29.5	yes	44	9.5	odd	6
45.5.h.a.29.18	yes	44	45.14	odd	6
135.5.h.a.44.5		44	9.2	odd	6
135.5.h.a.44.18		44	45.29	odd	6
135.5.h.a.89.5		44	45.4	even	6
135.5.h.a.89.18		44	9.4	even	3
405.5.d.a.404.9		44	1.1	even	1	trivial
405.5.d.a.404.10		44	15.14	odd	2	inner
405.5.d.a.404.35		44	3.2	odd	2	inner
405.5.d.a.404.36		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)

\(f(q)\)	\(=\)	\(q-5.23381 q^{2} +11.3928 q^{4} +(3.38042 + 24.7704i) q^{5} -57.8140i q^{7} +24.1135 q^{8} +(-17.6925 - 129.644i) q^{10} -169.160i q^{11} -114.462i q^{13} +302.588i q^{14} -308.489 q^{16} -209.860 q^{17} +26.9887 q^{19} +(38.5123 + 282.203i) q^{20} +885.349i q^{22} +917.283 q^{23} +(-602.145 + 167.469i) q^{25} +599.071i q^{26} -658.661i q^{28} +356.673i q^{29} -1258.54 q^{31} +1228.76 q^{32} +1098.36 q^{34} +(1432.08 - 195.436i) q^{35} -368.530i q^{37} -141.254 q^{38} +(81.5137 + 597.300i) q^{40} -1837.31i q^{41} +320.590i q^{43} -1927.19i q^{44} -4800.88 q^{46} +2448.58 q^{47} -941.461 q^{49} +(3151.51 - 876.500i) q^{50} -1304.04i q^{52} -1898.83 q^{53} +(4190.15 - 571.831i) q^{55} -1394.10i q^{56} -1866.76i q^{58} +4859.73i q^{59} +947.679 q^{61} +6586.95 q^{62} -1495.26 q^{64} +(2835.27 - 386.930i) q^{65} -2168.49i q^{67} -2390.88 q^{68} +(-7495.21 + 1022.87i) q^{70} +9078.87i q^{71} +984.836i q^{73} +1928.81i q^{74} +307.475 q^{76} -9779.80 q^{77} -10251.0 q^{79} +(-1042.82 - 7641.40i) q^{80} +9616.15i q^{82} -4780.77 q^{83} +(-709.415 - 5198.31i) q^{85} -1677.91i q^{86} -4079.02i q^{88} +858.461i q^{89} -6617.50 q^{91} +10450.4 q^{92} -12815.4 q^{94} +(91.2332 + 668.521i) q^{95} -4265.61i q^{97} +4927.43 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) 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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