LMFDB - Embedded newform 405.5.d.a.404.22

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			404.22
Character	$\chi$	$=$	405.404
Dual form			405.5.d.a.404.21

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.633003 q^{2} -15.5993 q^{4} +(-6.01314 - 24.2661i) q^{5} +84.5447i q^{7} +20.0025 q^{8} +(3.80633 + 15.3605i) q^{10} -171.443i q^{11} +146.899i q^{13} -53.5170i q^{14} +236.927 q^{16} +273.459 q^{17} +229.656 q^{19} +(93.8008 + 378.534i) q^{20} +108.524i q^{22} -389.261 q^{23} +(-552.684 + 291.830i) q^{25} -92.9873i q^{26} -1318.84i q^{28} -354.068i q^{29} +621.826 q^{31} -470.015 q^{32} -173.100 q^{34} +(2051.57 - 508.379i) q^{35} -730.131i q^{37} -145.373 q^{38} +(-120.278 - 485.381i) q^{40} -576.092i q^{41} +904.783i q^{43} +2674.39i q^{44} +246.404 q^{46} -792.834 q^{47} -4746.80 q^{49} +(349.851 - 184.730i) q^{50} -2291.52i q^{52} -3467.44 q^{53} +(-4160.24 + 1030.91i) q^{55} +1691.10i q^{56} +224.126i q^{58} +2813.71i q^{59} -5321.66 q^{61} -393.618 q^{62} -3493.32 q^{64} +(3564.65 - 883.321i) q^{65} +170.819i q^{67} -4265.77 q^{68} +(-1298.65 + 321.805i) q^{70} -2798.11i q^{71} +5866.22i q^{73} +462.175i q^{74} -3582.48 q^{76} +14494.6 q^{77} -2933.33 q^{79} +(-1424.68 - 5749.29i) q^{80} +364.668i q^{82} -6889.32 q^{83} +(-1644.35 - 6635.77i) q^{85} -572.731i q^{86} -3429.27i q^{88} +1031.26i q^{89} -12419.5 q^{91} +6072.21 q^{92} +501.866 q^{94} +(-1380.95 - 5572.85i) q^{95} -14053.5i q^{97} +3004.74 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.633003			−0.158251			−0.0791254	−	0.996865i	$-0.525213\pi$
$2$	−0.633003			−0.158251			−0.0791254	+	0.996865i	$0.525213\pi$
$3$		0			0
$3$		0			0
$4$	−15.5993			−0.974957
$5$	−6.01314	−	24.2661i	−0.240525	−	0.970643i
$5$	−6.01314	−	24.2661i	−0.240525	−	0.970643i
$6$		0			0
$7$			84.5447i			1.72540i	0.505715	+	0.862701i	$0.331229\pi$
$7$			84.5447i			1.72540i	−0.505715	+	0.862701i	$0.668771\pi$
$8$	20.0025			0.312538
$9$		0			0
$10$	3.80633	+	15.3605i	0.0380633	+	0.153605i
$11$		−	171.443i		−	1.41688i	−0.705771	−	0.708440i	$-0.749399\pi$
$11$		−	171.443i		−	1.41688i	0.705771	−	0.708440i	$-0.250601\pi$
$12$		0			0
$13$			146.899i			0.869222i	0.900618	+	0.434611i	$0.143114\pi$
$13$			146.899i			0.869222i	−0.900618	+	0.434611i	$0.856886\pi$
$14$		−	53.5170i		−	0.273046i
$15$		0			0
$16$	236.927			0.925497
$17$	273.459			0.946224			0.473112	−	0.881002i	$-0.343131\pi$
$17$	273.459			0.946224			0.473112	+	0.881002i	$0.343131\pi$
$18$		0			0
$19$	229.656			0.636167			0.318083	−	0.948063i	$-0.396961\pi$
$19$	229.656			0.636167			0.318083	+	0.948063i	$0.396961\pi$
$20$	93.8008	+	378.534i	0.234502	+	0.946335i
$21$		0			0
$22$			108.524i			0.224222i
$23$	−389.261			−0.735844			−0.367922	−	0.929857i	$-0.619931\pi$
$23$	−389.261			−0.735844			−0.367922	+	0.929857i	$0.619931\pi$
$24$		0			0
$25$	−552.684	+	291.830i	−0.884295	+	0.466929i
$26$		−	92.9873i		−	0.137555i
$27$		0			0
$28$		−	1318.84i		−	1.68219i
$29$		−	354.068i		−	0.421008i	−0.977593	−	0.210504i	$-0.932489\pi$
$29$		−	354.068i		−	0.421008i	0.977593	−	0.210504i	$-0.0675106\pi$
$30$		0			0
$31$	621.826			0.647061			0.323531	−	0.946218i	$-0.395130\pi$
$31$	621.826			0.647061			0.323531	+	0.946218i	$0.395130\pi$
$32$	−470.015			−0.458999
$33$		0			0
$34$	−173.100			−0.149741
$35$	2051.57	−	508.379i	1.67475	−	0.415003i
$36$		0			0
$37$		−	730.131i		−	0.533332i	−0.963789	−	0.266666i	$-0.914078\pi$
$37$		−	730.131i		−	0.533332i	0.963789	−	0.266666i	$-0.0859220\pi$
$38$	−145.373			−0.100674
$39$		0			0
$40$	−120.278	−	485.381i	−0.0751735	−	0.303363i
$41$		−	576.092i		−	0.342708i	−0.985210	−	0.171354i	$-0.945186\pi$
$41$		−	576.092i		−	0.342708i	0.985210	−	0.171354i	$-0.0548142\pi$
$42$		0			0
$43$			904.783i			0.489337i	0.969607	+	0.244668i	$0.0786791\pi$
$43$			904.783i			0.489337i	−0.969607	+	0.244668i	$0.921321\pi$
$44$			2674.39i			1.38140i
$45$		0			0
$46$	246.404			0.116448
$47$	−792.834			−0.358911			−0.179455	−	0.983766i	$-0.557434\pi$
$47$	−792.834			−0.358911			−0.179455	+	0.983766i	$0.557434\pi$
$48$		0			0
$49$	−4746.80			−1.97701
$50$	349.851	−	184.730i	0.139940	−	0.0738918i
$51$		0			0
$52$		−	2291.52i		−	0.847454i
$53$	−3467.44			−1.23440			−0.617202	−	0.786805i	$-0.711734\pi$
$53$	−3467.44			−1.23440			−0.617202	+	0.786805i	$0.711734\pi$
$54$		0			0
$55$	−4160.24	+	1030.91i	−1.37529	+	0.340796i
$56$			1691.10i			0.539254i
$57$		0			0
$58$			224.126i			0.0666249i
$59$			2813.71i			0.808305i	0.914692	+	0.404152i	$0.132433\pi$
$59$			2813.71i			0.808305i	−0.914692	+	0.404152i	$0.867567\pi$
$60$		0			0
$61$	−5321.66			−1.43017			−0.715084	−	0.699038i	$-0.753612\pi$
$61$	−5321.66			−1.43017			−0.715084	+	0.699038i	$0.753612\pi$
$62$	−393.618			−0.102398
$63$		0			0
$64$	−3493.32			−0.852860
$65$	3564.65	−	883.321i	0.843705	−	0.209070i
$66$		0			0
$67$			170.819i			0.0380529i	0.999819	+	0.0190264i	$0.00605667\pi$
$67$			170.819i			0.0380529i	−0.999819	+	0.0190264i	$0.993943\pi$
$68$	−4265.77			−0.922528
$69$		0			0
$70$	−1298.65	+	321.805i	−0.265030	+	0.0656746i
$71$		−	2798.11i		−	0.555071i	−0.960715	−	0.277535i	$-0.910482\pi$
$71$		−	2798.11i		−	0.555071i	0.960715	−	0.277535i	$-0.0895176\pi$
$72$		0			0
$73$			5866.22i			1.10081i	0.834898	+	0.550405i	$0.185527\pi$
$73$			5866.22i			1.10081i	−0.834898	+	0.550405i	$0.814473\pi$
$74$			462.175i			0.0844001i
$75$		0			0
$76$	−3582.48			−0.620235
$77$	14494.6			2.44469
$78$		0			0
$79$	−2933.33			−0.470010			−0.235005	−	0.971994i	$-0.575511\pi$
$79$	−2933.33			−0.470010			−0.235005	+	0.971994i	$0.575511\pi$
$80$	−1424.68	−	5749.29i	−0.222606	−	0.898327i
$81$		0			0
$82$			364.668i			0.0542338i
$83$	−6889.32			−1.00005			−0.500023	−	0.866012i	$-0.666675\pi$
$83$	−6889.32			−1.00005			−0.500023	+	0.866012i	$0.666675\pi$
$84$		0			0
$85$	−1644.35	−	6635.77i	−0.227591	−	0.918446i
$86$		−	572.731i		−	0.0774379i
$87$		0			0
$88$		−	3429.27i		−	0.442830i
$89$			1031.26i			0.130193i	0.997879	+	0.0650966i	$0.0207356\pi$
$89$			1031.26i			0.130193i	−0.997879	+	0.0650966i	$0.979264\pi$
$90$		0			0
$91$	−12419.5			−1.49976
$92$	6072.21			0.717416
$93$		0			0
$94$	501.866			0.0567979
$95$	−1380.95	−	5572.85i	−0.153014	−	0.617490i
$96$		0			0
$97$		−	14053.5i		−	1.49362i	−0.665036	−	0.746811i	$-0.731584\pi$
$97$		−	14053.5i		−	1.49362i	0.665036	−	0.746811i	$-0.268416\pi$
$98$	3004.74			0.312864
$99$		0			0
$100$	8621.49	−	4552.35i	0.862149	−	0.455235i
$101$		−	13622.3i		−	1.33539i	−0.744434	−	0.667696i	$-0.767281\pi$
$101$		−	13622.3i		−	1.33539i	0.744434	−	0.667696i	$-0.232719\pi$
$102$		0			0
$103$		−	16346.3i		−	1.54079i	−0.637564	−	0.770397i	$-0.720058\pi$
$103$		−	16346.3i		−	1.54079i	0.637564	−	0.770397i	$-0.279942\pi$
$104$			2938.33i			0.271665i
$105$		0			0
$106$	2194.90			0.195345
$107$	−15293.8			−1.33582			−0.667909	−	0.744243i	$-0.732810\pi$
$107$	−15293.8			−1.33582			−0.667909	+	0.744243i	$0.732810\pi$
$108$		0			0
$109$	−3110.73			−0.261824			−0.130912	−	0.991394i	$-0.541791\pi$
$109$	−3110.73			−0.261824			−0.130912	+	0.991394i	$0.541791\pi$
$110$	2633.44	−	652.568i	0.217640	−	0.0539312i
$111$		0			0
$112$			20030.9i			1.59685i
$113$	−8040.95			−0.629724			−0.314862	−	0.949137i	$-0.601958\pi$
$113$	−8040.95			−0.629724			−0.314862	+	0.949137i	$0.601958\pi$
$114$		0			0
$115$	2340.68	+	9445.84i	0.176989	+	0.714241i
$116$			5523.22i			0.410465i
$117$		0			0
$118$		−	1781.09i		−	0.127915i
$119$			23119.5i			1.63262i
$120$		0			0
$121$	−14751.5			−1.00755
$122$	3368.63			0.226325
$123$		0			0
$124$	−9700.05			−0.630857
$125$	10404.9	+	11656.7i	0.665916	+	0.746026i
$126$		0			0
$127$		−	8105.72i		−	0.502555i	−0.967915	−	0.251278i	$-0.919149\pi$
$127$		−	8105.72i		−	0.502555i	0.967915	−	0.251278i	$-0.0808507\pi$
$128$	9731.52			0.593965
$129$		0			0
$130$	−2256.44	+	559.145i	−0.133517	+	0.0330855i
$131$		−	5875.90i		−	0.342398i	−0.985236	−	0.171199i	$-0.945236\pi$
$131$		−	5875.90i		−	0.342398i	0.985236	−	0.171199i	$-0.0547642\pi$
$132$		0			0
$133$			19416.2i			1.09764i
$134$		−	108.129i		−	0.00602190i
$135$		0			0
$136$	5469.85			0.295731
$137$	−35049.6			−1.86742			−0.933710	−	0.358030i	$-0.883448\pi$
$137$	−35049.6			−1.86742			−0.933710	+	0.358030i	$0.883448\pi$
$138$		0			0
$139$	11987.1			0.620416			0.310208	−	0.950669i	$-0.399601\pi$
$139$	11987.1			0.620416			0.310208	+	0.950669i	$0.399601\pi$
$140$	−32003.0	+	7930.36i	−1.63281	+	0.404610i
$141$		0			0
$142$			1771.21i			0.0878404i
$143$	25184.7			1.23158
$144$		0			0
$145$	−8591.84	+	2129.06i	−0.408649	+	0.101263i
$146$		−	3713.33i		−	0.174204i
$147$		0			0
$148$			11389.5i			0.519975i
$149$			13052.2i			0.587910i	0.955819	+	0.293955i	$0.0949715\pi$
$149$			13052.2i			0.587910i	−0.955819	+	0.293955i	$0.905028\pi$
$150$		0			0
$151$	−28884.7			−1.26682			−0.633409	−	0.773817i	$-0.718345\pi$
$151$	−28884.7			−1.26682			−0.633409	+	0.773817i	$0.718345\pi$
$152$	4593.69			0.198826
$153$		0			0
$154$	−9175.10			−0.386874
$155$	−3739.12	−	15089.3i	−0.155635	−	0.628065i
$156$		0			0
$157$			20788.9i			0.843399i	0.906736	+	0.421700i	$0.138566\pi$
$157$			20788.9i			0.843399i	−0.906736	+	0.421700i	$0.861434\pi$
$158$	1856.81			0.0743794
$159$		0			0
$160$	2826.26	+	11405.4i	0.110401	+	0.445524i
$161$		−	32910.0i		−	1.26963i
$162$		0			0
$163$		−	24195.2i		−	0.910654i	−0.890324	−	0.455327i	$-0.849522\pi$
$163$		−	24195.2i		−	0.910654i	0.890324	−	0.455327i	$-0.150478\pi$
$164$			8986.64i			0.334125i
$165$		0			0
$166$	4360.96			0.158258
$167$	10368.5			0.371777			0.185889	−	0.982571i	$-0.440484\pi$
$167$	10368.5			0.371777			0.185889	+	0.982571i	$0.440484\pi$
$168$		0			0
$169$	6981.80			0.244452
$170$	1040.88	+	4200.46i	0.0360165	+	0.145345i
$171$		0			0
$172$		−	14114.0i		−	0.477082i
$173$	38227.9			1.27729			0.638644	−	0.769502i	$-0.279496\pi$
$173$	38227.9			1.27729			0.638644	+	0.769502i	$0.279496\pi$
$174$		0			0
$175$	−24672.7	−	46726.5i	−0.805640	−	1.52576i
$176$		−	40619.4i		−	1.31132i
$177$		0			0
$178$		−	652.791i		−	0.0206032i
$179$			29062.0i			0.907025i	0.891250	+	0.453512i	$0.149829\pi$
$179$			29062.0i			0.907025i	−0.891250	+	0.453512i	$0.850171\pi$
$180$		0			0
$181$	23849.6			0.727988			0.363994	−	0.931401i	$-0.381413\pi$
$181$	23849.6			0.727988			0.363994	+	0.931401i	$0.381413\pi$
$182$	7861.58			0.237338
$183$		0			0
$184$	−7786.18			−0.229979
$185$	−17717.4	+	4390.38i	−0.517674	+	0.128280i
$186$		0			0
$187$		−	46882.5i		−	1.34069i
$188$	12367.7			0.349923
$189$		0			0
$190$	874.148	+	3527.63i	0.0242146	+	0.0977183i
$191$			5746.67i			0.157525i	0.996893	+	0.0787626i	$0.0250969\pi$
$191$			5746.67i			0.157525i	−0.996893	+	0.0787626i	$0.974903\pi$
$192$		0			0
$193$			6266.80i			0.168241i	0.996456	+	0.0841204i	$0.0268080\pi$
$193$			6266.80i			0.168241i	−0.996456	+	0.0841204i	$0.973192\pi$
$194$			8895.91i			0.236367i
$195$		0			0
$196$	74046.9			1.92750
$197$	−55114.0			−1.42013			−0.710067	−	0.704134i	$-0.751335\pi$
$197$	−55114.0			−1.42013			−0.710067	+	0.704134i	$0.751335\pi$
$198$		0			0
$199$	10375.3			0.261995			0.130998	−	0.991383i	$-0.458182\pi$
$199$	10375.3			0.261995			0.130998	+	0.991383i	$0.458182\pi$
$200$	−11055.0	+	5837.33i	−0.276376	+	0.145933i
$201$		0			0
$202$			8622.98i			0.211327i
$203$	29934.6			0.726409
$204$		0			0
$205$	−13979.5	+	3464.12i	−0.332647	+	0.0824300i
$206$			10347.3i			0.243832i
$207$		0			0
$208$			34804.3i			0.804463i
$209$		−	39372.8i		−	0.901372i
$210$		0			0
$211$	51797.2			1.16343			0.581716	−	0.813392i	$-0.302381\pi$
$211$	51797.2			1.16343			0.581716	+	0.813392i	$0.302381\pi$
$212$	54089.7			1.20349
$213$		0			0
$214$	9681.01			0.211394
$215$	21955.5	−	5440.59i	0.474971	−	0.117698i
$216$		0			0
$217$			52572.1i			1.11644i
$218$	1969.10			0.0414339
$219$		0			0
$220$	64896.8	−	16081.4i	1.34084	−	0.332261i
$221$			40170.7i			0.822479i
$222$		0			0
$223$			25702.7i			0.516856i	0.966031	+	0.258428i	$0.0832045\pi$
$223$			25702.7i			0.516856i	−0.966031	+	0.258428i	$0.916796\pi$
$224$		−	39737.3i		−	0.791958i
$225$		0			0
$226$	5089.94			0.0996543
$227$	−34229.0			−0.664267			−0.332133	−	0.943232i	$-0.607768\pi$
$227$	−34229.0			−0.664267			−0.332133	+	0.943232i	$0.607768\pi$
$228$		0			0
$229$	5239.96			0.0999210			0.0499605	−	0.998751i	$-0.484090\pi$
$229$	5239.96			0.0999210			0.0499605	+	0.998751i	$0.484090\pi$
$230$	−1481.66	−	5979.25i	−0.0280087	−	0.113029i
$231$		0			0
$232$		−	7082.23i		−	0.131581i
$233$	−44606.8			−0.821654			−0.410827	−	0.911713i	$-0.634760\pi$
$233$	−44606.8			−0.821654			−0.410827	+	0.911713i	$0.634760\pi$
$234$		0			0
$235$	4767.42	+	19239.0i	0.0863272	+	0.348374i
$236$		−	43891.9i		−	0.788062i
$237$		0			0
$238$		−	14634.7i		−	0.258363i
$239$		−	36079.3i		−	0.631630i	−0.948821	−	0.315815i	$-0.897722\pi$
$239$		−	36079.3i		−	0.631630i	0.948821	−	0.315815i	$-0.102278\pi$
$240$		0			0
$241$	34036.5			0.586018			0.293009	−	0.956110i	$-0.405343\pi$
$241$	34036.5			0.586018			0.293009	+	0.956110i	$0.405343\pi$
$242$	9337.78			0.159446
$243$		0			0
$244$	83014.2			1.39435
$245$	28543.2	+	115186.i	0.475522	+	1.91897i
$246$		0			0
$247$			33736.2i			0.552970i
$248$	12438.0			0.202231
$249$		0			0
$250$	−6586.36	−	7378.70i	−0.105382	−	0.118059i
$251$		−	60480.3i		−	0.959990i	−0.877271	−	0.479995i	$-0.840639\pi$
$251$		−	60480.3i		−	0.959990i	0.877271	−	0.479995i	$-0.159361\pi$
$252$		0			0
$253$			66735.9i			1.04260i
$254$			5130.94i			0.0795298i
$255$		0			0
$256$	49733.0			0.758865
$257$	−15239.6			−0.230731			−0.115366	−	0.993323i	$-0.536804\pi$
$257$	−15239.6			−0.230731			−0.115366	+	0.993323i	$0.536804\pi$
$258$		0			0
$259$	61728.7			0.920211
$260$	−55606.1	+	13779.2i	−0.822575	+	0.203834i
$261$		0			0
$262$			3719.46i			0.0541848i
$263$	−6461.85			−0.0934212			−0.0467106	−	0.998908i	$-0.514874\pi$
$263$	−6461.85			−0.0934212			−0.0467106	+	0.998908i	$0.514874\pi$
$264$		0			0
$265$	20850.2	+	84141.1i	0.296906	+	1.19817i
$266$		−	12290.5i		−	0.173703i
$267$		0			0
$268$		−	2664.66i		−	0.0370999i
$269$			77325.2i			1.06860i	0.845294	+	0.534301i	$0.179425\pi$
$269$			77325.2i			1.06860i	−0.845294	+	0.534301i	$0.820575\pi$
$270$		0			0
$271$	−5856.99			−0.0797510			−0.0398755	−	0.999205i	$-0.512696\pi$
$271$	−5856.99			−0.0797510			−0.0398755	+	0.999205i	$0.512696\pi$
$272$	64789.9			0.875728
$273$		0			0
$274$	22186.5			0.295521
$275$	50032.2	+	94753.6i	0.661582	+	1.25294i
$276$		0			0
$277$			52293.4i			0.681533i	0.940148	+	0.340767i	$0.110687\pi$
$277$			52293.4i			0.681533i	−0.940148	+	0.340767i	$0.889313\pi$
$278$	−7587.84			−0.0981813
$279$		0			0
$280$	41036.4	−	10168.8i	0.523423	−	0.129704i
$281$		−	10482.5i		−	0.132756i	−0.997795	−	0.0663779i	$-0.978856\pi$
$281$		−	10482.5i		−	0.132756i	0.997795	−	0.0663779i	$-0.0211443\pi$
$282$		0			0
$283$			5558.31i			0.0694017i	0.999398	+	0.0347009i	$0.0110478\pi$
$283$			5558.31i			0.0694017i	−0.999398	+	0.0347009i	$0.988952\pi$
$284$			43648.6i			0.541170i
$285$		0			0
$286$	−15942.0			−0.194899
$287$	48705.5			0.591309
$288$		0			0
$289$	−8741.28			−0.104660
$290$	5438.66	−	1347.70i	0.0646690	−	0.0160250i
$291$		0			0
$292$		−	91509.0i		−	1.07324i
$293$	−150754.			−1.75603			−0.878017	−	0.478629i	$-0.841134\pi$
$293$	−150754.			−1.75603			−0.878017	+	0.478629i	$0.841134\pi$
$294$		0			0
$295$	68277.7	−	16919.2i	0.784575	−	0.194418i
$296$		−	14604.4i		−	0.166687i
$297$		0			0
$298$		−	8262.07i		−	0.0930371i
$299$		−	57181.9i		−	0.639612i
$300$		0			0
$301$	−76494.6			−0.844302
$302$	18284.1			0.200475
$303$		0			0
$304$	54411.8			0.588770
$305$	31999.9	+	129136.i	0.343992	+	1.38818i
$306$		0			0
$307$		−	140225.i		−	1.48781i	−0.668284	−	0.743906i	$-0.732971\pi$
$307$		−	140225.i		−	1.48781i	0.668284	−	0.743906i	$-0.267029\pi$
$308$	−226105.			−2.38347
$309$		0			0
$310$	2366.88	+	9551.55i	0.0246293	+	0.0993918i
$311$		−	190742.i		−	1.97208i	−0.166500	−	0.986041i	$-0.553247\pi$
$311$		−	190742.i		−	1.97208i	0.166500	−	0.986041i	$-0.446753\pi$
$312$		0			0
$313$		−	33669.6i		−	0.343676i	−0.985125	−	0.171838i	$-0.945029\pi$
$313$		−	33669.6i		−	0.343676i	0.985125	−	0.171838i	$-0.0549706\pi$
$314$		−	13159.5i		−	0.133469i
$315$		0			0
$316$	45757.9			0.458239
$317$	78496.1			0.781141			0.390571	−	0.920573i	$-0.372278\pi$
$317$	78496.1			0.781141			0.390571	+	0.920573i	$0.372278\pi$
$318$		0			0
$319$	−60702.3			−0.596519
$320$	21005.8	+	84769.0i	0.205135	+	0.827823i
$321$		0			0
$322$			20832.1i			0.200919i
$323$	62801.5			0.601956
$324$		0			0
$325$	−42869.5	−	81188.6i	−0.405865	−	0.768649i
$326$			15315.6i			0.144112i
$327$		0			0
$328$		−	11523.3i		−	0.107109i
$329$		−	67029.9i		−	0.619265i
$330$		0			0
$331$	−172493.			−1.57440			−0.787199	−	0.616699i	$-0.788469\pi$
$331$	−172493.			−1.57440			−0.787199	+	0.616699i	$0.788469\pi$
$332$	107469.			0.975001
$333$		0			0
$334$	−6563.29			−0.0588340
$335$	4145.12	−	1027.16i	0.0369358	−	0.00915269i
$336$		0			0
$337$		−	32299.9i		−	0.284408i	−0.989837	−	0.142204i	$-0.954581\pi$
$337$		−	32299.9i		−	0.284408i	0.989837	−	0.142204i	$-0.0454189\pi$
$338$	−4419.50			−0.0386848
$339$		0			0
$340$	25650.6	+	103513.i	0.221891	+	0.895445i
$341$		−	106607.i		−	0.916808i
$342$		0			0
$343$		−	198325.i		−	1.68574i
$344$			18097.9i			0.152936i
$345$		0			0
$346$	−24198.4			−0.202132
$347$	−232255.			−1.92888			−0.964440	−	0.264300i	$-0.914859\pi$
$347$	−232255.			−1.92888			−0.964440	+	0.264300i	$0.914859\pi$
$348$		0			0
$349$	−165955.			−1.36251			−0.681256	−	0.732045i	$-0.738566\pi$
$349$	−165955.			−1.36251			−0.681256	+	0.732045i	$0.738566\pi$
$350$	15617.9	+	29578.0i	0.127493	+	0.241453i
$351$		0			0
$352$			80580.6i			0.650347i
$353$	129181.			1.03669			0.518345	−	0.855172i	$-0.326548\pi$
$353$	129181.			1.03669			0.518345	+	0.855172i	$0.326548\pi$
$354$		0			0
$355$	−67899.2	+	16825.4i	−0.538775	+	0.133509i
$356$		−	16087.0i		−	0.126933i
$357$		0			0
$358$		−	18396.3i		−	0.143537i
$359$		−	205368.i		−	1.59347i	−0.604331	−	0.796734i	$-0.706559\pi$
$359$		−	205368.i		−	1.59347i	0.604331	−	0.796734i	$-0.293441\pi$
$360$		0			0
$361$	−77579.1			−0.595292
$362$	−15096.9			−0.115205
$363$		0			0
$364$	193736.			1.46220
$365$	142350.	−	35274.4i	1.06849	−	0.264773i
$366$		0			0
$367$			103715.i			0.770036i	0.922909	+	0.385018i	$0.125805\pi$
$367$			103715.i			0.770036i	−0.922909	+	0.385018i	$0.874195\pi$
$368$	−92226.6			−0.681021
$369$		0			0
$370$	11215.2	−	2779.12i	0.0819224	−	0.0203004i
$371$		−	293154.i		−	2.12984i
$372$		0			0
$373$			180353.i			1.29630i	0.761511	+	0.648152i	$0.224458\pi$
$373$			180353.i			1.29630i	−0.761511	+	0.648152i	$0.775542\pi$
$374$			29676.8i			0.212165i
$375$		0			0
$376$	−15858.6			−0.112173
$377$	52012.1			0.365950
$378$		0			0
$379$	−145168.			−1.01063			−0.505317	−	0.862934i	$-0.668624\pi$
$379$	−145168.			−1.01063			−0.505317	+	0.862934i	$0.668624\pi$
$380$	21541.9	+	86932.6i	0.149182	+	0.602026i
$381$		0			0
$382$		−	3637.66i		−	0.0249285i
$383$	113438.			0.773322			0.386661	−	0.922222i	$-0.373628\pi$
$383$	113438.			0.773322			0.386661	+	0.922222i	$0.373628\pi$
$384$		0			0
$385$	−87157.8	−	351726.i	−0.588010	−	2.37292i
$386$		−	3966.91i		−	0.0266242i
$387$		0			0
$388$			219225.i			1.45622i
$389$			264820.i			1.75005i	0.484076	+	0.875026i	$0.339156\pi$
$389$			264820.i			1.75005i	−0.484076	+	0.875026i	$0.660844\pi$
$390$		0			0
$391$	−106447.			−0.696273
$392$	−94947.8			−0.617892
$393$		0			0
$394$	34887.3			0.224737
$395$	17638.5	+	71180.4i	0.113049	+	0.456212i
$396$		0			0
$397$			82700.5i			0.524719i	0.964970	+	0.262360i	$0.0845007\pi$
$397$			82700.5i			0.524719i	−0.964970	+	0.262360i	$0.915499\pi$
$398$	−6567.59			−0.0414610
$399$		0			0
$400$	−130946.	+	69142.6i	−0.818413	+	0.432141i
$401$			24096.4i			0.149852i	0.997189	+	0.0749261i	$0.0238721\pi$
$401$			24096.4i			0.149852i	−0.997189	+	0.0749261i	$0.976128\pi$
$402$		0			0
$403$			91345.3i			0.562440i
$404$			212499.i			1.30195i
$405$		0			0
$406$	−18948.7			−0.114955
$407$	−125175.			−0.755667
$408$		0			0
$409$	−166616.			−0.996024			−0.498012	−	0.867170i	$-0.665936\pi$
$409$	−166616.			−0.996024			−0.498012	+	0.867170i	$0.665936\pi$
$410$	8849.06	−	2192.80i	0.0526417	−	0.0130446i
$411$		0			0
$412$			254991.i			1.50221i
$413$	−237884.			−1.39465
$414$		0			0
$415$	41426.4	+	167177.i	0.240537	+	0.970687i
$416$		−	69044.6i		−	0.398972i
$417$		0			0
$418$			24923.1i			0.142643i
$419$			107998.i			0.615161i	0.951522	+	0.307581i	$0.0995194\pi$
$419$			107998.i			0.615161i	−0.951522	+	0.307581i	$0.900481\pi$
$420$		0			0
$421$	258850.			1.46044			0.730222	−	0.683210i	$-0.239417\pi$
$421$	258850.			1.46044			0.730222	+	0.683210i	$0.239417\pi$
$422$	−32787.8			−0.184114
$423$		0			0
$424$	−69357.3			−0.385799
$425$	−151136.	+	79803.6i	−0.836741	+	0.441819i
$426$		0			0
$427$		−	449918.i		−	2.46762i
$428$	238572.			1.30236
$429$		0			0
$430$	−13897.9	+	3443.91i	−0.0751645	+	0.0186258i
$431$			81054.9i			0.436340i	0.975911	+	0.218170i	$0.0700087\pi$
$431$			81054.9i			0.436340i	−0.975911	+	0.218170i	$0.929991\pi$
$432$		0			0
$433$		−	173783.i		−	0.926897i	−0.886124	−	0.463448i	$-0.846612\pi$
$433$		−	173783.i		−	0.926897i	0.886124	−	0.463448i	$-0.153388\pi$
$434$		−	33278.3i		−	0.176678i
$435$		0			0
$436$	48525.3			0.255267
$437$	−89396.2			−0.468119
$438$		0			0
$439$	−258879.			−1.34328			−0.671642	−	0.740876i	$-0.734411\pi$
$439$	−258879.			−1.34328			−0.671642	+	0.740876i	$0.734411\pi$
$440$	−83215.0	+	20620.7i	−0.429829	+	0.106512i
$441$		0			0
$442$		−	25428.2i		−	0.130158i
$443$	192465.			0.980717			0.490358	−	0.871521i	$-0.336866\pi$
$443$	192465.			0.980717			0.490358	+	0.871521i	$0.336866\pi$
$444$		0			0
$445$	25024.6	−	6201.11i	0.126371	−	0.0313148i
$446$		−	16269.9i		−	0.0817929i
$447$		0			0
$448$		−	295341.i		−	1.47153i
$449$		−	56772.8i		−	0.281610i	−0.990037	−	0.140805i	$-0.955031\pi$
$449$		−	56772.8i		−	0.281610i	0.990037	−	0.140805i	$-0.0449690\pi$
$450$		0			0
$451$	−98766.7			−0.485576
$452$	125433.			0.613954
$453$		0			0
$454$	21667.1			0.105121
$455$	74680.1	+	301372.i	0.360730	+	1.45573i
$456$		0			0
$457$		−	313035.i		−	1.49886i	−0.662084	−	0.749429i	$-0.730328\pi$
$457$		−	313035.i		−	1.49886i	0.662084	−	0.749429i	$-0.269672\pi$
$458$	−3316.91			−0.0158126
$459$		0			0
$460$	−36513.0	−	147349.i	−0.172557	−	0.696354i
$461$		−	61769.5i		−	0.290651i	−0.989384	−	0.145326i	$-0.953577\pi$
$461$		−	61769.5i		−	0.290651i	0.989384	−	0.145326i	$-0.0464230\pi$
$462$		0			0
$463$		−	169715.i		−	0.791695i	−0.918316	−	0.395847i	$-0.870451\pi$
$463$		−	169715.i		−	0.791695i	0.918316	−	0.395847i	$-0.129549\pi$
$464$		−	83888.4i		−	0.389642i
$465$		0			0
$466$	28236.2			0.130027
$467$	76633.3			0.351386			0.175693	−	0.984445i	$-0.443783\pi$
$467$	76633.3			0.351386			0.175693	+	0.984445i	$0.443783\pi$
$468$		0			0
$469$	−14441.9			−0.0656565
$470$	−3017.79	−	12178.3i	−0.0136613	−	0.0551305i
$471$		0			0
$472$			56281.1i			0.252626i
$473$	155118.			0.693331
$474$		0			0
$475$	−126927.	+	67020.6i	−0.562559	+	0.297044i
$476$		−	360648.i		−	1.59173i
$477$		0			0
$478$			22838.3i			0.0999559i
$479$			35139.5i			0.153153i	0.997064	+	0.0765763i	$0.0243989\pi$
$479$			35139.5i			0.153153i	−0.997064	+	0.0765763i	$0.975601\pi$
$480$		0			0
$481$	107255.			0.463584
$482$	−21545.2			−0.0927379
$483$		0			0
$484$	230114.			0.982318
$485$	−341023.	+	84505.6i	−1.44977	+	0.359254i
$486$		0			0
$487$			290883.i			1.22648i	0.789896	+	0.613240i	$0.210134\pi$
$487$			290883.i			1.22648i	−0.789896	+	0.613240i	$0.789866\pi$
$488$	−106446.			−0.446983
$489$		0			0
$490$	−18067.9	−	72913.3i	−0.0752517	−	0.303679i
$491$		−	443607.i		−	1.84007i	−0.391832	−	0.920037i	$-0.628158\pi$
$491$		−	443607.i		−	1.84007i	0.391832	−	0.920037i	$-0.371842\pi$
$492$		0			0
$493$		−	96823.1i		−	0.398368i
$494$		−	21355.1i		−	0.0875080i
$495$		0			0
$496$	147327.			0.598853
$497$	236565.			0.957720
$498$		0			0
$499$	−289323.			−1.16193			−0.580967	−	0.813927i	$-0.697325\pi$
$499$	−289323.			−1.16193			−0.580967	+	0.813927i	$0.697325\pi$
$500$	−162310.	−	181836.i	−0.649240	−	0.727343i
$501$		0			0
$502$			38284.2i			0.151919i
$503$	349897.			1.38294			0.691472	−	0.722403i	$-0.256963\pi$
$503$	349897.			1.38294			0.691472	+	0.722403i	$0.256963\pi$
$504$		0			0
$505$	−330561.	+	81913.0i	−1.29619	+	0.321196i
$506$		−	42244.1i		−	0.164993i
$507$		0			0
$508$			126444.i			0.489970i
$509$			469357.i			1.81162i	0.423679	+	0.905812i	$0.360738\pi$
$509$			469357.i			1.81162i	−0.423679	+	0.905812i	$0.639262\pi$
$510$		0			0
$511$	−495958.			−1.89934
$512$	−187185.			−0.714056
$513$		0			0
$514$	9646.70			0.0365134
$515$	−396660.	+	98292.5i	−1.49556	+	0.370600i
$516$		0			0
$517$			135925.i			0.508534i
$518$	−39074.4			−0.145624
$519$		0			0
$520$	71301.8	−	17668.6i	0.263690	−	0.0653425i
$521$		−	430054.i		−	1.58434i	−0.610302	−	0.792169i	$-0.708952\pi$
$521$		−	430054.i		−	1.58434i	0.610302	−	0.792169i	$-0.291048\pi$
$522$		0			0
$523$			21279.4i			0.0777959i	0.999243	+	0.0388980i	$0.0123847\pi$
$523$			21279.4i			0.0777959i	−0.999243	+	0.0388980i	$0.987615\pi$
$524$			91660.0i			0.333824i
$525$		0			0
$526$	4090.37			0.0147840
$527$	170044.			0.612265
$528$		0			0
$529$	−128317.			−0.458534
$530$	−13198.2	−	53261.6i	−0.0469855	−	0.189611i
$531$		0			0
$532$		−	302879.i		−	1.07015i
$533$	84627.1			0.297890
$534$		0			0
$535$	91963.6	+	371120.i	0.321298	+	1.29660i
$536$			3416.81i			0.0118930i
$537$		0			0
$538$		−	48947.1i		−	0.169107i
$539$			813804.i			2.80119i
$540$		0			0
$541$	−127926.			−0.437083			−0.218541	−	0.975828i	$-0.570130\pi$
$541$	−127926.			−0.437083			−0.218541	+	0.975828i	$0.570130\pi$
$542$	3707.50			0.0126207
$543$		0			0
$544$	−128530.			−0.434316
$545$	18705.3	+	75485.2i	0.0629754	+	0.254138i
$546$		0			0
$547$			505163.i			1.68833i	0.536084	+	0.844165i	$0.319903\pi$
$547$			505163.i			1.68833i	−0.536084	+	0.844165i	$0.680097\pi$
$548$	546750.			1.82065
$549$		0			0
$550$	−31670.5	−	59979.3i	−0.104696	−	0.198279i
$551$		−	81313.9i		−	0.267832i
$552$		0			0
$553$		−	247998.i		−	0.810956i
$554$		−	33101.9i		−	0.107853i
$555$		0			0
$556$	−186990.			−0.604879
$557$	277756.			0.895268			0.447634	−	0.894217i	$-0.352267\pi$
$557$	277756.			0.895268			0.447634	+	0.894217i	$0.352267\pi$
$558$		0			0
$559$	−132911.			−0.425342
$560$	486072.	−	120449.i	1.54998	−	0.384084i
$561$		0			0
$562$			6635.47i			0.0210087i
$563$	−160015.			−0.504828			−0.252414	−	0.967619i	$-0.581225\pi$
$563$	−160015.			−0.504828			−0.252414	+	0.967619i	$0.581225\pi$
$564$		0			0
$565$	48351.3	+	195122.i	0.151465	+	0.611237i
$566$		−	3518.43i		−	0.0109829i
$567$		0			0
$568$		−	55969.1i		−	0.173481i
$569$		−	176175.i		−	0.544152i	−0.962276	−	0.272076i	$-0.912290\pi$
$569$		−	176175.i		−	0.544152i	0.962276	−	0.272076i	$-0.0877103\pi$
$570$		0			0
$571$	−455004.			−1.39554			−0.697770	−	0.716321i	$-0.745824\pi$
$571$	−455004.			−1.39554			−0.697770	+	0.716321i	$0.745824\pi$
$572$	−392863.			−1.20074
$573$		0			0
$574$	−30830.8			−0.0935751
$575$	215139.	−	113598.i	0.650703	−	0.343586i
$576$		0			0
$577$		−	356530.i		−	1.07089i	−0.844571	−	0.535444i	$-0.820144\pi$
$577$		−	356530.i		−	1.07089i	0.844571	−	0.535444i	$-0.179856\pi$
$578$	5533.26			0.0165625
$579$		0			0
$580$	134027.	−	33211.9i	0.398415	−	0.0987273i
$581$		−	582455.i		−	1.72548i
$582$		0			0
$583$			594467.i			1.74900i
$584$			117339.i			0.344046i
$585$		0			0
$586$	95427.6			0.277894
$587$	−506136.			−1.46889			−0.734447	−	0.678666i	$-0.762559\pi$
$587$	−506136.			−1.46889			−0.734447	+	0.678666i	$0.762559\pi$
$588$		0			0
$589$	142806.			0.411639
$590$	−43220.0	+	10709.9i	−0.124160	+	0.0307668i
$591$		0			0
$592$		−	172988.i		−	0.493597i
$593$	−412533.			−1.17314			−0.586570	−	0.809899i	$-0.699522\pi$
$593$	−412533.			−1.17314			−0.586570	+	0.809899i	$0.699522\pi$
$594$		0			0
$595$	561019.	−	139021.i	1.58469	−	0.392686i
$596$		−	203605.i		−	0.573186i
$597$		0			0
$598$			36196.3i			0.101219i
$599$		−	49852.4i		−	0.138942i	−0.997584	−	0.0694708i	$-0.977869\pi$
$599$		−	49852.4i		−	0.138942i	0.997584	−	0.0694708i	$-0.0221311\pi$
$600$		0			0
$601$	627522.			1.73732			0.868660	−	0.495409i	$-0.164982\pi$
$601$	627522.			1.73732			0.868660	+	0.495409i	$0.164982\pi$
$602$	48421.3			0.133611
$603$		0			0
$604$	450581.			1.23509
$605$	88703.1	+	357962.i	0.242342	+	0.977972i
$606$		0			0
$607$			324895.i			0.881792i	0.897558	+	0.440896i	$0.145339\pi$
$607$			324895.i			0.881792i	−0.897558	+	0.440896i	$0.854661\pi$
$608$	−107942.			−0.292000
$609$		0			0
$610$	−20256.0	−	81743.3i	−0.0544370	−	0.219681i
$611$		−	116466.i		−	0.311973i
$612$		0			0
$613$			53253.9i			0.141720i	0.997486	+	0.0708599i	$0.0225743\pi$
$613$			53253.9i			0.141720i	−0.997486	+	0.0708599i	$0.977426\pi$
$614$			88762.7i			0.235447i
$615$		0			0
$616$	289927.			0.764059
$617$	373316.			0.980632			0.490316	−	0.871545i	$-0.336881\pi$
$617$	373316.			0.980632			0.490316	+	0.871545i	$0.336881\pi$
$618$		0			0
$619$	502070.			1.31034			0.655169	−	0.755482i	$-0.272597\pi$
$619$	502070.			1.31034			0.655169	+	0.755482i	$0.272597\pi$
$620$	58327.7	+	235382.i	0.151737	+	0.612336i
$621$		0			0
$622$			120740.i			0.312084i
$623$	−87187.6			−0.224636
$624$		0			0
$625$	220295.	−	322580.i	0.563955	−	0.825805i
$626$			21313.0i			0.0543871i
$627$		0			0
$628$		−	324293.i		−	0.822278i
$629$		−	199661.i		−	0.504651i
$630$		0			0
$631$	−379130.			−0.952202			−0.476101	−	0.879391i	$-0.657950\pi$
$631$	−379130.			−0.952202			−0.476101	+	0.879391i	$0.657950\pi$
$632$	−58673.8			−0.146896
$633$		0			0
$634$	−49688.3			−0.123616
$635$	−196694.	+	48740.8i	−0.487802	+	0.120877i
$636$		0			0
$637$		−	697299.i		−	1.71846i
$638$	38424.8			0.0943996
$639$		0			0
$640$	−58517.0	−	236146.i	−0.142864	−	0.576528i
$641$		−	376044.i		−	0.915215i	−0.889154	−	0.457607i	$-0.848707\pi$
$641$		−	376044.i		−	0.915215i	0.889154	−	0.457607i	$-0.151293\pi$
$642$		0			0
$643$		−	162660.i		−	0.393421i	−0.980462	−	0.196711i	$-0.936974\pi$
$643$		−	162660.i		−	0.393421i	0.980462	−	0.196711i	$-0.0630260\pi$
$644$			513373.i			1.23783i
$645$		0			0
$646$	−39753.5			−0.0952600
$647$	−344019.			−0.821814			−0.410907	−	0.911677i	$-0.634788\pi$
$647$	−344019.			−0.821814			−0.410907	+	0.911677i	$0.634788\pi$
$648$		0			0
$649$	482389.			1.14527
$650$	27136.5	+	51392.6i	0.0642284	+	0.121639i
$651$		0			0
$652$			377428.i			0.887848i
$653$	−526564.			−1.23488			−0.617440	−	0.786618i	$-0.711830\pi$
$653$	−526564.			−1.23488			−0.617440	+	0.786618i	$0.711830\pi$
$654$		0			0
$655$	−142585.	+	35332.6i	−0.332347	+	0.0823556i
$656$		−	136492.i		−	0.317175i
$657$		0			0
$658$			42430.1i			0.0979992i
$659$			482106.i			1.11012i	0.831809	+	0.555062i	$0.187305\pi$
$659$			482106.i			1.11012i	−0.831809	+	0.555062i	$0.812695\pi$
$660$		0			0
$661$	371946.			0.851288			0.425644	−	0.904891i	$-0.360048\pi$
$661$	371946.			0.851288			0.425644	+	0.904891i	$0.360048\pi$
$662$	109188.			0.249150
$663$		0			0
$664$	−137803.			−0.312553
$665$	471155.	−	116752.i	1.06542	−	0.264011i
$666$		0			0
$667$			137825.i			0.309796i
$668$	−161741.			−0.362467
$669$		0			0
$670$	−2623.87	+	650.196i	−0.00584511	+	0.00144842i
$671$			912358.i			2.02638i
$672$		0			0
$673$			281183.i			0.620811i	0.950604	+	0.310406i	$0.100465\pi$
$673$			281183.i			0.620811i	−0.950604	+	0.310406i	$0.899535\pi$
$674$			20446.0i			0.0450078i
$675$		0			0
$676$	−108911.			−0.238330
$677$	269535.			0.588083			0.294041	−	0.955793i	$-0.405000\pi$
$677$	269535.			0.588083			0.294041	+	0.955793i	$0.405000\pi$
$678$		0			0
$679$	1.18815e6			2.57710
$680$	−32890.9	−	132732.i	−0.0711309	−	0.287050i
$681$		0			0
$682$			67482.8i			0.145086i
$683$	−425001.			−0.911063			−0.455531	−	0.890220i	$-0.650551\pi$
$683$	−425001.			−0.911063			−0.455531	+	0.890220i	$0.650551\pi$
$684$		0			0
$685$	210758.	+	850516.i	0.449162	+	1.81260i
$686$			125541.i			0.266769i
$687$		0			0
$688$			214368.i			0.452880i
$689$		−	509362.i		−	1.07297i
$690$		0			0
$691$	409423.			0.857465			0.428732	−	0.903432i	$-0.358960\pi$
$691$	409423.			0.857465			0.428732	+	0.903432i	$0.358960\pi$
$692$	−596329.			−1.24530
$693$		0			0
$694$	147018.			0.305247
$695$	−72079.8	−	290879.i	−0.149226	−	0.602202i
$696$		0			0
$697$		−	157537.i		−	0.324279i
$698$	105050.			0.215619
$699$		0			0
$700$	384877.	+	728901.i	0.785464	+	1.48755i
$701$			590764.i			1.20220i	0.799172	+	0.601102i	$0.205272\pi$
$701$			590764.i			1.20220i	−0.799172	+	0.601102i	$0.794728\pi$
$702$		0			0
$703$		−	167679.i		−	0.339288i
$704$			598903.i			1.20840i
$705$		0			0
$706$	−81771.9			−0.164057
$707$	1.15170e6			2.30409
$708$		0			0
$709$	287453.			0.571840			0.285920	−	0.958253i	$-0.407701\pi$
$709$	287453.			0.571840			0.285920	+	0.958253i	$0.407701\pi$
$710$	42980.4	−	10650.5i	0.0852616	−	0.0211278i
$711$		0			0
$712$			20627.7i			0.0406904i
$713$	−242053.			−0.476136
$714$		0			0
$715$	−151439.	−	611133.i	−0.296227	−	1.19543i
$716$		−	453347.i		−	0.884310i
$717$		0			0
$718$			129998.i			0.252167i
$719$		−	645625.i		−	1.24889i	−0.781070	−	0.624443i	$-0.785326\pi$
$719$		−	645625.i		−	1.24889i	0.781070	−	0.624443i	$-0.214674\pi$
$720$		0			0
$721$	1.38199e6			2.65849
$722$	49107.8			0.0942054
$723$		0			0
$724$	−372038.			−0.709757
$725$	103328.	+	195688.i	0.196581	+	0.372296i
$726$		0			0
$727$		−	534521.i		−	1.01134i	−0.862728	−	0.505669i	$-0.831246\pi$
$727$		−	534521.i		−	1.01134i	0.862728	−	0.505669i	$-0.168754\pi$
$728$	−248420.			−0.468732
$729$		0			0
$730$	−90108.0	+	22328.8i	−0.169090	+	0.0419005i
$731$			247421.i			0.463022i
$732$		0			0
$733$			188968.i			0.351706i	0.984416	+	0.175853i	$0.0562683\pi$
$733$			188968.i			0.351706i	−0.984416	+	0.175853i	$0.943732\pi$
$734$		−	65652.1i		−	0.121859i
$735$		0			0
$736$	182959.			0.337752
$737$	29285.7			0.0539164
$738$		0			0
$739$	−683135.			−1.25089			−0.625443	−	0.780270i	$-0.715082\pi$
$739$	−683135.			−1.25089			−0.625443	+	0.780270i	$0.715082\pi$
$740$	276379.	−	68486.8i	0.504710	−	0.125067i
$741$		0			0
$742$			185567.i			0.337049i
$743$	−301638.			−0.546397			−0.273198	−	0.961958i	$-0.588082\pi$
$743$	−301638.			−0.546397			−0.273198	+	0.961958i	$0.588082\pi$
$744$		0			0
$745$	316725.	−	78484.6i	0.570650	−	0.141407i
$746$		−	114164.i		−	0.205141i
$747$		0			0
$748$			731334.i			1.30711i
$749$		−	1.29301e6i		−	2.30482i
$750$		0			0
$751$	230008.			0.407814			0.203907	−	0.978990i	$-0.434636\pi$
$751$	230008.			0.407814			0.203907	+	0.978990i	$0.434636\pi$
$752$	−187844.			−0.332171
$753$		0			0
$754$	−32923.8			−0.0579119
$755$	173688.	+	700918.i	0.304702	+	1.22963i
$756$		0			0
$757$			897739.i			1.56660i	0.621644	+	0.783300i	$0.286465\pi$
$757$			897739.i			1.56660i	−0.621644	+	0.783300i	$0.713535\pi$
$758$	91892.1			0.159934
$759$		0			0
$760$	−27622.5	−	111471.i	−0.0478228	−	0.192989i
$761$		−	396078.i		−	0.683930i	−0.939713	−	0.341965i	$-0.888908\pi$
$761$		−	396078.i		−	0.683930i	0.939713	−	0.341965i	$-0.111092\pi$
$762$		0			0
$763$		−	262996.i		−	0.451752i
$764$		−	89644.1i		−	0.153580i
$765$		0			0
$766$	−71806.5			−0.122379
$767$	−413330.			−0.702597
$768$		0			0
$769$	192483.			0.325491			0.162745	−	0.986668i	$-0.447965\pi$
$769$	192483.			0.325491			0.162745	+	0.986668i	$0.447965\pi$
$770$	55171.1	+	222644.i	0.0930530	+	0.375516i
$771$		0			0
$772$		−	97757.8i		−	0.164028i
$773$	−542226.			−0.907447			−0.453723	−	0.891143i	$-0.649905\pi$
$773$	−542226.			−0.907447			−0.453723	+	0.891143i	$0.649905\pi$
$774$		0			0
$775$	−343673.	+	181468.i	−0.572193	+	0.302131i
$776$		−	281105.i		−	0.466815i
$777$		0			0
$778$		−	167632.i		−	0.276947i
$779$		−	132303.i		−	0.218019i
$780$		0			0
$781$	−479715.			−0.786469
$782$	67381.2			0.110186
$783$		0			0
$784$	−1.12465e6			−1.82972
$785$	504466.	−	125007.i	0.818639	−	0.202859i
$786$		0			0
$787$			984739.i			1.58991i	0.606670	+	0.794954i	$0.292505\pi$
$787$			984739.i			1.58991i	−0.606670	+	0.794954i	$0.707495\pi$
$788$	859740.			1.38457
$789$		0			0
$790$	−11165.2	−	45057.4i	−0.0178901	−	0.0721958i
$791$		−	679819.i		−	1.08653i
$792$		0			0
$793$		−	781744.i		−	1.24313i
$794$		−	52349.7i		−	0.0830373i
$795$		0			0
$796$	−161847.			−0.255434
$797$	911433.			1.43486			0.717428	−	0.696633i	$-0.245319\pi$
$797$	911433.			1.43486			0.717428	+	0.696633i	$0.245319\pi$
$798$		0			0
$799$	−216807.			−0.339610
$800$	259770.	−	137165.i	0.405891	−	0.214320i
$801$		0			0
$802$		−	15253.1i		−	0.0237142i
$803$	1.00572e6			1.55972
$804$		0			0
$805$	−798596.	+	197892.i	−1.23235	+	0.305377i
$806$		−	57821.9i		−	0.0890066i
$807$		0			0
$808$		−	272480.i		−	0.417361i
$809$		−	485562.i		−	0.741904i	−0.928652	−	0.370952i	$-0.879031\pi$
$809$		−	485562.i		−	0.741904i	0.928652	−	0.370952i	$-0.120969\pi$
$810$		0			0
$811$	157252.			0.239087			0.119543	−	0.992829i	$-0.461857\pi$
$811$	157252.			0.239087			0.119543	+	0.992829i	$0.461857\pi$
$812$	−466959.			−0.708217
$813$		0			0
$814$	79236.5			0.119585
$815$	−587122.	+	145489.i	−0.883920	+	0.219036i
$816$		0			0
$817$			207789.i			0.311300i
$818$	105468.			0.157622
$819$		0			0
$820$	218070.	−	54037.9i	0.324316	−	0.0803657i
$821$			568597.i			0.843564i	0.906697	+	0.421782i	$0.138595\pi$
$821$			568597.i			0.843564i	−0.906697	+	0.421782i	$0.861405\pi$
$822$		0			0
$823$		−	717994.i		−	1.06004i	−0.847986	−	0.530018i	$-0.822185\pi$
$823$		−	717994.i		−	1.06004i	0.847986	−	0.530018i	$-0.177815\pi$
$824$		−	326966.i		−	0.481558i
$825$		0			0
$826$	150581.			0.220705
$827$	−39891.9			−0.0583275			−0.0291638	−	0.999575i	$-0.509284\pi$
$827$	−39891.9			−0.0583275			−0.0291638	+	0.999575i	$0.509284\pi$
$828$		0			0
$829$	1.05219e6			1.53103			0.765515	−	0.643418i	$-0.222484\pi$
$829$	1.05219e6			1.53103			0.765515	+	0.643418i	$0.222484\pi$
$830$	−26223.0	−	105823.i	−0.0380651	−	0.153612i
$831$		0			0
$832$		−	513163.i		−	0.741325i
$833$	−1.29806e6			−1.87070
$834$		0			0
$835$	−62347.2	−	251603.i	−0.0894219	−	0.360863i
$836$			614189.i			0.878799i
$837$		0			0
$838$		−	68363.3i		−	0.0973498i
$839$			980894.i			1.39347i	0.717328	+	0.696736i	$0.245365\pi$
$839$			980894.i			1.39347i	−0.717328	+	0.696736i	$0.754635\pi$
$840$		0			0
$841$	581917.			0.822752
$842$	−163853.			−0.231116
$843$		0			0
$844$	−808000.			−1.13430
$845$	−41982.5	−	169421.i	−0.0587970	−	0.237276i
$846$		0			0
$847$		−	1.24717e6i		−	1.73843i
$848$	−821531.			−1.14244
$849$		0			0
$850$	95669.8	−	50515.9i	0.132415	−	0.0699182i
$851$			284212.i			0.392449i
$852$		0			0
$853$			702281.i			0.965190i	0.875844	+	0.482595i	$0.160306\pi$
$853$			702281.i			0.965190i	−0.875844	+	0.482595i	$0.839694\pi$
$854$			284799.i			0.390502i
$855$		0			0
$856$	−305913.			−0.417494
$857$	517726.			0.704917			0.352459	−	0.935827i	$-0.385346\pi$
$857$	517726.			0.704917			0.352459	+	0.935827i	$0.385346\pi$
$858$		0			0
$859$	−649610.			−0.880372			−0.440186	−	0.897906i	$-0.645088\pi$
$859$	−649610.			−0.880372			−0.440186	+	0.897906i	$0.645088\pi$
$860$	−342491.	+	84869.4i	−0.463076	+	0.114750i
$861$		0			0
$862$		−	51308.0i		−	0.0690511i
$863$	1.39346e6			1.87100			0.935499	−	0.353329i	$-0.114950\pi$
$863$	1.39346e6			1.87100			0.935499	+	0.353329i	$0.114950\pi$
$864$		0			0
$865$	−229870.	−	927642.i	−0.307220	−	1.23979i
$866$			110005.i			0.146682i
$867$		0			0
$868$		−	820088.i		−	1.08848i
$869$			502898.i			0.665948i
$870$		0			0
$871$	−25093.1			−0.0330764
$872$	−62222.3			−0.0818301
$873$		0			0
$874$	56588.1			0.0740802
$875$	−985509.	+	879683.i	−1.28720	+	1.14897i
$876$		0			0
$877$			742120.i			0.964884i	0.875928	+	0.482442i	$0.160250\pi$
$877$			742120.i			0.964884i	−0.875928	+	0.482442i	$0.839750\pi$
$878$	163871.			0.212576
$879$		0			0
$880$	−985674.	+	244250.i	−1.27282	+	0.315406i
$881$			737962.i			0.950784i	0.879774	+	0.475392i	$0.157694\pi$
$881$			737962.i			0.950784i	−0.879774	+	0.475392i	$0.842306\pi$
$882$		0			0
$883$		−	143168.i		−	0.183622i	−0.995776	−	0.0918109i	$-0.970734\pi$
$883$		−	143168.i		−	0.183622i	0.995776	−	0.0918109i	$-0.0292655\pi$
$884$		−	626635.i		−	0.801882i
$885$		0			0
$886$	−121831.			−0.155199
$887$	−1.00952e6			−1.28312			−0.641559	−	0.767074i	$-0.721712\pi$
$887$	−1.00952e6			−1.28312			−0.641559	+	0.767074i	$0.721712\pi$
$888$		0			0
$889$	685295.			0.867110
$890$	−15840.7	+	3925.32i	−0.0199983	+	0.00495559i
$891$		0			0
$892$		−	400945.i		−	0.503912i
$893$	−182079.			−0.228327
$894$		0			0
$895$	705220.	−	174754.i	0.880397	−	0.218163i
$896$			822748.i			1.02483i
$897$		0			0
$898$			35937.4i			0.0445649i
$899$		−	220169.i		−	0.272418i
$900$		0			0
$901$	−948202.			−1.16802
$902$	62519.6			0.0768428
$903$		0			0
$904$	−160839.			−0.196813
$905$	−143411.	−	578737.i	−0.175100	−	0.706617i
$906$		0			0
$907$		−	479343.i		−	0.582682i	−0.956619	−	0.291341i	$-0.905898\pi$
$907$		−	479343.i		−	0.582682i	0.956619	−	0.291341i	$-0.0941015\pi$
$908$	533949.			0.647631
$909$		0			0
$910$	−47272.8	−	190770.i	−0.0570858	−	0.230370i
$911$		−	871989.i		−	1.05069i	−0.850890	−	0.525344i	$-0.823937\pi$
$911$		−	871989.i		−	1.05069i	0.850890	−	0.525344i	$-0.176063\pi$
$912$		0			0
$913$			1.18112e6i			1.41695i
$914$			198152.i			0.237196i
$915$		0			0
$916$	−81739.7			−0.0974186
$917$	496776.			0.590775
$918$		0			0
$919$	411896.			0.487705			0.243852	−	0.969812i	$-0.421589\pi$
$919$	411896.			0.487705			0.243852	+	0.969812i	$0.421589\pi$
$920$	46819.4	+	188940.i	0.0553159	+	0.223228i
$921$		0			0
$922$			39100.3i			0.0459958i
$923$	411039.			0.482480
$924$		0			0
$925$	213074.	+	403532.i	0.249028	+	0.471622i
$926$			107430.i			0.125286i
$927$		0			0
$928$			166417.i			0.193243i
$929$		−	873650.i		−	1.01229i	−0.862448	−	0.506146i	$-0.831070\pi$
$929$		−	873650.i		−	1.01229i	0.862448	−	0.506146i	$-0.168930\pi$
$930$		0			0
$931$	−1.09013e6			−1.25771
$932$	695835.			0.801077
$933$		0			0
$934$	−48509.1			−0.0556070
$935$	−1.13765e6	+	281911.i	−1.30133	+	0.322469i
$936$		0			0
$937$			271368.i			0.309086i	0.987986	+	0.154543i	$0.0493906\pi$
$937$			271368.i			0.309086i	−0.987986	+	0.154543i	$0.950609\pi$
$938$	9141.75			0.0103902
$939$		0			0
$940$	−74368.4	−	300115.i	−0.0841653	−	0.339650i
$941$			640354.i			0.723171i	0.932339	+	0.361585i	$0.117764\pi$
$941$			640354.i			0.723171i	−0.932339	+	0.361585i	$0.882236\pi$
$942$		0			0
$943$			224250.i			0.252180i
$944$			666644.i			0.748084i
$945$		0			0
$946$	−98190.4			−0.109720
$947$	514318.			0.573498			0.286749	−	0.958006i	$-0.407426\pi$
$947$	514318.			0.573498			0.286749	+	0.958006i	$0.407426\pi$
$948$		0			0
$949$	−861739.			−0.956849
$950$	80345.4	−	42424.3i	0.0890254	−	0.0470075i
$951$		0			0
$952$			462447.i			0.510256i
$953$	926631.			1.02028			0.510142	−	0.860090i	$-0.329593\pi$
$953$	926631.			1.02028			0.510142	+	0.860090i	$0.329593\pi$
$954$		0			0
$955$	139449.	−	34555.5i	0.152901	−	0.0378888i
$956$			562813.i			0.615812i
$957$		0			0
$958$		−	22243.4i		−	0.0242365i
$959$		−	2.96326e6i		−	3.22205i
$960$		0			0
$961$	−536854.			−0.581312
$962$	−67892.9			−0.0733625
$963$		0			0
$964$	−530946.			−0.571343
$965$	152071.	−	37683.2i	0.163302	−	0.0404662i
$966$		0			0
$967$		−	273406.i		−	0.292385i	−0.989256	−	0.146193i	$-0.953298\pi$
$967$		−	273406.i		−	0.292385i	0.989256	−	0.146193i	$-0.0467019\pi$
$968$	−295067.			−0.314898
$969$		0			0
$970$	215869.	−	53492.3i	0.229428	−	0.0568523i
$971$			952676.i			1.01043i	0.862993	+	0.505215i	$0.168587\pi$
$971$			952676.i			1.01043i	−0.862993	+	0.505215i	$0.831413\pi$
$972$		0			0
$973$			1.01344e6i			1.07047i
$974$		−	184130.i		−	0.194091i
$975$		0			0
$976$	−1.26085e6			−1.32362
$977$	−1.10301e6			−1.15555			−0.577776	−	0.816196i	$-0.696079\pi$
$977$	−1.10301e6			−1.15555			−0.577776	+	0.816196i	$0.696079\pi$
$978$		0			0
$979$	176802.			0.184468
$980$	−445254.	−	1.79683e6i	−0.463613	−	1.87091i
$981$		0			0
$982$			280805.i			0.291193i
$983$	1.14462e6			1.18456			0.592278	−	0.805733i	$-0.298229\pi$
$983$	1.14462e6			1.18456			0.592278	+	0.805733i	$0.298229\pi$
$984$		0			0
$985$	331408.	+	1.33740e6i	0.341578	+	1.37844i
$986$			61289.3i			0.0630421i
$987$		0			0
$988$		−	526261.i		−	0.539122i
$989$		−	352197.i		−	0.360075i
$990$		0			0
$991$	−320938.			−0.326794			−0.163397	−	0.986560i	$-0.552245\pi$
$991$	−320938.			−0.326794			−0.163397	+	0.986560i	$0.552245\pi$
$992$	−292267.			−0.297000
$993$		0			0
$994$	−149747.			−0.151560
$995$	−62388.0	−	251767.i	−0.0630166	−	0.254304i
$996$		0			0
$997$			997037.i			1.00305i	0.865144	+	0.501523i	$0.167227\pi$
$997$			997037.i			1.00305i	−0.865144	+	0.501523i	$0.832773\pi$
$998$	183142.			0.183877
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.d.a.404.22		44
3.2	odd	2	inner	405.5.d.a.404.24		44
5.4	even	2	inner	405.5.d.a.404.23		44
9.2	odd	6		45.5.h.a.14.11	✓	44
9.4	even	3		45.5.h.a.29.12	yes	44
9.5	odd	6		135.5.h.a.89.11		44
9.7	even	3		135.5.h.a.44.12		44
15.14	odd	2	inner	405.5.d.a.404.21		44
45.4	even	6		45.5.h.a.29.11	yes	44
45.14	odd	6		135.5.h.a.89.12		44
45.29	odd	6		45.5.h.a.14.12	yes	44
45.34	even	6		135.5.h.a.44.11		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.h.a.14.11	✓	44	9.2	odd	6
45.5.h.a.14.12	yes	44	45.29	odd	6
45.5.h.a.29.11	yes	44	45.4	even	6
45.5.h.a.29.12	yes	44	9.4	even	3
135.5.h.a.44.11		44	45.34	even	6
135.5.h.a.44.12		44	9.7	even	3
135.5.h.a.89.11		44	9.5	odd	6
135.5.h.a.89.12		44	45.14	odd	6
405.5.d.a.404.21		44	15.14	odd	2	inner
405.5.d.a.404.22		44	1.1	even	1	trivial
405.5.d.a.404.23		44	5.4	even	2	inner
405.5.d.a.404.24		44	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.633003 q^{2} -15.5993 q^{4} +(-6.01314 - 24.2661i) q^{5} +84.5447i q^{7} +20.0025 q^{8} +(3.80633 + 15.3605i) q^{10} -171.443i q^{11} +146.899i q^{13} -53.5170i q^{14} +236.927 q^{16} +273.459 q^{17} +229.656 q^{19} +(93.8008 + 378.534i) q^{20} +108.524i q^{22} -389.261 q^{23} +(-552.684 + 291.830i) q^{25} -92.9873i q^{26} -1318.84i q^{28} -354.068i q^{29} +621.826 q^{31} -470.015 q^{32} -173.100 q^{34} +(2051.57 - 508.379i) q^{35} -730.131i q^{37} -145.373 q^{38} +(-120.278 - 485.381i) q^{40} -576.092i q^{41} +904.783i q^{43} +2674.39i q^{44} +246.404 q^{46} -792.834 q^{47} -4746.80 q^{49} +(349.851 - 184.730i) q^{50} -2291.52i q^{52} -3467.44 q^{53} +(-4160.24 + 1030.91i) q^{55} +1691.10i q^{56} +224.126i q^{58} +2813.71i q^{59} -5321.66 q^{61} -393.618 q^{62} -3493.32 q^{64} +(3564.65 - 883.321i) q^{65} +170.819i q^{67} -4265.77 q^{68} +(-1298.65 + 321.805i) q^{70} -2798.11i q^{71} +5866.22i q^{73} +462.175i q^{74} -3582.48 q^{76} +14494.6 q^{77} -2933.33 q^{79} +(-1424.68 - 5749.29i) q^{80} +364.668i q^{82} -6889.32 q^{83} +(-1644.35 - 6635.77i) q^{85} -572.731i q^{86} -3429.27i q^{88} +1031.26i q^{89} -12419.5 q^{91} +6072.21 q^{92} +501.866 q^{94} +(-1380.95 - 5572.85i) q^{95} -14053.5i q^{97} +3004.74 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) 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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