LMFDB - Embedded newform 405.3.d.b.404.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,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, 3); 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, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [24] 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:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.2
Character	$\chi$	$=$	405.404
Dual form			405.3.d.b.404.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.57411 q^{2} +8.77429 q^{4} +(3.06744 + 3.94852i) q^{5} -6.02185i q^{7} -17.0638 q^{8} +(-10.9634 - 14.1125i) q^{10} +15.2008i q^{11} +11.0271i q^{13} +21.5228i q^{14} +25.8909 q^{16} -26.8426 q^{17} +15.8915 q^{19} +(26.9146 + 34.6455i) q^{20} -54.3292i q^{22} +11.5342 q^{23} +(-6.18165 + 24.2237i) q^{25} -39.4120i q^{26} -52.8374i q^{28} -54.4382i q^{29} -31.5722 q^{31} -24.2818 q^{32} +95.9384 q^{34} +(23.7774 - 18.4716i) q^{35} +38.1480i q^{37} -56.7980 q^{38} +(-52.3423 - 67.3769i) q^{40} +37.6636i q^{41} +4.53892i q^{43} +133.376i q^{44} -41.2246 q^{46} +3.71811 q^{47} +12.7374 q^{49} +(22.0939 - 86.5782i) q^{50} +96.7548i q^{52} -56.1230 q^{53} +(-60.0205 + 46.6274i) q^{55} +102.756i q^{56} +194.568i q^{58} +50.0428i q^{59} -48.4417 q^{61} +112.842 q^{62} -16.7778 q^{64} +(-43.5407 + 33.8249i) q^{65} +69.3702i q^{67} -235.524 q^{68} +(-84.9831 + 66.0197i) q^{70} +38.5522i q^{71} +27.2700i q^{73} -136.345i q^{74} +139.437 q^{76} +91.5366 q^{77} -78.8985 q^{79} +(79.4189 + 102.231i) q^{80} -134.614i q^{82} -150.602 q^{83} +(-82.3379 - 105.988i) q^{85} -16.2226i q^{86} -259.383i q^{88} -3.50521i q^{89} +66.4034 q^{91} +101.204 q^{92} -13.2890 q^{94} +(48.7462 + 62.7479i) q^{95} +27.6546i q^{97} -45.5248 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91}+ \cdots + 384 q^{94}+O(q^{100}) $ 24 * q + 48 * q^4 + 12 * q^10 + 96 * q^16 + 48 * q^25 + 144 * q^34 + 72 * q^40 - 168 * q^46 - 288 * q^49 - 132 * q^55 - 360 * q^61 - 72 * q^64 - 156 * q^70 + 48 * q^76 - 480 * q^79 - 456 * q^85 - 48 * q^91 + 384 * 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 $

$2$	−3.57411			−1.78706			−0.893528	−	0.449007i	$-0.851778\pi$
$2$	−3.57411			−1.78706			−0.893528	+	0.449007i	$0.851778\pi$
$3$		0			0
$3$		0			0
$4$	8.77429			2.19357
$5$	3.06744	+	3.94852i	0.613488	+	0.789704i
$5$	3.06744	+	3.94852i	0.613488	+	0.789704i
$6$		0			0
$7$		−	6.02185i		−	0.860264i	−0.902766	−	0.430132i	$-0.858467\pi$
$7$		−	6.02185i		−	0.860264i	0.902766	−	0.430132i	$-0.141533\pi$
$8$	−17.0638			−2.13298
$9$		0			0
$10$	−10.9634	−	14.1125i	−1.09634	−	1.41125i
$11$			15.2008i			1.38189i	0.722909	+	0.690944i	$0.242805\pi$
$11$			15.2008i			1.38189i	−0.722909	+	0.690944i	$0.757195\pi$
$12$		0			0
$13$			11.0271i			0.848237i	0.905607	+	0.424119i	$0.139416\pi$
$13$			11.0271i			0.848237i	−0.905607	+	0.424119i	$0.860584\pi$
$14$			21.5228i			1.53734i
$15$		0			0
$16$	25.8909			1.61818
$17$	−26.8426			−1.57898			−0.789488	−	0.613767i	$-0.789654\pi$
$17$	−26.8426			−1.57898			−0.789488	+	0.613767i	$0.789654\pi$
$18$		0			0
$19$	15.8915			0.836395			0.418197	−	0.908356i	$-0.362662\pi$
$19$	15.8915			0.836395			0.418197	+	0.908356i	$0.362662\pi$
$20$	26.9146	+	34.6455i	1.34573	+	1.73227i
$21$		0			0
$22$		−	54.3292i		−	2.46951i
$23$	11.5342			0.501488			0.250744	−	0.968053i	$-0.419325\pi$
$23$	11.5342			0.501488			0.250744	+	0.968053i	$0.419325\pi$
$24$		0			0
$25$	−6.18165	+	24.2237i	−0.247266	+	0.968948i
$26$		−	39.4120i		−	1.51585i
$27$		0			0
$28$		−	52.8374i		−	1.88705i
$29$		−	54.4382i		−	1.87718i	−0.345037	−	0.938589i	$-0.612134\pi$
$29$		−	54.4382i		−	1.87718i	0.345037	−	0.938589i	$-0.387866\pi$
$30$		0			0
$31$	−31.5722			−1.01846			−0.509228	−	0.860631i	$-0.670069\pi$
$31$	−31.5722			−1.01846			−0.509228	+	0.860631i	$0.670069\pi$
$32$	−24.2818			−0.758807
$33$		0			0
$34$	95.9384			2.82172
$35$	23.7774	−	18.4716i	0.679354	−	0.527761i
$36$		0			0
$37$			38.1480i			1.03103i	0.856881	+	0.515514i	$0.172399\pi$
$37$			38.1480i			1.03103i	−0.856881	+	0.515514i	$0.827601\pi$
$38$	−56.7980			−1.49468
$39$		0			0
$40$	−52.3423	−	67.3769i	−1.30856	−	1.68442i
$41$			37.6636i			0.918625i	0.888275	+	0.459312i	$0.151904\pi$
$41$			37.6636i			0.918625i	−0.888275	+	0.459312i	$0.848096\pi$
$42$		0			0
$43$			4.53892i			0.105556i	0.998606	+	0.0527782i	$0.0168076\pi$
$43$			4.53892i			0.105556i	−0.998606	+	0.0527782i	$0.983192\pi$
$44$			133.376i			3.03127i
$45$		0			0
$46$	−41.2246			−0.896187
$47$	3.71811			0.0791088			0.0395544	−	0.999217i	$-0.487406\pi$
$47$	3.71811			0.0791088			0.0395544	+	0.999217i	$0.487406\pi$
$48$		0			0
$49$	12.7374			0.259947
$50$	22.0939	−	86.5782i	0.441878	−	1.73156i
$51$		0			0
$52$			96.7548i			1.86067i
$53$	−56.1230			−1.05892			−0.529462	−	0.848333i	$-0.677606\pi$
$53$	−56.1230			−1.05892			−0.529462	+	0.848333i	$0.677606\pi$
$54$		0			0
$55$	−60.0205	+	46.6274i	−1.09128	+	0.847771i
$56$			102.756i			1.83492i
$57$		0			0
$58$			194.568i			3.35462i
$59$			50.0428i			0.848184i	0.905619	+	0.424092i	$0.139407\pi$
$59$			50.0428i			0.848184i	−0.905619	+	0.424092i	$0.860593\pi$
$60$		0			0
$61$	−48.4417			−0.794126			−0.397063	−	0.917791i	$-0.629971\pi$
$61$	−48.4417			−0.794126			−0.397063	+	0.917791i	$0.629971\pi$
$62$	112.842			1.82004
$63$		0			0
$64$	−16.7778			−0.262153
$65$	−43.5407	+	33.8249i	−0.669857	+	0.520383i
$66$		0			0
$67$			69.3702i			1.03538i	0.855570	+	0.517688i	$0.173207\pi$
$67$			69.3702i			1.03538i	−0.855570	+	0.517688i	$0.826793\pi$
$68$	−235.524			−3.46359
$69$		0			0
$70$	−84.9831	+	66.0197i	−1.21404	+	0.943139i
$71$			38.5522i			0.542989i	0.962440	+	0.271494i	$0.0875179\pi$
$71$			38.5522i			0.542989i	−0.962440	+	0.271494i	$0.912482\pi$
$72$		0			0
$73$			27.2700i			0.373561i	0.982402	+	0.186781i	$0.0598054\pi$
$73$			27.2700i			0.373561i	−0.982402	+	0.186781i	$0.940195\pi$
$74$		−	136.345i		−	1.84250i
$75$		0			0
$76$	139.437			1.83469
$77$	91.5366			1.18879
$78$		0			0
$79$	−78.8985			−0.998716			−0.499358	−	0.866396i	$-0.666431\pi$
$79$	−78.8985			−0.998716			−0.499358	+	0.866396i	$0.666431\pi$
$80$	79.4189	+	102.231i	0.992736	+	1.27789i
$81$		0			0
$82$		−	134.614i		−	1.64163i
$83$	−150.602			−1.81448			−0.907242	−	0.420609i	$-0.861816\pi$
$83$	−150.602			−1.81448			−0.907242	+	0.420609i	$0.861816\pi$
$84$		0			0
$85$	−82.3379	−	105.988i	−0.968682	−	1.24692i
$86$		−	16.2226i		−	0.188635i
$87$		0			0
$88$		−	259.383i		−	2.94754i
$89$		−	3.50521i		−	0.0393843i	−0.999806	−	0.0196922i	$-0.993731\pi$
$89$		−	3.50521i		−	0.0393843i	0.999806	−	0.0196922i	$-0.00626862\pi$
$90$		0			0
$91$	66.4034			0.729708
$92$	101.204			1.10005
$93$		0			0
$94$	−13.2890			−0.141372
$95$	48.7462	+	62.7479i	0.513118	+	0.660504i
$96$		0			0
$97$			27.6546i			0.285099i	0.989788	+	0.142550i	$0.0455300\pi$
$97$			27.6546i			0.285099i	−0.989788	+	0.142550i	$0.954470\pi$
$98$	−45.5248			−0.464539
$99$		0			0
$100$	−54.2396	+	212.546i	−0.542396	+	2.12546i
$101$		−	36.0907i		−	0.357333i	−0.983910	−	0.178667i	$-0.942822\pi$
$101$		−	36.0907i		−	0.357333i	0.983910	−	0.178667i	$-0.0571784\pi$
$102$		0			0
$103$			28.0612i			0.272439i	0.990679	+	0.136219i	$0.0434952\pi$
$103$			28.0612i			0.272439i	−0.990679	+	0.136219i	$0.956505\pi$
$104$		−	188.164i		−	1.80927i
$105$		0			0
$106$	200.590			1.89236
$107$	91.8455			0.858369			0.429184	−	0.903217i	$-0.358801\pi$
$107$	91.8455			0.858369			0.429184	+	0.903217i	$0.358801\pi$
$108$		0			0
$109$	78.1779			0.717228			0.358614	−	0.933486i	$-0.383249\pi$
$109$	78.1779			0.717228			0.358614	+	0.933486i	$0.383249\pi$
$110$	214.520	−	166.652i	1.95018	−	1.51501i
$111$		0			0
$112$		−	155.911i		−	1.39206i
$113$	6.56726			0.0581174			0.0290587	−	0.999578i	$-0.490749\pi$
$113$	6.56726			0.0581174			0.0290587	+	0.999578i	$0.490749\pi$
$114$		0			0
$115$	35.3805	+	45.5431i	0.307656	+	0.396027i
$116$		−	477.656i		−	4.11772i
$117$		0			0
$118$		−	178.859i		−	1.51575i
$119$			161.642i			1.35833i
$120$		0			0
$121$	−110.063			−0.909613
$122$	173.136			1.41915
$123$		0			0
$124$	−277.023			−2.23406
$125$	−114.610	+	49.8963i	−0.916877	+	0.399170i
$126$		0			0
$127$			247.486i			1.94871i	0.225023	+	0.974353i	$0.427754\pi$
$127$			247.486i			1.94871i	−0.225023	+	0.974353i	$0.572246\pi$
$128$	157.093			1.22729
$129$		0			0
$130$	155.619	−	120.894i	1.19707	−	0.929954i
$131$			142.174i			1.08530i	0.839960	+	0.542648i	$0.182578\pi$
$131$			142.174i			1.08530i	−0.839960	+	0.542648i	$0.817422\pi$
$132$		0			0
$133$		−	95.6961i		−	0.719520i
$134$		−	247.937i		−	1.85027i
$135$		0			0
$136$	458.037			3.36792
$137$	72.8900			0.532044			0.266022	−	0.963967i	$-0.414291\pi$
$137$	72.8900			0.532044			0.266022	+	0.963967i	$0.414291\pi$
$138$		0			0
$139$	−66.9989			−0.482007			−0.241003	−	0.970524i	$-0.577476\pi$
$139$	−66.9989			−0.482007			−0.241003	+	0.970524i	$0.577476\pi$
$140$	208.630	−	162.075i	1.49021	−	1.15768i
$141$		0			0
$142$		−	137.790i		−	0.970352i
$143$	−167.620			−1.17217
$144$		0			0
$145$	214.950	−	166.986i	1.48242	−	1.15163i
$146$		−	97.4660i		−	0.667575i
$147$		0			0
$148$			334.721i			2.26163i
$149$		−	76.5668i		−	0.513871i	−0.966429	−	0.256936i	$-0.917287\pi$
$149$		−	76.5668i		−	0.513871i	0.966429	−	0.256936i	$-0.0827128\pi$
$150$		0			0
$151$	−39.9051			−0.264272			−0.132136	−	0.991232i	$-0.542184\pi$
$151$	−39.9051			−0.264272			−0.132136	+	0.991232i	$0.542184\pi$
$152$	−271.170			−1.78401
$153$		0			0
$154$	−327.162			−2.12443
$155$	−96.8456	−	124.663i	−0.624810	−	0.804280i
$156$		0			0
$157$		−	135.575i		−	0.863538i	−0.901984	−	0.431769i	$-0.857890\pi$
$157$		−	135.575i		−	0.863538i	0.901984	−	0.431769i	$-0.142110\pi$
$158$	281.992			1.78476
$159$		0			0
$160$	−74.4830	−	95.8773i	−0.465518	−	0.599233i
$161$		−	69.4572i		−	0.431411i
$162$		0			0
$163$		−	85.9303i		−	0.527180i	−0.964635	−	0.263590i	$-0.915093\pi$
$163$		−	85.9303i		−	0.527180i	0.964635	−	0.263590i	$-0.0849066\pi$
$164$			330.471i			2.01507i
$165$		0			0
$166$	538.269			3.24258
$167$	−111.718			−0.668970			−0.334485	−	0.942401i	$-0.608562\pi$
$167$	−111.718			−0.668970			−0.334485	+	0.942401i	$0.608562\pi$
$168$		0			0
$169$	47.4034			0.280494
$170$	294.285	+	378.815i	1.73109	+	2.22832i
$171$		0			0
$172$			39.8258i			0.231545i
$173$	−8.68986			−0.0502304			−0.0251152	−	0.999685i	$-0.507995\pi$
$173$	−8.68986			−0.0502304			−0.0251152	+	0.999685i	$0.507995\pi$
$174$		0			0
$175$	145.871	+	37.2249i	0.833550	+	0.212714i
$176$			393.562i			2.23615i
$177$		0			0
$178$			12.5280i			0.0703821i
$179$			57.4651i			0.321034i	0.987033	+	0.160517i	$0.0513162\pi$
$179$			57.4651i			0.321034i	−0.987033	+	0.160517i	$0.948684\pi$
$180$		0			0
$181$	−139.711			−0.771884			−0.385942	−	0.922523i	$-0.626124\pi$
$181$	−139.711			−0.771884			−0.385942	+	0.922523i	$0.626124\pi$
$182$	−237.333			−1.30403
$183$		0			0
$184$	−196.818			−1.06966
$185$	−150.628	+	117.017i	−0.814207	+	0.632522i
$186$		0			0
$187$		−	408.028i		−	2.18197i
$188$	32.6238			0.173531
$189$		0			0
$190$	−174.224	−	224.268i	−0.916970	−	1.18036i
$191$			31.4760i			0.164796i	0.996600	+	0.0823980i	$0.0262579\pi$
$191$			31.4760i			0.164796i	−0.996600	+	0.0823980i	$0.973742\pi$
$192$		0			0
$193$		−	50.0752i		−	0.259457i	−0.991550	−	0.129729i	$-0.958589\pi$
$193$		−	50.0752i		−	0.259457i	0.991550	−	0.129729i	$-0.0414106\pi$
$194$		−	98.8407i		−	0.509488i
$195$		0			0
$196$	111.761			0.570211
$197$	−37.4908			−0.190308			−0.0951542	−	0.995463i	$-0.530334\pi$
$197$	−37.4908			−0.190308			−0.0951542	+	0.995463i	$0.530334\pi$
$198$		0			0
$199$	40.5173			0.203604			0.101802	−	0.994805i	$-0.467539\pi$
$199$	40.5173			0.203604			0.101802	+	0.994805i	$0.467539\pi$
$200$	105.483	−	413.349i	0.527413	−	2.06675i
$201$		0			0
$202$			128.992i			0.638575i
$203$	−327.818			−1.61487
$204$		0			0
$205$	−148.716	+	115.531i	−0.725442	+	0.563565i
$206$		−	100.294i		−	0.486864i
$207$		0			0
$208$			285.502i			1.37260i
$209$			241.563i			1.15580i
$210$		0			0
$211$	172.473			0.817406			0.408703	−	0.912668i	$-0.365981\pi$
$211$	172.473			0.817406			0.408703	+	0.912668i	$0.365981\pi$
$212$	−492.439			−2.32283
$213$		0			0
$214$	−328.266			−1.53395
$215$	−17.9220	+	13.9229i	−0.0833583	+	0.0647575i
$216$		0			0
$217$			190.123i			0.876141i
$218$	−279.417			−1.28173
$219$		0			0
$220$	−526.637	+	409.122i	−2.39381	+	1.85965i
$221$		−	295.995i		−	1.33935i
$222$		0			0
$223$			299.356i			1.34240i	0.741276	+	0.671201i	$0.234221\pi$
$223$			299.356i			1.34240i	−0.741276	+	0.671201i	$0.765779\pi$
$224$			146.221i			0.652774i
$225$		0			0
$226$	−23.4721			−0.103859
$227$	298.716			1.31593			0.657965	−	0.753049i	$-0.271418\pi$
$227$	298.716			1.31593			0.657965	+	0.753049i	$0.271418\pi$
$228$		0			0
$229$	−203.310			−0.887816			−0.443908	−	0.896072i	$-0.646408\pi$
$229$	−203.310			−0.887816			−0.443908	+	0.896072i	$0.646408\pi$
$230$	−126.454	−	162.776i	−0.549799	−	0.707722i
$231$		0			0
$232$			928.924i			4.00398i
$233$	436.398			1.87295			0.936476	−	0.350733i	$-0.114067\pi$
$233$	436.398			1.87295			0.936476	+	0.350733i	$0.114067\pi$
$234$		0			0
$235$	11.4051	+	14.6810i	0.0485322	+	0.0624725i
$236$			439.090i			1.86055i
$237$		0			0
$238$		−	577.726i		−	2.42742i
$239$		−	239.697i		−	1.00292i	−0.865182	−	0.501458i	$-0.832797\pi$
$239$		−	239.697i		−	1.00292i	0.865182	−	0.501458i	$-0.167203\pi$
$240$		0			0
$241$	205.348			0.852067			0.426034	−	0.904707i	$-0.359911\pi$
$241$	205.348			0.852067			0.426034	+	0.904707i	$0.359911\pi$
$242$	393.378			1.62553
$243$		0			0
$244$	−425.041			−1.74197
$245$	39.0711	+	50.2938i	0.159474	+	0.205281i
$246$		0			0
$247$			175.237i			0.709461i
$248$	538.742			2.17235
$249$		0			0
$250$	409.628	−	178.335i	1.63851	−	0.713340i
$251$			357.277i			1.42341i	0.702477	+	0.711707i	$0.252077\pi$
$251$			357.277i			1.42341i	−0.702477	+	0.711707i	$0.747923\pi$
$252$		0			0
$253$			175.329i			0.692999i
$254$		−	884.542i		−	3.48245i
$255$		0			0
$256$	−494.357			−1.93108
$257$	280.513			1.09149			0.545746	−	0.837951i	$-0.316246\pi$
$257$	280.513			1.09149			0.545746	+	0.837951i	$0.316246\pi$
$258$		0			0
$259$	229.721			0.886955
$260$	−382.038	+	296.789i	−1.46938	+	1.14150i
$261$		0			0
$262$		−	508.145i		−	1.93949i
$263$	120.494			0.458153			0.229077	−	0.973408i	$-0.426429\pi$
$263$	120.494			0.458153			0.229077	+	0.973408i	$0.426429\pi$
$264$		0			0
$265$	−172.154	−	221.603i	−0.649637	−	0.836237i
$266$			342.029i			1.28582i
$267$		0			0
$268$			608.674i			2.27117i
$269$			308.065i			1.14522i	0.819827	+	0.572612i	$0.194070\pi$
$269$			308.065i			1.14522i	−0.819827	+	0.572612i	$0.805930\pi$
$270$		0			0
$271$	354.684			1.30880			0.654399	−	0.756149i	$-0.272922\pi$
$271$	354.684			1.30880			0.654399	+	0.756149i	$0.272922\pi$
$272$	−694.980			−2.55507
$273$		0			0
$274$	−260.517			−0.950792
$275$	−368.219	−	93.9658i	−1.33898	−	0.341694i
$276$		0			0
$277$		−	480.306i		−	1.73396i	−0.498346	−	0.866978i	$-0.666059\pi$
$277$		−	480.306i		−	1.73396i	0.498346	−	0.866978i	$-0.333941\pi$
$278$	239.462			0.861373
$279$		0			0
$280$	−405.733	+	315.197i	−1.44905	+	1.12570i
$281$		−	126.250i		−	0.449287i	−0.974441	−	0.224644i	$-0.927878\pi$
$281$		−	126.250i		−	0.449287i	0.974441	−	0.224644i	$-0.0721218\pi$
$282$		0			0
$283$		−	365.044i		−	1.28991i	−0.764221	−	0.644954i	$-0.776876\pi$
$283$		−	365.044i		−	1.28991i	0.764221	−	0.644954i	$-0.223124\pi$
$284$			338.268i			1.19109i
$285$		0			0
$286$	599.093			2.09473
$287$	226.804			0.790259
$288$		0			0
$289$	431.524			1.49316
$290$	−768.257	+	596.826i	−2.64916	+	2.05802i
$291$		0			0
$292$			239.275i			0.819433i
$293$	−194.169			−0.662693			−0.331347	−	0.943509i	$-0.607503\pi$
$293$	−194.169			−0.662693			−0.331347	+	0.943509i	$0.607503\pi$
$294$		0			0
$295$	−197.595	+	153.503i	−0.669814	+	0.520350i
$296$		−	650.951i		−	2.19916i
$297$		0			0
$298$			273.658i			0.918317i
$299$			127.189i			0.425380i
$300$		0			0
$301$	27.3327			0.0908063
$302$	142.625			0.472269
$303$		0			0
$304$	411.446			1.35344
$305$	−148.592	−	191.273i	−0.487187	−	0.627125i
$306$		0			0
$307$		−	492.555i		−	1.60441i	−0.597046	−	0.802207i	$-0.703659\pi$
$307$		−	492.555i		−	1.60441i	0.597046	−	0.802207i	$-0.296341\pi$
$308$	803.168			2.60769
$309$		0			0
$310$	346.137	+	445.561i	1.11657	+	1.43729i
$311$			186.362i			0.599235i	0.954059	+	0.299617i	$0.0968590\pi$
$311$			186.362i			0.599235i	−0.954059	+	0.299617i	$0.903141\pi$
$312$		0			0
$313$			218.646i			0.698548i	0.937021	+	0.349274i	$0.113572\pi$
$313$			218.646i			0.698548i	−0.937021	+	0.349274i	$0.886428\pi$
$314$			484.562i			1.54319i
$315$		0			0
$316$	−692.278			−2.19075
$317$	204.603			0.645435			0.322718	−	0.946495i	$-0.395404\pi$
$317$	204.603			0.645435			0.322718	+	0.946495i	$0.395404\pi$
$318$		0			0
$319$	827.502			2.59405
$320$	−51.4649	−	66.2476i	−0.160828	−	0.207024i
$321$		0			0
$322$			248.248i			0.770957i
$323$	−426.569			−1.32065
$324$		0			0
$325$	−267.117	−	68.1656i	−0.821897	−	0.209740i
$326$			307.125i			0.942100i
$327$		0			0
$328$		−	642.686i		−	1.95941i
$329$		−	22.3899i		−	0.0680544i
$330$		0			0
$331$	−596.610			−1.80245			−0.901224	−	0.433353i	$-0.857330\pi$
$331$	−596.610			−1.80245			−0.901224	+	0.433353i	$0.857330\pi$
$332$	−1321.43			−3.98020
$333$		0			0
$334$	399.293			1.19549
$335$	−273.910	+	212.789i	−0.817641	+	0.635190i
$336$		0			0
$337$			332.596i			0.986931i	0.869765	+	0.493466i	$0.164270\pi$
$337$			332.596i			0.986931i	−0.869765	+	0.493466i	$0.835730\pi$
$338$	−169.425			−0.501258
$339$		0			0
$340$	−722.457	−	929.973i	−2.12487	−	2.73522i
$341$		−	479.921i		−	1.40739i
$342$		0			0
$343$		−	371.773i		−	1.08389i
$344$		−	77.4514i		−	0.225149i
$345$		0			0
$346$	31.0585			0.0897645
$347$	−69.2057			−0.199440			−0.0997201	−	0.995016i	$-0.531795\pi$
$347$	−69.2057			−0.199440			−0.0997201	+	0.995016i	$0.531795\pi$
$348$		0			0
$349$	554.851			1.58983			0.794915	−	0.606721i	$-0.207515\pi$
$349$	554.851			1.58983			0.794915	+	0.606721i	$0.207515\pi$
$350$	−521.361	−	133.046i	−1.48960	−	0.380132i
$351$		0			0
$352$		−	369.102i		−	1.04859i
$353$	176.268			0.499343			0.249672	−	0.968331i	$-0.419677\pi$
$353$	176.268			0.499343			0.249672	+	0.968331i	$0.419677\pi$
$354$		0			0
$355$	−152.224	+	118.257i	−0.428801	+	0.333117i
$356$		−	30.7557i		−	0.0863924i
$357$		0			0
$358$		−	205.387i		−	0.573706i
$359$			122.033i			0.339924i	0.985451	+	0.169962i	$0.0543645\pi$
$359$			122.033i			0.339924i	−0.985451	+	0.169962i	$0.945636\pi$
$360$		0			0
$361$	−108.460			−0.300444
$362$	499.343			1.37940
$363$		0			0
$364$	582.642			1.60067
$365$	−107.676	+	83.6490i	−0.295003	+	0.229175i
$366$		0			0
$367$		−	672.393i		−	1.83213i	−0.401025	−	0.916067i	$-0.631346\pi$
$367$		−	672.393i		−	1.83213i	0.401025	−	0.916067i	$-0.368654\pi$
$368$	298.632			0.811499
$369$		0			0
$370$	538.362	−	418.231i	1.45503	−	1.13035i
$371$			337.964i			0.910954i
$372$		0			0
$373$		−	513.816i		−	1.37752i	−0.724988	−	0.688762i	$-0.758155\pi$
$373$		−	513.816i		−	1.37752i	0.724988	−	0.688762i	$-0.241845\pi$
$374$			1458.34i			3.89930i
$375$		0			0
$376$	−63.4453			−0.168737
$377$	600.294			1.59229
$378$		0			0
$379$	−496.340			−1.30960			−0.654802	−	0.755800i	$-0.727248\pi$
$379$	−496.340			−1.30960			−0.654802	+	0.755800i	$0.727248\pi$
$380$	427.713	+	550.568i	1.12556	+	1.44886i
$381$		0			0
$382$		−	112.499i		−	0.294500i
$383$	595.923			1.55594			0.777968	−	0.628304i	$-0.216251\pi$
$383$	595.923			1.55594			0.777968	+	0.628304i	$0.216251\pi$
$384$		0			0
$385$	280.783	+	361.434i	0.729306	+	0.938791i
$386$			178.974i			0.463664i
$387$		0			0
$388$			242.650i			0.625385i
$389$		−	439.285i		−	1.12927i	−0.825342	−	0.564633i	$-0.809018\pi$
$389$		−	439.285i		−	1.12927i	0.825342	−	0.564633i	$-0.190982\pi$
$390$		0			0
$391$	−309.608			−0.791836
$392$	−217.349			−0.554461
$393$		0			0
$394$	133.996			0.340092
$395$	−242.016	−	311.533i	−0.612700	−	0.788690i
$396$		0			0
$397$			405.932i			1.02250i	0.859432	+	0.511250i	$0.170817\pi$
$397$			405.932i			1.02250i	−0.859432	+	0.511250i	$0.829183\pi$
$398$	−144.813			−0.363853
$399$		0			0
$400$	−160.049	+	627.174i	−0.400122	+	1.56794i
$401$		−	328.446i		−	0.819066i	−0.912295	−	0.409533i	$-0.865692\pi$
$401$		−	328.446i		−	0.819066i	0.912295	−	0.409533i	$-0.134308\pi$
$402$		0			0
$403$		−	348.149i		−	0.863893i
$404$		−	316.670i		−	0.783836i
$405$		0			0
$406$	1171.66			2.88586
$407$	−579.879			−1.42476
$408$		0			0
$409$	−16.3136			−0.0398866			−0.0199433	−	0.999801i	$-0.506349\pi$
$409$	−16.3136			−0.0398866			−0.0199433	+	0.999801i	$0.506349\pi$
$410$	531.526	−	412.920i	1.29641	−	1.00712i
$411$		0			0
$412$			246.217i			0.597614i
$413$	301.350			0.729662
$414$		0			0
$415$	−461.963	−	594.656i	−1.11316	−	1.43291i
$416$		−	267.758i		−	0.643648i
$417$		0			0
$418$		−	863.373i		−	2.06549i
$419$		−	381.549i		−	0.910618i	−0.890333	−	0.455309i	$-0.849529\pi$
$419$		−	381.549i		−	0.910618i	0.890333	−	0.455309i	$-0.150471\pi$
$420$		0			0
$421$	423.338			1.00555			0.502776	−	0.864416i	$-0.332312\pi$
$421$	423.338			1.00555			0.502776	+	0.864416i	$0.332312\pi$
$422$	−616.436			−1.46075
$423$		0			0
$424$	957.674			2.25866
$425$	165.931	−	650.226i	0.390427	−	1.52994i
$426$		0			0
$427$			291.708i			0.683158i
$428$	805.878			1.88289
$429$		0			0
$430$	64.0554	−	49.7619i	0.148966	−	0.115725i
$431$		−	48.3081i		−	0.112084i	−0.998428	−	0.0560419i	$-0.982152\pi$
$431$		−	48.3081i		−	0.112084i	0.998428	−	0.0560419i	$-0.0178480\pi$
$432$		0			0
$433$			36.0601i			0.0832797i	0.999133	+	0.0416399i	$0.0132582\pi$
$433$			36.0601i			0.0832797i	−0.999133	+	0.0416399i	$0.986742\pi$
$434$		−	679.520i		−	1.56571i
$435$		0			0
$436$	685.955			1.57329
$437$	183.296			0.419441
$438$		0			0
$439$	−260.086			−0.592452			−0.296226	−	0.955118i	$-0.595728\pi$
$439$	−260.086			−0.592452			−0.296226	+	0.955118i	$0.595728\pi$
$440$	1024.18	−	795.642i	2.32768	−	1.80828i
$441$		0			0
$442$			1057.92i			2.39349i
$443$	−22.5294			−0.0508565			−0.0254282	−	0.999677i	$-0.508095\pi$
$443$	−22.5294			−0.0508565			−0.0254282	+	0.999677i	$0.508095\pi$
$444$		0			0
$445$	13.8404	−	10.7520i	0.0311020	−	0.0241618i
$446$		−	1069.93i		−	2.39895i
$447$		0			0
$448$			101.033i			0.225521i
$449$		−	275.640i		−	0.613898i	−0.951726	−	0.306949i	$-0.900692\pi$
$449$		−	275.640i		−	0.613898i	0.951726	−	0.306949i	$-0.0993080\pi$
$450$		0			0
$451$	−572.516			−1.26944
$452$	57.6230			0.127485
$453$		0			0
$454$	−1067.64			−2.35164
$455$	203.688	+	262.195i	0.447667	+	0.576253i
$456$		0			0
$457$		−	119.408i		−	0.261286i	−0.991429	−	0.130643i	$-0.958296\pi$
$457$		−	119.408i		−	0.261286i	0.991429	−	0.130643i	$-0.0417041\pi$
$458$	726.653			1.58658
$459$		0			0
$460$	310.438	+	399.608i	0.674866	+	0.868713i
$461$		−	90.3566i		−	0.196001i	−0.995186	−	0.0980007i	$-0.968755\pi$
$461$		−	90.3566i		−	0.196001i	0.995186	−	0.0980007i	$-0.0312447\pi$
$462$		0			0
$463$			610.436i			1.31844i	0.751951	+	0.659219i	$0.229113\pi$
$463$			610.436i			1.31844i	−0.751951	+	0.659219i	$0.770887\pi$
$464$		−	1409.46i		−	3.03762i
$465$		0			0
$466$	−1559.73			−3.34707
$467$	65.6533			0.140585			0.0702926	−	0.997526i	$-0.477607\pi$
$467$	65.6533			0.140585			0.0702926	+	0.997526i	$0.477607\pi$
$468$		0			0
$469$	417.736			0.890696
$470$	−40.7630	−	52.4717i	−0.0867299	−	0.111642i
$471$		0			0
$472$		−	853.923i		−	1.80916i
$473$	−68.9951			−0.145867
$474$		0			0
$475$	−98.2357	+	384.951i	−0.206812	+	0.810423i
$476$			1418.29i			2.97960i
$477$		0			0
$478$			856.704i			1.79227i
$479$		−	93.9812i		−	0.196203i	−0.995176	−	0.0981015i	$-0.968723\pi$
$479$		−	93.9812i		−	0.196203i	0.995176	−	0.0981015i	$-0.0312770\pi$
$480$		0			0
$481$	−420.661			−0.874555
$482$	−733.938			−1.52269
$483$		0			0
$484$	−965.725			−1.99530
$485$	−109.195	+	84.8288i	−0.225144	+	0.174905i
$486$		0			0
$487$			805.464i			1.65393i	0.562253	+	0.826965i	$0.309935\pi$
$487$			805.464i			1.65393i	−0.562253	+	0.826965i	$0.690065\pi$
$488$	826.601			1.69386
$489$		0			0
$490$	−139.645	−	179.756i	−0.284989	−	0.366849i
$491$			940.783i			1.91605i	0.286678	+	0.958027i	$0.407449\pi$
$491$			940.783i			1.91605i	−0.286678	+	0.958027i	$0.592551\pi$
$492$		0			0
$493$			1461.26i			2.96402i
$494$		−	626.316i		−	1.26785i
$495$		0			0
$496$	−817.433			−1.64805
$497$	232.155			0.467114
$498$		0			0
$499$	379.929			0.761380			0.380690	−	0.924703i	$-0.375686\pi$
$499$	379.929			0.761380			0.380690	+	0.924703i	$0.375686\pi$
$500$	−1005.62	+	437.804i	−2.01123	+	0.875609i
$501$		0			0
$502$		−	1276.95i		−	2.54372i
$503$	−637.608			−1.26761			−0.633805	−	0.773493i	$-0.718508\pi$
$503$	−637.608			−1.26761			−0.633805	+	0.773493i	$0.718508\pi$
$504$		0			0
$505$	142.505	−	110.706i	0.282188	−	0.219220i
$506$		−	626.645i		−	1.23843i
$507$		0			0
$508$			2171.51i			4.27463i
$509$		−	963.113i		−	1.89217i	−0.323922	−	0.946084i	$-0.605002\pi$
$509$		−	963.113i		−	1.89217i	0.323922	−	0.946084i	$-0.394998\pi$
$510$		0			0
$511$	164.216			0.321361
$512$	1138.52			2.22366
$513$		0			0
$514$	−1002.59			−1.95056
$515$	−110.800	+	86.0760i	−0.215146	+	0.167138i
$516$		0			0
$517$			56.5181i			0.109319i
$518$	−821.050			−1.58504
$519$		0			0
$520$	742.971	−	577.182i	1.42879	−	1.10997i
$521$		−	36.7876i		−	0.0706095i	−0.999377	−	0.0353048i	$-0.988760\pi$
$521$		−	36.7876i		−	0.0706095i	0.999377	−	0.0353048i	$-0.0112402\pi$
$522$		0			0
$523$		−	441.292i		−	0.843770i	−0.906649	−	0.421885i	$-0.861369\pi$
$523$		−	441.292i		−	0.843770i	0.906649	−	0.421885i	$-0.138631\pi$
$524$			1247.47i			2.38067i
$525$		0			0
$526$	−430.660			−0.818746
$527$	847.478			1.60812
$528$		0			0
$529$	−395.962			−0.748510
$530$	615.297	+	792.034i	1.16094	+	1.49440i
$531$		0			0
$532$		−	839.665i		−	1.57832i
$533$	−415.320			−0.779212
$534$		0			0
$535$	281.730	+	362.654i	0.526599	+	0.677858i
$536$		−	1183.72i		−	2.20844i
$537$		0			0
$538$		−	1101.06i		−	2.04658i
$539$			193.618i			0.359217i
$540$		0			0
$541$	−426.631			−0.788598			−0.394299	−	0.918982i	$-0.629013\pi$
$541$	−426.631			−0.788598			−0.394299	+	0.918982i	$0.629013\pi$
$542$	−1267.68			−2.33890
$543$		0			0
$544$	651.786			1.19814
$545$	239.806	+	308.687i	0.440011	+	0.566398i
$546$		0			0
$547$			602.928i			1.10224i	0.834424	+	0.551122i	$0.185800\pi$
$547$			602.928i			1.10224i	−0.834424	+	0.551122i	$0.814200\pi$
$548$	639.558			1.16708
$549$		0			0
$550$	1316.05	+	335.844i	2.39283	+	0.610626i
$551$		−	865.104i		−	1.57006i
$552$		0			0
$553$			475.115i			0.859159i
$554$			1716.67i			3.09868i
$555$		0			0
$556$	−587.868			−1.05732
$557$	527.490			0.947020			0.473510	−	0.880788i	$-0.342987\pi$
$557$	527.490			0.947020			0.473510	+	0.880788i	$0.342987\pi$
$558$		0			0
$559$	−50.0511			−0.0895368
$560$	615.619	−	478.248i	1.09932	−	0.854014i
$561$		0			0
$562$			451.231i			0.802902i
$563$	14.2805			0.0253650			0.0126825	−	0.999920i	$-0.495963\pi$
$563$	14.2805			0.0253650			0.0126825	+	0.999920i	$0.495963\pi$
$564$		0			0
$565$	20.1447	+	25.9310i	0.0356543	+	0.0458955i
$566$			1304.71i			2.30514i
$567$		0			0
$568$		−	657.849i		−	1.15818i
$569$			672.782i			1.18239i	0.806527	+	0.591197i	$0.201344\pi$
$569$			672.782i			1.18239i	−0.806527	+	0.591197i	$0.798656\pi$
$570$		0			0
$571$	857.665			1.50204			0.751020	−	0.660280i	$-0.229562\pi$
$571$	857.665			1.50204			0.751020	+	0.660280i	$0.229562\pi$
$572$	−1470.75			−2.57123
$573$		0			0
$574$	−810.625			−1.41224
$575$	−71.3005	+	279.401i	−0.124001	+	0.485915i
$576$		0			0
$577$		−	134.308i		−	0.232769i	−0.993204	−	0.116385i	$-0.962869\pi$
$577$		−	134.308i		−	0.232769i	0.993204	−	0.116385i	$-0.0371305\pi$
$578$	−1542.32			−2.66837
$579$		0			0
$580$	1886.04	−	1465.18i	3.25179	−	2.52617i
$581$			906.903i			1.56093i
$582$		0			0
$583$		−	853.112i		−	1.46331i
$584$		−	465.330i		−	0.796799i
$585$		0			0
$586$	693.983			1.18427
$587$	900.274			1.53369			0.766843	−	0.641835i	$-0.221826\pi$
$587$	900.274			1.53369			0.766843	+	0.641835i	$0.221826\pi$
$588$		0			0
$589$	−501.729			−0.851832
$590$	706.228	−	548.638i	1.19700	−	0.929895i
$591$		0			0
$592$			987.688i			1.66839i
$593$	−282.338			−0.476118			−0.238059	−	0.971251i	$-0.576511\pi$
$593$	−282.338			−0.476118			−0.238059	+	0.971251i	$0.576511\pi$
$594$		0			0
$595$	−638.246	+	495.826i	−1.07268	+	0.833322i
$596$		−	671.819i		−	1.12721i
$597$		0			0
$598$		−	454.587i		−	0.760179i
$599$		−	96.2914i		−	0.160754i	−0.996765	−	0.0803768i	$-0.974388\pi$
$599$		−	96.2914i		−	0.160754i	0.996765	−	0.0803768i	$-0.0256124\pi$
$600$		0			0
$601$	−537.809			−0.894857			−0.447428	−	0.894320i	$-0.647660\pi$
$601$	−537.809			−0.894857			−0.447428	+	0.894320i	$0.647660\pi$
$602$	−97.6901			−0.162276
$603$		0			0
$604$	−350.138			−0.579699
$605$	−337.612	−	434.587i	−0.558036	−	0.718325i
$606$		0			0
$607$			473.265i			0.779679i	0.920883	+	0.389839i	$0.127470\pi$
$607$			473.265i			0.779679i	−0.920883	+	0.389839i	$0.872530\pi$
$608$	−385.874			−0.634662
$609$		0			0
$610$	531.084	+	683.632i	0.870630	+	1.12071i
$611$			40.9999i			0.0671030i
$612$		0			0
$613$		−	109.517i		−	0.178658i	−0.996002	−	0.0893288i	$-0.971528\pi$
$613$		−	109.517i		−	0.178658i	0.996002	−	0.0893288i	$-0.0284722\pi$
$614$			1760.45i			2.86718i
$615$		0			0
$616$	−1561.97			−2.53566
$617$	1132.89			1.83613			0.918064	−	0.396432i	$-0.129752\pi$
$617$	1132.89			1.83613			0.918064	+	0.396432i	$0.129752\pi$
$618$		0			0
$619$	153.529			0.248027			0.124014	−	0.992281i	$-0.460423\pi$
$619$	153.529			0.248027			0.124014	+	0.992281i	$0.460423\pi$
$620$	−849.751	−	1093.83i	−1.37057	−	1.76424i
$621$		0			0
$622$		−	666.079i		−	1.07087i
$623$	−21.1078			−0.0338809
$624$		0			0
$625$	−548.574	−	299.485i	−0.877719	−	0.479176i
$626$		−	781.464i		−	1.24835i
$627$		0			0
$628$		−	1189.58i		−	1.89423i
$629$		−	1023.99i		−	1.62797i
$630$		0			0
$631$	−289.486			−0.458774			−0.229387	−	0.973335i	$-0.573672\pi$
$631$	−289.486			−0.458774			−0.229387	+	0.973335i	$0.573672\pi$
$632$	1346.31			2.13024
$633$		0			0
$634$	−731.274			−1.15343
$635$	−977.203	+	759.147i	−1.53890	+	1.19551i
$636$		0			0
$637$			140.456i			0.220496i
$638$	−2957.58			−4.63571
$639$		0			0
$640$	481.873	+	620.285i	0.752927	+	0.969196i
$641$		−	777.452i		−	1.21287i	−0.795132	−	0.606436i	$-0.792598\pi$
$641$		−	777.452i		−	1.21287i	0.795132	−	0.606436i	$-0.207402\pi$
$642$		0			0
$643$			292.731i			0.455258i	0.973748	+	0.227629i	$0.0730974\pi$
$643$			292.731i			0.455258i	−0.973748	+	0.227629i	$0.926903\pi$
$644$		−	609.438i		−	0.946332i
$645$		0			0
$646$	1524.60			2.36007
$647$	−504.594			−0.779899			−0.389949	−	0.920836i	$-0.627507\pi$
$647$	−504.594			−0.779899			−0.389949	+	0.920836i	$0.627507\pi$
$648$		0			0
$649$	−760.689			−1.17209
$650$	954.705	+	243.631i	1.46878	+	0.374818i
$651$		0			0
$652$		−	753.977i		−	1.15641i
$653$	221.350			0.338974			0.169487	−	0.985532i	$-0.445789\pi$
$653$	221.350			0.338974			0.169487	+	0.985532i	$0.445789\pi$
$654$		0			0
$655$	−561.376	+	436.109i	−0.857063	+	0.665816i
$656$			975.146i			1.48650i
$657$		0			0
$658$			80.0240i			0.121617i
$659$			768.534i			1.16621i	0.812396	+	0.583106i	$0.198163\pi$
$659$			768.534i			1.16621i	−0.812396	+	0.583106i	$0.801837\pi$
$660$		0			0
$661$	−383.835			−0.580689			−0.290345	−	0.956922i	$-0.593770\pi$
$661$	−383.835			−0.580689			−0.290345	+	0.956922i	$0.593770\pi$
$662$	2132.35			3.22108
$663$		0			0
$664$	2569.85			3.87026
$665$	377.858	−	293.542i	0.568208	−	0.441416i
$666$		0			0
$667$		−	627.901i		−	0.941381i
$668$	−980.245			−1.46743
$669$		0			0
$670$	978.984	−	760.531i	1.46117	−	1.13512i
$671$		−	736.351i		−	1.09739i
$672$		0			0
$673$			394.850i			0.586702i	0.956005	+	0.293351i	$0.0947704\pi$
$673$			394.850i			0.586702i	−0.956005	+	0.293351i	$0.905230\pi$
$674$		−	1188.73i		−	1.76370i
$675$		0			0
$676$	415.931			0.615283
$677$	606.978			0.896571			0.448285	−	0.893890i	$-0.352035\pi$
$677$	606.978			0.896571			0.448285	+	0.893890i	$0.352035\pi$
$678$		0			0
$679$	166.532			0.245260
$680$	1405.00	+	1808.57i	2.06618	+	2.65966i
$681$		0			0
$682$			1715.29i			2.51509i
$683$	711.162			1.04123			0.520616	−	0.853791i	$-0.325702\pi$
$683$	711.162			1.04123			0.520616	+	0.853791i	$0.325702\pi$
$684$		0			0
$685$	223.586	+	287.808i	0.326402	+	0.420157i
$686$			1328.76i			1.93697i
$687$		0			0
$688$			117.517i			0.170810i
$689$		−	618.873i		−	0.898219i
$690$		0			0
$691$	850.879			1.23137			0.615687	−	0.787991i	$-0.288879\pi$
$691$	850.879			1.23137			0.615687	+	0.787991i	$0.288879\pi$
$692$	−76.2473			−0.110184
$693$		0			0
$694$	247.349			0.356411
$695$	−205.515	−	264.547i	−0.295705	−	0.380643i
$696$		0			0
$697$		−	1010.99i		−	1.45049i
$698$	−1983.10			−2.84112
$699$		0			0
$700$	1279.92	+	326.622i	1.82845	+	0.466603i
$701$		−	1073.69i		−	1.53166i	−0.643043	−	0.765830i	$-0.722328\pi$
$701$		−	1073.69i		−	1.53166i	0.643043	−	0.765830i	$-0.277672\pi$
$702$		0			0
$703$			606.229i			0.862345i
$704$		−	255.036i		−	0.362266i
$705$		0			0
$706$	−630.002			−0.892355
$707$	−217.332			−0.307401
$708$		0			0
$709$	833.608			1.17575			0.587876	−	0.808951i	$-0.299964\pi$
$709$	833.608			1.17575			0.587876	+	0.808951i	$0.299964\pi$
$710$	544.067	−	422.662i	0.766291	−	0.595299i
$711$		0			0
$712$			59.8123i			0.0840060i
$713$	−364.160			−0.510743
$714$		0			0
$715$	−514.164	−	661.851i	−0.719111	−	0.925666i
$716$			504.216i			0.704212i
$717$		0			0
$718$		−	436.159i		−	0.607463i
$719$			433.030i			0.602267i	0.953582	+	0.301134i	$0.0973650\pi$
$719$			433.030i			0.602267i	−0.953582	+	0.301134i	$0.902635\pi$
$720$		0			0
$721$	168.980			0.234369
$722$	387.649			0.536910
$723$		0			0
$724$	−1225.86			−1.69318
$725$	1318.69	+	336.518i	1.81889	+	0.464162i
$726$		0			0
$727$		−	678.559i		−	0.933369i	−0.884424	−	0.466684i	$-0.845448\pi$
$727$		−	678.559i		−	0.933369i	0.884424	−	0.466684i	$-0.154552\pi$
$728$	−1133.10			−1.55645
$729$		0			0
$730$	384.847	−	298.971i	0.527187	−	0.409549i
$731$		−	121.836i		−	0.166671i
$732$		0			0
$733$			182.056i			0.248371i	0.992259	+	0.124186i	$0.0396318\pi$
$733$			182.056i			0.248371i	−0.992259	+	0.124186i	$0.960368\pi$
$734$			2403.21i			3.27413i
$735$		0			0
$736$	−280.072			−0.380532
$737$	−1054.48			−1.43077
$738$		0			0
$739$	727.990			0.985101			0.492551	−	0.870284i	$-0.336065\pi$
$739$	727.990			0.985101			0.492551	+	0.870284i	$0.336065\pi$
$740$	−1321.65	+	1026.74i	−1.78602	+	1.38748i
$741$		0			0
$742$		−	1207.92i		−	1.62793i
$743$	−394.282			−0.530663			−0.265331	−	0.964157i	$-0.585481\pi$
$743$	−394.282			−0.530663			−0.265331	+	0.964157i	$0.585481\pi$
$744$		0			0
$745$	302.326	−	234.864i	0.405806	−	0.315254i
$746$			1836.44i			2.46171i
$747$		0			0
$748$		−	3580.15i		−	4.78630i
$749$		−	553.079i		−	0.738423i
$750$		0			0
$751$	1339.18			1.78320			0.891600	−	0.452824i	$-0.149583\pi$
$751$	1339.18			1.78320			0.891600	+	0.452824i	$0.149583\pi$
$752$	96.2654			0.128013
$753$		0			0
$754$	−2145.52			−2.84552
$755$	−122.406	−	157.566i	−0.162128	−	0.208697i
$756$		0			0
$757$		−	185.595i		−	0.245171i	−0.992458	−	0.122586i	$-0.960881\pi$
$757$		−	185.595i		−	0.245171i	0.992458	−	0.122586i	$-0.0391186\pi$
$758$	1773.98			2.34034
$759$		0			0
$760$	−831.797	−	1070.72i	−1.09447	−	1.40884i
$761$		−	933.732i		−	1.22698i	−0.789703	−	0.613490i	$-0.789765\pi$
$761$		−	933.732i		−	1.22698i	0.789703	−	0.613490i	$-0.210235\pi$
$762$		0			0
$763$		−	470.775i		−	0.617005i
$764$			276.180i			0.361492i
$765$		0			0
$766$	−2129.90			−2.78054
$767$	−551.827			−0.719461
$768$		0			0
$769$	1058.93			1.37702			0.688511	−	0.725226i	$-0.258265\pi$
$769$	1058.93			1.37702			0.688511	+	0.725226i	$0.258265\pi$
$770$	−1003.55	−	1291.81i	−1.30331	−	1.67767i
$771$		0			0
$772$		−	439.374i		−	0.569137i
$773$	−79.0622			−0.102280			−0.0511399	−	0.998692i	$-0.516285\pi$
$773$	−79.0622			−0.102280			−0.0511399	+	0.998692i	$0.516285\pi$
$774$		0			0
$775$	195.168	−	764.794i	0.251830	−	0.986831i
$776$		−	471.894i		−	0.608111i
$777$		0			0
$778$			1570.05i			2.01806i
$779$			598.531i			0.768333i
$780$		0			0
$781$	−586.023			−0.750350
$782$	1106.57			1.41506
$783$		0			0
$784$	329.783			0.420641
$785$	535.323	−	415.869i	0.681940	−	0.529770i
$786$		0			0
$787$			966.245i			1.22776i	0.789401	+	0.613878i	$0.210391\pi$
$787$			966.245i			1.22776i	−0.789401	+	0.613878i	$0.789609\pi$
$788$	−328.955			−0.417455
$789$		0			0
$790$	864.994	+	1113.45i	1.09493	+	1.40943i
$791$		−	39.5470i		−	0.0499962i
$792$		0			0
$793$		−	534.171i		−	0.673608i
$794$		−	1450.85i		−	1.82726i
$795$		0			0
$796$	355.510			0.446621
$797$	−1275.81			−1.60077			−0.800384	−	0.599488i	$-0.795371\pi$
$797$	−1275.81			−1.60077			−0.800384	+	0.599488i	$0.795371\pi$
$798$		0			0
$799$	−99.8037			−0.124911
$800$	150.102	−	588.195i	0.187627	−	0.735244i
$801$		0			0
$802$			1173.90i			1.46372i
$803$	−414.524			−0.516220
$804$		0			0
$805$	274.253	−	213.056i	0.340688	−	0.264666i
$806$			1244.32i			1.54383i
$807$		0			0
$808$			615.845i			0.762185i
$809$			1533.89i			1.89603i	0.318220	+	0.948017i	$0.396915\pi$
$809$			1533.89i			1.89603i	−0.318220	+	0.948017i	$0.603085\pi$
$810$		0			0
$811$	−753.037			−0.928530			−0.464265	−	0.885696i	$-0.653681\pi$
$811$	−753.037			−0.928530			−0.464265	+	0.885696i	$0.653681\pi$
$812$	−2876.37			−3.54233
$813$		0			0
$814$	2072.55			2.54613
$815$	339.298	−	263.586i	0.416316	−	0.323418i
$816$		0			0
$817$			72.1303i			0.0882867i
$818$	58.3067			0.0712796
$819$		0			0
$820$	−1304.87	+	1013.70i	−1.59131	+	1.23622i
$821$			1543.26i			1.87973i	0.341551	+	0.939863i	$0.389048\pi$
$821$			1543.26i			1.87973i	−0.341551	+	0.939863i	$0.610952\pi$
$822$		0			0
$823$			1167.96i			1.41915i	0.704632	+	0.709573i	$0.251112\pi$
$823$			1167.96i			1.41915i	−0.704632	+	0.709573i	$0.748888\pi$
$824$		−	478.832i		−	0.581107i
$825$		0			0
$826$	−1077.06			−1.30395
$827$	432.942			0.523509			0.261755	−	0.965134i	$-0.415699\pi$
$827$	432.942			0.523509			0.261755	+	0.965134i	$0.415699\pi$
$828$		0			0
$829$	470.069			0.567031			0.283515	−	0.958968i	$-0.408499\pi$
$829$	470.069			0.567031			0.283515	+	0.958968i	$0.408499\pi$
$830$	1651.11	+	2125.37i	1.98929	+	2.56068i
$831$		0			0
$832$		−	185.010i		−	0.222368i
$833$	−341.904			−0.410449
$834$		0			0
$835$	−342.688	−	441.121i	−0.410405	−	0.528288i
$836$			2119.54i			2.53534i
$837$		0			0
$838$			1363.70i			1.62733i
$839$			1009.21i			1.20288i	0.798919	+	0.601439i	$0.205406\pi$
$839$			1009.21i			1.20288i	−0.798919	+	0.601439i	$0.794594\pi$
$840$		0			0
$841$	−2122.51			−2.52380
$842$	−1513.06			−1.79698
$843$		0			0
$844$	1513.32			1.79304
$845$	145.407	+	187.173i	0.172079	+	0.221507i
$846$		0			0
$847$			662.783i			0.782507i
$848$	−1453.08			−1.71353
$849$		0			0
$850$	−593.058	+	2323.98i	−0.697715	+	2.73410i
$851$			440.007i			0.517047i
$852$		0			0
$853$		−	900.755i		−	1.05598i	−0.849249	−	0.527992i	$-0.822945\pi$
$853$		−	900.755i		−	1.05598i	0.849249	−	0.527992i	$-0.177055\pi$
$854$		−	1042.60i		−	1.22084i
$855$		0			0
$856$	−1567.24			−1.83088
$857$	−111.824			−0.130483			−0.0652417	−	0.997869i	$-0.520782\pi$
$857$	−111.824			−0.130483			−0.0652417	+	0.997869i	$0.520782\pi$
$858$		0			0
$859$	180.456			0.210076			0.105038	−	0.994468i	$-0.466504\pi$
$859$	180.456			0.210076			0.105038	+	0.994468i	$0.466504\pi$
$860$	−157.253	+	122.163i	−0.182852	+	0.142050i
$861$		0			0
$862$			172.659i			0.200300i
$863$	451.735			0.523447			0.261724	−	0.965143i	$-0.415709\pi$
$863$	451.735			0.523447			0.261724	+	0.965143i	$0.415709\pi$
$864$		0			0
$865$	−26.6556	−	34.3121i	−0.0308157	−	0.0396672i
$866$		−	128.883i		−	0.148826i
$867$		0			0
$868$			1668.19i			1.92188i
$869$		−	1199.32i		−	1.38011i
$870$		0			0
$871$	−764.951			−0.878244
$872$	−1334.01			−1.52983
$873$		0			0
$874$	−655.120			−0.749566
$875$	300.468	+	690.161i	0.343392	+	0.788756i
$876$		0			0
$877$		−	736.225i		−	0.839481i	−0.907644	−	0.419741i	$-0.862121\pi$
$877$		−	736.225i		−	0.839481i	0.907644	−	0.419741i	$-0.137879\pi$
$878$	929.579			1.05875
$879$		0			0
$880$	−1553.99	+	1207.23i	−1.76590	+	1.37185i
$881$			52.0576i			0.0590892i	0.999563	+	0.0295446i	$0.00940571\pi$
$881$			52.0576i			0.0590892i	−0.999563	+	0.0295446i	$0.990594\pi$
$882$		0			0
$883$		−	637.785i		−	0.722294i	−0.932509	−	0.361147i	$-0.882385\pi$
$883$		−	637.785i		−	0.722294i	0.932509	−	0.361147i	$-0.117615\pi$
$884$		−	2597.15i		−	2.93795i
$885$		0			0
$886$	80.5227			0.0908834
$887$	−1346.54			−1.51809			−0.759043	−	0.651041i	$-0.774333\pi$
$887$	−1346.54			−1.51809			−0.759043	+	0.651041i	$0.774333\pi$
$888$		0			0
$889$	1490.32			1.67640
$890$	−49.4671	+	38.4289i	−0.0555810	+	0.0431785i
$891$		0			0
$892$			2626.63i			2.94465i
$893$	59.0864			0.0661661
$894$		0			0
$895$	−226.902	+	176.271i	−0.253522	+	0.196951i
$896$		−	945.990i		−	1.05579i
$897$		0			0
$898$			985.169i			1.09707i
$899$			1718.73i			1.91182i
$900$		0			0
$901$	1506.49			1.67202
$902$	2046.24			2.26855
$903$		0			0
$904$	−112.063			−0.123963
$905$	−428.555	−	551.652i	−0.473541	−	0.609560i
$906$		0			0
$907$			1073.34i			1.18339i	0.806161	+	0.591696i	$0.201541\pi$
$907$			1073.34i			1.18339i	−0.806161	+	0.591696i	$0.798459\pi$
$908$	2621.02			2.88658
$909$		0			0
$910$	−728.005	−	937.115i	−0.800005	−	1.02980i
$911$		−	83.7182i		−	0.0918970i	−0.998944	−	0.0459485i	$-0.985369\pi$
$911$		−	83.7182i		−	0.0918970i	0.998944	−	0.0459485i	$-0.0146310\pi$
$912$		0			0
$913$		−	2289.27i		−	2.50741i
$914$			426.776i			0.466932i
$915$		0			0
$916$	−1783.90			−1.94749
$917$	856.149			0.933641
$918$		0			0
$919$	1235.72			1.34463			0.672317	−	0.740264i	$-0.265299\pi$
$919$	1235.72			1.34463			0.672317	+	0.740264i	$0.265299\pi$
$920$	−603.727	−	777.140i	−0.656225	−	0.844717i
$921$		0			0
$922$			322.945i			0.350266i
$923$	−425.119			−0.460583
$924$		0			0
$925$	−924.085	−	235.818i	−0.999011	−	0.254938i
$926$		−	2181.77i		−	2.35612i
$927$		0			0
$928$			1321.86i			1.42442i
$929$		−	745.068i		−	0.802011i	−0.916076	−	0.401006i	$-0.868661\pi$
$929$		−	745.068i		−	0.802011i	0.916076	−	0.401006i	$-0.131339\pi$
$930$		0			0
$931$	202.416			0.217418
$932$	3829.08			4.10845
$933$		0			0
$934$	−234.652			−0.251234
$935$	1611.11	−	1251.60i	1.72311	−	1.33861i
$936$		0			0
$937$			353.460i			0.377225i	0.982052	+	0.188613i	$0.0603991\pi$
$937$			353.460i			0.377225i	−0.982052	+	0.188613i	$0.939601\pi$
$938$	−1493.04			−1.59172
$939$		0			0
$940$	100.071	+	128.816i	0.106459	+	0.137038i
$941$			253.779i			0.269690i	0.990867	+	0.134845i	$0.0430537\pi$
$941$			253.779i			0.269690i	−0.990867	+	0.134845i	$0.956946\pi$
$942$		0			0
$943$			434.420i			0.460679i
$944$			1295.66i			1.37252i
$945$		0			0
$946$	246.596			0.260673
$947$	−1574.72			−1.66285			−0.831424	−	0.555639i	$-0.812474\pi$
$947$	−1574.72			−1.66285			−0.831424	+	0.555639i	$0.812474\pi$
$948$		0			0
$949$	−300.708			−0.316869
$950$	351.105	−	1375.86i	0.369585	−	1.44827i
$951$		0			0
$952$		−	2758.23i		−	2.89730i
$953$	137.098			0.143860			0.0719300	−	0.997410i	$-0.477084\pi$
$953$	137.098			0.143860			0.0719300	+	0.997410i	$0.477084\pi$
$954$		0			0
$955$	−124.284	+	96.5508i	−0.130140	+	0.101100i
$956$		−	2103.17i		−	2.19997i
$957$		0			0
$958$			335.900i			0.350626i
$959$		−	438.932i		−	0.457698i
$960$		0			0
$961$	35.8009			0.0372538
$962$	1503.49			1.56288
$963$		0			0
$964$	1801.78			1.86907
$965$	197.723	−	153.603i	0.204894	−	0.159174i
$966$		0			0
$967$		−	377.253i		−	0.390127i	−0.980791	−	0.195064i	$-0.937509\pi$
$967$		−	377.253i		−	0.390127i	0.980791	−	0.195064i	$-0.0624913\pi$
$968$	1878.10			1.94019
$969$		0			0
$970$	390.275	−	303.188i	0.402345	−	0.312565i
$971$			855.490i			0.881040i	0.897743	+	0.440520i	$0.145206\pi$
$971$			855.490i			0.881040i	−0.897743	+	0.440520i	$0.854794\pi$
$972$		0			0
$973$			403.457i			0.414653i
$974$		−	2878.82i		−	2.95567i
$975$		0			0
$976$	−1254.20			−1.28504
$977$	−1074.71			−1.10001			−0.550004	−	0.835162i	$-0.685374\pi$
$977$	−1074.71			−1.10001			−0.550004	+	0.835162i	$0.685374\pi$
$978$		0			0
$979$	53.2818			0.0544247
$980$	342.821	+	441.292i	0.349818	+	0.450298i
$981$		0			0
$982$		−	3362.46i		−	3.42410i
$983$	1161.55			1.18164			0.590821	−	0.806803i	$-0.298804\pi$
$983$	1161.55			1.18164			0.590821	+	0.806803i	$0.298804\pi$
$984$		0			0
$985$	−115.001	−	148.033i	−0.116752	−	0.150287i
$986$		−	5222.71i		−	5.29687i
$987$		0			0
$988$			1537.58i			1.55625i
$989$			52.3529i			0.0529352i
$990$		0			0
$991$	463.882			0.468095			0.234047	−	0.972225i	$-0.424803\pi$
$991$	463.882			0.468095			0.234047	+	0.972225i	$0.424803\pi$
$992$	766.629			0.772812
$993$		0			0
$994$	−829.750			−0.834759
$995$	124.284	+	159.983i	0.124909	+	0.160787i
$996$		0			0
$997$			186.488i			0.187050i	0.995617	+	0.0935248i	$0.0298134\pi$
$997$			186.488i			0.187050i	−0.995617	+	0.0935248i	$0.970187\pi$
$998$	−1357.91			−1.36063
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.b.404.2	yes	24
3.2	odd	2	inner	405.3.d.b.404.23	yes	24
5.4	even	2	inner	405.3.d.b.404.24	yes	24
9.2	odd	6		405.3.h.k.134.2		48
9.4	even	3		405.3.h.k.269.24		48
9.5	odd	6		405.3.h.k.269.1		48
9.7	even	3		405.3.h.k.134.23		48
15.14	odd	2	inner	405.3.d.b.404.1	✓	24
45.4	even	6		405.3.h.k.269.2		48
45.14	odd	6		405.3.h.k.269.23		48
45.29	odd	6		405.3.h.k.134.24		48
45.34	even	6		405.3.h.k.134.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.3.d.b.404.1	✓	24	15.14	odd	2	inner
405.3.d.b.404.2	yes	24	1.1	even	1	trivial
405.3.d.b.404.23	yes	24	3.2	odd	2	inner
405.3.d.b.404.24	yes	24	5.4	even	2	inner
405.3.h.k.134.1		48	45.34	even	6
405.3.h.k.134.2		48	9.2	odd	6
405.3.h.k.134.23		48	9.7	even	3
405.3.h.k.134.24		48	45.29	odd	6
405.3.h.k.269.1		48	9.5	odd	6
405.3.h.k.269.2		48	45.4	even	6
405.3.h.k.269.23		48	45.14	odd	6
405.3.h.k.269.24		48	9.4	even	3

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.57411 q^{2} +8.77429 q^{4} +(3.06744 + 3.94852i) q^{5} -6.02185i q^{7} -17.0638 q^{8} +(-10.9634 - 14.1125i) q^{10} +15.2008i q^{11} +11.0271i q^{13} +21.5228i q^{14} +25.8909 q^{16} -26.8426 q^{17} +15.8915 q^{19} +(26.9146 + 34.6455i) q^{20} -54.3292i q^{22} +11.5342 q^{23} +(-6.18165 + 24.2237i) q^{25} -39.4120i q^{26} -52.8374i q^{28} -54.4382i q^{29} -31.5722 q^{31} -24.2818 q^{32} +95.9384 q^{34} +(23.7774 - 18.4716i) q^{35} +38.1480i q^{37} -56.7980 q^{38} +(-52.3423 - 67.3769i) q^{40} +37.6636i q^{41} +4.53892i q^{43} +133.376i q^{44} -41.2246 q^{46} +3.71811 q^{47} +12.7374 q^{49} +(22.0939 - 86.5782i) q^{50} +96.7548i q^{52} -56.1230 q^{53} +(-60.0205 + 46.6274i) q^{55} +102.756i q^{56} +194.568i q^{58} +50.0428i q^{59} -48.4417 q^{61} +112.842 q^{62} -16.7778 q^{64} +(-43.5407 + 33.8249i) q^{65} +69.3702i q^{67} -235.524 q^{68} +(-84.9831 + 66.0197i) q^{70} +38.5522i q^{71} +27.2700i q^{73} -136.345i q^{74} +139.437 q^{76} +91.5366 q^{77} -78.8985 q^{79} +(79.4189 + 102.231i) q^{80} -134.614i q^{82} -150.602 q^{83} +(-82.3379 - 105.988i) q^{85} -16.2226i q^{86} -259.383i q^{88} -3.50521i q^{89} +66.4034 q^{91} +101.204 q^{92} -13.2890 q^{94} +(48.7462 + 62.7479i) q^{95} +27.6546i q^{97} -45.5248 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91}+ \cdots + 384 q^{94}+O(q^{100}) \) 24 * q + 48 * q^4 + 12 * q^10 + 96 * q^16 + 48 * q^25 + 144 * q^34 + 72 * q^40 - 168 * q^46 - 288 * q^49 - 132 * q^55 - 360 * q^61 - 72 * q^64 - 156 * q^70 + 48 * q^76 - 480 * q^79 - 456 * q^85 - 48 * q^91 + 384 * q^94

Introduction

L-functions

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