LMFDB - Embedded newform 405.3.d.b.404.4

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			404.4
Character	$\chi$	$=$	405.404
Dual form			405.3.d.b.404.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.48400 q^{2} +8.13825 q^{4} +(-4.82085 + 1.32641i) q^{5} +12.7764i q^{7} -14.4177 q^{8} +(16.7958 - 4.62123i) q^{10} +14.1879i q^{11} +4.50087i q^{13} -44.5129i q^{14} +17.6781 q^{16} +9.20574 q^{17} -15.8342 q^{19} +(-39.2333 + 10.7947i) q^{20} -49.4307i q^{22} +4.24292 q^{23} +(21.4812 - 12.7889i) q^{25} -15.6810i q^{26} +103.977i q^{28} +26.5256i q^{29} +31.4358 q^{31} -3.91996 q^{32} -32.0728 q^{34} +(-16.9468 - 61.5930i) q^{35} +14.6355i q^{37} +55.1662 q^{38} +(69.5055 - 19.1238i) q^{40} +44.6491i q^{41} -65.3824i q^{43} +115.465i q^{44} -14.7823 q^{46} -56.5711 q^{47} -114.236 q^{49} +(-74.8407 + 44.5565i) q^{50} +36.6292i q^{52} +17.8840 q^{53} +(-18.8190 - 68.3978i) q^{55} -184.205i q^{56} -92.4153i q^{58} -48.3018i q^{59} -91.4697 q^{61} -109.522 q^{62} -57.0554 q^{64} +(-5.97002 - 21.6980i) q^{65} +50.6634i q^{67} +74.9187 q^{68} +(59.0425 + 214.590i) q^{70} -67.7587i q^{71} +52.6768i q^{73} -50.9899i q^{74} -128.862 q^{76} -181.270 q^{77} -54.8962 q^{79} +(-85.2237 + 23.4485i) q^{80} -155.558i q^{82} -2.55751 q^{83} +(-44.3795 + 12.2106i) q^{85} +227.792i q^{86} -204.557i q^{88} -82.7313i q^{89} -57.5048 q^{91} +34.5299 q^{92} +197.094 q^{94} +(76.3341 - 21.0026i) q^{95} -63.8998i q^{97} +397.997 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.48400			−1.74200			−0.871000	−	0.491283i	$-0.836528\pi$
$2$	−3.48400			−1.74200			−0.871000	+	0.491283i	$0.836528\pi$
$3$		0			0
$3$		0			0
$4$	8.13825			2.03456
$5$	−4.82085	+	1.32641i	−0.964171	+	0.265283i
$5$	−4.82085	+	1.32641i	−0.964171	+	0.265283i
$6$		0			0
$7$			12.7764i			1.82520i	0.408859	+	0.912598i	$0.365927\pi$
$7$			12.7764i			1.82520i	−0.408859	+	0.912598i	$0.634073\pi$
$8$	−14.4177			−1.80221
$9$		0			0
$10$	16.7958	−	4.62123i	1.67958	−	0.462123i
$11$			14.1879i			1.28981i	0.764263	+	0.644905i	$0.223103\pi$
$11$			14.1879i			1.28981i	−0.764263	+	0.644905i	$0.776897\pi$
$12$		0			0
$13$			4.50087i			0.346221i	0.984902	+	0.173110i	$0.0553818\pi$
$13$			4.50087i			0.346221i	−0.984902	+	0.173110i	$0.944618\pi$
$14$		−	44.5129i		−	3.17949i
$15$		0			0
$16$	17.6781			1.10488
$17$	9.20574			0.541514			0.270757	−	0.962648i	$-0.412726\pi$
$17$	9.20574			0.541514			0.270757	+	0.962648i	$0.412726\pi$
$18$		0			0
$19$	−15.8342			−0.833376			−0.416688	−	0.909049i	$-0.636809\pi$
$19$	−15.8342			−0.833376			−0.416688	+	0.909049i	$0.636809\pi$
$20$	−39.2333	+	10.7947i	−1.96167	+	0.539735i
$21$		0			0
$22$		−	49.4307i		−	2.24685i
$23$	4.24292			0.184475			0.0922373	−	0.995737i	$-0.470598\pi$
$23$	4.24292			0.184475			0.0922373	+	0.995737i	$0.470598\pi$
$24$		0			0
$25$	21.4812	−	12.7889i	0.859250	−	0.511556i
$26$		−	15.6810i		−	0.603117i
$27$		0			0
$28$			103.977i			3.71348i
$29$			26.5256i			0.914677i	0.889293	+	0.457338i	$0.151197\pi$
$29$			26.5256i			0.914677i	−0.889293	+	0.457338i	$0.848803\pi$
$30$		0			0
$31$	31.4358			1.01406			0.507028	−	0.861929i	$-0.330744\pi$
$31$	31.4358			1.01406			0.507028	+	0.861929i	$0.330744\pi$
$32$	−3.91996			−0.122499
$33$		0			0
$34$	−32.0728			−0.943318
$35$	−16.9468	−	61.5930i	−0.484193	−	1.75980i
$36$		0			0
$37$			14.6355i			0.395553i	0.980247	+	0.197777i	$0.0633720\pi$
$37$			14.6355i			0.395553i	−0.980247	+	0.197777i	$0.936628\pi$
$38$	55.1662			1.45174
$39$		0			0
$40$	69.5055	−	19.1238i	1.73764	−	0.478095i
$41$			44.6491i			1.08900i	0.838760	+	0.544502i	$0.183281\pi$
$41$			44.6491i			1.08900i	−0.838760	+	0.544502i	$0.816719\pi$
$42$		0			0
$43$		−	65.3824i		−	1.52052i	−0.649618	−	0.760261i	$-0.725071\pi$
$43$		−	65.3824i		−	1.52052i	0.649618	−	0.760261i	$-0.274929\pi$
$44$			115.465i			2.62420i
$45$		0			0
$46$	−14.7823			−0.321355
$47$	−56.5711			−1.20364			−0.601821	−	0.798631i	$-0.705558\pi$
$47$	−56.5711			−1.20364			−0.601821	+	0.798631i	$0.705558\pi$
$48$		0			0
$49$	−114.236			−2.33134
$50$	−74.8407	+	44.5565i	−1.49681	+	0.891130i
$51$		0			0
$52$			36.6292i			0.704408i
$53$	17.8840			0.337434			0.168717	−	0.985665i	$-0.446038\pi$
$53$	17.8840			0.337434			0.168717	+	0.985665i	$0.446038\pi$
$54$		0			0
$55$	−18.8190	−	68.3978i	−0.342164	−	1.24360i
$56$		−	184.205i		−	3.28938i
$57$		0			0
$58$		−	92.4153i		−	1.59337i
$59$		−	48.3018i		−	0.818674i	−0.912383	−	0.409337i	$-0.865760\pi$
$59$		−	48.3018i		−	0.818674i	0.912383	−	0.409337i	$-0.134240\pi$
$60$		0			0
$61$	−91.4697			−1.49950			−0.749752	−	0.661719i	$-0.769827\pi$
$61$	−91.4697			−1.49950			−0.749752	+	0.661719i	$0.769827\pi$
$62$	−109.522			−1.76649
$63$		0			0
$64$	−57.0554			−0.891491
$65$	−5.97002	−	21.6980i	−0.0918465	−	0.333816i
$66$		0			0
$67$			50.6634i			0.756170i	0.925771	+	0.378085i	$0.123417\pi$
$67$			50.6634i			0.756170i	−0.925771	+	0.378085i	$0.876583\pi$
$68$	74.9187			1.10175
$69$		0			0
$70$	59.0425	+	214.590i	0.843464	+	3.06557i
$71$		−	67.7587i		−	0.954348i	−0.878809	−	0.477174i	$-0.841661\pi$
$71$		−	67.7587i		−	0.954348i	0.878809	−	0.477174i	$-0.158339\pi$
$72$		0			0
$73$			52.6768i			0.721600i	0.932643	+	0.360800i	$0.117496\pi$
$73$			52.6768i			0.721600i	−0.932643	+	0.360800i	$0.882504\pi$
$74$		−	50.9899i		−	0.689053i
$75$		0			0
$76$	−128.862			−1.69556
$77$	−181.270			−2.35415
$78$		0			0
$79$	−54.8962			−0.694888			−0.347444	−	0.937701i	$-0.612950\pi$
$79$	−54.8962			−0.694888			−0.347444	+	0.937701i	$0.612950\pi$
$80$	−85.2237	+	23.4485i	−1.06530	+	0.293107i
$81$		0			0
$82$		−	155.558i		−	1.89704i
$83$	−2.55751			−0.0308134			−0.0154067	−	0.999881i	$-0.504904\pi$
$83$	−2.55751			−0.0308134			−0.0154067	+	0.999881i	$0.504904\pi$
$84$		0			0
$85$	−44.3795	+	12.2106i	−0.522112	+	0.143655i
$86$			227.792i			2.64875i
$87$		0			0
$88$		−	204.557i		−	2.32451i
$89$		−	82.7313i		−	0.929565i	−0.885425	−	0.464783i	$-0.846132\pi$
$89$		−	82.7313i		−	0.929565i	0.885425	−	0.464783i	$-0.153868\pi$
$90$		0			0
$91$	−57.5048			−0.631921
$92$	34.5299			0.375325
$93$		0			0
$94$	197.094			2.09674
$95$	76.3341	−	21.0026i	0.803517	−	0.221081i
$96$		0			0
$97$		−	63.8998i		−	0.658760i	−0.944197	−	0.329380i	$-0.893160\pi$
$97$		−	63.8998i		−	0.658760i	0.944197	−	0.329380i	$-0.106840\pi$
$98$	397.997			4.06119
$99$		0			0
$100$	174.820	−	104.079i	1.74820	−	1.04079i
$101$			80.6189i			0.798206i	0.916906	+	0.399103i	$0.130678\pi$
$101$			80.6189i			0.798206i	−0.916906	+	0.399103i	$0.869322\pi$
$102$		0			0
$103$			35.9731i			0.349253i	0.984635	+	0.174627i	$0.0558719\pi$
$103$			35.9731i			0.349253i	−0.984635	+	0.174627i	$0.944128\pi$
$104$		−	64.8921i		−	0.623962i
$105$		0			0
$106$	−62.3079			−0.587810
$107$	−21.6725			−0.202546			−0.101273	−	0.994859i	$-0.532292\pi$
$107$	−21.6725			−0.202546			−0.101273	+	0.994859i	$0.532292\pi$
$108$		0			0
$109$	−1.09218			−0.0100200			−0.00501000	−	0.999987i	$-0.501595\pi$
$109$	−1.09218			−0.0100200			−0.00501000	+	0.999987i	$0.501595\pi$
$110$	65.5655	+	238.298i	0.596050	+	2.16634i
$111$		0			0
$112$			225.862i			2.01663i
$113$	152.215			1.34703			0.673517	−	0.739172i	$-0.264783\pi$
$113$	152.215			1.34703			0.673517	+	0.739172i	$0.264783\pi$
$114$		0			0
$115$	−20.4545	+	5.62786i	−0.177865	+	0.0489379i
$116$			215.872i			1.86097i
$117$		0			0
$118$			168.283i			1.42613i
$119$			117.616i			0.988370i
$120$		0			0
$121$	−80.2966			−0.663609
$122$	318.680			2.61213
$123$		0			0
$124$	255.832			2.06316
$125$	−86.5946	+	90.1464i	−0.692757	+	0.721172i
$126$		0			0
$127$			114.497i			0.901549i	0.892638	+	0.450775i	$0.148852\pi$
$127$			114.497i			0.901549i	−0.892638	+	0.450775i	$0.851148\pi$
$128$	214.461			1.67548
$129$		0			0
$130$	20.7995	+	75.5960i	0.159997	+	0.581507i
$131$		−	171.106i		−	1.30615i	−0.757292	−	0.653077i	$-0.773478\pi$
$131$		−	171.106i		−	1.30615i	0.757292	−	0.653077i	$-0.226522\pi$
$132$		0			0
$133$		−	202.303i		−	1.52107i
$134$		−	176.511i		−	1.31725i
$135$		0			0
$136$	−132.725			−0.975922
$137$	−36.4207			−0.265844			−0.132922	−	0.991126i	$-0.542436\pi$
$137$	−36.4207			−0.265844			−0.132922	+	0.991126i	$0.542436\pi$
$138$		0			0
$139$	75.7842			0.545210			0.272605	−	0.962126i	$-0.412115\pi$
$139$	75.7842			0.545210			0.272605	+	0.962126i	$0.412115\pi$
$140$	−137.917	−	501.259i	−0.985121	−	3.58042i
$141$		0			0
$142$			236.071i			1.66247i
$143$	−63.8579			−0.446559
$144$		0			0
$145$	−35.1840	−	127.876i	−0.242648	−	0.881905i
$146$		−	183.526i		−	1.25703i
$147$		0			0
$148$			119.107i			0.804778i
$149$			191.417i			1.28468i	0.766422	+	0.642338i	$0.222035\pi$
$149$			191.417i			1.28468i	−0.766422	+	0.642338i	$0.777965\pi$
$150$		0			0
$151$	239.238			1.58436			0.792178	−	0.610290i	$-0.208947\pi$
$151$	239.238			1.58436			0.792178	+	0.610290i	$0.208947\pi$
$152$	228.292			1.50192
$153$		0			0
$154$	631.544			4.10094
$155$	−151.547	+	41.6968i	−0.977724	+	0.269012i
$156$		0			0
$157$		−	143.346i		−	0.913030i	−0.889716	−	0.456515i	$-0.849097\pi$
$157$		−	143.346i		−	0.913030i	0.889716	−	0.456515i	$-0.150903\pi$
$158$	191.258			1.21050
$159$		0			0
$160$	18.8976	−	5.19949i	0.118110	−	0.0324968i
$161$			54.2090i			0.336702i
$162$		0			0
$163$		−	138.957i		−	0.852496i	−0.904606	−	0.426248i	$-0.859835\pi$
$163$		−	138.957i		−	0.852496i	0.904606	−	0.426248i	$-0.140165\pi$
$164$			363.366i			2.21565i
$165$		0			0
$166$	8.91037			0.0536769
$167$	−205.580			−1.23102			−0.615510	−	0.788129i	$-0.711050\pi$
$167$	−205.580			−1.23102			−0.615510	+	0.788129i	$0.711050\pi$
$168$		0			0
$169$	148.742			0.880131
$170$	154.618	−	42.5418i	0.909519	−	0.250246i
$171$		0			0
$172$		−	532.099i		−	3.09360i
$173$	−47.5375			−0.274783			−0.137392	−	0.990517i	$-0.543872\pi$
$173$	−47.5375			−0.274783			−0.137392	+	0.990517i	$0.543872\pi$
$174$		0			0
$175$	163.396	+	274.452i	0.933690	+	1.56830i
$176$			250.816i			1.42509i
$177$		0			0
$178$			288.236i			1.61930i
$179$		−	124.298i		−	0.694402i	−0.937791	−	0.347201i	$-0.887132\pi$
$179$		−	124.298i		−	0.694402i	0.937791	−	0.347201i	$-0.112868\pi$
$180$		0			0
$181$	−12.0770			−0.0667237			−0.0333618	−	0.999443i	$-0.510621\pi$
$181$	−12.0770			−0.0667237			−0.0333618	+	0.999443i	$0.510621\pi$
$182$	200.347			1.10081
$183$		0			0
$184$	−61.1729			−0.332462
$185$	−19.4127	−	70.5554i	−0.104933	−	0.381381i
$186$		0			0
$187$			130.610i			0.698450i
$188$	−460.390			−2.44888
$189$		0			0
$190$	−265.948	+	73.1732i	−1.39973	+	0.385122i
$191$		−	285.715i		−	1.49589i	−0.663761	−	0.747945i	$-0.731041\pi$
$191$		−	285.715i		−	1.49589i	0.663761	−	0.747945i	$-0.268959\pi$
$192$		0			0
$193$			328.867i			1.70397i	0.523563	+	0.851987i	$0.324602\pi$
$193$			328.867i			1.70397i	−0.523563	+	0.851987i	$0.675398\pi$
$194$			222.627i			1.14756i
$195$		0			0
$196$	−929.678			−4.74326
$197$	−90.3147			−0.458450			−0.229225	−	0.973373i	$-0.573619\pi$
$197$	−90.3147			−0.458450			−0.229225	+	0.973373i	$0.573619\pi$
$198$		0			0
$199$	0.315816			0.00158702			0.000793508	−	1.00000i	$-0.499747\pi$
$199$	0.315816			0.00158702			0.000793508		1.00000i	$0.499747\pi$
$200$	−309.710	+	184.386i	−1.54855	+	0.921931i
$201$		0			0
$202$		−	280.876i		−	1.39048i
$203$	−338.901			−1.66946
$204$		0			0
$205$	−59.2232	−	215.247i	−0.288894	−	1.04998i
$206$		−	125.330i		−	0.608399i
$207$		0			0
$208$			79.5670i			0.382534i
$209$		−	224.653i		−	1.07490i
$210$		0			0
$211$	−328.374			−1.55627			−0.778137	−	0.628095i	$-0.783835\pi$
$211$	−328.374			−1.55627			−0.778137	+	0.628095i	$0.783835\pi$
$212$	145.545			0.686531
$213$		0			0
$214$	75.5069			0.352836
$215$	86.7242	+	315.199i	0.403368	+	1.46604i
$216$		0			0
$217$			401.635i			1.85085i
$218$	3.80515			0.0174548
$219$		0			0
$220$	−153.154	−	556.639i	−0.696155	−	2.53018i
$221$			41.4339i			0.187484i
$222$		0			0
$223$			50.5589i			0.226722i	0.993554	+	0.113361i	$0.0361616\pi$
$223$			50.5589i			0.226722i	−0.993554	+	0.113361i	$0.963838\pi$
$224$		−	50.0829i		−	0.223584i
$225$		0			0
$226$	−530.316			−2.34653
$227$	−301.013			−1.32605			−0.663024	−	0.748598i	$-0.730727\pi$
$227$	−301.013			−1.32605			−0.663024	+	0.748598i	$0.730727\pi$
$228$		0			0
$229$	279.425			1.22020			0.610099	−	0.792325i	$-0.291130\pi$
$229$	279.425			1.22020			0.610099	+	0.792325i	$0.291130\pi$
$230$	71.2634	−	19.6075i	0.309841	−	0.0852499i
$231$		0			0
$232$		−	382.438i		−	1.64844i
$233$	27.3269			0.117283			0.0586415	−	0.998279i	$-0.481323\pi$
$233$	27.3269			0.117283			0.0586415	+	0.998279i	$0.481323\pi$
$234$		0			0
$235$	272.721	−	75.0368i	1.16052	−	0.319305i
$236$		−	393.092i		−	1.66564i
$237$		0			0
$238$		−	409.774i		−	1.72174i
$239$			17.7233i			0.0741561i	0.999312	+	0.0370780i	$0.0118050\pi$
$239$			17.7233i			0.0741561i	−0.999312	+	0.0370780i	$0.988195\pi$
$240$		0			0
$241$	203.893			0.846029			0.423014	−	0.906123i	$-0.360972\pi$
$241$	203.893			0.846029			0.423014	+	0.906123i	$0.360972\pi$
$242$	279.753			1.15601
$243$		0			0
$244$	−744.404			−3.05083
$245$	550.713	−	151.524i	2.24781	−	0.618464i
$246$		0			0
$247$		−	71.2675i		−	0.288532i
$248$	−453.230			−1.82754
$249$		0			0
$250$	301.695	−	314.070i	1.20678	−	1.25628i
$251$			117.194i			0.466908i	0.972368	+	0.233454i	$0.0750029\pi$
$251$			117.194i			0.466908i	−0.972368	+	0.233454i	$0.924997\pi$
$252$		0			0
$253$			60.1981i			0.237937i
$254$		−	398.907i		−	1.57050i
$255$		0			0
$256$	−518.960			−2.02719
$257$	153.365			0.596749			0.298375	−	0.954449i	$-0.403556\pi$
$257$	153.365			0.596749			0.298375	+	0.954449i	$0.403556\pi$
$258$		0			0
$259$	−186.988			−0.721962
$260$	−48.5855	−	176.584i	−0.186867	−	0.679170i
$261$		0			0
$262$			596.133i			2.27532i
$263$	263.495			1.00188			0.500941	−	0.865481i	$-0.332987\pi$
$263$	263.495			1.00188			0.500941	+	0.865481i	$0.332987\pi$
$264$		0			0
$265$	−86.2162	+	23.7216i	−0.325344	+	0.0895155i
$266$			704.823i			2.64971i
$267$		0			0
$268$			412.311i			1.53848i
$269$		−	10.4219i		−	0.0387431i	−0.999812	−	0.0193715i	$-0.993833\pi$
$269$		−	10.4219i		−	0.0387431i	0.999812	−	0.0193715i	$-0.00616654\pi$
$270$		0			0
$271$	501.590			1.85089			0.925443	−	0.378886i	$-0.123693\pi$
$271$	501.590			1.85089			0.925443	+	0.378886i	$0.123693\pi$
$272$	162.740			0.598311
$273$		0			0
$274$	126.890			0.463101
$275$	181.448	+	304.774i	0.659810	+	1.10827i
$276$		0			0
$277$		−	351.272i		−	1.26813i	−0.773280	−	0.634065i	$-0.781385\pi$
$277$		−	351.272i		−	1.26813i	0.773280	−	0.634065i	$-0.218615\pi$
$278$	−264.032			−0.949756
$279$		0			0
$280$	244.333	+	888.027i	0.872617	+	3.17153i
$281$		−	292.122i		−	1.03958i	−0.854294	−	0.519791i	$-0.826010\pi$
$281$		−	292.122i		−	1.03958i	0.854294	−	0.519791i	$-0.173990\pi$
$282$		0			0
$283$		−	188.057i		−	0.664513i	−0.943189	−	0.332256i	$-0.892190\pi$
$283$		−	188.057i		−	0.664513i	0.943189	−	0.332256i	$-0.107810\pi$
$284$		−	551.437i		−	1.94168i
$285$		0			0
$286$	222.481			0.777906
$287$	−570.454			−1.98764
$288$		0			0
$289$	−204.254			−0.706762
$290$	122.581	+	445.521i	0.422693	+	1.53628i
$291$		0			0
$292$			428.697i			1.46814i
$293$	309.970			1.05792			0.528958	−	0.848648i	$-0.322583\pi$
$293$	309.970			1.05792			0.528958	+	0.848648i	$0.322583\pi$
$294$		0			0
$295$	64.0682	+	232.856i	0.217180	+	0.789341i
$296$		−	211.009i		−	0.712869i
$297$		0			0
$298$		−	666.895i		−	2.23790i
$299$			19.0968i			0.0638689i
$300$		0			0
$301$	835.350			2.77525
$302$	−833.504			−2.75995
$303$		0			0
$304$	−279.918			−0.920784
$305$	440.962	−	121.327i	1.44578	−	0.397793i
$306$		0			0
$307$			143.887i			0.468687i	0.972154	+	0.234344i	$0.0752941\pi$
$307$			143.887i			0.468687i	−0.972154	+	0.234344i	$0.924706\pi$
$308$	−1475.22			−4.78968
$309$		0			0
$310$	527.990	−	145.272i	1.70319	−	0.468619i
$311$			27.8635i			0.0895933i	0.998996	+	0.0447966i	$0.0142640\pi$
$311$			27.8635i			0.0895933i	−0.998996	+	0.0447966i	$0.985736\pi$
$312$		0			0
$313$			417.963i			1.33535i	0.744454	+	0.667673i	$0.232710\pi$
$313$			417.963i			1.33535i	−0.744454	+	0.667673i	$0.767290\pi$
$314$			499.416i			1.59050i
$315$		0			0
$316$	−446.759			−1.41379
$317$	−67.0688			−0.211573			−0.105787	−	0.994389i	$-0.533736\pi$
$317$	−67.0688			−0.211573			−0.105787	+	0.994389i	$0.533736\pi$
$318$		0			0
$319$	−376.343			−1.17976
$320$	275.056	−	75.6791i	0.859549	−	0.236497i
$321$		0			0
$322$		−	188.864i		−	0.586535i
$323$	−145.765			−0.451285
$324$		0			0
$325$	57.5612	+	96.6843i	0.177111	+	0.297490i
$326$			484.125i			1.48505i
$327$		0			0
$328$		−	643.736i		−	1.96261i
$329$		−	722.774i		−	2.19688i
$330$		0			0
$331$	−106.486			−0.321709			−0.160854	−	0.986978i	$-0.551425\pi$
$331$	−106.486			−0.321709			−0.160854	+	0.986978i	$0.551425\pi$
$332$	−20.8137			−0.0626918
$333$		0			0
$334$	716.242			2.14444
$335$	−67.2006	−	244.241i	−0.200599	−	0.729077i
$336$		0			0
$337$		−	532.879i		−	1.58124i	−0.612306	−	0.790621i	$-0.709758\pi$
$337$		−	532.879i		−	1.58124i	0.612306	−	0.790621i	$-0.290242\pi$
$338$	−518.218			−1.53319
$339$		0			0
$340$	−361.172	+	99.3732i	−1.06227	+	0.292274i
$341$			446.008i			1.30794i
$342$		0			0
$343$		−	833.474i		−	2.42995i
$344$			942.662i			2.74030i
$345$		0			0
$346$	165.621			0.478672
$347$	2.71126			0.00781344			0.00390672	−	0.999992i	$-0.498756\pi$
$347$	2.71126			0.00781344			0.00390672	+	0.999992i	$0.498756\pi$
$348$		0			0
$349$	−198.853			−0.569780			−0.284890	−	0.958560i	$-0.591957\pi$
$349$	−198.853			−0.569780			−0.284890	+	0.958560i	$0.591957\pi$
$350$	−569.270	−	956.192i	−1.62649	−	2.73198i
$351$		0			0
$352$		−	55.6160i		−	0.158000i
$353$	−251.151			−0.711475			−0.355737	−	0.934586i	$-0.615770\pi$
$353$	−251.151			−0.711475			−0.355737	+	0.934586i	$0.615770\pi$
$354$		0			0
$355$	89.8761	+	326.655i	0.253172	+	0.920154i
$356$		−	673.288i		−	1.89126i
$357$		0			0
$358$			433.054i			1.20965i
$359$			314.839i			0.876989i	0.898734	+	0.438495i	$0.144488\pi$
$359$			314.839i			0.876989i	−0.898734	+	0.438495i	$0.855512\pi$
$360$		0			0
$361$	−110.280			−0.305484
$362$	42.0762			0.116233
$363$		0			0
$364$	−467.988			−1.28568
$365$	−69.8712	−	253.947i	−0.191428	−	0.695745i
$366$		0			0
$367$			225.392i			0.614148i	0.951686	+	0.307074i	$0.0993499\pi$
$367$			225.392i			0.614148i	−0.951686	+	0.307074i	$0.900650\pi$
$368$	75.0069			0.203823
$369$		0			0
$370$	67.6338	+	245.815i	0.182794	+	0.664365i
$371$			228.493i			0.615883i
$372$		0			0
$373$			501.181i			1.34365i	0.740711	+	0.671824i	$0.234489\pi$
$373$			501.181i			1.34365i	−0.740711	+	0.671824i	$0.765511\pi$
$374$		−	455.046i		−	1.21670i
$375$		0			0
$376$	815.624			2.16921
$377$	−119.388			−0.316680
$378$		0			0
$379$	3.37369			0.00890156			0.00445078	−	0.999990i	$-0.498583\pi$
$379$	3.37369			0.00890156			0.00445078	+	0.999990i	$0.498583\pi$
$380$	621.226	−	170.925i	1.63481	−	0.449802i
$381$		0			0
$382$			995.430i			2.60584i
$383$	−415.059			−1.08371			−0.541853	−	0.840473i	$-0.682277\pi$
$383$	−415.059			−1.08371			−0.541853	+	0.840473i	$0.682277\pi$
$384$		0			0
$385$	873.876	−	240.439i	2.26981	−	0.624517i
$386$		−	1145.77i		−	2.96832i
$387$		0			0
$388$		−	520.032i		−	1.34029i
$389$			738.710i			1.89900i	0.313769	+	0.949499i	$0.398408\pi$
$389$			738.710i			1.89900i	−0.313769	+	0.949499i	$0.601592\pi$
$390$		0			0
$391$	39.0592			0.0998956
$392$	1647.01			4.20156
$393$		0			0
$394$	314.656			0.798620
$395$	264.646	−	72.8151i	0.669991	−	0.184342i
$396$		0			0
$397$			258.122i			0.650183i	0.945683	+	0.325091i	$0.105395\pi$
$397$			258.122i			0.650183i	−0.945683	+	0.325091i	$0.894605\pi$
$398$	−1.10030			−0.00276458
$399$		0			0
$400$	379.749	−	226.084i	0.949371	−	0.565210i
$401$			299.683i			0.747338i	0.927562	+	0.373669i	$0.121900\pi$
$401$			299.683i			0.747338i	−0.927562	+	0.373669i	$0.878100\pi$
$402$		0			0
$403$			141.488i			0.351088i
$404$			656.097i			1.62400i
$405$		0			0
$406$	1180.73			2.90821
$407$	−207.647			−0.510188
$408$		0			0
$409$	−549.380			−1.34323			−0.671614	−	0.740901i	$-0.734399\pi$
$409$	−549.380			−1.34323			−0.671614	+	0.740901i	$0.734399\pi$
$410$	206.334	+	749.920i	0.503253	+	1.82907i
$411$		0			0
$412$			292.758i			0.710578i
$413$	617.121			1.49424
$414$		0			0
$415$	12.3294	−	3.39232i	0.0297094	−	0.00817427i
$416$		−	17.6432i		−	0.0424116i
$417$		0			0
$418$			782.693i			1.87247i
$419$			247.992i			0.591866i	0.955209	+	0.295933i	$0.0956306\pi$
$419$			247.992i			0.591866i	−0.955209	+	0.295933i	$0.904369\pi$
$420$		0			0
$421$	−326.349			−0.775176			−0.387588	−	0.921833i	$-0.626692\pi$
$421$	−326.349			−0.775176			−0.387588	+	0.921833i	$0.626692\pi$
$422$	1144.05			2.71103
$423$		0			0
$424$	−257.846			−0.608127
$425$	197.751	−	117.731i	0.465296	−	0.277015i
$426$		0			0
$427$		−	1168.65i		−	2.73689i
$428$	−176.376			−0.412094
$429$		0			0
$430$	−302.147	−	1098.15i	−0.702668	−	2.55385i
$431$			96.3692i			0.223594i	0.993731	+	0.111797i	$0.0356607\pi$
$431$			96.3692i			0.223594i	−0.993731	+	0.111797i	$0.964339\pi$
$432$		0			0
$433$			672.095i			1.55218i	0.630620	+	0.776091i	$0.282800\pi$
$433$			672.095i			1.55218i	−0.630620	+	0.776091i	$0.717200\pi$
$434$		−	1399.30i		−	3.22418i
$435$		0			0
$436$	−8.88843			−0.0203863
$437$	−67.1830			−0.153737
$438$		0			0
$439$	103.753			0.236339			0.118170	−	0.992993i	$-0.462297\pi$
$439$	103.753			0.236339			0.118170	+	0.992993i	$0.462297\pi$
$440$	271.327	+	986.137i	0.616652	+	2.24122i
$441$		0			0
$442$		−	144.356i		−	0.326596i
$443$	227.758			0.514127			0.257063	−	0.966395i	$-0.417245\pi$
$443$	227.758			0.514127			0.257063	+	0.966395i	$0.417245\pi$
$444$		0			0
$445$	109.736	+	398.835i	0.246598	+	0.896260i
$446$		−	176.147i		−	0.394949i
$447$		0			0
$448$		−	728.961i		−	1.62715i
$449$		−	504.100i		−	1.12272i	−0.827573	−	0.561359i	$-0.810279\pi$
$449$		−	504.100i		−	1.12272i	0.827573	−	0.561359i	$-0.189721\pi$
$450$		0			0
$451$	−633.478			−1.40461
$452$	1238.76			2.74063
$453$		0			0
$454$	1048.73			2.30998
$455$	277.222	−	76.2752i	0.609279	−	0.167638i
$456$		0			0
$457$		−	179.987i		−	0.393844i	−0.980419	−	0.196922i	$-0.936905\pi$
$457$		−	179.987i		−	0.393844i	0.980419	−	0.196922i	$-0.0630946\pi$
$458$	−973.518			−2.12558
$459$		0			0
$460$	−166.464	+	45.8010i	−0.361877	+	0.0995673i
$461$			583.193i			1.26506i	0.774535	+	0.632531i	$0.217984\pi$
$461$			583.193i			1.26506i	−0.774535	+	0.632531i	$0.782016\pi$
$462$		0			0
$463$		−	15.7765i		−	0.0340746i	−0.999855	−	0.0170373i	$-0.994577\pi$
$463$		−	15.7765i		−	0.0340746i	0.999855	−	0.0170373i	$-0.00542340\pi$
$464$			468.924i			1.01061i
$465$		0			0
$466$	−95.2071			−0.204307
$467$	−400.171			−0.856896			−0.428448	−	0.903566i	$-0.640940\pi$
$467$	−400.171			−0.856896			−0.428448	+	0.903566i	$0.640940\pi$
$468$		0			0
$469$	−647.294			−1.38016
$470$	−950.161	+	261.428i	−2.02162	+	0.556230i
$471$		0			0
$472$			696.399i			1.47542i
$473$	927.640			1.96118
$474$		0			0
$475$	−340.137	+	202.501i	−0.716079	+	0.426319i
$476$			957.189i			2.01090i
$477$		0			0
$478$		−	61.7480i		−	0.129180i
$479$		−	162.148i		−	0.338514i	−0.985572	−	0.169257i	$-0.945863\pi$
$479$		−	162.148i		−	0.338514i	0.985572	−	0.169257i	$-0.0541368\pi$
$480$		0			0
$481$	−65.8723			−0.136949
$482$	−710.363			−1.47378
$483$		0			0
$484$	−653.474			−1.35015
$485$	84.7576	+	308.051i	0.174758	+	0.635157i
$486$		0			0
$487$		−	328.071i		−	0.673658i	−0.941566	−	0.336829i	$-0.890646\pi$
$487$		−	328.071i		−	0.673658i	0.941566	−	0.336829i	$-0.109354\pi$
$488$	1318.78			2.70242
$489$		0			0
$490$	−1918.68	+	527.909i	−3.91568	+	1.07736i
$491$			39.4500i			0.0803462i	0.999193	+	0.0401731i	$0.0127909\pi$
$491$			39.4500i			0.0803462i	−0.999193	+	0.0401731i	$0.987209\pi$
$492$		0			0
$493$			244.188i			0.495311i
$494$			248.296i			0.502623i
$495$		0			0
$496$	555.726			1.12041
$497$	865.710			1.74187
$498$		0			0
$499$	−75.7024			−0.151708			−0.0758541	−	0.997119i	$-0.524168\pi$
$499$	−75.7024			−0.151708			−0.0758541	+	0.997119i	$0.524168\pi$
$500$	−704.728	+	733.634i	−1.40946	+	1.46727i
$501$		0			0
$502$		−	408.304i		−	0.813354i
$503$	−139.725			−0.277784			−0.138892	−	0.990308i	$-0.544354\pi$
$503$	−139.725			−0.277784			−0.138892	+	0.990308i	$0.544354\pi$
$504$		0			0
$505$	−106.934	−	388.652i	−0.211751	−	0.769607i
$506$		−	209.730i		−	0.414486i
$507$		0			0
$508$			931.804i			1.83426i
$509$			801.430i			1.57452i	0.616622	+	0.787259i	$0.288501\pi$
$509$			801.430i			1.57452i	−0.616622	+	0.787259i	$0.711499\pi$
$510$		0			0
$511$	−673.018			−1.31706
$512$	950.213			1.85588
$513$		0			0
$514$	−534.322			−1.03954
$515$	−47.7152	−	173.421i	−0.0926510	−	0.336740i
$516$		0			0
$517$		−	802.626i		−	1.55247i
$518$	651.466			1.25766
$519$		0			0
$520$	86.0738	+	312.835i	0.165526	+	0.601606i
$521$		−	472.323i		−	0.906570i	−0.891366	−	0.453285i	$-0.850252\pi$
$521$		−	472.323i		−	0.906570i	0.891366	−	0.453285i	$-0.149748\pi$
$522$		0			0
$523$		−	843.303i		−	1.61243i	−0.591620	−	0.806217i	$-0.701511\pi$
$523$		−	843.303i		−	1.61243i	0.591620	−	0.806217i	$-0.298489\pi$
$524$		−	1392.50i		−	2.65745i
$525$		0			0
$526$	−918.016			−1.74528
$527$	289.390			0.549126
$528$		0			0
$529$	−510.998			−0.965969
$530$	300.377	−	82.6461i	0.566749	−	0.155936i
$531$		0			0
$532$		−	1646.39i		−	3.09472i
$533$	−200.960			−0.377036
$534$		0			0
$535$	104.480	−	28.7467i	0.195289	−	0.0537321i
$536$		−	730.448i		−	1.36278i
$537$		0			0
$538$			36.3098i			0.0674904i
$539$		−	1620.76i		−	3.00698i
$540$		0			0
$541$	797.318			1.47379			0.736893	−	0.676010i	$-0.236292\pi$
$541$	797.318			1.47379			0.736893	+	0.676010i	$0.236292\pi$
$542$	−1747.54			−3.22424
$543$		0			0
$544$	−36.0862			−0.0663349
$545$	5.26524	−	1.44868i	0.00966098	−	0.00265813i
$546$		0			0
$547$			882.252i			1.61289i	0.591308	+	0.806446i	$0.298612\pi$
$547$			882.252i			1.61289i	−0.591308	+	0.806446i	$0.701388\pi$
$548$	−296.401			−0.540877
$549$		0			0
$550$	−632.164	−	1061.83i	−1.14939	−	1.93060i
$551$		−	420.011i		−	0.762270i
$552$		0			0
$553$		−	701.374i		−	1.26831i
$554$			1223.83i			2.20908i
$555$		0			0
$556$	616.751			1.10926
$557$	589.979			1.05921			0.529604	−	0.848245i	$-0.322341\pi$
$557$	589.979			1.05921			0.529604	+	0.848245i	$0.322341\pi$
$558$		0			0
$559$	294.278			0.526436
$560$	−299.587	−	1088.85i	−0.534977	−	1.94437i
$561$		0			0
$562$			1017.75i			1.81095i
$563$	−786.155			−1.39637			−0.698183	−	0.715919i	$-0.746008\pi$
$563$	−786.155			−1.39637			−0.698183	+	0.715919i	$0.746008\pi$
$564$		0			0
$565$	−733.805	+	201.900i	−1.29877	+	0.357345i
$566$			655.191i			1.15758i
$567$		0			0
$568$			976.922i			1.71993i
$569$			1090.38i			1.91631i	0.286242	+	0.958157i	$0.407594\pi$
$569$			1090.38i			1.91631i	−0.286242	+	0.958157i	$0.592406\pi$
$570$		0			0
$571$	−2.68062			−0.00469461			−0.00234730	−	0.999997i	$-0.500747\pi$
$571$	−2.68062			−0.00469461			−0.00234730	+	0.999997i	$0.500747\pi$
$572$	−519.692			−0.908552
$573$		0			0
$574$	1987.46			3.46247
$575$	91.1431	−	54.2622i	0.158510	−	0.0943691i
$576$		0			0
$577$			802.089i			1.39010i	0.718960	+	0.695051i	$0.244618\pi$
$577$			802.089i			1.39010i	−0.718960	+	0.695051i	$0.755382\pi$
$578$	711.622			1.23118
$579$		0			0
$580$	−286.336	−	1040.69i	−0.493683	−	1.79429i
$581$		−	32.6757i		−	0.0562405i
$582$		0			0
$583$			253.737i			0.435226i
$584$		−	759.476i		−	1.30047i
$585$		0			0
$586$	−1079.93			−1.84289
$587$	−1109.11			−1.88945			−0.944727	−	0.327857i	$-0.893674\pi$
$587$	−1109.11			−1.88945			−0.944727	+	0.327857i	$0.893674\pi$
$588$		0			0
$589$	−497.759			−0.845091
$590$	−223.213	−	811.269i	−0.378328	−	1.37503i
$591$		0			0
$592$			258.728i			0.437040i
$593$	633.417			1.06816			0.534078	−	0.845435i	$-0.320659\pi$
$593$	633.417			1.06816			0.534078	+	0.845435i	$0.320659\pi$
$594$		0			0
$595$	−156.008	−	567.009i	−0.262198	−	0.952957i
$596$			1557.80i			2.61375i
$597$		0			0
$598$		−	66.5333i		−	0.111260i
$599$			524.561i			0.875729i	0.899041	+	0.437864i	$0.144265\pi$
$599$			524.561i			0.875729i	−0.899041	+	0.437864i	$0.855735\pi$
$600$		0			0
$601$	694.003			1.15475			0.577374	−	0.816480i	$-0.304078\pi$
$601$	694.003			1.15475			0.577374	+	0.816480i	$0.304078\pi$
$602$	−2910.36			−4.83448
$603$		0			0
$604$	1946.98			3.22347
$605$	387.098	−	106.507i	0.639832	−	0.176044i
$606$		0			0
$607$			899.771i			1.48233i	0.671326	+	0.741163i	$0.265725\pi$
$607$			899.771i			1.48233i	−0.671326	+	0.741163i	$0.734275\pi$
$608$	62.0693			0.102088
$609$		0			0
$610$	−1536.31	+	422.702i	−2.51854	+	0.692955i
$611$		−	254.619i		−	0.416726i
$612$		0			0
$613$			966.059i			1.57595i	0.615706	+	0.787976i	$0.288871\pi$
$613$			966.059i			1.57595i	−0.615706	+	0.787976i	$0.711129\pi$
$614$		−	501.302i		−	0.816453i
$615$		0			0
$616$	2613.49			4.24268
$617$	−279.257			−0.452605			−0.226302	−	0.974057i	$-0.572664\pi$
$617$	−279.257			−0.452605			−0.226302	+	0.974057i	$0.572664\pi$
$618$		0			0
$619$	−838.583			−1.35474			−0.677369	−	0.735643i	$-0.736880\pi$
$619$	−838.583			−1.35474			−0.677369	+	0.735643i	$0.736880\pi$
$620$	−1233.33	+	339.339i	−1.98924	+	0.547322i
$621$		0			0
$622$		−	97.0765i		−	0.156071i
$623$	1057.01			1.69664
$624$		0			0
$625$	297.888	−	549.443i	0.476621	−	0.879109i
$626$		−	1456.18i		−	2.32617i
$627$		0			0
$628$		−	1166.58i		−	1.85762i
$629$			134.730i			0.214198i
$630$		0			0
$631$	−544.887			−0.863529			−0.431764	−	0.901986i	$-0.642109\pi$
$631$	−544.887			−0.863529			−0.431764	+	0.901986i	$0.642109\pi$
$632$	791.475			1.25233
$633$		0			0
$634$	233.668			0.368561
$635$	−151.870	−	551.972i	−0.239166	−	0.869247i
$636$		0			0
$637$		−	514.160i		−	0.807158i
$638$	1311.18			2.05514
$639$		0			0
$640$	−1033.88	+	284.464i	−1.61544	+	0.444475i
$641$			0.293309i			0.000457581i	1.00000		0.000228790i	$7.28262e-5\pi$
$641$			0.293309i			0.000457581i	−1.00000		0.000228790i	$0.999927\pi$
$642$		0			0
$643$		−	155.418i		−	0.241707i	−0.992670	−	0.120854i	$-0.961437\pi$
$643$		−	155.418i		−	0.241707i	0.992670	−	0.120854i	$-0.0385632\pi$
$644$			441.167i			0.685042i
$645$		0			0
$646$	507.846			0.786139
$647$	657.668			1.01649			0.508244	−	0.861213i	$-0.330295\pi$
$647$	657.668			1.01649			0.508244	+	0.861213i	$0.330295\pi$
$648$		0			0
$649$	685.301			1.05593
$650$	−200.543	−	336.848i	−0.308528	−	0.518228i
$651$		0			0
$652$		−	1130.87i		−	1.73446i
$653$	1006.20			1.54089			0.770446	−	0.637506i	$-0.220034\pi$
$653$	1006.20			1.54089			0.770446	+	0.637506i	$0.220034\pi$
$654$		0			0
$655$	226.958	+	824.877i	0.346500	+	1.25935i
$656$			789.314i			1.20322i
$657$		0			0
$658$			2518.14i			3.82697i
$659$			1251.23i			1.89868i	0.314249	+	0.949340i	$0.398247\pi$
$659$			1251.23i			1.89868i	−0.314249	+	0.949340i	$0.601753\pi$
$660$		0			0
$661$	−797.169			−1.20600			−0.603002	−	0.797739i	$-0.706029\pi$
$661$	−797.169			−1.20600			−0.603002	+	0.797739i	$0.706029\pi$
$662$	370.996			0.560416
$663$		0			0
$664$	36.8734			0.0555322
$665$	268.338	+	975.273i	0.403515	+	1.46658i
$666$		0			0
$667$			112.546i			0.168735i
$668$	−1673.07			−2.50459
$669$		0			0
$670$	234.127	+	850.935i	0.349443	+	1.27005i
$671$		−	1297.76i		−	1.93407i
$672$		0			0
$673$		−	1250.84i		−	1.85861i	−0.369318	−	0.929303i	$-0.620409\pi$
$673$		−	1250.84i		−	1.85861i	0.369318	−	0.929303i	$-0.379591\pi$
$674$			1856.55i			2.75452i
$675$		0			0
$676$	1210.50			1.79068
$677$	−604.295			−0.892607			−0.446303	−	0.894882i	$-0.647260\pi$
$677$	−604.295			−0.892607			−0.446303	+	0.894882i	$0.647260\pi$
$678$		0			0
$679$	816.407			1.20237
$680$	639.850	−	176.049i	0.940955	−	0.258895i
$681$		0			0
$682$		−	1553.89i		−	2.27843i
$683$	707.854			1.03639			0.518195	−	0.855263i	$-0.326604\pi$
$683$	707.854			1.03639			0.518195	+	0.855263i	$0.326604\pi$
$684$		0			0
$685$	175.579	−	48.3089i	0.256319	−	0.0705240i
$686$			2903.82i			4.23298i
$687$		0			0
$688$		−	1155.84i		−	1.68000i
$689$			80.4936i			0.116827i
$690$		0			0
$691$	486.209			0.703631			0.351815	−	0.936069i	$-0.385565\pi$
$691$	486.209			0.703631			0.351815	+	0.936069i	$0.385565\pi$
$692$	−386.872			−0.559063
$693$		0			0
$694$	−9.44604			−0.0136110
$695$	−365.345	+	100.521i	−0.525676	+	0.144635i
$696$		0			0
$697$			411.028i			0.589711i
$698$	692.805			0.992557
$699$		0			0
$700$	1329.76	+	2233.56i	1.89965	+	3.19080i
$701$		−	646.017i		−	0.921564i	−0.887513	−	0.460782i	$-0.847569\pi$
$701$		−	646.017i		−	0.921564i	0.887513	−	0.460782i	$-0.152431\pi$
$702$		0			0
$703$		−	231.740i		−	0.329645i
$704$		−	809.497i		−	1.14985i
$705$		0			0
$706$	875.009			1.23939
$707$	−1030.02			−1.45688
$708$		0			0
$709$	639.268			0.901647			0.450823	−	0.892613i	$-0.351130\pi$
$709$	639.268			0.901647			0.450823	+	0.892613i	$0.351130\pi$
$710$	−313.128	−	1138.06i	−0.441026	−	1.60291i
$711$		0			0
$712$			1192.79i			1.67527i
$713$	133.379			0.187068
$714$		0			0
$715$	307.850	−	84.7021i	0.430559	−	0.118464i
$716$		−	1011.57i		−	1.41280i
$717$		0			0
$718$		−	1096.90i		−	1.52772i
$719$		−	1060.47i		−	1.47493i	−0.675386	−	0.737465i	$-0.736023\pi$
$719$		−	1060.47i		−	1.47493i	0.675386	−	0.737465i	$-0.263977\pi$
$720$		0			0
$721$	−459.606			−0.637456
$722$	384.214			0.532153
$723$		0			0
$724$	−98.2855			−0.135754
$725$	339.234	+	569.804i	0.467908	+	0.785936i
$726$		0			0
$727$			800.091i			1.10054i	0.834988	+	0.550269i	$0.185475\pi$
$727$			800.091i			1.10054i	−0.834988	+	0.550269i	$0.814525\pi$
$728$	829.085			1.13885
$729$		0			0
$730$	243.431	+	884.751i	0.333468	+	1.21199i
$731$		−	601.894i		−	0.823384i
$732$		0			0
$733$			9.68613i			0.0132144i	0.999978	+	0.00660718i	$0.00210315\pi$
$733$			9.68613i			0.0132144i	−0.999978	+	0.00660718i	$0.997897\pi$
$734$		−	785.267i		−	1.06985i
$735$		0			0
$736$	−16.6321			−0.0225979
$737$	−718.807			−0.975315
$738$		0			0
$739$	−1107.89			−1.49918			−0.749590	−	0.661902i	$-0.769749\pi$
$739$	−1107.89			−1.49918			−0.749590	+	0.661902i	$0.769749\pi$
$740$	−157.985	−	574.198i	−0.213494	−	0.775943i
$741$		0			0
$742$		−	796.069i		−	1.07287i
$743$	780.455			1.05041			0.525206	−	0.850975i	$-0.323988\pi$
$743$	780.455			1.05041			0.525206	+	0.850975i	$0.323988\pi$
$744$		0			0
$745$	−253.898	−	922.791i	−0.340802	−	1.23865i
$746$		−	1746.11i		−	2.34063i
$747$		0			0
$748$			1062.94i			1.42104i
$749$		−	276.895i		−	0.369687i
$750$		0			0
$751$	−73.3096			−0.0976160			−0.0488080	−	0.998808i	$-0.515542\pi$
$751$	−73.3096			−0.0976160			−0.0488080	+	0.998808i	$0.515542\pi$
$752$	−1000.07			−1.32988
$753$		0			0
$754$	415.949			0.551657
$755$	−1153.33	+	317.328i	−1.52759	+	0.420302i
$756$		0			0
$757$			95.5082i			0.126167i	0.998008	+	0.0630834i	$0.0200934\pi$
$757$			95.5082i			0.126167i	−0.998008	+	0.0630834i	$0.979907\pi$
$758$	−11.7539			−0.0155065
$759$		0			0
$760$	−1100.56	+	302.809i	−1.44811	+	0.398433i
$761$			789.348i			1.03725i	0.855001	+	0.518626i	$0.173556\pi$
$761$			789.348i			1.03725i	−0.855001	+	0.518626i	$0.826444\pi$
$762$		0			0
$763$		−	13.9541i		−	0.0182884i
$764$		−	2325.22i		−	3.04348i
$765$		0			0
$766$	1446.07			1.88781
$767$	217.400			0.283442
$768$		0			0
$769$	−30.0502			−0.0390770			−0.0195385	−	0.999809i	$-0.506220\pi$
$769$	−30.0502			−0.0390770			−0.0195385	+	0.999809i	$0.506220\pi$
$770$	−3044.58	+	837.689i	−3.95400	+	1.08791i
$771$		0			0
$772$			2676.40i			3.46684i
$773$	−1290.89			−1.66998			−0.834988	−	0.550269i	$-0.814525\pi$
$773$	−1290.89			−1.66998			−0.834988	+	0.550269i	$0.814525\pi$
$774$		0			0
$775$	675.279	−	402.029i	0.871328	−	0.518747i
$776$			921.285i			1.18722i
$777$		0			0
$778$		−	2573.67i		−	3.30805i
$779$		−	706.981i		−	0.907549i
$780$		0			0
$781$	961.354			1.23093
$782$	−136.082			−0.174018
$783$		0			0
$784$	−2019.47			−2.57586
$785$	190.136	+	691.049i	0.242211	+	0.880317i
$786$		0			0
$787$		−	85.6276i		−	0.108803i	−0.998519	−	0.0544013i	$-0.982675\pi$
$787$		−	85.6276i		−	0.108803i	0.998519	−	0.0544013i	$-0.0173250\pi$
$788$	−735.003			−0.932745
$789$		0			0
$790$	−922.028	+	253.688i	−1.16712	+	0.321124i
$791$			1944.75i			2.45860i
$792$		0			0
$793$		−	411.693i		−	0.519159i
$794$		−	899.299i		−	1.13262i
$795$		0			0
$796$	2.57019			0.00322889
$797$	757.641			0.950617			0.475308	−	0.879819i	$-0.342337\pi$
$797$	757.641			0.950617			0.475308	+	0.879819i	$0.342337\pi$
$798$		0			0
$799$	−520.780			−0.651789
$800$	−84.2057	+	50.1320i	−0.105257	+	0.0626650i
$801$		0			0
$802$		−	1044.09i		−	1.30186i
$803$	−747.373			−0.930726
$804$		0			0
$805$	−71.9037	−	261.334i	−0.0893213	−	0.324638i
$806$		−	492.945i		−	0.611594i
$807$		0			0
$808$		−	1162.34i		−	1.43853i
$809$			624.032i			0.771362i	0.922632	+	0.385681i	$0.126033\pi$
$809$			624.032i			0.771362i	−0.922632	+	0.385681i	$0.873967\pi$
$810$		0			0
$811$	972.993			1.19974			0.599872	−	0.800096i	$-0.295218\pi$
$811$	972.993			1.19974			0.599872	+	0.800096i	$0.295218\pi$
$812$	−2758.06			−3.39663
$813$		0			0
$814$	723.440			0.888748
$815$	184.314	+	669.890i	0.226152	+	0.821951i
$816$		0			0
$817$			1035.28i			1.26717i
$818$	1914.04			2.33990
$819$		0			0
$820$	−481.974	−	1751.73i	−0.587773	−	2.13626i
$821$		−	936.341i		−	1.14049i	−0.821475	−	0.570244i	$-0.806849\pi$
$821$		−	936.341i		−	1.14049i	0.821475	−	0.570244i	$-0.193151\pi$
$822$		0			0
$823$			256.883i			0.312130i	0.987747	+	0.156065i	$0.0498809\pi$
$823$			256.883i			0.312130i	−0.987747	+	0.156065i	$0.950119\pi$
$824$		−	518.648i		−	0.629428i
$825$		0			0
$826$	−2150.05			−2.60297
$827$	−13.1651			−0.0159191			−0.00795957	−	0.999968i	$-0.502534\pi$
$827$	−13.1651			−0.0159191			−0.00795957	+	0.999968i	$0.502534\pi$
$828$		0			0
$829$	−1494.14			−1.80234			−0.901171	−	0.433464i	$-0.857291\pi$
$829$	−1494.14			−1.80234			−0.901171	+	0.433464i	$0.857291\pi$
$830$	−42.9556	+	11.8188i	−0.0517537	+	0.0142396i
$831$		0			0
$832$		−	256.799i		−	0.308653i
$833$	−1051.62			−1.26245
$834$		0			0
$835$	991.073	−	272.685i	1.18691	−	0.326569i
$836$		−	1828.29i		−	2.18695i
$837$		0			0
$838$		−	864.004i		−	1.03103i
$839$		−	112.413i		−	0.133985i	−0.997753	−	0.0669923i	$-0.978660\pi$
$839$		−	112.413i		−	0.133985i	0.997753	−	0.0669923i	$-0.0213403\pi$
$840$		0			0
$841$	137.391			0.163366
$842$	1137.00			1.35036
$843$		0			0
$844$	−2672.39			−3.16634
$845$	−717.064	+	197.294i	−0.848597	+	0.233484i
$846$		0			0
$847$		−	1025.90i		−	1.21122i
$848$	316.156			0.372826
$849$		0			0
$850$	−688.964	+	410.176i	−0.810546	+	0.482560i
$851$			62.0970i			0.0729695i
$852$		0			0
$853$		−	62.4653i		−	0.0732302i	−0.999329	−	0.0366151i	$-0.988342\pi$
$853$		−	62.4653i		−	0.0732302i	0.999329	−	0.0366151i	$-0.0116576\pi$
$854$			4071.58i			4.76766i
$855$		0			0
$856$	312.467			0.365031
$857$	1527.56			1.78246			0.891228	−	0.453556i	$-0.149845\pi$
$857$	1527.56			1.78246			0.891228	+	0.453556i	$0.149845\pi$
$858$		0			0
$859$	−864.271			−1.00614			−0.503068	−	0.864247i	$-0.667796\pi$
$859$	−864.271			−1.00614			−0.503068	+	0.864247i	$0.667796\pi$
$860$	705.783	+	2565.17i	0.820678	+	2.98276i
$861$		0			0
$862$		−	335.750i		−	0.389501i
$863$	−593.430			−0.687636			−0.343818	−	0.939036i	$-0.611720\pi$
$863$	−593.430			−0.687636			−0.343818	+	0.939036i	$0.611720\pi$
$864$		0			0
$865$	229.171	−	63.0544i	0.264938	−	0.0728952i
$866$		−	2341.58i		−	2.70390i
$867$		0			0
$868$			3268.61i			3.76567i
$869$		−	778.862i		−	0.896274i
$870$		0			0
$871$	−228.029			−0.261802
$872$	15.7467			0.0180581
$873$		0			0
$874$	234.065			0.267809
$875$	−1151.74	−	1106.36i	−1.31628	−	1.26442i
$876$		0			0
$877$			80.1032i			0.0913377i	0.998957	+	0.0456689i	$0.0145419\pi$
$877$			80.1032i			0.0913377i	−0.998957	+	0.0456689i	$0.985458\pi$
$878$	−361.475			−0.411703
$879$		0			0
$880$	−332.686	−	1209.15i	−0.378052	−	1.37403i
$881$			612.391i			0.695109i	0.937660	+	0.347555i	$0.112988\pi$
$881$			612.391i			0.695109i	−0.937660	+	0.347555i	$0.887012\pi$
$882$		0			0
$883$			1106.80i			1.25345i	0.779239	+	0.626727i	$0.215606\pi$
$883$			1106.80i			1.25345i	−0.779239	+	0.626727i	$0.784394\pi$
$884$			337.199i			0.381447i
$885$		0			0
$886$	−793.509			−0.895608
$887$	−741.000			−0.835400			−0.417700	−	0.908585i	$-0.637164\pi$
$887$	−741.000			−0.835400			−0.417700	+	0.908585i	$0.637164\pi$
$888$		0			0
$889$	−1462.85			−1.64550
$890$	−382.320	−	1389.54i	−0.429573	−	1.56128i
$891$		0			0
$892$			411.461i			0.461279i
$893$	895.756			1.00309
$894$		0			0
$895$	164.871	+	599.222i	0.184213	+	0.669522i
$896$			2740.03i			3.05807i
$897$		0			0
$898$			1756.28i			1.95577i
$899$			833.853i			0.927534i
$900$		0			0
$901$	164.636			0.182725
$902$	2207.04			2.44682
$903$		0			0
$904$	−2194.58			−2.42764
$905$	58.2214	−	16.0191i	0.0643330	−	0.0177006i
$906$		0			0
$907$		−	295.839i		−	0.326174i	−0.986612	−	0.163087i	$-0.947855\pi$
$907$		−	295.839i		−	0.326174i	0.986612	−	0.163087i	$-0.0521450\pi$
$908$	−2449.72			−2.69793
$909$		0			0
$910$	−965.842	+	265.743i	−1.06136	+	0.292025i
$911$			695.520i			0.763469i	0.924272	+	0.381734i	$0.124673\pi$
$911$			695.520i			0.763469i	−0.924272	+	0.381734i	$0.875327\pi$
$912$		0			0
$913$		−	36.2857i		−	0.0397434i
$914$			627.073i			0.686076i
$915$		0			0
$916$	2274.03			2.48257
$917$	2186.11			2.38399
$918$		0			0
$919$	60.1197			0.0654186			0.0327093	−	0.999465i	$-0.489586\pi$
$919$	60.1197			0.0654186			0.0327093	+	0.999465i	$0.489586\pi$
$920$	294.906	−	81.1407i	0.320550	−	0.0881964i
$921$		0			0
$922$		−	2031.84i		−	2.20374i
$923$	304.973			0.330415
$924$		0			0
$925$	187.171	+	314.388i	0.202348	+	0.339879i
$926$			54.9654i			0.0593579i
$927$		0			0
$928$		−	103.979i		−	0.112047i
$929$		−	1427.23i		−	1.53631i	−0.640262	−	0.768156i	$-0.721174\pi$
$929$		−	1427.23i		−	1.53631i	0.640262	−	0.768156i	$-0.278826\pi$
$930$		0			0
$931$	1808.82			1.94288
$932$	222.394			0.238620
$933$		0			0
$934$	1394.19			1.49271
$935$	−173.243	−	629.653i	−0.185287	−	0.673425i
$936$		0			0
$937$		−	174.346i		−	0.186068i	−0.995663	−	0.0930341i	$-0.970343\pi$
$937$		−	174.346i		−	0.186068i	0.995663	−	0.0930341i	$-0.0296566\pi$
$938$	2255.17			2.40423
$939$		0			0
$940$	2219.47	−	610.668i	2.36114	−	0.649647i
$941$			459.422i			0.488228i	0.969747	+	0.244114i	$0.0784971\pi$
$941$			459.422i			0.488228i	−0.969747	+	0.244114i	$0.921503\pi$
$942$		0			0
$943$			189.442i			0.200893i
$944$		−	853.885i		−	0.904540i
$945$		0			0
$946$	−3231.90			−3.41638
$947$	853.975			0.901769			0.450884	−	0.892582i	$-0.351109\pi$
$947$	853.975			0.901769			0.450884	+	0.892582i	$0.351109\pi$
$948$		0			0
$949$	−237.091			−0.249833
$950$	1185.04	−	705.515i	1.24741	−	0.742647i
$951$		0			0
$952$		−	1695.75i		−	1.78125i
$953$	1103.28			1.15769			0.578843	−	0.815439i	$-0.303504\pi$
$953$	1103.28			1.15769			0.578843	+	0.815439i	$0.303504\pi$
$954$		0			0
$955$	378.976	+	1377.39i	0.396834	+	1.44229i
$956$			144.237i			0.150875i
$957$		0			0
$958$			564.925i			0.589692i
$959$		−	465.324i		−	0.485218i
$960$		0			0
$961$	27.2069			0.0283110
$962$	229.499			0.238565
$963$		0			0
$964$	1659.33			1.72130
$965$	−436.214	−	1585.42i	−0.452035	−	1.64292i
$966$		0			0
$967$			1023.22i			1.05814i	0.848579	+	0.529068i	$0.177459\pi$
$967$			1023.22i			1.05814i	−0.848579	+	0.529068i	$0.822541\pi$
$968$	1157.69			1.19596
$969$		0			0
$970$	−295.295	−	1073.25i	−0.304428	−	1.10644i
$971$			589.306i			0.606906i	0.952846	+	0.303453i	$0.0981395\pi$
$971$			589.306i			0.606906i	−0.952846	+	0.303453i	$0.901860\pi$
$972$		0			0
$973$			968.247i			0.995115i
$974$			1143.00i			1.17351i
$975$		0			0
$976$	−1617.01			−1.65678
$977$	−757.233			−0.775059			−0.387530	−	0.921857i	$-0.626672\pi$
$977$	−757.233			−0.775059			−0.387530	+	0.921857i	$0.626672\pi$
$978$		0			0
$979$	1173.78			1.19896
$980$	4481.84	−	1233.14i	4.57331	−	1.25830i
$981$		0			0
$982$		−	137.444i		−	0.139963i
$983$	−1403.90			−1.42818			−0.714091	−	0.700053i	$-0.753160\pi$
$983$	−1403.90			−1.42818			−0.714091	+	0.700053i	$0.753160\pi$
$984$		0			0
$985$	435.394	−	119.795i	0.442024	−	0.121619i
$986$		−	850.752i		−	0.862831i
$987$		0			0
$988$		−	579.993i		−	0.587037i
$989$		−	277.412i		−	0.280498i
$990$		0			0
$991$	1778.87			1.79502			0.897512	−	0.440990i	$-0.145373\pi$
$991$	1778.87			1.79502			0.897512	+	0.440990i	$0.145373\pi$
$992$	−123.227			−0.124221
$993$		0			0
$994$	−3016.13			−3.03434
$995$	−1.52250	+	0.418903i	−0.00153015	+	0.000421008i
$996$		0			0
$997$		−	1393.94i		−	1.39813i	−0.715056	−	0.699067i	$-0.753599\pi$
$997$		−	1393.94i		−	1.39813i	0.715056	−	0.699067i	$-0.246401\pi$
$998$	263.747			0.264276
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.b.404.4	yes	24
3.2	odd	2	inner	405.3.d.b.404.21	yes	24
5.4	even	2	inner	405.3.d.b.404.22	yes	24
9.2	odd	6		405.3.h.k.134.4		48
9.4	even	3		405.3.h.k.269.22		48
9.5	odd	6		405.3.h.k.269.3		48
9.7	even	3		405.3.h.k.134.21		48
15.14	odd	2	inner	405.3.d.b.404.3	✓	24
45.4	even	6		405.3.h.k.269.4		48
45.14	odd	6		405.3.h.k.269.21		48
45.29	odd	6		405.3.h.k.134.22		48
45.34	even	6		405.3.h.k.134.3		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.3.d.b.404.3	✓	24	15.14	odd	2	inner
405.3.d.b.404.4	yes	24	1.1	even	1	trivial
405.3.d.b.404.21	yes	24	3.2	odd	2	inner
405.3.d.b.404.22	yes	24	5.4	even	2	inner
405.3.h.k.134.3		48	45.34	even	6
405.3.h.k.134.4		48	9.2	odd	6
405.3.h.k.134.21		48	9.7	even	3
405.3.h.k.134.22		48	45.29	odd	6
405.3.h.k.269.3		48	9.5	odd	6
405.3.h.k.269.4		48	45.4	even	6
405.3.h.k.269.21		48	45.14	odd	6
405.3.h.k.269.22		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)

\(f(q)\)	\(=\)	\(q-3.48400 q^{2} +8.13825 q^{4} +(-4.82085 + 1.32641i) q^{5} +12.7764i q^{7} -14.4177 q^{8} +(16.7958 - 4.62123i) q^{10} +14.1879i q^{11} +4.50087i q^{13} -44.5129i q^{14} +17.6781 q^{16} +9.20574 q^{17} -15.8342 q^{19} +(-39.2333 + 10.7947i) q^{20} -49.4307i q^{22} +4.24292 q^{23} +(21.4812 - 12.7889i) q^{25} -15.6810i q^{26} +103.977i q^{28} +26.5256i q^{29} +31.4358 q^{31} -3.91996 q^{32} -32.0728 q^{34} +(-16.9468 - 61.5930i) q^{35} +14.6355i q^{37} +55.1662 q^{38} +(69.5055 - 19.1238i) q^{40} +44.6491i q^{41} -65.3824i q^{43} +115.465i q^{44} -14.7823 q^{46} -56.5711 q^{47} -114.236 q^{49} +(-74.8407 + 44.5565i) q^{50} +36.6292i q^{52} +17.8840 q^{53} +(-18.8190 - 68.3978i) q^{55} -184.205i q^{56} -92.4153i q^{58} -48.3018i q^{59} -91.4697 q^{61} -109.522 q^{62} -57.0554 q^{64} +(-5.97002 - 21.6980i) q^{65} +50.6634i q^{67} +74.9187 q^{68} +(59.0425 + 214.590i) q^{70} -67.7587i q^{71} +52.6768i q^{73} -50.9899i q^{74} -128.862 q^{76} -181.270 q^{77} -54.8962 q^{79} +(-85.2237 + 23.4485i) q^{80} -155.558i q^{82} -2.55751 q^{83} +(-44.3795 + 12.2106i) q^{85} +227.792i q^{86} -204.557i q^{88} -82.7313i q^{89} -57.5048 q^{91} +34.5299 q^{92} +197.094 q^{94} +(76.3341 - 21.0026i) q^{95} -63.8998i q^{97} +397.997 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91}+ \cdots + 384 q^{94}+O(q^{100}) \) 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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