LMFDB - Embedded newform 495.3.g.a.89.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,3,Mod(89,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.89"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	495.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.4877730858$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.3
Character	$\chi$	$=$	495.89
Dual form			495.3.g.a.89.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.53246 q^{2} +8.47824 q^{4} +(4.02218 + 2.97020i) q^{5} -11.9775i q^{7} -15.8192 q^{8} +(-14.2082 - 10.4921i) q^{10} -3.31662i q^{11} -8.88033i q^{13} +42.3098i q^{14} +21.9676 q^{16} -20.9112 q^{17} -32.1345 q^{19} +(34.1010 + 25.1821i) q^{20} +11.7158i q^{22} -25.5821 q^{23} +(7.35583 + 23.8933i) q^{25} +31.3694i q^{26} -101.548i q^{28} -6.07542i q^{29} +46.2603 q^{31} -14.3229 q^{32} +73.8677 q^{34} +(35.5754 - 48.1754i) q^{35} +12.9739i q^{37} +113.514 q^{38} +(-63.6276 - 46.9862i) q^{40} +43.8120i q^{41} +48.3388i q^{43} -28.1192i q^{44} +90.3677 q^{46} -37.3165 q^{47} -94.4594 q^{49} +(-25.9841 - 84.4022i) q^{50} -75.2896i q^{52} -82.3754 q^{53} +(9.85104 - 13.3401i) q^{55} +189.474i q^{56} +21.4611i q^{58} -48.1280i q^{59} -8.63837 q^{61} -163.412 q^{62} -37.2755 q^{64} +(26.3764 - 35.7183i) q^{65} +49.8698i q^{67} -177.290 q^{68} +(-125.669 + 170.178i) q^{70} +3.62072i q^{71} +122.698i q^{73} -45.8297i q^{74} -272.444 q^{76} -39.7247 q^{77} -20.9703 q^{79} +(88.3577 + 65.2483i) q^{80} -154.764i q^{82} +71.6603 q^{83} +(-84.1084 - 62.1103i) q^{85} -170.755i q^{86} +52.4663i q^{88} +83.5205i q^{89} -106.364 q^{91} -216.891 q^{92} +131.819 q^{94} +(-129.251 - 95.4460i) q^{95} -73.3915i q^{97} +333.674 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 72 q^{4} + 8 q^{10} + 184 q^{16} - 80 q^{19} + 32 q^{25} - 16 q^{31} - 160 q^{34} - 136 q^{40} + 560 q^{46} - 104 q^{49} - 96 q^{61} + 264 q^{64} - 872 q^{70} - 176 q^{76} - 672 q^{79} + 16 q^{85}+ \cdots + 400 q^{94}+O(q^{100}) $ 40 * q + 72 * q^4 + 8 * q^10 + 184 * q^16 - 80 * q^19 + 32 * q^25 - 16 * q^31 - 160 * q^34 - 136 * q^40 + 560 * q^46 - 104 * q^49 - 96 * q^61 + 264 * q^64 - 872 * q^70 - 176 * q^76 - 672 * q^79 + 16 * q^85 - 864 * q^91 + 400 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	−3.53246			−1.76623			−0.883114	−	0.469159i	$-0.844557\pi$
$2$	−3.53246			−1.76623			−0.883114	+	0.469159i	$0.844557\pi$
$3$		0			0
$3$		0			0
$4$	8.47824			2.11956
$5$	4.02218	+	2.97020i	0.804436	+	0.594040i
$5$	4.02218	+	2.97020i	0.804436	+	0.594040i
$6$		0			0
$7$		−	11.9775i		−	1.71106i	−0.517749	−	0.855532i	$-0.673230\pi$
$7$		−	11.9775i		−	1.71106i	0.517749	−	0.855532i	$-0.326770\pi$
$8$	−15.8192			−1.97740
$9$		0			0
$10$	−14.2082	−	10.4921i	−1.42082	−	1.04921i
$11$		−	3.31662i		−	0.301511i
$11$		−	3.31662i		−	0.301511i
$12$		0			0
$13$		−	8.88033i		−	0.683102i	−0.939863	−	0.341551i	$-0.889048\pi$
$13$		−	8.88033i		−	0.683102i	0.939863	−	0.341551i	$-0.110952\pi$
$14$			42.3098i			3.02213i
$15$		0			0
$16$	21.9676			1.37298
$17$	−20.9112			−1.23007			−0.615034	−	0.788500i	$-0.710858\pi$
$17$	−20.9112			−1.23007			−0.615034	+	0.788500i	$0.710858\pi$
$18$		0			0
$19$	−32.1345			−1.69129			−0.845646	−	0.533745i	$-0.820784\pi$
$19$	−32.1345			−1.69129			−0.845646	+	0.533745i	$0.820784\pi$
$20$	34.1010	+	25.1821i	1.70505	+	1.25910i
$21$		0			0
$22$			11.7158i			0.532538i
$23$	−25.5821			−1.11227			−0.556133	−	0.831093i	$-0.687715\pi$
$23$	−25.5821			−1.11227			−0.556133	+	0.831093i	$0.687715\pi$
$24$		0			0
$25$	7.35583	+	23.8933i	0.294233	+	0.955734i
$26$			31.3694i			1.20651i
$27$		0			0
$28$		−	101.548i		−	3.62671i
$29$		−	6.07542i		−	0.209497i	−0.994499	−	0.104749i	$-0.966596\pi$
$29$		−	6.07542i		−	0.209497i	0.994499	−	0.104749i	$-0.0334038\pi$
$30$		0			0
$31$	46.2603			1.49227			0.746133	−	0.665797i	$-0.231908\pi$
$31$	46.2603			1.49227			0.746133	+	0.665797i	$0.231908\pi$
$32$	−14.3229			−0.447591
$33$		0			0
$34$	73.8677			2.17258
$35$	35.5754	−	48.1754i	1.01644	−	1.37644i
$36$		0			0
$37$			12.9739i			0.350646i	0.984511	+	0.175323i	$0.0560969\pi$
$37$			12.9739i			0.350646i	−0.984511	+	0.175323i	$0.943903\pi$
$38$	113.514			2.98721
$39$		0			0
$40$	−63.6276	−	46.9862i	−1.59069	−	1.17465i
$41$			43.8120i			1.06859i	0.845300	+	0.534293i	$0.179422\pi$
$41$			43.8120i			1.06859i	−0.845300	+	0.534293i	$0.820578\pi$
$42$		0			0
$43$			48.3388i			1.12416i	0.827083	+	0.562079i	$0.189998\pi$
$43$			48.3388i			1.12416i	−0.827083	+	0.562079i	$0.810002\pi$
$44$		−	28.1192i		−	0.639072i
$45$		0			0
$46$	90.3677			1.96451
$47$	−37.3165			−0.793968			−0.396984	−	0.917826i	$-0.629943\pi$
$47$	−37.3165			−0.793968			−0.396984	+	0.917826i	$0.629943\pi$
$48$		0			0
$49$	−94.4594			−1.92774
$50$	−25.9841	−	84.4022i	−0.519683	−	1.68804i
$51$		0			0
$52$		−	75.2896i		−	1.44788i
$53$	−82.3754			−1.55425			−0.777126	−	0.629345i	$-0.783323\pi$
$53$	−82.3754			−1.55425			−0.777126	+	0.629345i	$0.783323\pi$
$54$		0			0
$55$	9.85104	−	13.3401i	0.179110	−	0.242546i
$56$			189.474i			3.38346i
$57$		0			0
$58$			21.4611i			0.370020i
$59$		−	48.1280i		−	0.815728i	−0.913043	−	0.407864i	$-0.866274\pi$
$59$		−	48.1280i		−	0.815728i	0.913043	−	0.407864i	$-0.133726\pi$
$60$		0			0
$61$	−8.63837			−0.141613			−0.0708063	−	0.997490i	$-0.522557\pi$
$61$	−8.63837			−0.141613			−0.0708063	+	0.997490i	$0.522557\pi$
$62$	−163.412			−2.63568
$63$		0			0
$64$	−37.2755			−0.582429
$65$	26.3764	−	35.7183i	0.405790	−	0.549512i
$66$		0			0
$67$			49.8698i			0.744325i	0.928168	+	0.372162i	$0.121384\pi$
$67$			49.8698i			0.744325i	−0.928168	+	0.372162i	$0.878616\pi$
$68$	−177.290			−2.60720
$69$		0			0
$70$	−125.669	+	170.178i	−1.79527	+	2.43111i
$71$			3.62072i			0.0509960i	0.999675	+	0.0254980i	$0.00811715\pi$
$71$			3.62072i			0.0509960i	−0.999675	+	0.0254980i	$0.991883\pi$
$72$		0			0
$73$			122.698i			1.68079i	0.541971	+	0.840397i	$0.317678\pi$
$73$			122.698i			1.68079i	−0.541971	+	0.840397i	$0.682322\pi$
$74$		−	45.8297i		−	0.619320i
$75$		0			0
$76$	−272.444			−3.58479
$77$	−39.7247			−0.515905
$78$		0			0
$79$	−20.9703			−0.265447			−0.132724	−	0.991153i	$-0.542372\pi$
$79$	−20.9703			−0.265447			−0.132724	+	0.991153i	$0.542372\pi$
$80$	88.3577	+	65.2483i	1.10447	+	0.815603i
$81$		0			0
$82$		−	154.764i		−	1.88737i
$83$	71.6603			0.863377			0.431689	−	0.902023i	$-0.357918\pi$
$83$	71.6603			0.863377			0.431689	+	0.902023i	$0.357918\pi$
$84$		0			0
$85$	−84.1084	−	62.1103i	−0.989511	−	0.730710i
$86$		−	170.755i		−	1.98552i
$87$		0			0
$88$			52.4663i			0.596208i
$89$			83.5205i			0.938433i	0.883083	+	0.469216i	$0.155464\pi$
$89$			83.5205i			0.938433i	−0.883083	+	0.469216i	$0.844536\pi$
$90$		0			0
$91$	−106.364			−1.16883
$92$	−216.891			−2.35751
$93$		0			0
$94$	131.819			1.40233
$95$	−129.251	−	95.4460i	−1.36053	−	1.00469i
$96$		0			0
$97$		−	73.3915i		−	0.756613i	−0.925680	−	0.378306i	$-0.876507\pi$
$97$		−	73.3915i		−	0.756613i	0.925680	−	0.378306i	$-0.123493\pi$
$98$	333.674			3.40483
$99$		0			0
$100$	62.3645	+	202.574i	0.623645	+	2.02574i
$101$		−	47.4513i		−	0.469815i	−0.972018	−	0.234907i	$-0.924521\pi$
$101$		−	47.4513i		−	0.469815i	0.972018	−	0.234907i	$-0.0754787\pi$
$102$		0			0
$103$		−	124.614i		−	1.20985i	−0.796283	−	0.604925i	$-0.793203\pi$
$103$		−	124.614i		−	1.20985i	0.796283	−	0.604925i	$-0.206797\pi$
$104$			140.480i			1.35077i
$105$		0			0
$106$	290.987			2.74516
$107$	−55.1994			−0.515882			−0.257941	−	0.966161i	$-0.583044\pi$
$107$	−55.1994			−0.515882			−0.257941	+	0.966161i	$0.583044\pi$
$108$		0			0
$109$	−153.102			−1.40460			−0.702302	−	0.711879i	$-0.747844\pi$
$109$	−153.102			−1.40460			−0.702302	+	0.711879i	$0.747844\pi$
$110$	−34.7984	+	47.1231i	−0.316349	+	0.428392i
$111$		0			0
$112$		−	263.116i		−	2.34925i
$113$	79.0747			0.699776			0.349888	−	0.936791i	$-0.386220\pi$
$113$	79.0747			0.699776			0.349888	+	0.936791i	$0.386220\pi$
$114$		0			0
$115$	−102.896	−	75.9840i	−0.894746	−	0.660730i
$116$		−	51.5089i		−	0.444042i
$117$		0			0
$118$			170.010i			1.44076i
$119$			250.462i			2.10473i
$120$		0			0
$121$	−11.0000			−0.0909091
$122$	30.5146			0.250120
$123$		0			0
$124$	392.206			3.16295
$125$	−41.3816	+	117.952i	−0.331053	+	0.943612i
$126$		0			0
$127$		−	145.813i		−	1.14813i	−0.818808	−	0.574067i	$-0.805365\pi$
$127$		−	145.813i		−	1.14813i	0.818808	−	0.574067i	$-0.194635\pi$
$128$	188.966			1.47629
$129$		0			0
$130$	−93.1733	+	126.173i	−0.716718	+	0.970563i
$131$		−	150.989i		−	1.15258i	−0.817244	−	0.576292i	$-0.804499\pi$
$131$		−	150.989i		−	1.15258i	0.817244	−	0.576292i	$-0.195501\pi$
$132$		0			0
$133$			384.890i			2.89391i
$134$		−	176.163i		−	1.31465i
$135$		0			0
$136$	330.798			2.43234
$137$	−121.013			−0.883304			−0.441652	−	0.897186i	$-0.645607\pi$
$137$	−121.013			−0.883304			−0.441652	+	0.897186i	$0.645607\pi$
$138$		0			0
$139$	98.6998			0.710070			0.355035	−	0.934853i	$-0.384469\pi$
$139$	98.6998			0.710070			0.355035	+	0.934853i	$0.384469\pi$
$140$	301.617	−	408.443i	2.15441	−	2.91745i
$141$		0			0
$142$		−	12.7900i		−	0.0900706i
$143$	−29.4527			−0.205963
$144$		0			0
$145$	18.0452	−	24.4364i	0.124450	−	0.168527i
$146$		−	433.425i		−	2.96867i
$147$		0			0
$148$			109.996i			0.743215i
$149$		−	234.493i		−	1.57378i	−0.617093	−	0.786890i	$-0.711690\pi$
$149$		−	234.493i		−	1.57378i	0.617093	−	0.786890i	$-0.288310\pi$
$150$		0			0
$151$	−128.844			−0.853268			−0.426634	−	0.904424i	$-0.640301\pi$
$151$	−128.844			−0.853268			−0.426634	+	0.904424i	$0.640301\pi$
$152$	508.342			3.34436
$153$		0			0
$154$	140.326			0.911207
$155$	186.067	+	137.402i	1.20043	+	0.886466i
$156$		0			0
$157$		−	102.290i		−	0.651527i	−0.945451	−	0.325763i	$-0.894379\pi$
$157$		−	102.290i		−	0.651527i	0.945451	−	0.325763i	$-0.105621\pi$
$158$	74.0768			0.468840
$159$		0			0
$160$	−57.6093	−	42.5419i	−0.360058	−	0.265887i
$161$			306.409i			1.90316i
$162$		0			0
$163$		−	84.0948i		−	0.515919i	−0.966156	−	0.257959i	$-0.916950\pi$
$163$		−	84.0948i		−	0.515919i	0.966156	−	0.257959i	$-0.0830501\pi$
$164$			371.449i			2.26493i
$165$		0			0
$166$	−253.137			−1.52492
$167$	−169.813			−1.01685			−0.508424	−	0.861107i	$-0.669772\pi$
$167$	−169.813			−1.01685			−0.508424	+	0.861107i	$0.669772\pi$
$168$		0			0
$169$	90.1397			0.533371
$170$	297.109	+	219.402i	1.74770	+	1.29060i
$171$		0			0
$172$			409.828i			2.38272i
$173$	−85.9443			−0.496788			−0.248394	−	0.968659i	$-0.579903\pi$
$173$	−85.9443			−0.496788			−0.248394	+	0.968659i	$0.579903\pi$
$174$		0			0
$175$	286.181	−	88.1041i	1.63532	−	0.503452i
$176$		−	72.8584i		−	0.413968i
$177$		0			0
$178$		−	295.033i		−	1.65749i
$179$		−	71.3623i		−	0.398672i	−0.979931	−	0.199336i	$-0.936121\pi$
$179$		−	71.3623i		−	0.398672i	0.979931	−	0.199336i	$-0.0638785\pi$
$180$		0			0
$181$	−26.5638			−0.146761			−0.0733807	−	0.997304i	$-0.523379\pi$
$181$	−26.5638			−0.146761			−0.0733807	+	0.997304i	$0.523379\pi$
$182$	375.725			2.06442
$183$		0			0
$184$	404.688			2.19939
$185$	−38.5351	+	52.1833i	−0.208298	+	0.282072i
$186$		0			0
$187$			69.3545i			0.370880i
$188$	−316.378			−1.68286
$189$		0			0
$190$	456.573	+	337.159i	2.40301	+	1.77452i
$191$		−	226.785i		−	1.18735i	−0.804703	−	0.593677i	$-0.797676\pi$
$191$		−	226.785i		−	1.18735i	0.804703	−	0.593677i	$-0.202324\pi$
$192$		0			0
$193$			45.4162i			0.235317i	0.993054	+	0.117659i	$0.0375388\pi$
$193$			45.4162i			0.235317i	−0.993054	+	0.117659i	$0.962461\pi$
$194$			259.252i			1.33635i
$195$		0			0
$196$	−800.850			−4.08597
$197$	−11.2017			−0.0568612			−0.0284306	−	0.999596i	$-0.509051\pi$
$197$	−11.2017			−0.0568612			−0.0284306	+	0.999596i	$0.509051\pi$
$198$		0			0
$199$	177.268			0.890794			0.445397	−	0.895333i	$-0.353063\pi$
$199$	177.268			0.890794			0.445397	+	0.895333i	$0.353063\pi$
$200$	−116.363	−	377.973i	−0.581816	−	1.88987i
$201$		0			0
$202$			167.620i			0.829800i
$203$	−72.7680			−0.358463
$204$		0			0
$205$	−130.130	+	176.220i	−0.634782	+	0.859608i
$206$			440.195i			2.13687i
$207$		0			0
$208$		−	195.080i		−	0.937884i
$209$			106.578i			0.509943i
$210$		0			0
$211$	363.589			1.72317			0.861585	−	0.507613i	$-0.169472\pi$
$211$	363.589			1.72317			0.861585	+	0.507613i	$0.169472\pi$
$212$	−698.399			−3.29433
$213$		0			0
$214$	194.989			0.911165
$215$	−143.576	+	194.427i	−0.667795	+	0.904313i
$216$		0			0
$217$		−	554.080i		−	2.55336i
$218$	540.825			2.48085
$219$		0			0
$220$	83.5195	−	113.100i	0.379634	−	0.514092i
$221$			185.698i			0.840262i
$222$		0			0
$223$		−	300.794i		−	1.34885i	−0.738342	−	0.674427i	$-0.764391\pi$
$223$		−	300.794i		−	1.34885i	0.738342	−	0.674427i	$-0.235609\pi$
$224$			171.552i			0.765858i
$225$		0			0
$226$	−279.328			−1.23596
$227$	111.606			0.491656			0.245828	−	0.969313i	$-0.420940\pi$
$227$	111.606			0.491656			0.245828	+	0.969313i	$0.420940\pi$
$228$		0			0
$229$	−56.8355			−0.248190			−0.124095	−	0.992270i	$-0.539603\pi$
$229$	−56.8355			−0.248190			−0.124095	+	0.992270i	$0.539603\pi$
$230$	363.475	+	268.410i	1.58033	+	1.16700i
$231$		0			0
$232$			96.1082i			0.414260i
$233$	193.708			0.831366			0.415683	−	0.909510i	$-0.363543\pi$
$233$	193.708			0.831366			0.415683	+	0.909510i	$0.363543\pi$
$234$		0			0
$235$	−150.093	−	110.837i	−0.638696	−	0.471648i
$236$		−	408.041i		−	1.72899i
$237$		0			0
$238$		−	884.748i		−	3.71743i
$239$		−	317.127i		−	1.32689i	−0.748224	−	0.663446i	$-0.769093\pi$
$239$		−	317.127i		−	1.32689i	0.748224	−	0.663446i	$-0.230907\pi$
$240$		0			0
$241$	−280.493			−1.16387			−0.581935	−	0.813235i	$-0.697704\pi$
$241$	−280.493			−1.16387			−0.581935	+	0.813235i	$0.697704\pi$
$242$	38.8570			0.160566
$243$		0			0
$244$	−73.2382			−0.300156
$245$	−379.933	−	280.563i	−1.55075	−	1.14516i
$246$		0			0
$247$			285.365i			1.15532i
$248$	−731.800			−2.95081
$249$		0			0
$250$	146.179	−	416.659i	0.584714	−	1.66663i
$251$			299.975i			1.19512i	0.801824	+	0.597560i	$0.203863\pi$
$251$			299.975i			1.19512i	−0.801824	+	0.597560i	$0.796137\pi$
$252$		0			0
$253$			84.8463i			0.335361i
$254$			515.078i			2.02787i
$255$		0			0
$256$	−518.411			−2.02504
$257$	−326.950			−1.27218			−0.636090	−	0.771615i	$-0.719449\pi$
$257$	−326.950			−1.27218			−0.636090	+	0.771615i	$0.719449\pi$
$258$		0			0
$259$	155.394			0.599978
$260$	223.625	−	302.828i	0.860097	−	1.16472i
$261$		0			0
$262$			533.360i			2.03573i
$263$	60.3698			0.229543			0.114771	−	0.993392i	$-0.463386\pi$
$263$	60.3698			0.229543			0.114771	+	0.993392i	$0.463386\pi$
$264$		0			0
$265$	−331.328	−	244.671i	−1.25030	−	0.923288i
$266$		−	1359.61i		−	5.11130i
$267$		0			0
$268$			422.808i			1.57764i
$269$		−	159.233i		−	0.591944i	−0.955197	−	0.295972i	$-0.904356\pi$
$269$		−	159.233i		−	0.591944i	0.955197	−	0.295972i	$-0.0956435\pi$
$270$		0			0
$271$	−302.681			−1.11690			−0.558452	−	0.829537i	$-0.688605\pi$
$271$	−302.681			−1.11690			−0.558452	+	0.829537i	$0.688605\pi$
$272$	−459.369			−1.68886
$273$		0			0
$274$	427.472			1.56012
$275$	79.2453	−	24.3965i	0.288165	−	0.0887146i
$276$		0			0
$277$		−	230.494i		−	0.832109i	−0.909340	−	0.416054i	$-0.863413\pi$
$277$		−	230.494i		−	0.832109i	0.909340	−	0.416054i	$-0.136587\pi$
$278$	−348.653			−1.25415
$279$		0			0
$280$	−562.775	+	762.097i	−2.00991	+	2.72177i
$281$			216.007i			0.768708i	0.923186	+	0.384354i	$0.125576\pi$
$281$			216.007i			0.768708i	−0.923186	+	0.384354i	$0.874424\pi$
$282$		0			0
$283$			475.434i			1.67998i	0.542601	+	0.839990i	$0.317439\pi$
$283$			475.434i			1.67998i	−0.542601	+	0.839990i	$0.682561\pi$
$284$			30.6973i			0.108089i
$285$		0			0
$286$	104.040			0.363778
$287$	524.756			1.82842
$288$		0			0
$289$	148.277			0.513068
$290$	−63.7439	+	86.3205i	−0.219807	+	0.297657i
$291$		0			0
$292$			1040.26i			3.56255i
$293$	−50.4047			−0.172030			−0.0860149	−	0.996294i	$-0.527413\pi$
$293$	−50.4047			−0.172030			−0.0860149	+	0.996294i	$0.527413\pi$
$294$		0			0
$295$	142.950	−	193.579i	0.484575	−	0.656201i
$296$		−	205.237i		−	0.693367i
$297$		0			0
$298$			828.337i			2.77966i
$299$			227.178i			0.759791i
$300$		0			0
$301$	578.976			1.92351
$302$	455.134			1.50707
$303$		0			0
$304$	−705.920			−2.32210
$305$	−34.7450	−	25.6577i	−0.113918	−	0.0841235i
$306$		0			0
$307$			281.282i			0.916228i	0.888893	+	0.458114i	$0.151475\pi$
$307$			281.282i			0.916228i	−0.888893	+	0.458114i	$0.848525\pi$
$308$	−336.796			−1.09349
$309$		0			0
$310$	−657.273	−	485.367i	−2.12024	−	1.56570i
$311$		−	219.149i		−	0.704659i	−0.935876	−	0.352330i	$-0.885390\pi$
$311$		−	219.149i		−	0.704659i	0.935876	−	0.352330i	$-0.114610\pi$
$312$		0			0
$313$		−	134.925i		−	0.431071i	−0.976496	−	0.215536i	$-0.930850\pi$
$313$		−	134.925i		−	0.431071i	0.976496	−	0.215536i	$-0.0691497\pi$
$314$			361.334i			1.15075i
$315$		0			0
$316$	−177.792			−0.562632
$317$	−204.752			−0.645904			−0.322952	−	0.946415i	$-0.604675\pi$
$317$	−204.752			−0.645904			−0.322952	+	0.946415i	$0.604675\pi$
$318$		0			0
$319$	−20.1499			−0.0631658
$320$	−149.929	−	110.716i	−0.468527	−	0.345986i
$321$		0			0
$322$		−	1082.37i		−	3.36141i
$323$	671.970			2.08040
$324$		0			0
$325$	212.181	−	65.3222i	0.652864	−	0.200991i
$326$			297.061i			0.911230i
$327$		0			0
$328$		−	693.071i		−	2.11302i
$329$			446.956i			1.35853i
$330$		0			0
$331$	−273.536			−0.826391			−0.413196	−	0.910642i	$-0.635587\pi$
$331$	−273.536			−0.826391			−0.413196	+	0.910642i	$0.635587\pi$
$332$	607.554			1.82998
$333$		0			0
$334$	599.859			1.79598
$335$	−148.123	+	200.585i	−0.442159	+	0.598761i
$336$		0			0
$337$			8.13691i			0.0241451i	0.999927	+	0.0120726i	$0.00384291\pi$
$337$			8.13691i			0.0241451i	−0.999927	+	0.0120726i	$0.996157\pi$
$338$	−318.415			−0.942055
$339$		0			0
$340$	−713.091	−	526.586i	−2.09733	−	1.54878i
$341$		−	153.428i		−	0.449935i
$342$		0			0
$343$			544.488i			1.58743i
$344$		−	764.681i		−	2.22291i
$345$		0			0
$346$	303.595			0.877441
$347$	−119.124			−0.343296			−0.171648	−	0.985158i	$-0.554909\pi$
$347$	−119.124			−0.343296			−0.171648	+	0.985158i	$0.554909\pi$
$348$		0			0
$349$	−300.806			−0.861907			−0.430954	−	0.902374i	$-0.641823\pi$
$349$	−300.806			−0.861907			−0.430954	+	0.902374i	$0.641823\pi$
$350$	−1010.92	+	311.224i	−2.88835	+	0.889211i
$351$		0			0
$352$			47.5037i			0.134954i
$353$	312.338			0.884809			0.442405	−	0.896816i	$-0.354126\pi$
$353$	312.338			0.884809			0.442405	+	0.896816i	$0.354126\pi$
$354$		0			0
$355$	−10.7543	+	14.5632i	−0.0302937	+	0.0410230i
$356$			708.107i			1.98907i
$357$		0			0
$358$			252.084i			0.704146i
$359$		−	534.106i		−	1.48776i	−0.668313	−	0.743880i	$-0.732983\pi$
$359$		−	534.106i		−	1.48776i	0.668313	−	0.743880i	$-0.267017\pi$
$360$		0			0
$361$	671.628			1.86047
$362$	93.8355			0.259214
$363$		0			0
$364$	−901.778			−2.47741
$365$	−364.438	+	493.513i	−0.998459	+	1.35209i
$366$		0			0
$367$			338.463i			0.922243i	0.887337	+	0.461122i	$0.152553\pi$
$367$			338.463i			0.922243i	−0.887337	+	0.461122i	$0.847447\pi$
$368$	−561.978			−1.52712
$369$		0			0
$370$	136.123	−	184.335i	0.367901	−	0.498203i
$371$			986.647i			2.65943i
$372$		0			0
$373$		−	202.496i		−	0.542884i	−0.962455	−	0.271442i	$-0.912499\pi$
$373$		−	202.496i		−	0.542884i	0.962455	−	0.271442i	$-0.0875006\pi$
$374$		−	244.992i		−	0.655058i
$375$		0			0
$376$	590.317			1.56999
$377$	−53.9517			−0.143108
$378$		0			0
$379$	−531.040			−1.40116			−0.700580	−	0.713574i	$-0.747075\pi$
$379$	−531.040			−1.40116			−0.700580	+	0.713574i	$0.747075\pi$
$380$	−1095.82	−	809.214i	−2.88374	−	2.12951i
$381$		0			0
$382$			801.107i			2.09714i
$383$	530.502			1.38512			0.692561	−	0.721359i	$-0.256482\pi$
$383$	530.502			1.38512			0.692561	+	0.721359i	$0.256482\pi$
$384$		0			0
$385$	−159.780	−	117.990i	−0.415013	−	0.306468i
$386$		−	160.431i		−	0.415624i
$387$		0			0
$388$		−	622.231i		−	1.60369i
$389$			486.382i			1.25034i	0.780488	+	0.625170i	$0.214970\pi$
$389$			486.382i			1.25034i	−0.780488	+	0.625170i	$0.785030\pi$
$390$		0			0
$391$	534.952			1.36816
$392$	1494.27			3.81192
$393$		0			0
$394$	39.5694			0.100430
$395$	−84.3464	−	62.2861i	−0.213535	−	0.157686i
$396$		0			0
$397$		−	617.209i		−	1.55468i	−0.629080	−	0.777341i	$-0.716568\pi$
$397$		−	617.209i		−	1.55468i	0.629080	−	0.777341i	$-0.283432\pi$
$398$	−626.191			−1.57334
$399$		0			0
$400$	161.590	+	524.880i	0.403975	+	1.31220i
$401$		−	90.3979i		−	0.225431i	−0.993627	−	0.112716i	$-0.964045\pi$
$401$		−	90.3979i		−	0.225431i	0.993627	−	0.112716i	$-0.0359549\pi$
$402$		0			0
$403$		−	410.806i		−	1.01937i
$404$		−	402.303i		−	0.995801i
$405$		0			0
$406$	257.050			0.633128
$407$	43.0295			0.105724
$408$		0			0
$409$	418.934			1.02429			0.512144	−	0.858900i	$-0.328851\pi$
$409$	418.934			1.02429			0.512144	+	0.858900i	$0.328851\pi$
$410$	459.680	−	622.488i	1.12117	−	1.51826i
$411$		0			0
$412$		−	1056.51i		−	2.56435i
$413$	−576.451			−1.39576
$414$		0			0
$415$	288.230	+	212.845i	0.694531	+	0.512881i
$416$			127.192i			0.305751i
$417$		0			0
$418$		−	376.483i		−	0.900676i
$419$			701.649i			1.67458i	0.546759	+	0.837290i	$0.315861\pi$
$419$			701.649i			1.67458i	−0.546759	+	0.837290i	$0.684139\pi$
$420$		0			0
$421$	282.781			0.671688			0.335844	−	0.941918i	$-0.390978\pi$
$421$	282.781			0.671688			0.335844	+	0.941918i	$0.390978\pi$
$422$	−1284.36			−3.04351
$423$		0			0
$424$	1303.11			3.07338
$425$	−153.819	−	499.638i	−0.361927	−	1.17562i
$426$		0			0
$427$			103.466i			0.242308i
$428$	−467.994			−1.09344
$429$		0			0
$430$	507.176	−	686.806i	1.17948	−	1.59722i
$431$		−	87.2912i		−	0.202532i	−0.994859	−	0.101266i	$-0.967711\pi$
$431$		−	87.2912i		−	0.202532i	0.994859	−	0.101266i	$-0.0322893\pi$
$432$		0			0
$433$		−	770.823i		−	1.78019i	−0.455774	−	0.890096i	$-0.650637\pi$
$433$		−	770.823i		−	1.78019i	0.455774	−	0.890096i	$-0.349363\pi$
$434$			1957.26i			4.50982i
$435$		0			0
$436$	−1298.03			−2.97714
$437$	822.069			1.88116
$438$		0			0
$439$	−736.791			−1.67834			−0.839170	−	0.543869i	$-0.816959\pi$
$439$	−736.791			−1.67834			−0.839170	+	0.543869i	$0.816959\pi$
$440$	−155.836	+	211.029i	−0.354172	+	0.479611i
$441$		0			0
$442$		−	655.970i		−	1.48410i
$443$	643.473			1.45253			0.726267	−	0.687412i	$-0.241253\pi$
$443$	643.473			1.45253			0.726267	+	0.687412i	$0.241253\pi$
$444$		0			0
$445$	−248.073	+	335.934i	−0.557467	+	0.754909i
$446$			1062.54i			2.38238i
$447$		0			0
$448$			446.465i			0.996574i
$449$		−	571.450i		−	1.27272i	−0.771393	−	0.636359i	$-0.780440\pi$
$449$		−	571.450i		−	1.27272i	0.771393	−	0.636359i	$-0.219560\pi$
$450$		0			0
$451$	145.308			0.322191
$452$	670.415			1.48322
$453$		0			0
$454$	−394.243			−0.868377
$455$	−427.814	−	315.922i	−0.940250	−	0.694333i
$456$		0			0
$457$		−	335.758i		−	0.734700i	−0.930083	−	0.367350i	$-0.880265\pi$
$457$		−	335.758i		−	0.734700i	0.930083	−	0.367350i	$-0.119735\pi$
$458$	200.769			0.438360
$459$		0			0
$460$	−872.375	−	644.211i	−1.89647	−	1.40046i
$461$			588.712i			1.27703i	0.769608	+	0.638517i	$0.220452\pi$
$461$			588.712i			1.27703i	−0.769608	+	0.638517i	$0.779548\pi$
$462$		0			0
$463$		−	108.334i		−	0.233983i	−0.993133	−	0.116992i	$-0.962675\pi$
$463$		−	108.334i		−	0.233983i	0.993133	−	0.116992i	$-0.0373251\pi$
$464$		−	133.463i		−	0.287635i
$465$		0			0
$466$	−684.266			−1.46838
$467$	−22.9583			−0.0491612			−0.0245806	−	0.999698i	$-0.507825\pi$
$467$	−22.9583			−0.0491612			−0.0245806	+	0.999698i	$0.507825\pi$
$468$		0			0
$469$	597.313			1.27359
$470$	530.199	+	391.528i	1.12808	+	0.833039i
$471$		0			0
$472$			761.346i			1.61302i
$473$	160.322			0.338947
$474$		0			0
$475$	−236.376	−	767.801i	−0.497634	−	1.61642i
$476$			2123.48i			4.46110i
$477$		0			0
$478$			1120.24i			2.34359i
$479$		−	654.037i		−	1.36542i	−0.730689	−	0.682711i	$-0.760801\pi$
$479$		−	654.037i		−	1.36542i	0.730689	−	0.682711i	$-0.239199\pi$
$480$		0			0
$481$	115.212			0.239527
$482$	990.828			2.05566
$483$		0			0
$484$	−93.2607			−0.192687
$485$	217.987	−	295.193i	0.449458	−	0.608646i
$486$		0			0
$487$			331.080i			0.679835i	0.940455	+	0.339918i	$0.110399\pi$
$487$			331.080i			0.679835i	−0.940455	+	0.339918i	$0.889601\pi$
$488$	136.652			0.280025
$489$		0			0
$490$	1342.09	+	991.078i	2.73897	+	2.02261i
$491$		−	496.116i		−	1.01042i	−0.862997	−	0.505210i	$-0.831415\pi$
$491$		−	496.116i		−	1.01042i	0.862997	−	0.505210i	$-0.168585\pi$
$492$		0			0
$493$			127.044i			0.257696i
$494$		−	1008.04i		−	2.04057i
$495$		0			0
$496$	1016.23			2.04885
$497$	43.3670			0.0872575
$498$		0			0
$499$	−360.827			−0.723100			−0.361550	−	0.932353i	$-0.617752\pi$
$499$	−360.827			−0.723100			−0.361550	+	0.932353i	$0.617752\pi$
$500$	−350.843	+	1000.02i	−0.701686	+	2.00004i
$501$		0			0
$502$		−	1059.65i		−	2.11085i
$503$	505.779			1.00553			0.502763	−	0.864425i	$-0.332317\pi$
$503$	505.779			1.00553			0.502763	+	0.864425i	$0.332317\pi$
$504$		0			0
$505$	140.940	−	190.857i	0.279089	−	0.377936i
$506$		−	299.716i		−	0.592323i
$507$		0			0
$508$		−	1236.24i		−	2.43354i
$509$		−	45.9736i		−	0.0903214i	−0.998980	−	0.0451607i	$-0.985620\pi$
$509$		−	45.9736i		−	0.0903214i	0.998980	−	0.0451607i	$-0.0143800\pi$
$510$		0			0
$511$	1469.61			2.87595
$512$	1075.40			2.10039
$513$		0			0
$514$	1154.94			2.24696
$515$	370.130	−	501.222i	0.718699	−	0.973246i
$516$		0			0
$517$			123.765i			0.239390i
$518$	−548.923			−1.05970
$519$		0			0
$520$	−417.253	+	565.034i	−0.802409	+	1.08660i
$521$		−	768.500i		−	1.47505i	−0.675321	−	0.737524i	$-0.735995\pi$
$521$		−	768.500i		−	1.47505i	0.675321	−	0.737524i	$-0.264005\pi$
$522$		0			0
$523$			313.175i			0.598805i	0.954127	+	0.299402i	$0.0967873\pi$
$523$			313.175i			0.598805i	−0.954127	+	0.299402i	$0.903213\pi$
$524$		−	1280.12i		−	2.44297i
$525$		0			0
$526$	−213.254			−0.405425
$527$	−967.356			−1.83559
$528$		0			0
$529$	125.444			0.237135
$530$	1170.40	+	864.291i	2.20831	+	1.63074i
$531$		0			0
$532$			3263.19i			6.13382i
$533$	389.065			0.729953
$534$		0			0
$535$	−222.022	−	163.953i	−0.414994	−	0.306454i
$536$		−	788.899i		−	1.47183i
$537$		0			0
$538$			562.483i			1.04551i
$539$			313.286i			0.581236i
$540$		0			0
$541$	−447.242			−0.826695			−0.413348	−	0.910573i	$-0.635641\pi$
$541$	−447.242			−0.826695			−0.413348	+	0.910573i	$0.635641\pi$
$542$	1069.21			1.97271
$543$		0			0
$544$	299.509			0.550568
$545$	−615.803	−	454.743i	−1.12991	−	0.834391i
$546$		0			0
$547$			869.109i			1.58886i	0.607353	+	0.794432i	$0.292231\pi$
$547$			869.109i			1.58886i	−0.607353	+	0.794432i	$0.707769\pi$
$548$	−1025.98			−1.87222
$549$		0			0
$550$	−279.930	+	86.1796i	−0.508964	+	0.156690i
$551$			195.231i			0.354321i
$552$		0			0
$553$			251.171i			0.454197i
$554$			814.210i			1.46969i
$555$		0			0
$556$	836.801			1.50504
$557$	259.488			0.465867			0.232934	−	0.972493i	$-0.425167\pi$
$557$	259.488			0.465867			0.232934	+	0.972493i	$0.425167\pi$
$558$		0			0
$559$	429.265			0.767916
$560$	781.508	−	1058.30i	1.39555	−	1.88982i
$561$		0			0
$562$		−	763.035i		−	1.35771i
$563$	74.5811			0.132471			0.0662355	−	0.997804i	$-0.478901\pi$
$563$	74.5811			0.132471			0.0662355	+	0.997804i	$0.478901\pi$
$564$		0			0
$565$	318.053	+	234.868i	0.562925	+	0.415695i
$566$		−	1679.45i		−	2.96723i
$567$		0			0
$568$		−	57.2768i		−	0.100839i
$569$			532.055i			0.935070i	0.883975	+	0.467535i	$0.154858\pi$
$569$			532.055i			0.935070i	−0.883975	+	0.467535i	$0.845142\pi$
$570$		0			0
$571$	751.794			1.31663			0.658313	−	0.752744i	$-0.271270\pi$
$571$	751.794			1.31663			0.658313	+	0.752744i	$0.271270\pi$
$572$	−249.707			−0.436551
$573$		0			0
$574$	−1853.68			−3.22940
$575$	−188.178	−	611.242i	−0.327265	−	1.06303i
$576$		0			0
$577$		−	9.96725i		−	0.0172743i	−0.999963	−	0.00863713i	$-0.997251\pi$
$577$		−	9.96725i		−	0.0172743i	0.999963	−	0.00863713i	$-0.00274932\pi$
$578$	−523.781			−0.906195
$579$		0			0
$580$	152.992	−	207.178i	0.263779	−	0.357203i
$581$		−	858.308i		−	1.47729i
$582$		0			0
$583$			273.208i			0.468625i
$584$		−	1940.98i		−	3.32360i
$585$		0			0
$586$	178.052			0.303844
$587$	−186.077			−0.316997			−0.158499	−	0.987359i	$-0.550665\pi$
$587$	−186.077			−0.316997			−0.158499	+	0.987359i	$0.550665\pi$
$588$		0			0
$589$	−1486.55			−2.52386
$590$	−504.963	+	683.810i	−0.855870	+	1.15900i
$591$		0			0
$592$			285.006i			0.481429i
$593$	114.489			0.193067			0.0965335	−	0.995330i	$-0.469224\pi$
$593$	114.489			0.193067			0.0965335	+	0.995330i	$0.469224\pi$
$594$		0			0
$595$	−743.924	+	1007.40i	−1.25029	+	1.69312i
$596$		−	1988.09i		−	3.33572i
$597$		0			0
$598$		−	802.495i		−	1.34196i
$599$		−	735.724i		−	1.22825i	−0.789208	−	0.614127i	$-0.789508\pi$
$599$		−	735.724i		−	1.22825i	0.789208	−	0.614127i	$-0.210492\pi$
$600$		0			0
$601$	−1066.26			−1.77415			−0.887075	−	0.461625i	$-0.847267\pi$
$601$	−1066.26			−1.77415			−0.887075	+	0.461625i	$0.847267\pi$
$602$	−2045.21			−3.39735
$603$		0			0
$604$	−1092.37			−1.80855
$605$	−44.2440	−	32.6722i	−0.0731305	−	0.0540036i
$606$		0			0
$607$			336.422i			0.554238i	0.960836	+	0.277119i	$0.0893796\pi$
$607$			336.422i			0.554238i	−0.960836	+	0.277119i	$0.910620\pi$
$608$	460.260			0.757007
$609$		0			0
$610$	122.735	+	90.6346i	0.201205	+	0.148581i
$611$			331.383i			0.542361i
$612$		0			0
$613$		−	56.5327i		−	0.0922231i	−0.998936	−	0.0461115i	$-0.985317\pi$
$613$		−	56.5327i		−	0.0922231i	0.998936	−	0.0461115i	$-0.0146830\pi$
$614$		−	993.617i		−	1.61827i
$615$		0			0
$616$	628.413			1.02015
$617$	107.194			0.173734			0.0868669	−	0.996220i	$-0.472315\pi$
$617$	107.194			0.173734			0.0868669	+	0.996220i	$0.472315\pi$
$618$		0			0
$619$	551.896			0.891593			0.445796	−	0.895134i	$-0.352921\pi$
$619$	551.896			0.891593			0.445796	+	0.895134i	$0.352921\pi$
$620$	1577.52	+	1164.93i	2.54439	+	1.87892i
$621$		0			0
$622$			774.134i			1.24459i
$623$	1000.36			1.60572
$624$		0			0
$625$	−516.784	+	351.511i	−0.826854	+	0.562417i
$626$			476.618i			0.761370i
$627$		0			0
$628$		−	867.237i		−	1.38095i
$629$		−	271.299i		−	0.431318i
$630$		0			0
$631$	187.206			0.296681			0.148341	−	0.988936i	$-0.452607\pi$
$631$	187.206			0.296681			0.148341	+	0.988936i	$0.452607\pi$
$632$	331.734			0.524895
$633$		0			0
$634$	723.276			1.14081
$635$	433.094	−	586.486i	0.682038	−	0.923600i
$636$		0			0
$637$			838.831i			1.31685i
$638$	71.1786			0.111565
$639$		0			0
$640$	760.053	+	561.266i	1.18758	+	0.876978i
$641$			980.284i			1.52930i	0.644444	+	0.764652i	$0.277089\pi$
$641$			980.284i			1.52930i	−0.644444	+	0.764652i	$0.722911\pi$
$642$		0			0
$643$			579.355i			0.901019i	0.892772	+	0.450509i	$0.148758\pi$
$643$			579.355i			0.901019i	−0.892772	+	0.450509i	$0.851242\pi$
$644$			2597.81i			4.03386i
$645$		0			0
$646$	−2373.71			−3.67447
$647$	−68.4383			−0.105778			−0.0528890	−	0.998600i	$-0.516843\pi$
$647$	−68.4383			−0.105778			−0.0528890	+	0.998600i	$0.516843\pi$
$648$		0			0
$649$	−159.622			−0.245951
$650$	−749.519	+	230.748i	−1.15311	+	0.354996i
$651$		0			0
$652$		−	712.976i		−	1.09352i
$653$	−1064.76			−1.63057			−0.815285	−	0.579060i	$-0.803420\pi$
$653$	−1064.76			−1.63057			−0.815285	+	0.579060i	$0.803420\pi$
$654$		0			0
$655$	448.466	−	607.303i	0.684681	−	0.927180i
$656$			962.446i			1.46714i
$657$		0			0
$658$		−	1578.85i		−	2.39947i
$659$		−	435.812i		−	0.661323i	−0.943749	−	0.330662i	$-0.892728\pi$
$659$		−	435.812i		−	0.661323i	0.943749	−	0.330662i	$-0.107272\pi$
$660$		0			0
$661$	−883.343			−1.33637			−0.668187	−	0.743994i	$-0.732929\pi$
$661$	−883.343			−1.33637			−0.668187	+	0.743994i	$0.732929\pi$
$662$	966.252			1.45960
$663$		0			0
$664$	−1133.61			−1.70724
$665$	−1143.20	+	1548.10i	−1.71910	+	2.32796i
$666$		0			0
$667$			155.422i			0.233017i
$668$	−1439.72			−2.15527
$669$		0			0
$670$	523.238	−	708.558i	0.780953	−	1.05755i
$671$			28.6502i			0.0426978i
$672$		0			0
$673$			1191.28i			1.77010i	0.465493	+	0.885052i	$0.345877\pi$
$673$			1191.28i			1.77010i	−0.465493	+	0.885052i	$0.654123\pi$
$674$		−	28.7433i		−	0.0426458i
$675$		0			0
$676$	764.227			1.13051
$677$	−98.6878			−0.145772			−0.0728861	−	0.997340i	$-0.523221\pi$
$677$	−98.6878			−0.145772			−0.0728861	+	0.997340i	$0.523221\pi$
$678$		0			0
$679$	−879.043			−1.29461
$680$	1330.53	+	982.535i	1.95666	+	1.44491i
$681$		0			0
$682$			541.977i			0.794688i
$683$	243.769			0.356910			0.178455	−	0.983948i	$-0.442890\pi$
$683$	243.769			0.356910			0.178455	+	0.983948i	$0.442890\pi$
$684$		0			0
$685$	−486.735	−	359.432i	−0.710561	−	0.524718i
$686$		−	1923.38i		−	2.80376i
$687$		0			0
$688$			1061.89i			1.54344i
$689$			731.521i			1.06171i
$690$		0			0
$691$	126.468			0.183022			0.0915109	−	0.995804i	$-0.470830\pi$
$691$	126.468			0.183022			0.0915109	+	0.995804i	$0.470830\pi$
$692$	−728.657			−1.05297
$693$		0			0
$694$	420.800			0.606340
$695$	396.988	+	293.158i	0.571206	+	0.421810i
$696$		0			0
$697$		−	916.160i		−	1.31443i
$698$	1062.58			1.52232
$699$		0			0
$700$	2426.32	−	746.968i	3.46617	−	1.06710i
$701$			564.200i			0.804851i	0.915453	+	0.402425i	$0.131833\pi$
$701$			564.200i			0.804851i	−0.915453	+	0.402425i	$0.868167\pi$
$702$		0			0
$703$		−	416.910i		−	0.593044i
$704$			123.629i			0.175609i
$705$		0			0
$706$	−1103.32			−1.56277
$707$	−568.346			−0.803883
$708$		0			0
$709$	937.330			1.32204			0.661022	−	0.750366i	$-0.270123\pi$
$709$	937.330			1.32204			0.661022	+	0.750366i	$0.270123\pi$
$710$	37.9889	−	51.4437i	0.0535055	−	0.0724560i
$711$		0			0
$712$		−	1321.23i		−	1.85566i
$713$	−1183.44			−1.65980
$714$		0			0
$715$	−118.464	−	87.4805i	−0.165684	−	0.122350i
$716$		−	605.027i		−	0.845010i
$717$		0			0
$718$			1886.70i			2.62772i
$719$		−	397.871i		−	0.553368i	−0.960961	−	0.276684i	$-0.910765\pi$
$719$		−	397.871i		−	0.553368i	0.960961	−	0.276684i	$-0.0892355\pi$
$720$		0			0
$721$	−1492.56			−2.07013
$722$	−2372.50			−3.28601
$723$		0			0
$724$	−225.214			−0.311070
$725$	145.162	−	44.6897i	0.200224	−	0.0616410i
$726$		0			0
$727$			830.580i			1.14248i	0.820785	+	0.571238i	$0.193537\pi$
$727$			830.580i			1.14248i	−0.820785	+	0.571238i	$0.806463\pi$
$728$	1682.59			2.31125
$729$		0			0
$730$	1287.36	−	1743.31i	1.76351	−	2.38810i
$731$		−	1010.82i		−	1.38279i
$732$		0			0
$733$			275.106i			0.375316i	0.982234	+	0.187658i	$0.0600896\pi$
$733$			275.106i			0.375316i	−0.982234	+	0.187658i	$0.939910\pi$
$734$		−	1195.61i		−	1.62889i
$735$		0			0
$736$	366.410			0.497840
$737$	165.399			0.224422
$738$		0			0
$739$	−299.185			−0.404851			−0.202425	−	0.979298i	$-0.564882\pi$
$739$	−299.185			−0.404851			−0.202425	+	0.979298i	$0.564882\pi$
$740$	−326.710	+	442.423i	−0.441499	+	0.597869i
$741$		0			0
$742$		−	3485.29i		−	4.69715i
$743$	1268.06			1.70668			0.853340	−	0.521355i	$-0.174573\pi$
$743$	1268.06			1.70668			0.853340	+	0.521355i	$0.174573\pi$
$744$		0			0
$745$	696.492	−	943.174i	0.934889	−	1.26601i
$746$			715.308i			0.958858i
$747$		0			0
$748$			588.004i			0.786102i
$749$			661.148i			0.882707i
$750$		0			0
$751$	−462.879			−0.616350			−0.308175	−	0.951330i	$-0.599718\pi$
$751$	−462.879			−0.616350			−0.308175	+	0.951330i	$0.599718\pi$
$752$	−819.755			−1.09010
$753$		0			0
$754$	190.582			0.252761
$755$	−518.232	−	382.691i	−0.686399	−	0.506876i
$756$		0			0
$757$		−	233.013i		−	0.307811i	−0.988086	−	0.153905i	$-0.950815\pi$
$757$		−	233.013i		−	0.307811i	0.988086	−	0.153905i	$-0.0491851\pi$
$758$	1875.87			2.47477
$759$		0			0
$760$	2044.64	+	1509.88i	2.69032	+	1.98668i
$761$			596.489i			0.783823i	0.920003	+	0.391911i	$0.128186\pi$
$761$			596.489i			0.783823i	−0.920003	+	0.391911i	$0.871814\pi$
$762$		0			0
$763$			1833.77i			2.40337i
$764$		−	1922.74i		−	2.51667i
$765$		0			0
$766$	−1873.97			−2.44644
$767$	−427.392			−0.557226
$768$		0			0
$769$	64.2763			0.0835843			0.0417922	−	0.999126i	$-0.486693\pi$
$769$	64.2763			0.0835843			0.0417922	+	0.999126i	$0.486693\pi$
$770$	564.415	+	416.796i	0.733007	+	0.541293i
$771$		0			0
$772$			385.050i			0.498769i
$773$	669.406			0.865985			0.432992	−	0.901398i	$-0.357458\pi$
$773$	669.406			0.865985			0.432992	+	0.901398i	$0.357458\pi$
$774$		0			0
$775$	340.282	+	1105.31i	0.439074	+	1.42621i
$776$			1160.99i			1.49613i
$777$		0			0
$778$		−	1718.12i		−	2.20839i
$779$		−	1407.88i		−	1.80729i
$780$		0			0
$781$	12.0086			0.0153759
$782$	−1889.69			−2.41649
$783$		0			0
$784$	−2075.05			−2.64675
$785$	303.821	−	411.427i	0.387033	−	0.524111i
$786$		0			0
$787$		−	1304.38i		−	1.65740i	−0.559690	−	0.828702i	$-0.689080\pi$
$787$		−	1304.38i		−	1.65740i	0.559690	−	0.828702i	$-0.310920\pi$
$788$	−94.9704			−0.120521
$789$		0			0
$790$	297.950	+	220.023i	0.377152	+	0.278510i
$791$		−	947.114i		−	1.19736i
$792$		0			0
$793$			76.7115i			0.0967359i
$794$			2180.26i			2.74592i
$795$		0			0
$796$	1502.92			1.88809
$797$	507.312			0.636527			0.318263	−	0.948002i	$-0.396900\pi$
$797$	507.312			0.636527			0.318263	+	0.948002i	$0.396900\pi$
$798$		0			0
$799$	780.331			0.976634
$800$	−105.357	−	342.222i	−0.131696	−	0.427778i
$801$		0			0
$802$			319.327i			0.398163i
$803$	406.943			0.506779
$804$		0			0
$805$	−910.095	+	1232.43i	−1.13055	+	1.53097i
$806$			1451.16i			1.80044i
$807$		0			0
$808$			750.641i			0.929011i
$809$		−	782.832i		−	0.967654i	−0.875164	−	0.483827i	$-0.839246\pi$
$809$		−	782.832i		−	0.967654i	0.875164	−	0.483827i	$-0.160754\pi$
$810$		0			0
$811$	−1182.73			−1.45836			−0.729178	−	0.684324i	$-0.760097\pi$
$811$	−1182.73			−1.45836			−0.729178	+	0.684324i	$0.760097\pi$
$812$	−616.945			−0.759785
$813$		0			0
$814$	−152.000			−0.186732
$815$	249.778	−	338.244i	0.306476	−	0.415023i
$816$		0			0
$817$		−	1553.35i		−	1.90128i
$818$	−1479.87			−1.80913
$819$		0			0
$820$	−1103.28	+	1494.03i	−1.34546	+	1.82199i
$821$		−	317.113i		−	0.386253i	−0.981174	−	0.193126i	$-0.938137\pi$
$821$		−	317.113i		−	0.386253i	0.981174	−	0.193126i	$-0.0618627\pi$
$822$		0			0
$823$			606.110i			0.736465i	0.929734	+	0.368232i	$0.120037\pi$
$823$			606.110i			0.736465i	−0.929734	+	0.368232i	$0.879963\pi$
$824$			1971.30i			2.39236i
$825$		0			0
$826$	2036.29			2.46524
$827$	−84.2509			−0.101875			−0.0509377	−	0.998702i	$-0.516221\pi$
$827$	−84.2509			−0.101875			−0.0509377	+	0.998702i	$0.516221\pi$
$828$		0			0
$829$	1297.66			1.56533			0.782666	−	0.622441i	$-0.213859\pi$
$829$	1297.66			1.56533			0.782666	+	0.622441i	$0.213859\pi$
$830$	−1018.16	−	751.867i	−1.22670	−	0.905864i
$831$		0			0
$832$			331.019i			0.397859i
$833$	1975.26			2.37126
$834$		0			0
$835$	−683.020	−	504.380i	−0.817988	−	0.604048i
$836$			903.596i			1.08086i
$837$		0			0
$838$		−	2478.54i		−	2.95769i
$839$		−	1523.93i		−	1.81637i	−0.418574	−	0.908183i	$-0.637470\pi$
$839$		−	1523.93i		−	1.81637i	0.418574	−	0.908183i	$-0.362530\pi$
$840$		0			0
$841$	804.089			0.956111
$842$	−998.911			−1.18635
$843$		0			0
$844$	3082.60			3.65237
$845$	362.558	+	267.733i	0.429063	+	0.316844i
$846$		0			0
$847$			131.752i			0.155551i
$848$	−1809.59			−2.13395
$849$		0			0
$850$	543.358	+	1764.95i	0.639245	+	2.07641i
$851$		−	331.900i		−	0.390011i
$852$		0			0
$853$			940.069i			1.10207i	0.834481	+	0.551037i	$0.185768\pi$
$853$			940.069i			1.10207i	−0.834481	+	0.551037i	$0.814232\pi$
$854$		−	365.488i		−	0.427972i
$855$		0			0
$856$	873.209			1.02010
$857$	403.572			0.470912			0.235456	−	0.971885i	$-0.424342\pi$
$857$	403.572			0.470912			0.235456	+	0.971885i	$0.424342\pi$
$858$		0			0
$859$	1546.38			1.80021			0.900107	−	0.435669i	$-0.143488\pi$
$859$	1546.38			1.80021			0.900107	+	0.435669i	$0.143488\pi$
$860$	−1217.27	+	1648.40i	−1.41543	+	1.91675i
$861$		0			0
$862$			308.352i			0.357717i
$863$	−1545.58			−1.79094			−0.895470	−	0.445122i	$-0.853160\pi$
$863$	−1545.58			−1.79094			−0.895470	+	0.445122i	$0.853160\pi$
$864$		0			0
$865$	−345.683	−	255.272i	−0.399634	−	0.295112i
$866$			2722.90i			3.14422i
$867$		0			0
$868$		−	4697.63i		−	5.41201i
$869$			69.5507i			0.0800354i
$870$		0			0
$871$	442.860			0.508450
$872$	2421.95			2.77746
$873$		0			0
$874$	−2903.92			−3.32257
$875$	1412.76	+	495.646i	1.61458	+	0.566452i
$876$		0			0
$877$			958.856i			1.09334i	0.837350	+	0.546668i	$0.184104\pi$
$877$			958.856i			1.09334i	−0.837350	+	0.546668i	$0.815896\pi$
$878$	2602.68			2.96433
$879$		0			0
$880$	216.404	−	293.049i	0.245914	−	0.333011i
$881$			431.579i			0.489874i	0.969539	+	0.244937i	$0.0787673\pi$
$881$			431.579i			0.489874i	−0.969539	+	0.244937i	$0.921233\pi$
$882$		0			0
$883$			517.909i			0.586534i	0.956031	+	0.293267i	$0.0947425\pi$
$883$			517.909i			0.586534i	−0.956031	+	0.293267i	$0.905258\pi$
$884$			1574.39i			1.78099i
$885$		0			0
$886$	−2273.04			−2.56551
$887$	−873.218			−0.984462			−0.492231	−	0.870465i	$-0.663819\pi$
$887$	−873.218			−0.984462			−0.492231	+	0.870465i	$0.663819\pi$
$888$		0			0
$889$	−1746.47			−1.96453
$890$	876.306	−	1186.67i	0.984613	−	1.33334i
$891$		0			0
$892$		−	2550.21i		−	2.85898i
$893$	1199.15			1.34283
$894$		0			0
$895$	211.960	−	287.032i	0.236827	−	0.320706i
$896$		−	2263.33i		−	2.52603i
$897$		0			0
$898$			2018.62i			2.24791i
$899$		−	281.050i		−	0.312626i
$900$		0			0
$901$	1722.56			1.91184
$902$	−513.294			−0.569062
$903$		0			0
$904$	−1250.90			−1.38374
$905$	−106.844	−	78.8998i	−0.118060	−	0.0871821i
$906$		0			0
$907$		−	1007.47i		−	1.11077i	−0.831592	−	0.555387i	$-0.812570\pi$
$907$		−	1007.47i		−	1.11077i	0.831592	−	0.555387i	$-0.187430\pi$
$908$	946.222			1.04210
$909$		0			0
$910$	1511.23	+	1115.98i	1.66070	+	1.22635i
$911$			939.296i			1.03106i	0.856871	+	0.515530i	$0.172405\pi$
$911$			939.296i			1.03106i	−0.856871	+	0.515530i	$0.827595\pi$
$912$		0			0
$913$		−	237.670i		−	0.260318i
$914$			1186.05i			1.29765i
$915$		0			0
$916$	−481.865			−0.526053
$917$	−1808.46			−1.97215
$918$		0			0
$919$	−668.911			−0.727868			−0.363934	−	0.931425i	$-0.618567\pi$
$919$	−668.911			−0.727868			−0.363934	+	0.931425i	$0.618567\pi$
$920$	1627.73	+	1202.01i	1.76927	+	1.30653i
$921$		0			0
$922$		−	2079.60i		−	2.25553i
$923$	32.1532			0.0348355
$924$		0			0
$925$	−309.990	+	95.4337i	−0.335124	+	0.103172i
$926$			382.686i			0.413268i
$927$		0			0
$928$			87.0177i			0.0937691i
$929$			613.010i			0.659860i	0.944005	+	0.329930i	$0.107025\pi$
$929$			613.010i			0.659860i	−0.944005	+	0.329930i	$0.892975\pi$
$930$		0			0
$931$	3035.41			3.26037
$932$	1642.30			1.76213
$933$		0			0
$934$	81.0992			0.0868299
$935$	−205.997	+	278.956i	−0.220317	+	0.298349i
$936$		0			0
$937$		−	525.834i		−	0.561189i	−0.959826	−	0.280594i	$-0.909468\pi$
$937$		−	525.834i		−	0.561189i	0.959826	−	0.280594i	$-0.0905315\pi$
$938$	−2109.98			−2.24945
$939$		0			0
$940$	−1272.53	−	939.706i	−1.35375	−	0.999688i
$941$		−	156.165i		−	0.165957i	−0.996551	−	0.0829783i	$-0.973557\pi$
$941$		−	156.165i		−	0.165957i	0.996551	−	0.0829783i	$-0.0264432\pi$
$942$		0			0
$943$		−	1120.80i		−	1.18855i
$944$		−	1057.26i		−	1.11998i
$945$		0			0
$946$	−566.330			−0.598657
$947$	516.506			0.545413			0.272706	−	0.962097i	$-0.412081\pi$
$947$	516.506			0.545413			0.272706	+	0.962097i	$0.412081\pi$
$948$		0			0
$949$	1089.60			1.14815
$950$	834.988	+	2712.22i	0.878934	+	2.85497i
$951$		0			0
$952$		−	3962.12i		−	4.16189i
$953$	−1407.50			−1.47691			−0.738456	−	0.674302i	$-0.764445\pi$
$953$	−1407.50			−1.47691			−0.738456	+	0.674302i	$0.764445\pi$
$954$		0			0
$955$	673.596	−	912.168i	0.705336	−	0.955150i
$956$		−	2688.68i		−	2.81243i
$957$		0			0
$958$			2310.36i			2.41165i
$959$			1449.42i			1.51139i
$960$		0			0
$961$	1179.01			1.22686
$962$	−406.983			−0.423059
$963$		0			0
$964$	−2378.09			−2.46689
$965$	−134.895	+	182.672i	−0.139788	+	0.189297i
$966$		0			0
$967$			472.714i			0.488846i	0.969669	+	0.244423i	$0.0785986\pi$
$967$			472.714i			0.488846i	−0.969669	+	0.244423i	$0.921401\pi$
$968$	174.011			0.179764
$969$		0			0
$970$	−770.030	+	1042.76i	−0.793846	+	1.07501i
$971$		−	44.7781i		−	0.0461154i	−0.999734	−	0.0230577i	$-0.992660\pi$
$971$		−	44.7781i		−	0.0461154i	0.999734	−	0.0230577i	$-0.00734015\pi$
$972$		0			0
$973$		−	1182.17i		−	1.21498i
$974$		−	1169.52i		−	1.20074i
$975$		0			0
$976$	−189.764			−0.194431
$977$	−671.923			−0.687741			−0.343870	−	0.939017i	$-0.611738\pi$
$977$	−671.923			−0.687741			−0.343870	+	0.939017i	$0.611738\pi$
$978$		0			0
$979$	277.006			0.282948
$980$	−3221.16	−	2378.68i	−3.28690	−	2.42723i
$981$		0			0
$982$			1752.51i			1.78463i
$983$	−778.514			−0.791977			−0.395989	−	0.918255i	$-0.629598\pi$
$983$	−778.514			−0.791977			−0.395989	+	0.918255i	$0.629598\pi$
$984$		0			0
$985$	−45.0551	−	33.2712i	−0.0457412	−	0.0337779i
$986$		−	448.777i		−	0.455150i
$987$		0			0
$988$			2419.40i			2.44878i
$989$		−	1236.61i		−	1.25036i
$990$		0			0
$991$	23.1742			0.0233846			0.0116923	−	0.999932i	$-0.496278\pi$
$991$	23.1742			0.0233846			0.0116923	+	0.999932i	$0.496278\pi$
$992$	−662.582			−0.667925
$993$		0			0
$994$	−153.192			−0.154117
$995$	713.003	+	526.521i	0.716586	+	0.529167i
$996$		0			0
$997$		−	878.080i		−	0.880722i	−0.897821	−	0.440361i	$-0.854850\pi$
$997$		−	878.080i		−	0.880722i	0.897821	−	0.440361i	$-0.145150\pi$
$998$	1274.60			1.27716
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.g.a.89.3	✓	40
3.2	odd	2	inner	495.3.g.a.89.37	yes	40
5.4	even	2	inner	495.3.g.a.89.38	yes	40
15.14	odd	2	inner	495.3.g.a.89.4	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.3.g.a.89.3	✓	40	1.1	even	1	trivial
495.3.g.a.89.4	yes	40	15.14	odd	2	inner
495.3.g.a.89.37	yes	40	3.2	odd	2	inner
495.3.g.a.89.38	yes	40	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	495.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.53246 q^{2} +8.47824 q^{4} +(4.02218 + 2.97020i) q^{5} -11.9775i q^{7} -15.8192 q^{8} +(-14.2082 - 10.4921i) q^{10} -3.31662i q^{11} -8.88033i q^{13} +42.3098i q^{14} +21.9676 q^{16} -20.9112 q^{17} -32.1345 q^{19} +(34.1010 + 25.1821i) q^{20} +11.7158i q^{22} -25.5821 q^{23} +(7.35583 + 23.8933i) q^{25} +31.3694i q^{26} -101.548i q^{28} -6.07542i q^{29} +46.2603 q^{31} -14.3229 q^{32} +73.8677 q^{34} +(35.5754 - 48.1754i) q^{35} +12.9739i q^{37} +113.514 q^{38} +(-63.6276 - 46.9862i) q^{40} +43.8120i q^{41} +48.3388i q^{43} -28.1192i q^{44} +90.3677 q^{46} -37.3165 q^{47} -94.4594 q^{49} +(-25.9841 - 84.4022i) q^{50} -75.2896i q^{52} -82.3754 q^{53} +(9.85104 - 13.3401i) q^{55} +189.474i q^{56} +21.4611i q^{58} -48.1280i q^{59} -8.63837 q^{61} -163.412 q^{62} -37.2755 q^{64} +(26.3764 - 35.7183i) q^{65} +49.8698i q^{67} -177.290 q^{68} +(-125.669 + 170.178i) q^{70} +3.62072i q^{71} +122.698i q^{73} -45.8297i q^{74} -272.444 q^{76} -39.7247 q^{77} -20.9703 q^{79} +(88.3577 + 65.2483i) q^{80} -154.764i q^{82} +71.6603 q^{83} +(-84.1084 - 62.1103i) q^{85} -170.755i q^{86} +52.4663i q^{88} +83.5205i q^{89} -106.364 q^{91} -216.891 q^{92} +131.819 q^{94} +(-129.251 - 95.4460i) q^{95} -73.3915i q^{97} +333.674 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 72 q^{4} + 8 q^{10} + 184 q^{16} - 80 q^{19} + 32 q^{25} - 16 q^{31} - 160 q^{34} - 136 q^{40} + 560 q^{46} - 104 q^{49} - 96 q^{61} + 264 q^{64} - 872 q^{70} - 176 q^{76} - 672 q^{79} + 16 q^{85}+ \cdots + 400 q^{94}+O(q^{100}) \) 40 * q + 72 * q^4 + 8 * q^10 + 184 * q^16 - 80 * q^19 + 32 * q^25 - 16 * q^31 - 160 * q^34 - 136 * q^40 + 560 * q^46 - 104 * q^49 - 96 * q^61 + 264 * q^64 - 872 * q^70 - 176 * q^76 - 672 * q^79 + 16 * q^85 - 864 * q^91 + 400 * q^94

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