LMFDB - Embedded newform 495.3.g.a.89.38

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			89.38
Character	$\chi$	$=$	495.89
Dual form			495.3.g.a.89.37

$q$-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	$-0.155443\pi$
$2$	3.53246			1.76623			0.883114	+	0.469159i	$0.155443\pi$
$3$		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	$0.326770\pi$
$7$			11.9775i			1.71106i	−0.517749	+	0.855532i	$0.673230\pi$
$8$	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	$0.110952\pi$
$13$			8.88033i			0.683102i	−0.939863	+	0.341551i	$0.889048\pi$
$14$			42.3098i			3.02213i
$15$		0			0
$16$	21.9676			1.37298
$17$	20.9112			1.23007			0.615034	−	0.788500i	$-0.289142\pi$
$17$	20.9112			1.23007			0.615034	+	0.788500i	$0.289142\pi$
$18$		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	$-0.312285\pi$
$23$	25.5821			1.11227			0.556133	+	0.831093i	$0.312285\pi$
$24$		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	$-0.943903\pi$
$37$		−	12.9739i		−	0.350646i	0.984511	−	0.175323i	$-0.0560969\pi$
$38$	−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	$-0.810002\pi$
$43$		−	48.3388i		−	1.12416i	0.827083	−	0.562079i	$-0.189998\pi$
$44$		−	28.1192i		−	0.639072i
$45$		0			0
$46$	90.3677			1.96451
$47$	37.3165			0.793968			0.396984	−	0.917826i	$-0.370057\pi$
$47$	37.3165			0.793968			0.396984	+	0.917826i	$0.370057\pi$
$48$		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	$-0.216677\pi$
$53$	82.3754			1.55425			0.777126	+	0.629345i	$0.216677\pi$
$54$		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	$-0.878616\pi$
$67$		−	49.8698i		−	0.744325i	0.928168	−	0.372162i	$-0.121384\pi$
$68$	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	$-0.682322\pi$
$73$		−	122.698i		−	1.68079i	0.541971	−	0.840397i	$-0.317678\pi$
$74$		−	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	$-0.642082\pi$
$83$	−71.6603			−0.863377			−0.431689	+	0.902023i	$0.642082\pi$
$84$		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	$0.123493\pi$
$97$			73.3915i			0.756613i	−0.925680	+	0.378306i	$0.876507\pi$
$98$	−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	$0.206797\pi$
$103$			124.614i			1.20985i	−0.796283	+	0.604925i	$0.793203\pi$
$104$			140.480i			1.35077i
$105$		0			0
$106$	290.987			2.74516
$107$	55.1994			0.515882			0.257941	−	0.966161i	$-0.416956\pi$
$107$	55.1994			0.515882			0.257941	+	0.966161i	$0.416956\pi$
$108$		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	$-0.613780\pi$
$113$	−79.0747			−0.699776			−0.349888	+	0.936791i	$0.613780\pi$
$114$		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	$0.194635\pi$
$127$			145.813i			1.14813i	−0.818808	+	0.574067i	$0.805365\pi$
$128$	−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	$-0.354393\pi$
$137$	121.013			0.883304			0.441652	+	0.897186i	$0.354393\pi$
$138$		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	$0.105621\pi$
$157$			102.290i			0.651527i	−0.945451	+	0.325763i	$0.894379\pi$
$158$	−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	$0.0830501\pi$
$163$			84.0948i			0.515919i	−0.966156	+	0.257959i	$0.916950\pi$
$164$			371.449i			2.26493i
$165$		0			0
$166$	−253.137			−1.52492
$167$	169.813			1.01685			0.508424	−	0.861107i	$-0.330228\pi$
$167$	169.813			1.01685			0.508424	+	0.861107i	$0.330228\pi$
$168$		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	$-0.420097\pi$
$173$	85.9443			0.496788			0.248394	+	0.968659i	$0.420097\pi$
$174$		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	$-0.962461\pi$
$193$		−	45.4162i		−	0.235317i	0.993054	−	0.117659i	$-0.0375388\pi$
$194$			259.252i			1.33635i
$195$		0			0
$196$	−800.850			−4.08597
$197$	11.2017			0.0568612			0.0284306	−	0.999596i	$-0.490949\pi$
$197$	11.2017			0.0568612			0.0284306	+	0.999596i	$0.490949\pi$
$198$		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	$0.235609\pi$
$223$			300.794i			1.34885i	−0.738342	+	0.674427i	$0.764391\pi$
$224$			171.552i			0.765858i
$225$		0			0
$226$	−279.328			−1.23596
$227$	−111.606			−0.491656			−0.245828	−	0.969313i	$-0.579060\pi$
$227$	−111.606			−0.491656			−0.245828	+	0.969313i	$0.579060\pi$
$228$		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	$-0.636457\pi$
$233$	−193.708			−0.831366			−0.415683	+	0.909510i	$0.636457\pi$
$234$		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	$-0.280551\pi$
$257$	326.950			1.27218			0.636090	+	0.771615i	$0.280551\pi$
$258$		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	$-0.536614\pi$
$263$	−60.3698			−0.229543			−0.114771	+	0.993392i	$0.536614\pi$
$264$		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	$0.136587\pi$
$277$			230.494i			0.832109i	−0.909340	+	0.416054i	$0.863413\pi$
$278$	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	$-0.682561\pi$
$283$		−	475.434i		−	1.67998i	0.542601	−	0.839990i	$-0.317439\pi$
$284$			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	$-0.472587\pi$
$293$	50.4047			0.172030			0.0860149	+	0.996294i	$0.472587\pi$
$294$		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	$-0.848525\pi$
$307$		−	281.282i		−	0.916228i	0.888893	−	0.458114i	$-0.151475\pi$
$308$	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	$0.0691497\pi$
$313$			134.925i			0.431071i	−0.976496	+	0.215536i	$0.930850\pi$
$314$			361.334i			1.15075i
$315$		0			0
$316$	−177.792			−0.562632
$317$	204.752			0.645904			0.322952	−	0.946415i	$-0.395325\pi$
$317$	204.752			0.645904			0.322952	+	0.946415i	$0.395325\pi$
$318$		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	$-0.996157\pi$
$337$		−	8.13691i		−	0.0241451i	0.999927	−	0.0120726i	$-0.00384291\pi$
$338$	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	$-0.445091\pi$
$347$	119.124			0.343296			0.171648	+	0.985158i	$0.445091\pi$
$348$		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	$-0.645874\pi$
$353$	−312.338			−0.884809			−0.442405	+	0.896816i	$0.645874\pi$
$354$		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	$-0.847447\pi$
$367$		−	338.463i		−	0.922243i	0.887337	−	0.461122i	$-0.152553\pi$
$368$	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	$0.0875006\pi$
$373$			202.496i			0.542884i	−0.962455	+	0.271442i	$0.912499\pi$
$374$		−	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	$-0.743518\pi$
$383$	−530.502			−1.38512			−0.692561	+	0.721359i	$0.743518\pi$
$384$		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	$0.283432\pi$
$397$			617.209i			1.55468i	−0.629080	+	0.777341i	$0.716568\pi$
$398$	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	$0.349363\pi$
$433$			770.823i			1.78019i	−0.455774	+	0.890096i	$0.650637\pi$
$434$			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	$-0.758747\pi$
$443$	−643.473			−1.45253			−0.726267	+	0.687412i	$0.758747\pi$
$444$		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	$0.119735\pi$
$457$			335.758i			0.734700i	−0.930083	+	0.367350i	$0.880265\pi$
$458$	−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	$0.0373251\pi$
$463$			108.334i			0.233983i	−0.993133	+	0.116992i	$0.962675\pi$
$464$		−	133.463i		−	0.287635i
$465$		0			0
$466$	−684.266			−1.46838
$467$	22.9583			0.0491612			0.0245806	−	0.999698i	$-0.492175\pi$
$467$	22.9583			0.0491612			0.0245806	+	0.999698i	$0.492175\pi$
$468$		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	$-0.889601\pi$
$487$		−	331.080i		−	0.679835i	0.940455	−	0.339918i	$-0.110399\pi$
$488$	−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	$-0.667683\pi$
$503$	−505.779			−1.00553			−0.502763	+	0.864425i	$0.667683\pi$
$504$		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	$-0.903213\pi$
$523$		−	313.175i		−	0.598805i	0.954127	−	0.299402i	$-0.0967873\pi$
$524$		−	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	$-0.707769\pi$
$547$		−	869.109i		−	1.58886i	0.607353	−	0.794432i	$-0.292231\pi$
$548$	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	$-0.574833\pi$
$557$	−259.488			−0.465867			−0.232934	+	0.972493i	$0.574833\pi$
$558$		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	$-0.521099\pi$
$563$	−74.5811			−0.132471			−0.0662355	+	0.997804i	$0.521099\pi$
$564$		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	$0.00274932\pi$
$577$			9.96725i			0.0172743i	−0.999963	+	0.00863713i	$0.997251\pi$
$578$	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	$-0.449335\pi$
$587$	186.077			0.316997			0.158499	+	0.987359i	$0.449335\pi$
$588$		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	$-0.530776\pi$
$593$	−114.489			−0.193067			−0.0965335	+	0.995330i	$0.530776\pi$
$594$		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	$-0.910620\pi$
$607$		−	336.422i		−	0.554238i	0.960836	−	0.277119i	$-0.0893796\pi$
$608$	−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	$0.0146830\pi$
$613$			56.5327i			0.0922231i	−0.998936	+	0.0461115i	$0.985317\pi$
$614$		−	993.617i		−	1.61827i
$615$		0			0
$616$	628.413			1.02015
$617$	−107.194			−0.173734			−0.0868669	−	0.996220i	$-0.527685\pi$
$617$	−107.194			−0.173734			−0.0868669	+	0.996220i	$0.527685\pi$
$618$		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	$-0.851242\pi$
$643$		−	579.355i		−	0.901019i	0.892772	−	0.450509i	$-0.148758\pi$
$644$			2597.81i			4.03386i
$645$		0			0
$646$	−2373.71			−3.67447
$647$	68.4383			0.105778			0.0528890	−	0.998600i	$-0.483157\pi$
$647$	68.4383			0.105778			0.0528890	+	0.998600i	$0.483157\pi$
$648$		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	$-0.196580\pi$
$653$	1064.76			1.63057			0.815285	+	0.579060i	$0.196580\pi$
$654$		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	$-0.654123\pi$
$673$		−	1191.28i		−	1.77010i	0.465493	−	0.885052i	$-0.345877\pi$
$674$		−	28.7433i		−	0.0426458i
$675$		0			0
$676$	764.227			1.13051
$677$	98.6878			0.145772			0.0728861	−	0.997340i	$-0.476779\pi$
$677$	98.6878			0.145772			0.0728861	+	0.997340i	$0.476779\pi$
$678$		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	$-0.557110\pi$
$683$	−243.769			−0.356910			−0.178455	+	0.983948i	$0.557110\pi$
$684$		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	$-0.806463\pi$
$727$		−	830.580i		−	1.14248i	0.820785	−	0.571238i	$-0.193537\pi$
$728$	−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	$-0.939910\pi$
$733$		−	275.106i		−	0.375316i	0.982234	−	0.187658i	$-0.0600896\pi$
$734$		−	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	$-0.825427\pi$
$743$	−1268.06			−1.70668			−0.853340	+	0.521355i	$0.825427\pi$
$744$		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	$0.0491851\pi$
$757$			233.013i			0.307811i	−0.988086	+	0.153905i	$0.950815\pi$
$758$	−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	$-0.642542\pi$
$773$	−669.406			−0.865985			−0.432992	+	0.901398i	$0.642542\pi$
$774$		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	$0.310920\pi$
$787$			1304.38i			1.65740i	−0.559690	+	0.828702i	$0.689080\pi$
$788$	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	$-0.603100\pi$
$797$	−507.312			−0.636527			−0.318263	+	0.948002i	$0.603100\pi$
$798$		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	$-0.879963\pi$
$823$		−	606.110i		−	0.736465i	0.929734	−	0.368232i	$-0.120037\pi$
$824$			1971.30i			2.39236i
$825$		0			0
$826$	2036.29			2.46524
$827$	84.2509			0.101875			0.0509377	−	0.998702i	$-0.483779\pi$
$827$	84.2509			0.101875			0.0509377	+	0.998702i	$0.483779\pi$
$828$		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	$-0.814232\pi$
$853$		−	940.069i		−	1.10207i	0.834481	−	0.551037i	$-0.185768\pi$
$854$		−	365.488i		−	0.427972i
$855$		0			0
$856$	873.209			1.02010
$857$	−403.572			−0.470912			−0.235456	−	0.971885i	$-0.575658\pi$
$857$	−403.572			−0.470912			−0.235456	+	0.971885i	$0.575658\pi$
$858$		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	$-0.146840\pi$
$863$	1545.58			1.79094			0.895470	+	0.445122i	$0.146840\pi$
$864$		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	$-0.815896\pi$
$877$		−	958.856i		−	1.09334i	0.837350	−	0.546668i	$-0.184104\pi$
$878$	−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	$-0.905258\pi$
$883$		−	517.909i		−	0.586534i	0.956031	−	0.293267i	$-0.0947425\pi$
$884$			1574.39i			1.78099i
$885$		0			0
$886$	−2273.04			−2.56551
$887$	873.218			0.984462			0.492231	−	0.870465i	$-0.336181\pi$
$887$	873.218			0.984462			0.492231	+	0.870465i	$0.336181\pi$
$888$		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	$0.187430\pi$
$907$			1007.47i			1.11077i	−0.831592	+	0.555387i	$0.812570\pi$
$908$	−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	$0.0905315\pi$
$937$			525.834i			0.561189i	−0.959826	+	0.280594i	$0.909468\pi$
$938$	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	$-0.587919\pi$
$947$	−516.506			−0.545413			−0.272706	+	0.962097i	$0.587919\pi$
$948$		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	$-0.235555\pi$
$953$	1407.50			1.47691			0.738456	+	0.674302i	$0.235555\pi$
$954$		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	$-0.921401\pi$
$967$		−	472.714i		−	0.488846i	0.969669	−	0.244423i	$-0.0785986\pi$
$968$	−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	$-0.388262\pi$
$977$	671.923			0.687741			0.343870	+	0.939017i	$0.388262\pi$
$978$		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	$-0.370402\pi$
$983$	778.514			0.791977			0.395989	+	0.918255i	$0.370402\pi$
$984$		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	$0.145150\pi$
$997$			878.080i			0.880722i	−0.897821	+	0.440361i	$0.854850\pi$
$998$	−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.89.38	yes	40
3.2	odd	2	inner	495.3.g.a.89.4	yes	40
5.4	even	2	inner	495.3.g.a.89.3	✓	40
15.14	odd	2	inner	495.3.g.a.89.37	yes	40

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more