LMFDB - Embedded newform 2034.3.d.a.2033.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2034,3,Mod(2033,2034)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2034.2033"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2034, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2034 = 2 \cdot 3^{2} \cdot 113 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2034.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$55.4224857709$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2033.2
Character	$\chi$	$=$	2034.2033
Dual form			2034.3.d.a.2033.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -2.00000 q^{4} +4.88752 q^{5} -5.78081 q^{7} -2.82843i q^{8} +6.91200i q^{10} +19.4325i q^{11} +6.00266 q^{13} -8.17529i q^{14} +4.00000 q^{16} -14.4422 q^{17} -21.2302i q^{19} -9.77504 q^{20} -27.4817 q^{22} -12.7142 q^{23} -1.11214 q^{25} +8.48905i q^{26} +11.5616 q^{28} -35.8938 q^{29} +27.2643 q^{31} +5.65685i q^{32} -20.4243i q^{34} -28.2538 q^{35} -27.0839i q^{37} +30.0240 q^{38} -13.8240i q^{40} -8.57245i q^{41} +6.75215i q^{43} -38.8650i q^{44} -17.9805i q^{46} +21.0518 q^{47} -15.5823 q^{49} -1.57281i q^{50} -12.0053 q^{52} +48.9382i q^{53} +94.9767i q^{55} +16.3506i q^{56} -50.7614i q^{58} +5.43196 q^{59} -16.8936 q^{61} +38.5576i q^{62} -8.00000 q^{64} +29.3381 q^{65} -8.46018i q^{67} +28.8843 q^{68} -39.9569i q^{70} -103.677 q^{71} -11.1320i q^{73} +38.3024 q^{74} +42.4604i q^{76} -112.335i q^{77} -65.6916i q^{79} +19.5501 q^{80} +12.1233 q^{82} +2.76043i q^{83} -70.5864 q^{85} -9.54898 q^{86} +54.9634 q^{88} +158.238 q^{89} -34.7002 q^{91} +25.4283 q^{92} +29.7717i q^{94} -103.763i q^{95} -160.589 q^{97} -22.0367i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4} - 8 q^{7} + 144 q^{16} + 36 q^{25} + 16 q^{28} + 64 q^{31} + 108 q^{49} - 80 q^{61} - 288 q^{64} + 248 q^{82} - 164 q^{85} + 516 q^{91} - 420 q^{97}+O(q^{100}) $ 36 * q - 72 * q^4 - 8 * q^7 + 144 * q^16 + 36 * q^25 + 16 * q^28 + 64 * q^31 + 108 * q^49 - 80 * q^61 - 288 * q^64 + 248 * q^82 - 164 * q^85 + 516 * q^91 - 420 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2034\mathbb{Z}\right)^\times$.

$n$	$227$	$1585$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$		0			0
$3$		0			0
$4$	−2.00000			−0.500000
$5$	4.88752			0.977504			0.488752	−	0.872423i	$-0.337452\pi$
$5$	4.88752			0.977504			0.488752	+	0.872423i	$0.337452\pi$
$6$		0			0
$7$	−5.78081			−0.825829			−0.412915	−	0.910770i	$-0.635489\pi$
$7$	−5.78081			−0.825829			−0.412915	+	0.910770i	$0.635489\pi$
$8$		−	2.82843i		−	0.353553i
$9$		0			0
$10$			6.91200i			0.691200i
$11$			19.4325i			1.76659i	0.468817	+	0.883295i	$0.344680\pi$
$11$			19.4325i			1.76659i	−0.468817	+	0.883295i	$0.655320\pi$
$12$		0			0
$13$	6.00266			0.461743			0.230872	−	0.972984i	$-0.425842\pi$
$13$	6.00266			0.461743			0.230872	+	0.972984i	$0.425842\pi$
$14$		−	8.17529i		−	0.583950i
$15$		0			0
$16$	4.00000			0.250000
$17$	−14.4422			−0.849540			−0.424770	−	0.905301i	$-0.639645\pi$
$17$	−14.4422			−0.849540			−0.424770	+	0.905301i	$0.639645\pi$
$18$		0			0
$19$		−	21.2302i		−	1.11738i	−0.829377	−	0.558689i	$-0.811304\pi$
$19$		−	21.2302i		−	1.11738i	0.829377	−	0.558689i	$-0.188696\pi$
$20$	−9.77504			−0.488752
$21$		0			0
$22$	−27.4817			−1.24917
$23$	−12.7142			−0.552790			−0.276395	−	0.961044i	$-0.589140\pi$
$23$	−12.7142			−0.552790			−0.276395	+	0.961044i	$0.589140\pi$
$24$		0			0
$25$	−1.11214			−0.0444857
$26$			8.48905i			0.326502i
$27$		0			0
$28$	11.5616			0.412915
$29$	−35.8938			−1.23772			−0.618858	−	0.785503i	$-0.712404\pi$
$29$	−35.8938			−1.23772			−0.618858	+	0.785503i	$0.712404\pi$
$30$		0			0
$31$	27.2643			0.879495			0.439747	−	0.898121i	$-0.355068\pi$
$31$	27.2643			0.879495			0.439747	+	0.898121i	$0.355068\pi$
$32$			5.65685i			0.176777i
$33$		0			0
$34$		−	20.4243i		−	0.600715i
$35$	−28.2538			−0.807252
$36$		0			0
$37$		−	27.0839i		−	0.731997i	−0.930615	−	0.365998i	$-0.880728\pi$
$37$		−	27.0839i		−	0.731997i	0.930615	−	0.365998i	$-0.119272\pi$
$38$	30.0240			0.790106
$39$		0			0
$40$		−	13.8240i		−	0.345600i
$41$		−	8.57245i		−	0.209084i	−0.994520	−	0.104542i	$-0.966662\pi$
$41$		−	8.57245i		−	0.209084i	0.994520	−	0.104542i	$-0.0333377\pi$
$42$		0			0
$43$			6.75215i			0.157027i	0.996913	+	0.0785133i	$0.0250173\pi$
$43$			6.75215i			0.157027i	−0.996913	+	0.0785133i	$0.974983\pi$
$44$		−	38.8650i		−	0.883295i
$45$		0			0
$46$		−	17.9805i		−	0.390881i
$47$	21.0518			0.447910			0.223955	−	0.974599i	$-0.428103\pi$
$47$	21.0518			0.447910			0.223955	+	0.974599i	$0.428103\pi$
$48$		0			0
$49$	−15.5823			−0.318006
$50$		−	1.57281i		−	0.0314562i
$51$		0			0
$52$	−12.0053			−0.230872
$53$			48.9382i			0.923363i	0.887046	+	0.461681i	$0.152754\pi$
$53$			48.9382i			0.923363i	−0.887046	+	0.461681i	$0.847246\pi$
$54$		0			0
$55$			94.9767i			1.72685i
$56$			16.3506i			0.291975i
$57$		0			0
$58$		−	50.7614i		−	0.875197i
$59$	5.43196			0.0920672			0.0460336	−	0.998940i	$-0.485342\pi$
$59$	5.43196			0.0920672			0.0460336	+	0.998940i	$0.485342\pi$
$60$		0			0
$61$	−16.8936			−0.276944			−0.138472	−	0.990366i	$-0.544219\pi$
$61$	−16.8936			−0.276944			−0.138472	+	0.990366i	$0.544219\pi$
$62$			38.5576i			0.621897i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	29.3381			0.451356
$66$		0			0
$67$		−	8.46018i		−	0.126271i	−0.998005	−	0.0631357i	$-0.979890\pi$
$67$		−	8.46018i		−	0.126271i	0.998005	−	0.0631357i	$-0.0201101\pi$
$68$	28.8843			0.424770
$69$		0			0
$70$		−	39.9569i		−	0.570813i
$71$	−103.677			−1.46024			−0.730119	−	0.683320i	$-0.760536\pi$
$71$	−103.677			−1.46024			−0.730119	+	0.683320i	$0.760536\pi$
$72$		0			0
$73$		−	11.1320i		−	0.152493i	−0.997089	−	0.0762465i	$-0.975706\pi$
$73$		−	11.1320i		−	0.152493i	0.997089	−	0.0762465i	$-0.0242936\pi$
$74$	38.3024			0.517600
$75$		0			0
$76$			42.4604i			0.558689i
$77$		−	112.335i		−	1.45890i
$78$		0			0
$79$		−	65.6916i		−	0.831540i	−0.909470	−	0.415770i	$-0.863512\pi$
$79$		−	65.6916i		−	0.831540i	0.909470	−	0.415770i	$-0.136488\pi$
$80$	19.5501			0.244376
$81$		0			0
$82$	12.1233			0.147845
$83$			2.76043i			0.0332582i	0.999862	+	0.0166291i	$0.00529346\pi$
$83$			2.76043i			0.0332582i	−0.999862	+	0.0166291i	$0.994707\pi$
$84$		0			0
$85$	−70.5864			−0.830428
$86$	−9.54898			−0.111035
$87$		0			0
$88$	54.9634			0.624584
$89$	158.238			1.77796			0.888979	−	0.457949i	$-0.151416\pi$
$89$	158.238			1.77796			0.888979	+	0.457949i	$0.151416\pi$
$90$		0			0
$91$	−34.7002			−0.381321
$92$	25.4283			0.276395
$93$		0			0
$94$			29.7717i			0.316720i
$95$		−	103.763i		−	1.09224i
$96$		0			0
$97$	−160.589			−1.65556			−0.827779	−	0.561055i	$-0.810396\pi$
$97$	−160.589			−1.65556			−0.827779	+	0.561055i	$0.810396\pi$
$98$		−	22.0367i		−	0.224864i
$99$		0			0
$100$	2.22429			0.0222429
$101$	121.014			1.19815			0.599077	−	0.800691i	$-0.295534\pi$
$101$	121.014			1.19815			0.599077	+	0.800691i	$0.295534\pi$
$102$		0			0
$103$		−	129.103i		−	1.25342i	−0.779251	−	0.626712i	$-0.784400\pi$
$103$		−	129.103i		−	1.25342i	0.779251	−	0.626712i	$-0.215600\pi$
$104$		−	16.9781i		−	0.163251i
$105$		0			0
$106$	−69.2091			−0.652916
$107$	2.36235			0.0220781			0.0110390	−	0.999939i	$-0.496486\pi$
$107$	2.36235			0.0220781			0.0110390	+	0.999939i	$0.496486\pi$
$108$		0			0
$109$	−138.384			−1.26958			−0.634789	−	0.772685i	$-0.718913\pi$
$109$	−138.384			−1.26958			−0.634789	+	0.772685i	$0.718913\pi$
$110$	−134.317			−1.22107
$111$		0			0
$112$	−23.1232			−0.206457
$113$	−39.5725	−	105.844i	−0.350199	−	0.936675i
$113$	−39.5725	−	105.844i	−0.350199	−	0.936675i
$114$		0			0
$115$	−62.1407			−0.540354
$116$	71.7875			0.618858
$117$		0			0
$118$			7.68196i			0.0651013i
$119$	83.4874			0.701575
$120$		0			0
$121$	−256.622			−2.12084
$122$		−	23.8912i		−	0.195829i
$123$		0			0
$124$	−54.5287			−0.439747
$125$	−127.624			−1.02099
$126$		0			0
$127$	−137.052			−1.07915			−0.539574	−	0.841938i	$-0.681415\pi$
$127$	−137.052			−1.07915			−0.539574	+	0.841938i	$0.681415\pi$
$128$		−	11.3137i		−	0.0883883i
$129$		0			0
$130$			41.4904i			0.319157i
$131$		−	85.6233i		−	0.653613i	−0.945091	−	0.326806i	$-0.894028\pi$
$131$		−	85.6233i		−	0.653613i	0.945091	−	0.326806i	$-0.105972\pi$
$132$		0			0
$133$			122.728i			0.922764i
$134$	11.9645			0.0892873
$135$		0			0
$136$			40.8486i			0.300358i
$137$	−219.055			−1.59894			−0.799471	−	0.600704i	$-0.794887\pi$
$137$	−219.055			−1.59894			−0.799471	+	0.600704i	$0.794887\pi$
$138$		0			0
$139$	3.13532			0.0225563			0.0112781	−	0.999936i	$-0.496410\pi$
$139$	3.13532			0.0225563			0.0112781	+	0.999936i	$0.496410\pi$
$140$	56.5076			0.403626
$141$		0			0
$142$		−	146.621i		−	1.03254i
$143$			116.647i			0.815711i
$144$		0			0
$145$	−175.431			−1.20987
$146$	15.7430			0.107829
$147$		0			0
$148$			54.1678i			0.365998i
$149$		−	181.287i		−	1.21669i	−0.793673	−	0.608345i	$-0.791834\pi$
$149$		−	181.287i		−	1.21669i	0.793673	−	0.608345i	$-0.208166\pi$
$150$		0			0
$151$			22.9345i			0.151884i	0.997112	+	0.0759419i	$0.0241964\pi$
$151$			22.9345i			0.151884i	−0.997112	+	0.0759419i	$0.975804\pi$
$152$	−60.0481			−0.395053
$153$		0			0
$154$	158.866			1.03160
$155$	133.255			0.859710
$156$		0			0
$157$	−75.1348			−0.478566			−0.239283	−	0.970950i	$-0.576912\pi$
$157$	−75.1348			−0.478566			−0.239283	+	0.970950i	$0.576912\pi$
$158$	92.9020			0.587987
$159$		0			0
$160$			27.6480i			0.172800i
$161$	73.4981			0.456510
$162$		0			0
$163$	238.010			1.46019			0.730093	−	0.683348i	$-0.239477\pi$
$163$	238.010			1.46019			0.730093	+	0.683348i	$0.239477\pi$
$164$			17.1449i			0.104542i
$165$		0			0
$166$	−3.90384			−0.0235171
$167$	76.7007			0.459285			0.229643	−	0.973275i	$-0.426244\pi$
$167$	76.7007			0.459285			0.229643	+	0.973275i	$0.426244\pi$
$168$		0			0
$169$	−132.968			−0.786793
$170$		−	99.8243i		−	0.587202i
$171$		0			0
$172$		−	13.5043i		−	0.0785133i
$173$			241.393i			1.39534i	0.716422	+	0.697668i	$0.245779\pi$
$173$			241.393i			1.39534i	−0.716422	+	0.697668i	$0.754221\pi$
$174$		0			0
$175$	6.42909			0.0367376
$176$			77.7300i			0.441648i
$177$		0			0
$178$			223.783i			1.25721i
$179$	−150.684			−0.841808			−0.420904	−	0.907105i	$-0.638287\pi$
$179$	−150.684			−0.841808			−0.420904	+	0.907105i	$0.638287\pi$
$180$		0			0
$181$		−	327.634i		−	1.81013i	−0.425270	−	0.905067i	$-0.639821\pi$
$181$		−	327.634i		−	1.81013i	0.425270	−	0.905067i	$-0.360179\pi$
$182$		−	49.0735i		−	0.269635i
$183$		0			0
$184$			35.9611i			0.195441i
$185$		−	132.373i		−	0.715530i
$186$		0			0
$187$		−	280.647i		−	1.50079i
$188$	−42.1036			−0.223955
$189$		0			0
$190$	146.743			0.772332
$191$	147.664			0.773108			0.386554	−	0.922267i	$-0.373665\pi$
$191$	147.664			0.773108			0.386554	+	0.922267i	$0.373665\pi$
$192$		0			0
$193$			13.3001i			0.0689125i	0.999406	+	0.0344563i	$0.0109699\pi$
$193$			13.3001i			0.0689125i	−0.999406	+	0.0344563i	$0.989030\pi$
$194$		−	227.107i		−	1.17066i
$195$		0			0
$196$	31.1646			0.159003
$197$	−165.935			−0.842309			−0.421154	−	0.906989i	$-0.638375\pi$
$197$	−165.935			−0.842309			−0.421154	+	0.906989i	$0.638375\pi$
$198$		0			0
$199$			181.810i			0.913618i	0.889565	+	0.456809i	$0.151008\pi$
$199$			181.810i			0.913618i	−0.889565	+	0.456809i	$0.848992\pi$
$200$			3.14562i			0.0157281i
$201$		0			0
$202$			171.139i			0.847223i
$203$	207.495			1.02214
$204$		0			0
$205$		−	41.8980i		−	0.204380i
$206$	182.579			0.886305
$207$		0			0
$208$	24.0106			0.115436
$209$	412.556			1.97395
$210$		0			0
$211$	−113.186			−0.536426			−0.268213	−	0.963360i	$-0.586433\pi$
$211$	−113.186			−0.536426			−0.268213	+	0.963360i	$0.586433\pi$
$212$		−	97.8765i		−	0.461681i
$213$		0			0
$214$			3.34087i			0.0156116i
$215$			33.0013i			0.153494i
$216$		0			0
$217$	−157.610			−0.726313
$218$		−	195.705i		−	0.897728i
$219$		0			0
$220$		−	189.953i		−	0.863425i
$221$	−86.6915			−0.392269
$222$		0			0
$223$			343.939i			1.54233i	0.636636	+	0.771164i	$0.280325\pi$
$223$			343.939i			1.54233i	−0.636636	+	0.771164i	$0.719675\pi$
$224$		−	32.7012i		−	0.145987i
$225$		0			0
$226$	149.686	−	55.9640i	0.662329	−	0.247628i
$227$		−	41.9165i		−	0.184654i	−0.995729	−	0.0923272i	$-0.970569\pi$
$227$		−	41.9165i		−	0.184654i	0.995729	−	0.0923272i	$-0.0294306\pi$
$228$		0			0
$229$		−	219.082i		−	0.956692i	−0.878171	−	0.478346i	$-0.841236\pi$
$229$		−	219.082i		−	0.956692i	0.878171	−	0.478346i	$-0.158764\pi$
$230$		−	87.8803i		−	0.382088i
$231$		0			0
$232$			101.523i			0.437599i
$233$		−	36.9921i		−	0.158765i	−0.996844	−	0.0793823i	$-0.974705\pi$
$233$		−	36.9921i		−	0.158765i	0.996844	−	0.0793823i	$-0.0252948\pi$
$234$		0			0
$235$	102.891			0.437834
$236$	−10.8639			−0.0460336
$237$		0			0
$238$			118.069i			0.496088i
$239$		−	363.056i		−	1.51906i	−0.650470	−	0.759532i	$-0.725428\pi$
$239$		−	363.056i		−	1.51906i	0.650470	−	0.759532i	$-0.274572\pi$
$240$		0			0
$241$	110.938			0.460322			0.230161	−	0.973153i	$-0.426075\pi$
$241$	110.938			0.460322			0.230161	+	0.973153i	$0.426075\pi$
$242$		−	362.918i		−	1.49966i
$243$		0			0
$244$	33.7872			0.138472
$245$	−76.1587			−0.310852
$246$		0			0
$247$		−	127.438i		−	0.515942i
$248$		−	77.1152i		−	0.310948i
$249$		0			0
$250$		−	180.487i		−	0.721948i
$251$			139.303i			0.554991i	0.960727	+	0.277495i	$0.0895043\pi$
$251$			139.303i			0.554991i	−0.960727	+	0.277495i	$0.910496\pi$
$252$		0			0
$253$		−	247.068i		−	0.976553i
$254$		−	193.820i		−	0.763073i
$255$		0			0
$256$	16.0000			0.0625000
$257$			60.1184i			0.233924i	0.993136	+	0.116962i	$0.0373155\pi$
$257$			60.1184i			0.233924i	−0.993136	+	0.116962i	$0.962684\pi$
$258$		0			0
$259$			156.567i			0.604504i
$260$	−58.6763			−0.225678
$261$		0			0
$262$	121.090			0.462174
$263$	395.273			1.50294			0.751470	−	0.659768i	$-0.229345\pi$
$263$	395.273			1.50294			0.751470	+	0.659768i	$0.229345\pi$
$264$		0			0
$265$			239.187i			0.902591i
$266$	−173.563			−0.652493
$267$		0			0
$268$			16.9204i			0.0631357i
$269$	−205.989			−0.765757			−0.382878	−	0.923799i	$-0.625067\pi$
$269$	−205.989			−0.765757			−0.382878	+	0.923799i	$0.625067\pi$
$270$		0			0
$271$		−	341.609i		−	1.26055i	−0.776372	−	0.630275i	$-0.782942\pi$
$271$		−	341.609i		−	1.26055i	0.776372	−	0.630275i	$-0.217058\pi$
$272$	−57.7687			−0.212385
$273$		0			0
$274$		−	309.791i		−	1.13062i
$275$		−	21.6117i		−	0.0785881i
$276$		0			0
$277$	−54.7984			−0.197828			−0.0989141	−	0.995096i	$-0.531537\pi$
$277$	−54.7984			−0.197828			−0.0989141	+	0.995096i	$0.531537\pi$
$278$			4.43401i			0.0159497i
$279$		0			0
$280$			79.9138i			0.285407i
$281$	−406.517			−1.44668			−0.723341	−	0.690491i	$-0.757394\pi$
$281$	−406.517			−1.44668			−0.723341	+	0.690491i	$0.757394\pi$
$282$		0			0
$283$	26.8163			0.0947572			0.0473786	−	0.998877i	$-0.484913\pi$
$283$	26.8163			0.0947572			0.0473786	+	0.998877i	$0.484913\pi$
$284$	207.354			0.730119
$285$		0			0
$286$	−164.963			−0.576795
$287$			49.5556i			0.172668i
$288$		0			0
$289$	−80.4237			−0.278283
$290$		−	248.098i		−	0.855509i
$291$		0			0
$292$			22.2640i			0.0762465i
$293$	−9.70552			−0.0331246			−0.0165623	−	0.999863i	$-0.505272\pi$
$293$	−9.70552			−0.0331246			−0.0165623	+	0.999863i	$0.505272\pi$
$294$		0			0
$295$	26.5488			0.0899960
$296$	−76.6048			−0.258800
$297$		0			0
$298$	256.378			0.860329
$299$	−76.3188			−0.255247
$300$		0			0
$301$		−	39.0328i		−	0.129677i
$302$	−32.4342			−0.107398
$303$		0			0
$304$		−	84.9208i		−	0.279345i
$305$	−82.5678			−0.270714
$306$		0			0
$307$	−80.4517			−0.262058			−0.131029	−	0.991379i	$-0.541828\pi$
$307$	−80.4517			−0.262058			−0.131029	+	0.991379i	$0.541828\pi$
$308$			224.671i			0.729451i
$309$		0			0
$310$			188.451i			0.607907i
$311$			74.2119i			0.238623i	0.992857	+	0.119312i	$0.0380688\pi$
$311$			74.2119i			0.238623i	−0.992857	+	0.119312i	$0.961931\pi$
$312$		0			0
$313$	86.9272			0.277723			0.138861	−	0.990312i	$-0.455656\pi$
$313$	86.9272			0.277723			0.138861	+	0.990312i	$0.455656\pi$
$314$		−	106.257i		−	0.338397i
$315$		0			0
$316$			131.383i			0.415770i
$317$		−	141.260i		−	0.445615i	−0.974862	−	0.222807i	$-0.928478\pi$
$317$		−	141.260i		−	0.445615i	0.974862	−	0.222807i	$-0.0715221\pi$
$318$		0			0
$319$		−	697.505i		−	2.18654i
$320$	−39.1002			−0.122188
$321$		0			0
$322$			103.942i			0.322801i
$323$			306.610i			0.949257i
$324$		0			0
$325$	−6.67582			−0.0205410
$326$			336.597i			1.03251i
$327$		0			0
$328$	−24.2465			−0.0739224
$329$	−121.696			−0.369897
$330$		0			0
$331$	−226.795			−0.685181			−0.342590	−	0.939485i	$-0.611304\pi$
$331$	−226.795			−0.685181			−0.342590	+	0.939485i	$0.611304\pi$
$332$		−	5.52087i		−	0.0166291i
$333$		0			0
$334$			108.471i			0.324764i
$335$		−	41.3493i		−	0.123431i
$336$		0			0
$337$	−117.711			−0.349289			−0.174645	−	0.984632i	$-0.555878\pi$
$337$	−117.711			−0.349289			−0.174645	+	0.984632i	$0.555878\pi$
$338$		−	188.045i		−	0.556347i
$339$		0			0
$340$	141.173			0.415214
$341$			529.814i			1.55371i
$342$		0			0
$343$	373.338			1.08845
$344$	19.0980			0.0555173
$345$		0			0
$346$	−341.381			−0.986651
$347$		−	94.4644i		−	0.272232i	−0.990693	−	0.136116i	$-0.956538\pi$
$347$		−	94.4644i		−	0.272232i	0.990693	−	0.136116i	$-0.0434619\pi$
$348$		0			0
$349$			68.4243i			0.196058i	0.995184	+	0.0980291i	$0.0312538\pi$
$349$			68.4243i			0.196058i	−0.995184	+	0.0980291i	$0.968746\pi$
$350$			9.09210i			0.0259774i
$351$		0			0
$352$	−109.927			−0.312292
$353$		−	465.553i		−	1.31885i	−0.751772	−	0.659423i	$-0.770800\pi$
$353$		−	465.553i		−	1.31885i	0.751772	−	0.659423i	$-0.229200\pi$
$354$		0			0
$355$	−506.723			−1.42739
$356$	−316.476			−0.888979
$357$		0			0
$358$		−	213.099i		−	0.595248i
$359$	401.765			1.11912			0.559561	−	0.828789i	$-0.310970\pi$
$359$	401.765			1.11912			0.559561	+	0.828789i	$0.310970\pi$
$360$		0			0
$361$	−89.7212			−0.248535
$362$	463.345			1.27996
$363$		0			0
$364$	69.4004			0.190661
$365$		−	54.4078i		−	0.149062i
$366$		0			0
$367$	156.846			0.427372			0.213686	−	0.976902i	$-0.431453\pi$
$367$	156.846			0.427372			0.213686	+	0.976902i	$0.431453\pi$
$368$	−50.8567			−0.138197
$369$		0			0
$370$	187.204			0.505956
$371$		−	282.902i		−	0.762540i
$372$		0			0
$373$			466.395i			1.25039i	0.780470	+	0.625194i	$0.214980\pi$
$373$			466.395i			1.25039i	−0.780470	+	0.625194i	$0.785020\pi$
$374$	396.895			1.06122
$375$		0			0
$376$		−	59.5434i		−	0.158360i
$377$	−215.458			−0.571507
$378$		0			0
$379$			267.668i			0.706247i	0.935577	+	0.353124i	$0.114881\pi$
$379$			267.668i			0.706247i	−0.935577	+	0.353124i	$0.885119\pi$
$380$			207.526i			0.546121i
$381$		0			0
$382$			208.828i			0.546670i
$383$			176.740i			0.461463i	0.973017	+	0.230732i	$0.0741119\pi$
$383$			176.740i			0.461463i	−0.973017	+	0.230732i	$0.925888\pi$
$384$		0			0
$385$		−	549.042i		−	1.42608i
$386$	−18.8092			−0.0487285
$387$		0			0
$388$	321.178			0.827779
$389$			92.5383i			0.237888i	0.992901	+	0.118944i	$0.0379508\pi$
$389$			92.5383i			0.237888i	−0.992901	+	0.118944i	$0.962049\pi$
$390$		0			0
$391$	183.620			0.469617
$392$			44.0734i			0.112432i
$393$		0			0
$394$		−	234.667i		−	0.595602i
$395$		−	321.069i		−	0.812833i
$396$		0			0
$397$			487.128i			1.22702i	0.789686	+	0.613511i	$0.210244\pi$
$397$			487.128i			1.22702i	−0.789686	+	0.613511i	$0.789756\pi$
$398$	−257.118			−0.646025
$399$		0			0
$400$	−4.44857			−0.0111214
$401$		−	166.816i		−	0.416000i	−0.978129	−	0.208000i	$-0.933305\pi$
$401$		−	166.816i		−	0.416000i	0.978129	−	0.208000i	$-0.0666954\pi$
$402$		0			0
$403$	163.659			0.406101
$404$	−242.027			−0.599077
$405$		0			0
$406$			293.442i			0.722764i
$407$	526.307			1.29314
$408$		0			0
$409$		−	82.3303i		−	0.201297i	−0.994922	−	0.100648i	$-0.967908\pi$
$409$		−	82.3303i		−	0.201297i	0.994922	−	0.100648i	$-0.0320917\pi$
$410$	59.2527			0.144519
$411$		0			0
$412$			258.205i			0.626712i
$413$	−31.4011			−0.0760318
$414$		0			0
$415$			13.4917i			0.0325101i
$416$			33.9562i			0.0816254i
$417$		0			0
$418$			583.442i			1.39579i
$419$	−537.355			−1.28247			−0.641236	−	0.767344i	$-0.721578\pi$
$419$	−537.355			−1.28247			−0.641236	+	0.767344i	$0.721578\pi$
$420$		0			0
$421$	−737.410			−1.75157			−0.875784	−	0.482703i	$-0.839655\pi$
$421$	−737.410			−1.75157			−0.875784	+	0.482703i	$0.839655\pi$
$422$		−	160.069i		−	0.379310i
$423$		0			0
$424$	138.418			0.326458
$425$	16.0618			0.0377924
$426$		0			0
$427$	97.6586			0.228709
$428$	−4.72471			−0.0110390
$429$		0			0
$430$	−46.6708			−0.108537
$431$	164.942			0.382697			0.191348	−	0.981522i	$-0.438714\pi$
$431$	164.942			0.382697			0.191348	+	0.981522i	$0.438714\pi$
$432$		0			0
$433$		−	8.95538i		−	0.0206822i	−0.999947	−	0.0103411i	$-0.996708\pi$
$433$		−	8.95538i		−	0.0206822i	0.999947	−	0.0103411i	$-0.00329173\pi$
$434$		−	222.894i		−	0.513581i
$435$		0			0
$436$	276.768			0.634789
$437$			269.924i			0.617675i
$438$		0			0
$439$	−58.6346			−0.133564			−0.0667821	−	0.997768i	$-0.521273\pi$
$439$	−58.6346			−0.133564			−0.0667821	+	0.997768i	$0.521273\pi$
$440$	268.635			0.610533
$441$		0			0
$442$		−	122.600i		−	0.277376i
$443$		−	65.9767i		−	0.148932i	−0.997224	−	0.0744659i	$-0.976275\pi$
$443$		−	65.9767i		−	0.148932i	0.997224	−	0.0744659i	$-0.0237252\pi$
$444$		0			0
$445$	773.392			1.73796
$446$	−486.404			−1.09059
$447$		0			0
$448$	46.2464			0.103229
$449$	625.557			1.39322			0.696612	−	0.717448i	$-0.254690\pi$
$449$	625.557			1.39322			0.696612	+	0.717448i	$0.254690\pi$
$450$		0			0
$451$	166.584			0.369366
$452$	79.1450	+	211.689i	0.175100	+	0.468338i
$453$		0			0
$454$	59.2789			0.130570
$455$	−169.598			−0.372743
$456$		0			0
$457$			356.885i			0.780929i	0.920618	+	0.390465i	$0.127686\pi$
$457$			356.885i			0.780929i	−0.920618	+	0.390465i	$0.872314\pi$
$458$	309.829			0.676483
$459$		0			0
$460$	124.281			0.270177
$461$			749.137i			1.62503i	0.582943	+	0.812513i	$0.301901\pi$
$461$			749.137i			1.62503i	−0.582943	+	0.812513i	$0.698099\pi$
$462$		0			0
$463$	49.6707			0.107280			0.0536400	−	0.998560i	$-0.482918\pi$
$463$	49.6707			0.107280			0.0536400	+	0.998560i	$0.482918\pi$
$464$	−143.575			−0.309429
$465$		0			0
$466$	52.3148			0.112263
$467$		−	36.5126i		−	0.0781854i	−0.999236	−	0.0390927i	$-0.987553\pi$
$467$		−	36.5126i		−	0.0781854i	0.999236	−	0.0390927i	$-0.0124468\pi$
$468$		0			0
$469$			48.9067i			0.104279i
$470$			145.510i			0.309595i
$471$		0			0
$472$		−	15.3639i		−	0.0325507i
$473$	−131.211			−0.277402
$474$		0			0
$475$			23.6110i			0.0497074i
$476$	−166.975			−0.350787
$477$		0			0
$478$	513.439			1.07414
$479$	−304.132			−0.634930			−0.317465	−	0.948270i	$-0.602832\pi$
$479$	−304.132			−0.634930			−0.317465	+	0.948270i	$0.602832\pi$
$480$		0			0
$481$		−	162.575i		−	0.337995i
$482$			156.890i			0.325497i
$483$		0			0
$484$	513.244			1.06042
$485$	−784.882			−1.61831
$486$		0			0
$487$			199.930i			0.410535i	0.978706	+	0.205267i	$0.0658064\pi$
$487$			199.930i			0.410535i	−0.978706	+	0.205267i	$0.934194\pi$
$488$			47.7823i			0.0979146i
$489$		0			0
$490$		−	107.705i		−	0.219806i
$491$	661.709			1.34768			0.673838	−	0.738879i	$-0.264645\pi$
$491$	661.709			1.34768			0.673838	+	0.738879i	$0.264645\pi$
$492$		0			0
$493$	518.384			1.05149
$494$	180.224			0.364826
$495$		0			0
$496$	109.057			0.219874
$497$	599.336			1.20591
$498$		0			0
$499$			492.508i			0.986990i	0.869749	+	0.493495i	$0.164281\pi$
$499$			492.508i			0.986990i	−0.869749	+	0.493495i	$0.835719\pi$
$500$	255.247			0.510495
$501$		0			0
$502$	−197.004			−0.392438
$503$		−	247.139i		−	0.491330i	−0.969355	−	0.245665i	$-0.920994\pi$
$503$		−	247.139i		−	0.491330i	0.969355	−	0.245665i	$-0.0790063\pi$
$504$		0			0
$505$	591.456			1.17120
$506$	349.407			0.690527
$507$		0			0
$508$	274.104			0.539574
$509$			762.096i			1.49724i	0.662998	+	0.748621i	$0.269284\pi$
$509$			762.096i			1.49724i	−0.662998	+	0.748621i	$0.730716\pi$
$510$		0			0
$511$			64.3518i			0.125933i
$512$			22.6274i			0.0441942i
$513$		0			0
$514$	−85.0203			−0.165409
$515$		−	630.992i		−	1.22523i
$516$		0			0
$517$			409.088i			0.791274i
$518$	−221.419			−0.427449
$519$		0			0
$520$		−	82.9808i		−	0.159578i
$521$		−	149.501i		−	0.286949i	−0.989654	−	0.143475i	$-0.954172\pi$
$521$		−	149.501i		−	0.286949i	0.989654	−	0.143475i	$-0.0458275\pi$
$522$		0			0
$523$		−	157.013i		−	0.300217i	−0.988670	−	0.150108i	$-0.952038\pi$
$523$		−	157.013i		−	0.300217i	0.988670	−	0.150108i	$-0.0479623\pi$
$524$			171.247i			0.326806i
$525$		0			0
$526$			559.001i			1.06274i
$527$	−393.756			−0.747166
$528$		0			0
$529$	−367.350			−0.694424
$530$	−338.261			−0.638228
$531$		0			0
$532$		−	245.455i		−	0.461382i
$533$		−	51.4575i		−	0.0965431i
$534$		0			0
$535$	11.5460			0.0215814
$536$	−23.9290			−0.0446437
$537$		0			0
$538$		−	291.312i		−	0.541472i
$539$		−	302.803i		−	0.561786i
$540$		0			0
$541$			847.808i			1.56711i	0.621321	+	0.783556i	$0.286596\pi$
$541$			847.808i			1.56711i	−0.621321	+	0.783556i	$0.713404\pi$
$542$	483.108			0.891343
$543$		0			0
$544$		−	81.6973i		−	0.150179i
$545$	−676.355			−1.24102
$546$		0			0
$547$	−772.163			−1.41163			−0.705817	−	0.708395i	$-0.749420\pi$
$547$	−772.163			−1.41163			−0.705817	+	0.708395i	$0.749420\pi$
$548$	438.110			0.799471
$549$		0			0
$550$	30.5636			0.0555702
$551$			762.032i			1.38300i
$552$		0			0
$553$			379.751i			0.686710i
$554$		−	77.4966i		−	0.139886i
$555$		0			0
$556$	−6.27064			−0.0112781
$557$			950.504i			1.70647i	0.521526	+	0.853236i	$0.325363\pi$
$557$			950.504i			1.70647i	−0.521526	+	0.853236i	$0.674637\pi$
$558$		0			0
$559$			40.5308i			0.0725060i
$560$	−113.015			−0.201813
$561$		0			0
$562$		−	574.902i		−	1.02296i
$563$			916.850i			1.62851i	0.580509	+	0.814254i	$0.302854\pi$
$563$			916.850i			1.62851i	−0.580509	+	0.814254i	$0.697146\pi$
$564$		0			0
$565$	−193.411	−	517.316i	−0.342321	−	0.915604i
$566$			37.9239i			0.0670034i
$567$		0			0
$568$			293.243i			0.516272i
$569$			456.318i			0.801964i	0.916086	+	0.400982i	$0.131331\pi$
$569$			456.318i			0.801964i	−0.916086	+	0.400982i	$0.868669\pi$
$570$		0			0
$571$		−	712.697i		−	1.24816i	−0.781362	−	0.624078i	$-0.785475\pi$
$571$		−	712.697i		−	1.24816i	0.781362	−	0.624078i	$-0.214525\pi$
$572$		−	233.293i		−	0.407855i
$573$		0			0
$574$	−70.0823			−0.122095
$575$	14.1400			0.0245913
$576$		0			0
$577$			204.476i			0.354378i	0.984177	+	0.177189i	$0.0567004\pi$
$577$			204.476i			0.354378i	−0.984177	+	0.177189i	$0.943300\pi$
$578$		−	113.736i		−	0.196775i
$579$		0			0
$580$	350.863			0.604936
$581$		−	15.9575i		−	0.0274656i
$582$		0			0
$583$	−950.992			−1.63120
$584$	−31.4860			−0.0539144
$585$		0			0
$586$		−	13.7257i		−	0.0234226i
$587$			20.7779i			0.0353968i	0.999843	+	0.0176984i	$0.00563386\pi$
$587$			20.7779i			0.0353968i	−0.999843	+	0.0176984i	$0.994366\pi$
$588$		0			0
$589$		−	578.827i		−	0.982729i
$590$			37.5457i			0.0636368i
$591$		0			0
$592$		−	108.336i		−	0.182999i
$593$			165.137i			0.278477i	0.990259	+	0.139239i	$0.0444655\pi$
$593$			165.137i			0.278477i	−0.990259	+	0.139239i	$0.955534\pi$
$594$		0			0
$595$	408.046			0.685792
$596$			362.573i			0.608345i
$597$		0			0
$598$		−	107.931i		−	0.180487i
$599$	708.441			1.18271			0.591353	−	0.806413i	$-0.298594\pi$
$599$	708.441			1.18271			0.591353	+	0.806413i	$0.298594\pi$
$600$		0			0
$601$	−424.001			−0.705493			−0.352746	−	0.935719i	$-0.614752\pi$
$601$	−424.001			−0.705493			−0.352746	+	0.935719i	$0.614752\pi$
$602$	55.2008			0.0916956
$603$		0			0
$604$		−	45.8689i		−	0.0759419i
$605$	−1254.24			−2.07313
$606$		0			0
$607$		−	937.106i		−	1.54383i	−0.635725	−	0.771916i	$-0.719299\pi$
$607$		−	937.106i		−	1.54383i	0.635725	−	0.771916i	$-0.280701\pi$
$608$	120.096			0.197527
$609$		0			0
$610$		−	116.768i		−	0.191424i
$611$	126.367			0.206819
$612$		0			0
$613$		−	336.976i		−	0.549715i	−0.961485	−	0.274858i	$-0.911369\pi$
$613$		−	336.976i		−	0.549715i	0.961485	−	0.274858i	$-0.0886307\pi$
$614$		−	113.776i		−	0.185303i
$615$		0			0
$616$	−317.733			−0.515800
$617$			445.267i			0.721665i	0.932631	+	0.360832i	$0.117507\pi$
$617$			445.267i			0.721665i	−0.932631	+	0.360832i	$0.882493\pi$
$618$		0			0
$619$			1002.91i			1.62022i	0.586281	+	0.810108i	$0.300591\pi$
$619$			1002.91i			1.62022i	−0.586281	+	0.810108i	$0.699409\pi$
$620$	−266.510			−0.429855
$621$		0			0
$622$	−104.951			−0.168732
$623$	−914.744			−1.46829
$624$		0			0
$625$	−595.960			−0.953535
$626$			122.934i			0.196380i
$627$		0			0
$628$	150.270			0.239283
$629$			391.150i			0.621860i
$630$		0			0
$631$			431.979i			0.684594i	0.939592	+	0.342297i	$0.111205\pi$
$631$			431.979i			0.684594i	−0.939592	+	0.342297i	$0.888795\pi$
$632$	−185.804			−0.293994
$633$		0			0
$634$	199.772			0.315097
$635$	−669.843			−1.05487
$636$		0			0
$637$	−93.5352			−0.146837
$638$	986.421			1.54611
$639$		0			0
$640$		−	55.2960i		−	0.0864000i
$641$	1200.09			1.87222			0.936108	−	0.351711i	$-0.114400\pi$
$641$	1200.09			1.87222			0.936108	+	0.351711i	$0.114400\pi$
$642$		0			0
$643$		−	95.3336i		−	0.148264i	−0.997248	−	0.0741319i	$-0.976381\pi$
$643$		−	95.3336i		−	0.148264i	0.997248	−	0.0741319i	$-0.0236186\pi$
$644$	−146.996			−0.228255
$645$		0			0
$646$	−433.612			−0.671226
$647$			894.999i			1.38331i	0.722230	+	0.691653i	$0.243117\pi$
$647$			894.999i			1.38331i	−0.722230	+	0.691653i	$0.756883\pi$
$648$		0			0
$649$			105.557i			0.162645i
$650$		−	9.44104i		−	0.0145247i
$651$		0			0
$652$	−476.020			−0.730093
$653$			447.580i			0.685421i	0.939441	+	0.342710i	$0.111345\pi$
$653$			447.580i			0.685421i	−0.939441	+	0.342710i	$0.888655\pi$
$654$		0			0
$655$		−	418.485i		−	0.638909i
$656$		−	34.2898i		−	0.0522710i
$657$		0			0
$658$		−	172.104i		−	0.261557i
$659$	−954.313			−1.44812			−0.724061	−	0.689736i	$-0.757727\pi$
$659$	−954.313			−1.44812			−0.724061	+	0.689736i	$0.757727\pi$
$660$		0			0
$661$		−	84.5909i		−	0.127974i	−0.997951	−	0.0639871i	$-0.979618\pi$
$661$		−	84.5909i		−	0.127974i	0.997951	−	0.0639871i	$-0.0203816\pi$
$662$		−	320.736i		−	0.484496i
$663$		0			0
$664$	7.80769			0.0117586
$665$			599.834i			0.902006i
$666$		0			0
$667$	456.359			0.684197
$668$	−153.401			−0.229643
$669$		0			0
$670$	58.4768			0.0872787
$671$		−	328.285i		−	0.489247i
$672$		0			0
$673$		−	1036.46i		−	1.54007i	−0.638004	−	0.770033i	$-0.720240\pi$
$673$		−	1036.46i		−	1.54007i	0.638004	−	0.770033i	$-0.279760\pi$
$674$		−	166.468i		−	0.246985i
$675$		0			0
$676$	265.936			0.393397
$677$			447.204i			0.660567i	0.943882	+	0.330283i	$0.107144\pi$
$677$			447.204i			0.660567i	−0.943882	+	0.330283i	$0.892856\pi$
$678$		0			0
$679$	928.334			1.36721
$680$			199.649i			0.293601i
$681$		0			0
$682$	−749.270			−1.09864
$683$	−449.319			−0.657862			−0.328931	−	0.944354i	$-0.606688\pi$
$683$	−449.319			−0.657862			−0.328931	+	0.944354i	$0.606688\pi$
$684$		0			0
$685$	−1070.64			−1.56297
$686$			527.979i			0.769649i
$687$		0			0
$688$			27.0086i			0.0392567i
$689$			293.760i			0.426356i
$690$		0			0
$691$	−690.035			−0.998603			−0.499302	−	0.866428i	$-0.666410\pi$
$691$	−690.035			−0.998603			−0.499302	+	0.866428i	$0.666410\pi$
$692$		−	482.786i		−	0.697668i
$693$		0			0
$694$	133.593			0.192497
$695$	15.3239			0.0220488
$696$		0			0
$697$			123.805i			0.177625i
$698$	−96.7666			−0.138634
$699$		0			0
$700$	−12.8582			−0.0183688
$701$	−1258.44			−1.79521			−0.897604	−	0.440802i	$-0.854694\pi$
$701$	−1258.44			−1.79521			−0.897604	+	0.440802i	$0.854694\pi$
$702$		0			0
$703$	−574.996			−0.817918
$704$		−	155.460i		−	0.220824i
$705$		0			0
$706$	658.391			0.932566
$707$	−699.556			−0.989471
$708$		0			0
$709$	−553.558			−0.780759			−0.390379	−	0.920654i	$-0.627656\pi$
$709$	−553.558			−0.780759			−0.390379	+	0.920654i	$0.627656\pi$
$710$		−	716.615i		−	1.00932i
$711$		0			0
$712$		−	447.565i		−	0.628603i
$713$	−346.643			−0.486176
$714$		0			0
$715$			570.113i			0.797361i
$716$	301.367			0.420904
$717$		0			0
$718$			568.181i			0.791339i
$719$			73.5865i			0.102346i	0.998690	+	0.0511728i	$0.0162959\pi$
$719$			73.5865i			0.102346i	−0.998690	+	0.0511728i	$0.983704\pi$
$720$		0			0
$721$			746.317i			1.03511i
$722$		−	126.885i		−	0.175741i
$723$		0			0
$724$			655.268i			0.905067i
$725$	39.9190			0.0550607
$726$		0			0
$727$	969.141			1.33307			0.666534	−	0.745474i	$-0.267777\pi$
$727$	969.141			1.33307			0.666534	+	0.745474i	$0.267777\pi$
$728$			98.1470i			0.134817i
$729$		0			0
$730$	76.9442			0.105403
$731$		−	97.5157i		−	0.133400i
$732$		0			0
$733$			639.495i			0.872435i	0.899841	+	0.436218i	$0.143682\pi$
$733$			639.495i			0.872435i	−0.899841	+	0.436218i	$0.856318\pi$
$734$			221.813i			0.302198i
$735$		0			0
$736$		−	71.9222i		−	0.0977203i
$737$	164.402			0.223070
$738$		0			0
$739$	739.707			1.00096			0.500479	−	0.865749i	$-0.333157\pi$
$739$	739.707			1.00096			0.500479	+	0.865749i	$0.333157\pi$
$740$			264.746i			0.357765i
$741$		0			0
$742$	400.084			0.539197
$743$	−660.701			−0.889234			−0.444617	−	0.895721i	$-0.646660\pi$
$743$	−660.701			−0.889234			−0.444617	+	0.895721i	$0.646660\pi$
$744$		0			0
$745$		−	886.042i		−	1.18932i
$746$	−659.581			−0.884157
$747$		0			0
$748$			561.295i			0.750394i
$749$	−13.6563			−0.0182327
$750$		0			0
$751$		−	565.842i		−	0.753451i	−0.926325	−	0.376725i	$-0.877050\pi$
$751$		−	565.842i		−	0.753451i	0.926325	−	0.376725i	$-0.122950\pi$
$752$	84.2071			0.111978
$753$		0			0
$754$		−	304.704i		−	0.404116i
$755$			112.093i			0.148467i
$756$		0			0
$757$			280.458i			0.370486i	0.982693	+	0.185243i	$0.0593072\pi$
$757$			280.458i			0.370486i	−0.982693	+	0.185243i	$0.940693\pi$
$758$	−378.539			−0.499392
$759$		0			0
$760$	−293.486			−0.386166
$761$		−	1115.78i		−	1.46621i	−0.680116	−	0.733104i	$-0.738071\pi$
$761$		−	1115.78i		−	1.46621i	0.680116	−	0.733104i	$-0.261929\pi$
$762$		0			0
$763$	799.972			1.04846
$764$	−295.327			−0.386554
$765$		0			0
$766$	−249.949			−0.326304
$767$	32.6062			0.0425114
$768$		0			0
$769$	1202.81			1.56412			0.782058	−	0.623205i	$-0.214170\pi$
$769$	1202.81			1.56412			0.782058	+	0.623205i	$0.214170\pi$
$770$	776.462			1.00839
$771$		0			0
$772$		−	26.6002i		−	0.0344563i
$773$			506.567i			0.655326i	0.944795	+	0.327663i	$0.106261\pi$
$773$			506.567i			0.655326i	−0.944795	+	0.327663i	$0.893739\pi$
$774$		0			0
$775$	−30.3219			−0.0391250
$776$			454.214i			0.585328i
$777$		0			0
$778$	−130.869			−0.168212
$779$	−181.995			−0.233626
$780$		0			0
$781$		−	2014.70i		−	2.57964i
$782$			259.678i			0.332069i
$783$		0			0
$784$	−62.3291			−0.0795015
$785$	−367.223			−0.467800
$786$		0			0
$787$	−331.878			−0.421700			−0.210850	−	0.977518i	$-0.567623\pi$
$787$	−331.878			−0.421700			−0.210850	+	0.977518i	$0.567623\pi$
$788$	331.870			0.421154
$789$		0			0
$790$	454.060			0.574760
$791$	228.761	+	611.865i	0.289205	+	0.773534i
$792$		0			0
$793$	−101.407			−0.127877
$794$	−688.903			−0.867636
$795$		0			0
$796$		−	363.620i		−	0.456809i
$797$	1221.80			1.53300			0.766499	−	0.642246i	$-0.221997\pi$
$797$	1221.80			1.53300			0.766499	+	0.642246i	$0.221997\pi$
$798$		0			0
$799$	−304.033			−0.380517
$800$		−	6.29123i		−	0.00786404i
$801$		0			0
$802$	235.914			0.294157
$803$	216.322			0.269392
$804$		0			0
$805$	359.223			0.446240
$806$			231.448i			0.287157i
$807$		0			0
$808$		−	342.278i		−	0.423612i
$809$		−	1052.81i		−	1.30138i	−0.759345	−	0.650688i	$-0.774481\pi$
$809$		−	1052.81i		−	1.30138i	0.759345	−	0.650688i	$-0.225519\pi$
$810$		0			0
$811$			748.572i			0.923023i	0.887134	+	0.461512i	$0.152693\pi$
$811$			748.572i			0.923023i	−0.887134	+	0.461512i	$0.847307\pi$
$812$	−414.990			−0.511071
$813$		0			0
$814$			744.311i			0.914387i
$815$	1163.28			1.42734
$816$		0			0
$817$	143.349			0.175458
$818$	116.433			0.142338
$819$		0			0
$820$			83.7960i			0.102190i
$821$			448.205i			0.545925i	0.962025	+	0.272963i	$0.0880035\pi$
$821$			448.205i			0.545925i	−0.962025	+	0.272963i	$0.911996\pi$
$822$		0			0
$823$	798.338			0.970034			0.485017	−	0.874505i	$-0.338813\pi$
$823$	798.338			0.970034			0.485017	+	0.874505i	$0.338813\pi$
$824$	−365.158			−0.443152
$825$		0			0
$826$		−	44.4079i		−	0.0537626i
$827$		−	717.357i		−	0.867421i	−0.901052	−	0.433711i	$-0.857204\pi$
$827$		−	717.357i		−	0.867421i	0.901052	−	0.433711i	$-0.142796\pi$
$828$		0			0
$829$			385.912i			0.465515i	0.972535	+	0.232758i	$0.0747749\pi$
$829$			385.912i			0.465515i	−0.972535	+	0.232758i	$0.925225\pi$
$830$	−19.0801			−0.0229881
$831$		0			0
$832$	−48.0213			−0.0577179
$833$	225.042			0.270159
$834$		0			0
$835$	374.876			0.448953
$836$	−825.111			−0.986975
$837$		0			0
$838$		−	759.935i		−	0.906844i
$839$	794.047			0.946421			0.473210	−	0.880950i	$-0.343095\pi$
$839$	794.047			0.946421			0.473210	+	0.880950i	$0.343095\pi$
$840$		0			0
$841$	447.362			0.531941
$842$		−	1042.86i		−	1.23855i
$843$		0			0
$844$	226.372			0.268213
$845$	−649.884			−0.769094
$846$		0			0
$847$	1483.48			1.75145
$848$			195.753i			0.230841i
$849$		0			0
$850$			22.7148i			0.0267233i
$851$			344.349i			0.404640i
$852$		0			0
$853$	−770.056			−0.902762			−0.451381	−	0.892331i	$-0.649068\pi$
$853$	−770.056			−0.902762			−0.451381	+	0.892331i	$0.649068\pi$
$854$			138.110i			0.161721i
$855$		0			0
$856$		−	6.68174i		−	0.00780578i
$857$	1185.85			1.38373			0.691864	−	0.722028i	$-0.256790\pi$
$857$	1185.85			1.38373			0.691864	+	0.722028i	$0.256790\pi$
$858$		0			0
$859$		−	630.498i		−	0.733990i	−0.930223	−	0.366995i	$-0.880387\pi$
$859$		−	630.498i		−	0.733990i	0.930223	−	0.366995i	$-0.119613\pi$
$860$		−	66.0025i		−	0.0767471i
$861$		0			0
$862$			233.264i			0.270608i
$863$			1415.46i			1.64016i	0.572251	+	0.820079i	$0.306070\pi$
$863$			1415.46i			1.64016i	−0.572251	+	0.820079i	$0.693930\pi$
$864$		0			0
$865$			1179.81i			1.36395i
$866$	12.6648			0.0146245
$867$		0			0
$868$	315.220			0.363156
$869$	1276.55			1.46899
$870$		0			0
$871$		−	50.7836i		−	0.0583049i
$872$			391.409i			0.448864i
$873$		0			0
$874$	−381.730			−0.436762
$875$	737.767			0.843163
$876$		0			0
$877$		−	948.293i		−	1.08129i	−0.841250	−	0.540646i	$-0.818180\pi$
$877$		−	948.293i		−	1.08129i	0.841250	−	0.540646i	$-0.181820\pi$
$878$		−	82.9219i		−	0.0944441i
$879$		0			0
$880$			379.907i			0.431712i
$881$	−976.880			−1.10883			−0.554416	−	0.832240i	$-0.687058\pi$
$881$	−976.880			−1.10883			−0.554416	+	0.832240i	$0.687058\pi$
$882$		0			0
$883$			1095.89i			1.24109i	0.784169	+	0.620547i	$0.213089\pi$
$883$			1095.89i			1.24109i	−0.784169	+	0.620547i	$0.786911\pi$
$884$	173.383			0.196135
$885$		0			0
$886$	93.3052			0.105311
$887$	−43.3224			−0.0488414			−0.0244207	−	0.999702i	$-0.507774\pi$
$887$	−43.3224			−0.0488414			−0.0244207	+	0.999702i	$0.507774\pi$
$888$		0			0
$889$	792.270			0.891192
$890$			1093.74i			1.22892i
$891$		0			0
$892$		−	687.878i		−	0.771164i
$893$		−	446.933i		−	0.500485i
$894$		0			0
$895$	−736.469			−0.822871
$896$			65.4023i			0.0729937i
$897$		0			0
$898$			884.672i			0.985158i
$899$	−978.620			−1.08856
$900$		0			0
$901$		−	706.774i		−	0.784433i
$902$			235.585i			0.261181i
$903$		0			0
$904$	−299.373	+	111.928i	−0.331165	+	0.123814i
$905$		−	1601.32i		−	1.76941i
$906$		0			0
$907$		−	263.495i		−	0.290512i	−0.989394	−	0.145256i	$-0.953599\pi$
$907$		−	263.495i		−	0.290512i	0.989394	−	0.145256i	$-0.0464006\pi$
$908$			83.8331i			0.0923272i
$909$		0			0
$910$		−	239.848i		−	0.263569i
$911$		−	501.664i		−	0.550674i	−0.961348	−	0.275337i	$-0.911211\pi$
$911$		−	501.664i		−	0.550674i	0.961348	−	0.275337i	$-0.0887894\pi$
$912$		0			0
$913$	−53.6421			−0.0587537
$914$	−504.711			−0.552200
$915$		0			0
$916$			438.165i			0.478346i
$917$			494.971i			0.539773i
$918$		0			0
$919$	494.486			0.538069			0.269035	−	0.963130i	$-0.413295\pi$
$919$	494.486			0.538069			0.269035	+	0.963130i	$0.413295\pi$
$920$			175.761i			0.191044i
$921$		0			0
$922$	−1059.44			−1.14907
$923$	−622.338			−0.674255
$924$		0			0
$925$			30.1212i			0.0325634i
$926$			70.2449i			0.0758584i
$927$		0			0
$928$		−	203.046i		−	0.218799i
$929$			1794.56i			1.93171i	0.259083	+	0.965855i	$0.416580\pi$
$929$			1794.56i			1.93171i	−0.259083	+	0.965855i	$0.583420\pi$
$930$		0			0
$931$			330.815i			0.355333i
$932$			73.9843i			0.0793823i
$933$		0			0
$934$	51.6366			0.0552854
$935$		−	1371.67i		−	1.46703i
$936$		0			0
$937$		−	929.616i		−	0.992119i	−0.868288	−	0.496060i	$-0.834780\pi$
$937$		−	929.616i		−	0.992119i	0.868288	−	0.496060i	$-0.165220\pi$
$938$	−69.1645			−0.0737361
$939$		0			0
$940$	−205.782			−0.218917
$941$	931.115			0.989495			0.494747	−	0.869037i	$-0.335261\pi$
$941$	931.115			0.989495			0.494747	+	0.869037i	$0.335261\pi$
$942$		0			0
$943$			108.991i			0.115579i
$944$	21.7279			0.0230168
$945$		0			0
$946$		−	185.560i		−	0.196153i
$947$	−1258.74			−1.32919			−0.664593	−	0.747205i	$-0.731395\pi$
$947$	−1258.74			−1.32919			−0.664593	+	0.747205i	$0.731395\pi$
$948$		0			0
$949$		−	66.8215i		−	0.0704126i
$950$	−33.3910			−0.0351485
$951$		0			0
$952$		−	236.138i		−	0.248044i
$953$			623.935i			0.654707i	0.944902	+	0.327353i	$0.106157\pi$
$953$			623.935i			0.654707i	−0.944902	+	0.327353i	$0.893843\pi$
$954$		0			0
$955$	721.709			0.755716
$956$			726.112i			0.759532i
$957$		0			0
$958$		−	430.107i		−	0.448964i
$959$	1266.32			1.32045
$960$		0			0
$961$	−217.656			−0.226489
$962$	229.916			0.238998
$963$		0			0
$964$	−221.875			−0.230161
$965$			65.0046i			0.0673623i
$966$		0			0
$967$	−511.263			−0.528711			−0.264355	−	0.964425i	$-0.585159\pi$
$967$	−511.263			−0.528711			−0.264355	+	0.964425i	$0.585159\pi$
$968$			725.836i			0.749830i
$969$		0			0
$970$		−	1109.99i		−	1.14432i
$971$	1546.73			1.59292			0.796460	−	0.604691i	$-0.206703\pi$
$971$	1546.73			1.59292			0.796460	+	0.604691i	$0.206703\pi$
$972$		0			0
$973$	−18.1247			−0.0186276
$974$	−282.744			−0.290292
$975$		0			0
$976$	−67.5744			−0.0692360
$977$	5.23144			0.00535459			0.00267730	−	0.999996i	$-0.499148\pi$
$977$	5.23144			0.00535459			0.00267730	+	0.999996i	$0.499148\pi$
$978$		0			0
$979$			3074.96i			3.14092i
$980$	152.317			0.155426
$981$		0			0
$982$			935.797i			0.952950i
$983$	−872.031			−0.887112			−0.443556	−	0.896247i	$-0.646283\pi$
$983$	−872.031			−0.887112			−0.443556	+	0.896247i	$0.646283\pi$
$984$		0			0
$985$	−811.010			−0.823360
$986$			733.106i			0.743515i
$987$		0			0
$988$			254.875i			0.257971i
$989$		−	85.8479i		−	0.0868027i
$990$		0			0
$991$	162.061			0.163533			0.0817664	−	0.996652i	$-0.473944\pi$
$991$	162.061			0.163533			0.0817664	+	0.996652i	$0.473944\pi$
$992$			154.230i			0.155474i
$993$		0			0
$994$			847.590i			0.852706i
$995$			888.600i			0.893065i
$996$		0			0
$997$			288.619i			0.289487i	0.989469	+	0.144744i	$0.0462357\pi$
$997$			288.619i			0.289487i	−0.989469	+	0.144744i	$0.953764\pi$
$998$	−696.511			−0.697907
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2034.3.d.a.2033.2	yes	36
3.2	odd	2	inner	2034.3.d.a.2033.35	yes	36
113.112	even	2	inner	2034.3.d.a.2033.36	yes	36
339.338	odd	2	inner	2034.3.d.a.2033.1	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2034.3.d.a.2033.1	✓	36	339.338	odd	2	inner
2034.3.d.a.2033.2	yes	36	1.1	even	1	trivial
2034.3.d.a.2033.35	yes	36	3.2	odd	2	inner
2034.3.d.a.2033.36	yes	36	113.112	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:	\( N \)	\(=\)	\( 2034 = 2 \cdot 3^{2} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2034.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +4.88752 q^{5} -5.78081 q^{7} -2.82843i q^{8} +6.91200i q^{10} +19.4325i q^{11} +6.00266 q^{13} -8.17529i q^{14} +4.00000 q^{16} -14.4422 q^{17} -21.2302i q^{19} -9.77504 q^{20} -27.4817 q^{22} -12.7142 q^{23} -1.11214 q^{25} +8.48905i q^{26} +11.5616 q^{28} -35.8938 q^{29} +27.2643 q^{31} +5.65685i q^{32} -20.4243i q^{34} -28.2538 q^{35} -27.0839i q^{37} +30.0240 q^{38} -13.8240i q^{40} -8.57245i q^{41} +6.75215i q^{43} -38.8650i q^{44} -17.9805i q^{46} +21.0518 q^{47} -15.5823 q^{49} -1.57281i q^{50} -12.0053 q^{52} +48.9382i q^{53} +94.9767i q^{55} +16.3506i q^{56} -50.7614i q^{58} +5.43196 q^{59} -16.8936 q^{61} +38.5576i q^{62} -8.00000 q^{64} +29.3381 q^{65} -8.46018i q^{67} +28.8843 q^{68} -39.9569i q^{70} -103.677 q^{71} -11.1320i q^{73} +38.3024 q^{74} +42.4604i q^{76} -112.335i q^{77} -65.6916i q^{79} +19.5501 q^{80} +12.1233 q^{82} +2.76043i q^{83} -70.5864 q^{85} -9.54898 q^{86} +54.9634 q^{88} +158.238 q^{89} -34.7002 q^{91} +25.4283 q^{92} +29.7717i q^{94} -103.763i q^{95} -160.589 q^{97} -22.0367i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} - 8 q^{7} + 144 q^{16} + 36 q^{25} + 16 q^{28} + 64 q^{31} + 108 q^{49} - 80 q^{61} - 288 q^{64} + 248 q^{82} - 164 q^{85} + 516 q^{91} - 420 q^{97}+O(q^{100}) \) 36 * q - 72 * q^4 - 8 * q^7 + 144 * q^16 + 36 * q^25 + 16 * q^28 + 64 * q^31 + 108 * q^49 - 80 * q^61 - 288 * q^64 + 248 * q^82 - 164 * q^85 + 516 * q^91 - 420 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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