LMFDB - Embedded newform 495.3.g.a.89.8

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			89.8
Character	$\chi$	$=$	495.89
Dual form			495.3.g.a.89.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.54676 q^{2} +2.48600 q^{4} +(4.87674 - 1.10334i) q^{5} -0.521381i q^{7} +3.85580 q^{8} +(-12.4199 + 2.80995i) q^{10} +3.31662i q^{11} +19.8675i q^{13} +1.32783i q^{14} -19.7638 q^{16} +29.1566 q^{17} -11.6374 q^{19} +(12.1236 - 2.74291i) q^{20} -8.44666i q^{22} -10.0456 q^{23} +(22.5653 - 10.7614i) q^{25} -50.5978i q^{26} -1.29615i q^{28} -17.0117i q^{29} -44.1549 q^{31} +34.9105 q^{32} -74.2550 q^{34} +(-0.575262 - 2.54264i) q^{35} +68.5012i q^{37} +29.6378 q^{38} +(18.8037 - 4.25426i) q^{40} +44.3605i q^{41} -66.6852i q^{43} +8.24513i q^{44} +25.5838 q^{46} +43.6257 q^{47} +48.7282 q^{49} +(-57.4684 + 27.4068i) q^{50} +49.3906i q^{52} -33.4856 q^{53} +(3.65937 + 16.1743i) q^{55} -2.01034i q^{56} +43.3246i q^{58} +43.1043i q^{59} +62.2008 q^{61} +112.452 q^{62} -9.85363 q^{64} +(21.9207 + 96.8888i) q^{65} +115.863i q^{67} +72.4834 q^{68} +(1.46506 + 6.47551i) q^{70} +11.7801i q^{71} +31.0848i q^{73} -174.456i q^{74} -28.9307 q^{76} +1.72923 q^{77} +117.132 q^{79} +(-96.3830 + 21.8062i) q^{80} -112.976i q^{82} +48.5317 q^{83} +(142.189 - 32.1697i) q^{85} +169.831i q^{86} +12.7882i q^{88} +127.200i q^{89} +10.3585 q^{91} -24.9734 q^{92} -111.104 q^{94} +(-56.7528 + 12.8401i) q^{95} +106.836i q^{97} -124.099 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.54676			−1.27338			−0.636691	−	0.771119i	$-0.719697\pi$
$2$	−2.54676			−1.27338			−0.636691	+	0.771119i	$0.719697\pi$
$3$		0			0
$3$		0			0
$4$	2.48600			0.621500
$5$	4.87674	−	1.10334i	0.975349	−	0.220668i
$5$	4.87674	−	1.10334i	0.975349	−	0.220668i
$6$		0			0
$7$		−	0.521381i		−	0.0744831i	−0.999306	−	0.0372415i	$-0.988143\pi$
$7$		−	0.521381i		−	0.0744831i	0.999306	−	0.0372415i	$-0.0118571\pi$
$8$	3.85580			0.481975
$9$		0			0
$10$	−12.4199	+	2.80995i	−1.24199	+	0.280995i
$11$			3.31662i			0.301511i
$11$			3.31662i			0.301511i
$12$		0			0
$13$			19.8675i			1.52827i	0.645056	+	0.764135i	$0.276834\pi$
$13$			19.8675i			1.52827i	−0.645056	+	0.764135i	$0.723166\pi$
$14$			1.32783i			0.0948453i
$15$		0			0
$16$	−19.7638			−1.23524
$17$	29.1566			1.71510			0.857548	−	0.514404i	$-0.171987\pi$
$17$	29.1566			1.71510			0.857548	+	0.514404i	$0.171987\pi$
$18$		0			0
$19$	−11.6374			−0.612497			−0.306248	−	0.951952i	$-0.599074\pi$
$19$	−11.6374			−0.612497			−0.306248	+	0.951952i	$0.599074\pi$
$20$	12.1236	−	2.74291i	0.606179	−	0.137145i
$21$		0			0
$22$		−	8.44666i		−	0.383939i
$23$	−10.0456			−0.436766			−0.218383	−	0.975863i	$-0.570078\pi$
$23$	−10.0456			−0.436766			−0.218383	+	0.975863i	$0.570078\pi$
$24$		0			0
$25$	22.5653	−	10.7614i	0.902611	−	0.430457i
$26$		−	50.5978i		−	1.94607i
$27$		0			0
$28$		−	1.29615i		−	0.0462912i
$29$		−	17.0117i		−	0.586609i	−0.956019	−	0.293304i	$-0.905245\pi$
$29$		−	17.0117i		−	0.586609i	0.956019	−	0.293304i	$-0.0947549\pi$
$30$		0			0
$31$	−44.1549			−1.42435			−0.712176	−	0.702001i	$-0.752290\pi$
$31$	−44.1549			−1.42435			−0.712176	+	0.702001i	$0.752290\pi$
$32$	34.9105			1.09095
$33$		0			0
$34$	−74.2550			−2.18397
$35$	−0.575262	−	2.54264i	−0.0164361	−	0.0726470i
$36$		0			0
$37$			68.5012i			1.85138i	0.378278	+	0.925692i	$0.376516\pi$
$37$			68.5012i			1.85138i	−0.378278	+	0.925692i	$0.623484\pi$
$38$	29.6378			0.779942
$39$		0			0
$40$	18.8037	−	4.25426i	0.470093	−	0.106357i
$41$			44.3605i			1.08196i	0.841034	+	0.540982i	$0.181947\pi$
$41$			44.3605i			1.08196i	−0.841034	+	0.540982i	$0.818053\pi$
$42$		0			0
$43$		−	66.6852i		−	1.55082i	−0.631459	−	0.775409i	$-0.717544\pi$
$43$		−	66.6852i		−	1.55082i	0.631459	−	0.775409i	$-0.282456\pi$
$44$			8.24513i			0.187389i
$45$		0			0
$46$	25.5838			0.556170
$47$	43.6257			0.928207			0.464104	−	0.885781i	$-0.346377\pi$
$47$	43.6257			0.928207			0.464104	+	0.885781i	$0.346377\pi$
$48$		0			0
$49$	48.7282			0.994452
$50$	−57.4684	+	27.4068i	−1.14937	+	0.548136i
$51$		0			0
$52$			49.3906i			0.949820i
$53$	−33.4856			−0.631804			−0.315902	−	0.948792i	$-0.602307\pi$
$53$	−33.4856			−0.631804			−0.315902	+	0.948792i	$0.602307\pi$
$54$		0			0
$55$	3.65937	+	16.1743i	0.0665340	+	0.294079i
$56$		−	2.01034i		−	0.0358989i
$57$		0			0
$58$			43.3246i			0.746977i
$59$			43.1043i			0.730581i	0.930894	+	0.365291i	$0.119030\pi$
$59$			43.1043i			0.730581i	−0.930894	+	0.365291i	$0.880970\pi$
$60$		0			0
$61$	62.2008			1.01968			0.509842	−	0.860268i	$-0.329704\pi$
$61$	62.2008			1.01968			0.509842	+	0.860268i	$0.329704\pi$
$62$	112.452			1.81374
$63$		0			0
$64$	−9.85363			−0.153963
$65$	21.9207	+	96.8888i	0.337241	+	1.49060i
$66$		0			0
$67$			115.863i			1.72930i	0.502373	+	0.864651i	$0.332460\pi$
$67$			115.863i			1.72930i	−0.502373	+	0.864651i	$0.667540\pi$
$68$	72.4834			1.06593
$69$		0			0
$70$	1.46506	+	6.47551i	0.0209294	+	0.0925073i
$71$			11.7801i			0.165918i	0.996553	+	0.0829588i	$0.0264370\pi$
$71$			11.7801i			0.165918i	−0.996553	+	0.0829588i	$0.973563\pi$
$72$		0			0
$73$			31.0848i			0.425820i	0.977072	+	0.212910i	$0.0682940\pi$
$73$			31.0848i			0.425820i	−0.977072	+	0.212910i	$0.931706\pi$
$74$		−	174.456i		−	2.35752i
$75$		0			0
$76$	−28.9307			−0.380667
$77$	1.72923			0.0224575
$78$		0			0
$79$	117.132			1.48269			0.741343	−	0.671126i	$-0.234189\pi$
$79$	117.132			1.48269			0.741343	+	0.671126i	$0.234189\pi$
$80$	−96.3830	+	21.8062i	−1.20479	+	0.272578i
$81$		0			0
$82$		−	112.976i		−	1.37775i
$83$	48.5317			0.584720			0.292360	−	0.956308i	$-0.405560\pi$
$83$	48.5317			0.584720			0.292360	+	0.956308i	$0.405560\pi$
$84$		0			0
$85$	142.189	−	32.1697i	1.67282	−	0.378468i
$86$			169.831i			1.97478i
$87$		0			0
$88$			12.7882i			0.145321i
$89$			127.200i			1.42921i	0.699526	+	0.714607i	$0.253394\pi$
$89$			127.200i			1.42921i	−0.699526	+	0.714607i	$0.746606\pi$
$90$		0			0
$91$	10.3585			0.113830
$92$	−24.9734			−0.271450
$93$		0			0
$94$	−111.104			−1.18196
$95$	−56.7528	+	12.8401i	−0.597398	+	0.135159i
$96$		0			0
$97$			106.836i			1.10140i	0.834703	+	0.550700i	$0.185639\pi$
$97$			106.836i			1.10140i	−0.834703	+	0.550700i	$0.814361\pi$
$98$	−124.099			−1.26632
$99$		0			0
$100$	56.0973	−	26.7529i	0.560973	−	0.267529i
$101$		−	155.874i		−	1.54331i	−0.636042	−	0.771654i	$-0.719430\pi$
$101$		−	155.874i		−	1.54331i	0.636042	−	0.771654i	$-0.280570\pi$
$102$		0			0
$103$		−	24.8424i		−	0.241189i	−0.992702	−	0.120594i	$-0.961520\pi$
$103$		−	24.8424i		−	0.241189i	0.992702	−	0.120594i	$-0.0384800\pi$
$104$			76.6051i			0.736587i
$105$		0			0
$106$	85.2799			0.804527
$107$	148.451			1.38740			0.693698	−	0.720266i	$-0.255980\pi$
$107$	148.451			1.38740			0.693698	+	0.720266i	$0.255980\pi$
$108$		0			0
$109$	−74.1845			−0.680592			−0.340296	−	0.940318i	$-0.610527\pi$
$109$	−74.1845			−0.680592			−0.340296	+	0.940318i	$0.610527\pi$
$110$	−9.31955	−	41.1922i	−0.0847232	−	0.374474i
$111$		0			0
$112$			10.3045i			0.0920043i
$113$	−13.5324			−0.119756			−0.0598778	−	0.998206i	$-0.519071\pi$
$113$	−13.5324			−0.119756			−0.0598778	+	0.998206i	$0.519071\pi$
$114$		0			0
$115$	−48.9899	+	11.0838i	−0.425999	+	0.0963805i
$116$		−	42.2910i		−	0.364577i
$117$		0			0
$118$		−	109.776i		−	0.930308i
$119$		−	15.2017i		−	0.127746i
$120$		0			0
$121$	−11.0000			−0.0909091
$122$	−158.411			−1.29845
$123$		0			0
$124$	−109.769			−0.885236
$125$	98.1715	−	77.3780i	0.785372	−	0.619024i
$126$		0			0
$127$			10.5789i			0.0832981i	0.999132	+	0.0416491i	$0.0132611\pi$
$127$			10.5789i			0.0832981i	−0.999132	+	0.0416491i	$0.986739\pi$
$128$	−114.547			−0.894901
$129$		0			0
$130$	−55.8267	−	246.753i	−0.429436	−	1.89810i
$131$			153.818i			1.17418i	0.809520	+	0.587092i	$0.199727\pi$
$131$			153.818i			1.17418i	−0.809520	+	0.587092i	$0.800273\pi$
$132$		0			0
$133$			6.06755i			0.0456206i
$134$		−	295.076i		−	2.20206i
$135$		0			0
$136$	112.422			0.826633
$137$	100.618			0.734440			0.367220	−	0.930134i	$-0.380310\pi$
$137$	100.618			0.734440			0.367220	+	0.930134i	$0.380310\pi$
$138$		0			0
$139$	−117.599			−0.846039			−0.423020	−	0.906121i	$-0.639030\pi$
$139$	−117.599			−0.846039			−0.423020	+	0.906121i	$0.639030\pi$
$140$	−1.43010	−	6.32101i	−0.0102150	−	0.0451501i
$141$		0			0
$142$		−	30.0012i		−	0.211276i
$143$	−65.8931			−0.460791
$144$		0			0
$145$	−18.7697	−	82.9615i	−0.129446	−	0.572148i
$146$		−	79.1657i		−	0.542231i
$147$		0			0
$148$			170.294i			1.15064i
$149$		−	145.756i		−	0.978226i	−0.872221	−	0.489113i	$-0.837321\pi$
$149$		−	145.756i		−	0.978226i	0.872221	−	0.489113i	$-0.162679\pi$
$150$		0			0
$151$	−32.1881			−0.213166			−0.106583	−	0.994304i	$-0.533991\pi$
$151$	−32.1881			−0.213166			−0.106583	+	0.994304i	$0.533991\pi$
$152$	−44.8716			−0.295208
$153$		0			0
$154$	−4.40393			−0.0285969
$155$	−215.332	+	48.7180i	−1.38924	+	0.314310i
$156$		0			0
$157$		−	66.1924i		−	0.421607i	−0.977528	−	0.210804i	$-0.932392\pi$
$157$		−	66.1924i		−	0.421607i	0.977528	−	0.210804i	$-0.0676081\pi$
$158$	−298.308			−1.88802
$159$		0			0
$160$	170.250	−	38.5183i	1.06406	−	0.240739i
$161$			5.23760i			0.0325317i
$162$		0			0
$163$		−	213.474i		−	1.30966i	−0.755776	−	0.654830i	$-0.772740\pi$
$163$		−	213.474i		−	1.30966i	0.755776	−	0.654830i	$-0.227260\pi$
$164$			110.280i			0.672440i
$165$		0			0
$166$	−123.599			−0.744571
$167$	185.973			1.11361			0.556805	−	0.830643i	$-0.312027\pi$
$167$	185.973			1.11361			0.556805	+	0.830643i	$0.312027\pi$
$168$		0			0
$169$	−225.718			−1.33561
$170$	−362.123	+	81.9287i	−2.13013	+	0.481934i
$171$		0			0
$172$		−	165.779i		−	0.963834i
$173$	−117.325			−0.678180			−0.339090	−	0.940754i	$-0.610119\pi$
$173$	−117.325			−0.678180			−0.339090	+	0.940754i	$0.610119\pi$
$174$		0			0
$175$	−5.61081	−	11.7651i	−0.0320618	−	0.0672292i
$176$		−	65.5491i		−	0.372438i
$177$		0			0
$178$		−	323.948i		−	1.81993i
$179$		−	336.458i		−	1.87966i	−0.341648	−	0.939828i	$-0.610985\pi$
$179$		−	336.458i		−	1.87966i	0.341648	−	0.939828i	$-0.389015\pi$
$180$		0			0
$181$	3.87423			0.0214046			0.0107023	−	0.999943i	$-0.496593\pi$
$181$	3.87423			0.0214046			0.0107023	+	0.999943i	$0.496593\pi$
$182$	−26.3808			−0.144949
$183$		0			0
$184$	−38.7339			−0.210510
$185$	75.5803	+	334.063i	0.408542	+	1.80575i
$186$		0			0
$187$			96.7016i			0.517121i
$188$	108.454			0.576881
$189$		0			0
$190$	144.536	−	32.7006i	0.760716	−	0.172109i
$191$		−	114.301i		−	0.598434i	−0.954185	−	0.299217i	$-0.903275\pi$
$191$		−	114.301i		−	0.598434i	0.954185	−	0.299217i	$-0.0967254\pi$
$192$		0			0
$193$			221.972i			1.15011i	0.818114	+	0.575057i	$0.195020\pi$
$193$			221.972i			1.15011i	−0.818114	+	0.575057i	$0.804980\pi$
$194$		−	272.085i		−	1.40250i
$195$		0			0
$196$	121.138			0.618052
$197$	71.3783			0.362326			0.181163	−	0.983453i	$-0.442014\pi$
$197$	71.3783			0.362326			0.181163	+	0.983453i	$0.442014\pi$
$198$		0			0
$199$	−284.564			−1.42997			−0.714985	−	0.699139i	$-0.753567\pi$
$199$	−284.564			−1.42997			−0.714985	+	0.699139i	$0.753567\pi$
$200$	87.0071	−	41.4939i	0.435036	−	0.207470i
$201$		0			0
$202$			396.974i			1.96522i
$203$	−8.86956			−0.0436924
$204$		0			0
$205$	48.9448	+	216.335i	0.238755	+	1.05529i
$206$			63.2678i			0.307125i
$207$		0			0
$208$		−	392.658i		−	1.88778i
$209$		−	38.5970i		−	0.184675i
$210$		0			0
$211$	384.597			1.82274			0.911368	−	0.411593i	$-0.135027\pi$
$211$	384.597			1.82274			0.911368	+	0.411593i	$0.135027\pi$
$212$	−83.2452			−0.392666
$213$		0			0
$214$	−378.070			−1.76668
$215$	−73.5766	−	325.207i	−0.342217	−	1.51259i
$216$		0			0
$217$			23.0216i			0.106090i
$218$	188.930			0.866653
$219$		0			0
$220$	9.09720	+	40.2094i	0.0413509	+	0.182770i
$221$			579.270i			2.62113i
$222$		0			0
$223$			94.4995i			0.423764i	0.977295	+	0.211882i	$0.0679593\pi$
$223$			94.4995i			0.423764i	−0.977295	+	0.211882i	$0.932041\pi$
$224$		−	18.2017i		−	0.0812576i
$225$		0			0
$226$	34.4638			0.152495
$227$	15.6432			0.0689128			0.0344564	−	0.999406i	$-0.489030\pi$
$227$	15.6432			0.0689128			0.0344564	+	0.999406i	$0.489030\pi$
$228$		0			0
$229$	−149.818			−0.654228			−0.327114	−	0.944985i	$-0.606076\pi$
$229$	−149.818			−0.654228			−0.327114	+	0.944985i	$0.606076\pi$
$230$	124.766	−	28.2277i	0.542460	−	0.122729i
$231$		0			0
$232$		−	65.5935i		−	0.282731i
$233$	136.083			0.584049			0.292024	−	0.956411i	$-0.405671\pi$
$233$	136.083			0.584049			0.292024	+	0.956411i	$0.405671\pi$
$234$		0			0
$235$	212.752	−	48.1341i	0.905326	−	0.204826i
$236$			107.157i			0.454056i
$237$		0			0
$238$			38.7152i			0.162669i
$239$			193.328i			0.808903i	0.914559	+	0.404452i	$0.132538\pi$
$239$			193.328i			0.808903i	−0.914559	+	0.404452i	$0.867462\pi$
$240$		0			0
$241$	−59.0015			−0.244820			−0.122410	−	0.992480i	$-0.539062\pi$
$241$	−59.0015			−0.244820			−0.122410	+	0.992480i	$0.539062\pi$
$242$	28.0144			0.115762
$243$		0			0
$244$	154.631			0.633734
$245$	237.635	−	53.7638i	0.969938	−	0.219444i
$246$		0			0
$247$		−	231.207i		−	0.936061i
$248$	−170.252			−0.686502
$249$		0			0
$250$	−250.020	+	197.063i	−1.00008	+	0.788253i
$251$		−	237.143i		−	0.944792i	−0.881386	−	0.472396i	$-0.843389\pi$
$251$		−	237.143i		−	0.944792i	0.881386	−	0.472396i	$-0.156611\pi$
$252$		0			0
$253$		−	33.3176i		−	0.131690i
$254$		−	26.9419i		−	0.106070i
$255$		0			0
$256$	331.139			1.29351
$257$	−168.744			−0.656591			−0.328296	−	0.944575i	$-0.606474\pi$
$257$	−168.744			−0.656591			−0.328296	+	0.944575i	$0.606474\pi$
$258$		0			0
$259$	35.7153			0.137897
$260$	54.4948	+	240.866i	0.209595	+	0.926406i
$261$		0			0
$262$		−	391.739i		−	1.49519i
$263$	28.5331			0.108491			0.0542454	−	0.998528i	$-0.482725\pi$
$263$	28.5331			0.108491			0.0542454	+	0.998528i	$0.482725\pi$
$264$		0			0
$265$	−163.301	+	36.9461i	−0.616229	+	0.139419i
$266$		−	15.4526i		−	0.0580925i
$267$		0			0
$268$			288.036i			1.07476i
$269$			17.4611i			0.0649113i	0.999473	+	0.0324557i	$0.0103328\pi$
$269$			17.4611i			0.0649113i	−0.999473	+	0.0324557i	$0.989667\pi$
$270$		0			0
$271$	−198.778			−0.733499			−0.366750	−	0.930320i	$-0.619529\pi$
$271$	−198.778			−0.733499			−0.366750	+	0.930320i	$0.619529\pi$
$272$	−576.246			−2.11855
$273$		0			0
$274$	−256.251			−0.935222
$275$	35.6916	+	74.8405i	0.129788	+	0.272147i
$276$		0			0
$277$			131.609i			0.475123i	0.971372	+	0.237562i	$0.0763482\pi$
$277$			131.609i			0.475123i	−0.971372	+	0.237562i	$0.923652\pi$
$278$	299.498			1.07733
$279$		0			0
$280$	−2.21809	−	9.80392i	−0.00792176	−	0.0350140i
$281$		−	88.5256i		−	0.315038i	−0.987516	−	0.157519i	$-0.949650\pi$
$281$		−	88.5256i		−	0.315038i	0.987516	−	0.157519i	$-0.0503495\pi$
$282$		0			0
$283$		−	158.497i		−	0.560059i	−0.959991	−	0.280030i	$-0.909656\pi$
$283$		−	158.497i		−	0.560059i	0.959991	−	0.280030i	$-0.0903443\pi$
$284$			29.2854i			0.103118i
$285$		0			0
$286$	167.814			0.586762
$287$	23.1287			0.0805879
$288$		0			0
$289$	561.109			1.94155
$290$	47.8019	+	211.283i	0.164834	+	0.728563i
$291$		0			0
$292$			77.2769i			0.264647i
$293$	211.430			0.721606			0.360803	−	0.932642i	$-0.382503\pi$
$293$	211.430			0.721606			0.360803	+	0.932642i	$0.382503\pi$
$294$		0			0
$295$	47.5588	+	210.209i	0.161216	+	0.712571i
$296$			264.127i			0.892320i
$297$		0			0
$298$			371.205i			1.24565i
$299$		−	199.581i		−	0.667497i
$300$		0			0
$301$	−34.7684			−0.115510
$302$	81.9755			0.271442
$303$		0			0
$304$	230.000			0.756579
$305$	303.337	−	68.6287i	0.994548	−	0.225012i
$306$		0			0
$307$			134.001i			0.436484i	0.975895	+	0.218242i	$0.0700322\pi$
$307$			134.001i			0.436484i	−0.975895	+	0.218242i	$0.929968\pi$
$308$	4.29886			0.0139573
$309$		0			0
$310$	548.400	−	124.073i	1.76903	−	0.400236i
$311$		−	142.049i		−	0.456749i	−0.973573	−	0.228374i	$-0.926659\pi$
$311$		−	142.049i		−	0.456749i	0.973573	−	0.228374i	$-0.0733410\pi$
$312$		0			0
$313$			230.891i			0.737671i	0.929495	+	0.368836i	$0.120243\pi$
$313$			230.891i			0.737671i	−0.929495	+	0.368836i	$0.879757\pi$
$314$			168.576i			0.536867i
$315$		0			0
$316$	291.191			0.921489
$317$	249.250			0.786278			0.393139	−	0.919479i	$-0.371389\pi$
$317$	249.250			0.786278			0.393139	+	0.919479i	$0.371389\pi$
$318$		0			0
$319$	56.4213			0.176869
$320$	−48.0536	+	10.8719i	−0.150168	+	0.0339748i
$321$		0			0
$322$		−	13.3389i		−	0.0414252i
$323$	−339.309			−1.05049
$324$		0			0
$325$	213.803	+	448.316i	0.657855	+	1.37943i
$326$			543.669i			1.66770i
$327$		0			0
$328$			171.045i			0.521479i
$329$		−	22.7456i		−	0.0691357i
$330$		0			0
$331$	−436.814			−1.31968			−0.659839	−	0.751407i	$-0.729376\pi$
$331$	−436.814			−1.31968			−0.659839	+	0.751407i	$0.729376\pi$
$332$	120.650			0.363403
$333$		0			0
$334$	−473.629			−1.41805
$335$	127.837	+	565.035i	0.381602	+	1.68667i
$336$		0			0
$337$		−	467.424i		−	1.38701i	−0.720450	−	0.693507i	$-0.756065\pi$
$337$		−	467.424i		−	1.38701i	0.720450	−	0.693507i	$-0.243935\pi$
$338$	574.850			1.70074
$339$		0			0
$340$	353.483	−	79.9740i	1.03966	−	0.235218i
$341$		−	146.445i		−	0.429459i
$342$		0			0
$343$		−	50.9536i		−	0.148553i
$344$		−	257.125i		−	0.747455i
$345$		0			0
$346$	298.799			0.863582
$347$	−237.902			−0.685596			−0.342798	−	0.939409i	$-0.611375\pi$
$347$	−237.902			−0.685596			−0.342798	+	0.939409i	$0.611375\pi$
$348$		0			0
$349$	125.577			0.359820			0.179910	−	0.983683i	$-0.442419\pi$
$349$	125.577			0.359820			0.179910	+	0.983683i	$0.442419\pi$
$350$	14.2894	+	29.9630i	0.0408269	+	0.0856084i
$351$		0			0
$352$			115.785i			0.328935i
$353$	−242.501			−0.686973			−0.343486	−	0.939158i	$-0.611608\pi$
$353$	−242.501			−0.686973			−0.343486	+	0.939158i	$0.611608\pi$
$354$		0			0
$355$	12.9975	+	57.4487i	0.0366128	+	0.161827i
$356$			316.219i			0.888257i
$357$		0			0
$358$			856.880i			2.39352i
$359$			475.810i			1.32538i	0.748895	+	0.662689i	$0.230585\pi$
$359$			475.810i			1.32538i	−0.748895	+	0.662689i	$0.769415\pi$
$360$		0			0
$361$	−225.570			−0.624847
$362$	−9.86676			−0.0272562
$363$		0			0
$364$	25.7514			0.0707455
$365$	34.2972	+	151.593i	0.0939649	+	0.415323i
$366$		0			0
$367$			508.971i			1.38684i	0.720533	+	0.693420i	$0.243897\pi$
$367$			508.971i			1.38684i	−0.720533	+	0.693420i	$0.756103\pi$
$368$	198.540			0.539510
$369$		0			0
$370$	−192.485	−	850.779i	−0.520230	−	2.29940i
$371$			17.4588i			0.0470587i
$372$		0			0
$373$		−	31.7821i		−	0.0852068i	−0.999092	−	0.0426034i	$-0.986435\pi$
$373$		−	31.7821i		−	0.0852068i	0.999092	−	0.0426034i	$-0.0135652\pi$
$374$		−	246.276i		−	0.658492i
$375$		0			0
$376$	168.212			0.447372
$377$	337.979			0.896496
$378$		0			0
$379$	−603.519			−1.59240			−0.796199	−	0.605035i	$-0.793159\pi$
$379$	−603.519			−1.59240			−0.796199	+	0.605035i	$0.793159\pi$
$380$	−141.088	+	31.9204i	−0.371283	+	0.0840012i
$381$		0			0
$382$			291.097i			0.762034i
$383$	−151.313			−0.395074			−0.197537	−	0.980295i	$-0.563294\pi$
$383$	−151.313			−0.395074			−0.197537	+	0.980295i	$0.563294\pi$
$384$		0			0
$385$	8.43300	−	1.90793i	0.0219039	−	0.00495566i
$386$		−	565.310i		−	1.46453i
$387$		0			0
$388$			265.594i			0.684520i
$389$		−	219.639i		−	0.564625i	−0.959323	−	0.282312i	$-0.908899\pi$
$389$		−	219.639i		−	0.564625i	0.959323	−	0.282312i	$-0.0911015\pi$
$390$		0			0
$391$	−292.897			−0.749096
$392$	187.886			0.479301
$393$		0			0
$394$	−181.784			−0.461380
$395$	571.224	−	129.237i	1.44614	−	0.327182i
$396$		0			0
$397$		−	532.971i		−	1.34250i	−0.741233	−	0.671248i	$-0.765758\pi$
$397$		−	532.971i		−	1.34250i	0.741233	−	0.671248i	$-0.234242\pi$
$398$	724.718			1.82090
$399$		0			0
$400$	−445.976	+	212.687i	−1.11494	+	0.531717i
$401$		−	349.126i		−	0.870639i	−0.900276	−	0.435319i	$-0.856635\pi$
$401$		−	349.126i		−	0.870639i	0.900276	−	0.435319i	$-0.143365\pi$
$402$		0			0
$403$		−	877.249i		−	2.17680i
$404$		−	387.503i		−	0.959166i
$405$		0			0
$406$	22.5887			0.0556371
$407$	−227.193			−0.558213
$408$		0			0
$409$	−668.482			−1.63443			−0.817215	−	0.576333i	$-0.804483\pi$
$409$	−668.482			−1.63443			−0.817215	+	0.576333i	$0.804483\pi$
$410$	−124.651	−	550.953i	−0.304026	−	1.34379i
$411$		0			0
$412$		−	61.7583i		−	0.149899i
$413$	22.4738			0.0544159
$414$		0			0
$415$	236.677	−	53.5471i	0.570306	−	0.129029i
$416$			693.585i			1.66727i
$417$		0			0
$418$			98.2975i			0.235161i
$419$			316.922i			0.756377i	0.925729	+	0.378188i	$0.123453\pi$
$419$			316.922i			0.756377i	−0.925729	+	0.378188i	$0.876547\pi$
$420$		0			0
$421$	−138.858			−0.329830			−0.164915	−	0.986308i	$-0.552735\pi$
$421$	−138.858			−0.329830			−0.164915	+	0.986308i	$0.552735\pi$
$422$	−979.478			−2.32104
$423$		0			0
$424$	−129.114			−0.304513
$425$	657.927	−	313.767i	1.54806	−	0.738276i
$426$		0			0
$427$		−	32.4303i		−	0.0759492i
$428$	369.050			0.862266
$429$		0			0
$430$	187.382	+	828.224i	0.435772	+	1.92610i
$431$		−	632.454i		−	1.46741i	−0.679468	−	0.733705i	$-0.737789\pi$
$431$		−	632.454i		−	1.46741i	0.679468	−	0.733705i	$-0.262211\pi$
$432$		0			0
$433$			593.067i			1.36967i	0.728699	+	0.684834i	$0.240125\pi$
$433$			593.067i			1.36967i	−0.728699	+	0.684834i	$0.759875\pi$
$434$		−	58.6305i		−	0.135093i
$435$		0			0
$436$	−184.423			−0.422988
$437$	116.905			0.267518
$438$		0			0
$439$	32.1254			0.0731787			0.0365893	−	0.999330i	$-0.488351\pi$
$439$	32.1254			0.0731787			0.0365893	+	0.999330i	$0.488351\pi$
$440$	14.1098	+	62.3649i	0.0320677	+	0.141739i
$441$		0			0
$442$		−	1475.26i		−	3.33770i
$443$	−43.0520			−0.0971828			−0.0485914	−	0.998819i	$-0.515473\pi$
$443$	−43.0520			−0.0971828			−0.0485914	+	0.998819i	$0.515473\pi$
$444$		0			0
$445$	140.345	+	620.322i	0.315382	+	1.39398i
$446$		−	240.668i		−	0.539614i
$447$		0			0
$448$			5.13750i			0.0114676i
$449$		−	91.5811i		−	0.203967i	−0.994786	−	0.101983i	$-0.967481\pi$
$449$		−	91.5811i		−	0.203967i	0.994786	−	0.101983i	$-0.0325189\pi$
$450$		0			0
$451$	−147.127			−0.326224
$452$	−33.6415			−0.0744281
$453$		0			0
$454$	−39.8395			−0.0877523
$455$	50.5160	−	11.4290i	0.111024	−	0.0251187i
$456$		0			0
$457$			370.296i			0.810276i	0.914256	+	0.405138i	$0.132777\pi$
$457$			370.296i			0.810276i	−0.914256	+	0.405138i	$0.867223\pi$
$458$	381.551			0.833081
$459$		0			0
$460$	−121.789	+	27.5542i	−0.264759	+	0.0599005i
$461$		−	544.300i		−	1.18069i	−0.807149	−	0.590347i	$-0.798991\pi$
$461$		−	544.300i		−	1.18069i	0.807149	−	0.590347i	$-0.201009\pi$
$462$		0			0
$463$		−	252.721i		−	0.545833i	−0.962038	−	0.272916i	$-0.912012\pi$
$463$		−	252.721i		−	0.545833i	0.962038	−	0.272916i	$-0.0879883\pi$
$464$			336.215i			0.724601i
$465$		0			0
$466$	−346.572			−0.743717
$467$	−612.912			−1.31245			−0.656223	−	0.754567i	$-0.727847\pi$
$467$	−612.912			−1.31245			−0.656223	+	0.754567i	$0.727847\pi$
$468$		0			0
$469$	60.4089			0.128804
$470$	−541.828	+	122.586i	−1.15282	+	0.260822i
$471$		0			0
$472$			166.201i			0.352122i
$473$	221.170			0.467589
$474$		0			0
$475$	−262.602	+	125.236i	−0.552846	+	0.263654i
$476$		−	37.7915i		−	0.0793939i
$477$		0			0
$478$		−	492.360i		−	1.03004i
$479$		−	481.367i		−	1.00494i	−0.864594	−	0.502471i	$-0.832425\pi$
$479$		−	481.367i		−	1.00494i	0.864594	−	0.502471i	$-0.167575\pi$
$480$		0			0
$481$	−1360.95			−2.82941
$482$	150.263			0.311749
$483$		0			0
$484$	−27.3460			−0.0565000
$485$	117.876	+	521.011i	0.243044	+	1.07425i
$486$		0			0
$487$		−	836.741i		−	1.71815i	−0.511846	−	0.859077i	$-0.671038\pi$
$487$		−	836.741i		−	1.71815i	0.511846	−	0.859077i	$-0.328962\pi$
$488$	239.834			0.491462
$489$		0			0
$490$	−605.199	+	136.924i	−1.23510	+	0.279436i
$491$			118.116i			0.240562i	0.992740	+	0.120281i	$0.0383795\pi$
$491$			118.116i			0.240562i	−0.992740	+	0.120281i	$0.961620\pi$
$492$		0			0
$493$		−	496.003i		−	1.00609i
$494$			588.829i			1.19196i
$495$		0			0
$496$	872.669			1.75941
$497$	6.14195			0.0123580
$498$		0			0
$499$	806.088			1.61541			0.807704	−	0.589589i	$-0.200710\pi$
$499$	806.088			1.61541			0.807704	+	0.589589i	$0.200710\pi$
$500$	244.054	−	192.362i	0.488109	−	0.384723i
$501$		0			0
$502$			603.947i			1.20308i
$503$	−636.754			−1.26591			−0.632957	−	0.774187i	$-0.718159\pi$
$503$	−636.754			−1.26591			−0.632957	+	0.774187i	$0.718159\pi$
$504$		0			0
$505$	−171.982	−	760.158i	−0.340559	−	1.50526i
$506$			84.8519i			0.167692i
$507$		0			0
$508$			26.2991i			0.0517698i
$509$		−	172.214i		−	0.338339i	−0.985587	−	0.169169i	$-0.945892\pi$
$509$		−	172.214i		−	0.338339i	0.985587	−	0.169169i	$-0.0541085\pi$
$510$		0			0
$511$	16.2070			0.0317163
$512$	−385.144			−0.752234
$513$		0			0
$514$	429.751			0.836091
$515$	−27.4097	−	121.150i	−0.0532227	−	0.235243i
$516$		0			0
$517$			144.690i			0.279865i
$518$	−90.9583			−0.175595
$519$		0			0
$520$	84.5216	+	373.583i	0.162542	+	0.718430i
$521$		−	636.699i		−	1.22207i	−0.791603	−	0.611036i	$-0.790753\pi$
$521$		−	636.699i		−	1.22207i	0.791603	−	0.611036i	$-0.209247\pi$
$522$		0			0
$523$		−	470.003i		−	0.898668i	−0.893364	−	0.449334i	$-0.851661\pi$
$523$		−	470.003i		−	0.898668i	0.893364	−	0.449334i	$-0.148339\pi$
$524$			382.392i			0.729756i
$525$		0			0
$526$	−72.6670			−0.138150
$527$	−1287.41			−2.44290
$528$		0			0
$529$	−428.085			−0.809235
$530$	415.888	−	94.0929i	0.784695	−	0.177534i
$531$		0			0
$532$			15.0839i			0.0283532i
$533$	−881.333			−1.65353
$534$		0			0
$535$	723.959	−	163.793i	1.35319	−	0.306154i
$536$			446.745i			0.833479i
$537$		0			0
$538$		−	44.4694i		−	0.0826569i
$539$			161.613i			0.299839i
$540$		0			0
$541$	432.219			0.798927			0.399463	−	0.916749i	$-0.369196\pi$
$541$	432.219			0.798927			0.399463	+	0.916749i	$0.369196\pi$
$542$	506.241			0.934024
$543$		0			0
$544$	1017.87			1.87109
$545$	−361.779	+	81.8509i	−0.663815	+	0.150185i
$546$		0			0
$547$			134.676i			0.246208i	0.992394	+	0.123104i	$0.0392849\pi$
$547$			134.676i			0.246208i	−0.992394	+	0.123104i	$0.960715\pi$
$548$	250.137			0.456454
$549$		0			0
$550$	−90.8981	−	190.601i	−0.165269	−	0.346547i
$551$			197.972i			0.359296i
$552$		0			0
$553$		−	61.0705i		−	0.110435i
$554$		−	335.177i		−	0.605013i
$555$		0			0
$556$	−292.352			−0.525813
$557$	−762.780			−1.36944			−0.684722	−	0.728804i	$-0.740076\pi$
$557$	−762.780			−1.36944			−0.684722	+	0.728804i	$0.740076\pi$
$558$		0			0
$559$	1324.87			2.37007
$560$	11.3694	+	50.2523i	0.0203024	+	0.0897363i
$561$		0			0
$562$			225.454i			0.401163i
$563$	−514.186			−0.913296			−0.456648	−	0.889647i	$-0.650950\pi$
$563$	−514.186			−0.913296			−0.456648	+	0.889647i	$0.650950\pi$
$564$		0			0
$565$	−65.9940	+	14.9309i	−0.116804	+	0.0264263i
$566$			403.654i			0.713169i
$567$		0			0
$568$			45.4218i			0.0799680i
$569$		−	335.611i		−	0.589826i	−0.955524	−	0.294913i	$-0.904709\pi$
$569$		−	335.611i		−	0.589826i	0.955524	−	0.294913i	$-0.0952907\pi$
$570$		0			0
$571$	930.967			1.63042			0.815208	−	0.579169i	$-0.196623\pi$
$571$	930.967			1.63042			0.815208	+	0.579169i	$0.196623\pi$
$572$	−163.810			−0.286382
$573$		0			0
$574$	−58.9034			−0.102619
$575$	−226.682	+	108.105i	−0.394230	+	0.188009i
$576$		0			0
$577$		−	57.3089i		−	0.0993221i	−0.998766	−	0.0496611i	$-0.984186\pi$
$577$		−	57.3089i		−	0.0993221i	0.998766	−	0.0496611i	$-0.0158141\pi$
$578$	−1429.01			−2.47234
$579$		0			0
$580$	−46.6614	−	206.242i	−0.0804507	−	0.355590i
$581$		−	25.3035i		−	0.0435517i
$582$		0			0
$583$		−	111.059i		−	0.190496i
$584$			119.857i			0.205234i
$585$		0			0
$586$	−538.463			−0.918879
$587$	−287.044			−0.489002			−0.244501	−	0.969649i	$-0.578624\pi$
$587$	−287.044			−0.489002			−0.244501	+	0.969649i	$0.578624\pi$
$588$		0			0
$589$	513.851			0.872412
$590$	−121.121	−	535.351i	−0.205290	−	0.907375i
$591$		0			0
$592$		−	1353.84i		−	2.28690i
$593$	784.320			1.32263			0.661316	−	0.750108i	$-0.269998\pi$
$593$	784.320			1.32263			0.661316	+	0.750108i	$0.269998\pi$
$594$		0			0
$595$	−16.7727	−	74.1349i	−0.0281894	−	0.124597i
$596$		−	362.349i		−	0.607967i
$597$		0			0
$598$			508.287i			0.849978i
$599$		−	181.459i		−	0.302936i	−0.988462	−	0.151468i	$-0.951600\pi$
$599$		−	181.459i		−	0.302936i	0.988462	−	0.151468i	$-0.0484001\pi$
$600$		0			0
$601$	295.753			0.492101			0.246051	−	0.969257i	$-0.420867\pi$
$601$	295.753			0.492101			0.246051	+	0.969257i	$0.420867\pi$
$602$	88.5469			0.147088
$603$		0			0
$604$	−80.0197			−0.132483
$605$	−53.6442	+	12.1368i	−0.0886681	+	0.0200608i
$606$		0			0
$607$		−	851.913i		−	1.40348i	−0.712433	−	0.701740i	$-0.752407\pi$
$607$		−	851.913i		−	1.40348i	0.712433	−	0.701740i	$-0.247593\pi$
$608$	−406.269			−0.668206
$609$		0			0
$610$	−772.528	+	174.781i	−1.26644	+	0.286526i
$611$			866.735i			1.41855i
$612$		0			0
$613$		−	29.3560i		−	0.0478891i	−0.999713	−	0.0239446i	$-0.992377\pi$
$613$		−	29.3560i		−	0.0478891i	0.999713	−	0.0239446i	$-0.00762252\pi$
$614$		−	341.268i		−	0.555810i
$615$		0			0
$616$	6.66755			0.0108239
$617$	609.377			0.987645			0.493822	−	0.869563i	$-0.335599\pi$
$617$	609.377			0.987645			0.493822	+	0.869563i	$0.335599\pi$
$618$		0			0
$619$	−11.3024			−0.0182591			−0.00912955	−	0.999958i	$-0.502906\pi$
$619$	−11.3024			−0.0182591			−0.00912955	+	0.999958i	$0.502906\pi$
$620$	−535.316	+	121.113i	−0.863413	+	0.195344i
$621$		0			0
$622$			361.765i			0.581615i
$623$	66.3197			0.106452
$624$		0			0
$625$	393.383	−	485.669i	0.629413	−	0.777071i
$626$		−	588.025i		−	0.939337i
$627$		0			0
$628$		−	164.554i		−	0.262029i
$629$			1997.26i			3.17530i
$630$		0			0
$631$	181.951			0.288354			0.144177	−	0.989552i	$-0.453947\pi$
$631$	181.951			0.288354			0.144177	+	0.989552i	$0.453947\pi$
$632$	451.638			0.714617
$633$		0			0
$634$	−634.781			−1.00123
$635$	11.6721	+	51.5904i	0.0183813	+	0.0812448i
$636$		0			0
$637$			968.107i			1.51979i
$638$	−143.692			−0.225222
$639$		0			0
$640$	−558.618	+	126.385i	−0.872840	+	0.197476i
$641$			689.024i			1.07492i	0.843289	+	0.537460i	$0.180616\pi$
$641$			689.024i			1.07492i	−0.843289	+	0.537460i	$0.819384\pi$
$642$		0			0
$643$			973.695i			1.51430i	0.653241	+	0.757150i	$0.273409\pi$
$643$			973.695i			1.51430i	−0.653241	+	0.757150i	$0.726591\pi$
$644$			13.0207i			0.0202184i
$645$		0			0
$646$	864.139			1.33768
$647$	364.273			0.563019			0.281510	−	0.959558i	$-0.409165\pi$
$647$	364.273			0.563019			0.281510	+	0.959558i	$0.409165\pi$
$648$		0			0
$649$	−142.961			−0.220278
$650$	−544.505	−	1141.75i	−0.837700	−	1.75654i
$651$		0			0
$652$		−	530.698i		−	0.813954i
$653$	−261.834			−0.400971			−0.200485	−	0.979697i	$-0.564252\pi$
$653$	−261.834			−0.400971			−0.200485	+	0.979697i	$0.564252\pi$
$654$		0			0
$655$	169.714	+	750.132i	0.259106	+	1.14524i
$656$		−	876.732i		−	1.33648i
$657$		0			0
$658$			57.9278i			0.0880361i
$659$			725.538i			1.10097i	0.834846	+	0.550484i	$0.185557\pi$
$659$			725.538i			1.10097i	−0.834846	+	0.550484i	$0.814443\pi$
$660$		0			0
$661$	−155.237			−0.234852			−0.117426	−	0.993082i	$-0.537464\pi$
$661$	−155.237			−0.234852			−0.117426	+	0.993082i	$0.537464\pi$
$662$	1112.46			1.68045
$663$		0			0
$664$	187.128			0.281820
$665$	6.69458	+	29.5899i	0.0100670	+	0.0444960i
$666$		0			0
$667$			170.893i			0.256211i
$668$	462.329			0.692109
$669$		0			0
$670$	−325.570	−	1439.01i	−0.485925	−	2.14778i
$671$			206.297i			0.307447i
$672$		0			0
$673$		−	967.081i		−	1.43697i	−0.695543	−	0.718485i	$-0.744836\pi$
$673$		−	967.081i		−	1.43697i	0.695543	−	0.718485i	$-0.255164\pi$
$674$			1190.42i			1.76620i
$675$		0			0
$676$	−561.135			−0.830081
$677$	974.015			1.43872			0.719361	−	0.694636i	$-0.244435\pi$
$677$	974.015			1.43872			0.719361	+	0.694636i	$0.244435\pi$
$678$		0			0
$679$	55.7022			0.0820356
$680$	548.254	−	124.040i	0.806255	−	0.182412i
$681$		0			0
$682$			372.962i			0.546864i
$683$	−1033.38			−1.51300			−0.756500	−	0.653994i	$-0.773092\pi$
$683$	−1033.38			−1.51300			−0.756500	+	0.653994i	$0.773092\pi$
$684$		0			0
$685$	490.689	−	111.016i	0.716335	−	0.162068i
$686$			129.767i			0.189164i
$687$		0			0
$688$			1317.95i			1.91563i
$689$		−	665.275i		−	0.965567i
$690$		0			0
$691$	113.736			0.164596			0.0822982	−	0.996608i	$-0.473774\pi$
$691$	113.736			0.164596			0.0822982	+	0.996608i	$0.473774\pi$
$692$	−291.671			−0.421489
$693$		0			0
$694$	605.880			0.873025
$695$	−573.502	+	129.752i	−0.825183	+	0.186694i
$696$		0			0
$697$			1293.40i			1.85567i
$698$	−319.816			−0.458189
$699$		0			0
$700$	−13.9485	−	29.2481i	−0.0199264	−	0.0417830i
$701$			1232.70i			1.75849i	0.476374	+	0.879243i	$0.341951\pi$
$701$			1232.70i			1.75849i	−0.476374	+	0.879243i	$0.658049\pi$
$702$		0			0
$703$		−	797.179i		−	1.13397i
$704$		−	32.6808i		−	0.0464216i
$705$		0			0
$706$	617.594			0.874779
$707$	−81.2699			−0.114950
$708$		0			0
$709$	1313.50			1.85261			0.926303	−	0.376779i	$-0.122968\pi$
$709$	1313.50			1.85261			0.926303	+	0.376779i	$0.122968\pi$
$710$	−33.1016	−	146.308i	−0.0466220	−	0.206068i
$711$		0			0
$712$			490.458i			0.688845i
$713$	443.564			0.622109
$714$		0			0
$715$	−321.344	+	72.7026i	−0.449432	+	0.101682i
$716$		−	836.436i		−	1.16821i
$717$		0			0
$718$		−	1211.78i		−	1.68771i
$719$			158.445i			0.220369i	0.993911	+	0.110184i	$0.0351441\pi$
$719$			158.445i			0.220369i	−0.993911	+	0.110184i	$0.964856\pi$
$720$		0			0
$721$	−12.9524			−0.0179645
$722$	574.473			0.795669
$723$		0			0
$724$	9.63135			0.0133030
$725$	−183.070	−	383.873i	−0.252510	−	0.529479i
$726$		0			0
$727$		−	364.192i		−	0.500952i	−0.968123	−	0.250476i	$-0.919413\pi$
$727$		−	364.192i		−	0.500952i	0.968123	−	0.250476i	$-0.0805871\pi$
$728$	39.9405			0.0548633
$729$		0			0
$730$	−87.3468	−	386.071i	−0.119653	−	0.528864i
$731$		−	1944.32i		−	2.65980i
$732$		0			0
$733$		−	40.1103i		−	0.0547208i	−0.999626	−	0.0273604i	$-0.991290\pi$
$733$		−	40.1103i		−	0.0547208i	0.999626	−	0.0273604i	$-0.00871017\pi$
$734$		−	1296.23i		−	1.76598i
$735$		0			0
$736$	−350.698			−0.476492
$737$	−384.275			−0.521404
$738$		0			0
$739$	599.976			0.811876			0.405938	−	0.913901i	$-0.366945\pi$
$739$	599.976			0.811876			0.405938	+	0.913901i	$0.366945\pi$
$740$	187.893	+	830.481i	0.253909	+	1.12227i
$741$		0			0
$742$		−	44.4633i		−	0.0599236i
$743$	−360.552			−0.485265			−0.242632	−	0.970118i	$-0.578011\pi$
$743$	−360.552			−0.485265			−0.242632	+	0.970118i	$0.578011\pi$
$744$		0			0
$745$	−160.818	−	710.813i	−0.215863	−	0.954111i
$746$			80.9416i			0.108501i
$747$		0			0
$748$			240.400i			0.321391i
$749$		−	77.3997i		−	0.103337i
$750$		0			0
$751$	−550.409			−0.732901			−0.366451	−	0.930437i	$-0.619427\pi$
$751$	−550.409			−0.732901			−0.366451	+	0.930437i	$0.619427\pi$
$752$	−862.210			−1.14656
$753$		0			0
$754$	−860.753			−1.14158
$755$	−156.973	+	35.5145i	−0.207911	+	0.0470391i
$756$		0			0
$757$			400.305i			0.528805i	0.964412	+	0.264402i	$0.0851747\pi$
$757$			400.305i			0.528805i	−0.964412	+	0.264402i	$0.914825\pi$
$758$	1537.02			2.02773
$759$		0			0
$760$	−218.827	+	49.5087i	−0.287931	+	0.0651431i
$761$			374.188i			0.491706i	0.969307	+	0.245853i	$0.0790680\pi$
$761$			374.188i			0.491706i	−0.969307	+	0.245853i	$0.920932\pi$
$762$		0			0
$763$			38.6784i			0.0506926i
$764$		−	284.152i		−	0.371927i
$765$		0			0
$766$	385.359			0.503080
$767$	−856.375			−1.11653
$768$		0			0
$769$	−642.958			−0.836096			−0.418048	−	0.908425i	$-0.637286\pi$
$769$	−642.958			−0.836096			−0.418048	+	0.908425i	$0.637286\pi$
$770$	−21.4768	+	4.85904i	−0.0278920	+	0.00631044i
$771$		0			0
$772$			551.822i			0.714796i
$773$	−1285.61			−1.66315			−0.831574	−	0.555414i	$-0.812560\pi$
$773$	−1285.61			−1.66315			−0.831574	+	0.555414i	$0.812560\pi$
$774$		0			0
$775$	−996.368	+	475.170i	−1.28564	+	0.613123i
$776$			411.937i			0.530847i
$777$		0			0
$778$			559.369i			0.718983i
$779$		−	516.243i		−	0.662699i
$780$		0			0
$781$	−39.0703			−0.0500260
$782$	745.938			0.953885
$783$		0			0
$784$	−963.054			−1.22838
$785$	−73.0328	−	322.803i	−0.0930354	−	0.411214i
$786$		0			0
$787$		−	1359.15i		−	1.72701i	−0.504342	−	0.863504i	$-0.668265\pi$
$787$		−	1359.15i		−	1.72701i	0.504342	−	0.863504i	$-0.331735\pi$
$788$	177.447			0.225186
$789$		0			0
$790$	−1454.77	+	329.136i	−1.84148	+	0.416627i
$791$			7.05553i			0.00891976i
$792$		0			0
$793$			1235.77i			1.55835i
$794$			1357.35i			1.70951i
$795$		0			0
$796$	−707.427			−0.888727
$797$	1506.17			1.88980			0.944898	−	0.327364i	$-0.106160\pi$
$797$	1506.17			1.88980			0.944898	+	0.327364i	$0.106160\pi$
$798$		0			0
$799$	1271.98			1.59196
$800$	787.766	−	375.687i	0.984707	−	0.469609i
$801$		0			0
$802$			889.141i			1.10866i
$803$	−103.097			−0.128389
$804$		0			0
$805$	5.77886	+	25.5424i	0.00717871	+	0.0317297i
$806$			2234.14i			2.77189i
$807$		0			0
$808$		−	601.019i		−	0.743835i
$809$		−	38.4397i		−	0.0475151i	−0.999718	−	0.0237576i	$-0.992437\pi$
$809$		−	38.4397i		−	0.0475151i	0.999718	−	0.0237576i	$-0.00756298\pi$
$810$		0			0
$811$	−413.331			−0.509656			−0.254828	−	0.966986i	$-0.582019\pi$
$811$	−413.331			−0.509656			−0.254828	+	0.966986i	$0.582019\pi$
$812$	−22.0497			−0.0271548
$813$		0			0
$814$	578.606			0.710818
$815$	−235.535	−	1041.06i	−0.289000	−	1.27737i
$816$		0			0
$817$			776.045i			0.949872i
$818$	1702.46			2.08125
$819$		0			0
$820$	121.677	+	537.808i	0.148386	+	0.655864i
$821$			226.821i			0.276275i	0.990413	+	0.138137i	$0.0441115\pi$
$821$			226.821i			0.276275i	−0.990413	+	0.138137i	$0.955888\pi$
$822$		0			0
$823$		−	87.6605i		−	0.106513i	−0.998581	−	0.0532567i	$-0.983040\pi$
$823$		−	87.6605i		−	0.106513i	0.998581	−	0.0532567i	$-0.0169601\pi$
$824$		−	95.7874i		−	0.116247i
$825$		0			0
$826$	−57.2354			−0.0692922
$827$	1135.36			1.37286			0.686432	−	0.727194i	$-0.259176\pi$
$827$	1135.36			1.37286			0.686432	+	0.727194i	$0.259176\pi$
$828$		0			0
$829$	−298.877			−0.360527			−0.180264	−	0.983618i	$-0.557695\pi$
$829$	−298.877			−0.360527			−0.180264	+	0.983618i	$0.557695\pi$
$830$	−602.760	+	136.372i	−0.726217	+	0.164303i
$831$		0			0
$832$		−	195.767i		−	0.235297i
$833$	1420.75			1.70558
$834$		0			0
$835$	906.942	−	205.192i	1.08616	−	0.245739i
$836$		−	95.9522i		−	0.114775i
$837$		0			0
$838$		−	807.125i		−	0.963156i
$839$			824.580i			0.982813i	0.870930	+	0.491407i	$0.163517\pi$
$839$			824.580i			0.982813i	−0.870930	+	0.491407i	$0.836483\pi$
$840$		0			0
$841$	551.604			0.655890
$842$	353.639			0.419999
$843$		0			0
$844$	956.109			1.13283
$845$	−1100.77	+	249.044i	−1.30268	+	0.294727i
$846$		0			0
$847$			5.73520i			0.00677119i
$848$	661.803			0.780428
$849$		0			0
$850$	−1675.59	+	799.091i	−1.97128	+	0.940107i
$851$		−	688.137i		−	0.808622i
$852$		0			0
$853$		−	1267.04i		−	1.48539i	−0.669628	−	0.742696i	$-0.733547\pi$
$853$		−	1267.04i		−	1.48539i	0.669628	−	0.742696i	$-0.266453\pi$
$854$			82.5923i			0.0967123i
$855$		0			0
$856$	572.398			0.668689
$857$	15.4870			0.0180712			0.00903558	−	0.999959i	$-0.497124\pi$
$857$	15.4870			0.0180712			0.00903558	+	0.999959i	$0.497124\pi$
$858$		0			0
$859$	783.182			0.911736			0.455868	−	0.890047i	$-0.349329\pi$
$859$	783.182			0.911736			0.455868	+	0.890047i	$0.349329\pi$
$860$	−182.911	−	808.464i	−0.212688	−	0.940074i
$861$		0			0
$862$			1610.71i			1.86857i
$863$	−437.554			−0.507015			−0.253508	−	0.967333i	$-0.581584\pi$
$863$	−437.554			−0.507015			−0.253508	+	0.967333i	$0.581584\pi$
$864$		0			0
$865$	−572.165	+	129.450i	−0.661463	+	0.149653i
$866$		−	1510.40i		−	1.74411i
$867$		0			0
$868$			57.2316i			0.0659350i
$869$			388.483i			0.447047i
$870$		0			0
$871$	−2301.91			−2.64284
$872$	−286.040			−0.328028
$873$		0			0
$874$	−297.730			−0.340652
$875$	−40.3434	−	51.1848i	−0.0461068	−	0.0584969i
$876$		0			0
$877$		−	564.528i		−	0.643704i	−0.946790	−	0.321852i	$-0.895695\pi$
$877$		−	564.528i		−	0.643704i	0.946790	−	0.321852i	$-0.104305\pi$
$878$	−81.8159			−0.0931843
$879$		0			0
$880$	−72.3231	−	319.666i	−0.0821853	−	0.363257i
$881$		−	1328.59i		−	1.50804i	−0.656850	−	0.754022i	$-0.728111\pi$
$881$		−	1328.59i		−	1.50804i	0.656850	−	0.754022i	$-0.271889\pi$
$882$		0			0
$883$			778.410i			0.881552i	0.897617	+	0.440776i	$0.145297\pi$
$883$			778.410i			0.881552i	−0.897617	+	0.440776i	$0.854703\pi$
$884$			1440.06i			1.62903i
$885$		0			0
$886$	109.643			0.123751
$887$	803.383			0.905730			0.452865	−	0.891579i	$-0.350402\pi$
$887$	803.383			0.905730			0.452865	+	0.891579i	$0.350402\pi$
$888$		0			0
$889$	5.51562			0.00620430
$890$	−357.426	−	1579.81i	−0.401602	−	1.77507i
$891$		0			0
$892$			234.926i			0.263370i
$893$	−507.692			−0.568524
$894$		0			0
$895$	−371.229	−	1640.82i	−0.414781	−	1.83332i
$896$			59.7228i			0.0666549i
$897$		0			0
$898$			233.235i			0.259728i
$899$			751.148i			0.835538i
$900$		0			0
$901$	−976.327			−1.08360
$902$	374.698			0.415408
$903$		0			0
$904$	−52.1781			−0.0577192
$905$	18.8937	−	4.27461i	0.0208770	−	0.00472332i
$906$		0			0
$907$		−	1372.85i		−	1.51362i	−0.653635	−	0.756810i	$-0.726757\pi$
$907$		−	1372.85i		−	1.51362i	0.653635	−	0.756810i	$-0.273243\pi$
$908$	38.8890			0.0428293
$909$		0			0
$910$	−128.652	+	29.1070i	−0.141376	+	0.0319857i
$911$		−	1688.79i		−	1.85377i	−0.375342	−	0.926887i	$-0.622475\pi$
$911$		−	1688.79i		−	1.85377i	0.375342	−	0.926887i	$-0.377525\pi$
$912$		0			0
$913$			160.962i			0.176300i
$914$		−	943.056i		−	1.03179i
$915$		0			0
$916$	−372.448			−0.406603
$917$	80.1980			0.0874569
$918$		0			0
$919$	−214.818			−0.233752			−0.116876	−	0.993147i	$-0.537288\pi$
$919$	−214.818			−0.233752			−0.116876	+	0.993147i	$0.537288\pi$
$920$	−188.895	+	42.7367i	−0.205321	+	0.0464530i
$921$		0			0
$922$			1386.20i			1.50347i
$923$	−234.042			−0.253567
$924$		0			0
$925$	737.171	+	1545.75i	0.796942	+	1.67108i
$926$			643.620i			0.695054i
$927$		0			0
$928$		−	593.886i		−	0.639963i
$929$			656.011i			0.706147i	0.935596	+	0.353074i	$0.114863\pi$
$929$			656.011i			0.706147i	−0.935596	+	0.353074i	$0.885137\pi$
$930$		0			0
$931$	−567.071			−0.609099
$932$	338.303			0.362986
$933$		0			0
$934$	1560.94			1.67124
$935$	106.695	+	471.589i	0.114112	+	0.504373i
$936$		0			0
$937$			1448.23i			1.54560i	0.634650	+	0.772800i	$0.281144\pi$
$937$			1448.23i			1.54560i	−0.634650	+	0.772800i	$0.718856\pi$
$938$	−153.847			−0.164016
$939$		0			0
$940$	528.900	−	119.661i	0.562660	−	0.127299i
$941$		−	522.010i		−	0.554740i	−0.960763	−	0.277370i	$-0.910537\pi$
$941$		−	522.010i		−	0.554740i	0.960763	−	0.277370i	$-0.0894627\pi$
$942$		0			0
$943$		−	445.629i		−	0.472565i
$944$		−	851.905i		−	0.902441i
$945$		0			0
$946$	−563.267			−0.595420
$947$	229.963			0.242833			0.121416	−	0.992602i	$-0.461256\pi$
$947$	229.963			0.242833			0.121416	+	0.992602i	$0.461256\pi$
$948$		0			0
$949$	−617.578			−0.650767
$950$	668.785	−	318.945i	0.703984	−	0.335732i
$951$		0			0
$952$		−	58.6148i		−	0.0615701i
$953$	1682.00			1.76496			0.882479	−	0.470352i	$-0.155873\pi$
$953$	1682.00			1.76496			0.882479	+	0.470352i	$0.155873\pi$
$954$		0			0
$955$	−126.113	−	557.416i	−0.132055	−	0.583681i
$956$			480.613i			0.502734i
$957$		0			0
$958$			1225.93i			1.27967i
$959$		−	52.4605i		−	0.0547033i
$960$		0			0
$961$	988.658			1.02878
$962$	3466.01			3.60292
$963$		0			0
$964$	−146.678			−0.152155
$965$	244.911	+	1082.50i	0.253794	+	1.12176i
$966$		0			0
$967$			720.519i			0.745107i	0.928011	+	0.372554i	$0.121518\pi$
$967$			720.519i			0.745107i	−0.928011	+	0.372554i	$0.878482\pi$
$968$	−42.4138			−0.0438159
$969$		0			0
$970$	−300.203	−	1326.89i	−0.309488	−	1.36793i
$971$		−	1031.47i		−	1.06227i	−0.847286	−	0.531137i	$-0.821765\pi$
$971$		−	1031.47i		−	1.06227i	0.847286	−	0.531137i	$-0.178235\pi$
$972$		0			0
$973$			61.3142i			0.0630156i
$974$			2130.98i			2.18787i
$975$		0			0
$976$	−1229.32			−1.25955
$977$	1321.30			1.35241			0.676204	−	0.736715i	$-0.263624\pi$
$977$	1321.30			1.35241			0.676204	+	0.736715i	$0.263624\pi$
$978$		0			0
$979$	−421.875			−0.430924
$980$	590.760	−	133.657i	0.602817	−	0.136385i
$981$		0			0
$982$		−	300.813i		−	0.306327i
$983$	−1211.86			−1.23282			−0.616410	−	0.787425i	$-0.711414\pi$
$983$	−1211.86			−1.23282			−0.616410	+	0.787425i	$0.711414\pi$
$984$		0			0
$985$	348.094	−	78.7547i	0.353395	−	0.0799540i
$986$			1263.20i			1.28114i
$987$		0			0
$988$		−	574.781i		−	0.581762i
$989$			669.894i			0.677345i
$990$		0			0
$991$	−369.725			−0.373083			−0.186541	−	0.982447i	$-0.559728\pi$
$991$	−369.725			−0.373083			−0.186541	+	0.982447i	$0.559728\pi$
$992$	−1541.47			−1.55390
$993$		0			0
$994$	−15.6421			−0.0157365
$995$	−1387.75	+	313.972i	−1.39472	+	0.315549i
$996$		0			0
$997$		−	335.786i		−	0.336796i	−0.985719	−	0.168398i	$-0.946141\pi$
$997$		−	335.786i		−	0.336796i	0.985719	−	0.168398i	$-0.0538594\pi$
$998$	−2052.92			−2.05703
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.g.a.89.8	yes	40
3.2	odd	2	inner	495.3.g.a.89.34	yes	40
5.4	even	2	inner	495.3.g.a.89.33	yes	40
15.14	odd	2	inner	495.3.g.a.89.7	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.3.g.a.89.7	✓	40	15.14	odd	2	inner
495.3.g.a.89.8	yes	40	1.1	even	1	trivial
495.3.g.a.89.33	yes	40	5.4	even	2	inner
495.3.g.a.89.34	yes	40	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.54676 q^{2} +2.48600 q^{4} +(4.87674 - 1.10334i) q^{5} -0.521381i q^{7} +3.85580 q^{8} +(-12.4199 + 2.80995i) q^{10} +3.31662i q^{11} +19.8675i q^{13} +1.32783i q^{14} -19.7638 q^{16} +29.1566 q^{17} -11.6374 q^{19} +(12.1236 - 2.74291i) q^{20} -8.44666i q^{22} -10.0456 q^{23} +(22.5653 - 10.7614i) q^{25} -50.5978i q^{26} -1.29615i q^{28} -17.0117i q^{29} -44.1549 q^{31} +34.9105 q^{32} -74.2550 q^{34} +(-0.575262 - 2.54264i) q^{35} +68.5012i q^{37} +29.6378 q^{38} +(18.8037 - 4.25426i) q^{40} +44.3605i q^{41} -66.6852i q^{43} +8.24513i q^{44} +25.5838 q^{46} +43.6257 q^{47} +48.7282 q^{49} +(-57.4684 + 27.4068i) q^{50} +49.3906i q^{52} -33.4856 q^{53} +(3.65937 + 16.1743i) q^{55} -2.01034i q^{56} +43.3246i q^{58} +43.1043i q^{59} +62.2008 q^{61} +112.452 q^{62} -9.85363 q^{64} +(21.9207 + 96.8888i) q^{65} +115.863i q^{67} +72.4834 q^{68} +(1.46506 + 6.47551i) q^{70} +11.7801i q^{71} +31.0848i q^{73} -174.456i q^{74} -28.9307 q^{76} +1.72923 q^{77} +117.132 q^{79} +(-96.3830 + 21.8062i) q^{80} -112.976i q^{82} +48.5317 q^{83} +(142.189 - 32.1697i) q^{85} +169.831i q^{86} +12.7882i q^{88} +127.200i q^{89} +10.3585 q^{91} -24.9734 q^{92} -111.104 q^{94} +(-56.7528 + 12.8401i) q^{95} +106.836i q^{97} -124.099 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 72 q^{4} + 8 q^{10} + 184 q^{16} - 80 q^{19} + 32 q^{25} - 16 q^{31} - 160 q^{34} - 136 q^{40} + 560 q^{46} - 104 q^{49} - 96 q^{61} + 264 q^{64} - 872 q^{70} - 176 q^{76} - 672 q^{79} + 16 q^{85}+ \cdots + 400 q^{94}+O(q^{100}) \) 40 * q + 72 * q^4 + 8 * q^10 + 184 * q^16 - 80 * q^19 + 32 * q^25 - 16 * q^31 - 160 * q^34 - 136 * q^40 + 560 * q^46 - 104 * q^49 - 96 * q^61 + 264 * q^64 - 872 * q^70 - 176 * q^76 - 672 * q^79 + 16 * q^85 - 864 * q^91 + 400 * q^94

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