LMFDB - Embedded newform 495.3.g.a.89.11

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			89.11
Character	$\chi$	$=$	495.89
Dual form			495.3.g.a.89.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.09140 q^{2} +0.373951 q^{4} +(-3.17303 + 3.86418i) q^{5} -2.11730i q^{7} +7.58352 q^{8} +(6.63607 - 8.08155i) q^{10} +3.31662i q^{11} -17.4883i q^{13} +4.42812i q^{14} -17.3560 q^{16} -17.3754 q^{17} -16.6823 q^{19} +(-1.18656 + 1.44502i) q^{20} -6.93639i q^{22} +17.6229 q^{23} +(-4.86380 - 24.5223i) q^{25} +36.5750i q^{26} -0.791768i q^{28} +38.5140i q^{29} +49.2598 q^{31} +5.96419 q^{32} +36.3388 q^{34} +(8.18164 + 6.71826i) q^{35} +12.4676i q^{37} +34.8893 q^{38} +(-24.0627 + 29.3041i) q^{40} -1.84757i q^{41} -19.5151i q^{43} +1.24026i q^{44} -36.8565 q^{46} +59.7968 q^{47} +44.5170 q^{49} +(10.1721 + 51.2859i) q^{50} -6.53977i q^{52} -5.90680 q^{53} +(-12.8160 - 10.5237i) q^{55} -16.0566i q^{56} -80.5481i q^{58} -80.0660i q^{59} +52.2868 q^{61} -103.022 q^{62} +56.9504 q^{64} +(67.5779 + 55.4908i) q^{65} +74.7426i q^{67} -6.49754 q^{68} +(-17.1111 - 14.0506i) q^{70} +75.9608i q^{71} +9.20374i q^{73} -26.0748i q^{74} -6.23836 q^{76} +7.02230 q^{77} +123.270 q^{79} +(55.0709 - 67.0666i) q^{80} +3.86401i q^{82} +65.5321 q^{83} +(55.1325 - 67.1416i) q^{85} +40.8139i q^{86} +25.1517i q^{88} +94.2528i q^{89} -37.0280 q^{91} +6.59010 q^{92} -125.059 q^{94} +(52.9333 - 64.4633i) q^{95} +92.6881i q^{97} -93.1029 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.09140			−1.04570			−0.522850	−	0.852425i	$-0.675131\pi$
$2$	−2.09140			−1.04570			−0.522850	+	0.852425i	$0.675131\pi$
$3$		0			0
$3$		0			0
$4$	0.373951			0.0934878
$5$	−3.17303	+	3.86418i	−0.634605	+	0.772836i
$5$	−3.17303	+	3.86418i	−0.634605	+	0.772836i
$6$		0			0
$7$		−	2.11730i		−	0.302472i	−0.988498	−	0.151236i	$-0.951675\pi$
$7$		−	2.11730i		−	0.302472i	0.988498	−	0.151236i	$-0.0483253\pi$
$8$	7.58352			0.947940
$9$		0			0
$10$	6.63607	−	8.08155i	0.663607	−	0.808155i
$11$			3.31662i			0.301511i
$11$			3.31662i			0.301511i
$12$		0			0
$13$		−	17.4883i		−	1.34525i	−0.739982	−	0.672627i	$-0.765166\pi$
$13$		−	17.4883i		−	1.34525i	0.739982	−	0.672627i	$-0.234834\pi$
$14$			4.42812i			0.316295i
$15$		0			0
$16$	−17.3560			−1.08475
$17$	−17.3754			−1.02208			−0.511040	−	0.859557i	$-0.670740\pi$
$17$	−17.3754			−1.02208			−0.511040	+	0.859557i	$0.670740\pi$
$18$		0			0
$19$	−16.6823			−0.878014			−0.439007	−	0.898484i	$-0.644670\pi$
$19$	−16.6823			−0.878014			−0.439007	+	0.898484i	$0.644670\pi$
$20$	−1.18656	+	1.44502i	−0.0593279	+	0.0722508i
$21$		0			0
$22$		−	6.93639i		−	0.315290i
$23$	17.6229			0.766213			0.383107	−	0.923704i	$-0.374854\pi$
$23$	17.6229			0.766213			0.383107	+	0.923704i	$0.374854\pi$
$24$		0			0
$25$	−4.86380	−	24.5223i	−0.194552	−	0.980892i
$26$			36.5750i			1.40673i
$27$		0			0
$28$		−	0.791768i		−	0.0282774i
$29$			38.5140i			1.32807i	0.747703	+	0.664034i	$0.231157\pi$
$29$			38.5140i			1.32807i	−0.747703	+	0.664034i	$0.768843\pi$
$30$		0			0
$31$	49.2598			1.58903			0.794513	−	0.607247i	$-0.207726\pi$
$31$	49.2598			1.58903			0.794513	+	0.607247i	$0.207726\pi$
$32$	5.96419			0.186381
$33$		0			0
$34$	36.3388			1.06879
$35$	8.18164	+	6.71826i	0.233761	+	0.191950i
$36$		0			0
$37$			12.4676i			0.336963i	0.985705	+	0.168481i	$0.0538863\pi$
$37$			12.4676i			0.336963i	−0.985705	+	0.168481i	$0.946114\pi$
$38$	34.8893			0.918139
$39$		0			0
$40$	−24.0627	+	29.3041i	−0.601568	+	0.732602i
$41$		−	1.84757i		−	0.0450628i	−0.999746	−	0.0225314i	$-0.992827\pi$
$41$		−	1.84757i		−	0.0450628i	0.999746	−	0.0225314i	$-0.00717257\pi$
$42$		0			0
$43$		−	19.5151i		−	0.453840i	−0.973913	−	0.226920i	$-0.927134\pi$
$43$		−	19.5151i		−	0.453840i	0.973913	−	0.226920i	$-0.0728656\pi$
$44$			1.24026i			0.0281876i
$45$		0			0
$46$	−36.8565			−0.801229
$47$	59.7968			1.27227			0.636136	−	0.771577i	$-0.280532\pi$
$47$	59.7968			1.27227			0.636136	+	0.771577i	$0.280532\pi$
$48$		0			0
$49$	44.5170			0.908511
$50$	10.1721	+	51.2859i	0.203443	+	1.02572i
$51$		0			0
$52$		−	6.53977i		−	0.125765i
$53$	−5.90680			−0.111449			−0.0557245	−	0.998446i	$-0.517747\pi$
$53$	−5.90680			−0.111449			−0.0557245	+	0.998446i	$0.517747\pi$
$54$		0			0
$55$	−12.8160	−	10.5237i	−0.233019	−	0.191341i
$56$		−	16.0566i		−	0.286725i
$57$		0			0
$58$		−	80.5481i		−	1.38876i
$59$		−	80.0660i		−	1.35705i	−0.734577	−	0.678525i	$-0.762619\pi$
$59$		−	80.0660i		−	1.35705i	0.734577	−	0.678525i	$-0.237381\pi$
$60$		0			0
$61$	52.2868			0.857160			0.428580	−	0.903504i	$-0.359014\pi$
$61$	52.2868			0.857160			0.428580	+	0.903504i	$0.359014\pi$
$62$	−103.022			−1.66164
$63$		0			0
$64$	56.9504			0.889849
$65$	67.5779	+	55.4908i	1.03966	+	0.853705i
$66$		0			0
$67$			74.7426i			1.11556i	0.829988	+	0.557781i	$0.188347\pi$
$67$			74.7426i			1.11556i	−0.829988	+	0.557781i	$0.811653\pi$
$68$	−6.49754			−0.0955520
$69$		0			0
$70$	−17.1111	−	14.0506i	−0.244444	−	0.200722i
$71$			75.9608i			1.06987i	0.844893	+	0.534935i	$0.179664\pi$
$71$			75.9608i			1.06987i	−0.844893	+	0.534935i	$0.820336\pi$
$72$		0			0
$73$			9.20374i			0.126079i	0.998011	+	0.0630393i	$0.0200794\pi$
$73$			9.20374i			0.126079i	−0.998011	+	0.0630393i	$0.979921\pi$
$74$		−	26.0748i		−	0.352362i
$75$		0			0
$76$	−6.23836			−0.0820836
$77$	7.02230			0.0911986
$78$		0			0
$79$	123.270			1.56038			0.780190	−	0.625543i	$-0.215123\pi$
$79$	123.270			1.56038			0.780190	+	0.625543i	$0.215123\pi$
$80$	55.0709	−	67.0666i	0.688387	−	0.838333i
$81$		0			0
$82$			3.86401i			0.0471221i
$83$	65.5321			0.789543			0.394772	−	0.918779i	$-0.370824\pi$
$83$	65.5321			0.789543			0.394772	+	0.918779i	$0.370824\pi$
$84$		0			0
$85$	55.1325	−	67.1416i	0.648618	−	0.789901i
$86$			40.8139i			0.474580i
$87$		0			0
$88$			25.1517i			0.285815i
$89$			94.2528i			1.05902i	0.848304	+	0.529510i	$0.177624\pi$
$89$			94.2528i			1.05902i	−0.848304	+	0.529510i	$0.822376\pi$
$90$		0			0
$91$	−37.0280			−0.406901
$92$	6.59010			0.0716316
$93$		0			0
$94$	−125.059			−1.33041
$95$	52.9333	−	64.4633i	0.557193	−	0.678561i
$96$		0			0
$97$			92.6881i			0.955548i	0.878483	+	0.477774i	$0.158556\pi$
$97$			92.6881i			0.955548i	−0.878483	+	0.477774i	$0.841444\pi$
$98$	−93.1029			−0.950030
$99$		0			0
$100$	−1.81882	−	9.17015i	−0.0181882	−	0.0917015i
$101$		−	137.609i		−	1.36247i	−0.732065	−	0.681234i	$-0.761443\pi$
$101$		−	137.609i		−	1.36247i	0.732065	−	0.681234i	$-0.238557\pi$
$102$		0			0
$103$			60.2306i			0.584763i	0.956302	+	0.292382i	$0.0944478\pi$
$103$			60.2306i			0.584763i	−0.956302	+	0.292382i	$0.905552\pi$
$104$		−	132.623i		−	1.27522i
$105$		0			0
$106$	12.3535			0.116542
$107$	−94.7911			−0.885898			−0.442949	−	0.896547i	$-0.646068\pi$
$107$	−94.7911			−0.885898			−0.442949	+	0.896547i	$0.646068\pi$
$108$		0			0
$109$	92.4906			0.848537			0.424269	−	0.905536i	$-0.360531\pi$
$109$	92.4906			0.848537			0.424269	+	0.905536i	$0.360531\pi$
$110$	26.8035	+	22.0093i	0.243668	+	0.200085i
$111$		0			0
$112$			36.7478i			0.328106i
$113$	213.972			1.89356			0.946781	−	0.321880i	$-0.104315\pi$
$113$	213.972			1.89356			0.946781	+	0.321880i	$0.104315\pi$
$114$		0			0
$115$	−55.9179	+	68.0981i	−0.486243	+	0.592157i
$116$			14.4023i			0.124158i
$117$		0			0
$118$			167.450i			1.41907i
$119$			36.7889i			0.309150i
$120$		0			0
$121$	−11.0000			−0.0909091
$122$	−109.353			−0.896332
$123$		0			0
$124$	18.4208			0.148555
$125$	110.192	+	59.0153i	0.881533	+	0.472123i
$126$		0			0
$127$			147.705i			1.16303i	0.813534	+	0.581517i	$0.197541\pi$
$127$			147.705i			1.16303i	−0.813534	+	0.581517i	$0.802459\pi$
$128$	−142.963			−1.11690
$129$		0			0
$130$	−141.332	−	116.053i	−1.08717	−	0.892719i
$131$			0.0345272i			0.000263566i	1.00000		0.000131783i	$4.19479e-5\pi$
$131$			0.0345272i			0.000263566i	−1.00000		0.000131783i	$0.999958\pi$
$132$		0			0
$133$			35.3214i			0.265574i
$134$		−	156.317i		−	1.16654i
$135$		0			0
$136$	−131.766			−0.968870
$137$	−222.917			−1.62713			−0.813565	−	0.581475i	$-0.802476\pi$
$137$	−222.917			−1.62713			−0.813565	+	0.581475i	$0.802476\pi$
$138$		0			0
$139$	156.754			1.12773			0.563865	−	0.825867i	$-0.309314\pi$
$139$	156.754			1.12773			0.563865	+	0.825867i	$0.309314\pi$
$140$	3.05953	+	2.51230i	0.0218538	+	0.0179450i
$141$		0			0
$142$		−	158.864i		−	1.11876i
$143$	58.0021			0.405609
$144$		0			0
$145$	−148.825	−	122.206i	−1.02638	−	0.842799i
$146$		−	19.2487i		−	0.131840i
$147$		0			0
$148$			4.66229i			0.0315019i
$149$		−	203.337i		−	1.36468i	−0.731035	−	0.682340i	$-0.760963\pi$
$149$		−	203.337i		−	1.36468i	0.731035	−	0.682340i	$-0.239037\pi$
$150$		0			0
$151$	−167.632			−1.11014			−0.555071	−	0.831803i	$-0.687309\pi$
$151$	−167.632			−1.11014			−0.555071	+	0.831803i	$0.687309\pi$
$152$	−126.510			−0.832304
$153$		0			0
$154$	−14.6864			−0.0953664
$155$	−156.303	+	190.349i	−1.00840	+	1.22806i
$156$		0			0
$157$		−	65.0769i		−	0.414503i	−0.978288	−	0.207251i	$-0.933548\pi$
$157$		−	65.0769i		−	0.414503i	0.978288	−	0.207251i	$-0.0664518\pi$
$158$	−257.807			−1.63169
$159$		0			0
$160$	−18.9245	+	23.0467i	−0.118278	+	0.144042i
$161$		−	37.3130i		−	0.231758i
$162$		0			0
$163$		−	116.690i		−	0.715892i	−0.933742	−	0.357946i	$-0.883477\pi$
$163$		−	116.690i		−	0.715892i	0.933742	−	0.357946i	$-0.116523\pi$
$164$		−	0.690903i		−	0.00421282i
$165$		0			0
$166$	−137.054			−0.825625
$167$	66.0772			0.395672			0.197836	−	0.980235i	$-0.436609\pi$
$167$	66.0772			0.395672			0.197836	+	0.980235i	$0.436609\pi$
$168$		0			0
$169$	−136.840			−0.809706
$170$	−115.304	+	140.420i	−0.678259	+	0.825999i
$171$		0			0
$172$		−	7.29770i		−	0.0424285i
$173$	3.28057			0.0189628			0.00948142	−	0.999955i	$-0.496982\pi$
$173$	3.28057			0.0189628			0.00948142	+	0.999955i	$0.496982\pi$
$174$		0			0
$175$	−51.9211	+	10.2981i	−0.296692	+	0.0588465i
$176$		−	57.5632i		−	0.327064i
$177$		0			0
$178$		−	197.120i		−	1.10742i
$179$		−	184.053i		−	1.02823i	−0.857721	−	0.514115i	$-0.828120\pi$
$179$		−	184.053i		−	1.02823i	0.857721	−	0.514115i	$-0.171880\pi$
$180$		0			0
$181$	24.9083			0.137615			0.0688074	−	0.997630i	$-0.478081\pi$
$181$	24.9083			0.137615			0.0688074	+	0.997630i	$0.478081\pi$
$182$	77.4403			0.425496
$183$		0			0
$184$	133.644			0.726324
$185$	−48.1772	−	39.5601i	−0.260417	−	0.213839i
$186$		0			0
$187$		−	57.6276i		−	0.308169i
$188$	22.3611			0.118942
$189$		0			0
$190$	−110.705	+	134.819i	−0.582656	+	0.709571i
$191$			267.905i			1.40264i	0.712845	+	0.701321i	$0.247406\pi$
$191$			267.905i			1.40264i	−0.712845	+	0.701321i	$0.752594\pi$
$192$		0			0
$193$		−	237.746i		−	1.23185i	−0.787806	−	0.615923i	$-0.788783\pi$
$193$		−	237.746i		−	1.23185i	0.787806	−	0.615923i	$-0.211217\pi$
$194$		−	193.848i		−	0.999216i
$195$		0			0
$196$	16.6472			0.0849347
$197$	−265.552			−1.34798			−0.673989	−	0.738741i	$-0.735421\pi$
$197$	−265.552			−1.34798			−0.673989	+	0.738741i	$0.735421\pi$
$198$		0			0
$199$	−14.4813			−0.0727702			−0.0363851	−	0.999338i	$-0.511584\pi$
$199$	−14.4813			−0.0727702			−0.0363851	+	0.999338i	$0.511584\pi$
$200$	−36.8847	−	185.965i	−0.184423	−	0.929826i
$201$		0			0
$202$			287.796i			1.42473i
$203$	81.5457			0.401703
$204$		0			0
$205$	7.13936	+	5.86240i	0.0348262	+	0.0285971i
$206$		−	125.966i		−	0.611487i
$207$		0			0
$208$			303.526i			1.45926i
$209$		−	55.3288i		−	0.264731i
$210$		0			0
$211$	−69.0358			−0.327184			−0.163592	−	0.986528i	$-0.552308\pi$
$211$	−69.0358			−0.327184			−0.163592	+	0.986528i	$0.552308\pi$
$212$	−2.20885			−0.0104191
$213$		0			0
$214$	198.246			0.926383
$215$	75.4100	+	61.9220i	0.350744	+	0.288009i
$216$		0			0
$217$		−	104.298i		−	0.480635i
$218$	−193.435			−0.887315
$219$		0			0
$220$	−4.79257	−	3.93537i	−0.0217844	−	0.0178880i
$221$			303.865i			1.37496i
$222$		0			0
$223$		−	293.230i		−	1.31493i	−0.753484	−	0.657466i	$-0.771628\pi$
$223$		−	293.230i		−	1.31493i	0.753484	−	0.657466i	$-0.228372\pi$
$224$		−	12.6280i		−	0.0563750i
$225$		0			0
$226$	−447.502			−1.98010
$227$	−66.3648			−0.292356			−0.146178	−	0.989258i	$-0.546697\pi$
$227$	−66.3648			−0.292356			−0.146178	+	0.989258i	$0.546697\pi$
$228$		0			0
$229$	346.543			1.51329			0.756645	−	0.653826i	$-0.226837\pi$
$229$	346.543			1.51329			0.756645	+	0.653826i	$0.226837\pi$
$230$	116.947	−	142.420i	0.508464	−	0.619219i
$231$		0			0
$232$			292.071i			1.25893i
$233$	324.146			1.39119			0.695593	−	0.718436i	$-0.255142\pi$
$233$	324.146			1.39119			0.695593	+	0.718436i	$0.255142\pi$
$234$		0			0
$235$	−189.737	+	231.066i	−0.807391	+	0.983258i
$236$		−	29.9408i		−	0.126868i
$237$		0			0
$238$		−	76.9403i		−	0.323278i
$239$			82.5697i			0.345480i	0.984967	+	0.172740i	$0.0552620\pi$
$239$			82.5697i			0.345480i	−0.984967	+	0.172740i	$0.944738\pi$
$240$		0			0
$241$	339.538			1.40887			0.704436	−	0.709767i	$-0.251200\pi$
$241$	339.538			1.40887			0.704436	+	0.709767i	$0.251200\pi$
$242$	23.0054			0.0950636
$243$		0			0
$244$	19.5527			0.0801340
$245$	−141.254	+	172.022i	−0.576546	+	0.702130i
$246$		0			0
$247$			291.744i			1.18115i
$248$	373.563			1.50630
$249$		0			0
$250$	−230.455	−	123.425i	−0.921819	−	0.493699i
$251$		−	116.328i		−	0.463457i	−0.972781	−	0.231728i	$-0.925562\pi$
$251$		−	116.328i		−	0.463457i	0.972781	−	0.231728i	$-0.0744380\pi$
$252$		0			0
$253$			58.4485i			0.231022i
$254$		−	308.911i		−	1.21618i
$255$		0			0
$256$	71.1907			0.278089
$257$	364.500			1.41829			0.709145	−	0.705063i	$-0.249081\pi$
$257$	364.500			1.41829			0.709145	+	0.705063i	$0.249081\pi$
$258$		0			0
$259$	26.3977			0.101922
$260$	25.2708	+	20.7509i	0.0971956	+	0.0798110i
$261$		0			0
$262$		−	0.0722101i		−	0.000275611i
$263$	−355.574			−1.35199			−0.675996	−	0.736906i	$-0.736286\pi$
$263$	−355.574			−1.35199			−0.675996	+	0.736906i	$0.736286\pi$
$264$		0			0
$265$	18.7424	−	22.8249i	0.0707261	−	0.0861318i
$266$		−	73.8712i		−	0.277711i
$267$		0			0
$268$			27.9501i			0.104291i
$269$			449.766i			1.67199i	0.548735	+	0.835996i	$0.315110\pi$
$269$			449.766i			1.67199i	−0.548735	+	0.835996i	$0.684890\pi$
$270$		0			0
$271$	49.9340			0.184258			0.0921291	−	0.995747i	$-0.470633\pi$
$271$	49.9340			0.184258			0.0921291	+	0.995747i	$0.470633\pi$
$272$	301.566			1.10870
$273$		0			0
$274$	466.208			1.70149
$275$	81.3313	−	16.1314i	0.295750	−	0.0586596i
$276$		0			0
$277$		−	525.573i		−	1.89737i	−0.316216	−	0.948687i	$-0.602412\pi$
$277$		−	525.573i		−	1.89737i	0.316216	−	0.948687i	$-0.397588\pi$
$278$	−327.836			−1.17927
$279$		0			0
$280$	62.0456	+	50.9480i	0.221591	+	0.181957i
$281$			148.724i			0.529268i	0.964349	+	0.264634i	$0.0852511\pi$
$281$			148.724i			0.529268i	−0.964349	+	0.264634i	$0.914749\pi$
$282$		0			0
$283$			225.872i			0.798134i	0.916922	+	0.399067i	$0.130666\pi$
$283$			225.872i			0.798134i	−0.916922	+	0.399067i	$0.869334\pi$
$284$			28.4056i			0.100020i
$285$		0			0
$286$	−121.306			−0.424145
$287$	−3.91187			−0.0136302
$288$		0			0
$289$	12.9032			0.0446477
$290$	311.252	+	255.581i	1.07328	+	0.881314i
$291$		0			0
$292$			3.44175i			0.0117868i
$293$	185.601			0.633450			0.316725	−	0.948517i	$-0.397417\pi$
$293$	185.601			0.633450			0.316725	+	0.948517i	$0.397417\pi$
$294$		0			0
$295$	309.389	+	254.052i	1.04878	+	0.861192i
$296$			94.5485i			0.319420i
$297$		0			0
$298$			425.259i			1.42704i
$299$		−	308.194i		−	1.03075i
$300$		0			0
$301$	−41.3194			−0.137274
$302$	350.584			1.16088
$303$		0			0
$304$	289.537			0.952424
$305$	−165.907	+	202.046i	−0.543958	+	0.662444i
$306$		0			0
$307$		−	111.872i		−	0.364403i	−0.983261	−	0.182202i	$-0.941678\pi$
$307$		−	111.872i		−	0.364403i	0.983261	−	0.182202i	$-0.0583224\pi$
$308$	2.62600			0.00852596
$309$		0			0
$310$	326.891	−	398.096i	1.05449	−	1.28418i
$311$		−	340.464i		−	1.09474i	−0.836891	−	0.547370i	$-0.815629\pi$
$311$		−	340.464i		−	1.09474i	0.836891	−	0.547370i	$-0.184371\pi$
$312$		0			0
$313$		−	104.675i		−	0.334425i	−0.985921	−	0.167213i	$-0.946523\pi$
$313$		−	104.675i		−	0.334425i	0.985921	−	0.167213i	$-0.0534767\pi$
$314$			136.102i			0.433445i
$315$		0			0
$316$	46.0970			0.145876
$317$	9.42536			0.0297330			0.0148665	−	0.999889i	$-0.495268\pi$
$317$	9.42536			0.0297330			0.0148665	+	0.999889i	$0.495268\pi$
$318$		0			0
$319$	−127.736			−0.400427
$320$	−180.705	+	220.067i	−0.564703	+	0.687708i
$321$		0			0
$322$			78.0364i			0.242349i
$323$	289.861			0.897401
$324$		0			0
$325$	−428.853	+	85.0595i	−1.31955	+	0.261722i
$326$			244.046i			0.748608i
$327$		0			0
$328$		−	14.0111i		−	0.0427168i
$329$		−	126.608i		−	0.384826i
$330$		0			0
$331$	495.716			1.49763			0.748816	−	0.662778i	$-0.230623\pi$
$331$	495.716			1.49763			0.748816	+	0.662778i	$0.230623\pi$
$332$	24.5058			0.0738127
$333$		0			0
$334$	−138.194			−0.413754
$335$	−288.819	−	237.160i	−0.862146	−	0.707941i
$336$		0			0
$337$		−	425.429i		−	1.26240i	−0.775619	−	0.631201i	$-0.782562\pi$
$337$		−	425.429i		−	1.26240i	0.775619	−	0.631201i	$-0.217438\pi$
$338$	286.188			0.846709
$339$		0			0
$340$	20.6169	−	25.1077i	0.0606378	−	0.0738461i
$341$			163.376i			0.479109i
$342$		0			0
$343$		−	198.004i		−	0.577271i
$344$		−	147.993i		−	0.430213i
$345$		0			0
$346$	−6.86099			−0.0198294
$347$	292.096			0.841775			0.420888	−	0.907113i	$-0.361719\pi$
$347$	292.096			0.841775			0.420888	+	0.907113i	$0.361719\pi$
$348$		0			0
$349$	526.327			1.50810			0.754050	−	0.656817i	$-0.228097\pi$
$349$	526.327			1.50810			0.754050	+	0.656817i	$0.228097\pi$
$350$	108.588	−	21.5375i	0.310251	−	0.0615357i
$351$		0			0
$352$			19.7810i			0.0561960i
$353$	115.541			0.327313			0.163657	−	0.986517i	$-0.447671\pi$
$353$	115.541			0.327313			0.163657	+	0.986517i	$0.447671\pi$
$354$		0			0
$355$	−293.526	−	241.026i	−0.826835	−	0.678946i
$356$			35.2459i			0.0990055i
$357$		0			0
$358$			384.929i			1.07522i
$359$		−	171.524i		−	0.477783i	−0.971046	−	0.238892i	$-0.923216\pi$
$359$		−	171.524i		−	0.477783i	0.971046	−	0.238892i	$-0.0767841\pi$
$360$		0			0
$361$	−82.7018			−0.229091
$362$	−52.0931			−0.143904
$363$		0			0
$364$	−13.8467			−0.0380403
$365$	−35.5649	−	29.2037i	−0.0974382	−	0.0800102i
$366$		0			0
$367$		−	536.339i		−	1.46141i	−0.682691	−	0.730707i	$-0.739190\pi$
$367$		−	536.339i		−	1.46141i	0.682691	−	0.730707i	$-0.260810\pi$
$368$	−305.862			−0.831148
$369$		0			0
$370$	100.758	+	82.7360i	0.272318	+	0.223611i
$371$			12.5065i			0.0337102i
$372$		0			0
$373$			341.058i			0.914365i	0.889373	+	0.457183i	$0.151141\pi$
$373$			341.058i			0.914365i	−0.889373	+	0.457183i	$0.848859\pi$
$374$			120.522i			0.322252i
$375$		0			0
$376$	453.470			1.20604
$377$	673.543			1.78659
$378$		0			0
$379$	−275.593			−0.727158			−0.363579	−	0.931563i	$-0.618445\pi$
$379$	−275.593			−0.727158			−0.363579	+	0.931563i	$0.618445\pi$
$380$	19.7945	−	24.1061i	0.0520907	−	0.0634372i
$381$		0			0
$382$		−	560.296i		−	1.46674i
$383$	−482.596			−1.26004			−0.630021	−	0.776578i	$-0.716954\pi$
$383$	−482.596			−1.26004			−0.630021	+	0.776578i	$0.716954\pi$
$384$		0			0
$385$	−22.2819	+	27.1354i	−0.0578752	+	0.0704816i
$386$			497.223i			1.28814i
$387$		0			0
$388$			34.6608i			0.0893321i
$389$			301.323i			0.774609i	0.921952	+	0.387304i	$0.126594\pi$
$389$			301.323i			0.774609i	−0.921952	+	0.387304i	$0.873406\pi$
$390$		0			0
$391$	−306.204			−0.783131
$392$	337.596			0.861213
$393$		0			0
$394$	555.375			1.40958
$395$	−391.139	+	476.338i	−0.990225	+	1.20592i
$396$		0			0
$397$			40.6659i			0.102433i	0.998688	+	0.0512165i	$0.0163099\pi$
$397$			40.6659i			0.102433i	−0.998688	+	0.0512165i	$0.983690\pi$
$398$	30.2861			0.0760958
$399$		0			0
$400$	84.4159	+	425.608i	0.211040	+	1.06402i
$401$			287.950i			0.718079i	0.933322	+	0.359039i	$0.116896\pi$
$401$			287.950i			0.718079i	−0.933322	+	0.359039i	$0.883104\pi$
$402$		0			0
$403$		−	861.470i		−	2.13764i
$404$		−	51.4592i		−	0.127374i
$405$		0			0
$406$	−170.545			−0.420061
$407$	−41.3504			−0.101598
$408$		0			0
$409$	143.744			0.351453			0.175726	−	0.984439i	$-0.443773\pi$
$409$	143.744			0.351453			0.175726	+	0.984439i	$0.443773\pi$
$410$	−14.9313	−	12.2606i	−0.0364177	−	0.0299040i
$411$		0			0
$412$			22.5233i			0.0546682i
$413$	−169.524			−0.410469
$414$		0			0
$415$	−207.935	+	253.228i	−0.501049	+	0.610188i
$416$		−	104.303i		−	0.250730i
$417$		0			0
$418$			115.715i			0.276829i
$419$			422.567i			1.00851i	0.863554	+	0.504257i	$0.168233\pi$
$419$			422.567i			1.00851i	−0.863554	+	0.504257i	$0.831767\pi$
$420$		0			0
$421$	−220.688			−0.524199			−0.262099	−	0.965041i	$-0.584415\pi$
$421$	−220.688			−0.524199			−0.262099	+	0.965041i	$0.584415\pi$
$422$	144.381			0.342136
$423$		0			0
$424$	−44.7943			−0.105647
$425$	84.5103	+	426.084i	0.198848	+	1.00255i
$426$		0			0
$427$		−	110.707i		−	0.259267i
$428$	−35.4472			−0.0828206
$429$		0			0
$430$	−157.712	−	129.504i	−0.366773	−	0.301171i
$431$		−	306.571i		−	0.711302i	−0.934619	−	0.355651i	$-0.884259\pi$
$431$		−	306.571i		−	0.711302i	0.934619	−	0.355651i	$-0.115741\pi$
$432$		0			0
$433$			311.097i			0.718470i	0.933247	+	0.359235i	$0.116962\pi$
$433$			311.097i			0.718470i	−0.933247	+	0.359235i	$0.883038\pi$
$434$			218.129i			0.502600i
$435$		0			0
$436$	34.5870			0.0793279
$437$	−293.990			−0.672746
$438$		0			0
$439$	−603.127			−1.37386			−0.686932	−	0.726721i	$-0.741043\pi$
$439$	−603.127			−1.37386			−0.686932	+	0.726721i	$0.741043\pi$
$440$	−97.1906	−	79.8070i	−0.220888	−	0.181379i
$441$		0			0
$442$		−	635.504i		−	1.43779i
$443$	−155.198			−0.350334			−0.175167	−	0.984539i	$-0.556047\pi$
$443$	−155.198			−0.350334			−0.175167	+	0.984539i	$0.556047\pi$
$444$		0			0
$445$	−364.210	−	299.067i	−0.818449	−	0.672060i
$446$			613.261i			1.37502i
$447$		0			0
$448$		−	120.581i		−	0.269154i
$449$		−	7.81951i		−	0.0174154i	−0.999962	−	0.00870769i	$-0.997228\pi$
$449$		−	7.81951i		−	0.0174154i	0.999962	−	0.00870769i	$-0.00277178\pi$
$450$		0			0
$451$	6.12771			0.0135869
$452$	80.0152			0.177025
$453$		0			0
$454$	138.795			0.305717
$455$	117.491	−	143.083i	0.258222	−	0.314468i
$456$		0			0
$457$			817.477i			1.78879i	0.447277	+	0.894395i	$0.352394\pi$
$457$			817.477i			1.78879i	−0.447277	+	0.894395i	$0.647606\pi$
$458$	−724.761			−1.58245
$459$		0			0
$460$	−20.9106	+	25.4654i	−0.0454578	+	0.0553595i
$461$		−	122.774i		−	0.266322i	−0.991094	−	0.133161i	$-0.957487\pi$
$461$		−	122.774i		−	0.266322i	0.991094	−	0.133161i	$-0.0425127\pi$
$462$		0			0
$463$			340.949i			0.736390i	0.929749	+	0.368195i	$0.120024\pi$
$463$			340.949i			0.736390i	−0.929749	+	0.368195i	$0.879976\pi$
$464$		−	668.447i		−	1.44062i
$465$		0			0
$466$	−677.920			−1.45476
$467$	489.141			1.04741			0.523706	−	0.851899i	$-0.324549\pi$
$467$	489.141			1.04741			0.523706	+	0.851899i	$0.324549\pi$
$468$		0			0
$469$	158.253			0.337426
$470$	396.816	−	483.251i	0.844288	−	1.02819i
$471$		0			0
$472$		−	607.182i		−	1.28640i
$473$	64.7243			0.136838
$474$		0			0
$475$	81.1392	+	409.088i	0.170819	+	0.861237i
$476$			13.7572i			0.0289018i
$477$		0			0
$478$		−	172.686i		−	0.361268i
$479$			402.120i			0.839498i	0.907640	+	0.419749i	$0.137882\pi$
$479$			402.120i			0.839498i	−0.907640	+	0.419749i	$0.862118\pi$
$480$		0			0
$481$	218.038			0.453300
$482$	−710.110			−1.47326
$483$		0			0
$484$	−4.11346			−0.00849889
$485$	−358.164	−	294.102i	−0.738482	−	0.606396i
$486$		0			0
$487$			230.832i			0.473987i	0.971511	+	0.236993i	$0.0761620\pi$
$487$			230.832i			0.473987i	−0.971511	+	0.236993i	$0.923838\pi$
$488$	396.518			0.812536
$489$		0			0
$490$	295.418	−	359.766i	0.602894	−	0.734217i
$491$		−	819.053i		−	1.66813i	−0.551665	−	0.834066i	$-0.686007\pi$
$491$		−	819.053i		−	1.66813i	0.551665	−	0.834066i	$-0.313993\pi$
$492$		0			0
$493$		−	669.194i		−	1.35739i
$494$		−	610.154i		−	1.23513i
$495$		0			0
$496$	−854.952			−1.72369
$497$	160.832			0.323606
$498$		0			0
$499$	−658.657			−1.31995			−0.659977	−	0.751286i	$-0.729434\pi$
$499$	−658.657			−1.31995			−0.659977	+	0.751286i	$0.729434\pi$
$500$	41.2063	+	22.0689i	0.0824126	+	0.0441377i
$501$		0			0
$502$			243.287i			0.484636i
$503$	213.933			0.425314			0.212657	−	0.977127i	$-0.431788\pi$
$503$	213.933			0.425314			0.212657	+	0.977127i	$0.431788\pi$
$504$		0			0
$505$	531.748	+	436.638i	1.05297	+	0.864630i
$506$		−	122.239i		−	0.241580i
$507$		0			0
$508$			55.2346i			0.108729i
$509$		−	167.021i		−	0.328136i	−0.986449	−	0.164068i	$-0.947538\pi$
$509$		−	167.021i		−	0.328136i	0.986449	−	0.164068i	$-0.0524617\pi$
$510$		0			0
$511$	19.4871			0.0381352
$512$	422.963			0.826099
$513$		0			0
$514$	−762.316			−1.48310
$515$	−232.742	−	191.113i	−0.451926	−	0.371094i
$516$		0			0
$517$			198.324i			0.383605i
$518$	−55.2082			−0.106580
$519$		0			0
$520$	512.478	+	420.815i	0.985535	+	0.809261i
$521$		−	166.892i		−	0.320329i	−0.987090	−	0.160165i	$-0.948797\pi$
$521$		−	166.892i		−	0.320329i	0.987090	−	0.160165i	$-0.0512025\pi$
$522$		0			0
$523$			349.507i			0.668273i	0.942525	+	0.334136i	$0.108445\pi$
$523$			349.507i			0.668273i	−0.942525	+	0.334136i	$0.891555\pi$
$524$			0.0129115i			2.46402e-5i
$525$		0			0
$526$	743.647			1.41378
$527$	−855.907			−1.62411
$528$		0			0
$529$	−218.433			−0.412918
$530$	−39.1979	+	47.7360i	−0.0739583	+	0.0900680i
$531$		0			0
$532$			13.2085i			0.0248280i
$533$	−32.3109			−0.0606208
$534$		0			0
$535$	300.775	−	366.290i	0.562196	−	0.684654i
$536$			566.812i			1.05748i
$537$		0			0
$538$		−	940.640i		−	1.74840i
$539$			147.646i			0.273926i
$540$		0			0
$541$	855.321			1.58100			0.790500	−	0.612462i	$-0.209821\pi$
$541$	855.321			1.58100			0.790500	+	0.612462i	$0.209821\pi$
$542$	−104.432			−0.192679
$543$		0			0
$544$	−103.630			−0.190496
$545$	−293.475	+	357.400i	−0.538486	+	0.655781i
$546$		0			0
$547$		−	176.906i		−	0.323411i	−0.986839	−	0.161706i	$-0.948300\pi$
$547$		−	176.906i		−	0.323411i	0.986839	−	0.161706i	$-0.0516995\pi$
$548$	−83.3600			−0.152117
$549$		0			0
$550$	−170.096	+	33.7372i	−0.309266	+	0.0613404i
$551$		−	642.500i		−	1.16606i
$552$		0			0
$553$		−	261.000i		−	0.471971i
$554$			1099.18i			1.98408i
$555$		0			0
$556$	58.6185			0.105429
$557$	−470.845			−0.845324			−0.422662	−	0.906287i	$-0.638904\pi$
$557$	−470.845			−0.845324			−0.422662	+	0.906287i	$0.638904\pi$
$558$		0			0
$559$	−341.286			−0.610530
$560$	−142.000	−	116.602i	−0.253572	−	0.208218i
$561$		0			0
$562$		−	311.042i		−	0.553455i
$563$	−904.919			−1.60732			−0.803658	−	0.595091i	$-0.797116\pi$
$563$	−904.919			−1.60732			−0.803658	+	0.595091i	$0.797116\pi$
$564$		0			0
$565$	−678.940	+	826.828i	−1.20166	+	1.46341i
$566$		−	472.388i		−	0.834608i
$567$		0			0
$568$			576.050i			1.01417i
$569$			36.7748i			0.0646305i	0.999478	+	0.0323153i	$0.0102881\pi$
$569$			36.7748i			0.0646305i	−0.999478	+	0.0323153i	$0.989712\pi$
$570$		0			0
$571$	532.647			0.932831			0.466416	−	0.884566i	$-0.345545\pi$
$571$	532.647			0.932831			0.466416	+	0.884566i	$0.345545\pi$
$572$	21.6900			0.0379195
$573$		0			0
$574$	8.18129			0.0142531
$575$	−85.7142	−	432.154i	−0.149068	−	0.751572i
$576$		0			0
$577$			977.288i			1.69374i	0.531801	+	0.846870i	$0.321516\pi$
$577$			977.288i			1.69374i	−0.531801	+	0.846870i	$0.678484\pi$
$578$	−26.9857			−0.0466881
$579$		0			0
$580$	−55.6533	−	45.6990i	−0.0959539	−	0.0787914i
$581$		−	138.751i		−	0.238815i
$582$		0			0
$583$		−	19.5906i		−	0.0336031i
$584$			69.7967i			0.119515i
$585$		0			0
$586$	−388.166			−0.662399
$587$	−543.985			−0.926721			−0.463361	−	0.886170i	$-0.653357\pi$
$587$	−543.985			−0.926721			−0.463361	+	0.886170i	$0.653357\pi$
$588$		0			0
$589$	−821.766			−1.39519
$590$	−647.057	−	531.323i	−1.09671	−	0.900548i
$591$		0			0
$592$		−	216.388i		−	0.365520i
$593$	1019.47			1.71917			0.859583	−	0.510997i	$-0.170724\pi$
$593$	1019.47			1.71917			0.859583	+	0.510997i	$0.170724\pi$
$594$		0			0
$595$	−142.159	−	116.732i	−0.238923	−	0.196188i
$596$		−	76.0382i		−	0.127581i
$597$		0			0
$598$			644.558i			1.07786i
$599$		−	743.115i		−	1.24059i	−0.784368	−	0.620296i	$-0.787012\pi$
$599$		−	743.115i		−	1.24059i	0.784368	−	0.620296i	$-0.212988\pi$
$600$		0			0
$601$	499.426			0.830991			0.415496	−	0.909595i	$-0.363608\pi$
$601$	499.426			0.830991			0.415496	+	0.909595i	$0.363608\pi$
$602$	86.4154			0.143547
$603$		0			0
$604$	−62.6860			−0.103785
$605$	34.9033	−	42.5060i	0.0576914	−	0.0702578i
$606$		0			0
$607$			704.828i			1.16117i	0.814201	+	0.580583i	$0.197175\pi$
$607$			704.828i			1.16117i	−0.814201	+	0.580583i	$0.802825\pi$
$608$	−99.4962			−0.163645
$609$		0			0
$610$	346.978	−	422.558i	0.568817	−	0.692718i
$611$		−	1045.74i		−	1.71153i
$612$		0			0
$613$			570.118i			0.930046i	0.885299	+	0.465023i	$0.153954\pi$
$613$			570.118i			0.930046i	−0.885299	+	0.465023i	$0.846046\pi$
$614$			233.969i			0.381056i
$615$		0			0
$616$	53.2537			0.0864508
$617$	691.267			1.12037			0.560184	−	0.828368i	$-0.310730\pi$
$617$	691.267			1.12037			0.560184	+	0.828368i	$0.310730\pi$
$618$		0			0
$619$	350.743			0.566628			0.283314	−	0.959027i	$-0.408566\pi$
$619$	350.743			0.566628			0.283314	+	0.959027i	$0.408566\pi$
$620$	−58.4496	+	71.1812i	−0.0942735	+	0.114808i
$621$		0			0
$622$			712.046i			1.14477i
$623$	199.562			0.320324
$624$		0			0
$625$	−577.687	+	238.543i	−0.924299	+	0.381669i
$626$			218.918i			0.349709i
$627$		0			0
$628$		−	24.3356i		−	0.0387509i
$629$		−	216.630i		−	0.344403i
$630$		0			0
$631$	−703.703			−1.11522			−0.557609	−	0.830103i	$-0.688281\pi$
$631$	−703.703			−1.11522			−0.557609	+	0.830103i	$0.688281\pi$
$632$	934.820			1.47915
$633$		0			0
$634$	−19.7122			−0.0310918
$635$	−570.760	−	468.673i	−0.898835	−	0.738067i
$636$		0			0
$637$		−	778.527i		−	1.22218i
$638$	267.148			0.418727
$639$		0			0
$640$	453.625	−	552.434i	0.708788	−	0.863178i
$641$		−	277.323i		−	0.432641i	−0.976322	−	0.216320i	$-0.930594\pi$
$641$		−	277.323i		−	0.432641i	0.976322	−	0.216320i	$-0.0694056\pi$
$642$		0			0
$643$			803.516i			1.24964i	0.780770	+	0.624818i	$0.214827\pi$
$643$			803.516i			1.24964i	−0.780770	+	0.624818i	$0.785173\pi$
$644$		−	13.9532i		−	0.0216665i
$645$		0			0
$646$	−606.214			−0.938412
$647$	336.726			0.520442			0.260221	−	0.965549i	$-0.416205\pi$
$647$	336.726			0.520442			0.260221	+	0.965549i	$0.416205\pi$
$648$		0			0
$649$	265.549			0.409166
$650$	896.903	−	177.893i	1.37985	−	0.273682i
$651$		0			0
$652$		−	43.6365i		−	0.0669272i
$653$	815.199			1.24839			0.624195	−	0.781268i	$-0.285427\pi$
$653$	815.199			1.24839			0.624195	+	0.781268i	$0.285427\pi$
$654$		0			0
$655$	−0.133419	−	0.109556i	−0.000203693	−	0.000167260i
$656$			32.0664i			0.0488818i
$657$		0			0
$658$			264.788i			0.402413i
$659$		−	1063.95i		−	1.61449i	−0.590213	−	0.807247i	$-0.700956\pi$
$659$		−	1063.95i		−	1.61449i	0.590213	−	0.807247i	$-0.299044\pi$
$660$		0			0
$661$	22.7560			0.0344266			0.0172133	−	0.999852i	$-0.494521\pi$
$661$	22.7560			0.0344266			0.0172133	+	0.999852i	$0.494521\pi$
$662$	−1036.74			−1.56607
$663$		0			0
$664$	496.964			0.748439
$665$	−136.488	−	112.076i	−0.205246	−	0.168535i
$666$		0			0
$667$			678.728i			1.01758i
$668$	24.7097			0.0369905
$669$		0			0
$670$	604.036	+	495.997i	0.901546	+	0.740294i
$671$			173.416i			0.258443i
$672$		0			0
$673$		−	18.3044i		−	0.0271982i	−0.999908	−	0.0135991i	$-0.995671\pi$
$673$		−	18.3044i		−	0.0271982i	0.999908	−	0.0135991i	$-0.00432887\pi$
$674$			889.743i			1.32009i
$675$		0			0
$676$	−51.1716			−0.0756976
$677$	72.3895			0.106927			0.0534634	−	0.998570i	$-0.482974\pi$
$677$	72.3895			0.106927			0.0534634	+	0.998570i	$0.482974\pi$
$678$		0			0
$679$	196.249			0.289026
$680$	418.098	−	509.169i	0.614850	−	0.748778i
$681$		0			0
$682$		−	341.685i		−	0.501005i
$683$	−52.9046			−0.0774592			−0.0387296	−	0.999250i	$-0.512331\pi$
$683$	−52.9046			−0.0774592			−0.0387296	+	0.999250i	$0.512331\pi$
$684$		0			0
$685$	707.321	−	861.391i	1.03258	−	1.25750i
$686$			414.105i			0.603652i
$687$		0			0
$688$			338.704i			0.492302i
$689$			103.300i			0.149927i
$690$		0			0
$691$	−157.056			−0.227288			−0.113644	−	0.993522i	$-0.536252\pi$
$691$	−157.056			−0.227288			−0.113644	+	0.993522i	$0.536252\pi$
$692$	1.22677			0.00177279
$693$		0			0
$694$	−610.889			−0.880244
$695$	−497.386	+	605.728i	−0.715664	+	0.871551i
$696$		0			0
$697$			32.1023i			0.0460578i
$698$	−1100.76			−1.57702
$699$		0			0
$700$	−19.4160	+	3.85100i	−0.0277371	+	0.00550143i
$701$		−	755.243i		−	1.07738i	−0.842504	−	0.538690i	$-0.818919\pi$
$701$		−	755.243i		−	1.07738i	0.842504	−	0.538690i	$-0.181081\pi$
$702$		0			0
$703$		−	207.988i		−	0.295858i
$704$			188.883i			0.268300i
$705$		0			0
$706$	−241.643			−0.342271
$707$	−291.361			−0.412108
$708$		0			0
$709$	378.768			0.534229			0.267114	−	0.963665i	$-0.413930\pi$
$709$	378.768			0.534229			0.267114	+	0.963665i	$0.413930\pi$
$710$	613.881	+	504.081i	0.864621	+	0.709973i
$711$		0			0
$712$			714.767i			1.00389i
$713$	868.101			1.21753
$714$		0			0
$715$	−184.042	+	224.131i	−0.257402	+	0.313469i
$716$		−	68.8269i		−	0.0961270i
$717$		0			0
$718$			358.726i			0.499618i
$719$		−	177.894i		−	0.247419i	−0.992318	−	0.123710i	$-0.960521\pi$
$719$		−	177.894i		−	0.247419i	0.992318	−	0.123710i	$-0.0394791\pi$
$720$		0			0
$721$	127.526			0.176874
$722$	172.962			0.239560
$723$		0			0
$724$	9.31448			0.0128653
$725$	944.451	−	187.324i	1.30269	−	0.258378i
$726$		0			0
$727$			1326.23i			1.82425i	0.409907	+	0.912127i	$0.365561\pi$
$727$			1326.23i			1.82425i	−0.409907	+	0.912127i	$0.634439\pi$
$728$	−280.802			−0.385717
$729$		0			0
$730$	74.3805	+	61.0767i	0.101891	+	0.0836667i
$731$			339.082i			0.463861i
$732$		0			0
$733$		−	388.490i		−	0.530000i	−0.964248	−	0.265000i	$-0.914628\pi$
$733$		−	388.490i		−	0.530000i	0.964248	−	0.265000i	$-0.0853720\pi$
$734$			1121.70i			1.52820i
$735$		0			0
$736$	105.106			0.142808
$737$	−247.893			−0.336354
$738$		0			0
$739$	−688.983			−0.932319			−0.466159	−	0.884701i	$-0.654363\pi$
$739$	−688.983			−0.932319			−0.466159	+	0.884701i	$0.654363\pi$
$740$	−18.0159	−	14.7936i	−0.0243458	−	0.0199913i
$741$		0			0
$742$		−	26.1560i		−	0.0352507i
$743$	−547.962			−0.737499			−0.368749	−	0.929529i	$-0.620214\pi$
$743$	−547.962			−0.737499			−0.368749	+	0.929529i	$0.620214\pi$
$744$		0			0
$745$	785.732	+	645.194i	1.05467	+	0.866033i
$746$		−	713.289i		−	0.956152i
$747$		0			0
$748$		−	21.5499i		−	0.0288100i
$749$			200.701i			0.267959i
$750$		0			0
$751$	908.961			1.21033			0.605167	−	0.796098i	$-0.293106\pi$
$751$	908.961			1.21033			0.605167	+	0.796098i	$0.293106\pi$
$752$	−1037.83			−1.38009
$753$		0			0
$754$	−1408.65			−1.86823
$755$	531.899	−	647.759i	0.704502	−	0.857958i
$756$		0			0
$757$		−	1495.35i		−	1.97536i	−0.156477	−	0.987682i	$-0.550014\pi$
$757$		−	1495.35i		−	1.97536i	0.156477	−	0.987682i	$-0.449986\pi$
$758$	576.375			0.760389
$759$		0			0
$760$	401.421	−	488.859i	0.528185	−	0.643235i
$761$			1361.43i			1.78900i	0.447067	+	0.894501i	$0.352469\pi$
$761$			1361.43i			1.78900i	−0.447067	+	0.894501i	$0.647531\pi$
$762$		0			0
$763$		−	195.830i		−	0.256659i
$764$			100.183i			0.131130i
$765$		0			0
$766$	1009.30			1.31763
$767$	−1400.22			−1.82558
$768$		0			0
$769$	−339.978			−0.442104			−0.221052	−	0.975262i	$-0.570949\pi$
$769$	−339.978			−0.442104			−0.221052	+	0.975262i	$0.570949\pi$
$770$	46.6004	−	56.7510i	0.0605200	−	0.0737026i
$771$		0			0
$772$		−	88.9056i		−	0.115163i
$773$	−544.045			−0.703810			−0.351905	−	0.936036i	$-0.614466\pi$
$773$	−544.045			−0.703810			−0.351905	+	0.936036i	$0.614466\pi$
$774$		0			0
$775$	−239.590	−	1207.96i	−0.309148	−	1.55866i
$776$			702.902i			0.905802i
$777$		0			0
$778$		−	630.186i		−	0.810008i
$779$			30.8217i			0.0395658i
$780$		0			0
$781$	−251.934			−0.322578
$782$	640.395			0.818920
$783$		0			0
$784$	−772.636			−0.985505
$785$	251.469	+	206.491i	0.320343	+	0.263046i
$786$		0			0
$787$			69.1884i			0.0879141i	0.999033	+	0.0439571i	$0.0139965\pi$
$787$			69.1884i			0.0879141i	−0.999033	+	0.0439571i	$0.986004\pi$
$788$	−99.3034			−0.126020
$789$		0			0
$790$	818.028	−	996.212i	1.03548	−	1.26103i
$791$		−	453.044i		−	0.572749i
$792$		0			0
$793$		−	914.406i		−	1.15310i
$794$		−	85.0486i		−	0.107114i
$795$		0			0
$796$	−5.41529			−0.00680313
$797$	312.876			0.392567			0.196284	−	0.980547i	$-0.437113\pi$
$797$	312.876			0.392567			0.196284	+	0.980547i	$0.437113\pi$
$798$		0			0
$799$	−1038.99			−1.30036
$800$	−29.0086	−	146.256i	−0.0362608	−	0.182820i
$801$		0			0
$802$		−	602.218i		−	0.750895i
$803$	−30.5254			−0.0380142
$804$		0			0
$805$	144.184	+	118.395i	0.179111	+	0.147075i
$806$			1801.68i			2.23533i
$807$		0			0
$808$		−	1043.56i		−	1.29154i
$809$			686.872i			0.849038i	0.905419	+	0.424519i	$0.139557\pi$
$809$			686.872i			0.849038i	−0.905419	+	0.424519i	$0.860443\pi$
$810$		0			0
$811$	900.127			1.10990			0.554949	−	0.831884i	$-0.312738\pi$
$811$	900.127			1.10990			0.554949	+	0.831884i	$0.312738\pi$
$812$	30.4941			0.0375543
$813$		0			0
$814$	86.4803			0.106241
$815$	450.913	+	370.262i	0.553268	+	0.454309i
$816$		0			0
$817$			325.557i			0.398478i
$818$	−300.626			−0.367514
$819$		0			0
$820$	2.66977	+	2.19225i	0.00325582	+	0.00267348i
$821$		−	1058.47i		−	1.28924i	−0.764503	−	0.644620i	$-0.777015\pi$
$821$		−	1058.47i		−	1.28924i	0.764503	−	0.644620i	$-0.222985\pi$
$822$		0			0
$823$			257.006i			0.312280i	0.987735	+	0.156140i	$0.0499051\pi$
$823$			257.006i			0.312280i	−0.987735	+	0.156140i	$0.950095\pi$
$824$			456.760i			0.554320i
$825$		0			0
$826$	354.542			0.429228
$827$	−1068.89			−1.29249			−0.646244	−	0.763131i	$-0.723661\pi$
$827$	−1068.89			−1.29249			−0.646244	+	0.763131i	$0.723661\pi$
$828$		0			0
$829$	−1079.37			−1.30202			−0.651009	−	0.759070i	$-0.725654\pi$
$829$	−1079.37			−1.30202			−0.651009	+	0.759070i	$0.725654\pi$
$830$	434.875	−	529.601i	0.523946	−	0.638073i
$831$		0			0
$832$		−	995.964i		−	1.19707i
$833$	−773.500			−0.928571
$834$		0			0
$835$	−209.665	+	255.334i	−0.251096	+	0.305790i
$836$		−	20.6903i		−	0.0247491i
$837$		0			0
$838$		−	883.757i		−	1.05460i
$839$			1026.52i			1.22350i	0.791052	+	0.611749i	$0.209534\pi$
$839$			1026.52i			1.22350i	−0.791052	+	0.611749i	$0.790466\pi$
$840$		0			0
$841$	−642.325			−0.763763
$842$	461.546			0.548155
$843$		0			0
$844$	−25.8160			−0.0305877
$845$	434.198	−	528.776i	0.513844	−	0.625770i
$846$		0			0
$847$			23.2903i			0.0274974i
$848$	102.518			0.120894
$849$		0			0
$850$	−176.745	−	891.112i	−0.207935	−	1.04837i
$851$			219.716i			0.258185i
$852$		0			0
$853$		−	1000.74i		−	1.17321i	−0.809875	−	0.586603i	$-0.800465\pi$
$853$		−	1000.74i		−	1.17321i	0.809875	−	0.586603i	$-0.199535\pi$
$854$			231.532i			0.271115i
$855$		0			0
$856$	−718.850			−0.839778
$857$	−730.751			−0.852685			−0.426342	−	0.904562i	$-0.640198\pi$
$857$	−730.751			−0.852685			−0.426342	+	0.904562i	$0.640198\pi$
$858$		0			0
$859$	−1047.38			−1.21930			−0.609652	−	0.792669i	$-0.708691\pi$
$859$	−1047.38			−1.21930			−0.609652	+	0.792669i	$0.708691\pi$
$860$	28.1997	+	23.1558i	0.0327903	+	0.0269254i
$861$		0			0
$862$			641.162i			0.743808i
$863$	703.978			0.815733			0.407867	−	0.913042i	$-0.366273\pi$
$863$	703.978			0.815733			0.407867	+	0.913042i	$0.366273\pi$
$864$		0			0
$865$	−10.4093	+	12.6767i	−0.0120339	+	0.0146552i
$866$		−	650.629i		−	0.751303i
$867$		0			0
$868$		−	39.0023i		−	0.0449336i
$869$			408.840i			0.470472i
$870$		0			0
$871$	1307.12			1.50071
$872$	701.404			0.804362
$873$		0			0
$874$	614.851			0.703490
$875$	124.953	−	233.309i	0.142804	−	0.266639i
$876$		0			0
$877$		−	592.692i		−	0.675817i	−0.941179	−	0.337909i	$-0.890281\pi$
$877$		−	592.692i		−	0.675817i	0.941179	−	0.337909i	$-0.109719\pi$
$878$	1261.38			1.43665
$879$		0			0
$880$	222.435	+	182.650i	0.252767	+	0.207556i
$881$			481.567i			0.546614i	0.961927	+	0.273307i	$0.0881175\pi$
$881$			481.567i			0.546614i	−0.961927	+	0.273307i	$0.911882\pi$
$882$		0			0
$883$			416.531i			0.471722i	0.971787	+	0.235861i	$0.0757910\pi$
$883$			416.531i			0.471722i	−0.971787	+	0.235861i	$0.924209\pi$
$884$			113.631i			0.128542i
$885$		0			0
$886$	324.581			0.366344
$887$	−1409.71			−1.58930			−0.794651	−	0.607066i	$-0.792346\pi$
$887$	−1409.71			−1.58930			−0.794651	+	0.607066i	$0.792346\pi$
$888$		0			0
$889$	312.737			0.351785
$890$	761.708	+	625.468i	0.855852	+	0.702773i
$891$		0			0
$892$		−	109.654i		−	0.122930i
$893$	−997.546			−1.11707
$894$		0			0
$895$	711.215	+	584.006i	0.794654	+	0.652521i
$896$			302.695i			0.337829i
$897$		0			0
$898$			16.3537i			0.0182113i
$899$			1897.19i			2.11033i
$900$		0			0
$901$	102.633			0.113910
$902$	−12.8155			−0.0142079
$903$		0			0
$904$	1622.66			1.79498
$905$	−79.0346	+	96.2501i	−0.0873311	+	0.106354i
$906$		0			0
$907$			1639.35i			1.80744i	0.428127	+	0.903718i	$0.359173\pi$
$907$			1639.35i			1.80744i	−0.428127	+	0.903718i	$0.640827\pi$
$908$	−24.8172			−0.0273317
$909$		0			0
$910$	−245.720	+	299.243i	−0.270022	+	0.328839i
$911$		−	1073.97i		−	1.17889i	−0.807808	−	0.589446i	$-0.799346\pi$
$911$		−	1073.97i		−	1.17889i	0.807808	−	0.589446i	$-0.200654\pi$
$912$		0			0
$913$			217.345i			0.238056i
$914$		−	1709.67i		−	1.87054i
$915$		0			0
$916$	129.590			0.141474
$917$	0.0731044			7.97213e−5
$918$		0			0
$919$	653.884			0.711517			0.355759	−	0.934578i	$-0.384223\pi$
$919$	653.884			0.711517			0.355759	+	0.934578i	$0.384223\pi$
$920$	−424.055	+	516.423i	−0.460929	+	0.561329i
$921$		0			0
$922$			256.770i			0.278492i
$923$	1328.43			1.43925
$924$		0			0
$925$	305.735	−	60.6400i	0.330524	−	0.0655568i
$926$		−	713.060i		−	0.770043i
$927$		0			0
$928$			229.705i			0.247526i
$929$			1604.50i			1.72713i	0.504239	+	0.863564i	$0.331773\pi$
$929$			1604.50i			1.72713i	−0.504239	+	0.863564i	$0.668227\pi$
$930$		0			0
$931$	−742.645			−0.797686
$932$	121.215			0.130059
$933$		0			0
$934$	−1022.99			−1.09528
$935$	222.683	+	182.854i	0.238164	+	0.195566i
$936$		0			0
$937$			977.381i			1.04310i	0.853222	+	0.521548i	$0.174645\pi$
$937$			977.381i			1.04310i	−0.853222	+	0.521548i	$0.825355\pi$
$938$	−330.970			−0.352846
$939$		0			0
$940$	−70.9523	+	86.4073i	−0.0754812	+	0.0919227i
$941$			929.571i			0.987854i	0.869503	+	0.493927i	$0.164439\pi$
$941$			929.571i			0.987854i	−0.869503	+	0.493927i	$0.835561\pi$
$942$		0			0
$943$		−	32.5596i		−	0.0345277i
$944$			1389.62i			1.47206i
$945$		0			0
$946$	−135.364			−0.143091
$947$	1064.91			1.12451			0.562256	−	0.826963i	$-0.309934\pi$
$947$	1064.91			1.12451			0.562256	+	0.826963i	$0.309934\pi$
$948$		0			0
$949$	160.958			0.169608
$950$	−169.695	−	855.566i	−0.178626	−	0.900596i
$951$		0			0
$952$			278.989i			0.293056i
$953$	1091.80			1.14564			0.572820	−	0.819681i	$-0.305849\pi$
$953$	1091.80			1.14564			0.572820	+	0.819681i	$0.305849\pi$
$954$		0			0
$955$	−1035.23	−	850.069i	−1.08401	−	0.890125i
$956$			30.8770i			0.0322981i
$957$		0			0
$958$		−	840.993i		−	0.877863i
$959$			471.982i			0.492161i
$960$		0			0
$961$	1465.53			1.52500
$962$	−456.004			−0.474016
$963$		0			0
$964$	126.971			0.131712
$965$	918.695	+	754.376i	0.952016	+	0.781737i
$966$		0			0
$967$		−	1645.98i		−	1.70215i	−0.525043	−	0.851076i	$-0.675951\pi$
$967$		−	1645.98i		−	1.70215i	0.525043	−	0.851076i	$-0.324049\pi$
$968$	−83.4187			−0.0861763
$969$		0			0
$970$	749.064	+	615.085i	0.772231	+	0.634108i
$971$		−	500.415i		−	0.515361i	−0.966230	−	0.257680i	$-0.917042\pi$
$971$		−	500.415i		−	0.515361i	0.966230	−	0.257680i	$-0.0829582\pi$
$972$		0			0
$973$		−	331.897i		−	0.341106i
$974$		−	482.761i		−	0.495648i
$975$		0			0
$976$	−907.487			−0.929803
$977$	1089.28			1.11493			0.557463	−	0.830202i	$-0.311775\pi$
$977$	1089.28			1.11493			0.557463	+	0.830202i	$0.311775\pi$
$978$		0			0
$979$	−312.601			−0.319307
$980$	−52.8220	+	64.3278i	−0.0539000	+	0.0656406i
$981$		0			0
$982$			1712.97i			1.74436i
$983$	1805.31			1.83654			0.918268	−	0.395959i	$-0.129588\pi$
$983$	1805.31			1.83654			0.918268	+	0.395959i	$0.129588\pi$
$984$		0			0
$985$	842.603	−	1026.14i	0.855435	−	1.04177i
$986$			1399.55i			1.41942i
$987$		0			0
$988$			109.098i			0.110423i
$989$		−	343.913i		−	0.347738i
$990$		0			0
$991$	−277.330			−0.279848			−0.139924	−	0.990162i	$-0.544686\pi$
$991$	−277.330			−0.279848			−0.139924	+	0.990162i	$0.544686\pi$
$992$	293.795			0.296164
$993$		0			0
$994$	−336.364			−0.338394
$995$	45.9495	−	55.9583i	0.0461804	−	0.0562395i
$996$		0			0
$997$		−	1333.87i		−	1.33788i	−0.743316	−	0.668940i	$-0.766748\pi$
$997$		−	1333.87i		−	1.33788i	0.743316	−	0.668940i	$-0.233252\pi$
$998$	1377.51			1.38027
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.g.a.89.11	✓	40
3.2	odd	2	inner	495.3.g.a.89.29	yes	40
5.4	even	2	inner	495.3.g.a.89.30	yes	40
15.14	odd	2	inner	495.3.g.a.89.12	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.3.g.a.89.11	✓	40	1.1	even	1	trivial
495.3.g.a.89.12	yes	40	15.14	odd	2	inner
495.3.g.a.89.29	yes	40	3.2	odd	2	inner
495.3.g.a.89.30	yes	40	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.09140 q^{2} +0.373951 q^{4} +(-3.17303 + 3.86418i) q^{5} -2.11730i q^{7} +7.58352 q^{8} +(6.63607 - 8.08155i) q^{10} +3.31662i q^{11} -17.4883i q^{13} +4.42812i q^{14} -17.3560 q^{16} -17.3754 q^{17} -16.6823 q^{19} +(-1.18656 + 1.44502i) q^{20} -6.93639i q^{22} +17.6229 q^{23} +(-4.86380 - 24.5223i) q^{25} +36.5750i q^{26} -0.791768i q^{28} +38.5140i q^{29} +49.2598 q^{31} +5.96419 q^{32} +36.3388 q^{34} +(8.18164 + 6.71826i) q^{35} +12.4676i q^{37} +34.8893 q^{38} +(-24.0627 + 29.3041i) q^{40} -1.84757i q^{41} -19.5151i q^{43} +1.24026i q^{44} -36.8565 q^{46} +59.7968 q^{47} +44.5170 q^{49} +(10.1721 + 51.2859i) q^{50} -6.53977i q^{52} -5.90680 q^{53} +(-12.8160 - 10.5237i) q^{55} -16.0566i q^{56} -80.5481i q^{58} -80.0660i q^{59} +52.2868 q^{61} -103.022 q^{62} +56.9504 q^{64} +(67.5779 + 55.4908i) q^{65} +74.7426i q^{67} -6.49754 q^{68} +(-17.1111 - 14.0506i) q^{70} +75.9608i q^{71} +9.20374i q^{73} -26.0748i q^{74} -6.23836 q^{76} +7.02230 q^{77} +123.270 q^{79} +(55.0709 - 67.0666i) q^{80} +3.86401i q^{82} +65.5321 q^{83} +(55.1325 - 67.1416i) q^{85} +40.8139i q^{86} +25.1517i q^{88} +94.2528i q^{89} -37.0280 q^{91} +6.59010 q^{92} -125.059 q^{94} +(52.9333 - 64.4633i) q^{95} +92.6881i q^{97} -93.1029 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 72 q^{4} + 8 q^{10} + 184 q^{16} - 80 q^{19} + 32 q^{25} - 16 q^{31} - 160 q^{34} - 136 q^{40} + 560 q^{46} - 104 q^{49} - 96 q^{61} + 264 q^{64} - 872 q^{70} - 176 q^{76} - 672 q^{79} + 16 q^{85}+ \cdots + 400 q^{94}+O(q^{100}) \) 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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