LMFDB - Embedded newform 648.4.i.n.433.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,4,Mod(217,648)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("648.217"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-8,0,30,0,0,0,46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-67})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 16x^{2} - 17x + 289 $ x^4 - x^3 - 16x^2 - 17x + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.1
Root			$-3.29436 + 2.47935i$ of defining polynomial
Character	$\chi$	$=$	648.433
Dual form			648.4.i.n.217.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.08872 + 15.7421i) q^{5} +(0.411277 + 0.712352i) q^{7} +(32.7662 + 56.7527i) q^{11} +(-15.9113 + 27.5591i) q^{13} +124.420 q^{17} +27.8872 q^{19} +(21.5887 - 37.3928i) q^{23} +(-102.710 - 177.899i) q^{25} +(-19.9113 - 34.4873i) q^{29} +(-147.420 + 255.338i) q^{31} -14.9519 q^{35} -104.177 q^{37} +(-153.823 + 266.428i) q^{41} +(180.541 + 312.706i) q^{43} +(-198.823 - 344.371i) q^{47} +(171.162 - 296.461i) q^{49} +107.161 q^{53} -1191.21 q^{55} +(-94.4677 + 163.623i) q^{59} +(49.5083 + 85.7509i) q^{61} +(-289.226 - 500.955i) q^{65} +(-212.524 + 368.102i) q^{67} -445.532 q^{71} -499.806 q^{73} +(-26.9519 + 46.6821i) q^{77} +(-285.138 - 493.873i) q^{79} +(-654.371 - 1133.40i) q^{83} +(-1130.82 + 1958.63i) q^{85} +1323.00 q^{89} -26.1757 q^{91} +(-253.459 + 439.004i) q^{95} +(-472.436 - 818.283i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} + 30 q^{7} + 46 q^{11} - 92 q^{13} + 44 q^{17} - 172 q^{19} + 58 q^{23} - 184 q^{25} - 108 q^{29} - 136 q^{31} + 564 q^{35} - 360 q^{37} - 672 q^{41} + 70 q^{43} - 852 q^{47} - 166 q^{49}+ \cdots - 472 q^{97}+O(q^{100}) $ 4 * q - 8 * q^5 + 30 * q^7 + 46 * q^11 - 92 * q^13 + 44 * q^17 - 172 * q^19 + 58 * q^23 - 184 * q^25 - 108 * q^29 - 136 * q^31 + 564 * q^35 - 360 * q^37 - 672 * q^41 + 70 * q^43 - 852 * q^47 - 166 * q^49 + 1336 * q^53 - 2780 * q^55 - 548 * q^59 - 284 * q^61 + 34 * q^65 - 1162 * q^67 - 1612 * q^71 - 3020 * q^73 + 516 * q^77 + 22 * q^79 - 1540 * q^83 - 3304 * q^85 + 5292 * q^89 - 3564 * q^91 - 1666 * q^95 - 472 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−9.08872	+	15.7421i	−0.812920	+	1.40802i	0.0978916	+	0.995197i	$0.468790\pi$
$5$	−9.08872	+	15.7421i	−0.812920	+	1.40802i	−0.910812	+	0.412822i	$0.864543\pi$
$6$		0			0
$7$	0.411277	+	0.712352i	0.0222068	+	0.0384634i	0.876915	−	0.480645i	$-0.159597\pi$
$7$	0.411277	+	0.712352i	0.0222068	+	0.0384634i	−0.854708	+	0.519108i	$0.826264\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	32.7662	+	56.7527i	0.898125	+	1.55560i	0.829889	+	0.557928i	$0.188404\pi$
$11$	32.7662	+	56.7527i	0.898125	+	1.55560i	0.0682357	+	0.997669i	$0.478263\pi$
$12$		0			0
$13$	−15.9113	+	27.5591i	−0.339461	+	0.587964i	−0.984331	−	0.176328i	$-0.943578\pi$
$13$	−15.9113	+	27.5591i	−0.339461	+	0.587964i	0.644870	+	0.764292i	$0.276911\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	124.420			1.77507			0.887535	−	0.460741i	$-0.152416\pi$
$17$	124.420			1.77507			0.887535	+	0.460741i	$0.152416\pi$
$18$		0			0
$19$	27.8872			0.336725			0.168362	−	0.985725i	$-0.446152\pi$
$19$	27.8872			0.336725			0.168362	+	0.985725i	$0.446152\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	21.5887	−	37.3928i	0.195720	−	0.338997i	−0.751416	−	0.659828i	$-0.770629\pi$
$23$	21.5887	−	37.3928i	0.195720	−	0.338997i	0.947136	+	0.320831i	$0.103962\pi$
$24$		0			0
$25$	−102.710	−	177.899i	−0.821678	−	1.42319i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−19.9113	−	34.4873i	−0.127498	−	0.220832i	0.795209	−	0.606336i	$-0.207361\pi$
$29$	−19.9113	−	34.4873i	−0.127498	−	0.220832i	−0.922706	+	0.385503i	$0.874028\pi$
$30$		0			0
$31$	−147.420	+	255.338i	−0.854108	+	1.47936i	0.0233627	+	0.999727i	$0.492563\pi$
$31$	−147.420	+	255.338i	−0.854108	+	1.47936i	−0.877470	+	0.479631i	$0.840771\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−14.9519			−0.0722096
$36$		0			0
$37$	−104.177			−0.462883			−0.231441	−	0.972849i	$-0.574344\pi$
$37$	−104.177			−0.462883			−0.231441	+	0.972849i	$0.574344\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−153.823	+	266.428i	−0.585928	+	1.01486i	0.408831	+	0.912610i	$0.365937\pi$
$41$	−153.823	+	266.428i	−0.585928	+	1.01486i	−0.994759	+	0.102247i	$0.967397\pi$
$42$		0			0
$43$	180.541	+	312.706i	0.640283	+	1.10900i	0.985369	+	0.170432i	$0.0545164\pi$
$43$	180.541	+	312.706i	0.640283	+	1.10900i	−0.345086	+	0.938571i	$0.612150\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−198.823	−	344.371i	−0.617048	−	1.06876i	−0.990022	−	0.140916i	$-0.954995\pi$
$47$	−198.823	−	344.371i	−0.617048	−	1.06876i	0.372974	−	0.927842i	$-0.378338\pi$
$48$		0			0
$49$	171.162	−	296.461i	0.499014	−	0.864317i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	107.161			0.277730			0.138865	−	0.990311i	$-0.455655\pi$
$53$	107.161			0.277730			0.138865	+	0.990311i	$0.455655\pi$
$54$		0			0
$55$	−1191.21			−2.92041
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−94.4677	+	163.623i	−0.208452	+	0.361049i	−0.951227	−	0.308492i	$-0.900176\pi$
$59$	−94.4677	+	163.623i	−0.208452	+	0.361049i	0.742775	+	0.669541i	$0.233509\pi$
$60$		0			0
$61$	49.5083	+	85.7509i	0.103916	+	0.179988i	0.913295	−	0.407299i	$-0.133529\pi$
$61$	49.5083	+	85.7509i	0.103916	+	0.179988i	−0.809379	+	0.587287i	$0.800196\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−289.226	−	500.955i	−0.551910	−	0.955935i
$66$		0			0
$67$	−212.524	+	368.102i	−0.387522	+	0.671207i	−0.992116	−	0.125327i	$-0.960002\pi$
$67$	−212.524	+	368.102i	−0.387522	+	0.671207i	0.604594	+	0.796534i	$0.293335\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−445.532			−0.744718			−0.372359	−	0.928089i	$-0.621451\pi$
$71$	−445.532			−0.744718			−0.372359	+	0.928089i	$0.621451\pi$
$72$		0			0
$73$	−499.806			−0.801341			−0.400670	−	0.916222i	$-0.631223\pi$
$73$	−499.806			−0.801341			−0.400670	+	0.916222i	$0.631223\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−26.9519	+	46.6821i	−0.0398890	+	0.0690898i
$78$		0			0
$79$	−285.138	−	493.873i	−0.406082	−	0.703355i	0.588365	−	0.808596i	$-0.299772\pi$
$79$	−285.138	−	493.873i	−0.406082	−	0.703355i	−0.994447	+	0.105241i	$0.966439\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−654.371	−	1133.40i	−0.865381	−	1.49888i	−0.866669	−	0.498884i	$-0.833743\pi$
$83$	−654.371	−	1133.40i	−0.865381	−	1.49888i	0.00128787	−	0.999999i	$-0.499590\pi$
$84$		0			0
$85$	−1130.82	+	1958.63i	−1.44299	+	2.49933i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1323.00			1.57570			0.787852	−	0.615864i	$-0.211193\pi$
$89$	1323.00			1.57570			0.787852	+	0.615864i	$0.211193\pi$
$90$		0			0
$91$	−26.1757			−0.0301534
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−253.459	+	439.004i	−0.273730	+	0.474115i
$96$		0			0
$97$	−472.436	−	818.283i	−0.494522	−	0.856537i	0.505458	−	0.862851i	$-0.331323\pi$
$97$	−472.436	−	818.283i	−0.494522	−	0.856537i	−0.999980	+	0.00631404i	$0.997990\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	699.050	+	1210.79i	0.688694	+	1.19285i	0.972261	+	0.233900i	$0.0751488\pi$
$101$	699.050	+	1210.79i	0.688694	+	1.19285i	−0.283567	+	0.958952i	$0.591518\pi$
$102$		0			0
$103$	480.275	−	831.861i	0.459446	−	0.795784i	−0.539486	−	0.841995i	$-0.681381\pi$
$103$	480.275	−	831.861i	0.459446	−	0.795784i	0.998932	+	0.0462110i	$0.0147147\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1935.87			1.74905			0.874523	−	0.484983i	$-0.161174\pi$
$107$	1935.87			1.74905			0.874523	+	0.484983i	$0.161174\pi$
$108$		0			0
$109$	−1305.40			−1.14711			−0.573555	−	0.819167i	$-0.694436\pi$
$109$	−1305.40			−1.14711			−0.573555	+	0.819167i	$0.694436\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−547.016	+	947.459i	−0.455389	+	0.788756i	−0.998710	−	0.0507683i	$-0.983833\pi$
$113$	−547.016	+	947.459i	−0.455389	+	0.788756i	0.543322	+	0.839524i	$0.317166\pi$
$114$		0			0
$115$	392.428	+	679.705i	0.318209	+	0.551155i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	51.1709	+	88.6305i	0.0394187	+	0.0682752i
$120$		0			0
$121$	−1481.74	+	2566.46i	−1.11326	+	1.92822i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1461.82			1.04600
$126$		0			0
$127$	−1301.05			−0.909050			−0.454525	−	0.890734i	$-0.650191\pi$
$127$	−1301.05			−0.909050			−0.454525	+	0.890734i	$0.650191\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−978.833	+	1695.39i	−0.652832	+	1.13074i	0.329601	+	0.944120i	$0.393086\pi$
$131$	−978.833	+	1695.39i	−0.652832	+	1.13074i	−0.982433	+	0.186618i	$0.940247\pi$
$132$		0			0
$133$	11.4694	+	19.8655i	0.00747760	+	0.0129516i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1276.98	+	2211.80i	0.796351	+	1.37932i	0.921978	+	0.387243i	$0.126573\pi$
$137$	1276.98	+	2211.80i	0.796351	+	1.37932i	−0.125626	+	0.992078i	$0.540094\pi$
$138$		0			0
$139$	−102.630	+	177.761i	−0.0626258	+	0.108471i	−0.895638	−	0.444783i	$-0.853281\pi$
$139$	−102.630	+	177.761i	−0.0626258	+	0.108471i	0.833013	+	0.553254i	$0.186614\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2085.41			−1.21951
$144$		0			0
$145$	723.872			0.414582
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−82.1202	+	142.236i	−0.0451513	+	0.0782044i	−0.887718	−	0.460388i	$-0.847710\pi$
$149$	−82.1202	+	142.236i	−0.0451513	+	0.0782044i	0.842567	+	0.538592i	$0.181044\pi$
$150$		0			0
$151$	−1713.86	−	2968.49i	−0.923654	−	1.59982i	−0.793712	−	0.608294i	$-0.791854\pi$
$151$	−1713.86	−	2968.49i	−0.923654	−	1.59982i	−0.129942	−	0.991522i	$-0.541479\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2679.71	−	4641.40i	−1.38864	−	2.40520i
$156$		0			0
$157$	494.042	−	855.706i	0.251139	−	0.434986i	−0.712700	−	0.701468i	$-0.752528\pi$
$157$	494.042	−	855.706i	0.251139	−	0.434986i	0.963840	+	0.266482i	$0.0858614\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	35.5157			0.0173853
$162$		0			0
$163$	−2845.45			−1.36732			−0.683660	−	0.729801i	$-0.739613\pi$
$163$	−2845.45			−1.36732			−0.683660	+	0.729801i	$0.739613\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1602.95	−	2776.38i	0.742752	−	1.28648i	−0.208485	−	0.978026i	$-0.566853\pi$
$167$	1602.95	−	2776.38i	0.742752	−	1.28648i	0.951238	−	0.308459i	$-0.0998133\pi$
$168$		0			0
$169$	592.163	+	1025.66i	0.269532	+	0.466844i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−892.817	−	1546.40i	−0.392368	−	0.679601i	0.600394	−	0.799705i	$-0.295011\pi$
$173$	−892.817	−	1546.40i	−0.392368	−	0.679601i	−0.992761	+	0.120104i	$0.961677\pi$
$174$		0			0
$175$	84.4843	−	146.331i	0.0364938	−	0.0632091i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1318.84			−0.550696			−0.275348	−	0.961345i	$-0.588793\pi$
$179$	−1318.84			−0.550696			−0.275348	+	0.961345i	$0.588793\pi$
$180$		0			0
$181$	3462.97			1.42210			0.711051	−	0.703141i	$-0.248220\pi$
$181$	3462.97			1.42210			0.711051	+	0.703141i	$0.248220\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	946.840	−	1639.97i	0.376287	−	0.651748i
$186$		0			0
$187$	4076.75	+	7061.14i	1.59423	+	2.76129i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−795.655	−	1378.12i	−0.301422	−	0.522078i	0.675036	−	0.737784i	$-0.264128\pi$
$191$	−795.655	−	1378.12i	−0.301422	−	0.522078i	−0.976458	+	0.215706i	$0.930795\pi$
$192$		0			0
$193$	382.228	−	662.038i	0.142556	−	0.246915i	−0.785902	−	0.618351i	$-0.787801\pi$
$193$	382.228	−	662.038i	0.142556	−	0.246915i	0.928459	+	0.371436i	$0.121134\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1191.63			0.430965			0.215483	−	0.976508i	$-0.430868\pi$
$197$	1191.63			0.430965			0.215483	+	0.976508i	$0.430868\pi$
$198$		0			0
$199$	−2708.20			−0.964719			−0.482360	−	0.875973i	$-0.660220\pi$
$199$	−2708.20			−0.964719			−0.482360	+	0.875973i	$0.660220\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	16.3781	−	28.3677i	0.00566264	−	0.00980798i
$204$		0			0
$205$	−2796.10	−	4842.99i	−0.952625	−	1.65000i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	913.758	+	1582.68i	0.302421	+	0.523808i
$210$		0			0
$211$	1649.48	−	2856.98i	0.538173	−	0.932144i	−0.460829	−	0.887489i	$-0.652448\pi$
$211$	1649.48	−	2856.98i	0.538173	−	0.932144i	0.999002	−	0.0446549i	$-0.0142188\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6563.54			−2.08200
$216$		0			0
$217$	−242.521			−0.0758682
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1979.67	+	3428.90i	−0.602567	+	1.04368i
$222$		0			0
$223$	1260.83	+	2183.83i	0.378617	+	0.655784i	0.990861	−	0.134885i	$-0.0430664\pi$
$223$	1260.83	+	2183.83i	0.378617	+	0.655784i	−0.612244	+	0.790669i	$0.709733\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1657.33	+	2870.59i	0.484586	+	0.839328i	0.999843	−	0.0177077i	$-0.00563684\pi$
$227$	1657.33	+	2870.59i	0.484586	+	0.839328i	−0.515257	+	0.857036i	$0.672304\pi$
$228$		0			0
$229$	2064.83	−	3576.39i	0.595842	−	1.03203i	−0.397585	−	0.917565i	$-0.630152\pi$
$229$	2064.83	−	3576.39i	0.595842	−	1.03203i	0.993427	−	0.114464i	$-0.0365151\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	986.257			0.277304			0.138652	−	0.990341i	$-0.455723\pi$
$233$	986.257			0.277304			0.138652	+	0.990341i	$0.455723\pi$
$234$		0			0
$235$	7228.17			2.00644
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−179.499	+	310.902i	−0.0485809	+	0.0841446i	−0.889293	−	0.457337i	$-0.848803\pi$
$239$	−179.499	+	310.902i	−0.0485809	+	0.0841446i	0.840712	+	0.541482i	$0.182137\pi$
$240$		0			0
$241$	−1021.02	−	1768.45i	−0.272902	−	0.472680i	0.696702	−	0.717361i	$-0.254650\pi$
$241$	−1021.02	−	1768.45i	−0.272902	−	0.472680i	−0.969604	+	0.244681i	$0.921317\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3111.28	+	5388.90i	0.811317	+	1.40524i
$246$		0			0
$247$	−443.721	+	768.548i	−0.114305	+	0.197982i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2609.69			−0.656264			−0.328132	−	0.944632i	$-0.606419\pi$
$251$	−2609.69			−0.656264			−0.328132	+	0.944632i	$0.606419\pi$
$252$		0			0
$253$	2829.52			0.703124
$254$		0			0
$255$		0			0
$256$		0			0
$257$	439.419	−	761.096i	0.106654	−	0.184731i	−0.807759	−	0.589514i	$-0.799319\pi$
$257$	439.419	−	761.096i	0.106654	−	0.184731i	0.914413	+	0.404783i	$0.132653\pi$
$258$		0			0
$259$	−42.8457	−	74.2110i	−0.0102792	−	0.0178040i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1608.67	+	2786.29i	0.377166	+	0.653271i	0.990649	−	0.136438i	$-0.0435653\pi$
$263$	1608.67	+	2786.29i	0.377166	+	0.653271i	−0.613483	+	0.789708i	$0.710232\pi$
$264$		0			0
$265$	−973.955	+	1686.94i	−0.225772	+	0.391049i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7978.38			1.80837			0.904183	−	0.427146i	$-0.140481\pi$
$269$	7978.38			1.80837			0.904183	+	0.427146i	$0.140481\pi$
$270$		0			0
$271$	4248.21			0.952251			0.476126	−	0.879377i	$-0.342041\pi$
$271$	4248.21			0.952251			0.476126	+	0.879377i	$0.342041\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6730.81	−	11658.1i	1.47594	−	2.55640i
$276$		0			0
$277$	−415.517	−	719.697i	−0.0901301	−	0.156110i	0.817436	−	0.576020i	$-0.195395\pi$
$277$	−415.517	−	719.697i	−0.0901301	−	0.156110i	−0.907566	+	0.419910i	$0.862062\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1481.00	+	2565.17i	0.314410	+	0.544573i	0.979312	−	0.202357i	$-0.0648601\pi$
$281$	1481.00	+	2565.17i	0.314410	+	0.544573i	−0.664902	+	0.746930i	$0.731527\pi$
$282$		0			0
$283$	2215.42	−	3837.22i	0.465346	−	0.806004i	−0.533871	−	0.845566i	$-0.679263\pi$
$283$	2215.42	−	3837.22i	0.465346	−	0.806004i	0.999217	+	0.0395625i	$0.0125964\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−253.054			−0.0520465
$288$		0			0
$289$	10567.2			2.15087
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1711.43	−	2964.29i	0.341239	−	0.591043i	−0.643424	−	0.765510i	$-0.722487\pi$
$293$	1711.43	−	2964.29i	0.341239	−	0.591043i	0.984663	+	0.174467i	$0.0558202\pi$
$294$		0			0
$295$	−1717.18	−	2974.24i	−0.338909	−	0.587007i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	687.008	+	1189.93i	0.132879	+	0.230153i
$300$		0			0
$301$	−148.504	+	257.217i	−0.0284374	+	0.0492549i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1799.87			−0.337902
$306$		0			0
$307$	139.734			0.0259774			0.0129887	−	0.999916i	$-0.495865\pi$
$307$	139.734			0.0259774			0.0129887	+	0.999916i	$0.495865\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3799.94	+	6581.69i	−0.692845	+	1.20004i	0.278057	+	0.960565i	$0.410310\pi$
$311$	−3799.94	+	6581.69i	−0.692845	+	1.20004i	−0.970902	+	0.239478i	$0.923024\pi$
$312$		0			0
$313$	1508.76	+	2613.25i	0.272460	+	0.471915i	0.969491	−	0.245126i	$-0.0788293\pi$
$313$	1508.76	+	2613.25i	0.272460	+	0.471915i	−0.697031	+	0.717041i	$0.745496\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	960.603	+	1663.81i	0.170198	+	0.294792i	0.938489	−	0.345309i	$-0.112226\pi$
$317$	960.603	+	1663.81i	0.170198	+	0.294792i	−0.768291	+	0.640101i	$0.778893\pi$
$318$		0			0
$319$	1304.83	−	2260.04i	0.229018	−	0.396670i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3469.72			0.597710
$324$		0			0
$325$	6536.98			1.11571
$326$		0			0
$327$		0			0
$328$		0			0
$329$	163.542	−	283.263i	0.0274054	−	0.0474675i
$330$		0			0
$331$	−456.206	−	790.172i	−0.0757563	−	0.131214i	0.825659	−	0.564170i	$-0.190804\pi$
$331$	−456.206	−	790.172i	−0.0757563	−	0.131214i	−0.901415	+	0.432956i	$0.857470\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3863.14	−	6691.16i	−0.630048	−	1.09128i
$336$		0			0
$337$	3007.31	−	5208.82i	0.486109	−	0.841965i	−0.513764	−	0.857932i	$-0.671749\pi$
$337$	3007.31	−	5208.82i	0.486109	−	0.841965i	0.999873	+	0.0159664i	$0.00508248\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−19321.5			−3.06838
$342$		0			0
$343$	563.715			0.0887398
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1369.92	+	2372.77i	−0.211934	+	0.367080i	−0.952320	−	0.305102i	$-0.901309\pi$
$347$	−1369.92	+	2372.77i	−0.211934	+	0.367080i	0.740386	+	0.672182i	$0.234643\pi$
$348$		0			0
$349$	3804.34	+	6589.32i	0.583501	+	1.01065i	0.995060	+	0.0992704i	$0.0316509\pi$
$349$	3804.34	+	6589.32i	0.583501	+	1.01065i	−0.411560	+	0.911383i	$0.635016\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4362.56	+	7556.17i	0.657777	+	1.13930i	0.981190	+	0.193046i	$0.0618366\pi$
$353$	4362.56	+	7556.17i	0.657777	+	1.13930i	−0.323412	+	0.946258i	$0.604830\pi$
$354$		0			0
$355$	4049.32	−	7013.63i	0.605396	−	1.04858i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10470.0			−1.53923			−0.769614	−	0.638509i	$-0.779551\pi$
$359$	−10470.0			−1.53923			−0.769614	+	0.638509i	$0.779551\pi$
$360$		0			0
$361$	−6081.30			−0.886616
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4542.60	−	7868.01i	0.651426	−	1.12830i
$366$		0			0
$367$	3947.07	+	6836.53i	0.561405	+	0.972381i	0.997374	+	0.0724198i	$0.0230721\pi$
$367$	3947.07	+	6836.53i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	44.0727	+	76.3362i	0.00616750	+	0.0106824i
$372$		0			0
$373$	134.970	−	233.775i	0.0187359	−	0.0324515i	−0.856505	−	0.516138i	$-0.827369\pi$
$373$	134.970	−	233.775i	0.0187359	−	0.0324515i	0.875241	+	0.483686i	$0.160702\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1267.26			0.173122
$378$		0			0
$379$	−8580.72			−1.16296			−0.581480	−	0.813561i	$-0.697526\pi$
$379$	−8580.72			−1.16296			−0.581480	+	0.813561i	$0.697526\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1707.46	+	2957.41i	−0.227800	+	0.394561i	−0.957156	−	0.289574i	$-0.906486\pi$
$383$	−1707.46	+	2957.41i	−0.227800	+	0.394561i	0.729356	+	0.684134i	$0.239820\pi$
$384$		0			0
$385$	−489.917	−	848.561i	−0.0648532	−	0.112329i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1652.00	−	2861.35i	−0.215321	−	0.372947i	0.738051	−	0.674745i	$-0.235746\pi$
$389$	−1652.00	−	2861.35i	−0.215321	−	0.372947i	−0.953372	+	0.301798i	$0.902413\pi$
$390$		0			0
$391$	2686.06	−	4652.39i	0.347417	−	0.601743i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10366.1			1.32045
$396$		0			0
$397$	−12756.1			−1.61262			−0.806311	−	0.591492i	$-0.798539\pi$
$397$	−12756.1			−1.61262			−0.806311	+	0.591492i	$0.798539\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2314.26	−	4008.42i	0.288201	−	0.499179i	−0.685179	−	0.728375i	$-0.740276\pi$
$401$	2314.26	−	4008.42i	0.288201	−	0.499179i	0.973380	+	0.229195i	$0.0736095\pi$
$402$		0			0
$403$	−4691.27	−	8125.51i	−0.579873	−	1.00437i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3413.50	−	5912.35i	−0.415727	−	0.720060i
$408$		0			0
$409$	1482.78	−	2568.24i	0.179263	−	0.310492i	−0.762365	−	0.647147i	$-0.775962\pi$
$409$	1482.78	−	2568.24i	0.179263	−	0.310492i	0.941628	+	0.336654i	$0.109295\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−155.409			−0.0185162
$414$		0			0
$415$	23789.6			2.81394
$416$		0			0
$417$		0			0
$418$		0			0
$419$	935.131	−	1619.69i	0.109031	−	0.188848i	−0.806347	−	0.591443i	$-0.798558\pi$
$419$	935.131	−	1619.69i	0.109031	−	0.188848i	0.915378	+	0.402595i	$0.131892\pi$
$420$		0			0
$421$	6325.31	+	10955.8i	0.732248	+	1.26829i	0.955920	+	0.293627i	$0.0948623\pi$
$421$	6325.31	+	10955.8i	0.732248	+	1.26829i	−0.223672	+	0.974665i	$0.571804\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12779.1	−	22134.1i	−1.45854	−	2.52626i
$426$		0			0
$427$	−40.7232	+	70.5347i	−0.00461530	+	0.00799394i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3266.47			0.365059			0.182530	−	0.983200i	$-0.441571\pi$
$431$	3266.47			0.365059			0.182530	+	0.983200i	$0.441571\pi$
$432$		0			0
$433$	6788.95			0.753478			0.376739	−	0.926319i	$-0.377045\pi$
$433$	6788.95			0.753478			0.376739	+	0.926319i	$0.377045\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	602.050	−	1042.78i	0.0659038	−	0.114149i
$438$		0			0
$439$	−2459.27	−	4259.59i	−0.267368	−	0.463096i	0.700813	−	0.713345i	$-0.252821\pi$
$439$	−2459.27	−	4259.59i	−0.267368	−	0.463096i	−0.968181	+	0.250249i	$0.919487\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7016.74	+	12153.4i	0.752540	+	1.30344i	0.946588	+	0.322446i	$0.104505\pi$
$443$	7016.74	+	12153.4i	0.752540	+	1.30344i	−0.194047	+	0.980992i	$0.562162\pi$
$444$		0			0
$445$	−12024.4	+	20826.8i	−1.28092	+	2.21862i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5374.04			−0.564848			−0.282424	−	0.959290i	$-0.591138\pi$
$449$	−5374.04			−0.564848			−0.282424	+	0.959290i	$0.591138\pi$
$450$		0			0
$451$	−20160.7			−2.10495
$452$		0			0
$453$		0			0
$454$		0			0
$455$	237.904	−	412.062i	0.0245123	−	0.0424566i
$456$		0			0
$457$	−6084.62	−	10538.9i	−0.622815	−	1.07875i	−0.988959	−	0.148189i	$-0.952655\pi$
$457$	−6084.62	−	10538.9i	−0.622815	−	1.07875i	0.366144	−	0.930558i	$-0.380678\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3093.04	+	5357.30i	0.312489	+	0.541246i	0.978900	−	0.204338i	$-0.0655041\pi$
$461$	3093.04	+	5357.30i	0.312489	+	0.541246i	−0.666412	+	0.745584i	$0.732171\pi$
$462$		0			0
$463$	−7598.89	+	13161.7i	−0.762743	+	1.32111i	0.178688	+	0.983906i	$0.442815\pi$
$463$	−7598.89	+	13161.7i	−0.762743	+	1.32111i	−0.941431	+	0.337204i	$0.890519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1981.82			0.196377			0.0981883	−	0.995168i	$-0.468695\pi$
$467$	1981.82			0.196377			0.0981883	+	0.995168i	$0.468695\pi$
$468$		0			0
$469$	−349.625			−0.0344225
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11831.3	+	20492.3i	−1.15011	+	1.99205i
$474$		0			0
$475$	−2864.29	−	4961.10i	−0.276679	−	0.479223i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4889.76	+	8469.31i	0.466427	+	0.807876i	0.999265	−	0.0383417i	$-0.0122075\pi$
$479$	4889.76	+	8469.31i	0.466427	+	0.807876i	−0.532837	+	0.846218i	$0.678874\pi$
$480$		0			0
$481$	1657.60	−	2871.04i	0.157131	−	0.272158i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	17175.4			1.60803
$486$		0			0
$487$	18668.9			1.73710			0.868549	−	0.495603i	$-0.165053\pi$
$487$	18668.9			1.73710			0.868549	+	0.495603i	$0.165053\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8003.11	−	13861.8i	0.735591	−	1.27408i	−0.218872	−	0.975753i	$-0.570238\pi$
$491$	8003.11	−	13861.8i	0.735591	−	1.27408i	0.954464	−	0.298328i	$-0.0964288\pi$
$492$		0			0
$493$	−2477.35	−	4290.90i	−0.226317	−	0.391993i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−183.237	−	317.376i	−0.0165378	−	0.0286444i
$498$		0			0
$499$	−4930.29	+	8539.50i	−0.442304	+	0.766094i	−0.997860	−	0.0653857i	$-0.979172\pi$
$499$	−4930.29	+	8539.50i	−0.442304	+	0.766094i	0.555556	+	0.831479i	$0.312506\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10512.0			0.931825			0.465913	−	0.884831i	$-0.345726\pi$
$503$	10512.0			0.931825			0.465913	+	0.884831i	$0.345726\pi$
$504$		0			0
$505$	−25413.9			−2.23941
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8333.63	+	14434.3i	−0.725700	+	1.25695i	0.232985	+	0.972480i	$0.425151\pi$
$509$	−8333.63	+	14434.3i	−0.725700	+	1.25695i	−0.958685	+	0.284470i	$0.908183\pi$
$510$		0			0
$511$	−205.558	−	356.038i	−0.0177952	−	0.0308223i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8730.18	+	15121.1i	0.746986	+	1.29382i
$516$		0			0
$517$	13029.3	−	22567.4i	1.10837	−	1.91976i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10923.9			0.918586			0.459293	−	0.888285i	$-0.348103\pi$
$521$	10923.9			0.918586			0.459293	+	0.888285i	$0.348103\pi$
$522$		0			0
$523$	−17488.0			−1.46213			−0.731067	−	0.682305i	$-0.760978\pi$
$523$	−17488.0			−1.46213			−0.731067	+	0.682305i	$0.760978\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−18341.9	+	31769.1i	−1.51610	+	2.62596i
$528$		0			0
$529$	5151.35	+	8922.41i	0.423387	+	0.733328i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4895.03	−	8478.43i	−0.397799	−	0.689009i
$534$		0			0
$535$	−17594.6	+	30474.8i	−1.42184	+	2.46269i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	22433.3			1.79271
$540$		0			0
$541$	−8724.14			−0.693309			−0.346655	−	0.937993i	$-0.612682\pi$
$541$	−8724.14			−0.693309			−0.346655	+	0.937993i	$0.612682\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11864.5	−	20549.9i	0.932510	−	1.61515i
$546$		0			0
$547$	479.917	+	831.241i	0.0375133	+	0.0649749i	0.884172	−	0.467161i	$-0.154723\pi$
$547$	479.917	+	831.241i	0.0375133	+	0.0649749i	−0.846659	+	0.532136i	$0.821390\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−555.270	−	961.757i	−0.0429316	−	0.0743597i
$552$		0			0
$553$	234.541	−	406.237i	0.0180356	−	0.0312386i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14212.0			−1.08112			−0.540558	−	0.841307i	$-0.681787\pi$
$557$	−14212.0			−1.08112			−0.540558	+	0.841307i	$0.681787\pi$
$558$		0			0
$559$	−11490.5			−0.869405
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3555.28	−	6157.93i	0.266141	−	0.460969i	−0.701721	−	0.712452i	$-0.747585\pi$
$563$	3555.28	−	6157.93i	0.266141	−	0.460969i	0.967862	+	0.251483i	$0.0809181\pi$
$564$		0			0
$565$	−9943.35	−	17222.4i	−0.740389	−	1.28239i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2429.59	−	4208.17i	−0.179005	−	0.310045i	0.762535	−	0.646947i	$-0.223954\pi$
$569$	−2429.59	−	4208.17i	−0.179005	−	0.310045i	−0.941540	+	0.336901i	$0.890621\pi$
$570$		0			0
$571$	−4290.14	+	7430.74i	−0.314425	+	0.544601i	−0.979315	−	0.202341i	$-0.935145\pi$
$571$	−4290.14	+	7430.74i	−0.314425	+	0.544601i	0.664890	+	0.746941i	$0.268478\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8869.49			−0.643276
$576$		0			0
$577$	−1029.99			−0.0743136			−0.0371568	−	0.999309i	$-0.511830\pi$
$577$	−1029.99			−0.0743136			−0.0371568	+	0.999309i	$0.511830\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	538.255	−	932.286i	0.0384348	−	0.0665709i
$582$		0			0
$583$	3511.25	+	6081.66i	0.249436	+	0.432036i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7770.72	−	13459.3i	−0.546391	−	0.946378i	−0.998518	−	0.0544239i	$-0.982668\pi$
$587$	−7770.72	−	13459.3i	−0.546391	−	0.946378i	0.452126	−	0.891954i	$-0.350666\pi$
$588$		0			0
$589$	−4111.12	+	7120.68i	−0.287599	+	0.498136i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12643.5			−0.875562			−0.437781	−	0.899082i	$-0.644235\pi$
$593$	−12643.5			−0.875562			−0.437781	+	0.899082i	$0.644235\pi$
$594$		0			0
$595$	−1860.31			−0.128177
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8581.46	+	14863.5i	−0.585357	+	1.01387i	0.409474	+	0.912322i	$0.365712\pi$
$599$	−8581.46	+	14863.5i	−0.585357	+	1.01387i	−0.994831	+	0.101546i	$0.967621\pi$
$600$		0			0
$601$	2005.09	+	3472.92i	0.136089	+	0.235713i	0.926013	−	0.377492i	$-0.123213\pi$
$601$	2005.09	+	3472.92i	0.136089	+	0.235713i	−0.789924	+	0.613205i	$0.789880\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−26934.3	−	46651.6i	−1.80998	−	3.13497i
$606$		0			0
$607$	−8660.58	+	15000.6i	−0.579114	+	1.00305i	0.416468	+	0.909151i	$0.363268\pi$
$607$	−8660.58	+	15000.6i	−0.579114	+	1.00305i	−0.995581	+	0.0939037i	$0.970065\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12654.1			0.837855
$612$		0			0
$613$	2474.79			0.163060			0.0815300	−	0.996671i	$-0.474019\pi$
$613$	2474.79			0.163060			0.0815300	+	0.996671i	$0.474019\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5922.19	+	10257.5i	−0.386416	+	0.669291i	−0.991964	−	0.126517i	$-0.959620\pi$
$617$	−5922.19	+	10257.5i	−0.386416	+	0.669291i	0.605549	+	0.795808i	$0.292954\pi$
$618$		0			0
$619$	3924.32	+	6797.12i	0.254817	+	0.441356i	0.964846	−	0.262817i	$-0.0846515\pi$
$619$	3924.32	+	6797.12i	0.254817	+	0.441356i	−0.710029	+	0.704173i	$0.751318\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	544.119	+	942.442i	0.0349914	+	0.0606069i
$624$		0			0
$625$	−447.377	+	774.880i	−0.0286322	+	0.0495924i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12961.7			−0.821649
$630$		0			0
$631$	−11346.0			−0.715811			−0.357906	−	0.933758i	$-0.616509\pi$
$631$	−11346.0			−0.715811			−0.357906	+	0.933758i	$0.616509\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11824.9	−	20481.3i	0.738985	−	1.27996i
$636$		0			0
$637$	5446.80	+	9434.14i	0.338791	+	0.586804i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8792.62	−	15229.3i	−0.541791	−	0.938409i	−0.998801	−	0.0489477i	$-0.984413\pi$
$641$	−8792.62	−	15229.3i	−0.541791	−	0.938409i	0.457011	−	0.889461i	$-0.348920\pi$
$642$		0			0
$643$	5322.44	−	9218.73i	0.326433	−	0.565399i	−0.655368	−	0.755309i	$-0.727487\pi$
$643$	5322.44	−	9218.73i	0.326433	−	0.565399i	0.981801	+	0.189911i	$0.0608199\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	7872.89			0.478385			0.239193	−	0.970972i	$-0.423117\pi$
$647$	7872.89			0.478385			0.239193	+	0.970972i	$0.423117\pi$
$648$		0			0
$649$	−12381.4			−0.748862
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1360.54	−	2356.53i	0.0815347	−	0.141222i	−0.822375	−	0.568946i	$-0.807351\pi$
$653$	1360.54	−	2356.53i	0.0815347	−	0.141222i	0.903909	+	0.427724i	$0.140685\pi$
$654$		0			0
$655$	−17792.7	−	30817.8i	−1.06140	−	1.83840i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1000.38	+	1732.71i	0.0591338	+	0.102423i	0.894077	−	0.447914i	$-0.147833\pi$
$659$	1000.38	+	1732.71i	0.0591338	+	0.102423i	−0.834943	+	0.550336i	$0.814499\pi$
$660$		0			0
$661$	13869.0	−	24021.8i	0.816100	−	1.41353i	−0.0924356	−	0.995719i	$-0.529465\pi$
$661$	13869.0	−	24021.8i	0.816100	−	1.41353i	0.908535	−	0.417808i	$-0.137201\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−416.968			−0.0243148
$666$		0			0
$667$	−1719.44			−0.0998153
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3244.39	+	5619.46i	−0.186659	+	0.323304i
$672$		0			0
$673$	12346.5	+	21384.8i	0.707166	+	1.22485i	0.965904	+	0.258900i	$0.0833601\pi$
$673$	12346.5	+	21384.8i	0.707166	+	1.22485i	−0.258738	+	0.965948i	$0.583307\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7382.36	+	12786.6i	0.419095	+	0.725893i	0.995849	−	0.0910246i	$-0.0290142\pi$
$677$	7382.36	+	12786.6i	0.419095	+	0.725893i	−0.576754	+	0.816918i	$0.695681\pi$
$678$		0			0
$679$	388.604	−	673.082i	0.0219635	−	0.0380420i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	571.548			0.0320200			0.0160100	−	0.999872i	$-0.494904\pi$
$683$	571.548			0.0320200			0.0160100	+	0.999872i	$0.494904\pi$
$684$		0			0
$685$	−46424.6			−2.58948
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1705.07	+	2953.26i	−0.0942784	+	0.163295i
$690$		0			0
$691$	1753.79	+	3037.66i	0.0965519	+	0.167233i	0.910255	−	0.414048i	$-0.135885\pi$
$691$	1753.79	+	3037.66i	0.0965519	+	0.167233i	−0.813703	+	0.581280i	$0.802552\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1865.56	−	3231.24i	−0.101819	−	0.176357i
$696$		0			0
$697$	−19138.5	+	33148.9i	−1.04006	+	1.80144i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21145.2			1.13929			0.569644	−	0.821891i	$-0.307081\pi$
$701$	21145.2			1.13929			0.569644	+	0.821891i	$0.307081\pi$
$702$		0			0
$703$	−2905.22			−0.155864
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−575.006	+	995.939i	−0.0305874	+	0.0529790i
$708$		0			0
$709$	7907.61	+	13696.4i	0.418867	+	0.725499i	0.995826	−	0.0912745i	$-0.0290941\pi$
$709$	7907.61	+	13696.4i	0.418867	+	0.725499i	−0.576959	+	0.816773i	$0.695761\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6365.20	+	11024.9i	0.334332	+	0.579080i
$714$		0			0
$715$	18953.7	−	32828.7i	0.991367	−	1.71710i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−33309.8			−1.72774			−0.863871	−	0.503713i	$-0.831967\pi$
$719$	−33309.8			−1.72774			−0.863871	+	0.503713i	$0.831967\pi$
$720$		0			0
$721$	790.104			0.0408114
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4090.17	+	7084.38i	−0.209524	+	0.362906i
$726$		0			0
$727$	−3518.65	−	6094.48i	−0.179504	−	0.310910i	0.762207	−	0.647334i	$-0.224116\pi$
$727$	−3518.65	−	6094.48i	−0.179504	−	0.310910i	−0.941711	+	0.336424i	$0.890783\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	22462.8	+	38906.7i	1.13655	+	1.96856i
$732$		0			0
$733$	−6330.63	+	10965.0i	−0.319000	+	0.552524i	−0.980280	−	0.197615i	$-0.936680\pi$
$733$	−6330.63	+	10965.0i	−0.319000	+	0.552524i	0.661280	+	0.750140i	$0.270014\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−27854.4			−1.39217
$738$		0			0
$739$	32709.4			1.62819			0.814096	−	0.580730i	$-0.197233\pi$
$739$	32709.4			1.62819			0.814096	+	0.580730i	$0.197233\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14690.6	+	25444.8i	−0.725364	+	1.25637i	0.233460	+	0.972366i	$0.424995\pi$
$743$	−14690.6	+	25444.8i	−0.725364	+	1.25637i	−0.958824	+	0.284001i	$0.908338\pi$
$744$		0			0
$745$	−1492.74	−	2585.49i	−0.0734089	−	0.127148i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	796.180	+	1379.02i	0.0388408	+	0.0672743i
$750$		0			0
$751$	−1112.82	+	1927.46i	−0.0540712	+	0.0936540i	−0.891794	−	0.452442i	$-0.850553\pi$
$751$	−1112.82	+	1927.46i	−0.0540712	+	0.0936540i	0.837723	+	0.546096i	$0.183886\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	62307.1			3.00343
$756$		0			0
$757$	32628.1			1.56656			0.783281	−	0.621668i	$-0.213545\pi$
$757$	32628.1			1.56656			0.783281	+	0.621668i	$0.213545\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9345.32	−	16186.6i	0.445161	−	0.771041i	−0.552903	−	0.833246i	$-0.686480\pi$
$761$	9345.32	−	16186.6i	0.445161	−	0.771041i	0.998063	+	0.0622046i	$0.0198131\pi$
$762$		0			0
$763$	−536.882	−	929.908i	−0.0254737	−	0.0441218i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3006.20	−	5206.89i	−0.141522	−	0.245124i
$768$		0			0
$769$	−3422.83	+	5928.52i	−0.160508	+	0.278008i	−0.935051	−	0.354513i	$-0.884647\pi$
$769$	−3422.83	+	5928.52i	−0.160508	+	0.278008i	0.774543	+	0.632521i	$0.217980\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	161.873			0.00753190			0.00376595	−	0.999993i	$-0.498801\pi$
$773$	161.873			0.00753190			0.00376595	+	0.999993i	$0.498801\pi$
$774$		0			0
$775$	60565.7			2.80721
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4289.69	+	7429.95i	−0.197296	+	0.341727i
$780$		0			0
$781$	−14598.4	−	25285.2i	−0.668849	−	1.15848i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8980.43	+	15554.6i	0.408312	+	0.707218i
$786$		0			0
$787$	−4979.56	+	8624.85i	−0.225543	+	0.390651i	−0.956482	−	0.291791i	$-0.905749\pi$
$787$	−4979.56	+	8624.85i	−0.225543	+	0.390651i	0.730939	+	0.682442i	$0.239082\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−899.899			−0.0404510
$792$		0			0
$793$	−3150.96			−0.141102
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2328.18	+	4032.53i	−0.103473	+	0.179221i	−0.913114	−	0.407705i	$-0.866329\pi$
$797$	−2328.18	+	4032.53i	−0.103473	+	0.179221i	0.809640	+	0.586927i	$0.199662\pi$
$798$		0			0
$799$	−24737.4	−	42846.5i	−1.09530	−	1.89712i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16376.7	−	28365.3i	−0.719704	−	1.24656i
$804$		0			0
$805$	−322.793	+	559.093i	−0.0141329	+	0.0244788i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−173.564			−0.00754289			−0.00377144	−	0.999993i	$-0.501200\pi$
$809$	−173.564			−0.00754289			−0.00377144	+	0.999993i	$0.501200\pi$
$810$		0			0
$811$	−22674.1			−0.981744			−0.490872	−	0.871232i	$-0.663322\pi$
$811$	−22674.1			−0.981744			−0.490872	+	0.871232i	$0.663322\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	25861.5	−	44793.5i	1.11152	−	1.92521i
$816$		0			0
$817$	5034.78	+	8720.49i	0.215599	+	0.373429i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	20416.1	+	35361.7i	0.867877	+	1.50321i	0.864162	+	0.503214i	$0.167849\pi$
$821$	20416.1	+	35361.7i	0.867877	+	1.50321i	0.00371541	+	0.999993i	$0.498817\pi$
$822$		0			0
$823$	2440.51	−	4227.08i	0.103367	−	0.179036i	−0.809703	−	0.586840i	$-0.800372\pi$
$823$	2440.51	−	4227.08i	0.103367	−	0.179036i	0.913070	+	0.407804i	$0.133705\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22213.4			0.934022			0.467011	−	0.884251i	$-0.345331\pi$
$827$	22213.4			0.934022			0.467011	+	0.884251i	$0.345331\pi$
$828$		0			0
$829$	35477.2			1.48634			0.743168	−	0.669104i	$-0.233322\pi$
$829$	35477.2			1.48634			0.743168	+	0.669104i	$0.233322\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	21295.9	−	36885.5i	0.885784	−	1.53422i
$834$		0			0
$835$	29137.5	+	50467.5i	1.20760	+	2.09162i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11752.6	+	20356.2i	0.483607	+	0.837631i	0.999823	−	0.0188271i	$-0.00599320\pi$
$839$	11752.6	+	20356.2i	0.483607	+	0.837631i	−0.516216	+	0.856458i	$0.672660\pi$
$840$		0			0
$841$	11401.6	−	19748.1i	0.467489	−	0.809714i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−21528.0			−0.876433
$846$		0			0
$847$	−2437.63			−0.0988876
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2249.06	+	3895.48i	−0.0905954	+	0.156916i
$852$		0			0
$853$	6217.04	+	10768.2i	0.249551	+	0.432236i	0.963401	−	0.268063i	$-0.0863835\pi$
$853$	6217.04	+	10768.2i	0.249551	+	0.432236i	−0.713850	+	0.700299i	$0.753050\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	397.933	+	689.240i	0.0158613	+	0.0274726i	0.873847	−	0.486201i	$-0.161618\pi$
$857$	397.933	+	689.240i	0.0158613	+	0.0274726i	−0.857986	+	0.513673i	$0.828284\pi$
$858$		0			0
$859$	4291.98	−	7433.92i	0.170478	−	0.295276i	−0.768109	−	0.640319i	$-0.778802\pi$
$859$	4291.98	−	7433.92i	0.170478	−	0.295276i	0.938587	+	0.345043i	$0.112136\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−37865.4			−1.49357			−0.746787	−	0.665063i	$-0.768405\pi$
$863$	−37865.4			−1.49357			−0.746787	+	0.665063i	$0.768405\pi$
$864$		0			0
$865$	32458.3			1.27585
$866$		0			0
$867$		0			0
$868$		0			0
$869$	18685.7	−	32364.6i	0.729425	−	1.26340i
$870$		0			0
$871$	−6763.06	−	11714.0i	−0.263097	−	0.455697i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	601.213	+	1041.33i	0.0232283	+	0.0402325i
$876$		0			0
$877$	−2302.28	+	3987.66i	−0.0886459	+	0.153539i	−0.906939	−	0.421262i	$-0.861587\pi$
$877$	−2302.28	+	3987.66i	−0.0886459	+	0.153539i	0.818293	+	0.574801i	$0.194921\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13021.9			0.497979			0.248990	−	0.968506i	$-0.419901\pi$
$881$	13021.9			0.497979			0.248990	+	0.968506i	$0.419901\pi$
$882$		0			0
$883$	−4301.10			−0.163923			−0.0819613	−	0.996636i	$-0.526118\pi$
$883$	−4301.10			−0.163923			−0.0819613	+	0.996636i	$0.526118\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7597.89	+	13159.9i	−0.287613	+	0.498160i	−0.973239	−	0.229794i	$-0.926195\pi$
$887$	−7597.89	+	13159.9i	−0.287613	+	0.498160i	0.685627	+	0.727953i	$0.259528\pi$
$888$		0			0
$889$	−535.091	−	926.804i	−0.0201871	−	0.0349651i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5544.61	−	9603.55i	−0.207775	−	0.359877i
$894$		0			0
$895$	11986.6	−	20761.3i	0.447672	−	0.775391i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11741.2			0.435587
$900$		0			0
$901$	13332.9			0.492990
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−31474.0	+	54514.5i	−1.15606	+	2.00235i
$906$		0			0
$907$	2328.99	+	4033.94i	0.0852624	+	0.147679i	0.905503	−	0.424340i	$-0.139494\pi$
$907$	2328.99	+	4033.94i	0.0852624	+	0.147679i	−0.820241	+	0.572019i	$0.806160\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6446.40	+	11165.5i	0.234444	+	0.406069i	0.959111	−	0.283030i	$-0.0913396\pi$
$911$	6446.40	+	11165.5i	0.234444	+	0.406069i	−0.724667	+	0.689099i	$0.758006\pi$
$912$		0			0
$913$	42882.5	−	74274.7i	1.55444	−	2.69237i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1610.28			−0.0579894
$918$		0			0
$919$	−36405.3			−1.30675			−0.653373	−	0.757036i	$-0.726647\pi$
$919$	−36405.3			−1.30675			−0.653373	+	0.757036i	$0.726647\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7088.99	−	12278.5i	0.252803	−	0.437867i
$924$		0			0
$925$	10700.0	+	18533.0i	0.380341	+	0.658770i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2139.85	−	3706.33i	−0.0755717	−	0.130894i	0.825763	−	0.564017i	$-0.190745\pi$
$929$	−2139.85	−	3706.33i	−0.0755717	−	0.130894i	−0.901335	+	0.433123i	$0.857412\pi$
$930$		0			0
$931$	4773.23	−	8267.47i	0.168030	−	0.291037i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−148210.			−5.18394
$936$		0			0
$937$	21006.6			0.732396			0.366198	−	0.930537i	$-0.380659\pi$
$937$	21006.6			0.732396			0.366198	+	0.930537i	$0.380659\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6764.54	−	11716.5i	0.234344	−	0.405896i	−0.724738	−	0.689025i	$-0.758039\pi$
$941$	6764.54	−	11716.5i	0.234344	−	0.405896i	0.959082	+	0.283129i	$0.0913724\pi$
$942$		0			0
$943$	6641.67	+	11503.7i	0.229356	+	0.397256i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2076.57	+	3596.73i	0.0712561	+	0.123419i	0.899452	−	0.437019i	$-0.143966\pi$
$947$	2076.57	+	3596.73i	0.0712561	+	0.123419i	−0.828196	+	0.560439i	$0.810633\pi$
$948$		0			0
$949$	7952.55	−	13774.2i	0.272024	−	0.471159i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−18387.5			−0.625005			−0.312503	−	0.949917i	$-0.601167\pi$
$953$	−18387.5			−0.625005			−0.312503	+	0.949917i	$0.601167\pi$
$954$		0			0
$955$	28926.0			0.980128
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1050.39	+	1819.32i	−0.0353689	+	0.0612607i
$960$		0			0
$961$	−28569.6	−	49483.9i	−0.959000	−	1.66104i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6947.93	+	12034.2i	0.231774	+	0.401444i
$966$		0			0
$967$	17227.0	−	29838.1i	0.572889	−	0.992273i	−0.423378	−	0.905953i	$-0.639156\pi$
$967$	17227.0	−	29838.1i	0.572889	−	0.992273i	0.996267	−	0.0863204i	$-0.0275109\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2689.17			−0.0888770			−0.0444385	−	0.999012i	$-0.514150\pi$
$971$	−2689.17			−0.0888770			−0.0444385	+	0.999012i	$0.514150\pi$
$972$		0			0
$973$	−168.838			−0.00556288
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−74.4796	+	129.002i	−0.00243891	+	0.00422431i	−0.867242	−	0.497886i	$-0.834110\pi$
$977$	−74.4796	+	129.002i	−0.00243891	+	0.00422431i	0.864803	+	0.502111i	$0.167443\pi$
$978$		0			0
$979$	43349.6	+	75083.8i	1.41518	+	2.45116i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16110.9	−	27904.9i	−0.522744	−	0.905419i	−0.999650	−	0.0264649i	$-0.991575\pi$
$983$	−16110.9	−	27904.9i	−0.522744	−	0.905419i	0.476906	−	0.878955i	$-0.341758\pi$
$984$		0			0
$985$	−10830.4	+	18758.8i	−0.350340	+	0.606807i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	15590.6			0.501265
$990$		0			0
$991$	2979.37			0.0955023			0.0477512	−	0.998859i	$-0.484795\pi$
$991$	2979.37			0.0955023			0.0477512	+	0.998859i	$0.484795\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	24614.1	−	42632.8i	0.784240	−	1.35834i
$996$		0			0
$997$	16046.9	+	27794.1i	0.509740	+	0.882896i	0.999936	+	0.0112838i	$0.00359182\pi$
$997$	16046.9	+	27794.1i	0.509740	+	0.882896i	−0.490196	+	0.871612i	$0.663075\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.n.433.1		4
3.2	odd	2		648.4.i.t.433.2		4
9.2	odd	6		648.4.i.t.217.2		4
9.4	even	3		648.4.a.f.1.2	yes	2
9.5	odd	6		648.4.a.c.1.1	✓	2
9.7	even	3	inner	648.4.i.n.217.1		4
36.23	even	6		1296.4.a.m.1.1		2
36.31	odd	6		1296.4.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.c.1.1	✓	2	9.5	odd	6
648.4.a.f.1.2	yes	2	9.4	even	3
648.4.i.n.217.1		4	9.7	even	3	inner
648.4.i.n.433.1		4	1.1	even	1	trivial
648.4.i.t.217.2		4	9.2	odd	6
648.4.i.t.433.2		4	3.2	odd	2
1296.4.a.m.1.1		2	36.23	even	6
1296.4.a.q.1.2		2	36.31	odd	6

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-9.08872 + 15.7421i) q^{5} +(0.411277 + 0.712352i) q^{7} +(32.7662 + 56.7527i) q^{11} +(-15.9113 + 27.5591i) q^{13} +124.420 q^{17} +27.8872 q^{19} +(21.5887 - 37.3928i) q^{23} +(-102.710 - 177.899i) q^{25} +(-19.9113 - 34.4873i) q^{29} +(-147.420 + 255.338i) q^{31} -14.9519 q^{35} -104.177 q^{37} +(-153.823 + 266.428i) q^{41} +(180.541 + 312.706i) q^{43} +(-198.823 - 344.371i) q^{47} +(171.162 - 296.461i) q^{49} +107.161 q^{53} -1191.21 q^{55} +(-94.4677 + 163.623i) q^{59} +(49.5083 + 85.7509i) q^{61} +(-289.226 - 500.955i) q^{65} +(-212.524 + 368.102i) q^{67} -445.532 q^{71} -499.806 q^{73} +(-26.9519 + 46.6821i) q^{77} +(-285.138 - 493.873i) q^{79} +(-654.371 - 1133.40i) q^{83} +(-1130.82 + 1958.63i) q^{85} +1323.00 q^{89} -26.1757 q^{91} +(-253.459 + 439.004i) q^{95} +(-472.436 - 818.283i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} + 30 q^{7} + 46 q^{11} - 92 q^{13} + 44 q^{17} - 172 q^{19} + 58 q^{23} - 184 q^{25} - 108 q^{29} - 136 q^{31} + 564 q^{35} - 360 q^{37} - 672 q^{41} + 70 q^{43} - 852 q^{47} - 166 q^{49}+ \cdots - 472 q^{97}+O(q^{100}) \) 4 * q - 8 * q^5 + 30 * q^7 + 46 * q^11 - 92 * q^13 + 44 * q^17 - 172 * q^19 + 58 * q^23 - 184 * q^25 - 108 * q^29 - 136 * q^31 + 564 * q^35 - 360 * q^37 - 672 * q^41 + 70 * q^43 - 852 * q^47 - 166 * q^49 + 1336 * q^53 - 2780 * q^55 - 548 * q^59 - 284 * q^61 + 34 * q^65 - 1162 * q^67 - 1612 * q^71 - 3020 * q^73 + 516 * q^77 + 22 * q^79 - 1540 * q^83 - 3304 * q^85 + 5292 * q^89 - 3564 * q^91 - 1666 * q^95 - 472 * q^97

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