LMFDB - Embedded newform 3042.3.d.a.3041.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,3,Mod(3041,3042)] 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("3042.3041"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3042.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,8,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$82.8884964184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3042.3041
Dual form			3042.3.d.a.3041.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} -4.24264 q^{5} +4.00000i q^{7} +2.82843 q^{8} -6.00000 q^{10} -16.9706 q^{11} +5.65685i q^{14} +4.00000 q^{16} -12.7279i q^{17} -16.0000i q^{19} -8.48528 q^{20} -24.0000 q^{22} -16.9706i q^{23} -7.00000 q^{25} +8.00000i q^{28} -4.24264i q^{29} +44.0000i q^{31} +5.65685 q^{32} -18.0000i q^{34} -16.9706i q^{35} +34.0000i q^{37} -22.6274i q^{38} -12.0000 q^{40} +46.6690 q^{41} +40.0000 q^{43} -33.9411 q^{44} -24.0000i q^{46} +84.8528 q^{47} +33.0000 q^{49} -9.89949 q^{50} -38.1838i q^{53} +72.0000 q^{55} +11.3137i q^{56} -6.00000i q^{58} -33.9411 q^{59} +50.0000 q^{61} +62.2254i q^{62} +8.00000 q^{64} +8.00000i q^{67} -25.4558i q^{68} -24.0000i q^{70} -50.9117 q^{71} +16.0000i q^{73} +48.0833i q^{74} -32.0000i q^{76} -67.8823i q^{77} -76.0000 q^{79} -16.9706 q^{80} +66.0000 q^{82} +118.794 q^{83} +54.0000i q^{85} +56.5685 q^{86} -48.0000 q^{88} -12.7279 q^{89} -33.9411i q^{92} +120.000 q^{94} +67.8823i q^{95} +176.000i q^{97} +46.6690 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 24 q^{10} + 16 q^{16} - 96 q^{22} - 28 q^{25} - 48 q^{40} + 160 q^{43} + 132 q^{49} + 288 q^{55} + 200 q^{61} + 32 q^{64} - 304 q^{79} + 264 q^{82} - 192 q^{88} + 480 q^{94}+O(q^{100}) $ 4 * q + 8 * q^4 - 24 * q^10 + 16 * q^16 - 96 * q^22 - 28 * q^25 - 48 * q^40 + 160 * q^43 + 132 * q^49 + 288 * q^55 + 200 * q^61 + 32 * q^64 - 304 * q^79 + 264 * q^82 - 192 * q^88 + 480 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	−4.24264	−0.848528	−0.424264	−	0.905539i	$-0.639467\pi$
$5$	−4.24264	−0.848528	−0.424264	+	0.905539i	$0.639467\pi$
$6$	0	0
$7$	4.00000i	0.571429i	0.958315	+	0.285714i	$0.0922308\pi$
$7$	4.00000i	0.571429i	−0.958315	+	0.285714i	$0.907769\pi$
$8$	2.82843	0.353553
$9$	0	0
$10$	−6.00000	−0.600000
$11$	−16.9706	−1.54278	−0.771389	−	0.636364i	$-0.780438\pi$
$11$	−16.9706	−1.54278	−0.771389	+	0.636364i	$0.780438\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	5.65685i	0.404061i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 12.7279i	− 0.748701i	−0.927287	−	0.374351i	$-0.877866\pi$
$17$	− 12.7279i	− 0.748701i	0.927287	−	0.374351i	$-0.122134\pi$
$18$	0	0
$19$	− 16.0000i	− 0.842105i	−0.907036	−	0.421053i	$-0.861661\pi$
$19$	− 16.0000i	− 0.842105i	0.907036	−	0.421053i	$-0.138339\pi$
$20$	−8.48528	−0.424264
$21$	0	0
$22$	−24.0000	−1.09091
$23$	− 16.9706i	− 0.737851i	−0.929459	−	0.368925i	$-0.879726\pi$
$23$	− 16.9706i	− 0.737851i	0.929459	−	0.368925i	$-0.120274\pi$
$24$	0	0
$25$	−7.00000	−0.280000
$26$	0	0
$27$	0	0
$28$	8.00000i	0.285714i
$29$	− 4.24264i	− 0.146298i	−0.997321	−	0.0731490i	$-0.976695\pi$
$29$	− 4.24264i	− 0.146298i	0.997321	−	0.0731490i	$-0.0233049\pi$
$30$	0	0
$31$	44.0000i	1.41935i	0.704527	+	0.709677i	$0.251159\pi$
$31$	44.0000i	1.41935i	−0.704527	+	0.709677i	$0.748841\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	− 18.0000i	− 0.529412i
$35$	− 16.9706i	− 0.484873i
$36$	0	0
$37$	34.0000i	0.918919i	0.888199	+	0.459459i	$0.151957\pi$
$37$	34.0000i	0.918919i	−0.888199	+	0.459459i	$0.848043\pi$
$38$	− 22.6274i	− 0.595458i
$39$	0	0
$40$	−12.0000	−0.300000
$41$	46.6690	1.13827	0.569135	−	0.822244i	$-0.307278\pi$
$41$	46.6690	1.13827	0.569135	+	0.822244i	$0.307278\pi$
$42$	0	0
$43$	40.0000	0.930233	0.465116	−	0.885250i	$-0.346013\pi$
$43$	40.0000	0.930233	0.465116	+	0.885250i	$0.346013\pi$
$44$	−33.9411	−0.771389
$45$	0	0
$46$	− 24.0000i	− 0.521739i
$47$	84.8528	1.80538	0.902690	−	0.430293i	$-0.141590\pi$
$47$	84.8528	1.80538	0.902690	+	0.430293i	$0.141590\pi$
$48$	0	0
$49$	33.0000	0.673469
$50$	−9.89949	−0.197990
$51$	0	0
$52$	0	0
$53$	− 38.1838i	− 0.720448i	−0.932866	−	0.360224i	$-0.882700\pi$
$53$	− 38.1838i	− 0.720448i	0.932866	−	0.360224i	$-0.117300\pi$
$54$	0	0
$55$	72.0000	1.30909
$56$	11.3137i	0.202031i
$57$	0	0
$58$	− 6.00000i	− 0.103448i
$59$	−33.9411	−0.575273	−0.287637	−	0.957740i	$-0.592870\pi$
$59$	−33.9411	−0.575273	−0.287637	+	0.957740i	$0.592870\pi$
$60$	0	0
$61$	50.0000	0.819672	0.409836	−	0.912159i	$-0.365586\pi$
$61$	50.0000	0.819672	0.409836	+	0.912159i	$0.365586\pi$
$62$	62.2254i	1.00364i
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	8.00000i	0.119403i	0.998216	+	0.0597015i	$0.0190149\pi$
$67$	8.00000i	0.119403i	−0.998216	+	0.0597015i	$0.980985\pi$
$68$	− 25.4558i	− 0.374351i
$69$	0	0
$70$	− 24.0000i	− 0.342857i
$71$	−50.9117	−0.717066	−0.358533	−	0.933517i	$-0.616723\pi$
$71$	−50.9117	−0.717066	−0.358533	+	0.933517i	$0.616723\pi$
$72$	0	0
$73$	16.0000i	0.219178i	0.993977	+	0.109589i	$0.0349535\pi$
$73$	16.0000i	0.219178i	−0.993977	+	0.109589i	$0.965047\pi$
$74$	48.0833i	0.649774i
$75$	0	0
$76$	− 32.0000i	− 0.421053i
$77$	− 67.8823i	− 0.881588i
$78$	0	0
$79$	−76.0000	−0.962025	−0.481013	−	0.876714i	$-0.659731\pi$
$79$	−76.0000	−0.962025	−0.481013	+	0.876714i	$0.659731\pi$
$80$	−16.9706	−0.212132
$81$	0	0
$82$	66.0000	0.804878
$83$	118.794	1.43125	0.715626	−	0.698484i	$-0.246141\pi$
$83$	118.794	1.43125	0.715626	+	0.698484i	$0.246141\pi$
$84$	0	0
$85$	54.0000i	0.635294i
$86$	56.5685	0.657774
$87$	0	0
$88$	−48.0000	−0.545455
$89$	−12.7279	−0.143010	−0.0715052	−	0.997440i	$-0.522780\pi$
$89$	−12.7279	−0.143010	−0.0715052	+	0.997440i	$0.522780\pi$
$90$	0	0
$91$	0	0
$92$	− 33.9411i	− 0.368925i
$93$	0	0
$94$	120.000	1.27660
$95$	67.8823i	0.714550i
$96$	0	0
$97$	176.000i	1.81443i	0.420664	+	0.907216i	$0.361797\pi$
$97$	176.000i	1.81443i	−0.420664	+	0.907216i	$0.638203\pi$
$98$	46.6690	0.476215
$99$	0	0
$100$	−14.0000	−0.140000
$101$	29.6985i	0.294044i	0.989133	+	0.147022i	$0.0469689\pi$
$101$	29.6985i	0.294044i	−0.989133	+	0.147022i	$0.953031\pi$
$102$	0	0
$103$	28.0000	0.271845	0.135922	−	0.990719i	$-0.456600\pi$
$103$	28.0000	0.271845	0.135922	+	0.990719i	$0.456600\pi$
$104$	0	0
$105$	0	0
$106$	− 54.0000i	− 0.509434i
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	56.0000i	0.513761i	0.966443	+	0.256881i	$0.0826948\pi$
$109$	56.0000i	0.513761i	−0.966443	+	0.256881i	$0.917305\pi$
$110$	101.823	0.925667
$111$	0	0
$112$	16.0000i	0.142857i
$113$	156.978i	1.38918i	0.719404	+	0.694592i	$0.244415\pi$
$113$	156.978i	1.38918i	−0.719404	+	0.694592i	$0.755585\pi$
$114$	0	0
$115$	72.0000i	0.626087i
$116$	− 8.48528i	− 0.0731490i
$117$	0	0
$118$	−48.0000	−0.406780
$119$	50.9117	0.427829
$120$	0	0
$121$	167.000	1.38017
$122$	70.7107	0.579596
$123$	0	0
$124$	88.0000i	0.709677i
$125$	135.765	1.08612
$126$	0	0
$127$	−92.0000	−0.724409	−0.362205	−	0.932099i	$-0.617976\pi$
$127$	−92.0000	−0.724409	−0.362205	+	0.932099i	$0.617976\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	0	0
$131$	169.706i	1.29546i	0.761869	+	0.647731i	$0.224282\pi$
$131$	169.706i	1.29546i	−0.761869	+	0.647731i	$0.775718\pi$
$132$	0	0
$133$	64.0000	0.481203
$134$	11.3137i	0.0844307i
$135$	0	0
$136$	− 36.0000i	− 0.264706i
$137$	−156.978	−1.14582	−0.572911	−	0.819617i	$-0.694186\pi$
$137$	−156.978	−1.14582	−0.572911	+	0.819617i	$0.694186\pi$
$138$	0	0
$139$	152.000	1.09353	0.546763	−	0.837288i	$-0.315860\pi$
$139$	152.000	1.09353	0.546763	+	0.837288i	$0.315860\pi$
$140$	− 33.9411i	− 0.242437i
$141$	0	0
$142$	−72.0000	−0.507042
$143$	0	0
$144$	0	0
$145$	18.0000i	0.124138i
$146$	22.6274i	0.154982i
$147$	0	0
$148$	68.0000i	0.459459i
$149$	275.772	1.85082	0.925408	−	0.378972i	$-0.123722\pi$
$149$	275.772	1.85082	0.925408	+	0.378972i	$0.123722\pi$
$150$	0	0
$151$	148.000i	0.980132i	0.871685	+	0.490066i	$0.163027\pi$
$151$	148.000i	0.980132i	−0.871685	+	0.490066i	$0.836973\pi$
$152$	− 45.2548i	− 0.297729i
$153$	0	0
$154$	− 96.0000i	− 0.623377i
$155$	− 186.676i	− 1.20436i
$156$	0	0
$157$	−82.0000	−0.522293	−0.261146	−	0.965299i	$-0.584101\pi$
$157$	−82.0000	−0.522293	−0.261146	+	0.965299i	$0.584101\pi$
$158$	−107.480	−0.680255
$159$	0	0
$160$	−24.0000	−0.150000
$161$	67.8823	0.421629
$162$	0	0
$163$	− 56.0000i	− 0.343558i	−0.985135	−	0.171779i	$-0.945048\pi$
$163$	− 56.0000i	− 0.343558i	0.985135	−	0.171779i	$-0.0549515\pi$
$164$	93.3381	0.569135
$165$	0	0
$166$	168.000	1.01205
$167$	−33.9411	−0.203240	−0.101620	−	0.994823i	$-0.532403\pi$
$167$	−33.9411	−0.203240	−0.101620	+	0.994823i	$0.532403\pi$
$168$	0	0
$169$	0	0
$170$	76.3675i	0.449221i
$171$	0	0
$172$	80.0000	0.465116
$173$	− 173.948i	− 1.00548i	−0.864437	−	0.502741i	$-0.832325\pi$
$173$	− 173.948i	− 1.00548i	0.864437	−	0.502741i	$-0.167675\pi$
$174$	0	0
$175$	− 28.0000i	− 0.160000i
$176$	−67.8823	−0.385695
$177$	0	0
$178$	−18.0000	−0.101124
$179$	− 203.647i	− 1.13769i	−0.822444	−	0.568846i	$-0.807390\pi$
$179$	− 203.647i	− 1.13769i	0.822444	−	0.568846i	$-0.192610\pi$
$180$	0	0
$181$	232.000	1.28177	0.640884	−	0.767638i	$-0.278568\pi$
$181$	232.000	1.28177	0.640884	+	0.767638i	$0.278568\pi$
$182$	0	0
$183$	0	0
$184$	− 48.0000i	− 0.260870i
$185$	− 144.250i	− 0.779729i
$186$	0	0
$187$	216.000i	1.15508i
$188$	169.706	0.902690
$189$	0	0
$190$	96.0000i	0.505263i
$191$	33.9411i	0.177702i	0.996045	+	0.0888511i	$0.0283195\pi$
$191$	33.9411i	0.177702i	−0.996045	+	0.0888511i	$0.971680\pi$
$192$	0	0
$193$	− 206.000i	− 1.06736i	−0.845687	−	0.533679i	$-0.820809\pi$
$193$	− 206.000i	− 1.06736i	0.845687	−	0.533679i	$-0.179191\pi$
$194$	248.902i	1.28300i
$195$	0	0
$196$	66.0000	0.336735
$197$	165.463	0.839914	0.419957	−	0.907544i	$-0.362045\pi$
$197$	165.463	0.839914	0.419957	+	0.907544i	$0.362045\pi$
$198$	0	0
$199$	−20.0000	−0.100503	−0.0502513	−	0.998737i	$-0.516002\pi$
$199$	−20.0000	−0.100503	−0.0502513	+	0.998737i	$0.516002\pi$
$200$	−19.7990	−0.0989949
$201$	0	0
$202$	42.0000i	0.207921i
$203$	16.9706	0.0835988
$204$	0	0
$205$	−198.000	−0.965854
$206$	39.5980	0.192223
$207$	0	0
$208$	0	0
$209$	271.529i	1.29918i
$210$	0	0
$211$	296.000	1.40284	0.701422	−	0.712746i	$-0.252549\pi$
$211$	296.000	1.40284	0.701422	+	0.712746i	$0.252549\pi$
$212$	− 76.3675i	− 0.360224i
$213$	0	0
$214$	0	0
$215$	−169.706	−0.789328
$216$	0	0
$217$	−176.000	−0.811060
$218$	79.1960i	0.363284i
$219$	0	0
$220$	144.000	0.654545
$221$	0	0
$222$	0	0
$223$	− 436.000i	− 1.95516i	−0.210571	−	0.977578i	$-0.567532\pi$
$223$	− 436.000i	− 1.95516i	0.210571	−	0.977578i	$-0.432468\pi$
$224$	22.6274i	0.101015i
$225$	0	0
$226$	222.000i	0.982301i
$227$	16.9706	0.0747602	0.0373801	−	0.999301i	$-0.488099\pi$
$227$	16.9706	0.0747602	0.0373801	+	0.999301i	$0.488099\pi$
$228$	0	0
$229$	− 8.00000i	− 0.0349345i	−0.999847	−	0.0174672i	$-0.994440\pi$
$229$	− 8.00000i	− 0.0349345i	0.999847	−	0.0174672i	$-0.00556028\pi$
$230$	101.823i	0.442710i
$231$	0	0
$232$	− 12.0000i	− 0.0517241i
$233$	− 12.7279i	− 0.0546263i	−0.999627	−	0.0273131i	$-0.991305\pi$
$233$	− 12.7279i	− 0.0546263i	0.999627	−	0.0273131i	$-0.00869512\pi$
$234$	0	0
$235$	−360.000	−1.53191
$236$	−67.8823	−0.287637
$237$	0	0
$238$	72.0000	0.302521
$239$	135.765	0.568052	0.284026	−	0.958817i	$-0.408330\pi$
$239$	135.765	0.568052	0.284026	+	0.958817i	$0.408330\pi$
$240$	0	0
$241$	− 32.0000i	− 0.132780i	−0.997794	−	0.0663900i	$-0.978852\pi$
$241$	− 32.0000i	− 0.132780i	0.997794	−	0.0663900i	$-0.0211482\pi$
$242$	236.174	0.975924
$243$	0	0
$244$	100.000	0.409836
$245$	−140.007	−0.571458
$246$	0	0
$247$	0	0
$248$	124.451i	0.501818i
$249$	0	0
$250$	192.000	0.768000
$251$	50.9117i	0.202835i	0.994844	+	0.101418i	$0.0323379\pi$
$251$	50.9117i	0.202835i	−0.994844	+	0.101418i	$0.967662\pi$
$252$	0	0
$253$	288.000i	1.13834i
$254$	−130.108	−0.512235
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 182.434i	− 0.709858i	−0.934893	−	0.354929i	$-0.884505\pi$
$257$	− 182.434i	− 0.709858i	0.934893	−	0.354929i	$-0.115495\pi$
$258$	0	0
$259$	−136.000	−0.525097
$260$	0	0
$261$	0	0
$262$	240.000i	0.916031i
$263$	− 373.352i	− 1.41959i	−0.704408	−	0.709795i	$-0.748787\pi$
$263$	− 373.352i	− 1.41959i	0.704408	−	0.709795i	$-0.251213\pi$
$264$	0	0
$265$	162.000i	0.611321i
$266$	90.5097	0.340262
$267$	0	0
$268$	16.0000i	0.0597015i
$269$	343.654i	1.27752i	0.769404	+	0.638762i	$0.220553\pi$
$269$	343.654i	1.27752i	−0.769404	+	0.638762i	$0.779447\pi$
$270$	0	0
$271$	− 380.000i	− 1.40221i	−0.713056	−	0.701107i	$-0.752690\pi$
$271$	− 380.000i	− 1.40221i	0.713056	−	0.701107i	$-0.247310\pi$
$272$	− 50.9117i	− 0.187175i
$273$	0	0
$274$	−222.000	−0.810219
$275$	118.794	0.431978
$276$	0	0
$277$	328.000	1.18412	0.592058	−	0.805896i	$-0.298316\pi$
$277$	328.000	1.18412	0.592058	+	0.805896i	$0.298316\pi$
$278$	214.960	0.773239
$279$	0	0
$280$	− 48.0000i	− 0.171429i
$281$	−284.257	−1.01159	−0.505795	−	0.862654i	$-0.668801\pi$
$281$	−284.257	−1.01159	−0.505795	+	0.862654i	$0.668801\pi$
$282$	0	0
$283$	208.000	0.734982	0.367491	−	0.930027i	$-0.380217\pi$
$283$	208.000	0.734982	0.367491	+	0.930027i	$0.380217\pi$
$284$	−101.823	−0.358533
$285$	0	0
$286$	0	0
$287$	186.676i	0.650440i
$288$	0	0
$289$	127.000	0.439446
$290$	25.4558i	0.0877788i
$291$	0	0
$292$	32.0000i	0.109589i
$293$	436.992	1.49144	0.745720	−	0.666259i	$-0.232106\pi$
$293$	436.992	1.49144	0.745720	+	0.666259i	$0.232106\pi$
$294$	0	0
$295$	144.000	0.488136
$296$	96.1665i	0.324887i
$297$	0	0
$298$	390.000	1.30872
$299$	0	0
$300$	0	0
$301$	160.000i	0.531561i
$302$	209.304i	0.693058i
$303$	0	0
$304$	− 64.0000i	− 0.210526i
$305$	−212.132	−0.695515
$306$	0	0
$307$	520.000i	1.69381i	0.531743	+	0.846906i	$0.321537\pi$
$307$	520.000i	1.69381i	−0.531743	+	0.846906i	$0.678463\pi$
$308$	− 135.765i	− 0.440794i
$309$	0	0
$310$	− 264.000i	− 0.851613i
$311$	− 373.352i	− 1.20049i	−0.799816	−	0.600245i	$-0.795070\pi$
$311$	− 373.352i	− 1.20049i	0.799816	−	0.600245i	$-0.204930\pi$
$312$	0	0
$313$	−94.0000	−0.300319	−0.150160	−	0.988662i	$-0.547979\pi$
$313$	−94.0000	−0.300319	−0.150160	+	0.988662i	$0.547979\pi$
$314$	−115.966	−0.369317
$315$	0	0
$316$	−152.000	−0.481013
$317$	335.169	1.05731	0.528657	−	0.848835i	$-0.322696\pi$
$317$	335.169	1.05731	0.528657	+	0.848835i	$0.322696\pi$
$318$	0	0
$319$	72.0000i	0.225705i
$320$	−33.9411	−0.106066
$321$	0	0
$322$	96.0000	0.298137
$323$	−203.647	−0.630485
$324$	0	0
$325$	0	0
$326$	− 79.1960i	− 0.242932i
$327$	0	0
$328$	132.000	0.402439
$329$	339.411i	1.03165i
$330$	0	0
$331$	536.000i	1.61934i	0.586889	+	0.809668i	$0.300353\pi$
$331$	536.000i	1.61934i	−0.586889	+	0.809668i	$0.699647\pi$
$332$	237.588	0.715626
$333$	0	0
$334$	−48.0000	−0.143713
$335$	− 33.9411i	− 0.101317i
$336$	0	0
$337$	208.000	0.617211	0.308605	−	0.951190i	$-0.400138\pi$
$337$	208.000	0.617211	0.308605	+	0.951190i	$0.400138\pi$
$338$	0	0
$339$	0	0
$340$	108.000i	0.317647i
$341$	− 746.705i	− 2.18975i
$342$	0	0
$343$	328.000i	0.956268i
$344$	113.137	0.328887
$345$	0	0
$346$	− 246.000i	− 0.710983i
$347$	− 288.500i	− 0.831411i	−0.909499	−	0.415705i	$-0.863535\pi$
$347$	− 288.500i	− 0.831411i	0.909499	−	0.415705i	$-0.136465\pi$
$348$	0	0
$349$	238.000i	0.681948i	0.940073	+	0.340974i	$0.110757\pi$
$349$	238.000i	0.681948i	−0.940073	+	0.340974i	$0.889243\pi$
$350$	− 39.5980i	− 0.113137i
$351$	0	0
$352$	−96.0000	−0.272727
$353$	−224.860	−0.636997	−0.318499	−	0.947923i	$-0.603179\pi$
$353$	−224.860	−0.636997	−0.318499	+	0.947923i	$0.603179\pi$
$354$	0	0
$355$	216.000	0.608451
$356$	−25.4558	−0.0715052
$357$	0	0
$358$	− 288.000i	− 0.804469i
$359$	560.029	1.55997	0.779984	−	0.625799i	$-0.215227\pi$
$359$	560.029	1.55997	0.779984	+	0.625799i	$0.215227\pi$
$360$	0	0
$361$	105.000	0.290859
$362$	328.098	0.906347
$363$	0	0
$364$	0	0
$365$	− 67.8823i	− 0.185979i
$366$	0	0
$367$	284.000	0.773842	0.386921	−	0.922113i	$-0.373539\pi$
$367$	284.000	0.773842	0.386921	+	0.922113i	$0.373539\pi$
$368$	− 67.8823i	− 0.184463i
$369$	0	0
$370$	− 204.000i	− 0.551351i
$371$	152.735	0.411685
$372$	0	0
$373$	−190.000	−0.509383	−0.254692	−	0.967022i	$-0.581974\pi$
$373$	−190.000	−0.509383	−0.254692	+	0.967022i	$0.581974\pi$
$374$	305.470i	0.816765i
$375$	0	0
$376$	240.000	0.638298
$377$	0	0
$378$	0	0
$379$	− 160.000i	− 0.422164i	−0.977468	−	0.211082i	$-0.932301\pi$
$379$	− 160.000i	− 0.422164i	0.977468	−	0.211082i	$-0.0676986\pi$
$380$	135.765i	0.357275i
$381$	0	0
$382$	48.0000i	0.125654i
$383$	−271.529	−0.708953	−0.354477	−	0.935065i	$-0.615341\pi$
$383$	−271.529	−0.708953	−0.354477	+	0.935065i	$0.615341\pi$
$384$	0	0
$385$	288.000i	0.748052i
$386$	− 291.328i	− 0.754736i
$387$	0	0
$388$	352.000i	0.907216i
$389$	− 403.051i	− 1.03612i	−0.855344	−	0.518060i	$-0.826654\pi$
$389$	− 403.051i	− 1.03612i	0.855344	−	0.518060i	$-0.173346\pi$
$390$	0	0
$391$	−216.000	−0.552430
$392$	93.3381	0.238107
$393$	0	0
$394$	234.000	0.593909
$395$	322.441	0.816306
$396$	0	0
$397$	− 146.000i	− 0.367758i	−0.982949	−	0.183879i	$-0.941135\pi$
$397$	− 146.000i	− 0.367758i	0.982949	−	0.183879i	$-0.0588655\pi$
$398$	−28.2843	−0.0710660
$399$	0	0
$400$	−28.0000	−0.0700000
$401$	−326.683	−0.814672	−0.407336	−	0.913278i	$-0.633542\pi$
$401$	−326.683	−0.814672	−0.407336	+	0.913278i	$0.633542\pi$
$402$	0	0
$403$	0	0
$404$	59.3970i	0.147022i
$405$	0	0
$406$	24.0000	0.0591133
$407$	− 576.999i	− 1.41769i
$408$	0	0
$409$	368.000i	0.899756i	0.893090	+	0.449878i	$0.148532\pi$
$409$	368.000i	0.899756i	−0.893090	+	0.449878i	$0.851468\pi$
$410$	−280.014	−0.682962
$411$	0	0
$412$	56.0000	0.135922
$413$	− 135.765i	− 0.328728i
$414$	0	0
$415$	−504.000	−1.21446
$416$	0	0
$417$	0	0
$418$	384.000i	0.918660i
$419$	− 390.323i	− 0.931558i	−0.884901	−	0.465779i	$-0.845774\pi$
$419$	− 390.323i	− 0.931558i	0.884901	−	0.465779i	$-0.154226\pi$
$420$	0	0
$421$	− 40.0000i	− 0.0950119i	−0.998871	−	0.0475059i	$-0.984873\pi$
$421$	− 40.0000i	− 0.0950119i	0.998871	−	0.0475059i	$-0.0151273\pi$
$422$	418.607	0.991960
$423$	0	0
$424$	− 108.000i	− 0.254717i
$425$	89.0955i	0.209636i
$426$	0	0
$427$	200.000i	0.468384i
$428$	0	0
$429$	0	0
$430$	−240.000	−0.558140
$431$	−152.735	−0.354374	−0.177187	−	0.984177i	$-0.556700\pi$
$431$	−152.735	−0.354374	−0.177187	+	0.984177i	$0.556700\pi$
$432$	0	0
$433$	−542.000	−1.25173	−0.625866	−	0.779931i	$-0.715254\pi$
$433$	−542.000	−1.25173	−0.625866	+	0.779931i	$0.715254\pi$
$434$	−248.902	−0.573506
$435$	0	0
$436$	112.000i	0.256881i
$437$	−271.529	−0.621348
$438$	0	0
$439$	4.00000	0.00911162	0.00455581	−	0.999990i	$-0.498550\pi$
$439$	4.00000	0.00911162	0.00455581	+	0.999990i	$0.498550\pi$
$440$	203.647	0.462834
$441$	0	0
$442$	0	0
$443$	− 322.441i	− 0.727857i	−0.931427	−	0.363929i	$-0.881435\pi$
$443$	− 322.441i	− 0.727857i	0.931427	−	0.363929i	$-0.118565\pi$
$444$	0	0
$445$	54.0000	0.121348
$446$	− 616.597i	− 1.38250i
$447$	0	0
$448$	32.0000i	0.0714286i
$449$	−216.375	−0.481904	−0.240952	−	0.970537i	$-0.577460\pi$
$449$	−216.375	−0.481904	−0.240952	+	0.970537i	$0.577460\pi$
$450$	0	0
$451$	−792.000	−1.75610
$452$	313.955i	0.694592i
$453$	0	0
$454$	24.0000	0.0528634
$455$	0	0
$456$	0	0
$457$	− 400.000i	− 0.875274i	−0.899152	−	0.437637i	$-0.855816\pi$
$457$	− 400.000i	− 0.875274i	0.899152	−	0.437637i	$-0.144184\pi$
$458$	− 11.3137i	− 0.0247024i
$459$	0	0
$460$	144.000i	0.313043i
$461$	−301.227	−0.653422	−0.326711	−	0.945124i	$-0.605940\pi$
$461$	−301.227	−0.653422	−0.326711	+	0.945124i	$0.605940\pi$
$462$	0	0
$463$	604.000i	1.30454i	0.757989	+	0.652268i	$0.226182\pi$
$463$	604.000i	1.30454i	−0.757989	+	0.652268i	$0.773818\pi$
$464$	− 16.9706i	− 0.0365745i
$465$	0	0
$466$	− 18.0000i	− 0.0386266i
$467$	356.382i	0.763130i	0.924342	+	0.381565i	$0.124615\pi$
$467$	356.382i	0.763130i	−0.924342	+	0.381565i	$0.875385\pi$
$468$	0	0
$469$	−32.0000	−0.0682303
$470$	−509.117	−1.08323
$471$	0	0
$472$	−96.0000	−0.203390
$473$	−678.823	−1.43514
$474$	0	0
$475$	112.000i	0.235789i
$476$	101.823	0.213915
$477$	0	0
$478$	192.000	0.401674
$479$	−526.087	−1.09830	−0.549152	−	0.835723i	$-0.685049\pi$
$479$	−526.087	−1.09830	−0.549152	+	0.835723i	$0.685049\pi$
$480$	0	0
$481$	0	0
$482$	− 45.2548i	− 0.0938897i
$483$	0	0
$484$	334.000	0.690083
$485$	− 746.705i	− 1.53960i
$486$	0	0
$487$	596.000i	1.22382i	0.790928	+	0.611910i	$0.209598\pi$
$487$	596.000i	1.22382i	−0.790928	+	0.611910i	$0.790402\pi$
$488$	141.421	0.289798
$489$	0	0
$490$	−198.000	−0.404082
$491$	− 271.529i	− 0.553012i	−0.961012	−	0.276506i	$-0.910823\pi$
$491$	− 271.529i	− 0.553012i	0.961012	−	0.276506i	$-0.0891766\pi$
$492$	0	0
$493$	−54.0000	−0.109533
$494$	0	0
$495$	0	0
$496$	176.000i	0.354839i
$497$	− 203.647i	− 0.409752i
$498$	0	0
$499$	224.000i	0.448898i	0.974486	+	0.224449i	$0.0720582\pi$
$499$	224.000i	0.448898i	−0.974486	+	0.224449i	$0.927942\pi$
$500$	271.529	0.543058
$501$	0	0
$502$	72.0000i	0.143426i
$503$	− 865.499i	− 1.72067i	−0.509726	−	0.860337i	$-0.670253\pi$
$503$	− 865.499i	− 1.72067i	0.509726	−	0.860337i	$-0.329747\pi$
$504$	0	0
$505$	− 126.000i	− 0.249505i
$506$	407.294i	0.804928i
$507$	0	0
$508$	−184.000	−0.362205
$509$	479.418	0.941883	0.470941	−	0.882164i	$-0.343914\pi$
$509$	479.418	0.941883	0.470941	+	0.882164i	$0.343914\pi$
$510$	0	0
$511$	−64.0000	−0.125245
$512$	22.6274	0.0441942
$513$	0	0
$514$	− 258.000i	− 0.501946i
$515$	−118.794	−0.230668
$516$	0	0
$517$	−1440.00	−2.78530
$518$	−192.333	−0.371299
$519$	0	0
$520$	0	0
$521$	521.845i	1.00162i	0.865557	+	0.500811i	$0.166965\pi$
$521$	521.845i	1.00162i	−0.865557	+	0.500811i	$0.833035\pi$
$522$	0	0
$523$	−736.000	−1.40727	−0.703633	−	0.710564i	$-0.748440\pi$
$523$	−736.000	−1.40727	−0.703633	+	0.710564i	$0.748440\pi$
$524$	339.411i	0.647731i
$525$	0	0
$526$	− 528.000i	− 1.00380i
$527$	560.029	1.06267
$528$	0	0
$529$	241.000	0.455577
$530$	229.103i	0.432269i
$531$	0	0
$532$	128.000	0.240602
$533$	0	0
$534$	0	0
$535$	0	0
$536$	22.6274i	0.0422153i
$537$	0	0
$538$	486.000i	0.903346i
$539$	−560.029	−1.03901
$540$	0	0
$541$	808.000i	1.49353i	0.665088	+	0.746765i	$0.268394\pi$
$541$	808.000i	1.49353i	−0.665088	+	0.746765i	$0.731606\pi$
$542$	− 537.401i	− 0.991515i
$543$	0	0
$544$	− 72.0000i	− 0.132353i
$545$	− 237.588i	− 0.435941i
$546$	0	0
$547$	536.000	0.979890	0.489945	−	0.871753i	$-0.337017\pi$
$547$	536.000	0.979890	0.489945	+	0.871753i	$0.337017\pi$
$548$	−313.955	−0.572911
$549$	0	0
$550$	168.000	0.305455
$551$	−67.8823	−0.123198
$552$	0	0
$553$	− 304.000i	− 0.549729i
$554$	463.862	0.837296
$555$	0	0
$556$	304.000	0.546763
$557$	165.463	0.297061	0.148531	−	0.988908i	$-0.452546\pi$
$557$	165.463	0.297061	0.148531	+	0.988908i	$0.452546\pi$
$558$	0	0
$559$	0	0
$560$	− 67.8823i	− 0.121218i
$561$	0	0
$562$	−402.000	−0.715302
$563$	− 322.441i	− 0.572719i	−0.958122	−	0.286359i	$-0.907555\pi$
$563$	− 322.441i	− 0.572719i	0.958122	−	0.286359i	$-0.0924451\pi$
$564$	0	0
$565$	− 666.000i	− 1.17876i
$566$	294.156	0.519711
$567$	0	0
$568$	−144.000	−0.253521
$569$	156.978i	0.275883i	0.990440	+	0.137942i	$0.0440487\pi$
$569$	156.978i	0.275883i	−0.990440	+	0.137942i	$0.955951\pi$
$570$	0	0
$571$	−368.000	−0.644483	−0.322242	−	0.946657i	$-0.604436\pi$
$571$	−368.000	−0.644483	−0.322242	+	0.946657i	$0.604436\pi$
$572$	0	0
$573$	0	0
$574$	264.000i	0.459930i
$575$	118.794i	0.206598i
$576$	0	0
$577$	− 142.000i	− 0.246101i	−0.992400	−	0.123050i	$-0.960732\pi$
$577$	− 142.000i	− 0.246101i	0.992400	−	0.123050i	$-0.0392676\pi$
$578$	179.605	0.310736
$579$	0	0
$580$	36.0000i	0.0620690i
$581$	475.176i	0.817858i
$582$	0	0
$583$	648.000i	1.11149i
$584$	45.2548i	0.0774912i
$585$	0	0
$586$	618.000	1.05461
$587$	373.352	0.636035	0.318017	−	0.948085i	$-0.396983\pi$
$587$	373.352	0.636035	0.318017	+	0.948085i	$0.396983\pi$
$588$	0	0
$589$	704.000	1.19525
$590$	203.647	0.345164
$591$	0	0
$592$	136.000i	0.229730i
$593$	−1107.33	−1.86733	−0.933667	−	0.358142i	$-0.883410\pi$
$593$	−1107.33	−1.86733	−0.933667	+	0.358142i	$0.883410\pi$
$594$	0	0
$595$	−216.000	−0.363025
$596$	551.543	0.925408
$597$	0	0
$598$	0	0
$599$	− 797.616i	− 1.33158i	−0.746139	−	0.665790i	$-0.768095\pi$
$599$	− 797.616i	− 1.33158i	0.746139	−	0.665790i	$-0.231905\pi$
$600$	0	0
$601$	158.000	0.262895	0.131448	−	0.991323i	$-0.458037\pi$
$601$	158.000	0.262895	0.131448	+	0.991323i	$0.458037\pi$
$602$	226.274i	0.375871i
$603$	0	0
$604$	296.000i	0.490066i
$605$	−708.521	−1.17111
$606$	0	0
$607$	332.000	0.546952	0.273476	−	0.961879i	$-0.411827\pi$
$607$	332.000	0.546952	0.273476	+	0.961879i	$0.411827\pi$
$608$	− 90.5097i	− 0.148865i
$609$	0	0
$610$	−300.000	−0.491803
$611$	0	0
$612$	0	0
$613$	578.000i	0.942904i	0.881892	+	0.471452i	$0.156270\pi$
$613$	578.000i	0.942904i	−0.881892	+	0.471452i	$0.843730\pi$
$614$	735.391i	1.19771i
$615$	0	0
$616$	− 192.000i	− 0.311688i
$617$	−55.1543	−0.0893911	−0.0446956	−	0.999001i	$-0.514232\pi$
$617$	−55.1543	−0.0893911	−0.0446956	+	0.999001i	$0.514232\pi$
$618$	0	0
$619$	− 896.000i	− 1.44750i	−0.690064	−	0.723748i	$-0.742418\pi$
$619$	− 896.000i	− 1.44750i	0.690064	−	0.723748i	$-0.257582\pi$
$620$	− 373.352i	− 0.602181i
$621$	0	0
$622$	− 528.000i	− 0.848875i
$623$	− 50.9117i	− 0.0817202i
$624$	0	0
$625$	−401.000	−0.641600
$626$	−132.936	−0.212358
$627$	0	0
$628$	−164.000	−0.261146
$629$	432.749	0.687996
$630$	0	0
$631$	− 20.0000i	− 0.0316957i	−0.999874	−	0.0158479i	$-0.994955\pi$
$631$	− 20.0000i	− 0.0316957i	0.999874	−	0.0158479i	$-0.00504474\pi$
$632$	−214.960	−0.340127
$633$	0	0
$634$	474.000	0.747634
$635$	390.323	0.614682
$636$	0	0
$637$	0	0
$638$	101.823i	0.159598i
$639$	0	0
$640$	−48.0000	−0.0750000
$641$	258.801i	0.403746i	0.979412	+	0.201873i	$0.0647028\pi$
$641$	258.801i	0.403746i	−0.979412	+	0.201873i	$0.935297\pi$
$642$	0	0
$643$	728.000i	1.13219i	0.824339	+	0.566096i	$0.191547\pi$
$643$	728.000i	1.13219i	−0.824339	+	0.566096i	$0.808453\pi$
$644$	135.765	0.210814
$645$	0	0
$646$	−288.000	−0.445820
$647$	− 458.205i	− 0.708200i	−0.935208	−	0.354100i	$-0.884787\pi$
$647$	− 458.205i	− 0.708200i	0.935208	−	0.354100i	$-0.115213\pi$
$648$	0	0
$649$	576.000	0.887519
$650$	0	0
$651$	0	0
$652$	− 112.000i	− 0.171779i
$653$	− 301.227i	− 0.461298i	−0.973037	−	0.230649i	$-0.925915\pi$
$653$	− 301.227i	− 0.461298i	0.973037	−	0.230649i	$-0.0740849\pi$
$654$	0	0
$655$	− 720.000i	− 1.09924i
$656$	186.676	0.284567
$657$	0	0
$658$	480.000i	0.729483i
$659$	1052.17i	1.59662i	0.602244	+	0.798312i	$0.294273\pi$
$659$	1052.17i	1.59662i	−0.602244	+	0.798312i	$0.705727\pi$
$660$	0	0
$661$	− 62.0000i	− 0.0937973i	−0.998900	−	0.0468986i	$-0.985066\pi$
$661$	− 62.0000i	− 0.0937973i	0.998900	−	0.0468986i	$-0.0149338\pi$
$662$	758.018i	1.14504i
$663$	0	0
$664$	336.000	0.506024
$665$	−271.529	−0.408314
$666$	0	0
$667$	−72.0000	−0.107946
$668$	−67.8823	−0.101620
$669$	0	0
$670$	− 48.0000i	− 0.0716418i
$671$	−848.528	−1.26457
$672$	0	0
$673$	670.000	0.995542	0.497771	−	0.867308i	$-0.334152\pi$
$673$	670.000	0.995542	0.497771	+	0.867308i	$0.334152\pi$
$674$	294.156	0.436434
$675$	0	0
$676$	0	0
$677$	1294.01i	1.91138i	0.294372	+	0.955691i	$0.404889\pi$
$677$	1294.01i	1.91138i	−0.294372	+	0.955691i	$0.595111\pi$
$678$	0	0
$679$	−704.000	−1.03682
$680$	152.735i	0.224610i
$681$	0	0
$682$	− 1056.00i	− 1.54839i
$683$	560.029	0.819954	0.409977	−	0.912096i	$-0.365537\pi$
$683$	560.029	0.819954	0.409977	+	0.912096i	$0.365537\pi$
$684$	0	0
$685$	666.000	0.972263
$686$	463.862i	0.676184i
$687$	0	0
$688$	160.000	0.232558
$689$	0	0
$690$	0	0
$691$	− 40.0000i	− 0.0578871i	−0.999581	−	0.0289436i	$-0.990786\pi$
$691$	− 40.0000i	− 0.0578871i	0.999581	−	0.0289436i	$-0.00921431\pi$
$692$	− 347.897i	− 0.502741i
$693$	0	0
$694$	− 408.000i	− 0.587896i
$695$	−644.881	−0.927887
$696$	0	0
$697$	− 594.000i	− 0.852224i
$698$	336.583i	0.482210i
$699$	0	0
$700$	− 56.0000i	− 0.0800000i
$701$	954.594i	1.36176i	0.732395	+	0.680880i	$0.238403\pi$
$701$	954.594i	1.36176i	−0.732395	+	0.680880i	$0.761597\pi$
$702$	0	0
$703$	544.000	0.773826
$704$	−135.765	−0.192847
$705$	0	0
$706$	−318.000	−0.450425
$707$	−118.794	−0.168025
$708$	0	0
$709$	− 968.000i	− 1.36530i	−0.730744	−	0.682652i	$-0.760827\pi$
$709$	− 968.000i	− 1.36530i	0.730744	−	0.682652i	$-0.239173\pi$
$710$	305.470	0.430240
$711$	0	0
$712$	−36.0000	−0.0505618
$713$	746.705	1.04727
$714$	0	0
$715$	0	0
$716$	− 407.294i	− 0.568846i
$717$	0	0
$718$	792.000	1.10306
$719$	1170.97i	1.62861i	0.580439	+	0.814304i	$0.302881\pi$
$719$	1170.97i	1.62861i	−0.580439	+	0.814304i	$0.697119\pi$
$720$	0	0
$721$	112.000i	0.155340i
$722$	148.492	0.205668
$723$	0	0
$724$	464.000	0.640884
$725$	29.6985i	0.0409634i
$726$	0	0
$727$	508.000	0.698762	0.349381	−	0.936981i	$-0.386392\pi$
$727$	508.000	0.698762	0.349381	+	0.936981i	$0.386392\pi$
$728$	0	0
$729$	0	0
$730$	− 96.0000i	− 0.131507i
$731$	− 509.117i	− 0.696466i
$732$	0	0
$733$	− 1144.00i	− 1.56071i	−0.625337	−	0.780355i	$-0.715039\pi$
$733$	− 1144.00i	− 1.56071i	0.625337	−	0.780355i	$-0.284961\pi$
$734$	401.637	0.547189
$735$	0	0
$736$	− 96.0000i	− 0.130435i
$737$	− 135.765i	− 0.184212i
$738$	0	0
$739$	304.000i	0.411367i	0.978619	+	0.205683i	$0.0659417\pi$
$739$	304.000i	0.411367i	−0.978619	+	0.205683i	$0.934058\pi$
$740$	− 288.500i	− 0.389864i
$741$	0	0
$742$	216.000	0.291105
$743$	848.528	1.14203	0.571015	−	0.820940i	$-0.306550\pi$
$743$	848.528	1.14203	0.571015	+	0.820940i	$0.306550\pi$
$744$	0	0
$745$	−1170.00	−1.57047
$746$	−268.701	−0.360188
$747$	0	0
$748$	432.000i	0.577540i
$749$	0	0
$750$	0	0
$751$	−188.000	−0.250333	−0.125166	−	0.992136i	$-0.539946\pi$
$751$	−188.000	−0.250333	−0.125166	+	0.992136i	$0.539946\pi$
$752$	339.411	0.451345
$753$	0	0
$754$	0	0
$755$	− 627.911i	− 0.831670i
$756$	0	0
$757$	−1240.00	−1.63804	−0.819022	−	0.573761i	$-0.805484\pi$
$757$	−1240.00	−1.63804	−0.819022	+	0.573761i	$0.805484\pi$
$758$	− 226.274i	− 0.298515i
$759$	0	0
$760$	192.000i	0.252632i
$761$	156.978	0.206278	0.103139	−	0.994667i	$-0.467111\pi$
$761$	156.978	0.206278	0.103139	+	0.994667i	$0.467111\pi$
$762$	0	0
$763$	−224.000	−0.293578
$764$	67.8823i	0.0888511i
$765$	0	0
$766$	−384.000	−0.501305
$767$	0	0
$768$	0	0
$769$	− 910.000i	− 1.18336i	−0.806175	−	0.591678i	$-0.798466\pi$
$769$	− 910.000i	− 1.18336i	0.806175	−	0.591678i	$-0.201534\pi$
$770$	407.294i	0.528953i
$771$	0	0
$772$	− 412.000i	− 0.533679i
$773$	−1387.34	−1.79475	−0.897376	−	0.441266i	$-0.854529\pi$
$773$	−1387.34	−1.79475	−0.897376	+	0.441266i	$0.854529\pi$
$774$	0	0
$775$	− 308.000i	− 0.397419i
$776$	497.803i	0.641499i
$777$	0	0
$778$	− 570.000i	− 0.732648i
$779$	− 746.705i	− 0.958543i
$780$	0	0
$781$	864.000	1.10627
$782$	−305.470	−0.390627
$783$	0	0
$784$	132.000	0.168367
$785$	347.897	0.443180
$786$	0	0
$787$	1360.00i	1.72808i	0.503422	+	0.864041i	$0.332074\pi$
$787$	1360.00i	1.72808i	−0.503422	+	0.864041i	$0.667926\pi$
$788$	330.926	0.419957
$789$	0	0
$790$	456.000	0.577215
$791$	−627.911	−0.793819
$792$	0	0
$793$	0	0
$794$	− 206.475i	− 0.260044i
$795$	0	0
$796$	−40.0000	−0.0502513
$797$	− 106.066i	− 0.133082i	−0.997784	−	0.0665408i	$-0.978804\pi$
$797$	− 106.066i	− 0.133082i	0.997784	−	0.0665408i	$-0.0211963\pi$
$798$	0	0
$799$	− 1080.00i	− 1.35169i
$800$	−39.5980	−0.0494975
$801$	0	0
$802$	−462.000	−0.576060
$803$	− 271.529i	− 0.338143i
$804$	0	0
$805$	−288.000	−0.357764
$806$	0	0
$807$	0	0
$808$	84.0000i	0.103960i
$809$	1107.33i	1.36876i	0.729124	+	0.684381i	$0.239928\pi$
$809$	1107.33i	1.36876i	−0.729124	+	0.684381i	$0.760072\pi$
$810$	0	0
$811$	− 160.000i	− 0.197287i	−0.995123	−	0.0986436i	$-0.968550\pi$
$811$	− 160.000i	− 0.197287i	0.995123	−	0.0986436i	$-0.0314504\pi$
$812$	33.9411	0.0417994
$813$	0	0
$814$	− 816.000i	− 1.00246i
$815$	237.588i	0.291519i
$816$	0	0
$817$	− 640.000i	− 0.783354i
$818$	520.431i	0.636223i
$819$	0	0
$820$	−396.000	−0.482927
$821$	436.992	0.532268	0.266134	−	0.963936i	$-0.414254\pi$
$821$	436.992	0.532268	0.266134	+	0.963936i	$0.414254\pi$
$822$	0	0
$823$	−332.000	−0.403402	−0.201701	−	0.979447i	$-0.564647\pi$
$823$	−332.000	−0.403402	−0.201701	+	0.979447i	$0.564647\pi$
$824$	79.1960	0.0961116
$825$	0	0
$826$	− 192.000i	− 0.232446i
$827$	−101.823	−0.123124	−0.0615619	−	0.998103i	$-0.519608\pi$
$827$	−101.823	−0.123124	−0.0615619	+	0.998103i	$0.519608\pi$
$828$	0	0
$829$	−632.000	−0.762364	−0.381182	−	0.924500i	$-0.624483\pi$
$829$	−632.000	−0.762364	−0.381182	+	0.924500i	$0.624483\pi$
$830$	−712.764	−0.858751
$831$	0	0
$832$	0	0
$833$	− 420.021i	− 0.504227i
$834$	0	0
$835$	144.000	0.172455
$836$	543.058i	0.649591i
$837$	0	0
$838$	− 552.000i	− 0.658711i
$839$	−729.734	−0.869767	−0.434883	−	0.900487i	$-0.643210\pi$
$839$	−729.734	−0.869767	−0.434883	+	0.900487i	$0.643210\pi$
$840$	0	0
$841$	823.000	0.978597
$842$	− 56.5685i	− 0.0671835i
$843$	0	0
$844$	592.000	0.701422
$845$	0	0
$846$	0	0
$847$	668.000i	0.788666i
$848$	− 152.735i	− 0.180112i
$849$	0	0
$850$	126.000i	0.148235i
$851$	576.999	0.678025
$852$	0	0
$853$	− 446.000i	− 0.522860i	−0.965222	−	0.261430i	$-0.915806\pi$
$853$	− 446.000i	− 0.522860i	0.965222	−	0.261430i	$-0.0841941\pi$
$854$	282.843i	0.331198i
$855$	0	0
$856$	0	0
$857$	− 428.507i	− 0.500008i	−0.968245	−	0.250004i	$-0.919568\pi$
$857$	− 428.507i	− 0.500008i	0.968245	−	0.250004i	$-0.0804319\pi$
$858$	0	0
$859$	728.000	0.847497	0.423749	−	0.905780i	$-0.360714\pi$
$859$	728.000	0.847497	0.423749	+	0.905780i	$0.360714\pi$
$860$	−339.411	−0.394664
$861$	0	0
$862$	−216.000	−0.250580
$863$	−916.410	−1.06189	−0.530945	−	0.847407i	$-0.678163\pi$
$863$	−916.410	−1.06189	−0.530945	+	0.847407i	$0.678163\pi$
$864$	0	0
$865$	738.000i	0.853179i
$866$	−766.504	−0.885108
$867$	0	0
$868$	−352.000	−0.405530
$869$	1289.76	1.48419
$870$	0	0
$871$	0	0
$872$	158.392i	0.181642i
$873$	0	0
$874$	−384.000	−0.439359
$875$	543.058i	0.620638i
$876$	0	0
$877$	− 910.000i	− 1.03763i	−0.854887	−	0.518814i	$-0.826374\pi$
$877$	− 910.000i	− 1.03763i	0.854887	−	0.518814i	$-0.173626\pi$
$878$	5.65685	0.00644289
$879$	0	0
$880$	288.000	0.327273
$881$	− 929.138i	− 1.05464i	−0.849667	−	0.527320i	$-0.823197\pi$
$881$	− 929.138i	− 1.05464i	0.849667	−	0.527320i	$-0.176803\pi$
$882$	0	0
$883$	−1064.00	−1.20498	−0.602492	−	0.798125i	$-0.705825\pi$
$883$	−1064.00	−1.20498	−0.602492	+	0.798125i	$0.705825\pi$
$884$	0	0
$885$	0	0
$886$	− 456.000i	− 0.514673i
$887$	1391.59i	1.56887i	0.620212	+	0.784434i	$0.287047\pi$
$887$	1391.59i	1.56887i	−0.620212	+	0.784434i	$0.712953\pi$
$888$	0	0
$889$	− 368.000i	− 0.413948i
$890$	76.3675	0.0858062
$891$	0	0
$892$	− 872.000i	− 0.977578i
$893$	− 1357.65i	− 1.52032i
$894$	0	0
$895$	864.000i	0.965363i
$896$	45.2548i	0.0505076i
$897$	0	0
$898$	−306.000	−0.340757
$899$	186.676	0.207649
$900$	0	0
$901$	−486.000	−0.539401
$902$	−1120.06	−1.24175
$903$	0	0
$904$	444.000i	0.491150i
$905$	−984.293	−1.08762
$906$	0	0
$907$	1768.00	1.94928	0.974642	−	0.223771i	$-0.0718369\pi$
$907$	1768.00	1.94928	0.974642	+	0.223771i	$0.0718369\pi$
$908$	33.9411	0.0373801
$909$	0	0
$910$	0	0
$911$	237.588i	0.260799i	0.991462	+	0.130399i	$0.0416260\pi$
$911$	237.588i	0.260799i	−0.991462	+	0.130399i	$0.958374\pi$
$912$	0	0
$913$	−2016.00	−2.20811
$914$	− 565.685i	− 0.618912i
$915$	0	0
$916$	− 16.0000i	− 0.0174672i
$917$	−678.823	−0.740264
$918$	0	0
$919$	380.000	0.413493	0.206746	−	0.978395i	$-0.433712\pi$
$919$	380.000	0.413493	0.206746	+	0.978395i	$0.433712\pi$
$920$	203.647i	0.221355i
$921$	0	0
$922$	−426.000	−0.462039
$923$	0	0
$924$	0	0
$925$	− 238.000i	− 0.257297i
$926$	854.185i	0.922446i
$927$	0	0
$928$	− 24.0000i	− 0.0258621i
$929$	666.095	0.717002	0.358501	−	0.933529i	$-0.383288\pi$
$929$	666.095	0.717002	0.358501	+	0.933529i	$0.383288\pi$
$930$	0	0
$931$	− 528.000i	− 0.567132i
$932$	− 25.4558i	− 0.0273131i
$933$	0	0
$934$	504.000i	0.539615i
$935$	− 916.410i	− 0.980118i
$936$	0	0
$937$	−178.000	−0.189968	−0.0949840	−	0.995479i	$-0.530280\pi$
$937$	−178.000	−0.189968	−0.0949840	+	0.995479i	$0.530280\pi$
$938$	−45.2548	−0.0482461
$939$	0	0
$940$	−720.000	−0.765957
$941$	−436.992	−0.464391	−0.232196	−	0.972669i	$-0.574591\pi$
$941$	−436.992	−0.464391	−0.232196	+	0.972669i	$0.574591\pi$
$942$	0	0
$943$	− 792.000i	− 0.839873i
$944$	−135.765	−0.143818
$945$	0	0
$946$	−960.000	−1.01480
$947$	−1798.88	−1.89956	−0.949778	−	0.312924i	$-0.898691\pi$
$947$	−1798.88	−1.89956	−0.949778	+	0.312924i	$0.898691\pi$
$948$	0	0
$949$	0	0
$950$	158.392i	0.166728i
$951$	0	0
$952$	144.000	0.151261
$953$	1310.98i	1.37563i	0.725886	+	0.687815i	$0.241430\pi$
$953$	1310.98i	1.37563i	−0.725886	+	0.687815i	$0.758570\pi$
$954$	0	0
$955$	− 144.000i	− 0.150785i
$956$	271.529	0.284026
$957$	0	0
$958$	−744.000	−0.776618
$959$	− 627.911i	− 0.654756i
$960$	0	0
$961$	−975.000	−1.01457
$962$	0	0
$963$	0	0
$964$	− 64.0000i	− 0.0663900i
$965$	873.984i	0.905683i
$966$	0	0
$967$	1700.00i	1.75801i	0.476808	+	0.879007i	$0.341794\pi$
$967$	1700.00i	1.75801i	−0.476808	+	0.879007i	$0.658206\pi$
$968$	472.347	0.487962
$969$	0	0
$970$	− 1056.00i	− 1.08866i
$971$	− 458.205i	− 0.471890i	−0.971766	−	0.235945i	$-0.924181\pi$
$971$	− 458.205i	− 0.471890i	0.971766	−	0.235945i	$-0.0758185\pi$
$972$	0	0
$973$	608.000i	0.624872i
$974$	842.871i	0.865371i
$975$	0	0
$976$	200.000	0.204918
$977$	759.433	0.777311	0.388655	−	0.921383i	$-0.372940\pi$
$977$	759.433	0.777311	0.388655	+	0.921383i	$0.372940\pi$
$978$	0	0
$979$	216.000	0.220633
$980$	−280.014	−0.285729
$981$	0	0
$982$	− 384.000i	− 0.391039i
$983$	1052.17	1.07037	0.535186	−	0.844734i	$-0.320242\pi$
$983$	1052.17	1.07037	0.535186	+	0.844734i	$0.320242\pi$
$984$	0	0
$985$	−702.000	−0.712690
$986$	−76.3675	−0.0774519
$987$	0	0
$988$	0	0
$989$	− 678.823i	− 0.686373i
$990$	0	0
$991$	−772.000	−0.779011	−0.389506	−	0.921024i	$-0.627354\pi$
$991$	−772.000	−0.779011	−0.389506	+	0.921024i	$0.627354\pi$
$992$	248.902i	0.250909i
$993$	0	0
$994$	− 288.000i	− 0.289738i
$995$	84.8528	0.0852792
$996$	0	0
$997$	194.000	0.194584	0.0972919	−	0.995256i	$-0.468982\pi$
$997$	194.000	0.194584	0.0972919	+	0.995256i	$0.468982\pi$
$998$	316.784i	0.317419i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.3.d.a.3041.4		4
3.2	odd	2	inner	3042.3.d.a.3041.2		4
13.5	odd	4		18.3.b.a.17.1	✓	2
13.8	odd	4		3042.3.c.e.1691.2		2
13.12	even	2	inner	3042.3.d.a.3041.1		4
39.5	even	4		18.3.b.a.17.2	yes	2
39.8	even	4		3042.3.c.e.1691.1		2
39.38	odd	2	inner	3042.3.d.a.3041.3		4
52.31	even	4		144.3.e.b.17.2		2
65.18	even	4		450.3.b.b.449.2		4
65.44	odd	4		450.3.d.f.251.2		2
65.57	even	4		450.3.b.b.449.3		4
91.5	even	12		882.3.s.d.557.1		4
91.18	odd	12		882.3.s.b.863.2		4
91.31	even	12		882.3.s.d.863.2		4
91.44	odd	12		882.3.s.b.557.1		4
91.83	even	4		882.3.b.a.197.1		2
104.5	odd	4		576.3.e.c.449.1		2
104.83	even	4		576.3.e.f.449.1		2
117.5	even	12		162.3.d.b.107.1		4
117.31	odd	12		162.3.d.b.107.2		4
117.70	odd	12		162.3.d.b.53.1		4
117.83	even	12		162.3.d.b.53.2		4
143.109	even	4		2178.3.c.d.485.2		2
156.83	odd	4		144.3.e.b.17.1		2
195.44	even	4		450.3.d.f.251.1		2
195.83	odd	4		450.3.b.b.449.4		4
195.122	odd	4		450.3.b.b.449.1		4
208.5	odd	4		2304.3.h.f.2177.1		4
208.83	even	4		2304.3.h.c.2177.4		4
208.109	odd	4		2304.3.h.f.2177.4		4
208.187	even	4		2304.3.h.c.2177.1		4
260.83	odd	4		3600.3.c.b.449.2		4
260.187	odd	4		3600.3.c.b.449.4		4
260.239	even	4		3600.3.l.d.1601.2		2
273.5	odd	12		882.3.s.d.557.2		4
273.44	even	12		882.3.s.b.557.2		4
273.83	odd	4		882.3.b.a.197.2		2
273.122	odd	12		882.3.s.d.863.1		4
273.200	even	12		882.3.s.b.863.1		4
312.5	even	4		576.3.e.c.449.2		2
312.83	odd	4		576.3.e.f.449.2		2
429.395	odd	4		2178.3.c.d.485.1		2
468.31	even	12		1296.3.q.f.593.2		4
468.83	odd	12		1296.3.q.f.1025.2		4
468.187	even	12		1296.3.q.f.1025.1		4
468.239	odd	12		1296.3.q.f.593.1		4
624.5	even	4		2304.3.h.f.2177.3		4
624.83	odd	4		2304.3.h.c.2177.2		4
624.317	even	4		2304.3.h.f.2177.2		4
624.395	odd	4		2304.3.h.c.2177.3		4
780.83	even	4		3600.3.c.b.449.1		4
780.239	odd	4		3600.3.l.d.1601.1		2
780.707	even	4		3600.3.c.b.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.b.a.17.1	✓	2	13.5	odd	4
18.3.b.a.17.2	yes	2	39.5	even	4
144.3.e.b.17.1		2	156.83	odd	4
144.3.e.b.17.2		2	52.31	even	4
162.3.d.b.53.1		4	117.70	odd	12
162.3.d.b.53.2		4	117.83	even	12
162.3.d.b.107.1		4	117.5	even	12
162.3.d.b.107.2		4	117.31	odd	12
450.3.b.b.449.1		4	195.122	odd	4
450.3.b.b.449.2		4	65.18	even	4
450.3.b.b.449.3		4	65.57	even	4
450.3.b.b.449.4		4	195.83	odd	4
450.3.d.f.251.1		2	195.44	even	4
450.3.d.f.251.2		2	65.44	odd	4
576.3.e.c.449.1		2	104.5	odd	4
576.3.e.c.449.2		2	312.5	even	4
576.3.e.f.449.1		2	104.83	even	4
576.3.e.f.449.2		2	312.83	odd	4
882.3.b.a.197.1		2	91.83	even	4
882.3.b.a.197.2		2	273.83	odd	4
882.3.s.b.557.1		4	91.44	odd	12
882.3.s.b.557.2		4	273.44	even	12
882.3.s.b.863.1		4	273.200	even	12
882.3.s.b.863.2		4	91.18	odd	12
882.3.s.d.557.1		4	91.5	even	12
882.3.s.d.557.2		4	273.5	odd	12
882.3.s.d.863.1		4	273.122	odd	12
882.3.s.d.863.2		4	91.31	even	12
1296.3.q.f.593.1		4	468.239	odd	12
1296.3.q.f.593.2		4	468.31	even	12
1296.3.q.f.1025.1		4	468.187	even	12
1296.3.q.f.1025.2		4	468.83	odd	12
2178.3.c.d.485.1		2	429.395	odd	4
2178.3.c.d.485.2		2	143.109	even	4
2304.3.h.c.2177.1		4	208.187	even	4
2304.3.h.c.2177.2		4	624.83	odd	4
2304.3.h.c.2177.3		4	624.395	odd	4
2304.3.h.c.2177.4		4	208.83	even	4
2304.3.h.f.2177.1		4	208.5	odd	4
2304.3.h.f.2177.2		4	624.317	even	4
2304.3.h.f.2177.3		4	624.5	even	4
2304.3.h.f.2177.4		4	208.109	odd	4
3042.3.c.e.1691.1		2	39.8	even	4
3042.3.c.e.1691.2		2	13.8	odd	4
3042.3.d.a.3041.1		4	13.12	even	2	inner
3042.3.d.a.3041.2		4	3.2	odd	2	inner
3042.3.d.a.3041.3		4	39.38	odd	2	inner
3042.3.d.a.3041.4		4	1.1	even	1	trivial
3600.3.c.b.449.1		4	780.83	even	4
3600.3.c.b.449.2		4	260.83	odd	4
3600.3.c.b.449.3		4	780.707	even	4
3600.3.c.b.449.4		4	260.187	odd	4
3600.3.l.d.1601.1		2	780.239	odd	4
3600.3.l.d.1601.2		2	260.239	even	4

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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3042.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} -4.24264 q^{5} +4.00000i q^{7} +2.82843 q^{8} -6.00000 q^{10} -16.9706 q^{11} +5.65685i q^{14} +4.00000 q^{16} -12.7279i q^{17} -16.0000i q^{19} -8.48528 q^{20} -24.0000 q^{22} -16.9706i q^{23} -7.00000 q^{25} +8.00000i q^{28} -4.24264i q^{29} +44.0000i q^{31} +5.65685 q^{32} -18.0000i q^{34} -16.9706i q^{35} +34.0000i q^{37} -22.6274i q^{38} -12.0000 q^{40} +46.6690 q^{41} +40.0000 q^{43} -33.9411 q^{44} -24.0000i q^{46} +84.8528 q^{47} +33.0000 q^{49} -9.89949 q^{50} -38.1838i q^{53} +72.0000 q^{55} +11.3137i q^{56} -6.00000i q^{58} -33.9411 q^{59} +50.0000 q^{61} +62.2254i q^{62} +8.00000 q^{64} +8.00000i q^{67} -25.4558i q^{68} -24.0000i q^{70} -50.9117 q^{71} +16.0000i q^{73} +48.0833i q^{74} -32.0000i q^{76} -67.8823i q^{77} -76.0000 q^{79} -16.9706 q^{80} +66.0000 q^{82} +118.794 q^{83} +54.0000i q^{85} +56.5685 q^{86} -48.0000 q^{88} -12.7279 q^{89} -33.9411i q^{92} +120.000 q^{94} +67.8823i q^{95} +176.000i q^{97} +46.6690 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 24 q^{10} + 16 q^{16} - 96 q^{22} - 28 q^{25} - 48 q^{40} + 160 q^{43} + 132 q^{49} + 288 q^{55} + 200 q^{61} + 32 q^{64} - 304 q^{79} + 264 q^{82} - 192 q^{88} + 480 q^{94}+O(q^{100}) \) 4 * q + 8 * q^4 - 24 * q^10 + 16 * q^16 - 96 * q^22 - 28 * q^25 - 48 * q^40 + 160 * q^43 + 132 * q^49 + 288 * q^55 + 200 * q^61 + 32 * q^64 - 304 * q^79 + 264 * q^82 - 192 * q^88 + 480 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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