LMFDB - Embedded newform 3042.3.c.e.1691.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,3,Mod(1691,3042)] 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("3042.1691"); 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, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3042.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-4,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$82.8884964184$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1691.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	3042.1691
Dual form			3042.3.c.e.1691.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} +4.00000 q^{7} -2.82843i q^{8} +6.00000 q^{10} +16.9706i q^{11} +5.65685i q^{14} +4.00000 q^{16} +12.7279i q^{17} +16.0000 q^{19} +8.48528i q^{20} -24.0000 q^{22} +16.9706i q^{23} +7.00000 q^{25} -8.00000 q^{28} -4.24264i q^{29} -44.0000 q^{31} +5.65685i q^{32} -18.0000 q^{34} -16.9706i q^{35} +34.0000 q^{37} +22.6274i q^{38} -12.0000 q^{40} +46.6690i q^{41} -40.0000 q^{43} -33.9411i q^{44} -24.0000 q^{46} -84.8528i q^{47} -33.0000 q^{49} +9.89949i q^{50} -38.1838i q^{53} +72.0000 q^{55} -11.3137i q^{56} +6.00000 q^{58} +33.9411i q^{59} +50.0000 q^{61} -62.2254i q^{62} -8.00000 q^{64} -8.00000 q^{67} -25.4558i q^{68} +24.0000 q^{70} -50.9117i q^{71} +16.0000 q^{73} +48.0833i q^{74} -32.0000 q^{76} +67.8823i q^{77} -76.0000 q^{79} -16.9706i q^{80} -66.0000 q^{82} +118.794i q^{83} +54.0000 q^{85} -56.5685i q^{86} +48.0000 q^{88} +12.7279i q^{89} -33.9411i q^{92} +120.000 q^{94} -67.8823i q^{95} -176.000 q^{97} -46.6690i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 8 q^{7} + 12 q^{10} + 8 q^{16} + 32 q^{19} - 48 q^{22} + 14 q^{25} - 16 q^{28} - 88 q^{31} - 36 q^{34} + 68 q^{37} - 24 q^{40} - 80 q^{43} - 48 q^{46} - 66 q^{49} + 144 q^{55} + 12 q^{58}+ \cdots - 352 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 8 * q^7 + 12 * q^10 + 8 * q^16 + 32 * q^19 - 48 * q^22 + 14 * q^25 - 16 * q^28 - 88 * q^31 - 36 * q^34 + 68 * q^37 - 24 * q^40 - 80 * q^43 - 48 * q^46 - 66 * q^49 + 144 * q^55 + 12 * q^58 + 100 * q^61 - 16 * q^64 - 16 * q^67 + 48 * q^70 + 32 * q^73 - 64 * q^76 - 152 * q^79 - 132 * q^82 + 108 * q^85 + 96 * q^88 + 240 * q^94 - 352 * q^97

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 4.24264i	− 0.848528i	−0.905539	−	0.424264i	$-0.860533\pi$
$5$	− 4.24264i	− 0.848528i	0.905539	−	0.424264i	$-0.139467\pi$
$6$	0	0
$7$	4.00000	0.571429	0.285714	−	0.958315i	$-0.407769\pi$
$7$	4.00000	0.571429	0.285714	+	0.958315i	$0.407769\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	6.00000	0.600000
$11$	16.9706i	1.54278i	0.636364	+	0.771389i	$0.280438\pi$
$11$	16.9706i	1.54278i	−0.636364	+	0.771389i	$0.719562\pi$
$12$	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	$0.122134\pi$
$17$	12.7279i	0.748701i	−0.927287	+	0.374351i	$0.877866\pi$
$18$	0	0
$19$	16.0000	0.842105	0.421053	−	0.907036i	$-0.361661\pi$
$19$	16.0000	0.842105	0.421053	+	0.907036i	$0.361661\pi$
$20$	8.48528i	0.424264i
$21$	0	0
$22$	−24.0000	−1.09091
$23$	16.9706i	0.737851i	0.929459	+	0.368925i	$0.120274\pi$
$23$	16.9706i	0.737851i	−0.929459	+	0.368925i	$0.879726\pi$
$24$	0	0
$25$	7.00000	0.280000
$26$	0	0
$27$	0	0
$28$	−8.00000	−0.285714
$29$	− 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
$31$	−44.0000	−1.41935	−0.709677	−	0.704527i	$-0.751159\pi$
$31$	−44.0000	−1.41935	−0.709677	+	0.704527i	$0.751159\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	−18.0000	−0.529412
$35$	− 16.9706i	− 0.484873i
$36$	0	0
$37$	34.0000	0.918919	0.459459	−	0.888199i	$-0.348043\pi$
$37$	34.0000	0.918919	0.459459	+	0.888199i	$0.348043\pi$
$38$	22.6274i	0.595458i
$39$	0	0
$40$	−12.0000	−0.300000
$41$	46.6690i	1.13827i	0.822244	+	0.569135i	$0.192722\pi$
$41$	46.6690i	1.13827i	−0.822244	+	0.569135i	$0.807278\pi$
$42$	0	0
$43$	−40.0000	−0.930233	−0.465116	−	0.885250i	$-0.653987\pi$
$43$	−40.0000	−0.930233	−0.465116	+	0.885250i	$0.653987\pi$
$44$	− 33.9411i	− 0.771389i
$45$	0	0
$46$	−24.0000	−0.521739
$47$	− 84.8528i	− 1.80538i	−0.430293	−	0.902690i	$-0.641590\pi$
$47$	− 84.8528i	− 1.80538i	0.430293	−	0.902690i	$-0.358410\pi$
$48$	0	0
$49$	−33.0000	−0.673469
$50$	9.89949i	0.197990i
$51$	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
$58$	6.00000	0.103448
$59$	33.9411i	0.575273i	0.957740	+	0.287637i	$0.0928695\pi$
$59$	33.9411i	0.575273i	−0.957740	+	0.287637i	$0.907130\pi$
$60$	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
$67$	−8.00000	−0.119403	−0.0597015	−	0.998216i	$-0.519015\pi$
$67$	−8.00000	−0.119403	−0.0597015	+	0.998216i	$0.519015\pi$
$68$	− 25.4558i	− 0.374351i
$69$	0	0
$70$	24.0000	0.342857
$71$	− 50.9117i	− 0.717066i	−0.933517	−	0.358533i	$-0.883277\pi$
$71$	− 50.9117i	− 0.717066i	0.933517	−	0.358533i	$-0.116723\pi$
$72$	0	0
$73$	16.0000	0.219178	0.109589	−	0.993977i	$-0.465047\pi$
$73$	16.0000	0.219178	0.109589	+	0.993977i	$0.465047\pi$
$74$	48.0833i	0.649774i
$75$	0	0
$76$	−32.0000	−0.421053
$77$	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	$0.659731\pi$
$80$	− 16.9706i	− 0.212132i
$81$	0	0
$82$	−66.0000	−0.804878
$83$	118.794i	1.43125i	0.698484	+	0.715626i	$0.253859\pi$
$83$	118.794i	1.43125i	−0.698484	+	0.715626i	$0.746141\pi$
$84$	0	0
$85$	54.0000	0.635294
$86$	− 56.5685i	− 0.657774i
$87$	0	0
$88$	48.0000	0.545455
$89$	12.7279i	0.143010i	0.997440	+	0.0715052i	$0.0227802\pi$
$89$	12.7279i	0.143010i	−0.997440	+	0.0715052i	$0.977220\pi$
$90$	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
$97$	−176.000	−1.81443	−0.907216	−	0.420664i	$-0.861797\pi$
$97$	−176.000	−1.81443	−0.907216	+	0.420664i	$0.861797\pi$
$98$	− 46.6690i	− 0.476215i
$99$	0	0
$100$	−14.0000	−0.140000
$101$	− 29.6985i	− 0.294044i	−0.989133	−	0.147022i	$-0.953031\pi$
$101$	− 29.6985i	− 0.294044i	0.989133	−	0.147022i	$-0.0469689\pi$
$102$	0	0
$103$	−28.0000	−0.271845	−0.135922	−	0.990719i	$-0.543400\pi$
$103$	−28.0000	−0.271845	−0.135922	+	0.990719i	$0.543400\pi$
$104$	0	0
$105$	0	0
$106$	54.0000	0.509434
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−56.0000	−0.513761	−0.256881	−	0.966443i	$-0.582695\pi$
$109$	−56.0000	−0.513761	−0.256881	+	0.966443i	$0.582695\pi$
$110$	101.823i	0.925667i
$111$	0	0
$112$	16.0000	0.142857
$113$	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
$115$	72.0000	0.626087
$116$	8.48528i	0.0731490i
$117$	0	0
$118$	−48.0000	−0.406780
$119$	50.9117i	0.427829i
$120$	0	0
$121$	−167.000	−1.38017
$122$	70.7107i	0.579596i
$123$	0	0
$124$	88.0000	0.709677
$125$	− 135.765i	− 1.08612i
$126$	0	0
$127$	92.0000	0.724409	0.362205	−	0.932099i	$-0.382024\pi$
$127$	92.0000	0.724409	0.362205	+	0.932099i	$0.382024\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	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
$136$	36.0000	0.264706
$137$	156.978i	1.14582i	0.819617	+	0.572911i	$0.194186\pi$
$137$	156.978i	1.14582i	−0.819617	+	0.572911i	$0.805814\pi$
$138$	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
$145$	−18.0000	−0.124138
$146$	22.6274i	0.154982i
$147$	0	0
$148$	−68.0000	−0.459459
$149$	275.772i	1.85082i	0.378972	+	0.925408i	$0.376278\pi$
$149$	275.772i	1.85082i	−0.378972	+	0.925408i	$0.623722\pi$
$150$	0	0
$151$	148.000	0.980132	0.490066	−	0.871685i	$-0.336973\pi$
$151$	148.000	0.980132	0.490066	+	0.871685i	$0.336973\pi$
$152$	− 45.2548i	− 0.297729i
$153$	0	0
$154$	−96.0000	−0.623377
$155$	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	$0.584101\pi$
$158$	− 107.480i	− 0.680255i
$159$	0	0
$160$	24.0000	0.150000
$161$	67.8823i	0.421629i
$162$	0	0
$163$	−56.0000	−0.343558	−0.171779	−	0.985135i	$-0.554952\pi$
$163$	−56.0000	−0.343558	−0.171779	+	0.985135i	$0.554952\pi$
$164$	− 93.3381i	− 0.569135i
$165$	0	0
$166$	−168.000	−1.01205
$167$	33.9411i	0.203240i	0.994823	+	0.101620i	$0.0324026\pi$
$167$	33.9411i	0.203240i	−0.994823	+	0.101620i	$0.967597\pi$
$168$	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	$0.167675\pi$
$173$	173.948i	1.00548i	−0.864437	+	0.502741i	$0.832325\pi$
$174$	0	0
$175$	28.0000	0.160000
$176$	67.8823i	0.385695i
$177$	0	0
$178$	−18.0000	−0.101124
$179$	203.647i	1.13769i	0.822444	+	0.568846i	$0.192610\pi$
$179$	203.647i	1.13769i	−0.822444	+	0.568846i	$0.807390\pi$
$180$	0	0
$181$	−232.000	−1.28177	−0.640884	−	0.767638i	$-0.721432\pi$
$181$	−232.000	−1.28177	−0.640884	+	0.767638i	$0.721432\pi$
$182$	0	0
$183$	0	0
$184$	48.0000	0.260870
$185$	− 144.250i	− 0.779729i
$186$	0	0
$187$	−216.000	−1.15508
$188$	169.706i	0.902690i
$189$	0	0
$190$	96.0000	0.505263
$191$	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
$193$	−206.000	−1.06736	−0.533679	−	0.845687i	$-0.679191\pi$
$193$	−206.000	−1.06736	−0.533679	+	0.845687i	$0.679191\pi$
$194$	− 248.902i	− 1.28300i
$195$	0	0
$196$	66.0000	0.336735
$197$	165.463i	0.839914i	0.907544	+	0.419957i	$0.137955\pi$
$197$	165.463i	0.839914i	−0.907544	+	0.419957i	$0.862045\pi$
$198$	0	0
$199$	20.0000	0.100503	0.0502513	−	0.998737i	$-0.483998\pi$
$199$	20.0000	0.100503	0.0502513	+	0.998737i	$0.483998\pi$
$200$	− 19.7990i	− 0.0989949i
$201$	0	0
$202$	42.0000	0.207921
$203$	− 16.9706i	− 0.0835988i
$204$	0	0
$205$	198.000	0.965854
$206$	− 39.5980i	− 0.192223i
$207$	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
$215$	169.706i	0.789328i
$216$	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
$223$	436.000	1.95516	0.977578	−	0.210571i	$-0.0675325\pi$
$223$	436.000	1.95516	0.977578	+	0.210571i	$0.0675325\pi$
$224$	22.6274i	0.101015i
$225$	0	0
$226$	−222.000	−0.982301
$227$	16.9706i	0.0747602i	0.999301	+	0.0373801i	$0.0119012\pi$
$227$	16.9706i	0.0747602i	−0.999301	+	0.0373801i	$0.988099\pi$
$228$	0	0
$229$	−8.00000	−0.0349345	−0.0174672	−	0.999847i	$-0.505560\pi$
$229$	−8.00000	−0.0349345	−0.0174672	+	0.999847i	$0.505560\pi$
$230$	101.823i	0.442710i
$231$	0	0
$232$	−12.0000	−0.0517241
$233$	12.7279i	0.0546263i	0.999627	+	0.0273131i	$0.00869512\pi$
$233$	12.7279i	0.0546263i	−0.999627	+	0.0273131i	$0.991305\pi$
$234$	0	0
$235$	−360.000	−1.53191
$236$	− 67.8823i	− 0.287637i
$237$	0	0
$238$	−72.0000	−0.302521
$239$	135.765i	0.568052i	0.958817	+	0.284026i	$0.0916703\pi$
$239$	135.765i	0.568052i	−0.958817	+	0.284026i	$0.908330\pi$
$240$	0	0
$241$	−32.0000	−0.132780	−0.0663900	−	0.997794i	$-0.521148\pi$
$241$	−32.0000	−0.132780	−0.0663900	+	0.997794i	$0.521148\pi$
$242$	− 236.174i	− 0.975924i
$243$	0	0
$244$	−100.000	−0.409836
$245$	140.007i	0.571458i
$246$	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	$-0.967662\pi$
$251$	− 50.9117i	− 0.202835i	0.994844	−	0.101418i	$-0.0323379\pi$
$252$	0	0
$253$	−288.000	−1.13834
$254$	130.108i	0.512235i
$255$	0	0
$256$	16.0000	0.0625000
$257$	182.434i	0.709858i	0.934893	+	0.354929i	$0.115495\pi$
$257$	182.434i	0.709858i	−0.934893	+	0.354929i	$0.884505\pi$
$258$	0	0
$259$	136.000	0.525097
$260$	0	0
$261$	0	0
$262$	−240.000	−0.916031
$263$	− 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
$265$	−162.000	−0.611321
$266$	90.5097i	0.340262i
$267$	0	0
$268$	16.0000	0.0597015
$269$	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
$271$	−380.000	−1.40221	−0.701107	−	0.713056i	$-0.747310\pi$
$271$	−380.000	−1.40221	−0.701107	+	0.713056i	$0.747310\pi$
$272$	50.9117i	0.187175i
$273$	0	0
$274$	−222.000	−0.810219
$275$	118.794i	0.431978i
$276$	0	0
$277$	−328.000	−1.18412	−0.592058	−	0.805896i	$-0.701684\pi$
$277$	−328.000	−1.18412	−0.592058	+	0.805896i	$0.701684\pi$
$278$	214.960i	0.773239i
$279$	0	0
$280$	−48.0000	−0.171429
$281$	284.257i	1.01159i	0.862654	+	0.505795i	$0.168801\pi$
$281$	284.257i	1.01159i	−0.862654	+	0.505795i	$0.831199\pi$
$282$	0	0
$283$	−208.000	−0.734982	−0.367491	−	0.930027i	$-0.619783\pi$
$283$	−208.000	−0.734982	−0.367491	+	0.930027i	$0.619783\pi$
$284$	101.823i	0.358533i
$285$	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
$292$	−32.0000	−0.109589
$293$	− 436.992i	− 1.49144i	−0.666259	−	0.745720i	$-0.732106\pi$
$293$	− 436.992i	− 1.49144i	0.666259	−	0.745720i	$-0.267894\pi$
$294$	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
$301$	−160.000	−0.531561
$302$	209.304i	0.693058i
$303$	0	0
$304$	64.0000	0.210526
$305$	− 212.132i	− 0.695515i
$306$	0	0
$307$	520.000	1.69381	0.846906	−	0.531743i	$-0.178463\pi$
$307$	520.000	1.69381	0.846906	+	0.531743i	$0.178463\pi$
$308$	− 135.765i	− 0.440794i
$309$	0	0
$310$	−264.000	−0.851613
$311$	373.352i	1.20049i	0.799816	+	0.600245i	$0.204930\pi$
$311$	373.352i	1.20049i	−0.799816	+	0.600245i	$0.795070\pi$
$312$	0	0
$313$	−94.0000	−0.300319	−0.150160	−	0.988662i	$-0.547979\pi$
$313$	−94.0000	−0.300319	−0.150160	+	0.988662i	$0.547979\pi$
$314$	− 115.966i	− 0.369317i
$315$	0	0
$316$	152.000	0.481013
$317$	335.169i	1.05731i	0.848835	+	0.528657i	$0.177304\pi$
$317$	335.169i	1.05731i	−0.848835	+	0.528657i	$0.822696\pi$
$318$	0	0
$319$	72.0000	0.225705
$320$	33.9411i	0.106066i
$321$	0	0
$322$	−96.0000	−0.298137
$323$	203.647i	0.630485i
$324$	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
$331$	−536.000	−1.61934	−0.809668	−	0.586889i	$-0.800353\pi$
$331$	−536.000	−1.61934	−0.809668	+	0.586889i	$0.800353\pi$
$332$	− 237.588i	− 0.715626i
$333$	0	0
$334$	−48.0000	−0.143713
$335$	33.9411i	0.101317i
$336$	0	0
$337$	−208.000	−0.617211	−0.308605	−	0.951190i	$-0.599862\pi$
$337$	−208.000	−0.617211	−0.308605	+	0.951190i	$0.599862\pi$
$338$	0	0
$339$	0	0
$340$	−108.000	−0.317647
$341$	− 746.705i	− 2.18975i
$342$	0	0
$343$	−328.000	−0.956268
$344$	113.137i	0.328887i
$345$	0	0
$346$	−246.000	−0.710983
$347$	− 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
$349$	238.000	0.681948	0.340974	−	0.940073i	$-0.389243\pi$
$349$	238.000	0.681948	0.340974	+	0.940073i	$0.389243\pi$
$350$	39.5980i	0.113137i
$351$	0	0
$352$	−96.0000	−0.272727
$353$	− 224.860i	− 0.636997i	−0.947923	−	0.318499i	$-0.896821\pi$
$353$	− 224.860i	− 0.636997i	0.947923	−	0.318499i	$-0.103179\pi$
$354$	0	0
$355$	−216.000	−0.608451
$356$	− 25.4558i	− 0.0715052i
$357$	0	0
$358$	−288.000	−0.804469
$359$	− 560.029i	− 1.55997i	−0.625799	−	0.779984i	$-0.715227\pi$
$359$	− 560.029i	− 1.55997i	0.625799	−	0.779984i	$-0.284773\pi$
$360$	0	0
$361$	−105.000	−0.290859
$362$	− 328.098i	− 0.906347i
$363$	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
$370$	204.000	0.551351
$371$	− 152.735i	− 0.411685i
$372$	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
$379$	160.000	0.422164	0.211082	−	0.977468i	$-0.432301\pi$
$379$	160.000	0.422164	0.211082	+	0.977468i	$0.432301\pi$
$380$	135.765i	0.357275i
$381$	0	0
$382$	−48.0000	−0.125654
$383$	− 271.529i	− 0.708953i	−0.935065	−	0.354477i	$-0.884659\pi$
$383$	− 271.529i	− 0.708953i	0.935065	−	0.354477i	$-0.115341\pi$
$384$	0	0
$385$	288.000	0.748052
$386$	− 291.328i	− 0.754736i
$387$	0	0
$388$	352.000	0.907216
$389$	403.051i	1.03612i	0.855344	+	0.518060i	$0.173346\pi$
$389$	403.051i	1.03612i	−0.855344	+	0.518060i	$0.826654\pi$
$390$	0	0
$391$	−216.000	−0.552430
$392$	93.3381i	0.238107i
$393$	0	0
$394$	−234.000	−0.593909
$395$	322.441i	0.816306i
$396$	0	0
$397$	−146.000	−0.367758	−0.183879	−	0.982949i	$-0.558865\pi$
$397$	−146.000	−0.367758	−0.183879	+	0.982949i	$0.558865\pi$
$398$	28.2843i	0.0710660i
$399$	0	0
$400$	28.0000	0.0700000
$401$	326.683i	0.814672i	0.913278	+	0.407336i	$0.133542\pi$
$401$	326.683i	0.814672i	−0.913278	+	0.407336i	$0.866458\pi$
$402$	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
$409$	−368.000	−0.899756	−0.449878	−	0.893090i	$-0.648532\pi$
$409$	−368.000	−0.899756	−0.449878	+	0.893090i	$0.648532\pi$
$410$	280.014i	0.682962i
$411$	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
$418$	−384.000	−0.918660
$419$	− 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
$421$	40.0000	0.0950119	0.0475059	−	0.998871i	$-0.484873\pi$
$421$	40.0000	0.0950119	0.0475059	+	0.998871i	$0.484873\pi$
$422$	418.607i	0.991960i
$423$	0	0
$424$	−108.000	−0.254717
$425$	89.0955i	0.209636i
$426$	0	0
$427$	200.000	0.468384
$428$	0	0
$429$	0	0
$430$	−240.000	−0.558140
$431$	− 152.735i	− 0.354374i	−0.984177	−	0.177187i	$-0.943300\pi$
$431$	− 152.735i	− 0.354374i	0.984177	−	0.177187i	$-0.0566997\pi$
$432$	0	0
$433$	542.000	1.25173	0.625866	−	0.779931i	$-0.284746\pi$
$433$	542.000	1.25173	0.625866	+	0.779931i	$0.284746\pi$
$434$	− 248.902i	− 0.573506i
$435$	0	0
$436$	112.000	0.256881
$437$	271.529i	0.621348i
$438$	0	0
$439$	−4.00000	−0.00911162	−0.00455581	−	0.999990i	$-0.501450\pi$
$439$	−4.00000	−0.00911162	−0.00455581	+	0.999990i	$0.501450\pi$
$440$	− 203.647i	− 0.462834i
$441$	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
$448$	−32.0000	−0.0714286
$449$	216.375i	0.481904i	0.970537	+	0.240952i	$0.0774596\pi$
$449$	216.375i	0.481904i	−0.970537	+	0.240952i	$0.922540\pi$
$450$	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
$457$	400.000	0.875274	0.437637	−	0.899152i	$-0.355816\pi$
$457$	400.000	0.875274	0.437637	+	0.899152i	$0.355816\pi$
$458$	− 11.3137i	− 0.0247024i
$459$	0	0
$460$	−144.000	−0.313043
$461$	− 301.227i	− 0.653422i	−0.945124	−	0.326711i	$-0.894060\pi$
$461$	− 301.227i	− 0.653422i	0.945124	−	0.326711i	$-0.105940\pi$
$462$	0	0
$463$	604.000	1.30454	0.652268	−	0.757989i	$-0.273818\pi$
$463$	604.000	1.30454	0.652268	+	0.757989i	$0.273818\pi$
$464$	− 16.9706i	− 0.0365745i
$465$	0	0
$466$	−18.0000	−0.0386266
$467$	− 356.382i	− 0.763130i	−0.924342	−	0.381565i	$-0.875385\pi$
$467$	− 356.382i	− 0.763130i	0.924342	−	0.381565i	$-0.124615\pi$
$468$	0	0
$469$	−32.0000	−0.0682303
$470$	− 509.117i	− 1.08323i
$471$	0	0
$472$	96.0000	0.203390
$473$	− 678.823i	− 1.43514i
$474$	0	0
$475$	112.000	0.235789
$476$	− 101.823i	− 0.213915i
$477$	0	0
$478$	−192.000	−0.401674
$479$	526.087i	1.09830i	0.835723	+	0.549152i	$0.185049\pi$
$479$	526.087i	1.09830i	−0.835723	+	0.549152i	$0.814951\pi$
$480$	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
$487$	−596.000	−1.22382	−0.611910	−	0.790928i	$-0.709598\pi$
$487$	−596.000	−1.22382	−0.611910	+	0.790928i	$0.709598\pi$
$488$	− 141.421i	− 0.289798i
$489$	0	0
$490$	−198.000	−0.404082
$491$	271.529i	0.553012i	0.961012	+	0.276506i	$0.0891766\pi$
$491$	271.529i	0.553012i	−0.961012	+	0.276506i	$0.910823\pi$
$492$	0	0
$493$	54.0000	0.109533
$494$	0	0
$495$	0	0
$496$	−176.000	−0.354839
$497$	− 203.647i	− 0.409752i
$498$	0	0
$499$	−224.000	−0.448898	−0.224449	−	0.974486i	$-0.572058\pi$
$499$	−224.000	−0.448898	−0.224449	+	0.974486i	$0.572058\pi$
$500$	271.529i	0.543058i
$501$	0	0
$502$	72.0000	0.143426
$503$	− 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
$505$	−126.000	−0.249505
$506$	− 407.294i	− 0.804928i
$507$	0	0
$508$	−184.000	−0.362205
$509$	479.418i	0.941883i	0.882164	+	0.470941i	$0.156086\pi$
$509$	479.418i	0.941883i	−0.882164	+	0.470941i	$0.843914\pi$
$510$	0	0
$511$	64.0000	0.125245
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−258.000	−0.501946
$515$	118.794i	0.230668i
$516$	0	0
$517$	1440.00	2.78530
$518$	192.333i	0.371299i
$519$	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
$526$	528.000	1.00380
$527$	− 560.029i	− 1.06267i
$528$	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
$538$	−486.000	−0.903346
$539$	− 560.029i	− 1.03901i
$540$	0	0
$541$	808.000	1.49353	0.746765	−	0.665088i	$-0.231606\pi$
$541$	808.000	1.49353	0.746765	+	0.665088i	$0.231606\pi$
$542$	− 537.401i	− 0.991515i
$543$	0	0
$544$	−72.0000	−0.132353
$545$	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	$0.337017\pi$
$548$	− 313.955i	− 0.572911i
$549$	0	0
$550$	−168.000	−0.305455
$551$	− 67.8823i	− 0.123198i
$552$	0	0
$553$	−304.000	−0.549729
$554$	− 463.862i	− 0.837296i
$555$	0	0
$556$	−304.000	−0.546763
$557$	− 165.463i	− 0.297061i	−0.988908	−	0.148531i	$-0.952546\pi$
$557$	− 165.463i	− 0.297061i	0.988908	−	0.148531i	$-0.0474543\pi$
$558$	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	$0.0924451\pi$
$563$	322.441i	0.572719i	−0.958122	+	0.286359i	$0.907555\pi$
$564$	0	0
$565$	666.000	1.17876
$566$	− 294.156i	− 0.519711i
$567$	0	0
$568$	−144.000	−0.253521
$569$	− 156.978i	− 0.275883i	−0.990440	−	0.137942i	$-0.955951\pi$
$569$	− 156.978i	− 0.275883i	0.990440	−	0.137942i	$-0.0440487\pi$
$570$	0	0
$571$	368.000	0.644483	0.322242	−	0.946657i	$-0.395564\pi$
$571$	368.000	0.644483	0.322242	+	0.946657i	$0.395564\pi$
$572$	0	0
$573$	0	0
$574$	−264.000	−0.459930
$575$	118.794i	0.206598i
$576$	0	0
$577$	142.000	0.246101	0.123050	−	0.992400i	$-0.460732\pi$
$577$	142.000	0.246101	0.123050	+	0.992400i	$0.460732\pi$
$578$	179.605i	0.310736i
$579$	0	0
$580$	36.0000	0.0620690
$581$	475.176i	0.817858i
$582$	0	0
$583$	648.000	1.11149
$584$	− 45.2548i	− 0.0774912i
$585$	0	0
$586$	618.000	1.05461
$587$	373.352i	0.636035i	0.948085	+	0.318017i	$0.103017\pi$
$587$	373.352i	0.636035i	−0.948085	+	0.318017i	$0.896983\pi$
$588$	0	0
$589$	−704.000	−1.19525
$590$	203.647i	0.345164i
$591$	0	0
$592$	136.000	0.229730
$593$	1107.33i	1.86733i	0.358142	+	0.933667i	$0.383410\pi$
$593$	1107.33i	1.86733i	−0.358142	+	0.933667i	$0.616590\pi$
$594$	0	0
$595$	216.000	0.363025
$596$	− 551.543i	− 0.925408i
$597$	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
$604$	−296.000	−0.490066
$605$	708.521i	1.17111i
$606$	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
$613$	−578.000	−0.942904	−0.471452	−	0.881892i	$-0.656270\pi$
$613$	−578.000	−0.942904	−0.471452	+	0.881892i	$0.656270\pi$
$614$	735.391i	1.19771i
$615$	0	0
$616$	192.000	0.311688
$617$	− 55.1543i	− 0.0893911i	−0.999001	−	0.0446956i	$-0.985768\pi$
$617$	− 55.1543i	− 0.0893911i	0.999001	−	0.0446956i	$-0.0142318\pi$
$618$	0	0
$619$	−896.000	−1.44750	−0.723748	−	0.690064i	$-0.757582\pi$
$619$	−896.000	−1.44750	−0.723748	+	0.690064i	$0.757582\pi$
$620$	− 373.352i	− 0.602181i
$621$	0	0
$622$	−528.000	−0.848875
$623$	50.9117i	0.0817202i
$624$	0	0
$625$	−401.000	−0.641600
$626$	− 132.936i	− 0.212358i
$627$	0	0
$628$	164.000	0.261146
$629$	432.749i	0.687996i
$630$	0	0
$631$	−20.0000	−0.0316957	−0.0158479	−	0.999874i	$-0.505045\pi$
$631$	−20.0000	−0.0316957	−0.0158479	+	0.999874i	$0.505045\pi$
$632$	214.960i	0.340127i
$633$	0	0
$634$	−474.000	−0.747634
$635$	− 390.323i	− 0.614682i
$636$	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	$-0.935297\pi$
$641$	− 258.801i	− 0.403746i	0.979412	−	0.201873i	$-0.0647028\pi$
$642$	0	0
$643$	−728.000	−1.13219	−0.566096	−	0.824339i	$-0.691547\pi$
$643$	−728.000	−1.13219	−0.566096	+	0.824339i	$0.691547\pi$
$644$	− 135.765i	− 0.210814i
$645$	0	0
$646$	−288.000	−0.445820
$647$	458.205i	0.708200i	0.935208	+	0.354100i	$0.115213\pi$
$647$	458.205i	0.708200i	−0.935208	+	0.354100i	$0.884787\pi$
$648$	0	0
$649$	−576.000	−0.887519
$650$	0	0
$651$	0	0
$652$	112.000	0.171779
$653$	− 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
$655$	720.000	1.09924
$656$	186.676i	0.284567i
$657$	0	0
$658$	480.000	0.729483
$659$	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
$661$	−62.0000	−0.0937973	−0.0468986	−	0.998900i	$-0.514934\pi$
$661$	−62.0000	−0.0937973	−0.0468986	+	0.998900i	$0.514934\pi$
$662$	− 758.018i	− 1.14504i
$663$	0	0
$664$	336.000	0.506024
$665$	− 271.529i	− 0.408314i
$666$	0	0
$667$	72.0000	0.107946
$668$	− 67.8823i	− 0.101620i
$669$	0	0
$670$	−48.0000	−0.0716418
$671$	848.528i	1.26457i
$672$	0	0
$673$	−670.000	−0.995542	−0.497771	−	0.867308i	$-0.665848\pi$
$673$	−670.000	−0.995542	−0.497771	+	0.867308i	$0.665848\pi$
$674$	− 294.156i	− 0.436434i
$675$	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
$682$	1056.00	1.54839
$683$	− 560.029i	− 0.819954i	−0.912096	−	0.409977i	$-0.865537\pi$
$683$	− 560.029i	− 0.819954i	0.912096	−	0.409977i	$-0.134463\pi$
$684$	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
$691$	40.0000	0.0578871	0.0289436	−	0.999581i	$-0.490786\pi$
$691$	40.0000	0.0578871	0.0289436	+	0.999581i	$0.490786\pi$
$692$	− 347.897i	− 0.502741i
$693$	0	0
$694$	408.000	0.587896
$695$	− 644.881i	− 0.927887i
$696$	0	0
$697$	−594.000	−0.852224
$698$	336.583i	0.482210i
$699$	0	0
$700$	−56.0000	−0.0800000
$701$	− 954.594i	− 1.36176i	−0.732395	−	0.680880i	$-0.761597\pi$
$701$	− 954.594i	− 1.36176i	0.732395	−	0.680880i	$-0.238403\pi$
$702$	0	0
$703$	544.000	0.773826
$704$	− 135.765i	− 0.192847i
$705$	0	0
$706$	318.000	0.450425
$707$	− 118.794i	− 0.168025i
$708$	0	0
$709$	−968.000	−1.36530	−0.682652	−	0.730744i	$-0.739173\pi$
$709$	−968.000	−1.36530	−0.682652	+	0.730744i	$0.739173\pi$
$710$	− 305.470i	− 0.430240i
$711$	0	0
$712$	36.0000	0.0505618
$713$	− 746.705i	− 1.04727i
$714$	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	$-0.697119\pi$
$719$	− 1170.97i	− 1.62861i	0.580439	−	0.814304i	$-0.302881\pi$
$720$	0	0
$721$	−112.000	−0.155340
$722$	− 148.492i	− 0.205668i
$723$	0	0
$724$	464.000	0.640884
$725$	− 29.6985i	− 0.0409634i
$726$	0	0
$727$	−508.000	−0.698762	−0.349381	−	0.936981i	$-0.613608\pi$
$727$	−508.000	−0.698762	−0.349381	+	0.936981i	$0.613608\pi$
$728$	0	0
$729$	0	0
$730$	96.0000	0.131507
$731$	− 509.117i	− 0.696466i
$732$	0	0
$733$	1144.00	1.56071	0.780355	−	0.625337i	$-0.215039\pi$
$733$	1144.00	1.56071	0.780355	+	0.625337i	$0.215039\pi$
$734$	401.637i	0.547189i
$735$	0	0
$736$	−96.0000	−0.130435
$737$	− 135.765i	− 0.184212i
$738$	0	0
$739$	304.000	0.411367	0.205683	−	0.978619i	$-0.434058\pi$
$739$	304.000	0.411367	0.205683	+	0.978619i	$0.434058\pi$
$740$	288.500i	0.389864i
$741$	0	0
$742$	216.000	0.291105
$743$	848.528i	1.14203i	0.820940	+	0.571015i	$0.193450\pi$
$743$	848.528i	1.14203i	−0.820940	+	0.571015i	$0.806550\pi$
$744$	0	0
$745$	1170.00	1.57047
$746$	− 268.701i	− 0.360188i
$747$	0	0
$748$	432.000	0.577540
$749$	0	0
$750$	0	0
$751$	188.000	0.250333	0.125166	−	0.992136i	$-0.460054\pi$
$751$	188.000	0.250333	0.125166	+	0.992136i	$0.460054\pi$
$752$	− 339.411i	− 0.451345i
$753$	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
$760$	−192.000	−0.252632
$761$	− 156.978i	− 0.206278i	−0.994667	−	0.103139i	$-0.967111\pi$
$761$	− 156.978i	− 0.206278i	0.994667	−	0.103139i	$-0.0328887\pi$
$762$	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
$769$	910.000	1.18336	0.591678	−	0.806175i	$-0.298466\pi$
$769$	910.000	1.18336	0.591678	+	0.806175i	$0.298466\pi$
$770$	407.294i	0.528953i
$771$	0	0
$772$	412.000	0.533679
$773$	− 1387.34i	− 1.79475i	−0.441266	−	0.897376i	$-0.645471\pi$
$773$	− 1387.34i	− 1.79475i	0.441266	−	0.897376i	$-0.354529\pi$
$774$	0	0
$775$	−308.000	−0.397419
$776$	497.803i	0.641499i
$777$	0	0
$778$	−570.000	−0.732648
$779$	746.705i	0.958543i
$780$	0	0
$781$	864.000	1.10627
$782$	− 305.470i	− 0.390627i
$783$	0	0
$784$	−132.000	−0.168367
$785$	347.897i	0.443180i
$786$	0	0
$787$	1360.00	1.72808	0.864041	−	0.503422i	$-0.167926\pi$
$787$	1360.00	1.72808	0.864041	+	0.503422i	$0.167926\pi$
$788$	− 330.926i	− 0.419957i
$789$	0	0
$790$	−456.000	−0.577215
$791$	627.911i	0.793819i
$792$	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	$0.0211963\pi$
$797$	106.066i	0.133082i	−0.997784	+	0.0665408i	$0.978804\pi$
$798$	0	0
$799$	1080.00	1.35169
$800$	39.5980i	0.0494975i
$801$	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
$808$	−84.0000	−0.103960
$809$	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
$811$	160.000	0.197287	0.0986436	−	0.995123i	$-0.468550\pi$
$811$	160.000	0.197287	0.0986436	+	0.995123i	$0.468550\pi$
$812$	33.9411i	0.0417994i
$813$	0	0
$814$	−816.000	−1.00246
$815$	237.588i	0.291519i
$816$	0	0
$817$	−640.000	−0.783354
$818$	− 520.431i	− 0.636223i
$819$	0	0
$820$	−396.000	−0.482927
$821$	436.992i	0.532268i	0.963936	+	0.266134i	$0.0857464\pi$
$821$	436.992i	0.532268i	−0.963936	+	0.266134i	$0.914254\pi$
$822$	0	0
$823$	332.000	0.403402	0.201701	−	0.979447i	$-0.435353\pi$
$823$	332.000	0.403402	0.201701	+	0.979447i	$0.435353\pi$
$824$	79.1960i	0.0961116i
$825$	0	0
$826$	−192.000	−0.232446
$827$	101.823i	0.123124i	0.998103	+	0.0615619i	$0.0196082\pi$
$827$	101.823i	0.123124i	−0.998103	+	0.0615619i	$0.980392\pi$
$828$	0	0
$829$	632.000	0.762364	0.381182	−	0.924500i	$-0.375517\pi$
$829$	632.000	0.762364	0.381182	+	0.924500i	$0.375517\pi$
$830$	712.764i	0.858751i
$831$	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
$838$	552.000	0.658711
$839$	729.734i	0.869767i	0.900487	+	0.434883i	$0.143210\pi$
$839$	729.734i	0.869767i	−0.900487	+	0.434883i	$0.856790\pi$
$840$	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
$847$	−668.000	−0.788666
$848$	− 152.735i	− 0.180112i
$849$	0	0
$850$	−126.000	−0.148235
$851$	576.999i	0.678025i
$852$	0	0
$853$	−446.000	−0.522860	−0.261430	−	0.965222i	$-0.584194\pi$
$853$	−446.000	−0.522860	−0.261430	+	0.965222i	$0.584194\pi$
$854$	282.843i	0.331198i
$855$	0	0
$856$	0	0
$857$	428.507i	0.500008i	0.968245	+	0.250004i	$0.0804319\pi$
$857$	428.507i	0.500008i	−0.968245	+	0.250004i	$0.919568\pi$
$858$	0	0
$859$	728.000	0.847497	0.423749	−	0.905780i	$-0.360714\pi$
$859$	728.000	0.847497	0.423749	+	0.905780i	$0.360714\pi$
$860$	− 339.411i	− 0.394664i
$861$	0	0
$862$	216.000	0.250580
$863$	− 916.410i	− 1.06189i	−0.847407	−	0.530945i	$-0.821837\pi$
$863$	− 916.410i	− 1.06189i	0.847407	−	0.530945i	$-0.178163\pi$
$864$	0	0
$865$	738.000	0.853179
$866$	766.504i	0.885108i
$867$	0	0
$868$	352.000	0.405530
$869$	− 1289.76i	− 1.48419i
$870$	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
$877$	910.000	1.03763	0.518814	−	0.854887i	$-0.326374\pi$
$877$	910.000	1.03763	0.518814	+	0.854887i	$0.326374\pi$
$878$	− 5.65685i	− 0.00644289i
$879$	0	0
$880$	288.000	0.327273
$881$	929.138i	1.05464i	0.849667	+	0.527320i	$0.176803\pi$
$881$	929.138i	1.05464i	−0.849667	+	0.527320i	$0.823197\pi$
$882$	0	0
$883$	1064.00	1.20498	0.602492	−	0.798125i	$-0.294175\pi$
$883$	1064.00	1.20498	0.602492	+	0.798125i	$0.294175\pi$
$884$	0	0
$885$	0	0
$886$	456.000	0.514673
$887$	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
$889$	368.000	0.413948
$890$	76.3675i	0.0858062i
$891$	0	0
$892$	−872.000	−0.977578
$893$	− 1357.65i	− 1.52032i
$894$	0	0
$895$	864.000	0.965363
$896$	− 45.2548i	− 0.0505076i
$897$	0	0
$898$	−306.000	−0.340757
$899$	186.676i	0.207649i
$900$	0	0
$901$	486.000	0.539401
$902$	− 1120.06i	− 1.24175i
$903$	0	0
$904$	444.000	0.491150
$905$	984.293i	1.08762i
$906$	0	0
$907$	−1768.00	−1.94928	−0.974642	−	0.223771i	$-0.928163\pi$
$907$	−1768.00	−1.94928	−0.974642	+	0.223771i	$0.928163\pi$
$908$	− 33.9411i	− 0.0373801i
$909$	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
$916$	16.0000	0.0174672
$917$	678.823i	0.740264i
$918$	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
$925$	238.000	0.257297
$926$	854.185i	0.922446i
$927$	0	0
$928$	24.0000	0.0258621
$929$	666.095i	0.717002i	0.933529	+	0.358501i	$0.116712\pi$
$929$	666.095i	0.717002i	−0.933529	+	0.358501i	$0.883288\pi$
$930$	0	0
$931$	−528.000	−0.567132
$932$	− 25.4558i	− 0.0273131i
$933$	0	0
$934$	504.000	0.539615
$935$	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	$0.530280\pi$
$938$	− 45.2548i	− 0.0482461i
$939$	0	0
$940$	720.000	0.765957
$941$	− 436.992i	− 0.464391i	−0.972669	−	0.232196i	$-0.925409\pi$
$941$	− 436.992i	− 0.464391i	0.972669	−	0.232196i	$-0.0745909\pi$
$942$	0	0
$943$	−792.000	−0.839873
$944$	135.765i	0.143818i
$945$	0	0
$946$	960.000	1.01480
$947$	1798.88i	1.89956i	0.312924	+	0.949778i	$0.398691\pi$
$947$	1798.88i	1.89956i	−0.312924	+	0.949778i	$0.601309\pi$
$948$	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	$-0.758570\pi$
$953$	− 1310.98i	− 1.37563i	0.725886	−	0.687815i	$-0.241430\pi$
$954$	0	0
$955$	144.000	0.150785
$956$	− 271.529i	− 0.284026i
$957$	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
$964$	64.0000	0.0663900
$965$	873.984i	0.905683i
$966$	0	0
$967$	−1700.00	−1.75801	−0.879007	−	0.476808i	$-0.841794\pi$
$967$	−1700.00	−1.75801	−0.879007	+	0.476808i	$0.841794\pi$
$968$	472.347i	0.487962i
$969$	0	0
$970$	−1056.00	−1.08866
$971$	− 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
$973$	608.000	0.624872
$974$	− 842.871i	− 0.865371i
$975$	0	0
$976$	200.000	0.204918
$977$	759.433i	0.777311i	0.921383	+	0.388655i	$0.127060\pi$
$977$	759.433i	0.777311i	−0.921383	+	0.388655i	$0.872940\pi$
$978$	0	0
$979$	−216.000	−0.220633
$980$	− 280.014i	− 0.285729i
$981$	0	0
$982$	−384.000	−0.391039
$983$	− 1052.17i	− 1.07037i	−0.844734	−	0.535186i	$-0.820242\pi$
$983$	− 1052.17i	− 1.07037i	0.844734	−	0.535186i	$-0.179758\pi$
$984$	0	0
$985$	702.000	0.712690
$986$	76.3675i	0.0774519i
$987$	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
$994$	288.000	0.289738
$995$	− 84.8528i	− 0.0852792i
$996$	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.c.e.1691.2		2
3.2	odd	2	inner	3042.3.c.e.1691.1		2
13.5	odd	4		3042.3.d.a.3041.4		4
13.8	odd	4		3042.3.d.a.3041.1		4
13.12	even	2		18.3.b.a.17.1	✓	2
39.5	even	4		3042.3.d.a.3041.2		4
39.8	even	4		3042.3.d.a.3041.3		4
39.38	odd	2		18.3.b.a.17.2	yes	2
52.51	odd	2		144.3.e.b.17.2		2
65.12	odd	4		450.3.b.b.449.3		4
65.38	odd	4		450.3.b.b.449.2		4
65.64	even	2		450.3.d.f.251.2		2
91.12	odd	6		882.3.s.d.557.1		4
91.25	even	6		882.3.s.b.863.2		4
91.38	odd	6		882.3.s.d.863.2		4
91.51	even	6		882.3.s.b.557.1		4
91.90	odd	2		882.3.b.a.197.1		2
104.51	odd	2		576.3.e.f.449.1		2
104.77	even	2		576.3.e.c.449.1		2
117.25	even	6		162.3.d.b.53.1		4
117.38	odd	6		162.3.d.b.53.2		4
117.77	odd	6		162.3.d.b.107.1		4
117.103	even	6		162.3.d.b.107.2		4
143.142	odd	2		2178.3.c.d.485.2		2
156.155	even	2		144.3.e.b.17.1		2
195.38	even	4		450.3.b.b.449.4		4
195.77	even	4		450.3.b.b.449.1		4
195.194	odd	2		450.3.d.f.251.1		2
208.51	odd	4		2304.3.h.c.2177.4		4
208.77	even	4		2304.3.h.f.2177.4		4
208.155	odd	4		2304.3.h.c.2177.1		4
208.181	even	4		2304.3.h.f.2177.1		4
260.103	even	4		3600.3.c.b.449.2		4
260.207	even	4		3600.3.c.b.449.4		4
260.259	odd	2		3600.3.l.d.1601.2		2
273.38	even	6		882.3.s.d.863.1		4
273.116	odd	6		882.3.s.b.863.1		4
273.194	even	6		882.3.s.d.557.2		4
273.233	odd	6		882.3.s.b.557.2		4
273.272	even	2		882.3.b.a.197.2		2
312.77	odd	2		576.3.e.c.449.2		2
312.155	even	2		576.3.e.f.449.2		2
429.428	even	2		2178.3.c.d.485.1		2
468.103	odd	6		1296.3.q.f.593.2		4
468.155	even	6		1296.3.q.f.1025.2		4
468.259	odd	6		1296.3.q.f.1025.1		4
468.311	even	6		1296.3.q.f.593.1		4
624.77	odd	4		2304.3.h.f.2177.2		4
624.155	even	4		2304.3.h.c.2177.3		4
624.389	odd	4		2304.3.h.f.2177.3		4
624.467	even	4		2304.3.h.c.2177.2		4
780.467	odd	4		3600.3.c.b.449.3		4
780.623	odd	4		3600.3.c.b.449.1		4
780.779	even	2		3600.3.l.d.1601.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.b.a.17.1	✓	2	13.12	even	2
18.3.b.a.17.2	yes	2	39.38	odd	2
144.3.e.b.17.1		2	156.155	even	2
144.3.e.b.17.2		2	52.51	odd	2
162.3.d.b.53.1		4	117.25	even	6
162.3.d.b.53.2		4	117.38	odd	6
162.3.d.b.107.1		4	117.77	odd	6
162.3.d.b.107.2		4	117.103	even	6
450.3.b.b.449.1		4	195.77	even	4
450.3.b.b.449.2		4	65.38	odd	4
450.3.b.b.449.3		4	65.12	odd	4
450.3.b.b.449.4		4	195.38	even	4
450.3.d.f.251.1		2	195.194	odd	2
450.3.d.f.251.2		2	65.64	even	2
576.3.e.c.449.1		2	104.77	even	2
576.3.e.c.449.2		2	312.77	odd	2
576.3.e.f.449.1		2	104.51	odd	2
576.3.e.f.449.2		2	312.155	even	2
882.3.b.a.197.1		2	91.90	odd	2
882.3.b.a.197.2		2	273.272	even	2
882.3.s.b.557.1		4	91.51	even	6
882.3.s.b.557.2		4	273.233	odd	6
882.3.s.b.863.1		4	273.116	odd	6
882.3.s.b.863.2		4	91.25	even	6
882.3.s.d.557.1		4	91.12	odd	6
882.3.s.d.557.2		4	273.194	even	6
882.3.s.d.863.1		4	273.38	even	6
882.3.s.d.863.2		4	91.38	odd	6
1296.3.q.f.593.1		4	468.311	even	6
1296.3.q.f.593.2		4	468.103	odd	6
1296.3.q.f.1025.1		4	468.259	odd	6
1296.3.q.f.1025.2		4	468.155	even	6
2178.3.c.d.485.1		2	429.428	even	2
2178.3.c.d.485.2		2	143.142	odd	2
2304.3.h.c.2177.1		4	208.155	odd	4
2304.3.h.c.2177.2		4	624.467	even	4
2304.3.h.c.2177.3		4	624.155	even	4
2304.3.h.c.2177.4		4	208.51	odd	4
2304.3.h.f.2177.1		4	208.181	even	4
2304.3.h.f.2177.2		4	624.77	odd	4
2304.3.h.f.2177.3		4	624.389	odd	4
2304.3.h.f.2177.4		4	208.77	even	4
3042.3.c.e.1691.1		2	3.2	odd	2	inner
3042.3.c.e.1691.2		2	1.1	even	1	trivial
3042.3.d.a.3041.1		4	13.8	odd	4
3042.3.d.a.3041.2		4	39.5	even	4
3042.3.d.a.3041.3		4	39.8	even	4
3042.3.d.a.3041.4		4	13.5	odd	4
3600.3.c.b.449.1		4	780.623	odd	4
3600.3.c.b.449.2		4	260.103	even	4
3600.3.c.b.449.3		4	780.467	odd	4
3600.3.c.b.449.4		4	260.207	even	4
3600.3.l.d.1601.1		2	780.779	even	2
3600.3.l.d.1601.2		2	260.259	odd	2

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} +4.00000 q^{7} -2.82843i q^{8} +6.00000 q^{10} +16.9706i q^{11} +5.65685i q^{14} +4.00000 q^{16} +12.7279i q^{17} +16.0000 q^{19} +8.48528i q^{20} -24.0000 q^{22} +16.9706i q^{23} +7.00000 q^{25} -8.00000 q^{28} -4.24264i q^{29} -44.0000 q^{31} +5.65685i q^{32} -18.0000 q^{34} -16.9706i q^{35} +34.0000 q^{37} +22.6274i q^{38} -12.0000 q^{40} +46.6690i q^{41} -40.0000 q^{43} -33.9411i q^{44} -24.0000 q^{46} -84.8528i q^{47} -33.0000 q^{49} +9.89949i q^{50} -38.1838i q^{53} +72.0000 q^{55} -11.3137i q^{56} +6.00000 q^{58} +33.9411i q^{59} +50.0000 q^{61} -62.2254i q^{62} -8.00000 q^{64} -8.00000 q^{67} -25.4558i q^{68} +24.0000 q^{70} -50.9117i q^{71} +16.0000 q^{73} +48.0833i q^{74} -32.0000 q^{76} +67.8823i q^{77} -76.0000 q^{79} -16.9706i q^{80} -66.0000 q^{82} +118.794i q^{83} +54.0000 q^{85} -56.5685i q^{86} +48.0000 q^{88} +12.7279i q^{89} -33.9411i q^{92} +120.000 q^{94} -67.8823i q^{95} -176.000 q^{97} -46.6690i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 8 q^{7} + 12 q^{10} + 8 q^{16} + 32 q^{19} - 48 q^{22} + 14 q^{25} - 16 q^{28} - 88 q^{31} - 36 q^{34} + 68 q^{37} - 24 q^{40} - 80 q^{43} - 48 q^{46} - 66 q^{49} + 144 q^{55} + 12 q^{58}+ \cdots - 352 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 8 * q^7 + 12 * q^10 + 8 * q^16 + 32 * q^19 - 48 * q^22 + 14 * q^25 - 16 * q^28 - 88 * q^31 - 36 * q^34 + 68 * q^37 - 24 * q^40 - 80 * q^43 - 48 * q^46 - 66 * q^49 + 144 * q^55 + 12 * q^58 + 100 * q^61 - 16 * q^64 - 16 * q^67 + 48 * q^70 + 32 * q^73 - 64 * q^76 - 152 * q^79 - 132 * q^82 + 108 * q^85 + 96 * q^88 + 240 * q^94 - 352 * q^97

Introduction

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