LMFDB - Embedded newform 882.3.b.a.197.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(197,882)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$24.0327593166$
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			197.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.3.b.a.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} +2.82843i q^{8} -6.00000 q^{10} -16.9706i q^{11} -8.00000 q^{13} +4.00000 q^{16} -12.7279i q^{17} +16.0000 q^{19} +8.48528i q^{20} -24.0000 q^{22} +16.9706i q^{23} +7.00000 q^{25} +11.3137i q^{26} -4.24264i q^{29} -44.0000 q^{31} -5.65685i q^{32} -18.0000 q^{34} -34.0000 q^{37} -22.6274i q^{38} +12.0000 q^{40} +46.6690i q^{41} -40.0000 q^{43} +33.9411i q^{44} +24.0000 q^{46} -84.8528i q^{47} -9.89949i q^{50} +16.0000 q^{52} -38.1838i q^{53} -72.0000 q^{55} -6.00000 q^{58} +33.9411i q^{59} -50.0000 q^{61} +62.2254i q^{62} -8.00000 q^{64} +33.9411i q^{65} +8.00000 q^{67} +25.4558i q^{68} +50.9117i q^{71} +16.0000 q^{73} +48.0833i q^{74} -32.0000 q^{76} -76.0000 q^{79} -16.9706i q^{80} +66.0000 q^{82} +118.794i q^{83} -54.0000 q^{85} +56.5685i q^{86} +48.0000 q^{88} +12.7279i q^{89} -33.9411i q^{92} -120.000 q^{94} -67.8823i q^{95} -176.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} - 12 q^{10} - 16 q^{13} + 8 q^{16} + 32 q^{19} - 48 q^{22} + 14 q^{25} - 88 q^{31} - 36 q^{34} - 68 q^{37} + 24 q^{40} - 80 q^{43} + 48 q^{46} + 32 q^{52} - 144 q^{55} - 12 q^{58} - 100 q^{61} - 16 q^{64} + 16 q^{67} + 32 q^{73} - 64 q^{76} - 152 q^{79} + 132 q^{82} - 108 q^{85} + 96 q^{88} - 240 q^{94} - 352 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 - 12 * q^10 - 16 * q^13 + 8 * q^16 + 32 * q^19 - 48 * q^22 + 14 * q^25 - 88 * q^31 - 36 * q^34 - 68 * q^37 + 24 * q^40 - 80 * q^43 + 48 * q^46 + 32 * q^52 - 144 * q^55 - 12 * q^58 - 100 * q^61 - 16 * q^64 + 16 * q^67 + 32 * q^73 - 64 * q^76 - 152 * q^79 + 132 * q^82 - 108 * q^85 + 96 * q^88 - 240 * q^94 - 352 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$1$	$-1$

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
$7$	0	0
$7$	0	0
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−6.00000	−0.600000
$11$	− 16.9706i	− 1.54278i	−0.636364	−	0.771389i	$-0.719562\pi$
$11$	− 16.9706i	− 1.54278i	0.636364	−	0.771389i	$-0.280438\pi$
$12$	0	0
$13$	−8.00000	−0.615385	−0.307692	−	0.951486i	$-0.599557\pi$
$13$	−8.00000	−0.615385	−0.307692	+	0.951486i	$0.599557\pi$
$14$	0	0
$15$	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
$19$	16.0000	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
$26$	11.3137i	0.435143i
$27$	0	0
$28$	0	0
$29$	− 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
$35$	0	0
$36$	0	0
$37$	−34.0000	−0.918919	−0.459459	−	0.888199i	$-0.651957\pi$
$37$	−34.0000	−0.918919	−0.459459	+	0.888199i	$0.651957\pi$
$38$	− 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
$49$	0	0
$50$	− 9.89949i	− 0.197990i
$51$	0	0
$52$	16.0000	0.307692
$53$	− 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
$56$	0	0
$57$	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	$-0.634414\pi$
$61$	−50.0000	−0.819672	−0.409836	+	0.912159i	$0.634414\pi$
$62$	62.2254i	1.00364i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	33.9411i	0.522171i
$66$	0	0
$67$	8.00000	0.119403	0.0597015	−	0.998216i	$-0.480985\pi$
$67$	8.00000	0.119403	0.0597015	+	0.998216i	$0.480985\pi$
$68$	25.4558i	0.374351i
$69$	0	0
$70$	0	0
$71$	50.9117i	0.717066i	0.933517	+	0.358533i	$0.116723\pi$
$71$	50.9117i	0.717066i	−0.933517	+	0.358533i	$0.883277\pi$
$72$	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
$77$	0	0
$78$	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	$0.861797\pi$
$98$	0	0
$99$	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	$0.456600\pi$
$104$	− 22.6274i	− 0.217571i
$105$	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	$-0.417305\pi$
$109$	56.0000	0.513761	0.256881	+	0.966443i	$0.417305\pi$
$110$	101.823i	0.925667i
$111$	0	0
$112$	0	0
$113$	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
$119$	0	0
$120$	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
$130$	48.0000	0.369231
$131$	− 169.706i	− 1.29546i	−0.761869	−	0.647731i	$-0.775718\pi$
$131$	− 169.706i	− 1.29546i	0.761869	−	0.647731i	$-0.224282\pi$
$132$	0	0
$133$	0	0
$134$	− 11.3137i	− 0.0844307i
$135$	0	0
$136$	36.0000	0.264706
$137$	− 156.978i	− 1.14582i	−0.819617	−	0.572911i	$-0.805814\pi$
$137$	− 156.978i	− 1.14582i	0.819617	−	0.572911i	$-0.194186\pi$
$138$	0	0
$139$	−152.000	−1.09353	−0.546763	−	0.837288i	$-0.684140\pi$
$139$	−152.000	−1.09353	−0.546763	+	0.837288i	$0.684140\pi$
$140$	0	0
$141$	0	0
$142$	72.0000	0.507042
$143$	135.765i	0.949402i
$144$	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	$-0.623722\pi$
$149$	− 275.772i	− 1.85082i	0.378972	−	0.925408i	$-0.376278\pi$
$150$	0	0
$151$	−148.000	−0.980132	−0.490066	−	0.871685i	$-0.663027\pi$
$151$	−148.000	−0.980132	−0.490066	+	0.871685i	$0.663027\pi$
$152$	45.2548i	0.297729i
$153$	0	0
$154$	0	0
$155$	186.676i	1.20436i
$156$	0	0
$157$	82.0000	0.522293	0.261146	−	0.965299i	$-0.415899\pi$
$157$	82.0000	0.522293	0.261146	+	0.965299i	$0.415899\pi$
$158$	107.480i	0.680255i
$159$	0	0
$160$	−24.0000	−0.150000
$161$	0	0
$162$	0	0
$163$	56.0000	0.343558	0.171779	−	0.985135i	$-0.445048\pi$
$163$	56.0000	0.343558	0.171779	+	0.985135i	$0.445048\pi$
$164$	− 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
$169$	−105.000	−0.621302
$170$	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
$175$	0	0
$176$	− 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	$-0.278568\pi$
$181$	232.000	1.28177	0.640884	+	0.767638i	$0.278568\pi$
$182$	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	$-0.320809\pi$
$193$	206.000	1.06736	0.533679	+	0.845687i	$0.320809\pi$
$194$	248.902i	1.28300i
$195$	0	0
$196$	0	0
$197$	− 165.463i	− 0.839914i	−0.907544	−	0.419957i	$-0.862045\pi$
$197$	− 165.463i	− 0.839914i	0.907544	−	0.419957i	$-0.137955\pi$
$198$	0	0
$199$	−20.0000	−0.100503	−0.0502513	−	0.998737i	$-0.516002\pi$
$199$	−20.0000	−0.100503	−0.0502513	+	0.998737i	$0.516002\pi$
$200$	19.7990i	0.0989949i
$201$	0	0
$202$	42.0000	0.207921
$203$	0	0
$204$	0	0
$205$	198.000	0.965854
$206$	− 39.5980i	− 0.192223i
$207$	0	0
$208$	−32.0000	−0.153846
$209$	− 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
$217$	0	0
$218$	− 79.1960i	− 0.363284i
$219$	0	0
$220$	144.000	0.654545
$221$	101.823i	0.460739i
$222$	0	0
$223$	436.000	1.95516	0.977578	−	0.210571i	$-0.0675325\pi$
$223$	436.000	1.95516	0.977578	+	0.210571i	$0.0675325\pi$
$224$	0	0
$225$	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
$238$	0	0
$239$	− 135.765i	− 0.568052i	−0.958817	−	0.284026i	$-0.908330\pi$
$239$	− 135.765i	− 0.568052i	0.958817	−	0.284026i	$-0.0916703\pi$
$240$	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
$245$	0	0
$246$	0	0
$247$	−128.000	−0.518219
$248$	− 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
$253$	288.000	1.13834
$254$	− 130.108i	− 0.512235i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 182.434i	− 0.709858i	−0.934893	−	0.354929i	$-0.884505\pi$
$257$	− 182.434i	− 0.709858i	0.934893	−	0.354929i	$-0.115495\pi$
$258$	0	0
$259$	0	0
$260$	− 67.8823i	− 0.261086i
$261$	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
$266$	0	0
$267$	0	0
$268$	−16.0000	−0.0597015
$269$	− 343.654i	− 1.27752i	−0.769404	−	0.638762i	$-0.779447\pi$
$269$	− 343.654i	− 1.27752i	0.769404	−	0.638762i	$-0.220553\pi$
$270$	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
$280$	0	0
$281$	− 284.257i	− 1.01159i	−0.862654	−	0.505795i	$-0.831199\pi$
$281$	− 284.257i	− 1.01159i	0.862654	−	0.505795i	$-0.168801\pi$
$282$	0	0
$283$	208.000	0.734982	0.367491	−	0.930027i	$-0.380217\pi$
$283$	208.000	0.734982	0.367491	+	0.930027i	$0.380217\pi$
$284$	− 101.823i	− 0.358533i
$285$	0	0
$286$	192.000	0.671329
$287$	0	0
$288$	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
$299$	− 135.765i	− 0.454062i
$300$	0	0
$301$	0	0
$302$	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	$0.178463\pi$
$308$	0	0
$309$	0	0
$310$	264.000	0.851613
$311$	− 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	$-0.452021\pi$
$313$	94.0000	0.300319	0.150160	+	0.988662i	$0.452021\pi$
$314$	− 115.966i	− 0.369317i
$315$	0	0
$316$	152.000	0.481013
$317$	− 335.169i	− 1.05731i	−0.848835	−	0.528657i	$-0.822696\pi$
$317$	− 335.169i	− 1.05731i	0.848835	−	0.528657i	$-0.177304\pi$
$318$	0	0
$319$	−72.0000	−0.225705
$320$	33.9411i	0.106066i
$321$	0	0
$322$	0	0
$323$	− 203.647i	− 0.630485i
$324$	0	0
$325$	−56.0000	−0.172308
$326$	− 79.1960i	− 0.242932i
$327$	0	0
$328$	−132.000	−0.402439
$329$	0	0
$330$	0	0
$331$	536.000	1.61934	0.809668	−	0.586889i	$-0.199647\pi$
$331$	536.000	1.61934	0.809668	+	0.586889i	$0.199647\pi$
$332$	− 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	$0.599862\pi$
$338$	148.492i	0.439327i
$339$	0	0
$340$	108.000	0.317647
$341$	746.705i	2.18975i
$342$	0	0
$343$	0	0
$344$	− 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	$0.389243\pi$
$350$	0	0
$351$	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	$0.284773\pi$
$359$	560.029i	1.55997i	−0.625799	+	0.779984i	$0.715227\pi$
$360$	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	$-0.626461\pi$
$367$	−284.000	−0.773842	−0.386921	+	0.922113i	$0.626461\pi$
$368$	67.8823i	0.184463i
$369$	0	0
$370$	204.000	0.551351
$371$	0	0
$372$	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
$377$	33.9411i	0.0900295i
$378$	0	0
$379$	−160.000	−0.422164	−0.211082	−	0.977468i	$-0.567699\pi$
$379$	−160.000	−0.422164	−0.211082	+	0.977468i	$0.567699\pi$
$380$	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
$385$	0	0
$386$	− 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
$392$	0	0
$393$	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	$-0.866458\pi$
$401$	− 326.683i	− 0.814672i	0.913278	−	0.407336i	$-0.133542\pi$
$402$	0	0
$403$	352.000	0.873449
$404$	− 59.3970i	− 0.147022i
$405$	0	0
$406$	0	0
$407$	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
$413$	0	0
$414$	0	0
$415$	504.000	1.21446
$416$	45.2548i	0.108786i
$417$	0	0
$418$	−384.000	−0.918660
$419$	390.323i	0.931558i	0.884901	+	0.465779i	$0.154226\pi$
$419$	390.323i	0.931558i	−0.884901	+	0.465779i	$0.845774\pi$
$420$	0	0
$421$	−40.0000	−0.0950119	−0.0475059	−	0.998871i	$-0.515127\pi$
$421$	−40.0000	−0.0950119	−0.0475059	+	0.998871i	$0.515127\pi$
$422$	− 418.607i	− 0.991960i
$423$	0	0
$424$	108.000	0.254717
$425$	− 89.0955i	− 0.209636i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	240.000	0.558140
$431$	152.735i	0.354374i	0.984177	+	0.177187i	$0.0566997\pi$
$431$	152.735i	0.354374i	−0.984177	+	0.177187i	$0.943300\pi$
$432$	0	0
$433$	−542.000	−1.25173	−0.625866	−	0.779931i	$-0.715254\pi$
$433$	−542.000	−1.25173	−0.625866	+	0.779931i	$0.715254\pi$
$434$	0	0
$435$	0	0
$436$	−112.000	−0.256881
$437$	271.529i	0.621348i
$438$	0	0
$439$	4.00000	0.00911162	0.00455581	−	0.999990i	$-0.498550\pi$
$439$	4.00000	0.00911162	0.00455581	+	0.999990i	$0.498550\pi$
$440$	− 203.647i	− 0.462834i
$441$	0	0
$442$	144.000	0.325792
$443$	− 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
$448$	0	0
$449$	− 216.375i	− 0.481904i	−0.970537	−	0.240952i	$-0.922540\pi$
$449$	− 216.375i	− 0.481904i	0.970537	−	0.240952i	$-0.0774596\pi$
$450$	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	$-0.644184\pi$
$457$	−400.000	−0.875274	−0.437637	+	0.899152i	$0.644184\pi$
$458$	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	$-0.726182\pi$
$463$	−604.000	−1.30454	−0.652268	+	0.757989i	$0.726182\pi$
$464$	− 16.9706i	− 0.0365745i
$465$	0	0
$466$	18.0000	0.0386266
$467$	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
$469$	0	0
$470$	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
$476$	0	0
$477$	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
$481$	272.000	0.565489
$482$	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	$-0.290402\pi$
$487$	596.000	1.22382	0.611910	+	0.790928i	$0.290402\pi$
$488$	− 141.421i	− 0.289798i
$489$	0	0
$490$	0	0
$491$	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
$494$	181.019i	0.366436i
$495$	0	0
$496$	−176.000	−0.354839
$497$	0	0
$498$	0	0
$499$	224.000	0.448898	0.224449	−	0.974486i	$-0.427942\pi$
$499$	224.000	0.448898	0.224449	+	0.974486i	$0.427942\pi$
$500$	271.529i	0.543058i
$501$	0	0
$502$	72.0000	0.143426
$503$	865.499i	1.72067i	0.509726	+	0.860337i	$0.329747\pi$
$503$	865.499i	1.72067i	−0.509726	+	0.860337i	$0.670253\pi$
$504$	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
$511$	0	0
$512$	− 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
$518$	0	0
$519$	0	0
$520$	−96.0000	−0.184615
$521$	− 521.845i	− 1.00162i	−0.865557	−	0.500811i	$-0.833035\pi$
$521$	− 521.845i	− 1.00162i	0.865557	−	0.500811i	$-0.166965\pi$
$522$	0	0
$523$	736.000	1.40727	0.703633	−	0.710564i	$-0.251560\pi$
$523$	736.000	1.40727	0.703633	+	0.710564i	$0.251560\pi$
$524$	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
$532$	0	0
$533$	− 373.352i	− 0.700474i
$534$	0	0
$535$	0	0
$536$	22.6274i	0.0422153i
$537$	0	0
$538$	−486.000	−0.903346
$539$	0	0
$540$	0	0
$541$	−808.000	−1.49353	−0.746765	−	0.665088i	$-0.768394\pi$
$541$	−808.000	−1.49353	−0.746765	+	0.665088i	$0.768394\pi$
$542$	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
$553$	0	0
$554$	463.862i	0.837296i
$555$	0	0
$556$	304.000	0.546763
$557$	165.463i	0.297061i	0.988908	+	0.148531i	$0.0474543\pi$
$557$	165.463i	0.297061i	−0.988908	+	0.148531i	$0.952546\pi$
$558$	0	0
$559$	320.000	0.572451
$560$	0	0
$561$	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
$565$	666.000	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	$0.395564\pi$
$572$	− 271.529i	− 0.474701i
$573$	0	0
$574$	0	0
$575$	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
$581$	0	0
$582$	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
$595$	0	0
$596$	551.543i	0.925408i
$597$	0	0
$598$	−192.000	−0.321070
$599$	− 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	$-0.541963\pi$
$601$	−158.000	−0.262895	−0.131448	+	0.991323i	$0.541963\pi$
$602$	0	0
$603$	0	0
$604$	296.000	0.490066
$605$	708.521i	1.17111i
$606$	0	0
$607$	−332.000	−0.546952	−0.273476	−	0.961879i	$-0.588173\pi$
$607$	−332.000	−0.546952	−0.273476	+	0.961879i	$0.588173\pi$
$608$	− 90.5097i	− 0.148865i
$609$	0	0
$610$	300.000	0.491803
$611$	678.823i	1.11100i
$612$	0	0
$613$	578.000	0.942904	0.471452	−	0.881892i	$-0.343730\pi$
$613$	578.000	0.942904	0.471452	+	0.881892i	$0.343730\pi$
$614$	− 735.391i	− 1.19771i
$615$	0	0
$616$	0	0
$617$	55.1543i	0.0893911i	0.999001	+	0.0446956i	$0.0142318\pi$
$617$	55.1543i	0.0893911i	−0.999001	+	0.0446956i	$0.985768\pi$
$618$	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
$623$	0	0
$624$	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	$-0.494955\pi$
$631$	20.0000	0.0316957	0.0158479	+	0.999874i	$0.494955\pi$
$632$	− 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	$0.691547\pi$
$644$	0	0
$645$	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
$650$	79.1960i	0.121840i
$651$	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
$658$	0	0
$659$	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
$665$	0	0
$666$	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
$676$	210.000	0.310651
$677$	− 1294.01i	− 1.91138i	−0.294372	−	0.955691i	$-0.595111\pi$
$677$	− 1294.01i	− 1.91138i	0.294372	−	0.955691i	$-0.404889\pi$
$678$	0	0
$679$	0	0
$680$	− 152.735i	− 0.224610i
$681$	0	0
$682$	1056.00	1.54839
$683$	560.029i	0.819954i	0.912096	+	0.409977i	$0.134463\pi$
$683$	560.029i	0.819954i	−0.912096	+	0.409977i	$0.865537\pi$
$684$	0	0
$685$	−666.000	−0.972263
$686$	0	0
$687$	0	0
$688$	−160.000	−0.232558
$689$	305.470i	0.443353i
$690$	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
$700$	0	0
$701$	− 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
$707$	0	0
$708$	0	0
$709$	968.000	1.36530	0.682652	−	0.730744i	$-0.260827\pi$
$709$	968.000	1.36530	0.682652	+	0.730744i	$0.260827\pi$
$710$	− 305.470i	− 0.430240i
$711$	0	0
$712$	−36.0000	−0.0505618
$713$	− 746.705i	− 1.04727i
$714$	0	0
$715$	576.000	0.805594
$716$	− 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
$721$	0	0
$722$	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	$-0.386392\pi$
$727$	508.000	0.698762	0.349381	+	0.936981i	$0.386392\pi$
$728$	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	$-0.565942\pi$
$739$	−304.000	−0.411367	−0.205683	+	0.978619i	$0.565942\pi$
$740$	− 288.500i	− 0.389864i
$741$	0	0
$742$	0	0
$743$	− 848.528i	− 1.14203i	−0.820940	−	0.571015i	$-0.806550\pi$
$743$	− 848.528i	− 1.14203i	0.820940	−	0.571015i	$-0.193450\pi$
$744$	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
$754$	48.0000	0.0636605
$755$	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
$763$	0	0
$764$	− 67.8823i	− 0.0888511i
$765$	0	0
$766$	−384.000	−0.501305
$767$	− 271.529i	− 0.354014i
$768$	0	0
$769$	910.000	1.18336	0.591678	−	0.806175i	$-0.298466\pi$
$769$	910.000	1.18336	0.591678	+	0.806175i	$0.298466\pi$
$770$	0	0
$771$	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
$784$	0	0
$785$	− 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
$791$	0	0
$792$	0	0
$793$	400.000	0.504414
$794$	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
$799$	−1080.00	−1.35169
$800$	− 39.5980i	− 0.0494975i
$801$	0	0
$802$	−462.000	−0.576060
$803$	− 271.529i	− 0.338143i
$804$	0	0
$805$	0	0
$806$	− 497.803i	− 0.617622i
$807$	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	$0.468550\pi$
$812$	0	0
$813$	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	$-0.914254\pi$
$821$	− 436.992i	− 0.532268i	0.963936	−	0.266134i	$-0.0857464\pi$
$822$	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
$826$	0	0
$827$	− 101.823i	− 0.123124i	−0.998103	−	0.0615619i	$-0.980392\pi$
$827$	− 101.823i	− 0.123124i	0.998103	−	0.0615619i	$-0.0196082\pi$
$828$	0	0
$829$	−632.000	−0.762364	−0.381182	−	0.924500i	$-0.624483\pi$
$829$	−632.000	−0.762364	−0.381182	+	0.924500i	$0.624483\pi$
$830$	− 712.764i	− 0.858751i
$831$	0	0
$832$	64.0000	0.0769231
$833$	0	0
$834$	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
$845$	445.477i	0.527192i
$846$	0	0
$847$	0	0
$848$	− 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	$0.584194\pi$
$854$	0	0
$855$	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	$-0.639286\pi$
$859$	−728.000	−0.847497	−0.423749	+	0.905780i	$0.639286\pi$
$860$	− 339.411i	− 0.394664i
$861$	0	0
$862$	216.000	0.250580
$863$	916.410i	1.06189i	0.847407	+	0.530945i	$0.178163\pi$
$863$	916.410i	1.06189i	−0.847407	+	0.530945i	$0.821837\pi$
$864$	0	0
$865$	−738.000	−0.853179
$866$	766.504i	0.885108i
$867$	0	0
$868$	0	0
$869$	1289.76i	1.48419i
$870$	0	0
$871$	−64.0000	−0.0734788
$872$	158.392i	0.181642i
$873$	0	0
$874$	384.000	0.439359
$875$	0	0
$876$	0	0
$877$	−910.000	−1.03763	−0.518814	−	0.854887i	$-0.673626\pi$
$877$	−910.000	−1.03763	−0.518814	+	0.854887i	$0.673626\pi$
$878$	− 5.65685i	− 0.00644289i
$879$	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	$-0.294175\pi$
$883$	1064.00	1.20498	0.602492	+	0.798125i	$0.294175\pi$
$884$	− 203.647i	− 0.230370i
$885$	0	0
$886$	−456.000	−0.514673
$887$	− 1391.59i	− 1.56887i	−0.620212	−	0.784434i	$-0.712953\pi$
$887$	− 1391.59i	− 1.56887i	0.620212	−	0.784434i	$-0.287047\pi$
$888$	0	0
$889$	0	0
$890$	− 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
$896$	0	0
$897$	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
$917$	0	0
$918$	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
$923$	− 407.294i	− 0.441271i
$924$	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
$931$	0	0
$932$	− 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	$-0.469720\pi$
$937$	178.000	0.189968	0.0949840	+	0.995479i	$0.469720\pi$
$938$	0	0
$939$	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	$-0.601309\pi$
$947$	− 1798.88i	− 1.89956i	0.312924	−	0.949778i	$-0.398691\pi$
$948$	0	0
$949$	−128.000	−0.134879
$950$	− 158.392i	− 0.166728i
$951$	0	0
$952$	0	0
$953$	− 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
$959$	0	0
$960$	0	0
$961$	975.000	1.01457
$962$	− 384.666i	− 0.399861i
$963$	0	0
$964$	64.0000	0.0663900
$965$	− 873.984i	− 0.905683i
$966$	0	0
$967$	1700.00	1.75801	0.879007	−	0.476808i	$-0.158206\pi$
$967$	1700.00	1.75801	0.879007	+	0.476808i	$0.158206\pi$
$968$	− 472.347i	− 0.487962i
$969$	0	0
$970$	1056.00	1.08866
$971$	458.205i	0.471890i	0.971766	+	0.235945i	$0.0758185\pi$
$971$	458.205i	0.471890i	−0.971766	+	0.235945i	$0.924181\pi$
$972$	0	0
$973$	0	0
$974$	− 842.871i	− 0.865371i
$975$	0	0
$976$	−200.000	−0.204918
$977$	− 759.433i	− 0.777311i	−0.921383	−	0.388655i	$-0.872940\pi$
$977$	− 759.433i	− 0.777311i	0.921383	−	0.388655i	$-0.127060\pi$
$978$	0	0
$979$	216.000	0.220633
$980$	0	0
$981$	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
$988$	256.000	0.259109
$989$	− 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
$994$	0	0
$995$	84.8528i	0.0852792i
$996$	0	0
$997$	−194.000	−0.194584	−0.0972919	−	0.995256i	$-0.531018\pi$
$997$	−194.000	−0.194584	−0.0972919	+	0.995256i	$0.531018\pi$
$998$	− 316.784i	− 0.317419i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.b.a.197.1		2
3.2	odd	2	inner	882.3.b.a.197.2		2
7.2	even	3		882.3.s.d.557.1		4
7.3	odd	6		882.3.s.b.863.2		4
7.4	even	3		882.3.s.d.863.2		4
7.5	odd	6		882.3.s.b.557.1		4
7.6	odd	2		18.3.b.a.17.1	✓	2
21.2	odd	6		882.3.s.d.557.2		4
21.5	even	6		882.3.s.b.557.2		4
21.11	odd	6		882.3.s.d.863.1		4
21.17	even	6		882.3.s.b.863.1		4
21.20	even	2		18.3.b.a.17.2	yes	2
28.27	even	2		144.3.e.b.17.2		2
35.13	even	4		450.3.b.b.449.2		4
35.27	even	4		450.3.b.b.449.3		4
35.34	odd	2		450.3.d.f.251.2		2
56.13	odd	2		576.3.e.c.449.1		2
56.27	even	2		576.3.e.f.449.1		2
63.13	odd	6		162.3.d.b.107.2		4
63.20	even	6		162.3.d.b.53.2		4
63.34	odd	6		162.3.d.b.53.1		4
63.41	even	6		162.3.d.b.107.1		4
77.76	even	2		2178.3.c.d.485.2		2
84.83	odd	2		144.3.e.b.17.1		2
91.34	even	4		3042.3.d.a.3041.4		4
91.83	even	4		3042.3.d.a.3041.1		4
91.90	odd	2		3042.3.c.e.1691.2		2
105.62	odd	4		450.3.b.b.449.1		4
105.83	odd	4		450.3.b.b.449.4		4
105.104	even	2		450.3.d.f.251.1		2
112.13	odd	4		2304.3.h.f.2177.4		4
112.27	even	4		2304.3.h.c.2177.1		4
112.69	odd	4		2304.3.h.f.2177.1		4
112.83	even	4		2304.3.h.c.2177.4		4
140.27	odd	4		3600.3.c.b.449.4		4
140.83	odd	4		3600.3.c.b.449.2		4
140.139	even	2		3600.3.l.d.1601.2		2
168.83	odd	2		576.3.e.f.449.2		2
168.125	even	2		576.3.e.c.449.2		2
231.230	odd	2		2178.3.c.d.485.1		2
252.83	odd	6		1296.3.q.f.1025.2		4
252.139	even	6		1296.3.q.f.593.2		4
252.167	odd	6		1296.3.q.f.593.1		4
252.223	even	6		1296.3.q.f.1025.1		4
273.83	odd	4		3042.3.d.a.3041.3		4
273.125	odd	4		3042.3.d.a.3041.2		4
273.272	even	2		3042.3.c.e.1691.1		2
336.83	odd	4		2304.3.h.c.2177.2		4
336.125	even	4		2304.3.h.f.2177.2		4
336.251	odd	4		2304.3.h.c.2177.3		4
336.293	even	4		2304.3.h.f.2177.3		4
420.83	even	4		3600.3.c.b.449.1		4
420.167	even	4		3600.3.c.b.449.3		4
420.419	odd	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	7.6	odd	2
18.3.b.a.17.2	yes	2	21.20	even	2
144.3.e.b.17.1		2	84.83	odd	2
144.3.e.b.17.2		2	28.27	even	2
162.3.d.b.53.1		4	63.34	odd	6
162.3.d.b.53.2		4	63.20	even	6
162.3.d.b.107.1		4	63.41	even	6
162.3.d.b.107.2		4	63.13	odd	6
450.3.b.b.449.1		4	105.62	odd	4
450.3.b.b.449.2		4	35.13	even	4
450.3.b.b.449.3		4	35.27	even	4
450.3.b.b.449.4		4	105.83	odd	4
450.3.d.f.251.1		2	105.104	even	2
450.3.d.f.251.2		2	35.34	odd	2
576.3.e.c.449.1		2	56.13	odd	2
576.3.e.c.449.2		2	168.125	even	2
576.3.e.f.449.1		2	56.27	even	2
576.3.e.f.449.2		2	168.83	odd	2
882.3.b.a.197.1		2	1.1	even	1	trivial
882.3.b.a.197.2		2	3.2	odd	2	inner
882.3.s.b.557.1		4	7.5	odd	6
882.3.s.b.557.2		4	21.5	even	6
882.3.s.b.863.1		4	21.17	even	6
882.3.s.b.863.2		4	7.3	odd	6
882.3.s.d.557.1		4	7.2	even	3
882.3.s.d.557.2		4	21.2	odd	6
882.3.s.d.863.1		4	21.11	odd	6
882.3.s.d.863.2		4	7.4	even	3
1296.3.q.f.593.1		4	252.167	odd	6
1296.3.q.f.593.2		4	252.139	even	6
1296.3.q.f.1025.1		4	252.223	even	6
1296.3.q.f.1025.2		4	252.83	odd	6
2178.3.c.d.485.1		2	231.230	odd	2
2178.3.c.d.485.2		2	77.76	even	2
2304.3.h.c.2177.1		4	112.27	even	4
2304.3.h.c.2177.2		4	336.83	odd	4
2304.3.h.c.2177.3		4	336.251	odd	4
2304.3.h.c.2177.4		4	112.83	even	4
2304.3.h.f.2177.1		4	112.69	odd	4
2304.3.h.f.2177.2		4	336.125	even	4
2304.3.h.f.2177.3		4	336.293	even	4
2304.3.h.f.2177.4		4	112.13	odd	4
3042.3.c.e.1691.1		2	273.272	even	2
3042.3.c.e.1691.2		2	91.90	odd	2
3042.3.d.a.3041.1		4	91.83	even	4
3042.3.d.a.3041.2		4	273.125	odd	4
3042.3.d.a.3041.3		4	273.83	odd	4
3042.3.d.a.3041.4		4	91.34	even	4
3600.3.c.b.449.1		4	420.83	even	4
3600.3.c.b.449.2		4	140.83	odd	4
3600.3.c.b.449.3		4	420.167	even	4
3600.3.c.b.449.4		4	140.27	odd	4
3600.3.l.d.1601.1		2	420.419	odd	2
3600.3.l.d.1601.2		2	140.139	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} +2.82843i q^{8} -6.00000 q^{10} -16.9706i q^{11} -8.00000 q^{13} +4.00000 q^{16} -12.7279i q^{17} +16.0000 q^{19} +8.48528i q^{20} -24.0000 q^{22} +16.9706i q^{23} +7.00000 q^{25} +11.3137i q^{26} -4.24264i q^{29} -44.0000 q^{31} -5.65685i q^{32} -18.0000 q^{34} -34.0000 q^{37} -22.6274i q^{38} +12.0000 q^{40} +46.6690i q^{41} -40.0000 q^{43} +33.9411i q^{44} +24.0000 q^{46} -84.8528i q^{47} -9.89949i q^{50} +16.0000 q^{52} -38.1838i q^{53} -72.0000 q^{55} -6.00000 q^{58} +33.9411i q^{59} -50.0000 q^{61} +62.2254i q^{62} -8.00000 q^{64} +33.9411i q^{65} +8.00000 q^{67} +25.4558i q^{68} +50.9117i q^{71} +16.0000 q^{73} +48.0833i q^{74} -32.0000 q^{76} -76.0000 q^{79} -16.9706i q^{80} +66.0000 q^{82} +118.794i q^{83} -54.0000 q^{85} +56.5685i q^{86} +48.0000 q^{88} +12.7279i q^{89} -33.9411i q^{92} -120.000 q^{94} -67.8823i q^{95} -176.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} - 12 q^{10} - 16 q^{13} + 8 q^{16} + 32 q^{19} - 48 q^{22} + 14 q^{25} - 88 q^{31} - 36 q^{34} - 68 q^{37} + 24 q^{40} - 80 q^{43} + 48 q^{46} + 32 q^{52} - 144 q^{55} - 12 q^{58} - 100 q^{61} - 16 q^{64} + 16 q^{67} + 32 q^{73} - 64 q^{76} - 152 q^{79} + 132 q^{82} - 108 q^{85} + 96 q^{88} - 240 q^{94} - 352 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 - 12 * q^10 - 16 * q^13 + 8 * q^16 + 32 * q^19 - 48 * q^22 + 14 * q^25 - 88 * q^31 - 36 * q^34 - 68 * q^37 + 24 * q^40 - 80 * q^43 + 48 * q^46 + 32 * q^52 - 144 * q^55 - 12 * q^58 - 100 * q^61 - 16 * q^64 + 16 * q^67 + 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more