LMFDB - Embedded newform 450.3.b.b.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,3,Mod(449,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("450.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	450.b (of order $2$, degree $1$, not 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:	$12.2616118962$
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:	$\Z[a_1, \ldots, a_{29}]$
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			449.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	450.449
Dual form			450.3.b.b.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -4.00000i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} -4.00000i q^{7} -2.82843 q^{8} +16.9706i q^{11} -8.00000i q^{13} +5.65685i q^{14} +4.00000 q^{16} +12.7279 q^{17} +16.0000 q^{19} -24.0000i q^{22} -16.9706 q^{23} +11.3137i q^{26} -8.00000i q^{28} -4.24264i q^{29} +44.0000 q^{31} -5.65685 q^{32} -18.0000 q^{34} -34.0000i q^{37} -22.6274 q^{38} +46.6690i q^{41} +40.0000i q^{43} +33.9411i q^{44} +24.0000 q^{46} +84.8528 q^{47} +33.0000 q^{49} -16.0000i q^{52} +38.1838 q^{53} +11.3137i q^{56} +6.00000i q^{58} -33.9411i q^{59} +50.0000 q^{61} -62.2254 q^{62} +8.00000 q^{64} +8.00000i q^{67} +25.4558 q^{68} -50.9117i q^{71} +16.0000i q^{73} +48.0833i q^{74} +32.0000 q^{76} +67.8823 q^{77} +76.0000 q^{79} -66.0000i q^{82} +118.794 q^{83} -56.5685i q^{86} -48.0000i q^{88} -12.7279i q^{89} -32.0000 q^{91} -33.9411 q^{92} -120.000 q^{94} +176.000i q^{97} -46.6690 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 16 q^{16} + 64 q^{19} + 176 q^{31} - 72 q^{34} + 96 q^{46} + 132 q^{49} + 200 q^{61} + 32 q^{64} + 128 q^{76} + 304 q^{79} - 128 q^{91} - 480 q^{94}+O(q^{100}) $ 4 * q + 8 * q^4 + 16 * q^16 + 64 * q^19 + 176 * q^31 - 72 * q^34 + 96 * q^46 + 132 * q^49 + 200 * q^61 + 32 * q^64 + 128 * q^76 + 304 * q^79 - 128 * q^91 - 480 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 0.571429i	−0.958315	−	0.285714i	$-0.907769\pi$
$7$	− 4.00000i	− 0.571429i	0.958315	−	0.285714i	$-0.0922308\pi$
$8$	−2.82843	−0.353553
$9$	0	0
$10$	0	0
$11$	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
$13$	− 8.00000i	− 0.615385i	−0.951486	−	0.307692i	$-0.900443\pi$
$13$	− 8.00000i	− 0.615385i	0.951486	−	0.307692i	$-0.0995567\pi$
$14$	5.65685i	0.404061i
$15$	0	0
$16$	4.00000	0.250000
$17$	12.7279	0.748701	0.374351	−	0.927287i	$-0.377866\pi$
$17$	12.7279	0.748701	0.374351	+	0.927287i	$0.377866\pi$
$18$	0	0
$19$	16.0000	0.842105	0.421053	−	0.907036i	$-0.361661\pi$
$19$	16.0000	0.842105	0.421053	+	0.907036i	$0.361661\pi$
$20$	0	0
$21$	0	0
$22$	− 24.0000i	− 1.09091i
$23$	−16.9706	−0.737851	−0.368925	−	0.929459i	$-0.620274\pi$
$23$	−16.9706	−0.737851	−0.368925	+	0.929459i	$0.620274\pi$
$24$	0	0
$25$	0	0
$26$	11.3137i	0.435143i
$27$	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
$31$	44.0000	1.41935	0.709677	−	0.704527i	$-0.248841\pi$
$31$	44.0000	1.41935	0.709677	+	0.704527i	$0.248841\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	−18.0000	−0.529412
$35$	0	0
$36$	0	0
$37$	− 34.0000i	− 0.918919i	−0.888199	−	0.459459i	$-0.848043\pi$
$37$	− 34.0000i	− 0.918919i	0.888199	−	0.459459i	$-0.151957\pi$
$38$	−22.6274	−0.595458
$39$	0	0
$40$	0	0
$41$	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
$43$	40.0000i	0.930233i	0.885250	+	0.465116i	$0.153987\pi$
$43$	40.0000i	0.930233i	−0.885250	+	0.465116i	$0.846013\pi$
$44$	33.9411i	0.771389i
$45$	0	0
$46$	24.0000	0.521739
$47$	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
$50$	0	0
$51$	0	0
$52$	− 16.0000i	− 0.307692i
$53$	38.1838	0.720448	0.360224	−	0.932866i	$-0.382700\pi$
$53$	38.1838	0.720448	0.360224	+	0.932866i	$0.382700\pi$
$54$	0	0
$55$	0	0
$56$	11.3137i	0.202031i
$57$	0	0
$58$	6.00000i	0.103448i
$59$	− 33.9411i	− 0.575273i	−0.957740	−	0.287637i	$-0.907130\pi$
$59$	− 33.9411i	− 0.575273i	0.957740	−	0.287637i	$-0.0928695\pi$
$60$	0	0
$61$	50.0000	0.819672	0.409836	−	0.912159i	$-0.365586\pi$
$61$	50.0000	0.819672	0.409836	+	0.912159i	$0.365586\pi$
$62$	−62.2254	−1.00364
$63$	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	$0.980985\pi$
$68$	25.4558	0.374351
$69$	0	0
$70$	0	0
$71$	− 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
$73$	16.0000i	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
$76$	32.0000	0.421053
$77$	67.8823	0.881588
$78$	0	0
$79$	76.0000	0.962025	0.481013	−	0.876714i	$-0.340269\pi$
$79$	76.0000	0.962025	0.481013	+	0.876714i	$0.340269\pi$
$80$	0	0
$81$	0	0
$82$	− 66.0000i	− 0.804878i
$83$	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
$85$	0	0
$86$	− 56.5685i	− 0.657774i
$87$	0	0
$88$	− 48.0000i	− 0.545455i
$89$	− 12.7279i	− 0.143010i	−0.997440	−	0.0715052i	$-0.977220\pi$
$89$	− 12.7279i	− 0.143010i	0.997440	−	0.0715052i	$-0.0227802\pi$
$90$	0	0
$91$	−32.0000	−0.351648
$92$	−33.9411	−0.368925
$93$	0	0
$94$	−120.000	−1.27660
$95$	0	0
$96$	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
$100$	0	0
$101$	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
$103$	28.0000i	0.271845i	0.990719	+	0.135922i	$0.0433998\pi$
$103$	28.0000i	0.271845i	−0.990719	+	0.135922i	$0.956600\pi$
$104$	22.6274i	0.217571i
$105$	0	0
$106$	−54.0000	−0.509434
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−56.0000	−0.513761	−0.256881	−	0.966443i	$-0.582695\pi$
$109$	−56.0000	−0.513761	−0.256881	+	0.966443i	$0.582695\pi$
$110$	0	0
$111$	0	0
$112$	− 16.0000i	− 0.142857i
$113$	−156.978	−1.38918	−0.694592	−	0.719404i	$-0.744415\pi$
$113$	−156.978	−1.38918	−0.694592	+	0.719404i	$0.744415\pi$
$114$	0	0
$115$	0	0
$116$	− 8.48528i	− 0.0731490i
$117$	0	0
$118$	48.0000i	0.406780i
$119$	− 50.9117i	− 0.427829i
$120$	0	0
$121$	−167.000	−1.38017
$122$	−70.7107	−0.579596
$123$	0	0
$124$	88.0000	0.709677
$125$	0	0
$126$	0	0
$127$	92.0000i	0.724409i	0.932099	+	0.362205i	$0.117976\pi$
$127$	92.0000i	0.724409i	−0.932099	+	0.362205i	$0.882024\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	0	0
$131$	− 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
$133$	− 64.0000i	− 0.481203i
$134$	− 11.3137i	− 0.0844307i
$135$	0	0
$136$	−36.0000	−0.264706
$137$	−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	$-0.684140\pi$
$139$	−152.000	−1.09353	−0.546763	+	0.837288i	$0.684140\pi$
$140$	0	0
$141$	0	0
$142$	72.0000i	0.507042i
$143$	135.765	0.949402
$144$	0	0
$145$	0	0
$146$	− 22.6274i	− 0.154982i
$147$	0	0
$148$	− 68.0000i	− 0.459459i
$149$	− 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	$0.663027\pi$
$152$	−45.2548	−0.297729
$153$	0	0
$154$	−96.0000	−0.623377
$155$	0	0
$156$	0	0
$157$	− 82.0000i	− 0.522293i	−0.965299	−	0.261146i	$-0.915899\pi$
$157$	− 82.0000i	− 0.522293i	0.965299	−	0.261146i	$-0.0841006\pi$
$158$	−107.480	−0.680255
$159$	0	0
$160$	0	0
$161$	67.8823i	0.421629i
$162$	0	0
$163$	− 56.0000i	− 0.343558i	−0.985135	−	0.171779i	$-0.945048\pi$
$163$	− 56.0000i	− 0.343558i	0.985135	−	0.171779i	$-0.0549515\pi$
$164$	93.3381i	0.569135i
$165$	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
$169$	105.000	0.621302
$170$	0	0
$171$	0	0
$172$	80.0000i	0.465116i
$173$	−173.948	−1.00548	−0.502741	−	0.864437i	$-0.667675\pi$
$173$	−173.948	−1.00548	−0.502741	+	0.864437i	$0.667675\pi$
$174$	0	0
$175$	0	0
$176$	67.8823i	0.385695i
$177$	0	0
$178$	18.0000i	0.101124i
$179$	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	$0.721432\pi$
$182$	45.2548	0.248653
$183$	0	0
$184$	48.0000	0.260870
$185$	0	0
$186$	0	0
$187$	216.000i	1.15508i
$188$	169.706	0.902690
$189$	0	0
$190$	0	0
$191$	− 33.9411i	− 0.177702i	−0.996045	−	0.0888511i	$-0.971680\pi$
$191$	− 33.9411i	− 0.177702i	0.996045	−	0.0888511i	$-0.0283195\pi$
$192$	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	$-0.637955\pi$
$197$	−165.463	−0.839914	−0.419957	+	0.907544i	$0.637955\pi$
$198$	0	0
$199$	−20.0000	−0.100503	−0.0502513	−	0.998737i	$-0.516002\pi$
$199$	−20.0000	−0.100503	−0.0502513	+	0.998737i	$0.516002\pi$
$200$	0	0
$201$	0	0
$202$	− 42.0000i	− 0.207921i
$203$	−16.9706	−0.0835988
$204$	0	0
$205$	0	0
$206$	− 39.5980i	− 0.192223i
$207$	0	0
$208$	− 32.0000i	− 0.153846i
$209$	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	$0.252549\pi$
$212$	76.3675	0.360224
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 176.000i	− 0.811060i
$218$	79.1960	0.363284
$219$	0	0
$220$	0	0
$221$	− 101.823i	− 0.460739i
$222$	0	0
$223$	436.000i	1.95516i	0.210571	+	0.977578i	$0.432468\pi$
$223$	436.000i	1.95516i	−0.210571	+	0.977578i	$0.567532\pi$
$224$	22.6274i	0.101015i
$225$	0	0
$226$	222.000	0.982301
$227$	−16.9706	−0.0747602	−0.0373801	−	0.999301i	$-0.511901\pi$
$227$	−16.9706	−0.0747602	−0.0373801	+	0.999301i	$0.511901\pi$
$228$	0	0
$229$	−8.00000	−0.0349345	−0.0174672	−	0.999847i	$-0.505560\pi$
$229$	−8.00000	−0.0349345	−0.0174672	+	0.999847i	$0.505560\pi$
$230$	0	0
$231$	0	0
$232$	12.0000i	0.0517241i
$233$	−12.7279	−0.0546263	−0.0273131	−	0.999627i	$-0.508695\pi$
$233$	−12.7279	−0.0546263	−0.0273131	+	0.999627i	$0.508695\pi$
$234$	0	0
$235$	0	0
$236$	− 67.8823i	− 0.287637i
$237$	0	0
$238$	72.0000i	0.302521i
$239$	− 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	$-0.478852\pi$
$241$	32.0000	0.132780	0.0663900	+	0.997794i	$0.478852\pi$
$242$	236.174	0.975924
$243$	0	0
$244$	100.000	0.409836
$245$	0	0
$246$	0	0
$247$	− 128.000i	− 0.518219i
$248$	−124.451	−0.501818
$249$	0	0
$250$	0	0
$251$	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
$254$	− 130.108i	− 0.512235i
$255$	0	0
$256$	16.0000	0.0625000
$257$	182.434	0.709858	0.354929	−	0.934893i	$-0.384505\pi$
$257$	182.434	0.709858	0.354929	+	0.934893i	$0.384505\pi$
$258$	0	0
$259$	−136.000	−0.525097
$260$	0	0
$261$	0	0
$262$	240.000i	0.916031i
$263$	373.352	1.41959	0.709795	−	0.704408i	$-0.248787\pi$
$263$	373.352	1.41959	0.709795	+	0.704408i	$0.248787\pi$
$264$	0	0
$265$	0	0
$266$	90.5097i	0.340262i
$267$	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
$271$	380.000	1.40221	0.701107	−	0.713056i	$-0.252690\pi$
$271$	380.000	1.40221	0.701107	+	0.713056i	$0.252690\pi$
$272$	50.9117	0.187175
$273$	0	0
$274$	222.000	0.810219
$275$	0	0
$276$	0	0
$277$	− 328.000i	− 1.18412i	−0.805896	−	0.592058i	$-0.798316\pi$
$277$	− 328.000i	− 1.18412i	0.805896	−	0.592058i	$-0.201684\pi$
$278$	214.960	0.773239
$279$	0	0
$280$	0	0
$281$	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
$283$	208.000i	0.734982i	0.930027	+	0.367491i	$0.119783\pi$
$283$	208.000i	0.734982i	−0.930027	+	0.367491i	$0.880217\pi$
$284$	− 101.823i	− 0.358533i
$285$	0	0
$286$	−192.000	−0.671329
$287$	186.676	0.650440
$288$	0	0
$289$	−127.000	−0.439446
$290$	0	0
$291$	0	0
$292$	32.0000i	0.109589i
$293$	−436.992	−1.49144	−0.745720	−	0.666259i	$-0.767894\pi$
$293$	−436.992	−1.49144	−0.745720	+	0.666259i	$0.767894\pi$
$294$	0	0
$295$	0	0
$296$	96.1665i	0.324887i
$297$	0	0
$298$	390.000i	1.30872i
$299$	135.765i	0.454062i
$300$	0	0
$301$	160.000	0.531561
$302$	209.304	0.693058
$303$	0	0
$304$	64.0000	0.210526
$305$	0	0
$306$	0	0
$307$	− 520.000i	− 1.69381i	−0.531743	−	0.846906i	$-0.678463\pi$
$307$	− 520.000i	− 1.69381i	0.531743	−	0.846906i	$-0.321537\pi$
$308$	135.765	0.440794
$309$	0	0
$310$	0	0
$311$	− 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
$313$	94.0000i	0.300319i	0.988662	+	0.150160i	$0.0479788\pi$
$313$	94.0000i	0.300319i	−0.988662	+	0.150160i	$0.952021\pi$
$314$	115.966i	0.369317i
$315$	0	0
$316$	152.000	0.481013
$317$	−335.169	−1.05731	−0.528657	−	0.848835i	$-0.677304\pi$
$317$	−335.169	−1.05731	−0.528657	+	0.848835i	$0.677304\pi$
$318$	0	0
$319$	72.0000	0.225705
$320$	0	0
$321$	0	0
$322$	− 96.0000i	− 0.298137i
$323$	203.647	0.630485
$324$	0	0
$325$	0	0
$326$	79.1960i	0.242932i
$327$	0	0
$328$	− 132.000i	− 0.402439i
$329$	− 339.411i	− 1.03165i
$330$	0	0
$331$	536.000	1.61934	0.809668	−	0.586889i	$-0.199647\pi$
$331$	536.000	1.61934	0.809668	+	0.586889i	$0.199647\pi$
$332$	237.588	0.715626
$333$	0	0
$334$	48.0000	0.143713
$335$	0	0
$336$	0	0
$337$	− 208.000i	− 0.617211i	−0.951190	−	0.308605i	$-0.900138\pi$
$337$	− 208.000i	− 0.617211i	0.951190	−	0.308605i	$-0.0998622\pi$
$338$	−148.492	−0.439327
$339$	0	0
$340$	0	0
$341$	746.705i	2.18975i
$342$	0	0
$343$	− 328.000i	− 0.956268i
$344$	− 113.137i	− 0.328887i
$345$	0	0
$346$	246.000	0.710983
$347$	−288.500	−0.831411	−0.415705	−	0.909499i	$-0.636465\pi$
$347$	−288.500	−0.831411	−0.415705	+	0.909499i	$0.636465\pi$
$348$	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
$352$	− 96.0000i	− 0.272727i
$353$	−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
$355$	0	0
$356$	− 25.4558i	− 0.0715052i
$357$	0	0
$358$	− 288.000i	− 0.804469i
$359$	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
$362$	328.098	0.906347
$363$	0	0
$364$	−64.0000	−0.175824
$365$	0	0
$366$	0	0
$367$	284.000i	0.773842i	0.922113	+	0.386921i	$0.126461\pi$
$367$	284.000i	0.773842i	−0.922113	+	0.386921i	$0.873539\pi$
$368$	−67.8823	−0.184463
$369$	0	0
$370$	0	0
$371$	− 152.735i	− 0.411685i
$372$	0	0
$373$	190.000i	0.509383i	0.967022	+	0.254692i	$0.0819740\pi$
$373$	190.000i	0.509383i	−0.967022	+	0.254692i	$0.918026\pi$
$374$	− 305.470i	− 0.816765i
$375$	0	0
$376$	−240.000	−0.638298
$377$	−33.9411	−0.0900295
$378$	0	0
$379$	160.000	0.422164	0.211082	−	0.977468i	$-0.432301\pi$
$379$	160.000	0.422164	0.211082	+	0.977468i	$0.432301\pi$
$380$	0	0
$381$	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
$385$	0	0
$386$	291.328i	0.754736i
$387$	0	0
$388$	352.000i	0.907216i
$389$	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
$392$	−93.3381	−0.238107
$393$	0	0
$394$	234.000	0.593909
$395$	0	0
$396$	0	0
$397$	146.000i	0.367758i	0.982949	+	0.183879i	$0.0588655\pi$
$397$	146.000i	0.367758i	−0.982949	+	0.183879i	$0.941135\pi$
$398$	28.2843	0.0710660
$399$	0	0
$400$	0	0
$401$	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
$403$	− 352.000i	− 0.873449i
$404$	59.3970i	0.147022i
$405$	0	0
$406$	24.0000	0.0591133
$407$	576.999	1.41769
$408$	0	0
$409$	−368.000	−0.899756	−0.449878	−	0.893090i	$-0.648532\pi$
$409$	−368.000	−0.899756	−0.449878	+	0.893090i	$0.648532\pi$
$410$	0	0
$411$	0	0
$412$	56.0000i	0.135922i
$413$	−135.765	−0.328728
$414$	0	0
$415$	0	0
$416$	45.2548i	0.108786i
$417$	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
$421$	−40.0000	−0.0950119	−0.0475059	−	0.998871i	$-0.515127\pi$
$421$	−40.0000	−0.0950119	−0.0475059	+	0.998871i	$0.515127\pi$
$422$	−418.607	−0.991960
$423$	0	0
$424$	−108.000	−0.254717
$425$	0	0
$426$	0	0
$427$	− 200.000i	− 0.468384i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 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
$433$	− 542.000i	− 1.25173i	−0.779931	−	0.625866i	$-0.784746\pi$
$433$	− 542.000i	− 1.25173i	0.779931	−	0.625866i	$-0.215254\pi$
$434$	248.902i	0.573506i
$435$	0	0
$436$	−112.000	−0.256881
$437$	−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	$0.498550\pi$
$440$	0	0
$441$	0	0
$442$	144.000i	0.325792i
$443$	322.441	0.727857	0.363929	−	0.931427i	$-0.381435\pi$
$443$	322.441	0.727857	0.363929	+	0.931427i	$0.381435\pi$
$444$	0	0
$445$	0	0
$446$	− 616.597i	− 1.38250i
$447$	0	0
$448$	− 32.0000i	− 0.0714286i
$449$	− 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
$452$	−313.955	−0.694592
$453$	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	$-0.144184\pi$
$458$	11.3137	0.0247024
$459$	0	0
$460$	0	0
$461$	− 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
$463$	604.000i	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
$466$	18.0000	0.0386266
$467$	−356.382	−0.763130	−0.381565	−	0.924342i	$-0.624615\pi$
$467$	−356.382	−0.763130	−0.381565	+	0.924342i	$0.624615\pi$
$468$	0	0
$469$	32.0000	0.0682303
$470$	0	0
$471$	0	0
$472$	96.0000i	0.203390i
$473$	−678.823	−1.43514
$474$	0	0
$475$	0	0
$476$	− 101.823i	− 0.213915i
$477$	0	0
$478$	192.000i	0.401674i
$479$	− 526.087i	− 1.09830i	−0.835723	−	0.549152i	$-0.814951\pi$
$479$	− 526.087i	− 1.09830i	0.835723	−	0.549152i	$-0.185049\pi$
$480$	0	0
$481$	−272.000	−0.565489
$482$	−45.2548	−0.0938897
$483$	0	0
$484$	−334.000	−0.690083
$485$	0	0
$486$	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
$490$	0	0
$491$	− 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
$493$	− 54.0000i	− 0.109533i
$494$	181.019i	0.366436i
$495$	0	0
$496$	176.000	0.354839
$497$	−203.647	−0.409752
$498$	0	0
$499$	−224.000	−0.448898	−0.224449	−	0.974486i	$-0.572058\pi$
$499$	−224.000	−0.448898	−0.224449	+	0.974486i	$0.572058\pi$
$500$	0	0
$501$	0	0
$502$	− 72.0000i	− 0.143426i
$503$	865.499	1.72067	0.860337	−	0.509726i	$-0.170253\pi$
$503$	865.499	1.72067	0.860337	+	0.509726i	$0.170253\pi$
$504$	0	0
$505$	0	0
$506$	407.294i	0.804928i
$507$	0	0
$508$	184.000i	0.362205i
$509$	− 479.418i	− 0.941883i	−0.882164	−	0.470941i	$-0.843914\pi$
$509$	− 479.418i	− 0.941883i	0.882164	−	0.470941i	$-0.156086\pi$
$510$	0	0
$511$	64.0000	0.125245
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	−258.000	−0.501946
$515$	0	0
$516$	0	0
$517$	1440.00i	2.78530i
$518$	192.333	0.371299
$519$	0	0
$520$	0	0
$521$	− 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
$523$	736.000i	1.40727i	0.710564	+	0.703633i	$0.248440\pi$
$523$	736.000i	1.40727i	−0.710564	+	0.703633i	$0.751560\pi$
$524$	− 339.411i	− 0.647731i
$525$	0	0
$526$	−528.000	−1.00380
$527$	560.029	1.06267
$528$	0	0
$529$	−241.000	−0.455577
$530$	0	0
$531$	0	0
$532$	− 128.000i	− 0.240602i
$533$	373.352	0.700474
$534$	0	0
$535$	0	0
$536$	− 22.6274i	− 0.0422153i
$537$	0	0
$538$	− 486.000i	− 0.903346i
$539$	560.029i	1.03901i
$540$	0	0
$541$	−808.000	−1.49353	−0.746765	−	0.665088i	$-0.768394\pi$
$541$	−808.000	−1.49353	−0.746765	+	0.665088i	$0.768394\pi$
$542$	−537.401	−0.991515
$543$	0	0
$544$	−72.0000	−0.132353
$545$	0	0
$546$	0	0
$547$	536.000i	0.979890i	0.871753	+	0.489945i	$0.162983\pi$
$547$	536.000i	0.979890i	−0.871753	+	0.489945i	$0.837017\pi$
$548$	−313.955	−0.572911
$549$	0	0
$550$	0	0
$551$	− 67.8823i	− 0.123198i
$552$	0	0
$553$	− 304.000i	− 0.549729i
$554$	463.862i	0.837296i
$555$	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
$559$	320.000	0.572451
$560$	0	0
$561$	0	0
$562$	− 402.000i	− 0.715302i
$563$	−322.441	−0.572719	−0.286359	−	0.958122i	$-0.592445\pi$
$563$	−322.441	−0.572719	−0.286359	+	0.958122i	$0.592445\pi$
$564$	0	0
$565$	0	0
$566$	− 294.156i	− 0.519711i
$567$	0	0
$568$	144.000i	0.253521i
$569$	− 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	$0.395564\pi$
$572$	271.529	0.474701
$573$	0	0
$574$	−264.000	−0.459930
$575$	0	0
$576$	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
$580$	0	0
$581$	− 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	$-0.603017\pi$
$587$	−373.352	−0.636035	−0.318017	+	0.948085i	$0.603017\pi$
$588$	0	0
$589$	704.000	1.19525
$590$	0	0
$591$	0	0
$592$	− 136.000i	− 0.229730i
$593$	1107.33	1.86733	0.933667	−	0.358142i	$-0.116590\pi$
$593$	1107.33	1.86733	0.933667	+	0.358142i	$0.116590\pi$
$594$	0	0
$595$	0	0
$596$	− 551.543i	− 0.925408i
$597$	0	0
$598$	− 192.000i	− 0.321070i
$599$	− 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	$0.458037\pi$
$602$	−226.274	−0.375871
$603$	0	0
$604$	−296.000	−0.490066
$605$	0	0
$606$	0	0
$607$	332.000i	0.546952i	0.961879	+	0.273476i	$0.0881734\pi$
$607$	332.000i	0.546952i	−0.961879	+	0.273476i	$0.911827\pi$
$608$	−90.5097	−0.148865
$609$	0	0
$610$	0	0
$611$	− 678.823i	− 1.11100i
$612$	0	0
$613$	− 578.000i	− 0.942904i	−0.881892	−	0.471452i	$-0.843730\pi$
$613$	− 578.000i	− 0.942904i	0.881892	−	0.471452i	$-0.156270\pi$
$614$	735.391i	1.19771i
$615$	0	0
$616$	−192.000	−0.311688
$617$	55.1543	0.0893911	0.0446956	−	0.999001i	$-0.485768\pi$
$617$	55.1543	0.0893911	0.0446956	+	0.999001i	$0.485768\pi$
$618$	0	0
$619$	−896.000	−1.44750	−0.723748	−	0.690064i	$-0.757582\pi$
$619$	−896.000	−1.44750	−0.723748	+	0.690064i	$0.757582\pi$
$620$	0	0
$621$	0	0
$622$	528.000i	0.848875i
$623$	−50.9117	−0.0817202
$624$	0	0
$625$	0	0
$626$	− 132.936i	− 0.212358i
$627$	0	0
$628$	− 164.000i	− 0.261146i
$629$	− 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	$0.494955\pi$
$632$	−214.960	−0.340127
$633$	0	0
$634$	474.000	0.747634
$635$	0	0
$636$	0	0
$637$	− 264.000i	− 0.414443i
$638$	−101.823	−0.159598
$639$	0	0
$640$	0	0
$641$	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	$-0.808453\pi$
$643$	− 728.000i	− 1.13219i	0.824339	−	0.566096i	$-0.191547\pi$
$644$	135.765i	0.210814i
$645$	0	0
$646$	−288.000	−0.445820
$647$	458.205	0.708200	0.354100	−	0.935208i	$-0.384787\pi$
$647$	458.205	0.708200	0.354100	+	0.935208i	$0.384787\pi$
$648$	0	0
$649$	576.000	0.887519
$650$	0	0
$651$	0	0
$652$	− 112.000i	− 0.171779i
$653$	301.227	0.461298	0.230649	−	0.973037i	$-0.425915\pi$
$653$	301.227	0.461298	0.230649	+	0.973037i	$0.425915\pi$
$654$	0	0
$655$	0	0
$656$	186.676i	0.284567i
$657$	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
$661$	62.0000	0.0937973	0.0468986	−	0.998900i	$-0.485066\pi$
$661$	62.0000	0.0937973	0.0468986	+	0.998900i	$0.485066\pi$
$662$	−758.018	−1.14504
$663$	0	0
$664$	−336.000	−0.506024
$665$	0	0
$666$	0	0
$667$	72.0000i	0.107946i
$668$	−67.8823	−0.101620
$669$	0	0
$670$	0	0
$671$	848.528i	1.26457i
$672$	0	0
$673$	670.000i	0.995542i	0.867308	+	0.497771i	$0.165848\pi$
$673$	670.000i	0.995542i	−0.867308	+	0.497771i	$0.834152\pi$
$674$	294.156i	0.436434i
$675$	0	0
$676$	210.000	0.310651
$677$	1294.01	1.91138	0.955691	−	0.294372i	$-0.0951105\pi$
$677$	1294.01	1.91138	0.955691	+	0.294372i	$0.0951105\pi$
$678$	0	0
$679$	704.000	1.03682
$680$	0	0
$681$	0	0
$682$	− 1056.00i	− 1.54839i
$683$	−560.029	−0.819954	−0.409977	−	0.912096i	$-0.634463\pi$
$683$	−560.029	−0.819954	−0.409977	+	0.912096i	$0.634463\pi$
$684$	0	0
$685$	0	0
$686$	463.862i	0.676184i
$687$	0	0
$688$	160.000i	0.232558i
$689$	− 305.470i	− 0.443353i
$690$	0	0
$691$	−40.0000	−0.0578871	−0.0289436	−	0.999581i	$-0.509214\pi$
$691$	−40.0000	−0.0578871	−0.0289436	+	0.999581i	$0.509214\pi$
$692$	−347.897	−0.502741
$693$	0	0
$694$	408.000	0.587896
$695$	0	0
$696$	0	0
$697$	594.000i	0.852224i
$698$	−336.583	−0.482210
$699$	0	0
$700$	0	0
$701$	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
$703$	− 544.000i	− 0.773826i
$704$	135.765i	0.192847i
$705$	0	0
$706$	318.000	0.450425
$707$	118.794	0.168025
$708$	0	0
$709$	−968.000	−1.36530	−0.682652	−	0.730744i	$-0.739173\pi$
$709$	−968.000	−1.36530	−0.682652	+	0.730744i	$0.739173\pi$
$710$	0	0
$711$	0	0
$712$	36.0000i	0.0505618i
$713$	−746.705	−1.04727
$714$	0	0
$715$	0	0
$716$	407.294i	0.568846i
$717$	0	0
$718$	− 792.000i	− 1.10306i
$719$	− 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
$722$	148.492	0.205668
$723$	0	0
$724$	−464.000	−0.640884
$725$	0	0
$726$	0	0
$727$	− 508.000i	− 0.698762i	−0.936981	−	0.349381i	$-0.886392\pi$
$727$	− 508.000i	− 0.698762i	0.936981	−	0.349381i	$-0.113608\pi$
$728$	90.5097	0.124326
$729$	0	0
$730$	0	0
$731$	509.117i	0.696466i
$732$	0	0
$733$	1144.00i	1.56071i	0.625337	+	0.780355i	$0.284961\pi$
$733$	1144.00i	1.56071i	−0.625337	+	0.780355i	$0.715039\pi$
$734$	− 401.637i	− 0.547189i
$735$	0	0
$736$	96.0000	0.130435
$737$	−135.765	−0.184212
$738$	0	0
$739$	304.000	0.411367	0.205683	−	0.978619i	$-0.434058\pi$
$739$	304.000	0.411367	0.205683	+	0.978619i	$0.434058\pi$
$740$	0	0
$741$	0	0
$742$	216.000i	0.291105i
$743$	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
$745$	0	0
$746$	− 268.701i	− 0.360188i
$747$	0	0
$748$	432.000i	0.577540i
$749$	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	$0.460054\pi$
$752$	339.411	0.451345
$753$	0	0
$754$	48.0000	0.0636605
$755$	0	0
$756$	0	0
$757$	− 1240.00i	− 1.63804i	−0.573761	−	0.819022i	$-0.694516\pi$
$757$	− 1240.00i	− 1.63804i	0.573761	−	0.819022i	$-0.305484\pi$
$758$	−226.274	−0.298515
$759$	0	0
$760$	0	0
$761$	− 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
$763$	224.000i	0.293578i
$764$	− 67.8823i	− 0.0888511i
$765$	0	0
$766$	384.000	0.501305
$767$	−271.529	−0.354014
$768$	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
$772$	− 412.000i	− 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
$775$	0	0
$776$	− 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
$785$	0	0
$786$	0	0
$787$	− 1360.00i	− 1.72808i	−0.503422	−	0.864041i	$-0.667926\pi$
$787$	− 1360.00i	− 1.72808i	0.503422	−	0.864041i	$-0.332074\pi$
$788$	−330.926	−0.419957
$789$	0	0
$790$	0	0
$791$	627.911i	0.793819i
$792$	0	0
$793$	− 400.000i	− 0.504414i
$794$	− 206.475i	− 0.260044i
$795$	0	0
$796$	−40.0000	−0.0502513
$797$	106.066	0.133082	0.0665408	−	0.997784i	$-0.478804\pi$
$797$	106.066	0.133082	0.0665408	+	0.997784i	$0.478804\pi$
$798$	0	0
$799$	1080.00	1.35169
$800$	0	0
$801$	0	0
$802$	− 462.000i	− 0.576060i
$803$	−271.529	−0.338143
$804$	0	0
$805$	0	0
$806$	497.803i	0.617622i
$807$	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
$811$	−160.000	−0.197287	−0.0986436	−	0.995123i	$-0.531450\pi$
$811$	−160.000	−0.197287	−0.0986436	+	0.995123i	$0.531450\pi$
$812$	−33.9411	−0.0417994
$813$	0	0
$814$	−816.000	−1.00246
$815$	0	0
$816$	0	0
$817$	640.000i	0.783354i
$818$	520.431	0.636223
$819$	0	0
$820$	0	0
$821$	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
$823$	− 332.000i	− 0.403402i	−0.979447	−	0.201701i	$-0.935353\pi$
$823$	− 332.000i	− 0.403402i	0.979447	−	0.201701i	$-0.0646470\pi$
$824$	− 79.1960i	− 0.0961116i
$825$	0	0
$826$	192.000	0.232446
$827$	−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	$0.624483\pi$
$830$	0	0
$831$	0	0
$832$	− 64.0000i	− 0.0769231i
$833$	420.021	0.504227
$834$	0	0
$835$	0	0
$836$	543.058i	0.649591i
$837$	0	0
$838$	552.000i	0.658711i
$839$	− 729.734i	− 0.869767i	−0.900487	−	0.434883i	$-0.856790\pi$
$839$	− 729.734i	− 0.869767i	0.900487	−	0.434883i	$-0.143210\pi$
$840$	0	0
$841$	823.000	0.978597
$842$	56.5685	0.0671835
$843$	0	0
$844$	592.000	0.701422
$845$	0	0
$846$	0	0
$847$	668.000i	0.788666i
$848$	152.735	0.180112
$849$	0	0
$850$	0	0
$851$	576.999i	0.678025i
$852$	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
$857$	428.507	0.500008	0.250004	−	0.968245i	$-0.419568\pi$
$857$	428.507	0.500008	0.250004	+	0.968245i	$0.419568\pi$
$858$	0	0
$859$	−728.000	−0.847497	−0.423749	−	0.905780i	$-0.639286\pi$
$859$	−728.000	−0.847497	−0.423749	+	0.905780i	$0.639286\pi$
$860$	0	0
$861$	0	0
$862$	216.000i	0.250580i
$863$	−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
$865$	0	0
$866$	766.504i	0.885108i
$867$	0	0
$868$	− 352.000i	− 0.405530i
$869$	1289.76i	1.48419i
$870$	0	0
$871$	64.0000	0.0734788
$872$	158.392	0.181642
$873$	0	0
$874$	384.000	0.439359
$875$	0	0
$876$	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
$880$	0	0
$881$	− 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
$883$	− 1064.00i	− 1.20498i	−0.798125	−	0.602492i	$-0.794175\pi$
$883$	− 1064.00i	− 1.20498i	0.798125	−	0.602492i	$-0.205825\pi$
$884$	− 203.647i	− 0.230370i
$885$	0	0
$886$	−456.000	−0.514673
$887$	1391.59	1.56887	0.784434	−	0.620212i	$-0.212953\pi$
$887$	1391.59	1.56887	0.784434	+	0.620212i	$0.212953\pi$
$888$	0	0
$889$	368.000	0.413948
$890$	0	0
$891$	0	0
$892$	872.000i	0.977578i
$893$	1357.65	1.52032
$894$	0	0
$895$	0	0
$896$	45.2548i	0.0505076i
$897$	0	0
$898$	306.000i	0.340757i
$899$	− 186.676i	− 0.207649i
$900$	0	0
$901$	486.000	0.539401
$902$	1120.06	1.24175
$903$	0	0
$904$	444.000	0.491150
$905$	0	0
$906$	0	0
$907$	− 1768.00i	− 1.94928i	−0.223771	−	0.974642i	$-0.571837\pi$
$907$	− 1768.00i	− 1.94928i	0.223771	−	0.974642i	$-0.428163\pi$
$908$	−33.9411	−0.0373801
$909$	0	0
$910$	0	0
$911$	− 237.588i	− 0.260799i	−0.991462	−	0.130399i	$-0.958374\pi$
$911$	− 237.588i	− 0.260799i	0.991462	−	0.130399i	$-0.0416260\pi$
$912$	0	0
$913$	2016.00i	2.20811i
$914$	565.685i	0.618912i
$915$	0	0
$916$	−16.0000	−0.0174672
$917$	−678.823	−0.740264
$918$	0	0
$919$	−380.000	−0.413493	−0.206746	−	0.978395i	$-0.566288\pi$
$919$	−380.000	−0.413493	−0.206746	+	0.978395i	$0.566288\pi$
$920$	0	0
$921$	0	0
$922$	426.000i	0.462039i
$923$	−407.294	−0.441271
$924$	0	0
$925$	0	0
$926$	− 854.185i	− 0.922446i
$927$	0	0
$928$	24.0000i	0.0258621i
$929$	− 666.095i	− 0.717002i	−0.933529	−	0.358501i	$-0.883288\pi$
$929$	− 666.095i	− 0.717002i	0.933529	−	0.358501i	$-0.116712\pi$
$930$	0	0
$931$	528.000	0.567132
$932$	−25.4558	−0.0273131
$933$	0	0
$934$	504.000	0.539615
$935$	0	0
$936$	0	0
$937$	− 178.000i	− 0.189968i	−0.995479	−	0.0949840i	$-0.969720\pi$
$937$	− 178.000i	− 0.189968i	0.995479	−	0.0949840i	$-0.0302800\pi$
$938$	−45.2548	−0.0482461
$939$	0	0
$940$	0	0
$941$	− 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
$943$	− 792.000i	− 0.839873i
$944$	− 135.765i	− 0.143818i
$945$	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
$949$	128.000	0.134879
$950$	0	0
$951$	0	0
$952$	144.000i	0.151261i
$953$	1310.98	1.37563	0.687815	−	0.725886i	$-0.258570\pi$
$953$	1310.98	1.37563	0.687815	+	0.725886i	$0.258570\pi$
$954$	0	0
$955$	0	0
$956$	− 271.529i	− 0.284026i
$957$	0	0
$958$	744.000i	0.776618i
$959$	627.911i	0.654756i
$960$	0	0
$961$	975.000	1.01457
$962$	384.666	0.399861
$963$	0	0
$964$	64.0000	0.0663900
$965$	0	0
$966$	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
$970$	0	0
$971$	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
$973$	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	$-0.627060\pi$
$977$	−759.433	−0.777311	−0.388655	+	0.921383i	$0.627060\pi$
$978$	0	0
$979$	216.000	0.220633
$980$	0	0
$981$	0	0
$982$	384.000i	0.391039i
$983$	−1052.17	−1.07037	−0.535186	−	0.844734i	$-0.679758\pi$
$983$	−1052.17	−1.07037	−0.535186	+	0.844734i	$0.679758\pi$
$984$	0	0
$985$	0	0
$986$	76.3675i	0.0774519i
$987$	0	0
$988$	− 256.000i	− 0.259109i
$989$	− 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	$0.627354\pi$
$992$	−248.902	−0.250909
$993$	0	0
$994$	288.000	0.289738
$995$	0	0
$996$	0	0
$997$	194.000i	0.194584i	0.995256	+	0.0972919i	$0.0310180\pi$
$997$	194.000i	0.194584i	−0.995256	+	0.0972919i	$0.968982\pi$
$998$	316.784	0.317419
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.3.b.b.449.1		4
3.2	odd	2	inner	450.3.b.b.449.3		4
4.3	odd	2		3600.3.c.b.449.3		4
5.2	odd	4		450.3.d.f.251.1		2
5.3	odd	4		18.3.b.a.17.2	yes	2
5.4	even	2	inner	450.3.b.b.449.4		4
12.11	even	2		3600.3.c.b.449.4		4
15.2	even	4		450.3.d.f.251.2		2
15.8	even	4		18.3.b.a.17.1	✓	2
15.14	odd	2	inner	450.3.b.b.449.2		4
20.3	even	4		144.3.e.b.17.1		2
20.7	even	4		3600.3.l.d.1601.1		2
20.19	odd	2		3600.3.c.b.449.1		4
35.3	even	12		882.3.s.d.863.1		4
35.13	even	4		882.3.b.a.197.2		2
35.18	odd	12		882.3.s.b.863.1		4
35.23	odd	12		882.3.s.b.557.2		4
35.33	even	12		882.3.s.d.557.2		4
40.3	even	4		576.3.e.f.449.2		2
40.13	odd	4		576.3.e.c.449.2		2
45.13	odd	12		162.3.d.b.107.1		4
45.23	even	12		162.3.d.b.107.2		4
45.38	even	12		162.3.d.b.53.1		4
45.43	odd	12		162.3.d.b.53.2		4
55.43	even	4		2178.3.c.d.485.1		2
60.23	odd	4		144.3.e.b.17.2		2
60.47	odd	4		3600.3.l.d.1601.2		2
60.59	even	2		3600.3.c.b.449.2		4
65.8	even	4		3042.3.d.a.3041.2		4
65.18	even	4		3042.3.d.a.3041.3		4
65.38	odd	4		3042.3.c.e.1691.1		2
80.3	even	4		2304.3.h.c.2177.2		4
80.13	odd	4		2304.3.h.f.2177.2		4
80.43	even	4		2304.3.h.c.2177.3		4
80.53	odd	4		2304.3.h.f.2177.3		4
105.23	even	12		882.3.s.b.557.1		4
105.38	odd	12		882.3.s.d.863.2		4
105.53	even	12		882.3.s.b.863.2		4
105.68	odd	12		882.3.s.d.557.1		4
105.83	odd	4		882.3.b.a.197.1		2
120.53	even	4		576.3.e.c.449.1		2
120.83	odd	4		576.3.e.f.449.1		2
165.98	odd	4		2178.3.c.d.485.2		2
180.23	odd	12		1296.3.q.f.593.2		4
180.43	even	12		1296.3.q.f.1025.2		4
180.83	odd	12		1296.3.q.f.1025.1		4
180.103	even	12		1296.3.q.f.593.1		4
195.8	odd	4		3042.3.d.a.3041.4		4
195.38	even	4		3042.3.c.e.1691.2		2
195.83	odd	4		3042.3.d.a.3041.1		4
240.53	even	4		2304.3.h.f.2177.1		4
240.83	odd	4		2304.3.h.c.2177.4		4
240.173	even	4		2304.3.h.f.2177.4		4
240.203	odd	4		2304.3.h.c.2177.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.b.a.17.1	✓	2	15.8	even	4
18.3.b.a.17.2	yes	2	5.3	odd	4
144.3.e.b.17.1		2	20.3	even	4
144.3.e.b.17.2		2	60.23	odd	4
162.3.d.b.53.1		4	45.38	even	12
162.3.d.b.53.2		4	45.43	odd	12
162.3.d.b.107.1		4	45.13	odd	12
162.3.d.b.107.2		4	45.23	even	12
450.3.b.b.449.1		4	1.1	even	1	trivial
450.3.b.b.449.2		4	15.14	odd	2	inner
450.3.b.b.449.3		4	3.2	odd	2	inner
450.3.b.b.449.4		4	5.4	even	2	inner
450.3.d.f.251.1		2	5.2	odd	4
450.3.d.f.251.2		2	15.2	even	4
576.3.e.c.449.1		2	120.53	even	4
576.3.e.c.449.2		2	40.13	odd	4
576.3.e.f.449.1		2	120.83	odd	4
576.3.e.f.449.2		2	40.3	even	4
882.3.b.a.197.1		2	105.83	odd	4
882.3.b.a.197.2		2	35.13	even	4
882.3.s.b.557.1		4	105.23	even	12
882.3.s.b.557.2		4	35.23	odd	12
882.3.s.b.863.1		4	35.18	odd	12
882.3.s.b.863.2		4	105.53	even	12
882.3.s.d.557.1		4	105.68	odd	12
882.3.s.d.557.2		4	35.33	even	12
882.3.s.d.863.1		4	35.3	even	12
882.3.s.d.863.2		4	105.38	odd	12
1296.3.q.f.593.1		4	180.103	even	12
1296.3.q.f.593.2		4	180.23	odd	12
1296.3.q.f.1025.1		4	180.83	odd	12
1296.3.q.f.1025.2		4	180.43	even	12
2178.3.c.d.485.1		2	55.43	even	4
2178.3.c.d.485.2		2	165.98	odd	4
2304.3.h.c.2177.1		4	240.203	odd	4
2304.3.h.c.2177.2		4	80.3	even	4
2304.3.h.c.2177.3		4	80.43	even	4
2304.3.h.c.2177.4		4	240.83	odd	4
2304.3.h.f.2177.1		4	240.53	even	4
2304.3.h.f.2177.2		4	80.13	odd	4
2304.3.h.f.2177.3		4	80.53	odd	4
2304.3.h.f.2177.4		4	240.173	even	4
3042.3.c.e.1691.1		2	65.38	odd	4
3042.3.c.e.1691.2		2	195.38	even	4
3042.3.d.a.3041.1		4	195.83	odd	4
3042.3.d.a.3041.2		4	65.8	even	4
3042.3.d.a.3041.3		4	65.18	even	4
3042.3.d.a.3041.4		4	195.8	odd	4
3600.3.c.b.449.1		4	20.19	odd	2
3600.3.c.b.449.2		4	60.59	even	2
3600.3.c.b.449.3		4	4.3	odd	2
3600.3.c.b.449.4		4	12.11	even	2
3600.3.l.d.1601.1		2	20.7	even	4
3600.3.l.d.1601.2		2	60.47	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -4.00000i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -4.00000i q^{7} -2.82843 q^{8} +16.9706i q^{11} -8.00000i q^{13} +5.65685i q^{14} +4.00000 q^{16} +12.7279 q^{17} +16.0000 q^{19} -24.0000i q^{22} -16.9706 q^{23} +11.3137i q^{26} -8.00000i q^{28} -4.24264i q^{29} +44.0000 q^{31} -5.65685 q^{32} -18.0000 q^{34} -34.0000i q^{37} -22.6274 q^{38} +46.6690i q^{41} +40.0000i q^{43} +33.9411i q^{44} +24.0000 q^{46} +84.8528 q^{47} +33.0000 q^{49} -16.0000i q^{52} +38.1838 q^{53} +11.3137i q^{56} +6.00000i q^{58} -33.9411i q^{59} +50.0000 q^{61} -62.2254 q^{62} +8.00000 q^{64} +8.00000i q^{67} +25.4558 q^{68} -50.9117i q^{71} +16.0000i q^{73} +48.0833i q^{74} +32.0000 q^{76} +67.8823 q^{77} +76.0000 q^{79} -66.0000i q^{82} +118.794 q^{83} -56.5685i q^{86} -48.0000i q^{88} -12.7279i q^{89} -32.0000 q^{91} -33.9411 q^{92} -120.000 q^{94} +176.000i q^{97} -46.6690 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 16 q^{16} + 64 q^{19} + 176 q^{31} - 72 q^{34} + 96 q^{46} + 132 q^{49} + 200 q^{61} + 32 q^{64} + 128 q^{76} + 304 q^{79} - 128 q^{91} - 480 q^{94}+O(q^{100}) \) 4 * q + 8 * q^4 + 16 * q^16 + 64 * q^19 + 176 * q^31 - 72 * q^34 + 96 * q^46 + 132 * q^49 + 200 * q^61 + 32 * q^64 + 128 * q^76 + 304 * q^79 - 128 * q^91 - 480 * q^94

Introduction

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