LMFDB - Embedded newform 3600.3.c.b.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(449,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3600.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.c (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:	$98.0928951697$
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}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3600.449
Dual form			3600.3.c.b.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{7} +O(q^{10})$	$q+4.00000i q^{7} +16.9706i q^{11} -8.00000i q^{13} -12.7279 q^{17} -16.0000 q^{19} -16.9706 q^{23} +4.24264i q^{29} -44.0000 q^{31} -34.0000i q^{37} -46.6690i q^{41} -40.0000i q^{43} +84.8528 q^{47} +33.0000 q^{49} -38.1838 q^{53} -33.9411i q^{59} +50.0000 q^{61} -8.00000i q^{67} -50.9117i q^{71} +16.0000i q^{73} -67.8823 q^{77} -76.0000 q^{79} +118.794 q^{83} +12.7279i q^{89} +32.0000 q^{91} +176.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 64 q^{19} - 176 q^{31} + 132 q^{49} + 200 q^{61} - 304 q^{79} + 128 q^{91}+O(q^{100}) $ 4 * q - 64 * q^19 - 176 * q^31 + 132 * q^49 + 200 * q^61 - 304 * q^79 + 128 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000i	0.571429i	0.958315	+	0.285714i	$0.0922308\pi$
$7$	4.00000i	0.571429i	−0.958315	+	0.285714i	$0.907769\pi$
$8$	0	0
$9$	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	$-0.0995567\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−12.7279	−0.748701	−0.374351	−	0.927287i	$-0.622134\pi$
$17$	−12.7279	−0.748701	−0.374351	+	0.927287i	$0.622134\pi$
$18$	0	0
$19$	−16.0000	−0.842105	−0.421053	−	0.907036i	$-0.638339\pi$
$19$	−16.0000	−0.842105	−0.421053	+	0.907036i	$0.638339\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−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
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.24264i	0.146298i	0.997321	+	0.0731490i	$0.0233049\pi$
$29$	4.24264i	0.146298i	−0.997321	+	0.0731490i	$0.976695\pi$
$30$	0	0
$31$	−44.0000	−1.41935	−0.709677	−	0.704527i	$-0.751159\pi$
$31$	−44.0000	−1.41935	−0.709677	+	0.704527i	$0.751159\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	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	$-0.151957\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 46.6690i	− 1.13827i	−0.822244	−	0.569135i	$-0.807278\pi$
$41$	− 46.6690i	− 1.13827i	0.822244	−	0.569135i	$-0.192722\pi$
$42$	0	0
$43$	− 40.0000i	− 0.930233i	−0.885250	−	0.465116i	$-0.846013\pi$
$43$	− 40.0000i	− 0.930233i	0.885250	−	0.465116i	$-0.153987\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	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
$52$	0	0
$53$	−38.1838	−0.720448	−0.360224	−	0.932866i	$-0.617300\pi$
$53$	−38.1838	−0.720448	−0.360224	+	0.932866i	$0.617300\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 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	$0.365586\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.119403i	−0.998216	−	0.0597015i	$-0.980985\pi$
$67$	− 8.00000i	− 0.119403i	0.998216	−	0.0597015i	$-0.0190149\pi$
$68$	0	0
$69$	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	$0.965047\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−67.8823	−0.881588
$78$	0	0
$79$	−76.0000	−0.962025	−0.481013	−	0.876714i	$-0.659731\pi$
$79$	−76.0000	−0.962025	−0.481013	+	0.876714i	$0.659731\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	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
$86$	0	0
$87$	0	0
$88$	0	0
$89$	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
$91$	32.0000	0.351648
$92$	0	0
$93$	0	0
$94$	0	0
$95$	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	$0.638203\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 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
$103$	− 28.0000i	− 0.271845i	−0.990719	−	0.135922i	$-0.956600\pi$
$103$	− 28.0000i	− 0.271845i	0.990719	−	0.135922i	$-0.0433998\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	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
$112$	0	0
$113$	156.978	1.38918	0.694592	−	0.719404i	$-0.255585\pi$
$113$	156.978	1.38918	0.694592	+	0.719404i	$0.255585\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 50.9117i	− 0.427829i
$120$	0	0
$121$	−167.000	−1.38017
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 92.0000i	− 0.724409i	−0.932099	−	0.362205i	$-0.882024\pi$
$127$	− 92.0000i	− 0.724409i	0.932099	−	0.362205i	$-0.117976\pi$
$128$	0	0
$129$	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
$134$	0	0
$135$	0	0
$136$	0	0
$137$	156.978	1.14582	0.572911	−	0.819617i	$-0.305814\pi$
$137$	156.978	1.14582	0.572911	+	0.819617i	$0.305814\pi$
$138$	0	0
$139$	152.000	1.09353	0.546763	−	0.837288i	$-0.315860\pi$
$139$	152.000	1.09353	0.546763	+	0.837288i	$0.315860\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	135.765	0.949402
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	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	$0.336973\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	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	$-0.0841006\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 67.8823i	− 0.421629i
$162$	0	0
$163$	56.0000i	0.343558i	0.985135	+	0.171779i	$0.0549515\pi$
$163$	56.0000i	0.343558i	−0.985135	+	0.171779i	$0.945048\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−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
$172$	0	0
$173$	173.948	1.00548	0.502741	−	0.864437i	$-0.332325\pi$
$173$	173.948	1.00548	0.502741	+	0.864437i	$0.332325\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	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
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 216.000i	− 1.15508i
$188$	0	0
$189$	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	$-0.179191\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	165.463	0.839914	0.419957	−	0.907544i	$-0.362045\pi$
$197$	165.463	0.839914	0.419957	+	0.907544i	$0.362045\pi$
$198$	0	0
$199$	20.0000	0.100503	0.0502513	−	0.998737i	$-0.483998\pi$
$199$	20.0000	0.100503	0.0502513	+	0.998737i	$0.483998\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−16.9706	−0.0835988
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 271.529i	− 1.29918i
$210$	0	0
$211$	−296.000	−1.40284	−0.701422	−	0.712746i	$-0.747451\pi$
$211$	−296.000	−1.40284	−0.701422	+	0.712746i	$0.747451\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 176.000i	− 0.811060i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	101.823i	0.460739i
$222$	0	0
$223$	− 436.000i	− 1.95516i	−0.210571	−	0.977578i	$-0.567532\pi$
$223$	− 436.000i	− 1.95516i	0.210571	−	0.977578i	$-0.432468\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−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
$232$	0	0
$233$	12.7279	0.0546263	0.0273131	−	0.999627i	$-0.491305\pi$
$233$	12.7279	0.0546263	0.0273131	+	0.999627i	$0.491305\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	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	$-0.478852\pi$
$241$	32.0000	0.132780	0.0663900	+	0.997794i	$0.478852\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	128.000i	0.518219i
$248$	0	0
$249$	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
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−182.434	−0.709858	−0.354929	−	0.934893i	$-0.615495\pi$
$257$	−182.434	−0.709858	−0.354929	+	0.934893i	$0.615495\pi$
$258$	0	0
$259$	136.000	0.525097
$260$	0	0
$261$	0	0
$262$	0	0
$263$	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
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 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	$0.747310\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	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	$-0.201684\pi$
$278$	0	0
$279$	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
$283$	− 208.000i	− 0.734982i	−0.930027	−	0.367491i	$-0.880217\pi$
$283$	− 208.000i	− 0.734982i	0.930027	−	0.367491i	$-0.119783\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	186.676	0.650440
$288$	0	0
$289$	−127.000	−0.439446
$290$	0	0
$291$	0	0
$292$	0	0
$293$	436.992	1.49144	0.745720	−	0.666259i	$-0.232106\pi$
$293$	436.992	1.49144	0.745720	+	0.666259i	$0.232106\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	135.765i	0.454062i
$300$	0	0
$301$	160.000	0.531561
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	520.000i	1.69381i	0.531743	+	0.846906i	$0.321537\pi$
$307$	520.000i	1.69381i	−0.531743	+	0.846906i	$0.678463\pi$
$308$	0	0
$309$	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	$0.952021\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	335.169	1.05731	0.528657	−	0.848835i	$-0.322696\pi$
$317$	335.169	1.05731	0.528657	+	0.848835i	$0.322696\pi$
$318$	0	0
$319$	−72.0000	−0.225705
$320$	0	0
$321$	0	0
$322$	0	0
$323$	203.647	0.630485
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	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	$0.800353\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	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	$-0.0998622\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 746.705i	− 2.18975i
$342$	0	0
$343$	328.000i	0.956268i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−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
$352$	0	0
$353$	224.860	0.636997	0.318499	−	0.947923i	$-0.396821\pi$
$353$	224.860	0.636997	0.318499	+	0.947923i	$0.396821\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	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
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 284.000i	− 0.773842i	−0.922113	−	0.386921i	$-0.873539\pi$
$367$	− 284.000i	− 0.773842i	0.922113	−	0.386921i	$-0.126461\pi$
$368$	0	0
$369$	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	$0.918026\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	33.9411	0.0900295
$378$	0	0
$379$	−160.000	−0.422164	−0.211082	−	0.977468i	$-0.567699\pi$
$379$	−160.000	−0.422164	−0.211082	+	0.977468i	$0.567699\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−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
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 403.051i	− 1.03612i	−0.855344	−	0.518060i	$-0.826654\pi$
$389$	− 403.051i	− 1.03612i	0.855344	−	0.518060i	$-0.173346\pi$
$390$	0	0
$391$	216.000	0.552430
$392$	0	0
$393$	0	0
$394$	0	0
$395$	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	$0.941135\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 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
$403$	352.000i	0.873449i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	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
$412$	0	0
$413$	135.765	0.328728
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 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	$0.515127\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	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	$-0.215254\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	271.529	0.621348
$438$	0	0
$439$	−4.00000	−0.00911162	−0.00455581	−	0.999990i	$-0.501450\pi$
$439$	−4.00000	−0.00911162	−0.00455581	+	0.999990i	$0.501450\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	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
$446$	0	0
$447$	0	0
$448$	0	0
$449$	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
$452$	0	0
$453$	0	0
$454$	0	0
$455$	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	$-0.144184\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	301.227i	0.653422i	0.945124	+	0.326711i	$0.105940\pi$
$461$	301.227i	0.653422i	−0.945124	+	0.326711i	$0.894060\pi$
$462$	0	0
$463$	− 604.000i	− 1.30454i	−0.757989	−	0.652268i	$-0.773818\pi$
$463$	− 604.000i	− 1.30454i	0.757989	−	0.652268i	$-0.226182\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−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
$472$	0	0
$473$	678.823	1.43514
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 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
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 596.000i	− 1.22382i	−0.790928	−	0.611910i	$-0.790402\pi$
$487$	− 596.000i	− 1.22382i	0.790928	−	0.611910i	$-0.209598\pi$
$488$	0	0
$489$	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
$494$	0	0
$495$	0	0
$496$	0	0
$497$	203.647	0.409752
$498$	0	0
$499$	224.000	0.448898	0.224449	−	0.974486i	$-0.427942\pi$
$499$	224.000	0.448898	0.224449	+	0.974486i	$0.427942\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	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
$506$	0	0
$507$	0	0
$508$	0	0
$509$	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
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1440.00i	2.78530i
$518$	0	0
$519$	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
$523$	− 736.000i	− 1.40727i	−0.710564	−	0.703633i	$-0.751560\pi$
$523$	− 736.000i	− 1.40727i	0.710564	−	0.703633i	$-0.248440\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	560.029	1.06267
$528$	0	0
$529$	−241.000	−0.455577
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−373.352	−0.700474
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	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	$0.768394\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 536.000i	− 0.979890i	−0.871753	−	0.489945i	$-0.837017\pi$
$547$	− 536.000i	− 0.979890i	0.871753	−	0.489945i	$-0.162983\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 67.8823i	− 0.123198i
$552$	0	0
$553$	− 304.000i	− 0.549729i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−165.463	−0.297061	−0.148531	−	0.988908i	$-0.547454\pi$
$557$	−165.463	−0.297061	−0.148531	+	0.988908i	$0.547454\pi$
$558$	0	0
$559$	−320.000	−0.572451
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−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
$566$	0	0
$567$	0	0
$568$	0	0
$569$	156.978i	0.275883i	0.990440	+	0.137942i	$0.0440487\pi$
$569$	156.978i	0.275883i	−0.990440	+	0.137942i	$0.955951\pi$
$570$	0	0
$571$	−368.000	−0.644483	−0.322242	−	0.946657i	$-0.604436\pi$
$571$	−368.000	−0.644483	−0.322242	+	0.946657i	$0.604436\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	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	$-0.0392676\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	475.176i	0.817858i
$582$	0	0
$583$	− 648.000i	− 1.11149i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−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
$592$	0	0
$593$	−1107.33	−1.86733	−0.933667	−	0.358142i	$-0.883410\pi$
$593$	−1107.33	−1.86733	−0.933667	+	0.358142i	$0.883410\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	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	$0.458037\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 332.000i	− 0.546952i	−0.961879	−	0.273476i	$-0.911827\pi$
$607$	− 332.000i	− 0.546952i	0.961879	−	0.273476i	$-0.0881734\pi$
$608$	0	0
$609$	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	$-0.156270\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−55.1543	−0.0893911	−0.0446956	−	0.999001i	$-0.514232\pi$
$617$	−55.1543	−0.0893911	−0.0446956	+	0.999001i	$0.514232\pi$
$618$	0	0
$619$	896.000	1.44750	0.723748	−	0.690064i	$-0.242418\pi$
$619$	896.000	1.44750	0.723748	+	0.690064i	$0.242418\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−50.9117	−0.0817202
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	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	$0.505045\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 264.000i	− 0.414443i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 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
$643$	728.000i	1.13219i	0.824339	+	0.566096i	$0.191547\pi$
$643$	728.000i	1.13219i	−0.824339	+	0.566096i	$0.808453\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	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
$652$	0	0
$653$	−301.227	−0.461298	−0.230649	−	0.973037i	$-0.574085\pi$
$653$	−301.227	−0.461298	−0.230649	+	0.973037i	$0.574085\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	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	$-0.485066\pi$
$661$	62.0000	0.0937973	0.0468986	+	0.998900i	$0.485066\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 72.0000i	− 0.107946i
$668$	0	0
$669$	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	$0.834152\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1294.01	−1.91138	−0.955691	−	0.294372i	$-0.904889\pi$
$677$	−1294.01	−1.91138	−0.955691	+	0.294372i	$0.904889\pi$
$678$	0	0
$679$	−704.000	−1.03682
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−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
$686$	0	0
$687$	0	0
$688$	0	0
$689$	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	$0.490786\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	594.000i	0.852224i
$698$	0	0
$699$	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
$703$	544.000i	0.773826i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	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
$712$	0	0
$713$	746.705	1.04727
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 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
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	508.000i	0.698762i	0.936981	+	0.349381i	$0.113608\pi$
$727$	508.000i	0.698762i	−0.936981	+	0.349381i	$0.886392\pi$
$728$	0	0
$729$	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	$0.715039\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	135.765	0.184212
$738$	0	0
$739$	−304.000	−0.411367	−0.205683	−	0.978619i	$-0.565942\pi$
$739$	−304.000	−0.411367	−0.205683	+	0.978619i	$0.565942\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	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
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−188.000	−0.250333	−0.125166	−	0.992136i	$-0.539946\pi$
$751$	−188.000	−0.250333	−0.125166	+	0.992136i	$0.539946\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	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	$-0.305484\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	156.978i	0.206278i	0.994667	+	0.103139i	$0.0328887\pi$
$761$	156.978i	0.206278i	−0.994667	+	0.103139i	$0.967111\pi$
$762$	0	0
$763$	− 224.000i	− 0.293578i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−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
$772$	0	0
$773$	1387.34	1.79475	0.897376	−	0.441266i	$-0.145471\pi$
$773$	1387.34	1.79475	0.897376	+	0.441266i	$0.145471\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	746.705i	0.958543i
$780$	0	0
$781$	864.000	1.10627
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1360.00i	1.72808i	0.503422	+	0.864041i	$0.332074\pi$
$787$	1360.00i	1.72808i	−0.503422	+	0.864041i	$0.667926\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	627.911i	0.793819i
$792$	0	0
$793$	− 400.000i	− 0.504414i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−106.066	−0.133082	−0.0665408	−	0.997784i	$-0.521196\pi$
$797$	−106.066	−0.133082	−0.0665408	+	0.997784i	$0.521196\pi$
$798$	0	0
$799$	−1080.00	−1.35169
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−271.529	−0.338143
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1107.33i	− 1.36876i	−0.729124	−	0.684381i	$-0.760072\pi$
$809$	− 1107.33i	− 1.36876i	0.729124	−	0.684381i	$-0.239928\pi$
$810$	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
$814$	0	0
$815$	0	0
$816$	0	0
$817$	640.000i	0.783354i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 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
$823$	332.000i	0.403402i	0.979447	+	0.201701i	$0.0646470\pi$
$823$	332.000i	0.403402i	−0.979447	+	0.201701i	$0.935353\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−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
$832$	0	0
$833$	−420.021	−0.504227
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 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
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 668.000i	− 0.788666i
$848$	0	0
$849$	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	$-0.0841941\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−428.507	−0.500008	−0.250004	−	0.968245i	$-0.580432\pi$
$857$	−428.507	−0.500008	−0.250004	+	0.968245i	$0.580432\pi$
$858$	0	0
$859$	728.000	0.847497	0.423749	−	0.905780i	$-0.360714\pi$
$859$	728.000	0.847497	0.423749	+	0.905780i	$0.360714\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−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
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1289.76i	− 1.48419i
$870$	0	0
$871$	−64.0000	−0.0734788
$872$	0	0
$873$	0	0
$874$	0	0
$875$	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	$-0.173626\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	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
$883$	1064.00i	1.20498i	0.798125	+	0.602492i	$0.205825\pi$
$883$	1064.00i	1.20498i	−0.798125	+	0.602492i	$0.794175\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	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
$892$	0	0
$893$	−1357.65	−1.52032
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 186.676i	− 0.207649i
$900$	0	0
$901$	486.000	0.539401
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1768.00i	1.94928i	0.223771	+	0.974642i	$0.428163\pi$
$907$	1768.00i	1.94928i	−0.223771	+	0.974642i	$0.571837\pi$
$908$	0	0
$909$	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
$914$	0	0
$915$	0	0
$916$	0	0
$917$	678.823	0.740264
$918$	0	0
$919$	380.000	0.413493	0.206746	−	0.978395i	$-0.433712\pi$
$919$	380.000	0.413493	0.206746	+	0.978395i	$0.433712\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−407.294	−0.441271
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	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
$932$	0	0
$933$	0	0
$934$	0	0
$935$	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	$-0.0302800\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	436.992i	0.464391i	0.972669	+	0.232196i	$0.0745909\pi$
$941$	436.992i	0.464391i	−0.972669	+	0.232196i	$0.925409\pi$
$942$	0	0
$943$	792.000i	0.839873i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−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
$952$	0	0
$953$	−1310.98	−1.37563	−0.687815	−	0.725886i	$-0.741430\pi$
$953$	−1310.98	−1.37563	−0.687815	+	0.725886i	$0.741430\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	627.911i	0.654756i
$960$	0	0
$961$	975.000	1.01457
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1700.00i	− 1.75801i	−0.476808	−	0.879007i	$-0.658206\pi$
$967$	− 1700.00i	− 1.75801i	0.476808	−	0.879007i	$-0.341794\pi$
$968$	0	0
$969$	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
$974$	0	0
$975$	0	0
$976$	0	0
$977$	759.433	0.777311	0.388655	−	0.921383i	$-0.372940\pi$
$977$	759.433	0.777311	0.388655	+	0.921383i	$0.372940\pi$
$978$	0	0
$979$	−216.000	−0.220633
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−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
$986$	0	0
$987$	0	0
$988$	0	0
$989$	678.823i	0.686373i
$990$	0	0
$991$	772.000	0.779011	0.389506	−	0.921024i	$-0.372646\pi$
$991$	772.000	0.779011	0.389506	+	0.921024i	$0.372646\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	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	$0.968982\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.c.b.449.4		4
3.2	odd	2	inner	3600.3.c.b.449.3		4
4.3	odd	2		450.3.b.b.449.3		4
5.2	odd	4		3600.3.l.d.1601.2		2
5.3	odd	4		144.3.e.b.17.2		2
5.4	even	2	inner	3600.3.c.b.449.2		4
12.11	even	2		450.3.b.b.449.1		4
15.2	even	4		3600.3.l.d.1601.1		2
15.8	even	4		144.3.e.b.17.1		2
15.14	odd	2	inner	3600.3.c.b.449.1		4
20.3	even	4		18.3.b.a.17.1	✓	2
20.7	even	4		450.3.d.f.251.2		2
20.19	odd	2		450.3.b.b.449.2		4
40.3	even	4		576.3.e.c.449.1		2
40.13	odd	4		576.3.e.f.449.1		2
45.13	odd	12		1296.3.q.f.593.2		4
45.23	even	12		1296.3.q.f.593.1		4
45.38	even	12		1296.3.q.f.1025.2		4
45.43	odd	12		1296.3.q.f.1025.1		4
60.23	odd	4		18.3.b.a.17.2	yes	2
60.47	odd	4		450.3.d.f.251.1		2
60.59	even	2		450.3.b.b.449.4		4
80.3	even	4		2304.3.h.f.2177.4		4
80.13	odd	4		2304.3.h.c.2177.4		4
80.43	even	4		2304.3.h.f.2177.1		4
80.53	odd	4		2304.3.h.c.2177.1		4
120.53	even	4		576.3.e.f.449.2		2
120.83	odd	4		576.3.e.c.449.2		2
140.3	odd	12		882.3.s.d.863.2		4
140.23	even	12		882.3.s.b.557.1		4
140.83	odd	4		882.3.b.a.197.1		2
140.103	odd	12		882.3.s.d.557.1		4
140.123	even	12		882.3.s.b.863.2		4
180.23	odd	12		162.3.d.b.107.1		4
180.43	even	12		162.3.d.b.53.1		4
180.83	odd	12		162.3.d.b.53.2		4
180.103	even	12		162.3.d.b.107.2		4
220.43	odd	4		2178.3.c.d.485.2		2
240.53	even	4		2304.3.h.c.2177.3		4
240.83	odd	4		2304.3.h.f.2177.2		4
240.173	even	4		2304.3.h.c.2177.2		4
240.203	odd	4		2304.3.h.f.2177.3		4
260.83	odd	4		3042.3.d.a.3041.1		4
260.103	even	4		3042.3.c.e.1691.2		2
260.203	odd	4		3042.3.d.a.3041.4		4
420.23	odd	12		882.3.s.b.557.2		4
420.83	even	4		882.3.b.a.197.2		2
420.143	even	12		882.3.s.d.863.1		4
420.263	odd	12		882.3.s.b.863.1		4
420.383	even	12		882.3.s.d.557.2		4
660.263	even	4		2178.3.c.d.485.1		2
780.83	even	4		3042.3.d.a.3041.3		4
780.203	even	4		3042.3.d.a.3041.2		4
780.623	odd	4		3042.3.c.e.1691.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.b.a.17.1	✓	2	20.3	even	4
18.3.b.a.17.2	yes	2	60.23	odd	4
144.3.e.b.17.1		2	15.8	even	4
144.3.e.b.17.2		2	5.3	odd	4
162.3.d.b.53.1		4	180.43	even	12
162.3.d.b.53.2		4	180.83	odd	12
162.3.d.b.107.1		4	180.23	odd	12
162.3.d.b.107.2		4	180.103	even	12
450.3.b.b.449.1		4	12.11	even	2
450.3.b.b.449.2		4	20.19	odd	2
450.3.b.b.449.3		4	4.3	odd	2
450.3.b.b.449.4		4	60.59	even	2
450.3.d.f.251.1		2	60.47	odd	4
450.3.d.f.251.2		2	20.7	even	4
576.3.e.c.449.1		2	40.3	even	4
576.3.e.c.449.2		2	120.83	odd	4
576.3.e.f.449.1		2	40.13	odd	4
576.3.e.f.449.2		2	120.53	even	4
882.3.b.a.197.1		2	140.83	odd	4
882.3.b.a.197.2		2	420.83	even	4
882.3.s.b.557.1		4	140.23	even	12
882.3.s.b.557.2		4	420.23	odd	12
882.3.s.b.863.1		4	420.263	odd	12
882.3.s.b.863.2		4	140.123	even	12
882.3.s.d.557.1		4	140.103	odd	12
882.3.s.d.557.2		4	420.383	even	12
882.3.s.d.863.1		4	420.143	even	12
882.3.s.d.863.2		4	140.3	odd	12
1296.3.q.f.593.1		4	45.23	even	12
1296.3.q.f.593.2		4	45.13	odd	12
1296.3.q.f.1025.1		4	45.43	odd	12
1296.3.q.f.1025.2		4	45.38	even	12
2178.3.c.d.485.1		2	660.263	even	4
2178.3.c.d.485.2		2	220.43	odd	4
2304.3.h.c.2177.1		4	80.53	odd	4
2304.3.h.c.2177.2		4	240.173	even	4
2304.3.h.c.2177.3		4	240.53	even	4
2304.3.h.c.2177.4		4	80.13	odd	4
2304.3.h.f.2177.1		4	80.43	even	4
2304.3.h.f.2177.2		4	240.83	odd	4
2304.3.h.f.2177.3		4	240.203	odd	4
2304.3.h.f.2177.4		4	80.3	even	4
3042.3.c.e.1691.1		2	780.623	odd	4
3042.3.c.e.1691.2		2	260.103	even	4
3042.3.d.a.3041.1		4	260.83	odd	4
3042.3.d.a.3041.2		4	780.203	even	4
3042.3.d.a.3041.3		4	780.83	even	4
3042.3.d.a.3041.4		4	260.203	odd	4
3600.3.c.b.449.1		4	15.14	odd	2	inner
3600.3.c.b.449.2		4	5.4	even	2	inner
3600.3.c.b.449.3		4	3.2	odd	2	inner
3600.3.c.b.449.4		4	1.1	even	1	trivial
3600.3.l.d.1601.1		2	15.2	even	4
3600.3.l.d.1601.2		2	5.2	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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{7} +O(q^{10})\)	\(q+4.00000i q^{7} +16.9706i q^{11} -8.00000i q^{13} -12.7279 q^{17} -16.0000 q^{19} -16.9706 q^{23} +4.24264i q^{29} -44.0000 q^{31} -34.0000i q^{37} -46.6690i q^{41} -40.0000i q^{43} +84.8528 q^{47} +33.0000 q^{49} -38.1838 q^{53} -33.9411i q^{59} +50.0000 q^{61} -8.00000i q^{67} -50.9117i q^{71} +16.0000i q^{73} -67.8823 q^{77} -76.0000 q^{79} +118.794 q^{83} +12.7279i q^{89} +32.0000 q^{91} +176.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 64 q^{19} - 176 q^{31} + 132 q^{49} + 200 q^{61} - 304 q^{79} + 128 q^{91}+O(q^{100}) \) 4 * q - 64 * q^19 - 176 * q^31 + 132 * q^49 + 200 * q^61 - 304 * q^79 + 128 * q^91

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