LMFDB - Embedded newform 3600.3.j.n.1999.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(1999,3600)]

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([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("3600.1999");

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.j (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:	$8$
Coefficient field:	8.0.3317760000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 25x^{4} + 625 $ x^8 - 25*x^4 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1999.7
Root			$2.15988 + 0.578737i$ of defining polynomial
Character	$\chi$	$=$	3600.1999
Dual form			3600.3.j.n.1999.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{7} +O(q^{10})$	$q+5.19615 q^{7} +18.9737i q^{11} -13.0000i q^{13} -32.8634i q^{17} +15.5885i q^{19} +18.9737 q^{23} -32.8634 q^{29} -32.9090i q^{31} -46.0000i q^{37} -32.8634 q^{41} -12.1244 q^{43} -37.9473 q^{47} -22.0000 q^{49} +32.8634i q^{53} +37.9473i q^{59} +59.0000 q^{61} +39.8372 q^{67} -94.8683i q^{71} +26.0000i q^{73} +98.5901i q^{77} -138.564i q^{79} -113.842 q^{83} +131.453 q^{89} -67.5500i q^{91} -23.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 176 q^{49} + 472 q^{61}+O(q^{100}) $ 8 * q - 176 * q^49 + 472 * q^61

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
$7$	5.19615	0.742307	0.371154	−	0.928571i	$-0.378962\pi$
$7$	5.19615	0.742307	0.371154	+	0.928571i	$0.378962\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	18.9737i	1.72488i	0.506160	+	0.862439i	$0.331064\pi$
$11$	18.9737i	1.72488i	−0.506160	+	0.862439i	$0.668936\pi$
$12$	0	0
$13$	− 13.0000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	− 13.0000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 32.8634i	− 1.93314i	−0.256406	−	0.966569i	$-0.582538\pi$
$17$	− 32.8634i	− 1.93314i	0.256406	−	0.966569i	$-0.417462\pi$
$18$	0	0
$19$	15.5885i	0.820445i	0.911985	+	0.410223i	$0.134549\pi$
$19$	15.5885i	0.820445i	−0.911985	+	0.410223i	$0.865451\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	18.9737	0.824942	0.412471	−	0.910971i	$-0.364666\pi$
$23$	18.9737	0.824942	0.412471	+	0.910971i	$0.364666\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−32.8634	−1.13322	−0.566610	−	0.823986i	$-0.691745\pi$
$29$	−32.8634	−1.13322	−0.566610	+	0.823986i	$0.691745\pi$
$30$	0	0
$31$	− 32.9090i	− 1.06158i	−0.847504	−	0.530790i	$-0.821895\pi$
$31$	− 32.9090i	− 1.06158i	0.847504	−	0.530790i	$-0.178105\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 46.0000i	− 1.24324i	−0.783318	−	0.621622i	$-0.786474\pi$
$37$	− 46.0000i	− 1.24324i	0.783318	−	0.621622i	$-0.213526\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−32.8634	−0.801545	−0.400773	−	0.916178i	$-0.631258\pi$
$41$	−32.8634	−0.801545	−0.400773	+	0.916178i	$0.631258\pi$
$42$	0	0
$43$	−12.1244	−0.281962	−0.140981	−	0.990012i	$-0.545026\pi$
$43$	−12.1244	−0.281962	−0.140981	+	0.990012i	$0.545026\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−37.9473	−0.807390	−0.403695	−	0.914894i	$-0.632274\pi$
$47$	−37.9473	−0.807390	−0.403695	+	0.914894i	$0.632274\pi$
$48$	0	0
$49$	−22.0000	−0.448980
$50$	0	0
$51$	0	0
$52$	0	0
$53$	32.8634i	0.620063i	0.950726	+	0.310032i	$0.100340\pi$
$53$	32.8634i	0.620063i	−0.950726	+	0.310032i	$0.899660\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	37.9473i	0.643175i	0.946880	+	0.321588i	$0.104216\pi$
$59$	37.9473i	0.643175i	−0.946880	+	0.321588i	$0.895784\pi$
$60$	0	0
$61$	59.0000	0.967213	0.483607	−	0.875285i	$-0.339327\pi$
$61$	59.0000	0.967213	0.483607	+	0.875285i	$0.339327\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	39.8372	0.594585	0.297292	−	0.954787i	$-0.403916\pi$
$67$	39.8372	0.594585	0.297292	+	0.954787i	$0.403916\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 94.8683i	− 1.33617i	−0.744083	−	0.668087i	$-0.767113\pi$
$71$	− 94.8683i	− 1.33617i	0.744083	−	0.668087i	$-0.232887\pi$
$72$	0	0
$73$	26.0000i	0.356164i	0.984016	+	0.178082i	$0.0569893\pi$
$73$	26.0000i	0.356164i	−0.984016	+	0.178082i	$0.943011\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	98.5901i	1.28039i
$78$	0	0
$79$	− 138.564i	− 1.75398i	−0.480513	−	0.876988i	$-0.659549\pi$
$79$	− 138.564i	− 1.75398i	0.480513	−	0.876988i	$-0.340451\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−113.842	−1.37159	−0.685795	−	0.727795i	$-0.740545\pi$
$83$	−113.842	−1.37159	−0.685795	+	0.727795i	$0.740545\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	131.453	1.47700	0.738502	−	0.674251i	$-0.235533\pi$
$89$	131.453	1.47700	0.738502	+	0.674251i	$0.235533\pi$
$90$	0	0
$91$	− 67.5500i	− 0.742307i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 23.0000i	− 0.237113i	−0.992947	−	0.118557i	$-0.962173\pi$
$97$	− 23.0000i	− 0.237113i	0.992947	−	0.118557i	$-0.0378267\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	98.5901	0.976139	0.488070	−	0.872805i	$-0.337701\pi$
$101$	98.5901	0.976139	0.488070	+	0.872805i	$0.337701\pi$
$102$	0	0
$103$	−138.564	−1.34528	−0.672641	−	0.739969i	$-0.734840\pi$
$103$	−138.564	−1.34528	−0.672641	+	0.739969i	$0.734840\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	132.816	1.24127	0.620634	−	0.784100i	$-0.286875\pi$
$107$	132.816	1.24127	0.620634	+	0.784100i	$0.286875\pi$
$108$	0	0
$109$	−109.000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−109.000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 131.453i	− 1.16330i	−0.813438	−	0.581652i	$-0.802406\pi$
$113$	− 131.453i	− 1.16330i	0.813438	−	0.581652i	$-0.197594\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 170.763i	− 1.43498i
$120$	0	0
$121$	−239.000	−1.97521
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	207.846	1.63658	0.818292	−	0.574803i	$-0.194921\pi$
$127$	207.846	1.63658	0.818292	+	0.574803i	$0.194921\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	132.816i	1.01386i	0.861987	+	0.506930i	$0.169220\pi$
$131$	132.816i	1.01386i	−0.861987	+	0.506930i	$0.830780\pi$
$132$	0	0
$133$	81.0000i	0.609023i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	65.7267i	0.479757i	0.970803	+	0.239878i	$0.0771076\pi$
$137$	65.7267i	0.479757i	−0.970803	+	0.239878i	$0.922892\pi$
$138$	0	0
$139$	− 69.2820i	− 0.498432i	−0.968448	−	0.249216i	$-0.919827\pi$
$139$	− 69.2820i	− 0.498432i	0.968448	−	0.249216i	$-0.0801729\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	246.658	1.72488
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−65.7267	−0.441119	−0.220559	−	0.975374i	$-0.570788\pi$
$149$	−65.7267	−0.441119	−0.220559	+	0.975374i	$0.570788\pi$
$150$	0	0
$151$	− 36.3731i	− 0.240881i	−0.992721	−	0.120441i	$-0.961569\pi$
$151$	− 36.3731i	− 0.240881i	0.992721	−	0.120441i	$-0.0384307\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 157.000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$157$	− 157.000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	98.5901	0.612361
$162$	0	0
$163$	81.4064	0.499426	0.249713	−	0.968320i	$-0.419664\pi$
$163$	81.4064	0.499426	0.249713	+	0.968320i	$0.419664\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.9737	0.113615	0.0568074	−	0.998385i	$-0.481908\pi$
$167$	18.9737	0.113615	0.0568074	+	0.998385i	$0.481908\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	32.8634i	0.189962i	0.995479	+	0.0949808i	$0.0302790\pi$
$173$	32.8634i	0.189962i	−0.995479	+	0.0949808i	$0.969721\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	151.789i	0.847985i	0.905666	+	0.423993i	$0.139372\pi$
$179$	151.789i	0.847985i	−0.905666	+	0.423993i	$0.860628\pi$
$180$	0	0
$181$	−179.000	−0.988950	−0.494475	−	0.869192i	$-0.664640\pi$
$181$	−179.000	−0.988950	−0.494475	+	0.869192i	$0.664640\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	623.538	3.33443
$188$	0	0
$189$	0	0
$190$	0	0
$191$	56.9210i	0.298016i	0.988836	+	0.149008i	$0.0476080\pi$
$191$	56.9210i	0.298016i	−0.988836	+	0.149008i	$0.952392\pi$
$192$	0	0
$193$	73.0000i	0.378238i	0.981954	+	0.189119i	$0.0605632\pi$
$193$	73.0000i	0.378238i	−0.981954	+	0.189119i	$0.939437\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 328.634i	− 1.66819i	−0.551620	−	0.834095i	$-0.685990\pi$
$197$	− 328.634i	− 1.66819i	0.551620	−	0.834095i	$-0.314010\pi$
$198$	0	0
$199$	− 226.899i	− 1.14019i	−0.821577	−	0.570097i	$-0.806906\pi$
$199$	− 226.899i	− 1.14019i	0.821577	−	0.570097i	$-0.193094\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−170.763	−0.841197
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−295.770	−1.41517
$210$	0	0
$211$	− 417.424i	− 1.97831i	−0.146862	−	0.989157i	$-0.546917\pi$
$211$	− 417.424i	− 1.97831i	0.146862	−	0.989157i	$-0.453083\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 171.000i	− 0.788018i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−427.224	−1.93314
$222$	0	0
$223$	−116.047	−0.520392	−0.260196	−	0.965556i	$-0.583787\pi$
$223$	−116.047	−0.520392	−0.260196	+	0.965556i	$0.583787\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−360.500	−1.58810	−0.794052	−	0.607850i	$-0.792032\pi$
$227$	−360.500	−1.58810	−0.794052	+	0.607850i	$0.792032\pi$
$228$	0	0
$229$	131.000	0.572052	0.286026	−	0.958222i	$-0.407666\pi$
$229$	131.000	0.572052	0.286026	+	0.958222i	$0.407666\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	427.224i	1.83358i	0.399372	+	0.916789i	$0.369228\pi$
$233$	427.224i	1.83358i	−0.399372	+	0.916789i	$0.630772\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	246.658i	1.03204i	0.856576	+	0.516020i	$0.172587\pi$
$239$	246.658i	1.03204i	−0.856576	+	0.516020i	$0.827413\pi$
$240$	0	0
$241$	−119.000	−0.493776	−0.246888	−	0.969044i	$-0.579408\pi$
$241$	−119.000	−0.493776	−0.246888	+	0.969044i	$0.579408\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	202.650	0.820445
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 265.631i	− 1.05829i	−0.848531	−	0.529146i	$-0.822512\pi$
$251$	− 265.631i	− 1.05829i	0.848531	−	0.529146i	$-0.177488\pi$
$252$	0	0
$253$	360.000i	1.42292i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 65.7267i	− 0.255746i	−0.991791	−	0.127873i	$-0.959185\pi$
$257$	− 65.7267i	− 0.255746i	0.991791	−	0.127873i	$-0.0408150\pi$
$258$	0	0
$259$	− 239.023i	− 0.922869i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−208.710	−0.793575	−0.396788	−	0.917910i	$-0.629875\pi$
$263$	−208.710	−0.793575	−0.396788	+	0.917910i	$0.629875\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	361.497	1.34385	0.671927	−	0.740617i	$-0.265467\pi$
$269$	361.497	1.34385	0.671927	+	0.740617i	$0.265467\pi$
$270$	0	0
$271$	− 277.128i	− 1.02261i	−0.859398	−	0.511307i	$-0.829162\pi$
$271$	− 277.128i	− 1.02261i	0.859398	−	0.511307i	$-0.170838\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 443.000i	− 1.59928i	−0.600481	−	0.799639i	$-0.705024\pi$
$277$	− 443.000i	− 1.59928i	0.600481	−	0.799639i	$-0.294976\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−164.317	−0.584757	−0.292379	−	0.956303i	$-0.594447\pi$
$281$	−164.317	−0.584757	−0.292379	+	0.956303i	$0.594447\pi$
$282$	0	0
$283$	−403.568	−1.42603	−0.713017	−	0.701146i	$-0.752672\pi$
$283$	−403.568	−1.42603	−0.713017	+	0.701146i	$0.752672\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−170.763	−0.594993
$288$	0	0
$289$	−791.000	−2.73702
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 394.360i	− 1.34594i	−0.739670	−	0.672970i	$-0.765018\pi$
$293$	− 394.360i	− 1.34594i	0.739670	−	0.672970i	$-0.234982\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 246.658i	− 0.824942i
$300$	0	0
$301$	−63.0000	−0.209302
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	445.137	1.44996	0.724979	−	0.688771i	$-0.241849\pi$
$307$	445.137	1.44996	0.724979	+	0.688771i	$0.241849\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	75.8947i	0.244034i	0.992528	+	0.122017i	$0.0389363\pi$
$311$	75.8947i	0.244034i	−0.992528	+	0.122017i	$0.961064\pi$
$312$	0	0
$313$	− 407.000i	− 1.30032i	−0.759798	−	0.650160i	$-0.774702\pi$
$313$	− 407.000i	− 1.30032i	0.759798	−	0.650160i	$-0.225298\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	197.180i	0.622019i	0.950407	+	0.311010i	$0.100667\pi$
$317$	197.180i	0.622019i	−0.950407	+	0.311010i	$0.899333\pi$
$318$	0	0
$319$	− 623.538i	− 1.95467i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	512.289	1.58603
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−197.180	−0.599332
$330$	0	0
$331$	− 138.564i	− 0.418623i	−0.977849	−	0.209311i	$-0.932878\pi$
$331$	− 138.564i	− 0.418623i	0.977849	−	0.209311i	$-0.0671222\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 383.000i	− 1.13650i	−0.822856	−	0.568249i	$-0.807621\pi$
$337$	− 383.000i	− 1.13650i	0.822856	−	0.568249i	$-0.192379\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	624.404	1.83110
$342$	0	0
$343$	−368.927	−1.07559
$344$	0	0
$345$	0	0
$346$	0	0
$347$	569.210	1.64037	0.820187	−	0.572095i	$-0.193869\pi$
$347$	569.210	1.64037	0.820187	+	0.572095i	$0.193869\pi$
$348$	0	0
$349$	−218.000	−0.624642	−0.312321	−	0.949977i	$-0.601106\pi$
$349$	−218.000	−0.624642	−0.312321	+	0.949977i	$0.601106\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 230.043i	− 0.651681i	−0.945425	−	0.325841i	$-0.894353\pi$
$353$	− 230.043i	− 0.651681i	0.945425	−	0.325841i	$-0.105647\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 208.710i	− 0.581366i	−0.956819	−	0.290683i	$-0.906118\pi$
$359$	− 208.710i	− 0.581366i	0.956819	−	0.290683i	$-0.0938825\pi$
$360$	0	0
$361$	118.000	0.326870
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	479.778	1.30730	0.653649	−	0.756798i	$-0.273237\pi$
$367$	479.778	1.30730	0.653649	+	0.756798i	$0.273237\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	170.763i	0.460278i
$372$	0	0
$373$	253.000i	0.678284i	0.940735	+	0.339142i	$0.110137\pi$
$373$	253.000i	0.678284i	−0.940735	+	0.339142i	$0.889863\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	427.224i	1.13322i
$378$	0	0
$379$	− 292.717i	− 0.772339i	−0.922428	−	0.386170i	$-0.873798\pi$
$379$	− 292.717i	− 0.772339i	0.922428	−	0.386170i	$-0.126202\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−417.421	−1.08987	−0.544936	−	0.838478i	$-0.683446\pi$
$383$	−417.421	−1.08987	−0.544936	+	0.838478i	$0.683446\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	230.043	0.591371	0.295686	−	0.955285i	$-0.404452\pi$
$389$	230.043	0.591371	0.295686	+	0.955285i	$0.404452\pi$
$390$	0	0
$391$	− 623.538i	− 1.59473i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 37.0000i	− 0.0931990i	−0.998914	−	0.0465995i	$-0.985162\pi$
$397$	− 37.0000i	− 0.0931990i	0.998914	−	0.0465995i	$-0.0148385\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	558.677	1.39321	0.696605	−	0.717455i	$-0.254693\pi$
$401$	558.677	1.39321	0.696605	+	0.717455i	$0.254693\pi$
$402$	0	0
$403$	−427.817	−1.06158
$404$	0	0
$405$	0	0
$406$	0	0
$407$	872.789	2.14444
$408$	0	0
$409$	−49.0000	−0.119804	−0.0599022	−	0.998204i	$-0.519079\pi$
$409$	−49.0000	−0.119804	−0.0599022	+	0.998204i	$0.519079\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	197.180i	0.477434i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 284.605i	− 0.679248i	−0.940561	−	0.339624i	$-0.889700\pi$
$419$	− 284.605i	− 0.679248i	0.940561	−	0.339624i	$-0.110300\pi$
$420$	0	0
$421$	−2.00000	−0.00475059	−0.00237530	−	0.999997i	$-0.500756\pi$
$421$	−2.00000	−0.00475059	−0.00237530	+	0.999997i	$0.500756\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	306.573	0.717970
$428$	0	0
$429$	0	0
$430$	0	0
$431$	322.552i	0.748381i	0.927352	+	0.374191i	$0.122079\pi$
$431$	322.552i	0.748381i	−0.927352	+	0.374191i	$0.877921\pi$
$432$	0	0
$433$	− 287.000i	− 0.662818i	−0.943487	−	0.331409i	$-0.892476\pi$
$433$	− 287.000i	− 0.662818i	0.943487	−	0.331409i	$-0.107524\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	295.770i	0.676820i
$438$	0	0
$439$	157.617i	0.359036i	0.983755	+	0.179518i	$0.0574537\pi$
$439$	157.617i	0.359036i	−0.983755	+	0.179518i	$0.942546\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	417.421	0.942259	0.471129	−	0.882064i	$-0.343846\pi$
$443$	417.421	0.942259	0.471129	+	0.882064i	$0.343846\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	328.634	0.731923	0.365962	−	0.930630i	$-0.380740\pi$
$449$	328.634	0.731923	0.365962	+	0.930630i	$0.380740\pi$
$450$	0	0
$451$	− 623.538i	− 1.38257i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 554.000i	− 1.21225i	−0.795368	−	0.606127i	$-0.792722\pi$
$457$	− 554.000i	− 1.21225i	0.795368	−	0.606127i	$-0.207278\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−230.043	−0.499010	−0.249505	−	0.968374i	$-0.580268\pi$
$461$	−230.043	−0.499010	−0.249505	+	0.968374i	$0.580268\pi$
$462$	0	0
$463$	277.128	0.598549	0.299274	−	0.954167i	$-0.403255\pi$
$463$	277.128	0.598549	0.299274	+	0.954167i	$0.403255\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−702.026	−1.50327	−0.751633	−	0.659581i	$-0.770734\pi$
$467$	−702.026	−1.50327	−0.751633	+	0.659581i	$0.770734\pi$
$468$	0	0
$469$	207.000	0.441365
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 230.043i	− 0.486350i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	796.894i	1.66366i	0.555029	+	0.831831i	$0.312707\pi$
$479$	796.894i	1.66366i	−0.555029	+	0.831831i	$0.687293\pi$
$480$	0	0
$481$	−598.000	−1.24324
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	895.470	1.83875	0.919374	−	0.393385i	$-0.128696\pi$
$487$	895.470	1.83875	0.919374	+	0.393385i	$0.128696\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 683.052i	− 1.39114i	−0.718456	−	0.695572i	$-0.755151\pi$
$491$	− 683.052i	− 1.39114i	0.718456	−	0.695572i	$-0.244849\pi$
$492$	0	0
$493$	1080.00i	2.19067i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 492.950i	− 0.991852i
$498$	0	0
$499$	− 261.540i	− 0.524128i	−0.965051	−	0.262064i	$-0.915597\pi$
$499$	− 261.540i	− 0.524128i	0.965051	−	0.262064i	$-0.0844031\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−739.973	−1.47112	−0.735560	−	0.677460i	$-0.763081\pi$
$503$	−739.973	−1.47112	−0.735560	+	0.677460i	$0.763081\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−591.540	−1.16216	−0.581081	−	0.813846i	$-0.697370\pi$
$509$	−591.540	−1.16216	−0.581081	+	0.813846i	$0.697370\pi$
$510$	0	0
$511$	135.100i	0.264383i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 720.000i	− 1.39265i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−953.037	−1.82925	−0.914623	−	0.404308i	$-0.867513\pi$
$521$	−953.037	−1.82925	−0.914623	+	0.404308i	$0.867513\pi$
$522$	0	0
$523$	774.227	1.48036	0.740178	−	0.672410i	$-0.234741\pi$
$523$	774.227	1.48036	0.740178	+	0.672410i	$0.234741\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1081.50	−2.05218
$528$	0	0
$529$	−169.000	−0.319471
$530$	0	0
$531$	0	0
$532$	0	0
$533$	427.224i	0.801545i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 417.421i	− 0.774435i
$540$	0	0
$541$	61.0000	0.112754	0.0563771	−	0.998410i	$-0.482045\pi$
$541$	61.0000	0.112754	0.0563771	+	0.998410i	$0.482045\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−69.2820	−0.126658	−0.0633291	−	0.997993i	$-0.520172\pi$
$547$	−69.2820	−0.126658	−0.0633291	+	0.997993i	$0.520172\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 512.289i	− 0.929744i
$552$	0	0
$553$	− 720.000i	− 1.30199i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 394.360i	− 0.708008i	−0.935244	−	0.354004i	$-0.884820\pi$
$557$	− 394.360i	− 0.708008i	0.935244	−	0.354004i	$-0.115180\pi$
$558$	0	0
$559$	157.617i	0.281962i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	208.710	0.370711	0.185356	−	0.982672i	$-0.440656\pi$
$563$	208.710	0.370711	0.185356	+	0.982672i	$0.440656\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	558.677	0.981858	0.490929	−	0.871200i	$-0.336658\pi$
$569$	558.677	0.981858	0.490929	+	0.871200i	$0.336658\pi$
$570$	0	0
$571$	691.088i	1.21031i	0.796107	+	0.605156i	$0.206889\pi$
$571$	691.088i	1.21031i	−0.796107	+	0.605156i	$0.793111\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	983.000i	1.70364i	0.523835	+	0.851820i	$0.324501\pi$
$577$	983.000i	1.70364i	−0.523835	+	0.851820i	$0.675499\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−591.540	−1.01814
$582$	0	0
$583$	−623.538	−1.06953
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−170.763	−0.290908	−0.145454	−	0.989365i	$-0.546464\pi$
$587$	−170.763	−0.290908	−0.145454	+	0.989365i	$0.546464\pi$
$588$	0	0
$589$	513.000	0.870968
$590$	0	0
$591$	0	0
$592$	0	0
$593$	98.5901i	0.166256i	0.996539	+	0.0831282i	$0.0264911\pi$
$593$	98.5901i	0.166256i	−0.996539	+	0.0831282i	$0.973509\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	113.842i	0.190053i	0.995475	+	0.0950267i	$0.0302937\pi$
$599$	113.842i	0.190053i	−0.995475	+	0.0950267i	$0.969706\pi$
$600$	0	0
$601$	−961.000	−1.59900	−0.799501	−	0.600665i	$-0.794903\pi$
$601$	−961.000	−1.59900	−0.799501	+	0.600665i	$0.794903\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	554.256	0.913108	0.456554	−	0.889696i	$-0.349084\pi$
$607$	554.256	0.913108	0.456554	+	0.889696i	$0.349084\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	493.315i	0.807390i
$612$	0	0
$613$	334.000i	0.544861i	0.962175	+	0.272431i	$0.0878275\pi$
$613$	334.000i	0.544861i	−0.962175	+	0.272431i	$0.912172\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 755.857i	− 1.22505i	−0.790450	−	0.612526i	$-0.790153\pi$
$617$	− 755.857i	− 1.22505i	0.790450	−	0.612526i	$-0.209847\pi$
$618$	0	0
$619$	− 400.104i	− 0.646371i	−0.946336	−	0.323186i	$-0.895246\pi$
$619$	− 400.104i	− 0.646371i	0.946336	−	0.323186i	$-0.104754\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	683.052	1.09639
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1511.71	−2.40336
$630$	0	0
$631$	− 517.883i	− 0.820734i	−0.911920	−	0.410367i	$-0.865401\pi$
$631$	− 517.883i	− 0.820734i	0.911920	−	0.410367i	$-0.134599\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	286.000i	0.448980i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	65.7267	0.102538	0.0512689	−	0.998685i	$-0.483673\pi$
$641$	65.7267	0.102538	0.0512689	+	0.998685i	$0.483673\pi$
$642$	0	0
$643$	−623.538	−0.969733	−0.484866	−	0.874588i	$-0.661132\pi$
$643$	−623.538	−0.969733	−0.484866	+	0.874588i	$0.661132\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	531.263	0.821117	0.410558	−	0.911834i	$-0.365334\pi$
$647$	531.263	0.821117	0.410558	+	0.911834i	$0.365334\pi$
$648$	0	0
$649$	−720.000	−1.10940
$650$	0	0
$651$	0	0
$652$	0	0
$653$	657.267i	1.00653i	0.864131	+	0.503267i	$0.167869\pi$
$653$	657.267i	1.00653i	−0.864131	+	0.503267i	$0.832131\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1081.50i	1.64112i	0.571559	+	0.820561i	$0.306339\pi$
$659$	1081.50i	1.64112i	−0.571559	+	0.820561i	$0.693661\pi$
$660$	0	0
$661$	482.000	0.729198	0.364599	−	0.931165i	$-0.381206\pi$
$661$	482.000	0.729198	0.364599	+	0.931165i	$0.381206\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−623.538	−0.934840
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1119.45i	1.66833i
$672$	0	0
$673$	214.000i	0.317979i	0.987280	+	0.158990i	$0.0508236\pi$
$673$	214.000i	0.317979i	−0.987280	+	0.158990i	$0.949176\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 32.8634i	− 0.0485426i	−0.999705	−	0.0242713i	$-0.992273\pi$
$677$	− 32.8634i	− 0.0485426i	0.999705	−	0.0242713i	$-0.00772656\pi$
$678$	0	0
$679$	− 119.512i	− 0.176011i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−758.947	−1.11120	−0.555598	−	0.831451i	$-0.687511\pi$
$683$	−758.947	−1.11120	−0.555598	+	0.831451i	$0.687511\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	427.224	0.620063
$690$	0	0
$691$	415.692i	0.601581i	0.953690	+	0.300790i	$0.0972504\pi$
$691$	415.692i	0.601581i	−0.953690	+	0.300790i	$0.902750\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1080.00i	1.54950i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1084.49	1.54706	0.773531	−	0.633758i	$-0.218489\pi$
$701$	1084.49	1.54706	0.773531	+	0.633758i	$0.218489\pi$
$702$	0	0
$703$	717.069	1.02001
$704$	0	0
$705$	0	0
$706$	0	0
$707$	512.289	0.724595
$708$	0	0
$709$	491.000	0.692525	0.346262	−	0.938138i	$-0.387451\pi$
$709$	491.000	0.692525	0.346262	+	0.938138i	$0.387451\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 624.404i	− 0.875742i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1005.60i	− 1.39862i	−0.714821	−	0.699308i	$-0.753492\pi$
$719$	− 1005.60i	− 1.39862i	0.714821	−	0.699308i	$-0.246508\pi$
$720$	0	0
$721$	−720.000	−0.998613
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−964.752	−1.32703	−0.663516	−	0.748162i	$-0.730937\pi$
$727$	−964.752	−1.32703	−0.663516	+	0.748162i	$0.730937\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	398.447i	0.545071i
$732$	0	0
$733$	26.0000i	0.0354707i	0.999843	+	0.0177353i	$0.00564563\pi$
$733$	26.0000i	0.0354707i	−0.999843	+	0.0177353i	$0.994354\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	755.857i	1.02559i
$738$	0	0
$739$	− 415.692i	− 0.562506i	−0.959634	−	0.281253i	$-0.909250\pi$
$739$	− 415.692i	− 0.562506i	0.959634	−	0.281253i	$-0.0907501\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1214.31	1.63434	0.817170	−	0.576397i	$-0.195542\pi$
$743$	1214.31	1.63434	0.817170	+	0.576397i	$0.195542\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	690.130	0.921402
$750$	0	0
$751$	1177.79i	1.56830i	0.620570	+	0.784151i	$0.286901\pi$
$751$	1177.79i	1.56830i	−0.620570	+	0.784151i	$0.713099\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 803.000i	− 1.06077i	−0.847758	−	0.530383i	$-0.822048\pi$
$757$	− 803.000i	− 1.06077i	0.847758	−	0.530383i	$-0.177952\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1183.08	1.55464	0.777320	−	0.629106i	$-0.216579\pi$
$761$	1183.08	1.55464	0.777320	+	0.629106i	$0.216579\pi$
$762$	0	0
$763$	−566.381	−0.742307
$764$	0	0
$765$	0	0
$766$	0	0
$767$	493.315	0.643175
$768$	0	0
$769$	−671.000	−0.872562	−0.436281	−	0.899811i	$-0.643705\pi$
$769$	−671.000	−0.872562	−0.436281	+	0.899811i	$0.643705\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1215.94i	− 1.57302i	−0.617578	−	0.786510i	$-0.711886\pi$
$773$	− 1215.94i	− 1.57302i	0.617578	−	0.786510i	$-0.288114\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 512.289i	− 0.657624i
$780$	0	0
$781$	1800.00	2.30474
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−524.811	−0.666851	−0.333425	−	0.942777i	$-0.608204\pi$
$787$	−524.811	−0.666851	−0.333425	+	0.942777i	$0.608204\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 683.052i	− 0.863530i
$792$	0	0
$793$	− 767.000i	− 0.967213i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	722.994i	0.907144i	0.891220	+	0.453572i	$0.149851\pi$
$797$	722.994i	0.907144i	−0.891220	+	0.453572i	$0.850149\pi$
$798$	0	0
$799$	1247.08i	1.56080i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−493.315	−0.614340
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−624.404	−0.771822	−0.385911	−	0.922536i	$-0.626113\pi$
$809$	−624.404	−0.771822	−0.385911	+	0.922536i	$0.626113\pi$
$810$	0	0
$811$	136.832i	0.168720i	0.996435	+	0.0843601i	$0.0268846\pi$
$811$	136.832i	0.168720i	−0.996435	+	0.0843601i	$0.973115\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 189.000i	− 0.231334i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−262.907	−0.320228	−0.160114	−	0.987099i	$-0.551186\pi$
$821$	−262.907	−0.320228	−0.160114	+	0.987099i	$0.551186\pi$
$822$	0	0
$823$	1477.44	1.79519	0.897594	−	0.440824i	$-0.145314\pi$
$823$	1477.44	1.79519	0.897594	+	0.440824i	$0.145314\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	664.078	0.802997	0.401498	−	0.915860i	$-0.368490\pi$
$827$	664.078	0.802997	0.401498	+	0.915860i	$0.368490\pi$
$828$	0	0
$829$	818.000	0.986731	0.493366	−	0.869822i	$-0.335767\pi$
$829$	818.000	0.986731	0.493366	+	0.869822i	$0.335767\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	722.994i	0.867940i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1005.60i	− 1.19857i	−0.800534	−	0.599287i	$-0.795451\pi$
$839$	− 1005.60i	− 1.19857i	0.800534	−	0.599287i	$-0.204549\pi$
$840$	0	0
$841$	239.000	0.284185
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1241.88	−1.46621
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 872.789i	− 1.02560i
$852$	0	0
$853$	1213.00i	1.42204i	0.703172	+	0.711020i	$0.251766\pi$
$853$	1213.00i	1.42204i	−0.703172	+	0.711020i	$0.748234\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	591.540i	0.690245i	0.938558	+	0.345123i	$0.112163\pi$
$857$	591.540i	0.690245i	−0.938558	+	0.345123i	$0.887837\pi$
$858$	0	0
$859$	554.256i	0.645234i	0.946530	+	0.322617i	$0.104563\pi$
$859$	554.256i	0.645234i	−0.946530	+	0.322617i	$0.895437\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−341.526	−0.395743	−0.197871	−	0.980228i	$-0.563403\pi$
$863$	−341.526	−0.395743	−0.197871	+	0.980228i	$0.563403\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2629.07	3.02540
$870$	0	0
$871$	− 517.883i	− 0.594585i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 37.0000i	− 0.0421893i	−0.999777	−	0.0210946i	$-0.993285\pi$
$877$	− 37.0000i	− 0.0421893i	0.999777	−	0.0210946i	$-0.00671513\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1511.71	−1.71591	−0.857954	−	0.513727i	$-0.828264\pi$
$881$	−1511.71	−1.71591	−0.857954	+	0.513727i	$0.828264\pi$
$882$	0	0
$883$	265.004	0.300118	0.150059	−	0.988677i	$-0.452054\pi$
$883$	265.004	0.300118	0.150059	+	0.988677i	$0.452054\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−872.789	−0.983978	−0.491989	−	0.870601i	$-0.663730\pi$
$887$	−872.789	−0.983978	−0.491989	+	0.870601i	$0.663730\pi$
$888$	0	0
$889$	1080.00	1.21485
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 591.540i	− 0.662419i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1081.50i	1.20300i
$900$	0	0
$901$	1080.00	1.19867
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−138.564	−0.152772	−0.0763859	−	0.997078i	$-0.524338\pi$
$907$	−138.564	−0.152772	−0.0763859	+	0.997078i	$0.524338\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1783.52i	− 1.95777i	−0.204422	−	0.978883i	$-0.565531\pi$
$911$	− 1783.52i	− 1.95777i	0.204422	−	0.978883i	$-0.434469\pi$
$912$	0	0
$913$	− 2160.00i	− 2.36583i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	690.130i	0.752596i
$918$	0	0
$919$	881.614i	0.959319i	0.877455	+	0.479659i	$0.159240\pi$
$919$	881.614i	0.959319i	−0.877455	+	0.479659i	$0.840760\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1233.29	−1.33617
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−821.584	−0.884374	−0.442187	−	0.896923i	$-0.645797\pi$
$929$	−821.584	−0.884374	−0.442187	+	0.896923i	$0.645797\pi$
$930$	0	0
$931$	− 342.946i	− 0.368363i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1703.00i	1.81750i	0.417338	+	0.908751i	$0.362963\pi$
$937$	1703.00i	1.81750i	−0.417338	+	0.908751i	$0.637037\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	262.907	0.279391	0.139695	−	0.990195i	$-0.455388\pi$
$941$	262.907	0.279391	0.139695	+	0.990195i	$0.455388\pi$
$942$	0	0
$943$	−623.538	−0.661228
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−626.131	−0.661173	−0.330587	−	0.943776i	$-0.607247\pi$
$947$	−626.131	−0.661173	−0.330587	+	0.943776i	$0.607247\pi$
$948$	0	0
$949$	338.000	0.356164
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 131.453i	− 0.137936i	−0.997619	−	0.0689682i	$-0.978029\pi$
$953$	− 131.453i	− 0.137936i	0.997619	−	0.0689682i	$-0.0219707\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	341.526i	0.356127i
$960$	0	0
$961$	−122.000	−0.126951
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1385.64	−1.43293	−0.716464	−	0.697624i	$-0.754240\pi$
$967$	−1385.64	−1.43293	−0.716464	+	0.697624i	$0.754240\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	986.631i	1.01610i	0.861328	+	0.508049i	$0.169633\pi$
$971$	986.631i	1.01610i	−0.861328	+	0.508049i	$0.830367\pi$
$972$	0	0
$973$	− 360.000i	− 0.369990i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	65.7267i	0.0672740i	0.999434	+	0.0336370i	$0.0107090\pi$
$977$	65.7267i	0.0672740i	−0.999434	+	0.0336370i	$0.989291\pi$
$978$	0	0
$979$	2494.15i	2.54765i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	151.789	0.154414	0.0772072	−	0.997015i	$-0.475400\pi$
$983$	151.789	0.154414	0.0772072	+	0.997015i	$0.475400\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−230.043	−0.232602
$990$	0	0
$991$	659.911i	0.665904i	0.942944	+	0.332952i	$0.108045\pi$
$991$	659.911i	0.665904i	−0.942944	+	0.332952i	$0.891955\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	554.000i	0.555667i	0.960629	+	0.277834i	$0.0896164\pi$
$997$	554.000i	0.555667i	−0.960629	+	0.277834i	$0.910384\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.j.n.1999.7		8
3.2	odd	2	inner	3600.3.j.n.1999.5		8
4.3	odd	2	inner	3600.3.j.n.1999.1		8
5.2	odd	4		3600.3.e.y.3151.4	yes	4
5.3	odd	4		3600.3.e.bi.3151.2	yes	4
5.4	even	2	inner	3600.3.j.n.1999.4		8
12.11	even	2	inner	3600.3.j.n.1999.3		8
15.2	even	4		3600.3.e.y.3151.3	yes	4
15.8	even	4		3600.3.e.bi.3151.1	yes	4
15.14	odd	2	inner	3600.3.j.n.1999.2		8
20.3	even	4		3600.3.e.bi.3151.3	yes	4
20.7	even	4		3600.3.e.y.3151.1	✓	4
20.19	odd	2	inner	3600.3.j.n.1999.6		8
60.23	odd	4		3600.3.e.bi.3151.4	yes	4
60.47	odd	4		3600.3.e.y.3151.2	yes	4
60.59	even	2	inner	3600.3.j.n.1999.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3600.3.e.y.3151.1	✓	4	20.7	even	4
3600.3.e.y.3151.2	yes	4	60.47	odd	4
3600.3.e.y.3151.3	yes	4	15.2	even	4
3600.3.e.y.3151.4	yes	4	5.2	odd	4
3600.3.e.bi.3151.1	yes	4	15.8	even	4
3600.3.e.bi.3151.2	yes	4	5.3	odd	4
3600.3.e.bi.3151.3	yes	4	20.3	even	4
3600.3.e.bi.3151.4	yes	4	60.23	odd	4
3600.3.j.n.1999.1		8	4.3	odd	2	inner
3600.3.j.n.1999.2		8	15.14	odd	2	inner
3600.3.j.n.1999.3		8	12.11	even	2	inner
3600.3.j.n.1999.4		8	5.4	even	2	inner
3600.3.j.n.1999.5		8	3.2	odd	2	inner
3600.3.j.n.1999.6		8	20.19	odd	2	inner
3600.3.j.n.1999.7		8	1.1	even	1	trivial
3600.3.j.n.1999.8		8	60.59	even	2	inner

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.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{7} +O(q^{10})\)	\(q+5.19615 q^{7} +18.9737i q^{11} -13.0000i q^{13} -32.8634i q^{17} +15.5885i q^{19} +18.9737 q^{23} -32.8634 q^{29} -32.9090i q^{31} -46.0000i q^{37} -32.8634 q^{41} -12.1244 q^{43} -37.9473 q^{47} -22.0000 q^{49} +32.8634i q^{53} +37.9473i q^{59} +59.0000 q^{61} +39.8372 q^{67} -94.8683i q^{71} +26.0000i q^{73} +98.5901i q^{77} -138.564i q^{79} -113.842 q^{83} +131.453 q^{89} -67.5500i q^{91} -23.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 176 q^{49} + 472 q^{61}+O(q^{100}) \) 8 * q - 176 * q^49 + 472 * q^61

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