LMFDB - Embedded newform 3600.3.e.bi.3151.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(3151,3600)]

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, 0]))

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

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

chi := DirichletCharacter("3600.3151");

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.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$98.0928951697$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} + 100 $ x^4 - 10*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3151.3
Root			$-2.73861 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	3600.3151
Dual form			3600.3.e.bi.3151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615i q^{7} +O(q^{10})$	$q+5.19615i q^{7} -18.9737i q^{11} +13.0000 q^{13} -32.8634 q^{17} +15.5885i q^{19} -18.9737i q^{23} +32.8634 q^{29} +32.9090i q^{31} -46.0000 q^{37} -32.8634 q^{41} +12.1244i q^{43} -37.9473i q^{47} +22.0000 q^{49} -32.8634 q^{53} +37.9473i q^{59} +59.0000 q^{61} +39.8372i q^{67} +94.8683i q^{71} -26.0000 q^{73} +98.5901 q^{77} -138.564i q^{79} +113.842i q^{83} -131.453 q^{89} +67.5500i q^{91} -23.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 52 q^{13} - 184 q^{37} + 88 q^{49} + 236 q^{61} - 104 q^{73} - 92 q^{97}+O(q^{100}) $ 4 * q + 52 * q^13 - 184 * q^37 + 88 * q^49 + 236 * q^61 - 104 * q^73 - 92 * q^97

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
$7$	5.19615i	0.742307i	0.928571	+	0.371154i	$0.121038\pi$
$7$	5.19615i	0.742307i	−0.928571	+	0.371154i	$0.878962\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 18.9737i	− 1.72488i	−0.506160	−	0.862439i	$-0.668936\pi$
$11$	− 18.9737i	− 1.72488i	0.506160	−	0.862439i	$-0.331064\pi$
$12$	0	0
$13$	13.0000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$13$	13.0000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−32.8634	−1.93314	−0.966569	−	0.256406i	$-0.917462\pi$
$17$	−32.8634	−1.93314	−0.966569	+	0.256406i	$0.917462\pi$
$18$	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
$23$	− 18.9737i	− 0.824942i	−0.910971	−	0.412471i	$-0.864666\pi$
$23$	− 18.9737i	− 0.824942i	0.910971	−	0.412471i	$-0.135334\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	32.8634	1.13322	0.566610	−	0.823986i	$-0.308255\pi$
$29$	32.8634	1.13322	0.566610	+	0.823986i	$0.308255\pi$
$30$	0	0
$31$	32.9090i	1.06158i	0.847504	+	0.530790i	$0.178105\pi$
$31$	32.9090i	1.06158i	−0.847504	+	0.530790i	$0.821895\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−46.0000	−1.24324	−0.621622	−	0.783318i	$-0.713526\pi$
$37$	−46.0000	−1.24324	−0.621622	+	0.783318i	$0.713526\pi$
$38$	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
$43$	12.1244i	0.281962i	0.990012	+	0.140981i	$0.0450256\pi$
$43$	12.1244i	0.281962i	−0.990012	+	0.140981i	$0.954974\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 37.9473i	− 0.807390i	−0.914894	−	0.403695i	$-0.867726\pi$
$47$	− 37.9473i	− 0.807390i	0.914894	−	0.403695i	$-0.132274\pi$
$48$	0	0
$49$	22.0000	0.448980
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−32.8634	−0.620063	−0.310032	−	0.950726i	$-0.600340\pi$
$53$	−32.8634	−0.620063	−0.310032	+	0.950726i	$0.600340\pi$
$54$	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
$67$	39.8372i	0.594585i	0.954787	+	0.297292i	$0.0960836\pi$
$67$	39.8372i	0.594585i	−0.954787	+	0.297292i	$0.903916\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	94.8683i	1.33617i	0.744083	+	0.668087i	$0.232887\pi$
$71$	94.8683i	1.33617i	−0.744083	+	0.668087i	$0.767113\pi$
$72$	0	0
$73$	−26.0000	−0.356164	−0.178082	−	0.984016i	$-0.556989\pi$
$73$	−26.0000	−0.356164	−0.178082	+	0.984016i	$0.556989\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	98.5901	1.28039
$78$	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
$83$	113.842i	1.37159i	0.727795	+	0.685795i	$0.240545\pi$
$83$	113.842i	1.37159i	−0.727795	+	0.685795i	$0.759455\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−131.453	−1.47700	−0.738502	−	0.674251i	$-0.764467\pi$
$89$	−131.453	−1.47700	−0.738502	+	0.674251i	$0.764467\pi$
$90$	0	0
$91$	67.5500i	0.742307i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−23.0000	−0.237113	−0.118557	−	0.992947i	$-0.537827\pi$
$97$	−23.0000	−0.237113	−0.118557	+	0.992947i	$0.537827\pi$
$98$	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
$103$	138.564i	1.34528i	0.739969	+	0.672641i	$0.234840\pi$
$103$	138.564i	1.34528i	−0.739969	+	0.672641i	$0.765160\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	132.816i	1.24127i	0.784100	+	0.620634i	$0.213125\pi$
$107$	132.816i	1.24127i	−0.784100	+	0.620634i	$0.786875\pi$
$108$	0	0
$109$	109.000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$109$	109.000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	131.453	1.16330	0.581652	−	0.813438i	$-0.302406\pi$
$113$	131.453	1.16330	0.581652	+	0.813438i	$0.302406\pi$
$114$	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
$127$	207.846i	1.63658i	0.574803	+	0.818292i	$0.305079\pi$
$127$	207.846i	1.63658i	−0.574803	+	0.818292i	$0.694921\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 132.816i	− 1.01386i	−0.861987	−	0.506930i	$-0.830780\pi$
$131$	− 132.816i	− 1.01386i	0.861987	−	0.506930i	$-0.169220\pi$
$132$	0	0
$133$	−81.0000	−0.609023
$134$	0	0
$135$	0	0
$136$	0	0
$137$	65.7267	0.479757	0.239878	−	0.970803i	$-0.422892\pi$
$137$	65.7267	0.479757	0.239878	+	0.970803i	$0.422892\pi$
$138$	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
$143$	− 246.658i	− 1.72488i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	65.7267	0.441119	0.220559	−	0.975374i	$-0.429212\pi$
$149$	65.7267	0.441119	0.220559	+	0.975374i	$0.429212\pi$
$150$	0	0
$151$	36.3731i	0.240881i	0.992721	+	0.120441i	$0.0384307\pi$
$151$	36.3731i	0.240881i	−0.992721	+	0.120441i	$0.961569\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−157.000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−157.000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	98.5901	0.612361
$162$	0	0
$163$	− 81.4064i	− 0.499426i	−0.968320	−	0.249713i	$-0.919664\pi$
$163$	− 81.4064i	− 0.499426i	0.968320	−	0.249713i	$-0.0803362\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.9737i	0.113615i	0.998385	+	0.0568074i	$0.0180921\pi$
$167$	18.9737i	0.113615i	−0.998385	+	0.0568074i	$0.981908\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−32.8634	−0.189962	−0.0949808	−	0.995479i	$-0.530279\pi$
$173$	−32.8634	−0.189962	−0.0949808	+	0.995479i	$0.530279\pi$
$174$	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
$187$	623.538i	3.33443i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 56.9210i	− 0.298016i	−0.988836	−	0.149008i	$-0.952392\pi$
$191$	− 56.9210i	− 0.298016i	0.988836	−	0.149008i	$-0.0476080\pi$
$192$	0	0
$193$	−73.0000	−0.378238	−0.189119	−	0.981954i	$-0.560563\pi$
$193$	−73.0000	−0.378238	−0.189119	+	0.981954i	$0.560563\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−328.634	−1.66819	−0.834095	−	0.551620i	$-0.814010\pi$
$197$	−328.634	−1.66819	−0.834095	+	0.551620i	$0.814010\pi$
$198$	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
$203$	170.763i	0.841197i
$204$	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	$0.453083\pi$
$211$	417.424i	1.97831i	−0.146862	+	0.989157i	$0.546917\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−171.000	−0.788018
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−427.224	−1.93314
$222$	0	0
$223$	116.047i	0.520392i	0.965556	+	0.260196i	$0.0837872\pi$
$223$	116.047i	0.520392i	−0.965556	+	0.260196i	$0.916213\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 360.500i	− 1.58810i	−0.607850	−	0.794052i	$-0.707968\pi$
$227$	− 360.500i	− 1.58810i	0.607850	−	0.794052i	$-0.292032\pi$
$228$	0	0
$229$	−131.000	−0.572052	−0.286026	−	0.958222i	$-0.592334\pi$
$229$	−131.000	−0.572052	−0.286026	+	0.958222i	$0.592334\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−427.224	−1.83358	−0.916789	−	0.399372i	$-0.869228\pi$
$233$	−427.224	−1.83358	−0.916789	+	0.399372i	$0.869228\pi$
$234$	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
$247$	202.650i	0.820445i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	265.631i	1.05829i	0.848531	+	0.529146i	$0.177488\pi$
$251$	265.631i	1.05829i	−0.848531	+	0.529146i	$0.822512\pi$
$252$	0	0
$253$	−360.000	−1.42292
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−65.7267	−0.255746	−0.127873	−	0.991791i	$-0.540815\pi$
$257$	−65.7267	−0.255746	−0.127873	+	0.991791i	$0.540815\pi$
$258$	0	0
$259$	− 239.023i	− 0.922869i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	208.710i	0.793575i	0.917910	+	0.396788i	$0.129875\pi$
$263$	208.710i	0.793575i	−0.917910	+	0.396788i	$0.870125\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−361.497	−1.34385	−0.671927	−	0.740617i	$-0.734533\pi$
$269$	−361.497	−1.34385	−0.671927	+	0.740617i	$0.734533\pi$
$270$	0	0
$271$	277.128i	1.02261i	0.859398	+	0.511307i	$0.170838\pi$
$271$	277.128i	1.02261i	−0.859398	+	0.511307i	$0.829162\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−443.000	−1.59928	−0.799639	−	0.600481i	$-0.794976\pi$
$277$	−443.000	−1.59928	−0.799639	+	0.600481i	$0.794976\pi$
$278$	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
$283$	403.568i	1.42603i	0.701146	+	0.713017i	$0.252672\pi$
$283$	403.568i	1.42603i	−0.701146	+	0.713017i	$0.747328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 170.763i	− 0.594993i
$288$	0	0
$289$	791.000	2.73702
$290$	0	0
$291$	0	0
$292$	0	0
$293$	394.360	1.34594	0.672970	−	0.739670i	$-0.265018\pi$
$293$	394.360	1.34594	0.672970	+	0.739670i	$0.265018\pi$
$294$	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
$307$	445.137i	1.44996i	0.688771	+	0.724979i	$0.258151\pi$
$307$	445.137i	1.44996i	−0.688771	+	0.724979i	$0.741849\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 75.8947i	− 0.244034i	−0.992528	−	0.122017i	$-0.961064\pi$
$311$	− 75.8947i	− 0.244034i	0.992528	−	0.122017i	$-0.0389363\pi$
$312$	0	0
$313$	407.000	1.30032	0.650160	−	0.759798i	$-0.274702\pi$
$313$	407.000	1.30032	0.650160	+	0.759798i	$0.274702\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	197.180	0.622019	0.311010	−	0.950407i	$-0.399333\pi$
$317$	197.180	0.622019	0.311010	+	0.950407i	$0.399333\pi$
$318$	0	0
$319$	− 623.538i	− 1.95467i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 512.289i	− 1.58603i
$324$	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	$0.0671222\pi$
$331$	138.564i	0.418623i	−0.977849	+	0.209311i	$0.932878\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−383.000	−1.13650	−0.568249	−	0.822856i	$-0.692379\pi$
$337$	−383.000	−1.13650	−0.568249	+	0.822856i	$0.692379\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	624.404	1.83110
$342$	0	0
$343$	368.927i	1.07559i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	569.210i	1.64037i	0.572095	+	0.820187i	$0.306131\pi$
$347$	569.210i	1.64037i	−0.572095	+	0.820187i	$0.693869\pi$
$348$	0	0
$349$	218.000	0.624642	0.312321	−	0.949977i	$-0.398894\pi$
$349$	218.000	0.624642	0.312321	+	0.949977i	$0.398894\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	230.043	0.651681	0.325841	−	0.945425i	$-0.394353\pi$
$353$	230.043	0.651681	0.325841	+	0.945425i	$0.394353\pi$
$354$	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
$367$	479.778i	1.30730i	0.756798	+	0.653649i	$0.226763\pi$
$367$	479.778i	1.30730i	−0.756798	+	0.653649i	$0.773237\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 170.763i	− 0.460278i
$372$	0	0
$373$	−253.000	−0.678284	−0.339142	−	0.940735i	$-0.610137\pi$
$373$	−253.000	−0.678284	−0.339142	+	0.940735i	$0.610137\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	427.224	1.13322
$378$	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
$383$	417.421i	1.08987i	0.838478	+	0.544936i	$0.183446\pi$
$383$	417.421i	1.08987i	−0.838478	+	0.544936i	$0.816554\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−230.043	−0.591371	−0.295686	−	0.955285i	$-0.595548\pi$
$389$	−230.043	−0.591371	−0.295686	+	0.955285i	$0.595548\pi$
$390$	0	0
$391$	623.538i	1.59473i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−37.0000	−0.0931990	−0.0465995	−	0.998914i	$-0.514838\pi$
$397$	−37.0000	−0.0931990	−0.0465995	+	0.998914i	$0.514838\pi$
$398$	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
$403$	427.817i	1.06158i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	872.789i	2.14444i
$408$	0	0
$409$	49.0000	0.119804	0.0599022	−	0.998204i	$-0.480921\pi$
$409$	49.0000	0.119804	0.0599022	+	0.998204i	$0.480921\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−197.180	−0.477434
$414$	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
$427$	306.573i	0.717970i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 322.552i	− 0.748381i	−0.927352	−	0.374191i	$-0.877921\pi$
$431$	− 322.552i	− 0.748381i	0.927352	−	0.374191i	$-0.122079\pi$
$432$	0	0
$433$	287.000	0.662818	0.331409	−	0.943487i	$-0.392476\pi$
$433$	287.000	0.662818	0.331409	+	0.943487i	$0.392476\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	295.770	0.676820
$438$	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
$443$	− 417.421i	− 0.942259i	−0.882064	−	0.471129i	$-0.843846\pi$
$443$	− 417.421i	− 0.942259i	0.882064	−	0.471129i	$-0.156154\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−328.634	−0.731923	−0.365962	−	0.930630i	$-0.619260\pi$
$449$	−328.634	−0.731923	−0.365962	+	0.930630i	$0.619260\pi$
$450$	0	0
$451$	623.538i	1.38257i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−554.000	−1.21225	−0.606127	−	0.795368i	$-0.707278\pi$
$457$	−554.000	−1.21225	−0.606127	+	0.795368i	$0.707278\pi$
$458$	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
$463$	− 277.128i	− 0.598549i	−0.954167	−	0.299274i	$-0.903255\pi$
$463$	− 277.128i	− 0.598549i	0.954167	−	0.299274i	$-0.0967446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 702.026i	− 1.50327i	−0.659581	−	0.751633i	$-0.729266\pi$
$467$	− 702.026i	− 1.50327i	0.659581	−	0.751633i	$-0.270734\pi$
$468$	0	0
$469$	−207.000	−0.441365
$470$	0	0
$471$	0	0
$472$	0	0
$473$	230.043	0.486350
$474$	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
$487$	895.470i	1.83875i	0.393385	+	0.919374i	$0.371304\pi$
$487$	895.470i	1.83875i	−0.393385	+	0.919374i	$0.628696\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	683.052i	1.39114i	0.718456	+	0.695572i	$0.244849\pi$
$491$	683.052i	1.39114i	−0.718456	+	0.695572i	$0.755151\pi$
$492$	0	0
$493$	−1080.00	−2.19067
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−492.950	−0.991852
$498$	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
$503$	739.973i	1.47112i	0.677460	+	0.735560i	$0.263081\pi$
$503$	739.973i	1.47112i	−0.677460	+	0.735560i	$0.736919\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	591.540	1.16216	0.581081	−	0.813846i	$-0.302630\pi$
$509$	591.540	1.16216	0.581081	+	0.813846i	$0.302630\pi$
$510$	0	0
$511$	− 135.100i	− 0.264383i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−720.000	−1.39265
$518$	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
$523$	− 774.227i	− 1.48036i	−0.672410	−	0.740178i	$-0.734741\pi$
$523$	− 774.227i	− 1.48036i	0.672410	−	0.740178i	$-0.265259\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1081.50i	− 2.05218i
$528$	0	0
$529$	169.000	0.319471
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−427.224	−0.801545
$534$	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
$547$	− 69.2820i	− 0.126658i	−0.997993	−	0.0633291i	$-0.979828\pi$
$547$	− 69.2820i	− 0.126658i	0.997993	−	0.0633291i	$-0.0201718\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	512.289i	0.929744i
$552$	0	0
$553$	720.000	1.30199
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−394.360	−0.708008	−0.354004	−	0.935244i	$-0.615180\pi$
$557$	−394.360	−0.708008	−0.354004	+	0.935244i	$0.615180\pi$
$558$	0	0
$559$	157.617i	0.281962i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 208.710i	− 0.370711i	−0.982672	−	0.185356i	$-0.940656\pi$
$563$	− 208.710i	− 0.370711i	0.982672	−	0.185356i	$-0.0593437\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−558.677	−0.981858	−0.490929	−	0.871200i	$-0.663342\pi$
$569$	−558.677	−0.981858	−0.490929	+	0.871200i	$0.663342\pi$
$570$	0	0
$571$	− 691.088i	− 1.21031i	−0.796107	−	0.605156i	$-0.793111\pi$
$571$	− 691.088i	− 1.21031i	0.796107	−	0.605156i	$-0.206889\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	983.000	1.70364	0.851820	−	0.523835i	$-0.175499\pi$
$577$	983.000	1.70364	0.851820	+	0.523835i	$0.175499\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−591.540	−1.01814
$582$	0	0
$583$	623.538i	1.06953i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 170.763i	− 0.290908i	−0.989365	−	0.145454i	$-0.953536\pi$
$587$	− 170.763i	− 0.290908i	0.989365	−	0.145454i	$-0.0464643\pi$
$588$	0	0
$589$	−513.000	−0.870968
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−98.5901	−0.166256	−0.0831282	−	0.996539i	$-0.526491\pi$
$593$	−98.5901	−0.166256	−0.0831282	+	0.996539i	$0.526491\pi$
$594$	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
$607$	554.256i	0.913108i	0.889696	+	0.456554i	$0.150916\pi$
$607$	554.256i	0.913108i	−0.889696	+	0.456554i	$0.849084\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 493.315i	− 0.807390i
$612$	0	0
$613$	−334.000	−0.544861	−0.272431	−	0.962175i	$-0.587828\pi$
$613$	−334.000	−0.544861	−0.272431	+	0.962175i	$0.587828\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−755.857	−1.22505	−0.612526	−	0.790450i	$-0.709847\pi$
$617$	−755.857	−1.22505	−0.612526	+	0.790450i	$0.709847\pi$
$618$	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
$623$	− 683.052i	− 1.09639i
$624$	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	$0.134599\pi$
$631$	517.883i	0.820734i	−0.911920	+	0.410367i	$0.865401\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	286.000	0.448980
$638$	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
$643$	623.538i	0.969733i	0.874588	+	0.484866i	$0.161132\pi$
$643$	623.538i	0.969733i	−0.874588	+	0.484866i	$0.838868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	531.263i	0.821117i	0.911834	+	0.410558i	$0.134666\pi$
$647$	531.263i	0.821117i	−0.911834	+	0.410558i	$0.865334\pi$
$648$	0	0
$649$	720.000	1.10940
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−657.267	−1.00653	−0.503267	−	0.864131i	$-0.667869\pi$
$653$	−657.267	−1.00653	−0.503267	+	0.864131i	$0.667869\pi$
$654$	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
$667$	− 623.538i	− 0.934840i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1119.45i	− 1.66833i
$672$	0	0
$673$	−214.000	−0.317979	−0.158990	−	0.987280i	$-0.550824\pi$
$673$	−214.000	−0.317979	−0.158990	+	0.987280i	$0.550824\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.8634	−0.0485426	−0.0242713	−	0.999705i	$-0.507727\pi$
$677$	−32.8634	−0.0485426	−0.0242713	+	0.999705i	$0.507727\pi$
$678$	0	0
$679$	− 119.512i	− 0.176011i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	758.947i	1.11120i	0.831451	+	0.555598i	$0.187511\pi$
$683$	758.947i	1.11120i	−0.831451	+	0.555598i	$0.812489\pi$
$684$	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	$-0.902750\pi$
$691$	− 415.692i	− 0.601581i	0.953690	−	0.300790i	$-0.0972504\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1080.00	1.54950
$698$	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
$703$	− 717.069i	− 1.02001i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	512.289i	0.724595i
$708$	0	0
$709$	−491.000	−0.692525	−0.346262	−	0.938138i	$-0.612549\pi$
$709$	−491.000	−0.692525	−0.346262	+	0.938138i	$0.612549\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	624.404	0.875742
$714$	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
$727$	− 964.752i	− 1.32703i	−0.748162	−	0.663516i	$-0.769063\pi$
$727$	− 964.752i	− 1.32703i	0.748162	−	0.663516i	$-0.230937\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 398.447i	− 0.545071i
$732$	0	0
$733$	−26.0000	−0.0354707	−0.0177353	−	0.999843i	$-0.505646\pi$
$733$	−26.0000	−0.0354707	−0.0177353	+	0.999843i	$0.505646\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	755.857	1.02559
$738$	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
$743$	− 1214.31i	− 1.63434i	−0.576397	−	0.817170i	$-0.695542\pi$
$743$	− 1214.31i	− 1.63434i	0.576397	−	0.817170i	$-0.304458\pi$
$744$	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	$-0.713099\pi$
$751$	− 1177.79i	− 1.56830i	0.620570	−	0.784151i	$-0.286901\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−803.000	−1.06077	−0.530383	−	0.847758i	$-0.677952\pi$
$757$	−803.000	−1.06077	−0.530383	+	0.847758i	$0.677952\pi$
$758$	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
$763$	566.381i	0.742307i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	493.315i	0.643175i
$768$	0	0
$769$	671.000	0.872562	0.436281	−	0.899811i	$-0.356295\pi$
$769$	671.000	0.872562	0.436281	+	0.899811i	$0.356295\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1215.94	1.57302	0.786510	−	0.617578i	$-0.211886\pi$
$773$	1215.94	1.57302	0.786510	+	0.617578i	$0.211886\pi$
$774$	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
$787$	− 524.811i	− 0.666851i	−0.942777	−	0.333425i	$-0.891796\pi$
$787$	− 524.811i	− 0.666851i	0.942777	−	0.333425i	$-0.108204\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	683.052i	0.863530i
$792$	0	0
$793$	767.000	0.967213
$794$	0	0
$795$	0	0
$796$	0	0
$797$	722.994	0.907144	0.453572	−	0.891220i	$-0.350149\pi$
$797$	722.994	0.907144	0.453572	+	0.891220i	$0.350149\pi$
$798$	0	0
$799$	1247.08i	1.56080i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	493.315i	0.614340i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	624.404	0.771822	0.385911	−	0.922536i	$-0.373887\pi$
$809$	624.404	0.771822	0.385911	+	0.922536i	$0.373887\pi$
$810$	0	0
$811$	− 136.832i	− 0.168720i	−0.996435	−	0.0843601i	$-0.973115\pi$
$811$	− 136.832i	− 0.168720i	0.996435	−	0.0843601i	$-0.0268846\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−189.000	−0.231334
$818$	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
$823$	− 1477.44i	− 1.79519i	−0.440824	−	0.897594i	$-0.645314\pi$
$823$	− 1477.44i	− 1.79519i	0.440824	−	0.897594i	$-0.354686\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	664.078i	0.802997i	0.915860	+	0.401498i	$0.131510\pi$
$827$	664.078i	0.802997i	−0.915860	+	0.401498i	$0.868490\pi$
$828$	0	0
$829$	−818.000	−0.986731	−0.493366	−	0.869822i	$-0.664233\pi$
$829$	−818.000	−0.986731	−0.493366	+	0.869822i	$0.664233\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−722.994	−0.867940
$834$	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
$847$	− 1241.88i	− 1.46621i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	872.789i	1.02560i
$852$	0	0
$853$	−1213.00	−1.42204	−0.711020	−	0.703172i	$-0.751766\pi$
$853$	−1213.00	−1.42204	−0.711020	+	0.703172i	$0.751766\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	591.540	0.690245	0.345123	−	0.938558i	$-0.387837\pi$
$857$	591.540	0.690245	0.345123	+	0.938558i	$0.387837\pi$
$858$	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
$863$	341.526i	0.395743i	0.980228	+	0.197871i	$0.0634028\pi$
$863$	341.526i	0.395743i	−0.980228	+	0.197871i	$0.936597\pi$
$864$	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
$877$	−37.0000	−0.0421893	−0.0210946	−	0.999777i	$-0.506715\pi$
$877$	−37.0000	−0.0421893	−0.0210946	+	0.999777i	$0.506715\pi$
$878$	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
$883$	− 265.004i	− 0.300118i	−0.988677	−	0.150059i	$-0.952054\pi$
$883$	− 265.004i	− 0.300118i	0.988677	−	0.150059i	$-0.0479463\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 872.789i	− 0.983978i	−0.870601	−	0.491989i	$-0.836270\pi$
$887$	− 872.789i	− 0.983978i	0.870601	−	0.491989i	$-0.163730\pi$
$888$	0	0
$889$	−1080.00	−1.21485
$890$	0	0
$891$	0	0
$892$	0	0
$893$	591.540	0.662419
$894$	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
$907$	− 138.564i	− 0.152772i	−0.997078	−	0.0763859i	$-0.975662\pi$
$907$	− 138.564i	− 0.152772i	0.997078	−	0.0763859i	$-0.0243381\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1783.52i	1.95777i	0.204422	+	0.978883i	$0.434469\pi$
$911$	1783.52i	1.95777i	−0.204422	+	0.978883i	$0.565531\pi$
$912$	0	0
$913$	2160.00	2.36583
$914$	0	0
$915$	0	0
$916$	0	0
$917$	690.130	0.752596
$918$	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
$923$	1233.29i	1.33617i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	821.584	0.884374	0.442187	−	0.896923i	$-0.354203\pi$
$929$	821.584	0.884374	0.442187	+	0.896923i	$0.354203\pi$
$930$	0	0
$931$	342.946i	0.368363i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1703.00	1.81750	0.908751	−	0.417338i	$-0.137037\pi$
$937$	1703.00	1.81750	0.908751	+	0.417338i	$0.137037\pi$
$938$	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
$943$	623.538i	0.661228i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 626.131i	− 0.661173i	−0.943776	−	0.330587i	$-0.892753\pi$
$947$	− 626.131i	− 0.661173i	0.943776	−	0.330587i	$-0.107247\pi$
$948$	0	0
$949$	−338.000	−0.356164
$950$	0	0
$951$	0	0
$952$	0	0
$953$	131.453	0.137936	0.0689682	−	0.997619i	$-0.478029\pi$
$953$	131.453	0.137936	0.0689682	+	0.997619i	$0.478029\pi$
$954$	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
$967$	− 1385.64i	− 1.43293i	−0.697624	−	0.716464i	$-0.745760\pi$
$967$	− 1385.64i	− 1.43293i	0.697624	−	0.716464i	$-0.254240\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 986.631i	− 1.01610i	−0.861328	−	0.508049i	$-0.830367\pi$
$971$	− 986.631i	− 1.01610i	0.861328	−	0.508049i	$-0.169633\pi$
$972$	0	0
$973$	360.000	0.369990
$974$	0	0
$975$	0	0
$976$	0	0
$977$	65.7267	0.0672740	0.0336370	−	0.999434i	$-0.489291\pi$
$977$	65.7267	0.0672740	0.0336370	+	0.999434i	$0.489291\pi$
$978$	0	0
$979$	2494.15i	2.54765i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 151.789i	− 0.154414i	−0.997015	−	0.0772072i	$-0.975400\pi$
$983$	− 151.789i	− 0.154414i	0.997015	−	0.0772072i	$-0.0246003\pi$
$984$	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	$-0.891955\pi$
$991$	− 659.911i	− 0.665904i	0.942944	−	0.332952i	$-0.108045\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	554.000	0.555667	0.277834	−	0.960629i	$-0.410384\pi$
$997$	554.000	0.555667	0.277834	+	0.960629i	$0.410384\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+5.19615i q^{7} +O(q^{10})\)	\(q+5.19615i q^{7} -18.9737i q^{11} +13.0000 q^{13} -32.8634 q^{17} +15.5885i q^{19} -18.9737i q^{23} +32.8634 q^{29} +32.9090i q^{31} -46.0000 q^{37} -32.8634 q^{41} +12.1244i q^{43} -37.9473i q^{47} +22.0000 q^{49} -32.8634 q^{53} +37.9473i q^{59} +59.0000 q^{61} +39.8372i q^{67} +94.8683i q^{71} -26.0000 q^{73} +98.5901 q^{77} -138.564i q^{79} +113.842i q^{83} -131.453 q^{89} +67.5500i q^{91} -23.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 52 q^{13} - 184 q^{37} + 88 q^{49} + 236 q^{61} - 104 q^{73} - 92 q^{97}+O(q^{100}) \) 4 * q + 52 * q^13 - 184 * q^37 + 88 * q^49 + 236 * q^61 - 104 * q^73 - 92 * q^97

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