LMFDB - Embedded newform 1600.2.d.e.801.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(801,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1600.801");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.d (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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			801.3
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1600.801
Dual form			1600.2.d.e.801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -3.46410 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -3.46410 q^{7} +2.00000 q^{9} +3.00000i q^{11} -3.46410i q^{13} -3.00000 q^{17} +1.00000i q^{19} -3.46410i q^{21} +5.00000i q^{27} +10.3923i q^{29} -6.92820 q^{31} -3.00000 q^{33} -10.3923i q^{37} +3.46410 q^{39} -9.00000 q^{41} -4.00000i q^{43} -10.3923 q^{47} +5.00000 q^{49} -3.00000i q^{51} -1.00000 q^{57} -12.0000i q^{59} -3.46410i q^{61} -6.92820 q^{63} +11.0000i q^{67} -10.3923 q^{71} -7.00000 q^{73} -10.3923i q^{77} +10.3923 q^{79} +1.00000 q^{81} +15.0000i q^{83} -10.3923 q^{87} -3.00000 q^{89} +12.0000i q^{91} -6.92820i q^{93} -14.0000 q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{9}+O(q^{10}) $ 4 * q + 8 * q^9	$ 4 q + 8 q^{9} - 12 q^{17} - 12 q^{33} - 36 q^{41} + 20 q^{49} - 4 q^{57} - 28 q^{73} + 4 q^{81} - 12 q^{89} - 56 q^{97}+O(q^{100}) $ 4 * q + 8 * q^9 - 12 * q^17 - 12 * q^33 - 36 * q^41 + 20 * q^49 - 4 * q^57 - 28 * q^73 + 4 * q^81 - 12 * q^89 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	0	0
$13$	− 3.46410i	− 0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	− 3.46410i	− 0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	1.00000i	0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$	1.00000i	0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$	0	0
$21$	− 3.46410i	− 0.755929i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	10.3923i	1.92980i	0.262613	+	0.964901i	$0.415416\pi$
$29$	10.3923i	1.92980i	−0.262613	+	0.964901i	$0.584584\pi$
$30$	0	0
$31$	−6.92820	−1.24434	−0.622171	−	0.782881i	$-0.713749\pi$
$31$	−6.92820	−1.24434	−0.622171	+	0.782881i	$0.713749\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.3923i	− 1.70848i	−0.519875	−	0.854242i	$-0.674022\pi$
$37$	− 10.3923i	− 1.70848i	0.519875	−	0.854242i	$-0.325978\pi$
$38$	0	0
$39$	3.46410	0.554700
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.3923	−1.51587	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923	−1.51587	−0.757937	+	0.652328i	$0.773792\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	− 3.00000i	− 0.420084i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	0	0
$61$	− 3.46410i	− 0.443533i	−0.975100	−	0.221766i	$-0.928818\pi$
$61$	− 3.46410i	− 0.443533i	0.975100	−	0.221766i	$-0.0711822\pi$
$62$	0	0
$63$	−6.92820	−0.872872
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.0000i	1.34386i	0.740613	+	0.671932i	$0.234535\pi$
$67$	11.0000i	1.34386i	−0.740613	+	0.671932i	$0.765465\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.3923	−1.23334	−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923	−1.23334	−0.616670	+	0.787222i	$0.711519\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.3923i	− 1.18431i
$78$	0	0
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0000i	1.64646i	0.567705	+	0.823232i	$0.307831\pi$
$83$	15.0000i	1.64646i	−0.567705	+	0.823232i	$0.692169\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−10.3923	−1.11417
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	12.0000i	1.25794i
$92$	0	0
$93$	− 6.92820i	− 0.718421i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	6.00000i	0.603023i
$100$	0	0
$101$	− 10.3923i	− 1.03407i	−0.855963	−	0.517036i	$-0.827035\pi$
$101$	− 10.3923i	− 1.03407i	0.855963	−	0.517036i	$-0.172965\pi$
$102$	0	0
$103$	−10.3923	−1.02398	−0.511992	−	0.858990i	$-0.671092\pi$
$103$	−10.3923	−1.02398	−0.511992	+	0.858990i	$0.671092\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.00000i	− 0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$	− 9.00000i	− 0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$	0	0
$109$	− 3.46410i	− 0.331801i	−0.986143	−	0.165900i	$-0.946947\pi$
$109$	− 3.46410i	− 0.331801i	0.986143	−	0.165900i	$-0.0530530\pi$
$110$	0	0
$111$	10.3923	0.986394
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.92820i	− 0.640513i
$118$	0	0
$119$	10.3923	0.952661
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	− 9.00000i	− 0.811503i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	− 3.46410i	− 0.300376i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	− 5.00000i	− 0.424094i	−0.977259	−	0.212047i	$-0.931987\pi$
$139$	− 5.00000i	− 0.424094i	0.977259	−	0.212047i	$-0.0680131\pi$
$140$	0	0
$141$	− 10.3923i	− 0.875190i
$142$	0	0
$143$	10.3923	0.869048
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.00000i	0.412393i
$148$	0	0
$149$	10.3923i	0.851371i	0.904871	+	0.425685i	$0.139967\pi$
$149$	10.3923i	0.851371i	−0.904871	+	0.425685i	$0.860033\pi$
$150$	0	0
$151$	3.46410	0.281905	0.140952	−	0.990016i	$-0.454984\pi$
$151$	3.46410	0.281905	0.140952	+	0.990016i	$0.454984\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.92820i	− 0.552931i	−0.961024	−	0.276465i	$-0.910837\pi$
$157$	− 6.92820i	− 0.552931i	0.961024	−	0.276465i	$-0.0891631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.0000i	1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$	19.0000i	1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.3923	0.804181	0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923	0.804181	0.402090	+	0.915600i	$0.368284\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.00000i	0.152944i
$172$	0	0
$173$	20.7846i	1.58022i	0.612962	+	0.790112i	$0.289978\pi$
$173$	20.7846i	1.58022i	−0.612962	+	0.790112i	$0.710022\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	− 3.00000i	− 0.224231i	−0.993695	−	0.112115i	$-0.964237\pi$
$179$	− 3.00000i	− 0.224231i	0.993695	−	0.112115i	$-0.0357626\pi$
$180$	0	0
$181$	13.8564i	1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$	13.8564i	1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$	0	0
$183$	3.46410	0.256074
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 9.00000i	− 0.658145i
$188$	0	0
$189$	− 17.3205i	− 1.25988i
$190$	0	0
$191$	−20.7846	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$191$	−20.7846	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.7846i	1.48084i	0.672143	+	0.740421i	$0.265374\pi$
$197$	20.7846i	1.48084i	−0.672143	+	0.740421i	$0.734626\pi$
$198$	0	0
$199$	−13.8564	−0.982255	−0.491127	−	0.871088i	$-0.663415\pi$
$199$	−13.8564	−0.982255	−0.491127	+	0.871088i	$0.663415\pi$
$200$	0	0
$201$	−11.0000	−0.775880
$202$	0	0
$203$	− 36.0000i	− 2.52670i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.00000	−0.207514
$210$	0	0
$211$	− 25.0000i	− 1.72107i	−0.509390	−	0.860535i	$-0.670129\pi$
$211$	− 25.0000i	− 1.72107i	0.509390	−	0.860535i	$-0.329871\pi$
$212$	0	0
$213$	− 10.3923i	− 0.712069i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	24.0000	1.62923
$218$	0	0
$219$	− 7.00000i	− 0.473016i
$220$	0	0
$221$	10.3923i	0.699062i
$222$	0	0
$223$	−13.8564	−0.927894	−0.463947	−	0.885863i	$-0.653567\pi$
$223$	−13.8564	−0.927894	−0.463947	+	0.885863i	$0.653567\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	6.92820i	0.457829i	0.973447	+	0.228914i	$0.0735176\pi$
$229$	6.92820i	0.457829i	−0.973447	+	0.228914i	$0.926482\pi$
$230$	0	0
$231$	10.3923	0.683763
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.3923i	0.675053i
$238$	0	0
$239$	−10.3923	−0.672222	−0.336111	−	0.941822i	$-0.609112\pi$
$239$	−10.3923	−0.672222	−0.336111	+	0.941822i	$0.609112\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.46410	0.220416
$248$	0	0
$249$	−15.0000	−0.950586
$250$	0	0
$251$	9.00000i	0.568075i	0.958813	+	0.284037i	$0.0916740\pi$
$251$	9.00000i	0.568075i	−0.958813	+	0.284037i	$0.908326\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	36.0000i	2.23693i
$260$	0	0
$261$	20.7846i	1.28654i
$262$	0	0
$263$	20.7846	1.28163	0.640817	−	0.767694i	$-0.278596\pi$
$263$	20.7846	1.28163	0.640817	+	0.767694i	$0.278596\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3.00000i	− 0.183597i
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	20.7846	1.26258	0.631288	−	0.775549i	$-0.282527\pi$
$271$	20.7846	1.26258	0.631288	+	0.775549i	$0.282527\pi$
$272$	0	0
$273$	−12.0000	−0.726273
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.3205i	1.04069i	0.853957	+	0.520344i	$0.174196\pi$
$277$	17.3205i	1.04069i	−0.853957	+	0.520344i	$0.825804\pi$
$278$	0	0
$279$	−13.8564	−0.829561
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 11.0000i	− 0.653882i	−0.945045	−	0.326941i	$-0.893982\pi$
$283$	− 11.0000i	− 0.653882i	0.945045	−	0.326941i	$-0.106018\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	31.1769	1.84032
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 14.0000i	− 0.820695i
$292$	0	0
$293$	10.3923i	0.607125i	0.952812	+	0.303562i	$0.0981761\pi$
$293$	10.3923i	0.607125i	−0.952812	+	0.303562i	$0.901824\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−15.0000	−0.870388
$298$	0	0
$299$	0	0
$300$	0	0
$301$	13.8564i	0.798670i
$302$	0	0
$303$	10.3923	0.597022
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 19.0000i	− 1.08439i	−0.840254	−	0.542194i	$-0.817594\pi$
$307$	− 19.0000i	− 1.08439i	0.840254	−	0.542194i	$-0.182406\pi$
$308$	0	0
$309$	− 10.3923i	− 0.591198i
$310$	0	0
$311$	10.3923	0.589294	0.294647	−	0.955606i	$-0.404798\pi$
$311$	10.3923	0.589294	0.294647	+	0.955606i	$0.404798\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.7846i	1.16738i	0.811977	+	0.583690i	$0.198392\pi$
$317$	20.7846i	1.16738i	−0.811977	+	0.583690i	$0.801608\pi$
$318$	0	0
$319$	−31.1769	−1.74557
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	− 3.00000i	− 0.166924i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.46410	0.191565
$328$	0	0
$329$	36.0000	1.98474
$330$	0	0
$331$	19.0000i	1.04433i	0.852843	+	0.522167i	$0.174876\pi$
$331$	19.0000i	1.04433i	−0.852843	+	0.522167i	$0.825124\pi$
$332$	0	0
$333$	− 20.7846i	− 1.13899i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	0	0
$339$	15.0000i	0.814688i
$340$	0	0
$341$	− 20.7846i	− 1.12555i
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 33.0000i	− 1.77153i	−0.464131	−	0.885766i	$-0.653633\pi$
$347$	− 33.0000i	− 1.77153i	0.464131	−	0.885766i	$-0.346367\pi$
$348$	0	0
$349$	− 10.3923i	− 0.556287i	−0.960539	−	0.278144i	$-0.910281\pi$
$349$	− 10.3923i	− 0.556287i	0.960539	−	0.278144i	$-0.0897191\pi$
$350$	0	0
$351$	17.3205	0.924500
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.3923i	0.550019i
$358$	0	0
$359$	10.3923	0.548485	0.274242	−	0.961661i	$-0.411573\pi$
$359$	10.3923	0.548485	0.274242	+	0.961661i	$0.411573\pi$
$360$	0	0
$361$	18.0000	0.947368
$362$	0	0
$363$	2.00000i	0.104973i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.92820	0.361649	0.180825	−	0.983515i	$-0.442123\pi$
$367$	6.92820	0.361649	0.180825	+	0.983515i	$0.442123\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	0	0
$372$	0	0
$373$	13.8564i	0.717458i	0.933442	+	0.358729i	$0.116790\pi$
$373$	13.8564i	0.717458i	−0.933442	+	0.358729i	$0.883210\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	36.0000	1.85409
$378$	0	0
$379$	− 19.0000i	− 0.975964i	−0.872854	−	0.487982i	$-0.837733\pi$
$379$	− 19.0000i	− 0.975964i	0.872854	−	0.487982i	$-0.162267\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.1769	1.59307	0.796533	−	0.604595i	$-0.206665\pi$
$383$	31.1769	1.59307	0.796533	+	0.604595i	$0.206665\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.00000i	− 0.406663i
$388$	0	0
$389$	− 20.7846i	− 1.05382i	−0.849921	−	0.526911i	$-0.823350\pi$
$389$	− 20.7846i	− 1.05382i	0.849921	−	0.526911i	$-0.176650\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.7128i	1.39087i	0.718591	+	0.695433i	$0.244787\pi$
$397$	27.7128i	1.39087i	−0.718591	+	0.695433i	$0.755213\pi$
$398$	0	0
$399$	3.46410	0.173422
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	0	0
$403$	24.0000i	1.19553i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	31.1769	1.54538
$408$	0	0
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	− 9.00000i	− 0.443937i
$412$	0	0
$413$	41.5692i	2.04549i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.00000	0.244851
$418$	0	0
$419$	21.0000i	1.02592i	0.858413	+	0.512959i	$0.171451\pi$
$419$	21.0000i	1.02592i	−0.858413	+	0.512959i	$0.828549\pi$
$420$	0	0
$421$	20.7846i	1.01298i	0.862246	+	0.506490i	$0.169057\pi$
$421$	20.7846i	1.01298i	−0.862246	+	0.506490i	$0.830943\pi$
$422$	0	0
$423$	−20.7846	−1.01058
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.0000i	0.580721i
$428$	0	0
$429$	10.3923i	0.501745i
$430$	0	0
$431$	−10.3923	−0.500580	−0.250290	−	0.968171i	$-0.580526\pi$
$431$	−10.3923	−0.500580	−0.250290	+	0.968171i	$0.580526\pi$
$432$	0	0
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−17.3205	−0.826663	−0.413331	−	0.910581i	$-0.635635\pi$
$439$	−17.3205	−0.826663	−0.413331	+	0.910581i	$0.635635\pi$
$440$	0	0
$441$	10.0000	0.476190
$442$	0	0
$443$	− 39.0000i	− 1.85295i	−0.376361	−	0.926473i	$-0.622825\pi$
$443$	− 39.0000i	− 1.85295i	0.376361	−	0.926473i	$-0.377175\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10.3923	−0.491539
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	− 27.0000i	− 1.27138i
$452$	0	0
$453$	3.46410i	0.162758i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	0	0
$459$	− 15.0000i	− 0.700140i
$460$	0	0
$461$	20.7846i	0.968036i	0.875058	+	0.484018i	$0.160823\pi$
$461$	20.7846i	0.968036i	−0.875058	+	0.484018i	$0.839177\pi$
$462$	0	0
$463$	−20.7846	−0.965943	−0.482971	−	0.875636i	$-0.660442\pi$
$463$	−20.7846	−0.965943	−0.482971	+	0.875636i	$0.660442\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	− 38.1051i	− 1.75953i
$470$	0	0
$471$	6.92820	0.319235
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	41.5692	1.89935	0.949673	−	0.313243i	$-0.101415\pi$
$479$	41.5692	1.89935	0.949673	+	0.313243i	$0.101415\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.8564	0.627894	0.313947	−	0.949441i	$-0.398349\pi$
$487$	13.8564	0.627894	0.313947	+	0.949441i	$0.398349\pi$
$488$	0	0
$489$	−19.0000	−0.859210
$490$	0	0
$491$	12.0000i	0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$	12.0000i	0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$	0	0
$493$	− 31.1769i	− 1.40414i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	36.0000	1.61482
$498$	0	0
$499$	− 16.0000i	− 0.716258i	−0.933672	−	0.358129i	$-0.883415\pi$
$499$	− 16.0000i	− 0.716258i	0.933672	−	0.358129i	$-0.116585\pi$
$500$	0	0
$501$	10.3923i	0.464294i
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	0	0
$509$	20.7846i	0.921262i	0.887592	+	0.460631i	$0.152377\pi$
$509$	20.7846i	0.921262i	−0.887592	+	0.460631i	$0.847623\pi$
$510$	0	0
$511$	24.2487	1.07270
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 31.1769i	− 1.37116i
$518$	0	0
$519$	−20.7846	−0.912343
$520$	0	0
$521$	−21.0000	−0.920027	−0.460013	−	0.887912i	$-0.652155\pi$
$521$	−21.0000	−0.920027	−0.460013	+	0.887912i	$0.652155\pi$
$522$	0	0
$523$	− 1.00000i	− 0.0437269i	−0.999761	−	0.0218635i	$-0.993040\pi$
$523$	− 1.00000i	− 0.0437269i	0.999761	−	0.0218635i	$-0.00695991\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.7846	0.905392
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 24.0000i	− 1.04151i
$532$	0	0
$533$	31.1769i	1.35042i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.00000	0.129460
$538$	0	0
$539$	15.0000i	0.646096i
$540$	0	0
$541$	− 6.92820i	− 0.297867i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	− 6.92820i	− 0.297867i	0.988847	−	0.148933i	$-0.0475840\pi$
$542$	0	0
$543$	−13.8564	−0.594635
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.00000i	0.0427569i	0.999771	+	0.0213785i	$0.00680549\pi$
$547$	1.00000i	0.0427569i	−0.999771	+	0.0213785i	$0.993195\pi$
$548$	0	0
$549$	− 6.92820i	− 0.295689i
$550$	0	0
$551$	−10.3923	−0.442727
$552$	0	0
$553$	−36.0000	−1.53088
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10.3923i	− 0.440336i	−0.975462	−	0.220168i	$-0.929339\pi$
$557$	− 10.3923i	− 0.440336i	0.975462	−	0.220168i	$-0.0706606\pi$
$558$	0	0
$559$	−13.8564	−0.586064
$560$	0	0
$561$	9.00000	0.379980
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.46410	−0.145479
$568$	0	0
$569$	3.00000	0.125767	0.0628833	−	0.998021i	$-0.479970\pi$
$569$	3.00000	0.125767	0.0628833	+	0.998021i	$0.479970\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	− 20.7846i	− 0.868290i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−17.0000	−0.707719	−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000	−0.707719	−0.353860	+	0.935299i	$0.615131\pi$
$578$	0	0
$579$	− 7.00000i	− 0.290910i
$580$	0	0
$581$	− 51.9615i	− 2.15573i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.0000i	− 0.619116i	−0.950881	−	0.309558i	$-0.899819\pi$
$587$	− 15.0000i	− 0.619116i	0.950881	−	0.309558i	$-0.100181\pi$
$588$	0	0
$589$	− 6.92820i	− 0.285472i
$590$	0	0
$591$	−20.7846	−0.854965
$592$	0	0
$593$	3.00000	0.123195	0.0615976	−	0.998101i	$-0.480380\pi$
$593$	3.00000	0.123195	0.0615976	+	0.998101i	$0.480380\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 13.8564i	− 0.567105i
$598$	0	0
$599$	31.1769	1.27385	0.636927	−	0.770924i	$-0.280205\pi$
$599$	31.1769	1.27385	0.636927	+	0.770924i	$0.280205\pi$
$600$	0	0
$601$	41.0000	1.67242	0.836212	−	0.548406i	$-0.184765\pi$
$601$	41.0000	1.67242	0.836212	+	0.548406i	$0.184765\pi$
$602$	0	0
$603$	22.0000i	0.895909i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.7846	−0.843621	−0.421811	−	0.906684i	$-0.638605\pi$
$607$	−20.7846	−0.843621	−0.421811	+	0.906684i	$0.638605\pi$
$608$	0	0
$609$	36.0000	1.45879
$610$	0	0
$611$	36.0000i	1.45640i
$612$	0	0
$613$	− 20.7846i	− 0.839482i	−0.907644	−	0.419741i	$-0.862121\pi$
$613$	− 20.7846i	− 0.839482i	0.907644	−	0.419741i	$-0.137879\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.3923	0.416359
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.00000i	− 0.119808i
$628$	0	0
$629$	31.1769i	1.24310i
$630$	0	0
$631$	−13.8564	−0.551615	−0.275807	−	0.961213i	$-0.588945\pi$
$631$	−13.8564	−0.551615	−0.275807	+	0.961213i	$0.588945\pi$
$632$	0	0
$633$	25.0000	0.993661
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 17.3205i	− 0.686264i
$638$	0	0
$639$	−20.7846	−0.822226
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−41.5692	−1.63425	−0.817127	−	0.576457i	$-0.804435\pi$
$647$	−41.5692	−1.63425	−0.817127	+	0.576457i	$0.804435\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	24.0000i	0.940634i
$652$	0	0
$653$	31.1769i	1.22005i	0.792383	+	0.610023i	$0.208840\pi$
$653$	31.1769i	1.22005i	−0.792383	+	0.610023i	$0.791160\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	− 33.0000i	− 1.28550i	−0.766077	−	0.642749i	$-0.777794\pi$
$659$	− 33.0000i	− 1.28550i	0.766077	−	0.642749i	$-0.222206\pi$
$660$	0	0
$661$	10.3923i	0.404214i	0.979363	+	0.202107i	$0.0647788\pi$
$661$	10.3923i	0.404214i	−0.979363	+	0.202107i	$0.935221\pi$
$662$	0	0
$663$	−10.3923	−0.403604
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 13.8564i	− 0.535720i
$670$	0	0
$671$	10.3923	0.401190
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.7846i	0.798817i	0.916773	+	0.399409i	$0.130785\pi$
$677$	20.7846i	0.798817i	−0.916773	+	0.399409i	$0.869215\pi$
$678$	0	0
$679$	48.4974	1.86116
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 9.00000i	− 0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$	− 9.00000i	− 0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.92820	−0.264327
$688$	0	0
$689$	0	0
$690$	0	0
$691$	37.0000i	1.40755i	0.710425	+	0.703773i	$0.248503\pi$
$691$	37.0000i	1.40755i	−0.710425	+	0.703773i	$0.751497\pi$
$692$	0	0
$693$	− 20.7846i	− 0.789542i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	27.0000	1.02270
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	− 31.1769i	− 1.17754i	−0.808302	−	0.588768i	$-0.799613\pi$
$701$	− 31.1769i	− 1.17754i	0.808302	−	0.588768i	$-0.200387\pi$
$702$	0	0
$703$	10.3923	0.391953
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36.0000i	1.35392i
$708$	0	0
$709$	34.6410i	1.30097i	0.759519	+	0.650485i	$0.225434\pi$
$709$	34.6410i	1.30097i	−0.759519	+	0.650485i	$0.774566\pi$
$710$	0	0
$711$	20.7846	0.779484
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 10.3923i	− 0.388108i
$718$	0	0
$719$	10.3923	0.387568	0.193784	−	0.981044i	$-0.437924\pi$
$719$	10.3923	0.387568	0.193784	+	0.981044i	$0.437924\pi$
$720$	0	0
$721$	36.0000	1.34071
$722$	0	0
$723$	5.00000i	0.185952i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−45.0333	−1.67019	−0.835097	−	0.550103i	$-0.814588\pi$
$727$	−45.0333	−1.67019	−0.835097	+	0.550103i	$0.814588\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	12.0000i	0.443836i
$732$	0	0
$733$	− 31.1769i	− 1.15155i	−0.817610	−	0.575773i	$-0.804701\pi$
$733$	− 31.1769i	− 1.15155i	0.817610	−	0.575773i	$-0.195299\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−33.0000	−1.21557
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	3.46410i	0.127257i
$742$	0	0
$743$	41.5692	1.52503	0.762513	−	0.646972i	$-0.223965\pi$
$743$	41.5692	1.52503	0.762513	+	0.646972i	$0.223965\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	30.0000i	1.09764i
$748$	0	0
$749$	31.1769i	1.13918i
$750$	0	0
$751$	−51.9615	−1.89610	−0.948051	−	0.318117i	$-0.896950\pi$
$751$	−51.9615	−1.89610	−0.948051	+	0.318117i	$0.896950\pi$
$752$	0	0
$753$	−9.00000	−0.327978
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 6.92820i	− 0.251810i	−0.992042	−	0.125905i	$-0.959817\pi$
$757$	− 6.92820i	− 0.251810i	0.992042	−	0.125905i	$-0.0401834\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	0	0
$763$	12.0000i	0.434429i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−41.5692	−1.50098
$768$	0	0
$769$	−7.00000	−0.252426	−0.126213	−	0.992003i	$-0.540282\pi$
$769$	−7.00000	−0.252426	−0.126213	+	0.992003i	$0.540282\pi$
$770$	0	0
$771$	6.00000i	0.216085i
$772$	0	0
$773$	− 10.3923i	− 0.373785i	−0.982380	−	0.186893i	$-0.940158\pi$
$773$	− 10.3923i	− 0.373785i	0.982380	−	0.186893i	$-0.0598416\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−36.0000	−1.29149
$778$	0	0
$779$	− 9.00000i	− 0.322458i
$780$	0	0
$781$	− 31.1769i	− 1.11560i
$782$	0	0
$783$	−51.9615	−1.85695
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16.0000i	0.570338i	0.958477	+	0.285169i	$0.0920498\pi$
$787$	16.0000i	0.570338i	−0.958477	+	0.285169i	$0.907950\pi$
$788$	0	0
$789$	20.7846i	0.739952i
$790$	0	0
$791$	−51.9615	−1.84754
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 20.7846i	− 0.736229i	−0.929781	−	0.368114i	$-0.880004\pi$
$797$	− 20.7846i	− 0.736229i	0.929781	−	0.368114i	$-0.119996\pi$
$798$	0	0
$799$	31.1769	1.10296
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	− 21.0000i	− 0.741074i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	− 28.0000i	− 0.983213i	−0.870817	−	0.491606i	$-0.836410\pi$
$811$	− 28.0000i	− 0.983213i	0.870817	−	0.491606i	$-0.163590\pi$
$812$	0	0
$813$	20.7846i	0.728948i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	24.0000i	0.838628i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−34.6410	−1.20751	−0.603755	−	0.797170i	$-0.706329\pi$
$823$	−34.6410	−1.20751	−0.603755	+	0.797170i	$0.706329\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.00000i	− 0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$	− 3.00000i	− 0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$	0	0
$829$	10.3923i	0.360940i	0.983581	+	0.180470i	$0.0577618\pi$
$829$	10.3923i	0.360940i	−0.983581	+	0.180470i	$0.942238\pi$
$830$	0	0
$831$	−17.3205	−0.600842
$832$	0	0
$833$	−15.0000	−0.519719
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 34.6410i	− 1.19737i
$838$	0	0
$839$	−41.5692	−1.43513	−0.717564	−	0.696492i	$-0.754743\pi$
$839$	−41.5692	−1.43513	−0.717564	+	0.696492i	$0.754743\pi$
$840$	0	0
$841$	−79.0000	−2.72414
$842$	0	0
$843$	18.0000i	0.619953i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.92820	−0.238056
$848$	0	0
$849$	11.0000	0.377519
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 55.4256i	− 1.89774i	−0.315671	−	0.948869i	$-0.602230\pi$
$853$	− 55.4256i	− 1.89774i	0.315671	−	0.948869i	$-0.397770\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.0000	1.33221	0.666107	−	0.745856i	$-0.267959\pi$
$857$	39.0000	1.33221	0.666107	+	0.745856i	$0.267959\pi$
$858$	0	0
$859$	− 13.0000i	− 0.443554i	−0.975097	−	0.221777i	$-0.928814\pi$
$859$	− 13.0000i	− 0.443554i	0.975097	−	0.221777i	$-0.0711857\pi$
$860$	0	0
$861$	31.1769i	1.06251i
$862$	0	0
$863$	10.3923	0.353758	0.176879	−	0.984233i	$-0.443400\pi$
$863$	10.3923	0.353758	0.176879	+	0.984233i	$0.443400\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.00000i	− 0.271694i
$868$	0	0
$869$	31.1769i	1.05760i
$870$	0	0
$871$	38.1051	1.29114
$872$	0	0
$873$	−28.0000	−0.947656
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 48.4974i	− 1.63764i	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	− 48.4974i	− 1.63764i	0.574049	−	0.818821i	$-0.305372\pi$
$878$	0	0
$879$	−10.3923	−0.350524
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	− 35.0000i	− 1.17784i	−0.808190	−	0.588922i	$-0.799553\pi$
$883$	− 35.0000i	− 1.17784i	0.808190	−	0.588922i	$-0.200447\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.00000i	0.100504i
$892$	0	0
$893$	− 10.3923i	− 0.347765i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 72.0000i	− 2.40133i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−13.8564	−0.461112
$904$	0	0
$905$	0	0
$906$	0	0
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	0	0
$909$	− 20.7846i	− 0.689382i
$910$	0	0
$911$	−20.7846	−0.688625	−0.344312	−	0.938855i	$-0.611888\pi$
$911$	−20.7846	−0.688625	−0.344312	+	0.938855i	$0.611888\pi$
$912$	0	0
$913$	−45.0000	−1.48928
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−27.7128	−0.914161	−0.457081	−	0.889425i	$-0.651105\pi$
$919$	−27.7128	−0.914161	−0.457081	+	0.889425i	$0.651105\pi$
$920$	0	0
$921$	19.0000	0.626071
$922$	0	0
$923$	36.0000i	1.18495i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−20.7846	−0.682656
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	5.00000i	0.163868i
$932$	0	0
$933$	10.3923i	0.340229i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−35.0000	−1.14340	−0.571700	−	0.820463i	$-0.693716\pi$
$937$	−35.0000	−1.14340	−0.571700	+	0.820463i	$0.693716\pi$
$938$	0	0
$939$	22.0000i	0.717943i
$940$	0	0
$941$	− 51.9615i	− 1.69390i	−0.531675	−	0.846949i	$-0.678437\pi$
$941$	− 51.9615i	− 1.69390i	0.531675	−	0.846949i	$-0.321563\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	24.2487i	0.787146i
$950$	0	0
$951$	−20.7846	−0.673987
$952$	0	0
$953$	39.0000	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 31.1769i	− 1.00781i
$958$	0	0
$959$	31.1769	1.00676
$960$	0	0
$961$	17.0000	0.548387
$962$	0	0
$963$	− 18.0000i	− 0.580042i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−20.7846	−0.668388	−0.334194	−	0.942504i	$-0.608464\pi$
$967$	−20.7846	−0.668388	−0.334194	+	0.942504i	$0.608464\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	− 57.0000i	− 1.82922i	−0.404341	−	0.914609i	$-0.632499\pi$
$971$	− 57.0000i	− 1.82922i	0.404341	−	0.914609i	$-0.367501\pi$
$972$	0	0
$973$	17.3205i	0.555270i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	0	0
$979$	− 9.00000i	− 0.287641i
$980$	0	0
$981$	− 6.92820i	− 0.221201i
$982$	0	0
$983$	10.3923	0.331463	0.165732	−	0.986171i	$-0.447001\pi$
$983$	10.3923	0.331463	0.165732	+	0.986171i	$0.447001\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	36.0000i	1.14589i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	24.2487	0.770286	0.385143	−	0.922857i	$-0.374152\pi$
$991$	24.2487	0.770286	0.385143	+	0.922857i	$0.374152\pi$
$992$	0	0
$993$	−19.0000	−0.602947
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	51.9615	1.64399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.d.e.801.3	yes	4
4.3	odd	2	inner	1600.2.d.e.801.2	yes	4
5.2	odd	4		1600.2.f.g.1249.2		4
5.3	odd	4		1600.2.f.c.1249.4		4
5.4	even	2		1600.2.d.f.801.2	yes	4
8.3	odd	2	inner	1600.2.d.e.801.4	yes	4
8.5	even	2	inner	1600.2.d.e.801.1	✓	4
16.3	odd	4		6400.2.a.cf.1.1		2
16.5	even	4		6400.2.a.cf.1.2		2
16.11	odd	4		6400.2.a.bb.1.1		2
16.13	even	4		6400.2.a.bb.1.2		2
20.3	even	4		1600.2.f.g.1249.1		4
20.7	even	4		1600.2.f.c.1249.3		4
20.19	odd	2		1600.2.d.f.801.3	yes	4
40.3	even	4		1600.2.f.c.1249.2		4
40.13	odd	4		1600.2.f.g.1249.3		4
40.19	odd	2		1600.2.d.f.801.1	yes	4
40.27	even	4		1600.2.f.g.1249.4		4
40.29	even	2		1600.2.d.f.801.4	yes	4
40.37	odd	4		1600.2.f.c.1249.1		4
80.19	odd	4		6400.2.a.ba.1.2		2
80.29	even	4		6400.2.a.cg.1.1		2
80.59	odd	4		6400.2.a.cg.1.2		2
80.69	even	4		6400.2.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.2.d.e.801.1	✓	4	8.5	even	2	inner
1600.2.d.e.801.2	yes	4	4.3	odd	2	inner
1600.2.d.e.801.3	yes	4	1.1	even	1	trivial
1600.2.d.e.801.4	yes	4	8.3	odd	2	inner
1600.2.d.f.801.1	yes	4	40.19	odd	2
1600.2.d.f.801.2	yes	4	5.4	even	2
1600.2.d.f.801.3	yes	4	20.19	odd	2
1600.2.d.f.801.4	yes	4	40.29	even	2
1600.2.f.c.1249.1		4	40.37	odd	4
1600.2.f.c.1249.2		4	40.3	even	4
1600.2.f.c.1249.3		4	20.7	even	4
1600.2.f.c.1249.4		4	5.3	odd	4
1600.2.f.g.1249.1		4	20.3	even	4
1600.2.f.g.1249.2		4	5.2	odd	4
1600.2.f.g.1249.3		4	40.13	odd	4
1600.2.f.g.1249.4		4	40.27	even	4
6400.2.a.ba.1.1		2	80.69	even	4
6400.2.a.ba.1.2		2	80.19	odd	4
6400.2.a.bb.1.1		2	16.11	odd	4
6400.2.a.bb.1.2		2	16.13	even	4
6400.2.a.cf.1.1		2	16.3	odd	4
6400.2.a.cf.1.2		2	16.5	even	4
6400.2.a.cg.1.1		2	80.29	even	4
6400.2.a.cg.1.2		2	80.59	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -3.46410 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -3.46410 q^{7} +2.00000 q^{9} +3.00000i q^{11} -3.46410i q^{13} -3.00000 q^{17} +1.00000i q^{19} -3.46410i q^{21} +5.00000i q^{27} +10.3923i q^{29} -6.92820 q^{31} -3.00000 q^{33} -10.3923i q^{37} +3.46410 q^{39} -9.00000 q^{41} -4.00000i q^{43} -10.3923 q^{47} +5.00000 q^{49} -3.00000i q^{51} -1.00000 q^{57} -12.0000i q^{59} -3.46410i q^{61} -6.92820 q^{63} +11.0000i q^{67} -10.3923 q^{71} -7.00000 q^{73} -10.3923i q^{77} +10.3923 q^{79} +1.00000 q^{81} +15.0000i q^{83} -10.3923 q^{87} -3.00000 q^{89} +12.0000i q^{91} -6.92820i q^{93} -14.0000 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{9}+O(q^{10}) \) 4 * q + 8 * q^9	\( 4 q + 8 q^{9} - 12 q^{17} - 12 q^{33} - 36 q^{41} + 20 q^{49} - 4 q^{57} - 28 q^{73} + 4 q^{81} - 12 q^{89} - 56 q^{97}+O(q^{100}) \) 4 * q + 8 * q^9 - 12 * q^17 - 12 * q^33 - 36 * q^41 + 20 * q^49 - 4 * q^57 - 28 * q^73 + 4 * q^81 - 12 * q^89 - 56 * q^97

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