LMFDB - Embedded newform 1600.2.d.i.801.1

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:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			801.1
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1600.801
Dual form			1600.2.d.i.801.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949i q^{3} -1.41421 q^{7} -3.00000 q^{9} -2.00000i q^{11} +5.65685i q^{13} -4.89898 q^{17} +6.00000i q^{19} +3.46410i q^{21} -7.07107 q^{23} +6.92820i q^{29} +6.92820 q^{31} -4.89898 q^{33} +2.82843i q^{37} +13.8564 q^{39} -4.00000 q^{41} -2.44949i q^{43} -4.24264 q^{47} -5.00000 q^{49} +12.0000i q^{51} +14.6969 q^{57} -2.00000i q^{59} +3.46410i q^{61} +4.24264 q^{63} -2.44949i q^{67} +17.3205i q^{69} -6.92820 q^{71} +4.89898 q^{73} +2.82843i q^{77} +6.92820 q^{79} -9.00000 q^{81} +12.2474i q^{83} +16.9706 q^{87} +2.00000 q^{89} -8.00000i q^{91} -16.9706i q^{93} -14.6969 q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9} - 32 q^{41} - 40 q^{49} - 72 q^{81} + 16 q^{89}+O(q^{100}) $ 8 * q - 24 * q^9 - 32 * q^41 - 40 * q^49 - 72 * q^81 + 16 * q^89

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
$3$	− 2.44949i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	− 2.44949i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	5.65685i	1.56893i	0.620174	+	0.784465i	$0.287062\pi$
$13$	5.65685i	1.56893i	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.89898	−1.18818	−0.594089	−	0.804400i	$-0.702487\pi$
$17$	−4.89898	−1.18818	−0.594089	+	0.804400i	$0.702487\pi$
$18$	0	0
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	0	0
$21$	3.46410i	0.755929i
$22$	0	0
$23$	−7.07107	−1.47442	−0.737210	−	0.675664i	$-0.763857\pi$
$23$	−7.07107	−1.47442	−0.737210	+	0.675664i	$0.763857\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.92820i	1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$	6.92820i	1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	6.92820	1.24434	0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	0	0
$33$	−4.89898	−0.852803
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.82843i	0.464991i	0.972598	+	0.232495i	$0.0746890\pi$
$37$	2.82843i	0.464991i	−0.972598	+	0.232495i	$0.925311\pi$
$38$	0	0
$39$	13.8564	2.21880
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	− 2.44949i	− 0.373544i	−0.982403	−	0.186772i	$-0.940197\pi$
$43$	− 2.44949i	− 0.373544i	0.982403	−	0.186772i	$-0.0598025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.24264	−0.618853	−0.309426	−	0.950923i	$-0.600137\pi$
$47$	−4.24264	−0.618853	−0.309426	+	0.950923i	$0.600137\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	12.0000i	1.68034i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.6969	1.94666
$58$	0	0
$59$	− 2.00000i	− 0.260378i	−0.991489	−	0.130189i	$-0.958442\pi$
$59$	− 2.00000i	− 0.260378i	0.991489	−	0.130189i	$-0.0415584\pi$
$60$	0	0
$61$	3.46410i	0.443533i	0.975100	+	0.221766i	$0.0711822\pi$
$61$	3.46410i	0.443533i	−0.975100	+	0.221766i	$0.928818\pi$
$62$	0	0
$63$	4.24264	0.534522
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.44949i	− 0.299253i	−0.988743	−	0.149626i	$-0.952193\pi$
$67$	− 2.44949i	− 0.299253i	0.988743	−	0.149626i	$-0.0478071\pi$
$68$	0	0
$69$	17.3205i	2.08514i
$70$	0	0
$71$	−6.92820	−0.822226	−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820	−0.822226	−0.411113	+	0.911584i	$0.634860\pi$
$72$	0	0
$73$	4.89898	0.573382	0.286691	−	0.958023i	$-0.407445\pi$
$73$	4.89898	0.573382	0.286691	+	0.958023i	$0.407445\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.82843i	0.322329i
$78$	0	0
$79$	6.92820	0.779484	0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820	0.779484	0.389742	+	0.920924i	$0.372564\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	12.2474i	1.34433i	0.740400	+	0.672166i	$0.234636\pi$
$83$	12.2474i	1.34433i	−0.740400	+	0.672166i	$0.765364\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.9706	1.81944
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	− 8.00000i	− 0.838628i
$92$	0	0
$93$	− 16.9706i	− 1.75977i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.6969	−1.49225	−0.746124	−	0.665807i	$-0.768087\pi$
$97$	−14.6969	−1.49225	−0.746124	+	0.665807i	$0.768087\pi$
$98$	0	0
$99$	6.00000i	0.603023i
$100$	0	0
$101$	− 6.92820i	− 0.689382i	−0.938716	−	0.344691i	$-0.887984\pi$
$101$	− 6.92820i	− 0.689382i	0.938716	−	0.344691i	$-0.112016\pi$
$102$	0	0
$103$	9.89949	0.975426	0.487713	−	0.873004i	$-0.337831\pi$
$103$	9.89949	0.975426	0.487713	+	0.873004i	$0.337831\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.44949i	− 0.236801i	−0.992966	−	0.118401i	$-0.962223\pi$
$107$	− 2.44949i	− 0.236801i	0.992966	−	0.118401i	$-0.0377767\pi$
$108$	0	0
$109$	3.46410i	0.331801i	0.986143	+	0.165900i	$0.0530530\pi$
$109$	3.46410i	0.331801i	−0.986143	+	0.165900i	$0.946947\pi$
$110$	0	0
$111$	6.92820	0.657596
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 16.9706i	− 1.56893i
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	9.79796i	0.883452i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.07107	−0.627456	−0.313728	−	0.949513i	$-0.601578\pi$
$127$	−7.07107	−0.627456	−0.313728	+	0.949513i	$0.601578\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	10.0000i	0.873704i	0.899533	+	0.436852i	$0.143907\pi$
$131$	10.0000i	0.873704i	−0.899533	+	0.436852i	$0.856093\pi$
$132$	0	0
$133$	− 8.48528i	− 0.735767i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.79796	0.837096	0.418548	−	0.908195i	$-0.362539\pi$
$137$	9.79796	0.837096	0.418548	+	0.908195i	$0.362539\pi$
$138$	0	0
$139$	10.0000i	0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$	10.0000i	0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$	0	0
$141$	10.3923i	0.875190i
$142$	0	0
$143$	11.3137	0.946100
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.2474i	1.01015i
$148$	0	0
$149$	− 10.3923i	− 0.851371i	−0.904871	−	0.425685i	$-0.860033\pi$
$149$	− 10.3923i	− 0.851371i	0.904871	−	0.425685i	$-0.139967\pi$
$150$	0	0
$151$	13.8564	1.12762	0.563809	−	0.825905i	$-0.309335\pi$
$151$	13.8564	1.12762	0.563809	+	0.825905i	$0.309335\pi$
$152$	0	0
$153$	14.6969	1.18818
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.82843i	− 0.225733i	−0.993610	−	0.112867i	$-0.963997\pi$
$157$	− 2.82843i	− 0.225733i	0.993610	−	0.112867i	$-0.0360032\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	10.0000	0.788110
$162$	0	0
$163$	− 22.0454i	− 1.72673i	−0.504580	−	0.863365i	$-0.668353\pi$
$163$	− 22.0454i	− 1.72673i	0.504580	−	0.863365i	$-0.331647\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.89949	−0.766046	−0.383023	−	0.923739i	$-0.625117\pi$
$167$	−9.89949	−0.766046	−0.383023	+	0.923739i	$0.625117\pi$
$168$	0	0
$169$	−19.0000	−1.46154
$170$	0	0
$171$	− 18.0000i	− 1.37649i
$172$	0	0
$173$	8.48528i	0.645124i	0.946548	+	0.322562i	$0.104544\pi$
$173$	8.48528i	0.645124i	−0.946548	+	0.322562i	$0.895456\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.89898	−0.368230
$178$	0	0
$179$	2.00000i	0.149487i	0.997203	+	0.0747435i	$0.0238138\pi$
$179$	2.00000i	0.149487i	−0.997203	+	0.0747435i	$0.976186\pi$
$180$	0	0
$181$	20.7846i	1.54491i	0.635071	+	0.772454i	$0.280971\pi$
$181$	20.7846i	1.54491i	−0.635071	+	0.772454i	$0.719029\pi$
$182$	0	0
$183$	8.48528	0.627250
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.79796i	0.716498i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.8564	−1.00261	−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564	−1.00261	−0.501307	+	0.865269i	$0.667147\pi$
$192$	0	0
$193$	4.89898	0.352636	0.176318	−	0.984333i	$-0.443581\pi$
$193$	4.89898	0.352636	0.176318	+	0.984333i	$0.443581\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.6274i	1.61214i	0.591822	+	0.806068i	$0.298409\pi$
$197$	22.6274i	1.61214i	−0.591822	+	0.806068i	$0.701591\pi$
$198$	0	0
$199$	−20.7846	−1.47338	−0.736691	−	0.676230i	$-0.763613\pi$
$199$	−20.7846	−1.47338	−0.736691	+	0.676230i	$0.763613\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	0	0
$203$	− 9.79796i	− 0.687682i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	21.2132	1.47442
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	10.0000i	0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$	10.0000i	0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$	0	0
$213$	16.9706i	1.16280i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.79796	−0.665129
$218$	0	0
$219$	− 12.0000i	− 0.810885i
$220$	0	0
$221$	− 27.7128i	− 1.86417i
$222$	0	0
$223$	−26.8701	−1.79935	−0.899676	−	0.436558i	$-0.856197\pi$
$223$	−26.8701	−1.79935	−0.899676	+	0.436558i	$0.856197\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 17.1464i	− 1.13805i	−0.822321	−	0.569024i	$-0.807321\pi$
$227$	− 17.1464i	− 1.13805i	0.822321	−	0.569024i	$-0.192679\pi$
$228$	0	0
$229$	− 6.92820i	− 0.457829i	−0.973447	−	0.228914i	$-0.926482\pi$
$229$	− 6.92820i	− 0.457829i	0.973447	−	0.228914i	$-0.0735176\pi$
$230$	0	0
$231$	6.92820	0.455842
$232$	0	0
$233$	−14.6969	−0.962828	−0.481414	−	0.876493i	$-0.659877\pi$
$233$	−14.6969	−0.962828	−0.481414	+	0.876493i	$0.659877\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 16.9706i	− 1.10236i
$238$	0	0
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	22.0454i	1.41421i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−33.9411	−2.15962
$248$	0	0
$249$	30.0000	1.90117
$250$	0	0
$251$	− 26.0000i	− 1.64111i	−0.571571	−	0.820553i	$-0.693666\pi$
$251$	− 26.0000i	− 1.64111i	0.571571	−	0.820553i	$-0.306334\pi$
$252$	0	0
$253$	14.1421i	0.889108i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.79796	0.611180	0.305590	−	0.952163i	$-0.401146\pi$
$257$	9.79796	0.611180	0.305590	+	0.952163i	$0.401146\pi$
$258$	0	0
$259$	− 4.00000i	− 0.248548i
$260$	0	0
$261$	− 20.7846i	− 1.28654i
$262$	0	0
$263$	−26.8701	−1.65688	−0.828439	−	0.560079i	$-0.810771\pi$
$263$	−26.8701	−1.65688	−0.828439	+	0.560079i	$0.810771\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.89898i	− 0.299813i
$268$	0	0
$269$	− 17.3205i	− 1.05605i	−0.849229	−	0.528025i	$-0.822933\pi$
$269$	− 17.3205i	− 1.05605i	0.849229	−	0.528025i	$-0.177067\pi$
$270$	0	0
$271$	−20.7846	−1.26258	−0.631288	−	0.775549i	$-0.717473\pi$
$271$	−20.7846	−1.26258	−0.631288	+	0.775549i	$0.717473\pi$
$272$	0	0
$273$	−19.5959	−1.18600
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.1421i	0.849719i	0.905259	+	0.424859i	$0.139676\pi$
$277$	14.1421i	0.849719i	−0.905259	+	0.424859i	$0.860324\pi$
$278$	0	0
$279$	−20.7846	−1.24434
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	2.44949i	0.145607i	0.997346	+	0.0728035i	$0.0231946\pi$
$283$	2.44949i	0.145607i	−0.997346	+	0.0728035i	$0.976805\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	36.0000i	2.11036i
$292$	0	0
$293$	− 2.82843i	− 0.165238i	−0.996581	−	0.0826192i	$-0.973671\pi$
$293$	− 2.82843i	− 0.165238i	0.996581	−	0.0826192i	$-0.0263285\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 40.0000i	− 2.31326i
$300$	0	0
$301$	3.46410i	0.199667i
$302$	0	0
$303$	−16.9706	−0.974933
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.44949i	− 0.139800i	−0.997554	−	0.0698999i	$-0.977732\pi$
$307$	− 2.44949i	− 0.139800i	0.997554	−	0.0698999i	$-0.0222680\pi$
$308$	0	0
$309$	− 24.2487i	− 1.37946i
$310$	0	0
$311$	−27.7128	−1.57145	−0.785725	−	0.618576i	$-0.787710\pi$
$311$	−27.7128	−1.57145	−0.785725	+	0.618576i	$0.787710\pi$
$312$	0	0
$313$	−29.3939	−1.66144	−0.830720	−	0.556690i	$-0.812071\pi$
$313$	−29.3939	−1.66144	−0.830720	+	0.556690i	$0.812071\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.6274i	1.27088i	0.772149	+	0.635441i	$0.219182\pi$
$317$	22.6274i	1.27088i	−0.772149	+	0.635441i	$0.780818\pi$
$318$	0	0
$319$	13.8564	0.775810
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	− 29.3939i	− 1.63552i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.48528	0.469237
$328$	0	0
$329$	6.00000	0.330791
$330$	0	0
$331$	− 6.00000i	− 0.329790i	−0.986311	−	0.164895i	$-0.947272\pi$
$331$	− 6.00000i	− 0.329790i	0.986311	−	0.164895i	$-0.0527285\pi$
$332$	0	0
$333$	− 8.48528i	− 0.464991i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.5959	1.06746	0.533729	−	0.845656i	$-0.320790\pi$
$337$	19.5959	1.06746	0.533729	+	0.845656i	$0.320790\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 13.8564i	− 0.750366i
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 17.1464i	− 0.920468i	−0.887798	−	0.460234i	$-0.847765\pi$
$347$	− 17.1464i	− 0.920468i	0.887798	−	0.460234i	$-0.152235\pi$
$348$	0	0
$349$	− 6.92820i	− 0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$	− 6.92820i	− 0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.79796	0.521493	0.260746	−	0.965407i	$-0.416031\pi$
$353$	9.79796	0.521493	0.260746	+	0.965407i	$0.416031\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 16.9706i	− 0.898177i
$358$	0	0
$359$	−27.7128	−1.46263	−0.731313	−	0.682042i	$-0.761092\pi$
$359$	−27.7128	−1.46263	−0.731313	+	0.682042i	$0.761092\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	− 17.1464i	− 0.899954i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.89949	0.516749	0.258375	−	0.966045i	$-0.416813\pi$
$367$	9.89949	0.516749	0.258375	+	0.966045i	$0.416813\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	0	0
$372$	0	0
$373$	19.7990i	1.02515i	0.858642	+	0.512576i	$0.171309\pi$
$373$	19.7990i	1.02515i	−0.858642	+	0.512576i	$0.828691\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−39.1918	−2.01848
$378$	0	0
$379$	− 14.0000i	− 0.719132i	−0.933120	−	0.359566i	$-0.882925\pi$
$379$	− 14.0000i	− 0.719132i	0.933120	−	0.359566i	$-0.117075\pi$
$380$	0	0
$381$	17.3205i	0.887357i
$382$	0	0
$383$	12.7279	0.650366	0.325183	−	0.945651i	$-0.394574\pi$
$383$	12.7279	0.650366	0.325183	+	0.945651i	$0.394574\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.34847i	0.373544i
$388$	0	0
$389$	3.46410i	0.175637i	0.996136	+	0.0878185i	$0.0279895\pi$
$389$	3.46410i	0.175637i	−0.996136	+	0.0878185i	$0.972010\pi$
$390$	0	0
$391$	34.6410	1.75187
$392$	0	0
$393$	24.4949	1.23560
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.3137i	0.567819i	0.958851	+	0.283909i	$0.0916315\pi$
$397$	11.3137i	0.567819i	−0.958851	+	0.283909i	$0.908369\pi$
$398$	0	0
$399$	−20.7846	−1.04053
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	39.1918i	1.95228i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.65685	0.280400
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	− 24.0000i	− 1.18383i
$412$	0	0
$413$	2.82843i	0.139178i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	24.4949	1.19952
$418$	0	0
$419$	26.0000i	1.27018i	0.772437	+	0.635092i	$0.219038\pi$
$419$	26.0000i	1.27018i	−0.772437	+	0.635092i	$0.780962\pi$
$420$	0	0
$421$	− 3.46410i	− 0.168830i	−0.996431	−	0.0844150i	$-0.973098\pi$
$421$	− 3.46410i	− 0.168830i	0.996431	−	0.0844150i	$-0.0269021\pi$
$422$	0	0
$423$	12.7279	0.618853
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.89898i	− 0.237078i
$428$	0	0
$429$	− 27.7128i	− 1.33799i
$430$	0	0
$431$	−6.92820	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$431$	−6.92820	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$432$	0	0
$433$	34.2929	1.64801	0.824005	−	0.566583i	$-0.191735\pi$
$433$	34.2929	1.64801	0.824005	+	0.566583i	$0.191735\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 42.4264i	− 2.02953i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	15.0000	0.714286
$442$	0	0
$443$	22.0454i	1.04741i	0.851900	+	0.523704i	$0.175450\pi$
$443$	22.0454i	1.04741i	−0.851900	+	0.523704i	$0.824550\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−25.4558	−1.20402
$448$	0	0
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	0	0
$451$	8.00000i	0.376705i
$452$	0	0
$453$	− 33.9411i	− 1.59469i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.5959	−0.916658	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	−19.5959	−0.916658	−0.458329	+	0.888783i	$0.651552\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 20.7846i	− 0.968036i	−0.875058	−	0.484018i	$-0.839177\pi$
$461$	− 20.7846i	− 0.968036i	0.875058	−	0.484018i	$-0.160823\pi$
$462$	0	0
$463$	−1.41421	−0.0657241	−0.0328620	−	0.999460i	$-0.510462\pi$
$463$	−1.41421	−0.0657241	−0.0328620	+	0.999460i	$0.510462\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.8434i	1.47354i	0.676146	+	0.736768i	$0.263649\pi$
$467$	31.8434i	1.47354i	−0.676146	+	0.736768i	$0.736351\pi$
$468$	0	0
$469$	3.46410i	0.159957i
$470$	0	0
$471$	−6.92820	−0.319235
$472$	0	0
$473$	−4.89898	−0.225255
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−41.5692	−1.89935	−0.949673	−	0.313243i	$-0.898585\pi$
$479$	−41.5692	−1.89935	−0.949673	+	0.313243i	$0.898585\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	− 24.4949i	− 1.11456i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.8701	1.21760	0.608799	−	0.793324i	$-0.291651\pi$
$487$	26.8701	1.21760	0.608799	+	0.793324i	$0.291651\pi$
$488$	0	0
$489$	−54.0000	−2.44196
$490$	0	0
$491$	− 38.0000i	− 1.71492i	−0.514554	−	0.857458i	$-0.672042\pi$
$491$	− 38.0000i	− 1.71492i	0.514554	−	0.857458i	$-0.327958\pi$
$492$	0	0
$493$	− 33.9411i	− 1.52863i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.79796	0.439499
$498$	0	0
$499$	− 6.00000i	− 0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$	− 6.00000i	− 0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$	0	0
$501$	24.2487i	1.08335i
$502$	0	0
$503$	21.2132	0.945850	0.472925	−	0.881103i	$-0.343198\pi$
$503$	21.2132	0.945850	0.472925	+	0.881103i	$0.343198\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	46.5403i	2.06693i
$508$	0	0
$509$	− 20.7846i	− 0.921262i	−0.887592	−	0.460631i	$-0.847623\pi$
$509$	− 20.7846i	− 0.921262i	0.887592	−	0.460631i	$-0.152377\pi$
$510$	0	0
$511$	−6.92820	−0.306486
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.48528i	0.373182i
$518$	0	0
$519$	20.7846	0.912343
$520$	0	0
$521$	34.0000	1.48957	0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000	1.48957	0.744784	+	0.667306i	$0.232553\pi$
$522$	0	0
$523$	2.44949i	0.107109i	0.998565	+	0.0535544i	$0.0170550\pi$
$523$	2.44949i	0.107109i	−0.998565	+	0.0535544i	$0.982945\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.9411	−1.47850
$528$	0	0
$529$	27.0000	1.17391
$530$	0	0
$531$	6.00000i	0.260378i
$532$	0	0
$533$	− 22.6274i	− 0.980102i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.89898	0.211407
$538$	0	0
$539$	10.0000i	0.430730i
$540$	0	0
$541$	6.92820i	0.297867i	0.988847	+	0.148933i	$0.0475840\pi$
$541$	6.92820i	0.297867i	−0.988847	+	0.148933i	$0.952416\pi$
$542$	0	0
$543$	50.9117	2.18483
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 26.9444i	− 1.15206i	−0.817429	−	0.576029i	$-0.804601\pi$
$547$	− 26.9444i	− 1.15206i	0.817429	−	0.576029i	$-0.195399\pi$
$548$	0	0
$549$	− 10.3923i	− 0.443533i
$550$	0	0
$551$	−41.5692	−1.77091
$552$	0	0
$553$	−9.79796	−0.416652
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 25.4558i	− 1.07860i	−0.842114	−	0.539299i	$-0.818689\pi$
$557$	− 25.4558i	− 1.07860i	0.842114	−	0.539299i	$-0.181311\pi$
$558$	0	0
$559$	13.8564	0.586064
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	7.34847i	0.309701i	0.987938	+	0.154851i	$0.0494896\pi$
$563$	7.34847i	0.309701i	−0.987938	+	0.154851i	$0.950510\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.7279	0.534522
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	30.0000i	1.25546i	0.778431	+	0.627730i	$0.216016\pi$
$571$	30.0000i	1.25546i	−0.778431	+	0.627730i	$0.783984\pi$
$572$	0	0
$573$	33.9411i	1.41791i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.3939	1.22368	0.611842	−	0.790980i	$-0.290429\pi$
$577$	29.3939	1.22368	0.611842	+	0.790980i	$0.290429\pi$
$578$	0	0
$579$	− 12.0000i	− 0.498703i
$580$	0	0
$581$	− 17.3205i	− 0.718576i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.2474i	0.505506i	0.967531	+	0.252753i	$0.0813361\pi$
$587$	12.2474i	0.505506i	−0.967531	+	0.252753i	$0.918664\pi$
$588$	0	0
$589$	41.5692i	1.71283i
$590$	0	0
$591$	55.4256	2.27991
$592$	0	0
$593$	29.3939	1.20706	0.603531	−	0.797340i	$-0.293760\pi$
$593$	29.3939	1.20706	0.603531	+	0.797340i	$0.293760\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	50.9117i	2.08368i
$598$	0	0
$599$	20.7846	0.849236	0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846	0.849236	0.424618	+	0.905373i	$0.360408\pi$
$600$	0	0
$601$	−24.0000	−0.978980	−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000	−0.978980	−0.489490	+	0.872009i	$0.662817\pi$
$602$	0	0
$603$	7.34847i	0.299253i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.0122	1.66463	0.832317	−	0.554300i	$-0.187014\pi$
$607$	41.0122	1.66463	0.832317	+	0.554300i	$0.187014\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	− 24.0000i	− 0.970936i
$612$	0	0
$613$	5.65685i	0.228478i	0.993453	+	0.114239i	$0.0364430\pi$
$613$	5.65685i	0.228478i	−0.993453	+	0.114239i	$0.963557\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.4949	−0.986127	−0.493064	−	0.869993i	$-0.664123\pi$
$617$	−24.4949	−0.986127	−0.493064	+	0.869993i	$0.664123\pi$
$618$	0	0
$619$	14.0000i	0.562708i	0.959604	+	0.281354i	$0.0907834\pi$
$619$	14.0000i	0.562708i	−0.959604	+	0.281354i	$0.909217\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.82843	−0.113319
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 29.3939i	− 1.17388i
$628$	0	0
$629$	− 13.8564i	− 0.552491i
$630$	0	0
$631$	13.8564	0.551615	0.275807	−	0.961213i	$-0.411055\pi$
$631$	13.8564	0.551615	0.275807	+	0.961213i	$0.411055\pi$
$632$	0	0
$633$	24.4949	0.973585
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 28.2843i	− 1.12066i
$638$	0	0
$639$	20.7846	0.822226
$640$	0	0
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	2.44949i	0.0965984i	0.998833	+	0.0482992i	$0.0153801\pi$
$643$	2.44949i	0.0965984i	−0.998833	+	0.0482992i	$0.984620\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.24264	0.166795	0.0833977	−	0.996516i	$-0.473423\pi$
$647$	4.24264	0.166795	0.0833977	+	0.996516i	$0.473423\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	24.0000i	0.940634i
$652$	0	0
$653$	5.65685i	0.221370i	0.993856	+	0.110685i	$0.0353044\pi$
$653$	5.65685i	0.221370i	−0.993856	+	0.110685i	$0.964696\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−14.6969	−0.573382
$658$	0	0
$659$	2.00000i	0.0779089i	0.999241	+	0.0389545i	$0.0124027\pi$
$659$	2.00000i	0.0779089i	−0.999241	+	0.0389545i	$0.987597\pi$
$660$	0	0
$661$	− 10.3923i	− 0.404214i	−0.979363	−	0.202107i	$-0.935221\pi$
$661$	− 10.3923i	− 0.404214i	0.979363	−	0.202107i	$-0.0647788\pi$
$662$	0	0
$663$	−67.8823	−2.63633
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 48.9898i	− 1.89689i
$668$	0	0
$669$	65.8179i	2.54467i
$670$	0	0
$671$	6.92820	0.267460
$672$	0	0
$673$	−24.4949	−0.944209	−0.472104	−	0.881543i	$-0.656505\pi$
$673$	−24.4949	−0.944209	−0.472104	+	0.881543i	$0.656505\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5.65685i	− 0.217411i	−0.994074	−	0.108705i	$-0.965330\pi$
$677$	− 5.65685i	− 0.217411i	0.994074	−	0.108705i	$-0.0346705\pi$
$678$	0	0
$679$	20.7846	0.797640
$680$	0	0
$681$	−42.0000	−1.60944
$682$	0	0
$683$	− 2.44949i	− 0.0937271i	−0.998901	−	0.0468636i	$-0.985077\pi$
$683$	− 2.44949i	− 0.0937271i	0.998901	−	0.0468636i	$-0.0149226\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16.9706	−0.647467
$688$	0	0
$689$	0	0
$690$	0	0
$691$	42.0000i	1.59776i	0.601494	+	0.798878i	$0.294573\pi$
$691$	42.0000i	1.59776i	−0.601494	+	0.798878i	$0.705427\pi$
$692$	0	0
$693$	− 8.48528i	− 0.322329i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.5959	0.742248
$698$	0	0
$699$	36.0000i	1.36165i
$700$	0	0
$701$	− 3.46410i	− 0.130837i	−0.997858	−	0.0654187i	$-0.979162\pi$
$701$	− 3.46410i	− 0.130837i	0.997858	−	0.0654187i	$-0.0208383\pi$
$702$	0	0
$703$	−16.9706	−0.640057
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.79796i	0.368490i
$708$	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
$713$	−48.9898	−1.83468
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.9706i	0.633777i
$718$	0	0
$719$	6.92820	0.258378	0.129189	−	0.991620i	$-0.458763\pi$
$719$	6.92820	0.258378	0.129189	+	0.991620i	$0.458763\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−18.3848	−0.681854	−0.340927	−	0.940090i	$-0.610741\pi$
$727$	−18.3848	−0.681854	−0.340927	+	0.940090i	$0.610741\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	12.0000i	0.443836i
$732$	0	0
$733$	− 19.7990i	− 0.731292i	−0.930754	−	0.365646i	$-0.880848\pi$
$733$	− 19.7990i	− 0.731292i	0.930754	−	0.365646i	$-0.119152\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.89898	−0.180456
$738$	0	0
$739$	− 30.0000i	− 1.10357i	−0.833987	−	0.551784i	$-0.813947\pi$
$739$	− 30.0000i	− 1.10357i	0.833987	−	0.551784i	$-0.186053\pi$
$740$	0	0
$741$	83.1384i	3.05417i
$742$	0	0
$743$	−4.24264	−0.155647	−0.0778237	−	0.996967i	$-0.524797\pi$
$743$	−4.24264	−0.155647	−0.0778237	+	0.996967i	$0.524797\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 36.7423i	− 1.34433i
$748$	0	0
$749$	3.46410i	0.126576i
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	−63.6867	−2.32087
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.82843i	− 0.102801i	−0.998678	−	0.0514005i	$-0.983632\pi$
$757$	− 2.82843i	− 0.102801i	0.998678	−	0.0514005i	$-0.0163685\pi$
$758$	0	0
$759$	34.6410	1.25739
$760$	0	0
$761$	38.0000	1.37750	0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000	1.37750	0.688749	+	0.724999i	$0.258160\pi$
$762$	0	0
$763$	− 4.89898i	− 0.177355i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.3137	0.408514
$768$	0	0
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	− 24.0000i	− 0.864339i
$772$	0	0
$773$	45.2548i	1.62770i	0.581073	+	0.813852i	$0.302633\pi$
$773$	45.2548i	1.62770i	−0.581073	+	0.813852i	$0.697367\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−9.79796	−0.351500
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	13.8564i	0.495821i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	46.5403i	1.65898i	0.558520	+	0.829491i	$0.311370\pi$
$787$	46.5403i	1.65898i	−0.558520	+	0.829491i	$0.688630\pi$
$788$	0	0
$789$	65.8179i	2.34318i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−19.5959	−0.695871
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.9411i	1.20226i	0.799153	+	0.601128i	$0.205282\pi$
$797$	33.9411i	1.20226i	−0.799153	+	0.601128i	$0.794718\pi$
$798$	0	0
$799$	20.7846	0.735307
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	− 9.79796i	− 0.345762i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−42.4264	−1.49348
$808$	0	0
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	0	0
$811$	22.0000i	0.772524i	0.922389	+	0.386262i	$0.126234\pi$
$811$	22.0000i	0.772524i	−0.922389	+	0.386262i	$0.873766\pi$
$812$	0	0
$813$	50.9117i	1.78555i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.6969	0.514181
$818$	0	0
$819$	24.0000i	0.838628i
$820$	0	0
$821$	− 51.9615i	− 1.81347i	−0.421701	−	0.906735i	$-0.638567\pi$
$821$	− 51.9615i	− 1.81347i	0.421701	−	0.906735i	$-0.361433\pi$
$822$	0	0
$823$	−35.3553	−1.23241	−0.616205	−	0.787586i	$-0.711331\pi$
$823$	−35.3553	−1.23241	−0.616205	+	0.787586i	$0.711331\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.8434i	1.10730i	0.832749	+	0.553651i	$0.186766\pi$
$827$	31.8434i	1.10730i	−0.832749	+	0.553651i	$0.813234\pi$
$828$	0	0
$829$	− 45.0333i	− 1.56407i	−0.623233	−	0.782036i	$-0.714181\pi$
$829$	− 45.0333i	− 1.56407i	0.623233	−	0.782036i	$-0.285819\pi$
$830$	0	0
$831$	34.6410	1.20168
$832$	0	0
$833$	24.4949	0.848698
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	41.5692	1.43513	0.717564	−	0.696492i	$-0.245257\pi$
$839$	41.5692	1.43513	0.717564	+	0.696492i	$0.245257\pi$
$840$	0	0
$841$	−19.0000	−0.655172
$842$	0	0
$843$	− 19.5959i	− 0.674919i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.89949	−0.340151
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	− 20.0000i	− 0.685591i
$852$	0	0
$853$	5.65685i	0.193687i	0.995300	+	0.0968435i	$0.0308746\pi$
$853$	5.65685i	0.193687i	−0.995300	+	0.0968435i	$0.969125\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.1918	1.33877	0.669384	−	0.742917i	$-0.266558\pi$
$857$	39.1918	1.33877	0.669384	+	0.742917i	$0.266558\pi$
$858$	0	0
$859$	22.0000i	0.750630i	0.926897	+	0.375315i	$0.122466\pi$
$859$	22.0000i	0.750630i	−0.926897	+	0.375315i	$0.877534\pi$
$860$	0	0
$861$	− 13.8564i	− 0.472225i
$862$	0	0
$863$	−24.0416	−0.818387	−0.409193	−	0.912448i	$-0.634190\pi$
$863$	−24.0416	−0.818387	−0.409193	+	0.912448i	$0.634190\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 17.1464i	− 0.582323i
$868$	0	0
$869$	− 13.8564i	− 0.470046i
$870$	0	0
$871$	13.8564	0.469506
$872$	0	0
$873$	44.0908	1.49225
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.7696i	1.24162i	0.783961	+	0.620810i	$0.213196\pi$
$877$	36.7696i	1.24162i	−0.783961	+	0.620810i	$0.786804\pi$
$878$	0	0
$879$	−6.92820	−0.233682
$880$	0	0
$881$	52.0000	1.75192	0.875962	−	0.482380i	$-0.160227\pi$
$881$	52.0000	1.75192	0.875962	+	0.482380i	$0.160227\pi$
$882$	0	0
$883$	− 12.2474i	− 0.412159i	−0.978535	−	0.206080i	$-0.933929\pi$
$883$	− 12.2474i	− 0.412159i	0.978535	−	0.206080i	$-0.0660706\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.4975	1.66196	0.830981	−	0.556300i	$-0.187780\pi$
$887$	49.4975	1.66196	0.830981	+	0.556300i	$0.187780\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	0	0
$891$	18.0000i	0.603023i
$892$	0	0
$893$	− 25.4558i	− 0.851847i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−97.9796	−3.27144
$898$	0	0
$899$	48.0000i	1.60089i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	8.48528	0.282372
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26.9444i	0.894674i	0.894366	+	0.447337i	$0.147627\pi$
$907$	26.9444i	0.894674i	−0.894366	+	0.447337i	$0.852373\pi$
$908$	0	0
$909$	20.7846i	0.689382i
$910$	0	0
$911$	−48.4974	−1.60679	−0.803396	−	0.595446i	$-0.796976\pi$
$911$	−48.4974	−1.60679	−0.803396	+	0.595446i	$0.796976\pi$
$912$	0	0
$913$	24.4949	0.810663
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.1421i	− 0.467014i
$918$	0	0
$919$	−6.92820	−0.228540	−0.114270	−	0.993450i	$-0.536453\pi$
$919$	−6.92820	−0.228540	−0.114270	+	0.993450i	$0.536453\pi$
$920$	0	0
$921$	−6.00000	−0.197707
$922$	0	0
$923$	− 39.1918i	− 1.29001i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−29.6985	−0.975426
$928$	0	0
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	0	0
$931$	− 30.0000i	− 0.983210i
$932$	0	0
$933$	67.8823i	2.22237i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.4949	0.800213	0.400107	−	0.916469i	$-0.368973\pi$
$937$	24.4949	0.800213	0.400107	+	0.916469i	$0.368973\pi$
$938$	0	0
$939$	72.0000i	2.34963i
$940$	0	0
$941$	34.6410i	1.12926i	0.825342	+	0.564632i	$0.190982\pi$
$941$	34.6410i	1.12926i	−0.825342	+	0.564632i	$0.809018\pi$
$942$	0	0
$943$	28.2843	0.921063
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41.6413i	− 1.35316i	−0.736369	−	0.676581i	$-0.763461\pi$
$947$	− 41.6413i	− 1.35316i	0.736369	−	0.676581i	$-0.236539\pi$
$948$	0	0
$949$	27.7128i	0.899596i
$950$	0	0
$951$	55.4256	1.79730
$952$	0	0
$953$	39.1918	1.26955	0.634774	−	0.772698i	$-0.281093\pi$
$953$	39.1918	1.26955	0.634774	+	0.772698i	$0.281093\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 33.9411i	− 1.09716i
$958$	0	0
$959$	−13.8564	−0.447447
$960$	0	0
$961$	17.0000	0.548387
$962$	0	0
$963$	7.34847i	0.236801i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15.5563	−0.500258	−0.250129	−	0.968212i	$-0.580473\pi$
$967$	−15.5563	−0.500258	−0.250129	+	0.968212i	$0.580473\pi$
$968$	0	0
$969$	−72.0000	−2.31297
$970$	0	0
$971$	− 22.0000i	− 0.706014i	−0.935621	−	0.353007i	$-0.885159\pi$
$971$	− 22.0000i	− 0.706014i	0.935621	−	0.353007i	$-0.114841\pi$
$972$	0	0
$973$	− 14.1421i	− 0.453376i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.89898	0.156732	0.0783661	−	0.996925i	$-0.475030\pi$
$977$	4.89898	0.156732	0.0783661	+	0.996925i	$0.475030\pi$
$978$	0	0
$979$	− 4.00000i	− 0.127841i
$980$	0	0
$981$	− 10.3923i	− 0.331801i
$982$	0	0
$983$	46.6690	1.48851	0.744256	−	0.667895i	$-0.232804\pi$
$983$	46.6690	1.48851	0.744256	+	0.667895i	$0.232804\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 14.6969i	− 0.467809i
$988$	0	0
$989$	17.3205i	0.550760i
$990$	0	0
$991$	27.7128	0.880327	0.440163	−	0.897918i	$-0.354921\pi$
$991$	27.7128	0.880327	0.440163	+	0.897918i	$0.354921\pi$
$992$	0	0
$993$	−14.6969	−0.466393
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 56.5685i	− 1.79154i	−0.444514	−	0.895772i	$-0.646624\pi$
$997$	− 56.5685i	− 1.79154i	0.444514	−	0.895772i	$-0.353376\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.d.i.801.1		8
4.3	odd	2	inner	1600.2.d.i.801.8		8
5.2	odd	4		320.2.f.b.289.2	yes	8
5.3	odd	4		320.2.f.b.289.8	yes	8
5.4	even	2	inner	1600.2.d.i.801.7		8
8.3	odd	2	inner	1600.2.d.i.801.3		8
8.5	even	2	inner	1600.2.d.i.801.6		8
15.2	even	4		2880.2.d.g.289.5		8
15.8	even	4		2880.2.d.g.289.2		8
16.3	odd	4		6400.2.a.cu.1.1		4
16.5	even	4		6400.2.a.cu.1.2		4
16.11	odd	4		6400.2.a.ct.1.3		4
16.13	even	4		6400.2.a.ct.1.4		4
20.3	even	4		320.2.f.b.289.4	yes	8
20.7	even	4		320.2.f.b.289.6	yes	8
20.19	odd	2	inner	1600.2.d.i.801.2		8
40.3	even	4		320.2.f.b.289.5	yes	8
40.13	odd	4		320.2.f.b.289.1	✓	8
40.19	odd	2	inner	1600.2.d.i.801.5		8
40.27	even	4		320.2.f.b.289.3	yes	8
40.29	even	2	inner	1600.2.d.i.801.4		8
40.37	odd	4		320.2.f.b.289.7	yes	8
60.23	odd	4		2880.2.d.g.289.1		8
60.47	odd	4		2880.2.d.g.289.6		8
80.3	even	4		1280.2.c.h.769.2		4
80.13	odd	4		1280.2.c.g.769.4		4
80.19	odd	4		6400.2.a.cu.1.4		4
80.27	even	4		1280.2.c.g.769.1		4
80.29	even	4		6400.2.a.ct.1.1		4
80.37	odd	4		1280.2.c.h.769.3		4
80.43	even	4		1280.2.c.g.769.3		4
80.53	odd	4		1280.2.c.h.769.1		4
80.59	odd	4		6400.2.a.ct.1.2		4
80.67	even	4		1280.2.c.h.769.4		4
80.69	even	4		6400.2.a.cu.1.3		4
80.77	odd	4		1280.2.c.g.769.2		4
120.53	even	4		2880.2.d.g.289.8		8
120.77	even	4		2880.2.d.g.289.3		8
120.83	odd	4		2880.2.d.g.289.7		8
120.107	odd	4		2880.2.d.g.289.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.f.b.289.1	✓	8	40.13	odd	4
320.2.f.b.289.2	yes	8	5.2	odd	4
320.2.f.b.289.3	yes	8	40.27	even	4
320.2.f.b.289.4	yes	8	20.3	even	4
320.2.f.b.289.5	yes	8	40.3	even	4
320.2.f.b.289.6	yes	8	20.7	even	4
320.2.f.b.289.7	yes	8	40.37	odd	4
320.2.f.b.289.8	yes	8	5.3	odd	4
1280.2.c.g.769.1		4	80.27	even	4
1280.2.c.g.769.2		4	80.77	odd	4
1280.2.c.g.769.3		4	80.43	even	4
1280.2.c.g.769.4		4	80.13	odd	4
1280.2.c.h.769.1		4	80.53	odd	4
1280.2.c.h.769.2		4	80.3	even	4
1280.2.c.h.769.3		4	80.37	odd	4
1280.2.c.h.769.4		4	80.67	even	4
1600.2.d.i.801.1		8	1.1	even	1	trivial
1600.2.d.i.801.2		8	20.19	odd	2	inner
1600.2.d.i.801.3		8	8.3	odd	2	inner
1600.2.d.i.801.4		8	40.29	even	2	inner
1600.2.d.i.801.5		8	40.19	odd	2	inner
1600.2.d.i.801.6		8	8.5	even	2	inner
1600.2.d.i.801.7		8	5.4	even	2	inner
1600.2.d.i.801.8		8	4.3	odd	2	inner
2880.2.d.g.289.1		8	60.23	odd	4
2880.2.d.g.289.2		8	15.8	even	4
2880.2.d.g.289.3		8	120.77	even	4
2880.2.d.g.289.4		8	120.107	odd	4
2880.2.d.g.289.5		8	15.2	even	4
2880.2.d.g.289.6		8	60.47	odd	4
2880.2.d.g.289.7		8	120.83	odd	4
2880.2.d.g.289.8		8	120.53	even	4
6400.2.a.ct.1.1		4	80.29	even	4
6400.2.a.ct.1.2		4	80.59	odd	4
6400.2.a.ct.1.3		4	16.11	odd	4
6400.2.a.ct.1.4		4	16.13	even	4
6400.2.a.cu.1.1		4	16.3	odd	4
6400.2.a.cu.1.2		4	16.5	even	4
6400.2.a.cu.1.3		4	80.69	even	4
6400.2.a.cu.1.4		4	80.19	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)

\(f(q)\)	\(=\)	\(q-2.44949i q^{3} -1.41421 q^{7} -3.00000 q^{9} -2.00000i q^{11} +5.65685i q^{13} -4.89898 q^{17} +6.00000i q^{19} +3.46410i q^{21} -7.07107 q^{23} +6.92820i q^{29} +6.92820 q^{31} -4.89898 q^{33} +2.82843i q^{37} +13.8564 q^{39} -4.00000 q^{41} -2.44949i q^{43} -4.24264 q^{47} -5.00000 q^{49} +12.0000i q^{51} +14.6969 q^{57} -2.00000i q^{59} +3.46410i q^{61} +4.24264 q^{63} -2.44949i q^{67} +17.3205i q^{69} -6.92820 q^{71} +4.89898 q^{73} +2.82843i q^{77} +6.92820 q^{79} -9.00000 q^{81} +12.2474i q^{83} +16.9706 q^{87} +2.00000 q^{89} -8.00000i q^{91} -16.9706i q^{93} -14.6969 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} - 32 q^{41} - 40 q^{49} - 72 q^{81} + 16 q^{89}+O(q^{100}) \) 8 * q - 24 * q^9 - 32 * q^41 - 40 * q^49 - 72 * q^81 + 16 * q^89

Introduction

L-functions

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