LMFDB - Embedded newform 9280.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9280,2,Mod(1,9280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.a (trivial)

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:	yes
Analytic conductor:	$74.1011730757$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{3} -1.00000 q^{5} -4.82843 q^{7} +1.00000 q^{9} -0.828427 q^{11} +2.00000 q^{13} -2.00000 q^{15} +2.82843 q^{17} +4.82843 q^{19} -9.65685 q^{21} -3.17157 q^{23} +1.00000 q^{25} -4.00000 q^{27} -1.00000 q^{29} +6.48528 q^{31} -1.65685 q^{33} +4.82843 q^{35} +8.48528 q^{37} +4.00000 q^{39} -6.00000 q^{41} +6.00000 q^{43} -1.00000 q^{45} -11.6569 q^{47} +16.3137 q^{49} +5.65685 q^{51} +3.65685 q^{53} +0.828427 q^{55} +9.65685 q^{57} +3.65685 q^{61} -4.82843 q^{63} -2.00000 q^{65} -6.48528 q^{67} -6.34315 q^{69} -15.3137 q^{71} +8.48528 q^{73} +2.00000 q^{75} +4.00000 q^{77} -2.48528 q^{79} -11.0000 q^{81} -7.17157 q^{83} -2.82843 q^{85} -2.00000 q^{87} -7.65685 q^{89} -9.65685 q^{91} +12.9706 q^{93} -4.82843 q^{95} -12.4853 q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 4 q^{15} + 4 q^{19} - 8 q^{21} - 12 q^{23} + 2 q^{25} - 8 q^{27} - 2 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} + 8 q^{39} - 12 q^{41} + 12 q^{43}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^13 - 4 * q^15 + 4 * q^19 - 8 * q^21 - 12 * q^23 + 2 * q^25 - 8 * q^27 - 2 * q^29 - 4 * q^31 + 8 * q^33 + 4 * q^35 + 8 * q^39 - 12 * q^41 + 12 * q^43 - 2 * q^45 - 12 * q^47 + 10 * q^49 - 4 * q^53 - 4 * q^55 + 8 * q^57 - 4 * q^61 - 4 * q^63 - 4 * q^65 + 4 * q^67 - 24 * q^69 - 8 * q^71 + 4 * q^75 + 8 * q^77 + 12 * q^79 - 22 * q^81 - 20 * q^83 - 4 * q^87 - 4 * q^89 - 8 * q^91 - 8 * q^93 - 4 * q^95 - 8 * q^97 + 4 * q^99

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	4.82843	1.10772	0.553859	−	0.832611i	$-0.313155\pi$
$19$	4.82843	1.10772	0.553859	+	0.832611i	$0.313155\pi$
$20$	0	0
$21$	−9.65685	−2.10730
$22$	0	0
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.48528	1.16479	0.582395	−	0.812906i	$-0.302116\pi$
$31$	6.48528	1.16479	0.582395	+	0.812906i	$0.302116\pi$
$32$	0	0
$33$	−1.65685	−0.288421
$34$	0	0
$35$	4.82843	0.816153
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−11.6569	−1.70033	−0.850163	−	0.526519i	$-0.823497\pi$
$47$	−11.6569	−1.70033	−0.850163	+	0.526519i	$0.823497\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	0	0
$51$	5.65685	0.792118
$52$	0	0
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	0	0
$55$	0.828427	0.111705
$56$	0	0
$57$	9.65685	1.27908
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	3.65685	0.468212	0.234106	−	0.972211i	$-0.424784\pi$
$61$	3.65685	0.468212	0.234106	+	0.972211i	$0.424784\pi$
$62$	0	0
$63$	−4.82843	−0.608325
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−6.48528	−0.792303	−0.396152	−	0.918185i	$-0.629655\pi$
$67$	−6.48528	−0.792303	−0.396152	+	0.918185i	$0.629655\pi$
$68$	0	0
$69$	−6.34315	−0.763625
$70$	0	0
$71$	−15.3137	−1.81740	−0.908701	−	0.417447i	$-0.862925\pi$
$71$	−15.3137	−1.81740	−0.908701	+	0.417447i	$0.862925\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	0	0
$75$	2.00000	0.230940
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	−2.48528	−0.279616	−0.139808	−	0.990179i	$-0.544649\pi$
$79$	−2.48528	−0.279616	−0.139808	+	0.990179i	$0.544649\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−7.17157	−0.787182	−0.393591	−	0.919286i	$-0.628767\pi$
$83$	−7.17157	−0.787182	−0.393591	+	0.919286i	$0.628767\pi$
$84$	0	0
$85$	−2.82843	−0.306786
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−7.65685	−0.811625	−0.405812	−	0.913956i	$-0.633011\pi$
$89$	−7.65685	−0.811625	−0.405812	+	0.913956i	$0.633011\pi$
$90$	0	0
$91$	−9.65685	−1.01231
$92$	0	0
$93$	12.9706	1.34498
$94$	0	0
$95$	−4.82843	−0.495386
$96$	0	0
$97$	−12.4853	−1.26769	−0.633844	−	0.773461i	$-0.718524\pi$
$97$	−12.4853	−1.26769	−0.633844	+	0.773461i	$0.718524\pi$
$98$	0	0
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	−15.6569	−1.55792	−0.778958	−	0.627077i	$-0.784251\pi$
$101$	−15.6569	−1.55792	−0.778958	+	0.627077i	$0.784251\pi$
$102$	0	0
$103$	16.1421	1.59053	0.795266	−	0.606261i	$-0.207331\pi$
$103$	16.1421	1.59053	0.795266	+	0.606261i	$0.207331\pi$
$104$	0	0
$105$	9.65685	0.942412
$106$	0	0
$107$	−20.1421	−1.94721	−0.973607	−	0.228232i	$-0.926706\pi$
$107$	−20.1421	−1.94721	−0.973607	+	0.228232i	$0.926706\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	16.9706	1.61077
$112$	0	0
$113$	−2.82843	−0.266076	−0.133038	−	0.991111i	$-0.542473\pi$
$113$	−2.82843	−0.266076	−0.133038	+	0.991111i	$0.542473\pi$
$114$	0	0
$115$	3.17157	0.295751
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−13.6569	−1.25192
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	−12.0000	−1.08200
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	12.1421	1.06086	0.530432	−	0.847728i	$-0.322030\pi$
$131$	12.1421	1.06086	0.530432	+	0.847728i	$0.322030\pi$
$132$	0	0
$133$	−23.3137	−2.02155
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−5.17157	−0.441837	−0.220919	−	0.975292i	$-0.570906\pi$
$137$	−5.17157	−0.441837	−0.220919	+	0.975292i	$0.570906\pi$
$138$	0	0
$139$	−21.6569	−1.83691	−0.918455	−	0.395525i	$-0.870563\pi$
$139$	−21.6569	−1.83691	−0.918455	+	0.395525i	$0.870563\pi$
$140$	0	0
$141$	−23.3137	−1.96337
$142$	0	0
$143$	−1.65685	−0.138553
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	32.6274	2.69106
$148$	0	0
$149$	−9.31371	−0.763009	−0.381504	−	0.924367i	$-0.624594\pi$
$149$	−9.31371	−0.763009	−0.381504	+	0.924367i	$0.624594\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	2.82843	0.228665
$154$	0	0
$155$	−6.48528	−0.520910
$156$	0	0
$157$	−0.485281	−0.0387297	−0.0193648	−	0.999812i	$-0.506164\pi$
$157$	−0.485281	−0.0387297	−0.0193648	+	0.999812i	$0.506164\pi$
$158$	0	0
$159$	7.31371	0.580015
$160$	0	0
$161$	15.3137	1.20689
$162$	0	0
$163$	8.34315	0.653486	0.326743	−	0.945113i	$-0.394049\pi$
$163$	8.34315	0.653486	0.326743	+	0.945113i	$0.394049\pi$
$164$	0	0
$165$	1.65685	0.128986
$166$	0	0
$167$	−2.48528	−0.192317	−0.0961584	−	0.995366i	$-0.530656\pi$
$167$	−2.48528	−0.192317	−0.0961584	+	0.995366i	$0.530656\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	4.82843	0.369239
$172$	0	0
$173$	−17.3137	−1.31634	−0.658168	−	0.752871i	$-0.728669\pi$
$173$	−17.3137	−1.31634	−0.658168	+	0.752871i	$0.728669\pi$
$174$	0	0
$175$	−4.82843	−0.364995
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.3137	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$179$	23.3137	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	7.31371	0.540645
$184$	0	0
$185$	−8.48528	−0.623850
$186$	0	0
$187$	−2.34315	−0.171348
$188$	0	0
$189$	19.3137	1.40487
$190$	0	0
$191$	−20.8284	−1.50709	−0.753546	−	0.657395i	$-0.771658\pi$
$191$	−20.8284	−1.50709	−0.753546	+	0.657395i	$0.771658\pi$
$192$	0	0
$193$	4.48528	0.322858	0.161429	−	0.986884i	$-0.448390\pi$
$193$	4.48528	0.322858	0.161429	+	0.986884i	$0.448390\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	19.6569	1.40049	0.700246	−	0.713901i	$-0.253073\pi$
$197$	19.6569	1.40049	0.700246	+	0.713901i	$0.253073\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	0	0
$201$	−12.9706	−0.914873
$202$	0	0
$203$	4.82843	0.338889
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−3.17157	−0.220440
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	0.828427	0.0570313	0.0285156	−	0.999593i	$-0.490922\pi$
$211$	0.828427	0.0570313	0.0285156	+	0.999593i	$0.490922\pi$
$212$	0	0
$213$	−30.6274	−2.09856
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	−31.3137	−2.12571
$218$	0	0
$219$	16.9706	1.14676
$220$	0	0
$221$	5.65685	0.380521
$222$	0	0
$223$	−17.7990	−1.19191	−0.595954	−	0.803018i	$-0.703226\pi$
$223$	−17.7990	−1.19191	−0.595954	+	0.803018i	$0.703226\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−20.1421	−1.33688	−0.668440	−	0.743766i	$-0.733038\pi$
$227$	−20.1421	−1.33688	−0.668440	+	0.743766i	$0.733038\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	11.6569	0.760409
$236$	0	0
$237$	−4.97056	−0.322873
$238$	0	0
$239$	−0.686292	−0.0443925	−0.0221963	−	0.999754i	$-0.507066\pi$
$239$	−0.686292	−0.0443925	−0.0221963	+	0.999754i	$0.507066\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	−16.3137	−1.04224
$246$	0	0
$247$	9.65685	0.614451
$248$	0	0
$249$	−14.3431	−0.908960
$250$	0	0
$251$	−8.82843	−0.557245	−0.278623	−	0.960401i	$-0.589878\pi$
$251$	−8.82843	−0.557245	−0.278623	+	0.960401i	$0.589878\pi$
$252$	0	0
$253$	2.62742	0.165184
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	6.68629	0.417079	0.208540	−	0.978014i	$-0.433129\pi$
$257$	6.68629	0.417079	0.208540	+	0.978014i	$0.433129\pi$
$258$	0	0
$259$	−40.9706	−2.54579
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−19.6569	−1.21209	−0.606047	−	0.795429i	$-0.707246\pi$
$263$	−19.6569	−1.21209	−0.606047	+	0.795429i	$0.707246\pi$
$264$	0	0
$265$	−3.65685	−0.224639
$266$	0	0
$267$	−15.3137	−0.937184
$268$	0	0
$269$	21.3137	1.29952	0.649760	−	0.760140i	$-0.274869\pi$
$269$	21.3137	1.29952	0.649760	+	0.760140i	$0.274869\pi$
$270$	0	0
$271$	−9.79899	−0.595246	−0.297623	−	0.954683i	$-0.596194\pi$
$271$	−9.79899	−0.595246	−0.297623	+	0.954683i	$0.596194\pi$
$272$	0	0
$273$	−19.3137	−1.16892
$274$	0	0
$275$	−0.828427	−0.0499560
$276$	0	0
$277$	3.65685	0.219719	0.109860	−	0.993947i	$-0.464960\pi$
$277$	3.65685	0.219719	0.109860	+	0.993947i	$0.464960\pi$
$278$	0	0
$279$	6.48528	0.388264
$280$	0	0
$281$	−29.3137	−1.74871	−0.874355	−	0.485288i	$-0.838715\pi$
$281$	−29.3137	−1.74871	−0.874355	+	0.485288i	$0.838715\pi$
$282$	0	0
$283$	−4.82843	−0.287020	−0.143510	−	0.989649i	$-0.545839\pi$
$283$	−4.82843	−0.287020	−0.143510	+	0.989649i	$0.545839\pi$
$284$	0	0
$285$	−9.65685	−0.572023
$286$	0	0
$287$	28.9706	1.71008
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	−24.9706	−1.46380
$292$	0	0
$293$	−8.48528	−0.495715	−0.247858	−	0.968796i	$-0.579727\pi$
$293$	−8.48528	−0.495715	−0.247858	+	0.968796i	$0.579727\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.31371	0.192281
$298$	0	0
$299$	−6.34315	−0.366834
$300$	0	0
$301$	−28.9706	−1.66984
$302$	0	0
$303$	−31.3137	−1.79893
$304$	0	0
$305$	−3.65685	−0.209391
$306$	0	0
$307$	22.9706	1.31100	0.655500	−	0.755195i	$-0.272458\pi$
$307$	22.9706	1.31100	0.655500	+	0.755195i	$0.272458\pi$
$308$	0	0
$309$	32.2843	1.83659
$310$	0	0
$311$	14.4853	0.821385	0.410692	−	0.911774i	$-0.365287\pi$
$311$	14.4853	0.821385	0.410692	+	0.911774i	$0.365287\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	4.82843	0.272051
$316$	0	0
$317$	−2.82843	−0.158860	−0.0794301	−	0.996840i	$-0.525310\pi$
$317$	−2.82843	−0.158860	−0.0794301	+	0.996840i	$0.525310\pi$
$318$	0	0
$319$	0.828427	0.0463830
$320$	0	0
$321$	−40.2843	−2.24845
$322$	0	0
$323$	13.6569	0.759888
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	56.2843	3.10305
$330$	0	0
$331$	−21.7990	−1.19818	−0.599090	−	0.800681i	$-0.704471\pi$
$331$	−21.7990	−1.19818	−0.599090	+	0.800681i	$0.704471\pi$
$332$	0	0
$333$	8.48528	0.464991
$334$	0	0
$335$	6.48528	0.354329
$336$	0	0
$337$	−1.17157	−0.0638196	−0.0319098	−	0.999491i	$-0.510159\pi$
$337$	−1.17157	−0.0638196	−0.0319098	+	0.999491i	$0.510159\pi$
$338$	0	0
$339$	−5.65685	−0.307238
$340$	0	0
$341$	−5.37258	−0.290942
$342$	0	0
$343$	−44.9706	−2.42818
$344$	0	0
$345$	6.34315	0.341503
$346$	0	0
$347$	8.14214	0.437093	0.218546	−	0.975827i	$-0.429869\pi$
$347$	8.14214	0.437093	0.218546	+	0.975827i	$0.429869\pi$
$348$	0	0
$349$	−20.6274	−1.10416	−0.552080	−	0.833791i	$-0.686166\pi$
$349$	−20.6274	−1.10416	−0.552080	+	0.833791i	$0.686166\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	0	0
$353$	−4.34315	−0.231162	−0.115581	−	0.993298i	$-0.536873\pi$
$353$	−4.34315	−0.231162	−0.115581	+	0.993298i	$0.536873\pi$
$354$	0	0
$355$	15.3137	0.812767
$356$	0	0
$357$	−27.3137	−1.44559
$358$	0	0
$359$	−3.85786	−0.203610	−0.101805	−	0.994804i	$-0.532462\pi$
$359$	−3.85786	−0.203610	−0.101805	+	0.994804i	$0.532462\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	0	0
$363$	−20.6274	−1.08266
$364$	0	0
$365$	−8.48528	−0.444140
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−17.6569	−0.916698
$372$	0	0
$373$	6.97056	0.360922	0.180461	−	0.983582i	$-0.442241\pi$
$373$	6.97056	0.360922	0.180461	+	0.983582i	$0.442241\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−22.4853	−1.15499	−0.577496	−	0.816394i	$-0.695970\pi$
$379$	−22.4853	−1.15499	−0.577496	+	0.816394i	$0.695970\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	0	0
$383$	2.48528	0.126992	0.0634960	−	0.997982i	$-0.479775\pi$
$383$	2.48528	0.126992	0.0634960	+	0.997982i	$0.479775\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	29.3137	1.48626	0.743132	−	0.669145i	$-0.233339\pi$
$389$	29.3137	1.48626	0.743132	+	0.669145i	$0.233339\pi$
$390$	0	0
$391$	−8.97056	−0.453661
$392$	0	0
$393$	24.2843	1.22498
$394$	0	0
$395$	2.48528	0.125048
$396$	0	0
$397$	19.6569	0.986549	0.493275	−	0.869874i	$-0.335800\pi$
$397$	19.6569	0.986549	0.493275	+	0.869874i	$0.335800\pi$
$398$	0	0
$399$	−46.6274	−2.33429
$400$	0	0
$401$	−6.68629	−0.333897	−0.166949	−	0.985966i	$-0.553391\pi$
$401$	−6.68629	−0.333897	−0.166949	+	0.985966i	$0.553391\pi$
$402$	0	0
$403$	12.9706	0.646110
$404$	0	0
$405$	11.0000	0.546594
$406$	0	0
$407$	−7.02944	−0.348436
$408$	0	0
$409$	−2.97056	−0.146885	−0.0734424	−	0.997299i	$-0.523399\pi$
$409$	−2.97056	−0.146885	−0.0734424	+	0.997299i	$0.523399\pi$
$410$	0	0
$411$	−10.3431	−0.510190
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.17157	0.352039
$416$	0	0
$417$	−43.3137	−2.12108
$418$	0	0
$419$	−28.9706	−1.41530	−0.707652	−	0.706561i	$-0.750246\pi$
$419$	−28.9706	−1.41530	−0.707652	+	0.706561i	$0.750246\pi$
$420$	0	0
$421$	−18.9706	−0.924569	−0.462284	−	0.886732i	$-0.652970\pi$
$421$	−18.9706	−0.924569	−0.462284	+	0.886732i	$0.652970\pi$
$422$	0	0
$423$	−11.6569	−0.566776
$424$	0	0
$425$	2.82843	0.137199
$426$	0	0
$427$	−17.6569	−0.854475
$428$	0	0
$429$	−3.31371	−0.159987
$430$	0	0
$431$	3.31371	0.159616	0.0798079	−	0.996810i	$-0.474569\pi$
$431$	3.31371	0.159616	0.0798079	+	0.996810i	$0.474569\pi$
$432$	0	0
$433$	−29.1716	−1.40190	−0.700948	−	0.713212i	$-0.747240\pi$
$433$	−29.1716	−1.40190	−0.700948	+	0.713212i	$0.747240\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	−15.3137	−0.732554
$438$	0	0
$439$	−10.3431	−0.493651	−0.246826	−	0.969060i	$-0.579388\pi$
$439$	−10.3431	−0.493651	−0.246826	+	0.969060i	$0.579388\pi$
$440$	0	0
$441$	16.3137	0.776843
$442$	0	0
$443$	7.65685	0.363788	0.181894	−	0.983318i	$-0.441777\pi$
$443$	7.65685	0.363788	0.181894	+	0.983318i	$0.441777\pi$
$444$	0	0
$445$	7.65685	0.362970
$446$	0	0
$447$	−18.6274	−0.881047
$448$	0	0
$449$	11.6569	0.550121	0.275060	−	0.961427i	$-0.411302\pi$
$449$	11.6569	0.550121	0.275060	+	0.961427i	$0.411302\pi$
$450$	0	0
$451$	4.97056	0.234055
$452$	0	0
$453$	−24.0000	−1.12762
$454$	0	0
$455$	9.65685	0.452720
$456$	0	0
$457$	19.6569	0.919509	0.459754	−	0.888046i	$-0.347937\pi$
$457$	19.6569	0.919509	0.459754	+	0.888046i	$0.347937\pi$
$458$	0	0
$459$	−11.3137	−0.528079
$460$	0	0
$461$	35.6569	1.66071	0.830353	−	0.557238i	$-0.188139\pi$
$461$	35.6569	1.66071	0.830353	+	0.557238i	$0.188139\pi$
$462$	0	0
$463$	21.7990	1.01308	0.506542	−	0.862215i	$-0.330923\pi$
$463$	21.7990	1.01308	0.506542	+	0.862215i	$0.330923\pi$
$464$	0	0
$465$	−12.9706	−0.601495
$466$	0	0
$467$	−10.9706	−0.507657	−0.253829	−	0.967249i	$-0.581690\pi$
$467$	−10.9706	−0.507657	−0.253829	+	0.967249i	$0.581690\pi$
$468$	0	0
$469$	31.3137	1.44593
$470$	0	0
$471$	−0.970563	−0.0447212
$472$	0	0
$473$	−4.97056	−0.228547
$474$	0	0
$475$	4.82843	0.221543
$476$	0	0
$477$	3.65685	0.167436
$478$	0	0
$479$	7.17157	0.327678	0.163839	−	0.986487i	$-0.447612\pi$
$479$	7.17157	0.327678	0.163839	+	0.986487i	$0.447612\pi$
$480$	0	0
$481$	16.9706	0.773791
$482$	0	0
$483$	30.6274	1.39360
$484$	0	0
$485$	12.4853	0.566927
$486$	0	0
$487$	9.79899	0.444035	0.222017	−	0.975043i	$-0.428736\pi$
$487$	9.79899	0.444035	0.222017	+	0.975043i	$0.428736\pi$
$488$	0	0
$489$	16.6863	0.754580
$490$	0	0
$491$	7.45584	0.336478	0.168239	−	0.985746i	$-0.446192\pi$
$491$	7.45584	0.336478	0.168239	+	0.985746i	$0.446192\pi$
$492$	0	0
$493$	−2.82843	−0.127386
$494$	0	0
$495$	0.828427	0.0372350
$496$	0	0
$497$	73.9411	3.31671
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	0	0
$501$	−4.97056	−0.222068
$502$	0	0
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	15.6569	0.696721
$506$	0	0
$507$	−18.0000	−0.799408
$508$	0	0
$509$	−0.627417	−0.0278098	−0.0139049	−	0.999903i	$-0.504426\pi$
$509$	−0.627417	−0.0278098	−0.0139049	+	0.999903i	$0.504426\pi$
$510$	0	0
$511$	−40.9706	−1.81243
$512$	0	0
$513$	−19.3137	−0.852721
$514$	0	0
$515$	−16.1421	−0.711307
$516$	0	0
$517$	9.65685	0.424708
$518$	0	0
$519$	−34.6274	−1.51997
$520$	0	0
$521$	−21.3137	−0.933771	−0.466885	−	0.884318i	$-0.654624\pi$
$521$	−21.3137	−0.933771	−0.466885	+	0.884318i	$0.654624\pi$
$522$	0	0
$523$	−2.48528	−0.108674	−0.0543369	−	0.998523i	$-0.517304\pi$
$523$	−2.48528	−0.108674	−0.0543369	+	0.998523i	$0.517304\pi$
$524$	0	0
$525$	−9.65685	−0.421460
$526$	0	0
$527$	18.3431	0.799040
$528$	0	0
$529$	−12.9411	−0.562658
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	20.1421	0.870820
$536$	0	0
$537$	46.6274	2.01212
$538$	0	0
$539$	−13.5147	−0.582120
$540$	0	0
$541$	5.02944	0.216232	0.108116	−	0.994138i	$-0.465518\pi$
$541$	5.02944	0.216232	0.108116	+	0.994138i	$0.465518\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	2.48528	0.106263	0.0531315	−	0.998588i	$-0.483080\pi$
$547$	2.48528	0.106263	0.0531315	+	0.998588i	$0.483080\pi$
$548$	0	0
$549$	3.65685	0.156071
$550$	0	0
$551$	−4.82843	−0.205698
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	−16.9706	−0.720360
$556$	0	0
$557$	27.9411	1.18390	0.591952	−	0.805973i	$-0.298358\pi$
$557$	27.9411	1.18390	0.591952	+	0.805973i	$0.298358\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	−4.68629	−0.197855
$562$	0	0
$563$	−7.65685	−0.322698	−0.161349	−	0.986897i	$-0.551584\pi$
$563$	−7.65685	−0.322698	−0.161349	+	0.986897i	$0.551584\pi$
$564$	0	0
$565$	2.82843	0.118993
$566$	0	0
$567$	53.1127	2.23052
$568$	0	0
$569$	27.6569	1.15944	0.579718	−	0.814817i	$-0.303163\pi$
$569$	27.6569	1.15944	0.579718	+	0.814817i	$0.303163\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	−41.6569	−1.74024
$574$	0	0
$575$	−3.17157	−0.132264
$576$	0	0
$577$	23.7990	0.990765	0.495382	−	0.868675i	$-0.335028\pi$
$577$	23.7990	0.990765	0.495382	+	0.868675i	$0.335028\pi$
$578$	0	0
$579$	8.97056	0.372804
$580$	0	0
$581$	34.6274	1.43659
$582$	0	0
$583$	−3.02944	−0.125466
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	29.7990	1.22994	0.614968	−	0.788552i	$-0.289169\pi$
$587$	29.7990	1.22994	0.614968	+	0.788552i	$0.289169\pi$
$588$	0	0
$589$	31.3137	1.29026
$590$	0	0
$591$	39.3137	1.61715
$592$	0	0
$593$	−7.65685	−0.314429	−0.157215	−	0.987564i	$-0.550251\pi$
$593$	−7.65685	−0.314429	−0.157215	+	0.987564i	$0.550251\pi$
$594$	0	0
$595$	13.6569	0.559876
$596$	0	0
$597$	24.0000	0.982255
$598$	0	0
$599$	−37.7990	−1.54442	−0.772212	−	0.635364i	$-0.780850\pi$
$599$	−37.7990	−1.54442	−0.772212	+	0.635364i	$0.780850\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−6.48528	−0.264101
$604$	0	0
$605$	10.3137	0.419312
$606$	0	0
$607$	9.02944	0.366494	0.183247	−	0.983067i	$-0.441339\pi$
$607$	9.02944	0.366494	0.183247	+	0.983067i	$0.441339\pi$
$608$	0	0
$609$	9.65685	0.391315
$610$	0	0
$611$	−23.3137	−0.943172
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−9.17157	−0.369234	−0.184617	−	0.982811i	$-0.559104\pi$
$617$	−9.17157	−0.369234	−0.184617	+	0.982811i	$0.559104\pi$
$618$	0	0
$619$	9.79899	0.393855	0.196927	−	0.980418i	$-0.436904\pi$
$619$	9.79899	0.393855	0.196927	+	0.980418i	$0.436904\pi$
$620$	0	0
$621$	12.6863	0.509083
$622$	0	0
$623$	36.9706	1.48119
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	24.0000	0.956943
$630$	0	0
$631$	36.9706	1.47177	0.735887	−	0.677104i	$-0.236765\pi$
$631$	36.9706	1.47177	0.735887	+	0.677104i	$0.236765\pi$
$632$	0	0
$633$	1.65685	0.0658540
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	32.6274	1.29275
$638$	0	0
$639$	−15.3137	−0.605801
$640$	0	0
$641$	0.627417	0.0247815	0.0123907	−	0.999923i	$-0.496056\pi$
$641$	0.627417	0.0247815	0.0123907	+	0.999923i	$0.496056\pi$
$642$	0	0
$643$	−19.4558	−0.767264	−0.383632	−	0.923486i	$-0.625327\pi$
$643$	−19.4558	−0.767264	−0.383632	+	0.923486i	$0.625327\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	−41.1127	−1.61631	−0.808153	−	0.588972i	$-0.799533\pi$
$647$	−41.1127	−1.61631	−0.808153	+	0.588972i	$0.799533\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−62.6274	−2.45456
$652$	0	0
$653$	17.1716	0.671976	0.335988	−	0.941866i	$-0.390930\pi$
$653$	17.1716	0.671976	0.335988	+	0.941866i	$0.390930\pi$
$654$	0	0
$655$	−12.1421	−0.474432
$656$	0	0
$657$	8.48528	0.331042
$658$	0	0
$659$	−1.79899	−0.0700787	−0.0350393	−	0.999386i	$-0.511156\pi$
$659$	−1.79899	−0.0700787	−0.0350393	+	0.999386i	$0.511156\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	11.3137	0.439388
$664$	0	0
$665$	23.3137	0.904067
$666$	0	0
$667$	3.17157	0.122804
$668$	0	0
$669$	−35.5980	−1.37630
$670$	0	0
$671$	−3.02944	−0.116950
$672$	0	0
$673$	22.9706	0.885450	0.442725	−	0.896657i	$-0.354012\pi$
$673$	22.9706	0.885450	0.442725	+	0.896657i	$0.354012\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	36.7696	1.41317	0.706584	−	0.707629i	$-0.250235\pi$
$677$	36.7696	1.41317	0.706584	+	0.707629i	$0.250235\pi$
$678$	0	0
$679$	60.2843	2.31350
$680$	0	0
$681$	−40.2843	−1.54370
$682$	0	0
$683$	−11.8579	−0.453729	−0.226864	−	0.973926i	$-0.572847\pi$
$683$	−11.8579	−0.453729	−0.226864	+	0.973926i	$0.572847\pi$
$684$	0	0
$685$	5.17157	0.197596
$686$	0	0
$687$	4.00000	0.152610
$688$	0	0
$689$	7.31371	0.278630
$690$	0	0
$691$	44.9706	1.71076	0.855380	−	0.518000i	$-0.173323\pi$
$691$	44.9706	1.71076	0.855380	+	0.518000i	$0.173323\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	21.6569	0.821491
$696$	0	0
$697$	−16.9706	−0.642806
$698$	0	0
$699$	36.0000	1.36165
$700$	0	0
$701$	−6.68629	−0.252538	−0.126269	−	0.991996i	$-0.540300\pi$
$701$	−6.68629	−0.252538	−0.126269	+	0.991996i	$0.540300\pi$
$702$	0	0
$703$	40.9706	1.54523
$704$	0	0
$705$	23.3137	0.878045
$706$	0	0
$707$	75.5980	2.84315
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	−2.48528	−0.0932053
$712$	0	0
$713$	−20.5685	−0.770298
$714$	0	0
$715$	1.65685	0.0619628
$716$	0	0
$717$	−1.37258	−0.0512601
$718$	0	0
$719$	−34.6274	−1.29138	−0.645692	−	0.763598i	$-0.723431\pi$
$719$	−34.6274	−1.29138	−0.645692	+	0.763598i	$0.723431\pi$
$720$	0	0
$721$	−77.9411	−2.90268
$722$	0	0
$723$	20.0000	0.743808
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−23.9411	−0.887927	−0.443964	−	0.896045i	$-0.646428\pi$
$727$	−23.9411	−0.887927	−0.443964	+	0.896045i	$0.646428\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	16.9706	0.627679
$732$	0	0
$733$	22.8284	0.843187	0.421594	−	0.906785i	$-0.361471\pi$
$733$	22.8284	0.843187	0.421594	+	0.906785i	$0.361471\pi$
$734$	0	0
$735$	−32.6274	−1.20348
$736$	0	0
$737$	5.37258	0.197902
$738$	0	0
$739$	−14.4853	−0.532850	−0.266425	−	0.963856i	$-0.585842\pi$
$739$	−14.4853	−0.532850	−0.266425	+	0.963856i	$0.585842\pi$
$740$	0	0
$741$	19.3137	0.709507
$742$	0	0
$743$	−52.6274	−1.93071	−0.965356	−	0.260935i	$-0.915969\pi$
$743$	−52.6274	−1.93071	−0.965356	+	0.260935i	$0.915969\pi$
$744$	0	0
$745$	9.31371	0.341228
$746$	0	0
$747$	−7.17157	−0.262394
$748$	0	0
$749$	97.2548	3.55361
$750$	0	0
$751$	16.1421	0.589035	0.294517	−	0.955646i	$-0.404841\pi$
$751$	16.1421	0.589035	0.294517	+	0.955646i	$0.404841\pi$
$752$	0	0
$753$	−17.6569	−0.643452
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	−19.5147	−0.709275	−0.354637	−	0.935004i	$-0.615396\pi$
$757$	−19.5147	−0.709275	−0.354637	+	0.935004i	$0.615396\pi$
$758$	0	0
$759$	5.25483	0.190738
$760$	0	0
$761$	8.62742	0.312744	0.156372	−	0.987698i	$-0.450020\pi$
$761$	8.62742	0.312744	0.156372	+	0.987698i	$0.450020\pi$
$762$	0	0
$763$	9.65685	0.349602
$764$	0	0
$765$	−2.82843	−0.102262
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−15.6569	−0.564601	−0.282300	−	0.959326i	$-0.591097\pi$
$769$	−15.6569	−0.564601	−0.282300	+	0.959326i	$0.591097\pi$
$770$	0	0
$771$	13.3726	0.481602
$772$	0	0
$773$	8.48528	0.305194	0.152597	−	0.988288i	$-0.451236\pi$
$773$	8.48528	0.305194	0.152597	+	0.988288i	$0.451236\pi$
$774$	0	0
$775$	6.48528	0.232958
$776$	0	0
$777$	−81.9411	−2.93962
$778$	0	0
$779$	−28.9706	−1.03798
$780$	0	0
$781$	12.6863	0.453951
$782$	0	0
$783$	4.00000	0.142948
$784$	0	0
$785$	0.485281	0.0173204
$786$	0	0
$787$	−17.7990	−0.634465	−0.317233	−	0.948348i	$-0.602754\pi$
$787$	−17.7990	−0.634465	−0.317233	+	0.948348i	$0.602754\pi$
$788$	0	0
$789$	−39.3137	−1.39961
$790$	0	0
$791$	13.6569	0.485582
$792$	0	0
$793$	7.31371	0.259717
$794$	0	0
$795$	−7.31371	−0.259391
$796$	0	0
$797$	5.85786	0.207496	0.103748	−	0.994604i	$-0.466916\pi$
$797$	5.85786	0.207496	0.103748	+	0.994604i	$0.466916\pi$
$798$	0	0
$799$	−32.9706	−1.16641
$800$	0	0
$801$	−7.65685	−0.270542
$802$	0	0
$803$	−7.02944	−0.248063
$804$	0	0
$805$	−15.3137	−0.539737
$806$	0	0
$807$	42.6274	1.50056
$808$	0	0
$809$	42.2843	1.48664	0.743318	−	0.668938i	$-0.233251\pi$
$809$	42.2843	1.48664	0.743318	+	0.668938i	$0.233251\pi$
$810$	0	0
$811$	−37.6569	−1.32231	−0.661155	−	0.750249i	$-0.729934\pi$
$811$	−37.6569	−1.32231	−0.661155	+	0.750249i	$0.729934\pi$
$812$	0	0
$813$	−19.5980	−0.687331
$814$	0	0
$815$	−8.34315	−0.292248
$816$	0	0
$817$	28.9706	1.01355
$818$	0	0
$819$	−9.65685	−0.337438
$820$	0	0
$821$	22.6863	0.791757	0.395879	−	0.918303i	$-0.370440\pi$
$821$	22.6863	0.791757	0.395879	+	0.918303i	$0.370440\pi$
$822$	0	0
$823$	−30.9706	−1.07957	−0.539783	−	0.841804i	$-0.681494\pi$
$823$	−30.9706	−1.07957	−0.539783	+	0.841804i	$0.681494\pi$
$824$	0	0
$825$	−1.65685	−0.0576843
$826$	0	0
$827$	−17.3137	−0.602057	−0.301028	−	0.953615i	$-0.597330\pi$
$827$	−17.3137	−0.602057	−0.301028	+	0.953615i	$0.597330\pi$
$828$	0	0
$829$	−20.6274	−0.716420	−0.358210	−	0.933641i	$-0.616613\pi$
$829$	−20.6274	−0.716420	−0.358210	+	0.933641i	$0.616613\pi$
$830$	0	0
$831$	7.31371	0.253710
$832$	0	0
$833$	46.1421	1.59873
$834$	0	0
$835$	2.48528	0.0860067
$836$	0	0
$837$	−25.9411	−0.896656
$838$	0	0
$839$	2.48528	0.0858014	0.0429007	−	0.999079i	$-0.486340\pi$
$839$	2.48528	0.0858014	0.0429007	+	0.999079i	$0.486340\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−58.6274	−2.01924
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	49.7990	1.71111
$848$	0	0
$849$	−9.65685	−0.331422
$850$	0	0
$851$	−26.9117	−0.922521
$852$	0	0
$853$	−51.1127	−1.75007	−0.875033	−	0.484064i	$-0.839160\pi$
$853$	−51.1127	−1.75007	−0.875033	+	0.484064i	$0.839160\pi$
$854$	0	0
$855$	−4.82843	−0.165129
$856$	0	0
$857$	3.37258	0.115205	0.0576026	−	0.998340i	$-0.481654\pi$
$857$	3.37258	0.115205	0.0576026	+	0.998340i	$0.481654\pi$
$858$	0	0
$859$	56.4264	1.92524	0.962622	−	0.270848i	$-0.0873041\pi$
$859$	56.4264	1.92524	0.962622	+	0.270848i	$0.0873041\pi$
$860$	0	0
$861$	57.9411	1.97463
$862$	0	0
$863$	−36.1421	−1.23029	−0.615146	−	0.788413i	$-0.710903\pi$
$863$	−36.1421	−1.23029	−0.615146	+	0.788413i	$0.710903\pi$
$864$	0	0
$865$	17.3137	0.588684
$866$	0	0
$867$	−18.0000	−0.611312
$868$	0	0
$869$	2.05887	0.0698425
$870$	0	0
$871$	−12.9706	−0.439491
$872$	0	0
$873$	−12.4853	−0.422563
$874$	0	0
$875$	4.82843	0.163231
$876$	0	0
$877$	−38.2843	−1.29277	−0.646384	−	0.763012i	$-0.723720\pi$
$877$	−38.2843	−1.29277	−0.646384	+	0.763012i	$0.723720\pi$
$878$	0	0
$879$	−16.9706	−0.572403
$880$	0	0
$881$	29.3137	0.987604	0.493802	−	0.869574i	$-0.335607\pi$
$881$	29.3137	0.987604	0.493802	+	0.869574i	$0.335607\pi$
$882$	0	0
$883$	14.4853	0.487469	0.243734	−	0.969842i	$-0.421628\pi$
$883$	14.4853	0.487469	0.243734	+	0.969842i	$0.421628\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.68629	−0.224504	−0.112252	−	0.993680i	$-0.535806\pi$
$887$	−6.68629	−0.224504	−0.112252	+	0.993680i	$0.535806\pi$
$888$	0	0
$889$	−28.9706	−0.971641
$890$	0	0
$891$	9.11270	0.305287
$892$	0	0
$893$	−56.2843	−1.88348
$894$	0	0
$895$	−23.3137	−0.779291
$896$	0	0
$897$	−12.6863	−0.423583
$898$	0	0
$899$	−6.48528	−0.216296
$900$	0	0
$901$	10.3431	0.344580
$902$	0	0
$903$	−57.9411	−1.92816
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	0	0
$909$	−15.6569	−0.519305
$910$	0	0
$911$	32.1421	1.06492	0.532458	−	0.846456i	$-0.321268\pi$
$911$	32.1421	1.06492	0.532458	+	0.846456i	$0.321268\pi$
$912$	0	0
$913$	5.94113	0.196623
$914$	0	0
$915$	−7.31371	−0.241784
$916$	0	0
$917$	−58.6274	−1.93605
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	45.9411	1.51381
$922$	0	0
$923$	−30.6274	−1.00811
$924$	0	0
$925$	8.48528	0.278994
$926$	0	0
$927$	16.1421	0.530177
$928$	0	0
$929$	−4.62742	−0.151821	−0.0759103	−	0.997115i	$-0.524186\pi$
$929$	−4.62742	−0.151821	−0.0759103	+	0.997115i	$0.524186\pi$
$930$	0	0
$931$	78.7696	2.58157
$932$	0	0
$933$	28.9706	0.948454
$934$	0	0
$935$	2.34315	0.0766291
$936$	0	0
$937$	19.6569	0.642161	0.321081	−	0.947052i	$-0.395954\pi$
$937$	19.6569	0.642161	0.321081	+	0.947052i	$0.395954\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	0	0
$941$	27.9411	0.910855	0.455427	−	0.890273i	$-0.349486\pi$
$941$	27.9411	0.910855	0.455427	+	0.890273i	$0.349486\pi$
$942$	0	0
$943$	19.0294	0.619684
$944$	0	0
$945$	−19.3137	−0.628275
$946$	0	0
$947$	−44.9117	−1.45943	−0.729717	−	0.683749i	$-0.760348\pi$
$947$	−44.9117	−1.45943	−0.729717	+	0.683749i	$0.760348\pi$
$948$	0	0
$949$	16.9706	0.550888
$950$	0	0
$951$	−5.65685	−0.183436
$952$	0	0
$953$	29.3137	0.949564	0.474782	−	0.880103i	$-0.342527\pi$
$953$	29.3137	0.949564	0.474782	+	0.880103i	$0.342527\pi$
$954$	0	0
$955$	20.8284	0.673992
$956$	0	0
$957$	1.65685	0.0535585
$958$	0	0
$959$	24.9706	0.806342
$960$	0	0
$961$	11.0589	0.356738
$962$	0	0
$963$	−20.1421	−0.649071
$964$	0	0
$965$	−4.48528	−0.144386
$966$	0	0
$967$	14.9706	0.481421	0.240710	−	0.970597i	$-0.422620\pi$
$967$	14.9706	0.481421	0.240710	+	0.970597i	$0.422620\pi$
$968$	0	0
$969$	27.3137	0.877443
$970$	0	0
$971$	−28.1421	−0.903124	−0.451562	−	0.892240i	$-0.649133\pi$
$971$	−28.1421	−0.903124	−0.451562	+	0.892240i	$0.649133\pi$
$972$	0	0
$973$	104.569	3.35231
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−2.68629	−0.0859421	−0.0429710	−	0.999076i	$-0.513682\pi$
$977$	−2.68629	−0.0859421	−0.0429710	+	0.999076i	$0.513682\pi$
$978$	0	0
$979$	6.34315	0.202728
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−9.31371	−0.297061	−0.148531	−	0.988908i	$-0.547454\pi$
$983$	−9.31371	−0.297061	−0.148531	+	0.988908i	$0.547454\pi$
$984$	0	0
$985$	−19.6569	−0.626319
$986$	0	0
$987$	112.569	3.58310
$988$	0	0
$989$	−19.0294	−0.605101
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	0	0
$993$	−43.5980	−1.38354
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	−6.82843	−0.216258	−0.108129	−	0.994137i	$-0.534486\pi$
$997$	−6.82843	−0.216258	−0.108129	+	0.994137i	$0.534486\pi$
$998$	0	0
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.be.1.1		2
4.3	odd	2		9280.2.a.w.1.2		2
8.3	odd	2		2320.2.a.k.1.2		2
8.5	even	2		145.2.a.b.1.2	✓	2
24.5	odd	2		1305.2.a.n.1.1		2
40.13	odd	4		725.2.b.c.349.2		4
40.29	even	2		725.2.a.c.1.1		2
40.37	odd	4		725.2.b.c.349.3		4
56.13	odd	2		7105.2.a.e.1.2		2
120.29	odd	2		6525.2.a.p.1.2		2
232.173	even	2		4205.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.2	✓	2	8.5	even	2
725.2.a.c.1.1		2	40.29	even	2
725.2.b.c.349.2		4	40.13	odd	4
725.2.b.c.349.3		4	40.37	odd	4
1305.2.a.n.1.1		2	24.5	odd	2
2320.2.a.k.1.2		2	8.3	odd	2
4205.2.a.d.1.1		2	232.173	even	2
6525.2.a.p.1.2		2	120.29	odd	2
7105.2.a.e.1.2		2	56.13	odd	2
9280.2.a.w.1.2		2	4.3	odd	2
9280.2.a.be.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{3} -1.00000 q^{5} -4.82843 q^{7} +1.00000 q^{9} -0.828427 q^{11} +2.00000 q^{13} -2.00000 q^{15} +2.82843 q^{17} +4.82843 q^{19} -9.65685 q^{21} -3.17157 q^{23} +1.00000 q^{25} -4.00000 q^{27} -1.00000 q^{29} +6.48528 q^{31} -1.65685 q^{33} +4.82843 q^{35} +8.48528 q^{37} +4.00000 q^{39} -6.00000 q^{41} +6.00000 q^{43} -1.00000 q^{45} -11.6569 q^{47} +16.3137 q^{49} +5.65685 q^{51} +3.65685 q^{53} +0.828427 q^{55} +9.65685 q^{57} +3.65685 q^{61} -4.82843 q^{63} -2.00000 q^{65} -6.48528 q^{67} -6.34315 q^{69} -15.3137 q^{71} +8.48528 q^{73} +2.00000 q^{75} +4.00000 q^{77} -2.48528 q^{79} -11.0000 q^{81} -7.17157 q^{83} -2.82843 q^{85} -2.00000 q^{87} -7.65685 q^{89} -9.65685 q^{91} +12.9706 q^{93} -4.82843 q^{95} -12.4853 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 4 q^{15} + 4 q^{19} - 8 q^{21} - 12 q^{23} + 2 q^{25} - 8 q^{27} - 2 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} + 8 q^{39} - 12 q^{41} + 12 q^{43}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^13 - 4 * q^15 + 4 * q^19 - 8 * q^21 - 12 * q^23 + 2 * q^25 - 8 * q^27 - 2 * q^29 - 4 * q^31 + 8 * q^33 + 4 * q^35 + 8 * q^39 - 12 * q^41 + 12 * q^43 - 2 * q^45 - 12 * q^47 + 10 * q^49 - 4 * q^53 - 4 * q^55 + 8 * q^57 - 4 * q^61 - 4 * q^63 - 4 * q^65 + 4 * q^67 - 24 * q^69 - 8 * q^71 + 4 * q^75 + 8 * q^77 + 12 * q^79 - 22 * q^81 - 20 * q^83 - 4 * q^87 - 4 * q^89 - 8 * q^91 - 8 * q^93 - 4 * q^95 - 8 * q^97 + 4 * q^99

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