LMFDB - Embedded newform 2280.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2280,2,Mod(1,2280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2280.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:	$18.2058916609$
Analytic rank:	$0$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +2.82843 q^{11} +0.828427 q^{13} +1.00000 q^{15} -7.65685 q^{17} -1.00000 q^{19} +2.82843 q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.828427 q^{29} -2.82843 q^{33} +2.82843 q^{35} -4.82843 q^{37} -0.828427 q^{39} -0.828427 q^{41} +2.82843 q^{43} -1.00000 q^{45} +4.00000 q^{47} +1.00000 q^{49} +7.65685 q^{51} -2.00000 q^{53} -2.82843 q^{55} +1.00000 q^{57} +8.00000 q^{59} +9.31371 q^{61} -2.82843 q^{63} -0.828427 q^{65} -1.65685 q^{67} -4.00000 q^{69} +8.00000 q^{71} -0.343146 q^{73} -1.00000 q^{75} -8.00000 q^{77} +13.6569 q^{79} +1.00000 q^{81} +8.00000 q^{83} +7.65685 q^{85} +0.828427 q^{87} +7.17157 q^{89} -2.34315 q^{91} +1.00000 q^{95} -4.82843 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{13} + 2 q^{15} - 4 q^{17} - 2 q^{19} + 8 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 4 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 4 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 4 q^{65} + 8 q^{67} - 8 q^{69} + 16 q^{71} - 12 q^{73} - 2 q^{75} - 16 q^{77} + 16 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} - 4 q^{87} + 20 q^{89} - 16 q^{91} + 2 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^13 + 2 * q^15 - 4 * q^17 - 2 * q^19 + 8 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 - 4 * q^37 + 4 * q^39 + 4 * q^41 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 4 * q^51 - 4 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 4 * q^65 + 8 * q^67 - 8 * q^69 + 16 * q^71 - 12 * q^73 - 2 * q^75 - 16 * q^77 + 16 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 - 4 * q^87 + 20 * q^89 - 16 * q^91 + 2 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	0.828427	0.229764	0.114882	−	0.993379i	$-0.463351\pi$
$13$	0.828427	0.229764	0.114882	+	0.993379i	$0.463351\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	−4.82843	−0.793789	−0.396894	−	0.917864i	$-0.629912\pi$
$37$	−4.82843	−0.793789	−0.396894	+	0.917864i	$0.629912\pi$
$38$	0	0
$39$	−0.828427	−0.132655
$40$	0	0
$41$	−0.828427	−0.129379	−0.0646893	−	0.997905i	$-0.520606\pi$
$41$	−0.828427	−0.129379	−0.0646893	+	0.997905i	$0.520606\pi$
$42$	0	0
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.65685	1.07217
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−2.82843	−0.381385
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	9.31371	1.19250	0.596249	−	0.802799i	$-0.296657\pi$
$61$	9.31371	1.19250	0.596249	+	0.802799i	$0.296657\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	−0.828427	−0.102754
$66$	0	0
$67$	−1.65685	−0.202417	−0.101208	−	0.994865i	$-0.532271\pi$
$67$	−1.65685	−0.202417	−0.101208	+	0.994865i	$0.532271\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−0.343146	−0.0401622	−0.0200811	−	0.999798i	$-0.506392\pi$
$73$	−0.343146	−0.0401622	−0.0200811	+	0.999798i	$0.506392\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	7.65685	0.830502
$86$	0	0
$87$	0.828427	0.0888167
$88$	0	0
$89$	7.17157	0.760185	0.380093	−	0.924948i	$-0.375892\pi$
$89$	7.17157	0.760185	0.380093	+	0.924948i	$0.375892\pi$
$90$	0	0
$91$	−2.34315	−0.245628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−4.82843	−0.490252	−0.245126	−	0.969491i	$-0.578829\pi$
$97$	−4.82843	−0.490252	−0.245126	+	0.969491i	$0.578829\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	0	0
$101$	−3.65685	−0.363871	−0.181935	−	0.983311i	$-0.558236\pi$
$101$	−3.65685	−0.363871	−0.181935	+	0.983311i	$0.558236\pi$
$102$	0	0
$103$	−1.65685	−0.163255	−0.0816274	−	0.996663i	$-0.526012\pi$
$103$	−1.65685	−0.163255	−0.0816274	+	0.996663i	$0.526012\pi$
$104$	0	0
$105$	−2.82843	−0.276026
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	4.82843	0.458294
$112$	0	0
$113$	7.65685	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$113$	7.65685	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0.828427	0.0765881
$118$	0	0
$119$	21.6569	1.98528
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0.828427	0.0746968
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−2.82843	−0.249029
$130$	0	0
$131$	5.17157	0.451842	0.225921	−	0.974146i	$-0.427461\pi$
$131$	5.17157	0.451842	0.225921	+	0.974146i	$0.427461\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	22.9706	1.96251	0.981254	−	0.192720i	$-0.0617309\pi$
$137$	22.9706	1.96251	0.981254	+	0.192720i	$0.0617309\pi$
$138$	0	0
$139$	−1.65685	−0.140533	−0.0702663	−	0.997528i	$-0.522385\pi$
$139$	−1.65685	−0.140533	−0.0702663	+	0.997528i	$0.522385\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	2.34315	0.195944
$144$	0	0
$145$	0.828427	0.0687971
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	15.6569	1.28266	0.641330	−	0.767265i	$-0.278383\pi$
$149$	15.6569	1.28266	0.641330	+	0.767265i	$0.278383\pi$
$150$	0	0
$151$	−3.31371	−0.269666	−0.134833	−	0.990868i	$-0.543050\pi$
$151$	−3.31371	−0.269666	−0.134833	+	0.990868i	$0.543050\pi$
$152$	0	0
$153$	−7.65685	−0.619020
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	−2.82843	−0.221540	−0.110770	−	0.993846i	$-0.535332\pi$
$163$	−2.82843	−0.221540	−0.110770	+	0.993846i	$0.535332\pi$
$164$	0	0
$165$	2.82843	0.220193
$166$	0	0
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	11.6569	0.886254	0.443127	−	0.896459i	$-0.353869\pi$
$173$	11.6569	0.886254	0.443127	+	0.896459i	$0.353869\pi$
$174$	0	0
$175$	−2.82843	−0.213809
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−11.3137	−0.845626	−0.422813	−	0.906217i	$-0.638957\pi$
$179$	−11.3137	−0.845626	−0.422813	+	0.906217i	$0.638957\pi$
$180$	0	0
$181$	−5.31371	−0.394965	−0.197482	−	0.980306i	$-0.563277\pi$
$181$	−5.31371	−0.394965	−0.197482	+	0.980306i	$0.563277\pi$
$182$	0	0
$183$	−9.31371	−0.688489
$184$	0	0
$185$	4.82843	0.354993
$186$	0	0
$187$	−21.6569	−1.58371
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	18.8284	1.36238	0.681189	−	0.732108i	$-0.261463\pi$
$191$	18.8284	1.36238	0.681189	+	0.732108i	$0.261463\pi$
$192$	0	0
$193$	−13.7990	−0.993273	−0.496637	−	0.867959i	$-0.665432\pi$
$193$	−13.7990	−0.993273	−0.496637	+	0.867959i	$0.665432\pi$
$194$	0	0
$195$	0.828427	0.0593249
$196$	0	0
$197$	−2.68629	−0.191390	−0.0956952	−	0.995411i	$-0.530507\pi$
$197$	−2.68629	−0.191390	−0.0956952	+	0.995411i	$0.530507\pi$
$198$	0	0
$199$	8.97056	0.635906	0.317953	−	0.948106i	$-0.397005\pi$
$199$	8.97056	0.635906	0.317953	+	0.948106i	$0.397005\pi$
$200$	0	0
$201$	1.65685	0.116865
$202$	0	0
$203$	2.34315	0.164457
$204$	0	0
$205$	0.828427	0.0578599
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−2.82843	−0.195646
$210$	0	0
$211$	−18.6274	−1.28236	−0.641182	−	0.767389i	$-0.721556\pi$
$211$	−18.6274	−1.28236	−0.641182	+	0.767389i	$0.721556\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−2.82843	−0.192897
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.343146	0.0231876
$220$	0	0
$221$	−6.34315	−0.426686
$222$	0	0
$223$	12.9706	0.868573	0.434287	−	0.900775i	$-0.357001\pi$
$223$	12.9706	0.868573	0.434287	+	0.900775i	$0.357001\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	1.65685	0.109969	0.0549846	−	0.998487i	$-0.482489\pi$
$227$	1.65685	0.109969	0.0549846	+	0.998487i	$0.482489\pi$
$228$	0	0
$229$	−24.6274	−1.62743	−0.813713	−	0.581267i	$-0.802557\pi$
$229$	−24.6274	−1.62743	−0.813713	+	0.581267i	$0.802557\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	0.343146	0.0224802	0.0112401	−	0.999937i	$-0.496422\pi$
$233$	0.343146	0.0224802	0.0112401	+	0.999937i	$0.496422\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	−13.6569	−0.887108
$238$	0	0
$239$	27.7990	1.79817	0.899084	−	0.437777i	$-0.144234\pi$
$239$	27.7990	1.79817	0.899084	+	0.437777i	$0.144234\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−0.828427	−0.0527116
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−8.48528	−0.535586	−0.267793	−	0.963476i	$-0.586294\pi$
$251$	−8.48528	−0.535586	−0.267793	+	0.963476i	$0.586294\pi$
$252$	0	0
$253$	11.3137	0.711287
$254$	0	0
$255$	−7.65685	−0.479491
$256$	0	0
$257$	10.9706	0.684325	0.342162	−	0.939641i	$-0.388841\pi$
$257$	10.9706	0.684325	0.342162	+	0.939641i	$0.388841\pi$
$258$	0	0
$259$	13.6569	0.848596
$260$	0	0
$261$	−0.828427	−0.0512784
$262$	0	0
$263$	9.65685	0.595467	0.297734	−	0.954649i	$-0.403769\pi$
$263$	9.65685	0.595467	0.297734	+	0.954649i	$0.403769\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	−7.17157	−0.438893
$268$	0	0
$269$	18.4853	1.12707	0.563534	−	0.826093i	$-0.309441\pi$
$269$	18.4853	1.12707	0.563534	+	0.826093i	$0.309441\pi$
$270$	0	0
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	0	0
$273$	2.34315	0.141814
$274$	0	0
$275$	2.82843	0.170561
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.14214	0.485719	0.242860	−	0.970061i	$-0.421915\pi$
$281$	8.14214	0.485719	0.242860	+	0.970061i	$0.421915\pi$
$282$	0	0
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	2.34315	0.138312
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	4.82843	0.283047
$292$	0	0
$293$	−7.65685	−0.447318	−0.223659	−	0.974667i	$-0.571800\pi$
$293$	−7.65685	−0.447318	−0.223659	+	0.974667i	$0.571800\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	−2.82843	−0.164122
$298$	0	0
$299$	3.31371	0.191637
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	3.65685	0.210081
$304$	0	0
$305$	−9.31371	−0.533301
$306$	0	0
$307$	−6.34315	−0.362022	−0.181011	−	0.983481i	$-0.557937\pi$
$307$	−6.34315	−0.362022	−0.181011	+	0.983481i	$0.557937\pi$
$308$	0	0
$309$	1.65685	0.0942551
$310$	0	0
$311$	30.1421	1.70920	0.854602	−	0.519284i	$-0.173801\pi$
$311$	30.1421	1.70920	0.854602	+	0.519284i	$0.173801\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	2.82843	0.159364
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−2.34315	−0.131191
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	7.65685	0.426039
$324$	0	0
$325$	0.828427	0.0459529
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	−11.3137	−0.623745
$330$	0	0
$331$	−25.6569	−1.41023	−0.705114	−	0.709094i	$-0.749104\pi$
$331$	−25.6569	−1.41023	−0.705114	+	0.709094i	$0.749104\pi$
$332$	0	0
$333$	−4.82843	−0.264596
$334$	0	0
$335$	1.65685	0.0905236
$336$	0	0
$337$	14.4853	0.789064	0.394532	−	0.918882i	$-0.370907\pi$
$337$	14.4853	0.789064	0.394532	+	0.918882i	$0.370907\pi$
$338$	0	0
$339$	−7.65685	−0.415863
$340$	0	0
$341$	0	0
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	4.68629	0.251573	0.125787	−	0.992057i	$-0.459855\pi$
$347$	4.68629	0.251573	0.125787	+	0.992057i	$0.459855\pi$
$348$	0	0
$349$	−5.31371	−0.284436	−0.142218	−	0.989835i	$-0.545423\pi$
$349$	−5.31371	−0.284436	−0.142218	+	0.989835i	$0.545423\pi$
$350$	0	0
$351$	−0.828427	−0.0442182
$352$	0	0
$353$	−20.3431	−1.08276	−0.541378	−	0.840779i	$-0.682097\pi$
$353$	−20.3431	−1.08276	−0.541378	+	0.840779i	$0.682097\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	−21.6569	−1.14620
$358$	0	0
$359$	−8.48528	−0.447836	−0.223918	−	0.974608i	$-0.571885\pi$
$359$	−8.48528	−0.447836	−0.223918	+	0.974608i	$0.571885\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.00000	0.157459
$364$	0	0
$365$	0.343146	0.0179611
$366$	0	0
$367$	6.14214	0.320617	0.160308	−	0.987067i	$-0.448751\pi$
$367$	6.14214	0.320617	0.160308	+	0.987067i	$0.448751\pi$
$368$	0	0
$369$	−0.828427	−0.0431262
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	16.8284	0.871343	0.435671	−	0.900106i	$-0.356511\pi$
$373$	16.8284	0.871343	0.435671	+	0.900106i	$0.356511\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−0.686292	−0.0353458
$378$	0	0
$379$	9.65685	0.496039	0.248020	−	0.968755i	$-0.420220\pi$
$379$	9.65685	0.496039	0.248020	+	0.968755i	$0.420220\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	−14.6274	−0.747426	−0.373713	−	0.927544i	$-0.621916\pi$
$383$	−14.6274	−0.747426	−0.373713	+	0.927544i	$0.621916\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	2.82843	0.143777
$388$	0	0
$389$	−12.6274	−0.640235	−0.320118	−	0.947378i	$-0.603722\pi$
$389$	−12.6274	−0.640235	−0.320118	+	0.947378i	$0.603722\pi$
$390$	0	0
$391$	−30.6274	−1.54890
$392$	0	0
$393$	−5.17157	−0.260871
$394$	0	0
$395$	−13.6569	−0.687151
$396$	0	0
$397$	−28.6274	−1.43677	−0.718384	−	0.695646i	$-0.755118\pi$
$397$	−28.6274	−1.43677	−0.718384	+	0.695646i	$0.755118\pi$
$398$	0	0
$399$	−2.82843	−0.141598
$400$	0	0
$401$	12.8284	0.640621	0.320311	−	0.947313i	$-0.396213\pi$
$401$	12.8284	0.640621	0.320311	+	0.947313i	$0.396213\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−13.6569	−0.676945
$408$	0	0
$409$	26.9706	1.33361	0.666804	−	0.745233i	$-0.267662\pi$
$409$	26.9706	1.33361	0.666804	+	0.745233i	$0.267662\pi$
$410$	0	0
$411$	−22.9706	−1.13305
$412$	0	0
$413$	−22.6274	−1.11342
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	1.65685	0.0811365
$418$	0	0
$419$	−35.7990	−1.74890	−0.874448	−	0.485120i	$-0.838776\pi$
$419$	−35.7990	−1.74890	−0.874448	+	0.485120i	$0.838776\pi$
$420$	0	0
$421$	−15.6569	−0.763068	−0.381534	−	0.924355i	$-0.624604\pi$
$421$	−15.6569	−0.763068	−0.381534	+	0.924355i	$0.624604\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	−7.65685	−0.371412
$426$	0	0
$427$	−26.3431	−1.27483
$428$	0	0
$429$	−2.34315	−0.113128
$430$	0	0
$431$	32.9706	1.58814	0.794068	−	0.607829i	$-0.207959\pi$
$431$	32.9706	1.58814	0.794068	+	0.607829i	$0.207959\pi$
$432$	0	0
$433$	−18.4853	−0.888346	−0.444173	−	0.895941i	$-0.646502\pi$
$433$	−18.4853	−0.888346	−0.444173	+	0.895941i	$0.646502\pi$
$434$	0	0
$435$	−0.828427	−0.0397200
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−20.2843	−0.968115	−0.484058	−	0.875036i	$-0.660838\pi$
$439$	−20.2843	−0.968115	−0.484058	+	0.875036i	$0.660838\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−2.34315	−0.111326	−0.0556631	−	0.998450i	$-0.517727\pi$
$443$	−2.34315	−0.111326	−0.0556631	+	0.998450i	$0.517727\pi$
$444$	0	0
$445$	−7.17157	−0.339965
$446$	0	0
$447$	−15.6569	−0.740544
$448$	0	0
$449$	34.4853	1.62746	0.813731	−	0.581242i	$-0.197433\pi$
$449$	34.4853	1.62746	0.813731	+	0.581242i	$0.197433\pi$
$450$	0	0
$451$	−2.34315	−0.110334
$452$	0	0
$453$	3.31371	0.155692
$454$	0	0
$455$	2.34315	0.109848
$456$	0	0
$457$	5.31371	0.248565	0.124282	−	0.992247i	$-0.460337\pi$
$457$	5.31371	0.248565	0.124282	+	0.992247i	$0.460337\pi$
$458$	0	0
$459$	7.65685	0.357391
$460$	0	0
$461$	−26.2843	−1.22418	−0.612090	−	0.790788i	$-0.709671\pi$
$461$	−26.2843	−1.22418	−0.612090	+	0.790788i	$0.709671\pi$
$462$	0	0
$463$	14.1421	0.657241	0.328620	−	0.944462i	$-0.393416\pi$
$463$	14.1421	0.657241	0.328620	+	0.944462i	$0.393416\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.6569	−1.00216	−0.501080	−	0.865401i	$-0.667064\pi$
$467$	−21.6569	−1.00216	−0.501080	+	0.865401i	$0.667064\pi$
$468$	0	0
$469$	4.68629	0.216393
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−19.7990	−0.904639	−0.452319	−	0.891856i	$-0.649403\pi$
$479$	−19.7990	−0.904639	−0.452319	+	0.891856i	$0.649403\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	11.3137	0.514792
$484$	0	0
$485$	4.82843	0.219248
$486$	0	0
$487$	−17.6569	−0.800108	−0.400054	−	0.916491i	$-0.631009\pi$
$487$	−17.6569	−0.800108	−0.400054	+	0.916491i	$0.631009\pi$
$488$	0	0
$489$	2.82843	0.127906
$490$	0	0
$491$	24.4853	1.10501	0.552503	−	0.833511i	$-0.313673\pi$
$491$	24.4853	1.10501	0.552503	+	0.833511i	$0.313673\pi$
$492$	0	0
$493$	6.34315	0.285681
$494$	0	0
$495$	−2.82843	−0.127128
$496$	0	0
$497$	−22.6274	−1.01498
$498$	0	0
$499$	−1.65685	−0.0741710	−0.0370855	−	0.999312i	$-0.511807\pi$
$499$	−1.65685	−0.0741710	−0.0370855	+	0.999312i	$0.511807\pi$
$500$	0	0
$501$	5.65685	0.252730
$502$	0	0
$503$	−36.9706	−1.64844	−0.824218	−	0.566273i	$-0.808385\pi$
$503$	−36.9706	−1.64844	−0.824218	+	0.566273i	$0.808385\pi$
$504$	0	0
$505$	3.65685	0.162728
$506$	0	0
$507$	12.3137	0.546871
$508$	0	0
$509$	38.7696	1.71843	0.859215	−	0.511615i	$-0.170952\pi$
$509$	38.7696	1.71843	0.859215	+	0.511615i	$0.170952\pi$
$510$	0	0
$511$	0.970563	0.0429352
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	1.65685	0.0730097
$516$	0	0
$517$	11.3137	0.497576
$518$	0	0
$519$	−11.6569	−0.511679
$520$	0	0
$521$	−11.1716	−0.489435	−0.244718	−	0.969594i	$-0.578695\pi$
$521$	−11.1716	−0.489435	−0.244718	+	0.969594i	$0.578695\pi$
$522$	0	0
$523$	26.6274	1.16434	0.582168	−	0.813069i	$-0.302205\pi$
$523$	26.6274	1.16434	0.582168	+	0.813069i	$0.302205\pi$
$524$	0	0
$525$	2.82843	0.123443
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	−0.686292	−0.0297266
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	11.3137	0.488223
$538$	0	0
$539$	2.82843	0.121829
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	5.31371	0.228033
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	−6.34315	−0.271213	−0.135607	−	0.990763i	$-0.543298\pi$
$547$	−6.34315	−0.271213	−0.135607	+	0.990763i	$0.543298\pi$
$548$	0	0
$549$	9.31371	0.397499
$550$	0	0
$551$	0.828427	0.0352922
$552$	0	0
$553$	−38.6274	−1.64260
$554$	0	0
$555$	−4.82843	−0.204955
$556$	0	0
$557$	−28.6274	−1.21298	−0.606491	−	0.795090i	$-0.707424\pi$
$557$	−28.6274	−1.21298	−0.606491	+	0.795090i	$0.707424\pi$
$558$	0	0
$559$	2.34315	0.0991045
$560$	0	0
$561$	21.6569	0.914353
$562$	0	0
$563$	12.9706	0.546644	0.273322	−	0.961923i	$-0.411878\pi$
$563$	12.9706	0.546644	0.273322	+	0.961923i	$0.411878\pi$
$564$	0	0
$565$	−7.65685	−0.322126
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	37.7990	1.58462	0.792308	−	0.610121i	$-0.208879\pi$
$569$	37.7990	1.58462	0.792308	+	0.610121i	$0.208879\pi$
$570$	0	0
$571$	−28.9706	−1.21238	−0.606190	−	0.795320i	$-0.707303\pi$
$571$	−28.9706	−1.21238	−0.606190	+	0.795320i	$0.707303\pi$
$572$	0	0
$573$	−18.8284	−0.786569
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	13.7990	0.573466
$580$	0	0
$581$	−22.6274	−0.938743
$582$	0	0
$583$	−5.65685	−0.234283
$584$	0	0
$585$	−0.828427	−0.0342512
$586$	0	0
$587$	8.97056	0.370255	0.185127	−	0.982715i	$-0.440730\pi$
$587$	8.97056	0.370255	0.185127	+	0.982715i	$0.440730\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	2.68629	0.110499
$592$	0	0
$593$	5.02944	0.206534	0.103267	−	0.994654i	$-0.467070\pi$
$593$	5.02944	0.206534	0.103267	+	0.994654i	$0.467070\pi$
$594$	0	0
$595$	−21.6569	−0.887844
$596$	0	0
$597$	−8.97056	−0.367141
$598$	0	0
$599$	0.970563	0.0396561	0.0198281	−	0.999803i	$-0.493688\pi$
$599$	0.970563	0.0396561	0.0198281	+	0.999803i	$0.493688\pi$
$600$	0	0
$601$	−19.6569	−0.801820	−0.400910	−	0.916117i	$-0.631306\pi$
$601$	−19.6569	−0.801820	−0.400910	+	0.916117i	$0.631306\pi$
$602$	0	0
$603$	−1.65685	−0.0674723
$604$	0	0
$605$	3.00000	0.121967
$606$	0	0
$607$	17.6569	0.716670	0.358335	−	0.933593i	$-0.383345\pi$
$607$	17.6569	0.716670	0.358335	+	0.933593i	$0.383345\pi$
$608$	0	0
$609$	−2.34315	−0.0949491
$610$	0	0
$611$	3.31371	0.134058
$612$	0	0
$613$	40.6274	1.64093	0.820463	−	0.571700i	$-0.193716\pi$
$613$	40.6274	1.64093	0.820463	+	0.571700i	$0.193716\pi$
$614$	0	0
$615$	−0.828427	−0.0334054
$616$	0	0
$617$	−42.9706	−1.72993	−0.864965	−	0.501832i	$-0.832659\pi$
$617$	−42.9706	−1.72993	−0.864965	+	0.501832i	$0.832659\pi$
$618$	0	0
$619$	−49.6569	−1.99588	−0.997939	−	0.0641737i	$-0.979559\pi$
$619$	−49.6569	−1.99588	−0.997939	+	0.0641737i	$0.979559\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−20.2843	−0.812672
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.82843	0.112956
$628$	0	0
$629$	36.9706	1.47411
$630$	0	0
$631$	−22.6274	−0.900783	−0.450392	−	0.892831i	$-0.648716\pi$
$631$	−22.6274	−0.900783	−0.450392	+	0.892831i	$0.648716\pi$
$632$	0	0
$633$	18.6274	0.740373
$634$	0	0
$635$	−12.0000	−0.476205
$636$	0	0
$637$	0.828427	0.0328235
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	23.1716	0.915222	0.457611	−	0.889152i	$-0.348705\pi$
$641$	23.1716	0.915222	0.457611	+	0.889152i	$0.348705\pi$
$642$	0	0
$643$	−44.7696	−1.76554	−0.882769	−	0.469807i	$-0.844324\pi$
$643$	−44.7696	−1.76554	−0.882769	+	0.469807i	$0.844324\pi$
$644$	0	0
$645$	2.82843	0.111369
$646$	0	0
$647$	−6.34315	−0.249375	−0.124687	−	0.992196i	$-0.539793\pi$
$647$	−6.34315	−0.249375	−0.124687	+	0.992196i	$0.539793\pi$
$648$	0	0
$649$	22.6274	0.888204
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.6274	0.963745	0.481873	−	0.876241i	$-0.339957\pi$
$653$	24.6274	0.963745	0.481873	+	0.876241i	$0.339957\pi$
$654$	0	0
$655$	−5.17157	−0.202070
$656$	0	0
$657$	−0.343146	−0.0133874
$658$	0	0
$659$	−40.9706	−1.59599	−0.797993	−	0.602666i	$-0.794105\pi$
$659$	−40.9706	−1.59599	−0.797993	+	0.602666i	$0.794105\pi$
$660$	0	0
$661$	−6.68629	−0.260067	−0.130033	−	0.991510i	$-0.541508\pi$
$661$	−6.68629	−0.260067	−0.130033	+	0.991510i	$0.541508\pi$
$662$	0	0
$663$	6.34315	0.246347
$664$	0	0
$665$	−2.82843	−0.109682
$666$	0	0
$667$	−3.31371	−0.128307
$668$	0	0
$669$	−12.9706	−0.501471
$670$	0	0
$671$	26.3431	1.01697
$672$	0	0
$673$	13.1127	0.505457	0.252729	−	0.967537i	$-0.418672\pi$
$673$	13.1127	0.505457	0.252729	+	0.967537i	$0.418672\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	13.6569	0.524102
$680$	0	0
$681$	−1.65685	−0.0634908
$682$	0	0
$683$	42.6274	1.63109	0.815546	−	0.578692i	$-0.196437\pi$
$683$	42.6274	1.63109	0.815546	+	0.578692i	$0.196437\pi$
$684$	0	0
$685$	−22.9706	−0.877660
$686$	0	0
$687$	24.6274	0.939595
$688$	0	0
$689$	−1.65685	−0.0631211
$690$	0	0
$691$	1.65685	0.0630297	0.0315149	−	0.999503i	$-0.489967\pi$
$691$	1.65685	0.0630297	0.0315149	+	0.999503i	$0.489967\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	1.65685	0.0628481
$696$	0	0
$697$	6.34315	0.240264
$698$	0	0
$699$	−0.343146	−0.0129790
$700$	0	0
$701$	22.2843	0.841665	0.420833	−	0.907138i	$-0.361738\pi$
$701$	22.2843	0.841665	0.420833	+	0.907138i	$0.361738\pi$
$702$	0	0
$703$	4.82843	0.182108
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	10.3431	0.388994
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	13.6569	0.512172
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−2.34315	−0.0876287
$716$	0	0
$717$	−27.7990	−1.03817
$718$	0	0
$719$	−17.4558	−0.650993	−0.325497	−	0.945543i	$-0.605531\pi$
$719$	−17.4558	−0.650993	−0.325497	+	0.945543i	$0.605531\pi$
$720$	0	0
$721$	4.68629	0.174527
$722$	0	0
$723$	22.0000	0.818189
$724$	0	0
$725$	−0.828427	−0.0307670
$726$	0	0
$727$	−16.4853	−0.611405	−0.305703	−	0.952127i	$-0.598891\pi$
$727$	−16.4853	−0.611405	−0.305703	+	0.952127i	$0.598891\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.6569	−0.801008
$732$	0	0
$733$	−30.9706	−1.14392	−0.571962	−	0.820280i	$-0.693817\pi$
$733$	−30.9706	−1.14392	−0.571962	+	0.820280i	$0.693817\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−4.68629	−0.172622
$738$	0	0
$739$	0.686292	0.0252456	0.0126228	−	0.999920i	$-0.495982\pi$
$739$	0.686292	0.0252456	0.0126228	+	0.999920i	$0.495982\pi$
$740$	0	0
$741$	0.828427	0.0304330
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	−15.6569	−0.573623
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	−11.3137	−0.413394
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	8.48528	0.309221
$754$	0	0
$755$	3.31371	0.120598
$756$	0	0
$757$	20.3431	0.739384	0.369692	−	0.929154i	$-0.379463\pi$
$757$	20.3431	0.739384	0.369692	+	0.929154i	$0.379463\pi$
$758$	0	0
$759$	−11.3137	−0.410662
$760$	0	0
$761$	41.5980	1.50793	0.753963	−	0.656917i	$-0.228140\pi$
$761$	41.5980	1.50793	0.753963	+	0.656917i	$0.228140\pi$
$762$	0	0
$763$	−39.5980	−1.43354
$764$	0	0
$765$	7.65685	0.276834
$766$	0	0
$767$	6.62742	0.239302
$768$	0	0
$769$	40.6274	1.46506	0.732531	−	0.680734i	$-0.238339\pi$
$769$	40.6274	1.46506	0.732531	+	0.680734i	$0.238339\pi$
$770$	0	0
$771$	−10.9706	−0.395095
$772$	0	0
$773$	28.6274	1.02966	0.514828	−	0.857293i	$-0.327856\pi$
$773$	28.6274	1.02966	0.514828	+	0.857293i	$0.327856\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−13.6569	−0.489937
$778$	0	0
$779$	0.828427	0.0296815
$780$	0	0
$781$	22.6274	0.809673
$782$	0	0
$783$	0.828427	0.0296056
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	45.9411	1.63762	0.818812	−	0.574062i	$-0.194633\pi$
$787$	45.9411	1.63762	0.818812	+	0.574062i	$0.194633\pi$
$788$	0	0
$789$	−9.65685	−0.343793
$790$	0	0
$791$	−21.6569	−0.770029
$792$	0	0
$793$	7.71573	0.273994
$794$	0	0
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	−47.6569	−1.68809	−0.844046	−	0.536270i	$-0.819833\pi$
$797$	−47.6569	−1.68809	−0.844046	+	0.536270i	$0.819833\pi$
$798$	0	0
$799$	−30.6274	−1.08352
$800$	0	0
$801$	7.17157	0.253395
$802$	0	0
$803$	−0.970563	−0.0342504
$804$	0	0
$805$	11.3137	0.398756
$806$	0	0
$807$	−18.4853	−0.650713
$808$	0	0
$809$	−10.6863	−0.375710	−0.187855	−	0.982197i	$-0.560154\pi$
$809$	−10.6863	−0.375710	−0.187855	+	0.982197i	$0.560154\pi$
$810$	0	0
$811$	49.2548	1.72957	0.864786	−	0.502141i	$-0.167454\pi$
$811$	49.2548	1.72957	0.864786	+	0.502141i	$0.167454\pi$
$812$	0	0
$813$	−16.9706	−0.595184
$814$	0	0
$815$	2.82843	0.0990755
$816$	0	0
$817$	−2.82843	−0.0989541
$818$	0	0
$819$	−2.34315	−0.0818761
$820$	0	0
$821$	28.3431	0.989183	0.494591	−	0.869126i	$-0.335318\pi$
$821$	28.3431	0.989183	0.494591	+	0.869126i	$0.335318\pi$
$822$	0	0
$823$	18.8284	0.656318	0.328159	−	0.944623i	$-0.393572\pi$
$823$	18.8284	0.656318	0.328159	+	0.944623i	$0.393572\pi$
$824$	0	0
$825$	−2.82843	−0.0984732
$826$	0	0
$827$	1.65685	0.0576145	0.0288072	−	0.999585i	$-0.490829\pi$
$827$	1.65685	0.0576145	0.0288072	+	0.999585i	$0.490829\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	−7.65685	−0.265294
$834$	0	0
$835$	5.65685	0.195764
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.62742	0.228804	0.114402	−	0.993435i	$-0.463505\pi$
$839$	6.62742	0.228804	0.114402	+	0.993435i	$0.463505\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	−8.14214	−0.280430
$844$	0	0
$845$	12.3137	0.423604
$846$	0	0
$847$	8.48528	0.291558
$848$	0	0
$849$	−8.48528	−0.291214
$850$	0	0
$851$	−19.3137	−0.662065
$852$	0	0
$853$	54.2843	1.85866	0.929329	−	0.369253i	$-0.120386\pi$
$853$	54.2843	1.85866	0.929329	+	0.369253i	$0.120386\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−28.6274	−0.977894	−0.488947	−	0.872314i	$-0.662619\pi$
$857$	−28.6274	−0.977894	−0.488947	+	0.872314i	$0.662619\pi$
$858$	0	0
$859$	−24.6863	−0.842285	−0.421143	−	0.906994i	$-0.638371\pi$
$859$	−24.6863	−0.842285	−0.421143	+	0.906994i	$0.638371\pi$
$860$	0	0
$861$	−2.34315	−0.0798542
$862$	0	0
$863$	38.6274	1.31489	0.657446	−	0.753501i	$-0.271637\pi$
$863$	38.6274	1.31489	0.657446	+	0.753501i	$0.271637\pi$
$864$	0	0
$865$	−11.6569	−0.396345
$866$	0	0
$867$	−41.6274	−1.41374
$868$	0	0
$869$	38.6274	1.31035
$870$	0	0
$871$	−1.37258	−0.0465082
$872$	0	0
$873$	−4.82843	−0.163417
$874$	0	0
$875$	2.82843	0.0956183
$876$	0	0
$877$	8.82843	0.298115	0.149057	−	0.988829i	$-0.452376\pi$
$877$	8.82843	0.298115	0.149057	+	0.988829i	$0.452376\pi$
$878$	0	0
$879$	7.65685	0.258259
$880$	0	0
$881$	6.28427	0.211722	0.105861	−	0.994381i	$-0.466240\pi$
$881$	6.28427	0.211722	0.105861	+	0.994381i	$0.466240\pi$
$882$	0	0
$883$	−25.4558	−0.856657	−0.428329	−	0.903623i	$-0.640897\pi$
$883$	−25.4558	−0.856657	−0.428329	+	0.903623i	$0.640897\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	−6.62742	−0.222527	−0.111263	−	0.993791i	$-0.535490\pi$
$887$	−6.62742	−0.222527	−0.111263	+	0.993791i	$0.535490\pi$
$888$	0	0
$889$	−33.9411	−1.13835
$890$	0	0
$891$	2.82843	0.0947559
$892$	0	0
$893$	−4.00000	−0.133855
$894$	0	0
$895$	11.3137	0.378176
$896$	0	0
$897$	−3.31371	−0.110642
$898$	0	0
$899$	0	0
$900$	0	0
$901$	15.3137	0.510174
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	5.31371	0.176634
$906$	0	0
$907$	0.686292	0.0227879	0.0113940	−	0.999935i	$-0.496373\pi$
$907$	0.686292	0.0227879	0.0113940	+	0.999935i	$0.496373\pi$
$908$	0	0
$909$	−3.65685	−0.121290
$910$	0	0
$911$	9.37258	0.310528	0.155264	−	0.987873i	$-0.450377\pi$
$911$	9.37258	0.310528	0.155264	+	0.987873i	$0.450377\pi$
$912$	0	0
$913$	22.6274	0.748858
$914$	0	0
$915$	9.31371	0.307902
$916$	0	0
$917$	−14.6274	−0.483040
$918$	0	0
$919$	11.3137	0.373205	0.186602	−	0.982436i	$-0.440252\pi$
$919$	11.3137	0.373205	0.186602	+	0.982436i	$0.440252\pi$
$920$	0	0
$921$	6.34315	0.209014
$922$	0	0
$923$	6.62742	0.218144
$924$	0	0
$925$	−4.82843	−0.158758
$926$	0	0
$927$	−1.65685	−0.0544182
$928$	0	0
$929$	−51.2548	−1.68162	−0.840808	−	0.541333i	$-0.817920\pi$
$929$	−51.2548	−1.68162	−0.840808	+	0.541333i	$0.817920\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−30.1421	−0.986809
$934$	0	0
$935$	21.6569	0.708255
$936$	0	0
$937$	7.65685	0.250139	0.125069	−	0.992148i	$-0.460085\pi$
$937$	7.65685	0.250139	0.125069	+	0.992148i	$0.460085\pi$
$938$	0	0
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	−34.7696	−1.13346	−0.566728	−	0.823905i	$-0.691791\pi$
$941$	−34.7696	−1.13346	−0.566728	+	0.823905i	$0.691791\pi$
$942$	0	0
$943$	−3.31371	−0.107909
$944$	0	0
$945$	−2.82843	−0.0920087
$946$	0	0
$947$	−30.6274	−0.995257	−0.497629	−	0.867390i	$-0.665796\pi$
$947$	−30.6274	−0.995257	−0.497629	+	0.867390i	$0.665796\pi$
$948$	0	0
$949$	−0.284271	−0.00922784
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−19.6569	−0.636748	−0.318374	−	0.947965i	$-0.603137\pi$
$953$	−19.6569	−0.636748	−0.318374	+	0.947965i	$0.603137\pi$
$954$	0	0
$955$	−18.8284	−0.609274
$956$	0	0
$957$	2.34315	0.0757431
$958$	0	0
$959$	−64.9706	−2.09801
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	13.7990	0.444205
$966$	0	0
$967$	1.45584	0.0468168	0.0234084	−	0.999726i	$-0.492548\pi$
$967$	1.45584	0.0468168	0.0234084	+	0.999726i	$0.492548\pi$
$968$	0	0
$969$	−7.65685	−0.245974
$970$	0	0
$971$	50.9117	1.63383	0.816917	−	0.576755i	$-0.195681\pi$
$971$	50.9117	1.63383	0.816917	+	0.576755i	$0.195681\pi$
$972$	0	0
$973$	4.68629	0.150236
$974$	0	0
$975$	−0.828427	−0.0265309
$976$	0	0
$977$	−15.9411	−0.510002	−0.255001	−	0.966941i	$-0.582076\pi$
$977$	−15.9411	−0.510002	−0.255001	+	0.966941i	$0.582076\pi$
$978$	0	0
$979$	20.2843	0.648288
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	−18.3431	−0.585055	−0.292528	−	0.956257i	$-0.594496\pi$
$983$	−18.3431	−0.585055	−0.292528	+	0.956257i	$0.594496\pi$
$984$	0	0
$985$	2.68629	0.0855924
$986$	0	0
$987$	11.3137	0.360119
$988$	0	0
$989$	11.3137	0.359755
$990$	0	0
$991$	24.9706	0.793216	0.396608	−	0.917988i	$-0.370187\pi$
$991$	24.9706	0.793216	0.396608	+	0.917988i	$0.370187\pi$
$992$	0	0
$993$	25.6569	0.814196
$994$	0	0
$995$	−8.97056	−0.284386
$996$	0	0
$997$	53.3137	1.68846	0.844231	−	0.535979i	$-0.180058\pi$
$997$	53.3137	1.68846	0.844231	+	0.535979i	$0.180058\pi$
$998$	0	0
$999$	4.82843	0.152765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2280.2.a.l.1.1	✓	2
3.2	odd	2		6840.2.a.ba.1.1		2
4.3	odd	2		4560.2.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.l.1.1	✓	2	1.1	even	1	trivial
4560.2.a.bm.1.2		2	4.3	odd	2
6840.2.a.ba.1.1		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +2.82843 q^{11} +0.828427 q^{13} +1.00000 q^{15} -7.65685 q^{17} -1.00000 q^{19} +2.82843 q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.828427 q^{29} -2.82843 q^{33} +2.82843 q^{35} -4.82843 q^{37} -0.828427 q^{39} -0.828427 q^{41} +2.82843 q^{43} -1.00000 q^{45} +4.00000 q^{47} +1.00000 q^{49} +7.65685 q^{51} -2.00000 q^{53} -2.82843 q^{55} +1.00000 q^{57} +8.00000 q^{59} +9.31371 q^{61} -2.82843 q^{63} -0.828427 q^{65} -1.65685 q^{67} -4.00000 q^{69} +8.00000 q^{71} -0.343146 q^{73} -1.00000 q^{75} -8.00000 q^{77} +13.6569 q^{79} +1.00000 q^{81} +8.00000 q^{83} +7.65685 q^{85} +0.828427 q^{87} +7.17157 q^{89} -2.34315 q^{91} +1.00000 q^{95} -4.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{13} + 2 q^{15} - 4 q^{17} - 2 q^{19} + 8 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 4 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 4 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 4 q^{65} + 8 q^{67} - 8 q^{69} + 16 q^{71} - 12 q^{73} - 2 q^{75} - 16 q^{77} + 16 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} - 4 q^{87} + 20 q^{89} - 16 q^{91} + 2 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^13 + 2 * q^15 - 4 * q^17 - 2 * q^19 + 8 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 - 4 * q^37 + 4 * q^39 + 4 * q^41 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 4 * q^51 - 4 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 4 * q^65 + 8 * q^67 - 8 * q^69 + 16 * q^71 - 12 * q^73 - 2 * q^75 - 16 * q^77 + 16 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 - 4 * q^87 + 20 * q^89 - 16 * q^91 + 2 * q^95 - 4 * q^97

Introduction

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