LMFDB - Embedded newform 3584.2.a.o.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3584.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 44x^{7} + 86x^{6} - 236x^{5} - 58x^{4} + 368x^{3} - 194x^{2} - 12x + 18 $ x^10 - 2x^9 - 19x^8 + 44x^7 + 86x^6 - 236x^5 - 58x^4 + 368x^3 - 194x^2 - 12*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.40586$ of defining polynomial
Character	$\chi$	$=$	3584.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.99347 q^{3} +0.369776 q^{5} -1.00000 q^{7} +5.96088 q^{9} +O(q^{10})$	$q+2.99347 q^{3} +0.369776 q^{5} -1.00000 q^{7} +5.96088 q^{9} +3.93465 q^{11} +6.07457 q^{13} +1.10691 q^{15} +6.75635 q^{17} +0.117109 q^{19} -2.99347 q^{21} -8.67135 q^{23} -4.86327 q^{25} +8.86331 q^{27} -6.27619 q^{29} +0.808080 q^{31} +11.7783 q^{33} -0.369776 q^{35} +10.1743 q^{37} +18.1840 q^{39} -7.27490 q^{41} -0.0365776 q^{43} +2.20419 q^{45} -4.06779 q^{47} +1.00000 q^{49} +20.2250 q^{51} +2.53918 q^{53} +1.45494 q^{55} +0.350562 q^{57} -5.53975 q^{59} -6.02663 q^{61} -5.96088 q^{63} +2.24623 q^{65} +4.88072 q^{67} -25.9574 q^{69} -2.35733 q^{71} -6.82414 q^{73} -14.5581 q^{75} -3.93465 q^{77} -6.35733 q^{79} +8.64944 q^{81} +8.65033 q^{83} +2.49834 q^{85} -18.7876 q^{87} -3.56443 q^{89} -6.07457 q^{91} +2.41897 q^{93} +0.0433041 q^{95} -0.211290 q^{97} +23.4540 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{7} + 22 q^{9}+O(q^{10}) $ 10 * q - 10 * q^7 + 22 * q^9	$ 10 q - 10 q^{7} + 22 q^{9} - 16 q^{15} + 16 q^{17} - 8 q^{23} + 30 q^{25} + 8 q^{31} + 12 q^{33} + 20 q^{41} + 24 q^{47} + 10 q^{49} + 32 q^{55} + 28 q^{57} - 22 q^{63} + 32 q^{65} + 48 q^{73} - 40 q^{79} + 62 q^{81} + 8 q^{87} + 16 q^{89} + 32 q^{95} + 32 q^{97}+O(q^{100}) $ 10 * q - 10 * q^7 + 22 * q^9 - 16 * q^15 + 16 * q^17 - 8 * q^23 + 30 * q^25 + 8 * q^31 + 12 * q^33 + 20 * q^41 + 24 * q^47 + 10 * q^49 + 32 * q^55 + 28 * q^57 - 22 * q^63 + 32 * q^65 + 48 * q^73 - 40 * q^79 + 62 * q^81 + 8 * q^87 + 16 * q^89 + 32 * q^95 + 32 * 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
$3$	2.99347	1.72828	0.864141	−	0.503250i	$-0.167862\pi$
$3$	2.99347	1.72828	0.864141	+	0.503250i	$0.167862\pi$
$4$	0	0
$5$	0.369776	0.165369	0.0826844	−	0.996576i	$-0.473651\pi$
$5$	0.369776	0.165369	0.0826844	+	0.996576i	$0.473651\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.96088	1.98696
$10$	0	0
$11$	3.93465	1.18634	0.593170	−	0.805077i	$-0.297876\pi$
$11$	3.93465	1.18634	0.593170	+	0.805077i	$0.297876\pi$
$12$	0	0
$13$	6.07457	1.68478	0.842391	−	0.538867i	$-0.181147\pi$
$13$	6.07457	1.68478	0.842391	+	0.538867i	$0.181147\pi$
$14$	0	0
$15$	1.10691	0.285804
$16$	0	0
$17$	6.75635	1.63866	0.819328	−	0.573325i	$-0.194347\pi$
$17$	6.75635	1.63866	0.819328	+	0.573325i	$0.194347\pi$
$18$	0	0
$19$	0.117109	0.0268666	0.0134333	−	0.999910i	$-0.495724\pi$
$19$	0.117109	0.0268666	0.0134333	+	0.999910i	$0.495724\pi$
$20$	0	0
$21$	−2.99347	−0.653229
$22$	0	0
$23$	−8.67135	−1.80810	−0.904050	−	0.427426i	$-0.859420\pi$
$23$	−8.67135	−1.80810	−0.904050	+	0.427426i	$0.859420\pi$
$24$	0	0
$25$	−4.86327	−0.972653
$26$	0	0
$27$	8.86331	1.70574
$28$	0	0
$29$	−6.27619	−1.16546	−0.582730	−	0.812666i	$-0.698015\pi$
$29$	−6.27619	−1.16546	−0.582730	+	0.812666i	$0.698015\pi$
$30$	0	0
$31$	0.808080	0.145136	0.0725678	−	0.997363i	$-0.476881\pi$
$31$	0.808080	0.145136	0.0725678	+	0.997363i	$0.476881\pi$
$32$	0	0
$33$	11.7783	2.05033
$34$	0	0
$35$	−0.369776	−0.0625035
$36$	0	0
$37$	10.1743	1.67264	0.836319	−	0.548243i	$-0.184703\pi$
$37$	10.1743	1.67264	0.836319	+	0.548243i	$0.184703\pi$
$38$	0	0
$39$	18.1840	2.91178
$40$	0	0
$41$	−7.27490	−1.13615	−0.568074	−	0.822977i	$-0.692311\pi$
$41$	−7.27490	−1.13615	−0.568074	+	0.822977i	$0.692311\pi$
$42$	0	0
$43$	−0.0365776	−0.00557804	−0.00278902	−	0.999996i	$-0.500888\pi$
$43$	−0.0365776	−0.00557804	−0.00278902	+	0.999996i	$0.500888\pi$
$44$	0	0
$45$	2.20419	0.328581
$46$	0	0
$47$	−4.06779	−0.593349	−0.296674	−	0.954979i	$-0.595878\pi$
$47$	−4.06779	−0.593349	−0.296674	+	0.954979i	$0.595878\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	20.2250	2.83206
$52$	0	0
$53$	2.53918	0.348784	0.174392	−	0.984676i	$-0.444204\pi$
$53$	2.53918	0.348784	0.174392	+	0.984676i	$0.444204\pi$
$54$	0	0
$55$	1.45494	0.196184
$56$	0	0
$57$	0.350562	0.0464331
$58$	0	0
$59$	−5.53975	−0.721213	−0.360607	−	0.932718i	$-0.617430\pi$
$59$	−5.53975	−0.721213	−0.360607	+	0.932718i	$0.617430\pi$
$60$	0	0
$61$	−6.02663	−0.771631	−0.385815	−	0.922576i	$-0.626080\pi$
$61$	−6.02663	−0.771631	−0.385815	+	0.922576i	$0.626080\pi$
$62$	0	0
$63$	−5.96088	−0.751000
$64$	0	0
$65$	2.24623	0.278610
$66$	0	0
$67$	4.88072	0.596275	0.298138	−	0.954523i	$-0.403635\pi$
$67$	4.88072	0.596275	0.298138	+	0.954523i	$0.403635\pi$
$68$	0	0
$69$	−25.9574	−3.12491
$70$	0	0
$71$	−2.35733	−0.279763	−0.139882	−	0.990168i	$-0.544672\pi$
$71$	−2.35733	−0.279763	−0.139882	+	0.990168i	$0.544672\pi$
$72$	0	0
$73$	−6.82414	−0.798706	−0.399353	−	0.916797i	$-0.630765\pi$
$73$	−6.82414	−0.798706	−0.399353	+	0.916797i	$0.630765\pi$
$74$	0	0
$75$	−14.5581	−1.68102
$76$	0	0
$77$	−3.93465	−0.448395
$78$	0	0
$79$	−6.35733	−0.715255	−0.357627	−	0.933864i	$-0.616414\pi$
$79$	−6.35733	−0.715255	−0.357627	+	0.933864i	$0.616414\pi$
$80$	0	0
$81$	8.64944	0.961049
$82$	0	0
$83$	8.65033	0.949497	0.474748	−	0.880122i	$-0.342539\pi$
$83$	8.65033	0.949497	0.474748	+	0.880122i	$0.342539\pi$
$84$	0	0
$85$	2.49834	0.270983
$86$	0	0
$87$	−18.7876	−2.01424
$88$	0	0
$89$	−3.56443	−0.377829	−0.188915	−	0.981994i	$-0.560497\pi$
$89$	−3.56443	−0.377829	−0.188915	+	0.981994i	$0.560497\pi$
$90$	0	0
$91$	−6.07457	−0.636788
$92$	0	0
$93$	2.41897	0.250835
$94$	0	0
$95$	0.0433041	0.00444290
$96$	0	0
$97$	−0.211290	−0.0214532	−0.0107266	−	0.999942i	$-0.503414\pi$
$97$	−0.211290	−0.0214532	−0.0107266	+	0.999942i	$0.503414\pi$
$98$	0	0
$99$	23.4540	2.35721
$100$	0	0
$101$	−9.84907	−0.980019	−0.490010	−	0.871717i	$-0.663007\pi$
$101$	−9.84907	−0.980019	−0.490010	+	0.871717i	$0.663007\pi$
$102$	0	0
$103$	11.6587	1.14877	0.574385	−	0.818585i	$-0.305241\pi$
$103$	11.6587	1.14877	0.574385	+	0.818585i	$0.305241\pi$
$104$	0	0
$105$	−1.10691	−0.108024
$106$	0	0
$107$	10.4269	1.00801	0.504005	−	0.863701i	$-0.331859\pi$
$107$	10.4269	1.00801	0.504005	+	0.863701i	$0.331859\pi$
$108$	0	0
$109$	13.2092	1.26521	0.632607	−	0.774473i	$-0.281985\pi$
$109$	13.2092	1.26521	0.632607	+	0.774473i	$0.281985\pi$
$110$	0	0
$111$	30.4564	2.89079
$112$	0	0
$113$	19.3760	1.82274	0.911369	−	0.411590i	$-0.135027\pi$
$113$	19.3760	1.82274	0.911369	+	0.411590i	$0.135027\pi$
$114$	0	0
$115$	−3.20645	−0.299003
$116$	0	0
$117$	36.2098	3.34759
$118$	0	0
$119$	−6.75635	−0.619354
$120$	0	0
$121$	4.48145	0.407405
$122$	0	0
$123$	−21.7772	−1.96359
$124$	0	0
$125$	−3.64720	−0.326215
$126$	0	0
$127$	−18.5931	−1.64987	−0.824936	−	0.565227i	$-0.808789\pi$
$127$	−18.5931	−1.64987	−0.824936	+	0.565227i	$0.808789\pi$
$128$	0	0
$129$	−0.109494	−0.00964043
$130$	0	0
$131$	3.06663	0.267933	0.133966	−	0.990986i	$-0.457229\pi$
$131$	3.06663	0.267933	0.133966	+	0.990986i	$0.457229\pi$
$132$	0	0
$133$	−0.117109	−0.0101546
$134$	0	0
$135$	3.27744	0.282077
$136$	0	0
$137$	6.90239	0.589711	0.294855	−	0.955542i	$-0.404729\pi$
$137$	6.90239	0.589711	0.294855	+	0.955542i	$0.404729\pi$
$138$	0	0
$139$	−18.9927	−1.61094	−0.805472	−	0.592634i	$-0.798088\pi$
$139$	−18.9927	−1.61094	−0.805472	+	0.592634i	$0.798088\pi$
$140$	0	0
$141$	−12.1768	−1.02547
$142$	0	0
$143$	23.9013	1.99873
$144$	0	0
$145$	−2.32078	−0.192731
$146$	0	0
$147$	2.99347	0.246897
$148$	0	0
$149$	12.6726	1.03818	0.519090	−	0.854720i	$-0.326271\pi$
$149$	12.6726	1.03818	0.519090	+	0.854720i	$0.326271\pi$
$150$	0	0
$151$	9.87845	0.803897	0.401949	−	0.915662i	$-0.368333\pi$
$151$	9.87845	0.803897	0.401949	+	0.915662i	$0.368333\pi$
$152$	0	0
$153$	40.2738	3.25594
$154$	0	0
$155$	0.298809	0.0240009
$156$	0	0
$157$	11.0877	0.884898	0.442449	−	0.896794i	$-0.354110\pi$
$157$	11.0877	0.884898	0.442449	+	0.896794i	$0.354110\pi$
$158$	0	0
$159$	7.60097	0.602796
$160$	0	0
$161$	8.67135	0.683398
$162$	0	0
$163$	−10.6611	−0.835045	−0.417523	−	0.908667i	$-0.637102\pi$
$163$	−10.6611	−0.835045	−0.417523	+	0.908667i	$0.637102\pi$
$164$	0	0
$165$	4.35532	0.339061
$166$	0	0
$167$	13.6840	1.05890	0.529448	−	0.848342i	$-0.322399\pi$
$167$	13.6840	1.05890	0.529448	+	0.848342i	$0.322399\pi$
$168$	0	0
$169$	23.9004	1.83849
$170$	0	0
$171$	0.698072	0.0533829
$172$	0	0
$173$	0.242470	0.0184346	0.00921731	−	0.999958i	$-0.497066\pi$
$173$	0.242470	0.0184346	0.00921731	+	0.999958i	$0.497066\pi$
$174$	0	0
$175$	4.86327	0.367628
$176$	0	0
$177$	−16.5831	−1.24646
$178$	0	0
$179$	0.776129	0.0580106	0.0290053	−	0.999579i	$-0.490766\pi$
$179$	0.776129	0.0580106	0.0290053	+	0.999579i	$0.490766\pi$
$180$	0	0
$181$	−15.0614	−1.11951	−0.559754	−	0.828659i	$-0.689104\pi$
$181$	−15.0614	−1.11951	−0.559754	+	0.828659i	$0.689104\pi$
$182$	0	0
$183$	−18.0406	−1.33360
$184$	0	0
$185$	3.76220	0.276602
$186$	0	0
$187$	26.5839	1.94400
$188$	0	0
$189$	−8.86331	−0.644711
$190$	0	0
$191$	−15.4209	−1.11582	−0.557910	−	0.829902i	$-0.688397\pi$
$191$	−15.4209	−1.11582	−0.557910	+	0.829902i	$0.688397\pi$
$192$	0	0
$193$	9.75377	0.702092	0.351046	−	0.936358i	$-0.385826\pi$
$193$	9.75377	0.702092	0.351046	+	0.936358i	$0.385826\pi$
$194$	0	0
$195$	6.72402	0.481517
$196$	0	0
$197$	−21.3127	−1.51847	−0.759234	−	0.650817i	$-0.774426\pi$
$197$	−21.3127	−1.51847	−0.759234	+	0.650817i	$0.774426\pi$
$198$	0	0
$199$	−12.2290	−0.866892	−0.433446	−	0.901180i	$-0.642702\pi$
$199$	−12.2290	−0.866892	−0.433446	+	0.901180i	$0.642702\pi$
$200$	0	0
$201$	14.6103	1.03053
$202$	0	0
$203$	6.27619	0.440502
$204$	0	0
$205$	−2.69008	−0.187884
$206$	0	0
$207$	−51.6888	−3.59262
$208$	0	0
$209$	0.460782	0.0318730
$210$	0	0
$211$	−23.6543	−1.62843	−0.814213	−	0.580566i	$-0.802831\pi$
$211$	−23.6543	−1.62843	−0.814213	+	0.580566i	$0.802831\pi$
$212$	0	0
$213$	−7.05659	−0.483509
$214$	0	0
$215$	−0.0135255	−0.000922434 0
$216$	0	0
$217$	−0.808080	−0.0548561
$218$	0	0
$219$	−20.4279	−1.38039
$220$	0	0
$221$	41.0419	2.76078
$222$	0	0
$223$	10.2530	0.686591	0.343296	−	0.939227i	$-0.388457\pi$
$223$	10.2530	0.686591	0.343296	+	0.939227i	$0.388457\pi$
$224$	0	0
$225$	−28.9893	−1.93262
$226$	0	0
$227$	−11.9299	−0.791818	−0.395909	−	0.918290i	$-0.629570\pi$
$227$	−11.9299	−0.791818	−0.395909	+	0.918290i	$0.629570\pi$
$228$	0	0
$229$	5.89932	0.389838	0.194919	−	0.980819i	$-0.437556\pi$
$229$	5.89932	0.389838	0.194919	+	0.980819i	$0.437556\pi$
$230$	0	0
$231$	−11.7783	−0.774953
$232$	0	0
$233$	8.61032	0.564081	0.282040	−	0.959403i	$-0.408989\pi$
$233$	8.61032	0.564081	0.282040	+	0.959403i	$0.408989\pi$
$234$	0	0
$235$	−1.50417	−0.0981213
$236$	0	0
$237$	−19.0305	−1.23616
$238$	0	0
$239$	18.0485	1.16746	0.583729	−	0.811949i	$-0.301593\pi$
$239$	18.0485	1.16746	0.583729	+	0.811949i	$0.301593\pi$
$240$	0	0
$241$	15.9635	1.02830	0.514148	−	0.857701i	$-0.328108\pi$
$241$	15.9635	1.02830	0.514148	+	0.857701i	$0.328108\pi$
$242$	0	0
$243$	−0.698072	−0.0447813
$244$	0	0
$245$	0.369776	0.0236241
$246$	0	0
$247$	0.711386	0.0452644
$248$	0	0
$249$	25.8945	1.64100
$250$	0	0
$251$	−20.6242	−1.30179	−0.650895	−	0.759168i	$-0.725606\pi$
$251$	−20.6242	−1.30179	−0.650895	+	0.759168i	$0.725606\pi$
$252$	0	0
$253$	−34.1187	−2.14502
$254$	0	0
$255$	7.47870	0.468334
$256$	0	0
$257$	−5.66877	−0.353608	−0.176804	−	0.984246i	$-0.556576\pi$
$257$	−5.66877	−0.353608	−0.176804	+	0.984246i	$0.556576\pi$
$258$	0	0
$259$	−10.1743	−0.632198
$260$	0	0
$261$	−37.4116	−2.31572
$262$	0	0
$263$	11.2910	0.696231	0.348115	−	0.937452i	$-0.386822\pi$
$263$	11.2910	0.696231	0.348115	+	0.937452i	$0.386822\pi$
$264$	0	0
$265$	0.938928	0.0576779
$266$	0	0
$267$	−10.6700	−0.652995
$268$	0	0
$269$	−0.242470	−0.0147836	−0.00739182	−	0.999973i	$-0.502353\pi$
$269$	−0.242470	−0.0147836	−0.00739182	+	0.999973i	$0.502353\pi$
$270$	0	0
$271$	−2.25299	−0.136859	−0.0684297	−	0.997656i	$-0.521799\pi$
$271$	−2.25299	−0.136859	−0.0684297	+	0.997656i	$0.521799\pi$
$272$	0	0
$273$	−18.1840	−1.10055
$274$	0	0
$275$	−19.1352	−1.15390
$276$	0	0
$277$	17.5105	1.05211	0.526053	−	0.850452i	$-0.323671\pi$
$277$	17.5105	1.05211	0.526053	+	0.850452i	$0.323671\pi$
$278$	0	0
$279$	4.81687	0.288378
$280$	0	0
$281$	−32.7772	−1.95532	−0.977661	−	0.210190i	$-0.932592\pi$
$281$	−32.7772	−1.95532	−0.977661	+	0.210190i	$0.932592\pi$
$282$	0	0
$283$	−24.7190	−1.46939	−0.734696	−	0.678396i	$-0.762675\pi$
$283$	−24.7190	−1.46939	−0.734696	+	0.678396i	$0.762675\pi$
$284$	0	0
$285$	0.129629	0.00767859
$286$	0	0
$287$	7.27490	0.429424
$288$	0	0
$289$	28.6483	1.68519
$290$	0	0
$291$	−0.632490	−0.0370772
$292$	0	0
$293$	15.7531	0.920303	0.460152	−	0.887840i	$-0.347795\pi$
$293$	15.7531	0.920303	0.460152	+	0.887840i	$0.347795\pi$
$294$	0	0
$295$	−2.04846	−0.119266
$296$	0	0
$297$	34.8740	2.02359
$298$	0	0
$299$	−52.6747	−3.04625
$300$	0	0
$301$	0.0365776	0.00210830
$302$	0	0
$303$	−29.4829	−1.69375
$304$	0	0
$305$	−2.22850	−0.127604
$306$	0	0
$307$	−15.7863	−0.900971	−0.450485	−	0.892784i	$-0.648749\pi$
$307$	−15.7863	−0.900971	−0.450485	+	0.892784i	$0.648749\pi$
$308$	0	0
$309$	34.9001	1.98540
$310$	0	0
$311$	14.2186	0.806261	0.403130	−	0.915143i	$-0.367922\pi$
$311$	14.2186	0.806261	0.403130	+	0.915143i	$0.367922\pi$
$312$	0	0
$313$	26.8794	1.51931	0.759656	−	0.650325i	$-0.225367\pi$
$313$	26.8794	1.51931	0.759656	+	0.650325i	$0.225367\pi$
$314$	0	0
$315$	−2.20419	−0.124192
$316$	0	0
$317$	0.832075	0.0467340	0.0233670	−	0.999727i	$-0.492561\pi$
$317$	0.832075	0.0467340	0.0233670	+	0.999727i	$0.492561\pi$
$318$	0	0
$319$	−24.6946	−1.38263
$320$	0	0
$321$	31.2127	1.74212
$322$	0	0
$323$	0.791229	0.0440252
$324$	0	0
$325$	−29.5422	−1.63871
$326$	0	0
$327$	39.5414	2.18665
$328$	0	0
$329$	4.06779	0.224265
$330$	0	0
$331$	−13.4164	−0.737433	−0.368716	−	0.929542i	$-0.620203\pi$
$331$	−13.4164	−0.737433	−0.368716	+	0.929542i	$0.620203\pi$
$332$	0	0
$333$	60.6475	3.32346
$334$	0	0
$335$	1.80477	0.0986054
$336$	0	0
$337$	14.6613	0.798653	0.399326	−	0.916809i	$-0.369244\pi$
$337$	14.6613	0.798653	0.399326	+	0.916809i	$0.369244\pi$
$338$	0	0
$339$	58.0014	3.15021
$340$	0	0
$341$	3.17951	0.172180
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−9.59843	−0.516762
$346$	0	0
$347$	−6.02974	−0.323693	−0.161847	−	0.986816i	$-0.551745\pi$
$347$	−6.02974	−0.323693	−0.161847	+	0.986816i	$0.551745\pi$
$348$	0	0
$349$	−24.9754	−1.33690	−0.668452	−	0.743756i	$-0.733043\pi$
$349$	−24.9754	−1.33690	−0.668452	+	0.743756i	$0.733043\pi$
$350$	0	0
$351$	53.8408	2.87381
$352$	0	0
$353$	−22.2968	−1.18674	−0.593370	−	0.804930i	$-0.702203\pi$
$353$	−22.2968	−1.18674	−0.593370	+	0.804930i	$0.702203\pi$
$354$	0	0
$355$	−0.871682	−0.0462641
$356$	0	0
$357$	−20.2250	−1.07042
$358$	0	0
$359$	−26.4896	−1.39807	−0.699035	−	0.715087i	$-0.746387\pi$
$359$	−26.4896	−1.39807	−0.699035	+	0.715087i	$0.746387\pi$
$360$	0	0
$361$	−18.9863	−0.999278
$362$	0	0
$363$	13.4151	0.704110
$364$	0	0
$365$	−2.52340	−0.132081
$366$	0	0
$367$	10.2138	0.533157	0.266579	−	0.963813i	$-0.414107\pi$
$367$	10.2138	0.533157	0.266579	+	0.963813i	$0.414107\pi$
$368$	0	0
$369$	−43.3648	−2.25748
$370$	0	0
$371$	−2.53918	−0.131828
$372$	0	0
$373$	16.9320	0.876708	0.438354	−	0.898802i	$-0.355562\pi$
$373$	16.9320	0.876708	0.438354	+	0.898802i	$0.355562\pi$
$374$	0	0
$375$	−10.9178	−0.563792
$376$	0	0
$377$	−38.1251	−1.96354
$378$	0	0
$379$	1.34665	0.0691729	0.0345865	−	0.999402i	$-0.488989\pi$
$379$	1.34665	0.0691729	0.0345865	+	0.999402i	$0.488989\pi$
$380$	0	0
$381$	−55.6579	−2.85144
$382$	0	0
$383$	−14.0473	−0.717784	−0.358892	−	0.933379i	$-0.616845\pi$
$383$	−14.0473	−0.717784	−0.358892	+	0.933379i	$0.616845\pi$
$384$	0	0
$385$	−1.45494	−0.0741505
$386$	0	0
$387$	−0.218035	−0.0110833
$388$	0	0
$389$	−4.17440	−0.211651	−0.105825	−	0.994385i	$-0.533748\pi$
$389$	−4.17440	−0.211651	−0.105825	+	0.994385i	$0.533748\pi$
$390$	0	0
$391$	−58.5867	−2.96285
$392$	0	0
$393$	9.17987	0.463063
$394$	0	0
$395$	−2.35079	−0.118281
$396$	0	0
$397$	21.4578	1.07694	0.538469	−	0.842645i	$-0.319003\pi$
$397$	21.4578	1.07694	0.538469	+	0.842645i	$0.319003\pi$
$398$	0	0
$399$	−0.350562	−0.0175501
$400$	0	0
$401$	−0.747050	−0.0373059	−0.0186530	−	0.999826i	$-0.505938\pi$
$401$	−0.747050	−0.0373059	−0.0186530	+	0.999826i	$0.505938\pi$
$402$	0	0
$403$	4.90874	0.244522
$404$	0	0
$405$	3.19835	0.158927
$406$	0	0
$407$	40.0321	1.98432
$408$	0	0
$409$	7.69274	0.380382	0.190191	−	0.981747i	$-0.439089\pi$
$409$	7.69274	0.380382	0.190191	+	0.981747i	$0.439089\pi$
$410$	0	0
$411$	20.6621	1.01919
$412$	0	0
$413$	5.53975	0.272593
$414$	0	0
$415$	3.19868	0.157017
$416$	0	0
$417$	−56.8542	−2.78416
$418$	0	0
$419$	16.5196	0.807036	0.403518	−	0.914972i	$-0.367787\pi$
$419$	16.5196	0.807036	0.403518	+	0.914972i	$0.367787\pi$
$420$	0	0
$421$	17.5105	0.853411	0.426705	−	0.904391i	$-0.359674\pi$
$421$	17.5105	0.853411	0.426705	+	0.904391i	$0.359674\pi$
$422$	0	0
$423$	−24.2476	−1.17896
$424$	0	0
$425$	−32.8579	−1.59384
$426$	0	0
$427$	6.02663	0.291649
$428$	0	0
$429$	71.5478	3.45436
$430$	0	0
$431$	−27.3912	−1.31939	−0.659693	−	0.751535i	$-0.729314\pi$
$431$	−27.3912	−1.31939	−0.659693	+	0.751535i	$0.729314\pi$
$432$	0	0
$433$	−28.9749	−1.39245	−0.696223	−	0.717826i	$-0.745137\pi$
$433$	−28.9749	−1.39245	−0.696223	+	0.717826i	$0.745137\pi$
$434$	0	0
$435$	−6.94720	−0.333093
$436$	0	0
$437$	−1.01549	−0.0485776
$438$	0	0
$439$	34.1407	1.62945	0.814724	−	0.579848i	$-0.196888\pi$
$439$	34.1407	1.62945	0.814724	+	0.579848i	$0.196888\pi$
$440$	0	0
$441$	5.96088	0.283851
$442$	0	0
$443$	−18.9122	−0.898546	−0.449273	−	0.893394i	$-0.648317\pi$
$443$	−18.9122	−0.898546	−0.449273	+	0.893394i	$0.648317\pi$
$444$	0	0
$445$	−1.31804	−0.0624811
$446$	0	0
$447$	37.9351	1.79427
$448$	0	0
$449$	−28.8293	−1.36054	−0.680270	−	0.732962i	$-0.738137\pi$
$449$	−28.8293	−1.36054	−0.680270	+	0.732962i	$0.738137\pi$
$450$	0	0
$451$	−28.6242	−1.34786
$452$	0	0
$453$	29.5709	1.38936
$454$	0	0
$455$	−2.24623	−0.105305
$456$	0	0
$457$	−24.0906	−1.12691	−0.563456	−	0.826146i	$-0.690528\pi$
$457$	−24.0906	−1.12691	−0.563456	+	0.826146i	$0.690528\pi$
$458$	0	0
$459$	59.8836	2.79513
$460$	0	0
$461$	−21.2107	−0.987882	−0.493941	−	0.869496i	$-0.664444\pi$
$461$	−21.2107	−0.987882	−0.493941	+	0.869496i	$0.664444\pi$
$462$	0	0
$463$	15.5565	0.722973	0.361486	−	0.932377i	$-0.382269\pi$
$463$	15.5565	0.722973	0.361486	+	0.932377i	$0.382269\pi$
$464$	0	0
$465$	0.894475	0.0414803
$466$	0	0
$467$	2.50838	0.116074	0.0580371	−	0.998314i	$-0.481516\pi$
$467$	2.50838	0.116074	0.0580371	+	0.998314i	$0.481516\pi$
$468$	0	0
$469$	−4.88072	−0.225371
$470$	0	0
$471$	33.1909	1.52935
$472$	0	0
$473$	−0.143920	−0.00661746
$474$	0	0
$475$	−0.569532	−0.0261319
$476$	0	0
$477$	15.1358	0.693019
$478$	0	0
$479$	−25.1184	−1.14769	−0.573845	−	0.818964i	$-0.694549\pi$
$479$	−25.1184	−1.14769	−0.573845	+	0.818964i	$0.694549\pi$
$480$	0	0
$481$	61.8042	2.81803
$482$	0	0
$483$	25.9574	1.18110
$484$	0	0
$485$	−0.0781298	−0.00354769
$486$	0	0
$487$	−17.0804	−0.773987	−0.386993	−	0.922082i	$-0.626486\pi$
$487$	−17.0804	−0.773987	−0.386993	+	0.922082i	$0.626486\pi$
$488$	0	0
$489$	−31.9138	−1.44319
$490$	0	0
$491$	−12.8459	−0.579727	−0.289864	−	0.957068i	$-0.593610\pi$
$491$	−12.8459	−0.579727	−0.289864	+	0.957068i	$0.593610\pi$
$492$	0	0
$493$	−42.4041	−1.90979
$494$	0	0
$495$	8.67271	0.389809
$496$	0	0
$497$	2.35733	0.105740
$498$	0	0
$499$	−3.94086	−0.176417	−0.0882086	−	0.996102i	$-0.528114\pi$
$499$	−3.94086	−0.176417	−0.0882086	+	0.996102i	$0.528114\pi$
$500$	0	0
$501$	40.9625	1.83007
$502$	0	0
$503$	−38.0573	−1.69689	−0.848447	−	0.529281i	$-0.822462\pi$
$503$	−38.0573	−1.69689	−0.848447	+	0.529281i	$0.822462\pi$
$504$	0	0
$505$	−3.64195	−0.162065
$506$	0	0
$507$	71.5451	3.17743
$508$	0	0
$509$	−16.0223	−0.710176	−0.355088	−	0.934833i	$-0.615549\pi$
$509$	−16.0223	−0.710176	−0.355088	+	0.934833i	$0.615549\pi$
$510$	0	0
$511$	6.82414	0.301882
$512$	0	0
$513$	1.03797	0.0458276
$514$	0	0
$515$	4.31112	0.189971
$516$	0	0
$517$	−16.0053	−0.703914
$518$	0	0
$519$	0.725826	0.0318602
$520$	0	0
$521$	−17.2497	−0.755722	−0.377861	−	0.925862i	$-0.623340\pi$
$521$	−17.2497	−0.755722	−0.377861	+	0.925862i	$0.623340\pi$
$522$	0	0
$523$	36.6235	1.60143	0.800717	−	0.599043i	$-0.204452\pi$
$523$	36.6235	1.60143	0.800717	+	0.599043i	$0.204452\pi$
$524$	0	0
$525$	14.5581	0.635366
$526$	0	0
$527$	5.45967	0.237827
$528$	0	0
$529$	52.1922	2.26923
$530$	0	0
$531$	−33.0217	−1.43302
$532$	0	0
$533$	−44.1919	−1.91416
$534$	0	0
$535$	3.85563	0.166693
$536$	0	0
$537$	2.32332	0.100259
$538$	0	0
$539$	3.93465	0.169477
$540$	0	0
$541$	26.2577	1.12891	0.564454	−	0.825465i	$-0.309087\pi$
$541$	26.2577	1.12891	0.564454	+	0.825465i	$0.309087\pi$
$542$	0	0
$543$	−45.0860	−1.93483
$544$	0	0
$545$	4.88445	0.209227
$546$	0	0
$547$	3.75319	0.160475	0.0802374	−	0.996776i	$-0.474432\pi$
$547$	3.75319	0.160475	0.0802374	+	0.996776i	$0.474432\pi$
$548$	0	0
$549$	−35.9240	−1.53320
$550$	0	0
$551$	−0.734998	−0.0313120
$552$	0	0
$553$	6.35733	0.270341
$554$	0	0
$555$	11.2620	0.478047
$556$	0	0
$557$	4.15042	0.175859	0.0879294	−	0.996127i	$-0.471975\pi$
$557$	4.15042	0.175859	0.0879294	+	0.996127i	$0.471975\pi$
$558$	0	0
$559$	−0.222193	−0.00939778
$560$	0	0
$561$	79.5781	3.35979
$562$	0	0
$563$	3.13556	0.132148	0.0660740	−	0.997815i	$-0.478953\pi$
$563$	3.13556	0.132148	0.0660740	+	0.997815i	$0.478953\pi$
$564$	0	0
$565$	7.16477	0.301424
$566$	0	0
$567$	−8.64944	−0.363242
$568$	0	0
$569$	15.9020	0.666647	0.333323	−	0.942813i	$-0.391830\pi$
$569$	15.9020	0.666647	0.333323	+	0.942813i	$0.391830\pi$
$570$	0	0
$571$	8.50038	0.355730	0.177865	−	0.984055i	$-0.443081\pi$
$571$	8.50038	0.355730	0.177865	+	0.984055i	$0.443081\pi$
$572$	0	0
$573$	−46.1621	−1.92845
$574$	0	0
$575$	42.1711	1.75865
$576$	0	0
$577$	−32.4207	−1.34969	−0.674846	−	0.737959i	$-0.735790\pi$
$577$	−32.4207	−1.34969	−0.674846	+	0.737959i	$0.735790\pi$
$578$	0	0
$579$	29.1976	1.21341
$580$	0	0
$581$	−8.65033	−0.358876
$582$	0	0
$583$	9.99079	0.413776
$584$	0	0
$585$	13.3895	0.553587
$586$	0	0
$587$	18.3326	0.756666	0.378333	−	0.925670i	$-0.376497\pi$
$587$	18.3326	0.756666	0.378333	+	0.925670i	$0.376497\pi$
$588$	0	0
$589$	0.0946334	0.00389930
$590$	0	0
$591$	−63.7991	−2.62434
$592$	0	0
$593$	−24.3751	−1.00096	−0.500482	−	0.865747i	$-0.666844\pi$
$593$	−24.3751	−1.00096	−0.500482	+	0.865747i	$0.666844\pi$
$594$	0	0
$595$	−2.49834	−0.102422
$596$	0	0
$597$	−36.6072	−1.49823
$598$	0	0
$599$	41.8673	1.71065	0.855325	−	0.518091i	$-0.173357\pi$
$599$	41.8673	1.71065	0.855325	+	0.518091i	$0.173357\pi$
$600$	0	0
$601$	28.9937	1.18268	0.591340	−	0.806423i	$-0.298599\pi$
$601$	28.9937	1.18268	0.591340	+	0.806423i	$0.298599\pi$
$602$	0	0
$603$	29.0934	1.18478
$604$	0	0
$605$	1.65713	0.0673720
$606$	0	0
$607$	32.7589	1.32964	0.664821	−	0.747003i	$-0.268508\pi$
$607$	32.7589	1.32964	0.664821	+	0.747003i	$0.268508\pi$
$608$	0	0
$609$	18.7876	0.761312
$610$	0	0
$611$	−24.7101	−0.999663
$612$	0	0
$613$	−1.94718	−0.0786459	−0.0393230	−	0.999227i	$-0.512520\pi$
$613$	−1.94718	−0.0786459	−0.0393230	+	0.999227i	$0.512520\pi$
$614$	0	0
$615$	−8.05269	−0.324716
$616$	0	0
$617$	−1.14258	−0.0459985	−0.0229992	−	0.999735i	$-0.507322\pi$
$617$	−1.14258	−0.0459985	−0.0229992	+	0.999735i	$0.507322\pi$
$618$	0	0
$619$	34.4805	1.38589	0.692943	−	0.720992i	$-0.256314\pi$
$619$	34.4805	1.38589	0.692943	+	0.720992i	$0.256314\pi$
$620$	0	0
$621$	−76.8568	−3.08416
$622$	0	0
$623$	3.56443	0.142806
$624$	0	0
$625$	22.9677	0.918707
$626$	0	0
$627$	1.37934	0.0550855
$628$	0	0
$629$	68.7409	2.74088
$630$	0	0
$631$	−7.20195	−0.286705	−0.143352	−	0.989672i	$-0.545788\pi$
$631$	−7.20195	−0.286705	−0.143352	+	0.989672i	$0.545788\pi$
$632$	0	0
$633$	−70.8084	−2.81438
$634$	0	0
$635$	−6.87528	−0.272837
$636$	0	0
$637$	6.07457	0.240683
$638$	0	0
$639$	−14.0517	−0.555878
$640$	0	0
$641$	17.6807	0.698345	0.349173	−	0.937058i	$-0.386463\pi$
$641$	17.6807	0.698345	0.349173	+	0.937058i	$0.386463\pi$
$642$	0	0
$643$	−36.9615	−1.45762	−0.728810	−	0.684716i	$-0.759926\pi$
$643$	−36.9615	−1.45762	−0.728810	+	0.684716i	$0.759926\pi$
$644$	0	0
$645$	−0.0404883	−0.00159423
$646$	0	0
$647$	−18.6129	−0.731747	−0.365873	−	0.930665i	$-0.619230\pi$
$647$	−18.6129	−0.731747	−0.365873	+	0.930665i	$0.619230\pi$
$648$	0	0
$649$	−21.7969	−0.855605
$650$	0	0
$651$	−2.41897	−0.0948068
$652$	0	0
$653$	−28.0079	−1.09604	−0.548018	−	0.836467i	$-0.684617\pi$
$653$	−28.0079	−1.09604	−0.548018	+	0.836467i	$0.684617\pi$
$654$	0	0
$655$	1.13397	0.0443077
$656$	0	0
$657$	−40.6779	−1.58700
$658$	0	0
$659$	40.9629	1.59569	0.797843	−	0.602865i	$-0.205974\pi$
$659$	40.9629	1.59569	0.797843	+	0.602865i	$0.205974\pi$
$660$	0	0
$661$	−16.8098	−0.653825	−0.326912	−	0.945055i	$-0.606008\pi$
$661$	−16.8098	−0.653825	−0.326912	+	0.945055i	$0.606008\pi$
$662$	0	0
$663$	122.858	4.77140
$664$	0	0
$665$	−0.0433041	−0.00167926
$666$	0	0
$667$	54.4230	2.10727
$668$	0	0
$669$	30.6920	1.18662
$670$	0	0
$671$	−23.7127	−0.915417
$672$	0	0
$673$	−8.69460	−0.335152	−0.167576	−	0.985859i	$-0.553594\pi$
$673$	−8.69460	−0.335152	−0.167576	+	0.985859i	$0.553594\pi$
$674$	0	0
$675$	−43.1046	−1.65910
$676$	0	0
$677$	39.2305	1.50775	0.753876	−	0.657017i	$-0.228182\pi$
$677$	39.2305	1.50775	0.753876	+	0.657017i	$0.228182\pi$
$678$	0	0
$679$	0.211290	0.00810855
$680$	0	0
$681$	−35.7119	−1.36848
$682$	0	0
$683$	−41.6065	−1.59203	−0.796015	−	0.605277i	$-0.793062\pi$
$683$	−41.6065	−1.59203	−0.796015	+	0.605277i	$0.793062\pi$
$684$	0	0
$685$	2.55234	0.0975198
$686$	0	0
$687$	17.6595	0.673751
$688$	0	0
$689$	15.4244	0.587624
$690$	0	0
$691$	−15.0503	−0.572541	−0.286271	−	0.958149i	$-0.592416\pi$
$691$	−15.0503	−0.572541	−0.286271	+	0.958149i	$0.592416\pi$
$692$	0	0
$693$	−23.4540	−0.890942
$694$	0	0
$695$	−7.02306	−0.266400
$696$	0	0
$697$	−49.1518	−1.86176
$698$	0	0
$699$	25.7747	0.974890
$700$	0	0
$701$	32.4666	1.22625	0.613123	−	0.789987i	$-0.289913\pi$
$701$	32.4666	1.22625	0.613123	+	0.789987i	$0.289913\pi$
$702$	0	0
$703$	1.19150	0.0449381
$704$	0	0
$705$	−4.50270	−0.169581
$706$	0	0
$707$	9.84907	0.370412
$708$	0	0
$709$	−40.9483	−1.53785	−0.768923	−	0.639342i	$-0.779207\pi$
$709$	−40.9483	−1.53785	−0.768923	+	0.639342i	$0.779207\pi$
$710$	0	0
$711$	−37.8952	−1.42118
$712$	0	0
$713$	−7.00714	−0.262420
$714$	0	0
$715$	8.83812	0.330527
$716$	0	0
$717$	54.0276	2.01770
$718$	0	0
$719$	42.4750	1.58405	0.792025	−	0.610489i	$-0.209027\pi$
$719$	42.4750	1.58405	0.792025	+	0.610489i	$0.209027\pi$
$720$	0	0
$721$	−11.6587	−0.434194
$722$	0	0
$723$	47.7862	1.77719
$724$	0	0
$725$	30.5228	1.13359
$726$	0	0
$727$	2.84716	0.105595	0.0527977	−	0.998605i	$-0.483186\pi$
$727$	2.84716	0.105595	0.0527977	+	0.998605i	$0.483186\pi$
$728$	0	0
$729$	−28.0380	−1.03844
$730$	0	0
$731$	−0.247131	−0.00914049
$732$	0	0
$733$	16.3001	0.602058	0.301029	−	0.953615i	$-0.402670\pi$
$733$	16.3001	0.602058	0.301029	+	0.953615i	$0.402670\pi$
$734$	0	0
$735$	1.10691	0.0408291
$736$	0	0
$737$	19.2039	0.707386
$738$	0	0
$739$	−1.70803	−0.0628308	−0.0314154	−	0.999506i	$-0.510001\pi$
$739$	−1.70803	−0.0628308	−0.0314154	+	0.999506i	$0.510001\pi$
$740$	0	0
$741$	2.12951	0.0782297
$742$	0	0
$743$	−20.6504	−0.757591	−0.378796	−	0.925480i	$-0.623662\pi$
$743$	−20.6504	−0.757591	−0.378796	+	0.925480i	$0.623662\pi$
$744$	0	0
$745$	4.68602	0.171683
$746$	0	0
$747$	51.5635	1.88661
$748$	0	0
$749$	−10.4269	−0.380992
$750$	0	0
$751$	32.5679	1.18842	0.594210	−	0.804310i	$-0.297465\pi$
$751$	32.5679	1.18842	0.594210	+	0.804310i	$0.297465\pi$
$752$	0	0
$753$	−61.7380	−2.24986
$754$	0	0
$755$	3.65281	0.132940
$756$	0	0
$757$	−17.4453	−0.634062	−0.317031	−	0.948415i	$-0.602686\pi$
$757$	−17.4453	−0.634062	−0.317031	+	0.948415i	$0.602686\pi$
$758$	0	0
$759$	−102.133	−3.70721
$760$	0	0
$761$	−3.14810	−0.114119	−0.0570593	−	0.998371i	$-0.518172\pi$
$761$	−3.14810	−0.114119	−0.0570593	+	0.998371i	$0.518172\pi$
$762$	0	0
$763$	−13.2092	−0.478206
$764$	0	0
$765$	14.8923	0.538431
$766$	0	0
$767$	−33.6516	−1.21509
$768$	0	0
$769$	46.3568	1.67167	0.835834	−	0.548982i	$-0.184985\pi$
$769$	46.3568	1.67167	0.835834	+	0.548982i	$0.184985\pi$
$770$	0	0
$771$	−16.9693	−0.611134
$772$	0	0
$773$	−3.39055	−0.121950	−0.0609748	−	0.998139i	$-0.519421\pi$
$773$	−3.39055	−0.121950	−0.0609748	+	0.998139i	$0.519421\pi$
$774$	0	0
$775$	−3.92991	−0.141167
$776$	0	0
$777$	−30.4564	−1.09262
$778$	0	0
$779$	−0.851956	−0.0305245
$780$	0	0
$781$	−9.27524	−0.331894
$782$	0	0
$783$	−55.6278	−1.98798
$784$	0	0
$785$	4.09998	0.146335
$786$	0	0
$787$	−0.966714	−0.0344596	−0.0172298	−	0.999852i	$-0.505485\pi$
$787$	−0.966714	−0.0344596	−0.0172298	+	0.999852i	$0.505485\pi$
$788$	0	0
$789$	33.7992	1.20328
$790$	0	0
$791$	−19.3760	−0.688930
$792$	0	0
$793$	−36.6092	−1.30003
$794$	0	0
$795$	2.81066	0.0996837
$796$	0	0
$797$	13.8418	0.490301	0.245150	−	0.969485i	$-0.421163\pi$
$797$	13.8418	0.490301	0.245150	+	0.969485i	$0.421163\pi$
$798$	0	0
$799$	−27.4834	−0.972294
$800$	0	0
$801$	−21.2471	−0.750731
$802$	0	0
$803$	−26.8506	−0.947537
$804$	0	0
$805$	3.20645	0.113013
$806$	0	0
$807$	−0.725826	−0.0255503
$808$	0	0
$809$	0.856545	0.0301145	0.0150573	−	0.999887i	$-0.495207\pi$
$809$	0.856545	0.0301145	0.0150573	+	0.999887i	$0.495207\pi$
$810$	0	0
$811$	25.6209	0.899671	0.449836	−	0.893111i	$-0.351483\pi$
$811$	25.6209	0.899671	0.449836	+	0.893111i	$0.351483\pi$
$812$	0	0
$813$	−6.74427	−0.236532
$814$	0	0
$815$	−3.94223	−0.138090
$816$	0	0
$817$	−0.00428357	−0.000149863 0
$818$	0	0
$819$	−36.2098	−1.26527
$820$	0	0
$821$	−33.7705	−1.17860	−0.589299	−	0.807915i	$-0.700596\pi$
$821$	−33.7705	−1.17860	−0.589299	+	0.807915i	$0.700596\pi$
$822$	0	0
$823$	−8.12997	−0.283393	−0.141696	−	0.989910i	$-0.545256\pi$
$823$	−8.12997	−0.283393	−0.141696	+	0.989910i	$0.545256\pi$
$824$	0	0
$825$	−57.2808	−1.99426
$826$	0	0
$827$	−19.2110	−0.668033	−0.334016	−	0.942567i	$-0.608404\pi$
$827$	−19.2110	−0.668033	−0.334016	+	0.942567i	$0.608404\pi$
$828$	0	0
$829$	33.0789	1.14888	0.574439	−	0.818547i	$-0.305220\pi$
$829$	33.0789	1.14888	0.574439	+	0.818547i	$0.305220\pi$
$830$	0	0
$831$	52.4173	1.81834
$832$	0	0
$833$	6.75635	0.234094
$834$	0	0
$835$	5.06000	0.175108
$836$	0	0
$837$	7.16227	0.247564
$838$	0	0
$839$	7.13889	0.246462	0.123231	−	0.992378i	$-0.460674\pi$
$839$	7.13889	0.246462	0.123231	+	0.992378i	$0.460674\pi$
$840$	0	0
$841$	10.3906	0.358295
$842$	0	0
$843$	−98.1175	−3.37935
$844$	0	0
$845$	8.83778	0.304029
$846$	0	0
$847$	−4.48145	−0.153985
$848$	0	0
$849$	−73.9957	−2.53953
$850$	0	0
$851$	−88.2245	−3.02430
$852$	0	0
$853$	−3.34699	−0.114599	−0.0572993	−	0.998357i	$-0.518249\pi$
$853$	−3.34699	−0.114599	−0.0572993	+	0.998357i	$0.518249\pi$
$854$	0	0
$855$	0.258130	0.00882787
$856$	0	0
$857$	13.9947	0.478050	0.239025	−	0.971013i	$-0.423172\pi$
$857$	13.9947	0.478050	0.239025	+	0.971013i	$0.423172\pi$
$858$	0	0
$859$	−35.0781	−1.19685	−0.598424	−	0.801179i	$-0.704206\pi$
$859$	−35.0781	−1.19685	−0.598424	+	0.801179i	$0.704206\pi$
$860$	0	0
$861$	21.7772	0.742165
$862$	0	0
$863$	−4.18861	−0.142582	−0.0712910	−	0.997456i	$-0.522712\pi$
$863$	−4.18861	−0.142582	−0.0712910	+	0.997456i	$0.522712\pi$
$864$	0	0
$865$	0.0896594	0.00304851
$866$	0	0
$867$	85.7579	2.91249
$868$	0	0
$869$	−25.0138	−0.848536
$870$	0	0
$871$	29.6483	1.00459
$872$	0	0
$873$	−1.25947	−0.0426267
$874$	0	0
$875$	3.64720	0.123298
$876$	0	0
$877$	−0.663532	−0.0224059	−0.0112029	−	0.999937i	$-0.503566\pi$
$877$	−0.663532	−0.0224059	−0.0112029	+	0.999937i	$0.503566\pi$
$878$	0	0
$879$	47.1563	1.59054
$880$	0	0
$881$	−26.5838	−0.895631	−0.447816	−	0.894126i	$-0.647798\pi$
$881$	−26.5838	−0.895631	−0.447816	+	0.894126i	$0.647798\pi$
$882$	0	0
$883$	−38.1327	−1.28327	−0.641634	−	0.767011i	$-0.721743\pi$
$883$	−38.1327	−1.28327	−0.641634	+	0.767011i	$0.721743\pi$
$884$	0	0
$885$	−6.13202	−0.206126
$886$	0	0
$887$	−50.6497	−1.70065	−0.850326	−	0.526257i	$-0.823595\pi$
$887$	−50.6497	−1.70065	−0.850326	+	0.526257i	$0.823595\pi$
$888$	0	0
$889$	18.5931	0.623593
$890$	0	0
$891$	34.0325	1.14013
$892$	0	0
$893$	−0.476375	−0.0159413
$894$	0	0
$895$	0.286994	0.00959315
$896$	0	0
$897$	−157.680	−5.26479
$898$	0	0
$899$	−5.07167	−0.169150
$900$	0	0
$901$	17.1556	0.571536
$902$	0	0
$903$	0.109494	0.00364374
$904$	0	0
$905$	−5.56936	−0.185132
$906$	0	0
$907$	36.3370	1.20655	0.603276	−	0.797533i	$-0.293862\pi$
$907$	36.3370	1.20655	0.603276	+	0.797533i	$0.293862\pi$
$908$	0	0
$909$	−58.7091	−1.94726
$910$	0	0
$911$	−25.1125	−0.832015	−0.416008	−	0.909361i	$-0.636571\pi$
$911$	−25.1125	−0.832015	−0.416008	+	0.909361i	$0.636571\pi$
$912$	0	0
$913$	34.0360	1.12643
$914$	0	0
$915$	−6.67096	−0.220535
$916$	0	0
$917$	−3.06663	−0.101269
$918$	0	0
$919$	13.4558	0.443866	0.221933	−	0.975062i	$-0.428763\pi$
$919$	13.4558	0.443866	0.221933	+	0.975062i	$0.428763\pi$
$920$	0	0
$921$	−47.2558	−1.55713
$922$	0	0
$923$	−14.3197	−0.471340
$924$	0	0
$925$	−49.4801	−1.62690
$926$	0	0
$927$	69.4963	2.28256
$928$	0	0
$929$	−12.3637	−0.405641	−0.202820	−	0.979216i	$-0.565011\pi$
$929$	−12.3637	−0.405641	−0.202820	+	0.979216i	$0.565011\pi$
$930$	0	0
$931$	0.117109	0.00383809
$932$	0	0
$933$	42.5629	1.39345
$934$	0	0
$935$	9.83007	0.321478
$936$	0	0
$937$	41.9442	1.37026	0.685129	−	0.728422i	$-0.259746\pi$
$937$	41.9442	1.37026	0.685129	+	0.728422i	$0.259746\pi$
$938$	0	0
$939$	80.4627	2.62580
$940$	0	0
$941$	15.5943	0.508360	0.254180	−	0.967157i	$-0.418194\pi$
$941$	15.5943	0.508360	0.254180	+	0.967157i	$0.418194\pi$
$942$	0	0
$943$	63.0832	2.05427
$944$	0	0
$945$	−3.27744	−0.106615
$946$	0	0
$947$	13.8044	0.448583	0.224291	−	0.974522i	$-0.427993\pi$
$947$	13.8044	0.448583	0.224291	+	0.974522i	$0.427993\pi$
$948$	0	0
$949$	−41.4537	−1.34564
$950$	0	0
$951$	2.49080	0.0807696
$952$	0	0
$953$	−4.06472	−0.131669	−0.0658346	−	0.997831i	$-0.520971\pi$
$953$	−4.06472	−0.131669	−0.0658346	+	0.997831i	$0.520971\pi$
$954$	0	0
$955$	−5.70229	−0.184522
$956$	0	0
$957$	−73.9226	−2.38958
$958$	0	0
$959$	−6.90239	−0.222890
$960$	0	0
$961$	−30.3470	−0.978936
$962$	0	0
$963$	62.1536	2.00287
$964$	0	0
$965$	3.60671	0.116104
$966$	0	0
$967$	54.3821	1.74881	0.874406	−	0.485195i	$-0.161251\pi$
$967$	54.3821	1.74881	0.874406	+	0.485195i	$0.161251\pi$
$968$	0	0
$969$	2.36852	0.0760879
$970$	0	0
$971$	18.5975	0.596821	0.298410	−	0.954438i	$-0.403544\pi$
$971$	18.5975	0.596821	0.298410	+	0.954438i	$0.403544\pi$
$972$	0	0
$973$	18.9927	0.608879
$974$	0	0
$975$	−88.4339	−2.83215
$976$	0	0
$977$	−2.81062	−0.0899197	−0.0449598	−	0.998989i	$-0.514316\pi$
$977$	−2.81062	−0.0899197	−0.0449598	+	0.998989i	$0.514316\pi$
$978$	0	0
$979$	−14.0248	−0.448234
$980$	0	0
$981$	78.7385	2.51393
$982$	0	0
$983$	44.4447	1.41757	0.708783	−	0.705427i	$-0.249245\pi$
$983$	44.4447	1.41757	0.708783	+	0.705427i	$0.249245\pi$
$984$	0	0
$985$	−7.88093	−0.251107
$986$	0	0
$987$	12.1768	0.387593
$988$	0	0
$989$	0.317177	0.0100857
$990$	0	0
$991$	−43.6101	−1.38532	−0.692660	−	0.721264i	$-0.743561\pi$
$991$	−43.6101	−1.38532	−0.692660	+	0.721264i	$0.743561\pi$
$992$	0	0
$993$	−40.1617	−1.27449
$994$	0	0
$995$	−4.52200	−0.143357
$996$	0	0
$997$	8.23286	0.260737	0.130369	−	0.991466i	$-0.458384\pi$
$997$	8.23286	0.260737	0.130369	+	0.991466i	$0.458384\pi$
$998$	0	0
$999$	90.1776	2.85309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.a.o.1.9	yes	10
4.3	odd	2		3584.2.a.p.1.2	yes	10
8.3	odd	2		3584.2.a.p.1.9	yes	10
8.5	even	2	inner	3584.2.a.o.1.2	✓	10
16.3	odd	4		3584.2.b.g.1793.2		10
16.5	even	4		3584.2.b.h.1793.2		10
16.11	odd	4		3584.2.b.g.1793.9		10
16.13	even	4		3584.2.b.h.1793.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.o.1.2	✓	10	8.5	even	2	inner
3584.2.a.o.1.9	yes	10	1.1	even	1	trivial
3584.2.a.p.1.2	yes	10	4.3	odd	2
3584.2.a.p.1.9	yes	10	8.3	odd	2
3584.2.b.g.1793.2		10	16.3	odd	4
3584.2.b.g.1793.9		10	16.11	odd	4
3584.2.b.h.1793.2		10	16.5	even	4
3584.2.b.h.1793.9		10	16.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.a (trivial)

\(f(q)\)	\(=\)	\(q+2.99347 q^{3} +0.369776 q^{5} -1.00000 q^{7} +5.96088 q^{9} +O(q^{10})\)	\(q+2.99347 q^{3} +0.369776 q^{5} -1.00000 q^{7} +5.96088 q^{9} +3.93465 q^{11} +6.07457 q^{13} +1.10691 q^{15} +6.75635 q^{17} +0.117109 q^{19} -2.99347 q^{21} -8.67135 q^{23} -4.86327 q^{25} +8.86331 q^{27} -6.27619 q^{29} +0.808080 q^{31} +11.7783 q^{33} -0.369776 q^{35} +10.1743 q^{37} +18.1840 q^{39} -7.27490 q^{41} -0.0365776 q^{43} +2.20419 q^{45} -4.06779 q^{47} +1.00000 q^{49} +20.2250 q^{51} +2.53918 q^{53} +1.45494 q^{55} +0.350562 q^{57} -5.53975 q^{59} -6.02663 q^{61} -5.96088 q^{63} +2.24623 q^{65} +4.88072 q^{67} -25.9574 q^{69} -2.35733 q^{71} -6.82414 q^{73} -14.5581 q^{75} -3.93465 q^{77} -6.35733 q^{79} +8.64944 q^{81} +8.65033 q^{83} +2.49834 q^{85} -18.7876 q^{87} -3.56443 q^{89} -6.07457 q^{91} +2.41897 q^{93} +0.0433041 q^{95} -0.211290 q^{97} +23.4540 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{7} + 22 q^{9}+O(q^{10}) \) 10 * q - 10 * q^7 + 22 * q^9	\( 10 q - 10 q^{7} + 22 q^{9} - 16 q^{15} + 16 q^{17} - 8 q^{23} + 30 q^{25} + 8 q^{31} + 12 q^{33} + 20 q^{41} + 24 q^{47} + 10 q^{49} + 32 q^{55} + 28 q^{57} - 22 q^{63} + 32 q^{65} + 48 q^{73} - 40 q^{79} + 62 q^{81} + 8 q^{87} + 16 q^{89} + 32 q^{95} + 32 q^{97}+O(q^{100}) \) 10 * q - 10 * q^7 + 22 * q^9 - 16 * q^15 + 16 * q^17 - 8 * q^23 + 30 * q^25 + 8 * q^31 + 12 * q^33 + 20 * q^41 + 24 * q^47 + 10 * q^49 + 32 * q^55 + 28 * q^57 - 22 * q^63 + 32 * q^65 + 48 * q^73 - 40 * q^79 + 62 * q^81 + 8 * q^87 + 16 * q^89 + 32 * q^95 + 32 * q^97

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