LMFDB - Embedded newform 896.2.a.l.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [896,2,Mod(1,896)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("896.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(896, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2,0,2,0,-3,0,7,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	896.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.10278 q^{3} +2.52444 q^{5} -1.00000 q^{7} +6.62721 q^{9} -3.62721 q^{11} -4.72999 q^{13} +7.83276 q^{15} +4.20555 q^{17} +7.10278 q^{19} -3.10278 q^{21} -0.578337 q^{23} +1.37279 q^{25} +11.2544 q^{27} +8.20555 q^{29} -5.04888 q^{31} -11.2544 q^{33} -2.52444 q^{35} -3.04888 q^{37} -14.6761 q^{39} -0.205550 q^{41} -4.78389 q^{43} +16.7300 q^{45} -6.20555 q^{47} +1.00000 q^{49} +13.0489 q^{51} -2.00000 q^{53} -9.15667 q^{55} +22.0383 q^{57} -12.5628 q^{59} +10.5244 q^{61} -6.62721 q^{63} -11.9406 q^{65} -6.57834 q^{67} -1.79445 q^{69} -9.25443 q^{73} +4.25945 q^{75} +3.62721 q^{77} -5.15667 q^{79} +15.0383 q^{81} -5.94610 q^{83} +10.6167 q^{85} +25.4600 q^{87} -10.4111 q^{89} +4.72999 q^{91} -15.6655 q^{93} +17.9305 q^{95} -9.36222 q^{97} -24.0383 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{5} - 3 q^{7} + 7 q^{9} + 2 q^{11} + 6 q^{13} - 4 q^{15} - 2 q^{17} + 14 q^{19} - 2 q^{21} + 17 q^{25} + 8 q^{27} + 10 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} + 2 q^{37} - 20 q^{39}+ \cdots - 30 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^5 - 3 * q^7 + 7 * q^9 + 2 * q^11 + 6 * q^13 - 4 * q^15 - 2 * q^17 + 14 * q^19 - 2 * q^21 + 17 * q^25 + 8 * q^27 + 10 * q^29 - 4 * q^31 - 8 * q^33 - 2 * q^35 + 2 * q^37 - 20 * q^39 + 14 * q^41 + 2 * q^43 + 30 * q^45 - 4 * q^47 + 3 * q^49 + 28 * q^51 - 6 * q^53 - 24 * q^55 + 24 * q^57 + 10 * q^59 + 26 * q^61 - 7 * q^63 - 16 * q^65 - 18 * q^67 - 20 * q^69 - 2 * q^73 + 2 * q^75 - 2 * q^77 - 12 * q^79 + 3 * q^81 - 14 * q^83 - 12 * q^85 + 36 * q^87 - 2 * q^89 - 6 * q^91 + 8 * q^93 + 4 * q^95 - 10 * q^97 - 30 * 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
$3$	3.10278	1.79139	0.895694	−	0.444671i	$-0.146679\pi$
$3$	3.10278	1.79139	0.895694	+	0.444671i	$0.146679\pi$
$4$	0	0
$5$	2.52444	1.12896	0.564481	−	0.825446i	$-0.309076\pi$
$5$	2.52444	1.12896	0.564481	+	0.825446i	$0.309076\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.62721	2.20907
$10$	0	0
$11$	−3.62721	−1.09365	−0.546823	−	0.837248i	$-0.684163\pi$
$11$	−3.62721	−1.09365	−0.546823	+	0.837248i	$0.684163\pi$
$12$	0	0
$13$	−4.72999	−1.31186	−0.655931	−	0.754821i	$-0.727724\pi$
$13$	−4.72999	−1.31186	−0.655931	+	0.754821i	$0.727724\pi$
$14$	0	0
$15$	7.83276	2.02241
$16$	0	0
$17$	4.20555	1.02000	0.509998	−	0.860176i	$-0.329646\pi$
$17$	4.20555	1.02000	0.509998	+	0.860176i	$0.329646\pi$
$18$	0	0
$19$	7.10278	1.62949	0.814744	−	0.579821i	$-0.196877\pi$
$19$	7.10278	1.62949	0.814744	+	0.579821i	$0.196877\pi$
$20$	0	0
$21$	−3.10278	−0.677081
$22$	0	0
$23$	−0.578337	−0.120592	−0.0602958	−	0.998181i	$-0.519204\pi$
$23$	−0.578337	−0.120592	−0.0602958	+	0.998181i	$0.519204\pi$
$24$	0	0
$25$	1.37279	0.274557
$26$	0	0
$27$	11.2544	2.16592
$28$	0	0
$29$	8.20555	1.52373	0.761866	−	0.647734i	$-0.224283\pi$
$29$	8.20555	1.52373	0.761866	+	0.647734i	$0.224283\pi$
$30$	0	0
$31$	−5.04888	−0.906805	−0.453402	−	0.891306i	$-0.649790\pi$
$31$	−5.04888	−0.906805	−0.453402	+	0.891306i	$0.649790\pi$
$32$	0	0
$33$	−11.2544	−1.95914
$34$	0	0
$35$	−2.52444	−0.426708
$36$	0	0
$37$	−3.04888	−0.501232	−0.250616	−	0.968087i	$-0.580633\pi$
$37$	−3.04888	−0.501232	−0.250616	+	0.968087i	$0.580633\pi$
$38$	0	0
$39$	−14.6761	−2.35006
$40$	0	0
$41$	−0.205550	−0.0321015	−0.0160508	−	0.999871i	$-0.505109\pi$
$41$	−0.205550	−0.0321015	−0.0160508	+	0.999871i	$0.505109\pi$
$42$	0	0
$43$	−4.78389	−0.729536	−0.364768	−	0.931098i	$-0.618852\pi$
$43$	−4.78389	−0.729536	−0.364768	+	0.931098i	$0.618852\pi$
$44$	0	0
$45$	16.7300	2.49396
$46$	0	0
$47$	−6.20555	−0.905173	−0.452586	−	0.891721i	$-0.649499\pi$
$47$	−6.20555	−0.905173	−0.452586	+	0.891721i	$0.649499\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	13.0489	1.82721
$52$	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
$55$	−9.15667	−1.23469
$56$	0	0
$57$	22.0383	2.91905
$58$	0	0
$59$	−12.5628	−1.63553	−0.817765	−	0.575552i	$-0.804787\pi$
$59$	−12.5628	−1.63553	−0.817765	+	0.575552i	$0.804787\pi$
$60$	0	0
$61$	10.5244	1.34752	0.673758	−	0.738952i	$-0.264679\pi$
$61$	10.5244	1.34752	0.673758	+	0.738952i	$0.264679\pi$
$62$	0	0
$63$	−6.62721	−0.834950
$64$	0	0
$65$	−11.9406	−1.48104
$66$	0	0
$67$	−6.57834	−0.803672	−0.401836	−	0.915712i	$-0.631628\pi$
$67$	−6.57834	−0.803672	−0.401836	+	0.915712i	$0.631628\pi$
$68$	0	0
$69$	−1.79445	−0.216026
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−9.25443	−1.08315	−0.541574	−	0.840653i	$-0.682172\pi$
$73$	−9.25443	−1.08315	−0.541574	+	0.840653i	$0.682172\pi$
$74$	0	0
$75$	4.25945	0.491839
$76$	0	0
$77$	3.62721	0.413359
$78$	0	0
$79$	−5.15667	−0.580171	−0.290086	−	0.957001i	$-0.593684\pi$
$79$	−5.15667	−0.580171	−0.290086	+	0.957001i	$0.593684\pi$
$80$	0	0
$81$	15.0383	1.67092
$82$	0	0
$83$	−5.94610	−0.652669	−0.326335	−	0.945254i	$-0.605814\pi$
$83$	−5.94610	−0.652669	−0.326335	+	0.945254i	$0.605814\pi$
$84$	0	0
$85$	10.6167	1.15154
$86$	0	0
$87$	25.4600	2.72960
$88$	0	0
$89$	−10.4111	−1.10357	−0.551787	−	0.833985i	$-0.686054\pi$
$89$	−10.4111	−1.10357	−0.551787	+	0.833985i	$0.686054\pi$
$90$	0	0
$91$	4.72999	0.495837
$92$	0	0
$93$	−15.6655	−1.62444
$94$	0	0
$95$	17.9305	1.83963
$96$	0	0
$97$	−9.36222	−0.950590	−0.475295	−	0.879827i	$-0.657659\pi$
$97$	−9.36222	−0.950590	−0.475295	+	0.879827i	$0.657659\pi$
$98$	0	0
$99$	−24.0383	−2.41594
$100$	0	0
$101$	4.31889	0.429745	0.214873	−	0.976642i	$-0.431066\pi$
$101$	4.31889	0.429745	0.214873	+	0.976642i	$0.431066\pi$
$102$	0	0
$103$	−5.04888	−0.497481	−0.248740	−	0.968570i	$-0.580017\pi$
$103$	−5.04888	−0.497481	−0.248740	+	0.968570i	$0.580017\pi$
$104$	0	0
$105$	−7.83276	−0.764399
$106$	0	0
$107$	9.83276	0.950569	0.475285	−	0.879832i	$-0.342345\pi$
$107$	9.83276	0.950569	0.475285	+	0.879832i	$0.342345\pi$
$108$	0	0
$109$	19.4600	1.86393	0.931964	−	0.362551i	$-0.118094\pi$
$109$	19.4600	1.86393	0.931964	+	0.362551i	$0.118094\pi$
$110$	0	0
$111$	−9.45998	−0.897901
$112$	0	0
$113$	8.78389	0.826319	0.413159	−	0.910659i	$-0.364425\pi$
$113$	8.78389	0.826319	0.413159	+	0.910659i	$0.364425\pi$
$114$	0	0
$115$	−1.45998	−0.136143
$116$	0	0
$117$	−31.3466	−2.89800
$118$	0	0
$119$	−4.20555	−0.385522
$120$	0	0
$121$	2.15667	0.196061
$122$	0	0
$123$	−0.637776	−0.0575063
$124$	0	0
$125$	−9.15667	−0.818998
$126$	0	0
$127$	−14.6761	−1.30229	−0.651146	−	0.758952i	$-0.725712\pi$
$127$	−14.6761	−1.30229	−0.651146	+	0.758952i	$0.725712\pi$
$128$	0	0
$129$	−14.8433	−1.30688
$130$	0	0
$131$	4.15165	0.362731	0.181366	−	0.983416i	$-0.441948\pi$
$131$	4.15165	0.362731	0.181366	+	0.983416i	$0.441948\pi$
$132$	0	0
$133$	−7.10278	−0.615889
$134$	0	0
$135$	28.4111	2.44524
$136$	0	0
$137$	5.25443	0.448916	0.224458	−	0.974484i	$-0.427939\pi$
$137$	5.25443	0.448916	0.224458	+	0.974484i	$0.427939\pi$
$138$	0	0
$139$	8.89722	0.754653	0.377326	−	0.926080i	$-0.376843\pi$
$139$	8.89722	0.754653	0.377326	+	0.926080i	$0.376843\pi$
$140$	0	0
$141$	−19.2544	−1.62152
$142$	0	0
$143$	17.1567	1.43471
$144$	0	0
$145$	20.7144	1.72024
$146$	0	0
$147$	3.10278	0.255913
$148$	0	0
$149$	21.6655	1.77491	0.887455	−	0.460895i	$-0.152472\pi$
$149$	21.6655	1.77491	0.887455	+	0.460895i	$0.152472\pi$
$150$	0	0
$151$	4.98944	0.406035	0.203017	−	0.979175i	$-0.434925\pi$
$151$	4.98944	0.406035	0.203017	+	0.979175i	$0.434925\pi$
$152$	0	0
$153$	27.8711	2.25324
$154$	0	0
$155$	−12.7456	−1.02375
$156$	0	0
$157$	20.0922	1.60353	0.801767	−	0.597637i	$-0.203894\pi$
$157$	20.0922	1.60353	0.801767	+	0.597637i	$0.203894\pi$
$158$	0	0
$159$	−6.20555	−0.492132
$160$	0	0
$161$	0.578337	0.0455793
$162$	0	0
$163$	16.0383	1.25622	0.628109	−	0.778126i	$-0.283829\pi$
$163$	16.0383	1.25622	0.628109	+	0.778126i	$0.283829\pi$
$164$	0	0
$165$	−28.4111	−2.21180
$166$	0	0
$167$	−1.79445	−0.138859	−0.0694294	−	0.997587i	$-0.522118\pi$
$167$	−1.79445	−0.138859	−0.0694294	+	0.997587i	$0.522118\pi$
$168$	0	0
$169$	9.37279	0.720984
$170$	0	0
$171$	47.0716	3.59966
$172$	0	0
$173$	−22.8277	−1.73556	−0.867780	−	0.496948i	$-0.834454\pi$
$173$	−22.8277	−1.73556	−0.867780	+	0.496948i	$0.834454\pi$
$174$	0	0
$175$	−1.37279	−0.103773
$176$	0	0
$177$	−38.9794	−2.92987
$178$	0	0
$179$	3.51941	0.263053	0.131527	−	0.991313i	$-0.458012\pi$
$179$	3.51941	0.263053	0.131527	+	0.991313i	$0.458012\pi$
$180$	0	0
$181$	−10.9355	−0.812832	−0.406416	−	0.913688i	$-0.633222\pi$
$181$	−10.9355	−0.812832	−0.406416	+	0.913688i	$0.633222\pi$
$182$	0	0
$183$	32.6550	2.41392
$184$	0	0
$185$	−7.69670	−0.565872
$186$	0	0
$187$	−15.2544	−1.11551
$188$	0	0
$189$	−11.2544	−0.818639
$190$	0	0
$191$	0.745574	0.0539478	0.0269739	−	0.999636i	$-0.491413\pi$
$191$	0.745574	0.0539478	0.0269739	+	0.999636i	$0.491413\pi$
$192$	0	0
$193$	−14.1361	−1.01754	−0.508768	−	0.860904i	$-0.669899\pi$
$193$	−14.1361	−1.01754	−0.508768	+	0.860904i	$0.669899\pi$
$194$	0	0
$195$	−37.0489	−2.65313
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	0.637776	0.0452107	0.0226054	−	0.999744i	$-0.492804\pi$
$199$	0.637776	0.0452107	0.0226054	+	0.999744i	$0.492804\pi$
$200$	0	0
$201$	−20.4111	−1.43969
$202$	0	0
$203$	−8.20555	−0.575917
$204$	0	0
$205$	−0.518898	−0.0362414
$206$	0	0
$207$	−3.83276	−0.266395
$208$	0	0
$209$	−25.7633	−1.78208
$210$	0	0
$211$	−0.676089	−0.0465439	−0.0232719	−	0.999729i	$-0.507408\pi$
$211$	−0.676089	−0.0465439	−0.0232719	+	0.999729i	$0.507408\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−12.0766	−0.823619
$216$	0	0
$217$	5.04888	0.342740
$218$	0	0
$219$	−28.7144	−1.94034
$220$	0	0
$221$	−19.8922	−1.33809
$222$	0	0
$223$	23.6655	1.58476	0.792380	−	0.610027i	$-0.208842\pi$
$223$	23.6655	1.58476	0.792380	+	0.610027i	$0.208842\pi$
$224$	0	0
$225$	9.09775	0.606517
$226$	0	0
$227$	−8.89722	−0.590530	−0.295265	−	0.955415i	$-0.595408\pi$
$227$	−8.89722	−0.590530	−0.295265	+	0.955415i	$0.595408\pi$
$228$	0	0
$229$	7.68111	0.507582	0.253791	−	0.967259i	$-0.418322\pi$
$229$	7.68111	0.507582	0.253791	+	0.967259i	$0.418322\pi$
$230$	0	0
$231$	11.2544	0.740487
$232$	0	0
$233$	5.66553	0.371161	0.185580	−	0.982629i	$-0.440583\pi$
$233$	5.66553	0.371161	0.185580	+	0.982629i	$0.440583\pi$
$234$	0	0
$235$	−15.6655	−1.02191
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−11.4217	−0.738806	−0.369403	−	0.929269i	$-0.620438\pi$
$239$	−11.4217	−0.738806	−0.369403	+	0.929269i	$0.620438\pi$
$240$	0	0
$241$	17.5577	1.13099	0.565496	−	0.824751i	$-0.308685\pi$
$241$	17.5577	1.13099	0.565496	+	0.824751i	$0.308685\pi$
$242$	0	0
$243$	12.8972	0.827357
$244$	0	0
$245$	2.52444	0.161280
$246$	0	0
$247$	−33.5960	−2.13766
$248$	0	0
$249$	−18.4494	−1.16918
$250$	0	0
$251$	−12.1517	−0.767005	−0.383503	−	0.923540i	$-0.625282\pi$
$251$	−12.1517	−0.767005	−0.383503	+	0.923540i	$0.625282\pi$
$252$	0	0
$253$	2.09775	0.131885
$254$	0	0
$255$	32.9411	2.06285
$256$	0	0
$257$	6.41110	0.399913	0.199957	−	0.979805i	$-0.435920\pi$
$257$	6.41110	0.399913	0.199957	+	0.979805i	$0.435920\pi$
$258$	0	0
$259$	3.04888	0.189448
$260$	0	0
$261$	54.3799	3.36603
$262$	0	0
$263$	−10.3133	−0.635948	−0.317974	−	0.948099i	$-0.603003\pi$
$263$	−10.3133	−0.635948	−0.317974	+	0.948099i	$0.603003\pi$
$264$	0	0
$265$	−5.04888	−0.310150
$266$	0	0
$267$	−32.3033	−1.97693
$268$	0	0
$269$	15.6811	0.956094	0.478047	−	0.878334i	$-0.341345\pi$
$269$	15.6811	0.956094	0.478047	+	0.878334i	$0.341345\pi$
$270$	0	0
$271$	25.7633	1.56501	0.782504	−	0.622646i	$-0.213942\pi$
$271$	25.7633	1.56501	0.782504	+	0.622646i	$0.213942\pi$
$272$	0	0
$273$	14.6761	0.888237
$274$	0	0
$275$	−4.97939	−0.300269
$276$	0	0
$277$	17.2544	1.03672	0.518359	−	0.855163i	$-0.326543\pi$
$277$	17.2544	1.03672	0.518359	+	0.855163i	$0.326543\pi$
$278$	0	0
$279$	−33.4600	−2.00320
$280$	0	0
$281$	−32.5089	−1.93932	−0.969658	−	0.244466i	$-0.921387\pi$
$281$	−32.5089	−1.93932	−0.969658	+	0.244466i	$0.921387\pi$
$282$	0	0
$283$	22.9950	1.36691	0.683455	−	0.729993i	$-0.260477\pi$
$283$	22.9950	1.36691	0.683455	+	0.729993i	$0.260477\pi$
$284$	0	0
$285$	55.6344	3.29549
$286$	0	0
$287$	0.205550	0.0121332
$288$	0	0
$289$	0.686652	0.0403913
$290$	0	0
$291$	−29.0489	−1.70288
$292$	0	0
$293$	8.72999	0.510011	0.255006	−	0.966940i	$-0.417923\pi$
$293$	8.72999	0.510011	0.255006	+	0.966940i	$0.417923\pi$
$294$	0	0
$295$	−31.7139	−1.84645
$296$	0	0
$297$	−40.8222	−2.36874
$298$	0	0
$299$	2.73553	0.158200
$300$	0	0
$301$	4.78389	0.275739
$302$	0	0
$303$	13.4005	0.769841
$304$	0	0
$305$	26.5683	1.52130
$306$	0	0
$307$	26.6605	1.52160	0.760798	−	0.648989i	$-0.224808\pi$
$307$	26.6605	1.52160	0.760798	+	0.648989i	$0.224808\pi$
$308$	0	0
$309$	−15.6655	−0.891181
$310$	0	0
$311$	−10.0978	−0.572591	−0.286295	−	0.958141i	$-0.592424\pi$
$311$	−10.0978	−0.572591	−0.286295	+	0.958141i	$0.592424\pi$
$312$	0	0
$313$	−0.205550	−0.0116184	−0.00580919	−	0.999983i	$-0.501849\pi$
$313$	−0.205550	−0.0116184	−0.00580919	+	0.999983i	$0.501849\pi$
$314$	0	0
$315$	−16.7300	−0.942628
$316$	0	0
$317$	−30.4111	−1.70806	−0.854029	−	0.520226i	$-0.825848\pi$
$317$	−30.4111	−1.70806	−0.854029	+	0.520226i	$0.825848\pi$
$318$	0	0
$319$	−29.7633	−1.66642
$320$	0	0
$321$	30.5089	1.70284
$322$	0	0
$323$	29.8711	1.66207
$324$	0	0
$325$	−6.49327	−0.360182
$326$	0	0
$327$	60.3799	3.33902
$328$	0	0
$329$	6.20555	0.342123
$330$	0	0
$331$	−2.47054	−0.135793	−0.0678965	−	0.997692i	$-0.521629\pi$
$331$	−2.47054	−0.135793	−0.0678965	+	0.997692i	$0.521629\pi$
$332$	0	0
$333$	−20.2056	−1.10726
$334$	0	0
$335$	−16.6066	−0.907316
$336$	0	0
$337$	−11.2927	−0.615155	−0.307577	−	0.951523i	$-0.599518\pi$
$337$	−11.2927	−0.615155	−0.307577	+	0.951523i	$0.599518\pi$
$338$	0	0
$339$	27.2544	1.48026
$340$	0	0
$341$	18.3133	0.991723
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−4.52998	−0.243886
$346$	0	0
$347$	−2.68614	−0.144199	−0.0720997	−	0.997397i	$-0.522970\pi$
$347$	−2.68614	−0.144199	−0.0720997	+	0.997397i	$0.522970\pi$
$348$	0	0
$349$	17.4756	0.935445	0.467723	−	0.883875i	$-0.345075\pi$
$349$	17.4756	0.935445	0.467723	+	0.883875i	$0.345075\pi$
$350$	0	0
$351$	−53.2333	−2.84138
$352$	0	0
$353$	1.05892	0.0563608	0.0281804	−	0.999603i	$-0.491029\pi$
$353$	1.05892	0.0563608	0.0281804	+	0.999603i	$0.491029\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−13.0489	−0.690620
$358$	0	0
$359$	11.8328	0.624509	0.312255	−	0.949998i	$-0.398916\pi$
$359$	11.8328	0.624509	0.312255	+	0.949998i	$0.398916\pi$
$360$	0	0
$361$	31.4494	1.65523
$362$	0	0
$363$	6.69167	0.351222
$364$	0	0
$365$	−23.3622	−1.22283
$366$	0	0
$367$	19.2544	1.00507	0.502536	−	0.864556i	$-0.332400\pi$
$367$	19.2544	1.00507	0.502536	+	0.864556i	$0.332400\pi$
$368$	0	0
$369$	−1.36222	−0.0709146
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−9.66553	−0.500462	−0.250231	−	0.968186i	$-0.580507\pi$
$373$	−9.66553	−0.500462	−0.250231	+	0.968186i	$0.580507\pi$
$374$	0	0
$375$	−28.4111	−1.46714
$376$	0	0
$377$	−38.8122	−1.99893
$378$	0	0
$379$	12.0383	0.618367	0.309183	−	0.951002i	$-0.399944\pi$
$379$	12.0383	0.618367	0.309183	+	0.951002i	$0.399944\pi$
$380$	0	0
$381$	−45.5366	−2.33291
$382$	0	0
$383$	31.0278	1.58544	0.792722	−	0.609583i	$-0.208663\pi$
$383$	31.0278	1.58544	0.792722	+	0.609583i	$0.208663\pi$
$384$	0	0
$385$	9.15667	0.466667
$386$	0	0
$387$	−31.7038	−1.61160
$388$	0	0
$389$	−13.1466	−0.666560	−0.333280	−	0.942828i	$-0.608156\pi$
$389$	−13.1466	−0.666560	−0.333280	+	0.942828i	$0.608156\pi$
$390$	0	0
$391$	−2.43223	−0.123003
$392$	0	0
$393$	12.8816	0.649793
$394$	0	0
$395$	−13.0177	−0.654992
$396$	0	0
$397$	1.47556	0.0740563	0.0370282	−	0.999314i	$-0.488211\pi$
$397$	1.47556	0.0740563	0.0370282	+	0.999314i	$0.488211\pi$
$398$	0	0
$399$	−22.0383	−1.10330
$400$	0	0
$401$	−12.9794	−0.648160	−0.324080	−	0.946030i	$-0.605055\pi$
$401$	−12.9794	−0.648160	−0.324080	+	0.946030i	$0.605055\pi$
$402$	0	0
$403$	23.8811	1.18960
$404$	0	0
$405$	37.9633	1.88641
$406$	0	0
$407$	11.0589	0.548170
$408$	0	0
$409$	−1.36222	−0.0673577	−0.0336788	−	0.999433i	$-0.510722\pi$
$409$	−1.36222	−0.0673577	−0.0336788	+	0.999433i	$0.510722\pi$
$410$	0	0
$411$	16.3033	0.804183
$412$	0	0
$413$	12.5628	0.618173
$414$	0	0
$415$	−15.0106	−0.736840
$416$	0	0
$417$	27.6061	1.35188
$418$	0	0
$419$	−25.3083	−1.23639	−0.618196	−	0.786024i	$-0.712136\pi$
$419$	−25.3083	−1.23639	−0.618196	+	0.786024i	$0.712136\pi$
$420$	0	0
$421$	−1.05892	−0.0516087	−0.0258044	−	0.999667i	$-0.508215\pi$
$421$	−1.05892	−0.0516087	−0.0258044	+	0.999667i	$0.508215\pi$
$422$	0	0
$423$	−41.1255	−1.99959
$424$	0	0
$425$	5.77332	0.280047
$426$	0	0
$427$	−10.5244	−0.509313
$428$	0	0
$429$	53.2333	2.57013
$430$	0	0
$431$	12.2439	0.589766	0.294883	−	0.955533i	$-0.404719\pi$
$431$	12.2439	0.589766	0.294883	+	0.955533i	$0.404719\pi$
$432$	0	0
$433$	37.0278	1.77944	0.889720	−	0.456507i	$-0.150899\pi$
$433$	37.0278	1.77944	0.889720	+	0.456507i	$0.150899\pi$
$434$	0	0
$435$	64.2721	3.08161
$436$	0	0
$437$	−4.10780	−0.196503
$438$	0	0
$439$	−8.33447	−0.397783	−0.198891	−	0.980022i	$-0.563734\pi$
$439$	−8.33447	−0.397783	−0.198891	+	0.980022i	$0.563734\pi$
$440$	0	0
$441$	6.62721	0.315582
$442$	0	0
$443$	27.5960	1.31113	0.655564	−	0.755140i	$-0.272431\pi$
$443$	27.5960	1.31113	0.655564	+	0.755140i	$0.272431\pi$
$444$	0	0
$445$	−26.2822	−1.24589
$446$	0	0
$447$	67.2233	3.17955
$448$	0	0
$449$	−1.66553	−0.0786010	−0.0393005	−	0.999227i	$-0.512513\pi$
$449$	−1.66553	−0.0786010	−0.0393005	+	0.999227i	$0.512513\pi$
$450$	0	0
$451$	0.745574	0.0351077
$452$	0	0
$453$	15.4811	0.727366
$454$	0	0
$455$	11.9406	0.559782
$456$	0	0
$457$	25.7250	1.20336	0.601682	−	0.798736i	$-0.294498\pi$
$457$	25.7250	1.20336	0.601682	+	0.798736i	$0.294498\pi$
$458$	0	0
$459$	47.3311	2.20922
$460$	0	0
$461$	−12.4267	−0.578768	−0.289384	−	0.957213i	$-0.593451\pi$
$461$	−12.4267	−0.578768	−0.289384	+	0.957213i	$0.593451\pi$
$462$	0	0
$463$	−4.33447	−0.201440	−0.100720	−	0.994915i	$-0.532115\pi$
$463$	−4.33447	−0.201440	−0.100720	+	0.994915i	$0.532115\pi$
$464$	0	0
$465$	−39.5466	−1.83393
$466$	0	0
$467$	10.6917	0.494752	0.247376	−	0.968920i	$-0.420432\pi$
$467$	10.6917	0.494752	0.247376	+	0.968920i	$0.420432\pi$
$468$	0	0
$469$	6.57834	0.303759
$470$	0	0
$471$	62.3416	2.87255
$472$	0	0
$473$	17.3522	0.797854
$474$	0	0
$475$	9.75060	0.447388
$476$	0	0
$477$	−13.2544	−0.606878
$478$	0	0
$479$	−2.95112	−0.134840	−0.0674202	−	0.997725i	$-0.521477\pi$
$479$	−2.95112	−0.134840	−0.0674202	+	0.997725i	$0.521477\pi$
$480$	0	0
$481$	14.4211	0.657548
$482$	0	0
$483$	1.79445	0.0816503
$484$	0	0
$485$	−23.6344	−1.07318
$486$	0	0
$487$	5.93051	0.268737	0.134369	−	0.990931i	$-0.457099\pi$
$487$	5.93051	0.268737	0.134369	+	0.990931i	$0.457099\pi$
$488$	0	0
$489$	49.7633	2.25037
$490$	0	0
$491$	21.3028	0.961381	0.480691	−	0.876890i	$-0.340386\pi$
$491$	21.3028	0.961381	0.480691	+	0.876890i	$0.340386\pi$
$492$	0	0
$493$	34.5089	1.55420
$494$	0	0
$495$	−60.6832	−2.72751
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.6761	1.64185	0.820924	−	0.571038i	$-0.193459\pi$
$499$	36.6761	1.64185	0.820924	+	0.571038i	$0.193459\pi$
$500$	0	0
$501$	−5.56777	−0.248750
$502$	0	0
$503$	−20.7456	−0.924999	−0.462500	−	0.886619i	$-0.653047\pi$
$503$	−20.7456	−0.924999	−0.462500	+	0.886619i	$0.653047\pi$
$504$	0	0
$505$	10.9028	0.485167
$506$	0	0
$507$	29.0816	1.29156
$508$	0	0
$509$	−21.0333	−0.932284	−0.466142	−	0.884710i	$-0.654356\pi$
$509$	−21.0333	−0.932284	−0.466142	+	0.884710i	$0.654356\pi$
$510$	0	0
$511$	9.25443	0.409392
$512$	0	0
$513$	79.9377	3.52933
$514$	0	0
$515$	−12.7456	−0.561637
$516$	0	0
$517$	22.5089	0.989938
$518$	0	0
$519$	−70.8293	−3.10906
$520$	0	0
$521$	−12.7355	−0.557954	−0.278977	−	0.960298i	$-0.589995\pi$
$521$	−12.7355	−0.557954	−0.278977	+	0.960298i	$0.589995\pi$
$522$	0	0
$523$	−0.151651	−0.00663123	−0.00331562	−	0.999995i	$-0.501055\pi$
$523$	−0.151651	−0.00663123	−0.00331562	+	0.999995i	$0.501055\pi$
$524$	0	0
$525$	−4.25945	−0.185898
$526$	0	0
$527$	−21.2333	−0.924937
$528$	0	0
$529$	−22.6655	−0.985458
$530$	0	0
$531$	−83.2560	−3.61300
$532$	0	0
$533$	0.972250	0.0421128
$534$	0	0
$535$	24.8222	1.07316
$536$	0	0
$537$	10.9200	0.471231
$538$	0	0
$539$	−3.62721	−0.156235
$540$	0	0
$541$	11.5678	0.497337	0.248669	−	0.968589i	$-0.420007\pi$
$541$	11.5678	0.497337	0.248669	+	0.968589i	$0.420007\pi$
$542$	0	0
$543$	−33.9305	−1.45610
$544$	0	0
$545$	49.1255	2.10431
$546$	0	0
$547$	−10.0594	−0.430111	−0.215055	−	0.976602i	$-0.568993\pi$
$547$	−10.0594	−0.430111	−0.215055	+	0.976602i	$0.568993\pi$
$548$	0	0
$549$	69.7477	2.97676
$550$	0	0
$551$	58.2822	2.48290
$552$	0	0
$553$	5.15667	0.219284
$554$	0	0
$555$	−23.8811	−1.01370
$556$	0	0
$557$	4.50885	0.191046	0.0955231	−	0.995427i	$-0.469548\pi$
$557$	4.50885	0.191046	0.0955231	+	0.995427i	$0.469548\pi$
$558$	0	0
$559$	22.6277	0.957051
$560$	0	0
$561$	−47.3311	−1.99832
$562$	0	0
$563$	24.9739	1.05252	0.526261	−	0.850323i	$-0.323593\pi$
$563$	24.9739	1.05252	0.526261	+	0.850323i	$0.323593\pi$
$564$	0	0
$565$	22.1744	0.932883
$566$	0	0
$567$	−15.0383	−0.631550
$568$	0	0
$569$	−21.9406	−0.919796	−0.459898	−	0.887972i	$-0.652114\pi$
$569$	−21.9406	−0.919796	−0.459898	+	0.887972i	$0.652114\pi$
$570$	0	0
$571$	19.2161	0.804169	0.402085	−	0.915602i	$-0.368286\pi$
$571$	19.2161	0.804169	0.402085	+	0.915602i	$0.368286\pi$
$572$	0	0
$573$	2.31335	0.0966415
$574$	0	0
$575$	−0.793934	−0.0331093
$576$	0	0
$577$	−0.432226	−0.0179938	−0.00899689	−	0.999960i	$-0.502864\pi$
$577$	−0.432226	−0.0179938	−0.00899689	+	0.999960i	$0.502864\pi$
$578$	0	0
$579$	−43.8610	−1.82280
$580$	0	0
$581$	5.94610	0.246686
$582$	0	0
$583$	7.25443	0.300448
$584$	0	0
$585$	−79.1326	−3.27173
$586$	0	0
$587$	13.3083	0.549293	0.274647	−	0.961545i	$-0.411439\pi$
$587$	13.3083	0.549293	0.274647	+	0.961545i	$0.411439\pi$
$588$	0	0
$589$	−35.8610	−1.47763
$590$	0	0
$591$	−31.0278	−1.27631
$592$	0	0
$593$	7.68665	0.315653	0.157826	−	0.987467i	$-0.449551\pi$
$593$	7.68665	0.315653	0.157826	+	0.987467i	$0.449551\pi$
$594$	0	0
$595$	−10.6167	−0.435240
$596$	0	0
$597$	1.97887	0.0809899
$598$	0	0
$599$	37.0177	1.51250	0.756251	−	0.654281i	$-0.227029\pi$
$599$	37.0177	1.51250	0.756251	+	0.654281i	$0.227029\pi$
$600$	0	0
$601$	9.78440	0.399114	0.199557	−	0.979886i	$-0.436050\pi$
$601$	9.78440	0.399114	0.199557	+	0.979886i	$0.436050\pi$
$602$	0	0
$603$	−43.5960	−1.77537
$604$	0	0
$605$	5.44439	0.221346
$606$	0	0
$607$	−40.6066	−1.64817	−0.824086	−	0.566465i	$-0.808311\pi$
$607$	−40.6066	−1.64817	−0.824086	+	0.566465i	$0.808311\pi$
$608$	0	0
$609$	−25.4600	−1.03169
$610$	0	0
$611$	29.3522	1.18746
$612$	0	0
$613$	24.8122	1.00215	0.501077	−	0.865403i	$-0.332937\pi$
$613$	24.8122	1.00215	0.501077	+	0.865403i	$0.332937\pi$
$614$	0	0
$615$	−1.61003	−0.0649225
$616$	0	0
$617$	−40.4494	−1.62843	−0.814216	−	0.580562i	$-0.802833\pi$
$617$	−40.4494	−1.62843	−0.814216	+	0.580562i	$0.802833\pi$
$618$	0	0
$619$	−5.20053	−0.209027	−0.104513	−	0.994523i	$-0.533329\pi$
$619$	−5.20053	−0.209027	−0.104513	+	0.994523i	$0.533329\pi$
$620$	0	0
$621$	−6.50885	−0.261191
$622$	0	0
$623$	10.4111	0.417112
$624$	0	0
$625$	−29.9794	−1.19918
$626$	0	0
$627$	−79.9377	−3.19240
$628$	0	0
$629$	−12.8222	−0.511255
$630$	0	0
$631$	43.7422	1.74135	0.870674	−	0.491861i	$-0.163683\pi$
$631$	43.7422	1.74135	0.870674	+	0.491861i	$0.163683\pi$
$632$	0	0
$633$	−2.09775	−0.0833781
$634$	0	0
$635$	−37.0489	−1.47024
$636$	0	0
$637$	−4.72999	−0.187409
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.21611	0.285019	0.142510	−	0.989793i	$-0.454483\pi$
$641$	7.21611	0.285019	0.142510	+	0.989793i	$0.454483\pi$
$642$	0	0
$643$	20.3472	0.802413	0.401207	−	0.915988i	$-0.368591\pi$
$643$	20.3472	0.802413	0.401207	+	0.915988i	$0.368591\pi$
$644$	0	0
$645$	−37.4711	−1.47542
$646$	0	0
$647$	−11.5577	−0.454381	−0.227191	−	0.973850i	$-0.572954\pi$
$647$	−11.5577	−0.454381	−0.227191	+	0.973850i	$0.572954\pi$
$648$	0	0
$649$	45.5678	1.78869
$650$	0	0
$651$	15.6655	0.613980
$652$	0	0
$653$	−14.3033	−0.559731	−0.279866	−	0.960039i	$-0.590290\pi$
$653$	−14.3033	−0.559731	−0.279866	+	0.960039i	$0.590290\pi$
$654$	0	0
$655$	10.4806	0.409510
$656$	0	0
$657$	−61.3311	−2.39275
$658$	0	0
$659$	−19.7038	−0.767553	−0.383776	−	0.923426i	$-0.625377\pi$
$659$	−19.7038	−0.767553	−0.383776	+	0.923426i	$0.625377\pi$
$660$	0	0
$661$	−29.7789	−1.15826	−0.579132	−	0.815234i	$-0.696608\pi$
$661$	−29.7789	−1.15826	−0.579132	+	0.815234i	$0.696608\pi$
$662$	0	0
$663$	−61.7210	−2.39705
$664$	0	0
$665$	−17.9305	−0.695316
$666$	0	0
$667$	−4.74557	−0.183749
$668$	0	0
$669$	73.4288	2.83892
$670$	0	0
$671$	−38.1744	−1.47371
$672$	0	0
$673$	32.9200	1.26897	0.634485	−	0.772935i	$-0.281212\pi$
$673$	32.9200	1.26897	0.634485	+	0.772935i	$0.281212\pi$
$674$	0	0
$675$	15.4499	0.594668
$676$	0	0
$677$	22.7089	0.872772	0.436386	−	0.899759i	$-0.356258\pi$
$677$	22.7089	0.872772	0.436386	+	0.899759i	$0.356258\pi$
$678$	0	0
$679$	9.36222	0.359289
$680$	0	0
$681$	−27.6061	−1.05787
$682$	0	0
$683$	23.5194	0.899945	0.449973	−	0.893042i	$-0.351434\pi$
$683$	23.5194	0.899945	0.449973	+	0.893042i	$0.351434\pi$
$684$	0	0
$685$	13.2645	0.506809
$686$	0	0
$687$	23.8328	0.909277
$688$	0	0
$689$	9.45998	0.360396
$690$	0	0
$691$	−35.1794	−1.33829	−0.669144	−	0.743133i	$-0.733339\pi$
$691$	−35.1794	−1.33829	−0.669144	+	0.743133i	$0.733339\pi$
$692$	0	0
$693$	24.0383	0.913140
$694$	0	0
$695$	22.4605	0.851975
$696$	0	0
$697$	−0.864451	−0.0327434
$698$	0	0
$699$	17.5789	0.664893
$700$	0	0
$701$	1.48110	0.0559404	0.0279702	−	0.999609i	$-0.491096\pi$
$701$	1.48110	0.0559404	0.0279702	+	0.999609i	$0.491096\pi$
$702$	0	0
$703$	−21.6555	−0.816752
$704$	0	0
$705$	−48.6066	−1.83063
$706$	0	0
$707$	−4.31889	−0.162428
$708$	0	0
$709$	−29.3622	−1.10272	−0.551361	−	0.834267i	$-0.685891\pi$
$709$	−29.3622	−1.10272	−0.551361	+	0.834267i	$0.685891\pi$
$710$	0	0
$711$	−34.1744	−1.28164
$712$	0	0
$713$	2.91995	0.109353
$714$	0	0
$715$	43.3110	1.61974
$716$	0	0
$717$	−35.4389	−1.32349
$718$	0	0
$719$	−13.3833	−0.499115	−0.249557	−	0.968360i	$-0.580285\pi$
$719$	−13.3833	−0.499115	−0.249557	+	0.968360i	$0.580285\pi$
$720$	0	0
$721$	5.04888	0.188030
$722$	0	0
$723$	54.4777	2.02605
$724$	0	0
$725$	11.2645	0.418352
$726$	0	0
$727$	−21.5366	−0.798748	−0.399374	−	0.916788i	$-0.630773\pi$
$727$	−21.5366	−0.798748	−0.399374	+	0.916788i	$0.630773\pi$
$728$	0	0
$729$	−5.09775	−0.188806
$730$	0	0
$731$	−20.1189	−0.744124
$732$	0	0
$733$	−30.2978	−1.11907	−0.559537	−	0.828806i	$-0.689021\pi$
$733$	−30.2978	−1.11907	−0.559537	+	0.828806i	$0.689021\pi$
$734$	0	0
$735$	7.83276	0.288916
$736$	0	0
$737$	23.8610	0.878932
$738$	0	0
$739$	−12.9794	−0.477455	−0.238727	−	0.971087i	$-0.576730\pi$
$739$	−12.9794	−0.477455	−0.238727	+	0.971087i	$0.576730\pi$
$740$	0	0
$741$	−104.241	−3.82939
$742$	0	0
$743$	−50.0071	−1.83458	−0.917292	−	0.398215i	$-0.869630\pi$
$743$	−50.0071	−1.83458	−0.917292	+	0.398215i	$0.869630\pi$
$744$	0	0
$745$	54.6933	2.00381
$746$	0	0
$747$	−39.4061	−1.44179
$748$	0	0
$749$	−9.83276	−0.359281
$750$	0	0
$751$	−22.8917	−0.835329	−0.417665	−	0.908601i	$-0.637151\pi$
$751$	−22.8917	−0.835329	−0.417665	+	0.908601i	$0.637151\pi$
$752$	0	0
$753$	−37.7038	−1.37400
$754$	0	0
$755$	12.5955	0.458398
$756$	0	0
$757$	−29.0278	−1.05503	−0.527516	−	0.849545i	$-0.676876\pi$
$757$	−29.0278	−1.05503	−0.527516	+	0.849545i	$0.676876\pi$
$758$	0	0
$759$	6.50885	0.236256
$760$	0	0
$761$	36.8122	1.33444	0.667220	−	0.744861i	$-0.267484\pi$
$761$	36.8122	1.33444	0.667220	+	0.744861i	$0.267484\pi$
$762$	0	0
$763$	−19.4600	−0.704498
$764$	0	0
$765$	70.3588	2.54383
$766$	0	0
$767$	59.4217	2.14559
$768$	0	0
$769$	−47.8711	−1.72628	−0.863138	−	0.504969i	$-0.831504\pi$
$769$	−47.8711	−1.72628	−0.863138	+	0.504969i	$0.831504\pi$
$770$	0	0
$771$	19.8922	0.716400
$772$	0	0
$773$	41.2489	1.48362	0.741810	−	0.670610i	$-0.233968\pi$
$773$	41.2489	1.48362	0.741810	+	0.670610i	$0.233968\pi$
$774$	0	0
$775$	−6.93103	−0.248970
$776$	0	0
$777$	9.45998	0.339375
$778$	0	0
$779$	−1.45998	−0.0523091
$780$	0	0
$781$	0	0
$782$	0	0
$783$	92.3488	3.30028
$784$	0	0
$785$	50.7215	1.81033
$786$	0	0
$787$	−6.79947	−0.242375	−0.121188	−	0.992630i	$-0.538670\pi$
$787$	−6.79947	−0.242375	−0.121188	+	0.992630i	$0.538670\pi$
$788$	0	0
$789$	−32.0000	−1.13923
$790$	0	0
$791$	−8.78389	−0.312319
$792$	0	0
$793$	−49.7805	−1.76776
$794$	0	0
$795$	−15.6655	−0.555599
$796$	0	0
$797$	6.18996	0.219260	0.109630	−	0.993972i	$-0.465033\pi$
$797$	6.18996	0.219260	0.109630	+	0.993972i	$0.465033\pi$
$798$	0	0
$799$	−26.0978	−0.923272
$800$	0	0
$801$	−68.9966	−2.43787
$802$	0	0
$803$	33.5678	1.18458
$804$	0	0
$805$	1.45998	0.0514574
$806$	0	0
$807$	48.6550	1.71274
$808$	0	0
$809$	30.3517	1.06711	0.533554	−	0.845766i	$-0.320856\pi$
$809$	30.3517	1.06711	0.533554	+	0.845766i	$0.320856\pi$
$810$	0	0
$811$	40.0328	1.40574	0.702870	−	0.711318i	$-0.251901\pi$
$811$	40.0328	1.40574	0.702870	+	0.711318i	$0.251901\pi$
$812$	0	0
$813$	79.9377	2.80354
$814$	0	0
$815$	40.4877	1.41822
$816$	0	0
$817$	−33.9789	−1.18877
$818$	0	0
$819$	31.3466	1.09534
$820$	0	0
$821$	−51.9789	−1.81408	−0.907038	−	0.421050i	$-0.861662\pi$
$821$	−51.9789	−1.81408	−0.907038	+	0.421050i	$0.861662\pi$
$822$	0	0
$823$	−26.9200	−0.938371	−0.469185	−	0.883100i	$-0.655452\pi$
$823$	−26.9200	−0.938371	−0.469185	+	0.883100i	$0.655452\pi$
$824$	0	0
$825$	−15.4499	−0.537898
$826$	0	0
$827$	−47.5960	−1.65508	−0.827538	−	0.561409i	$-0.810259\pi$
$827$	−47.5960	−1.65508	−0.827538	+	0.561409i	$0.810259\pi$
$828$	0	0
$829$	8.21109	0.285183	0.142591	−	0.989782i	$-0.454456\pi$
$829$	8.21109	0.285183	0.142591	+	0.989782i	$0.454456\pi$
$830$	0	0
$831$	53.5366	1.85716
$832$	0	0
$833$	4.20555	0.145714
$834$	0	0
$835$	−4.52998	−0.156766
$836$	0	0
$837$	−56.8222	−1.96406
$838$	0	0
$839$	5.04888	0.174307	0.0871533	−	0.996195i	$-0.472223\pi$
$839$	5.04888	0.174307	0.0871533	+	0.996195i	$0.472223\pi$
$840$	0	0
$841$	38.3311	1.32176
$842$	0	0
$843$	−100.868	−3.47407
$844$	0	0
$845$	23.6610	0.813964
$846$	0	0
$847$	−2.15667	−0.0741042
$848$	0	0
$849$	71.3482	2.44867
$850$	0	0
$851$	1.76328	0.0604444
$852$	0	0
$853$	−1.14109	−0.0390701	−0.0195351	−	0.999809i	$-0.506219\pi$
$853$	−1.14109	−0.0390701	−0.0195351	+	0.999809i	$0.506219\pi$
$854$	0	0
$855$	118.829	4.06388
$856$	0	0
$857$	−3.45998	−0.118191	−0.0590953	−	0.998252i	$-0.518822\pi$
$857$	−3.45998	−0.118191	−0.0590953	+	0.998252i	$0.518822\pi$
$858$	0	0
$859$	−4.15165	−0.141653	−0.0708263	−	0.997489i	$-0.522564\pi$
$859$	−4.15165	−0.141653	−0.0708263	+	0.997489i	$0.522564\pi$
$860$	0	0
$861$	0.637776	0.0217353
$862$	0	0
$863$	13.7633	0.468507	0.234254	−	0.972175i	$-0.424735\pi$
$863$	13.7633	0.468507	0.234254	+	0.972175i	$0.424735\pi$
$864$	0	0
$865$	−57.6272	−1.95938
$866$	0	0
$867$	2.13053	0.0723564
$868$	0	0
$869$	18.7044	0.634502
$870$	0	0
$871$	31.1155	1.05431
$872$	0	0
$873$	−62.0455	−2.09992
$874$	0	0
$875$	9.15667	0.309552
$876$	0	0
$877$	−53.9688	−1.82240	−0.911199	−	0.411967i	$-0.864842\pi$
$877$	−53.9688	−1.82240	−0.911199	+	0.411967i	$0.864842\pi$
$878$	0	0
$879$	27.0872	0.913628
$880$	0	0
$881$	56.1744	1.89256	0.946281	−	0.323344i	$-0.104807\pi$
$881$	56.1744	1.89256	0.946281	+	0.323344i	$0.104807\pi$
$882$	0	0
$883$	−52.5371	−1.76801	−0.884007	−	0.467473i	$-0.845165\pi$
$883$	−52.5371	−1.76801	−0.884007	+	0.467473i	$0.845165\pi$
$884$	0	0
$885$	−98.4011	−3.30772
$886$	0	0
$887$	44.4988	1.49412	0.747062	−	0.664755i	$-0.231464\pi$
$887$	44.4988	1.49412	0.747062	+	0.664755i	$0.231464\pi$
$888$	0	0
$889$	14.6761	0.492220
$890$	0	0
$891$	−54.5472	−1.82740
$892$	0	0
$893$	−44.0766	−1.47497
$894$	0	0
$895$	8.88454	0.296978
$896$	0	0
$897$	8.48773	0.283397
$898$	0	0
$899$	−41.4288	−1.38173
$900$	0	0
$901$	−8.41110	−0.280214
$902$	0	0
$903$	14.8433	0.493955
$904$	0	0
$905$	−27.6061	−0.917657
$906$	0	0
$907$	44.4182	1.47488	0.737442	−	0.675411i	$-0.236034\pi$
$907$	44.4182	1.47488	0.737442	+	0.675411i	$0.236034\pi$
$908$	0	0
$909$	28.6222	0.949338
$910$	0	0
$911$	48.9894	1.62309	0.811546	−	0.584288i	$-0.198626\pi$
$911$	48.9894	1.62309	0.811546	+	0.584288i	$0.198626\pi$
$912$	0	0
$913$	21.5678	0.713789
$914$	0	0
$915$	82.4354	2.72523
$916$	0	0
$917$	−4.15165	−0.137100
$918$	0	0
$919$	−34.9200	−1.15190	−0.575951	−	0.817484i	$-0.695368\pi$
$919$	−34.9200	−1.15190	−0.575951	+	0.817484i	$0.695368\pi$
$920$	0	0
$921$	82.7215	2.72577
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−4.18546	−0.137617
$926$	0	0
$927$	−33.4600	−1.09897
$928$	0	0
$929$	3.98995	0.130906	0.0654531	−	0.997856i	$-0.479151\pi$
$929$	3.98995	0.130906	0.0654531	+	0.997856i	$0.479151\pi$
$930$	0	0
$931$	7.10278	0.232784
$932$	0	0
$933$	−31.3311	−1.02573
$934$	0	0
$935$	−38.5089	−1.25937
$936$	0	0
$937$	−47.9789	−1.56740	−0.783701	−	0.621139i	$-0.786670\pi$
$937$	−47.9789	−1.56740	−0.783701	+	0.621139i	$0.786670\pi$
$938$	0	0
$939$	−0.637776	−0.0208130
$940$	0	0
$941$	−26.1900	−0.853768	−0.426884	−	0.904306i	$-0.640389\pi$
$941$	−26.1900	−0.853768	−0.426884	+	0.904306i	$0.640389\pi$
$942$	0	0
$943$	0.118877	0.00387118
$944$	0	0
$945$	−28.4111	−0.924213
$946$	0	0
$947$	−57.3905	−1.86494	−0.932470	−	0.361247i	$-0.882351\pi$
$947$	−57.3905	−1.86494	−0.932470	+	0.361247i	$0.882351\pi$
$948$	0	0
$949$	43.7733	1.42094
$950$	0	0
$951$	−94.3588	−3.05979
$952$	0	0
$953$	−3.68665	−0.119422	−0.0597112	−	0.998216i	$-0.519018\pi$
$953$	−3.68665	−0.119422	−0.0597112	+	0.998216i	$0.519018\pi$
$954$	0	0
$955$	1.88216	0.0609051
$956$	0	0
$957$	−92.3488	−2.98521
$958$	0	0
$959$	−5.25443	−0.169674
$960$	0	0
$961$	−5.50885	−0.177705
$962$	0	0
$963$	65.1638	2.09987
$964$	0	0
$965$	−35.6856	−1.14876
$966$	0	0
$967$	27.6172	0.888108	0.444054	−	0.896000i	$-0.353540\pi$
$967$	27.6172	0.888108	0.444054	+	0.896000i	$0.353540\pi$
$968$	0	0
$969$	92.6832	2.97741
$970$	0	0
$971$	9.94610	0.319186	0.159593	−	0.987183i	$-0.448982\pi$
$971$	9.94610	0.319186	0.159593	+	0.987183i	$0.448982\pi$
$972$	0	0
$973$	−8.89722	−0.285232
$974$	0	0
$975$	−20.1471	−0.645225
$976$	0	0
$977$	−21.2544	−0.679989	−0.339995	−	0.940427i	$-0.610425\pi$
$977$	−21.2544	−0.679989	−0.339995	+	0.940427i	$0.610425\pi$
$978$	0	0
$979$	37.7633	1.20692
$980$	0	0
$981$	128.965	4.11755
$982$	0	0
$983$	16.1844	0.516203	0.258101	−	0.966118i	$-0.416903\pi$
$983$	16.1844	0.516203	0.258101	+	0.966118i	$0.416903\pi$
$984$	0	0
$985$	−25.2444	−0.804353
$986$	0	0
$987$	19.2544	0.612875
$988$	0	0
$989$	2.76670	0.0879759
$990$	0	0
$991$	16.0766	0.510691	0.255345	−	0.966850i	$-0.417811\pi$
$991$	16.0766	0.510691	0.255345	+	0.966850i	$0.417811\pi$
$992$	0	0
$993$	−7.66553	−0.243258
$994$	0	0
$995$	1.61003	0.0510412
$996$	0	0
$997$	29.4444	0.932513	0.466257	−	0.884650i	$-0.345602\pi$
$997$	29.4444	0.932513	0.466257	+	0.884650i	$0.345602\pi$
$998$	0	0
$999$	−34.3133	−1.08563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.a.l.1.3	yes	3
3.2	odd	2		8064.2.a.bu.1.2		3
4.3	odd	2		896.2.a.j.1.1	yes	3
7.6	odd	2		6272.2.a.u.1.1		3
8.3	odd	2		896.2.a.k.1.3	yes	3
8.5	even	2		896.2.a.i.1.1	✓	3
12.11	even	2		8064.2.a.cb.1.2		3
16.3	odd	4		1792.2.b.o.897.1		6
16.5	even	4		1792.2.b.p.897.1		6
16.11	odd	4		1792.2.b.o.897.6		6
16.13	even	4		1792.2.b.p.897.6		6
24.5	odd	2		8064.2.a.ce.1.2		3
24.11	even	2		8064.2.a.ch.1.2		3
28.27	even	2		6272.2.a.w.1.3		3
56.13	odd	2		6272.2.a.x.1.3		3
56.27	even	2		6272.2.a.v.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.a.i.1.1	✓	3	8.5	even	2
896.2.a.j.1.1	yes	3	4.3	odd	2
896.2.a.k.1.3	yes	3	8.3	odd	2
896.2.a.l.1.3	yes	3	1.1	even	1	trivial
1792.2.b.o.897.1		6	16.3	odd	4
1792.2.b.o.897.6		6	16.11	odd	4
1792.2.b.p.897.1		6	16.5	even	4
1792.2.b.p.897.6		6	16.13	even	4
6272.2.a.u.1.1		3	7.6	odd	2
6272.2.a.v.1.1		3	56.27	even	2
6272.2.a.w.1.3		3	28.27	even	2
6272.2.a.x.1.3		3	56.13	odd	2
8064.2.a.bu.1.2		3	3.2	odd	2
8064.2.a.cb.1.2		3	12.11	even	2
8064.2.a.ce.1.2		3	24.5	odd	2
8064.2.a.ch.1.2		3	24.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.a (trivial)

\(f(q)\)	\(=\)	\(q+3.10278 q^{3} +2.52444 q^{5} -1.00000 q^{7} +6.62721 q^{9} -3.62721 q^{11} -4.72999 q^{13} +7.83276 q^{15} +4.20555 q^{17} +7.10278 q^{19} -3.10278 q^{21} -0.578337 q^{23} +1.37279 q^{25} +11.2544 q^{27} +8.20555 q^{29} -5.04888 q^{31} -11.2544 q^{33} -2.52444 q^{35} -3.04888 q^{37} -14.6761 q^{39} -0.205550 q^{41} -4.78389 q^{43} +16.7300 q^{45} -6.20555 q^{47} +1.00000 q^{49} +13.0489 q^{51} -2.00000 q^{53} -9.15667 q^{55} +22.0383 q^{57} -12.5628 q^{59} +10.5244 q^{61} -6.62721 q^{63} -11.9406 q^{65} -6.57834 q^{67} -1.79445 q^{69} -9.25443 q^{73} +4.25945 q^{75} +3.62721 q^{77} -5.15667 q^{79} +15.0383 q^{81} -5.94610 q^{83} +10.6167 q^{85} +25.4600 q^{87} -10.4111 q^{89} +4.72999 q^{91} -15.6655 q^{93} +17.9305 q^{95} -9.36222 q^{97} -24.0383 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{5} - 3 q^{7} + 7 q^{9} + 2 q^{11} + 6 q^{13} - 4 q^{15} - 2 q^{17} + 14 q^{19} - 2 q^{21} + 17 q^{25} + 8 q^{27} + 10 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} + 2 q^{37} - 20 q^{39}+ \cdots - 30 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^5 - 3 * q^7 + 7 * q^9 + 2 * q^11 + 6 * q^13 - 4 * q^15 - 2 * q^17 + 14 * q^19 - 2 * q^21 + 17 * q^25 + 8 * q^27 + 10 * q^29 - 4 * q^31 - 8 * q^33 - 2 * q^35 + 2 * q^37 - 20 * q^39 + 14 * q^41 + 2 * q^43 + 30 * q^45 - 4 * q^47 + 3 * q^49 + 28 * q^51 - 6 * q^53 - 24 * q^55 + 24 * q^57 + 10 * q^59 + 26 * q^61 - 7 * q^63 - 16 * q^65 - 18 * q^67 - 20 * q^69 - 2 * q^73 + 2 * q^75 - 2 * q^77 - 12 * q^79 + 3 * q^81 - 14 * q^83 - 12 * q^85 + 36 * q^87 - 2 * q^89 - 6 * q^91 + 8 * q^93 + 4 * q^95 - 10 * q^97 - 30 * q^99

Introduction

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