LMFDB - Embedded newform 1568.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, 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("1568.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.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:	$12.5205430369$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{3} -3.82843 q^{5} +2.82843 q^{9} +O(q^{10})$	$q-2.41421 q^{3} -3.82843 q^{5} +2.82843 q^{9} +0.414214 q^{11} +2.82843 q^{13} +9.24264 q^{15} +5.82843 q^{17} -3.58579 q^{19} +3.24264 q^{23} +9.65685 q^{25} +0.414214 q^{27} +2.82843 q^{29} -8.41421 q^{31} -1.00000 q^{33} -2.65685 q^{37} -6.82843 q^{39} +1.17157 q^{41} -1.65685 q^{43} -10.8284 q^{45} -7.58579 q^{47} -14.0711 q^{51} -1.00000 q^{53} -1.58579 q^{55} +8.65685 q^{57} +8.89949 q^{59} +2.65685 q^{61} -10.8284 q^{65} -11.2426 q^{67} -7.82843 q^{69} +2.34315 q^{71} +3.34315 q^{73} -23.3137 q^{75} -8.07107 q^{79} -9.48528 q^{81} -15.3137 q^{83} -22.3137 q^{85} -6.82843 q^{87} +9.00000 q^{89} +20.3137 q^{93} +13.7279 q^{95} +6.82843 q^{97} +1.17157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{11} + 10 q^{15} + 6 q^{17} - 10 q^{19} - 2 q^{23} + 8 q^{25} - 2 q^{27} - 14 q^{31} - 2 q^{33} + 6 q^{37} - 8 q^{39} + 8 q^{41} + 8 q^{43} - 16 q^{45} - 18 q^{47} - 14 q^{51} - 2 q^{53} - 6 q^{55} + 6 q^{57} - 2 q^{59} - 6 q^{61} - 16 q^{65} - 14 q^{67} - 10 q^{69} + 16 q^{71} + 18 q^{73} - 24 q^{75} - 2 q^{79} - 2 q^{81} - 8 q^{83} - 22 q^{85} - 8 q^{87} + 18 q^{89} + 18 q^{93} + 2 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^11 + 10 * q^15 + 6 * q^17 - 10 * q^19 - 2 * q^23 + 8 * q^25 - 2 * q^27 - 14 * q^31 - 2 * q^33 + 6 * q^37 - 8 * q^39 + 8 * q^41 + 8 * q^43 - 16 * q^45 - 18 * q^47 - 14 * q^51 - 2 * q^53 - 6 * q^55 + 6 * q^57 - 2 * q^59 - 6 * q^61 - 16 * q^65 - 14 * q^67 - 10 * q^69 + 16 * q^71 + 18 * q^73 - 24 * q^75 - 2 * q^79 - 2 * q^81 - 8 * q^83 - 22 * q^85 - 8 * q^87 + 18 * q^89 + 18 * q^93 + 2 * q^95 + 8 * q^97 + 8 * q^99

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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	0	0
$5$	−3.82843	−1.71212	−0.856062	−	0.516873i	$-0.827096\pi$
$5$	−3.82843	−1.71212	−0.856062	+	0.516873i	$0.827096\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	0.414214	0.124890	0.0624450	−	0.998048i	$-0.480110\pi$
$11$	0.414214	0.124890	0.0624450	+	0.998048i	$0.480110\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	9.24264	2.38644
$16$	0	0
$17$	5.82843	1.41360	0.706801	−	0.707413i	$-0.250138\pi$
$17$	5.82843	1.41360	0.706801	+	0.707413i	$0.250138\pi$
$18$	0	0
$19$	−3.58579	−0.822636	−0.411318	−	0.911492i	$-0.634931\pi$
$19$	−3.58579	−0.822636	−0.411318	+	0.911492i	$0.634931\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.24264	0.676137	0.338069	−	0.941121i	$-0.390226\pi$
$23$	3.24264	0.676137	0.338069	+	0.941121i	$0.390226\pi$
$24$	0	0
$25$	9.65685	1.93137
$26$	0	0
$27$	0.414214	0.0797154
$28$	0	0
$29$	2.82843	0.525226	0.262613	−	0.964901i	$-0.415416\pi$
$29$	2.82843	0.525226	0.262613	+	0.964901i	$0.415416\pi$
$30$	0	0
$31$	−8.41421	−1.51124	−0.755619	−	0.655012i	$-0.772664\pi$
$31$	−8.41421	−1.51124	−0.755619	+	0.655012i	$0.772664\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.65685	−0.436784	−0.218392	−	0.975861i	$-0.570081\pi$
$37$	−2.65685	−0.436784	−0.218392	+	0.975861i	$0.570081\pi$
$38$	0	0
$39$	−6.82843	−1.09342
$40$	0	0
$41$	1.17157	0.182969	0.0914845	−	0.995807i	$-0.470839\pi$
$41$	1.17157	0.182969	0.0914845	+	0.995807i	$0.470839\pi$
$42$	0	0
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	0	0
$45$	−10.8284	−1.61421
$46$	0	0
$47$	−7.58579	−1.10650	−0.553250	−	0.833015i	$-0.686613\pi$
$47$	−7.58579	−1.10650	−0.553250	+	0.833015i	$0.686613\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−14.0711	−1.97034
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	−1.58579	−0.213827
$56$	0	0
$57$	8.65685	1.14663
$58$	0	0
$59$	8.89949	1.15862	0.579308	−	0.815109i	$-0.303323\pi$
$59$	8.89949	1.15862	0.579308	+	0.815109i	$0.303323\pi$
$60$	0	0
$61$	2.65685	0.340175	0.170088	−	0.985429i	$-0.445595\pi$
$61$	2.65685	0.340175	0.170088	+	0.985429i	$0.445595\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.8284	−1.34310
$66$	0	0
$67$	−11.2426	−1.37351	−0.686754	−	0.726890i	$-0.740965\pi$
$67$	−11.2426	−1.37351	−0.686754	+	0.726890i	$0.740965\pi$
$68$	0	0
$69$	−7.82843	−0.942432
$70$	0	0
$71$	2.34315	0.278080	0.139040	−	0.990287i	$-0.455598\pi$
$71$	2.34315	0.278080	0.139040	+	0.990287i	$0.455598\pi$
$72$	0	0
$73$	3.34315	0.391286	0.195643	−	0.980675i	$-0.437321\pi$
$73$	3.34315	0.391286	0.195643	+	0.980675i	$0.437321\pi$
$74$	0	0
$75$	−23.3137	−2.69204
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.07107	−0.908066	−0.454033	−	0.890985i	$-0.650015\pi$
$79$	−8.07107	−0.908066	−0.454033	+	0.890985i	$0.650015\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−15.3137	−1.68090	−0.840449	−	0.541891i	$-0.817709\pi$
$83$	−15.3137	−1.68090	−0.840449	+	0.541891i	$0.817709\pi$
$84$	0	0
$85$	−22.3137	−2.42026
$86$	0	0
$87$	−6.82843	−0.732084
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	20.3137	2.10643
$94$	0	0
$95$	13.7279	1.40846
$96$	0	0
$97$	6.82843	0.693322	0.346661	−	0.937991i	$-0.387315\pi$
$97$	6.82843	0.693322	0.346661	+	0.937991i	$0.387315\pi$
$98$	0	0
$99$	1.17157	0.117748
$100$	0	0
$101$	3.48528	0.346798	0.173399	−	0.984852i	$-0.444525\pi$
$101$	3.48528	0.346798	0.173399	+	0.984852i	$0.444525\pi$
$102$	0	0
$103$	−5.58579	−0.550384	−0.275192	−	0.961389i	$-0.588741\pi$
$103$	−5.58579	−0.550384	−0.275192	+	0.961389i	$0.588741\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.89949	0.666999	0.333500	−	0.942750i	$-0.391770\pi$
$107$	6.89949	0.666999	0.333500	+	0.942750i	$0.391770\pi$
$108$	0	0
$109$	−13.8284	−1.32452	−0.662262	−	0.749273i	$-0.730403\pi$
$109$	−13.8284	−1.32452	−0.662262	+	0.749273i	$0.730403\pi$
$110$	0	0
$111$	6.41421	0.608810
$112$	0	0
$113$	−10.1421	−0.954092	−0.477046	−	0.878878i	$-0.658292\pi$
$113$	−10.1421	−0.954092	−0.477046	+	0.878878i	$0.658292\pi$
$114$	0	0
$115$	−12.4142	−1.15763
$116$	0	0
$117$	8.00000	0.739600
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.8284	−0.984402
$122$	0	0
$123$	−2.82843	−0.255031
$124$	0	0
$125$	−17.8284	−1.59462
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	17.7279	1.54890	0.774448	−	0.632638i	$-0.218028\pi$
$131$	17.7279	1.54890	0.774448	+	0.632638i	$0.218028\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.58579	−0.136483
$136$	0	0
$137$	−17.1421	−1.46455	−0.732276	−	0.681008i	$-0.761542\pi$
$137$	−17.1421	−1.46455	−0.732276	+	0.681008i	$0.761542\pi$
$138$	0	0
$139$	7.31371	0.620341	0.310170	−	0.950681i	$-0.399614\pi$
$139$	7.31371	0.620341	0.310170	+	0.950681i	$0.399614\pi$
$140$	0	0
$141$	18.3137	1.54229
$142$	0	0
$143$	1.17157	0.0979718
$144$	0	0
$145$	−10.8284	−0.899252
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.8284	0.969023	0.484511	−	0.874785i	$-0.338997\pi$
$149$	11.8284	0.969023	0.484511	+	0.874785i	$0.338997\pi$
$150$	0	0
$151$	−18.8995	−1.53802	−0.769010	−	0.639237i	$-0.779250\pi$
$151$	−18.8995	−1.53802	−0.769010	+	0.639237i	$0.779250\pi$
$152$	0	0
$153$	16.4853	1.33276
$154$	0	0
$155$	32.2132	2.58743
$156$	0	0
$157$	−20.6569	−1.64860	−0.824298	−	0.566156i	$-0.808430\pi$
$157$	−20.6569	−1.64860	−0.824298	+	0.566156i	$0.808430\pi$
$158$	0	0
$159$	2.41421	0.191460
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.2426	1.19390	0.596948	−	0.802280i	$-0.296380\pi$
$163$	15.2426	1.19390	0.596948	+	0.802280i	$0.296380\pi$
$164$	0	0
$165$	3.82843	0.298043
$166$	0	0
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−10.1421	−0.775588
$172$	0	0
$173$	−11.0000	−0.836315	−0.418157	−	0.908375i	$-0.637324\pi$
$173$	−11.0000	−0.836315	−0.418157	+	0.908375i	$0.637324\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−21.4853	−1.61493
$178$	0	0
$179$	8.41421	0.628908	0.314454	−	0.949273i	$-0.398179\pi$
$179$	8.41421	0.628908	0.314454	+	0.949273i	$0.398179\pi$
$180$	0	0
$181$	−13.3137	−0.989600	−0.494800	−	0.869007i	$-0.664759\pi$
$181$	−13.3137	−0.989600	−0.494800	+	0.869007i	$0.664759\pi$
$182$	0	0
$183$	−6.41421	−0.474152
$184$	0	0
$185$	10.1716	0.747829
$186$	0	0
$187$	2.41421	0.176545
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.8995	0.788660	0.394330	−	0.918969i	$-0.370977\pi$
$191$	10.8995	0.788660	0.394330	+	0.918969i	$0.370977\pi$
$192$	0	0
$193$	19.1421	1.37788	0.688941	−	0.724818i	$-0.258076\pi$
$193$	19.1421	1.37788	0.688941	+	0.724818i	$0.258076\pi$
$194$	0	0
$195$	26.1421	1.87208
$196$	0	0
$197$	−13.1716	−0.938436	−0.469218	−	0.883082i	$-0.655464\pi$
$197$	−13.1716	−0.938436	−0.469218	+	0.883082i	$0.655464\pi$
$198$	0	0
$199$	−2.75736	−0.195464	−0.0977320	−	0.995213i	$-0.531159\pi$
$199$	−2.75736	−0.195464	−0.0977320	+	0.995213i	$0.531159\pi$
$200$	0	0
$201$	27.1421	1.91446
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.48528	−0.313266
$206$	0	0
$207$	9.17157	0.637468
$208$	0	0
$209$	−1.48528	−0.102739
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	−5.65685	−0.387601
$214$	0	0
$215$	6.34315	0.432599
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8.07107	−0.545392
$220$	0	0
$221$	16.4853	1.10892
$222$	0	0
$223$	−13.6569	−0.914531	−0.457265	−	0.889330i	$-0.651171\pi$
$223$	−13.6569	−0.914531	−0.457265	+	0.889330i	$0.651171\pi$
$224$	0	0
$225$	27.3137	1.82091
$226$	0	0
$227$	1.92893	0.128028	0.0640139	−	0.997949i	$-0.479610\pi$
$227$	1.92893	0.128028	0.0640139	+	0.997949i	$0.479610\pi$
$228$	0	0
$229$	9.82843	0.649481	0.324740	−	0.945803i	$-0.394723\pi$
$229$	9.82843	0.649481	0.324740	+	0.945803i	$0.394723\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.82843	0.512857	0.256429	−	0.966563i	$-0.417454\pi$
$233$	7.82843	0.512857	0.256429	+	0.966563i	$0.417454\pi$
$234$	0	0
$235$	29.0416	1.89447
$236$	0	0
$237$	19.4853	1.26571
$238$	0	0
$239$	−21.3137	−1.37867	−0.689335	−	0.724443i	$-0.742097\pi$
$239$	−21.3137	−1.37867	−0.689335	+	0.724443i	$0.742097\pi$
$240$	0	0
$241$	12.3137	0.793196	0.396598	−	0.917992i	$-0.370191\pi$
$241$	12.3137	0.793196	0.396598	+	0.917992i	$0.370191\pi$
$242$	0	0
$243$	21.6569	1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.1421	−0.645329
$248$	0	0
$249$	36.9706	2.34291
$250$	0	0
$251$	−2.97056	−0.187500	−0.0937501	−	0.995596i	$-0.529885\pi$
$251$	−2.97056	−0.187500	−0.0937501	+	0.995596i	$0.529885\pi$
$252$	0	0
$253$	1.34315	0.0844428
$254$	0	0
$255$	53.8701	3.37347
$256$	0	0
$257$	−21.4853	−1.34022	−0.670108	−	0.742264i	$-0.733752\pi$
$257$	−21.4853	−1.34022	−0.670108	+	0.742264i	$0.733752\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	−8.89949	−0.548766	−0.274383	−	0.961620i	$-0.588474\pi$
$263$	−8.89949	−0.548766	−0.274383	+	0.961620i	$0.588474\pi$
$264$	0	0
$265$	3.82843	0.235178
$266$	0	0
$267$	−21.7279	−1.32973
$268$	0	0
$269$	−1.34315	−0.0818930	−0.0409465	−	0.999161i	$-0.513037\pi$
$269$	−1.34315	−0.0818930	−0.0409465	+	0.999161i	$0.513037\pi$
$270$	0	0
$271$	−12.2132	−0.741899	−0.370950	−	0.928653i	$-0.620968\pi$
$271$	−12.2132	−0.741899	−0.370950	+	0.928653i	$0.620968\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−12.3137	−0.739859	−0.369930	−	0.929060i	$-0.620618\pi$
$277$	−12.3137	−0.739859	−0.369930	+	0.929060i	$0.620618\pi$
$278$	0	0
$279$	−23.7990	−1.42481
$280$	0	0
$281$	4.48528	0.267569	0.133785	−	0.991010i	$-0.457287\pi$
$281$	4.48528	0.267569	0.133785	+	0.991010i	$0.457287\pi$
$282$	0	0
$283$	−8.41421	−0.500173	−0.250087	−	0.968223i	$-0.580459\pi$
$283$	−8.41421	−0.500173	−0.250087	+	0.968223i	$0.580459\pi$
$284$	0	0
$285$	−33.1421	−1.96317
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.9706	0.998268
$290$	0	0
$291$	−16.4853	−0.966384
$292$	0	0
$293$	16.6274	0.971384	0.485692	−	0.874130i	$-0.338568\pi$
$293$	16.6274	0.971384	0.485692	+	0.874130i	$0.338568\pi$
$294$	0	0
$295$	−34.0711	−1.98369
$296$	0	0
$297$	0.171573	0.00995567
$298$	0	0
$299$	9.17157	0.530406
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−8.41421	−0.483384
$304$	0	0
$305$	−10.1716	−0.582423
$306$	0	0
$307$	−25.6569	−1.46431	−0.732157	−	0.681136i	$-0.761486\pi$
$307$	−25.6569	−1.46431	−0.732157	+	0.681136i	$0.761486\pi$
$308$	0	0
$309$	13.4853	0.767151
$310$	0	0
$311$	27.2426	1.54479	0.772394	−	0.635143i	$-0.219059\pi$
$311$	27.2426	1.54479	0.772394	+	0.635143i	$0.219059\pi$
$312$	0	0
$313$	−27.2843	−1.54220	−0.771099	−	0.636715i	$-0.780293\pi$
$313$	−27.2843	−1.54220	−0.771099	+	0.636715i	$0.780293\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.3137	−1.81492	−0.907459	−	0.420140i	$-0.861981\pi$
$317$	−32.3137	−1.81492	−0.907459	+	0.420140i	$0.861981\pi$
$318$	0	0
$319$	1.17157	0.0655955
$320$	0	0
$321$	−16.6569	−0.929695
$322$	0	0
$323$	−20.8995	−1.16288
$324$	0	0
$325$	27.3137	1.51509
$326$	0	0
$327$	33.3848	1.84618
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.5858	1.40632	0.703161	−	0.711031i	$-0.251771\pi$
$331$	25.5858	1.40632	0.703161	+	0.711031i	$0.251771\pi$
$332$	0	0
$333$	−7.51472	−0.411804
$334$	0	0
$335$	43.0416	2.35162
$336$	0	0
$337$	14.8284	0.807756	0.403878	−	0.914813i	$-0.367662\pi$
$337$	14.8284	0.807756	0.403878	+	0.914813i	$0.367662\pi$
$338$	0	0
$339$	24.4853	1.32986
$340$	0	0
$341$	−3.48528	−0.188739
$342$	0	0
$343$	0	0
$344$	0	0
$345$	29.9706	1.61356
$346$	0	0
$347$	15.1005	0.810638	0.405319	−	0.914175i	$-0.367161\pi$
$347$	15.1005	0.810638	0.405319	+	0.914175i	$0.367161\pi$
$348$	0	0
$349$	−10.8284	−0.579632	−0.289816	−	0.957082i	$-0.593594\pi$
$349$	−10.8284	−0.579632	−0.289816	+	0.957082i	$0.593594\pi$
$350$	0	0
$351$	1.17157	0.0625339
$352$	0	0
$353$	−7.82843	−0.416665	−0.208333	−	0.978058i	$-0.566804\pi$
$353$	−7.82843	−0.416665	−0.208333	+	0.978058i	$0.566804\pi$
$354$	0	0
$355$	−8.97056	−0.476108
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.3848	−1.33976	−0.669879	−	0.742471i	$-0.733654\pi$
$359$	−25.3848	−1.33976	−0.669879	+	0.742471i	$0.733654\pi$
$360$	0	0
$361$	−6.14214	−0.323270
$362$	0	0
$363$	26.1421	1.37211
$364$	0	0
$365$	−12.7990	−0.669930
$366$	0	0
$367$	−29.7279	−1.55178	−0.775892	−	0.630865i	$-0.782700\pi$
$367$	−29.7279	−1.55178	−0.775892	+	0.630865i	$0.782700\pi$
$368$	0	0
$369$	3.31371	0.172505
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14.3137	0.741136	0.370568	−	0.928805i	$-0.379163\pi$
$373$	14.3137	0.741136	0.370568	+	0.928805i	$0.379163\pi$
$374$	0	0
$375$	43.0416	2.22266
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−1.02944	−0.0528786	−0.0264393	−	0.999650i	$-0.508417\pi$
$379$	−1.02944	−0.0528786	−0.0264393	+	0.999650i	$0.508417\pi$
$380$	0	0
$381$	−13.6569	−0.699662
$382$	0	0
$383$	2.75736	0.140894	0.0704472	−	0.997516i	$-0.477557\pi$
$383$	2.75736	0.140894	0.0704472	+	0.997516i	$0.477557\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.68629	−0.238218
$388$	0	0
$389$	23.8284	1.20815	0.604075	−	0.796928i	$-0.293543\pi$
$389$	23.8284	1.20815	0.604075	+	0.796928i	$0.293543\pi$
$390$	0	0
$391$	18.8995	0.955789
$392$	0	0
$393$	−42.7990	−2.15892
$394$	0	0
$395$	30.8995	1.55472
$396$	0	0
$397$	9.82843	0.493275	0.246637	−	0.969108i	$-0.420674\pi$
$397$	9.82843	0.493275	0.246637	+	0.969108i	$0.420674\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.17157	0.108443	0.0542216	−	0.998529i	$-0.482732\pi$
$401$	2.17157	0.108443	0.0542216	+	0.998529i	$0.482732\pi$
$402$	0	0
$403$	−23.7990	−1.18551
$404$	0	0
$405$	36.3137	1.80444
$406$	0	0
$407$	−1.10051	−0.0545500
$408$	0	0
$409$	36.4558	1.80263	0.901313	−	0.433169i	$-0.142605\pi$
$409$	36.4558	1.80263	0.901313	+	0.433169i	$0.142605\pi$
$410$	0	0
$411$	41.3848	2.04136
$412$	0	0
$413$	0	0
$414$	0	0
$415$	58.6274	2.87791
$416$	0	0
$417$	−17.6569	−0.864660
$418$	0	0
$419$	1.65685	0.0809426	0.0404713	−	0.999181i	$-0.487114\pi$
$419$	1.65685	0.0809426	0.0404713	+	0.999181i	$0.487114\pi$
$420$	0	0
$421$	0.485281	0.0236512	0.0118256	−	0.999930i	$-0.496236\pi$
$421$	0.485281	0.0236512	0.0118256	+	0.999930i	$0.496236\pi$
$422$	0	0
$423$	−21.4558	−1.04322
$424$	0	0
$425$	56.2843	2.73019
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.82843	−0.136558
$430$	0	0
$431$	−21.2426	−1.02322	−0.511611	−	0.859217i	$-0.670951\pi$
$431$	−21.2426	−1.02322	−0.511611	+	0.859217i	$0.670951\pi$
$432$	0	0
$433$	−28.4853	−1.36892	−0.684458	−	0.729053i	$-0.739961\pi$
$433$	−28.4853	−1.36892	−0.684458	+	0.729053i	$0.739961\pi$
$434$	0	0
$435$	26.1421	1.25342
$436$	0	0
$437$	−11.6274	−0.556215
$438$	0	0
$439$	36.6985	1.75152	0.875762	−	0.482744i	$-0.160360\pi$
$439$	36.6985	1.75152	0.875762	+	0.482744i	$0.160360\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.0711	−1.23867	−0.619337	−	0.785125i	$-0.712598\pi$
$443$	−26.0711	−1.23867	−0.619337	+	0.785125i	$0.712598\pi$
$444$	0	0
$445$	−34.4558	−1.63336
$446$	0	0
$447$	−28.5563	−1.35067
$448$	0	0
$449$	6.82843	0.322253	0.161127	−	0.986934i	$-0.448487\pi$
$449$	6.82843	0.322253	0.161127	+	0.986934i	$0.448487\pi$
$450$	0	0
$451$	0.485281	0.0228510
$452$	0	0
$453$	45.6274	2.14376
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.2843	−1.74408	−0.872042	−	0.489430i	$-0.837205\pi$
$457$	−37.2843	−1.74408	−0.872042	+	0.489430i	$0.837205\pi$
$458$	0	0
$459$	2.41421	0.112686
$460$	0	0
$461$	−25.4558	−1.18560	−0.592798	−	0.805351i	$-0.701977\pi$
$461$	−25.4558	−1.18560	−0.592798	+	0.805351i	$0.701977\pi$
$462$	0	0
$463$	11.3137	0.525793	0.262896	−	0.964824i	$-0.415322\pi$
$463$	11.3137	0.525793	0.262896	+	0.964824i	$0.415322\pi$
$464$	0	0
$465$	−77.7696	−3.60648
$466$	0	0
$467$	−18.5563	−0.858685	−0.429343	−	0.903142i	$-0.641255\pi$
$467$	−18.5563	−0.858685	−0.429343	+	0.903142i	$0.641255\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	49.8701	2.29789
$472$	0	0
$473$	−0.686292	−0.0315557
$474$	0	0
$475$	−34.6274	−1.58881
$476$	0	0
$477$	−2.82843	−0.129505
$478$	0	0
$479$	−30.6985	−1.40265	−0.701325	−	0.712842i	$-0.747408\pi$
$479$	−30.6985	−1.40265	−0.701325	+	0.712842i	$0.747408\pi$
$480$	0	0
$481$	−7.51472	−0.342642
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−26.1421	−1.18705
$486$	0	0
$487$	13.7279	0.622072	0.311036	−	0.950398i	$-0.399324\pi$
$487$	13.7279	0.622072	0.311036	+	0.950398i	$0.399324\pi$
$488$	0	0
$489$	−36.7990	−1.66411
$490$	0	0
$491$	−7.65685	−0.345549	−0.172774	−	0.984961i	$-0.555273\pi$
$491$	−7.65685	−0.345549	−0.172774	+	0.984961i	$0.555273\pi$
$492$	0	0
$493$	16.4853	0.742460
$494$	0	0
$495$	−4.48528	−0.201598
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.5563	1.18883	0.594413	−	0.804160i	$-0.297385\pi$
$499$	26.5563	1.18883	0.594413	+	0.804160i	$0.297385\pi$
$500$	0	0
$501$	4.82843	0.215718
$502$	0	0
$503$	21.6569	0.965631	0.482816	−	0.875722i	$-0.339614\pi$
$503$	21.6569	0.965631	0.482816	+	0.875722i	$0.339614\pi$
$504$	0	0
$505$	−13.3431	−0.593762
$506$	0	0
$507$	12.0711	0.536095
$508$	0	0
$509$	−24.5147	−1.08660	−0.543298	−	0.839540i	$-0.682825\pi$
$509$	−24.5147	−1.08660	−0.543298	+	0.839540i	$0.682825\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.48528	−0.0655768
$514$	0	0
$515$	21.3848	0.942326
$516$	0	0
$517$	−3.14214	−0.138191
$518$	0	0
$519$	26.5563	1.16569
$520$	0	0
$521$	−7.00000	−0.306676	−0.153338	−	0.988174i	$-0.549002\pi$
$521$	−7.00000	−0.306676	−0.153338	+	0.988174i	$0.549002\pi$
$522$	0	0
$523$	7.72792	0.337918	0.168959	−	0.985623i	$-0.445959\pi$
$523$	7.72792	0.337918	0.168959	+	0.985623i	$0.445959\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−49.0416	−2.13629
$528$	0	0
$529$	−12.4853	−0.542838
$530$	0	0
$531$	25.1716	1.09235
$532$	0	0
$533$	3.31371	0.143533
$534$	0	0
$535$	−26.4142	−1.14199
$536$	0	0
$537$	−20.3137	−0.876601
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−20.1716	−0.867244	−0.433622	−	0.901095i	$-0.642765\pi$
$541$	−20.1716	−0.867244	−0.433622	+	0.901095i	$0.642765\pi$
$542$	0	0
$543$	32.1421	1.37935
$544$	0	0
$545$	52.9411	2.26775
$546$	0	0
$547$	−22.9706	−0.982150	−0.491075	−	0.871117i	$-0.663396\pi$
$547$	−22.9706	−0.982150	−0.491075	+	0.871117i	$0.663396\pi$
$548$	0	0
$549$	7.51472	0.320720
$550$	0	0
$551$	−10.1421	−0.432070
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−24.5563	−1.04236
$556$	0	0
$557$	−5.82843	−0.246958	−0.123479	−	0.992347i	$-0.539405\pi$
$557$	−5.82843	−0.246958	−0.123479	+	0.992347i	$0.539405\pi$
$558$	0	0
$559$	−4.68629	−0.198209
$560$	0	0
$561$	−5.82843	−0.246076
$562$	0	0
$563$	−0.0710678	−0.00299515	−0.00149758	−	0.999999i	$-0.500477\pi$
$563$	−0.0710678	−0.00299515	−0.00149758	+	0.999999i	$0.500477\pi$
$564$	0	0
$565$	38.8284	1.63352
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.3431	1.06244	0.531220	−	0.847234i	$-0.321734\pi$
$569$	25.3431	1.06244	0.531220	+	0.847234i	$0.321734\pi$
$570$	0	0
$571$	−15.5858	−0.652245	−0.326122	−	0.945328i	$-0.605742\pi$
$571$	−15.5858	−0.652245	−0.326122	+	0.945328i	$0.605742\pi$
$572$	0	0
$573$	−26.3137	−1.09927
$574$	0	0
$575$	31.3137	1.30587
$576$	0	0
$577$	−39.0000	−1.62359	−0.811796	−	0.583942i	$-0.801510\pi$
$577$	−39.0000	−1.62359	−0.811796	+	0.583942i	$0.801510\pi$
$578$	0	0
$579$	−46.2132	−1.92056
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.414214	−0.0171550
$584$	0	0
$585$	−30.6274	−1.26629
$586$	0	0
$587$	4.97056	0.205157	0.102579	−	0.994725i	$-0.467291\pi$
$587$	4.97056	0.205157	0.102579	+	0.994725i	$0.467291\pi$
$588$	0	0
$589$	30.1716	1.24320
$590$	0	0
$591$	31.7990	1.30804
$592$	0	0
$593$	11.4853	0.471644	0.235822	−	0.971796i	$-0.424222\pi$
$593$	11.4853	0.471644	0.235822	+	0.971796i	$0.424222\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.65685	0.272447
$598$	0	0
$599$	−17.8701	−0.730151	−0.365075	−	0.930978i	$-0.618957\pi$
$599$	−17.8701	−0.730151	−0.365075	+	0.930978i	$0.618957\pi$
$600$	0	0
$601$	2.14214	0.0873795	0.0436898	−	0.999045i	$-0.486089\pi$
$601$	2.14214	0.0873795	0.0436898	+	0.999045i	$0.486089\pi$
$602$	0	0
$603$	−31.7990	−1.29495
$604$	0	0
$605$	41.4558	1.68542
$606$	0	0
$607$	1.38478	0.0562063	0.0281032	−	0.999605i	$-0.491053\pi$
$607$	1.38478	0.0562063	0.0281032	+	0.999605i	$0.491053\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.4558	−0.868011
$612$	0	0
$613$	21.4853	0.867782	0.433891	−	0.900965i	$-0.357140\pi$
$613$	21.4853	0.867782	0.433891	+	0.900965i	$0.357140\pi$
$614$	0	0
$615$	10.8284	0.436644
$616$	0	0
$617$	−43.1127	−1.73565	−0.867826	−	0.496868i	$-0.834483\pi$
$617$	−43.1127	−1.73565	−0.867826	+	0.496868i	$0.834483\pi$
$618$	0	0
$619$	−32.0711	−1.28905	−0.644523	−	0.764585i	$-0.722944\pi$
$619$	−32.0711	−1.28905	−0.644523	+	0.764585i	$0.722944\pi$
$620$	0	0
$621$	1.34315	0.0538986
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	3.58579	0.143203
$628$	0	0
$629$	−15.4853	−0.617439
$630$	0	0
$631$	18.3431	0.730229	0.365115	−	0.930963i	$-0.381030\pi$
$631$	18.3431	0.730229	0.365115	+	0.930963i	$0.381030\pi$
$632$	0	0
$633$	28.9706	1.15148
$634$	0	0
$635$	−21.6569	−0.859426
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.62742	0.262177
$640$	0	0
$641$	−17.0000	−0.671460	−0.335730	−	0.941958i	$-0.608983\pi$
$641$	−17.0000	−0.671460	−0.335730	+	0.941958i	$0.608983\pi$
$642$	0	0
$643$	−44.9706	−1.77347	−0.886733	−	0.462282i	$-0.847031\pi$
$643$	−44.9706	−1.77347	−0.886733	+	0.462282i	$0.847031\pi$
$644$	0	0
$645$	−15.3137	−0.602977
$646$	0	0
$647$	25.8701	1.01706	0.508528	−	0.861045i	$-0.330190\pi$
$647$	25.8701	1.01706	0.508528	+	0.861045i	$0.330190\pi$
$648$	0	0
$649$	3.68629	0.144700
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.9411	1.13255	0.566277	−	0.824215i	$-0.308383\pi$
$653$	28.9411	1.13255	0.566277	+	0.824215i	$0.308383\pi$
$654$	0	0
$655$	−67.8701	−2.65190
$656$	0	0
$657$	9.45584	0.368908
$658$	0	0
$659$	−10.6274	−0.413985	−0.206993	−	0.978342i	$-0.566368\pi$
$659$	−10.6274	−0.413985	−0.206993	+	0.978342i	$0.566368\pi$
$660$	0	0
$661$	19.3431	0.752361	0.376181	−	0.926546i	$-0.377237\pi$
$661$	19.3431	0.752361	0.376181	+	0.926546i	$0.377237\pi$
$662$	0	0
$663$	−39.7990	−1.54566
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.17157	0.355125
$668$	0	0
$669$	32.9706	1.27472
$670$	0	0
$671$	1.10051	0.0424845
$672$	0	0
$673$	26.1421	1.00771	0.503853	−	0.863790i	$-0.331915\pi$
$673$	26.1421	1.00771	0.503853	+	0.863790i	$0.331915\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−20.7990	−0.799370	−0.399685	−	0.916653i	$-0.630881\pi$
$677$	−20.7990	−0.799370	−0.399685	+	0.916653i	$0.630881\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4.65685	−0.178451
$682$	0	0
$683$	−48.5563	−1.85796	−0.928979	−	0.370134i	$-0.879312\pi$
$683$	−48.5563	−1.85796	−0.928979	+	0.370134i	$0.879312\pi$
$684$	0	0
$685$	65.6274	2.50749
$686$	0	0
$687$	−23.7279	−0.905277
$688$	0	0
$689$	−2.82843	−0.107754
$690$	0	0
$691$	21.0416	0.800461	0.400231	−	0.916414i	$-0.368930\pi$
$691$	21.0416	0.800461	0.400231	+	0.916414i	$0.368930\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.0000	−1.06210
$696$	0	0
$697$	6.82843	0.258645
$698$	0	0
$699$	−18.8995	−0.714845
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	9.52691	0.359314
$704$	0	0
$705$	−70.1127	−2.64060
$706$	0	0
$707$	0	0
$708$	0	0
$709$	42.5980	1.59980	0.799900	−	0.600133i	$-0.204886\pi$
$709$	42.5980	1.59980	0.799900	+	0.600133i	$0.204886\pi$
$710$	0	0
$711$	−22.8284	−0.856133
$712$	0	0
$713$	−27.2843	−1.02180
$714$	0	0
$715$	−4.48528	−0.167740
$716$	0	0
$717$	51.4558	1.92165
$718$	0	0
$719$	17.0416	0.635546	0.317773	−	0.948167i	$-0.397065\pi$
$719$	17.0416	0.635546	0.317773	+	0.948167i	$0.397065\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−29.7279	−1.10559
$724$	0	0
$725$	27.3137	1.01441
$726$	0	0
$727$	−45.6569	−1.69332	−0.846659	−	0.532135i	$-0.821390\pi$
$727$	−45.6569	−1.69332	−0.846659	+	0.532135i	$0.821390\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	−9.65685	−0.357172
$732$	0	0
$733$	−46.3137	−1.71064	−0.855318	−	0.518104i	$-0.826638\pi$
$733$	−46.3137	−1.71064	−0.855318	+	0.518104i	$0.826638\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.65685	−0.171537
$738$	0	0
$739$	52.6985	1.93855	0.969273	−	0.245989i	$-0.0791128\pi$
$739$	52.6985	1.93855	0.969273	+	0.245989i	$0.0791128\pi$
$740$	0	0
$741$	24.4853	0.899489
$742$	0	0
$743$	37.6569	1.38150	0.690748	−	0.723096i	$-0.257281\pi$
$743$	37.6569	1.38150	0.690748	+	0.723096i	$0.257281\pi$
$744$	0	0
$745$	−45.2843	−1.65909
$746$	0	0
$747$	−43.3137	−1.58477
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−3.72792	−0.136034	−0.0680169	−	0.997684i	$-0.521667\pi$
$751$	−3.72792	−0.136034	−0.0680169	+	0.997684i	$0.521667\pi$
$752$	0	0
$753$	7.17157	0.261347
$754$	0	0
$755$	72.3553	2.63328
$756$	0	0
$757$	22.1421	0.804770	0.402385	−	0.915471i	$-0.368181\pi$
$757$	22.1421	0.804770	0.402385	+	0.915471i	$0.368181\pi$
$758$	0	0
$759$	−3.24264	−0.117700
$760$	0	0
$761$	−20.1127	−0.729085	−0.364542	−	0.931187i	$-0.618775\pi$
$761$	−20.1127	−0.729085	−0.364542	+	0.931187i	$0.618775\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−63.1127	−2.28184
$766$	0	0
$767$	25.1716	0.908893
$768$	0	0
$769$	−42.1421	−1.51968	−0.759842	−	0.650108i	$-0.774724\pi$
$769$	−42.1421	−1.51968	−0.759842	+	0.650108i	$0.774724\pi$
$770$	0	0
$771$	51.8701	1.86805
$772$	0	0
$773$	−19.1421	−0.688495	−0.344247	−	0.938879i	$-0.611866\pi$
$773$	−19.1421	−0.688495	−0.344247	+	0.938879i	$0.611866\pi$
$774$	0	0
$775$	−81.2548	−2.91876
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.20101	−0.150517
$780$	0	0
$781$	0.970563	0.0347295
$782$	0	0
$783$	1.17157	0.0418686
$784$	0	0
$785$	79.0833	2.82260
$786$	0	0
$787$	−3.78680	−0.134985	−0.0674924	−	0.997720i	$-0.521500\pi$
$787$	−3.78680	−0.134985	−0.0674924	+	0.997720i	$0.521500\pi$
$788$	0	0
$789$	21.4853	0.764896
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.51472	0.266855
$794$	0	0
$795$	−9.24264	−0.327803
$796$	0	0
$797$	15.1127	0.535319	0.267660	−	0.963514i	$-0.413750\pi$
$797$	15.1127	0.535319	0.267660	+	0.963514i	$0.413750\pi$
$798$	0	0
$799$	−44.2132	−1.56415
$800$	0	0
$801$	25.4558	0.899438
$802$	0	0
$803$	1.38478	0.0488677
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.24264	0.114146
$808$	0	0
$809$	15.9706	0.561495	0.280748	−	0.959782i	$-0.409418\pi$
$809$	15.9706	0.561495	0.280748	+	0.959782i	$0.409418\pi$
$810$	0	0
$811$	6.34315	0.222738	0.111369	−	0.993779i	$-0.464476\pi$
$811$	6.34315	0.222738	0.111369	+	0.993779i	$0.464476\pi$
$812$	0	0
$813$	29.4853	1.03409
$814$	0	0
$815$	−58.3553	−2.04410
$816$	0	0
$817$	5.94113	0.207854
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.1421	−0.458664	−0.229332	−	0.973348i	$-0.573654\pi$
$821$	−13.1421	−0.458664	−0.229332	+	0.973348i	$0.573654\pi$
$822$	0	0
$823$	34.0122	1.18559	0.592795	−	0.805353i	$-0.298024\pi$
$823$	34.0122	1.18559	0.592795	+	0.805353i	$0.298024\pi$
$824$	0	0
$825$	−9.65685	−0.336209
$826$	0	0
$827$	30.3431	1.05513	0.527567	−	0.849513i	$-0.323104\pi$
$827$	30.3431	1.05513	0.527567	+	0.849513i	$0.323104\pi$
$828$	0	0
$829$	−40.7990	−1.41701	−0.708504	−	0.705707i	$-0.750629\pi$
$829$	−40.7990	−1.41701	−0.708504	+	0.705707i	$0.750629\pi$
$830$	0	0
$831$	29.7279	1.03125
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.65685	0.264976
$836$	0	0
$837$	−3.48528	−0.120469
$838$	0	0
$839$	−19.3137	−0.666783	−0.333392	−	0.942788i	$-0.608193\pi$
$839$	−19.3137	−0.666783	−0.333392	+	0.942788i	$0.608193\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	−10.8284	−0.372951
$844$	0	0
$845$	19.1421	0.658509
$846$	0	0
$847$	0	0
$848$	0	0
$849$	20.3137	0.697165
$850$	0	0
$851$	−8.61522	−0.295326
$852$	0	0
$853$	23.5147	0.805129	0.402564	−	0.915392i	$-0.368119\pi$
$853$	23.5147	0.805129	0.402564	+	0.915392i	$0.368119\pi$
$854$	0	0
$855$	38.8284	1.32790
$856$	0	0
$857$	−36.6569	−1.25217	−0.626087	−	0.779753i	$-0.715345\pi$
$857$	−36.6569	−1.25217	−0.626087	+	0.779753i	$0.715345\pi$
$858$	0	0
$859$	−36.5563	−1.24729	−0.623643	−	0.781709i	$-0.714348\pi$
$859$	−36.5563	−1.24729	−0.623643	+	0.781709i	$0.714348\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.38478	−0.115219	−0.0576096	−	0.998339i	$-0.518348\pi$
$863$	−3.38478	−0.115219	−0.0576096	+	0.998339i	$0.518348\pi$
$864$	0	0
$865$	42.1127	1.43187
$866$	0	0
$867$	−40.9706	−1.39143
$868$	0	0
$869$	−3.34315	−0.113408
$870$	0	0
$871$	−31.7990	−1.07747
$872$	0	0
$873$	19.3137	0.653670
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33.4264	−1.12873	−0.564365	−	0.825526i	$-0.690879\pi$
$877$	−33.4264	−1.12873	−0.564365	+	0.825526i	$0.690879\pi$
$878$	0	0
$879$	−40.1421	−1.35396
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	49.2548	1.65756	0.828779	−	0.559577i	$-0.189036\pi$
$883$	49.2548	1.65756	0.828779	+	0.559577i	$0.189036\pi$
$884$	0	0
$885$	82.2548	2.76497
$886$	0	0
$887$	−50.8995	−1.70904	−0.854519	−	0.519420i	$-0.826148\pi$
$887$	−50.8995	−1.70904	−0.854519	+	0.519420i	$0.826148\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.92893	−0.131624
$892$	0	0
$893$	27.2010	0.910247
$894$	0	0
$895$	−32.2132	−1.07677
$896$	0	0
$897$	−22.1421	−0.739304
$898$	0	0
$899$	−23.7990	−0.793741
$900$	0	0
$901$	−5.82843	−0.194173
$902$	0	0
$903$	0	0
$904$	0	0
$905$	50.9706	1.69432
$906$	0	0
$907$	−39.5269	−1.31247	−0.656235	−	0.754557i	$-0.727852\pi$
$907$	−39.5269	−1.31247	−0.656235	+	0.754557i	$0.727852\pi$
$908$	0	0
$909$	9.85786	0.326965
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	−6.34315	−0.209927
$914$	0	0
$915$	24.5563	0.811808
$916$	0	0
$917$	0	0
$918$	0	0
$919$	24.5563	0.810039	0.405020	−	0.914308i	$-0.367265\pi$
$919$	24.5563	0.810039	0.405020	+	0.914308i	$0.367265\pi$
$920$	0	0
$921$	61.9411	2.04103
$922$	0	0
$923$	6.62742	0.218144
$924$	0	0
$925$	−25.6569	−0.843592
$926$	0	0
$927$	−15.7990	−0.518907
$928$	0	0
$929$	30.6569	1.00582	0.502909	−	0.864339i	$-0.332263\pi$
$929$	30.6569	1.00582	0.502909	+	0.864339i	$0.332263\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−65.7696	−2.15320
$934$	0	0
$935$	−9.24264	−0.302267
$936$	0	0
$937$	28.6274	0.935217	0.467608	−	0.883936i	$-0.345116\pi$
$937$	28.6274	0.935217	0.467608	+	0.883936i	$0.345116\pi$
$938$	0	0
$939$	65.8701	2.14959
$940$	0	0
$941$	−52.2548	−1.70346	−0.851729	−	0.523982i	$-0.824446\pi$
$941$	−52.2548	−1.70346	−0.851729	+	0.523982i	$0.824446\pi$
$942$	0	0
$943$	3.79899	0.123712
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.2132	0.786823	0.393412	−	0.919362i	$-0.371295\pi$
$947$	24.2132	0.786823	0.393412	+	0.919362i	$0.371295\pi$
$948$	0	0
$949$	9.45584	0.306950
$950$	0	0
$951$	78.0122	2.52972
$952$	0	0
$953$	35.1127	1.13741	0.568706	−	0.822541i	$-0.307444\pi$
$953$	35.1127	1.13741	0.568706	+	0.822541i	$0.307444\pi$
$954$	0	0
$955$	−41.7279	−1.35028
$956$	0	0
$957$	−2.82843	−0.0914301
$958$	0	0
$959$	0	0
$960$	0	0
$961$	39.7990	1.28384
$962$	0	0
$963$	19.5147	0.628853
$964$	0	0
$965$	−73.2843	−2.35910
$966$	0	0
$967$	−29.6569	−0.953700	−0.476850	−	0.878985i	$-0.658222\pi$
$967$	−29.6569	−0.953700	−0.476850	+	0.878985i	$0.658222\pi$
$968$	0	0
$969$	50.4558	1.62088
$970$	0	0
$971$	38.4142	1.23277	0.616385	−	0.787445i	$-0.288596\pi$
$971$	38.4142	1.23277	0.616385	+	0.787445i	$0.288596\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−65.9411	−2.11181
$976$	0	0
$977$	−51.4853	−1.64716	−0.823580	−	0.567200i	$-0.808027\pi$
$977$	−51.4853	−1.64716	−0.823580	+	0.567200i	$0.808027\pi$
$978$	0	0
$979$	3.72792	0.119145
$980$	0	0
$981$	−39.1127	−1.24877
$982$	0	0
$983$	60.8406	1.94051	0.970257	−	0.242076i	$-0.0778284\pi$
$983$	60.8406	1.94051	0.970257	+	0.242076i	$0.0778284\pi$
$984$	0	0
$985$	50.4264	1.60672
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.37258	−0.170838
$990$	0	0
$991$	14.6985	0.466913	0.233456	−	0.972367i	$-0.424996\pi$
$991$	14.6985	0.466913	0.233456	+	0.972367i	$0.424996\pi$
$992$	0	0
$993$	−61.7696	−1.96020
$994$	0	0
$995$	10.5563	0.334659
$996$	0	0
$997$	−11.2843	−0.357376	−0.178688	−	0.983906i	$-0.557185\pi$
$997$	−11.2843	−0.357376	−0.178688	+	0.983906i	$0.557185\pi$
$998$	0	0
$999$	−1.10051	−0.0348184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.a.j.1.1		2
4.3	odd	2		1568.2.a.u.1.2		2
7.2	even	3		1568.2.i.x.1537.2		4
7.3	odd	6		224.2.i.a.65.1	✓	4
7.4	even	3		1568.2.i.x.961.2		4
7.5	odd	6		224.2.i.a.193.1	yes	4
7.6	odd	2		1568.2.a.w.1.2		2
8.3	odd	2		3136.2.a.be.1.1		2
8.5	even	2		3136.2.a.bx.1.2		2
21.5	even	6		2016.2.s.q.865.2		4
21.17	even	6		2016.2.s.q.289.2		4
28.3	even	6		224.2.i.d.65.2	yes	4
28.11	odd	6		1568.2.i.o.961.1		4
28.19	even	6		224.2.i.d.193.2	yes	4
28.23	odd	6		1568.2.i.o.1537.1		4
28.27	even	2		1568.2.a.l.1.1		2
56.3	even	6		448.2.i.g.65.1		4
56.5	odd	6		448.2.i.j.193.2		4
56.13	odd	2		3136.2.a.bd.1.1		2
56.19	even	6		448.2.i.g.193.1		4
56.27	even	2		3136.2.a.bw.1.2		2
56.45	odd	6		448.2.i.j.65.2		4
84.47	odd	6		2016.2.s.s.865.2		4
84.59	odd	6		2016.2.s.s.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.i.a.65.1	✓	4	7.3	odd	6
224.2.i.a.193.1	yes	4	7.5	odd	6
224.2.i.d.65.2	yes	4	28.3	even	6
224.2.i.d.193.2	yes	4	28.19	even	6
448.2.i.g.65.1		4	56.3	even	6
448.2.i.g.193.1		4	56.19	even	6
448.2.i.j.65.2		4	56.45	odd	6
448.2.i.j.193.2		4	56.5	odd	6
1568.2.a.j.1.1		2	1.1	even	1	trivial
1568.2.a.l.1.1		2	28.27	even	2
1568.2.a.u.1.2		2	4.3	odd	2
1568.2.a.w.1.2		2	7.6	odd	2
1568.2.i.o.961.1		4	28.11	odd	6
1568.2.i.o.1537.1		4	28.23	odd	6
1568.2.i.x.961.2		4	7.4	even	3
1568.2.i.x.1537.2		4	7.2	even	3
2016.2.s.q.289.2		4	21.17	even	6
2016.2.s.q.865.2		4	21.5	even	6
2016.2.s.s.289.2		4	84.59	odd	6
2016.2.s.s.865.2		4	84.47	odd	6
3136.2.a.bd.1.1		2	56.13	odd	2
3136.2.a.be.1.1		2	8.3	odd	2
3136.2.a.bw.1.2		2	56.27	even	2
3136.2.a.bx.1.2		2	8.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{3} -3.82843 q^{5} +2.82843 q^{9} +O(q^{10})\)	\(q-2.41421 q^{3} -3.82843 q^{5} +2.82843 q^{9} +0.414214 q^{11} +2.82843 q^{13} +9.24264 q^{15} +5.82843 q^{17} -3.58579 q^{19} +3.24264 q^{23} +9.65685 q^{25} +0.414214 q^{27} +2.82843 q^{29} -8.41421 q^{31} -1.00000 q^{33} -2.65685 q^{37} -6.82843 q^{39} +1.17157 q^{41} -1.65685 q^{43} -10.8284 q^{45} -7.58579 q^{47} -14.0711 q^{51} -1.00000 q^{53} -1.58579 q^{55} +8.65685 q^{57} +8.89949 q^{59} +2.65685 q^{61} -10.8284 q^{65} -11.2426 q^{67} -7.82843 q^{69} +2.34315 q^{71} +3.34315 q^{73} -23.3137 q^{75} -8.07107 q^{79} -9.48528 q^{81} -15.3137 q^{83} -22.3137 q^{85} -6.82843 q^{87} +9.00000 q^{89} +20.3137 q^{93} +13.7279 q^{95} +6.82843 q^{97} +1.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{11} + 10 q^{15} + 6 q^{17} - 10 q^{19} - 2 q^{23} + 8 q^{25} - 2 q^{27} - 14 q^{31} - 2 q^{33} + 6 q^{37} - 8 q^{39} + 8 q^{41} + 8 q^{43} - 16 q^{45} - 18 q^{47} - 14 q^{51} - 2 q^{53} - 6 q^{55} + 6 q^{57} - 2 q^{59} - 6 q^{61} - 16 q^{65} - 14 q^{67} - 10 q^{69} + 16 q^{71} + 18 q^{73} - 24 q^{75} - 2 q^{79} - 2 q^{81} - 8 q^{83} - 22 q^{85} - 8 q^{87} + 18 q^{89} + 18 q^{93} + 2 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^11 + 10 * q^15 + 6 * q^17 - 10 * q^19 - 2 * q^23 + 8 * q^25 - 2 * q^27 - 14 * q^31 - 2 * q^33 + 6 * q^37 - 8 * q^39 + 8 * q^41 + 8 * q^43 - 16 * q^45 - 18 * q^47 - 14 * q^51 - 2 * q^53 - 6 * q^55 + 6 * q^57 - 2 * q^59 - 6 * q^61 - 16 * q^65 - 14 * q^67 - 10 * q^69 + 16 * q^71 + 18 * q^73 - 24 * q^75 - 2 * q^79 - 2 * q^81 - 8 * q^83 - 22 * q^85 - 8 * q^87 + 18 * q^89 + 18 * q^93 + 2 * q^95 + 8 * q^97 + 8 * q^99

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