LMFDB - Embedded newform 1568.2.a.j.1.2

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			1.2
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)

$f(q)$	$=$	$q+0.414214 q^{3} +1.82843 q^{5} -2.82843 q^{9} +O(q^{10})$	$q+0.414214 q^{3} +1.82843 q^{5} -2.82843 q^{9} -2.41421 q^{11} -2.82843 q^{13} +0.757359 q^{15} +0.171573 q^{17} -6.41421 q^{19} -5.24264 q^{23} -1.65685 q^{25} -2.41421 q^{27} -2.82843 q^{29} -5.58579 q^{31} -1.00000 q^{33} +8.65685 q^{37} -1.17157 q^{39} +6.82843 q^{41} +9.65685 q^{43} -5.17157 q^{45} -10.4142 q^{47} +0.0710678 q^{51} -1.00000 q^{53} -4.41421 q^{55} -2.65685 q^{57} -10.8995 q^{59} -8.65685 q^{61} -5.17157 q^{65} -2.75736 q^{67} -2.17157 q^{69} +13.6569 q^{71} +14.6569 q^{73} -0.686292 q^{75} +6.07107 q^{79} +7.48528 q^{81} +7.31371 q^{83} +0.313708 q^{85} -1.17157 q^{87} +9.00000 q^{89} -2.31371 q^{93} -11.7279 q^{95} +1.17157 q^{97} +6.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	0	0
$5$	1.82843	0.817697	0.408849	−	0.912602i	$-0.365930\pi$
$5$	1.82843	0.817697	0.408849	+	0.912602i	$0.365930\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−2.41421	−0.727913	−0.363956	−	0.931416i	$-0.618574\pi$
$11$	−2.41421	−0.727913	−0.363956	+	0.931416i	$0.618574\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	0.757359	0.195549
$16$	0	0
$17$	0.171573	0.0416125	0.0208063	−	0.999784i	$-0.493377\pi$
$17$	0.171573	0.0416125	0.0208063	+	0.999784i	$0.493377\pi$
$18$	0	0
$19$	−6.41421	−1.47152	−0.735761	−	0.677242i	$-0.763175\pi$
$19$	−6.41421	−1.47152	−0.735761	+	0.677242i	$0.763175\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.24264	−1.09317	−0.546583	−	0.837405i	$-0.684072\pi$
$23$	−5.24264	−1.09317	−0.546583	+	0.837405i	$0.684072\pi$
$24$	0	0
$25$	−1.65685	−0.331371
$26$	0	0
$27$	−2.41421	−0.464616
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	0	0
$31$	−5.58579	−1.00324	−0.501618	−	0.865089i	$-0.667262\pi$
$31$	−5.58579	−1.00324	−0.501618	+	0.865089i	$0.667262\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.65685	1.42318	0.711589	−	0.702596i	$-0.247976\pi$
$37$	8.65685	1.42318	0.711589	+	0.702596i	$0.247976\pi$
$38$	0	0
$39$	−1.17157	−0.187602
$40$	0	0
$41$	6.82843	1.06642	0.533211	−	0.845983i	$-0.320985\pi$
$41$	6.82843	1.06642	0.533211	+	0.845983i	$0.320985\pi$
$42$	0	0
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	0	0
$45$	−5.17157	−0.770933
$46$	0	0
$47$	−10.4142	−1.51907	−0.759535	−	0.650467i	$-0.774573\pi$
$47$	−10.4142	−1.51907	−0.759535	+	0.650467i	$0.774573\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.0710678	0.00995148
$52$	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
$55$	−4.41421	−0.595212
$56$	0	0
$57$	−2.65685	−0.351909
$58$	0	0
$59$	−10.8995	−1.41899	−0.709497	−	0.704709i	$-0.751078\pi$
$59$	−10.8995	−1.41899	−0.709497	+	0.704709i	$0.751078\pi$
$60$	0	0
$61$	−8.65685	−1.10840	−0.554198	−	0.832385i	$-0.686975\pi$
$61$	−8.65685	−1.10840	−0.554198	+	0.832385i	$0.686975\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.17157	−0.641455
$66$	0	0
$67$	−2.75736	−0.336865	−0.168433	−	0.985713i	$-0.553871\pi$
$67$	−2.75736	−0.336865	−0.168433	+	0.985713i	$0.553871\pi$
$68$	0	0
$69$	−2.17157	−0.261427
$70$	0	0
$71$	13.6569	1.62077	0.810385	−	0.585897i	$-0.199258\pi$
$71$	13.6569	1.62077	0.810385	+	0.585897i	$0.199258\pi$
$72$	0	0
$73$	14.6569	1.71546	0.857728	−	0.514105i	$-0.171876\pi$
$73$	14.6569	1.71546	0.857728	+	0.514105i	$0.171876\pi$
$74$	0	0
$75$	−0.686292	−0.0792461
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.07107	0.683048	0.341524	−	0.939873i	$-0.389057\pi$
$79$	6.07107	0.683048	0.341524	+	0.939873i	$0.389057\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	7.31371	0.802784	0.401392	−	0.915906i	$-0.368527\pi$
$83$	7.31371	0.802784	0.401392	+	0.915906i	$0.368527\pi$
$84$	0	0
$85$	0.313708	0.0340265
$86$	0	0
$87$	−1.17157	−0.125606
$88$	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
$93$	−2.31371	−0.239920
$94$	0	0
$95$	−11.7279	−1.20326
$96$	0	0
$97$	1.17157	0.118955	0.0594776	−	0.998230i	$-0.481057\pi$
$97$	1.17157	0.118955	0.0594776	+	0.998230i	$0.481057\pi$
$98$	0	0
$99$	6.82843	0.686283
$100$	0	0
$101$	−13.4853	−1.34184	−0.670918	−	0.741532i	$-0.734100\pi$
$101$	−13.4853	−1.34184	−0.670918	+	0.741532i	$0.734100\pi$
$102$	0	0
$103$	−8.41421	−0.829077	−0.414539	−	0.910032i	$-0.636057\pi$
$103$	−8.41421	−0.829077	−0.414539	+	0.910032i	$0.636057\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.8995	−1.24704	−0.623521	−	0.781807i	$-0.714298\pi$
$107$	−12.8995	−1.24704	−0.623521	+	0.781807i	$0.714298\pi$
$108$	0	0
$109$	−8.17157	−0.782695	−0.391347	−	0.920243i	$-0.627991\pi$
$109$	−8.17157	−0.782695	−0.391347	+	0.920243i	$0.627991\pi$
$110$	0	0
$111$	3.58579	0.340348
$112$	0	0
$113$	18.1421	1.70667	0.853334	−	0.521364i	$-0.174577\pi$
$113$	18.1421	1.70667	0.853334	+	0.521364i	$0.174577\pi$
$114$	0	0
$115$	−9.58579	−0.893879
$116$	0	0
$117$	8.00000	0.739600
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.17157	−0.470143
$122$	0	0
$123$	2.82843	0.255031
$124$	0	0
$125$	−12.1716	−1.08866
$126$	0	0
$127$	−5.65685	−0.501965	−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685	−0.501965	−0.250982	+	0.967992i	$0.580754\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−7.72792	−0.675192	−0.337596	−	0.941291i	$-0.609614\pi$
$131$	−7.72792	−0.675192	−0.337596	+	0.941291i	$0.609614\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.41421	−0.379915
$136$	0	0
$137$	11.1421	0.951937	0.475968	−	0.879462i	$-0.342098\pi$
$137$	11.1421	0.951937	0.475968	+	0.879462i	$0.342098\pi$
$138$	0	0
$139$	−15.3137	−1.29889	−0.649446	−	0.760408i	$-0.724999\pi$
$139$	−15.3137	−1.29889	−0.649446	+	0.760408i	$0.724999\pi$
$140$	0	0
$141$	−4.31371	−0.363280
$142$	0	0
$143$	6.82843	0.571022
$144$	0	0
$145$	−5.17157	−0.429476
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.17157	0.505595	0.252797	−	0.967519i	$-0.418649\pi$
$149$	6.17157	0.505595	0.252797	+	0.967519i	$0.418649\pi$
$150$	0	0
$151$	0.899495	0.0731999	0.0365999	−	0.999330i	$-0.488347\pi$
$151$	0.899495	0.0731999	0.0365999	+	0.999330i	$0.488347\pi$
$152$	0	0
$153$	−0.485281	−0.0392327
$154$	0	0
$155$	−10.2132	−0.820344
$156$	0	0
$157$	−9.34315	−0.745664	−0.372832	−	0.927899i	$-0.621613\pi$
$157$	−9.34315	−0.745664	−0.372832	+	0.927899i	$0.621613\pi$
$158$	0	0
$159$	−0.414214	−0.0328493
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.75736	0.529277	0.264639	−	0.964348i	$-0.414747\pi$
$163$	6.75736	0.529277	0.264639	+	0.964348i	$0.414747\pi$
$164$	0	0
$165$	−1.82843	−0.142343
$166$	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
$171$	18.1421	1.38736
$172$	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
$177$	−4.51472	−0.339347
$178$	0	0
$179$	5.58579	0.417501	0.208751	−	0.977969i	$-0.433060\pi$
$179$	5.58579	0.417501	0.208751	+	0.977969i	$0.433060\pi$
$180$	0	0
$181$	9.31371	0.692283	0.346141	−	0.938182i	$-0.387492\pi$
$181$	9.31371	0.692283	0.346141	+	0.938182i	$0.387492\pi$
$182$	0	0
$183$	−3.58579	−0.265069
$184$	0	0
$185$	15.8284	1.16373
$186$	0	0
$187$	−0.414214	−0.0302903
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.89949	−0.643945	−0.321972	−	0.946749i	$-0.604346\pi$
$191$	−8.89949	−0.643945	−0.321972	+	0.946749i	$0.604346\pi$
$192$	0	0
$193$	−9.14214	−0.658065	−0.329033	−	0.944319i	$-0.606723\pi$
$193$	−9.14214	−0.658065	−0.329033	+	0.944319i	$0.606723\pi$
$194$	0	0
$195$	−2.14214	−0.153402
$196$	0	0
$197$	−18.8284	−1.34147	−0.670735	−	0.741697i	$-0.734021\pi$
$197$	−18.8284	−1.34147	−0.670735	+	0.741697i	$0.734021\pi$
$198$	0	0
$199$	−11.2426	−0.796970	−0.398485	−	0.917175i	$-0.630464\pi$
$199$	−11.2426	−0.796970	−0.398485	+	0.917175i	$0.630464\pi$
$200$	0	0
$201$	−1.14214	−0.0805600
$202$	0	0
$203$	0	0
$204$	0	0
$205$	12.4853	0.872010
$206$	0	0
$207$	14.8284	1.03065
$208$	0	0
$209$	15.4853	1.07114
$210$	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
$215$	17.6569	1.20419
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.07107	0.410245
$220$	0	0
$221$	−0.485281	−0.0326436
$222$	0	0
$223$	−2.34315	−0.156909	−0.0784543	−	0.996918i	$-0.524998\pi$
$223$	−2.34315	−0.156909	−0.0784543	+	0.996918i	$0.524998\pi$
$224$	0	0
$225$	4.68629	0.312419
$226$	0	0
$227$	16.0711	1.06667	0.533337	−	0.845903i	$-0.320938\pi$
$227$	16.0711	1.06667	0.533337	+	0.845903i	$0.320938\pi$
$228$	0	0
$229$	4.17157	0.275665	0.137833	−	0.990456i	$-0.455986\pi$
$229$	4.17157	0.275665	0.137833	+	0.990456i	$0.455986\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.17157	0.142264	0.0711322	−	0.997467i	$-0.477339\pi$
$233$	2.17157	0.142264	0.0711322	+	0.997467i	$0.477339\pi$
$234$	0	0
$235$	−19.0416	−1.24214
$236$	0	0
$237$	2.51472	0.163349
$238$	0	0
$239$	1.31371	0.0849767	0.0424884	−	0.999097i	$-0.486471\pi$
$239$	1.31371	0.0849767	0.0424884	+	0.999097i	$0.486471\pi$
$240$	0	0
$241$	−10.3137	−0.664364	−0.332182	−	0.943215i	$-0.607785\pi$
$241$	−10.3137	−0.664364	−0.332182	+	0.943215i	$0.607785\pi$
$242$	0	0
$243$	10.3431	0.663513
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.1421	1.15436
$248$	0	0
$249$	3.02944	0.191983
$250$	0	0
$251$	30.9706	1.95484	0.977422	−	0.211295i	$-0.0677681\pi$
$251$	30.9706	1.95484	0.977422	+	0.211295i	$0.0677681\pi$
$252$	0	0
$253$	12.6569	0.795730
$254$	0	0
$255$	0.129942	0.00813730
$256$	0	0
$257$	−4.51472	−0.281620	−0.140810	−	0.990037i	$-0.544971\pi$
$257$	−4.51472	−0.281620	−0.140810	+	0.990037i	$0.544971\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	10.8995	0.672092	0.336046	−	0.941846i	$-0.390910\pi$
$263$	10.8995	0.672092	0.336046	+	0.941846i	$0.390910\pi$
$264$	0	0
$265$	−1.82843	−0.112319
$266$	0	0
$267$	3.72792	0.228145
$268$	0	0
$269$	−12.6569	−0.771702	−0.385851	−	0.922561i	$-0.626092\pi$
$269$	−12.6569	−0.771702	−0.385851	+	0.922561i	$0.626092\pi$
$270$	0	0
$271$	30.2132	1.83532	0.917661	−	0.397365i	$-0.130075\pi$
$271$	30.2132	1.83532	0.917661	+	0.397365i	$0.130075\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	10.3137	0.619691	0.309845	−	0.950787i	$-0.399723\pi$
$277$	10.3137	0.619691	0.309845	+	0.950787i	$0.399723\pi$
$278$	0	0
$279$	15.7990	0.945861
$280$	0	0
$281$	−12.4853	−0.744809	−0.372405	−	0.928070i	$-0.621467\pi$
$281$	−12.4853	−0.744809	−0.372405	+	0.928070i	$0.621467\pi$
$282$	0	0
$283$	−5.58579	−0.332041	−0.166020	−	0.986122i	$-0.553092\pi$
$283$	−5.58579	−0.332041	−0.166020	+	0.986122i	$0.553092\pi$
$284$	0	0
$285$	−4.85786	−0.287755
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9706	−0.998268
$290$	0	0
$291$	0.485281	0.0284477
$292$	0	0
$293$	−28.6274	−1.67243	−0.836216	−	0.548401i	$-0.815237\pi$
$293$	−28.6274	−1.67243	−0.836216	+	0.548401i	$0.815237\pi$
$294$	0	0
$295$	−19.9289	−1.16031
$296$	0	0
$297$	5.82843	0.338200
$298$	0	0
$299$	14.8284	0.857550
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−5.58579	−0.320895
$304$	0	0
$305$	−15.8284	−0.906333
$306$	0	0
$307$	−14.3431	−0.818607	−0.409303	−	0.912398i	$-0.634228\pi$
$307$	−14.3431	−0.818607	−0.409303	+	0.912398i	$0.634228\pi$
$308$	0	0
$309$	−3.48528	−0.198271
$310$	0	0
$311$	18.7574	1.06363	0.531816	−	0.846860i	$-0.321510\pi$
$311$	18.7574	1.06363	0.531816	+	0.846860i	$0.321510\pi$
$312$	0	0
$313$	29.2843	1.65524	0.827622	−	0.561285i	$-0.189693\pi$
$313$	29.2843	1.65524	0.827622	+	0.561285i	$0.189693\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.68629	−0.544036	−0.272018	−	0.962292i	$-0.587691\pi$
$317$	−9.68629	−0.544036	−0.272018	+	0.962292i	$0.587691\pi$
$318$	0	0
$319$	6.82843	0.382319
$320$	0	0
$321$	−5.34315	−0.298225
$322$	0	0
$323$	−1.10051	−0.0612337
$324$	0	0
$325$	4.68629	0.259949
$326$	0	0
$327$	−3.38478	−0.187179
$328$	0	0
$329$	0	0
$330$	0	0
$331$	28.4142	1.56179	0.780893	−	0.624665i	$-0.214764\pi$
$331$	28.4142	1.56179	0.780893	+	0.624665i	$0.214764\pi$
$332$	0	0
$333$	−24.4853	−1.34179
$334$	0	0
$335$	−5.04163	−0.275454
$336$	0	0
$337$	9.17157	0.499607	0.249804	−	0.968296i	$-0.419634\pi$
$337$	9.17157	0.499607	0.249804	+	0.968296i	$0.419634\pi$
$338$	0	0
$339$	7.51472	0.408143
$340$	0	0
$341$	13.4853	0.730269
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.97056	−0.213768
$346$	0	0
$347$	34.8995	1.87350	0.936752	−	0.349995i	$-0.113817\pi$
$347$	34.8995	1.87350	0.936752	+	0.349995i	$0.113817\pi$
$348$	0	0
$349$	−5.17157	−0.276828	−0.138414	−	0.990374i	$-0.544200\pi$
$349$	−5.17157	−0.276828	−0.138414	+	0.990374i	$0.544200\pi$
$350$	0	0
$351$	6.82843	0.364474
$352$	0	0
$353$	−2.17157	−0.115581	−0.0577906	−	0.998329i	$-0.518406\pi$
$353$	−2.17157	−0.115581	−0.0577906	+	0.998329i	$0.518406\pi$
$354$	0	0
$355$	24.9706	1.32530
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.3848	0.600866	0.300433	−	0.953803i	$-0.402869\pi$
$359$	11.3848	0.600866	0.300433	+	0.953803i	$0.402869\pi$
$360$	0	0
$361$	22.1421	1.16538
$362$	0	0
$363$	−2.14214	−0.112433
$364$	0	0
$365$	26.7990	1.40272
$366$	0	0
$367$	−4.27208	−0.223001	−0.111500	−	0.993764i	$-0.535566\pi$
$367$	−4.27208	−0.223001	−0.111500	+	0.993764i	$0.535566\pi$
$368$	0	0
$369$	−19.3137	−1.00543
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8.31371	−0.430468	−0.215234	−	0.976563i	$-0.569051\pi$
$373$	−8.31371	−0.430468	−0.215234	+	0.976563i	$0.569051\pi$
$374$	0	0
$375$	−5.04163	−0.260349
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−34.9706	−1.79632	−0.898159	−	0.439672i	$-0.855095\pi$
$379$	−34.9706	−1.79632	−0.898159	+	0.439672i	$0.855095\pi$
$380$	0	0
$381$	−2.34315	−0.120043
$382$	0	0
$383$	11.2426	0.574472	0.287236	−	0.957860i	$-0.407264\pi$
$383$	11.2426	0.574472	0.287236	+	0.957860i	$0.407264\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−27.3137	−1.38843
$388$	0	0
$389$	18.1716	0.921335	0.460668	−	0.887573i	$-0.347610\pi$
$389$	18.1716	0.921335	0.460668	+	0.887573i	$0.347610\pi$
$390$	0	0
$391$	−0.899495	−0.0454894
$392$	0	0
$393$	−3.20101	−0.161470
$394$	0	0
$395$	11.1005	0.558527
$396$	0	0
$397$	4.17157	0.209365	0.104683	−	0.994506i	$-0.466617\pi$
$397$	4.17157	0.209365	0.104683	+	0.994506i	$0.466617\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.82843	0.390933	0.195466	−	0.980710i	$-0.437378\pi$
$401$	7.82843	0.390933	0.195466	+	0.980710i	$0.437378\pi$
$402$	0	0
$403$	15.7990	0.787004
$404$	0	0
$405$	13.6863	0.680077
$406$	0	0
$407$	−20.8995	−1.03595
$408$	0	0
$409$	−14.4558	−0.714795	−0.357398	−	0.933952i	$-0.616336\pi$
$409$	−14.4558	−0.714795	−0.357398	+	0.933952i	$0.616336\pi$
$410$	0	0
$411$	4.61522	0.227652
$412$	0	0
$413$	0	0
$414$	0	0
$415$	13.3726	0.656434
$416$	0	0
$417$	−6.34315	−0.310625
$418$	0	0
$419$	−9.65685	−0.471768	−0.235884	−	0.971781i	$-0.575799\pi$
$419$	−9.65685	−0.471768	−0.235884	+	0.971781i	$0.575799\pi$
$420$	0	0
$421$	−16.4853	−0.803443	−0.401722	−	0.915762i	$-0.631588\pi$
$421$	−16.4853	−0.803443	−0.401722	+	0.915762i	$0.631588\pi$
$422$	0	0
$423$	29.4558	1.43219
$424$	0	0
$425$	−0.284271	−0.0137892
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.82843	0.136558
$430$	0	0
$431$	−12.7574	−0.614500	−0.307250	−	0.951629i	$-0.599409\pi$
$431$	−12.7574	−0.614500	−0.307250	+	0.951629i	$0.599409\pi$
$432$	0	0
$433$	−11.5147	−0.553362	−0.276681	−	0.960962i	$-0.589235\pi$
$433$	−11.5147	−0.553362	−0.276681	+	0.960962i	$0.589235\pi$
$434$	0	0
$435$	−2.14214	−0.102708
$436$	0	0
$437$	33.6274	1.60862
$438$	0	0
$439$	−22.6985	−1.08334	−0.541670	−	0.840591i	$-0.682208\pi$
$439$	−22.6985	−1.08334	−0.541670	+	0.840591i	$0.682208\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11.9289	−0.566761	−0.283380	−	0.959008i	$-0.591456\pi$
$443$	−11.9289	−0.566761	−0.283380	+	0.959008i	$0.591456\pi$
$444$	0	0
$445$	16.4558	0.780082
$446$	0	0
$447$	2.55635	0.120911
$448$	0	0
$449$	1.17157	0.0552899	0.0276450	−	0.999618i	$-0.491199\pi$
$449$	1.17157	0.0552899	0.0276450	+	0.999618i	$0.491199\pi$
$450$	0	0
$451$	−16.4853	−0.776262
$452$	0	0
$453$	0.372583	0.0175055
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.2843	0.902080	0.451040	−	0.892504i	$-0.351053\pi$
$457$	19.2843	0.902080	0.451040	+	0.892504i	$0.351053\pi$
$458$	0	0
$459$	−0.414214	−0.0193338
$460$	0	0
$461$	25.4558	1.18560	0.592798	−	0.805351i	$-0.298023\pi$
$461$	25.4558	1.18560	0.592798	+	0.805351i	$0.298023\pi$
$462$	0	0
$463$	−11.3137	−0.525793	−0.262896	−	0.964824i	$-0.584678\pi$
$463$	−11.3137	−0.525793	−0.262896	+	0.964824i	$0.584678\pi$
$464$	0	0
$465$	−4.23045	−0.196182
$466$	0	0
$467$	12.5563	0.581039	0.290519	−	0.956869i	$-0.406172\pi$
$467$	12.5563	0.581039	0.290519	+	0.956869i	$0.406172\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.87006	−0.178323
$472$	0	0
$473$	−23.3137	−1.07197
$474$	0	0
$475$	10.6274	0.487619
$476$	0	0
$477$	2.82843	0.129505
$478$	0	0
$479$	28.6985	1.31127	0.655634	−	0.755079i	$-0.272402\pi$
$479$	28.6985	1.31127	0.655634	+	0.755079i	$0.272402\pi$
$480$	0	0
$481$	−24.4853	−1.11643
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.14214	0.0972694
$486$	0	0
$487$	−11.7279	−0.531443	−0.265721	−	0.964050i	$-0.585610\pi$
$487$	−11.7279	−0.531443	−0.265721	+	0.964050i	$0.585610\pi$
$488$	0	0
$489$	2.79899	0.126575
$490$	0	0
$491$	3.65685	0.165032	0.0825158	−	0.996590i	$-0.473705\pi$
$491$	3.65685	0.165032	0.0825158	+	0.996590i	$0.473705\pi$
$492$	0	0
$493$	−0.485281	−0.0218560
$494$	0	0
$495$	12.4853	0.561172
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.55635	−0.203970	−0.101985	−	0.994786i	$-0.532519\pi$
$499$	−4.55635	−0.203970	−0.101985	+	0.994786i	$0.532519\pi$
$500$	0	0
$501$	−0.828427	−0.0370114
$502$	0	0
$503$	10.3431	0.461178	0.230589	−	0.973051i	$-0.425935\pi$
$503$	10.3431	0.461178	0.230589	+	0.973051i	$0.425935\pi$
$504$	0	0
$505$	−24.6569	−1.09722
$506$	0	0
$507$	−2.07107	−0.0919794
$508$	0	0
$509$	−41.4853	−1.83880	−0.919401	−	0.393321i	$-0.871326\pi$
$509$	−41.4853	−1.83880	−0.919401	+	0.393321i	$0.871326\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	15.4853	0.683692
$514$	0	0
$515$	−15.3848	−0.677934
$516$	0	0
$517$	25.1421	1.10575
$518$	0	0
$519$	−4.55635	−0.200002
$520$	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
$523$	−17.7279	−0.775188	−0.387594	−	0.921830i	$-0.626694\pi$
$523$	−17.7279	−0.775188	−0.387594	+	0.921830i	$0.626694\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.958369	−0.0417472
$528$	0	0
$529$	4.48528	0.195012
$530$	0	0
$531$	30.8284	1.33784
$532$	0	0
$533$	−19.3137	−0.836570
$534$	0	0
$535$	−23.5858	−1.01970
$536$	0	0
$537$	2.31371	0.0998439
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−25.8284	−1.11045	−0.555225	−	0.831700i	$-0.687368\pi$
$541$	−25.8284	−1.11045	−0.555225	+	0.831700i	$0.687368\pi$
$542$	0	0
$543$	3.85786	0.165557
$544$	0	0
$545$	−14.9411	−0.640007
$546$	0	0
$547$	10.9706	0.469067	0.234534	−	0.972108i	$-0.424644\pi$
$547$	10.9706	0.469067	0.234534	+	0.972108i	$0.424644\pi$
$548$	0	0
$549$	24.4853	1.04501
$550$	0	0
$551$	18.1421	0.772881
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.55635	0.278302
$556$	0	0
$557$	−0.171573	−0.00726978	−0.00363489	−	0.999993i	$-0.501157\pi$
$557$	−0.171573	−0.00726978	−0.00363489	+	0.999993i	$0.501157\pi$
$558$	0	0
$559$	−27.3137	−1.15525
$560$	0	0
$561$	−0.171573	−0.00724381
$562$	0	0
$563$	14.0711	0.593025	0.296512	−	0.955029i	$-0.404176\pi$
$563$	14.0711	0.593025	0.296512	+	0.955029i	$0.404176\pi$
$564$	0	0
$565$	33.1716	1.39554
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.6569	1.53674	0.768368	−	0.640009i	$-0.221069\pi$
$569$	36.6569	1.53674	0.768368	+	0.640009i	$0.221069\pi$
$570$	0	0
$571$	−18.4142	−0.770611	−0.385305	−	0.922789i	$-0.625904\pi$
$571$	−18.4142	−0.770611	−0.385305	+	0.922789i	$0.625904\pi$
$572$	0	0
$573$	−3.68629	−0.153997
$574$	0	0
$575$	8.68629	0.362243
$576$	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
$579$	−3.78680	−0.157374
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.41421	0.0999865
$584$	0	0
$585$	14.6274	0.604769
$586$	0	0
$587$	−28.9706	−1.19574	−0.597872	−	0.801592i	$-0.703987\pi$
$587$	−28.9706	−1.19574	−0.597872	+	0.801592i	$0.703987\pi$
$588$	0	0
$589$	35.8284	1.47628
$590$	0	0
$591$	−7.79899	−0.320808
$592$	0	0
$593$	−5.48528	−0.225254	−0.112627	−	0.993637i	$-0.535926\pi$
$593$	−5.48528	−0.225254	−0.112627	+	0.993637i	$0.535926\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.65685	−0.190592
$598$	0	0
$599$	35.8701	1.46561	0.732805	−	0.680438i	$-0.238211\pi$
$599$	35.8701	1.46561	0.732805	+	0.680438i	$0.238211\pi$
$600$	0	0
$601$	−26.1421	−1.06636	−0.533180	−	0.846002i	$-0.679003\pi$
$601$	−26.1421	−1.06636	−0.533180	+	0.846002i	$0.679003\pi$
$602$	0	0
$603$	7.79899	0.317599
$604$	0	0
$605$	−9.45584	−0.384435
$606$	0	0
$607$	−35.3848	−1.43622	−0.718112	−	0.695928i	$-0.754993\pi$
$607$	−35.3848	−1.43622	−0.718112	+	0.695928i	$0.754993\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.4558	1.19166
$612$	0	0
$613$	4.51472	0.182348	0.0911739	−	0.995835i	$-0.470938\pi$
$613$	4.51472	0.182348	0.0911739	+	0.995835i	$0.470938\pi$
$614$	0	0
$615$	5.17157	0.208538
$616$	0	0
$617$	19.1127	0.769448	0.384724	−	0.923032i	$-0.374297\pi$
$617$	19.1127	0.769448	0.384724	+	0.923032i	$0.374297\pi$
$618$	0	0
$619$	−17.9289	−0.720625	−0.360312	−	0.932832i	$-0.617330\pi$
$619$	−17.9289	−0.720625	−0.360312	+	0.932832i	$0.617330\pi$
$620$	0	0
$621$	12.6569	0.507902
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	6.41421	0.256159
$628$	0	0
$629$	1.48528	0.0592220
$630$	0	0
$631$	29.6569	1.18062	0.590310	−	0.807176i	$-0.299005\pi$
$631$	29.6569	1.18062	0.590310	+	0.807176i	$0.299005\pi$
$632$	0	0
$633$	−4.97056	−0.197562
$634$	0	0
$635$	−10.3431	−0.410455
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−38.6274	−1.52808
$640$	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
$643$	−11.0294	−0.434959	−0.217479	−	0.976065i	$-0.569783\pi$
$643$	−11.0294	−0.434959	−0.217479	+	0.976065i	$0.569783\pi$
$644$	0	0
$645$	7.31371	0.287977
$646$	0	0
$647$	−27.8701	−1.09569	−0.547843	−	0.836581i	$-0.684551\pi$
$647$	−27.8701	−1.09569	−0.547843	+	0.836581i	$0.684551\pi$
$648$	0	0
$649$	26.3137	1.03290
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−38.9411	−1.52388	−0.761942	−	0.647645i	$-0.775754\pi$
$653$	−38.9411	−1.52388	−0.761942	+	0.647645i	$0.775754\pi$
$654$	0	0
$655$	−14.1299	−0.552103
$656$	0	0
$657$	−41.4558	−1.61735
$658$	0	0
$659$	34.6274	1.34889	0.674446	−	0.738324i	$-0.264382\pi$
$659$	34.6274	1.34889	0.674446	+	0.738324i	$0.264382\pi$
$660$	0	0
$661$	30.6569	1.19241	0.596207	−	0.802831i	$-0.296674\pi$
$661$	30.6569	1.19241	0.596207	+	0.802831i	$0.296674\pi$
$662$	0	0
$663$	−0.201010	−0.00780659
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.8284	0.574159
$668$	0	0
$669$	−0.970563	−0.0375241
$670$	0	0
$671$	20.8995	0.806816
$672$	0	0
$673$	−2.14214	−0.0825733	−0.0412866	−	0.999147i	$-0.513146\pi$
$673$	−2.14214	−0.0825733	−0.0412866	+	0.999147i	$0.513146\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	18.7990	0.722504	0.361252	−	0.932468i	$-0.382349\pi$
$677$	18.7990	0.722504	0.361252	+	0.932468i	$0.382349\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6.65685	0.255091
$682$	0	0
$683$	−17.4437	−0.667463	−0.333731	−	0.942668i	$-0.608308\pi$
$683$	−17.4437	−0.667463	−0.333731	+	0.942668i	$0.608308\pi$
$684$	0	0
$685$	20.3726	0.778396
$686$	0	0
$687$	1.72792	0.0659243
$688$	0	0
$689$	2.82843	0.107754
$690$	0	0
$691$	−27.0416	−1.02871	−0.514356	−	0.857577i	$-0.671969\pi$
$691$	−27.0416	−1.02871	−0.514356	+	0.857577i	$0.671969\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.0000	−1.06210
$696$	0	0
$697$	1.17157	0.0443765
$698$	0	0
$699$	0.899495	0.0340220
$700$	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
$703$	−55.5269	−2.09424
$704$	0	0
$705$	−7.88730	−0.297053
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36.5980	−1.37447	−0.687233	−	0.726437i	$-0.741175\pi$
$709$	−36.5980	−1.37447	−0.687233	+	0.726437i	$0.741175\pi$
$710$	0	0
$711$	−17.1716	−0.643984
$712$	0	0
$713$	29.2843	1.09670
$714$	0	0
$715$	12.4853	0.466923
$716$	0	0
$717$	0.544156	0.0203219
$718$	0	0
$719$	−31.0416	−1.15766	−0.578829	−	0.815449i	$-0.696490\pi$
$719$	−31.0416	−1.15766	−0.578829	+	0.815449i	$0.696490\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.27208	−0.158880
$724$	0	0
$725$	4.68629	0.174044
$726$	0	0
$727$	−34.3431	−1.27372	−0.636858	−	0.770981i	$-0.719766\pi$
$727$	−34.3431	−1.27372	−0.636858	+	0.770981i	$0.719766\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	1.65685	0.0612810
$732$	0	0
$733$	−23.6863	−0.874873	−0.437437	−	0.899249i	$-0.644114\pi$
$733$	−23.6863	−0.874873	−0.437437	+	0.899249i	$0.644114\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.65685	0.245208
$738$	0	0
$739$	−6.69848	−0.246408	−0.123204	−	0.992381i	$-0.539317\pi$
$739$	−6.69848	−0.246408	−0.123204	+	0.992381i	$0.539317\pi$
$740$	0	0
$741$	7.51472	0.276060
$742$	0	0
$743$	26.3431	0.966436	0.483218	−	0.875500i	$-0.339468\pi$
$743$	26.3431	0.966436	0.483218	+	0.875500i	$0.339468\pi$
$744$	0	0
$745$	11.2843	0.413424
$746$	0	0
$747$	−20.6863	−0.756872
$748$	0	0
$749$	0	0
$750$	0	0
$751$	21.7279	0.792863	0.396432	−	0.918064i	$-0.370248\pi$
$751$	21.7279	0.792863	0.396432	+	0.918064i	$0.370248\pi$
$752$	0	0
$753$	12.8284	0.467494
$754$	0	0
$755$	1.64466	0.0598553
$756$	0	0
$757$	−6.14214	−0.223240	−0.111620	−	0.993751i	$-0.535604\pi$
$757$	−6.14214	−0.223240	−0.111620	+	0.993751i	$0.535604\pi$
$758$	0	0
$759$	5.24264	0.190296
$760$	0	0
$761$	42.1127	1.52658	0.763292	−	0.646054i	$-0.223582\pi$
$761$	42.1127	1.52658	0.763292	+	0.646054i	$0.223582\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.887302	−0.0320805
$766$	0	0
$767$	30.8284	1.11315
$768$	0	0
$769$	−13.8579	−0.499727	−0.249864	−	0.968281i	$-0.580386\pi$
$769$	−13.8579	−0.499727	−0.249864	+	0.968281i	$0.580386\pi$
$770$	0	0
$771$	−1.87006	−0.0673485
$772$	0	0
$773$	9.14214	0.328820	0.164410	−	0.986392i	$-0.447428\pi$
$773$	9.14214	0.328820	0.164410	+	0.986392i	$0.447428\pi$
$774$	0	0
$775$	9.25483	0.332443
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−43.7990	−1.56926
$780$	0	0
$781$	−32.9706	−1.17978
$782$	0	0
$783$	6.82843	0.244028
$784$	0	0
$785$	−17.0833	−0.609728
$786$	0	0
$787$	−46.2132	−1.64732	−0.823661	−	0.567082i	$-0.808072\pi$
$787$	−46.2132	−1.64732	−0.823661	+	0.567082i	$0.808072\pi$
$788$	0	0
$789$	4.51472	0.160728
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.4853	0.869498
$794$	0	0
$795$	−0.757359	−0.0268608
$796$	0	0
$797$	−47.1127	−1.66882	−0.834409	−	0.551146i	$-0.814191\pi$
$797$	−47.1127	−1.66882	−0.834409	+	0.551146i	$0.814191\pi$
$798$	0	0
$799$	−1.78680	−0.0632123
$800$	0	0
$801$	−25.4558	−0.899438
$802$	0	0
$803$	−35.3848	−1.24870
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.24264	−0.184550
$808$	0	0
$809$	−17.9706	−0.631811	−0.315906	−	0.948791i	$-0.602308\pi$
$809$	−17.9706	−0.631811	−0.315906	+	0.948791i	$0.602308\pi$
$810$	0	0
$811$	17.6569	0.620016	0.310008	−	0.950734i	$-0.399668\pi$
$811$	17.6569	0.620016	0.310008	+	0.950734i	$0.399668\pi$
$812$	0	0
$813$	12.5147	0.438910
$814$	0	0
$815$	12.3553	0.432789
$816$	0	0
$817$	−61.9411	−2.16705
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.1421	0.528464	0.264232	−	0.964459i	$-0.414882\pi$
$821$	15.1421	0.528464	0.264232	+	0.964459i	$0.414882\pi$
$822$	0	0
$823$	−48.0122	−1.67360	−0.836800	−	0.547509i	$-0.815576\pi$
$823$	−48.0122	−1.67360	−0.836800	+	0.547509i	$0.815576\pi$
$824$	0	0
$825$	1.65685	0.0576843
$826$	0	0
$827$	41.6569	1.44855	0.724275	−	0.689511i	$-0.242174\pi$
$827$	41.6569	1.44855	0.724275	+	0.689511i	$0.242174\pi$
$828$	0	0
$829$	−1.20101	−0.0417128	−0.0208564	−	0.999782i	$-0.506639\pi$
$829$	−1.20101	−0.0417128	−0.0208564	+	0.999782i	$0.506639\pi$
$830$	0	0
$831$	4.27208	0.148197
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3.65685	−0.126551
$836$	0	0
$837$	13.4853	0.466120
$838$	0	0
$839$	3.31371	0.114402	0.0572010	−	0.998363i	$-0.481782\pi$
$839$	3.31371	0.114402	0.0572010	+	0.998363i	$0.481782\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	−5.17157	−0.178118
$844$	0	0
$845$	−9.14214	−0.314499
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.31371	−0.0794063
$850$	0	0
$851$	−45.3848	−1.55577
$852$	0	0
$853$	40.4853	1.38619	0.693095	−	0.720846i	$-0.256247\pi$
$853$	40.4853	1.38619	0.693095	+	0.720846i	$0.256247\pi$
$854$	0	0
$855$	33.1716	1.13444
$856$	0	0
$857$	−25.3431	−0.865705	−0.432853	−	0.901465i	$-0.642493\pi$
$857$	−25.3431	−0.865705	−0.432853	+	0.901465i	$0.642493\pi$
$858$	0	0
$859$	−5.44365	−0.185735	−0.0928675	−	0.995678i	$-0.529603\pi$
$859$	−5.44365	−0.185735	−0.0928675	+	0.995678i	$0.529603\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.3848	1.13643	0.568216	−	0.822880i	$-0.307634\pi$
$863$	33.3848	1.13643	0.568216	+	0.822880i	$0.307634\pi$
$864$	0	0
$865$	−20.1127	−0.683852
$866$	0	0
$867$	−7.02944	−0.238732
$868$	0	0
$869$	−14.6569	−0.497200
$870$	0	0
$871$	7.79899	0.264259
$872$	0	0
$873$	−3.31371	−0.112152
$874$	0	0
$875$	0	0
$876$	0	0
$877$	51.4264	1.73655	0.868273	−	0.496086i	$-0.165230\pi$
$877$	51.4264	1.73655	0.868273	+	0.496086i	$0.165230\pi$
$878$	0	0
$879$	−11.8579	−0.399956
$880$	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
$883$	−41.2548	−1.38834	−0.694168	−	0.719813i	$-0.744227\pi$
$883$	−41.2548	−1.38834	−0.694168	+	0.719813i	$0.744227\pi$
$884$	0	0
$885$	−8.25483	−0.277483
$886$	0	0
$887$	−31.1005	−1.04425	−0.522126	−	0.852868i	$-0.674861\pi$
$887$	−31.1005	−1.04425	−0.522126	+	0.852868i	$0.674861\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−18.0711	−0.605404
$892$	0	0
$893$	66.7990	2.23534
$894$	0	0
$895$	10.2132	0.341390
$896$	0	0
$897$	6.14214	0.205080
$898$	0	0
$899$	15.7990	0.526926
$900$	0	0
$901$	−0.171573	−0.00571592
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.0294	0.566078
$906$	0	0
$907$	25.5269	0.847607	0.423804	−	0.905754i	$-0.360695\pi$
$907$	25.5269	0.847607	0.423804	+	0.905754i	$0.360695\pi$
$908$	0	0
$909$	38.1421	1.26509
$910$	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
$913$	−17.6569	−0.584357
$914$	0	0
$915$	−6.55635	−0.216746
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−6.55635	−0.216274	−0.108137	−	0.994136i	$-0.534489\pi$
$919$	−6.55635	−0.216274	−0.108137	+	0.994136i	$0.534489\pi$
$920$	0	0
$921$	−5.94113	−0.195767
$922$	0	0
$923$	−38.6274	−1.27144
$924$	0	0
$925$	−14.3431	−0.471600
$926$	0	0
$927$	23.7990	0.781661
$928$	0	0
$929$	19.3431	0.634628	0.317314	−	0.948321i	$-0.397219\pi$
$929$	19.3431	0.634628	0.317314	+	0.948321i	$0.397219\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	7.76955	0.254364
$934$	0	0
$935$	−0.757359	−0.0247683
$936$	0	0
$937$	−16.6274	−0.543194	−0.271597	−	0.962411i	$-0.587552\pi$
$937$	−16.6274	−0.543194	−0.271597	+	0.962411i	$0.587552\pi$
$938$	0	0
$939$	12.1299	0.395846
$940$	0	0
$941$	38.2548	1.24707	0.623536	−	0.781795i	$-0.285695\pi$
$941$	38.2548	1.24707	0.623536	+	0.781795i	$0.285695\pi$
$942$	0	0
$943$	−35.7990	−1.16578
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.2132	−0.591850	−0.295925	−	0.955211i	$-0.595628\pi$
$947$	−18.2132	−0.591850	−0.295925	+	0.955211i	$0.595628\pi$
$948$	0	0
$949$	−41.4558	−1.34571
$950$	0	0
$951$	−4.01219	−0.130104
$952$	0	0
$953$	−27.1127	−0.878266	−0.439133	−	0.898422i	$-0.644714\pi$
$953$	−27.1127	−0.878266	−0.439133	+	0.898422i	$0.644714\pi$
$954$	0	0
$955$	−16.2721	−0.526552
$956$	0	0
$957$	2.82843	0.0914301
$958$	0	0
$959$	0	0
$960$	0	0
$961$	0.201010	0.00648420
$962$	0	0
$963$	36.4853	1.17572
$964$	0	0
$965$	−16.7157	−0.538098
$966$	0	0
$967$	−18.3431	−0.589876	−0.294938	−	0.955516i	$-0.595299\pi$
$967$	−18.3431	−0.589876	−0.294938	+	0.955516i	$0.595299\pi$
$968$	0	0
$969$	−0.455844	−0.0146438
$970$	0	0
$971$	35.5858	1.14200	0.571001	−	0.820949i	$-0.306555\pi$
$971$	35.5858	1.14200	0.571001	+	0.820949i	$0.306555\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.94113	0.0621658
$976$	0	0
$977$	−34.5147	−1.10422	−0.552112	−	0.833770i	$-0.686178\pi$
$977$	−34.5147	−1.10422	−0.552112	+	0.833770i	$0.686178\pi$
$978$	0	0
$979$	−21.7279	−0.694427
$980$	0	0
$981$	23.1127	0.737932
$982$	0	0
$983$	−26.8406	−0.856083	−0.428041	−	0.903759i	$-0.640796\pi$
$983$	−26.8406	−0.856083	−0.428041	+	0.903759i	$0.640796\pi$
$984$	0	0
$985$	−34.4264	−1.09692
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−50.6274	−1.60986
$990$	0	0
$991$	−44.6985	−1.41989	−0.709947	−	0.704255i	$-0.751281\pi$
$991$	−44.6985	−1.41989	−0.709947	+	0.704255i	$0.751281\pi$
$992$	0	0
$993$	11.7696	0.373495
$994$	0	0
$995$	−20.5563	−0.651680
$996$	0	0
$997$	45.2843	1.43417	0.717084	−	0.696987i	$-0.245477\pi$
$997$	45.2843	1.43417	0.717084	+	0.696987i	$0.245477\pi$
$998$	0	0
$999$	−20.8995	−0.661231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.a.j.1.2		2
4.3	odd	2		1568.2.a.u.1.1		2
7.2	even	3		1568.2.i.x.1537.1		4
7.3	odd	6		224.2.i.a.65.2	✓	4
7.4	even	3		1568.2.i.x.961.1		4
7.5	odd	6		224.2.i.a.193.2	yes	4
7.6	odd	2		1568.2.a.w.1.1		2
8.3	odd	2		3136.2.a.be.1.2		2
8.5	even	2		3136.2.a.bx.1.1		2
21.5	even	6		2016.2.s.q.865.1		4
21.17	even	6		2016.2.s.q.289.1		4
28.3	even	6		224.2.i.d.65.1	yes	4
28.11	odd	6		1568.2.i.o.961.2		4
28.19	even	6		224.2.i.d.193.1	yes	4
28.23	odd	6		1568.2.i.o.1537.2		4
28.27	even	2		1568.2.a.l.1.2		2
56.3	even	6		448.2.i.g.65.2		4
56.5	odd	6		448.2.i.j.193.1		4
56.13	odd	2		3136.2.a.bd.1.2		2
56.19	even	6		448.2.i.g.193.2		4
56.27	even	2		3136.2.a.bw.1.1		2
56.45	odd	6		448.2.i.j.65.1		4
84.47	odd	6		2016.2.s.s.865.1		4
84.59	odd	6		2016.2.s.s.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.i.a.65.2	✓	4	7.3	odd	6
224.2.i.a.193.2	yes	4	7.5	odd	6
224.2.i.d.65.1	yes	4	28.3	even	6
224.2.i.d.193.1	yes	4	28.19	even	6
448.2.i.g.65.2		4	56.3	even	6
448.2.i.g.193.2		4	56.19	even	6
448.2.i.j.65.1		4	56.45	odd	6
448.2.i.j.193.1		4	56.5	odd	6
1568.2.a.j.1.2		2	1.1	even	1	trivial
1568.2.a.l.1.2		2	28.27	even	2
1568.2.a.u.1.1		2	4.3	odd	2
1568.2.a.w.1.1		2	7.6	odd	2
1568.2.i.o.961.2		4	28.11	odd	6
1568.2.i.o.1537.2		4	28.23	odd	6
1568.2.i.x.961.1		4	7.4	even	3
1568.2.i.x.1537.1		4	7.2	even	3
2016.2.s.q.289.1		4	21.17	even	6
2016.2.s.q.865.1		4	21.5	even	6
2016.2.s.s.289.1		4	84.59	odd	6
2016.2.s.s.865.1		4	84.47	odd	6
3136.2.a.bd.1.2		2	56.13	odd	2
3136.2.a.be.1.2		2	8.3	odd	2
3136.2.a.bw.1.1		2	56.27	even	2
3136.2.a.bx.1.1		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)

\(f(q)\)	\(=\)	\(q+0.414214 q^{3} +1.82843 q^{5} -2.82843 q^{9} +O(q^{10})\)	\(q+0.414214 q^{3} +1.82843 q^{5} -2.82843 q^{9} -2.41421 q^{11} -2.82843 q^{13} +0.757359 q^{15} +0.171573 q^{17} -6.41421 q^{19} -5.24264 q^{23} -1.65685 q^{25} -2.41421 q^{27} -2.82843 q^{29} -5.58579 q^{31} -1.00000 q^{33} +8.65685 q^{37} -1.17157 q^{39} +6.82843 q^{41} +9.65685 q^{43} -5.17157 q^{45} -10.4142 q^{47} +0.0710678 q^{51} -1.00000 q^{53} -4.41421 q^{55} -2.65685 q^{57} -10.8995 q^{59} -8.65685 q^{61} -5.17157 q^{65} -2.75736 q^{67} -2.17157 q^{69} +13.6569 q^{71} +14.6569 q^{73} -0.686292 q^{75} +6.07107 q^{79} +7.48528 q^{81} +7.31371 q^{83} +0.313708 q^{85} -1.17157 q^{87} +9.00000 q^{89} -2.31371 q^{93} -11.7279 q^{95} +1.17157 q^{97} +6.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5}+O(q^{10}) \) 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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