LMFDB - Embedded newform 6728.2.a.z.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6728,2,Mod(1,6728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6728, 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("6728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6728 = 2^{3} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6728.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:	$53.7233504799$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 18 x^{10} + 83 x^{9} + 83 x^{8} - 577 x^{7} + 121 x^{6} + 1416 x^{5} - 1289 x^{4} + \cdots + 64 $ x^12 - 4x^11 - 18x^10 + 83x^9 + 83x^8 - 577x^7 + 121x^6 + 1416x^5 - 1289x^4 - 465x^3 + 1056x^2 - 464*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.642693$ of defining polynomial
Character	$\chi$	$=$	6728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.642693 q^{3} -2.00467 q^{5} -3.04346 q^{7} -2.58695 q^{9} +O(q^{10})$	$q-0.642693 q^{3} -2.00467 q^{5} -3.04346 q^{7} -2.58695 q^{9} -5.21658 q^{11} +2.25114 q^{13} +1.28839 q^{15} +6.39356 q^{17} +0.375146 q^{19} +1.95601 q^{21} +3.84489 q^{23} -0.981284 q^{25} +3.59069 q^{27} +9.60018 q^{31} +3.35266 q^{33} +6.10115 q^{35} -2.61926 q^{37} -1.44679 q^{39} -9.89670 q^{41} +9.78274 q^{43} +5.18598 q^{45} +5.18221 q^{47} +2.26267 q^{49} -4.10909 q^{51} +5.80333 q^{53} +10.4575 q^{55} -0.241104 q^{57} +1.95491 q^{59} +3.10249 q^{61} +7.87327 q^{63} -4.51280 q^{65} -2.51389 q^{67} -2.47108 q^{69} +3.72815 q^{71} -11.0612 q^{73} +0.630665 q^{75} +15.8765 q^{77} -5.15734 q^{79} +5.45312 q^{81} +1.57610 q^{83} -12.8170 q^{85} -17.9607 q^{89} -6.85126 q^{91} -6.16997 q^{93} -0.752045 q^{95} -4.11002 q^{97} +13.4950 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9	$ 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9} - 3 q^{11} - 3 q^{13} + 3 q^{15} - 8 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 12 q^{25} - 7 q^{27} - 29 q^{31} - 46 q^{33} + 17 q^{35} - 38 q^{37} + 10 q^{39} - 11 q^{41} - 9 q^{43} - 54 q^{45} - 34 q^{47} + 33 q^{49} + 17 q^{51} - 15 q^{53} - 2 q^{55} - q^{57} + 57 q^{59} - 37 q^{61} + 9 q^{63} - 59 q^{65} + 33 q^{67} + 21 q^{69} - 21 q^{71} - 13 q^{73} - 13 q^{75} + 3 q^{77} - 32 q^{79} + 36 q^{81} + 48 q^{83} - 17 q^{85} - 20 q^{89} - 2 q^{91} - 37 q^{93} - 7 q^{97} + 16 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9 - 3 * q^11 - 3 * q^13 + 3 * q^15 - 8 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 12 * q^25 - 7 * q^27 - 29 * q^31 - 46 * q^33 + 17 * q^35 - 38 * q^37 + 10 * q^39 - 11 * q^41 - 9 * q^43 - 54 * q^45 - 34 * q^47 + 33 * q^49 + 17 * q^51 - 15 * q^53 - 2 * q^55 - q^57 + 57 * q^59 - 37 * q^61 + 9 * q^63 - 59 * q^65 + 33 * q^67 + 21 * q^69 - 21 * q^71 - 13 * q^73 - 13 * q^75 + 3 * q^77 - 32 * q^79 + 36 * q^81 + 48 * q^83 - 17 * q^85 - 20 * q^89 - 2 * q^91 - 37 * q^93 - 7 * q^97 + 16 * 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
$3$	−0.642693	−0.371059	−0.185529	−	0.982639i	$-0.559400\pi$
$3$	−0.642693	−0.371059	−0.185529	+	0.982639i	$0.559400\pi$
$4$	0	0
$5$	−2.00467	−0.896517	−0.448259	−	0.893904i	$-0.647956\pi$
$5$	−2.00467	−0.896517	−0.448259	+	0.893904i	$0.647956\pi$
$6$	0	0
$7$	−3.04346	−1.15032	−0.575160	−	0.818041i	$-0.695060\pi$
$7$	−3.04346	−1.15032	−0.575160	+	0.818041i	$0.695060\pi$
$8$	0	0
$9$	−2.58695	−0.862315
$10$	0	0
$11$	−5.21658	−1.57286	−0.786430	−	0.617680i	$-0.788073\pi$
$11$	−5.21658	−1.57286	−0.786430	+	0.617680i	$0.788073\pi$
$12$	0	0
$13$	2.25114	0.624354	0.312177	−	0.950024i	$-0.398942\pi$
$13$	2.25114	0.624354	0.312177	+	0.950024i	$0.398942\pi$
$14$	0	0
$15$	1.28839	0.332661
$16$	0	0
$17$	6.39356	1.55067	0.775333	−	0.631553i	$-0.217582\pi$
$17$	6.39356	1.55067	0.775333	+	0.631553i	$0.217582\pi$
$18$	0	0
$19$	0.375146	0.0860644	0.0430322	−	0.999074i	$-0.486298\pi$
$19$	0.375146	0.0860644	0.0430322	+	0.999074i	$0.486298\pi$
$20$	0	0
$21$	1.95601	0.426837
$22$	0	0
$23$	3.84489	0.801715	0.400857	−	0.916141i	$-0.368712\pi$
$23$	3.84489	0.801715	0.400857	+	0.916141i	$0.368712\pi$
$24$	0	0
$25$	−0.981284	−0.196257
$26$	0	0
$27$	3.59069	0.691029
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	9.60018	1.72424	0.862121	−	0.506702i	$-0.169135\pi$
$31$	9.60018	1.72424	0.862121	+	0.506702i	$0.169135\pi$
$32$	0	0
$33$	3.35266	0.583624
$34$	0	0
$35$	6.10115	1.03128
$36$	0	0
$37$	−2.61926	−0.430604	−0.215302	−	0.976548i	$-0.569074\pi$
$37$	−2.61926	−0.430604	−0.215302	+	0.976548i	$0.569074\pi$
$38$	0	0
$39$	−1.44679	−0.231672
$40$	0	0
$41$	−9.89670	−1.54560	−0.772802	−	0.634647i	$-0.781145\pi$
$41$	−9.89670	−1.54560	−0.772802	+	0.634647i	$0.781145\pi$
$42$	0	0
$43$	9.78274	1.49185	0.745927	−	0.666027i	$-0.232007\pi$
$43$	9.78274	1.49185	0.745927	+	0.666027i	$0.232007\pi$
$44$	0	0
$45$	5.18598	0.773080
$46$	0	0
$47$	5.18221	0.755903	0.377951	−	0.925825i	$-0.376629\pi$
$47$	5.18221	0.755903	0.377951	+	0.925825i	$0.376629\pi$
$48$	0	0
$49$	2.26267	0.323238
$50$	0	0
$51$	−4.10909	−0.575388
$52$	0	0
$53$	5.80333	0.797148	0.398574	−	0.917136i	$-0.369505\pi$
$53$	5.80333	0.797148	0.398574	+	0.917136i	$0.369505\pi$
$54$	0	0
$55$	10.4575	1.41010
$56$	0	0
$57$	−0.241104	−0.0319350
$58$	0	0
$59$	1.95491	0.254508	0.127254	−	0.991870i	$-0.459384\pi$
$59$	1.95491	0.254508	0.127254	+	0.991870i	$0.459384\pi$
$60$	0	0
$61$	3.10249	0.397233	0.198616	−	0.980077i	$-0.436355\pi$
$61$	3.10249	0.397233	0.198616	+	0.980077i	$0.436355\pi$
$62$	0	0
$63$	7.87327	0.991939
$64$	0	0
$65$	−4.51280	−0.559744
$66$	0	0
$67$	−2.51389	−0.307121	−0.153560	−	0.988139i	$-0.549074\pi$
$67$	−2.51389	−0.307121	−0.153560	+	0.988139i	$0.549074\pi$
$68$	0	0
$69$	−2.47108	−0.297483
$70$	0	0
$71$	3.72815	0.442450	0.221225	−	0.975223i	$-0.428994\pi$
$71$	3.72815	0.442450	0.221225	+	0.975223i	$0.428994\pi$
$72$	0	0
$73$	−11.0612	−1.29461	−0.647306	−	0.762230i	$-0.724104\pi$
$73$	−11.0612	−1.29461	−0.647306	+	0.762230i	$0.724104\pi$
$74$	0	0
$75$	0.630665	0.0728229
$76$	0	0
$77$	15.8765	1.80929
$78$	0	0
$79$	−5.15734	−0.580246	−0.290123	−	0.956989i	$-0.593696\pi$
$79$	−5.15734	−0.580246	−0.290123	+	0.956989i	$0.593696\pi$
$80$	0	0
$81$	5.45312	0.605903
$82$	0	0
$83$	1.57610	0.172999	0.0864997	−	0.996252i	$-0.472432\pi$
$83$	1.57610	0.172999	0.0864997	+	0.996252i	$0.472432\pi$
$84$	0	0
$85$	−12.8170	−1.39020
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.9607	−1.90383	−0.951913	−	0.306369i	$-0.900886\pi$
$89$	−17.9607	−1.90383	−0.951913	+	0.306369i	$0.900886\pi$
$90$	0	0
$91$	−6.85126	−0.718207
$92$	0	0
$93$	−6.16997	−0.639796
$94$	0	0
$95$	−0.752045	−0.0771582
$96$	0	0
$97$	−4.11002	−0.417309	−0.208655	−	0.977989i	$-0.566908\pi$
$97$	−4.11002	−0.417309	−0.208655	+	0.977989i	$0.566908\pi$
$98$	0	0
$99$	13.4950	1.35630
$100$	0	0
$101$	10.4964	1.04443	0.522214	−	0.852815i	$-0.325106\pi$
$101$	10.4964	1.04443	0.522214	+	0.852815i	$0.325106\pi$
$102$	0	0
$103$	7.69163	0.757879	0.378939	−	0.925421i	$-0.376289\pi$
$103$	7.69163	0.757879	0.378939	+	0.925421i	$0.376289\pi$
$104$	0	0
$105$	−3.92117	−0.382667
$106$	0	0
$107$	4.82447	0.466399	0.233199	−	0.972429i	$-0.425081\pi$
$107$	4.82447	0.466399	0.233199	+	0.972429i	$0.425081\pi$
$108$	0	0
$109$	−16.7362	−1.60304	−0.801518	−	0.597970i	$-0.795974\pi$
$109$	−16.7362	−1.60304	−0.801518	+	0.597970i	$0.795974\pi$
$110$	0	0
$111$	1.68338	0.159779
$112$	0	0
$113$	4.16053	0.391390	0.195695	−	0.980665i	$-0.437304\pi$
$113$	4.16053	0.391390	0.195695	+	0.980665i	$0.437304\pi$
$114$	0	0
$115$	−7.70774	−0.718751
$116$	0	0
$117$	−5.82358	−0.538390
$118$	0	0
$119$	−19.4586	−1.78376
$120$	0	0
$121$	16.2127	1.47389
$122$	0	0
$123$	6.36054	0.573510
$124$	0	0
$125$	11.9905	1.07246
$126$	0	0
$127$	−7.18091	−0.637202	−0.318601	−	0.947889i	$-0.603213\pi$
$127$	−7.18091	−0.637202	−0.318601	+	0.947889i	$0.603213\pi$
$128$	0	0
$129$	−6.28730	−0.553566
$130$	0	0
$131$	13.1474	1.14869	0.574346	−	0.818613i	$-0.305257\pi$
$131$	13.1474	1.14869	0.574346	+	0.818613i	$0.305257\pi$
$132$	0	0
$133$	−1.14174	−0.0990017
$134$	0	0
$135$	−7.19816	−0.619519
$136$	0	0
$137$	0.288027	0.0246078	0.0123039	−	0.999924i	$-0.496083\pi$
$137$	0.288027	0.0246078	0.0123039	+	0.999924i	$0.496083\pi$
$138$	0	0
$139$	6.42076	0.544602	0.272301	−	0.962212i	$-0.412215\pi$
$139$	6.42076	0.544602	0.272301	+	0.962212i	$0.412215\pi$
$140$	0	0
$141$	−3.33057	−0.280484
$142$	0	0
$143$	−11.7433	−0.982021
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.45420	−0.119940
$148$	0	0
$149$	−12.1293	−0.993674	−0.496837	−	0.867844i	$-0.665505\pi$
$149$	−12.1293	−0.993674	−0.496837	+	0.867844i	$0.665505\pi$
$150$	0	0
$151$	−20.0065	−1.62811	−0.814053	−	0.580791i	$-0.802743\pi$
$151$	−20.0065	−1.62811	−0.814053	+	0.580791i	$0.802743\pi$
$152$	0	0
$153$	−16.5398	−1.33716
$154$	0	0
$155$	−19.2452	−1.54581
$156$	0	0
$157$	−0.892328	−0.0712155	−0.0356078	−	0.999366i	$-0.511337\pi$
$157$	−0.892328	−0.0712155	−0.0356078	+	0.999366i	$0.511337\pi$
$158$	0	0
$159$	−3.72976	−0.295789
$160$	0	0
$161$	−11.7018	−0.922229
$162$	0	0
$163$	−16.8115	−1.31678	−0.658388	−	0.752679i	$-0.728761\pi$
$163$	−16.8115	−1.31678	−0.658388	+	0.752679i	$0.728761\pi$
$164$	0	0
$165$	−6.72099	−0.523229
$166$	0	0
$167$	5.27518	0.408206	0.204103	−	0.978949i	$-0.434572\pi$
$167$	5.27518	0.408206	0.204103	+	0.978949i	$0.434572\pi$
$168$	0	0
$169$	−7.93237	−0.610182
$170$	0	0
$171$	−0.970482	−0.0742146
$172$	0	0
$173$	−14.4168	−1.09609	−0.548043	−	0.836450i	$-0.684627\pi$
$173$	−14.4168	−1.09609	−0.548043	+	0.836450i	$0.684627\pi$
$174$	0	0
$175$	2.98650	0.225758
$176$	0	0
$177$	−1.25641	−0.0944376
$178$	0	0
$179$	13.4756	1.00722	0.503608	−	0.863932i	$-0.332006\pi$
$179$	13.4756	1.00722	0.503608	+	0.863932i	$0.332006\pi$
$180$	0	0
$181$	−9.36509	−0.696102	−0.348051	−	0.937476i	$-0.613156\pi$
$181$	−9.36509	−0.696102	−0.348051	+	0.937476i	$0.613156\pi$
$182$	0	0
$183$	−1.99395	−0.147397
$184$	0	0
$185$	5.25076	0.386044
$186$	0	0
$187$	−33.3525	−2.43898
$188$	0	0
$189$	−10.9281	−0.794905
$190$	0	0
$191$	−23.7366	−1.71752	−0.858760	−	0.512378i	$-0.828765\pi$
$191$	−23.7366	−1.71752	−0.858760	+	0.512378i	$0.828765\pi$
$192$	0	0
$193$	9.13451	0.657516	0.328758	−	0.944414i	$-0.393370\pi$
$193$	9.13451	0.657516	0.328758	+	0.944414i	$0.393370\pi$
$194$	0	0
$195$	2.90035	0.207698
$196$	0	0
$197$	16.0375	1.14262	0.571311	−	0.820734i	$-0.306435\pi$
$197$	16.0375	1.14262	0.571311	+	0.820734i	$0.306435\pi$
$198$	0	0
$199$	8.69482	0.616359	0.308180	−	0.951328i	$-0.400280\pi$
$199$	8.69482	0.616359	0.308180	+	0.951328i	$0.400280\pi$
$200$	0	0
$201$	1.61566	0.113960
$202$	0	0
$203$	0	0
$204$	0	0
$205$	19.8396	1.38566
$206$	0	0
$207$	−9.94652	−0.691331
$208$	0	0
$209$	−1.95698	−0.135367
$210$	0	0
$211$	−18.5617	−1.27784	−0.638919	−	0.769274i	$-0.720618\pi$
$211$	−18.5617	−1.27784	−0.638919	+	0.769274i	$0.720618\pi$
$212$	0	0
$213$	−2.39606	−0.164175
$214$	0	0
$215$	−19.6112	−1.33747
$216$	0	0
$217$	−29.2178	−1.98343
$218$	0	0
$219$	7.10894	0.480378
$220$	0	0
$221$	14.3928	0.968164
$222$	0	0
$223$	15.9522	1.06824	0.534119	−	0.845409i	$-0.320643\pi$
$223$	15.9522	1.06824	0.534119	+	0.845409i	$0.320643\pi$
$224$	0	0
$225$	2.53853	0.169235
$226$	0	0
$227$	21.9510	1.45694	0.728470	−	0.685078i	$-0.240232\pi$
$227$	21.9510	1.45694	0.728470	+	0.685078i	$0.240232\pi$
$228$	0	0
$229$	−6.88197	−0.454773	−0.227387	−	0.973805i	$-0.573018\pi$
$229$	−6.88197	−0.454773	−0.227387	+	0.973805i	$0.573018\pi$
$230$	0	0
$231$	−10.2037	−0.671354
$232$	0	0
$233$	6.25867	0.410019	0.205009	−	0.978760i	$-0.434278\pi$
$233$	6.25867	0.410019	0.205009	+	0.978760i	$0.434278\pi$
$234$	0	0
$235$	−10.3886	−0.677680
$236$	0	0
$237$	3.31458	0.215305
$238$	0	0
$239$	−26.1271	−1.69002	−0.845009	−	0.534751i	$-0.820405\pi$
$239$	−26.1271	−1.69002	−0.845009	+	0.534751i	$0.820405\pi$
$240$	0	0
$241$	26.0299	1.67674	0.838368	−	0.545105i	$-0.183510\pi$
$241$	26.0299	1.67674	0.838368	+	0.545105i	$0.183510\pi$
$242$	0	0
$243$	−14.2768	−0.915854
$244$	0	0
$245$	−4.53591	−0.289788
$246$	0	0
$247$	0.844506	0.0537346
$248$	0	0
$249$	−1.01295	−0.0641930
$250$	0	0
$251$	−4.66383	−0.294378	−0.147189	−	0.989108i	$-0.547023\pi$
$251$	−4.66383	−0.294378	−0.147189	+	0.989108i	$0.547023\pi$
$252$	0	0
$253$	−20.0572	−1.26098
$254$	0	0
$255$	8.23739	0.515846
$256$	0	0
$257$	16.5375	1.03158	0.515792	−	0.856714i	$-0.327498\pi$
$257$	16.5375	1.03158	0.515792	+	0.856714i	$0.327498\pi$
$258$	0	0
$259$	7.97162	0.495332
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.0146	−1.04916	−0.524582	−	0.851360i	$-0.675778\pi$
$263$	−17.0146	−1.04916	−0.524582	+	0.851360i	$0.675778\pi$
$264$	0	0
$265$	−11.6338	−0.714657
$266$	0	0
$267$	11.5432	0.706432
$268$	0	0
$269$	−27.4509	−1.67371	−0.836857	−	0.547422i	$-0.815609\pi$
$269$	−27.4509	−1.67371	−0.836857	+	0.547422i	$0.815609\pi$
$270$	0	0
$271$	−15.6866	−0.952893	−0.476446	−	0.879204i	$-0.658075\pi$
$271$	−15.6866	−0.952893	−0.476446	+	0.879204i	$0.658075\pi$
$272$	0	0
$273$	4.40326	0.266497
$274$	0	0
$275$	5.11895	0.308684
$276$	0	0
$277$	12.0836	0.726033	0.363016	−	0.931783i	$-0.381747\pi$
$277$	12.0836	0.726033	0.363016	+	0.931783i	$0.381747\pi$
$278$	0	0
$279$	−24.8351	−1.48684
$280$	0	0
$281$	−30.8853	−1.84247	−0.921233	−	0.389012i	$-0.872816\pi$
$281$	−30.8853	−1.84247	−0.921233	+	0.389012i	$0.872816\pi$
$282$	0	0
$283$	26.7220	1.58846	0.794230	−	0.607618i	$-0.207875\pi$
$283$	26.7220	1.58846	0.794230	+	0.607618i	$0.207875\pi$
$284$	0	0
$285$	0.483334	0.0286302
$286$	0	0
$287$	30.1202	1.77794
$288$	0	0
$289$	23.8776	1.40456
$290$	0	0
$291$	2.64148	0.154846
$292$	0	0
$293$	−6.89515	−0.402819	−0.201410	−	0.979507i	$-0.564552\pi$
$293$	−6.89515	−0.402819	−0.201410	+	0.979507i	$0.564552\pi$
$294$	0	0
$295$	−3.91897	−0.228171
$296$	0	0
$297$	−18.7311	−1.08689
$298$	0	0
$299$	8.65538	0.500554
$300$	0	0
$301$	−29.7734	−1.71611
$302$	0	0
$303$	−6.74594	−0.387544
$304$	0	0
$305$	−6.21947	−0.356126
$306$	0	0
$307$	27.1314	1.54847	0.774236	−	0.632897i	$-0.218134\pi$
$307$	27.1314	1.54847	0.774236	+	0.632897i	$0.218134\pi$
$308$	0	0
$309$	−4.94336	−0.281218
$310$	0	0
$311$	6.62654	0.375757	0.187878	−	0.982192i	$-0.439839\pi$
$311$	6.62654	0.375757	0.187878	+	0.982192i	$0.439839\pi$
$312$	0	0
$313$	23.5353	1.33029	0.665147	−	0.746713i	$-0.268369\pi$
$313$	23.5353	1.33029	0.665147	+	0.746713i	$0.268369\pi$
$314$	0	0
$315$	−15.7833	−0.889291
$316$	0	0
$317$	20.4734	1.14990	0.574951	−	0.818188i	$-0.305021\pi$
$317$	20.4734	1.14990	0.574951	+	0.818188i	$0.305021\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.10065	−0.173061
$322$	0	0
$323$	2.39852	0.133457
$324$	0	0
$325$	−2.20901	−0.122534
$326$	0	0
$327$	10.7562	0.594821
$328$	0	0
$329$	−15.7719	−0.869531
$330$	0	0
$331$	−22.7736	−1.25175	−0.625875	−	0.779923i	$-0.715258\pi$
$331$	−22.7736	−1.25175	−0.625875	+	0.779923i	$0.715258\pi$
$332$	0	0
$333$	6.77588	0.371316
$334$	0	0
$335$	5.03953	0.275339
$336$	0	0
$337$	−33.4549	−1.82240	−0.911202	−	0.411961i	$-0.864844\pi$
$337$	−33.4549	−1.82240	−0.911202	+	0.411961i	$0.864844\pi$
$338$	0	0
$339$	−2.67394	−0.145229
$340$	0	0
$341$	−50.0801	−2.71199
$342$	0	0
$343$	14.4179	0.778493
$344$	0	0
$345$	4.95371	0.266699
$346$	0	0
$347$	23.9717	1.28687	0.643435	−	0.765501i	$-0.277509\pi$
$347$	23.9717	1.28687	0.643435	+	0.765501i	$0.277509\pi$
$348$	0	0
$349$	14.5018	0.776266	0.388133	−	0.921603i	$-0.373120\pi$
$349$	14.5018	0.776266	0.388133	+	0.921603i	$0.373120\pi$
$350$	0	0
$351$	8.08315	0.431447
$352$	0	0
$353$	11.1215	0.591937	0.295968	−	0.955198i	$-0.404358\pi$
$353$	11.1215	0.591937	0.295968	+	0.955198i	$0.404358\pi$
$354$	0	0
$355$	−7.47373	−0.396664
$356$	0	0
$357$	12.5059	0.661881
$358$	0	0
$359$	12.6458	0.667421	0.333710	−	0.942676i	$-0.391699\pi$
$359$	12.6458	0.667421	0.333710	+	0.942676i	$0.391699\pi$
$360$	0	0
$361$	−18.8593	−0.992593
$362$	0	0
$363$	−10.4198	−0.546899
$364$	0	0
$365$	22.1740	1.16064
$366$	0	0
$367$	9.36739	0.488974	0.244487	−	0.969653i	$-0.421380\pi$
$367$	9.36739	0.488974	0.244487	+	0.969653i	$0.421380\pi$
$368$	0	0
$369$	25.6022	1.33280
$370$	0	0
$371$	−17.6622	−0.916976
$372$	0	0
$373$	−2.15206	−0.111429	−0.0557146	−	0.998447i	$-0.517744\pi$
$373$	−2.15206	−0.111429	−0.0557146	+	0.998447i	$0.517744\pi$
$374$	0	0
$375$	−7.70622	−0.397948
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−11.0068	−0.565381	−0.282690	−	0.959211i	$-0.591227\pi$
$379$	−11.0068	−0.565381	−0.282690	+	0.959211i	$0.591227\pi$
$380$	0	0
$381$	4.61512	0.236440
$382$	0	0
$383$	9.32280	0.476373	0.238186	−	0.971219i	$-0.423447\pi$
$383$	9.32280	0.476373	0.238186	+	0.971219i	$0.423447\pi$
$384$	0	0
$385$	−31.8272	−1.62206
$386$	0	0
$387$	−25.3074	−1.28645
$388$	0	0
$389$	−22.6038	−1.14606	−0.573030	−	0.819535i	$-0.694232\pi$
$389$	−22.6038	−1.14606	−0.573030	+	0.819535i	$0.694232\pi$
$390$	0	0
$391$	24.5825	1.24319
$392$	0	0
$393$	−8.44972	−0.426232
$394$	0	0
$395$	10.3388	0.520200
$396$	0	0
$397$	8.03395	0.403213	0.201606	−	0.979467i	$-0.435384\pi$
$397$	8.03395	0.403213	0.201606	+	0.979467i	$0.435384\pi$
$398$	0	0
$399$	0.733790	0.0367355
$400$	0	0
$401$	−6.72656	−0.335908	−0.167954	−	0.985795i	$-0.553716\pi$
$401$	−6.72656	−0.335908	−0.167954	+	0.985795i	$0.553716\pi$
$402$	0	0
$403$	21.6113	1.07654
$404$	0	0
$405$	−10.9317	−0.543202
$406$	0	0
$407$	13.6636	0.677279
$408$	0	0
$409$	3.26109	0.161251	0.0806253	−	0.996744i	$-0.474308\pi$
$409$	3.26109	0.161251	0.0806253	+	0.996744i	$0.474308\pi$
$410$	0	0
$411$	−0.185113	−0.00913094
$412$	0	0
$413$	−5.94971	−0.292766
$414$	0	0
$415$	−3.15957	−0.155097
$416$	0	0
$417$	−4.12658	−0.202079
$418$	0	0
$419$	1.35327	0.0661113	0.0330557	−	0.999454i	$-0.489476\pi$
$419$	1.35327	0.0661113	0.0330557	+	0.999454i	$0.489476\pi$
$420$	0	0
$421$	12.4639	0.607452	0.303726	−	0.952759i	$-0.401769\pi$
$421$	12.4639	0.607452	0.303726	+	0.952759i	$0.401769\pi$
$422$	0	0
$423$	−13.4061	−0.651826
$424$	0	0
$425$	−6.27390	−0.304329
$426$	0	0
$427$	−9.44230	−0.456945
$428$	0	0
$429$	7.54731	0.364388
$430$	0	0
$431$	9.77156	0.470680	0.235340	−	0.971913i	$-0.424380\pi$
$431$	9.77156	0.470680	0.235340	+	0.971913i	$0.424380\pi$
$432$	0	0
$433$	−6.18507	−0.297235	−0.148618	−	0.988895i	$-0.547482\pi$
$433$	−6.18507	−0.297235	−0.148618	+	0.988895i	$0.547482\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.44239	0.0689991
$438$	0	0
$439$	6.74856	0.322091	0.161046	−	0.986947i	$-0.448513\pi$
$439$	6.74856	0.322091	0.161046	+	0.986947i	$0.448513\pi$
$440$	0	0
$441$	−5.85339	−0.278733
$442$	0	0
$443$	−8.39757	−0.398981	−0.199490	−	0.979900i	$-0.563929\pi$
$443$	−8.39757	−0.398981	−0.199490	+	0.979900i	$0.563929\pi$
$444$	0	0
$445$	36.0052	1.70681
$446$	0	0
$447$	7.79544	0.368712
$448$	0	0
$449$	1.54831	0.0730692	0.0365346	−	0.999332i	$-0.488368\pi$
$449$	1.54831	0.0730692	0.0365346	+	0.999332i	$0.488368\pi$
$450$	0	0
$451$	51.6269	2.43102
$452$	0	0
$453$	12.8580	0.604123
$454$	0	0
$455$	13.7345	0.643885
$456$	0	0
$457$	−26.1936	−1.22529	−0.612643	−	0.790360i	$-0.709894\pi$
$457$	−26.1936	−1.22529	−0.612643	+	0.790360i	$0.709894\pi$
$458$	0	0
$459$	22.9573	1.07155
$460$	0	0
$461$	26.6503	1.24123	0.620615	−	0.784116i	$-0.286883\pi$
$461$	26.6503	1.24123	0.620615	+	0.784116i	$0.286883\pi$
$462$	0	0
$463$	−15.5366	−0.722049	−0.361024	−	0.932556i	$-0.617573\pi$
$463$	−15.5366	−0.722049	−0.361024	+	0.932556i	$0.617573\pi$
$464$	0	0
$465$	12.3688	0.573588
$466$	0	0
$467$	−1.15114	−0.0532685	−0.0266342	−	0.999645i	$-0.508479\pi$
$467$	−1.15114	−0.0532685	−0.0266342	+	0.999645i	$0.508479\pi$
$468$	0	0
$469$	7.65093	0.353287
$470$	0	0
$471$	0.573493	0.0264252
$472$	0	0
$473$	−51.0325	−2.34648
$474$	0	0
$475$	−0.368125	−0.0168907
$476$	0	0
$477$	−15.0129	−0.687393
$478$	0	0
$479$	4.16004	0.190077	0.0950385	−	0.995474i	$-0.469703\pi$
$479$	4.16004	0.190077	0.0950385	+	0.995474i	$0.469703\pi$
$480$	0	0
$481$	−5.89632	−0.268849
$482$	0	0
$483$	7.52065	0.342201
$484$	0	0
$485$	8.23924	0.374125
$486$	0	0
$487$	−32.9099	−1.49129	−0.745646	−	0.666343i	$-0.767859\pi$
$487$	−32.9099	−1.49129	−0.745646	+	0.666343i	$0.767859\pi$
$488$	0	0
$489$	10.8046	0.488601
$490$	0	0
$491$	29.0268	1.30996	0.654981	−	0.755645i	$-0.272676\pi$
$491$	29.0268	1.30996	0.654981	+	0.755645i	$0.272676\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−27.0531	−1.21595
$496$	0	0
$497$	−11.3465	−0.508960
$498$	0	0
$499$	13.8963	0.622086	0.311043	−	0.950396i	$-0.399322\pi$
$499$	13.8963	0.622086	0.311043	+	0.950396i	$0.399322\pi$
$500$	0	0
$501$	−3.39032	−0.151469
$502$	0	0
$503$	−1.06767	−0.0476051	−0.0238026	−	0.999717i	$-0.507577\pi$
$503$	−1.06767	−0.0476051	−0.0238026	+	0.999717i	$0.507577\pi$
$504$	0	0
$505$	−21.0418	−0.936347
$506$	0	0
$507$	5.09808	0.226414
$508$	0	0
$509$	−30.2348	−1.34013	−0.670066	−	0.742301i	$-0.733734\pi$
$509$	−30.2348	−1.34013	−0.670066	+	0.742301i	$0.733734\pi$
$510$	0	0
$511$	33.6643	1.48922
$512$	0	0
$513$	1.34703	0.0594730
$514$	0	0
$515$	−15.4192	−0.679451
$516$	0	0
$517$	−27.0334	−1.18893
$518$	0	0
$519$	9.26555	0.406712
$520$	0	0
$521$	9.87168	0.432486	0.216243	−	0.976340i	$-0.430620\pi$
$521$	9.87168	0.432486	0.216243	+	0.976340i	$0.430620\pi$
$522$	0	0
$523$	−26.0370	−1.13852	−0.569260	−	0.822158i	$-0.692770\pi$
$523$	−26.0370	−1.13852	−0.569260	+	0.822158i	$0.692770\pi$
$524$	0	0
$525$	−1.91940	−0.0837697
$526$	0	0
$527$	61.3793	2.67372
$528$	0	0
$529$	−8.21684	−0.357254
$530$	0	0
$531$	−5.05726	−0.219466
$532$	0	0
$533$	−22.2789	−0.965004
$534$	0	0
$535$	−9.67148	−0.418135
$536$	0	0
$537$	−8.66070	−0.373737
$538$	0	0
$539$	−11.8034	−0.508408
$540$	0	0
$541$	−11.8556	−0.509712	−0.254856	−	0.966979i	$-0.582028\pi$
$541$	−11.8556	−0.509712	−0.254856	+	0.966979i	$0.582028\pi$
$542$	0	0
$543$	6.01888	0.258295
$544$	0	0
$545$	33.5506	1.43715
$546$	0	0
$547$	25.2628	1.08016	0.540079	−	0.841614i	$-0.318394\pi$
$547$	25.2628	1.08016	0.540079	+	0.841614i	$0.318394\pi$
$548$	0	0
$549$	−8.02597	−0.342540
$550$	0	0
$551$	0	0
$552$	0	0
$553$	15.6962	0.667469
$554$	0	0
$555$	−3.37463	−0.143245
$556$	0	0
$557$	−42.3699	−1.79527	−0.897634	−	0.440741i	$-0.854716\pi$
$557$	−42.3699	−1.79527	−0.897634	+	0.440741i	$0.854716\pi$
$558$	0	0
$559$	22.0223	0.931445
$560$	0	0
$561$	21.4354	0.905005
$562$	0	0
$563$	−5.16150	−0.217531	−0.108766	−	0.994067i	$-0.534690\pi$
$563$	−5.16150	−0.217531	−0.108766	+	0.994067i	$0.534690\pi$
$564$	0	0
$565$	−8.34050	−0.350888
$566$	0	0
$567$	−16.5964	−0.696983
$568$	0	0
$569$	18.4523	0.773561	0.386780	−	0.922172i	$-0.373587\pi$
$569$	18.4523	0.773561	0.386780	+	0.922172i	$0.373587\pi$
$570$	0	0
$571$	−16.9405	−0.708940	−0.354470	−	0.935067i	$-0.615339\pi$
$571$	−16.9405	−0.708940	−0.354470	+	0.935067i	$0.615339\pi$
$572$	0	0
$573$	15.2553	0.637301
$574$	0	0
$575$	−3.77293	−0.157342
$576$	0	0
$577$	−27.5192	−1.14564	−0.572820	−	0.819681i	$-0.694151\pi$
$577$	−27.5192	−1.14564	−0.572820	+	0.819681i	$0.694151\pi$
$578$	0	0
$579$	−5.87068	−0.243977
$580$	0	0
$581$	−4.79680	−0.199005
$582$	0	0
$583$	−30.2735	−1.25380
$584$	0	0
$585$	11.6744	0.482676
$586$	0	0
$587$	15.1996	0.627354	0.313677	−	0.949530i	$-0.398439\pi$
$587$	15.1996	0.627354	0.313677	+	0.949530i	$0.398439\pi$
$588$	0	0
$589$	3.60147	0.148396
$590$	0	0
$591$	−10.3072	−0.423980
$592$	0	0
$593$	24.1143	0.990255	0.495128	−	0.868820i	$-0.335121\pi$
$593$	24.1143	0.990255	0.495128	+	0.868820i	$0.335121\pi$
$594$	0	0
$595$	39.0080	1.59917
$596$	0	0
$597$	−5.58810	−0.228706
$598$	0	0
$599$	−2.13064	−0.0870558	−0.0435279	−	0.999052i	$-0.513860\pi$
$599$	−2.13064	−0.0870558	−0.0435279	+	0.999052i	$0.513860\pi$
$600$	0	0
$601$	−3.18865	−0.130068	−0.0650339	−	0.997883i	$-0.520716\pi$
$601$	−3.18865	−0.130068	−0.0650339	+	0.997883i	$0.520716\pi$
$602$	0	0
$603$	6.50330	0.264835
$604$	0	0
$605$	−32.5013	−1.32136
$606$	0	0
$607$	−34.5502	−1.40235	−0.701174	−	0.712990i	$-0.747341\pi$
$607$	−34.5502	−1.40235	−0.701174	+	0.712990i	$0.747341\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.6659	0.471951
$612$	0	0
$613$	−44.9441	−1.81528	−0.907638	−	0.419754i	$-0.862116\pi$
$613$	−44.9441	−1.81528	−0.907638	+	0.419754i	$0.862116\pi$
$614$	0	0
$615$	−12.7508	−0.514162
$616$	0	0
$617$	37.4322	1.50696	0.753481	−	0.657469i	$-0.228373\pi$
$617$	37.4322	1.50696	0.753481	+	0.657469i	$0.228373\pi$
$618$	0	0
$619$	−27.0205	−1.08604	−0.543022	−	0.839719i	$-0.682720\pi$
$619$	−27.0205	−1.08604	−0.543022	+	0.839719i	$0.682720\pi$
$620$	0	0
$621$	13.8058	0.554008
$622$	0	0
$623$	54.6626	2.19001
$624$	0	0
$625$	−19.1307	−0.765226
$626$	0	0
$627$	1.25774	0.0502292
$628$	0	0
$629$	−16.7464	−0.667722
$630$	0	0
$631$	−44.3906	−1.76716	−0.883581	−	0.468278i	$-0.844875\pi$
$631$	−44.3906	−1.76716	−0.883581	+	0.468278i	$0.844875\pi$
$632$	0	0
$633$	11.9295	0.474154
$634$	0	0
$635$	14.3954	0.571263
$636$	0	0
$637$	5.09358	0.201815
$638$	0	0
$639$	−9.64453	−0.381532
$640$	0	0
$641$	−9.69885	−0.383082	−0.191541	−	0.981485i	$-0.561348\pi$
$641$	−9.69885	−0.383082	−0.191541	+	0.981485i	$0.561348\pi$
$642$	0	0
$643$	−47.0623	−1.85596	−0.927978	−	0.372636i	$-0.878454\pi$
$643$	−47.0623	−1.85596	−0.927978	+	0.372636i	$0.878454\pi$
$644$	0	0
$645$	12.6040	0.496282
$646$	0	0
$647$	−27.1225	−1.06629	−0.533147	−	0.846023i	$-0.678991\pi$
$647$	−27.1225	−1.06629	−0.533147	+	0.846023i	$0.678991\pi$
$648$	0	0
$649$	−10.1980	−0.400306
$650$	0	0
$651$	18.7781	0.735970
$652$	0	0
$653$	−18.7094	−0.732156	−0.366078	−	0.930584i	$-0.619300\pi$
$653$	−18.7094	−0.732156	−0.366078	+	0.930584i	$0.619300\pi$
$654$	0	0
$655$	−26.3562	−1.02982
$656$	0	0
$657$	28.6146	1.11636
$658$	0	0
$659$	−1.78625	−0.0695822	−0.0347911	−	0.999395i	$-0.511077\pi$
$659$	−1.78625	−0.0695822	−0.0347911	+	0.999395i	$0.511077\pi$
$660$	0	0
$661$	0.534460	0.0207881	0.0103940	−	0.999946i	$-0.496691\pi$
$661$	0.534460	0.0207881	0.0103940	+	0.999946i	$0.496691\pi$
$662$	0	0
$663$	−9.25015	−0.359246
$664$	0	0
$665$	2.28882	0.0887567
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−10.2524	−0.396380
$670$	0	0
$671$	−16.1844	−0.624791
$672$	0	0
$673$	−15.8447	−0.610767	−0.305383	−	0.952229i	$-0.598785\pi$
$673$	−15.8447	−0.610767	−0.305383	+	0.952229i	$0.598785\pi$
$674$	0	0
$675$	−3.52349	−0.135619
$676$	0	0
$677$	11.1743	0.429464	0.214732	−	0.976673i	$-0.431112\pi$
$677$	11.1743	0.429464	0.214732	+	0.976673i	$0.431112\pi$
$678$	0	0
$679$	12.5087	0.480039
$680$	0	0
$681$	−14.1078	−0.540611
$682$	0	0
$683$	33.7101	1.28988	0.644940	−	0.764233i	$-0.276882\pi$
$683$	33.7101	1.28988	0.644940	+	0.764233i	$0.276882\pi$
$684$	0	0
$685$	−0.577399	−0.0220613
$686$	0	0
$687$	4.42299	0.168748
$688$	0	0
$689$	13.0641	0.497703
$690$	0	0
$691$	−15.3939	−0.585613	−0.292806	−	0.956172i	$-0.594589\pi$
$691$	−15.3939	−0.585613	−0.292806	+	0.956172i	$0.594589\pi$
$692$	0	0
$693$	−41.0716	−1.56018
$694$	0	0
$695$	−12.8715	−0.488245
$696$	0	0
$697$	−63.2751	−2.39671
$698$	0	0
$699$	−4.02240	−0.152141
$700$	0	0
$701$	−20.4093	−0.770849	−0.385424	−	0.922739i	$-0.625945\pi$
$701$	−20.4093	−0.770849	−0.385424	+	0.922739i	$0.625945\pi$
$702$	0	0
$703$	−0.982605	−0.0370596
$704$	0	0
$705$	6.67670	0.251459
$706$	0	0
$707$	−31.9453	−1.20143
$708$	0	0
$709$	−6.36326	−0.238977	−0.119489	−	0.992836i	$-0.538125\pi$
$709$	−6.36326	−0.238977	−0.119489	+	0.992836i	$0.538125\pi$
$710$	0	0
$711$	13.3417	0.500355
$712$	0	0
$713$	36.9116	1.38235
$714$	0	0
$715$	23.5414	0.880399
$716$	0	0
$717$	16.7917	0.627097
$718$	0	0
$719$	−13.4279	−0.500777	−0.250388	−	0.968145i	$-0.580558\pi$
$719$	−13.4279	−0.500777	−0.250388	+	0.968145i	$0.580558\pi$
$720$	0	0
$721$	−23.4092	−0.871804
$722$	0	0
$723$	−16.7293	−0.622168
$724$	0	0
$725$	0	0
$726$	0	0
$727$	16.0514	0.595313	0.297657	−	0.954673i	$-0.403795\pi$
$727$	16.0514	0.595313	0.297657	+	0.954673i	$0.403795\pi$
$728$	0	0
$729$	−7.18380	−0.266067
$730$	0	0
$731$	62.5465	2.31337
$732$	0	0
$733$	10.2275	0.377761	0.188880	−	0.982000i	$-0.439514\pi$
$733$	10.2275	0.377761	0.188880	+	0.982000i	$0.439514\pi$
$734$	0	0
$735$	2.91519	0.107529
$736$	0	0
$737$	13.1139	0.483058
$738$	0	0
$739$	−48.8385	−1.79655	−0.898277	−	0.439429i	$-0.855181\pi$
$739$	−48.8385	−1.79655	−0.898277	+	0.439429i	$0.855181\pi$
$740$	0	0
$741$	−0.542758	−0.0199387
$742$	0	0
$743$	−36.1787	−1.32727	−0.663633	−	0.748058i	$-0.730987\pi$
$743$	−36.1787	−1.32727	−0.663633	+	0.748058i	$0.730987\pi$
$744$	0	0
$745$	24.3154	0.890846
$746$	0	0
$747$	−4.07729	−0.149180
$748$	0	0
$749$	−14.6831	−0.536508
$750$	0	0
$751$	15.0704	0.549927	0.274963	−	0.961455i	$-0.411334\pi$
$751$	15.0704	0.549927	0.274963	+	0.961455i	$0.411334\pi$
$752$	0	0
$753$	2.99741	0.109232
$754$	0	0
$755$	40.1065	1.45962
$756$	0	0
$757$	18.6028	0.676132	0.338066	−	0.941122i	$-0.390227\pi$
$757$	18.6028	0.676132	0.338066	+	0.941122i	$0.390227\pi$
$758$	0	0
$759$	12.8906	0.467899
$760$	0	0
$761$	1.94238	0.0704114	0.0352057	−	0.999380i	$-0.488791\pi$
$761$	1.94238	0.0704114	0.0352057	+	0.999380i	$0.488791\pi$
$762$	0	0
$763$	50.9360	1.84401
$764$	0	0
$765$	33.1569	1.19879
$766$	0	0
$767$	4.40079	0.158903
$768$	0	0
$769$	−26.7488	−0.964587	−0.482293	−	0.876010i	$-0.660196\pi$
$769$	−26.7488	−0.964587	−0.482293	+	0.876010i	$0.660196\pi$
$770$	0	0
$771$	−10.6286	−0.382778
$772$	0	0
$773$	16.4141	0.590375	0.295188	−	0.955439i	$-0.404618\pi$
$773$	16.4141	0.590375	0.295188	+	0.955439i	$0.404618\pi$
$774$	0	0
$775$	−9.42050	−0.338394
$776$	0	0
$777$	−5.12331	−0.183798
$778$	0	0
$779$	−3.71271	−0.133021
$780$	0	0
$781$	−19.4482	−0.695912
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.78883	0.0638459
$786$	0	0
$787$	53.5786	1.90987	0.954936	−	0.296813i	$-0.0959237\pi$
$787$	53.5786	1.90987	0.954936	+	0.296813i	$0.0959237\pi$
$788$	0	0
$789$	10.9351	0.389301
$790$	0	0
$791$	−12.6624	−0.450224
$792$	0	0
$793$	6.98413	0.248014
$794$	0	0
$795$	7.47694	0.265180
$796$	0	0
$797$	−32.5725	−1.15378	−0.576889	−	0.816822i	$-0.695734\pi$
$797$	−32.5725	−1.15378	−0.576889	+	0.816822i	$0.695734\pi$
$798$	0	0
$799$	33.1327	1.17215
$800$	0	0
$801$	46.4632	1.64170
$802$	0	0
$803$	57.7015	2.03624
$804$	0	0
$805$	23.4582	0.826794
$806$	0	0
$807$	17.6425	0.621046
$808$	0	0
$809$	−15.6409	−0.549903	−0.274952	−	0.961458i	$-0.588662\pi$
$809$	−15.6409	−0.549903	−0.274952	+	0.961458i	$0.588662\pi$
$810$	0	0
$811$	−11.1465	−0.391407	−0.195704	−	0.980663i	$-0.562699\pi$
$811$	−11.1465	−0.391407	−0.195704	+	0.980663i	$0.562699\pi$
$812$	0	0
$813$	10.0817	0.353579
$814$	0	0
$815$	33.7015	1.18051
$816$	0	0
$817$	3.66996	0.128396
$818$	0	0
$819$	17.7238	0.619321
$820$	0	0
$821$	13.8965	0.484992	0.242496	−	0.970152i	$-0.422034\pi$
$821$	13.8965	0.484992	0.242496	+	0.970152i	$0.422034\pi$
$822$	0	0
$823$	39.5713	1.37937	0.689684	−	0.724110i	$-0.257749\pi$
$823$	39.5713	1.37937	0.689684	+	0.724110i	$0.257749\pi$
$824$	0	0
$825$	−3.28991	−0.114540
$826$	0	0
$827$	−1.67915	−0.0583897	−0.0291949	−	0.999574i	$-0.509294\pi$
$827$	−1.67915	−0.0583897	−0.0291949	+	0.999574i	$0.509294\pi$
$828$	0	0
$829$	−4.99177	−0.173371	−0.0866856	−	0.996236i	$-0.527628\pi$
$829$	−4.99177	−0.173371	−0.0866856	+	0.996236i	$0.527628\pi$
$830$	0	0
$831$	−7.76604	−0.269401
$832$	0	0
$833$	14.4665	0.501234
$834$	0	0
$835$	−10.5750	−0.365964
$836$	0	0
$837$	34.4713	1.19150
$838$	0	0
$839$	11.8500	0.409109	0.204554	−	0.978855i	$-0.434425\pi$
$839$	11.8500	0.409109	0.204554	+	0.978855i	$0.434425\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	19.8498	0.683663
$844$	0	0
$845$	15.9018	0.547039
$846$	0	0
$847$	−49.3429	−1.69544
$848$	0	0
$849$	−17.1741	−0.589412
$850$	0	0
$851$	−10.0708	−0.345221
$852$	0	0
$853$	−43.2144	−1.47963	−0.739816	−	0.672809i	$-0.765088\pi$
$853$	−43.2144	−1.47963	−0.739816	+	0.672809i	$0.765088\pi$
$854$	0	0
$855$	1.94550	0.0665347
$856$	0	0
$857$	10.5346	0.359857	0.179928	−	0.983680i	$-0.442413\pi$
$857$	10.5346	0.359857	0.179928	+	0.983680i	$0.442413\pi$
$858$	0	0
$859$	−40.8646	−1.39428	−0.697142	−	0.716933i	$-0.745545\pi$
$859$	−40.8646	−1.39428	−0.697142	+	0.716933i	$0.745545\pi$
$860$	0	0
$861$	−19.3581	−0.659721
$862$	0	0
$863$	18.1773	0.618763	0.309381	−	0.950938i	$-0.399878\pi$
$863$	18.1773	0.618763	0.309381	+	0.950938i	$0.399878\pi$
$864$	0	0
$865$	28.9009	0.982660
$866$	0	0
$867$	−15.3459	−0.521176
$868$	0	0
$869$	26.9037	0.912645
$870$	0	0
$871$	−5.65912	−0.191752
$872$	0	0
$873$	10.6324	0.359852
$874$	0	0
$875$	−36.4927	−1.23368
$876$	0	0
$877$	−27.7717	−0.937784	−0.468892	−	0.883255i	$-0.655347\pi$
$877$	−27.7717	−0.937784	−0.468892	+	0.883255i	$0.655347\pi$
$878$	0	0
$879$	4.43147	0.149470
$880$	0	0
$881$	−33.1499	−1.11685	−0.558425	−	0.829555i	$-0.688594\pi$
$881$	−33.1499	−1.11685	−0.558425	+	0.829555i	$0.688594\pi$
$882$	0	0
$883$	50.5600	1.70148	0.850740	−	0.525586i	$-0.176154\pi$
$883$	50.5600	1.70148	0.850740	+	0.525586i	$0.176154\pi$
$884$	0	0
$885$	2.51869	0.0846649
$886$	0	0
$887$	−28.5515	−0.958666	−0.479333	−	0.877633i	$-0.659121\pi$
$887$	−28.5515	−0.958666	−0.479333	+	0.877633i	$0.659121\pi$
$888$	0	0
$889$	21.8548	0.732987
$890$	0	0
$891$	−28.4467	−0.953000
$892$	0	0
$893$	1.94408	0.0650563
$894$	0	0
$895$	−27.0142	−0.902987
$896$	0	0
$897$	−5.56275	−0.185735
$898$	0	0
$899$	0	0
$900$	0	0
$901$	37.1039	1.23611
$902$	0	0
$903$	19.1352	0.636779
$904$	0	0
$905$	18.7739	0.624067
$906$	0	0
$907$	41.9252	1.39210	0.696051	−	0.717992i	$-0.254939\pi$
$907$	41.9252	1.39210	0.696051	+	0.717992i	$0.254939\pi$
$908$	0	0
$909$	−27.1535	−0.900625
$910$	0	0
$911$	54.3057	1.79923	0.899615	−	0.436684i	$-0.143847\pi$
$911$	54.3057	1.79923	0.899615	+	0.436684i	$0.143847\pi$
$912$	0	0
$913$	−8.22186	−0.272104
$914$	0	0
$915$	3.99721	0.132144
$916$	0	0
$917$	−40.0135	−1.32136
$918$	0	0
$919$	−17.2760	−0.569883	−0.284941	−	0.958545i	$-0.591974\pi$
$919$	−17.2760	−0.569883	−0.284941	+	0.958545i	$0.591974\pi$
$920$	0	0
$921$	−17.4372	−0.574575
$922$	0	0
$923$	8.39259	0.276246
$924$	0	0
$925$	2.57024	0.0845089
$926$	0	0
$927$	−19.8978	−0.653530
$928$	0	0
$929$	−16.6181	−0.545222	−0.272611	−	0.962124i	$-0.587887\pi$
$929$	−16.6181	−0.545222	−0.272611	+	0.962124i	$0.587887\pi$
$930$	0	0
$931$	0.848830	0.0278193
$932$	0	0
$933$	−4.25883	−0.139428
$934$	0	0
$935$	66.8609	2.18659
$936$	0	0
$937$	24.6142	0.804111	0.402055	−	0.915615i	$-0.368296\pi$
$937$	24.6142	0.804111	0.402055	+	0.915615i	$0.368296\pi$
$938$	0	0
$939$	−15.1260	−0.493617
$940$	0	0
$941$	−23.0391	−0.751052	−0.375526	−	0.926812i	$-0.622538\pi$
$941$	−23.0391	−0.751052	−0.375526	+	0.926812i	$0.622538\pi$
$942$	0	0
$943$	−38.0517	−1.23913
$944$	0	0
$945$	21.9073	0.712646
$946$	0	0
$947$	−38.0543	−1.23660	−0.618299	−	0.785943i	$-0.712178\pi$
$947$	−38.0543	−1.23660	−0.618299	+	0.785943i	$0.712178\pi$
$948$	0	0
$949$	−24.9003	−0.808296
$950$	0	0
$951$	−13.1581	−0.426681
$952$	0	0
$953$	−29.3692	−0.951361	−0.475681	−	0.879618i	$-0.657798\pi$
$953$	−29.3692	−0.951361	−0.475681	+	0.879618i	$0.657798\pi$
$954$	0	0
$955$	47.5841	1.53979
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.876598	−0.0283068
$960$	0	0
$961$	61.1634	1.97301
$962$	0	0
$963$	−12.4806	−0.402183
$964$	0	0
$965$	−18.3117	−0.589475
$966$	0	0
$967$	−27.0798	−0.870827	−0.435414	−	0.900230i	$-0.643398\pi$
$967$	−27.0798	−0.870827	−0.435414	+	0.900230i	$0.643398\pi$
$968$	0	0
$969$	−1.54151	−0.0495204
$970$	0	0
$971$	39.7852	1.27677	0.638384	−	0.769718i	$-0.279603\pi$
$971$	39.7852	1.27677	0.638384	+	0.769718i	$0.279603\pi$
$972$	0	0
$973$	−19.5413	−0.626467
$974$	0	0
$975$	1.41971	0.0454673
$976$	0	0
$977$	−14.9350	−0.477812	−0.238906	−	0.971043i	$-0.576789\pi$
$977$	−14.9350	−0.477812	−0.238906	+	0.971043i	$0.576789\pi$
$978$	0	0
$979$	93.6932	2.99445
$980$	0	0
$981$	43.2956	1.38232
$982$	0	0
$983$	−44.2130	−1.41018	−0.705088	−	0.709120i	$-0.749092\pi$
$983$	−44.2130	−1.41018	−0.705088	+	0.709120i	$0.749092\pi$
$984$	0	0
$985$	−32.1499	−1.02438
$986$	0	0
$987$	10.1365	0.322647
$988$	0	0
$989$	37.6136	1.19604
$990$	0	0
$991$	−35.6299	−1.13182	−0.565910	−	0.824467i	$-0.691475\pi$
$991$	−35.6299	−1.13182	−0.565910	+	0.824467i	$0.691475\pi$
$992$	0	0
$993$	14.6364	0.464473
$994$	0	0
$995$	−17.4303	−0.552577
$996$	0	0
$997$	60.4950	1.91590	0.957949	−	0.286940i	$-0.0926380\pi$
$997$	60.4950	1.91590	0.957949	+	0.286940i	$0.0926380\pi$
$998$	0	0
$999$	−9.40495	−0.297560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6728.2.a.z.1.6		12
29.23	even	7		232.2.m.d.65.3	yes	24
29.24	even	7		232.2.m.d.25.3	✓	24
29.28	even	2		6728.2.a.bb.1.7		12
116.23	odd	14		464.2.u.i.65.2		24
116.111	odd	14		464.2.u.i.257.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.m.d.25.3	✓	24	29.24	even	7
232.2.m.d.65.3	yes	24	29.23	even	7
464.2.u.i.65.2		24	116.23	odd	14
464.2.u.i.257.2		24	116.111	odd	14
6728.2.a.z.1.6		12	1.1	even	1	trivial
6728.2.a.bb.1.7		12	29.28	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:	\( N \)	\(=\)	\( 6728 = 2^{3} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6728.a (trivial)

\(f(q)\)	\(=\)	\(q-0.642693 q^{3} -2.00467 q^{5} -3.04346 q^{7} -2.58695 q^{9} +O(q^{10})\)	\(q-0.642693 q^{3} -2.00467 q^{5} -3.04346 q^{7} -2.58695 q^{9} -5.21658 q^{11} +2.25114 q^{13} +1.28839 q^{15} +6.39356 q^{17} +0.375146 q^{19} +1.95601 q^{21} +3.84489 q^{23} -0.981284 q^{25} +3.59069 q^{27} +9.60018 q^{31} +3.35266 q^{33} +6.10115 q^{35} -2.61926 q^{37} -1.44679 q^{39} -9.89670 q^{41} +9.78274 q^{43} +5.18598 q^{45} +5.18221 q^{47} +2.26267 q^{49} -4.10909 q^{51} +5.80333 q^{53} +10.4575 q^{55} -0.241104 q^{57} +1.95491 q^{59} +3.10249 q^{61} +7.87327 q^{63} -4.51280 q^{65} -2.51389 q^{67} -2.47108 q^{69} +3.72815 q^{71} -11.0612 q^{73} +0.630665 q^{75} +15.8765 q^{77} -5.15734 q^{79} +5.45312 q^{81} +1.57610 q^{83} -12.8170 q^{85} -17.9607 q^{89} -6.85126 q^{91} -6.16997 q^{93} -0.752045 q^{95} -4.11002 q^{97} +13.4950 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9	\( 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9} - 3 q^{11} - 3 q^{13} + 3 q^{15} - 8 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 12 q^{25} - 7 q^{27} - 29 q^{31} - 46 q^{33} + 17 q^{35} - 38 q^{37} + 10 q^{39} - 11 q^{41} - 9 q^{43} - 54 q^{45} - 34 q^{47} + 33 q^{49} + 17 q^{51} - 15 q^{53} - 2 q^{55} - q^{57} + 57 q^{59} - 37 q^{61} + 9 q^{63} - 59 q^{65} + 33 q^{67} + 21 q^{69} - 21 q^{71} - 13 q^{73} - 13 q^{75} + 3 q^{77} - 32 q^{79} + 36 q^{81} + 48 q^{83} - 17 q^{85} - 20 q^{89} - 2 q^{91} - 37 q^{93} - 7 q^{97} + 16 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9 - 3 * q^11 - 3 * q^13 + 3 * q^15 - 8 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 12 * q^25 - 7 * q^27 - 29 * q^31 - 46 * q^33 + 17 * q^35 - 38 * q^37 + 10 * q^39 - 11 * q^41 - 9 * q^43 - 54 * q^45 - 34 * q^47 + 33 * q^49 + 17 * q^51 - 15 * q^53 - 2 * q^55 - q^57 + 57 * q^59 - 37 * q^61 + 9 * q^63 - 59 * q^65 + 33 * q^67 + 21 * q^69 - 21 * q^71 - 13 * q^73 - 13 * q^75 + 3 * q^77 - 32 * q^79 + 36 * q^81 + 48 * q^83 - 17 * q^85 - 20 * q^89 - 2 * q^91 - 37 * q^93 - 7 * q^97 + 16 * q^99

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