LMFDB - Embedded newform 7728.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.12311 q^{11} +0.561553 q^{13} +2.56155 q^{15} +7.12311 q^{17} -3.12311 q^{19} +1.00000 q^{21} -1.00000 q^{23} +1.56155 q^{25} +1.00000 q^{27} +3.43845 q^{29} +5.12311 q^{31} +3.12311 q^{33} +2.56155 q^{35} +2.56155 q^{37} +0.561553 q^{39} -0.561553 q^{41} +5.68466 q^{43} +2.56155 q^{45} -6.56155 q^{47} +1.00000 q^{49} +7.12311 q^{51} -2.24621 q^{53} +8.00000 q^{55} -3.12311 q^{57} -13.1231 q^{59} +9.12311 q^{61} +1.00000 q^{63} +1.43845 q^{65} -7.12311 q^{67} -1.00000 q^{69} -15.3693 q^{71} +0.876894 q^{73} +1.56155 q^{75} +3.12311 q^{77} +2.24621 q^{79} +1.00000 q^{81} +0.876894 q^{83} +18.2462 q^{85} +3.43845 q^{87} +14.0000 q^{89} +0.561553 q^{91} +5.12311 q^{93} -8.00000 q^{95} -16.5616 q^{97} +3.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 3 q^{13} + q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{23} - q^{25} + 2 q^{27} + 11 q^{29} + 2 q^{31} - 2 q^{33} + q^{35} + q^{37} - 3 q^{39} + 3 q^{41} - q^{43} + q^{45} - 9 q^{47} + 2 q^{49} + 6 q^{51} + 12 q^{53} + 16 q^{55} + 2 q^{57} - 18 q^{59} + 10 q^{61} + 2 q^{63} + 7 q^{65} - 6 q^{67} - 2 q^{69} - 6 q^{71} + 10 q^{73} - q^{75} - 2 q^{77} - 12 q^{79} + 2 q^{81} + 10 q^{83} + 20 q^{85} + 11 q^{87} + 28 q^{89} - 3 q^{91} + 2 q^{93} - 16 q^{95} - 29 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 - 3 * q^13 + q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 2 * q^23 - q^25 + 2 * q^27 + 11 * q^29 + 2 * q^31 - 2 * q^33 + q^35 + q^37 - 3 * q^39 + 3 * q^41 - q^43 + q^45 - 9 * q^47 + 2 * q^49 + 6 * q^51 + 12 * q^53 + 16 * q^55 + 2 * q^57 - 18 * q^59 + 10 * q^61 + 2 * q^63 + 7 * q^65 - 6 * q^67 - 2 * q^69 - 6 * q^71 + 10 * q^73 - q^75 - 2 * q^77 - 12 * q^79 + 2 * q^81 + 10 * q^83 + 20 * q^85 + 11 * q^87 + 28 * q^89 - 3 * q^91 + 2 * q^93 - 16 * q^95 - 29 * q^97 - 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.56155	1.14556	0.572781	−	0.819709i	$-0.305865\pi$
$5$	2.56155	1.14556	0.572781	+	0.819709i	$0.305865\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.12311	0.941652	0.470826	−	0.882226i	$-0.343956\pi$
$11$	3.12311	0.941652	0.470826	+	0.882226i	$0.343956\pi$
$12$	0	0
$13$	0.561553	0.155747	0.0778734	−	0.996963i	$-0.475187\pi$
$13$	0.561553	0.155747	0.0778734	+	0.996963i	$0.475187\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	7.12311	1.72761	0.863803	−	0.503829i	$-0.168076\pi$
$17$	7.12311	1.72761	0.863803	+	0.503829i	$0.168076\pi$
$18$	0	0
$19$	−3.12311	−0.716490	−0.358245	−	0.933628i	$-0.616625\pi$
$19$	−3.12311	−0.716490	−0.358245	+	0.933628i	$0.616625\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.56155	0.312311
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.43845	0.638504	0.319252	−	0.947670i	$-0.396568\pi$
$29$	3.43845	0.638504	0.319252	+	0.947670i	$0.396568\pi$
$30$	0	0
$31$	5.12311	0.920137	0.460068	−	0.887883i	$-0.347825\pi$
$31$	5.12311	0.920137	0.460068	+	0.887883i	$0.347825\pi$
$32$	0	0
$33$	3.12311	0.543663
$34$	0	0
$35$	2.56155	0.432981
$36$	0	0
$37$	2.56155	0.421117	0.210558	−	0.977581i	$-0.432472\pi$
$37$	2.56155	0.421117	0.210558	+	0.977581i	$0.432472\pi$
$38$	0	0
$39$	0.561553	0.0899204
$40$	0	0
$41$	−0.561553	−0.0876998	−0.0438499	−	0.999038i	$-0.513962\pi$
$41$	−0.561553	−0.0876998	−0.0438499	+	0.999038i	$0.513962\pi$
$42$	0	0
$43$	5.68466	0.866902	0.433451	−	0.901177i	$-0.357296\pi$
$43$	5.68466	0.866902	0.433451	+	0.901177i	$0.357296\pi$
$44$	0	0
$45$	2.56155	0.381854
$46$	0	0
$47$	−6.56155	−0.957101	−0.478550	−	0.878060i	$-0.658838\pi$
$47$	−6.56155	−0.957101	−0.478550	+	0.878060i	$0.658838\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.12311	0.997434
$52$	0	0
$53$	−2.24621	−0.308541	−0.154270	−	0.988029i	$-0.549303\pi$
$53$	−2.24621	−0.308541	−0.154270	+	0.988029i	$0.549303\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	−3.12311	−0.413665
$58$	0	0
$59$	−13.1231	−1.70848	−0.854241	−	0.519877i	$-0.825978\pi$
$59$	−13.1231	−1.70848	−0.854241	+	0.519877i	$0.825978\pi$
$60$	0	0
$61$	9.12311	1.16809	0.584047	−	0.811720i	$-0.301468\pi$
$61$	9.12311	1.16809	0.584047	+	0.811720i	$0.301468\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	1.43845	0.178417
$66$	0	0
$67$	−7.12311	−0.870226	−0.435113	−	0.900376i	$-0.643292\pi$
$67$	−7.12311	−0.870226	−0.435113	+	0.900376i	$0.643292\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−15.3693	−1.82400	−0.912001	−	0.410188i	$-0.865463\pi$
$71$	−15.3693	−1.82400	−0.912001	+	0.410188i	$0.865463\pi$
$72$	0	0
$73$	0.876894	0.102633	0.0513164	−	0.998682i	$-0.483658\pi$
$73$	0.876894	0.102633	0.0513164	+	0.998682i	$0.483658\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	3.12311	0.355911
$78$	0	0
$79$	2.24621	0.252719	0.126359	−	0.991985i	$-0.459671\pi$
$79$	2.24621	0.252719	0.126359	+	0.991985i	$0.459671\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.876894	0.0962517	0.0481258	−	0.998841i	$-0.484675\pi$
$83$	0.876894	0.0962517	0.0481258	+	0.998841i	$0.484675\pi$
$84$	0	0
$85$	18.2462	1.97908
$86$	0	0
$87$	3.43845	0.368640
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0.561553	0.0588667
$92$	0	0
$93$	5.12311	0.531241
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−16.5616	−1.68157	−0.840785	−	0.541368i	$-0.817906\pi$
$97$	−16.5616	−1.68157	−0.840785	+	0.541368i	$0.817906\pi$
$98$	0	0
$99$	3.12311	0.313884
$100$	0	0
$101$	−0.246211	−0.0244989	−0.0122495	−	0.999925i	$-0.503899\pi$
$101$	−0.246211	−0.0244989	−0.0122495	+	0.999925i	$0.503899\pi$
$102$	0	0
$103$	9.93087	0.978518	0.489259	−	0.872139i	$-0.337267\pi$
$103$	9.93087	0.978518	0.489259	+	0.872139i	$0.337267\pi$
$104$	0	0
$105$	2.56155	0.249982
$106$	0	0
$107$	−3.12311	−0.301922	−0.150961	−	0.988540i	$-0.548237\pi$
$107$	−3.12311	−0.301922	−0.150961	+	0.988540i	$0.548237\pi$
$108$	0	0
$109$	14.5616	1.39474	0.697372	−	0.716709i	$-0.254353\pi$
$109$	14.5616	1.39474	0.697372	+	0.716709i	$0.254353\pi$
$110$	0	0
$111$	2.56155	0.243132
$112$	0	0
$113$	9.68466	0.911056	0.455528	−	0.890221i	$-0.349450\pi$
$113$	9.68466	0.911056	0.455528	+	0.890221i	$0.349450\pi$
$114$	0	0
$115$	−2.56155	−0.238866
$116$	0	0
$117$	0.561553	0.0519156
$118$	0	0
$119$	7.12311	0.652974
$120$	0	0
$121$	−1.24621	−0.113292
$122$	0	0
$123$	−0.561553	−0.0506335
$124$	0	0
$125$	−8.80776	−0.787790
$126$	0	0
$127$	4.31534	0.382925	0.191462	−	0.981500i	$-0.438677\pi$
$127$	4.31534	0.382925	0.191462	+	0.981500i	$0.438677\pi$
$128$	0	0
$129$	5.68466	0.500506
$130$	0	0
$131$	−13.1231	−1.14657	−0.573286	−	0.819356i	$-0.694331\pi$
$131$	−13.1231	−1.14657	−0.573286	+	0.819356i	$0.694331\pi$
$132$	0	0
$133$	−3.12311	−0.270808
$134$	0	0
$135$	2.56155	0.220463
$136$	0	0
$137$	−4.56155	−0.389720	−0.194860	−	0.980831i	$-0.562425\pi$
$137$	−4.56155	−0.389720	−0.194860	+	0.980831i	$0.562425\pi$
$138$	0	0
$139$	16.8078	1.42562	0.712808	−	0.701359i	$-0.247423\pi$
$139$	16.8078	1.42562	0.712808	+	0.701359i	$0.247423\pi$
$140$	0	0
$141$	−6.56155	−0.552582
$142$	0	0
$143$	1.75379	0.146659
$144$	0	0
$145$	8.80776	0.731445
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	10.2462	0.839402	0.419701	−	0.907662i	$-0.362135\pi$
$149$	10.2462	0.839402	0.419701	+	0.907662i	$0.362135\pi$
$150$	0	0
$151$	−19.6847	−1.60191	−0.800957	−	0.598721i	$-0.795676\pi$
$151$	−19.6847	−1.60191	−0.800957	+	0.598721i	$0.795676\pi$
$152$	0	0
$153$	7.12311	0.575869
$154$	0	0
$155$	13.1231	1.05407
$156$	0	0
$157$	−2.24621	−0.179267	−0.0896336	−	0.995975i	$-0.528570\pi$
$157$	−2.24621	−0.179267	−0.0896336	+	0.995975i	$0.528570\pi$
$158$	0	0
$159$	−2.24621	−0.178136
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	−2.24621	−0.173817	−0.0869085	−	0.996216i	$-0.527699\pi$
$167$	−2.24621	−0.173817	−0.0869085	+	0.996216i	$0.527699\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	−3.12311	−0.238830
$172$	0	0
$173$	−14.4924	−1.10184	−0.550919	−	0.834559i	$-0.685723\pi$
$173$	−14.4924	−1.10184	−0.550919	+	0.834559i	$0.685723\pi$
$174$	0	0
$175$	1.56155	0.118042
$176$	0	0
$177$	−13.1231	−0.986393
$178$	0	0
$179$	−16.8078	−1.25627	−0.628136	−	0.778104i	$-0.716182\pi$
$179$	−16.8078	−1.25627	−0.628136	+	0.778104i	$0.716182\pi$
$180$	0	0
$181$	−6.87689	−0.511156	−0.255578	−	0.966789i	$-0.582266\pi$
$181$	−6.87689	−0.511156	−0.255578	+	0.966789i	$0.582266\pi$
$182$	0	0
$183$	9.12311	0.674399
$184$	0	0
$185$	6.56155	0.482415
$186$	0	0
$187$	22.2462	1.62680
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	−9.68466	−0.697117	−0.348558	−	0.937287i	$-0.613329\pi$
$193$	−9.68466	−0.697117	−0.348558	+	0.937287i	$0.613329\pi$
$194$	0	0
$195$	1.43845	0.103009
$196$	0	0
$197$	−3.93087	−0.280063	−0.140031	−	0.990147i	$-0.544720\pi$
$197$	−3.93087	−0.280063	−0.140031	+	0.990147i	$0.544720\pi$
$198$	0	0
$199$	−24.1771	−1.71387	−0.856934	−	0.515426i	$-0.827634\pi$
$199$	−24.1771	−1.71387	−0.856934	+	0.515426i	$0.827634\pi$
$200$	0	0
$201$	−7.12311	−0.502425
$202$	0	0
$203$	3.43845	0.241332
$204$	0	0
$205$	−1.43845	−0.100466
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−9.75379	−0.674684
$210$	0	0
$211$	−16.4924	−1.13539	−0.567693	−	0.823241i	$-0.692164\pi$
$211$	−16.4924	−1.13539	−0.567693	+	0.823241i	$0.692164\pi$
$212$	0	0
$213$	−15.3693	−1.05309
$214$	0	0
$215$	14.5616	0.993090
$216$	0	0
$217$	5.12311	0.347779
$218$	0	0
$219$	0.876894	0.0592550
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	5.12311	0.343069	0.171534	−	0.985178i	$-0.445128\pi$
$223$	5.12311	0.343069	0.171534	+	0.985178i	$0.445128\pi$
$224$	0	0
$225$	1.56155	0.104104
$226$	0	0
$227$	26.8078	1.77929	0.889647	−	0.456649i	$-0.150951\pi$
$227$	26.8078	1.77929	0.889647	+	0.456649i	$0.150951\pi$
$228$	0	0
$229$	−14.2462	−0.941416	−0.470708	−	0.882289i	$-0.656001\pi$
$229$	−14.2462	−0.941416	−0.470708	+	0.882289i	$0.656001\pi$
$230$	0	0
$231$	3.12311	0.205485
$232$	0	0
$233$	14.4924	0.949430	0.474715	−	0.880140i	$-0.342551\pi$
$233$	14.4924	0.949430	0.474715	+	0.880140i	$0.342551\pi$
$234$	0	0
$235$	−16.8078	−1.09642
$236$	0	0
$237$	2.24621	0.145907
$238$	0	0
$239$	28.4924	1.84302	0.921511	−	0.388353i	$-0.126956\pi$
$239$	28.4924	1.84302	0.921511	+	0.388353i	$0.126956\pi$
$240$	0	0
$241$	17.0540	1.09854	0.549272	−	0.835644i	$-0.314905\pi$
$241$	17.0540	1.09854	0.549272	+	0.835644i	$0.314905\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.56155	0.163652
$246$	0	0
$247$	−1.75379	−0.111591
$248$	0	0
$249$	0.876894	0.0555709
$250$	0	0
$251$	4.56155	0.287923	0.143961	−	0.989583i	$-0.454016\pi$
$251$	4.56155	0.287923	0.143961	+	0.989583i	$0.454016\pi$
$252$	0	0
$253$	−3.12311	−0.196348
$254$	0	0
$255$	18.2462	1.14262
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	2.56155	0.159167
$260$	0	0
$261$	3.43845	0.212835
$262$	0	0
$263$	5.93087	0.365713	0.182857	−	0.983140i	$-0.441466\pi$
$263$	5.93087	0.365713	0.182857	+	0.983140i	$0.441466\pi$
$264$	0	0
$265$	−5.75379	−0.353452
$266$	0	0
$267$	14.0000	0.856786
$268$	0	0
$269$	−4.24621	−0.258896	−0.129448	−	0.991586i	$-0.541321\pi$
$269$	−4.24621	−0.258896	−0.129448	+	0.991586i	$0.541321\pi$
$270$	0	0
$271$	2.24621	0.136448	0.0682238	−	0.997670i	$-0.478267\pi$
$271$	2.24621	0.136448	0.0682238	+	0.997670i	$0.478267\pi$
$272$	0	0
$273$	0.561553	0.0339867
$274$	0	0
$275$	4.87689	0.294088
$276$	0	0
$277$	4.87689	0.293024	0.146512	−	0.989209i	$-0.453195\pi$
$277$	4.87689	0.293024	0.146512	+	0.989209i	$0.453195\pi$
$278$	0	0
$279$	5.12311	0.306712
$280$	0	0
$281$	−23.9309	−1.42760	−0.713798	−	0.700352i	$-0.753027\pi$
$281$	−23.9309	−1.42760	−0.713798	+	0.700352i	$0.753027\pi$
$282$	0	0
$283$	−22.4924	−1.33704	−0.668518	−	0.743696i	$-0.733071\pi$
$283$	−22.4924	−1.33704	−0.668518	+	0.743696i	$0.733071\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−0.561553	−0.0331474
$288$	0	0
$289$	33.7386	1.98463
$290$	0	0
$291$	−16.5616	−0.970855
$292$	0	0
$293$	−13.6155	−0.795428	−0.397714	−	0.917510i	$-0.630196\pi$
$293$	−13.6155	−0.795428	−0.397714	+	0.917510i	$0.630196\pi$
$294$	0	0
$295$	−33.6155	−1.95717
$296$	0	0
$297$	3.12311	0.181221
$298$	0	0
$299$	−0.561553	−0.0324754
$300$	0	0
$301$	5.68466	0.327658
$302$	0	0
$303$	−0.246211	−0.0141445
$304$	0	0
$305$	23.3693	1.33812
$306$	0	0
$307$	25.9309	1.47995	0.739976	−	0.672633i	$-0.234837\pi$
$307$	25.9309	1.47995	0.739976	+	0.672633i	$0.234837\pi$
$308$	0	0
$309$	9.93087	0.564947
$310$	0	0
$311$	30.7386	1.74303	0.871514	−	0.490371i	$-0.163139\pi$
$311$	30.7386	1.74303	0.871514	+	0.490371i	$0.163139\pi$
$312$	0	0
$313$	−12.2462	−0.692197	−0.346098	−	0.938198i	$-0.612494\pi$
$313$	−12.2462	−0.692197	−0.346098	+	0.938198i	$0.612494\pi$
$314$	0	0
$315$	2.56155	0.144327
$316$	0	0
$317$	2.80776	0.157700	0.0788499	−	0.996887i	$-0.474875\pi$
$317$	2.80776	0.157700	0.0788499	+	0.996887i	$0.474875\pi$
$318$	0	0
$319$	10.7386	0.601248
$320$	0	0
$321$	−3.12311	−0.174315
$322$	0	0
$323$	−22.2462	−1.23781
$324$	0	0
$325$	0.876894	0.0486413
$326$	0	0
$327$	14.5616	0.805256
$328$	0	0
$329$	−6.56155	−0.361750
$330$	0	0
$331$	−8.49242	−0.466786	−0.233393	−	0.972383i	$-0.574983\pi$
$331$	−8.49242	−0.466786	−0.233393	+	0.972383i	$0.574983\pi$
$332$	0	0
$333$	2.56155	0.140372
$334$	0	0
$335$	−18.2462	−0.996897
$336$	0	0
$337$	2.63068	0.143302	0.0716512	−	0.997430i	$-0.477173\pi$
$337$	2.63068	0.143302	0.0716512	+	0.997430i	$0.477173\pi$
$338$	0	0
$339$	9.68466	0.525998
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.56155	−0.137909
$346$	0	0
$347$	13.9309	0.747848	0.373924	−	0.927459i	$-0.378012\pi$
$347$	13.9309	0.747848	0.373924	+	0.927459i	$0.378012\pi$
$348$	0	0
$349$	0.246211	0.0131794	0.00658969	−	0.999978i	$-0.497902\pi$
$349$	0.246211	0.0131794	0.00658969	+	0.999978i	$0.497902\pi$
$350$	0	0
$351$	0.561553	0.0299735
$352$	0	0
$353$	−13.1922	−0.702152	−0.351076	−	0.936347i	$-0.614184\pi$
$353$	−13.1922	−0.702152	−0.351076	+	0.936347i	$0.614184\pi$
$354$	0	0
$355$	−39.3693	−2.08951
$356$	0	0
$357$	7.12311	0.376995
$358$	0	0
$359$	10.5616	0.557417	0.278709	−	0.960376i	$-0.410094\pi$
$359$	10.5616	0.557417	0.278709	+	0.960376i	$0.410094\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	−1.24621	−0.0654091
$364$	0	0
$365$	2.24621	0.117572
$366$	0	0
$367$	22.5616	1.17770	0.588852	−	0.808241i	$-0.299580\pi$
$367$	22.5616	1.17770	0.588852	+	0.808241i	$0.299580\pi$
$368$	0	0
$369$	−0.561553	−0.0292333
$370$	0	0
$371$	−2.24621	−0.116617
$372$	0	0
$373$	6.87689	0.356072	0.178036	−	0.984024i	$-0.443026\pi$
$373$	6.87689	0.356072	0.178036	+	0.984024i	$0.443026\pi$
$374$	0	0
$375$	−8.80776	−0.454831
$376$	0	0
$377$	1.93087	0.0994448
$378$	0	0
$379$	17.0540	0.876004	0.438002	−	0.898974i	$-0.355686\pi$
$379$	17.0540	0.876004	0.438002	+	0.898974i	$0.355686\pi$
$380$	0	0
$381$	4.31534	0.221082
$382$	0	0
$383$	−10.2462	−0.523557	−0.261778	−	0.965128i	$-0.584309\pi$
$383$	−10.2462	−0.523557	−0.261778	+	0.965128i	$0.584309\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	0	0
$387$	5.68466	0.288967
$388$	0	0
$389$	−4.63068	−0.234785	−0.117392	−	0.993086i	$-0.537454\pi$
$389$	−4.63068	−0.234785	−0.117392	+	0.993086i	$0.537454\pi$
$390$	0	0
$391$	−7.12311	−0.360231
$392$	0	0
$393$	−13.1231	−0.661973
$394$	0	0
$395$	5.75379	0.289505
$396$	0	0
$397$	−24.2462	−1.21688	−0.608441	−	0.793599i	$-0.708205\pi$
$397$	−24.2462	−1.21688	−0.608441	+	0.793599i	$0.708205\pi$
$398$	0	0
$399$	−3.12311	−0.156351
$400$	0	0
$401$	−0.246211	−0.0122952	−0.00614760	−	0.999981i	$-0.501957\pi$
$401$	−0.246211	−0.0122952	−0.00614760	+	0.999981i	$0.501957\pi$
$402$	0	0
$403$	2.87689	0.143308
$404$	0	0
$405$	2.56155	0.127285
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−7.61553	−0.376564	−0.188282	−	0.982115i	$-0.560292\pi$
$409$	−7.61553	−0.376564	−0.188282	+	0.982115i	$0.560292\pi$
$410$	0	0
$411$	−4.56155	−0.225005
$412$	0	0
$413$	−13.1231	−0.645746
$414$	0	0
$415$	2.24621	0.110262
$416$	0	0
$417$	16.8078	0.823080
$418$	0	0
$419$	33.8617	1.65425	0.827127	−	0.562015i	$-0.189974\pi$
$419$	33.8617	1.65425	0.827127	+	0.562015i	$0.189974\pi$
$420$	0	0
$421$	−12.1771	−0.593475	−0.296737	−	0.954959i	$-0.595899\pi$
$421$	−12.1771	−0.593475	−0.296737	+	0.954959i	$0.595899\pi$
$422$	0	0
$423$	−6.56155	−0.319034
$424$	0	0
$425$	11.1231	0.539550
$426$	0	0
$427$	9.12311	0.441498
$428$	0	0
$429$	1.75379	0.0846737
$430$	0	0
$431$	15.6847	0.755503	0.377752	−	0.925907i	$-0.376697\pi$
$431$	15.6847	0.755503	0.377752	+	0.925907i	$0.376697\pi$
$432$	0	0
$433$	2.31534	0.111268	0.0556341	−	0.998451i	$-0.482282\pi$
$433$	2.31534	0.111268	0.0556341	+	0.998451i	$0.482282\pi$
$434$	0	0
$435$	8.80776	0.422300
$436$	0	0
$437$	3.12311	0.149398
$438$	0	0
$439$	−17.6155	−0.840743	−0.420372	−	0.907352i	$-0.638100\pi$
$439$	−17.6155	−0.840743	−0.420372	+	0.907352i	$0.638100\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−37.9309	−1.80215	−0.901075	−	0.433663i	$-0.857221\pi$
$443$	−37.9309	−1.80215	−0.901075	+	0.433663i	$0.857221\pi$
$444$	0	0
$445$	35.8617	1.70001
$446$	0	0
$447$	10.2462	0.484629
$448$	0	0
$449$	16.2462	0.766706	0.383353	−	0.923602i	$-0.374769\pi$
$449$	16.2462	0.766706	0.383353	+	0.923602i	$0.374769\pi$
$450$	0	0
$451$	−1.75379	−0.0825827
$452$	0	0
$453$	−19.6847	−0.924866
$454$	0	0
$455$	1.43845	0.0674354
$456$	0	0
$457$	−17.3693	−0.812502	−0.406251	−	0.913761i	$-0.633164\pi$
$457$	−17.3693	−0.812502	−0.406251	+	0.913761i	$0.633164\pi$
$458$	0	0
$459$	7.12311	0.332478
$460$	0	0
$461$	−8.73863	−0.406999	−0.203499	−	0.979075i	$-0.565231\pi$
$461$	−8.73863	−0.406999	−0.203499	+	0.979075i	$0.565231\pi$
$462$	0	0
$463$	9.43845	0.438642	0.219321	−	0.975653i	$-0.429616\pi$
$463$	9.43845	0.438642	0.219321	+	0.975653i	$0.429616\pi$
$464$	0	0
$465$	13.1231	0.608569
$466$	0	0
$467$	9.68466	0.448153	0.224076	−	0.974572i	$-0.428064\pi$
$467$	9.68466	0.448153	0.224076	+	0.974572i	$0.428064\pi$
$468$	0	0
$469$	−7.12311	−0.328914
$470$	0	0
$471$	−2.24621	−0.103500
$472$	0	0
$473$	17.7538	0.816320
$474$	0	0
$475$	−4.87689	−0.223767
$476$	0	0
$477$	−2.24621	−0.102847
$478$	0	0
$479$	9.61553	0.439345	0.219672	−	0.975574i	$-0.429501\pi$
$479$	9.61553	0.439345	0.219672	+	0.975574i	$0.429501\pi$
$480$	0	0
$481$	1.43845	0.0655875
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−42.4233	−1.92634
$486$	0	0
$487$	1.43845	0.0651823	0.0325911	−	0.999469i	$-0.489624\pi$
$487$	1.43845	0.0651823	0.0325911	+	0.999469i	$0.489624\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	−22.7386	−1.02618	−0.513090	−	0.858335i	$-0.671499\pi$
$491$	−22.7386	−1.02618	−0.513090	+	0.858335i	$0.671499\pi$
$492$	0	0
$493$	24.4924	1.10308
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	−15.3693	−0.689408
$498$	0	0
$499$	−16.6307	−0.744492	−0.372246	−	0.928134i	$-0.621412\pi$
$499$	−16.6307	−0.744492	−0.372246	+	0.928134i	$0.621412\pi$
$500$	0	0
$501$	−2.24621	−0.100353
$502$	0	0
$503$	−42.1080	−1.87750	−0.938750	−	0.344598i	$-0.888015\pi$
$503$	−42.1080	−1.87750	−0.938750	+	0.344598i	$0.888015\pi$
$504$	0	0
$505$	−0.630683	−0.0280650
$506$	0	0
$507$	−12.6847	−0.563345
$508$	0	0
$509$	21.3693	0.947178	0.473589	−	0.880746i	$-0.342958\pi$
$509$	21.3693	0.947178	0.473589	+	0.880746i	$0.342958\pi$
$510$	0	0
$511$	0.876894	0.0387915
$512$	0	0
$513$	−3.12311	−0.137888
$514$	0	0
$515$	25.4384	1.12095
$516$	0	0
$517$	−20.4924	−0.901256
$518$	0	0
$519$	−14.4924	−0.636147
$520$	0	0
$521$	−16.2462	−0.711759	−0.355880	−	0.934532i	$-0.615819\pi$
$521$	−16.2462	−0.711759	−0.355880	+	0.934532i	$0.615819\pi$
$522$	0	0
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	0	0
$525$	1.56155	0.0681518
$526$	0	0
$527$	36.4924	1.58963
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−13.1231	−0.569494
$532$	0	0
$533$	−0.315342	−0.0136590
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	−16.8078	−0.725309
$538$	0	0
$539$	3.12311	0.134522
$540$	0	0
$541$	9.36932	0.402818	0.201409	−	0.979507i	$-0.435448\pi$
$541$	9.36932	0.402818	0.201409	+	0.979507i	$0.435448\pi$
$542$	0	0
$543$	−6.87689	−0.295116
$544$	0	0
$545$	37.3002	1.59776
$546$	0	0
$547$	−37.6155	−1.60832	−0.804162	−	0.594410i	$-0.797386\pi$
$547$	−37.6155	−1.60832	−0.804162	+	0.594410i	$0.797386\pi$
$548$	0	0
$549$	9.12311	0.389365
$550$	0	0
$551$	−10.7386	−0.457481
$552$	0	0
$553$	2.24621	0.0955186
$554$	0	0
$555$	6.56155	0.278522
$556$	0	0
$557$	36.9848	1.56710	0.783549	−	0.621330i	$-0.213407\pi$
$557$	36.9848	1.56710	0.783549	+	0.621330i	$0.213407\pi$
$558$	0	0
$559$	3.19224	0.135017
$560$	0	0
$561$	22.2462	0.939236
$562$	0	0
$563$	−15.4384	−0.650653	−0.325326	−	0.945602i	$-0.605474\pi$
$563$	−15.4384	−0.650653	−0.325326	+	0.945602i	$0.605474\pi$
$564$	0	0
$565$	24.8078	1.04367
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	25.5464	1.07096	0.535480	−	0.844548i	$-0.320131\pi$
$569$	25.5464	1.07096	0.535480	+	0.844548i	$0.320131\pi$
$570$	0	0
$571$	−18.6307	−0.779670	−0.389835	−	0.920885i	$-0.627468\pi$
$571$	−18.6307	−0.779670	−0.389835	+	0.920885i	$0.627468\pi$
$572$	0	0
$573$	20.0000	0.835512
$574$	0	0
$575$	−1.56155	−0.0651213
$576$	0	0
$577$	−36.2462	−1.50895	−0.754475	−	0.656329i	$-0.772108\pi$
$577$	−36.2462	−1.50895	−0.754475	+	0.656329i	$0.772108\pi$
$578$	0	0
$579$	−9.68466	−0.402481
$580$	0	0
$581$	0.876894	0.0363797
$582$	0	0
$583$	−7.01515	−0.290538
$584$	0	0
$585$	1.43845	0.0594725
$586$	0	0
$587$	−40.9848	−1.69163	−0.845813	−	0.533480i	$-0.820884\pi$
$587$	−40.9848	−1.69163	−0.845813	+	0.533480i	$0.820884\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−3.93087	−0.161694
$592$	0	0
$593$	−23.3002	−0.956824	−0.478412	−	0.878136i	$-0.658787\pi$
$593$	−23.3002	−0.956824	−0.478412	+	0.878136i	$0.658787\pi$
$594$	0	0
$595$	18.2462	0.748022
$596$	0	0
$597$	−24.1771	−0.989502
$598$	0	0
$599$	−38.7386	−1.58282	−0.791409	−	0.611287i	$-0.790652\pi$
$599$	−38.7386	−1.58282	−0.791409	+	0.611287i	$0.790652\pi$
$600$	0	0
$601$	21.3693	0.871673	0.435836	−	0.900026i	$-0.356453\pi$
$601$	21.3693	0.871673	0.435836	+	0.900026i	$0.356453\pi$
$602$	0	0
$603$	−7.12311	−0.290075
$604$	0	0
$605$	−3.19224	−0.129783
$606$	0	0
$607$	6.38447	0.259138	0.129569	−	0.991570i	$-0.458641\pi$
$607$	6.38447	0.259138	0.129569	+	0.991570i	$0.458641\pi$
$608$	0	0
$609$	3.43845	0.139333
$610$	0	0
$611$	−3.68466	−0.149065
$612$	0	0
$613$	−16.8078	−0.678859	−0.339430	−	0.940631i	$-0.610234\pi$
$613$	−16.8078	−0.678859	−0.339430	+	0.940631i	$0.610234\pi$
$614$	0	0
$615$	−1.43845	−0.0580038
$616$	0	0
$617$	48.2462	1.94232	0.971160	−	0.238430i	$-0.0766328\pi$
$617$	48.2462	1.94232	0.971160	+	0.238430i	$0.0766328\pi$
$618$	0	0
$619$	8.24621	0.331443	0.165722	−	0.986173i	$-0.447005\pi$
$619$	8.24621	0.331443	0.165722	+	0.986173i	$0.447005\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	14.0000	0.560898
$624$	0	0
$625$	−30.3693	−1.21477
$626$	0	0
$627$	−9.75379	−0.389529
$628$	0	0
$629$	18.2462	0.727524
$630$	0	0
$631$	24.4924	0.975028	0.487514	−	0.873115i	$-0.337904\pi$
$631$	24.4924	0.975028	0.487514	+	0.873115i	$0.337904\pi$
$632$	0	0
$633$	−16.4924	−0.655515
$634$	0	0
$635$	11.0540	0.438664
$636$	0	0
$637$	0.561553	0.0222495
$638$	0	0
$639$	−15.3693	−0.608001
$640$	0	0
$641$	−25.6847	−1.01448	−0.507242	−	0.861804i	$-0.669335\pi$
$641$	−25.6847	−1.01448	−0.507242	+	0.861804i	$0.669335\pi$
$642$	0	0
$643$	38.0000	1.49857	0.749287	−	0.662246i	$-0.230396\pi$
$643$	38.0000	1.49857	0.749287	+	0.662246i	$0.230396\pi$
$644$	0	0
$645$	14.5616	0.573361
$646$	0	0
$647$	36.4924	1.43467	0.717333	−	0.696731i	$-0.245363\pi$
$647$	36.4924	1.43467	0.717333	+	0.696731i	$0.245363\pi$
$648$	0	0
$649$	−40.9848	−1.60880
$650$	0	0
$651$	5.12311	0.200790
$652$	0	0
$653$	−33.5464	−1.31277	−0.656386	−	0.754425i	$-0.727916\pi$
$653$	−33.5464	−1.31277	−0.656386	+	0.754425i	$0.727916\pi$
$654$	0	0
$655$	−33.6155	−1.31347
$656$	0	0
$657$	0.876894	0.0342109
$658$	0	0
$659$	35.6155	1.38738	0.693692	−	0.720272i	$-0.255983\pi$
$659$	35.6155	1.38738	0.693692	+	0.720272i	$0.255983\pi$
$660$	0	0
$661$	41.1231	1.59950	0.799752	−	0.600331i	$-0.204964\pi$
$661$	41.1231	1.59950	0.799752	+	0.600331i	$0.204964\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	−3.43845	−0.133137
$668$	0	0
$669$	5.12311	0.198071
$670$	0	0
$671$	28.4924	1.09994
$672$	0	0
$673$	10.3153	0.397627	0.198814	−	0.980037i	$-0.436291\pi$
$673$	10.3153	0.397627	0.198814	+	0.980037i	$0.436291\pi$
$674$	0	0
$675$	1.56155	0.0601042
$676$	0	0
$677$	27.3693	1.05189	0.525944	−	0.850519i	$-0.323712\pi$
$677$	27.3693	1.05189	0.525944	+	0.850519i	$0.323712\pi$
$678$	0	0
$679$	−16.5616	−0.635574
$680$	0	0
$681$	26.8078	1.02728
$682$	0	0
$683$	40.4924	1.54940	0.774700	−	0.632329i	$-0.217901\pi$
$683$	40.4924	1.54940	0.774700	+	0.632329i	$0.217901\pi$
$684$	0	0
$685$	−11.6847	−0.446448
$686$	0	0
$687$	−14.2462	−0.543527
$688$	0	0
$689$	−1.26137	−0.0480542
$690$	0	0
$691$	12.9460	0.492490	0.246245	−	0.969208i	$-0.420803\pi$
$691$	12.9460	0.492490	0.246245	+	0.969208i	$0.420803\pi$
$692$	0	0
$693$	3.12311	0.118637
$694$	0	0
$695$	43.0540	1.63313
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	14.4924	0.548154
$700$	0	0
$701$	17.7538	0.670551	0.335276	−	0.942120i	$-0.391171\pi$
$701$	17.7538	0.670551	0.335276	+	0.942120i	$0.391171\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−16.8078	−0.633017
$706$	0	0
$707$	−0.246211	−0.00925973
$708$	0	0
$709$	35.3693	1.32832	0.664161	−	0.747589i	$-0.268789\pi$
$709$	35.3693	1.32832	0.664161	+	0.747589i	$0.268789\pi$
$710$	0	0
$711$	2.24621	0.0842395
$712$	0	0
$713$	−5.12311	−0.191862
$714$	0	0
$715$	4.49242	0.168007
$716$	0	0
$717$	28.4924	1.06407
$718$	0	0
$719$	−14.5616	−0.543054	−0.271527	−	0.962431i	$-0.587529\pi$
$719$	−14.5616	−0.543054	−0.271527	+	0.962431i	$0.587529\pi$
$720$	0	0
$721$	9.93087	0.369845
$722$	0	0
$723$	17.0540	0.634244
$724$	0	0
$725$	5.36932	0.199411
$726$	0	0
$727$	18.2462	0.676715	0.338357	−	0.941018i	$-0.390129\pi$
$727$	18.2462	0.676715	0.338357	+	0.941018i	$0.390129\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.4924	1.49767
$732$	0	0
$733$	42.2462	1.56040	0.780200	−	0.625531i	$-0.215117\pi$
$733$	42.2462	1.56040	0.780200	+	0.625531i	$0.215117\pi$
$734$	0	0
$735$	2.56155	0.0944843
$736$	0	0
$737$	−22.2462	−0.819450
$738$	0	0
$739$	17.6155	0.647998	0.323999	−	0.946057i	$-0.394973\pi$
$739$	17.6155	0.647998	0.323999	+	0.946057i	$0.394973\pi$
$740$	0	0
$741$	−1.75379	−0.0644270
$742$	0	0
$743$	−40.9848	−1.50359	−0.751794	−	0.659398i	$-0.770811\pi$
$743$	−40.9848	−1.50359	−0.751794	+	0.659398i	$0.770811\pi$
$744$	0	0
$745$	26.2462	0.961587
$746$	0	0
$747$	0.876894	0.0320839
$748$	0	0
$749$	−3.12311	−0.114116
$750$	0	0
$751$	8.49242	0.309893	0.154946	−	0.987923i	$-0.450479\pi$
$751$	8.49242	0.309893	0.154946	+	0.987923i	$0.450479\pi$
$752$	0	0
$753$	4.56155	0.166232
$754$	0	0
$755$	−50.4233	−1.83509
$756$	0	0
$757$	−0.630683	−0.0229226	−0.0114613	−	0.999934i	$-0.503648\pi$
$757$	−0.630683	−0.0229226	−0.0114613	+	0.999934i	$0.503648\pi$
$758$	0	0
$759$	−3.12311	−0.113362
$760$	0	0
$761$	−52.7386	−1.91177	−0.955887	−	0.293735i	$-0.905102\pi$
$761$	−52.7386	−1.91177	−0.955887	+	0.293735i	$0.905102\pi$
$762$	0	0
$763$	14.5616	0.527164
$764$	0	0
$765$	18.2462	0.659693
$766$	0	0
$767$	−7.36932	−0.266091
$768$	0	0
$769$	−1.82292	−0.0657361	−0.0328681	−	0.999460i	$-0.510464\pi$
$769$	−1.82292	−0.0657361	−0.0328681	+	0.999460i	$0.510464\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	21.3002	0.766114	0.383057	−	0.923725i	$-0.374871\pi$
$773$	21.3002	0.766114	0.383057	+	0.923725i	$0.374871\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	2.56155	0.0918952
$778$	0	0
$779$	1.75379	0.0628360
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	3.43845	0.122880
$784$	0	0
$785$	−5.75379	−0.205362
$786$	0	0
$787$	19.1231	0.681665	0.340833	−	0.940124i	$-0.389291\pi$
$787$	19.1231	0.681665	0.340833	+	0.940124i	$0.389291\pi$
$788$	0	0
$789$	5.93087	0.211145
$790$	0	0
$791$	9.68466	0.344347
$792$	0	0
$793$	5.12311	0.181927
$794$	0	0
$795$	−5.75379	−0.204066
$796$	0	0
$797$	−34.4233	−1.21934	−0.609668	−	0.792657i	$-0.708697\pi$
$797$	−34.4233	−1.21934	−0.609668	+	0.792657i	$0.708697\pi$
$798$	0	0
$799$	−46.7386	−1.65349
$800$	0	0
$801$	14.0000	0.494666
$802$	0	0
$803$	2.73863	0.0966443
$804$	0	0
$805$	−2.56155	−0.0902829
$806$	0	0
$807$	−4.24621	−0.149474
$808$	0	0
$809$	−33.8617	−1.19052	−0.595258	−	0.803535i	$-0.702950\pi$
$809$	−33.8617	−1.19052	−0.595258	+	0.803535i	$0.702950\pi$
$810$	0	0
$811$	12.3153	0.432450	0.216225	−	0.976344i	$-0.430626\pi$
$811$	12.3153	0.432450	0.216225	+	0.976344i	$0.430626\pi$
$812$	0	0
$813$	2.24621	0.0787781
$814$	0	0
$815$	20.4924	0.717818
$816$	0	0
$817$	−17.7538	−0.621126
$818$	0	0
$819$	0.561553	0.0196222
$820$	0	0
$821$	43.4773	1.51737	0.758684	−	0.651459i	$-0.225843\pi$
$821$	43.4773	1.51737	0.758684	+	0.651459i	$0.225843\pi$
$822$	0	0
$823$	−35.6847	−1.24389	−0.621944	−	0.783061i	$-0.713657\pi$
$823$	−35.6847	−1.24389	−0.621944	+	0.783061i	$0.713657\pi$
$824$	0	0
$825$	4.87689	0.169792
$826$	0	0
$827$	0.738634	0.0256848	0.0128424	−	0.999918i	$-0.495912\pi$
$827$	0.738634	0.0256848	0.0128424	+	0.999918i	$0.495912\pi$
$828$	0	0
$829$	−24.7386	−0.859208	−0.429604	−	0.903017i	$-0.641347\pi$
$829$	−24.7386	−0.859208	−0.429604	+	0.903017i	$0.641347\pi$
$830$	0	0
$831$	4.87689	0.169178
$832$	0	0
$833$	7.12311	0.246801
$834$	0	0
$835$	−5.75379	−0.199118
$836$	0	0
$837$	5.12311	0.177080
$838$	0	0
$839$	33.6155	1.16054	0.580268	−	0.814425i	$-0.302948\pi$
$839$	33.6155	1.16054	0.580268	+	0.814425i	$0.302948\pi$
$840$	0	0
$841$	−17.1771	−0.592313
$842$	0	0
$843$	−23.9309	−0.824223
$844$	0	0
$845$	−32.4924	−1.11777
$846$	0	0
$847$	−1.24621	−0.0428203
$848$	0	0
$849$	−22.4924	−0.771938
$850$	0	0
$851$	−2.56155	−0.0878089
$852$	0	0
$853$	−4.56155	−0.156185	−0.0780923	−	0.996946i	$-0.524883\pi$
$853$	−4.56155	−0.156185	−0.0780923	+	0.996946i	$0.524883\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	13.6847	0.467459	0.233730	−	0.972302i	$-0.424907\pi$
$857$	13.6847	0.467459	0.233730	+	0.972302i	$0.424907\pi$
$858$	0	0
$859$	−12.1771	−0.415477	−0.207738	−	0.978184i	$-0.566610\pi$
$859$	−12.1771	−0.415477	−0.207738	+	0.978184i	$0.566610\pi$
$860$	0	0
$861$	−0.561553	−0.0191377
$862$	0	0
$863$	38.7386	1.31868	0.659339	−	0.751846i	$-0.270836\pi$
$863$	38.7386	1.31868	0.659339	+	0.751846i	$0.270836\pi$
$864$	0	0
$865$	−37.1231	−1.26222
$866$	0	0
$867$	33.7386	1.14582
$868$	0	0
$869$	7.01515	0.237973
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	−16.5616	−0.560524
$874$	0	0
$875$	−8.80776	−0.297757
$876$	0	0
$877$	21.8617	0.738218	0.369109	−	0.929386i	$-0.379663\pi$
$877$	21.8617	0.738218	0.369109	+	0.929386i	$0.379663\pi$
$878$	0	0
$879$	−13.6155	−0.459240
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	17.1231	0.576238	0.288119	−	0.957595i	$-0.406970\pi$
$883$	17.1231	0.576238	0.288119	+	0.957595i	$0.406970\pi$
$884$	0	0
$885$	−33.6155	−1.12997
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	4.31534	0.144732
$890$	0	0
$891$	3.12311	0.104628
$892$	0	0
$893$	20.4924	0.685753
$894$	0	0
$895$	−43.0540	−1.43914
$896$	0	0
$897$	−0.561553	−0.0187497
$898$	0	0
$899$	17.6155	0.587511
$900$	0	0
$901$	−16.0000	−0.533037
$902$	0	0
$903$	5.68466	0.189174
$904$	0	0
$905$	−17.6155	−0.585560
$906$	0	0
$907$	17.5464	0.582619	0.291309	−	0.956629i	$-0.405909\pi$
$907$	17.5464	0.582619	0.291309	+	0.956629i	$0.405909\pi$
$908$	0	0
$909$	−0.246211	−0.00816631
$910$	0	0
$911$	39.0540	1.29392	0.646958	−	0.762526i	$-0.276041\pi$
$911$	39.0540	1.29392	0.646958	+	0.762526i	$0.276041\pi$
$912$	0	0
$913$	2.73863	0.0906355
$914$	0	0
$915$	23.3693	0.772566
$916$	0	0
$917$	−13.1231	−0.433363
$918$	0	0
$919$	1.26137	0.0416086	0.0208043	−	0.999784i	$-0.493377\pi$
$919$	1.26137	0.0416086	0.0208043	+	0.999784i	$0.493377\pi$
$920$	0	0
$921$	25.9309	0.854451
$922$	0	0
$923$	−8.63068	−0.284082
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	9.93087	0.326173
$928$	0	0
$929$	−40.5616	−1.33078	−0.665391	−	0.746495i	$-0.731735\pi$
$929$	−40.5616	−1.33078	−0.665391	+	0.746495i	$0.731735\pi$
$930$	0	0
$931$	−3.12311	−0.102356
$932$	0	0
$933$	30.7386	1.00634
$934$	0	0
$935$	56.9848	1.86360
$936$	0	0
$937$	18.1771	0.593819	0.296910	−	0.954906i	$-0.404044\pi$
$937$	18.1771	0.593819	0.296910	+	0.954906i	$0.404044\pi$
$938$	0	0
$939$	−12.2462	−0.399640
$940$	0	0
$941$	39.0540	1.27312	0.636562	−	0.771226i	$-0.280356\pi$
$941$	39.0540	1.27312	0.636562	+	0.771226i	$0.280356\pi$
$942$	0	0
$943$	0.561553	0.0182867
$944$	0	0
$945$	2.56155	0.0833273
$946$	0	0
$947$	26.0691	0.847133	0.423566	−	0.905865i	$-0.360778\pi$
$947$	26.0691	0.847133	0.423566	+	0.905865i	$0.360778\pi$
$948$	0	0
$949$	0.492423	0.0159847
$950$	0	0
$951$	2.80776	0.0910480
$952$	0	0
$953$	50.0000	1.61966	0.809829	−	0.586665i	$-0.199560\pi$
$953$	50.0000	1.61966	0.809829	+	0.586665i	$0.199560\pi$
$954$	0	0
$955$	51.2311	1.65780
$956$	0	0
$957$	10.7386	0.347131
$958$	0	0
$959$	−4.56155	−0.147300
$960$	0	0
$961$	−4.75379	−0.153348
$962$	0	0
$963$	−3.12311	−0.100641
$964$	0	0
$965$	−24.8078	−0.798590
$966$	0	0
$967$	26.2462	0.844021	0.422011	−	0.906591i	$-0.361324\pi$
$967$	26.2462	0.844021	0.422011	+	0.906591i	$0.361324\pi$
$968$	0	0
$969$	−22.2462	−0.714651
$970$	0	0
$971$	−39.6155	−1.27132	−0.635661	−	0.771968i	$-0.719273\pi$
$971$	−39.6155	−1.27132	−0.635661	+	0.771968i	$0.719273\pi$
$972$	0	0
$973$	16.8078	0.538832
$974$	0	0
$975$	0.876894	0.0280831
$976$	0	0
$977$	−46.8078	−1.49751	−0.748757	−	0.662845i	$-0.769349\pi$
$977$	−46.8078	−1.49751	−0.748757	+	0.662845i	$0.769349\pi$
$978$	0	0
$979$	43.7235	1.39741
$980$	0	0
$981$	14.5616	0.464915
$982$	0	0
$983$	23.8617	0.761071	0.380536	−	0.924766i	$-0.375740\pi$
$983$	23.8617	0.761071	0.380536	+	0.924766i	$0.375740\pi$
$984$	0	0
$985$	−10.0691	−0.320829
$986$	0	0
$987$	−6.56155	−0.208857
$988$	0	0
$989$	−5.68466	−0.180762
$990$	0	0
$991$	−10.2462	−0.325482	−0.162741	−	0.986669i	$-0.552033\pi$
$991$	−10.2462	−0.325482	−0.162741	+	0.986669i	$0.552033\pi$
$992$	0	0
$993$	−8.49242	−0.269499
$994$	0	0
$995$	−61.9309	−1.96334
$996$	0	0
$997$	−47.4773	−1.50362	−0.751810	−	0.659380i	$-0.770819\pi$
$997$	−47.4773	−1.50362	−0.751810	+	0.659380i	$0.770819\pi$
$998$	0	0
$999$	2.56155	0.0810439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bm.1.2		2
4.3	odd	2		966.2.a.l.1.2	✓	2
12.11	even	2		2898.2.a.ba.1.1		2
28.27	even	2		6762.2.a.bw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.l.1.2	✓	2	4.3	odd	2
2898.2.a.ba.1.1		2	12.11	even	2
6762.2.a.bw.1.1		2	28.27	even	2
7728.2.a.bm.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.12311 q^{11} +0.561553 q^{13} +2.56155 q^{15} +7.12311 q^{17} -3.12311 q^{19} +1.00000 q^{21} -1.00000 q^{23} +1.56155 q^{25} +1.00000 q^{27} +3.43845 q^{29} +5.12311 q^{31} +3.12311 q^{33} +2.56155 q^{35} +2.56155 q^{37} +0.561553 q^{39} -0.561553 q^{41} +5.68466 q^{43} +2.56155 q^{45} -6.56155 q^{47} +1.00000 q^{49} +7.12311 q^{51} -2.24621 q^{53} +8.00000 q^{55} -3.12311 q^{57} -13.1231 q^{59} +9.12311 q^{61} +1.00000 q^{63} +1.43845 q^{65} -7.12311 q^{67} -1.00000 q^{69} -15.3693 q^{71} +0.876894 q^{73} +1.56155 q^{75} +3.12311 q^{77} +2.24621 q^{79} +1.00000 q^{81} +0.876894 q^{83} +18.2462 q^{85} +3.43845 q^{87} +14.0000 q^{89} +0.561553 q^{91} +5.12311 q^{93} -8.00000 q^{95} -16.5616 q^{97} +3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 3 q^{13} + q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{23} - q^{25} + 2 q^{27} + 11 q^{29} + 2 q^{31} - 2 q^{33} + q^{35} + q^{37} - 3 q^{39} + 3 q^{41} - q^{43} + q^{45} - 9 q^{47} + 2 q^{49} + 6 q^{51} + 12 q^{53} + 16 q^{55} + 2 q^{57} - 18 q^{59} + 10 q^{61} + 2 q^{63} + 7 q^{65} - 6 q^{67} - 2 q^{69} - 6 q^{71} + 10 q^{73} - q^{75} - 2 q^{77} - 12 q^{79} + 2 q^{81} + 10 q^{83} + 20 q^{85} + 11 q^{87} + 28 q^{89} - 3 q^{91} + 2 q^{93} - 16 q^{95} - 29 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 - 3 * q^13 + q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 2 * q^23 - q^25 + 2 * q^27 + 11 * q^29 + 2 * q^31 - 2 * q^33 + q^35 + q^37 - 3 * q^39 + 3 * q^41 - q^43 + q^45 - 9 * q^47 + 2 * q^49 + 6 * q^51 + 12 * q^53 + 16 * q^55 + 2 * q^57 - 18 * q^59 + 10 * q^61 + 2 * q^63 + 7 * q^65 - 6 * q^67 - 2 * q^69 - 6 * q^71 + 10 * q^73 - q^75 - 2 * q^77 - 12 * q^79 + 2 * q^81 + 10 * q^83 + 20 * q^85 + 11 * q^87 + 28 * q^89 - 3 * q^91 + 2 * q^93 - 16 * q^95 - 29 * q^97 - 2 * q^99

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