LMFDB - Embedded newform 3048.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3048,2,Mod(1,3048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3048 = 2^{3} \cdot 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3048.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:	$24.3384025361$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} - 3x^{5} + 68x^{4} - 6x^{3} - 84x^{2} + 52x - 8 $ x^8 - 16x^6 - 3x^5 + 68x^4 - 6x^3 - 84x^2 + 52x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80769$ of defining polynomial
Character	$\chi$	$=$	3048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.37975 q^{5} -0.0538072 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.37975 q^{5} -0.0538072 q^{7} +1.00000 q^{9} -1.79855 q^{11} +3.12572 q^{13} -4.37975 q^{15} +4.85202 q^{17} +3.63048 q^{19} -0.0538072 q^{21} -8.81086 q^{23} +14.1822 q^{25} +1.00000 q^{27} -4.82328 q^{29} +4.67261 q^{31} -1.79855 q^{33} +0.235662 q^{35} -3.90374 q^{37} +3.12572 q^{39} -9.25486 q^{41} +2.09579 q^{43} -4.37975 q^{45} -5.77571 q^{47} -6.99710 q^{49} +4.85202 q^{51} +12.6004 q^{53} +7.87718 q^{55} +3.63048 q^{57} +12.6571 q^{59} -13.2847 q^{61} -0.0538072 q^{63} -13.6899 q^{65} -2.44390 q^{67} -8.81086 q^{69} -12.0205 q^{71} -12.8813 q^{73} +14.1822 q^{75} +0.0967748 q^{77} -7.31853 q^{79} +1.00000 q^{81} -4.17978 q^{83} -21.2506 q^{85} -4.82328 q^{87} -12.5922 q^{89} -0.168186 q^{91} +4.67261 q^{93} -15.9006 q^{95} +1.30625 q^{97} -1.79855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 4 q^{5} - 7 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 4 * q^5 - 7 * q^7 + 8 * q^9	$ 8 q + 8 q^{3} - 4 q^{5} - 7 q^{7} + 8 q^{9} - 8 q^{11} - 10 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{21} - 20 q^{23} + 4 q^{25} + 8 q^{27} - 5 q^{29} - 14 q^{31} - 8 q^{33} - 19 q^{35} - 20 q^{37} - 10 q^{39} - 18 q^{41} - 10 q^{43} - 4 q^{45} - 23 q^{47} + 7 q^{49} - 10 q^{51} - 11 q^{55} - 19 q^{59} - 24 q^{61} - 7 q^{63} - 11 q^{65} + 4 q^{67} - 20 q^{69} - 47 q^{71} - 12 q^{73} + 4 q^{75} - 9 q^{77} - 35 q^{79} + 8 q^{81} - 19 q^{83} - 29 q^{85} - 5 q^{87} - 9 q^{89} + 6 q^{91} - 14 q^{93} - 39 q^{95} + 8 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 4 * q^5 - 7 * q^7 + 8 * q^9 - 8 * q^11 - 10 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^21 - 20 * q^23 + 4 * q^25 + 8 * q^27 - 5 * q^29 - 14 * q^31 - 8 * q^33 - 19 * q^35 - 20 * q^37 - 10 * q^39 - 18 * q^41 - 10 * q^43 - 4 * q^45 - 23 * q^47 + 7 * q^49 - 10 * q^51 - 11 * q^55 - 19 * q^59 - 24 * q^61 - 7 * q^63 - 11 * q^65 + 4 * q^67 - 20 * q^69 - 47 * q^71 - 12 * q^73 + 4 * q^75 - 9 * q^77 - 35 * q^79 + 8 * q^81 - 19 * q^83 - 29 * q^85 - 5 * q^87 - 9 * q^89 + 6 * q^91 - 14 * q^93 - 39 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−4.37975	−1.95868	−0.979342	−	0.202210i	$-0.935188\pi$
$5$	−4.37975	−1.95868	−0.979342	+	0.202210i	$0.935188\pi$
$6$	0	0
$7$	−0.0538072	−0.0203372	−0.0101686	−	0.999948i	$-0.503237\pi$
$7$	−0.0538072	−0.0203372	−0.0101686	+	0.999948i	$0.503237\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.79855	−0.542282	−0.271141	−	0.962540i	$-0.587401\pi$
$11$	−1.79855	−0.542282	−0.271141	+	0.962540i	$0.587401\pi$
$12$	0	0
$13$	3.12572	0.866919	0.433460	−	0.901173i	$-0.357293\pi$
$13$	3.12572	0.866919	0.433460	+	0.901173i	$0.357293\pi$
$14$	0	0
$15$	−4.37975	−1.13085
$16$	0	0
$17$	4.85202	1.17679	0.588394	−	0.808574i	$-0.299760\pi$
$17$	4.85202	1.17679	0.588394	+	0.808574i	$0.299760\pi$
$18$	0	0
$19$	3.63048	0.832889	0.416445	−	0.909161i	$-0.363276\pi$
$19$	3.63048	0.832889	0.416445	+	0.909161i	$0.363276\pi$
$20$	0	0
$21$	−0.0538072	−0.0117417
$22$	0	0
$23$	−8.81086	−1.83719	−0.918595	−	0.395199i	$-0.870676\pi$
$23$	−8.81086	−1.83719	−0.918595	+	0.395199i	$0.870676\pi$
$24$	0	0
$25$	14.1822	2.83644
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.82328	−0.895660	−0.447830	−	0.894119i	$-0.647803\pi$
$29$	−4.82328	−0.895660	−0.447830	+	0.894119i	$0.647803\pi$
$30$	0	0
$31$	4.67261	0.839225	0.419612	−	0.907703i	$-0.362166\pi$
$31$	4.67261	0.839225	0.419612	+	0.907703i	$0.362166\pi$
$32$	0	0
$33$	−1.79855	−0.313087
$34$	0	0
$35$	0.235662	0.0398342
$36$	0	0
$37$	−3.90374	−0.641771	−0.320885	−	0.947118i	$-0.603980\pi$
$37$	−3.90374	−0.641771	−0.320885	+	0.947118i	$0.603980\pi$
$38$	0	0
$39$	3.12572	0.500516
$40$	0	0
$41$	−9.25486	−1.44537	−0.722683	−	0.691180i	$-0.757091\pi$
$41$	−9.25486	−1.44537	−0.722683	+	0.691180i	$0.757091\pi$
$42$	0	0
$43$	2.09579	0.319604	0.159802	−	0.987149i	$-0.448914\pi$
$43$	2.09579	0.319604	0.159802	+	0.987149i	$0.448914\pi$
$44$	0	0
$45$	−4.37975	−0.652895
$46$	0	0
$47$	−5.77571	−0.842474	−0.421237	−	0.906951i	$-0.638404\pi$
$47$	−5.77571	−0.842474	−0.421237	+	0.906951i	$0.638404\pi$
$48$	0	0
$49$	−6.99710	−0.999586
$50$	0	0
$51$	4.85202	0.679419
$52$	0	0
$53$	12.6004	1.73080	0.865401	−	0.501080i	$-0.167064\pi$
$53$	12.6004	1.73080	0.865401	+	0.501080i	$0.167064\pi$
$54$	0	0
$55$	7.87718	1.06216
$56$	0	0
$57$	3.63048	0.480869
$58$	0	0
$59$	12.6571	1.64782	0.823910	−	0.566720i	$-0.191788\pi$
$59$	12.6571	1.64782	0.823910	+	0.566720i	$0.191788\pi$
$60$	0	0
$61$	−13.2847	−1.70093	−0.850465	−	0.526032i	$-0.823679\pi$
$61$	−13.2847	−1.70093	−0.850465	+	0.526032i	$0.823679\pi$
$62$	0	0
$63$	−0.0538072	−0.00677908
$64$	0	0
$65$	−13.6899	−1.69802
$66$	0	0
$67$	−2.44390	−0.298569	−0.149285	−	0.988794i	$-0.547697\pi$
$67$	−2.44390	−0.298569	−0.149285	+	0.988794i	$0.547697\pi$
$68$	0	0
$69$	−8.81086	−1.06070
$70$	0	0
$71$	−12.0205	−1.42657	−0.713283	−	0.700876i	$-0.752793\pi$
$71$	−12.0205	−1.42657	−0.713283	+	0.700876i	$0.752793\pi$
$72$	0	0
$73$	−12.8813	−1.50765	−0.753823	−	0.657077i	$-0.771793\pi$
$73$	−12.8813	−1.50765	−0.753823	+	0.657077i	$0.771793\pi$
$74$	0	0
$75$	14.1822	1.63762
$76$	0	0
$77$	0.0967748	0.0110285
$78$	0	0
$79$	−7.31853	−0.823399	−0.411700	−	0.911320i	$-0.635065\pi$
$79$	−7.31853	−0.823399	−0.411700	+	0.911320i	$0.635065\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.17978	−0.458790	−0.229395	−	0.973333i	$-0.573675\pi$
$83$	−4.17978	−0.458790	−0.229395	+	0.973333i	$0.573675\pi$
$84$	0	0
$85$	−21.2506	−2.30496
$86$	0	0
$87$	−4.82328	−0.517110
$88$	0	0
$89$	−12.5922	−1.33477	−0.667383	−	0.744715i	$-0.732586\pi$
$89$	−12.5922	−1.33477	−0.667383	+	0.744715i	$0.732586\pi$
$90$	0	0
$91$	−0.168186	−0.0176307
$92$	0	0
$93$	4.67261	0.484527
$94$	0	0
$95$	−15.9006	−1.63137
$96$	0	0
$97$	1.30625	0.132629	0.0663147	−	0.997799i	$-0.478876\pi$
$97$	1.30625	0.132629	0.0663147	+	0.997799i	$0.478876\pi$
$98$	0	0
$99$	−1.79855	−0.180761
$100$	0	0
$101$	5.52045	0.549305	0.274652	−	0.961544i	$-0.411437\pi$
$101$	5.52045	0.549305	0.274652	+	0.961544i	$0.411437\pi$
$102$	0	0
$103$	−13.4376	−1.32404	−0.662022	−	0.749484i	$-0.730302\pi$
$103$	−13.4376	−1.32404	−0.662022	+	0.749484i	$0.730302\pi$
$104$	0	0
$105$	0.235662	0.0229983
$106$	0	0
$107$	7.01057	0.677737	0.338869	−	0.940834i	$-0.389956\pi$
$107$	7.01057	0.677737	0.338869	+	0.940834i	$0.389956\pi$
$108$	0	0
$109$	17.7795	1.70296	0.851481	−	0.524385i	$-0.175705\pi$
$109$	17.7795	1.70296	0.851481	+	0.524385i	$0.175705\pi$
$110$	0	0
$111$	−3.90374	−0.370526
$112$	0	0
$113$	18.0777	1.70060	0.850302	−	0.526295i	$-0.176419\pi$
$113$	18.0777	1.70060	0.850302	+	0.526295i	$0.176419\pi$
$114$	0	0
$115$	38.5894	3.59848
$116$	0	0
$117$	3.12572	0.288973
$118$	0	0
$119$	−0.261074	−0.0239326
$120$	0	0
$121$	−7.76523	−0.705930
$122$	0	0
$123$	−9.25486	−0.834483
$124$	0	0
$125$	−40.2158	−3.59701
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	0	0
$129$	2.09579	0.184524
$130$	0	0
$131$	−14.7670	−1.29019	−0.645097	−	0.764100i	$-0.723183\pi$
$131$	−14.7670	−1.29019	−0.645097	+	0.764100i	$0.723183\pi$
$132$	0	0
$133$	−0.195346	−0.0169387
$134$	0	0
$135$	−4.37975	−0.376949
$136$	0	0
$137$	−2.97601	−0.254258	−0.127129	−	0.991886i	$-0.540576\pi$
$137$	−2.97601	−0.254258	−0.127129	+	0.991886i	$0.540576\pi$
$138$	0	0
$139$	−22.6962	−1.92507	−0.962535	−	0.271157i	$-0.912594\pi$
$139$	−22.6962	−1.92507	−0.962535	+	0.271157i	$0.912594\pi$
$140$	0	0
$141$	−5.77571	−0.486402
$142$	0	0
$143$	−5.62175	−0.470115
$144$	0	0
$145$	21.1248	1.75432
$146$	0	0
$147$	−6.99710	−0.577111
$148$	0	0
$149$	−3.12018	−0.255615	−0.127807	−	0.991799i	$-0.540794\pi$
$149$	−3.12018	−0.255615	−0.127807	+	0.991799i	$0.540794\pi$
$150$	0	0
$151$	8.28853	0.674511	0.337256	−	0.941413i	$-0.390501\pi$
$151$	8.28853	0.674511	0.337256	+	0.941413i	$0.390501\pi$
$152$	0	0
$153$	4.85202	0.392263
$154$	0	0
$155$	−20.4649	−1.64378
$156$	0	0
$157$	−15.0747	−1.20309	−0.601547	−	0.798837i	$-0.705449\pi$
$157$	−15.0747	−1.20309	−0.601547	+	0.798837i	$0.705449\pi$
$158$	0	0
$159$	12.6004	0.999279
$160$	0	0
$161$	0.474088	0.0373634
$162$	0	0
$163$	−12.1487	−0.951562	−0.475781	−	0.879564i	$-0.657834\pi$
$163$	−12.1487	−0.951562	−0.475781	+	0.879564i	$0.657834\pi$
$164$	0	0
$165$	7.87718	0.613238
$166$	0	0
$167$	−21.6518	−1.67547	−0.837734	−	0.546079i	$-0.816120\pi$
$167$	−21.6518	−1.67547	−0.837734	+	0.546079i	$0.816120\pi$
$168$	0	0
$169$	−3.22987	−0.248451
$170$	0	0
$171$	3.63048	0.277630
$172$	0	0
$173$	13.4764	1.02459	0.512296	−	0.858809i	$-0.328795\pi$
$173$	13.4764	1.02459	0.512296	+	0.858809i	$0.328795\pi$
$174$	0	0
$175$	−0.763106	−0.0576854
$176$	0	0
$177$	12.6571	0.951370
$178$	0	0
$179$	−7.68819	−0.574642	−0.287321	−	0.957834i	$-0.592765\pi$
$179$	−7.68819	−0.574642	−0.287321	+	0.957834i	$0.592765\pi$
$180$	0	0
$181$	6.61531	0.491712	0.245856	−	0.969306i	$-0.420931\pi$
$181$	6.61531	0.491712	0.245856	+	0.969306i	$0.420931\pi$
$182$	0	0
$183$	−13.2847	−0.982032
$184$	0	0
$185$	17.0974	1.25703
$186$	0	0
$187$	−8.72658	−0.638151
$188$	0	0
$189$	−0.0538072	−0.00391390
$190$	0	0
$191$	−18.3520	−1.32790	−0.663951	−	0.747776i	$-0.731122\pi$
$191$	−18.3520	−1.32790	−0.663951	+	0.747776i	$0.731122\pi$
$192$	0	0
$193$	−13.5441	−0.974925	−0.487463	−	0.873144i	$-0.662077\pi$
$193$	−13.5441	−0.974925	−0.487463	+	0.873144i	$0.662077\pi$
$194$	0	0
$195$	−13.6899	−0.980353
$196$	0	0
$197$	−9.73483	−0.693578	−0.346789	−	0.937943i	$-0.612728\pi$
$197$	−9.73483	−0.693578	−0.346789	+	0.937943i	$0.612728\pi$
$198$	0	0
$199$	13.4701	0.954873	0.477437	−	0.878666i	$-0.341566\pi$
$199$	13.4701	0.954873	0.477437	+	0.878666i	$0.341566\pi$
$200$	0	0
$201$	−2.44390	−0.172379
$202$	0	0
$203$	0.259527	0.0182152
$204$	0	0
$205$	40.5340	2.83102
$206$	0	0
$207$	−8.81086	−0.612397
$208$	0	0
$209$	−6.52958	−0.451661
$210$	0	0
$211$	26.4635	1.82183	0.910913	−	0.412600i	$-0.135379\pi$
$211$	26.4635	1.82183	0.910913	+	0.412600i	$0.135379\pi$
$212$	0	0
$213$	−12.0205	−0.823628
$214$	0	0
$215$	−9.17902	−0.626004
$216$	0	0
$217$	−0.251420	−0.0170675
$218$	0	0
$219$	−12.8813	−0.870440
$220$	0	0
$221$	15.1661	1.02018
$222$	0	0
$223$	−2.44596	−0.163793	−0.0818967	−	0.996641i	$-0.526098\pi$
$223$	−2.44596	−0.163793	−0.0818967	+	0.996641i	$0.526098\pi$
$224$	0	0
$225$	14.1822	0.945481
$226$	0	0
$227$	−2.72395	−0.180795	−0.0903975	−	0.995906i	$-0.528814\pi$
$227$	−2.72395	−0.180795	−0.0903975	+	0.995906i	$0.528814\pi$
$228$	0	0
$229$	14.9160	0.985677	0.492839	−	0.870121i	$-0.335959\pi$
$229$	14.9160	0.985677	0.492839	+	0.870121i	$0.335959\pi$
$230$	0	0
$231$	0.0967748	0.00636732
$232$	0	0
$233$	3.89955	0.255468	0.127734	−	0.991808i	$-0.459230\pi$
$233$	3.89955	0.255468	0.127734	+	0.991808i	$0.459230\pi$
$234$	0	0
$235$	25.2962	1.65014
$236$	0	0
$237$	−7.31853	−0.475390
$238$	0	0
$239$	20.7106	1.33966	0.669828	−	0.742516i	$-0.266368\pi$
$239$	20.7106	1.33966	0.669828	+	0.742516i	$0.266368\pi$
$240$	0	0
$241$	−22.8309	−1.47067	−0.735333	−	0.677706i	$-0.762974\pi$
$241$	−22.8309	−1.47067	−0.735333	+	0.677706i	$0.762974\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	30.6456	1.95787
$246$	0	0
$247$	11.3479	0.722047
$248$	0	0
$249$	−4.17978	−0.264883
$250$	0	0
$251$	28.2481	1.78300	0.891501	−	0.453018i	$-0.149653\pi$
$251$	28.2481	1.78300	0.891501	+	0.453018i	$0.149653\pi$
$252$	0	0
$253$	15.8467	0.996276
$254$	0	0
$255$	−21.2506	−1.33077
$256$	0	0
$257$	14.3311	0.893947	0.446974	−	0.894547i	$-0.352502\pi$
$257$	14.3311	0.893947	0.446974	+	0.894547i	$0.352502\pi$
$258$	0	0
$259$	0.210049	0.0130518
$260$	0	0
$261$	−4.82328	−0.298553
$262$	0	0
$263$	−11.0759	−0.682970	−0.341485	−	0.939887i	$-0.610930\pi$
$263$	−11.0759	−0.682970	−0.341485	+	0.939887i	$0.610930\pi$
$264$	0	0
$265$	−55.1867	−3.39009
$266$	0	0
$267$	−12.5922	−0.770627
$268$	0	0
$269$	7.29590	0.444839	0.222419	−	0.974951i	$-0.428605\pi$
$269$	7.29590	0.444839	0.222419	+	0.974951i	$0.428605\pi$
$270$	0	0
$271$	−5.25888	−0.319454	−0.159727	−	0.987161i	$-0.551061\pi$
$271$	−5.25888	−0.319454	−0.159727	+	0.987161i	$0.551061\pi$
$272$	0	0
$273$	−0.168186	−0.0101791
$274$	0	0
$275$	−25.5074	−1.53815
$276$	0	0
$277$	15.9700	0.959543	0.479772	−	0.877393i	$-0.340719\pi$
$277$	15.9700	0.959543	0.479772	+	0.877393i	$0.340719\pi$
$278$	0	0
$279$	4.67261	0.279742
$280$	0	0
$281$	2.28065	0.136052	0.0680261	−	0.997684i	$-0.478330\pi$
$281$	2.28065	0.136052	0.0680261	+	0.997684i	$0.478330\pi$
$282$	0	0
$283$	5.80178	0.344880	0.172440	−	0.985020i	$-0.444835\pi$
$283$	5.80178	0.344880	0.172440	+	0.985020i	$0.444835\pi$
$284$	0	0
$285$	−15.9006	−0.941870
$286$	0	0
$287$	0.497978	0.0293947
$288$	0	0
$289$	6.54212	0.384830
$290$	0	0
$291$	1.30625	0.0765737
$292$	0	0
$293$	−3.64830	−0.213136	−0.106568	−	0.994305i	$-0.533986\pi$
$293$	−3.64830	−0.213136	−0.106568	+	0.994305i	$0.533986\pi$
$294$	0	0
$295$	−55.4352	−3.22756
$296$	0	0
$297$	−1.79855	−0.104362
$298$	0	0
$299$	−27.5403	−1.59270
$300$	0	0
$301$	−0.112769	−0.00649987
$302$	0	0
$303$	5.52045	0.317141
$304$	0	0
$305$	58.1836	3.33158
$306$	0	0
$307$	15.9566	0.910689	0.455344	−	0.890315i	$-0.349516\pi$
$307$	15.9566	0.910689	0.455344	+	0.890315i	$0.349516\pi$
$308$	0	0
$309$	−13.4376	−0.764437
$310$	0	0
$311$	7.53925	0.427512	0.213756	−	0.976887i	$-0.431430\pi$
$311$	7.53925	0.427512	0.213756	+	0.976887i	$0.431430\pi$
$312$	0	0
$313$	27.1451	1.53433	0.767165	−	0.641450i	$-0.221667\pi$
$313$	27.1451	1.53433	0.767165	+	0.641450i	$0.221667\pi$
$314$	0	0
$315$	0.235662	0.0132781
$316$	0	0
$317$	17.4848	0.982045	0.491022	−	0.871147i	$-0.336623\pi$
$317$	17.4848	0.982045	0.491022	+	0.871147i	$0.336623\pi$
$318$	0	0
$319$	8.67488	0.485700
$320$	0	0
$321$	7.01057	0.391292
$322$	0	0
$323$	17.6152	0.980134
$324$	0	0
$325$	44.3297	2.45897
$326$	0	0
$327$	17.7795	0.983206
$328$	0	0
$329$	0.310775	0.0171336
$330$	0	0
$331$	−22.4321	−1.23298	−0.616489	−	0.787364i	$-0.711445\pi$
$331$	−22.4321	−1.23298	−0.616489	+	0.787364i	$0.711445\pi$
$332$	0	0
$333$	−3.90374	−0.213924
$334$	0	0
$335$	10.7037	0.584803
$336$	0	0
$337$	−5.76102	−0.313823	−0.156911	−	0.987613i	$-0.550154\pi$
$337$	−5.76102	−0.313823	−0.156911	+	0.987613i	$0.550154\pi$
$338$	0	0
$339$	18.0777	0.981844
$340$	0	0
$341$	−8.40390	−0.455097
$342$	0	0
$343$	0.753146	0.0406660
$344$	0	0
$345$	38.5894	2.07758
$346$	0	0
$347$	−35.4731	−1.90430	−0.952148	−	0.305636i	$-0.901131\pi$
$347$	−35.4731	−1.90430	−0.952148	+	0.305636i	$0.901131\pi$
$348$	0	0
$349$	11.8514	0.634392	0.317196	−	0.948360i	$-0.397259\pi$
$349$	11.8514	0.634392	0.317196	+	0.948360i	$0.397259\pi$
$350$	0	0
$351$	3.12572	0.166839
$352$	0	0
$353$	25.1689	1.33960	0.669802	−	0.742540i	$-0.266379\pi$
$353$	25.1689	1.33960	0.669802	+	0.742540i	$0.266379\pi$
$354$	0	0
$355$	52.6466	2.79419
$356$	0	0
$357$	−0.261074	−0.0138175
$358$	0	0
$359$	−0.0463052	−0.00244390	−0.00122195	−	0.999999i	$-0.500389\pi$
$359$	−0.0463052	−0.00244390	−0.00122195	+	0.999999i	$0.500389\pi$
$360$	0	0
$361$	−5.81962	−0.306296
$362$	0	0
$363$	−7.76523	−0.407569
$364$	0	0
$365$	56.4171	2.95300
$366$	0	0
$367$	0.0755775	0.00394512	0.00197256	−	0.999998i	$-0.499372\pi$
$367$	0.0755775	0.00394512	0.00197256	+	0.999998i	$0.499372\pi$
$368$	0	0
$369$	−9.25486	−0.481789
$370$	0	0
$371$	−0.677994	−0.0351997
$372$	0	0
$373$	−28.4717	−1.47421	−0.737104	−	0.675780i	$-0.763807\pi$
$373$	−28.4717	−1.47421	−0.737104	+	0.675780i	$0.763807\pi$
$374$	0	0
$375$	−40.2158	−2.07674
$376$	0	0
$377$	−15.0762	−0.776465
$378$	0	0
$379$	−15.3730	−0.789660	−0.394830	−	0.918754i	$-0.629196\pi$
$379$	−15.3730	−0.789660	−0.394830	+	0.918754i	$0.629196\pi$
$380$	0	0
$381$	−1.00000	−0.0512316
$382$	0	0
$383$	−22.3629	−1.14269	−0.571344	−	0.820710i	$-0.693578\pi$
$383$	−22.3629	−1.14269	−0.571344	+	0.820710i	$0.693578\pi$
$384$	0	0
$385$	−0.423850	−0.0216014
$386$	0	0
$387$	2.09579	0.106535
$388$	0	0
$389$	−24.3677	−1.23549	−0.617745	−	0.786379i	$-0.711953\pi$
$389$	−24.3677	−1.23549	−0.617745	+	0.786379i	$0.711953\pi$
$390$	0	0
$391$	−42.7505	−2.16198
$392$	0	0
$393$	−14.7670	−0.744894
$394$	0	0
$395$	32.0533	1.61278
$396$	0	0
$397$	15.8214	0.794051	0.397026	−	0.917808i	$-0.370042\pi$
$397$	15.8214	0.794051	0.397026	+	0.917808i	$0.370042\pi$
$398$	0	0
$399$	−0.195346	−0.00977954
$400$	0	0
$401$	4.48935	0.224187	0.112094	−	0.993698i	$-0.464244\pi$
$401$	4.48935	0.224187	0.112094	+	0.993698i	$0.464244\pi$
$402$	0	0
$403$	14.6053	0.727540
$404$	0	0
$405$	−4.37975	−0.217632
$406$	0	0
$407$	7.02105	0.348021
$408$	0	0
$409$	0.966937	0.0478119	0.0239060	−	0.999714i	$-0.492390\pi$
$409$	0.966937	0.0478119	0.0239060	+	0.999714i	$0.492390\pi$
$410$	0	0
$411$	−2.97601	−0.146796
$412$	0	0
$413$	−0.681046	−0.0335121
$414$	0	0
$415$	18.3064	0.898625
$416$	0	0
$417$	−22.6962	−1.11144
$418$	0	0
$419$	−1.60480	−0.0783997	−0.0391999	−	0.999231i	$-0.512481\pi$
$419$	−1.60480	−0.0783997	−0.0391999	+	0.999231i	$0.512481\pi$
$420$	0	0
$421$	−20.3508	−0.991839	−0.495919	−	0.868369i	$-0.665169\pi$
$421$	−20.3508	−0.991839	−0.495919	+	0.868369i	$0.665169\pi$
$422$	0	0
$423$	−5.77571	−0.280825
$424$	0	0
$425$	68.8124	3.33789
$426$	0	0
$427$	0.714813	0.0345922
$428$	0	0
$429$	−5.62175	−0.271421
$430$	0	0
$431$	−12.2184	−0.588542	−0.294271	−	0.955722i	$-0.595077\pi$
$431$	−12.2184	−0.588542	−0.294271	+	0.955722i	$0.595077\pi$
$432$	0	0
$433$	15.1815	0.729574	0.364787	−	0.931091i	$-0.381142\pi$
$433$	15.1815	0.729574	0.364787	+	0.931091i	$0.381142\pi$
$434$	0	0
$435$	21.1248	1.01285
$436$	0	0
$437$	−31.9876	−1.53018
$438$	0	0
$439$	−22.7038	−1.08359	−0.541797	−	0.840509i	$-0.682256\pi$
$439$	−22.7038	−1.08359	−0.541797	+	0.840509i	$0.682256\pi$
$440$	0	0
$441$	−6.99710	−0.333195
$442$	0	0
$443$	−11.8279	−0.561959	−0.280979	−	0.959714i	$-0.590659\pi$
$443$	−11.8279	−0.561959	−0.280979	+	0.959714i	$0.590659\pi$
$444$	0	0
$445$	55.1505	2.61438
$446$	0	0
$447$	−3.12018	−0.147579
$448$	0	0
$449$	−1.40007	−0.0660733	−0.0330367	−	0.999454i	$-0.510518\pi$
$449$	−1.40007	−0.0660733	−0.0330367	+	0.999454i	$0.510518\pi$
$450$	0	0
$451$	16.6453	0.783796
$452$	0	0
$453$	8.28853	0.389429
$454$	0	0
$455$	0.736615	0.0345330
$456$	0	0
$457$	−5.83588	−0.272991	−0.136495	−	0.990641i	$-0.543584\pi$
$457$	−5.83588	−0.272991	−0.136495	+	0.990641i	$0.543584\pi$
$458$	0	0
$459$	4.85202	0.226473
$460$	0	0
$461$	3.89486	0.181402	0.0907008	−	0.995878i	$-0.471089\pi$
$461$	3.89486	0.181402	0.0907008	+	0.995878i	$0.471089\pi$
$462$	0	0
$463$	7.54816	0.350793	0.175396	−	0.984498i	$-0.443879\pi$
$463$	7.54816	0.350793	0.175396	+	0.984498i	$0.443879\pi$
$464$	0	0
$465$	−20.4649	−0.949035
$466$	0	0
$467$	−21.5358	−0.996556	−0.498278	−	0.867017i	$-0.666034\pi$
$467$	−21.5358	−0.996556	−0.498278	+	0.867017i	$0.666034\pi$
$468$	0	0
$469$	0.131499	0.00607207
$470$	0	0
$471$	−15.0747	−0.694607
$472$	0	0
$473$	−3.76937	−0.173316
$474$	0	0
$475$	51.4883	2.36244
$476$	0	0
$477$	12.6004	0.576934
$478$	0	0
$479$	−23.0213	−1.05187	−0.525936	−	0.850524i	$-0.676285\pi$
$479$	−23.0213	−1.05187	−0.525936	+	0.850524i	$0.676285\pi$
$480$	0	0
$481$	−12.2020	−0.556363
$482$	0	0
$483$	0.474088	0.0215718
$484$	0	0
$485$	−5.72105	−0.259779
$486$	0	0
$487$	−8.89605	−0.403118	−0.201559	−	0.979476i	$-0.564601\pi$
$487$	−8.89605	−0.403118	−0.201559	+	0.979476i	$0.564601\pi$
$488$	0	0
$489$	−12.1487	−0.549385
$490$	0	0
$491$	31.6286	1.42738	0.713688	−	0.700463i	$-0.247023\pi$
$491$	31.6286	1.42738	0.713688	+	0.700463i	$0.247023\pi$
$492$	0	0
$493$	−23.4026	−1.05400
$494$	0	0
$495$	7.87718	0.354053
$496$	0	0
$497$	0.646788	0.0290124
$498$	0	0
$499$	−9.62746	−0.430984	−0.215492	−	0.976506i	$-0.569136\pi$
$499$	−9.62746	−0.430984	−0.215492	+	0.976506i	$0.569136\pi$
$500$	0	0
$501$	−21.6518	−0.967332
$502$	0	0
$503$	22.0345	0.982471	0.491236	−	0.871027i	$-0.336545\pi$
$503$	22.0345	0.982471	0.491236	+	0.871027i	$0.336545\pi$
$504$	0	0
$505$	−24.1782	−1.07592
$506$	0	0
$507$	−3.22987	−0.143443
$508$	0	0
$509$	−41.4308	−1.83639	−0.918195	−	0.396129i	$-0.870353\pi$
$509$	−41.4308	−1.83639	−0.918195	+	0.396129i	$0.870353\pi$
$510$	0	0
$511$	0.693109	0.0306614
$512$	0	0
$513$	3.63048	0.160290
$514$	0	0
$515$	58.8533	2.59339
$516$	0	0
$517$	10.3879	0.456858
$518$	0	0
$519$	13.4764	0.591549
$520$	0	0
$521$	−3.23822	−0.141869	−0.0709344	−	0.997481i	$-0.522598\pi$
$521$	−3.23822	−0.141869	−0.0709344	+	0.997481i	$0.522598\pi$
$522$	0	0
$523$	−21.1323	−0.924049	−0.462025	−	0.886867i	$-0.652877\pi$
$523$	−21.1323	−0.924049	−0.462025	+	0.886867i	$0.652877\pi$
$524$	0	0
$525$	−0.763106	−0.0333047
$526$	0	0
$527$	22.6716	0.987590
$528$	0	0
$529$	54.6312	2.37527
$530$	0	0
$531$	12.6571	0.549273
$532$	0	0
$533$	−28.9281	−1.25302
$534$	0	0
$535$	−30.7045	−1.32747
$536$	0	0
$537$	−7.68819	−0.331770
$538$	0	0
$539$	12.5846	0.542058
$540$	0	0
$541$	44.5294	1.91447	0.957235	−	0.289311i	$-0.0934263\pi$
$541$	44.5294	1.91447	0.957235	+	0.289311i	$0.0934263\pi$
$542$	0	0
$543$	6.61531	0.283890
$544$	0	0
$545$	−77.8696	−3.33557
$546$	0	0
$547$	12.7311	0.544341	0.272170	−	0.962249i	$-0.412259\pi$
$547$	12.7311	0.544341	0.272170	+	0.962249i	$0.412259\pi$
$548$	0	0
$549$	−13.2847	−0.566977
$550$	0	0
$551$	−17.5108	−0.745985
$552$	0	0
$553$	0.393790	0.0167457
$554$	0	0
$555$	17.0974	0.725744
$556$	0	0
$557$	−7.79188	−0.330152	−0.165076	−	0.986281i	$-0.552787\pi$
$557$	−7.79188	−0.330152	−0.165076	+	0.986281i	$0.552787\pi$
$558$	0	0
$559$	6.55084	0.277071
$560$	0	0
$561$	−8.72658	−0.368437
$562$	0	0
$563$	−1.25756	−0.0530000	−0.0265000	−	0.999649i	$-0.508436\pi$
$563$	−1.25756	−0.0530000	−0.0265000	+	0.999649i	$0.508436\pi$
$564$	0	0
$565$	−79.1757	−3.33095
$566$	0	0
$567$	−0.0538072	−0.00225969
$568$	0	0
$569$	−12.8572	−0.539002	−0.269501	−	0.963000i	$-0.586859\pi$
$569$	−12.8572	−0.539002	−0.269501	+	0.963000i	$0.586859\pi$
$570$	0	0
$571$	−10.2343	−0.428291	−0.214146	−	0.976802i	$-0.568697\pi$
$571$	−10.2343	−0.428291	−0.214146	+	0.976802i	$0.568697\pi$
$572$	0	0
$573$	−18.3520	−0.766665
$574$	0	0
$575$	−124.958	−5.21109
$576$	0	0
$577$	1.40494	0.0584885	0.0292442	−	0.999572i	$-0.490690\pi$
$577$	1.40494	0.0584885	0.0292442	+	0.999572i	$0.490690\pi$
$578$	0	0
$579$	−13.5441	−0.562873
$580$	0	0
$581$	0.224902	0.00933052
$582$	0	0
$583$	−22.6624	−0.938583
$584$	0	0
$585$	−13.6899	−0.566007
$586$	0	0
$587$	24.8660	1.02633	0.513164	−	0.858291i	$-0.328473\pi$
$587$	24.8660	1.02633	0.513164	+	0.858291i	$0.328473\pi$
$588$	0	0
$589$	16.9638	0.698981
$590$	0	0
$591$	−9.73483	−0.400437
$592$	0	0
$593$	−5.95991	−0.244744	−0.122372	−	0.992484i	$-0.539050\pi$
$593$	−5.95991	−0.244744	−0.122372	+	0.992484i	$0.539050\pi$
$594$	0	0
$595$	1.14344	0.0468764
$596$	0	0
$597$	13.4701	0.551296
$598$	0	0
$599$	−14.3166	−0.584961	−0.292481	−	0.956271i	$-0.594481\pi$
$599$	−14.3166	−0.584961	−0.292481	+	0.956271i	$0.594481\pi$
$600$	0	0
$601$	47.7302	1.94696	0.973478	−	0.228781i	$-0.0734739\pi$
$601$	47.7302	1.94696	0.973478	+	0.228781i	$0.0734739\pi$
$602$	0	0
$603$	−2.44390	−0.0995231
$604$	0	0
$605$	34.0098	1.38269
$606$	0	0
$607$	29.5341	1.19875	0.599376	−	0.800468i	$-0.295416\pi$
$607$	29.5341	1.19875	0.599376	+	0.800468i	$0.295416\pi$
$608$	0	0
$609$	0.259527	0.0105166
$610$	0	0
$611$	−18.0533	−0.730357
$612$	0	0
$613$	−38.7890	−1.56667	−0.783337	−	0.621598i	$-0.786484\pi$
$613$	−38.7890	−1.56667	−0.783337	+	0.621598i	$0.786484\pi$
$614$	0	0
$615$	40.5340	1.63449
$616$	0	0
$617$	25.8573	1.04097	0.520487	−	0.853869i	$-0.325750\pi$
$617$	25.8573	1.04097	0.520487	+	0.853869i	$0.325750\pi$
$618$	0	0
$619$	15.3185	0.615702	0.307851	−	0.951435i	$-0.400390\pi$
$619$	15.3185	0.615702	0.307851	+	0.951435i	$0.400390\pi$
$620$	0	0
$621$	−8.81086	−0.353568
$622$	0	0
$623$	0.677549	0.0271454
$624$	0	0
$625$	105.224	4.20897
$626$	0	0
$627$	−6.52958	−0.260766
$628$	0	0
$629$	−18.9410	−0.755228
$630$	0	0
$631$	−7.57047	−0.301376	−0.150688	−	0.988581i	$-0.548149\pi$
$631$	−7.57047	−0.301376	−0.150688	+	0.988581i	$0.548149\pi$
$632$	0	0
$633$	26.4635	1.05183
$634$	0	0
$635$	4.37975	0.173805
$636$	0	0
$637$	−21.8710	−0.866561
$638$	0	0
$639$	−12.0205	−0.475522
$640$	0	0
$641$	28.3061	1.11803	0.559013	−	0.829159i	$-0.311180\pi$
$641$	28.3061	1.11803	0.559013	+	0.829159i	$0.311180\pi$
$642$	0	0
$643$	−31.2242	−1.23136	−0.615682	−	0.787995i	$-0.711119\pi$
$643$	−31.2242	−1.23136	−0.615682	+	0.787995i	$0.711119\pi$
$644$	0	0
$645$	−9.17902	−0.361424
$646$	0	0
$647$	−11.6820	−0.459267	−0.229634	−	0.973277i	$-0.573753\pi$
$647$	−11.6820	−0.459267	−0.229634	+	0.973277i	$0.573753\pi$
$648$	0	0
$649$	−22.7645	−0.893583
$650$	0	0
$651$	−0.251420	−0.00985393
$652$	0	0
$653$	14.7230	0.576155	0.288077	−	0.957607i	$-0.406984\pi$
$653$	14.7230	0.576155	0.288077	+	0.957607i	$0.406984\pi$
$654$	0	0
$655$	64.6756	2.52708
$656$	0	0
$657$	−12.8813	−0.502549
$658$	0	0
$659$	−13.2194	−0.514955	−0.257477	−	0.966284i	$-0.582891\pi$
$659$	−13.2194	−0.514955	−0.257477	+	0.966284i	$0.582891\pi$
$660$	0	0
$661$	5.92347	0.230396	0.115198	−	0.993343i	$-0.463250\pi$
$661$	5.92347	0.230396	0.115198	+	0.993343i	$0.463250\pi$
$662$	0	0
$663$	15.1661	0.589001
$664$	0	0
$665$	0.855567	0.0331775
$666$	0	0
$667$	42.4972	1.64550
$668$	0	0
$669$	−2.44596	−0.0945661
$670$	0	0
$671$	23.8931	0.922384
$672$	0	0
$673$	16.9333	0.652730	0.326365	−	0.945244i	$-0.394176\pi$
$673$	16.9333	0.652730	0.326365	+	0.945244i	$0.394176\pi$
$674$	0	0
$675$	14.1822	0.545874
$676$	0	0
$677$	−25.7300	−0.988883	−0.494441	−	0.869211i	$-0.664627\pi$
$677$	−25.7300	−0.988883	−0.494441	+	0.869211i	$0.664627\pi$
$678$	0	0
$679$	−0.0702857	−0.00269732
$680$	0	0
$681$	−2.72395	−0.104382
$682$	0	0
$683$	31.6865	1.21245	0.606226	−	0.795293i	$-0.292683\pi$
$683$	31.6865	1.21245	0.606226	+	0.795293i	$0.292683\pi$
$684$	0	0
$685$	13.0342	0.498011
$686$	0	0
$687$	14.9160	0.569081
$688$	0	0
$689$	39.3854	1.50047
$690$	0	0
$691$	−49.5360	−1.88444	−0.942220	−	0.334995i	$-0.891265\pi$
$691$	−49.5360	−1.88444	−0.942220	+	0.334995i	$0.891265\pi$
$692$	0	0
$693$	0.0967748	0.00367617
$694$	0	0
$695$	99.4039	3.77060
$696$	0	0
$697$	−44.9048	−1.70089
$698$	0	0
$699$	3.89955	0.147495
$700$	0	0
$701$	16.1385	0.609543	0.304772	−	0.952425i	$-0.401420\pi$
$701$	16.1385	0.609543	0.304772	+	0.952425i	$0.401420\pi$
$702$	0	0
$703$	−14.1724	−0.534524
$704$	0	0
$705$	25.2962	0.952709
$706$	0	0
$707$	−0.297040	−0.0111713
$708$	0	0
$709$	−8.28699	−0.311224	−0.155612	−	0.987818i	$-0.549735\pi$
$709$	−8.28699	−0.311224	−0.155612	+	0.987818i	$0.549735\pi$
$710$	0	0
$711$	−7.31853	−0.274466
$712$	0	0
$713$	−41.1697	−1.54182
$714$	0	0
$715$	24.6219	0.920806
$716$	0	0
$717$	20.7106	0.773451
$718$	0	0
$719$	−28.8234	−1.07493	−0.537465	−	0.843286i	$-0.680618\pi$
$719$	−28.8234	−1.07493	−0.537465	+	0.843286i	$0.680618\pi$
$720$	0	0
$721$	0.723039	0.0269274
$722$	0	0
$723$	−22.8309	−0.849089
$724$	0	0
$725$	−68.4048	−2.54049
$726$	0	0
$727$	21.8850	0.811668	0.405834	−	0.913947i	$-0.366981\pi$
$727$	21.8850	0.811668	0.405834	+	0.913947i	$0.366981\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.1688	0.376107
$732$	0	0
$733$	−22.6598	−0.836957	−0.418479	−	0.908227i	$-0.637436\pi$
$733$	−22.6598	−0.836957	−0.418479	+	0.908227i	$0.637436\pi$
$734$	0	0
$735$	30.6456	1.13038
$736$	0	0
$737$	4.39546	0.161909
$738$	0	0
$739$	−27.3851	−1.00738	−0.503689	−	0.863885i	$-0.668024\pi$
$739$	−27.3851	−1.00738	−0.503689	+	0.863885i	$0.668024\pi$
$740$	0	0
$741$	11.3479	0.416874
$742$	0	0
$743$	−31.6979	−1.16288	−0.581442	−	0.813588i	$-0.697511\pi$
$743$	−31.6979	−1.16288	−0.581442	+	0.813588i	$0.697511\pi$
$744$	0	0
$745$	13.6656	0.500669
$746$	0	0
$747$	−4.17978	−0.152930
$748$	0	0
$749$	−0.377219	−0.0137833
$750$	0	0
$751$	−35.2936	−1.28788	−0.643941	−	0.765076i	$-0.722702\pi$
$751$	−35.2936	−1.28788	−0.643941	+	0.765076i	$0.722702\pi$
$752$	0	0
$753$	28.2481	1.02942
$754$	0	0
$755$	−36.3017	−1.32115
$756$	0	0
$757$	33.2411	1.20817	0.604084	−	0.796920i	$-0.293539\pi$
$757$	33.2411	1.20817	0.604084	+	0.796920i	$0.293539\pi$
$758$	0	0
$759$	15.8467	0.575200
$760$	0	0
$761$	−11.2855	−0.409101	−0.204550	−	0.978856i	$-0.565573\pi$
$761$	−11.2855	−0.409101	−0.204550	+	0.978856i	$0.565573\pi$
$762$	0	0
$763$	−0.956664	−0.0346335
$764$	0	0
$765$	−21.2506	−0.768319
$766$	0	0
$767$	39.5627	1.42853
$768$	0	0
$769$	−5.54460	−0.199944	−0.0999718	−	0.994990i	$-0.531875\pi$
$769$	−5.54460	−0.199944	−0.0999718	+	0.994990i	$0.531875\pi$
$770$	0	0
$771$	14.3311	0.516121
$772$	0	0
$773$	46.7569	1.68173	0.840865	−	0.541245i	$-0.182047\pi$
$773$	46.7569	1.68173	0.840865	+	0.541245i	$0.182047\pi$
$774$	0	0
$775$	66.2679	2.38041
$776$	0	0
$777$	0.210049	0.00753548
$778$	0	0
$779$	−33.5996	−1.20383
$780$	0	0
$781$	21.6193	0.773601
$782$	0	0
$783$	−4.82328	−0.172370
$784$	0	0
$785$	66.0236	2.35648
$786$	0	0
$787$	−0.363524	−0.0129582	−0.00647912	−	0.999979i	$-0.502062\pi$
$787$	−0.363524	−0.0129582	−0.00647912	+	0.999979i	$0.502062\pi$
$788$	0	0
$789$	−11.0759	−0.394313
$790$	0	0
$791$	−0.972710	−0.0345856
$792$	0	0
$793$	−41.5242	−1.47457
$794$	0	0
$795$	−55.1867	−1.95727
$796$	0	0
$797$	40.5630	1.43681	0.718407	−	0.695623i	$-0.244872\pi$
$797$	40.5630	1.43681	0.718407	+	0.695623i	$0.244872\pi$
$798$	0	0
$799$	−28.0239	−0.991413
$800$	0	0
$801$	−12.5922	−0.444922
$802$	0	0
$803$	23.1677	0.817570
$804$	0	0
$805$	−2.07639	−0.0731830
$806$	0	0
$807$	7.29590	0.256828
$808$	0	0
$809$	20.1274	0.707643	0.353821	−	0.935313i	$-0.384882\pi$
$809$	20.1274	0.707643	0.353821	+	0.935313i	$0.384882\pi$
$810$	0	0
$811$	17.7185	0.622179	0.311090	−	0.950381i	$-0.399306\pi$
$811$	17.7185	0.622179	0.311090	+	0.950381i	$0.399306\pi$
$812$	0	0
$813$	−5.25888	−0.184437
$814$	0	0
$815$	53.2084	1.86381
$816$	0	0
$817$	7.60871	0.266195
$818$	0	0
$819$	−0.168186	−0.00587691
$820$	0	0
$821$	17.7638	0.619960	0.309980	−	0.950743i	$-0.399678\pi$
$821$	17.7638	0.619960	0.309980	+	0.950743i	$0.399678\pi$
$822$	0	0
$823$	−43.2256	−1.50675	−0.753374	−	0.657592i	$-0.771575\pi$
$823$	−43.2256	−1.50675	−0.753374	+	0.657592i	$0.771575\pi$
$824$	0	0
$825$	−25.5074	−0.888053
$826$	0	0
$827$	28.7660	1.00029	0.500147	−	0.865941i	$-0.333279\pi$
$827$	28.7660	1.00029	0.500147	+	0.865941i	$0.333279\pi$
$828$	0	0
$829$	17.6020	0.611344	0.305672	−	0.952137i	$-0.401119\pi$
$829$	17.6020	0.611344	0.305672	+	0.952137i	$0.401119\pi$
$830$	0	0
$831$	15.9700	0.553993
$832$	0	0
$833$	−33.9501	−1.17630
$834$	0	0
$835$	94.8296	3.28171
$836$	0	0
$837$	4.67261	0.161509
$838$	0	0
$839$	−50.8558	−1.75574	−0.877869	−	0.478901i	$-0.841035\pi$
$839$	−50.8558	−1.75574	−0.877869	+	0.478901i	$0.841035\pi$
$840$	0	0
$841$	−5.73601	−0.197793
$842$	0	0
$843$	2.28065	0.0785497
$844$	0	0
$845$	14.1460	0.486638
$846$	0	0
$847$	0.417826	0.0143567
$848$	0	0
$849$	5.80178	0.199117
$850$	0	0
$851$	34.3953	1.17906
$852$	0	0
$853$	−2.04228	−0.0699263	−0.0349632	−	0.999389i	$-0.511131\pi$
$853$	−2.04228	−0.0699263	−0.0349632	+	0.999389i	$0.511131\pi$
$854$	0	0
$855$	−15.9006	−0.543789
$856$	0	0
$857$	48.4144	1.65380	0.826902	−	0.562346i	$-0.190101\pi$
$857$	48.4144	1.65380	0.826902	+	0.562346i	$0.190101\pi$
$858$	0	0
$859$	−25.1124	−0.856825	−0.428413	−	0.903583i	$-0.640927\pi$
$859$	−25.1124	−0.856825	−0.428413	+	0.903583i	$0.640927\pi$
$860$	0	0
$861$	0.497978	0.0169711
$862$	0	0
$863$	38.2670	1.30262	0.651311	−	0.758811i	$-0.274219\pi$
$863$	38.2670	1.30262	0.651311	+	0.758811i	$0.274219\pi$
$864$	0	0
$865$	−59.0233	−2.00685
$866$	0	0
$867$	6.54212	0.222182
$868$	0	0
$869$	13.1627	0.446515
$870$	0	0
$871$	−7.63894	−0.258835
$872$	0	0
$873$	1.30625	0.0442098
$874$	0	0
$875$	2.16390	0.0731533
$876$	0	0
$877$	24.7840	0.836897	0.418448	−	0.908241i	$-0.362574\pi$
$877$	24.7840	0.836897	0.418448	+	0.908241i	$0.362574\pi$
$878$	0	0
$879$	−3.64830	−0.123054
$880$	0	0
$881$	−31.6882	−1.06760	−0.533801	−	0.845610i	$-0.679237\pi$
$881$	−31.6882	−1.06760	−0.533801	+	0.845610i	$0.679237\pi$
$882$	0	0
$883$	−25.9387	−0.872908	−0.436454	−	0.899727i	$-0.643766\pi$
$883$	−25.9387	−0.872908	−0.436454	+	0.899727i	$0.643766\pi$
$884$	0	0
$885$	−55.4352	−1.86343
$886$	0	0
$887$	−1.43119	−0.0480547	−0.0240274	−	0.999711i	$-0.507649\pi$
$887$	−1.43119	−0.0480547	−0.0240274	+	0.999711i	$0.507649\pi$
$888$	0	0
$889$	0.0538072	0.00180464
$890$	0	0
$891$	−1.79855	−0.0602536
$892$	0	0
$893$	−20.9686	−0.701687
$894$	0	0
$895$	33.6724	1.12554
$896$	0	0
$897$	−27.5403	−0.919543
$898$	0	0
$899$	−22.5373	−0.751660
$900$	0	0
$901$	61.1376	2.03679
$902$	0	0
$903$	−0.112769	−0.00375270
$904$	0	0
$905$	−28.9734	−0.963109
$906$	0	0
$907$	−50.5274	−1.67773	−0.838867	−	0.544336i	$-0.816782\pi$
$907$	−50.5274	−1.67773	−0.838867	+	0.544336i	$0.816782\pi$
$908$	0	0
$909$	5.52045	0.183102
$910$	0	0
$911$	4.82746	0.159941	0.0799704	−	0.996797i	$-0.474517\pi$
$911$	4.82746	0.159941	0.0799704	+	0.996797i	$0.474517\pi$
$912$	0	0
$913$	7.51752	0.248794
$914$	0	0
$915$	58.1836	1.92349
$916$	0	0
$917$	0.794569	0.0262390
$918$	0	0
$919$	46.6972	1.54040	0.770199	−	0.637803i	$-0.220157\pi$
$919$	46.6972	1.54040	0.770199	+	0.637803i	$0.220157\pi$
$920$	0	0
$921$	15.9566	0.525786
$922$	0	0
$923$	−37.5726	−1.23672
$924$	0	0
$925$	−55.3637	−1.82035
$926$	0	0
$927$	−13.4376	−0.441348
$928$	0	0
$929$	54.5443	1.78954	0.894770	−	0.446526i	$-0.147339\pi$
$929$	54.5443	1.78954	0.894770	+	0.446526i	$0.147339\pi$
$930$	0	0
$931$	−25.4028	−0.832545
$932$	0	0
$933$	7.53925	0.246824
$934$	0	0
$935$	38.2203	1.24994
$936$	0	0
$937$	19.3213	0.631200	0.315600	−	0.948892i	$-0.397794\pi$
$937$	19.3213	0.631200	0.315600	+	0.948892i	$0.397794\pi$
$938$	0	0
$939$	27.1451	0.885845
$940$	0	0
$941$	−7.45696	−0.243090	−0.121545	−	0.992586i	$-0.538785\pi$
$941$	−7.45696	−0.243090	−0.121545	+	0.992586i	$0.538785\pi$
$942$	0	0
$943$	81.5432	2.65541
$944$	0	0
$945$	0.235662	0.00766610
$946$	0	0
$947$	−30.6916	−0.997342	−0.498671	−	0.866791i	$-0.666178\pi$
$947$	−30.6916	−0.997342	−0.498671	+	0.866791i	$0.666178\pi$
$948$	0	0
$949$	−40.2635	−1.30701
$950$	0	0
$951$	17.4848	0.566984
$952$	0	0
$953$	15.4761	0.501321	0.250660	−	0.968075i	$-0.419352\pi$
$953$	15.4761	0.501321	0.250660	+	0.968075i	$0.419352\pi$
$954$	0	0
$955$	80.3771	2.60094
$956$	0	0
$957$	8.67488	0.280419
$958$	0	0
$959$	0.160131	0.00517090
$960$	0	0
$961$	−9.16675	−0.295702
$962$	0	0
$963$	7.01057	0.225912
$964$	0	0
$965$	59.3198	1.90957
$966$	0	0
$967$	36.5944	1.17680	0.588398	−	0.808571i	$-0.299759\pi$
$967$	36.5944	1.17680	0.588398	+	0.808571i	$0.299759\pi$
$968$	0	0
$969$	17.6152	0.565881
$970$	0	0
$971$	25.6350	0.822665	0.411333	−	0.911485i	$-0.365063\pi$
$971$	25.6350	0.822665	0.411333	+	0.911485i	$0.365063\pi$
$972$	0	0
$973$	1.22122	0.0391506
$974$	0	0
$975$	44.3297	1.41969
$976$	0	0
$977$	−44.5028	−1.42377	−0.711885	−	0.702296i	$-0.752158\pi$
$977$	−44.5028	−1.42377	−0.711885	+	0.702296i	$0.752158\pi$
$978$	0	0
$979$	22.6476	0.723819
$980$	0	0
$981$	17.7795	0.567654
$982$	0	0
$983$	−15.8646	−0.506002	−0.253001	−	0.967466i	$-0.581418\pi$
$983$	−15.8646	−0.506002	−0.253001	+	0.967466i	$0.581418\pi$
$984$	0	0
$985$	42.6361	1.35850
$986$	0	0
$987$	0.310775	0.00989208
$988$	0	0
$989$	−18.4657	−0.587174
$990$	0	0
$991$	−1.46080	−0.0464037	−0.0232019	−	0.999731i	$-0.507386\pi$
$991$	−1.46080	−0.0464037	−0.0232019	+	0.999731i	$0.507386\pi$
$992$	0	0
$993$	−22.4321	−0.711860
$994$	0	0
$995$	−58.9959	−1.87030
$996$	0	0
$997$	6.34102	0.200822	0.100411	−	0.994946i	$-0.467984\pi$
$997$	6.34102	0.200822	0.100411	+	0.994946i	$0.467984\pi$
$998$	0	0
$999$	−3.90374	−0.123509

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3048.2.a.l.1.1	✓	8
3.2	odd	2		9144.2.a.x.1.8		8
4.3	odd	2		6096.2.a.bi.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3048.2.a.l.1.1	✓	8	1.1	even	1	trivial
6096.2.a.bi.1.1		8	4.3	odd	2
9144.2.a.x.1.8		8	3.2	odd	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:	\( N \)	\(=\)	\( 3048 = 2^{3} \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3048.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.37975 q^{5} -0.0538072 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.37975 q^{5} -0.0538072 q^{7} +1.00000 q^{9} -1.79855 q^{11} +3.12572 q^{13} -4.37975 q^{15} +4.85202 q^{17} +3.63048 q^{19} -0.0538072 q^{21} -8.81086 q^{23} +14.1822 q^{25} +1.00000 q^{27} -4.82328 q^{29} +4.67261 q^{31} -1.79855 q^{33} +0.235662 q^{35} -3.90374 q^{37} +3.12572 q^{39} -9.25486 q^{41} +2.09579 q^{43} -4.37975 q^{45} -5.77571 q^{47} -6.99710 q^{49} +4.85202 q^{51} +12.6004 q^{53} +7.87718 q^{55} +3.63048 q^{57} +12.6571 q^{59} -13.2847 q^{61} -0.0538072 q^{63} -13.6899 q^{65} -2.44390 q^{67} -8.81086 q^{69} -12.0205 q^{71} -12.8813 q^{73} +14.1822 q^{75} +0.0967748 q^{77} -7.31853 q^{79} +1.00000 q^{81} -4.17978 q^{83} -21.2506 q^{85} -4.82328 q^{87} -12.5922 q^{89} -0.168186 q^{91} +4.67261 q^{93} -15.9006 q^{95} +1.30625 q^{97} -1.79855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 4 q^{5} - 7 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 4 * q^5 - 7 * q^7 + 8 * q^9	\( 8 q + 8 q^{3} - 4 q^{5} - 7 q^{7} + 8 q^{9} - 8 q^{11} - 10 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{21} - 20 q^{23} + 4 q^{25} + 8 q^{27} - 5 q^{29} - 14 q^{31} - 8 q^{33} - 19 q^{35} - 20 q^{37} - 10 q^{39} - 18 q^{41} - 10 q^{43} - 4 q^{45} - 23 q^{47} + 7 q^{49} - 10 q^{51} - 11 q^{55} - 19 q^{59} - 24 q^{61} - 7 q^{63} - 11 q^{65} + 4 q^{67} - 20 q^{69} - 47 q^{71} - 12 q^{73} + 4 q^{75} - 9 q^{77} - 35 q^{79} + 8 q^{81} - 19 q^{83} - 29 q^{85} - 5 q^{87} - 9 q^{89} + 6 q^{91} - 14 q^{93} - 39 q^{95} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 4 * q^5 - 7 * q^7 + 8 * q^9 - 8 * q^11 - 10 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^21 - 20 * q^23 + 4 * q^25 + 8 * q^27 - 5 * q^29 - 14 * q^31 - 8 * q^33 - 19 * q^35 - 20 * q^37 - 10 * q^39 - 18 * q^41 - 10 * q^43 - 4 * q^45 - 23 * q^47 + 7 * q^49 - 10 * q^51 - 11 * q^55 - 19 * q^59 - 24 * q^61 - 7 * q^63 - 11 * q^65 + 4 * q^67 - 20 * q^69 - 47 * q^71 - 12 * q^73 + 4 * q^75 - 9 * q^77 - 35 * q^79 + 8 * q^81 - 19 * q^83 - 29 * q^85 - 5 * q^87 - 9 * q^89 + 6 * q^91 - 14 * q^93 - 39 * q^95 + 8 * q^97 - 8 * q^99

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