LMFDB - Embedded newform 4840.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.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:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.477260$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.294963 q^{3} +1.00000 q^{5} -4.39026 q^{7} -2.91300 q^{9} +5.96281 q^{13} +0.294963 q^{15} -1.66785 q^{17} +7.69351 q^{19} -1.29496 q^{21} -0.904706 q^{23} +1.00000 q^{25} -1.74411 q^{27} -4.73899 q^{29} -5.12608 q^{31} -4.39026 q^{35} -0.184976 q^{37} +1.75881 q^{39} -2.62237 q^{41} +3.59822 q^{43} -2.91300 q^{45} -0.776180 q^{47} +12.2744 q^{49} -0.491953 q^{51} -9.59554 q^{53} +2.26930 q^{57} -11.2538 q^{59} +13.1898 q^{61} +12.7888 q^{63} +5.96281 q^{65} -7.79954 q^{67} -0.266855 q^{69} -6.97072 q^{71} +12.7541 q^{73} +0.294963 q^{75} -10.0313 q^{79} +8.22454 q^{81} +4.09529 q^{83} -1.66785 q^{85} -1.39783 q^{87} -0.466291 q^{89} -26.1783 q^{91} -1.51200 q^{93} +7.69351 q^{95} +7.09963 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9} + 3 q^{13} - 2 q^{15} + 11 q^{17} + 2 q^{19} - 2 q^{21} - 11 q^{23} + 4 q^{25} + q^{27} + 4 q^{29} - 17 q^{31} - 7 q^{35} - 3 q^{37} + q^{39} + 13 q^{41} - 7 q^{43}+ \cdots + 2 q^{97}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 - 7 * q^7 - 4 * q^9 + 3 * q^13 - 2 * q^15 + 11 * q^17 + 2 * q^19 - 2 * q^21 - 11 * q^23 + 4 * q^25 + q^27 + 4 * q^29 - 17 * q^31 - 7 * q^35 - 3 * q^37 + q^39 + 13 * q^41 - 7 * q^43 - 4 * q^45 - q^47 - 5 * q^49 - q^51 - 15 * q^53 + 17 * q^57 - 17 * q^59 + 4 * q^61 + 15 * q^63 + 3 * q^65 + 7 * q^67 + 4 * q^69 - 15 * q^71 + 7 * q^73 - 2 * q^75 - 12 * q^79 - 8 * q^81 + 9 * q^83 + 11 * q^85 - 23 * q^87 - 12 * q^89 - 24 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97

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
$3$	0.294963	0.170297	0.0851485	−	0.996368i	$-0.472864\pi$
$3$	0.294963	0.170297	0.0851485	+	0.996368i	$0.472864\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.39026	−1.65936	−0.829681	−	0.558238i	$-0.811477\pi$
$7$	−4.39026	−1.65936	−0.829681	+	0.558238i	$0.811477\pi$
$8$	0	0
$9$	−2.91300	−0.970999
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.96281	1.65379	0.826893	−	0.562359i	$-0.190106\pi$
$13$	5.96281	1.65379	0.826893	+	0.562359i	$0.190106\pi$
$14$	0	0
$15$	0.294963	0.0761591
$16$	0	0
$17$	−1.66785	−0.404513	−0.202256	−	0.979333i	$-0.564827\pi$
$17$	−1.66785	−0.404513	−0.202256	+	0.979333i	$0.564827\pi$
$18$	0	0
$19$	7.69351	1.76501	0.882506	−	0.470301i	$-0.155855\pi$
$19$	7.69351	1.76501	0.882506	+	0.470301i	$0.155855\pi$
$20$	0	0
$21$	−1.29496	−0.282584
$22$	0	0
$23$	−0.904706	−0.188644	−0.0943221	−	0.995542i	$-0.530068\pi$
$23$	−0.904706	−0.188644	−0.0943221	+	0.995542i	$0.530068\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.74411	−0.335655
$28$	0	0
$29$	−4.73899	−0.880008	−0.440004	−	0.897996i	$-0.645023\pi$
$29$	−4.73899	−0.880008	−0.440004	+	0.897996i	$0.645023\pi$
$30$	0	0
$31$	−5.12608	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$31$	−5.12608	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.39026	−0.742089
$36$	0	0
$37$	−0.184976	−0.0304098	−0.0152049	−	0.999884i	$-0.504840\pi$
$37$	−0.184976	−0.0304098	−0.0152049	+	0.999884i	$0.504840\pi$
$38$	0	0
$39$	1.75881	0.281635
$40$	0	0
$41$	−2.62237	−0.409545	−0.204773	−	0.978810i	$-0.565645\pi$
$41$	−2.62237	−0.409545	−0.204773	+	0.978810i	$0.565645\pi$
$42$	0	0
$43$	3.59822	0.548723	0.274361	−	0.961627i	$-0.411534\pi$
$43$	3.59822	0.548723	0.274361	+	0.961627i	$0.411534\pi$
$44$	0	0
$45$	−2.91300	−0.434244
$46$	0	0
$47$	−0.776180	−0.113217	−0.0566087	−	0.998396i	$-0.518029\pi$
$47$	−0.776180	−0.113217	−0.0566087	+	0.998396i	$0.518029\pi$
$48$	0	0
$49$	12.2744	1.75348
$50$	0	0
$51$	−0.491953	−0.0688872
$52$	0	0
$53$	−9.59554	−1.31805	−0.659024	−	0.752122i	$-0.729031\pi$
$53$	−9.59554	−1.31805	−0.659024	+	0.752122i	$0.729031\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.26930	0.300576
$58$	0	0
$59$	−11.2538	−1.46512	−0.732561	−	0.680701i	$-0.761675\pi$
$59$	−11.2538	−1.46512	−0.732561	+	0.680701i	$0.761675\pi$
$60$	0	0
$61$	13.1898	1.68878	0.844390	−	0.535729i	$-0.179963\pi$
$61$	13.1898	1.68878	0.844390	+	0.535729i	$0.179963\pi$
$62$	0	0
$63$	12.7888	1.61124
$64$	0	0
$65$	5.96281	0.739596
$66$	0	0
$67$	−7.79954	−0.952866	−0.476433	−	0.879211i	$-0.658070\pi$
$67$	−7.79954	−0.952866	−0.476433	+	0.879211i	$0.658070\pi$
$68$	0	0
$69$	−0.266855	−0.0321255
$70$	0	0
$71$	−6.97072	−0.827273	−0.413636	−	0.910442i	$-0.635742\pi$
$71$	−6.97072	−0.827273	−0.413636	+	0.910442i	$0.635742\pi$
$72$	0	0
$73$	12.7541	1.49275	0.746375	−	0.665526i	$-0.231793\pi$
$73$	12.7541	1.49275	0.746375	+	0.665526i	$0.231793\pi$
$74$	0	0
$75$	0.294963	0.0340594
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0313	−1.12861	−0.564303	−	0.825568i	$-0.690855\pi$
$79$	−10.0313	−1.12861	−0.564303	+	0.825568i	$0.690855\pi$
$80$	0	0
$81$	8.22454	0.913838
$82$	0	0
$83$	4.09529	0.449517	0.224758	−	0.974415i	$-0.427841\pi$
$83$	4.09529	0.449517	0.224758	+	0.974415i	$0.427841\pi$
$84$	0	0
$85$	−1.66785	−0.180904
$86$	0	0
$87$	−1.39783	−0.149863
$88$	0	0
$89$	−0.466291	−0.0494267	−0.0247134	−	0.999695i	$-0.507867\pi$
$89$	−0.466291	−0.0494267	−0.0247134	+	0.999695i	$0.507867\pi$
$90$	0	0
$91$	−26.1783	−2.74423
$92$	0	0
$93$	−1.51200	−0.156787
$94$	0	0
$95$	7.69351	0.789338
$96$	0	0
$97$	7.09963	0.720858	0.360429	−	0.932787i	$-0.382630\pi$
$97$	7.09963	0.720858	0.360429	+	0.932787i	$0.382630\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.2851	−1.62043	−0.810214	−	0.586135i	$-0.800649\pi$
$101$	−16.2851	−1.62043	−0.810214	+	0.586135i	$0.800649\pi$
$102$	0	0
$103$	3.19059	0.314378	0.157189	−	0.987569i	$-0.449757\pi$
$103$	3.19059	0.314378	0.157189	+	0.987569i	$0.449757\pi$
$104$	0	0
$105$	−1.29496	−0.126375
$106$	0	0
$107$	−13.1644	−1.27265	−0.636324	−	0.771422i	$-0.719546\pi$
$107$	−13.1644	−1.27265	−0.636324	+	0.771422i	$0.719546\pi$
$108$	0	0
$109$	1.33532	0.127900	0.0639502	−	0.997953i	$-0.479630\pi$
$109$	1.33532	0.127900	0.0639502	+	0.997953i	$0.479630\pi$
$110$	0	0
$111$	−0.0545610	−0.00517870
$112$	0	0
$113$	−1.67448	−0.157522	−0.0787611	−	0.996894i	$-0.525096\pi$
$113$	−1.67448	−0.157522	−0.0787611	+	0.996894i	$0.525096\pi$
$114$	0	0
$115$	−0.904706	−0.0843643
$116$	0	0
$117$	−17.3696	−1.60582
$118$	0	0
$119$	7.32228	0.671232
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−0.773501	−0.0697443
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.901470	0.0799926	0.0399963	−	0.999200i	$-0.487265\pi$
$127$	0.901470	0.0799926	0.0399963	+	0.999200i	$0.487265\pi$
$128$	0	0
$129$	1.06134	0.0934458
$130$	0	0
$131$	−14.5527	−1.27148	−0.635739	−	0.771904i	$-0.719305\pi$
$131$	−14.5527	−1.27148	−0.635739	+	0.771904i	$0.719305\pi$
$132$	0	0
$133$	−33.7765	−2.92879
$134$	0	0
$135$	−1.74411	−0.150109
$136$	0	0
$137$	−20.5895	−1.75908	−0.879540	−	0.475824i	$-0.842150\pi$
$137$	−20.5895	−1.75908	−0.879540	+	0.475824i	$0.842150\pi$
$138$	0	0
$139$	−14.9841	−1.27094	−0.635469	−	0.772126i	$-0.719193\pi$
$139$	−14.9841	−1.27094	−0.635469	+	0.772126i	$0.719193\pi$
$140$	0	0
$141$	−0.228944	−0.0192806
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.73899	−0.393552
$146$	0	0
$147$	3.62048	0.298612
$148$	0	0
$149$	10.9626	0.898089	0.449045	−	0.893509i	$-0.351764\pi$
$149$	10.9626	0.898089	0.449045	+	0.893509i	$0.351764\pi$
$150$	0	0
$151$	−5.91172	−0.481089	−0.240544	−	0.970638i	$-0.577326\pi$
$151$	−5.91172	−0.481089	−0.240544	+	0.970638i	$0.577326\pi$
$152$	0	0
$153$	4.85844	0.392781
$154$	0	0
$155$	−5.12608	−0.411737
$156$	0	0
$157$	−4.42628	−0.353256	−0.176628	−	0.984278i	$-0.556519\pi$
$157$	−4.42628	−0.353256	−0.176628	+	0.984278i	$0.556519\pi$
$158$	0	0
$159$	−2.83033	−0.224460
$160$	0	0
$161$	3.97189	0.313029
$162$	0	0
$163$	−7.38464	−0.578410	−0.289205	−	0.957267i	$-0.593391\pi$
$163$	−7.38464	−0.578410	−0.289205	+	0.957267i	$0.593391\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.4876	−1.50799	−0.753996	−	0.656879i	$-0.771876\pi$
$167$	−19.4876	−1.50799	−0.753996	+	0.656879i	$0.771876\pi$
$168$	0	0
$169$	22.5551	1.73501
$170$	0	0
$171$	−22.4112	−1.71383
$172$	0	0
$173$	9.50805	0.722883	0.361442	−	0.932395i	$-0.382285\pi$
$173$	9.50805	0.722883	0.361442	+	0.932395i	$0.382285\pi$
$174$	0	0
$175$	−4.39026	−0.331872
$176$	0	0
$177$	−3.31946	−0.249506
$178$	0	0
$179$	7.21253	0.539090	0.269545	−	0.962988i	$-0.413127\pi$
$179$	7.21253	0.539090	0.269545	+	0.962988i	$0.413127\pi$
$180$	0	0
$181$	−23.3895	−1.73853	−0.869263	−	0.494350i	$-0.835406\pi$
$181$	−23.3895	−1.73853	−0.869263	+	0.494350i	$0.835406\pi$
$182$	0	0
$183$	3.89050	0.287594
$184$	0	0
$185$	−0.184976	−0.0135997
$186$	0	0
$187$	0	0
$188$	0	0
$189$	7.65711	0.556973
$190$	0	0
$191$	−4.16572	−0.301421	−0.150710	−	0.988578i	$-0.548156\pi$
$191$	−4.16572	−0.301421	−0.150710	+	0.988578i	$0.548156\pi$
$192$	0	0
$193$	−23.1660	−1.66753	−0.833763	−	0.552122i	$-0.813818\pi$
$193$	−23.1660	−1.66753	−0.833763	+	0.552122i	$0.813818\pi$
$194$	0	0
$195$	1.75881	0.125951
$196$	0	0
$197$	10.1507	0.723209	0.361604	−	0.932332i	$-0.382229\pi$
$197$	10.1507	0.723209	0.361604	+	0.932332i	$0.382229\pi$
$198$	0	0
$199$	−17.0131	−1.20603	−0.603014	−	0.797731i	$-0.706034\pi$
$199$	−17.0131	−1.20603	−0.603014	+	0.797731i	$0.706034\pi$
$200$	0	0
$201$	−2.30058	−0.162270
$202$	0	0
$203$	20.8054	1.46025
$204$	0	0
$205$	−2.62237	−0.183154
$206$	0	0
$207$	2.63541	0.183173
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−6.01620	−0.414173	−0.207086	−	0.978323i	$-0.566398\pi$
$211$	−6.01620	−0.414173	−0.207086	+	0.978323i	$0.566398\pi$
$212$	0	0
$213$	−2.05611	−0.140882
$214$	0	0
$215$	3.59822	0.245396
$216$	0	0
$217$	22.5048	1.52773
$218$	0	0
$219$	3.76197	0.254211
$220$	0	0
$221$	−9.94506	−0.668977
$222$	0	0
$223$	2.49407	0.167016	0.0835078	−	0.996507i	$-0.473388\pi$
$223$	2.49407	0.167016	0.0835078	+	0.996507i	$0.473388\pi$
$224$	0	0
$225$	−2.91300	−0.194200
$226$	0	0
$227$	−24.9953	−1.65899	−0.829497	−	0.558512i	$-0.811373\pi$
$227$	−24.9953	−1.65899	−0.829497	+	0.558512i	$0.811373\pi$
$228$	0	0
$229$	−2.77082	−0.183101	−0.0915506	−	0.995800i	$-0.529182\pi$
$229$	−2.77082	−0.183101	−0.0915506	+	0.995800i	$0.529182\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.60025	−0.628933	−0.314467	−	0.949269i	$-0.601826\pi$
$233$	−9.60025	−0.628933	−0.314467	+	0.949269i	$0.601826\pi$
$234$	0	0
$235$	−0.776180	−0.0506324
$236$	0	0
$237$	−2.95885	−0.192198
$238$	0	0
$239$	14.7692	0.955341	0.477671	−	0.878539i	$-0.341481\pi$
$239$	14.7692	0.955341	0.477671	+	0.878539i	$0.341481\pi$
$240$	0	0
$241$	−10.4338	−0.672099	−0.336049	−	0.941844i	$-0.609091\pi$
$241$	−10.4338	−0.672099	−0.336049	+	0.941844i	$0.609091\pi$
$242$	0	0
$243$	7.65828	0.491279
$244$	0	0
$245$	12.2744	0.784180
$246$	0	0
$247$	45.8749	2.91895
$248$	0	0
$249$	1.20796	0.0765513
$250$	0	0
$251$	−6.73394	−0.425042	−0.212521	−	0.977156i	$-0.568167\pi$
$251$	−6.73394	−0.425042	−0.212521	+	0.977156i	$0.568167\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.491953	−0.0308073
$256$	0	0
$257$	7.74192	0.482928	0.241464	−	0.970410i	$-0.422372\pi$
$257$	7.74192	0.482928	0.241464	+	0.970410i	$0.422372\pi$
$258$	0	0
$259$	0.812091	0.0504609
$260$	0	0
$261$	13.8047	0.854487
$262$	0	0
$263$	18.1618	1.11990	0.559952	−	0.828525i	$-0.310819\pi$
$263$	18.1618	1.11990	0.559952	+	0.828525i	$0.310819\pi$
$264$	0	0
$265$	−9.59554	−0.589449
$266$	0	0
$267$	−0.137538	−0.00841721
$268$	0	0
$269$	−10.7267	−0.654021	−0.327011	−	0.945021i	$-0.606041\pi$
$269$	−10.7267	−0.654021	−0.327011	+	0.945021i	$0.606041\pi$
$270$	0	0
$271$	−18.7875	−1.14126	−0.570630	−	0.821207i	$-0.693301\pi$
$271$	−18.7875	−1.14126	−0.570630	+	0.821207i	$0.693301\pi$
$272$	0	0
$273$	−7.72162	−0.467334
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.8467	1.73323	0.866614	−	0.498978i	$-0.166291\pi$
$277$	28.8467	1.73323	0.866614	+	0.498978i	$0.166291\pi$
$278$	0	0
$279$	14.9323	0.893971
$280$	0	0
$281$	27.0542	1.61392	0.806959	−	0.590608i	$-0.201112\pi$
$281$	27.0542	1.61392	0.806959	+	0.590608i	$0.201112\pi$
$282$	0	0
$283$	12.8731	0.765228	0.382614	−	0.923908i	$-0.375024\pi$
$283$	12.8731	0.765228	0.382614	+	0.923908i	$0.375024\pi$
$284$	0	0
$285$	2.26930	0.134422
$286$	0	0
$287$	11.5129	0.679583
$288$	0	0
$289$	−14.2183	−0.836370
$290$	0	0
$291$	2.09413	0.122760
$292$	0	0
$293$	5.88873	0.344024	0.172012	−	0.985095i	$-0.444973\pi$
$293$	5.88873	0.344024	0.172012	+	0.985095i	$0.444973\pi$
$294$	0	0
$295$	−11.2538	−0.655223
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.39459	−0.311977
$300$	0	0
$301$	−15.7971	−0.910529
$302$	0	0
$303$	−4.80350	−0.275954
$304$	0	0
$305$	13.1898	0.755246
$306$	0	0
$307$	6.85844	0.391432	0.195716	−	0.980661i	$-0.437297\pi$
$307$	6.85844	0.391432	0.195716	+	0.980661i	$0.437297\pi$
$308$	0	0
$309$	0.941105	0.0535376
$310$	0	0
$311$	−14.2693	−0.809138	−0.404569	−	0.914508i	$-0.632578\pi$
$311$	−14.2693	−0.809138	−0.404569	+	0.914508i	$0.632578\pi$
$312$	0	0
$313$	1.59973	0.0904220	0.0452110	−	0.998977i	$-0.485604\pi$
$313$	1.59973	0.0904220	0.0452110	+	0.998977i	$0.485604\pi$
$314$	0	0
$315$	12.7888	0.720568
$316$	0	0
$317$	−30.8556	−1.73302	−0.866511	−	0.499158i	$-0.833643\pi$
$317$	−30.8556	−1.73302	−0.866511	+	0.499158i	$0.833643\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.88300	−0.216728
$322$	0	0
$323$	−12.8316	−0.713970
$324$	0	0
$325$	5.96281	0.330757
$326$	0	0
$327$	0.393870	0.0217810
$328$	0	0
$329$	3.40763	0.187869
$330$	0	0
$331$	−1.16816	−0.0642079	−0.0321040	−	0.999485i	$-0.510221\pi$
$331$	−1.16816	−0.0642079	−0.0321040	+	0.999485i	$0.510221\pi$
$332$	0	0
$333$	0.538833	0.0295279
$334$	0	0
$335$	−7.79954	−0.426134
$336$	0	0
$337$	14.0291	0.764212	0.382106	−	0.924119i	$-0.375199\pi$
$337$	14.0291	0.764212	0.382106	+	0.924119i	$0.375199\pi$
$338$	0	0
$339$	−0.493910	−0.0268255
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−23.1558	−1.25029
$344$	0	0
$345$	−0.266855	−0.0143670
$346$	0	0
$347$	33.3514	1.79040	0.895198	−	0.445669i	$-0.147034\pi$
$347$	33.3514	1.79040	0.895198	+	0.445669i	$0.147034\pi$
$348$	0	0
$349$	−35.3268	−1.89100	−0.945500	−	0.325623i	$-0.894426\pi$
$349$	−35.3268	−1.89100	−0.945500	+	0.325623i	$0.894426\pi$
$350$	0	0
$351$	−10.3998	−0.555102
$352$	0	0
$353$	11.7558	0.625698	0.312849	−	0.949803i	$-0.398717\pi$
$353$	11.7558	0.625698	0.312849	+	0.949803i	$0.398717\pi$
$354$	0	0
$355$	−6.97072	−0.369968
$356$	0	0
$357$	2.15980	0.114309
$358$	0	0
$359$	−2.72101	−0.143609	−0.0718047	−	0.997419i	$-0.522876\pi$
$359$	−2.72101	−0.143609	−0.0718047	+	0.997419i	$0.522876\pi$
$360$	0	0
$361$	40.1901	2.11527
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.7541	0.667578
$366$	0	0
$367$	3.10358	0.162006	0.0810029	−	0.996714i	$-0.474188\pi$
$367$	3.10358	0.162006	0.0810029	+	0.996714i	$0.474188\pi$
$368$	0	0
$369$	7.63895	0.397668
$370$	0	0
$371$	42.1269	2.18712
$372$	0	0
$373$	−27.4604	−1.42185	−0.710924	−	0.703269i	$-0.751723\pi$
$373$	−27.4604	−1.42185	−0.710924	+	0.703269i	$0.751723\pi$
$374$	0	0
$375$	0.294963	0.0152318
$376$	0	0
$377$	−28.2577	−1.45535
$378$	0	0
$379$	13.9435	0.716229	0.358114	−	0.933678i	$-0.383420\pi$
$379$	13.9435	0.716229	0.358114	+	0.933678i	$0.383420\pi$
$380$	0	0
$381$	0.265900	0.0136225
$382$	0	0
$383$	8.15608	0.416756	0.208378	−	0.978048i	$-0.433182\pi$
$383$	8.15608	0.416756	0.208378	+	0.978048i	$0.433182\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.4816	−0.532809
$388$	0	0
$389$	−3.10121	−0.157237	−0.0786187	−	0.996905i	$-0.525051\pi$
$389$	−3.10121	−0.157237	−0.0786187	+	0.996905i	$0.525051\pi$
$390$	0	0
$391$	1.50891	0.0763090
$392$	0	0
$393$	−4.29252	−0.216529
$394$	0	0
$395$	−10.0313	−0.504728
$396$	0	0
$397$	23.1546	1.16209	0.581047	−	0.813870i	$-0.302643\pi$
$397$	23.1546	1.16209	0.581047	+	0.813870i	$0.302643\pi$
$398$	0	0
$399$	−9.96281	−0.498764
$400$	0	0
$401$	−8.24375	−0.411673	−0.205837	−	0.978586i	$-0.565992\pi$
$401$	−8.24375	−0.411673	−0.205837	+	0.978586i	$0.565992\pi$
$402$	0	0
$403$	−30.5658	−1.52259
$404$	0	0
$405$	8.22454	0.408681
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.18649	0.108115	0.0540574	−	0.998538i	$-0.482785\pi$
$409$	2.18649	0.108115	0.0540574	+	0.998538i	$0.482785\pi$
$410$	0	0
$411$	−6.07314	−0.299566
$412$	0	0
$413$	49.4071	2.43117
$414$	0	0
$415$	4.09529	0.201030
$416$	0	0
$417$	−4.41977	−0.216437
$418$	0	0
$419$	10.7348	0.524429	0.262215	−	0.965010i	$-0.415547\pi$
$419$	10.7348	0.524429	0.262215	+	0.965010i	$0.415547\pi$
$420$	0	0
$421$	18.7885	0.915696	0.457848	−	0.889031i	$-0.348620\pi$
$421$	18.7885	0.915696	0.457848	+	0.889031i	$0.348620\pi$
$422$	0	0
$423$	2.26101	0.109934
$424$	0	0
$425$	−1.66785	−0.0809025
$426$	0	0
$427$	−57.9066	−2.80230
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.97878	−0.432493	−0.216246	−	0.976339i	$-0.569381\pi$
$431$	−8.97878	−0.432493	−0.216246	+	0.976339i	$0.569381\pi$
$432$	0	0
$433$	31.5059	1.51408	0.757039	−	0.653370i	$-0.226645\pi$
$433$	31.5059	1.51408	0.757039	+	0.653370i	$0.226645\pi$
$434$	0	0
$435$	−1.39783	−0.0670206
$436$	0	0
$437$	−6.96037	−0.332959
$438$	0	0
$439$	27.4750	1.31131	0.655655	−	0.755060i	$-0.272393\pi$
$439$	27.4750	1.31131	0.655655	+	0.755060i	$0.272393\pi$
$440$	0	0
$441$	−35.7552	−1.70263
$442$	0	0
$443$	−30.1960	−1.43466	−0.717328	−	0.696736i	$-0.754635\pi$
$443$	−30.1960	−1.43466	−0.717328	+	0.696736i	$0.754635\pi$
$444$	0	0
$445$	−0.466291	−0.0221043
$446$	0	0
$447$	3.23355	0.152942
$448$	0	0
$449$	−27.3820	−1.29224	−0.646118	−	0.763237i	$-0.723609\pi$
$449$	−27.3820	−1.29224	−0.646118	+	0.763237i	$0.723609\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−1.74374	−0.0819279
$454$	0	0
$455$	−26.1783	−1.22726
$456$	0	0
$457$	14.2884	0.668385	0.334193	−	0.942505i	$-0.391536\pi$
$457$	14.2884	0.668385	0.334193	+	0.942505i	$0.391536\pi$
$458$	0	0
$459$	2.90892	0.135777
$460$	0	0
$461$	8.78738	0.409269	0.204635	−	0.978838i	$-0.434399\pi$
$461$	8.78738	0.409269	0.204635	+	0.978838i	$0.434399\pi$
$462$	0	0
$463$	−13.8394	−0.643172	−0.321586	−	0.946880i	$-0.604216\pi$
$463$	−13.8394	−0.643172	−0.321586	+	0.946880i	$0.604216\pi$
$464$	0	0
$465$	−1.51200	−0.0701175
$466$	0	0
$467$	24.8893	1.15174	0.575871	−	0.817541i	$-0.304663\pi$
$467$	24.8893	1.15174	0.575871	+	0.817541i	$0.304663\pi$
$468$	0	0
$469$	34.2420	1.58115
$470$	0	0
$471$	−1.30559	−0.0601583
$472$	0	0
$473$	0	0
$474$	0	0
$475$	7.69351	0.353002
$476$	0	0
$477$	27.9518	1.27982
$478$	0	0
$479$	−28.5906	−1.30634	−0.653169	−	0.757212i	$-0.726561\pi$
$479$	−28.5906	−1.30634	−0.653169	+	0.757212i	$0.726561\pi$
$480$	0	0
$481$	−1.10297	−0.0502913
$482$	0	0
$483$	1.17156	0.0533079
$484$	0	0
$485$	7.09963	0.322377
$486$	0	0
$487$	4.65371	0.210880	0.105440	−	0.994426i	$-0.466375\pi$
$487$	4.65371	0.210880	0.105440	+	0.994426i	$0.466375\pi$
$488$	0	0
$489$	−2.17820	−0.0985014
$490$	0	0
$491$	24.9360	1.12535	0.562673	−	0.826680i	$-0.309773\pi$
$491$	24.9360	1.12535	0.562673	+	0.826680i	$0.309773\pi$
$492$	0	0
$493$	7.90392	0.355974
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.6033	1.37274
$498$	0	0
$499$	15.3848	0.688719	0.344359	−	0.938838i	$-0.388096\pi$
$499$	15.3848	0.688719	0.344359	+	0.938838i	$0.388096\pi$
$500$	0	0
$501$	−5.74810	−0.256806
$502$	0	0
$503$	−32.7511	−1.46030	−0.730149	−	0.683288i	$-0.760549\pi$
$503$	−32.7511	−1.46030	−0.730149	+	0.683288i	$0.760549\pi$
$504$	0	0
$505$	−16.2851	−0.724677
$506$	0	0
$507$	6.65292	0.295467
$508$	0	0
$509$	15.7017	0.695964	0.347982	−	0.937501i	$-0.386867\pi$
$509$	15.7017	0.695964	0.347982	+	0.937501i	$0.386867\pi$
$510$	0	0
$511$	−55.9936	−2.47701
$512$	0	0
$513$	−13.4184	−0.592435
$514$	0	0
$515$	3.19059	0.140594
$516$	0	0
$517$	0	0
$518$	0	0
$519$	2.80452	0.123105
$520$	0	0
$521$	−33.6243	−1.47311	−0.736553	−	0.676380i	$-0.763548\pi$
$521$	−33.6243	−1.47311	−0.736553	+	0.676380i	$0.763548\pi$
$522$	0	0
$523$	−22.9574	−1.00386	−0.501928	−	0.864910i	$-0.667376\pi$
$523$	−22.9574	−1.00386	−0.501928	+	0.864910i	$0.667376\pi$
$524$	0	0
$525$	−1.29496	−0.0565168
$526$	0	0
$527$	8.54952	0.372423
$528$	0	0
$529$	−22.1815	−0.964413
$530$	0	0
$531$	32.7823	1.42263
$532$	0	0
$533$	−15.6367	−0.677300
$534$	0	0
$535$	−13.1644	−0.569145
$536$	0	0
$537$	2.12743	0.0918053
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.7777	−0.807314	−0.403657	−	0.914910i	$-0.632261\pi$
$541$	−18.7777	−0.807314	−0.403657	+	0.914910i	$0.632261\pi$
$542$	0	0
$543$	−6.89902	−0.296065
$544$	0	0
$545$	1.33532	0.0571988
$546$	0	0
$547$	19.2785	0.824291	0.412146	−	0.911118i	$-0.364780\pi$
$547$	19.2785	0.824291	0.412146	+	0.911118i	$0.364780\pi$
$548$	0	0
$549$	−38.4218	−1.63980
$550$	0	0
$551$	−36.4595	−1.55323
$552$	0	0
$553$	44.0399	1.87277
$554$	0	0
$555$	−0.0545610	−0.00231598
$556$	0	0
$557$	−17.9380	−0.760057	−0.380028	−	0.924975i	$-0.624086\pi$
$557$	−17.9380	−0.760057	−0.380028	+	0.924975i	$0.624086\pi$
$558$	0	0
$559$	21.4555	0.907470
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.81288	−0.0764038	−0.0382019	−	0.999270i	$-0.512163\pi$
$563$	−1.81288	−0.0764038	−0.0382019	+	0.999270i	$0.512163\pi$
$564$	0	0
$565$	−1.67448	−0.0704460
$566$	0	0
$567$	−36.1079	−1.51639
$568$	0	0
$569$	8.53549	0.357826	0.178913	−	0.983865i	$-0.442742\pi$
$569$	8.53549	0.357826	0.178913	+	0.983865i	$0.442742\pi$
$570$	0	0
$571$	−32.2491	−1.34958	−0.674792	−	0.738008i	$-0.735767\pi$
$571$	−32.2491	−1.34958	−0.674792	+	0.738008i	$0.735767\pi$
$572$	0	0
$573$	−1.22873	−0.0513310
$574$	0	0
$575$	−0.904706	−0.0377288
$576$	0	0
$577$	29.9652	1.24747	0.623734	−	0.781637i	$-0.285615\pi$
$577$	29.9652	1.24747	0.623734	+	0.781637i	$0.285615\pi$
$578$	0	0
$579$	−6.83312	−0.283975
$580$	0	0
$581$	−17.9794	−0.745911
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−17.3696	−0.718147
$586$	0	0
$587$	9.80339	0.404629	0.202315	−	0.979321i	$-0.435154\pi$
$587$	9.80339	0.404629	0.202315	+	0.979321i	$0.435154\pi$
$588$	0	0
$589$	−39.4376	−1.62500
$590$	0	0
$591$	2.99409	0.123160
$592$	0	0
$593$	15.4666	0.635136	0.317568	−	0.948235i	$-0.397134\pi$
$593$	15.4666	0.635136	0.317568	+	0.948235i	$0.397134\pi$
$594$	0	0
$595$	7.32228	0.300184
$596$	0	0
$597$	−5.01824	−0.205383
$598$	0	0
$599$	−0.278191	−0.0113666	−0.00568328	−	0.999984i	$-0.501809\pi$
$599$	−0.278191	−0.0113666	−0.00568328	+	0.999984i	$0.501809\pi$
$600$	0	0
$601$	30.0055	1.22395	0.611974	−	0.790878i	$-0.290376\pi$
$601$	30.0055	1.22395	0.611974	+	0.790878i	$0.290376\pi$
$602$	0	0
$603$	22.7200	0.925232
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−22.2971	−0.905011	−0.452505	−	0.891762i	$-0.649470\pi$
$607$	−22.2971	−0.905011	−0.452505	+	0.891762i	$0.649470\pi$
$608$	0	0
$609$	6.13682	0.248676
$610$	0	0
$611$	−4.62821	−0.187237
$612$	0	0
$613$	−28.3748	−1.14605	−0.573024	−	0.819539i	$-0.694230\pi$
$613$	−28.3748	−1.14605	−0.573024	+	0.819539i	$0.694230\pi$
$614$	0	0
$615$	−0.773501	−0.0311906
$616$	0	0
$617$	−15.1603	−0.610332	−0.305166	−	0.952299i	$-0.598712\pi$
$617$	−15.1603	−0.610332	−0.305166	+	0.952299i	$0.598712\pi$
$618$	0	0
$619$	−29.5238	−1.18666	−0.593331	−	0.804959i	$-0.702187\pi$
$619$	−29.5238	−1.18666	−0.593331	+	0.804959i	$0.702187\pi$
$620$	0	0
$621$	1.57791	0.0633194
$622$	0	0
$623$	2.04714	0.0820167
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.308511	0.0123011
$630$	0	0
$631$	−45.6894	−1.81887	−0.909433	−	0.415849i	$-0.863484\pi$
$631$	−45.6894	−1.81887	−0.909433	+	0.415849i	$0.863484\pi$
$632$	0	0
$633$	−1.77456	−0.0705323
$634$	0	0
$635$	0.901470	0.0357738
$636$	0	0
$637$	73.1897	2.89988
$638$	0	0
$639$	20.3057	0.803281
$640$	0	0
$641$	36.1151	1.42646	0.713231	−	0.700929i	$-0.247231\pi$
$641$	36.1151	1.42646	0.713231	+	0.700929i	$0.247231\pi$
$642$	0	0
$643$	−33.4972	−1.32100	−0.660500	−	0.750826i	$-0.729656\pi$
$643$	−33.4972	−1.32100	−0.660500	+	0.750826i	$0.729656\pi$
$644$	0	0
$645$	1.06134	0.0417902
$646$	0	0
$647$	39.2066	1.54137	0.770686	−	0.637215i	$-0.219914\pi$
$647$	39.2066	1.54137	0.770686	+	0.637215i	$0.219914\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.63808	0.260167
$652$	0	0
$653$	24.7595	0.968913	0.484456	−	0.874815i	$-0.339017\pi$
$653$	24.7595	0.968913	0.484456	+	0.874815i	$0.339017\pi$
$654$	0	0
$655$	−14.5527	−0.568622
$656$	0	0
$657$	−37.1525	−1.44946
$658$	0	0
$659$	24.8813	0.969238	0.484619	−	0.874725i	$-0.338958\pi$
$659$	24.8813	0.969238	0.484619	+	0.874725i	$0.338958\pi$
$660$	0	0
$661$	−5.94396	−0.231193	−0.115597	−	0.993296i	$-0.536878\pi$
$661$	−5.94396	−0.231193	−0.115597	+	0.993296i	$0.536878\pi$
$662$	0	0
$663$	−2.93342	−0.113925
$664$	0	0
$665$	−33.7765	−1.30980
$666$	0	0
$667$	4.28739	0.166009
$668$	0	0
$669$	0.735660	0.0284422
$670$	0	0
$671$	0	0
$672$	0	0
$673$	37.8500	1.45901	0.729505	−	0.683975i	$-0.239750\pi$
$673$	37.8500	1.45901	0.729505	+	0.683975i	$0.239750\pi$
$674$	0	0
$675$	−1.74411	−0.0671310
$676$	0	0
$677$	0.190100	0.00730612	0.00365306	−	0.999993i	$-0.498837\pi$
$677$	0.190100	0.00730612	0.00365306	+	0.999993i	$0.498837\pi$
$678$	0	0
$679$	−31.1692	−1.19616
$680$	0	0
$681$	−7.37267	−0.282521
$682$	0	0
$683$	23.0773	0.883030	0.441515	−	0.897254i	$-0.354441\pi$
$683$	23.0773	0.883030	0.441515	+	0.897254i	$0.354441\pi$
$684$	0	0
$685$	−20.5895	−0.786685
$686$	0	0
$687$	−0.817290	−0.0311816
$688$	0	0
$689$	−57.2164	−2.17977
$690$	0	0
$691$	46.7082	1.77686	0.888432	−	0.459008i	$-0.151795\pi$
$691$	46.7082	1.77686	0.888432	+	0.459008i	$0.151795\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.9841	−0.568381
$696$	0	0
$697$	4.37371	0.165666
$698$	0	0
$699$	−2.83172	−0.107105
$700$	0	0
$701$	4.17239	0.157589	0.0787946	−	0.996891i	$-0.474893\pi$
$701$	4.17239	0.157589	0.0787946	+	0.996891i	$0.474893\pi$
$702$	0	0
$703$	−1.42311	−0.0536737
$704$	0	0
$705$	−0.228944	−0.00862254
$706$	0	0
$707$	71.4957	2.68887
$708$	0	0
$709$	−8.74277	−0.328342	−0.164171	−	0.986432i	$-0.552495\pi$
$709$	−8.74277	−0.328342	−0.164171	+	0.986432i	$0.552495\pi$
$710$	0	0
$711$	29.2211	1.09588
$712$	0	0
$713$	4.63760	0.173679
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.35637	0.162692
$718$	0	0
$719$	−7.40160	−0.276033	−0.138017	−	0.990430i	$-0.544073\pi$
$719$	−7.40160	−0.276033	−0.138017	+	0.990430i	$0.544073\pi$
$720$	0	0
$721$	−14.0075	−0.521667
$722$	0	0
$723$	−3.07758	−0.114456
$724$	0	0
$725$	−4.73899	−0.176002
$726$	0	0
$727$	18.0187	0.668275	0.334137	−	0.942524i	$-0.391555\pi$
$727$	18.0187	0.668275	0.334137	+	0.942524i	$0.391555\pi$
$728$	0	0
$729$	−22.4147	−0.830175
$730$	0	0
$731$	−6.00128	−0.221965
$732$	0	0
$733$	−36.3939	−1.34424	−0.672120	−	0.740442i	$-0.734616\pi$
$733$	−36.3939	−1.34424	−0.672120	+	0.740442i	$0.734616\pi$
$734$	0	0
$735$	3.62048	0.133543
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−12.4319	−0.457313	−0.228657	−	0.973507i	$-0.573433\pi$
$739$	−12.4319	−0.457313	−0.228657	+	0.973507i	$0.573433\pi$
$740$	0	0
$741$	13.5314	0.497089
$742$	0	0
$743$	37.3696	1.37096	0.685479	−	0.728092i	$-0.259593\pi$
$743$	37.3696	1.37096	0.685479	+	0.728092i	$0.259593\pi$
$744$	0	0
$745$	10.9626	0.401638
$746$	0	0
$747$	−11.9296	−0.436480
$748$	0	0
$749$	57.7950	2.11178
$750$	0	0
$751$	−7.89958	−0.288260	−0.144130	−	0.989559i	$-0.546038\pi$
$751$	−7.89958	−0.288260	−0.144130	+	0.989559i	$0.546038\pi$
$752$	0	0
$753$	−1.98626	−0.0723834
$754$	0	0
$755$	−5.91172	−0.215149
$756$	0	0
$757$	−3.17036	−0.115229	−0.0576143	−	0.998339i	$-0.518349\pi$
$757$	−3.17036	−0.115229	−0.0576143	+	0.998339i	$0.518349\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	46.7689	1.69537	0.847685	−	0.530499i	$-0.177996\pi$
$761$	46.7689	1.69537	0.847685	+	0.530499i	$0.177996\pi$
$762$	0	0
$763$	−5.86239	−0.212233
$764$	0	0
$765$	4.85844	0.175657
$766$	0	0
$767$	−67.1044	−2.42300
$768$	0	0
$769$	−1.95610	−0.0705388	−0.0352694	−	0.999378i	$-0.511229\pi$
$769$	−1.95610	−0.0705388	−0.0352694	+	0.999378i	$0.511229\pi$
$770$	0	0
$771$	2.28358	0.0822411
$772$	0	0
$773$	−20.4428	−0.735277	−0.367639	−	0.929969i	$-0.619834\pi$
$773$	−20.4428	−0.735277	−0.367639	+	0.929969i	$0.619834\pi$
$774$	0	0
$775$	−5.12608	−0.184134
$776$	0	0
$777$	0.239537	0.00859333
$778$	0	0
$779$	−20.1752	−0.722852
$780$	0	0
$781$	0	0
$782$	0	0
$783$	8.26534	0.295379
$784$	0	0
$785$	−4.42628	−0.157981
$786$	0	0
$787$	−2.59196	−0.0923933	−0.0461967	−	0.998932i	$-0.514710\pi$
$787$	−2.59196	−0.0923933	−0.0461967	+	0.998932i	$0.514710\pi$
$788$	0	0
$789$	5.35706	0.190716
$790$	0	0
$791$	7.35141	0.261386
$792$	0	0
$793$	78.6483	2.79288
$794$	0	0
$795$	−2.83033	−0.100381
$796$	0	0
$797$	22.3867	0.792977	0.396488	−	0.918040i	$-0.370229\pi$
$797$	22.3867	0.792977	0.396488	+	0.918040i	$0.370229\pi$
$798$	0	0
$799$	1.29455	0.0457979
$800$	0	0
$801$	1.35830	0.0479933
$802$	0	0
$803$	0	0
$804$	0	0
$805$	3.97189	0.139991
$806$	0	0
$807$	−3.16399	−0.111378
$808$	0	0
$809$	11.9561	0.420354	0.210177	−	0.977663i	$-0.432596\pi$
$809$	11.9561	0.420354	0.210177	+	0.977663i	$0.432596\pi$
$810$	0	0
$811$	18.4696	0.648554	0.324277	−	0.945962i	$-0.394879\pi$
$811$	18.4696	0.648554	0.324277	+	0.945962i	$0.394879\pi$
$812$	0	0
$813$	−5.54162	−0.194353
$814$	0	0
$815$	−7.38464	−0.258673
$816$	0	0
$817$	27.6829	0.968503
$818$	0	0
$819$	76.2572	2.66464
$820$	0	0
$821$	55.4734	1.93604	0.968018	−	0.250881i	$-0.0807201\pi$
$821$	55.4734	1.93604	0.968018	+	0.250881i	$0.0807201\pi$
$822$	0	0
$823$	43.7133	1.52375	0.761874	−	0.647725i	$-0.224279\pi$
$823$	43.7133	1.52375	0.761874	+	0.647725i	$0.224279\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.7396	1.10369	0.551846	−	0.833946i	$-0.313924\pi$
$827$	31.7396	1.10369	0.551846	+	0.833946i	$0.313924\pi$
$828$	0	0
$829$	−35.0302	−1.21665	−0.608325	−	0.793688i	$-0.708158\pi$
$829$	−35.0302	−1.21665	−0.608325	+	0.793688i	$0.708158\pi$
$830$	0	0
$831$	8.50870	0.295164
$832$	0	0
$833$	−20.4718	−0.709304
$834$	0	0
$835$	−19.4876	−0.674394
$836$	0	0
$837$	8.94047	0.309028
$838$	0	0
$839$	−15.8542	−0.547349	−0.273674	−	0.961822i	$-0.588239\pi$
$839$	−15.8542	−0.547349	−0.273674	+	0.961822i	$0.588239\pi$
$840$	0	0
$841$	−6.54197	−0.225585
$842$	0	0
$843$	7.97998	0.274845
$844$	0	0
$845$	22.5551	0.775919
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.79710	0.130316
$850$	0	0
$851$	0.167349	0.00573664
$852$	0	0
$853$	−22.5865	−0.773346	−0.386673	−	0.922217i	$-0.626376\pi$
$853$	−22.5865	−0.773346	−0.386673	+	0.922217i	$0.626376\pi$
$854$	0	0
$855$	−22.4112	−0.766446
$856$	0	0
$857$	40.9108	1.39749	0.698744	−	0.715372i	$-0.253743\pi$
$857$	40.9108	1.39749	0.698744	+	0.715372i	$0.253743\pi$
$858$	0	0
$859$	15.9523	0.544284	0.272142	−	0.962257i	$-0.412268\pi$
$859$	15.9523	0.544284	0.272142	+	0.962257i	$0.412268\pi$
$860$	0	0
$861$	3.39587	0.115731
$862$	0	0
$863$	7.28894	0.248118	0.124059	−	0.992275i	$-0.460409\pi$
$863$	7.28894	0.248118	0.124059	+	0.992275i	$0.460409\pi$
$864$	0	0
$865$	9.50805	0.323283
$866$	0	0
$867$	−4.19387	−0.142431
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−46.5072	−1.57584
$872$	0	0
$873$	−20.6812	−0.699952
$874$	0	0
$875$	−4.39026	−0.148418
$876$	0	0
$877$	7.14718	0.241343	0.120671	−	0.992692i	$-0.461495\pi$
$877$	7.14718	0.241343	0.120671	+	0.992692i	$0.461495\pi$
$878$	0	0
$879$	1.73696	0.0585861
$880$	0	0
$881$	−17.5763	−0.592160	−0.296080	−	0.955163i	$-0.595680\pi$
$881$	−17.5763	−0.592160	−0.296080	+	0.955163i	$0.595680\pi$
$882$	0	0
$883$	−6.26855	−0.210954	−0.105477	−	0.994422i	$-0.533637\pi$
$883$	−6.26855	−0.210954	−0.105477	+	0.994422i	$0.533637\pi$
$884$	0	0
$885$	−3.31946	−0.111582
$886$	0	0
$887$	−24.6894	−0.828989	−0.414495	−	0.910052i	$-0.636042\pi$
$887$	−24.6894	−0.828989	−0.414495	+	0.910052i	$0.636042\pi$
$888$	0	0
$889$	−3.95769	−0.132737
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.97155	−0.199830
$894$	0	0
$895$	7.21253	0.241088
$896$	0	0
$897$	−1.59120	−0.0531288
$898$	0	0
$899$	24.2924	0.810199
$900$	0	0
$901$	16.0039	0.533167
$902$	0	0
$903$	−4.65956	−0.155060
$904$	0	0
$905$	−23.3895	−0.777492
$906$	0	0
$907$	25.4380	0.844655	0.422327	−	0.906443i	$-0.361213\pi$
$907$	25.4380	0.844655	0.422327	+	0.906443i	$0.361213\pi$
$908$	0	0
$909$	47.4384	1.57343
$910$	0	0
$911$	−24.5296	−0.812701	−0.406350	−	0.913717i	$-0.633199\pi$
$911$	−24.5296	−0.812701	−0.406350	+	0.913717i	$0.633199\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.89050	0.128616
$916$	0	0
$917$	63.8903	2.10984
$918$	0	0
$919$	21.2250	0.700148	0.350074	−	0.936722i	$-0.386156\pi$
$919$	21.2250	0.700148	0.350074	+	0.936722i	$0.386156\pi$
$920$	0	0
$921$	2.02298	0.0666596
$922$	0	0
$923$	−41.5651	−1.36813
$924$	0	0
$925$	−0.184976	−0.00608196
$926$	0	0
$927$	−9.29417	−0.305261
$928$	0	0
$929$	34.2535	1.12382	0.561910	−	0.827198i	$-0.310067\pi$
$929$	34.2535	1.12382	0.561910	+	0.827198i	$0.310067\pi$
$930$	0	0
$931$	94.4329	3.09491
$932$	0	0
$933$	−4.20891	−0.137794
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34.9322	1.14119	0.570593	−	0.821233i	$-0.306713\pi$
$937$	34.9322	1.14119	0.570593	+	0.821233i	$0.306713\pi$
$938$	0	0
$939$	0.471860	0.0153986
$940$	0	0
$941$	−18.7014	−0.609649	−0.304825	−	0.952409i	$-0.598598\pi$
$941$	−18.7014	−0.609649	−0.304825	+	0.952409i	$0.598598\pi$
$942$	0	0
$943$	2.37247	0.0772583
$944$	0	0
$945$	7.65711	0.249086
$946$	0	0
$947$	12.7418	0.414052	0.207026	−	0.978335i	$-0.433621\pi$
$947$	12.7418	0.414052	0.207026	+	0.978335i	$0.433621\pi$
$948$	0	0
$949$	76.0501	2.46869
$950$	0	0
$951$	−9.10125	−0.295128
$952$	0	0
$953$	5.77686	0.187131	0.0935654	−	0.995613i	$-0.470174\pi$
$953$	5.77686	0.187131	0.0935654	+	0.995613i	$0.470174\pi$
$954$	0	0
$955$	−4.16572	−0.134799
$956$	0	0
$957$	0	0
$958$	0	0
$959$	90.3933	2.91895
$960$	0	0
$961$	−4.72330	−0.152364
$962$	0	0
$963$	38.3478	1.23574
$964$	0	0
$965$	−23.1660	−0.745741
$966$	0	0
$967$	−36.2900	−1.16701	−0.583504	−	0.812110i	$-0.698319\pi$
$967$	−36.2900	−1.16701	−0.583504	+	0.812110i	$0.698319\pi$
$968$	0	0
$969$	−3.78485	−0.121587
$970$	0	0
$971$	−6.98883	−0.224282	−0.112141	−	0.993692i	$-0.535771\pi$
$971$	−6.98883	−0.224282	−0.112141	+	0.993692i	$0.535771\pi$
$972$	0	0
$973$	65.7842	2.10895
$974$	0	0
$975$	1.75881	0.0563269
$976$	0	0
$977$	25.9994	0.831795	0.415897	−	0.909412i	$-0.363468\pi$
$977$	25.9994	0.831795	0.415897	+	0.909412i	$0.363468\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−3.88978	−0.124191
$982$	0	0
$983$	28.6461	0.913669	0.456834	−	0.889552i	$-0.348983\pi$
$983$	28.6461	0.913669	0.456834	+	0.889552i	$0.348983\pi$
$984$	0	0
$985$	10.1507	0.323429
$986$	0	0
$987$	1.00512	0.0319935
$988$	0	0
$989$	−3.25533	−0.103513
$990$	0	0
$991$	27.9673	0.888409	0.444205	−	0.895925i	$-0.353486\pi$
$991$	27.9673	0.888409	0.444205	+	0.895925i	$0.353486\pi$
$992$	0	0
$993$	−0.344564	−0.0109344
$994$	0	0
$995$	−17.0131	−0.539352
$996$	0	0
$997$	13.0579	0.413547	0.206773	−	0.978389i	$-0.433704\pi$
$997$	13.0579	0.413547	0.206773	+	0.978389i	$0.433704\pi$
$998$	0	0
$999$	0.322619	0.0102072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.y.1.3		4
4.3	odd	2		9680.2.a.cu.1.2		4
11.5	even	5		440.2.y.a.201.2	yes	8
11.9	even	5		440.2.y.a.81.2	✓	8
11.10	odd	2		4840.2.a.z.1.3		4
44.27	odd	10		880.2.bo.d.641.1		8
44.31	odd	10		880.2.bo.d.81.1		8
44.43	even	2		9680.2.a.ct.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.a.81.2	✓	8	11.9	even	5
440.2.y.a.201.2	yes	8	11.5	even	5
880.2.bo.d.81.1		8	44.31	odd	10
880.2.bo.d.641.1		8	44.27	odd	10
4840.2.a.y.1.3		4	1.1	even	1	trivial
4840.2.a.z.1.3		4	11.10	odd	2
9680.2.a.ct.1.2		4	44.43	even	2
9680.2.a.cu.1.2		4	4.3	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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q+0.294963 q^{3} +1.00000 q^{5} -4.39026 q^{7} -2.91300 q^{9} +5.96281 q^{13} +0.294963 q^{15} -1.66785 q^{17} +7.69351 q^{19} -1.29496 q^{21} -0.904706 q^{23} +1.00000 q^{25} -1.74411 q^{27} -4.73899 q^{29} -5.12608 q^{31} -4.39026 q^{35} -0.184976 q^{37} +1.75881 q^{39} -2.62237 q^{41} +3.59822 q^{43} -2.91300 q^{45} -0.776180 q^{47} +12.2744 q^{49} -0.491953 q^{51} -9.59554 q^{53} +2.26930 q^{57} -11.2538 q^{59} +13.1898 q^{61} +12.7888 q^{63} +5.96281 q^{65} -7.79954 q^{67} -0.266855 q^{69} -6.97072 q^{71} +12.7541 q^{73} +0.294963 q^{75} -10.0313 q^{79} +8.22454 q^{81} +4.09529 q^{83} -1.66785 q^{85} -1.39783 q^{87} -0.466291 q^{89} -26.1783 q^{91} -1.51200 q^{93} +7.69351 q^{95} +7.09963 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9} + 3 q^{13} - 2 q^{15} + 11 q^{17} + 2 q^{19} - 2 q^{21} - 11 q^{23} + 4 q^{25} + q^{27} + 4 q^{29} - 17 q^{31} - 7 q^{35} - 3 q^{37} + q^{39} + 13 q^{41} - 7 q^{43}+ \cdots + 2 q^{97}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 - 7 * q^7 - 4 * q^9 + 3 * q^13 - 2 * q^15 + 11 * q^17 + 2 * q^19 - 2 * q^21 - 11 * q^23 + 4 * q^25 + q^27 + 4 * q^29 - 17 * q^31 - 7 * q^35 - 3 * q^37 + q^39 + 13 * q^41 - 7 * q^43 - 4 * q^45 - q^47 - 5 * q^49 - q^51 - 15 * q^53 + 17 * q^57 - 17 * q^59 + 4 * q^61 + 15 * q^63 + 3 * q^65 + 7 * q^67 + 4 * q^69 - 15 * q^71 + 7 * q^73 - 2 * q^75 - 12 * q^79 - 8 * q^81 + 9 * q^83 + 11 * q^85 - 23 * q^87 - 12 * q^89 - 24 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more