LMFDB - Embedded newform 784.4.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -14.1421 q^{5} -25.0000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} -14.1421 q^{5} -25.0000 q^{9} -54.0000 q^{11} -22.6274 q^{13} +20.0000 q^{15} -32.5269 q^{17} -63.6396 q^{19} -28.0000 q^{23} +75.0000 q^{25} +73.5391 q^{27} +282.000 q^{29} -274.357 q^{31} +76.3675 q^{33} +146.000 q^{37} +32.0000 q^{39} -340.825 q^{41} -10.0000 q^{43} +353.553 q^{45} -506.288 q^{47} +46.0000 q^{51} +598.000 q^{53} +763.675 q^{55} +90.0000 q^{57} +575.585 q^{59} -466.690 q^{61} +320.000 q^{65} -916.000 q^{67} +39.5980 q^{69} -420.000 q^{71} +702.864 q^{73} -106.066 q^{75} +292.000 q^{79} +571.000 q^{81} +1152.58 q^{83} +460.000 q^{85} -398.808 q^{87} -445.477 q^{89} +388.000 q^{93} +900.000 q^{95} +589.727 q^{97} +1350.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 50 q^{9}+O(q^{10}) $ 2 * q - 50 * q^9	$ 2 q - 50 q^{9} - 108 q^{11} + 40 q^{15} - 56 q^{23} + 150 q^{25} + 564 q^{29} + 292 q^{37} + 64 q^{39} - 20 q^{43} + 92 q^{51} + 1196 q^{53} + 180 q^{57} + 640 q^{65} - 1832 q^{67} - 840 q^{71} + 584 q^{79} + 1142 q^{81} + 920 q^{85} + 776 q^{93} + 1800 q^{95} + 2700 q^{99}+O(q^{100}) $ 2 * q - 50 * q^9 - 108 * q^11 + 40 * q^15 - 56 * q^23 + 150 * q^25 + 564 * q^29 + 292 * q^37 + 64 * q^39 - 20 * q^43 + 92 * q^51 + 1196 * q^53 + 180 * q^57 + 640 * q^65 - 1832 * q^67 - 840 * q^71 + 584 * q^79 + 1142 * q^81 + 920 * q^85 + 776 * q^93 + 1800 * q^95 + 2700 * 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
$3$	−1.41421	−0.272166	−0.136083	−	0.990697i	$-0.543451\pi$
$3$	−1.41421	−0.272166	−0.136083	+	0.990697i	$0.543451\pi$
$4$	0	0
$5$	−14.1421	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−14.1421	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−25.0000	−0.925926
$10$	0	0
$11$	−54.0000	−1.48015	−0.740073	−	0.672526i	$-0.765209\pi$
$11$	−54.0000	−1.48015	−0.740073	+	0.672526i	$0.765209\pi$
$12$	0	0
$13$	−22.6274	−0.482747	−0.241374	−	0.970432i	$-0.577598\pi$
$13$	−22.6274	−0.482747	−0.241374	+	0.970432i	$0.577598\pi$
$14$	0	0
$15$	20.0000	0.344265
$16$	0	0
$17$	−32.5269	−0.464055	−0.232027	−	0.972709i	$-0.574536\pi$
$17$	−32.5269	−0.464055	−0.232027	+	0.972709i	$0.574536\pi$
$18$	0	0
$19$	−63.6396	−0.768417	−0.384209	−	0.923246i	$-0.625526\pi$
$19$	−63.6396	−0.768417	−0.384209	+	0.923246i	$0.625526\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−28.0000	−0.253844	−0.126922	−	0.991913i	$-0.540510\pi$
$23$	−28.0000	−0.253844	−0.126922	+	0.991913i	$0.540510\pi$
$24$	0	0
$25$	75.0000	0.600000
$26$	0	0
$27$	73.5391	0.524171
$28$	0	0
$29$	282.000	1.80573	0.902864	−	0.429927i	$-0.141461\pi$
$29$	282.000	1.80573	0.902864	+	0.429927i	$0.141461\pi$
$30$	0	0
$31$	−274.357	−1.58955	−0.794775	−	0.606904i	$-0.792411\pi$
$31$	−274.357	−1.58955	−0.794775	+	0.606904i	$0.792411\pi$
$32$	0	0
$33$	76.3675	0.402845
$34$	0	0
$35$	0	0
$36$	0	0
$37$	146.000	0.648710	0.324355	−	0.945936i	$-0.394853\pi$
$37$	146.000	0.648710	0.324355	+	0.945936i	$0.394853\pi$
$38$	0	0
$39$	32.0000	0.131387
$40$	0	0
$41$	−340.825	−1.29824	−0.649122	−	0.760684i	$-0.724863\pi$
$41$	−340.825	−1.29824	−0.649122	+	0.760684i	$0.724863\pi$
$42$	0	0
$43$	−10.0000	−0.0354648	−0.0177324	−	0.999843i	$-0.505645\pi$
$43$	−10.0000	−0.0354648	−0.0177324	+	0.999843i	$0.505645\pi$
$44$	0	0
$45$	353.553	1.17121
$46$	0	0
$47$	−506.288	−1.57127	−0.785636	−	0.618689i	$-0.787664\pi$
$47$	−506.288	−1.57127	−0.785636	+	0.618689i	$0.787664\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	46.0000	0.126300
$52$	0	0
$53$	598.000	1.54984	0.774921	−	0.632058i	$-0.217790\pi$
$53$	598.000	1.54984	0.774921	+	0.632058i	$0.217790\pi$
$54$	0	0
$55$	763.675	1.87225
$56$	0	0
$57$	90.0000	0.209137
$58$	0	0
$59$	575.585	1.27008	0.635040	−	0.772479i	$-0.280983\pi$
$59$	575.585	1.27008	0.635040	+	0.772479i	$0.280983\pi$
$60$	0	0
$61$	−466.690	−0.979567	−0.489784	−	0.871844i	$-0.662924\pi$
$61$	−466.690	−0.979567	−0.489784	+	0.871844i	$0.662924\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	320.000	0.610633
$66$	0	0
$67$	−916.000	−1.67026	−0.835128	−	0.550055i	$-0.814607\pi$
$67$	−916.000	−1.67026	−0.835128	+	0.550055i	$0.814607\pi$
$68$	0	0
$69$	39.5980	0.0690875
$70$	0	0
$71$	−420.000	−0.702040	−0.351020	−	0.936368i	$-0.614165\pi$
$71$	−420.000	−0.702040	−0.351020	+	0.936368i	$0.614165\pi$
$72$	0	0
$73$	702.864	1.12690	0.563452	−	0.826149i	$-0.309473\pi$
$73$	702.864	1.12690	0.563452	+	0.826149i	$0.309473\pi$
$74$	0	0
$75$	−106.066	−0.163299
$76$	0	0
$77$	0	0
$78$	0	0
$79$	292.000	0.415855	0.207928	−	0.978144i	$-0.433328\pi$
$79$	292.000	0.415855	0.207928	+	0.978144i	$0.433328\pi$
$80$	0	0
$81$	571.000	0.783265
$82$	0	0
$83$	1152.58	1.52425	0.762124	−	0.647431i	$-0.224157\pi$
$83$	1152.58	1.52425	0.762124	+	0.647431i	$0.224157\pi$
$84$	0	0
$85$	460.000	0.586988
$86$	0	0
$87$	−398.808	−0.491457
$88$	0	0
$89$	−445.477	−0.530567	−0.265284	−	0.964170i	$-0.585466\pi$
$89$	−445.477	−0.530567	−0.265284	+	0.964170i	$0.585466\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	388.000	0.432621
$94$	0	0
$95$	900.000	0.971979
$96$	0	0
$97$	589.727	0.617296	0.308648	−	0.951176i	$-0.400123\pi$
$97$	589.727	0.617296	0.308648	+	0.951176i	$0.400123\pi$
$98$	0	0
$99$	1350.00	1.37051
$100$	0	0
$101$	183.848	0.181124	0.0905621	−	0.995891i	$-0.471134\pi$
$101$	183.848	0.181124	0.0905621	+	0.995891i	$0.471134\pi$
$102$	0	0
$103$	−1886.56	−1.80474	−0.902371	−	0.430961i	$-0.858175\pi$
$103$	−1886.56	−1.80474	−0.902371	+	0.430961i	$0.858175\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1340.00	1.21068	0.605340	−	0.795967i	$-0.293037\pi$
$107$	1340.00	1.21068	0.605340	+	0.795967i	$0.293037\pi$
$108$	0	0
$109$	−538.000	−0.472762	−0.236381	−	0.971660i	$-0.575961\pi$
$109$	−538.000	−0.472762	−0.236381	+	0.971660i	$0.575961\pi$
$110$	0	0
$111$	−206.475	−0.176556
$112$	0	0
$113$	2068.00	1.72160	0.860801	−	0.508941i	$-0.169963\pi$
$113$	2068.00	1.72160	0.860801	+	0.508941i	$0.169963\pi$
$114$	0	0
$115$	395.980	0.321090
$116$	0	0
$117$	565.685	0.446988
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1585.00	1.19083
$122$	0	0
$123$	482.000	0.353337
$124$	0	0
$125$	707.107	0.505964
$126$	0	0
$127$	944.000	0.659578	0.329789	−	0.944055i	$-0.393022\pi$
$127$	944.000	0.659578	0.329789	+	0.944055i	$0.393022\pi$
$128$	0	0
$129$	14.1421	0.00965229
$130$	0	0
$131$	453.963	0.302770	0.151385	−	0.988475i	$-0.451627\pi$
$131$	453.963	0.302770	0.151385	+	0.988475i	$0.451627\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1040.00	−0.663029
$136$	0	0
$137$	−1300.00	−0.810704	−0.405352	−	0.914161i	$-0.632851\pi$
$137$	−1300.00	−0.810704	−0.405352	+	0.914161i	$0.632851\pi$
$138$	0	0
$139$	−2795.90	−1.70608	−0.853040	−	0.521845i	$-0.825244\pi$
$139$	−2795.90	−1.70608	−0.853040	+	0.521845i	$0.825244\pi$
$140$	0	0
$141$	716.000	0.427646
$142$	0	0
$143$	1221.88	0.714537
$144$	0	0
$145$	−3988.08	−2.28408
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2190.00	−1.20411	−0.602053	−	0.798456i	$-0.705650\pi$
$149$	−2190.00	−1.20411	−0.602053	+	0.798456i	$0.705650\pi$
$150$	0	0
$151$	−1184.00	−0.638096	−0.319048	−	0.947738i	$-0.603363\pi$
$151$	−1184.00	−0.638096	−0.319048	+	0.947738i	$0.603363\pi$
$152$	0	0
$153$	813.173	0.429681
$154$	0	0
$155$	3880.00	2.01064
$156$	0	0
$157$	−848.528	−0.431337	−0.215669	−	0.976467i	$-0.569193\pi$
$157$	−848.528	−0.431337	−0.215669	+	0.976467i	$0.569193\pi$
$158$	0	0
$159$	−845.700	−0.421814
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1822.00	0.875522	0.437761	−	0.899091i	$-0.355772\pi$
$163$	1822.00	0.875522	0.437761	+	0.899091i	$0.355772\pi$
$164$	0	0
$165$	−1080.00	−0.509563
$166$	0	0
$167$	1530.18	0.709035	0.354517	−	0.935049i	$-0.384645\pi$
$167$	1530.18	0.709035	0.354517	+	0.935049i	$0.384645\pi$
$168$	0	0
$169$	−1685.00	−0.766955
$170$	0	0
$171$	1590.99	0.711497
$172$	0	0
$173$	−1747.97	−0.768182	−0.384091	−	0.923295i	$-0.625485\pi$
$173$	−1747.97	−0.768182	−0.384091	+	0.923295i	$0.625485\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−814.000	−0.345672
$178$	0	0
$179$	3580.00	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	3580.00	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	2918.94	1.19869	0.599345	−	0.800491i	$-0.295428\pi$
$181$	2918.94	1.19869	0.599345	+	0.800491i	$0.295428\pi$
$182$	0	0
$183$	660.000	0.266604
$184$	0	0
$185$	−2064.75	−0.820560
$186$	0	0
$187$	1756.45	0.686869
$188$	0	0
$189$	0	0
$190$	0	0
$191$	484.000	0.183356	0.0916780	−	0.995789i	$-0.470777\pi$
$191$	484.000	0.183356	0.0916780	+	0.995789i	$0.470777\pi$
$192$	0	0
$193$	−800.000	−0.298369	−0.149185	−	0.988809i	$-0.547665\pi$
$193$	−800.000	−0.298369	−0.149185	+	0.988809i	$0.547665\pi$
$194$	0	0
$195$	−452.548	−0.166193
$196$	0	0
$197$	−1686.00	−0.609759	−0.304880	−	0.952391i	$-0.598616\pi$
$197$	−1686.00	−0.609759	−0.304880	+	0.952391i	$0.598616\pi$
$198$	0	0
$199$	110.309	0.0392943	0.0196472	−	0.999807i	$-0.493746\pi$
$199$	110.309	0.0392943	0.0196472	+	0.999807i	$0.493746\pi$
$200$	0	0
$201$	1295.42	0.454586
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4820.00	1.64216
$206$	0	0
$207$	700.000	0.235040
$208$	0	0
$209$	3436.54	1.13737
$210$	0	0
$211$	−3804.00	−1.24113	−0.620564	−	0.784156i	$-0.713096\pi$
$211$	−3804.00	−1.24113	−0.620564	+	0.784156i	$0.713096\pi$
$212$	0	0
$213$	593.970	0.191071
$214$	0	0
$215$	141.421	0.0448598
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−994.000	−0.306705
$220$	0	0
$221$	736.000	0.224021
$222$	0	0
$223$	−1335.02	−0.400894	−0.200447	−	0.979705i	$-0.564239\pi$
$223$	−1335.02	−0.400894	−0.200447	+	0.979705i	$0.564239\pi$
$224$	0	0
$225$	−1875.00	−0.555556
$226$	0	0
$227$	668.923	0.195586	0.0977929	−	0.995207i	$-0.468822\pi$
$227$	668.923	0.195586	0.0977929	+	0.995207i	$0.468822\pi$
$228$	0	0
$229$	−1001.26	−0.288932	−0.144466	−	0.989510i	$-0.546146\pi$
$229$	−1001.26	−0.288932	−0.144466	+	0.989510i	$0.546146\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	680.000	0.191194	0.0955972	−	0.995420i	$-0.469524\pi$
$233$	680.000	0.191194	0.0955972	+	0.995420i	$0.469524\pi$
$234$	0	0
$235$	7160.00	1.98752
$236$	0	0
$237$	−412.950	−0.113181
$238$	0	0
$239$	−1028.00	−0.278225	−0.139113	−	0.990277i	$-0.544425\pi$
$239$	−1028.00	−0.278225	−0.139113	+	0.990277i	$0.544425\pi$
$240$	0	0
$241$	2815.70	0.752594	0.376297	−	0.926499i	$-0.377197\pi$
$241$	2815.70	0.752594	0.376297	+	0.926499i	$0.377197\pi$
$242$	0	0
$243$	−2793.07	−0.737348
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1440.00	0.370951
$248$	0	0
$249$	−1630.00	−0.414848
$250$	0	0
$251$	−3904.64	−0.981908	−0.490954	−	0.871185i	$-0.663352\pi$
$251$	−3904.64	−0.981908	−0.490954	+	0.871185i	$0.663352\pi$
$252$	0	0
$253$	1512.00	0.375726
$254$	0	0
$255$	−650.538	−0.159758
$256$	0	0
$257$	6979.14	1.69396	0.846979	−	0.531627i	$-0.178419\pi$
$257$	6979.14	1.69396	0.846979	+	0.531627i	$0.178419\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−7050.00	−1.67197
$262$	0	0
$263$	−7580.00	−1.77720	−0.888599	−	0.458686i	$-0.848320\pi$
$263$	−7580.00	−1.77720	−0.888599	+	0.458686i	$0.848320\pi$
$264$	0	0
$265$	−8457.00	−1.96041
$266$	0	0
$267$	630.000	0.144402
$268$	0	0
$269$	893.783	0.202583	0.101292	−	0.994857i	$-0.467702\pi$
$269$	893.783	0.202583	0.101292	+	0.994857i	$0.467702\pi$
$270$	0	0
$271$	5543.72	1.24265	0.621323	−	0.783555i	$-0.286596\pi$
$271$	5543.72	1.24265	0.621323	+	0.783555i	$0.286596\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4050.00	−0.888088
$276$	0	0
$277$	3446.00	0.747473	0.373737	−	0.927535i	$-0.378076\pi$
$277$	3446.00	0.747473	0.373737	+	0.927535i	$0.378076\pi$
$278$	0	0
$279$	6858.94	1.47181
$280$	0	0
$281$	−5216.00	−1.10733	−0.553666	−	0.832739i	$-0.686772\pi$
$281$	−5216.00	−1.10733	−0.553666	+	0.832739i	$0.686772\pi$
$282$	0	0
$283$	733.977	0.154171	0.0770855	−	0.997024i	$-0.475439\pi$
$283$	733.977	0.154171	0.0770855	+	0.997024i	$0.475439\pi$
$284$	0	0
$285$	−1272.79	−0.264539
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3855.00	−0.784653
$290$	0	0
$291$	−834.000	−0.168007
$292$	0	0
$293$	−828.729	−0.165238	−0.0826192	−	0.996581i	$-0.526329\pi$
$293$	−828.729	−0.165238	−0.0826192	+	0.996581i	$0.526329\pi$
$294$	0	0
$295$	−8140.00	−1.60654
$296$	0	0
$297$	−3971.11	−0.775849
$298$	0	0
$299$	633.568	0.122542
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−260.000	−0.0492957
$304$	0	0
$305$	6600.00	1.23907
$306$	0	0
$307$	−8551.75	−1.58982	−0.794909	−	0.606729i	$-0.792481\pi$
$307$	−8551.75	−1.58982	−0.794909	+	0.606729i	$0.792481\pi$
$308$	0	0
$309$	2668.00	0.491188
$310$	0	0
$311$	−8541.85	−1.55744	−0.778720	−	0.627372i	$-0.784131\pi$
$311$	−8541.85	−1.55744	−0.778720	+	0.627372i	$0.784131\pi$
$312$	0	0
$313$	−2558.31	−0.461995	−0.230997	−	0.972954i	$-0.574199\pi$
$313$	−2558.31	−0.461995	−0.230997	+	0.972954i	$0.574199\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3170.00	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	3170.00	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	−15228.0	−2.67274
$320$	0	0
$321$	−1895.05	−0.329505
$322$	0	0
$323$	2070.00	0.356588
$324$	0	0
$325$	−1697.06	−0.289648
$326$	0	0
$327$	760.847	0.128670
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6610.00	−1.09764	−0.548819	−	0.835941i	$-0.684923\pi$
$331$	−6610.00	−1.09764	−0.548819	+	0.835941i	$0.684923\pi$
$332$	0	0
$333$	−3650.00	−0.600657
$334$	0	0
$335$	12954.2	2.11273
$336$	0	0
$337$	8866.00	1.43312	0.716561	−	0.697525i	$-0.245715\pi$
$337$	8866.00	1.43312	0.716561	+	0.697525i	$0.245715\pi$
$338$	0	0
$339$	−2924.59	−0.468561
$340$	0	0
$341$	14815.3	2.35277
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−560.000	−0.0873895
$346$	0	0
$347$	1046.00	0.161822	0.0809110	−	0.996721i	$-0.474217\pi$
$347$	1046.00	0.161822	0.0809110	+	0.996721i	$0.474217\pi$
$348$	0	0
$349$	−10182.3	−1.56174	−0.780871	−	0.624692i	$-0.785224\pi$
$349$	−10182.3	−1.56174	−0.780871	+	0.624692i	$0.785224\pi$
$350$	0	0
$351$	−1664.00	−0.253042
$352$	0	0
$353$	−3845.25	−0.579779	−0.289889	−	0.957060i	$-0.593618\pi$
$353$	−3845.25	−0.579779	−0.289889	+	0.957060i	$0.593618\pi$
$354$	0	0
$355$	5939.70	0.888018
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1904.00	0.279914	0.139957	−	0.990158i	$-0.455303\pi$
$359$	1904.00	0.279914	0.139957	+	0.990158i	$0.455303\pi$
$360$	0	0
$361$	−2809.00	−0.409535
$362$	0	0
$363$	−2241.53	−0.324104
$364$	0	0
$365$	−9940.00	−1.42543
$366$	0	0
$367$	−11579.6	−1.64700	−0.823500	−	0.567316i	$-0.807982\pi$
$367$	−11579.6	−1.64700	−0.823500	+	0.567316i	$0.807982\pi$
$368$	0	0
$369$	8520.64	1.20208
$370$	0	0
$371$	0	0
$372$	0	0
$373$	50.0000	0.00694076	0.00347038	−	0.999994i	$-0.498895\pi$
$373$	50.0000	0.00694076	0.00347038	+	0.999994i	$0.498895\pi$
$374$	0	0
$375$	−1000.00	−0.137706
$376$	0	0
$377$	−6380.93	−0.871710
$378$	0	0
$379$	−8830.00	−1.19675	−0.598373	−	0.801218i	$-0.704186\pi$
$379$	−8830.00	−1.19675	−0.598373	+	0.801218i	$0.704186\pi$
$380$	0	0
$381$	−1335.02	−0.179514
$382$	0	0
$383$	−3917.37	−0.522633	−0.261316	−	0.965253i	$-0.584157\pi$
$383$	−3917.37	−0.522633	−0.261316	+	0.965253i	$0.584157\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	250.000	0.0328378
$388$	0	0
$389$	−1278.00	−0.166574	−0.0832869	−	0.996526i	$-0.526542\pi$
$389$	−1278.00	−0.166574	−0.0832869	+	0.996526i	$0.526542\pi$
$390$	0	0
$391$	910.754	0.117797
$392$	0	0
$393$	−642.000	−0.0824036
$394$	0	0
$395$	−4129.50	−0.526020
$396$	0	0
$397$	1878.08	0.237425	0.118713	−	0.992929i	$-0.462123\pi$
$397$	1878.08	0.237425	0.118713	+	0.992929i	$0.462123\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−114.000	−0.0141967	−0.00709836	−	0.999975i	$-0.502259\pi$
$401$	−114.000	−0.0141967	−0.00709836	+	0.999975i	$0.502259\pi$
$402$	0	0
$403$	6208.00	0.767351
$404$	0	0
$405$	−8075.16	−0.990760
$406$	0	0
$407$	−7884.00	−0.960185
$408$	0	0
$409$	16355.4	1.97731	0.988657	−	0.150191i	$-0.0479887\pi$
$409$	16355.4	1.97731	0.988657	+	0.150191i	$0.0479887\pi$
$410$	0	0
$411$	1838.48	0.220646
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16300.0	−1.92804
$416$	0	0
$417$	3954.00	0.464336
$418$	0	0
$419$	−6684.99	−0.779434	−0.389717	−	0.920935i	$-0.627427\pi$
$419$	−6684.99	−0.779434	−0.389717	+	0.920935i	$0.627427\pi$
$420$	0	0
$421$	1014.00	0.117386	0.0586928	−	0.998276i	$-0.481307\pi$
$421$	1014.00	0.117386	0.0586928	+	0.998276i	$0.481307\pi$
$422$	0	0
$423$	12657.2	1.45488
$424$	0	0
$425$	−2439.52	−0.278433
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1728.00	−0.194472
$430$	0	0
$431$	−3036.00	−0.339302	−0.169651	−	0.985504i	$-0.554264\pi$
$431$	−3036.00	−0.339302	−0.169651	+	0.985504i	$0.554264\pi$
$432$	0	0
$433$	14884.6	1.65198	0.825991	−	0.563683i	$-0.190616\pi$
$433$	14884.6	1.65198	0.825991	+	0.563683i	$0.190616\pi$
$434$	0	0
$435$	5640.00	0.621649
$436$	0	0
$437$	1781.91	0.195058
$438$	0	0
$439$	6296.08	0.684500	0.342250	−	0.939609i	$-0.388811\pi$
$439$	6296.08	0.684500	0.342250	+	0.939609i	$0.388811\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10380.0	1.11325	0.556624	−	0.830765i	$-0.312096\pi$
$443$	10380.0	1.11325	0.556624	+	0.830765i	$0.312096\pi$
$444$	0	0
$445$	6300.00	0.671121
$446$	0	0
$447$	3097.13	0.327716
$448$	0	0
$449$	−4898.00	−0.514813	−0.257406	−	0.966303i	$-0.582868\pi$
$449$	−4898.00	−0.514813	−0.257406	+	0.966303i	$0.582868\pi$
$450$	0	0
$451$	18404.6	1.92159
$452$	0	0
$453$	1674.43	0.173668
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1464.00	0.149853	0.0749267	−	0.997189i	$-0.476128\pi$
$457$	1464.00	0.149853	0.0749267	+	0.997189i	$0.476128\pi$
$458$	0	0
$459$	−2392.00	−0.243244
$460$	0	0
$461$	−2687.01	−0.271467	−0.135734	−	0.990745i	$-0.543339\pi$
$461$	−2687.01	−0.271467	−0.135734	+	0.990745i	$0.543339\pi$
$462$	0	0
$463$	−13760.0	−1.38117	−0.690585	−	0.723252i	$-0.742647\pi$
$463$	−13760.0	−1.38117	−0.690585	+	0.723252i	$0.742647\pi$
$464$	0	0
$465$	−5487.15	−0.547227
$466$	0	0
$467$	14683.8	1.45500	0.727499	−	0.686109i	$-0.240682\pi$
$467$	14683.8	1.45500	0.727499	+	0.686109i	$0.240682\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1200.00	0.117395
$472$	0	0
$473$	540.000	0.0524931
$474$	0	0
$475$	−4772.97	−0.461050
$476$	0	0
$477$	−14950.0	−1.43504
$478$	0	0
$479$	−6875.91	−0.655883	−0.327942	−	0.944698i	$-0.606355\pi$
$479$	−6875.91	−0.655883	−0.327942	+	0.944698i	$0.606355\pi$
$480$	0	0
$481$	−3303.60	−0.313163
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8340.00	−0.780825
$486$	0	0
$487$	−3276.00	−0.304825	−0.152412	−	0.988317i	$-0.548704\pi$
$487$	−3276.00	−0.304825	−0.152412	+	0.988317i	$0.548704\pi$
$488$	0	0
$489$	−2576.70	−0.238287
$490$	0	0
$491$	15956.0	1.46657	0.733283	−	0.679923i	$-0.237987\pi$
$491$	15956.0	1.46657	0.733283	+	0.679923i	$0.237987\pi$
$492$	0	0
$493$	−9172.59	−0.837957
$494$	0	0
$495$	−19091.9	−1.73357
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10108.0	0.906806	0.453403	−	0.891306i	$-0.350210\pi$
$499$	10108.0	0.906806	0.453403	+	0.891306i	$0.350210\pi$
$500$	0	0
$501$	−2164.00	−0.192975
$502$	0	0
$503$	3908.89	0.346498	0.173249	−	0.984878i	$-0.444573\pi$
$503$	3908.89	0.346498	0.173249	+	0.984878i	$0.444573\pi$
$504$	0	0
$505$	−2600.00	−0.229106
$506$	0	0
$507$	2382.95	0.208739
$508$	0	0
$509$	−3552.50	−0.309356	−0.154678	−	0.987965i	$-0.549434\pi$
$509$	−3552.50	−0.309356	−0.154678	+	0.987965i	$0.549434\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4680.00	−0.402782
$514$	0	0
$515$	26680.0	2.28284
$516$	0	0
$517$	27339.6	2.32571
$518$	0	0
$519$	2472.00	0.209073
$520$	0	0
$521$	−15238.2	−1.28137	−0.640687	−	0.767802i	$-0.721350\pi$
$521$	−15238.2	−1.28137	−0.640687	+	0.767802i	$0.721350\pi$
$522$	0	0
$523$	17175.6	1.43602	0.718009	−	0.696034i	$-0.245054\pi$
$523$	17175.6	1.43602	0.718009	+	0.696034i	$0.245054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8924.00	0.737639
$528$	0	0
$529$	−11383.0	−0.935563
$530$	0	0
$531$	−14389.6	−1.17600
$532$	0	0
$533$	7712.00	0.626724
$534$	0	0
$535$	−18950.5	−1.53140
$536$	0	0
$537$	−5062.88	−0.406852
$538$	0	0
$539$	0	0
$540$	0	0
$541$	13666.0	1.08604	0.543020	−	0.839720i	$-0.317281\pi$
$541$	13666.0	1.08604	0.543020	+	0.839720i	$0.317281\pi$
$542$	0	0
$543$	−4128.00	−0.326242
$544$	0	0
$545$	7608.47	0.598002
$546$	0	0
$547$	5650.00	0.441639	0.220820	−	0.975315i	$-0.429127\pi$
$547$	5650.00	0.441639	0.220820	+	0.975315i	$0.429127\pi$
$548$	0	0
$549$	11667.3	0.907007
$550$	0	0
$551$	−17946.4	−1.38755
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2920.00	0.223328
$556$	0	0
$557$	−11430.0	−0.869488	−0.434744	−	0.900554i	$-0.643161\pi$
$557$	−11430.0	−0.869488	−0.434744	+	0.900554i	$0.643161\pi$
$558$	0	0
$559$	226.274	0.0171205
$560$	0	0
$561$	−2484.00	−0.186942
$562$	0	0
$563$	12851.0	0.961995	0.480998	−	0.876722i	$-0.340275\pi$
$563$	12851.0	0.961995	0.480998	+	0.876722i	$0.340275\pi$
$564$	0	0
$565$	−29245.9	−2.17767
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3770.00	0.277762	0.138881	−	0.990309i	$-0.455649\pi$
$569$	3770.00	0.277762	0.138881	+	0.990309i	$0.455649\pi$
$570$	0	0
$571$	3370.00	0.246988	0.123494	−	0.992345i	$-0.460590\pi$
$571$	3370.00	0.246988	0.123494	+	0.992345i	$0.460590\pi$
$572$	0	0
$573$	−684.479	−0.0499032
$574$	0	0
$575$	−2100.00	−0.152306
$576$	0	0
$577$	2125.56	0.153359	0.0766797	−	0.997056i	$-0.475568\pi$
$577$	2125.56	0.153359	0.0766797	+	0.997056i	$0.475568\pi$
$578$	0	0
$579$	1131.37	0.0812058
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−32292.0	−2.29399
$584$	0	0
$585$	−8000.00	−0.565400
$586$	0	0
$587$	−11094.5	−0.780101	−0.390050	−	0.920793i	$-0.627542\pi$
$587$	−11094.5	−0.780101	−0.390050	+	0.920793i	$0.627542\pi$
$588$	0	0
$589$	17460.0	1.22144
$590$	0	0
$591$	2384.36	0.165955
$592$	0	0
$593$	19489.3	1.34963	0.674813	−	0.737988i	$-0.264224\pi$
$593$	19489.3	1.34963	0.674813	+	0.737988i	$0.264224\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−156.000	−0.0106946
$598$	0	0
$599$	23952.0	1.63381	0.816905	−	0.576772i	$-0.195688\pi$
$599$	23952.0	1.63381	0.816905	+	0.576772i	$0.195688\pi$
$600$	0	0
$601$	11193.5	0.759721	0.379861	−	0.925044i	$-0.375972\pi$
$601$	11193.5	0.759721	0.379861	+	0.925044i	$0.375972\pi$
$602$	0	0
$603$	22900.0	1.54653
$604$	0	0
$605$	−22415.3	−1.50630
$606$	0	0
$607$	−12597.8	−0.842388	−0.421194	−	0.906971i	$-0.638389\pi$
$607$	−12597.8	−0.842388	−0.421194	+	0.906971i	$0.638389\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11456.0	0.758527
$612$	0	0
$613$	−10230.0	−0.674039	−0.337019	−	0.941498i	$-0.609419\pi$
$613$	−10230.0	−0.674039	−0.337019	+	0.941498i	$0.609419\pi$
$614$	0	0
$615$	−6816.51	−0.446940
$616$	0	0
$617$	9046.00	0.590240	0.295120	−	0.955460i	$-0.404640\pi$
$617$	9046.00	0.590240	0.295120	+	0.955460i	$0.404640\pi$
$618$	0	0
$619$	9668.98	0.627834	0.313917	−	0.949450i	$-0.398359\pi$
$619$	9668.98	0.627834	0.313917	+	0.949450i	$0.398359\pi$
$620$	0	0
$621$	−2059.09	−0.133057
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19375.0	−1.24000
$626$	0	0
$627$	−4860.00	−0.309553
$628$	0	0
$629$	−4748.93	−0.301037
$630$	0	0
$631$	20036.0	1.26406	0.632029	−	0.774945i	$-0.282222\pi$
$631$	20036.0	1.26406	0.632029	+	0.774945i	$0.282222\pi$
$632$	0	0
$633$	5379.67	0.337792
$634$	0	0
$635$	−13350.2	−0.834308
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10500.0	0.650037
$640$	0	0
$641$	9690.00	0.597086	0.298543	−	0.954396i	$-0.403499\pi$
$641$	9690.00	0.597086	0.298543	+	0.954396i	$0.403499\pi$
$642$	0	0
$643$	−4311.94	−0.264458	−0.132229	−	0.991219i	$-0.542213\pi$
$643$	−4311.94	−0.264458	−0.132229	+	0.991219i	$0.542213\pi$
$644$	0	0
$645$	−200.000	−0.0122093
$646$	0	0
$647$	12940.1	0.786284	0.393142	−	0.919478i	$-0.371388\pi$
$647$	12940.1	0.786284	0.393142	+	0.919478i	$0.371388\pi$
$648$	0	0
$649$	−31081.6	−1.87991
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21910.0	1.31302	0.656512	−	0.754316i	$-0.272031\pi$
$653$	21910.0	1.31302	0.656512	+	0.754316i	$0.272031\pi$
$654$	0	0
$655$	−6420.00	−0.382977
$656$	0	0
$657$	−17571.6	−1.04343
$658$	0	0
$659$	23322.0	1.37860	0.689299	−	0.724477i	$-0.257919\pi$
$659$	23322.0	1.37860	0.689299	+	0.724477i	$0.257919\pi$
$660$	0	0
$661$	30052.0	1.76836	0.884182	−	0.467142i	$-0.154716\pi$
$661$	30052.0	1.76836	0.884182	+	0.467142i	$0.154716\pi$
$662$	0	0
$663$	−1040.86	−0.0609709
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7896.00	−0.458372
$668$	0	0
$669$	1888.00	0.109110
$670$	0	0
$671$	25201.3	1.44990
$672$	0	0
$673$	6820.00	0.390627	0.195313	−	0.980741i	$-0.437428\pi$
$673$	6820.00	0.390627	0.195313	+	0.980741i	$0.437428\pi$
$674$	0	0
$675$	5515.43	0.314502
$676$	0	0
$677$	−16614.2	−0.943183	−0.471591	−	0.881817i	$-0.656320\pi$
$677$	−16614.2	−0.943183	−0.471591	+	0.881817i	$0.656320\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−946.000	−0.0532317
$682$	0	0
$683$	−11140.0	−0.624100	−0.312050	−	0.950066i	$-0.601016\pi$
$683$	−11140.0	−0.624100	−0.312050	+	0.950066i	$0.601016\pi$
$684$	0	0
$685$	18384.8	1.02547
$686$	0	0
$687$	1416.00	0.0786372
$688$	0	0
$689$	−13531.2	−0.748182
$690$	0	0
$691$	10274.3	0.565631	0.282816	−	0.959174i	$-0.408732\pi$
$691$	10274.3	0.565631	0.282816	+	0.959174i	$0.408732\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	39540.0	2.15804
$696$	0	0
$697$	11086.0	0.602457
$698$	0	0
$699$	−961.665	−0.0520365
$700$	0	0
$701$	12326.0	0.664118	0.332059	−	0.943259i	$-0.392257\pi$
$701$	12326.0	0.664118	0.332059	+	0.943259i	$0.392257\pi$
$702$	0	0
$703$	−9291.38	−0.498480
$704$	0	0
$705$	−10125.8	−0.540934
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−7982.00	−0.422807	−0.211404	−	0.977399i	$-0.567803\pi$
$709$	−7982.00	−0.422807	−0.211404	+	0.977399i	$0.567803\pi$
$710$	0	0
$711$	−7300.00	−0.385051
$712$	0	0
$713$	7682.01	0.403497
$714$	0	0
$715$	−17280.0	−0.903826
$716$	0	0
$717$	1453.81	0.0757233
$718$	0	0
$719$	15242.4	0.790606	0.395303	−	0.918551i	$-0.370640\pi$
$719$	15242.4	0.790606	0.395303	+	0.918551i	$0.370640\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3982.00	−0.204830
$724$	0	0
$725$	21150.0	1.08344
$726$	0	0
$727$	27076.5	1.38131	0.690655	−	0.723184i	$-0.257322\pi$
$727$	27076.5	1.38131	0.690655	+	0.723184i	$0.257322\pi$
$728$	0	0
$729$	−11467.0	−0.582584
$730$	0	0
$731$	325.269	0.0164576
$732$	0	0
$733$	17573.0	0.885504	0.442752	−	0.896644i	$-0.354002\pi$
$733$	17573.0	0.885504	0.442752	+	0.896644i	$0.354002\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	49464.0	2.47223
$738$	0	0
$739$	10758.0	0.535507	0.267753	−	0.963487i	$-0.413719\pi$
$739$	10758.0	0.535507	0.267753	+	0.963487i	$0.413719\pi$
$740$	0	0
$741$	−2036.47	−0.100960
$742$	0	0
$743$	−27648.0	−1.36515	−0.682575	−	0.730815i	$-0.739140\pi$
$743$	−27648.0	−1.36515	−0.682575	+	0.730815i	$0.739140\pi$
$744$	0	0
$745$	30971.3	1.52309
$746$	0	0
$747$	−28814.6	−1.41134
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4236.00	−0.205824	−0.102912	−	0.994690i	$-0.532816\pi$
$751$	−4236.00	−0.205824	−0.102912	+	0.994690i	$0.532816\pi$
$752$	0	0
$753$	5522.00	0.267242
$754$	0	0
$755$	16744.3	0.807135
$756$	0	0
$757$	7850.00	0.376900	0.188450	−	0.982083i	$-0.439654\pi$
$757$	7850.00	0.376900	0.188450	+	0.982083i	$0.439654\pi$
$758$	0	0
$759$	−2138.29	−0.102260
$760$	0	0
$761$	9476.65	0.451417	0.225708	−	0.974195i	$-0.427530\pi$
$761$	9476.65	0.451417	0.225708	+	0.974195i	$0.427530\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−11500.0	−0.543508
$766$	0	0
$767$	−13024.0	−0.613128
$768$	0	0
$769$	6045.76	0.283506	0.141753	−	0.989902i	$-0.454726\pi$
$769$	6045.76	0.283506	0.141753	+	0.989902i	$0.454726\pi$
$770$	0	0
$771$	−9870.00	−0.461037
$772$	0	0
$773$	−14292.0	−0.665005	−0.332503	−	0.943102i	$-0.607893\pi$
$773$	−14292.0	−0.665005	−0.332503	+	0.943102i	$0.607893\pi$
$774$	0	0
$775$	−20576.8	−0.953730
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21690.0	0.997593
$780$	0	0
$781$	22680.0	1.03912
$782$	0	0
$783$	20738.0	0.946509
$784$	0	0
$785$	12000.0	0.545603
$786$	0	0
$787$	16527.9	0.748611	0.374305	−	0.927306i	$-0.377881\pi$
$787$	16527.9	0.748611	0.374305	+	0.927306i	$0.377881\pi$
$788$	0	0
$789$	10719.7	0.483692
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10560.0	0.472883
$794$	0	0
$795$	11960.0	0.533557
$796$	0	0
$797$	−3099.96	−0.137774	−0.0688871	−	0.997624i	$-0.521945\pi$
$797$	−3099.96	−0.137774	−0.0688871	+	0.997624i	$0.521945\pi$
$798$	0	0
$799$	16468.0	0.729156
$800$	0	0
$801$	11136.9	0.491266
$802$	0	0
$803$	−37954.7	−1.66798
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1264.00	−0.0551362
$808$	0	0
$809$	−21488.0	−0.933842	−0.466921	−	0.884299i	$-0.654637\pi$
$809$	−21488.0	−0.933842	−0.466921	+	0.884299i	$0.654637\pi$
$810$	0	0
$811$	637.810	0.0276160	0.0138080	−	0.999905i	$-0.495605\pi$
$811$	637.810	0.0276160	0.0138080	+	0.999905i	$0.495605\pi$
$812$	0	0
$813$	−7840.00	−0.338205
$814$	0	0
$815$	−25767.0	−1.10746
$816$	0	0
$817$	636.396	0.0272518
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6534.00	0.277757	0.138878	−	0.990309i	$-0.455650\pi$
$821$	6534.00	0.277757	0.138878	+	0.990309i	$0.455650\pi$
$822$	0	0
$823$	−38632.0	−1.63624	−0.818121	−	0.575046i	$-0.804984\pi$
$823$	−38632.0	−1.63624	−0.818121	+	0.575046i	$0.804984\pi$
$824$	0	0
$825$	5727.56	0.241707
$826$	0	0
$827$	10564.0	0.444191	0.222096	−	0.975025i	$-0.428710\pi$
$827$	10564.0	0.444191	0.222096	+	0.975025i	$0.428710\pi$
$828$	0	0
$829$	33712.0	1.41238	0.706192	−	0.708020i	$-0.250411\pi$
$829$	33712.0	1.41238	0.706192	+	0.708020i	$0.250411\pi$
$830$	0	0
$831$	−4873.38	−0.203436
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−21640.0	−0.896866
$836$	0	0
$837$	−20176.0	−0.833195
$838$	0	0
$839$	26165.8	1.07669	0.538345	−	0.842725i	$-0.319050\pi$
$839$	26165.8	1.07669	0.538345	+	0.842725i	$0.319050\pi$
$840$	0	0
$841$	55135.0	2.26065
$842$	0	0
$843$	7376.54	0.301378
$844$	0	0
$845$	23829.5	0.970130
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1038.00	−0.0419600
$850$	0	0
$851$	−4088.00	−0.164671
$852$	0	0
$853$	−44802.3	−1.79836	−0.899180	−	0.437579i	$-0.855836\pi$
$853$	−44802.3	−1.79836	−0.899180	+	0.437579i	$0.855836\pi$
$854$	0	0
$855$	−22500.0	−0.899981
$856$	0	0
$857$	39364.6	1.56904	0.784522	−	0.620101i	$-0.212909\pi$
$857$	39364.6	1.56904	0.784522	+	0.620101i	$0.212909\pi$
$858$	0	0
$859$	14884.6	0.591218	0.295609	−	0.955309i	$-0.404478\pi$
$859$	14884.6	0.591218	0.295609	+	0.955309i	$0.404478\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8252.00	−0.325494	−0.162747	−	0.986668i	$-0.552035\pi$
$863$	−8252.00	−0.325494	−0.162747	+	0.986668i	$0.552035\pi$
$864$	0	0
$865$	24720.0	0.971682
$866$	0	0
$867$	5451.79	0.213555
$868$	0	0
$869$	−15768.0	−0.615527
$870$	0	0
$871$	20726.7	0.806312
$872$	0	0
$873$	−14743.2	−0.571570
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19190.0	−0.738883	−0.369441	−	0.929254i	$-0.620451\pi$
$877$	−19190.0	−0.738883	−0.369441	+	0.929254i	$0.620451\pi$
$878$	0	0
$879$	1172.00	0.0449722
$880$	0	0
$881$	−11433.9	−0.437251	−0.218626	−	0.975809i	$-0.570157\pi$
$881$	−11433.9	−0.437251	−0.218626	+	0.975809i	$0.570157\pi$
$882$	0	0
$883$	−40580.0	−1.54658	−0.773288	−	0.634056i	$-0.781389\pi$
$883$	−40580.0	−1.54658	−0.773288	+	0.634056i	$0.781389\pi$
$884$	0	0
$885$	11511.7	0.437245
$886$	0	0
$887$	28236.2	1.06886	0.534430	−	0.845213i	$-0.320526\pi$
$887$	28236.2	1.06886	0.534430	+	0.845213i	$0.320526\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−30834.0	−1.15935
$892$	0	0
$893$	32220.0	1.20739
$894$	0	0
$895$	−50628.8	−1.89088
$896$	0	0
$897$	−896.000	−0.0333518
$898$	0	0
$899$	−77368.8	−2.87029
$900$	0	0
$901$	−19451.1	−0.719212
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−41280.0	−1.51624
$906$	0	0
$907$	15236.0	0.557776	0.278888	−	0.960324i	$-0.410034\pi$
$907$	15236.0	0.557776	0.278888	+	0.960324i	$0.410034\pi$
$908$	0	0
$909$	−4596.19	−0.167708
$910$	0	0
$911$	38568.0	1.40265	0.701325	−	0.712841i	$-0.252592\pi$
$911$	38568.0	1.40265	0.701325	+	0.712841i	$0.252592\pi$
$912$	0	0
$913$	−62239.5	−2.25611
$914$	0	0
$915$	−9333.81	−0.337231
$916$	0	0
$917$	0	0
$918$	0	0
$919$	24032.0	0.862614	0.431307	−	0.902205i	$-0.358053\pi$
$919$	24032.0	0.862614	0.431307	+	0.902205i	$0.358053\pi$
$920$	0	0
$921$	12094.0	0.432694
$922$	0	0
$923$	9503.52	0.338908
$924$	0	0
$925$	10950.0	0.389226
$926$	0	0
$927$	47164.0	1.67106
$928$	0	0
$929$	−7533.52	−0.266057	−0.133028	−	0.991112i	$-0.542470\pi$
$929$	−7533.52	−0.266057	−0.133028	+	0.991112i	$0.542470\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12080.0	0.423882
$934$	0	0
$935$	−24840.0	−0.868829
$936$	0	0
$937$	−51509.9	−1.79590	−0.897948	−	0.440101i	$-0.854943\pi$
$937$	−51509.9	−1.79590	−0.897948	+	0.440101i	$0.854943\pi$
$938$	0	0
$939$	3618.00	0.125739
$940$	0	0
$941$	−47362.0	−1.64076	−0.820381	−	0.571817i	$-0.806239\pi$
$941$	−47362.0	−1.64076	−0.820381	+	0.571817i	$0.806239\pi$
$942$	0	0
$943$	9543.11	0.329551
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10986.0	0.376977	0.188488	−	0.982075i	$-0.439641\pi$
$947$	10986.0	0.376977	0.188488	+	0.982075i	$0.439641\pi$
$948$	0	0
$949$	−15904.0	−0.544010
$950$	0	0
$951$	−4483.06	−0.152863
$952$	0	0
$953$	−36938.0	−1.25555	−0.627775	−	0.778395i	$-0.716034\pi$
$953$	−36938.0	−1.25555	−0.627775	+	0.778395i	$0.716034\pi$
$954$	0	0
$955$	−6844.79	−0.231929
$956$	0	0
$957$	21535.6	0.727428
$958$	0	0
$959$	0	0
$960$	0	0
$961$	45481.0	1.52667
$962$	0	0
$963$	−33500.0	−1.12100
$964$	0	0
$965$	11313.7	0.377411
$966$	0	0
$967$	32460.0	1.07947	0.539733	−	0.841836i	$-0.318525\pi$
$967$	32460.0	1.07947	0.539733	+	0.841836i	$0.318525\pi$
$968$	0	0
$969$	−2927.42	−0.0970509
$970$	0	0
$971$	16561.9	0.547369	0.273684	−	0.961820i	$-0.411758\pi$
$971$	16561.9	0.547369	0.273684	+	0.961820i	$0.411758\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2400.00	0.0788323
$976$	0	0
$977$	14460.0	0.473507	0.236754	−	0.971570i	$-0.423917\pi$
$977$	14460.0	0.473507	0.236754	+	0.971570i	$0.423917\pi$
$978$	0	0
$979$	24055.8	0.785317
$980$	0	0
$981$	13450.0	0.437743
$982$	0	0
$983$	−24966.5	−0.810080	−0.405040	−	0.914299i	$-0.632742\pi$
$983$	−24966.5	−0.810080	−0.405040	+	0.914299i	$0.632742\pi$
$984$	0	0
$985$	23843.6	0.771291
$986$	0	0
$987$	0	0
$988$	0	0
$989$	280.000	0.00900251
$990$	0	0
$991$	8480.00	0.271822	0.135911	−	0.990721i	$-0.456604\pi$
$991$	8480.00	0.271822	0.135911	+	0.990721i	$0.456604\pi$
$992$	0	0
$993$	9347.95	0.298739
$994$	0	0
$995$	−1560.00	−0.0497038
$996$	0	0
$997$	−12108.5	−0.384634	−0.192317	−	0.981333i	$-0.561600\pi$
$997$	−12108.5	−0.384634	−0.192317	+	0.981333i	$0.561600\pi$
$998$	0	0
$999$	10736.7	0.340034

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.v.1.1		2
4.3	odd	2		392.4.a.f.1.2	yes	2
7.6	odd	2	inner	784.4.a.v.1.2		2
28.3	even	6		392.4.i.k.177.2		4
28.11	odd	6		392.4.i.k.177.1		4
28.19	even	6		392.4.i.k.361.2		4
28.23	odd	6		392.4.i.k.361.1		4
28.27	even	2		392.4.a.f.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.4.a.f.1.1	✓	2	28.27	even	2
392.4.a.f.1.2	yes	2	4.3	odd	2
392.4.i.k.177.1		4	28.11	odd	6
392.4.i.k.177.2		4	28.3	even	6
392.4.i.k.361.1		4	28.23	odd	6
392.4.i.k.361.2		4	28.19	even	6
784.4.a.v.1.1		2	1.1	even	1	trivial
784.4.a.v.1.2		2	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -14.1421 q^{5} -25.0000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} -14.1421 q^{5} -25.0000 q^{9} -54.0000 q^{11} -22.6274 q^{13} +20.0000 q^{15} -32.5269 q^{17} -63.6396 q^{19} -28.0000 q^{23} +75.0000 q^{25} +73.5391 q^{27} +282.000 q^{29} -274.357 q^{31} +76.3675 q^{33} +146.000 q^{37} +32.0000 q^{39} -340.825 q^{41} -10.0000 q^{43} +353.553 q^{45} -506.288 q^{47} +46.0000 q^{51} +598.000 q^{53} +763.675 q^{55} +90.0000 q^{57} +575.585 q^{59} -466.690 q^{61} +320.000 q^{65} -916.000 q^{67} +39.5980 q^{69} -420.000 q^{71} +702.864 q^{73} -106.066 q^{75} +292.000 q^{79} +571.000 q^{81} +1152.58 q^{83} +460.000 q^{85} -398.808 q^{87} -445.477 q^{89} +388.000 q^{93} +900.000 q^{95} +589.727 q^{97} +1350.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 50 q^{9}+O(q^{10}) \) 2 * q - 50 * q^9	\( 2 q - 50 q^{9} - 108 q^{11} + 40 q^{15} - 56 q^{23} + 150 q^{25} + 564 q^{29} + 292 q^{37} + 64 q^{39} - 20 q^{43} + 92 q^{51} + 1196 q^{53} + 180 q^{57} + 640 q^{65} - 1832 q^{67} - 840 q^{71} + 584 q^{79} + 1142 q^{81} + 920 q^{85} + 776 q^{93} + 1800 q^{95} + 2700 q^{99}+O(q^{100}) \) 2 * q - 50 * q^9 - 108 * q^11 + 40 * q^15 - 56 * q^23 + 150 * q^25 + 564 * q^29 + 292 * q^37 + 64 * q^39 - 20 * q^43 + 92 * q^51 + 1196 * q^53 + 180 * q^57 + 640 * q^65 - 1832 * q^67 - 840 * q^71 + 584 * q^79 + 1142 * q^81 + 920 * q^85 + 776 * q^93 + 1800 * q^95 + 2700 * q^99

Introduction

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