LMFDB - Embedded newform 1380.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,4,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1380.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.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:	$81.4226358079$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 345x^{3} - 1858x^{2} + 6144x + 4608 $ x^5 - 345x^3 - 1858x^2 + 6144*x + 4608
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.77455$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -5.00000 q^{5} -8.87971 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -5.00000 q^{5} -8.87971 q^{7} +9.00000 q^{9} -31.0336 q^{11} +8.98811 q^{13} -15.0000 q^{15} +45.3787 q^{17} +46.4194 q^{19} -26.6391 q^{21} +23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +184.061 q^{29} -31.1244 q^{31} -93.1008 q^{33} +44.3986 q^{35} -230.329 q^{37} +26.9643 q^{39} -145.752 q^{41} +445.006 q^{43} -45.0000 q^{45} -450.141 q^{47} -264.151 q^{49} +136.136 q^{51} +45.3571 q^{53} +155.168 q^{55} +139.258 q^{57} -262.216 q^{59} -694.783 q^{61} -79.9174 q^{63} -44.9406 q^{65} -798.374 q^{67} +69.0000 q^{69} +400.029 q^{71} -652.639 q^{73} +75.0000 q^{75} +275.569 q^{77} +38.2625 q^{79} +81.0000 q^{81} +316.038 q^{83} -226.893 q^{85} +552.182 q^{87} -778.171 q^{89} -79.8119 q^{91} -93.3731 q^{93} -232.097 q^{95} -1588.04 q^{97} -279.302 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 15 q^{3} - 25 q^{5} - 5 q^{7} + 45 q^{9}+O(q^{10}) $ 5 * q + 15 * q^3 - 25 * q^5 - 5 * q^7 + 45 * q^9	$ 5 q + 15 q^{3} - 25 q^{5} - 5 q^{7} + 45 q^{9} - 18 q^{11} + 8 q^{13} - 75 q^{15} - 31 q^{17} - 84 q^{19} - 15 q^{21} + 115 q^{23} + 125 q^{25} + 135 q^{27} - 65 q^{29} - 81 q^{31} - 54 q^{33} + 25 q^{35} - 373 q^{37} + 24 q^{39} - 339 q^{41} - 796 q^{43} - 225 q^{45} - 382 q^{47} - 180 q^{49} - 93 q^{51} - 341 q^{53} + 90 q^{55} - 252 q^{57} + 457 q^{59} - 780 q^{61} - 45 q^{63} - 40 q^{65} - 299 q^{67} + 345 q^{69} + 11 q^{71} - 1092 q^{73} + 375 q^{75} - 468 q^{77} - 1290 q^{79} + 405 q^{81} - 659 q^{83} + 155 q^{85} - 195 q^{87} - 1574 q^{89} - 4068 q^{91} - 243 q^{93} + 420 q^{95} - 1462 q^{97} - 162 q^{99}+O(q^{100}) $ 5 * q + 15 * q^3 - 25 * q^5 - 5 * q^7 + 45 * q^9 - 18 * q^11 + 8 * q^13 - 75 * q^15 - 31 * q^17 - 84 * q^19 - 15 * q^21 + 115 * q^23 + 125 * q^25 + 135 * q^27 - 65 * q^29 - 81 * q^31 - 54 * q^33 + 25 * q^35 - 373 * q^37 + 24 * q^39 - 339 * q^41 - 796 * q^43 - 225 * q^45 - 382 * q^47 - 180 * q^49 - 93 * q^51 - 341 * q^53 + 90 * q^55 - 252 * q^57 + 457 * q^59 - 780 * q^61 - 45 * q^63 - 40 * q^65 - 299 * q^67 + 345 * q^69 + 11 * q^71 - 1092 * q^73 + 375 * q^75 - 468 * q^77 - 1290 * q^79 + 405 * q^81 - 659 * q^83 + 155 * q^85 - 195 * q^87 - 1574 * q^89 - 4068 * q^91 - 243 * q^93 + 420 * q^95 - 1462 * q^97 - 162 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−8.87971	−0.479459	−0.239730	−	0.970840i	$-0.577059\pi$
$7$	−8.87971	−0.479459	−0.239730	+	0.970840i	$0.577059\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−31.0336	−0.850635	−0.425317	−	0.905044i	$-0.639837\pi$
$11$	−31.0336	−0.850635	−0.425317	+	0.905044i	$0.639837\pi$
$12$	0	0
$13$	8.98811	0.191758	0.0958790	−	0.995393i	$-0.469434\pi$
$13$	8.98811	0.191758	0.0958790	+	0.995393i	$0.469434\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	45.3787	0.647409	0.323704	−	0.946158i	$-0.395072\pi$
$17$	45.3787	0.647409	0.323704	+	0.946158i	$0.395072\pi$
$18$	0	0
$19$	46.4194	0.560491	0.280246	−	0.959928i	$-0.409584\pi$
$19$	46.4194	0.560491	0.280246	+	0.959928i	$0.409584\pi$
$20$	0	0
$21$	−26.6391	−0.276816
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	184.061	1.17859	0.589296	−	0.807917i	$-0.299405\pi$
$29$	184.061	1.17859	0.589296	+	0.807917i	$0.299405\pi$
$30$	0	0
$31$	−31.1244	−0.180326	−0.0901629	−	0.995927i	$-0.528739\pi$
$31$	−31.1244	−0.180326	−0.0901629	+	0.995927i	$0.528739\pi$
$32$	0	0
$33$	−93.1008	−0.491114
$34$	0	0
$35$	44.3986	0.214421
$36$	0	0
$37$	−230.329	−1.02340	−0.511701	−	0.859164i	$-0.670984\pi$
$37$	−230.329	−1.02340	−0.511701	+	0.859164i	$0.670984\pi$
$38$	0	0
$39$	26.9643	0.110712
$40$	0	0
$41$	−145.752	−0.555187	−0.277594	−	0.960699i	$-0.589537\pi$
$41$	−145.752	−0.555187	−0.277594	+	0.960699i	$0.589537\pi$
$42$	0	0
$43$	445.006	1.57821	0.789103	−	0.614261i	$-0.210546\pi$
$43$	445.006	1.57821	0.789103	+	0.614261i	$0.210546\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−450.141	−1.39702	−0.698509	−	0.715601i	$-0.746153\pi$
$47$	−450.141	−1.39702	−0.698509	+	0.715601i	$0.746153\pi$
$48$	0	0
$49$	−264.151	−0.770119
$50$	0	0
$51$	136.136	0.373782
$52$	0	0
$53$	45.3571	0.117552	0.0587762	−	0.998271i	$-0.481280\pi$
$53$	45.3571	0.117552	0.0587762	+	0.998271i	$0.481280\pi$
$54$	0	0
$55$	155.168	0.380415
$56$	0	0
$57$	139.258	0.323600
$58$	0	0
$59$	−262.216	−0.578603	−0.289301	−	0.957238i	$-0.593423\pi$
$59$	−262.216	−0.578603	−0.289301	+	0.957238i	$0.593423\pi$
$60$	0	0
$61$	−694.783	−1.45832	−0.729162	−	0.684341i	$-0.760090\pi$
$61$	−694.783	−1.45832	−0.729162	+	0.684341i	$0.760090\pi$
$62$	0	0
$63$	−79.9174	−0.159820
$64$	0	0
$65$	−44.9406	−0.0857568
$66$	0	0
$67$	−798.374	−1.45577	−0.727887	−	0.685697i	$-0.759498\pi$
$67$	−798.374	−1.45577	−0.727887	+	0.685697i	$0.759498\pi$
$68$	0	0
$69$	69.0000	0.120386
$70$	0	0
$71$	400.029	0.668658	0.334329	−	0.942456i	$-0.391490\pi$
$71$	400.029	0.668658	0.334329	+	0.942456i	$0.391490\pi$
$72$	0	0
$73$	−652.639	−1.04638	−0.523189	−	0.852217i	$-0.675258\pi$
$73$	−652.639	−1.04638	−0.523189	+	0.852217i	$0.675258\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	275.569	0.407845
$78$	0	0
$79$	38.2625	0.0544920	0.0272460	−	0.999629i	$-0.491326\pi$
$79$	38.2625	0.0544920	0.0272460	+	0.999629i	$0.491326\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	316.038	0.417948	0.208974	−	0.977921i	$-0.432988\pi$
$83$	316.038	0.417948	0.208974	+	0.977921i	$0.432988\pi$
$84$	0	0
$85$	−226.893	−0.289530
$86$	0	0
$87$	552.182	0.680461
$88$	0	0
$89$	−778.171	−0.926809	−0.463404	−	0.886147i	$-0.653372\pi$
$89$	−778.171	−0.926809	−0.463404	+	0.886147i	$0.653372\pi$
$90$	0	0
$91$	−79.8119	−0.0919402
$92$	0	0
$93$	−93.3731	−0.104111
$94$	0	0
$95$	−232.097	−0.250659
$96$	0	0
$97$	−1588.04	−1.66228	−0.831139	−	0.556065i	$-0.812310\pi$
$97$	−1588.04	−1.66228	−0.831139	+	0.556065i	$0.812310\pi$
$98$	0	0
$99$	−279.302	−0.283545
$100$	0	0
$101$	11.1303	0.0109654	0.00548271	−	0.999985i	$-0.498255\pi$
$101$	11.1303	0.0109654	0.00548271	+	0.999985i	$0.498255\pi$
$102$	0	0
$103$	−555.060	−0.530988	−0.265494	−	0.964113i	$-0.585535\pi$
$103$	−555.060	−0.530988	−0.265494	+	0.964113i	$0.585535\pi$
$104$	0	0
$105$	133.196	0.123796
$106$	0	0
$107$	−1108.75	−1.00175	−0.500875	−	0.865520i	$-0.666988\pi$
$107$	−1108.75	−1.00175	−0.500875	+	0.865520i	$0.666988\pi$
$108$	0	0
$109$	1818.22	1.59774	0.798872	−	0.601500i	$-0.205430\pi$
$109$	1818.22	1.59774	0.798872	+	0.601500i	$0.205430\pi$
$110$	0	0
$111$	−690.987	−0.590861
$112$	0	0
$113$	−1204.44	−1.00269	−0.501346	−	0.865247i	$-0.667162\pi$
$113$	−1204.44	−1.00269	−0.501346	+	0.865247i	$0.667162\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	80.8930	0.0639193
$118$	0	0
$119$	−402.950	−0.310406
$120$	0	0
$121$	−367.916	−0.276421
$122$	0	0
$123$	−437.257	−0.320538
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1604.73	1.12123	0.560615	−	0.828076i	$-0.310565\pi$
$127$	1604.73	1.12123	0.560615	+	0.828076i	$0.310565\pi$
$128$	0	0
$129$	1335.02	0.911177
$130$	0	0
$131$	−1085.96	−0.724283	−0.362141	−	0.932123i	$-0.617954\pi$
$131$	−1085.96	−0.724283	−0.362141	+	0.932123i	$0.617954\pi$
$132$	0	0
$133$	−412.191	−0.268733
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	791.616	0.493667	0.246833	−	0.969058i	$-0.420610\pi$
$137$	791.616	0.493667	0.246833	+	0.969058i	$0.420610\pi$
$138$	0	0
$139$	−364.330	−0.222317	−0.111158	−	0.993803i	$-0.535456\pi$
$139$	−364.330	−0.222317	−0.111158	+	0.993803i	$0.535456\pi$
$140$	0	0
$141$	−1350.42	−0.806569
$142$	0	0
$143$	−278.933	−0.163116
$144$	0	0
$145$	−920.303	−0.527083
$146$	0	0
$147$	−792.452	−0.444628
$148$	0	0
$149$	−577.916	−0.317750	−0.158875	−	0.987299i	$-0.550787\pi$
$149$	−577.916	−0.317750	−0.158875	+	0.987299i	$0.550787\pi$
$150$	0	0
$151$	−2351.24	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$151$	−2351.24	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$152$	0	0
$153$	408.408	0.215803
$154$	0	0
$155$	155.622	0.0806442
$156$	0	0
$157$	2275.18	1.15656	0.578278	−	0.815840i	$-0.303725\pi$
$157$	2275.18	1.15656	0.578278	+	0.815840i	$0.303725\pi$
$158$	0	0
$159$	136.071	0.0678689
$160$	0	0
$161$	−204.233	−0.0999742
$162$	0	0
$163$	−650.942	−0.312796	−0.156398	−	0.987694i	$-0.549988\pi$
$163$	−650.942	−0.312796	−0.156398	+	0.987694i	$0.549988\pi$
$164$	0	0
$165$	465.504	0.219633
$166$	0	0
$167$	−717.479	−0.332456	−0.166228	−	0.986087i	$-0.553159\pi$
$167$	−717.479	−0.332456	−0.166228	+	0.986087i	$0.553159\pi$
$168$	0	0
$169$	−2116.21	−0.963229
$170$	0	0
$171$	417.774	0.186830
$172$	0	0
$173$	−2180.26	−0.958161	−0.479081	−	0.877771i	$-0.659030\pi$
$173$	−2180.26	−0.958161	−0.479081	+	0.877771i	$0.659030\pi$
$174$	0	0
$175$	−221.993	−0.0958919
$176$	0	0
$177$	−786.647	−0.334056
$178$	0	0
$179$	3465.20	1.44693	0.723466	−	0.690360i	$-0.242548\pi$
$179$	3465.20	1.44693	0.723466	+	0.690360i	$0.242548\pi$
$180$	0	0
$181$	−2954.81	−1.21342	−0.606711	−	0.794922i	$-0.707512\pi$
$181$	−2954.81	−1.21342	−0.606711	+	0.794922i	$0.707512\pi$
$182$	0	0
$183$	−2084.35	−0.841964
$184$	0	0
$185$	1151.65	0.457679
$186$	0	0
$187$	−1408.26	−0.550708
$188$	0	0
$189$	−239.752	−0.0922720
$190$	0	0
$191$	4317.12	1.63548	0.817738	−	0.575590i	$-0.195228\pi$
$191$	4317.12	1.63548	0.817738	+	0.575590i	$0.195228\pi$
$192$	0	0
$193$	3410.02	1.27181	0.635904	−	0.771769i	$-0.280628\pi$
$193$	3410.02	1.27181	0.635904	+	0.771769i	$0.280628\pi$
$194$	0	0
$195$	−134.822	−0.0495117
$196$	0	0
$197$	−350.150	−0.126635	−0.0633176	−	0.997993i	$-0.520168\pi$
$197$	−350.150	−0.126635	−0.0633176	+	0.997993i	$0.520168\pi$
$198$	0	0
$199$	−3283.96	−1.16982	−0.584908	−	0.811099i	$-0.698869\pi$
$199$	−3283.96	−1.16982	−0.584908	+	0.811099i	$0.698869\pi$
$200$	0	0
$201$	−2395.12	−0.840492
$202$	0	0
$203$	−1634.40	−0.565087
$204$	0	0
$205$	728.761	0.248287
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	−1440.56	−0.476773
$210$	0	0
$211$	−2264.37	−0.738794	−0.369397	−	0.929272i	$-0.620436\pi$
$211$	−2264.37	−0.738794	−0.369397	+	0.929272i	$0.620436\pi$
$212$	0	0
$213$	1200.09	0.386050
$214$	0	0
$215$	−2225.03	−0.705795
$216$	0	0
$217$	276.375	0.0864589
$218$	0	0
$219$	−1957.92	−0.604127
$220$	0	0
$221$	407.869	0.124146
$222$	0	0
$223$	1606.69	0.482474	0.241237	−	0.970466i	$-0.422447\pi$
$223$	1606.69	0.482474	0.241237	+	0.970466i	$0.422447\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3496.22	1.02226	0.511129	−	0.859504i	$-0.329228\pi$
$227$	3496.22	1.02226	0.511129	+	0.859504i	$0.329228\pi$
$228$	0	0
$229$	−1774.73	−0.512128	−0.256064	−	0.966660i	$-0.582426\pi$
$229$	−1774.73	−0.512128	−0.256064	+	0.966660i	$0.582426\pi$
$230$	0	0
$231$	826.708	0.235469
$232$	0	0
$233$	5804.08	1.63192	0.815961	−	0.578106i	$-0.196208\pi$
$233$	5804.08	1.63192	0.815961	+	0.578106i	$0.196208\pi$
$234$	0	0
$235$	2250.71	0.624765
$236$	0	0
$237$	114.788	0.0314610
$238$	0	0
$239$	1232.28	0.333512	0.166756	−	0.985998i	$-0.446671\pi$
$239$	1232.28	0.333512	0.166756	+	0.985998i	$0.446671\pi$
$240$	0	0
$241$	1332.19	0.356073	0.178037	−	0.984024i	$-0.443025\pi$
$241$	1332.19	0.356073	0.178037	+	0.984024i	$0.443025\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1320.75	0.344408
$246$	0	0
$247$	417.222	0.107479
$248$	0	0
$249$	948.114	0.241302
$250$	0	0
$251$	−6192.99	−1.55736	−0.778682	−	0.627419i	$-0.784111\pi$
$251$	−6192.99	−1.55736	−0.778682	+	0.627419i	$0.784111\pi$
$252$	0	0
$253$	−713.773	−0.177370
$254$	0	0
$255$	−680.680	−0.167160
$256$	0	0
$257$	703.870	0.170841	0.0854206	−	0.996345i	$-0.472777\pi$
$257$	703.870	0.170841	0.0854206	+	0.996345i	$0.472777\pi$
$258$	0	0
$259$	2045.26	0.490680
$260$	0	0
$261$	1656.54	0.392864
$262$	0	0
$263$	−7600.06	−1.78190	−0.890950	−	0.454102i	$-0.849960\pi$
$263$	−7600.06	−1.78190	−0.890950	+	0.454102i	$0.849960\pi$
$264$	0	0
$265$	−226.785	−0.0525710
$266$	0	0
$267$	−2334.51	−0.535093
$268$	0	0
$269$	1169.24	0.265018	0.132509	−	0.991182i	$-0.457697\pi$
$269$	1169.24	0.265018	0.132509	+	0.991182i	$0.457697\pi$
$270$	0	0
$271$	1851.50	0.415020	0.207510	−	0.978233i	$-0.433464\pi$
$271$	1851.50	0.415020	0.207510	+	0.978233i	$0.433464\pi$
$272$	0	0
$273$	−239.436	−0.0530817
$274$	0	0
$275$	−775.840	−0.170127
$276$	0	0
$277$	3449.10	0.748146	0.374073	−	0.927399i	$-0.377961\pi$
$277$	3449.10	0.748146	0.374073	+	0.927399i	$0.377961\pi$
$278$	0	0
$279$	−280.119	−0.0601086
$280$	0	0
$281$	4671.92	0.991826	0.495913	−	0.868372i	$-0.334833\pi$
$281$	4671.92	0.991826	0.495913	+	0.868372i	$0.334833\pi$
$282$	0	0
$283$	−8123.89	−1.70641	−0.853207	−	0.521573i	$-0.825345\pi$
$283$	−8123.89	−1.70641	−0.853207	+	0.521573i	$0.825345\pi$
$284$	0	0
$285$	−696.290	−0.144718
$286$	0	0
$287$	1294.24	0.266190
$288$	0	0
$289$	−2853.77	−0.580862
$290$	0	0
$291$	−4764.12	−0.959716
$292$	0	0
$293$	−7422.00	−1.47986	−0.739928	−	0.672686i	$-0.765141\pi$
$293$	−7422.00	−1.47986	−0.739928	+	0.672686i	$0.765141\pi$
$294$	0	0
$295$	1311.08	0.258759
$296$	0	0
$297$	−837.907	−0.163705
$298$	0	0
$299$	206.727	0.0399843
$300$	0	0
$301$	−3951.53	−0.756685
$302$	0	0
$303$	33.3909	0.00633088
$304$	0	0
$305$	3473.91	0.652183
$306$	0	0
$307$	−4721.03	−0.877666	−0.438833	−	0.898569i	$-0.644608\pi$
$307$	−4721.03	−0.877666	−0.438833	+	0.898569i	$0.644608\pi$
$308$	0	0
$309$	−1665.18	−0.306566
$310$	0	0
$311$	6440.30	1.17426	0.587131	−	0.809492i	$-0.300257\pi$
$311$	6440.30	1.17426	0.587131	+	0.809492i	$0.300257\pi$
$312$	0	0
$313$	−10361.9	−1.87120	−0.935601	−	0.353058i	$-0.885142\pi$
$313$	−10361.9	−1.87120	−0.935601	+	0.353058i	$0.885142\pi$
$314$	0	0
$315$	399.587	0.0714736
$316$	0	0
$317$	−6114.54	−1.08336	−0.541682	−	0.840583i	$-0.682212\pi$
$317$	−6114.54	−1.08336	−0.541682	+	0.840583i	$0.682212\pi$
$318$	0	0
$319$	−5712.06	−1.00255
$320$	0	0
$321$	−3326.26	−0.578361
$322$	0	0
$323$	2106.45	0.362867
$324$	0	0
$325$	224.703	0.0383516
$326$	0	0
$327$	5454.67	0.922458
$328$	0	0
$329$	3997.13	0.669814
$330$	0	0
$331$	−1896.20	−0.314877	−0.157439	−	0.987529i	$-0.550324\pi$
$331$	−1896.20	−0.314877	−0.157439	+	0.987529i	$0.550324\pi$
$332$	0	0
$333$	−2072.96	−0.341134
$334$	0	0
$335$	3991.87	0.651042
$336$	0	0
$337$	3003.53	0.485498	0.242749	−	0.970089i	$-0.421951\pi$
$337$	3003.53	0.485498	0.242749	+	0.970089i	$0.421951\pi$
$338$	0	0
$339$	−3613.32	−0.578904
$340$	0	0
$341$	965.901	0.153391
$342$	0	0
$343$	5391.32	0.848700
$344$	0	0
$345$	−345.000	−0.0538382
$346$	0	0
$347$	5590.23	0.864840	0.432420	−	0.901672i	$-0.357660\pi$
$347$	5590.23	0.864840	0.432420	+	0.901672i	$0.357660\pi$
$348$	0	0
$349$	9891.75	1.51717	0.758586	−	0.651573i	$-0.225890\pi$
$349$	9891.75	1.51717	0.758586	+	0.651573i	$0.225890\pi$
$350$	0	0
$351$	242.679	0.0369038
$352$	0	0
$353$	−1842.13	−0.277752	−0.138876	−	0.990310i	$-0.544349\pi$
$353$	−1842.13	−0.277752	−0.138876	+	0.990310i	$0.544349\pi$
$354$	0	0
$355$	−2000.15	−0.299033
$356$	0	0
$357$	−1208.85	−0.179213
$358$	0	0
$359$	−5740.37	−0.843914	−0.421957	−	0.906616i	$-0.638657\pi$
$359$	−5740.37	−0.843914	−0.421957	+	0.906616i	$0.638657\pi$
$360$	0	0
$361$	−4704.24	−0.685850
$362$	0	0
$363$	−1103.75	−0.159592
$364$	0	0
$365$	3263.19	0.467954
$366$	0	0
$367$	198.578	0.0282443	0.0141222	−	0.999900i	$-0.495505\pi$
$367$	198.578	0.0282443	0.0141222	+	0.999900i	$0.495505\pi$
$368$	0	0
$369$	−1311.77	−0.185062
$370$	0	0
$371$	−402.758	−0.0563616
$372$	0	0
$373$	−3909.72	−0.542728	−0.271364	−	0.962477i	$-0.587475\pi$
$373$	−3909.72	−0.542728	−0.271364	+	0.962477i	$0.587475\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	1654.36	0.226005
$378$	0	0
$379$	−9188.80	−1.24537	−0.622687	−	0.782471i	$-0.713959\pi$
$379$	−9188.80	−1.24537	−0.622687	+	0.782471i	$0.713959\pi$
$380$	0	0
$381$	4814.18	0.647343
$382$	0	0
$383$	1546.04	0.206264	0.103132	−	0.994668i	$-0.467114\pi$
$383$	1546.04	0.206264	0.103132	+	0.994668i	$0.467114\pi$
$384$	0	0
$385$	−1377.85	−0.182394
$386$	0	0
$387$	4005.06	0.526068
$388$	0	0
$389$	52.3008	0.00681686	0.00340843	−	0.999994i	$-0.498915\pi$
$389$	52.3008	0.00681686	0.00340843	+	0.999994i	$0.498915\pi$
$390$	0	0
$391$	1043.71	0.134994
$392$	0	0
$393$	−3257.89	−0.418165
$394$	0	0
$395$	−191.313	−0.0243696
$396$	0	0
$397$	−15516.9	−1.96165	−0.980823	−	0.194901i	$-0.937561\pi$
$397$	−15516.9	−1.96165	−0.980823	+	0.194901i	$0.937561\pi$
$398$	0	0
$399$	−1236.57	−0.155153
$400$	0	0
$401$	−11917.5	−1.48412	−0.742061	−	0.670332i	$-0.766152\pi$
$401$	−11917.5	−1.48412	−0.742061	+	0.670332i	$0.766152\pi$
$402$	0	0
$403$	−279.749	−0.0345789
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	7147.94	0.870541
$408$	0	0
$409$	2286.57	0.276440	0.138220	−	0.990402i	$-0.455862\pi$
$409$	2286.57	0.276440	0.138220	+	0.990402i	$0.455862\pi$
$410$	0	0
$411$	2374.85	0.285019
$412$	0	0
$413$	2328.40	0.277417
$414$	0	0
$415$	−1580.19	−0.186912
$416$	0	0
$417$	−1092.99	−0.128355
$418$	0	0
$419$	7602.70	0.886435	0.443218	−	0.896414i	$-0.353837\pi$
$419$	7602.70	0.886435	0.443218	+	0.896414i	$0.353837\pi$
$420$	0	0
$421$	−7950.32	−0.920368	−0.460184	−	0.887824i	$-0.652217\pi$
$421$	−7950.32	−0.920368	−0.460184	+	0.887824i	$0.652217\pi$
$422$	0	0
$423$	−4051.27	−0.465673
$424$	0	0
$425$	1134.47	0.129482
$426$	0	0
$427$	6169.47	0.699208
$428$	0	0
$429$	−836.800	−0.0941751
$430$	0	0
$431$	7006.80	0.783076	0.391538	−	0.920162i	$-0.371943\pi$
$431$	7006.80	0.783076	0.391538	+	0.920162i	$0.371943\pi$
$432$	0	0
$433$	2852.98	0.316641	0.158321	−	0.987388i	$-0.449392\pi$
$433$	2852.98	0.316641	0.158321	+	0.987388i	$0.449392\pi$
$434$	0	0
$435$	−2760.91	−0.304311
$436$	0	0
$437$	1067.65	0.116870
$438$	0	0
$439$	10828.6	1.17726	0.588632	−	0.808401i	$-0.299667\pi$
$439$	10828.6	1.17726	0.588632	+	0.808401i	$0.299667\pi$
$440$	0	0
$441$	−2377.36	−0.256706
$442$	0	0
$443$	14362.6	1.54038	0.770190	−	0.637814i	$-0.220161\pi$
$443$	14362.6	1.54038	0.770190	+	0.637814i	$0.220161\pi$
$444$	0	0
$445$	3890.86	0.414481
$446$	0	0
$447$	−1733.75	−0.183453
$448$	0	0
$449$	−18692.0	−1.96465	−0.982327	−	0.187170i	$-0.940068\pi$
$449$	−18692.0	−1.96465	−0.982327	+	0.187170i	$0.940068\pi$
$450$	0	0
$451$	4523.22	0.472262
$452$	0	0
$453$	−7053.73	−0.731596
$454$	0	0
$455$	399.059	0.0411169
$456$	0	0
$457$	8242.33	0.843676	0.421838	−	0.906671i	$-0.361385\pi$
$457$	8242.33	0.843676	0.421838	+	0.906671i	$0.361385\pi$
$458$	0	0
$459$	1225.22	0.124594
$460$	0	0
$461$	16597.2	1.67681	0.838404	−	0.545049i	$-0.183489\pi$
$461$	16597.2	1.67681	0.838404	+	0.545049i	$0.183489\pi$
$462$	0	0
$463$	−340.173	−0.0341451	−0.0170726	−	0.999854i	$-0.505435\pi$
$463$	−340.173	−0.0341451	−0.0170726	+	0.999854i	$0.505435\pi$
$464$	0	0
$465$	466.866	0.0465599
$466$	0	0
$467$	−938.423	−0.0929872	−0.0464936	−	0.998919i	$-0.514805\pi$
$467$	−938.423	−0.0929872	−0.0464936	+	0.998919i	$0.514805\pi$
$468$	0	0
$469$	7089.33	0.697985
$470$	0	0
$471$	6825.55	0.667738
$472$	0	0
$473$	−13810.1	−1.34248
$474$	0	0
$475$	1160.48	0.112098
$476$	0	0
$477$	408.214	0.0391841
$478$	0	0
$479$	93.2123	0.00889139	0.00444570	−	0.999990i	$-0.498585\pi$
$479$	93.2123	0.00889139	0.00444570	+	0.999990i	$0.498585\pi$
$480$	0	0
$481$	−2070.22	−0.196245
$482$	0	0
$483$	−612.700	−0.0577201
$484$	0	0
$485$	7940.19	0.743393
$486$	0	0
$487$	−8742.63	−0.813484	−0.406742	−	0.913543i	$-0.633335\pi$
$487$	−8742.63	−0.813484	−0.406742	+	0.913543i	$0.633335\pi$
$488$	0	0
$489$	−1952.83	−0.180593
$490$	0	0
$491$	11069.3	1.01742	0.508708	−	0.860939i	$-0.330123\pi$
$491$	11069.3	1.01742	0.508708	+	0.860939i	$0.330123\pi$
$492$	0	0
$493$	8352.43	0.763031
$494$	0	0
$495$	1396.51	0.126805
$496$	0	0
$497$	−3552.15	−0.320595
$498$	0	0
$499$	−6681.60	−0.599418	−0.299709	−	0.954031i	$-0.596890\pi$
$499$	−6681.60	−0.599418	−0.299709	+	0.954031i	$0.596890\pi$
$500$	0	0
$501$	−2152.44	−0.191944
$502$	0	0
$503$	1414.81	0.125414	0.0627070	−	0.998032i	$-0.480027\pi$
$503$	1414.81	0.125414	0.0627070	+	0.998032i	$0.480027\pi$
$504$	0	0
$505$	−55.6515	−0.00490388
$506$	0	0
$507$	−6348.64	−0.556120
$508$	0	0
$509$	−3381.02	−0.294423	−0.147211	−	0.989105i	$-0.547030\pi$
$509$	−3381.02	−0.294423	−0.147211	+	0.989105i	$0.547030\pi$
$510$	0	0
$511$	5795.24	0.501696
$512$	0	0
$513$	1253.32	0.107867
$514$	0	0
$515$	2775.30	0.237465
$516$	0	0
$517$	13969.5	1.18835
$518$	0	0
$519$	−6540.77	−0.553195
$520$	0	0
$521$	1070.53	0.0900204	0.0450102	−	0.998987i	$-0.485668\pi$
$521$	1070.53	0.0900204	0.0450102	+	0.998987i	$0.485668\pi$
$522$	0	0
$523$	−599.115	−0.0500907	−0.0250454	−	0.999686i	$-0.507973\pi$
$523$	−599.115	−0.0500907	−0.0250454	+	0.999686i	$0.507973\pi$
$524$	0	0
$525$	−665.978	−0.0553632
$526$	0	0
$527$	−1412.38	−0.116745
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−2359.94	−0.192868
$532$	0	0
$533$	−1310.04	−0.106462
$534$	0	0
$535$	5543.77	0.447996
$536$	0	0
$537$	10395.6	0.835387
$538$	0	0
$539$	8197.54	0.655090
$540$	0	0
$541$	3534.72	0.280904	0.140452	−	0.990087i	$-0.455144\pi$
$541$	3534.72	0.280904	0.140452	+	0.990087i	$0.455144\pi$
$542$	0	0
$543$	−8864.44	−0.700570
$544$	0	0
$545$	−9091.12	−0.714533
$546$	0	0
$547$	18940.5	1.48050	0.740252	−	0.672329i	$-0.234706\pi$
$547$	18940.5	1.48050	0.740252	+	0.672329i	$0.234706\pi$
$548$	0	0
$549$	−6253.04	−0.486108
$550$	0	0
$551$	8543.97	0.660591
$552$	0	0
$553$	−339.760	−0.0261267
$554$	0	0
$555$	3454.94	0.264241
$556$	0	0
$557$	6317.35	0.480565	0.240282	−	0.970703i	$-0.422760\pi$
$557$	6317.35	0.480565	0.240282	+	0.970703i	$0.422760\pi$
$558$	0	0
$559$	3999.77	0.302634
$560$	0	0
$561$	−4224.79	−0.317952
$562$	0	0
$563$	12468.8	0.933387	0.466694	−	0.884419i	$-0.345445\pi$
$563$	12468.8	0.933387	0.466694	+	0.884419i	$0.345445\pi$
$564$	0	0
$565$	6022.20	0.448417
$566$	0	0
$567$	−719.257	−0.0532733
$568$	0	0
$569$	20458.2	1.50730	0.753649	−	0.657277i	$-0.228292\pi$
$569$	20458.2	1.50730	0.753649	+	0.657277i	$0.228292\pi$
$570$	0	0
$571$	−21667.4	−1.58801	−0.794005	−	0.607911i	$-0.792008\pi$
$571$	−21667.4	−1.58801	−0.794005	+	0.607911i	$0.792008\pi$
$572$	0	0
$573$	12951.4	0.944243
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−258.473	−0.0186489	−0.00932443	−	0.999957i	$-0.502968\pi$
$577$	−258.473	−0.0186489	−0.00932443	+	0.999957i	$0.502968\pi$
$578$	0	0
$579$	10230.1	0.734278
$580$	0	0
$581$	−2806.33	−0.200389
$582$	0	0
$583$	−1407.59	−0.0999941
$584$	0	0
$585$	−404.465	−0.0285856
$586$	0	0
$587$	26564.0	1.86783	0.933914	−	0.357497i	$-0.116370\pi$
$587$	26564.0	1.86783	0.933914	+	0.357497i	$0.116370\pi$
$588$	0	0
$589$	−1444.77	−0.101071
$590$	0	0
$591$	−1050.45	−0.0731129
$592$	0	0
$593$	8325.44	0.576534	0.288267	−	0.957550i	$-0.406921\pi$
$593$	8325.44	0.576534	0.288267	+	0.957550i	$0.406921\pi$
$594$	0	0
$595$	2014.75	0.138818
$596$	0	0
$597$	−9851.87	−0.675394
$598$	0	0
$599$	7977.78	0.544179	0.272090	−	0.962272i	$-0.412285\pi$
$599$	7977.78	0.544179	0.272090	+	0.962272i	$0.412285\pi$
$600$	0	0
$601$	−15798.7	−1.07229	−0.536143	−	0.844127i	$-0.680119\pi$
$601$	−15798.7	−1.07229	−0.536143	+	0.844127i	$0.680119\pi$
$602$	0	0
$603$	−7185.37	−0.485258
$604$	0	0
$605$	1839.58	0.123619
$606$	0	0
$607$	8424.73	0.563343	0.281672	−	0.959511i	$-0.409111\pi$
$607$	8424.73	0.563343	0.281672	+	0.959511i	$0.409111\pi$
$608$	0	0
$609$	−4903.21	−0.326253
$610$	0	0
$611$	−4045.92	−0.267889
$612$	0	0
$613$	−1619.73	−0.106722	−0.0533608	−	0.998575i	$-0.516993\pi$
$613$	−1619.73	−0.106722	−0.0533608	+	0.998575i	$0.516993\pi$
$614$	0	0
$615$	2186.28	0.143349
$616$	0	0
$617$	11780.1	0.768634	0.384317	−	0.923201i	$-0.374437\pi$
$617$	11780.1	0.768634	0.384317	+	0.923201i	$0.374437\pi$
$618$	0	0
$619$	−11366.2	−0.738038	−0.369019	−	0.929422i	$-0.620306\pi$
$619$	−11366.2	−0.738038	−0.369019	+	0.929422i	$0.620306\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	0	0
$623$	6909.94	0.444367
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−4321.68	−0.275265
$628$	0	0
$629$	−10452.0	−0.662559
$630$	0	0
$631$	17044.3	1.07532	0.537658	−	0.843163i	$-0.319309\pi$
$631$	17044.3	1.07532	0.537658	+	0.843163i	$0.319309\pi$
$632$	0	0
$633$	−6793.11	−0.426543
$634$	0	0
$635$	−8023.63	−0.501430
$636$	0	0
$637$	−2374.22	−0.147676
$638$	0	0
$639$	3600.26	0.222886
$640$	0	0
$641$	−4306.28	−0.265347	−0.132674	−	0.991160i	$-0.542356\pi$
$641$	−4306.28	−0.265347	−0.132674	+	0.991160i	$0.542356\pi$
$642$	0	0
$643$	−16286.7	−0.998885	−0.499443	−	0.866347i	$-0.666462\pi$
$643$	−16286.7	−0.998885	−0.499443	+	0.866347i	$0.666462\pi$
$644$	0	0
$645$	−6675.10	−0.407491
$646$	0	0
$647$	18723.4	1.13770	0.568852	−	0.822440i	$-0.307388\pi$
$647$	18723.4	1.13770	0.568852	+	0.822440i	$0.307388\pi$
$648$	0	0
$649$	8137.49	0.492179
$650$	0	0
$651$	829.126	0.0499171
$652$	0	0
$653$	−8091.76	−0.484923	−0.242462	−	0.970161i	$-0.577955\pi$
$653$	−8091.76	−0.484923	−0.242462	+	0.970161i	$0.577955\pi$
$654$	0	0
$655$	5429.82	0.323909
$656$	0	0
$657$	−5873.75	−0.348793
$658$	0	0
$659$	11037.9	0.652466	0.326233	−	0.945289i	$-0.394221\pi$
$659$	11037.9	0.652466	0.326233	+	0.945289i	$0.394221\pi$
$660$	0	0
$661$	−22000.8	−1.29460	−0.647300	−	0.762235i	$-0.724102\pi$
$661$	−22000.8	−1.29460	−0.647300	+	0.762235i	$0.724102\pi$
$662$	0	0
$663$	1223.61	0.0716756
$664$	0	0
$665$	2060.95	0.120181
$666$	0	0
$667$	4233.39	0.245754
$668$	0	0
$669$	4820.06	0.278557
$670$	0	0
$671$	21561.6	1.24050
$672$	0	0
$673$	18342.6	1.05060	0.525302	−	0.850916i	$-0.323952\pi$
$673$	18342.6	1.05060	0.525302	+	0.850916i	$0.323952\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	9560.23	0.542732	0.271366	−	0.962476i	$-0.412525\pi$
$677$	9560.23	0.542732	0.271366	+	0.962476i	$0.412525\pi$
$678$	0	0
$679$	14101.3	0.796995
$680$	0	0
$681$	10488.7	0.590201
$682$	0	0
$683$	6407.19	0.358952	0.179476	−	0.983762i	$-0.442560\pi$
$683$	6407.19	0.358952	0.179476	+	0.983762i	$0.442560\pi$
$684$	0	0
$685$	−3958.08	−0.220774
$686$	0	0
$687$	−5324.18	−0.295677
$688$	0	0
$689$	407.675	0.0225416
$690$	0	0
$691$	32260.3	1.77603	0.888016	−	0.459812i	$-0.152083\pi$
$691$	32260.3	1.77603	0.888016	+	0.459812i	$0.152083\pi$
$692$	0	0
$693$	2480.12	0.135948
$694$	0	0
$695$	1821.65	0.0994232
$696$	0	0
$697$	−6614.05	−0.359433
$698$	0	0
$699$	17412.2	0.942191
$700$	0	0
$701$	−13740.1	−0.740311	−0.370155	−	0.928970i	$-0.620696\pi$
$701$	−13740.1	−0.740311	−0.370155	+	0.928970i	$0.620696\pi$
$702$	0	0
$703$	−10691.7	−0.573608
$704$	0	0
$705$	6752.12	0.360709
$706$	0	0
$707$	−98.8339	−0.00525747
$708$	0	0
$709$	−31371.7	−1.66176	−0.830882	−	0.556449i	$-0.812163\pi$
$709$	−31371.7	−1.66176	−0.830882	+	0.556449i	$0.812163\pi$
$710$	0	0
$711$	344.363	0.0181640
$712$	0	0
$713$	−715.860	−0.0376005
$714$	0	0
$715$	1394.67	0.0729477
$716$	0	0
$717$	3696.83	0.192553
$718$	0	0
$719$	68.9055	0.00357405	0.00178702	−	0.999998i	$-0.499431\pi$
$719$	68.9055	0.00357405	0.00178702	+	0.999998i	$0.499431\pi$
$720$	0	0
$721$	4928.78	0.254587
$722$	0	0
$723$	3996.56	0.205579
$724$	0	0
$725$	4601.51	0.235718
$726$	0	0
$727$	10544.5	0.537929	0.268965	−	0.963150i	$-0.413319\pi$
$727$	10544.5	0.537929	0.268965	+	0.963150i	$0.413319\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	20193.8	1.02174
$732$	0	0
$733$	9779.97	0.492813	0.246406	−	0.969167i	$-0.420750\pi$
$733$	9779.97	0.492813	0.246406	+	0.969167i	$0.420750\pi$
$734$	0	0
$735$	3962.26	0.198844
$736$	0	0
$737$	24776.4	1.23833
$738$	0	0
$739$	−9843.77	−0.489999	−0.244999	−	0.969523i	$-0.578788\pi$
$739$	−9843.77	−0.489999	−0.244999	+	0.969523i	$0.578788\pi$
$740$	0	0
$741$	1251.67	0.0620528
$742$	0	0
$743$	25430.4	1.25565	0.627827	−	0.778353i	$-0.283944\pi$
$743$	25430.4	1.25565	0.627827	+	0.778353i	$0.283944\pi$
$744$	0	0
$745$	2889.58	0.142102
$746$	0	0
$747$	2844.34	0.139316
$748$	0	0
$749$	9845.41	0.480299
$750$	0	0
$751$	3054.57	0.148419	0.0742097	−	0.997243i	$-0.476357\pi$
$751$	3054.57	0.148419	0.0742097	+	0.997243i	$0.476357\pi$
$752$	0	0
$753$	−18579.0	−0.899144
$754$	0	0
$755$	11756.2	0.566692
$756$	0	0
$757$	2759.37	0.132485	0.0662424	−	0.997804i	$-0.478899\pi$
$757$	2759.37	0.132485	0.0662424	+	0.997804i	$0.478899\pi$
$758$	0	0
$759$	−2141.32	−0.102404
$760$	0	0
$761$	−12035.2	−0.573291	−0.286645	−	0.958037i	$-0.592540\pi$
$761$	−12035.2	−0.573291	−0.286645	+	0.958037i	$0.592540\pi$
$762$	0	0
$763$	−16145.3	−0.766054
$764$	0	0
$765$	−2042.04	−0.0965100
$766$	0	0
$767$	−2356.82	−0.110952
$768$	0	0
$769$	−40812.4	−1.91383	−0.956914	−	0.290371i	$-0.906221\pi$
$769$	−40812.4	−1.91383	−0.956914	+	0.290371i	$0.906221\pi$
$770$	0	0
$771$	2111.61	0.0986352
$772$	0	0
$773$	40428.0	1.88110	0.940552	−	0.339650i	$-0.110309\pi$
$773$	40428.0	1.88110	0.940552	+	0.339650i	$0.110309\pi$
$774$	0	0
$775$	−778.109	−0.0360652
$776$	0	0
$777$	6135.77	0.283294
$778$	0	0
$779$	−6765.73	−0.311178
$780$	0	0
$781$	−12414.3	−0.568784
$782$	0	0
$783$	4969.63	0.226820
$784$	0	0
$785$	−11375.9	−0.517228
$786$	0	0
$787$	21170.5	0.958890	0.479445	−	0.877572i	$-0.340838\pi$
$787$	21170.5	0.958890	0.479445	+	0.877572i	$0.340838\pi$
$788$	0	0
$789$	−22800.2	−1.02878
$790$	0	0
$791$	10695.1	0.480750
$792$	0	0
$793$	−6244.78	−0.279645
$794$	0	0
$795$	−680.356	−0.0303519
$796$	0	0
$797$	24841.0	1.10403	0.552016	−	0.833834i	$-0.313859\pi$
$797$	24841.0	1.10403	0.552016	+	0.833834i	$0.313859\pi$
$798$	0	0
$799$	−20426.8	−0.904442
$800$	0	0
$801$	−7003.54	−0.308936
$802$	0	0
$803$	20253.7	0.890085
$804$	0	0
$805$	1021.17	0.0447098
$806$	0	0
$807$	3507.73	0.153008
$808$	0	0
$809$	−21896.3	−0.951586	−0.475793	−	0.879557i	$-0.657839\pi$
$809$	−21896.3	−0.951586	−0.475793	+	0.879557i	$0.657839\pi$
$810$	0	0
$811$	−27351.7	−1.18428	−0.592138	−	0.805837i	$-0.701716\pi$
$811$	−27351.7	−1.18428	−0.592138	+	0.805837i	$0.701716\pi$
$812$	0	0
$813$	5554.50	0.239612
$814$	0	0
$815$	3254.71	0.139886
$816$	0	0
$817$	20656.9	0.884570
$818$	0	0
$819$	−718.307	−0.0306467
$820$	0	0
$821$	12356.5	0.525269	0.262634	−	0.964895i	$-0.415409\pi$
$821$	12356.5	0.525269	0.262634	+	0.964895i	$0.415409\pi$
$822$	0	0
$823$	38454.0	1.62870	0.814351	−	0.580372i	$-0.197093\pi$
$823$	38454.0	1.62870	0.814351	+	0.580372i	$0.197093\pi$
$824$	0	0
$825$	−2327.52	−0.0982228
$826$	0	0
$827$	10702.1	0.449996	0.224998	−	0.974359i	$-0.427762\pi$
$827$	10702.1	0.449996	0.224998	+	0.974359i	$0.427762\pi$
$828$	0	0
$829$	−14281.3	−0.598322	−0.299161	−	0.954203i	$-0.596707\pi$
$829$	−14281.3	−0.598322	−0.299161	+	0.954203i	$0.596707\pi$
$830$	0	0
$831$	10347.3	0.431942
$832$	0	0
$833$	−11986.8	−0.498582
$834$	0	0
$835$	3587.40	0.148679
$836$	0	0
$837$	−840.358	−0.0347037
$838$	0	0
$839$	5770.71	0.237458	0.118729	−	0.992927i	$-0.462118\pi$
$839$	5770.71	0.237458	0.118729	+	0.992927i	$0.462118\pi$
$840$	0	0
$841$	9489.28	0.389080
$842$	0	0
$843$	14015.8	0.572631
$844$	0	0
$845$	10581.1	0.430769
$846$	0	0
$847$	3266.99	0.132533
$848$	0	0
$849$	−24371.7	−0.985198
$850$	0	0
$851$	−5297.57	−0.213394
$852$	0	0
$853$	−43030.5	−1.72724	−0.863621	−	0.504142i	$-0.831809\pi$
$853$	−43030.5	−1.72724	−0.863621	+	0.504142i	$0.831809\pi$
$854$	0	0
$855$	−2088.87	−0.0835531
$856$	0	0
$857$	23510.5	0.937109	0.468555	−	0.883434i	$-0.344775\pi$
$857$	23510.5	0.937109	0.468555	+	0.883434i	$0.344775\pi$
$858$	0	0
$859$	−5764.55	−0.228968	−0.114484	−	0.993425i	$-0.536522\pi$
$859$	−5764.55	−0.228968	−0.114484	+	0.993425i	$0.536522\pi$
$860$	0	0
$861$	3882.72	0.153685
$862$	0	0
$863$	−12810.2	−0.505289	−0.252644	−	0.967559i	$-0.581300\pi$
$863$	−12810.2	−0.505289	−0.252644	+	0.967559i	$0.581300\pi$
$864$	0	0
$865$	10901.3	0.428503
$866$	0	0
$867$	−8561.32	−0.335361
$868$	0	0
$869$	−1187.42	−0.0463528
$870$	0	0
$871$	−7175.88	−0.279156
$872$	0	0
$873$	−14292.3	−0.554092
$874$	0	0
$875$	1109.96	0.0428842
$876$	0	0
$877$	39298.0	1.51311	0.756555	−	0.653930i	$-0.226881\pi$
$877$	39298.0	1.51311	0.756555	+	0.653930i	$0.226881\pi$
$878$	0	0
$879$	−22266.0	−0.854396
$880$	0	0
$881$	−18834.3	−0.720255	−0.360128	−	0.932903i	$-0.617267\pi$
$881$	−18834.3	−0.720255	−0.360128	+	0.932903i	$0.617267\pi$
$882$	0	0
$883$	−16662.5	−0.635036	−0.317518	−	0.948252i	$-0.602849\pi$
$883$	−16662.5	−0.635036	−0.317518	+	0.948252i	$0.602849\pi$
$884$	0	0
$885$	3933.23	0.149395
$886$	0	0
$887$	40705.4	1.54087	0.770435	−	0.637518i	$-0.220039\pi$
$887$	40705.4	1.54087	0.770435	+	0.637518i	$0.220039\pi$
$888$	0	0
$889$	−14249.5	−0.537585
$890$	0	0
$891$	−2513.72	−0.0945150
$892$	0	0
$893$	−20895.3	−0.783016
$894$	0	0
$895$	−17326.0	−0.647088
$896$	0	0
$897$	620.180	0.0230850
$898$	0	0
$899$	−5728.77	−0.212531
$900$	0	0
$901$	2058.25	0.0761044
$902$	0	0
$903$	−11854.6	−0.436873
$904$	0	0
$905$	14774.1	0.542659
$906$	0	0
$907$	−18842.8	−0.689818	−0.344909	−	0.938636i	$-0.612090\pi$
$907$	−18842.8	−0.689818	−0.344909	+	0.938636i	$0.612090\pi$
$908$	0	0
$909$	100.173	0.00365514
$910$	0	0
$911$	−17033.4	−0.619474	−0.309737	−	0.950822i	$-0.600241\pi$
$911$	−17033.4	−0.619474	−0.309737	+	0.950822i	$0.600241\pi$
$912$	0	0
$913$	−9807.79	−0.355521
$914$	0	0
$915$	10421.7	0.376538
$916$	0	0
$917$	9643.04	0.347264
$918$	0	0
$919$	−11941.0	−0.428615	−0.214308	−	0.976766i	$-0.568750\pi$
$919$	−11941.0	−0.428615	−0.214308	+	0.976766i	$0.568750\pi$
$920$	0	0
$921$	−14163.1	−0.506721
$922$	0	0
$923$	3595.51	0.128221
$924$	0	0
$925$	−5758.23	−0.204680
$926$	0	0
$927$	−4995.54	−0.176996
$928$	0	0
$929$	−25354.7	−0.895437	−0.447719	−	0.894174i	$-0.647763\pi$
$929$	−25354.7	−0.895437	−0.447719	+	0.894174i	$0.647763\pi$
$930$	0	0
$931$	−12261.7	−0.431645
$932$	0	0
$933$	19320.9	0.677961
$934$	0	0
$935$	7041.32	0.246284
$936$	0	0
$937$	−7804.96	−0.272120	−0.136060	−	0.990701i	$-0.543444\pi$
$937$	−7804.96	−0.272120	−0.136060	+	0.990701i	$0.543444\pi$
$938$	0	0
$939$	−31085.6	−1.08034
$940$	0	0
$941$	−19058.1	−0.660231	−0.330115	−	0.943941i	$-0.607088\pi$
$941$	−19058.1	−0.660231	−0.330115	+	0.943941i	$0.607088\pi$
$942$	0	0
$943$	−3352.30	−0.115765
$944$	0	0
$945$	1198.76	0.0412653
$946$	0	0
$947$	−26981.7	−0.925860	−0.462930	−	0.886395i	$-0.653202\pi$
$947$	−26981.7	−0.925860	−0.462930	+	0.886395i	$0.653202\pi$
$948$	0	0
$949$	−5865.99	−0.200651
$950$	0	0
$951$	−18343.6	−0.625481
$952$	0	0
$953$	−31699.8	−1.07750	−0.538751	−	0.842465i	$-0.681104\pi$
$953$	−31699.8	−1.07750	−0.538751	+	0.842465i	$0.681104\pi$
$954$	0	0
$955$	−21585.6	−0.731407
$956$	0	0
$957$	−17136.2	−0.578823
$958$	0	0
$959$	−7029.33	−0.236693
$960$	0	0
$961$	−28822.3	−0.967483
$962$	0	0
$963$	−9978.78	−0.333917
$964$	0	0
$965$	−17050.1	−0.568769
$966$	0	0
$967$	−9091.67	−0.302346	−0.151173	−	0.988507i	$-0.548305\pi$
$967$	−9091.67	−0.302346	−0.151173	+	0.988507i	$0.548305\pi$
$968$	0	0
$969$	6319.35	0.209501
$970$	0	0
$971$	−4741.90	−0.156720	−0.0783599	−	0.996925i	$-0.524968\pi$
$971$	−4741.90	−0.156720	−0.0783599	+	0.996925i	$0.524968\pi$
$972$	0	0
$973$	3235.15	0.106592
$974$	0	0
$975$	674.108	0.0221423
$976$	0	0
$977$	−45757.6	−1.49838	−0.749189	−	0.662356i	$-0.769557\pi$
$977$	−45757.6	−1.49838	−0.749189	+	0.662356i	$0.769557\pi$
$978$	0	0
$979$	24149.4	0.788375
$980$	0	0
$981$	16364.0	0.532582
$982$	0	0
$983$	1119.35	0.0363193	0.0181596	−	0.999835i	$-0.494219\pi$
$983$	1119.35	0.0363193	0.0181596	+	0.999835i	$0.494219\pi$
$984$	0	0
$985$	1750.75	0.0566330
$986$	0	0
$987$	11991.4	0.386717
$988$	0	0
$989$	10235.1	0.329079
$990$	0	0
$991$	13942.7	0.446927	0.223463	−	0.974712i	$-0.428264\pi$
$991$	13942.7	0.446927	0.223463	+	0.974712i	$0.428264\pi$
$992$	0	0
$993$	−5688.59	−0.181794
$994$	0	0
$995$	16419.8	0.523158
$996$	0	0
$997$	43183.9	1.37176	0.685881	−	0.727713i	$-0.259417\pi$
$997$	43183.9	1.37176	0.685881	+	0.727713i	$0.259417\pi$
$998$	0	0
$999$	−6218.88	−0.196954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.g.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.g.1.2	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -5.00000 q^{5} -8.87971 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -5.00000 q^{5} -8.87971 q^{7} +9.00000 q^{9} -31.0336 q^{11} +8.98811 q^{13} -15.0000 q^{15} +45.3787 q^{17} +46.4194 q^{19} -26.6391 q^{21} +23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +184.061 q^{29} -31.1244 q^{31} -93.1008 q^{33} +44.3986 q^{35} -230.329 q^{37} +26.9643 q^{39} -145.752 q^{41} +445.006 q^{43} -45.0000 q^{45} -450.141 q^{47} -264.151 q^{49} +136.136 q^{51} +45.3571 q^{53} +155.168 q^{55} +139.258 q^{57} -262.216 q^{59} -694.783 q^{61} -79.9174 q^{63} -44.9406 q^{65} -798.374 q^{67} +69.0000 q^{69} +400.029 q^{71} -652.639 q^{73} +75.0000 q^{75} +275.569 q^{77} +38.2625 q^{79} +81.0000 q^{81} +316.038 q^{83} -226.893 q^{85} +552.182 q^{87} -778.171 q^{89} -79.8119 q^{91} -93.3731 q^{93} -232.097 q^{95} -1588.04 q^{97} -279.302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15 q^{3} - 25 q^{5} - 5 q^{7} + 45 q^{9}+O(q^{10}) \) 5 * q + 15 * q^3 - 25 * q^5 - 5 * q^7 + 45 * q^9	\( 5 q + 15 q^{3} - 25 q^{5} - 5 q^{7} + 45 q^{9} - 18 q^{11} + 8 q^{13} - 75 q^{15} - 31 q^{17} - 84 q^{19} - 15 q^{21} + 115 q^{23} + 125 q^{25} + 135 q^{27} - 65 q^{29} - 81 q^{31} - 54 q^{33} + 25 q^{35} - 373 q^{37} + 24 q^{39} - 339 q^{41} - 796 q^{43} - 225 q^{45} - 382 q^{47} - 180 q^{49} - 93 q^{51} - 341 q^{53} + 90 q^{55} - 252 q^{57} + 457 q^{59} - 780 q^{61} - 45 q^{63} - 40 q^{65} - 299 q^{67} + 345 q^{69} + 11 q^{71} - 1092 q^{73} + 375 q^{75} - 468 q^{77} - 1290 q^{79} + 405 q^{81} - 659 q^{83} + 155 q^{85} - 195 q^{87} - 1574 q^{89} - 4068 q^{91} - 243 q^{93} + 420 q^{95} - 1462 q^{97} - 162 q^{99}+O(q^{100}) \) 5 * q + 15 * q^3 - 25 * q^5 - 5 * q^7 + 45 * q^9 - 18 * q^11 + 8 * q^13 - 75 * q^15 - 31 * q^17 - 84 * q^19 - 15 * q^21 + 115 * q^23 + 125 * q^25 + 135 * q^27 - 65 * q^29 - 81 * q^31 - 54 * q^33 + 25 * q^35 - 373 * q^37 + 24 * q^39 - 339 * q^41 - 796 * q^43 - 225 * q^45 - 382 * q^47 - 180 * q^49 - 93 * q^51 - 341 * q^53 + 90 * q^55 - 252 * q^57 + 457 * q^59 - 780 * q^61 - 45 * q^63 - 40 * q^65 - 299 * q^67 + 345 * q^69 + 11 * q^71 - 1092 * q^73 + 375 * q^75 - 468 * q^77 - 1290 * q^79 + 405 * q^81 - 659 * q^83 + 155 * q^85 - 195 * q^87 - 1574 * q^89 - 4068 * q^91 - 243 * q^93 + 420 * q^95 - 1462 * q^97 - 162 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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