LMFDB - Embedded newform 1725.4.a.bf.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1725,4,Mod(1,1725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1725.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1725 = 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1725.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,1,-33,39,0,-3,-19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$101.778294760$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 63 x^{9} + 64 x^{8} + 1341 x^{7} - 1301 x^{6} - 11425 x^{5} + 10222 x^{4} + \cdots + 37920 $ x^11 - x^10 - 63x^9 + 64x^8 + 1341x^7 - 1301x^6 - 11425x^5 + 10222x^4 + 38938x^3 - 34600x^2 - 42408*x + 37920
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			1.5
Root			$-1.43196$ of defining polynomial
Character	$\chi$	$=$	1725.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.43196 q^{2} -3.00000 q^{3} -5.94948 q^{4} +4.29589 q^{6} +18.6070 q^{7} +19.9752 q^{8} +9.00000 q^{9} -68.7311 q^{11} +17.8484 q^{12} -11.3366 q^{13} -26.6446 q^{14} +18.9921 q^{16} +105.452 q^{17} -12.8877 q^{18} -12.7765 q^{19} -55.8210 q^{21} +98.4205 q^{22} -23.0000 q^{23} -59.9255 q^{24} +16.2336 q^{26} -27.0000 q^{27} -110.702 q^{28} -61.3625 q^{29} -86.3869 q^{31} -186.997 q^{32} +206.193 q^{33} -151.003 q^{34} -53.5453 q^{36} -13.2096 q^{37} +18.2955 q^{38} +34.0098 q^{39} +261.964 q^{41} +79.9337 q^{42} +88.3435 q^{43} +408.914 q^{44} +32.9352 q^{46} -406.521 q^{47} -56.9764 q^{48} +3.22106 q^{49} -316.356 q^{51} +67.4469 q^{52} +288.459 q^{53} +38.6630 q^{54} +371.678 q^{56} +38.3295 q^{57} +87.8689 q^{58} +623.379 q^{59} -473.003 q^{61} +123.703 q^{62} +167.463 q^{63} +115.836 q^{64} -295.262 q^{66} +99.9886 q^{67} -627.384 q^{68} +69.0000 q^{69} -134.061 q^{71} +179.776 q^{72} -439.726 q^{73} +18.9157 q^{74} +76.0136 q^{76} -1278.88 q^{77} -48.7008 q^{78} +855.021 q^{79} +81.0000 q^{81} -375.123 q^{82} +835.479 q^{83} +332.106 q^{84} -126.505 q^{86} +184.088 q^{87} -1372.91 q^{88} +1405.99 q^{89} -210.940 q^{91} +136.838 q^{92} +259.161 q^{93} +582.124 q^{94} +560.992 q^{96} +408.145 q^{97} -4.61244 q^{98} -618.580 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + q^{2} - 33 q^{3} + 39 q^{4} - 3 q^{6} - 19 q^{7} - 18 q^{8} + 99 q^{9} - 41 q^{11} - 117 q^{12} - 57 q^{13} + 8 q^{14} + 227 q^{16} - 66 q^{17} + 9 q^{18} - 215 q^{19} + 57 q^{21} + 342 q^{22} - 253 q^{23}+ \cdots - 369 q^{99}+O(q^{100}) $ 11 * q + q^2 - 33 * q^3 + 39 * q^4 - 3 * q^6 - 19 * q^7 - 18 * q^8 + 99 * q^9 - 41 * q^11 - 117 * q^12 - 57 * q^13 + 8 * q^14 + 227 * q^16 - 66 * q^17 + 9 * q^18 - 215 * q^19 + 57 * q^21 + 342 * q^22 - 253 * q^23 + 54 * q^24 + 364 * q^26 - 297 * q^27 - 657 * q^28 + 513 * q^29 + 340 * q^31 - 263 * q^32 + 123 * q^33 - 125 * q^34 + 351 * q^36 - 20 * q^37 - 124 * q^38 + 171 * q^39 + 515 * q^41 - 24 * q^42 - 1401 * q^43 - 879 * q^44 - 23 * q^46 + 152 * q^47 - 681 * q^48 + 1108 * q^49 + 198 * q^51 - 697 * q^52 - 794 * q^53 - 27 * q^54 + 1113 * q^56 + 645 * q^57 - 3058 * q^58 + 1332 * q^59 + 148 * q^61 - 984 * q^62 - 171 * q^63 + 1422 * q^64 - 1026 * q^66 - 56 * q^67 - 2277 * q^68 + 759 * q^69 - 198 * q^71 - 162 * q^72 + 205 * q^73 + 2901 * q^74 - 4522 * q^76 + 851 * q^77 - 1092 * q^78 - 645 * q^79 + 891 * q^81 - 37 * q^82 - 2591 * q^83 + 1971 * q^84 - 567 * q^86 - 1539 * q^87 + 2247 * q^88 + 4130 * q^89 - 1623 * q^91 - 897 * q^92 - 1020 * q^93 + 49 * q^94 + 789 * q^96 - 2348 * q^97 - 2804 * q^98 - 369 * q^99

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 $

$2$	−1.43196	−0.506276	−0.253138	−	0.967430i	$-0.581463\pi$
$2$	−1.43196	−0.506276	−0.253138	+	0.967430i	$0.581463\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−5.94948	−0.743685
$5$	0	0
$5$	0	0
$6$	4.29589	0.292298
$7$	18.6070	1.00468	0.502342	−	0.864669i	$-0.332472\pi$
$7$	18.6070	1.00468	0.502342	+	0.864669i	$0.332472\pi$
$8$	19.9752	0.882785
$9$	9.00000	0.333333
$10$	0	0
$11$	−68.7311	−1.88393	−0.941964	−	0.335713i	$-0.891023\pi$
$11$	−68.7311	−1.88393	−0.941964	+	0.335713i	$0.891023\pi$
$12$	17.8484	0.429367
$13$	−11.3366	−0.241862	−0.120931	−	0.992661i	$-0.538588\pi$
$13$	−11.3366	−0.241862	−0.120931	+	0.992661i	$0.538588\pi$
$14$	−26.6446	−0.508647
$15$	0	0
$16$	18.9921	0.296752
$17$	105.452	1.50446	0.752231	−	0.658900i	$-0.228978\pi$
$17$	105.452	1.50446	0.752231	+	0.658900i	$0.228978\pi$
$18$	−12.8877	−0.168759
$19$	−12.7765	−0.154270	−0.0771351	−	0.997021i	$-0.524577\pi$
$19$	−12.7765	−0.154270	−0.0771351	+	0.997021i	$0.524577\pi$
$20$	0	0
$21$	−55.8210	−0.580055
$22$	98.4205	0.953787
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−59.9255	−0.509676
$25$	0	0
$26$	16.2336	0.122449
$27$	−27.0000	−0.192450
$28$	−110.702	−0.747169
$29$	−61.3625	−0.392922	−0.196461	−	0.980512i	$-0.562945\pi$
$29$	−61.3625	−0.392922	−0.196461	+	0.980512i	$0.562945\pi$
$30$	0	0
$31$	−86.3869	−0.500501	−0.250251	−	0.968181i	$-0.580513\pi$
$31$	−86.3869	−0.500501	−0.250251	+	0.968181i	$0.580513\pi$
$32$	−186.997	−1.03302
$33$	206.193	1.08769
$34$	−151.003	−0.761673
$35$	0	0
$36$	−53.5453	−0.247895
$37$	−13.2096	−0.0586933	−0.0293466	−	0.999569i	$-0.509343\pi$
$37$	−13.2096	−0.0586933	−0.0293466	+	0.999569i	$0.509343\pi$
$38$	18.2955	0.0781032
$39$	34.0098	0.139639
$40$	0	0
$41$	261.964	0.997850	0.498925	−	0.866645i	$-0.333728\pi$
$41$	261.964	0.997850	0.498925	+	0.866645i	$0.333728\pi$
$42$	79.9337	0.293668
$43$	88.3435	0.313308	0.156654	−	0.987654i	$-0.449929\pi$
$43$	88.3435	0.313308	0.156654	+	0.987654i	$0.449929\pi$
$44$	408.914	1.40105
$45$	0	0
$46$	32.9352	0.105566
$47$	−406.521	−1.26164	−0.630821	−	0.775928i	$-0.717282\pi$
$47$	−406.521	−1.26164	−0.630821	+	0.775928i	$0.717282\pi$
$48$	−56.9764	−0.171330
$49$	3.22106	0.00939083
$50$	0	0
$51$	−316.356	−0.868601
$52$	67.4469	0.179869
$53$	288.459	0.747603	0.373801	−	0.927509i	$-0.378054\pi$
$53$	288.459	0.747603	0.373801	+	0.927509i	$0.378054\pi$
$54$	38.6630	0.0974328
$55$	0	0
$56$	371.678	0.886921
$57$	38.3295	0.0890679
$58$	87.8689	0.198927
$59$	623.379	1.37554	0.687771	−	0.725927i	$-0.258589\pi$
$59$	623.379	1.37554	0.687771	+	0.725927i	$0.258589\pi$
$60$	0	0
$61$	−473.003	−0.992816	−0.496408	−	0.868089i	$-0.665348\pi$
$61$	−473.003	−0.992816	−0.496408	+	0.868089i	$0.665348\pi$
$62$	123.703	0.253392
$63$	167.463	0.334895
$64$	115.836	0.226243
$65$	0	0
$66$	−295.262	−0.550669
$67$	99.9886	0.182322	0.0911608	−	0.995836i	$-0.470942\pi$
$67$	99.9886	0.182322	0.0911608	+	0.995836i	$0.470942\pi$
$68$	−627.384	−1.11885
$69$	69.0000	0.120386
$70$	0	0
$71$	−134.061	−0.224086	−0.112043	−	0.993703i	$-0.535739\pi$
$71$	−134.061	−0.224086	−0.112043	+	0.993703i	$0.535739\pi$
$72$	179.776	0.294262
$73$	−439.726	−0.705013	−0.352507	−	0.935809i	$-0.614671\pi$
$73$	−439.726	−0.705013	−0.352507	+	0.935809i	$0.614671\pi$
$74$	18.9157	0.0297150
$75$	0	0
$76$	76.0136	0.114728
$77$	−1278.88	−1.89275
$78$	−48.7008	−0.0706959
$79$	855.021	1.21769	0.608844	−	0.793290i	$-0.291634\pi$
$79$	855.021	1.21769	0.608844	+	0.793290i	$0.291634\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−375.123	−0.505187
$83$	835.479	1.10489	0.552444	−	0.833550i	$-0.313695\pi$
$83$	835.479	1.10489	0.552444	+	0.833550i	$0.313695\pi$
$84$	332.106	0.431378
$85$	0	0
$86$	−126.505	−0.158620
$87$	184.088	0.226853
$88$	−1372.91	−1.66310
$89$	1405.99	1.67455	0.837274	−	0.546783i	$-0.184148\pi$
$89$	1405.99	1.67455	0.837274	+	0.546783i	$0.184148\pi$
$90$	0	0
$91$	−210.940	−0.242995
$92$	136.838	0.155069
$93$	259.161	0.288965
$94$	582.124	0.638739
$95$	0	0
$96$	560.992	0.596417
$97$	408.145	0.427226	0.213613	−	0.976918i	$-0.431477\pi$
$97$	408.145	0.427226	0.213613	+	0.976918i	$0.431477\pi$
$98$	−4.61244	−0.00475435
$99$	−618.580	−0.627976
$100$	0	0
$101$	−326.973	−0.322129	−0.161064	−	0.986944i	$-0.551493\pi$
$101$	−326.973	−0.322129	−0.161064	+	0.986944i	$0.551493\pi$
$102$	453.010	0.439752
$103$	1009.53	0.965743	0.482872	−	0.875691i	$-0.339594\pi$
$103$	1009.53	0.965743	0.482872	+	0.875691i	$0.339594\pi$
$104$	−226.450	−0.213512
$105$	0	0
$106$	−413.063	−0.378493
$107$	−1235.84	−1.11657	−0.558285	−	0.829649i	$-0.688540\pi$
$107$	−1235.84	−1.11657	−0.558285	+	0.829649i	$0.688540\pi$
$108$	160.636	0.143122
$109$	450.807	0.396142	0.198071	−	0.980188i	$-0.436532\pi$
$109$	450.807	0.396142	0.198071	+	0.980188i	$0.436532\pi$
$110$	0	0
$111$	39.6289	0.0338866
$112$	353.387	0.298142
$113$	1522.28	1.26729	0.633644	−	0.773625i	$-0.281558\pi$
$113$	1522.28	1.26729	0.633644	+	0.773625i	$0.281558\pi$
$114$	−54.8865	−0.0450929
$115$	0	0
$116$	365.075	0.292210
$117$	−102.029	−0.0806207
$118$	−892.656	−0.696404
$119$	1962.15	1.51151
$120$	0	0
$121$	3392.97	2.54919
$122$	677.323	0.502639
$123$	−785.891	−0.576109
$124$	513.957	0.372215
$125$	0	0
$126$	−239.801	−0.169549
$127$	−855.601	−0.597813	−0.298907	−	0.954282i	$-0.596622\pi$
$127$	−855.601	−0.597813	−0.298907	+	0.954282i	$0.596622\pi$
$128$	1330.10	0.918483
$129$	−265.031	−0.180889
$130$	0	0
$131$	−543.334	−0.362377	−0.181188	−	0.983448i	$-0.557994\pi$
$131$	−543.334	−0.362377	−0.181188	+	0.983448i	$0.557994\pi$
$132$	−1226.74	−0.808896
$133$	−237.733	−0.154993
$134$	−143.180	−0.0923050
$135$	0	0
$136$	2106.42	1.32812
$137$	−270.992	−0.168996	−0.0844978	−	0.996424i	$-0.526929\pi$
$137$	−270.992	−0.168996	−0.0844978	+	0.996424i	$0.526929\pi$
$138$	−98.8055	−0.0609484
$139$	−1413.32	−0.862418	−0.431209	−	0.902252i	$-0.641913\pi$
$139$	−1413.32	−0.862418	−0.431209	+	0.902252i	$0.641913\pi$
$140$	0	0
$141$	1219.56	0.728410
$142$	191.970	0.113449
$143$	779.177	0.455651
$144$	170.929	0.0989174
$145$	0	0
$146$	629.671	0.356931
$147$	−9.66317	−0.00542180
$148$	78.5905	0.0436493
$149$	1722.96	0.947316	0.473658	−	0.880709i	$-0.342933\pi$
$149$	1722.96	0.947316	0.473658	+	0.880709i	$0.342933\pi$
$150$	0	0
$151$	−911.514	−0.491245	−0.245622	−	0.969366i	$-0.578992\pi$
$151$	−911.514	−0.491245	−0.245622	+	0.969366i	$0.578992\pi$
$152$	−255.213	−0.136187
$153$	949.067	0.501487
$154$	1831.31	0.958255
$155$	0	0
$156$	−202.341	−0.103848
$157$	735.909	0.374089	0.187044	−	0.982351i	$-0.440109\pi$
$157$	735.909	0.374089	0.187044	+	0.982351i	$0.440109\pi$
$158$	−1224.36	−0.616486
$159$	−865.378	−0.431629
$160$	0	0
$161$	−427.961	−0.209491
$162$	−115.989	−0.0562529
$163$	−3637.20	−1.74778	−0.873888	−	0.486127i	$-0.838409\pi$
$163$	−3637.20	−1.74778	−0.873888	+	0.486127i	$0.838409\pi$
$164$	−1558.55	−0.742086
$165$	0	0
$166$	−1196.38	−0.559378
$167$	−986.252	−0.456997	−0.228498	−	0.973544i	$-0.573382\pi$
$167$	−986.252	−0.456997	−0.228498	+	0.973544i	$0.573382\pi$
$168$	−1115.03	−0.512064
$169$	−2068.48	−0.941503
$170$	0	0
$171$	−114.989	−0.0514234
$172$	−525.598	−0.233003
$173$	−1945.48	−0.854983	−0.427491	−	0.904019i	$-0.640603\pi$
$173$	−1945.48	−0.854983	−0.427491	+	0.904019i	$0.640603\pi$
$174$	−263.607	−0.114850
$175$	0	0
$176$	−1305.35	−0.559060
$177$	−1870.14	−0.794170
$178$	−2013.33	−0.847783
$179$	−2914.59	−1.21702	−0.608510	−	0.793546i	$-0.708233\pi$
$179$	−2914.59	−1.21702	−0.608510	+	0.793546i	$0.708233\pi$
$180$	0	0
$181$	1969.82	0.808926	0.404463	−	0.914554i	$-0.367458\pi$
$181$	1969.82	0.808926	0.404463	+	0.914554i	$0.367458\pi$
$182$	302.059	0.123023
$183$	1419.01	0.573203
$184$	−459.428	−0.184073
$185$	0	0
$186$	−371.109	−0.146296
$187$	−7247.83	−2.83430
$188$	2418.59	0.938265
$189$	−502.389	−0.193352
$190$	0	0
$191$	−4820.60	−1.82621	−0.913106	−	0.407722i	$-0.866323\pi$
$191$	−4820.60	−1.82621	−0.913106	+	0.407722i	$0.866323\pi$
$192$	−347.509	−0.130621
$193$	2248.72	0.838686	0.419343	−	0.907828i	$-0.362261\pi$
$193$	2248.72	0.838686	0.419343	+	0.907828i	$0.362261\pi$
$194$	−584.450	−0.216294
$195$	0	0
$196$	−19.1636	−0.00698382
$197$	−946.861	−0.342442	−0.171221	−	0.985233i	$-0.554771\pi$
$197$	−946.861	−0.342442	−0.171221	+	0.985233i	$0.554771\pi$
$198$	885.785	0.317929
$199$	672.254	0.239471	0.119736	−	0.992806i	$-0.461795\pi$
$199$	672.254	0.239471	0.119736	+	0.992806i	$0.461795\pi$
$200$	0	0
$201$	−299.966	−0.105263
$202$	468.214	0.163086
$203$	−1141.77	−0.394762
$204$	1882.15	0.645966
$205$	0	0
$206$	−1445.60	−0.488932
$207$	−207.000	−0.0695048
$208$	−215.306	−0.0717731
$209$	878.144	0.290634
$210$	0	0
$211$	−1101.67	−0.359440	−0.179720	−	0.983718i	$-0.557519\pi$
$211$	−1101.67	−0.359440	−0.179720	+	0.983718i	$0.557519\pi$
$212$	−1716.18	−0.555981
$213$	402.183	0.129376
$214$	1769.68	0.565292
$215$	0	0
$216$	−539.329	−0.169892
$217$	−1607.40	−0.502846
$218$	−645.539	−0.200557
$219$	1319.18	0.407040
$220$	0	0
$221$	−1195.47	−0.363872
$222$	−56.7472	−0.0171560
$223$	−5675.68	−1.70436	−0.852179	−	0.523250i	$-0.824719\pi$
$223$	−5675.68	−1.70436	−0.852179	+	0.523250i	$0.824719\pi$
$224$	−3479.46	−1.03786
$225$	0	0
$226$	−2179.84	−0.641597
$227$	−3247.62	−0.949570	−0.474785	−	0.880102i	$-0.657474\pi$
$227$	−3247.62	−0.949570	−0.474785	+	0.880102i	$0.657474\pi$
$228$	−228.041	−0.0662385
$229$	1900.46	0.548410	0.274205	−	0.961671i	$-0.411585\pi$
$229$	1900.46	0.548410	0.274205	+	0.961671i	$0.411585\pi$
$230$	0	0
$231$	3836.64	1.09278
$232$	−1225.73	−0.346866
$233$	−3634.00	−1.02176	−0.510882	−	0.859651i	$-0.670681\pi$
$233$	−3634.00	−1.02176	−0.510882	+	0.859651i	$0.670681\pi$
$234$	146.102	0.0408163
$235$	0	0
$236$	−3708.78	−1.02297
$237$	−2565.06	−0.703033
$238$	−2809.72	−0.765241
$239$	2673.25	0.723507	0.361754	−	0.932274i	$-0.382178\pi$
$239$	2673.25	0.723507	0.361754	+	0.932274i	$0.382178\pi$
$240$	0	0
$241$	−6547.13	−1.74995	−0.874974	−	0.484170i	$-0.839122\pi$
$241$	−6547.13	−1.74995	−0.874974	+	0.484170i	$0.839122\pi$
$242$	−4858.61	−1.29059
$243$	−243.000	−0.0641500
$244$	2814.12	0.738343
$245$	0	0
$246$	1125.37	0.291670
$247$	144.842	0.0373121
$248$	−1725.59	−0.441835
$249$	−2506.44	−0.637907
$250$	0	0
$251$	1024.23	0.257564	0.128782	−	0.991673i	$-0.458893\pi$
$251$	1024.23	0.257564	0.128782	+	0.991673i	$0.458893\pi$
$252$	−996.319	−0.249056
$253$	1580.82	0.392826
$254$	1225.19	0.302658
$255$	0	0
$256$	−2831.35	−0.691248
$257$	−1513.16	−0.367269	−0.183635	−	0.982995i	$-0.558786\pi$
$257$	−1513.16	−0.367269	−0.183635	+	0.982995i	$0.558786\pi$
$258$	379.514	0.0915795
$259$	−245.792	−0.0589682
$260$	0	0
$261$	−552.263	−0.130974
$262$	778.035	0.183462
$263$	1096.56	0.257097	0.128549	−	0.991703i	$-0.458968\pi$
$263$	1096.56	0.257097	0.128549	+	0.991703i	$0.458968\pi$
$264$	4118.74	0.960194
$265$	0	0
$266$	340.425	0.0784691
$267$	−4217.98	−0.966801
$268$	−594.880	−0.135590
$269$	−731.322	−0.165760	−0.0828801	−	0.996560i	$-0.526412\pi$
$269$	−731.322	−0.165760	−0.0828801	+	0.996560i	$0.526412\pi$
$270$	0	0
$271$	−4623.37	−1.03634	−0.518172	−	0.855276i	$-0.673387\pi$
$271$	−4623.37	−1.03634	−0.518172	+	0.855276i	$0.673387\pi$
$272$	2002.76	0.446452
$273$	632.821	0.140293
$274$	388.051	0.0855584
$275$	0	0
$276$	−410.514	−0.0895291
$277$	−4836.27	−1.04904	−0.524519	−	0.851399i	$-0.675755\pi$
$277$	−4836.27	−1.04904	−0.524519	+	0.851399i	$0.675755\pi$
$278$	2023.82	0.436622
$279$	−777.482	−0.166834
$280$	0	0
$281$	−3034.06	−0.644117	−0.322059	−	0.946720i	$-0.604375\pi$
$281$	−3034.06	−0.644117	−0.322059	+	0.946720i	$0.604375\pi$
$282$	−1746.37	−0.368776
$283$	474.672	0.0997043	0.0498522	−	0.998757i	$-0.484125\pi$
$283$	474.672	0.0997043	0.0498522	+	0.998757i	$0.484125\pi$
$284$	797.593	0.166649
$285$	0	0
$286$	−1115.75	−0.230685
$287$	4874.36	1.00252
$288$	−1682.98	−0.344341
$289$	6207.11	1.26341
$290$	0	0
$291$	−1224.44	−0.246659
$292$	2616.14	0.524308
$293$	−2821.19	−0.562511	−0.281255	−	0.959633i	$-0.590751\pi$
$293$	−2821.19	−0.562511	−0.281255	+	0.959633i	$0.590751\pi$
$294$	13.8373	0.00274493
$295$	0	0
$296$	−263.865	−0.0518136
$297$	1855.74	0.362562
$298$	−2467.21	−0.479603
$299$	260.742	0.0504317
$300$	0	0
$301$	1643.81	0.314776
$302$	1305.26	0.248705
$303$	980.919	0.185981
$304$	−242.653	−0.0457800
$305$	0	0
$306$	−1359.03	−0.253891
$307$	1325.64	0.246445	0.123222	−	0.992379i	$-0.460677\pi$
$307$	1325.64	0.246445	0.123222	+	0.992379i	$0.460677\pi$
$308$	7608.68	1.40761
$309$	−3028.58	−0.557572
$310$	0	0
$311$	8916.76	1.62580	0.812899	−	0.582405i	$-0.197888\pi$
$311$	8916.76	1.62580	0.812899	+	0.582405i	$0.197888\pi$
$312$	679.351	0.123271
$313$	7712.14	1.39270	0.696352	−	0.717701i	$-0.254805\pi$
$313$	7712.14	1.39270	0.696352	+	0.717701i	$0.254805\pi$
$314$	−1053.80	−0.189392
$315$	0	0
$316$	−5086.93	−0.905576
$317$	8284.00	1.46775	0.733874	−	0.679286i	$-0.237710\pi$
$317$	8284.00	1.46775	0.733874	+	0.679286i	$0.237710\pi$
$318$	1239.19	0.218523
$319$	4217.51	0.740237
$320$	0	0
$321$	3707.51	0.644652
$322$	612.825	0.106060
$323$	−1347.31	−0.232094
$324$	−481.908	−0.0826317
$325$	0	0
$326$	5208.34	0.884857
$327$	−1352.42	−0.228713
$328$	5232.77	0.880888
$329$	−7564.15	−1.26755
$330$	0	0
$331$	642.736	0.106731	0.0533655	−	0.998575i	$-0.483005\pi$
$331$	642.736	0.106731	0.0533655	+	0.998575i	$0.483005\pi$
$332$	−4970.66	−0.821689
$333$	−118.887	−0.0195644
$334$	1412.28	0.231366
$335$	0	0
$336$	−1060.16	−0.172133
$337$	−977.788	−0.158052	−0.0790260	−	0.996873i	$-0.525181\pi$
$337$	−977.788	−0.158052	−0.0790260	+	0.996873i	$0.525181\pi$
$338$	2961.99	0.476660
$339$	−4566.83	−0.731669
$340$	0	0
$341$	5937.47	0.942909
$342$	164.660	0.0260344
$343$	−6322.27	−0.995250
$344$	1764.67	0.276584
$345$	0	0
$346$	2785.85	0.432857
$347$	−12315.0	−1.90520	−0.952600	−	0.304224i	$-0.901603\pi$
$347$	−12315.0	−1.90520	−0.952600	+	0.304224i	$0.901603\pi$
$348$	−1095.22	−0.168708
$349$	−6810.69	−1.04461	−0.522304	−	0.852760i	$-0.674927\pi$
$349$	−6810.69	−1.04461	−0.522304	+	0.852760i	$0.674927\pi$
$350$	0	0
$351$	306.088	0.0465464
$352$	12852.5	1.94614
$353$	−6361.18	−0.959125	−0.479563	−	0.877508i	$-0.659205\pi$
$353$	−6361.18	−0.959125	−0.479563	+	0.877508i	$0.659205\pi$
$354$	2677.97	0.402069
$355$	0	0
$356$	−8364.92	−1.24534
$357$	−5886.44	−0.872670
$358$	4173.59	0.616148
$359$	10335.4	1.51944	0.759722	−	0.650248i	$-0.225335\pi$
$359$	10335.4	1.51944	0.759722	+	0.650248i	$0.225335\pi$
$360$	0	0
$361$	−6695.76	−0.976201
$362$	−2820.71	−0.409540
$363$	−10178.9	−1.47177
$364$	1254.99	0.180712
$365$	0	0
$366$	−2031.97	−0.290199
$367$	−1041.57	−0.148146	−0.0740729	−	0.997253i	$-0.523600\pi$
$367$	−1041.57	−0.148146	−0.0740729	+	0.997253i	$0.523600\pi$
$368$	−436.819	−0.0618771
$369$	2357.67	0.332617
$370$	0	0
$371$	5367.37	0.751105
$372$	−1541.87	−0.214899
$373$	−7841.80	−1.08856	−0.544280	−	0.838903i	$-0.683197\pi$
$373$	−7841.80	−1.08856	−0.544280	+	0.838903i	$0.683197\pi$
$374$	10378.6	1.43494
$375$	0	0
$376$	−8120.32	−1.11376
$377$	695.642	0.0950329
$378$	719.404	0.0978892
$379$	−5607.04	−0.759933	−0.379966	−	0.925000i	$-0.624064\pi$
$379$	−5607.04	−0.759933	−0.379966	+	0.925000i	$0.624064\pi$
$380$	0	0
$381$	2566.80	0.345148
$382$	6902.93	0.924567
$383$	8351.48	1.11421	0.557103	−	0.830444i	$-0.311913\pi$
$383$	8351.48	1.11421	0.557103	+	0.830444i	$0.311913\pi$
$384$	−3990.31	−0.530286
$385$	0	0
$386$	−3220.08	−0.424606
$387$	795.092	0.104436
$388$	−2428.25	−0.317721
$389$	4957.48	0.646155	0.323078	−	0.946373i	$-0.395283\pi$
$389$	4957.48	0.646155	0.323078	+	0.946373i	$0.395283\pi$
$390$	0	0
$391$	−2425.39	−0.313702
$392$	64.3411	0.00829009
$393$	1630.00	0.209218
$394$	1355.87	0.173370
$395$	0	0
$396$	3680.23	0.467016
$397$	480.923	0.0607981	0.0303991	−	0.999538i	$-0.490322\pi$
$397$	480.923	0.0607981	0.0303991	+	0.999538i	$0.490322\pi$
$398$	−962.643	−0.121239
$399$	713.198	0.0894852
$400$	0	0
$401$	−10821.6	−1.34764	−0.673822	−	0.738894i	$-0.735349\pi$
$401$	−10821.6	−1.34764	−0.673822	+	0.738894i	$0.735349\pi$
$402$	429.540	0.0532923
$403$	979.334	0.121052
$404$	1945.32	0.239562
$405$	0	0
$406$	1634.98	0.199859
$407$	907.913	0.110574
$408$	−6319.26	−0.766789
$409$	−12067.4	−1.45891	−0.729457	−	0.684026i	$-0.760227\pi$
$409$	−12067.4	−1.45891	−0.729457	+	0.684026i	$0.760227\pi$
$410$	0	0
$411$	812.976	0.0975697
$412$	−6006.15	−0.718209
$413$	11599.2	1.38199
$414$	296.417	0.0351886
$415$	0	0
$416$	2119.91	0.249849
$417$	4239.96	0.497918
$418$	−1257.47	−0.147141
$419$	11540.0	1.34551	0.672754	−	0.739866i	$-0.265111\pi$
$419$	11540.0	1.34551	0.672754	+	0.739866i	$0.265111\pi$
$420$	0	0
$421$	−12174.9	−1.40943	−0.704715	−	0.709490i	$-0.748925\pi$
$421$	−12174.9	−1.40943	−0.704715	+	0.709490i	$0.748925\pi$
$422$	1577.55	0.181976
$423$	−3658.69	−0.420548
$424$	5762.02	0.659973
$425$	0	0
$426$	−575.911	−0.0655000
$427$	−8801.17	−0.997467
$428$	7352.59	0.830376
$429$	−2337.53	−0.263070
$430$	0	0
$431$	−3476.64	−0.388547	−0.194274	−	0.980947i	$-0.562235\pi$
$431$	−3476.64	−0.388547	−0.194274	+	0.980947i	$0.562235\pi$
$432$	−512.788	−0.0571100
$433$	−8653.90	−0.960461	−0.480231	−	0.877142i	$-0.659447\pi$
$433$	−8653.90	−0.960461	−0.480231	+	0.877142i	$0.659447\pi$
$434$	2301.74	0.254579
$435$	0	0
$436$	−2682.07	−0.294605
$437$	293.860	0.0321676
$438$	−1889.01	−0.206074
$439$	−10136.3	−1.10201	−0.551004	−	0.834503i	$-0.685755\pi$
$439$	−10136.3	−1.10201	−0.551004	+	0.834503i	$0.685755\pi$
$440$	0	0
$441$	28.9895	0.00313028
$442$	1711.86	0.184220
$443$	15996.8	1.71564	0.857821	−	0.513949i	$-0.171818\pi$
$443$	15996.8	1.71564	0.857821	+	0.513949i	$0.171818\pi$
$444$	−235.771	−0.0252009
$445$	0	0
$446$	8127.37	0.862875
$447$	−5168.87	−0.546933
$448$	2155.37	0.227302
$449$	2414.04	0.253732	0.126866	−	0.991920i	$-0.459508\pi$
$449$	2414.04	0.253732	0.126866	+	0.991920i	$0.459508\pi$
$450$	0	0
$451$	−18005.1	−1.87988
$452$	−9056.74	−0.942463
$453$	2734.54	0.283620
$454$	4650.48	0.480744
$455$	0	0
$456$	765.638	0.0786278
$457$	−6917.84	−0.708102	−0.354051	−	0.935226i	$-0.615196\pi$
$457$	−6917.84	−0.708102	−0.354051	+	0.935226i	$0.615196\pi$
$458$	−2721.39	−0.277647
$459$	−2847.20	−0.289534
$460$	0	0
$461$	18529.2	1.87200	0.935998	−	0.352004i	$-0.114500\pi$
$461$	18529.2	1.87200	0.935998	+	0.352004i	$0.114500\pi$
$462$	−5493.94	−0.553249
$463$	−8678.49	−0.871110	−0.435555	−	0.900162i	$-0.643448\pi$
$463$	−8678.49	−0.871110	−0.435555	+	0.900162i	$0.643448\pi$
$464$	−1165.41	−0.116600
$465$	0	0
$466$	5203.75	0.517294
$467$	−1360.77	−0.134838	−0.0674188	−	0.997725i	$-0.521476\pi$
$467$	−1360.77	−0.134838	−0.0674188	+	0.997725i	$0.521476\pi$
$468$	607.022	0.0599564
$469$	1860.49	0.183176
$470$	0	0
$471$	−2207.73	−0.215980
$472$	12452.1	1.21431
$473$	−6071.95	−0.590251
$474$	3673.08	0.355928
$475$	0	0
$476$	−11673.7	−1.12409
$477$	2596.13	0.249201
$478$	−3828.00	−0.366294
$479$	−3486.24	−0.332548	−0.166274	−	0.986080i	$-0.553174\pi$
$479$	−3486.24	−0.332548	−0.166274	+	0.986080i	$0.553174\pi$
$480$	0	0
$481$	149.752	0.0141957
$482$	9375.25	0.885956
$483$	1283.88	0.120950
$484$	−20186.4	−1.89579
$485$	0	0
$486$	347.967	0.0324776
$487$	−9282.68	−0.863734	−0.431867	−	0.901937i	$-0.642145\pi$
$487$	−9282.68	−0.863734	−0.431867	+	0.901937i	$0.642145\pi$
$488$	−9448.30	−0.876444
$489$	10911.6	1.00908
$490$	0	0
$491$	−9680.79	−0.889792	−0.444896	−	0.895582i	$-0.646759\pi$
$491$	−9680.79	−0.889792	−0.444896	+	0.895582i	$0.646759\pi$
$492$	4675.64	0.428444
$493$	−6470.79	−0.591136
$494$	−207.409	−0.0188902
$495$	0	0
$496$	−1640.67	−0.148525
$497$	−2494.47	−0.225136
$498$	3589.13	0.322957
$499$	14191.6	1.27315	0.636577	−	0.771213i	$-0.280350\pi$
$499$	14191.6	1.27315	0.636577	+	0.771213i	$0.280350\pi$
$500$	0	0
$501$	2958.76	0.263847
$502$	−1466.66	−0.130399
$503$	9532.07	0.844959	0.422479	−	0.906373i	$-0.361160\pi$
$503$	9532.07	0.844959	0.422479	+	0.906373i	$0.361160\pi$
$504$	3345.10	0.295640
$505$	0	0
$506$	−2263.67	−0.198878
$507$	6205.44	0.543577
$508$	5090.38	0.444585
$509$	−14514.0	−1.26390	−0.631949	−	0.775010i	$-0.717745\pi$
$509$	−14514.0	−1.26390	−0.631949	+	0.775010i	$0.717745\pi$
$510$	0	0
$511$	−8181.98	−0.708316
$512$	−6586.44	−0.568520
$513$	344.966	0.0296893
$514$	2166.79	0.185939
$515$	0	0
$516$	1576.79	0.134524
$517$	27940.7	2.37685
$518$	351.965	0.0298542
$519$	5836.43	0.493624
$520$	0	0
$521$	−176.251	−0.0148209	−0.00741045	−	0.999973i	$-0.502359\pi$
$521$	−176.251	−0.0148209	−0.00741045	+	0.999973i	$0.502359\pi$
$522$	790.820	0.0663089
$523$	−20899.1	−1.74733	−0.873663	−	0.486531i	$-0.838262\pi$
$523$	−20899.1	−1.74733	−0.873663	+	0.486531i	$0.838262\pi$
$524$	3232.56	0.269494
$525$	0	0
$526$	−1570.23	−0.130162
$527$	−9109.66	−0.752985
$528$	3916.05	0.322773
$529$	529.000	0.0434783
$530$	0	0
$531$	5610.41	0.458514
$532$	1414.39	0.115266
$533$	−2969.78	−0.241342
$534$	6039.99	0.489468
$535$	0	0
$536$	1997.29	0.160951
$537$	8743.77	0.702647
$538$	1047.23	0.0839204
$539$	−221.387	−0.0176917
$540$	0	0
$541$	479.151	0.0380782	0.0190391	−	0.999819i	$-0.493939\pi$
$541$	479.151	0.0380782	0.0190391	+	0.999819i	$0.493939\pi$
$542$	6620.49	0.524676
$543$	−5909.46	−0.467034
$544$	−19719.2	−1.55414
$545$	0	0
$546$	−906.177	−0.0710271
$547$	1390.37	0.108680	0.0543400	−	0.998522i	$-0.482695\pi$
$547$	1390.37	0.108680	0.0543400	+	0.998522i	$0.482695\pi$
$548$	1612.26	0.125679
$549$	−4257.02	−0.330939
$550$	0	0
$551$	783.999	0.0606161
$552$	1378.29	0.106275
$553$	15909.4	1.22339
$554$	6925.37	0.531102
$555$	0	0
$556$	8408.51	0.641368
$557$	17367.7	1.32117	0.660587	−	0.750749i	$-0.270307\pi$
$557$	17367.7	1.32117	0.660587	+	0.750749i	$0.270307\pi$
$558$	1113.33	0.0844639
$559$	−1001.52	−0.0757774
$560$	0	0
$561$	21743.5	1.63638
$562$	4344.67	0.326101
$563$	−8121.32	−0.607945	−0.303972	−	0.952681i	$-0.598313\pi$
$563$	−8121.32	−0.607945	−0.303972	+	0.952681i	$0.598313\pi$
$564$	−7255.77	−0.541707
$565$	0	0
$566$	−679.713	−0.0504779
$567$	1507.17	0.111632
$568$	−2677.89	−0.197820
$569$	14677.2	1.08137	0.540686	−	0.841224i	$-0.318165\pi$
$569$	14677.2	1.08137	0.540686	+	0.841224i	$0.318165\pi$
$570$	0	0
$571$	17082.8	1.25200	0.626000	−	0.779823i	$-0.284691\pi$
$571$	17082.8	1.25200	0.626000	+	0.779823i	$0.284691\pi$
$572$	−4635.70	−0.338861
$573$	14461.8	1.05436
$574$	−6979.91	−0.507554
$575$	0	0
$576$	1042.53	0.0754142
$577$	15897.2	1.14698	0.573492	−	0.819211i	$-0.305588\pi$
$577$	15897.2	1.14698	0.573492	+	0.819211i	$0.305588\pi$
$578$	−8888.36	−0.639632
$579$	−6746.16	−0.484215
$580$	0	0
$581$	15545.8	1.11006
$582$	1753.35	0.124877
$583$	−19826.1	−1.40843
$584$	−8783.58	−0.622375
$585$	0	0
$586$	4039.84	0.284786
$587$	9471.02	0.665947	0.332974	−	0.942936i	$-0.391948\pi$
$587$	9471.02	0.665947	0.332974	+	0.942936i	$0.391948\pi$
$588$	57.4908	0.00403211
$589$	1103.72	0.0772124
$590$	0	0
$591$	2840.58	0.197709
$592$	−250.879	−0.0174174
$593$	−11374.1	−0.787656	−0.393828	−	0.919184i	$-0.628849\pi$
$593$	−11374.1	−0.787656	−0.393828	+	0.919184i	$0.628849\pi$
$594$	−2657.35	−0.183556
$595$	0	0
$596$	−10250.7	−0.704505
$597$	−2016.76	−0.138259
$598$	−373.373	−0.0255324
$599$	21021.4	1.43391	0.716954	−	0.697120i	$-0.245536\pi$
$599$	21021.4	1.43391	0.716954	+	0.697120i	$0.245536\pi$
$600$	0	0
$601$	−13017.7	−0.883533	−0.441767	−	0.897130i	$-0.645648\pi$
$601$	−13017.7	−0.883533	−0.441767	+	0.897130i	$0.645648\pi$
$602$	−2353.88	−0.159363
$603$	899.897	0.0607739
$604$	5423.04	0.365331
$605$	0	0
$606$	−1404.64	−0.0941578
$607$	−1909.10	−0.127658	−0.0638288	−	0.997961i	$-0.520331\pi$
$607$	−1909.10	−0.127658	−0.0638288	+	0.997961i	$0.520331\pi$
$608$	2389.17	0.159365
$609$	3425.32	0.227916
$610$	0	0
$611$	4608.57	0.305144
$612$	−5646.46	−0.372949
$613$	−23227.4	−1.53042	−0.765208	−	0.643784i	$-0.777364\pi$
$613$	−23227.4	−1.53042	−0.765208	+	0.643784i	$0.777364\pi$
$614$	−1898.28	−0.124769
$615$	0	0
$616$	−25545.8	−1.67090
$617$	16400.0	1.07008	0.535041	−	0.844826i	$-0.320296\pi$
$617$	16400.0	1.07008	0.535041	+	0.844826i	$0.320296\pi$
$618$	4336.81	0.282285
$619$	−21863.8	−1.41968	−0.709840	−	0.704363i	$-0.751233\pi$
$619$	−21863.8	−1.41968	−0.709840	+	0.704363i	$0.751233\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	−12768.5	−0.823102
$623$	26161.3	1.68239
$624$	645.919	0.0414382
$625$	0	0
$626$	−11043.5	−0.705092
$627$	−2634.43	−0.167798
$628$	−4378.28	−0.278204
$629$	−1392.98	−0.0883018
$630$	0	0
$631$	30880.1	1.94821	0.974103	−	0.226107i	$-0.0725998\pi$
$631$	30880.1	1.94821	0.974103	+	0.226107i	$0.0725998\pi$
$632$	17079.2	1.07496
$633$	3305.00	0.207523
$634$	−11862.4	−0.743085
$635$	0	0
$636$	5148.55	0.320996
$637$	−36.5158	−0.00227129
$638$	−6039.33	−0.374764
$639$	−1206.55	−0.0746953
$640$	0	0
$641$	21191.9	1.30582	0.652908	−	0.757437i	$-0.273549\pi$
$641$	21191.9	1.30582	0.652908	+	0.757437i	$0.273549\pi$
$642$	−5309.03	−0.326372
$643$	7942.85	0.487147	0.243573	−	0.969882i	$-0.421680\pi$
$643$	7942.85	0.487147	0.243573	+	0.969882i	$0.421680\pi$
$644$	2546.15	0.155795
$645$	0	0
$646$	1929.30	0.117503
$647$	−16915.1	−1.02782	−0.513910	−	0.857844i	$-0.671804\pi$
$647$	−16915.1	−1.02782	−0.513910	+	0.857844i	$0.671804\pi$
$648$	1617.99	0.0980873
$649$	−42845.5	−2.59142
$650$	0	0
$651$	4822.21	0.290318
$652$	21639.4	1.29979
$653$	2556.08	0.153181	0.0765905	−	0.997063i	$-0.475597\pi$
$653$	2556.08	0.153181	0.0765905	+	0.997063i	$0.475597\pi$
$654$	1936.62	0.115792
$655$	0	0
$656$	4975.25	0.296114
$657$	−3957.53	−0.235004
$658$	10831.6	0.641731
$659$	−24703.6	−1.46026	−0.730132	−	0.683306i	$-0.760542\pi$
$659$	−24703.6	−1.46026	−0.730132	+	0.683306i	$0.760542\pi$
$660$	0	0
$661$	−25803.1	−1.51834	−0.759171	−	0.650891i	$-0.774395\pi$
$661$	−25803.1	−1.51834	−0.759171	+	0.650891i	$0.774395\pi$
$662$	−920.375	−0.0540353
$663$	3586.40	0.210082
$664$	16688.8	0.975379
$665$	0	0
$666$	170.242	0.00990499
$667$	1411.34	0.0819298
$668$	5867.69	0.339862
$669$	17027.0	0.984012
$670$	0	0
$671$	32510.0	1.87040
$672$	10438.4	0.599210
$673$	18128.9	1.03836	0.519181	−	0.854664i	$-0.326237\pi$
$673$	18128.9	1.03836	0.519181	+	0.854664i	$0.326237\pi$
$674$	1400.16	0.0800179
$675$	0	0
$676$	12306.4	0.700181
$677$	−15712.9	−0.892017	−0.446008	−	0.895029i	$-0.647155\pi$
$677$	−15712.9	−0.892017	−0.446008	+	0.895029i	$0.647155\pi$
$678$	6539.53	0.370426
$679$	7594.37	0.429227
$680$	0	0
$681$	9742.87	0.548234
$682$	−8502.24	−0.477372
$683$	−35118.2	−1.96744	−0.983720	−	0.179705i	$-0.942486\pi$
$683$	−35118.2	−1.96744	−0.983720	+	0.179705i	$0.942486\pi$
$684$	684.122	0.0382428
$685$	0	0
$686$	9053.27	0.503871
$687$	−5701.38	−0.316625
$688$	1677.83	0.0929749
$689$	−3270.15	−0.180817
$690$	0	0
$691$	7211.94	0.397041	0.198520	−	0.980097i	$-0.436386\pi$
$691$	7211.94	0.397041	0.198520	+	0.980097i	$0.436386\pi$
$692$	11574.6	0.635838
$693$	−11509.9	−0.630918
$694$	17634.7	0.964557
$695$	0	0
$696$	3677.18	0.200263
$697$	27624.6	1.50123
$698$	9752.67	0.528859
$699$	10902.0	0.589916
$700$	0	0
$701$	−31816.2	−1.71424	−0.857118	−	0.515119i	$-0.827748\pi$
$701$	−31816.2	−1.71424	−0.857118	+	0.515119i	$0.827748\pi$
$702$	−438.307	−0.0235653
$703$	168.773	0.00905462
$704$	−7961.56	−0.426225
$705$	0	0
$706$	9108.98	0.485582
$707$	−6083.99	−0.323638
$708$	11126.3	0.590612
$709$	−15248.0	−0.807689	−0.403845	−	0.914828i	$-0.632326\pi$
$709$	−15248.0	−0.807689	−0.403845	+	0.914828i	$0.632326\pi$
$710$	0	0
$711$	7695.19	0.405896
$712$	28084.9	1.47827
$713$	1986.90	0.104362
$714$	8429.17	0.441812
$715$	0	0
$716$	17340.3	0.905080
$717$	−8019.75	−0.417717
$718$	−14799.9	−0.769257
$719$	7259.00	0.376516	0.188258	−	0.982120i	$-0.439716\pi$
$719$	7259.00	0.376516	0.188258	+	0.982120i	$0.439716\pi$
$720$	0	0
$721$	18784.3	0.970267
$722$	9588.09	0.494227
$723$	19641.4	1.01033
$724$	−11719.4	−0.601586
$725$	0	0
$726$	14575.8	0.745123
$727$	−32005.8	−1.63278	−0.816388	−	0.577503i	$-0.804027\pi$
$727$	−32005.8	−1.63278	−0.816388	+	0.577503i	$0.804027\pi$
$728$	−4213.56	−0.214512
$729$	729.000	0.0370370
$730$	0	0
$731$	9315.99	0.471360
$732$	−8442.36	−0.426282
$733$	−9976.20	−0.502700	−0.251350	−	0.967896i	$-0.580875\pi$
$733$	−9976.20	−0.502700	−0.251350	+	0.967896i	$0.580875\pi$
$734$	1491.49	0.0750026
$735$	0	0
$736$	4300.94	0.215400
$737$	−6872.33	−0.343481
$738$	−3376.10	−0.168396
$739$	−6263.89	−0.311801	−0.155901	−	0.987773i	$-0.549828\pi$
$739$	−6263.89	−0.311801	−0.155901	+	0.987773i	$0.549828\pi$
$740$	0	0
$741$	−434.527	−0.0215422
$742$	−7685.88	−0.380266
$743$	34425.9	1.69981	0.849907	−	0.526932i	$-0.176658\pi$
$743$	34425.9	1.69981	0.849907	+	0.526932i	$0.176658\pi$
$744$	5176.77	0.255094
$745$	0	0
$746$	11229.2	0.551112
$747$	7519.31	0.368296
$748$	43120.8	2.10783
$749$	−22995.3	−1.12180
$750$	0	0
$751$	2341.14	0.113754	0.0568771	−	0.998381i	$-0.481886\pi$
$751$	2341.14	0.113754	0.0568771	+	0.998381i	$0.481886\pi$
$752$	−7720.71	−0.374395
$753$	−3072.68	−0.148705
$754$	−996.135	−0.0481128
$755$	0	0
$756$	2988.96	0.143793
$757$	2620.34	0.125810	0.0629049	−	0.998020i	$-0.479964\pi$
$757$	2620.34	0.125810	0.0629049	+	0.998020i	$0.479964\pi$
$758$	8029.08	0.384735
$759$	−4742.45	−0.226798
$760$	0	0
$761$	−20483.8	−0.975737	−0.487869	−	0.872917i	$-0.662225\pi$
$761$	−20483.8	−0.975737	−0.487869	+	0.872917i	$0.662225\pi$
$762$	−3675.57	−0.174740
$763$	8388.17	0.397998
$764$	28680.1	1.35813
$765$	0	0
$766$	−11959.0	−0.564095
$767$	−7067.00	−0.332692
$768$	8494.06	0.399092
$769$	21318.2	0.999681	0.499840	−	0.866118i	$-0.333392\pi$
$769$	21318.2	0.999681	0.499840	+	0.866118i	$0.333392\pi$
$770$	0	0
$771$	4539.47	0.212043
$772$	−13378.7	−0.623718
$773$	−36354.5	−1.69157	−0.845783	−	0.533528i	$-0.820866\pi$
$773$	−36354.5	−1.69157	−0.845783	+	0.533528i	$0.820866\pi$
$774$	−1138.54	−0.0528735
$775$	0	0
$776$	8152.77	0.377149
$777$	737.376	0.0340453
$778$	−7098.94	−0.327133
$779$	−3346.98	−0.153939
$780$	0	0
$781$	9214.16	0.422162
$782$	3473.08	0.158820
$783$	1656.79	0.0756178
$784$	61.1747	0.00278675
$785$	0	0
$786$	−2334.11	−0.105922
$787$	−37671.2	−1.70627	−0.853135	−	0.521690i	$-0.825302\pi$
$787$	−37671.2	−1.70627	−0.853135	+	0.521690i	$0.825302\pi$
$788$	5633.33	0.254669
$789$	−3289.67	−0.148435
$790$	0	0
$791$	28325.0	1.27322
$792$	−12356.2	−0.554368
$793$	5362.24	0.240125
$794$	−688.665	−0.0307806
$795$	0	0
$796$	−3999.56	−0.178091
$797$	8415.26	0.374007	0.187004	−	0.982359i	$-0.440122\pi$
$797$	8415.26	0.374007	0.187004	+	0.982359i	$0.440122\pi$
$798$	−1021.27	−0.0453042
$799$	−42868.5	−1.89809
$800$	0	0
$801$	12653.9	0.558183
$802$	15496.2	0.682280
$803$	30222.8	1.32819
$804$	1784.64	0.0782828
$805$	0	0
$806$	−1402.37	−0.0612858
$807$	2193.97	0.0957017
$808$	−6531.33	−0.284371
$809$	−19394.6	−0.842864	−0.421432	−	0.906860i	$-0.638472\pi$
$809$	−19394.6	−0.842864	−0.421432	+	0.906860i	$0.638472\pi$
$810$	0	0
$811$	−25798.6	−1.11703	−0.558514	−	0.829495i	$-0.688628\pi$
$811$	−25798.6	−1.11703	−0.558514	+	0.829495i	$0.688628\pi$
$812$	6792.96	0.293579
$813$	13870.1	0.598334
$814$	−1300.10	−0.0559809
$815$	0	0
$816$	−6008.27	−0.257759
$817$	−1128.72	−0.0483341
$818$	17280.1	0.738613
$819$	−1898.46	−0.0809984
$820$	0	0
$821$	14553.1	0.618643	0.309321	−	0.950958i	$-0.399898\pi$
$821$	14553.1	0.618643	0.309321	+	0.950958i	$0.399898\pi$
$822$	−1164.15	−0.0493972
$823$	30812.3	1.30504	0.652521	−	0.757771i	$-0.273711\pi$
$823$	30812.3	1.30504	0.652521	+	0.757771i	$0.273711\pi$
$824$	20165.4	0.852544
$825$	0	0
$826$	−16609.7	−0.699666
$827$	−16709.8	−0.702609	−0.351304	−	0.936261i	$-0.614262\pi$
$827$	−16709.8	−0.702609	−0.351304	+	0.936261i	$0.614262\pi$
$828$	1231.54	0.0516897
$829$	−26582.3	−1.11368	−0.556841	−	0.830619i	$-0.687987\pi$
$829$	−26582.3	−1.11368	−0.556841	+	0.830619i	$0.687987\pi$
$830$	0	0
$831$	14508.8	0.605662
$832$	−1313.19	−0.0547195
$833$	339.667	0.0141282
$834$	−6071.47	−0.252084
$835$	0	0
$836$	−5224.50	−0.216140
$837$	2332.45	0.0963215
$838$	−16524.9	−0.681198
$839$	−10884.1	−0.447868	−0.223934	−	0.974604i	$-0.571890\pi$
$839$	−10884.1	−0.447868	−0.223934	+	0.974604i	$0.571890\pi$
$840$	0	0
$841$	−20623.6	−0.845613
$842$	17434.1	0.713561
$843$	9102.19	0.371881
$844$	6554.34	0.267310
$845$	0	0
$846$	5239.11	0.212913
$847$	63133.0	2.56113
$848$	5478.46	0.221853
$849$	−1424.02	−0.0575643
$850$	0	0
$851$	303.822	0.0122384
$852$	−2392.78	−0.0962151
$853$	2246.92	0.0901912	0.0450956	−	0.998983i	$-0.485641\pi$
$853$	2246.92	0.0901912	0.0450956	+	0.998983i	$0.485641\pi$
$854$	12603.0	0.504993
$855$	0	0
$856$	−24686.0	−0.985691
$857$	22125.6	0.881907	0.440954	−	0.897530i	$-0.354640\pi$
$857$	22125.6	0.881907	0.440954	+	0.897530i	$0.354640\pi$
$858$	3347.26	0.133186
$859$	−28298.7	−1.12403	−0.562013	−	0.827128i	$-0.689973\pi$
$859$	−28298.7	−1.12403	−0.562013	+	0.827128i	$0.689973\pi$
$860$	0	0
$861$	−14623.1	−0.578808
$862$	4978.42	0.196712
$863$	27286.2	1.07628	0.538142	−	0.842854i	$-0.319126\pi$
$863$	27286.2	1.07628	0.538142	+	0.842854i	$0.319126\pi$
$864$	5048.93	0.198806
$865$	0	0
$866$	12392.1	0.486258
$867$	−18621.3	−0.729428
$868$	9563.21	0.373959
$869$	−58766.6	−2.29404
$870$	0	0
$871$	−1133.53	−0.0440967
$872$	9004.94	0.349708
$873$	3673.31	0.142409
$874$	−420.797	−0.0162857
$875$	0	0
$876$	−7848.41	−0.302709
$877$	1748.46	0.0673220	0.0336610	−	0.999433i	$-0.489283\pi$
$877$	1748.46	0.0673220	0.0336610	+	0.999433i	$0.489283\pi$
$878$	14514.9	0.557920
$879$	8463.57	0.324766
$880$	0	0
$881$	−18917.6	−0.723441	−0.361721	−	0.932287i	$-0.617811\pi$
$881$	−18917.6	−0.723441	−0.361721	+	0.932287i	$0.617811\pi$
$882$	−41.5119	−0.00158478
$883$	−33861.1	−1.29050	−0.645252	−	0.763969i	$-0.723248\pi$
$883$	−33861.1	−1.29050	−0.645252	+	0.763969i	$0.723248\pi$
$884$	7112.40	0.270606
$885$	0	0
$886$	−22906.8	−0.868588
$887$	−17068.9	−0.646131	−0.323065	−	0.946377i	$-0.604713\pi$
$887$	−17068.9	−0.646131	−0.323065	+	0.946377i	$0.604713\pi$
$888$	791.594	0.0299146
$889$	−15920.2	−0.600614
$890$	0	0
$891$	−5567.22	−0.209325
$892$	33767.4	1.26751
$893$	5193.92	0.194634
$894$	7401.64	0.276899
$895$	0	0
$896$	24749.3	0.922785
$897$	−782.225	−0.0291168
$898$	−3456.82	−0.128458
$899$	5300.92	0.196658
$900$	0	0
$901$	30418.6	1.12474
$902$	25782.6	0.951737
$903$	−4931.43	−0.181736
$904$	30407.7	1.11874
$905$	0	0
$906$	−3915.77	−0.143590
$907$	13775.7	0.504315	0.252158	−	0.967686i	$-0.418860\pi$
$907$	13775.7	0.504315	0.252158	+	0.967686i	$0.418860\pi$
$908$	19321.7	0.706181
$909$	−2942.76	−0.107376
$910$	0	0
$911$	12815.8	0.466087	0.233044	−	0.972466i	$-0.425131\pi$
$911$	12815.8	0.466087	0.233044	+	0.972466i	$0.425131\pi$
$912$	727.960	0.0264311
$913$	−57423.4	−2.08153
$914$	9906.09	0.358495
$915$	0	0
$916$	−11306.7	−0.407844
$917$	−10109.8	−0.364074
$918$	4077.09	0.146584
$919$	−10845.7	−0.389301	−0.194650	−	0.980873i	$-0.562357\pi$
$919$	−10845.7	−0.389301	−0.194650	+	0.980873i	$0.562357\pi$
$920$	0	0
$921$	−3976.93	−0.142285
$922$	−26533.1	−0.947747
$923$	1519.80	0.0541979
$924$	−22826.0	−0.812685
$925$	0	0
$926$	12427.3	0.441022
$927$	9085.73	0.321914
$928$	11474.6	0.405897
$929$	−20136.2	−0.711137	−0.355568	−	0.934650i	$-0.615713\pi$
$929$	−20136.2	−0.711137	−0.355568	+	0.934650i	$0.615713\pi$
$930$	0	0
$931$	−41.1539	−0.00144873
$932$	21620.4	0.759870
$933$	−26750.3	−0.938655
$934$	1948.58	0.0682650
$935$	0	0
$936$	−2038.05	−0.0711708
$937$	53199.9	1.85482	0.927409	−	0.374049i	$-0.122031\pi$
$937$	53199.9	1.85482	0.927409	+	0.374049i	$0.122031\pi$
$938$	−2664.15	−0.0927374
$939$	−23136.4	−0.804077
$940$	0	0
$941$	27234.1	0.943470	0.471735	−	0.881740i	$-0.343628\pi$
$941$	27234.1	0.943470	0.471735	+	0.881740i	$0.343628\pi$
$942$	3161.39	0.109346
$943$	−6025.17	−0.208066
$944$	11839.3	0.408195
$945$	0	0
$946$	8694.81	0.298830
$947$	16989.4	0.582978	0.291489	−	0.956574i	$-0.405849\pi$
$947$	16989.4	0.582978	0.291489	+	0.956574i	$0.405849\pi$
$948$	15260.8	0.522835
$949$	4984.99	0.170516
$950$	0	0
$951$	−24852.0	−0.847405
$952$	39194.2	1.33434
$953$	15399.3	0.523432	0.261716	−	0.965145i	$-0.415712\pi$
$953$	15399.3	0.523432	0.261716	+	0.965145i	$0.415712\pi$
$954$	−3717.57	−0.126164
$955$	0	0
$956$	−15904.5	−0.538062
$957$	−12652.5	−0.427376
$958$	4992.17	0.168361
$959$	−5042.35	−0.169787
$960$	0	0
$961$	−22328.3	−0.749498
$962$	−214.440	−0.00718693
$963$	−11122.5	−0.372190
$964$	38952.0	1.30141
$965$	0	0
$966$	−1838.48	−0.0612339
$967$	50180.9	1.66878	0.834389	−	0.551176i	$-0.185821\pi$
$967$	50180.9	1.66878	0.834389	+	0.551176i	$0.185821\pi$
$968$	67775.1	2.25039
$969$	4041.92	0.133999
$970$	0	0
$971$	7978.69	0.263696	0.131848	−	0.991270i	$-0.457909\pi$
$971$	7978.69	0.263696	0.131848	+	0.991270i	$0.457909\pi$
$972$	1445.72	0.0477074
$973$	−26297.7	−0.866458
$974$	13292.5	0.437288
$975$	0	0
$976$	−8983.33	−0.294620
$977$	7051.67	0.230914	0.115457	−	0.993312i	$-0.463167\pi$
$977$	7051.67	0.230914	0.115457	+	0.993312i	$0.463167\pi$
$978$	−15625.0	−0.510872
$979$	−96635.4	−3.15473
$980$	0	0
$981$	4057.26	0.132047
$982$	13862.5	0.450480
$983$	−7154.32	−0.232134	−0.116067	−	0.993241i	$-0.537029\pi$
$983$	−7154.32	−0.232134	−0.116067	+	0.993241i	$0.537029\pi$
$984$	−15698.3	−0.508581
$985$	0	0
$986$	9265.94	0.299278
$987$	22692.4	0.731822
$988$	−861.736	−0.0277484
$989$	−2031.90	−0.0653293
$990$	0	0
$991$	−56414.3	−1.80833	−0.904167	−	0.427178i	$-0.859508\pi$
$991$	−56414.3	−1.80833	−0.904167	+	0.427178i	$0.859508\pi$
$992$	16154.1	0.517030
$993$	−1928.21	−0.0616212
$994$	3572.00	0.113981
$995$	0	0
$996$	14912.0	0.474402
$997$	−37934.5	−1.20501	−0.602507	−	0.798114i	$-0.705831\pi$
$997$	−37934.5	−1.20501	−0.602507	+	0.798114i	$0.705831\pi$
$998$	−20321.9	−0.644567
$999$	356.660	0.0112955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.4.a.bf.1.5	yes	11
5.4	even	2		1725.4.a.be.1.7	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1725.4.a.be.1.7	✓	11	5.4	even	2
1725.4.a.bf.1.5	yes	11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1725.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43196 q^{2} -3.00000 q^{3} -5.94948 q^{4} +4.29589 q^{6} +18.6070 q^{7} +19.9752 q^{8} +9.00000 q^{9} -68.7311 q^{11} +17.8484 q^{12} -11.3366 q^{13} -26.6446 q^{14} +18.9921 q^{16} +105.452 q^{17} -12.8877 q^{18} -12.7765 q^{19} -55.8210 q^{21} +98.4205 q^{22} -23.0000 q^{23} -59.9255 q^{24} +16.2336 q^{26} -27.0000 q^{27} -110.702 q^{28} -61.3625 q^{29} -86.3869 q^{31} -186.997 q^{32} +206.193 q^{33} -151.003 q^{34} -53.5453 q^{36} -13.2096 q^{37} +18.2955 q^{38} +34.0098 q^{39} +261.964 q^{41} +79.9337 q^{42} +88.3435 q^{43} +408.914 q^{44} +32.9352 q^{46} -406.521 q^{47} -56.9764 q^{48} +3.22106 q^{49} -316.356 q^{51} +67.4469 q^{52} +288.459 q^{53} +38.6630 q^{54} +371.678 q^{56} +38.3295 q^{57} +87.8689 q^{58} +623.379 q^{59} -473.003 q^{61} +123.703 q^{62} +167.463 q^{63} +115.836 q^{64} -295.262 q^{66} +99.9886 q^{67} -627.384 q^{68} +69.0000 q^{69} -134.061 q^{71} +179.776 q^{72} -439.726 q^{73} +18.9157 q^{74} +76.0136 q^{76} -1278.88 q^{77} -48.7008 q^{78} +855.021 q^{79} +81.0000 q^{81} -375.123 q^{82} +835.479 q^{83} +332.106 q^{84} -126.505 q^{86} +184.088 q^{87} -1372.91 q^{88} +1405.99 q^{89} -210.940 q^{91} +136.838 q^{92} +259.161 q^{93} +582.124 q^{94} +560.992 q^{96} +408.145 q^{97} -4.61244 q^{98} -618.580 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + q^{2} - 33 q^{3} + 39 q^{4} - 3 q^{6} - 19 q^{7} - 18 q^{8} + 99 q^{9} - 41 q^{11} - 117 q^{12} - 57 q^{13} + 8 q^{14} + 227 q^{16} - 66 q^{17} + 9 q^{18} - 215 q^{19} + 57 q^{21} + 342 q^{22} - 253 q^{23}+ \cdots - 369 q^{99}+O(q^{100}) \) 11 * q + q^2 - 33 * q^3 + 39 * q^4 - 3 * q^6 - 19 * q^7 - 18 * q^8 + 99 * q^9 - 41 * q^11 - 117 * q^12 - 57 * q^13 + 8 * q^14 + 227 * q^16 - 66 * q^17 + 9 * q^18 - 215 * q^19 + 57 * q^21 + 342 * q^22 - 253 * q^23 + 54 * q^24 + 364 * q^26 - 297 * q^27 - 657 * q^28 + 513 * q^29 + 340 * q^31 - 263 * q^32 + 123 * q^33 - 125 * q^34 + 351 * q^36 - 20 * q^37 - 124 * q^38 + 171 * q^39 + 515 * q^41 - 24 * q^42 - 1401 * q^43 - 879 * q^44 - 23 * q^46 + 152 * q^47 - 681 * q^48 + 1108 * q^49 + 198 * q^51 - 697 * q^52 - 794 * q^53 - 27 * q^54 + 1113 * q^56 + 645 * q^57 - 3058 * q^58 + 1332 * q^59 + 148 * q^61 - 984 * q^62 - 171 * q^63 + 1422 * q^64 - 1026 * q^66 - 56 * q^67 - 2277 * q^68 + 759 * q^69 - 198 * q^71 - 162 * q^72 + 205 * q^73 + 2901 * q^74 - 4522 * q^76 + 851 * q^77 - 1092 * q^78 - 645 * q^79 + 891 * q^81 - 37 * q^82 - 2591 * q^83 + 1971 * q^84 - 567 * q^86 - 1539 * q^87 + 2247 * q^88 + 4130 * q^89 - 1623 * q^91 - 897 * q^92 - 1020 * q^93 + 49 * q^94 + 789 * q^96 - 2348 * q^97 - 2804 * q^98 - 369 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more