LMFDB - Embedded newform 531.4.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 34x^{5} + 25x^{4} + 315x^{3} - 146x^{2} - 736x + 512 $ x^7 - x^6 - 34x^5 + 25x^4 + 315x^3 - 146x^2 - 736*x + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.775001$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.77500 q^{2} -4.84937 q^{4} -6.23028 q^{5} -18.0779 q^{7} -22.8076 q^{8} +O(q^{10})$	\(q+1.77500 q^{2} -4.84937 q^{4} -6.23028 q^{5} -18.0779 q^{7} -22.8076 q^{8} -11.0588 q^{10} -13.3400 q^{11} -66.3734 q^{13} -32.0883 q^{14} -1.68865 q^{16} +97.6720 q^{17} +109.769 q^{19} +30.2129 q^{20} -23.6785 q^{22} +147.736 q^{23} -86.1836 q^{25} -117.813 q^{26} +87.6663 q^{28} +173.639 q^{29} +148.613 q^{31} +179.464 q^{32} +173.368 q^{34} +112.630 q^{35} -446.332 q^{37} +194.840 q^{38} +142.098 q^{40} -182.883 q^{41} -223.613 q^{43} +64.6904 q^{44} +262.231 q^{46} +529.113 q^{47} -16.1904 q^{49} -152.976 q^{50} +321.869 q^{52} -398.029 q^{53} +83.1117 q^{55} +412.314 q^{56} +308.210 q^{58} +59.0000 q^{59} +788.427 q^{61} +263.789 q^{62} +332.058 q^{64} +413.525 q^{65} +288.375 q^{67} -473.648 q^{68} +199.919 q^{70} +139.803 q^{71} +549.077 q^{73} -792.240 q^{74} -532.310 q^{76} +241.158 q^{77} -190.617 q^{79} +10.5208 q^{80} -324.617 q^{82} -410.618 q^{83} -608.524 q^{85} -396.914 q^{86} +304.253 q^{88} -1291.81 q^{89} +1199.89 q^{91} -716.425 q^{92} +939.176 q^{94} -683.891 q^{95} +1488.11 q^{97} -28.7380 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8}+O(q^{10}) $ 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8	$ 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8} - 79 q^{10} + 131 q^{11} - 123 q^{13} + 117 q^{14} + 202 q^{16} + 235 q^{17} - 80 q^{19} - 61 q^{20} + 688 q^{22} + 274 q^{23} + 193 q^{25} + 180 q^{26} - 118 q^{28} + 406 q^{29} - 346 q^{31} + 854 q^{32} + 178 q^{34} + 424 q^{35} - 157 q^{37} + 129 q^{38} - 590 q^{40} + 825 q^{41} - 815 q^{43} + 1690 q^{44} + 1457 q^{46} + 1196 q^{47} + 914 q^{49} - 713 q^{50} + 1030 q^{52} + 900 q^{53} - 1044 q^{55} - 2172 q^{56} + 1242 q^{58} + 413 q^{59} + 420 q^{61} - 646 q^{62} + 3541 q^{64} - 190 q^{65} + 1316 q^{67} + 611 q^{68} + 4658 q^{70} + 173 q^{71} - 418 q^{73} - 660 q^{74} + 1540 q^{76} + 753 q^{77} + 2635 q^{79} - 6155 q^{80} - 125 q^{82} - 457 q^{83} + 1270 q^{85} - 3482 q^{86} + 7685 q^{88} - 592 q^{89} + 3179 q^{91} + 3500 q^{92} + 2064 q^{94} + 2250 q^{95} - 1906 q^{97} - 2994 q^{98}+O(q^{100}) $ 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8 - 79 * q^10 + 131 * q^11 - 123 * q^13 + 117 * q^14 + 202 * q^16 + 235 * q^17 - 80 * q^19 - 61 * q^20 + 688 * q^22 + 274 * q^23 + 193 * q^25 + 180 * q^26 - 118 * q^28 + 406 * q^29 - 346 * q^31 + 854 * q^32 + 178 * q^34 + 424 * q^35 - 157 * q^37 + 129 * q^38 - 590 * q^40 + 825 * q^41 - 815 * q^43 + 1690 * q^44 + 1457 * q^46 + 1196 * q^47 + 914 * q^49 - 713 * q^50 + 1030 * q^52 + 900 * q^53 - 1044 * q^55 - 2172 * q^56 + 1242 * q^58 + 413 * q^59 + 420 * q^61 - 646 * q^62 + 3541 * q^64 - 190 * q^65 + 1316 * q^67 + 611 * q^68 + 4658 * q^70 + 173 * q^71 - 418 * q^73 - 660 * q^74 + 1540 * q^76 + 753 * q^77 + 2635 * q^79 - 6155 * q^80 - 125 * q^82 - 457 * q^83 + 1270 * q^85 - 3482 * q^86 + 7685 * q^88 - 592 * q^89 + 3179 * q^91 + 3500 * q^92 + 2064 * q^94 + 2250 * q^95 - 1906 * q^97 - 2994 * q^98

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 $

$2$	1.77500	0.627558	0.313779	−	0.949496i	$-0.398405\pi$
$2$	1.77500	0.627558	0.313779	+	0.949496i	$0.398405\pi$
$3$	0	0
$3$	0	0
$4$	−4.84937	−0.606171
$5$	−6.23028	−0.557253	−0.278627	−	0.960399i	$-0.589879\pi$
$5$	−6.23028	−0.557253	−0.278627	+	0.960399i	$0.589879\pi$
$6$	0	0
$7$	−18.0779	−0.976114	−0.488057	−	0.872812i	$-0.662294\pi$
$7$	−18.0779	−0.976114	−0.488057	+	0.872812i	$0.662294\pi$
$8$	−22.8076	−1.00797
$9$	0	0
$10$	−11.0588	−0.349709
$11$	−13.3400	−0.365650	−0.182825	−	0.983145i	$-0.558524\pi$
$11$	−13.3400	−0.365650	−0.182825	+	0.983145i	$0.558524\pi$
$12$	0	0
$13$	−66.3734	−1.41605	−0.708026	−	0.706187i	$-0.750414\pi$
$13$	−66.3734	−1.41605	−0.708026	+	0.706187i	$0.750414\pi$
$14$	−32.0883	−0.612568
$15$	0	0
$16$	−1.68865	−0.0263851
$17$	97.6720	1.39347	0.696733	−	0.717330i	$-0.254636\pi$
$17$	97.6720	1.39347	0.696733	+	0.717330i	$0.254636\pi$
$18$	0	0
$19$	109.769	1.32540	0.662702	−	0.748883i	$-0.269409\pi$
$19$	109.769	1.32540	0.662702	+	0.748883i	$0.269409\pi$
$20$	30.2129	0.337791
$21$	0	0
$22$	−23.6785	−0.229466
$23$	147.736	1.33935	0.669674	−	0.742655i	$-0.266434\pi$
$23$	147.736	1.33935	0.669674	+	0.742655i	$0.266434\pi$
$24$	0	0
$25$	−86.1836	−0.689469
$26$	−117.813	−0.888654
$27$	0	0
$28$	87.6663	0.591692
$29$	173.639	1.11186	0.555931	−	0.831228i	$-0.312362\pi$
$29$	173.639	1.11186	0.555931	+	0.831228i	$0.312362\pi$
$30$	0	0
$31$	148.613	0.861024	0.430512	−	0.902585i	$-0.358333\pi$
$31$	148.613	0.861024	0.430512	+	0.902585i	$0.358333\pi$
$32$	179.464	0.991407
$33$	0	0
$34$	173.368	0.874481
$35$	112.630	0.543943
$36$	0	0
$37$	−446.332	−1.98315	−0.991574	−	0.129539i	$-0.958650\pi$
$37$	−446.332	−1.98315	−0.991574	+	0.129539i	$0.958650\pi$
$38$	194.840	0.831768
$39$	0	0
$40$	142.098	0.561692
$41$	−182.883	−0.696622	−0.348311	−	0.937379i	$-0.613245\pi$
$41$	−182.883	−0.696622	−0.348311	+	0.937379i	$0.613245\pi$
$42$	0	0
$43$	−223.613	−0.793040	−0.396520	−	0.918026i	$-0.629782\pi$
$43$	−223.613	−0.793040	−0.396520	+	0.918026i	$0.629782\pi$
$44$	64.6904	0.221647
$45$	0	0
$46$	262.231	0.840519
$47$	529.113	1.64211	0.821053	−	0.570851i	$-0.193387\pi$
$47$	529.113	1.64211	0.821053	+	0.570851i	$0.193387\pi$
$48$	0	0
$49$	−16.1904	−0.0472023
$50$	−152.976	−0.432681
$51$	0	0
$52$	321.869	0.858369
$53$	−398.029	−1.03157	−0.515787	−	0.856717i	$-0.672500\pi$
$53$	−398.029	−1.03157	−0.515787	+	0.856717i	$0.672500\pi$
$54$	0	0
$55$	83.1117	0.203760
$56$	412.314	0.983889
$57$	0	0
$58$	308.210	0.697758
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	788.427	1.65488	0.827440	−	0.561554i	$-0.189796\pi$
$61$	788.427	1.65488	0.827440	+	0.561554i	$0.189796\pi$
$62$	263.789	0.540342
$63$	0	0
$64$	332.058	0.648550
$65$	413.525	0.789099
$66$	0	0
$67$	288.375	0.525829	0.262915	−	0.964819i	$-0.415316\pi$
$67$	288.375	0.525829	0.262915	+	0.964819i	$0.415316\pi$
$68$	−473.648	−0.844680
$69$	0	0
$70$	199.919	0.341355
$71$	139.803	0.233684	0.116842	−	0.993150i	$-0.462723\pi$
$71$	139.803	0.233684	0.116842	+	0.993150i	$0.462723\pi$
$72$	0	0
$73$	549.077	0.880336	0.440168	−	0.897915i	$-0.354919\pi$
$73$	549.077	0.880336	0.440168	+	0.897915i	$0.354919\pi$
$74$	−792.240	−1.24454
$75$	0	0
$76$	−532.310	−0.803422
$77$	241.158	0.356916
$78$	0	0
$79$	−190.617	−0.271469	−0.135735	−	0.990745i	$-0.543339\pi$
$79$	−190.617	−0.271469	−0.135735	+	0.990745i	$0.543339\pi$
$80$	10.5208	0.0147032
$81$	0	0
$82$	−324.617	−0.437171
$83$	−410.618	−0.543026	−0.271513	−	0.962435i	$-0.587524\pi$
$83$	−410.618	−0.543026	−0.271513	+	0.962435i	$0.587524\pi$
$84$	0	0
$85$	−608.524	−0.776514
$86$	−396.914	−0.497679
$87$	0	0
$88$	304.253	0.368562
$89$	−1291.81	−1.53856	−0.769281	−	0.638911i	$-0.779385\pi$
$89$	−1291.81	−1.53856	−0.769281	+	0.638911i	$0.779385\pi$
$90$	0	0
$91$	1199.89	1.38223
$92$	−716.425	−0.811875
$93$	0	0
$94$	939.176	1.03052
$95$	−683.891	−0.738586
$96$	0	0
$97$	1488.11	1.55768	0.778839	−	0.627225i	$-0.215809\pi$
$97$	1488.11	1.55768	0.778839	+	0.627225i	$0.215809\pi$
$98$	−28.7380	−0.0296222
$99$	0	0
$100$	417.936	0.417936
$101$	1287.68	1.26861	0.634303	−	0.773085i	$-0.281287\pi$
$101$	1287.68	1.26861	0.634303	+	0.773085i	$0.281287\pi$
$102$	0	0
$103$	−469.022	−0.448680	−0.224340	−	0.974511i	$-0.572023\pi$
$103$	−469.022	−0.448680	−0.224340	+	0.974511i	$0.572023\pi$
$104$	1513.82	1.42733
$105$	0	0
$106$	−706.501	−0.647372
$107$	1237.22	1.11782	0.558908	−	0.829230i	$-0.311221\pi$
$107$	1237.22	1.11782	0.558908	+	0.829230i	$0.311221\pi$
$108$	0	0
$109$	2145.12	1.88500	0.942501	−	0.334204i	$-0.108468\pi$
$109$	2145.12	1.88500	0.942501	+	0.334204i	$0.108468\pi$
$110$	147.523	0.127871
$111$	0	0
$112$	30.5272	0.0257549
$113$	−1114.62	−0.927919	−0.463960	−	0.885856i	$-0.653572\pi$
$113$	−1114.62	−0.927919	−0.463960	+	0.885856i	$0.653572\pi$
$114$	0	0
$115$	−920.435	−0.746357
$116$	−842.041	−0.673979
$117$	0	0
$118$	104.725	0.0817011
$119$	−1765.70	−1.36018
$120$	0	0
$121$	−1153.05	−0.866300
$122$	1399.46	1.03853
$123$	0	0
$124$	−720.681	−0.521928
$125$	1315.73	0.941462
$126$	0	0
$127$	−1267.75	−0.885782	−0.442891	−	0.896576i	$-0.646047\pi$
$127$	−1267.75	−0.885782	−0.442891	+	0.896576i	$0.646047\pi$
$128$	−846.308	−0.584404
$129$	0	0
$130$	734.007	0.495205
$131$	2119.49	1.41359	0.706796	−	0.707418i	$-0.250140\pi$
$131$	2119.49	1.41359	0.706796	+	0.707418i	$0.250140\pi$
$132$	0	0
$133$	−1984.39	−1.29375
$134$	511.865	0.329988
$135$	0	0
$136$	−2227.67	−1.40457
$137$	845.583	0.527322	0.263661	−	0.964615i	$-0.415070\pi$
$137$	845.583	0.527322	0.263661	+	0.964615i	$0.415070\pi$
$138$	0	0
$139$	−1390.93	−0.848754	−0.424377	−	0.905486i	$-0.639507\pi$
$139$	−1390.93	−0.848754	−0.424377	+	0.905486i	$0.639507\pi$
$140$	−546.186	−0.329722
$141$	0	0
$142$	248.151	0.146650
$143$	885.418	0.517779
$144$	0	0
$145$	−1081.82	−0.619589
$146$	974.612	0.552462
$147$	0	0
$148$	2164.43	1.20213
$149$	1007.80	0.554110	0.277055	−	0.960854i	$-0.410642\pi$
$149$	1007.80	0.554110	0.277055	+	0.960854i	$0.410642\pi$
$150$	0	0
$151$	−1926.18	−1.03808	−0.519041	−	0.854750i	$-0.673711\pi$
$151$	−1926.18	−1.03808	−0.519041	+	0.854750i	$0.673711\pi$
$152$	−2503.57	−1.33596
$153$	0	0
$154$	428.056	0.223985
$155$	−925.903	−0.479808
$156$	0	0
$157$	−1235.58	−0.628092	−0.314046	−	0.949408i	$-0.601685\pi$
$157$	−1235.58	−0.628092	−0.314046	+	0.949408i	$0.601685\pi$
$158$	−338.345	−0.170363
$159$	0	0
$160$	−1118.11	−0.552465
$161$	−2670.75	−1.30736
$162$	0	0
$163$	3664.53	1.76091	0.880454	−	0.474131i	$-0.157238\pi$
$163$	3664.53	1.76091	0.880454	+	0.474131i	$0.157238\pi$
$164$	886.867	0.422272
$165$	0	0
$166$	−728.847	−0.340780
$167$	−3953.29	−1.83183	−0.915913	−	0.401377i	$-0.868532\pi$
$167$	−3953.29	−1.83183	−0.915913	+	0.401377i	$0.868532\pi$
$168$	0	0
$169$	2208.43	1.00520
$170$	−1080.13	−0.487307
$171$	0	0
$172$	1084.38	0.480718
$173$	2225.27	0.977945	0.488973	−	0.872299i	$-0.337372\pi$
$173$	2225.27	0.977945	0.488973	+	0.872299i	$0.337372\pi$
$174$	0	0
$175$	1558.02	0.673000
$176$	22.5265	0.00964772
$177$	0	0
$178$	−2292.97	−0.965536
$179$	2130.61	0.889662	0.444831	−	0.895615i	$-0.353264\pi$
$179$	2130.61	0.889662	0.444831	+	0.895615i	$0.353264\pi$
$180$	0	0
$181$	1429.58	0.587072	0.293536	−	0.955948i	$-0.405168\pi$
$181$	1429.58	0.587072	0.293536	+	0.955948i	$0.405168\pi$
$182$	2129.81	0.867427
$183$	0	0
$184$	−3369.50	−1.35002
$185$	2780.77	1.10512
$186$	0	0
$187$	−1302.94	−0.509521
$188$	−2565.86	−0.995398
$189$	0	0
$190$	−1213.91	−0.463506
$191$	920.502	0.348718	0.174359	−	0.984682i	$-0.444215\pi$
$191$	920.502	0.348718	0.174359	+	0.984682i	$0.444215\pi$
$192$	0	0
$193$	−1870.05	−0.697458	−0.348729	−	0.937224i	$-0.613387\pi$
$193$	−1870.05	−0.697458	−0.348729	+	0.937224i	$0.613387\pi$
$194$	2641.40	0.977532
$195$	0	0
$196$	78.5132	0.0286127
$197$	476.512	0.172335	0.0861676	−	0.996281i	$-0.472538\pi$
$197$	476.512	0.172335	0.0861676	+	0.996281i	$0.472538\pi$
$198$	0	0
$199$	4164.08	1.48334	0.741668	−	0.670767i	$-0.234035\pi$
$199$	4164.08	1.48334	0.741668	+	0.670767i	$0.234035\pi$
$200$	1965.64	0.694960
$201$	0	0
$202$	2285.64	0.796123
$203$	−3139.03	−1.08530
$204$	0	0
$205$	1139.41	0.388195
$206$	−832.514	−0.281573
$207$	0	0
$208$	112.081	0.0373627
$209$	−1464.31	−0.484634
$210$	0	0
$211$	−2499.21	−0.815415	−0.407707	−	0.913113i	$-0.633672\pi$
$211$	−2499.21	−0.815415	−0.407707	+	0.913113i	$0.633672\pi$
$212$	1930.19	0.625311
$213$	0	0
$214$	2196.06	0.701494
$215$	1393.18	0.441924
$216$	0	0
$217$	−2686.61	−0.840457
$218$	3807.59	1.18295
$219$	0	0
$220$	−403.040	−0.123513
$221$	−6482.82	−1.97322
$222$	0	0
$223$	2567.55	0.771013	0.385506	−	0.922705i	$-0.374027\pi$
$223$	2567.55	0.771013	0.385506	+	0.922705i	$0.374027\pi$
$224$	−3244.33	−0.967726
$225$	0	0
$226$	−1978.46	−0.582323
$227$	969.018	0.283330	0.141665	−	0.989915i	$-0.454754\pi$
$227$	969.018	0.283330	0.141665	+	0.989915i	$0.454754\pi$
$228$	0	0
$229$	4217.16	1.21693	0.608467	−	0.793579i	$-0.291785\pi$
$229$	4217.16	1.21693	0.608467	+	0.793579i	$0.291785\pi$
$230$	−1633.77	−0.468382
$231$	0	0
$232$	−3960.30	−1.12072
$233$	−382.087	−0.107431	−0.0537154	−	0.998556i	$-0.517106\pi$
$233$	−382.087	−0.107431	−0.0537154	+	0.998556i	$0.517106\pi$
$234$	0	0
$235$	−3296.52	−0.915070
$236$	−286.113	−0.0789168
$237$	0	0
$238$	−3134.12	−0.853593
$239$	4069.70	1.10145	0.550726	−	0.834686i	$-0.314351\pi$
$239$	4069.70	1.10145	0.550726	+	0.834686i	$0.314351\pi$
$240$	0	0
$241$	4573.11	1.22232	0.611162	−	0.791506i	$-0.290703\pi$
$241$	4573.11	1.22232	0.611162	+	0.791506i	$0.290703\pi$
$242$	−2046.66	−0.543653
$243$	0	0
$244$	−3823.37	−1.00314
$245$	100.871	0.0263036
$246$	0	0
$247$	−7285.73	−1.87684
$248$	−3389.52	−0.867882
$249$	0	0
$250$	2335.43	0.590822
$251$	−4263.66	−1.07219	−0.536095	−	0.844158i	$-0.680101\pi$
$251$	−4263.66	−1.07219	−0.536095	+	0.844158i	$0.680101\pi$
$252$	0	0
$253$	−1970.79	−0.489733
$254$	−2250.25	−0.555879
$255$	0	0
$256$	−4158.66	−1.01530
$257$	−3758.60	−0.912277	−0.456138	−	0.889909i	$-0.650768\pi$
$257$	−3758.60	−0.912277	−0.456138	+	0.889909i	$0.650768\pi$
$258$	0	0
$259$	8068.73	1.93578
$260$	−2005.34	−0.478329
$261$	0	0
$262$	3762.09	0.887110
$263$	2832.72	0.664157	0.332078	−	0.943252i	$-0.392250\pi$
$263$	2832.72	0.664157	0.332078	+	0.943252i	$0.392250\pi$
$264$	0	0
$265$	2479.83	0.574848
$266$	−3522.29	−0.811900
$267$	0	0
$268$	−1398.44	−0.318743
$269$	7351.53	1.66629	0.833143	−	0.553058i	$-0.186539\pi$
$269$	7351.53	1.66629	0.833143	+	0.553058i	$0.186539\pi$
$270$	0	0
$271$	−4053.05	−0.908507	−0.454253	−	0.890872i	$-0.650094\pi$
$271$	−4053.05	−0.908507	−0.454253	+	0.890872i	$0.650094\pi$
$272$	−164.934	−0.0367668
$273$	0	0
$274$	1500.91	0.330925
$275$	1149.69	0.252104
$276$	0	0
$277$	3054.14	0.662475	0.331238	−	0.943547i	$-0.392534\pi$
$277$	3054.14	0.662475	0.331238	+	0.943547i	$0.392534\pi$
$278$	−2468.90	−0.532642
$279$	0	0
$280$	−2568.83	−0.548275
$281$	3512.24	0.745632	0.372816	−	0.927905i	$-0.378392\pi$
$281$	3512.24	0.745632	0.372816	+	0.927905i	$0.378392\pi$
$282$	0	0
$283$	−7955.47	−1.67104	−0.835518	−	0.549463i	$-0.814832\pi$
$283$	−7955.47	−1.67104	−0.835518	+	0.549463i	$0.814832\pi$
$284$	−677.958	−0.141653
$285$	0	0
$286$	1571.62	0.324936
$287$	3306.13	0.679982
$288$	0	0
$289$	4626.82	0.941750
$290$	−1920.24	−0.388828
$291$	0	0
$292$	−2662.68	−0.533635
$293$	−4933.26	−0.983632	−0.491816	−	0.870699i	$-0.663667\pi$
$293$	−4933.26	−0.983632	−0.491816	+	0.870699i	$0.663667\pi$
$294$	0	0
$295$	−367.587	−0.0725482
$296$	10179.8	1.99895
$297$	0	0
$298$	1788.85	0.347736
$299$	−9805.72	−1.89659
$300$	0	0
$301$	4042.46	0.774098
$302$	−3418.97	−0.651456
$303$	0	0
$304$	−185.361	−0.0349710
$305$	−4912.12	−0.922188
$306$	0	0
$307$	−4362.63	−0.811037	−0.405518	−	0.914087i	$-0.632909\pi$
$307$	−4362.63	−0.811037	−0.405518	+	0.914087i	$0.632909\pi$
$308$	−1169.47	−0.216352
$309$	0	0
$310$	−1643.48	−0.301107
$311$	3542.33	0.645875	0.322937	−	0.946420i	$-0.395330\pi$
$311$	3542.33	0.645875	0.322937	+	0.946420i	$0.395330\pi$
$312$	0	0
$313$	−1808.65	−0.326616	−0.163308	−	0.986575i	$-0.552216\pi$
$313$	−1808.65	−0.326616	−0.163308	+	0.986575i	$0.552216\pi$
$314$	−2193.16	−0.394164
$315$	0	0
$316$	924.372	0.164557
$317$	−302.078	−0.0535217	−0.0267609	−	0.999642i	$-0.508519\pi$
$317$	−302.078	−0.0535217	−0.0267609	+	0.999642i	$0.508519\pi$
$318$	0	0
$319$	−2316.34	−0.406552
$320$	−2068.81	−0.361407
$321$	0	0
$322$	−4740.58	−0.820442
$323$	10721.3	1.84691
$324$	0	0
$325$	5720.30	0.976323
$326$	6504.54	1.10507
$327$	0	0
$328$	4171.13	0.702171
$329$	−9565.23	−1.60288
$330$	0	0
$331$	−4133.17	−0.686343	−0.343172	−	0.939273i	$-0.611501\pi$
$331$	−4133.17	−0.686343	−0.343172	+	0.939273i	$0.611501\pi$
$332$	1991.24	0.329167
$333$	0	0
$334$	−7017.10	−1.14958
$335$	−1796.66	−0.293020
$336$	0	0
$337$	−3246.37	−0.524750	−0.262375	−	0.964966i	$-0.584506\pi$
$337$	−3246.37	−0.524750	−0.262375	+	0.964966i	$0.584506\pi$
$338$	3919.96	0.630821
$339$	0	0
$340$	2950.96	0.470701
$341$	−1982.50	−0.314833
$342$	0	0
$343$	6493.40	1.02219
$344$	5100.10	0.799357
$345$	0	0
$346$	3949.87	0.613717
$347$	−463.911	−0.0717696	−0.0358848	−	0.999356i	$-0.511425\pi$
$347$	−463.911	−0.0717696	−0.0358848	+	0.999356i	$0.511425\pi$
$348$	0	0
$349$	−6606.06	−1.01322	−0.506611	−	0.862175i	$-0.669102\pi$
$349$	−6606.06	−1.01322	−0.506611	+	0.862175i	$0.669102\pi$
$350$	2765.48	0.422346
$351$	0	0
$352$	−2394.04	−0.362508
$353$	6284.55	0.947572	0.473786	−	0.880640i	$-0.342887\pi$
$353$	6284.55	0.947572	0.473786	+	0.880640i	$0.342887\pi$
$354$	0	0
$355$	−871.014	−0.130221
$356$	6264.48	0.932632
$357$	0	0
$358$	3781.84	0.558314
$359$	−5773.93	−0.848848	−0.424424	−	0.905464i	$-0.639523\pi$
$359$	−5773.93	−0.848848	−0.424424	+	0.905464i	$0.639523\pi$
$360$	0	0
$361$	5190.19	0.756698
$362$	2537.51	0.368421
$363$	0	0
$364$	−5818.71	−0.837866
$365$	−3420.90	−0.490570
$366$	0	0
$367$	−669.964	−0.0952911	−0.0476456	−	0.998864i	$-0.515172\pi$
$367$	−669.964	−0.0952911	−0.0476456	+	0.998864i	$0.515172\pi$
$368$	−249.474	−0.0353389
$369$	0	0
$370$	4935.88	0.693524
$371$	7195.51	1.00693
$372$	0	0
$373$	3345.79	0.464446	0.232223	−	0.972663i	$-0.425400\pi$
$373$	3345.79	0.464446	0.232223	+	0.972663i	$0.425400\pi$
$374$	−2312.72	−0.319754
$375$	0	0
$376$	−12067.8	−1.65519
$377$	−11525.0	−1.57445
$378$	0	0
$379$	6881.01	0.932595	0.466298	−	0.884628i	$-0.345588\pi$
$379$	6881.01	0.932595	0.466298	+	0.884628i	$0.345588\pi$
$380$	3316.44	0.447710
$381$	0	0
$382$	1633.89	0.218841
$383$	4080.85	0.544444	0.272222	−	0.962235i	$-0.412242\pi$
$383$	4080.85	0.544444	0.272222	+	0.962235i	$0.412242\pi$
$384$	0	0
$385$	−1502.48	−0.198893
$386$	−3319.35	−0.437695
$387$	0	0
$388$	−7216.40	−0.944219
$389$	4698.80	0.612439	0.306220	−	0.951961i	$-0.400936\pi$
$389$	4698.80	0.612439	0.306220	+	0.951961i	$0.400936\pi$
$390$	0	0
$391$	14429.6	1.86634
$392$	369.265	0.0475783
$393$	0	0
$394$	845.809	0.108150
$395$	1187.60	0.151277
$396$	0	0
$397$	10293.0	1.30124	0.650620	−	0.759404i	$-0.274509\pi$
$397$	10293.0	1.30124	0.650620	+	0.759404i	$0.274509\pi$
$398$	7391.25	0.930879
$399$	0	0
$400$	145.534	0.0181917
$401$	−7673.35	−0.955583	−0.477792	−	0.878473i	$-0.658563\pi$
$401$	−7673.35	−0.955583	−0.477792	+	0.878473i	$0.658563\pi$
$402$	0	0
$403$	−9863.97	−1.21925
$404$	−6244.45	−0.768992
$405$	0	0
$406$	−5571.78	−0.681091
$407$	5954.05	0.725138
$408$	0	0
$409$	3707.61	0.448239	0.224119	−	0.974562i	$-0.428049\pi$
$409$	3707.61	0.448239	0.224119	+	0.974562i	$0.428049\pi$
$410$	2022.46	0.243615
$411$	0	0
$412$	2274.46	0.271977
$413$	−1066.59	−0.127079
$414$	0	0
$415$	2558.27	0.302603
$416$	−11911.6	−1.40388
$417$	0	0
$418$	−2599.16	−0.304136
$419$	11357.0	1.32416	0.662082	−	0.749431i	$-0.269673\pi$
$419$	11357.0	1.32416	0.662082	+	0.749431i	$0.269673\pi$
$420$	0	0
$421$	3814.03	0.441531	0.220766	−	0.975327i	$-0.429144\pi$
$421$	3814.03	0.441531	0.220766	+	0.975327i	$0.429144\pi$
$422$	−4436.10	−0.511720
$423$	0	0
$424$	9078.10	1.03979
$425$	−8417.72	−0.960752
$426$	0	0
$427$	−14253.1	−1.61535
$428$	−5999.72	−0.677588
$429$	0	0
$430$	2472.89	0.277333
$431$	1307.09	0.146079	0.0730396	−	0.997329i	$-0.476730\pi$
$431$	1307.09	0.146079	0.0730396	+	0.997329i	$0.476730\pi$
$432$	0	0
$433$	7915.74	0.878536	0.439268	−	0.898356i	$-0.355238\pi$
$433$	7915.74	0.878536	0.439268	+	0.898356i	$0.355238\pi$
$434$	−4768.74	−0.527435
$435$	0	0
$436$	−10402.5	−1.14263
$437$	16216.8	1.77518
$438$	0	0
$439$	2570.45	0.279456	0.139728	−	0.990190i	$-0.455377\pi$
$439$	2570.45	0.279456	0.139728	+	0.990190i	$0.455377\pi$
$440$	−1895.58	−0.205383
$441$	0	0
$442$	−11507.0	−1.23831
$443$	633.121	0.0679017	0.0339509	−	0.999424i	$-0.489191\pi$
$443$	633.121	0.0679017	0.0339509	+	0.999424i	$0.489191\pi$
$444$	0	0
$445$	8048.36	0.857369
$446$	4557.40	0.483855
$447$	0	0
$448$	−6002.90	−0.633059
$449$	−10146.6	−1.06648	−0.533238	−	0.845965i	$-0.679025\pi$
$449$	−10146.6	−1.06648	−0.533238	+	0.845965i	$0.679025\pi$
$450$	0	0
$451$	2439.65	0.254720
$452$	5405.22	0.562478
$453$	0	0
$454$	1720.01	0.177806
$455$	−7475.65	−0.770250
$456$	0	0
$457$	−7173.92	−0.734315	−0.367157	−	0.930159i	$-0.619669\pi$
$457$	−7173.92	−0.734315	−0.367157	+	0.930159i	$0.619669\pi$
$458$	7485.47	0.763697
$459$	0	0
$460$	4463.53	0.452420
$461$	15959.8	1.61242	0.806208	−	0.591632i	$-0.201516\pi$
$461$	15959.8	1.61242	0.806208	+	0.591632i	$0.201516\pi$
$462$	0	0
$463$	10486.8	1.05262	0.526308	−	0.850294i	$-0.323576\pi$
$463$	10486.8	1.05262	0.526308	+	0.850294i	$0.323576\pi$
$464$	−293.216	−0.0293366
$465$	0	0
$466$	−678.205	−0.0674190
$467$	11871.2	1.17630	0.588150	−	0.808752i	$-0.299856\pi$
$467$	11871.2	1.17630	0.588150	+	0.808752i	$0.299856\pi$
$468$	0	0
$469$	−5213.20	−0.513269
$470$	−5851.33	−0.574259
$471$	0	0
$472$	−1345.65	−0.131226
$473$	2983.00	0.289975
$474$	0	0
$475$	−9460.27	−0.913825
$476$	8562.54	0.824503
$477$	0	0
$478$	7223.72	0.691225
$479$	9570.85	0.912950	0.456475	−	0.889736i	$-0.349112\pi$
$479$	9570.85	0.912950	0.456475	+	0.889736i	$0.349112\pi$
$480$	0	0
$481$	29624.6	2.80824
$482$	8117.28	0.767078
$483$	0	0
$484$	5591.54	0.525126
$485$	−9271.34	−0.868021
$486$	0	0
$487$	11208.2	1.04290	0.521451	−	0.853281i	$-0.325391\pi$
$487$	11208.2	1.04290	0.521451	+	0.853281i	$0.325391\pi$
$488$	−17982.2	−1.66806
$489$	0	0
$490$	179.046	0.0165071
$491$	−7720.19	−0.709587	−0.354793	−	0.934945i	$-0.615449\pi$
$491$	−7720.19	−0.709587	−0.354793	+	0.934945i	$0.615449\pi$
$492$	0	0
$493$	16959.7	1.54934
$494$	−12932.2	−1.17783
$495$	0	0
$496$	−250.956	−0.0227182
$497$	−2527.35	−0.228103
$498$	0	0
$499$	−3679.73	−0.330115	−0.165057	−	0.986284i	$-0.552781\pi$
$499$	−3679.73	−0.330115	−0.165057	+	0.986284i	$0.552781\pi$
$500$	−6380.48	−0.570687
$501$	0	0
$502$	−7568.00	−0.672861
$503$	−15306.9	−1.35686	−0.678431	−	0.734664i	$-0.737340\pi$
$503$	−15306.9	−1.35686	−0.678431	+	0.734664i	$0.737340\pi$
$504$	0	0
$505$	−8022.62	−0.706935
$506$	−3498.15	−0.307336
$507$	0	0
$508$	6147.77	0.536935
$509$	9361.34	0.815195	0.407597	−	0.913162i	$-0.366367\pi$
$509$	9361.34	0.815195	0.407597	+	0.913162i	$0.366367\pi$
$510$	0	0
$511$	−9926.14	−0.859308
$512$	−611.164	−0.0527537
$513$	0	0
$514$	−6671.52	−0.572506
$515$	2922.14	0.250029
$516$	0	0
$517$	−7058.34	−0.600436
$518$	14322.0	1.21481
$519$	0	0
$520$	−9431.53	−0.795385
$521$	5933.23	0.498924	0.249462	−	0.968385i	$-0.419746\pi$
$521$	5933.23	0.498924	0.249462	+	0.968385i	$0.419746\pi$
$522$	0	0
$523$	17269.3	1.44385	0.721925	−	0.691972i	$-0.243258\pi$
$523$	17269.3	1.44385	0.721925	+	0.691972i	$0.243258\pi$
$524$	−10278.2	−0.856879
$525$	0	0
$526$	5028.09	0.416797
$527$	14515.4	1.19981
$528$	0	0
$529$	9658.83	0.793855
$530$	4401.70	0.360750
$531$	0	0
$532$	9623.03	0.784231
$533$	12138.6	0.986453
$534$	0	0
$535$	−7708.21	−0.622907
$536$	−6577.15	−0.530018
$537$	0	0
$538$	13049.0	1.04569
$539$	215.979	0.0172595
$540$	0	0
$541$	−1132.58	−0.0900067	−0.0450034	−	0.998987i	$-0.514330\pi$
$541$	−1132.58	−0.0900067	−0.0450034	+	0.998987i	$0.514330\pi$
$542$	−7194.17	−0.570141
$543$	0	0
$544$	17528.6	1.38149
$545$	−13364.7	−1.05042
$546$	0	0
$547$	8546.77	0.668069	0.334034	−	0.942561i	$-0.391590\pi$
$547$	8546.77	0.668069	0.334034	+	0.942561i	$0.391590\pi$
$548$	−4100.55	−0.319647
$549$	0	0
$550$	2040.69	0.158210
$551$	19060.2	1.47367
$552$	0	0
$553$	3445.95	0.264985
$554$	5421.11	0.415741
$555$	0	0
$556$	6745.12	0.514490
$557$	16689.2	1.26956	0.634778	−	0.772695i	$-0.281092\pi$
$557$	16689.2	1.26956	0.634778	+	0.772695i	$0.281092\pi$
$558$	0	0
$559$	14842.0	1.12299
$560$	−190.193	−0.0143520
$561$	0	0
$562$	6234.23	0.467927
$563$	6844.25	0.512346	0.256173	−	0.966631i	$-0.417538\pi$
$563$	6844.25	0.512346	0.256173	+	0.966631i	$0.417538\pi$
$564$	0	0
$565$	6944.41	0.517086
$566$	−14121.0	−1.04867
$567$	0	0
$568$	−3188.58	−0.235546
$569$	12047.3	0.887611	0.443806	−	0.896123i	$-0.353628\pi$
$569$	12047.3	0.887611	0.443806	+	0.896123i	$0.353628\pi$
$570$	0	0
$571$	−14990.8	−1.09868	−0.549338	−	0.835600i	$-0.685120\pi$
$571$	−14990.8	−1.09868	−0.549338	+	0.835600i	$0.685120\pi$
$572$	−4293.72	−0.313863
$573$	0	0
$574$	5868.39	0.426728
$575$	−12732.4	−0.923439
$576$	0	0
$577$	−5773.70	−0.416572	−0.208286	−	0.978068i	$-0.566788\pi$
$577$	−5773.70	−0.416572	−0.208286	+	0.978068i	$0.566788\pi$
$578$	8212.61	0.591002
$579$	0	0
$580$	5246.16	0.375577
$581$	7423.10	0.530055
$582$	0	0
$583$	5309.69	0.377195
$584$	−12523.1	−0.887348
$585$	0	0
$586$	−8756.54	−0.617286
$587$	1595.79	0.112207	0.0561034	−	0.998425i	$-0.482132\pi$
$587$	1595.79	0.112207	0.0561034	+	0.998425i	$0.482132\pi$
$588$	0	0
$589$	16313.1	1.14121
$590$	−652.467	−0.0455282
$591$	0	0
$592$	753.698	0.0523256
$593$	−5301.19	−0.367106	−0.183553	−	0.983010i	$-0.558760\pi$
$593$	−5301.19	−0.367106	−0.183553	+	0.983010i	$0.558760\pi$
$594$	0	0
$595$	11000.8	0.757966
$596$	−4887.20	−0.335885
$597$	0	0
$598$	−17405.2	−1.19022
$599$	−12610.9	−0.860211	−0.430105	−	0.902779i	$-0.641524\pi$
$599$	−12610.9	−0.860211	−0.430105	+	0.902779i	$0.641524\pi$
$600$	0	0
$601$	−27751.9	−1.88357	−0.941783	−	0.336221i	$-0.890851\pi$
$601$	−27751.9	−1.88357	−0.941783	+	0.336221i	$0.890851\pi$
$602$	7175.37	0.485791
$603$	0	0
$604$	9340.76	0.629255
$605$	7183.80	0.482749
$606$	0	0
$607$	−20229.0	−1.35267	−0.676334	−	0.736595i	$-0.736433\pi$
$607$	−20229.0	−1.35267	−0.676334	+	0.736595i	$0.736433\pi$
$608$	19699.5	1.31402
$609$	0	0
$610$	−8719.02	−0.578726
$611$	−35119.0	−2.32531
$612$	0	0
$613$	−21903.1	−1.44316	−0.721580	−	0.692331i	$-0.756584\pi$
$613$	−21903.1	−1.44316	−0.721580	+	0.692331i	$0.756584\pi$
$614$	−7743.67	−0.508972
$615$	0	0
$616$	−5500.25	−0.359759
$617$	−8023.42	−0.523519	−0.261759	−	0.965133i	$-0.584303\pi$
$617$	−8023.42	−0.523519	−0.261759	+	0.965133i	$0.584303\pi$
$618$	0	0
$619$	−6262.61	−0.406649	−0.203324	−	0.979111i	$-0.565175\pi$
$619$	−6262.61	−0.406649	−0.203324	+	0.979111i	$0.565175\pi$
$620$	4490.05	0.290846
$621$	0	0
$622$	6287.64	0.405324
$623$	23353.2	1.50181
$624$	0	0
$625$	2575.56	0.164836
$626$	−3210.35	−0.204970
$627$	0	0
$628$	5991.81	0.380731
$629$	−43594.1	−2.76345
$630$	0	0
$631$	−6018.53	−0.379705	−0.189852	−	0.981813i	$-0.560801\pi$
$631$	−6018.53	−0.379705	−0.189852	+	0.981813i	$0.560801\pi$
$632$	4347.53	0.273632
$633$	0	0
$634$	−536.189	−0.0335880
$635$	7898.42	0.493605
$636$	0	0
$637$	1074.61	0.0668409
$638$	−4111.51	−0.255135
$639$	0	0
$640$	5272.74	0.325661
$641$	−3669.06	−0.226083	−0.113041	−	0.993590i	$-0.536059\pi$
$641$	−3669.06	−0.226083	−0.113041	+	0.993590i	$0.536059\pi$
$642$	0	0
$643$	−9099.99	−0.558116	−0.279058	−	0.960274i	$-0.590022\pi$
$643$	−9099.99	−0.558116	−0.279058	+	0.960274i	$0.590022\pi$
$644$	12951.4	0.792482
$645$	0	0
$646$	19030.4	1.15904
$647$	1873.56	0.113844	0.0569222	−	0.998379i	$-0.481871\pi$
$647$	1873.56	0.113844	0.0569222	+	0.998379i	$0.481871\pi$
$648$	0	0
$649$	−787.058	−0.0476036
$650$	10153.5	0.612699
$651$	0	0
$652$	−17770.7	−1.06741
$653$	23009.5	1.37892	0.689458	−	0.724325i	$-0.257849\pi$
$653$	23009.5	1.37892	0.689458	+	0.724325i	$0.257849\pi$
$654$	0	0
$655$	−13205.0	−0.787729
$656$	308.825	0.0183805
$657$	0	0
$658$	−16978.3	−1.00590
$659$	10038.1	0.593368	0.296684	−	0.954976i	$-0.404119\pi$
$659$	10038.1	0.593368	0.296684	+	0.954976i	$0.404119\pi$
$660$	0	0
$661$	−21406.5	−1.25963	−0.629816	−	0.776744i	$-0.716870\pi$
$661$	−21406.5	−1.25963	−0.629816	+	0.776744i	$0.716870\pi$
$662$	−7336.38	−0.430720
$663$	0	0
$664$	9365.23	0.547351
$665$	12363.3	0.720944
$666$	0	0
$667$	25652.7	1.48917
$668$	19171.0	1.11040
$669$	0	0
$670$	−3189.07	−0.183887
$671$	−10517.6	−0.605107
$672$	0	0
$673$	−934.831	−0.0535439	−0.0267720	−	0.999642i	$-0.508523\pi$
$673$	−934.831	−0.0535439	−0.0267720	+	0.999642i	$0.508523\pi$
$674$	−5762.30	−0.329311
$675$	0	0
$676$	−10709.5	−0.609324
$677$	3937.51	0.223531	0.111766	−	0.993735i	$-0.464349\pi$
$677$	3937.51	0.223531	0.111766	+	0.993735i	$0.464349\pi$
$678$	0	0
$679$	−26901.9	−1.52047
$680$	13879.0	0.782699
$681$	0	0
$682$	−3518.93	−0.197576
$683$	12374.6	0.693268	0.346634	−	0.938001i	$-0.387325\pi$
$683$	12374.6	0.693268	0.346634	+	0.938001i	$0.387325\pi$
$684$	0	0
$685$	−5268.22	−0.293852
$686$	11525.8	0.641482
$687$	0	0
$688$	377.605	0.0209245
$689$	26418.5	1.46076
$690$	0	0
$691$	−5887.06	−0.324102	−0.162051	−	0.986782i	$-0.551811\pi$
$691$	−5887.06	−0.324102	−0.162051	+	0.986782i	$0.551811\pi$
$692$	−10791.2	−0.592802
$693$	0	0
$694$	−823.443	−0.0450396
$695$	8665.87	0.472971
$696$	0	0
$697$	−17862.5	−0.970720
$698$	−11725.8	−0.635855
$699$	0	0
$700$	−7555.40	−0.407953
$701$	−7809.40	−0.420766	−0.210383	−	0.977619i	$-0.567471\pi$
$701$	−7809.40	−0.420766	−0.210383	+	0.977619i	$0.567471\pi$
$702$	0	0
$703$	−48993.3	−2.62847
$704$	−4429.64	−0.237142
$705$	0	0
$706$	11155.1	0.594656
$707$	−23278.6	−1.23830
$708$	0	0
$709$	13649.5	0.723018	0.361509	−	0.932369i	$-0.382262\pi$
$709$	13649.5	0.723018	0.361509	+	0.932369i	$0.382262\pi$
$710$	−1546.05	−0.0817215
$711$	0	0
$712$	29463.2	1.55082
$713$	21955.5	1.15321
$714$	0	0
$715$	−5516.41	−0.288534
$716$	−10332.1	−0.539288
$717$	0	0
$718$	−10248.7	−0.532701
$719$	−475.869	−0.0246828	−0.0123414	−	0.999924i	$-0.503928\pi$
$719$	−475.869	−0.0246828	−0.0123414	+	0.999924i	$0.503928\pi$
$720$	0	0
$721$	8478.92	0.437963
$722$	9212.60	0.474872
$723$	0	0
$724$	−6932.57	−0.355866
$725$	−14964.9	−0.766594
$726$	0	0
$727$	−8057.43	−0.411050	−0.205525	−	0.978652i	$-0.565890\pi$
$727$	−8057.43	−0.411050	−0.205525	+	0.978652i	$0.565890\pi$
$728$	−27366.7	−1.39324
$729$	0	0
$730$	−6072.11	−0.307861
$731$	−21840.8	−1.10508
$732$	0	0
$733$	28918.2	1.45719	0.728594	−	0.684946i	$-0.240174\pi$
$733$	28918.2	1.45719	0.728594	+	0.684946i	$0.240174\pi$
$734$	−1189.19	−0.0598007
$735$	0	0
$736$	26513.2	1.32784
$737$	−3846.91	−0.192270
$738$	0	0
$739$	−25403.3	−1.26451	−0.632256	−	0.774760i	$-0.717871\pi$
$739$	−25403.3	−1.26451	−0.632256	+	0.774760i	$0.717871\pi$
$740$	−13485.0	−0.669890
$741$	0	0
$742$	12772.0	0.631909
$743$	30468.3	1.50441	0.752204	−	0.658930i	$-0.228991\pi$
$743$	30468.3	1.50441	0.752204	+	0.658930i	$0.228991\pi$
$744$	0	0
$745$	−6278.89	−0.308779
$746$	5938.77	0.291466
$747$	0	0
$748$	6318.44	0.308857
$749$	−22366.3	−1.09112
$750$	0	0
$751$	−1951.23	−0.0948088	−0.0474044	−	0.998876i	$-0.515095\pi$
$751$	−1951.23	−0.0948088	−0.0474044	+	0.998876i	$0.515095\pi$
$752$	−893.485	−0.0433272
$753$	0	0
$754$	−20456.9	−0.988061
$755$	12000.6	0.578474
$756$	0	0
$757$	13529.5	0.649588	0.324794	−	0.945785i	$-0.394705\pi$
$757$	13529.5	0.649588	0.324794	+	0.945785i	$0.394705\pi$
$758$	12213.8	0.585257
$759$	0	0
$760$	15597.9	0.744469
$761$	−31379.5	−1.49475	−0.747376	−	0.664402i	$-0.768686\pi$
$761$	−31379.5	−1.49475	−0.747376	+	0.664402i	$0.768686\pi$
$762$	0	0
$763$	−38779.2	−1.83998
$764$	−4463.86	−0.211383
$765$	0	0
$766$	7243.52	0.341670
$767$	−3916.03	−0.184354
$768$	0	0
$769$	−1805.38	−0.0846604	−0.0423302	−	0.999104i	$-0.513478\pi$
$769$	−1805.38	−0.0846604	−0.0423302	+	0.999104i	$0.513478\pi$
$770$	−2666.91	−0.124817
$771$	0	0
$772$	9068.58	0.422779
$773$	−27743.2	−1.29089	−0.645443	−	0.763808i	$-0.723327\pi$
$773$	−27743.2	−1.29089	−0.645443	+	0.763808i	$0.723327\pi$
$774$	0	0
$775$	−12808.0	−0.593649
$776$	−33940.3	−1.57008
$777$	0	0
$778$	8340.38	0.384341
$779$	−20074.8	−0.923306
$780$	0	0
$781$	−1864.97	−0.0854467
$782$	25612.6	1.17123
$783$	0	0
$784$	27.3399	0.00124544
$785$	7698.04	0.350006
$786$	0	0
$787$	−2980.67	−0.135005	−0.0675027	−	0.997719i	$-0.521503\pi$
$787$	−2980.67	−0.135005	−0.0675027	+	0.997719i	$0.521503\pi$
$788$	−2310.78	−0.104465
$789$	0	0
$790$	2107.99	0.0949352
$791$	20150.0	0.905755
$792$	0	0
$793$	−52330.6	−2.34340
$794$	18270.1	0.816603
$795$	0	0
$796$	−20193.2	−0.899156
$797$	−26151.2	−1.16226	−0.581131	−	0.813810i	$-0.697390\pi$
$797$	−26151.2	−1.16226	−0.581131	+	0.813810i	$0.697390\pi$
$798$	0	0
$799$	51679.5	2.28822
$800$	−15466.8	−0.683544
$801$	0	0
$802$	−13620.2	−0.599684
$803$	−7324.66	−0.321895
$804$	0	0
$805$	16639.5	0.728529
$806$	−17508.6	−0.765152
$807$	0	0
$808$	−29369.0	−1.27871
$809$	30368.8	1.31979	0.659895	−	0.751358i	$-0.270601\pi$
$809$	30368.8	1.31979	0.659895	+	0.751358i	$0.270601\pi$
$810$	0	0
$811$	−5497.72	−0.238041	−0.119020	−	0.992892i	$-0.537975\pi$
$811$	−5497.72	−0.238041	−0.119020	+	0.992892i	$0.537975\pi$
$812$	15222.3	0.657880
$813$	0	0
$814$	10568.4	0.455066
$815$	−22831.0	−0.981272
$816$	0	0
$817$	−24545.8	−1.05110
$818$	6581.02	0.281296
$819$	0	0
$820$	−5525.43	−0.235313
$821$	33287.7	1.41504	0.707521	−	0.706692i	$-0.249814\pi$
$821$	33287.7	1.41504	0.707521	+	0.706692i	$0.249814\pi$
$822$	0	0
$823$	−13109.6	−0.555253	−0.277626	−	0.960689i	$-0.589548\pi$
$823$	−13109.6	−0.555253	−0.277626	+	0.960689i	$0.589548\pi$
$824$	10697.3	0.452254
$825$	0	0
$826$	−1893.21	−0.0797495
$827$	−31629.7	−1.32995	−0.664977	−	0.746864i	$-0.731559\pi$
$827$	−31629.7	−1.32995	−0.664977	+	0.746864i	$0.731559\pi$
$828$	0	0
$829$	23175.2	0.970938	0.485469	−	0.874254i	$-0.338649\pi$
$829$	23175.2	0.970938	0.485469	+	0.874254i	$0.338649\pi$
$830$	4540.92	0.189901
$831$	0	0
$832$	−22039.8	−0.918380
$833$	−1581.35	−0.0657748
$834$	0	0
$835$	24630.1	1.02079
$836$	7100.99	0.293771
$837$	0	0
$838$	20158.7	0.830990
$839$	−24731.6	−1.01767	−0.508837	−	0.860863i	$-0.669924\pi$
$839$	−24731.6	−1.01767	−0.508837	+	0.860863i	$0.669924\pi$
$840$	0	0
$841$	5761.61	0.236238
$842$	6769.91	0.277086
$843$	0	0
$844$	12119.6	0.494281
$845$	−13759.1	−0.560151
$846$	0	0
$847$	20844.6	0.845607
$848$	672.130	0.0272182
$849$	0	0
$850$	−14941.5	−0.602927
$851$	−65939.1	−2.65613
$852$	0	0
$853$	26779.6	1.07493	0.537466	−	0.843285i	$-0.319382\pi$
$853$	26779.6	1.07493	0.537466	+	0.843285i	$0.319382\pi$
$854$	−25299.2	−1.01373
$855$	0	0
$856$	−28218.0	−1.12672
$857$	−32339.7	−1.28904	−0.644518	−	0.764589i	$-0.722942\pi$
$857$	−32339.7	−1.28904	−0.644518	+	0.764589i	$0.722942\pi$
$858$	0	0
$859$	41682.2	1.65562	0.827810	−	0.561009i	$-0.189587\pi$
$859$	41682.2	1.65562	0.827810	+	0.561009i	$0.189587\pi$
$860$	−6756.02	−0.267882
$861$	0	0
$862$	2320.08	0.0916732
$863$	15307.7	0.603802	0.301901	−	0.953339i	$-0.402379\pi$
$863$	15307.7	0.603802	0.301901	+	0.953339i	$0.402379\pi$
$864$	0	0
$865$	−13864.1	−0.544963
$866$	14050.4	0.551332
$867$	0	0
$868$	13028.4	0.509461
$869$	2542.82	0.0992628
$870$	0	0
$871$	−19140.4	−0.744601
$872$	−48925.1	−1.90002
$873$	0	0
$874$	28784.8	1.11403
$875$	−23785.7	−0.918974
$876$	0	0
$877$	−42064.0	−1.61961	−0.809807	−	0.586696i	$-0.800428\pi$
$877$	−42064.0	−1.61961	−0.809807	+	0.586696i	$0.800428\pi$
$878$	4562.56	0.175374
$879$	0	0
$880$	−140.346	−0.00537623
$881$	983.391	0.0376065	0.0188032	−	0.999823i	$-0.494014\pi$
$881$	983.391	0.0376065	0.0188032	+	0.999823i	$0.494014\pi$
$882$	0	0
$883$	3067.46	0.116906	0.0584531	−	0.998290i	$-0.481383\pi$
$883$	3067.46	0.116906	0.0584531	+	0.998290i	$0.481383\pi$
$884$	31437.6	1.19611
$885$	0	0
$886$	1123.79	0.0426123
$887$	−15178.8	−0.574583	−0.287291	−	0.957843i	$-0.592755\pi$
$887$	−15178.8	−0.574583	−0.287291	+	0.957843i	$0.592755\pi$
$888$	0	0
$889$	22918.2	0.864624
$890$	14285.9	0.538048
$891$	0	0
$892$	−12451.0	−0.467366
$893$	58080.1	2.17646
$894$	0	0
$895$	−13274.3	−0.495767
$896$	15299.4	0.570445
$897$	0	0
$898$	−18010.2	−0.669275
$899$	25805.1	0.957340
$900$	0	0
$901$	−38876.2	−1.43746
$902$	4330.38	0.159851
$903$	0	0
$904$	25421.9	0.935310
$905$	−8906.70	−0.327148
$906$	0	0
$907$	−2893.09	−0.105914	−0.0529568	−	0.998597i	$-0.516865\pi$
$907$	−2893.09	−0.105914	−0.0529568	+	0.998597i	$0.516865\pi$
$908$	−4699.13	−0.171747
$909$	0	0
$910$	−13269.3	−0.483377
$911$	2299.05	0.0836124	0.0418062	−	0.999126i	$-0.486689\pi$
$911$	2299.05	0.0836124	0.0418062	+	0.999126i	$0.486689\pi$
$912$	0	0
$913$	5477.63	0.198557
$914$	−12733.7	−0.460825
$915$	0	0
$916$	−20450.6	−0.737671
$917$	−38315.8	−1.37983
$918$	0	0
$919$	−45794.9	−1.64378	−0.821891	−	0.569645i	$-0.807081\pi$
$919$	−45794.9	−1.64378	−0.821891	+	0.569645i	$0.807081\pi$
$920$	20993.0	0.752301
$921$	0	0
$922$	28328.7	1.01188
$923$	−9279.22	−0.330909
$924$	0	0
$925$	38466.5	1.36732
$926$	18614.0	0.660578
$927$	0	0
$928$	31162.0	1.10231
$929$	43137.0	1.52344	0.761721	−	0.647905i	$-0.224354\pi$
$929$	43137.0	1.52344	0.761721	+	0.647905i	$0.224354\pi$
$930$	0	0
$931$	−1777.20	−0.0625621
$932$	1852.88	0.0651214
$933$	0	0
$934$	21071.4	0.738197
$935$	8117.69	0.283932
$936$	0	0
$937$	20420.6	0.711966	0.355983	−	0.934492i	$-0.384146\pi$
$937$	20420.6	0.711966	0.355983	+	0.934492i	$0.384146\pi$
$938$	−9253.44	−0.322106
$939$	0	0
$940$	15986.1	0.554689
$941$	21235.7	0.735668	0.367834	−	0.929892i	$-0.380100\pi$
$941$	21235.7	0.735668	0.367834	+	0.929892i	$0.380100\pi$
$942$	0	0
$943$	−27018.3	−0.933020
$944$	−99.6303	−0.00343505
$945$	0	0
$946$	5294.82	0.181976
$947$	21631.9	0.742283	0.371141	−	0.928576i	$-0.378966\pi$
$947$	21631.9	0.742283	0.371141	+	0.928576i	$0.378966\pi$
$948$	0	0
$949$	−36444.1	−1.24660
$950$	−16792.0	−0.573478
$951$	0	0
$952$	40271.5	1.37102
$953$	−869.752	−0.0295635	−0.0147818	−	0.999891i	$-0.504705\pi$
$953$	−869.752	−0.0295635	−0.0147818	+	0.999891i	$0.504705\pi$
$954$	0	0
$955$	−5734.99	−0.194324
$956$	−19735.5	−0.667668
$957$	0	0
$958$	16988.3	0.572929
$959$	−15286.4	−0.514726
$960$	0	0
$961$	−7705.09	−0.258638
$962$	52583.6	1.76233
$963$	0	0
$964$	−22176.7	−0.740937
$965$	11651.0	0.388661
$966$	0	0
$967$	45869.5	1.52540	0.762700	−	0.646752i	$-0.223873\pi$
$967$	45869.5	1.52540	0.762700	+	0.646752i	$0.223873\pi$
$968$	26298.3	0.873200
$969$	0	0
$970$	−16456.6	−0.544733
$971$	−4456.05	−0.147272	−0.0736361	−	0.997285i	$-0.523460\pi$
$971$	−4456.05	−0.147272	−0.0736361	+	0.997285i	$0.523460\pi$
$972$	0	0
$973$	25145.0	0.828481
$974$	19894.6	0.654481
$975$	0	0
$976$	−1331.38	−0.0436642
$977$	34528.0	1.13065	0.565326	−	0.824867i	$-0.308750\pi$
$977$	34528.0	1.13065	0.565326	+	0.824867i	$0.308750\pi$
$978$	0	0
$979$	17232.7	0.562575
$980$	−489.159	−0.0159445
$981$	0	0
$982$	−13703.3	−0.445307
$983$	−17284.8	−0.560834	−0.280417	−	0.959878i	$-0.590473\pi$
$983$	−17284.8	−0.560834	−0.280417	+	0.959878i	$0.590473\pi$
$984$	0	0
$985$	−2968.80	−0.0960344
$986$	30103.5	0.972303
$987$	0	0
$988$	35331.2	1.13769
$989$	−33035.7	−1.06216
$990$	0	0
$991$	−49628.1	−1.59081	−0.795403	−	0.606081i	$-0.792741\pi$
$991$	−49628.1	−1.59081	−0.795403	+	0.606081i	$0.792741\pi$
$992$	26670.7	0.853625
$993$	0	0
$994$	−4486.04	−0.143148
$995$	−25943.4	−0.826594
$996$	0	0
$997$	−1153.92	−0.0366549	−0.0183274	−	0.999832i	$-0.505834\pi$
$997$	−1153.92	−0.0366549	−0.0183274	+	0.999832i	$0.505834\pi$
$998$	−6531.53	−0.207166
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.d.1.4		7
3.2	odd	2		177.4.a.a.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.4	✓	7	3.2	odd	2
531.4.a.d.1.4		7	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+1.77500 q^{2} -4.84937 q^{4} -6.23028 q^{5} -18.0779 q^{7} -22.8076 q^{8} +O(q^{10})\)	\(q+1.77500 q^{2} -4.84937 q^{4} -6.23028 q^{5} -18.0779 q^{7} -22.8076 q^{8} -11.0588 q^{10} -13.3400 q^{11} -66.3734 q^{13} -32.0883 q^{14} -1.68865 q^{16} +97.6720 q^{17} +109.769 q^{19} +30.2129 q^{20} -23.6785 q^{22} +147.736 q^{23} -86.1836 q^{25} -117.813 q^{26} +87.6663 q^{28} +173.639 q^{29} +148.613 q^{31} +179.464 q^{32} +173.368 q^{34} +112.630 q^{35} -446.332 q^{37} +194.840 q^{38} +142.098 q^{40} -182.883 q^{41} -223.613 q^{43} +64.6904 q^{44} +262.231 q^{46} +529.113 q^{47} -16.1904 q^{49} -152.976 q^{50} +321.869 q^{52} -398.029 q^{53} +83.1117 q^{55} +412.314 q^{56} +308.210 q^{58} +59.0000 q^{59} +788.427 q^{61} +263.789 q^{62} +332.058 q^{64} +413.525 q^{65} +288.375 q^{67} -473.648 q^{68} +199.919 q^{70} +139.803 q^{71} +549.077 q^{73} -792.240 q^{74} -532.310 q^{76} +241.158 q^{77} -190.617 q^{79} +10.5208 q^{80} -324.617 q^{82} -410.618 q^{83} -608.524 q^{85} -396.914 q^{86} +304.253 q^{88} -1291.81 q^{89} +1199.89 q^{91} -716.425 q^{92} +939.176 q^{94} -683.891 q^{95} +1488.11 q^{97} -28.7380 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8}+O(q^{10}) \) 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8	\( 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8} - 79 q^{10} + 131 q^{11} - 123 q^{13} + 117 q^{14} + 202 q^{16} + 235 q^{17} - 80 q^{19} - 61 q^{20} + 688 q^{22} + 274 q^{23} + 193 q^{25} + 180 q^{26} - 118 q^{28} + 406 q^{29} - 346 q^{31} + 854 q^{32} + 178 q^{34} + 424 q^{35} - 157 q^{37} + 129 q^{38} - 590 q^{40} + 825 q^{41} - 815 q^{43} + 1690 q^{44} + 1457 q^{46} + 1196 q^{47} + 914 q^{49} - 713 q^{50} + 1030 q^{52} + 900 q^{53} - 1044 q^{55} - 2172 q^{56} + 1242 q^{58} + 413 q^{59} + 420 q^{61} - 646 q^{62} + 3541 q^{64} - 190 q^{65} + 1316 q^{67} + 611 q^{68} + 4658 q^{70} + 173 q^{71} - 418 q^{73} - 660 q^{74} + 1540 q^{76} + 753 q^{77} + 2635 q^{79} - 6155 q^{80} - 125 q^{82} - 457 q^{83} + 1270 q^{85} - 3482 q^{86} + 7685 q^{88} - 592 q^{89} + 3179 q^{91} + 3500 q^{92} + 2064 q^{94} + 2250 q^{95} - 1906 q^{97} - 2994 q^{98}+O(q^{100}) \) 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8 - 79 * q^10 + 131 * q^11 - 123 * q^13 + 117 * q^14 + 202 * q^16 + 235 * q^17 - 80 * q^19 - 61 * q^20 + 688 * q^22 + 274 * q^23 + 193 * q^25 + 180 * q^26 - 118 * q^28 + 406 * q^29 - 346 * q^31 + 854 * q^32 + 178 * q^34 + 424 * q^35 - 157 * q^37 + 129 * q^38 - 590 * q^40 + 825 * q^41 - 815 * q^43 + 1690 * q^44 + 1457 * q^46 + 1196 * q^47 + 914 * q^49 - 713 * q^50 + 1030 * q^52 + 900 * q^53 - 1044 * q^55 - 2172 * q^56 + 1242 * q^58 + 413 * q^59 + 420 * q^61 - 646 * q^62 + 3541 * q^64 - 190 * q^65 + 1316 * q^67 + 611 * q^68 + 4658 * q^70 + 173 * q^71 - 418 * q^73 - 660 * q^74 + 1540 * q^76 + 753 * q^77 + 2635 * q^79 - 6155 * q^80 - 125 * q^82 - 457 * q^83 + 1270 * q^85 - 3482 * q^86 + 7685 * q^88 - 592 * q^89 + 3179 * q^91 + 3500 * q^92 + 2064 * q^94 + 2250 * q^95 - 1906 * q^97 - 2994 * q^98

Introduction

L-functions

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