LMFDB - Embedded newform 531.4.a.f.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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.780043$ 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)

$f(q)$	$=$	$q-0.780043 q^{2} -7.39153 q^{4} +21.9196 q^{5} +32.1153 q^{7} +12.0061 q^{8} +O(q^{10})$	\(q-0.780043 q^{2} -7.39153 q^{4} +21.9196 q^{5} +32.1153 q^{7} +12.0061 q^{8} -17.0983 q^{10} +35.3089 q^{11} +34.5155 q^{13} -25.0513 q^{14} +49.7670 q^{16} +27.9810 q^{17} +74.3473 q^{19} -162.020 q^{20} -27.5424 q^{22} -128.081 q^{23} +355.471 q^{25} -26.9235 q^{26} -237.381 q^{28} -42.7099 q^{29} -211.536 q^{31} -134.869 q^{32} -21.8263 q^{34} +703.955 q^{35} -165.794 q^{37} -57.9940 q^{38} +263.168 q^{40} -377.795 q^{41} -393.343 q^{43} -260.987 q^{44} +99.9085 q^{46} -261.503 q^{47} +688.391 q^{49} -277.282 q^{50} -255.122 q^{52} +113.158 q^{53} +773.957 q^{55} +385.578 q^{56} +33.3155 q^{58} +59.0000 q^{59} +337.123 q^{61} +165.007 q^{62} -292.933 q^{64} +756.567 q^{65} +183.638 q^{67} -206.822 q^{68} -549.115 q^{70} +168.306 q^{71} +805.659 q^{73} +129.326 q^{74} -549.540 q^{76} +1133.95 q^{77} +797.644 q^{79} +1090.88 q^{80} +294.696 q^{82} -251.649 q^{83} +613.333 q^{85} +306.824 q^{86} +423.920 q^{88} -653.582 q^{89} +1108.47 q^{91} +946.713 q^{92} +203.983 q^{94} +1629.67 q^{95} -1574.87 q^{97} -536.974 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * 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 $

$2$	−0.780043	−0.275787	−0.137893	−	0.990447i	$-0.544033\pi$
$2$	−0.780043	−0.275787	−0.137893	+	0.990447i	$0.544033\pi$
$3$	0	0
$3$	0	0
$4$	−7.39153	−0.923942
$5$	21.9196	1.96055	0.980276	−	0.197633i	$-0.0633253\pi$
$5$	21.9196	1.96055	0.980276	+	0.197633i	$0.0633253\pi$
$6$	0	0
$7$	32.1153	1.73406	0.867031	−	0.498255i	$-0.166026\pi$
$7$	32.1153	1.73406	0.867031	+	0.498255i	$0.166026\pi$
$8$	12.0061	0.530598
$9$	0	0
$10$	−17.0983	−0.540694
$11$	35.3089	0.967820	0.483910	−	0.875118i	$-0.339216\pi$
$11$	35.3089	0.967820	0.483910	+	0.875118i	$0.339216\pi$
$12$	0	0
$13$	34.5155	0.736375	0.368187	−	0.929752i	$-0.379978\pi$
$13$	34.5155	0.736375	0.368187	+	0.929752i	$0.379978\pi$
$14$	−25.0513	−0.478231
$15$	0	0
$16$	49.7670	0.777610
$17$	27.9810	0.399199	0.199599	−	0.979878i	$-0.436036\pi$
$17$	27.9810	0.399199	0.199599	+	0.979878i	$0.436036\pi$
$18$	0	0
$19$	74.3473	0.897707	0.448853	−	0.893605i	$-0.351833\pi$
$19$	74.3473	0.897707	0.448853	+	0.893605i	$0.351833\pi$
$20$	−162.020	−1.81144
$21$	0	0
$22$	−27.5424	−0.266912
$23$	−128.081	−1.16116	−0.580580	−	0.814203i	$-0.697174\pi$
$23$	−128.081	−1.16116	−0.580580	+	0.814203i	$0.697174\pi$
$24$	0	0
$25$	355.471	2.84377
$26$	−26.9235	−0.203082
$27$	0	0
$28$	−237.381	−1.60217
$29$	−42.7099	−0.273484	−0.136742	−	0.990607i	$-0.543663\pi$
$29$	−42.7099	−0.273484	−0.136742	+	0.990607i	$0.543663\pi$
$30$	0	0
$31$	−211.536	−1.22558	−0.612790	−	0.790246i	$-0.709953\pi$
$31$	−211.536	−1.22558	−0.612790	+	0.790246i	$0.709953\pi$
$32$	−134.869	−0.745052
$33$	0	0
$34$	−21.8263	−0.110094
$35$	703.955	3.39972
$36$	0	0
$37$	−165.794	−0.736658	−0.368329	−	0.929695i	$-0.620070\pi$
$37$	−165.794	−0.736658	−0.368329	+	0.929695i	$0.620070\pi$
$38$	−57.9940	−0.247576
$39$	0	0
$40$	263.168	1.04026
$41$	−377.795	−1.43906	−0.719532	−	0.694459i	$-0.755644\pi$
$41$	−377.795	−1.43906	−0.719532	+	0.694459i	$0.755644\pi$
$42$	0	0
$43$	−393.343	−1.39498	−0.697491	−	0.716594i	$-0.745700\pi$
$43$	−393.343	−1.39498	−0.697491	+	0.716594i	$0.745700\pi$
$44$	−260.987	−0.894209
$45$	0	0
$46$	99.9085	0.320233
$47$	−261.503	−0.811577	−0.405789	−	0.913967i	$-0.633003\pi$
$47$	−261.503	−0.811577	−0.405789	+	0.913967i	$0.633003\pi$
$48$	0	0
$49$	688.391	2.00697
$50$	−277.282	−0.784273
$51$	0	0
$52$	−255.122	−0.680367
$53$	113.158	0.293273	0.146636	−	0.989190i	$-0.453155\pi$
$53$	113.158	0.293273	0.146636	+	0.989190i	$0.453155\pi$
$54$	0	0
$55$	773.957	1.89746
$56$	385.578	0.920089
$57$	0	0
$58$	33.3155	0.0754232
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	337.123	0.707610	0.353805	−	0.935319i	$-0.384888\pi$
$61$	337.123	0.707610	0.353805	+	0.935319i	$0.384888\pi$
$62$	165.007	0.337998
$63$	0	0
$64$	−292.933	−0.572135
$65$	756.567	1.44370
$66$	0	0
$67$	183.638	0.334850	0.167425	−	0.985885i	$-0.446455\pi$
$67$	183.638	0.334850	0.167425	+	0.985885i	$0.446455\pi$
$68$	−206.822	−0.368837
$69$	0	0
$70$	−549.115	−0.937597
$71$	168.306	0.281327	0.140664	−	0.990057i	$-0.455076\pi$
$71$	168.306	0.281327	0.140664	+	0.990057i	$0.455076\pi$
$72$	0	0
$73$	805.659	1.29172	0.645858	−	0.763457i	$-0.276500\pi$
$73$	805.659	1.29172	0.645858	+	0.763457i	$0.276500\pi$
$74$	129.326	0.203161
$75$	0	0
$76$	−549.540	−0.829429
$77$	1133.95	1.67826
$78$	0	0
$79$	797.644	1.13597	0.567987	−	0.823038i	$-0.307722\pi$
$79$	797.644	1.13597	0.567987	+	0.823038i	$0.307722\pi$
$80$	1090.88	1.52454
$81$	0	0
$82$	294.696	0.396875
$83$	−251.649	−0.332796	−0.166398	−	0.986059i	$-0.553214\pi$
$83$	−251.649	−0.332796	−0.166398	+	0.986059i	$0.553214\pi$
$84$	0	0
$85$	613.333	0.782650
$86$	306.824	0.384717
$87$	0	0
$88$	423.920	0.513523
$89$	−653.582	−0.778422	−0.389211	−	0.921149i	$-0.627252\pi$
$89$	−653.582	−0.778422	−0.389211	+	0.921149i	$0.627252\pi$
$90$	0	0
$91$	1108.47	1.27692
$92$	946.713	1.07284
$93$	0	0
$94$	203.983	0.223822
$95$	1629.67	1.76000
$96$	0	0
$97$	−1574.87	−1.64849	−0.824246	−	0.566231i	$-0.808401\pi$
$97$	−1574.87	−1.64849	−0.824246	+	0.566231i	$0.808401\pi$
$98$	−536.974	−0.553495
$99$	0	0
$100$	−2627.47	−2.62747
$101$	−1309.13	−1.28973	−0.644867	−	0.764295i	$-0.723087\pi$
$101$	−1309.13	−1.28973	−0.644867	+	0.764295i	$0.723087\pi$
$102$	0	0
$103$	855.586	0.818479	0.409240	−	0.912427i	$-0.365794\pi$
$103$	855.586	0.818479	0.409240	+	0.912427i	$0.365794\pi$
$104$	414.395	0.390719
$105$	0	0
$106$	−88.2681	−0.0808807
$107$	−661.333	−0.597509	−0.298755	−	0.954330i	$-0.596571\pi$
$107$	−661.333	−0.597509	−0.298755	+	0.954330i	$0.596571\pi$
$108$	0	0
$109$	979.622	0.860832	0.430416	−	0.902631i	$-0.358367\pi$
$109$	979.622	0.860832	0.430416	+	0.902631i	$0.358367\pi$
$110$	−603.720	−0.523295
$111$	0	0
$112$	1598.28	1.34842
$113$	−1814.46	−1.51053	−0.755267	−	0.655417i	$-0.772493\pi$
$113$	−1814.46	−1.51053	−0.755267	+	0.655417i	$0.772493\pi$
$114$	0	0
$115$	−2807.48	−2.27652
$116$	315.692	0.252683
$117$	0	0
$118$	−46.0225	−0.0359044
$119$	898.616	0.692235
$120$	0	0
$121$	−84.2847	−0.0633244
$122$	−262.970	−0.195149
$123$	0	0
$124$	1563.57	1.13236
$125$	5051.83	3.61480
$126$	0	0
$127$	−2057.03	−1.43726	−0.718629	−	0.695393i	$-0.755230\pi$
$127$	−2057.03	−1.43726	−0.718629	+	0.695393i	$0.755230\pi$
$128$	1307.45	0.902839
$129$	0	0
$130$	−590.154	−0.398154
$131$	−2007.37	−1.33881	−0.669407	−	0.742896i	$-0.733452\pi$
$131$	−2007.37	−1.33881	−0.669407	+	0.742896i	$0.733452\pi$
$132$	0	0
$133$	2387.68	1.55668
$134$	−143.245	−0.0923471
$135$	0	0
$136$	335.941	0.211814
$137$	1424.64	0.888434	0.444217	−	0.895919i	$-0.353482\pi$
$137$	1424.64	0.888434	0.444217	+	0.895919i	$0.353482\pi$
$138$	0	0
$139$	455.829	0.278150	0.139075	−	0.990282i	$-0.455587\pi$
$139$	455.829	0.278150	0.139075	+	0.990282i	$0.455587\pi$
$140$	−5203.31	−3.14114
$141$	0	0
$142$	−131.286	−0.0775863
$143$	1218.70	0.712678
$144$	0	0
$145$	−936.185	−0.536179
$146$	−628.449	−0.356238
$147$	0	0
$148$	1225.47	0.680629
$149$	−2370.80	−1.30351	−0.651756	−	0.758429i	$-0.725967\pi$
$149$	−2370.80	−1.30351	−0.651756	+	0.758429i	$0.725967\pi$
$150$	0	0
$151$	413.879	0.223053	0.111526	−	0.993761i	$-0.464426\pi$
$151$	413.879	0.223053	0.111526	+	0.993761i	$0.464426\pi$
$152$	892.617	0.476321
$153$	0	0
$154$	−884.532	−0.462842
$155$	−4636.79	−2.40281
$156$	0	0
$157$	−833.858	−0.423880	−0.211940	−	0.977283i	$-0.567978\pi$
$157$	−833.858	−0.423880	−0.211940	+	0.977283i	$0.567978\pi$
$158$	−622.196	−0.313286
$159$	0	0
$160$	−2956.28	−1.46071
$161$	−4113.35	−2.01352
$162$	0	0
$163$	3278.36	1.57534	0.787672	−	0.616095i	$-0.211286\pi$
$163$	3278.36	1.57534	0.787672	+	0.616095i	$0.211286\pi$
$164$	2792.48	1.32961
$165$	0	0
$166$	196.297	0.0917808
$167$	−938.229	−0.434745	−0.217372	−	0.976089i	$-0.569749\pi$
$167$	−938.229	−0.434745	−0.217372	+	0.976089i	$0.569749\pi$
$168$	0	0
$169$	−1005.68	−0.457752
$170$	−478.426	−0.215845
$171$	0	0
$172$	2907.41	1.28888
$173$	1635.88	0.718924	0.359462	−	0.933160i	$-0.382960\pi$
$173$	1635.88	0.718924	0.359462	+	0.933160i	$0.382960\pi$
$174$	0	0
$175$	11416.0	4.93126
$176$	1757.22	0.752587
$177$	0	0
$178$	509.822	0.214678
$179$	1356.18	0.566287	0.283144	−	0.959078i	$-0.408623\pi$
$179$	1356.18	0.566287	0.283144	+	0.959078i	$0.408623\pi$
$180$	0	0
$181$	−2778.89	−1.14118	−0.570590	−	0.821235i	$-0.693285\pi$
$181$	−2778.89	−1.14118	−0.570590	+	0.821235i	$0.693285\pi$
$182$	−864.657	−0.352157
$183$	0	0
$184$	−1537.74	−0.616109
$185$	−3634.14	−1.44426
$186$	0	0
$187$	987.976	0.386353
$188$	1932.91	0.749850
$189$	0	0
$190$	−1271.21	−0.485385
$191$	3200.32	1.21239	0.606197	−	0.795315i	$-0.292694\pi$
$191$	3200.32	1.21239	0.606197	+	0.795315i	$0.292694\pi$
$192$	0	0
$193$	945.056	0.352470	0.176235	−	0.984348i	$-0.443608\pi$
$193$	945.056	0.352470	0.176235	+	0.984348i	$0.443608\pi$
$194$	1228.47	0.454632
$195$	0	0
$196$	−5088.26	−1.85432
$197$	135.114	0.0488653	0.0244327	−	0.999701i	$-0.492222\pi$
$197$	135.114	0.0488653	0.0244327	+	0.999701i	$0.492222\pi$
$198$	0	0
$199$	−1993.07	−0.709976	−0.354988	−	0.934871i	$-0.615515\pi$
$199$	−1993.07	−0.709976	−0.354988	+	0.934871i	$0.615515\pi$
$200$	4267.80	1.50889
$201$	0	0
$202$	1021.18	0.355691
$203$	−1371.64	−0.474238
$204$	0	0
$205$	−8281.12	−2.82136
$206$	−667.393	−0.225726
$207$	0	0
$208$	1717.73	0.572612
$209$	2625.12	0.868819
$210$	0	0
$211$	1231.14	0.401684	0.200842	−	0.979624i	$-0.435632\pi$
$211$	1231.14	0.401684	0.200842	+	0.979624i	$0.435632\pi$
$212$	−836.412	−0.270967
$213$	0	0
$214$	515.868	0.164785
$215$	−8621.93	−2.73493
$216$	0	0
$217$	−6793.53	−2.12523
$218$	−764.147	−0.237406
$219$	0	0
$220$	−5720.73	−1.75314
$221$	965.777	0.293960
$222$	0	0
$223$	4085.96	1.22698	0.613489	−	0.789703i	$-0.289765\pi$
$223$	4085.96	1.22698	0.613489	+	0.789703i	$0.289765\pi$
$224$	−4331.35	−1.29197
$225$	0	0
$226$	1415.36	0.416585
$227$	−5198.03	−1.51985	−0.759924	−	0.650012i	$-0.774764\pi$
$227$	−5198.03	−1.51985	−0.759924	+	0.650012i	$0.774764\pi$
$228$	0	0
$229$	−1237.19	−0.357014	−0.178507	−	0.983939i	$-0.557127\pi$
$229$	−1237.19	−0.357014	−0.178507	+	0.983939i	$0.557127\pi$
$230$	2189.96	0.627833
$231$	0	0
$232$	−512.777	−0.145110
$233$	5514.40	1.55047	0.775236	−	0.631671i	$-0.217631\pi$
$233$	5514.40	1.55047	0.775236	+	0.631671i	$0.217631\pi$
$234$	0	0
$235$	−5732.05	−1.59114
$236$	−436.100	−0.120287
$237$	0	0
$238$	−700.959	−0.190909
$239$	3600.71	0.974522	0.487261	−	0.873256i	$-0.337996\pi$
$239$	3600.71	0.974522	0.487261	+	0.873256i	$0.337996\pi$
$240$	0	0
$241$	6157.76	1.64588	0.822938	−	0.568131i	$-0.192333\pi$
$241$	6157.76	1.64588	0.822938	+	0.568131i	$0.192333\pi$
$242$	65.7457	0.0174640
$243$	0	0
$244$	−2491.86	−0.653790
$245$	15089.3	3.93477
$246$	0	0
$247$	2566.13	0.661049
$248$	−2539.71	−0.650289
$249$	0	0
$250$	−3940.65	−0.996913
$251$	−6014.95	−1.51259	−0.756296	−	0.654230i	$-0.772993\pi$
$251$	−6014.95	−1.51259	−0.756296	+	0.654230i	$0.772993\pi$
$252$	0	0
$253$	−4522.39	−1.12379
$254$	1604.57	0.396377
$255$	0	0
$256$	1323.60	0.323144
$257$	3933.41	0.954706	0.477353	−	0.878712i	$-0.341596\pi$
$257$	3933.41	0.954706	0.477353	+	0.878712i	$0.341596\pi$
$258$	0	0
$259$	−5324.52	−1.27741
$260$	−5592.19	−1.33390
$261$	0	0
$262$	1565.83	0.369227
$263$	−1974.57	−0.462955	−0.231477	−	0.972840i	$-0.574356\pi$
$263$	−1974.57	−0.462955	−0.231477	+	0.972840i	$0.574356\pi$
$264$	0	0
$265$	2480.38	0.574977
$266$	−1862.49	−0.429311
$267$	0	0
$268$	−1357.36	−0.309381
$269$	−334.493	−0.0758155	−0.0379078	−	0.999281i	$-0.512069\pi$
$269$	−334.493	−0.0758155	−0.0379078	+	0.999281i	$0.512069\pi$
$270$	0	0
$271$	−4193.71	−0.940036	−0.470018	−	0.882657i	$-0.655753\pi$
$271$	−4193.71	−0.940036	−0.470018	+	0.882657i	$0.655753\pi$
$272$	1392.53	0.310421
$273$	0	0
$274$	−1111.28	−0.245018
$275$	12551.3	2.75225
$276$	0	0
$277$	2515.63	0.545666	0.272833	−	0.962061i	$-0.412039\pi$
$277$	2515.63	0.545666	0.272833	+	0.962061i	$0.412039\pi$
$278$	−355.566	−0.0767101
$279$	0	0
$280$	8451.72	1.80388
$281$	616.348	0.130848	0.0654239	−	0.997858i	$-0.479160\pi$
$281$	616.348	0.130848	0.0654239	+	0.997858i	$0.479160\pi$
$282$	0	0
$283$	−3782.80	−0.794573	−0.397286	−	0.917695i	$-0.630048\pi$
$283$	−3782.80	−0.794573	−0.397286	+	0.917695i	$0.630048\pi$
$284$	−1244.04	−0.259930
$285$	0	0
$286$	−950.640	−0.196547
$287$	−12133.0	−2.49543
$288$	0	0
$289$	−4130.07	−0.840640
$290$	730.264	0.147871
$291$	0	0
$292$	−5955.06	−1.19347
$293$	−1298.66	−0.258938	−0.129469	−	0.991583i	$-0.541327\pi$
$293$	−1298.66	−0.258938	−0.129469	+	0.991583i	$0.541327\pi$
$294$	0	0
$295$	1293.26	0.255242
$296$	−1990.53	−0.390869
$297$	0	0
$298$	1849.32	0.359491
$299$	−4420.77	−0.855049
$300$	0	0
$301$	−12632.3	−2.41898
$302$	−322.843	−0.0615150
$303$	0	0
$304$	3700.04	0.698066
$305$	7389.62	1.38731
$306$	0	0
$307$	−2669.82	−0.496335	−0.248167	−	0.968717i	$-0.579828\pi$
$307$	−2669.82	−0.496335	−0.248167	+	0.968717i	$0.579828\pi$
$308$	−8381.65	−1.55061
$309$	0	0
$310$	3616.89	0.662664
$311$	−3834.46	−0.699139	−0.349570	−	0.936910i	$-0.613672\pi$
$311$	−3834.46	−0.699139	−0.349570	+	0.936910i	$0.613672\pi$
$312$	0	0
$313$	6620.03	1.19548	0.597741	−	0.801689i	$-0.296065\pi$
$313$	6620.03	1.19548	0.597741	+	0.801689i	$0.296065\pi$
$314$	650.445	0.116900
$315$	0	0
$316$	−5895.81	−1.04957
$317$	2662.49	0.471736	0.235868	−	0.971785i	$-0.424207\pi$
$317$	2662.49	0.471736	0.235868	+	0.971785i	$0.424207\pi$
$318$	0	0
$319$	−1508.04	−0.264683
$320$	−6420.98	−1.12170
$321$	0	0
$322$	3208.59	0.555303
$323$	2080.31	0.358364
$324$	0	0
$325$	12269.2	2.09408
$326$	−2557.26	−0.434459
$327$	0	0
$328$	−4535.82	−0.763564
$329$	−8398.24	−1.40732
$330$	0	0
$331$	−6571.43	−1.09123	−0.545617	−	0.838035i	$-0.683705\pi$
$331$	−6571.43	−1.09123	−0.545617	+	0.838035i	$0.683705\pi$
$332$	1860.07	0.307484
$333$	0	0
$334$	731.859	0.119897
$335$	4025.27	0.656490
$336$	0	0
$337$	5603.74	0.905802	0.452901	−	0.891561i	$-0.350389\pi$
$337$	5603.74	0.905802	0.452901	+	0.891561i	$0.350389\pi$
$338$	784.474	0.126242
$339$	0	0
$340$	−4533.47	−0.723123
$341$	−7469.09	−1.18614
$342$	0	0
$343$	11092.3	1.74615
$344$	−4722.49	−0.740174
$345$	0	0
$346$	−1276.06	−0.198270
$347$	−653.011	−0.101024	−0.0505122	−	0.998723i	$-0.516085\pi$
$347$	−653.011	−0.101024	−0.0505122	+	0.998723i	$0.516085\pi$
$348$	0	0
$349$	1891.17	0.290063	0.145031	−	0.989427i	$-0.453672\pi$
$349$	1891.17	0.290063	0.145031	+	0.989427i	$0.453672\pi$
$350$	−8904.99	−1.35998
$351$	0	0
$352$	−4762.06	−0.721076
$353$	10774.6	1.62457	0.812285	−	0.583261i	$-0.198224\pi$
$353$	10774.6	1.62457	0.812285	+	0.583261i	$0.198224\pi$
$354$	0	0
$355$	3689.20	0.551557
$356$	4830.97	0.719216
$357$	0	0
$358$	−1057.88	−0.156175
$359$	−2797.37	−0.411253	−0.205626	−	0.978631i	$-0.565923\pi$
$359$	−2797.37	−0.411253	−0.205626	+	0.978631i	$0.565923\pi$
$360$	0	0
$361$	−1331.48	−0.194122
$362$	2167.66	0.314722
$363$	0	0
$364$	−8193.32	−1.17980
$365$	17659.8	2.53248
$366$	0	0
$367$	3198.93	0.454994	0.227497	−	0.973779i	$-0.426946\pi$
$367$	3198.93	0.454994	0.227497	+	0.973779i	$0.426946\pi$
$368$	−6374.20	−0.902930
$369$	0	0
$370$	2834.79	0.398307
$371$	3634.10	0.508553
$372$	0	0
$373$	263.678	0.0366025	0.0183013	−	0.999833i	$-0.494174\pi$
$373$	263.678	0.0366025	0.0183013	+	0.999833i	$0.494174\pi$
$374$	−770.663	−0.106551
$375$	0	0
$376$	−3139.62	−0.430621
$377$	−1474.15	−0.201387
$378$	0	0
$379$	1657.03	0.224580	0.112290	−	0.993675i	$-0.464181\pi$
$379$	1657.03	0.224580	0.112290	+	0.993675i	$0.464181\pi$
$380$	−12045.7	−1.62614
$381$	0	0
$382$	−2496.39	−0.334362
$383$	−12320.1	−1.64367	−0.821837	−	0.569723i	$-0.807051\pi$
$383$	−12320.1	−1.64367	−0.821837	+	0.569723i	$0.807051\pi$
$384$	0	0
$385$	24855.9	3.29032
$386$	−737.184	−0.0972064
$387$	0	0
$388$	11640.7	1.52311
$389$	4911.58	0.640172	0.320086	−	0.947389i	$-0.396288\pi$
$389$	4911.58	0.640172	0.320086	+	0.947389i	$0.396288\pi$
$390$	0	0
$391$	−3583.82	−0.463534
$392$	8264.85	1.06489
$393$	0	0
$394$	−105.395	−0.0134764
$395$	17484.1	2.22714
$396$	0	0
$397$	15177.7	1.91875	0.959377	−	0.282126i	$-0.0910397\pi$
$397$	15177.7	1.91875	0.959377	+	0.282126i	$0.0910397\pi$
$398$	1554.68	0.195802
$399$	0	0
$400$	17690.7	2.21134
$401$	6165.32	0.767785	0.383892	−	0.923378i	$-0.374583\pi$
$401$	6165.32	0.767785	0.383892	+	0.923378i	$0.374583\pi$
$402$	0	0
$403$	−7301.26	−0.902486
$404$	9676.47	1.19164
$405$	0	0
$406$	1069.94	0.130788
$407$	−5853.99	−0.712953
$408$	0	0
$409$	−7801.41	−0.943166	−0.471583	−	0.881822i	$-0.656317\pi$
$409$	−7801.41	−0.943166	−0.471583	+	0.881822i	$0.656317\pi$
$410$	6459.63	0.778093
$411$	0	0
$412$	−6324.09	−0.756227
$413$	1894.80	0.225756
$414$	0	0
$415$	−5516.06	−0.652465
$416$	−4655.06	−0.548638
$417$	0	0
$418$	−2047.70	−0.239609
$419$	8108.97	0.945463	0.472732	−	0.881207i	$-0.343268\pi$
$419$	8108.97	0.945463	0.472732	+	0.881207i	$0.343268\pi$
$420$	0	0
$421$	9487.25	1.09829	0.549145	−	0.835727i	$-0.314953\pi$
$421$	9487.25	1.09829	0.549145	+	0.835727i	$0.314953\pi$
$422$	−960.343	−0.110779
$423$	0	0
$424$	1358.58	0.155610
$425$	9946.41	1.13523
$426$	0	0
$427$	10826.8	1.22704
$428$	4888.27	0.552064
$429$	0	0
$430$	6725.47	0.754258
$431$	−11388.1	−1.27273	−0.636366	−	0.771387i	$-0.719563\pi$
$431$	−11388.1	−1.27273	−0.636366	+	0.771387i	$0.719563\pi$
$432$	0	0
$433$	15467.8	1.71671	0.858353	−	0.513060i	$-0.171488\pi$
$433$	15467.8	1.71671	0.858353	+	0.513060i	$0.171488\pi$
$434$	5299.24	0.586110
$435$	0	0
$436$	−7240.91	−0.795359
$437$	−9522.46	−1.04238
$438$	0	0
$439$	8989.92	0.977370	0.488685	−	0.872460i	$-0.337477\pi$
$439$	8989.92	0.977370	0.488685	+	0.872460i	$0.337477\pi$
$440$	9292.17	1.00679
$441$	0	0
$442$	−753.347	−0.0810703
$443$	−12756.2	−1.36809	−0.684045	−	0.729439i	$-0.739781\pi$
$443$	−12756.2	−1.36809	−0.684045	+	0.729439i	$0.739781\pi$
$444$	0	0
$445$	−14326.3	−1.52614
$446$	−3187.22	−0.338384
$447$	0	0
$448$	−9407.62	−0.992116
$449$	7974.32	0.838154	0.419077	−	0.907951i	$-0.362354\pi$
$449$	7974.32	0.838154	0.419077	+	0.907951i	$0.362354\pi$
$450$	0	0
$451$	−13339.5	−1.39275
$452$	13411.7	1.39565
$453$	0	0
$454$	4054.69	0.419154
$455$	24297.4	2.50347
$456$	0	0
$457$	−2012.86	−0.206034	−0.103017	−	0.994680i	$-0.532850\pi$
$457$	−2012.86	−0.206034	−0.103017	+	0.994680i	$0.532850\pi$
$458$	965.065	0.0984596
$459$	0	0
$460$	20751.6	2.10337
$461$	12665.8	1.27962	0.639810	−	0.768533i	$-0.279013\pi$
$461$	12665.8	1.27962	0.639810	+	0.768533i	$0.279013\pi$
$462$	0	0
$463$	−3474.03	−0.348709	−0.174354	−	0.984683i	$-0.555784\pi$
$463$	−3474.03	−0.348709	−0.174354	+	0.984683i	$0.555784\pi$
$464$	−2125.54	−0.212664
$465$	0	0
$466$	−4301.46	−0.427600
$467$	−317.104	−0.0314214	−0.0157107	−	0.999877i	$-0.505001\pi$
$467$	−317.104	−0.0314214	−0.0157107	+	0.999877i	$0.505001\pi$
$468$	0	0
$469$	5897.58	0.580650
$470$	4471.24	0.438815
$471$	0	0
$472$	708.357	0.0690779
$473$	−13888.5	−1.35009
$474$	0	0
$475$	26428.3	2.55287
$476$	−6642.15	−0.639585
$477$	0	0
$478$	−2808.71	−0.268760
$479$	−19601.8	−1.86979	−0.934895	−	0.354923i	$-0.884507\pi$
$479$	−19601.8	−1.86979	−0.934895	+	0.354923i	$0.884507\pi$
$480$	0	0
$481$	−5722.46	−0.542457
$482$	−4803.31	−0.453911
$483$	0	0
$484$	622.993	0.0585080
$485$	−34520.6	−3.23196
$486$	0	0
$487$	−13669.1	−1.27189	−0.635943	−	0.771736i	$-0.719389\pi$
$487$	−13669.1	−1.27189	−0.635943	+	0.771736i	$0.719389\pi$
$488$	4047.52	0.375456
$489$	0	0
$490$	−11770.3	−1.08516
$491$	13294.5	1.22194	0.610970	−	0.791653i	$-0.290779\pi$
$491$	13294.5	1.22194	0.610970	+	0.791653i	$0.290779\pi$
$492$	0	0
$493$	−1195.06	−0.109174
$494$	−2001.69	−0.182308
$495$	0	0
$496$	−10527.5	−0.953023
$497$	5405.19	0.487839
$498$	0	0
$499$	4779.40	0.428768	0.214384	−	0.976749i	$-0.431226\pi$
$499$	4779.40	0.428768	0.214384	+	0.976749i	$0.431226\pi$
$500$	−37340.8	−3.33986
$501$	0	0
$502$	4691.92	0.417152
$503$	6846.81	0.606927	0.303463	−	0.952843i	$-0.401857\pi$
$503$	6846.81	0.606927	0.303463	+	0.952843i	$0.401857\pi$
$504$	0	0
$505$	−28695.6	−2.52859
$506$	3527.65	0.309928
$507$	0	0
$508$	15204.6	1.32794
$509$	−2306.50	−0.200852	−0.100426	−	0.994945i	$-0.532021\pi$
$509$	−2306.50	−0.200852	−0.100426	+	0.994945i	$0.532021\pi$
$510$	0	0
$511$	25874.0	2.23992
$512$	−11492.1	−0.991958
$513$	0	0
$514$	−3068.23	−0.263295
$515$	18754.1	1.60467
$516$	0	0
$517$	−9233.37	−0.785461
$518$	4153.35	0.352293
$519$	0	0
$520$	9083.38	0.766024
$521$	2031.49	0.170828	0.0854139	−	0.996346i	$-0.472779\pi$
$521$	2031.49	0.170828	0.0854139	+	0.996346i	$0.472779\pi$
$522$	0	0
$523$	−16135.1	−1.34902	−0.674512	−	0.738264i	$-0.735646\pi$
$523$	−16135.1	−1.34902	−0.674512	+	0.738264i	$0.735646\pi$
$524$	14837.5	1.23699
$525$	0	0
$526$	1540.25	0.127677
$527$	−5918.98	−0.489250
$528$	0	0
$529$	4237.69	0.348293
$530$	−1934.81	−0.158571
$531$	0	0
$532$	−17648.6	−1.43828
$533$	−13039.8	−1.05969
$534$	0	0
$535$	−14496.2	−1.17145
$536$	2204.76	0.177670
$537$	0	0
$538$	260.919	0.0209089
$539$	24306.3	1.94239
$540$	0	0
$541$	4833.44	0.384114	0.192057	−	0.981384i	$-0.438484\pi$
$541$	4833.44	0.384114	0.192057	+	0.981384i	$0.438484\pi$
$542$	3271.27	0.259249
$543$	0	0
$544$	−3773.76	−0.297424
$545$	21473.0	1.68771
$546$	0	0
$547$	−6411.62	−0.501172	−0.250586	−	0.968094i	$-0.580623\pi$
$547$	−6411.62	−0.501172	−0.250586	+	0.968094i	$0.580623\pi$
$548$	−10530.3	−0.820861
$549$	0	0
$550$	−9790.52	−0.759035
$551$	−3175.36	−0.245508
$552$	0	0
$553$	25616.5	1.96985
$554$	−1962.30	−0.150488
$555$	0	0
$556$	−3369.27	−0.256995
$557$	4099.19	0.311828	0.155914	−	0.987771i	$-0.450168\pi$
$557$	4099.19	0.311828	0.155914	+	0.987771i	$0.450168\pi$
$558$	0	0
$559$	−13576.4	−1.02723
$560$	35033.8	2.64365
$561$	0	0
$562$	−480.777	−0.0360861
$563$	4795.83	0.359006	0.179503	−	0.983757i	$-0.442551\pi$
$563$	4795.83	0.359006	0.179503	+	0.983757i	$0.442551\pi$
$564$	0	0
$565$	−39772.4	−2.96148
$566$	2950.74	0.219133
$567$	0	0
$568$	2020.69	0.149271
$569$	−14582.8	−1.07441	−0.537207	−	0.843451i	$-0.680520\pi$
$569$	−14582.8	−1.07441	−0.537207	+	0.843451i	$0.680520\pi$
$570$	0	0
$571$	−17268.6	−1.26562	−0.632809	−	0.774308i	$-0.718098\pi$
$571$	−17268.6	−1.26562	−0.632809	+	0.774308i	$0.718098\pi$
$572$	−9008.08	−0.658473
$573$	0	0
$574$	9464.24	0.688205
$575$	−45529.0	−3.30207
$576$	0	0
$577$	−10005.8	−0.721918	−0.360959	−	0.932582i	$-0.617551\pi$
$577$	−10005.8	−0.721918	−0.360959	+	0.932582i	$0.617551\pi$
$578$	3221.63	0.231837
$579$	0	0
$580$	6919.85	0.495398
$581$	−8081.78	−0.577089
$582$	0	0
$583$	3995.48	0.283835
$584$	9672.79	0.685382
$585$	0	0
$586$	1013.01	0.0714116
$587$	1313.62	0.0923662	0.0461831	−	0.998933i	$-0.485294\pi$
$587$	1313.62	0.0923662	0.0461831	+	0.998933i	$0.485294\pi$
$588$	0	0
$589$	−15727.1	−1.10021
$590$	−1008.80	−0.0703924
$591$	0	0
$592$	−8251.07	−0.572833
$593$	−18128.5	−1.25539	−0.627697	−	0.778458i	$-0.716002\pi$
$593$	−18128.5	−1.25539	−0.627697	+	0.778458i	$0.716002\pi$
$594$	0	0
$595$	19697.3	1.35716
$596$	17523.8	1.20437
$597$	0	0
$598$	3448.39	0.235811
$599$	−636.072	−0.0433876	−0.0216938	−	0.999765i	$-0.506906\pi$
$599$	−636.072	−0.0433876	−0.0216938	+	0.999765i	$0.506906\pi$
$600$	0	0
$601$	12022.0	0.815950	0.407975	−	0.912993i	$-0.366235\pi$
$601$	12022.0	0.815950	0.407975	+	0.912993i	$0.366235\pi$
$602$	9853.74	0.667123
$603$	0	0
$604$	−3059.20	−0.206088
$605$	−1847.49	−0.124151
$606$	0	0
$607$	4317.86	0.288726	0.144363	−	0.989525i	$-0.453887\pi$
$607$	4317.86	0.288726	0.144363	+	0.989525i	$0.453887\pi$
$608$	−10027.1	−0.668838
$609$	0	0
$610$	−5764.22	−0.382600
$611$	−9025.90	−0.597625
$612$	0	0
$613$	7724.56	0.508959	0.254480	−	0.967078i	$-0.418096\pi$
$613$	7724.56	0.508959	0.254480	+	0.967078i	$0.418096\pi$
$614$	2082.57	0.136883
$615$	0	0
$616$	13614.3	0.890480
$617$	13543.2	0.883676	0.441838	−	0.897095i	$-0.354327\pi$
$617$	13543.2	0.883676	0.441838	+	0.897095i	$0.354327\pi$
$618$	0	0
$619$	24983.1	1.62222	0.811112	−	0.584891i	$-0.198863\pi$
$619$	24983.1	1.62222	0.811112	+	0.584891i	$0.198863\pi$
$620$	34273.0	2.22006
$621$	0	0
$622$	2991.04	0.192813
$623$	−20990.0	−1.34983
$624$	0	0
$625$	66300.6	4.24324
$626$	−5163.90	−0.329698
$627$	0	0
$628$	6163.49	0.391640
$629$	−4639.07	−0.294073
$630$	0	0
$631$	2921.67	0.184327	0.0921633	−	0.995744i	$-0.470622\pi$
$631$	2921.67	0.184327	0.0921633	+	0.995744i	$0.470622\pi$
$632$	9576.55	0.602745
$633$	0	0
$634$	−2076.85	−0.130098
$635$	−45089.3	−2.81782
$636$	0	0
$637$	23760.1	1.47788
$638$	1176.33	0.0729960
$639$	0	0
$640$	28658.8	1.77006
$641$	−7252.88	−0.446914	−0.223457	−	0.974714i	$-0.571734\pi$
$641$	−7252.88	−0.446914	−0.223457	+	0.974714i	$0.571734\pi$
$642$	0	0
$643$	21945.7	1.34596	0.672981	−	0.739659i	$-0.265013\pi$
$643$	21945.7	1.34596	0.672981	+	0.739659i	$0.265013\pi$
$644$	30404.0	1.86038
$645$	0	0
$646$	−1622.73	−0.0988319
$647$	4657.90	0.283031	0.141515	−	0.989936i	$-0.454803\pi$
$647$	4657.90	0.283031	0.141515	+	0.989936i	$0.454803\pi$
$648$	0	0
$649$	2083.22	0.125999
$650$	−9570.53	−0.577519
$651$	0	0
$652$	−24232.1	−1.45553
$653$	6157.88	0.369030	0.184515	−	0.982830i	$-0.440929\pi$
$653$	6157.88	0.369030	0.184515	+	0.982830i	$0.440929\pi$
$654$	0	0
$655$	−44000.8	−2.62482
$656$	−18801.7	−1.11903
$657$	0	0
$658$	6550.98	0.388121
$659$	5471.34	0.323419	0.161709	−	0.986838i	$-0.448299\pi$
$659$	5471.34	0.323419	0.161709	+	0.986838i	$0.448299\pi$
$660$	0	0
$661$	14162.9	0.833396	0.416698	−	0.909045i	$-0.363187\pi$
$661$	14162.9	0.833396	0.416698	+	0.909045i	$0.363187\pi$
$662$	5126.00	0.300948
$663$	0	0
$664$	−3021.31	−0.176581
$665$	52337.1	3.05195
$666$	0	0
$667$	5470.32	0.317558
$668$	6934.95	0.401679
$669$	0	0
$670$	−3139.88	−0.181051
$671$	11903.4	0.684839
$672$	0	0
$673$	−9280.39	−0.531550	−0.265775	−	0.964035i	$-0.585628\pi$
$673$	−9280.39	−0.531550	−0.265775	+	0.964035i	$0.585628\pi$
$674$	−4371.16	−0.249808
$675$	0	0
$676$	7433.53	0.422936
$677$	−17040.6	−0.967391	−0.483695	−	0.875236i	$-0.660706\pi$
$677$	−17040.6	−0.967391	−0.483695	+	0.875236i	$0.660706\pi$
$678$	0	0
$679$	−50577.4	−2.85859
$680$	7363.70	0.415272
$681$	0	0
$682$	5826.21	0.327122
$683$	−20830.7	−1.16700	−0.583502	−	0.812112i	$-0.698318\pi$
$683$	−20830.7	−1.16700	−0.583502	+	0.812112i	$0.698318\pi$
$684$	0	0
$685$	31227.7	1.74182
$686$	−8652.47	−0.481564
$687$	0	0
$688$	−19575.5	−1.08475
$689$	3905.71	0.215959
$690$	0	0
$691$	13181.3	0.725671	0.362835	−	0.931853i	$-0.381809\pi$
$691$	13181.3	0.725671	0.362835	+	0.931853i	$0.381809\pi$
$692$	−12091.7	−0.664244
$693$	0	0
$694$	509.376	0.0278612
$695$	9991.60	0.545328
$696$	0	0
$697$	−10571.1	−0.574473
$698$	−1475.19	−0.0799955
$699$	0	0
$700$	−84382.0	−4.55620
$701$	2796.51	0.150675	0.0753373	−	0.997158i	$-0.475997\pi$
$701$	2796.51	0.150675	0.0753373	+	0.997158i	$0.475997\pi$
$702$	0	0
$703$	−12326.3	−0.661303
$704$	−10343.1	−0.553723
$705$	0	0
$706$	−8404.63	−0.448035
$707$	−42043.0	−2.23648
$708$	0	0
$709$	−8280.22	−0.438604	−0.219302	−	0.975657i	$-0.570378\pi$
$709$	−8280.22	−0.438604	−0.219302	+	0.975657i	$0.570378\pi$
$710$	−2877.74	−0.152112
$711$	0	0
$712$	−7846.94	−0.413029
$713$	27093.7	1.42309
$714$	0	0
$715$	26713.5	1.39724
$716$	−10024.2	−0.523216
$717$	0	0
$718$	2182.07	0.113418
$719$	30936.1	1.60462	0.802310	−	0.596907i	$-0.203604\pi$
$719$	30936.1	1.60462	0.802310	+	0.596907i	$0.203604\pi$
$720$	0	0
$721$	27477.4	1.41929
$722$	1038.61	0.0535363
$723$	0	0
$724$	20540.3	1.05438
$725$	−15182.1	−0.777723
$726$	0	0
$727$	24053.2	1.22708	0.613538	−	0.789665i	$-0.289746\pi$
$727$	24053.2	1.22708	0.613538	+	0.789665i	$0.289746\pi$
$728$	13308.4	0.677530
$729$	0	0
$730$	−13775.4	−0.698424
$731$	−11006.1	−0.556875
$732$	0	0
$733$	−4234.34	−0.213368	−0.106684	−	0.994293i	$-0.534023\pi$
$733$	−4234.34	−0.213368	−0.106684	+	0.994293i	$0.534023\pi$
$734$	−2495.30	−0.125481
$735$	0	0
$736$	17274.1	0.865125
$737$	6484.04	0.324074
$738$	0	0
$739$	−29248.6	−1.45592	−0.727961	−	0.685619i	$-0.759532\pi$
$739$	−29248.6	−1.45592	−0.727961	+	0.685619i	$0.759532\pi$
$740$	26861.9	1.33441
$741$	0	0
$742$	−2834.75	−0.140252
$743$	5051.56	0.249426	0.124713	−	0.992193i	$-0.460199\pi$
$743$	5051.56	0.249426	0.124713	+	0.992193i	$0.460199\pi$
$744$	0	0
$745$	−51967.0	−2.55560
$746$	−205.680	−0.0100945
$747$	0	0
$748$	−7302.66	−0.356967
$749$	−21238.9	−1.03612
$750$	0	0
$751$	12132.8	0.589526	0.294763	−	0.955570i	$-0.404759\pi$
$751$	12132.8	0.589526	0.294763	+	0.955570i	$0.404759\pi$
$752$	−13014.2	−0.631090
$753$	0	0
$754$	1149.90	0.0555397
$755$	9072.07	0.437307
$756$	0	0
$757$	−14667.2	−0.704214	−0.352107	−	0.935960i	$-0.614535\pi$
$757$	−14667.2	−0.704214	−0.352107	+	0.935960i	$0.614535\pi$
$758$	−1292.55	−0.0619362
$759$	0	0
$760$	19565.8	0.933852
$761$	2328.04	0.110895	0.0554477	−	0.998462i	$-0.482341\pi$
$761$	2328.04	0.110895	0.0554477	+	0.998462i	$0.482341\pi$
$762$	0	0
$763$	31460.8	1.49274
$764$	−23655.3	−1.12018
$765$	0	0
$766$	9610.19	0.453303
$767$	2036.41	0.0958678
$768$	0	0
$769$	25456.4	1.19373	0.596867	−	0.802340i	$-0.296412\pi$
$769$	25456.4	1.19373	0.596867	+	0.802340i	$0.296412\pi$
$770$	−19388.6	−0.907425
$771$	0	0
$772$	−6985.41	−0.325661
$773$	−2331.11	−0.108466	−0.0542331	−	0.998528i	$-0.517271\pi$
$773$	−2331.11	−0.108466	−0.0542331	+	0.998528i	$0.517271\pi$
$774$	0	0
$775$	−75194.8	−3.48526
$776$	−18908.0	−0.874686
$777$	0	0
$778$	−3831.24	−0.176551
$779$	−28088.0	−1.29186
$780$	0	0
$781$	5942.69	0.272274
$782$	2795.54	0.127836
$783$	0	0
$784$	34259.2	1.56064
$785$	−18277.9	−0.831038
$786$	0	0
$787$	−15041.1	−0.681268	−0.340634	−	0.940196i	$-0.610642\pi$
$787$	−15041.1	−0.681268	−0.340634	+	0.940196i	$0.610642\pi$
$788$	−998.699	−0.0451487
$789$	0	0
$790$	−13638.3	−0.614214
$791$	−58272.0	−2.61936
$792$	0	0
$793$	11636.0	0.521066
$794$	−11839.2	−0.529167
$795$	0	0
$796$	14731.9	0.655976
$797$	−38360.4	−1.70489	−0.852444	−	0.522818i	$-0.824881\pi$
$797$	−38360.4	−1.70489	−0.852444	+	0.522818i	$0.824881\pi$
$798$	0	0
$799$	−7317.10	−0.323981
$800$	−47941.9	−2.11875
$801$	0	0
$802$	−4809.22	−0.211745
$803$	28446.9	1.25015
$804$	0	0
$805$	−90163.1	−3.94762
$806$	5695.29	0.248894
$807$	0	0
$808$	−15717.5	−0.684330
$809$	−20027.1	−0.870351	−0.435175	−	0.900346i	$-0.643314\pi$
$809$	−20027.1	−0.870351	−0.435175	+	0.900346i	$0.643314\pi$
$810$	0	0
$811$	−27584.7	−1.19436	−0.597182	−	0.802106i	$-0.703713\pi$
$811$	−27584.7	−1.19436	−0.597182	+	0.802106i	$0.703713\pi$
$812$	10138.5	0.438168
$813$	0	0
$814$	4566.36	0.196623
$815$	71860.5	3.08854
$816$	0	0
$817$	−29244.0	−1.25228
$818$	6085.43	0.260113
$819$	0	0
$820$	61210.2	2.60677
$821$	5945.56	0.252742	0.126371	−	0.991983i	$-0.459667\pi$
$821$	5945.56	0.252742	0.126371	+	0.991983i	$0.459667\pi$
$822$	0	0
$823$	−5880.56	−0.249069	−0.124534	−	0.992215i	$-0.539744\pi$
$823$	−5880.56	−0.249069	−0.124534	+	0.992215i	$0.539744\pi$
$824$	10272.2	0.434283
$825$	0	0
$826$	−1478.03	−0.0622604
$827$	−37847.0	−1.59138	−0.795689	−	0.605705i	$-0.792891\pi$
$827$	−37847.0	−1.59138	−0.795689	+	0.605705i	$0.792891\pi$
$828$	0	0
$829$	−38100.9	−1.59626	−0.798130	−	0.602485i	$-0.794177\pi$
$829$	−38100.9	−1.59626	−0.798130	+	0.602485i	$0.794177\pi$
$830$	4302.76	0.179941
$831$	0	0
$832$	−10110.7	−0.421305
$833$	19261.8	0.801180
$834$	0	0
$835$	−20565.7	−0.852340
$836$	−19403.6	−0.802738
$837$	0	0
$838$	−6325.34	−0.260746
$839$	15102.5	0.621450	0.310725	−	0.950500i	$-0.399428\pi$
$839$	15102.5	0.621450	0.310725	+	0.950500i	$0.399428\pi$
$840$	0	0
$841$	−22564.9	−0.925207
$842$	−7400.46	−0.302894
$843$	0	0
$844$	−9100.03	−0.371133
$845$	−22044.2	−0.897447
$846$	0	0
$847$	−2706.83	−0.109808
$848$	5631.54	0.228052
$849$	0	0
$850$	−7758.63	−0.313081
$851$	21235.0	0.855378
$852$	0	0
$853$	32185.9	1.29194	0.645970	−	0.763363i	$-0.276453\pi$
$853$	32185.9	1.29194	0.645970	+	0.763363i	$0.276453\pi$
$854$	−8445.36	−0.338401
$855$	0	0
$856$	−7940.00	−0.317037
$857$	9750.86	0.388661	0.194331	−	0.980936i	$-0.437747\pi$
$857$	9750.86	0.388661	0.194331	+	0.980936i	$0.437747\pi$
$858$	0	0
$859$	11337.4	0.450323	0.225162	−	0.974321i	$-0.427709\pi$
$859$	11337.4	0.450323	0.225162	+	0.974321i	$0.427709\pi$
$860$	63729.3	2.52692
$861$	0	0
$862$	8883.24	0.351003
$863$	−25245.6	−0.995794	−0.497897	−	0.867236i	$-0.665894\pi$
$863$	−25245.6	−0.995794	−0.497897	+	0.867236i	$0.665894\pi$
$864$	0	0
$865$	35857.9	1.40949
$866$	−12065.5	−0.473445
$867$	0	0
$868$	50214.6	1.96359
$869$	28163.9	1.09942
$870$	0	0
$871$	6338.35	0.246575
$872$	11761.4	0.456756
$873$	0	0
$874$	7427.92	0.287475
$875$	162241.	6.26828
$876$	0	0
$877$	−9052.34	−0.348547	−0.174274	−	0.984697i	$-0.555758\pi$
$877$	−9052.34	−0.348547	−0.174274	+	0.984697i	$0.555758\pi$
$878$	−7012.52	−0.269546
$879$	0	0
$880$	38517.6	1.47549
$881$	32080.2	1.22680	0.613400	−	0.789772i	$-0.289801\pi$
$881$	32080.2	1.22680	0.613400	+	0.789772i	$0.289801\pi$
$882$	0	0
$883$	−39328.6	−1.49888	−0.749440	−	0.662072i	$-0.769677\pi$
$883$	−39328.6	−1.49888	−0.749440	+	0.662072i	$0.769677\pi$
$884$	−7138.57	−0.271602
$885$	0	0
$886$	9950.36	0.377301
$887$	−50922.9	−1.92765	−0.963823	−	0.266542i	$-0.914119\pi$
$887$	−50922.9	−1.92765	−0.963823	+	0.266542i	$0.914119\pi$
$888$	0	0
$889$	−66062.1	−2.49230
$890$	11175.1	0.420888
$891$	0	0
$892$	−30201.5	−1.13366
$893$	−19442.0	−0.728558
$894$	0	0
$895$	29726.9	1.11024
$896$	41989.1	1.56558
$897$	0	0
$898$	−6220.31	−0.231152
$899$	9034.67	0.335176
$900$	0	0
$901$	3166.27	0.117074
$902$	10405.4	0.384103
$903$	0	0
$904$	−21784.5	−0.801486
$905$	−60912.3	−2.23734
$906$	0	0
$907$	−43998.4	−1.61074	−0.805370	−	0.592772i	$-0.798034\pi$
$907$	−43998.4	−1.61074	−0.805370	+	0.592772i	$0.798034\pi$
$908$	38421.4	1.40425
$909$	0	0
$910$	−18953.0	−0.690423
$911$	28063.3	1.02061	0.510306	−	0.859993i	$-0.329532\pi$
$911$	28063.3	1.02061	0.510306	+	0.859993i	$0.329532\pi$
$912$	0	0
$913$	−8885.45	−0.322087
$914$	1570.11	0.0568214
$915$	0	0
$916$	9144.77	0.329860
$917$	−64467.2	−2.32159
$918$	0	0
$919$	29137.9	1.04589	0.522944	−	0.852367i	$-0.324834\pi$
$919$	29137.9	1.04589	0.522944	+	0.852367i	$0.324834\pi$
$920$	−33706.8	−1.20791
$921$	0	0
$922$	−9879.87	−0.352902
$923$	5809.16	0.207162
$924$	0	0
$925$	−58934.9	−2.09488
$926$	2709.89	0.0961692
$927$	0	0
$928$	5760.23	0.203760
$929$	9282.01	0.327807	0.163904	−	0.986476i	$-0.447591\pi$
$929$	9282.01	0.327807	0.163904	+	0.986476i	$0.447591\pi$
$930$	0	0
$931$	51180.0	1.80167
$932$	−40759.8	−1.43255
$933$	0	0
$934$	247.354	0.00866561
$935$	21656.1	0.757465
$936$	0	0
$937$	−26659.3	−0.929478	−0.464739	−	0.885448i	$-0.653852\pi$
$937$	−26659.3	−0.929478	−0.464739	+	0.885448i	$0.653852\pi$
$938$	−4600.36	−0.160135
$939$	0	0
$940$	42368.6	1.47012
$941$	56714.2	1.96475	0.982375	−	0.186923i	$-0.0598515\pi$
$941$	56714.2	1.96475	0.982375	+	0.186923i	$0.0598515\pi$
$942$	0	0
$943$	48388.2	1.67098
$944$	2936.26	0.101236
$945$	0	0
$946$	10833.6	0.372337
$947$	3737.90	0.128264	0.0641318	−	0.997941i	$-0.479572\pi$
$947$	3737.90	0.128264	0.0641318	+	0.997941i	$0.479572\pi$
$948$	0	0
$949$	27807.7	0.951188
$950$	−20615.2	−0.704047
$951$	0	0
$952$	10788.8	0.367298
$953$	29404.8	0.999489	0.499745	−	0.866173i	$-0.333427\pi$
$953$	29404.8	0.999489	0.499745	+	0.866173i	$0.333427\pi$
$954$	0	0
$955$	70149.9	2.37696
$956$	−26614.8	−0.900402
$957$	0	0
$958$	15290.3	0.515663
$959$	45752.8	1.54060
$960$	0	0
$961$	14956.4	0.502044
$962$	4463.76	0.149602
$963$	0	0
$964$	−45515.3	−1.52069
$965$	20715.3	0.691035
$966$	0	0
$967$	52879.0	1.75851	0.879253	−	0.476356i	$-0.158043\pi$
$967$	52879.0	1.75851	0.879253	+	0.476356i	$0.158043\pi$
$968$	−1011.93	−0.0335997
$969$	0	0
$970$	26927.5	0.891330
$971$	−25027.8	−0.827167	−0.413584	−	0.910466i	$-0.635723\pi$
$971$	−25027.8	−0.827167	−0.413584	+	0.910466i	$0.635723\pi$
$972$	0	0
$973$	14639.1	0.482330
$974$	10662.5	0.350769
$975$	0	0
$976$	16777.6	0.550244
$977$	29643.3	0.970701	0.485350	−	0.874320i	$-0.338692\pi$
$977$	29643.3	0.970701	0.485350	+	0.874320i	$0.338692\pi$
$978$	0	0
$979$	−23077.2	−0.753372
$980$	−111533.	−3.63550
$981$	0	0
$982$	−10370.3	−0.336995
$983$	14434.3	0.468345	0.234173	−	0.972195i	$-0.424762\pi$
$983$	14434.3	0.468345	0.234173	+	0.972195i	$0.424762\pi$
$984$	0	0
$985$	2961.65	0.0958030
$986$	932.201	0.0301088
$987$	0	0
$988$	−18967.6	−0.610771
$989$	50379.6	1.61980
$990$	0	0
$991$	15333.7	0.491516	0.245758	−	0.969331i	$-0.420963\pi$
$991$	15333.7	0.491516	0.245758	+	0.969331i	$0.420963\pi$
$992$	28529.6	0.913120
$993$	0	0
$994$	−4216.28	−0.134539
$995$	−43687.4	−1.39194
$996$	0	0
$997$	3787.12	0.120300	0.0601501	−	0.998189i	$-0.480842\pi$
$997$	3787.12	0.120300	0.0601501	+	0.998189i	$0.480842\pi$
$998$	−3728.14	−0.118249
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.f.1.4		8
3.2	odd	2		177.4.a.c.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.5	✓	8	3.2	odd	2
531.4.a.f.1.4		8	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)

\(f(q)\)	\(=\)	\(q-0.780043 q^{2} -7.39153 q^{4} +21.9196 q^{5} +32.1153 q^{7} +12.0061 q^{8} +O(q^{10})\)	\(q-0.780043 q^{2} -7.39153 q^{4} +21.9196 q^{5} +32.1153 q^{7} +12.0061 q^{8} -17.0983 q^{10} +35.3089 q^{11} +34.5155 q^{13} -25.0513 q^{14} +49.7670 q^{16} +27.9810 q^{17} +74.3473 q^{19} -162.020 q^{20} -27.5424 q^{22} -128.081 q^{23} +355.471 q^{25} -26.9235 q^{26} -237.381 q^{28} -42.7099 q^{29} -211.536 q^{31} -134.869 q^{32} -21.8263 q^{34} +703.955 q^{35} -165.794 q^{37} -57.9940 q^{38} +263.168 q^{40} -377.795 q^{41} -393.343 q^{43} -260.987 q^{44} +99.9085 q^{46} -261.503 q^{47} +688.391 q^{49} -277.282 q^{50} -255.122 q^{52} +113.158 q^{53} +773.957 q^{55} +385.578 q^{56} +33.3155 q^{58} +59.0000 q^{59} +337.123 q^{61} +165.007 q^{62} -292.933 q^{64} +756.567 q^{65} +183.638 q^{67} -206.822 q^{68} -549.115 q^{70} +168.306 q^{71} +805.659 q^{73} +129.326 q^{74} -549.540 q^{76} +1133.95 q^{77} +797.644 q^{79} +1090.88 q^{80} +294.696 q^{82} -251.649 q^{83} +613.333 q^{85} +306.824 q^{86} +423.920 q^{88} -653.582 q^{89} +1108.47 q^{91} +946.713 q^{92} +203.983 q^{94} +1629.67 q^{95} -1574.87 q^{97} -536.974 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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