LMFDB - Embedded newform 531.6.a.d.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,6,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, 6, names="a")

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

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
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:	$85.1638083207$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + \cdots - 50564480 $ x^12 - 4x^11 - 283x^10 + 1045x^9 + 27968x^8 - 94393x^7 - 1130486x^6 + 3566264x^5 + 15496192x^4 - 53008480x^3 - 16576192x^2 + 120303168*x - 50564480
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.87969$ 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)

$f(q)$	$=$	$q+2.87969 q^{2} -23.7074 q^{4} +77.5334 q^{5} +45.2937 q^{7} -160.420 q^{8} +O(q^{10})$	\(q+2.87969 q^{2} -23.7074 q^{4} +77.5334 q^{5} +45.2937 q^{7} -160.420 q^{8} +223.272 q^{10} -155.365 q^{11} +523.303 q^{13} +130.432 q^{14} +296.678 q^{16} -1585.45 q^{17} -2043.33 q^{19} -1838.12 q^{20} -447.403 q^{22} +1063.14 q^{23} +2886.43 q^{25} +1506.95 q^{26} -1073.80 q^{28} -2231.64 q^{29} -8697.85 q^{31} +5987.78 q^{32} -4565.61 q^{34} +3511.78 q^{35} +15093.5 q^{37} -5884.16 q^{38} -12437.9 q^{40} +864.137 q^{41} +11392.8 q^{43} +3683.30 q^{44} +3061.51 q^{46} -5535.68 q^{47} -14755.5 q^{49} +8312.01 q^{50} -12406.2 q^{52} -18099.6 q^{53} -12046.0 q^{55} -7266.01 q^{56} -6426.42 q^{58} +3481.00 q^{59} -44361.9 q^{61} -25047.1 q^{62} +7749.20 q^{64} +40573.5 q^{65} -33357.0 q^{67} +37587.0 q^{68} +10112.8 q^{70} -2860.19 q^{71} -63785.8 q^{73} +43464.5 q^{74} +48442.2 q^{76} -7037.06 q^{77} +82957.4 q^{79} +23002.5 q^{80} +2488.44 q^{82} +67941.5 q^{83} -122926. q^{85} +32807.6 q^{86} +24923.6 q^{88} -22542.4 q^{89} +23702.3 q^{91} -25204.3 q^{92} -15941.0 q^{94} -158427. q^{95} +26284.2 q^{97} -42491.1 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * 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 $

$2$	2.87969	0.509061	0.254531	−	0.967065i	$-0.418079\pi$
$2$	2.87969	0.509061	0.254531	+	0.967065i	$0.418079\pi$
$3$	0	0
$3$	0	0
$4$	−23.7074	−0.740857
$5$	77.5334	1.38696	0.693480	−	0.720476i	$-0.256077\pi$
$5$	77.5334	1.38696	0.693480	+	0.720476i	$0.256077\pi$
$6$	0	0
$7$	45.2937	0.349376	0.174688	−	0.984624i	$-0.444108\pi$
$7$	45.2937	0.349376	0.174688	+	0.984624i	$0.444108\pi$
$8$	−160.420	−0.886203
$9$	0	0
$10$	223.272	0.706048
$11$	−155.365	−0.387143	−0.193572	−	0.981086i	$-0.562007\pi$
$11$	−155.365	−0.387143	−0.193572	+	0.981086i	$0.562007\pi$
$12$	0	0
$13$	523.303	0.858806	0.429403	−	0.903113i	$-0.358724\pi$
$13$	523.303	0.858806	0.429403	+	0.903113i	$0.358724\pi$
$14$	130.432	0.177854
$15$	0	0
$16$	296.678	0.289725
$17$	−1585.45	−1.33055	−0.665275	−	0.746598i	$-0.731686\pi$
$17$	−1585.45	−1.33055	−0.665275	+	0.746598i	$0.731686\pi$
$18$	0	0
$19$	−2043.33	−1.29854	−0.649270	−	0.760558i	$-0.724926\pi$
$19$	−2043.33	−1.29854	−0.649270	+	0.760558i	$0.724926\pi$
$20$	−1838.12	−1.02754
$21$	0	0
$22$	−447.403	−0.197080
$23$	1063.14	0.419055	0.209527	−	0.977803i	$-0.432807\pi$
$23$	1063.14	0.419055	0.209527	+	0.977803i	$0.432807\pi$
$24$	0	0
$25$	2886.43	0.923657
$26$	1506.95	0.437185
$27$	0	0
$28$	−1073.80	−0.258837
$29$	−2231.64	−0.492753	−0.246376	−	0.969174i	$-0.579240\pi$
$29$	−2231.64	−0.492753	−0.246376	+	0.969174i	$0.579240\pi$
$30$	0	0
$31$	−8697.85	−1.62558	−0.812788	−	0.582559i	$-0.802051\pi$
$31$	−8697.85	−1.62558	−0.812788	+	0.582559i	$0.802051\pi$
$32$	5987.78	1.03369
$33$	0	0
$34$	−4565.61	−0.677332
$35$	3511.78	0.484570
$36$	0	0
$37$	15093.5	1.81253	0.906265	−	0.422711i	$-0.138921\pi$
$37$	15093.5	1.81253	0.906265	+	0.422711i	$0.138921\pi$
$38$	−5884.16	−0.661037
$39$	0	0
$40$	−12437.9	−1.22913
$41$	864.137	0.0802829	0.0401414	−	0.999194i	$-0.487219\pi$
$41$	864.137	0.0802829	0.0401414	+	0.999194i	$0.487219\pi$
$42$	0	0
$43$	11392.8	0.939633	0.469817	−	0.882764i	$-0.344320\pi$
$43$	11392.8	0.939633	0.469817	+	0.882764i	$0.344320\pi$
$44$	3683.30	0.286818
$45$	0	0
$46$	3061.51	0.213324
$47$	−5535.68	−0.365533	−0.182767	−	0.983156i	$-0.558505\pi$
$47$	−5535.68	−0.365533	−0.182767	+	0.983156i	$0.558505\pi$
$48$	0	0
$49$	−14755.5	−0.877937
$50$	8312.01	0.470198
$51$	0	0
$52$	−12406.2	−0.636252
$53$	−18099.6	−0.885071	−0.442536	−	0.896751i	$-0.645921\pi$
$53$	−18099.6	−0.885071	−0.442536	+	0.896751i	$0.645921\pi$
$54$	0	0
$55$	−12046.0	−0.536952
$56$	−7266.01	−0.309618
$57$	0	0
$58$	−6426.42	−0.250841
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−44361.9	−1.52646	−0.763230	−	0.646127i	$-0.776387\pi$
$61$	−44361.9	−1.52646	−0.763230	+	0.646127i	$0.776387\pi$
$62$	−25047.1	−0.827518
$63$	0	0
$64$	7749.20	0.236487
$65$	40573.5	1.19113
$66$	0	0
$67$	−33357.0	−0.907821	−0.453910	−	0.891047i	$-0.649971\pi$
$67$	−33357.0	−0.907821	−0.453910	+	0.891047i	$0.649971\pi$
$68$	37587.0	0.985747
$69$	0	0
$70$	10112.8	0.246676
$71$	−2860.19	−0.0673363	−0.0336681	−	0.999433i	$-0.510719\pi$
$71$	−2860.19	−0.0673363	−0.0336681	+	0.999433i	$0.510719\pi$
$72$	0	0
$73$	−63785.8	−1.40093	−0.700466	−	0.713686i	$-0.747024\pi$
$73$	−63785.8	−1.40093	−0.700466	+	0.713686i	$0.747024\pi$
$74$	43464.5	0.922689
$75$	0	0
$76$	48442.2	0.962032
$77$	−7037.06	−0.135258
$78$	0	0
$79$	82957.4	1.49550	0.747751	−	0.663979i	$-0.231133\pi$
$79$	82957.4	1.49550	0.747751	+	0.663979i	$0.231133\pi$
$80$	23002.5	0.401837
$81$	0	0
$82$	2488.44	0.0408689
$83$	67941.5	1.08253	0.541265	−	0.840852i	$-0.317946\pi$
$83$	67941.5	1.08253	0.541265	+	0.840852i	$0.317946\pi$
$84$	0	0
$85$	−122926.	−1.84542
$86$	32807.6	0.478331
$87$	0	0
$88$	24923.6	0.343087
$89$	−22542.4	−0.301665	−0.150833	−	0.988559i	$-0.548195\pi$
$89$	−22542.4	−0.301665	−0.150833	+	0.988559i	$0.548195\pi$
$90$	0	0
$91$	23702.3	0.300046
$92$	−25204.3	−0.310459
$93$	0	0
$94$	−15941.0	−0.186079
$95$	−158427.	−1.80102
$96$	0	0
$97$	26284.2	0.283639	0.141820	−	0.989893i	$-0.454705\pi$
$97$	26284.2	0.283639	0.141820	+	0.989893i	$0.454705\pi$
$98$	−42491.1	−0.446924
$99$	0	0
$100$	−68429.7	−0.684297
$101$	61695.6	0.601798	0.300899	−	0.953656i	$-0.402713\pi$
$101$	61695.6	0.601798	0.300899	+	0.953656i	$0.402713\pi$
$102$	0	0
$103$	−65169.6	−0.605274	−0.302637	−	0.953106i	$-0.597867\pi$
$103$	−65169.6	−0.605274	−0.302637	+	0.953106i	$0.597867\pi$
$104$	−83948.2	−0.761076
$105$	0	0
$106$	−52121.0	−0.450556
$107$	−153050.	−1.29233	−0.646167	−	0.763196i	$-0.723629\pi$
$107$	−153050.	−1.29233	−0.646167	+	0.763196i	$0.723629\pi$
$108$	0	0
$109$	−203743.	−1.64254	−0.821271	−	0.570539i	$-0.806734\pi$
$109$	−203743.	−1.64254	−0.821271	+	0.570539i	$0.806734\pi$
$110$	−34688.6	−0.273342
$111$	0	0
$112$	13437.7	0.101223
$113$	−240950.	−1.77514	−0.887568	−	0.460678i	$-0.847606\pi$
$113$	−240950.	−1.77514	−0.887568	+	0.460678i	$0.847606\pi$
$114$	0	0
$115$	82428.8	0.581212
$116$	52906.4	0.365059
$117$	0	0
$118$	10024.2	0.0662741
$119$	−71811.1	−0.464862
$120$	0	0
$121$	−136913.	−0.850120
$122$	−127748.	−0.777061
$123$	0	0
$124$	206203.	1.20432
$125$	−18497.3	−0.105885
$126$	0	0
$127$	−150990.	−0.830689	−0.415344	−	0.909664i	$-0.636339\pi$
$127$	−150990.	−0.830689	−0.415344	+	0.909664i	$0.636339\pi$
$128$	−169294.	−0.913304
$129$	0	0
$130$	116839.	0.606358
$131$	250601.	1.27586	0.637932	−	0.770092i	$-0.279790\pi$
$131$	250601.	1.27586	0.637932	+	0.770092i	$0.279790\pi$
$132$	0	0
$133$	−92550.2	−0.453679
$134$	−96057.7	−0.462136
$135$	0	0
$136$	254338.	1.17914
$137$	15685.5	0.0713996	0.0356998	−	0.999363i	$-0.488634\pi$
$137$	15685.5	0.0713996	0.0356998	+	0.999363i	$0.488634\pi$
$138$	0	0
$139$	67099.0	0.294563	0.147282	−	0.989095i	$-0.452948\pi$
$139$	67099.0	0.294563	0.147282	+	0.989095i	$0.452948\pi$
$140$	−83255.1	−0.358997
$141$	0	0
$142$	−8236.45	−0.0342783
$143$	−81303.0	−0.332481
$144$	0	0
$145$	−173027.	−0.683428
$146$	−183683.	−0.713160
$147$	0	0
$148$	−357827.	−1.34282
$149$	−109381.	−0.403623	−0.201811	−	0.979424i	$-0.564683\pi$
$149$	−109381.	−0.403623	−0.201811	+	0.979424i	$0.564683\pi$
$150$	0	0
$151$	−325298.	−1.16102	−0.580509	−	0.814254i	$-0.697146\pi$
$151$	−325298.	−1.16102	−0.580509	+	0.814254i	$0.697146\pi$
$152$	327791.	1.15077
$153$	0	0
$154$	−20264.5	−0.0688549
$155$	−674374.	−2.25461
$156$	0	0
$157$	−244345.	−0.791142	−0.395571	−	0.918435i	$-0.629453\pi$
$157$	−244345.	−0.791142	−0.395571	+	0.918435i	$0.629453\pi$
$158$	238891.	0.761303
$159$	0	0
$160$	464253.	1.43369
$161$	48153.5	0.146408
$162$	0	0
$163$	86224.9	0.254193	0.127097	−	0.991890i	$-0.459434\pi$
$163$	86224.9	0.254193	0.127097	+	0.991890i	$0.459434\pi$
$164$	−20486.5	−0.0594781
$165$	0	0
$166$	195650.	0.551074
$167$	−161244.	−0.447396	−0.223698	−	0.974659i	$-0.571813\pi$
$167$	−161244.	−0.447396	−0.223698	+	0.974659i	$0.571813\pi$
$168$	0	0
$169$	−97447.0	−0.262453
$170$	−353987.	−0.939432
$171$	0	0
$172$	−270093.	−0.696133
$173$	−348734.	−0.885887	−0.442944	−	0.896549i	$-0.646066\pi$
$173$	−348734.	−0.885887	−0.442944	+	0.896549i	$0.646066\pi$
$174$	0	0
$175$	130737.	0.322703
$176$	−46093.5	−0.112165
$177$	0	0
$178$	−64915.0	−0.153566
$179$	306324.	0.714577	0.357288	−	0.933994i	$-0.383701\pi$
$179$	306324.	0.714577	0.357288	+	0.933994i	$0.383701\pi$
$180$	0	0
$181$	525696.	1.19272	0.596360	−	0.802717i	$-0.296613\pi$
$181$	525696.	1.19272	0.596360	+	0.802717i	$0.296613\pi$
$182$	68255.3	0.152742
$183$	0	0
$184$	−170549.	−0.371367
$185$	1.17025e6	2.51391
$186$	0	0
$187$	246324.	0.515114
$188$	131237.	0.270808
$189$	0	0
$190$	−456219.	−0.916831
$191$	540574.	1.07219	0.536095	−	0.844158i	$-0.319899\pi$
$191$	540574.	1.07219	0.536095	+	0.844158i	$0.319899\pi$
$192$	0	0
$193$	−646535.	−1.24939	−0.624696	−	0.780868i	$-0.714777\pi$
$193$	−646535.	−1.24939	−0.624696	+	0.780868i	$0.714777\pi$
$194$	75690.4	0.144390
$195$	0	0
$196$	349814.	0.650425
$197$	628338.	1.15353	0.576764	−	0.816911i	$-0.304315\pi$
$197$	628338.	1.15353	0.576764	+	0.816911i	$0.304315\pi$
$198$	0	0
$199$	−196577.	−0.351884	−0.175942	−	0.984401i	$-0.556297\pi$
$199$	−196577.	−0.351884	−0.175942	+	0.984401i	$0.556297\pi$
$200$	−463040.	−0.818547
$201$	0	0
$202$	177664.	0.306352
$203$	−101079.	−0.172156
$204$	0	0
$205$	66999.5	0.111349
$206$	−187668.	−0.308121
$207$	0	0
$208$	155253.	0.248817
$209$	317463.	0.502721
$210$	0	0
$211$	−869514.	−1.34453	−0.672265	−	0.740310i	$-0.734679\pi$
$211$	−869514.	−1.34453	−0.672265	+	0.740310i	$0.734679\pi$
$212$	429094.	0.655711
$213$	0	0
$214$	−440737.	−0.657877
$215$	883321.	1.30323
$216$	0	0
$217$	−393958.	−0.567937
$218$	−586716.	−0.836154
$219$	0	0
$220$	285579.	0.397804
$221$	−829673.	−1.14268
$222$	0	0
$223$	−108640.	−0.146294	−0.0731472	−	0.997321i	$-0.523304\pi$
$223$	−108640.	−0.146294	−0.0731472	+	0.997321i	$0.523304\pi$
$224$	271209.	0.361146
$225$	0	0
$226$	−693861.	−0.903653
$227$	1.10451e6	1.42268	0.711338	−	0.702850i	$-0.248090\pi$
$227$	1.10451e6	1.42268	0.711338	+	0.702850i	$0.248090\pi$
$228$	0	0
$229$	113945.	0.143584	0.0717920	−	0.997420i	$-0.477128\pi$
$229$	113945.	0.143584	0.0717920	+	0.997420i	$0.477128\pi$
$230$	237369.	0.295872
$231$	0	0
$232$	357999.	0.436679
$233$	374241.	0.451608	0.225804	−	0.974173i	$-0.427499\pi$
$233$	374241.	0.451608	0.225804	+	0.974173i	$0.427499\pi$
$234$	0	0
$235$	−429200.	−0.506980
$236$	−82525.5	−0.0964513
$237$	0	0
$238$	−206793.	−0.236643
$239$	1.06804e6	1.20946	0.604731	−	0.796430i	$-0.293281\pi$
$239$	1.06804e6	1.20946	0.604731	+	0.796430i	$0.293281\pi$
$240$	0	0
$241$	−330951.	−0.367046	−0.183523	−	0.983015i	$-0.558750\pi$
$241$	−330951.	−0.367046	−0.183523	+	0.983015i	$0.558750\pi$
$242$	−394266.	−0.432763
$243$	0	0
$244$	1.05170e6	1.13089
$245$	−1.14404e6	−1.21766
$246$	0	0
$247$	−1.06928e6	−1.11519
$248$	1.39531e6	1.44059
$249$	0	0
$250$	−53266.3	−0.0539017
$251$	−903692.	−0.905391	−0.452695	−	0.891665i	$-0.649537\pi$
$251$	−903692.	−0.905391	−0.452695	+	0.891665i	$0.649537\pi$
$252$	0	0
$253$	−165175.	−0.162234
$254$	−434803.	−0.422872
$255$	0	0
$256$	−735487.	−0.701415
$257$	−721353.	−0.681264	−0.340632	−	0.940197i	$-0.610641\pi$
$257$	−721353.	−0.681264	−0.340632	+	0.940197i	$0.610641\pi$
$258$	0	0
$259$	683640.	0.633254
$260$	−961892.	−0.882455
$261$	0	0
$262$	721652.	0.649493
$263$	−601446.	−0.536176	−0.268088	−	0.963394i	$-0.586392\pi$
$263$	−601446.	−0.536176	−0.268088	+	0.963394i	$0.586392\pi$
$264$	0	0
$265$	−1.40332e6	−1.22756
$266$	−266516.	−0.230950
$267$	0	0
$268$	790808.	0.672565
$269$	−1.22319e6	−1.03065	−0.515326	−	0.856994i	$-0.672329\pi$
$269$	−1.22319e6	−1.03065	−0.515326	+	0.856994i	$0.672329\pi$
$270$	0	0
$271$	−1.75129e6	−1.44855	−0.724277	−	0.689509i	$-0.757826\pi$
$271$	−1.75129e6	−1.44855	−0.724277	+	0.689509i	$0.757826\pi$
$272$	−470370.	−0.385494
$273$	0	0
$274$	45169.2	0.0363468
$275$	−448450.	−0.357588
$276$	0	0
$277$	638822.	0.500243	0.250121	−	0.968214i	$-0.419529\pi$
$277$	638822.	0.500243	0.250121	+	0.968214i	$0.419529\pi$
$278$	193224.	0.149951
$279$	0	0
$280$	−563358.	−0.429427
$281$	1.72109e6	1.30028	0.650141	−	0.759814i	$-0.274710\pi$
$281$	1.72109e6	1.30028	0.650141	+	0.759814i	$0.274710\pi$
$282$	0	0
$283$	−480652.	−0.356750	−0.178375	−	0.983963i	$-0.557084\pi$
$283$	−480652.	−0.356750	−0.178375	+	0.983963i	$0.557084\pi$
$284$	67807.7	0.0498865
$285$	0	0
$286$	−234127.	−0.169253
$287$	39140.0	0.0280489
$288$	0	0
$289$	1.09381e6	0.770364
$290$	−498262.	−0.347907
$291$	0	0
$292$	1.51220e6	1.03789
$293$	2.08942e6	1.42186	0.710931	−	0.703261i	$-0.248274\pi$
$293$	2.08942e6	1.42186	0.710931	+	0.703261i	$0.248274\pi$
$294$	0	0
$295$	269894.	0.180567
$296$	−2.42129e6	−1.60627
$297$	0	0
$298$	−314982.	−0.205469
$299$	556344.	0.359886
$300$	0	0
$301$	516021.	0.328285
$302$	−936756.	−0.591029
$303$	0	0
$304$	−606213.	−0.376220
$305$	−3.43953e6	−2.11714
$306$	0	0
$307$	2.82729e6	1.71208	0.856040	−	0.516910i	$-0.172918\pi$
$307$	2.82729e6	1.71208	0.856040	+	0.516910i	$0.172918\pi$
$308$	166830.	0.100207
$309$	0	0
$310$	−1.94198e6	−1.14773
$311$	1.90959e6	1.11954	0.559769	−	0.828648i	$-0.310890\pi$
$311$	1.90959e6	1.11954	0.559769	+	0.828648i	$0.310890\pi$
$312$	0	0
$313$	−2.15361e6	−1.24253	−0.621264	−	0.783602i	$-0.713380\pi$
$313$	−2.15361e6	−1.24253	−0.621264	+	0.783602i	$0.713380\pi$
$314$	−703637.	−0.402740
$315$	0	0
$316$	−1.96670e6	−1.10795
$317$	2.27397e6	1.27097	0.635486	−	0.772113i	$-0.280800\pi$
$317$	2.27397e6	1.27097	0.635486	+	0.772113i	$0.280800\pi$
$318$	0	0
$319$	346719.	0.190766
$320$	600822.	0.327998
$321$	0	0
$322$	138667.	0.0745304
$323$	3.23961e6	1.72777
$324$	0	0
$325$	1.51048e6	0.793242
$326$	248301.	0.129400
$327$	0	0
$328$	−138625.	−0.0711469
$329$	−250732.	−0.127708
$330$	0	0
$331$	3.45401e6	1.73282	0.866410	−	0.499334i	$-0.166422\pi$
$331$	3.45401e6	1.73282	0.866410	+	0.499334i	$0.166422\pi$
$332$	−1.61072e6	−0.801999
$333$	0	0
$334$	−464332.	−0.227752
$335$	−2.58628e6	−1.25911
$336$	0	0
$337$	2.83240e6	1.35856	0.679281	−	0.733878i	$-0.262292\pi$
$337$	2.83240e6	1.35856	0.679281	+	0.733878i	$0.262292\pi$
$338$	−280617.	−0.133605
$339$	0	0
$340$	2.91425e6	1.36719
$341$	1.35134e6	0.629331
$342$	0	0
$343$	−1.42958e6	−0.656106
$344$	−1.82763e6	−0.832706
$345$	0	0
$346$	−1.00424e6	−0.450971
$347$	152475.	0.0679791	0.0339895	−	0.999422i	$-0.489179\pi$
$347$	152475.	0.0679791	0.0339895	+	0.999422i	$0.489179\pi$
$348$	0	0
$349$	3.07524e6	1.35150	0.675748	−	0.737132i	$-0.263821\pi$
$349$	3.07524e6	1.35150	0.675748	+	0.737132i	$0.263821\pi$
$350$	376482.	0.164276
$351$	0	0
$352$	−930291.	−0.400186
$353$	−2.89392e6	−1.23609	−0.618045	−	0.786142i	$-0.712075\pi$
$353$	−2.89392e6	−1.23609	−0.618045	+	0.786142i	$0.712075\pi$
$354$	0	0
$355$	−221760.	−0.0933927
$356$	534422.	0.223491
$357$	0	0
$358$	882118.	0.363764
$359$	3.98179e6	1.63058	0.815291	−	0.579052i	$-0.196577\pi$
$359$	3.98179e6	1.63058	0.815291	+	0.579052i	$0.196577\pi$
$360$	0	0
$361$	1.69912e6	0.686207
$362$	1.51384e6	0.607167
$363$	0	0
$364$	−561921.	−0.222291
$365$	−4.94553e6	−1.94304
$366$	0	0
$367$	1.99313e6	0.772451	0.386225	−	0.922404i	$-0.373779\pi$
$367$	1.99313e6	0.772451	0.386225	+	0.922404i	$0.373779\pi$
$368$	315410.	0.121411
$369$	0	0
$370$	3.36995e6	1.27973
$371$	−819796.	−0.309223
$372$	0	0
$373$	1.35506e6	0.504298	0.252149	−	0.967688i	$-0.418863\pi$
$373$	1.35506e6	0.504298	0.252149	+	0.967688i	$0.418863\pi$
$374$	709336.	0.262224
$375$	0	0
$376$	888034.	0.323936
$377$	−1.16782e6	−0.423179
$378$	0	0
$379$	54757.5	0.0195815	0.00979074	−	0.999952i	$-0.496883\pi$
$379$	54757.5	0.0195815	0.00979074	+	0.999952i	$0.496883\pi$
$380$	3.75589e6	1.33430
$381$	0	0
$382$	1.55668e6	0.545810
$383$	2.22595e6	0.775388	0.387694	−	0.921788i	$-0.373272\pi$
$383$	2.22595e6	0.775388	0.387694	+	0.921788i	$0.373272\pi$
$384$	0	0
$385$	−545607.	−0.187598
$386$	−1.86182e6	−0.636017
$387$	0	0
$388$	−623131.	−0.210136
$389$	−3.54979e6	−1.18940	−0.594702	−	0.803947i	$-0.702730\pi$
$389$	−3.54979e6	−1.18940	−0.594702	+	0.803947i	$0.702730\pi$
$390$	0	0
$391$	−1.68556e6	−0.557573
$392$	2.36707e6	0.778030
$393$	0	0
$394$	1.80942e6	0.587216
$395$	6.43197e6	2.07420
$396$	0	0
$397$	−3.38798e6	−1.07886	−0.539430	−	0.842031i	$-0.681360\pi$
$397$	−3.38798e6	−1.07886	−0.539430	+	0.842031i	$0.681360\pi$
$398$	−566079.	−0.179130
$399$	0	0
$400$	856341.	0.267607
$401$	−1.54937e6	−0.481165	−0.240583	−	0.970629i	$-0.577339\pi$
$401$	−1.54937e6	−0.481165	−0.240583	+	0.970629i	$0.577339\pi$
$402$	0	0
$403$	−4.55161e6	−1.39605
$404$	−1.46264e6	−0.445846
$405$	0	0
$406$	−291077.	−0.0876379
$407$	−2.34500e6	−0.701708
$408$	0	0
$409$	−1.40903e6	−0.416497	−0.208248	−	0.978076i	$-0.566776\pi$
$409$	−1.40903e6	−0.416497	−0.208248	+	0.978076i	$0.566776\pi$
$410$	192937.	0.0566835
$411$	0	0
$412$	1.54500e6	0.448421
$413$	157667.	0.0454849
$414$	0	0
$415$	5.26774e6	1.50143
$416$	3.13342e6	0.887739
$417$	0	0
$418$	914193.	0.255916
$419$	1.78943e6	0.497942	0.248971	−	0.968511i	$-0.419908\pi$
$419$	1.78943e6	0.497942	0.248971	+	0.968511i	$0.419908\pi$
$420$	0	0
$421$	4.50624e6	1.23911	0.619554	−	0.784954i	$-0.287314\pi$
$421$	4.50624e6	1.23911	0.619554	+	0.784954i	$0.287314\pi$
$422$	−2.50393e6	−0.684449
$423$	0	0
$424$	2.90353e6	0.784353
$425$	−4.57630e6	−1.22897
$426$	0	0
$427$	−2.00931e6	−0.533308
$428$	3.62843e6	0.957434
$429$	0	0
$430$	2.54369e6	0.663426
$431$	2.68824e6	0.697067	0.348534	−	0.937296i	$-0.386680\pi$
$431$	2.68824e6	0.697067	0.348534	+	0.937296i	$0.386680\pi$
$432$	0	0
$433$	1.53010e6	0.392194	0.196097	−	0.980584i	$-0.437173\pi$
$433$	1.53010e6	0.392194	0.196097	+	0.980584i	$0.437173\pi$
$434$	−1.13447e6	−0.289115
$435$	0	0
$436$	4.83022e6	1.21689
$437$	−2.17235e6	−0.544159
$438$	0	0
$439$	−698534.	−0.172992	−0.0864960	−	0.996252i	$-0.527567\pi$
$439$	−698534.	−0.172992	−0.0864960	+	0.996252i	$0.527567\pi$
$440$	1.93241e6	0.475848
$441$	0	0
$442$	−2.38920e6	−0.581696
$443$	1.73221e6	0.419365	0.209682	−	0.977770i	$-0.432757\pi$
$443$	1.73221e6	0.419365	0.209682	+	0.977770i	$0.432757\pi$
$444$	0	0
$445$	−1.74779e6	−0.418397
$446$	−312849.	−0.0744728
$447$	0	0
$448$	350990.	0.0826228
$449$	−2.98288e6	−0.698265	−0.349133	−	0.937073i	$-0.613524\pi$
$449$	−2.98288e6	−0.698265	−0.349133	+	0.937073i	$0.613524\pi$
$450$	0	0
$451$	−134257.	−0.0310810
$452$	5.71231e6	1.31512
$453$	0	0
$454$	3.18065e6	0.724229
$455$	1.83772e6	0.416152
$456$	0	0
$457$	−3.19426e6	−0.715451	−0.357725	−	0.933827i	$-0.616448\pi$
$457$	−3.19426e6	−0.715451	−0.357725	+	0.933827i	$0.616448\pi$
$458$	328125.	0.0730931
$459$	0	0
$460$	−1.95417e6	−0.430595
$461$	7.83598e6	1.71728	0.858640	−	0.512580i	$-0.171310\pi$
$461$	7.83598e6	1.71728	0.858640	+	0.512580i	$0.171310\pi$
$462$	0	0
$463$	−763425.	−0.165506	−0.0827530	−	0.996570i	$-0.526371\pi$
$463$	−763425.	−0.165506	−0.0827530	+	0.996570i	$0.526371\pi$
$464$	−662079.	−0.142763
$465$	0	0
$466$	1.07770e6	0.229896
$467$	−2.85947e6	−0.606727	−0.303363	−	0.952875i	$-0.598110\pi$
$467$	−2.85947e6	−0.606727	−0.303363	+	0.952875i	$0.598110\pi$
$468$	0	0
$469$	−1.51086e6	−0.317171
$470$	−1.23596e6	−0.258084
$471$	0	0
$472$	−558421.	−0.115374
$473$	−1.77004e6	−0.363773
$474$	0	0
$475$	−5.89794e6	−1.19941
$476$	1.70246e6	0.344396
$477$	0	0
$478$	3.07562e6	0.615690
$479$	−5.27842e6	−1.05115	−0.525575	−	0.850747i	$-0.676150\pi$
$479$	−5.27842e6	−1.05115	−0.525575	+	0.850747i	$0.676150\pi$
$480$	0	0
$481$	7.89846e6	1.55661
$482$	−953034.	−0.186849
$483$	0	0
$484$	3.24585e6	0.629817
$485$	2.03791e6	0.393396
$486$	0	0
$487$	−5.59895e6	−1.06975	−0.534877	−	0.844930i	$-0.679642\pi$
$487$	−5.59895e6	−1.06975	−0.534877	+	0.844930i	$0.679642\pi$
$488$	7.11652e6	1.35275
$489$	0	0
$490$	−3.29448e6	−0.619865
$491$	−4.25242e6	−0.796035	−0.398018	−	0.917378i	$-0.630302\pi$
$491$	−4.25242e6	−0.796035	−0.398018	+	0.917378i	$0.630302\pi$
$492$	0	0
$493$	3.53816e6	0.655633
$494$	−3.07920e6	−0.567702
$495$	0	0
$496$	−2.58046e6	−0.470970
$497$	−129549.	−0.0235257
$498$	0	0
$499$	−45082.1	−0.00810500	−0.00405250	−	0.999992i	$-0.501290\pi$
$499$	−45082.1	−0.00810500	−0.00405250	+	0.999992i	$0.501290\pi$
$500$	438522.	0.0784453
$501$	0	0
$502$	−2.60235e6	−0.460899
$503$	5.54321e6	0.976881	0.488440	−	0.872597i	$-0.337566\pi$
$503$	5.54321e6	0.976881	0.488440	+	0.872597i	$0.337566\pi$
$504$	0	0
$505$	4.78347e6	0.834669
$506$	−475651.	−0.0825871
$507$	0	0
$508$	3.57958e6	0.615421
$509$	8.38202e6	1.43402	0.717008	−	0.697065i	$-0.245511\pi$
$509$	8.38202e6	1.43402	0.717008	+	0.697065i	$0.245511\pi$
$510$	0	0
$511$	−2.88910e6	−0.489452
$512$	3.29942e6	0.556241
$513$	0	0
$514$	−2.07727e6	−0.346805
$515$	−5.05282e6	−0.839490
$516$	0	0
$517$	860052.	0.141514
$518$	1.96867e6	0.322365
$519$	0	0
$520$	−6.50879e6	−1.05558
$521$	6.69577e6	1.08070	0.540351	−	0.841440i	$-0.318291\pi$
$521$	6.69577e6	1.08070	0.540351	+	0.841440i	$0.318291\pi$
$522$	0	0
$523$	−4.06164e6	−0.649303	−0.324651	−	0.945834i	$-0.605247\pi$
$523$	−4.06164e6	−0.649303	−0.324651	+	0.945834i	$0.605247\pi$
$524$	−5.94110e6	−0.945233
$525$	0	0
$526$	−1.73198e6	−0.272946
$527$	1.37900e7	2.16291
$528$	0	0
$529$	−5.30608e6	−0.824393
$530$	−4.04112e6	−0.624903
$531$	0	0
$532$	2.19413e6	0.336111
$533$	452205.	0.0689474
$534$	0	0
$535$	−1.18665e7	−1.79242
$536$	5.35113e6	0.804513
$537$	0	0
$538$	−3.52239e6	−0.524665
$539$	2.29249e6	0.339887
$540$	0	0
$541$	−2.10875e6	−0.309765	−0.154883	−	0.987933i	$-0.549500\pi$
$541$	−2.10875e6	−0.309765	−0.154883	+	0.987933i	$0.549500\pi$
$542$	−5.04317e6	−0.737403
$543$	0	0
$544$	−9.49334e6	−1.37538
$545$	−1.57969e7	−2.27814
$546$	0	0
$547$	−7.48493e6	−1.06960	−0.534798	−	0.844980i	$-0.679612\pi$
$547$	−7.48493e6	−1.06960	−0.534798	+	0.844980i	$0.679612\pi$
$548$	−371862.	−0.0528969
$549$	0	0
$550$	−1.29140e6	−0.182034
$551$	4.55999e6	0.639860
$552$	0	0
$553$	3.75745e6	0.522493
$554$	1.83961e6	0.254654
$555$	0	0
$556$	−1.59074e6	−0.218229
$557$	−7.42234e6	−1.01368	−0.506842	−	0.862039i	$-0.669187\pi$
$557$	−7.42234e6	−1.01368	−0.506842	+	0.862039i	$0.669187\pi$
$558$	0	0
$559$	5.96187e6	0.806962
$560$	1.04187e6	0.140392
$561$	0	0
$562$	4.95619e6	0.661923
$563$	818283.	0.108801	0.0544005	−	0.998519i	$-0.482675\pi$
$563$	818283.	0.108801	0.0544005	+	0.998519i	$0.482675\pi$
$564$	0	0
$565$	−1.86817e7	−2.46204
$566$	−1.38413e6	−0.181608
$567$	0	0
$568$	458831.	0.0596736
$569$	−1.44362e7	−1.86928	−0.934638	−	0.355599i	$-0.884277\pi$
$569$	−1.44362e7	−1.86928	−0.934638	+	0.355599i	$0.884277\pi$
$570$	0	0
$571$	7.48418e6	0.960625	0.480312	−	0.877098i	$-0.340523\pi$
$571$	7.48418e6	0.960625	0.480312	+	0.877098i	$0.340523\pi$
$572$	1.92748e6	0.246321
$573$	0	0
$574$	112711.	0.0142786
$575$	3.06868e6	0.387063
$576$	0	0
$577$	−1.08673e7	−1.35888	−0.679441	−	0.733730i	$-0.737778\pi$
$577$	−1.08673e7	−1.35888	−0.679441	+	0.733730i	$0.737778\pi$
$578$	3.14982e6	0.392163
$579$	0	0
$580$	4.10201e6	0.506322
$581$	3.07732e6	0.378210
$582$	0	0
$583$	2.81204e6	0.342649
$584$	1.02325e7	1.24151
$585$	0	0
$586$	6.01689e6	0.723815
$587$	−1.25762e7	−1.50644	−0.753222	−	0.657766i	$-0.771501\pi$
$587$	−1.25762e7	−1.50644	−0.753222	+	0.657766i	$0.771501\pi$
$588$	0	0
$589$	1.77726e7	2.11088
$590$	777209.	0.0919196
$591$	0	0
$592$	4.47791e6	0.525135
$593$	1.44755e7	1.69042	0.845212	−	0.534430i	$-0.179474\pi$
$593$	1.44755e7	1.69042	0.845212	+	0.534430i	$0.179474\pi$
$594$	0	0
$595$	−5.56776e6	−0.644745
$596$	2.59314e6	0.299026
$597$	0	0
$598$	1.60210e6	0.183204
$599$	−898434.	−0.102310	−0.0511551	−	0.998691i	$-0.516290\pi$
$599$	−898434.	−0.102310	−0.0511551	+	0.998691i	$0.516290\pi$
$600$	0	0
$601$	1.02932e7	1.16242	0.581210	−	0.813754i	$-0.302580\pi$
$601$	1.02932e7	1.16242	0.581210	+	0.813754i	$0.302580\pi$
$602$	1.48598e6	0.167117
$603$	0	0
$604$	7.71197e6	0.860148
$605$	−1.06153e7	−1.17908
$606$	0	0
$607$	1.82491e6	0.201034	0.100517	−	0.994935i	$-0.467950\pi$
$607$	1.82491e6	0.201034	0.100517	+	0.994935i	$0.467950\pi$
$608$	−1.22350e7	−1.34229
$609$	0	0
$610$	−9.90476e6	−1.07775
$611$	−2.89684e6	−0.313922
$612$	0	0
$613$	−621392.	−0.0667905	−0.0333953	−	0.999442i	$-0.510632\pi$
$613$	−621392.	−0.0667905	−0.0333953	+	0.999442i	$0.510632\pi$
$614$	8.14170e6	0.871554
$615$	0	0
$616$	1.12888e6	0.119866
$617$	−5.04840e6	−0.533877	−0.266938	−	0.963714i	$-0.586012\pi$
$617$	−5.04840e6	−0.533877	−0.266938	+	0.963714i	$0.586012\pi$
$618$	0	0
$619$	1.49642e7	1.56974	0.784869	−	0.619662i	$-0.212730\pi$
$619$	1.49642e7	1.56974	0.784869	+	0.619662i	$0.212730\pi$
$620$	1.59877e7	1.67034
$621$	0	0
$622$	5.49902e6	0.569914
$623$	−1.02103e6	−0.105395
$624$	0	0
$625$	−1.04542e7	−1.07051
$626$	−6.20172e6	−0.632523
$627$	0	0
$628$	5.79279e6	0.586123
$629$	−2.39300e7	−2.41166
$630$	0	0
$631$	1.72533e7	1.72504	0.862519	−	0.506024i	$-0.168885\pi$
$631$	1.72533e7	1.72504	0.862519	+	0.506024i	$0.168885\pi$
$632$	−1.33080e7	−1.32532
$633$	0	0
$634$	6.54831e6	0.647002
$635$	−1.17068e7	−1.15213
$636$	0	0
$637$	−7.72159e6	−0.753977
$638$	998442.	0.0971116
$639$	0	0
$640$	−1.31259e7	−1.26672
$641$	1.83730e7	1.76618	0.883092	−	0.469200i	$-0.155457\pi$
$641$	1.83730e7	1.76618	0.883092	+	0.469200i	$0.155457\pi$
$642$	0	0
$643$	1.64115e7	1.56538	0.782690	−	0.622411i	$-0.213847\pi$
$643$	1.64115e7	1.56538	0.782690	+	0.622411i	$0.213847\pi$
$644$	−1.14160e6	−0.108467
$645$	0	0
$646$	9.32907e6	0.879543
$647$	6.16024e6	0.578545	0.289272	−	0.957247i	$-0.406587\pi$
$647$	6.16024e6	0.578545	0.289272	+	0.957247i	$0.406587\pi$
$648$	0	0
$649$	−540826.	−0.0504018
$650$	4.34970e6	0.403809
$651$	0	0
$652$	−2.04417e6	−0.188321
$653$	−1.38748e7	−1.27334	−0.636671	−	0.771135i	$-0.719689\pi$
$653$	−1.38748e7	−1.27334	−0.636671	+	0.771135i	$0.719689\pi$
$654$	0	0
$655$	1.94300e7	1.76957
$656$	256371.	0.0232600
$657$	0	0
$658$	−722029.	−0.0650114
$659$	−1.76364e7	−1.58196	−0.790982	−	0.611839i	$-0.790430\pi$
$659$	−1.76364e7	−1.58196	−0.790982	+	0.611839i	$0.790430\pi$
$660$	0	0
$661$	6.64665e6	0.591697	0.295849	−	0.955235i	$-0.404398\pi$
$661$	6.64665e6	0.591697	0.295849	+	0.955235i	$0.404398\pi$
$662$	9.94646e6	0.882111
$663$	0	0
$664$	−1.08992e7	−0.959341
$665$	−7.17573e6	−0.629234
$666$	0	0
$667$	−2.37254e6	−0.206490
$668$	3.82267e6	0.331456
$669$	0	0
$670$	−7.44768e6	−0.640965
$671$	6.89228e6	0.590958
$672$	0	0
$673$	−1.52418e7	−1.29717	−0.648586	−	0.761141i	$-0.724639\pi$
$673$	−1.52418e7	−1.29717	−0.648586	+	0.761141i	$0.724639\pi$
$674$	8.15642e6	0.691591
$675$	0	0
$676$	2.31022e6	0.194440
$677$	4.32639e6	0.362789	0.181394	−	0.983410i	$-0.441939\pi$
$677$	4.32639e6	0.362789	0.181394	+	0.983410i	$0.441939\pi$
$678$	0	0
$679$	1.19051e6	0.0990967
$680$	1.97197e7	1.63542
$681$	0	0
$682$	3.89144e6	0.320368
$683$	−6.89334e6	−0.565428	−0.282714	−	0.959204i	$-0.591235\pi$
$683$	−6.89334e6	−0.565428	−0.282714	+	0.959204i	$0.591235\pi$
$684$	0	0
$685$	1.21615e6	0.0990284
$686$	−4.11675e6	−0.333998
$687$	0	0
$688$	3.37999e6	0.272235
$689$	−9.47155e6	−0.760104
$690$	0	0
$691$	−1.77702e7	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$691$	−1.77702e7	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$692$	8.26757e6	0.656315
$693$	0	0
$694$	439080.	0.0346055
$695$	5.20241e6	0.408548
$696$	0	0
$697$	−1.37005e6	−0.106820
$698$	8.85571e6	0.687995
$699$	0	0
$700$	−3.09944e6	−0.239077
$701$	2.94835e6	0.226613	0.113306	−	0.993560i	$-0.463856\pi$
$701$	2.94835e6	0.226613	0.113306	+	0.993560i	$0.463856\pi$
$702$	0	0
$703$	−3.08410e7	−2.35364
$704$	−1.20396e6	−0.0915543
$705$	0	0
$706$	−8.33359e6	−0.629246
$707$	2.79442e6	0.210254
$708$	0	0
$709$	−1.10740e7	−0.827346	−0.413673	−	0.910425i	$-0.635754\pi$
$709$	−1.10740e7	−0.827346	−0.413673	+	0.910425i	$0.635754\pi$
$710$	−638600.	−0.0475426
$711$	0	0
$712$	3.61625e6	0.267336
$713$	−9.24702e6	−0.681205
$714$	0	0
$715$	−6.30370e6	−0.461137
$716$	−7.26216e6	−0.529399
$717$	0	0
$718$	1.14663e7	0.830066
$719$	5.21161e6	0.375967	0.187984	−	0.982172i	$-0.439805\pi$
$719$	5.21161e6	0.375967	0.187984	+	0.982172i	$0.439805\pi$
$720$	0	0
$721$	−2.95177e6	−0.211468
$722$	4.89292e6	0.349322
$723$	0	0
$724$	−1.24629e7	−0.883634
$725$	−6.44147e6	−0.455135
$726$	0	0
$727$	−6.91096e6	−0.484956	−0.242478	−	0.970157i	$-0.577960\pi$
$727$	−6.91096e6	−0.484956	−0.242478	+	0.970157i	$0.577960\pi$
$728$	−3.80232e6	−0.265901
$729$	0	0
$730$	−1.42416e7	−0.989124
$731$	−1.80627e7	−1.25023
$732$	0	0
$733$	9.42723e6	0.648073	0.324037	−	0.946045i	$-0.394960\pi$
$733$	9.42723e6	0.648073	0.324037	+	0.946045i	$0.394960\pi$
$734$	5.73959e6	0.393225
$735$	0	0
$736$	6.36584e6	0.433173
$737$	5.18251e6	0.351457
$738$	0	0
$739$	−1.88305e7	−1.26839	−0.634193	−	0.773175i	$-0.718668\pi$
$739$	−1.88305e7	−1.26839	−0.634193	+	0.773175i	$0.718668\pi$
$740$	−2.77436e7	−1.86244
$741$	0	0
$742$	−2.36076e6	−0.157413
$743$	−2.36519e6	−0.157179	−0.0785893	−	0.996907i	$-0.525042\pi$
$743$	−2.36519e6	−0.157179	−0.0785893	+	0.996907i	$0.525042\pi$
$744$	0	0
$745$	−8.48067e6	−0.559808
$746$	3.90216e6	0.256719
$747$	0	0
$748$	−5.83971e6	−0.381625
$749$	−6.93222e6	−0.451510
$750$	0	0
$751$	−1.67747e7	−1.08531	−0.542657	−	0.839955i	$-0.682581\pi$
$751$	−1.67747e7	−1.08531	−0.542657	+	0.839955i	$0.682581\pi$
$752$	−1.64232e6	−0.105904
$753$	0	0
$754$	−3.36297e6	−0.215424
$755$	−2.52215e7	−1.61029
$756$	0	0
$757$	−9.79306e6	−0.621124	−0.310562	−	0.950553i	$-0.600517\pi$
$757$	−9.79306e6	−0.621124	−0.310562	+	0.950553i	$0.600517\pi$
$758$	157684.	0.00996817
$759$	0	0
$760$	2.54148e7	1.59607
$761$	−1.83642e7	−1.14950	−0.574751	−	0.818328i	$-0.694901\pi$
$761$	−1.83642e7	−1.14950	−0.574751	+	0.818328i	$0.694901\pi$
$762$	0	0
$763$	−9.22827e6	−0.573864
$764$	−1.28156e7	−0.794338
$765$	0	0
$766$	6.41005e6	0.394720
$767$	1.82162e6	0.111807
$768$	0	0
$769$	−1.44796e6	−0.0882962	−0.0441481	−	0.999025i	$-0.514057\pi$
$769$	−1.44796e6	−0.0882962	−0.0441481	+	0.999025i	$0.514057\pi$
$770$	−1.57118e6	−0.0954989
$771$	0	0
$772$	1.53277e7	0.925620
$773$	136665.	0.00822637	0.00411318	−	0.999992i	$-0.498691\pi$
$773$	136665.	0.00822637	0.00411318	+	0.999992i	$0.498691\pi$
$774$	0	0
$775$	−2.51057e7	−1.50148
$776$	−4.21651e6	−0.251362
$777$	0	0
$778$	−1.02223e7	−0.605479
$779$	−1.76572e6	−0.104251
$780$	0	0
$781$	444374.	0.0260688
$782$	−4.85388e6	−0.283839
$783$	0	0
$784$	−4.37763e6	−0.254360
$785$	−1.89449e7	−1.09728
$786$	0	0
$787$	2.64152e7	1.52026	0.760129	−	0.649772i	$-0.225136\pi$
$787$	2.64152e7	1.52026	0.760129	+	0.649772i	$0.225136\pi$
$788$	−1.48963e7	−0.854598
$789$	0	0
$790$	1.85220e7	1.05590
$791$	−1.09135e7	−0.620189
$792$	0	0
$793$	−2.32147e7	−1.31093
$794$	−9.75633e6	−0.549206
$795$	0	0
$796$	4.66032e6	0.260695
$797$	1.25477e7	0.699708	0.349854	−	0.936804i	$-0.386231\pi$
$797$	1.25477e7	0.699708	0.349854	+	0.936804i	$0.386231\pi$
$798$	0	0
$799$	8.77657e6	0.486360
$800$	1.72833e7	0.954776
$801$	0	0
$802$	−4.46170e6	−0.244943
$803$	9.91009e6	0.542361
$804$	0	0
$805$	3.73351e6	0.203061
$806$	−1.31072e7	−0.710677
$807$	0	0
$808$	−9.89720e6	−0.533315
$809$	−2.17849e7	−1.17027	−0.585134	−	0.810937i	$-0.698958\pi$
$809$	−2.17849e7	−1.17027	−0.585134	+	0.810937i	$0.698958\pi$
$810$	0	0
$811$	−2.63202e7	−1.40520	−0.702599	−	0.711586i	$-0.747977\pi$
$811$	−2.63202e7	−1.40520	−0.702599	+	0.711586i	$0.747977\pi$
$812$	2.39633e6	0.127543
$813$	0	0
$814$	−6.75286e6	−0.357213
$815$	6.68531e6	0.352556
$816$	0	0
$817$	−2.32793e7	−1.22015
$818$	−4.05756e6	−0.212022
$819$	0	0
$820$	−1.58838e6	−0.0824937
$821$	3.00385e7	1.55532	0.777661	−	0.628684i	$-0.216406\pi$
$821$	3.00385e7	1.55532	0.777661	+	0.628684i	$0.216406\pi$
$822$	0	0
$823$	8.67425e6	0.446408	0.223204	−	0.974772i	$-0.428348\pi$
$823$	8.67425e6	0.446408	0.223204	+	0.974772i	$0.428348\pi$
$824$	1.04545e7	0.536395
$825$	0	0
$826$	454033.	0.0231546
$827$	5.30575e6	0.269763	0.134882	−	0.990862i	$-0.456935\pi$
$827$	5.30575e6	0.269763	0.134882	+	0.990862i	$0.456935\pi$
$828$	0	0
$829$	−3221.11	−0.000162787 0	−8.13934e−5	−	1.00000i	$-0.500026\pi$
$829$	−3221.11	−0.000162787 0	−8.13934e−5		1.00000i	$0.500026\pi$
$830$	1.51694e7	0.764318
$831$	0	0
$832$	4.05518e6	0.203096
$833$	2.33941e7	1.16814
$834$	0	0
$835$	−1.25018e7	−0.620520
$836$	−7.52622e6	−0.372444
$837$	0	0
$838$	5.15298e6	0.253483
$839$	1.48233e7	0.727008	0.363504	−	0.931593i	$-0.381580\pi$
$839$	1.48233e7	0.727008	0.363504	+	0.931593i	$0.381580\pi$
$840$	0	0
$841$	−1.55309e7	−0.757195
$842$	1.29766e7	0.630782
$843$	0	0
$844$	2.06139e7	0.996104
$845$	−7.55540e6	−0.364012
$846$	0	0
$847$	−6.20128e6	−0.297011
$848$	−5.36975e6	−0.256427
$849$	0	0
$850$	−1.31783e7	−0.625622
$851$	1.60465e7	0.759549
$852$	0	0
$853$	−1.50069e7	−0.706183	−0.353092	−	0.935589i	$-0.614870\pi$
$853$	−1.50069e7	−0.706183	−0.353092	+	0.935589i	$0.614870\pi$
$854$	−5.78619e6	−0.271486
$855$	0	0
$856$	2.45523e7	1.14527
$857$	−9.04584e6	−0.420724	−0.210362	−	0.977624i	$-0.567464\pi$
$857$	−9.04584e6	−0.420724	−0.210362	+	0.977624i	$0.567464\pi$
$858$	0	0
$859$	5.42003e6	0.250622	0.125311	−	0.992118i	$-0.460007\pi$
$859$	5.42003e6	0.250622	0.125311	+	0.992118i	$0.460007\pi$
$860$	−2.09412e7	−0.965509
$861$	0	0
$862$	7.74129e6	0.354850
$863$	2.45589e7	1.12249	0.561245	−	0.827650i	$-0.310322\pi$
$863$	2.45589e7	1.12249	0.561245	+	0.827650i	$0.310322\pi$
$864$	0	0
$865$	−2.70385e7	−1.22869
$866$	4.40622e6	0.199651
$867$	0	0
$868$	9.33972e6	0.420760
$869$	−1.28887e7	−0.578974
$870$	0	0
$871$	−1.74558e7	−0.779641
$872$	3.26844e7	1.45562
$873$	0	0
$874$	−6.25568e6	−0.277010
$875$	−837810.	−0.0369935
$876$	0	0
$877$	1.94355e7	0.853289	0.426644	−	0.904420i	$-0.359696\pi$
$877$	1.94355e7	0.853289	0.426644	+	0.904420i	$0.359696\pi$
$878$	−2.01156e6	−0.0880636
$879$	0	0
$880$	−3.57378e6	−0.155568
$881$	4.44437e7	1.92917	0.964584	−	0.263775i	$-0.0849675\pi$
$881$	4.44437e7	1.92917	0.964584	+	0.263775i	$0.0849675\pi$
$882$	0	0
$883$	−1.10995e7	−0.479072	−0.239536	−	0.970888i	$-0.576995\pi$
$883$	−1.10995e7	−0.479072	−0.239536	+	0.970888i	$0.576995\pi$
$884$	1.96694e7	0.846565
$885$	0	0
$886$	4.98823e6	0.213482
$887$	1.92347e7	0.820872	0.410436	−	0.911889i	$-0.365377\pi$
$887$	1.92347e7	0.820872	0.410436	+	0.911889i	$0.365377\pi$
$888$	0	0
$889$	−6.83889e6	−0.290223
$890$	−5.03308e6	−0.212990
$891$	0	0
$892$	2.57557e6	0.108383
$893$	1.13113e7	0.474659
$894$	0	0
$895$	2.37504e7	0.991089
$896$	−7.66793e6	−0.319086
$897$	0	0
$898$	−8.58977e6	−0.355460
$899$	1.94105e7	0.801008
$900$	0	0
$901$	2.86960e7	1.17763
$902$	−386617.	−0.0158221
$903$	0	0
$904$	3.86532e7	1.57313
$905$	4.07590e7	1.65425
$906$	0	0
$907$	−1.01314e6	−0.0408931	−0.0204466	−	0.999791i	$-0.506509\pi$
$907$	−1.01314e6	−0.0408931	−0.0204466	+	0.999791i	$0.506509\pi$
$908$	−2.61851e7	−1.05400
$909$	0	0
$910$	5.29206e6	0.211847
$911$	3.28323e7	1.31071	0.655354	−	0.755322i	$-0.272520\pi$
$911$	3.28323e7	1.31071	0.655354	+	0.755322i	$0.272520\pi$
$912$	0	0
$913$	−1.05557e7	−0.419094
$914$	−9.19846e6	−0.364208
$915$	0	0
$916$	−2.70134e6	−0.106375
$917$	1.13507e7	0.445756
$918$	0	0
$919$	6.66545e6	0.260340	0.130170	−	0.991492i	$-0.458448\pi$
$919$	6.66545e6	0.260340	0.130170	+	0.991492i	$0.458448\pi$
$920$	−1.32232e7	−0.515071
$921$	0	0
$922$	2.25652e7	0.874201
$923$	−1.49675e6	−0.0578288
$924$	0	0
$925$	4.35662e7	1.67416
$926$	−2.19842e6	−0.0842528
$927$	0	0
$928$	−1.33626e7	−0.509354
$929$	−1.25443e7	−0.476877	−0.238439	−	0.971158i	$-0.576636\pi$
$929$	−1.25443e7	−0.476877	−0.238439	+	0.971158i	$0.576636\pi$
$930$	0	0
$931$	3.01504e7	1.14004
$932$	−8.87229e6	−0.334577
$933$	0	0
$934$	−8.23437e6	−0.308861
$935$	1.90983e7	0.714442
$936$	0	0
$937$	−3.57395e7	−1.32984	−0.664921	−	0.746914i	$-0.731535\pi$
$937$	−3.57395e7	−1.32984	−0.664921	+	0.746914i	$0.731535\pi$
$938$	−4.35081e6	−0.161459
$939$	0	0
$940$	1.01752e7	0.375599
$941$	−2.36120e7	−0.869277	−0.434639	−	0.900605i	$-0.643124\pi$
$941$	−2.36120e7	−0.869277	−0.434639	+	0.900605i	$0.643124\pi$
$942$	0	0
$943$	918698.	0.0336429
$944$	1.03274e6	0.0377190
$945$	0	0
$946$	−5.09716e6	−0.185183
$947$	5.01198e7	1.81608	0.908038	−	0.418887i	$-0.137580\pi$
$947$	5.01198e7	1.81608	0.908038	+	0.418887i	$0.137580\pi$
$948$	0	0
$949$	−3.33793e7	−1.20313
$950$	−1.69842e7	−0.610571
$951$	0	0
$952$	1.15199e7	0.411962
$953$	−3.11140e7	−1.10975	−0.554874	−	0.831935i	$-0.687233\pi$
$953$	−3.11140e7	−1.10975	−0.554874	+	0.831935i	$0.687233\pi$
$954$	0	0
$955$	4.19125e7	1.48708
$956$	−2.53204e7	−0.896038
$957$	0	0
$958$	−1.52002e7	−0.535100
$959$	710453.	0.0249453
$960$	0	0
$961$	4.70234e7	1.64250
$962$	2.27451e7	0.792410
$963$	0	0
$964$	7.84598e6	0.271928
$965$	−5.01280e7	−1.73286
$966$	0	0
$967$	−1.64097e7	−0.564331	−0.282166	−	0.959366i	$-0.591053\pi$
$967$	−1.64097e7	−0.564331	−0.282166	+	0.959366i	$0.591053\pi$
$968$	2.19635e7	0.753379
$969$	0	0
$970$	5.86853e6	0.200263
$971$	2.88792e7	0.982962	0.491481	−	0.870888i	$-0.336456\pi$
$971$	2.88792e7	0.982962	0.491481	+	0.870888i	$0.336456\pi$
$972$	0	0
$973$	3.03916e6	0.102913
$974$	−1.61232e7	−0.544571
$975$	0	0
$976$	−1.31612e7	−0.442253
$977$	−2.68826e7	−0.901021	−0.450510	−	0.892771i	$-0.648758\pi$
$977$	−2.68826e7	−0.901021	−0.450510	+	0.892771i	$0.648758\pi$
$978$	0	0
$979$	3.50230e6	0.116788
$980$	2.71223e7	0.902113
$981$	0	0
$982$	−1.22456e7	−0.405231
$983$	1.32072e7	0.435942	0.217971	−	0.975955i	$-0.430056\pi$
$983$	1.32072e7	0.435942	0.217971	+	0.975955i	$0.430056\pi$
$984$	0	0
$985$	4.87172e7	1.59990
$986$	1.01888e7	0.333757
$987$	0	0
$988$	2.53499e7	0.826199
$989$	1.21121e7	0.393758
$990$	0	0
$991$	2.08572e6	0.0674639	0.0337320	−	0.999431i	$-0.489261\pi$
$991$	2.08572e6	0.0674639	0.0337320	+	0.999431i	$0.489261\pi$
$992$	−5.20807e7	−1.68034
$993$	0	0
$994$	−373059.	−0.0119760
$995$	−1.52413e7	−0.488048
$996$	0	0
$997$	3.49916e6	0.111487	0.0557437	−	0.998445i	$-0.482247\pi$
$997$	3.49916e6	0.111487	0.0557437	+	0.998445i	$0.482247\pi$
$998$	−129822.	−0.00412594
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.d.1.8		12
3.2	odd	2		177.6.a.b.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.b.1.5	✓	12	3.2	odd	2
531.6.a.d.1.8		12	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+2.87969 q^{2} -23.7074 q^{4} +77.5334 q^{5} +45.2937 q^{7} -160.420 q^{8} +O(q^{10})\)	\(q+2.87969 q^{2} -23.7074 q^{4} +77.5334 q^{5} +45.2937 q^{7} -160.420 q^{8} +223.272 q^{10} -155.365 q^{11} +523.303 q^{13} +130.432 q^{14} +296.678 q^{16} -1585.45 q^{17} -2043.33 q^{19} -1838.12 q^{20} -447.403 q^{22} +1063.14 q^{23} +2886.43 q^{25} +1506.95 q^{26} -1073.80 q^{28} -2231.64 q^{29} -8697.85 q^{31} +5987.78 q^{32} -4565.61 q^{34} +3511.78 q^{35} +15093.5 q^{37} -5884.16 q^{38} -12437.9 q^{40} +864.137 q^{41} +11392.8 q^{43} +3683.30 q^{44} +3061.51 q^{46} -5535.68 q^{47} -14755.5 q^{49} +8312.01 q^{50} -12406.2 q^{52} -18099.6 q^{53} -12046.0 q^{55} -7266.01 q^{56} -6426.42 q^{58} +3481.00 q^{59} -44361.9 q^{61} -25047.1 q^{62} +7749.20 q^{64} +40573.5 q^{65} -33357.0 q^{67} +37587.0 q^{68} +10112.8 q^{70} -2860.19 q^{71} -63785.8 q^{73} +43464.5 q^{74} +48442.2 q^{76} -7037.06 q^{77} +82957.4 q^{79} +23002.5 q^{80} +2488.44 q^{82} +67941.5 q^{83} -122926. q^{85} +32807.6 q^{86} +24923.6 q^{88} -22542.4 q^{89} +23702.3 q^{91} -25204.3 q^{92} -15941.0 q^{94} -158427. q^{95} +26284.2 q^{97} -42491.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more