LMFDB - Embedded newform 531.6.a.d.1.9

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			1.9
Root			$4.14510$ 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)

$f(q)$	$=$	$q+4.14510 q^{2} -14.8181 q^{4} +34.9684 q^{5} -110.249 q^{7} -194.066 q^{8} +O(q^{10})$	\(q+4.14510 q^{2} -14.8181 q^{4} +34.9684 q^{5} -110.249 q^{7} -194.066 q^{8} +144.948 q^{10} -176.487 q^{11} +818.757 q^{13} -456.993 q^{14} -330.244 q^{16} +1403.91 q^{17} +2005.24 q^{19} -518.166 q^{20} -731.559 q^{22} -846.668 q^{23} -1902.21 q^{25} +3393.83 q^{26} +1633.68 q^{28} -3649.85 q^{29} +5743.05 q^{31} +4841.22 q^{32} +5819.34 q^{34} -3855.23 q^{35} -12323.7 q^{37} +8311.92 q^{38} -6786.18 q^{40} -11909.1 q^{41} -13792.7 q^{43} +2615.21 q^{44} -3509.53 q^{46} -7639.87 q^{47} -4652.21 q^{49} -7884.86 q^{50} -12132.4 q^{52} -6035.29 q^{53} -6171.49 q^{55} +21395.5 q^{56} -15129.0 q^{58} +3481.00 q^{59} -40205.0 q^{61} +23805.6 q^{62} +30635.1 q^{64} +28630.6 q^{65} -54222.1 q^{67} -20803.3 q^{68} -15980.3 q^{70} +15242.5 q^{71} +68611.8 q^{73} -51082.8 q^{74} -29713.8 q^{76} +19457.5 q^{77} -85026.3 q^{79} -11548.1 q^{80} -49364.6 q^{82} -62362.4 q^{83} +49092.4 q^{85} -57172.3 q^{86} +34250.2 q^{88} +136085. q^{89} -90267.0 q^{91} +12546.0 q^{92} -31668.1 q^{94} +70120.0 q^{95} +77595.8 q^{97} -19283.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.14510	0.732758	0.366379	−	0.930466i	$-0.380597\pi$
$2$	4.14510	0.732758	0.366379	+	0.930466i	$0.380597\pi$
$3$	0	0
$3$	0	0
$4$	−14.8181	−0.463066
$5$	34.9684	0.625534	0.312767	−	0.949830i	$-0.398744\pi$
$5$	34.9684	0.625534	0.312767	+	0.949830i	$0.398744\pi$
$6$	0	0
$7$	−110.249	−0.850411	−0.425205	−	0.905097i	$-0.639798\pi$
$7$	−110.249	−0.850411	−0.425205	+	0.905097i	$0.639798\pi$
$8$	−194.066	−1.07207
$9$	0	0
$10$	144.948	0.458365
$11$	−176.487	−0.439777	−0.219888	−	0.975525i	$-0.570569\pi$
$11$	−176.487	−0.439777	−0.219888	+	0.975525i	$0.570569\pi$
$12$	0	0
$13$	818.757	1.34368	0.671841	−	0.740695i	$-0.265504\pi$
$13$	818.757	1.34368	0.671841	+	0.740695i	$0.265504\pi$
$14$	−456.993	−0.623145
$15$	0	0
$16$	−330.244	−0.322504
$17$	1403.91	1.17819	0.589096	−	0.808063i	$-0.299484\pi$
$17$	1403.91	1.17819	0.589096	+	0.808063i	$0.299484\pi$
$18$	0	0
$19$	2005.24	1.27433	0.637165	−	0.770728i	$-0.280107\pi$
$19$	2005.24	1.27433	0.637165	+	0.770728i	$0.280107\pi$
$20$	−518.166	−0.289664
$21$	0	0
$22$	−731.559	−0.322250
$23$	−846.668	−0.333729	−0.166864	−	0.985980i	$-0.553364\pi$
$23$	−846.668	−0.333729	−0.166864	+	0.985980i	$0.553364\pi$
$24$	0	0
$25$	−1902.21	−0.608707
$26$	3393.83	0.984594
$27$	0	0
$28$	1633.68	0.393796
$29$	−3649.85	−0.805897	−0.402949	−	0.915223i	$-0.632015\pi$
$29$	−3649.85	−0.805897	−0.402949	+	0.915223i	$0.632015\pi$
$30$	0	0
$31$	5743.05	1.07334	0.536672	−	0.843791i	$-0.319681\pi$
$31$	5743.05	1.07334	0.536672	+	0.843791i	$0.319681\pi$
$32$	4841.22	0.835756
$33$	0	0
$34$	5819.34	0.863330
$35$	−3855.23	−0.531961
$36$	0	0
$37$	−12323.7	−1.47991	−0.739955	−	0.672656i	$-0.765153\pi$
$37$	−12323.7	−1.47991	−0.739955	+	0.672656i	$0.765153\pi$
$38$	8311.92	0.933775
$39$	0	0
$40$	−6786.18	−0.670618
$41$	−11909.1	−1.10642	−0.553211	−	0.833041i	$-0.686598\pi$
$41$	−11909.1	−1.10642	−0.553211	+	0.833041i	$0.686598\pi$
$42$	0	0
$43$	−13792.7	−1.13757	−0.568787	−	0.822485i	$-0.692587\pi$
$43$	−13792.7	−1.13757	−0.568787	+	0.822485i	$0.692587\pi$
$44$	2615.21	0.203646
$45$	0	0
$46$	−3509.53	−0.244542
$47$	−7639.87	−0.504477	−0.252239	−	0.967665i	$-0.581167\pi$
$47$	−7639.87	−0.504477	−0.252239	+	0.967665i	$0.581167\pi$
$48$	0	0
$49$	−4652.21	−0.276802
$50$	−7884.86	−0.446035
$51$	0	0
$52$	−12132.4	−0.622214
$53$	−6035.29	−0.295127	−0.147563	−	0.989053i	$-0.547143\pi$
$53$	−6035.29	−0.295127	−0.147563	+	0.989053i	$0.547143\pi$
$54$	0	0
$55$	−6171.49	−0.275095
$56$	21395.5	0.911702
$57$	0	0
$58$	−15129.0	−0.590527
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−40205.0	−1.38342	−0.691712	−	0.722174i	$-0.743143\pi$
$61$	−40205.0	−1.38342	−0.691712	+	0.722174i	$0.743143\pi$
$62$	23805.6	0.786501
$63$	0	0
$64$	30635.1	0.934910
$65$	28630.6	0.840519
$66$	0	0
$67$	−54222.1	−1.47567	−0.737836	−	0.674980i	$-0.764152\pi$
$67$	−54222.1	−1.47567	−0.737836	+	0.674980i	$0.764152\pi$
$68$	−20803.3	−0.545581
$69$	0	0
$70$	−15980.3	−0.389798
$71$	15242.5	0.358848	0.179424	−	0.983772i	$-0.442577\pi$
$71$	15242.5	0.358848	0.179424	+	0.983772i	$0.442577\pi$
$72$	0	0
$73$	68611.8	1.50693	0.753463	−	0.657490i	$-0.228382\pi$
$73$	68611.8	1.50693	0.753463	+	0.657490i	$0.228382\pi$
$74$	−51082.8	−1.08442
$75$	0	0
$76$	−29713.8	−0.590099
$77$	19457.5	0.373991
$78$	0	0
$79$	−85026.3	−1.53280	−0.766401	−	0.642363i	$-0.777954\pi$
$79$	−85026.3	−1.53280	−0.766401	+	0.642363i	$0.777954\pi$
$80$	−11548.1	−0.201737
$81$	0	0
$82$	−49364.6	−0.810739
$83$	−62362.4	−0.993636	−0.496818	−	0.867855i	$-0.665498\pi$
$83$	−62362.4	−0.993636	−0.496818	+	0.867855i	$0.665498\pi$
$84$	0	0
$85$	49092.4	0.737000
$86$	−57172.3	−0.833566
$87$	0	0
$88$	34250.2	0.471473
$89$	136085.	1.82111	0.910555	−	0.413389i	$-0.135655\pi$
$89$	136085.	1.82111	0.910555	+	0.413389i	$0.135655\pi$
$90$	0	0
$91$	−90267.0	−1.14268
$92$	12546.0	0.154539
$93$	0	0
$94$	−31668.1	−0.369660
$95$	70120.0	0.797137
$96$	0	0
$97$	77595.8	0.837353	0.418677	−	0.908135i	$-0.362494\pi$
$97$	77595.8	0.837353	0.418677	+	0.908135i	$0.362494\pi$
$98$	−19283.9	−0.202829
$99$	0	0
$100$	28187.2	0.281872
$101$	−172130.	−1.67901	−0.839503	−	0.543355i	$-0.817154\pi$
$101$	−172130.	−1.67901	−0.839503	+	0.543355i	$0.817154\pi$
$102$	0	0
$103$	58120.5	0.539804	0.269902	−	0.962888i	$-0.413009\pi$
$103$	58120.5	0.539804	0.269902	+	0.962888i	$0.413009\pi$
$104$	−158893.	−1.44053
$105$	0	0
$106$	−25016.9	−0.216256
$107$	142810.	1.20587	0.602935	−	0.797791i	$-0.293998\pi$
$107$	142810.	1.20587	0.602935	+	0.797791i	$0.293998\pi$
$108$	0	0
$109$	15204.1	0.122573	0.0612866	−	0.998120i	$-0.480480\pi$
$109$	15204.1	0.122573	0.0612866	+	0.998120i	$0.480480\pi$
$110$	−25581.5	−0.201578
$111$	0	0
$112$	36409.0	0.274261
$113$	−145209.	−1.06978	−0.534892	−	0.844920i	$-0.679648\pi$
$113$	−145209.	−1.06978	−0.534892	+	0.844920i	$0.679648\pi$
$114$	0	0
$115$	−29606.7	−0.208759
$116$	54083.9	0.373184
$117$	0	0
$118$	14429.1	0.0953969
$119$	−154779.	−1.00195
$120$	0	0
$121$	−129903.	−0.806597
$122$	−166654.	−1.01371
$123$	0	0
$124$	−85101.2	−0.497029
$125$	−175794.	−1.00630
$126$	0	0
$127$	−21609.6	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$127$	−21609.6	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$128$	−27933.1	−0.150693
$129$	0	0
$130$	118677.	0.615897
$131$	−3428.28	−0.0174541	−0.00872707	−	0.999962i	$-0.502778\pi$
$131$	−3428.28	−0.0174541	−0.00872707	+	0.999962i	$0.502778\pi$
$132$	0	0
$133$	−221075.	−1.08370
$134$	−224756.	−1.08131
$135$	0	0
$136$	−272451.	−1.26311
$137$	−53929.4	−0.245484	−0.122742	−	0.992439i	$-0.539169\pi$
$137$	−53929.4	−0.245484	−0.122742	+	0.992439i	$0.539169\pi$
$138$	0	0
$139$	−46431.4	−0.203833	−0.101917	−	0.994793i	$-0.532497\pi$
$139$	−46431.4	−0.203833	−0.101917	+	0.994793i	$0.532497\pi$
$140$	57127.2	0.246333
$141$	0	0
$142$	63181.7	0.262948
$143$	−144500.	−0.590920
$144$	0	0
$145$	−127629.	−0.504116
$146$	284403.	1.10421
$147$	0	0
$148$	182613.	0.685296
$149$	−395001.	−1.45758	−0.728790	−	0.684737i	$-0.759917\pi$
$149$	−395001.	−1.45758	−0.728790	+	0.684737i	$0.759917\pi$
$150$	0	0
$151$	359613.	1.28349	0.641746	−	0.766918i	$-0.278211\pi$
$151$	359613.	1.28349	0.641746	+	0.766918i	$0.278211\pi$
$152$	−389148.	−1.36617
$153$	0	0
$154$	80653.5	0.274045
$155$	200826.	0.671413
$156$	0	0
$157$	−602958.	−1.95226	−0.976130	−	0.217185i	$-0.930312\pi$
$157$	−602958.	−1.95226	−0.976130	+	0.217185i	$0.930312\pi$
$158$	−352443.	−1.12317
$159$	0	0
$160$	169290.	0.522794
$161$	93344.2	0.283807
$162$	0	0
$163$	−571386.	−1.68446	−0.842230	−	0.539118i	$-0.818758\pi$
$163$	−571386.	−1.68446	−0.842230	+	0.539118i	$0.818758\pi$
$164$	176471.	0.512346
$165$	0	0
$166$	−258498.	−0.728094
$167$	484312.	1.34380	0.671899	−	0.740643i	$-0.265479\pi$
$167$	484312.	1.34380	0.671899	+	0.740643i	$0.265479\pi$
$168$	0	0
$169$	299070.	0.805483
$170$	203493.	0.540042
$171$	0	0
$172$	204382.	0.526772
$173$	79449.3	0.201825	0.100912	−	0.994895i	$-0.467824\pi$
$173$	79449.3	0.201825	0.100912	+	0.994895i	$0.467824\pi$
$174$	0	0
$175$	209716.	0.517651
$176$	58283.9	0.141830
$177$	0	0
$178$	564087.	1.33443
$179$	417149.	0.973103	0.486552	−	0.873652i	$-0.338255\pi$
$179$	417149.	0.973103	0.486552	+	0.873652i	$0.338255\pi$
$180$	0	0
$181$	−349894.	−0.793854	−0.396927	−	0.917850i	$-0.629923\pi$
$181$	−349894.	−0.793854	−0.396927	+	0.917850i	$0.629923\pi$
$182$	−374166.	−0.837309
$183$	0	0
$184$	164310.	0.357782
$185$	−430939.	−0.925734
$186$	0	0
$187$	−247772.	−0.518141
$188$	113209.	0.233606
$189$	0	0
$190$	290655.	0.584108
$191$	162965.	0.323229	0.161614	−	0.986854i	$-0.448330\pi$
$191$	162965.	0.323229	0.161614	+	0.986854i	$0.448330\pi$
$192$	0	0
$193$	−408375.	−0.789162	−0.394581	−	0.918861i	$-0.629110\pi$
$193$	−408375.	−0.789162	−0.394581	+	0.918861i	$0.629110\pi$
$194$	321642.	0.613577
$195$	0	0
$196$	68936.9	0.128178
$197$	639162.	1.17340	0.586699	−	0.809805i	$-0.300427\pi$
$197$	639162.	1.17340	0.586699	+	0.809805i	$0.300427\pi$
$198$	0	0
$199$	−137487.	−0.246111	−0.123055	−	0.992400i	$-0.539269\pi$
$199$	−137487.	−0.246111	−0.123055	+	0.992400i	$0.539269\pi$
$200$	369154.	0.652578
$201$	0	0
$202$	−713495.	−1.23030
$203$	402391.	0.685344
$204$	0	0
$205$	−416444.	−0.692105
$206$	240916.	0.395546
$207$	0	0
$208$	−270389.	−0.433343
$209$	−353899.	−0.560420
$210$	0	0
$211$	−146412.	−0.226397	−0.113199	−	0.993572i	$-0.536110\pi$
$211$	−146412.	−0.226397	−0.113199	+	0.993572i	$0.536110\pi$
$212$	89431.6	0.136663
$213$	0	0
$214$	591964.	0.883610
$215$	−482310.	−0.711591
$216$	0	0
$217$	−633165.	−0.912782
$218$	63022.7	0.0898164
$219$	0	0
$220$	91449.8	0.127387
$221$	1.14946e6	1.58312
$222$	0	0
$223$	212713.	0.286439	0.143220	−	0.989691i	$-0.454254\pi$
$223$	212713.	0.286439	0.143220	+	0.989691i	$0.454254\pi$
$224$	−533738.	−0.710736
$225$	0	0
$226$	−601905.	−0.783893
$227$	−56603.0	−0.0729079	−0.0364540	−	0.999335i	$-0.511606\pi$
$227$	−56603.0	−0.0729079	−0.0364540	+	0.999335i	$0.511606\pi$
$228$	0	0
$229$	−893019.	−1.12531	−0.562655	−	0.826692i	$-0.690220\pi$
$229$	−893019.	−1.12531	−0.562655	+	0.826692i	$0.690220\pi$
$230$	−122723.	−0.152970
$231$	0	0
$232$	708311.	0.863981
$233$	1.08789e6	1.31280	0.656398	−	0.754415i	$-0.272079\pi$
$233$	1.08789e6	1.31280	0.656398	+	0.754415i	$0.272079\pi$
$234$	0	0
$235$	−267154.	−0.315568
$236$	−51581.9	−0.0602861
$237$	0	0
$238$	−641575.	−0.734185
$239$	330858.	0.374668	0.187334	−	0.982296i	$-0.440015\pi$
$239$	330858.	0.374668	0.187334	+	0.982296i	$0.440015\pi$
$240$	0	0
$241$	53121.9	0.0589157	0.0294578	−	0.999566i	$-0.490622\pi$
$241$	53121.9	0.0589157	0.0294578	+	0.999566i	$0.490622\pi$
$242$	−538462.	−0.591040
$243$	0	0
$244$	595762.	0.640616
$245$	−162680.	−0.173149
$246$	0	0
$247$	1.64180e6	1.71229
$248$	−1.11453e6	−1.15070
$249$	0	0
$250$	−728683.	−0.737375
$251$	1.41412e6	1.41678	0.708390	−	0.705821i	$-0.249422\pi$
$251$	1.41412e6	1.41678	0.708390	+	0.705821i	$0.249422\pi$
$252$	0	0
$253$	149426.	0.146766
$254$	−89573.9	−0.0871158
$255$	0	0
$256$	−1.09611e6	−1.04533
$257$	−4655.81	−0.00439706	−0.00219853	−	0.999998i	$-0.500700\pi$
$257$	−4655.81	−0.00439706	−0.00219853	+	0.999998i	$0.500700\pi$
$258$	0	0
$259$	1.35867e6	1.25853
$260$	−424252.	−0.389216
$261$	0	0
$262$	−14210.6	−0.0127897
$263$	147942.	0.131887	0.0659433	−	0.997823i	$-0.478994\pi$
$263$	147942.	0.131887	0.0659433	+	0.997823i	$0.478994\pi$
$264$	0	0
$265$	−211045.	−0.184612
$266$	−916379.	−0.794092
$267$	0	0
$268$	803470.	0.683333
$269$	−1.11352e6	−0.938245	−0.469122	−	0.883133i	$-0.655430\pi$
$269$	−1.11352e6	−0.938245	−0.469122	+	0.883133i	$0.655430\pi$
$270$	0	0
$271$	−427188.	−0.353343	−0.176671	−	0.984270i	$-0.556533\pi$
$271$	−427188.	−0.353343	−0.176671	+	0.984270i	$0.556533\pi$
$272$	−463632.	−0.379971
$273$	0	0
$274$	−223543.	−0.179881
$275$	335716.	0.267695
$276$	0	0
$277$	−1.69355e6	−1.32616	−0.663082	−	0.748547i	$-0.730752\pi$
$277$	−1.69355e6	−1.32616	−0.663082	+	0.748547i	$0.730752\pi$
$278$	−192463.	−0.149360
$279$	0	0
$280$	748168.	0.570301
$281$	−2.23022e6	−1.68493	−0.842466	−	0.538750i	$-0.818897\pi$
$281$	−2.23022e6	−1.68493	−0.842466	+	0.538750i	$0.818897\pi$
$282$	0	0
$283$	−1.84337e6	−1.36819	−0.684097	−	0.729391i	$-0.739803\pi$
$283$	−1.84337e6	−1.36819	−0.684097	+	0.729391i	$0.739803\pi$
$284$	−225865.	−0.166170
$285$	0	0
$286$	−598969.	−0.433001
$287$	1.31297e6	0.940913
$288$	0	0
$289$	551100.	0.388138
$290$	−529037.	−0.369395
$291$	0	0
$292$	−1.01670e6	−0.697806
$293$	−326532.	−0.222206	−0.111103	−	0.993809i	$-0.535438\pi$
$293$	−326532.	−0.222206	−0.111103	+	0.993809i	$0.535438\pi$
$294$	0	0
$295$	121725.	0.0814376
$296$	2.39160e6	1.58657
$297$	0	0
$298$	−1.63732e6	−1.06805
$299$	−693216.	−0.448426
$300$	0	0
$301$	1.52063e6	0.967404
$302$	1.49063e6	0.940488
$303$	0	0
$304$	−662217.	−0.410976
$305$	−1.40590e6	−0.865379
$306$	0	0
$307$	−1.14971e6	−0.696216	−0.348108	−	0.937454i	$-0.613176\pi$
$307$	−1.14971e6	−0.696216	−0.348108	+	0.937454i	$0.613176\pi$
$308$	−288324.	−0.173182
$309$	0	0
$310$	832443.	0.491983
$311$	1.82214e6	1.06827	0.534133	−	0.845400i	$-0.320638\pi$
$311$	1.82214e6	1.06827	0.534133	+	0.845400i	$0.320638\pi$
$312$	0	0
$313$	−1.40324e6	−0.809603	−0.404801	−	0.914405i	$-0.632659\pi$
$313$	−1.40324e6	−0.809603	−0.404801	+	0.914405i	$0.632659\pi$
$314$	−2.49932e6	−1.43053
$315$	0	0
$316$	1.25993e6	0.709788
$317$	−2.02187e6	−1.13007	−0.565034	−	0.825068i	$-0.691137\pi$
$317$	−2.02187e6	−1.13007	−0.565034	+	0.825068i	$0.691137\pi$
$318$	0	0
$319$	644152.	0.354415
$320$	1.07126e6	0.584818
$321$	0	0
$322$	386921.	0.207961
$323$	2.81517e6	1.50141
$324$	0	0
$325$	−1.55745e6	−0.817909
$326$	−2.36846e6	−1.23430
$327$	0	0
$328$	2.31116e6	1.18616
$329$	842287.	0.429013
$330$	0	0
$331$	2.67961e6	1.34432	0.672159	−	0.740407i	$-0.265367\pi$
$331$	2.67961e6	1.34432	0.672159	+	0.740407i	$0.265367\pi$
$332$	924093.	0.460119
$333$	0	0
$334$	2.00752e6	0.984678
$335$	−1.89606e6	−0.923083
$336$	0	0
$337$	3.08781e6	1.48107	0.740536	−	0.672017i	$-0.234572\pi$
$337$	3.08781e6	1.48107	0.740536	+	0.672017i	$0.234572\pi$
$338$	1.23968e6	0.590224
$339$	0	0
$340$	−727457.	−0.341280
$341$	−1.01358e6	−0.472031
$342$	0	0
$343$	2.36585e6	1.08581
$344$	2.67670e6	1.21956
$345$	0	0
$346$	329326.	0.147889
$347$	−1.59388e6	−0.710612	−0.355306	−	0.934750i	$-0.615623\pi$
$347$	−1.59388e6	−0.710612	−0.355306	+	0.934750i	$0.615623\pi$
$348$	0	0
$349$	4.02815e6	1.77028	0.885141	−	0.465323i	$-0.154062\pi$
$349$	4.02815e6	1.77028	0.885141	+	0.465323i	$0.154062\pi$
$350$	869296.	0.379313
$351$	0	0
$352$	−854414.	−0.367546
$353$	1.83928e6	0.785618	0.392809	−	0.919620i	$-0.371503\pi$
$353$	1.83928e6	0.785618	0.392809	+	0.919620i	$0.371503\pi$
$354$	0	0
$355$	533006.	0.224471
$356$	−2.01653e6	−0.843294
$357$	0	0
$358$	1.72913e6	0.713049
$359$	−2.26891e6	−0.929140	−0.464570	−	0.885536i	$-0.653791\pi$
$359$	−2.26891e6	−0.929140	−0.464570	+	0.885536i	$0.653791\pi$
$360$	0	0
$361$	1.54488e6	0.623916
$362$	−1.45035e6	−0.581703
$363$	0	0
$364$	1.33759e6	0.529137
$365$	2.39925e6	0.942634
$366$	0	0
$367$	−4.64590e6	−1.80055	−0.900273	−	0.435325i	$-0.856633\pi$
$367$	−4.64590e6	−1.80055	−0.900273	+	0.435325i	$0.856633\pi$
$368$	279607.	0.107629
$369$	0	0
$370$	−1.78629e6	−0.678339
$371$	665383.	0.250979
$372$	0	0
$373$	−417633.	−0.155426	−0.0777128	−	0.996976i	$-0.524762\pi$
$373$	−417633.	−0.155426	−0.0777128	+	0.996976i	$0.524762\pi$
$374$	−1.02704e6	−0.379672
$375$	0	0
$376$	1.48264e6	0.540836
$377$	−2.98834e6	−1.08287
$378$	0	0
$379$	−137274.	−0.0490897	−0.0245449	−	0.999699i	$-0.507814\pi$
$379$	−137274.	−0.0490897	−0.0245449	+	0.999699i	$0.507814\pi$
$380$	−1.03905e6	−0.369127
$381$	0	0
$382$	675505.	0.236848
$383$	3.45107e6	1.20214	0.601072	−	0.799195i	$-0.294740\pi$
$383$	3.45107e6	1.20214	0.601072	+	0.799195i	$0.294740\pi$
$384$	0	0
$385$	680399.	0.233944
$386$	−1.69276e6	−0.578264
$387$	0	0
$388$	−1.14982e6	−0.387750
$389$	4.61323e6	1.54572	0.772860	−	0.634576i	$-0.218825\pi$
$389$	4.61323e6	1.54572	0.772860	+	0.634576i	$0.218825\pi$
$390$	0	0
$391$	−1.18864e6	−0.393197
$392$	902835.	0.296752
$393$	0	0
$394$	2.64939e6	0.859816
$395$	−2.97324e6	−0.958819
$396$	0	0
$397$	2.12195e6	0.675707	0.337854	−	0.941199i	$-0.390299\pi$
$397$	2.12195e6	0.675707	0.337854	+	0.941199i	$0.390299\pi$
$398$	−569900.	−0.180339
$399$	0	0
$400$	628193.	0.196310
$401$	138495.	0.0430105	0.0215052	−	0.999769i	$-0.493154\pi$
$401$	138495.	0.0430105	0.0215052	+	0.999769i	$0.493154\pi$
$402$	0	0
$403$	4.70217e6	1.44223
$404$	2.55064e6	0.777491
$405$	0	0
$406$	1.66795e6	0.502191
$407$	2.17497e6	0.650830
$408$	0	0
$409$	−379673.	−0.112228	−0.0561141	−	0.998424i	$-0.517871\pi$
$409$	−379673.	−0.112228	−0.0561141	+	0.998424i	$0.517871\pi$
$410$	−1.72620e6	−0.507145
$411$	0	0
$412$	−861236.	−0.249965
$413$	−383776.	−0.110714
$414$	0	0
$415$	−2.18071e6	−0.621553
$416$	3.96378e6	1.12299
$417$	0	0
$418$	−1.46695e6	−0.410652
$419$	−2.94021e6	−0.818170	−0.409085	−	0.912496i	$-0.634152\pi$
$419$	−2.94021e6	−0.818170	−0.409085	+	0.912496i	$0.634152\pi$
$420$	0	0
$421$	−2.46278e6	−0.677204	−0.338602	−	0.940930i	$-0.609954\pi$
$421$	−2.46278e6	−0.677204	−0.338602	+	0.940930i	$0.609954\pi$
$422$	−606894.	−0.165894
$423$	0	0
$424$	1.17124e6	0.316397
$425$	−2.67053e6	−0.717174
$426$	0	0
$427$	4.43255e6	1.17648
$428$	−2.11618e6	−0.558397
$429$	0	0
$430$	−1.99923e6	−0.521424
$431$	−894076.	−0.231836	−0.115918	−	0.993259i	$-0.536981\pi$
$431$	−894076.	−0.231836	−0.115918	+	0.993259i	$0.536981\pi$
$432$	0	0
$433$	−2.26975e6	−0.581779	−0.290890	−	0.956757i	$-0.593951\pi$
$433$	−2.26975e6	−0.581779	−0.290890	+	0.956757i	$0.593951\pi$
$434$	−2.62453e6	−0.668848
$435$	0	0
$436$	−225297.	−0.0567595
$437$	−1.69777e6	−0.425281
$438$	0	0
$439$	−5.23628e6	−1.29677	−0.648383	−	0.761314i	$-0.724555\pi$
$439$	−5.23628e6	−1.29677	−0.648383	+	0.761314i	$0.724555\pi$
$440$	1.19768e6	0.294922
$441$	0	0
$442$	4.76463e6	1.16004
$443$	−87708.4	−0.0212340	−0.0106170	−	0.999944i	$-0.503380\pi$
$443$	−87708.4	−0.0212340	−0.0106170	+	0.999944i	$0.503380\pi$
$444$	0	0
$445$	4.75869e6	1.13917
$446$	881719.	0.209891
$447$	0	0
$448$	−3.37749e6	−0.795058
$449$	2.20132e6	0.515308	0.257654	−	0.966237i	$-0.417051\pi$
$449$	2.20132e6	0.515308	0.257654	+	0.966237i	$0.417051\pi$
$450$	0	0
$451$	2.10181e6	0.486578
$452$	2.15172e6	0.495381
$453$	0	0
$454$	−234625.	−0.0534239
$455$	−3.15649e6	−0.714787
$456$	0	0
$457$	6.51177e6	1.45851	0.729254	−	0.684243i	$-0.239867\pi$
$457$	6.51177e6	1.45851	0.729254	+	0.684243i	$0.239867\pi$
$458$	−3.70166e6	−0.824580
$459$	0	0
$460$	438715.	0.0966691
$461$	−3.62362e6	−0.794128	−0.397064	−	0.917791i	$-0.629971\pi$
$461$	−3.62362e6	−0.794128	−0.397064	+	0.917791i	$0.629971\pi$
$462$	0	0
$463$	−1.17608e6	−0.254967	−0.127484	−	0.991841i	$-0.540690\pi$
$463$	−1.17608e6	−0.254967	−0.127484	+	0.991841i	$0.540690\pi$
$464$	1.20534e6	0.259905
$465$	0	0
$466$	4.50944e6	0.961961
$467$	2.36743e6	0.502324	0.251162	−	0.967945i	$-0.419187\pi$
$467$	2.36743e6	0.502324	0.251162	+	0.967945i	$0.419187\pi$
$468$	0	0
$469$	5.97792e6	1.25493
$470$	−1.10738e6	−0.231235
$471$	0	0
$472$	−675544.	−0.139572
$473$	2.43425e6	0.500278
$474$	0	0
$475$	−3.81438e6	−0.775693
$476$	2.29353e6	0.463968
$477$	0	0
$478$	1.37144e6	0.274541
$479$	9.93768e6	1.97900	0.989501	−	0.144525i	$-0.0461654\pi$
$479$	9.93768e6	1.97900	0.989501	+	0.144525i	$0.0461654\pi$
$480$	0	0
$481$	−1.00901e7	−1.98853
$482$	220196.	0.0431709
$483$	0	0
$484$	1.92492e6	0.373508
$485$	2.71340e6	0.523793
$486$	0	0
$487$	1.20632e6	0.230483	0.115242	−	0.993337i	$-0.463236\pi$
$487$	1.20632e6	0.230483	0.115242	+	0.993337i	$0.463236\pi$
$488$	7.80241e6	1.48313
$489$	0	0
$490$	−674327.	−0.126876
$491$	7.58775e6	1.42040	0.710198	−	0.704002i	$-0.248605\pi$
$491$	7.58775e6	1.42040	0.710198	+	0.704002i	$0.248605\pi$
$492$	0	0
$493$	−5.12405e6	−0.949502
$494$	6.80544e6	1.25470
$495$	0	0
$496$	−1.89661e6	−0.346157
$497$	−1.68047e6	−0.305168
$498$	0	0
$499$	1.92717e6	0.346472	0.173236	−	0.984880i	$-0.444578\pi$
$499$	1.92717e6	0.346472	0.173236	+	0.984880i	$0.444578\pi$
$500$	2.60493e6	0.465984
$501$	0	0
$502$	5.86168e6	1.03816
$503$	8.43785e6	1.48700	0.743502	−	0.668734i	$-0.233164\pi$
$503$	8.43785e6	1.48700	0.743502	+	0.668734i	$0.233164\pi$
$504$	0	0
$505$	−6.01910e6	−1.05028
$506$	619388.	0.107544
$507$	0	0
$508$	320213.	0.0550528
$509$	−6.51732e6	−1.11500	−0.557500	−	0.830177i	$-0.688239\pi$
$509$	−6.51732e6	−1.11500	−0.557500	+	0.830177i	$0.688239\pi$
$510$	0	0
$511$	−7.56437e6	−1.28151
$512$	−3.64963e6	−0.615282
$513$	0	0
$514$	−19298.8	−0.00322198
$515$	2.03238e6	0.337666
$516$	0	0
$517$	1.34834e6	0.221857
$518$	5.63182e6	0.922198
$519$	0	0
$520$	−5.55623e6	−0.901098
$521$	252152.	0.0406975	0.0203487	−	0.999793i	$-0.493522\pi$
$521$	252152.	0.0406975	0.0203487	+	0.999793i	$0.493522\pi$
$522$	0	0
$523$	−7.96238e6	−1.27288	−0.636442	−	0.771325i	$-0.719594\pi$
$523$	−7.96238e6	−1.27288	−0.636442	+	0.771325i	$0.719594\pi$
$524$	50800.7	0.00808242
$525$	0	0
$526$	613233.	0.0966410
$527$	8.06272e6	1.26461
$528$	0	0
$529$	−5.71950e6	−0.888625
$530$	−874802.	−0.135276
$531$	0	0
$532$	3.27591e6	0.501826
$533$	−9.75069e6	−1.48668
$534$	0	0
$535$	4.99385e6	0.754312
$536$	1.05227e7	1.58203
$537$	0	0
$538$	−4.61564e6	−0.687506
$539$	821056.	0.121731
$540$	0	0
$541$	−1.58336e6	−0.232587	−0.116294	−	0.993215i	$-0.537101\pi$
$541$	−1.58336e6	−0.232587	−0.116294	+	0.993215i	$0.537101\pi$
$542$	−1.77074e6	−0.258915
$543$	0	0
$544$	6.79662e6	0.984681
$545$	531664.	0.0766737
$546$	0	0
$547$	−5.75298e6	−0.822099	−0.411050	−	0.911613i	$-0.634838\pi$
$547$	−5.75298e6	−0.822099	−0.411050	+	0.911613i	$0.634838\pi$
$548$	799132.	0.113676
$549$	0	0
$550$	1.39158e6	0.196156
$551$	−7.31881e6	−1.02698
$552$	0	0
$553$	9.37405e6	1.30351
$554$	−7.01992e6	−0.971757
$555$	0	0
$556$	688026.	0.0943882
$557$	−1.88965e6	−0.258074	−0.129037	−	0.991640i	$-0.541189\pi$
$557$	−1.88965e6	−0.258074	−0.129037	+	0.991640i	$0.541189\pi$
$558$	0	0
$559$	−1.12929e7	−1.52854
$560$	1.27316e6	0.171559
$561$	0	0
$562$	−9.24450e6	−1.23465
$563$	−1.46556e7	−1.94864	−0.974322	−	0.225159i	$-0.927710\pi$
$563$	−1.46556e7	−1.94864	−0.974322	+	0.225159i	$0.927710\pi$
$564$	0	0
$565$	−5.07772e6	−0.669187
$566$	−7.64098e6	−1.00255
$567$	0	0
$568$	−2.95805e6	−0.384711
$569$	−3.30520e6	−0.427973	−0.213987	−	0.976837i	$-0.568645\pi$
$569$	−3.30520e6	−0.427973	−0.213987	+	0.976837i	$0.568645\pi$
$570$	0	0
$571$	9.00066e6	1.15527	0.577636	−	0.816295i	$-0.303975\pi$
$571$	9.00066e6	1.15527	0.577636	+	0.816295i	$0.303975\pi$
$572$	2.14122e6	0.273635
$573$	0	0
$574$	5.44239e6	0.689461
$575$	1.61054e6	0.203143
$576$	0	0
$577$	4.23516e6	0.529579	0.264789	−	0.964306i	$-0.414698\pi$
$577$	4.23516e6	0.529579	0.264789	+	0.964306i	$0.414698\pi$
$578$	2.28437e6	0.284411
$579$	0	0
$580$	1.89123e6	0.233439
$581$	6.87537e6	0.844999
$582$	0	0
$583$	1.06515e6	0.129790
$584$	−1.33152e7	−1.61553
$585$	0	0
$586$	−1.35351e6	−0.162823
$587$	1.09727e7	1.31438	0.657188	−	0.753727i	$-0.271746\pi$
$587$	1.09727e7	1.31438	0.657188	+	0.753727i	$0.271746\pi$
$588$	0	0
$589$	1.15162e7	1.36779
$590$	504563.	0.0596740
$591$	0	0
$592$	4.06981e6	0.477276
$593$	1.11353e7	1.30036	0.650182	−	0.759779i	$-0.274693\pi$
$593$	1.11353e7	1.30036	0.650182	+	0.759779i	$0.274693\pi$
$594$	0	0
$595$	−5.41238e6	−0.626752
$596$	5.85317e6	0.674956
$597$	0	0
$598$	−2.87345e6	−0.328587
$599$	−6.17540e6	−0.703231	−0.351615	−	0.936145i	$-0.614368\pi$
$599$	−6.17540e6	−0.703231	−0.351615	+	0.936145i	$0.614368\pi$
$600$	0	0
$601$	1.17003e7	1.32133	0.660666	−	0.750680i	$-0.270274\pi$
$601$	1.17003e7	1.32133	0.660666	+	0.750680i	$0.270274\pi$
$602$	6.30318e6	0.708873
$603$	0	0
$604$	−5.32879e6	−0.594341
$605$	−4.54251e6	−0.504554
$606$	0	0
$607$	7.60296e6	0.837551	0.418775	−	0.908090i	$-0.362459\pi$
$607$	7.60296e6	0.837551	0.418775	+	0.908090i	$0.362459\pi$
$608$	9.70779e6	1.06503
$609$	0	0
$610$	−5.82762e6	−0.634113
$611$	−6.25520e6	−0.677857
$612$	0	0
$613$	−5.69046e6	−0.611640	−0.305820	−	0.952089i	$-0.598931\pi$
$613$	−5.69046e6	−0.611640	−0.305820	+	0.952089i	$0.598931\pi$
$614$	−4.76569e6	−0.510158
$615$	0	0
$616$	−3.77604e6	−0.400945
$617$	−1.04223e7	−1.10218	−0.551089	−	0.834446i	$-0.685788\pi$
$617$	−1.04223e7	−1.10218	−0.551089	+	0.834446i	$0.685788\pi$
$618$	0	0
$619$	5.95972e6	0.625172	0.312586	−	0.949889i	$-0.398805\pi$
$619$	5.95972e6	0.625172	0.312586	+	0.949889i	$0.398805\pi$
$620$	−2.97586e6	−0.310909
$621$	0	0
$622$	7.55294e6	0.782781
$623$	−1.50032e7	−1.54869
$624$	0	0
$625$	−202820.	−0.0207688
$626$	−5.81659e6	−0.593243
$627$	0	0
$628$	8.93470e6	0.904026
$629$	−1.73013e7	−1.74362
$630$	0	0
$631$	−8.46233e6	−0.846090	−0.423045	−	0.906109i	$-0.639039\pi$
$631$	−8.46233e6	−0.846090	−0.423045	+	0.906109i	$0.639039\pi$
$632$	1.65007e7	1.64327
$633$	0	0
$634$	−8.38085e6	−0.828065
$635$	−755652.	−0.0743683
$636$	0	0
$637$	−3.80903e6	−0.371934
$638$	2.67008e6	0.259700
$639$	0	0
$640$	−976775.	−0.0942638
$641$	−1.51067e7	−1.45219	−0.726096	−	0.687593i	$-0.758667\pi$
$641$	−1.51067e7	−1.45219	−0.726096	+	0.687593i	$0.758667\pi$
$642$	0	0
$643$	−2.65460e6	−0.253205	−0.126602	−	0.991954i	$-0.540407\pi$
$643$	−2.65460e6	−0.253205	−0.126602	+	0.991954i	$0.540407\pi$
$644$	−1.38318e6	−0.131421
$645$	0	0
$646$	1.16692e7	1.10017
$647$	9.81722e6	0.921993	0.460997	−	0.887402i	$-0.347492\pi$
$647$	9.81722e6	0.921993	0.460997	+	0.887402i	$0.347492\pi$
$648$	0	0
$649$	−614353.	−0.0572540
$650$	−6.45578e6	−0.599329
$651$	0	0
$652$	8.46687e6	0.780017
$653$	1.23492e7	1.13333	0.566664	−	0.823949i	$-0.308234\pi$
$653$	1.23492e7	1.13333	0.566664	+	0.823949i	$0.308234\pi$
$654$	0	0
$655$	−119882.	−0.0109182
$656$	3.93292e6	0.356825
$657$	0	0
$658$	3.49137e6	0.314362
$659$	−1.03862e7	−0.931627	−0.465814	−	0.884883i	$-0.654238\pi$
$659$	−1.03862e7	−0.931627	−0.465814	+	0.884883i	$0.654238\pi$
$660$	0	0
$661$	5.33228e6	0.474689	0.237345	−	0.971426i	$-0.423723\pi$
$661$	5.33228e6	0.474689	0.237345	+	0.971426i	$0.423723\pi$
$662$	1.11073e7	0.985060
$663$	0	0
$664$	1.21024e7	1.06525
$665$	−7.73064e6	−0.677893
$666$	0	0
$667$	3.09021e6	0.268951
$668$	−7.17659e6	−0.622267
$669$	0	0
$670$	−7.85938e6	−0.676396
$671$	7.09567e6	0.608397
$672$	0	0
$673$	1.40382e7	1.19474	0.597371	−	0.801965i	$-0.296212\pi$
$673$	1.40382e7	1.19474	0.597371	+	0.801965i	$0.296212\pi$
$674$	1.27993e7	1.08527
$675$	0	0
$676$	−4.43166e6	−0.372992
$677$	−2.13392e7	−1.78940	−0.894698	−	0.446671i	$-0.852610\pi$
$677$	−2.13392e7	−1.78940	−0.894698	+	0.446671i	$0.852610\pi$
$678$	0	0
$679$	−8.55484e6	−0.712094
$680$	−9.52717e6	−0.790117
$681$	0	0
$682$	−4.20138e6	−0.345885
$683$	−6.51986e6	−0.534794	−0.267397	−	0.963586i	$-0.586164\pi$
$683$	−6.51986e6	−0.534794	−0.267397	+	0.963586i	$0.586164\pi$
$684$	0	0
$685$	−1.88583e6	−0.153559
$686$	9.80670e6	0.795633
$687$	0	0
$688$	4.55497e6	0.366872
$689$	−4.94144e6	−0.396557
$690$	0	0
$691$	8.62667e6	0.687302	0.343651	−	0.939097i	$-0.388336\pi$
$691$	8.62667e6	0.687302	0.343651	+	0.939097i	$0.388336\pi$
$692$	−1.17729e6	−0.0934583
$693$	0	0
$694$	−6.60681e6	−0.520707
$695$	−1.62363e6	−0.127505
$696$	0	0
$697$	−1.67193e7	−1.30358
$698$	1.66971e7	1.29719
$699$	0	0
$700$	−3.10760e6	−0.239707
$701$	−2.00997e7	−1.54488	−0.772440	−	0.635088i	$-0.780964\pi$
$701$	−2.00997e7	−1.54488	−0.772440	+	0.635088i	$0.780964\pi$
$702$	0	0
$703$	−2.47119e7	−1.88589
$704$	−5.40672e6	−0.411152
$705$	0	0
$706$	7.62401e6	0.575667
$707$	1.89771e7	1.42784
$708$	0	0
$709$	−5.65773e6	−0.422695	−0.211347	−	0.977411i	$-0.567785\pi$
$709$	−5.65773e6	−0.422695	−0.211347	+	0.977411i	$0.567785\pi$
$710$	2.20937e6	0.164483
$711$	0	0
$712$	−2.64095e7	−1.95236
$713$	−4.86246e6	−0.358206
$714$	0	0
$715$	−5.05295e6	−0.369641
$716$	−6.18137e6	−0.450611
$717$	0	0
$718$	−9.40487e6	−0.680834
$719$	−1.35569e7	−0.977996	−0.488998	−	0.872285i	$-0.662637\pi$
$719$	−1.35569e7	−0.977996	−0.488998	+	0.872285i	$0.662637\pi$
$720$	0	0
$721$	−6.40771e6	−0.459055
$722$	6.40368e6	0.457179
$723$	0	0
$724$	5.18478e6	0.367607
$725$	6.94277e6	0.490555
$726$	0	0
$727$	−1.97077e7	−1.38293	−0.691465	−	0.722410i	$-0.743034\pi$
$727$	−1.97077e7	−1.38293	−0.691465	+	0.722410i	$0.743034\pi$
$728$	1.75177e7	1.22504
$729$	0	0
$730$	9.94513e6	0.690722
$731$	−1.93637e7	−1.34028
$732$	0	0
$733$	−475415.	−0.0326823	−0.0163412	−	0.999866i	$-0.505202\pi$
$733$	−475415.	−0.0326823	−0.0163412	+	0.999866i	$0.505202\pi$
$734$	−1.92577e7	−1.31936
$735$	0	0
$736$	−4.09890e6	−0.278916
$737$	9.56953e6	0.648966
$738$	0	0
$739$	1.72886e7	1.16453	0.582264	−	0.813000i	$-0.302167\pi$
$739$	1.72886e7	1.16453	0.582264	+	0.813000i	$0.302167\pi$
$740$	6.38570e6	0.428676
$741$	0	0
$742$	2.75808e6	0.183907
$743$	−2.09012e7	−1.38899	−0.694494	−	0.719499i	$-0.744372\pi$
$743$	−2.09012e7	−1.38899	−0.694494	+	0.719499i	$0.744372\pi$
$744$	0	0
$745$	−1.38126e7	−0.911767
$746$	−1.73113e6	−0.113889
$747$	0	0
$748$	3.67152e6	0.239934
$749$	−1.57447e7	−1.02548
$750$	0	0
$751$	−1.10366e7	−0.714063	−0.357031	−	0.934092i	$-0.616211\pi$
$751$	−1.10366e7	−0.714063	−0.357031	+	0.934092i	$0.616211\pi$
$752$	2.52302e6	0.162696
$753$	0	0
$754$	−1.23870e7	−0.793481
$755$	1.25751e7	0.802868
$756$	0	0
$757$	−4.46889e6	−0.283439	−0.141720	−	0.989907i	$-0.545263\pi$
$757$	−4.46889e6	−0.283439	−0.141720	+	0.989907i	$0.545263\pi$
$758$	−569016.	−0.0359709
$759$	0	0
$760$	−1.36079e7	−0.854589
$761$	−1.50552e7	−0.942379	−0.471190	−	0.882032i	$-0.656175\pi$
$761$	−1.50552e7	−0.942379	−0.471190	+	0.882032i	$0.656175\pi$
$762$	0	0
$763$	−1.67624e6	−0.104237
$764$	−2.41483e6	−0.149676
$765$	0	0
$766$	1.43050e7	0.880881
$767$	2.85009e6	0.174933
$768$	0	0
$769$	−2.11014e7	−1.28675	−0.643377	−	0.765549i	$-0.722467\pi$
$769$	−2.11014e7	−1.28675	−0.643377	+	0.765549i	$0.722467\pi$
$770$	2.82032e6	0.171424
$771$	0	0
$772$	6.05135e6	0.365434
$773$	2.11957e7	1.27585	0.637924	−	0.770099i	$-0.279793\pi$
$773$	2.11957e7	1.27585	0.637924	+	0.770099i	$0.279793\pi$
$774$	0	0
$775$	−1.09245e7	−0.653352
$776$	−1.50587e7	−0.897704
$777$	0	0
$778$	1.91223e7	1.13264
$779$	−2.38806e7	−1.40995
$780$	0	0
$781$	−2.69011e6	−0.157813
$782$	−4.92705e6	−0.288118
$783$	0	0
$784$	1.53636e6	0.0892696
$785$	−2.10845e7	−1.22121
$786$	0	0
$787$	9.79595e6	0.563780	0.281890	−	0.959447i	$-0.409039\pi$
$787$	9.79595e6	0.563780	0.281890	+	0.959447i	$0.409039\pi$
$788$	−9.47117e6	−0.543360
$789$	0	0
$790$	−1.23244e7	−0.702582
$791$	1.60091e7	0.909756
$792$	0	0
$793$	−3.29181e7	−1.85888
$794$	8.79569e6	0.495130
$795$	0	0
$796$	2.03730e6	0.113965
$797$	8.45495e6	0.471482	0.235741	−	0.971816i	$-0.424248\pi$
$797$	8.45495e6	0.471482	0.235741	+	0.971816i	$0.424248\pi$
$798$	0	0
$799$	−1.07257e7	−0.594371
$800$	−9.20901e6	−0.508730
$801$	0	0
$802$	574078.	0.0315163
$803$	−1.21091e7	−0.662711
$804$	0	0
$805$	3.26410e6	0.177531
$806$	1.94910e7	1.05681
$807$	0	0
$808$	3.34045e7	1.80002
$809$	2.42480e7	1.30258	0.651291	−	0.758828i	$-0.274228\pi$
$809$	2.42480e7	1.30258	0.651291	+	0.758828i	$0.274228\pi$
$810$	0	0
$811$	4.71726e6	0.251848	0.125924	−	0.992040i	$-0.459811\pi$
$811$	4.71726e6	0.251848	0.125924	+	0.992040i	$0.459811\pi$
$812$	−5.96268e6	−0.317359
$813$	0	0
$814$	9.01548e6	0.476901
$815$	−1.99805e7	−1.05369
$816$	0	0
$817$	−2.76577e7	−1.44964
$818$	−1.57379e6	−0.0822361
$819$	0	0
$820$	6.17091e6	0.320490
$821$	2.51555e7	1.30249	0.651246	−	0.758867i	$-0.274247\pi$
$821$	2.51555e7	1.30249	0.651246	+	0.758867i	$0.274247\pi$
$822$	0	0
$823$	3.88001e6	0.199679	0.0998397	−	0.995004i	$-0.468167\pi$
$823$	3.88001e6	0.199679	0.0998397	+	0.995004i	$0.468167\pi$
$824$	−1.12792e7	−0.578710
$825$	0	0
$826$	−1.59079e6	−0.0811266
$827$	1.75611e7	0.892871	0.446435	−	0.894816i	$-0.352693\pi$
$827$	1.75611e7	0.892871	0.446435	+	0.894816i	$0.352693\pi$
$828$	0	0
$829$	−1.13811e7	−0.575173	−0.287587	−	0.957755i	$-0.592853\pi$
$829$	−1.13811e7	−0.575173	−0.287587	+	0.957755i	$0.592853\pi$
$830$	−9.03928e6	−0.455448
$831$	0	0
$832$	2.50827e7	1.25622
$833$	−6.53127e6	−0.326126
$834$	0	0
$835$	1.69356e7	0.840591
$836$	5.24412e6	0.259512
$837$	0	0
$838$	−1.21875e7	−0.599520
$839$	−1.12946e7	−0.553943	−0.276972	−	0.960878i	$-0.589331\pi$
$839$	−1.12946e7	−0.553943	−0.276972	+	0.960878i	$0.589331\pi$
$840$	0	0
$841$	−7.18977e6	−0.350530
$842$	−1.02085e7	−0.496227
$843$	0	0
$844$	2.16955e6	0.104837
$845$	1.04580e7	0.503857
$846$	0	0
$847$	1.43217e7	0.685938
$848$	1.99312e6	0.0951795
$849$	0	0
$850$	−1.10696e7	−0.525515
$851$	1.04341e7	0.493889
$852$	0	0
$853$	1.36639e7	0.642985	0.321493	−	0.946912i	$-0.395815\pi$
$853$	1.36639e7	0.642985	0.321493	+	0.946912i	$0.395815\pi$
$854$	1.83734e7	0.862073
$855$	0	0
$856$	−2.77146e7	−1.29278
$857$	2.02331e7	0.941044	0.470522	−	0.882388i	$-0.344066\pi$
$857$	2.02331e7	0.941044	0.470522	+	0.882388i	$0.344066\pi$
$858$	0	0
$859$	2.02393e7	0.935861	0.467931	−	0.883765i	$-0.345000\pi$
$859$	2.02393e7	0.935861	0.467931	+	0.883765i	$0.345000\pi$
$860$	7.14693e6	0.329514
$861$	0	0
$862$	−3.70604e6	−0.169880
$863$	2.16287e7	0.988560	0.494280	−	0.869303i	$-0.335432\pi$
$863$	2.16287e7	0.988560	0.494280	+	0.869303i	$0.335432\pi$
$864$	0	0
$865$	2.77822e6	0.126248
$866$	−9.40835e6	−0.426303
$867$	0	0
$868$	9.38231e6	0.422679
$869$	1.50061e7	0.674090
$870$	0	0
$871$	−4.43948e7	−1.98283
$872$	−2.95060e6	−0.131407
$873$	0	0
$874$	−7.03744e6	−0.311628
$875$	1.93810e7	0.855769
$876$	0	0
$877$	−1.99308e7	−0.875034	−0.437517	−	0.899210i	$-0.644142\pi$
$877$	−1.99308e7	−0.875034	−0.437517	+	0.899210i	$0.644142\pi$
$878$	−2.17049e7	−0.950216
$879$	0	0
$880$	2.03810e6	0.0887192
$881$	9.32845e6	0.404920	0.202460	−	0.979290i	$-0.435106\pi$
$881$	9.32845e6	0.404920	0.202460	+	0.979290i	$0.435106\pi$
$882$	0	0
$883$	−1.36534e7	−0.589302	−0.294651	−	0.955605i	$-0.595203\pi$
$883$	−1.36534e7	−0.589302	−0.294651	+	0.955605i	$0.595203\pi$
$884$	−1.70328e7	−0.733088
$885$	0	0
$886$	−363560.	−0.0155594
$887$	1.94974e7	0.832085	0.416042	−	0.909345i	$-0.363417\pi$
$887$	1.94974e7	0.832085	0.416042	+	0.909345i	$0.363417\pi$
$888$	0	0
$889$	2.38243e6	0.101103
$890$	1.97252e7	0.834733
$891$	0	0
$892$	−3.15201e6	−0.132640
$893$	−1.53198e7	−0.642870
$894$	0	0
$895$	1.45871e7	0.608709
$896$	3.07959e6	0.128151
$897$	0	0
$898$	9.12468e6	0.377596
$899$	−2.09613e7	−0.865004
$900$	0	0
$901$	−8.47299e6	−0.347716
$902$	8.71223e6	0.356544
$903$	0	0
$904$	2.81801e7	1.14689
$905$	−1.22353e7	−0.496583
$906$	0	0
$907$	−2.93504e7	−1.18467	−0.592333	−	0.805693i	$-0.701793\pi$
$907$	−2.93504e7	−1.18467	−0.592333	+	0.805693i	$0.701793\pi$
$908$	838750.	0.0337612
$909$	0	0
$910$	−1.30840e7	−0.523765
$911$	4.70892e7	1.87986	0.939929	−	0.341370i	$-0.110891\pi$
$911$	4.70892e7	1.87986	0.939929	+	0.341370i	$0.110891\pi$
$912$	0	0
$913$	1.10062e7	0.436978
$914$	2.69920e7	1.06873
$915$	0	0
$916$	1.32329e7	0.521093
$917$	377964.	0.0148432
$918$	0	0
$919$	658785.	0.0257309	0.0128654	−	0.999917i	$-0.495905\pi$
$919$	658785.	0.0257309	0.0128654	+	0.999917i	$0.495905\pi$
$920$	5.74564e6	0.223805
$921$	0	0
$922$	−1.50203e7	−0.581903
$923$	1.24799e7	0.482177
$924$	0	0
$925$	2.34422e7	0.900832
$926$	−4.87498e6	−0.186829
$927$	0	0
$928$	−1.76697e7	−0.673533
$929$	−4.12467e7	−1.56801	−0.784006	−	0.620753i	$-0.786827\pi$
$929$	−4.12467e7	−1.56801	−0.784006	+	0.620753i	$0.786827\pi$
$930$	0	0
$931$	−9.32878e6	−0.352737
$932$	−1.61205e7	−0.607911
$933$	0	0
$934$	9.81323e6	0.368082
$935$	−8.66420e6	−0.324115
$936$	0	0
$937$	−2.66656e7	−0.992208	−0.496104	−	0.868263i	$-0.665236\pi$
$937$	−2.66656e7	−0.992208	−0.496104	+	0.868263i	$0.665236\pi$
$938$	2.47791e7	0.919557
$939$	0	0
$940$	3.95872e6	0.146129
$941$	6.50842e6	0.239608	0.119804	−	0.992798i	$-0.461773\pi$
$941$	6.50842e6	0.239608	0.119804	+	0.992798i	$0.461773\pi$
$942$	0	0
$943$	1.00831e7	0.369245
$944$	−1.14958e6	−0.0419864
$945$	0	0
$946$	1.00902e7	0.366583
$947$	3.13849e7	1.13722	0.568612	−	0.822606i	$-0.307481\pi$
$947$	3.13849e7	1.13722	0.568612	+	0.822606i	$0.307481\pi$
$948$	0	0
$949$	5.61764e7	2.02483
$950$	−1.58110e7	−0.568395
$951$	0	0
$952$	3.00374e7	1.07416
$953$	−1.79573e7	−0.640485	−0.320242	−	0.947336i	$-0.603764\pi$
$953$	−1.79573e7	−0.640485	−0.320242	+	0.947336i	$0.603764\pi$
$954$	0	0
$955$	5.69862e6	0.202191
$956$	−4.90268e6	−0.173496
$957$	0	0
$958$	4.11927e7	1.45013
$959$	5.94565e6	0.208763
$960$	0	0
$961$	4.35351e6	0.152066
$962$	−4.18244e7	−1.45711
$963$	0	0
$964$	−787166.	−0.0272818
$965$	−1.42802e7	−0.493648
$966$	0	0
$967$	2.71548e7	0.933858	0.466929	−	0.884295i	$-0.345360\pi$
$967$	2.71548e7	0.933858	0.466929	+	0.884295i	$0.345360\pi$
$968$	2.52098e7	0.864730
$969$	0	0
$970$	1.12473e7	0.383813
$971$	5.34589e7	1.81958	0.909792	−	0.415065i	$-0.136241\pi$
$971$	5.34589e7	1.81958	0.909792	+	0.415065i	$0.136241\pi$
$972$	0	0
$973$	5.11900e6	0.173342
$974$	5.00031e6	0.168888
$975$	0	0
$976$	1.32774e7	0.446159
$977$	−5.95524e6	−0.199601	−0.0998006	−	0.995007i	$-0.531820\pi$
$977$	−5.95524e6	−0.199601	−0.0998006	+	0.995007i	$0.531820\pi$
$978$	0	0
$979$	−2.40173e7	−0.800881
$980$	2.41062e6	0.0801794
$981$	0	0
$982$	3.14520e7	1.04081
$983$	3.53361e7	1.16636	0.583182	−	0.812341i	$-0.301807\pi$
$983$	3.53361e7	1.16636	0.583182	+	0.812341i	$0.301807\pi$
$984$	0	0
$985$	2.23505e7	0.734000
$986$	−2.12397e7	−0.695755
$987$	0	0
$988$	−2.43284e7	−0.792906
$989$	1.16779e7	0.379641
$990$	0	0
$991$	7.02723e6	0.227300	0.113650	−	0.993521i	$-0.463746\pi$
$991$	7.02723e6	0.227300	0.113650	+	0.993521i	$0.463746\pi$
$992$	2.78034e7	0.897053
$993$	0	0
$994$	−6.96571e6	−0.223614
$995$	−4.80772e6	−0.153951
$996$	0	0
$997$	2.73040e6	0.0869939	0.0434970	−	0.999054i	$-0.486150\pi$
$997$	2.73040e6	0.0869939	0.0434970	+	0.999054i	$0.486150\pi$
$998$	7.98831e6	0.253880
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.d.1.9		12
3.2	odd	2		177.6.a.b.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.b.1.4	✓	12	3.2	odd	2
531.6.a.d.1.9		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)

\(f(q)\)	\(=\)	\(q+4.14510 q^{2} -14.8181 q^{4} +34.9684 q^{5} -110.249 q^{7} -194.066 q^{8} +O(q^{10})\)	\(q+4.14510 q^{2} -14.8181 q^{4} +34.9684 q^{5} -110.249 q^{7} -194.066 q^{8} +144.948 q^{10} -176.487 q^{11} +818.757 q^{13} -456.993 q^{14} -330.244 q^{16} +1403.91 q^{17} +2005.24 q^{19} -518.166 q^{20} -731.559 q^{22} -846.668 q^{23} -1902.21 q^{25} +3393.83 q^{26} +1633.68 q^{28} -3649.85 q^{29} +5743.05 q^{31} +4841.22 q^{32} +5819.34 q^{34} -3855.23 q^{35} -12323.7 q^{37} +8311.92 q^{38} -6786.18 q^{40} -11909.1 q^{41} -13792.7 q^{43} +2615.21 q^{44} -3509.53 q^{46} -7639.87 q^{47} -4652.21 q^{49} -7884.86 q^{50} -12132.4 q^{52} -6035.29 q^{53} -6171.49 q^{55} +21395.5 q^{56} -15129.0 q^{58} +3481.00 q^{59} -40205.0 q^{61} +23805.6 q^{62} +30635.1 q^{64} +28630.6 q^{65} -54222.1 q^{67} -20803.3 q^{68} -15980.3 q^{70} +15242.5 q^{71} +68611.8 q^{73} -51082.8 q^{74} -29713.8 q^{76} +19457.5 q^{77} -85026.3 q^{79} -11548.1 q^{80} -49364.6 q^{82} -62362.4 q^{83} +49092.4 q^{85} -57172.3 q^{86} +34250.2 q^{88} +136085. q^{89} -90267.0 q^{91} +12546.0 q^{92} -31668.1 q^{94} +70120.0 q^{95} +77595.8 q^{97} -19283.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) \) 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

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Groups

Database

Properties

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