LMFDB - Embedded newform 531.6.a.d.1.10

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			1.10
Root			$8.80731$ 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)

$f(q)$	$=$	$q+8.80731 q^{2} +45.5687 q^{4} +88.4760 q^{5} -134.279 q^{7} +119.504 q^{8} +O(q^{10})$	\(q+8.80731 q^{2} +45.5687 q^{4} +88.4760 q^{5} -134.279 q^{7} +119.504 q^{8} +779.236 q^{10} -423.269 q^{11} -713.936 q^{13} -1182.64 q^{14} -405.693 q^{16} -924.109 q^{17} -64.3204 q^{19} +4031.74 q^{20} -3727.86 q^{22} -3703.37 q^{23} +4703.01 q^{25} -6287.86 q^{26} -6118.92 q^{28} -4990.17 q^{29} +3097.37 q^{31} -7397.18 q^{32} -8138.92 q^{34} -11880.5 q^{35} -4978.34 q^{37} -566.489 q^{38} +10573.2 q^{40} -996.199 q^{41} +7715.17 q^{43} -19287.8 q^{44} -32616.7 q^{46} +17274.4 q^{47} +1223.84 q^{49} +41420.8 q^{50} -32533.1 q^{52} +34756.3 q^{53} -37449.1 q^{55} -16046.8 q^{56} -43950.0 q^{58} +3481.00 q^{59} +33316.0 q^{61} +27279.5 q^{62} -52167.1 q^{64} -63166.3 q^{65} +50498.8 q^{67} -42110.4 q^{68} -104635. q^{70} +16903.5 q^{71} -84572.2 q^{73} -43845.8 q^{74} -2930.99 q^{76} +56836.1 q^{77} +9984.38 q^{79} -35894.1 q^{80} -8773.83 q^{82} -11880.4 q^{83} -81761.5 q^{85} +67949.9 q^{86} -50582.1 q^{88} +8140.62 q^{89} +95866.6 q^{91} -168758. q^{92} +152141. q^{94} -5690.81 q^{95} -137348. q^{97} +10778.7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	8.80731	1.55693	0.778463	−	0.627690i	$-0.215999\pi$
$2$	8.80731	1.55693	0.778463	+	0.627690i	$0.215999\pi$
$3$	0	0
$3$	0	0
$4$	45.5687	1.42402
$5$	88.4760	1.58271	0.791354	−	0.611359i	$-0.209377\pi$
$5$	88.4760	1.58271	0.791354	+	0.611359i	$0.209377\pi$
$6$	0	0
$7$	−134.279	−1.03577	−0.517884	−	0.855451i	$-0.673280\pi$
$7$	−134.279	−1.03577	−0.517884	+	0.855451i	$0.673280\pi$
$8$	119.504	0.660170
$9$	0	0
$10$	779.236	2.46416
$11$	−423.269	−1.05471	−0.527357	−	0.849644i	$-0.676817\pi$
$11$	−423.269	−1.05471	−0.527357	+	0.849644i	$0.676817\pi$
$12$	0	0
$13$	−713.936	−1.17166	−0.585829	−	0.810434i	$-0.699231\pi$
$13$	−713.936	−1.17166	−0.585829	+	0.810434i	$0.699231\pi$
$14$	−1182.64	−1.61262
$15$	0	0
$16$	−405.693	−0.396184
$17$	−924.109	−0.775534	−0.387767	−	0.921757i	$-0.626754\pi$
$17$	−924.109	−0.775534	−0.387767	+	0.921757i	$0.626754\pi$
$18$	0	0
$19$	−64.3204	−0.0408756	−0.0204378	−	0.999791i	$-0.506506\pi$
$19$	−64.3204	−0.0408756	−0.0204378	+	0.999791i	$0.506506\pi$
$20$	4031.74	2.25381
$21$	0	0
$22$	−3727.86	−1.64211
$23$	−3703.37	−1.45975	−0.729873	−	0.683583i	$-0.760421\pi$
$23$	−3703.37	−1.45975	−0.729873	+	0.683583i	$0.760421\pi$
$24$	0	0
$25$	4703.01	1.50496
$26$	−6287.86	−1.82419
$27$	0	0
$28$	−6118.92	−1.47496
$29$	−4990.17	−1.10185	−0.550923	−	0.834556i	$-0.685724\pi$
$29$	−4990.17	−1.10185	−0.550923	+	0.834556i	$0.685724\pi$
$30$	0	0
$31$	3097.37	0.578880	0.289440	−	0.957196i	$-0.406531\pi$
$31$	3097.37	0.578880	0.289440	+	0.957196i	$0.406531\pi$
$32$	−7397.18	−1.27700
$33$	0	0
$34$	−8138.92	−1.20745
$35$	−11880.5	−1.63932
$36$	0	0
$37$	−4978.34	−0.597834	−0.298917	−	0.954279i	$-0.596625\pi$
$37$	−4978.34	−0.597834	−0.298917	+	0.954279i	$0.596625\pi$
$38$	−566.489	−0.0636404
$39$	0	0
$40$	10573.2	1.04486
$41$	−996.199	−0.0925522	−0.0462761	−	0.998929i	$-0.514735\pi$
$41$	−996.199	−0.0925522	−0.0462761	+	0.998929i	$0.514735\pi$
$42$	0	0
$43$	7715.17	0.636318	0.318159	−	0.948037i	$-0.396935\pi$
$43$	7715.17	0.636318	0.318159	+	0.948037i	$0.396935\pi$
$44$	−19287.8	−1.50193
$45$	0	0
$46$	−32616.7	−2.27272
$47$	17274.4	1.14067	0.570333	−	0.821414i	$-0.306814\pi$
$47$	17274.4	1.14067	0.570333	+	0.821414i	$0.306814\pi$
$48$	0	0
$49$	1223.84	0.0728171
$50$	41420.8	2.34312
$51$	0	0
$52$	−32533.1	−1.66847
$53$	34756.3	1.69959	0.849796	−	0.527112i	$-0.176725\pi$
$53$	34756.3	1.69959	0.849796	+	0.527112i	$0.176725\pi$
$54$	0	0
$55$	−37449.1	−1.66930
$56$	−16046.8	−0.683784
$57$	0	0
$58$	−43950.0	−1.71549
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	33316.0	1.14638	0.573190	−	0.819422i	$-0.305706\pi$
$61$	33316.0	1.14638	0.573190	+	0.819422i	$0.305706\pi$
$62$	27279.5	0.901274
$63$	0	0
$64$	−52167.1	−1.59201
$65$	−63166.3	−1.85439
$66$	0	0
$67$	50498.8	1.37434	0.687169	−	0.726497i	$-0.258853\pi$
$67$	50498.8	1.37434	0.687169	+	0.726497i	$0.258853\pi$
$68$	−42110.4	−1.10438
$69$	0	0
$70$	−104635.	−2.55230
$71$	16903.5	0.397951	0.198975	−	0.980004i	$-0.436239\pi$
$71$	16903.5	0.397951	0.198975	+	0.980004i	$0.436239\pi$
$72$	0	0
$73$	−84572.2	−1.85746	−0.928732	−	0.370752i	$-0.879100\pi$
$73$	−84572.2	−1.85746	−0.928732	+	0.370752i	$0.879100\pi$
$74$	−43845.8	−0.930783
$75$	0	0
$76$	−2930.99	−0.0582078
$77$	56836.1	1.09244
$78$	0	0
$79$	9984.38	0.179992	0.0899960	−	0.995942i	$-0.471315\pi$
$79$	9984.38	0.179992	0.0899960	+	0.995942i	$0.471315\pi$
$80$	−35894.1	−0.627044
$81$	0	0
$82$	−8773.83	−0.144097
$83$	−11880.4	−0.189293	−0.0946467	−	0.995511i	$-0.530172\pi$
$83$	−11880.4	−0.189293	−0.0946467	+	0.995511i	$0.530172\pi$
$84$	0	0
$85$	−81761.5	−1.22744
$86$	67949.9	0.990701
$87$	0	0
$88$	−50582.1	−0.696290
$89$	8140.62	0.108939	0.0544694	−	0.998515i	$-0.482653\pi$
$89$	8140.62	0.108939	0.0544694	+	0.998515i	$0.482653\pi$
$90$	0	0
$91$	95866.6	1.21357
$92$	−168758.	−2.07871
$93$	0	0
$94$	152141.	1.77593
$95$	−5690.81	−0.0646942
$96$	0	0
$97$	−137348.	−1.48215	−0.741077	−	0.671420i	$-0.765685\pi$
$97$	−137348.	−1.48215	−0.741077	+	0.671420i	$0.765685\pi$
$98$	10778.7	0.113371
$99$	0	0
$100$	214310.	2.14310
$101$	32965.3	0.321553	0.160777	−	0.986991i	$-0.448600\pi$
$101$	32965.3	0.321553	0.160777	+	0.986991i	$0.448600\pi$
$102$	0	0
$103$	168560.	1.56553	0.782764	−	0.622319i	$-0.213809\pi$
$103$	168560.	1.56553	0.782764	+	0.622319i	$0.213809\pi$
$104$	−85318.0	−0.773494
$105$	0	0
$106$	306110.	2.64614
$107$	36729.3	0.310137	0.155069	−	0.987904i	$-0.450440\pi$
$107$	36729.3	0.310137	0.155069	+	0.987904i	$0.450440\pi$
$108$	0	0
$109$	171480.	1.38244	0.691222	−	0.722642i	$-0.257073\pi$
$109$	171480.	1.38244	0.691222	+	0.722642i	$0.257073\pi$
$110$	−329826.	−2.59898
$111$	0	0
$112$	54476.0	0.410356
$113$	−10840.2	−0.0798624	−0.0399312	−	0.999202i	$-0.512714\pi$
$113$	−10840.2	−0.0798624	−0.0399312	+	0.999202i	$0.512714\pi$
$114$	0	0
$115$	−327659.	−2.31035
$116$	−227396.	−1.56905
$117$	0	0
$118$	30658.2	0.202695
$119$	124088.	0.803274
$120$	0	0
$121$	18105.3	0.112420
$122$	293425.	1.78483
$123$	0	0
$124$	141143.	0.824337
$125$	139616.	0.799208
$126$	0	0
$127$	−266071.	−1.46382	−0.731911	−	0.681400i	$-0.761371\pi$
$127$	−266071.	−1.46382	−0.731911	+	0.681400i	$0.761371\pi$
$128$	−222742.	−1.20165
$129$	0	0
$130$	−556325.	−2.88715
$131$	−344872.	−1.75582	−0.877910	−	0.478826i	$-0.841062\pi$
$131$	−344872.	−1.75582	−0.877910	+	0.478826i	$0.841062\pi$
$132$	0	0
$133$	8636.87	0.0423377
$134$	444758.	2.13975
$135$	0	0
$136$	−110434.	−0.511985
$137$	−299475.	−1.36320	−0.681600	−	0.731725i	$-0.738716\pi$
$137$	−299475.	−1.36320	−0.681600	+	0.731725i	$0.738716\pi$
$138$	0	0
$139$	−279442.	−1.22675	−0.613373	−	0.789793i	$-0.710188\pi$
$139$	−279442.	−1.22675	−0.613373	+	0.789793i	$0.710188\pi$
$140$	−541377.	−2.33443
$141$	0	0
$142$	148874.	0.619581
$143$	302187.	1.23576
$144$	0	0
$145$	−441511.	−1.74390
$146$	−744853.	−2.89193
$147$	0	0
$148$	−226856.	−0.851328
$149$	−522957.	−1.92974	−0.964872	−	0.262719i	$-0.915381\pi$
$149$	−522957.	−1.92974	−0.964872	+	0.262719i	$0.915381\pi$
$150$	0	0
$151$	−225544.	−0.804988	−0.402494	−	0.915423i	$-0.631857\pi$
$151$	−225544.	−0.804988	−0.402494	+	0.915423i	$0.631857\pi$
$152$	−7686.52	−0.0269849
$153$	0	0
$154$	500573.	1.70085
$155$	274043.	0.916197
$156$	0	0
$157$	169574.	0.549047	0.274524	−	0.961580i	$-0.411480\pi$
$157$	169574.	0.549047	0.274524	+	0.961580i	$0.411480\pi$
$158$	87935.5	0.280234
$159$	0	0
$160$	−654473.	−2.02112
$161$	497284.	1.51196
$162$	0	0
$163$	247010.	0.728192	0.364096	−	0.931361i	$-0.381378\pi$
$163$	247010.	0.728192	0.364096	+	0.931361i	$0.381378\pi$
$164$	−45395.5	−0.131796
$165$	0	0
$166$	−104634.	−0.294716
$167$	37137.6	0.103044	0.0515220	−	0.998672i	$-0.483593\pi$
$167$	37137.6	0.103044	0.0515220	+	0.998672i	$0.483593\pi$
$168$	0	0
$169$	138412.	0.372785
$170$	−720099.	−1.91104
$171$	0	0
$172$	351570.	0.906131
$173$	49966.1	0.126929	0.0634644	−	0.997984i	$-0.479785\pi$
$173$	49966.1	0.126929	0.0634644	+	0.997984i	$0.479785\pi$
$174$	0	0
$175$	−631515.	−1.55879
$176$	171717.	0.417861
$177$	0	0
$178$	71697.0	0.169610
$179$	53182.0	0.124060	0.0620300	−	0.998074i	$-0.480243\pi$
$179$	53182.0	0.124060	0.0620300	+	0.998074i	$0.480243\pi$
$180$	0	0
$181$	−228956.	−0.519463	−0.259732	−	0.965681i	$-0.583634\pi$
$181$	−228956.	−0.519463	−0.259732	+	0.965681i	$0.583634\pi$
$182$	844327.	1.88944
$183$	0	0
$184$	−442566.	−0.963681
$185$	−440464.	−0.946195
$186$	0	0
$187$	391146.	0.817966
$188$	787171.	1.62433
$189$	0	0
$190$	−50120.7	−0.100724
$191$	−383018.	−0.759689	−0.379845	−	0.925050i	$-0.624023\pi$
$191$	−383018.	−0.759689	−0.379845	+	0.925050i	$0.624023\pi$
$192$	0	0
$193$	−671956.	−1.29852	−0.649258	−	0.760568i	$-0.724921\pi$
$193$	−671956.	−1.29852	−0.649258	+	0.760568i	$0.724921\pi$
$194$	−1.20967e6	−2.30761
$195$	0	0
$196$	55768.6	0.103693
$197$	517980.	0.950927	0.475464	−	0.879735i	$-0.342280\pi$
$197$	517980.	0.950927	0.475464	+	0.879735i	$0.342280\pi$
$198$	0	0
$199$	437764.	0.783624	0.391812	−	0.920045i	$-0.371848\pi$
$199$	437764.	0.783624	0.391812	+	0.920045i	$0.371848\pi$
$200$	562026.	0.993531
$201$	0	0
$202$	290335.	0.500635
$203$	670075.	1.14126
$204$	0	0
$205$	−88139.8	−0.146483
$206$	1.48456e6	2.43741
$207$	0	0
$208$	289639.	0.464193
$209$	27224.8	0.0431121
$210$	0	0
$211$	795724.	1.23043	0.615214	−	0.788360i	$-0.289070\pi$
$211$	795724.	1.23043	0.615214	+	0.788360i	$0.289070\pi$
$212$	1.58380e6	2.42025
$213$	0	0
$214$	323487.	0.482861
$215$	682608.	1.00711
$216$	0	0
$217$	−415911.	−0.599586
$218$	1.51028e6	2.15237
$219$	0	0
$220$	−1.70651e6	−2.37712
$221$	659755.	0.908662
$222$	0	0
$223$	−1.44232e6	−1.94222	−0.971110	−	0.238632i	$-0.923301\pi$
$223$	−1.44232e6	−1.94222	−0.971110	+	0.238632i	$0.923301\pi$
$224$	993285.	1.32268
$225$	0	0
$226$	−95473.3	−0.124340
$227$	421642.	0.543099	0.271549	−	0.962425i	$-0.412464\pi$
$227$	421642.	0.543099	0.271549	+	0.962425i	$0.412464\pi$
$228$	0	0
$229$	−1.41705e6	−1.78565	−0.892823	−	0.450407i	$-0.851279\pi$
$229$	−1.41705e6	−1.78565	−0.892823	+	0.450407i	$0.851279\pi$
$230$	−2.88580e6	−3.59705
$231$	0	0
$232$	−596344.	−0.727406
$233$	−673154.	−0.812315	−0.406158	−	0.913803i	$-0.633132\pi$
$233$	−673154.	−0.812315	−0.406158	+	0.913803i	$0.633132\pi$
$234$	0	0
$235$	1.52837e6	1.80534
$236$	158625.	0.185392
$237$	0	0
$238$	1.09289e6	1.25064
$239$	760337.	0.861016	0.430508	−	0.902587i	$-0.358334\pi$
$239$	760337.	0.861016	0.430508	+	0.902587i	$0.358334\pi$
$240$	0	0
$241$	470539.	0.521859	0.260929	−	0.965358i	$-0.415971\pi$
$241$	470539.	0.521859	0.260929	+	0.965358i	$0.415971\pi$
$242$	159459.	0.175029
$243$	0	0
$244$	1.51817e6	1.63247
$245$	108280.	0.115248
$246$	0	0
$247$	45920.7	0.0478923
$248$	370147.	0.382159
$249$	0	0
$250$	1.22964e6	1.24431
$251$	−962815.	−0.964625	−0.482313	−	0.875999i	$-0.660203\pi$
$251$	−962815.	−0.964625	−0.482313	+	0.875999i	$0.660203\pi$
$252$	0	0
$253$	1.56752e6	1.53961
$254$	−2.34337e6	−2.27906
$255$	0	0
$256$	−292409.	−0.278863
$257$	931917.	0.880125	0.440062	−	0.897967i	$-0.354956\pi$
$257$	931917.	0.880125	0.440062	+	0.897967i	$0.354956\pi$
$258$	0	0
$259$	668486.	0.619217
$260$	−2.87840e6	−2.64070
$261$	0	0
$262$	−3.03740e6	−2.73368
$263$	1.26124e6	1.12437	0.562184	−	0.827012i	$-0.309961\pi$
$263$	1.26124e6	1.12437	0.562184	+	0.827012i	$0.309961\pi$
$264$	0	0
$265$	3.07510e6	2.68996
$266$	76067.6	0.0659167
$267$	0	0
$268$	2.30116e6	1.95709
$269$	688569.	0.580185	0.290093	−	0.956999i	$-0.406314\pi$
$269$	688569.	0.580185	0.290093	+	0.956999i	$0.406314\pi$
$270$	0	0
$271$	−733019.	−0.606306	−0.303153	−	0.952942i	$-0.598039\pi$
$271$	−733019.	−0.606306	−0.303153	+	0.952942i	$0.598039\pi$
$272$	374905.	0.307255
$273$	0	0
$274$	−2.63757e6	−2.12240
$275$	−1.99064e6	−1.58730
$276$	0	0
$277$	−176065.	−0.137871	−0.0689356	−	0.997621i	$-0.521960\pi$
$277$	−176065.	−0.137871	−0.0689356	+	0.997621i	$0.521960\pi$
$278$	−2.46113e6	−1.90995
$279$	0	0
$280$	−1.41976e6	−1.08223
$281$	1.04068e6	0.786232	0.393116	−	0.919489i	$-0.371397\pi$
$281$	1.04068e6	0.786232	0.393116	+	0.919489i	$0.371397\pi$
$282$	0	0
$283$	715219.	0.530851	0.265426	−	0.964131i	$-0.414488\pi$
$283$	715219.	0.530851	0.265426	+	0.964131i	$0.414488\pi$
$284$	770268.	0.566691
$285$	0	0
$286$	2.66145e6	1.92399
$287$	133769.	0.0958626
$288$	0	0
$289$	−565879.	−0.398546
$290$	−3.88852e6	−2.71512
$291$	0	0
$292$	−3.85384e6	−2.64507
$293$	−2.03867e6	−1.38732	−0.693662	−	0.720301i	$-0.744004\pi$
$293$	−2.03867e6	−1.38732	−0.693662	+	0.720301i	$0.744004\pi$
$294$	0	0
$295$	307985.	0.206051
$296$	−594930.	−0.394672
$297$	0	0
$298$	−4.60584e6	−3.00447
$299$	2.64397e6	1.71032
$300$	0	0
$301$	−1.03599e6	−0.659079
$302$	−1.98644e6	−1.25331
$303$	0	0
$304$	26094.3	0.0161943
$305$	2.94767e6	1.81439
$306$	0	0
$307$	−1.35132e6	−0.818301	−0.409151	−	0.912467i	$-0.634175\pi$
$307$	−1.35132e6	−0.818301	−0.409151	+	0.912467i	$0.634175\pi$
$308$	2.58994e6	1.55566
$309$	0	0
$310$	2.41358e6	1.42645
$311$	1.87621e6	1.09997	0.549985	−	0.835175i	$-0.314633\pi$
$311$	1.87621e6	1.09997	0.549985	+	0.835175i	$0.314633\pi$
$312$	0	0
$313$	2.45653e6	1.41730	0.708649	−	0.705561i	$-0.249305\pi$
$313$	2.45653e6	1.41730	0.708649	+	0.705561i	$0.249305\pi$
$314$	1.49349e6	0.854826
$315$	0	0
$316$	454975.	0.256312
$317$	1.75984e6	0.983615	0.491808	−	0.870704i	$-0.336336\pi$
$317$	1.75984e6	0.983615	0.491808	+	0.870704i	$0.336336\pi$
$318$	0	0
$319$	2.11218e6	1.16213
$320$	−4.61553e6	−2.51969
$321$	0	0
$322$	4.37974e6	2.35401
$323$	59439.1	0.0317005
$324$	0	0
$325$	−3.35765e6	−1.76330
$326$	2.17549e6	1.13374
$327$	0	0
$328$	−119049.	−0.0611002
$329$	−2.31959e6	−1.18147
$330$	0	0
$331$	−2.62035e6	−1.31459	−0.657293	−	0.753635i	$-0.728299\pi$
$331$	−2.62035e6	−1.31459	−0.657293	+	0.753635i	$0.728299\pi$
$332$	−541374.	−0.269558
$333$	0	0
$334$	327082.	0.160432
$335$	4.46793e6	2.17518
$336$	0	0
$337$	2.56955e6	1.23249	0.616244	−	0.787555i	$-0.288653\pi$
$337$	2.56955e6	1.23249	0.616244	+	0.787555i	$0.288653\pi$
$338$	1.21904e6	0.580398
$339$	0	0
$340$	−3.72577e6	−1.74791
$341$	−1.31102e6	−0.610552
$342$	0	0
$343$	2.09249e6	0.960347
$344$	921991.	0.420079
$345$	0	0
$346$	440067.	0.197619
$347$	−1.69640e6	−0.756318	−0.378159	−	0.925741i	$-0.623443\pi$
$347$	−1.69640e6	−0.756318	−0.378159	+	0.925741i	$0.623443\pi$
$348$	0	0
$349$	−2.36923e6	−1.04122	−0.520612	−	0.853794i	$-0.674296\pi$
$349$	−2.36923e6	−1.04122	−0.520612	+	0.853794i	$0.674296\pi$
$350$	−5.56195e6	−2.42693
$351$	0	0
$352$	3.13099e6	1.34687
$353$	−4.33080e6	−1.84983	−0.924914	−	0.380175i	$-0.875864\pi$
$353$	−4.33080e6	−1.84983	−0.924914	+	0.380175i	$0.875864\pi$
$354$	0	0
$355$	1.49555e6	0.629840
$356$	370957.	0.155131
$357$	0	0
$358$	468390.	0.193152
$359$	1.01517e6	0.415723	0.207862	−	0.978158i	$-0.433350\pi$
$359$	1.01517e6	0.415723	0.207862	+	0.978158i	$0.433350\pi$
$360$	0	0
$361$	−2.47196e6	−0.998329
$362$	−2.01648e6	−0.808766
$363$	0	0
$364$	4.36852e6	1.72815
$365$	−7.48261e6	−2.93982
$366$	0	0
$367$	−2.21426e6	−0.858151	−0.429076	−	0.903269i	$-0.641161\pi$
$367$	−2.21426e6	−0.858151	−0.429076	+	0.903269i	$0.641161\pi$
$368$	1.50243e6	0.578328
$369$	0	0
$370$	−3.87930e6	−1.47316
$371$	−4.66705e6	−1.76038
$372$	0	0
$373$	−2.01840e6	−0.751166	−0.375583	−	0.926789i	$-0.622557\pi$
$373$	−2.01840e6	−0.751166	−0.375583	+	0.926789i	$0.622557\pi$
$374$	3.44495e6	1.27351
$375$	0	0
$376$	2.06435e6	0.753033
$377$	3.56267e6	1.29099
$378$	0	0
$379$	4.76169e6	1.70280	0.851398	−	0.524520i	$-0.175755\pi$
$379$	4.76169e6	1.70280	0.851398	+	0.524520i	$0.175755\pi$
$380$	−259323.	−0.0921259
$381$	0	0
$382$	−3.37336e6	−1.18278
$383$	−1.00203e6	−0.349046	−0.174523	−	0.984653i	$-0.555838\pi$
$383$	−1.00203e6	−0.349046	−0.174523	+	0.984653i	$0.555838\pi$
$384$	0	0
$385$	5.02863e6	1.72901
$386$	−5.91812e6	−2.02170
$387$	0	0
$388$	−6.25877e6	−2.11062
$389$	5.08858e6	1.70499	0.852496	−	0.522734i	$-0.175088\pi$
$389$	5.08858e6	1.70499	0.852496	+	0.522734i	$0.175088\pi$
$390$	0	0
$391$	3.42232e6	1.13208
$392$	146253.	0.0480717
$393$	0	0
$394$	4.56201e6	1.48052
$395$	883378.	0.284875
$396$	0	0
$397$	−4.16172e6	−1.32525	−0.662624	−	0.748953i	$-0.730557\pi$
$397$	−4.16172e6	−1.32525	−0.662624	+	0.748953i	$0.730557\pi$
$398$	3.85552e6	1.22004
$399$	0	0
$400$	−1.90798e6	−0.596243
$401$	−6.12721e6	−1.90284	−0.951420	−	0.307897i	$-0.900375\pi$
$401$	−6.12721e6	−1.90284	−0.951420	+	0.307897i	$0.900375\pi$
$402$	0	0
$403$	−2.21132e6	−0.678250
$404$	1.50218e6	0.457899
$405$	0	0
$406$	5.90156e6	1.77685
$407$	2.10718e6	0.630543
$408$	0	0
$409$	−1.39272e6	−0.411675	−0.205837	−	0.978586i	$-0.565992\pi$
$409$	−1.39272e6	−0.411675	−0.205837	+	0.978586i	$0.565992\pi$
$410$	−776274.	−0.228063
$411$	0	0
$412$	7.68104e6	2.22934
$413$	−467425.	−0.134846
$414$	0	0
$415$	−1.05113e6	−0.299596
$416$	5.28111e6	1.49621
$417$	0	0
$418$	239777.	0.0671223
$419$	4.89970e6	1.36343	0.681717	−	0.731616i	$-0.261233\pi$
$419$	4.89970e6	1.36343	0.681717	+	0.731616i	$0.261233\pi$
$420$	0	0
$421$	−1.47517e6	−0.405637	−0.202818	−	0.979216i	$-0.565010\pi$
$421$	−1.47517e6	−0.405637	−0.202818	+	0.979216i	$0.565010\pi$
$422$	7.00819e6	1.91569
$423$	0	0
$424$	4.15351e6	1.12202
$425$	−4.34609e6	−1.16715
$426$	0	0
$427$	−4.47364e6	−1.18739
$428$	1.67371e6	0.441642
$429$	0	0
$430$	6.01194e6	1.56799
$431$	6.48538e6	1.68168	0.840838	−	0.541286i	$-0.182063\pi$
$431$	6.48538e6	1.68168	0.840838	+	0.541286i	$0.182063\pi$
$432$	0	0
$433$	5.86229e6	1.50262	0.751308	−	0.659952i	$-0.229423\pi$
$433$	5.86229e6	1.50262	0.751308	+	0.659952i	$0.229423\pi$
$434$	−3.66306e6	−0.933511
$435$	0	0
$436$	7.81413e6	1.96863
$437$	238202.	0.0596680
$438$	0	0
$439$	−4.42815e6	−1.09663	−0.548316	−	0.836271i	$-0.684731\pi$
$439$	−4.42815e6	−1.09663	−0.548316	+	0.836271i	$0.684731\pi$
$440$	−4.47530e6	−1.10202
$441$	0	0
$442$	5.81067e6	1.41472
$443$	7.50913e6	1.81794	0.908971	−	0.416859i	$-0.136869\pi$
$443$	7.50913e6	1.81794	0.908971	+	0.416859i	$0.136869\pi$
$444$	0	0
$445$	720250.	0.172418
$446$	−1.27029e7	−3.02389
$447$	0	0
$448$	7.00494e6	1.64896
$449$	−1.95240e6	−0.457040	−0.228520	−	0.973539i	$-0.573389\pi$
$449$	−1.95240e6	−0.457040	−0.228520	+	0.973539i	$0.573389\pi$
$450$	0	0
$451$	421660.	0.0976160
$452$	−493975.	−0.113726
$453$	0	0
$454$	3.71353e6	0.845565
$455$	8.48190e6	1.92072
$456$	0	0
$457$	−5.94530e6	−1.33163	−0.665814	−	0.746118i	$-0.731916\pi$
$457$	−5.94530e6	−1.33163	−0.665814	+	0.746118i	$0.731916\pi$
$458$	−1.24804e7	−2.78012
$459$	0	0
$460$	−1.49310e7	−3.28999
$461$	−6.52618e6	−1.43023	−0.715116	−	0.699006i	$-0.753626\pi$
$461$	−6.52618e6	−1.43023	−0.715116	+	0.699006i	$0.753626\pi$
$462$	0	0
$463$	−1.21024e6	−0.262373	−0.131187	−	0.991358i	$-0.541879\pi$
$463$	−1.21024e6	−0.262373	−0.131187	+	0.991358i	$0.541879\pi$
$464$	2.02448e6	0.436534
$465$	0	0
$466$	−5.92868e6	−1.26472
$467$	3.93565e6	0.835073	0.417536	−	0.908660i	$-0.362894\pi$
$467$	3.93565e6	0.835073	0.417536	+	0.908660i	$0.362894\pi$
$468$	0	0
$469$	−6.78092e6	−1.42350
$470$	1.34608e7	2.81078
$471$	0	0
$472$	415992.	0.0859469
$473$	−3.26559e6	−0.671133
$474$	0	0
$475$	−302499.	−0.0615163
$476$	5.65455e6	1.14388
$477$	0	0
$478$	6.69652e6	1.34054
$479$	242633.	0.0483183	0.0241592	−	0.999708i	$-0.492309\pi$
$479$	242633.	0.0483183	0.0241592	+	0.999708i	$0.492309\pi$
$480$	0	0
$481$	3.55422e6	0.700457
$482$	4.14418e6	0.812496
$483$	0	0
$484$	825035.	0.160088
$485$	−1.21520e7	−2.34582
$486$	0	0
$487$	−2.57797e6	−0.492556	−0.246278	−	0.969199i	$-0.579208\pi$
$487$	−2.57797e6	−0.492556	−0.246278	+	0.969199i	$0.579208\pi$
$488$	3.98139e6	0.756807
$489$	0	0
$490$	953657.	0.179433
$491$	287405.	0.0538010	0.0269005	−	0.999638i	$-0.491436\pi$
$491$	287405.	0.0538010	0.0269005	+	0.999638i	$0.491436\pi$
$492$	0	0
$493$	4.61147e6	0.854519
$494$	404437.	0.0745648
$495$	0	0
$496$	−1.25658e6	−0.229343
$497$	−2.26978e6	−0.412185
$498$	0	0
$499$	4.52048e6	0.812706	0.406353	−	0.913716i	$-0.366800\pi$
$499$	4.52048e6	0.812706	0.406353	+	0.913716i	$0.366800\pi$
$500$	6.36211e6	1.13809
$501$	0	0
$502$	−8.47981e6	−1.50185
$503$	2.75026e6	0.484679	0.242339	−	0.970192i	$-0.422085\pi$
$503$	2.75026e6	0.484679	0.242339	+	0.970192i	$0.422085\pi$
$504$	0	0
$505$	2.91664e6	0.508925
$506$	1.38056e7	2.39706
$507$	0	0
$508$	−1.21245e7	−2.08451
$509$	−2.32843e6	−0.398354	−0.199177	−	0.979964i	$-0.563827\pi$
$509$	−2.32843e6	−0.398354	−0.199177	+	0.979964i	$0.563827\pi$
$510$	0	0
$511$	1.13563e7	1.92390
$512$	4.55240e6	0.767477
$513$	0	0
$514$	8.20768e6	1.37029
$515$	1.49135e7	2.47777
$516$	0	0
$517$	−7.31171e6	−1.20307
$518$	5.88757e6	0.964076
$519$	0	0
$520$	−7.54860e6	−1.22422
$521$	−1.70375e6	−0.274986	−0.137493	−	0.990503i	$-0.543904\pi$
$521$	−1.70375e6	−0.274986	−0.137493	+	0.990503i	$0.543904\pi$
$522$	0	0
$523$	−4.39690e6	−0.702898	−0.351449	−	0.936207i	$-0.614311\pi$
$523$	−4.39690e6	−0.702898	−0.351449	+	0.936207i	$0.614311\pi$
$524$	−1.57154e7	−2.50032
$525$	0	0
$526$	1.11081e7	1.75056
$527$	−2.86231e6	−0.448941
$528$	0	0
$529$	7.27858e6	1.13086
$530$	2.70834e7	4.18806
$531$	0	0
$532$	393571.	0.0602898
$533$	711223.	0.108440
$534$	0	0
$535$	3.24967e6	0.490856
$536$	6.03478e6	0.907298
$537$	0	0
$538$	6.06444e6	0.903306
$539$	−518012.	−0.0768011
$540$	0	0
$541$	−2.96004e6	−0.434815	−0.217407	−	0.976081i	$-0.569760\pi$
$541$	−2.96004e6	−0.434815	−0.217407	+	0.976081i	$0.569760\pi$
$542$	−6.45593e6	−0.943975
$543$	0	0
$544$	6.83580e6	0.990358
$545$	1.51719e7	2.18801
$546$	0	0
$547$	−3.64363e6	−0.520673	−0.260337	−	0.965518i	$-0.583834\pi$
$547$	−3.64363e6	−0.520673	−0.260337	+	0.965518i	$0.583834\pi$
$548$	−1.36467e7	−1.94123
$549$	0	0
$550$	−1.75321e7	−2.47132
$551$	320970.	0.0450386
$552$	0	0
$553$	−1.34069e6	−0.186430
$554$	−1.55066e6	−0.214655
$555$	0	0
$556$	−1.27338e7	−1.74691
$557$	5.39423e6	0.736701	0.368350	−	0.929687i	$-0.379923\pi$
$557$	5.39423e6	0.736701	0.368350	+	0.929687i	$0.379923\pi$
$558$	0	0
$559$	−5.50814e6	−0.745548
$560$	4.81982e6	0.649473
$561$	0	0
$562$	9.16558e6	1.22411
$563$	8.18538e6	1.08835	0.544174	−	0.838973i	$-0.316843\pi$
$563$	8.18538e6	1.08835	0.544174	+	0.838973i	$0.316843\pi$
$564$	0	0
$565$	−959101.	−0.126399
$566$	6.29915e6	0.826496
$567$	0	0
$568$	2.02002e6	0.262715
$569$	−1.23194e7	−1.59518	−0.797590	−	0.603201i	$-0.793892\pi$
$569$	−1.23194e7	−1.59518	−0.797590	+	0.603201i	$0.793892\pi$
$570$	0	0
$571$	1.09533e7	1.40590	0.702948	−	0.711241i	$-0.251866\pi$
$571$	1.09533e7	1.40590	0.702948	+	0.711241i	$0.251866\pi$
$572$	1.37703e7	1.75975
$573$	0	0
$574$	1.17814e6	0.149251
$575$	−1.74170e7	−2.19686
$576$	0	0
$577$	−803138.	−0.100427	−0.0502135	−	0.998739i	$-0.515990\pi$
$577$	−803138.	−0.100427	−0.0502135	+	0.998739i	$0.515990\pi$
$578$	−4.98387e6	−0.620508
$579$	0	0
$580$	−2.01191e7	−2.48335
$581$	1.59529e6	0.196064
$582$	0	0
$583$	−1.47113e7	−1.79258
$584$	−1.01067e7	−1.22624
$585$	0	0
$586$	−1.79552e7	−2.15996
$587$	−6.51982e6	−0.780980	−0.390490	−	0.920607i	$-0.627694\pi$
$587$	−6.51982e6	−0.780980	−0.390490	+	0.920607i	$0.627694\pi$
$588$	0	0
$589$	−199224.	−0.0236621
$590$	2.71252e6	0.320806
$591$	0	0
$592$	2.01968e6	0.236852
$593$	2.02927e6	0.236975	0.118488	−	0.992956i	$-0.462195\pi$
$593$	2.02927e6	0.236975	0.118488	+	0.992956i	$0.462195\pi$
$594$	0	0
$595$	1.09789e7	1.27135
$596$	−2.38304e7	−2.74800
$597$	0	0
$598$	2.32862e7	2.66285
$599$	−667254.	−0.0759843	−0.0379922	−	0.999278i	$-0.512096\pi$
$599$	−667254.	−0.0759843	−0.0379922	+	0.999278i	$0.512096\pi$
$600$	0	0
$601$	−7.42028e6	−0.837981	−0.418991	−	0.907991i	$-0.637616\pi$
$601$	−7.42028e6	−0.837981	−0.418991	+	0.907991i	$0.637616\pi$
$602$	−9.12424e6	−1.02614
$603$	0	0
$604$	−1.02778e7	−1.14632
$605$	1.60189e6	0.177927
$606$	0	0
$607$	−2.14891e6	−0.236727	−0.118363	−	0.992970i	$-0.537765\pi$
$607$	−2.14891e6	−0.236727	−0.118363	+	0.992970i	$0.537765\pi$
$608$	475789.	0.0521982
$609$	0	0
$610$	2.59611e7	2.82487
$611$	−1.23328e7	−1.33647
$612$	0	0
$613$	1.69964e7	1.82686	0.913429	−	0.406999i	$-0.133425\pi$
$613$	1.69964e7	1.82686	0.913429	+	0.406999i	$0.133425\pi$
$614$	−1.19015e7	−1.27404
$615$	0	0
$616$	6.79211e6	0.721196
$617$	3.94765e6	0.417470	0.208735	−	0.977972i	$-0.433065\pi$
$617$	3.94765e6	0.417470	0.208735	+	0.977972i	$0.433065\pi$
$618$	0	0
$619$	−4.73731e6	−0.496942	−0.248471	−	0.968639i	$-0.579928\pi$
$619$	−4.73731e6	−0.496942	−0.248471	+	0.968639i	$0.579928\pi$
$620$	1.24878e7	1.30468
$621$	0	0
$622$	1.65244e7	1.71257
$623$	−1.09311e6	−0.112835
$624$	0	0
$625$	−2.34425e6	−0.240051
$626$	2.16354e7	2.20663
$627$	0	0
$628$	7.72726e6	0.781855
$629$	4.60053e6	0.463640
$630$	0	0
$631$	−1.34194e7	−1.34172	−0.670859	−	0.741585i	$-0.734074\pi$
$631$	−1.34194e7	−1.34172	−0.670859	+	0.741585i	$0.734074\pi$
$632$	1.19317e6	0.118825
$633$	0	0
$634$	1.54995e7	1.53142
$635$	−2.35409e7	−2.31680
$636$	0	0
$637$	−873742.	−0.0853168
$638$	1.86027e7	1.80935
$639$	0	0
$640$	−1.97073e7	−1.90185
$641$	−1.36524e7	−1.31239	−0.656196	−	0.754591i	$-0.727835\pi$
$641$	−1.36524e7	−1.31239	−0.656196	+	0.754591i	$0.727835\pi$
$642$	0	0
$643$	−918910.	−0.0876487	−0.0438244	−	0.999039i	$-0.513954\pi$
$643$	−918910.	−0.0876487	−0.0438244	+	0.999039i	$0.513954\pi$
$644$	2.26606e7	2.15306
$645$	0	0
$646$	523498.	0.0493553
$647$	1.68376e7	1.58132	0.790658	−	0.612258i	$-0.209738\pi$
$647$	1.68376e7	1.58132	0.790658	+	0.612258i	$0.209738\pi$
$648$	0	0
$649$	−1.47340e6	−0.137312
$650$	−2.95718e7	−2.74533
$651$	0	0
$652$	1.12559e7	1.03696
$653$	−7.85260e6	−0.720661	−0.360330	−	0.932825i	$-0.617336\pi$
$653$	−7.85260e6	−0.720661	−0.360330	+	0.932825i	$0.617336\pi$
$654$	0	0
$655$	−3.05129e7	−2.77895
$656$	404151.	0.0366677
$657$	0	0
$658$	−2.04293e7	−1.83946
$659$	−1.37264e7	−1.23124	−0.615622	−	0.788042i	$-0.711095\pi$
$659$	−1.37264e7	−1.23124	−0.615622	+	0.788042i	$0.711095\pi$
$660$	0	0
$661$	−1.55788e7	−1.38686	−0.693428	−	0.720526i	$-0.743901\pi$
$661$	−1.55788e7	−1.38686	−0.693428	+	0.720526i	$0.743901\pi$
$662$	−2.30782e7	−2.04672
$663$	0	0
$664$	−1.41975e6	−0.124966
$665$	764156.	0.0670082
$666$	0	0
$667$	1.84804e7	1.60841
$668$	1.69231e6	0.146737
$669$	0	0
$670$	3.93504e7	3.38659
$671$	−1.41016e7	−1.20910
$672$	0	0
$673$	9.65010e6	0.821285	0.410643	−	0.911796i	$-0.365304\pi$
$673$	9.65010e6	0.821285	0.410643	+	0.911796i	$0.365304\pi$
$674$	2.26308e7	1.91889
$675$	0	0
$676$	6.30727e6	0.530853
$677$	−2.85548e6	−0.239446	−0.119723	−	0.992807i	$-0.538201\pi$
$677$	−2.85548e6	−0.239446	−0.119723	+	0.992807i	$0.538201\pi$
$678$	0	0
$679$	1.84430e7	1.53517
$680$	−9.77080e6	−0.810322
$681$	0	0
$682$	−1.15465e7	−0.950585
$683$	4.39043e6	0.360127	0.180063	−	0.983655i	$-0.442370\pi$
$683$	4.39043e6	0.360127	0.180063	+	0.983655i	$0.442370\pi$
$684$	0	0
$685$	−2.64964e7	−2.15755
$686$	1.84292e7	1.49519
$687$	0	0
$688$	−3.12999e6	−0.252099
$689$	−2.48138e7	−1.99134
$690$	0	0
$691$	−8.79380e6	−0.700618	−0.350309	−	0.936634i	$-0.613923\pi$
$691$	−8.79380e6	−0.700618	−0.350309	+	0.936634i	$0.613923\pi$
$692$	2.27689e6	0.180749
$693$	0	0
$694$	−1.49407e7	−1.17753
$695$	−2.47239e7	−1.94158
$696$	0	0
$697$	920597.	0.0717774
$698$	−2.08666e7	−1.62111
$699$	0	0
$700$	−2.87773e7	−2.21975
$701$	−1.57360e7	−1.20948	−0.604742	−	0.796421i	$-0.706724\pi$
$701$	−1.57360e7	−1.20948	−0.604742	+	0.796421i	$0.706724\pi$
$702$	0	0
$703$	320209.	0.0244368
$704$	2.20807e7	1.67912
$705$	0	0
$706$	−3.81427e7	−2.88005
$707$	−4.42654e6	−0.333055
$708$	0	0
$709$	2.11378e7	1.57922	0.789611	−	0.613607i	$-0.210282\pi$
$709$	2.11378e7	1.57922	0.789611	+	0.613607i	$0.210282\pi$
$710$	1.31718e7	0.980615
$711$	0	0
$712$	972834.	0.0719182
$713$	−1.14707e7	−0.845017
$714$	0	0
$715$	2.67363e7	1.95585
$716$	2.42343e6	0.176664
$717$	0	0
$718$	8.94095e6	0.647250
$719$	2.29229e7	1.65366	0.826831	−	0.562451i	$-0.190141\pi$
$719$	2.29229e7	1.65366	0.826831	+	0.562451i	$0.190141\pi$
$720$	0	0
$721$	−2.26340e7	−1.62152
$722$	−2.17713e7	−1.55433
$723$	0	0
$724$	−1.04332e7	−0.739727
$725$	−2.34688e7	−1.65824
$726$	0	0
$727$	6.81555e6	0.478261	0.239130	−	0.970987i	$-0.423138\pi$
$727$	6.81555e6	0.478261	0.239130	+	0.970987i	$0.423138\pi$
$728$	1.14564e7	0.801161
$729$	0	0
$730$	−6.59016e7	−4.57709
$731$	−7.12966e6	−0.493487
$732$	0	0
$733$	1.68755e7	1.16011	0.580053	−	0.814578i	$-0.303032\pi$
$733$	1.68755e7	1.16011	0.580053	+	0.814578i	$0.303032\pi$
$734$	−1.95017e7	−1.33608
$735$	0	0
$736$	2.73945e7	1.86410
$737$	−2.13745e7	−1.44953
$738$	0	0
$739$	6.09722e6	0.410696	0.205348	−	0.978689i	$-0.434167\pi$
$739$	6.09722e6	0.410696	0.205348	+	0.978689i	$0.434167\pi$
$740$	−2.00714e7	−1.34740
$741$	0	0
$742$	−4.11041e7	−2.74079
$743$	−5.82391e6	−0.387028	−0.193514	−	0.981097i	$-0.561989\pi$
$743$	−5.82391e6	−0.387028	−0.193514	+	0.981097i	$0.561989\pi$
$744$	0	0
$745$	−4.62691e7	−3.05422
$746$	−1.77767e7	−1.16951
$747$	0	0
$748$	1.78240e7	1.16480
$749$	−4.93198e6	−0.321230
$750$	0	0
$751$	−1.45028e7	−0.938321	−0.469160	−	0.883113i	$-0.655443\pi$
$751$	−1.45028e7	−0.938321	−0.469160	+	0.883113i	$0.655443\pi$
$752$	−7.00810e6	−0.451914
$753$	0	0
$754$	3.13775e7	2.00997
$755$	−1.99553e7	−1.27406
$756$	0	0
$757$	2.63506e7	1.67128	0.835642	−	0.549274i	$-0.185096\pi$
$757$	2.63506e7	1.67128	0.835642	+	0.549274i	$0.185096\pi$
$758$	4.19376e7	2.65113
$759$	0	0
$760$	−680072.	−0.0427092
$761$	8.31384e6	0.520403	0.260202	−	0.965554i	$-0.416211\pi$
$761$	8.31384e6	0.520403	0.260202	+	0.965554i	$0.416211\pi$
$762$	0	0
$763$	−2.30262e7	−1.43189
$764$	−1.74536e7	−1.08181
$765$	0	0
$766$	−8.82517e6	−0.543439
$767$	−2.48521e6	−0.152537
$768$	0	0
$769$	1.06046e7	0.646666	0.323333	−	0.946285i	$-0.395197\pi$
$769$	1.06046e7	0.646666	0.323333	+	0.946285i	$0.395197\pi$
$770$	4.42887e7	2.69194
$771$	0	0
$772$	−3.06202e7	−1.84912
$773$	2.65938e7	1.60078	0.800390	−	0.599479i	$-0.204626\pi$
$773$	2.65938e7	1.60078	0.800390	+	0.599479i	$0.204626\pi$
$774$	0	0
$775$	1.45669e7	0.871192
$776$	−1.64136e7	−0.978474
$777$	0	0
$778$	4.48167e7	2.65455
$779$	64075.9	0.00378313
$780$	0	0
$781$	−7.15470e6	−0.419724
$782$	3.01414e7	1.76257
$783$	0	0
$784$	−496502.	−0.0288490
$785$	1.50032e7	0.868981
$786$	0	0
$787$	−3.05442e7	−1.75789	−0.878945	−	0.476923i	$-0.841752\pi$
$787$	−3.05442e7	−1.75789	−0.878945	+	0.476923i	$0.841752\pi$
$788$	2.36037e7	1.35414
$789$	0	0
$790$	7.78018e6	0.443529
$791$	1.45562e6	0.0827190
$792$	0	0
$793$	−2.37855e7	−1.34317
$794$	−3.66536e7	−2.06331
$795$	0	0
$796$	1.99483e7	1.11590
$797$	2.58869e7	1.44356	0.721779	−	0.692123i	$-0.243325\pi$
$797$	2.58869e7	1.44356	0.721779	+	0.692123i	$0.243325\pi$
$798$	0	0
$799$	−1.59634e7	−0.884625
$800$	−3.47890e7	−1.92184
$801$	0	0
$802$	−5.39643e7	−2.96258
$803$	3.57967e7	1.95909
$804$	0	0
$805$	4.39977e7	2.39299
$806$	−1.94758e7	−1.05599
$807$	0	0
$808$	3.93947e6	0.212280
$809$	−6.74823e6	−0.362509	−0.181254	−	0.983436i	$-0.558016\pi$
$809$	−6.74823e6	−0.362509	−0.181254	+	0.983436i	$0.558016\pi$
$810$	0	0
$811$	−2.22287e7	−1.18676	−0.593378	−	0.804924i	$-0.702206\pi$
$811$	−2.22287e7	−1.18676	−0.593378	+	0.804924i	$0.702206\pi$
$812$	3.05345e7	1.62517
$813$	0	0
$814$	1.85585e7	0.981709
$815$	2.18545e7	1.15251
$816$	0	0
$817$	−496243.	−0.0260099
$818$	−1.22661e7	−0.640947
$819$	0	0
$820$	−4.01641e6	−0.208595
$821$	1.57133e7	0.813599	0.406799	−	0.913517i	$-0.366645\pi$
$821$	1.57133e7	0.813599	0.406799	+	0.913517i	$0.366645\pi$
$822$	0	0
$823$	−2.22414e7	−1.14462	−0.572311	−	0.820036i	$-0.693953\pi$
$823$	−2.22414e7	−1.14462	−0.572311	+	0.820036i	$0.693953\pi$
$824$	2.01435e7	1.03351
$825$	0	0
$826$	−4.11676e6	−0.209945
$827$	−1.20770e7	−0.614038	−0.307019	−	0.951703i	$-0.599332\pi$
$827$	−1.20770e7	−0.614038	−0.307019	+	0.951703i	$0.599332\pi$
$828$	0	0
$829$	−1.91099e6	−0.0965765	−0.0482882	−	0.998833i	$-0.515377\pi$
$829$	−1.91099e6	−0.0965765	−0.0482882	+	0.998833i	$0.515377\pi$
$830$	−9.25762e6	−0.466449
$831$	0	0
$832$	3.72440e7	1.86530
$833$	−1.13096e6	−0.0564722
$834$	0	0
$835$	3.28579e6	0.163088
$836$	1.24060e6	0.0613925
$837$	0	0
$838$	4.31532e7	2.12277
$839$	−3.50301e7	−1.71805	−0.859025	−	0.511933i	$-0.828930\pi$
$839$	−3.50301e7	−1.71805	−0.859025	+	0.511933i	$0.828930\pi$
$840$	0	0
$841$	4.39070e6	0.214064
$842$	−1.29923e7	−0.631547
$843$	0	0
$844$	3.62601e7	1.75216
$845$	1.22462e7	0.590009
$846$	0	0
$847$	−2.43116e6	−0.116441
$848$	−1.41004e7	−0.673352
$849$	0	0
$850$	−3.82774e7	−1.81717
$851$	1.84366e7	0.872685
$852$	0	0
$853$	−2.10044e7	−0.988412	−0.494206	−	0.869345i	$-0.664541\pi$
$853$	−2.10044e7	−0.988412	−0.494206	+	0.869345i	$0.664541\pi$
$854$	−3.94008e7	−1.84867
$855$	0	0
$856$	4.38929e6	0.204743
$857$	−2.13586e7	−0.993390	−0.496695	−	0.867925i	$-0.665453\pi$
$857$	−2.13586e7	−0.993390	−0.496695	+	0.867925i	$0.665453\pi$
$858$	0	0
$859$	2.37853e7	1.09983	0.549915	−	0.835221i	$-0.314660\pi$
$859$	2.37853e7	1.09983	0.549915	+	0.835221i	$0.314660\pi$
$860$	3.11055e7	1.43414
$861$	0	0
$862$	5.71188e7	2.61825
$863$	345138.	0.0157749	0.00788744	−	0.999969i	$-0.497489\pi$
$863$	345138.	0.0157749	0.00788744	+	0.999969i	$0.497489\pi$
$864$	0	0
$865$	4.42080e6	0.200891
$866$	5.16310e7	2.33946
$867$	0	0
$868$	−1.89525e7	−0.853823
$869$	−4.22607e6	−0.189840
$870$	0	0
$871$	−3.60529e7	−1.61026
$872$	2.04925e7	0.912649
$873$	0	0
$874$	2.09792e6	0.0928988
$875$	−1.87475e7	−0.827794
$876$	0	0
$877$	−2.10330e7	−0.923425	−0.461712	−	0.887030i	$-0.652765\pi$
$877$	−2.10330e7	−0.923425	−0.461712	+	0.887030i	$0.652765\pi$
$878$	−3.90001e7	−1.70738
$879$	0	0
$880$	1.51928e7	0.661352
$881$	3.40176e7	1.47660	0.738302	−	0.674470i	$-0.235628\pi$
$881$	3.40176e7	1.47660	0.738302	+	0.674470i	$0.235628\pi$
$882$	0	0
$883$	9.95646e6	0.429737	0.214869	−	0.976643i	$-0.431068\pi$
$883$	9.95646e6	0.429737	0.214869	+	0.976643i	$0.431068\pi$
$884$	3.00642e7	1.29395
$885$	0	0
$886$	6.61352e7	2.83040
$887$	1.73362e7	0.739850	0.369925	−	0.929062i	$-0.379383\pi$
$887$	1.73362e7	0.739850	0.369925	+	0.929062i	$0.379383\pi$
$888$	0	0
$889$	3.57277e7	1.51618
$890$	6.34346e6	0.268443
$891$	0	0
$892$	−6.57245e7	−2.76576
$893$	−1.11110e6	−0.0466254
$894$	0	0
$895$	4.70533e6	0.196351
$896$	2.99095e7	1.24463
$897$	0	0
$898$	−1.71954e7	−0.711577
$899$	−1.54564e7	−0.637836
$900$	0	0
$901$	−3.21187e7	−1.31809
$902$	3.71369e6	0.151981
$903$	0	0
$904$	−1.29545e6	−0.0527228
$905$	−2.02571e7	−0.822158
$906$	0	0
$907$	7.69488e6	0.310587	0.155294	−	0.987868i	$-0.450368\pi$
$907$	7.69488e6	0.310587	0.155294	+	0.987868i	$0.450368\pi$
$908$	1.92137e7	0.773384
$909$	0	0
$910$	7.47027e7	2.99042
$911$	1.61662e7	0.645376	0.322688	−	0.946505i	$-0.395413\pi$
$911$	1.61662e7	0.645376	0.322688	+	0.946505i	$0.395413\pi$
$912$	0	0
$913$	5.02859e6	0.199650
$914$	−5.23621e7	−2.07325
$915$	0	0
$916$	−6.45730e7	−2.54280
$917$	4.63091e7	1.81862
$918$	0	0
$919$	−3.62199e7	−1.41468	−0.707341	−	0.706873i	$-0.750105\pi$
$919$	−3.62199e7	−1.41468	−0.707341	+	0.706873i	$0.750105\pi$
$920$	−3.91564e7	−1.52522
$921$	0	0
$922$	−5.74780e7	−2.22677
$923$	−1.20680e7	−0.466263
$924$	0	0
$925$	−2.34132e7	−0.899717
$926$	−1.06590e7	−0.408496
$927$	0	0
$928$	3.69132e7	1.40706
$929$	2.11322e6	0.0803350	0.0401675	−	0.999193i	$-0.487211\pi$
$929$	2.11322e6	0.0803350	0.0401675	+	0.999193i	$0.487211\pi$
$930$	0	0
$931$	−78717.6	−0.00297644
$932$	−3.06747e7	−1.15675
$933$	0	0
$934$	3.46625e7	1.30015
$935$	3.46071e7	1.29460
$936$	0	0
$937$	−1.46913e6	−0.0546652	−0.0273326	−	0.999626i	$-0.508701\pi$
$937$	−1.46913e6	−0.0546652	−0.0273326	+	0.999626i	$0.508701\pi$
$938$	−5.97217e7	−2.21628
$939$	0	0
$940$	6.96458e7	2.57084
$941$	3.58757e7	1.32077	0.660384	−	0.750928i	$-0.270393\pi$
$941$	3.58757e7	1.32077	0.660384	+	0.750928i	$0.270393\pi$
$942$	0	0
$943$	3.68929e6	0.135103
$944$	−1.41222e6	−0.0515788
$945$	0	0
$946$	−2.87611e7	−1.04491
$947$	2.42642e7	0.879208	0.439604	−	0.898192i	$-0.355119\pi$
$947$	2.42642e7	0.879208	0.439604	+	0.898192i	$0.355119\pi$
$948$	0	0
$949$	6.03792e7	2.17631
$950$	−2.66420e6	−0.0957764
$951$	0	0
$952$	1.48290e7	0.530298
$953$	3.58333e7	1.27807	0.639036	−	0.769177i	$-0.279334\pi$
$953$	3.58333e7	1.27807	0.639036	+	0.769177i	$0.279334\pi$
$954$	0	0
$955$	−3.38879e7	−1.20237
$956$	3.46475e7	1.22611
$957$	0	0
$958$	2.13695e6	0.0752281
$959$	4.02132e7	1.41196
$960$	0	0
$961$	−1.90355e7	−0.664898
$962$	3.13031e7	1.09056
$963$	0	0
$964$	2.14418e7	0.743138
$965$	−5.94520e7	−2.05517
$966$	0	0
$967$	−3.27974e7	−1.12791	−0.563953	−	0.825807i	$-0.690720\pi$
$967$	−3.27974e7	−1.12791	−0.563953	+	0.825807i	$0.690720\pi$
$968$	2.16365e6	0.0742161
$969$	0	0
$970$	−1.07027e8	−3.65226
$971$	−5.64198e7	−1.92036	−0.960182	−	0.279376i	$-0.909872\pi$
$971$	−5.64198e7	−1.92036	−0.960182	+	0.279376i	$0.909872\pi$
$972$	0	0
$973$	3.75232e7	1.27063
$974$	−2.27050e7	−0.766874
$975$	0	0
$976$	−1.35161e7	−0.454178
$977$	3.59696e7	1.20559	0.602794	−	0.797897i	$-0.294054\pi$
$977$	3.59696e7	1.20559	0.602794	+	0.797897i	$0.294054\pi$
$978$	0	0
$979$	−3.44567e6	−0.114899
$980$	4.93419e6	0.164116
$981$	0	0
$982$	2.53126e6	0.0837642
$983$	119682.	0.00395044	0.00197522	−	0.999998i	$-0.499371\pi$
$983$	119682.	0.00395044	0.00197522	+	0.999998i	$0.499371\pi$
$984$	0	0
$985$	4.58288e7	1.50504
$986$	4.06146e7	1.33042
$987$	0	0
$988$	2.09254e6	0.0681997
$989$	−2.85721e7	−0.928863
$990$	0	0
$991$	−7.51751e6	−0.243159	−0.121579	−	0.992582i	$-0.538796\pi$
$991$	−7.51751e6	−0.243159	−0.121579	+	0.992582i	$0.538796\pi$
$992$	−2.29118e7	−0.739230
$993$	0	0
$994$	−1.99906e7	−0.641742
$995$	3.87316e7	1.24025
$996$	0	0
$997$	3.03530e7	0.967084	0.483542	−	0.875321i	$-0.339350\pi$
$997$	3.03530e7	0.967084	0.483542	+	0.875321i	$0.339350\pi$
$998$	3.98133e7	1.26532
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.d.1.10		12
3.2	odd	2		177.6.a.b.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.b.1.3	✓	12	3.2	odd	2
531.6.a.d.1.10		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)

\(f(q)\)	\(=\)	\(q+8.80731 q^{2} +45.5687 q^{4} +88.4760 q^{5} -134.279 q^{7} +119.504 q^{8} +O(q^{10})\)	\(q+8.80731 q^{2} +45.5687 q^{4} +88.4760 q^{5} -134.279 q^{7} +119.504 q^{8} +779.236 q^{10} -423.269 q^{11} -713.936 q^{13} -1182.64 q^{14} -405.693 q^{16} -924.109 q^{17} -64.3204 q^{19} +4031.74 q^{20} -3727.86 q^{22} -3703.37 q^{23} +4703.01 q^{25} -6287.86 q^{26} -6118.92 q^{28} -4990.17 q^{29} +3097.37 q^{31} -7397.18 q^{32} -8138.92 q^{34} -11880.5 q^{35} -4978.34 q^{37} -566.489 q^{38} +10573.2 q^{40} -996.199 q^{41} +7715.17 q^{43} -19287.8 q^{44} -32616.7 q^{46} +17274.4 q^{47} +1223.84 q^{49} +41420.8 q^{50} -32533.1 q^{52} +34756.3 q^{53} -37449.1 q^{55} -16046.8 q^{56} -43950.0 q^{58} +3481.00 q^{59} +33316.0 q^{61} +27279.5 q^{62} -52167.1 q^{64} -63166.3 q^{65} +50498.8 q^{67} -42110.4 q^{68} -104635. q^{70} +16903.5 q^{71} -84572.2 q^{73} -43845.8 q^{74} -2930.99 q^{76} +56836.1 q^{77} +9984.38 q^{79} -35894.1 q^{80} -8773.83 q^{82} -11880.4 q^{83} -81761.5 q^{85} +67949.9 q^{86} -50582.1 q^{88} +8140.62 q^{89} +95866.6 q^{91} -168758. q^{92} +152141. q^{94} -5690.81 q^{95} -137348. q^{97} +10778.7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) \) 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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Database

Properties

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