LMFDB - Embedded newform 531.6.a.e.1.7

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:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + \cdots - 6400833792 $ x^13 - 331x^11 - 41x^10 + 41990x^9 + 7229x^8 - 2592364x^7 - 312100x^6 + 81977088x^5 - 3773728x^4 - 1245415104x^3 + 453320896x^2 + 6872784896*x - 6400833792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.05732$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.05732 q^{2} -30.8821 q^{4} -33.7931 q^{5} -85.6226 q^{7} +66.4863 q^{8} +O(q^{10})$	\(q-1.05732 q^{2} -30.8821 q^{4} -33.7931 q^{5} -85.6226 q^{7} +66.4863 q^{8} +35.7300 q^{10} +33.3696 q^{11} -1043.41 q^{13} +90.5302 q^{14} +917.930 q^{16} -522.840 q^{17} -1300.57 q^{19} +1043.60 q^{20} -35.2822 q^{22} -2439.93 q^{23} -1983.02 q^{25} +1103.22 q^{26} +2644.20 q^{28} -7069.65 q^{29} +2519.18 q^{31} -3098.10 q^{32} +552.807 q^{34} +2893.45 q^{35} -6580.49 q^{37} +1375.12 q^{38} -2246.78 q^{40} -19615.8 q^{41} +18389.9 q^{43} -1030.52 q^{44} +2579.77 q^{46} -18005.2 q^{47} -9475.78 q^{49} +2096.69 q^{50} +32222.7 q^{52} -16541.3 q^{53} -1127.66 q^{55} -5692.73 q^{56} +7474.86 q^{58} -3481.00 q^{59} -29130.6 q^{61} -2663.57 q^{62} -26098.1 q^{64} +35260.1 q^{65} -23137.8 q^{67} +16146.4 q^{68} -3059.30 q^{70} +35612.7 q^{71} +76489.3 q^{73} +6957.66 q^{74} +40164.4 q^{76} -2857.19 q^{77} +61578.6 q^{79} -31019.7 q^{80} +20740.1 q^{82} +11802.0 q^{83} +17668.4 q^{85} -19443.9 q^{86} +2218.62 q^{88} -34947.8 q^{89} +89339.6 q^{91} +75350.0 q^{92} +19037.2 q^{94} +43950.4 q^{95} -89284.3 q^{97} +10018.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 246 q^{4} + 14 q^{5} + 373 q^{7} - 123 q^{8}+O(q^{10}) $ 13 * q + 246 * q^4 + 14 * q^5 + 373 * q^7 - 123 * q^8	$ 13 q + 246 q^{4} + 14 q^{5} + 373 q^{7} - 123 q^{8} + 137 q^{10} - 250 q^{11} + 1054 q^{13} + 575 q^{14} + 922 q^{16} - 271 q^{17} + 671 q^{19} + 5491 q^{20} + 1094 q^{22} - 3975 q^{23} + 15569 q^{25} - 4622 q^{26} + 21214 q^{28} + 10613 q^{29} + 25597 q^{31} - 15966 q^{32} + 31796 q^{34} - 6729 q^{35} + 17585 q^{37} - 34903 q^{38} + 31382 q^{40} - 12537 q^{41} + 26644 q^{43} - 6654 q^{44} + 149005 q^{46} - 52087 q^{47} + 95384 q^{49} - 121821 q^{50} + 263630 q^{52} - 20014 q^{53} + 120932 q^{55} - 126688 q^{56} + 86066 q^{58} - 45253 q^{59} - 11667 q^{61} - 164794 q^{62} + 151893 q^{64} + 28674 q^{65} + 1106 q^{67} + 4043 q^{68} + 56066 q^{70} - 21230 q^{71} + 81131 q^{73} - 102042 q^{74} + 73900 q^{76} + 104655 q^{77} - 13470 q^{79} + 191969 q^{80} + 79909 q^{82} + 76149 q^{83} + 10035 q^{85} + 321496 q^{86} - 276779 q^{88} + 190205 q^{89} + 80601 q^{91} - 45672 q^{92} + 36768 q^{94} - 9875 q^{95} + 160850 q^{97} + 116644 q^{98}+O(q^{100}) $ 13 * q + 246 * q^4 + 14 * q^5 + 373 * q^7 - 123 * q^8 + 137 * q^10 - 250 * q^11 + 1054 * q^13 + 575 * q^14 + 922 * q^16 - 271 * q^17 + 671 * q^19 + 5491 * q^20 + 1094 * q^22 - 3975 * q^23 + 15569 * q^25 - 4622 * q^26 + 21214 * q^28 + 10613 * q^29 + 25597 * q^31 - 15966 * q^32 + 31796 * q^34 - 6729 * q^35 + 17585 * q^37 - 34903 * q^38 + 31382 * q^40 - 12537 * q^41 + 26644 * q^43 - 6654 * q^44 + 149005 * q^46 - 52087 * q^47 + 95384 * q^49 - 121821 * q^50 + 263630 * q^52 - 20014 * q^53 + 120932 * q^55 - 126688 * q^56 + 86066 * q^58 - 45253 * q^59 - 11667 * q^61 - 164794 * q^62 + 151893 * q^64 + 28674 * q^65 + 1106 * q^67 + 4043 * q^68 + 56066 * q^70 - 21230 * q^71 + 81131 * q^73 - 102042 * q^74 + 73900 * q^76 + 104655 * q^77 - 13470 * q^79 + 191969 * q^80 + 79909 * q^82 + 76149 * q^83 + 10035 * q^85 + 321496 * q^86 - 276779 * q^88 + 190205 * q^89 + 80601 * q^91 - 45672 * q^92 + 36768 * q^94 - 9875 * q^95 + 160850 * q^97 + 116644 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.05732	−0.186909	−0.0934545	−	0.995624i	$-0.529791\pi$
$2$	−1.05732	−0.186909	−0.0934545	+	0.995624i	$0.529791\pi$
$3$	0	0
$3$	0	0
$4$	−30.8821	−0.965065
$5$	−33.7931	−0.604510	−0.302255	−	0.953227i	$-0.597739\pi$
$5$	−33.7931	−0.604510	−0.302255	+	0.953227i	$0.597739\pi$
$6$	0	0
$7$	−85.6226	−0.660455	−0.330227	−	0.943901i	$-0.607125\pi$
$7$	−85.6226	−0.660455	−0.330227	+	0.943901i	$0.607125\pi$
$8$	66.4863	0.367288
$9$	0	0
$10$	35.7300	0.112988
$11$	33.3696	0.0831513	0.0415757	−	0.999135i	$-0.486762\pi$
$11$	33.3696	0.0831513	0.0415757	+	0.999135i	$0.486762\pi$
$12$	0	0
$13$	−1043.41	−1.71237	−0.856184	−	0.516671i	$-0.827171\pi$
$13$	−1043.41	−1.71237	−0.856184	+	0.516671i	$0.827171\pi$
$14$	90.5302	0.123445
$15$	0	0
$16$	917.930	0.896416
$17$	−522.840	−0.438780	−0.219390	−	0.975637i	$-0.570407\pi$
$17$	−522.840	−0.438780	−0.219390	+	0.975637i	$0.570407\pi$
$18$	0	0
$19$	−1300.57	−0.826514	−0.413257	−	0.910614i	$-0.635609\pi$
$19$	−1300.57	−0.826514	−0.413257	+	0.910614i	$0.635609\pi$
$20$	1043.60	0.583391
$21$	0	0
$22$	−35.2822	−0.0155417
$23$	−2439.93	−0.961739	−0.480869	−	0.876792i	$-0.659679\pi$
$23$	−2439.93	−0.961739	−0.480869	+	0.876792i	$0.659679\pi$
$24$	0	0
$25$	−1983.02	−0.634568
$26$	1103.22	0.320057
$27$	0	0
$28$	2644.20	0.637382
$29$	−7069.65	−1.56100	−0.780500	−	0.625156i	$-0.785035\pi$
$29$	−7069.65	−1.56100	−0.780500	+	0.625156i	$0.785035\pi$
$30$	0	0
$31$	2519.18	0.470821	0.235410	−	0.971896i	$-0.424357\pi$
$31$	2519.18	0.470821	0.235410	+	0.971896i	$0.424357\pi$
$32$	−3098.10	−0.534836
$33$	0	0
$34$	552.807	0.0820118
$35$	2893.45	0.399251
$36$	0	0
$37$	−6580.49	−0.790231	−0.395115	−	0.918632i	$-0.629295\pi$
$37$	−6580.49	−0.790231	−0.395115	+	0.918632i	$0.629295\pi$
$38$	1375.12	0.154483
$39$	0	0
$40$	−2246.78	−0.222029
$41$	−19615.8	−1.82241	−0.911206	−	0.411951i	$-0.864847\pi$
$41$	−19615.8	−1.82241	−0.911206	+	0.411951i	$0.864847\pi$
$42$	0	0
$43$	18389.9	1.51673	0.758364	−	0.651831i	$-0.225999\pi$
$43$	18389.9	1.51673	0.758364	+	0.651831i	$0.225999\pi$
$44$	−1030.52	−0.0802464
$45$	0	0
$46$	2579.77	0.179758
$47$	−18005.2	−1.18892	−0.594461	−	0.804124i	$-0.702635\pi$
$47$	−18005.2	−1.18892	−0.594461	+	0.804124i	$0.702635\pi$
$48$	0	0
$49$	−9475.78	−0.563799
$50$	2096.69	0.118606
$51$	0	0
$52$	32222.7	1.65255
$53$	−16541.3	−0.808870	−0.404435	−	0.914567i	$-0.632532\pi$
$53$	−16541.3	−0.808870	−0.404435	+	0.914567i	$0.632532\pi$
$54$	0	0
$55$	−1127.66	−0.0502658
$56$	−5692.73	−0.242577
$57$	0	0
$58$	7474.86	0.291765
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	−29130.6	−1.00236	−0.501182	−	0.865342i	$-0.667101\pi$
$61$	−29130.6	−1.00236	−0.501182	+	0.865342i	$0.667101\pi$
$62$	−2663.57	−0.0880006
$63$	0	0
$64$	−26098.1	−0.796450
$65$	35260.1	1.03514
$66$	0	0
$67$	−23137.8	−0.629702	−0.314851	−	0.949141i	$-0.601955\pi$
$67$	−23137.8	−0.629702	−0.314851	+	0.949141i	$0.601955\pi$
$68$	16146.4	0.423451
$69$	0	0
$70$	−3059.30	−0.0746237
$71$	35612.7	0.838414	0.419207	−	0.907891i	$-0.362308\pi$
$71$	35612.7	0.838414	0.419207	+	0.907891i	$0.362308\pi$
$72$	0	0
$73$	76489.3	1.67994	0.839970	−	0.542633i	$-0.182573\pi$
$73$	76489.3	1.67994	0.839970	+	0.542633i	$0.182573\pi$
$74$	6957.66	0.147701
$75$	0	0
$76$	40164.4	0.797640
$77$	−2857.19	−0.0549177
$78$	0	0
$79$	61578.6	1.11010	0.555050	−	0.831817i	$-0.312699\pi$
$79$	61578.6	1.11010	0.555050	+	0.831817i	$0.312699\pi$
$80$	−31019.7	−0.541892
$81$	0	0
$82$	20740.1	0.340625
$83$	11802.0	0.188045	0.0940224	−	0.995570i	$-0.470027\pi$
$83$	11802.0	0.188045	0.0940224	+	0.995570i	$0.470027\pi$
$84$	0	0
$85$	17668.4	0.265247
$86$	−19443.9	−0.283490
$87$	0	0
$88$	2218.62	0.0305405
$89$	−34947.8	−0.467675	−0.233838	−	0.972276i	$-0.575128\pi$
$89$	−34947.8	−0.467675	−0.233838	+	0.972276i	$0.575128\pi$
$90$	0	0
$91$	89339.6	1.13094
$92$	75350.0	0.928140
$93$	0	0
$94$	19037.2	0.222220
$95$	43950.4	0.499636
$96$	0	0
$97$	−89284.3	−0.963487	−0.481744	−	0.876312i	$-0.659996\pi$
$97$	−89284.3	−0.963487	−0.481744	+	0.876312i	$0.659996\pi$
$98$	10018.9	0.105379
$99$	0	0
$100$	61239.9	0.612399
$101$	25180.8	0.245621	0.122811	−	0.992430i	$-0.460809\pi$
$101$	25180.8	0.245621	0.122811	+	0.992430i	$0.460809\pi$
$102$	0	0
$103$	133786.	1.24256	0.621282	−	0.783587i	$-0.286612\pi$
$103$	133786.	1.24256	0.621282	+	0.783587i	$0.286612\pi$
$104$	−69372.5	−0.628933
$105$	0	0
$106$	17489.3	0.151185
$107$	71670.6	0.605176	0.302588	−	0.953122i	$-0.402149\pi$
$107$	71670.6	0.605176	0.302588	+	0.953122i	$0.402149\pi$
$108$	0	0
$109$	−101459.	−0.817948	−0.408974	−	0.912546i	$-0.634113\pi$
$109$	−101459.	−0.817948	−0.408974	+	0.912546i	$0.634113\pi$
$110$	1192.30	0.00939512
$111$	0	0
$112$	−78595.5	−0.592042
$113$	−234915.	−1.73067	−0.865337	−	0.501190i	$-0.832896\pi$
$113$	−234915.	−1.73067	−0.865337	+	0.501190i	$0.832896\pi$
$114$	0	0
$115$	82452.7	0.581380
$116$	218325.	1.50647
$117$	0	0
$118$	3680.52	0.0243335
$119$	44766.9	0.289794
$120$	0	0
$121$	−159937.	−0.993086
$122$	30800.3	0.187351
$123$	0	0
$124$	−77797.6	−0.454372
$125$	172616.	0.988112
$126$	0	0
$127$	28866.5	0.158813	0.0794064	−	0.996842i	$-0.474698\pi$
$127$	28866.5	0.158813	0.0794064	+	0.996842i	$0.474698\pi$
$128$	126733.	0.683700
$129$	0	0
$130$	−37281.1	−0.193478
$131$	−119417.	−0.607977	−0.303989	−	0.952676i	$-0.598318\pi$
$131$	−119417.	−0.607977	−0.303989	+	0.952676i	$0.598318\pi$
$132$	0	0
$133$	111358.	0.545875
$134$	24464.0	0.117697
$135$	0	0
$136$	−34761.7	−0.161159
$137$	199935.	0.910097	0.455049	−	0.890467i	$-0.349622\pi$
$137$	199935.	0.910097	0.455049	+	0.890467i	$0.349622\pi$
$138$	0	0
$139$	13511.9	0.0593172	0.0296586	−	0.999560i	$-0.490558\pi$
$139$	13511.9	0.0593172	0.0296586	+	0.999560i	$0.490558\pi$
$140$	−89355.9	−0.385304
$141$	0	0
$142$	−37653.9	−0.156707
$143$	−34818.2	−0.142386
$144$	0	0
$145$	238905.	0.943640
$146$	−80873.4	−0.313996
$147$	0	0
$148$	203219.	0.762624
$149$	448024.	1.65324	0.826619	−	0.562762i	$-0.190261\pi$
$149$	448024.	1.65324	0.826619	+	0.562762i	$0.190261\pi$
$150$	0	0
$151$	−221792.	−0.791597	−0.395799	−	0.918337i	$-0.629532\pi$
$151$	−221792.	−0.791597	−0.395799	+	0.918337i	$0.629532\pi$
$152$	−86470.2	−0.303569
$153$	0	0
$154$	3020.95	0.0102646
$155$	−85131.0	−0.284616
$156$	0	0
$157$	135887.	0.439974	0.219987	−	0.975503i	$-0.429398\pi$
$157$	135887.	0.439974	0.219987	+	0.975503i	$0.429398\pi$
$158$	−65108.0	−0.207487
$159$	0	0
$160$	104695.	0.323314
$161$	208913.	0.635185
$162$	0	0
$163$	301769.	0.889623	0.444812	−	0.895624i	$-0.353271\pi$
$163$	301769.	0.889623	0.444812	+	0.895624i	$0.353271\pi$
$164$	605777.	1.75875
$165$	0	0
$166$	−12478.5	−0.0351473
$167$	−572098.	−1.58737	−0.793687	−	0.608326i	$-0.791841\pi$
$167$	−572098.	−1.58737	−0.793687	+	0.608326i	$0.791841\pi$
$168$	0	0
$169$	717414.	1.93221
$170$	−18681.1	−0.0495770
$171$	0	0
$172$	−567918.	−1.46374
$173$	279838.	0.710873	0.355436	−	0.934700i	$-0.384332\pi$
$173$	279838.	0.710873	0.355436	+	0.934700i	$0.384332\pi$
$174$	0	0
$175$	169792.	0.419103
$176$	30630.9	0.0745381
$177$	0	0
$178$	36950.8	0.0874127
$179$	−421817.	−0.983991	−0.491995	−	0.870598i	$-0.663732\pi$
$179$	−421817.	−0.983991	−0.491995	+	0.870598i	$0.663732\pi$
$180$	0	0
$181$	−257718.	−0.584720	−0.292360	−	0.956308i	$-0.594441\pi$
$181$	−257718.	−0.584720	−0.292360	+	0.956308i	$0.594441\pi$
$182$	−94460.2	−0.211383
$183$	0	0
$184$	−162222.	−0.353235
$185$	222375.	0.477702
$186$	0	0
$187$	−17447.0	−0.0364851
$188$	556038.	1.14739
$189$	0	0
$190$	−46469.5	−0.0933864
$191$	256126.	0.508008	0.254004	−	0.967203i	$-0.418252\pi$
$191$	256126.	0.508008	0.254004	+	0.967203i	$0.418252\pi$
$192$	0	0
$193$	−788156.	−1.52307	−0.761534	−	0.648125i	$-0.775553\pi$
$193$	−788156.	−1.52307	−0.761534	+	0.648125i	$0.775553\pi$
$194$	94401.8	0.180084
$195$	0	0
$196$	292632.	0.544103
$197$	−953753.	−1.75094	−0.875468	−	0.483275i	$-0.839447\pi$
$197$	−953753.	−1.75094	−0.875468	+	0.483275i	$0.839447\pi$
$198$	0	0
$199$	694899.	1.24391	0.621955	−	0.783053i	$-0.286339\pi$
$199$	694899.	1.24391	0.621955	+	0.783053i	$0.286339\pi$
$200$	−131844.	−0.233069
$201$	0	0
$202$	−26624.1	−0.0459088
$203$	605321.	1.03097
$204$	0	0
$205$	662879.	1.10167
$206$	−141454.	−0.232246
$207$	0	0
$208$	−957778.	−1.53499
$209$	−43399.5	−0.0687257
$210$	0	0
$211$	605261.	0.935915	0.467957	−	0.883751i	$-0.344990\pi$
$211$	605261.	0.935915	0.467957	+	0.883751i	$0.344990\pi$
$212$	510828.	0.780612
$213$	0	0
$214$	−75778.5	−0.113113
$215$	−621452.	−0.916877
$216$	0	0
$217$	−215699.	−0.310956
$218$	107275.	0.152882
$219$	0	0
$220$	34824.6	0.0485097
$221$	545537.	0.751352
$222$	0	0
$223$	1.35183e6	1.82037	0.910186	−	0.414199i	$-0.135938\pi$
$223$	1.35183e6	1.82037	0.910186	+	0.414199i	$0.135938\pi$
$224$	265268.	0.353235
$225$	0	0
$226$	248380.	0.323479
$227$	−289071.	−0.372341	−0.186170	−	0.982517i	$-0.559608\pi$
$227$	−289071.	−0.372341	−0.186170	+	0.982517i	$0.559608\pi$
$228$	0	0
$229$	851974.	1.07359	0.536794	−	0.843713i	$-0.319635\pi$
$229$	851974.	1.07359	0.536794	+	0.843713i	$0.319635\pi$
$230$	−87178.7	−0.108665
$231$	0	0
$232$	−470035.	−0.573337
$233$	−1.51708e6	−1.83071	−0.915356	−	0.402646i	$-0.868090\pi$
$233$	−1.51708e6	−1.83071	−0.915356	+	0.402646i	$0.868090\pi$
$234$	0	0
$235$	608452.	0.718715
$236$	107501.	0.125641
$237$	0	0
$238$	−47332.8	−0.0541651
$239$	166996.	0.189109	0.0945543	−	0.995520i	$-0.469857\pi$
$239$	166996.	0.189109	0.0945543	+	0.995520i	$0.469857\pi$
$240$	0	0
$241$	−228585.	−0.253516	−0.126758	−	0.991934i	$-0.540457\pi$
$241$	−228585.	−0.253516	−0.126758	+	0.991934i	$0.540457\pi$
$242$	169105.	0.185617
$243$	0	0
$244$	899614.	0.967346
$245$	320216.	0.340822
$246$	0	0
$247$	1.35703e6	1.41530
$248$	167491.	0.172927
$249$	0	0
$250$	−182510.	−0.184687
$251$	−1.34491e6	−1.34743	−0.673717	−	0.738989i	$-0.735303\pi$
$251$	−1.34491e6	−1.34743	−0.673717	+	0.738989i	$0.735303\pi$
$252$	0	0
$253$	−81419.3	−0.0799698
$254$	−30521.1	−0.0296835
$255$	0	0
$256$	701141.	0.668660
$257$	193589.	0.182830	0.0914151	−	0.995813i	$-0.470861\pi$
$257$	193589.	0.182830	0.0914151	+	0.995813i	$0.470861\pi$
$258$	0	0
$259$	563438.	0.521912
$260$	−1.08891e6	−0.998981
$261$	0	0
$262$	126261.	0.113636
$263$	1.29250e6	1.15224	0.576118	−	0.817367i	$-0.304567\pi$
$263$	1.29250e6	1.15224	0.576118	+	0.817367i	$0.304567\pi$
$264$	0	0
$265$	558981.	0.488970
$266$	−117741.	−0.102029
$267$	0	0
$268$	714544.	0.607704
$269$	2.00558e6	1.68990	0.844948	−	0.534848i	$-0.179631\pi$
$269$	2.00558e6	1.68990	0.844948	+	0.534848i	$0.179631\pi$
$270$	0	0
$271$	1.28831e6	1.06561	0.532803	−	0.846240i	$-0.321139\pi$
$271$	1.28831e6	1.06561	0.532803	+	0.846240i	$0.321139\pi$
$272$	−479930.	−0.393329
$273$	0	0
$274$	−211395.	−0.170105
$275$	−66172.7	−0.0527651
$276$	0	0
$277$	1.13601e6	0.889572	0.444786	−	0.895637i	$-0.353280\pi$
$277$	1.13601e6	0.889572	0.444786	+	0.895637i	$0.353280\pi$
$278$	−14286.4	−0.0110869
$279$	0	0
$280$	192375.	0.146640
$281$	−1.45769e6	−1.10128	−0.550642	−	0.834742i	$-0.685617\pi$
$281$	−1.45769e6	−1.10128	−0.550642	+	0.834742i	$0.685617\pi$
$282$	0	0
$283$	−2.50369e6	−1.85830	−0.929148	−	0.369709i	$-0.879457\pi$
$283$	−2.50369e6	−1.85830	−0.929148	+	0.369709i	$0.879457\pi$
$284$	−1.09979e6	−0.809124
$285$	0	0
$286$	36813.9	0.0266132
$287$	1.67956e6	1.20362
$288$	0	0
$289$	−1.14650e6	−0.807472
$290$	−252599.	−0.176375
$291$	0	0
$292$	−2.36215e6	−1.62125
$293$	−1.43176e6	−0.974322	−0.487161	−	0.873312i	$-0.661968\pi$
$293$	−1.43176e6	−0.974322	−0.487161	+	0.873312i	$0.661968\pi$
$294$	0	0
$295$	117634.	0.0787005
$296$	−437512.	−0.290242
$297$	0	0
$298$	−473703.	−0.309005
$299$	2.54585e6	1.64685
$300$	0	0
$301$	−1.57459e6	−1.00173
$302$	234505.	0.147957
$303$	0	0
$304$	−1.19383e6	−0.740900
$305$	984415.	0.605938
$306$	0	0
$307$	−2.16520e6	−1.31115	−0.655574	−	0.755131i	$-0.727573\pi$
$307$	−2.16520e6	−1.31115	−0.655574	+	0.755131i	$0.727573\pi$
$308$	88235.9	0.0529991
$309$	0	0
$310$	90010.5	0.0531972
$311$	−1.28475e6	−0.753216	−0.376608	−	0.926373i	$-0.622910\pi$
$311$	−1.28475e6	−0.753216	−0.376608	+	0.926373i	$0.622910\pi$
$312$	0	0
$313$	−191903.	−0.110719	−0.0553594	−	0.998466i	$-0.517630\pi$
$313$	−191903.	−0.110719	−0.0553594	+	0.998466i	$0.517630\pi$
$314$	−143675.	−0.0822351
$315$	0	0
$316$	−1.90167e6	−1.07132
$317$	−2.80920e6	−1.57013	−0.785064	−	0.619414i	$-0.787370\pi$
$317$	−2.80920e6	−1.57013	−0.785064	+	0.619414i	$0.787370\pi$
$318$	0	0
$319$	−235911.	−0.129799
$320$	881935.	0.481462
$321$	0	0
$322$	−220887.	−0.118722
$323$	679991.	0.362658
$324$	0	0
$325$	2.06911e6	1.08661
$326$	−319066.	−0.166279
$327$	0	0
$328$	−1.30418e6	−0.669351
$329$	1.54165e6	0.785230
$330$	0	0
$331$	−104756.	−0.0525546	−0.0262773	−	0.999655i	$-0.508365\pi$
$331$	−104756.	−0.0525546	−0.0262773	+	0.999655i	$0.508365\pi$
$332$	−364471.	−0.181475
$333$	0	0
$334$	604889.	0.296694
$335$	781899.	0.380661
$336$	0	0
$337$	−226616.	−0.108697	−0.0543483	−	0.998522i	$-0.517308\pi$
$337$	−226616.	−0.108697	−0.0543483	+	0.998522i	$0.517308\pi$
$338$	−758534.	−0.361147
$339$	0	0
$340$	−545637.	−0.255980
$341$	84064.1	0.0391493
$342$	0	0
$343$	2.25040e6	1.03282
$344$	1.22267e6	0.557077
$345$	0	0
$346$	−295878.	−0.132869
$347$	1.32731e6	0.591766	0.295883	−	0.955224i	$-0.404386\pi$
$347$	1.32731e6	0.591766	0.295883	+	0.955224i	$0.404386\pi$
$348$	0	0
$349$	−321857.	−0.141449	−0.0707245	−	0.997496i	$-0.522531\pi$
$349$	−321857.	−0.141449	−0.0707245	+	0.997496i	$0.522531\pi$
$350$	−179524.	−0.0783342
$351$	0	0
$352$	−103382.	−0.0444723
$353$	−982643.	−0.419719	−0.209860	−	0.977732i	$-0.567301\pi$
$353$	−982643.	−0.419719	−0.209860	+	0.977732i	$0.567301\pi$
$354$	0	0
$355$	−1.20346e6	−0.506830
$356$	1.07926e6	0.451337
$357$	0	0
$358$	445994.	0.183917
$359$	−937903.	−0.384080	−0.192040	−	0.981387i	$-0.561510\pi$
$359$	−937903.	−0.384080	−0.192040	+	0.981387i	$0.561510\pi$
$360$	0	0
$361$	−784613.	−0.316875
$362$	272489.	0.109289
$363$	0	0
$364$	−2.75899e6	−1.09143
$365$	−2.58481e6	−1.01554
$366$	0	0
$367$	−4.14787e6	−1.60753	−0.803766	−	0.594946i	$-0.797174\pi$
$367$	−4.14787e6	−1.60753	−0.803766	+	0.594946i	$0.797174\pi$
$368$	−2.23968e6	−0.862118
$369$	0	0
$370$	−235121.	−0.0892868
$371$	1.41630e6	0.534222
$372$	0	0
$373$	−2.43392e6	−0.905805	−0.452902	−	0.891560i	$-0.649611\pi$
$373$	−2.43392e6	−0.905805	−0.452902	+	0.891560i	$0.649611\pi$
$374$	18447.0	0.00681939
$375$	0	0
$376$	−1.19710e6	−0.436677
$377$	7.37655e6	2.67301
$378$	0	0
$379$	−196709.	−0.0703440	−0.0351720	−	0.999381i	$-0.511198\pi$
$379$	−196709.	−0.0703440	−0.0351720	+	0.999381i	$0.511198\pi$
$380$	−1.35728e6	−0.482181
$381$	0	0
$382$	−270807.	−0.0949513
$383$	−3.93318e6	−1.37008	−0.685041	−	0.728504i	$-0.740216\pi$
$383$	−3.93318e6	−1.37008	−0.685041	+	0.728504i	$0.740216\pi$
$384$	0	0
$385$	96553.3	0.0331983
$386$	833331.	0.284675
$387$	0	0
$388$	2.75729e6	0.929828
$389$	−2.56292e6	−0.858739	−0.429369	−	0.903129i	$-0.641264\pi$
$389$	−2.56292e6	−0.858739	−0.429369	+	0.903129i	$0.641264\pi$
$390$	0	0
$391$	1.27569e6	0.421991
$392$	−630009.	−0.207077
$393$	0	0
$394$	1.00842e6	0.327266
$395$	−2.08093e6	−0.671066
$396$	0	0
$397$	−4.68608e6	−1.49222	−0.746110	−	0.665822i	$-0.768081\pi$
$397$	−4.68608e6	−1.49222	−0.746110	+	0.665822i	$0.768081\pi$
$398$	−734729.	−0.232498
$399$	0	0
$400$	−1.82028e6	−0.568837
$401$	−1.79185e6	−0.556468	−0.278234	−	0.960513i	$-0.589749\pi$
$401$	−1.79185e6	−0.556468	−0.278234	+	0.960513i	$0.589749\pi$
$402$	0	0
$403$	−2.62854e6	−0.806218
$404$	−777635.	−0.237040
$405$	0	0
$406$	−640016.	−0.192697
$407$	−219588.	−0.0657087
$408$	0	0
$409$	736446.	0.217687	0.108843	−	0.994059i	$-0.465285\pi$
$409$	736446.	0.217687	0.108843	+	0.994059i	$0.465285\pi$
$410$	−700874.	−0.205911
$411$	0	0
$412$	−4.13160e6	−1.19915
$413$	298052.	0.0859839
$414$	0	0
$415$	−398827.	−0.113675
$416$	3.23260e6	0.915837
$417$	0	0
$418$	45887.0	0.0128455
$419$	−5.71624e6	−1.59065	−0.795327	−	0.606181i	$-0.792701\pi$
$419$	−5.71624e6	−1.59065	−0.795327	+	0.606181i	$0.792701\pi$
$420$	0	0
$421$	−547892.	−0.150657	−0.0753286	−	0.997159i	$-0.524001\pi$
$421$	−547892.	−0.150657	−0.0753286	+	0.997159i	$0.524001\pi$
$422$	−639952.	−0.174931
$423$	0	0
$424$	−1.09977e6	−0.297088
$425$	1.03680e6	0.278435
$426$	0	0
$427$	2.49424e6	0.662016
$428$	−2.21334e6	−0.584034
$429$	0	0
$430$	657071.	0.171373
$431$	1.20482e6	0.312412	0.156206	−	0.987724i	$-0.450074\pi$
$431$	1.20482e6	0.312412	0.156206	+	0.987724i	$0.450074\pi$
$432$	0	0
$433$	−4.71811e6	−1.20934	−0.604669	−	0.796477i	$-0.706695\pi$
$433$	−4.71811e6	−1.20934	−0.604669	+	0.796477i	$0.706695\pi$
$434$	228062.	0.0581204
$435$	0	0
$436$	3.13328e6	0.789373
$437$	3.17330e6	0.794890
$438$	0	0
$439$	−3.47079e6	−0.859541	−0.429770	−	0.902938i	$-0.641406\pi$
$439$	−3.47079e6	−0.859541	−0.429770	+	0.902938i	$0.641406\pi$
$440$	−74974.1	−0.0184620
$441$	0	0
$442$	−576806.	−0.140435
$443$	2.07936e6	0.503409	0.251705	−	0.967804i	$-0.419009\pi$
$443$	2.07936e6	0.503409	0.251705	+	0.967804i	$0.419009\pi$
$444$	0	0
$445$	1.18099e6	0.282714
$446$	−1.42931e6	−0.340244
$447$	0	0
$448$	2.23458e6	0.526019
$449$	3.92965e6	0.919893	0.459947	−	0.887947i	$-0.347869\pi$
$449$	3.92965e6	0.919893	0.459947	+	0.887947i	$0.347869\pi$
$450$	0	0
$451$	−654571.	−0.151536
$452$	7.25468e6	1.67021
$453$	0	0
$454$	305640.	0.0695938
$455$	−3.01906e6	−0.683666
$456$	0	0
$457$	−490813.	−0.109932	−0.0549662	−	0.998488i	$-0.517505\pi$
$457$	−490813.	−0.109932	−0.0549662	+	0.998488i	$0.517505\pi$
$458$	−900806.	−0.200663
$459$	0	0
$460$	−2.54631e6	−0.561070
$461$	6.08206e6	1.33290	0.666452	−	0.745548i	$-0.267812\pi$
$461$	6.08206e6	1.33290	0.666452	+	0.745548i	$0.267812\pi$
$462$	0	0
$463$	8.92213e6	1.93427	0.967133	−	0.254271i	$-0.0818354\pi$
$463$	8.92213e6	1.93427	0.967133	+	0.254271i	$0.0818354\pi$
$464$	−6.48944e6	−1.39930
$465$	0	0
$466$	1.60404e6	0.342176
$467$	−1.91638e6	−0.406621	−0.203310	−	0.979114i	$-0.565170\pi$
$467$	−1.91638e6	−0.406621	−0.203310	+	0.979114i	$0.565170\pi$
$468$	0	0
$469$	1.98112e6	0.415890
$470$	−643327.	−0.134334
$471$	0	0
$472$	−231439.	−0.0478169
$473$	613663.	0.126118
$474$	0	0
$475$	2.57907e6	0.524479
$476$	−1.38249e6	−0.279670
$477$	0	0
$478$	−176568.	−0.0353461
$479$	2.38919e6	0.475786	0.237893	−	0.971291i	$-0.423543\pi$
$479$	2.38919e6	0.475786	0.237893	+	0.971291i	$0.423543\pi$
$480$	0	0
$481$	6.86616e6	1.35317
$482$	241686.	0.0473843
$483$	0	0
$484$	4.93920e6	0.958392
$485$	3.01720e6	0.582437
$486$	0	0
$487$	−876099.	−0.167390	−0.0836952	−	0.996491i	$-0.526672\pi$
$487$	−876099.	−0.167390	−0.0836952	+	0.996491i	$0.526672\pi$
$488$	−1.93679e6	−0.368156
$489$	0	0
$490$	−338570.	−0.0637027
$491$	−2.47004e6	−0.462381	−0.231190	−	0.972909i	$-0.574262\pi$
$491$	−2.47004e6	−0.462381	−0.231190	+	0.972909i	$0.574262\pi$
$492$	0	0
$493$	3.69629e6	0.684935
$494$	−1.43481e6	−0.264532
$495$	0	0
$496$	2.31243e6	0.422051
$497$	−3.04925e6	−0.553735
$498$	0	0
$499$	−1.91170e6	−0.343691	−0.171846	−	0.985124i	$-0.554973\pi$
$499$	−1.91170e6	−0.343691	−0.171846	+	0.985124i	$0.554973\pi$
$500$	−5.33075e6	−0.953593
$501$	0	0
$502$	1.42199e6	0.251848
$503$	−4.85098e6	−0.854888	−0.427444	−	0.904042i	$-0.640586\pi$
$503$	−4.85098e6	−0.854888	−0.427444	+	0.904042i	$0.640586\pi$
$504$	0	0
$505$	−850938.	−0.148480
$506$	86086.0	0.0149471
$507$	0	0
$508$	−891459.	−0.153265
$509$	−7.45465e6	−1.27536	−0.637680	−	0.770301i	$-0.720106\pi$
$509$	−7.45465e6	−1.27536	−0.637680	+	0.770301i	$0.720106\pi$
$510$	0	0
$511$	−6.54921e6	−1.10952
$512$	−4.79679e6	−0.808679
$513$	0	0
$514$	−204685.	−0.0341726
$515$	−4.52106e6	−0.751142
$516$	0	0
$517$	−600826.	−0.0988605
$518$	−595733.	−0.0975500
$519$	0	0
$520$	2.34432e6	0.380196
$521$	−3.66620e6	−0.591727	−0.295864	−	0.955230i	$-0.595607\pi$
$521$	−3.66620e6	−0.591727	−0.295864	+	0.955230i	$0.595607\pi$
$522$	0	0
$523$	9.66704e6	1.54539	0.772697	−	0.634774i	$-0.218907\pi$
$523$	9.66704e6	1.54539	0.772697	+	0.634774i	$0.218907\pi$
$524$	3.68784e6	0.586738
$525$	0	0
$526$	−1.36658e6	−0.215363
$527$	−1.31713e6	−0.206586
$528$	0	0
$529$	−483104.	−0.0750587
$530$	−591020.	−0.0913928
$531$	0	0
$532$	−3.43897e6	−0.526805
$533$	2.04674e7	3.12064
$534$	0	0
$535$	−2.42197e6	−0.365835
$536$	−1.53835e6	−0.231282
$537$	0	0
$538$	−2.12054e6	−0.315857
$539$	−316203.	−0.0468807
$540$	0	0
$541$	7.32547e6	1.07608	0.538038	−	0.842921i	$-0.319166\pi$
$541$	7.32547e6	1.07608	0.538038	+	0.842921i	$0.319166\pi$
$542$	−1.36215e6	−0.199171
$543$	0	0
$544$	1.61981e6	0.234675
$545$	3.42863e6	0.494458
$546$	0	0
$547$	6.36157e6	0.909067	0.454533	−	0.890730i	$-0.349806\pi$
$547$	6.36157e6	0.909067	0.454533	+	0.890730i	$0.349806\pi$
$548$	−6.17441e6	−0.878303
$549$	0	0
$550$	69965.5	0.00986228
$551$	9.19458e6	1.29019
$552$	0	0
$553$	−5.27251e6	−0.733170
$554$	−1.20112e6	−0.166269
$555$	0	0
$556$	−417277.	−0.0572449
$557$	−4.74906e6	−0.648589	−0.324295	−	0.945956i	$-0.605127\pi$
$557$	−4.74906e6	−0.648589	−0.324295	+	0.945956i	$0.605127\pi$
$558$	0	0
$559$	−1.91882e7	−2.59720
$560$	2.65599e6	0.357895
$561$	0	0
$562$	1.54124e6	0.205840
$563$	−3.15470e6	−0.419457	−0.209729	−	0.977760i	$-0.567258\pi$
$563$	−3.15470e6	−0.419457	−0.209729	+	0.977760i	$0.567258\pi$
$564$	0	0
$565$	7.93853e6	1.04621
$566$	2.64720e6	0.347332
$567$	0	0
$568$	2.36775e6	0.307940
$569$	−8.64099e6	−1.11888	−0.559439	−	0.828872i	$-0.688983\pi$
$569$	−8.64099e6	−1.11888	−0.559439	+	0.828872i	$0.688983\pi$
$570$	0	0
$571$	6.75178e6	0.866619	0.433310	−	0.901245i	$-0.357346\pi$
$571$	6.75178e6	0.866619	0.433310	+	0.901245i	$0.357346\pi$
$572$	1.07526e6	0.137411
$573$	0	0
$574$	−1.77582e6	−0.224967
$575$	4.83843e6	0.610288
$576$	0	0
$577$	−80122.8	−0.0100188	−0.00500941	−	0.999987i	$-0.501595\pi$
$577$	−80122.8	−0.0100188	−0.00500941	+	0.999987i	$0.501595\pi$
$578$	1.21221e6	0.150924
$579$	0	0
$580$	−7.37790e6	−0.910674
$581$	−1.01052e6	−0.124195
$582$	0	0
$583$	−551975.	−0.0672586
$584$	5.08549e6	0.617022
$585$	0	0
$586$	1.51383e6	0.182110
$587$	−8.20721e6	−0.983106	−0.491553	−	0.870848i	$-0.663571\pi$
$587$	−8.20721e6	−0.983106	−0.491553	+	0.870848i	$0.663571\pi$
$588$	0	0
$589$	−3.27638e6	−0.389140
$590$	−124376.	−0.0147098
$591$	0	0
$592$	−6.04043e6	−0.708375
$593$	−3.34469e6	−0.390588	−0.195294	−	0.980745i	$-0.562566\pi$
$593$	−3.34469e6	−0.390588	−0.195294	+	0.980745i	$0.562566\pi$
$594$	0	0
$595$	−1.51281e6	−0.175183
$596$	−1.38359e7	−1.59548
$597$	0	0
$598$	−2.69177e6	−0.307811
$599$	1.54909e7	1.76405	0.882024	−	0.471204i	$-0.156181\pi$
$599$	1.54909e7	1.76405	0.882024	+	0.471204i	$0.156181\pi$
$600$	0	0
$601$	−6.02417e6	−0.680317	−0.340158	−	0.940368i	$-0.610481\pi$
$601$	−6.02417e6	−0.680317	−0.340158	+	0.940368i	$0.610481\pi$
$602$	1.66484e6	0.187232
$603$	0	0
$604$	6.84941e6	0.763943
$605$	5.40479e6	0.600330
$606$	0	0
$607$	−1.80813e7	−1.99186	−0.995930	−	0.0901349i	$-0.971270\pi$
$607$	−1.80813e7	−1.99186	−0.995930	+	0.0901349i	$0.971270\pi$
$608$	4.02930e6	0.442050
$609$	0	0
$610$	−1.04084e6	−0.113255
$611$	1.87868e7	2.03587
$612$	0	0
$613$	1.63982e7	1.76256	0.881281	−	0.472593i	$-0.156682\pi$
$613$	1.63982e7	1.76256	0.881281	+	0.472593i	$0.156682\pi$
$614$	2.28930e6	0.245065
$615$	0	0
$616$	−189964.	−0.0201706
$617$	1.39449e7	1.47470	0.737350	−	0.675511i	$-0.236077\pi$
$617$	1.39449e7	1.47470	0.737350	+	0.675511i	$0.236077\pi$
$618$	0	0
$619$	3.47959e6	0.365007	0.182504	−	0.983205i	$-0.441580\pi$
$619$	3.47959e6	0.365007	0.182504	+	0.983205i	$0.441580\pi$
$620$	2.62902e6	0.274673
$621$	0	0
$622$	1.35839e6	0.140783
$623$	2.99232e6	0.308878
$624$	0	0
$625$	363714.	0.0372443
$626$	202902.	0.0206943
$627$	0	0
$628$	−4.19646e6	−0.424604
$629$	3.44054e6	0.346737
$630$	0	0
$631$	1.90874e6	0.190842	0.0954210	−	0.995437i	$-0.469580\pi$
$631$	1.90874e6	0.190842	0.0954210	+	0.995437i	$0.469580\pi$
$632$	4.09413e6	0.407726
$633$	0	0
$634$	2.97022e6	0.293471
$635$	−975491.	−0.0960039
$636$	0	0
$637$	9.88714e6	0.965432
$638$	249433.	0.0242606
$639$	0	0
$640$	−4.28271e6	−0.413303
$641$	−1.01908e7	−0.979632	−0.489816	−	0.871826i	$-0.662936\pi$
$641$	−1.01908e7	−0.979632	−0.489816	+	0.871826i	$0.662936\pi$
$642$	0	0
$643$	1.55416e7	1.48241	0.741205	−	0.671279i	$-0.234255\pi$
$643$	1.55416e7	1.48241	0.741205	+	0.671279i	$0.234255\pi$
$644$	−6.45166e6	−0.612995
$645$	0	0
$646$	−718966.	−0.0677839
$647$	3.01771e6	0.283411	0.141705	−	0.989909i	$-0.454741\pi$
$647$	3.01771e6	0.283411	0.141705	+	0.989909i	$0.454741\pi$
$648$	0	0
$649$	−116160.	−0.0108254
$650$	−2.18771e6	−0.203098
$651$	0	0
$652$	−9.31927e6	−0.858544
$653$	5.97669e6	0.548502	0.274251	−	0.961658i	$-0.411570\pi$
$653$	5.97669e6	0.548502	0.274251	+	0.961658i	$0.411570\pi$
$654$	0	0
$655$	4.03547e6	0.367528
$656$	−1.80059e7	−1.63364
$657$	0	0
$658$	−1.63001e6	−0.146766
$659$	3.91411e6	0.351091	0.175546	−	0.984471i	$-0.443831\pi$
$659$	3.91411e6	0.351091	0.175546	+	0.984471i	$0.443831\pi$
$660$	0	0
$661$	7.35500e6	0.654755	0.327378	−	0.944894i	$-0.393835\pi$
$661$	7.35500e6	0.654755	0.327378	+	0.944894i	$0.393835\pi$
$662$	110761.	0.00982293
$663$	0	0
$664$	784673.	0.0690667
$665$	−3.76314e6	−0.329987
$666$	0	0
$667$	1.72494e7	1.50127
$668$	1.76676e7	1.53192
$669$	0	0
$670$	−826715.	−0.0711490
$671$	−972077.	−0.0833478
$672$	0	0
$673$	−1.68761e6	−0.143627	−0.0718134	−	0.997418i	$-0.522879\pi$
$673$	−1.68761e6	−0.143627	−0.0718134	+	0.997418i	$0.522879\pi$
$674$	239605.	0.0203164
$675$	0	0
$676$	−2.21553e7	−1.86470
$677$	8.90194e6	0.746471	0.373236	−	0.927737i	$-0.378248\pi$
$677$	8.90194e6	0.746471	0.373236	+	0.927737i	$0.378248\pi$
$678$	0	0
$679$	7.64475e6	0.636340
$680$	1.17471e6	0.0974220
$681$	0	0
$682$	−88882.3	−0.00731736
$683$	−6.04714e6	−0.496019	−0.248010	−	0.968758i	$-0.579776\pi$
$683$	−6.04714e6	−0.496019	−0.248010	+	0.968758i	$0.579776\pi$
$684$	0	0
$685$	−6.75643e6	−0.550163
$686$	−2.37938e6	−0.193043
$687$	0	0
$688$	1.68806e7	1.35962
$689$	1.72593e7	1.38508
$690$	0	0
$691$	−3.65486e6	−0.291189	−0.145595	−	0.989344i	$-0.546509\pi$
$691$	−3.65486e6	−0.291189	−0.145595	+	0.989344i	$0.546509\pi$
$692$	−8.64199e6	−0.686039
$693$	0	0
$694$	−1.40339e6	−0.110606
$695$	−456610.	−0.0358578
$696$	0	0
$697$	1.02559e7	0.799637
$698$	340305.	0.0264381
$699$	0	0
$700$	−5.24352e6	−0.404462
$701$	1.63664e7	1.25793	0.628966	−	0.777432i	$-0.283478\pi$
$701$	1.63664e7	1.25793	0.628966	+	0.777432i	$0.283478\pi$
$702$	0	0
$703$	8.55840e6	0.653137
$704$	−870882.	−0.0662258
$705$	0	0
$706$	1.03896e6	0.0784493
$707$	−2.15604e6	−0.162222
$708$	0	0
$709$	−2.45108e7	−1.83123	−0.915613	−	0.402061i	$-0.868294\pi$
$709$	−2.45108e7	−1.83123	−0.915613	+	0.402061i	$0.868294\pi$
$710$	1.27244e6	0.0947310
$711$	0	0
$712$	−2.32355e6	−0.171772
$713$	−6.14662e6	−0.452806
$714$	0	0
$715$	1.17662e6	0.0860735
$716$	1.30266e7	0.949615
$717$	0	0
$718$	991661.	0.0717880
$719$	−1.16640e7	−0.841446	−0.420723	−	0.907189i	$-0.638224\pi$
$719$	−1.16640e7	−0.841446	−0.420723	+	0.907189i	$0.638224\pi$
$720$	0	0
$721$	−1.14551e7	−0.820657
$722$	829584.	0.0592267
$723$	0	0
$724$	7.95886e6	0.564293
$725$	1.40193e7	0.990560
$726$	0	0
$727$	1.76785e7	1.24054	0.620268	−	0.784390i	$-0.287024\pi$
$727$	1.76785e7	1.24054	0.620268	+	0.784390i	$0.287024\pi$
$728$	5.93985e6	0.415382
$729$	0	0
$730$	2.73297e6	0.189813
$731$	−9.61497e6	−0.665510
$732$	0	0
$733$	1.31251e6	0.0902279	0.0451140	−	0.998982i	$-0.485635\pi$
$733$	1.31251e6	0.0902279	0.0451140	+	0.998982i	$0.485635\pi$
$734$	4.38561e6	0.300462
$735$	0	0
$736$	7.55914e6	0.514373
$737$	−772099.	−0.0523606
$738$	0	0
$739$	2.62184e6	0.176602	0.0883010	−	0.996094i	$-0.471856\pi$
$739$	2.62184e6	0.176602	0.0883010	+	0.996094i	$0.471856\pi$
$740$	−6.86741e6	−0.461014
$741$	0	0
$742$	−1.49748e6	−0.0998509
$743$	−7.41645e6	−0.492861	−0.246430	−	0.969161i	$-0.579258\pi$
$743$	−7.41645e6	−0.492861	−0.246430	+	0.969161i	$0.579258\pi$
$744$	0	0
$745$	−1.51401e7	−0.999399
$746$	2.57343e6	0.169303
$747$	0	0
$748$	538798.	0.0352105
$749$	−6.13662e6	−0.399691
$750$	0	0
$751$	1.89341e7	1.22503	0.612514	−	0.790460i	$-0.290158\pi$
$751$	1.89341e7	1.22503	0.612514	+	0.790460i	$0.290158\pi$
$752$	−1.65275e7	−1.06577
$753$	0	0
$754$	−7.79935e6	−0.499609
$755$	7.49506e6	0.478528
$756$	0	0
$757$	−3.95313e6	−0.250727	−0.125364	−	0.992111i	$-0.540010\pi$
$757$	−3.95313e6	−0.250727	−0.125364	+	0.992111i	$0.540010\pi$
$758$	207984.	0.0131479
$759$	0	0
$760$	2.92210e6	0.183510
$761$	−2.27391e7	−1.42335	−0.711675	−	0.702509i	$-0.752063\pi$
$761$	−2.27391e7	−1.42335	−0.711675	+	0.702509i	$0.752063\pi$
$762$	0	0
$763$	8.68721e6	0.540218
$764$	−7.90971e6	−0.490261
$765$	0	0
$766$	4.15862e6	0.256081
$767$	3.63212e6	0.222931
$768$	0	0
$769$	−6.06544e6	−0.369867	−0.184934	−	0.982751i	$-0.559207\pi$
$769$	−6.06544e6	−0.369867	−0.184934	+	0.982751i	$0.559207\pi$
$770$	−102087.	−0.00620505
$771$	0	0
$772$	2.43399e7	1.46986
$773$	−2.81598e7	−1.69504	−0.847521	−	0.530762i	$-0.821906\pi$
$773$	−2.81598e7	−1.69504	−0.847521	+	0.530762i	$0.821906\pi$
$774$	0	0
$775$	−4.99560e6	−0.298768
$776$	−5.93618e6	−0.353878
$777$	0	0
$778$	2.70982e6	0.160506
$779$	2.55118e7	1.50625
$780$	0	0
$781$	1.18838e6	0.0697153
$782$	−1.34881e6	−0.0788740
$783$	0	0
$784$	−8.69810e6	−0.505399
$785$	−4.59203e6	−0.265969
$786$	0	0
$787$	1.35985e7	0.782627	0.391313	−	0.920257i	$-0.372021\pi$
$787$	1.35985e7	0.782627	0.391313	+	0.920257i	$0.372021\pi$
$788$	2.94539e7	1.68977
$789$	0	0
$790$	2.20020e6	0.125428
$791$	2.01141e7	1.14303
$792$	0	0
$793$	3.03952e7	1.71642
$794$	4.95467e6	0.278909
$795$	0	0
$796$	−2.14599e7	−1.20045
$797$	−3.47277e6	−0.193656	−0.0968279	−	0.995301i	$-0.530870\pi$
$797$	−3.47277e6	−0.193656	−0.0968279	+	0.995301i	$0.530870\pi$
$798$	0	0
$799$	9.41384e6	0.521675
$800$	6.14361e6	0.339390
$801$	0	0
$802$	1.89455e6	0.104009
$803$	2.55242e6	0.139689
$804$	0	0
$805$	−7.05981e6	−0.383976
$806$	2.77920e6	0.150689
$807$	0	0
$808$	1.67418e6	0.0902138
$809$	4.11626e6	0.221122	0.110561	−	0.993869i	$-0.464735\pi$
$809$	4.11626e6	0.221122	0.110561	+	0.993869i	$0.464735\pi$
$810$	0	0
$811$	−1.46103e7	−0.780022	−0.390011	−	0.920810i	$-0.627529\pi$
$811$	−1.46103e7	−0.780022	−0.390011	+	0.920810i	$0.627529\pi$
$812$	−1.86936e7	−0.994953
$813$	0	0
$814$	232174.	0.0122815
$815$	−1.01977e7	−0.537786
$816$	0	0
$817$	−2.39174e7	−1.25360
$818$	−778656.	−0.0406876
$819$	0	0
$820$	−2.04711e7	−1.06318
$821$	2.36586e7	1.22498	0.612492	−	0.790477i	$-0.290167\pi$
$821$	2.36586e7	1.22498	0.612492	+	0.790477i	$0.290167\pi$
$822$	0	0
$823$	2.73951e6	0.140985	0.0704925	−	0.997512i	$-0.477543\pi$
$823$	2.73951e6	0.140985	0.0704925	+	0.997512i	$0.477543\pi$
$824$	8.89495e6	0.456379
$825$	0	0
$826$	−315135.	−0.0160712
$827$	−2.71855e7	−1.38221	−0.691103	−	0.722756i	$-0.742875\pi$
$827$	−2.71855e7	−1.38221	−0.691103	+	0.722756i	$0.742875\pi$
$828$	0	0
$829$	−2.81195e6	−0.142109	−0.0710545	−	0.997472i	$-0.522636\pi$
$829$	−2.81195e6	−0.142109	−0.0710545	+	0.997472i	$0.522636\pi$
$830$	421687.	0.0212469
$831$	0	0
$832$	2.72310e7	1.36382
$833$	4.95432e6	0.247384
$834$	0	0
$835$	1.93330e7	0.959583
$836$	1.34027e6	0.0663248
$837$	0	0
$838$	6.04388e6	0.297308
$839$	2.10069e6	0.103028	0.0515142	−	0.998672i	$-0.483595\pi$
$839$	2.10069e6	0.103028	0.0515142	+	0.998672i	$0.483595\pi$
$840$	0	0
$841$	2.94688e7	1.43672
$842$	579296.	0.0281592
$843$	0	0
$844$	−1.86917e7	−0.903219
$845$	−2.42437e7	−1.16804
$846$	0	0
$847$	1.36943e7	0.655888
$848$	−1.51837e7	−0.725084
$849$	0	0
$850$	−1.09623e6	−0.0520421
$851$	1.60559e7	0.759995
$852$	0	0
$853$	1.91254e7	0.899991	0.449996	−	0.893031i	$-0.351426\pi$
$853$	1.91254e7	0.899991	0.449996	+	0.893031i	$0.351426\pi$
$854$	−2.63720e6	−0.123737
$855$	0	0
$856$	4.76511e6	0.222274
$857$	1.08164e7	0.503072	0.251536	−	0.967848i	$-0.419064\pi$
$857$	1.08164e7	0.503072	0.251536	+	0.967848i	$0.419064\pi$
$858$	0	0
$859$	−4.44481e6	−0.205528	−0.102764	−	0.994706i	$-0.532769\pi$
$859$	−4.44481e6	−0.205528	−0.102764	+	0.994706i	$0.532769\pi$
$860$	1.91917e7	0.884846
$861$	0	0
$862$	−1.27387e6	−0.0583926
$863$	−3.43899e7	−1.57183	−0.785913	−	0.618337i	$-0.787807\pi$
$863$	−3.43899e7	−1.57183	−0.785913	+	0.618337i	$0.787807\pi$
$864$	0	0
$865$	−9.45661e6	−0.429730
$866$	4.98853e6	0.226036
$867$	0	0
$868$	6.66123e6	0.300092
$869$	2.05485e6	0.0923062
$870$	0	0
$871$	2.41423e7	1.07828
$872$	−6.74566e6	−0.300423
$873$	0	0
$874$	−3.35518e6	−0.148572
$875$	−1.47798e7	−0.652604
$876$	0	0
$877$	2.34437e7	1.02926	0.514632	−	0.857411i	$-0.327929\pi$
$877$	2.34437e7	1.02926	0.514632	+	0.857411i	$0.327929\pi$
$878$	3.66972e6	0.160656
$879$	0	0
$880$	−1.03511e6	−0.0450590
$881$	−1.52635e7	−0.662542	−0.331271	−	0.943536i	$-0.607477\pi$
$881$	−1.52635e7	−0.662542	−0.331271	+	0.943536i	$0.607477\pi$
$882$	0	0
$883$	1.88369e7	0.813034	0.406517	−	0.913643i	$-0.366743\pi$
$883$	1.88369e7	0.813034	0.406517	+	0.913643i	$0.366743\pi$
$884$	−1.68473e7	−0.725104
$885$	0	0
$886$	−2.19855e6	−0.0940917
$887$	−3.45462e7	−1.47432	−0.737160	−	0.675718i	$-0.763834\pi$
$887$	−3.45462e7	−1.47432	−0.737160	+	0.675718i	$0.763834\pi$
$888$	0	0
$889$	−2.47163e6	−0.104889
$890$	−1.24868e6	−0.0528418
$891$	0	0
$892$	−4.17474e7	−1.75678
$893$	2.34171e7	0.982661
$894$	0	0
$895$	1.42545e7	0.594832
$896$	−1.08512e7	−0.451553
$897$	0	0
$898$	−4.15488e6	−0.171936
$899$	−1.78097e7	−0.734951
$900$	0	0
$901$	8.64843e6	0.354916
$902$	692089.	0.0283234
$903$	0	0
$904$	−1.56186e7	−0.635656
$905$	8.70909e6	0.353469
$906$	0	0
$907$	−1.49504e7	−0.603440	−0.301720	−	0.953397i	$-0.597561\pi$
$907$	−1.49504e7	−0.603440	−0.301720	+	0.953397i	$0.597561\pi$
$908$	8.92713e6	0.359333
$909$	0	0
$910$	3.19211e6	0.127783
$911$	4.82835e7	1.92754	0.963768	−	0.266741i	$-0.0859467\pi$
$911$	4.82835e7	1.92754	0.963768	+	0.266741i	$0.0859467\pi$
$912$	0	0
$913$	393829.	0.0156362
$914$	518945.	0.0205473
$915$	0	0
$916$	−2.63107e7	−1.03608
$917$	1.02248e7	0.401541
$918$	0	0
$919$	2.12186e7	0.828758	0.414379	−	0.910105i	$-0.363999\pi$
$919$	2.12186e7	0.828758	0.414379	+	0.910105i	$0.363999\pi$
$920$	5.48198e6	0.213534
$921$	0	0
$922$	−6.43067e6	−0.249132
$923$	−3.71587e7	−1.43567
$924$	0	0
$925$	1.30493e7	0.501455
$926$	−9.43352e6	−0.361532
$927$	0	0
$928$	2.19025e7	0.834879
$929$	−1.47484e6	−0.0560667	−0.0280333	−	0.999607i	$-0.508924\pi$
$929$	−1.47484e6	−0.0560667	−0.0280333	+	0.999607i	$0.508924\pi$
$930$	0	0
$931$	1.23239e7	0.465988
$932$	4.68507e7	1.76676
$933$	0	0
$934$	2.02622e6	0.0760011
$935$	589587.	0.0220556
$936$	0	0
$937$	−2.87501e7	−1.06977	−0.534885	−	0.844925i	$-0.679645\pi$
$937$	−2.87501e7	−1.06977	−0.534885	+	0.844925i	$0.679645\pi$
$938$	−2.09467e6	−0.0777335
$939$	0	0
$940$	−1.87903e7	−0.693607
$941$	−4.42692e7	−1.62978	−0.814888	−	0.579618i	$-0.803202\pi$
$941$	−4.42692e7	−1.62978	−0.814888	+	0.579618i	$0.803202\pi$
$942$	0	0
$943$	4.78611e7	1.75268
$944$	−3.19531e6	−0.116703
$945$	0	0
$946$	−648836.	−0.0235726
$947$	8.37611e6	0.303506	0.151753	−	0.988418i	$-0.451508\pi$
$947$	8.37611e6	0.303506	0.151753	+	0.988418i	$0.451508\pi$
$948$	0	0
$949$	−7.98098e7	−2.87668
$950$	−2.72689e6	−0.0980299
$951$	0	0
$952$	2.97638e6	0.106438
$953$	−1.04083e7	−0.371232	−0.185616	−	0.982622i	$-0.559428\pi$
$953$	−1.04083e7	−0.371232	−0.185616	+	0.982622i	$0.559428\pi$
$954$	0	0
$955$	−8.65531e6	−0.307096
$956$	−5.15719e6	−0.182502
$957$	0	0
$958$	−2.52613e6	−0.0889287
$959$	−1.71190e7	−0.601078
$960$	0	0
$961$	−2.22829e7	−0.778328
$962$	−7.25970e6	−0.252919
$963$	0	0
$964$	7.05917e6	0.244659
$965$	2.66343e7	0.920709
$966$	0	0
$967$	−3.41560e7	−1.17463	−0.587315	−	0.809358i	$-0.699815\pi$
$967$	−3.41560e7	−1.17463	−0.587315	+	0.809358i	$0.699815\pi$
$968$	−1.06336e7	−0.364749
$969$	0	0
$970$	−3.19013e6	−0.108863
$971$	2.80041e7	0.953176	0.476588	−	0.879127i	$-0.341873\pi$
$971$	2.80041e7	0.953176	0.476588	+	0.879127i	$0.341873\pi$
$972$	0	0
$973$	−1.15693e6	−0.0391763
$974$	926314.	0.0312868
$975$	0	0
$976$	−2.67399e7	−0.898534
$977$	4.46833e6	0.149764	0.0748822	−	0.997192i	$-0.476142\pi$
$977$	4.46833e6	0.149764	0.0748822	+	0.997192i	$0.476142\pi$
$978$	0	0
$979$	−1.16619e6	−0.0388878
$980$	−9.88894e6	−0.328916
$981$	0	0
$982$	2.61161e6	0.0864231
$983$	2.39250e7	0.789711	0.394856	−	0.918743i	$-0.370795\pi$
$983$	2.39250e7	0.789711	0.394856	+	0.918743i	$0.370795\pi$
$984$	0	0
$985$	3.22303e7	1.05846
$986$	−3.90815e6	−0.128020
$987$	0	0
$988$	−4.19079e7	−1.36585
$989$	−4.48700e7	−1.45870
$990$	0	0
$991$	−7.93514e6	−0.256667	−0.128334	−	0.991731i	$-0.540963\pi$
$991$	−7.93514e6	−0.256667	−0.128334	+	0.991731i	$0.540963\pi$
$992$	−7.80469e6	−0.251812
$993$	0	0
$994$	3.22402e6	0.103498
$995$	−2.34828e7	−0.751956
$996$	0	0
$997$	−7.07884e6	−0.225540	−0.112770	−	0.993621i	$-0.535972\pi$
$997$	−7.07884e6	−0.225540	−0.112770	+	0.993621i	$0.535972\pi$
$998$	2.02127e6	0.0642390
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.e.1.7		13
3.2	odd	2		177.6.a.d.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.d.1.7	✓	13	3.2	odd	2
531.6.a.e.1.7		13	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)

\(f(q)\)	\(=\)	\(q-1.05732 q^{2} -30.8821 q^{4} -33.7931 q^{5} -85.6226 q^{7} +66.4863 q^{8} +O(q^{10})\)	\(q-1.05732 q^{2} -30.8821 q^{4} -33.7931 q^{5} -85.6226 q^{7} +66.4863 q^{8} +35.7300 q^{10} +33.3696 q^{11} -1043.41 q^{13} +90.5302 q^{14} +917.930 q^{16} -522.840 q^{17} -1300.57 q^{19} +1043.60 q^{20} -35.2822 q^{22} -2439.93 q^{23} -1983.02 q^{25} +1103.22 q^{26} +2644.20 q^{28} -7069.65 q^{29} +2519.18 q^{31} -3098.10 q^{32} +552.807 q^{34} +2893.45 q^{35} -6580.49 q^{37} +1375.12 q^{38} -2246.78 q^{40} -19615.8 q^{41} +18389.9 q^{43} -1030.52 q^{44} +2579.77 q^{46} -18005.2 q^{47} -9475.78 q^{49} +2096.69 q^{50} +32222.7 q^{52} -16541.3 q^{53} -1127.66 q^{55} -5692.73 q^{56} +7474.86 q^{58} -3481.00 q^{59} -29130.6 q^{61} -2663.57 q^{62} -26098.1 q^{64} +35260.1 q^{65} -23137.8 q^{67} +16146.4 q^{68} -3059.30 q^{70} +35612.7 q^{71} +76489.3 q^{73} +6957.66 q^{74} +40164.4 q^{76} -2857.19 q^{77} +61578.6 q^{79} -31019.7 q^{80} +20740.1 q^{82} +11802.0 q^{83} +17668.4 q^{85} -19443.9 q^{86} +2218.62 q^{88} -34947.8 q^{89} +89339.6 q^{91} +75350.0 q^{92} +19037.2 q^{94} +43950.4 q^{95} -89284.3 q^{97} +10018.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 246 q^{4} + 14 q^{5} + 373 q^{7} - 123 q^{8}+O(q^{10}) \) 13 * q + 246 * q^4 + 14 * q^5 + 373 * q^7 - 123 * q^8	\( 13 q + 246 q^{4} + 14 q^{5} + 373 q^{7} - 123 q^{8} + 137 q^{10} - 250 q^{11} + 1054 q^{13} + 575 q^{14} + 922 q^{16} - 271 q^{17} + 671 q^{19} + 5491 q^{20} + 1094 q^{22} - 3975 q^{23} + 15569 q^{25} - 4622 q^{26} + 21214 q^{28} + 10613 q^{29} + 25597 q^{31} - 15966 q^{32} + 31796 q^{34} - 6729 q^{35} + 17585 q^{37} - 34903 q^{38} + 31382 q^{40} - 12537 q^{41} + 26644 q^{43} - 6654 q^{44} + 149005 q^{46} - 52087 q^{47} + 95384 q^{49} - 121821 q^{50} + 263630 q^{52} - 20014 q^{53} + 120932 q^{55} - 126688 q^{56} + 86066 q^{58} - 45253 q^{59} - 11667 q^{61} - 164794 q^{62} + 151893 q^{64} + 28674 q^{65} + 1106 q^{67} + 4043 q^{68} + 56066 q^{70} - 21230 q^{71} + 81131 q^{73} - 102042 q^{74} + 73900 q^{76} + 104655 q^{77} - 13470 q^{79} + 191969 q^{80} + 79909 q^{82} + 76149 q^{83} + 10035 q^{85} + 321496 q^{86} - 276779 q^{88} + 190205 q^{89} + 80601 q^{91} - 45672 q^{92} + 36768 q^{94} - 9875 q^{95} + 160850 q^{97} + 116644 q^{98}+O(q^{100}) \) 13 * q + 246 * q^4 + 14 * q^5 + 373 * q^7 - 123 * q^8 + 137 * q^10 - 250 * q^11 + 1054 * q^13 + 575 * q^14 + 922 * q^16 - 271 * q^17 + 671 * q^19 + 5491 * q^20 + 1094 * q^22 - 3975 * q^23 + 15569 * q^25 - 4622 * q^26 + 21214 * q^28 + 10613 * q^29 + 25597 * q^31 - 15966 * q^32 + 31796 * q^34 - 6729 * q^35 + 17585 * q^37 - 34903 * q^38 + 31382 * q^40 - 12537 * q^41 + 26644 * q^43 - 6654 * q^44 + 149005 * q^46 - 52087 * q^47 + 95384 * q^49 - 121821 * q^50 + 263630 * q^52 - 20014 * q^53 + 120932 * q^55 - 126688 * q^56 + 86066 * q^58 - 45253 * q^59 - 11667 * q^61 - 164794 * q^62 + 151893 * q^64 + 28674 * q^65 + 1106 * q^67 + 4043 * q^68 + 56066 * q^70 - 21230 * q^71 + 81131 * q^73 - 102042 * q^74 + 73900 * q^76 + 104655 * q^77 - 13470 * q^79 + 191969 * q^80 + 79909 * q^82 + 76149 * q^83 + 10035 * q^85 + 321496 * q^86 - 276779 * q^88 + 190205 * q^89 + 80601 * q^91 - 45672 * q^92 + 36768 * q^94 - 9875 * q^95 + 160850 * q^97 + 116644 * q^98

Introduction

L-functions

Modular forms

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Database

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