LMFDB - Embedded newform 531.6.a.e.1.1

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			1.1
Root			$10.6702$ 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)

$f(q)$	$=$	$q-10.6702 q^{2} +81.8525 q^{4} +61.2778 q^{5} +186.685 q^{7} -531.935 q^{8} +O(q^{10})$	\(q-10.6702 q^{2} +81.8525 q^{4} +61.2778 q^{5} +186.685 q^{7} -531.935 q^{8} -653.844 q^{10} +424.892 q^{11} +983.052 q^{13} -1991.96 q^{14} +3056.56 q^{16} -1043.17 q^{17} -302.603 q^{19} +5015.74 q^{20} -4533.67 q^{22} -4122.49 q^{23} +629.966 q^{25} -10489.3 q^{26} +15280.7 q^{28} +8426.58 q^{29} +10049.9 q^{31} -15592.1 q^{32} +11130.8 q^{34} +11439.6 q^{35} +10289.5 q^{37} +3228.83 q^{38} -32595.8 q^{40} -1605.50 q^{41} -4917.05 q^{43} +34778.5 q^{44} +43987.7 q^{46} +24046.6 q^{47} +18044.3 q^{49} -6721.85 q^{50} +80465.3 q^{52} +6856.60 q^{53} +26036.4 q^{55} -99304.4 q^{56} -89913.0 q^{58} -3481.00 q^{59} -31403.4 q^{61} -107234. q^{62} +68560.1 q^{64} +60239.3 q^{65} -61147.0 q^{67} -85386.3 q^{68} -122063. q^{70} +7454.90 q^{71} -4324.62 q^{73} -109790. q^{74} -24768.9 q^{76} +79321.0 q^{77} -30676.1 q^{79} +187299. q^{80} +17130.9 q^{82} +29240.6 q^{83} -63923.3 q^{85} +52465.8 q^{86} -226015. q^{88} -23916.5 q^{89} +183521. q^{91} -337436. q^{92} -256582. q^{94} -18542.9 q^{95} +31653.6 q^{97} -192536. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−10.6702	−1.88624	−0.943119	−	0.332456i	$-0.892123\pi$
$2$	−10.6702	−1.88624	−0.943119	+	0.332456i	$0.892123\pi$
$3$	0	0
$3$	0	0
$4$	81.8525	2.55789
$5$	61.2778	1.09617	0.548085	−	0.836423i	$-0.315357\pi$
$5$	61.2778	1.09617	0.548085	+	0.836423i	$0.315357\pi$
$6$	0	0
$7$	186.685	1.44001	0.720004	−	0.693970i	$-0.244140\pi$
$7$	186.685	1.44001	0.720004	+	0.693970i	$0.244140\pi$
$8$	−531.935	−2.93855
$9$	0	0
$10$	−653.844	−2.06764
$11$	424.892	1.05876	0.529379	−	0.848385i	$-0.322425\pi$
$11$	424.892	1.05876	0.529379	+	0.848385i	$0.322425\pi$
$12$	0	0
$13$	983.052	1.61331	0.806656	−	0.591021i	$-0.201275\pi$
$13$	983.052	1.61331	0.806656	+	0.591021i	$0.201275\pi$
$14$	−1991.96	−2.71620
$15$	0	0
$16$	3056.56	2.98492
$17$	−1043.17	−0.875455	−0.437727	−	0.899108i	$-0.644217\pi$
$17$	−1043.17	−0.875455	−0.437727	+	0.899108i	$0.644217\pi$
$18$	0	0
$19$	−302.603	−0.192305	−0.0961523	−	0.995367i	$-0.530654\pi$
$19$	−302.603	−0.192305	−0.0961523	+	0.995367i	$0.530654\pi$
$20$	5015.74	2.80389
$21$	0	0
$22$	−4533.67	−1.99707
$23$	−4122.49	−1.62495	−0.812475	−	0.582996i	$-0.801880\pi$
$23$	−4122.49	−1.62495	−0.812475	+	0.582996i	$0.801880\pi$
$24$	0	0
$25$	629.966	0.201589
$26$	−10489.3	−3.04309
$27$	0	0
$28$	15280.7	3.68338
$29$	8426.58	1.86061	0.930307	−	0.366782i	$-0.119540\pi$
$29$	8426.58	1.86061	0.930307	+	0.366782i	$0.119540\pi$
$30$	0	0
$31$	10049.9	1.87827	0.939137	−	0.343544i	$-0.111628\pi$
$31$	10049.9	1.87827	0.939137	+	0.343544i	$0.111628\pi$
$32$	−15592.1	−2.69171
$33$	0	0
$34$	11130.8	1.65132
$35$	11439.6	1.57849
$36$	0	0
$37$	10289.5	1.23563	0.617815	−	0.786323i	$-0.288018\pi$
$37$	10289.5	1.23563	0.617815	+	0.786323i	$0.288018\pi$
$38$	3228.83	0.362732
$39$	0	0
$40$	−32595.8	−3.22116
$41$	−1605.50	−0.149159	−0.0745796	−	0.997215i	$-0.523761\pi$
$41$	−1605.50	−0.149159	−0.0745796	+	0.997215i	$0.523761\pi$
$42$	0	0
$43$	−4917.05	−0.405540	−0.202770	−	0.979226i	$-0.564994\pi$
$43$	−4917.05	−0.405540	−0.202770	+	0.979226i	$0.564994\pi$
$44$	34778.5	2.70819
$45$	0	0
$46$	43987.7	3.06504
$47$	24046.6	1.58785	0.793926	−	0.608015i	$-0.208034\pi$
$47$	24046.6	1.58785	0.793926	+	0.608015i	$0.208034\pi$
$48$	0	0
$49$	18044.3	1.07362
$50$	−6721.85	−0.380245
$51$	0	0
$52$	80465.3	4.12668
$53$	6856.60	0.335289	0.167644	−	0.985848i	$-0.446384\pi$
$53$	6856.60	0.335289	0.167644	+	0.985848i	$0.446384\pi$
$54$	0	0
$55$	26036.4	1.16058
$56$	−99304.4	−4.23154
$57$	0	0
$58$	−89913.0	−3.50956
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	−31403.4	−1.08057	−0.540283	−	0.841483i	$-0.681683\pi$
$61$	−31403.4	−1.08057	−0.540283	+	0.841483i	$0.681683\pi$
$62$	−107234.	−3.54287
$63$	0	0
$64$	68560.1	2.09229
$65$	60239.3	1.76846
$66$	0	0
$67$	−61147.0	−1.66413	−0.832066	−	0.554676i	$-0.812842\pi$
$67$	−61147.0	−1.66413	−0.832066	+	0.554676i	$0.812842\pi$
$68$	−85386.3	−2.23932
$69$	0	0
$70$	−122063.	−2.97741
$71$	7454.90	0.175508	0.0877538	−	0.996142i	$-0.472031\pi$
$71$	7454.90	0.175508	0.0877538	+	0.996142i	$0.472031\pi$
$72$	0	0
$73$	−4324.62	−0.0949820	−0.0474910	−	0.998872i	$-0.515123\pi$
$73$	−4324.62	−0.0949820	−0.0474910	+	0.998872i	$0.515123\pi$
$74$	−109790.	−2.33069
$75$	0	0
$76$	−24768.9	−0.491894
$77$	79321.0	1.52462
$78$	0	0
$79$	−30676.1	−0.553010	−0.276505	−	0.961013i	$-0.589176\pi$
$79$	−30676.1	−0.553010	−0.276505	+	0.961013i	$0.589176\pi$
$80$	187299.	3.27198
$81$	0	0
$82$	17130.9	0.281350
$83$	29240.6	0.465898	0.232949	−	0.972489i	$-0.425163\pi$
$83$	29240.6	0.465898	0.232949	+	0.972489i	$0.425163\pi$
$84$	0	0
$85$	−63923.3	−0.959648
$86$	52465.8	0.764944
$87$	0	0
$88$	−226015.	−3.11122
$89$	−23916.5	−0.320053	−0.160027	−	0.987113i	$-0.551158\pi$
$89$	−23916.5	−0.320053	−0.160027	+	0.987113i	$0.551158\pi$
$90$	0	0
$91$	183521.	2.32318
$92$	−337436.	−4.15645
$93$	0	0
$94$	−256582.	−2.99506
$95$	−18542.9	−0.210799
$96$	0	0
$97$	31653.6	0.341581	0.170790	−	0.985307i	$-0.445368\pi$
$97$	31653.6	0.341581	0.170790	+	0.985307i	$0.445368\pi$
$98$	−192536.	−2.02510
$99$	0	0
$100$	51564.3	0.515643
$101$	90035.5	0.878234	0.439117	−	0.898430i	$-0.355291\pi$
$101$	90035.5	0.878234	0.439117	+	0.898430i	$0.355291\pi$
$102$	0	0
$103$	98739.3	0.917059	0.458529	−	0.888679i	$-0.348376\pi$
$103$	98739.3	0.917059	0.458529	+	0.888679i	$0.348376\pi$
$104$	−522920.	−4.74080
$105$	0	0
$106$	−73161.1	−0.632435
$107$	98628.3	0.832803	0.416401	−	0.909181i	$-0.363291\pi$
$107$	98628.3	0.832803	0.416401	+	0.909181i	$0.363291\pi$
$108$	0	0
$109$	57081.3	0.460180	0.230090	−	0.973169i	$-0.426098\pi$
$109$	57081.3	0.460180	0.230090	+	0.973169i	$0.426098\pi$
$110$	−277813.	−2.18913
$111$	0	0
$112$	570614.	4.29830
$113$	−70846.0	−0.521938	−0.260969	−	0.965347i	$-0.584042\pi$
$113$	−70846.0	−0.521938	−0.260969	+	0.965347i	$0.584042\pi$
$114$	0	0
$115$	−252617.	−1.78122
$116$	689737.	4.75925
$117$	0	0
$118$	37142.9	0.245567
$119$	−194745.	−1.26066
$120$	0	0
$121$	19482.1	0.120969
$122$	335079.	2.03821
$123$	0	0
$124$	822612.	4.80442
$125$	−152890.	−0.875194
$126$	0	0
$127$	−347695.	−1.91289	−0.956444	−	0.291916i	$-0.905707\pi$
$127$	−347695.	−1.91289	−0.956444	+	0.291916i	$0.905707\pi$
$128$	−232602.	−1.25484
$129$	0	0
$130$	−642763.	−3.33574
$131$	−107422.	−0.546909	−0.273455	−	0.961885i	$-0.588166\pi$
$131$	−107422.	−0.546909	−0.273455	+	0.961885i	$0.588166\pi$
$132$	0	0
$133$	−56491.5	−0.276920
$134$	652449.	3.13895
$135$	0	0
$136$	554900.	2.57257
$137$	−191277.	−0.870686	−0.435343	−	0.900265i	$-0.643373\pi$
$137$	−191277.	−0.870686	−0.435343	+	0.900265i	$0.643373\pi$
$138$	0	0
$139$	23303.1	0.102300	0.0511502	−	0.998691i	$-0.483711\pi$
$139$	23303.1	0.102300	0.0511502	+	0.998691i	$0.483711\pi$
$140$	936364.	4.03761
$141$	0	0
$142$	−79545.1	−0.331049
$143$	417691.	1.70811
$144$	0	0
$145$	516362.	2.03955
$146$	46144.5	0.179159
$147$	0	0
$148$	842219.	3.16061
$149$	16697.0	0.0616132	0.0308066	−	0.999525i	$-0.490192\pi$
$149$	16697.0	0.0616132	0.0308066	+	0.999525i	$0.490192\pi$
$150$	0	0
$151$	−141376.	−0.504583	−0.252291	−	0.967651i	$-0.581184\pi$
$151$	−141376.	−0.504583	−0.252291	+	0.967651i	$0.581184\pi$
$152$	160965.	0.565097
$153$	0	0
$154$	−846369.	−2.87579
$155$	615837.	2.05891
$156$	0	0
$157$	−6851.37	−0.0221834	−0.0110917	−	0.999938i	$-0.503531\pi$
$157$	−6851.37	−0.0221834	−0.0110917	+	0.999938i	$0.503531\pi$
$158$	327319.	1.04311
$159$	0	0
$160$	−955447.	−2.95057
$161$	−769608.	−2.33994
$162$	0	0
$163$	−275831.	−0.813158	−0.406579	−	0.913616i	$-0.633278\pi$
$163$	−275831.	−0.813158	−0.406579	+	0.913616i	$0.633278\pi$
$164$	−131414.	−0.381533
$165$	0	0
$166$	−312002.	−0.878794
$167$	−546461.	−1.51624	−0.758120	−	0.652115i	$-0.773882\pi$
$167$	−546461.	−1.51624	−0.758120	+	0.652115i	$0.773882\pi$
$168$	0	0
$169$	595099.	1.60277
$170$	682072.	1.81012
$171$	0	0
$172$	−402473.	−1.03733
$173$	563301.	1.43095	0.715476	−	0.698637i	$-0.246210\pi$
$173$	563301.	1.43095	0.715476	+	0.698637i	$0.246210\pi$
$174$	0	0
$175$	117605.	0.290290
$176$	1.29871e6	3.16031
$177$	0	0
$178$	255193.	0.603696
$179$	−603180.	−1.40707	−0.703533	−	0.710663i	$-0.748395\pi$
$179$	−603180.	−1.40707	−0.703533	+	0.710663i	$0.748395\pi$
$180$	0	0
$181$	158219.	0.358973	0.179487	−	0.983760i	$-0.442556\pi$
$181$	158219.	0.358973	0.179487	+	0.983760i	$0.442556\pi$
$182$	−1.95820e6	−4.38207
$183$	0	0
$184$	2.19290e6	4.77501
$185$	630516.	1.35446
$186$	0	0
$187$	−443235.	−0.926895
$188$	1.96828e6	4.06155
$189$	0	0
$190$	197855.	0.397616
$191$	512314.	1.01614	0.508070	−	0.861316i	$-0.330359\pi$
$191$	512314.	1.01614	0.508070	+	0.861316i	$0.330359\pi$
$192$	0	0
$193$	−779315.	−1.50598	−0.752991	−	0.658030i	$-0.771390\pi$
$193$	−779315.	−1.50598	−0.752991	+	0.658030i	$0.771390\pi$
$194$	−337749.	−0.644302
$195$	0	0
$196$	1.47697e6	2.74620
$197$	375338.	0.689059	0.344530	−	0.938775i	$-0.388038\pi$
$197$	375338.	0.689059	0.344530	+	0.938775i	$0.388038\pi$
$198$	0	0
$199$	373163.	0.667984	0.333992	−	0.942576i	$-0.391604\pi$
$199$	373163.	0.667984	0.333992	+	0.942576i	$0.391604\pi$
$200$	−335101.	−0.592381
$201$	0	0
$202$	−960694.	−1.65656
$203$	1.57312e6	2.67930
$204$	0	0
$205$	−98381.4	−0.163504
$206$	−1.05357e6	−1.72979
$207$	0	0
$208$	3.00476e6	4.81560
$209$	−128574.	−0.203604
$210$	0	0
$211$	968282.	1.49725	0.748627	−	0.662991i	$-0.230713\pi$
$211$	968282.	1.49725	0.748627	+	0.662991i	$0.230713\pi$
$212$	561230.	0.857633
$213$	0	0
$214$	−1.05238e6	−1.57086
$215$	−301306.	−0.444541
$216$	0	0
$217$	1.87617e6	2.70473
$218$	−609068.	−0.868009
$219$	0	0
$220$	2.13115e6	2.96864
$221$	−1.02549e6	−1.41238
$222$	0	0
$223$	−423194.	−0.569872	−0.284936	−	0.958547i	$-0.591972\pi$
$223$	−423194.	−0.569872	−0.284936	+	0.958547i	$0.591972\pi$
$224$	−2.91081e6	−3.87608
$225$	0	0
$226$	755939.	0.984500
$227$	−185430.	−0.238845	−0.119423	−	0.992844i	$-0.538104\pi$
$227$	−185430.	−0.238845	−0.119423	+	0.992844i	$0.538104\pi$
$228$	0	0
$229$	418097.	0.526851	0.263426	−	0.964680i	$-0.415148\pi$
$229$	418097.	0.526851	0.263426	+	0.964680i	$0.415148\pi$
$230$	2.69547e6	3.35981
$231$	0	0
$232$	−4.48239e6	−5.46751
$233$	−137239.	−0.165611	−0.0828055	−	0.996566i	$-0.526388\pi$
$233$	−137239.	−0.165611	−0.0828055	+	0.996566i	$0.526388\pi$
$234$	0	0
$235$	1.47352e6	1.74056
$236$	−284929.	−0.333009
$237$	0	0
$238$	2.07796e6	2.37791
$239$	207578.	0.235064	0.117532	−	0.993069i	$-0.462502\pi$
$239$	207578.	0.235064	0.117532	+	0.993069i	$0.462502\pi$
$240$	0	0
$241$	−507933.	−0.563331	−0.281666	−	0.959513i	$-0.590887\pi$
$241$	−507933.	−0.563331	−0.281666	+	0.959513i	$0.590887\pi$
$242$	−207877.	−0.228176
$243$	0	0
$244$	−2.57044e6	−2.76397
$245$	1.10572e6	1.17687
$246$	0	0
$247$	−297475.	−0.310247
$248$	−5.34591e6	−5.51941
$249$	0	0
$250$	1.63136e6	1.65082
$251$	58645.3	0.0587556	0.0293778	−	0.999568i	$-0.490647\pi$
$251$	58645.3	0.0587556	0.0293778	+	0.999568i	$0.490647\pi$
$252$	0	0
$253$	−1.75161e6	−1.72043
$254$	3.70997e6	3.60816
$255$	0	0
$256$	287983.	0.274642
$257$	−1586.01	−0.00149787	−0.000748935	−	1.00000i	$-0.500238\pi$
$257$	−1586.01	−0.00149787	−0.000748935		1.00000i	$0.500238\pi$
$258$	0	0
$259$	1.92089e6	1.77932
$260$	4.93074e6	4.52354
$261$	0	0
$262$	1.14621e6	1.03160
$263$	1.39407e6	1.24278	0.621391	−	0.783501i	$-0.286568\pi$
$263$	1.39407e6	1.24278	0.621391	+	0.783501i	$0.286568\pi$
$264$	0	0
$265$	420157.	0.367534
$266$	602774.	0.522337
$267$	0	0
$268$	−5.00504e6	−4.25667
$269$	−749950.	−0.631904	−0.315952	−	0.948775i	$-0.602324\pi$
$269$	−749950.	−0.631904	−0.315952	+	0.948775i	$0.602324\pi$
$270$	0	0
$271$	66861.0	0.0553031	0.0276516	−	0.999618i	$-0.491197\pi$
$271$	66861.0	0.0553031	0.0276516	+	0.999618i	$0.491197\pi$
$272$	−3.18852e6	−2.61316
$273$	0	0
$274$	2.04096e6	1.64232
$275$	267668.	0.213434
$276$	0	0
$277$	−398436.	−0.312003	−0.156002	−	0.987757i	$-0.549860\pi$
$277$	−398436.	−0.312003	−0.156002	+	0.987757i	$0.549860\pi$
$278$	−248649.	−0.192963
$279$	0	0
$280$	−6.08515e6	−4.63849
$281$	−406894.	−0.307408	−0.153704	−	0.988117i	$-0.549120\pi$
$281$	−406894.	−0.307408	−0.153704	+	0.988117i	$0.549120\pi$
$282$	0	0
$283$	2.31675e6	1.71954	0.859772	−	0.510678i	$-0.170605\pi$
$283$	2.31675e6	1.71954	0.859772	+	0.510678i	$0.170605\pi$
$284$	610203.	0.448930
$285$	0	0
$286$	−4.45683e6	−3.22189
$287$	−299723.	−0.214790
$288$	0	0
$289$	−331649.	−0.233579
$290$	−5.50967e6	−3.84707
$291$	0	0
$292$	−353981.	−0.242954
$293$	−1.71243e6	−1.16531	−0.582657	−	0.812718i	$-0.697987\pi$
$293$	−1.71243e6	−1.16531	−0.582657	+	0.812718i	$0.697987\pi$
$294$	0	0
$295$	−213308.	−0.142709
$296$	−5.47333e6	−3.63097
$297$	0	0
$298$	−178160.	−0.116217
$299$	−4.05263e6	−2.62155
$300$	0	0
$301$	−917940.	−0.583980
$302$	1.50850e6	0.951763
$303$	0	0
$304$	−924924.	−0.574014
$305$	−1.92433e6	−1.18448
$306$	0	0
$307$	518531.	0.313999	0.157000	−	0.987599i	$-0.449818\pi$
$307$	518531.	0.313999	0.157000	+	0.987599i	$0.449818\pi$
$308$	6.49262e6	3.89981
$309$	0	0
$310$	−6.57109e6	−3.88359
$311$	50432.7	0.0295673	0.0147836	−	0.999891i	$-0.495294\pi$
$311$	50432.7	0.0295673	0.0147836	+	0.999891i	$0.495294\pi$
$312$	0	0
$313$	2.84888e6	1.64366	0.821832	−	0.569730i	$-0.192952\pi$
$313$	2.84888e6	1.64366	0.821832	+	0.569730i	$0.192952\pi$
$314$	73105.2	0.0418432
$315$	0	0
$316$	−2.51092e6	−1.41454
$317$	−1.74502e6	−0.975331	−0.487666	−	0.873031i	$-0.662151\pi$
$317$	−1.74502e6	−0.975331	−0.487666	+	0.873031i	$0.662151\pi$
$318$	0	0
$319$	3.58038e6	1.96994
$320$	4.20121e6	2.29351
$321$	0	0
$322$	8.21185e6	4.41368
$323$	315667.	0.168354
$324$	0	0
$325$	619290.	0.325226
$326$	2.94317e6	1.53381
$327$	0	0
$328$	854021.	0.438313
$329$	4.48915e6	2.28652
$330$	0	0
$331$	−515970.	−0.258854	−0.129427	−	0.991589i	$-0.541314\pi$
$331$	−515970.	−0.258854	−0.129427	+	0.991589i	$0.541314\pi$
$332$	2.39341e6	1.19172
$333$	0	0
$334$	5.83083e6	2.85999
$335$	−3.74695e6	−1.82417
$336$	0	0
$337$	−2.01691e6	−0.967414	−0.483707	−	0.875230i	$-0.660710\pi$
$337$	−2.01691e6	−0.967414	−0.483707	+	0.875230i	$0.660710\pi$
$338$	−6.34981e6	−3.02321
$339$	0	0
$340$	−5.23228e6	−2.45467
$341$	4.27013e6	1.98864
$342$	0	0
$343$	230992.	0.106014
$344$	2.61555e6	1.19170
$345$	0	0
$346$	−6.01052e6	−2.69912
$347$	−1.32354e6	−0.590084	−0.295042	−	0.955484i	$-0.595334\pi$
$347$	−1.32354e6	−0.590084	−0.295042	+	0.955484i	$0.595334\pi$
$348$	0	0
$349$	−1.20078e6	−0.527714	−0.263857	−	0.964562i	$-0.584995\pi$
$349$	−1.20078e6	−0.527714	−0.263857	+	0.964562i	$0.584995\pi$
$350$	−1.25487e6	−0.547556
$351$	0	0
$352$	−6.62494e6	−2.84987
$353$	2.56612e6	1.09607	0.548037	−	0.836454i	$-0.315375\pi$
$353$	2.56612e6	1.09607	0.548037	+	0.836454i	$0.315375\pi$
$354$	0	0
$355$	456820.	0.192386
$356$	−1.95762e6	−0.818661
$357$	0	0
$358$	6.43603e6	2.65406
$359$	−1.98664e6	−0.813547	−0.406773	−	0.913529i	$-0.633346\pi$
$359$	−1.98664e6	−0.813547	−0.406773	+	0.913529i	$0.633346\pi$
$360$	0	0
$361$	−2.38453e6	−0.963019
$362$	−1.68822e6	−0.677109
$363$	0	0
$364$	1.50217e7	5.94244
$365$	−265003.	−0.104116
$366$	0	0
$367$	−1.10023e6	−0.426400	−0.213200	−	0.977009i	$-0.568389\pi$
$367$	−1.10023e6	−0.426400	−0.213200	+	0.977009i	$0.568389\pi$
$368$	−1.26006e7	−4.85035
$369$	0	0
$370$	−6.72771e6	−2.55484
$371$	1.28003e6	0.482818
$372$	0	0
$373$	−2.20944e6	−0.822263	−0.411131	−	0.911576i	$-0.634866\pi$
$373$	−2.20944e6	−0.822263	−0.411131	+	0.911576i	$0.634866\pi$
$374$	4.72940e6	1.74834
$375$	0	0
$376$	−1.27913e7	−4.66599
$377$	8.28377e6	3.00175
$378$	0	0
$379$	−560573.	−0.200463	−0.100231	−	0.994964i	$-0.531958\pi$
$379$	−560573.	−0.200463	−0.100231	+	0.994964i	$0.531958\pi$
$380$	−1.51778e6	−0.539200
$381$	0	0
$382$	−5.46648e6	−1.91668
$383$	3.40354e6	1.18559	0.592795	−	0.805354i	$-0.298025\pi$
$383$	3.40354e6	1.18559	0.592795	+	0.805354i	$0.298025\pi$
$384$	0	0
$385$	4.86061e6	1.67124
$386$	8.31543e6	2.84064
$387$	0	0
$388$	2.59092e6	0.873726
$389$	3.10419e6	1.04010	0.520048	−	0.854137i	$-0.325914\pi$
$389$	3.10419e6	1.04010	0.520048	+	0.854137i	$0.325914\pi$
$390$	0	0
$391$	4.30047e6	1.42257
$392$	−9.59842e6	−3.15489
$393$	0	0
$394$	−4.00492e6	−1.29973
$395$	−1.87976e6	−0.606193
$396$	0	0
$397$	−3.18351e6	−1.01375	−0.506875	−	0.862020i	$-0.669199\pi$
$397$	−3.18351e6	−1.01375	−0.506875	+	0.862020i	$0.669199\pi$
$398$	−3.98171e6	−1.25998
$399$	0	0
$400$	1.92553e6	0.601728
$401$	−311471.	−0.0967289	−0.0483644	−	0.998830i	$-0.515401\pi$
$401$	−311471.	−0.0967289	−0.0483644	+	0.998830i	$0.515401\pi$
$402$	0	0
$403$	9.87961e6	3.03024
$404$	7.36963e6	2.24643
$405$	0	0
$406$	−1.67854e7	−5.05379
$407$	4.37191e6	1.30823
$408$	0	0
$409$	−559438.	−0.165365	−0.0826825	−	0.996576i	$-0.526349\pi$
$409$	−559438.	−0.165365	−0.0826825	+	0.996576i	$0.526349\pi$
$410$	1.04975e6	0.308407
$411$	0	0
$412$	8.08206e6	2.34574
$413$	−649851.	−0.187473
$414$	0	0
$415$	1.79180e6	0.510703
$416$	−1.53278e7	−4.34257
$417$	0	0
$418$	1.37190e6	0.384046
$419$	−142773.	−0.0397294	−0.0198647	−	0.999803i	$-0.506324\pi$
$419$	−142773.	−0.0397294	−0.0198647	+	0.999803i	$0.506324\pi$
$420$	0	0
$421$	−3.82502e6	−1.05179	−0.525895	−	0.850550i	$-0.676270\pi$
$421$	−3.82502e6	−1.05179	−0.525895	+	0.850550i	$0.676270\pi$
$422$	−1.03317e7	−2.82418
$423$	0	0
$424$	−3.64727e6	−0.985265
$425$	−657163.	−0.176482
$426$	0	0
$427$	−5.86254e6	−1.55602
$428$	8.07298e6	2.13022
$429$	0	0
$430$	3.21499e6	0.838509
$431$	4.27196e6	1.10773	0.553865	−	0.832606i	$-0.313152\pi$
$431$	4.27196e6	1.10773	0.553865	+	0.832606i	$0.313152\pi$
$432$	0	0
$433$	−3.40429e6	−0.872583	−0.436292	−	0.899805i	$-0.643708\pi$
$433$	−3.40429e6	−0.872583	−0.436292	+	0.899805i	$0.643708\pi$
$434$	−2.00191e7	−5.10176
$435$	0	0
$436$	4.67225e6	1.17709
$437$	1.24748e6	0.312486
$438$	0	0
$439$	6.97376e6	1.72705	0.863527	−	0.504303i	$-0.168250\pi$
$439$	6.97376e6	1.72705	0.863527	+	0.504303i	$0.168250\pi$
$440$	−1.38497e7	−3.41042
$441$	0	0
$442$	1.09422e7	2.66409
$443$	−4.62500e6	−1.11970	−0.559851	−	0.828593i	$-0.689142\pi$
$443$	−4.62500e6	−1.11970	−0.559851	+	0.828593i	$0.689142\pi$
$444$	0	0
$445$	−1.46555e6	−0.350833
$446$	4.51555e6	1.07491
$447$	0	0
$448$	1.27992e7	3.01291
$449$	4.22998e6	0.990199	0.495100	−	0.868836i	$-0.335132\pi$
$449$	4.22998e6	0.990199	0.495100	+	0.868836i	$0.335132\pi$
$450$	0	0
$451$	−682163.	−0.157924
$452$	−5.79893e6	−1.33506
$453$	0	0
$454$	1.97857e6	0.450519
$455$	1.12458e7	2.54660
$456$	0	0
$457$	8.46533e6	1.89607	0.948033	−	0.318173i	$-0.103069\pi$
$457$	8.46533e6	1.89607	0.948033	+	0.318173i	$0.103069\pi$
$458$	−4.46116e6	−0.993767
$459$	0	0
$460$	−2.06774e7	−4.55617
$461$	5.61651e6	1.23088	0.615438	−	0.788185i	$-0.288979\pi$
$461$	5.61651e6	1.23088	0.615438	+	0.788185i	$0.288979\pi$
$462$	0	0
$463$	692123.	0.150048	0.0750241	−	0.997182i	$-0.476097\pi$
$463$	692123.	0.150048	0.0750241	+	0.997182i	$0.476097\pi$
$464$	2.57563e7	5.55378
$465$	0	0
$466$	1.46437e6	0.312382
$467$	−6.20510e6	−1.31661	−0.658304	−	0.752752i	$-0.728726\pi$
$467$	−6.20510e6	−1.31661	−0.658304	+	0.752752i	$0.728726\pi$
$468$	0	0
$469$	−1.14152e7	−2.39636
$470$	−1.57228e7	−3.28310
$471$	0	0
$472$	1.85167e6	0.382567
$473$	−2.08921e6	−0.429368
$474$	0	0
$475$	−190630.	−0.0387665
$476$	−1.59404e7	−3.22463
$477$	0	0
$478$	−2.21489e6	−0.443387
$479$	4.54829e6	0.905752	0.452876	−	0.891573i	$-0.350398\pi$
$479$	4.54829e6	0.905752	0.452876	+	0.891573i	$0.350398\pi$
$480$	0	0
$481$	1.01151e7	1.99346
$482$	5.41973e6	1.06258
$483$	0	0
$484$	1.59466e6	0.309425
$485$	1.93966e6	0.374430
$486$	0	0
$487$	−3.94237e6	−0.753242	−0.376621	−	0.926367i	$-0.622914\pi$
$487$	−3.94237e6	−0.753242	−0.376621	+	0.926367i	$0.622914\pi$
$488$	1.67045e7	3.17530
$489$	0	0
$490$	−1.17982e7	−2.21986
$491$	−557676.	−0.104395	−0.0521973	−	0.998637i	$-0.516622\pi$
$491$	−557676.	−0.104395	−0.0521973	+	0.998637i	$0.516622\pi$
$492$	0	0
$493$	−8.79037e6	−1.62888
$494$	3.17411e6	0.585200
$495$	0	0
$496$	3.07182e7	5.60649
$497$	1.39172e6	0.252732
$498$	0	0
$499$	−4.02783e6	−0.724136	−0.362068	−	0.932152i	$-0.617929\pi$
$499$	−4.02783e6	−0.724136	−0.362068	+	0.932152i	$0.617929\pi$
$500$	−1.25144e7	−2.23865
$501$	0	0
$502$	−625755.	−0.110827
$503$	−9.17895e6	−1.61761	−0.808803	−	0.588079i	$-0.799884\pi$
$503$	−9.17895e6	−1.61761	−0.808803	+	0.588079i	$0.799884\pi$
$504$	0	0
$505$	5.51717e6	0.962694
$506$	1.86900e7	3.24514
$507$	0	0
$508$	−2.84597e7	−4.89296
$509$	8.23834e6	1.40944	0.704718	−	0.709487i	$-0.251073\pi$
$509$	8.23834e6	1.40944	0.704718	+	0.709487i	$0.251073\pi$
$510$	0	0
$511$	−807343.	−0.136775
$512$	4.37045e6	0.736803
$513$	0	0
$514$	16923.0	0.00282534
$515$	6.05053e6	1.00525
$516$	0	0
$517$	1.02172e7	1.68115
$518$	−2.04962e7	−3.35621
$519$	0	0
$520$	−3.20434e7	−5.19673
$521$	4.78219e6	0.771850	0.385925	−	0.922530i	$-0.373882\pi$
$521$	4.78219e6	0.771850	0.385925	+	0.922530i	$0.373882\pi$
$522$	0	0
$523$	−1.22841e7	−1.96376	−0.981881	−	0.189501i	$-0.939313\pi$
$523$	−1.22841e7	−1.96376	−0.981881	+	0.189501i	$0.939313\pi$
$524$	−8.79277e6	−1.39893
$525$	0	0
$526$	−1.48749e7	−2.34418
$527$	−1.04838e7	−1.64434
$528$	0	0
$529$	1.05586e7	1.64046
$530$	−4.48315e6	−0.693256
$531$	0	0
$532$	−4.62398e6	−0.708331
$533$	−1.57829e6	−0.240640
$534$	0	0
$535$	6.04373e6	0.912894
$536$	3.25262e7	4.89014
$537$	0	0
$538$	8.00209e6	1.19192
$539$	7.66689e6	1.13670
$540$	0	0
$541$	−7.08864e6	−1.04128	−0.520642	−	0.853775i	$-0.674307\pi$
$541$	−7.08864e6	−1.04128	−0.520642	+	0.853775i	$0.674307\pi$
$542$	−713418.	−0.104315
$543$	0	0
$544$	1.62652e7	2.35647
$545$	3.49782e6	0.504436
$546$	0	0
$547$	1.55919e6	0.222807	0.111404	−	0.993775i	$-0.464465\pi$
$547$	1.55919e6	0.222807	0.111404	+	0.993775i	$0.464465\pi$
$548$	−1.56565e7	−2.22712
$549$	0	0
$550$	−2.85606e6	−0.402588
$551$	−2.54991e6	−0.357805
$552$	0	0
$553$	−5.72678e6	−0.796338
$554$	4.25138e6	0.588512
$555$	0	0
$556$	1.90742e6	0.261674
$557$	−2.40850e6	−0.328934	−0.164467	−	0.986383i	$-0.552590\pi$
$557$	−2.40850e6	−0.328934	−0.164467	+	0.986383i	$0.552590\pi$
$558$	0	0
$559$	−4.83372e6	−0.654262
$560$	3.49659e7	4.71167
$561$	0	0
$562$	4.34163e6	0.579845
$563$	1.96787e6	0.261652	0.130826	−	0.991405i	$-0.458237\pi$
$563$	1.96787e6	0.261652	0.130826	+	0.991405i	$0.458237\pi$
$564$	0	0
$565$	−4.34129e6	−0.572133
$566$	−2.47201e7	−3.24347
$567$	0	0
$568$	−3.96552e6	−0.515739
$569$	−5.44328e6	−0.704823	−0.352411	−	0.935845i	$-0.614638\pi$
$569$	−5.44328e6	−0.704823	−0.352411	+	0.935845i	$0.614638\pi$
$570$	0	0
$571$	1.26908e7	1.62892	0.814461	−	0.580218i	$-0.197033\pi$
$571$	1.26908e7	1.62892	0.814461	+	0.580218i	$0.197033\pi$
$572$	3.41891e7	4.36915
$573$	0	0
$574$	3.19809e6	0.405146
$575$	−2.59703e6	−0.327573
$576$	0	0
$577$	4.17811e6	0.522445	0.261222	−	0.965279i	$-0.415874\pi$
$577$	4.17811e6	0.522445	0.261222	+	0.965279i	$0.415874\pi$
$578$	3.53875e6	0.440585
$579$	0	0
$580$	4.22655e7	5.21695
$581$	5.45878e6	0.670896
$582$	0	0
$583$	2.91331e6	0.354990
$584$	2.30042e6	0.279110
$585$	0	0
$586$	1.82719e7	2.19806
$587$	−1.07114e6	−0.128307	−0.0641537	−	0.997940i	$-0.520435\pi$
$587$	−1.07114e6	−0.128307	−0.0641537	+	0.997940i	$0.520435\pi$
$588$	0	0
$589$	−3.04114e6	−0.361201
$590$	2.27603e6	0.269183
$591$	0	0
$592$	3.14503e7	3.68826
$593$	7.74188e6	0.904085	0.452043	−	0.891996i	$-0.350695\pi$
$593$	7.74188e6	0.904085	0.452043	+	0.891996i	$0.350695\pi$
$594$	0	0
$595$	−1.19335e7	−1.38190
$596$	1.36670e6	0.157600
$597$	0	0
$598$	4.32422e7	4.94487
$599$	−3.00377e6	−0.342058	−0.171029	−	0.985266i	$-0.554709\pi$
$599$	−3.00377e6	−0.342058	−0.171029	+	0.985266i	$0.554709\pi$
$600$	0	0
$601$	5.32845e6	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$601$	5.32845e6	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$602$	9.79458e6	1.10153
$603$	0	0
$604$	−1.15720e7	−1.29067
$605$	1.19382e6	0.132602
$606$	0	0
$607$	1.59774e7	1.76009	0.880046	−	0.474888i	$-0.157512\pi$
$607$	1.59774e7	1.76009	0.880046	+	0.474888i	$0.157512\pi$
$608$	4.71821e6	0.517629
$609$	0	0
$610$	2.05329e7	2.23422
$611$	2.36391e7	2.56170
$612$	0	0
$613$	8.04897e6	0.865145	0.432573	−	0.901599i	$-0.357606\pi$
$613$	8.04897e6	0.865145	0.432573	+	0.901599i	$0.357606\pi$
$614$	−5.53281e6	−0.592277
$615$	0	0
$616$	−4.21936e7	−4.48018
$617$	1.00376e7	1.06150	0.530748	−	0.847529i	$-0.321911\pi$
$617$	1.00376e7	1.06150	0.530748	+	0.847529i	$0.321911\pi$
$618$	0	0
$619$	−3.31323e6	−0.347556	−0.173778	−	0.984785i	$-0.555598\pi$
$619$	−3.31323e6	−0.347556	−0.173778	+	0.984785i	$0.555598\pi$
$620$	5.04078e7	5.26646
$621$	0	0
$622$	−538126.	−0.0557709
$623$	−4.46485e6	−0.460879
$624$	0	0
$625$	−1.13374e7	−1.16095
$626$	−3.03980e7	−3.10034
$627$	0	0
$628$	−560802.	−0.0567427
$629$	−1.07337e7	−1.08174
$630$	0	0
$631$	1.30632e7	1.30610	0.653048	−	0.757316i	$-0.273490\pi$
$631$	1.30632e7	1.30610	0.653048	+	0.757316i	$0.273490\pi$
$632$	1.63177e7	1.62505
$633$	0	0
$634$	1.86197e7	1.83971
$635$	−2.13060e7	−2.09685
$636$	0	0
$637$	1.77385e7	1.73208
$638$	−3.82033e7	−3.71577
$639$	0	0
$640$	−1.42534e7	−1.37552
$641$	3.24010e6	0.311468	0.155734	−	0.987799i	$-0.450226\pi$
$641$	3.24010e6	0.311468	0.155734	+	0.987799i	$0.450226\pi$
$642$	0	0
$643$	−1.43038e7	−1.36435	−0.682174	−	0.731189i	$-0.738966\pi$
$643$	−1.43038e7	−1.36435	−0.682174	+	0.731189i	$0.738966\pi$
$644$	−6.29944e7	−5.98531
$645$	0	0
$646$	−3.36823e6	−0.317556
$647$	1.69491e7	1.59179	0.795896	−	0.605434i	$-0.207000\pi$
$647$	1.69491e7	1.59179	0.795896	+	0.605434i	$0.207000\pi$
$648$	0	0
$649$	−1.47905e6	−0.137839
$650$	−6.60793e6	−0.613454
$651$	0	0
$652$	−2.25775e7	−2.07997
$653$	−4.66123e6	−0.427777	−0.213889	−	0.976858i	$-0.568613\pi$
$653$	−4.66123e6	−0.427777	−0.213889	+	0.976858i	$0.568613\pi$
$654$	0	0
$655$	−6.58258e6	−0.599506
$656$	−4.90730e6	−0.445228
$657$	0	0
$658$	−4.79000e7	−4.31291
$659$	−1.84395e7	−1.65400	−0.827001	−	0.562201i	$-0.809955\pi$
$659$	−1.84395e7	−1.65400	−0.827001	+	0.562201i	$0.809955\pi$
$660$	0	0
$661$	−1.59041e6	−0.141581	−0.0707905	−	0.997491i	$-0.522552\pi$
$661$	−1.59041e6	−0.141581	−0.0707905	+	0.997491i	$0.522552\pi$
$662$	5.50549e6	0.488260
$663$	0	0
$664$	−1.55541e7	−1.36907
$665$	−3.46168e6	−0.303551
$666$	0	0
$667$	−3.47385e7	−3.02341
$668$	−4.47292e7	−3.87838
$669$	0	0
$670$	3.99806e7	3.44082
$671$	−1.33430e7	−1.14406
$672$	0	0
$673$	−1.37250e7	−1.16809	−0.584044	−	0.811722i	$-0.698530\pi$
$673$	−1.37250e7	−1.16809	−0.584044	+	0.811722i	$0.698530\pi$
$674$	2.15208e7	1.82477
$675$	0	0
$676$	4.87104e7	4.09972
$677$	−1.55559e7	−1.30444	−0.652219	−	0.758031i	$-0.726162\pi$
$677$	−1.55559e7	−1.30444	−0.652219	+	0.758031i	$0.726162\pi$
$678$	0	0
$679$	5.90925e6	0.491878
$680$	3.40030e7	2.81998
$681$	0	0
$682$	−4.55630e7	−3.75104
$683$	1.03021e7	0.845033	0.422517	−	0.906355i	$-0.361147\pi$
$683$	1.03021e7	0.845033	0.422517	+	0.906355i	$0.361147\pi$
$684$	0	0
$685$	−1.17210e7	−0.954420
$686$	−2.46472e6	−0.199967
$687$	0	0
$688$	−1.50292e7	−1.21050
$689$	6.74040e6	0.540926
$690$	0	0
$691$	5.02324e6	0.400211	0.200106	−	0.979774i	$-0.435871\pi$
$691$	5.02324e6	0.400211	0.200106	+	0.979774i	$0.435871\pi$
$692$	4.61076e7	3.66022
$693$	0	0
$694$	1.41224e7	1.11304
$695$	1.42797e6	0.112139
$696$	0	0
$697$	1.67481e6	0.130582
$698$	1.28125e7	0.995393
$699$	0	0
$700$	9.62630e6	0.742530
$701$	1.22395e7	0.940739	0.470370	−	0.882469i	$-0.344121\pi$
$701$	1.22395e7	0.940739	0.470370	+	0.882469i	$0.344121\pi$
$702$	0	0
$703$	−3.11363e6	−0.237617
$704$	2.91306e7	2.21523
$705$	0	0
$706$	−2.73809e7	−2.06745
$707$	1.68083e7	1.26466
$708$	0	0
$709$	3.66774e6	0.274021	0.137010	−	0.990570i	$-0.456251\pi$
$709$	3.66774e6	0.274021	0.137010	+	0.990570i	$0.456251\pi$
$710$	−4.87435e6	−0.362886
$711$	0	0
$712$	1.27220e7	0.940493
$713$	−4.14307e7	−3.05210
$714$	0	0
$715$	2.55952e7	1.87238
$716$	−4.93718e7	−3.59912
$717$	0	0
$718$	2.11978e7	1.53454
$719$	−1.96481e7	−1.41742	−0.708709	−	0.705501i	$-0.750722\pi$
$719$	−1.96481e7	−1.41742	−0.708709	+	0.705501i	$0.750722\pi$
$720$	0	0
$721$	1.84332e7	1.32057
$722$	2.54433e7	1.81648
$723$	0	0
$724$	1.29506e7	0.918215
$725$	5.30846e6	0.375080
$726$	0	0
$727$	1.64307e6	0.115297	0.0576486	−	0.998337i	$-0.481640\pi$
$727$	1.64307e6	0.115297	0.0576486	+	0.998337i	$0.481640\pi$
$728$	−9.76214e7	−6.82679
$729$	0	0
$730$	2.82763e6	0.196388
$731$	5.12933e6	0.355032
$732$	0	0
$733$	−4.63590e6	−0.318694	−0.159347	−	0.987223i	$-0.550939\pi$
$733$	−4.63590e6	−0.318694	−0.159347	+	0.987223i	$0.550939\pi$
$734$	1.17396e7	0.804291
$735$	0	0
$736$	6.42781e7	4.37390
$737$	−2.59809e7	−1.76191
$738$	0	0
$739$	2.52043e7	1.69771	0.848855	−	0.528625i	$-0.177292\pi$
$739$	2.52043e7	1.69771	0.848855	+	0.528625i	$0.177292\pi$
$740$	5.16093e7	3.46457
$741$	0	0
$742$	−1.36581e7	−0.910710
$743$	−2.59323e7	−1.72333	−0.861666	−	0.507475i	$-0.830579\pi$
$743$	−2.59323e7	−1.72333	−0.861666	+	0.507475i	$0.830579\pi$
$744$	0	0
$745$	1.02316e6	0.0675386
$746$	2.35751e7	1.55098
$747$	0	0
$748$	−3.62799e7	−2.37090
$749$	1.84124e7	1.19924
$750$	0	0
$751$	2.11649e7	1.36936	0.684678	−	0.728846i	$-0.259943\pi$
$751$	2.11649e7	1.36936	0.684678	+	0.728846i	$0.259943\pi$
$752$	7.34999e7	4.73961
$753$	0	0
$754$	−8.83892e7	−5.66201
$755$	−8.66319e6	−0.553109
$756$	0	0
$757$	1.49940e7	0.950992	0.475496	−	0.879718i	$-0.342269\pi$
$757$	1.49940e7	0.950992	0.475496	+	0.879718i	$0.342269\pi$
$758$	5.98140e6	0.378120
$759$	0	0
$760$	9.86360e6	0.619443
$761$	2.32932e7	1.45803	0.729015	−	0.684497i	$-0.239978\pi$
$761$	2.32932e7	1.45803	0.729015	+	0.684497i	$0.239978\pi$
$762$	0	0
$763$	1.06562e7	0.662663
$764$	4.19342e7	2.59917
$765$	0	0
$766$	−3.63164e7	−2.23630
$767$	−3.42201e6	−0.210035
$768$	0	0
$769$	−5.03552e6	−0.307064	−0.153532	−	0.988144i	$-0.549065\pi$
$769$	−5.03552e6	−0.307064	−0.153532	+	0.988144i	$0.549065\pi$
$770$	−5.18636e7	−3.15236
$771$	0	0
$772$	−6.37890e7	−3.85214
$773$	1.34665e7	0.810599	0.405300	−	0.914184i	$-0.367167\pi$
$773$	1.34665e7	0.810599	0.405300	+	0.914184i	$0.367167\pi$
$774$	0	0
$775$	6.33112e6	0.378640
$776$	−1.68376e7	−1.00375
$777$	0	0
$778$	−3.31222e7	−1.96187
$779$	485829.	0.0286840
$780$	0	0
$781$	3.16753e6	0.185820
$782$	−4.58867e7	−2.68331
$783$	0	0
$784$	5.51535e7	3.20467
$785$	−419837.	−0.0243168
$786$	0	0
$787$	6.43085e6	0.370111	0.185055	−	0.982728i	$-0.440754\pi$
$787$	6.43085e6	0.370111	0.185055	+	0.982728i	$0.440754\pi$
$788$	3.07223e7	1.76254
$789$	0	0
$790$	2.00574e7	1.14342
$791$	−1.32259e7	−0.751595
$792$	0	0
$793$	−3.08711e7	−1.74329
$794$	3.39686e7	1.91217
$795$	0	0
$796$	3.05444e7	1.70863
$797$	−2.19054e7	−1.22153	−0.610765	−	0.791812i	$-0.709138\pi$
$797$	−2.19054e7	−1.22153	−0.610765	+	0.791812i	$0.709138\pi$
$798$	0	0
$799$	−2.50848e7	−1.39009
$800$	−9.82247e6	−0.542620
$801$	0	0
$802$	3.32344e6	0.182454
$803$	−1.83750e6	−0.100563
$804$	0	0
$805$	−4.71599e7	−2.56497
$806$	−1.05417e8	−5.71575
$807$	0	0
$808$	−4.78930e7	−2.58074
$809$	3.19147e7	1.71443	0.857214	−	0.514961i	$-0.172194\pi$
$809$	3.19147e7	1.71443	0.857214	+	0.514961i	$0.172194\pi$
$810$	0	0
$811$	7.21928e6	0.385426	0.192713	−	0.981255i	$-0.438271\pi$
$811$	7.21928e6	0.385426	0.192713	+	0.981255i	$0.438271\pi$
$812$	1.28764e8	6.85335
$813$	0	0
$814$	−4.66490e7	−2.46764
$815$	−1.69023e7	−0.891359
$816$	0	0
$817$	1.48792e6	0.0779872
$818$	5.96930e6	0.311918
$819$	0	0
$820$	−8.05277e6	−0.418225
$821$	−1.80572e7	−0.934958	−0.467479	−	0.884004i	$-0.654838\pi$
$821$	−1.80572e7	−0.934958	−0.467479	+	0.884004i	$0.654838\pi$
$822$	0	0
$823$	−4.54084e6	−0.233688	−0.116844	−	0.993150i	$-0.537278\pi$
$823$	−4.54084e6	−0.233688	−0.116844	+	0.993150i	$0.537278\pi$
$824$	−5.25229e7	−2.69483
$825$	0	0
$826$	6.93402e6	0.353619
$827$	−2.72523e7	−1.38561	−0.692803	−	0.721127i	$-0.743624\pi$
$827$	−2.72523e7	−1.38561	−0.692803	+	0.721127i	$0.743624\pi$
$828$	0	0
$829$	1.54051e7	0.778537	0.389269	−	0.921124i	$-0.372728\pi$
$829$	1.54051e7	0.778537	0.389269	+	0.921124i	$0.372728\pi$
$830$	−1.91188e7	−0.963308
$831$	0	0
$832$	6.73982e7	3.37552
$833$	−1.88234e7	−0.939906
$834$	0	0
$835$	−3.34859e7	−1.66206
$836$	−1.05241e7	−0.520797
$837$	0	0
$838$	1.52341e6	0.0749390
$839$	−1.50266e7	−0.736979	−0.368489	−	0.929632i	$-0.620125\pi$
$839$	−1.50266e7	−0.736979	−0.368489	+	0.929632i	$0.620125\pi$
$840$	0	0
$841$	5.04961e7	2.46188
$842$	4.08137e7	1.98393
$843$	0	0
$844$	7.92564e7	3.82982
$845$	3.64663e7	1.75691
$846$	0	0
$847$	3.63702e6	0.174196
$848$	2.09576e7	1.00081
$849$	0	0
$850$	7.01205e6	0.332887
$851$	−4.24182e7	−2.00784
$852$	0	0
$853$	3.68547e7	1.73428	0.867142	−	0.498060i	$-0.165954\pi$
$853$	3.68547e7	1.73428	0.867142	+	0.498060i	$0.165954\pi$
$854$	6.25543e7	2.93503
$855$	0	0
$856$	−5.24639e7	−2.44724
$857$	3.27506e7	1.52324	0.761618	−	0.648026i	$-0.224405\pi$
$857$	3.27506e7	1.52324	0.761618	+	0.648026i	$0.224405\pi$
$858$	0	0
$859$	3.34950e7	1.54881	0.774403	−	0.632693i	$-0.218050\pi$
$859$	3.34950e7	1.54881	0.774403	+	0.632693i	$0.218050\pi$
$860$	−2.46627e7	−1.13709
$861$	0	0
$862$	−4.55826e7	−2.08944
$863$	−1.12755e7	−0.515358	−0.257679	−	0.966231i	$-0.582958\pi$
$863$	−1.12755e7	−0.515358	−0.257679	+	0.966231i	$0.582958\pi$
$864$	0	0
$865$	3.45178e7	1.56857
$866$	3.63244e7	1.64590
$867$	0	0
$868$	1.53569e8	6.91840
$869$	−1.30340e7	−0.585503
$870$	0	0
$871$	−6.01107e7	−2.68476
$872$	−3.03636e7	−1.35226
$873$	0	0
$874$	−1.33108e7	−0.589422
$875$	−2.85423e7	−1.26029
$876$	0	0
$877$	−3.74225e6	−0.164299	−0.0821493	−	0.996620i	$-0.526178\pi$
$877$	−3.74225e6	−0.164299	−0.0821493	+	0.996620i	$0.526178\pi$
$878$	−7.44112e7	−3.25763
$879$	0	0
$880$	7.95818e7	3.46423
$881$	−1.20043e7	−0.521073	−0.260537	−	0.965464i	$-0.583899\pi$
$881$	−1.20043e7	−0.521073	−0.260537	+	0.965464i	$0.583899\pi$
$882$	0	0
$883$	−1.38235e7	−0.596647	−0.298324	−	0.954465i	$-0.596427\pi$
$883$	−1.38235e7	−0.596647	−0.298324	+	0.954465i	$0.596427\pi$
$884$	−8.39392e7	−3.61272
$885$	0	0
$886$	4.93495e7	2.11202
$887$	−1.58414e7	−0.676059	−0.338030	−	0.941135i	$-0.609760\pi$
$887$	−1.58414e7	−0.676059	−0.338030	+	0.941135i	$0.609760\pi$
$888$	0	0
$889$	−6.49095e7	−2.75457
$890$	1.56376e7	0.661754
$891$	0	0
$892$	−3.46395e7	−1.45767
$893$	−7.27659e6	−0.305351
$894$	0	0
$895$	−3.69615e7	−1.54238
$896$	−4.34234e7	−1.80698
$897$	0	0
$898$	−4.51346e7	−1.86775
$899$	8.46865e7	3.49474
$900$	0	0
$901$	−7.15262e6	−0.293530
$902$	7.27880e6	0.297881
$903$	0	0
$904$	3.76855e7	1.53374
$905$	9.69531e6	0.393496
$906$	0	0
$907$	−3.64791e7	−1.47240	−0.736200	−	0.676764i	$-0.763382\pi$
$907$	−3.64791e7	−1.47240	−0.736200	+	0.676764i	$0.763382\pi$
$908$	−1.51780e7	−0.610940
$909$	0	0
$910$	−1.19994e8	−4.80349
$911$	3.55848e7	1.42059	0.710294	−	0.703905i	$-0.248562\pi$
$911$	3.55848e7	1.42059	0.710294	+	0.703905i	$0.248562\pi$
$912$	0	0
$913$	1.24241e7	0.493273
$914$	−9.03265e7	−3.57643
$915$	0	0
$916$	3.42223e7	1.34763
$917$	−2.00541e7	−0.787553
$918$	0	0
$919$	3.19136e7	1.24649	0.623243	−	0.782028i	$-0.285815\pi$
$919$	3.19136e7	1.24649	0.623243	+	0.782028i	$0.285815\pi$
$920$	1.34376e8	5.23422
$921$	0	0
$922$	−5.99291e7	−2.32172
$923$	7.32856e6	0.283149
$924$	0	0
$925$	6.48202e6	0.249090
$926$	−7.38507e6	−0.283026
$927$	0	0
$928$	−1.31388e8	−5.00824
$929$	3.37958e6	0.128477	0.0642383	−	0.997935i	$-0.479538\pi$
$929$	3.37958e6	0.128477	0.0642383	+	0.997935i	$0.479538\pi$
$930$	0	0
$931$	−5.46028e6	−0.206462
$932$	−1.12334e7	−0.423615
$933$	0	0
$934$	6.62095e7	2.48344
$935$	−2.71605e7	−1.01603
$936$	0	0
$937$	3.48856e7	1.29807	0.649034	−	0.760760i	$-0.275173\pi$
$937$	3.48856e7	1.29807	0.649034	+	0.760760i	$0.275173\pi$
$938$	1.21802e8	4.52011
$939$	0	0
$940$	1.20612e8	4.45215
$941$	4.74186e7	1.74572	0.872859	−	0.487972i	$-0.162263\pi$
$941$	4.74186e7	1.74572	0.872859	+	0.487972i	$0.162263\pi$
$942$	0	0
$943$	6.61865e6	0.242376
$944$	−1.06399e7	−0.388603
$945$	0	0
$946$	2.22923e7	0.809891
$947$	−3.31556e7	−1.20139	−0.600693	−	0.799480i	$-0.705108\pi$
$947$	−3.31556e7	−1.20139	−0.600693	+	0.799480i	$0.705108\pi$
$948$	0	0
$949$	−4.25133e6	−0.153236
$950$	2.03405e6	0.0731229
$951$	0	0
$952$	1.03592e8	3.70452
$953$	−4.68158e7	−1.66978	−0.834892	−	0.550413i	$-0.814470\pi$
$953$	−4.68158e7	−1.66978	−0.834892	+	0.550413i	$0.814470\pi$
$954$	0	0
$955$	3.13935e7	1.11386
$956$	1.69908e7	0.601269
$957$	0	0
$958$	−4.85310e7	−1.70846
$959$	−3.57086e7	−1.25379
$960$	0	0
$961$	7.23719e7	2.52791
$962$	−1.07930e8	−3.76013
$963$	0	0
$964$	−4.15756e7	−1.44094
$965$	−4.77547e7	−1.65081
$966$	0	0
$967$	−3.49546e7	−1.20209	−0.601047	−	0.799214i	$-0.705250\pi$
$967$	−3.49546e7	−1.20209	−0.601047	+	0.799214i	$0.705250\pi$
$968$	−1.03632e7	−0.355473
$969$	0	0
$970$	−2.06965e7	−0.706265
$971$	−5.44571e6	−0.185356	−0.0926780	−	0.995696i	$-0.529543\pi$
$971$	−5.44571e6	−0.185356	−0.0926780	+	0.995696i	$0.529543\pi$
$972$	0	0
$973$	4.35035e6	0.147313
$974$	4.20657e7	1.42079
$975$	0	0
$976$	−9.59862e7	−3.22540
$977$	−2.76151e7	−0.925573	−0.462786	−	0.886470i	$-0.653150\pi$
$977$	−2.76151e7	−0.925573	−0.462786	+	0.886470i	$0.653150\pi$
$978$	0	0
$979$	−1.01619e7	−0.338859
$980$	9.05057e7	3.01031
$981$	0	0
$982$	5.95050e6	0.196913
$983$	8.85517e6	0.292289	0.146145	−	0.989263i	$-0.453313\pi$
$983$	8.85517e6	0.292289	0.146145	+	0.989263i	$0.453313\pi$
$984$	0	0
$985$	2.29999e7	0.755326
$986$	9.37948e7	3.07246
$987$	0	0
$988$	−2.43491e7	−0.793579
$989$	2.02705e7	0.658982
$990$	0	0
$991$	3.63535e7	1.17588	0.587938	−	0.808906i	$-0.299940\pi$
$991$	3.63535e7	1.17588	0.587938	+	0.808906i	$0.299940\pi$
$992$	−1.56699e8	−5.05577
$993$	0	0
$994$	−1.48499e7	−0.476713
$995$	2.28666e7	0.732224
$996$	0	0
$997$	3.59054e6	0.114399	0.0571994	−	0.998363i	$-0.481783\pi$
$997$	3.59054e6	0.114399	0.0571994	+	0.998363i	$0.481783\pi$
$998$	4.29777e7	1.36589
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.e.1.1		13
3.2	odd	2		177.6.a.d.1.13	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.d.1.13	✓	13	3.2	odd	2
531.6.a.e.1.1		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)

\(f(q)\)	\(=\)	\(q-10.6702 q^{2} +81.8525 q^{4} +61.2778 q^{5} +186.685 q^{7} -531.935 q^{8} +O(q^{10})\)	\(q-10.6702 q^{2} +81.8525 q^{4} +61.2778 q^{5} +186.685 q^{7} -531.935 q^{8} -653.844 q^{10} +424.892 q^{11} +983.052 q^{13} -1991.96 q^{14} +3056.56 q^{16} -1043.17 q^{17} -302.603 q^{19} +5015.74 q^{20} -4533.67 q^{22} -4122.49 q^{23} +629.966 q^{25} -10489.3 q^{26} +15280.7 q^{28} +8426.58 q^{29} +10049.9 q^{31} -15592.1 q^{32} +11130.8 q^{34} +11439.6 q^{35} +10289.5 q^{37} +3228.83 q^{38} -32595.8 q^{40} -1605.50 q^{41} -4917.05 q^{43} +34778.5 q^{44} +43987.7 q^{46} +24046.6 q^{47} +18044.3 q^{49} -6721.85 q^{50} +80465.3 q^{52} +6856.60 q^{53} +26036.4 q^{55} -99304.4 q^{56} -89913.0 q^{58} -3481.00 q^{59} -31403.4 q^{61} -107234. q^{62} +68560.1 q^{64} +60239.3 q^{65} -61147.0 q^{67} -85386.3 q^{68} -122063. q^{70} +7454.90 q^{71} -4324.62 q^{73} -109790. q^{74} -24768.9 q^{76} +79321.0 q^{77} -30676.1 q^{79} +187299. q^{80} +17130.9 q^{82} +29240.6 q^{83} -63923.3 q^{85} +52465.8 q^{86} -226015. q^{88} -23916.5 q^{89} +183521. q^{91} -337436. q^{92} -256582. q^{94} -18542.9 q^{95} +31653.6 q^{97} -192536. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 246 q^{4} + 14 q^{5} + 373 q^{7} - 123 q^{8}+O(q^{10}) \) 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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