LMFDB - Embedded newform 531.6.a.c.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} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + \cdots + 49172480 $ x^12 - 2x^11 - 269x^10 + 143x^9 + 25384x^8 + 8539x^7 - 1009736x^6 - 720516x^5 + 15565376x^4 + 6775664x^3 - 75006848x^2 + 21512960*x + 49172480
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			$7.96013$ 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)

$f(q)$	$=$	$q+5.96013 q^{2} +3.52316 q^{4} -65.6425 q^{5} +119.026 q^{7} -169.726 q^{8} +O(q^{10})$	\(q+5.96013 q^{2} +3.52316 q^{4} -65.6425 q^{5} +119.026 q^{7} -169.726 q^{8} -391.238 q^{10} -237.620 q^{11} +999.008 q^{13} +709.408 q^{14} -1124.33 q^{16} +1276.40 q^{17} +1426.51 q^{19} -231.269 q^{20} -1416.25 q^{22} -546.573 q^{23} +1183.94 q^{25} +5954.22 q^{26} +419.346 q^{28} +1547.67 q^{29} -7569.31 q^{31} -1269.92 q^{32} +7607.51 q^{34} -7813.14 q^{35} -1012.21 q^{37} +8502.16 q^{38} +11141.2 q^{40} -13264.9 q^{41} -21794.0 q^{43} -837.174 q^{44} -3257.65 q^{46} -23038.8 q^{47} -2639.91 q^{49} +7056.46 q^{50} +3519.67 q^{52} -19051.5 q^{53} +15598.0 q^{55} -20201.7 q^{56} +9224.33 q^{58} +3481.00 q^{59} -7235.44 q^{61} -45114.1 q^{62} +28409.6 q^{64} -65577.4 q^{65} +69873.8 q^{67} +4496.96 q^{68} -46567.3 q^{70} -17826.0 q^{71} +9089.91 q^{73} -6032.93 q^{74} +5025.81 q^{76} -28282.9 q^{77} +31488.7 q^{79} +73803.8 q^{80} -79060.3 q^{82} +102025. q^{83} -83786.1 q^{85} -129895. q^{86} +40330.2 q^{88} -104185. q^{89} +118908. q^{91} -1925.67 q^{92} -137314. q^{94} -93639.5 q^{95} -117345. q^{97} -15734.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * 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 $

$2$	5.96013	1.05361	0.526806	−	0.849985i	$-0.323389\pi$
$2$	5.96013	1.05361	0.526806	+	0.849985i	$0.323389\pi$
$3$	0	0
$3$	0	0
$4$	3.52316	0.110099
$5$	−65.6425	−1.17425	−0.587125	−	0.809496i	$-0.699740\pi$
$5$	−65.6425	−1.17425	−0.587125	+	0.809496i	$0.699740\pi$
$6$	0	0
$7$	119.026	0.918111	0.459055	−	0.888408i	$-0.348188\pi$
$7$	119.026	0.918111	0.459055	+	0.888408i	$0.348188\pi$
$8$	−169.726	−0.937611
$9$	0	0
$10$	−391.238	−1.23720
$11$	−237.620	−0.592109	−0.296054	−	0.955171i	$-0.595671\pi$
$11$	−237.620	−0.592109	−0.296054	+	0.955171i	$0.595671\pi$
$12$	0	0
$13$	999.008	1.63950	0.819749	−	0.572723i	$-0.194113\pi$
$13$	999.008	1.63950	0.819749	+	0.572723i	$0.194113\pi$
$14$	709.408	0.967333
$15$	0	0
$16$	−1124.33	−1.09798
$17$	1276.40	1.07118	0.535592	−	0.844477i	$-0.320088\pi$
$17$	1276.40	1.07118	0.535592	+	0.844477i	$0.320088\pi$
$18$	0	0
$19$	1426.51	0.906546	0.453273	−	0.891372i	$-0.350256\pi$
$19$	1426.51	0.906546	0.453273	+	0.891372i	$0.350256\pi$
$20$	−231.269	−0.129283
$21$	0	0
$22$	−1416.25	−0.623853
$23$	−546.573	−0.215441	−0.107721	−	0.994181i	$-0.534355\pi$
$23$	−546.573	−0.215441	−0.107721	+	0.994181i	$0.534355\pi$
$24$	0	0
$25$	1183.94	0.378862
$26$	5954.22	1.72739
$27$	0	0
$28$	419.346	0.101083
$29$	1547.67	0.341731	0.170865	−	0.985294i	$-0.445344\pi$
$29$	1547.67	0.341731	0.170865	+	0.985294i	$0.445344\pi$
$30$	0	0
$31$	−7569.31	−1.41466	−0.707330	−	0.706883i	$-0.750101\pi$
$31$	−7569.31	−1.41466	−0.707330	+	0.706883i	$0.750101\pi$
$32$	−1269.92	−0.219231
$33$	0	0
$34$	7607.51	1.12861
$35$	−7813.14	−1.07809
$36$	0	0
$37$	−1012.21	−0.121554	−0.0607769	−	0.998151i	$-0.519358\pi$
$37$	−1012.21	−0.121554	−0.0607769	+	0.998151i	$0.519358\pi$
$38$	8502.16	0.955148
$39$	0	0
$40$	11141.2	1.10099
$41$	−13264.9	−1.23237	−0.616187	−	0.787600i	$-0.711324\pi$
$41$	−13264.9	−1.23237	−0.616187	+	0.787600i	$0.711324\pi$
$42$	0	0
$43$	−21794.0	−1.79749	−0.898743	−	0.438476i	$-0.855518\pi$
$43$	−21794.0	−1.79749	−0.898743	+	0.438476i	$0.855518\pi$
$44$	−837.174	−0.0651905
$45$	0	0
$46$	−3257.65	−0.226992
$47$	−23038.8	−1.52130	−0.760650	−	0.649163i	$-0.775119\pi$
$47$	−23038.8	−1.52130	−0.760650	+	0.649163i	$0.775119\pi$
$48$	0	0
$49$	−2639.91	−0.157072
$50$	7056.46	0.399174
$51$	0	0
$52$	3519.67	0.180507
$53$	−19051.5	−0.931624	−0.465812	−	0.884884i	$-0.654238\pi$
$53$	−19051.5	−0.931624	−0.465812	+	0.884884i	$0.654238\pi$
$54$	0	0
$55$	15598.0	0.695283
$56$	−20201.7	−0.860831
$57$	0	0
$58$	9224.33	0.360052
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−7235.44	−0.248966	−0.124483	−	0.992222i	$-0.539727\pi$
$61$	−7235.44	−0.248966	−0.124483	+	0.992222i	$0.539727\pi$
$62$	−45114.1	−1.49050
$63$	0	0
$64$	28409.6	0.866992
$65$	−65577.4	−1.92518
$66$	0	0
$67$	69873.8	1.90164	0.950818	−	0.309752i	$-0.100246\pi$
$67$	69873.8	1.90164	0.950818	+	0.309752i	$0.100246\pi$
$68$	4496.96	0.117936
$69$	0	0
$70$	−46567.3	−1.13589
$71$	−17826.0	−0.419671	−0.209836	−	0.977737i	$-0.567293\pi$
$71$	−17826.0	−0.419671	−0.209836	+	0.977737i	$0.567293\pi$
$72$	0	0
$73$	9089.91	0.199642	0.0998212	−	0.995005i	$-0.468173\pi$
$73$	9089.91	0.199642	0.0998212	+	0.995005i	$0.468173\pi$
$74$	−6032.93	−0.128071
$75$	0	0
$76$	5025.81	0.0998096
$77$	−28282.9	−0.543622
$78$	0	0
$79$	31488.7	0.567658	0.283829	−	0.958875i	$-0.408395\pi$
$79$	31488.7	0.567658	0.283829	+	0.958875i	$0.408395\pi$
$80$	73803.8	1.28930
$81$	0	0
$82$	−79060.3	−1.29845
$83$	102025.	1.62560	0.812798	−	0.582546i	$-0.197944\pi$
$83$	102025.	1.62560	0.812798	+	0.582546i	$0.197944\pi$
$84$	0	0
$85$	−83786.1	−1.25784
$86$	−129895.	−1.89385
$87$	0	0
$88$	40330.2	0.555168
$89$	−104185.	−1.39422	−0.697111	−	0.716963i	$-0.745532\pi$
$89$	−104185.	−1.39422	−0.697111	+	0.716963i	$0.745532\pi$
$90$	0	0
$91$	118908.	1.50524
$92$	−1925.67	−0.0237198
$93$	0	0
$94$	−137314.	−1.60286
$95$	−93639.5	−1.06451
$96$	0	0
$97$	−117345.	−1.26629	−0.633147	−	0.774032i	$-0.718237\pi$
$97$	−117345.	−1.26629	−0.633147	+	0.774032i	$0.718237\pi$
$98$	−15734.2	−0.165493
$99$	0	0
$100$	4171.22	0.0417122
$101$	−109234.	−1.06550	−0.532751	−	0.846272i	$-0.678842\pi$
$101$	−109234.	−1.06550	−0.532751	+	0.846272i	$0.678842\pi$
$102$	0	0
$103$	−162469.	−1.50896	−0.754481	−	0.656322i	$-0.772111\pi$
$103$	−162469.	−1.50896	−0.754481	+	0.656322i	$0.772111\pi$
$104$	−169557.	−1.53721
$105$	0	0
$106$	−113550.	−0.981570
$107$	9555.85	0.0806882	0.0403441	−	0.999186i	$-0.487155\pi$
$107$	9555.85	0.0806882	0.0403441	+	0.999186i	$0.487155\pi$
$108$	0	0
$109$	−80849.6	−0.651796	−0.325898	−	0.945405i	$-0.605667\pi$
$109$	−80849.6	−0.651796	−0.325898	+	0.945405i	$0.605667\pi$
$110$	92966.1	0.732559
$111$	0	0
$112$	−133824.	−1.00806
$113$	131306.	0.967359	0.483679	−	0.875245i	$-0.339300\pi$
$113$	131306.	0.967359	0.483679	+	0.875245i	$0.339300\pi$
$114$	0	0
$115$	35878.5	0.252982
$116$	5452.70	0.0376241
$117$	0	0
$118$	20747.2	0.137169
$119$	151924.	0.983466
$120$	0	0
$121$	−104588.	−0.649407
$122$	−43124.1	−0.262314
$123$	0	0
$124$	−26667.9	−0.155752
$125$	127416.	0.729371
$126$	0	0
$127$	−234331.	−1.28920	−0.644600	−	0.764520i	$-0.722976\pi$
$127$	−234331.	−1.28920	−0.644600	+	0.764520i	$0.722976\pi$
$128$	209963.	1.13270
$129$	0	0
$130$	−390850.	−2.02839
$131$	−135156.	−0.688108	−0.344054	−	0.938950i	$-0.611800\pi$
$131$	−135156.	−0.688108	−0.344054	+	0.938950i	$0.611800\pi$
$132$	0	0
$133$	169791.	0.832309
$134$	416457.	2.00359
$135$	0	0
$136$	−216638.	−1.00435
$137$	133343.	0.606971	0.303486	−	0.952836i	$-0.401850\pi$
$137$	133343.	0.606971	0.303486	+	0.952836i	$0.401850\pi$
$138$	0	0
$139$	197409.	0.866622	0.433311	−	0.901244i	$-0.357345\pi$
$139$	197409.	0.866622	0.433311	+	0.901244i	$0.357345\pi$
$140$	−27527.0	−0.118697
$141$	0	0
$142$	−106246.	−0.442171
$143$	−237384.	−0.970761
$144$	0	0
$145$	−101593.	−0.401277
$146$	54177.1	0.210346
$147$	0	0
$148$	−3566.20	−0.0133829
$149$	−94205.2	−0.347624	−0.173812	−	0.984779i	$-0.555608\pi$
$149$	−94205.2	−0.347624	−0.173812	+	0.984779i	$0.555608\pi$
$150$	0	0
$151$	−32019.8	−0.114281	−0.0571407	−	0.998366i	$-0.518198\pi$
$151$	−32019.8	−0.114281	−0.0571407	+	0.998366i	$0.518198\pi$
$152$	−242115.	−0.849987
$153$	0	0
$154$	−168570.	−0.572766
$155$	496869.	1.66116
$156$	0	0
$157$	487099.	1.57713	0.788565	−	0.614951i	$-0.210824\pi$
$157$	487099.	1.57713	0.788565	+	0.614951i	$0.210824\pi$
$158$	187677.	0.598091
$159$	0	0
$160$	83361.0	0.257432
$161$	−65056.2	−0.197799
$162$	0	0
$163$	349125.	1.02923	0.514615	−	0.857421i	$-0.327935\pi$
$163$	349125.	1.02923	0.514615	+	0.857421i	$0.327935\pi$
$164$	−46734.2	−0.135683
$165$	0	0
$166$	608084.	1.71275
$167$	−68091.2	−0.188930	−0.0944648	−	0.995528i	$-0.530114\pi$
$167$	−68091.2	−0.188930	−0.0944648	+	0.995528i	$0.530114\pi$
$168$	0	0
$169$	626725.	1.68795
$170$	−499376.	−1.32527
$171$	0	0
$172$	−76783.7	−0.197901
$173$	−15052.9	−0.0382388	−0.0191194	−	0.999817i	$-0.506086\pi$
$173$	−15052.9	−0.0382388	−0.0191194	+	0.999817i	$0.506086\pi$
$174$	0	0
$175$	140920.	0.347837
$176$	267163.	0.650122
$177$	0	0
$178$	−620959.	−1.46897
$179$	−716413.	−1.67121	−0.835604	−	0.549332i	$-0.814882\pi$
$179$	−716413.	−1.67121	−0.835604	+	0.549332i	$0.814882\pi$
$180$	0	0
$181$	−702418.	−1.59367	−0.796837	−	0.604194i	$-0.793495\pi$
$181$	−702418.	−1.59367	−0.796837	+	0.604194i	$0.793495\pi$
$182$	708704.	1.58594
$183$	0	0
$184$	92767.5	0.202000
$185$	66444.3	0.142734
$186$	0	0
$187$	−303298.	−0.634258
$188$	−81169.3	−0.167493
$189$	0	0
$190$	−558104.	−1.12158
$191$	28509.8	0.0565472	0.0282736	−	0.999600i	$-0.490999\pi$
$191$	28509.8	0.0565472	0.0282736	+	0.999600i	$0.490999\pi$
$192$	0	0
$193$	−310039.	−0.599133	−0.299567	−	0.954075i	$-0.596842\pi$
$193$	−310039.	−0.599133	−0.299567	+	0.954075i	$0.596842\pi$
$194$	−699390.	−1.33418
$195$	0	0
$196$	−9300.85	−0.0172935
$197$	−664034.	−1.21906	−0.609530	−	0.792763i	$-0.708642\pi$
$197$	−664034.	−1.21906	−0.609530	+	0.792763i	$0.708642\pi$
$198$	0	0
$199$	326015.	0.583586	0.291793	−	0.956481i	$-0.405748\pi$
$199$	326015.	0.583586	0.291793	+	0.956481i	$0.405748\pi$
$200$	−200946.	−0.355225
$201$	0	0
$202$	−651048.	−1.12263
$203$	184213.	0.313747
$204$	0	0
$205$	870739.	1.44712
$206$	−968338.	−1.58986
$207$	0	0
$208$	−1.12321e6	−1.80013
$209$	−338967.	−0.536774
$210$	0	0
$211$	−775314.	−1.19887	−0.599434	−	0.800424i	$-0.704608\pi$
$211$	−775314.	−1.19887	−0.599434	+	0.800424i	$0.704608\pi$
$212$	−67121.7	−0.102571
$213$	0	0
$214$	56954.1	0.0850141
$215$	1.43061e6	2.11070
$216$	0	0
$217$	−900942.	−1.29882
$218$	−481874.	−0.686740
$219$	0	0
$220$	54954.2	0.0765499
$221$	1.27513e6	1.75620
$222$	0	0
$223$	−855406.	−1.15189	−0.575944	−	0.817489i	$-0.695365\pi$
$223$	−855406.	−1.15189	−0.575944	+	0.817489i	$0.695365\pi$
$224$	−151153.	−0.201279
$225$	0	0
$226$	782599.	1.01922
$227$	−255022.	−0.328483	−0.164241	−	0.986420i	$-0.552518\pi$
$227$	−255022.	−0.328483	−0.164241	+	0.986420i	$0.552518\pi$
$228$	0	0
$229$	504531.	0.635769	0.317884	−	0.948129i	$-0.397028\pi$
$229$	504531.	0.635769	0.317884	+	0.948129i	$0.397028\pi$
$230$	213840.	0.266545
$231$	0	0
$232$	−262680.	−0.320410
$233$	−315787.	−0.381070	−0.190535	−	0.981680i	$-0.561022\pi$
$233$	−315787.	−0.381070	−0.190535	+	0.981680i	$0.561022\pi$
$234$	0	0
$235$	1.51232e6	1.78638
$236$	12264.1	0.0143336
$237$	0	0
$238$	905488.	1.03619
$239$	1.31913e6	1.49380	0.746901	−	0.664935i	$-0.231541\pi$
$239$	1.31913e6	1.49380	0.746901	+	0.664935i	$0.231541\pi$
$240$	0	0
$241$	−101157.	−0.112190	−0.0560950	−	0.998425i	$-0.517865\pi$
$241$	−101157.	−0.112190	−0.0560950	+	0.998425i	$0.517865\pi$
$242$	−623356.	−0.684223
$243$	0	0
$244$	−25491.6	−0.0274109
$245$	173291.	0.184442
$246$	0	0
$247$	1.42509e6	1.48628
$248$	1.28471e6	1.32640
$249$	0	0
$250$	759415.	0.768474
$251$	−1.78020e6	−1.78354	−0.891771	−	0.452487i	$-0.850537\pi$
$251$	−1.78020e6	−1.78354	−0.891771	+	0.452487i	$0.850537\pi$
$252$	0	0
$253$	129877.	0.127565
$254$	−1.39664e6	−1.35832
$255$	0	0
$256$	342297.	0.326440
$257$	381509.	0.360307	0.180153	−	0.983639i	$-0.442341\pi$
$257$	381509.	0.360307	0.180153	+	0.983639i	$0.442341\pi$
$258$	0	0
$259$	−120479.	−0.111600
$260$	−231040.	−0.211960
$261$	0	0
$262$	−805547.	−0.724999
$263$	225601.	0.201118	0.100559	−	0.994931i	$-0.467937\pi$
$263$	225601.	0.201118	0.100559	+	0.994931i	$0.467937\pi$
$264$	0	0
$265$	1.25059e6	1.09396
$266$	1.01197e6	0.876931
$267$	0	0
$268$	246177.	0.209368
$269$	562259.	0.473757	0.236879	−	0.971539i	$-0.423876\pi$
$269$	562259.	0.473757	0.236879	+	0.971539i	$0.423876\pi$
$270$	0	0
$271$	−1.36135e6	−1.12602	−0.563012	−	0.826449i	$-0.690358\pi$
$271$	−1.36135e6	−1.12602	−0.563012	+	0.826449i	$0.690358\pi$
$272$	−1.43509e6	−1.17614
$273$	0	0
$274$	794741.	0.639512
$275$	−281329.	−0.224327
$276$	0	0
$277$	−2.03987e6	−1.59736	−0.798681	−	0.601755i	$-0.794468\pi$
$277$	−2.03987e6	−1.59736	−0.798681	+	0.601755i	$0.794468\pi$
$278$	1.17658e6	0.913083
$279$	0	0
$280$	1.32609e6	1.01083
$281$	667020.	0.503933	0.251967	−	0.967736i	$-0.418923\pi$
$281$	667020.	0.503933	0.251967	+	0.967736i	$0.418923\pi$
$282$	0	0
$283$	1.41041e6	1.04684	0.523418	−	0.852076i	$-0.324657\pi$
$283$	1.41041e6	1.04684	0.523418	+	0.852076i	$0.324657\pi$
$284$	−62804.0	−0.0462053
$285$	0	0
$286$	−1.41484e6	−1.02281
$287$	−1.57886e6	−1.13146
$288$	0	0
$289$	209339.	0.147437
$290$	−605508.	−0.422790
$291$	0	0
$292$	32025.2	0.0219804
$293$	−928970.	−0.632168	−0.316084	−	0.948731i	$-0.602368\pi$
$293$	−928970.	−0.632168	−0.316084	+	0.948731i	$0.602368\pi$
$294$	0	0
$295$	−228502.	−0.152874
$296$	171799.	0.113970
$297$	0	0
$298$	−561475.	−0.366261
$299$	−546031.	−0.353215
$300$	0	0
$301$	−2.59404e6	−1.65029
$302$	−190842.	−0.120408
$303$	0	0
$304$	−1.60386e6	−0.995366
$305$	474952.	0.292348
$306$	0	0
$307$	−756619.	−0.458175	−0.229087	−	0.973406i	$-0.573574\pi$
$307$	−756619.	−0.458175	−0.229087	+	0.973406i	$0.573574\pi$
$308$	−99645.1	−0.0598521
$309$	0	0
$310$	2.96140e6	1.75022
$311$	1.45073e6	0.850521	0.425260	−	0.905071i	$-0.360182\pi$
$311$	1.45073e6	0.850521	0.425260	+	0.905071i	$0.360182\pi$
$312$	0	0
$313$	−401791.	−0.231814	−0.115907	−	0.993260i	$-0.536977\pi$
$313$	−401791.	−0.231814	−0.115907	+	0.993260i	$0.536977\pi$
$314$	2.90317e6	1.66168
$315$	0	0
$316$	110940.	0.0624985
$317$	1.81917e6	1.01678	0.508388	−	0.861128i	$-0.330241\pi$
$317$	1.81917e6	1.01678	0.508388	+	0.861128i	$0.330241\pi$
$318$	0	0
$319$	−367758.	−0.202342
$320$	−1.86488e6	−1.01807
$321$	0	0
$322$	−387743.	−0.208403
$323$	1.82079e6	0.971078
$324$	0	0
$325$	1.18277e6	0.621143
$326$	2.08083e6	1.08441
$327$	0	0
$328$	2.25139e6	1.15549
$329$	−2.74220e6	−1.39672
$330$	0	0
$331$	3.27514e6	1.64309	0.821543	−	0.570147i	$-0.193114\pi$
$331$	3.27514e6	1.64309	0.821543	+	0.570147i	$0.193114\pi$
$332$	359451.	0.178976
$333$	0	0
$334$	−405833.	−0.199059
$335$	−4.58669e6	−2.23299
$336$	0	0
$337$	1.40437e6	0.673606	0.336803	−	0.941575i	$-0.390654\pi$
$337$	1.40437e6	0.673606	0.336803	+	0.941575i	$0.390654\pi$
$338$	3.73536e6	1.77845
$339$	0	0
$340$	−295192.	−0.138486
$341$	1.79862e6	0.837633
$342$	0	0
$343$	−2.31468e6	−1.06232
$344$	3.69900e6	1.68534
$345$	0	0
$346$	−89717.1	−0.0402888
$347$	−4.10924e6	−1.83205	−0.916025	−	0.401120i	$-0.868621\pi$
$347$	−4.10924e6	−1.83205	−0.916025	+	0.401120i	$0.868621\pi$
$348$	0	0
$349$	65008.4	0.0285697	0.0142849	−	0.999898i	$-0.495453\pi$
$349$	65008.4	0.0285697	0.0142849	+	0.999898i	$0.495453\pi$
$350$	839899.	0.366486
$351$	0	0
$352$	301759.	0.129809
$353$	−2.87105e6	−1.22632	−0.613160	−	0.789959i	$-0.710102\pi$
$353$	−2.87105e6	−1.22632	−0.613160	+	0.789959i	$0.710102\pi$
$354$	0	0
$355$	1.17015e6	0.492799
$356$	−367062.	−0.153502
$357$	0	0
$358$	−4.26991e6	−1.76081
$359$	2.59406e6	1.06229	0.531146	−	0.847281i	$-0.321762\pi$
$359$	2.59406e6	1.06229	0.531146	+	0.847281i	$0.321762\pi$
$360$	0	0
$361$	−441179.	−0.178175
$362$	−4.18651e6	−1.67911
$363$	0	0
$364$	418930.	0.165725
$365$	−596685.	−0.234430
$366$	0	0
$367$	1.69904e6	0.658472	0.329236	−	0.944248i	$-0.393209\pi$
$367$	1.69904e6	0.658472	0.329236	+	0.944248i	$0.393209\pi$
$368$	614528.	0.236550
$369$	0	0
$370$	396017.	0.150387
$371$	−2.26762e6	−0.855334
$372$	0	0
$373$	1.30896e6	0.487142	0.243571	−	0.969883i	$-0.421681\pi$
$373$	1.30896e6	0.487142	0.243571	+	0.969883i	$0.421681\pi$
$374$	−1.80770e6	−0.668262
$375$	0	0
$376$	3.91027e6	1.42639
$377$	1.54614e6	0.560266
$378$	0	0
$379$	−2.85503e6	−1.02097	−0.510485	−	0.859887i	$-0.670534\pi$
$379$	−2.85503e6	−1.02097	−0.510485	+	0.859887i	$0.670534\pi$
$380$	−329907.	−0.117201
$381$	0	0
$382$	169922.	0.0595788
$383$	1.29291e6	0.450371	0.225186	−	0.974316i	$-0.427701\pi$
$383$	1.29291e6	0.450371	0.225186	+	0.974316i	$0.427701\pi$
$384$	0	0
$385$	1.85656e6	0.638347
$386$	−1.84788e6	−0.631254
$387$	0	0
$388$	−413425.	−0.139417
$389$	514685.	0.172452	0.0862259	−	0.996276i	$-0.472519\pi$
$389$	514685.	0.172452	0.0862259	+	0.996276i	$0.472519\pi$
$390$	0	0
$391$	−697646.	−0.230777
$392$	448061.	0.147273
$393$	0	0
$394$	−3.95773e6	−1.28442
$395$	−2.06700e6	−0.666572
$396$	0	0
$397$	−3.92612e6	−1.25022	−0.625112	−	0.780535i	$-0.714947\pi$
$397$	−3.92612e6	−1.25022	−0.625112	+	0.780535i	$0.714947\pi$
$398$	1.94309e6	0.614873
$399$	0	0
$400$	−1.33114e6	−0.415982
$401$	2.41292e6	0.749346	0.374673	−	0.927157i	$-0.377755\pi$
$401$	2.41292e6	0.749346	0.374673	+	0.927157i	$0.377755\pi$
$402$	0	0
$403$	−7.56181e6	−2.31933
$404$	−384849.	−0.117310
$405$	0	0
$406$	1.09793e6	0.330567
$407$	240523.	0.0719730
$408$	0	0
$409$	3.67029e6	1.08491	0.542454	−	0.840086i	$-0.317495\pi$
$409$	3.67029e6	1.08491	0.542454	+	0.840086i	$0.317495\pi$
$410$	5.18972e6	1.52470
$411$	0	0
$412$	−572405.	−0.166135
$413$	414328.	0.119528
$414$	0	0
$415$	−6.69720e6	−1.90885
$416$	−1.26866e6	−0.359429
$417$	0	0
$418$	−2.02029e6	−0.565551
$419$	1.90475e6	0.530034	0.265017	−	0.964244i	$-0.414622\pi$
$419$	1.90475e6	0.530034	0.265017	+	0.964244i	$0.414622\pi$
$420$	0	0
$421$	4.91363e6	1.35113	0.675565	−	0.737300i	$-0.263900\pi$
$421$	4.91363e6	1.35113	0.675565	+	0.737300i	$0.263900\pi$
$422$	−4.62097e6	−1.26314
$423$	0	0
$424$	3.23354e6	0.873500
$425$	1.51118e6	0.405831
$426$	0	0
$427$	−861202.	−0.228578
$428$	33666.8	0.00888367
$429$	0	0
$430$	8.52664e6	2.22386
$431$	3.65588e6	0.947979	0.473990	−	0.880530i	$-0.342813\pi$
$431$	3.65588e6	0.947979	0.473990	+	0.880530i	$0.342813\pi$
$432$	0	0
$433$	−5.06298e6	−1.29774	−0.648869	−	0.760900i	$-0.724758\pi$
$433$	−5.06298e6	−1.29774	−0.648869	+	0.760900i	$0.724758\pi$
$434$	−5.36973e6	−1.36845
$435$	0	0
$436$	−284846.	−0.0717619
$437$	−779690.	−0.195307
$438$	0	0
$439$	−3.94551e6	−0.977108	−0.488554	−	0.872534i	$-0.662475\pi$
$439$	−3.94551e6	−0.977108	−0.488554	+	0.872534i	$0.662475\pi$
$440$	−2.64738e6	−0.651905
$441$	0	0
$442$	7.59996e6	1.85036
$443$	3.37423e6	0.816894	0.408447	−	0.912782i	$-0.366070\pi$
$443$	3.37423e6	0.816894	0.408447	+	0.912782i	$0.366070\pi$
$444$	0	0
$445$	6.83900e6	1.63716
$446$	−5.09833e6	−1.21364
$447$	0	0
$448$	3.38147e6	0.795995
$449$	5.64111e6	1.32053	0.660266	−	0.751032i	$-0.270444\pi$
$449$	5.64111e6	1.32053	0.660266	+	0.751032i	$0.270444\pi$
$450$	0	0
$451$	3.15200e6	0.729700
$452$	462611.	0.106505
$453$	0	0
$454$	−1.51996e6	−0.346094
$455$	−7.80539e6	−1.76753
$456$	0	0
$457$	−134096.	−0.0300348	−0.0150174	−	0.999887i	$-0.504780\pi$
$457$	−134096.	−0.0300348	−0.0150174	+	0.999887i	$0.504780\pi$
$458$	3.00707e6	0.669854
$459$	0	0
$460$	126406.	0.0278530
$461$	−4.00954e6	−0.878703	−0.439351	−	0.898315i	$-0.644792\pi$
$461$	−4.00954e6	−0.878703	−0.439351	+	0.898315i	$0.644792\pi$
$462$	0	0
$463$	8.02971e6	1.74079	0.870397	−	0.492351i	$-0.163862\pi$
$463$	8.02971e6	1.74079	0.870397	+	0.492351i	$0.163862\pi$
$464$	−1.74009e6	−0.375212
$465$	0	0
$466$	−1.88213e6	−0.401500
$467$	1.17884e6	0.250129	0.125065	−	0.992149i	$-0.460086\pi$
$467$	1.17884e6	0.250129	0.125065	+	0.992149i	$0.460086\pi$
$468$	0	0
$469$	8.31677e6	1.74591
$470$	9.01365e6	1.88216
$471$	0	0
$472$	−590815.	−0.122067
$473$	5.17869e6	1.06431
$474$	0	0
$475$	1.68890e6	0.343456
$476$	535254.	0.108278
$477$	0	0
$478$	7.86219e6	1.57389
$479$	7.03708e6	1.40137	0.700686	−	0.713470i	$-0.252877\pi$
$479$	7.03708e6	1.40137	0.700686	+	0.713470i	$0.252877\pi$
$480$	0	0
$481$	−1.01121e6	−0.199287
$482$	−602910.	−0.118205
$483$	0	0
$484$	−368479.	−0.0714990
$485$	7.70281e6	1.48694
$486$	0	0
$487$	−1.74859e6	−0.334093	−0.167046	−	0.985949i	$-0.553423\pi$
$487$	−1.74859e6	−0.334093	−0.167046	+	0.985949i	$0.553423\pi$
$488$	1.22804e6	0.233433
$489$	0	0
$490$	1.03284e6	0.194330
$491$	−5.71036e6	−1.06896	−0.534478	−	0.845183i	$-0.679492\pi$
$491$	−5.71036e6	−1.06896	−0.534478	+	0.845183i	$0.679492\pi$
$492$	0	0
$493$	1.97545e6	0.366057
$494$	8.49373e6	1.56596
$495$	0	0
$496$	8.51040e6	1.55327
$497$	−2.12176e6	−0.385305
$498$	0	0
$499$	−3.18564e6	−0.572724	−0.286362	−	0.958121i	$-0.592446\pi$
$499$	−3.18564e6	−0.572724	−0.286362	+	0.958121i	$0.592446\pi$
$500$	448907.	0.0803029
$501$	0	0
$502$	−1.06102e7	−1.87916
$503$	−7.57762e6	−1.33541	−0.667703	−	0.744428i	$-0.732722\pi$
$503$	−7.57762e6	−1.33541	−0.667703	+	0.744428i	$0.732722\pi$
$504$	0	0
$505$	7.17039e6	1.25116
$506$	774083.	0.134404
$507$	0	0
$508$	−825586.	−0.141939
$509$	6.23348e6	1.06644	0.533220	−	0.845977i	$-0.320982\pi$
$509$	6.23348e6	1.06644	0.533220	+	0.845977i	$0.320982\pi$
$510$	0	0
$511$	1.08193e6	0.183294
$512$	−4.67867e6	−0.788764
$513$	0	0
$514$	2.27385e6	0.379624
$515$	1.06649e7	1.77190
$516$	0	0
$517$	5.47448e6	0.900775
$518$	−718073.	−0.117583
$519$	0	0
$520$	1.11302e7	1.80507
$521$	4.48711e6	0.724224	0.362112	−	0.932135i	$-0.382056\pi$
$521$	4.48711e6	0.724224	0.362112	+	0.932135i	$0.382056\pi$
$522$	0	0
$523$	−6.66955e6	−1.06621	−0.533104	−	0.846050i	$-0.678975\pi$
$523$	−6.66955e6	−1.06621	−0.533104	+	0.846050i	$0.678975\pi$
$524$	−476176.	−0.0757599
$525$	0	0
$526$	1.34461e6	0.211901
$527$	−9.66147e6	−1.51536
$528$	0	0
$529$	−6.13760e6	−0.953585
$530$	7.45369e6	1.15261
$531$	0	0
$532$	598200.	0.0916363
$533$	−1.32517e7	−2.02048
$534$	0	0
$535$	−627270.	−0.0947481
$536$	−1.18594e7	−1.78299
$537$	0	0
$538$	3.35114e6	0.499157
$539$	627297.	0.0930039
$540$	0	0
$541$	−5.56886e6	−0.818038	−0.409019	−	0.912526i	$-0.634129\pi$
$541$	−5.56886e6	−0.818038	−0.409019	+	0.912526i	$0.634129\pi$
$542$	−8.11385e6	−1.18639
$543$	0	0
$544$	−1.62093e6	−0.234837
$545$	5.30717e6	0.765371
$546$	0	0
$547$	1.15071e7	1.64437	0.822184	−	0.569222i	$-0.192756\pi$
$547$	1.15071e7	1.64437	0.822184	+	0.569222i	$0.192756\pi$
$548$	469788.	0.0668268
$549$	0	0
$550$	−1.67676e6	−0.236354
$551$	2.20776e6	0.309794
$552$	0	0
$553$	3.74796e6	0.521173
$554$	−1.21579e7	−1.68300
$555$	0	0
$556$	695503.	0.0954140
$557$	−3.88277e6	−0.530277	−0.265139	−	0.964210i	$-0.585418\pi$
$557$	−3.88277e6	−0.530277	−0.265139	+	0.964210i	$0.585418\pi$
$558$	0	0
$559$	−2.17724e7	−2.94697
$560$	8.78454e6	1.18372
$561$	0	0
$562$	3.97553e6	0.530950
$563$	−981269.	−0.130472	−0.0652360	−	0.997870i	$-0.520780\pi$
$563$	−981269.	−0.130472	−0.0652360	+	0.997870i	$0.520780\pi$
$564$	0	0
$565$	−8.61924e6	−1.13592
$566$	8.40621e6	1.10296
$567$	0	0
$568$	3.02554e6	0.393488
$569$	8.63190e6	1.11770	0.558850	−	0.829268i	$-0.311243\pi$
$569$	8.63190e6	1.11770	0.558850	+	0.829268i	$0.311243\pi$
$570$	0	0
$571$	1.06342e7	1.36494	0.682469	−	0.730915i	$-0.260906\pi$
$571$	1.06342e7	1.36494	0.682469	+	0.730915i	$0.260906\pi$
$572$	−836344.	−0.106880
$573$	0	0
$574$	−9.41019e6	−1.19212
$575$	−647112.	−0.0816225
$576$	0	0
$577$	3.83248e6	0.479225	0.239613	−	0.970869i	$-0.422980\pi$
$577$	3.83248e6	0.479225	0.239613	+	0.970869i	$0.422980\pi$
$578$	1.24769e6	0.155341
$579$	0	0
$580$	−357929.	−0.0441801
$581$	1.21436e7	1.49248
$582$	0	0
$583$	4.52703e6	0.551622
$584$	−1.54279e6	−0.187187
$585$	0	0
$586$	−5.53678e6	−0.666060
$587$	3.74170e6	0.448202	0.224101	−	0.974566i	$-0.428055\pi$
$587$	3.74170e6	0.448202	0.224101	+	0.974566i	$0.428055\pi$
$588$	0	0
$589$	−1.07977e7	−1.28245
$590$	−1.36190e6	−0.161070
$591$	0	0
$592$	1.13806e6	0.133463
$593$	411565.	0.0480620	0.0240310	−	0.999711i	$-0.492350\pi$
$593$	411565.	0.0480620	0.0240310	+	0.999711i	$0.492350\pi$
$594$	0	0
$595$	−9.97269e6	−1.15483
$596$	−331900.	−0.0382729
$597$	0	0
$598$	−3.25442e6	−0.372152
$599$	−6.06004e6	−0.690094	−0.345047	−	0.938585i	$-0.612137\pi$
$599$	−6.06004e6	−0.690094	−0.345047	+	0.938585i	$0.612137\pi$
$600$	0	0
$601$	6.69768e6	0.756376	0.378188	−	0.925729i	$-0.376547\pi$
$601$	6.69768e6	0.756376	0.378188	+	0.925729i	$0.376547\pi$
$602$	−1.54608e7	−1.73877
$603$	0	0
$604$	−112811.	−0.0125822
$605$	6.86540e6	0.762566
$606$	0	0
$607$	−1.00236e6	−0.110421	−0.0552106	−	0.998475i	$-0.517583\pi$
$607$	−1.00236e6	−0.110421	−0.0552106	+	0.998475i	$0.517583\pi$
$608$	−1.81155e6	−0.198743
$609$	0	0
$610$	2.83078e6	0.308022
$611$	−2.30159e7	−2.49417
$612$	0	0
$613$	−1.24680e7	−1.34013	−0.670064	−	0.742303i	$-0.733733\pi$
$613$	−1.24680e7	−1.34013	−0.670064	+	0.742303i	$0.733733\pi$
$614$	−4.50955e6	−0.482739
$615$	0	0
$616$	4.80033e6	0.509705
$617$	7.98964e6	0.844917	0.422459	−	0.906382i	$-0.361167\pi$
$617$	7.98964e6	0.844917	0.422459	+	0.906382i	$0.361167\pi$
$618$	0	0
$619$	−8.83016e6	−0.926279	−0.463140	−	0.886285i	$-0.653277\pi$
$619$	−8.83016e6	−0.926279	−0.463140	+	0.886285i	$0.653277\pi$
$620$	1.75055e6	0.182892
$621$	0	0
$622$	8.64653e6	0.896119
$623$	−1.24007e7	−1.28005
$624$	0	0
$625$	−1.20637e7	−1.23533
$626$	−2.39473e6	−0.244242
$627$	0	0
$628$	1.71613e6	0.173640
$629$	−1.29199e6	−0.130207
$630$	0	0
$631$	−8.18563e6	−0.818425	−0.409212	−	0.912439i	$-0.634196\pi$
$631$	−8.18563e6	−0.818425	−0.409212	+	0.912439i	$0.634196\pi$
$632$	−5.34444e6	−0.532242
$633$	0	0
$634$	1.08425e7	1.07129
$635$	1.53821e7	1.51384
$636$	0	0
$637$	−2.63730e6	−0.257520
$638$	−2.19189e6	−0.213190
$639$	0	0
$640$	−1.37825e7	−1.33008
$641$	1.08929e7	1.04713	0.523564	−	0.851986i	$-0.324602\pi$
$641$	1.08929e7	1.04713	0.523564	+	0.851986i	$0.324602\pi$
$642$	0	0
$643$	4.87772e6	0.465254	0.232627	−	0.972566i	$-0.425268\pi$
$643$	4.87772e6	0.465254	0.232627	+	0.972566i	$0.425268\pi$
$644$	−229204.	−0.0217774
$645$	0	0
$646$	1.08522e7	1.02314
$647$	5.09261e6	0.478277	0.239138	−	0.970985i	$-0.423135\pi$
$647$	5.09261e6	0.478277	0.239138	+	0.970985i	$0.423135\pi$
$648$	0	0
$649$	−827156.	−0.0770860
$650$	7.04946e6	0.654444
$651$	0	0
$652$	1.23002e6	0.113317
$653$	1.57557e7	1.44595	0.722976	−	0.690873i	$-0.242774\pi$
$653$	1.57557e7	1.44595	0.722976	+	0.690873i	$0.242774\pi$
$654$	0	0
$655$	8.87198e6	0.808011
$656$	1.49141e7	1.35312
$657$	0	0
$658$	−1.63439e7	−1.47160
$659$	−1.52866e7	−1.37119	−0.685596	−	0.727983i	$-0.740458\pi$
$659$	−1.52866e7	−1.37119	−0.685596	+	0.727983i	$0.740458\pi$
$660$	0	0
$661$	−7.43781e6	−0.662127	−0.331064	−	0.943608i	$-0.607408\pi$
$661$	−7.43781e6	−0.662127	−0.331064	+	0.943608i	$0.607408\pi$
$662$	1.95203e7	1.73117
$663$	0	0
$664$	−1.73163e7	−1.52418
$665$	−1.11455e7	−0.977339
$666$	0	0
$667$	−845916.	−0.0736229
$668$	−239896.	−0.0208009
$669$	0	0
$670$	−2.73373e7	−2.35271
$671$	1.71929e6	0.147415
$672$	0	0
$673$	−2.00568e7	−1.70697	−0.853483	−	0.521121i	$-0.825514\pi$
$673$	−2.00568e7	−1.70697	−0.853483	+	0.521121i	$0.825514\pi$
$674$	8.37021e6	0.709719
$675$	0	0
$676$	2.20805e6	0.185841
$677$	−2.08686e6	−0.174993	−0.0874965	−	0.996165i	$-0.527887\pi$
$677$	−2.08686e6	−0.174993	−0.0874965	+	0.996165i	$0.527887\pi$
$678$	0	0
$679$	−1.39670e7	−1.16260
$680$	1.42207e7	1.17936
$681$	0	0
$682$	1.07200e7	0.882540
$683$	−5.45861e6	−0.447745	−0.223872	−	0.974618i	$-0.571870\pi$
$683$	−5.45861e6	−0.447745	−0.223872	+	0.974618i	$0.571870\pi$
$684$	0	0
$685$	−8.75296e6	−0.712736
$686$	−1.37958e7	−1.11927
$687$	0	0
$688$	2.45036e7	1.97360
$689$	−1.90326e7	−1.52739
$690$	0	0
$691$	655073.	0.0521908	0.0260954	−	0.999659i	$-0.491693\pi$
$691$	655073.	0.0521908	0.0260954	+	0.999659i	$0.491693\pi$
$692$	−53033.7	−0.00421004
$693$	0	0
$694$	−2.44916e7	−1.93027
$695$	−1.29584e7	−1.01763
$696$	0	0
$697$	−1.69313e7	−1.32010
$698$	387459.	0.0301014
$699$	0	0
$700$	496482.	0.0382965
$701$	2.15464e7	1.65607	0.828037	−	0.560674i	$-0.189458\pi$
$701$	2.15464e7	1.65607	0.828037	+	0.560674i	$0.189458\pi$
$702$	0	0
$703$	−1.44393e6	−0.110194
$704$	−6.75069e6	−0.513354
$705$	0	0
$706$	−1.71118e7	−1.29207
$707$	−1.30016e7	−0.978248
$708$	0	0
$709$	1.65045e7	1.23307	0.616533	−	0.787329i	$-0.288537\pi$
$709$	1.65045e7	1.23307	0.616533	+	0.787329i	$0.288537\pi$
$710$	6.97423e6	0.519219
$711$	0	0
$712$	1.76829e7	1.30724
$713$	4.13719e6	0.304776
$714$	0	0
$715$	1.55825e7	1.13992
$716$	−2.52404e6	−0.183998
$717$	0	0
$718$	1.54609e7	1.11924
$719$	−3.65458e6	−0.263642	−0.131821	−	0.991274i	$-0.542082\pi$
$719$	−3.65458e6	−0.263642	−0.131821	+	0.991274i	$0.542082\pi$
$720$	0	0
$721$	−1.93380e7	−1.38539
$722$	−2.62948e6	−0.187727
$723$	0	0
$724$	−2.47473e6	−0.175462
$725$	1.83236e6	0.129469
$726$	0	0
$727$	1.40192e7	0.983757	0.491878	−	0.870664i	$-0.336310\pi$
$727$	1.40192e7	0.983757	0.491878	+	0.870664i	$0.336310\pi$
$728$	−2.01817e7	−1.41133
$729$	0	0
$730$	−3.55632e6	−0.246998
$731$	−2.78178e7	−1.92544
$732$	0	0
$733$	−2.41229e7	−1.65833	−0.829163	−	0.559007i	$-0.811183\pi$
$733$	−2.41229e7	−1.65833	−0.829163	+	0.559007i	$0.811183\pi$
$734$	1.01265e7	0.693774
$735$	0	0
$736$	694106.	0.0472315
$737$	−1.66034e7	−1.12597
$738$	0	0
$739$	−1.00291e7	−0.675537	−0.337768	−	0.941229i	$-0.609672\pi$
$739$	−1.00291e7	−0.675537	−0.337768	+	0.941229i	$0.609672\pi$
$740$	234094.	0.0157149
$741$	0	0
$742$	−1.35153e7	−0.901190
$743$	−4.51165e6	−0.299822	−0.149911	−	0.988699i	$-0.547899\pi$
$743$	−4.51165e6	−0.299822	−0.149911	+	0.988699i	$0.547899\pi$
$744$	0	0
$745$	6.18387e6	0.408197
$746$	7.80160e6	0.513259
$747$	0	0
$748$	−1.06857e6	−0.0698310
$749$	1.13739e6	0.0740807
$750$	0	0
$751$	1.58055e7	1.02260	0.511302	−	0.859401i	$-0.329163\pi$
$751$	1.58055e7	1.02260	0.511302	+	0.859401i	$0.329163\pi$
$752$	2.59031e7	1.67035
$753$	0	0
$754$	9.21518e6	0.590304
$755$	2.10186e6	0.134195
$756$	0	0
$757$	7.71419e6	0.489272	0.244636	−	0.969615i	$-0.421331\pi$
$757$	7.71419e6	0.489272	0.244636	+	0.969615i	$0.421331\pi$
$758$	−1.70163e7	−1.07571
$759$	0	0
$760$	1.58930e7	0.998097
$761$	1.26944e7	0.794603	0.397302	−	0.917688i	$-0.369947\pi$
$761$	1.26944e7	0.794603	0.397302	+	0.917688i	$0.369947\pi$
$762$	0	0
$763$	−9.62316e6	−0.598421
$764$	100445.	0.00622578
$765$	0	0
$766$	7.70590e6	0.474517
$767$	3.47755e6	0.213444
$768$	0	0
$769$	−3.29528e6	−0.200944	−0.100472	−	0.994940i	$-0.532035\pi$
$769$	−3.29528e6	−0.200944	−0.100472	+	0.994940i	$0.532035\pi$
$770$	1.10653e7	0.672571
$771$	0	0
$772$	−1.09232e6	−0.0659639
$773$	5.51865e6	0.332188	0.166094	−	0.986110i	$-0.446884\pi$
$773$	5.51865e6	0.332188	0.166094	+	0.986110i	$0.446884\pi$
$774$	0	0
$775$	−8.96164e6	−0.535961
$776$	1.99164e7	1.18729
$777$	0	0
$778$	3.06759e6	0.181697
$779$	−1.89224e7	−1.11720
$780$	0	0
$781$	4.23583e6	0.248491
$782$	−4.15806e6	−0.243150
$783$	0	0
$784$	2.96813e6	0.172462
$785$	−3.19744e7	−1.85194
$786$	0	0
$787$	1.63062e7	0.938459	0.469230	−	0.883076i	$-0.344532\pi$
$787$	1.63062e7	0.938459	0.469230	+	0.883076i	$0.344532\pi$
$788$	−2.33950e6	−0.134217
$789$	0	0
$790$	−1.23196e7	−0.702309
$791$	1.56287e7	0.888143
$792$	0	0
$793$	−7.22826e6	−0.408179
$794$	−2.34002e7	−1.31725
$795$	0	0
$796$	1.14860e6	0.0642521
$797$	−3.19084e7	−1.77934	−0.889671	−	0.456602i	$-0.849066\pi$
$797$	−3.19084e7	−1.77934	−0.889671	+	0.456602i	$0.849066\pi$
$798$	0	0
$799$	−2.94067e7	−1.62959
$800$	−1.50352e6	−0.0830584
$801$	0	0
$802$	1.43813e7	0.789520
$803$	−2.15995e6	−0.118210
$804$	0	0
$805$	4.27045e6	0.232265
$806$	−4.50694e7	−2.44368
$807$	0	0
$808$	1.85398e7	0.999025
$809$	−1.70126e7	−0.913900	−0.456950	−	0.889492i	$-0.651058\pi$
$809$	−1.70126e7	−0.913900	−0.456950	+	0.889492i	$0.651058\pi$
$810$	0	0
$811$	1.05241e7	0.561865	0.280932	−	0.959728i	$-0.409356\pi$
$811$	1.05241e7	0.561865	0.280932	+	0.959728i	$0.409356\pi$
$812$	649010.	0.0345431
$813$	0	0
$814$	1.43355e6	0.0758317
$815$	−2.29175e7	−1.20857
$816$	0	0
$817$	−3.10892e7	−1.62950
$818$	2.18754e7	1.14307
$819$	0	0
$820$	3.06775e6	0.159326
$821$	−3.31358e7	−1.71569	−0.857847	−	0.513906i	$-0.828198\pi$
$821$	−3.31358e7	−1.71569	−0.857847	+	0.513906i	$0.828198\pi$
$822$	0	0
$823$	2.83322e7	1.45808	0.729039	−	0.684472i	$-0.239967\pi$
$823$	2.83322e7	1.45808	0.729039	+	0.684472i	$0.239967\pi$
$824$	2.75752e7	1.41482
$825$	0	0
$826$	2.46945e6	0.125936
$827$	−8.10406e6	−0.412039	−0.206020	−	0.978548i	$-0.566051\pi$
$827$	−8.10406e6	−0.412039	−0.206020	+	0.978548i	$0.566051\pi$
$828$	0	0
$829$	−2.88681e7	−1.45892	−0.729461	−	0.684022i	$-0.760229\pi$
$829$	−2.88681e7	−1.45892	−0.729461	+	0.684022i	$0.760229\pi$
$830$	−3.99162e7	−2.01119
$831$	0	0
$832$	2.83814e7	1.42143
$833$	−3.36959e6	−0.168253
$834$	0	0
$835$	4.46968e6	0.221851
$836$	−1.19423e6	−0.0590981
$837$	0	0
$838$	1.13526e7	0.558450
$839$	3.01221e7	1.47734	0.738671	−	0.674066i	$-0.235454\pi$
$839$	3.01221e7	1.47734	0.738671	+	0.674066i	$0.235454\pi$
$840$	0	0
$841$	−1.81159e7	−0.883220
$842$	2.92859e7	1.42357
$843$	0	0
$844$	−2.73156e6	−0.131994
$845$	−4.11398e7	−1.98208
$846$	0	0
$847$	−1.24486e7	−0.596228
$848$	2.14202e7	1.02290
$849$	0	0
$850$	9.00686e6	0.427589
$851$	553249.	0.0261877
$852$	0	0
$853$	−3.03837e7	−1.42977	−0.714887	−	0.699240i	$-0.753522\pi$
$853$	−3.03837e7	−1.42977	−0.714887	+	0.699240i	$0.753522\pi$
$854$	−5.13288e6	−0.240833
$855$	0	0
$856$	−1.62187e6	−0.0756541
$857$	3.42703e7	1.59392	0.796958	−	0.604035i	$-0.206441\pi$
$857$	3.42703e7	1.59392	0.796958	+	0.604035i	$0.206441\pi$
$858$	0	0
$859$	−7.21405e6	−0.333577	−0.166789	−	0.985993i	$-0.553340\pi$
$859$	−7.21405e6	−0.333577	−0.166789	+	0.985993i	$0.553340\pi$
$860$	5.04028e6	0.232385
$861$	0	0
$862$	2.17895e7	0.998802
$863$	3.87581e7	1.77148	0.885739	−	0.464184i	$-0.153652\pi$
$863$	3.87581e7	1.77148	0.885739	+	0.464184i	$0.153652\pi$
$864$	0	0
$865$	988108.	0.0449019
$866$	−3.01761e7	−1.36731
$867$	0	0
$868$	−3.17416e6	−0.142998
$869$	−7.48235e6	−0.336115
$870$	0	0
$871$	6.98045e7	3.11773
$872$	1.37222e7	0.611131
$873$	0	0
$874$	−4.64706e6	−0.205778
$875$	1.51657e7	0.669644
$876$	0	0
$877$	−4.06345e6	−0.178400	−0.0892002	−	0.996014i	$-0.528431\pi$
$877$	−4.06345e6	−0.178400	−0.0892002	+	0.996014i	$0.528431\pi$
$878$	−2.35158e7	−1.02949
$879$	0	0
$880$	−1.75373e7	−0.763405
$881$	4.47103e7	1.94074	0.970372	−	0.241616i	$-0.0776775\pi$
$881$	4.47103e7	1.94074	0.970372	+	0.241616i	$0.0776775\pi$
$882$	0	0
$883$	−6.18489e6	−0.266950	−0.133475	−	0.991052i	$-0.542614\pi$
$883$	−6.18489e6	−0.266950	−0.133475	+	0.991052i	$0.542614\pi$
$884$	4.49250e6	0.193356
$885$	0	0
$886$	2.01109e7	0.860690
$887$	−4.16533e7	−1.77763	−0.888813	−	0.458270i	$-0.848469\pi$
$887$	−4.16533e7	−1.77763	−0.888813	+	0.458270i	$0.848469\pi$
$888$	0	0
$889$	−2.78914e7	−1.18363
$890$	4.07613e7	1.72494
$891$	0	0
$892$	−3.01374e6	−0.126822
$893$	−3.28650e7	−1.37913
$894$	0	0
$895$	4.70272e7	1.96242
$896$	2.49909e7	1.03995
$897$	0	0
$898$	3.36218e7	1.39133
$899$	−1.17148e7	−0.483433
$900$	0	0
$901$	−2.43174e7	−0.997941
$902$	1.87863e7	0.768821
$903$	0	0
$904$	−2.22860e7	−0.907006
$905$	4.61085e7	1.87137
$906$	0	0
$907$	2.48069e7	1.00128	0.500639	−	0.865656i	$-0.333098\pi$
$907$	2.48069e7	1.00128	0.500639	+	0.865656i	$0.333098\pi$
$908$	−898483.	−0.0361656
$909$	0	0
$910$	−4.65212e7	−1.86229
$911$	−2.87038e7	−1.14589	−0.572946	−	0.819593i	$-0.694200\pi$
$911$	−2.87038e7	−1.14589	−0.572946	+	0.819593i	$0.694200\pi$
$912$	0	0
$913$	−2.42432e7	−0.962529
$914$	−799228.	−0.0316450
$915$	0	0
$916$	1.77754e6	0.0699974
$917$	−1.60870e7	−0.631760
$918$	0	0
$919$	3.98850e7	1.55783	0.778917	−	0.627127i	$-0.215769\pi$
$919$	3.98850e7	1.55783	0.778917	+	0.627127i	$0.215769\pi$
$920$	−6.08950e6	−0.237198
$921$	0	0
$922$	−2.38974e7	−0.925812
$923$	−1.78084e7	−0.688050
$924$	0	0
$925$	−1.19840e6	−0.0460521
$926$	4.78581e7	1.83412
$927$	0	0
$928$	−1.96542e6	−0.0749180
$929$	3.09058e7	1.17490	0.587451	−	0.809260i	$-0.300132\pi$
$929$	3.09058e7	1.17490	0.587451	+	0.809260i	$0.300132\pi$
$930$	0	0
$931$	−3.76585e6	−0.142393
$932$	−1.11257e6	−0.0419553
$933$	0	0
$934$	7.02606e6	0.263539
$935$	1.99093e7	0.744777
$936$	0	0
$937$	−3.92068e7	−1.45886	−0.729428	−	0.684058i	$-0.760214\pi$
$937$	−3.92068e7	−1.45886	−0.729428	+	0.684058i	$0.760214\pi$
$938$	4.95690e7	1.83951
$939$	0	0
$940$	5.32816e6	0.196679
$941$	4.21602e7	1.55213	0.776066	−	0.630652i	$-0.217212\pi$
$941$	4.21602e7	1.55213	0.776066	+	0.630652i	$0.217212\pi$
$942$	0	0
$943$	7.25021e6	0.265504
$944$	−3.91379e6	−0.142944
$945$	0	0
$946$	3.08657e7	1.12137
$947$	−5.32912e7	−1.93099	−0.965497	−	0.260416i	$-0.916140\pi$
$947$	−5.32912e7	−1.93099	−0.965497	+	0.260416i	$0.916140\pi$
$948$	0	0
$949$	9.08090e6	0.327313
$950$	1.00661e7	0.361869
$951$	0	0
$952$	−2.57854e7	−0.922109
$953$	−3.66273e7	−1.30639	−0.653194	−	0.757190i	$-0.726572\pi$
$953$	−3.66273e7	−1.30639	−0.653194	+	0.757190i	$0.726572\pi$
$954$	0	0
$955$	−1.87146e6	−0.0664005
$956$	4.64751e6	0.164466
$957$	0	0
$958$	4.19419e7	1.47650
$959$	1.58712e7	0.557267
$960$	0	0
$961$	2.86654e7	1.00127
$962$	−6.02695e6	−0.209971
$963$	0	0
$964$	−356393.	−0.0123520
$965$	2.03518e7	0.703532
$966$	0	0
$967$	−3.44677e7	−1.18535	−0.592675	−	0.805442i	$-0.701928\pi$
$967$	−3.44677e7	−1.18535	−0.592675	+	0.805442i	$0.701928\pi$
$968$	1.77512e7	0.608891
$969$	0	0
$970$	4.59098e7	1.56666
$971$	−9.21316e6	−0.313589	−0.156794	−	0.987631i	$-0.550116\pi$
$971$	−9.21316e6	−0.313589	−0.156794	+	0.987631i	$0.550116\pi$
$972$	0	0
$973$	2.34967e7	0.795655
$974$	−1.04219e7	−0.352004
$975$	0	0
$976$	8.13501e6	0.273359
$977$	−3.66542e6	−0.122854	−0.0614268	−	0.998112i	$-0.519565\pi$
$977$	−3.66542e6	−0.122854	−0.0614268	+	0.998112i	$0.519565\pi$
$978$	0	0
$979$	2.47566e7	0.825531
$980$	610531.	0.0203069
$981$	0	0
$982$	−3.40345e7	−1.12626
$983$	7.59961e6	0.250846	0.125423	−	0.992103i	$-0.459971\pi$
$983$	7.59961e6	0.250846	0.125423	+	0.992103i	$0.459971\pi$
$984$	0	0
$985$	4.35889e7	1.43148
$986$	1.17739e7	0.385682
$987$	0	0
$988$	5.02083e6	0.163638
$989$	1.19120e7	0.387252
$990$	0	0
$991$	3.53749e7	1.14422	0.572112	−	0.820175i	$-0.306124\pi$
$991$	3.53749e7	1.14422	0.572112	+	0.820175i	$0.306124\pi$
$992$	9.61245e6	0.310138
$993$	0	0
$994$	−1.26459e7	−0.405962
$995$	−2.14005e7	−0.685276
$996$	0	0
$997$	3.58278e7	1.14152	0.570759	−	0.821118i	$-0.306649\pi$
$997$	3.58278e7	1.14152	0.570759	+	0.821118i	$0.306649\pi$
$998$	−1.89868e7	−0.603429
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+5.96013 q^{2} +3.52316 q^{4} -65.6425 q^{5} +119.026 q^{7} -169.726 q^{8} +O(q^{10})\)	\(q+5.96013 q^{2} +3.52316 q^{4} -65.6425 q^{5} +119.026 q^{7} -169.726 q^{8} -391.238 q^{10} -237.620 q^{11} +999.008 q^{13} +709.408 q^{14} -1124.33 q^{16} +1276.40 q^{17} +1426.51 q^{19} -231.269 q^{20} -1416.25 q^{22} -546.573 q^{23} +1183.94 q^{25} +5954.22 q^{26} +419.346 q^{28} +1547.67 q^{29} -7569.31 q^{31} -1269.92 q^{32} +7607.51 q^{34} -7813.14 q^{35} -1012.21 q^{37} +8502.16 q^{38} +11141.2 q^{40} -13264.9 q^{41} -21794.0 q^{43} -837.174 q^{44} -3257.65 q^{46} -23038.8 q^{47} -2639.91 q^{49} +7056.46 q^{50} +3519.67 q^{52} -19051.5 q^{53} +15598.0 q^{55} -20201.7 q^{56} +9224.33 q^{58} +3481.00 q^{59} -7235.44 q^{61} -45114.1 q^{62} +28409.6 q^{64} -65577.4 q^{65} +69873.8 q^{67} +4496.96 q^{68} -46567.3 q^{70} -17826.0 q^{71} +9089.91 q^{73} -6032.93 q^{74} +5025.81 q^{76} -28282.9 q^{77} +31488.7 q^{79} +73803.8 q^{80} -79060.3 q^{82} +102025. q^{83} -83786.1 q^{85} -129895. q^{86} +40330.2 q^{88} -104185. q^{89} +118908. q^{91} -1925.67 q^{92} -137314. q^{94} -93639.5 q^{95} -117345. q^{97} -15734.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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