LMFDB - Embedded newform 531.6.a.b.1.4

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + \cdots - 14846072 $ x^11 - 5x^10 - 238x^9 + 1067x^8 + 20782x^7 - 79077x^6 - 813818x^5 + 2364885x^4 + 13849341x^3 - 23890558x^2 - 74443300x - 14846072
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.75393$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.75393 q^{2} -9.40015 q^{4} +8.06966 q^{5} -28.8581 q^{7} +196.813 q^{8} +O(q^{10})$	\(q-4.75393 q^{2} -9.40015 q^{4} +8.06966 q^{5} -28.8581 q^{7} +196.813 q^{8} -38.3626 q^{10} -556.758 q^{11} -1129.41 q^{13} +137.189 q^{14} -634.832 q^{16} -447.456 q^{17} -1350.14 q^{19} -75.8560 q^{20} +2646.79 q^{22} +4466.66 q^{23} -3059.88 q^{25} +5369.15 q^{26} +271.270 q^{28} +2943.74 q^{29} +1113.75 q^{31} -3280.08 q^{32} +2127.18 q^{34} -232.875 q^{35} -5875.21 q^{37} +6418.45 q^{38} +1588.22 q^{40} +638.732 q^{41} -9932.25 q^{43} +5233.61 q^{44} -21234.2 q^{46} -25092.0 q^{47} -15974.2 q^{49} +14546.5 q^{50} +10616.6 q^{52} -19396.9 q^{53} -4492.85 q^{55} -5679.66 q^{56} -13994.3 q^{58} -3481.00 q^{59} +7814.71 q^{61} -5294.71 q^{62} +35907.9 q^{64} -9113.98 q^{65} +36824.7 q^{67} +4206.15 q^{68} +1107.07 q^{70} -24174.9 q^{71} -78229.0 q^{73} +27930.3 q^{74} +12691.5 q^{76} +16067.0 q^{77} +4205.35 q^{79} -5122.88 q^{80} -3036.49 q^{82} +36905.9 q^{83} -3610.82 q^{85} +47217.2 q^{86} -109578. q^{88} -6625.63 q^{89} +32592.7 q^{91} -41987.2 q^{92} +119286. q^{94} -10895.1 q^{95} +88314.2 q^{97} +75940.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.75393	−0.840384	−0.420192	−	0.907435i	$-0.638037\pi$
$2$	−4.75393	−0.840384	−0.420192	+	0.907435i	$0.638037\pi$
$3$	0	0
$3$	0	0
$4$	−9.40015	−0.293755
$5$	8.06966	0.144355	0.0721773	−	0.997392i	$-0.477005\pi$
$5$	8.06966	0.144355	0.0721773	+	0.997392i	$0.477005\pi$
$6$	0	0
$7$	−28.8581	−0.222599	−0.111299	−	0.993787i	$-0.535501\pi$
$7$	−28.8581	−0.222599	−0.111299	+	0.993787i	$0.535501\pi$
$8$	196.813	1.08725
$9$	0	0
$10$	−38.3626	−0.121313
$11$	−556.758	−1.38735	−0.693673	−	0.720290i	$-0.744009\pi$
$11$	−556.758	−1.38735	−0.693673	+	0.720290i	$0.744009\pi$
$12$	0	0
$13$	−1129.41	−1.85351	−0.926754	−	0.375670i	$-0.877413\pi$
$13$	−1129.41	−1.85351	−0.926754	+	0.375670i	$0.877413\pi$
$14$	137.189	0.187068
$15$	0	0
$16$	−634.832	−0.619954
$17$	−447.456	−0.375516	−0.187758	−	0.982215i	$-0.560122\pi$
$17$	−447.456	−0.375516	−0.187758	+	0.982215i	$0.560122\pi$
$18$	0	0
$19$	−1350.14	−0.858012	−0.429006	−	0.903302i	$-0.641136\pi$
$19$	−1350.14	−0.858012	−0.429006	+	0.903302i	$0.641136\pi$
$20$	−75.8560	−0.0424048
$21$	0	0
$22$	2646.79	1.16590
$23$	4466.66	1.76061	0.880304	−	0.474409i	$-0.157338\pi$
$23$	4466.66	1.76061	0.880304	+	0.474409i	$0.157338\pi$
$24$	0	0
$25$	−3059.88	−0.979162
$26$	5369.15	1.55766
$27$	0	0
$28$	271.270	0.0653894
$29$	2943.74	0.649987	0.324994	−	0.945716i	$-0.394638\pi$
$29$	2943.74	0.649987	0.324994	+	0.945716i	$0.394638\pi$
$30$	0	0
$31$	1113.75	0.208154	0.104077	−	0.994569i	$-0.466811\pi$
$31$	1113.75	0.208154	0.104077	+	0.994569i	$0.466811\pi$
$32$	−3280.08	−0.566252
$33$	0	0
$34$	2127.18	0.315578
$35$	−232.875	−0.0321331
$36$	0	0
$37$	−5875.21	−0.705536	−0.352768	−	0.935711i	$-0.614759\pi$
$37$	−5875.21	−0.705536	−0.352768	+	0.935711i	$0.614759\pi$
$38$	6418.45	0.721060
$39$	0	0
$40$	1588.22	0.156950
$41$	638.732	0.0593415	0.0296708	−	0.999560i	$-0.490554\pi$
$41$	638.732	0.0593415	0.0296708	+	0.999560i	$0.490554\pi$
$42$	0	0
$43$	−9932.25	−0.819175	−0.409587	−	0.912271i	$-0.634327\pi$
$43$	−9932.25	−0.819175	−0.409587	+	0.912271i	$0.634327\pi$
$44$	5233.61	0.407540
$45$	0	0
$46$	−21234.2	−1.47959
$47$	−25092.0	−1.65688	−0.828440	−	0.560078i	$-0.810771\pi$
$47$	−25092.0	−1.65688	−0.828440	+	0.560078i	$0.810771\pi$
$48$	0	0
$49$	−15974.2	−0.950450
$50$	14546.5	0.822872
$51$	0	0
$52$	10616.6	0.544476
$53$	−19396.9	−0.948509	−0.474255	−	0.880388i	$-0.657282\pi$
$53$	−19396.9	−0.948509	−0.474255	+	0.880388i	$0.657282\pi$
$54$	0	0
$55$	−4492.85	−0.200270
$56$	−5679.66	−0.242021
$57$	0	0
$58$	−13994.3	−0.546239
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	7814.71	0.268898	0.134449	−	0.990920i	$-0.457073\pi$
$61$	7814.71	0.268898	0.134449	+	0.990920i	$0.457073\pi$
$62$	−5294.71	−0.174929
$63$	0	0
$64$	35907.9	1.09582
$65$	−9113.98	−0.267562
$66$	0	0
$67$	36824.7	1.00220	0.501098	−	0.865391i	$-0.332930\pi$
$67$	36824.7	1.00220	0.501098	+	0.865391i	$0.332930\pi$
$68$	4206.15	0.110310
$69$	0	0
$70$	1107.07	0.0270042
$71$	−24174.9	−0.569141	−0.284570	−	0.958655i	$-0.591851\pi$
$71$	−24174.9	−0.569141	−0.284570	+	0.958655i	$0.591851\pi$
$72$	0	0
$73$	−78229.0	−1.71815	−0.859075	−	0.511850i	$-0.828960\pi$
$73$	−78229.0	−1.71815	−0.859075	+	0.511850i	$0.828960\pi$
$74$	27930.3	0.592921
$75$	0	0
$76$	12691.5	0.252045
$77$	16067.0	0.308822
$78$	0	0
$79$	4205.35	0.0758115	0.0379057	−	0.999281i	$-0.487931\pi$
$79$	4205.35	0.0758115	0.0379057	+	0.999281i	$0.487931\pi$
$80$	−5122.88	−0.0894931
$81$	0	0
$82$	−3036.49	−0.0498697
$83$	36905.9	0.588032	0.294016	−	0.955801i	$-0.405008\pi$
$83$	36905.9	0.588032	0.294016	+	0.955801i	$0.405008\pi$
$84$	0	0
$85$	−3610.82	−0.0542074
$86$	47217.2	0.688421
$87$	0	0
$88$	−109578.	−1.50839
$89$	−6625.63	−0.0886650	−0.0443325	−	0.999017i	$-0.514116\pi$
$89$	−6625.63	−0.0886650	−0.0443325	+	0.999017i	$0.514116\pi$
$90$	0	0
$91$	32592.7	0.412588
$92$	−41987.2	−0.517187
$93$	0	0
$94$	119286.	1.39242
$95$	−10895.1	−0.123858
$96$	0	0
$97$	88314.2	0.953018	0.476509	−	0.879170i	$-0.341902\pi$
$97$	88314.2	0.953018	0.476509	+	0.879170i	$0.341902\pi$
$98$	75940.3	0.798743
$99$	0	0
$100$	28763.3	0.287633
$101$	123051.	1.20028	0.600138	−	0.799896i	$-0.295112\pi$
$101$	123051.	1.20028	0.600138	+	0.799896i	$0.295112\pi$
$102$	0	0
$103$	87267.1	0.810509	0.405254	−	0.914204i	$-0.367183\pi$
$103$	87267.1	0.810509	0.405254	+	0.914204i	$0.367183\pi$
$104$	−222284.	−2.01523
$105$	0	0
$106$	92211.3	0.797112
$107$	2531.40	0.0213748	0.0106874	−	0.999943i	$-0.496598\pi$
$107$	2531.40	0.0213748	0.0106874	+	0.999943i	$0.496598\pi$
$108$	0	0
$109$	−158752.	−1.27983	−0.639914	−	0.768446i	$-0.721030\pi$
$109$	−158752.	−1.27983	−0.639914	+	0.768446i	$0.721030\pi$
$110$	21358.7	0.168304
$111$	0	0
$112$	18320.1	0.138001
$113$	45254.3	0.333399	0.166699	−	0.986008i	$-0.446689\pi$
$113$	45254.3	0.333399	0.166699	+	0.986008i	$0.446689\pi$
$114$	0	0
$115$	36044.4	0.254152
$116$	−27671.6	−0.190937
$117$	0	0
$118$	16548.4	0.109409
$119$	12912.7	0.0835893
$120$	0	0
$121$	148929.	0.924731
$122$	−37150.6	−0.225978
$123$	0	0
$124$	−10469.5	−0.0611463
$125$	−49909.9	−0.285701
$126$	0	0
$127$	−60121.1	−0.330764	−0.165382	−	0.986230i	$-0.552886\pi$
$127$	−60121.1	−0.330764	−0.165382	+	0.986230i	$0.552886\pi$
$128$	−65741.1	−0.354660
$129$	0	0
$130$	43327.2	0.224855
$131$	−339921.	−1.73061	−0.865307	−	0.501243i	$-0.832876\pi$
$131$	−339921.	−1.73061	−0.865307	+	0.501243i	$0.832876\pi$
$132$	0	0
$133$	38962.4	0.190992
$134$	−175062.	−0.842229
$135$	0	0
$136$	−88065.4	−0.408280
$137$	−67355.2	−0.306598	−0.153299	−	0.988180i	$-0.548990\pi$
$137$	−67355.2	−0.306598	−0.153299	+	0.988180i	$0.548990\pi$
$138$	0	0
$139$	308422.	1.35397	0.676983	−	0.735999i	$-0.263287\pi$
$139$	308422.	1.35397	0.676983	+	0.735999i	$0.263287\pi$
$140$	2189.06	0.00943926
$141$	0	0
$142$	114926.	0.478297
$143$	628810.	2.57146
$144$	0	0
$145$	23755.0	0.0938286
$146$	371895.	1.44391
$147$	0	0
$148$	55227.8	0.207254
$149$	−386065.	−1.42460	−0.712302	−	0.701873i	$-0.752347\pi$
$149$	−386065.	−1.42460	−0.712302	+	0.701873i	$0.752347\pi$
$150$	0	0
$151$	361671.	1.29084	0.645418	−	0.763829i	$-0.276683\pi$
$151$	361671.	1.29084	0.645418	+	0.763829i	$0.276683\pi$
$152$	−265725.	−0.932875
$153$	0	0
$154$	−76381.3	−0.259529
$155$	8987.62	0.0300480
$156$	0	0
$157$	−125710.	−0.407026	−0.203513	−	0.979072i	$-0.565236\pi$
$157$	−125710.	−0.407026	−0.203513	+	0.979072i	$0.565236\pi$
$158$	−19992.0	−0.0637107
$159$	0	0
$160$	−26469.1	−0.0817410
$161$	−128899.	−0.391909
$162$	0	0
$163$	312195.	0.920358	0.460179	−	0.887826i	$-0.347785\pi$
$163$	312195.	0.920358	0.460179	+	0.887826i	$0.347785\pi$
$164$	−6004.17	−0.0174318
$165$	0	0
$166$	−175448.	−0.494173
$167$	696281.	1.93194	0.965969	−	0.258656i	$-0.0832798\pi$
$167$	696281.	1.93194	0.965969	+	0.258656i	$0.0832798\pi$
$168$	0	0
$169$	904280.	2.43549
$170$	17165.6	0.0455550
$171$	0	0
$172$	93364.6	0.240636
$173$	467449.	1.18746	0.593730	−	0.804664i	$-0.297655\pi$
$173$	467449.	1.18746	0.593730	+	0.804664i	$0.297655\pi$
$174$	0	0
$175$	88302.3	0.217960
$176$	353448.	0.860091
$177$	0	0
$178$	31497.8	0.0745127
$179$	205080.	0.478400	0.239200	−	0.970970i	$-0.423115\pi$
$179$	205080.	0.478400	0.239200	+	0.970970i	$0.423115\pi$
$180$	0	0
$181$	184862.	0.419421	0.209711	−	0.977763i	$-0.432748\pi$
$181$	184862.	0.419421	0.209711	+	0.977763i	$0.432748\pi$
$182$	−154943.	−0.346733
$183$	0	0
$184$	879098.	1.91422
$185$	−47411.0	−0.101847
$186$	0	0
$187$	249125.	0.520971
$188$	235869.	0.486716
$189$	0	0
$190$	51794.7	0.104088
$191$	272484.	0.540453	0.270226	−	0.962797i	$-0.412901\pi$
$191$	272484.	0.540453	0.270226	+	0.962797i	$0.412901\pi$
$192$	0	0
$193$	−270438.	−0.522606	−0.261303	−	0.965257i	$-0.584152\pi$
$193$	−270438.	−0.522606	−0.261303	+	0.965257i	$0.584152\pi$
$194$	−419840.	−0.800901
$195$	0	0
$196$	150160.	0.279199
$197$	707478.	1.29882	0.649408	−	0.760440i	$-0.275017\pi$
$197$	707478.	1.29882	0.649408	+	0.760440i	$0.275017\pi$
$198$	0	0
$199$	−1.02712e6	−1.83861	−0.919303	−	0.393550i	$-0.871247\pi$
$199$	−1.02712e6	−1.83861	−0.919303	+	0.393550i	$0.871247\pi$
$200$	−602226.	−1.06459
$201$	0	0
$202$	−584975.	−1.00869
$203$	−84950.8	−0.144686
$204$	0	0
$205$	5154.35	0.00856622
$206$	−414862.	−0.681138
$207$	0	0
$208$	716988.	1.14909
$209$	751699.	1.19036
$210$	0	0
$211$	−806769.	−1.24751	−0.623753	−	0.781621i	$-0.714393\pi$
$211$	−806769.	−1.24751	−0.623753	+	0.781621i	$0.714393\pi$
$212$	182333.	0.278629
$213$	0	0
$214$	−12034.1	−0.0179630
$215$	−80149.9	−0.118252
$216$	0	0
$217$	−32140.8	−0.0463349
$218$	754694.	1.07555
$219$	0	0
$220$	42233.5	0.0588302
$221$	505363.	0.696021
$222$	0	0
$223$	−337189.	−0.454058	−0.227029	−	0.973888i	$-0.572901\pi$
$223$	−337189.	−0.454058	−0.227029	+	0.973888i	$0.572901\pi$
$224$	94656.9	0.126047
$225$	0	0
$226$	−215136.	−0.280183
$227$	259818.	0.334661	0.167331	−	0.985901i	$-0.446485\pi$
$227$	259818.	0.334661	0.167331	+	0.985901i	$0.446485\pi$
$228$	0	0
$229$	227371.	0.286515	0.143257	−	0.989685i	$-0.454242\pi$
$229$	227371.	0.286515	0.143257	+	0.989685i	$0.454242\pi$
$230$	−171353.	−0.213585
$231$	0	0
$232$	579368.	0.706699
$233$	−265219.	−0.320048	−0.160024	−	0.987113i	$-0.551157\pi$
$233$	−265219.	−0.320048	−0.160024	+	0.987113i	$0.551157\pi$
$234$	0	0
$235$	−202484.	−0.239178
$236$	32721.9	0.0382436
$237$	0	0
$238$	−61386.2	−0.0702471
$239$	245730.	0.278268	0.139134	−	0.990274i	$-0.455568\pi$
$239$	245730.	0.278268	0.139134	+	0.990274i	$0.455568\pi$
$240$	0	0
$241$	1.23701e6	1.37193	0.685963	−	0.727636i	$-0.259381\pi$
$241$	1.23701e6	1.37193	0.685963	+	0.727636i	$0.259381\pi$
$242$	−707997.	−0.777129
$243$	0	0
$244$	−73459.4	−0.0789902
$245$	−128906.	−0.137202
$246$	0	0
$247$	1.52486e6	1.59033
$248$	219202.	0.226316
$249$	0	0
$250$	237268.	0.240099
$251$	412168.	0.412943	0.206472	−	0.978453i	$-0.433802\pi$
$251$	412168.	0.412943	0.206472	+	0.978453i	$0.433802\pi$
$252$	0	0
$253$	−2.48685e6	−2.44258
$254$	285812.	0.277969
$255$	0	0
$256$	−836524.	−0.797772
$257$	1.57045e6	1.48317	0.741585	−	0.670859i	$-0.234074\pi$
$257$	1.57045e6	1.48317	0.741585	+	0.670859i	$0.234074\pi$
$258$	0	0
$259$	169547.	0.157051
$260$	85672.7	0.0785976
$261$	0	0
$262$	1.61596e6	1.45438
$263$	−1.39983e6	−1.24792	−0.623959	−	0.781457i	$-0.714477\pi$
$263$	−1.39983e6	−1.24792	−0.623959	+	0.781457i	$0.714477\pi$
$264$	0	0
$265$	−156526.	−0.136922
$266$	−185224.	−0.160507
$267$	0	0
$268$	−346158.	−0.294400
$269$	1.66101e6	1.39956	0.699780	−	0.714358i	$-0.253281\pi$
$269$	1.66101e6	1.39956	0.699780	+	0.714358i	$0.253281\pi$
$270$	0	0
$271$	−2.02912e6	−1.67836	−0.839178	−	0.543856i	$-0.816964\pi$
$271$	−2.02912e6	−1.67836	−0.839178	+	0.543856i	$0.816964\pi$
$272$	284060.	0.232802
$273$	0	0
$274$	320202.	0.257660
$275$	1.70361e6	1.35844
$276$	0	0
$277$	−1.82153e6	−1.42638	−0.713191	−	0.700970i	$-0.752751\pi$
$277$	−1.82153e6	−1.42638	−0.713191	+	0.700970i	$0.752751\pi$
$278$	−1.46621e6	−1.13785
$279$	0	0
$280$	−45833.0	−0.0349368
$281$	1.98485e6	1.49955	0.749777	−	0.661691i	$-0.230161\pi$
$281$	1.98485e6	1.49955	0.749777	+	0.661691i	$0.230161\pi$
$282$	0	0
$283$	2.17263e6	1.61258	0.806289	−	0.591522i	$-0.201473\pi$
$283$	2.17263e6	1.61258	0.806289	+	0.591522i	$0.201473\pi$
$284$	227248.	0.167188
$285$	0	0
$286$	−2.98932e6	−2.16101
$287$	−18432.6	−0.0132093
$288$	0	0
$289$	−1.21964e6	−0.858988
$290$	−112930.	−0.0788521
$291$	0	0
$292$	735365.	0.504714
$293$	1.73848e6	1.18305	0.591523	−	0.806288i	$-0.298527\pi$
$293$	1.73848e6	1.18305	0.591523	+	0.806288i	$0.298527\pi$
$294$	0	0
$295$	−28090.5	−0.0187934
$296$	−1.15632e6	−0.767094
$297$	0	0
$298$	1.83532e6	1.19721
$299$	−5.04470e6	−3.26330
$300$	0	0
$301$	286626.	0.182347
$302$	−1.71936e6	−1.08480
$303$	0	0
$304$	857110.	0.531928
$305$	63062.1	0.0388167
$306$	0	0
$307$	1.20069e6	0.727082	0.363541	−	0.931578i	$-0.381568\pi$
$307$	1.20069e6	0.727082	0.363541	+	0.931578i	$0.381568\pi$
$308$	−151032.	−0.0907178
$309$	0	0
$310$	−42726.5	−0.0252519
$311$	−1.20148e6	−0.704395	−0.352197	−	0.935926i	$-0.614565\pi$
$311$	−1.20148e6	−0.704395	−0.352197	+	0.935926i	$0.614565\pi$
$312$	0	0
$313$	−1.09781e6	−0.633385	−0.316693	−	0.948528i	$-0.602572\pi$
$313$	−1.09781e6	−0.633385	−0.316693	+	0.948528i	$0.602572\pi$
$314$	597619.	0.342058
$315$	0	0
$316$	−39531.0	−0.0222700
$317$	−36761.0	−0.0205466	−0.0102733	−	0.999947i	$-0.503270\pi$
$317$	−36761.0	−0.0205466	−0.0102733	+	0.999947i	$0.503270\pi$
$318$	0	0
$319$	−1.63895e6	−0.901758
$320$	289765.	0.158187
$321$	0	0
$322$	612778.	0.329354
$323$	604127.	0.322197
$324$	0	0
$325$	3.45587e6	1.81488
$326$	−1.48415e6	−0.773454
$327$	0	0
$328$	125711.	0.0645191
$329$	724108.	0.368819
$330$	0	0
$331$	2.44095e6	1.22459	0.612293	−	0.790631i	$-0.290247\pi$
$331$	2.44095e6	1.22459	0.612293	+	0.790631i	$0.290247\pi$
$332$	−346921.	−0.172737
$333$	0	0
$334$	−3.31007e6	−1.62357
$335$	297163.	0.144671
$336$	0	0
$337$	−532207.	−0.255274	−0.127637	−	0.991821i	$-0.540739\pi$
$337$	−532207.	−0.255274	−0.127637	+	0.991821i	$0.540739\pi$
$338$	−4.29888e6	−2.04675
$339$	0	0
$340$	33942.3	0.0159237
$341$	−620092.	−0.288782
$342$	0	0
$343$	946003.	0.434168
$344$	−1.95480e6	−0.890648
$345$	0	0
$346$	−2.22222e6	−0.997923
$347$	3.36260e6	1.49917	0.749587	−	0.661906i	$-0.230252\pi$
$347$	3.36260e6	1.49917	0.749587	+	0.661906i	$0.230252\pi$
$348$	0	0
$349$	3.19465e6	1.40398	0.701989	−	0.712188i	$-0.252296\pi$
$349$	3.19465e6	1.40398	0.701989	+	0.712188i	$0.252296\pi$
$350$	−419783.	−0.183170
$351$	0	0
$352$	1.82621e6	0.785587
$353$	−3.86212e6	−1.64964	−0.824821	−	0.565394i	$-0.808724\pi$
$353$	−3.86212e6	−1.64964	−0.824821	+	0.565394i	$0.808724\pi$
$354$	0	0
$355$	−195084.	−0.0821580
$356$	62281.9	0.0260458
$357$	0	0
$358$	−974936.	−0.402039
$359$	−3.20790e6	−1.31367	−0.656833	−	0.754036i	$-0.728104\pi$
$359$	−3.20790e6	−1.31367	−0.656833	+	0.754036i	$0.728104\pi$
$360$	0	0
$361$	−653232.	−0.263815
$362$	−878819.	−0.352475
$363$	0	0
$364$	−306376.	−0.121200
$365$	−631282.	−0.248023
$366$	0	0
$367$	−3.71516e6	−1.43983	−0.719917	−	0.694060i	$-0.755820\pi$
$367$	−3.71516e6	−1.43983	−0.719917	+	0.694060i	$0.755820\pi$
$368$	−2.83558e6	−1.09150
$369$	0	0
$370$	225388.	0.0855908
$371$	559756.	0.211137
$372$	0	0
$373$	−106590.	−0.0396684	−0.0198342	−	0.999803i	$-0.506314\pi$
$373$	−106590.	−0.0396684	−0.0198342	+	0.999803i	$0.506314\pi$
$374$	−1.18432e6	−0.437815
$375$	0	0
$376$	−4.93845e6	−1.80144
$377$	−3.32470e6	−1.20476
$378$	0	0
$379$	2.90777e6	1.03983	0.519914	−	0.854219i	$-0.325964\pi$
$379$	2.90777e6	1.03983	0.519914	+	0.854219i	$0.325964\pi$
$380$	102416.	0.0363838
$381$	0	0
$382$	−1.29537e6	−0.454188
$383$	−4.43725e6	−1.54567	−0.772835	−	0.634608i	$-0.781162\pi$
$383$	−4.43725e6	−1.54567	−0.772835	+	0.634608i	$0.781162\pi$
$384$	0	0
$385$	129655.	0.0445798
$386$	1.28564e6	0.439190
$387$	0	0
$388$	−830167.	−0.279954
$389$	2.41971e6	0.810755	0.405378	−	0.914149i	$-0.367140\pi$
$389$	2.41971e6	0.810755	0.405378	+	0.914149i	$0.367140\pi$
$390$	0	0
$391$	−1.99863e6	−0.661137
$392$	−3.14394e6	−1.03338
$393$	0	0
$394$	−3.36330e6	−1.09150
$395$	33935.8	0.0109437
$396$	0	0
$397$	−435661.	−0.138731	−0.0693654	−	0.997591i	$-0.522097\pi$
$397$	−435661.	−0.138731	−0.0693654	+	0.997591i	$0.522097\pi$
$398$	4.88286e6	1.54514
$399$	0	0
$400$	1.94251e6	0.607035
$401$	−2.31919e6	−0.720237	−0.360119	−	0.932906i	$-0.617264\pi$
$401$	−2.31919e6	−0.720237	−0.360119	+	0.932906i	$0.617264\pi$
$402$	0	0
$403$	−1.25789e6	−0.385815
$404$	−1.15670e6	−0.352587
$405$	0	0
$406$	403850.	0.121592
$407$	3.27107e6	0.978823
$408$	0	0
$409$	−293179.	−0.0866612	−0.0433306	−	0.999061i	$-0.513797\pi$
$409$	−293179.	−0.0866612	−0.0433306	+	0.999061i	$0.513797\pi$
$410$	−24503.4	−0.00719891
$411$	0	0
$412$	−820324.	−0.238091
$413$	100455.	0.0289799
$414$	0	0
$415$	297818.	0.0848851
$416$	3.70456e6	1.04955
$417$	0	0
$418$	−3.57353e6	−1.00036
$419$	−989556.	−0.275363	−0.137681	−	0.990477i	$-0.543965\pi$
$419$	−989556.	−0.275363	−0.137681	+	0.990477i	$0.543965\pi$
$420$	0	0
$421$	−1.03799e6	−0.285421	−0.142711	−	0.989764i	$-0.545582\pi$
$421$	−1.03799e6	−0.285421	−0.142711	+	0.989764i	$0.545582\pi$
$422$	3.83532e6	1.04838
$423$	0	0
$424$	−3.81756e6	−1.03127
$425$	1.36916e6	0.367691
$426$	0	0
$427$	−225518.	−0.0598564
$428$	−23795.6	−0.00627895
$429$	0	0
$430$	381027.	0.0993767
$431$	6.08956e6	1.57904	0.789519	−	0.613726i	$-0.210330\pi$
$431$	6.08956e6	1.57904	0.789519	+	0.613726i	$0.210330\pi$
$432$	0	0
$433$	679052.	0.174054	0.0870269	−	0.996206i	$-0.472263\pi$
$433$	679052.	0.174054	0.0870269	+	0.996206i	$0.472263\pi$
$434$	152795.	0.0389391
$435$	0	0
$436$	1.49229e6	0.375956
$437$	−6.03059e6	−1.51062
$438$	0	0
$439$	−6.18297e6	−1.53121	−0.765606	−	0.643309i	$-0.777561\pi$
$439$	−6.18297e6	−1.53121	−0.765606	+	0.643309i	$0.777561\pi$
$440$	−884254.	−0.217743
$441$	0	0
$442$	−2.40246e6	−0.584925
$443$	−226949.	−0.0549437	−0.0274719	−	0.999623i	$-0.508746\pi$
$443$	−226949.	−0.0549437	−0.0274719	+	0.999623i	$0.508746\pi$
$444$	0	0
$445$	−53466.6	−0.0127992
$446$	1.60297e6	0.381583
$447$	0	0
$448$	−1.03623e6	−0.243929
$449$	418434.	0.0979515	0.0489758	−	0.998800i	$-0.484404\pi$
$449$	418434.	0.0979515	0.0489758	+	0.998800i	$0.484404\pi$
$450$	0	0
$451$	−355619.	−0.0823273
$452$	−425397.	−0.0979374
$453$	0	0
$454$	−1.23516e6	−0.281244
$455$	263012.	0.0595590
$456$	0	0
$457$	6.90035e6	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$457$	6.90035e6	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$458$	−1.08091e6	−0.240782
$459$	0	0
$460$	−338823.	−0.0746583
$461$	−373717.	−0.0819013	−0.0409507	−	0.999161i	$-0.513039\pi$
$461$	−373717.	−0.0819013	−0.0409507	+	0.999161i	$0.513039\pi$
$462$	0	0
$463$	−5.19758e6	−1.12680	−0.563402	−	0.826183i	$-0.690508\pi$
$463$	−5.19758e6	−1.12680	−0.563402	+	0.826183i	$0.690508\pi$
$464$	−1.86878e6	−0.402962
$465$	0	0
$466$	1.26083e6	0.268963
$467$	6.54913e6	1.38961	0.694803	−	0.719200i	$-0.255492\pi$
$467$	6.54913e6	1.38961	0.694803	+	0.719200i	$0.255492\pi$
$468$	0	0
$469$	−1.06269e6	−0.223087
$470$	962596.	0.201001
$471$	0	0
$472$	−685107.	−0.141548
$473$	5.52986e6	1.13648
$474$	0	0
$475$	4.13125e6	0.840133
$476$	−121382.	−0.0245548
$477$	0	0
$478$	−1.16818e6	−0.233852
$479$	2.70999e6	0.539671	0.269836	−	0.962906i	$-0.413031\pi$
$479$	2.70999e6	0.539671	0.269836	+	0.962906i	$0.413031\pi$
$480$	0	0
$481$	6.63554e6	1.30772
$482$	−5.88066e6	−1.15294
$483$	0	0
$484$	−1.39995e6	−0.271644
$485$	712666.	0.137572
$486$	0	0
$487$	−3.13993e6	−0.599925	−0.299963	−	0.953951i	$-0.596974\pi$
$487$	−3.13993e6	−0.599925	−0.299963	+	0.953951i	$0.596974\pi$
$488$	1.53804e6	0.292360
$489$	0	0
$490$	612812.	0.115302
$491$	51374.3	0.00961705	0.00480853	−	0.999988i	$-0.498469\pi$
$491$	51374.3	0.00961705	0.00480853	+	0.999988i	$0.498469\pi$
$492$	0	0
$493$	−1.31720e6	−0.244081
$494$	−7.24908e6	−1.33649
$495$	0	0
$496$	−707048.	−0.129046
$497$	697643.	0.126690
$498$	0	0
$499$	4.49962e6	0.808955	0.404477	−	0.914548i	$-0.367453\pi$
$499$	4.49962e6	0.808955	0.404477	+	0.914548i	$0.367453\pi$
$500$	469160.	0.0839260
$501$	0	0
$502$	−1.95942e6	−0.347031
$503$	−1.70528e6	−0.300521	−0.150260	−	0.988646i	$-0.548011\pi$
$503$	−1.70528e6	−0.300521	−0.150260	+	0.988646i	$0.548011\pi$
$504$	0	0
$505$	992979.	0.173265
$506$	1.18223e7	2.05270
$507$	0	0
$508$	565147.	0.0971634
$509$	−616751.	−0.105515	−0.0527576	−	0.998607i	$-0.516801\pi$
$509$	−616751.	−0.105515	−0.0527576	+	0.998607i	$0.516801\pi$
$510$	0	0
$511$	2.25754e6	0.382458
$512$	6.08049e6	1.02509
$513$	0	0
$514$	−7.46580e6	−1.24643
$515$	704216.	0.117001
$516$	0	0
$517$	1.39702e7	2.29867
$518$	−806017.	−0.131983
$519$	0	0
$520$	−1.79375e6	−0.290907
$521$	−1.06106e7	−1.71255	−0.856277	−	0.516517i	$-0.827228\pi$
$521$	−1.06106e7	−1.71255	−0.856277	+	0.516517i	$0.827228\pi$
$522$	0	0
$523$	−8.38687e6	−1.34074	−0.670371	−	0.742026i	$-0.733865\pi$
$523$	−8.38687e6	−1.34074	−0.670371	+	0.742026i	$0.733865\pi$
$524$	3.19531e6	0.508376
$525$	0	0
$526$	6.65470e6	1.04873
$527$	−498356.	−0.0781652
$528$	0	0
$529$	1.35147e7	2.09974
$530$	744114.	0.115067
$531$	0	0
$532$	−366252.	−0.0561049
$533$	−721391.	−0.109990
$534$	0	0
$535$	20427.6	0.00308555
$536$	7.24760e6	1.08964
$537$	0	0
$538$	−7.89633e6	−1.17617
$539$	8.89377e6	1.31860
$540$	0	0
$541$	−3.78138e6	−0.555465	−0.277733	−	0.960658i	$-0.589583\pi$
$541$	−3.78138e6	−0.555465	−0.277733	+	0.960658i	$0.589583\pi$
$542$	9.64629e6	1.41046
$543$	0	0
$544$	1.46769e6	0.212636
$545$	−1.28107e6	−0.184749
$546$	0	0
$547$	−3.06265e6	−0.437652	−0.218826	−	0.975764i	$-0.570223\pi$
$547$	−3.06265e6	−0.437652	−0.218826	+	0.975764i	$0.570223\pi$
$548$	633149.	0.0900647
$549$	0	0
$550$	−8.09886e6	−1.14161
$551$	−3.97445e6	−0.557697
$552$	0	0
$553$	−121359.	−0.0168755
$554$	8.65941e6	1.19871
$555$	0	0
$556$	−2.89921e6	−0.397734
$557$	−2.10567e6	−0.287576	−0.143788	−	0.989608i	$-0.545928\pi$
$557$	−2.10567e6	−0.287576	−0.143788	+	0.989608i	$0.545928\pi$
$558$	0	0
$559$	1.12176e7	1.51835
$560$	147837.	0.0199210
$561$	0	0
$562$	−9.43584e6	−1.26020
$563$	3.85100e6	0.512039	0.256019	−	0.966672i	$-0.417589\pi$
$563$	3.85100e6	0.512039	0.256019	+	0.966672i	$0.417589\pi$
$564$	0	0
$565$	365187.	0.0481276
$566$	−1.03286e7	−1.35518
$567$	0	0
$568$	−4.75795e6	−0.618799
$569$	−7.28749e6	−0.943621	−0.471810	−	0.881700i	$-0.656399\pi$
$569$	−7.28749e6	−0.943621	−0.471810	+	0.881700i	$0.656399\pi$
$570$	0	0
$571$	9.75125e6	1.25161	0.625807	−	0.779978i	$-0.284770\pi$
$571$	9.75125e6	1.25161	0.625807	+	0.779978i	$0.284770\pi$
$572$	−5.91091e6	−0.755377
$573$	0	0
$574$	87627.2	0.0111009
$575$	−1.36674e7	−1.72392
$576$	0	0
$577$	−3.33348e6	−0.416829	−0.208415	−	0.978041i	$-0.566830\pi$
$577$	−3.33348e6	−0.416829	−0.208415	+	0.978041i	$0.566830\pi$
$578$	5.79808e6	0.721880
$579$	0	0
$580$	−223301.	−0.0275626
$581$	−1.06504e6	−0.130895
$582$	0	0
$583$	1.07994e7	1.31591
$584$	−1.53965e7	−1.86806
$585$	0	0
$586$	−8.26464e6	−0.994214
$587$	−1.65752e6	−0.198548	−0.0992738	−	0.995060i	$-0.531652\pi$
$587$	−1.65752e6	−0.198548	−0.0992738	+	0.995060i	$0.531652\pi$
$588$	0	0
$589$	−1.50372e6	−0.178599
$590$	133540.	0.0157936
$591$	0	0
$592$	3.72977e6	0.437400
$593$	1.21542e7	1.41935	0.709677	−	0.704527i	$-0.248841\pi$
$593$	1.21542e7	1.41935	0.709677	+	0.704527i	$0.248841\pi$
$594$	0	0
$595$	104201.	0.0120665
$596$	3.62906e6	0.418484
$597$	0	0
$598$	2.39821e7	2.74243
$599$	−6.59980e6	−0.751560	−0.375780	−	0.926709i	$-0.622625\pi$
$599$	−6.59980e6	−0.751560	−0.375780	+	0.926709i	$0.622625\pi$
$600$	0	0
$601$	−2.88369e6	−0.325658	−0.162829	−	0.986654i	$-0.552062\pi$
$601$	−2.88369e6	−0.325658	−0.162829	+	0.986654i	$0.552062\pi$
$602$	−1.36260e6	−0.153242
$603$	0	0
$604$	−3.39976e6	−0.379189
$605$	1.20181e6	0.133489
$606$	0	0
$607$	−1.26616e7	−1.39482	−0.697410	−	0.716673i	$-0.745664\pi$
$607$	−1.26616e7	−1.39482	−0.697410	+	0.716673i	$0.745664\pi$
$608$	4.42855e6	0.485851
$609$	0	0
$610$	−299793.	−0.0326209
$611$	2.83393e7	3.07104
$612$	0	0
$613$	2.13203e6	0.229162	0.114581	−	0.993414i	$-0.463447\pi$
$613$	2.13203e6	0.229162	0.114581	+	0.993414i	$0.463447\pi$
$614$	−5.70798e6	−0.611028
$615$	0	0
$616$	3.16220e6	0.335767
$617$	−1.12243e7	−1.18698	−0.593492	−	0.804840i	$-0.702251\pi$
$617$	−1.12243e7	−1.18698	−0.593492	+	0.804840i	$0.702251\pi$
$618$	0	0
$619$	−1.62425e7	−1.70383	−0.851916	−	0.523679i	$-0.824559\pi$
$619$	−1.62425e7	−1.70383	−0.851916	+	0.523679i	$0.824559\pi$
$620$	−84485.0	−0.00882674
$621$	0	0
$622$	5.71176e6	0.591962
$623$	191203.	0.0197367
$624$	0	0
$625$	9.15937e6	0.937920
$626$	5.21893e6	0.532287
$627$	0	0
$628$	1.18170e6	0.119566
$629$	2.62890e6	0.264940
$630$	0	0
$631$	1.11288e7	1.11270	0.556348	−	0.830950i	$-0.312202\pi$
$631$	1.11288e7	1.11270	0.556348	+	0.830950i	$0.312202\pi$
$632$	827670.	0.0824261
$633$	0	0
$634$	174759.	0.0172670
$635$	−485157.	−0.0477472
$636$	0	0
$637$	1.80415e7	1.76167
$638$	7.79147e6	0.757823
$639$	0	0
$640$	−530509.	−0.0511968
$641$	−2.41239e6	−0.231901	−0.115950	−	0.993255i	$-0.536991\pi$
$641$	−2.41239e6	−0.231901	−0.115950	+	0.993255i	$0.536991\pi$
$642$	0	0
$643$	7.10405e6	0.677608	0.338804	−	0.940857i	$-0.389978\pi$
$643$	7.10405e6	0.677608	0.338804	+	0.940857i	$0.389978\pi$
$644$	1.21167e6	0.115125
$645$	0	0
$646$	−2.87198e6	−0.270769
$647$	4.34524e6	0.408088	0.204044	−	0.978962i	$-0.434591\pi$
$647$	4.34524e6	0.408088	0.204044	+	0.978962i	$0.434591\pi$
$648$	0	0
$649$	1.93808e6	0.180617
$650$	−1.64290e7	−1.52520
$651$	0	0
$652$	−2.93468e6	−0.270359
$653$	1.74836e7	1.60453	0.802263	−	0.596971i	$-0.203629\pi$
$653$	1.74836e7	1.60453	0.802263	+	0.596971i	$0.203629\pi$
$654$	0	0
$655$	−2.74305e6	−0.249822
$656$	−405488.	−0.0367890
$657$	0	0
$658$	−3.44236e6	−0.309950
$659$	1.73557e7	1.55678	0.778392	−	0.627779i	$-0.216036\pi$
$659$	1.73557e7	1.55678	0.778392	+	0.627779i	$0.216036\pi$
$660$	0	0
$661$	5.32811e6	0.474318	0.237159	−	0.971471i	$-0.423784\pi$
$661$	5.32811e6	0.474318	0.237159	+	0.971471i	$0.423784\pi$
$662$	−1.16041e7	−1.02912
$663$	0	0
$664$	7.26358e6	0.639338
$665$	314413.	0.0275706
$666$	0	0
$667$	1.31487e7	1.14437
$668$	−6.54514e6	−0.567516
$669$	0	0
$670$	−1.41269e6	−0.121580
$671$	−4.35091e6	−0.373055
$672$	0	0
$673$	9.87967e6	0.840823	0.420412	−	0.907333i	$-0.361886\pi$
$673$	9.87967e6	0.840823	0.420412	+	0.907333i	$0.361886\pi$
$674$	2.53008e6	0.214528
$675$	0	0
$676$	−8.50036e6	−0.715436
$677$	−1.67635e7	−1.40570	−0.702849	−	0.711339i	$-0.748089\pi$
$677$	−1.67635e7	−1.40570	−0.702849	+	0.711339i	$0.748089\pi$
$678$	0	0
$679$	−2.54858e6	−0.212141
$680$	−710658.	−0.0589370
$681$	0	0
$682$	2.94787e6	0.242688
$683$	−4.71929e6	−0.387102	−0.193551	−	0.981090i	$-0.562000\pi$
$683$	−4.71929e6	−0.387102	−0.193551	+	0.981090i	$0.562000\pi$
$684$	0	0
$685$	−543534.	−0.0442589
$686$	−4.49723e6	−0.364868
$687$	0	0
$688$	6.30532e6	0.507850
$689$	2.19071e7	1.75807
$690$	0	0
$691$	−1.44431e7	−1.15071	−0.575353	−	0.817905i	$-0.695135\pi$
$691$	−1.44431e7	−1.15071	−0.575353	+	0.817905i	$0.695135\pi$
$692$	−4.39409e6	−0.348822
$693$	0	0
$694$	−1.59856e7	−1.25988
$695$	2.48886e6	0.195451
$696$	0	0
$697$	−285804.	−0.0222837
$698$	−1.51872e7	−1.17988
$699$	0	0
$700$	−830055.	−0.0640268
$701$	40301.8	0.00309763	0.00154881	−	0.999999i	$-0.499507\pi$
$701$	40301.8	0.00309763	0.00154881	+	0.999999i	$0.499507\pi$
$702$	0	0
$703$	7.93233e6	0.605358
$704$	−1.99920e7	−1.52029
$705$	0	0
$706$	1.83603e7	1.38633
$707$	−3.55102e6	−0.267180
$708$	0	0
$709$	2.32913e7	1.74011	0.870056	−	0.492953i	$-0.164082\pi$
$709$	2.32913e7	1.74011	0.870056	+	0.492953i	$0.164082\pi$
$710$	927414.	0.0690443
$711$	0	0
$712$	−1.30401e6	−0.0964011
$713$	4.97476e6	0.366478
$714$	0	0
$715$	5.07428e6	0.371201
$716$	−1.92778e6	−0.140532
$717$	0	0
$718$	1.52501e7	1.10398
$719$	−2.22695e7	−1.60653	−0.803265	−	0.595622i	$-0.796906\pi$
$719$	−2.22695e7	−1.60653	−0.803265	+	0.595622i	$0.796906\pi$
$720$	0	0
$721$	−2.51836e6	−0.180418
$722$	3.10542e6	0.221706
$723$	0	0
$724$	−1.73773e6	−0.123207
$725$	−9.00750e6	−0.636443
$726$	0	0
$727$	−2.97421e6	−0.208706	−0.104353	−	0.994540i	$-0.533277\pi$
$727$	−2.97421e6	−0.208706	−0.104353	+	0.994540i	$0.533277\pi$
$728$	6.41468e6	0.448587
$729$	0	0
$730$	3.00107e6	0.208434
$731$	4.44425e6	0.307613
$732$	0	0
$733$	6.38381e6	0.438854	0.219427	−	0.975629i	$-0.429581\pi$
$733$	6.38381e6	0.438854	0.219427	+	0.975629i	$0.429581\pi$
$734$	1.76616e7	1.21001
$735$	0	0
$736$	−1.46510e7	−0.996948
$737$	−2.05025e7	−1.39039
$738$	0	0
$739$	−2.18800e7	−1.47379	−0.736897	−	0.676005i	$-0.763710\pi$
$739$	−2.18800e7	−1.47379	−0.736897	+	0.676005i	$0.763710\pi$
$740$	445670.	0.0299181
$741$	0	0
$742$	−2.66104e6	−0.177436
$743$	1.15290e7	0.766157	0.383079	−	0.923716i	$-0.374864\pi$
$743$	1.15290e7	0.766157	0.383079	+	0.923716i	$0.374864\pi$
$744$	0	0
$745$	−3.11541e6	−0.205648
$746$	506722.	0.0333367
$747$	0	0
$748$	−2.34181e6	−0.153038
$749$	−73051.5	−0.00475800
$750$	0	0
$751$	−1.87579e6	−0.121363	−0.0606813	−	0.998157i	$-0.519327\pi$
$751$	−1.87579e6	−0.121363	−0.0606813	+	0.998157i	$0.519327\pi$
$752$	1.59292e7	1.02719
$753$	0	0
$754$	1.58054e7	1.01246
$755$	2.91856e6	0.186338
$756$	0	0
$757$	−1.76953e7	−1.12232	−0.561161	−	0.827707i	$-0.689645\pi$
$757$	−1.76953e7	−1.12232	−0.561161	+	0.827707i	$0.689645\pi$
$758$	−1.38233e7	−0.873855
$759$	0	0
$760$	−2.14431e6	−0.134665
$761$	−8.98396e6	−0.562349	−0.281175	−	0.959657i	$-0.590724\pi$
$761$	−8.98396e6	−0.562349	−0.281175	+	0.959657i	$0.590724\pi$
$762$	0	0
$763$	4.58127e6	0.284888
$764$	−2.56139e6	−0.158760
$765$	0	0
$766$	2.10944e7	1.29896
$767$	3.93149e6	0.241306
$768$	0	0
$769$	−1.93851e7	−1.18210	−0.591049	−	0.806636i	$-0.701286\pi$
$769$	−1.93851e7	−1.18210	−0.591049	+	0.806636i	$0.701286\pi$
$770$	−616372.	−0.0374641
$771$	0	0
$772$	2.54216e6	0.153518
$773$	−1.11176e7	−0.669207	−0.334604	−	0.942359i	$-0.608602\pi$
$773$	−1.11176e7	−0.669207	−0.334604	+	0.942359i	$0.608602\pi$
$774$	0	0
$775$	−3.40796e6	−0.203817
$776$	1.73814e7	1.03617
$777$	0	0
$778$	−1.15031e7	−0.681346
$779$	−862374.	−0.0509158
$780$	0	0
$781$	1.34596e7	0.789596
$782$	9.50136e6	0.555609
$783$	0	0
$784$	1.01409e7	0.589235
$785$	−1.01444e6	−0.0587560
$786$	0	0
$787$	3.09807e7	1.78302	0.891508	−	0.453006i	$-0.149648\pi$
$787$	3.09807e7	1.78302	0.891508	+	0.453006i	$0.149648\pi$
$788$	−6.65040e6	−0.381533
$789$	0	0
$790$	−161328.	−0.00919693
$791$	−1.30595e6	−0.0742141
$792$	0	0
$793$	−8.82603e6	−0.498405
$794$	2.07110e6	0.116587
$795$	0	0
$796$	9.65509e6	0.540099
$797$	−1.87042e6	−0.104302	−0.0521511	−	0.998639i	$-0.516608\pi$
$797$	−1.87042e6	−0.104302	−0.0521511	+	0.998639i	$0.516608\pi$
$798$	0	0
$799$	1.12276e7	0.622185
$800$	1.00367e7	0.554452
$801$	0	0
$802$	1.10253e7	0.605276
$803$	4.35547e7	2.38367
$804$	0	0
$805$	−1.04017e6	−0.0565739
$806$	5.97991e6	0.324233
$807$	0	0
$808$	2.42181e7	1.30500
$809$	−2.51924e7	−1.35331	−0.676656	−	0.736299i	$-0.736572\pi$
$809$	−2.51924e7	−1.35331	−0.676656	+	0.736299i	$0.736572\pi$
$810$	0	0
$811$	−4.70372e6	−0.251125	−0.125562	−	0.992086i	$-0.540073\pi$
$811$	−4.70372e6	−0.251125	−0.125562	+	0.992086i	$0.540073\pi$
$812$	798550.	0.0425023
$813$	0	0
$814$	−1.55504e7	−0.822587
$815$	2.51931e6	0.132858
$816$	0	0
$817$	1.34099e7	0.702862
$818$	1.39375e6	0.0728287
$819$	0	0
$820$	−48451.6	−0.00251637
$821$	2.56227e6	0.132668	0.0663341	−	0.997797i	$-0.478870\pi$
$821$	2.56227e6	0.132668	0.0663341	+	0.997797i	$0.478870\pi$
$822$	0	0
$823$	−1.26515e7	−0.651092	−0.325546	−	0.945526i	$-0.605548\pi$
$823$	−1.26515e7	−0.651092	−0.325546	+	0.945526i	$0.605548\pi$
$824$	1.71753e7	0.881226
$825$	0	0
$826$	−477556.	−0.0243542
$827$	−2.53615e7	−1.28947	−0.644735	−	0.764406i	$-0.723032\pi$
$827$	−2.53615e7	−1.28947	−0.644735	+	0.764406i	$0.723032\pi$
$828$	0	0
$829$	−3.29273e7	−1.66406	−0.832032	−	0.554728i	$-0.812822\pi$
$829$	−3.29273e7	−1.66406	−0.832032	+	0.554728i	$0.812822\pi$
$830$	−1.41581e6	−0.0713361
$831$	0	0
$832$	−4.05548e7	−2.03111
$833$	7.14776e6	0.356909
$834$	0	0
$835$	5.61875e6	0.278884
$836$	−7.06609e6	−0.349674
$837$	0	0
$838$	4.70428e6	0.231411
$839$	−7.45341e6	−0.365553	−0.182776	−	0.983155i	$-0.558508\pi$
$839$	−7.45341e6	−0.365553	−0.182776	+	0.983155i	$0.558508\pi$
$840$	0	0
$841$	−1.18455e7	−0.577516
$842$	4.93451e6	0.239863
$843$	0	0
$844$	7.58375e6	0.366461
$845$	7.29723e6	0.351574
$846$	0	0
$847$	−4.29780e6	−0.205844
$848$	1.23138e7	0.588032
$849$	0	0
$850$	−6.50890e6	−0.309001
$851$	−2.62425e7	−1.24217
$852$	0	0
$853$	8.47401e6	0.398764	0.199382	−	0.979922i	$-0.436106\pi$
$853$	8.47401e6	0.398764	0.199382	+	0.979922i	$0.436106\pi$
$854$	1.07210e6	0.0503024
$855$	0	0
$856$	498214.	0.0232398
$857$	−400864.	−0.0186443	−0.00932213	−	0.999957i	$-0.502967\pi$
$857$	−400864.	−0.0186443	−0.00932213	+	0.999957i	$0.502967\pi$
$858$	0	0
$859$	−3.38025e7	−1.56302	−0.781512	−	0.623890i	$-0.785551\pi$
$859$	−3.38025e7	−1.56302	−0.781512	+	0.623890i	$0.785551\pi$
$860$	753421.	0.0347369
$861$	0	0
$862$	−2.89493e7	−1.32700
$863$	1.70141e7	0.777648	0.388824	−	0.921312i	$-0.372881\pi$
$863$	1.70141e7	0.777648	0.388824	+	0.921312i	$0.372881\pi$
$864$	0	0
$865$	3.77216e6	0.171415
$866$	−3.22817e6	−0.146272
$867$	0	0
$868$	302129.	0.0136111
$869$	−2.34137e6	−0.105177
$870$	0	0
$871$	−4.15903e7	−1.85758
$872$	−3.12444e7	−1.39149
$873$	0	0
$874$	2.86690e7	1.26950
$875$	1.44030e6	0.0635967
$876$	0	0
$877$	2.44197e7	1.07212	0.536058	−	0.844181i	$-0.319913\pi$
$877$	2.44197e7	1.07212	0.536058	+	0.844181i	$0.319913\pi$
$878$	2.93934e7	1.28681
$879$	0	0
$880$	2.85221e6	0.124158
$881$	−2.05768e7	−0.893180	−0.446590	−	0.894739i	$-0.647362\pi$
$881$	−2.05768e7	−0.893180	−0.446590	+	0.894739i	$0.647362\pi$
$882$	0	0
$883$	−1.69532e7	−0.731727	−0.365864	−	0.930668i	$-0.619226\pi$
$883$	−1.69532e7	−0.731727	−0.365864	+	0.930668i	$0.619226\pi$
$884$	−4.75048e6	−0.204459
$885$	0	0
$886$	1.07890e6	0.0461738
$887$	2.25912e7	0.964118	0.482059	−	0.876139i	$-0.339889\pi$
$887$	2.25912e7	0.964118	0.482059	+	0.876139i	$0.339889\pi$
$888$	0	0
$889$	1.73498e6	0.0736276
$890$	254177.	0.0107562
$891$	0	0
$892$	3.16962e6	0.133381
$893$	3.38776e7	1.42162
$894$	0	0
$895$	1.65493e6	0.0690592
$896$	1.89716e6	0.0789469
$897$	0	0
$898$	−1.98921e6	−0.0823169
$899$	3.27861e6	0.135298
$900$	0	0
$901$	8.67924e6	0.356180
$902$	1.69059e6	0.0691865
$903$	0	0
$904$	8.90665e6	0.362488
$905$	1.49177e6	0.0605454
$906$	0	0
$907$	−3.60866e6	−0.145656	−0.0728279	−	0.997345i	$-0.523202\pi$
$907$	−3.60866e6	−0.145656	−0.0728279	+	0.997345i	$0.523202\pi$
$908$	−2.44233e6	−0.0983082
$909$	0	0
$910$	−1.25034e6	−0.0500524
$911$	3.83128e7	1.52950	0.764748	−	0.644329i	$-0.222863\pi$
$911$	3.83128e7	1.52950	0.764748	+	0.644329i	$0.222863\pi$
$912$	0	0
$913$	−2.05477e7	−0.815804
$914$	−3.28038e7	−1.29885
$915$	0	0
$916$	−2.13732e6	−0.0841650
$917$	9.80948e6	0.385232
$918$	0	0
$919$	−4.24918e7	−1.65965	−0.829824	−	0.558026i	$-0.811559\pi$
$919$	−4.24918e7	−1.65965	−0.829824	+	0.558026i	$0.811559\pi$
$920$	7.09402e6	0.276327
$921$	0	0
$922$	1.77663e6	0.0688286
$923$	2.73035e7	1.05491
$924$	0	0
$925$	1.79774e7	0.690834
$926$	2.47089e7	0.946948
$927$	0	0
$928$	−9.65571e6	−0.368056
$929$	2.93992e7	1.11762	0.558812	−	0.829294i	$-0.311257\pi$
$929$	2.93992e7	1.11762	0.558812	+	0.829294i	$0.311257\pi$
$930$	0	0
$931$	2.15674e7	0.815498
$932$	2.49310e6	0.0940156
$933$	0	0
$934$	−3.11341e7	−1.16780
$935$	2.01035e6	0.0752045
$936$	0	0
$937$	−8.24040e6	−0.306619	−0.153310	−	0.988178i	$-0.548993\pi$
$937$	−8.24040e6	−0.306619	−0.153310	+	0.988178i	$0.548993\pi$
$938$	5.05196e6	0.187479
$939$	0	0
$940$	1.90338e6	0.0702597
$941$	−1.23397e7	−0.454286	−0.227143	−	0.973861i	$-0.572938\pi$
$941$	−1.23397e7	−0.454286	−0.227143	+	0.973861i	$0.572938\pi$
$942$	0	0
$943$	2.85299e6	0.104477
$944$	2.20985e6	0.0807111
$945$	0	0
$946$	−2.62886e7	−0.955079
$947$	−2.68109e7	−0.971485	−0.485743	−	0.874102i	$-0.661451\pi$
$947$	−2.68109e7	−0.971485	−0.485743	+	0.874102i	$0.661451\pi$
$948$	0	0
$949$	8.83529e7	3.18460
$950$	−1.96397e7	−0.706034
$951$	0	0
$952$	2.54140e6	0.0908826
$953$	−2.06693e6	−0.0737214	−0.0368607	−	0.999320i	$-0.511736\pi$
$953$	−2.06693e6	−0.0737214	−0.0368607	+	0.999320i	$0.511736\pi$
$954$	0	0
$955$	2.19885e6	0.0780168
$956$	−2.30990e6	−0.0817424
$957$	0	0
$958$	−1.28831e7	−0.453531
$959$	1.94374e6	0.0682484
$960$	0	0
$961$	−2.73887e7	−0.956672
$962$	−3.15449e7	−1.09898
$963$	0	0
$964$	−1.16281e7	−0.403010
$965$	−2.18234e6	−0.0754406
$966$	0	0
$967$	1.24136e6	0.0426904	0.0213452	−	0.999772i	$-0.493205\pi$
$967$	1.24136e6	0.0426904	0.0213452	+	0.999772i	$0.493205\pi$
$968$	2.93112e7	1.00541
$969$	0	0
$970$	−3.38796e6	−0.115614
$971$	−2.08971e7	−0.711277	−0.355639	−	0.934624i	$-0.615737\pi$
$971$	−2.08971e7	−0.711277	−0.355639	+	0.934624i	$0.615737\pi$
$972$	0	0
$973$	−8.90046e6	−0.301391
$974$	1.49270e7	0.504167
$975$	0	0
$976$	−4.96103e6	−0.166705
$977$	−4.79889e7	−1.60844	−0.804220	−	0.594332i	$-0.797416\pi$
$977$	−4.79889e7	−1.60844	−0.804220	+	0.594332i	$0.797416\pi$
$978$	0	0
$979$	3.68888e6	0.123009
$980$	1.21174e6	0.0403036
$981$	0	0
$982$	−244230.	−0.00808202
$983$	5.71930e7	1.88781	0.943907	−	0.330212i	$-0.107120\pi$
$983$	5.71930e7	1.88781	0.943907	+	0.330212i	$0.107120\pi$
$984$	0	0
$985$	5.70911e6	0.187490
$986$	6.26186e6	0.205121
$987$	0	0
$988$	−1.43339e7	−0.467167
$989$	−4.43639e7	−1.44225
$990$	0	0
$991$	5.03427e7	1.62837	0.814184	−	0.580607i	$-0.197185\pi$
$991$	5.03427e7	1.62837	0.814184	+	0.580607i	$0.197185\pi$
$992$	−3.65320e6	−0.117868
$993$	0	0
$994$	−3.31655e6	−0.106468
$995$	−8.28852e6	−0.265411
$996$	0	0
$997$	−2.10591e7	−0.670968	−0.335484	−	0.942046i	$-0.608900\pi$
$997$	−2.10591e7	−0.670968	−0.335484	+	0.942046i	$0.608900\pi$
$998$	−2.13909e7	−0.679833
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.b.1.4		11
3.2	odd	2		177.6.a.a.1.8	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.a.1.8	✓	11	3.2	odd	2
531.6.a.b.1.4		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-4.75393 q^{2} -9.40015 q^{4} +8.06966 q^{5} -28.8581 q^{7} +196.813 q^{8} +O(q^{10})\)	\(q-4.75393 q^{2} -9.40015 q^{4} +8.06966 q^{5} -28.8581 q^{7} +196.813 q^{8} -38.3626 q^{10} -556.758 q^{11} -1129.41 q^{13} +137.189 q^{14} -634.832 q^{16} -447.456 q^{17} -1350.14 q^{19} -75.8560 q^{20} +2646.79 q^{22} +4466.66 q^{23} -3059.88 q^{25} +5369.15 q^{26} +271.270 q^{28} +2943.74 q^{29} +1113.75 q^{31} -3280.08 q^{32} +2127.18 q^{34} -232.875 q^{35} -5875.21 q^{37} +6418.45 q^{38} +1588.22 q^{40} +638.732 q^{41} -9932.25 q^{43} +5233.61 q^{44} -21234.2 q^{46} -25092.0 q^{47} -15974.2 q^{49} +14546.5 q^{50} +10616.6 q^{52} -19396.9 q^{53} -4492.85 q^{55} -5679.66 q^{56} -13994.3 q^{58} -3481.00 q^{59} +7814.71 q^{61} -5294.71 q^{62} +35907.9 q^{64} -9113.98 q^{65} +36824.7 q^{67} +4206.15 q^{68} +1107.07 q^{70} -24174.9 q^{71} -78229.0 q^{73} +27930.3 q^{74} +12691.5 q^{76} +16067.0 q^{77} +4205.35 q^{79} -5122.88 q^{80} -3036.49 q^{82} +36905.9 q^{83} -3610.82 q^{85} +47217.2 q^{86} -109578. q^{88} -6625.63 q^{89} +32592.7 q^{91} -41987.2 q^{92} +119286. q^{94} -10895.1 q^{95} +88314.2 q^{97} +75940.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more