LMFDB - Embedded newform 531.6.a.b.1.8

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			1.8
Root			$-5.62527$ 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)

$f(q)$	$=$	$q+6.62527 q^{2} +11.8942 q^{4} +85.2025 q^{5} -103.010 q^{7} -133.206 q^{8} +O(q^{10})$	\(q+6.62527 q^{2} +11.8942 q^{4} +85.2025 q^{5} -103.010 q^{7} -133.206 q^{8} +564.490 q^{10} -579.703 q^{11} +435.487 q^{13} -682.469 q^{14} -1263.14 q^{16} -424.138 q^{17} +1540.54 q^{19} +1013.42 q^{20} -3840.69 q^{22} +4164.00 q^{23} +4134.46 q^{25} +2885.22 q^{26} -1225.22 q^{28} +8418.14 q^{29} +7073.40 q^{31} -4106.07 q^{32} -2810.03 q^{34} -8776.70 q^{35} +12420.5 q^{37} +10206.5 q^{38} -11349.5 q^{40} +2421.33 q^{41} -11711.3 q^{43} -6895.12 q^{44} +27587.6 q^{46} +9436.30 q^{47} -6195.96 q^{49} +27391.9 q^{50} +5179.79 q^{52} +23764.2 q^{53} -49392.1 q^{55} +13721.5 q^{56} +55772.5 q^{58} -3481.00 q^{59} +5566.88 q^{61} +46863.2 q^{62} +13216.7 q^{64} +37104.6 q^{65} -57888.7 q^{67} -5044.80 q^{68} -58148.0 q^{70} +11960.0 q^{71} +25056.7 q^{73} +82289.1 q^{74} +18323.5 q^{76} +59715.1 q^{77} +17801.5 q^{79} -107623. q^{80} +16042.0 q^{82} +2768.37 q^{83} -36137.6 q^{85} -77590.7 q^{86} +77219.9 q^{88} -17994.8 q^{89} -44859.5 q^{91} +49527.6 q^{92} +62518.1 q^{94} +131257. q^{95} -166125. q^{97} -41049.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	6.62527	1.17119	0.585597	−	0.810602i	$-0.300860\pi$
$2$	6.62527	1.17119	0.585597	+	0.810602i	$0.300860\pi$
$3$	0	0
$3$	0	0
$4$	11.8942	0.371695
$5$	85.2025	1.52415	0.762074	−	0.647490i	$-0.224181\pi$
$5$	85.2025	1.52415	0.762074	+	0.647490i	$0.224181\pi$
$6$	0	0
$7$	−103.010	−0.794573	−0.397287	−	0.917695i	$-0.630048\pi$
$7$	−103.010	−0.794573	−0.397287	+	0.917695i	$0.630048\pi$
$8$	−133.206	−0.735867
$9$	0	0
$10$	564.490	1.78507
$11$	−579.703	−1.44452	−0.722260	−	0.691622i	$-0.756897\pi$
$11$	−579.703	−1.44452	−0.722260	+	0.691622i	$0.756897\pi$
$12$	0	0
$13$	435.487	0.714689	0.357344	−	0.933973i	$-0.383682\pi$
$13$	435.487	0.714689	0.357344	+	0.933973i	$0.383682\pi$
$14$	−682.469	−0.930599
$15$	0	0
$16$	−1263.14	−1.23354
$17$	−424.138	−0.355946	−0.177973	−	0.984035i	$-0.556954\pi$
$17$	−424.138	−0.355946	−0.177973	+	0.984035i	$0.556954\pi$
$18$	0	0
$19$	1540.54	0.979012	0.489506	−	0.872000i	$-0.337177\pi$
$19$	1540.54	0.979012	0.489506	+	0.872000i	$0.337177\pi$
$20$	1013.42	0.566518
$21$	0	0
$22$	−3840.69	−1.69181
$23$	4164.00	1.64131	0.820656	−	0.571423i	$-0.193609\pi$
$23$	4164.00	1.64131	0.820656	+	0.571423i	$0.193609\pi$
$24$	0	0
$25$	4134.46	1.32303
$26$	2885.22	0.837039
$27$	0	0
$28$	−1225.22	−0.295339
$29$	8418.14	1.85875	0.929376	−	0.369135i	$-0.120346\pi$
$29$	8418.14	1.85875	0.929376	+	0.369135i	$0.120346\pi$
$30$	0	0
$31$	7073.40	1.32198	0.660989	−	0.750396i	$-0.270137\pi$
$31$	7073.40	1.32198	0.660989	+	0.750396i	$0.270137\pi$
$32$	−4106.07	−0.708845
$33$	0	0
$34$	−2810.03	−0.416882
$35$	−8776.70	−1.21105
$36$	0	0
$37$	12420.5	1.49154	0.745769	−	0.666205i	$-0.232082\pi$
$37$	12420.5	1.49154	0.745769	+	0.666205i	$0.232082\pi$
$38$	10206.5	1.14661
$39$	0	0
$40$	−11349.5	−1.12157
$41$	2421.33	0.224954	0.112477	−	0.993654i	$-0.464122\pi$
$41$	2421.33	0.224954	0.112477	+	0.993654i	$0.464122\pi$
$42$	0	0
$43$	−11711.3	−0.965905	−0.482953	−	0.875646i	$-0.660436\pi$
$43$	−11711.3	−0.965905	−0.482953	+	0.875646i	$0.660436\pi$
$44$	−6895.12	−0.536921
$45$	0	0
$46$	27587.6	1.92229
$47$	9436.30	0.623099	0.311550	−	0.950230i	$-0.399152\pi$
$47$	9436.30	0.623099	0.311550	+	0.950230i	$0.399152\pi$
$48$	0	0
$49$	−6195.96	−0.368654
$50$	27391.9	1.54952
$51$	0	0
$52$	5179.79	0.265646
$53$	23764.2	1.16207	0.581037	−	0.813877i	$-0.302647\pi$
$53$	23764.2	1.16207	0.581037	+	0.813877i	$0.302647\pi$
$54$	0	0
$55$	−49392.1	−2.20166
$56$	13721.5	0.584700
$57$	0	0
$58$	55772.5	2.17696
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	5566.88	0.191552	0.0957761	−	0.995403i	$-0.469467\pi$
$61$	5566.88	0.191552	0.0957761	+	0.995403i	$0.469467\pi$
$62$	46863.2	1.54829
$63$	0	0
$64$	13216.7	0.403343
$65$	37104.6	1.08929
$66$	0	0
$67$	−57888.7	−1.57546	−0.787729	−	0.616022i	$-0.788743\pi$
$67$	−57888.7	−1.57546	−0.787729	+	0.616022i	$0.788743\pi$
$68$	−5044.80	−0.132304
$69$	0	0
$70$	−58148.0	−1.41837
$71$	11960.0	0.281569	0.140784	−	0.990040i	$-0.455038\pi$
$71$	11960.0	0.281569	0.140784	+	0.990040i	$0.455038\pi$
$72$	0	0
$73$	25056.7	0.550322	0.275161	−	0.961398i	$-0.411269\pi$
$73$	25056.7	0.550322	0.275161	+	0.961398i	$0.411269\pi$
$74$	82289.1	1.74688
$75$	0	0
$76$	18323.5	0.363894
$77$	59715.1	1.14778
$78$	0	0
$79$	17801.5	0.320914	0.160457	−	0.987043i	$-0.448703\pi$
$79$	17801.5	0.320914	0.160457	+	0.987043i	$0.448703\pi$
$80$	−107623.	−1.88009
$81$	0	0
$82$	16042.0	0.263465
$83$	2768.37	0.0441092	0.0220546	−	0.999757i	$-0.492979\pi$
$83$	2768.37	0.0441092	0.0220546	+	0.999757i	$0.492979\pi$
$84$	0	0
$85$	−36137.6	−0.542515
$86$	−77590.7	−1.13126
$87$	0	0
$88$	77219.9	1.06297
$89$	−17994.8	−0.240809	−0.120404	−	0.992725i	$-0.538419\pi$
$89$	−17994.8	−0.240809	−0.120404	+	0.992725i	$0.538419\pi$
$90$	0	0
$91$	−44859.5	−0.567872
$92$	49527.6	0.610067
$93$	0	0
$94$	62518.1	0.729770
$95$	131257.	1.49216
$96$	0	0
$97$	−166125.	−1.79269	−0.896346	−	0.443355i	$-0.853788\pi$
$97$	−166125.	−1.79269	−0.896346	+	0.443355i	$0.853788\pi$
$98$	−41049.9	−0.431765
$99$	0	0
$100$	49176.3	0.491763
$101$	38127.7	0.371909	0.185954	−	0.982558i	$-0.440462\pi$
$101$	38127.7	0.371909	0.185954	+	0.982558i	$0.440462\pi$
$102$	0	0
$103$	81615.4	0.758018	0.379009	−	0.925393i	$-0.376265\pi$
$103$	81615.4	0.758018	0.379009	+	0.925393i	$0.376265\pi$
$104$	−58009.5	−0.525916
$105$	0	0
$106$	157444.	1.36101
$107$	−121265.	−1.02395	−0.511973	−	0.859002i	$-0.671085\pi$
$107$	−121265.	−1.02395	−0.511973	+	0.859002i	$0.671085\pi$
$108$	0	0
$109$	−153096.	−1.23424	−0.617118	−	0.786870i	$-0.711700\pi$
$109$	−153096.	−1.23424	−0.617118	+	0.786870i	$0.711700\pi$
$110$	−327236.	−2.57857
$111$	0	0
$112$	130116.	0.980136
$113$	54428.5	0.400987	0.200494	−	0.979695i	$-0.435745\pi$
$113$	54428.5	0.400987	0.200494	+	0.979695i	$0.435745\pi$
$114$	0	0
$115$	354783.	2.50160
$116$	100127.	0.690889
$117$	0	0
$118$	−23062.6	−0.152476
$119$	43690.4	0.282825
$120$	0	0
$121$	175004.	1.08664
$122$	36882.1	0.224345
$123$	0	0
$124$	84132.8	0.491373
$125$	86008.6	0.492342
$126$	0	0
$127$	−203111.	−1.11744	−0.558721	−	0.829356i	$-0.688708\pi$
$127$	−203111.	−1.11744	−0.558721	+	0.829356i	$0.688708\pi$
$128$	218959.	1.18124
$129$	0	0
$130$	245828.	1.27577
$131$	282282.	1.43716	0.718579	−	0.695445i	$-0.244793\pi$
$131$	282282.	1.43716	0.718579	+	0.695445i	$0.244793\pi$
$132$	0	0
$133$	−158690.	−0.777896
$134$	−383529.	−1.84517
$135$	0	0
$136$	56497.7	0.261929
$137$	290437.	1.32206	0.661029	−	0.750360i	$-0.270120\pi$
$137$	290437.	1.32206	0.661029	+	0.750360i	$0.270120\pi$
$138$	0	0
$139$	429789.	1.88677	0.943383	−	0.331705i	$-0.107624\pi$
$139$	429789.	1.88677	0.943383	+	0.331705i	$0.107624\pi$
$140$	−104392.	−0.450140
$141$	0	0
$142$	79238.2	0.329772
$143$	−252453.	−1.03238
$144$	0	0
$145$	717247.	2.83301
$146$	166008.	0.644534
$147$	0	0
$148$	147732.	0.554397
$149$	383936.	1.41675	0.708376	−	0.705835i	$-0.249428\pi$
$149$	383936.	1.41675	0.708376	+	0.705835i	$0.249428\pi$
$150$	0	0
$151$	−91551.4	−0.326755	−0.163378	−	0.986564i	$-0.552239\pi$
$151$	−91551.4	−0.326755	−0.163378	+	0.986564i	$0.552239\pi$
$152$	−205209.	−0.720422
$153$	0	0
$154$	395629.	1.34427
$155$	602671.	2.01489
$156$	0	0
$157$	−18734.2	−0.0606578	−0.0303289	−	0.999540i	$-0.509655\pi$
$157$	−18734.2	−0.0606578	−0.0303289	+	0.999540i	$0.509655\pi$
$158$	117940.	0.375852
$159$	0	0
$160$	−349847.	−1.08039
$161$	−428933.	−1.30414
$162$	0	0
$163$	85172.5	0.251091	0.125545	−	0.992088i	$-0.459932\pi$
$163$	85172.5	0.251091	0.125545	+	0.992088i	$0.459932\pi$
$164$	28799.9	0.0836143
$165$	0	0
$166$	18341.2	0.0516604
$167$	−541622.	−1.50281	−0.751407	−	0.659839i	$-0.770625\pi$
$167$	−541622.	−1.50281	−0.751407	+	0.659839i	$0.770625\pi$
$168$	0	0
$169$	−181644.	−0.489220
$170$	−239421.	−0.635390
$171$	0	0
$172$	−139297.	−0.359022
$173$	−238752.	−0.606502	−0.303251	−	0.952911i	$-0.598072\pi$
$173$	−238752.	−0.606502	−0.303251	+	0.952911i	$0.598072\pi$
$174$	0	0
$175$	−425890.	−1.05124
$176$	732247.	1.78187
$177$	0	0
$178$	−119221.	−0.282034
$179$	−284533.	−0.663744	−0.331872	−	0.943324i	$-0.607680\pi$
$179$	−284533.	−0.663744	−0.331872	+	0.943324i	$0.607680\pi$
$180$	0	0
$181$	−402281.	−0.912711	−0.456355	−	0.889798i	$-0.650845\pi$
$181$	−402281.	−0.912711	−0.456355	+	0.889798i	$0.650845\pi$
$182$	−297206.	−0.665089
$183$	0	0
$184$	−554670.	−1.20779
$185$	1.05826e6	2.27332
$186$	0	0
$187$	245874.	0.514172
$188$	112238.	0.231603
$189$	0	0
$190$	869617.	1.74761
$191$	624305.	1.23826	0.619132	−	0.785287i	$-0.287485\pi$
$191$	624305.	1.23826	0.619132	+	0.785287i	$0.287485\pi$
$192$	0	0
$193$	835697.	1.61494	0.807469	−	0.589910i	$-0.200837\pi$
$193$	835697.	1.61494	0.807469	+	0.589910i	$0.200837\pi$
$194$	−1.10062e6	−2.09959
$195$	0	0
$196$	−73696.3	−0.137027
$197$	724552.	1.33016	0.665080	−	0.746772i	$-0.268397\pi$
$197$	724552.	1.33016	0.665080	+	0.746772i	$0.268397\pi$
$198$	0	0
$199$	−691882.	−1.23851	−0.619255	−	0.785190i	$-0.712565\pi$
$199$	−691882.	−1.23851	−0.619255	+	0.785190i	$0.712565\pi$
$200$	−550736.	−0.973572
$201$	0	0
$202$	252606.	0.435577
$203$	−867152.	−1.47691
$204$	0	0
$205$	206303.	0.342863
$206$	540725.	0.887786
$207$	0	0
$208$	−550082.	−0.881595
$209$	−893053.	−1.41420
$210$	0	0
$211$	249343.	0.385559	0.192779	−	0.981242i	$-0.438250\pi$
$211$	249343.	0.385559	0.192779	+	0.981242i	$0.438250\pi$
$212$	282657.	0.431937
$213$	0	0
$214$	−803416.	−1.19924
$215$	−997833.	−1.47218
$216$	0	0
$217$	−728630.	−1.05041
$218$	−1.01430e6	−1.44553
$219$	0	0
$220$	−587482.	−0.818347
$221$	−184706.	−0.254391
$222$	0	0
$223$	−542135.	−0.730038	−0.365019	−	0.931000i	$-0.618938\pi$
$223$	−542135.	−0.730038	−0.365019	+	0.931000i	$0.618938\pi$
$224$	422966.	0.563229
$225$	0	0
$226$	360604.	0.469634
$227$	480667.	0.619127	0.309563	−	0.950879i	$-0.399817\pi$
$227$	480667.	0.619127	0.309563	+	0.950879i	$0.399817\pi$
$228$	0	0
$229$	−623840.	−0.786112	−0.393056	−	0.919514i	$-0.628582\pi$
$229$	−623840.	−0.786112	−0.393056	+	0.919514i	$0.628582\pi$
$230$	2.35053e6	2.92986
$231$	0	0
$232$	−1.12135e6	−1.36779
$233$	−645676.	−0.779157	−0.389579	−	0.920993i	$-0.627379\pi$
$233$	−645676.	−0.779157	−0.389579	+	0.920993i	$0.627379\pi$
$234$	0	0
$235$	803996.	0.949695
$236$	−41403.9	−0.0483906
$237$	0	0
$238$	289461.	0.331243
$239$	−1.10502e6	−1.25134	−0.625669	−	0.780089i	$-0.715174\pi$
$239$	−1.10502e6	−1.25134	−0.625669	+	0.780089i	$0.715174\pi$
$240$	0	0
$241$	1.44502e6	1.60262	0.801309	−	0.598250i	$-0.204137\pi$
$241$	1.44502e6	1.60262	0.801309	+	0.598250i	$0.204137\pi$
$242$	1.15945e6	1.27266
$243$	0	0
$244$	66213.8	0.0711990
$245$	−527911.	−0.561883
$246$	0	0
$247$	670883.	0.699688
$248$	−942220.	−0.972799
$249$	0	0
$250$	569830.	0.576628
$251$	−528809.	−0.529803	−0.264902	−	0.964275i	$-0.585340\pi$
$251$	−528809.	−0.529803	−0.264902	+	0.964275i	$0.585340\pi$
$252$	0	0
$253$	−2.41388e6	−2.37091
$254$	−1.34567e6	−1.30874
$255$	0	0
$256$	1.02773e6	0.980116
$257$	1.15172e6	1.08771	0.543856	−	0.839179i	$-0.316964\pi$
$257$	1.15172e6	1.08771	0.543856	+	0.839179i	$0.316964\pi$
$258$	0	0
$259$	−1.27943e6	−1.18514
$260$	441331.	0.404884
$261$	0	0
$262$	1.87019e6	1.68319
$263$	−452277.	−0.403195	−0.201598	−	0.979468i	$-0.564613\pi$
$263$	−452277.	−0.403195	−0.201598	+	0.979468i	$0.564613\pi$
$264$	0	0
$265$	2.02477e6	1.77117
$266$	−1.05137e6	−0.911067
$267$	0	0
$268$	−688543.	−0.585590
$269$	1.71384e6	1.44407	0.722036	−	0.691856i	$-0.243207\pi$
$269$	1.71384e6	1.44407	0.722036	+	0.691856i	$0.243207\pi$
$270$	0	0
$271$	1.32879e6	1.09909	0.549546	−	0.835463i	$-0.314801\pi$
$271$	1.32879e6	1.09909	0.549546	+	0.835463i	$0.314801\pi$
$272$	535746.	0.439073
$273$	0	0
$274$	1.92422e6	1.54839
$275$	−2.39676e6	−1.91114
$276$	0	0
$277$	−1.54297e6	−1.20825	−0.604126	−	0.796889i	$-0.706478\pi$
$277$	−1.54297e6	−1.20825	−0.604126	+	0.796889i	$0.706478\pi$
$278$	2.84747e6	2.20977
$279$	0	0
$280$	1.16911e6	0.891169
$281$	5515.62	0.00416705	0.00208352	−	0.999998i	$-0.499337\pi$
$281$	5515.62	0.00416705	0.00208352	+	0.999998i	$0.499337\pi$
$282$	0	0
$283$	−944988.	−0.701391	−0.350696	−	0.936489i	$-0.614055\pi$
$283$	−944988.	−0.701391	−0.350696	+	0.936489i	$0.614055\pi$
$284$	142255.	0.104658
$285$	0	0
$286$	−1.67257e6	−1.20912
$287$	−249421.	−0.178742
$288$	0	0
$289$	−1.23996e6	−0.873302
$290$	4.75196e6	3.31801
$291$	0	0
$292$	298031.	0.204552
$293$	1.52734e6	1.03936	0.519681	−	0.854361i	$-0.326051\pi$
$293$	1.52734e6	1.03936	0.519681	+	0.854361i	$0.326051\pi$
$294$	0	0
$295$	−296590.	−0.198427
$296$	−1.65448e6	−1.09757
$297$	0	0
$298$	2.54368e6	1.65929
$299$	1.81337e6	1.17303
$300$	0	0
$301$	1.20638e6	0.767482
$302$	−606553.	−0.382694
$303$	0	0
$304$	−1.94592e6	−1.20765
$305$	474312.	0.291954
$306$	0	0
$307$	−2.82858e6	−1.71286	−0.856431	−	0.516261i	$-0.827323\pi$
$307$	−2.82858e6	−1.71286	−0.856431	+	0.516261i	$0.827323\pi$
$308$	710266.	0.426623
$309$	0	0
$310$	3.99286e6	2.35983
$311$	−490910.	−0.287806	−0.143903	−	0.989592i	$-0.545965\pi$
$311$	−490910.	−0.287806	−0.143903	+	0.989592i	$0.545965\pi$
$312$	0	0
$313$	3.05776e6	1.76418	0.882089	−	0.471083i	$-0.156137\pi$
$313$	3.05776e6	1.76418	0.882089	+	0.471083i	$0.156137\pi$
$314$	−124119.	−0.0710420
$315$	0	0
$316$	211735.	0.119282
$317$	3.44305e6	1.92440	0.962199	−	0.272348i	$-0.0878002\pi$
$317$	3.44305e6	1.92440	0.962199	+	0.272348i	$0.0878002\pi$
$318$	0	0
$319$	−4.88002e6	−2.68500
$320$	1.12610e6	0.614754
$321$	0	0
$322$	−2.84180e6	−1.52740
$323$	−653399.	−0.348476
$324$	0	0
$325$	1.80050e6	0.945553
$326$	564291.	0.294076
$327$	0	0
$328$	−322536.	−0.165536
$329$	−972032.	−0.495098
$330$	0	0
$331$	−2.53352e6	−1.27102	−0.635512	−	0.772091i	$-0.719211\pi$
$331$	−2.53352e6	−1.27102	−0.635512	+	0.772091i	$0.719211\pi$
$332$	32927.7	0.0163952
$333$	0	0
$334$	−3.58840e6	−1.76009
$335$	−4.93226e6	−2.40123
$336$	0	0
$337$	−2.44618e6	−1.17331	−0.586657	−	0.809835i	$-0.699556\pi$
$337$	−2.44618e6	−1.17331	−0.586657	+	0.809835i	$0.699556\pi$
$338$	−1.20344e6	−0.572972
$339$	0	0
$340$	−429829.	−0.201650
$341$	−4.10047e6	−1.90962
$342$	0	0
$343$	2.36953e6	1.08750
$344$	1.56002e6	0.710778
$345$	0	0
$346$	−1.58180e6	−0.710332
$347$	−92368.8	−0.0411815	−0.0205907	−	0.999788i	$-0.506555\pi$
$347$	−92368.8	−0.0411815	−0.0205907	+	0.999788i	$0.506555\pi$
$348$	0	0
$349$	519444.	0.228284	0.114142	−	0.993464i	$-0.463588\pi$
$349$	519444.	0.228284	0.114142	+	0.993464i	$0.463588\pi$
$350$	−2.82164e6	−1.23121
$351$	0	0
$352$	2.38030e6	1.02394
$353$	1.65548e6	0.707109	0.353555	−	0.935414i	$-0.384973\pi$
$353$	1.65548e6	0.707109	0.353555	+	0.935414i	$0.384973\pi$
$354$	0	0
$355$	1.01902e6	0.429153
$356$	−214035.	−0.0895074
$357$	0	0
$358$	−1.88511e6	−0.777373
$359$	−2.51488e6	−1.02987	−0.514934	−	0.857230i	$-0.672184\pi$
$359$	−2.51488e6	−1.02987	−0.514934	+	0.857230i	$0.672184\pi$
$360$	0	0
$361$	−102848.	−0.0415361
$362$	−2.66522e6	−1.06896
$363$	0	0
$364$	−533570.	−0.211075
$365$	2.13489e6	0.838773
$366$	0	0
$367$	−792348.	−0.307080	−0.153540	−	0.988142i	$-0.549067\pi$
$367$	−792348.	−0.307080	−0.153540	+	0.988142i	$0.549067\pi$
$368$	−5.25972e6	−2.02462
$369$	0	0
$370$	7.01124e6	2.66250
$371$	−2.44795e6	−0.923352
$372$	0	0
$373$	846687.	0.315102	0.157551	−	0.987511i	$-0.449640\pi$
$373$	846687.	0.315102	0.157551	+	0.987511i	$0.449640\pi$
$374$	1.62898e6	0.602195
$375$	0	0
$376$	−1.25697e6	−0.458518
$377$	3.66599e6	1.32843
$378$	0	0
$379$	1.63333e6	0.584085	0.292043	−	0.956405i	$-0.405665\pi$
$379$	1.63333e6	0.584085	0.292043	+	0.956405i	$0.405665\pi$
$380$	1.56121e6	0.554628
$381$	0	0
$382$	4.13619e6	1.45025
$383$	4.23461e6	1.47508	0.737541	−	0.675302i	$-0.235987\pi$
$383$	4.23461e6	1.47508	0.737541	+	0.675302i	$0.235987\pi$
$384$	0	0
$385$	5.08787e6	1.74938
$386$	5.53672e6	1.89141
$387$	0	0
$388$	−1.97593e6	−0.666335
$389$	−4.32595e6	−1.44946	−0.724732	−	0.689031i	$-0.758037\pi$
$389$	−4.32595e6	−1.44946	−0.724732	+	0.689031i	$0.758037\pi$
$390$	0	0
$391$	−1.76611e6	−0.584219
$392$	825340.	0.271280
$393$	0	0
$394$	4.80035e6	1.55787
$395$	1.51673e6	0.489120
$396$	0	0
$397$	3.34065e6	1.06379	0.531894	−	0.846811i	$-0.321480\pi$
$397$	3.34065e6	1.06379	0.531894	+	0.846811i	$0.321480\pi$
$398$	−4.58391e6	−1.45054
$399$	0	0
$400$	−5.22241e6	−1.63200
$401$	600538.	0.186500	0.0932502	−	0.995643i	$-0.470274\pi$
$401$	600538.	0.186500	0.0932502	+	0.995643i	$0.470274\pi$
$402$	0	0
$403$	3.08037e6	0.944802
$404$	453500.	0.138237
$405$	0	0
$406$	−5.74512e6	−1.72975
$407$	−7.20019e6	−2.15456
$408$	0	0
$409$	−1.87846e6	−0.555256	−0.277628	−	0.960689i	$-0.589548\pi$
$409$	−1.87846e6	−0.555256	−0.277628	+	0.960689i	$0.589548\pi$
$410$	1.36681e6	0.401559
$411$	0	0
$412$	970754.	0.281751
$413$	358577.	0.103445
$414$	0	0
$415$	235872.	0.0672289
$416$	−1.78814e6	−0.506604
$417$	0	0
$418$	−5.91672e6	−1.65630
$419$	4.65041e6	1.29407	0.647033	−	0.762462i	$-0.276010\pi$
$419$	4.65041e6	1.29407	0.647033	+	0.762462i	$0.276010\pi$
$420$	0	0
$421$	−1.24723e6	−0.342959	−0.171479	−	0.985188i	$-0.554855\pi$
$421$	−1.24723e6	−0.342959	−0.171479	+	0.985188i	$0.554855\pi$
$422$	1.65196e6	0.451564
$423$	0	0
$424$	−3.16554e6	−0.855131
$425$	−1.75358e6	−0.470927
$426$	0	0
$427$	−573443.	−0.152202
$428$	−1.44236e6	−0.380596
$429$	0	0
$430$	−6.61092e6	−1.72421
$431$	−975574.	−0.252969	−0.126484	−	0.991969i	$-0.540369\pi$
$431$	−975574.	−0.252969	−0.126484	+	0.991969i	$0.540369\pi$
$432$	0	0
$433$	2.56758e6	0.658119	0.329060	−	0.944309i	$-0.393268\pi$
$433$	2.56758e6	0.658119	0.329060	+	0.944309i	$0.393268\pi$
$434$	−4.82737e6	−1.23023
$435$	0	0
$436$	−1.82096e6	−0.458760
$437$	6.41479e6	1.60686
$438$	0	0
$439$	875156.	0.216733	0.108366	−	0.994111i	$-0.465438\pi$
$439$	875156.	0.216733	0.108366	+	0.994111i	$0.465438\pi$
$440$	6.57933e6	1.62013
$441$	0	0
$442$	−1.22373e6	−0.297941
$443$	−4.39426e6	−1.06384	−0.531920	−	0.846795i	$-0.678529\pi$
$443$	−4.39426e6	−1.06384	−0.531920	+	0.846795i	$0.678529\pi$
$444$	0	0
$445$	−1.53320e6	−0.367028
$446$	−3.59180e6	−0.855016
$447$	0	0
$448$	−1.36145e6	−0.320485
$449$	−2.22268e6	−0.520308	−0.260154	−	0.965567i	$-0.583773\pi$
$449$	−2.22268e6	−0.520308	−0.260154	+	0.965567i	$0.583773\pi$
$450$	0	0
$451$	−1.40365e6	−0.324951
$452$	647386.	0.149045
$453$	0	0
$454$	3.18455e6	0.725118
$455$	−3.82214e6	−0.865521
$456$	0	0
$457$	−3.49053e6	−0.781810	−0.390905	−	0.920431i	$-0.627838\pi$
$457$	−3.49053e6	−0.781810	−0.390905	+	0.920431i	$0.627838\pi$
$458$	−4.13311e6	−0.920690
$459$	0	0
$460$	4.21987e6	0.929833
$461$	−341191.	−0.0747732	−0.0373866	−	0.999301i	$-0.511903\pi$
$461$	−341191.	−0.0747732	−0.0373866	+	0.999301i	$0.511903\pi$
$462$	0	0
$463$	−6.72986e6	−1.45899	−0.729497	−	0.683984i	$-0.760246\pi$
$463$	−6.72986e6	−1.45899	−0.729497	+	0.683984i	$0.760246\pi$
$464$	−1.06333e7	−2.29284
$465$	0	0
$466$	−4.27778e6	−0.912544
$467$	2.11594e6	0.448964	0.224482	−	0.974478i	$-0.427931\pi$
$467$	2.11594e6	0.448964	0.224482	+	0.974478i	$0.427931\pi$
$468$	0	0
$469$	5.96311e6	1.25182
$470$	5.32669e6	1.11228
$471$	0	0
$472$	463691.	0.0958017
$473$	6.78908e6	1.39527
$474$	0	0
$475$	6.36929e6	1.29526
$476$	519664.	0.105125
$477$	0	0
$478$	−7.32105e6	−1.46556
$479$	8.71672e6	1.73586	0.867929	−	0.496688i	$-0.165451\pi$
$479$	8.71672e6	1.73586	0.867929	+	0.496688i	$0.165451\pi$
$480$	0	0
$481$	5.40896e6	1.06599
$482$	9.57363e6	1.87698
$483$	0	0
$484$	2.08154e6	0.403898
$485$	−1.41543e7	−2.73233
$486$	0	0
$487$	−8.19716e6	−1.56618	−0.783089	−	0.621910i	$-0.786357\pi$
$487$	−8.19716e6	−1.56618	−0.783089	+	0.621910i	$0.786357\pi$
$488$	−741542.	−0.140957
$489$	0	0
$490$	−3.49756e6	−0.658074
$491$	6.65738e6	1.24623	0.623117	−	0.782128i	$-0.285866\pi$
$491$	6.65738e6	1.24623	0.623117	+	0.782128i	$0.285866\pi$
$492$	0	0
$493$	−3.57045e6	−0.661616
$494$	4.44479e6	0.819471
$495$	0	0
$496$	−8.93472e6	−1.63071
$497$	−1.23200e6	−0.223727
$498$	0	0
$499$	−2.16723e6	−0.389630	−0.194815	−	0.980840i	$-0.562411\pi$
$499$	−2.16723e6	−0.389630	−0.194815	+	0.980840i	$0.562411\pi$
$500$	1.02301e6	0.183001
$501$	0	0
$502$	−3.50351e6	−0.620503
$503$	3.01053e6	0.530547	0.265273	−	0.964173i	$-0.414538\pi$
$503$	3.01053e6	0.530547	0.265273	+	0.964173i	$0.414538\pi$
$504$	0	0
$505$	3.24857e6	0.566844
$506$	−1.59926e7	−2.77679
$507$	0	0
$508$	−2.41586e6	−0.415348
$509$	−8.34439e6	−1.42758	−0.713790	−	0.700360i	$-0.753023\pi$
$509$	−8.34439e6	−1.42758	−0.713790	+	0.700360i	$0.753023\pi$
$510$	0	0
$511$	−2.58109e6	−0.437271
$512$	−197715.	−0.0333322
$513$	0	0
$514$	7.63045e6	1.27392
$515$	6.95384e6	1.15533
$516$	0	0
$517$	−5.47025e6	−0.900079
$518$	−8.47659e6	−1.38802
$519$	0	0
$520$	−4.94256e6	−0.801573
$521$	−7.03207e6	−1.13498	−0.567491	−	0.823380i	$-0.692086\pi$
$521$	−7.03207e6	−1.13498	−0.567491	+	0.823380i	$0.692086\pi$
$522$	0	0
$523$	4.76023e6	0.760982	0.380491	−	0.924785i	$-0.375755\pi$
$523$	4.76023e6	0.760982	0.380491	+	0.924785i	$0.375755\pi$
$524$	3.35753e6	0.534185
$525$	0	0
$526$	−2.99646e6	−0.472220
$527$	−3.00010e6	−0.470553
$528$	0	0
$529$	1.09025e7	1.69390
$530$	1.34146e7	2.07439
$531$	0	0
$532$	−1.88750e6	−0.289140
$533$	1.05446e6	0.160772
$534$	0	0
$535$	−1.03321e7	−1.56065
$536$	7.71113e6	1.15933
$537$	0	0
$538$	1.13546e7	1.69129
$539$	3.59182e6	0.532528
$540$	0	0
$541$	1.92671e6	0.283024	0.141512	−	0.989937i	$-0.454804\pi$
$541$	1.92671e6	0.283024	0.141512	+	0.989937i	$0.454804\pi$
$542$	8.80362e6	1.28725
$543$	0	0
$544$	1.74154e6	0.252311
$545$	−1.30442e7	−1.88116
$546$	0	0
$547$	2.45466e6	0.350771	0.175385	−	0.984500i	$-0.443883\pi$
$547$	2.45466e6	0.350771	0.175385	+	0.984500i	$0.443883\pi$
$548$	3.45453e6	0.491403
$549$	0	0
$550$	−1.58792e7	−2.23832
$551$	1.29685e7	1.81974
$552$	0	0
$553$	−1.83373e6	−0.254989
$554$	−1.02226e7	−1.41510
$555$	0	0
$556$	5.11201e6	0.701302
$557$	−7.38777e6	−1.00896	−0.504482	−	0.863422i	$-0.668316\pi$
$557$	−7.38777e6	−1.00896	−0.504482	+	0.863422i	$0.668316\pi$
$558$	0	0
$559$	−5.10013e6	−0.690322
$560$	1.10862e7	1.49387
$561$	0	0
$562$	36542.5	0.00488042
$563$	−2.38312e6	−0.316866	−0.158433	−	0.987370i	$-0.550644\pi$
$563$	−2.38312e6	−0.316866	−0.158433	+	0.987370i	$0.550644\pi$
$564$	0	0
$565$	4.63745e6	0.611164
$566$	−6.26081e6	−0.821465
$567$	0	0
$568$	−1.59314e6	−0.207197
$569$	7.30870e6	0.946367	0.473184	−	0.880964i	$-0.343105\pi$
$569$	7.30870e6	0.946367	0.473184	+	0.880964i	$0.343105\pi$
$570$	0	0
$571$	2.52181e6	0.323684	0.161842	−	0.986817i	$-0.448256\pi$
$571$	2.52181e6	0.323684	0.161842	+	0.986817i	$0.448256\pi$
$572$	−3.00274e6	−0.383731
$573$	0	0
$574$	−1.65248e6	−0.209342
$575$	1.72159e7	2.17150
$576$	0	0
$577$	−758503.	−0.0948457	−0.0474229	−	0.998875i	$-0.515101\pi$
$577$	−758503.	−0.0948457	−0.0474229	+	0.998875i	$0.515101\pi$
$578$	−8.21510e6	−1.02281
$579$	0	0
$580$	8.53111e6	1.05302
$581$	−285169.	−0.0350480
$582$	0	0
$583$	−1.37762e7	−1.67864
$584$	−3.33771e6	−0.404964
$585$	0	0
$586$	1.01190e7	1.21729
$587$	−4.83844e6	−0.579576	−0.289788	−	0.957091i	$-0.593585\pi$
$587$	−4.83844e6	−0.579576	−0.289788	+	0.957091i	$0.593585\pi$
$588$	0	0
$589$	1.08968e7	1.29423
$590$	−1.96499e6	−0.232397
$591$	0	0
$592$	−1.56888e7	−1.83987
$593$	3.38920e6	0.395785	0.197893	−	0.980224i	$-0.436590\pi$
$593$	3.38920e6	0.395785	0.197893	+	0.980224i	$0.436590\pi$
$594$	0	0
$595$	3.72253e6	0.431068
$596$	4.56663e6	0.526600
$597$	0	0
$598$	1.20141e7	1.37384
$599$	−7.50330e6	−0.854447	−0.427224	−	0.904146i	$-0.640508\pi$
$599$	−7.50330e6	−0.854447	−0.427224	+	0.904146i	$0.640508\pi$
$600$	0	0
$601$	1.43644e7	1.62219	0.811095	−	0.584915i	$-0.198872\pi$
$601$	1.43644e7	1.62219	0.811095	+	0.584915i	$0.198872\pi$
$602$	7.99261e6	0.898871
$603$	0	0
$604$	−1.08893e6	−0.121453
$605$	1.49108e7	1.65620
$606$	0	0
$607$	−7.75401e6	−0.854191	−0.427095	−	0.904207i	$-0.640463\pi$
$607$	−7.75401e6	−0.854191	−0.427095	+	0.904207i	$0.640463\pi$
$608$	−6.32555e6	−0.693968
$609$	0	0
$610$	3.14244e6	0.341935
$611$	4.10939e6	0.445322
$612$	0	0
$613$	2.52805e6	0.271728	0.135864	−	0.990727i	$-0.456619\pi$
$613$	2.52805e6	0.271728	0.135864	+	0.990727i	$0.456619\pi$
$614$	−1.87401e7	−2.00609
$615$	0	0
$616$	−7.95442e6	−0.844611
$617$	−8.69599e6	−0.919615	−0.459808	−	0.888018i	$-0.652082\pi$
$617$	−8.69599e6	−0.919615	−0.459808	+	0.888018i	$0.652082\pi$
$618$	0	0
$619$	−1.01187e6	−0.106145	−0.0530723	−	0.998591i	$-0.516901\pi$
$619$	−1.01187e6	−0.106145	−0.0530723	+	0.998591i	$0.516901\pi$
$620$	7.16832e6	0.748925
$621$	0	0
$622$	−3.25241e6	−0.337077
$623$	1.85364e6	0.191340
$624$	0	0
$625$	−5.59205e6	−0.572626
$626$	2.02585e7	2.06619
$627$	0	0
$628$	−222829.	−0.0225462
$629$	−5.26800e6	−0.530907
$630$	0	0
$631$	1.14790e7	1.14771	0.573854	−	0.818958i	$-0.305448\pi$
$631$	1.14790e7	1.14771	0.573854	+	0.818958i	$0.305448\pi$
$632$	−2.37126e6	−0.236150
$633$	0	0
$634$	2.28111e7	2.25384
$635$	−1.73056e7	−1.70315
$636$	0	0
$637$	−2.69826e6	−0.263473
$638$	−3.23315e7	−3.14466
$639$	0	0
$640$	1.86558e7	1.80038
$641$	−1.67427e7	−1.60946	−0.804728	−	0.593643i	$-0.797689\pi$
$641$	−1.67427e7	−1.60946	−0.804728	+	0.593643i	$0.797689\pi$
$642$	0	0
$643$	−8.05866e6	−0.768662	−0.384331	−	0.923195i	$-0.625568\pi$
$643$	−8.05866e6	−0.768662	−0.384331	+	0.923195i	$0.625568\pi$
$644$	−5.10183e6	−0.484743
$645$	0	0
$646$	−4.32895e6	−0.408132
$647$	−3.80079e6	−0.356955	−0.178477	−	0.983944i	$-0.557117\pi$
$647$	−3.80079e6	−0.356955	−0.178477	+	0.983944i	$0.557117\pi$
$648$	0	0
$649$	2.01794e6	0.188060
$650$	1.19288e7	1.10743
$651$	0	0
$652$	1.01306e6	0.0933291
$653$	7.58623e6	0.696215	0.348107	−	0.937455i	$-0.386824\pi$
$653$	7.58623e6	0.696215	0.348107	+	0.937455i	$0.386824\pi$
$654$	0	0
$655$	2.40511e7	2.19044
$656$	−3.05848e6	−0.277489
$657$	0	0
$658$	−6.43998e6	−0.579856
$659$	−1.09734e7	−0.984299	−0.492149	−	0.870511i	$-0.663789\pi$
$659$	−1.09734e7	−0.984299	−0.492149	+	0.870511i	$0.663789\pi$
$660$	0	0
$661$	−5.85236e6	−0.520988	−0.260494	−	0.965476i	$-0.583885\pi$
$661$	−5.85236e6	−0.520988	−0.260494	+	0.965476i	$0.583885\pi$
$662$	−1.67852e7	−1.48862
$663$	0	0
$664$	−368764.	−0.0324585
$665$	−1.35208e7	−1.18563
$666$	0	0
$667$	3.50531e7	3.05079
$668$	−6.44219e6	−0.558589
$669$	0	0
$670$	−3.26776e7	−2.81231
$671$	−3.22713e6	−0.276701
$672$	0	0
$673$	6.43218e6	0.547420	0.273710	−	0.961812i	$-0.411749\pi$
$673$	6.43218e6	0.547420	0.273710	+	0.961812i	$0.411749\pi$
$674$	−1.62066e7	−1.37418
$675$	0	0
$676$	−2.16052e6	−0.181841
$677$	2.35723e7	1.97665	0.988325	−	0.152363i	$-0.0486882\pi$
$677$	2.35723e7	1.97665	0.988325	+	0.152363i	$0.0486882\pi$
$678$	0	0
$679$	1.71125e7	1.42442
$680$	4.81375e6	0.399219
$681$	0	0
$682$	−2.71667e7	−2.23654
$683$	−2.00530e7	−1.64486	−0.822428	−	0.568870i	$-0.807381\pi$
$683$	−2.00530e7	−1.64486	−0.822428	+	0.568870i	$0.807381\pi$
$684$	0	0
$685$	2.47460e7	2.01501
$686$	1.56988e7	1.27367
$687$	0	0
$688$	1.47931e7	1.19148
$689$	1.03490e7	0.830520
$690$	0	0
$691$	2.03136e7	1.61842	0.809211	−	0.587518i	$-0.199895\pi$
$691$	2.03136e7	1.61842	0.809211	+	0.587518i	$0.199895\pi$
$692$	−2.83978e6	−0.225434
$693$	0	0
$694$	−611969.	−0.0482315
$695$	3.66191e7	2.87571
$696$	0	0
$697$	−1.02698e6	−0.0800716
$698$	3.44146e6	0.267365
$699$	0	0
$700$	−5.06564e6	−0.390742
$701$	−1.59318e7	−1.22453	−0.612265	−	0.790653i	$-0.709741\pi$
$701$	−1.59318e7	−1.22453	−0.612265	+	0.790653i	$0.709741\pi$
$702$	0	0
$703$	1.91342e7	1.46023
$704$	−7.66178e6	−0.582637
$705$	0	0
$706$	1.09680e7	0.828162
$707$	−3.92753e6	−0.295509
$708$	0	0
$709$	2.13021e7	1.59150	0.795749	−	0.605626i	$-0.207077\pi$
$709$	2.13021e7	1.59150	0.795749	+	0.605626i	$0.207077\pi$
$710$	6.75129e6	0.502621
$711$	0	0
$712$	2.39702e6	0.177203
$713$	2.94536e7	2.16978
$714$	0	0
$715$	−2.15096e7	−1.57350
$716$	−3.38431e6	−0.246710
$717$	0	0
$718$	−1.66618e7	−1.20618
$719$	1.46154e7	1.05436	0.527180	−	0.849753i	$-0.323249\pi$
$719$	1.46154e7	1.05436	0.527180	+	0.849753i	$0.323249\pi$
$720$	0	0
$721$	−8.40720e6	−0.602300
$722$	−681393.	−0.0486469
$723$	0	0
$724$	−4.78483e6	−0.339250
$725$	3.48045e7	2.45918
$726$	0	0
$727$	−6.13626e6	−0.430594	−0.215297	−	0.976549i	$-0.569072\pi$
$727$	−6.13626e6	−0.430594	−0.215297	+	0.976549i	$0.569072\pi$
$728$	5.97556e6	0.417878
$729$	0	0
$730$	1.41443e7	0.982365
$731$	4.96721e6	0.343810
$732$	0	0
$733$	1.06072e7	0.729190	0.364595	−	0.931166i	$-0.381207\pi$
$733$	1.06072e7	0.729190	0.364595	+	0.931166i	$0.381207\pi$
$734$	−5.24953e6	−0.359650
$735$	0	0
$736$	−1.70977e7	−1.16344
$737$	3.35582e7	2.27578
$738$	0	0
$739$	−75441.2	−0.00508156	−0.00254078	−	0.999997i	$-0.500809\pi$
$739$	−75441.2	−0.00508156	−0.00254078	+	0.999997i	$0.500809\pi$
$740$	1.25872e7	0.844984
$741$	0	0
$742$	−1.62183e7	−1.08142
$743$	3.43706e6	0.228410	0.114205	−	0.993457i	$-0.463568\pi$
$743$	3.43706e6	0.228410	0.114205	+	0.993457i	$0.463568\pi$
$744$	0	0
$745$	3.27123e7	2.15934
$746$	5.60953e6	0.369045
$747$	0	0
$748$	2.92448e6	0.191115
$749$	1.24915e7	0.813600
$750$	0	0
$751$	8.47000e6	0.548004	0.274002	−	0.961729i	$-0.411653\pi$
$751$	8.47000e6	0.548004	0.274002	+	0.961729i	$0.411653\pi$
$752$	−1.19194e7	−0.768616
$753$	0	0
$754$	2.42882e7	1.55585
$755$	−7.80040e6	−0.498023
$756$	0	0
$757$	−9.70756e6	−0.615702	−0.307851	−	0.951435i	$-0.599610\pi$
$757$	−9.70756e6	−0.615702	−0.307851	+	0.951435i	$0.599610\pi$
$758$	1.08213e7	0.684077
$759$	0	0
$760$	−1.74843e7	−1.09803
$761$	4.60781e6	0.288425	0.144212	−	0.989547i	$-0.453935\pi$
$761$	4.60781e6	0.288425	0.144212	+	0.989547i	$0.453935\pi$
$762$	0	0
$763$	1.57704e7	0.980691
$764$	7.42564e6	0.460257
$765$	0	0
$766$	2.80554e7	1.72761
$767$	−1.51593e6	−0.0930445
$768$	0	0
$769$	−1.09236e7	−0.666114	−0.333057	−	0.942907i	$-0.608080\pi$
$769$	−1.09236e7	−0.666114	−0.333057	+	0.942907i	$0.608080\pi$
$770$	3.37086e7	2.04887
$771$	0	0
$772$	9.93999e6	0.600265
$773$	−6.36770e6	−0.383295	−0.191648	−	0.981464i	$-0.561383\pi$
$773$	−6.36770e6	−0.383295	−0.191648	+	0.981464i	$0.561383\pi$
$774$	0	0
$775$	2.92447e7	1.74901
$776$	2.21289e7	1.31918
$777$	0	0
$778$	−2.86606e7	−1.69760
$779$	3.73014e6	0.220233
$780$	0	0
$781$	−6.93323e6	−0.406732
$782$	−1.17009e7	−0.684233
$783$	0	0
$784$	7.82638e6	0.454748
$785$	−1.59620e6	−0.0924514
$786$	0	0
$787$	−2.17413e7	−1.25126	−0.625631	−	0.780119i	$-0.715159\pi$
$787$	−2.17413e7	−1.25126	−0.625631	+	0.780119i	$0.715159\pi$
$788$	8.61799e6	0.494414
$789$	0	0
$790$	1.00487e7	0.572854
$791$	−5.60668e6	−0.318614
$792$	0	0
$793$	2.42430e6	0.136900
$794$	2.21328e7	1.24590
$795$	0	0
$796$	−8.22942e6	−0.460348
$797$	1.29730e7	0.723424	0.361712	−	0.932290i	$-0.382192\pi$
$797$	1.29730e7	0.723424	0.361712	+	0.932290i	$0.382192\pi$
$798$	0	0
$799$	−4.00229e6	−0.221790
$800$	−1.69764e7	−0.937822
$801$	0	0
$802$	3.97873e6	0.218428
$803$	−1.45254e7	−0.794951
$804$	0	0
$805$	−3.65461e7	−1.98770
$806$	2.04083e7	1.10655
$807$	0	0
$808$	−5.07884e6	−0.273675
$809$	1.50725e7	0.809682	0.404841	−	0.914387i	$-0.367327\pi$
$809$	1.50725e7	0.809682	0.404841	+	0.914387i	$0.367327\pi$
$810$	0	0
$811$	−9.99366e6	−0.533547	−0.266773	−	0.963759i	$-0.585958\pi$
$811$	−9.99366e6	−0.533547	−0.266773	+	0.963759i	$0.585958\pi$
$812$	−1.03141e7	−0.548962
$813$	0	0
$814$	−4.77032e7	−2.52340
$815$	7.25691e6	0.382699
$816$	0	0
$817$	−1.80417e7	−0.945633
$818$	−1.24453e7	−0.650312
$819$	0	0
$820$	2.45382e6	0.127441
$821$	−2.35975e7	−1.22182	−0.610912	−	0.791698i	$-0.709197\pi$
$821$	−2.35975e7	−1.22182	−0.610912	+	0.791698i	$0.709197\pi$
$822$	0	0
$823$	−7.42657e6	−0.382198	−0.191099	−	0.981571i	$-0.561205\pi$
$823$	−7.42657e6	−0.382198	−0.191099	+	0.981571i	$0.561205\pi$
$824$	−1.08717e7	−0.557800
$825$	0	0
$826$	2.37567e6	0.121154
$827$	2.03546e7	1.03490	0.517451	−	0.855713i	$-0.326881\pi$
$827$	2.03546e7	1.03490	0.517451	+	0.855713i	$0.326881\pi$
$828$	0	0
$829$	1.84957e7	0.934725	0.467362	−	0.884066i	$-0.345204\pi$
$829$	1.84957e7	0.934725	0.467362	+	0.884066i	$0.345204\pi$
$830$	1.56272e6	0.0787381
$831$	0	0
$832$	5.75572e6	0.288264
$833$	2.62794e6	0.131221
$834$	0	0
$835$	−4.61476e7	−2.29051
$836$	−1.06222e7	−0.525652
$837$	0	0
$838$	3.08103e7	1.51560
$839$	−4.89142e6	−0.239900	−0.119950	−	0.992780i	$-0.538273\pi$
$839$	−4.89142e6	−0.239900	−0.119950	+	0.992780i	$0.538273\pi$
$840$	0	0
$841$	5.03540e7	2.45496
$842$	−8.26325e6	−0.401671
$843$	0	0
$844$	2.96574e6	0.143310
$845$	−1.54765e7	−0.745644
$846$	0	0
$847$	−1.80272e7	−0.863413
$848$	−3.00176e7	−1.43346
$849$	0	0
$850$	−1.16179e7	−0.551547
$851$	5.17189e7	2.44808
$852$	0	0
$853$	−2.49648e7	−1.17478	−0.587389	−	0.809305i	$-0.699844\pi$
$853$	−2.49648e7	−1.17478	−0.587389	+	0.809305i	$0.699844\pi$
$854$	−3.79922e6	−0.178258
$855$	0	0
$856$	1.61533e7	0.753488
$857$	4.44419e6	0.206700	0.103350	−	0.994645i	$-0.467044\pi$
$857$	4.44419e6	0.206700	0.103350	+	0.994645i	$0.467044\pi$
$858$	0	0
$859$	−2.83513e7	−1.31096	−0.655480	−	0.755212i	$-0.727534\pi$
$859$	−2.83513e7	−1.31096	−0.655480	+	0.755212i	$0.727534\pi$
$860$	−1.18685e7	−0.547203
$861$	0	0
$862$	−6.46345e6	−0.296276
$863$	−1.98739e7	−0.908358	−0.454179	−	0.890911i	$-0.650067\pi$
$863$	−1.98739e7	−0.908358	−0.454179	+	0.890911i	$0.650067\pi$
$864$	0	0
$865$	−2.03423e7	−0.924400
$866$	1.70109e7	0.770785
$867$	0	0
$868$	−8.66651e6	−0.390431
$869$	−1.03196e7	−0.463566
$870$	0	0
$871$	−2.52098e7	−1.12596
$872$	2.03934e7	0.908234
$873$	0	0
$874$	4.24997e7	1.88195
$875$	−8.85973e6	−0.391202
$876$	0	0
$877$	2.39535e7	1.05165	0.525823	−	0.850594i	$-0.323757\pi$
$877$	2.39535e7	1.05165	0.525823	+	0.850594i	$0.323757\pi$
$878$	5.79815e6	0.253836
$879$	0	0
$880$	6.23893e7	2.71583
$881$	−1.44336e7	−0.626522	−0.313261	−	0.949667i	$-0.601421\pi$
$881$	−1.44336e7	−0.626522	−0.313261	+	0.949667i	$0.601421\pi$
$882$	0	0
$883$	−1.96597e7	−0.848547	−0.424274	−	0.905534i	$-0.639471\pi$
$883$	−1.96597e7	−0.848547	−0.424274	+	0.905534i	$0.639471\pi$
$884$	−2.19694e6	−0.0945558
$885$	0	0
$886$	−2.91131e7	−1.24596
$887$	−430936.	−0.0183909	−0.00919547	−	0.999958i	$-0.502927\pi$
$887$	−430936.	−0.0183909	−0.00919547	+	0.999958i	$0.502927\pi$
$888$	0	0
$889$	2.09225e7	0.887889
$890$	−1.01579e7	−0.429861
$891$	0	0
$892$	−6.44829e6	−0.271352
$893$	1.45370e7	0.610021
$894$	0	0
$895$	−2.42430e7	−1.01164
$896$	−2.25549e7	−0.938580
$897$	0	0
$898$	−1.47258e7	−0.609381
$899$	5.95449e7	2.45723
$900$	0	0
$901$	−1.00793e7	−0.413636
$902$	−9.29956e6	−0.380580
$903$	0	0
$904$	−7.25021e6	−0.295073
$905$	−3.42753e7	−1.39111
$906$	0	0
$907$	−1.15056e7	−0.464398	−0.232199	−	0.972668i	$-0.574592\pi$
$907$	−1.15056e7	−0.464398	−0.232199	+	0.972668i	$0.574592\pi$
$908$	5.71717e6	0.230126
$909$	0	0
$910$	−2.53227e7	−1.01369
$911$	−3.19861e7	−1.27693	−0.638463	−	0.769653i	$-0.720429\pi$
$911$	−3.19861e7	−1.27693	−0.638463	+	0.769653i	$0.720429\pi$
$912$	0	0
$913$	−1.60483e6	−0.0637166
$914$	−2.31257e7	−0.915652
$915$	0	0
$916$	−7.42011e6	−0.292194
$917$	−2.90778e7	−1.14193
$918$	0	0
$919$	3.83417e7	1.49756	0.748778	−	0.662821i	$-0.230641\pi$
$919$	3.83417e7	1.49756	0.748778	+	0.662821i	$0.230641\pi$
$920$	−4.72593e7	−1.84085
$921$	0	0
$922$	−2.26049e6	−0.0875739
$923$	5.20842e6	0.201234
$924$	0	0
$925$	5.13520e7	1.97335
$926$	−4.45872e7	−1.70877
$927$	0	0
$928$	−3.45655e7	−1.31757
$929$	−3.81222e7	−1.44923	−0.724617	−	0.689151i	$-0.757984\pi$
$929$	−3.81222e7	−1.44923	−0.724617	+	0.689151i	$0.757984\pi$
$930$	0	0
$931$	−9.54510e6	−0.360916
$932$	−7.67983e6	−0.289609
$933$	0	0
$934$	1.40187e7	0.525824
$935$	2.09490e7	0.783674
$936$	0	0
$937$	−1.63559e7	−0.608591	−0.304295	−	0.952578i	$-0.598421\pi$
$937$	−1.63559e7	−0.608591	−0.304295	+	0.952578i	$0.598421\pi$
$938$	3.95072e7	1.46612
$939$	0	0
$940$	9.56293e6	0.352997
$941$	−1.57758e7	−0.580789	−0.290395	−	0.956907i	$-0.593787\pi$
$941$	−1.57758e7	−0.580789	−0.290395	+	0.956907i	$0.593787\pi$
$942$	0	0
$943$	1.00824e7	0.369220
$944$	4.39700e6	0.160593
$945$	0	0
$946$	4.49795e7	1.63413
$947$	2.76570e7	1.00214	0.501072	−	0.865405i	$-0.332939\pi$
$947$	2.76570e7	1.00214	0.501072	+	0.865405i	$0.332939\pi$
$948$	0	0
$949$	1.09119e7	0.393309
$950$	4.21983e7	1.51700
$951$	0	0
$952$	−5.81982e6	−0.208122
$953$	−1.15464e6	−0.0411825	−0.0205912	−	0.999788i	$-0.506555\pi$
$953$	−1.15464e6	−0.0411825	−0.0205912	+	0.999788i	$0.506555\pi$
$954$	0	0
$955$	5.31923e7	1.88730
$956$	−1.31434e7	−0.465116
$957$	0	0
$958$	5.77507e7	2.03303
$959$	−2.99179e7	−1.05047
$960$	0	0
$961$	2.14039e7	0.747625
$962$	3.58358e7	1.24848
$963$	0	0
$964$	1.71874e7	0.595686
$965$	7.12035e7	2.46140
$966$	0	0
$967$	−4.77090e7	−1.64072	−0.820360	−	0.571848i	$-0.806227\pi$
$967$	−4.77090e7	−1.64072	−0.820360	+	0.571848i	$0.806227\pi$
$968$	−2.33116e7	−0.799621
$969$	0	0
$970$	−9.37759e7	−3.20009
$971$	−3.61364e7	−1.22998	−0.614988	−	0.788537i	$-0.710839\pi$
$971$	−3.61364e7	−1.22998	−0.614988	+	0.788537i	$0.710839\pi$
$972$	0	0
$973$	−4.42725e7	−1.49917
$974$	−5.43084e7	−1.83430
$975$	0	0
$976$	−7.03176e6	−0.236287
$977$	−2.58953e7	−0.867930	−0.433965	−	0.900930i	$-0.642886\pi$
$977$	−2.58953e7	−0.867930	−0.433965	+	0.900930i	$0.642886\pi$
$978$	0	0
$979$	1.04316e7	0.347853
$980$	−6.27911e6	−0.208849
$981$	0	0
$982$	4.41070e7	1.45958
$983$	3.78204e7	1.24837	0.624183	−	0.781278i	$-0.285432\pi$
$983$	3.78204e7	1.24837	0.624183	+	0.781278i	$0.285432\pi$
$984$	0	0
$985$	6.17336e7	2.02736
$986$	−2.36552e7	−0.774880
$987$	0	0
$988$	7.97965e6	0.260071
$989$	−4.87659e7	−1.58535
$990$	0	0
$991$	−3.64574e7	−1.17924	−0.589619	−	0.807681i	$-0.700722\pi$
$991$	−3.64574e7	−1.17924	−0.589619	+	0.807681i	$0.700722\pi$
$992$	−2.90439e7	−0.937077
$993$	0	0
$994$	−8.16232e6	−0.262028
$995$	−5.89501e7	−1.88767
$996$	0	0
$997$	1.17735e6	0.0375119	0.0187559	−	0.999824i	$-0.494029\pi$
$997$	1.17735e6	0.0375119	0.0187559	+	0.999824i	$0.494029\pi$
$998$	−1.43585e7	−0.456333
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+6.62527 q^{2} +11.8942 q^{4} +85.2025 q^{5} -103.010 q^{7} -133.206 q^{8} +O(q^{10})\)	\(q+6.62527 q^{2} +11.8942 q^{4} +85.2025 q^{5} -103.010 q^{7} -133.206 q^{8} +564.490 q^{10} -579.703 q^{11} +435.487 q^{13} -682.469 q^{14} -1263.14 q^{16} -424.138 q^{17} +1540.54 q^{19} +1013.42 q^{20} -3840.69 q^{22} +4164.00 q^{23} +4134.46 q^{25} +2885.22 q^{26} -1225.22 q^{28} +8418.14 q^{29} +7073.40 q^{31} -4106.07 q^{32} -2810.03 q^{34} -8776.70 q^{35} +12420.5 q^{37} +10206.5 q^{38} -11349.5 q^{40} +2421.33 q^{41} -11711.3 q^{43} -6895.12 q^{44} +27587.6 q^{46} +9436.30 q^{47} -6195.96 q^{49} +27391.9 q^{50} +5179.79 q^{52} +23764.2 q^{53} -49392.1 q^{55} +13721.5 q^{56} +55772.5 q^{58} -3481.00 q^{59} +5566.88 q^{61} +46863.2 q^{62} +13216.7 q^{64} +37104.6 q^{65} -57888.7 q^{67} -5044.80 q^{68} -58148.0 q^{70} +11960.0 q^{71} +25056.7 q^{73} +82289.1 q^{74} +18323.5 q^{76} +59715.1 q^{77} +17801.5 q^{79} -107623. q^{80} +16042.0 q^{82} +2768.37 q^{83} -36137.6 q^{85} -77590.7 q^{86} +77219.9 q^{88} -17994.8 q^{89} -44859.5 q^{91} +49527.6 q^{92} +62518.1 q^{94} +131257. q^{95} -166125. q^{97} -41049.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) \) 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

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Database

Properties

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