LMFDB - Embedded newform 531.6.a.b.1.6

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			1.6
Root			$-0.216241$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.21624 q^{2} -30.5208 q^{4} -0.914506 q^{5} +61.7584 q^{7} -76.0403 q^{8} +O(q^{10})$	\(q+1.21624 q^{2} -30.5208 q^{4} -0.914506 q^{5} +61.7584 q^{7} -76.0403 q^{8} -1.11226 q^{10} +311.532 q^{11} -150.260 q^{13} +75.1131 q^{14} +884.181 q^{16} -1232.93 q^{17} -266.956 q^{19} +27.9114 q^{20} +378.898 q^{22} +1920.26 q^{23} -3124.16 q^{25} -182.752 q^{26} -1884.91 q^{28} +2981.52 q^{29} -8356.31 q^{31} +3508.67 q^{32} -1499.54 q^{34} -56.4784 q^{35} +8500.15 q^{37} -324.682 q^{38} +69.5393 q^{40} -13953.0 q^{41} -11289.0 q^{43} -9508.19 q^{44} +2335.50 q^{46} +24855.4 q^{47} -12992.9 q^{49} -3799.74 q^{50} +4586.04 q^{52} +35346.4 q^{53} -284.898 q^{55} -4696.13 q^{56} +3626.25 q^{58} -3481.00 q^{59} -46879.0 q^{61} -10163.3 q^{62} -24026.4 q^{64} +137.413 q^{65} +37659.9 q^{67} +37629.8 q^{68} -68.6913 q^{70} +56413.7 q^{71} +66253.6 q^{73} +10338.2 q^{74} +8147.69 q^{76} +19239.7 q^{77} -51579.2 q^{79} -808.588 q^{80} -16970.2 q^{82} +28845.7 q^{83} +1127.52 q^{85} -13730.1 q^{86} -23689.0 q^{88} +9518.85 q^{89} -9279.79 q^{91} -58607.8 q^{92} +30230.2 q^{94} +244.132 q^{95} +96360.6 q^{97} -15802.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.21624	0.215003	0.107502	−	0.994205i	$-0.465715\pi$
$2$	1.21624	0.215003	0.107502	+	0.994205i	$0.465715\pi$
$3$	0	0
$3$	0	0
$4$	−30.5208	−0.953774
$5$	−0.914506	−0.0163592	−0.00817959	−	0.999967i	$-0.502604\pi$
$5$	−0.914506	−0.0163592	−0.00817959	+	0.999967i	$0.502604\pi$
$6$	0	0
$7$	61.7584	0.476377	0.238188	−	0.971219i	$-0.423446\pi$
$7$	61.7584	0.476377	0.238188	+	0.971219i	$0.423446\pi$
$8$	−76.0403	−0.420067
$9$	0	0
$10$	−1.11226	−0.00351727
$11$	311.532	0.776284	0.388142	−	0.921599i	$-0.373117\pi$
$11$	311.532	0.776284	0.388142	+	0.921599i	$0.373117\pi$
$12$	0	0
$13$	−150.260	−0.246595	−0.123297	−	0.992370i	$-0.539347\pi$
$13$	−150.260	−0.246595	−0.123297	+	0.992370i	$0.539347\pi$
$14$	75.1131	0.102422
$15$	0	0
$16$	884.181	0.863458
$17$	−1232.93	−1.03470	−0.517350	−	0.855774i	$-0.673082\pi$
$17$	−1232.93	−1.03470	−0.517350	+	0.855774i	$0.673082\pi$
$18$	0	0
$19$	−266.956	−0.169651	−0.0848253	−	0.996396i	$-0.527033\pi$
$19$	−266.956	−0.169651	−0.0848253	+	0.996396i	$0.527033\pi$
$20$	27.9114	0.0156029
$21$	0	0
$22$	378.898	0.166904
$23$	1920.26	0.756903	0.378451	−	0.925621i	$-0.376457\pi$
$23$	1920.26	0.756903	0.378451	+	0.925621i	$0.376457\pi$
$24$	0	0
$25$	−3124.16	−0.999732
$26$	−182.752	−0.0530187
$27$	0	0
$28$	−1884.91	−0.454356
$29$	2981.52	0.658329	0.329165	−	0.944273i	$-0.393233\pi$
$29$	2981.52	0.658329	0.329165	+	0.944273i	$0.393233\pi$
$30$	0	0
$31$	−8356.31	−1.56175	−0.780873	−	0.624689i	$-0.785226\pi$
$31$	−8356.31	−1.56175	−0.780873	+	0.624689i	$0.785226\pi$
$32$	3508.67	0.605713
$33$	0	0
$34$	−1499.54	−0.222464
$35$	−56.4784	−0.00779313
$36$	0	0
$37$	8500.15	1.02076	0.510378	−	0.859950i	$-0.329505\pi$
$37$	8500.15	1.02076	0.510378	+	0.859950i	$0.329505\pi$
$38$	−324.682	−0.0364754
$39$	0	0
$40$	69.5393	0.00687195
$41$	−13953.0	−1.29631	−0.648155	−	0.761509i	$-0.724459\pi$
$41$	−13953.0	−1.29631	−0.648155	+	0.761509i	$0.724459\pi$
$42$	0	0
$43$	−11289.0	−0.931071	−0.465535	−	0.885029i	$-0.654138\pi$
$43$	−11289.0	−0.931071	−0.465535	+	0.885029i	$0.654138\pi$
$44$	−9508.19	−0.740400
$45$	0	0
$46$	2335.50	0.162736
$47$	24855.4	1.64126	0.820628	−	0.571463i	$-0.193624\pi$
$47$	24855.4	1.64126	0.820628	+	0.571463i	$0.193624\pi$
$48$	0	0
$49$	−12992.9	−0.773065
$50$	−3799.74	−0.214946
$51$	0	0
$52$	4586.04	0.235196
$53$	35346.4	1.72844	0.864222	−	0.503111i	$-0.167811\pi$
$53$	35346.4	1.72844	0.864222	+	0.503111i	$0.167811\pi$
$54$	0	0
$55$	−284.898	−0.0126994
$56$	−4696.13	−0.200110
$57$	0	0
$58$	3626.25	0.141543
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	−46879.0	−1.61307	−0.806536	−	0.591185i	$-0.798660\pi$
$61$	−46879.0	−1.61307	−0.806536	+	0.591185i	$0.798660\pi$
$62$	−10163.3	−0.335780
$63$	0	0
$64$	−24026.4	−0.733228
$65$	137.413	0.00403409
$66$	0	0
$67$	37659.9	1.02492	0.512462	−	0.858710i	$-0.328734\pi$
$67$	37659.9	1.02492	0.512462	+	0.858710i	$0.328734\pi$
$68$	37629.8	0.986870
$69$	0	0
$70$	−68.6913	−0.00167555
$71$	56413.7	1.32812	0.664062	−	0.747677i	$-0.268831\pi$
$71$	56413.7	1.32812	0.664062	+	0.747677i	$0.268831\pi$
$72$	0	0
$73$	66253.6	1.45513	0.727566	−	0.686037i	$-0.240651\pi$
$73$	66253.6	1.45513	0.727566	+	0.686037i	$0.240651\pi$
$74$	10338.2	0.219466
$75$	0	0
$76$	8147.69	0.161808
$77$	19239.7	0.369804
$78$	0	0
$79$	−51579.2	−0.929837	−0.464918	−	0.885354i	$-0.653916\pi$
$79$	−51579.2	−0.929837	−0.464918	+	0.885354i	$0.653916\pi$
$80$	−808.588	−0.0141255
$81$	0	0
$82$	−16970.2	−0.278710
$83$	28845.7	0.459606	0.229803	−	0.973237i	$-0.426192\pi$
$83$	28845.7	0.459606	0.229803	+	0.973237i	$0.426192\pi$
$84$	0	0
$85$	1127.52	0.0169269
$86$	−13730.1	−0.200183
$87$	0	0
$88$	−23689.0	−0.326092
$89$	9518.85	0.127382	0.0636912	−	0.997970i	$-0.479713\pi$
$89$	9518.85	0.127382	0.0636912	+	0.997970i	$0.479713\pi$
$90$	0	0
$91$	−9279.79	−0.117472
$92$	−58607.8	−0.721914
$93$	0	0
$94$	30230.2	0.352875
$95$	244.132	0.00277534
$96$	0	0
$97$	96360.6	1.03985	0.519924	−	0.854212i	$-0.325960\pi$
$97$	96360.6	1.03985	0.519924	+	0.854212i	$0.325960\pi$
$98$	−15802.5	−0.166211
$99$	0	0
$100$	95351.8	0.953518
$101$	2640.59	0.0257571	0.0128785	−	0.999917i	$-0.495901\pi$
$101$	2640.59	0.0257571	0.0128785	+	0.999917i	$0.495901\pi$
$102$	0	0
$103$	117451.	1.09085	0.545425	−	0.838160i	$-0.316368\pi$
$103$	117451.	1.09085	0.545425	+	0.838160i	$0.316368\pi$
$104$	11425.8	0.103586
$105$	0	0
$106$	42989.7	0.371621
$107$	228868.	1.93253	0.966263	−	0.257558i	$-0.0829177\pi$
$107$	228868.	1.93253	0.966263	+	0.257558i	$0.0829177\pi$
$108$	0	0
$109$	163809.	1.32060	0.660301	−	0.751001i	$-0.270429\pi$
$109$	163809.	1.32060	0.660301	+	0.751001i	$0.270429\pi$
$110$	−346.504	−0.00273040
$111$	0	0
$112$	54605.6	0.411331
$113$	−16245.7	−0.119686	−0.0598430	−	0.998208i	$-0.519060\pi$
$113$	−16245.7	−0.119686	−0.0598430	+	0.998208i	$0.519060\pi$
$114$	0	0
$115$	−1756.09	−0.0123823
$116$	−90998.4	−0.627897
$117$	0	0
$118$	−4233.73	−0.0279910
$119$	−76143.5	−0.492908
$120$	0	0
$121$	−63998.8	−0.397382
$122$	−57016.2	−0.346816
$123$	0	0
$124$	255041.	1.48955
$125$	5714.90	0.0327140
$126$	0	0
$127$	285457.	1.57047	0.785237	−	0.619195i	$-0.212541\pi$
$127$	285457.	1.57047	0.785237	+	0.619195i	$0.212541\pi$
$128$	−141499.	−0.763360
$129$	0	0
$130$	167.128	0.000867341 0
$131$	−153837.	−0.783217	−0.391609	−	0.920132i	$-0.628081\pi$
$131$	−153837.	−0.783217	−0.391609	+	0.920132i	$0.628081\pi$
$132$	0	0
$133$	−16486.8	−0.0808176
$134$	45803.5	0.220362
$135$	0	0
$136$	93752.1	0.434644
$137$	281641.	1.28202	0.641010	−	0.767532i	$-0.278516\pi$
$137$	281641.	1.28202	0.641010	+	0.767532i	$0.278516\pi$
$138$	0	0
$139$	171456.	0.752689	0.376344	−	0.926480i	$-0.377181\pi$
$139$	171456.	0.752689	0.376344	+	0.926480i	$0.377181\pi$
$140$	1723.76	0.00743289
$141$	0	0
$142$	68612.7	0.285551
$143$	−46810.7	−0.191428
$144$	0	0
$145$	−2726.62	−0.0107697
$146$	80580.4	0.312858
$147$	0	0
$148$	−259431.	−0.973570
$149$	−246577.	−0.909885	−0.454942	−	0.890521i	$-0.650340\pi$
$149$	−246577.	−0.909885	−0.454942	+	0.890521i	$0.650340\pi$
$150$	0	0
$151$	161787.	0.577433	0.288717	−	0.957415i	$-0.406771\pi$
$151$	161787.	0.577433	0.288717	+	0.957415i	$0.406771\pi$
$152$	20299.4	0.0712646
$153$	0	0
$154$	23400.1	0.0795090
$155$	7641.90	0.0255489
$156$	0	0
$157$	−212289.	−0.687351	−0.343675	−	0.939089i	$-0.611672\pi$
$157$	−212289.	−0.687351	−0.343675	+	0.939089i	$0.611672\pi$
$158$	−62732.7	−0.199918
$159$	0	0
$160$	−3208.70	−0.00990897
$161$	118592.	0.360571
$162$	0	0
$163$	428153.	1.26221	0.631103	−	0.775699i	$-0.282603\pi$
$163$	428153.	1.26221	0.631103	+	0.775699i	$0.282603\pi$
$164$	425857.	1.23639
$165$	0	0
$166$	35083.3	0.0988166
$167$	−260524.	−0.722863	−0.361432	−	0.932399i	$-0.617712\pi$
$167$	−260524.	−0.722863	−0.361432	+	0.932399i	$0.617712\pi$
$168$	0	0
$169$	−348715.	−0.939191
$170$	1371.33	0.00363932
$171$	0	0
$172$	344547.	0.888031
$173$	−375123.	−0.952924	−0.476462	−	0.879195i	$-0.658081\pi$
$173$	−375123.	−0.952924	−0.476462	+	0.879195i	$0.658081\pi$
$174$	0	0
$175$	−192943.	−0.476249
$176$	275451.	0.670289
$177$	0	0
$178$	11577.2	0.0273876
$179$	569801.	1.32920	0.664601	−	0.747199i	$-0.268602\pi$
$179$	569801.	1.32920	0.664601	+	0.747199i	$0.268602\pi$
$180$	0	0
$181$	−532297.	−1.20770	−0.603848	−	0.797099i	$-0.706367\pi$
$181$	−532297.	−1.20770	−0.603848	+	0.797099i	$0.706367\pi$
$182$	−11286.5	−0.0252569
$183$	0	0
$184$	−146017.	−0.317950
$185$	−7773.43	−0.0166987
$186$	0	0
$187$	−384096.	−0.803222
$188$	−758606.	−1.56539
$189$	0	0
$190$	296.924	0.000596707 0
$191$	756470.	1.50040	0.750202	−	0.661209i	$-0.229956\pi$
$191$	756470.	1.50040	0.750202	+	0.661209i	$0.229956\pi$
$192$	0	0
$193$	715629.	1.38291	0.691456	−	0.722418i	$-0.256969\pi$
$193$	715629.	1.38291	0.691456	+	0.722418i	$0.256969\pi$
$194$	117198.	0.223571
$195$	0	0
$196$	396553.	0.737329
$197$	−330788.	−0.607273	−0.303637	−	0.952788i	$-0.598201\pi$
$197$	−330788.	−0.607273	−0.303637	+	0.952788i	$0.598201\pi$
$198$	0	0
$199$	−145072.	−0.259687	−0.129844	−	0.991534i	$-0.541448\pi$
$199$	−145072.	−0.259687	−0.129844	+	0.991534i	$0.541448\pi$
$200$	237562.	0.419955
$201$	0	0
$202$	3211.59	0.00553785
$203$	184134.	0.313613
$204$	0	0
$205$	12760.1	0.0212065
$206$	142849.	0.234536
$207$	0	0
$208$	−132857.	−0.212924
$209$	−83165.2	−0.131697
$210$	0	0
$211$	836441.	1.29339	0.646695	−	0.762749i	$-0.276151\pi$
$211$	836441.	1.29339	0.646695	+	0.762749i	$0.276151\pi$
$212$	−1.07880e6	−1.64854
$213$	0	0
$214$	278358.	0.415499
$215$	10323.8	0.0152315
$216$	0	0
$217$	−516072.	−0.743980
$218$	199231.	0.283933
$219$	0	0
$220$	8695.29	0.0121123
$221$	185259.	0.255152
$222$	0	0
$223$	734281.	0.988781	0.494391	−	0.869240i	$-0.335391\pi$
$223$	734281.	0.988781	0.494391	+	0.869240i	$0.335391\pi$
$224$	216690.	0.288548
$225$	0	0
$226$	−19758.7	−0.0257329
$227$	−370264.	−0.476922	−0.238461	−	0.971152i	$-0.576643\pi$
$227$	−370264.	−0.476922	−0.238461	+	0.971152i	$0.576643\pi$
$228$	0	0
$229$	−690656.	−0.870309	−0.435154	−	0.900356i	$-0.643306\pi$
$229$	−690656.	−0.870309	−0.435154	+	0.900356i	$0.643306\pi$
$230$	−2135.83	−0.00266223
$231$	0	0
$232$	−226716.	−0.276543
$233$	924763.	1.11594	0.557970	−	0.829861i	$-0.311581\pi$
$233$	924763.	1.11594	0.557970	+	0.829861i	$0.311581\pi$
$234$	0	0
$235$	−22730.4	−0.0268496
$236$	106243.	0.124171
$237$	0	0
$238$	−92608.9	−0.105977
$239$	568443.	0.643713	0.321856	−	0.946788i	$-0.395693\pi$
$239$	568443.	0.643713	0.321856	+	0.946788i	$0.395693\pi$
$240$	0	0
$241$	−292121.	−0.323981	−0.161990	−	0.986792i	$-0.551791\pi$
$241$	−292121.	−0.323981	−0.161990	+	0.986792i	$0.551791\pi$
$242$	−77838.0	−0.0854384
$243$	0	0
$244$	1.43078e6	1.53851
$245$	11882.1	0.0126467
$246$	0	0
$247$	40112.7	0.0418350
$248$	635417.	0.656039
$249$	0	0
$250$	6950.69	0.00703360
$251$	1.76450e6	1.76781	0.883907	−	0.467663i	$-0.154904\pi$
$251$	1.76450e6	1.76781	0.883907	+	0.467663i	$0.154904\pi$
$252$	0	0
$253$	598222.	0.587572
$254$	347184.	0.337657
$255$	0	0
$256$	596748.	0.569103
$257$	824695.	0.778862	0.389431	−	0.921056i	$-0.372672\pi$
$257$	824695.	0.778862	0.389431	+	0.921056i	$0.372672\pi$
$258$	0	0
$259$	524955.	0.486265
$260$	−4193.96	−0.00384761
$261$	0	0
$262$	−187103.	−0.168394
$263$	915456.	0.816109	0.408055	−	0.912958i	$-0.366207\pi$
$263$	915456.	0.816109	0.408055	+	0.912958i	$0.366207\pi$
$264$	0	0
$265$	−32324.4	−0.0282759
$266$	−20051.9	−0.0173760
$267$	0	0
$268$	−1.14941e6	−0.977546
$269$	−1.00139e6	−0.843769	−0.421884	−	0.906650i	$-0.638631\pi$
$269$	−1.00139e6	−0.843769	−0.421884	+	0.906650i	$0.638631\pi$
$270$	0	0
$271$	−2.25170e6	−1.86246	−0.931232	−	0.364428i	$-0.881265\pi$
$271$	−2.25170e6	−1.86246	−0.931232	+	0.364428i	$0.881265\pi$
$272$	−1.09013e6	−0.893421
$273$	0	0
$274$	342544.	0.275638
$275$	−973277.	−0.776077
$276$	0	0
$277$	5282.15	0.00413629	0.00206815	−	0.999998i	$-0.499342\pi$
$277$	5282.15	0.00413629	0.00206815	+	0.999998i	$0.499342\pi$
$278$	208532.	0.161830
$279$	0	0
$280$	4294.63	0.00327364
$281$	2.48246e6	1.87550	0.937748	−	0.347317i	$-0.112907\pi$
$281$	2.48246e6	1.87550	0.937748	+	0.347317i	$0.112907\pi$
$282$	0	0
$283$	−87005.1	−0.0645771	−0.0322886	−	0.999479i	$-0.510280\pi$
$283$	−87005.1	−0.0645771	−0.0322886	+	0.999479i	$0.510280\pi$
$284$	−1.72179e6	−1.26673
$285$	0	0
$286$	−56933.1	−0.0411576
$287$	−861716.	−0.617532
$288$	0	0
$289$	100250.	0.0706059
$290$	−3316.23	−0.00231552
$291$	0	0
$292$	−2.02211e6	−1.38787
$293$	−165504.	−0.112626	−0.0563131	−	0.998413i	$-0.517935\pi$
$293$	−165504.	−0.112626	−0.0563131	+	0.998413i	$0.517935\pi$
$294$	0	0
$295$	3183.39	0.00212978
$296$	−646354.	−0.428786
$297$	0	0
$298$	−299897.	−0.195628
$299$	−288538.	−0.186648
$300$	0	0
$301$	−697188.	−0.443541
$302$	196772.	0.124150
$303$	0	0
$304$	−236037.	−0.146486
$305$	42871.1	0.0263885
$306$	0	0
$307$	746565.	0.452087	0.226043	−	0.974117i	$-0.427421\pi$
$307$	746565.	0.452087	0.226043	+	0.974117i	$0.427421\pi$
$308$	−587210.	−0.352709
$309$	0	0
$310$	9294.39	0.00549309
$311$	−638206.	−0.374162	−0.187081	−	0.982344i	$-0.559903\pi$
$311$	−638206.	−0.374162	−0.187081	+	0.982344i	$0.559903\pi$
$312$	0	0
$313$	1.55232e6	0.895615	0.447808	−	0.894130i	$-0.352205\pi$
$313$	1.55232e6	0.895615	0.447808	+	0.894130i	$0.352205\pi$
$314$	−258195.	−0.147782
$315$	0	0
$316$	1.57424e6	0.886854
$317$	−1.97404e6	−1.10334	−0.551669	−	0.834063i	$-0.686009\pi$
$317$	−1.97404e6	−1.10334	−0.551669	+	0.834063i	$0.686009\pi$
$318$	0	0
$319$	928840.	0.511051
$320$	21972.3	0.0119950
$321$	0	0
$322$	144237.	0.0775239
$323$	329137.	0.175538
$324$	0	0
$325$	469436.	0.246529
$326$	520737.	0.271378
$327$	0	0
$328$	1.06099e6	0.544537
$329$	1.53503e6	0.781857
$330$	0	0
$331$	−2.40736e6	−1.20774	−0.603868	−	0.797084i	$-0.706375\pi$
$331$	−2.40736e6	−1.20774	−0.603868	+	0.797084i	$0.706375\pi$
$332$	−880391.	−0.438360
$333$	0	0
$334$	−316860.	−0.155418
$335$	−34440.2	−0.0167669
$336$	0	0
$337$	1.90636e6	0.914389	0.457194	−	0.889367i	$-0.348854\pi$
$337$	1.90636e6	0.914389	0.457194	+	0.889367i	$0.348854\pi$
$338$	−424121.	−0.201929
$339$	0	0
$340$	−34412.7	−0.0161444
$341$	−2.60326e6	−1.21236
$342$	0	0
$343$	−1.84039e6	−0.844647
$344$	858416.	0.391112
$345$	0	0
$346$	−456240.	−0.204882
$347$	−3.72382e6	−1.66022	−0.830110	−	0.557600i	$-0.811722\pi$
$347$	−3.72382e6	−1.66022	−0.830110	+	0.557600i	$0.811722\pi$
$348$	0	0
$349$	−99542.3	−0.0437466	−0.0218733	−	0.999761i	$-0.506963\pi$
$349$	−99542.3	−0.0437466	−0.0218733	+	0.999761i	$0.506963\pi$
$350$	−234665.	−0.102395
$351$	0	0
$352$	1.09306e6	0.470206
$353$	−528656.	−0.225806	−0.112903	−	0.993606i	$-0.536015\pi$
$353$	−528656.	−0.225806	−0.112903	+	0.993606i	$0.536015\pi$
$354$	0	0
$355$	−51590.7	−0.0217270
$356$	−290523.	−0.121494
$357$	0	0
$358$	693016.	0.285782
$359$	1.49351e6	0.611605	0.305803	−	0.952095i	$-0.401075\pi$
$359$	1.49351e6	0.611605	0.305803	+	0.952095i	$0.401075\pi$
$360$	0	0
$361$	−2.40483e6	−0.971219
$362$	−647402.	−0.259658
$363$	0	0
$364$	283226.	0.112042
$365$	−60589.3	−0.0238048
$366$	0	0
$367$	154208.	0.0597642	0.0298821	−	0.999553i	$-0.490487\pi$
$367$	154208.	0.0597642	0.0298821	+	0.999553i	$0.490487\pi$
$368$	1.69786e6	0.653554
$369$	0	0
$370$	−9454.36	−0.00359028
$371$	2.18293e6	0.823390
$372$	0	0
$373$	−443333.	−0.164990	−0.0824951	−	0.996591i	$-0.526289\pi$
$373$	−443333.	−0.164990	−0.0824951	+	0.996591i	$0.526289\pi$
$374$	−467153.	−0.172695
$375$	0	0
$376$	−1.89001e6	−0.689438
$377$	−448003.	−0.162341
$378$	0	0
$379$	2.52046e6	0.901324	0.450662	−	0.892695i	$-0.351188\pi$
$379$	2.52046e6	0.901324	0.450662	+	0.892695i	$0.351188\pi$
$380$	−7451.11	−0.00264705
$381$	0	0
$382$	920049.	0.322591
$383$	−3.56075e6	−1.24035	−0.620175	−	0.784463i	$-0.712938\pi$
$383$	−3.56075e6	−1.24035	−0.620175	+	0.784463i	$0.712938\pi$
$384$	0	0
$385$	−17594.8	−0.00604969
$386$	870377.	0.297330
$387$	0	0
$388$	−2.94100e6	−0.991780
$389$	−1.66815e6	−0.558933	−0.279467	−	0.960155i	$-0.590158\pi$
$389$	−1.66815e6	−0.558933	−0.279467	+	0.960155i	$0.590158\pi$
$390$	0	0
$391$	−2.36754e6	−0.783168
$392$	987984.	0.324739
$393$	0	0
$394$	−402318.	−0.130566
$395$	47169.4	0.0152114
$396$	0	0
$397$	−4.98663e6	−1.58793	−0.793964	−	0.607964i	$-0.791986\pi$
$397$	−4.98663e6	−1.58793	−0.793964	+	0.607964i	$0.791986\pi$
$398$	−176443.	−0.0558336
$399$	0	0
$400$	−2.76233e6	−0.863227
$401$	−1.72114e6	−0.534509	−0.267255	−	0.963626i	$-0.586116\pi$
$401$	−1.72114e6	−0.534509	−0.267255	+	0.963626i	$0.586116\pi$
$402$	0	0
$403$	1.25562e6	0.385119
$404$	−80592.7	−0.0245664
$405$	0	0
$406$	223951.	0.0674277
$407$	2.64807e6	0.792397
$408$	0	0
$409$	4.38543e6	1.29630	0.648148	−	0.761515i	$-0.275544\pi$
$409$	4.38543e6	1.29630	0.648148	+	0.761515i	$0.275544\pi$
$410$	15519.4	0.00455947
$411$	0	0
$412$	−3.58470e6	−1.04042
$413$	−214981.	−0.0620190
$414$	0	0
$415$	−26379.5	−0.00751877
$416$	−527211.	−0.149366
$417$	0	0
$418$	−101149.	−0.0283153
$419$	−5.77647e6	−1.60741	−0.803706	−	0.595026i	$-0.797142\pi$
$419$	−5.77647e6	−1.60741	−0.803706	+	0.595026i	$0.797142\pi$
$420$	0	0
$421$	−624265.	−0.171658	−0.0858290	−	0.996310i	$-0.527354\pi$
$421$	−624265.	−0.171658	−0.0858290	+	0.996310i	$0.527354\pi$
$422$	1.01731e6	0.278083
$423$	0	0
$424$	−2.68775e6	−0.726062
$425$	3.85186e6	1.03442
$426$	0	0
$427$	−2.89517e6	−0.768431
$428$	−6.98522e6	−1.84319
$429$	0	0
$430$	12556.2	0.00327483
$431$	−1.51781e6	−0.393573	−0.196787	−	0.980446i	$-0.563051\pi$
$431$	−1.51781e6	−0.393573	−0.196787	+	0.980446i	$0.563051\pi$
$432$	0	0
$433$	6.06106e6	1.55356	0.776781	−	0.629770i	$-0.216851\pi$
$433$	6.06106e6	1.55356	0.776781	+	0.629770i	$0.216851\pi$
$434$	−627668.	−0.159958
$435$	0	0
$436$	−4.99958e6	−1.25956
$437$	−512624.	−0.128409
$438$	0	0
$439$	4.36235e6	1.08034	0.540168	−	0.841557i	$-0.318361\pi$
$439$	4.36235e6	1.08034	0.540168	+	0.841557i	$0.318361\pi$
$440$	21663.7	0.00533459
$441$	0	0
$442$	225320.	0.0548585
$443$	6.36182e6	1.54018	0.770091	−	0.637934i	$-0.220211\pi$
$443$	6.36182e6	1.54018	0.770091	+	0.637934i	$0.220211\pi$
$444$	0	0
$445$	−8705.04	−0.00208387
$446$	893063.	0.212591
$447$	0	0
$448$	−1.48383e6	−0.349293
$449$	4.13291e6	0.967475	0.483737	−	0.875213i	$-0.339279\pi$
$449$	4.13291e6	0.967475	0.483737	+	0.875213i	$0.339279\pi$
$450$	0	0
$451$	−4.34681e6	−1.00630
$452$	495832.	0.114153
$453$	0	0
$454$	−450330.	−0.102540
$455$	8486.42	0.00192175
$456$	0	0
$457$	−4.22694e6	−0.946751	−0.473375	−	0.880861i	$-0.656965\pi$
$457$	−4.22694e6	−0.946751	−0.473375	+	0.880861i	$0.656965\pi$
$458$	−840005.	−0.187119
$459$	0	0
$460$	53597.1	0.0118099
$461$	−5.51414e6	−1.20844	−0.604220	−	0.796817i	$-0.706515\pi$
$461$	−5.51414e6	−1.20844	−0.604220	+	0.796817i	$0.706515\pi$
$462$	0	0
$463$	−1.86964e6	−0.405327	−0.202664	−	0.979248i	$-0.564960\pi$
$463$	−1.86964e6	−0.405327	−0.202664	+	0.979248i	$0.564960\pi$
$464$	2.63621e6	0.568440
$465$	0	0
$466$	1.12473e6	0.239930
$467$	−4.32669e6	−0.918045	−0.459022	−	0.888425i	$-0.651800\pi$
$467$	−4.32669e6	−0.918045	−0.459022	+	0.888425i	$0.651800\pi$
$468$	0	0
$469$	2.32581e6	0.488250
$470$	−27645.7	−0.00577274
$471$	0	0
$472$	264696.	0.0546881
$473$	−3.51687e6	−0.722776
$474$	0	0
$475$	834013.	0.169605
$476$	2.32396e6	0.470122
$477$	0	0
$478$	691364.	0.138400
$479$	3.34586e6	0.666298	0.333149	−	0.942874i	$-0.391889\pi$
$479$	3.34586e6	0.666298	0.333149	+	0.942874i	$0.391889\pi$
$480$	0	0
$481$	−1.27723e6	−0.251713
$482$	−355289.	−0.0696569
$483$	0	0
$484$	1.95329e6	0.379013
$485$	−88122.3	−0.0170111
$486$	0	0
$487$	1.41911e6	0.271139	0.135570	−	0.990768i	$-0.456714\pi$
$487$	1.41911e6	0.271139	0.135570	+	0.990768i	$0.456714\pi$
$488$	3.56469e6	0.677599
$489$	0	0
$490$	14451.5	0.00271908
$491$	3.66412e6	0.685908	0.342954	−	0.939352i	$-0.388573\pi$
$491$	3.66412e6	0.685908	0.342954	+	0.939352i	$0.388573\pi$
$492$	0	0
$493$	−3.67600e6	−0.681174
$494$	48786.7	0.00899464
$495$	0	0
$496$	−7.38849e6	−1.34850
$497$	3.48402e6	0.632688
$498$	0	0
$499$	−1.94309e6	−0.349335	−0.174667	−	0.984627i	$-0.555885\pi$
$499$	−1.94309e6	−0.349335	−0.174667	+	0.984627i	$0.555885\pi$
$500$	−174423.	−0.0312017
$501$	0	0
$502$	2.14605e6	0.380085
$503$	−2.10114e6	−0.370285	−0.185142	−	0.982712i	$-0.559275\pi$
$503$	−2.10114e6	−0.370285	−0.185142	+	0.982712i	$0.559275\pi$
$504$	0	0
$505$	−2414.83	−0.000421365 0
$506$	727582.	0.126330
$507$	0	0
$508$	−8.71235e6	−1.49788
$509$	2.83832e6	0.485587	0.242793	−	0.970078i	$-0.421936\pi$
$509$	2.83832e6	0.485587	0.242793	+	0.970078i	$0.421936\pi$
$510$	0	0
$511$	4.09172e6	0.693192
$512$	5.25376e6	0.885718
$513$	0	0
$514$	1.00303e6	0.167458
$515$	−107410.	−0.0178454
$516$	0	0
$517$	7.74326e6	1.27408
$518$	638472.	0.104548
$519$	0	0
$520$	−10449.0	−0.00169459
$521$	−6.04781e6	−0.976121	−0.488060	−	0.872810i	$-0.662295\pi$
$521$	−6.04781e6	−0.976121	−0.488060	+	0.872810i	$0.662295\pi$
$522$	0	0
$523$	1.00745e7	1.61053	0.805265	−	0.592915i	$-0.202023\pi$
$523$	1.00745e7	1.61053	0.805265	+	0.592915i	$0.202023\pi$
$524$	4.69522e6	0.747012
$525$	0	0
$526$	1.11342e6	0.175466
$527$	1.03027e7	1.61594
$528$	0	0
$529$	−2.74895e6	−0.427098
$530$	−39314.3	−0.00607940
$531$	0	0
$532$	503188.	0.0770817
$533$	2.09658e6	0.319663
$534$	0	0
$535$	−209301.	−0.0316145
$536$	−2.86367e6	−0.430537
$537$	0	0
$538$	−1.21793e6	−0.181413
$539$	−4.04770e6	−0.600118
$540$	0	0
$541$	2.90078e6	0.426110	0.213055	−	0.977040i	$-0.431659\pi$
$541$	2.90078e6	0.426110	0.213055	+	0.977040i	$0.431659\pi$
$542$	−2.73861e6	−0.400435
$543$	0	0
$544$	−4.32593e6	−0.626732
$545$	−149804.	−0.0216040
$546$	0	0
$547$	−6.58103e6	−0.940428	−0.470214	−	0.882552i	$-0.655823\pi$
$547$	−6.58103e6	−0.940428	−0.470214	+	0.882552i	$0.655823\pi$
$548$	−8.59591e6	−1.22276
$549$	0	0
$550$	−1.18374e6	−0.166859
$551$	−795935.	−0.111686
$552$	0	0
$553$	−3.18545e6	−0.442953
$554$	6424.37	0.000889316 0
$555$	0	0
$556$	−5.23296e6	−0.717895
$557$	8.57629e6	1.17128	0.585641	−	0.810571i	$-0.300843\pi$
$557$	8.57629e6	1.17128	0.585641	+	0.810571i	$0.300843\pi$
$558$	0	0
$559$	1.69628e6	0.229597
$560$	−49937.1	−0.00672904
$561$	0	0
$562$	3.01927e6	0.403237
$563$	6.42359e6	0.854096	0.427048	−	0.904229i	$-0.359554\pi$
$563$	6.42359e6	0.854096	0.427048	+	0.904229i	$0.359554\pi$
$564$	0	0
$565$	14856.8	0.00195796
$566$	−105819.	−0.0138843
$567$	0	0
$568$	−4.28972e6	−0.557902
$569$	−1.17161e7	−1.51706	−0.758531	−	0.651638i	$-0.774082\pi$
$569$	−1.17161e7	−1.51706	−0.758531	+	0.651638i	$0.774082\pi$
$570$	0	0
$571$	−4.99504e6	−0.641134	−0.320567	−	0.947226i	$-0.603873\pi$
$571$	−4.99504e6	−0.641134	−0.320567	+	0.947226i	$0.603873\pi$
$572$	1.42870e6	0.182579
$573$	0	0
$574$	−1.04805e6	−0.132771
$575$	−5.99920e6	−0.756700
$576$	0	0
$577$	495976.	0.0620185	0.0310093	−	0.999519i	$-0.490128\pi$
$577$	495976.	0.0620185	0.0310093	+	0.999519i	$0.490128\pi$
$578$	121929.	0.0151805
$579$	0	0
$580$	83218.5	0.0102719
$581$	1.78146e6	0.218945
$582$	0	0
$583$	1.10115e7	1.34176
$584$	−5.03795e6	−0.611254
$585$	0	0
$586$	−201293.	−0.0242150
$587$	202428.	0.0242480	0.0121240	−	0.999927i	$-0.496141\pi$
$587$	202428.	0.0242480	0.0121240	+	0.999927i	$0.496141\pi$
$588$	0	0
$589$	2.23077e6	0.264951
$590$	3871.77	0.000457910 0
$591$	0	0
$592$	7.51567e6	0.881380
$593$	−7.95364e6	−0.928815	−0.464407	−	0.885622i	$-0.653733\pi$
$593$	−7.95364e6	−0.928815	−0.464407	+	0.885622i	$0.653733\pi$
$594$	0	0
$595$	69633.7	0.00806356
$596$	7.52571e6	0.867824
$597$	0	0
$598$	−350931.	−0.0401300
$599$	1.68592e7	1.91986	0.959930	−	0.280241i	$-0.0904144\pi$
$599$	1.68592e7	1.91986	0.959930	+	0.280241i	$0.0904144\pi$
$600$	0	0
$601$	5.03561e6	0.568677	0.284339	−	0.958724i	$-0.408226\pi$
$601$	5.03561e6	0.568677	0.284339	+	0.958724i	$0.408226\pi$
$602$	−847948.	−0.0953626
$603$	0	0
$604$	−4.93787e6	−0.550741
$605$	58527.3	0.00650085
$606$	0	0
$607$	9.93216e6	1.09414	0.547069	−	0.837088i	$-0.315744\pi$
$607$	9.93216e6	1.09414	0.547069	+	0.837088i	$0.315744\pi$
$608$	−936659.	−0.102760
$609$	0	0
$610$	52141.6	0.00567362
$611$	−3.73477e6	−0.404725
$612$	0	0
$613$	1.09871e7	1.18095	0.590477	−	0.807054i	$-0.298940\pi$
$613$	1.09871e7	1.18095	0.590477	+	0.807054i	$0.298940\pi$
$614$	908003.	0.0972000
$615$	0	0
$616$	−1.46299e6	−0.155343
$617$	−6.30095e6	−0.666336	−0.333168	−	0.942867i	$-0.608118\pi$
$617$	−6.30095e6	−0.666336	−0.333168	+	0.942867i	$0.608118\pi$
$618$	0	0
$619$	5.08049e6	0.532941	0.266470	−	0.963843i	$-0.414143\pi$
$619$	5.08049e6	0.532941	0.266470	+	0.963843i	$0.414143\pi$
$620$	−233236.	−0.0243679
$621$	0	0
$622$	−776212.	−0.0804460
$623$	587869.	0.0606820
$624$	0	0
$625$	9.75779e6	0.999197
$626$	1.88800e6	0.192560
$627$	0	0
$628$	6.47922e6	0.655577
$629$	−1.04801e7	−1.05618
$630$	0	0
$631$	−7.62504e6	−0.762376	−0.381188	−	0.924498i	$-0.624485\pi$
$631$	−7.62504e6	−0.762376	−0.381188	+	0.924498i	$0.624485\pi$
$632$	3.92210e6	0.390594
$633$	0	0
$634$	−2.40091e6	−0.237221
$635$	−261052.	−0.0256917
$636$	0	0
$637$	1.95231e6	0.190634
$638$	1.12969e6	0.109878
$639$	0	0
$640$	129402.	0.0124879
$641$	7.34805e6	0.706362	0.353181	−	0.935555i	$-0.385100\pi$
$641$	7.34805e6	0.706362	0.353181	+	0.935555i	$0.385100\pi$
$642$	0	0
$643$	−4.91274e6	−0.468594	−0.234297	−	0.972165i	$-0.575279\pi$
$643$	−4.91274e6	−0.468594	−0.234297	+	0.972165i	$0.575279\pi$
$644$	−3.61952e6	−0.343903
$645$	0	0
$646$	400310.	0.0377411
$647$	−5.02847e6	−0.472254	−0.236127	−	0.971722i	$-0.575878\pi$
$647$	−5.02847e6	−0.472254	−0.236127	+	0.971722i	$0.575878\pi$
$648$	0	0
$649$	−1.08444e6	−0.101064
$650$	570947.	0.0530045
$651$	0	0
$652$	−1.30676e7	−1.20386
$653$	−5.06290e6	−0.464639	−0.232320	−	0.972639i	$-0.574632\pi$
$653$	−5.06290e6	−0.464639	−0.232320	+	0.972639i	$0.574632\pi$
$654$	0	0
$655$	140685.	0.0128128
$656$	−1.23370e7	−1.11931
$657$	0	0
$658$	1.86697e6	0.168102
$659$	176951.	0.0158723	0.00793615	−	0.999969i	$-0.497474\pi$
$659$	176951.	0.0158723	0.00793615	+	0.999969i	$0.497474\pi$
$660$	0	0
$661$	1.83120e7	1.63016	0.815082	−	0.579346i	$-0.196692\pi$
$661$	1.83120e7	1.63016	0.815082	+	0.579346i	$0.196692\pi$
$662$	−2.92794e6	−0.259667
$663$	0	0
$664$	−2.19343e6	−0.193065
$665$	15077.2	0.00132211
$666$	0	0
$667$	5.72530e6	0.498291
$668$	7.95138e6	0.689448
$669$	0	0
$670$	−41887.5	−0.00360494
$671$	−1.46043e7	−1.25220
$672$	0	0
$673$	1.31732e7	1.12112	0.560561	−	0.828113i	$-0.310586\pi$
$673$	1.31732e7	1.12112	0.560561	+	0.828113i	$0.310586\pi$
$674$	2.31860e6	0.196596
$675$	0	0
$676$	1.06430e7	0.895776
$677$	5.29079e6	0.443658	0.221829	−	0.975086i	$-0.428797\pi$
$677$	5.29079e6	0.443658	0.221829	+	0.975086i	$0.428797\pi$
$678$	0	0
$679$	5.95107e6	0.495360
$680$	−85736.8	−0.00711042
$681$	0	0
$682$	−3.16619e6	−0.260661
$683$	1.28101e7	1.05076	0.525378	−	0.850869i	$-0.323924\pi$
$683$	1.28101e7	1.05076	0.525378	+	0.850869i	$0.323924\pi$
$684$	0	0
$685$	−257563.	−0.0209728
$686$	−2.23836e6	−0.181602
$687$	0	0
$688$	−9.98148e6	−0.803940
$689$	−5.31113e6	−0.426225
$690$	0	0
$691$	−2.41425e7	−1.92348	−0.961739	−	0.273969i	$-0.911664\pi$
$691$	−2.41425e7	−1.92348	−0.961739	+	0.273969i	$0.911664\pi$
$692$	1.14490e7	0.908874
$693$	0	0
$694$	−4.52907e6	−0.356952
$695$	−156797.	−0.0123134
$696$	0	0
$697$	1.72030e7	1.34129
$698$	−121067.	−0.00940565
$699$	0	0
$700$	5.88877e6	0.454234
$701$	1.34383e7	1.03288	0.516441	−	0.856323i	$-0.327257\pi$
$701$	1.34383e7	1.03288	0.516441	+	0.856323i	$0.327257\pi$
$702$	0	0
$703$	−2.26916e6	−0.173172
$704$	−7.48499e6	−0.569193
$705$	0	0
$706$	−642973.	−0.0485491
$707$	163078.	0.0122701
$708$	0	0
$709$	4.97439e6	0.371642	0.185821	−	0.982584i	$-0.440506\pi$
$709$	4.97439e6	0.371642	0.185821	+	0.982584i	$0.440506\pi$
$710$	−62746.7	−0.00467138
$711$	0	0
$712$	−723816.	−0.0535092
$713$	−1.60463e7	−1.18209
$714$	0	0
$715$	42808.6	0.00313160
$716$	−1.73908e7	−1.26776
$717$	0	0
$718$	1.81647e6	0.131497
$719$	2.75626e6	0.198837	0.0994185	−	0.995046i	$-0.468302\pi$
$719$	2.75626e6	0.198837	0.0994185	+	0.995046i	$0.468302\pi$
$720$	0	0
$721$	7.25360e6	0.519655
$722$	−2.92486e6	−0.208815
$723$	0	0
$724$	1.62461e7	1.15187
$725$	−9.31477e6	−0.658153
$726$	0	0
$727$	−1.11365e7	−0.781470	−0.390735	−	0.920503i	$-0.627779\pi$
$727$	−1.11365e7	−0.781470	−0.390735	+	0.920503i	$0.627779\pi$
$728$	705638.	0.0493462
$729$	0	0
$730$	−73691.2	−0.00511810
$731$	1.39185e7	0.963380
$732$	0	0
$733$	2.89889e6	0.199284	0.0996419	−	0.995023i	$-0.468230\pi$
$733$	2.89889e6	0.199284	0.0996419	+	0.995023i	$0.468230\pi$
$734$	187554.	0.0128495
$735$	0	0
$736$	6.73755e6	0.458466
$737$	1.17322e7	0.795633
$738$	0	0
$739$	2.76654e7	1.86348	0.931741	−	0.363123i	$-0.118290\pi$
$739$	2.76654e7	1.86348	0.931741	+	0.363123i	$0.118290\pi$
$740$	237251.	0.0159268
$741$	0	0
$742$	2.65497e6	0.177031
$743$	1.08048e7	0.718033	0.359017	−	0.933331i	$-0.383112\pi$
$743$	1.08048e7	0.718033	0.359017	+	0.933331i	$0.383112\pi$
$744$	0	0
$745$	225496.	0.0148850
$746$	−539200.	−0.0354734
$747$	0	0
$748$	1.17229e7	0.766092
$749$	1.41345e7	0.920611
$750$	0	0
$751$	−2.36060e7	−1.52729	−0.763645	−	0.645636i	$-0.776592\pi$
$751$	−2.36060e7	−1.52729	−0.763645	+	0.645636i	$0.776592\pi$
$752$	2.19767e7	1.41716
$753$	0	0
$754$	−544879.	−0.0349037
$755$	−147955.	−0.00944633
$756$	0	0
$757$	2.12366e7	1.34693	0.673464	−	0.739220i	$-0.264806\pi$
$757$	2.12366e7	1.34693	0.673464	+	0.739220i	$0.264806\pi$
$758$	3.06548e6	0.193787
$759$	0	0
$760$	−18563.9	−0.00116583
$761$	2.37074e7	1.48396	0.741982	−	0.670420i	$-0.233886\pi$
$761$	2.37074e7	1.48396	0.741982	+	0.670420i	$0.233886\pi$
$762$	0	0
$763$	1.01166e7	0.629104
$764$	−2.30880e7	−1.43105
$765$	0	0
$766$	−4.33073e6	−0.266679
$767$	523054.	0.0321039
$768$	0	0
$769$	1.33593e7	0.814644	0.407322	−	0.913285i	$-0.366463\pi$
$769$	1.33593e7	0.814644	0.407322	+	0.913285i	$0.366463\pi$
$770$	−21399.5	−0.00130070
$771$	0	0
$772$	−2.18415e7	−1.31899
$773$	1.28438e7	0.773117	0.386558	−	0.922265i	$-0.373664\pi$
$773$	1.28438e7	0.773117	0.386558	+	0.922265i	$0.373664\pi$
$774$	0	0
$775$	2.61065e7	1.56133
$776$	−7.32729e6	−0.436806
$777$	0	0
$778$	−2.02887e6	−0.120172
$779$	3.72484e6	0.219920
$780$	0	0
$781$	1.75747e7	1.03100
$782$	−2.87950e6	−0.168384
$783$	0	0
$784$	−1.14881e7	−0.667509
$785$	194139.	0.0112445
$786$	0	0
$787$	−5.78070e6	−0.332693	−0.166347	−	0.986067i	$-0.553197\pi$
$787$	−5.78070e6	−0.332693	−0.166347	+	0.986067i	$0.553197\pi$
$788$	1.00959e7	0.579201
$789$	0	0
$790$	57369.4	0.00327049
$791$	−1.00331e6	−0.0570157
$792$	0	0
$793$	7.04403e6	0.397776
$794$	−6.06495e6	−0.341410
$795$	0	0
$796$	4.42771e6	0.247683
$797$	4.78906e6	0.267057	0.133529	−	0.991045i	$-0.457369\pi$
$797$	4.78906e6	0.267057	0.133529	+	0.991045i	$0.457369\pi$
$798$	0	0
$799$	−3.06449e7	−1.69821
$800$	−1.09616e7	−0.605551
$801$	0	0
$802$	−2.09332e6	−0.114921
$803$	2.06401e7	1.12960
$804$	0	0
$805$	−108453.	−0.00589864
$806$	1.52713e6	0.0828017
$807$	0	0
$808$	−200791.	−0.0108197
$809$	1.17461e7	0.630987	0.315494	−	0.948928i	$-0.397830\pi$
$809$	1.17461e7	0.630987	0.315494	+	0.948928i	$0.397830\pi$
$810$	0	0
$811$	2.70056e6	0.144179	0.0720893	−	0.997398i	$-0.477033\pi$
$811$	2.70056e6	0.144179	0.0720893	+	0.997398i	$0.477033\pi$
$812$	−5.61991e6	−0.299116
$813$	0	0
$814$	3.22069e6	0.170368
$815$	−391548.	−0.0206486
$816$	0	0
$817$	3.01365e6	0.157957
$818$	5.33374e6	0.278708
$819$	0	0
$820$	−389448.	−0.0202262
$821$	−2.05695e7	−1.06504	−0.532520	−	0.846418i	$-0.678755\pi$
$821$	−2.05695e7	−1.06504	−0.532520	+	0.846418i	$0.678755\pi$
$822$	0	0
$823$	−3.85189e7	−1.98232	−0.991162	−	0.132654i	$-0.957650\pi$
$823$	−3.85189e7	−1.98232	−0.991162	+	0.132654i	$0.957650\pi$
$824$	−8.93103e6	−0.458230
$825$	0	0
$826$	−261469.	−0.0133343
$827$	−726695.	−0.0369478	−0.0184739	−	0.999829i	$-0.505881\pi$
$827$	−726695.	−0.0369478	−0.0184739	+	0.999829i	$0.505881\pi$
$828$	0	0
$829$	1.92752e7	0.974120	0.487060	−	0.873369i	$-0.338069\pi$
$829$	1.92752e7	0.974120	0.487060	+	0.873369i	$0.338069\pi$
$830$	−32083.8	−0.00161656
$831$	0	0
$832$	3.61020e6	0.180810
$833$	1.60193e7	0.799891
$834$	0	0
$835$	238250.	0.0118254
$836$	2.53827e6	0.125609
$837$	0	0
$838$	−7.02558e6	−0.345599
$839$	2.47117e7	1.21199	0.605994	−	0.795470i	$-0.292776\pi$
$839$	2.47117e7	1.21199	0.605994	+	0.795470i	$0.292776\pi$
$840$	0	0
$841$	−1.16217e7	−0.566602
$842$	−759257.	−0.0369070
$843$	0	0
$844$	−2.55288e7	−1.23360
$845$	318902.	0.0153644
$846$	0	0
$847$	−3.95246e6	−0.189304
$848$	3.12526e7	1.49244
$849$	0	0
$850$	4.68479e6	0.222404
$851$	1.63225e7	0.772613
$852$	0	0
$853$	2.66298e7	1.25313	0.626563	−	0.779370i	$-0.284461\pi$
$853$	2.66298e7	1.25313	0.626563	+	0.779370i	$0.284461\pi$
$854$	−3.52123e6	−0.165215
$855$	0	0
$856$	−1.74032e7	−0.811791
$857$	−5.12844e6	−0.238525	−0.119262	−	0.992863i	$-0.538053\pi$
$857$	−5.12844e6	−0.238525	−0.119262	+	0.992863i	$0.538053\pi$
$858$	0	0
$859$	2.30168e7	1.06430	0.532148	−	0.846651i	$-0.321385\pi$
$859$	2.30168e7	1.06430	0.532148	+	0.846651i	$0.321385\pi$
$860$	−315091.	−0.0145274
$861$	0	0
$862$	−1.84603e6	−0.0846194
$863$	3.81545e7	1.74389	0.871944	−	0.489606i	$-0.162859\pi$
$863$	3.81545e7	1.74389	0.871944	+	0.489606i	$0.162859\pi$
$864$	0	0
$865$	343052.	0.0155891
$866$	7.37171e6	0.334021
$867$	0	0
$868$	1.57509e7	0.709589
$869$	−1.60686e7	−0.721818
$870$	0	0
$871$	−5.65876e6	−0.252741
$872$	−1.24561e7	−0.554742
$873$	0	0
$874$	−623474.	−0.0276083
$875$	352943.	0.0155842
$876$	0	0
$877$	−3.38162e7	−1.48466	−0.742328	−	0.670036i	$-0.766278\pi$
$877$	−3.38162e7	−1.48466	−0.742328	+	0.670036i	$0.766278\pi$
$878$	5.30566e6	0.232275
$879$	0	0
$880$	−251901.	−0.0109654
$881$	−2.86617e7	−1.24412	−0.622059	−	0.782970i	$-0.713704\pi$
$881$	−2.86617e7	−1.24412	−0.622059	+	0.782970i	$0.713704\pi$
$882$	0	0
$883$	−4.01713e7	−1.73386	−0.866930	−	0.498430i	$-0.833910\pi$
$883$	−4.01713e7	−1.73386	−0.866930	+	0.498430i	$0.833910\pi$
$884$	−5.65425e6	−0.243357
$885$	0	0
$886$	7.73751e6	0.331144
$887$	1.88137e7	0.802906	0.401453	−	0.915880i	$-0.368505\pi$
$887$	1.88137e7	0.802906	0.401453	+	0.915880i	$0.368505\pi$
$888$	0	0
$889$	1.76293e7	0.748138
$890$	−10587.4	−0.000448039 0
$891$	0	0
$892$	−2.24108e7	−0.943074
$893$	−6.63529e6	−0.278440
$894$	0	0
$895$	−521086.	−0.0217446
$896$	−8.73876e6	−0.363647
$897$	0	0
$898$	5.02661e6	0.208010
$899$	−2.49146e7	−1.02814
$900$	0	0
$901$	−4.35795e7	−1.78842
$902$	−5.28677e6	−0.216359
$903$	0	0
$904$	1.23533e6	0.0502762
$905$	486789.	0.0197569
$906$	0	0
$907$	−8.85338e6	−0.357348	−0.178674	−	0.983908i	$-0.557181\pi$
$907$	−8.85338e6	−0.357348	−0.178674	+	0.983908i	$0.557181\pi$
$908$	1.13007e7	0.454875
$909$	0	0
$910$	10321.5	0.000413181 0
$911$	−1.12187e7	−0.447864	−0.223932	−	0.974605i	$-0.571889\pi$
$911$	−1.12187e7	−0.447864	−0.223932	+	0.974605i	$0.571889\pi$
$912$	0	0
$913$	8.98634e6	0.356785
$914$	−5.14098e6	−0.203554
$915$	0	0
$916$	2.10794e7	0.830078
$917$	−9.50072e6	−0.373107
$918$	0	0
$919$	2.15141e7	0.840299	0.420149	−	0.907455i	$-0.361978\pi$
$919$	2.15141e7	0.840299	0.420149	+	0.907455i	$0.361978\pi$
$920$	133533.	0.00520140
$921$	0	0
$922$	−6.70652e6	−0.259818
$923$	−8.47671e6	−0.327509
$924$	0	0
$925$	−2.65558e7	−1.02048
$926$	−2.27393e6	−0.0871466
$927$	0	0
$928$	1.04612e7	0.398759
$929$	−1.34111e7	−0.509831	−0.254916	−	0.966963i	$-0.582048\pi$
$929$	−1.34111e7	−0.509831	−0.254916	+	0.966963i	$0.582048\pi$
$930$	0	0
$931$	3.46853e6	0.131151
$932$	−2.82245e7	−1.06435
$933$	0	0
$934$	−5.26230e6	−0.197382
$935$	351258.	0.0131401
$936$	0	0
$937$	−3.24101e7	−1.20596	−0.602978	−	0.797758i	$-0.706019\pi$
$937$	−3.24101e7	−1.20596	−0.602978	+	0.797758i	$0.706019\pi$
$938$	2.82875e6	0.104975
$939$	0	0
$940$	693749.	0.0256084
$941$	3.91775e6	0.144232	0.0721162	−	0.997396i	$-0.477025\pi$
$941$	3.91775e6	0.144232	0.0721162	+	0.997396i	$0.477025\pi$
$942$	0	0
$943$	−2.67934e7	−0.981180
$944$	−3.07783e6	−0.112413
$945$	0	0
$946$	−4.27736e6	−0.155399
$947$	3.42000e7	1.23923	0.619614	−	0.784907i	$-0.287289\pi$
$947$	3.42000e7	1.23923	0.619614	+	0.784907i	$0.287289\pi$
$948$	0	0
$949$	−9.95525e6	−0.358828
$950$	1.01436e6	0.0364656
$951$	0	0
$952$	5.78998e6	0.207054
$953$	−4.25014e7	−1.51590	−0.757951	−	0.652312i	$-0.773799\pi$
$953$	−4.25014e7	−1.51590	−0.757951	+	0.652312i	$0.773799\pi$
$954$	0	0
$955$	−691796.	−0.0245454
$956$	−1.73493e7	−0.613956
$957$	0	0
$958$	4.06937e6	0.143256
$959$	1.73937e7	0.610725
$960$	0	0
$961$	4.11988e7	1.43905
$962$	−1.55342e6	−0.0541191
$963$	0	0
$964$	8.91574e6	0.309005
$965$	−654447.	−0.0226233
$966$	0	0
$967$	2.74628e7	0.944450	0.472225	−	0.881478i	$-0.343451\pi$
$967$	2.74628e7	0.944450	0.472225	+	0.881478i	$0.343451\pi$
$968$	4.86649e6	0.166927
$969$	0	0
$970$	−107178.	−0.00365743
$971$	−4.50046e7	−1.53183	−0.765913	−	0.642945i	$-0.777713\pi$
$971$	−4.50046e7	−1.53183	−0.765913	+	0.642945i	$0.777713\pi$
$972$	0	0
$973$	1.05888e7	0.358564
$974$	1.72598e6	0.0582958
$975$	0	0
$976$	−4.14495e7	−1.39282
$977$	1.02124e7	0.342287	0.171144	−	0.985246i	$-0.445254\pi$
$977$	1.02124e7	0.342287	0.171144	+	0.985246i	$0.445254\pi$
$978$	0	0
$979$	2.96543e6	0.0988850
$980$	−362650.	−0.0120621
$981$	0	0
$982$	4.45645e6	0.147472
$983$	−1.34161e7	−0.442835	−0.221418	−	0.975179i	$-0.571068\pi$
$983$	−1.34161e7	−0.442835	−0.221418	+	0.975179i	$0.571068\pi$
$984$	0	0
$985$	302507.	0.00993449
$986$	−4.47090e6	−0.146455
$987$	0	0
$988$	−1.22427e6	−0.0399011
$989$	−2.16777e7	−0.704730
$990$	0	0
$991$	−4.37148e7	−1.41398	−0.706992	−	0.707222i	$-0.749948\pi$
$991$	−4.37148e7	−1.41398	−0.706992	+	0.707222i	$0.749948\pi$
$992$	−2.93195e7	−0.945971
$993$	0	0
$994$	4.23741e6	0.136030
$995$	132669.	0.00424827
$996$	0	0
$997$	−3.11427e7	−0.992243	−0.496121	−	0.868253i	$-0.665243\pi$
$997$	−3.11427e7	−0.992243	−0.496121	+	0.868253i	$0.665243\pi$
$998$	−2.36327e6	−0.0751081
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.b.1.6		11
3.2	odd	2		177.6.a.a.1.6	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.a.1.6	✓	11	3.2	odd	2
531.6.a.b.1.6		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)

\(f(q)\)	\(=\)	\(q+1.21624 q^{2} -30.5208 q^{4} -0.914506 q^{5} +61.7584 q^{7} -76.0403 q^{8} +O(q^{10})\)	\(q+1.21624 q^{2} -30.5208 q^{4} -0.914506 q^{5} +61.7584 q^{7} -76.0403 q^{8} -1.11226 q^{10} +311.532 q^{11} -150.260 q^{13} +75.1131 q^{14} +884.181 q^{16} -1232.93 q^{17} -266.956 q^{19} +27.9114 q^{20} +378.898 q^{22} +1920.26 q^{23} -3124.16 q^{25} -182.752 q^{26} -1884.91 q^{28} +2981.52 q^{29} -8356.31 q^{31} +3508.67 q^{32} -1499.54 q^{34} -56.4784 q^{35} +8500.15 q^{37} -324.682 q^{38} +69.5393 q^{40} -13953.0 q^{41} -11289.0 q^{43} -9508.19 q^{44} +2335.50 q^{46} +24855.4 q^{47} -12992.9 q^{49} -3799.74 q^{50} +4586.04 q^{52} +35346.4 q^{53} -284.898 q^{55} -4696.13 q^{56} +3626.25 q^{58} -3481.00 q^{59} -46879.0 q^{61} -10163.3 q^{62} -24026.4 q^{64} +137.413 q^{65} +37659.9 q^{67} +37629.8 q^{68} -68.6913 q^{70} +56413.7 q^{71} +66253.6 q^{73} +10338.2 q^{74} +8147.69 q^{76} +19239.7 q^{77} -51579.2 q^{79} -808.588 q^{80} -16970.2 q^{82} +28845.7 q^{83} +1127.52 q^{85} -13730.1 q^{86} -23689.0 q^{88} +9518.85 q^{89} -9279.79 q^{91} -58607.8 q^{92} +30230.2 q^{94} +244.132 q^{95} +96360.6 q^{97} -15802.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) \) 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