LMFDB - Embedded newform 531.6.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,6,Mod(1,531)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$85.1638083207$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + \cdots + 49172480 $ x^12 - 2x^11 - 269x^10 + 143x^9 + 25384x^8 + 8539x^7 - 1009736x^6 - 720516x^5 + 15565376x^4 + 6775664x^3 - 75006848x^2 + 21512960*x + 49172480
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-8.49197$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.4920 q^{2} +78.0815 q^{4} -46.0665 q^{5} +183.145 q^{7} -483.486 q^{8} +O(q^{10})$	\(q-10.4920 q^{2} +78.0815 q^{4} -46.0665 q^{5} +183.145 q^{7} -483.486 q^{8} +483.328 q^{10} +536.597 q^{11} -601.772 q^{13} -1921.55 q^{14} +2574.11 q^{16} +1948.89 q^{17} -1829.58 q^{19} -3596.94 q^{20} -5629.96 q^{22} +143.630 q^{23} -1002.88 q^{25} +6313.78 q^{26} +14300.3 q^{28} -4181.48 q^{29} +6331.93 q^{31} -11536.0 q^{32} -20447.7 q^{34} -8436.86 q^{35} -1983.22 q^{37} +19195.9 q^{38} +22272.5 q^{40} -18079.8 q^{41} -16207.9 q^{43} +41898.3 q^{44} -1506.96 q^{46} -10082.4 q^{47} +16735.2 q^{49} +10522.2 q^{50} -46987.3 q^{52} -17087.2 q^{53} -24719.1 q^{55} -88548.1 q^{56} +43872.0 q^{58} +3481.00 q^{59} -10827.9 q^{61} -66434.5 q^{62} +38663.6 q^{64} +27721.5 q^{65} +53212.3 q^{67} +152172. q^{68} +88519.3 q^{70} +82455.8 q^{71} -25989.7 q^{73} +20807.9 q^{74} -142856. q^{76} +98275.1 q^{77} +18631.5 q^{79} -118580. q^{80} +189692. q^{82} -105042. q^{83} -89778.4 q^{85} +170053. q^{86} -259437. q^{88} +61443.6 q^{89} -110212. q^{91} +11214.9 q^{92} +105784. q^{94} +84282.2 q^{95} +170505. q^{97} -175585. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−10.4920	−1.85474	−0.927368	−	0.374150i	$-0.877935\pi$
$2$	−10.4920	−1.85474	−0.927368	+	0.374150i	$0.877935\pi$
$3$	0	0
$3$	0	0
$4$	78.0815	2.44005
$5$	−46.0665	−0.824062	−0.412031	−	0.911170i	$-0.635181\pi$
$5$	−46.0665	−0.824062	−0.412031	+	0.911170i	$0.635181\pi$
$6$	0	0
$7$	183.145	1.41270	0.706351	−	0.707862i	$-0.250340\pi$
$7$	183.145	1.41270	0.706351	+	0.707862i	$0.250340\pi$
$8$	−483.486	−2.67091
$9$	0	0
$10$	483.328	1.52842
$11$	536.597	1.33711	0.668554	−	0.743664i	$-0.266914\pi$
$11$	536.597	1.33711	0.668554	+	0.743664i	$0.266914\pi$
$12$	0	0
$13$	−601.772	−0.987583	−0.493792	−	0.869580i	$-0.664389\pi$
$13$	−601.772	−0.987583	−0.493792	+	0.869580i	$0.664389\pi$
$14$	−1921.55	−2.62019
$15$	0	0
$16$	2574.11	2.51378
$17$	1948.89	1.63555	0.817776	−	0.575536i	$-0.195207\pi$
$17$	1948.89	1.63555	0.817776	+	0.575536i	$0.195207\pi$
$18$	0	0
$19$	−1829.58	−1.16270	−0.581349	−	0.813654i	$-0.697475\pi$
$19$	−1829.58	−1.16270	−0.581349	+	0.813654i	$0.697475\pi$
$20$	−3596.94	−2.01075
$21$	0	0
$22$	−5629.96	−2.47998
$23$	143.630	0.0566143	0.0283071	−	0.999599i	$-0.490988\pi$
$23$	143.630	0.0566143	0.0283071	+	0.999599i	$0.490988\pi$
$24$	0	0
$25$	−1002.88	−0.320921
$26$	6313.78	1.83171
$27$	0	0
$28$	14300.3	3.44706
$29$	−4181.48	−0.923284	−0.461642	−	0.887066i	$-0.652739\pi$
$29$	−4181.48	−0.923284	−0.461642	+	0.887066i	$0.652739\pi$
$30$	0	0
$31$	6331.93	1.18340	0.591701	−	0.806158i	$-0.298457\pi$
$31$	6331.93	1.18340	0.591701	+	0.806158i	$0.298457\pi$
$32$	−11536.0	−1.99150
$33$	0	0
$34$	−20447.7	−3.03352
$35$	−8436.86	−1.16415
$36$	0	0
$37$	−1983.22	−0.238159	−0.119079	−	0.992885i	$-0.537994\pi$
$37$	−1983.22	−0.238159	−0.119079	+	0.992885i	$0.537994\pi$
$38$	19195.9	2.15650
$39$	0	0
$40$	22272.5	2.20099
$41$	−18079.8	−1.67970	−0.839852	−	0.542815i	$-0.817359\pi$
$41$	−18079.8	−1.67970	−0.839852	+	0.542815i	$0.817359\pi$
$42$	0	0
$43$	−16207.9	−1.33677	−0.668383	−	0.743817i	$-0.733013\pi$
$43$	−16207.9	−1.33677	−0.668383	+	0.743817i	$0.733013\pi$
$44$	41898.3	3.26261
$45$	0	0
$46$	−1506.96	−0.105005
$47$	−10082.4	−0.665763	−0.332881	−	0.942969i	$-0.608021\pi$
$47$	−10082.4	−0.665763	−0.332881	+	0.942969i	$0.608021\pi$
$48$	0	0
$49$	16735.2	0.995726
$50$	10522.2	0.595224
$51$	0	0
$52$	−46987.3	−2.40975
$53$	−17087.2	−0.835565	−0.417783	−	0.908547i	$-0.637193\pi$
$53$	−17087.2	−0.835565	−0.417783	+	0.908547i	$0.637193\pi$
$54$	0	0
$55$	−24719.1	−1.10186
$56$	−88548.1	−3.77320
$57$	0	0
$58$	43872.0	1.71245
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−10827.9	−0.372582	−0.186291	−	0.982495i	$-0.559647\pi$
$61$	−10827.9	−0.372582	−0.186291	+	0.982495i	$0.559647\pi$
$62$	−66434.5	−2.19490
$63$	0	0
$64$	38663.6	1.17992
$65$	27721.5	0.813830
$66$	0	0
$67$	53212.3	1.44819	0.724094	−	0.689701i	$-0.242258\pi$
$67$	53212.3	1.44819	0.724094	+	0.689701i	$0.242258\pi$
$68$	152172.	3.99082
$69$	0	0
$70$	88519.3	2.15920
$71$	82455.8	1.94122	0.970611	−	0.240653i	$-0.0773616\pi$
$71$	82455.8	1.94122	0.970611	+	0.240653i	$0.0773616\pi$
$72$	0	0
$73$	−25989.7	−0.570813	−0.285406	−	0.958407i	$-0.592129\pi$
$73$	−25989.7	−0.570813	−0.285406	+	0.958407i	$0.592129\pi$
$74$	20807.9	0.441722
$75$	0	0
$76$	−142856.	−2.83704
$77$	98275.1	1.88893
$78$	0	0
$79$	18631.5	0.335877	0.167938	−	0.985798i	$-0.446289\pi$
$79$	18631.5	0.335877	0.167938	+	0.985798i	$0.446289\pi$
$80$	−118580.	−2.07151
$81$	0	0
$82$	189692.	3.11541
$83$	−105042.	−1.67367	−0.836833	−	0.547458i	$-0.815596\pi$
$83$	−105042.	−1.67367	−0.836833	+	0.547458i	$0.815596\pi$
$84$	0	0
$85$	−89778.4	−1.34780
$86$	170053.	2.47935
$87$	0	0
$88$	−259437.	−3.57129
$89$	61443.6	0.822246	0.411123	−	0.911580i	$-0.365137\pi$
$89$	61443.6	0.822246	0.411123	+	0.911580i	$0.365137\pi$
$90$	0	0
$91$	−110212.	−1.39516
$92$	11214.9	0.138142
$93$	0	0
$94$	105784.	1.23481
$95$	84282.2	0.958136
$96$	0	0
$97$	170505.	1.83996	0.919980	−	0.391966i	$-0.128205\pi$
$97$	170505.	1.83996	0.919980	+	0.391966i	$0.128205\pi$
$98$	−175585.	−1.84681
$99$	0	0
$100$	−78306.3	−0.783063
$101$	−38589.6	−0.376415	−0.188208	−	0.982129i	$-0.560268\pi$
$101$	−38589.6	−0.376415	−0.188208	+	0.982129i	$0.560268\pi$
$102$	0	0
$103$	−8597.79	−0.0798534	−0.0399267	−	0.999203i	$-0.512712\pi$
$103$	−8597.79	−0.0798534	−0.0399267	+	0.999203i	$0.512712\pi$
$104$	290948.	2.63774
$105$	0	0
$106$	179278.	1.54975
$107$	45162.4	0.381345	0.190672	−	0.981654i	$-0.438933\pi$
$107$	45162.4	0.381345	0.190672	+	0.981654i	$0.438933\pi$
$108$	0	0
$109$	74420.5	0.599966	0.299983	−	0.953945i	$-0.403019\pi$
$109$	74420.5	0.599966	0.299983	+	0.953945i	$0.403019\pi$
$110$	259352.	2.04366
$111$	0	0
$112$	471437.	3.55123
$113$	−119180.	−0.878026	−0.439013	−	0.898481i	$-0.644672\pi$
$113$	−119180.	−0.878026	−0.439013	+	0.898481i	$0.644672\pi$
$114$	0	0
$115$	−6616.54	−0.0466537
$116$	−326496.	−2.25286
$117$	0	0
$118$	−36522.6	−0.241466
$119$	356929.	2.31055
$120$	0	0
$121$	126885.	0.787857
$122$	113607.	0.691041
$123$	0	0
$124$	494407.	2.88756
$125$	190157.	1.08852
$126$	0	0
$127$	187431.	1.03118	0.515588	−	0.856837i	$-0.327574\pi$
$127$	187431.	1.03118	0.515588	+	0.856837i	$0.327574\pi$
$128$	−36506.0	−0.196942
$129$	0	0
$130$	−290854.	−1.50944
$131$	−200431.	−1.02044	−0.510220	−	0.860044i	$-0.670436\pi$
$131$	−200431.	−1.02044	−0.510220	+	0.860044i	$0.670436\pi$
$132$	0	0
$133$	−335078.	−1.64255
$134$	−558302.	−2.68601
$135$	0	0
$136$	−942260.	−4.36841
$137$	−65393.6	−0.297669	−0.148834	−	0.988862i	$-0.547552\pi$
$137$	−65393.6	−0.297669	−0.148834	+	0.988862i	$0.547552\pi$
$138$	0	0
$139$	−102562.	−0.450245	−0.225123	−	0.974330i	$-0.572278\pi$
$139$	−102562.	−0.450245	−0.225123	+	0.974330i	$0.572278\pi$
$140$	−658762.	−2.84059
$141$	0	0
$142$	−865124.	−3.60046
$143$	−322909.	−1.32051
$144$	0	0
$145$	192626.	0.760843
$146$	272683.	1.05871
$147$	0	0
$148$	−154853.	−0.581119
$149$	−412586.	−1.52247	−0.761235	−	0.648476i	$-0.775407\pi$
$149$	−412586.	−1.52247	−0.761235	+	0.648476i	$0.775407\pi$
$150$	0	0
$151$	−344222.	−1.22856	−0.614280	−	0.789088i	$-0.710553\pi$
$151$	−344222.	−1.22856	−0.614280	+	0.789088i	$0.710553\pi$
$152$	884575.	3.10546
$153$	0	0
$154$	−1.03110e6	−3.50348
$155$	−291690.	−0.975197
$156$	0	0
$157$	−69151.6	−0.223900	−0.111950	−	0.993714i	$-0.535710\pi$
$157$	−69151.6	−0.223900	−0.111950	+	0.993714i	$0.535710\pi$
$158$	−195481.	−0.622963
$159$	0	0
$160$	531422.	1.64112
$161$	26305.2	0.0799791
$162$	0	0
$163$	−435248.	−1.28312	−0.641561	−	0.767072i	$-0.721713\pi$
$163$	−435248.	−1.28312	−0.641561	+	0.767072i	$0.721713\pi$
$164$	−1.41169e6	−4.09856
$165$	0	0
$166$	1.10210e6	3.10421
$167$	282385.	0.783521	0.391761	−	0.920067i	$-0.371866\pi$
$167$	282385.	0.783521	0.391761	+	0.920067i	$0.371866\pi$
$168$	0	0
$169$	−9163.31	−0.0246795
$170$	941952.	2.49981
$171$	0	0
$172$	−1.26554e6	−3.26177
$173$	542664.	1.37853	0.689265	−	0.724510i	$-0.257934\pi$
$173$	542664.	1.37853	0.689265	+	0.724510i	$0.257934\pi$
$174$	0	0
$175$	−183672.	−0.453366
$176$	1.38126e6	3.36120
$177$	0	0
$178$	−644665.	−1.52505
$179$	36141.3	0.0843084	0.0421542	−	0.999111i	$-0.486578\pi$
$179$	36141.3	0.0843084	0.0421542	+	0.999111i	$0.486578\pi$
$180$	0	0
$181$	−597328.	−1.35524	−0.677620	−	0.735412i	$-0.736989\pi$
$181$	−597328.	−1.35524	−0.677620	+	0.735412i	$0.736989\pi$
$182$	1.15634e6	2.58765
$183$	0	0
$184$	−69443.2	−0.151212
$185$	91360.0	0.196258
$186$	0	0
$187$	1.04577e6	2.18691
$188$	−787250.	−1.62449
$189$	0	0
$190$	−884287.	−1.77709
$191$	−653350.	−1.29587	−0.647936	−	0.761695i	$-0.724368\pi$
$191$	−653350.	−1.29587	−0.647936	+	0.761695i	$0.724368\pi$
$192$	0	0
$193$	−339340.	−0.655755	−0.327878	−	0.944720i	$-0.606333\pi$
$193$	−339340.	−0.655755	−0.327878	+	0.944720i	$0.606333\pi$
$194$	−1.78894e6	−3.41264
$195$	0	0
$196$	1.30671e6	2.42962
$197$	375530.	0.689411	0.344706	−	0.938711i	$-0.387979\pi$
$197$	375530.	0.689411	0.344706	+	0.938711i	$0.387979\pi$
$198$	0	0
$199$	−314088.	−0.562235	−0.281118	−	0.959673i	$-0.590705\pi$
$199$	−314088.	−0.562235	−0.281118	+	0.959673i	$0.590705\pi$
$200$	484878.	0.857151
$201$	0	0
$202$	404882.	0.698151
$203$	−765818.	−1.30432
$204$	0	0
$205$	832871.	1.38418
$206$	90207.8	0.148107
$207$	0	0
$208$	−1.54903e6	−2.48257
$209$	−981746.	−1.55465
$210$	0	0
$211$	−383881.	−0.593595	−0.296797	−	0.954941i	$-0.595919\pi$
$211$	−383881.	−0.593595	−0.296797	+	0.954941i	$0.595919\pi$
$212$	−1.33419e6	−2.03882
$213$	0	0
$214$	−473843.	−0.707294
$215$	746641.	1.10158
$216$	0	0
$217$	1.15966e6	1.67179
$218$	−780818.	−1.11278
$219$	0	0
$220$	−1.93011e6	−2.68859
$221$	−1.17279e6	−1.61524
$222$	0	0
$223$	−507820.	−0.683829	−0.341915	−	0.939731i	$-0.611075\pi$
$223$	−507820.	−0.683829	−0.341915	+	0.939731i	$0.611075\pi$
$224$	−2.11276e6	−2.81339
$225$	0	0
$226$	1.25043e6	1.62851
$227$	−1.23528e6	−1.59111	−0.795554	−	0.605882i	$-0.792820\pi$
$227$	−1.23528e6	−1.59111	−0.795554	+	0.605882i	$0.792820\pi$
$228$	0	0
$229$	941944.	1.18696	0.593481	−	0.804848i	$-0.297753\pi$
$229$	941944.	1.18696	0.593481	+	0.804848i	$0.297753\pi$
$230$	69420.5	0.0865303
$231$	0	0
$232$	2.02169e6	2.46601
$233$	−249966.	−0.301641	−0.150821	−	0.988561i	$-0.548192\pi$
$233$	−249966.	−0.301641	−0.150821	+	0.988561i	$0.548192\pi$
$234$	0	0
$235$	464461.	0.548630
$236$	271802.	0.317667
$237$	0	0
$238$	−3.74489e6	−4.28546
$239$	−1.41078e6	−1.59759	−0.798794	−	0.601604i	$-0.794528\pi$
$239$	−1.41078e6	−1.59759	−0.798794	+	0.601604i	$0.794528\pi$
$240$	0	0
$241$	25730.6	0.0285369	0.0142685	−	0.999898i	$-0.495458\pi$
$241$	25730.6	0.0285369	0.0142685	+	0.999898i	$0.495458\pi$
$242$	−1.33128e6	−1.46127
$243$	0	0
$244$	−845462.	−0.909117
$245$	−770930.	−0.820540
$246$	0	0
$247$	1.10099e6	1.14826
$248$	−3.06140e6	−3.16076
$249$	0	0
$250$	−1.99512e6	−2.01892
$251$	1.03590e6	1.03785	0.518925	−	0.854820i	$-0.326332\pi$
$251$	1.03590e6	1.03785	0.518925	+	0.854820i	$0.326332\pi$
$252$	0	0
$253$	77071.5	0.0756994
$254$	−1.96652e6	−1.91256
$255$	0	0
$256$	−854215.	−0.814643
$257$	−1.04974e6	−0.991403	−0.495702	−	0.868493i	$-0.665089\pi$
$257$	−1.04974e6	−0.991403	−0.495702	+	0.868493i	$0.665089\pi$
$258$	0	0
$259$	−363217.	−0.336447
$260$	2.16454e6	1.98578
$261$	0	0
$262$	2.10292e6	1.89265
$263$	1.38970e6	1.23889	0.619445	−	0.785040i	$-0.287358\pi$
$263$	1.38970e6	1.23889	0.619445	+	0.785040i	$0.287358\pi$
$264$	0	0
$265$	787146.	0.688558
$266$	3.51563e6	3.04649
$267$	0	0
$268$	4.15490e6	3.53365
$269$	1.65552e6	1.39494	0.697469	−	0.716615i	$-0.254310\pi$
$269$	1.65552e6	1.39494	0.697469	+	0.716615i	$0.254310\pi$
$270$	0	0
$271$	890523.	0.736583	0.368292	−	0.929710i	$-0.379943\pi$
$271$	890523.	0.736583	0.368292	+	0.929710i	$0.379943\pi$
$272$	5.01666e6	4.11142
$273$	0	0
$274$	686107.	0.552097
$275$	−538142.	−0.429106
$276$	0	0
$277$	−748295.	−0.585967	−0.292984	−	0.956117i	$-0.594648\pi$
$277$	−748295.	−0.585967	−0.292984	+	0.956117i	$0.594648\pi$
$278$	1.07608e6	0.835087
$279$	0	0
$280$	4.07910e6	3.10935
$281$	−110769.	−0.0836861	−0.0418430	−	0.999124i	$-0.513323\pi$
$281$	−110769.	−0.0836861	−0.0418430	+	0.999124i	$0.513323\pi$
$282$	0	0
$283$	−220603.	−0.163736	−0.0818682	−	0.996643i	$-0.526089\pi$
$283$	−220603.	−0.163736	−0.0818682	+	0.996643i	$0.526089\pi$
$284$	6.43827e6	4.73667
$285$	0	0
$286$	3.38795e6	2.44919
$287$	−3.31122e6	−2.37292
$288$	0	0
$289$	2.37830e6	1.67503
$290$	−2.02103e6	−1.41116
$291$	0	0
$292$	−2.02931e6	−1.39281
$293$	−1.55524e6	−1.05835	−0.529173	−	0.848514i	$-0.677498\pi$
$293$	−1.55524e6	−1.05835	−0.529173	+	0.848514i	$0.677498\pi$
$294$	0	0
$295$	−160357.	−0.107284
$296$	958859.	0.636100
$297$	0	0
$298$	4.32884e6	2.82378
$299$	−86432.6	−0.0559113
$300$	0	0
$301$	−2.96840e6	−1.88845
$302$	3.61157e6	2.27865
$303$	0	0
$304$	−4.70954e6	−2.92277
$305$	498805.	0.307031
$306$	0	0
$307$	−1.00187e6	−0.606686	−0.303343	−	0.952881i	$-0.598103\pi$
$307$	−1.00187e6	−0.606686	−0.303343	+	0.952881i	$0.598103\pi$
$308$	7.67347e6	4.60909
$309$	0	0
$310$	3.06040e6	1.80873
$311$	−2.66702e6	−1.56360	−0.781800	−	0.623529i	$-0.785698\pi$
$311$	−2.66702e6	−1.56360	−0.781800	+	0.623529i	$0.785698\pi$
$312$	0	0
$313$	1.88511e6	1.08762	0.543808	−	0.839210i	$-0.316982\pi$
$313$	1.88511e6	1.08762	0.543808	+	0.839210i	$0.316982\pi$
$314$	725537.	0.415275
$315$	0	0
$316$	1.45477e6	0.819555
$317$	2.77185e6	1.54925	0.774624	−	0.632422i	$-0.217939\pi$
$317$	2.77185e6	1.54925	0.774624	+	0.632422i	$0.217939\pi$
$318$	0	0
$319$	−2.24377e6	−1.23453
$320$	−1.78110e6	−0.972327
$321$	0	0
$322$	−275993.	−0.148340
$323$	−3.56564e6	−1.90165
$324$	0	0
$325$	603505.	0.316936
$326$	4.56661e6	2.37985
$327$	0	0
$328$	8.74131e6	4.48634
$329$	−1.84654e6	−0.940524
$330$	0	0
$331$	−3.55141e6	−1.78168	−0.890842	−	0.454312i	$-0.849885\pi$
$331$	−3.55141e6	−1.78168	−0.890842	+	0.454312i	$0.849885\pi$
$332$	−8.20186e6	−4.08383
$333$	0	0
$334$	−2.96278e6	−1.45323
$335$	−2.45130e6	−1.19340
$336$	0	0
$337$	3.41206e6	1.63660	0.818298	−	0.574794i	$-0.194918\pi$
$337$	3.41206e6	1.63660	0.818298	+	0.574794i	$0.194918\pi$
$338$	96141.2	0.0457739
$339$	0	0
$340$	−7.01003e6	−3.28869
$341$	3.39770e6	1.58234
$342$	0	0
$343$	−13155.6	−0.00603774
$344$	7.83629e6	3.57038
$345$	0	0
$346$	−5.69362e6	−2.55681
$347$	111495.	0.0497088	0.0248544	−	0.999691i	$-0.492088\pi$
$347$	111495.	0.0497088	0.0248544	+	0.999691i	$0.492088\pi$
$348$	0	0
$349$	−691993.	−0.304115	−0.152058	−	0.988372i	$-0.548590\pi$
$349$	−691993.	−0.304115	−0.152058	+	0.988372i	$0.548590\pi$
$350$	1.92709e6	0.840875
$351$	0	0
$352$	−6.19017e6	−2.66285
$353$	2.81613e6	1.20286	0.601432	−	0.798924i	$-0.294597\pi$
$353$	2.81613e6	1.20286	0.601432	+	0.798924i	$0.294597\pi$
$354$	0	0
$355$	−3.79845e6	−1.59969
$356$	4.79761e6	2.00632
$357$	0	0
$358$	−379193.	−0.156370
$359$	−2.09260e6	−0.856940	−0.428470	−	0.903556i	$-0.640947\pi$
$359$	−2.09260e6	−0.856940	−0.428470	+	0.903556i	$0.640947\pi$
$360$	0	0
$361$	871258.	0.351867
$362$	6.26715e6	2.51361
$363$	0	0
$364$	−8.60549e6	−3.40426
$365$	1.19725e6	0.470385
$366$	0	0
$367$	3.12044e6	1.20935	0.604674	−	0.796473i	$-0.293304\pi$
$367$	3.12044e6	1.20935	0.604674	+	0.796473i	$0.293304\pi$
$368$	369720.	0.142316
$369$	0	0
$370$	−958547.	−0.364006
$371$	−3.12943e6	−1.18040
$372$	0	0
$373$	−696892.	−0.259354	−0.129677	−	0.991556i	$-0.541394\pi$
$373$	−696892.	−0.259354	−0.129677	+	0.991556i	$0.541394\pi$
$374$	−1.09722e7	−4.05614
$375$	0	0
$376$	4.87470e6	1.77819
$377$	2.51630e6	0.911820
$378$	0	0
$379$	−5.00770e6	−1.79077	−0.895386	−	0.445291i	$-0.853100\pi$
$379$	−5.00770e6	−1.79077	−0.895386	+	0.445291i	$0.853100\pi$
$380$	6.58088e6	2.33790
$381$	0	0
$382$	6.85493e6	2.40350
$383$	−359356.	−0.125178	−0.0625890	−	0.998039i	$-0.519936\pi$
$383$	−359356.	−0.125178	−0.0625890	+	0.998039i	$0.519936\pi$
$384$	0	0
$385$	−4.52719e6	−1.55660
$386$	3.56035e6	1.21625
$387$	0	0
$388$	1.33133e7	4.48959
$389$	−2.58030e6	−0.864561	−0.432280	−	0.901739i	$-0.642291\pi$
$389$	−2.58030e6	−0.864561	−0.432280	+	0.901739i	$0.642291\pi$
$390$	0	0
$391$	279919.	0.0925956
$392$	−8.09122e6	−2.65949
$393$	0	0
$394$	−3.94005e6	−1.27868
$395$	−858287.	−0.276783
$396$	0	0
$397$	−2.84643e6	−0.906409	−0.453204	−	0.891407i	$-0.649719\pi$
$397$	−2.84643e6	−0.906409	−0.453204	+	0.891407i	$0.649719\pi$
$398$	3.29540e6	1.04280
$399$	0	0
$400$	−2.58153e6	−0.806727
$401$	2.13470e6	0.662944	0.331472	−	0.943465i	$-0.392455\pi$
$401$	2.13470e6	0.662944	0.331472	+	0.943465i	$0.392455\pi$
$402$	0	0
$403$	−3.81038e6	−1.16871
$404$	−3.01314e6	−0.918471
$405$	0	0
$406$	8.03495e6	2.41918
$407$	−1.06419e6	−0.318444
$408$	0	0
$409$	1.68398e6	0.497771	0.248885	−	0.968533i	$-0.419936\pi$
$409$	1.68398e6	0.497771	0.248885	+	0.968533i	$0.419936\pi$
$410$	−8.73846e6	−2.56729
$411$	0	0
$412$	−671328.	−0.194846
$413$	637528.	0.183918
$414$	0	0
$415$	4.83893e6	1.37921
$416$	6.94203e6	1.96677
$417$	0	0
$418$	1.03005e7	2.88347
$419$	−6.14268e6	−1.70932	−0.854659	−	0.519190i	$-0.826234\pi$
$419$	−6.14268e6	−1.70932	−0.854659	+	0.519190i	$0.826234\pi$
$420$	0	0
$421$	−1.00451e6	−0.276217	−0.138108	−	0.990417i	$-0.544102\pi$
$421$	−1.00451e6	−0.276217	−0.138108	+	0.990417i	$0.544102\pi$
$422$	4.02766e6	1.10096
$423$	0	0
$424$	8.26141e6	2.23172
$425$	−1.95450e6	−0.524884
$426$	0	0
$427$	−1.98309e6	−0.526347
$428$	3.52635e6	0.930499
$429$	0	0
$430$	−7.83374e6	−2.04314
$431$	−2.54886e6	−0.660927	−0.330464	−	0.943819i	$-0.607205\pi$
$431$	−2.54886e6	−0.660927	−0.330464	+	0.943819i	$0.607205\pi$
$432$	0	0
$433$	−4.98914e6	−1.27881	−0.639404	−	0.768871i	$-0.720819\pi$
$433$	−4.98914e6	−1.27881	−0.639404	+	0.768871i	$0.720819\pi$
$434$	−1.21672e7	−3.10074
$435$	0	0
$436$	5.81087e6	1.46395
$437$	−262783.	−0.0658253
$438$	0	0
$439$	−996560.	−0.246798	−0.123399	−	0.992357i	$-0.539380\pi$
$439$	−996560.	−0.246798	−0.123399	+	0.992357i	$0.539380\pi$
$440$	1.19514e7	2.94297
$441$	0	0
$442$	1.23048e7	2.99585
$443$	6.09227e6	1.47493	0.737463	−	0.675388i	$-0.236024\pi$
$443$	6.09227e6	1.47493	0.737463	+	0.675388i	$0.236024\pi$
$444$	0	0
$445$	−2.83049e6	−0.677582
$446$	5.32804e6	1.26832
$447$	0	0
$448$	7.08105e6	1.66687
$449$	−1.80157e6	−0.421732	−0.210866	−	0.977515i	$-0.567628\pi$
$449$	−1.80157e6	−0.421732	−0.210866	+	0.977515i	$0.567628\pi$
$450$	0	0
$451$	−9.70154e6	−2.24595
$452$	−9.30576e6	−2.14242
$453$	0	0
$454$	1.29605e7	2.95109
$455$	5.07706e6	1.14970
$456$	0	0
$457$	−3.51589e6	−0.787489	−0.393744	−	0.919220i	$-0.628820\pi$
$457$	−3.51589e6	−0.787489	−0.393744	+	0.919220i	$0.628820\pi$
$458$	−9.88286e6	−2.20150
$459$	0	0
$460$	−516629.	−0.113837
$461$	−6.94554e6	−1.52214	−0.761068	−	0.648672i	$-0.775325\pi$
$461$	−6.94554e6	−1.52214	−0.761068	+	0.648672i	$0.775325\pi$
$462$	0	0
$463$	−4.78918e6	−1.03827	−0.519133	−	0.854693i	$-0.673745\pi$
$463$	−4.78918e6	−1.03827	−0.519133	+	0.854693i	$0.673745\pi$
$464$	−1.07636e7	−2.32094
$465$	0	0
$466$	2.62264e6	0.559465
$467$	−2.62555e6	−0.557094	−0.278547	−	0.960423i	$-0.589853\pi$
$467$	−2.62555e6	−0.557094	−0.278547	+	0.960423i	$0.589853\pi$
$468$	0	0
$469$	9.74557e6	2.04586
$470$	−4.87311e6	−1.01756
$471$	0	0
$472$	−1.68301e6	−0.347723
$473$	−8.69711e6	−1.78740
$474$	0	0
$475$	1.83485e6	0.373135
$476$	2.78696e7	5.63784
$477$	0	0
$478$	1.48019e7	2.96311
$479$	2.10029e6	0.418255	0.209128	−	0.977888i	$-0.432938\pi$
$479$	2.10029e6	0.418255	0.209128	+	0.977888i	$0.432938\pi$
$480$	0	0
$481$	1.19345e6	0.235202
$482$	−269965.	−0.0529285
$483$	0	0
$484$	9.90739e6	1.92241
$485$	−7.85458e6	−1.51624
$486$	0	0
$487$	−3.94358e6	−0.753474	−0.376737	−	0.926320i	$-0.622954\pi$
$487$	−3.94358e6	−0.753474	−0.376737	+	0.926320i	$0.622954\pi$
$488$	5.23516e6	0.995131
$489$	0	0
$490$	8.08858e6	1.52189
$491$	−377373.	−0.0706427	−0.0353213	−	0.999376i	$-0.511245\pi$
$491$	−377373.	−0.0706427	−0.0353213	+	0.999376i	$0.511245\pi$
$492$	0	0
$493$	−8.14924e6	−1.51008
$494$	−1.15515e7	−2.12972
$495$	0	0
$496$	1.62991e7	2.97482
$497$	1.51014e7	2.74237
$498$	0	0
$499$	−2.96483e6	−0.533026	−0.266513	−	0.963831i	$-0.585872\pi$
$499$	−2.96483e6	−0.533026	−0.266513	+	0.963831i	$0.585872\pi$
$500$	1.48477e7	2.65604
$501$	0	0
$502$	−1.08687e7	−1.92494
$503$	3.33200e6	0.587198	0.293599	−	0.955929i	$-0.405147\pi$
$503$	3.33200e6	0.587198	0.293599	+	0.955929i	$0.405147\pi$
$504$	0	0
$505$	1.77769e6	0.310190
$506$	−808632.	−0.140402
$507$	0	0
$508$	1.46349e7	2.51612
$509$	3.15713e6	0.540129	0.270065	−	0.962842i	$-0.412955\pi$
$509$	3.15713e6	0.540129	0.270065	+	0.962842i	$0.412955\pi$
$510$	0	0
$511$	−4.75988e6	−0.806388
$512$	1.01306e7	1.70789
$513$	0	0
$514$	1.10139e7	1.83879
$515$	396070.	0.0658042
$516$	0	0
$517$	−5.41019e6	−0.890197
$518$	3.81087e6	0.624021
$519$	0	0
$520$	−1.34030e7	−2.17367
$521$	9.78221e6	1.57886	0.789429	−	0.613842i	$-0.210377\pi$
$521$	9.78221e6	1.57886	0.789429	+	0.613842i	$0.210377\pi$
$522$	0	0
$523$	1.09469e7	1.75000	0.874999	−	0.484124i	$-0.160862\pi$
$523$	1.09469e7	1.75000	0.874999	+	0.484124i	$0.160862\pi$
$524$	−1.56500e7	−2.48992
$525$	0	0
$526$	−1.45807e7	−2.29781
$527$	1.23402e7	1.93552
$528$	0	0
$529$	−6.41571e6	−0.996795
$530$	−8.25871e6	−1.27709
$531$	0	0
$532$	−2.61634e7	−4.00789
$533$	1.08799e7	1.65885
$534$	0	0
$535$	−2.08047e6	−0.314252
$536$	−2.57274e7	−3.86798
$537$	0	0
$538$	−1.73697e7	−2.58724
$539$	8.98004e6	1.33139
$540$	0	0
$541$	−9.12573e6	−1.34052	−0.670262	−	0.742125i	$-0.733818\pi$
$541$	−9.12573e6	−1.34052	−0.670262	+	0.742125i	$0.733818\pi$
$542$	−9.34334e6	−1.36617
$543$	0	0
$544$	−2.24823e7	−3.25720
$545$	−3.42829e6	−0.494409
$546$	0	0
$547$	6.24984e6	0.893100	0.446550	−	0.894759i	$-0.352652\pi$
$547$	6.24984e6	0.893100	0.446550	+	0.894759i	$0.352652\pi$
$548$	−5.10603e6	−0.726326
$549$	0	0
$550$	5.64617e6	0.795879
$551$	7.65035e6	1.07350
$552$	0	0
$553$	3.41227e6	0.474493
$554$	7.85109e6	1.08681
$555$	0	0
$556$	−8.00819e6	−1.09862
$557$	8.18033e6	1.11721	0.558603	−	0.829435i	$-0.311338\pi$
$557$	8.18033e6	1.11721	0.558603	+	0.829435i	$0.311338\pi$
$558$	0	0
$559$	9.75346e6	1.32017
$560$	−2.17174e7	−2.92643
$561$	0	0
$562$	1.16219e6	0.155216
$563$	−6.48080e6	−0.861704	−0.430852	−	0.902423i	$-0.641787\pi$
$563$	−6.48080e6	−0.861704	−0.430852	+	0.902423i	$0.641787\pi$
$564$	0	0
$565$	5.49020e6	0.723548
$566$	2.31456e6	0.303688
$567$	0	0
$568$	−3.98662e7	−5.18483
$569$	708193.	0.0917004	0.0458502	−	0.998948i	$-0.485400\pi$
$569$	708193.	0.0917004	0.0458502	+	0.998948i	$0.485400\pi$
$570$	0	0
$571$	4.84800e6	0.622260	0.311130	−	0.950367i	$-0.399292\pi$
$571$	4.84800e6	0.622260	0.311130	+	0.950367i	$0.399292\pi$
$572$	−2.52132e7	−3.22209
$573$	0	0
$574$	3.47412e7	4.40114
$575$	−144044.	−0.0181687
$576$	0	0
$577$	5.03751e6	0.629907	0.314954	−	0.949107i	$-0.398011\pi$
$577$	5.03751e6	0.629907	0.314954	+	0.949107i	$0.398011\pi$
$578$	−2.49531e7	−3.10674
$579$	0	0
$580$	1.50405e7	1.85649
$581$	−1.92380e7	−2.36439
$582$	0	0
$583$	−9.16892e6	−1.11724
$584$	1.25656e7	1.52459
$585$	0	0
$586$	1.63175e7	1.96295
$587$	714405.	0.0855755	0.0427878	−	0.999084i	$-0.486376\pi$
$587$	714405.	0.0855755	0.0427878	+	0.999084i	$0.486376\pi$
$588$	0	0
$589$	−1.15848e7	−1.37594
$590$	1.68247e6	0.198983
$591$	0	0
$592$	−5.10504e6	−0.598680
$593$	−1.42859e7	−1.66829	−0.834145	−	0.551545i	$-0.814038\pi$
$593$	−1.42859e7	−1.66829	−0.834145	+	0.551545i	$0.814038\pi$
$594$	0	0
$595$	−1.64425e7	−1.90404
$596$	−3.22153e7	−3.71490
$597$	0	0
$598$	906849.	0.103701
$599$	−1.22307e7	−1.39279	−0.696395	−	0.717659i	$-0.745214\pi$
$599$	−1.22307e7	−1.39279	−0.696395	+	0.717659i	$0.745214\pi$
$600$	0	0
$601$	1.62678e7	1.83714	0.918571	−	0.395257i	$-0.129344\pi$
$601$	1.62678e7	1.83714	0.918571	+	0.395257i	$0.129344\pi$
$602$	3.11444e7	3.50258
$603$	0	0
$604$	−2.68774e7	−2.99774
$605$	−5.84515e6	−0.649243
$606$	0	0
$607$	−4.79971e6	−0.528741	−0.264371	−	0.964421i	$-0.585164\pi$
$607$	−4.79971e6	−0.528741	−0.264371	+	0.964421i	$0.585164\pi$
$608$	2.11060e7	2.31551
$609$	0	0
$610$	−5.23345e6	−0.569461
$611$	6.06731e6	0.657496
$612$	0	0
$613$	7.16201e6	0.769810	0.384905	−	0.922956i	$-0.374234\pi$
$613$	7.16201e6	0.769810	0.384905	+	0.922956i	$0.374234\pi$
$614$	1.05116e7	1.12524
$615$	0	0
$616$	−4.75147e7	−5.04517
$617$	1.31807e6	0.139388	0.0696942	−	0.997568i	$-0.477798\pi$
$617$	1.31807e6	0.139388	0.0696942	+	0.997568i	$0.477798\pi$
$618$	0	0
$619$	8.49355e6	0.890969	0.445485	−	0.895290i	$-0.353031\pi$
$619$	8.49355e6	0.890969	0.445485	+	0.895290i	$0.353031\pi$
$620$	−2.27756e7	−2.37953
$621$	0	0
$622$	2.79823e7	2.90007
$623$	1.12531e7	1.16159
$624$	0	0
$625$	−5.62586e6	−0.576088
$626$	−1.97785e7	−2.01724
$627$	0	0
$628$	−5.39947e6	−0.546326
$629$	−3.86507e6	−0.389521
$630$	0	0
$631$	2.71872e6	0.271826	0.135913	−	0.990721i	$-0.456603\pi$
$631$	2.71872e6	0.271826	0.135913	+	0.990721i	$0.456603\pi$
$632$	−9.00806e6	−0.897096
$633$	0	0
$634$	−2.90821e7	−2.87345
$635$	−8.63430e6	−0.849753
$636$	0	0
$637$	−1.00708e7	−0.983362
$638$	2.35416e7	2.28973
$639$	0	0
$640$	1.68170e6	0.162293
$641$	9.82087e6	0.944071	0.472036	−	0.881579i	$-0.343519\pi$
$641$	9.82087e6	0.944071	0.472036	+	0.881579i	$0.343519\pi$
$642$	0	0
$643$	1.88105e7	1.79421	0.897103	−	0.441822i	$-0.145668\pi$
$643$	1.88105e7	1.79421	0.897103	+	0.441822i	$0.145668\pi$
$644$	2.05395e6	0.195153
$645$	0	0
$646$	3.74106e7	3.52707
$647$	−1.01896e7	−0.956964	−0.478482	−	0.878097i	$-0.658813\pi$
$647$	−1.01896e7	−0.956964	−0.478482	+	0.878097i	$0.658813\pi$
$648$	0	0
$649$	1.86789e6	0.174077
$650$	−6.33195e6	−0.587834
$651$	0	0
$652$	−3.39848e7	−3.13088
$653$	7.39824e6	0.678962	0.339481	−	0.940613i	$-0.389749\pi$
$653$	7.39824e6	0.678962	0.339481	+	0.940613i	$0.389749\pi$
$654$	0	0
$655$	9.23317e6	0.840907
$656$	−4.65394e7	−4.22241
$657$	0	0
$658$	1.93739e7	1.74442
$659$	−2.69270e6	−0.241532	−0.120766	−	0.992681i	$-0.538535\pi$
$659$	−2.69270e6	−0.241532	−0.120766	+	0.992681i	$0.538535\pi$
$660$	0	0
$661$	−1.86231e7	−1.65786	−0.828930	−	0.559352i	$-0.811050\pi$
$661$	−1.86231e7	−1.65786	−0.828930	+	0.559352i	$0.811050\pi$
$662$	3.72613e7	3.30456
$663$	0	0
$664$	5.07865e7	4.47021
$665$	1.54359e7	1.35356
$666$	0	0
$667$	−600587.	−0.0522711
$668$	2.20491e7	1.91183
$669$	0	0
$670$	2.57190e7	2.21344
$671$	−5.81024e6	−0.498182
$672$	0	0
$673$	−4.21307e6	−0.358559	−0.179280	−	0.983798i	$-0.557377\pi$
$673$	−4.21307e6	−0.358559	−0.179280	+	0.983798i	$0.557377\pi$
$674$	−3.57992e7	−3.03546
$675$	0	0
$676$	−715485.	−0.0602191
$677$	208397.	0.0174751	0.00873756	−	0.999962i	$-0.497219\pi$
$677$	208397.	0.0174751	0.00873756	+	0.999962i	$0.497219\pi$
$678$	0	0
$679$	3.12272e7	2.59931
$680$	4.34066e7	3.59984
$681$	0	0
$682$	−3.56485e7	−2.93482
$683$	4.15227e6	0.340591	0.170296	−	0.985393i	$-0.445528\pi$
$683$	4.15227e6	0.340591	0.170296	+	0.985393i	$0.445528\pi$
$684$	0	0
$685$	3.01245e6	0.245298
$686$	138028.	0.0111984
$687$	0	0
$688$	−4.17210e7	−3.36034
$689$	1.02826e7	0.825190
$690$	0	0
$691$	8.97043e6	0.714691	0.357345	−	0.933972i	$-0.383682\pi$
$691$	8.97043e6	0.714691	0.357345	+	0.933972i	$0.383682\pi$
$692$	4.23720e7	3.36368
$693$	0	0
$694$	−1.16981e6	−0.0921967
$695$	4.72467e6	0.371030
$696$	0	0
$697$	−3.52354e7	−2.74724
$698$	7.26037e6	0.564053
$699$	0	0
$700$	−1.43414e7	−1.10623
$701$	−1.23886e7	−0.952195	−0.476097	−	0.879393i	$-0.657949\pi$
$701$	−1.23886e7	−0.952195	−0.476097	+	0.879393i	$0.657949\pi$
$702$	0	0
$703$	3.62846e6	0.276907
$704$	2.07468e7	1.57768
$705$	0	0
$706$	−2.95468e7	−2.23100
$707$	−7.06751e6	−0.531763
$708$	0	0
$709$	−7.98264e6	−0.596391	−0.298195	−	0.954505i	$-0.596385\pi$
$709$	−7.98264e6	−0.596391	−0.298195	+	0.954505i	$0.596385\pi$
$710$	3.98532e7	2.96700
$711$	0	0
$712$	−2.97071e7	−2.19614
$713$	909457.	0.0669974
$714$	0	0
$715$	1.48753e7	1.08818
$716$	2.82197e6	0.205717
$717$	0	0
$718$	2.19555e7	1.58940
$719$	−1.40569e7	−1.01407	−0.507035	−	0.861926i	$-0.669258\pi$
$719$	−1.40569e7	−1.01407	−0.507035	+	0.861926i	$0.669258\pi$
$720$	0	0
$721$	−1.57464e6	−0.112809
$722$	−9.14121e6	−0.652621
$723$	0	0
$724$	−4.66403e7	−3.30685
$725$	4.19352e6	0.296301
$726$	0	0
$727$	−1.08044e7	−0.758166	−0.379083	−	0.925363i	$-0.623760\pi$
$727$	−1.08044e7	−0.758166	−0.379083	+	0.925363i	$0.623760\pi$
$728$	5.32858e7	3.72635
$729$	0	0
$730$	−1.25615e7	−0.872440
$731$	−3.15874e7	−2.18635
$732$	0	0
$733$	−1.13233e7	−0.778419	−0.389210	−	0.921149i	$-0.627252\pi$
$733$	−1.13233e7	−0.778419	−0.389210	+	0.921149i	$0.627252\pi$
$734$	−3.27396e7	−2.24302
$735$	0	0
$736$	−1.65692e6	−0.112747
$737$	2.85535e7	1.93638
$738$	0	0
$739$	7.10841e6	0.478808	0.239404	−	0.970920i	$-0.423048\pi$
$739$	7.10841e6	0.478808	0.239404	+	0.970920i	$0.423048\pi$
$740$	7.13353e6	0.478878
$741$	0	0
$742$	3.28339e7	2.18934
$743$	−2.17570e7	−1.44586	−0.722932	−	0.690919i	$-0.757206\pi$
$743$	−2.17570e7	−1.44586	−0.722932	+	0.690919i	$0.757206\pi$
$744$	0	0
$745$	1.90064e7	1.25461
$746$	7.31178e6	0.481034
$747$	0	0
$748$	8.16551e7	5.33616
$749$	8.27128e6	0.538726
$750$	0	0
$751$	1.14558e7	0.741180	0.370590	−	0.928797i	$-0.379155\pi$
$751$	1.14558e7	0.741180	0.370590	+	0.928797i	$0.379155\pi$
$752$	−2.59533e7	−1.67358
$753$	0	0
$754$	−2.64009e7	−1.69118
$755$	1.58571e7	1.01241
$756$	0	0
$757$	−2.26799e7	−1.43847	−0.719236	−	0.694766i	$-0.755508\pi$
$757$	−2.26799e7	−1.43847	−0.719236	+	0.694766i	$0.755508\pi$
$758$	5.25407e7	3.32141
$759$	0	0
$760$	−4.07493e7	−2.55909
$761$	−1.63222e7	−1.02169	−0.510844	−	0.859673i	$-0.670667\pi$
$761$	−1.63222e7	−1.02169	−0.510844	+	0.859673i	$0.670667\pi$
$762$	0	0
$763$	1.36298e7	0.847573
$764$	−5.10145e7	−3.16199
$765$	0	0
$766$	3.77035e6	0.232172
$767$	−2.09477e6	−0.128572
$768$	0	0
$769$	−2.90386e7	−1.77076	−0.885379	−	0.464870i	$-0.846101\pi$
$769$	−2.90386e7	−1.77076	−0.885379	+	0.464870i	$0.846101\pi$
$770$	4.74992e7	2.88708
$771$	0	0
$772$	−2.64962e7	−1.60007
$773$	−7.51073e6	−0.452099	−0.226050	−	0.974116i	$-0.572581\pi$
$773$	−7.51073e6	−0.452099	−0.226050	+	0.974116i	$0.572581\pi$
$774$	0	0
$775$	−6.35016e6	−0.379779
$776$	−8.24369e7	−4.91436
$777$	0	0
$778$	2.70724e7	1.60353
$779$	3.30783e7	1.95299
$780$	0	0
$781$	4.42455e7	2.59562
$782$	−2.93690e6	−0.171740
$783$	0	0
$784$	4.30782e7	2.50304
$785$	3.18557e6	0.184507
$786$	0	0
$787$	2.97607e7	1.71280	0.856399	−	0.516315i	$-0.172697\pi$
$787$	2.97607e7	1.71280	0.856399	+	0.516315i	$0.172697\pi$
$788$	2.93219e7	1.68220
$789$	0	0
$790$	9.00513e6	0.513360
$791$	−2.18272e7	−1.24039
$792$	0	0
$793$	6.51596e6	0.367955
$794$	2.98647e7	1.68115
$795$	0	0
$796$	−2.45244e7	−1.37188
$797$	−1.18682e7	−0.661817	−0.330909	−	0.943663i	$-0.607355\pi$
$797$	−1.18682e7	−0.661817	−0.330909	+	0.943663i	$0.607355\pi$
$798$	0	0
$799$	−1.96495e7	−1.08889
$800$	1.15692e7	0.639114
$801$	0	0
$802$	−2.23973e7	−1.22959
$803$	−1.39460e7	−0.763238
$804$	0	0
$805$	−1.21179e6	−0.0659078
$806$	3.99784e7	2.16764
$807$	0	0
$808$	1.86576e7	1.00537
$809$	1.20080e7	0.645057	0.322528	−	0.946560i	$-0.395467\pi$
$809$	1.20080e7	0.645057	0.322528	+	0.946560i	$0.395467\pi$
$810$	0	0
$811$	3.34804e7	1.78747	0.893734	−	0.448597i	$-0.148076\pi$
$811$	3.34804e7	1.78747	0.893734	+	0.448597i	$0.148076\pi$
$812$	−5.97963e7	−3.18261
$813$	0	0
$814$	1.11655e7	0.590630
$815$	2.00503e7	1.05737
$816$	0	0
$817$	2.96536e7	1.55426
$818$	−1.76683e7	−0.923233
$819$	0	0
$820$	6.50318e7	3.37747
$821$	2.19093e7	1.13441	0.567207	−	0.823575i	$-0.308024\pi$
$821$	2.19093e7	1.13441	0.567207	+	0.823575i	$0.308024\pi$
$822$	0	0
$823$	−1.77211e7	−0.911994	−0.455997	−	0.889981i	$-0.650717\pi$
$823$	−1.77211e7	−0.911994	−0.455997	+	0.889981i	$0.650717\pi$
$824$	4.15691e6	0.213281
$825$	0	0
$826$	−6.68893e6	−0.341120
$827$	−1.64575e7	−0.836759	−0.418380	−	0.908272i	$-0.637402\pi$
$827$	−1.64575e7	−0.836759	−0.418380	+	0.908272i	$0.637402\pi$
$828$	0	0
$829$	−7.42705e6	−0.375345	−0.187672	−	0.982232i	$-0.560094\pi$
$829$	−7.42705e6	−0.375345	−0.187672	+	0.982232i	$0.560094\pi$
$830$	−5.07699e7	−2.55806
$831$	0	0
$832$	−2.32667e7	−1.16527
$833$	3.26150e7	1.62856
$834$	0	0
$835$	−1.30085e7	−0.645670
$836$	−7.66562e7	−3.79343
$837$	0	0
$838$	6.44488e7	3.17033
$839$	−728562.	−0.0357324	−0.0178662	−	0.999840i	$-0.505687\pi$
$839$	−728562.	−0.0357324	−0.0178662	+	0.999840i	$0.505687\pi$
$840$	0	0
$841$	−3.02636e6	−0.147547
$842$	1.05393e7	0.512309
$843$	0	0
$844$	−2.99740e7	−1.44840
$845$	422122.	0.0203374
$846$	0	0
$847$	2.32384e7	1.11301
$848$	−4.39843e7	−2.10043
$849$	0	0
$850$	2.05065e7	0.973521
$851$	−284850.	−0.0134832
$852$	0	0
$853$	325772.	0.0153300	0.00766499	−	0.999971i	$-0.497560\pi$
$853$	325772.	0.0153300	0.00766499	+	0.999971i	$0.497560\pi$
$854$	2.08065e7	0.976235
$855$	0	0
$856$	−2.18354e7	−1.01854
$857$	−6.77652e6	−0.315177	−0.157589	−	0.987505i	$-0.550372\pi$
$857$	−6.77652e6	−0.315177	−0.157589	+	0.987505i	$0.550372\pi$
$858$	0	0
$859$	−1.74661e7	−0.807631	−0.403815	−	0.914840i	$-0.632316\pi$
$859$	−1.74661e7	−0.807631	−0.403815	+	0.914840i	$0.632316\pi$
$860$	5.82989e7	2.68790
$861$	0	0
$862$	2.67426e7	1.22585
$863$	1.22281e7	0.558897	0.279449	−	0.960161i	$-0.409848\pi$
$863$	1.22281e7	0.558897	0.279449	+	0.960161i	$0.409848\pi$
$864$	0	0
$865$	−2.49986e7	−1.13599
$866$	5.23459e7	2.37185
$867$	0	0
$868$	9.05483e7	4.07926
$869$	9.99760e6	0.449103
$870$	0	0
$871$	−3.20217e7	−1.43021
$872$	−3.59813e7	−1.60245
$873$	0	0
$874$	2.75711e6	0.122089
$875$	3.48263e7	1.53776
$876$	0	0
$877$	−2.26822e7	−0.995834	−0.497917	−	0.867225i	$-0.665902\pi$
$877$	−2.26822e7	−0.995834	−0.497917	+	0.867225i	$0.665902\pi$
$878$	1.04559e7	0.457746
$879$	0	0
$880$	−6.36299e7	−2.76984
$881$	1.07389e7	0.466145	0.233073	−	0.972459i	$-0.425122\pi$
$881$	1.07389e7	0.466145	0.233073	+	0.972459i	$0.425122\pi$
$882$	0	0
$883$	1.63107e7	0.703997	0.351999	−	0.936001i	$-0.385502\pi$
$883$	1.63107e7	0.703997	0.351999	+	0.936001i	$0.385502\pi$
$884$	−9.15729e7	−3.94127
$885$	0	0
$886$	−6.39200e7	−2.73560
$887$	−1.28289e6	−0.0547493	−0.0273747	−	0.999625i	$-0.508715\pi$
$887$	−1.28289e6	−0.0547493	−0.0273747	+	0.999625i	$0.508715\pi$
$888$	0	0
$889$	3.43271e7	1.45674
$890$	2.96974e7	1.25674
$891$	0	0
$892$	−3.96514e7	−1.66858
$893$	1.84466e7	0.774081
$894$	0	0
$895$	−1.66490e6	−0.0694754
$896$	−6.68590e6	−0.278221
$897$	0	0
$898$	1.89021e7	0.782201
$899$	−2.64769e7	−1.09262
$900$	0	0
$901$	−3.33010e7	−1.36661
$902$	1.01788e8	4.16564
$903$	0	0
$904$	5.76219e7	2.34513
$905$	2.75168e7	1.11680
$906$	0	0
$907$	−2.80898e7	−1.13379	−0.566893	−	0.823791i	$-0.691855\pi$
$907$	−2.80898e7	−1.13379	−0.566893	+	0.823791i	$0.691855\pi$
$908$	−9.64523e7	−3.88238
$909$	0	0
$910$	−5.32684e7	−2.13239
$911$	−3.24554e7	−1.29566	−0.647829	−	0.761785i	$-0.724323\pi$
$911$	−3.24554e7	−1.29566	−0.647829	+	0.761785i	$0.724323\pi$
$912$	0	0
$913$	−5.63654e7	−2.23787
$914$	3.68886e7	1.46058
$915$	0	0
$916$	7.35484e7	2.89624
$917$	−3.67081e7	−1.44158
$918$	0	0
$919$	−2.73570e7	−1.06851	−0.534256	−	0.845323i	$-0.679408\pi$
$919$	−2.73570e7	−1.06851	−0.534256	+	0.845323i	$0.679408\pi$
$920$	3.19900e6	0.124608
$921$	0	0
$922$	7.28724e7	2.82316
$923$	−4.96196e7	−1.91712
$924$	0	0
$925$	1.98893e6	0.0764302
$926$	5.02480e7	1.92571
$927$	0	0
$928$	4.82375e7	1.83872
$929$	2.68916e7	1.02230	0.511148	−	0.859493i	$-0.329220\pi$
$929$	2.68916e7	1.02230	0.511148	+	0.859493i	$0.329220\pi$
$930$	0	0
$931$	−3.06183e7	−1.15773
$932$	−1.95177e7	−0.736019
$933$	0	0
$934$	2.75472e7	1.03326
$935$	−4.81748e7	−1.80215
$936$	0	0
$937$	−3.27080e7	−1.21704	−0.608520	−	0.793538i	$-0.708237\pi$
$937$	−3.27080e7	−1.21704	−0.608520	+	0.793538i	$0.708237\pi$
$938$	−1.02250e8	−3.79453
$939$	0	0
$940$	3.62658e7	1.33868
$941$	−2.24893e7	−0.827947	−0.413973	−	0.910289i	$-0.635859\pi$
$941$	−2.24893e7	−0.827947	−0.413973	+	0.910289i	$0.635859\pi$
$942$	0	0
$943$	−2.59680e6	−0.0950953
$944$	8.96049e6	0.327267
$945$	0	0
$946$	9.12498e7	3.31516
$947$	2.60237e7	0.942963	0.471482	−	0.881876i	$-0.343719\pi$
$947$	2.60237e7	0.942963	0.471482	+	0.881876i	$0.343719\pi$
$948$	0	0
$949$	1.56399e7	0.563725
$950$	−1.92512e7	−0.692066
$951$	0	0
$952$	−1.72570e8	−6.17126
$953$	4.54892e7	1.62247	0.811233	−	0.584722i	$-0.198797\pi$
$953$	4.54892e7	1.62247	0.811233	+	0.584722i	$0.198797\pi$
$954$	0	0
$955$	3.00975e7	1.06788
$956$	−1.10156e8	−3.89819
$957$	0	0
$958$	−2.20362e7	−0.775753
$959$	−1.19765e7	−0.420517
$960$	0	0
$961$	1.14642e7	0.400439
$962$	−1.25216e7	−0.436237
$963$	0	0
$964$	2.00908e6	0.0696315
$965$	1.56322e7	0.540383
$966$	0	0
$967$	3.16445e7	1.08826	0.544129	−	0.839002i	$-0.316860\pi$
$967$	3.16445e7	1.08826	0.544129	+	0.839002i	$0.316860\pi$
$968$	−6.13472e7	−2.10429
$969$	0	0
$970$	8.24100e7	2.81223
$971$	2.31685e7	0.788587	0.394293	−	0.918985i	$-0.370989\pi$
$971$	2.31685e7	0.788587	0.394293	+	0.918985i	$0.370989\pi$
$972$	0	0
$973$	−1.87837e7	−0.636063
$974$	4.13760e7	1.39750
$975$	0	0
$976$	−2.78724e7	−0.936590
$977$	1.39480e7	0.467492	0.233746	−	0.972298i	$-0.424902\pi$
$977$	1.39480e7	0.467492	0.233746	+	0.972298i	$0.424902\pi$
$978$	0	0
$979$	3.29704e7	1.09943
$980$	−6.01954e7	−2.00216
$981$	0	0
$982$	3.95939e6	0.131024
$983$	9.57136e6	0.315929	0.157965	−	0.987445i	$-0.449507\pi$
$983$	9.57136e6	0.315929	0.157965	+	0.987445i	$0.449507\pi$
$984$	0	0
$985$	−1.72993e7	−0.568118
$986$	8.55016e7	2.80080
$987$	0	0
$988$	8.59669e7	2.80181
$989$	−2.32794e6	−0.0756801
$990$	0	0
$991$	−3.03725e7	−0.982417	−0.491209	−	0.871042i	$-0.663445\pi$
$991$	−3.03725e7	−0.982417	−0.491209	+	0.871042i	$0.663445\pi$
$992$	−7.30451e7	−2.35674
$993$	0	0
$994$	−1.58443e8	−5.08637
$995$	1.44689e7	0.463317
$996$	0	0
$997$	−2.69504e7	−0.858671	−0.429336	−	0.903145i	$-0.641252\pi$
$997$	−2.69504e7	−0.858671	−0.429336	+	0.903145i	$0.641252\pi$
$998$	3.11069e7	0.988623
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.c.1.2		12
3.2	odd	2		177.6.a.c.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.c.1.11	✓	12	3.2	odd	2
531.6.a.c.1.2		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-10.4920 q^{2} +78.0815 q^{4} -46.0665 q^{5} +183.145 q^{7} -483.486 q^{8} +O(q^{10})\)	\(q-10.4920 q^{2} +78.0815 q^{4} -46.0665 q^{5} +183.145 q^{7} -483.486 q^{8} +483.328 q^{10} +536.597 q^{11} -601.772 q^{13} -1921.55 q^{14} +2574.11 q^{16} +1948.89 q^{17} -1829.58 q^{19} -3596.94 q^{20} -5629.96 q^{22} +143.630 q^{23} -1002.88 q^{25} +6313.78 q^{26} +14300.3 q^{28} -4181.48 q^{29} +6331.93 q^{31} -11536.0 q^{32} -20447.7 q^{34} -8436.86 q^{35} -1983.22 q^{37} +19195.9 q^{38} +22272.5 q^{40} -18079.8 q^{41} -16207.9 q^{43} +41898.3 q^{44} -1506.96 q^{46} -10082.4 q^{47} +16735.2 q^{49} +10522.2 q^{50} -46987.3 q^{52} -17087.2 q^{53} -24719.1 q^{55} -88548.1 q^{56} +43872.0 q^{58} +3481.00 q^{59} -10827.9 q^{61} -66434.5 q^{62} +38663.6 q^{64} +27721.5 q^{65} +53212.3 q^{67} +152172. q^{68} +88519.3 q^{70} +82455.8 q^{71} -25989.7 q^{73} +20807.9 q^{74} -142856. q^{76} +98275.1 q^{77} +18631.5 q^{79} -118580. q^{80} +189692. q^{82} -105042. q^{83} -89778.4 q^{85} +170053. q^{86} -259437. q^{88} +61443.6 q^{89} -110212. q^{91} +11214.9 q^{92} +105784. q^{94} +84282.2 q^{95} +170505. q^{97} -175585. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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