LMFDB - Embedded newform 531.6.a.c.1.12

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			1.12
Root			$12.3715$ 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)

$f(q)$	$=$	$q+10.3715 q^{2} +75.5670 q^{4} -33.2592 q^{5} -56.9512 q^{7} +451.853 q^{8} +O(q^{10})$	\(q+10.3715 q^{2} +75.5670 q^{4} -33.2592 q^{5} -56.9512 q^{7} +451.853 q^{8} -344.946 q^{10} -432.466 q^{11} +178.063 q^{13} -590.667 q^{14} +2268.23 q^{16} -1658.66 q^{17} -168.835 q^{19} -2513.30 q^{20} -4485.30 q^{22} -143.318 q^{23} -2018.83 q^{25} +1846.77 q^{26} -4303.63 q^{28} -275.265 q^{29} -2273.75 q^{31} +9065.52 q^{32} -17202.7 q^{34} +1894.15 q^{35} -1120.19 q^{37} -1751.06 q^{38} -15028.3 q^{40} +2114.42 q^{41} +3840.06 q^{43} -32680.2 q^{44} -1486.42 q^{46} -14809.7 q^{47} -13563.6 q^{49} -20938.2 q^{50} +13455.7 q^{52} -9651.99 q^{53} +14383.5 q^{55} -25733.6 q^{56} -2854.90 q^{58} +3481.00 q^{59} +29254.1 q^{61} -23582.1 q^{62} +21439.3 q^{64} -5922.23 q^{65} +15092.5 q^{67} -125340. q^{68} +19645.1 q^{70} -13271.8 q^{71} +42497.1 q^{73} -11618.0 q^{74} -12758.3 q^{76} +24629.5 q^{77} -55585.7 q^{79} -75439.4 q^{80} +21929.6 q^{82} -81043.5 q^{83} +55165.7 q^{85} +39827.0 q^{86} -195411. q^{88} -107408. q^{89} -10140.9 q^{91} -10830.1 q^{92} -153598. q^{94} +5615.31 q^{95} -92436.4 q^{97} -140674. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	10.3715	1.83343	0.916715	−	0.399541i	$-0.130830\pi$
$2$	10.3715	1.83343	0.916715	+	0.399541i	$0.130830\pi$
$3$	0	0
$3$	0	0
$4$	75.5670	2.36147
$5$	−33.2592	−0.594958	−0.297479	−	0.954728i	$-0.596146\pi$
$5$	−33.2592	−0.594958	−0.297479	+	0.954728i	$0.596146\pi$
$6$	0	0
$7$	−56.9512	−0.439297	−0.219648	−	0.975579i	$-0.570491\pi$
$7$	−56.9512	−0.439297	−0.219648	+	0.975579i	$0.570491\pi$
$8$	451.853	2.49616
$9$	0	0
$10$	−344.946	−1.09081
$11$	−432.466	−1.07763	−0.538816	−	0.842423i	$-0.681128\pi$
$11$	−432.466	−1.07763	−0.538816	+	0.842423i	$0.681128\pi$
$12$	0	0
$13$	178.063	0.292224	0.146112	−	0.989268i	$-0.453324\pi$
$13$	178.063	0.292224	0.146112	+	0.989268i	$0.453324\pi$
$14$	−590.667	−0.805420
$15$	0	0
$16$	2268.23	2.21507
$17$	−1658.66	−1.39199	−0.695993	−	0.718048i	$-0.745036\pi$
$17$	−1658.66	−1.39199	−0.695993	+	0.718048i	$0.745036\pi$
$18$	0	0
$19$	−168.835	−0.107295	−0.0536473	−	0.998560i	$-0.517085\pi$
$19$	−168.835	−0.107295	−0.0536473	+	0.998560i	$0.517085\pi$
$20$	−2513.30	−1.40498
$21$	0	0
$22$	−4485.30	−1.97576
$23$	−143.318	−0.0564914	−0.0282457	−	0.999601i	$-0.508992\pi$
$23$	−143.318	−0.0564914	−0.0282457	+	0.999601i	$0.508992\pi$
$24$	0	0
$25$	−2018.83	−0.646025
$26$	1846.77	0.535772
$27$	0	0
$28$	−4303.63	−1.03739
$29$	−275.265	−0.0607794	−0.0303897	−	0.999538i	$-0.509675\pi$
$29$	−275.265	−0.0607794	−0.0303897	+	0.999538i	$0.509675\pi$
$30$	0	0
$31$	−2273.75	−0.424950	−0.212475	−	0.977166i	$-0.568152\pi$
$31$	−2273.75	−0.424950	−0.212475	+	0.977166i	$0.568152\pi$
$32$	9065.52	1.56501
$33$	0	0
$34$	−17202.7	−2.55211
$35$	1894.15	0.261363
$36$	0	0
$37$	−1120.19	−0.134520	−0.0672601	−	0.997735i	$-0.521426\pi$
$37$	−1120.19	−0.134520	−0.0672601	+	0.997735i	$0.521426\pi$
$38$	−1751.06	−0.196717
$39$	0	0
$40$	−15028.3	−1.48511
$41$	2114.42	0.196441	0.0982203	−	0.995165i	$-0.468685\pi$
$41$	2114.42	0.196441	0.0982203	+	0.995165i	$0.468685\pi$
$42$	0	0
$43$	3840.06	0.316714	0.158357	−	0.987382i	$-0.449380\pi$
$43$	3840.06	0.316714	0.158357	+	0.987382i	$0.449380\pi$
$44$	−32680.2	−2.54479
$45$	0	0
$46$	−1486.42	−0.103573
$47$	−14809.7	−0.977917	−0.488959	−	0.872307i	$-0.662623\pi$
$47$	−14809.7	−0.977917	−0.488959	+	0.872307i	$0.662623\pi$
$48$	0	0
$49$	−13563.6	−0.807019
$50$	−20938.2	−1.18444
$51$	0	0
$52$	13455.7	0.690077
$53$	−9651.99	−0.471984	−0.235992	−	0.971755i	$-0.575834\pi$
$53$	−9651.99	−0.471984	−0.235992	+	0.971755i	$0.575834\pi$
$54$	0	0
$55$	14383.5	0.641146
$56$	−25733.6	−1.09655
$57$	0	0
$58$	−2854.90	−0.111435
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	29254.1	1.00661	0.503307	−	0.864108i	$-0.332117\pi$
$61$	29254.1	1.00661	0.503307	+	0.864108i	$0.332117\pi$
$62$	−23582.1	−0.779117
$63$	0	0
$64$	21439.3	0.654275
$65$	−5922.23	−0.173861
$66$	0	0
$67$	15092.5	0.410747	0.205374	−	0.978684i	$-0.434159\pi$
$67$	15092.5	0.410747	0.205374	+	0.978684i	$0.434159\pi$
$68$	−125340.	−3.28713
$69$	0	0
$70$	19645.1	0.479191
$71$	−13271.8	−0.312453	−0.156226	−	0.987721i	$-0.549933\pi$
$71$	−13271.8	−0.312453	−0.156226	+	0.987721i	$0.549933\pi$
$72$	0	0
$73$	42497.1	0.933366	0.466683	−	0.884425i	$-0.345449\pi$
$73$	42497.1	0.933366	0.466683	+	0.884425i	$0.345449\pi$
$74$	−11618.0	−0.246634
$75$	0	0
$76$	−12758.3	−0.253373
$77$	24629.5	0.473400
$78$	0	0
$79$	−55585.7	−1.00206	−0.501032	−	0.865429i	$-0.667046\pi$
$79$	−55585.7	−1.00206	−0.501032	+	0.865429i	$0.667046\pi$
$80$	−75439.4	−1.31787
$81$	0	0
$82$	21929.6	0.360160
$83$	−81043.5	−1.29129	−0.645643	−	0.763639i	$-0.723411\pi$
$83$	−81043.5	−1.29129	−0.645643	+	0.763639i	$0.723411\pi$
$84$	0	0
$85$	55165.7	0.828174
$86$	39827.0	0.580673
$87$	0	0
$88$	−195411.	−2.68994
$89$	−107408.	−1.43734	−0.718672	−	0.695349i	$-0.755250\pi$
$89$	−107408.	−1.43734	−0.718672	+	0.695349i	$0.755250\pi$
$90$	0	0
$91$	−10140.9	−0.128373
$92$	−10830.1	−0.133403
$93$	0	0
$94$	−153598.	−1.79294
$95$	5615.31	0.0638358
$96$	0	0
$97$	−92436.4	−0.997502	−0.498751	−	0.866745i	$-0.666208\pi$
$97$	−92436.4	−0.997502	−0.498751	+	0.866745i	$0.666208\pi$
$98$	−140674.	−1.47961
$99$	0	0
$100$	−152557.	−1.52557
$101$	134769.	1.31458	0.657291	−	0.753637i	$-0.271702\pi$
$101$	134769.	1.31458	0.657291	+	0.753637i	$0.271702\pi$
$102$	0	0
$103$	166146.	1.54311	0.771553	−	0.636165i	$-0.219480\pi$
$103$	166146.	1.54311	0.771553	+	0.636165i	$0.219480\pi$
$104$	80458.3	0.729437
$105$	0	0
$106$	−100105.	−0.865350
$107$	−120010.	−1.01335	−0.506674	−	0.862138i	$-0.669125\pi$
$107$	−120010.	−1.01335	−0.506674	+	0.862138i	$0.669125\pi$
$108$	0	0
$109$	188031.	1.51587	0.757937	−	0.652328i	$-0.226207\pi$
$109$	188031.	1.51587	0.757937	+	0.652328i	$0.226207\pi$
$110$	149177.	1.17550
$111$	0	0
$112$	−129178.	−0.973071
$113$	−114073.	−0.840401	−0.420201	−	0.907431i	$-0.638040\pi$
$113$	−114073.	−0.840401	−0.420201	+	0.907431i	$0.638040\pi$
$114$	0	0
$115$	4766.65	0.0336100
$116$	−20801.0	−0.143529
$117$	0	0
$118$	36103.0	0.238692
$119$	94462.7	0.611495
$120$	0	0
$121$	25976.1	0.161291
$122$	303408.	1.84556
$123$	0	0
$124$	−171820.	−1.00351
$125$	171079.	0.979316
$126$	0	0
$127$	305300.	1.67964	0.839822	−	0.542861i	$-0.182659\pi$
$127$	305300.	1.67964	0.839822	+	0.542861i	$0.182659\pi$
$128$	−67740.1	−0.365444
$129$	0	0
$130$	−61422.1	−0.318762
$131$	152752.	0.777695	0.388848	−	0.921302i	$-0.372873\pi$
$131$	152752.	0.777695	0.388848	+	0.921302i	$0.372873\pi$
$132$	0	0
$133$	9615.35	0.0471342
$134$	156531.	0.753077
$135$	0	0
$136$	−749471.	−3.47462
$137$	128339.	0.584196	0.292098	−	0.956388i	$-0.405647\pi$
$137$	128339.	0.584196	0.292098	+	0.956388i	$0.405647\pi$
$138$	0	0
$139$	155483.	0.682566	0.341283	−	0.939961i	$-0.389139\pi$
$139$	155483.	0.682566	0.341283	+	0.939961i	$0.389139\pi$
$140$	143135.	0.617201
$141$	0	0
$142$	−137648.	−0.572860
$143$	−77006.3	−0.314910
$144$	0	0
$145$	9155.09	0.0361612
$146$	440756.	1.71126
$147$	0	0
$148$	−84649.5	−0.317665
$149$	−444140.	−1.63891	−0.819454	−	0.573146i	$-0.805723\pi$
$149$	−444140.	−1.63891	−0.819454	+	0.573146i	$0.805723\pi$
$150$	0	0
$151$	−68940.1	−0.246054	−0.123027	−	0.992403i	$-0.539260\pi$
$151$	−68940.1	−0.246054	−0.123027	+	0.992403i	$0.539260\pi$
$152$	−76288.6	−0.267825
$153$	0	0
$154$	255443.	0.867946
$155$	75623.0	0.252828
$156$	0	0
$157$	−69283.9	−0.224328	−0.112164	−	0.993690i	$-0.535778\pi$
$157$	−69283.9	−0.224328	−0.112164	+	0.993690i	$0.535778\pi$
$158$	−576504.	−1.83721
$159$	0	0
$160$	−301512.	−0.931117
$161$	8162.15	0.0248165
$162$	0	0
$163$	−267280.	−0.787946	−0.393973	−	0.919122i	$-0.628900\pi$
$163$	−267280.	−0.787946	−0.393973	+	0.919122i	$0.628900\pi$
$164$	159780.	0.463888
$165$	0	0
$166$	−840538.	−2.36749
$167$	−387809.	−1.07604	−0.538018	−	0.842934i	$-0.680827\pi$
$167$	−387809.	−1.07604	−0.538018	+	0.842934i	$0.680827\pi$
$168$	0	0
$169$	−339587.	−0.914605
$170$	572148.	1.51840
$171$	0	0
$172$	290182.	0.747910
$173$	190512.	0.483958	0.241979	−	0.970281i	$-0.422203\pi$
$173$	190512.	0.483958	0.241979	+	0.970281i	$0.422203\pi$
$174$	0	0
$175$	114975.	0.283796
$176$	−980932.	−2.38703
$177$	0	0
$178$	−1.11397e6	−2.63527
$179$	173534.	0.404811	0.202406	−	0.979302i	$-0.435124\pi$
$179$	173534.	0.404811	0.202406	+	0.979302i	$0.435124\pi$
$180$	0	0
$181$	717762.	1.62849	0.814243	−	0.580524i	$-0.197152\pi$
$181$	717762.	1.62849	0.814243	+	0.580524i	$0.197152\pi$
$182$	−105176.	−0.235363
$183$	0	0
$184$	−64758.8	−0.141011
$185$	37256.6	0.0800339
$186$	0	0
$187$	717315.	1.50005
$188$	−1.11913e6	−2.30932
$189$	0	0
$190$	58238.9	0.117039
$191$	−336630.	−0.667681	−0.333841	−	0.942630i	$-0.608345\pi$
$191$	−336630.	−0.667681	−0.333841	+	0.942630i	$0.608345\pi$
$192$	0	0
$193$	293107.	0.566413	0.283206	−	0.959059i	$-0.408602\pi$
$193$	293107.	0.566413	0.283206	+	0.959059i	$0.408602\pi$
$194$	−958699.	−1.82885
$195$	0	0
$196$	−1.02496e6	−1.90575
$197$	254135.	0.466551	0.233276	−	0.972411i	$-0.425056\pi$
$197$	254135.	0.466551	0.233276	+	0.972411i	$0.425056\pi$
$198$	0	0
$199$	65290.6	0.116874	0.0584370	−	0.998291i	$-0.481388\pi$
$199$	65290.6	0.116874	0.0584370	+	0.998291i	$0.481388\pi$
$200$	−912213.	−1.61258
$201$	0	0
$202$	1.39775e6	2.41020
$203$	15676.7	0.0267002
$204$	0	0
$205$	−70323.8	−0.116874
$206$	1.72317e6	2.82918
$207$	0	0
$208$	403888.	0.647295
$209$	73015.4	0.115624
$210$	0	0
$211$	93142.8	0.144027	0.0720134	−	0.997404i	$-0.477058\pi$
$211$	93142.8	0.144027	0.0720134	+	0.997404i	$0.477058\pi$
$212$	−729372.	−1.11458
$213$	0	0
$214$	−1.24468e6	−1.85790
$215$	−127717.	−0.188432
$216$	0	0
$217$	129493.	0.186679
$218$	1.95015e6	2.77925
$219$	0	0
$220$	1.08692e6	1.51405
$221$	−295346.	−0.406772
$222$	0	0
$223$	30528.4	0.0411095	0.0205548	−	0.999789i	$-0.493457\pi$
$223$	30528.4	0.0411095	0.0205548	+	0.999789i	$0.493457\pi$
$224$	−516292.	−0.687504
$225$	0	0
$226$	−1.18310e6	−1.54082
$227$	777882.	1.00196	0.500979	−	0.865460i	$-0.332973\pi$
$227$	777882.	1.00196	0.500979	+	0.865460i	$0.332973\pi$
$228$	0	0
$229$	−763186.	−0.961705	−0.480852	−	0.876802i	$-0.659673\pi$
$229$	−763186.	−0.961705	−0.480852	+	0.876802i	$0.659673\pi$
$230$	49437.1	0.0616216
$231$	0	0
$232$	−124379.	−0.151715
$233$	1.27683e6	1.54079	0.770393	−	0.637570i	$-0.220060\pi$
$233$	1.27683e6	1.54079	0.770393	+	0.637570i	$0.220060\pi$
$234$	0	0
$235$	492559.	0.581820
$236$	263049.	0.307437
$237$	0	0
$238$	979715.	1.12113
$239$	−71356.0	−0.0808045	−0.0404023	−	0.999183i	$-0.512864\pi$
$239$	−71356.0	−0.0808045	−0.0404023	+	0.999183i	$0.512864\pi$
$240$	0	0
$241$	−1.25147e6	−1.38797	−0.693984	−	0.719991i	$-0.744146\pi$
$241$	−1.25147e6	−1.38797	−0.693984	+	0.719991i	$0.744146\pi$
$242$	269410.	0.295716
$243$	0	0
$244$	2.21065e6	2.37709
$245$	451113.	0.480142
$246$	0	0
$247$	−30063.3	−0.0313540
$248$	−1.02740e6	−1.06074
$249$	0	0
$250$	1.77434e6	1.79551
$251$	1.22669e6	1.22900	0.614501	−	0.788916i	$-0.289358\pi$
$251$	1.22669e6	1.22900	0.614501	+	0.788916i	$0.289358\pi$
$252$	0	0
$253$	61980.3	0.0608769
$254$	3.16640e6	3.07951
$255$	0	0
$256$	−1.38862e6	−1.32429
$257$	1.22989e6	1.16154	0.580770	−	0.814068i	$-0.302752\pi$
$257$	1.22989e6	1.16154	0.580770	+	0.814068i	$0.302752\pi$
$258$	0	0
$259$	63796.2	0.0590943
$260$	−447525.	−0.410567
$261$	0	0
$262$	1.58426e6	1.42585
$263$	1.03240e6	0.920364	0.460182	−	0.887825i	$-0.347784\pi$
$263$	1.03240e6	0.920364	0.460182	+	0.887825i	$0.347784\pi$
$264$	0	0
$265$	321017.	0.280811
$266$	99725.1	0.0864173
$267$	0	0
$268$	1.14050e6	0.969967
$269$	565251.	0.476278	0.238139	−	0.971231i	$-0.423463\pi$
$269$	565251.	0.476278	0.238139	+	0.971231i	$0.423463\pi$
$270$	0	0
$271$	−1.35862e6	−1.12376	−0.561880	−	0.827219i	$-0.689922\pi$
$271$	−1.35862e6	−1.12376	−0.561880	+	0.827219i	$0.689922\pi$
$272$	−3.76222e6	−3.08334
$273$	0	0
$274$	1.33107e6	1.07108
$275$	873075.	0.696177
$276$	0	0
$277$	219170.	0.171625	0.0858127	−	0.996311i	$-0.472651\pi$
$277$	219170.	0.171625	0.0858127	+	0.996311i	$0.472651\pi$
$278$	1.61258e6	1.25144
$279$	0	0
$280$	855877.	0.652404
$281$	−1.54978e6	−1.17086	−0.585431	−	0.810722i	$-0.699075\pi$
$281$	−1.54978e6	−1.17086	−0.585431	+	0.810722i	$0.699075\pi$
$282$	0	0
$283$	−1.39782e6	−1.03749	−0.518746	−	0.854928i	$-0.673601\pi$
$283$	−1.39782e6	−1.03749	−0.518746	+	0.854928i	$0.673601\pi$
$284$	−1.00291e6	−0.737847
$285$	0	0
$286$	−798667.	−0.577365
$287$	−120419.	−0.0862957
$288$	0	0
$289$	1.33130e6	0.937628
$290$	94951.6	0.0662990
$291$	0	0
$292$	3.21138e6	2.20411
$293$	−2.33918e6	−1.59182	−0.795910	−	0.605415i	$-0.793007\pi$
$293$	−2.33918e6	−1.59182	−0.795910	+	0.605415i	$0.793007\pi$
$294$	0	0
$295$	−115775.	−0.0774569
$296$	−506162.	−0.335784
$297$	0	0
$298$	−4.60638e6	−3.00482
$299$	−25519.7	−0.0165081
$300$	0	0
$301$	−218696.	−0.139131
$302$	−715009.	−0.451122
$303$	0	0
$304$	−382956.	−0.237665
$305$	−972968.	−0.598893
$306$	0	0
$307$	1.73450e6	1.05034	0.525170	−	0.850998i	$-0.324002\pi$
$307$	1.73450e6	1.05034	0.525170	+	0.850998i	$0.324002\pi$
$308$	1.86118e6	1.11792
$309$	0	0
$310$	784320.	0.463542
$311$	1.36982e6	0.803086	0.401543	−	0.915840i	$-0.368474\pi$
$311$	1.36982e6	0.803086	0.401543	+	0.915840i	$0.368474\pi$
$312$	0	0
$313$	−2.30372e6	−1.32914	−0.664569	−	0.747227i	$-0.731385\pi$
$313$	−2.30372e6	−1.32914	−0.664569	+	0.747227i	$0.731385\pi$
$314$	−718575.	−0.411290
$315$	0	0
$316$	−4.20045e6	−2.36634
$317$	−2.31279e6	−1.29267	−0.646337	−	0.763052i	$-0.723700\pi$
$317$	−2.31279e6	−1.29267	−0.646337	+	0.763052i	$0.723700\pi$
$318$	0	0
$319$	119043.	0.0654978
$320$	−713053.	−0.389266
$321$	0	0
$322$	84653.3	0.0454993
$323$	280040.	0.149353
$324$	0	0
$325$	−359479.	−0.188784
$326$	−2.77208e6	−1.44465
$327$	0	0
$328$	955406.	0.490347
$329$	843431.	0.429596
$330$	0	0
$331$	1.01902e6	0.511224	0.255612	−	0.966779i	$-0.417723\pi$
$331$	1.01902e6	0.511224	0.255612	+	0.966779i	$0.417723\pi$
$332$	−6.12421e6	−3.04933
$333$	0	0
$334$	−4.02214e6	−1.97284
$335$	−501965.	−0.244377
$336$	0	0
$337$	1.96163e6	0.940899	0.470450	−	0.882427i	$-0.344092\pi$
$337$	1.96163e6	0.940899	0.470450	+	0.882427i	$0.344092\pi$
$338$	−3.52201e6	−1.67687
$339$	0	0
$340$	4.16870e6	1.95571
$341$	983320.	0.457940
$342$	0	0
$343$	1.72964e6	0.793817
$344$	1.73514e6	0.790569
$345$	0	0
$346$	1.97589e6	0.887304
$347$	−567054.	−0.252814	−0.126407	−	0.991978i	$-0.540344\pi$
$347$	−567054.	−0.252814	−0.126407	+	0.991978i	$0.540344\pi$
$348$	0	0
$349$	2.79317e6	1.22753	0.613767	−	0.789487i	$-0.289653\pi$
$349$	2.79317e6	1.22753	0.613767	+	0.789487i	$0.289653\pi$
$350$	1.19245e6	0.520321
$351$	0	0
$352$	−3.92053e6	−1.68651
$353$	−2.82854e6	−1.20816	−0.604081	−	0.796923i	$-0.706460\pi$
$353$	−2.82854e6	−1.20816	−0.604081	+	0.796923i	$0.706460\pi$
$354$	0	0
$355$	441409.	0.185896
$356$	−8.11648e6	−3.39424
$357$	0	0
$358$	1.79980e6	0.742194
$359$	200396.	0.0820641	0.0410321	−	0.999158i	$-0.486935\pi$
$359$	200396.	0.0820641	0.0410321	+	0.999158i	$0.486935\pi$
$360$	0	0
$361$	−2.44759e6	−0.988488
$362$	7.44424e6	2.98572
$363$	0	0
$364$	−766318.	−0.303148
$365$	−1.41342e6	−0.555313
$366$	0	0
$367$	−1.05311e6	−0.408140	−0.204070	−	0.978956i	$-0.565417\pi$
$367$	−1.05311e6	−0.408140	−0.204070	+	0.978956i	$0.565417\pi$
$368$	−325079.	−0.125132
$369$	0	0
$370$	386405.	0.146737
$371$	549693.	0.207341
$372$	0	0
$373$	−1.20089e6	−0.446922	−0.223461	−	0.974713i	$-0.571735\pi$
$373$	−1.20089e6	−0.446922	−0.223461	+	0.974713i	$0.571735\pi$
$374$	7.43959e6	2.75024
$375$	0	0
$376$	−6.69182e6	−2.44104
$377$	−49014.5	−0.0177612
$378$	0	0
$379$	−3.95845e6	−1.41556	−0.707778	−	0.706435i	$-0.750302\pi$
$379$	−3.95845e6	−1.41556	−0.707778	+	0.706435i	$0.750302\pi$
$380$	424332.	0.150746
$381$	0	0
$382$	−3.49134e6	−1.22415
$383$	3.58271e6	1.24800	0.624000	−	0.781424i	$-0.285506\pi$
$383$	3.58271e6	1.24800	0.624000	+	0.781424i	$0.285506\pi$
$384$	0	0
$385$	−819156.	−0.281653
$386$	3.03994e6	1.03848
$387$	0	0
$388$	−6.98514e6	−2.35557
$389$	2.75954e6	0.924619	0.462309	−	0.886719i	$-0.347021\pi$
$389$	2.75954e6	0.924619	0.462309	+	0.886719i	$0.347021\pi$
$390$	0	0
$391$	237716.	0.0786352
$392$	−6.12874e6	−2.01445
$393$	0	0
$394$	2.63575e6	0.855390
$395$	1.84873e6	0.596186
$396$	0	0
$397$	4.77985e6	1.52208	0.761041	−	0.648704i	$-0.224689\pi$
$397$	4.77985e6	1.52208	0.761041	+	0.648704i	$0.224689\pi$
$398$	677158.	0.214280
$399$	0	0
$400$	−4.57916e6	−1.43099
$401$	4.79242e6	1.48831	0.744156	−	0.668005i	$-0.232852\pi$
$401$	4.79242e6	1.48831	0.744156	+	0.668005i	$0.232852\pi$
$402$	0	0
$403$	−404871.	−0.124181
$404$	1.01841e7	3.10435
$405$	0	0
$406$	162590.	0.0489529
$407$	484445.	0.144963
$408$	0	0
$409$	804704.	0.237864	0.118932	−	0.992902i	$-0.462053\pi$
$409$	804704.	0.237864	0.118932	+	0.992902i	$0.462053\pi$
$410$	−729360.	−0.214280
$411$	0	0
$412$	1.25551e7	3.64400
$413$	−198247.	−0.0571915
$414$	0	0
$415$	2.69544e6	0.768262
$416$	1.61423e6	0.457334
$417$	0	0
$418$	757276.	0.211989
$419$	−5.46503e6	−1.52075	−0.760374	−	0.649486i	$-0.774984\pi$
$419$	−5.46503e6	−1.52075	−0.760374	+	0.649486i	$0.774984\pi$
$420$	0	0
$421$	−1.96598e6	−0.540597	−0.270298	−	0.962777i	$-0.587122\pi$
$421$	−1.96598e6	−0.540597	−0.270298	+	0.962777i	$0.587122\pi$
$422$	966026.	0.264063
$423$	0	0
$424$	−4.36128e6	−1.17815
$425$	3.34855e6	0.899258
$426$	0	0
$427$	−1.66606e6	−0.442202
$428$	−9.06880e6	−2.39299
$429$	0	0
$430$	−1.32461e6	−0.345476
$431$	−6.18098e6	−1.60274	−0.801372	−	0.598167i	$-0.795896\pi$
$431$	−6.18098e6	−1.60274	−0.801372	+	0.598167i	$0.795896\pi$
$432$	0	0
$433$	−6.81186e6	−1.74601	−0.873003	−	0.487715i	$-0.837831\pi$
$433$	−6.81186e6	−1.74601	−0.873003	+	0.487715i	$0.837831\pi$
$434$	1.34303e6	0.342264
$435$	0	0
$436$	1.42089e7	3.57969
$437$	24197.1	0.00606122
$438$	0	0
$439$	5.97720e6	1.48026	0.740128	−	0.672466i	$-0.234765\pi$
$439$	5.97720e6	1.48026	0.740128	+	0.672466i	$0.234765\pi$
$440$	6.49921e6	1.60040
$441$	0	0
$442$	−3.06317e6	−0.745787
$443$	490983.	0.118866	0.0594330	−	0.998232i	$-0.481071\pi$
$443$	490983.	0.118866	0.0594330	+	0.998232i	$0.481071\pi$
$444$	0	0
$445$	3.57229e6	0.855159
$446$	316624.	0.0753715
$447$	0	0
$448$	−1.22099e6	−0.287421
$449$	−7.25924e6	−1.69932	−0.849660	−	0.527331i	$-0.823193\pi$
$449$	−7.25924e6	−1.69932	−0.849660	+	0.527331i	$0.823193\pi$
$450$	0	0
$451$	−914415.	−0.211691
$452$	−8.62015e6	−1.98458
$453$	0	0
$454$	8.06777e6	1.83702
$455$	337278.	0.0763765
$456$	0	0
$457$	−2.78034e6	−0.622740	−0.311370	−	0.950289i	$-0.600788\pi$
$457$	−2.78034e6	−0.622740	−0.311370	+	0.950289i	$0.600788\pi$
$458$	−7.91535e6	−1.76322
$459$	0	0
$460$	360201.	0.0793690
$461$	6.01371e6	1.31792	0.658962	−	0.752176i	$-0.270996\pi$
$461$	6.01371e6	1.31792	0.658962	+	0.752176i	$0.270996\pi$
$462$	0	0
$463$	−7.10787e6	−1.54094	−0.770472	−	0.637474i	$-0.779979\pi$
$463$	−7.10787e6	−1.54094	−0.770472	+	0.637474i	$0.779979\pi$
$464$	−624364.	−0.134630
$465$	0	0
$466$	1.32425e7	2.82492
$467$	−5.51961e6	−1.17116	−0.585579	−	0.810615i	$-0.699133\pi$
$467$	−5.51961e6	−1.17116	−0.585579	+	0.810615i	$0.699133\pi$
$468$	0	0
$469$	−859537.	−0.180440
$470$	5.10855e6	1.06673
$471$	0	0
$472$	1.57290e6	0.324972
$473$	−1.66070e6	−0.341301
$474$	0	0
$475$	340849.	0.0693150
$476$	7.13826e6	1.44403
$477$	0	0
$478$	−740065.	−0.148150
$479$	3.18017e6	0.633304	0.316652	−	0.948542i	$-0.397441\pi$
$479$	3.18017e6	0.633304	0.316652	+	0.948542i	$0.397441\pi$
$480$	0	0
$481$	−199465.	−0.0393100
$482$	−1.29796e7	−2.54474
$483$	0	0
$484$	1.96293e6	0.380884
$485$	3.07436e6	0.593472
$486$	0	0
$487$	−4.01718e6	−0.767537	−0.383768	−	0.923429i	$-0.625374\pi$
$487$	−4.01718e6	−0.767537	−0.383768	+	0.923429i	$0.625374\pi$
$488$	1.32186e7	2.51267
$489$	0	0
$490$	4.67869e6	0.880308
$491$	−2.25211e6	−0.421585	−0.210793	−	0.977531i	$-0.567604\pi$
$491$	−2.25211e6	−0.421585	−0.210793	+	0.977531i	$0.567604\pi$
$492$	0	0
$493$	456571.	0.0846041
$494$	−311800.	−0.0574855
$495$	0	0
$496$	−5.15738e6	−0.941293
$497$	755845.	0.137259
$498$	0	0
$499$	4.87566e6	0.876561	0.438281	−	0.898838i	$-0.355588\pi$
$499$	4.87566e6	0.876561	0.438281	+	0.898838i	$0.355588\pi$
$500$	1.29280e7	2.31262
$501$	0	0
$502$	1.27226e7	2.25329
$503$	3.03011e6	0.533996	0.266998	−	0.963697i	$-0.413968\pi$
$503$	3.03011e6	0.533996	0.266998	+	0.963697i	$0.413968\pi$
$504$	0	0
$505$	−4.48232e6	−0.782122
$506$	642826.	0.111614
$507$	0	0
$508$	2.30706e7	3.96643
$509$	6.31223e6	1.07991	0.539956	−	0.841693i	$-0.318441\pi$
$509$	6.31223e6	1.07991	0.539956	+	0.841693i	$0.318441\pi$
$510$	0	0
$511$	−2.42026e6	−0.410024
$512$	−1.22343e7	−2.06255
$513$	0	0
$514$	1.27558e7	2.12960
$515$	−5.52586e6	−0.918083
$516$	0	0
$517$	6.40470e6	1.05384
$518$	661659.	0.108345
$519$	0	0
$520$	−2.67598e6	−0.433984
$521$	1.00384e7	1.62021	0.810106	−	0.586283i	$-0.199409\pi$
$521$	1.00384e7	1.62021	0.810106	+	0.586283i	$0.199409\pi$
$522$	0	0
$523$	4.29564e6	0.686710	0.343355	−	0.939206i	$-0.388437\pi$
$523$	4.29564e6	0.686710	0.343355	+	0.939206i	$0.388437\pi$
$524$	1.15430e7	1.83650
$525$	0	0
$526$	1.07075e7	1.68742
$527$	3.77138e6	0.591525
$528$	0	0
$529$	−6.41580e6	−0.996809
$530$	3.32942e6	0.514847
$531$	0	0
$532$	726603.	0.111306
$533$	376500.	0.0574046
$534$	0	0
$535$	3.99144e6	0.602899
$536$	6.81960e6	1.02529
$537$	0	0
$538$	5.86247e6	0.873223
$539$	5.86578e6	0.869669
$540$	0	0
$541$	−6.40724e6	−0.941190	−0.470595	−	0.882349i	$-0.655961\pi$
$541$	−6.40724e6	−0.941190	−0.470595	+	0.882349i	$0.655961\pi$
$542$	−1.40908e7	−2.06033
$543$	0	0
$544$	−1.50366e7	−2.17848
$545$	−6.25375e6	−0.901882
$546$	0	0
$547$	2.87214e6	0.410429	0.205214	−	0.978717i	$-0.434211\pi$
$547$	2.87214e6	0.410429	0.205214	+	0.978717i	$0.434211\pi$
$548$	9.69822e6	1.37956
$549$	0	0
$550$	9.05505e6	1.27639
$551$	46474.4	0.00652130
$552$	0	0
$553$	3.16567e6	0.440203
$554$	2.27311e6	0.314663
$555$	0	0
$556$	1.17494e7	1.61186
$557$	1.93992e6	0.264939	0.132470	−	0.991187i	$-0.457709\pi$
$557$	1.93992e6	0.264939	0.132470	+	0.991187i	$0.457709\pi$
$558$	0	0
$559$	683774.	0.0925514
$560$	4.29636e6	0.578937
$561$	0	0
$562$	−1.60735e7	−2.14669
$563$	−2.17438e6	−0.289110	−0.144555	−	0.989497i	$-0.546175\pi$
$563$	−2.17438e6	−0.289110	−0.144555	+	0.989497i	$0.546175\pi$
$564$	0	0
$565$	3.79397e6	0.500004
$566$	−1.44974e7	−1.90217
$567$	0	0
$568$	−5.99690e6	−0.779931
$569$	4.07789e6	0.528025	0.264012	−	0.964519i	$-0.414954\pi$
$569$	4.07789e6	0.528025	0.264012	+	0.964519i	$0.414954\pi$
$570$	0	0
$571$	7.41602e6	0.951876	0.475938	−	0.879479i	$-0.342109\pi$
$571$	7.41602e6	0.951876	0.475938	+	0.879479i	$0.342109\pi$
$572$	−5.81913e6	−0.743649
$573$	0	0
$574$	−1.24892e6	−0.158217
$575$	289335.	0.0364948
$576$	0	0
$577$	1.29497e6	0.161927	0.0809636	−	0.996717i	$-0.474200\pi$
$577$	1.29497e6	0.161927	0.0809636	+	0.996717i	$0.474200\pi$
$578$	1.38075e7	1.71908
$579$	0	0
$580$	691823.	0.0853935
$581$	4.61552e6	0.567258
$582$	0	0
$583$	4.17416e6	0.508625
$584$	1.92024e7	2.32983
$585$	0	0
$586$	−2.42607e7	−2.91849
$587$	−9.79730e6	−1.17358	−0.586788	−	0.809741i	$-0.699608\pi$
$587$	−9.79730e6	−1.17358	−0.586788	+	0.809741i	$0.699608\pi$
$588$	0	0
$589$	383888.	0.0455949
$590$	−1.20076e6	−0.142012
$591$	0	0
$592$	−2.54085e6	−0.297971
$593$	−2.89314e6	−0.337857	−0.168929	−	0.985628i	$-0.554031\pi$
$593$	−2.89314e6	−0.337857	−0.168929	+	0.985628i	$0.554031\pi$
$594$	0	0
$595$	−3.14175e6	−0.363814
$596$	−3.35623e7	−3.87023
$597$	0	0
$598$	−264676.	−0.0302665
$599$	−1.07146e6	−0.122013	−0.0610066	−	0.998137i	$-0.519431\pi$
$599$	−1.07146e6	−0.122013	−0.0610066	+	0.998137i	$0.519431\pi$
$600$	0	0
$601$	−8.77640e6	−0.991129	−0.495564	−	0.868571i	$-0.665039\pi$
$601$	−8.77640e6	−0.991129	−0.495564	+	0.868571i	$0.665039\pi$
$602$	−2.26820e6	−0.255088
$603$	0	0
$604$	−5.20960e6	−0.581048
$605$	−863943.	−0.0959614
$606$	0	0
$607$	−1.38832e7	−1.52939	−0.764696	−	0.644391i	$-0.777111\pi$
$607$	−1.38832e7	−1.52939	−0.764696	+	0.644391i	$0.777111\pi$
$608$	−1.53058e6	−0.167917
$609$	0	0
$610$	−1.00911e7	−1.09803
$611$	−2.63706e6	−0.285771
$612$	0	0
$613$	−2.96162e6	−0.318330	−0.159165	−	0.987252i	$-0.550880\pi$
$613$	−2.96162e6	−0.318330	−0.159165	+	0.987252i	$0.550880\pi$
$614$	1.79893e7	1.92572
$615$	0	0
$616$	1.11289e7	1.18168
$617$	1.63962e7	1.73393	0.866964	−	0.498370i	$-0.166068\pi$
$617$	1.63962e7	1.73393	0.866964	+	0.498370i	$0.166068\pi$
$618$	0	0
$619$	−1.35257e7	−1.41884	−0.709418	−	0.704788i	$-0.751042\pi$
$619$	−1.35257e7	−1.41884	−0.709418	+	0.704788i	$0.751042\pi$
$620$	5.71460e6	0.597045
$621$	0	0
$622$	1.42070e7	1.47240
$623$	6.11700e6	0.631420
$624$	0	0
$625$	618876.	0.0633729
$626$	−2.38930e7	−2.43688
$627$	0	0
$628$	−5.23558e6	−0.529743
$629$	1.85802e6	0.187250
$630$	0	0
$631$	−1.19825e7	−1.19805	−0.599025	−	0.800731i	$-0.704445\pi$
$631$	−1.19825e7	−1.19805	−0.599025	+	0.800731i	$0.704445\pi$
$632$	−2.51166e7	−2.50131
$633$	0	0
$634$	−2.39870e7	−2.37003
$635$	−1.01540e7	−0.999318
$636$	0	0
$637$	−2.41517e6	−0.235830
$638$	1.23465e6	0.120086
$639$	0	0
$640$	2.25298e6	0.217424
$641$	7.00957e6	0.673823	0.336912	−	0.941536i	$-0.390618\pi$
$641$	7.00957e6	0.673823	0.336912	+	0.941536i	$0.390618\pi$
$642$	0	0
$643$	605033.	0.0577101	0.0288550	−	0.999584i	$-0.490814\pi$
$643$	605033.	0.0577101	0.0288550	+	0.999584i	$0.490814\pi$
$644$	616789.	0.0586033
$645$	0	0
$646$	2.90442e6	0.273828
$647$	−1.46414e7	−1.37506	−0.687531	−	0.726155i	$-0.741305\pi$
$647$	−1.46414e7	−1.37506	−0.687531	+	0.726155i	$0.741305\pi$
$648$	0	0
$649$	−1.50542e6	−0.140296
$650$	−3.72831e6	−0.346122
$651$	0	0
$652$	−2.01975e7	−1.86071
$653$	−3.99497e6	−0.366632	−0.183316	−	0.983054i	$-0.558683\pi$
$653$	−3.99497e6	−0.366632	−0.183316	+	0.983054i	$0.558683\pi$
$654$	0	0
$655$	−5.08041e6	−0.462696
$656$	4.79598e6	0.435129
$657$	0	0
$658$	8.74761e6	0.787634
$659$	−274018.	−0.0245791	−0.0122895	−	0.999924i	$-0.503912\pi$
$659$	−274018.	−0.0245791	−0.0122895	+	0.999924i	$0.503912\pi$
$660$	0	0
$661$	−1.92782e7	−1.71618	−0.858091	−	0.513497i	$-0.828350\pi$
$661$	−1.92782e7	−1.71618	−0.858091	+	0.513497i	$0.828350\pi$
$662$	1.05687e7	0.937294
$663$	0	0
$664$	−3.66197e7	−3.22326
$665$	−319799.	−0.0280429
$666$	0	0
$667$	39450.5	0.00343351
$668$	−2.93056e7	−2.54102
$669$	0	0
$670$	−5.20610e6	−0.448049
$671$	−1.26514e7	−1.08476
$672$	0	0
$673$	−1.19222e7	−1.01466	−0.507329	−	0.861753i	$-0.669367\pi$
$673$	−1.19222e7	−1.01466	−0.507329	+	0.861753i	$0.669367\pi$
$674$	2.03450e7	1.72507
$675$	0	0
$676$	−2.56615e7	−2.15981
$677$	3.81809e6	0.320166	0.160083	−	0.987104i	$-0.448824\pi$
$677$	3.81809e6	0.320166	0.160083	+	0.987104i	$0.448824\pi$
$678$	0	0
$679$	5.26436e6	0.438199
$680$	2.49268e7	2.06725
$681$	0	0
$682$	1.01985e7	0.839602
$683$	−1.71086e7	−1.40334	−0.701672	−	0.712500i	$-0.747563\pi$
$683$	−1.71086e7	−1.40334	−0.701672	+	0.712500i	$0.747563\pi$
$684$	0	0
$685$	−4.26846e6	−0.347572
$686$	1.79389e7	1.45541
$687$	0	0
$688$	8.71014e6	0.701543
$689$	−1.71866e6	−0.137925
$690$	0	0
$691$	1.16432e6	0.0927639	0.0463820	−	0.998924i	$-0.485231\pi$
$691$	1.16432e6	0.0927639	0.0463820	+	0.998924i	$0.485231\pi$
$692$	1.43965e7	1.14285
$693$	0	0
$694$	−5.88117e6	−0.463516
$695$	−5.17122e6	−0.406098
$696$	0	0
$697$	−3.50710e6	−0.273443
$698$	2.89692e7	2.25060
$699$	0	0
$700$	8.68829e6	0.670177
$701$	2.35243e6	0.180810	0.0904050	−	0.995905i	$-0.471184\pi$
$701$	2.35243e6	0.180810	0.0904050	+	0.995905i	$0.471184\pi$
$702$	0	0
$703$	189127.	0.0144333
$704$	−9.27177e6	−0.705068
$705$	0	0
$706$	−2.93360e7	−2.21508
$707$	−7.67528e6	−0.577492
$708$	0	0
$709$	−2.37540e7	−1.77468	−0.887342	−	0.461112i	$-0.847451\pi$
$709$	−2.37540e7	−1.77468	−0.887342	+	0.461112i	$0.847451\pi$
$710$	4.57805e6	0.340828
$711$	0	0
$712$	−4.85325e7	−3.58784
$713$	325870.	0.0240060
$714$	0	0
$715$	2.56116e6	0.187358
$716$	1.31135e7	0.955949
$717$	0	0
$718$	2.07840e6	0.150459
$719$	−7.28156e6	−0.525294	−0.262647	−	0.964892i	$-0.584595\pi$
$719$	−7.28156e6	−0.525294	−0.262647	+	0.964892i	$0.584595\pi$
$720$	0	0
$721$	−9.46219e6	−0.677881
$722$	−2.53851e7	−1.81232
$723$	0	0
$724$	5.42392e7	3.84562
$725$	555713.	0.0392650
$726$	0	0
$727$	400663.	0.0281153	0.0140577	−	0.999901i	$-0.495525\pi$
$727$	400663.	0.0281153	0.0140577	+	0.999901i	$0.495525\pi$
$728$	−4.58220e6	−0.320439
$729$	0	0
$730$	−1.46592e7	−1.01813
$731$	−6.36936e6	−0.440862
$732$	0	0
$733$	2.06252e7	1.41787	0.708937	−	0.705272i	$-0.249175\pi$
$733$	2.06252e7	1.41787	0.708937	+	0.705272i	$0.249175\pi$
$734$	−1.09223e7	−0.748297
$735$	0	0
$736$	−1.29925e6	−0.0884097
$737$	−6.52701e6	−0.442634
$738$	0	0
$739$	−1.36694e7	−0.920744	−0.460372	−	0.887726i	$-0.652284\pi$
$739$	−1.36694e7	−0.920744	−0.460372	+	0.887726i	$0.652284\pi$
$740$	2.81537e6	0.188998
$741$	0	0
$742$	5.70111e6	0.380145
$743$	−1.13736e7	−0.755833	−0.377917	−	0.925840i	$-0.623359\pi$
$743$	−1.13736e7	−0.755833	−0.377917	+	0.925840i	$0.623359\pi$
$744$	0	0
$745$	1.47717e7	0.975081
$746$	−1.24550e7	−0.819400
$747$	0	0
$748$	5.42053e7	3.54232
$749$	6.83472e6	0.445160
$750$	0	0
$751$	−1.25684e6	−0.0813171	−0.0406585	−	0.999173i	$-0.512946\pi$
$751$	−1.25684e6	−0.0813171	−0.0406585	+	0.999173i	$0.512946\pi$
$752$	−3.35918e7	−2.16615
$753$	0	0
$754$	−508352.	−0.0325639
$755$	2.29289e6	0.146392
$756$	0	0
$757$	−1.34819e7	−0.855087	−0.427544	−	0.903995i	$-0.640621\pi$
$757$	−1.34819e7	−0.855087	−0.427544	+	0.903995i	$0.640621\pi$
$758$	−4.10549e7	−2.59533
$759$	0	0
$760$	2.53729e6	0.159344
$761$	865364.	0.0541673	0.0270837	−	0.999633i	$-0.491378\pi$
$761$	865364.	0.0541673	0.0270837	+	0.999633i	$0.491378\pi$
$762$	0	0
$763$	−1.07086e7	−0.665918
$764$	−2.54381e7	−1.57671
$765$	0	0
$766$	3.71579e7	2.28812
$767$	619837.	0.0380443
$768$	0	0
$769$	−2.08760e7	−1.27301	−0.636503	−	0.771274i	$-0.719620\pi$
$769$	−2.08760e7	−1.27301	−0.636503	+	0.771274i	$0.719620\pi$
$770$	−8.49584e6	−0.516392
$771$	0	0
$772$	2.21492e7	1.33757
$773$	−2.75458e7	−1.65809	−0.829043	−	0.559185i	$-0.811114\pi$
$773$	−2.75458e7	−1.65809	−0.829043	+	0.559185i	$0.811114\pi$
$774$	0	0
$775$	4.59031e6	0.274529
$776$	−4.17677e7	−2.48992
$777$	0	0
$778$	2.86204e7	1.69522
$779$	−356988.	−0.0210770
$780$	0	0
$781$	5.73961e6	0.336709
$782$	2.46546e6	0.144172
$783$	0	0
$784$	−3.07652e7	−1.78760
$785$	2.30433e6	0.133466
$786$	0	0
$787$	1.86224e7	1.07176	0.535881	−	0.844293i	$-0.319979\pi$
$787$	1.86224e7	1.07176	0.535881	+	0.844293i	$0.319979\pi$
$788$	1.92042e7	1.10175
$789$	0	0
$790$	1.91741e7	1.09307
$791$	6.49659e6	0.369185
$792$	0	0
$793$	5.20908e6	0.294156
$794$	4.95740e7	2.79063
$795$	0	0
$796$	4.93381e6	0.275994
$797$	1.01313e7	0.564965	0.282482	−	0.959272i	$-0.408842\pi$
$797$	1.01313e7	0.564965	0.282482	+	0.959272i	$0.408842\pi$
$798$	0	0
$799$	2.45643e7	1.36125
$800$	−1.83017e7	−1.01104
$801$	0	0
$802$	4.97044e7	2.72872
$803$	−1.83785e7	−1.00582
$804$	0	0
$805$	−271466.	−0.0147648
$806$	−4.19910e6	−0.227677
$807$	0	0
$808$	6.08960e7	3.28141
$809$	−1.76196e7	−0.946507	−0.473254	−	0.880926i	$-0.656921\pi$
$809$	−1.76196e7	−0.946507	−0.473254	+	0.880926i	$0.656921\pi$
$810$	0	0
$811$	−1.37418e6	−0.0733654	−0.0366827	−	0.999327i	$-0.511679\pi$
$811$	−1.37418e6	−0.0733654	−0.0366827	+	0.999327i	$0.511679\pi$
$812$	1.18464e6	0.0630516
$813$	0	0
$814$	5.02440e6	0.265780
$815$	8.88950e6	0.468795
$816$	0	0
$817$	−648337.	−0.0339817
$818$	8.34595e6	0.436107
$819$	0	0
$820$	−5.31416e6	−0.275994
$821$	1.13057e7	0.585382	0.292691	−	0.956207i	$-0.405449\pi$
$821$	1.13057e7	0.585382	0.292691	+	0.956207i	$0.405449\pi$
$822$	0	0
$823$	2.63250e7	1.35478	0.677389	−	0.735625i	$-0.263111\pi$
$823$	2.63250e7	1.35478	0.677389	+	0.735625i	$0.263111\pi$
$824$	7.50734e7	3.85184
$825$	0	0
$826$	−2.05611e6	−0.104857
$827$	−2.02274e6	−0.102843	−0.0514217	−	0.998677i	$-0.516375\pi$
$827$	−2.02274e6	−0.102843	−0.0514217	+	0.998677i	$0.516375\pi$
$828$	0	0
$829$	−1.96942e7	−0.995297	−0.497649	−	0.867379i	$-0.665803\pi$
$829$	−1.96942e7	−0.995297	−0.497649	+	0.867379i	$0.665803\pi$
$830$	2.79556e7	1.40855
$831$	0	0
$832$	3.81754e6	0.191195
$833$	2.24973e7	1.12336
$834$	0	0
$835$	1.28982e7	0.640196
$836$	5.51756e6	0.273043
$837$	0	0
$838$	−5.66803e7	−2.78819
$839$	3.39690e6	0.166601	0.0833006	−	0.996524i	$-0.473454\pi$
$839$	3.39690e6	0.166601	0.0833006	+	0.996524i	$0.473454\pi$
$840$	0	0
$841$	−2.04354e7	−0.996306
$842$	−2.03900e7	−0.991147
$843$	0	0
$844$	7.03852e6	0.340115
$845$	1.12944e7	0.544152
$846$	0	0
$847$	−1.47937e6	−0.0708546
$848$	−2.18929e7	−1.04548
$849$	0	0
$850$	3.47293e7	1.64873
$851$	160544.	0.00759923
$852$	0	0
$853$	−8.12673e6	−0.382422	−0.191211	−	0.981549i	$-0.561242\pi$
$853$	−8.12673e6	−0.382422	−0.191211	+	0.981549i	$0.561242\pi$
$854$	−1.72794e7	−0.810746
$855$	0	0
$856$	−5.42269e7	−2.52948
$857$	−2.17376e7	−1.01102	−0.505509	−	0.862821i	$-0.668695\pi$
$857$	−2.17376e7	−1.01102	−0.505509	+	0.862821i	$0.668695\pi$
$858$	0	0
$859$	180961.	0.00836764	0.00418382	−	0.999991i	$-0.498668\pi$
$859$	180961.	0.00836764	0.00418382	+	0.999991i	$0.498668\pi$
$860$	−9.65122e6	−0.444975
$861$	0	0
$862$	−6.41057e7	−2.93852
$863$	−3.07976e7	−1.40764	−0.703818	−	0.710381i	$-0.748523\pi$
$863$	−3.07976e7	−1.40764	−0.703818	+	0.710381i	$0.748523\pi$
$864$	0	0
$865$	−6.33629e6	−0.287935
$866$	−7.06488e7	−3.20118
$867$	0	0
$868$	9.78538e6	0.440837
$869$	2.40389e7	1.07986
$870$	0	0
$871$	2.68742e6	0.120030
$872$	8.49624e7	3.78386
$873$	0	0
$874$	250959.	0.0111128
$875$	−9.74318e6	−0.430210
$876$	0	0
$877$	1.10924e7	0.486998	0.243499	−	0.969901i	$-0.421705\pi$
$877$	1.10924e7	0.486998	0.243499	+	0.969901i	$0.421705\pi$
$878$	6.19923e7	2.71395
$879$	0	0
$880$	3.26250e7	1.42018
$881$	−3.34548e7	−1.45217	−0.726086	−	0.687603i	$-0.758663\pi$
$881$	−3.34548e7	−1.45217	−0.726086	+	0.687603i	$0.758663\pi$
$882$	0	0
$883$	−2.73782e7	−1.18169	−0.590844	−	0.806786i	$-0.701205\pi$
$883$	−2.73782e7	−1.18169	−0.590844	+	0.806786i	$0.701205\pi$
$884$	−2.23184e7	−0.960578
$885$	0	0
$886$	5.09221e6	0.217932
$887$	2.42168e7	1.03350	0.516748	−	0.856138i	$-0.327143\pi$
$887$	2.42168e7	1.03350	0.516748	+	0.856138i	$0.327143\pi$
$888$	0	0
$889$	−1.73872e7	−0.737862
$890$	3.70499e7	1.56788
$891$	0	0
$892$	2.30694e6	0.0970788
$893$	2.50040e6	0.104925
$894$	0	0
$895$	−5.77161e6	−0.240846
$896$	3.85788e6	0.160538
$897$	0	0
$898$	−7.52888e7	−3.11559
$899$	625884.	0.0258282
$900$	0	0
$901$	1.60094e7	0.656996
$902$	−9.48381e6	−0.388120
$903$	0	0
$904$	−5.15442e7	−2.09778
$905$	−2.38722e7	−0.968881
$906$	0	0
$907$	−2.31796e7	−0.935595	−0.467797	−	0.883836i	$-0.654952\pi$
$907$	−2.31796e7	−0.935595	−0.467797	+	0.883836i	$0.654952\pi$
$908$	5.87822e7	2.36609
$909$	0	0
$910$	3.49806e6	0.140031
$911$	−1.41080e7	−0.563209	−0.281605	−	0.959531i	$-0.590867\pi$
$911$	−1.41080e7	−0.563209	−0.281605	+	0.959531i	$0.590867\pi$
$912$	0	0
$913$	3.50486e7	1.39153
$914$	−2.88361e7	−1.14175
$915$	0	0
$916$	−5.76717e7	−2.27104
$917$	−8.69943e6	−0.341639
$918$	0	0
$919$	3.48252e7	1.36020	0.680102	−	0.733117i	$-0.261935\pi$
$919$	3.48252e7	1.36020	0.680102	+	0.733117i	$0.261935\pi$
$920$	2.15382e6	0.0838959
$921$	0	0
$922$	6.23709e7	2.41632
$923$	−2.36322e6	−0.0913060
$924$	0	0
$925$	2.26147e6	0.0869034
$926$	−7.37189e7	−2.82521
$927$	0	0
$928$	−2.49542e6	−0.0951204
$929$	−8.18223e6	−0.311052	−0.155526	−	0.987832i	$-0.549707\pi$
$929$	−8.18223e6	−0.311052	−0.155526	+	0.987832i	$0.549707\pi$
$930$	0	0
$931$	2.29000e6	0.0865888
$932$	9.64860e7	3.63852
$933$	0	0
$934$	−5.72463e7	−2.14724
$935$	−2.38573e7	−0.892467
$936$	0	0
$937$	−3.29144e7	−1.22472	−0.612361	−	0.790578i	$-0.709780\pi$
$937$	−3.29144e7	−1.22472	−0.612361	+	0.790578i	$0.709780\pi$
$938$	−8.91465e6	−0.330824
$939$	0	0
$940$	3.72212e7	1.37395
$941$	−2.12414e7	−0.782004	−0.391002	−	0.920390i	$-0.627872\pi$
$941$	−2.12414e7	−0.782004	−0.391002	+	0.920390i	$0.627872\pi$
$942$	0	0
$943$	−303035.	−0.0110972
$944$	7.89570e6	0.288377
$945$	0	0
$946$	−1.72239e7	−0.625752
$947$	−4.07231e7	−1.47559	−0.737795	−	0.675025i	$-0.764133\pi$
$947$	−4.07231e7	−1.47559	−0.737795	+	0.675025i	$0.764133\pi$
$948$	0	0
$949$	7.56716e6	0.272752
$950$	3.53509e6	0.127084
$951$	0	0
$952$	4.26833e7	1.52639
$953$	3.41302e7	1.21733	0.608663	−	0.793429i	$-0.291706\pi$
$953$	3.41302e7	1.21733	0.608663	+	0.793429i	$0.291706\pi$
$954$	0	0
$955$	1.11960e7	0.397242
$956$	−5.39216e6	−0.190817
$957$	0	0
$958$	3.29830e7	1.16112
$959$	−7.30908e6	−0.256635
$960$	0	0
$961$	−2.34592e7	−0.819417
$962$	−2.06874e6	−0.0720722
$963$	0	0
$964$	−9.45702e7	−3.27764
$965$	−9.74849e6	−0.336992
$966$	0	0
$967$	5.84049e6	0.200855	0.100428	−	0.994944i	$-0.467979\pi$
$967$	5.84049e6	0.200855	0.100428	+	0.994944i	$0.467979\pi$
$968$	1.17374e7	0.402608
$969$	0	0
$970$	3.18855e7	1.08809
$971$	3.95959e7	1.34773	0.673864	−	0.738856i	$-0.264633\pi$
$971$	3.95959e7	1.34773	0.673864	+	0.738856i	$0.264633\pi$
$972$	0	0
$973$	−8.85492e6	−0.299849
$974$	−4.16640e7	−1.40723
$975$	0	0
$976$	6.63550e7	2.22971
$977$	−2.78950e7	−0.934955	−0.467477	−	0.884005i	$-0.654837\pi$
$977$	−2.78950e7	−0.934955	−0.467477	+	0.884005i	$0.654837\pi$
$978$	0	0
$979$	4.64502e7	1.54893
$980$	3.40892e7	1.13384
$981$	0	0
$982$	−2.33576e7	−0.772948
$983$	2.02573e7	0.668650	0.334325	−	0.942458i	$-0.391492\pi$
$983$	2.02573e7	0.668650	0.334325	+	0.942458i	$0.391492\pi$
$984$	0	0
$985$	−8.45233e6	−0.277579
$986$	4.73531e6	0.155116
$987$	0	0
$988$	−2.27179e6	−0.0740416
$989$	−550352.	−0.0178916
$990$	0	0
$991$	4.46979e6	0.144578	0.0722891	−	0.997384i	$-0.476970\pi$
$991$	4.46979e6	0.144578	0.0722891	+	0.997384i	$0.476970\pi$
$992$	−2.06127e7	−0.665053
$993$	0	0
$994$	7.83921e6	0.251655
$995$	−2.17151e6	−0.0695351
$996$	0	0
$997$	−482842.	−0.0153839	−0.00769196	−	0.999970i	$-0.502448\pi$
$997$	−482842.	−0.0153839	−0.00769196	+	0.999970i	$0.502448\pi$
$998$	5.05677e7	1.60711
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.c.1.12		12
3.2	odd	2		177.6.a.c.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.c.1.1	✓	12	3.2	odd	2
531.6.a.c.1.12		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)

\(f(q)\)	\(=\)	\(q+10.3715 q^{2} +75.5670 q^{4} -33.2592 q^{5} -56.9512 q^{7} +451.853 q^{8} +O(q^{10})\)	\(q+10.3715 q^{2} +75.5670 q^{4} -33.2592 q^{5} -56.9512 q^{7} +451.853 q^{8} -344.946 q^{10} -432.466 q^{11} +178.063 q^{13} -590.667 q^{14} +2268.23 q^{16} -1658.66 q^{17} -168.835 q^{19} -2513.30 q^{20} -4485.30 q^{22} -143.318 q^{23} -2018.83 q^{25} +1846.77 q^{26} -4303.63 q^{28} -275.265 q^{29} -2273.75 q^{31} +9065.52 q^{32} -17202.7 q^{34} +1894.15 q^{35} -1120.19 q^{37} -1751.06 q^{38} -15028.3 q^{40} +2114.42 q^{41} +3840.06 q^{43} -32680.2 q^{44} -1486.42 q^{46} -14809.7 q^{47} -13563.6 q^{49} -20938.2 q^{50} +13455.7 q^{52} -9651.99 q^{53} +14383.5 q^{55} -25733.6 q^{56} -2854.90 q^{58} +3481.00 q^{59} +29254.1 q^{61} -23582.1 q^{62} +21439.3 q^{64} -5922.23 q^{65} +15092.5 q^{67} -125340. q^{68} +19645.1 q^{70} -13271.8 q^{71} +42497.1 q^{73} -11618.0 q^{74} -12758.3 q^{76} +24629.5 q^{77} -55585.7 q^{79} -75439.4 q^{80} +21929.6 q^{82} -81043.5 q^{83} +55165.7 q^{85} +39827.0 q^{86} -195411. q^{88} -107408. q^{89} -10140.9 q^{91} -10830.1 q^{92} -153598. q^{94} +5615.31 q^{95} -92436.4 q^{97} -140674. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) \) 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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