LMFDB - Embedded newform 207.6.a.h.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.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:	$33.1994507013$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 251 x^{8} + 296 x^{7} + 21169 x^{6} - 9042 x^{5} - 685949 x^{4} - 270764 x^{3} + \cdots + 5446216 $ x^10 - 2x^9 - 251x^8 + 296x^7 + 21169x^6 - 9042x^5 - 685949x^4 - 270764x^3 + 6230110x^2 + 11573432*x + 5446216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$11.3536$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.3536 q^{2} +75.1975 q^{4} -71.3591 q^{5} -210.967 q^{7} +447.250 q^{8} +O(q^{10})$	\(q+10.3536 q^{2} +75.1975 q^{4} -71.3591 q^{5} -210.967 q^{7} +447.250 q^{8} -738.825 q^{10} -170.188 q^{11} +605.525 q^{13} -2184.27 q^{14} +2224.34 q^{16} -1337.03 q^{17} -1794.65 q^{19} -5366.03 q^{20} -1762.07 q^{22} -529.000 q^{23} +1967.12 q^{25} +6269.38 q^{26} -15864.2 q^{28} -3006.00 q^{29} -9214.34 q^{31} +8717.97 q^{32} -13843.1 q^{34} +15054.4 q^{35} +6021.35 q^{37} -18581.1 q^{38} -31915.4 q^{40} -6606.07 q^{41} +22934.8 q^{43} -12797.7 q^{44} -5477.07 q^{46} +20179.5 q^{47} +27700.0 q^{49} +20366.9 q^{50} +45534.0 q^{52} +3604.41 q^{53} +12144.5 q^{55} -94355.0 q^{56} -31123.0 q^{58} +37190.1 q^{59} -31681.4 q^{61} -95401.8 q^{62} +19083.7 q^{64} -43209.8 q^{65} -3782.79 q^{67} -100541. q^{68} +155868. q^{70} -6321.88 q^{71} +12637.8 q^{73} +62342.8 q^{74} -134953. q^{76} +35904.1 q^{77} +79327.3 q^{79} -158727. q^{80} -68396.7 q^{82} -114368. q^{83} +95409.1 q^{85} +237459. q^{86} -76116.8 q^{88} -63683.3 q^{89} -127746. q^{91} -39779.5 q^{92} +208931. q^{94} +128065. q^{95} +151168. q^{97} +286795. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8}+O(q^{10}) $ 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8	$ 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8} - 250 q^{10} - 460 q^{11} + 464 q^{13} - 3676 q^{14} + 4612 q^{16} - 4756 q^{17} - 1780 q^{19} - 10314 q^{20} - 4214 q^{22} - 5290 q^{23} + 1330 q^{25} + 5152 q^{26} + 7072 q^{28} - 4048 q^{29} + 2816 q^{31} - 27436 q^{32} + 420 q^{34} - 9452 q^{35} + 2872 q^{37} - 31038 q^{38} + 2618 q^{40} - 34056 q^{41} + 7316 q^{43} - 33562 q^{44} + 4232 q^{46} - 49300 q^{47} + 45118 q^{49} - 44764 q^{50} - 25120 q^{52} - 86676 q^{53} - 2120 q^{55} - 290684 q^{56} - 87408 q^{58} - 67100 q^{59} - 40432 q^{61} - 230992 q^{62} + 136776 q^{64} - 184000 q^{65} - 50108 q^{67} - 270592 q^{68} + 117456 q^{70} - 238584 q^{71} - 13804 q^{73} - 150074 q^{74} - 197622 q^{76} - 116248 q^{77} - 9228 q^{79} - 313010 q^{80} - 68604 q^{82} - 155300 q^{83} + 80444 q^{85} + 80914 q^{86} - 237738 q^{88} - 213732 q^{89} - 264352 q^{91} - 101568 q^{92} + 140280 q^{94} + 123612 q^{95} + 42516 q^{97} + 151388 q^{98}+O(q^{100}) $ 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8 - 250 * q^10 - 460 * q^11 + 464 * q^13 - 3676 * q^14 + 4612 * q^16 - 4756 * q^17 - 1780 * q^19 - 10314 * q^20 - 4214 * q^22 - 5290 * q^23 + 1330 * q^25 + 5152 * q^26 + 7072 * q^28 - 4048 * q^29 + 2816 * q^31 - 27436 * q^32 + 420 * q^34 - 9452 * q^35 + 2872 * q^37 - 31038 * q^38 + 2618 * q^40 - 34056 * q^41 + 7316 * q^43 - 33562 * q^44 + 4232 * q^46 - 49300 * q^47 + 45118 * q^49 - 44764 * q^50 - 25120 * q^52 - 86676 * q^53 - 2120 * q^55 - 290684 * q^56 - 87408 * q^58 - 67100 * q^59 - 40432 * q^61 - 230992 * q^62 + 136776 * q^64 - 184000 * q^65 - 50108 * q^67 - 270592 * q^68 + 117456 * q^70 - 238584 * q^71 - 13804 * q^73 - 150074 * q^74 - 197622 * q^76 - 116248 * q^77 - 9228 * q^79 - 313010 * q^80 - 68604 * q^82 - 155300 * q^83 + 80444 * q^85 + 80914 * q^86 - 237738 * q^88 - 213732 * q^89 - 264352 * q^91 - 101568 * q^92 + 140280 * q^94 + 123612 * q^95 + 42516 * q^97 + 151388 * 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 $

$2$	10.3536	1.83028	0.915139	−	0.403137i	$-0.132080\pi$
$2$	10.3536	1.83028	0.915139	+	0.403137i	$0.132080\pi$
$3$	0	0
$3$	0	0
$4$	75.1975	2.34992
$5$	−71.3591	−1.27651	−0.638255	−	0.769825i	$-0.720344\pi$
$5$	−71.3591	−1.27651	−0.638255	+	0.769825i	$0.720344\pi$
$6$	0	0
$7$	−210.967	−1.62731	−0.813653	−	0.581351i	$-0.802524\pi$
$7$	−210.967	−1.62731	−0.813653	+	0.581351i	$0.802524\pi$
$8$	447.250	2.47073
$9$	0	0
$10$	−738.825	−2.33637
$11$	−170.188	−0.424080	−0.212040	−	0.977261i	$-0.568011\pi$
$11$	−170.188	−0.424080	−0.212040	+	0.977261i	$0.568011\pi$
$12$	0	0
$13$	605.525	0.993743	0.496871	−	0.867824i	$-0.334482\pi$
$13$	605.525	0.993743	0.496871	+	0.867824i	$0.334482\pi$
$14$	−2184.27	−2.97842
$15$	0	0
$16$	2224.34	2.17221
$17$	−1337.03	−1.12207	−0.561033	−	0.827794i	$-0.689596\pi$
$17$	−1337.03	−1.12207	−0.561033	+	0.827794i	$0.689596\pi$
$18$	0	0
$19$	−1794.65	−1.14050	−0.570251	−	0.821471i	$-0.693154\pi$
$19$	−1794.65	−1.14050	−0.570251	+	0.821471i	$0.693154\pi$
$20$	−5366.03	−2.99970
$21$	0	0
$22$	−1762.07	−0.776185
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	1967.12	0.629480
$26$	6269.38	1.81883
$27$	0	0
$28$	−15864.2	−3.82404
$29$	−3006.00	−0.663734	−0.331867	−	0.943326i	$-0.607679\pi$
$29$	−3006.00	−0.663734	−0.331867	+	0.943326i	$0.607679\pi$
$30$	0	0
$31$	−9214.34	−1.72211	−0.861053	−	0.508515i	$-0.830195\pi$
$31$	−9214.34	−1.72211	−0.861053	+	0.508515i	$0.830195\pi$
$32$	8717.97	1.50501
$33$	0	0
$34$	−13843.1	−2.05369
$35$	15054.4	2.07727
$36$	0	0
$37$	6021.35	0.723085	0.361543	−	0.932356i	$-0.382250\pi$
$37$	6021.35	0.723085	0.361543	+	0.932356i	$0.382250\pi$
$38$	−18581.1	−2.08744
$39$	0	0
$40$	−31915.4	−3.15392
$41$	−6606.07	−0.613739	−0.306869	−	0.951752i	$-0.599281\pi$
$41$	−6606.07	−0.613739	−0.306869	+	0.951752i	$0.599281\pi$
$42$	0	0
$43$	22934.8	1.89158	0.945790	−	0.324779i	$-0.105290\pi$
$43$	22934.8	1.89158	0.945790	+	0.324779i	$0.105290\pi$
$44$	−12797.7	−0.996555
$45$	0	0
$46$	−5477.07	−0.381640
$47$	20179.5	1.33250	0.666249	−	0.745729i	$-0.267899\pi$
$47$	20179.5	1.33250	0.666249	+	0.745729i	$0.267899\pi$
$48$	0	0
$49$	27700.0	1.64812
$50$	20366.9	1.15212
$51$	0	0
$52$	45534.0	2.33522
$53$	3604.41	0.176256	0.0881281	−	0.996109i	$-0.471912\pi$
$53$	3604.41	0.176256	0.0881281	+	0.996109i	$0.471912\pi$
$54$	0	0
$55$	12144.5	0.541343
$56$	−94355.0	−4.02064
$57$	0	0
$58$	−31123.0	−1.21482
$59$	37190.1	1.39090	0.695452	−	0.718572i	$-0.255204\pi$
$59$	37190.1	1.39090	0.695452	+	0.718572i	$0.255204\pi$
$60$	0	0
$61$	−31681.4	−1.09013	−0.545067	−	0.838392i	$-0.683496\pi$
$61$	−31681.4	−1.09013	−0.545067	+	0.838392i	$0.683496\pi$
$62$	−95401.8	−3.15194
$63$	0	0
$64$	19083.7	0.582388
$65$	−43209.8	−1.26852
$66$	0	0
$67$	−3782.79	−0.102950	−0.0514749	−	0.998674i	$-0.516392\pi$
$67$	−3782.79	−0.102950	−0.0514749	+	0.998674i	$0.516392\pi$
$68$	−100541.	−2.63677
$69$	0	0
$70$	155868.	3.80199
$71$	−6321.88	−0.148833	−0.0744167	−	0.997227i	$-0.523709\pi$
$71$	−6321.88	−0.148833	−0.0744167	+	0.997227i	$0.523709\pi$
$72$	0	0
$73$	12637.8	0.277564	0.138782	−	0.990323i	$-0.455681\pi$
$73$	12637.8	0.277564	0.138782	+	0.990323i	$0.455681\pi$
$74$	62342.8	1.32345
$75$	0	0
$76$	−134953.	−2.68009
$77$	35904.1	0.690108
$78$	0	0
$79$	79327.3	1.43006	0.715032	−	0.699092i	$-0.246412\pi$
$79$	79327.3	1.43006	0.715032	+	0.699092i	$0.246412\pi$
$80$	−158727.	−2.77285
$81$	0	0
$82$	−68396.7	−1.12331
$83$	−114368.	−1.82226	−0.911130	−	0.412118i	$-0.864789\pi$
$83$	−114368.	−1.82226	−0.911130	+	0.412118i	$0.864789\pi$
$84$	0	0
$85$	95409.1	1.43233
$86$	237459.	3.46212
$87$	0	0
$88$	−76116.8	−1.04779
$89$	−63683.3	−0.852217	−0.426109	−	0.904672i	$-0.640116\pi$
$89$	−63683.3	−0.852217	−0.426109	+	0.904672i	$0.640116\pi$
$90$	0	0
$91$	−127746.	−1.61712
$92$	−39779.5	−0.489992
$93$	0	0
$94$	208931.	2.43884
$95$	128065.	1.45586
$96$	0	0
$97$	151168.	1.63129	0.815644	−	0.578553i	$-0.196383\pi$
$97$	151168.	1.63129	0.815644	+	0.578553i	$0.196383\pi$
$98$	286795.	3.01652
$99$	0	0
$100$	147923.	1.47923
$101$	−188855.	−1.84215	−0.921074	−	0.389387i	$-0.872687\pi$
$101$	−188855.	−1.84215	−0.921074	+	0.389387i	$0.872687\pi$
$102$	0	0
$103$	143432.	1.33215	0.666074	−	0.745886i	$-0.267974\pi$
$103$	143432.	1.33215	0.666074	+	0.745886i	$0.267974\pi$
$104$	270821.	2.45527
$105$	0	0
$106$	37318.7	0.322598
$107$	−88610.8	−0.748216	−0.374108	−	0.927385i	$-0.622051\pi$
$107$	−88610.8	−0.748216	−0.374108	+	0.927385i	$0.622051\pi$
$108$	0	0
$109$	−94789.3	−0.764176	−0.382088	−	0.924126i	$-0.624795\pi$
$109$	−94789.3	−0.764176	−0.382088	+	0.924126i	$0.624795\pi$
$110$	125739.	0.990809
$111$	0	0
$112$	−469262.	−3.53485
$113$	−84969.6	−0.625990	−0.312995	−	0.949755i	$-0.601332\pi$
$113$	−84969.6	−0.625990	−0.312995	+	0.949755i	$0.601332\pi$
$114$	0	0
$115$	37749.0	0.266171
$116$	−226044.	−1.55972
$117$	0	0
$118$	385052.	2.54574
$119$	282069.	1.82594
$120$	0	0
$121$	−132087.	−0.820156
$122$	−328018.	−1.99525
$123$	0	0
$124$	−692895.	−4.04681
$125$	82625.0	0.472973
$126$	0	0
$127$	−826.453	−0.00454683	−0.00227342	−	0.999997i	$-0.500724\pi$
$127$	−826.453	−0.00454683	−0.00227342	+	0.999997i	$0.500724\pi$
$128$	−81389.9	−0.439082
$129$	0	0
$130$	−447378.	−2.32175
$131$	−262321.	−1.33554	−0.667768	−	0.744370i	$-0.732750\pi$
$131$	−262321.	−1.33554	−0.667768	+	0.744370i	$0.732750\pi$
$132$	0	0
$133$	378612.	1.85594
$134$	−39165.6	−0.188427
$135$	0	0
$136$	−597986.	−2.77232
$137$	12970.6	0.0590415	0.0295207	−	0.999564i	$-0.490602\pi$
$137$	12970.6	0.0590415	0.0295207	+	0.999564i	$0.490602\pi$
$138$	0	0
$139$	−89440.8	−0.392644	−0.196322	−	0.980539i	$-0.562900\pi$
$139$	−89440.8	−0.392644	−0.196322	+	0.980539i	$0.562900\pi$
$140$	1.13205e6	4.88143
$141$	0	0
$142$	−65454.4	−0.272407
$143$	−103053.	−0.421427
$144$	0	0
$145$	214506.	0.847264
$146$	130847.	0.508020
$147$	0	0
$148$	452790.	1.69919
$149$	−317159.	−1.17034	−0.585170	−	0.810911i	$-0.698972\pi$
$149$	−317159.	−1.17034	−0.585170	+	0.810911i	$0.698972\pi$
$150$	0	0
$151$	−96753.6	−0.345322	−0.172661	−	0.984981i	$-0.555237\pi$
$151$	−96753.6	−0.345322	−0.172661	+	0.984981i	$0.555237\pi$
$152$	−802658.	−2.81787
$153$	0	0
$154$	371737.	1.26309
$155$	657527.	2.19829
$156$	0	0
$157$	133196.	0.431261	0.215631	−	0.976475i	$-0.430819\pi$
$157$	133196.	0.431261	0.215631	+	0.976475i	$0.430819\pi$
$158$	821325.	2.61741
$159$	0	0
$160$	−622107.	−1.92117
$161$	111601.	0.339317
$162$	0	0
$163$	45781.6	0.134965	0.0674827	−	0.997720i	$-0.478503\pi$
$163$	45781.6	0.134965	0.0674827	+	0.997720i	$0.478503\pi$
$164$	−496760.	−1.44224
$165$	0	0
$166$	−1.18413e6	−3.33525
$167$	38039.4	0.105546	0.0527731	−	0.998607i	$-0.483194\pi$
$167$	38039.4	0.105546	0.0527731	+	0.998607i	$0.483194\pi$
$168$	0	0
$169$	−4631.97	−0.0124752
$170$	987830.	2.62156
$171$	0	0
$172$	1.72464e6	4.44506
$173$	−685175.	−1.74055	−0.870275	−	0.492566i	$-0.836059\pi$
$173$	−685175.	−1.74055	−0.870275	+	0.492566i	$0.836059\pi$
$174$	0	0
$175$	−414998.	−1.02436
$176$	−378557.	−0.921190
$177$	0	0
$178$	−659352.	−1.55980
$179$	−326298.	−0.761170	−0.380585	−	0.924746i	$-0.624277\pi$
$179$	−326298.	−0.761170	−0.380585	+	0.924746i	$0.624277\pi$
$180$	0	0
$181$	−359932.	−0.816627	−0.408313	−	0.912842i	$-0.633883\pi$
$181$	−359932.	−0.816627	−0.408313	+	0.912842i	$0.633883\pi$
$182$	−1.32263e6	−2.95979
$183$	0	0
$184$	−236595.	−0.515183
$185$	−429678.	−0.923026
$186$	0	0
$187$	227547.	0.475846
$188$	1.51745e6	3.13126
$189$	0	0
$190$	1.32593e6	2.66463
$191$	310829.	0.616508	0.308254	−	0.951304i	$-0.400255\pi$
$191$	310829.	0.616508	0.308254	+	0.951304i	$0.400255\pi$
$192$	0	0
$193$	−136244.	−0.263284	−0.131642	−	0.991297i	$-0.542025\pi$
$193$	−136244.	−0.263284	−0.131642	+	0.991297i	$0.542025\pi$
$194$	1.56514e6	2.98571
$195$	0	0
$196$	2.08297e6	3.87296
$197$	823205.	1.51127	0.755635	−	0.654992i	$-0.227328\pi$
$197$	823205.	1.51127	0.755635	+	0.654992i	$0.227328\pi$
$198$	0	0
$199$	654175.	1.17101	0.585506	−	0.810668i	$-0.300896\pi$
$199$	654175.	1.17101	0.585506	+	0.810668i	$0.300896\pi$
$200$	879797.	1.55528
$201$	0	0
$202$	−1.95533e6	−3.37165
$203$	634167.	1.08010
$204$	0	0
$205$	471403.	0.783444
$206$	1.48504e6	2.43820
$207$	0	0
$208$	1.34689e6	2.15862
$209$	305428.	0.483664
$210$	0	0
$211$	812196.	1.25590	0.627950	−	0.778254i	$-0.283894\pi$
$211$	812196.	1.25590	0.627950	+	0.778254i	$0.283894\pi$
$212$	271043.	0.414188
$213$	0	0
$214$	−917442.	−1.36944
$215$	−1.63661e6	−2.41462
$216$	0	0
$217$	1.94392e6	2.80239
$218$	−981413.	−1.39865
$219$	0	0
$220$	913235.	1.27211
$221$	−809604.	−1.11504
$222$	0	0
$223$	−252062.	−0.339426	−0.169713	−	0.985494i	$-0.554284\pi$
$223$	−252062.	−0.339426	−0.169713	+	0.985494i	$0.554284\pi$
$224$	−1.83920e6	−2.44912
$225$	0	0
$226$	−879744.	−1.14574
$227$	−999309.	−1.28717	−0.643584	−	0.765376i	$-0.722553\pi$
$227$	−999309.	−1.28717	−0.643584	+	0.765376i	$0.722553\pi$
$228$	0	0
$229$	963621.	1.21428	0.607138	−	0.794596i	$-0.292317\pi$
$229$	963621.	1.21428	0.607138	+	0.794596i	$0.292317\pi$
$230$	390839.	0.487167
$231$	0	0
$232$	−1.34444e6	−1.63991
$233$	86902.2	0.104868	0.0524338	−	0.998624i	$-0.483302\pi$
$233$	86902.2	0.104868	0.0524338	+	0.998624i	$0.483302\pi$
$234$	0	0
$235$	−1.43999e6	−1.70095
$236$	2.79660e6	3.26852
$237$	0	0
$238$	2.92043e6	3.34199
$239$	−1.42740e6	−1.61641	−0.808206	−	0.588900i	$-0.799561\pi$
$239$	−1.42740e6	−1.61641	−0.808206	+	0.588900i	$0.799561\pi$
$240$	0	0
$241$	−137084.	−0.152035	−0.0760176	−	0.997106i	$-0.524221\pi$
$241$	−137084.	−0.152035	−0.0760176	+	0.997106i	$0.524221\pi$
$242$	−1.36758e6	−1.50111
$243$	0	0
$244$	−2.38236e6	−2.56173
$245$	−1.97665e6	−2.10385
$246$	0	0
$247$	−1.08671e6	−1.13336
$248$	−4.12112e6	−4.25486
$249$	0	0
$250$	855468.	0.865672
$251$	947694.	0.949476	0.474738	−	0.880127i	$-0.342543\pi$
$251$	947694.	0.949476	0.474738	+	0.880127i	$0.342543\pi$
$252$	0	0
$253$	90029.6	0.0884269
$254$	−8556.78	−0.00832197
$255$	0	0
$256$	−1.45336e6	−1.38603
$257$	−1.25100e6	−1.18148	−0.590739	−	0.806862i	$-0.701164\pi$
$257$	−1.25100e6	−1.18148	−0.590739	+	0.806862i	$0.701164\pi$
$258$	0	0
$259$	−1.27030e6	−1.17668
$260$	−3.24926e6	−2.98093
$261$	0	0
$262$	−2.71598e6	−2.44440
$263$	305663.	0.272492	0.136246	−	0.990675i	$-0.456496\pi$
$263$	305663.	0.272492	0.136246	+	0.990675i	$0.456496\pi$
$264$	0	0
$265$	−257208.	−0.224993
$266$	3.92000e6	3.39689
$267$	0	0
$268$	−284456.	−0.241924
$269$	712.235	0.000600127 0	0.000300063	−	1.00000i	$-0.499904\pi$
$269$	712.235	0.000600127 0	0.000300063		1.00000i	$0.499904\pi$
$270$	0	0
$271$	−726592.	−0.600990	−0.300495	−	0.953783i	$-0.597152\pi$
$271$	−726592.	−0.600990	−0.300495	+	0.953783i	$0.597152\pi$
$272$	−2.97401e6	−2.43736
$273$	0	0
$274$	134292.	0.108062
$275$	−334782.	−0.266950
$276$	0	0
$277$	−650868.	−0.509675	−0.254837	−	0.966984i	$-0.582022\pi$
$277$	−650868.	−0.509675	−0.254837	+	0.966984i	$0.582022\pi$
$278$	−926036.	−0.718648
$279$	0	0
$280$	6.73309e6	5.13238
$281$	667888.	0.504589	0.252294	−	0.967650i	$-0.418815\pi$
$281$	667888.	0.504589	0.252294	+	0.967650i	$0.418815\pi$
$282$	0	0
$283$	−1.91825e6	−1.42377	−0.711883	−	0.702298i	$-0.752157\pi$
$283$	−1.91825e6	−1.42377	−0.711883	+	0.702298i	$0.752157\pi$
$284$	−475390.	−0.349747
$285$	0	0
$286$	−1.06698e6	−0.771328
$287$	1.39366e6	0.998740
$288$	0	0
$289$	367787.	0.259031
$290$	2.22091e6	1.55073
$291$	0	0
$292$	950328.	0.652254
$293$	357387.	0.243203	0.121602	−	0.992579i	$-0.461197\pi$
$293$	357387.	0.243203	0.121602	+	0.992579i	$0.461197\pi$
$294$	0	0
$295$	−2.65385e6	−1.77550
$296$	2.69305e6	1.78655
$297$	0	0
$298$	−3.28375e6	−2.14205
$299$	−320323.	−0.207210
$300$	0	0
$301$	−4.83849e6	−3.07818
$302$	−1.00175e6	−0.632036
$303$	0	0
$304$	−3.99191e6	−2.47741
$305$	2.26076e6	1.39157
$306$	0	0
$307$	242810.	0.147035	0.0735176	−	0.997294i	$-0.476577\pi$
$307$	242810.	0.147035	0.0735176	+	0.997294i	$0.476577\pi$
$308$	2.69990e6	1.62170
$309$	0	0
$310$	6.80779e6	4.02348
$311$	−510526.	−0.299307	−0.149654	−	0.988738i	$-0.547816\pi$
$311$	−510526.	−0.299307	−0.149654	+	0.988738i	$0.547816\pi$
$312$	0	0
$313$	268272.	0.154780	0.0773901	−	0.997001i	$-0.475341\pi$
$313$	268272.	0.154780	0.0773901	+	0.997001i	$0.475341\pi$
$314$	1.37906e6	0.789328
$315$	0	0
$316$	5.96522e6	3.36054
$317$	1.17566e6	0.657102	0.328551	−	0.944486i	$-0.393440\pi$
$317$	1.17566e6	0.657102	0.328551	+	0.944486i	$0.393440\pi$
$318$	0	0
$319$	511586.	0.281477
$320$	−1.36180e6	−0.743424
$321$	0	0
$322$	1.15548e6	0.621044
$323$	2.39950e6	1.27972
$324$	0	0
$325$	1.19114e6	0.625541
$326$	474006.	0.247024
$327$	0	0
$328$	−2.95457e6	−1.51638
$329$	−4.25721e6	−2.16838
$330$	0	0
$331$	3.74833e6	1.88047	0.940237	−	0.340520i	$-0.110603\pi$
$331$	3.74833e6	1.88047	0.940237	+	0.340520i	$0.110603\pi$
$332$	−8.60021e6	−4.28217
$333$	0	0
$334$	393846.	0.193179
$335$	269937.	0.131417
$336$	0	0
$337$	−3.05514e6	−1.46540	−0.732701	−	0.680551i	$-0.761740\pi$
$337$	−3.05514e6	−1.46540	−0.732701	+	0.680551i	$0.761740\pi$
$338$	−47957.7	−0.0228332
$339$	0	0
$340$	7.17453e6	3.36586
$341$	1.56817e6	0.730312
$342$	0	0
$343$	−2.29806e6	−1.05469
$344$	1.02576e7	4.67359
$345$	0	0
$346$	−7.09405e6	−3.18569
$347$	−4.00997e6	−1.78779	−0.893896	−	0.448274i	$-0.852039\pi$
$347$	−4.00997e6	−1.78779	−0.893896	+	0.448274i	$0.852039\pi$
$348$	0	0
$349$	1.17172e6	0.514944	0.257472	−	0.966286i	$-0.417111\pi$
$349$	1.17172e6	0.514944	0.257472	+	0.966286i	$0.417111\pi$
$350$	−4.29673e6	−1.87486
$351$	0	0
$352$	−1.48370e6	−0.638247
$353$	2.17711e6	0.929917	0.464958	−	0.885333i	$-0.346069\pi$
$353$	2.17711e6	0.929917	0.464958	+	0.885333i	$0.346069\pi$
$354$	0	0
$355$	451124.	0.189988
$356$	−4.78882e6	−2.00264
$357$	0	0
$358$	−3.37837e6	−1.39315
$359$	1.01162e6	0.414269	0.207135	−	0.978312i	$-0.433586\pi$
$359$	1.01162e6	0.414269	0.207135	+	0.978312i	$0.433586\pi$
$360$	0	0
$361$	744669.	0.300743
$362$	−3.72660e6	−1.49465
$363$	0	0
$364$	−9.60616e6	−3.80011
$365$	−901820.	−0.354313
$366$	0	0
$367$	−1.83005e6	−0.709249	−0.354625	−	0.935009i	$-0.615391\pi$
$367$	−1.83005e6	−0.709249	−0.354625	+	0.935009i	$0.615391\pi$
$368$	−1.17668e6	−0.452937
$369$	0	0
$370$	−4.44872e6	−1.68939
$371$	−760411.	−0.286823
$372$	0	0
$373$	206897.	0.0769986	0.0384993	−	0.999259i	$-0.487742\pi$
$373$	206897.	0.0769986	0.0384993	+	0.999259i	$0.487742\pi$
$374$	2.35593e6	0.870931
$375$	0	0
$376$	9.02530e6	3.29225
$377$	−1.82021e6	−0.659581
$378$	0	0
$379$	4.49105e6	1.60602	0.803008	−	0.595968i	$-0.203232\pi$
$379$	4.49105e6	1.60602	0.803008	+	0.595968i	$0.203232\pi$
$380$	9.63014e6	3.42116
$381$	0	0
$382$	3.21821e6	1.12838
$383$	−2.30581e6	−0.803204	−0.401602	−	0.915814i	$-0.631547\pi$
$383$	−2.30581e6	−0.803204	−0.401602	+	0.915814i	$0.631547\pi$
$384$	0	0
$385$	−2.56208e6	−0.880930
$386$	−1.41062e6	−0.481883
$387$	0	0
$388$	1.13675e7	3.83340
$389$	1.17176e6	0.392612	0.196306	−	0.980543i	$-0.437105\pi$
$389$	1.17176e6	0.392612	0.196306	+	0.980543i	$0.437105\pi$
$390$	0	0
$391$	707288.	0.233967
$392$	1.23888e7	4.07207
$393$	0	0
$394$	8.52315e6	2.76605
$395$	−5.66073e6	−1.82549
$396$	0	0
$397$	1.03765e6	0.330428	0.165214	−	0.986258i	$-0.447169\pi$
$397$	1.03765e6	0.330428	0.165214	+	0.986258i	$0.447169\pi$
$398$	6.77308e6	2.14328
$399$	0	0
$400$	4.37556e6	1.36736
$401$	6.13691e6	1.90585	0.952925	−	0.303205i	$-0.0980566\pi$
$401$	6.13691e6	1.90585	0.952925	+	0.303205i	$0.0980566\pi$
$402$	0	0
$403$	−5.57952e6	−1.71133
$404$	−1.42014e7	−4.32890
$405$	0	0
$406$	6.56592e6	1.97688
$407$	−1.02476e6	−0.306646
$408$	0	0
$409$	−2.13261e6	−0.630380	−0.315190	−	0.949029i	$-0.602068\pi$
$409$	−2.13261e6	−0.630380	−0.315190	+	0.949029i	$0.602068\pi$
$410$	4.88073e6	1.43392
$411$	0	0
$412$	1.07857e7	3.13044
$413$	−7.84588e6	−2.26343
$414$	0	0
$415$	8.16122e6	2.32614
$416$	5.27895e6	1.49560
$417$	0	0
$418$	3.16229e6	0.885240
$419$	−6.05469e6	−1.68483	−0.842417	−	0.538826i	$-0.818868\pi$
$419$	−6.05469e6	−1.68483	−0.842417	+	0.538826i	$0.818868\pi$
$420$	0	0
$421$	−4.62140e6	−1.27077	−0.635386	−	0.772194i	$-0.719159\pi$
$421$	−4.62140e6	−1.27077	−0.635386	+	0.772194i	$0.719159\pi$
$422$	8.40917e6	2.29865
$423$	0	0
$424$	1.61207e6	0.435482
$425$	−2.63010e6	−0.706318
$426$	0	0
$427$	6.68373e6	1.77398
$428$	−6.66331e6	−1.75825
$429$	0	0
$430$	−1.69448e7	−4.41943
$431$	1.49369e6	0.387319	0.193659	−	0.981069i	$-0.437964\pi$
$431$	1.49369e6	0.387319	0.193659	+	0.981069i	$0.437964\pi$
$432$	0	0
$433$	−7.44271e6	−1.90770	−0.953852	−	0.300276i	$-0.902921\pi$
$433$	−7.44271e6	−1.90770	−0.953852	+	0.300276i	$0.902921\pi$
$434$	2.01266e7	5.12916
$435$	0	0
$436$	−7.12792e6	−1.79575
$437$	949370.	0.237811
$438$	0	0
$439$	−2.11446e6	−0.523647	−0.261823	−	0.965116i	$-0.584324\pi$
$439$	−2.11446e6	−0.523647	−0.261823	+	0.965116i	$0.584324\pi$
$440$	5.43163e6	1.33751
$441$	0	0
$442$	−8.38234e6	−2.04084
$443$	2.16454e6	0.524031	0.262015	−	0.965064i	$-0.415613\pi$
$443$	2.16454e6	0.524031	0.262015	+	0.965064i	$0.415613\pi$
$444$	0	0
$445$	4.54438e6	1.08786
$446$	−2.60975e6	−0.621244
$447$	0	0
$448$	−4.02602e6	−0.947723
$449$	5.65548e6	1.32390	0.661948	−	0.749550i	$-0.269730\pi$
$449$	5.65548e6	1.32390	0.661948	+	0.749550i	$0.269730\pi$
$450$	0	0
$451$	1.12428e6	0.260274
$452$	−6.38950e6	−1.47103
$453$	0	0
$454$	−1.03465e7	−2.35588
$455$	9.11583e6	2.06427
$456$	0	0
$457$	213062.	0.0477216	0.0238608	−	0.999715i	$-0.492404\pi$
$457$	213062.	0.0477216	0.0238608	+	0.999715i	$0.492404\pi$
$458$	9.97697e6	2.22246
$459$	0	0
$460$	2.83863e6	0.625481
$461$	2.84868e6	0.624296	0.312148	−	0.950033i	$-0.398951\pi$
$461$	2.84868e6	0.624296	0.312148	+	0.950033i	$0.398951\pi$
$462$	0	0
$463$	−981530.	−0.212790	−0.106395	−	0.994324i	$-0.533931\pi$
$463$	−981530.	−0.212790	−0.106395	+	0.994324i	$0.533931\pi$
$464$	−6.68637e6	−1.44177
$465$	0	0
$466$	899753.	0.191937
$467$	1.12083e6	0.237820	0.118910	−	0.992905i	$-0.462060\pi$
$467$	1.12083e6	0.237820	0.118910	+	0.992905i	$0.462060\pi$
$468$	0	0
$469$	798043.	0.167531
$470$	−1.49092e7	−3.11321
$471$	0	0
$472$	1.66333e7	3.43655
$473$	−3.90324e6	−0.802182
$474$	0	0
$475$	−3.53030e6	−0.717923
$476$	2.12108e7	4.29082
$477$	0	0
$478$	−1.47788e7	−2.95848
$479$	8.75292e6	1.74307	0.871534	−	0.490335i	$-0.163126\pi$
$479$	8.75292e6	1.74307	0.871534	+	0.490335i	$0.163126\pi$
$480$	0	0
$481$	3.64608e6	0.718561
$482$	−1.41932e6	−0.278267
$483$	0	0
$484$	−9.93260e6	−1.92730
$485$	−1.07872e7	−2.08236
$486$	0	0
$487$	4.58029e6	0.875126	0.437563	−	0.899188i	$-0.355842\pi$
$487$	4.58029e6	0.875126	0.437563	+	0.899188i	$0.355842\pi$
$488$	−1.41695e7	−2.69343
$489$	0	0
$490$	−2.04655e7	−3.85062
$491$	3.32030e6	0.621546	0.310773	−	0.950484i	$-0.399412\pi$
$491$	3.32030e6	0.621546	0.310773	+	0.950484i	$0.399412\pi$
$492$	0	0
$493$	4.01911e6	0.744753
$494$	−1.12513e7	−2.07437
$495$	0	0
$496$	−2.04958e7	−3.74077
$497$	1.33371e6	0.242197
$498$	0	0
$499$	9.33617e6	1.67848	0.839242	−	0.543758i	$-0.182999\pi$
$499$	9.33617e6	1.67848	0.839242	+	0.543758i	$0.182999\pi$
$500$	6.21319e6	1.11145
$501$	0	0
$502$	9.81207e6	1.73781
$503$	7.75961e6	1.36748	0.683738	−	0.729728i	$-0.260353\pi$
$503$	7.75961e6	1.36748	0.683738	+	0.729728i	$0.260353\pi$
$504$	0	0
$505$	1.34765e7	2.35152
$506$	932133.	0.161846
$507$	0	0
$508$	−62147.2	−0.0106847
$509$	−8.78414e6	−1.50281	−0.751406	−	0.659840i	$-0.770624\pi$
$509$	−8.78414e6	−1.50281	−0.751406	+	0.659840i	$0.770624\pi$
$510$	0	0
$511$	−2.66615e6	−0.451681
$512$	−1.24430e7	−2.09774
$513$	0	0
$514$	−1.29524e7	−2.16244
$515$	−1.02352e7	−1.70050
$516$	0	0
$517$	−3.43432e6	−0.565086
$518$	−1.31523e7	−2.15365
$519$	0	0
$520$	−1.93256e7	−3.13418
$521$	6.42822e6	1.03752	0.518760	−	0.854920i	$-0.326394\pi$
$521$	6.42822e6	1.03752	0.518760	+	0.854920i	$0.326394\pi$
$522$	0	0
$523$	−4.75955e6	−0.760872	−0.380436	−	0.924807i	$-0.624226\pi$
$523$	−4.75955e6	−0.760872	−0.380436	+	0.924807i	$0.624226\pi$
$524$	−1.97259e7	−3.13840
$525$	0	0
$526$	3.16472e6	0.498736
$527$	1.23198e7	1.93232
$528$	0	0
$529$	279841.	0.0434783
$530$	−2.66303e6	−0.411800
$531$	0	0
$532$	2.84706e7	4.36132
$533$	−4.00014e6	−0.609898
$534$	0	0
$535$	6.32319e6	0.955106
$536$	−1.69185e6	−0.254361
$537$	0	0
$538$	7374.22	0.00109840
$539$	−4.71422e6	−0.698936
$540$	0	0
$541$	7.85812e6	1.15432	0.577159	−	0.816632i	$-0.304161\pi$
$541$	7.85812e6	1.15432	0.577159	+	0.816632i	$0.304161\pi$
$542$	−7.52286e6	−1.09998
$543$	0	0
$544$	−1.16562e7	−1.68872
$545$	6.76408e6	0.975479
$546$	0	0
$547$	1.00196e7	1.43180	0.715902	−	0.698201i	$-0.246016\pi$
$547$	1.00196e7	1.43180	0.715902	+	0.698201i	$0.246016\pi$
$548$	975353.	0.138743
$549$	0	0
$550$	−3.46620e6	−0.488593
$551$	5.39472e6	0.756990
$552$	0	0
$553$	−1.67354e7	−2.32715
$554$	−6.73884e6	−0.932847
$555$	0	0
$556$	−6.72572e6	−0.922682
$557$	5.74512e6	0.784623	0.392312	−	0.919832i	$-0.371675\pi$
$557$	5.74512e6	0.784623	0.392312	+	0.919832i	$0.371675\pi$
$558$	0	0
$559$	1.38876e7	1.87974
$560$	3.34861e7	4.51227
$561$	0	0
$562$	6.91506e6	0.923539
$563$	−1.23215e6	−0.163829	−0.0819146	−	0.996639i	$-0.526103\pi$
$563$	−1.23215e6	−0.163829	−0.0819146	+	0.996639i	$0.526103\pi$
$564$	0	0
$565$	6.06336e6	0.799084
$566$	−1.98608e7	−2.60589
$567$	0	0
$568$	−2.82746e6	−0.367728
$569$	−8.80266e6	−1.13981	−0.569906	−	0.821710i	$-0.693020\pi$
$569$	−8.80266e6	−1.13981	−0.569906	+	0.821710i	$0.693020\pi$
$570$	0	0
$571$	−7.49333e6	−0.961800	−0.480900	−	0.876776i	$-0.659690\pi$
$571$	−7.49333e6	−0.961800	−0.480900	+	0.876776i	$0.659690\pi$
$572$	−7.74935e6	−0.990320
$573$	0	0
$574$	1.44294e7	1.82797
$575$	−1.04061e6	−0.131256
$576$	0	0
$577$	−1.36469e7	−1.70646	−0.853228	−	0.521538i	$-0.825358\pi$
$577$	−1.36469e7	−1.70646	−0.853228	+	0.521538i	$0.825358\pi$
$578$	3.80793e6	0.474099
$579$	0	0
$580$	1.61303e7	1.99100
$581$	2.41279e7	2.96537
$582$	0	0
$583$	−613429.	−0.0747468
$584$	5.65224e6	0.685786
$585$	0	0
$586$	3.70024e6	0.445130
$587$	−2.92745e6	−0.350666	−0.175333	−	0.984509i	$-0.556100\pi$
$587$	−2.92745e6	−0.350666	−0.175333	+	0.984509i	$0.556100\pi$
$588$	0	0
$589$	1.65365e7	1.96406
$590$	−2.74770e7	−3.24967
$591$	0	0
$592$	1.33935e7	1.57069
$593$	−5.29722e6	−0.618603	−0.309301	−	0.950964i	$-0.600095\pi$
$593$	−5.29722e6	−0.618603	−0.309301	+	0.950964i	$0.600095\pi$
$594$	0	0
$595$	−2.01282e7	−2.33084
$596$	−2.38496e7	−2.75020
$597$	0	0
$598$	−3.31650e6	−0.379252
$599$	1.35436e7	1.54229	0.771147	−	0.636657i	$-0.219683\pi$
$599$	1.35436e7	1.54229	0.771147	+	0.636657i	$0.219683\pi$
$600$	0	0
$601$	−3.10785e6	−0.350973	−0.175486	−	0.984482i	$-0.556150\pi$
$601$	−3.10785e6	−0.350973	−0.175486	+	0.984482i	$0.556150\pi$
$602$	−5.00959e7	−5.63392
$603$	0	0
$604$	−7.27562e6	−0.811480
$605$	9.42561e6	1.04694
$606$	0	0
$607$	−5.80425e6	−0.639402	−0.319701	−	0.947518i	$-0.603582\pi$
$607$	−5.80425e6	−0.639402	−0.319701	+	0.947518i	$0.603582\pi$
$608$	−1.56457e7	−1.71647
$609$	0	0
$610$	2.34070e7	2.54696
$611$	1.22192e7	1.32416
$612$	0	0
$613$	−2.53122e6	−0.272069	−0.136035	−	0.990704i	$-0.543436\pi$
$613$	−2.53122e6	−0.272069	−0.136035	+	0.990704i	$0.543436\pi$
$614$	2.51397e6	0.269115
$615$	0	0
$616$	1.60581e7	1.70507
$617$	−1.03128e7	−1.09059	−0.545296	−	0.838244i	$-0.683583\pi$
$617$	−1.03128e7	−1.09059	−0.545296	+	0.838244i	$0.683583\pi$
$618$	0	0
$619$	−4.15021e6	−0.435355	−0.217677	−	0.976021i	$-0.569848\pi$
$619$	−4.15021e6	−0.435355	−0.217677	+	0.976021i	$0.569848\pi$
$620$	4.94444e7	5.16580
$621$	0	0
$622$	−5.28580e6	−0.547816
$623$	1.34351e7	1.38682
$624$	0	0
$625$	−1.20433e7	−1.23323
$626$	2.77759e6	0.283291
$627$	0	0
$628$	1.00160e7	1.01343
$629$	−8.05071e6	−0.811349
$630$	0	0
$631$	1.70512e7	1.70483	0.852416	−	0.522864i	$-0.175136\pi$
$631$	1.70512e7	1.70483	0.852416	+	0.522864i	$0.175136\pi$
$632$	3.54792e7	3.53330
$633$	0	0
$634$	1.21723e7	1.20268
$635$	58975.0	0.00580408
$636$	0	0
$637$	1.67730e7	1.63781
$638$	5.29677e6	0.515181
$639$	0	0
$640$	5.80791e6	0.560493
$641$	−1.64702e7	−1.58327	−0.791633	−	0.610997i	$-0.790769\pi$
$641$	−1.64702e7	−1.58327	−0.791633	+	0.610997i	$0.790769\pi$
$642$	0	0
$643$	6.28693e6	0.599668	0.299834	−	0.953991i	$-0.403069\pi$
$643$	6.28693e6	0.599668	0.299834	+	0.953991i	$0.403069\pi$
$644$	8.39215e6	0.797367
$645$	0	0
$646$	2.48435e7	2.34224
$647$	5.21472e6	0.489746	0.244873	−	0.969555i	$-0.421254\pi$
$647$	5.21472e6	0.489746	0.244873	+	0.969555i	$0.421254\pi$
$648$	0	0
$649$	−6.32932e6	−0.589855
$650$	1.23327e7	1.14491
$651$	0	0
$652$	3.44266e6	0.317158
$653$	−4.72335e6	−0.433479	−0.216739	−	0.976230i	$-0.569542\pi$
$653$	−4.72335e6	−0.433479	−0.216739	+	0.976230i	$0.569542\pi$
$654$	0	0
$655$	1.87190e7	1.70483
$656$	−1.46941e7	−1.33317
$657$	0	0
$658$	−4.40776e7	−3.96874
$659$	−5.27529e6	−0.473187	−0.236593	−	0.971609i	$-0.576031\pi$
$659$	−5.27529e6	−0.473187	−0.236593	+	0.971609i	$0.576031\pi$
$660$	0	0
$661$	2.64941e6	0.235855	0.117928	−	0.993022i	$-0.462375\pi$
$661$	2.64941e6	0.235855	0.117928	+	0.993022i	$0.462375\pi$
$662$	3.88088e7	3.44179
$663$	0	0
$664$	−5.11513e7	−4.50232
$665$	−2.70174e7	−2.36913
$666$	0	0
$667$	1.59018e6	0.138398
$668$	2.86047e6	0.248025
$669$	0	0
$670$	2.79482e6	0.240529
$671$	5.39181e6	0.462305
$672$	0	0
$673$	−2.58779e6	−0.220237	−0.110119	−	0.993918i	$-0.535123\pi$
$673$	−2.58779e6	−0.220237	−0.110119	+	0.993918i	$0.535123\pi$
$674$	−3.16318e7	−2.68209
$675$	0	0
$676$	−348313.	−0.0293158
$677$	−1.06620e7	−0.894064	−0.447032	−	0.894518i	$-0.647519\pi$
$677$	−1.06620e7	−0.894064	−0.447032	+	0.894518i	$0.647519\pi$
$678$	0	0
$679$	−3.18915e7	−2.65461
$680$	4.26718e7	3.53890
$681$	0	0
$682$	1.62363e7	1.33667
$683$	1.21075e7	0.993119	0.496559	−	0.868003i	$-0.334596\pi$
$683$	1.21075e7	0.993119	0.496559	+	0.868003i	$0.334596\pi$
$684$	0	0
$685$	−925568.	−0.0753671
$686$	−2.37932e7	−1.93038
$687$	0	0
$688$	5.10149e7	4.10890
$689$	2.18256e6	0.175153
$690$	0	0
$691$	1.66501e7	1.32654	0.663271	−	0.748379i	$-0.269168\pi$
$691$	1.66501e7	1.32654	0.663271	+	0.748379i	$0.269168\pi$
$692$	−5.15235e7	−4.09015
$693$	0	0
$694$	−4.15177e7	−3.27216
$695$	6.38242e6	0.501214
$696$	0	0
$697$	8.83250e6	0.688655
$698$	1.21315e7	0.942490
$699$	0	0
$700$	−3.12068e7	−2.40716
$701$	569003.	0.0437340	0.0218670	−	0.999761i	$-0.493039\pi$
$701$	569003.	0.0437340	0.0218670	+	0.999761i	$0.493039\pi$
$702$	0	0
$703$	−1.08062e7	−0.824679
$704$	−3.24782e6	−0.246979
$705$	0	0
$706$	2.25410e7	1.70201
$707$	3.98421e7	2.99774
$708$	0	0
$709$	−1.70513e6	−0.127392	−0.0636959	−	0.997969i	$-0.520289\pi$
$709$	−1.70513e6	−0.127392	−0.0636959	+	0.997969i	$0.520289\pi$
$710$	4.67077e6	0.347730
$711$	0	0
$712$	−2.84824e7	−2.10560
$713$	4.87439e6	0.359084
$714$	0	0
$715$	7.35380e6	0.537956
$716$	−2.45368e7	−1.78869
$717$	0	0
$718$	1.04740e7	0.758228
$719$	7.59885e6	0.548183	0.274092	−	0.961704i	$-0.411623\pi$
$719$	7.59885e6	0.548183	0.274092	+	0.961704i	$0.411623\pi$
$720$	0	0
$721$	−3.02593e7	−2.16781
$722$	7.71002e6	0.550443
$723$	0	0
$724$	−2.70660e7	−1.91901
$725$	−5.91318e6	−0.417807
$726$	0	0
$727$	7.24703e6	0.508539	0.254269	−	0.967133i	$-0.418165\pi$
$727$	7.24703e6	0.508539	0.254269	+	0.967133i	$0.418165\pi$
$728$	−5.71343e7	−3.99548
$729$	0	0
$730$	−9.33710e6	−0.648493
$731$	−3.06645e7	−2.12248
$732$	0	0
$733$	4.57843e6	0.314743	0.157372	−	0.987539i	$-0.449698\pi$
$733$	4.57843e6	0.314743	0.157372	+	0.987539i	$0.449698\pi$
$734$	−1.89477e7	−1.29812
$735$	0	0
$736$	−4.61181e6	−0.313817
$737$	643787.	0.0436590
$738$	0	0
$739$	−1.79403e7	−1.20842	−0.604211	−	0.796825i	$-0.706511\pi$
$739$	−1.79403e7	−1.20842	−0.604211	+	0.796825i	$0.706511\pi$
$740$	−3.23107e7	−2.16904
$741$	0	0
$742$	−7.87301e6	−0.524966
$743$	−1.58719e7	−1.05477	−0.527383	−	0.849628i	$-0.676827\pi$
$743$	−1.58719e7	−1.05477	−0.527383	+	0.849628i	$0.676827\pi$
$744$	0	0
$745$	2.26322e7	1.49395
$746$	2.14214e6	0.140929
$747$	0	0
$748$	1.71109e7	1.11820
$749$	1.86939e7	1.21758
$750$	0	0
$751$	−2.95699e7	−1.91316	−0.956578	−	0.291478i	$-0.905853\pi$
$751$	−2.95699e7	−1.91316	−0.956578	+	0.291478i	$0.905853\pi$
$752$	4.48862e7	2.89446
$753$	0	0
$754$	−1.88458e7	−1.20722
$755$	6.90425e6	0.440808
$756$	0	0
$757$	9.81021e6	0.622212	0.311106	−	0.950375i	$-0.399301\pi$
$757$	9.81021e6	0.622212	0.311106	+	0.950375i	$0.399301\pi$
$758$	4.64986e7	2.93946
$759$	0	0
$760$	5.72769e7	3.59704
$761$	−1.09443e7	−0.685056	−0.342528	−	0.939508i	$-0.611283\pi$
$761$	−1.09443e7	−0.685056	−0.342528	+	0.939508i	$0.611283\pi$
$762$	0	0
$763$	1.99974e7	1.24355
$764$	2.33736e7	1.44874
$765$	0	0
$766$	−2.38735e7	−1.47009
$767$	2.25196e7	1.38220
$768$	0	0
$769$	4.68373e6	0.285612	0.142806	−	0.989751i	$-0.454388\pi$
$769$	4.68373e6	0.285612	0.142806	+	0.989751i	$0.454388\pi$
$770$	−2.65269e7	−1.61235
$771$	0	0
$772$	−1.02452e7	−0.618697
$773$	−1.32674e7	−0.798612	−0.399306	−	0.916818i	$-0.630749\pi$
$773$	−1.32674e7	−0.798612	−0.399306	+	0.916818i	$0.630749\pi$
$774$	0	0
$775$	−1.81258e7	−1.08403
$776$	6.76100e7	4.03048
$777$	0	0
$778$	1.21319e7	0.718589
$779$	1.18556e7	0.699970
$780$	0	0
$781$	1.07591e6	0.0631173
$782$	7.32299e6	0.428225
$783$	0	0
$784$	6.16142e7	3.58006
$785$	−9.50472e6	−0.550510
$786$	0	0
$787$	1.05678e7	0.608204	0.304102	−	0.952639i	$-0.401644\pi$
$787$	1.05678e7	0.608204	0.304102	+	0.952639i	$0.401644\pi$
$788$	6.19029e7	3.55137
$789$	0	0
$790$	−5.86091e7	−3.34116
$791$	1.79258e7	1.01868
$792$	0	0
$793$	−1.91839e7	−1.08331
$794$	1.07435e7	0.604775
$795$	0	0
$796$	4.91923e7	2.75178
$797$	−6.69757e6	−0.373483	−0.186742	−	0.982409i	$-0.559793\pi$
$797$	−6.69757e6	−0.373483	−0.186742	+	0.982409i	$0.559793\pi$
$798$	0	0
$799$	−2.69806e7	−1.49515
$800$	1.71493e7	0.947376
$801$	0	0
$802$	6.35393e7	3.48824
$803$	−2.15080e6	−0.117709
$804$	0	0
$805$	−7.96378e6	−0.433141
$806$	−5.77682e7	−3.13221
$807$	0	0
$808$	−8.44654e7	−4.55145
$809$	2.87439e7	1.54409	0.772047	−	0.635566i	$-0.219233\pi$
$809$	2.87439e7	1.54409	0.772047	+	0.635566i	$0.219233\pi$
$810$	0	0
$811$	6.11987e6	0.326731	0.163365	−	0.986566i	$-0.447765\pi$
$811$	6.11987e6	0.326731	0.163365	+	0.986566i	$0.447765\pi$
$812$	4.76877e7	2.53815
$813$	0	0
$814$	−1.06100e7	−0.561248
$815$	−3.26694e6	−0.172285
$816$	0	0
$817$	−4.11600e7	−2.15735
$818$	−2.20802e7	−1.15377
$819$	0	0
$820$	3.54483e7	1.84103
$821$	2.55802e7	1.32448	0.662241	−	0.749291i	$-0.269605\pi$
$821$	2.55802e7	1.32448	0.662241	+	0.749291i	$0.269605\pi$
$822$	0	0
$823$	−1.38465e7	−0.712589	−0.356295	−	0.934374i	$-0.615960\pi$
$823$	−1.38465e7	−0.712589	−0.356295	+	0.934374i	$0.615960\pi$
$824$	6.41499e7	3.29138
$825$	0	0
$826$	−8.12332e7	−4.14270
$827$	−6.70325e6	−0.340817	−0.170409	−	0.985373i	$-0.554509\pi$
$827$	−6.70325e6	−0.340817	−0.170409	+	0.985373i	$0.554509\pi$
$828$	0	0
$829$	−1.66094e7	−0.839395	−0.419698	−	0.907664i	$-0.637864\pi$
$829$	−1.66094e7	−0.839395	−0.419698	+	0.907664i	$0.637864\pi$
$830$	8.44982e7	4.25748
$831$	0	0
$832$	1.15557e7	0.578744
$833$	−3.70357e7	−1.84930
$834$	0	0
$835$	−2.71446e6	−0.134731
$836$	2.29674e7	1.13657
$837$	0	0
$838$	−6.26880e7	−3.08372
$839$	1.49591e7	0.733668	0.366834	−	0.930286i	$-0.380442\pi$
$839$	1.49591e7	0.733668	0.366834	+	0.930286i	$0.380442\pi$
$840$	0	0
$841$	−1.14751e7	−0.559457
$842$	−4.78482e7	−2.32587
$843$	0	0
$844$	6.10751e7	2.95126
$845$	330533.	0.0159248
$846$	0	0
$847$	2.78660e7	1.33464
$848$	8.01744e6	0.382865
$849$	0	0
$850$	−2.72311e7	−1.29276
$851$	−3.18529e6	−0.150774
$852$	0	0
$853$	3.51212e7	1.65271	0.826356	−	0.563148i	$-0.190410\pi$
$853$	3.51212e7	1.65271	0.826356	+	0.563148i	$0.190410\pi$
$854$	6.92008e7	3.24688
$855$	0	0
$856$	−3.96312e7	−1.84864
$857$	−2.92814e7	−1.36188	−0.680941	−	0.732338i	$-0.738429\pi$
$857$	−2.92814e7	−1.36188	−0.680941	+	0.732338i	$0.738429\pi$
$858$	0	0
$859$	−2.71049e7	−1.25333	−0.626664	−	0.779289i	$-0.715580\pi$
$859$	−2.71049e7	−1.25333	−0.626664	+	0.779289i	$0.715580\pi$
$860$	−1.23069e8	−5.67417
$861$	0	0
$862$	1.54651e7	0.708901
$863$	−1.47872e7	−0.675863	−0.337931	−	0.941171i	$-0.609727\pi$
$863$	−1.47872e7	−0.675863	−0.337931	+	0.941171i	$0.609727\pi$
$864$	0	0
$865$	4.88935e7	2.22183
$866$	−7.70590e7	−3.49163
$867$	0	0
$868$	1.46178e8	6.58540
$869$	−1.35006e7	−0.606462
$870$	0	0
$871$	−2.29058e6	−0.102306
$872$	−4.23946e7	−1.88807
$873$	0	0
$874$	9.82941e6	0.435260
$875$	−1.74311e7	−0.769671
$876$	0	0
$877$	−1.07960e7	−0.473982	−0.236991	−	0.971512i	$-0.576161\pi$
$877$	−1.07960e7	−0.473982	−0.236991	+	0.971512i	$0.576161\pi$
$878$	−2.18923e7	−0.958420
$879$	0	0
$880$	2.70135e7	1.17591
$881$	−6.00778e6	−0.260780	−0.130390	−	0.991463i	$-0.541623\pi$
$881$	−6.00778e6	−0.260780	−0.130390	+	0.991463i	$0.541623\pi$
$882$	0	0
$883$	1.06315e7	0.458875	0.229437	−	0.973323i	$-0.426311\pi$
$883$	1.06315e7	0.458875	0.229437	+	0.973323i	$0.426311\pi$
$884$	−6.08802e7	−2.62027
$885$	0	0
$886$	2.24109e7	0.959123
$887$	−3.31806e6	−0.141604	−0.0708020	−	0.997490i	$-0.522556\pi$
$887$	−3.31806e6	−0.141604	−0.0708020	+	0.997490i	$0.522556\pi$
$888$	0	0
$889$	174354.	0.00739908
$890$	4.70508e7	1.99110
$891$	0	0
$892$	−1.89544e7	−0.797623
$893$	−3.62152e7	−1.51972
$894$	0	0
$895$	2.32843e7	0.971642
$896$	1.71706e7	0.714520
$897$	0	0
$898$	5.85547e7	2.42310
$899$	2.76983e7	1.14302
$900$	0	0
$901$	−4.81920e6	−0.197771
$902$	1.16403e7	0.476375
$903$	0	0
$904$	−3.80027e7	−1.54665
$905$	2.56844e7	1.04243
$906$	0	0
$907$	−4.63923e7	−1.87253	−0.936263	−	0.351301i	$-0.885739\pi$
$907$	−4.63923e7	−1.87253	−0.936263	+	0.351301i	$0.885739\pi$
$908$	−7.51455e7	−3.02474
$909$	0	0
$910$	9.43818e7	3.77820
$911$	−1.40524e7	−0.560987	−0.280494	−	0.959856i	$-0.590498\pi$
$911$	−1.40524e7	−0.560987	−0.280494	+	0.959856i	$0.590498\pi$
$912$	0	0
$913$	1.94642e7	0.772785
$914$	2.20596e6	0.0873439
$915$	0	0
$916$	7.24619e7	2.85345
$917$	5.53411e7	2.17332
$918$	0	0
$919$	6.17102e6	0.241028	0.120514	−	0.992712i	$-0.461546\pi$
$919$	6.17102e6	0.241028	0.120514	+	0.992712i	$0.461546\pi$
$920$	1.68832e7	0.657637
$921$	0	0
$922$	2.94941e7	1.14264
$923$	−3.82806e6	−0.147902
$924$	0	0
$925$	1.18447e7	0.455168
$926$	−1.01624e7	−0.389465
$927$	0	0
$928$	−2.62062e7	−0.998930
$929$	−4.76811e6	−0.181262	−0.0906310	−	0.995885i	$-0.528888\pi$
$929$	−4.76811e6	−0.181262	−0.0906310	+	0.995885i	$0.528888\pi$
$930$	0	0
$931$	−4.97118e7	−1.87969
$932$	6.53483e6	0.246430
$933$	0	0
$934$	1.16047e7	0.435277
$935$	−1.62375e7	−0.607422
$936$	0	0
$937$	−5.20047e6	−0.193506	−0.0967528	−	0.995308i	$-0.530846\pi$
$937$	−5.20047e6	−0.193506	−0.0967528	+	0.995308i	$0.530846\pi$
$938$	8.26264e6	0.306628
$939$	0	0
$940$	−1.08284e8	−3.99709
$941$	−2.95702e7	−1.08863	−0.544316	−	0.838881i	$-0.683211\pi$
$941$	−2.95702e7	−1.08863	−0.544316	+	0.838881i	$0.683211\pi$
$942$	0	0
$943$	3.49461e6	0.127973
$944$	8.27235e7	3.02133
$945$	0	0
$946$	−4.04127e7	−1.46822
$947$	3.00430e7	1.08860	0.544299	−	0.838891i	$-0.316796\pi$
$947$	3.00430e7	1.08860	0.544299	+	0.838891i	$0.316796\pi$
$948$	0	0
$949$	7.65249e6	0.275827
$950$	−3.65514e7	−1.31400
$951$	0	0
$952$	1.26155e8	4.51142
$953$	−2.08186e7	−0.742540	−0.371270	−	0.928525i	$-0.621078\pi$
$953$	−2.08186e7	−0.742540	−0.371270	+	0.928525i	$0.621078\pi$
$954$	0	0
$955$	−2.21805e7	−0.786979
$956$	−1.07337e8	−3.79844
$957$	0	0
$958$	9.06245e7	3.19030
$959$	−2.73636e6	−0.0960785
$960$	0	0
$961$	5.62749e7	1.96565
$962$	3.77501e7	1.31517
$963$	0	0
$964$	−1.03084e7	−0.357271
$965$	9.72226e6	0.336085
$966$	0	0
$967$	−1.35812e7	−0.467058	−0.233529	−	0.972350i	$-0.575027\pi$
$967$	−1.35812e7	−0.467058	−0.233529	+	0.972350i	$0.575027\pi$
$968$	−5.90759e7	−2.02639
$969$	0	0
$970$	−1.11687e8	−3.81130
$971$	809935.	0.0275678	0.0137839	−	0.999905i	$-0.495612\pi$
$971$	809935.	0.0275678	0.0137839	+	0.999905i	$0.495612\pi$
$972$	0	0
$973$	1.88690e7	0.638951
$974$	4.74226e7	1.60172
$975$	0	0
$976$	−7.04703e7	−2.36800
$977$	−5.74719e6	−0.192628	−0.0963140	−	0.995351i	$-0.530705\pi$
$977$	−5.74719e6	−0.192628	−0.0963140	+	0.995351i	$0.530705\pi$
$978$	0	0
$979$	1.08381e7	0.361409
$980$	−1.48639e8	−4.94387
$981$	0	0
$982$	3.43771e7	1.13760
$983$	1.84120e7	0.607740	0.303870	−	0.952714i	$-0.401721\pi$
$983$	1.84120e7	0.607740	0.303870	+	0.952714i	$0.401721\pi$
$984$	0	0
$985$	−5.87432e7	−1.92915
$986$	4.16123e7	1.36311
$987$	0	0
$988$	−8.17176e7	−2.66332
$989$	−1.21325e7	−0.394422
$990$	0	0
$991$	1.13190e6	0.0366120	0.0183060	−	0.999832i	$-0.494173\pi$
$991$	1.13190e6	0.0366120	0.0183060	+	0.999832i	$0.494173\pi$
$992$	−8.03304e7	−2.59180
$993$	0	0
$994$	1.38087e7	0.443289
$995$	−4.66813e7	−1.49481
$996$	0	0
$997$	3.72081e7	1.18549	0.592747	−	0.805389i	$-0.298043\pi$
$997$	3.72081e7	1.18549	0.592747	+	0.805389i	$0.298043\pi$
$998$	9.66631e7	3.07209
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.h.1.10	✓	10
3.2	odd	2		207.6.a.i.1.1	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.10	✓	10	1.1	even	1	trivial
207.6.a.i.1.1	yes	10	3.2	odd	2

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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+10.3536 q^{2} +75.1975 q^{4} -71.3591 q^{5} -210.967 q^{7} +447.250 q^{8} +O(q^{10})\)	\(q+10.3536 q^{2} +75.1975 q^{4} -71.3591 q^{5} -210.967 q^{7} +447.250 q^{8} -738.825 q^{10} -170.188 q^{11} +605.525 q^{13} -2184.27 q^{14} +2224.34 q^{16} -1337.03 q^{17} -1794.65 q^{19} -5366.03 q^{20} -1762.07 q^{22} -529.000 q^{23} +1967.12 q^{25} +6269.38 q^{26} -15864.2 q^{28} -3006.00 q^{29} -9214.34 q^{31} +8717.97 q^{32} -13843.1 q^{34} +15054.4 q^{35} +6021.35 q^{37} -18581.1 q^{38} -31915.4 q^{40} -6606.07 q^{41} +22934.8 q^{43} -12797.7 q^{44} -5477.07 q^{46} +20179.5 q^{47} +27700.0 q^{49} +20366.9 q^{50} +45534.0 q^{52} +3604.41 q^{53} +12144.5 q^{55} -94355.0 q^{56} -31123.0 q^{58} +37190.1 q^{59} -31681.4 q^{61} -95401.8 q^{62} +19083.7 q^{64} -43209.8 q^{65} -3782.79 q^{67} -100541. q^{68} +155868. q^{70} -6321.88 q^{71} +12637.8 q^{73} +62342.8 q^{74} -134953. q^{76} +35904.1 q^{77} +79327.3 q^{79} -158727. q^{80} -68396.7 q^{82} -114368. q^{83} +95409.1 q^{85} +237459. q^{86} -76116.8 q^{88} -63683.3 q^{89} -127746. q^{91} -39779.5 q^{92} +208931. q^{94} +128065. q^{95} +151168. q^{97} +286795. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8}+O(q^{10}) \) 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8} - 250 q^{10} - 460 q^{11} + 464 q^{13} - 3676 q^{14} + 4612 q^{16} - 4756 q^{17} - 1780 q^{19} - 10314 q^{20} - 4214 q^{22} - 5290 q^{23} + 1330 q^{25} + 5152 q^{26} + 7072 q^{28} - 4048 q^{29} + 2816 q^{31} - 27436 q^{32} + 420 q^{34} - 9452 q^{35} + 2872 q^{37} - 31038 q^{38} + 2618 q^{40} - 34056 q^{41} + 7316 q^{43} - 33562 q^{44} + 4232 q^{46} - 49300 q^{47} + 45118 q^{49} - 44764 q^{50} - 25120 q^{52} - 86676 q^{53} - 2120 q^{55} - 290684 q^{56} - 87408 q^{58} - 67100 q^{59} - 40432 q^{61} - 230992 q^{62} + 136776 q^{64} - 184000 q^{65} - 50108 q^{67} - 270592 q^{68} + 117456 q^{70} - 238584 q^{71} - 13804 q^{73} - 150074 q^{74} - 197622 q^{76} - 116248 q^{77} - 9228 q^{79} - 313010 q^{80} - 68604 q^{82} - 155300 q^{83} + 80444 q^{85} + 80914 q^{86} - 237738 q^{88} - 213732 q^{89} - 264352 q^{91} - 101568 q^{92} + 140280 q^{94} + 123612 q^{95} + 42516 q^{97} + 151388 q^{98}+O(q^{100}) \) 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8 - 250 * q^10 - 460 * q^11 + 464 * q^13 - 3676 * q^14 + 4612 * q^16 - 4756 * q^17 - 1780 * q^19 - 10314 * q^20 - 4214 * q^22 - 5290 * q^23 + 1330 * q^25 + 5152 * q^26 + 7072 * q^28 - 4048 * q^29 + 2816 * q^31 - 27436 * q^32 + 420 * q^34 - 9452 * q^35 + 2872 * q^37 - 31038 * q^38 + 2618 * q^40 - 34056 * q^41 + 7316 * q^43 - 33562 * q^44 + 4232 * q^46 - 49300 * q^47 + 45118 * q^49 - 44764 * q^50 - 25120 * q^52 - 86676 * q^53 - 2120 * q^55 - 290684 * q^56 - 87408 * q^58 - 67100 * q^59 - 40432 * q^61 - 230992 * q^62 + 136776 * q^64 - 184000 * q^65 - 50108 * q^67 - 270592 * q^68 + 117456 * q^70 - 238584 * q^71 - 13804 * q^73 - 150074 * q^74 - 197622 * q^76 - 116248 * q^77 - 9228 * q^79 - 313010 * q^80 - 68604 * q^82 - 155300 * q^83 + 80444 * q^85 + 80914 * q^86 - 237738 * q^88 - 213732 * q^89 - 264352 * q^91 - 101568 * q^92 + 140280 * q^94 + 123612 * q^95 + 42516 * q^97 + 151388 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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