LMFDB - Embedded newform 531.6.a.b.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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + \cdots - 14846072 $ x^11 - 5x^10 - 238x^9 + 1067x^8 + 20782x^7 - 79077x^6 - 813818x^5 + 2364885x^4 + 13849341x^3 - 23890558x^2 - 74443300x - 14846072
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.66878$ 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)

$f(q)$	$=$	$q-7.66878 q^{2} +26.8102 q^{4} +109.801 q^{5} +156.301 q^{7} +39.7996 q^{8} +O(q^{10})$	\(q-7.66878 q^{2} +26.8102 q^{4} +109.801 q^{5} +156.301 q^{7} +39.7996 q^{8} -842.036 q^{10} +426.905 q^{11} -123.969 q^{13} -1198.64 q^{14} -1163.14 q^{16} +852.404 q^{17} +232.142 q^{19} +2943.77 q^{20} -3273.84 q^{22} +3642.90 q^{23} +8931.16 q^{25} +950.692 q^{26} +4190.45 q^{28} +8173.34 q^{29} -9358.70 q^{31} +7646.28 q^{32} -6536.90 q^{34} +17161.9 q^{35} +6317.09 q^{37} -1780.24 q^{38} +4370.02 q^{40} -13017.8 q^{41} +23355.0 q^{43} +11445.4 q^{44} -27936.6 q^{46} +8861.12 q^{47} +7622.89 q^{49} -68491.1 q^{50} -3323.64 q^{52} -23012.3 q^{53} +46874.4 q^{55} +6220.71 q^{56} -62679.6 q^{58} -3481.00 q^{59} +13699.8 q^{61} +71769.8 q^{62} -21417.1 q^{64} -13611.9 q^{65} -1424.88 q^{67} +22853.1 q^{68} -131611. q^{70} -34543.2 q^{71} -50120.2 q^{73} -48444.4 q^{74} +6223.76 q^{76} +66725.5 q^{77} -69723.6 q^{79} -127713. q^{80} +99830.8 q^{82} -49883.0 q^{83} +93594.5 q^{85} -179104. q^{86} +16990.7 q^{88} +50735.3 q^{89} -19376.5 q^{91} +97666.8 q^{92} -67954.0 q^{94} +25489.3 q^{95} +21922.7 q^{97} -58458.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−7.66878	−1.35566	−0.677831	−	0.735218i	$-0.737080\pi$
$2$	−7.66878	−1.35566	−0.677831	+	0.735218i	$0.737080\pi$
$3$	0	0
$3$	0	0
$4$	26.8102	0.837818
$5$	109.801	1.96417	0.982086	−	0.188434i	$-0.0603410\pi$
$5$	109.801	1.96417	0.982086	+	0.188434i	$0.0603410\pi$
$6$	0	0
$7$	156.301	1.20563	0.602817	−	0.797879i	$-0.294045\pi$
$7$	156.301	1.20563	0.602817	+	0.797879i	$0.294045\pi$
$8$	39.7996	0.219864
$9$	0	0
$10$	−842.036	−2.66275
$11$	426.905	1.06377	0.531887	−	0.846815i	$-0.321483\pi$
$11$	426.905	1.06377	0.531887	+	0.846815i	$0.321483\pi$
$12$	0	0
$13$	−123.969	−0.203449	−0.101724	−	0.994813i	$-0.532436\pi$
$13$	−123.969	−0.203449	−0.101724	+	0.994813i	$0.532436\pi$
$14$	−1198.64	−1.63443
$15$	0	0
$16$	−1163.14	−1.13588
$17$	852.404	0.715358	0.357679	−	0.933845i	$-0.383568\pi$
$17$	852.404	0.715358	0.357679	+	0.933845i	$0.383568\pi$
$18$	0	0
$19$	232.142	0.147526	0.0737631	−	0.997276i	$-0.476499\pi$
$19$	232.142	0.147526	0.0737631	+	0.997276i	$0.476499\pi$
$20$	2943.77	1.64562
$21$	0	0
$22$	−3273.84	−1.44212
$23$	3642.90	1.43591	0.717956	−	0.696089i	$-0.245078\pi$
$23$	3642.90	1.43591	0.717956	+	0.696089i	$0.245078\pi$
$24$	0	0
$25$	8931.16	2.85797
$26$	950.692	0.275808
$27$	0	0
$28$	4190.45	1.01010
$29$	8173.34	1.80470	0.902349	−	0.431005i	$-0.141841\pi$
$29$	8173.34	1.80470	0.902349	+	0.431005i	$0.141841\pi$
$30$	0	0
$31$	−9358.70	−1.74909	−0.874544	−	0.484947i	$-0.838839\pi$
$31$	−9358.70	−1.74909	−0.874544	+	0.484947i	$0.838839\pi$
$32$	7646.28	1.32000
$33$	0	0
$34$	−6536.90	−0.969783
$35$	17161.9	2.36807
$36$	0	0
$37$	6317.09	0.758600	0.379300	−	0.925274i	$-0.376165\pi$
$37$	6317.09	0.758600	0.379300	+	0.925274i	$0.376165\pi$
$38$	−1780.24	−0.199996
$39$	0	0
$40$	4370.02	0.431850
$41$	−13017.8	−1.20942	−0.604712	−	0.796444i	$-0.706712\pi$
$41$	−13017.8	−1.20942	−0.604712	+	0.796444i	$0.706712\pi$
$42$	0	0
$43$	23355.0	1.92623	0.963117	−	0.269084i	$-0.0867208\pi$
$43$	23355.0	1.92623	0.963117	+	0.269084i	$0.0867208\pi$
$44$	11445.4	0.891250
$45$	0	0
$46$	−27936.6	−1.94661
$47$	8861.12	0.585119	0.292559	−	0.956247i	$-0.405493\pi$
$47$	8861.12	0.585119	0.292559	+	0.956247i	$0.405493\pi$
$48$	0	0
$49$	7622.89	0.453554
$50$	−68491.1	−3.87444
$51$	0	0
$52$	−3323.64	−0.170453
$53$	−23012.3	−1.12531	−0.562653	−	0.826693i	$-0.690219\pi$
$53$	−23012.3	−1.12531	−0.562653	+	0.826693i	$0.690219\pi$
$54$	0	0
$55$	46874.4	2.08944
$56$	6220.71	0.265075
$57$	0	0
$58$	−62679.6	−2.44656
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	13699.8	0.471399	0.235699	−	0.971826i	$-0.424262\pi$
$61$	13699.8	0.471399	0.235699	+	0.971826i	$0.424262\pi$
$62$	71769.8	2.37117
$63$	0	0
$64$	−21417.1	−0.653599
$65$	−13611.9	−0.399609
$66$	0	0
$67$	−1424.88	−0.0387785	−0.0193893	−	0.999812i	$-0.506172\pi$
$67$	−1424.88	−0.0387785	−0.0193893	+	0.999812i	$0.506172\pi$
$68$	22853.1	0.599340
$69$	0	0
$70$	−131611.	−3.21031
$71$	−34543.2	−0.813235	−0.406618	−	0.913598i	$-0.633292\pi$
$71$	−34543.2	−0.813235	−0.406618	+	0.913598i	$0.633292\pi$
$72$	0	0
$73$	−50120.2	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$73$	−50120.2	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$74$	−48444.4	−1.02840
$75$	0	0
$76$	6223.76	0.123600
$77$	66725.5	1.28252
$78$	0	0
$79$	−69723.6	−1.25693	−0.628466	−	0.777837i	$-0.716317\pi$
$79$	−69723.6	−1.25693	−0.628466	+	0.777837i	$0.716317\pi$
$80$	−127713.	−2.23106
$81$	0	0
$82$	99830.8	1.63957
$83$	−49883.0	−0.794799	−0.397399	−	0.917646i	$-0.630087\pi$
$83$	−49883.0	−0.794799	−0.397399	+	0.917646i	$0.630087\pi$
$84$	0	0
$85$	93594.5	1.40509
$86$	−179104.	−2.61132
$87$	0	0
$88$	16990.7	0.233886
$89$	50735.3	0.678946	0.339473	−	0.940616i	$-0.389751\pi$
$89$	50735.3	0.678946	0.339473	+	0.940616i	$0.389751\pi$
$90$	0	0
$91$	−19376.5	−0.245285
$92$	97666.8	1.20303
$93$	0	0
$94$	−67954.0	−0.793223
$95$	25489.3	0.289767
$96$	0	0
$97$	21922.7	0.236573	0.118286	−	0.992980i	$-0.462260\pi$
$97$	21922.7	0.236573	0.118286	+	0.992980i	$0.462260\pi$
$98$	−58458.2	−0.614866
$99$	0	0
$100$	239446.	2.39446
$101$	−136749.	−1.33389	−0.666944	−	0.745108i	$-0.732398\pi$
$101$	−136749.	−1.33389	−0.666944	+	0.745108i	$0.732398\pi$
$102$	0	0
$103$	−45802.7	−0.425400	−0.212700	−	0.977118i	$-0.568226\pi$
$103$	−45802.7	−0.425400	−0.212700	+	0.977118i	$0.568226\pi$
$104$	−4933.93	−0.0447311
$105$	0	0
$106$	176476.	1.52553
$107$	−42126.7	−0.355711	−0.177856	−	0.984057i	$-0.556916\pi$
$107$	−42126.7	−0.355711	−0.177856	+	0.984057i	$0.556916\pi$
$108$	0	0
$109$	−184019.	−1.48353	−0.741765	−	0.670660i	$-0.766011\pi$
$109$	−184019.	−1.48353	−0.741765	+	0.670660i	$0.766011\pi$
$110$	−359470.	−2.83257
$111$	0	0
$112$	−181800.	−1.36945
$113$	180222.	1.32773	0.663867	−	0.747851i	$-0.268914\pi$
$113$	180222.	1.32773	0.663867	+	0.747851i	$0.268914\pi$
$114$	0	0
$115$	399992.	2.82038
$116$	219129.	1.51201
$117$	0	0
$118$	26695.0	0.176492
$119$	133231.	0.862460
$120$	0	0
$121$	21197.0	0.131616
$122$	−105060.	−0.639057
$123$	0	0
$124$	−250908.	−1.46542
$125$	637519.	3.64937
$126$	0	0
$127$	−25072.4	−0.137939	−0.0689694	−	0.997619i	$-0.521971\pi$
$127$	−25072.4	−0.137939	−0.0689694	+	0.997619i	$0.521971\pi$
$128$	−80437.6	−0.433945
$129$	0	0
$130$	104387.	0.541734
$131$	−28027.7	−0.142695	−0.0713475	−	0.997452i	$-0.522730\pi$
$131$	−28027.7	−0.142695	−0.0713475	+	0.997452i	$0.522730\pi$
$132$	0	0
$133$	36283.9	0.177863
$134$	10927.1	0.0525705
$135$	0	0
$136$	33925.4	0.157281
$137$	−193823.	−0.882277	−0.441139	−	0.897439i	$-0.645425\pi$
$137$	−193823.	−0.882277	−0.441139	+	0.897439i	$0.645425\pi$
$138$	0	0
$139$	−385667.	−1.69307	−0.846536	−	0.532331i	$-0.821316\pi$
$139$	−385667.	−1.69307	−0.846536	+	0.532331i	$0.821316\pi$
$140$	460113.	1.98401
$141$	0	0
$142$	264904.	1.10247
$143$	−52923.1	−0.216424
$144$	0	0
$145$	897438.	3.54474
$146$	384361.	1.49230
$147$	0	0
$148$	169362.	0.635569
$149$	97076.0	0.358217	0.179109	−	0.983829i	$-0.442679\pi$
$149$	97076.0	0.358217	0.179109	+	0.983829i	$0.442679\pi$
$150$	0	0
$151$	28312.8	0.101051	0.0505255	−	0.998723i	$-0.483910\pi$
$151$	28312.8	0.101051	0.0505255	+	0.998723i	$0.483910\pi$
$152$	9239.15	0.0324357
$153$	0	0
$154$	−511703.	−1.73867
$155$	−1.02759e6	−3.43551
$156$	0	0
$157$	−147457.	−0.477439	−0.238719	−	0.971089i	$-0.576728\pi$
$157$	−147457.	−0.477439	−0.238719	+	0.971089i	$0.576728\pi$
$158$	534695.	1.70398
$159$	0	0
$160$	839565.	2.59271
$161$	569388.	1.73118
$162$	0	0
$163$	−140469.	−0.414105	−0.207053	−	0.978330i	$-0.566387\pi$
$163$	−140469.	−0.414105	−0.207053	+	0.978330i	$0.566387\pi$
$164$	−349010.	−1.01328
$165$	0	0
$166$	382542.	1.07748
$167$	33450.1	0.0928125	0.0464063	−	0.998923i	$-0.485223\pi$
$167$	33450.1	0.0928125	0.0464063	+	0.998923i	$0.485223\pi$
$168$	0	0
$169$	−355925.	−0.958609
$170$	−717755.	−1.90482
$171$	0	0
$172$	626152.	1.61383
$173$	−359847.	−0.914118	−0.457059	−	0.889436i	$-0.651097\pi$
$173$	−359847.	−0.914118	−0.457059	+	0.889436i	$0.651097\pi$
$174$	0	0
$175$	1.39595e6	3.44567
$176$	−496550.	−1.20832
$177$	0	0
$178$	−389078.	−0.920422
$179$	−51692.7	−0.120586	−0.0602930	−	0.998181i	$-0.519204\pi$
$179$	−51692.7	−0.120586	−0.0602930	+	0.998181i	$0.519204\pi$
$180$	0	0
$181$	−649652.	−1.47395	−0.736977	−	0.675918i	$-0.763748\pi$
$181$	−649652.	−1.47395	−0.736977	+	0.675918i	$0.763748\pi$
$182$	148594.	0.332523
$183$	0	0
$184$	144986.	0.315705
$185$	693620.	1.49002
$186$	0	0
$187$	363896.	0.760980
$188$	237568.	0.490223
$189$	0	0
$190$	−195472.	−0.392826
$191$	−76790.3	−0.152308	−0.0761540	−	0.997096i	$-0.524264\pi$
$191$	−76790.3	−0.152308	−0.0761540	+	0.997096i	$0.524264\pi$
$192$	0	0
$193$	496266.	0.959006	0.479503	−	0.877540i	$-0.340817\pi$
$193$	496266.	0.959006	0.479503	+	0.877540i	$0.340817\pi$
$194$	−168120.	−0.320712
$195$	0	0
$196$	204371.	0.379996
$197$	442784.	0.812879	0.406439	−	0.913678i	$-0.366770\pi$
$197$	442784.	0.812879	0.406439	+	0.913678i	$0.366770\pi$
$198$	0	0
$199$	730332.	1.30734	0.653668	−	0.756781i	$-0.273229\pi$
$199$	730332.	1.30734	0.653668	+	0.756781i	$0.273229\pi$
$200$	355457.	0.628365
$201$	0	0
$202$	1.04869e6	1.80830
$203$	1.27750e6	2.17581
$204$	0	0
$205$	−1.42936e6	−2.37552
$206$	351251.	0.576699
$207$	0	0
$208$	144194.	0.231093
$209$	99102.5	0.156935
$210$	0	0
$211$	960167.	1.48471	0.742353	−	0.670008i	$-0.233710\pi$
$211$	960167.	1.48471	0.742353	+	0.670008i	$0.233710\pi$
$212$	−616964.	−0.942801
$213$	0	0
$214$	323060.	0.482224
$215$	2.56439e6	3.78345
$216$	0	0
$217$	−1.46277e6	−2.10876
$218$	1.41120e6	2.01117
$219$	0	0
$220$	1.25671e6	1.75057
$221$	−105672.	−0.145539
$222$	0	0
$223$	−114848.	−0.154655	−0.0773273	−	0.997006i	$-0.524639\pi$
$223$	−114848.	−0.154655	−0.0773273	+	0.997006i	$0.524639\pi$
$224$	1.19512e6	1.59144
$225$	0	0
$226$	−1.38208e6	−1.79996
$227$	316011.	0.407040	0.203520	−	0.979071i	$-0.434762\pi$
$227$	316011.	0.407040	0.203520	+	0.979071i	$0.434762\pi$
$228$	0	0
$229$	−110133.	−0.138781	−0.0693903	−	0.997590i	$-0.522105\pi$
$229$	−110133.	−0.138781	−0.0693903	+	0.997590i	$0.522105\pi$
$230$	−3.06745e6	−3.82348
$231$	0	0
$232$	325296.	0.396788
$233$	−8645.45	−0.0104327	−0.00521636	−	0.999986i	$-0.501660\pi$
$233$	−8645.45	−0.0104327	−0.00521636	+	0.999986i	$0.501660\pi$
$234$	0	0
$235$	972956.	1.14927
$236$	−93326.2	−0.109075
$237$	0	0
$238$	−1.02172e6	−1.16920
$239$	−623677.	−0.706261	−0.353131	−	0.935574i	$-0.614883\pi$
$239$	−623677.	−0.706261	−0.353131	+	0.935574i	$0.614883\pi$
$240$	0	0
$241$	−1.73477e6	−1.92398	−0.961988	−	0.273092i	$-0.911954\pi$
$241$	−1.73477e6	−1.92398	−0.961988	+	0.273092i	$0.911954\pi$
$242$	−162555.	−0.178427
$243$	0	0
$244$	367293.	0.394946
$245$	836997.	0.890859
$246$	0	0
$247$	−28778.4	−0.0300141
$248$	−372473.	−0.384561
$249$	0	0
$250$	−4.88900e6	−4.94732
$251$	−1.44679e6	−1.44951	−0.724755	−	0.689006i	$-0.758047\pi$
$251$	−1.44679e6	−1.44951	−0.724755	+	0.689006i	$0.758047\pi$
$252$	0	0
$253$	1.55517e6	1.52749
$254$	192275.	0.186998
$255$	0	0
$256$	1.30221e6	1.24188
$257$	177900.	0.168013	0.0840064	−	0.996465i	$-0.473228\pi$
$257$	177900.	0.168013	0.0840064	+	0.996465i	$0.473228\pi$
$258$	0	0
$259$	987365.	0.914594
$260$	−364937.	−0.334799
$261$	0	0
$262$	214938.	0.193446
$263$	662979.	0.591031	0.295515	−	0.955338i	$-0.404509\pi$
$263$	662979.	0.591031	0.295515	+	0.955338i	$0.404509\pi$
$264$	0	0
$265$	−2.52676e6	−2.21029
$266$	−278253.	−0.241122
$267$	0	0
$268$	−38201.3	−0.0324893
$269$	1.10386e6	0.930105	0.465052	−	0.885283i	$-0.346035\pi$
$269$	1.10386e6	0.930105	0.465052	+	0.885283i	$0.346035\pi$
$270$	0	0
$271$	−1.65544e6	−1.36927	−0.684635	−	0.728886i	$-0.740039\pi$
$271$	−1.65544e6	−1.36927	−0.684635	+	0.728886i	$0.740039\pi$
$272$	−991466.	−0.812560
$273$	0	0
$274$	1.48639e6	1.19607
$275$	3.81276e6	3.04024
$276$	0	0
$277$	−87836.5	−0.0687821	−0.0343911	−	0.999408i	$-0.510949\pi$
$277$	−87836.5	−0.0687821	−0.0343911	+	0.999408i	$0.510949\pi$
$278$	2.95760e6	2.29523
$279$	0	0
$280$	683037.	0.520654
$281$	800455.	0.604743	0.302372	−	0.953190i	$-0.402222\pi$
$281$	800455.	0.604743	0.302372	+	0.953190i	$0.402222\pi$
$282$	0	0
$283$	−2.09590e6	−1.55563	−0.777813	−	0.628495i	$-0.783671\pi$
$283$	−2.09590e6	−1.55563	−0.777813	+	0.628495i	$0.783671\pi$
$284$	−926108.	−0.681343
$285$	0	0
$286$	405855.	0.293397
$287$	−2.03469e6	−1.45812
$288$	0	0
$289$	−693264.	−0.488263
$290$	−6.88225e6	−4.80547
$291$	0	0
$292$	−1.34373e6	−0.922264
$293$	−1.40390e6	−0.955363	−0.477681	−	0.878533i	$-0.658523\pi$
$293$	−1.40390e6	−0.955363	−0.477681	+	0.878533i	$0.658523\pi$
$294$	0	0
$295$	−382216.	−0.255713
$296$	251418.	0.166789
$297$	0	0
$298$	−744455.	−0.485621
$299$	−451607.	−0.292135
$300$	0	0
$301$	3.65040e6	2.32233
$302$	−217125.	−0.136991
$303$	0	0
$304$	−270013.	−0.167572
$305$	1.50424e6	0.925908
$306$	0	0
$307$	−1.33666e6	−0.809422	−0.404711	−	0.914445i	$-0.632628\pi$
$307$	−1.33666e6	−0.809422	−0.404711	+	0.914445i	$0.632628\pi$
$308$	1.78892e6	1.07452
$309$	0	0
$310$	7.88037e6	4.65739
$311$	2.02561e6	1.18756	0.593778	−	0.804629i	$-0.297636\pi$
$311$	2.02561e6	1.18756	0.593778	+	0.804629i	$0.297636\pi$
$312$	0	0
$313$	513441.	0.296231	0.148115	−	0.988970i	$-0.452679\pi$
$313$	513441.	0.296231	0.148115	+	0.988970i	$0.452679\pi$
$314$	1.13082e6	0.647245
$315$	0	0
$316$	−1.86930e6	−1.05308
$317$	−4929.11	−0.00275499	−0.00137750	−	0.999999i	$-0.500438\pi$
$317$	−4929.11	−0.00275499	−0.00137750	+	0.999999i	$0.500438\pi$
$318$	0	0
$319$	3.48924e6	1.91979
$320$	−2.35161e6	−1.28378
$321$	0	0
$322$	−4.36651e6	−2.34690
$323$	197879.	0.105534
$324$	0	0
$325$	−1.10719e6	−0.581451
$326$	1.07722e6	0.561386
$327$	0	0
$328$	−518104.	−0.265909
$329$	1.38500e6	0.705439
$330$	0	0
$331$	−1.97213e6	−0.989386	−0.494693	−	0.869068i	$-0.664719\pi$
$331$	−1.97213e6	−0.989386	−0.494693	+	0.869068i	$0.664719\pi$
$332$	−1.33737e6	−0.665897
$333$	0	0
$334$	−256522.	−0.125822
$335$	−156453.	−0.0761676
$336$	0	0
$337$	2.30193e6	1.10412	0.552062	−	0.833803i	$-0.313841\pi$
$337$	2.30193e6	1.10412	0.552062	+	0.833803i	$0.313841\pi$
$338$	2.72951e6	1.29955
$339$	0	0
$340$	2.50928e6	1.17721
$341$	−3.99528e6	−1.86063
$342$	0	0
$343$	−1.43548e6	−0.658814
$344$	929521.	0.423509
$345$	0	0
$346$	2.75958e6	1.23923
$347$	−947055.	−0.422232	−0.211116	−	0.977461i	$-0.567710\pi$
$347$	−947055.	−0.422232	−0.211116	+	0.977461i	$0.567710\pi$
$348$	0	0
$349$	1.96092e6	0.861781	0.430891	−	0.902404i	$-0.358199\pi$
$349$	1.96092e6	0.861781	0.430891	+	0.902404i	$0.358199\pi$
$350$	−1.07052e7	−4.67116
$351$	0	0
$352$	3.26423e6	1.40419
$353$	−2.49167e6	−1.06427	−0.532137	−	0.846658i	$-0.678611\pi$
$353$	−2.49167e6	−1.06427	−0.532137	+	0.846658i	$0.678611\pi$
$354$	0	0
$355$	−3.79286e6	−1.59733
$356$	1.36022e6	0.568834
$357$	0	0
$358$	396420.	0.163474
$359$	−282969.	−0.115878	−0.0579392	−	0.998320i	$-0.518453\pi$
$359$	−282969.	−0.115878	−0.0579392	+	0.998320i	$0.518453\pi$
$360$	0	0
$361$	−2.42221e6	−0.978236
$362$	4.98203e6	1.99818
$363$	0	0
$364$	−519486.	−0.205504
$365$	−5.50323e6	−2.16215
$366$	0	0
$367$	1.02276e6	0.396378	0.198189	−	0.980164i	$-0.436494\pi$
$367$	1.02276e6	0.396378	0.198189	+	0.980164i	$0.436494\pi$
$368$	−4.23720e6	−1.63102
$369$	0	0
$370$	−5.31922e6	−2.01996
$371$	−3.59684e6	−1.35671
$372$	0	0
$373$	3.02229e6	1.12477	0.562385	−	0.826876i	$-0.309884\pi$
$373$	3.02229e6	1.12477	0.562385	+	0.826876i	$0.309884\pi$
$374$	−2.79064e6	−1.03163
$375$	0	0
$376$	352669.	0.128646
$377$	−1.01324e6	−0.367164
$378$	0	0
$379$	−529457.	−0.189336	−0.0946679	−	0.995509i	$-0.530179\pi$
$379$	−529457.	−0.189336	−0.0946679	+	0.995509i	$0.530179\pi$
$380$	683372.	0.242772
$381$	0	0
$382$	588887.	0.206478
$383$	2.95011e6	1.02764	0.513820	−	0.857898i	$-0.328230\pi$
$383$	2.95011e6	1.02764	0.513820	+	0.857898i	$0.328230\pi$
$384$	0	0
$385$	7.32650e6	2.51910
$386$	−3.80575e6	−1.30009
$387$	0	0
$388$	587751.	0.198205
$389$	2.96852e6	0.994641	0.497321	−	0.867567i	$-0.334317\pi$
$389$	2.96852e6	0.994641	0.497321	+	0.867567i	$0.334317\pi$
$390$	0	0
$391$	3.10522e6	1.02719
$392$	303388.	0.0997202
$393$	0	0
$394$	−3.39561e6	−1.10199
$395$	−7.65569e6	−2.46883
$396$	0	0
$397$	1.04019e6	0.331234	0.165617	−	0.986190i	$-0.447038\pi$
$397$	1.04019e6	0.331234	0.165617	+	0.986190i	$0.447038\pi$
$398$	−5.60075e6	−1.77231
$399$	0	0
$400$	−1.03882e7	−3.24631
$401$	−3.31007e6	−1.02796	−0.513980	−	0.857802i	$-0.671829\pi$
$401$	−3.31007e6	−1.02796	−0.513980	+	0.857802i	$0.671829\pi$
$402$	0	0
$403$	1.16019e6	0.355850
$404$	−3.66625e6	−1.11756
$405$	0	0
$406$	−9.79686e6	−2.94966
$407$	2.69680e6	0.806979
$408$	0	0
$409$	−5.21191e6	−1.54059	−0.770297	−	0.637685i	$-0.779892\pi$
$409$	−5.21191e6	−1.54059	−0.770297	+	0.637685i	$0.779892\pi$
$410$	1.09615e7	3.22040
$411$	0	0
$412$	−1.22798e6	−0.356408
$413$	−544083.	−0.156960
$414$	0	0
$415$	−5.47718e6	−1.56112
$416$	−947903.	−0.268553
$417$	0	0
$418$	−759995.	−0.212750
$419$	3.41630e6	0.950649	0.475325	−	0.879810i	$-0.342331\pi$
$419$	3.41630e6	0.950649	0.475325	+	0.879810i	$0.342331\pi$
$420$	0	0
$421$	5.70486e6	1.56870	0.784350	−	0.620319i	$-0.212997\pi$
$421$	5.70486e6	1.56870	0.784350	+	0.620319i	$0.212997\pi$
$422$	−7.36331e6	−2.01276
$423$	0	0
$424$	−915881.	−0.247414
$425$	7.61296e6	2.04447
$426$	0	0
$427$	2.14128e6	0.568335
$428$	−1.12942e6	−0.298021
$429$	0	0
$430$	−1.96658e7	−5.12908
$431$	2.31113e6	0.599281	0.299640	−	0.954052i	$-0.403133\pi$
$431$	2.31113e6	0.599281	0.299640	+	0.954052i	$0.403133\pi$
$432$	0	0
$433$	−55388.5	−0.0141971	−0.00709856	−	0.999975i	$-0.502260\pi$
$433$	−55388.5	−0.0141971	−0.00709856	+	0.999975i	$0.502260\pi$
$434$	1.12177e7	2.85876
$435$	0	0
$436$	−4.93358e6	−1.24293
$437$	845669.	0.211835
$438$	0	0
$439$	−5.60645e6	−1.38844	−0.694219	−	0.719764i	$-0.744250\pi$
$439$	−5.60645e6	−1.38844	−0.694219	+	0.719764i	$0.744250\pi$
$440$	1.86558e6	0.459392
$441$	0	0
$442$	810374.	0.197301
$443$	−5.33809e6	−1.29234	−0.646170	−	0.763194i	$-0.723630\pi$
$443$	−5.33809e6	−1.29234	−0.646170	+	0.763194i	$0.723630\pi$
$444$	0	0
$445$	5.57077e6	1.33357
$446$	880747.	0.209659
$447$	0	0
$448$	−3.34751e6	−0.788001
$449$	3.68121e6	0.861737	0.430869	−	0.902415i	$-0.358207\pi$
$449$	3.68121e6	0.861737	0.430869	+	0.902415i	$0.358207\pi$
$450$	0	0
$451$	−5.55737e6	−1.28655
$452$	4.83177e6	1.11240
$453$	0	0
$454$	−2.42342e6	−0.551809
$455$	−2.12755e6	−0.481782
$456$	0	0
$457$	3.14852e6	0.705205	0.352603	−	0.935773i	$-0.385297\pi$
$457$	3.14852e6	0.705205	0.352603	+	0.935773i	$0.385297\pi$
$458$	844585.	0.188139
$459$	0	0
$460$	1.07239e7	2.36296
$461$	3.05916e6	0.670425	0.335213	−	0.942143i	$-0.391192\pi$
$461$	3.05916e6	0.670425	0.335213	+	0.942143i	$0.391192\pi$
$462$	0	0
$463$	8.35625e6	1.81159	0.905793	−	0.423721i	$-0.139276\pi$
$463$	8.35625e6	1.81159	0.905793	+	0.423721i	$0.139276\pi$
$464$	−9.50674e6	−2.04992
$465$	0	0
$466$	66300.1	0.0141432
$467$	−1.16583e6	−0.247367	−0.123683	−	0.992322i	$-0.539471\pi$
$467$	−1.16583e6	−0.247367	−0.123683	+	0.992322i	$0.539471\pi$
$468$	0	0
$469$	−222710.	−0.0467527
$470$	−7.46138e6	−1.55803
$471$	0	0
$472$	−138542.	−0.0286238
$473$	9.97037e6	2.04908
$474$	0	0
$475$	2.07330e6	0.421626
$476$	3.57196e6	0.722585
$477$	0	0
$478$	4.78284e6	0.957451
$479$	1.23399e6	0.245737	0.122869	−	0.992423i	$-0.460791\pi$
$479$	1.23399e6	0.245737	0.122869	+	0.992423i	$0.460791\pi$
$480$	0	0
$481$	−783124.	−0.154336
$482$	1.33036e7	2.60826
$483$	0	0
$484$	568294.	0.110271
$485$	2.40712e6	0.464669
$486$	0	0
$487$	1.60158e6	0.306004	0.153002	−	0.988226i	$-0.451106\pi$
$487$	1.60158e6	0.306004	0.153002	+	0.988226i	$0.451106\pi$
$488$	545245.	0.103644
$489$	0	0
$490$	−6.41875e6	−1.20770
$491$	−1.78594e6	−0.334321	−0.167161	−	0.985930i	$-0.553460\pi$
$491$	−1.78594e6	−0.334321	−0.167161	+	0.985930i	$0.553460\pi$
$492$	0	0
$493$	6.96699e6	1.29101
$494$	220695.	0.0406889
$495$	0	0
$496$	1.08855e7	1.98675
$497$	−5.39912e6	−0.980465
$498$	0	0
$499$	7.98618e6	1.43578	0.717889	−	0.696157i	$-0.245108\pi$
$499$	7.98618e6	1.43578	0.717889	+	0.696157i	$0.245108\pi$
$500$	1.70920e7	3.05751
$501$	0	0
$502$	1.10951e7	1.96505
$503$	3.77866e6	0.665914	0.332957	−	0.942942i	$-0.391954\pi$
$503$	3.77866e6	0.665914	0.332957	+	0.942942i	$0.391954\pi$
$504$	0	0
$505$	−1.50151e7	−2.61999
$506$	−1.19263e7	−2.07075
$507$	0	0
$508$	−672196.	−0.115568
$509$	5.46200e6	0.934452	0.467226	−	0.884138i	$-0.345253\pi$
$509$	5.46200e6	0.934452	0.467226	+	0.884138i	$0.345253\pi$
$510$	0	0
$511$	−7.83382e6	−1.32715
$512$	−7.41233e6	−1.24963
$513$	0	0
$514$	−1.36427e6	−0.227768
$515$	−5.02916e6	−0.835559
$516$	0	0
$517$	3.78286e6	0.622434
$518$	−7.57189e6	−1.23988
$519$	0	0
$520$	−541748.	−0.0878595
$521$	3.79145e6	0.611944	0.305972	−	0.952041i	$-0.401019\pi$
$521$	3.79145e6	0.611944	0.305972	+	0.952041i	$0.401019\pi$
$522$	0	0
$523$	2.52314e6	0.403355	0.201678	−	0.979452i	$-0.435361\pi$
$523$	2.52314e6	0.403355	0.201678	+	0.979452i	$0.435361\pi$
$524$	−751426.	−0.119552
$525$	0	0
$526$	−5.08424e6	−0.801238
$527$	−7.97740e6	−1.25122
$528$	0	0
$529$	6.83438e6	1.06184
$530$	1.93772e7	2.99641
$531$	0	0
$532$	972778.	0.149017
$533$	1.61381e6	0.246056
$534$	0	0
$535$	−4.62553e6	−0.698678
$536$	−56709.6	−0.00852599
$537$	0	0
$538$	−8.46523e6	−1.26091
$539$	3.25425e6	0.482480
$540$	0	0
$541$	3.18382e6	0.467687	0.233844	−	0.972274i	$-0.424870\pi$
$541$	3.18382e6	0.467687	0.233844	+	0.972274i	$0.424870\pi$
$542$	1.26952e7	1.85627
$543$	0	0
$544$	6.51772e6	0.944275
$545$	−2.02054e7	−2.91391
$546$	0	0
$547$	1.08153e7	1.54550	0.772749	−	0.634712i	$-0.218881\pi$
$547$	1.08153e7	1.54550	0.772749	+	0.634712i	$0.218881\pi$
$548$	−5.19644e6	−0.739188
$549$	0	0
$550$	−2.92392e7	−4.12153
$551$	1.89737e6	0.266240
$552$	0	0
$553$	−1.08978e7	−1.51540
$554$	673599.	0.0932453
$555$	0	0
$556$	−1.03398e7	−1.41849
$557$	300600.	0.0410536	0.0205268	−	0.999789i	$-0.493466\pi$
$557$	300600.	0.0410536	0.0205268	+	0.999789i	$0.493466\pi$
$558$	0	0
$559$	−2.89530e6	−0.391890
$560$	−1.99617e7	−2.68984
$561$	0	0
$562$	−6.13851e6	−0.819827
$563$	9.21092e6	1.22471	0.612353	−	0.790584i	$-0.290223\pi$
$563$	9.21092e6	1.22471	0.612353	+	0.790584i	$0.290223\pi$
$564$	0	0
$565$	1.97884e7	2.60790
$566$	1.60730e7	2.10890
$567$	0	0
$568$	−1.37480e6	−0.178801
$569$	1.44126e6	0.186621	0.0933106	−	0.995637i	$-0.470255\pi$
$569$	1.44126e6	0.186621	0.0933106	+	0.995637i	$0.470255\pi$
$570$	0	0
$571$	4.03264e6	0.517606	0.258803	−	0.965930i	$-0.416672\pi$
$571$	4.03264e6	0.517606	0.258803	+	0.965930i	$0.416672\pi$
$572$	−1.41888e6	−0.181324
$573$	0	0
$574$	1.56036e7	1.97672
$575$	3.25353e7	4.10379
$576$	0	0
$577$	2.52617e6	0.315880	0.157940	−	0.987449i	$-0.449515\pi$
$577$	2.52617e6	0.315880	0.157940	+	0.987449i	$0.449515\pi$
$578$	5.31649e6	0.661919
$579$	0	0
$580$	2.40605e7	2.96985
$581$	−7.79674e6	−0.958237
$582$	0	0
$583$	−9.82407e6	−1.19707
$584$	−1.99477e6	−0.242025
$585$	0	0
$586$	1.07662e7	1.29515
$587$	−5.01886e6	−0.601188	−0.300594	−	0.953752i	$-0.597185\pi$
$587$	−5.01886e6	−0.601188	−0.300594	+	0.953752i	$0.597185\pi$
$588$	0	0
$589$	−2.17255e6	−0.258036
$590$	2.93113e6	0.346661
$591$	0	0
$592$	−7.34766e6	−0.861678
$593$	1.51902e7	1.77390	0.886948	−	0.461870i	$-0.152821\pi$
$593$	1.51902e7	1.77390	0.886948	+	0.461870i	$0.152821\pi$
$594$	0	0
$595$	1.46289e7	1.69402
$596$	2.60263e6	0.300121
$597$	0	0
$598$	3.46328e6	0.396036
$599$	−7.13948e6	−0.813017	−0.406509	−	0.913647i	$-0.633254\pi$
$599$	−7.13948e6	−0.813017	−0.406509	+	0.913647i	$0.633254\pi$
$600$	0	0
$601$	8.80337e6	0.994175	0.497088	−	0.867700i	$-0.334403\pi$
$601$	8.80337e6	0.994175	0.497088	+	0.867700i	$0.334403\pi$
$602$	−2.79941e7	−3.14830
$603$	0	0
$604$	759072.	0.0846624
$605$	2.32744e6	0.258517
$606$	0	0
$607$	−9.90444e6	−1.09108	−0.545542	−	0.838084i	$-0.683676\pi$
$607$	−9.90444e6	−1.09108	−0.545542	+	0.838084i	$0.683676\pi$
$608$	1.77502e6	0.194735
$609$	0	0
$610$	−1.15357e7	−1.25522
$611$	−1.09851e6	−0.119042
$612$	0	0
$613$	7.13725e6	0.767149	0.383574	−	0.923510i	$-0.374693\pi$
$613$	7.13725e6	0.767149	0.383574	+	0.923510i	$0.374693\pi$
$614$	1.02506e7	1.09730
$615$	0	0
$616$	2.65565e6	0.281981
$617$	6.15006e6	0.650379	0.325189	−	0.945649i	$-0.394572\pi$
$617$	6.15006e6	0.650379	0.325189	+	0.945649i	$0.394572\pi$
$618$	0	0
$619$	−749330.	−0.0786043	−0.0393022	−	0.999227i	$-0.512513\pi$
$619$	−749330.	−0.0786043	−0.0393022	+	0.999227i	$0.512513\pi$
$620$	−2.75499e7	−2.87833
$621$	0	0
$622$	−1.55339e7	−1.60992
$623$	7.92997e6	0.818561
$624$	0	0
$625$	4.20901e7	4.31003
$626$	−3.93747e6	−0.401588
$627$	0	0
$628$	−3.95336e6	−0.400007
$629$	5.38472e6	0.542670
$630$	0	0
$631$	−2.06221e6	−0.206186	−0.103093	−	0.994672i	$-0.532874\pi$
$631$	−2.06221e6	−0.206186	−0.103093	+	0.994672i	$0.532874\pi$
$632$	−2.77497e6	−0.276354
$633$	0	0
$634$	37800.2	0.00373483
$635$	−2.75296e6	−0.270936
$636$	0	0
$637$	−945003.	−0.0922751
$638$	−2.67582e7	−2.60259
$639$	0	0
$640$	−8.83210e6	−0.852342
$641$	3.59859e6	0.345929	0.172964	−	0.984928i	$-0.444665\pi$
$641$	3.59859e6	0.345929	0.172964	+	0.984928i	$0.444665\pi$
$642$	0	0
$643$	1.37456e7	1.31110	0.655549	−	0.755153i	$-0.272437\pi$
$643$	1.37456e7	1.31110	0.655549	+	0.755153i	$0.272437\pi$
$644$	1.52654e7	1.45042
$645$	0	0
$646$	−1.51749e6	−0.143068
$647$	−5.03818e6	−0.473165	−0.236583	−	0.971611i	$-0.576027\pi$
$647$	−5.03818e6	−0.473165	−0.236583	+	0.971611i	$0.576027\pi$
$648$	0	0
$649$	−1.48606e6	−0.138492
$650$	8.49078e6	0.788251
$651$	0	0
$652$	−3.76599e6	−0.346945
$653$	−1.53399e7	−1.40780	−0.703898	−	0.710301i	$-0.748559\pi$
$653$	−1.53399e7	−1.40780	−0.703898	+	0.710301i	$0.748559\pi$
$654$	0	0
$655$	−3.07745e6	−0.280277
$656$	1.51415e7	1.37376
$657$	0	0
$658$	−1.06212e7	−0.956337
$659$	3.56478e6	0.319756	0.159878	−	0.987137i	$-0.448890\pi$
$659$	3.56478e6	0.319756	0.159878	+	0.987137i	$0.448890\pi$
$660$	0	0
$661$	−1.62245e7	−1.44434	−0.722168	−	0.691718i	$-0.756854\pi$
$661$	−1.62245e7	−1.44434	−0.722168	+	0.691718i	$0.756854\pi$
$662$	1.51238e7	1.34127
$663$	0	0
$664$	−1.98532e6	−0.174748
$665$	3.98399e6	0.349353
$666$	0	0
$667$	2.97747e7	2.59139
$668$	896804.	0.0777600
$669$	0	0
$670$	1.19980e6	0.103258
$671$	5.84850e6	0.501462
$672$	0	0
$673$	3.40408e6	0.289709	0.144854	−	0.989453i	$-0.453729\pi$
$673$	3.40408e6	0.289709	0.144854	+	0.989453i	$0.453729\pi$
$674$	−1.76530e7	−1.49682
$675$	0	0
$676$	−9.54240e6	−0.803139
$677$	−1.17767e7	−0.987531	−0.493766	−	0.869595i	$-0.664380\pi$
$677$	−1.17767e7	−0.987531	−0.493766	+	0.869595i	$0.664380\pi$
$678$	0	0
$679$	3.42653e6	0.285220
$680$	3.72502e6	0.308928
$681$	0	0
$682$	3.06389e7	2.52239
$683$	−1.47519e7	−1.21003	−0.605015	−	0.796214i	$-0.706833\pi$
$683$	−1.47519e7	−1.21003	−0.605015	+	0.796214i	$0.706833\pi$
$684$	0	0
$685$	−2.12819e7	−1.73294
$686$	1.10084e7	0.893128
$687$	0	0
$688$	−2.71652e7	−2.18797
$689$	2.85282e6	0.228942
$690$	0	0
$691$	−1.18621e6	−0.0945079	−0.0472539	−	0.998883i	$-0.515047\pi$
$691$	−1.18621e6	−0.0945079	−0.0472539	+	0.998883i	$0.515047\pi$
$692$	−9.64755e6	−0.765864
$693$	0	0
$694$	7.26275e6	0.572404
$695$	−4.23465e7	−3.32549
$696$	0	0
$697$	−1.10964e7	−0.865171
$698$	−1.50379e7	−1.16828
$699$	0	0
$700$	3.74256e7	2.88684
$701$	1.36391e7	1.04831	0.524155	−	0.851623i	$-0.324381\pi$
$701$	1.36391e7	1.04831	0.524155	+	0.851623i	$0.324381\pi$
$702$	0	0
$703$	1.46646e6	0.111913
$704$	−9.14308e6	−0.695282
$705$	0	0
$706$	1.91081e7	1.44280
$707$	−2.13739e7	−1.60818
$708$	0	0
$709$	−1.43919e7	−1.07524	−0.537618	−	0.843189i	$-0.680676\pi$
$709$	−1.43919e7	−1.07524	−0.537618	+	0.843189i	$0.680676\pi$
$710$	2.90866e7	2.16544
$711$	0	0
$712$	2.01925e6	0.149276
$713$	−3.40928e7	−2.51153
$714$	0	0
$715$	−5.81098e6	−0.425093
$716$	−1.38589e6	−0.101029
$717$	0	0
$718$	2.17003e6	0.157092
$719$	−1.64117e7	−1.18395	−0.591974	−	0.805957i	$-0.701651\pi$
$719$	−1.64117e7	−1.18395	−0.591974	+	0.805957i	$0.701651\pi$
$720$	0	0
$721$	−7.15899e6	−0.512877
$722$	1.85754e7	1.32616
$723$	0	0
$724$	−1.74173e7	−1.23491
$725$	7.29974e7	5.15778
$726$	0	0
$727$	−1.31438e7	−0.922328	−0.461164	−	0.887315i	$-0.652568\pi$
$727$	−1.31438e7	−0.922328	−0.461164	+	0.887315i	$0.652568\pi$
$728$	−771176.	−0.0539293
$729$	0	0
$730$	4.22030e7	2.93114
$731$	1.99079e7	1.37795
$732$	0	0
$733$	2.02772e6	0.139395	0.0696977	−	0.997568i	$-0.477797\pi$
$733$	2.02772e6	0.139395	0.0696977	+	0.997568i	$0.477797\pi$
$734$	−7.84333e6	−0.537354
$735$	0	0
$736$	2.78546e7	1.89541
$737$	−608288.	−0.0412516
$738$	0	0
$739$	−1.39245e7	−0.937929	−0.468964	−	0.883217i	$-0.655373\pi$
$739$	−1.39245e7	−0.937929	−0.468964	+	0.883217i	$0.655373\pi$
$740$	1.85961e7	1.24837
$741$	0	0
$742$	2.75834e7	1.83924
$743$	1.13430e6	0.0753800	0.0376900	−	0.999289i	$-0.488000\pi$
$743$	1.13430e6	0.0753800	0.0376900	+	0.999289i	$0.488000\pi$
$744$	0	0
$745$	1.06590e7	0.703600
$746$	−2.31772e7	−1.52481
$747$	0	0
$748$	9.75611e6	0.637562
$749$	−6.58442e6	−0.428858
$750$	0	0
$751$	−2.62894e7	−1.70091	−0.850456	−	0.526047i	$-0.823674\pi$
$751$	−2.62894e7	−1.70091	−0.850456	+	0.526047i	$0.823674\pi$
$752$	−1.03067e7	−0.664624
$753$	0	0
$754$	7.77033e6	0.497750
$755$	3.10876e6	0.198482
$756$	0	0
$757$	2.04231e7	1.29534	0.647668	−	0.761923i	$-0.275744\pi$
$757$	2.04231e7	1.29534	0.647668	+	0.761923i	$0.275744\pi$
$758$	4.06029e6	0.256675
$759$	0	0
$760$	1.01446e6	0.0637093
$761$	−671201.	−0.0420137	−0.0210068	−	0.999779i	$-0.506687\pi$
$761$	−671201.	−0.0420137	−0.0210068	+	0.999779i	$0.506687\pi$
$762$	0	0
$763$	−2.87623e7	−1.78860
$764$	−2.05876e6	−0.127606
$765$	0	0
$766$	−2.26237e7	−1.39313
$767$	431537.	0.0264868
$768$	0	0
$769$	5.15936e6	0.314616	0.157308	−	0.987550i	$-0.449719\pi$
$769$	5.15936e6	0.314616	0.157308	+	0.987550i	$0.449719\pi$
$770$	−5.61853e7	−3.41504
$771$	0	0
$772$	1.33050e7	0.803472
$773$	−1.30533e7	−0.785727	−0.392863	−	0.919597i	$-0.628515\pi$
$773$	−1.30533e7	−0.785727	−0.392863	+	0.919597i	$0.628515\pi$
$774$	0	0
$775$	−8.35841e7	−4.99884
$776$	872515.	0.0520138
$777$	0	0
$778$	−2.27650e7	−1.34840
$779$	−3.02198e6	−0.178422
$780$	0	0
$781$	−1.47467e7	−0.865099
$782$	−2.38133e7	−1.39252
$783$	0	0
$784$	−8.86649e6	−0.515183
$785$	−1.61909e7	−0.937772
$786$	0	0
$787$	−1.77189e7	−1.01977	−0.509883	−	0.860244i	$-0.670311\pi$
$787$	−1.77189e7	−1.01977	−0.509883	+	0.860244i	$0.670311\pi$
$788$	1.18711e7	0.681045
$789$	0	0
$790$	5.87098e7	3.34690
$791$	2.81688e7	1.60076
$792$	0	0
$793$	−1.69835e6	−0.0959056
$794$	−7.97695e6	−0.449041
$795$	0	0
$796$	1.95803e7	1.09531
$797$	−1.81052e7	−1.00962	−0.504809	−	0.863231i	$-0.668437\pi$
$797$	−1.81052e7	−1.00962	−0.504809	+	0.863231i	$0.668437\pi$
$798$	0	0
$799$	7.55326e6	0.418569
$800$	6.82901e7	3.77253
$801$	0	0
$802$	2.53842e7	1.39357
$803$	−2.13966e7	−1.17100
$804$	0	0
$805$	6.25191e7	3.40034
$806$	−8.89725e6	−0.482412
$807$	0	0
$808$	−5.44254e6	−0.293274
$809$	1.50822e7	0.810204	0.405102	−	0.914272i	$-0.367236\pi$
$809$	1.50822e7	0.810204	0.405102	+	0.914272i	$0.367236\pi$
$810$	0	0
$811$	3.23070e7	1.72482	0.862411	−	0.506209i	$-0.168954\pi$
$811$	3.23070e7	1.72482	0.862411	+	0.506209i	$0.168954\pi$
$812$	3.42500e7	1.82293
$813$	0	0
$814$	−2.06811e7	−1.09399
$815$	−1.54235e7	−0.813374
$816$	0	0
$817$	5.42167e6	0.284170
$818$	3.99690e7	2.08852
$819$	0	0
$820$	−3.83215e7	−1.99025
$821$	3.33802e7	1.72835	0.864175	−	0.503191i	$-0.167841\pi$
$821$	3.33802e7	1.72835	0.864175	+	0.503191i	$0.167841\pi$
$822$	0	0
$823$	3.38256e7	1.74079	0.870393	−	0.492357i	$-0.163865\pi$
$823$	3.38256e7	1.74079	0.870393	+	0.492357i	$0.163865\pi$
$824$	−1.82293e6	−0.0935301
$825$	0	0
$826$	4.17245e6	0.212785
$827$	−3.25009e7	−1.65246	−0.826231	−	0.563332i	$-0.809519\pi$
$827$	−3.25009e7	−1.65246	−0.826231	+	0.563332i	$0.809519\pi$
$828$	0	0
$829$	−508321.	−0.0256893	−0.0128446	−	0.999918i	$-0.504089\pi$
$829$	−508321.	−0.0256893	−0.0128446	+	0.999918i	$0.504089\pi$
$830$	4.20033e7	2.11635
$831$	0	0
$832$	2.65506e6	0.132974
$833$	6.49778e6	0.324454
$834$	0	0
$835$	3.67284e6	0.182300
$836$	2.65696e6	0.131483
$837$	0	0
$838$	−2.61988e7	−1.28876
$839$	583710.	0.0286281	0.0143140	−	0.999898i	$-0.495444\pi$
$839$	583710.	0.0286281	0.0143140	+	0.999898i	$0.495444\pi$
$840$	0	0
$841$	4.62924e7	2.25694
$842$	−4.37493e7	−2.12663
$843$	0	0
$844$	2.57423e7	1.24391
$845$	−3.90807e7	−1.88287
$846$	0	0
$847$	3.31310e6	0.158681
$848$	2.67665e7	1.27821
$849$	0	0
$850$	−5.83821e7	−2.77161
$851$	2.30125e7	1.08928
$852$	0	0
$853$	9.83319e6	0.462724	0.231362	−	0.972868i	$-0.425682\pi$
$853$	9.83319e6	0.462724	0.231362	+	0.972868i	$0.425682\pi$
$854$	−1.64210e7	−0.770469
$855$	0	0
$856$	−1.67662e6	−0.0782081
$857$	−1.07762e7	−0.501201	−0.250601	−	0.968091i	$-0.580628\pi$
$857$	−1.07762e7	−0.501201	−0.250601	+	0.968091i	$0.580628\pi$
$858$	0	0
$859$	−3.47029e7	−1.60466	−0.802329	−	0.596883i	$-0.796406\pi$
$859$	−3.47029e7	−1.60466	−0.802329	+	0.596883i	$0.796406\pi$
$860$	6.87518e7	3.16985
$861$	0	0
$862$	−1.77235e7	−0.812422
$863$	2.34351e7	1.07112	0.535561	−	0.844496i	$-0.320100\pi$
$863$	2.34351e7	1.07112	0.535561	+	0.844496i	$0.320100\pi$
$864$	0	0
$865$	−3.95114e7	−1.79548
$866$	424762.	0.0192465
$867$	0	0
$868$	−3.92172e7	−1.76676
$869$	−2.97654e7	−1.33709
$870$	0	0
$871$	176641.	0.00788944
$872$	−7.32389e6	−0.326175
$873$	0	0
$874$	−6.48525e6	−0.287176
$875$	9.96447e7	4.39981
$876$	0	0
$877$	5.93447e6	0.260545	0.130273	−	0.991478i	$-0.458415\pi$
$877$	5.93447e6	0.260545	0.130273	+	0.991478i	$0.458415\pi$
$878$	4.29946e7	1.88225
$879$	0	0
$880$	−5.45215e7	−2.37335
$881$	7.45973e6	0.323805	0.161902	−	0.986807i	$-0.448237\pi$
$881$	7.45973e6	0.323805	0.161902	+	0.986807i	$0.448237\pi$
$882$	0	0
$883$	−2.46902e7	−1.06567	−0.532835	−	0.846219i	$-0.678873\pi$
$883$	−2.46902e7	−1.06567	−0.532835	+	0.846219i	$0.678873\pi$
$884$	−2.83308e6	−0.121935
$885$	0	0
$886$	4.09366e7	1.75198
$887$	−2.56140e7	−1.09312	−0.546562	−	0.837419i	$-0.684064\pi$
$887$	−2.56140e7	−1.09312	−0.546562	+	0.837419i	$0.684064\pi$
$888$	0	0
$889$	−3.91883e6	−0.166304
$890$	−4.27210e7	−1.80787
$891$	0	0
$892$	−3.07911e6	−0.129572
$893$	2.05704e6	0.0863204
$894$	0	0
$895$	−5.67589e6	−0.236852
$896$	−1.25725e7	−0.523179
$897$	0	0
$898$	−2.82304e7	−1.16822
$899$	−7.64919e7	−3.15658
$900$	0	0
$901$	−1.96158e7	−0.804996
$902$	4.26183e7	1.74413
$903$	0	0
$904$	7.17275e6	0.291921
$905$	−7.13321e7	−2.89510
$906$	0	0
$907$	−1.05309e7	−0.425055	−0.212528	−	0.977155i	$-0.568170\pi$
$907$	−1.05309e7	−0.425055	−0.212528	+	0.977155i	$0.568170\pi$
$908$	8.47231e6	0.341026
$909$	0	0
$910$	1.63157e7	0.653133
$911$	−8.73649e6	−0.348772	−0.174386	−	0.984677i	$-0.555794\pi$
$911$	−8.73649e6	−0.348772	−0.174386	+	0.984677i	$0.555794\pi$
$912$	0	0
$913$	−2.12953e7	−0.845487
$914$	−2.41453e7	−0.956019
$915$	0	0
$916$	−2.95268e6	−0.116273
$917$	−4.38074e6	−0.172038
$918$	0	0
$919$	−1.11596e7	−0.435874	−0.217937	−	0.975963i	$-0.569933\pi$
$919$	−1.11596e7	−0.435874	−0.217937	+	0.975963i	$0.569933\pi$
$920$	1.59195e7	0.620099
$921$	0	0
$922$	−2.34600e7	−0.908870
$923$	4.28229e6	0.165452
$924$	0	0
$925$	5.64189e7	2.16806
$926$	−6.40822e7	−2.45590
$927$	0	0
$928$	6.24957e7	2.38221
$929$	−334818.	−0.0127283	−0.00636415	−	0.999980i	$-0.502026\pi$
$929$	−334818.	−0.0127283	−0.00636415	+	0.999980i	$0.502026\pi$
$930$	0	0
$931$	1.76959e6	0.0669112
$932$	−231786.	−0.00874073
$933$	0	0
$934$	8.94046e6	0.335345
$935$	3.99560e7	1.49469
$936$	0	0
$937$	−4.10034e6	−0.152571	−0.0762853	−	0.997086i	$-0.524306\pi$
$937$	−4.10034e6	−0.152571	−0.0762853	+	0.997086i	$0.524306\pi$
$938$	1.70791e6	0.0633808
$939$	0	0
$940$	2.60851e7	0.962882
$941$	760066.	0.0279819	0.0139910	−	0.999902i	$-0.495546\pi$
$941$	760066.	0.0279819	0.0139910	+	0.999902i	$0.495546\pi$
$942$	0	0
$943$	−4.74226e7	−1.73663
$944$	4.04889e6	0.147879
$945$	0	0
$946$	−7.64606e7	−2.77786
$947$	−2.02978e7	−0.735486	−0.367743	−	0.929927i	$-0.619869\pi$
$947$	−2.02978e7	−0.735486	−0.367743	+	0.929927i	$0.619869\pi$
$948$	0	0
$949$	6.21336e6	0.223955
$950$	−1.58996e7	−0.571582
$951$	0	0
$952$	5.30256e6	0.189624
$953$	7.64595e6	0.272709	0.136354	−	0.990660i	$-0.456461\pi$
$953$	7.64595e6	0.272709	0.136354	+	0.990660i	$0.456461\pi$
$954$	0	0
$955$	−8.43161e6	−0.299159
$956$	−1.67209e7	−0.591718
$957$	0	0
$958$	−9.46316e6	−0.333137
$959$	−3.02947e7	−1.06370
$960$	0	0
$961$	5.89562e7	2.05931
$962$	6.00561e6	0.209228
$963$	0	0
$964$	−4.65095e7	−1.61194
$965$	5.44903e7	1.88365
$966$	0	0
$967$	−362592.	−0.0124696	−0.00623479	−	0.999981i	$-0.501985\pi$
$967$	−362592.	−0.0124696	−0.00623479	+	0.999981i	$0.501985\pi$
$968$	843631.	0.0289377
$969$	0	0
$970$	−1.84597e7	−0.629934
$971$	4.16836e7	1.41879	0.709393	−	0.704813i	$-0.248969\pi$
$971$	4.16836e7	1.41879	0.709393	+	0.704813i	$0.248969\pi$
$972$	0	0
$973$	−6.02800e7	−2.04123
$974$	−1.22822e7	−0.414838
$975$	0	0
$976$	−1.59347e7	−0.535452
$977$	2.63657e7	0.883695	0.441847	−	0.897090i	$-0.354323\pi$
$977$	2.63657e7	0.883695	0.441847	+	0.897090i	$0.354323\pi$
$978$	0	0
$979$	2.16592e7	0.722246
$980$	2.24400e7	0.746377
$981$	0	0
$982$	1.36960e7	0.453227
$983$	−3.95927e6	−0.130687	−0.0653433	−	0.997863i	$-0.520814\pi$
$983$	−3.95927e6	−0.130687	−0.0653433	+	0.997863i	$0.520814\pi$
$984$	0	0
$985$	4.86179e7	1.59663
$986$	−5.34283e7	−1.75017
$987$	0	0
$988$	−771555.	−0.0251463
$989$	8.50800e7	2.76590
$990$	0	0
$991$	−3.87210e7	−1.25246	−0.626228	−	0.779640i	$-0.715402\pi$
$991$	−3.87210e7	−1.25246	−0.626228	+	0.779640i	$0.715402\pi$
$992$	−7.15592e7	−2.30880
$993$	0	0
$994$	4.14047e7	1.32918
$995$	8.01908e7	2.56783
$996$	0	0
$997$	3.15392e7	1.00488	0.502438	−	0.864613i	$-0.332436\pi$
$997$	3.15392e7	1.00488	0.502438	+	0.864613i	$0.332436\pi$
$998$	−6.12442e7	−1.94643
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.b.1.2		11
3.2	odd	2		177.6.a.a.1.10	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.a.1.10	✓	11	3.2	odd	2
531.6.a.b.1.2		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-7.66878 q^{2} +26.8102 q^{4} +109.801 q^{5} +156.301 q^{7} +39.7996 q^{8} +O(q^{10})\)	\(q-7.66878 q^{2} +26.8102 q^{4} +109.801 q^{5} +156.301 q^{7} +39.7996 q^{8} -842.036 q^{10} +426.905 q^{11} -123.969 q^{13} -1198.64 q^{14} -1163.14 q^{16} +852.404 q^{17} +232.142 q^{19} +2943.77 q^{20} -3273.84 q^{22} +3642.90 q^{23} +8931.16 q^{25} +950.692 q^{26} +4190.45 q^{28} +8173.34 q^{29} -9358.70 q^{31} +7646.28 q^{32} -6536.90 q^{34} +17161.9 q^{35} +6317.09 q^{37} -1780.24 q^{38} +4370.02 q^{40} -13017.8 q^{41} +23355.0 q^{43} +11445.4 q^{44} -27936.6 q^{46} +8861.12 q^{47} +7622.89 q^{49} -68491.1 q^{50} -3323.64 q^{52} -23012.3 q^{53} +46874.4 q^{55} +6220.71 q^{56} -62679.6 q^{58} -3481.00 q^{59} +13699.8 q^{61} +71769.8 q^{62} -21417.1 q^{64} -13611.9 q^{65} -1424.88 q^{67} +22853.1 q^{68} -131611. q^{70} -34543.2 q^{71} -50120.2 q^{73} -48444.4 q^{74} +6223.76 q^{76} +66725.5 q^{77} -69723.6 q^{79} -127713. q^{80} +99830.8 q^{82} -49883.0 q^{83} +93594.5 q^{85} -179104. q^{86} +16990.7 q^{88} +50735.3 q^{89} -19376.5 q^{91} +97666.8 q^{92} -67954.0 q^{94} +25489.3 q^{95} +21922.7 q^{97} -58458.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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