LMFDB - Embedded newform 531.6.a.c.1.7

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			1.7
Root			$1.38446$ 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)

$f(q)$	$=$	$q-0.615542 q^{2} -31.6211 q^{4} -61.9372 q^{5} -209.534 q^{7} +39.1615 q^{8} +O(q^{10})$	\(q-0.615542 q^{2} -31.6211 q^{4} -61.9372 q^{5} -209.534 q^{7} +39.1615 q^{8} +38.1250 q^{10} -1.45515 q^{11} +351.120 q^{13} +128.977 q^{14} +987.770 q^{16} -689.107 q^{17} +2735.73 q^{19} +1958.52 q^{20} +0.895706 q^{22} +1310.96 q^{23} +711.222 q^{25} -216.129 q^{26} +6625.69 q^{28} -508.995 q^{29} +4218.33 q^{31} -1861.18 q^{32} +424.174 q^{34} +12977.9 q^{35} +4022.89 q^{37} -1683.95 q^{38} -2425.55 q^{40} -7333.96 q^{41} +5250.80 q^{43} +46.0135 q^{44} -806.951 q^{46} -25394.3 q^{47} +27097.3 q^{49} -437.787 q^{50} -11102.8 q^{52} +16562.5 q^{53} +90.1280 q^{55} -8205.64 q^{56} +313.308 q^{58} +3481.00 q^{59} -4267.66 q^{61} -2596.56 q^{62} -30463.0 q^{64} -21747.4 q^{65} +13390.6 q^{67} +21790.3 q^{68} -7988.46 q^{70} -54112.6 q^{71} -32575.3 q^{73} -2476.26 q^{74} -86506.7 q^{76} +304.903 q^{77} +59592.4 q^{79} -61179.7 q^{80} +4514.36 q^{82} -94095.2 q^{83} +42681.4 q^{85} -3232.09 q^{86} -56.9858 q^{88} +136311. q^{89} -73571.4 q^{91} -41454.0 q^{92} +15631.3 q^{94} -169443. q^{95} +135175. q^{97} -16679.6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.615542	−0.108813	−0.0544067	−	0.998519i	$-0.517327\pi$
$2$	−0.615542	−0.108813	−0.0544067	+	0.998519i	$0.517327\pi$
$3$	0	0
$3$	0	0
$4$	−31.6211	−0.988160
$5$	−61.9372	−1.10797	−0.553984	−	0.832528i	$-0.686893\pi$
$5$	−61.9372	−1.10797	−0.553984	+	0.832528i	$0.686893\pi$
$6$	0	0
$7$	−209.534	−1.61625	−0.808125	−	0.589011i	$-0.799518\pi$
$7$	−209.534	−1.61625	−0.808125	+	0.589011i	$0.799518\pi$
$8$	39.1615	0.216339
$9$	0	0
$10$	38.1250	0.120562
$11$	−1.45515	−0.00362599	−0.00181299	−	0.999998i	$-0.500577\pi$
$11$	−1.45515	−0.00362599	−0.00181299	+	0.999998i	$0.500577\pi$
$12$	0	0
$13$	351.120	0.576231	0.288116	−	0.957596i	$-0.406971\pi$
$13$	351.120	0.576231	0.288116	+	0.957596i	$0.406971\pi$
$14$	128.977	0.175870
$15$	0	0
$16$	987.770	0.964619
$17$	−689.107	−0.578315	−0.289157	−	0.957282i	$-0.593375\pi$
$17$	−689.107	−0.578315	−0.289157	+	0.957282i	$0.593375\pi$
$18$	0	0
$19$	2735.73	1.73856	0.869278	−	0.494323i	$-0.164584\pi$
$19$	2735.73	1.73856	0.869278	+	0.494323i	$0.164584\pi$
$20$	1958.52	1.09485
$21$	0	0
$22$	0.895706	0.000394556 0
$23$	1310.96	0.516737	0.258369	−	0.966046i	$-0.416815\pi$
$23$	1310.96	0.516737	0.258369	+	0.966046i	$0.416815\pi$
$24$	0	0
$25$	711.222	0.227591
$26$	−216.129	−0.0627017
$27$	0	0
$28$	6625.69	1.59711
$29$	−508.995	−0.112388	−0.0561938	−	0.998420i	$-0.517896\pi$
$29$	−508.995	−0.112388	−0.0561938	+	0.998420i	$0.517896\pi$
$30$	0	0
$31$	4218.33	0.788381	0.394191	−	0.919029i	$-0.371025\pi$
$31$	4218.33	0.788381	0.394191	+	0.919029i	$0.371025\pi$
$32$	−1861.18	−0.321302
$33$	0	0
$34$	424.174	0.0629285
$35$	12977.9	1.79075
$36$	0	0
$37$	4022.89	0.483096	0.241548	−	0.970389i	$-0.422345\pi$
$37$	4022.89	0.483096	0.241548	+	0.970389i	$0.422345\pi$
$38$	−1683.95	−0.189178
$39$	0	0
$40$	−2425.55	−0.239696
$41$	−7333.96	−0.681364	−0.340682	−	0.940179i	$-0.610658\pi$
$41$	−7333.96	−0.681364	−0.340682	+	0.940179i	$0.610658\pi$
$42$	0	0
$43$	5250.80	0.433066	0.216533	−	0.976275i	$-0.430525\pi$
$43$	5250.80	0.433066	0.216533	+	0.976275i	$0.430525\pi$
$44$	46.0135	0.00358305
$45$	0	0
$46$	−806.951	−0.0562280
$47$	−25394.3	−1.67684	−0.838420	−	0.545024i	$-0.816521\pi$
$47$	−25394.3	−1.67684	−0.838420	+	0.545024i	$0.816521\pi$
$48$	0	0
$49$	27097.3	1.61227
$50$	−437.787	−0.0247650
$51$	0	0
$52$	−11102.8	−0.569408
$53$	16562.5	0.809911	0.404956	−	0.914336i	$-0.367287\pi$
$53$	16562.5	0.809911	0.404956	+	0.914336i	$0.367287\pi$
$54$	0	0
$55$	90.1280	0.00401747
$56$	−8205.64	−0.349657
$57$	0	0
$58$	313.308	0.0122293
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−4267.66	−0.146847	−0.0734235	−	0.997301i	$-0.523392\pi$
$61$	−4267.66	−0.146847	−0.0734235	+	0.997301i	$0.523392\pi$
$62$	−2596.56	−0.0857865
$63$	0	0
$64$	−30463.0	−0.929657
$65$	−21747.4	−0.638445
$66$	0	0
$67$	13390.6	0.364429	0.182215	−	0.983259i	$-0.441673\pi$
$67$	13390.6	0.364429	0.182215	+	0.983259i	$0.441673\pi$
$68$	21790.3	0.571467
$69$	0	0
$70$	−7988.46	−0.194858
$71$	−54112.6	−1.27395	−0.636976	−	0.770884i	$-0.719815\pi$
$71$	−54112.6	−1.27395	−0.636976	+	0.770884i	$0.719815\pi$
$72$	0	0
$73$	−32575.3	−0.715453	−0.357726	−	0.933826i	$-0.616448\pi$
$73$	−32575.3	−0.715453	−0.357726	+	0.933826i	$0.616448\pi$
$74$	−2476.26	−0.0525674
$75$	0	0
$76$	−86506.7	−1.71797
$77$	304.903	0.00586050
$78$	0	0
$79$	59592.4	1.07429	0.537147	−	0.843489i	$-0.319502\pi$
$79$	59592.4	1.07429	0.537147	+	0.843489i	$0.319502\pi$
$80$	−61179.7	−1.06877
$81$	0	0
$82$	4514.36	0.0741415
$83$	−94095.2	−1.49924	−0.749622	−	0.661867i	$-0.769764\pi$
$83$	−94095.2	−1.49924	−0.749622	+	0.661867i	$0.769764\pi$
$84$	0	0
$85$	42681.4	0.640754
$86$	−3232.09	−0.0471235
$87$	0	0
$88$	−56.9858	−0.000784441 0
$89$	136311.	1.82414	0.912068	−	0.410040i	$-0.134485\pi$
$89$	136311.	1.82414	0.912068	+	0.410040i	$0.134485\pi$
$90$	0	0
$91$	−73571.4	−0.931334
$92$	−41454.0	−0.510619
$93$	0	0
$94$	15631.3	0.182463
$95$	−169443.	−1.92626
$96$	0	0
$97$	135175.	1.45870	0.729350	−	0.684141i	$-0.239823\pi$
$97$	135175.	1.45870	0.729350	+	0.684141i	$0.239823\pi$
$98$	−16679.6	−0.175436
$99$	0	0
$100$	−22489.6	−0.224896
$101$	25496.8	0.248704	0.124352	−	0.992238i	$-0.460315\pi$
$101$	25496.8	0.248704	0.124352	+	0.992238i	$0.460315\pi$
$102$	0	0
$103$	62339.3	0.578987	0.289493	−	0.957180i	$-0.406513\pi$
$103$	62339.3	0.578987	0.289493	+	0.957180i	$0.406513\pi$
$104$	13750.4	0.124661
$105$	0	0
$106$	−10194.9	−0.0881293
$107$	16258.7	0.137286	0.0686429	−	0.997641i	$-0.478133\pi$
$107$	16258.7	0.137286	0.0686429	+	0.997641i	$0.478133\pi$
$108$	0	0
$109$	−209396.	−1.68812	−0.844059	−	0.536250i	$-0.819840\pi$
$109$	−209396.	−1.68812	−0.844059	+	0.536250i	$0.819840\pi$
$110$	−55.4776	−0.000437155 0
$111$	0	0
$112$	−206971.	−1.55907
$113$	143342.	1.05603	0.528017	−	0.849234i	$-0.322936\pi$
$113$	143342.	1.05603	0.528017	+	0.849234i	$0.322936\pi$
$114$	0	0
$115$	−81197.2	−0.572528
$116$	16095.0	0.111057
$117$	0	0
$118$	−2142.70	−0.0141663
$119$	144391.	0.934702
$120$	0	0
$121$	−161049.	−0.999987
$122$	2626.92	0.0159789
$123$	0	0
$124$	−133388.	−0.779046
$125$	149503.	0.855804
$126$	0	0
$127$	79745.4	0.438729	0.219365	−	0.975643i	$-0.429602\pi$
$127$	79745.4	0.438729	0.219365	+	0.975643i	$0.429602\pi$
$128$	78309.0	0.422461
$129$	0	0
$130$	13386.4	0.0694714
$131$	5118.41	0.0260589	0.0130295	−	0.999915i	$-0.495852\pi$
$131$	5118.41	0.0260589	0.0130295	+	0.999915i	$0.495852\pi$
$132$	0	0
$133$	−573227.	−2.80994
$134$	−8242.49	−0.0396548
$135$	0	0
$136$	−26986.4	−0.125112
$137$	237403.	1.08065	0.540324	−	0.841457i	$-0.318302\pi$
$137$	237403.	1.08065	0.540324	+	0.841457i	$0.318302\pi$
$138$	0	0
$139$	−60191.4	−0.264239	−0.132120	−	0.991234i	$-0.542178\pi$
$139$	−60191.4	−0.264239	−0.132120	+	0.991234i	$0.542178\pi$
$140$	−410377.	−1.76955
$141$	0	0
$142$	33308.6	0.138623
$143$	−510.932	−0.00208941
$144$	0	0
$145$	31525.7	0.124522
$146$	20051.4	0.0778509
$147$	0	0
$148$	−127208.	−0.477376
$149$	226647.	0.836343	0.418171	−	0.908368i	$-0.362671\pi$
$149$	226647.	0.836343	0.418171	+	0.908368i	$0.362671\pi$
$150$	0	0
$151$	−294492.	−1.05107	−0.525534	−	0.850773i	$-0.676134\pi$
$151$	−294492.	−1.05107	−0.525534	+	0.850773i	$0.676134\pi$
$152$	107135.	0.376117
$153$	0	0
$154$	−187.681	−0.000637702 0
$155$	−261272.	−0.873500
$156$	0	0
$157$	527580.	1.70820	0.854101	−	0.520108i	$-0.174108\pi$
$157$	527580.	1.70820	0.854101	+	0.520108i	$0.174108\pi$
$158$	−36681.6	−0.116898
$159$	0	0
$160$	115276.	0.355992
$161$	−274690.	−0.835177
$162$	0	0
$163$	138581.	0.408540	0.204270	−	0.978915i	$-0.434518\pi$
$163$	138581.	0.408540	0.204270	+	0.978915i	$0.434518\pi$
$164$	231908.	0.673296
$165$	0	0
$166$	57919.5	0.163138
$167$	−201512.	−0.559126	−0.279563	−	0.960127i	$-0.590190\pi$
$167$	−201512.	−0.559126	−0.279563	+	0.960127i	$0.590190\pi$
$168$	0	0
$169$	−248008.	−0.667958
$170$	−26272.2	−0.0697227
$171$	0	0
$172$	−166036.	−0.427939
$173$	−463291.	−1.17690	−0.588449	−	0.808535i	$-0.700261\pi$
$173$	−463291.	−1.17690	−0.588449	+	0.808535i	$0.700261\pi$
$174$	0	0
$175$	−149025.	−0.367844
$176$	−1437.35	−0.00349770
$177$	0	0
$178$	−83905.4	−0.198490
$179$	235004.	0.548205	0.274103	−	0.961700i	$-0.411619\pi$
$179$	235004.	0.548205	0.274103	+	0.961700i	$0.411619\pi$
$180$	0	0
$181$	345261.	0.783341	0.391670	−	0.920106i	$-0.371897\pi$
$181$	345261.	0.783341	0.391670	+	0.920106i	$0.371897\pi$
$182$	45286.3	0.101342
$183$	0	0
$184$	51339.1	0.111790
$185$	−249167.	−0.535255
$186$	0	0
$187$	1002.75	0.00209696
$188$	802996.	1.65699
$189$	0	0
$190$	104300.	0.209603
$191$	−973228.	−1.93033	−0.965164	−	0.261646i	$-0.915735\pi$
$191$	−973228.	−1.93033	−0.965164	+	0.261646i	$0.915735\pi$
$192$	0	0
$193$	−493451.	−0.953567	−0.476783	−	0.879021i	$-0.658197\pi$
$193$	−493451.	−0.953567	−0.476783	+	0.879021i	$0.658197\pi$
$194$	−83205.7	−0.158726
$195$	0	0
$196$	−856848.	−1.59318
$197$	−339146.	−0.622616	−0.311308	−	0.950309i	$-0.600767\pi$
$197$	−339146.	−0.622616	−0.311308	+	0.950309i	$0.600767\pi$
$198$	0	0
$199$	−589702.	−1.05560	−0.527801	−	0.849368i	$-0.676983\pi$
$199$	−589702.	−1.05560	−0.527801	+	0.849368i	$0.676983\pi$
$200$	27852.5	0.0492368
$201$	0	0
$202$	−15694.4	−0.0270623
$203$	106652.	0.181646
$204$	0	0
$205$	454245.	0.754929
$206$	−38372.4	−0.0630015
$207$	0	0
$208$	346825.	0.555844
$209$	−3980.89	−0.00630398
$210$	0	0
$211$	779769.	1.20576	0.602879	−	0.797833i	$-0.294020\pi$
$211$	779769.	1.20576	0.602879	+	0.797833i	$0.294020\pi$
$212$	−523726.	−0.800322
$213$	0	0
$214$	−10007.9	−0.0149385
$215$	−325220.	−0.479823
$216$	0	0
$217$	−883882.	−1.27422
$218$	128892.	0.183690
$219$	0	0
$220$	−2849.95	−0.00396991
$221$	−241959.	−0.333243
$222$	0	0
$223$	148653.	0.200176	0.100088	−	0.994979i	$-0.468088\pi$
$223$	148653.	0.200176	0.100088	+	0.994979i	$0.468088\pi$
$224$	389980.	0.519305
$225$	0	0
$226$	−88233.2	−0.114911
$227$	705507.	0.908733	0.454367	−	0.890815i	$-0.349866\pi$
$227$	705507.	0.908733	0.454367	+	0.890815i	$0.349866\pi$
$228$	0	0
$229$	221449.	0.279052	0.139526	−	0.990218i	$-0.455442\pi$
$229$	221449.	0.279052	0.139526	+	0.990218i	$0.455442\pi$
$230$	49980.3	0.0622987
$231$	0	0
$232$	−19933.0	−0.0243138
$233$	303806.	0.366612	0.183306	−	0.983056i	$-0.441320\pi$
$233$	303806.	0.366612	0.183306	+	0.983056i	$0.441320\pi$
$234$	0	0
$235$	1.57285e6	1.85788
$236$	−110073.	−0.128647
$237$	0	0
$238$	−88878.8	−0.101708
$239$	−1.12989e6	−1.27950	−0.639751	−	0.768583i	$-0.720962\pi$
$239$	−1.12989e6	−1.27950	−0.639751	+	0.768583i	$0.720962\pi$
$240$	0	0
$241$	−1.30930e6	−1.45210	−0.726049	−	0.687643i	$-0.758645\pi$
$241$	−1.30930e6	−1.45210	−0.726049	+	0.687643i	$0.758645\pi$
$242$	99132.3	0.108812
$243$	0	0
$244$	134948.	0.145108
$245$	−1.67834e6	−1.78634
$246$	0	0
$247$	960568.	1.00181
$248$	165196.	0.170557
$249$	0	0
$250$	−92025.2	−0.0931230
$251$	948490.	0.950273	0.475136	−	0.879912i	$-0.342399\pi$
$251$	948490.	0.950273	0.475136	+	0.879912i	$0.342399\pi$
$252$	0	0
$253$	−1907.64	−0.00187368
$254$	−49086.7	−0.0477396
$255$	0	0
$256$	926614.	0.883688
$257$	−1.82976e6	−1.72807	−0.864036	−	0.503431i	$-0.832071\pi$
$257$	−1.82976e6	−1.72807	−0.864036	+	0.503431i	$0.832071\pi$
$258$	0	0
$259$	−842930.	−0.780804
$260$	687676.	0.630886
$261$	0	0
$262$	−3150.60	−0.00283556
$263$	−1.23847e6	−1.10407	−0.552033	−	0.833822i	$-0.686148\pi$
$263$	−1.23847e6	−1.10407	−0.552033	+	0.833822i	$0.686148\pi$
$264$	0	0
$265$	−1.02584e6	−0.897355
$266$	352845.	0.305760
$267$	0	0
$268$	−423426.	−0.360115
$269$	−1.19822e6	−1.00962	−0.504808	−	0.863232i	$-0.668437\pi$
$269$	−1.19822e6	−1.00962	−0.504808	+	0.863232i	$0.668437\pi$
$270$	0	0
$271$	−907388.	−0.750533	−0.375267	−	0.926917i	$-0.622449\pi$
$271$	−907388.	−0.750533	−0.375267	+	0.926917i	$0.622449\pi$
$272$	−680679.	−0.557854
$273$	0	0
$274$	−146131.	−0.117589
$275$	−1034.94	−0.000825243 0
$276$	0	0
$277$	196862.	0.154157	0.0770783	−	0.997025i	$-0.475441\pi$
$277$	196862.	0.154157	0.0770783	+	0.997025i	$0.475441\pi$
$278$	37050.3	0.0287528
$279$	0	0
$280$	508235.	0.387409
$281$	−2.31407e6	−1.74828	−0.874140	−	0.485674i	$-0.838574\pi$
$281$	−2.31407e6	−1.74828	−0.874140	+	0.485674i	$0.838574\pi$
$282$	0	0
$283$	42804.6	0.0317705	0.0158852	−	0.999874i	$-0.494943\pi$
$283$	42804.6	0.0317705	0.0158852	+	0.999874i	$0.494943\pi$
$284$	1.71110e6	1.25887
$285$	0	0
$286$	314.500	0.000227356 0
$287$	1.53671e6	1.10125
$288$	0	0
$289$	−944988.	−0.665552
$290$	−19405.4	−0.0135496
$291$	0	0
$292$	1.03007e6	0.706981
$293$	943192.	0.641846	0.320923	−	0.947105i	$-0.396007\pi$
$293$	943192.	0.641846	0.320923	+	0.947105i	$0.396007\pi$
$294$	0	0
$295$	−215604.	−0.144245
$296$	157542.	0.104512
$297$	0	0
$298$	−139511.	−0.0910054
$299$	460304.	0.297760
$300$	0	0
$301$	−1.10022e6	−0.699944
$302$	181272.	0.114370
$303$	0	0
$304$	2.70227e6	1.67704
$305$	264327.	0.162702
$306$	0	0
$307$	−1.05157e6	−0.636784	−0.318392	−	0.947959i	$-0.603143\pi$
$307$	−1.05157e6	−0.636784	−0.318392	+	0.947959i	$0.603143\pi$
$308$	−9641.37	−0.00579111
$309$	0	0
$310$	160824.	0.0950486
$311$	616781.	0.361601	0.180801	−	0.983520i	$-0.442131\pi$
$311$	616781.	0.361601	0.180801	+	0.983520i	$0.442131\pi$
$312$	0	0
$313$	1.22803e6	0.708516	0.354258	−	0.935148i	$-0.384733\pi$
$313$	1.22803e6	0.708516	0.354258	+	0.935148i	$0.384733\pi$
$314$	−324747.	−0.185875
$315$	0	0
$316$	−1.88438e6	−1.06157
$317$	2.46036e6	1.37515	0.687577	−	0.726112i	$-0.258675\pi$
$317$	2.46036e6	1.37515	0.687577	+	0.726112i	$0.258675\pi$
$318$	0	0
$319$	740.664	0.000407516 0
$320$	1.88679e6	1.03003
$321$	0	0
$322$	169083.	0.0908785
$323$	−1.88521e6	−1.00543
$324$	0	0
$325$	249724.	0.131145
$326$	−85302.3	−0.0444546
$327$	0	0
$328$	−287209.	−0.147405
$329$	5.32096e6	2.71019
$330$	0	0
$331$	58317.0	0.0292567	0.0146283	−	0.999893i	$-0.495343\pi$
$331$	58317.0	0.0292567	0.0146283	+	0.999893i	$0.495343\pi$
$332$	2.97539e6	1.48149
$333$	0	0
$334$	124039.	0.0608405
$335$	−829378.	−0.403776
$336$	0	0
$337$	2.05912e6	0.987658	0.493829	−	0.869559i	$-0.335597\pi$
$337$	2.05912e6	0.987658	0.493829	+	0.869559i	$0.335597\pi$
$338$	152659.	0.0726828
$339$	0	0
$340$	−1.34963e6	−0.633167
$341$	−6138.30	−0.00285866
$342$	0	0
$343$	−2.15617e6	−0.989574
$344$	205629.	0.0936890
$345$	0	0
$346$	285175.	0.128062
$347$	3.05058e6	1.36006	0.680030	−	0.733184i	$-0.261967\pi$
$347$	3.05058e6	1.36006	0.680030	+	0.733184i	$0.261967\pi$
$348$	0	0
$349$	−2.54878e6	−1.12013	−0.560065	−	0.828449i	$-0.689224\pi$
$349$	−2.54878e6	−1.12013	−0.560065	+	0.828449i	$0.689224\pi$
$350$	91731.2	0.0400264
$351$	0	0
$352$	2708.30	0.00116504
$353$	139978.	0.0597891	0.0298945	−	0.999553i	$-0.490483\pi$
$353$	139978.	0.0597891	0.0298945	+	0.999553i	$0.490483\pi$
$354$	0	0
$355$	3.35159e6	1.41150
$356$	−4.31032e6	−1.80254
$357$	0	0
$358$	−144655.	−0.0596521
$359$	587724.	0.240679	0.120339	−	0.992733i	$-0.461602\pi$
$359$	587724.	0.240679	0.120339	+	0.992733i	$0.461602\pi$
$360$	0	0
$361$	5.00810e6	2.02258
$362$	−212522.	−0.0852380
$363$	0	0
$364$	2.32641e6	0.920307
$365$	2.01762e6	0.792698
$366$	0	0
$367$	360316.	0.139643	0.0698214	−	0.997560i	$-0.477757\pi$
$367$	360316.	0.139643	0.0698214	+	0.997560i	$0.477757\pi$
$368$	1.29493e6	0.498455
$369$	0	0
$370$	153372.	0.0582429
$371$	−3.47041e6	−1.30902
$372$	0	0
$373$	5.26924e6	1.96099	0.980497	−	0.196535i	$-0.0629690\pi$
$373$	5.26924e6	1.96099	0.980497	+	0.196535i	$0.0629690\pi$
$374$	−617.237	−0.000228178 0
$375$	0	0
$376$	−994478.	−0.362765
$377$	−178718.	−0.0647612
$378$	0	0
$379$	3.43445e6	1.22817	0.614086	−	0.789239i	$-0.289525\pi$
$379$	3.43445e6	1.22817	0.614086	+	0.789239i	$0.289525\pi$
$380$	5.35799e6	1.90346
$381$	0	0
$382$	599062.	0.210046
$383$	3.55421e6	1.23807	0.619036	−	0.785363i	$-0.287524\pi$
$383$	3.55421e6	1.23807	0.619036	+	0.785363i	$0.287524\pi$
$384$	0	0
$385$	−18884.8	−0.00649324
$386$	303740.	0.103761
$387$	0	0
$388$	−4.27437e6	−1.44143
$389$	−4.52272e6	−1.51539	−0.757697	−	0.652606i	$-0.773676\pi$
$389$	−4.52272e6	−1.51539	−0.757697	+	0.652606i	$0.773676\pi$
$390$	0	0
$391$	−903392.	−0.298837
$392$	1.06117e6	0.348795
$393$	0	0
$394$	208758.	0.0677491
$395$	−3.69099e6	−1.19028
$396$	0	0
$397$	407622.	0.129802	0.0649011	−	0.997892i	$-0.479327\pi$
$397$	407622.	0.129802	0.0649011	+	0.997892i	$0.479327\pi$
$398$	362987.	0.114864
$399$	0	0
$400$	702524.	0.219539
$401$	−3.00347e6	−0.932743	−0.466372	−	0.884589i	$-0.654439\pi$
$401$	−3.00347e6	−0.932743	−0.466372	+	0.884589i	$0.654439\pi$
$402$	0	0
$403$	1.48114e6	0.454290
$404$	−806237.	−0.245759
$405$	0	0
$406$	−65648.5	−0.0197656
$407$	−5853.91	−0.00175170
$408$	0	0
$409$	−4.77645e6	−1.41188	−0.705938	−	0.708274i	$-0.749474\pi$
$409$	−4.77645e6	−1.41188	−0.705938	+	0.708274i	$0.749474\pi$
$410$	−279607.	−0.0821464
$411$	0	0
$412$	−1.97124e6	−0.572131
$413$	−729387.	−0.210418
$414$	0	0
$415$	5.82799e6	1.66111
$416$	−653497.	−0.185144
$417$	0	0
$418$	2450.41	0.000685958 0
$419$	2.95949e6	0.823535	0.411768	−	0.911289i	$-0.364912\pi$
$419$	2.95949e6	0.823535	0.411768	+	0.911289i	$0.364912\pi$
$420$	0	0
$421$	1.17118e6	0.322047	0.161024	−	0.986951i	$-0.448520\pi$
$421$	1.17118e6	0.322047	0.161024	+	0.986951i	$0.448520\pi$
$422$	−479981.	−0.131203
$423$	0	0
$424$	648614.	0.175215
$425$	−490108.	−0.131619
$426$	0	0
$427$	894218.	0.237342
$428$	−514117.	−0.135660
$429$	0	0
$430$	200187.	0.0522112
$431$	2.43477e6	0.631341	0.315671	−	0.948869i	$-0.397771\pi$
$431$	2.43477e6	0.631341	0.315671	+	0.948869i	$0.397771\pi$
$432$	0	0
$433$	−229760.	−0.0588917	−0.0294459	−	0.999566i	$-0.509374\pi$
$433$	−229760.	−0.0588917	−0.0294459	+	0.999566i	$0.509374\pi$
$434$	544066.	0.138652
$435$	0	0
$436$	6.62134e6	1.66813
$437$	3.58643e6	0.898377
$438$	0	0
$439$	−2.63343e6	−0.652168	−0.326084	−	0.945341i	$-0.605729\pi$
$439$	−2.63343e6	−0.652168	−0.326084	+	0.945341i	$0.605729\pi$
$440$	3529.54	0.000869134 0
$441$	0	0
$442$	148936.	0.0362613
$443$	5.30123e6	1.28342	0.641708	−	0.766949i	$-0.278226\pi$
$443$	5.30123e6	1.28342	0.641708	+	0.766949i	$0.278226\pi$
$444$	0	0
$445$	−8.44275e6	−2.02108
$446$	−91502.4	−0.0217819
$447$	0	0
$448$	6.38302e6	1.50256
$449$	−2.32172e6	−0.543494	−0.271747	−	0.962369i	$-0.587601\pi$
$449$	−2.32172e6	−0.543494	−0.271747	+	0.962369i	$0.587601\pi$
$450$	0	0
$451$	10672.0	0.00247062
$452$	−4.53264e6	−1.04353
$453$	0	0
$454$	−434269.	−0.0988824
$455$	4.55681e6	1.03189
$456$	0	0
$457$	−3.05394e6	−0.684022	−0.342011	−	0.939696i	$-0.611108\pi$
$457$	−3.05394e6	−0.684022	−0.342011	+	0.939696i	$0.611108\pi$
$458$	−136311.	−0.0303647
$459$	0	0
$460$	2.56755e6	0.565749
$461$	−4.33222e6	−0.949419	−0.474710	−	0.880143i	$-0.657447\pi$
$461$	−4.33222e6	−0.949419	−0.474710	+	0.880143i	$0.657447\pi$
$462$	0	0
$463$	−6.38768e6	−1.38481	−0.692406	−	0.721508i	$-0.743449\pi$
$463$	−6.38768e6	−1.38481	−0.692406	+	0.721508i	$0.743449\pi$
$464$	−502770.	−0.108411
$465$	0	0
$466$	−187006.	−0.0398924
$467$	−595087.	−0.126266	−0.0631332	−	0.998005i	$-0.520109\pi$
$467$	−595087.	−0.126266	−0.0631332	+	0.998005i	$0.520109\pi$
$468$	0	0
$469$	−2.80578e6	−0.589009
$470$	−968158.	−0.202163
$471$	0	0
$472$	136321.	0.0281649
$473$	−7640.71	−0.00157029
$474$	0	0
$475$	1.94571e6	0.395680
$476$	−4.56581e6	−0.923635
$477$	0	0
$478$	695494.	0.139227
$479$	−854216.	−0.170110	−0.0850548	−	0.996376i	$-0.527107\pi$
$479$	−854216.	−0.170110	−0.0850548	+	0.996376i	$0.527107\pi$
$480$	0	0
$481$	1.41251e6	0.278375
$482$	805928.	0.158008
$483$	0	0
$484$	5.09254e6	0.988147
$485$	−8.37235e6	−1.61619
$486$	0	0
$487$	−6.99300e6	−1.33611	−0.668054	−	0.744113i	$-0.732872\pi$
$487$	−6.99300e6	−1.33611	−0.668054	+	0.744113i	$0.732872\pi$
$488$	−167128.	−0.0317687
$489$	0	0
$490$	1.03309e6	0.194378
$491$	−6.08880e6	−1.13980	−0.569899	−	0.821715i	$-0.693018\pi$
$491$	−6.08880e6	−1.13980	−0.569899	+	0.821715i	$0.693018\pi$
$492$	0	0
$493$	350752.	0.0649954
$494$	−591270.	−0.109010
$495$	0	0
$496$	4.16674e6	0.760487
$497$	1.13384e7	2.05902
$498$	0	0
$499$	1.04489e7	1.87853	0.939264	−	0.343196i	$-0.111510\pi$
$499$	1.04489e7	1.87853	0.939264	+	0.343196i	$0.111510\pi$
$500$	−4.72744e6	−0.845671
$501$	0	0
$502$	−583835.	−0.103403
$503$	−3.92622e6	−0.691918	−0.345959	−	0.938250i	$-0.612446\pi$
$503$	−3.92622e6	−0.691918	−0.345959	+	0.938250i	$0.612446\pi$
$504$	0	0
$505$	−1.57920e6	−0.275556
$506$	1174.23	0.000203882 0
$507$	0	0
$508$	−2.52164e6	−0.433534
$509$	5.65513e6	0.967493	0.483746	−	0.875208i	$-0.339276\pi$
$509$	5.65513e6	0.967493	0.483746	+	0.875208i	$0.339276\pi$
$510$	0	0
$511$	6.82561e6	1.15635
$512$	−3.07626e6	−0.518618
$513$	0	0
$514$	1.12629e6	0.188037
$515$	−3.86112e6	−0.641498
$516$	0	0
$517$	36952.5	0.00608020
$518$	518859.	0.0849620
$519$	0	0
$520$	−851659.	−0.138120
$521$	1.00145e7	1.61634	0.808172	−	0.588947i	$-0.200457\pi$
$521$	1.00145e7	1.61634	0.808172	+	0.588947i	$0.200457\pi$
$522$	0	0
$523$	−9.06776e6	−1.44959	−0.724796	−	0.688964i	$-0.758066\pi$
$523$	−9.06776e6	−1.44959	−0.724796	+	0.688964i	$0.758066\pi$
$524$	−161850.	−0.0257504
$525$	0	0
$526$	762328.	0.120137
$527$	−2.90688e6	−0.455933
$528$	0	0
$529$	−4.71773e6	−0.732983
$530$	631447.	0.0976443
$531$	0	0
$532$	1.81261e7	2.77667
$533$	−2.57510e6	−0.392623
$534$	0	0
$535$	−1.00702e6	−0.152108
$536$	524396.	0.0788402
$537$	0	0
$538$	737555.	0.109860
$539$	−39430.7	−0.00584605
$540$	0	0
$541$	1.13999e7	1.67459	0.837296	−	0.546750i	$-0.184135\pi$
$541$	1.13999e7	1.67459	0.837296	+	0.546750i	$0.184135\pi$
$542$	558536.	0.0816681
$543$	0	0
$544$	1.28255e6	0.185814
$545$	1.29694e7	1.87038
$546$	0	0
$547$	9.29969e6	1.32892	0.664462	−	0.747322i	$-0.268661\pi$
$547$	9.29969e6	1.32892	0.664462	+	0.747322i	$0.268661\pi$
$548$	−7.50694e6	−1.06785
$549$	0	0
$550$	637.046	8.97975e−5 0
$551$	−1.39247e6	−0.195392
$552$	0	0
$553$	−1.24866e7	−1.73633
$554$	−121177.	−0.0167743
$555$	0	0
$556$	1.90332e6	0.261110
$557$	5.55080e6	0.758084	0.379042	−	0.925379i	$-0.376254\pi$
$557$	5.55080e6	0.758084	0.379042	+	0.925379i	$0.376254\pi$
$558$	0	0
$559$	1.84366e6	0.249546
$560$	1.28192e7	1.72739
$561$	0	0
$562$	1.42441e6	0.190236
$563$	−361373.	−0.0480490	−0.0240245	−	0.999711i	$-0.507648\pi$
$563$	−361373.	−0.0480490	−0.0240245	+	0.999711i	$0.507648\pi$
$564$	0	0
$565$	−8.87823e6	−1.17005
$566$	−26348.0	−0.00345706
$567$	0	0
$568$	−2.11913e6	−0.275605
$569$	−6.95770e6	−0.900917	−0.450459	−	0.892797i	$-0.648739\pi$
$569$	−6.95770e6	−0.900917	−0.450459	+	0.892797i	$0.648739\pi$
$570$	0	0
$571$	−1.46705e7	−1.88302	−0.941511	−	0.336982i	$-0.890594\pi$
$571$	−1.46705e7	−1.88302	−0.941511	+	0.336982i	$0.890594\pi$
$572$	16156.2	0.00206467
$573$	0	0
$574$	−945911.	−0.119831
$575$	932384.	0.117605
$576$	0	0
$577$	−1.16193e7	−1.45291	−0.726457	−	0.687212i	$-0.758834\pi$
$577$	−1.16193e7	−1.45291	−0.726457	+	0.687212i	$0.758834\pi$
$578$	581680.	0.0724210
$579$	0	0
$580$	−996879.	−0.123047
$581$	1.97161e7	2.42315
$582$	0	0
$583$	−24101.0	−0.00293673
$584$	−1.27570e6	−0.154780
$585$	0	0
$586$	−580574.	−0.0698415
$587$	−1.26219e7	−1.51192	−0.755959	−	0.654619i	$-0.772829\pi$
$587$	−1.26219e7	−1.51192	−0.755959	+	0.654619i	$0.772829\pi$
$588$	0	0
$589$	1.15402e7	1.37064
$590$	132713.	0.0156958
$591$	0	0
$592$	3.97369e6	0.466004
$593$	−8.70200e6	−1.01621	−0.508103	−	0.861296i	$-0.669653\pi$
$593$	−8.70200e6	−1.01621	−0.508103	+	0.861296i	$0.669653\pi$
$594$	0	0
$595$	−8.94319e6	−1.03562
$596$	−7.16683e6	−0.826440
$597$	0	0
$598$	−283336.	−0.0324003
$599$	−314477.	−0.0358115	−0.0179057	−	0.999840i	$-0.505700\pi$
$599$	−314477.	−0.0358115	−0.0179057	+	0.999840i	$0.505700\pi$
$600$	0	0
$601$	−1.33321e7	−1.50561	−0.752803	−	0.658246i	$-0.771299\pi$
$601$	−1.33321e7	−1.50561	−0.752803	+	0.658246i	$0.771299\pi$
$602$	677231.	0.0761633
$603$	0	0
$604$	9.31216e6	1.03862
$605$	9.97492e6	1.10795
$606$	0	0
$607$	4.65927e6	0.513270	0.256635	−	0.966508i	$-0.417386\pi$
$607$	4.65927e6	0.513270	0.256635	+	0.966508i	$0.417386\pi$
$608$	−5.09168e6	−0.558602
$609$	0	0
$610$	−162704.	−0.0177041
$611$	−8.91644e6	−0.966248
$612$	0	0
$613$	3.95032e6	0.424601	0.212301	−	0.977204i	$-0.431904\pi$
$613$	3.95032e6	0.424601	0.212301	+	0.977204i	$0.431904\pi$
$614$	647286.	0.0692907
$615$	0	0
$616$	11940.4	0.00126785
$617$	−9.77175e6	−1.03338	−0.516689	−	0.856173i	$-0.672836\pi$
$617$	−9.77175e6	−1.03338	−0.516689	+	0.856173i	$0.672836\pi$
$618$	0	0
$619$	1.29372e7	1.35710	0.678552	−	0.734553i	$-0.262608\pi$
$619$	1.29372e7	1.35710	0.678552	+	0.734553i	$0.262608\pi$
$620$	8.26170e6	0.863158
$621$	0	0
$622$	−379655.	−0.0393471
$623$	−2.85618e7	−2.94826
$624$	0	0
$625$	−1.14824e7	−1.17579
$626$	−755907.	−0.0770961
$627$	0	0
$628$	−1.66827e7	−1.68798
$629$	−2.77220e6	−0.279382
$630$	0	0
$631$	−1.78566e7	−1.78536	−0.892680	−	0.450690i	$-0.851178\pi$
$631$	−1.78566e7	−1.78536	−0.892680	+	0.450690i	$0.851178\pi$
$632$	2.33373e6	0.232411
$633$	0	0
$634$	−1.51446e6	−0.149635
$635$	−4.93921e6	−0.486097
$636$	0	0
$637$	9.51441e6	0.929038
$638$	−455.910	−4.43432e−5 0
$639$	0	0
$640$	−4.85025e6	−0.468073
$641$	1.43521e7	1.37966	0.689828	−	0.723973i	$-0.257686\pi$
$641$	1.43521e7	1.37966	0.689828	+	0.723973i	$0.257686\pi$
$642$	0	0
$643$	−1.09103e6	−0.104066	−0.0520329	−	0.998645i	$-0.516570\pi$
$643$	−1.09103e6	−0.104066	−0.0520329	+	0.998645i	$0.516570\pi$
$644$	8.68601e6	0.825288
$645$	0	0
$646$	1.16043e6	0.109405
$647$	6.11094e6	0.573914	0.286957	−	0.957943i	$-0.407356\pi$
$647$	6.11094e6	0.573914	0.286957	+	0.957943i	$0.407356\pi$
$648$	0	0
$649$	−5065.38	−0.000472063 0
$650$	−153716.	−0.0142704
$651$	0	0
$652$	−4.38208e6	−0.403702
$653$	−9.26459e6	−0.850243	−0.425122	−	0.905136i	$-0.639769\pi$
$653$	−9.26459e6	−0.850243	−0.425122	+	0.905136i	$0.639769\pi$
$654$	0	0
$655$	−317020.	−0.0288724
$656$	−7.24427e6	−0.657256
$657$	0	0
$658$	−3.27528e6	−0.294906
$659$	−1.26244e7	−1.13239	−0.566196	−	0.824271i	$-0.691585\pi$
$659$	−1.26244e7	−1.13239	−0.566196	+	0.824271i	$0.691585\pi$
$660$	0	0
$661$	6.62521e6	0.589788	0.294894	−	0.955530i	$-0.404716\pi$
$661$	6.62521e6	0.589788	0.294894	+	0.955530i	$0.404716\pi$
$662$	−35896.5	−0.00318352
$663$	0	0
$664$	−3.68490e6	−0.324344
$665$	3.55041e7	3.11332
$666$	0	0
$667$	−667272.	−0.0580748
$668$	6.37203e6	0.552506
$669$	0	0
$670$	510517.	0.0439363
$671$	6210.08	0.000532465 0
$672$	0	0
$673$	−4.38372e6	−0.373083	−0.186541	−	0.982447i	$-0.559728\pi$
$673$	−4.38372e6	−0.373083	−0.186541	+	0.982447i	$0.559728\pi$
$674$	−1.26747e6	−0.107470
$675$	0	0
$676$	7.84229e6	0.660049
$677$	−1.24543e7	−1.04435	−0.522177	−	0.852837i	$-0.674880\pi$
$677$	−1.24543e7	−1.04435	−0.522177	+	0.852837i	$0.674880\pi$
$678$	0	0
$679$	−2.83236e7	−2.35762
$680$	1.67147e6	0.138620
$681$	0	0
$682$	3778.38	0.000311061 0
$683$	5.99410e6	0.491668	0.245834	−	0.969312i	$-0.420938\pi$
$683$	5.99410e6	0.491668	0.245834	+	0.969312i	$0.420938\pi$
$684$	0	0
$685$	−1.47041e7	−1.19732
$686$	1.32722e6	0.107679
$687$	0	0
$688$	5.18658e6	0.417744
$689$	5.81544e6	0.466696
$690$	0	0
$691$	−1.41412e7	−1.12666	−0.563329	−	0.826232i	$-0.690480\pi$
$691$	−1.41412e7	−1.12666	−0.563329	+	0.826232i	$0.690480\pi$
$692$	1.46498e7	1.16296
$693$	0	0
$694$	−1.87776e6	−0.147993
$695$	3.72809e6	0.292768
$696$	0	0
$697$	5.05388e6	0.394043
$698$	1.56888e6	0.121885
$699$	0	0
$700$	4.71234e6	0.363489
$701$	−1.48140e7	−1.13861	−0.569307	−	0.822125i	$-0.692788\pi$
$701$	−1.48140e7	−1.13861	−0.569307	+	0.822125i	$0.692788\pi$
$702$	0	0
$703$	1.10055e7	0.839890
$704$	44328.3	0.00337092
$705$	0	0
$706$	−86162.1	−0.00650586
$707$	−5.34244e6	−0.401968
$708$	0	0
$709$	−7.25689e6	−0.542170	−0.271085	−	0.962555i	$-0.587382\pi$
$709$	−7.25689e6	−0.542170	−0.271085	+	0.962555i	$0.587382\pi$
$710$	−2.06304e6	−0.153590
$711$	0	0
$712$	5.33815e6	0.394631
$713$	5.53006e6	0.407386
$714$	0	0
$715$	31645.7	0.00231499
$716$	−7.43109e6	−0.541714
$717$	0	0
$718$	−361769.	−0.0261891
$719$	4.63110e6	0.334089	0.167044	−	0.985949i	$-0.446578\pi$
$719$	4.63110e6	0.334089	0.167044	+	0.985949i	$0.446578\pi$
$720$	0	0
$721$	−1.30622e7	−0.935787
$722$	−3.08270e6	−0.220084
$723$	0	0
$724$	−1.09175e7	−0.774066
$725$	−362009.	−0.0255784
$726$	0	0
$727$	−1.27291e7	−0.893224	−0.446612	−	0.894728i	$-0.647369\pi$
$727$	−1.27291e7	−0.893224	−0.446612	+	0.894728i	$0.647369\pi$
$728$	−2.88116e6	−0.201483
$729$	0	0
$730$	−1.24193e6	−0.0862562
$731$	−3.61836e6	−0.250449
$732$	0	0
$733$	−7.43767e6	−0.511302	−0.255651	−	0.966769i	$-0.582290\pi$
$733$	−7.43767e6	−0.511302	−0.255651	+	0.966769i	$0.582290\pi$
$734$	−221790.	−0.0151950
$735$	0	0
$736$	−2.43993e6	−0.166029
$737$	−19485.4	−0.00132142
$738$	0	0
$739$	1.96771e6	0.132541	0.0662705	−	0.997802i	$-0.478890\pi$
$739$	1.96771e6	0.132541	0.0662705	+	0.997802i	$0.478890\pi$
$740$	7.87892e6	0.528917
$741$	0	0
$742$	2.13618e6	0.142439
$743$	−1.30961e7	−0.870300	−0.435150	−	0.900358i	$-0.643305\pi$
$743$	−1.30961e7	−0.870300	−0.435150	+	0.900358i	$0.643305\pi$
$744$	0	0
$745$	−1.40379e7	−0.926640
$746$	−3.24344e6	−0.213383
$747$	0	0
$748$	−31708.2	−0.00207213
$749$	−3.40674e6	−0.221888
$750$	0	0
$751$	−2.49792e7	−1.61614	−0.808068	−	0.589089i	$-0.799487\pi$
$751$	−2.49792e7	−1.61614	−0.808068	+	0.589089i	$0.799487\pi$
$752$	−2.50837e7	−1.61751
$753$	0	0
$754$	110008.	0.00704689
$755$	1.82400e7	1.16455
$756$	0	0
$757$	−553214.	−0.0350876	−0.0175438	−	0.999846i	$-0.505585\pi$
$757$	−553214.	−0.0350876	−0.0175438	+	0.999846i	$0.505585\pi$
$758$	−2.11405e6	−0.133642
$759$	0	0
$760$	−6.63565e6	−0.416725
$761$	6.91848e6	0.433061	0.216530	−	0.976276i	$-0.430526\pi$
$761$	6.91848e6	0.433061	0.216530	+	0.976276i	$0.430526\pi$
$762$	0	0
$763$	4.38756e7	2.72842
$764$	3.07745e7	1.90747
$765$	0	0
$766$	−2.18776e6	−0.134719
$767$	1.22225e6	0.0750189
$768$	0	0
$769$	−1.29851e7	−0.791825	−0.395912	−	0.918288i	$-0.629572\pi$
$769$	−1.29851e7	−0.791825	−0.395912	+	0.918288i	$0.629572\pi$
$770$	11624.4	0.000706552 0
$771$	0	0
$772$	1.56035e7	0.942276
$773$	5.40829e6	0.325545	0.162773	−	0.986664i	$-0.447956\pi$
$773$	5.40829e6	0.325545	0.162773	+	0.986664i	$0.447956\pi$
$774$	0	0
$775$	3.00017e6	0.179429
$776$	5.29364e6	0.315573
$777$	0	0
$778$	2.78392e6	0.164895
$779$	−2.00637e7	−1.18459
$780$	0	0
$781$	78742.0	0.00461933
$782$	556075.	0.0325175
$783$	0	0
$784$	2.67659e7	1.55522
$785$	−3.26768e7	−1.89263
$786$	0	0
$787$	4.47738e6	0.257684	0.128842	−	0.991665i	$-0.458874\pi$
$787$	4.47738e6	0.257684	0.128842	+	0.991665i	$0.458874\pi$
$788$	1.07242e7	0.615244
$789$	0	0
$790$	2.27196e6	0.129519
$791$	−3.00350e7	−1.70682
$792$	0	0
$793$	−1.49846e6	−0.0846178
$794$	−250909.	−0.0141242
$795$	0	0
$796$	1.86470e7	1.04310
$797$	2.50186e7	1.39514	0.697570	−	0.716516i	$-0.254264\pi$
$797$	2.50186e7	1.39514	0.697570	+	0.716516i	$0.254264\pi$
$798$	0	0
$799$	1.74994e7	0.969742
$800$	−1.32371e6	−0.0731255
$801$	0	0
$802$	1.84876e6	0.101495
$803$	47401.9	0.00259422
$804$	0	0
$805$	1.70136e7	0.925348
$806$	−911703.	−0.0494328
$807$	0	0
$808$	998492.	0.0538042
$809$	3.51699e7	1.88929	0.944646	−	0.328090i	$-0.106405\pi$
$809$	3.51699e7	1.88929	0.944646	+	0.328090i	$0.106405\pi$
$810$	0	0
$811$	−2.65046e6	−0.141504	−0.0707521	−	0.997494i	$-0.522540\pi$
$811$	−2.65046e6	−0.141504	−0.0707521	+	0.997494i	$0.522540\pi$
$812$	−3.37244e6	−0.179496
$813$	0	0
$814$	3603.32	0.000190609 0
$815$	−8.58332e6	−0.452649
$816$	0	0
$817$	1.43648e7	0.752910
$818$	2.94010e6	0.153631
$819$	0	0
$820$	−1.43637e7	−0.745990
$821$	−9.39154e6	−0.486272	−0.243136	−	0.969992i	$-0.578176\pi$
$821$	−9.39154e6	−0.486272	−0.243136	+	0.969992i	$0.578176\pi$
$822$	0	0
$823$	−1.99606e6	−0.102725	−0.0513623	−	0.998680i	$-0.516356\pi$
$823$	−1.99606e6	−0.102725	−0.0513623	+	0.998680i	$0.516356\pi$
$824$	2.44130e6	0.125257
$825$	0	0
$826$	448968.	0.0228963
$827$	−6.93283e6	−0.352490	−0.176245	−	0.984346i	$-0.556395\pi$
$827$	−6.93283e6	−0.352490	−0.176245	+	0.984346i	$0.556395\pi$
$828$	0	0
$829$	−3.01802e7	−1.52523	−0.762616	−	0.646851i	$-0.776085\pi$
$829$	−3.01802e7	−1.52523	−0.762616	+	0.646851i	$0.776085\pi$
$830$	−3.58738e6	−0.180751
$831$	0	0
$832$	−1.06962e7	−0.535697
$833$	−1.86730e7	−0.932397
$834$	0	0
$835$	1.24811e7	0.619493
$836$	125880.	0.00622934
$837$	0	0
$838$	−1.82169e6	−0.0896117
$839$	1.09044e7	0.534809	0.267404	−	0.963584i	$-0.413834\pi$
$839$	1.09044e7	0.534809	0.267404	+	0.963584i	$0.413834\pi$
$840$	0	0
$841$	−2.02521e7	−0.987369
$842$	−720913.	−0.0350431
$843$	0	0
$844$	−2.46572e7	−1.19148
$845$	1.53609e7	0.740075
$846$	0	0
$847$	3.37452e7	1.61623
$848$	1.63600e7	0.781256
$849$	0	0
$850$	301682.	0.0143220
$851$	5.27384e6	0.249634
$852$	0	0
$853$	8.47613e6	0.398864	0.199432	−	0.979912i	$-0.436090\pi$
$853$	8.47613e6	0.398864	0.199432	+	0.979912i	$0.436090\pi$
$854$	−550429.	−0.0258260
$855$	0	0
$856$	636713.	0.0297002
$857$	−2.96420e7	−1.37865	−0.689327	−	0.724450i	$-0.742094\pi$
$857$	−2.96420e7	−1.37865	−0.689327	+	0.724450i	$0.742094\pi$
$858$	0	0
$859$	−58116.2	−0.00268729	−0.00134364	−	0.999999i	$-0.500428\pi$
$859$	−58116.2	−0.00268729	−0.00134364	+	0.999999i	$0.500428\pi$
$860$	1.02838e7	0.474142
$861$	0	0
$862$	−1.49870e6	−0.0686984
$863$	−6.44367e6	−0.294514	−0.147257	−	0.989098i	$-0.547044\pi$
$863$	−6.44367e6	−0.294514	−0.147257	+	0.989098i	$0.547044\pi$
$864$	0	0
$865$	2.86950e7	1.30396
$866$	141427.	0.00640821
$867$	0	0
$868$	2.79493e7	1.25913
$869$	−86715.9	−0.00389538
$870$	0	0
$871$	4.70171e6	0.209996
$872$	−8.20027e6	−0.365205
$873$	0	0
$874$	−2.20760e6	−0.0977555
$875$	−3.13259e7	−1.38319
$876$	0	0
$877$	−1.86307e7	−0.817957	−0.408979	−	0.912544i	$-0.634115\pi$
$877$	−1.86307e7	−0.817957	−0.408979	+	0.912544i	$0.634115\pi$
$878$	1.62098e6	0.0709647
$879$	0	0
$880$	89025.7	0.00387533
$881$	−6.95060e6	−0.301705	−0.150853	−	0.988556i	$-0.548202\pi$
$881$	−6.95060e6	−0.301705	−0.150853	+	0.988556i	$0.548202\pi$
$882$	0	0
$883$	−1.75323e7	−0.756722	−0.378361	−	0.925658i	$-0.623512\pi$
$883$	−1.75323e7	−0.756722	−0.378361	+	0.925658i	$0.623512\pi$
$884$	7.65101e6	0.329297
$885$	0	0
$886$	−3.26313e6	−0.139653
$887$	−4.14544e6	−0.176914	−0.0884568	−	0.996080i	$-0.528194\pi$
$887$	−4.14544e6	−0.176914	−0.0884568	+	0.996080i	$0.528194\pi$
$888$	0	0
$889$	−1.67093e7	−0.709096
$890$	5.19687e6	0.219921
$891$	0	0
$892$	−4.70058e6	−0.197806
$893$	−6.94719e7	−2.91528
$894$	0	0
$895$	−1.45555e7	−0.607393
$896$	−1.64084e7	−0.682803
$897$	0	0
$898$	1.42912e6	0.0591395
$899$	−2.14711e6	−0.0886042
$900$	0	0
$901$	−1.14134e7	−0.468384
$902$	−6569.07	−0.000268836 0
$903$	0	0
$904$	5.61349e6	0.228461
$905$	−2.13845e7	−0.867916
$906$	0	0
$907$	6.52626e6	0.263418	0.131709	−	0.991288i	$-0.457953\pi$
$907$	6.52626e6	0.263418	0.131709	+	0.991288i	$0.457953\pi$
$908$	−2.23089e7	−0.897974
$909$	0	0
$910$	−2.80491e6	−0.112283
$911$	−4.12964e6	−0.164860	−0.0824302	−	0.996597i	$-0.526268\pi$
$911$	−4.12964e6	−0.164860	−0.0824302	+	0.996597i	$0.526268\pi$
$912$	0	0
$913$	136923.	0.00543623
$914$	1.87983e6	0.0744308
$915$	0	0
$916$	−7.00248e6	−0.275748
$917$	−1.07248e6	−0.0421178
$918$	0	0
$919$	−3.37200e7	−1.31704	−0.658520	−	0.752563i	$-0.728817\pi$
$919$	−3.37200e7	−1.31704	−0.658520	+	0.752563i	$0.728817\pi$
$920$	−3.17980e6	−0.123860
$921$	0	0
$922$	2.66666e6	0.103310
$923$	−1.90000e7	−0.734090
$924$	0	0
$925$	2.86117e6	0.109948
$926$	3.93188e6	0.150686
$927$	0	0
$928$	947331.	0.0361104
$929$	2.85561e7	1.08557	0.542787	−	0.839870i	$-0.317369\pi$
$929$	2.85561e7	1.08557	0.542787	+	0.839870i	$0.317369\pi$
$930$	0	0
$931$	7.41309e7	2.80301
$932$	−9.60669e6	−0.362272
$933$	0	0
$934$	366301.	0.0137395
$935$	−62107.9	−0.00232337
$936$	0	0
$937$	1.16758e7	0.434446	0.217223	−	0.976122i	$-0.430300\pi$
$937$	1.16758e7	0.434446	0.217223	+	0.976122i	$0.430300\pi$
$938$	1.72708e6	0.0640922
$939$	0	0
$940$	−4.97354e7	−1.83589
$941$	1.56740e7	0.577040	0.288520	−	0.957474i	$-0.406837\pi$
$941$	1.56740e7	0.577040	0.288520	+	0.957474i	$0.406837\pi$
$942$	0	0
$943$	−9.61453e6	−0.352086
$944$	3.43843e6	0.125583
$945$	0	0
$946$	4703.18	0.000170869 0
$947$	2.91932e7	1.05781	0.528904	−	0.848681i	$-0.322603\pi$
$947$	2.91932e7	1.05781	0.528904	+	0.848681i	$0.322603\pi$
$948$	0	0
$949$	−1.14378e7	−0.412266
$950$	−1.19767e6	−0.0430553
$951$	0	0
$952$	5.65457e6	0.202212
$953$	4.07979e7	1.45514	0.727571	−	0.686032i	$-0.240649\pi$
$953$	4.07979e7	1.45514	0.727571	+	0.686032i	$0.240649\pi$
$954$	0	0
$955$	6.02790e7	2.13874
$956$	3.57283e7	1.26435
$957$	0	0
$958$	525806.	0.0185102
$959$	−4.97439e7	−1.74660
$960$	0	0
$961$	−1.08349e7	−0.378455
$962$	−869462.	−0.0302910
$963$	0	0
$964$	4.14015e7	1.43490
$965$	3.05630e7	1.05652
$966$	0	0
$967$	−2.00499e7	−0.689519	−0.344760	−	0.938691i	$-0.612040\pi$
$967$	−2.00499e7	−0.689519	−0.344760	+	0.938691i	$0.612040\pi$
$968$	−6.30691e6	−0.216336
$969$	0	0
$970$	5.15353e6	0.175863
$971$	5.07323e7	1.72678	0.863388	−	0.504540i	$-0.168338\pi$
$971$	5.07323e7	1.72678	0.863388	+	0.504540i	$0.168338\pi$
$972$	0	0
$973$	1.26121e7	0.427077
$974$	4.30449e6	0.145386
$975$	0	0
$976$	−4.21546e6	−0.141651
$977$	−4.01142e7	−1.34450	−0.672252	−	0.740322i	$-0.734673\pi$
$977$	−4.01142e7	−1.34450	−0.672252	+	0.740322i	$0.734673\pi$
$978$	0	0
$979$	−198354.	−0.00661429
$980$	5.30708e7	1.76519
$981$	0	0
$982$	3.74791e6	0.124025
$983$	5.36333e7	1.77032	0.885159	−	0.465289i	$-0.154050\pi$
$983$	5.36333e7	1.77032	0.885159	+	0.465289i	$0.154050\pi$
$984$	0	0
$985$	2.10057e7	0.689839
$986$	−215903.	−0.00707238
$987$	0	0
$988$	−3.03742e7	−0.989949
$989$	6.88359e6	0.223782
$990$	0	0
$991$	4.34550e7	1.40558	0.702790	−	0.711397i	$-0.251937\pi$
$991$	4.34550e7	1.40558	0.702790	+	0.711397i	$0.251937\pi$
$992$	−7.85107e6	−0.253308
$993$	0	0
$994$	−6.97927e6	−0.224050
$995$	3.65245e7	1.16957
$996$	0	0
$997$	3.48232e7	1.10951	0.554754	−	0.832015i	$-0.312812\pi$
$997$	3.48232e7	1.10951	0.554754	+	0.832015i	$0.312812\pi$
$998$	−6.43171e6	−0.204409
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.c.1.7		12
3.2	odd	2		177.6.a.c.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.c.1.6	✓	12	3.2	odd	2
531.6.a.c.1.7		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)

\(f(q)\)	\(=\)	\(q-0.615542 q^{2} -31.6211 q^{4} -61.9372 q^{5} -209.534 q^{7} +39.1615 q^{8} +O(q^{10})\)	\(q-0.615542 q^{2} -31.6211 q^{4} -61.9372 q^{5} -209.534 q^{7} +39.1615 q^{8} +38.1250 q^{10} -1.45515 q^{11} +351.120 q^{13} +128.977 q^{14} +987.770 q^{16} -689.107 q^{17} +2735.73 q^{19} +1958.52 q^{20} +0.895706 q^{22} +1310.96 q^{23} +711.222 q^{25} -216.129 q^{26} +6625.69 q^{28} -508.995 q^{29} +4218.33 q^{31} -1861.18 q^{32} +424.174 q^{34} +12977.9 q^{35} +4022.89 q^{37} -1683.95 q^{38} -2425.55 q^{40} -7333.96 q^{41} +5250.80 q^{43} +46.0135 q^{44} -806.951 q^{46} -25394.3 q^{47} +27097.3 q^{49} -437.787 q^{50} -11102.8 q^{52} +16562.5 q^{53} +90.1280 q^{55} -8205.64 q^{56} +313.308 q^{58} +3481.00 q^{59} -4267.66 q^{61} -2596.56 q^{62} -30463.0 q^{64} -21747.4 q^{65} +13390.6 q^{67} +21790.3 q^{68} -7988.46 q^{70} -54112.6 q^{71} -32575.3 q^{73} -2476.26 q^{74} -86506.7 q^{76} +304.903 q^{77} +59592.4 q^{79} -61179.7 q^{80} +4514.36 q^{82} -94095.2 q^{83} +42681.4 q^{85} -3232.09 q^{86} -56.9858 q^{88} +136311. q^{89} -73571.4 q^{91} -41454.0 q^{92} +15631.3 q^{94} -169443. q^{95} +135175. q^{97} -16679.6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) \) 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

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Groups

Database

Properties

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