LMFDB - Embedded newform 531.6.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$85.1638083207$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + \cdots - 50564480 $ x^12 - 4x^11 - 283x^10 + 1045x^9 + 27968x^8 - 94393x^7 - 1130486x^6 + 3566264x^5 + 15496192x^4 - 53008480x^3 - 16576192x^2 + 120303168*x - 50564480
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-8.78000$ 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)

$f(q)$	$=$	$q-8.78000 q^{2} +45.0883 q^{4} +17.5207 q^{5} +12.9559 q^{7} -114.916 q^{8} +O(q^{10})$	\(q-8.78000 q^{2} +45.0883 q^{4} +17.5207 q^{5} +12.9559 q^{7} -114.916 q^{8} -153.831 q^{10} +434.226 q^{11} -637.700 q^{13} -113.753 q^{14} -433.869 q^{16} +850.752 q^{17} +1822.12 q^{19} +789.977 q^{20} -3812.51 q^{22} -715.863 q^{23} -2818.03 q^{25} +5599.00 q^{26} +584.162 q^{28} -1133.86 q^{29} -9559.39 q^{31} +7486.66 q^{32} -7469.60 q^{34} +226.997 q^{35} -107.111 q^{37} -15998.2 q^{38} -2013.40 q^{40} +6614.88 q^{41} -2995.66 q^{43} +19578.5 q^{44} +6285.27 q^{46} -10180.6 q^{47} -16639.1 q^{49} +24742.3 q^{50} -28752.8 q^{52} -11009.5 q^{53} +7607.93 q^{55} -1488.84 q^{56} +9955.28 q^{58} +3481.00 q^{59} +2294.87 q^{61} +83931.4 q^{62} -51849.1 q^{64} -11172.9 q^{65} +23569.8 q^{67} +38359.0 q^{68} -1993.03 q^{70} -56039.7 q^{71} +56572.1 q^{73} +940.434 q^{74} +82156.5 q^{76} +5625.81 q^{77} +65746.0 q^{79} -7601.67 q^{80} -58078.6 q^{82} +13649.8 q^{83} +14905.7 q^{85} +26301.9 q^{86} -49899.4 q^{88} -77607.3 q^{89} -8262.00 q^{91} -32277.1 q^{92} +89386.0 q^{94} +31924.8 q^{95} +41043.1 q^{97} +146092. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * 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 $

$2$	−8.78000	−1.55210	−0.776049	−	0.630672i	$-0.782779\pi$
$2$	−8.78000	−1.55210	−0.776049	+	0.630672i	$0.782779\pi$
$3$	0	0
$3$	0	0
$4$	45.0883	1.40901
$5$	17.5207	0.313419	0.156710	−	0.987645i	$-0.449911\pi$
$5$	17.5207	0.313419	0.156710	+	0.987645i	$0.449911\pi$
$6$	0	0
$7$	12.9559	0.0999364	0.0499682	−	0.998751i	$-0.484088\pi$
$7$	12.9559	0.0999364	0.0499682	+	0.998751i	$0.484088\pi$
$8$	−114.916	−0.634825
$9$	0	0
$10$	−153.831	−0.486457
$11$	434.226	1.08202	0.541009	−	0.841017i	$-0.318042\pi$
$11$	434.226	1.08202	0.541009	+	0.841017i	$0.318042\pi$
$12$	0	0
$13$	−637.700	−1.04655	−0.523273	−	0.852165i	$-0.675289\pi$
$13$	−637.700	−1.04655	−0.523273	+	0.852165i	$0.675289\pi$
$14$	−113.753	−0.155111
$15$	0	0
$16$	−433.869	−0.423700
$17$	850.752	0.713971	0.356986	−	0.934110i	$-0.383804\pi$
$17$	850.752	0.713971	0.356986	+	0.934110i	$0.383804\pi$
$18$	0	0
$19$	1822.12	1.15796	0.578980	−	0.815341i	$-0.303451\pi$
$19$	1822.12	1.15796	0.578980	+	0.815341i	$0.303451\pi$
$20$	789.977	0.441611
$21$	0	0
$22$	−3812.51	−1.67940
$23$	−715.863	−0.282170	−0.141085	−	0.989998i	$-0.545059\pi$
$23$	−715.863	−0.282170	−0.141085	+	0.989998i	$0.545059\pi$
$24$	0	0
$25$	−2818.03	−0.901768
$26$	5599.00	1.62434
$27$	0	0
$28$	584.162	0.140811
$29$	−1133.86	−0.250360	−0.125180	−	0.992134i	$-0.539951\pi$
$29$	−1133.86	−0.250360	−0.125180	+	0.992134i	$0.539951\pi$
$30$	0	0
$31$	−9559.39	−1.78659	−0.893297	−	0.449467i	$-0.851614\pi$
$31$	−9559.39	−1.78659	−0.893297	+	0.449467i	$0.851614\pi$
$32$	7486.66	1.29245
$33$	0	0
$34$	−7469.60	−1.10815
$35$	226.997	0.0313220
$36$	0	0
$37$	−107.111	−0.0128626	−0.00643132	−	0.999979i	$-0.502047\pi$
$37$	−107.111	−0.0128626	−0.00643132	+	0.999979i	$0.502047\pi$
$38$	−15998.2	−1.79727
$39$	0	0
$40$	−2013.40	−0.198966
$41$	6614.88	0.614557	0.307279	−	0.951620i	$-0.400582\pi$
$41$	6614.88	0.614557	0.307279	+	0.951620i	$0.400582\pi$
$42$	0	0
$43$	−2995.66	−0.247070	−0.123535	−	0.992340i	$-0.539423\pi$
$43$	−2995.66	−0.247070	−0.123535	+	0.992340i	$0.539423\pi$
$44$	19578.5	1.52457
$45$	0	0
$46$	6285.27	0.437955
$47$	−10180.6	−0.672249	−0.336125	−	0.941817i	$-0.609116\pi$
$47$	−10180.6	−0.672249	−0.336125	+	0.941817i	$0.609116\pi$
$48$	0	0
$49$	−16639.1	−0.990013
$50$	24742.3	1.39963
$51$	0	0
$52$	−28752.8	−1.47459
$53$	−11009.5	−0.538367	−0.269183	−	0.963089i	$-0.586754\pi$
$53$	−11009.5	−0.538367	−0.269183	+	0.963089i	$0.586754\pi$
$54$	0	0
$55$	7607.93	0.339125
$56$	−1488.84	−0.0634421
$57$	0	0
$58$	9955.28	0.388583
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	2294.87	0.0789648	0.0394824	−	0.999220i	$-0.487429\pi$
$61$	2294.87	0.0789648	0.0394824	+	0.999220i	$0.487429\pi$
$62$	83931.4	2.77297
$63$	0	0
$64$	−51849.1	−1.58231
$65$	−11172.9	−0.328007
$66$	0	0
$67$	23569.8	0.641459	0.320729	−	0.947171i	$-0.396072\pi$
$67$	23569.8	0.641459	0.320729	+	0.947171i	$0.396072\pi$
$68$	38359.0	1.00599
$69$	0	0
$70$	−1993.03	−0.0486148
$71$	−56039.7	−1.31932	−0.659660	−	0.751564i	$-0.729300\pi$
$71$	−56039.7	−1.31932	−0.659660	+	0.751564i	$0.729300\pi$
$72$	0	0
$73$	56572.1	1.24250	0.621248	−	0.783614i	$-0.286626\pi$
$73$	56572.1	1.24250	0.621248	+	0.783614i	$0.286626\pi$
$74$	940.434	0.0199641
$75$	0	0
$76$	82156.5	1.63158
$77$	5625.81	0.108133
$78$	0	0
$79$	65746.0	1.18523	0.592614	−	0.805487i	$-0.298096\pi$
$79$	65746.0	1.18523	0.592614	+	0.805487i	$0.298096\pi$
$80$	−7601.67	−0.132796
$81$	0	0
$82$	−58078.6	−0.953853
$83$	13649.8	0.217486	0.108743	−	0.994070i	$-0.465318\pi$
$83$	13649.8	0.217486	0.108743	+	0.994070i	$0.465318\pi$
$84$	0	0
$85$	14905.7	0.223772
$86$	26301.9	0.383478
$87$	0	0
$88$	−49899.4	−0.686892
$89$	−77607.3	−1.03855	−0.519275	−	0.854607i	$-0.673798\pi$
$89$	−77607.3	−1.03855	−0.519275	+	0.854607i	$0.673798\pi$
$90$	0	0
$91$	−8262.00	−0.104588
$92$	−32277.1	−0.397580
$93$	0	0
$94$	89386.0	1.04340
$95$	31924.8	0.362927
$96$	0	0
$97$	41043.1	0.442906	0.221453	−	0.975171i	$-0.428920\pi$
$97$	41043.1	0.442906	0.221453	+	0.975171i	$0.428920\pi$
$98$	146092.	1.53660
$99$	0	0
$100$	−127060.	−1.27060
$101$	175206.	1.70901	0.854505	−	0.519444i	$-0.173861\pi$
$101$	175206.	1.70901	0.854505	+	0.519444i	$0.173861\pi$
$102$	0	0
$103$	6862.58	0.0637374	0.0318687	−	0.999492i	$-0.489854\pi$
$103$	6862.58	0.0637374	0.0318687	+	0.999492i	$0.489854\pi$
$104$	73281.7	0.664373
$105$	0	0
$106$	96663.4	0.835598
$107$	231771.	1.95704	0.978520	−	0.206152i	$-0.0660941\pi$
$107$	231771.	1.95704	0.978520	+	0.206152i	$0.0660941\pi$
$108$	0	0
$109$	138570.	1.11713	0.558565	−	0.829461i	$-0.311352\pi$
$109$	138570.	1.11713	0.558565	+	0.829461i	$0.311352\pi$
$110$	−66797.6	−0.526356
$111$	0	0
$112$	−5621.17	−0.0423430
$113$	−26310.1	−0.193832	−0.0969162	−	0.995293i	$-0.530898\pi$
$113$	−26310.1	−0.193832	−0.0969162	+	0.995293i	$0.530898\pi$
$114$	0	0
$115$	−12542.4	−0.0884373
$116$	−51123.8	−0.352759
$117$	0	0
$118$	−30563.2	−0.202066
$119$	11022.3	0.0713517
$120$	0	0
$121$	27501.6	0.170763
$122$	−20149.0	−0.122561
$123$	0	0
$124$	−431017.	−2.51733
$125$	−104126.	−0.596051
$126$	0	0
$127$	−328460.	−1.80706	−0.903531	−	0.428522i	$-0.859035\pi$
$127$	−328460.	−1.80706	−0.903531	+	0.428522i	$0.859035\pi$
$128$	215661.	1.16345
$129$	0	0
$130$	98098.3	0.509100
$131$	−57771.7	−0.294128	−0.147064	−	0.989127i	$-0.546982\pi$
$131$	−57771.7	−0.294128	−0.147064	+	0.989127i	$0.546982\pi$
$132$	0	0
$133$	23607.3	0.115722
$134$	−206943.	−0.995608
$135$	0	0
$136$	−97764.7	−0.453247
$137$	−116139.	−0.528662	−0.264331	−	0.964432i	$-0.585151\pi$
$137$	−116139.	−0.528662	−0.264331	+	0.964432i	$0.585151\pi$
$138$	0	0
$139$	−433033.	−1.90101	−0.950504	−	0.310713i	$-0.899432\pi$
$139$	−433033.	−1.90101	−0.950504	+	0.310713i	$0.899432\pi$
$140$	10234.9	0.0441330
$141$	0	0
$142$	492029.	2.04771
$143$	−276906.	−1.13238
$144$	0	0
$145$	−19866.0	−0.0784675
$146$	−496703.	−1.92848
$147$	0	0
$148$	−4829.46	−0.0181236
$149$	−283234.	−1.04515	−0.522576	−	0.852593i	$-0.675029\pi$
$149$	−283234.	−1.04515	−0.522576	+	0.852593i	$0.675029\pi$
$150$	0	0
$151$	−118520.	−0.423009	−0.211505	−	0.977377i	$-0.567836\pi$
$151$	−118520.	−0.423009	−0.211505	+	0.977377i	$0.567836\pi$
$152$	−209390.	−0.735102
$153$	0	0
$154$	−49394.6	−0.167833
$155$	−167487.	−0.559953
$156$	0	0
$157$	−189854.	−0.614712	−0.307356	−	0.951595i	$-0.599444\pi$
$157$	−189854.	−0.614712	−0.307356	+	0.951595i	$0.599444\pi$
$158$	−577250.	−1.83959
$159$	0	0
$160$	131171.	0.405078
$161$	−9274.67	−0.0281990
$162$	0	0
$163$	127496.	0.375861	0.187931	−	0.982182i	$-0.439822\pi$
$163$	127496.	0.375861	0.187931	+	0.982182i	$0.439822\pi$
$164$	298254.	0.865917
$165$	0	0
$166$	−119845.	−0.337559
$167$	−201919.	−0.560255	−0.280128	−	0.959963i	$-0.590377\pi$
$167$	−201919.	−0.560255	−0.280128	+	0.959963i	$0.590377\pi$
$168$	0	0
$169$	35368.4	0.0952574
$170$	−130872.	−0.347317
$171$	0	0
$172$	−135069.	−0.348125
$173$	299080.	0.759752	0.379876	−	0.925037i	$-0.375967\pi$
$173$	299080.	0.759752	0.379876	+	0.925037i	$0.375967\pi$
$174$	0	0
$175$	−36510.2	−0.0901195
$176$	−188397.	−0.458451
$177$	0	0
$178$	681392.	1.61193
$179$	−659619.	−1.53872	−0.769362	−	0.638813i	$-0.779426\pi$
$179$	−659619.	−1.53872	−0.769362	+	0.638813i	$0.779426\pi$
$180$	0	0
$181$	−242677.	−0.550595	−0.275297	−	0.961359i	$-0.588776\pi$
$181$	−242677.	−0.550595	−0.275297	+	0.961359i	$0.588776\pi$
$182$	72540.3	0.162331
$183$	0	0
$184$	82263.7	0.179128
$185$	−1876.66	−0.00403139
$186$	0	0
$187$	369419.	0.772530
$188$	−459028.	−0.947207
$189$	0	0
$190$	−280300.	−0.563299
$191$	−202851.	−0.402341	−0.201171	−	0.979556i	$-0.564475\pi$
$191$	−202851.	−0.402341	−0.201171	+	0.979556i	$0.564475\pi$
$192$	0	0
$193$	−240767.	−0.465268	−0.232634	−	0.972564i	$-0.574734\pi$
$193$	−240767.	−0.465268	−0.232634	+	0.972564i	$0.574734\pi$
$194$	−360359.	−0.687433
$195$	0	0
$196$	−750231.	−1.39494
$197$	372003.	0.682937	0.341469	−	0.939893i	$-0.389076\pi$
$197$	372003.	0.682937	0.341469	+	0.939893i	$0.389076\pi$
$198$	0	0
$199$	−927046.	−1.65947	−0.829733	−	0.558160i	$-0.811508\pi$
$199$	−927046.	−1.65947	−0.829733	+	0.558160i	$0.811508\pi$
$200$	323835.	0.572465
$201$	0	0
$202$	−1.53830e6	−2.65255
$203$	−14690.2	−0.0250200
$204$	0	0
$205$	115897.	0.192614
$206$	−60253.4	−0.0989267
$207$	0	0
$208$	276678.	0.443421
$209$	791214.	1.25293
$210$	0	0
$211$	−372955.	−0.576701	−0.288350	−	0.957525i	$-0.593107\pi$
$211$	−372955.	−0.576701	−0.288350	+	0.957525i	$0.593107\pi$
$212$	−496400.	−0.758564
$213$	0	0
$214$	−2.03495e6	−3.03752
$215$	−52485.9	−0.0774366
$216$	0	0
$217$	−123851.	−0.178546
$218$	−1.21665e6	−1.73390
$219$	0	0
$220$	343029.	0.477831
$221$	−542525.	−0.747203
$222$	0	0
$223$	−632099.	−0.851183	−0.425592	−	0.904915i	$-0.639934\pi$
$223$	−632099.	−0.851183	−0.425592	+	0.904915i	$0.639934\pi$
$224$	96996.7	0.129163
$225$	0	0
$226$	231003.	0.300847
$227$	890035.	1.14642	0.573208	−	0.819410i	$-0.305699\pi$
$227$	890035.	1.14642	0.573208	+	0.819410i	$0.305699\pi$
$228$	0	0
$229$	259318.	0.326771	0.163386	−	0.986562i	$-0.447758\pi$
$229$	259318.	0.326771	0.163386	+	0.986562i	$0.447758\pi$
$230$	110122.	0.137263
$231$	0	0
$232$	130298.	0.158934
$233$	476260.	0.574717	0.287358	−	0.957823i	$-0.407223\pi$
$233$	476260.	0.574717	0.287358	+	0.957823i	$0.407223\pi$
$234$	0	0
$235$	−178372.	−0.210696
$236$	156952.	0.183438
$237$	0	0
$238$	−96775.7	−0.110745
$239$	400093.	0.453071	0.226535	−	0.974003i	$-0.427260\pi$
$239$	400093.	0.453071	0.226535	+	0.974003i	$0.427260\pi$
$240$	0	0
$241$	66354.4	0.0735914	0.0367957	−	0.999323i	$-0.488285\pi$
$241$	66354.4	0.0735914	0.0367957	+	0.999323i	$0.488285\pi$
$242$	−241464.	−0.265041
$243$	0	0
$244$	103472.	0.111262
$245$	−291529.	−0.310289
$246$	0	0
$247$	−1.16197e6	−1.21186
$248$	1.09852e6	1.13417
$249$	0	0
$250$	914224.	0.925129
$251$	378287.	0.378998	0.189499	−	0.981881i	$-0.439314\pi$
$251$	378287.	0.378998	0.189499	+	0.981881i	$0.439314\pi$
$252$	0	0
$253$	−310846.	−0.305313
$254$	2.88388e6	2.80474
$255$	0	0
$256$	−234337.	−0.223481
$257$	6758.28	0.00638268	0.00319134	−	0.999995i	$-0.498984\pi$
$257$	6758.28	0.00638268	0.00319134	+	0.999995i	$0.498984\pi$
$258$	0	0
$259$	−1387.72	−0.00128545
$260$	−503769.	−0.462166
$261$	0	0
$262$	507235.	0.456516
$263$	−1.53754e6	−1.37068	−0.685342	−	0.728222i	$-0.740347\pi$
$263$	−1.53754e6	−1.37068	−0.685342	+	0.728222i	$0.740347\pi$
$264$	0	0
$265$	−192894.	−0.168734
$266$	−207272.	−0.179613
$267$	0	0
$268$	1.06272e6	0.903822
$269$	−760328.	−0.640649	−0.320325	−	0.947308i	$-0.603792\pi$
$269$	−760328.	−0.640649	−0.320325	+	0.947308i	$0.603792\pi$
$270$	0	0
$271$	−621403.	−0.513985	−0.256992	−	0.966413i	$-0.582732\pi$
$271$	−621403.	−0.513985	−0.256992	+	0.966413i	$0.582732\pi$
$272$	−369115.	−0.302510
$273$	0	0
$274$	1.01970e6	0.820536
$275$	−1.22366e6	−0.975730
$276$	0	0
$277$	−317260.	−0.248437	−0.124219	−	0.992255i	$-0.539642\pi$
$277$	−317260.	−0.248437	−0.124219	+	0.992255i	$0.539642\pi$
$278$	3.80203e6	2.95055
$279$	0	0
$280$	−26085.4	−0.0198840
$281$	368043.	0.278057	0.139028	−	0.990288i	$-0.455602\pi$
$281$	368043.	0.278057	0.139028	+	0.990288i	$0.455602\pi$
$282$	0	0
$283$	−1.66530e6	−1.23602	−0.618010	−	0.786170i	$-0.712061\pi$
$283$	−1.66530e6	−1.23602	−0.618010	+	0.786170i	$0.712061\pi$
$284$	−2.52674e6	−1.85894
$285$	0	0
$286$	2.43124e6	1.75757
$287$	85701.9	0.0614166
$288$	0	0
$289$	−696078.	−0.490245
$290$	174423.	0.121789
$291$	0	0
$292$	2.55074e6	1.75069
$293$	−1.24407e6	−0.846598	−0.423299	−	0.905990i	$-0.639128\pi$
$293$	−1.24407e6	−0.846598	−0.423299	+	0.905990i	$0.639128\pi$
$294$	0	0
$295$	60989.4	0.0408037
$296$	12308.7	0.00816552
$297$	0	0
$298$	2.48679e6	1.62218
$299$	456506.	0.295303
$300$	0	0
$301$	−38811.5	−0.0246913
$302$	1.04061e6	0.656552
$303$	0	0
$304$	−790562.	−0.490628
$305$	40207.6	0.0247491
$306$	0	0
$307$	1.88311e6	1.14033	0.570165	−	0.821530i	$-0.306879\pi$
$307$	1.88311e6	1.14033	0.570165	+	0.821530i	$0.306879\pi$
$308$	253658.	0.152361
$309$	0	0
$310$	1.47053e6	0.869102
$311$	3.10201e6	1.81862	0.909310	−	0.416119i	$-0.136610\pi$
$311$	3.10201e6	1.81862	0.909310	+	0.416119i	$0.136610\pi$
$312$	0	0
$313$	−2.33451e6	−1.34690	−0.673449	−	0.739234i	$-0.735188\pi$
$313$	−2.33451e6	−1.34690	−0.673449	+	0.739234i	$0.735188\pi$
$314$	1.66692e6	0.954094
$315$	0	0
$316$	2.96438e6	1.67000
$317$	103565.	0.0578846	0.0289423	−	0.999581i	$-0.490786\pi$
$317$	103565.	0.0578846	0.0289423	+	0.999581i	$0.490786\pi$
$318$	0	0
$319$	−492352.	−0.270894
$320$	−908430.	−0.495926
$321$	0	0
$322$	81431.6	0.0437676
$323$	1.55018e6	0.826751
$324$	0	0
$325$	1.79706e6	0.943742
$326$	−1.11942e6	−0.583374
$327$	0	0
$328$	−760152.	−0.390136
$329$	−131900.	−0.0671822
$330$	0	0
$331$	1.75769e6	0.881804	0.440902	−	0.897555i	$-0.354659\pi$
$331$	1.75769e6	0.881804	0.440902	+	0.897555i	$0.354659\pi$
$332$	615446.	0.306439
$333$	0	0
$334$	1.77285e6	0.869572
$335$	412958.	0.201045
$336$	0	0
$337$	−620309.	−0.297532	−0.148766	−	0.988872i	$-0.547530\pi$
$337$	−620309.	−0.297532	−0.148766	+	0.988872i	$0.547530\pi$
$338$	−310534.	−0.147849
$339$	0	0
$340$	672075.	0.315297
$341$	−4.15094e6	−1.93313
$342$	0	0
$343$	−433326.	−0.198875
$344$	344247.	0.156846
$345$	0	0
$346$	−2.62592e6	−1.17921
$347$	−495722.	−0.221011	−0.110506	−	0.993876i	$-0.535247\pi$
$347$	−495722.	−0.221011	−0.110506	+	0.993876i	$0.535247\pi$
$348$	0	0
$349$	−1.71145e6	−0.752142	−0.376071	−	0.926591i	$-0.622725\pi$
$349$	−1.71145e6	−0.752142	−0.376071	+	0.926591i	$0.622725\pi$
$350$	320559.	0.139874
$351$	0	0
$352$	3.25091e6	1.39845
$353$	2.42249e6	1.03472	0.517362	−	0.855767i	$-0.326914\pi$
$353$	2.42249e6	1.03472	0.517362	+	0.855767i	$0.326914\pi$
$354$	0	0
$355$	−981853.	−0.413500
$356$	−3.49918e6	−1.46333
$357$	0	0
$358$	5.79146e6	2.38825
$359$	−2.09761e6	−0.858989	−0.429495	−	0.903069i	$-0.641308\pi$
$359$	−2.09761e6	−0.858989	−0.429495	+	0.903069i	$0.641308\pi$
$360$	0	0
$361$	844035.	0.340873
$362$	2.13070e6	0.854577
$363$	0	0
$364$	−372520.	−0.147366
$365$	991180.	0.389422
$366$	0	0
$367$	−4.92708e6	−1.90952	−0.954760	−	0.297376i	$-0.903888\pi$
$367$	−4.92708e6	−1.90952	−0.954760	+	0.297376i	$0.903888\pi$
$368$	310590.	0.119555
$369$	0	0
$370$	16477.0	0.00625712
$371$	−142638.	−0.0538024
$372$	0	0
$373$	1.78212e6	0.663231	0.331616	−	0.943415i	$-0.392406\pi$
$373$	1.78212e6	0.663231	0.331616	+	0.943415i	$0.392406\pi$
$374$	−3.24350e6	−1.19904
$375$	0	0
$376$	1.16991e6	0.426761
$377$	723062.	0.262013
$378$	0	0
$379$	399928.	0.143016	0.0715079	−	0.997440i	$-0.477219\pi$
$379$	399928.	0.143016	0.0715079	+	0.997440i	$0.477219\pi$
$380$	1.43944e6	0.511368
$381$	0	0
$382$	1.78103e6	0.624473
$383$	−1.37113e6	−0.477618	−0.238809	−	0.971067i	$-0.576757\pi$
$383$	−1.37113e6	−0.477618	−0.238809	+	0.971067i	$0.576757\pi$
$384$	0	0
$385$	98567.9	0.0338909
$386$	2.11393e6	0.722142
$387$	0	0
$388$	1.85057e6	0.624059
$389$	−3.29557e6	−1.10422	−0.552111	−	0.833771i	$-0.686178\pi$
$389$	−3.29557e6	−1.10422	−0.552111	+	0.833771i	$0.686178\pi$
$390$	0	0
$391$	−609022.	−0.201461
$392$	1.91210e6	0.628485
$393$	0	0
$394$	−3.26618e6	−1.05999
$395$	1.15191e6	0.371473
$396$	0	0
$397$	2.44883e6	0.779798	0.389899	−	0.920858i	$-0.372510\pi$
$397$	2.44883e6	0.779798	0.389899	+	0.920858i	$0.372510\pi$
$398$	8.13946e6	2.57566
$399$	0	0
$400$	1.22265e6	0.382079
$401$	−4.98383e6	−1.54775	−0.773877	−	0.633336i	$-0.781685\pi$
$401$	−4.98383e6	−1.54775	−0.773877	+	0.633336i	$0.781685\pi$
$402$	0	0
$403$	6.09602e6	1.86975
$404$	7.89973e6	2.40801
$405$	0	0
$406$	128980.	0.0388336
$407$	−46510.4	−0.0139176
$408$	0	0
$409$	−1.36543e6	−0.403611	−0.201805	−	0.979426i	$-0.564681\pi$
$409$	−1.36543e6	−0.403611	−0.201805	+	0.979426i	$0.564681\pi$
$410$	−1.01758e6	−0.298956
$411$	0	0
$412$	309422.	0.0898066
$413$	45099.6	0.0130106
$414$	0	0
$415$	239153.	0.0681641
$416$	−4.77425e6	−1.35261
$417$	0	0
$418$	−6.94686e6	−1.94468
$419$	−2.31238e6	−0.643465	−0.321732	−	0.946831i	$-0.604265\pi$
$419$	−2.31238e6	−0.643465	−0.321732	+	0.946831i	$0.604265\pi$
$420$	0	0
$421$	3.96201e6	1.08946	0.544728	−	0.838613i	$-0.316633\pi$
$421$	3.96201e6	1.08946	0.544728	+	0.838613i	$0.316633\pi$
$422$	3.27455e6	0.895097
$423$	0	0
$424$	1.26516e6	0.341768
$425$	−2.39744e6	−0.643837
$426$	0	0
$427$	29732.2	0.00789146
$428$	1.04502e7	2.75749
$429$	0	0
$430$	460826.	0.120189
$431$	3.23219e6	0.838115	0.419057	−	0.907960i	$-0.362361\pi$
$431$	3.23219e6	0.838115	0.419057	+	0.907960i	$0.362361\pi$
$432$	0	0
$433$	−2.25277e6	−0.577426	−0.288713	−	0.957416i	$-0.593227\pi$
$433$	−2.25277e6	−0.577426	−0.288713	+	0.957416i	$0.593227\pi$
$434$	1.08741e6	0.277121
$435$	0	0
$436$	6.24790e6	1.57405
$437$	−1.30439e6	−0.326741
$438$	0	0
$439$	−6.68557e6	−1.65568	−0.827841	−	0.560963i	$-0.810431\pi$
$439$	−6.68557e6	−1.65568	−0.827841	+	0.560963i	$0.810431\pi$
$440$	−874270.	−0.215285
$441$	0	0
$442$	4.76337e6	1.15973
$443$	6.04955e6	1.46458	0.732291	−	0.680992i	$-0.238451\pi$
$443$	6.04955e6	1.46458	0.732291	+	0.680992i	$0.238451\pi$
$444$	0	0
$445$	−1.35973e6	−0.325501
$446$	5.54983e6	1.32112
$447$	0	0
$448$	−671753.	−0.158130
$449$	514041.	0.120332	0.0601661	−	0.998188i	$-0.480837\pi$
$449$	514041.	0.120332	0.0601661	+	0.998188i	$0.480837\pi$
$450$	0	0
$451$	2.87235e6	0.664962
$452$	−1.18628e6	−0.273112
$453$	0	0
$454$	−7.81450e6	−1.77935
$455$	−144756.	−0.0327799
$456$	0	0
$457$	7.09850e6	1.58992	0.794962	−	0.606660i	$-0.207491\pi$
$457$	7.09850e6	1.58992	0.794962	+	0.606660i	$0.207491\pi$
$458$	−2.27681e6	−0.507182
$459$	0	0
$460$	−565515.	−0.124609
$461$	−538077.	−0.117921	−0.0589606	−	0.998260i	$-0.518779\pi$
$461$	−538077.	−0.117921	−0.0589606	+	0.998260i	$0.518779\pi$
$462$	0	0
$463$	−5.69755e6	−1.23520	−0.617598	−	0.786494i	$-0.711894\pi$
$463$	−5.69755e6	−1.23520	−0.617598	+	0.786494i	$0.711894\pi$
$464$	491946.	0.106077
$465$	0	0
$466$	−4.18156e6	−0.892017
$467$	−6.84726e6	−1.45286	−0.726431	−	0.687239i	$-0.758822\pi$
$467$	−6.84726e6	−1.45286	−0.726431	+	0.687239i	$0.758822\pi$
$468$	0	0
$469$	305369.	0.0641051
$470$	1.56610e6	0.327021
$471$	0	0
$472$	−400021.	−0.0826471
$473$	−1.30079e6	−0.267335
$474$	0	0
$475$	−5.13479e6	−1.04421
$476$	496977.	0.100535
$477$	0	0
$478$	−3.51281e6	−0.703210
$479$	−2.13269e6	−0.424706	−0.212353	−	0.977193i	$-0.568113\pi$
$479$	−2.13269e6	−0.424706	−0.212353	+	0.977193i	$0.568113\pi$
$480$	0	0
$481$	68304.7	0.0134613
$482$	−582591.	−0.114221
$483$	0	0
$484$	1.24000e6	0.240607
$485$	719103.	0.138815
$486$	0	0
$487$	1.44011e6	0.275152	0.137576	−	0.990491i	$-0.456069\pi$
$487$	1.44011e6	0.275152	0.137576	+	0.990491i	$0.456069\pi$
$488$	−263716.	−0.0501288
$489$	0	0
$490$	2.55962e6	0.481599
$491$	−193566.	−0.0362347	−0.0181174	−	0.999836i	$-0.505767\pi$
$491$	−193566.	−0.0362347	−0.0181174	+	0.999836i	$0.505767\pi$
$492$	0	0
$493$	−964633.	−0.178750
$494$	1.02021e7	1.88092
$495$	0	0
$496$	4.14752e6	0.756979
$497$	−726047.	−0.131848
$498$	0	0
$499$	1.09525e7	1.96908	0.984539	−	0.175165i	$-0.0560460\pi$
$499$	1.09525e7	1.96908	0.984539	+	0.175165i	$0.0560460\pi$
$500$	−4.69486e6	−0.839842
$501$	0	0
$502$	−3.32136e6	−0.588243
$503$	1.09972e6	0.193804	0.0969022	−	0.995294i	$-0.469107\pi$
$503$	1.09972e6	0.193804	0.0969022	+	0.995294i	$0.469107\pi$
$504$	0	0
$505$	3.06972e6	0.535636
$506$	2.72923e6	0.473875
$507$	0	0
$508$	−1.48097e7	−2.54617
$509$	−5.13320e6	−0.878201	−0.439100	−	0.898438i	$-0.644703\pi$
$509$	−5.13320e6	−0.878201	−0.439100	+	0.898438i	$0.644703\pi$
$510$	0	0
$511$	732944.	0.124171
$512$	−4.84369e6	−0.816586
$513$	0	0
$514$	−59337.6	−0.00990655
$515$	120237.	0.0199765
$516$	0	0
$517$	−4.42070e6	−0.727386
$518$	12184.2	0.00199514
$519$	0	0
$520$	1.28394e6	0.208227
$521$	−9.05184e6	−1.46097	−0.730487	−	0.682926i	$-0.760707\pi$
$521$	−9.05184e6	−1.46097	−0.730487	+	0.682926i	$0.760707\pi$
$522$	0	0
$523$	974287.	0.155752	0.0778758	−	0.996963i	$-0.475186\pi$
$523$	974287.	0.155752	0.0778758	+	0.996963i	$0.475186\pi$
$524$	−2.60483e6	−0.414430
$525$	0	0
$526$	1.34996e7	2.12744
$527$	−8.13267e6	−1.27558
$528$	0	0
$529$	−5.92388e6	−0.920380
$530$	1.69361e6	0.261892
$531$	0	0
$532$	1.06441e6	0.163054
$533$	−4.21831e6	−0.643162
$534$	0	0
$535$	4.06078e6	0.613374
$536$	−2.70854e6	−0.407214
$537$	0	0
$538$	6.67568e6	0.994351
$539$	−7.22516e6	−1.07121
$540$	0	0
$541$	6.38853e6	0.938443	0.469222	−	0.883080i	$-0.344535\pi$
$541$	6.38853e6	0.938443	0.469222	+	0.883080i	$0.344535\pi$
$542$	5.45592e6	0.797755
$543$	0	0
$544$	6.36930e6	0.922771
$545$	2.42784e6	0.350130
$546$	0	0
$547$	−4.31658e6	−0.616838	−0.308419	−	0.951251i	$-0.599800\pi$
$547$	−4.31658e6	−0.616838	−0.308419	+	0.951251i	$0.599800\pi$
$548$	−5.23653e6	−0.744891
$549$	0	0
$550$	1.07437e7	1.51443
$551$	−2.06603e6	−0.289907
$552$	0	0
$553$	851801.	0.118447
$554$	2.78554e6	0.385599
$555$	0	0
$556$	−1.95247e7	−2.67854
$557$	−6.96213e6	−0.950833	−0.475416	−	0.879761i	$-0.657703\pi$
$557$	−6.96213e6	−0.950833	−0.475416	+	0.879761i	$0.657703\pi$
$558$	0	0
$559$	1.91033e6	0.258570
$560$	−98486.7	−0.0132711
$561$	0	0
$562$	−3.23142e6	−0.431571
$563$	442629.	0.0588530	0.0294265	−	0.999567i	$-0.490632\pi$
$563$	442629.	0.0588530	0.0294265	+	0.999567i	$0.490632\pi$
$564$	0	0
$565$	−460970.	−0.0607508
$566$	1.46213e7	1.91843
$567$	0	0
$568$	6.43983e6	0.837537
$569$	−4.56709e6	−0.591369	−0.295685	−	0.955286i	$-0.595548\pi$
$569$	−4.56709e6	−0.591369	−0.295685	+	0.955286i	$0.595548\pi$
$570$	0	0
$571$	−1.20733e7	−1.54966	−0.774831	−	0.632169i	$-0.782165\pi$
$571$	−1.20733e7	−1.54966	−0.774831	+	0.632169i	$0.782165\pi$
$572$	−1.24852e7	−1.59554
$573$	0	0
$574$	−752463.	−0.0953247
$575$	2.01732e6	0.254452
$576$	0	0
$577$	−3.82492e6	−0.478280	−0.239140	−	0.970985i	$-0.576866\pi$
$577$	−3.82492e6	−0.478280	−0.239140	+	0.970985i	$0.576866\pi$
$578$	6.11156e6	0.760908
$579$	0	0
$580$	−895723.	−0.110562
$581$	176846.	0.0217347
$582$	0	0
$583$	−4.78062e6	−0.582522
$584$	−6.50101e6	−0.788767
$585$	0	0
$586$	1.09230e7	1.31400
$587$	1.24579e7	1.49227	0.746136	−	0.665793i	$-0.231907\pi$
$587$	1.24579e7	1.49227	0.746136	+	0.665793i	$0.231907\pi$
$588$	0	0
$589$	−1.74184e7	−2.06880
$590$	−535487.	−0.0633314
$591$	0	0
$592$	46472.1	0.00544990
$593$	−8.12876e6	−0.949265	−0.474633	−	0.880184i	$-0.657419\pi$
$593$	−8.12876e6	−0.949265	−0.474633	+	0.880184i	$0.657419\pi$
$594$	0	0
$595$	193118.	0.0223630
$596$	−1.27705e7	−1.47263
$597$	0	0
$598$	−4.00812e6	−0.458340
$599$	2.18535e6	0.248860	0.124430	−	0.992228i	$-0.460290\pi$
$599$	2.18535e6	0.248860	0.124430	+	0.992228i	$0.460290\pi$
$600$	0	0
$601$	−3.56350e6	−0.402430	−0.201215	−	0.979547i	$-0.564489\pi$
$601$	−3.56350e6	−0.402430	−0.201215	+	0.979547i	$0.564489\pi$
$602$	340765.	0.0383234
$603$	0	0
$604$	−5.34388e6	−0.596025
$605$	481845.	0.0535204
$606$	0	0
$607$	1.07019e6	0.117893	0.0589464	−	0.998261i	$-0.481226\pi$
$607$	1.07019e6	0.117893	0.0589464	+	0.998261i	$0.481226\pi$
$608$	1.36416e7	1.49660
$609$	0	0
$610$	−353023.	−0.0384130
$611$	6.49220e6	0.703540
$612$	0	0
$613$	−1.33424e7	−1.43411	−0.717057	−	0.697015i	$-0.754511\pi$
$613$	−1.33424e7	−1.43411	−0.717057	+	0.697015i	$0.754511\pi$
$614$	−1.65337e7	−1.76991
$615$	0	0
$616$	−646493.	−0.0686455
$617$	791902.	0.0837449	0.0418724	−	0.999123i	$-0.486668\pi$
$617$	791902.	0.0837449	0.0418724	+	0.999123i	$0.486668\pi$
$618$	0	0
$619$	9.53469e6	1.00018	0.500092	−	0.865972i	$-0.333299\pi$
$619$	9.53469e6	1.00018	0.500092	+	0.865972i	$0.333299\pi$
$620$	−7.55170e6	−0.788979
$621$	0	0
$622$	−2.72356e7	−2.82268
$623$	−1.00547e6	−0.103789
$624$	0	0
$625$	6.98198e6	0.714955
$626$	2.04970e7	2.09052
$627$	0	0
$628$	−8.56022e6	−0.866136
$629$	−91124.9	−0.00918355
$630$	0	0
$631$	−1.67624e6	−0.167595	−0.0837977	−	0.996483i	$-0.526705\pi$
$631$	−1.67624e6	−0.167595	−0.0837977	+	0.996483i	$0.526705\pi$
$632$	−7.55524e6	−0.752412
$633$	0	0
$634$	−909296.	−0.0898426
$635$	−5.75484e6	−0.566368
$636$	0	0
$637$	1.06108e7	1.03609
$638$	4.32285e6	0.420454
$639$	0	0
$640$	3.77853e6	0.364647
$641$	4.71863e6	0.453598	0.226799	−	0.973942i	$-0.427174\pi$
$641$	4.71863e6	0.453598	0.226799	+	0.973942i	$0.427174\pi$
$642$	0	0
$643$	−1.25582e7	−1.19785	−0.598923	−	0.800807i	$-0.704404\pi$
$643$	−1.25582e7	−1.19785	−0.598923	+	0.800807i	$0.704404\pi$
$644$	−418179.	−0.0397327
$645$	0	0
$646$	−1.36105e7	−1.28320
$647$	−1.79155e6	−0.168256	−0.0841278	−	0.996455i	$-0.526810\pi$
$647$	−1.79155e6	−0.168256	−0.0841278	+	0.996455i	$0.526810\pi$
$648$	0	0
$649$	1.51154e6	0.140867
$650$	−1.57781e7	−1.46478
$651$	0	0
$652$	5.74859e6	0.529593
$653$	−1.71252e7	−1.57164	−0.785821	−	0.618454i	$-0.787759\pi$
$653$	−1.71252e7	−1.57164	−0.785821	+	0.618454i	$0.787759\pi$
$654$	0	0
$655$	−1.01220e6	−0.0921854
$656$	−2.86999e6	−0.260388
$657$	0	0
$658$	1.15808e6	0.104273
$659$	1.46762e7	1.31643	0.658216	−	0.752829i	$-0.271311\pi$
$659$	1.46762e7	1.31643	0.658216	+	0.752829i	$0.271311\pi$
$660$	0	0
$661$	−1.49380e7	−1.32981	−0.664904	−	0.746929i	$-0.731528\pi$
$661$	−1.49380e7	−1.32981	−0.664904	+	0.746929i	$0.731528\pi$
$662$	−1.54325e7	−1.36865
$663$	0	0
$664$	−1.56857e6	−0.138065
$665$	413616.	0.0362696
$666$	0	0
$667$	811688.	0.0706438
$668$	−9.10419e6	−0.789406
$669$	0	0
$670$	−3.62577e6	−0.312042
$671$	996493.	0.0854413
$672$	0	0
$673$	8.30140e6	0.706502	0.353251	−	0.935529i	$-0.385076\pi$
$673$	8.30140e6	0.706502	0.353251	+	0.935529i	$0.385076\pi$
$674$	5.44631e6	0.461799
$675$	0	0
$676$	1.59470e6	0.134219
$677$	287812.	0.0241345	0.0120672	−	0.999927i	$-0.496159\pi$
$677$	287812.	0.0241345	0.0120672	+	0.999927i	$0.496159\pi$
$678$	0	0
$679$	531752.	0.0442624
$680$	−1.71290e6	−0.142056
$681$	0	0
$682$	3.64452e7	3.00040
$683$	2.12554e7	1.74348	0.871742	−	0.489965i	$-0.162990\pi$
$683$	2.12554e7	1.74348	0.871742	+	0.489965i	$0.162990\pi$
$684$	0	0
$685$	−2.03484e6	−0.165693
$686$	3.80460e6	0.308673
$687$	0	0
$688$	1.29972e6	0.104684
$689$	7.02076e6	0.563425
$690$	0	0
$691$	−2.41473e6	−0.192386	−0.0961932	−	0.995363i	$-0.530667\pi$
$691$	−2.41473e6	−0.192386	−0.0961932	+	0.995363i	$0.530667\pi$
$692$	1.34850e7	1.07050
$693$	0	0
$694$	4.35243e6	0.343031
$695$	−7.58702e6	−0.595812
$696$	0	0
$697$	5.62762e6	0.438776
$698$	1.50265e7	1.16740
$699$	0	0
$700$	−1.64618e6	−0.126979
$701$	−1.06511e7	−0.818654	−0.409327	−	0.912388i	$-0.634236\pi$
$701$	−1.06511e7	−0.818654	−0.409327	+	0.912388i	$0.634236\pi$
$702$	0	0
$703$	−195170.	−0.0148944
$704$	−2.25142e7	−1.71209
$705$	0	0
$706$	−2.12694e7	−1.60599
$707$	2.26995e6	0.170792
$708$	0	0
$709$	1.04662e7	0.781943	0.390971	−	0.920403i	$-0.372139\pi$
$709$	1.04662e7	0.781943	0.390971	+	0.920403i	$0.372139\pi$
$710$	8.62067e6	0.641793
$711$	0	0
$712$	8.91828e6	0.659297
$713$	6.84321e6	0.504122
$714$	0	0
$715$	−4.85158e6	−0.354910
$716$	−2.97411e7	−2.16808
$717$	0	0
$718$	1.84170e7	1.33324
$719$	1.40608e7	1.01435	0.507175	−	0.861843i	$-0.330690\pi$
$719$	1.40608e7	1.01435	0.507175	+	0.861843i	$0.330690\pi$
$720$	0	0
$721$	88911.1	0.00636968
$722$	−7.41063e6	−0.529069
$723$	0	0
$724$	−1.09419e7	−0.775793
$725$	3.19525e6	0.225766
$726$	0	0
$727$	4.54577e6	0.318986	0.159493	−	0.987199i	$-0.449014\pi$
$727$	4.54577e6	0.318986	0.159493	+	0.987199i	$0.449014\pi$
$728$	949432.	0.0663950
$729$	0	0
$730$	−8.70256e6	−0.604421
$731$	−2.54856e6	−0.176401
$732$	0	0
$733$	−6.72753e6	−0.462483	−0.231241	−	0.972896i	$-0.574279\pi$
$733$	−6.72753e6	−0.462483	−0.231241	+	0.972896i	$0.574279\pi$
$734$	4.32597e7	2.96376
$735$	0	0
$736$	−5.35942e6	−0.364690
$737$	1.02346e7	0.694070
$738$	0	0
$739$	7.96547e6	0.536538	0.268269	−	0.963344i	$-0.413548\pi$
$739$	7.96547e6	0.536538	0.268269	+	0.963344i	$0.413548\pi$
$740$	−84615.3	−0.00568028
$741$	0	0
$742$	1.25237e6	0.0835067
$743$	−3.70267e6	−0.246061	−0.123031	−	0.992403i	$-0.539261\pi$
$743$	−3.70267e6	−0.246061	−0.123031	+	0.992403i	$0.539261\pi$
$744$	0	0
$745$	−4.96244e6	−0.327570
$746$	−1.56470e7	−1.02940
$747$	0	0
$748$	1.66565e7	1.08850
$749$	3.00281e6	0.195580
$750$	0	0
$751$	−2.71608e7	−1.75728	−0.878642	−	0.477480i	$-0.841550\pi$
$751$	−2.71608e7	−1.75728	−0.878642	+	0.477480i	$0.841550\pi$
$752$	4.41706e6	0.284832
$753$	0	0
$754$	−6.34848e6	−0.406670
$755$	−2.07655e6	−0.132579
$756$	0	0
$757$	2.64905e7	1.68016	0.840078	−	0.542465i	$-0.182509\pi$
$757$	2.64905e7	1.68016	0.840078	+	0.542465i	$0.182509\pi$
$758$	−3.51137e6	−0.221975
$759$	0	0
$760$	−3.66866e6	−0.230395
$761$	2.94938e7	1.84616	0.923080	−	0.384607i	$-0.125663\pi$
$761$	2.94938e7	1.84616	0.923080	+	0.384607i	$0.125663\pi$
$762$	0	0
$763$	1.79531e6	0.111642
$764$	−9.14623e6	−0.566903
$765$	0	0
$766$	1.20385e7	0.741310
$767$	−2.21983e6	−0.136249
$768$	0	0
$769$	9.58663e6	0.584588	0.292294	−	0.956329i	$-0.405581\pi$
$769$	9.58663e6	0.584588	0.292294	+	0.956329i	$0.405581\pi$
$770$	−865426.	−0.0526021
$771$	0	0
$772$	−1.08558e7	−0.655567
$773$	−6.41914e6	−0.386392	−0.193196	−	0.981160i	$-0.561885\pi$
$773$	−6.41914e6	−0.386392	−0.193196	+	0.981160i	$0.561885\pi$
$774$	0	0
$775$	2.69386e7	1.61109
$776$	−4.71649e6	−0.281167
$777$	0	0
$778$	2.89351e7	1.71386
$779$	1.20531e7	0.711633
$780$	0	0
$781$	−2.43339e7	−1.42753
$782$	5.34721e6	0.312687
$783$	0	0
$784$	7.21920e6	0.419468
$785$	−3.32638e6	−0.192663
$786$	0	0
$787$	3.15258e7	1.81438	0.907191	−	0.420719i	$-0.138222\pi$
$787$	3.15258e7	1.81438	0.907191	+	0.420719i	$0.138222\pi$
$788$	1.67730e7	0.962265
$789$	0	0
$790$	−1.01138e7	−0.576563
$791$	−340872.	−0.0193709
$792$	0	0
$793$	−1.46344e6	−0.0826402
$794$	−2.15007e7	−1.21032
$795$	0	0
$796$	−4.17990e7	−2.33821
$797$	2.66338e7	1.48521	0.742605	−	0.669730i	$-0.233590\pi$
$797$	2.66338e7	1.48521	0.742605	+	0.669730i	$0.233590\pi$
$798$	0	0
$799$	−8.66120e6	−0.479967
$800$	−2.10976e7	−1.16549
$801$	0	0
$802$	4.37580e7	2.40227
$803$	2.45651e7	1.34440
$804$	0	0
$805$	−162498.	−0.00883811
$806$	−5.35230e7	−2.90204
$807$	0	0
$808$	−2.01338e7	−1.08492
$809$	4.02522e6	0.216231	0.108116	−	0.994138i	$-0.465518\pi$
$809$	4.02522e6	0.216231	0.108116	+	0.994138i	$0.465518\pi$
$810$	0	0
$811$	−2.10486e7	−1.12375	−0.561876	−	0.827222i	$-0.689920\pi$
$811$	−2.10486e7	−1.12375	−0.561876	+	0.827222i	$0.689920\pi$
$812$	−662357.	−0.0352535
$813$	0	0
$814$	408361.	0.0216015
$815$	2.23382e6	0.117802
$816$	0	0
$817$	−5.45846e6	−0.286098
$818$	1.19885e7	0.626444
$819$	0	0
$820$	5.22561e6	0.271395
$821$	−7.16356e6	−0.370912	−0.185456	−	0.982653i	$-0.559376\pi$
$821$	−7.16356e6	−0.370912	−0.185456	+	0.982653i	$0.559376\pi$
$822$	0	0
$823$	−1.93310e7	−0.994846	−0.497423	−	0.867508i	$-0.665720\pi$
$823$	−1.93310e7	−0.994846	−0.497423	+	0.867508i	$0.665720\pi$
$824$	−788617.	−0.0404621
$825$	0	0
$826$	−395974.	−0.0201938
$827$	−8.87409e6	−0.451191	−0.225595	−	0.974221i	$-0.572433\pi$
$827$	−8.87409e6	−0.451191	−0.225595	+	0.974221i	$0.572433\pi$
$828$	0	0
$829$	8.50913e6	0.430030	0.215015	−	0.976611i	$-0.431020\pi$
$829$	8.50913e6	0.430030	0.215015	+	0.976611i	$0.431020\pi$
$830$	−2.09976e6	−0.105797
$831$	0	0
$832$	3.30642e7	1.65596
$833$	−1.41558e7	−0.706841
$834$	0	0
$835$	−3.53776e6	−0.175595
$836$	3.56745e7	1.76540
$837$	0	0
$838$	2.03027e7	0.998721
$839$	3.61604e7	1.77349	0.886744	−	0.462261i	$-0.152962\pi$
$839$	3.61604e7	1.77349	0.886744	+	0.462261i	$0.152962\pi$
$840$	0	0
$841$	−1.92255e7	−0.937320
$842$	−3.47864e7	−1.69094
$843$	0	0
$844$	−1.68159e7	−0.812578
$845$	619678.	0.0298555
$846$	0	0
$847$	356308.	0.0170654
$848$	4.77668e6	0.228106
$849$	0	0
$850$	2.10495e7	0.999298
$851$	76676.8	0.00362944
$852$	0	0
$853$	1.02908e7	0.484257	0.242128	−	0.970244i	$-0.422154\pi$
$853$	1.02908e7	0.484257	0.242128	+	0.970244i	$0.422154\pi$
$854$	−261049.	−0.0122483
$855$	0	0
$856$	−2.66341e7	−1.24238
$857$	−1.01966e7	−0.474244	−0.237122	−	0.971480i	$-0.576204\pi$
$857$	−1.01966e7	−0.474244	−0.237122	+	0.971480i	$0.576204\pi$
$858$	0	0
$859$	−3.60575e7	−1.66730	−0.833648	−	0.552296i	$-0.813752\pi$
$859$	−3.60575e7	−1.66730	−0.833648	+	0.552296i	$0.813752\pi$
$860$	−2.36650e6	−0.109109
$861$	0	0
$862$	−2.83786e7	−1.30084
$863$	3.91789e7	1.79071	0.895354	−	0.445355i	$-0.146923\pi$
$863$	3.91789e7	1.79071	0.895354	+	0.445355i	$0.146923\pi$
$864$	0	0
$865$	5.24008e6	0.238121
$866$	1.97793e7	0.896222
$867$	0	0
$868$	−5.58423e6	−0.251573
$869$	2.85487e7	1.28244
$870$	0	0
$871$	−1.50305e7	−0.671316
$872$	−1.59239e7	−0.709181
$873$	0	0
$874$	1.14525e7	0.507135
$875$	−1.34905e6	−0.0595671
$876$	0	0
$877$	−3.90462e6	−0.171427	−0.0857137	−	0.996320i	$-0.527317\pi$
$877$	−3.90462e6	−0.171427	−0.0857137	+	0.996320i	$0.527317\pi$
$878$	5.86993e7	2.56978
$879$	0	0
$880$	−3.30084e6	−0.143687
$881$	−3.06209e7	−1.32916	−0.664582	−	0.747215i	$-0.731390\pi$
$881$	−3.06209e7	−1.32916	−0.664582	+	0.747215i	$0.731390\pi$
$882$	0	0
$883$	2.72635e7	1.17674	0.588368	−	0.808593i	$-0.299771\pi$
$883$	2.72635e7	1.17674	0.588368	+	0.808593i	$0.299771\pi$
$884$	−2.44615e7	−1.05282
$885$	0	0
$886$	−5.31150e7	−2.27318
$887$	−7.45001e6	−0.317942	−0.158971	−	0.987283i	$-0.550818\pi$
$887$	−7.45001e6	−0.317942	−0.158971	+	0.987283i	$0.550818\pi$
$888$	0	0
$889$	−4.25551e6	−0.180591
$890$	1.19384e7	0.505210
$891$	0	0
$892$	−2.85003e7	−1.19933
$893$	−1.85504e7	−0.778438
$894$	0	0
$895$	−1.15570e7	−0.482266
$896$	2.79410e6	0.116271
$897$	0	0
$898$	−4.51327e6	−0.186767
$899$	1.08390e7	0.447291
$900$	0	0
$901$	−9.36636e6	−0.384378
$902$	−2.52193e7	−1.03209
$903$	0	0
$904$	3.02344e6	0.123050
$905$	−4.25186e6	−0.172567
$906$	0	0
$907$	8.74302e6	0.352893	0.176447	−	0.984310i	$-0.443540\pi$
$907$	8.74302e6	0.352893	0.176447	+	0.984310i	$0.443540\pi$
$908$	4.01302e7	1.61531
$909$	0	0
$910$	1.27095e6	0.0508776
$911$	−2.92020e7	−1.16578	−0.582891	−	0.812551i	$-0.698078\pi$
$911$	−2.92020e7	−1.16578	−0.582891	+	0.812551i	$0.698078\pi$
$912$	0	0
$913$	5.92709e6	0.235323
$914$	−6.23248e7	−2.46772
$915$	0	0
$916$	1.16922e7	0.460424
$917$	−748486.	−0.0293941
$918$	0	0
$919$	3.40665e7	1.33057	0.665287	−	0.746588i	$-0.268309\pi$
$919$	3.40665e7	1.33057	0.665287	+	0.746588i	$0.268309\pi$
$920$	1.44132e6	0.0561422
$921$	0	0
$922$	4.72431e6	0.183025
$923$	3.57365e7	1.38073
$924$	0	0
$925$	301842.	0.0115991
$926$	5.00245e7	1.91715
$927$	0	0
$928$	−8.48882e6	−0.323577
$929$	−7.22466e6	−0.274649	−0.137325	−	0.990526i	$-0.543850\pi$
$929$	−7.22466e6	−0.274649	−0.137325	+	0.990526i	$0.543850\pi$
$930$	0	0
$931$	−3.03186e7	−1.14640
$932$	2.14737e7	0.809782
$933$	0	0
$934$	6.01189e7	2.25499
$935$	6.47247e6	0.242126
$936$	0	0
$937$	2.02690e7	0.754194	0.377097	−	0.926174i	$-0.376922\pi$
$937$	2.02690e7	0.754194	0.377097	+	0.926174i	$0.376922\pi$
$938$	−2.68114e6	−0.0994974
$939$	0	0
$940$	−8.04248e6	−0.296873
$941$	1.37966e7	0.507923	0.253962	−	0.967214i	$-0.418266\pi$
$941$	1.37966e7	0.507923	0.253962	+	0.967214i	$0.418266\pi$
$942$	0	0
$943$	−4.73534e6	−0.173409
$944$	−1.51030e6	−0.0551610
$945$	0	0
$946$	1.14210e7	0.414930
$947$	−3.23524e7	−1.17228	−0.586139	−	0.810210i	$-0.699353\pi$
$947$	−3.23524e7	−1.17228	−0.586139	+	0.810210i	$0.699353\pi$
$948$	0	0
$949$	−3.60760e7	−1.30033
$950$	4.50835e7	1.62072
$951$	0	0
$952$	−1.26663e6	−0.0452958
$953$	1.28170e7	0.457146	0.228573	−	0.973527i	$-0.426594\pi$
$953$	1.28170e7	0.457146	0.228573	+	0.973527i	$0.426594\pi$
$954$	0	0
$955$	−3.55409e6	−0.126101
$956$	1.80395e7	0.638381
$957$	0	0
$958$	1.87250e7	0.659186
$959$	−1.50470e6	−0.0528326
$960$	0	0
$961$	6.27527e7	2.19192
$962$	−599715.	−0.0208933
$963$	0	0
$964$	2.99181e6	0.103691
$965$	−4.21839e6	−0.145824
$966$	0	0
$967$	−4.53561e7	−1.55980	−0.779901	−	0.625903i	$-0.784731\pi$
$967$	−4.53561e7	−1.55980	−0.779901	+	0.625903i	$0.784731\pi$
$968$	−3.16036e6	−0.108405
$969$	0	0
$970$	−6.31372e6	−0.215455
$971$	6.25506e6	0.212904	0.106452	−	0.994318i	$-0.466051\pi$
$971$	6.25506e6	0.212904	0.106452	+	0.994318i	$0.466051\pi$
$972$	0	0
$973$	−5.61035e6	−0.189980
$974$	−1.26442e7	−0.427063
$975$	0	0
$976$	−995672.	−0.0334574
$977$	−6.26249e6	−0.209899	−0.104950	−	0.994478i	$-0.533468\pi$
$977$	−6.26249e6	−0.209899	−0.104950	+	0.994478i	$0.533468\pi$
$978$	0	0
$979$	−3.36991e7	−1.12373
$980$	−1.31445e7	−0.437200
$981$	0	0
$982$	1.69951e6	0.0562398
$983$	−2.95453e7	−0.975226	−0.487613	−	0.873060i	$-0.662132\pi$
$983$	−2.95453e7	−0.975226	−0.487613	+	0.873060i	$0.662132\pi$
$984$	0	0
$985$	6.51774e6	0.214046
$986$	8.46948e6	0.277437
$987$	0	0
$988$	−5.23912e7	−1.70752
$989$	2.14448e6	0.0697158
$990$	0	0
$991$	−2.48240e7	−0.802947	−0.401474	−	0.915871i	$-0.631502\pi$
$991$	−2.48240e7	−0.802947	−0.401474	+	0.915871i	$0.631502\pi$
$992$	−7.15679e7	−2.30908
$993$	0	0
$994$	6.37469e6	0.204641
$995$	−1.62425e7	−0.520109
$996$	0	0
$997$	−2.40550e7	−0.766421	−0.383211	−	0.923661i	$-0.625182\pi$
$997$	−2.40550e7	−0.766421	−0.383211	+	0.923661i	$0.625182\pi$
$998$	−9.61631e7	−3.05620
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-8.78000 q^{2} +45.0883 q^{4} +17.5207 q^{5} +12.9559 q^{7} -114.916 q^{8} +O(q^{10})\)	\(q-8.78000 q^{2} +45.0883 q^{4} +17.5207 q^{5} +12.9559 q^{7} -114.916 q^{8} -153.831 q^{10} +434.226 q^{11} -637.700 q^{13} -113.753 q^{14} -433.869 q^{16} +850.752 q^{17} +1822.12 q^{19} +789.977 q^{20} -3812.51 q^{22} -715.863 q^{23} -2818.03 q^{25} +5599.00 q^{26} +584.162 q^{28} -1133.86 q^{29} -9559.39 q^{31} +7486.66 q^{32} -7469.60 q^{34} +226.997 q^{35} -107.111 q^{37} -15998.2 q^{38} -2013.40 q^{40} +6614.88 q^{41} -2995.66 q^{43} +19578.5 q^{44} +6285.27 q^{46} -10180.6 q^{47} -16639.1 q^{49} +24742.3 q^{50} -28752.8 q^{52} -11009.5 q^{53} +7607.93 q^{55} -1488.84 q^{56} +9955.28 q^{58} +3481.00 q^{59} +2294.87 q^{61} +83931.4 q^{62} -51849.1 q^{64} -11172.9 q^{65} +23569.8 q^{67} +38359.0 q^{68} -1993.03 q^{70} -56039.7 q^{71} +56572.1 q^{73} +940.434 q^{74} +82156.5 q^{76} +5625.81 q^{77} +65746.0 q^{79} -7601.67 q^{80} -58078.6 q^{82} +13649.8 q^{83} +14905.7 q^{85} +26301.9 q^{86} -49899.4 q^{88} -77607.3 q^{89} -8262.00 q^{91} -32277.1 q^{92} +89386.0 q^{94} +31924.8 q^{95} +41043.1 q^{97} +146092. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * q^98

Introduction

L-functions

Modular forms

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Database

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