LMFDB - Embedded newform 531.6.a.c.1.4

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			1.4
Root			$-5.43261$ 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)

$f(q)$	$=$	$q-7.43261 q^{2} +23.2437 q^{4} -48.3022 q^{5} +120.542 q^{7} +65.0821 q^{8} +O(q^{10})$	\(q-7.43261 q^{2} +23.2437 q^{4} -48.3022 q^{5} +120.542 q^{7} +65.0821 q^{8} +359.012 q^{10} +192.015 q^{11} +600.307 q^{13} -895.942 q^{14} -1227.53 q^{16} -2195.23 q^{17} -393.685 q^{19} -1122.72 q^{20} -1427.17 q^{22} +212.558 q^{23} -791.894 q^{25} -4461.85 q^{26} +2801.84 q^{28} +320.477 q^{29} -1196.80 q^{31} +7041.12 q^{32} +16316.3 q^{34} -5822.45 q^{35} +1866.98 q^{37} +2926.10 q^{38} -3143.61 q^{40} +10116.5 q^{41} +18212.6 q^{43} +4463.14 q^{44} -1579.86 q^{46} -19027.8 q^{47} -2276.61 q^{49} +5885.84 q^{50} +13953.4 q^{52} +18428.0 q^{53} -9274.74 q^{55} +7845.13 q^{56} -2381.98 q^{58} +3481.00 q^{59} -16435.9 q^{61} +8895.35 q^{62} -13053.0 q^{64} -28996.2 q^{65} +10398.4 q^{67} -51025.3 q^{68} +43276.0 q^{70} +16382.6 q^{71} -68804.3 q^{73} -13876.6 q^{74} -9150.69 q^{76} +23145.9 q^{77} -3498.36 q^{79} +59292.4 q^{80} -75192.2 q^{82} +95365.4 q^{83} +106035. q^{85} -135367. q^{86} +12496.7 q^{88} +19992.9 q^{89} +72362.2 q^{91} +4940.64 q^{92} +141426. q^{94} +19015.8 q^{95} -99691.0 q^{97} +16921.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−7.43261	−1.31391	−0.656956	−	0.753929i	$-0.728156\pi$
$2$	−7.43261	−1.31391	−0.656956	+	0.753929i	$0.728156\pi$
$3$	0	0
$3$	0	0
$4$	23.2437	0.726366
$5$	−48.3022	−0.864057	−0.432028	−	0.901860i	$-0.642202\pi$
$5$	−48.3022	−0.864057	−0.432028	+	0.901860i	$0.642202\pi$
$6$	0	0
$7$	120.542	0.929808	0.464904	−	0.885361i	$-0.346089\pi$
$7$	120.542	0.929808	0.464904	+	0.885361i	$0.346089\pi$
$8$	65.0821	0.359531
$9$	0	0
$10$	359.012	1.13529
$11$	192.015	0.478468	0.239234	−	0.970962i	$-0.423104\pi$
$11$	192.015	0.478468	0.239234	+	0.970962i	$0.423104\pi$
$12$	0	0
$13$	600.307	0.985178	0.492589	−	0.870262i	$-0.336051\pi$
$13$	600.307	0.985178	0.492589	+	0.870262i	$0.336051\pi$
$14$	−895.942	−1.22169
$15$	0	0
$16$	−1227.53	−1.19876
$17$	−2195.23	−1.84229	−0.921146	−	0.389218i	$-0.872745\pi$
$17$	−2195.23	−1.84229	−0.921146	+	0.389218i	$0.872745\pi$
$18$	0	0
$19$	−393.685	−0.250187	−0.125093	−	0.992145i	$-0.539923\pi$
$19$	−393.685	−0.250187	−0.125093	+	0.992145i	$0.539923\pi$
$20$	−1122.72	−0.627621
$21$	0	0
$22$	−1427.17	−0.628665
$23$	212.558	0.0837834	0.0418917	−	0.999122i	$-0.486662\pi$
$23$	212.558	0.0837834	0.0418917	+	0.999122i	$0.486662\pi$
$24$	0	0
$25$	−791.894	−0.253406
$26$	−4461.85	−1.29444
$27$	0	0
$28$	2801.84	0.675381
$29$	320.477	0.0707623	0.0353812	−	0.999374i	$-0.488735\pi$
$29$	320.477	0.0707623	0.0353812	+	0.999374i	$0.488735\pi$
$30$	0	0
$31$	−1196.80	−0.223675	−0.111837	−	0.993727i	$-0.535674\pi$
$31$	−1196.80	−0.223675	−0.111837	+	0.993727i	$0.535674\pi$
$32$	7041.12	1.21553
$33$	0	0
$34$	16316.3	2.42061
$35$	−5822.45	−0.803407
$36$	0	0
$37$	1866.98	0.224200	0.112100	−	0.993697i	$-0.464242\pi$
$37$	1866.98	0.224200	0.112100	+	0.993697i	$0.464242\pi$
$38$	2926.10	0.328724
$39$	0	0
$40$	−3143.61	−0.310655
$41$	10116.5	0.939879	0.469939	−	0.882699i	$-0.344276\pi$
$41$	10116.5	0.939879	0.469939	+	0.882699i	$0.344276\pi$
$42$	0	0
$43$	18212.6	1.50211	0.751053	−	0.660242i	$-0.229546\pi$
$43$	18212.6	1.50211	0.751053	+	0.660242i	$0.229546\pi$
$44$	4463.14	0.347543
$45$	0	0
$46$	−1579.86	−0.110084
$47$	−19027.8	−1.25645	−0.628224	−	0.778033i	$-0.716218\pi$
$47$	−19027.8	−1.25645	−0.628224	+	0.778033i	$0.716218\pi$
$48$	0	0
$49$	−2276.61	−0.135456
$50$	5885.84	0.332953
$51$	0	0
$52$	13953.4	0.715600
$53$	18428.0	0.901131	0.450566	−	0.892743i	$-0.351222\pi$
$53$	18428.0	0.901131	0.450566	+	0.892743i	$0.351222\pi$
$54$	0	0
$55$	−9274.74	−0.413424
$56$	7845.13	0.334295
$57$	0	0
$58$	−2381.98	−0.0929755
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−16435.9	−0.565547	−0.282773	−	0.959187i	$-0.591254\pi$
$61$	−16435.9	−0.565547	−0.282773	+	0.959187i	$0.591254\pi$
$62$	8895.35	0.293889
$63$	0	0
$64$	−13053.0	−0.398345
$65$	−28996.2	−0.851250
$66$	0	0
$67$	10398.4	0.282995	0.141497	−	0.989939i	$-0.454808\pi$
$67$	10398.4	0.282995	0.141497	+	0.989939i	$0.454808\pi$
$68$	−51025.3	−1.33818
$69$	0	0
$70$	43276.0	1.05561
$71$	16382.6	0.385688	0.192844	−	0.981229i	$-0.438229\pi$
$71$	16382.6	0.385688	0.192844	+	0.981229i	$0.438229\pi$
$72$	0	0
$73$	−68804.3	−1.51115	−0.755577	−	0.655060i	$-0.772643\pi$
$73$	−68804.3	−1.51115	−0.755577	+	0.655060i	$0.772643\pi$
$74$	−13876.6	−0.294580
$75$	0	0
$76$	−9150.69	−0.181727
$77$	23145.9	0.444884
$78$	0	0
$79$	−3498.36	−0.0630662	−0.0315331	−	0.999503i	$-0.510039\pi$
$79$	−3498.36	−0.0630662	−0.0315331	+	0.999503i	$0.510039\pi$
$80$	59292.4	1.03580
$81$	0	0
$82$	−75192.2	−1.23492
$83$	95365.4	1.51948	0.759741	−	0.650226i	$-0.225326\pi$
$83$	95365.4	1.51948	0.759741	+	0.650226i	$0.225326\pi$
$84$	0	0
$85$	106035.	1.59184
$86$	−135367.	−1.97364
$87$	0	0
$88$	12496.7	0.172024
$89$	19992.9	0.267548	0.133774	−	0.991012i	$-0.457290\pi$
$89$	19992.9	0.267548	0.133774	+	0.991012i	$0.457290\pi$
$90$	0	0
$91$	72362.2	0.916027
$92$	4940.64	0.0608574
$93$	0	0
$94$	141426.	1.65086
$95$	19015.8	0.216176
$96$	0	0
$97$	−99691.0	−1.07579	−0.537894	−	0.843012i	$-0.680780\pi$
$97$	−99691.0	−1.07579	−0.537894	+	0.843012i	$0.680780\pi$
$98$	16921.2	0.177978
$99$	0	0
$100$	−18406.6	−0.184066
$101$	77792.2	0.758809	0.379405	−	0.925231i	$-0.376129\pi$
$101$	77792.2	0.758809	0.379405	+	0.925231i	$0.376129\pi$
$102$	0	0
$103$	−53809.1	−0.499761	−0.249880	−	0.968277i	$-0.580391\pi$
$103$	−53809.1	−0.499761	−0.249880	+	0.968277i	$0.580391\pi$
$104$	39069.2	0.354202
$105$	0	0
$106$	−136968.	−1.18401
$107$	90583.1	0.764870	0.382435	−	0.923982i	$-0.375086\pi$
$107$	90583.1	0.764870	0.382435	+	0.923982i	$0.375086\pi$
$108$	0	0
$109$	206631.	1.66583	0.832913	−	0.553405i	$-0.186672\pi$
$109$	206631.	1.66583	0.832913	+	0.553405i	$0.186672\pi$
$110$	68935.6	0.543202
$111$	0	0
$112$	−147969.	−1.11462
$113$	−83781.9	−0.617240	−0.308620	−	0.951185i	$-0.599867\pi$
$113$	−83781.9	−0.617240	−0.308620	+	0.951185i	$0.599867\pi$
$114$	0	0
$115$	−10267.0	−0.0723936
$116$	7449.08	0.0513993
$117$	0	0
$118$	−25872.9	−0.171057
$119$	−264618.	−1.71298
$120$	0	0
$121$	−124181.	−0.771068
$122$	122162.	0.743079
$123$	0	0
$124$	−27818.1	−0.162470
$125$	189195.	1.08301
$126$	0	0
$127$	−230499.	−1.26812	−0.634059	−	0.773285i	$-0.718612\pi$
$127$	−230499.	−1.26812	−0.634059	+	0.773285i	$0.718612\pi$
$128$	−128298.	−0.692142
$129$	0	0
$130$	215517.	1.11847
$131$	−186771.	−0.950892	−0.475446	−	0.879745i	$-0.657713\pi$
$131$	−186771.	−0.950892	−0.475446	+	0.879745i	$0.657713\pi$
$132$	0	0
$133$	−47455.6	−0.232626
$134$	−77287.0	−0.371830
$135$	0	0
$136$	−142870.	−0.662361
$137$	−111807.	−0.508941	−0.254471	−	0.967081i	$-0.581901\pi$
$137$	−111807.	−0.508941	−0.254471	+	0.967081i	$0.581901\pi$
$138$	0	0
$139$	195432.	0.857942	0.428971	−	0.903318i	$-0.358876\pi$
$139$	195432.	0.857942	0.428971	+	0.903318i	$0.358876\pi$
$140$	−135335.	−0.583568
$141$	0	0
$142$	−121765.	−0.506760
$143$	115268.	0.471376
$144$	0	0
$145$	−15479.8	−0.0611427
$146$	511396.	1.98552
$147$	0	0
$148$	43395.6	0.162851
$149$	371327.	1.37022	0.685111	−	0.728438i	$-0.259754\pi$
$149$	371327.	1.37022	0.685111	+	0.728438i	$0.259754\pi$
$150$	0	0
$151$	−405248.	−1.44637	−0.723184	−	0.690655i	$-0.757322\pi$
$151$	−405248.	−1.44637	−0.723184	+	0.690655i	$0.757322\pi$
$152$	−25621.8	−0.0899500
$153$	0	0
$154$	−172034.	−0.584538
$155$	57808.1	0.193268
$156$	0	0
$157$	93561.5	0.302934	0.151467	−	0.988462i	$-0.451600\pi$
$157$	93561.5	0.302934	0.151467	+	0.988462i	$0.451600\pi$
$158$	26001.9	0.0828634
$159$	0	0
$160$	−340102.	−1.05029
$161$	25622.2	0.0779026
$162$	0	0
$163$	−314860.	−0.928216	−0.464108	−	0.885779i	$-0.653625\pi$
$163$	−314860.	−0.928216	−0.464108	+	0.885779i	$0.653625\pi$
$164$	235146.	0.682696
$165$	0	0
$166$	−708814.	−1.99647
$167$	−82055.5	−0.227676	−0.113838	−	0.993499i	$-0.536314\pi$
$167$	−82055.5	−0.227676	−0.113838	+	0.993499i	$0.536314\pi$
$168$	0	0
$169$	−10924.8	−0.0294238
$170$	−788114.	−2.09154
$171$	0	0
$172$	423328.	1.09108
$173$	138981.	0.353053	0.176527	−	0.984296i	$-0.443514\pi$
$173$	138981.	0.353053	0.176527	+	0.984296i	$0.443514\pi$
$174$	0	0
$175$	−95456.5	−0.235619
$176$	−235704.	−0.573568
$177$	0	0
$178$	−148600.	−0.351535
$179$	−377160.	−0.879819	−0.439909	−	0.898042i	$-0.644989\pi$
$179$	−377160.	−0.879819	−0.439909	+	0.898042i	$0.644989\pi$
$180$	0	0
$181$	−395990.	−0.898438	−0.449219	−	0.893422i	$-0.648298\pi$
$181$	−395990.	−0.898438	−0.449219	+	0.893422i	$0.648298\pi$
$182$	−537840.	−1.20358
$183$	0	0
$184$	13833.7	0.0301228
$185$	−90179.5	−0.193722
$186$	0	0
$187$	−421517.	−0.881478
$188$	−442277.	−0.912640
$189$	0	0
$190$	−141337.	−0.284036
$191$	−611684.	−1.21323	−0.606616	−	0.794995i	$-0.707473\pi$
$191$	−611684.	−1.21323	−0.606616	+	0.794995i	$0.707473\pi$
$192$	0	0
$193$	374856.	0.724388	0.362194	−	0.932103i	$-0.382028\pi$
$193$	374856.	0.724388	0.362194	+	0.932103i	$0.382028\pi$
$194$	740965.	1.41349
$195$	0	0
$196$	−52916.9	−0.0983908
$197$	−527822.	−0.968997	−0.484498	−	0.874792i	$-0.660998\pi$
$197$	−527822.	−0.968997	−0.484498	+	0.874792i	$0.660998\pi$
$198$	0	0
$199$	167623.	0.300055	0.150028	−	0.988682i	$-0.452064\pi$
$199$	167623.	0.300055	0.150028	+	0.988682i	$0.452064\pi$
$200$	−51538.1	−0.0911074
$201$	0	0
$202$	−578199.	−0.997009
$203$	38631.0	0.0657954
$204$	0	0
$205$	−488651.	−0.812108
$206$	399942.	0.656642
$207$	0	0
$208$	−736894.	−1.18099
$209$	−75593.3	−0.119706
$210$	0	0
$211$	−256097.	−0.396003	−0.198001	−	0.980202i	$-0.563445\pi$
$211$	−256097.	−0.396003	−0.198001	+	0.980202i	$0.563445\pi$
$212$	428335.	0.654551
$213$	0	0
$214$	−673269.	−1.00497
$215$	−879709.	−1.29790
$216$	0	0
$217$	−144265.	−0.207975
$218$	−1.53581e6	−2.18875
$219$	0	0
$220$	−215579.	−0.300297
$221$	−1.31781e6	−1.81499
$222$	0	0
$223$	615040.	0.828212	0.414106	−	0.910229i	$-0.364094\pi$
$223$	615040.	0.828212	0.414106	+	0.910229i	$0.364094\pi$
$224$	848751.	1.13021
$225$	0	0
$226$	622718.	0.810999
$227$	−1.04368e6	−1.34432	−0.672161	−	0.740405i	$-0.734634\pi$
$227$	−1.04368e6	−1.34432	−0.672161	+	0.740405i	$0.734634\pi$
$228$	0	0
$229$	−608125.	−0.766309	−0.383154	−	0.923684i	$-0.625162\pi$
$229$	−608125.	−0.766309	−0.383154	+	0.923684i	$0.625162\pi$
$230$	76310.9	0.0951189
$231$	0	0
$232$	20857.3	0.0254413
$233$	−1.00513e6	−1.21292	−0.606462	−	0.795113i	$-0.707412\pi$
$233$	−1.00513e6	−1.21292	−0.606462	+	0.795113i	$0.707412\pi$
$234$	0	0
$235$	919086.	1.08564
$236$	80911.3	0.0945648
$237$	0	0
$238$	1.96680e6	2.25070
$239$	−1.22727e6	−1.38978	−0.694891	−	0.719115i	$-0.744547\pi$
$239$	−1.22727e6	−1.38978	−0.694891	+	0.719115i	$0.744547\pi$
$240$	0	0
$241$	1.34998e6	1.49721	0.748607	−	0.663014i	$-0.230723\pi$
$241$	1.34998e6	1.49721	0.748607	+	0.663014i	$0.230723\pi$
$242$	922991.	1.01312
$243$	0	0
$244$	−382031.	−0.410794
$245$	109965.	0.117042
$246$	0	0
$247$	−236332.	−0.246479
$248$	−77890.3	−0.0804182
$249$	0	0
$250$	−1.40621e6	−1.42299
$251$	−448506.	−0.449350	−0.224675	−	0.974434i	$-0.572132\pi$
$251$	−448506.	−0.449350	−0.224675	+	0.974434i	$0.572132\pi$
$252$	0	0
$253$	40814.3	0.0400877
$254$	1.71321e6	1.66620
$255$	0	0
$256$	1.37128e6	1.30776
$257$	−656292.	−0.619818	−0.309909	−	0.950766i	$-0.600299\pi$
$257$	−656292.	−0.619818	−0.309909	+	0.950766i	$0.600299\pi$
$258$	0	0
$259$	225050.	0.208463
$260$	−673978.	−0.618319
$261$	0	0
$262$	1.38820e6	1.24939
$263$	−1.25395e6	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$263$	−1.25395e6	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$264$	0	0
$265$	−890113.	−0.778628
$266$	352719.	0.305650
$267$	0	0
$268$	241697.	0.205558
$269$	−295417.	−0.248917	−0.124459	−	0.992225i	$-0.539719\pi$
$269$	−295417.	−0.248917	−0.124459	+	0.992225i	$0.539719\pi$
$270$	0	0
$271$	1.36641e6	1.13021	0.565103	−	0.825020i	$-0.308836\pi$
$271$	1.36641e6	1.13021	0.565103	+	0.825020i	$0.308836\pi$
$272$	2.69471e6	2.20846
$273$	0	0
$274$	831018.	0.668704
$275$	−152055.	−0.121247
$276$	0	0
$277$	−180534.	−0.141371	−0.0706853	−	0.997499i	$-0.522519\pi$
$277$	−180534.	−0.141371	−0.0706853	+	0.997499i	$0.522519\pi$
$278$	−1.45257e6	−1.12726
$279$	0	0
$280$	−378937.	−0.288850
$281$	−748821.	−0.565734	−0.282867	−	0.959159i	$-0.591285\pi$
$281$	−748821.	−0.565734	−0.282867	+	0.959159i	$0.591285\pi$
$282$	0	0
$283$	−1.43420e6	−1.06450	−0.532248	−	0.846589i	$-0.678652\pi$
$283$	−1.43420e6	−1.06450	−0.532248	+	0.846589i	$0.678652\pi$
$284$	380792.	0.280151
$285$	0	0
$286$	−856741.	−0.619347
$287$	1.21947e6	0.873907
$288$	0	0
$289$	3.39919e6	2.39404
$290$	115055.	0.0803361
$291$	0	0
$292$	−1.59927e6	−1.09765
$293$	1.99988e6	1.36093	0.680465	−	0.732781i	$-0.261778\pi$
$293$	1.99988e6	1.36093	0.680465	+	0.732781i	$0.261778\pi$
$294$	0	0
$295$	−168140.	−0.112491
$296$	121507.	0.0806070
$297$	0	0
$298$	−2.75993e6	−1.80035
$299$	127600.	0.0825416
$300$	0	0
$301$	2.19538e6	1.39667
$302$	3.01205e6	1.90040
$303$	0	0
$304$	483259.	0.299914
$305$	793890.	0.488665
$306$	0	0
$307$	2.66083e6	1.61128	0.805639	−	0.592407i	$-0.201822\pi$
$307$	2.66083e6	1.61128	0.805639	+	0.592407i	$0.201822\pi$
$308$	537996.	0.323148
$309$	0	0
$310$	−429665.	−0.253937
$311$	2.11645e6	1.24082	0.620408	−	0.784279i	$-0.286967\pi$
$311$	2.11645e6	1.24082	0.620408	+	0.784279i	$0.286967\pi$
$312$	0	0
$313$	725870.	0.418792	0.209396	−	0.977831i	$-0.432850\pi$
$313$	725870.	0.418792	0.209396	+	0.977831i	$0.432850\pi$
$314$	−695407.	−0.398029
$315$	0	0
$316$	−81314.8	−0.0458091
$317$	1.20622e6	0.674186	0.337093	−	0.941471i	$-0.390556\pi$
$317$	1.20622e6	0.674186	0.337093	+	0.941471i	$0.390556\pi$
$318$	0	0
$319$	61536.4	0.0338575
$320$	630487.	0.344192
$321$	0	0
$322$	−190440.	−0.102357
$323$	864229.	0.460917
$324$	0	0
$325$	−475379.	−0.249650
$326$	2.34023e6	1.21959
$327$	0	0
$328$	658405.	0.337916
$329$	−2.29365e6	−1.16826
$330$	0	0
$331$	−1.12789e6	−0.565845	−0.282923	−	0.959143i	$-0.591304\pi$
$331$	−1.12789e6	−0.565845	−0.282923	+	0.959143i	$0.591304\pi$
$332$	2.21665e6	1.10370
$333$	0	0
$334$	609887.	0.299146
$335$	−502264.	−0.244523
$336$	0	0
$337$	−2.51669e6	−1.20713	−0.603567	−	0.797312i	$-0.706255\pi$
$337$	−2.51669e6	−1.20713	−0.603567	+	0.797312i	$0.706255\pi$
$338$	81200.1	0.0386603
$339$	0	0
$340$	2.46464e6	1.15626
$341$	−229803.	−0.107021
$342$	0	0
$343$	−2.30038e6	−1.05576
$344$	1.18531e6	0.540054
$345$	0	0
$346$	−1.03299e6	−0.463881
$347$	−3.97910e6	−1.77403	−0.887015	−	0.461741i	$-0.847225\pi$
$347$	−3.97910e6	−1.77403	−0.887015	+	0.461741i	$0.847225\pi$
$348$	0	0
$349$	−1.16212e6	−0.510726	−0.255363	−	0.966845i	$-0.582195\pi$
$349$	−1.16212e6	−0.510726	−0.255363	+	0.966845i	$0.582195\pi$
$350$	709491.	0.309583
$351$	0	0
$352$	1.35200e6	0.581593
$353$	2.71947e6	1.16158	0.580788	−	0.814055i	$-0.302744\pi$
$353$	2.71947e6	1.16158	0.580788	+	0.814055i	$0.302744\pi$
$354$	0	0
$355$	−791315.	−0.333256
$356$	464710.	0.194338
$357$	0	0
$358$	2.80329e6	1.15601
$359$	2.66436e6	1.09108	0.545540	−	0.838085i	$-0.316325\pi$
$359$	2.66436e6	1.09108	0.545540	+	0.838085i	$0.316325\pi$
$360$	0	0
$361$	−2.32111e6	−0.937407
$362$	2.94324e6	1.18047
$363$	0	0
$364$	1.68197e6	0.665371
$365$	3.32340e6	1.30572
$366$	0	0
$367$	−4.17455e6	−1.61787	−0.808936	−	0.587897i	$-0.799956\pi$
$367$	−4.17455e6	−1.61787	−0.808936	+	0.587897i	$0.799956\pi$
$368$	−260921.	−0.100436
$369$	0	0
$370$	670269.	0.254533
$371$	2.22135e6	0.837879
$372$	0	0
$373$	−2.61953e6	−0.974879	−0.487440	−	0.873157i	$-0.662069\pi$
$373$	−2.61953e6	−0.974879	−0.487440	+	0.873157i	$0.662069\pi$
$374$	3.13297e6	1.15818
$375$	0	0
$376$	−1.23837e6	−0.451732
$377$	192385.	0.0697135
$378$	0	0
$379$	1.25240e6	0.447864	0.223932	−	0.974605i	$-0.428111\pi$
$379$	1.25240e6	0.447864	0.223932	+	0.974605i	$0.428111\pi$
$380$	441999.	0.157023
$381$	0	0
$382$	4.54641e6	1.59408
$383$	−5.09439e6	−1.77458	−0.887289	−	0.461214i	$-0.847414\pi$
$383$	−5.09439e6	−1.77458	−0.887289	+	0.461214i	$0.847414\pi$
$384$	0	0
$385$	−1.11800e6	−0.384405
$386$	−2.78616e6	−0.951783
$387$	0	0
$388$	−2.31719e6	−0.781416
$389$	−2.50585e6	−0.839615	−0.419808	−	0.907613i	$-0.637902\pi$
$389$	−2.50585e6	−0.839615	−0.419808	+	0.907613i	$0.637902\pi$
$390$	0	0
$391$	−466615.	−0.154353
$392$	−148167.	−0.0487008
$393$	0	0
$394$	3.92310e6	1.27318
$395$	168978.	0.0544927
$396$	0	0
$397$	3.27595e6	1.04318	0.521592	−	0.853195i	$-0.325338\pi$
$397$	3.27595e6	1.04318	0.521592	+	0.853195i	$0.325338\pi$
$398$	−1.24588e6	−0.394246
$399$	0	0
$400$	972073.	0.303773
$401$	−6.06788e6	−1.88441	−0.942206	−	0.335033i	$-0.891252\pi$
$401$	−6.06788e6	−1.88441	−0.942206	+	0.335033i	$0.891252\pi$
$402$	0	0
$403$	−718447.	−0.220360
$404$	1.80818e6	0.551173
$405$	0	0
$406$	−287129.	−0.0864494
$407$	358488.	0.107273
$408$	0	0
$409$	−325513.	−0.0962189	−0.0481095	−	0.998842i	$-0.515320\pi$
$409$	−325513.	−0.0962189	−0.0481095	+	0.998842i	$0.515320\pi$
$410$	3.63195e6	1.06704
$411$	0	0
$412$	−1.25072e6	−0.363009
$413$	419607.	0.121051
$414$	0	0
$415$	−4.60636e6	−1.31292
$416$	4.22683e6	1.19752
$417$	0	0
$418$	561855.	0.157284
$419$	−5.62708e6	−1.56584	−0.782921	−	0.622121i	$-0.786271\pi$
$419$	−5.62708e6	−1.56584	−0.782921	+	0.622121i	$0.786271\pi$
$420$	0	0
$421$	1.40296e6	0.385780	0.192890	−	0.981220i	$-0.438214\pi$
$421$	1.40296e6	0.385780	0.192890	+	0.981220i	$0.438214\pi$
$422$	1.90347e6	0.520313
$423$	0	0
$424$	1.19933e6	0.323985
$425$	1.73839e6	0.466848
$426$	0	0
$427$	−1.98122e6	−0.525850
$428$	2.10549e6	0.555576
$429$	0	0
$430$	6.53853e6	1.70533
$431$	1.55865e6	0.404163	0.202082	−	0.979369i	$-0.435229\pi$
$431$	1.55865e6	0.404163	0.202082	+	0.979369i	$0.435229\pi$
$432$	0	0
$433$	2.65215e6	0.679795	0.339898	−	0.940462i	$-0.389608\pi$
$433$	2.65215e6	0.679795	0.339898	+	0.940462i	$0.389608\pi$
$434$	1.07226e6	0.273261
$435$	0	0
$436$	4.80287e6	1.21000
$437$	−83680.9	−0.0209615
$438$	0	0
$439$	2.58237e6	0.639525	0.319763	−	0.947498i	$-0.396397\pi$
$439$	2.58237e6	0.639525	0.319763	+	0.947498i	$0.396397\pi$
$440$	−603620.	−0.148639
$441$	0	0
$442$	9.79479e6	2.38473
$443$	−4.79743e6	−1.16145	−0.580723	−	0.814101i	$-0.697230\pi$
$443$	−4.79743e6	−1.16145	−0.580723	+	0.814101i	$0.697230\pi$
$444$	0	0
$445$	−965704.	−0.231177
$446$	−4.57136e6	−1.08820
$447$	0	0
$448$	−1.57343e6	−0.370384
$449$	−4.66201e6	−1.09133	−0.545667	−	0.838002i	$-0.683724\pi$
$449$	−4.66201e6	−1.09133	−0.545667	+	0.838002i	$0.683724\pi$
$450$	0	0
$451$	1.94252e6	0.449702
$452$	−1.94740e6	−0.448342
$453$	0	0
$454$	7.75728e6	1.76632
$455$	−3.49526e6	−0.791499
$456$	0	0
$457$	−3.52146e6	−0.788738	−0.394369	−	0.918952i	$-0.629037\pi$
$457$	−3.52146e6	−0.788738	−0.394369	+	0.918952i	$0.629037\pi$
$458$	4.51995e6	1.00686
$459$	0	0
$460$	−238644.	−0.0525843
$461$	3.24447e6	0.711037	0.355518	−	0.934669i	$-0.384304\pi$
$461$	3.24447e6	0.711037	0.355518	+	0.934669i	$0.384304\pi$
$462$	0	0
$463$	3.08362e6	0.668511	0.334255	−	0.942483i	$-0.391515\pi$
$463$	3.08362e6	0.668511	0.334255	+	0.942483i	$0.391515\pi$
$464$	−393395.	−0.0848269
$465$	0	0
$466$	7.47076e6	1.59367
$467$	4.71258e6	0.999923	0.499962	−	0.866048i	$-0.333347\pi$
$467$	4.71258e6	0.999923	0.499962	+	0.866048i	$0.333347\pi$
$468$	0	0
$469$	1.25344e6	0.263131
$470$	−6.83121e6	−1.42644
$471$	0	0
$472$	226551.	0.0468070
$473$	3.49709e6	0.718710
$474$	0	0
$475$	311756.	0.0633989
$476$	−6.15070e6	−1.24425
$477$	0	0
$478$	9.12184e6	1.82605
$479$	1.95384e6	0.389089	0.194545	−	0.980894i	$-0.437677\pi$
$479$	1.95384e6	0.389089	0.194545	+	0.980894i	$0.437677\pi$
$480$	0	0
$481$	1.12076e6	0.220877
$482$	−1.00339e7	−1.96721
$483$	0	0
$484$	−2.88643e6	−0.560078
$485$	4.81530e6	0.929542
$486$	0	0
$487$	−3.00669e6	−0.574468	−0.287234	−	0.957860i	$-0.592736\pi$
$487$	−3.00669e6	−0.574468	−0.287234	+	0.957860i	$0.592736\pi$
$488$	−1.06968e6	−0.203332
$489$	0	0
$490$	−817331.	−0.153783
$491$	−600072.	−0.112331	−0.0561655	−	0.998421i	$-0.517887\pi$
$491$	−600072.	−0.112331	−0.0561655	+	0.998421i	$0.517887\pi$
$492$	0	0
$493$	−703522.	−0.130365
$494$	1.75656e6	0.323851
$495$	0	0
$496$	1.46911e6	0.268132
$497$	1.97479e6	0.358616
$498$	0	0
$499$	−9.20598e6	−1.65508	−0.827540	−	0.561407i	$-0.810260\pi$
$499$	−9.20598e6	−1.65508	−0.827540	+	0.561407i	$0.810260\pi$
$500$	4.39759e6	0.786664
$501$	0	0
$502$	3.33357e6	0.590406
$503$	5.84921e6	1.03081	0.515403	−	0.856948i	$-0.327642\pi$
$503$	5.84921e6	1.03081	0.515403	+	0.856948i	$0.327642\pi$
$504$	0	0
$505$	−3.75754e6	−0.655654
$506$	−303357.	−0.0526717
$507$	0	0
$508$	−5.35765e6	−0.921118
$509$	−7.98607e6	−1.36628	−0.683139	−	0.730289i	$-0.739386\pi$
$509$	−7.98607e6	−1.36628	−0.683139	+	0.730289i	$0.739386\pi$
$510$	0	0
$511$	−8.29381e6	−1.40508
$512$	−6.08669e6	−1.02614
$513$	0	0
$514$	4.87796e6	0.814387
$515$	2.59910e6	0.431822
$516$	0	0
$517$	−3.65362e6	−0.601170
$518$	−1.67271e6	−0.273903
$519$	0	0
$520$	−1.88713e6	−0.306051
$521$	2.58297e6	0.416893	0.208446	−	0.978034i	$-0.433159\pi$
$521$	2.58297e6	0.416893	0.208446	+	0.978034i	$0.433159\pi$
$522$	0	0
$523$	−4.44025e6	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$523$	−4.44025e6	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$524$	−4.34125e6	−0.690695
$525$	0	0
$526$	9.32011e6	1.46878
$527$	2.62725e6	0.412074
$528$	0	0
$529$	−6.39116e6	−0.992980
$530$	6.61586e6	1.02305
$531$	0	0
$532$	−1.10304e6	−0.168971
$533$	6.07302e6	0.925948
$534$	0	0
$535$	−4.37537e6	−0.660891
$536$	676748.	0.101745
$537$	0	0
$538$	2.19572e6	0.327056
$539$	−437143.	−0.0648115
$540$	0	0
$541$	4.63492e6	0.680846	0.340423	−	0.940272i	$-0.389430\pi$
$541$	4.63492e6	0.680846	0.340423	+	0.940272i	$0.389430\pi$
$542$	−1.01560e7	−1.48499
$543$	0	0
$544$	−1.54569e7	−2.23936
$545$	−9.98074e6	−1.43937
$546$	0	0
$547$	−1.30458e7	−1.86425	−0.932124	−	0.362138i	$-0.882047\pi$
$547$	−1.30458e7	−1.86425	−0.932124	+	0.362138i	$0.882047\pi$
$548$	−2.59881e6	−0.369677
$549$	0	0
$550$	1.13017e6	0.159308
$551$	−126167.	−0.0177038
$552$	0	0
$553$	−421699.	−0.0586395
$554$	1.34184e6	0.185749
$555$	0	0
$556$	4.54256e6	0.623180
$557$	5.71867e6	0.781011	0.390506	−	0.920600i	$-0.372300\pi$
$557$	5.71867e6	0.781011	0.390506	+	0.920600i	$0.372300\pi$
$558$	0	0
$559$	1.09331e7	1.47984
$560$	7.14723e6	0.963091
$561$	0	0
$562$	5.56570e6	0.743325
$563$	−5.22161e6	−0.694278	−0.347139	−	0.937814i	$-0.612847\pi$
$563$	−5.22161e6	−0.694278	−0.347139	+	0.937814i	$0.612847\pi$
$564$	0	0
$565$	4.04685e6	0.533330
$566$	1.06599e7	1.39865
$567$	0	0
$568$	1.06621e6	0.138667
$569$	845736.	0.109510	0.0547550	−	0.998500i	$-0.482562\pi$
$569$	845736.	0.109510	0.0547550	+	0.998500i	$0.482562\pi$
$570$	0	0
$571$	−4.22254e6	−0.541981	−0.270990	−	0.962582i	$-0.587351\pi$
$571$	−4.22254e6	−0.541981	−0.270990	+	0.962582i	$0.587351\pi$
$572$	2.67925e6	0.342392
$573$	0	0
$574$	−9.06382e6	−1.14824
$575$	−168324.	−0.0212312
$576$	0	0
$577$	−8.12269e6	−1.01569	−0.507844	−	0.861449i	$-0.669557\pi$
$577$	−8.12269e6	−1.01569	−0.507844	+	0.861449i	$0.669557\pi$
$578$	−2.52649e7	−3.14555
$579$	0	0
$580$	−359807.	−0.0444119
$581$	1.14955e7	1.41283
$582$	0	0
$583$	3.53844e6	0.431163
$584$	−4.47793e6	−0.543307
$585$	0	0
$586$	−1.48644e7	−1.78814
$587$	−1.23310e7	−1.47708	−0.738539	−	0.674210i	$-0.764484\pi$
$587$	−1.23310e7	−1.47708	−0.738539	+	0.674210i	$0.764484\pi$
$588$	0	0
$589$	471162.	0.0559605
$590$	1.24972e6	0.147803
$591$	0	0
$592$	−2.29178e6	−0.268762
$593$	1.45287e7	1.69664	0.848320	−	0.529484i	$-0.177614\pi$
$593$	1.45287e7	1.69664	0.848320	+	0.529484i	$0.177614\pi$
$594$	0	0
$595$	1.27816e7	1.48011
$596$	8.63102e6	0.995283
$597$	0	0
$598$	−948402.	−0.108452
$599$	−8.46504e6	−0.963967	−0.481983	−	0.876180i	$-0.660083\pi$
$599$	−8.46504e6	−0.963967	−0.481983	+	0.876180i	$0.660083\pi$
$600$	0	0
$601$	4.80633e6	0.542785	0.271392	−	0.962469i	$-0.412516\pi$
$601$	4.80633e6	0.542785	0.271392	+	0.962469i	$0.412516\pi$
$602$	−1.63174e7	−1.83510
$603$	0	0
$604$	−9.41948e6	−1.05059
$605$	5.99824e6	0.666247
$606$	0	0
$607$	−7.55460e6	−0.832223	−0.416111	−	0.909314i	$-0.636607\pi$
$607$	−7.55460e6	−0.832223	−0.416111	+	0.909314i	$0.636607\pi$
$608$	−2.77198e6	−0.304110
$609$	0	0
$610$	−5.90068e6	−0.642062
$611$	−1.14225e7	−1.23782
$612$	0	0
$613$	1.78145e7	1.91480	0.957399	−	0.288768i	$-0.0932456\pi$
$613$	1.78145e7	1.91480	0.957399	+	0.288768i	$0.0932456\pi$
$614$	−1.97769e7	−2.11708
$615$	0	0
$616$	1.50638e6	0.159950
$617$	−1.13067e6	−0.119570	−0.0597851	−	0.998211i	$-0.519042\pi$
$617$	−1.13067e6	−0.119570	−0.0597851	+	0.998211i	$0.519042\pi$
$618$	0	0
$619$	39879.3	0.00418332	0.00209166	−	0.999998i	$-0.499334\pi$
$619$	39879.3	0.00418332	0.00209166	+	0.999998i	$0.499334\pi$
$620$	1.34368e6	0.140383
$621$	0	0
$622$	−1.57308e7	−1.63032
$623$	2.40999e6	0.248768
$624$	0	0
$625$	−6.66386e6	−0.682379
$626$	−5.39511e6	−0.550256
$627$	0	0
$628$	2.17472e6	0.220041
$629$	−4.09846e6	−0.413042
$630$	0	0
$631$	7.70574e6	0.770443	0.385222	−	0.922824i	$-0.374125\pi$
$631$	7.70574e6	0.770443	0.385222	+	0.922824i	$0.374125\pi$
$632$	−227681.	−0.0226743
$633$	0	0
$634$	−8.96539e6	−0.885821
$635$	1.11336e7	1.09573
$636$	0	0
$637$	−1.36667e6	−0.133449
$638$	−457376.	−0.0444858
$639$	0	0
$640$	6.19709e6	0.598050
$641$	−1.60378e7	−1.54170	−0.770848	−	0.637019i	$-0.780167\pi$
$641$	−1.60378e7	−1.54170	−0.770848	+	0.637019i	$0.780167\pi$
$642$	0	0
$643$	2.62458e6	0.250341	0.125170	−	0.992135i	$-0.460052\pi$
$643$	2.62458e6	0.250341	0.125170	+	0.992135i	$0.460052\pi$
$644$	595555.	0.0565858
$645$	0	0
$646$	−6.42348e6	−0.605605
$647$	−1.06120e6	−0.0996634	−0.0498317	−	0.998758i	$-0.515868\pi$
$647$	−1.06120e6	−0.0996634	−0.0498317	+	0.998758i	$0.515868\pi$
$648$	0	0
$649$	668404.	0.0622912
$650$	3.53331e6	0.328018
$651$	0	0
$652$	−7.31852e6	−0.674224
$653$	1.18303e7	1.08571	0.542854	−	0.839827i	$-0.317344\pi$
$653$	1.18303e7	1.08571	0.542854	+	0.839827i	$0.317344\pi$
$654$	0	0
$655$	9.02145e6	0.821624
$656$	−1.24183e7	−1.12669
$657$	0	0
$658$	1.70478e7	1.53499
$659$	−3.08114e6	−0.276375	−0.138187	−	0.990406i	$-0.544128\pi$
$659$	−3.08114e6	−0.276375	−0.138187	+	0.990406i	$0.544128\pi$
$660$	0	0
$661$	−1.52738e7	−1.35970	−0.679850	−	0.733351i	$-0.737955\pi$
$661$	−1.52738e7	−1.35970	−0.679850	+	0.733351i	$0.737955\pi$
$662$	8.38318e6	0.743471
$663$	0	0
$664$	6.20658e6	0.546301
$665$	2.29221e6	0.201002
$666$	0	0
$667$	68120.0	0.00592871
$668$	−1.90727e6	−0.165376
$669$	0	0
$670$	3.73314e6	0.321282
$671$	−3.15593e6	−0.270596
$672$	0	0
$673$	1.59289e6	0.135565	0.0677824	−	0.997700i	$-0.478408\pi$
$673$	1.59289e6	0.135565	0.0677824	+	0.997700i	$0.478408\pi$
$674$	1.87056e7	1.58607
$675$	0	0
$676$	−253934.	−0.0213724
$677$	1.45086e7	1.21662	0.608308	−	0.793701i	$-0.291849\pi$
$677$	1.45086e7	1.21662	0.608308	+	0.793701i	$0.291849\pi$
$678$	0	0
$679$	−1.20170e7	−1.00028
$680$	6.90096e6	0.572318
$681$	0	0
$682$	1.70804e6	0.140617
$683$	−1.08149e7	−0.887094	−0.443547	−	0.896251i	$-0.646280\pi$
$683$	−1.08149e7	−0.887094	−0.443547	+	0.896251i	$0.646280\pi$
$684$	0	0
$685$	5.40053e6	0.439754
$686$	1.70978e7	1.38717
$687$	0	0
$688$	−2.23565e7	−1.80066
$689$	1.10624e7	0.887775
$690$	0	0
$691$	1.24968e7	0.995639	0.497820	−	0.867281i	$-0.334134\pi$
$691$	1.24968e7	0.995639	0.497820	+	0.867281i	$0.334134\pi$
$692$	3.23043e6	0.256446
$693$	0	0
$694$	2.95751e7	2.33092
$695$	−9.43978e6	−0.741310
$696$	0	0
$697$	−2.22081e7	−1.73153
$698$	8.63759e6	0.671049
$699$	0	0
$700$	−2.21876e6	−0.171146
$701$	4.63586e6	0.356316	0.178158	−	0.984002i	$-0.442986\pi$
$701$	4.63586e6	0.356316	0.178158	+	0.984002i	$0.442986\pi$
$702$	0	0
$703$	−735003.	−0.0560920
$704$	−2.50636e6	−0.190595
$705$	0	0
$706$	−2.02128e7	−1.52621
$707$	9.37723e6	0.705547
$708$	0	0
$709$	−1.99355e7	−1.48940	−0.744699	−	0.667401i	$-0.767407\pi$
$709$	−1.99355e7	−1.48940	−0.744699	+	0.667401i	$0.767407\pi$
$710$	5.88153e6	0.437870
$711$	0	0
$712$	1.30118e6	0.0961919
$713$	−254390.	−0.0187403
$714$	0	0
$715$	−5.56769e6	−0.407296
$716$	−8.76660e6	−0.639070
$717$	0	0
$718$	−1.98031e7	−1.43358
$719$	−4.81963e6	−0.347689	−0.173845	−	0.984773i	$-0.555619\pi$
$719$	−4.81963e6	−0.347689	−0.173845	+	0.984773i	$0.555619\pi$
$720$	0	0
$721$	−6.48625e6	−0.464682
$722$	1.72519e7	1.23167
$723$	0	0
$724$	−9.20428e6	−0.652595
$725$	−253784.	−0.0179316
$726$	0	0
$727$	3.77212e6	0.264697	0.132349	−	0.991203i	$-0.457748\pi$
$727$	3.77212e6	0.264697	0.132349	+	0.991203i	$0.457748\pi$
$728$	4.70949e6	0.329340
$729$	0	0
$730$	−2.47016e7	−1.71560
$731$	−3.99809e7	−2.76732
$732$	0	0
$733$	−8.67096e6	−0.596084	−0.298042	−	0.954553i	$-0.596333\pi$
$733$	−8.67096e6	−0.596084	−0.298042	+	0.954553i	$0.596333\pi$
$734$	3.10278e7	2.12574
$735$	0	0
$736$	1.49665e6	0.101841
$737$	1.99664e6	0.135404
$738$	0	0
$739$	2.67260e7	1.80021	0.900103	−	0.435678i	$-0.143491\pi$
$739$	2.67260e7	1.80021	0.900103	+	0.435678i	$0.143491\pi$
$740$	−2.09611e6	−0.140713
$741$	0	0
$742$	−1.65104e7	−1.10090
$743$	−882576.	−0.0586516	−0.0293258	−	0.999570i	$-0.509336\pi$
$743$	−882576.	−0.0586516	−0.0293258	+	0.999570i	$0.509336\pi$
$744$	0	0
$745$	−1.79359e7	−1.18395
$746$	1.94699e7	1.28091
$747$	0	0
$748$	−9.79762e6	−0.640275
$749$	1.09191e7	0.711183
$750$	0	0
$751$	1.50079e7	0.971005	0.485502	−	0.874235i	$-0.338637\pi$
$751$	1.50079e7	0.971005	0.485502	+	0.874235i	$0.338637\pi$
$752$	2.33572e7	1.50618
$753$	0	0
$754$	−1.42992e6	−0.0915974
$755$	1.95744e7	1.24974
$756$	0	0
$757$	−2.73466e7	−1.73446	−0.867228	−	0.497910i	$-0.834101\pi$
$757$	−2.73466e7	−1.73446	−0.867228	+	0.497910i	$0.834101\pi$
$758$	−9.30862e6	−0.588454
$759$	0	0
$760$	1.23759e6	0.0777219
$761$	−1.85952e7	−1.16397	−0.581983	−	0.813201i	$-0.697723\pi$
$761$	−1.85952e7	−1.16397	−0.581983	+	0.813201i	$0.697723\pi$
$762$	0	0
$763$	2.49077e7	1.54890
$764$	−1.42178e7	−0.881250
$765$	0	0
$766$	3.78646e7	2.33164
$767$	2.08967e6	0.128259
$768$	0	0
$769$	1.69589e6	0.103414	0.0517072	−	0.998662i	$-0.483534\pi$
$769$	1.69589e6	0.103414	0.0517072	+	0.998662i	$0.483534\pi$
$770$	8.30964e6	0.505074
$771$	0	0
$772$	8.71305e6	0.526171
$773$	8.47019e6	0.509853	0.254926	−	0.966960i	$-0.417949\pi$
$773$	8.47019e6	0.509853	0.254926	+	0.966960i	$0.417949\pi$
$774$	0	0
$775$	947739.	0.0566806
$776$	−6.48810e6	−0.386780
$777$	0	0
$778$	1.86250e7	1.10318
$779$	−3.98272e6	−0.235145
$780$	0	0
$781$	3.14570e6	0.184539
$782$	3.46817e6	0.202807
$783$	0	0
$784$	2.79461e6	0.162379
$785$	−4.51923e6	−0.261752
$786$	0	0
$787$	−7.45024e6	−0.428779	−0.214389	−	0.976748i	$-0.568776\pi$
$787$	−7.45024e6	−0.428779	−0.214389	+	0.976748i	$0.568776\pi$
$788$	−1.22686e7	−0.703846
$789$	0	0
$790$	−1.25595e6	−0.0715987
$791$	−1.00992e7	−0.573915
$792$	0	0
$793$	−9.86658e6	−0.557165
$794$	−2.43489e7	−1.37065
$795$	0	0
$796$	3.89618e6	0.217950
$797$	−1.44140e7	−0.803783	−0.401892	−	0.915687i	$-0.631647\pi$
$797$	−1.44140e7	−0.803783	−0.401892	+	0.915687i	$0.631647\pi$
$798$	0	0
$799$	4.17705e7	2.31474
$800$	−5.57582e6	−0.308023
$801$	0	0
$802$	4.51002e7	2.47595
$803$	−1.32114e7	−0.723039
$804$	0	0
$805$	−1.23761e6	−0.0673122
$806$	5.33994e6	0.289533
$807$	0	0
$808$	5.06288e6	0.272816
$809$	2.19181e7	1.17742	0.588710	−	0.808344i	$-0.299636\pi$
$809$	2.19181e7	1.17742	0.588710	+	0.808344i	$0.299636\pi$
$810$	0	0
$811$	−1.90616e6	−0.101767	−0.0508836	−	0.998705i	$-0.516204\pi$
$811$	−1.90616e6	−0.101767	−0.0508836	+	0.998705i	$0.516204\pi$
$812$	897927.	0.0477915
$813$	0	0
$814$	−2.66451e6	−0.140947
$815$	1.52085e7	0.802031
$816$	0	0
$817$	−7.17002e6	−0.375807
$818$	2.41941e6	0.126423
$819$	0	0
$820$	−1.13581e7	−0.589888
$821$	5.58709e6	0.289286	0.144643	−	0.989484i	$-0.453797\pi$
$821$	5.58709e6	0.289286	0.144643	+	0.989484i	$0.453797\pi$
$822$	0	0
$823$	7.03605e6	0.362101	0.181050	−	0.983474i	$-0.442050\pi$
$823$	7.03605e6	0.362101	0.181050	+	0.983474i	$0.442050\pi$
$824$	−3.50201e6	−0.179680
$825$	0	0
$826$	−3.11877e6	−0.159050
$827$	1.36355e7	0.693276	0.346638	−	0.937999i	$-0.387323\pi$
$827$	1.36355e7	0.693276	0.346638	+	0.937999i	$0.387323\pi$
$828$	0	0
$829$	8.78197e6	0.443819	0.221909	−	0.975067i	$-0.428771\pi$
$829$	8.78197e6	0.443819	0.221909	+	0.975067i	$0.428771\pi$
$830$	3.42373e7	1.72506
$831$	0	0
$832$	−7.83578e6	−0.392440
$833$	4.99770e6	0.249550
$834$	0	0
$835$	3.96346e6	0.196725
$836$	−1.75707e6	−0.0869507
$837$	0	0
$838$	4.18239e7	2.05738
$839$	−2.47541e7	−1.21407	−0.607034	−	0.794676i	$-0.707641\pi$
$839$	−2.47541e7	−1.21407	−0.607034	+	0.794676i	$0.707641\pi$
$840$	0	0
$841$	−2.04084e7	−0.994993
$842$	−1.04277e7	−0.506882
$843$	0	0
$844$	−5.95264e6	−0.287643
$845$	527694.	0.0254238
$846$	0	0
$847$	−1.49691e7	−0.716946
$848$	−2.26209e7	−1.08024
$849$	0	0
$850$	−1.29208e7	−0.613397
$851$	396843.	0.0187843
$852$	0	0
$853$	3.15345e7	1.48393	0.741964	−	0.670439i	$-0.233894\pi$
$853$	3.15345e7	1.48393	0.741964	+	0.670439i	$0.233894\pi$
$854$	1.47256e7	0.690921
$855$	0	0
$856$	5.89534e6	0.274995
$857$	3.48098e7	1.61901	0.809505	−	0.587113i	$-0.199735\pi$
$857$	3.48098e7	1.61901	0.809505	+	0.587113i	$0.199735\pi$
$858$	0	0
$859$	−3.28394e7	−1.51849	−0.759246	−	0.650803i	$-0.774432\pi$
$859$	−3.28394e7	−1.51849	−0.759246	+	0.650803i	$0.774432\pi$
$860$	−2.04477e7	−0.942754
$861$	0	0
$862$	−1.15849e7	−0.531035
$863$	−2.13743e7	−0.976935	−0.488468	−	0.872582i	$-0.662444\pi$
$863$	−2.13743e7	−0.976935	−0.488468	+	0.872582i	$0.662444\pi$
$864$	0	0
$865$	−6.71309e6	−0.305058
$866$	−1.97124e7	−0.893191
$867$	0	0
$868$	−3.35325e6	−0.151066
$869$	−671736.	−0.0301752
$870$	0	0
$871$	6.24221e6	0.278800
$872$	1.34480e7	0.598916
$873$	0	0
$874$	621967.	0.0275416
$875$	2.28059e7	1.00700
$876$	0	0
$877$	1.54465e7	0.678158	0.339079	−	0.940758i	$-0.389885\pi$
$877$	1.54465e7	0.678158	0.339079	+	0.940758i	$0.389885\pi$
$878$	−1.91938e7	−0.840280
$879$	0	0
$880$	1.13850e7	0.495595
$881$	−4.43148e6	−0.192357	−0.0961787	−	0.995364i	$-0.530662\pi$
$881$	−4.43148e6	−0.192357	−0.0961787	+	0.995364i	$0.530662\pi$
$882$	0	0
$883$	6.63288e6	0.286286	0.143143	−	0.989702i	$-0.454279\pi$
$883$	6.63288e6	0.286286	0.143143	+	0.989702i	$0.454279\pi$
$884$	−3.06309e7	−1.31834
$885$	0	0
$886$	3.56574e7	1.52604
$887$	1.88116e7	0.802818	0.401409	−	0.915899i	$-0.368521\pi$
$887$	1.88116e7	0.802818	0.401409	+	0.915899i	$0.368521\pi$
$888$	0	0
$889$	−2.77848e7	−1.17911
$890$	7.17770e6	0.303746
$891$	0	0
$892$	1.42958e7	0.601585
$893$	7.49096e6	0.314347
$894$	0	0
$895$	1.82177e7	0.760213
$896$	−1.54653e7	−0.643560
$897$	0	0
$898$	3.46509e7	1.43392
$899$	−383547.	−0.0158278
$900$	0	0
$901$	−4.04537e7	−1.66015
$902$	−1.44380e7	−0.590869
$903$	0	0
$904$	−5.45270e6	−0.221917
$905$	1.91272e7	0.776301
$906$	0	0
$907$	8.97079e6	0.362087	0.181043	−	0.983475i	$-0.442053\pi$
$907$	8.97079e6	0.362087	0.181043	+	0.983475i	$0.442053\pi$
$908$	−2.42590e7	−0.976470
$909$	0	0
$910$	2.59789e7	1.03996
$911$	−4.68204e7	−1.86913	−0.934564	−	0.355794i	$-0.884211\pi$
$911$	−4.68204e7	−1.86913	−0.934564	+	0.355794i	$0.884211\pi$
$912$	0	0
$913$	1.83116e7	0.727024
$914$	2.61737e7	1.03633
$915$	0	0
$916$	−1.41351e7	−0.556621
$917$	−2.25138e7	−0.884147
$918$	0	0
$919$	−2.58264e7	−1.00873	−0.504366	−	0.863490i	$-0.668274\pi$
$919$	−2.58264e7	−1.00873	−0.504366	+	0.863490i	$0.668274\pi$
$920$	−668200.	−0.0260278
$921$	0	0
$922$	−2.41149e7	−0.934240
$923$	9.83457e6	0.379971
$924$	0	0
$925$	−1.47845e6	−0.0568137
$926$	−2.29193e7	−0.878364
$927$	0	0
$928$	2.25652e6	0.0860139
$929$	−4.55842e7	−1.73291	−0.866453	−	0.499259i	$-0.833606\pi$
$929$	−4.55842e7	−1.73291	−0.866453	+	0.499259i	$0.833606\pi$
$930$	0	0
$931$	896268.	0.0338894
$932$	−2.33630e7	−0.881026
$933$	0	0
$934$	−3.50268e7	−1.31381
$935$	2.03602e7	0.761647
$936$	0	0
$937$	1.83075e6	0.0681208	0.0340604	−	0.999420i	$-0.489156\pi$
$937$	1.83075e6	0.0681208	0.0340604	+	0.999420i	$0.489156\pi$
$938$	−9.31634e6	−0.345731
$939$	0	0
$940$	2.13630e7	0.788573
$941$	−6.42585e6	−0.236568	−0.118284	−	0.992980i	$-0.537739\pi$
$941$	−6.42585e6	−0.236568	−0.118284	+	0.992980i	$0.537739\pi$
$942$	0	0
$943$	2.15035e6	0.0787463
$944$	−4.27303e6	−0.156065
$945$	0	0
$946$	−2.59925e7	−0.944322
$947$	2.39260e7	0.866953	0.433477	−	0.901165i	$-0.357287\pi$
$947$	2.39260e7	0.866953	0.433477	+	0.901165i	$0.357287\pi$
$948$	0	0
$949$	−4.13037e7	−1.48876
$950$	−2.31716e6	−0.0833006
$951$	0	0
$952$	−1.72219e7	−0.615869
$953$	−3.98437e7	−1.42111	−0.710554	−	0.703643i	$-0.751555\pi$
$953$	−3.98437e7	−1.42111	−0.710554	+	0.703643i	$0.751555\pi$
$954$	0	0
$955$	2.95457e7	1.04830
$956$	−2.85264e7	−1.00949
$957$	0	0
$958$	−1.45221e7	−0.511230
$959$	−1.34774e7	−0.473218
$960$	0	0
$961$	−2.71968e7	−0.949970
$962$	−8.33019e6	−0.290213
$963$	0	0
$964$	3.13785e7	1.08752
$965$	−1.81064e7	−0.625913
$966$	0	0
$967$	4.45202e7	1.53105	0.765527	−	0.643403i	$-0.222478\pi$
$967$	4.45202e7	1.53105	0.765527	+	0.643403i	$0.222478\pi$
$968$	−8.08198e6	−0.277223
$969$	0	0
$970$	−3.57903e7	−1.22134
$971$	−4.02063e7	−1.36850	−0.684252	−	0.729245i	$-0.739871\pi$
$971$	−4.02063e7	−1.36850	−0.684252	+	0.729245i	$0.739871\pi$
$972$	0	0
$973$	2.35577e7	0.797722
$974$	2.23475e7	0.754801
$975$	0	0
$976$	2.01755e7	0.677954
$977$	1.46923e7	0.492440	0.246220	−	0.969214i	$-0.420811\pi$
$977$	1.46923e7	0.492440	0.246220	+	0.969214i	$0.420811\pi$
$978$	0	0
$979$	3.83894e6	0.128013
$980$	2.55601e6	0.0850152
$981$	0	0
$982$	4.46010e6	0.147593
$983$	550436.	0.0181687	0.00908433	−	0.999959i	$-0.497108\pi$
$983$	550436.	0.0181687	0.00908433	+	0.999959i	$0.497108\pi$
$984$	0	0
$985$	2.54950e7	0.837268
$986$	5.22900e6	0.171288
$987$	0	0
$988$	−5.49322e6	−0.179034
$989$	3.87124e6	0.125852
$990$	0	0
$991$	−3.24918e7	−1.05097	−0.525485	−	0.850803i	$-0.676116\pi$
$991$	−3.24918e7	−1.05097	−0.525485	+	0.850803i	$0.676116\pi$
$992$	−8.42681e6	−0.271884
$993$	0	0
$994$	−1.46778e7	−0.471190
$995$	−8.09657e6	−0.259265
$996$	0	0
$997$	619302.	0.0197317	0.00986586	−	0.999951i	$-0.496860\pi$
$997$	619302.	0.0197317	0.00986586	+	0.999951i	$0.496860\pi$
$998$	6.84245e7	2.17463
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.c.1.4		12
3.2	odd	2		177.6.a.c.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.c.1.9	✓	12	3.2	odd	2
531.6.a.c.1.4		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)

\(f(q)\)	\(=\)	\(q-7.43261 q^{2} +23.2437 q^{4} -48.3022 q^{5} +120.542 q^{7} +65.0821 q^{8} +O(q^{10})\)	\(q-7.43261 q^{2} +23.2437 q^{4} -48.3022 q^{5} +120.542 q^{7} +65.0821 q^{8} +359.012 q^{10} +192.015 q^{11} +600.307 q^{13} -895.942 q^{14} -1227.53 q^{16} -2195.23 q^{17} -393.685 q^{19} -1122.72 q^{20} -1427.17 q^{22} +212.558 q^{23} -791.894 q^{25} -4461.85 q^{26} +2801.84 q^{28} +320.477 q^{29} -1196.80 q^{31} +7041.12 q^{32} +16316.3 q^{34} -5822.45 q^{35} +1866.98 q^{37} +2926.10 q^{38} -3143.61 q^{40} +10116.5 q^{41} +18212.6 q^{43} +4463.14 q^{44} -1579.86 q^{46} -19027.8 q^{47} -2276.61 q^{49} +5885.84 q^{50} +13953.4 q^{52} +18428.0 q^{53} -9274.74 q^{55} +7845.13 q^{56} -2381.98 q^{58} +3481.00 q^{59} -16435.9 q^{61} +8895.35 q^{62} -13053.0 q^{64} -28996.2 q^{65} +10398.4 q^{67} -51025.3 q^{68} +43276.0 q^{70} +16382.6 q^{71} -68804.3 q^{73} -13876.6 q^{74} -9150.69 q^{76} +23145.9 q^{77} -3498.36 q^{79} +59292.4 q^{80} -75192.2 q^{82} +95365.4 q^{83} +106035. q^{85} -135367. q^{86} +12496.7 q^{88} +19992.9 q^{89} +72362.2 q^{91} +4940.64 q^{92} +141426. q^{94} +19015.8 q^{95} -99691.0 q^{97} +16921.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) \) 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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