LMFDB - Embedded newform 387.6.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,6,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, 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("387.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	387.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:	$62.0685382676$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 $ x^10 - 2x^9 - 256x^8 + 266x^7 + 21986x^6 - 10450x^5 - 719484x^4 + 384582x^3 + 8437093x^2 - 5752252*x - 22734604
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$5.31531$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.31531 q^{2} -13.3781 q^{4} +52.5837 q^{5} -174.859 q^{7} -195.821 q^{8} +O(q^{10})$	\(q+4.31531 q^{2} -13.3781 q^{4} +52.5837 q^{5} -174.859 q^{7} -195.821 q^{8} +226.915 q^{10} +447.981 q^{11} +669.141 q^{13} -754.569 q^{14} -416.925 q^{16} +849.648 q^{17} -1288.87 q^{19} -703.471 q^{20} +1933.18 q^{22} +378.254 q^{23} -359.956 q^{25} +2887.55 q^{26} +2339.28 q^{28} -765.100 q^{29} -7094.59 q^{31} +4467.10 q^{32} +3666.49 q^{34} -9194.71 q^{35} -7908.22 q^{37} -5561.87 q^{38} -10297.0 q^{40} -12855.9 q^{41} +1849.00 q^{43} -5993.15 q^{44} +1632.28 q^{46} -26785.7 q^{47} +13768.5 q^{49} -1553.32 q^{50} -8951.86 q^{52} -30000.7 q^{53} +23556.5 q^{55} +34240.9 q^{56} -3301.64 q^{58} -1247.64 q^{59} -48441.4 q^{61} -30615.3 q^{62} +32618.5 q^{64} +35185.9 q^{65} +67004.8 q^{67} -11366.7 q^{68} -39678.0 q^{70} +74553.1 q^{71} -18066.8 q^{73} -34126.4 q^{74} +17242.7 q^{76} -78333.4 q^{77} -63230.5 q^{79} -21923.5 q^{80} -55477.3 q^{82} +88066.7 q^{83} +44677.6 q^{85} +7979.00 q^{86} -87724.0 q^{88} -50373.5 q^{89} -117005. q^{91} -5060.33 q^{92} -115588. q^{94} -67773.5 q^{95} +17502.1 q^{97} +59415.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8}+O(q^{10}) $ 10 * q - 8 * q^2 + 202 * q^4 - 138 * q^5 + 60 * q^7 - 294 * q^8	$ 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8} - 17 q^{10} - 745 q^{11} + 1917 q^{13} - 1936 q^{14} + 5354 q^{16} - 4017 q^{17} - 2404 q^{19} - 1311 q^{20} - 5836 q^{22} - 1733 q^{23} + 7120 q^{25} + 1484 q^{26} - 15028 q^{28} - 6996 q^{29} - 4899 q^{31} + 7554 q^{32} - 27033 q^{34} - 7084 q^{35} + 1466 q^{37} - 13905 q^{38} - 93211 q^{40} - 10297 q^{41} + 18490 q^{43} + 36140 q^{44} + 17991 q^{46} - 48592 q^{47} + 29458 q^{49} - 983 q^{50} + 14232 q^{52} - 127165 q^{53} + 106672 q^{55} + 7780 q^{56} - 10305 q^{58} - 99372 q^{59} + 17408 q^{61} - 28265 q^{62} + 47202 q^{64} - 54484 q^{65} - 2021 q^{67} - 192151 q^{68} - 33194 q^{70} - 11286 q^{71} + 49892 q^{73} + 125431 q^{74} - 249803 q^{76} - 98144 q^{77} - 91524 q^{79} - 12251 q^{80} - 158909 q^{82} + 105203 q^{83} - 87212 q^{85} - 14792 q^{86} - 461824 q^{88} + 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 7259 q^{94} + 305340 q^{95} + 108383 q^{97} - 354656 q^{98}+O(q^{100}) $ 10 * q - 8 * q^2 + 202 * q^4 - 138 * q^5 + 60 * q^7 - 294 * q^8 - 17 * q^10 - 745 * q^11 + 1917 * q^13 - 1936 * q^14 + 5354 * q^16 - 4017 * q^17 - 2404 * q^19 - 1311 * q^20 - 5836 * q^22 - 1733 * q^23 + 7120 * q^25 + 1484 * q^26 - 15028 * q^28 - 6996 * q^29 - 4899 * q^31 + 7554 * q^32 - 27033 * q^34 - 7084 * q^35 + 1466 * q^37 - 13905 * q^38 - 93211 * q^40 - 10297 * q^41 + 18490 * q^43 + 36140 * q^44 + 17991 * q^46 - 48592 * q^47 + 29458 * q^49 - 983 * q^50 + 14232 * q^52 - 127165 * q^53 + 106672 * q^55 + 7780 * q^56 - 10305 * q^58 - 99372 * q^59 + 17408 * q^61 - 28265 * q^62 + 47202 * q^64 - 54484 * q^65 - 2021 * q^67 - 192151 * q^68 - 33194 * q^70 - 11286 * q^71 + 49892 * q^73 + 125431 * q^74 - 249803 * q^76 - 98144 * q^77 - 91524 * q^79 - 12251 * q^80 - 158909 * q^82 + 105203 * q^83 - 87212 * q^85 - 14792 * q^86 - 461824 * q^88 + 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 7259 * q^94 + 305340 * q^95 + 108383 * q^97 - 354656 * 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 $

$2$	4.31531	0.762846	0.381423	−	0.924401i	$-0.375434\pi$
$2$	4.31531	0.762846	0.381423	+	0.924401i	$0.375434\pi$
$3$	0	0
$3$	0	0
$4$	−13.3781	−0.418067
$5$	52.5837	0.940646	0.470323	−	0.882494i	$-0.344137\pi$
$5$	52.5837	0.940646	0.470323	+	0.882494i	$0.344137\pi$
$6$	0	0
$7$	−174.859	−1.34878	−0.674391	−	0.738374i	$-0.735594\pi$
$7$	−174.859	−1.34878	−0.674391	+	0.738374i	$0.735594\pi$
$8$	−195.821	−1.08177
$9$	0	0
$10$	226.915	0.717567
$11$	447.981	1.11629	0.558147	−	0.829742i	$-0.311513\pi$
$11$	447.981	1.11629	0.558147	+	0.829742i	$0.311513\pi$
$12$	0	0
$13$	669.141	1.09814	0.549072	−	0.835775i	$-0.314981\pi$
$13$	669.141	1.09814	0.549072	+	0.835775i	$0.314981\pi$
$14$	−754.569	−1.02891
$15$	0	0
$16$	−416.925	−0.407154
$17$	849.648	0.713045	0.356522	−	0.934287i	$-0.383962\pi$
$17$	849.648	0.713045	0.356522	+	0.934287i	$0.383962\pi$
$18$	0	0
$19$	−1288.87	−0.819078	−0.409539	−	0.912293i	$-0.634310\pi$
$19$	−1288.87	−0.819078	−0.409539	+	0.912293i	$0.634310\pi$
$20$	−703.471	−0.393253
$21$	0	0
$22$	1933.18	0.851559
$23$	378.254	0.149095	0.0745476	−	0.997217i	$-0.476249\pi$
$23$	378.254	0.149095	0.0745476	+	0.997217i	$0.476249\pi$
$24$	0	0
$25$	−359.956	−0.115186
$26$	2887.55	0.837715
$27$	0	0
$28$	2339.28	0.563881
$29$	−765.100	−0.168936	−0.0844682	−	0.996426i	$-0.526919\pi$
$29$	−765.100	−0.168936	−0.0844682	+	0.996426i	$0.526919\pi$
$30$	0	0
$31$	−7094.59	−1.32594	−0.662969	−	0.748647i	$-0.730704\pi$
$31$	−7094.59	−1.32594	−0.662969	+	0.748647i	$0.730704\pi$
$32$	4467.10	0.771170
$33$	0	0
$34$	3666.49	0.543943
$35$	−9194.71	−1.26873
$36$	0	0
$37$	−7908.22	−0.949674	−0.474837	−	0.880074i	$-0.657493\pi$
$37$	−7908.22	−0.949674	−0.474837	+	0.880074i	$0.657493\pi$
$38$	−5561.87	−0.624830
$39$	0	0
$40$	−10297.0	−1.01756
$41$	−12855.9	−1.19438	−0.597192	−	0.802098i	$-0.703717\pi$
$41$	−12855.9	−1.19438	−0.597192	+	0.802098i	$0.703717\pi$
$42$	0	0
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	−5993.15	−0.466685
$45$	0	0
$46$	1632.28	0.113737
$47$	−26785.7	−1.76871	−0.884357	−	0.466811i	$-0.845403\pi$
$47$	−26785.7	−1.76871	−0.884357	+	0.466811i	$0.845403\pi$
$48$	0	0
$49$	13768.5	0.819215
$50$	−1553.32	−0.0878690
$51$	0	0
$52$	−8951.86	−0.459098
$53$	−30000.7	−1.46704	−0.733520	−	0.679668i	$-0.762124\pi$
$53$	−30000.7	−1.46704	−0.733520	+	0.679668i	$0.762124\pi$
$54$	0	0
$55$	23556.5	1.05004
$56$	34240.9	1.45907
$57$	0	0
$58$	−3301.64	−0.128872
$59$	−1247.64	−0.0466614	−0.0233307	−	0.999728i	$-0.507427\pi$
$59$	−1247.64	−0.0466614	−0.0233307	+	0.999728i	$0.507427\pi$
$60$	0	0
$61$	−48441.4	−1.66683	−0.833417	−	0.552644i	$-0.813619\pi$
$61$	−48441.4	−1.66683	−0.833417	+	0.552644i	$0.813619\pi$
$62$	−30615.3	−1.01149
$63$	0	0
$64$	32618.5	0.995438
$65$	35185.9	1.03296
$66$	0	0
$67$	67004.8	1.82356	0.911778	−	0.410683i	$-0.134710\pi$
$67$	67004.8	1.82356	0.911778	+	0.410683i	$0.134710\pi$
$68$	−11366.7	−0.298100
$69$	0	0
$70$	−39678.0	−0.967842
$71$	74553.1	1.75517	0.877587	−	0.479418i	$-0.159152\pi$
$71$	74553.1	1.75517	0.877587	+	0.479418i	$0.159152\pi$
$72$	0	0
$73$	−18066.8	−0.396802	−0.198401	−	0.980121i	$-0.563575\pi$
$73$	−18066.8	−0.396802	−0.198401	+	0.980121i	$0.563575\pi$
$74$	−34126.4	−0.724454
$75$	0	0
$76$	17242.7	0.342429
$77$	−78333.4	−1.50564
$78$	0	0
$79$	−63230.5	−1.13988	−0.569939	−	0.821687i	$-0.693033\pi$
$79$	−63230.5	−1.13988	−0.569939	+	0.821687i	$0.693033\pi$
$80$	−21923.5	−0.382987
$81$	0	0
$82$	−55477.3	−0.911131
$83$	88066.7	1.40319	0.701595	−	0.712576i	$-0.252472\pi$
$83$	88066.7	1.40319	0.701595	+	0.712576i	$0.252472\pi$
$84$	0	0
$85$	44677.6	0.670723
$86$	7979.00	0.116333
$87$	0	0
$88$	−87724.0	−1.20757
$89$	−50373.5	−0.674104	−0.337052	−	0.941486i	$-0.609430\pi$
$89$	−50373.5	−0.674104	−0.337052	+	0.941486i	$0.609430\pi$
$90$	0	0
$91$	−117005.	−1.48116
$92$	−5060.33	−0.0623317
$93$	0	0
$94$	−115588.	−1.34926
$95$	−67773.5	−0.770462
$96$	0	0
$97$	17502.1	0.188869	0.0944345	−	0.995531i	$-0.469896\pi$
$97$	17502.1	0.188869	0.0944345	+	0.995531i	$0.469896\pi$
$98$	59415.5	0.624934
$99$	0	0
$100$	4815.54	0.0481554
$101$	−182669.	−1.78181	−0.890906	−	0.454188i	$-0.849929\pi$
$101$	−182669.	−1.78181	−0.890906	+	0.454188i	$0.849929\pi$
$102$	0	0
$103$	−110578.	−1.02702	−0.513508	−	0.858085i	$-0.671654\pi$
$103$	−110578.	−1.02702	−0.513508	+	0.858085i	$0.671654\pi$
$104$	−131032.	−1.18794
$105$	0	0
$106$	−129462.	−1.11913
$107$	43935.4	0.370984	0.185492	−	0.982646i	$-0.440612\pi$
$107$	43935.4	0.370984	0.185492	+	0.982646i	$0.440612\pi$
$108$	0	0
$109$	−78170.4	−0.630197	−0.315098	−	0.949059i	$-0.602038\pi$
$109$	−78170.4	−0.630197	−0.315098	+	0.949059i	$0.602038\pi$
$110$	101654.	0.801015
$111$	0	0
$112$	72903.0	0.549162
$113$	−23097.5	−0.170165	−0.0850823	−	0.996374i	$-0.527115\pi$
$113$	−23097.5	−0.170165	−0.0850823	+	0.996374i	$0.527115\pi$
$114$	0	0
$115$	19890.0	0.140246
$116$	10235.6	0.0706267
$117$	0	0
$118$	−5383.93	−0.0355955
$119$	−148568.	−0.961743
$120$	0	0
$121$	39636.3	0.246110
$122$	−209040.	−1.27154
$123$	0	0
$124$	94912.4	0.554330
$125$	−183252.	−1.04899
$126$	0	0
$127$	218127.	1.20005	0.600027	−	0.799980i	$-0.295156\pi$
$127$	218127.	1.20005	0.600027	+	0.799980i	$0.295156\pi$
$128$	−2188.25	−0.0118052
$129$	0	0
$130$	151838.	0.787993
$131$	−107160.	−0.545575	−0.272787	−	0.962074i	$-0.587946\pi$
$131$	−107160.	−0.545575	−0.272787	+	0.962074i	$0.587946\pi$
$132$	0	0
$133$	225370.	1.10476
$134$	289146.	1.39109
$135$	0	0
$136$	−166379.	−0.771348
$137$	168770.	0.768234	0.384117	−	0.923284i	$-0.374506\pi$
$137$	168770.	0.768234	0.384117	+	0.923284i	$0.374506\pi$
$138$	0	0
$139$	294636.	1.29345	0.646725	−	0.762723i	$-0.276138\pi$
$139$	294636.	1.29345	0.646725	+	0.762723i	$0.276138\pi$
$140$	123008.	0.530412
$141$	0	0
$142$	321720.	1.33893
$143$	299763.	1.22585
$144$	0	0
$145$	−40231.8	−0.158909
$146$	−77963.8	−0.302699
$147$	0	0
$148$	105797.	0.397027
$149$	−121995.	−0.450169	−0.225084	−	0.974339i	$-0.572266\pi$
$149$	−121995.	−0.450169	−0.225084	+	0.974339i	$0.572266\pi$
$150$	0	0
$151$	−515468.	−1.83975	−0.919877	−	0.392208i	$-0.871711\pi$
$151$	−515468.	−1.83975	−0.919877	+	0.392208i	$0.871711\pi$
$152$	252387.	0.886050
$153$	0	0
$154$	−338033.	−1.14857
$155$	−373060.	−1.24724
$156$	0	0
$157$	178197.	0.576969	0.288484	−	0.957485i	$-0.406849\pi$
$157$	178197.	0.576969	0.288484	+	0.957485i	$0.406849\pi$
$158$	−272859.	−0.869552
$159$	0	0
$160$	234896.	0.725398
$161$	−66140.9	−0.201097
$162$	0	0
$163$	−111853.	−0.329747	−0.164873	−	0.986315i	$-0.552722\pi$
$163$	−111853.	−0.329747	−0.164873	+	0.986315i	$0.552722\pi$
$164$	171988.	0.499332
$165$	0	0
$166$	380035.	1.07042
$167$	128666.	0.357003	0.178501	−	0.983940i	$-0.442875\pi$
$167$	128666.	0.357003	0.178501	+	0.983940i	$0.442875\pi$
$168$	0	0
$169$	76457.2	0.205922
$170$	192798.	0.511658
$171$	0	0
$172$	−24736.2	−0.0637546
$173$	−553314.	−1.40558	−0.702792	−	0.711396i	$-0.748063\pi$
$173$	−553314.	−1.40558	−0.702792	+	0.711396i	$0.748063\pi$
$174$	0	0
$175$	62941.4	0.155361
$176$	−186775.	−0.454503
$177$	0	0
$178$	−217377.	−0.514238
$179$	302733.	0.706200	0.353100	−	0.935586i	$-0.385128\pi$
$179$	302733.	0.706200	0.353100	+	0.935586i	$0.385128\pi$
$180$	0	0
$181$	−702706.	−1.59433	−0.797164	−	0.603763i	$-0.793667\pi$
$181$	−702706.	−1.59433	−0.797164	+	0.603763i	$0.793667\pi$
$182$	−504913.	−1.12990
$183$	0	0
$184$	−74069.9	−0.161286
$185$	−415843.	−0.893306
$186$	0	0
$187$	380627.	0.795967
$188$	358342.	0.739440
$189$	0	0
$190$	−292464.	−0.587743
$191$	−105355.	−0.208965	−0.104482	−	0.994527i	$-0.533319\pi$
$191$	−105355.	−0.208965	−0.104482	+	0.994527i	$0.533319\pi$
$192$	0	0
$193$	11272.6	0.0217836	0.0108918	−	0.999941i	$-0.496533\pi$
$193$	11272.6	0.0217836	0.0108918	+	0.999941i	$0.496533\pi$
$194$	75526.9	0.144078
$195$	0	0
$196$	−184197.	−0.342486
$197$	−44453.7	−0.0816098	−0.0408049	−	0.999167i	$-0.512992\pi$
$197$	−44453.7	−0.0816098	−0.0408049	+	0.999167i	$0.512992\pi$
$198$	0	0
$199$	1.01040e6	1.80867	0.904333	−	0.426827i	$-0.140369\pi$
$199$	1.01040e6	1.80867	0.904333	+	0.426827i	$0.140369\pi$
$200$	70486.7	0.124604
$201$	0	0
$202$	−788273.	−1.35925
$203$	133784.	0.227858
$204$	0	0
$205$	−676013.	−1.12349
$206$	−477180.	−0.783455
$207$	0	0
$208$	−278982.	−0.447114
$209$	−577390.	−0.914331
$210$	0	0
$211$	758827.	1.17337	0.586687	−	0.809814i	$-0.300432\pi$
$211$	758827.	1.17337	0.586687	+	0.809814i	$0.300432\pi$
$212$	401354.	0.613321
$213$	0	0
$214$	189595.	0.283003
$215$	97227.2	0.143447
$216$	0	0
$217$	1.24055e6	1.78840
$218$	−337329.	−0.480743
$219$	0	0
$220$	−315142.	−0.438985
$221$	568535.	0.783026
$222$	0	0
$223$	−517178.	−0.696431	−0.348215	−	0.937415i	$-0.613212\pi$
$223$	−517178.	−0.696431	−0.348215	+	0.937415i	$0.613212\pi$
$224$	−781110.	−1.04014
$225$	0	0
$226$	−99672.9	−0.129809
$227$	−627619.	−0.808409	−0.404205	−	0.914669i	$-0.632452\pi$
$227$	−627619.	−0.808409	−0.404205	+	0.914669i	$0.632452\pi$
$228$	0	0
$229$	800425.	1.00863	0.504315	−	0.863520i	$-0.331745\pi$
$229$	800425.	1.00863	0.504315	+	0.863520i	$0.331745\pi$
$230$	85831.3	0.106986
$231$	0	0
$232$	149822.	0.182750
$233$	−102393.	−0.123560	−0.0617802	−	0.998090i	$-0.519678\pi$
$233$	−102393.	−0.123560	−0.0617802	+	0.998090i	$0.519678\pi$
$234$	0	0
$235$	−1.40849e6	−1.66373
$236$	16691.0	0.0195076
$237$	0	0
$238$	−641118.	−0.733661
$239$	879306.	0.995739	0.497869	−	0.867252i	$-0.334116\pi$
$239$	879306.	0.995739	0.497869	+	0.867252i	$0.334116\pi$
$240$	0	0
$241$	196740.	0.218198	0.109099	−	0.994031i	$-0.465203\pi$
$241$	196740.	0.218198	0.109099	+	0.994031i	$0.465203\pi$
$242$	171043.	0.187744
$243$	0	0
$244$	648056.	0.696848
$245$	724001.	0.770591
$246$	0	0
$247$	−862436.	−0.899466
$248$	1.38927e6	1.43435
$249$	0	0
$250$	−790788.	−0.800221
$251$	798778.	0.800280	0.400140	−	0.916454i	$-0.368962\pi$
$251$	798778.	0.800280	0.400140	+	0.916454i	$0.368962\pi$
$252$	0	0
$253$	169451.	0.166434
$254$	941287.	0.915456
$255$	0	0
$256$	−1.05324e6	−1.00444
$257$	93933.3	0.0887129	0.0443565	−	0.999016i	$-0.485876\pi$
$257$	93933.3	0.0887129	0.0443565	+	0.999016i	$0.485876\pi$
$258$	0	0
$259$	1.38282e6	1.28090
$260$	−470722.	−0.431848
$261$	0	0
$262$	−462428.	−0.416189
$263$	898502.	0.800995	0.400497	−	0.916298i	$-0.368837\pi$
$263$	898502.	0.800995	0.400497	+	0.916298i	$0.368837\pi$
$264$	0	0
$265$	−1.57755e6	−1.37996
$266$	972541.	0.842760
$267$	0	0
$268$	−896400.	−0.762368
$269$	921887.	0.776778	0.388389	−	0.921495i	$-0.373032\pi$
$269$	921887.	0.776778	0.388389	+	0.921495i	$0.373032\pi$
$270$	0	0
$271$	−227722.	−0.188357	−0.0941784	−	0.995555i	$-0.530022\pi$
$271$	−227722.	−0.188357	−0.0941784	+	0.995555i	$0.530022\pi$
$272$	−354240.	−0.290319
$273$	0	0
$274$	728294.	0.586044
$275$	−161253.	−0.128581
$276$	0	0
$277$	1.76180e6	1.37961	0.689806	−	0.723994i	$-0.257696\pi$
$277$	1.76180e6	1.37961	0.689806	+	0.723994i	$0.257696\pi$
$278$	1.27145e6	0.986702
$279$	0	0
$280$	1.80051e6	1.37246
$281$	270410.	0.204295	0.102147	−	0.994769i	$-0.467429\pi$
$281$	270410.	0.204295	0.102147	+	0.994769i	$0.467429\pi$
$282$	0	0
$283$	2.04589e6	1.51851	0.759254	−	0.650795i	$-0.225564\pi$
$283$	2.04589e6	1.51851	0.759254	+	0.650795i	$0.225564\pi$
$284$	−997381.	−0.733779
$285$	0	0
$286$	1.29357e6	0.935135
$287$	2.24797e6	1.61096
$288$	0	0
$289$	−697955.	−0.491567
$290$	−173612.	−0.121223
$291$	0	0
$292$	241700.	0.165890
$293$	−1.15172e6	−0.783751	−0.391875	−	0.920018i	$-0.628174\pi$
$293$	−1.15172e6	−0.783751	−0.391875	+	0.920018i	$0.628174\pi$
$294$	0	0
$295$	−65605.3	−0.0438919
$296$	1.54859e6	1.02732
$297$	0	0
$298$	−526445.	−0.343409
$299$	253105.	0.163728
$300$	0	0
$301$	−323314.	−0.205687
$302$	−2.22440e6	−1.40345
$303$	0	0
$304$	537363.	0.333490
$305$	−2.54723e6	−1.56790
$306$	0	0
$307$	−1.88053e6	−1.13877	−0.569384	−	0.822072i	$-0.692818\pi$
$307$	−1.88053e6	−1.13877	−0.569384	+	0.822072i	$0.692818\pi$
$308$	1.04795e6	0.629456
$309$	0	0
$310$	−1.60987e6	−0.951450
$311$	−1.94805e6	−1.14209	−0.571043	−	0.820920i	$-0.693461\pi$
$311$	−1.94805e6	−1.14209	−0.571043	+	0.820920i	$0.693461\pi$
$312$	0	0
$313$	−944428.	−0.544889	−0.272445	−	0.962171i	$-0.587832\pi$
$313$	−944428.	−0.544889	−0.272445	+	0.962171i	$0.587832\pi$
$314$	768977.	0.440138
$315$	0	0
$316$	845906.	0.476545
$317$	−34149.2	−0.0190868	−0.00954339	−	0.999954i	$-0.503038\pi$
$317$	−34149.2	−0.0190868	−0.00954339	+	0.999954i	$0.503038\pi$
$318$	0	0
$319$	−342751.	−0.188583
$320$	1.71520e6	0.936354
$321$	0	0
$322$	−285418.	−0.153406
$323$	−1.09509e6	−0.584039
$324$	0	0
$325$	−240861.	−0.126491
$326$	−482682.	−0.251546
$327$	0	0
$328$	2.51746e6	1.29204
$329$	4.68370e6	2.38561
$330$	0	0
$331$	1.73996e6	0.872908	0.436454	−	0.899727i	$-0.356234\pi$
$331$	1.73996e6	0.872908	0.436454	+	0.899727i	$0.356234\pi$
$332$	−1.17817e6	−0.586627
$333$	0	0
$334$	555232.	0.272338
$335$	3.52336e6	1.71532
$336$	0	0
$337$	−1.09453e6	−0.524990	−0.262495	−	0.964933i	$-0.584545\pi$
$337$	−1.09453e6	−0.524990	−0.262495	+	0.964933i	$0.584545\pi$
$338$	329936.	0.157086
$339$	0	0
$340$	−597703.	−0.280407
$341$	−3.17825e6	−1.48014
$342$	0	0
$343$	531301.	0.243840
$344$	−362072.	−0.164968
$345$	0	0
$346$	−2.38772e6	−1.07224
$347$	561799.	0.250471	0.125236	−	0.992127i	$-0.460031\pi$
$347$	561799.	0.250471	0.125236	+	0.992127i	$0.460031\pi$
$348$	0	0
$349$	2.06827e6	0.908957	0.454478	−	0.890758i	$-0.349826\pi$
$349$	2.06827e6	0.908957	0.454478	+	0.890758i	$0.349826\pi$
$350$	271611.	0.118516
$351$	0	0
$352$	2.00118e6	0.860852
$353$	−568169.	−0.242684	−0.121342	−	0.992611i	$-0.538720\pi$
$353$	−568169.	−0.242684	−0.121342	+	0.992611i	$0.538720\pi$
$354$	0	0
$355$	3.92028e6	1.65100
$356$	673903.	0.281821
$357$	0	0
$358$	1.30639e6	0.538722
$359$	−2.57455e6	−1.05430	−0.527151	−	0.849771i	$-0.676740\pi$
$359$	−2.57455e6	−1.05430	−0.527151	+	0.849771i	$0.676740\pi$
$360$	0	0
$361$	−814914.	−0.329112
$362$	−3.03239e6	−1.21623
$363$	0	0
$364$	1.56531e6	0.619223
$365$	−950019.	−0.373250
$366$	0	0
$367$	272102.	0.105455	0.0527274	−	0.998609i	$-0.483209\pi$
$367$	272102.	0.105455	0.0527274	+	0.998609i	$0.483209\pi$
$368$	−157704.	−0.0607047
$369$	0	0
$370$	−1.79449e6	−0.681455
$371$	5.24589e6	1.97872
$372$	0	0
$373$	−268967.	−0.100098	−0.0500491	−	0.998747i	$-0.515938\pi$
$373$	−268967.	−0.100098	−0.0500491	+	0.998747i	$0.515938\pi$
$374$	1.64252e6	0.607200
$375$	0	0
$376$	5.24518e6	1.91333
$377$	−511960.	−0.185517
$378$	0	0
$379$	−522052.	−0.186688	−0.0933438	−	0.995634i	$-0.529756\pi$
$379$	−522052.	−0.186688	−0.0933438	+	0.995634i	$0.529756\pi$
$380$	906683.	0.322104
$381$	0	0
$382$	−454640.	−0.159408
$383$	−2.00529e6	−0.698523	−0.349262	−	0.937025i	$-0.613568\pi$
$383$	−2.00529e6	−0.698523	−0.349262	+	0.937025i	$0.613568\pi$
$384$	0	0
$385$	−4.11906e6	−1.41627
$386$	48644.5	0.0166175
$387$	0	0
$388$	−234145.	−0.0789598
$389$	1.10827e6	0.371340	0.185670	−	0.982612i	$-0.440554\pi$
$389$	1.10827e6	0.371340	0.185670	+	0.982612i	$0.440554\pi$
$390$	0	0
$391$	321383.	0.106312
$392$	−2.69616e6	−0.886198
$393$	0	0
$394$	−191831.	−0.0622557
$395$	−3.32489e6	−1.07222
$396$	0	0
$397$	−3.77816e6	−1.20311	−0.601554	−	0.798832i	$-0.705452\pi$
$397$	−3.77816e6	−1.20311	−0.601554	+	0.798832i	$0.705452\pi$
$398$	4.36016e6	1.37973
$399$	0	0
$400$	150075.	0.0468983
$401$	−1.64922e6	−0.512173	−0.256086	−	0.966654i	$-0.582433\pi$
$401$	−1.64922e6	−0.512173	−0.256086	+	0.966654i	$0.582433\pi$
$402$	0	0
$403$	−4.74729e6	−1.45607
$404$	2.44377e6	0.744916
$405$	0	0
$406$	577320.	0.173821
$407$	−3.54273e6	−1.06011
$408$	0	0
$409$	−2.14199e6	−0.633153	−0.316577	−	0.948567i	$-0.602533\pi$
$409$	−2.14199e6	−0.633153	−0.316577	+	0.948567i	$0.602533\pi$
$410$	−2.91720e6	−0.857051
$411$	0	0
$412$	1.47933e6	0.429361
$413$	218160.	0.0629361
$414$	0	0
$415$	4.63087e6	1.31990
$416$	2.98912e6	0.846857
$417$	0	0
$418$	−2.49161e6	−0.697493
$419$	−6.49479e6	−1.80730	−0.903650	−	0.428272i	$-0.859123\pi$
$419$	−6.49479e6	−1.80730	−0.903650	+	0.428272i	$0.859123\pi$
$420$	0	0
$421$	3.30359e6	0.908408	0.454204	−	0.890898i	$-0.349924\pi$
$421$	3.30359e6	0.908408	0.454204	+	0.890898i	$0.349924\pi$
$422$	3.27457e6	0.895103
$423$	0	0
$424$	5.87476e6	1.58699
$425$	−305836.	−0.0821327
$426$	0	0
$427$	8.47040e6	2.24820
$428$	−587773.	−0.155096
$429$	0	0
$430$	419565.	0.109428
$431$	2.23343e6	0.579133	0.289567	−	0.957158i	$-0.406489\pi$
$431$	2.23343e6	0.579133	0.289567	+	0.957158i	$0.406489\pi$
$432$	0	0
$433$	−3.81775e6	−0.978561	−0.489280	−	0.872127i	$-0.662741\pi$
$433$	−3.81775e6	−0.978561	−0.489280	+	0.872127i	$0.662741\pi$
$434$	5.35336e6	1.36427
$435$	0	0
$436$	1.04577e6	0.263464
$437$	−487520.	−0.122121
$438$	0	0
$439$	4.94254e6	1.22402	0.612011	−	0.790850i	$-0.290361\pi$
$439$	4.94254e6	1.22402	0.612011	+	0.790850i	$0.290361\pi$
$440$	−4.61285e6	−1.13589
$441$	0	0
$442$	2.45340e6	0.597328
$443$	740033.	0.179160	0.0895802	−	0.995980i	$-0.471447\pi$
$443$	740033.	0.179160	0.0895802	+	0.995980i	$0.471447\pi$
$444$	0	0
$445$	−2.64882e6	−0.634093
$446$	−2.23178e6	−0.531269
$447$	0	0
$448$	−5.70363e6	−1.34263
$449$	1.75356e6	0.410492	0.205246	−	0.978710i	$-0.434201\pi$
$449$	1.75356e6	0.410492	0.205246	+	0.978710i	$0.434201\pi$
$450$	0	0
$451$	−5.75922e6	−1.33328
$452$	309002.	0.0711402
$453$	0	0
$454$	−2.70837e6	−0.616692
$455$	−6.15256e6	−1.39325
$456$	0	0
$457$	1.43140e6	0.320605	0.160302	−	0.987068i	$-0.448753\pi$
$457$	1.43140e6	0.320605	0.160302	+	0.987068i	$0.448753\pi$
$458$	3.45408e6	0.769429
$459$	0	0
$460$	−266091.	−0.0586321
$461$	8.02421e6	1.75853	0.879265	−	0.476332i	$-0.158034\pi$
$461$	8.02421e6	1.75853	0.879265	+	0.476332i	$0.158034\pi$
$462$	0	0
$463$	−5.42639e6	−1.17641	−0.588205	−	0.808712i	$-0.700165\pi$
$463$	−5.42639e6	−1.17641	−0.588205	+	0.808712i	$0.700165\pi$
$464$	318990.	0.0687831
$465$	0	0
$466$	−441856.	−0.0942575
$467$	4.68842e6	0.994796	0.497398	−	0.867522i	$-0.334289\pi$
$467$	4.68842e6	0.994796	0.497398	+	0.867522i	$0.334289\pi$
$468$	0	0
$469$	−1.17164e7	−2.45958
$470$	−6.07806e6	−1.26917
$471$	0	0
$472$	244313.	0.0504768
$473$	828318.	0.170233
$474$	0	0
$475$	463936.	0.0943461
$476$	1.98757e6	0.402072
$477$	0	0
$478$	3.79448e6	0.759595
$479$	−2.64873e6	−0.527471	−0.263735	−	0.964595i	$-0.584955\pi$
$479$	−2.64873e6	−0.527471	−0.263735	+	0.964595i	$0.584955\pi$
$480$	0	0
$481$	−5.29172e6	−1.04288
$482$	848994.	0.166451
$483$	0	0
$484$	−530260.	−0.102890
$485$	920324.	0.177659
$486$	0	0
$487$	2.23774e6	0.427550	0.213775	−	0.976883i	$-0.431424\pi$
$487$	2.23774e6	0.427550	0.213775	+	0.976883i	$0.431424\pi$
$488$	9.48583e6	1.80312
$489$	0	0
$490$	3.12428e6	0.587842
$491$	−3.10094e6	−0.580484	−0.290242	−	0.956953i	$-0.593736\pi$
$491$	−3.10094e6	−0.580484	−0.290242	+	0.956953i	$0.593736\pi$
$492$	0	0
$493$	−650066.	−0.120459
$494$	−3.72168e6	−0.686153
$495$	0	0
$496$	2.95792e6	0.539861
$497$	−1.30363e7	−2.36735
$498$	0	0
$499$	5.58008e6	1.00320	0.501601	−	0.865099i	$-0.332744\pi$
$499$	5.58008e6	1.00320	0.501601	+	0.865099i	$0.332744\pi$
$500$	2.45157e6	0.438550
$501$	0	0
$502$	3.44697e6	0.610490
$503$	−5.10876e6	−0.900317	−0.450158	−	0.892949i	$-0.648632\pi$
$503$	−5.10876e6	−0.900317	−0.450158	+	0.892949i	$0.648632\pi$
$504$	0	0
$505$	−9.60542e6	−1.67605
$506$	731231.	0.126963
$507$	0	0
$508$	−2.91814e6	−0.501703
$509$	−2.93415e6	−0.501982	−0.250991	−	0.967989i	$-0.580756\pi$
$509$	−2.93415e6	−0.501982	−0.250991	+	0.967989i	$0.580756\pi$
$510$	0	0
$511$	3.15914e6	0.535200
$512$	−4.47501e6	−0.754430
$513$	0	0
$514$	405351.	0.0676743
$515$	−5.81462e6	−0.966058
$516$	0	0
$517$	−1.19995e7	−1.97440
$518$	5.96729e6	0.977131
$519$	0	0
$520$	−6.89013e6	−1.11743
$521$	9.22452e6	1.48885	0.744423	−	0.667709i	$-0.232725\pi$
$521$	9.22452e6	1.48885	0.744423	+	0.667709i	$0.232725\pi$
$522$	0	0
$523$	−4.64359e6	−0.742335	−0.371168	−	0.928566i	$-0.621042\pi$
$523$	−4.64359e6	−0.742335	−0.371168	+	0.928566i	$0.621042\pi$
$524$	1.43360e6	0.228087
$525$	0	0
$526$	3.87731e6	0.611035
$527$	−6.02791e6	−0.945453
$528$	0	0
$529$	−6.29327e6	−0.977771
$530$	−6.80761e6	−1.05270
$531$	0	0
$532$	−3.01503e6	−0.461862
$533$	−8.60244e6	−1.31161
$534$	0	0
$535$	2.31028e6	0.348964
$536$	−1.31209e7	−1.97266
$537$	0	0
$538$	3.97823e6	0.592562
$539$	6.16805e6	0.914484
$540$	0	0
$541$	3.62594e6	0.532632	0.266316	−	0.963886i	$-0.414193\pi$
$541$	3.62594e6	0.532632	0.266316	+	0.963886i	$0.414193\pi$
$542$	−982690.	−0.143687
$543$	0	0
$544$	3.79546e6	0.549879
$545$	−4.11049e6	−0.592792
$546$	0	0
$547$	−5.56425e6	−0.795131	−0.397565	−	0.917574i	$-0.630145\pi$
$547$	−5.56425e6	−0.795131	−0.397565	+	0.917574i	$0.630145\pi$
$548$	−2.25782e6	−0.321173
$549$	0	0
$550$	−695858.	−0.0980876
$551$	986114.	0.138372
$552$	0	0
$553$	1.10564e7	1.53745
$554$	7.60271e6	1.05243
$555$	0	0
$556$	−3.94169e6	−0.540748
$557$	−6.83963e6	−0.934103	−0.467052	−	0.884230i	$-0.654684\pi$
$557$	−6.83963e6	−0.934103	−0.467052	+	0.884230i	$0.654684\pi$
$558$	0	0
$559$	1.23724e6	0.167465
$560$	3.83351e6	0.516567
$561$	0	0
$562$	1.16690e6	0.155845
$563$	5.62439e6	0.747833	0.373916	−	0.927462i	$-0.378015\pi$
$563$	5.62439e6	0.747833	0.373916	+	0.927462i	$0.378015\pi$
$564$	0	0
$565$	−1.21455e6	−0.160065
$566$	8.82866e6	1.15839
$567$	0	0
$568$	−1.45990e7	−1.89869
$569$	4.17021e6	0.539979	0.269990	−	0.962863i	$-0.412980\pi$
$569$	4.17021e6	0.539979	0.269990	+	0.962863i	$0.412980\pi$
$570$	0	0
$571$	−8.60607e6	−1.10462	−0.552312	−	0.833637i	$-0.686254\pi$
$571$	−8.60607e6	−1.10462	−0.552312	+	0.833637i	$0.686254\pi$
$572$	−4.01027e6	−0.512487
$573$	0	0
$574$	9.70069e6	1.22892
$575$	−136155.	−0.0171737
$576$	0	0
$577$	−3.23538e6	−0.404563	−0.202282	−	0.979327i	$-0.564836\pi$
$577$	−3.23538e6	−0.404563	−0.202282	+	0.979327i	$0.564836\pi$
$578$	−3.01189e6	−0.374990
$579$	0	0
$580$	538226.	0.0664347
$581$	−1.53992e7	−1.89260
$582$	0	0
$583$	−1.34398e7	−1.63765
$584$	3.53785e6	0.429247
$585$	0	0
$586$	−4.97003e6	−0.597881
$587$	1.12970e7	1.35322	0.676609	−	0.736342i	$-0.263449\pi$
$587$	1.12970e7	1.35322	0.676609	+	0.736342i	$0.263449\pi$
$588$	0	0
$589$	9.14401e6	1.08605
$590$	−283107.	−0.0334827
$591$	0	0
$592$	3.29714e6	0.386663
$593$	−1.61368e7	−1.88444	−0.942218	−	0.335000i	$-0.891264\pi$
$593$	−1.61368e7	−1.88444	−0.942218	+	0.335000i	$0.891264\pi$
$594$	0	0
$595$	−7.81227e6	−0.904659
$596$	1.63206e6	0.188201
$597$	0	0
$598$	1.09223e6	0.124899
$599$	−1.14121e7	−1.29957	−0.649785	−	0.760118i	$-0.725141\pi$
$599$	−1.14121e7	−1.29957	−0.649785	+	0.760118i	$0.725141\pi$
$600$	0	0
$601$	1.15192e7	1.30088	0.650438	−	0.759559i	$-0.274585\pi$
$601$	1.15192e7	1.30088	0.650438	+	0.759559i	$0.274585\pi$
$602$	−1.39520e6	−0.156908
$603$	0	0
$604$	6.89600e6	0.769139
$605$	2.08422e6	0.231503
$606$	0	0
$607$	1.48986e7	1.64125	0.820625	−	0.571466i	$-0.193625\pi$
$607$	1.48986e7	1.64125	0.820625	+	0.571466i	$0.193625\pi$
$608$	−5.75751e6	−0.631648
$609$	0	0
$610$	−1.09921e7	−1.19607
$611$	−1.79234e7	−1.94230
$612$	0	0
$613$	−5.39999e6	−0.580419	−0.290210	−	0.956963i	$-0.593725\pi$
$613$	−5.39999e6	−0.580419	−0.290210	+	0.956963i	$0.593725\pi$
$614$	−8.11508e6	−0.868704
$615$	0	0
$616$	1.53393e7	1.62875
$617$	1.26228e7	1.33489	0.667443	−	0.744660i	$-0.267389\pi$
$617$	1.26228e7	1.33489	0.667443	+	0.744660i	$0.267389\pi$
$618$	0	0
$619$	1.43557e7	1.50591	0.752953	−	0.658074i	$-0.228629\pi$
$619$	1.43557e7	1.50591	0.752953	+	0.658074i	$0.228629\pi$
$620$	4.99084e6	0.521428
$621$	0	0
$622$	−8.40642e6	−0.871235
$623$	8.80824e6	0.909220
$624$	0	0
$625$	−8.51120e6	−0.871546
$626$	−4.07550e6	−0.415666
$627$	0	0
$628$	−2.38395e6	−0.241211
$629$	−6.71920e6	−0.677160
$630$	0	0
$631$	−6.76873e6	−0.676759	−0.338379	−	0.941010i	$-0.609879\pi$
$631$	−6.76873e6	−0.676759	−0.338379	+	0.941010i	$0.609879\pi$
$632$	1.23818e7	1.23308
$633$	0	0
$634$	−147364.	−0.0145603
$635$	1.14699e7	1.12883
$636$	0	0
$637$	9.21310e6	0.899616
$638$	−1.47907e6	−0.143859
$639$	0	0
$640$	−115066.	−0.0111045
$641$	1.51643e6	0.145773	0.0728867	−	0.997340i	$-0.476779\pi$
$641$	1.51643e6	0.145773	0.0728867	+	0.997340i	$0.476779\pi$
$642$	0	0
$643$	6.58876e6	0.628458	0.314229	−	0.949347i	$-0.398254\pi$
$643$	6.58876e6	0.628458	0.314229	+	0.949347i	$0.398254\pi$
$644$	884842.	0.0840719
$645$	0	0
$646$	−4.72563e6	−0.445532
$647$	−9.99129e6	−0.938341	−0.469171	−	0.883108i	$-0.655447\pi$
$647$	−9.99129e6	−0.938341	−0.469171	+	0.883108i	$0.655447\pi$
$648$	0	0
$649$	−558918.	−0.0520878
$650$	−1.03939e6	−0.0964929
$651$	0	0
$652$	1.49639e6	0.137856
$653$	1.32714e7	1.21796	0.608980	−	0.793186i	$-0.291579\pi$
$653$	1.32714e7	1.21796	0.608980	+	0.793186i	$0.291579\pi$
$654$	0	0
$655$	−5.63486e6	−0.513192
$656$	5.35997e6	0.486298
$657$	0	0
$658$	2.02116e7	1.81985
$659$	−3.69381e6	−0.331330	−0.165665	−	0.986182i	$-0.552977\pi$
$659$	−3.69381e6	−0.331330	−0.165665	+	0.986182i	$0.552977\pi$
$660$	0	0
$661$	8.97800e6	0.799238	0.399619	−	0.916681i	$-0.369142\pi$
$661$	8.97800e6	0.799238	0.399619	+	0.916681i	$0.369142\pi$
$662$	7.50845e6	0.665894
$663$	0	0
$664$	−1.72453e7	−1.51792
$665$	1.18508e7	1.03919
$666$	0	0
$667$	−289402.	−0.0251876
$668$	−1.72131e6	−0.149251
$669$	0	0
$670$	1.52044e7	1.30852
$671$	−2.17009e7	−1.86068
$672$	0	0
$673$	8.59992e6	0.731909	0.365954	−	0.930633i	$-0.380743\pi$
$673$	8.59992e6	0.731909	0.365954	+	0.930633i	$0.380743\pi$
$674$	−4.72322e6	−0.400487
$675$	0	0
$676$	−1.02285e6	−0.0860889
$677$	8.01663e6	0.672233	0.336117	−	0.941820i	$-0.390886\pi$
$677$	8.01663e6	0.672233	0.336117	+	0.941820i	$0.390886\pi$
$678$	0	0
$679$	−3.06039e6	−0.254743
$680$	−8.74880e6	−0.725565
$681$	0	0
$682$	−1.37151e7	−1.12911
$683$	−2.81320e6	−0.230754	−0.115377	−	0.993322i	$-0.536808\pi$
$683$	−2.81320e6	−0.230754	−0.115377	+	0.993322i	$0.536808\pi$
$684$	0	0
$685$	8.87454e6	0.722636
$686$	2.29273e6	0.186012
$687$	0	0
$688$	−770895.	−0.0620904
$689$	−2.00747e7	−1.61102
$690$	0	0
$691$	1.04922e7	0.835935	0.417967	−	0.908462i	$-0.362743\pi$
$691$	1.04922e7	0.835935	0.417967	+	0.908462i	$0.362743\pi$
$692$	7.40231e6	0.587628
$693$	0	0
$694$	2.42434e6	0.191071
$695$	1.54931e7	1.21668
$696$	0	0
$697$	−1.09230e7	−0.851650
$698$	8.92521e6	0.693394
$699$	0	0
$700$	−842038.	−0.0649511
$701$	2.37537e7	1.82573	0.912865	−	0.408263i	$-0.133865\pi$
$701$	2.37537e7	1.82573	0.912865	+	0.408263i	$0.133865\pi$
$702$	0	0
$703$	1.01927e7	0.777856
$704$	1.46125e7	1.11120
$705$	0	0
$706$	−2.45182e6	−0.185130
$707$	3.19413e7	2.40328
$708$	0	0
$709$	−1.72000e7	−1.28503	−0.642516	−	0.766273i	$-0.722109\pi$
$709$	−1.72000e7	−1.28503	−0.642516	+	0.766273i	$0.722109\pi$
$710$	1.69172e7	1.25945
$711$	0	0
$712$	9.86417e6	0.729223
$713$	−2.68356e6	−0.197691
$714$	0	0
$715$	1.57626e7	1.15309
$716$	−4.05001e6	−0.295239
$717$	0	0
$718$	−1.11100e7	−0.804270
$719$	−2.22034e7	−1.60176	−0.800878	−	0.598827i	$-0.795634\pi$
$719$	−2.22034e7	−1.60176	−0.800878	+	0.598827i	$0.795634\pi$
$720$	0	0
$721$	1.93356e7	1.38522
$722$	−3.51660e6	−0.251062
$723$	0	0
$724$	9.40090e6	0.666535
$725$	275402.	0.0194591
$726$	0	0
$727$	−1.95017e7	−1.36847	−0.684236	−	0.729261i	$-0.739864\pi$
$727$	−1.95017e7	−1.36847	−0.684236	+	0.729261i	$0.739864\pi$
$728$	2.29120e7	1.60227
$729$	0	0
$730$	−4.09962e6	−0.284732
$731$	1.57100e6	0.108738
$732$	0	0
$733$	1.07906e7	0.741798	0.370899	−	0.928673i	$-0.379050\pi$
$733$	1.07906e7	0.741798	0.370899	+	0.928673i	$0.379050\pi$
$734$	1.17420e6	0.0804457
$735$	0	0
$736$	1.68970e6	0.114978
$737$	3.00169e7	2.03562
$738$	0	0
$739$	−1.44802e7	−0.975358	−0.487679	−	0.873023i	$-0.662156\pi$
$739$	−1.44802e7	−0.975358	−0.487679	+	0.873023i	$0.662156\pi$
$740$	5.56321e6	0.373462
$741$	0	0
$742$	2.26376e7	1.50946
$743$	−5.44908e6	−0.362119	−0.181059	−	0.983472i	$-0.557953\pi$
$743$	−5.44908e6	−0.362119	−0.181059	+	0.983472i	$0.557953\pi$
$744$	0	0
$745$	−6.41493e6	−0.423449
$746$	−1.16067e6	−0.0763594
$747$	0	0
$748$	−5.09207e6	−0.332767
$749$	−7.68248e6	−0.500376
$750$	0	0
$751$	−2.78004e6	−0.179867	−0.0899335	−	0.995948i	$-0.528665\pi$
$751$	−2.78004e6	−0.179867	−0.0899335	+	0.995948i	$0.528665\pi$
$752$	1.11676e7	0.720139
$753$	0	0
$754$	−2.20926e6	−0.141521
$755$	−2.71052e7	−1.73056
$756$	0	0
$757$	2.46778e7	1.56519	0.782595	−	0.622531i	$-0.213895\pi$
$757$	2.46778e7	1.56519	0.782595	+	0.622531i	$0.213895\pi$
$758$	−2.25281e6	−0.142414
$759$	0	0
$760$	1.32714e7	0.833459
$761$	8.63006e6	0.540197	0.270098	−	0.962833i	$-0.412944\pi$
$761$	8.63006e6	0.540197	0.270098	+	0.962833i	$0.412944\pi$
$762$	0	0
$763$	1.36688e7	0.849999
$764$	1.40946e6	0.0873611
$765$	0	0
$766$	−8.65345e6	−0.532865
$767$	−834845.	−0.0512410
$768$	0	0
$769$	−1.79104e6	−0.109217	−0.0546083	−	0.998508i	$-0.517391\pi$
$769$	−1.79104e6	−0.109217	−0.0546083	+	0.998508i	$0.517391\pi$
$770$	−1.77750e7	−1.08040
$771$	0	0
$772$	−150806.	−0.00910698
$773$	2.64220e7	1.59044	0.795219	−	0.606322i	$-0.207356\pi$
$773$	2.64220e7	1.59044	0.795219	+	0.606322i	$0.207356\pi$
$774$	0	0
$775$	2.55374e6	0.152729
$776$	−3.42727e6	−0.204312
$777$	0	0
$778$	4.78253e6	0.283275
$779$	1.65696e7	0.978293
$780$	0	0
$781$	3.33984e7	1.95929
$782$	1.38686e6	0.0810993
$783$	0	0
$784$	−5.74045e6	−0.333546
$785$	9.37028e6	0.542723
$786$	0	0
$787$	4.11413e6	0.236778	0.118389	−	0.992967i	$-0.462227\pi$
$787$	4.11413e6	0.236778	0.118389	+	0.992967i	$0.462227\pi$
$788$	594708.	0.0341184
$789$	0	0
$790$	−1.43479e7	−0.817940
$791$	4.03880e6	0.229515
$792$	0	0
$793$	−3.24142e7	−1.83043
$794$	−1.63039e7	−0.917786
$795$	0	0
$796$	−1.35172e7	−0.756143
$797$	−3.23381e6	−0.180330	−0.0901651	−	0.995927i	$-0.528739\pi$
$797$	−3.23381e6	−0.180330	−0.0901651	+	0.995927i	$0.528739\pi$
$798$	0	0
$799$	−2.27584e7	−1.26117
$800$	−1.60796e6	−0.0888279
$801$	0	0
$802$	−7.11687e6	−0.390709
$803$	−8.09359e6	−0.442948
$804$	0	0
$805$	−3.47793e6	−0.189161
$806$	−2.04860e7	−1.11076
$807$	0	0
$808$	3.57704e7	1.92750
$809$	1.52689e7	0.820231	0.410115	−	0.912034i	$-0.365488\pi$
$809$	1.52689e7	0.820231	0.410115	+	0.912034i	$0.365488\pi$
$810$	0	0
$811$	−3.04480e7	−1.62558	−0.812788	−	0.582560i	$-0.802051\pi$
$811$	−3.04480e7	−1.62558	−0.812788	+	0.582560i	$0.802051\pi$
$812$	−1.78978e6	−0.0952600
$813$	0	0
$814$	−1.52880e7	−0.808703
$815$	−5.88167e6	−0.310175
$816$	0	0
$817$	−2.38312e6	−0.124908
$818$	−9.24334e6	−0.482998
$819$	0	0
$820$	9.04378e6	0.469695
$821$	2.25035e6	0.116518	0.0582590	−	0.998302i	$-0.481445\pi$
$821$	2.25035e6	0.116518	0.0582590	+	0.998302i	$0.481445\pi$
$822$	0	0
$823$	−1.66620e7	−0.857485	−0.428743	−	0.903427i	$-0.641043\pi$
$823$	−1.66620e7	−0.857485	−0.428743	+	0.903427i	$0.641043\pi$
$824$	2.16535e7	1.11099
$825$	0	0
$826$	941428.	0.0480106
$827$	−3.70905e7	−1.88581	−0.942907	−	0.333056i	$-0.891920\pi$
$827$	−3.70905e7	−1.88581	−0.942907	+	0.333056i	$0.891920\pi$
$828$	0	0
$829$	2.88276e7	1.45688	0.728438	−	0.685111i	$-0.240246\pi$
$829$	2.88276e7	1.45688	0.728438	+	0.685111i	$0.240246\pi$
$830$	1.99836e7	1.00688
$831$	0	0
$832$	2.18264e7	1.09313
$833$	1.16984e7	0.584137
$834$	0	0
$835$	6.76572e6	0.335813
$836$	7.72439e6	0.382251
$837$	0	0
$838$	−2.80270e7	−1.37869
$839$	7.10869e6	0.348646	0.174323	−	0.984689i	$-0.444226\pi$
$839$	7.10869e6	0.348646	0.174323	+	0.984689i	$0.444226\pi$
$840$	0	0
$841$	−1.99258e7	−0.971460
$842$	1.42560e7	0.692975
$843$	0	0
$844$	−1.01517e7	−0.490548
$845$	4.02040e6	0.193699
$846$	0	0
$847$	−6.93075e6	−0.331949
$848$	1.25081e7	0.597311
$849$	0	0
$850$	−1.31978e6	−0.0626546
$851$	−2.99131e6	−0.141592
$852$	0	0
$853$	6.01755e6	0.283170	0.141585	−	0.989926i	$-0.454780\pi$
$853$	6.01755e6	0.283170	0.141585	+	0.989926i	$0.454780\pi$
$854$	3.65524e7	1.71503
$855$	0	0
$856$	−8.60345e6	−0.401317
$857$	−1.40484e7	−0.653392	−0.326696	−	0.945129i	$-0.605935\pi$
$857$	−1.40484e7	−0.653392	−0.326696	+	0.945129i	$0.605935\pi$
$858$	0	0
$859$	2.38780e7	1.10412	0.552060	−	0.833805i	$-0.313842\pi$
$859$	2.38780e7	1.10412	0.552060	+	0.833805i	$0.313842\pi$
$860$	−1.30072e6	−0.0599704
$861$	0	0
$862$	9.63792e6	0.441789
$863$	−3.12051e7	−1.42626	−0.713131	−	0.701031i	$-0.752723\pi$
$863$	−3.12051e7	−1.42626	−0.713131	+	0.701031i	$0.752723\pi$
$864$	0	0
$865$	−2.90953e7	−1.32216
$866$	−1.64748e7	−0.746491
$867$	0	0
$868$	−1.65963e7	−0.747671
$869$	−2.83261e7	−1.27244
$870$	0	0
$871$	4.48357e7	2.00253
$872$	1.53074e7	0.681725
$873$	0	0
$874$	−2.10380e6	−0.0931591
$875$	3.20432e7	1.41487
$876$	0	0
$877$	−2.97111e6	−0.130443	−0.0652213	−	0.997871i	$-0.520775\pi$
$877$	−2.97111e6	−0.130443	−0.0652213	+	0.997871i	$0.520775\pi$
$878$	2.13286e7	0.933739
$879$	0	0
$880$	−9.82131e6	−0.427526
$881$	−3.55211e7	−1.54187	−0.770934	−	0.636915i	$-0.780210\pi$
$881$	−3.55211e7	−1.54187	−0.770934	+	0.636915i	$0.780210\pi$
$882$	0	0
$883$	1.87105e7	0.807576	0.403788	−	0.914853i	$-0.367693\pi$
$883$	1.87105e7	0.807576	0.403788	+	0.914853i	$0.367693\pi$
$884$	−7.60593e6	−0.327357
$885$	0	0
$886$	3.19347e6	0.136672
$887$	−9.33897e6	−0.398557	−0.199278	−	0.979943i	$-0.563860\pi$
$887$	−9.33897e6	−0.398557	−0.199278	+	0.979943i	$0.563860\pi$
$888$	0	0
$889$	−3.81415e7	−1.61861
$890$	−1.14305e7	−0.483715
$891$	0	0
$892$	6.91887e6	0.291154
$893$	3.45232e7	1.44871
$894$	0	0
$895$	1.59188e7	0.664284
$896$	382635.	0.0159226
$897$	0	0
$898$	7.56715e6	0.313142
$899$	5.42807e6	0.223999
$900$	0	0
$901$	−2.54901e7	−1.04607
$902$	−2.48528e7	−1.01709
$903$	0	0
$904$	4.52297e6	0.184078
$905$	−3.69509e7	−1.49970
$906$	0	0
$907$	3.89432e7	1.57186	0.785930	−	0.618315i	$-0.212184\pi$
$907$	3.89432e7	1.57186	0.785930	+	0.618315i	$0.212184\pi$
$908$	8.39637e6	0.337969
$909$	0	0
$910$	−2.65502e7	−1.06283
$911$	−4.28351e7	−1.71003	−0.855015	−	0.518604i	$-0.826452\pi$
$911$	−4.28351e7	−1.71003	−0.855015	+	0.518604i	$0.826452\pi$
$912$	0	0
$913$	3.94522e7	1.56637
$914$	6.17692e6	0.244572
$915$	0	0
$916$	−1.07082e7	−0.421675
$917$	1.87378e7	0.735862
$918$	0	0
$919$	881759.	0.0344398	0.0172199	−	0.999852i	$-0.494518\pi$
$919$	881759.	0.0344398	0.0172199	+	0.999852i	$0.494518\pi$
$920$	−3.89487e6	−0.151713
$921$	0	0
$922$	3.46269e7	1.34149
$923$	4.98866e7	1.92743
$924$	0	0
$925$	2.84661e6	0.109389
$926$	−2.34165e7	−0.897419
$927$	0	0
$928$	−3.41778e6	−0.130279
$929$	1.81365e7	0.689468	0.344734	−	0.938700i	$-0.387969\pi$
$929$	1.81365e7	0.689468	0.344734	+	0.938700i	$0.387969\pi$
$930$	0	0
$931$	−1.77459e7	−0.671000
$932$	1.36982e6	0.0516565
$933$	0	0
$934$	2.02320e7	0.758876
$935$	2.00148e7	0.748723
$936$	0	0
$937$	−1.75363e7	−0.652512	−0.326256	−	0.945281i	$-0.605787\pi$
$937$	−1.75363e7	−0.652512	−0.326256	+	0.945281i	$0.605787\pi$
$938$	−5.05597e7	−1.87628
$939$	0	0
$940$	1.88429e7	0.695551
$941$	−7.69478e6	−0.283284	−0.141642	−	0.989918i	$-0.545238\pi$
$941$	−7.69478e6	−0.283284	−0.141642	+	0.989918i	$0.545238\pi$
$942$	0	0
$943$	−4.86281e6	−0.178077
$944$	520171.	0.0189984
$945$	0	0
$946$	3.57444e6	0.129862
$947$	3.00889e7	1.09026	0.545131	−	0.838351i	$-0.316480\pi$
$947$	3.00889e7	1.09026	0.545131	+	0.838351i	$0.316480\pi$
$948$	0	0
$949$	−1.20892e7	−0.435746
$950$	2.00203e6	0.0719715
$951$	0	0
$952$	2.90927e7	1.04038
$953$	−2.31755e7	−0.826601	−0.413301	−	0.910595i	$-0.635624\pi$
$953$	−2.31755e7	−0.826601	−0.413301	+	0.910595i	$0.635624\pi$
$954$	0	0
$955$	−5.53997e6	−0.196562
$956$	−1.17635e7	−0.416285
$957$	0	0
$958$	−1.14301e7	−0.402379
$959$	−2.95109e7	−1.03618
$960$	0	0
$961$	2.17041e7	0.758112
$962$	−2.28354e7	−0.795556
$963$	0	0
$964$	−2.63202e6	−0.0912212
$965$	592752.	0.0204906
$966$	0	0
$967$	−3.96944e7	−1.36510	−0.682548	−	0.730841i	$-0.739128\pi$
$967$	−3.96944e7	−1.36510	−0.682548	+	0.730841i	$0.739128\pi$
$968$	−7.76160e6	−0.266234
$969$	0	0
$970$	3.97148e6	0.135526
$971$	6.90625e6	0.235068	0.117534	−	0.993069i	$-0.462501\pi$
$971$	6.90625e6	0.235068	0.117534	+	0.993069i	$0.462501\pi$
$972$	0	0
$973$	−5.15197e7	−1.74458
$974$	9.65654e6	0.326155
$975$	0	0
$976$	2.01965e7	0.678658
$977$	1.38767e7	0.465104	0.232552	−	0.972584i	$-0.425292\pi$
$977$	1.38767e7	0.465104	0.232552	+	0.972584i	$0.425292\pi$
$978$	0	0
$979$	−2.25664e7	−0.752498
$980$	−9.68578e6	−0.322158
$981$	0	0
$982$	−1.33815e7	−0.442819
$983$	−1.20330e7	−0.397183	−0.198591	−	0.980082i	$-0.563637\pi$
$983$	−1.20330e7	−0.397183	−0.198591	+	0.980082i	$0.563637\pi$
$984$	0	0
$985$	−2.33754e6	−0.0767659
$986$	−2.80523e6	−0.0918918
$987$	0	0
$988$	1.15378e7	0.376037
$989$	699391.	0.0227368
$990$	0	0
$991$	9.67955e6	0.313091	0.156546	−	0.987671i	$-0.449964\pi$
$991$	9.67955e6	0.313091	0.156546	+	0.987671i	$0.449964\pi$
$992$	−3.16922e7	−1.02252
$993$	0	0
$994$	−5.62554e7	−1.80592
$995$	5.31303e7	1.70131
$996$	0	0
$997$	−273483.	−0.00871351	−0.00435676	−	0.999991i	$-0.501387\pi$
$997$	−273483.	−0.00871351	−0.00435676	+	0.999991i	$0.501387\pi$
$998$	2.40797e7	0.765289
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.6.a.e.1.8		10
3.2	odd	2		43.6.a.b.1.3	✓	10
12.11	even	2		688.6.a.h.1.9		10
15.14	odd	2		1075.6.a.b.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.3	✓	10	3.2	odd	2
387.6.a.e.1.8		10	1.1	even	1	trivial
688.6.a.h.1.9		10	12.11	even	2
1075.6.a.b.1.8		10	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q+4.31531 q^{2} -13.3781 q^{4} +52.5837 q^{5} -174.859 q^{7} -195.821 q^{8} +O(q^{10})\)	\(q+4.31531 q^{2} -13.3781 q^{4} +52.5837 q^{5} -174.859 q^{7} -195.821 q^{8} +226.915 q^{10} +447.981 q^{11} +669.141 q^{13} -754.569 q^{14} -416.925 q^{16} +849.648 q^{17} -1288.87 q^{19} -703.471 q^{20} +1933.18 q^{22} +378.254 q^{23} -359.956 q^{25} +2887.55 q^{26} +2339.28 q^{28} -765.100 q^{29} -7094.59 q^{31} +4467.10 q^{32} +3666.49 q^{34} -9194.71 q^{35} -7908.22 q^{37} -5561.87 q^{38} -10297.0 q^{40} -12855.9 q^{41} +1849.00 q^{43} -5993.15 q^{44} +1632.28 q^{46} -26785.7 q^{47} +13768.5 q^{49} -1553.32 q^{50} -8951.86 q^{52} -30000.7 q^{53} +23556.5 q^{55} +34240.9 q^{56} -3301.64 q^{58} -1247.64 q^{59} -48441.4 q^{61} -30615.3 q^{62} +32618.5 q^{64} +35185.9 q^{65} +67004.8 q^{67} -11366.7 q^{68} -39678.0 q^{70} +74553.1 q^{71} -18066.8 q^{73} -34126.4 q^{74} +17242.7 q^{76} -78333.4 q^{77} -63230.5 q^{79} -21923.5 q^{80} -55477.3 q^{82} +88066.7 q^{83} +44677.6 q^{85} +7979.00 q^{86} -87724.0 q^{88} -50373.5 q^{89} -117005. q^{91} -5060.33 q^{92} -115588. q^{94} -67773.5 q^{95} +17502.1 q^{97} +59415.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8}+O(q^{10}) \) 10 * q - 8 * q^2 + 202 * q^4 - 138 * q^5 + 60 * q^7 - 294 * q^8	\( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8} - 17 q^{10} - 745 q^{11} + 1917 q^{13} - 1936 q^{14} + 5354 q^{16} - 4017 q^{17} - 2404 q^{19} - 1311 q^{20} - 5836 q^{22} - 1733 q^{23} + 7120 q^{25} + 1484 q^{26} - 15028 q^{28} - 6996 q^{29} - 4899 q^{31} + 7554 q^{32} - 27033 q^{34} - 7084 q^{35} + 1466 q^{37} - 13905 q^{38} - 93211 q^{40} - 10297 q^{41} + 18490 q^{43} + 36140 q^{44} + 17991 q^{46} - 48592 q^{47} + 29458 q^{49} - 983 q^{50} + 14232 q^{52} - 127165 q^{53} + 106672 q^{55} + 7780 q^{56} - 10305 q^{58} - 99372 q^{59} + 17408 q^{61} - 28265 q^{62} + 47202 q^{64} - 54484 q^{65} - 2021 q^{67} - 192151 q^{68} - 33194 q^{70} - 11286 q^{71} + 49892 q^{73} + 125431 q^{74} - 249803 q^{76} - 98144 q^{77} - 91524 q^{79} - 12251 q^{80} - 158909 q^{82} + 105203 q^{83} - 87212 q^{85} - 14792 q^{86} - 461824 q^{88} + 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 7259 q^{94} + 305340 q^{95} + 108383 q^{97} - 354656 q^{98}+O(q^{100}) \) 10 * q - 8 * q^2 + 202 * q^4 - 138 * q^5 + 60 * q^7 - 294 * q^8 - 17 * q^10 - 745 * q^11 + 1917 * q^13 - 1936 * q^14 + 5354 * q^16 - 4017 * q^17 - 2404 * q^19 - 1311 * q^20 - 5836 * q^22 - 1733 * q^23 + 7120 * q^25 + 1484 * q^26 - 15028 * q^28 - 6996 * q^29 - 4899 * q^31 + 7554 * q^32 - 27033 * q^34 - 7084 * q^35 + 1466 * q^37 - 13905 * q^38 - 93211 * q^40 - 10297 * q^41 + 18490 * q^43 + 36140 * q^44 + 17991 * q^46 - 48592 * q^47 + 29458 * q^49 - 983 * q^50 + 14232 * q^52 - 127165 * q^53 + 106672 * q^55 + 7780 * q^56 - 10305 * q^58 - 99372 * q^59 + 17408 * q^61 - 28265 * q^62 + 47202 * q^64 - 54484 * q^65 - 2021 * q^67 - 192151 * q^68 - 33194 * q^70 - 11286 * q^71 + 49892 * q^73 + 125431 * q^74 - 249803 * q^76 - 98144 * q^77 - 91524 * q^79 - 12251 * q^80 - 158909 * q^82 + 105203 * q^83 - 87212 * q^85 - 14792 * q^86 - 461824 * q^88 + 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 7259 * q^94 + 305340 * q^95 + 108383 * q^97 - 354656 * q^98

Introduction

L-functions

Modular forms

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Database

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