LMFDB - Embedded newform 387.6.a.e.1.9

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			1.9
Root			$9.86547$ 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)

$f(q)$	$=$	$q+8.86547 q^{2} +46.5966 q^{4} +37.8251 q^{5} -124.747 q^{7} +129.406 q^{8} +O(q^{10})$	\(q+8.86547 q^{2} +46.5966 q^{4} +37.8251 q^{5} -124.747 q^{7} +129.406 q^{8} +335.337 q^{10} -590.079 q^{11} +434.774 q^{13} -1105.94 q^{14} -343.849 q^{16} -1925.58 q^{17} +654.129 q^{19} +1762.52 q^{20} -5231.33 q^{22} -2805.03 q^{23} -1694.26 q^{25} +3854.47 q^{26} -5812.80 q^{28} +1456.23 q^{29} +4419.85 q^{31} -7189.36 q^{32} -17071.2 q^{34} -4718.58 q^{35} +3753.09 q^{37} +5799.17 q^{38} +4894.78 q^{40} -1972.85 q^{41} +1849.00 q^{43} -27495.7 q^{44} -24867.9 q^{46} -2204.84 q^{47} -1245.12 q^{49} -15020.4 q^{50} +20259.0 q^{52} -24984.2 q^{53} -22319.8 q^{55} -16143.0 q^{56} +12910.1 q^{58} +42756.2 q^{59} -21022.8 q^{61} +39184.0 q^{62} -52733.9 q^{64} +16445.4 q^{65} -25272.1 q^{67} -89725.4 q^{68} -41832.4 q^{70} -48082.8 q^{71} +58801.2 q^{73} +33272.9 q^{74} +30480.2 q^{76} +73610.8 q^{77} +92704.7 q^{79} -13006.1 q^{80} -17490.3 q^{82} +1849.64 q^{83} -72835.2 q^{85} +16392.3 q^{86} -76359.5 q^{88} +70380.3 q^{89} -54236.8 q^{91} -130705. q^{92} -19546.9 q^{94} +24742.5 q^{95} -88842.1 q^{97} -11038.6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	8.86547	1.56721	0.783604	−	0.621260i	$-0.213379\pi$
$2$	8.86547	1.56721	0.783604	+	0.621260i	$0.213379\pi$
$3$	0	0
$3$	0	0
$4$	46.5966	1.45614
$5$	37.8251	0.676636	0.338318	−	0.941032i	$-0.390142\pi$
$5$	37.8251	0.676636	0.338318	+	0.941032i	$0.390142\pi$
$6$	0	0
$7$	−124.747	−0.962246	−0.481123	−	0.876653i	$-0.659771\pi$
$7$	−124.747	−0.962246	−0.481123	+	0.876653i	$0.659771\pi$
$8$	129.406	0.714872
$9$	0	0
$10$	335.337	1.06043
$11$	−590.079	−1.47038	−0.735188	−	0.677863i	$-0.762906\pi$
$11$	−590.079	−1.47038	−0.735188	+	0.677863i	$0.762906\pi$
$12$	0	0
$13$	434.774	0.713518	0.356759	−	0.934196i	$-0.383882\pi$
$13$	434.774	0.713518	0.356759	+	0.934196i	$0.383882\pi$
$14$	−1105.94	−1.50804
$15$	0	0
$16$	−343.849	−0.335790
$17$	−1925.58	−1.61599	−0.807995	−	0.589189i	$-0.799447\pi$
$17$	−1925.58	−1.61599	−0.807995	+	0.589189i	$0.799447\pi$
$18$	0	0
$19$	654.129	0.415700	0.207850	−	0.978161i	$-0.433353\pi$
$19$	654.129	0.415700	0.207850	+	0.978161i	$0.433353\pi$
$20$	1762.52	0.985279
$21$	0	0
$22$	−5231.33	−2.30439
$23$	−2805.03	−1.10565	−0.552825	−	0.833297i	$-0.686450\pi$
$23$	−2805.03	−1.10565	−0.552825	+	0.833297i	$0.686450\pi$
$24$	0	0
$25$	−1694.26	−0.542163
$26$	3854.47	1.11823
$27$	0	0
$28$	−5812.80	−1.40117
$29$	1456.23	0.321540	0.160770	−	0.986992i	$-0.448602\pi$
$29$	1456.23	0.321540	0.160770	+	0.986992i	$0.448602\pi$
$30$	0	0
$31$	4419.85	0.826044	0.413022	−	0.910721i	$-0.364473\pi$
$31$	4419.85	0.826044	0.413022	+	0.910721i	$0.364473\pi$
$32$	−7189.36	−1.24112
$33$	0	0
$34$	−17071.2	−2.53259
$35$	−4718.58	−0.651090
$36$	0	0
$37$	3753.09	0.450697	0.225348	−	0.974278i	$-0.427648\pi$
$37$	3753.09	0.450697	0.225348	+	0.974278i	$0.427648\pi$
$38$	5799.17	0.651488
$39$	0	0
$40$	4894.78	0.483708
$41$	−1972.85	−0.183288	−0.0916442	−	0.995792i	$-0.529212\pi$
$41$	−1972.85	−0.183288	−0.0916442	+	0.995792i	$0.529212\pi$
$42$	0	0
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	−27495.7	−2.14108
$45$	0	0
$46$	−24867.9	−1.73279
$47$	−2204.84	−0.145590	−0.0727950	−	0.997347i	$-0.523192\pi$
$47$	−2204.84	−0.145590	−0.0727950	+	0.997347i	$0.523192\pi$
$48$	0	0
$49$	−1245.12	−0.0740834
$50$	−15020.4	−0.849683
$51$	0	0
$52$	20259.0	1.03898
$53$	−24984.2	−1.22173	−0.610865	−	0.791735i	$-0.709178\pi$
$53$	−24984.2	−1.22173	−0.610865	+	0.791735i	$0.709178\pi$
$54$	0	0
$55$	−22319.8	−0.994910
$56$	−16143.0	−0.687882
$57$	0	0
$58$	12910.1	0.503920
$59$	42756.2	1.59907	0.799537	−	0.600616i	$-0.205078\pi$
$59$	42756.2	1.59907	0.799537	+	0.600616i	$0.205078\pi$
$60$	0	0
$61$	−21022.8	−0.723379	−0.361689	−	0.932299i	$-0.617800\pi$
$61$	−21022.8	−0.723379	−0.361689	+	0.932299i	$0.617800\pi$
$62$	39184.0	1.29458
$63$	0	0
$64$	−52733.9	−1.60931
$65$	16445.4	0.482792
$66$	0	0
$67$	−25272.1	−0.687788	−0.343894	−	0.939008i	$-0.611746\pi$
$67$	−25272.1	−0.687788	−0.343894	+	0.939008i	$0.611746\pi$
$68$	−89725.4	−2.35311
$69$	0	0
$70$	−41832.4	−1.02039
$71$	−48082.8	−1.13199	−0.565997	−	0.824407i	$-0.691509\pi$
$71$	−48082.8	−1.13199	−0.565997	+	0.824407i	$0.691509\pi$
$72$	0	0
$73$	58801.2	1.29145	0.645727	−	0.763568i	$-0.276554\pi$
$73$	58801.2	1.29145	0.645727	+	0.763568i	$0.276554\pi$
$74$	33272.9	0.706336
$75$	0	0
$76$	30480.2	0.605318
$77$	73610.8	1.41486
$78$	0	0
$79$	92704.7	1.67122	0.835611	−	0.549322i	$-0.185114\pi$
$79$	92704.7	1.67122	0.835611	+	0.549322i	$0.185114\pi$
$80$	−13006.1	−0.227208
$81$	0	0
$82$	−17490.3	−0.287251
$83$	1849.64	0.0294708	0.0147354	−	0.999891i	$-0.495309\pi$
$83$	1849.64	0.0294708	0.0147354	+	0.999891i	$0.495309\pi$
$84$	0	0
$85$	−72835.2	−1.09344
$86$	16392.3	0.238997
$87$	0	0
$88$	−76359.5	−1.05113
$89$	70380.3	0.941838	0.470919	−	0.882176i	$-0.343922\pi$
$89$	70380.3	0.941838	0.470919	+	0.882176i	$0.343922\pi$
$90$	0	0
$91$	−54236.8	−0.686579
$92$	−130705.	−1.60999
$93$	0	0
$94$	−19546.9	−0.228170
$95$	24742.5	0.281277
$96$	0	0
$97$	−88842.1	−0.958714	−0.479357	−	0.877620i	$-0.659130\pi$
$97$	−88842.1	−0.958714	−0.479357	+	0.877620i	$0.659130\pi$
$98$	−11038.6	−0.116104
$99$	0	0
$100$	−78946.8	−0.789468
$101$	−179221.	−1.74818	−0.874090	−	0.485763i	$-0.838542\pi$
$101$	−179221.	−1.74818	−0.874090	+	0.485763i	$0.838542\pi$
$102$	0	0
$103$	123443.	1.14650	0.573248	−	0.819382i	$-0.305683\pi$
$103$	123443.	1.14650	0.573248	+	0.819382i	$0.305683\pi$
$104$	56262.1	0.510074
$105$	0	0
$106$	−221496.	−1.91471
$107$	−98991.7	−0.835871	−0.417936	−	0.908477i	$-0.637246\pi$
$107$	−98991.7	−0.835871	−0.417936	+	0.908477i	$0.637246\pi$
$108$	0	0
$109$	46356.0	0.373715	0.186857	−	0.982387i	$-0.440170\pi$
$109$	46356.0	0.373715	0.186857	+	0.982387i	$0.440170\pi$
$110$	−197876.	−1.55923
$111$	0	0
$112$	42894.2	0.323113
$113$	−185076.	−1.36350	−0.681750	−	0.731586i	$-0.738781\pi$
$113$	−185076.	−1.36350	−0.681750	+	0.731586i	$0.738781\pi$
$114$	0	0
$115$	−106101.	−0.748123
$116$	67855.3	0.468208
$117$	0	0
$118$	379054.	2.50608
$119$	240211.	1.55498
$120$	0	0
$121$	187142.	1.16201
$122$	−186377.	−1.13369
$123$	0	0
$124$	205950.	1.20284
$125$	−182289.	−1.04348
$126$	0	0
$127$	−237332.	−1.30571	−0.652856	−	0.757482i	$-0.726429\pi$
$127$	−237332.	−1.30571	−0.652856	+	0.757482i	$0.726429\pi$
$128$	−237451.	−1.28100
$129$	0	0
$130$	145796.	0.756636
$131$	283496.	1.44334	0.721671	−	0.692236i	$-0.243374\pi$
$131$	283496.	1.44334	0.721671	+	0.692236i	$0.243374\pi$
$132$	0	0
$133$	−81600.9	−0.400005
$134$	−224049.	−1.07791
$135$	0	0
$136$	−249181.	−1.15523
$137$	−330992.	−1.50666	−0.753331	−	0.657642i	$-0.771554\pi$
$137$	−330992.	−1.50666	−0.753331	+	0.657642i	$0.771554\pi$
$138$	0	0
$139$	105975.	0.465230	0.232615	−	0.972569i	$-0.425272\pi$
$139$	105975.	0.465230	0.232615	+	0.972569i	$0.425272\pi$
$140$	−219870.	−0.948081
$141$	0	0
$142$	−426277.	−1.77407
$143$	−256551.	−1.04914
$144$	0	0
$145$	55082.0	0.217565
$146$	521300.	2.02398
$147$	0	0
$148$	174881.	0.656279
$149$	196700.	0.725836	0.362918	−	0.931821i	$-0.381780\pi$
$149$	196700.	0.725836	0.362918	+	0.931821i	$0.381780\pi$
$150$	0	0
$151$	−412513.	−1.47230	−0.736148	−	0.676820i	$-0.763357\pi$
$151$	−412513.	−1.47230	−0.736148	+	0.676820i	$0.763357\pi$
$152$	84648.0	0.297172
$153$	0	0
$154$	652594.	2.21739
$155$	167181.	0.558931
$156$	0	0
$157$	410971.	1.33064	0.665322	−	0.746557i	$-0.268294\pi$
$157$	410971.	1.33064	0.665322	+	0.746557i	$0.268294\pi$
$158$	821871.	2.61915
$159$	0	0
$160$	−271938.	−0.839790
$161$	349920.	1.06391
$162$	0	0
$163$	−82488.7	−0.243179	−0.121589	−	0.992580i	$-0.538799\pi$
$163$	−82488.7	−0.243179	−0.121589	+	0.992580i	$0.538799\pi$
$164$	−91928.2	−0.266894
$165$	0	0
$166$	16397.9	0.0461868
$167$	380336.	1.05530	0.527650	−	0.849462i	$-0.323073\pi$
$167$	380336.	1.05530	0.527650	+	0.849462i	$0.323073\pi$
$168$	0	0
$169$	−182265.	−0.490892
$170$	−645718.	−1.71365
$171$	0	0
$172$	86157.1	0.222060
$173$	−91482.2	−0.232392	−0.116196	−	0.993226i	$-0.537070\pi$
$173$	−91482.2	−0.232392	−0.116196	+	0.993226i	$0.537070\pi$
$174$	0	0
$175$	211354.	0.521694
$176$	202898.	0.493738
$177$	0	0
$178$	623955.	1.47606
$179$	607572.	1.41731	0.708656	−	0.705554i	$-0.249302\pi$
$179$	607572.	1.41731	0.708656	+	0.705554i	$0.249302\pi$
$180$	0	0
$181$	351519.	0.797540	0.398770	−	0.917051i	$-0.369437\pi$
$181$	351519.	0.797540	0.398770	+	0.917051i	$0.369437\pi$
$182$	−480835.	−1.07601
$183$	0	0
$184$	−362987.	−0.790398
$185$	141961.	0.304958
$186$	0	0
$187$	1.13624e6	2.37611
$188$	−102738.	−0.212000
$189$	0	0
$190$	219354.	0.440820
$191$	123308.	0.244573	0.122286	−	0.992495i	$-0.460977\pi$
$191$	123308.	0.244573	0.122286	+	0.992495i	$0.460977\pi$
$192$	0	0
$193$	−163481.	−0.315917	−0.157959	−	0.987446i	$-0.550491\pi$
$193$	−163481.	−0.315917	−0.157959	+	0.987446i	$0.550491\pi$
$194$	−787627.	−1.50251
$195$	0	0
$196$	−58018.3	−0.107876
$197$	−581124.	−1.06685	−0.533425	−	0.845848i	$-0.679095\pi$
$197$	−581124.	−1.06685	−0.533425	+	0.845848i	$0.679095\pi$
$198$	0	0
$199$	351526.	0.629253	0.314627	−	0.949216i	$-0.398121\pi$
$199$	351526.	0.629253	0.314627	+	0.949216i	$0.398121\pi$
$200$	−219247.	−0.387577
$201$	0	0
$202$	−1.58888e6	−2.73976
$203$	−181660.	−0.309400
$204$	0	0
$205$	−74623.4	−0.124020
$206$	1.09438e6	1.79680
$207$	0	0
$208$	−149497.	−0.239592
$209$	−385988.	−0.611235
$210$	0	0
$211$	327480.	0.506383	0.253192	−	0.967416i	$-0.418520\pi$
$211$	327480.	0.506383	0.253192	+	0.967416i	$0.418520\pi$
$212$	−1.16418e6	−1.77901
$213$	0	0
$214$	−877608.	−1.30998
$215$	69938.6	0.103186
$216$	0	0
$217$	−551364.	−0.794857
$218$	410968.	0.585689
$219$	0	0
$220$	−1.04003e6	−1.44873
$221$	−837191.	−1.15304
$222$	0	0
$223$	619602.	0.834355	0.417177	−	0.908825i	$-0.363019\pi$
$223$	619602.	0.834355	0.417177	+	0.908825i	$0.363019\pi$
$224$	896853.	1.19427
$225$	0	0
$226$	−1.64079e6	−2.13689
$227$	757446.	0.975635	0.487817	−	0.872946i	$-0.337793\pi$
$227$	757446.	0.975635	0.487817	+	0.872946i	$0.337793\pi$
$228$	0	0
$229$	1.21739e6	1.53405	0.767025	−	0.641617i	$-0.221736\pi$
$229$	1.21739e6	1.53405	0.767025	+	0.641617i	$0.221736\pi$
$230$	−940632.	−1.17247
$231$	0	0
$232$	188444.	0.229860
$233$	−379715.	−0.458213	−0.229107	−	0.973401i	$-0.573580\pi$
$233$	−379715.	−0.458213	−0.229107	+	0.973401i	$0.573580\pi$
$234$	0	0
$235$	−83398.2	−0.0985115
$236$	1.99229e6	2.32848
$237$	0	0
$238$	2.12958e6	2.43698
$239$	−281756.	−0.319064	−0.159532	−	0.987193i	$-0.550999\pi$
$239$	−281756.	−0.319064	−0.159532	+	0.987193i	$0.550999\pi$
$240$	0	0
$241$	−1.12524e6	−1.24796	−0.623981	−	0.781439i	$-0.714486\pi$
$241$	−1.12524e6	−1.24796	−0.623981	+	0.781439i	$0.714486\pi$
$242$	1.65911e6	1.82111
$243$	0	0
$244$	−979590.	−1.05334
$245$	−47096.8	−0.0501275
$246$	0	0
$247$	284398.	0.296609
$248$	571953.	0.590515
$249$	0	0
$250$	−1.61608e6	−1.63536
$251$	675567.	0.676837	0.338419	−	0.940996i	$-0.390108\pi$
$251$	675567.	0.676837	0.338419	+	0.940996i	$0.390108\pi$
$252$	0	0
$253$	1.65519e6	1.62572
$254$	−2.10406e6	−2.04632
$255$	0	0
$256$	−417633.	−0.398286
$257$	−1.85623e6	−1.75306	−0.876532	−	0.481343i	$-0.840149\pi$
$257$	−1.85623e6	−1.75306	−0.876532	+	0.481343i	$0.840149\pi$
$258$	0	0
$259$	−468188.	−0.433681
$260$	766298.	0.703014
$261$	0	0
$262$	2.51333e6	2.26202
$263$	1.54160e6	1.37430	0.687149	−	0.726516i	$-0.258862\pi$
$263$	1.54160e6	1.37430	0.687149	+	0.726516i	$0.258862\pi$
$264$	0	0
$265$	−945029.	−0.826667
$266$	−723430.	−0.626892
$267$	0	0
$268$	−1.17759e6	−1.00152
$269$	−1.16775e6	−0.983945	−0.491973	−	0.870611i	$-0.663724\pi$
$269$	−1.16775e6	−0.983945	−0.491973	+	0.870611i	$0.663724\pi$
$270$	0	0
$271$	−1.20256e6	−0.994684	−0.497342	−	0.867555i	$-0.665691\pi$
$271$	−1.20256e6	−0.994684	−0.497342	+	0.867555i	$0.665691\pi$
$272$	662109.	0.542634
$273$	0	0
$274$	−2.93440e6	−2.36125
$275$	999748.	0.797184
$276$	0	0
$277$	−504463.	−0.395030	−0.197515	−	0.980300i	$-0.563287\pi$
$277$	−504463.	−0.395030	−0.197515	+	0.980300i	$0.563287\pi$
$278$	939521.	0.729112
$279$	0	0
$280$	−610610.	−0.465446
$281$	−1.52712e6	−1.15374	−0.576870	−	0.816836i	$-0.695726\pi$
$281$	−1.52712e6	−1.15374	−0.576870	+	0.816836i	$0.695726\pi$
$282$	0	0
$283$	1.81916e6	1.35022	0.675110	−	0.737717i	$-0.264096\pi$
$283$	1.81916e6	1.35022	0.675110	+	0.737717i	$0.264096\pi$
$284$	−2.24049e6	−1.64834
$285$	0	0
$286$	−2.27444e6	−1.64422
$287$	246108.	0.176369
$288$	0	0
$289$	2.28799e6	1.61143
$290$	488328.	0.340970
$291$	0	0
$292$	2.73993e6	1.88054
$293$	2.35975e6	1.60582	0.802911	−	0.596099i	$-0.203283\pi$
$293$	2.35975e6	1.60582	0.802911	+	0.596099i	$0.203283\pi$
$294$	0	0
$295$	1.61726e6	1.08199
$296$	485671.	0.322190
$297$	0	0
$298$	1.74384e6	1.13754
$299$	−1.21955e6	−0.788901
$300$	0	0
$301$	−230658.	−0.146741
$302$	−3.65712e6	−2.30740
$303$	0	0
$304$	−224922.	−0.139588
$305$	−795189.	−0.489464
$306$	0	0
$307$	−2.59790e6	−1.57317	−0.786587	−	0.617480i	$-0.788154\pi$
$307$	−2.59790e6	−1.57317	−0.786587	+	0.617480i	$0.788154\pi$
$308$	3.43001e6	2.06024
$309$	0	0
$310$	1.48214e6	0.875962
$311$	2.21471e6	1.29842	0.649212	−	0.760607i	$-0.275099\pi$
$311$	2.21471e6	1.29842	0.649212	+	0.760607i	$0.275099\pi$
$312$	0	0
$313$	−2.43630e6	−1.40563	−0.702813	−	0.711374i	$-0.748073\pi$
$313$	−2.43630e6	−1.40563	−0.702813	+	0.711374i	$0.748073\pi$
$314$	3.64345e6	2.08540
$315$	0	0
$316$	4.31972e6	2.43354
$317$	2.59108e6	1.44821	0.724106	−	0.689689i	$-0.242253\pi$
$317$	2.59108e6	1.44821	0.724106	+	0.689689i	$0.242253\pi$
$318$	0	0
$319$	−859290.	−0.472784
$320$	−1.99467e6	−1.08892
$321$	0	0
$322$	3.10220e6	1.66737
$323$	−1.25958e6	−0.671767
$324$	0	0
$325$	−736620.	−0.386843
$326$	−731301.	−0.381112
$327$	0	0
$328$	−255298.	−0.131028
$329$	275047.	0.140093
$330$	0	0
$331$	−2.49362e6	−1.25101	−0.625503	−	0.780221i	$-0.715106\pi$
$331$	−2.49362e6	−1.25101	−0.625503	+	0.780221i	$0.715106\pi$
$332$	86186.7	0.0429136
$333$	0	0
$334$	3.37186e6	1.65388
$335$	−955921.	−0.465383
$336$	0	0
$337$	−159199.	−0.0763600	−0.0381800	−	0.999271i	$-0.512156\pi$
$337$	−159199.	−0.0763600	−0.0381800	+	0.999271i	$0.512156\pi$
$338$	−1.61586e6	−0.769331
$339$	0	0
$340$	−3.39387e6	−1.59220
$341$	−2.60806e6	−1.21460
$342$	0	0
$343$	2.25195e6	1.03353
$344$	239271.	0.109017
$345$	0	0
$346$	−811033.	−0.364207
$347$	−4.35264e6	−1.94057	−0.970285	−	0.241966i	$-0.922208\pi$
$347$	−4.35264e6	−1.94057	−0.970285	+	0.241966i	$0.922208\pi$
$348$	0	0
$349$	−502730.	−0.220938	−0.110469	−	0.993880i	$-0.535235\pi$
$349$	−502730.	−0.220938	−0.110469	+	0.993880i	$0.535235\pi$
$350$	1.87376e6	0.817604
$351$	0	0
$352$	4.24229e6	1.82492
$353$	990932.	0.423260	0.211630	−	0.977350i	$-0.432123\pi$
$353$	990932.	0.423260	0.211630	+	0.977350i	$0.432123\pi$
$354$	0	0
$355$	−1.81874e6	−0.765948
$356$	3.27948e6	1.37145
$357$	0	0
$358$	5.38641e6	2.22122
$359$	1.78805e6	0.732223	0.366112	−	0.930571i	$-0.380689\pi$
$359$	1.78805e6	0.732223	0.366112	+	0.930571i	$0.380689\pi$
$360$	0	0
$361$	−2.04821e6	−0.827194
$362$	3.11638e6	1.24991
$363$	0	0
$364$	−2.52725e6	−0.999758
$365$	2.22416e6	0.873844
$366$	0	0
$367$	−68021.3	−0.0263621	−0.0131810	−	0.999913i	$-0.504196\pi$
$367$	−68021.3	−0.0263621	−0.0131810	+	0.999913i	$0.504196\pi$
$368$	964507.	0.371267
$369$	0	0
$370$	1.25855e6	0.477933
$371$	3.11671e6	1.17560
$372$	0	0
$373$	−949477.	−0.353356	−0.176678	−	0.984269i	$-0.556535\pi$
$373$	−949477.	−0.353356	−0.176678	+	0.984269i	$0.556535\pi$
$374$	1.00733e7	3.72387
$375$	0	0
$376$	−285318.	−0.104078
$377$	633130.	0.229424
$378$	0	0
$379$	2.53069e6	0.904982	0.452491	−	0.891769i	$-0.350535\pi$
$379$	2.53069e6	0.904982	0.452491	+	0.891769i	$0.350535\pi$
$380$	1.15292e6	0.409580
$381$	0	0
$382$	1.09318e6	0.383297
$383$	−500631.	−0.174390	−0.0871948	−	0.996191i	$-0.527790\pi$
$383$	−500631.	−0.174390	−0.0871948	+	0.996191i	$0.527790\pi$
$384$	0	0
$385$	2.78434e6	0.957348
$386$	−1.44933e6	−0.495108
$387$	0	0
$388$	−4.13974e6	−1.39603
$389$	4.08546e6	1.36889	0.684443	−	0.729066i	$-0.260045\pi$
$389$	4.08546e6	1.36889	0.684443	+	0.729066i	$0.260045\pi$
$390$	0	0
$391$	5.40130e6	1.78672
$392$	−161125.	−0.0529601
$393$	0	0
$394$	−5.15194e6	−1.67198
$395$	3.50657e6	1.13081
$396$	0	0
$397$	953651.	0.303678	0.151839	−	0.988405i	$-0.451480\pi$
$397$	953651.	0.303678	0.151839	+	0.988405i	$0.451480\pi$
$398$	3.11645e6	0.986171
$399$	0	0
$400$	582570.	0.182053
$401$	−4.81115e6	−1.49413	−0.747064	−	0.664752i	$-0.768537\pi$
$401$	−4.81115e6	−1.49413	−0.747064	+	0.664752i	$0.768537\pi$
$402$	0	0
$403$	1.92163e6	0.589397
$404$	−8.35110e6	−2.54560
$405$	0	0
$406$	−1.61051e6	−0.484894
$407$	−2.21462e6	−0.662694
$408$	0	0
$409$	−1.42551e6	−0.421370	−0.210685	−	0.977554i	$-0.567569\pi$
$409$	−1.42551e6	−0.421370	−0.210685	+	0.977554i	$0.567569\pi$
$410$	−661572.	−0.194365
$411$	0	0
$412$	5.75201e6	1.66946
$413$	−5.33372e6	−1.53870
$414$	0	0
$415$	69962.7	0.0199410
$416$	−3.12575e6	−0.885565
$417$	0	0
$418$	−3.42197e6	−0.957933
$419$	−2.91631e6	−0.811520	−0.405760	−	0.913980i	$-0.632993\pi$
$419$	−2.91631e6	−0.811520	−0.405760	+	0.913980i	$0.632993\pi$
$420$	0	0
$421$	6.91223e6	1.90070	0.950349	−	0.311186i	$-0.100726\pi$
$421$	6.91223e6	1.90070	0.950349	+	0.311186i	$0.100726\pi$
$422$	2.90327e6	0.793608
$423$	0	0
$424$	−3.23309e6	−0.873380
$425$	3.26243e6	0.876131
$426$	0	0
$427$	2.62254e6	0.696068
$428$	−4.61268e6	−1.21715
$429$	0	0
$430$	620039.	0.161714
$431$	−2.58575e6	−0.670493	−0.335246	−	0.942131i	$-0.608820\pi$
$431$	−2.58575e6	−0.670493	−0.335246	+	0.942131i	$0.608820\pi$
$432$	0	0
$433$	−3.71200e6	−0.951455	−0.475727	−	0.879593i	$-0.657815\pi$
$433$	−3.71200e6	−0.951455	−0.475727	+	0.879593i	$0.657815\pi$
$434$	−4.88810e6	−1.24571
$435$	0	0
$436$	2.16003e6	0.544182
$437$	−1.83485e6	−0.459619
$438$	0	0
$439$	−415772.	−0.102966	−0.0514830	−	0.998674i	$-0.516395\pi$
$439$	−415772.	−0.102966	−0.0514830	+	0.998674i	$0.516395\pi$
$440$	−2.88831e6	−0.711233
$441$	0	0
$442$	−7.42209e6	−1.80705
$443$	602736.	0.145921	0.0729605	−	0.997335i	$-0.476755\pi$
$443$	602736.	0.145921	0.0729605	+	0.997335i	$0.476755\pi$
$444$	0	0
$445$	2.66214e6	0.637282
$446$	5.49307e6	1.30761
$447$	0	0
$448$	6.57841e6	1.54855
$449$	−1.84907e6	−0.432849	−0.216424	−	0.976299i	$-0.569440\pi$
$449$	−1.84907e6	−0.432849	−0.216424	+	0.976299i	$0.569440\pi$
$450$	0	0
$451$	1.16414e6	0.269503
$452$	−8.62393e6	−1.98545
$453$	0	0
$454$	6.71512e6	1.52902
$455$	−2.05151e6	−0.464564
$456$	0	0
$457$	−7.35472e6	−1.64731	−0.823655	−	0.567091i	$-0.808069\pi$
$457$	−7.35472e6	−1.64731	−0.823655	+	0.567091i	$0.808069\pi$
$458$	1.07927e7	2.40418
$459$	0	0
$460$	−4.94392e6	−1.08937
$461$	3.51032e6	0.769298	0.384649	−	0.923063i	$-0.374322\pi$
$461$	3.51032e6	0.769298	0.384649	+	0.923063i	$0.374322\pi$
$462$	0	0
$463$	−6.86607e6	−1.48852	−0.744262	−	0.667888i	$-0.767199\pi$
$463$	−6.86607e6	−1.48852	−0.744262	+	0.667888i	$0.767199\pi$
$464$	−500723.	−0.107970
$465$	0	0
$466$	−3.36635e6	−0.718116
$467$	−5.66490e6	−1.20199	−0.600993	−	0.799254i	$-0.705228\pi$
$467$	−5.66490e6	−1.20199	−0.600993	+	0.799254i	$0.705228\pi$
$468$	0	0
$469$	3.15263e6	0.661821
$470$	−739364.	−0.154388
$471$	0	0
$472$	5.53289e6	1.14313
$473$	−1.09106e6	−0.224230
$474$	0	0
$475$	−1.10827e6	−0.225377
$476$	1.11930e7	2.26427
$477$	0	0
$478$	−2.49790e6	−0.500041
$479$	−4.55961e6	−0.908006	−0.454003	−	0.891000i	$-0.650004\pi$
$479$	−4.55961e6	−0.908006	−0.454003	+	0.891000i	$0.650004\pi$
$480$	0	0
$481$	1.63174e6	0.321580
$482$	−9.97576e6	−1.95582
$483$	0	0
$484$	8.72019e6	1.69205
$485$	−3.36046e6	−0.648701
$486$	0	0
$487$	−9.17550e6	−1.75310	−0.876552	−	0.481308i	$-0.840162\pi$
$487$	−9.17550e6	−1.75310	−0.876552	+	0.481308i	$0.840162\pi$
$488$	−2.72047e6	−0.517123
$489$	0	0
$490$	−417535.	−0.0785603
$491$	−3.41449e6	−0.639179	−0.319589	−	0.947556i	$-0.603545\pi$
$491$	−3.41449e6	−0.639179	−0.319589	+	0.947556i	$0.603545\pi$
$492$	0	0
$493$	−2.80408e6	−0.519605
$494$	2.52132e6	0.464848
$495$	0	0
$496$	−1.51976e6	−0.277377
$497$	5.99820e6	1.08926
$498$	0	0
$499$	−1.99364e6	−0.358423	−0.179211	−	0.983811i	$-0.557355\pi$
$499$	−1.99364e6	−0.358423	−0.179211	+	0.983811i	$0.557355\pi$
$500$	−8.49405e6	−1.51946
$501$	0	0
$502$	5.98922e6	1.06074
$503$	−9.15685e6	−1.61371	−0.806856	−	0.590748i	$-0.798833\pi$
$503$	−9.15685e6	−1.61371	−0.806856	+	0.590748i	$0.798833\pi$
$504$	0	0
$505$	−6.77907e6	−1.18288
$506$	1.46740e7	2.54785
$507$	0	0
$508$	−1.10589e7	−1.90130
$509$	−6.56697e6	−1.12349	−0.561746	−	0.827309i	$-0.689870\pi$
$509$	−6.56697e6	−1.12349	−0.561746	+	0.827309i	$0.689870\pi$
$510$	0	0
$511$	−7.33529e6	−1.24270
$512$	3.89593e6	0.656805
$513$	0	0
$514$	−1.64563e7	−2.74742
$515$	4.66924e6	0.775761
$516$	0	0
$517$	1.30103e6	0.214072
$518$	−4.15070e6	−0.679669
$519$	0	0
$520$	2.12812e6	0.345134
$521$	−6.45805e6	−1.04234	−0.521168	−	0.853454i	$-0.674503\pi$
$521$	−6.45805e6	−1.04234	−0.521168	+	0.853454i	$0.674503\pi$
$522$	0	0
$523$	184405.	0.0294794	0.0147397	−	0.999891i	$-0.495308\pi$
$523$	184405.	0.0294794	0.0147397	+	0.999891i	$0.495308\pi$
$524$	1.32100e7	2.10171
$525$	0	0
$526$	1.36670e7	2.15381
$527$	−8.51076e6	−1.33488
$528$	0	0
$529$	1.43185e6	0.222463
$530$	−8.37813e6	−1.29556
$531$	0	0
$532$	−3.80232e6	−0.582465
$533$	−857745.	−0.130780
$534$	0	0
$535$	−3.74437e6	−0.565581
$536$	−3.27035e6	−0.491680
$537$	0	0
$538$	−1.03527e7	−1.54205
$539$	734719.	0.108931
$540$	0	0
$541$	−7.10369e6	−1.04350	−0.521748	−	0.853100i	$-0.674720\pi$
$541$	−7.10369e6	−1.04350	−0.521748	+	0.853100i	$0.674720\pi$
$542$	−1.06613e7	−1.55888
$543$	0	0
$544$	1.38437e7	2.00565
$545$	1.75342e6	0.252869
$546$	0	0
$547$	−9.75210e6	−1.39357	−0.696787	−	0.717279i	$-0.745388\pi$
$547$	−9.75210e6	−1.39357	−0.696787	+	0.717279i	$0.745388\pi$
$548$	−1.54231e7	−2.19392
$549$	0	0
$550$	8.86324e6	1.24935
$551$	952562.	0.133664
$552$	0	0
$553$	−1.15647e7	−1.60813
$554$	−4.47230e6	−0.619094
$555$	0	0
$556$	4.93809e6	0.677441
$557$	2.28586e6	0.312185	0.156092	−	0.987742i	$-0.450110\pi$
$557$	2.28586e6	0.312185	0.156092	+	0.987742i	$0.450110\pi$
$558$	0	0
$559$	803896.	0.108810
$560$	1.62248e6	0.218630
$561$	0	0
$562$	−1.35386e7	−1.80815
$563$	−2.31099e6	−0.307275	−0.153638	−	0.988127i	$-0.549099\pi$
$563$	−2.31099e6	−0.307275	−0.153638	+	0.988127i	$0.549099\pi$
$564$	0	0
$565$	−7.00054e6	−0.922593
$566$	1.61277e7	2.11608
$567$	0	0
$568$	−6.22218e6	−0.809230
$569$	−1.17226e7	−1.51790	−0.758949	−	0.651151i	$-0.774287\pi$
$569$	−1.17226e7	−1.51790	−0.758949	+	0.651151i	$0.774287\pi$
$570$	0	0
$571$	−5.10057e6	−0.654679	−0.327340	−	0.944907i	$-0.606152\pi$
$571$	−5.10057e6	−0.654679	−0.327340	+	0.944907i	$0.606152\pi$
$572$	−1.19544e7	−1.52770
$573$	0	0
$574$	2.18186e6	0.276406
$575$	4.75245e6	0.599443
$576$	0	0
$577$	599429.	0.0749546	0.0374773	−	0.999297i	$-0.488068\pi$
$577$	599429.	0.0749546	0.0374773	+	0.999297i	$0.488068\pi$
$578$	2.02841e7	2.52544
$579$	0	0
$580$	2.56663e6	0.316806
$581$	−230737.	−0.0283581
$582$	0	0
$583$	1.47426e7	1.79640
$584$	7.60920e6	0.923224
$585$	0	0
$586$	2.09203e7	2.51666
$587$	−4.32755e6	−0.518379	−0.259189	−	0.965826i	$-0.583455\pi$
$587$	−4.32755e6	−0.518379	−0.259189	+	0.965826i	$0.583455\pi$
$588$	0	0
$589$	2.89115e6	0.343386
$590$	1.43377e7	1.69571
$591$	0	0
$592$	−1.29050e6	−0.151340
$593$	1.59099e7	1.85794	0.928968	−	0.370160i	$-0.120697\pi$
$593$	1.59099e7	1.85794	0.928968	+	0.370160i	$0.120697\pi$
$594$	0	0
$595$	9.08599e6	1.05216
$596$	9.16555e6	1.05692
$597$	0	0
$598$	−1.08119e7	−1.23637
$599$	−73202.1	−0.00833598	−0.00416799	−	0.999991i	$-0.501327\pi$
$599$	−73202.1	−0.00833598	−0.00416799	+	0.999991i	$0.501327\pi$
$600$	0	0
$601$	−3.77328e6	−0.426121	−0.213060	−	0.977039i	$-0.568343\pi$
$601$	−3.77328e6	−0.426121	−0.213060	+	0.977039i	$0.568343\pi$
$602$	−2.04489e6	−0.229974
$603$	0	0
$604$	−1.92217e7	−2.14387
$605$	7.07868e6	0.786256
$606$	0	0
$607$	3.39314e6	0.373792	0.186896	−	0.982380i	$-0.440157\pi$
$607$	3.39314e6	0.373792	0.186896	+	0.982380i	$0.440157\pi$
$608$	−4.70277e6	−0.515935
$609$	0	0
$610$	−7.04973e6	−0.767093
$611$	−958605.	−0.103881
$612$	0	0
$613$	5.93491e6	0.637915	0.318957	−	0.947769i	$-0.396667\pi$
$613$	5.93491e6	0.637915	0.318957	+	0.947769i	$0.396667\pi$
$614$	−2.30316e7	−2.46549
$615$	0	0
$616$	9.52564e6	1.01145
$617$	1.07871e7	1.14076	0.570379	−	0.821382i	$-0.306796\pi$
$617$	1.07871e7	1.14076	0.570379	+	0.821382i	$0.306796\pi$
$618$	0	0
$619$	9.97743e6	1.04663	0.523313	−	0.852140i	$-0.324696\pi$
$619$	9.97743e6	1.04663	0.523313	+	0.852140i	$0.324696\pi$
$620$	7.79008e6	0.813884
$621$	0	0
$622$	1.96345e7	2.03490
$623$	−8.77976e6	−0.906280
$624$	0	0
$625$	−1.60054e6	−0.163895
$626$	−2.15990e7	−2.20291
$627$	0	0
$628$	1.91498e7	1.93761
$629$	−7.22687e6	−0.728322
$630$	0	0
$631$	−3.42994e6	−0.342936	−0.171468	−	0.985190i	$-0.554851\pi$
$631$	−3.42994e6	−0.342936	−0.171468	+	0.985190i	$0.554851\pi$
$632$	1.19965e7	1.19471
$633$	0	0
$634$	2.29711e7	2.26965
$635$	−8.97711e6	−0.883492
$636$	0	0
$637$	−541345.	−0.0528598
$638$	−7.61801e6	−0.740952
$639$	0	0
$640$	−8.98163e6	−0.866772
$641$	1.41725e7	1.36239	0.681193	−	0.732104i	$-0.261462\pi$
$641$	1.41725e7	1.36239	0.681193	+	0.732104i	$0.261462\pi$
$642$	0	0
$643$	4.26802e6	0.407098	0.203549	−	0.979065i	$-0.434752\pi$
$643$	4.26802e6	0.407098	0.203549	+	0.979065i	$0.434752\pi$
$644$	1.63051e7	1.54920
$645$	0	0
$646$	−1.11667e7	−1.05280
$647$	−1.26038e7	−1.18370	−0.591849	−	0.806049i	$-0.701602\pi$
$647$	−1.26038e7	−1.18370	−0.591849	+	0.806049i	$0.701602\pi$
$648$	0	0
$649$	−2.52295e7	−2.35124
$650$	−6.53048e6	−0.606264
$651$	0	0
$652$	−3.84369e6	−0.354103
$653$	−1.67724e7	−1.53926	−0.769631	−	0.638489i	$-0.779560\pi$
$653$	−1.67724e7	−1.53926	−0.769631	+	0.638489i	$0.779560\pi$
$654$	0	0
$655$	1.07233e7	0.976618
$656$	678364.	0.0615465
$657$	0	0
$658$	2.43842e6	0.219556
$659$	1.72113e7	1.54383	0.771915	−	0.635725i	$-0.219299\pi$
$659$	1.72113e7	1.54383	0.771915	+	0.635725i	$0.219299\pi$
$660$	0	0
$661$	5.67723e6	0.505397	0.252699	−	0.967545i	$-0.418682\pi$
$661$	5.67723e6	0.505397	0.252699	+	0.967545i	$0.418682\pi$
$662$	−2.21071e7	−1.96059
$663$	0	0
$664$	239353.	0.0210678
$665$	−3.08656e6	−0.270658
$666$	0	0
$667$	−4.08476e6	−0.355510
$668$	1.77223e7	1.53667
$669$	0	0
$670$	−8.47469e6	−0.729352
$671$	1.24051e7	1.06364
$672$	0	0
$673$	4.92989e6	0.419565	0.209783	−	0.977748i	$-0.432724\pi$
$673$	4.92989e6	0.419565	0.209783	+	0.977748i	$0.432724\pi$
$674$	−1.41137e6	−0.119672
$675$	0	0
$676$	−8.49292e6	−0.714809
$677$	1.25498e7	1.05236	0.526180	−	0.850373i	$-0.323624\pi$
$677$	1.25498e7	1.05236	0.526180	+	0.850373i	$0.323624\pi$
$678$	0	0
$679$	1.10828e7	0.922519
$680$	−9.42528e6	−0.781668
$681$	0	0
$682$	−2.31217e7	−1.90352
$683$	7.08503e6	0.581152	0.290576	−	0.956852i	$-0.406153\pi$
$683$	7.08503e6	0.581152	0.290576	+	0.956852i	$0.406153\pi$
$684$	0	0
$685$	−1.25198e7	−1.01946
$686$	1.99646e7	1.61976
$687$	0	0
$688$	−635777.	−0.0512075
$689$	−1.08625e7	−0.871726
$690$	0	0
$691$	−9.40053e6	−0.748958	−0.374479	−	0.927235i	$-0.622178\pi$
$691$	−9.40053e6	−0.748958	−0.374479	+	0.927235i	$0.622178\pi$
$692$	−4.26276e6	−0.338396
$693$	0	0
$694$	−3.85882e7	−3.04128
$695$	4.00853e6	0.314791
$696$	0	0
$697$	3.79888e6	0.296192
$698$	−4.45694e6	−0.346256
$699$	0	0
$700$	9.84839e6	0.759662
$701$	−2.14131e7	−1.64583	−0.822914	−	0.568167i	$-0.807653\pi$
$701$	−2.14131e7	−1.64583	−0.822914	+	0.568167i	$0.807653\pi$
$702$	0	0
$703$	2.45501e6	0.187355
$704$	3.11172e7	2.36629
$705$	0	0
$706$	8.78508e6	0.663337
$707$	2.23574e7	1.68218
$708$	0	0
$709$	1.70494e7	1.27377	0.636887	−	0.770957i	$-0.280222\pi$
$709$	1.70494e7	1.27377	0.636887	+	0.770957i	$0.280222\pi$
$710$	−1.61240e7	−1.20040
$711$	0	0
$712$	9.10761e6	0.673293
$713$	−1.23978e7	−0.913316
$714$	0	0
$715$	−9.70406e6	−0.709886
$716$	2.83108e7	2.06381
$717$	0	0
$718$	1.58519e7	1.14755
$719$	1.59376e7	1.14974	0.574872	−	0.818243i	$-0.305052\pi$
$719$	1.59376e7	1.14974	0.574872	+	0.818243i	$0.305052\pi$
$720$	0	0
$721$	−1.53991e7	−1.10321
$722$	−1.81584e7	−1.29639
$723$	0	0
$724$	1.63796e7	1.16133
$725$	−2.46723e6	−0.174327
$726$	0	0
$727$	8.26202e6	0.579763	0.289881	−	0.957063i	$-0.406384\pi$
$727$	8.26202e6	0.579763	0.289881	+	0.957063i	$0.406384\pi$
$728$	−7.01855e6	−0.490816
$729$	0	0
$730$	1.97182e7	1.36950
$731$	−3.56039e6	−0.246436
$732$	0	0
$733$	6.20168e6	0.426334	0.213167	−	0.977016i	$-0.431622\pi$
$733$	6.20168e6	0.426334	0.213167	+	0.977016i	$0.431622\pi$
$734$	−603041.	−0.0413149
$735$	0	0
$736$	2.01664e7	1.37225
$737$	1.49126e7	1.01131
$738$	0	0
$739$	−1.19003e7	−0.801578	−0.400789	−	0.916170i	$-0.631264\pi$
$739$	−1.19003e7	−0.801578	−0.400789	+	0.916170i	$0.631264\pi$
$740$	6.61490e6	0.444062
$741$	0	0
$742$	2.76311e7	1.84242
$743$	−7.41722e6	−0.492912	−0.246456	−	0.969154i	$-0.579266\pi$
$743$	−7.41722e6	−0.492912	−0.246456	+	0.969154i	$0.579266\pi$
$744$	0	0
$745$	7.44020e6	0.491127
$746$	−8.41756e6	−0.553782
$747$	0	0
$748$	5.29451e7	3.45996
$749$	1.23489e7	0.804314
$750$	0	0
$751$	1.46691e7	0.949084	0.474542	−	0.880233i	$-0.342614\pi$
$751$	1.46691e7	0.949084	0.474542	+	0.880233i	$0.342614\pi$
$752$	758131.	0.0488877
$753$	0	0
$754$	5.61299e6	0.359556
$755$	−1.56033e7	−0.996209
$756$	0	0
$757$	2.41775e6	0.153346	0.0766729	−	0.997056i	$-0.475570\pi$
$757$	2.41775e6	0.153346	0.0766729	+	0.997056i	$0.475570\pi$
$758$	2.24357e7	1.41830
$759$	0	0
$760$	3.20182e6	0.201077
$761$	−2.32422e7	−1.45484	−0.727422	−	0.686191i	$-0.759282\pi$
$761$	−2.32422e7	−1.45484	−0.727422	+	0.686191i	$0.759282\pi$
$762$	0	0
$763$	−5.78279e6	−0.359605
$764$	5.74574e6	0.356133
$765$	0	0
$766$	−4.43833e6	−0.273305
$767$	1.85893e7	1.14097
$768$	0	0
$769$	−2.55432e7	−1.55761	−0.778806	−	0.627265i	$-0.784174\pi$
$769$	−2.55432e7	−1.55761	−0.778806	+	0.627265i	$0.784174\pi$
$770$	2.46844e7	1.50036
$771$	0	0
$772$	−7.61765e6	−0.460021
$773$	−1.72159e7	−1.03629	−0.518146	−	0.855292i	$-0.673378\pi$
$773$	−1.72159e7	−1.03629	−0.518146	+	0.855292i	$0.673378\pi$
$774$	0	0
$775$	−7.48838e6	−0.447851
$776$	−1.14967e7	−0.685358
$777$	0	0
$778$	3.62196e7	2.14533
$779$	−1.29050e6	−0.0761930
$780$	0	0
$781$	2.83727e7	1.66446
$782$	4.78851e7	2.80016
$783$	0	0
$784$	428134.	0.0248765
$785$	1.55450e7	0.900362
$786$	0	0
$787$	−1.48812e7	−0.856449	−0.428225	−	0.903672i	$-0.640861\pi$
$787$	−1.48812e7	−0.856449	−0.428225	+	0.903672i	$0.640861\pi$
$788$	−2.70784e7	−1.55349
$789$	0	0
$790$	3.10874e7	1.77221
$791$	2.30878e7	1.31202
$792$	0	0
$793$	−9.14016e6	−0.516144
$794$	8.45456e6	0.475927
$795$	0	0
$796$	1.63799e7	0.916283
$797$	−5.81135e6	−0.324064	−0.162032	−	0.986785i	$-0.551805\pi$
$797$	−5.81135e6	−0.324064	−0.162032	+	0.986785i	$0.551805\pi$
$798$	0	0
$799$	4.24559e6	0.235272
$800$	1.21807e7	0.672893
$801$	0	0
$802$	−4.26531e7	−2.34161
$803$	−3.46973e7	−1.89892
$804$	0	0
$805$	1.32358e7	0.719878
$806$	1.70362e7	0.923708
$807$	0	0
$808$	−2.31922e7	−1.24972
$809$	2.41530e7	1.29748	0.648740	−	0.761010i	$-0.275296\pi$
$809$	2.41530e7	1.29748	0.648740	+	0.761010i	$0.275296\pi$
$810$	0	0
$811$	2.14467e7	1.14501	0.572505	−	0.819901i	$-0.305972\pi$
$811$	2.14467e7	1.14501	0.572505	+	0.819901i	$0.305972\pi$
$812$	−8.46476e6	−0.450531
$813$	0	0
$814$	−1.96336e7	−1.03858
$815$	−3.12014e6	−0.164543
$816$	0	0
$817$	1.20949e6	0.0633936
$818$	−1.26379e7	−0.660375
$819$	0	0
$820$	−3.47720e6	−0.180590
$821$	1.06313e7	0.550465	0.275232	−	0.961378i	$-0.411245\pi$
$821$	1.06313e7	0.550465	0.275232	+	0.961378i	$0.411245\pi$
$822$	0	0
$823$	−871518.	−0.0448515	−0.0224257	−	0.999749i	$-0.507139\pi$
$823$	−871518.	−0.0448515	−0.0224257	+	0.999749i	$0.507139\pi$
$824$	1.59742e7	0.819597
$825$	0	0
$826$	−4.72859e7	−2.41147
$827$	1.70354e7	0.866141	0.433071	−	0.901360i	$-0.357430\pi$
$827$	1.70354e7	0.866141	0.433071	+	0.901360i	$0.357430\pi$
$828$	0	0
$829$	5.65731e6	0.285906	0.142953	−	0.989729i	$-0.454340\pi$
$829$	5.65731e6	0.285906	0.142953	+	0.989729i	$0.454340\pi$
$830$	620252.	0.0312517
$831$	0	0
$832$	−2.29273e7	−1.14827
$833$	2.39758e6	0.119718
$834$	0	0
$835$	1.43862e7	0.714054
$836$	−1.79857e7	−0.890046
$837$	0	0
$838$	−2.58545e7	−1.27182
$839$	−3.42452e7	−1.67956	−0.839779	−	0.542929i	$-0.817315\pi$
$839$	−3.42452e7	−1.67956	−0.839779	+	0.542929i	$0.817315\pi$
$840$	0	0
$841$	−1.83905e7	−0.896612
$842$	6.12802e7	2.97879
$843$	0	0
$844$	1.52595e7	0.737366
$845$	−6.89419e6	−0.332155
$846$	0	0
$847$	−2.33455e7	−1.11814
$848$	8.59078e6	0.410245
$849$	0	0
$850$	2.89230e7	1.37308
$851$	−1.05275e7	−0.498313
$852$	0	0
$853$	−2.21968e7	−1.04452	−0.522261	−	0.852785i	$-0.674911\pi$
$853$	−2.21968e7	−1.04452	−0.522261	+	0.852785i	$0.674911\pi$
$854$	2.32500e7	1.09088
$855$	0	0
$856$	−1.28101e7	−0.597541
$857$	−3.15803e7	−1.46881	−0.734403	−	0.678714i	$-0.762538\pi$
$857$	−3.15803e7	−1.46881	−0.734403	+	0.678714i	$0.762538\pi$
$858$	0	0
$859$	−1.22037e6	−0.0564297	−0.0282148	−	0.999602i	$-0.508982\pi$
$859$	−1.22037e6	−0.0564297	−0.0282148	+	0.999602i	$0.508982\pi$
$860$	3.25890e6	0.150254
$861$	0	0
$862$	−2.29239e7	−1.05080
$863$	2.43543e7	1.11314	0.556570	−	0.830801i	$-0.312117\pi$
$863$	2.43543e7	1.11314	0.556570	+	0.830801i	$0.312117\pi$
$864$	0	0
$865$	−3.46032e6	−0.157245
$866$	−3.29086e7	−1.49113
$867$	0	0
$868$	−2.56917e7	−1.15743
$869$	−5.47031e7	−2.45732
$870$	0	0
$871$	−1.09877e7	−0.490749
$872$	5.99873e6	0.267158
$873$	0	0
$874$	−1.62668e7	−0.720318
$875$	2.27401e7	1.00409
$876$	0	0
$877$	2.52016e6	0.110644	0.0553222	−	0.998469i	$-0.482381\pi$
$877$	2.52016e6	0.110644	0.0553222	+	0.998469i	$0.482381\pi$
$878$	−3.68601e6	−0.161369
$879$	0	0
$880$	7.67465e6	0.334081
$881$	−1.04311e6	−0.0452785	−0.0226393	−	0.999744i	$-0.507207\pi$
$881$	−1.04311e6	−0.0452785	−0.0226393	+	0.999744i	$0.507207\pi$
$882$	0	0
$883$	−3.57753e7	−1.54412	−0.772061	−	0.635548i	$-0.780774\pi$
$883$	−3.57753e7	−1.54412	−0.772061	+	0.635548i	$0.780774\pi$
$884$	−3.90102e7	−1.67899
$885$	0	0
$886$	5.34354e6	0.228689
$887$	1.66616e7	0.711061	0.355531	−	0.934665i	$-0.384300\pi$
$887$	1.66616e7	0.711061	0.355531	+	0.934665i	$0.384300\pi$
$888$	0	0
$889$	2.96065e7	1.25642
$890$	2.36012e7	0.998754
$891$	0	0
$892$	2.88713e7	1.21494
$893$	−1.44225e6	−0.0605218
$894$	0	0
$895$	2.29815e7	0.959004
$896$	2.96214e7	1.23264
$897$	0	0
$898$	−1.63928e7	−0.678365
$899$	6.43631e6	0.265606
$900$	0	0
$901$	4.81090e7	1.97430
$902$	1.03206e7	0.422368
$903$	0	0
$904$	−2.39499e7	−0.974727
$905$	1.32963e7	0.539645
$906$	0	0
$907$	3.59470e7	1.45092	0.725462	−	0.688262i	$-0.241626\pi$
$907$	3.59470e7	1.45092	0.725462	+	0.688262i	$0.241626\pi$
$908$	3.52944e7	1.42066
$909$	0	0
$910$	−1.81876e7	−0.728069
$911$	−7.56404e6	−0.301966	−0.150983	−	0.988536i	$-0.548244\pi$
$911$	−7.56404e6	−0.301966	−0.150983	+	0.988536i	$0.548244\pi$
$912$	0	0
$913$	−1.09143e6	−0.0433331
$914$	−6.52030e7	−2.58168
$915$	0	0
$916$	5.67260e7	2.23380
$917$	−3.53654e7	−1.38885
$918$	0	0
$919$	−2.22167e7	−0.867741	−0.433870	−	0.900975i	$-0.642852\pi$
$919$	−2.22167e7	−0.867741	−0.433870	+	0.900975i	$0.642852\pi$
$920$	−1.37300e7	−0.534812
$921$	0	0
$922$	3.11207e7	1.20565
$923$	−2.09051e7	−0.807698
$924$	0	0
$925$	−6.35871e6	−0.244351
$926$	−6.08709e7	−2.33283
$927$	0	0
$928$	−1.04694e7	−0.399071
$929$	−9.18809e6	−0.349290	−0.174645	−	0.984631i	$-0.555878\pi$
$929$	−9.18809e6	−0.349290	−0.174645	+	0.984631i	$0.555878\pi$
$930$	0	0
$931$	−814470.	−0.0307965
$932$	−1.76934e7	−0.667224
$933$	0	0
$934$	−5.02220e7	−1.88376
$935$	4.29785e7	1.60777
$936$	0	0
$937$	4.06654e7	1.51313	0.756565	−	0.653919i	$-0.226876\pi$
$937$	4.06654e7	1.51313	0.756565	+	0.653919i	$0.226876\pi$
$938$	2.79495e7	1.03721
$939$	0	0
$940$	−3.88607e6	−0.143447
$941$	2.60105e7	0.957580	0.478790	−	0.877930i	$-0.341076\pi$
$941$	2.60105e7	0.957580	0.478790	+	0.877930i	$0.341076\pi$
$942$	0	0
$943$	5.53391e6	0.202653
$944$	−1.47017e7	−0.536954
$945$	0	0
$946$	−9.67273e6	−0.351416
$947$	2.03967e7	0.739067	0.369534	−	0.929217i	$-0.379517\pi$
$947$	2.03967e7	0.739067	0.369534	+	0.929217i	$0.379517\pi$
$948$	0	0
$949$	2.55652e7	0.921475
$950$	−9.82530e6	−0.353213
$951$	0	0
$952$	3.10846e7	1.11161
$953$	4.87462e6	0.173864	0.0869318	−	0.996214i	$-0.472294\pi$
$953$	4.87462e6	0.173864	0.0869318	+	0.996214i	$0.472294\pi$
$954$	0	0
$955$	4.66414e6	0.165487
$956$	−1.31289e7	−0.464603
$957$	0	0
$958$	−4.04231e7	−1.42304
$959$	4.12903e7	1.44978
$960$	0	0
$961$	−9.09409e6	−0.317651
$962$	1.44662e7	0.503984
$963$	0	0
$964$	−5.24322e7	−1.81721
$965$	−6.18368e6	−0.213761
$966$	0	0
$967$	−1.26016e7	−0.433371	−0.216686	−	0.976241i	$-0.569525\pi$
$967$	−1.26016e7	−0.433371	−0.216686	+	0.976241i	$0.569525\pi$
$968$	2.42173e7	0.830686
$969$	0	0
$970$	−2.97921e7	−1.01665
$971$	1.39978e6	0.0476443	0.0238221	−	0.999716i	$-0.492416\pi$
$971$	1.39978e6	0.0476443	0.0238221	+	0.999716i	$0.492416\pi$
$972$	0	0
$973$	−1.32201e7	−0.447665
$974$	−8.13452e7	−2.74748
$975$	0	0
$976$	7.22867e6	0.242904
$977$	−1.43010e6	−0.0479325	−0.0239662	−	0.999713i	$-0.507629\pi$
$977$	−1.43010e6	−0.0479325	−0.0239662	+	0.999713i	$0.507629\pi$
$978$	0	0
$979$	−4.15300e7	−1.38486
$980$	−2.19455e6	−0.0729928
$981$	0	0
$982$	−3.02711e7	−1.00173
$983$	−4.26255e7	−1.40697	−0.703487	−	0.710708i	$-0.748375\pi$
$983$	−4.26255e7	−1.40697	−0.703487	+	0.710708i	$0.748375\pi$
$984$	0	0
$985$	−2.19811e7	−0.721869
$986$	−2.48595e7	−0.814329
$987$	0	0
$988$	1.32520e7	0.431905
$989$	−5.18650e6	−0.168610
$990$	0	0
$991$	1.88273e7	0.608980	0.304490	−	0.952516i	$-0.401514\pi$
$991$	1.88273e7	0.608980	0.304490	+	0.952516i	$0.401514\pi$
$992$	−3.17759e7	−1.02522
$993$	0	0
$994$	5.31769e7	1.70709
$995$	1.32965e7	0.425775
$996$	0	0
$997$	2.10574e7	0.670915	0.335458	−	0.942055i	$-0.391109\pi$
$997$	2.10574e7	0.670915	0.335458	+	0.942055i	$0.391109\pi$
$998$	−1.76746e7	−0.561724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.6.a.e.1.9		10
3.2	odd	2		43.6.a.b.1.2	✓	10
12.11	even	2		688.6.a.h.1.6		10
15.14	odd	2		1075.6.a.b.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.2	✓	10	3.2	odd	2
387.6.a.e.1.9		10	1.1	even	1	trivial
688.6.a.h.1.6		10	12.11	even	2
1075.6.a.b.1.9		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)

\(f(q)\)	\(=\)	\(q+8.86547 q^{2} +46.5966 q^{4} +37.8251 q^{5} -124.747 q^{7} +129.406 q^{8} +O(q^{10})\)	\(q+8.86547 q^{2} +46.5966 q^{4} +37.8251 q^{5} -124.747 q^{7} +129.406 q^{8} +335.337 q^{10} -590.079 q^{11} +434.774 q^{13} -1105.94 q^{14} -343.849 q^{16} -1925.58 q^{17} +654.129 q^{19} +1762.52 q^{20} -5231.33 q^{22} -2805.03 q^{23} -1694.26 q^{25} +3854.47 q^{26} -5812.80 q^{28} +1456.23 q^{29} +4419.85 q^{31} -7189.36 q^{32} -17071.2 q^{34} -4718.58 q^{35} +3753.09 q^{37} +5799.17 q^{38} +4894.78 q^{40} -1972.85 q^{41} +1849.00 q^{43} -27495.7 q^{44} -24867.9 q^{46} -2204.84 q^{47} -1245.12 q^{49} -15020.4 q^{50} +20259.0 q^{52} -24984.2 q^{53} -22319.8 q^{55} -16143.0 q^{56} +12910.1 q^{58} +42756.2 q^{59} -21022.8 q^{61} +39184.0 q^{62} -52733.9 q^{64} +16445.4 q^{65} -25272.1 q^{67} -89725.4 q^{68} -41832.4 q^{70} -48082.8 q^{71} +58801.2 q^{73} +33272.9 q^{74} +30480.2 q^{76} +73610.8 q^{77} +92704.7 q^{79} -13006.1 q^{80} -17490.3 q^{82} +1849.64 q^{83} -72835.2 q^{85} +16392.3 q^{86} -76359.5 q^{88} +70380.3 q^{89} -54236.8 q^{91} -130705. q^{92} -19546.9 q^{94} +24742.5 q^{95} -88842.1 q^{97} -11038.6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8}+O(q^{10}) \) 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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