LMFDB - Embedded newform 387.6.a.e.1.10

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			1.10
Root			$11.5305$ 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)

$f(q)$	$=$	$q+10.5305 q^{2} +78.8905 q^{4} -86.8464 q^{5} -19.8137 q^{7} +493.778 q^{8} +O(q^{10})$	\(q+10.5305 q^{2} +78.8905 q^{4} -86.8464 q^{5} -19.8137 q^{7} +493.778 q^{8} -914.532 q^{10} +85.3712 q^{11} -229.081 q^{13} -208.647 q^{14} +2675.21 q^{16} -1356.51 q^{17} -2795.35 q^{19} -6851.35 q^{20} +898.997 q^{22} +1856.11 q^{23} +4417.29 q^{25} -2412.33 q^{26} -1563.11 q^{28} -7312.96 q^{29} -2937.36 q^{31} +12370.3 q^{32} -14284.7 q^{34} +1720.75 q^{35} +2577.36 q^{37} -29436.3 q^{38} -42882.8 q^{40} +3532.54 q^{41} +1849.00 q^{43} +6734.97 q^{44} +19545.7 q^{46} +7065.73 q^{47} -16414.4 q^{49} +46516.1 q^{50} -18072.3 q^{52} +3852.63 q^{53} -7414.18 q^{55} -9783.57 q^{56} -77008.8 q^{58} -27996.1 q^{59} -39244.4 q^{61} -30931.7 q^{62} +44658.1 q^{64} +19894.9 q^{65} -14809.1 q^{67} -107016. q^{68} +18120.3 q^{70} -8956.13 q^{71} +35168.6 q^{73} +27140.8 q^{74} -220526. q^{76} -1691.52 q^{77} -13263.6 q^{79} -232332. q^{80} +37199.3 q^{82} +9812.47 q^{83} +117808. q^{85} +19470.8 q^{86} +42154.4 q^{88} +87124.9 q^{89} +4538.95 q^{91} +146429. q^{92} +74405.4 q^{94} +242766. q^{95} +83982.4 q^{97} -172851. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	10.5305	1.86154	0.930769	−	0.365607i	$-0.119138\pi$
$2$	10.5305	1.86154	0.930769	+	0.365607i	$0.119138\pi$
$3$	0	0
$3$	0	0
$4$	78.8905	2.46533
$5$	−86.8464	−1.55356	−0.776778	−	0.629775i	$-0.783147\pi$
$5$	−86.8464	−1.55356	−0.776778	+	0.629775i	$0.783147\pi$
$6$	0	0
$7$	−19.8137	−0.152834	−0.0764171	−	0.997076i	$-0.524348\pi$
$7$	−19.8137	−0.152834	−0.0764171	+	0.997076i	$0.524348\pi$
$8$	493.778	2.72776
$9$	0	0
$10$	−914.532	−2.89200
$11$	85.3712	0.212730	0.106365	−	0.994327i	$-0.466079\pi$
$11$	85.3712	0.212730	0.106365	+	0.994327i	$0.466079\pi$
$12$	0	0
$13$	−229.081	−0.375951	−0.187975	−	0.982174i	$-0.560193\pi$
$13$	−229.081	−0.375951	−0.187975	+	0.982174i	$0.560193\pi$
$14$	−208.647	−0.284507
$15$	0	0
$16$	2675.21	2.61251
$17$	−1356.51	−1.13841	−0.569207	−	0.822194i	$-0.692750\pi$
$17$	−1356.51	−1.13841	−0.569207	+	0.822194i	$0.692750\pi$
$18$	0	0
$19$	−2795.35	−1.77645	−0.888223	−	0.459413i	$-0.848060\pi$
$19$	−2795.35	−1.77645	−0.888223	+	0.459413i	$0.848060\pi$
$20$	−6851.35	−3.83002
$21$	0	0
$22$	898.997	0.396006
$23$	1856.11	0.731618	0.365809	−	0.930690i	$-0.380792\pi$
$23$	1856.11	0.731618	0.365809	+	0.930690i	$0.380792\pi$
$24$	0	0
$25$	4417.29	1.41353
$26$	−2412.33	−0.699847
$27$	0	0
$28$	−1563.11	−0.376786
$29$	−7312.96	−1.61472	−0.807362	−	0.590057i	$-0.799105\pi$
$29$	−7312.96	−1.61472	−0.807362	+	0.590057i	$0.799105\pi$
$30$	0	0
$31$	−2937.36	−0.548975	−0.274488	−	0.961591i	$-0.588508\pi$
$31$	−2937.36	−0.548975	−0.274488	+	0.961591i	$0.588508\pi$
$32$	12370.3	2.13553
$33$	0	0
$34$	−14284.7	−2.11920
$35$	1720.75	0.237436
$36$	0	0
$37$	2577.36	0.309507	0.154754	−	0.987953i	$-0.450542\pi$
$37$	2577.36	0.309507	0.154754	+	0.987953i	$0.450542\pi$
$38$	−29436.3	−3.30692
$39$	0	0
$40$	−42882.8	−4.23773
$41$	3532.54	0.328192	0.164096	−	0.986444i	$-0.447529\pi$
$41$	3532.54	0.328192	0.164096	+	0.986444i	$0.447529\pi$
$42$	0	0
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	6734.97	0.524450
$45$	0	0
$46$	19545.7	1.36194
$47$	7065.73	0.466566	0.233283	−	0.972409i	$-0.425053\pi$
$47$	7065.73	0.466566	0.233283	+	0.972409i	$0.425053\pi$
$48$	0	0
$49$	−16414.4	−0.976642
$50$	46516.1	2.63135
$51$	0	0
$52$	−18072.3	−0.926842
$53$	3852.63	0.188394	0.0941972	−	0.995554i	$-0.469972\pi$
$53$	3852.63	0.188394	0.0941972	+	0.995554i	$0.469972\pi$
$54$	0	0
$55$	−7414.18	−0.330488
$56$	−9783.57	−0.416896
$57$	0	0
$58$	−77008.8	−3.00587
$59$	−27996.1	−1.04705	−0.523524	−	0.852011i	$-0.675383\pi$
$59$	−27996.1	−1.04705	−0.523524	+	0.852011i	$0.675383\pi$
$60$	0	0
$61$	−39244.4	−1.35037	−0.675186	−	0.737648i	$-0.735937\pi$
$61$	−39244.4	−1.35037	−0.675186	+	0.737648i	$0.735937\pi$
$62$	−30931.7	−1.02194
$63$	0	0
$64$	44658.1	1.36286
$65$	19894.9	0.584060
$66$	0	0
$67$	−14809.1	−0.403034	−0.201517	−	0.979485i	$-0.564587\pi$
$67$	−14809.1	−0.403034	−0.201517	+	0.979485i	$0.564587\pi$
$68$	−107016.	−2.80656
$69$	0	0
$70$	18120.3	0.441997
$71$	−8956.13	−0.210850	−0.105425	−	0.994427i	$-0.533620\pi$
$71$	−8956.13	−0.210850	−0.105425	+	0.994427i	$0.533620\pi$
$72$	0	0
$73$	35168.6	0.772411	0.386205	−	0.922413i	$-0.373786\pi$
$73$	35168.6	0.772411	0.386205	+	0.922413i	$0.373786\pi$
$74$	27140.8	0.576160
$75$	0	0
$76$	−220526.	−4.37952
$77$	−1691.52	−0.0325125
$78$	0	0
$79$	−13263.6	−0.239108	−0.119554	−	0.992828i	$-0.538146\pi$
$79$	−13263.6	−0.239108	−0.119554	+	0.992828i	$0.538146\pi$
$80$	−232332.	−4.05868
$81$	0	0
$82$	37199.3	0.610942
$83$	9812.47	0.156345	0.0781724	−	0.996940i	$-0.475092\pi$
$83$	9812.47	0.156345	0.0781724	+	0.996940i	$0.475092\pi$
$84$	0	0
$85$	117808.	1.76859
$86$	19470.8	0.283882
$87$	0	0
$88$	42154.4	0.580278
$89$	87124.9	1.16592	0.582958	−	0.812502i	$-0.301895\pi$
$89$	87124.9	1.16592	0.582958	+	0.812502i	$0.301895\pi$
$90$	0	0
$91$	4538.95	0.0574581
$92$	146429.	1.80368
$93$	0	0
$94$	74405.4	0.868530
$95$	242766.	2.75981
$96$	0	0
$97$	83982.4	0.906272	0.453136	−	0.891441i	$-0.350305\pi$
$97$	83982.4	0.906272	0.453136	+	0.891441i	$0.350305\pi$
$98$	−172851.	−1.81806
$99$	0	0
$100$	348482.	3.48482
$101$	−74775.4	−0.729382	−0.364691	−	0.931129i	$-0.618825\pi$
$101$	−74775.4	−0.729382	−0.364691	+	0.931129i	$0.618825\pi$
$102$	0	0
$103$	29400.7	0.273064	0.136532	−	0.990636i	$-0.456404\pi$
$103$	29400.7	0.273064	0.136532	+	0.990636i	$0.456404\pi$
$104$	−113115.	−1.02551
$105$	0	0
$106$	40570.0	0.350704
$107$	199822.	1.68727	0.843635	−	0.536917i	$-0.180411\pi$
$107$	199822.	1.68727	0.843635	+	0.536917i	$0.180411\pi$
$108$	0	0
$109$	−39465.7	−0.318166	−0.159083	−	0.987265i	$-0.550854\pi$
$109$	−39465.7	−0.318166	−0.159083	+	0.987265i	$0.550854\pi$
$110$	−78074.7	−0.615217
$111$	0	0
$112$	−53005.8	−0.399281
$113$	−41487.8	−0.305650	−0.152825	−	0.988253i	$-0.548837\pi$
$113$	−41487.8	−0.305650	−0.152825	+	0.988253i	$0.548837\pi$
$114$	0	0
$115$	−161196.	−1.13661
$116$	−576923.	−3.98082
$117$	0	0
$118$	−294811.	−1.94912
$119$	26877.5	0.173989
$120$	0	0
$121$	−153763.	−0.954746
$122$	−413262.	−2.51377
$123$	0	0
$124$	−231730.	−1.35340
$125$	−112231.	−0.642448
$126$	0	0
$127$	357769.	1.96831	0.984156	−	0.177305i	$-0.0567379\pi$
$127$	357769.	1.96831	0.984156	+	0.177305i	$0.0567379\pi$
$128$	74420.5	0.401483
$129$	0	0
$130$	209502.	1.08725
$131$	61207.9	0.311623	0.155811	−	0.987787i	$-0.450201\pi$
$131$	61207.9	0.311623	0.155811	+	0.987787i	$0.450201\pi$
$132$	0	0
$133$	55386.2	0.271502
$134$	−155946.	−0.750263
$135$	0	0
$136$	−669814.	−3.10533
$137$	309777.	1.41009	0.705047	−	0.709161i	$-0.250926\pi$
$137$	309777.	1.41009	0.705047	+	0.709161i	$0.250926\pi$
$138$	0	0
$139$	−193740.	−0.850517	−0.425259	−	0.905072i	$-0.639817\pi$
$139$	−193740.	−0.850517	−0.425259	+	0.905072i	$0.639817\pi$
$140$	135751.	0.585358
$141$	0	0
$142$	−94312.1	−0.392506
$143$	−19556.9	−0.0799762
$144$	0	0
$145$	635104.	2.50856
$146$	370342.	1.43787
$147$	0	0
$148$	203329.	0.763037
$149$	455017.	1.67904	0.839521	−	0.543327i	$-0.182836\pi$
$149$	455017.	1.67904	0.839521	+	0.543327i	$0.182836\pi$
$150$	0	0
$151$	−505868.	−1.80549	−0.902744	−	0.430179i	$-0.858451\pi$
$151$	−505868.	−1.80549	−0.902744	+	0.430179i	$0.858451\pi$
$152$	−1.38028e6	−4.84572
$153$	0	0
$154$	−17812.5	−0.0605232
$155$	255099.	0.852864
$156$	0	0
$157$	348495.	1.12836	0.564179	−	0.825652i	$-0.309193\pi$
$157$	348495.	1.12836	0.564179	+	0.825652i	$0.309193\pi$
$158$	−139672.	−0.445108
$159$	0	0
$160$	−1.07432e6	−3.31766
$161$	−36776.4	−0.111816
$162$	0	0
$163$	104915.	0.309291	0.154645	−	0.987970i	$-0.450576\pi$
$163$	104915.	0.309291	0.154645	+	0.987970i	$0.450576\pi$
$164$	278684.	0.809100
$165$	0	0
$166$	103330.	0.291042
$167$	−186982.	−0.518809	−0.259405	−	0.965769i	$-0.583526\pi$
$167$	−186982.	−0.518809	−0.259405	+	0.965769i	$0.583526\pi$
$168$	0	0
$169$	−318815.	−0.858661
$170$	1.24057e6	3.29230
$171$	0	0
$172$	145868.	0.375959
$173$	−491718.	−1.24911	−0.624555	−	0.780981i	$-0.714720\pi$
$173$	−491718.	−1.24911	−0.624555	+	0.780981i	$0.714720\pi$
$174$	0	0
$175$	−87522.9	−0.216036
$176$	228386.	0.555761
$177$	0	0
$178$	917464.	2.17040
$179$	−374509.	−0.873634	−0.436817	−	0.899550i	$-0.643894\pi$
$179$	−374509.	−0.873634	−0.436817	+	0.899550i	$0.643894\pi$
$180$	0	0
$181$	460794.	1.04547	0.522733	−	0.852496i	$-0.324912\pi$
$181$	460794.	1.04547	0.522733	+	0.852496i	$0.324912\pi$
$182$	47797.2	0.106961
$183$	0	0
$184$	916507.	1.99568
$185$	−223834.	−0.480837
$186$	0	0
$187$	−115807.	−0.242175
$188$	557419.	1.15024
$189$	0	0
$190$	2.55643e6	5.13749
$191$	22464.4	0.0445565	0.0222783	−	0.999752i	$-0.492908\pi$
$191$	22464.4	0.0445565	0.0222783	+	0.999752i	$0.492908\pi$
$192$	0	0
$193$	830124.	1.60417	0.802083	−	0.597212i	$-0.203725\pi$
$193$	830124.	1.60417	0.802083	+	0.597212i	$0.203725\pi$
$194$	884372.	1.68706
$195$	0	0
$196$	−1.29494e6	−2.40774
$197$	−802058.	−1.47245	−0.736224	−	0.676738i	$-0.763393\pi$
$197$	−802058.	−1.47245	−0.736224	+	0.676738i	$0.763393\pi$
$198$	0	0
$199$	−187945.	−0.336432	−0.168216	−	0.985750i	$-0.553801\pi$
$199$	−187945.	−0.336432	−0.168216	+	0.985750i	$0.553801\pi$
$200$	2.18116e6	3.85579
$201$	0	0
$202$	−787419.	−1.35777
$203$	144897.	0.246785
$204$	0	0
$205$	−306788.	−0.509864
$206$	309602.	0.508319
$207$	0	0
$208$	−612841.	−0.982176
$209$	−238642.	−0.377904
$210$	0	0
$211$	327613.	0.506589	0.253294	−	0.967389i	$-0.418486\pi$
$211$	327613.	0.506589	0.253294	+	0.967389i	$0.418486\pi$
$212$	303936.	0.464454
$213$	0	0
$214$	2.10422e6	3.14092
$215$	−160579.	−0.236915
$216$	0	0
$217$	58200.0	0.0839022
$218$	−415592.	−0.592278
$219$	0	0
$220$	−584908.	−0.814762
$221$	310751.	0.427988
$222$	0	0
$223$	−166692.	−0.224467	−0.112234	−	0.993682i	$-0.535801\pi$
$223$	−166692.	−0.224467	−0.112234	+	0.993682i	$0.535801\pi$
$224$	−245101.	−0.326382
$225$	0	0
$226$	−436886.	−0.568980
$227$	155259.	0.199982	0.0999912	−	0.994988i	$-0.468119\pi$
$227$	155259.	0.199982	0.0999912	+	0.994988i	$0.468119\pi$
$228$	0	0
$229$	−1.01590e6	−1.28016	−0.640079	−	0.768309i	$-0.721098\pi$
$229$	−1.01590e6	−1.28016	−0.640079	+	0.768309i	$0.721098\pi$
$230$	−1.69747e6	−2.11584
$231$	0	0
$232$	−3.61098e6	−4.40458
$233$	−23299.8	−0.0281166	−0.0140583	−	0.999901i	$-0.504475\pi$
$233$	−23299.8	−0.0281166	−0.0140583	+	0.999901i	$0.504475\pi$
$234$	0	0
$235$	−613633.	−0.724835
$236$	−2.20862e6	−2.58132
$237$	0	0
$238$	283032.	0.323887
$239$	−4871.64	−0.00551671	−0.00275836	−	0.999996i	$-0.500878\pi$
$239$	−4871.64	−0.00551671	−0.00275836	+	0.999996i	$0.500878\pi$
$240$	0	0
$241$	918873.	1.01909	0.509545	−	0.860444i	$-0.329814\pi$
$241$	918873.	1.01909	0.509545	+	0.860444i	$0.329814\pi$
$242$	−1.61919e6	−1.77730
$243$	0	0
$244$	−3.09601e6	−3.32911
$245$	1.42553e6	1.51727
$246$	0	0
$247$	640362.	0.667856
$248$	−1.45040e6	−1.49748
$249$	0	0
$250$	−1.18184e6	−1.19594
$251$	−1.47118e6	−1.47395	−0.736973	−	0.675922i	$-0.763746\pi$
$251$	−1.47118e6	−1.47395	−0.736973	+	0.675922i	$0.763746\pi$
$252$	0	0
$253$	158458.	0.155637
$254$	3.76747e6	3.66409
$255$	0	0
$256$	−645377.	−0.615480
$257$	813506.	0.768295	0.384147	−	0.923272i	$-0.374495\pi$
$257$	813506.	0.768295	0.384147	+	0.923272i	$0.374495\pi$
$258$	0	0
$259$	−51067.0	−0.0473033
$260$	1.56952e6	1.43990
$261$	0	0
$262$	644546.	0.580098
$263$	−691865.	−0.616782	−0.308391	−	0.951260i	$-0.599791\pi$
$263$	−691865.	−0.616782	−0.308391	+	0.951260i	$0.599791\pi$
$264$	0	0
$265$	−334587.	−0.292681
$266$	583242.	0.505411
$267$	0	0
$268$	−1.16830e6	−0.993610
$269$	759828.	0.640228	0.320114	−	0.947379i	$-0.396279\pi$
$269$	759828.	0.640228	0.320114	+	0.947379i	$0.396279\pi$
$270$	0	0
$271$	954020.	0.789104	0.394552	−	0.918874i	$-0.370900\pi$
$271$	954020.	0.789104	0.394552	+	0.918874i	$0.370900\pi$
$272$	−3.62895e6	−2.97412
$273$	0	0
$274$	3.26209e6	2.62494
$275$	377110.	0.300702
$276$	0	0
$277$	1.03398e6	0.809680	0.404840	−	0.914387i	$-0.367327\pi$
$277$	1.03398e6	0.809680	0.404840	+	0.914387i	$0.367327\pi$
$278$	−2.04017e6	−1.58327
$279$	0	0
$280$	849668.	0.647670
$281$	−2.05117e6	−1.54966	−0.774829	−	0.632171i	$-0.782164\pi$
$281$	−2.05117e6	−1.54966	−0.774829	+	0.632171i	$0.782164\pi$
$282$	0	0
$283$	−2.37044e6	−1.75940	−0.879698	−	0.475533i	$-0.842255\pi$
$283$	−2.37044e6	−1.75940	−0.879698	+	0.475533i	$0.842255\pi$
$284$	−706553.	−0.519815
$285$	0	0
$286$	−205943.	−0.148879
$287$	−69992.7	−0.0501589
$288$	0	0
$289$	420259.	0.295987
$290$	6.68793e6	4.66979
$291$	0	0
$292$	2.77447e6	1.90425
$293$	−1.48623e6	−1.01138	−0.505692	−	0.862714i	$-0.668763\pi$
$293$	−1.48623e6	−1.01138	−0.505692	+	0.862714i	$0.668763\pi$
$294$	0	0
$295$	2.43136e6	1.62665
$296$	1.27264e6	0.844263
$297$	0	0
$298$	4.79153e6	3.12560
$299$	−425200.	−0.275052
$300$	0	0
$301$	−36635.5	−0.0233070
$302$	−5.32702e6	−3.36099
$303$	0	0
$304$	−7.47815e6	−4.64098
$305$	3.40824e6	2.09788
$306$	0	0
$307$	−1.62571e6	−0.984456	−0.492228	−	0.870466i	$-0.663817\pi$
$307$	−1.62571e6	−0.984456	−0.492228	+	0.870466i	$0.663817\pi$
$308$	−133445.	−0.0801539
$309$	0	0
$310$	2.68631e6	1.58764
$311$	−2.07269e6	−1.21516	−0.607581	−	0.794258i	$-0.707860\pi$
$311$	−2.07269e6	−1.21516	−0.607581	+	0.794258i	$0.707860\pi$
$312$	0	0
$313$	−345918.	−0.199578	−0.0997889	−	0.995009i	$-0.531817\pi$
$313$	−345918.	−0.199578	−0.0997889	+	0.995009i	$0.531817\pi$
$314$	3.66981e6	2.10048
$315$	0	0
$316$	−1.04637e6	−0.589479
$317$	−574435.	−0.321065	−0.160532	−	0.987031i	$-0.551321\pi$
$317$	−574435.	−0.321065	−0.160532	+	0.987031i	$0.551321\pi$
$318$	0	0
$319$	−624316.	−0.343501
$320$	−3.87839e6	−2.11727
$321$	0	0
$322$	−387272.	−0.208150
$323$	3.79191e6	2.02233
$324$	0	0
$325$	−1.01192e6	−0.531419
$326$	1.10480e6	0.575757
$327$	0	0
$328$	1.74429e6	0.895229
$329$	−139998.	−0.0713072
$330$	0	0
$331$	−1.12898e6	−0.566393	−0.283196	−	0.959062i	$-0.591395\pi$
$331$	−1.12898e6	−0.566393	−0.283196	+	0.959062i	$0.591395\pi$
$332$	774111.	0.385441
$333$	0	0
$334$	−1.96900e6	−0.965784
$335$	1.28612e6	0.626135
$336$	0	0
$337$	−370207.	−0.177570	−0.0887850	−	0.996051i	$-0.528298\pi$
$337$	−370207.	−0.177570	−0.0887850	+	0.996051i	$0.528298\pi$
$338$	−3.35726e6	−1.59843
$339$	0	0
$340$	9.29392e6	4.36015
$341$	−250766.	−0.116784
$342$	0	0
$343$	658239.	0.302098
$344$	912996.	0.415980
$345$	0	0
$346$	−5.17801e6	−2.32527
$347$	−372676.	−0.166153	−0.0830764	−	0.996543i	$-0.526475\pi$
$347$	−372676.	−0.166153	−0.0830764	+	0.996543i	$0.526475\pi$
$348$	0	0
$349$	−2.14587e6	−0.943060	−0.471530	−	0.881850i	$-0.656298\pi$
$349$	−2.14587e6	−0.943060	−0.471530	+	0.881850i	$0.656298\pi$
$350$	−921656.	−0.402160
$351$	0	0
$352$	1.05607e6	0.454292
$353$	2.87740e6	1.22903	0.614517	−	0.788904i	$-0.289351\pi$
$353$	2.87740e6	1.22903	0.614517	+	0.788904i	$0.289351\pi$
$354$	0	0
$355$	777807.	0.327568
$356$	6.87332e6	2.87436
$357$	0	0
$358$	−3.94375e6	−1.62630
$359$	−647359.	−0.265100	−0.132550	−	0.991176i	$-0.542316\pi$
$359$	−647359.	−0.265100	−0.132550	+	0.991176i	$0.542316\pi$
$360$	0	0
$361$	5.33787e6	2.15576
$362$	4.85237e6	1.94618
$363$	0	0
$364$	358080.	0.141653
$365$	−3.05427e6	−1.19998
$366$	0	0
$367$	3.06021e6	1.18600	0.593001	−	0.805202i	$-0.297943\pi$
$367$	3.06021e6	1.18600	0.593001	+	0.805202i	$0.297943\pi$
$368$	4.96549e6	1.91136
$369$	0	0
$370$	−2.35708e6	−0.895096
$371$	−76335.0	−0.0287931
$372$	0	0
$373$	3.54156e6	1.31802	0.659010	−	0.752134i	$-0.270975\pi$
$373$	3.54156e6	1.31802	0.659010	+	0.752134i	$0.270975\pi$
$374$	−1.21950e6	−0.450819
$375$	0	0
$376$	3.48890e6	1.27268
$377$	1.67526e6	0.607057
$378$	0	0
$379$	−3.92757e6	−1.40451	−0.702257	−	0.711923i	$-0.747824\pi$
$379$	−3.92757e6	−1.40451	−0.702257	+	0.711923i	$0.747824\pi$
$380$	1.91519e7	6.80383
$381$	0	0
$382$	236560.	0.0829437
$383$	1.29688e6	0.451754	0.225877	−	0.974156i	$-0.427475\pi$
$383$	1.29688e6	0.451754	0.225877	+	0.974156i	$0.427475\pi$
$384$	0	0
$385$	146902.	0.0505099
$386$	8.74158e6	2.98622
$387$	0	0
$388$	6.62541e6	2.23426
$389$	100541.	0.0336874	0.0168437	−	0.999858i	$-0.494638\pi$
$389$	100541.	0.0336874	0.0168437	+	0.999858i	$0.494638\pi$
$390$	0	0
$391$	−2.51783e6	−0.832884
$392$	−8.10508e6	−2.66405
$393$	0	0
$394$	−8.44603e6	−2.74102
$395$	1.15190e6	0.371467
$396$	0	0
$397$	3.36979e6	1.07307	0.536533	−	0.843879i	$-0.319734\pi$
$397$	3.36979e6	1.07307	0.536533	+	0.843879i	$0.319734\pi$
$398$	−1.97914e6	−0.626281
$399$	0	0
$400$	1.18172e7	3.69287
$401$	−4.73407e6	−1.47019	−0.735095	−	0.677964i	$-0.762862\pi$
$401$	−4.73407e6	−1.47019	−0.735095	+	0.677964i	$0.762862\pi$
$402$	0	0
$403$	672894.	0.206388
$404$	−5.89906e6	−1.79817
$405$	0	0
$406$	1.52583e6	0.459400
$407$	220032.	0.0658416
$408$	0	0
$409$	−84169.8	−0.0248799	−0.0124399	−	0.999923i	$-0.503960\pi$
$409$	−84169.8	−0.0248799	−0.0124399	+	0.999923i	$0.503960\pi$
$410$	−3.23062e6	−0.949131
$411$	0	0
$412$	2.31943e6	0.673191
$413$	554705.	0.160025
$414$	0	0
$415$	−852178.	−0.242890
$416$	−2.83380e6	−0.802854
$417$	0	0
$418$	−2.51301e6	−0.703483
$419$	1.45191e6	0.404022	0.202011	−	0.979383i	$-0.435252\pi$
$419$	1.45191e6	0.404022	0.202011	+	0.979383i	$0.435252\pi$
$420$	0	0
$421$	4.69910e6	1.29214	0.646070	−	0.763278i	$-0.276411\pi$
$421$	4.69910e6	1.29214	0.646070	+	0.763278i	$0.276411\pi$
$422$	3.44992e6	0.943034
$423$	0	0
$424$	1.90235e6	0.513896
$425$	−5.99210e6	−1.60919
$426$	0	0
$427$	777577.	0.206383
$428$	1.57641e7	4.15967
$429$	0	0
$430$	−1.69097e6	−0.441026
$431$	−3.34282e6	−0.866803	−0.433401	−	0.901201i	$-0.642687\pi$
$431$	−3.34282e6	−0.866803	−0.433401	+	0.901201i	$0.642687\pi$
$432$	0	0
$433$	629642.	0.161389	0.0806945	−	0.996739i	$-0.474286\pi$
$433$	629642.	0.161389	0.0806945	+	0.996739i	$0.474286\pi$
$434$	612872.	0.156187
$435$	0	0
$436$	−3.11347e6	−0.784383
$437$	−5.18847e6	−1.29968
$438$	0	0
$439$	6.24963e6	1.54772	0.773861	−	0.633355i	$-0.218323\pi$
$439$	6.24963e6	1.54772	0.773861	+	0.633355i	$0.218323\pi$
$440$	−3.66096e6	−0.901494
$441$	0	0
$442$	3.27235e6	0.796716
$443$	−6.58305e6	−1.59374	−0.796871	−	0.604149i	$-0.793513\pi$
$443$	−6.58305e6	−1.59374	−0.796871	+	0.604149i	$0.793513\pi$
$444$	0	0
$445$	−7.56648e6	−1.81131
$446$	−1.75534e6	−0.417855
$447$	0	0
$448$	−884842.	−0.208291
$449$	−1.62106e6	−0.379476	−0.189738	−	0.981835i	$-0.560764\pi$
$449$	−1.62106e6	−0.379476	−0.189738	+	0.981835i	$0.560764\pi$
$450$	0	0
$451$	301577.	0.0698163
$452$	−3.27300e6	−0.753528
$453$	0	0
$454$	1.63495e6	0.372275
$455$	−394191.	−0.0892644
$456$	0	0
$457$	−6.01194e6	−1.34656	−0.673278	−	0.739390i	$-0.735114\pi$
$457$	−6.01194e6	−1.34656	−0.673278	+	0.739390i	$0.735114\pi$
$458$	−1.06979e7	−2.38306
$459$	0	0
$460$	−1.27169e7	−2.80211
$461$	1.97072e6	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$461$	1.97072e6	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$462$	0	0
$463$	−925538.	−0.200651	−0.100326	−	0.994955i	$-0.531988\pi$
$463$	−925538.	−0.200651	−0.100326	+	0.994955i	$0.531988\pi$
$464$	−1.95637e7	−4.21848
$465$	0	0
$466$	−245357.	−0.0523401
$467$	−1.87994e6	−0.398890	−0.199445	−	0.979909i	$-0.563914\pi$
$467$	−1.87994e6	−0.398890	−0.199445	+	0.979909i	$0.563914\pi$
$468$	0	0
$469$	293423.	0.0615973
$470$	−6.46184e6	−1.34931
$471$	0	0
$472$	−1.38238e7	−2.85610
$473$	157851.	0.0324411
$474$	0	0
$475$	−1.23479e7	−2.51107
$476$	2.12038e6	0.428939
$477$	0	0
$478$	−51300.6	−0.0102696
$479$	−8.18229e6	−1.62943	−0.814716	−	0.579860i	$-0.803107\pi$
$479$	−8.18229e6	−1.62943	−0.814716	+	0.579860i	$0.803107\pi$
$480$	0	0
$481$	−590425.	−0.116360
$482$	9.67615e6	1.89708
$483$	0	0
$484$	−1.21304e7	−2.35376
$485$	−7.29356e6	−1.40794
$486$	0	0
$487$	−1.84424e6	−0.352367	−0.176184	−	0.984357i	$-0.556375\pi$
$487$	−1.84424e6	−0.352367	−0.176184	+	0.984357i	$0.556375\pi$
$488$	−1.93780e7	−3.68350
$489$	0	0
$490$	1.50115e7	2.82445
$491$	4.29014e6	0.803096	0.401548	−	0.915838i	$-0.368472\pi$
$491$	4.29014e6	0.803096	0.401548	+	0.915838i	$0.368472\pi$
$492$	0	0
$493$	9.92009e6	1.83822
$494$	6.74330e6	1.24324
$495$	0	0
$496$	−7.85806e6	−1.43420
$497$	177454.	0.0322252
$498$	0	0
$499$	−7.80338e6	−1.40291	−0.701457	−	0.712711i	$-0.747467\pi$
$499$	−7.80338e6	−1.40291	−0.701457	+	0.712711i	$0.747467\pi$
$500$	−8.85396e6	−1.58384
$501$	0	0
$502$	−1.54922e7	−2.74381
$503$	−673490.	−0.118689	−0.0593446	−	0.998238i	$-0.518901\pi$
$503$	−673490.	−0.118689	−0.0593446	+	0.998238i	$0.518901\pi$
$504$	0	0
$505$	6.49397e6	1.13314
$506$	1.66864e6	0.289725
$507$	0	0
$508$	2.82246e7	4.85253
$509$	−9.44081e6	−1.61516	−0.807579	−	0.589760i	$-0.799222\pi$
$509$	−9.44081e6	−1.61516	−0.807579	+	0.589760i	$0.799222\pi$
$510$	0	0
$511$	−696821.	−0.118051
$512$	−9.17757e6	−1.54722
$513$	0	0
$514$	8.56659e6	1.43021
$515$	−2.55334e6	−0.424220
$516$	0	0
$517$	603210.	0.0992527
$518$	−537759.	−0.0880569
$519$	0	0
$520$	9.82365e6	1.59318
$521$	1.01037e7	1.63074	0.815369	−	0.578941i	$-0.196534\pi$
$521$	1.01037e7	1.63074	0.815369	+	0.578941i	$0.196534\pi$
$522$	0	0
$523$	1.22441e6	0.195737	0.0978687	−	0.995199i	$-0.468797\pi$
$523$	1.22441e6	0.195737	0.0978687	+	0.995199i	$0.468797\pi$
$524$	4.82872e6	0.768252
$525$	0	0
$526$	−7.28565e6	−1.14816
$527$	3.98455e6	0.624961
$528$	0	0
$529$	−2.99120e6	−0.464735
$530$	−3.52336e6	−0.544838
$531$	0	0
$532$	4.36944e6	0.669340
$533$	−809239.	−0.123384
$534$	0	0
$535$	−1.73539e7	−2.62127
$536$	−7.31240e6	−1.09938
$537$	0	0
$538$	8.00133e6	1.19181
$539$	−1.40132e6	−0.207761
$540$	0	0
$541$	−2.66077e6	−0.390853	−0.195426	−	0.980718i	$-0.562609\pi$
$541$	−2.66077e6	−0.390853	−0.195426	+	0.980718i	$0.562609\pi$
$542$	1.00463e7	1.46895
$543$	0	0
$544$	−1.67804e7	−2.43112
$545$	3.42745e6	0.494288
$546$	0	0
$547$	7.44368e6	1.06370	0.531850	−	0.846839i	$-0.321497\pi$
$547$	7.44368e6	1.06370	0.531850	+	0.846839i	$0.321497\pi$
$548$	2.44385e7	3.47634
$549$	0	0
$550$	3.97114e6	0.559768
$551$	2.04423e7	2.86847
$552$	0	0
$553$	262801.	0.0365438
$554$	1.08883e7	1.50725
$555$	0	0
$556$	−1.52843e7	−2.09680
$557$	−5.55246e6	−0.758311	−0.379155	−	0.925333i	$-0.623785\pi$
$557$	−5.55246e6	−0.758311	−0.379155	+	0.925333i	$0.623785\pi$
$558$	0	0
$559$	−423571.	−0.0573320
$560$	4.60337e6	0.620305
$561$	0	0
$562$	−2.15998e7	−2.88475
$563$	2.05400e6	0.273105	0.136552	−	0.990633i	$-0.456398\pi$
$563$	2.05400e6	0.273105	0.136552	+	0.990633i	$0.456398\pi$
$564$	0	0
$565$	3.60307e6	0.474845
$566$	−2.49618e7	−3.27518
$567$	0	0
$568$	−4.42234e6	−0.575150
$569$	−5.13151e6	−0.664453	−0.332227	−	0.943200i	$-0.607800\pi$
$569$	−5.13151e6	−0.664453	−0.332227	+	0.943200i	$0.607800\pi$
$570$	0	0
$571$	−1.13394e7	−1.45546	−0.727730	−	0.685863i	$-0.759425\pi$
$571$	−1.13394e7	−1.45546	−0.727730	+	0.685863i	$0.759425\pi$
$572$	−1.54286e6	−0.197167
$573$	0	0
$574$	−737055.	−0.0933728
$575$	8.19899e6	1.03417
$576$	0	0
$577$	−9.19195e6	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$577$	−9.19195e6	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$578$	4.42552e6	0.550992
$579$	0	0
$580$	5.01037e7	6.18443
$581$	−194421.	−0.0238948
$582$	0	0
$583$	328904.	0.0400772
$584$	1.73655e7	2.10695
$585$	0	0
$586$	−1.56506e7	−1.88273
$587$	−7.32138e6	−0.876997	−0.438498	−	0.898732i	$-0.644490\pi$
$587$	−7.32138e6	−0.876997	−0.438498	+	0.898732i	$0.644490\pi$
$588$	0	0
$589$	8.21094e6	0.975225
$590$	2.56033e7	3.02807
$591$	0	0
$592$	6.89498e6	0.808591
$593$	−8.89330e6	−1.03855	−0.519274	−	0.854608i	$-0.673797\pi$
$593$	−8.89330e6	−1.03855	−0.519274	+	0.854608i	$0.673797\pi$
$594$	0	0
$595$	−2.33421e6	−0.270301
$596$	3.58965e7	4.13939
$597$	0	0
$598$	−4.47755e6	−0.512021
$599$	−7.52622e6	−0.857057	−0.428529	−	0.903528i	$-0.640968\pi$
$599$	−7.52622e6	−0.857057	−0.428529	+	0.903528i	$0.640968\pi$
$600$	0	0
$601$	1.53849e7	1.73743	0.868716	−	0.495311i	$-0.164946\pi$
$601$	1.53849e7	1.73743	0.868716	+	0.495311i	$0.164946\pi$
$602$	−385789.	−0.0433869
$603$	0	0
$604$	−3.99081e7	−4.45112
$605$	1.33537e7	1.48325
$606$	0	0
$607$	−1.10155e7	−1.21348	−0.606740	−	0.794900i	$-0.707523\pi$
$607$	−1.10155e7	−1.21348	−0.606740	+	0.794900i	$0.707523\pi$
$608$	−3.45793e7	−3.79365
$609$	0	0
$610$	3.58903e7	3.90528
$611$	−1.61863e6	−0.175406
$612$	0	0
$613$	−8.89814e6	−0.956419	−0.478209	−	0.878246i	$-0.658714\pi$
$613$	−8.89814e6	−0.956419	−0.478209	+	0.878246i	$0.658714\pi$
$614$	−1.71194e7	−1.83260
$615$	0	0
$616$	−835235.	−0.0886864
$617$	−7.40581e6	−0.783177	−0.391588	−	0.920140i	$-0.628074\pi$
$617$	−7.40581e6	−0.783177	−0.391588	+	0.920140i	$0.628074\pi$
$618$	0	0
$619$	1.41475e7	1.48406	0.742032	−	0.670364i	$-0.233862\pi$
$619$	1.41475e7	1.48406	0.742032	+	0.670364i	$0.233862\pi$
$620$	2.01249e7	2.10259
$621$	0	0
$622$	−2.18264e7	−2.26207
$623$	−1.72627e6	−0.178192
$624$	0	0
$625$	−4.05719e6	−0.415456
$626$	−3.64267e6	−0.371522
$627$	0	0
$628$	2.74929e7	2.78177
$629$	−3.49621e6	−0.352347
$630$	0	0
$631$	9.57148e6	0.956986	0.478493	−	0.878091i	$-0.341183\pi$
$631$	9.57148e6	0.956986	0.478493	+	0.878091i	$0.341183\pi$
$632$	−6.54927e6	−0.652229
$633$	0	0
$634$	−6.04906e6	−0.597675
$635$	−3.10710e7	−3.05788
$636$	0	0
$637$	3.76023e6	0.367169
$638$	−6.57433e6	−0.639440
$639$	0	0
$640$	−6.46315e6	−0.623727
$641$	637566.	0.0612887	0.0306443	−	0.999530i	$-0.490244\pi$
$641$	637566.	0.0612887	0.0306443	+	0.999530i	$0.490244\pi$
$642$	0	0
$643$	−1.89702e7	−1.80944	−0.904718	−	0.426010i	$-0.859919\pi$
$643$	−1.89702e7	−1.80944	−0.904718	+	0.426010i	$0.859919\pi$
$644$	−2.90131e6	−0.275664
$645$	0	0
$646$	3.99306e7	3.76465
$647$	−2.02807e7	−1.90468	−0.952338	−	0.305044i	$-0.901329\pi$
$647$	−2.02807e7	−1.90468	−0.952338	+	0.305044i	$0.901329\pi$
$648$	0	0
$649$	−2.39006e6	−0.222739
$650$	−1.06560e7	−0.989258
$651$	0	0
$652$	8.27676e6	0.762503
$653$	3.76628e6	0.345645	0.172822	−	0.984953i	$-0.444711\pi$
$653$	3.76628e6	0.345645	0.172822	+	0.984953i	$0.444711\pi$
$654$	0	0
$655$	−5.31568e6	−0.484123
$656$	9.45029e6	0.857404
$657$	0	0
$658$	−1.47425e6	−0.132741
$659$	1.70994e7	1.53379	0.766897	−	0.641770i	$-0.221800\pi$
$659$	1.70994e7	1.53379	0.766897	+	0.641770i	$0.221800\pi$
$660$	0	0
$661$	875240.	0.0779154	0.0389577	−	0.999241i	$-0.487596\pi$
$661$	875240.	0.0779154	0.0389577	+	0.999241i	$0.487596\pi$
$662$	−1.18887e7	−1.05436
$663$	0	0
$664$	4.84518e6	0.426472
$665$	−4.81009e6	−0.421793
$666$	0	0
$667$	−1.35737e7	−1.18136
$668$	−1.47511e7	−1.27904
$669$	0	0
$670$	1.35434e7	1.16557
$671$	−3.35034e6	−0.287265
$672$	0	0
$673$	−1.59835e7	−1.36030	−0.680149	−	0.733074i	$-0.738085\pi$
$673$	−1.59835e7	−1.36030	−0.680149	+	0.733074i	$0.738085\pi$
$674$	−3.89844e6	−0.330553
$675$	0	0
$676$	−2.51515e7	−2.11688
$677$	2.27744e7	1.90974	0.954871	−	0.297020i	$-0.0959926\pi$
$677$	2.27744e7	1.90974	0.954871	+	0.297020i	$0.0959926\pi$
$678$	0	0
$679$	−1.66400e6	−0.138509
$680$	5.81709e7	4.82429
$681$	0	0
$682$	−2.64068e6	−0.217398
$683$	1.16349e7	0.954354	0.477177	−	0.878807i	$-0.341660\pi$
$683$	1.16349e7	0.954354	0.477177	+	0.878807i	$0.341660\pi$
$684$	0	0
$685$	−2.69030e7	−2.19066
$686$	6.93156e6	0.562368
$687$	0	0
$688$	4.94647e6	0.398404
$689$	−882566.	−0.0708271
$690$	0	0
$691$	−5.50626e6	−0.438694	−0.219347	−	0.975647i	$-0.570393\pi$
$691$	−5.50626e6	−0.438694	−0.219347	+	0.975647i	$0.570393\pi$
$692$	−3.87918e7	−3.07946
$693$	0	0
$694$	−3.92444e6	−0.309300
$695$	1.68257e7	1.32133
$696$	0	0
$697$	−4.79192e6	−0.373618
$698$	−2.25970e7	−1.75554
$699$	0	0
$700$	−6.90473e6	−0.532600
$701$	1.26842e7	0.974916	0.487458	−	0.873146i	$-0.337924\pi$
$701$	1.26842e7	0.974916	0.487458	+	0.873146i	$0.337924\pi$
$702$	0	0
$703$	−7.20462e6	−0.549823
$704$	3.81251e6	0.289921
$705$	0	0
$706$	3.03004e7	2.28789
$707$	1.48158e6	0.111475
$708$	0	0
$709$	919145.	0.0686702	0.0343351	−	0.999410i	$-0.489069\pi$
$709$	919145.	0.0686702	0.0343351	+	0.999410i	$0.489069\pi$
$710$	8.19067e6	0.609780
$711$	0	0
$712$	4.30203e7	3.18034
$713$	−5.45207e6	−0.401640
$714$	0	0
$715$	1.69845e6	0.124247
$716$	−2.95452e7	−2.15379
$717$	0	0
$718$	−6.81699e6	−0.493493
$719$	−7.21135e6	−0.520228	−0.260114	−	0.965578i	$-0.583760\pi$
$719$	−7.21135e6	−0.520228	−0.260114	+	0.965578i	$0.583760\pi$
$720$	0	0
$721$	−582536.	−0.0417335
$722$	5.62102e7	4.01303
$723$	0	0
$724$	3.63522e7	2.57742
$725$	−3.23035e7	−2.28247
$726$	0	0
$727$	−4.15671e6	−0.291685	−0.145842	−	0.989308i	$-0.546589\pi$
$727$	−4.15671e6	−0.291685	−0.145842	+	0.989308i	$0.546589\pi$
$728$	2.24123e6	0.156732
$729$	0	0
$730$	−3.21628e7	−2.23381
$731$	−2.50818e6	−0.173607
$732$	0	0
$733$	4.05351e6	0.278658	0.139329	−	0.990246i	$-0.455505\pi$
$733$	4.05351e6	0.278658	0.139329	+	0.990246i	$0.455505\pi$
$734$	3.22254e7	2.20779
$735$	0	0
$736$	2.29606e7	1.56239
$737$	−1.26427e6	−0.0857375
$738$	0	0
$739$	1.27176e7	0.856629	0.428315	−	0.903630i	$-0.359108\pi$
$739$	1.27176e7	0.856629	0.428315	+	0.903630i	$0.359108\pi$
$740$	−1.76584e7	−1.18542
$741$	0	0
$742$	−803842.	−0.0535995
$743$	1.86660e6	0.124045	0.0620224	−	0.998075i	$-0.480245\pi$
$743$	1.86660e6	0.124045	0.0620224	+	0.998075i	$0.480245\pi$
$744$	0	0
$745$	−3.95165e7	−2.60848
$746$	3.72942e7	2.45355
$747$	0	0
$748$	−9.13605e6	−0.597041
$749$	−3.95922e6	−0.257873
$750$	0	0
$751$	2.61334e7	1.69081	0.845407	−	0.534122i	$-0.179358\pi$
$751$	2.61334e7	1.69081	0.845407	+	0.534122i	$0.179358\pi$
$752$	1.89023e7	1.21891
$753$	0	0
$754$	1.76413e7	1.13006
$755$	4.39328e7	2.80492
$756$	0	0
$757$	1.20591e6	0.0764846	0.0382423	−	0.999268i	$-0.487824\pi$
$757$	1.20591e6	0.0764846	0.0382423	+	0.999268i	$0.487824\pi$
$758$	−4.13591e7	−2.61456
$759$	0	0
$760$	1.19872e8	7.52810
$761$	1.63050e6	0.102061	0.0510303	−	0.998697i	$-0.483749\pi$
$761$	1.63050e6	0.102061	0.0510303	+	0.998697i	$0.483749\pi$
$762$	0	0
$763$	781961.	0.0486266
$764$	1.77223e6	0.109846
$765$	0	0
$766$	1.36567e7	0.840958
$767$	6.41337e6	0.393639
$768$	0	0
$769$	−3.03077e6	−0.184815	−0.0924074	−	0.995721i	$-0.529456\pi$
$769$	−3.03077e6	−0.184815	−0.0924074	+	0.995721i	$0.529456\pi$
$770$	1.54695e6	0.0940262
$771$	0	0
$772$	6.54888e7	3.95480
$773$	−1.56440e7	−0.941669	−0.470834	−	0.882222i	$-0.656047\pi$
$773$	−1.56440e7	−0.941669	−0.470834	+	0.882222i	$0.656047\pi$
$774$	0	0
$775$	−1.29752e7	−0.775995
$776$	4.14686e7	2.47210
$777$	0	0
$778$	1.05874e6	0.0627104
$779$	−9.87468e6	−0.583015
$780$	0	0
$781$	−764595.	−0.0448543
$782$	−2.65139e7	−1.55045
$783$	0	0
$784$	−4.39120e7	−2.55149
$785$	−3.02655e7	−1.75297
$786$	0	0
$787$	2.77750e7	1.59852	0.799260	−	0.600986i	$-0.205225\pi$
$787$	2.77750e7	1.59852	0.799260	+	0.600986i	$0.205225\pi$
$788$	−6.32747e7	−3.63007
$789$	0	0
$790$	1.21300e7	0.691500
$791$	822028.	0.0467138
$792$	0	0
$793$	8.99016e6	0.507673
$794$	3.54854e7	1.99756
$795$	0	0
$796$	−1.48270e7	−0.829414
$797$	646020.	0.0360247	0.0180123	−	0.999838i	$-0.494266\pi$
$797$	646020.	0.0360247	0.0180123	+	0.999838i	$0.494266\pi$
$798$	0	0
$799$	−9.58473e6	−0.531145
$800$	5.46432e7	3.01864
$801$	0	0
$802$	−4.98519e7	−2.73682
$803$	3.00239e6	0.164315
$804$	0	0
$805$	3.19390e6	0.173713
$806$	7.08588e6	0.384199
$807$	0	0
$808$	−3.69224e7	−1.98958
$809$	1.67691e6	0.0900823	0.0450412	−	0.998985i	$-0.485658\pi$
$809$	1.67691e6	0.0900823	0.0450412	+	0.998985i	$0.485658\pi$
$810$	0	0
$811$	1.09231e7	0.583169	0.291585	−	0.956545i	$-0.405817\pi$
$811$	1.09231e7	0.583169	0.291585	+	0.956545i	$0.405817\pi$
$812$	1.14310e7	0.608406
$813$	0	0
$814$	2.31704e6	0.122567
$815$	−9.11145e6	−0.480500
$816$	0	0
$817$	−5.16860e6	−0.270905
$818$	−886346.	−0.0463148
$819$	0	0
$820$	−2.42027e7	−1.25698
$821$	−1.94364e7	−1.00637	−0.503186	−	0.864178i	$-0.667839\pi$
$821$	−1.94364e7	−1.00637	−0.503186	+	0.864178i	$0.667839\pi$
$822$	0	0
$823$	−3.08462e7	−1.58746	−0.793730	−	0.608271i	$-0.791864\pi$
$823$	−3.08462e7	−1.58746	−0.793730	+	0.608271i	$0.791864\pi$
$824$	1.45174e7	0.744853
$825$	0	0
$826$	5.84130e6	0.297892
$827$	1.97925e7	1.00632	0.503161	−	0.864193i	$-0.332170\pi$
$827$	1.97925e7	1.00632	0.503161	+	0.864193i	$0.332170\pi$
$828$	0	0
$829$	−1.75755e7	−0.888223	−0.444111	−	0.895972i	$-0.646481\pi$
$829$	−1.75755e7	−0.888223	−0.444111	+	0.895972i	$0.646481\pi$
$830$	−8.97382e6	−0.452150
$831$	0	0
$832$	−1.02303e7	−0.512367
$833$	2.22663e7	1.11182
$834$	0	0
$835$	1.62387e7	0.805999
$836$	−1.88266e7	−0.931657
$837$	0	0
$838$	1.52893e7	0.752103
$839$	1.50238e7	0.736842	0.368421	−	0.929659i	$-0.379899\pi$
$839$	1.50238e7	0.736842	0.368421	+	0.929659i	$0.379899\pi$
$840$	0	0
$841$	3.29682e7	1.60733
$842$	4.94837e7	2.40537
$843$	0	0
$844$	2.58456e7	1.24891
$845$	2.76879e7	1.33398
$846$	0	0
$847$	3.04661e6	0.145918
$848$	1.03066e7	0.492183
$849$	0	0
$850$	−6.30995e7	−2.99556
$851$	4.78387e6	0.226441
$852$	0	0
$853$	3.88390e7	1.82766	0.913829	−	0.406099i	$-0.133111\pi$
$853$	3.88390e7	1.82766	0.913829	+	0.406099i	$0.133111\pi$
$854$	8.18824e6	0.384190
$855$	0	0
$856$	9.86679e7	4.60248
$857$	3.57330e6	0.166195	0.0830975	−	0.996541i	$-0.473519\pi$
$857$	3.57330e6	0.166195	0.0830975	+	0.996541i	$0.473519\pi$
$858$	0	0
$859$	3.27850e6	0.151598	0.0757988	−	0.997123i	$-0.475849\pi$
$859$	3.27850e6	0.151598	0.0757988	+	0.997123i	$0.475849\pi$
$860$	−1.26681e7	−0.584073
$861$	0	0
$862$	−3.52014e7	−1.61359
$863$	−1.93441e7	−0.884139	−0.442070	−	0.896981i	$-0.645756\pi$
$863$	−1.93441e7	−0.884139	−0.442070	+	0.896981i	$0.645756\pi$
$864$	0	0
$865$	4.27039e7	1.94056
$866$	6.63042e6	0.300432
$867$	0	0
$868$	4.59142e6	0.206846
$869$	−1.13233e6	−0.0508655
$870$	0	0
$871$	3.39248e6	0.151521
$872$	−1.94873e7	−0.867881
$873$	0	0
$874$	−5.46370e7	−2.41940
$875$	2.22371e6	0.0981880
$876$	0	0
$877$	−2.57531e7	−1.13066	−0.565329	−	0.824866i	$-0.691251\pi$
$877$	−2.57531e7	−1.13066	−0.565329	+	0.824866i	$0.691251\pi$
$878$	6.58114e7	2.88115
$879$	0	0
$880$	−1.98345e7	−0.863405
$881$	−4.39384e6	−0.190724	−0.0953618	−	0.995443i	$-0.530401\pi$
$881$	−4.39384e6	−0.190724	−0.0953618	+	0.995443i	$0.530401\pi$
$882$	0	0
$883$	2.21529e7	0.956156	0.478078	−	0.878317i	$-0.341334\pi$
$883$	2.21529e7	0.956156	0.478078	+	0.878317i	$0.341334\pi$
$884$	2.45153e7	1.05513
$885$	0	0
$886$	−6.93226e7	−2.96681
$887$	4.36187e7	1.86150	0.930750	−	0.365655i	$-0.119155\pi$
$887$	4.36187e7	1.86150	0.930750	+	0.365655i	$0.119155\pi$
$888$	0	0
$889$	−7.08874e6	−0.300825
$890$	−7.96785e7	−3.37183
$891$	0	0
$892$	−1.31504e7	−0.553385
$893$	−1.97512e7	−0.828828
$894$	0	0
$895$	3.25247e7	1.35724
$896$	−1.47455e6	−0.0613604
$897$	0	0
$898$	−1.70705e7	−0.706409
$899$	2.14808e7	0.886443
$900$	0	0
$901$	−5.22613e6	−0.214471
$902$	3.17574e6	0.129966
$903$	0	0
$904$	−2.04858e7	−0.833742
$905$	−4.00183e7	−1.62419
$906$	0	0
$907$	1.38621e7	0.559515	0.279758	−	0.960071i	$-0.409746\pi$
$907$	1.38621e7	0.559515	0.279758	+	0.960071i	$0.409746\pi$
$908$	1.22484e7	0.493022
$909$	0	0
$910$	−4.15101e6	−0.166169
$911$	2.31037e7	0.922329	0.461164	−	0.887315i	$-0.347432\pi$
$911$	2.31037e7	0.922329	0.461164	+	0.887315i	$0.347432\pi$
$912$	0	0
$913$	837702.	0.0332593
$914$	−6.33085e7	−2.50666
$915$	0	0
$916$	−8.01451e7	−3.15601
$917$	−1.21275e6	−0.0476266
$918$	0	0
$919$	−2.27313e7	−0.887842	−0.443921	−	0.896066i	$-0.646413\pi$
$919$	−2.27313e7	−0.887842	−0.443921	+	0.896066i	$0.646413\pi$
$920$	−7.95953e7	−3.10040
$921$	0	0
$922$	2.07526e7	0.803978
$923$	2.05168e6	0.0792694
$924$	0	0
$925$	1.13850e7	0.437499
$926$	−9.74634e6	−0.373520
$927$	0	0
$928$	−9.04635e7	−3.44829
$929$	3.32467e7	1.26389	0.631945	−	0.775013i	$-0.282257\pi$
$929$	3.32467e7	1.26389	0.631945	+	0.775013i	$0.282257\pi$
$930$	0	0
$931$	4.58840e7	1.73495
$932$	−1.83813e6	−0.0693165
$933$	0	0
$934$	−1.97967e7	−0.742548
$935$	1.00574e7	0.376233
$936$	0	0
$937$	1.72533e7	0.641981	0.320991	−	0.947082i	$-0.395984\pi$
$937$	1.72533e7	0.641981	0.320991	+	0.947082i	$0.395984\pi$
$938$	3.08988e6	0.114666
$939$	0	0
$940$	−4.84098e7	−1.78696
$941$	−1.12167e6	−0.0412945	−0.0206473	−	0.999787i	$-0.506573\pi$
$941$	−1.12167e6	−0.0412945	−0.0206473	+	0.999787i	$0.506573\pi$
$942$	0	0
$943$	6.55679e6	0.240111
$944$	−7.48954e7	−2.73543
$945$	0	0
$946$	1.66225e6	0.0603903
$947$	−3.89113e7	−1.40994	−0.704970	−	0.709237i	$-0.749040\pi$
$947$	−3.89113e7	−1.40994	−0.704970	+	0.709237i	$0.749040\pi$
$948$	0	0
$949$	−8.05647e6	−0.290388
$950$	−1.30029e8	−4.67445
$951$	0	0
$952$	1.32715e7	0.474600
$953$	1.79943e7	0.641803	0.320902	−	0.947113i	$-0.396014\pi$
$953$	1.79943e7	0.641803	0.320902	+	0.947113i	$0.396014\pi$
$954$	0	0
$955$	−1.95095e6	−0.0692210
$956$	−384326.	−0.0136005
$957$	0	0
$958$	−8.61632e7	−3.03325
$959$	−6.13783e6	−0.215510
$960$	0	0
$961$	−2.00011e7	−0.698626
$962$	−6.21744e6	−0.216608
$963$	0	0
$964$	7.24903e7	2.51239
$965$	−7.20932e7	−2.49216
$966$	0	0
$967$	2.31110e7	0.794791	0.397395	−	0.917648i	$-0.369914\pi$
$967$	2.31110e7	0.794791	0.397395	+	0.917648i	$0.369914\pi$
$968$	−7.59247e7	−2.60432
$969$	0	0
$970$	−7.68045e7	−2.62094
$971$	3.50927e7	1.19445	0.597226	−	0.802073i	$-0.296270\pi$
$971$	3.50927e7	1.19445	0.597226	+	0.802073i	$0.296270\pi$
$972$	0	0
$973$	3.83871e6	0.129988
$974$	−1.94207e7	−0.655945
$975$	0	0
$976$	−1.04987e8	−3.52786
$977$	−7.20029e6	−0.241331	−0.120666	−	0.992693i	$-0.538503\pi$
$977$	−7.20029e6	−0.241331	−0.120666	+	0.992693i	$0.538503\pi$
$978$	0	0
$979$	7.43795e6	0.248026
$980$	1.12461e8	3.74056
$981$	0	0
$982$	4.51771e7	1.49499
$983$	3.28324e7	1.08373	0.541863	−	0.840467i	$-0.317719\pi$
$983$	3.28324e7	1.08373	0.541863	+	0.840467i	$0.317719\pi$
$984$	0	0
$985$	6.96558e7	2.28753
$986$	1.04463e8	3.42193
$987$	0	0
$988$	5.05184e7	1.64648
$989$	3.43195e6	0.111571
$990$	0	0
$991$	5.19670e7	1.68090	0.840452	−	0.541885i	$-0.182289\pi$
$991$	5.19670e7	1.68090	0.840452	+	0.541885i	$0.182289\pi$
$992$	−3.63360e7	−1.17235
$993$	0	0
$994$	1.86867e6	0.0599884
$995$	1.63223e7	0.522665
$996$	0	0
$997$	−5.30582e6	−0.169050	−0.0845249	−	0.996421i	$-0.526937\pi$
$997$	−5.30582e6	−0.169050	−0.0845249	+	0.996421i	$0.526937\pi$
$998$	−8.21731e7	−2.61158
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.6.a.e.1.10		10
3.2	odd	2		43.6.a.b.1.1	✓	10
12.11	even	2		688.6.a.h.1.1		10
15.14	odd	2		1075.6.a.b.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.1	✓	10	3.2	odd	2
387.6.a.e.1.10		10	1.1	even	1	trivial
688.6.a.h.1.1		10	12.11	even	2
1075.6.a.b.1.10		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)

\(f(q)\)	\(=\)	\(q+10.5305 q^{2} +78.8905 q^{4} -86.8464 q^{5} -19.8137 q^{7} +493.778 q^{8} +O(q^{10})\)	\(q+10.5305 q^{2} +78.8905 q^{4} -86.8464 q^{5} -19.8137 q^{7} +493.778 q^{8} -914.532 q^{10} +85.3712 q^{11} -229.081 q^{13} -208.647 q^{14} +2675.21 q^{16} -1356.51 q^{17} -2795.35 q^{19} -6851.35 q^{20} +898.997 q^{22} +1856.11 q^{23} +4417.29 q^{25} -2412.33 q^{26} -1563.11 q^{28} -7312.96 q^{29} -2937.36 q^{31} +12370.3 q^{32} -14284.7 q^{34} +1720.75 q^{35} +2577.36 q^{37} -29436.3 q^{38} -42882.8 q^{40} +3532.54 q^{41} +1849.00 q^{43} +6734.97 q^{44} +19545.7 q^{46} +7065.73 q^{47} -16414.4 q^{49} +46516.1 q^{50} -18072.3 q^{52} +3852.63 q^{53} -7414.18 q^{55} -9783.57 q^{56} -77008.8 q^{58} -27996.1 q^{59} -39244.4 q^{61} -30931.7 q^{62} +44658.1 q^{64} +19894.9 q^{65} -14809.1 q^{67} -107016. q^{68} +18120.3 q^{70} -8956.13 q^{71} +35168.6 q^{73} +27140.8 q^{74} -220526. q^{76} -1691.52 q^{77} -13263.6 q^{79} -232332. q^{80} +37199.3 q^{82} +9812.47 q^{83} +117808. q^{85} +19470.8 q^{86} +42154.4 q^{88} +87124.9 q^{89} +4538.95 q^{91} +146429. q^{92} +74405.4 q^{94} +242766. q^{95} +83982.4 q^{97} -172851. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8}+O(q^{10}) \) 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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