LMFDB - Embedded newform 784.6.a.s.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(1,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-14,0,70,0,0,0,244] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{79}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 79 $ x^2 - 79
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.88819$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+10.7764 q^{3} +106.106 q^{5} -126.869 q^{9} +93.4347 q^{11} +661.131 q^{13} +1143.43 q^{15} +455.919 q^{17} -1106.28 q^{19} -748.390 q^{23} +8133.39 q^{25} -3985.86 q^{27} +2804.78 q^{29} +359.299 q^{31} +1006.89 q^{33} -6812.82 q^{37} +7124.60 q^{39} +2319.39 q^{41} +19965.7 q^{43} -13461.6 q^{45} +14209.5 q^{47} +4913.17 q^{51} +26144.2 q^{53} +9913.94 q^{55} -11921.7 q^{57} -4904.35 q^{59} -13203.3 q^{61} +70149.6 q^{65} +59658.1 q^{67} -8064.94 q^{69} -8906.43 q^{71} -10480.6 q^{73} +87648.6 q^{75} +7230.02 q^{79} -12123.9 q^{81} +100461. q^{83} +48375.6 q^{85} +30225.4 q^{87} +20071.8 q^{89} +3871.94 q^{93} -117382. q^{95} -23320.9 q^{97} -11854.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{3} + 70 q^{5} + 244 q^{9} - 62 q^{11} + 1820 q^{13} + 2038 q^{15} + 1694 q^{17} - 826 q^{19} + 2734 q^{23} + 6312 q^{25} - 7154 q^{27} - 2852 q^{29} + 2674 q^{31} + 4858 q^{33} - 9146 q^{37}+ \cdots - 69500 q^{99}+O(q^{100}) $ 2 * q - 14 * q^3 + 70 * q^5 + 244 * q^9 - 62 * q^11 + 1820 * q^13 + 2038 * q^15 + 1694 * q^17 - 826 * q^19 + 2734 * q^23 + 6312 * q^25 - 7154 * q^27 - 2852 * q^29 + 2674 * q^31 + 4858 * q^33 - 9146 * q^37 - 21588 * q^39 + 6132 * q^41 + 16040 * q^43 - 26852 * q^45 + 25326 * q^47 - 25762 * q^51 + 14958 * q^53 + 15526 * q^55 - 18866 * q^57 + 1106 * q^59 - 28042 * q^61 + 28308 * q^65 + 102642 * q^67 - 94346 * q^69 + 11056 * q^71 + 35070 * q^73 + 132776 * q^75 - 101762 * q^79 - 23750 * q^81 + 44632 * q^83 + 3674 * q^85 + 170380 * q^87 - 75474 * q^89 - 53478 * q^93 - 127502 * q^95 - 8316 * q^97 - 69500 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	10.7764	0.691306	0.345653	−	0.938362i	$-0.387657\pi$
$3$	10.7764	0.691306	0.345653	+	0.938362i	$0.387657\pi$
$4$	0	0
$5$	106.106	1.89807	0.949037	−	0.315165i	$-0.102060\pi$
$5$	106.106	1.89807	0.949037	+	0.315165i	$0.102060\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−126.869	−0.522096
$10$	0	0
$11$	93.4347	0.232823	0.116412	−	0.993201i	$-0.462861\pi$
$11$	93.4347	0.232823	0.116412	+	0.993201i	$0.462861\pi$
$12$	0	0
$13$	661.131	1.08500	0.542499	−	0.840057i	$-0.317478\pi$
$13$	661.131	1.08500	0.542499	+	0.840057i	$0.317478\pi$
$14$	0	0
$15$	1143.43	1.31215
$16$	0	0
$17$	455.919	0.382618	0.191309	−	0.981530i	$-0.438727\pi$
$17$	455.919	0.382618	0.191309	+	0.981530i	$0.438727\pi$
$18$	0	0
$19$	−1106.28	−0.703041	−0.351521	−	0.936180i	$-0.614335\pi$
$19$	−1106.28	−0.703041	−0.351521	+	0.936180i	$0.614335\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−748.390	−0.294991	−0.147495	−	0.989063i	$-0.547121\pi$
$23$	−748.390	−0.294991	−0.147495	+	0.989063i	$0.547121\pi$
$24$	0	0
$25$	8133.39	2.60268
$26$	0	0
$27$	−3985.86	−1.05223
$28$	0	0
$29$	2804.78	0.619304	0.309652	−	0.950850i	$-0.399787\pi$
$29$	2804.78	0.619304	0.309652	+	0.950850i	$0.399787\pi$
$30$	0	0
$31$	359.299	0.0671508	0.0335754	−	0.999436i	$-0.489311\pi$
$31$	359.299	0.0671508	0.0335754	+	0.999436i	$0.489311\pi$
$32$	0	0
$33$	1006.89	0.160952
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6812.82	−0.818131	−0.409066	−	0.912505i	$-0.634145\pi$
$37$	−6812.82	−0.818131	−0.409066	+	0.912505i	$0.634145\pi$
$38$	0	0
$39$	7124.60	0.750065
$40$	0	0
$41$	2319.39	0.215484	0.107742	−	0.994179i	$-0.465638\pi$
$41$	2319.39	0.215484	0.107742	+	0.994179i	$0.465638\pi$
$42$	0	0
$43$	19965.7	1.64670	0.823349	−	0.567535i	$-0.192103\pi$
$43$	19965.7	1.64670	0.823349	+	0.567535i	$0.192103\pi$
$44$	0	0
$45$	−13461.6	−0.990978
$46$	0	0
$47$	14209.5	0.938287	0.469143	−	0.883122i	$-0.344563\pi$
$47$	14209.5	0.938287	0.469143	+	0.883122i	$0.344563\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4913.17	0.264506
$52$	0	0
$53$	26144.2	1.27846	0.639228	−	0.769017i	$-0.279254\pi$
$53$	26144.2	1.27846	0.639228	+	0.769017i	$0.279254\pi$
$54$	0	0
$55$	9913.94	0.441916
$56$	0	0
$57$	−11921.7	−0.486016
$58$	0	0
$59$	−4904.35	−0.183422	−0.0917110	−	0.995786i	$-0.529234\pi$
$59$	−4904.35	−0.183422	−0.0917110	+	0.995786i	$0.529234\pi$
$60$	0	0
$61$	−13203.3	−0.454315	−0.227158	−	0.973858i	$-0.572943\pi$
$61$	−13203.3	−0.454315	−0.227158	+	0.973858i	$0.572943\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	70149.6	2.05941
$66$	0	0
$67$	59658.1	1.62361	0.811807	−	0.583926i	$-0.198484\pi$
$67$	59658.1	1.62361	0.811807	+	0.583926i	$0.198484\pi$
$68$	0	0
$69$	−8064.94	−0.203929
$70$	0	0
$71$	−8906.43	−0.209680	−0.104840	−	0.994489i	$-0.533433\pi$
$71$	−8906.43	−0.209680	−0.104840	+	0.994489i	$0.533433\pi$
$72$	0	0
$73$	−10480.6	−0.230186	−0.115093	−	0.993355i	$-0.536717\pi$
$73$	−10480.6	−0.230186	−0.115093	+	0.993355i	$0.536717\pi$
$74$	0	0
$75$	87648.6	1.79925
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7230.02	0.130338	0.0651691	−	0.997874i	$-0.479241\pi$
$79$	7230.02	0.130338	0.0651691	+	0.997874i	$0.479241\pi$
$80$	0	0
$81$	−12123.9	−0.205319
$82$	0	0
$83$	100461.	1.60067	0.800336	−	0.599552i	$-0.204655\pi$
$83$	100461.	1.60067	0.800336	+	0.599552i	$0.204655\pi$
$84$	0	0
$85$	48375.6	0.726238
$86$	0	0
$87$	30225.4	0.428128
$88$	0	0
$89$	20071.8	0.268603	0.134302	−	0.990940i	$-0.457121\pi$
$89$	20071.8	0.268603	0.134302	+	0.990940i	$0.457121\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3871.94	0.0464217
$94$	0	0
$95$	−117382.	−1.33442
$96$	0	0
$97$	−23320.9	−0.251662	−0.125831	−	0.992052i	$-0.540160\pi$
$97$	−23320.9	−0.251662	−0.125831	+	0.992052i	$0.540160\pi$
$98$	0	0
$99$	−11854.0	−0.121556
$100$	0	0
$101$	−68974.1	−0.672795	−0.336398	−	0.941720i	$-0.609209\pi$
$101$	−68974.1	−0.672795	−0.336398	+	0.941720i	$0.609209\pi$
$102$	0	0
$103$	−113725.	−1.05624	−0.528121	−	0.849169i	$-0.677103\pi$
$103$	−113725.	−1.05624	−0.528121	+	0.849169i	$0.677103\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	121287.	1.02413	0.512066	−	0.858946i	$-0.328880\pi$
$107$	121287.	1.02413	0.512066	+	0.858946i	$0.328880\pi$
$108$	0	0
$109$	−61373.9	−0.494786	−0.247393	−	0.968915i	$-0.579574\pi$
$109$	−61373.9	−0.494786	−0.247393	+	0.968915i	$0.579574\pi$
$110$	0	0
$111$	−73417.7	−0.565579
$112$	0	0
$113$	−242939.	−1.78979	−0.894893	−	0.446281i	$-0.852748\pi$
$113$	−242939.	−1.78979	−0.894893	+	0.446281i	$0.852748\pi$
$114$	0	0
$115$	−79408.4	−0.559914
$116$	0	0
$117$	−83877.3	−0.566474
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−152321.	−0.945793
$122$	0	0
$123$	24994.7	0.148965
$124$	0	0
$125$	531418.	3.04201
$126$	0	0
$127$	−201922.	−1.11090	−0.555448	−	0.831551i	$-0.687453\pi$
$127$	−201922.	−1.11090	−0.555448	+	0.831551i	$0.687453\pi$
$128$	0	0
$129$	215159.	1.13837
$130$	0	0
$131$	−283521.	−1.44347	−0.721734	−	0.692171i	$-0.756655\pi$
$131$	−283521.	−1.44347	−0.721734	+	0.692171i	$0.756655\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−422922.	−1.99722
$136$	0	0
$137$	188471.	0.857911	0.428955	−	0.903326i	$-0.358882\pi$
$137$	188471.	0.857911	0.428955	+	0.903326i	$0.358882\pi$
$138$	0	0
$139$	211579.	0.928830	0.464415	−	0.885618i	$-0.346265\pi$
$139$	211579.	0.928830	0.464415	+	0.885618i	$0.346265\pi$
$140$	0	0
$141$	153128.	0.648643
$142$	0	0
$143$	61772.5	0.252613
$144$	0	0
$145$	297603.	1.17548
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−293324.	−1.08238	−0.541192	−	0.840899i	$-0.682027\pi$
$149$	−293324.	−1.08238	−0.541192	+	0.840899i	$0.682027\pi$
$150$	0	0
$151$	204476.	0.729794	0.364897	−	0.931048i	$-0.381104\pi$
$151$	204476.	0.729794	0.364897	+	0.931048i	$0.381104\pi$
$152$	0	0
$153$	−57842.2	−0.199764
$154$	0	0
$155$	38123.6	0.127457
$156$	0	0
$157$	505915.	1.63805	0.819027	−	0.573755i	$-0.194514\pi$
$157$	505915.	1.63805	0.819027	+	0.573755i	$0.194514\pi$
$158$	0	0
$159$	281740.	0.883804
$160$	0	0
$161$	0	0
$162$	0	0
$163$	284200.	0.837829	0.418914	−	0.908026i	$-0.362411\pi$
$163$	284200.	0.837829	0.418914	+	0.908026i	$0.362411\pi$
$164$	0	0
$165$	106837.	0.305499
$166$	0	0
$167$	13145.7	0.0364747	0.0182373	−	0.999834i	$-0.494195\pi$
$167$	13145.7	0.0364747	0.0182373	+	0.999834i	$0.494195\pi$
$168$	0	0
$169$	65800.6	0.177220
$170$	0	0
$171$	140353.	0.367055
$172$	0	0
$173$	51453.0	0.130706	0.0653530	−	0.997862i	$-0.479183\pi$
$173$	51453.0	0.130706	0.0653530	+	0.997862i	$0.479183\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−52851.2	−0.126801
$178$	0	0
$179$	497735.	1.16109	0.580544	−	0.814229i	$-0.302840\pi$
$179$	497735.	1.16109	0.580544	+	0.814229i	$0.302840\pi$
$180$	0	0
$181$	−227120.	−0.515299	−0.257650	−	0.966238i	$-0.582948\pi$
$181$	−227120.	−0.515299	−0.257650	+	0.966238i	$0.582948\pi$
$182$	0	0
$183$	−142284.	−0.314071
$184$	0	0
$185$	−722879.	−1.55287
$186$	0	0
$187$	42598.7	0.0890825
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−502778.	−0.997225	−0.498612	−	0.866825i	$-0.666157\pi$
$191$	−502778.	−0.997225	−0.498612	+	0.866825i	$0.666157\pi$
$192$	0	0
$193$	140322.	0.271165	0.135582	−	0.990766i	$-0.456709\pi$
$193$	140322.	0.271165	0.135582	+	0.990766i	$0.456709\pi$
$194$	0	0
$195$	755960.	1.42368
$196$	0	0
$197$	378993.	0.695769	0.347885	−	0.937537i	$-0.386900\pi$
$197$	378993.	0.695769	0.347885	+	0.937537i	$0.386900\pi$
$198$	0	0
$199$	461022.	0.825256	0.412628	−	0.910900i	$-0.364611\pi$
$199$	461022.	0.825256	0.412628	+	0.910900i	$0.364611\pi$
$200$	0	0
$201$	642899.	1.12241
$202$	0	0
$203$	0	0
$204$	0	0
$205$	246100.	0.409004
$206$	0	0
$207$	94947.9	0.154014
$208$	0	0
$209$	−103365.	−0.163684
$210$	0	0
$211$	−202139.	−0.312567	−0.156284	−	0.987712i	$-0.549951\pi$
$211$	−202139.	−0.312567	−0.156284	+	0.987712i	$0.549951\pi$
$212$	0	0
$213$	−95979.1	−0.144953
$214$	0	0
$215$	2.11848e6	3.12556
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−112943.	−0.159129
$220$	0	0
$221$	301422.	0.415140
$222$	0	0
$223$	−694340.	−0.934997	−0.467498	−	0.883994i	$-0.654845\pi$
$223$	−694340.	−0.934997	−0.467498	+	0.883994i	$0.654845\pi$
$224$	0	0
$225$	−1.03188e6	−1.35885
$226$	0	0
$227$	−1.26697e6	−1.63194	−0.815968	−	0.578096i	$-0.803796\pi$
$227$	−1.26697e6	−1.63194	−0.815968	+	0.578096i	$0.803796\pi$
$228$	0	0
$229$	856477.	1.07926	0.539631	−	0.841902i	$-0.318564\pi$
$229$	856477.	1.07926	0.539631	+	0.841902i	$0.318564\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−284358.	−0.343143	−0.171571	−	0.985172i	$-0.554884\pi$
$233$	−284358.	−0.343143	−0.171571	+	0.985172i	$0.554884\pi$
$234$	0	0
$235$	1.50771e6	1.78094
$236$	0	0
$237$	77913.5	0.0901035
$238$	0	0
$239$	427941.	0.484606	0.242303	−	0.970201i	$-0.422097\pi$
$239$	427941.	0.484606	0.242303	+	0.970201i	$0.422097\pi$
$240$	0	0
$241$	809581.	0.897879	0.448940	−	0.893562i	$-0.351802\pi$
$241$	809581.	0.897879	0.448940	+	0.893562i	$0.351802\pi$
$242$	0	0
$243$	837912.	0.910296
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−731395.	−0.762798
$248$	0	0
$249$	1.08261e6	1.10655
$250$	0	0
$251$	1.26305e6	1.26542	0.632711	−	0.774388i	$-0.281942\pi$
$251$	1.26305e6	1.26542	0.632711	+	0.774388i	$0.281942\pi$
$252$	0	0
$253$	−69925.6	−0.0686808
$254$	0	0
$255$	521314.	0.502052
$256$	0	0
$257$	56138.2	0.0530183	0.0265091	−	0.999649i	$-0.491561\pi$
$257$	56138.2	0.0530183	0.0265091	+	0.999649i	$0.491561\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−355841.	−0.323336
$262$	0	0
$263$	−1.54275e6	−1.37533	−0.687664	−	0.726029i	$-0.741364\pi$
$263$	−1.54275e6	−1.37533	−0.687664	+	0.726029i	$0.741364\pi$
$264$	0	0
$265$	2.77405e6	2.42660
$266$	0	0
$267$	216302.	0.185687
$268$	0	0
$269$	−1.20535e6	−1.01563	−0.507813	−	0.861467i	$-0.669546\pi$
$269$	−1.20535e6	−1.01563	−0.507813	+	0.861467i	$0.669546\pi$
$270$	0	0
$271$	−1.76489e6	−1.45980	−0.729900	−	0.683554i	$-0.760433\pi$
$271$	−1.76489e6	−1.45980	−0.729900	+	0.683554i	$0.760433\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	759941.	0.605966
$276$	0	0
$277$	1.48319e6	1.16144	0.580719	−	0.814104i	$-0.302771\pi$
$277$	1.48319e6	1.16144	0.580719	+	0.814104i	$0.302771\pi$
$278$	0	0
$279$	−45584.0	−0.0350592
$280$	0	0
$281$	−1.26812e6	−0.958061	−0.479031	−	0.877798i	$-0.659012\pi$
$281$	−1.26812e6	−0.958061	−0.479031	+	0.877798i	$0.659012\pi$
$282$	0	0
$283$	1.44657e6	1.07368	0.536838	−	0.843685i	$-0.319619\pi$
$283$	1.44657e6	1.07368	0.536838	+	0.843685i	$0.319619\pi$
$284$	0	0
$285$	−1.26496e6	−0.922495
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.21199e6	−0.853603
$290$	0	0
$291$	−251316.	−0.173975
$292$	0	0
$293$	367016.	0.249756	0.124878	−	0.992172i	$-0.460146\pi$
$293$	367016.	0.249756	0.124878	+	0.992172i	$0.460146\pi$
$294$	0	0
$295$	−520379.	−0.348149
$296$	0	0
$297$	−372417.	−0.244985
$298$	0	0
$299$	−494784.	−0.320064
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−743292.	−0.465107
$304$	0	0
$305$	−1.40094e6	−0.862324
$306$	0	0
$307$	131281.	0.0794979	0.0397490	−	0.999210i	$-0.487344\pi$
$307$	131281.	0.0794979	0.0397490	+	0.999210i	$0.487344\pi$
$308$	0	0
$309$	−1.22555e6	−0.730186
$310$	0	0
$311$	2.33471e6	1.36878	0.684389	−	0.729118i	$-0.260069\pi$
$311$	2.33471e6	1.36878	0.684389	+	0.729118i	$0.260069\pi$
$312$	0	0
$313$	−772801.	−0.445869	−0.222934	−	0.974833i	$-0.571564\pi$
$313$	−772801.	−0.445869	−0.222934	+	0.974833i	$0.571564\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	524336.	0.293063	0.146532	−	0.989206i	$-0.453189\pi$
$317$	524336.	0.293063	0.146532	+	0.989206i	$0.453189\pi$
$318$	0	0
$319$	262064.	0.144188
$320$	0	0
$321$	1.30704e6	0.707988
$322$	0	0
$323$	−504374.	−0.268996
$324$	0	0
$325$	5.37723e6	2.82391
$326$	0	0
$327$	−661389.	−0.342048
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−631635.	−0.316881	−0.158440	−	0.987369i	$-0.550647\pi$
$331$	−631635.	−0.316881	−0.158440	+	0.987369i	$0.550647\pi$
$332$	0	0
$333$	864339.	0.427143
$334$	0	0
$335$	6.33006e6	3.08174
$336$	0	0
$337$	3.33479e6	1.59954	0.799768	−	0.600309i	$-0.204956\pi$
$337$	3.33479e6	1.59954	0.799768	+	0.600309i	$0.204956\pi$
$338$	0	0
$339$	−2.61800e6	−1.23729
$340$	0	0
$341$	33571.0	0.0156343
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−855735.	−0.387072
$346$	0	0
$347$	650351.	0.289951	0.144975	−	0.989435i	$-0.453690\pi$
$347$	650351.	0.289951	0.144975	+	0.989435i	$0.453690\pi$
$348$	0	0
$349$	−601041.	−0.264144	−0.132072	−	0.991240i	$-0.542163\pi$
$349$	−601041.	−0.264144	−0.132072	+	0.991240i	$0.542163\pi$
$350$	0	0
$351$	−2.63517e6	−1.14167
$352$	0	0
$353$	−2.88889e6	−1.23394	−0.616970	−	0.786987i	$-0.711640\pi$
$353$	−2.88889e6	−1.23394	−0.616970	+	0.786987i	$0.711640\pi$
$354$	0	0
$355$	−945021.	−0.397989
$356$	0	0
$357$	0	0
$358$	0	0
$359$	97830.5	0.0400625	0.0200313	−	0.999799i	$-0.493623\pi$
$359$	97830.5	0.0400625	0.0200313	+	0.999799i	$0.493623\pi$
$360$	0	0
$361$	−1.25225e6	−0.505733
$362$	0	0
$363$	−1.64147e6	−0.653832
$364$	0	0
$365$	−1.11205e6	−0.436910
$366$	0	0
$367$	−2.14775e6	−0.832374	−0.416187	−	0.909279i	$-0.636634\pi$
$367$	−2.14775e6	−0.832374	−0.416187	+	0.909279i	$0.636634\pi$
$368$	0	0
$369$	−294260.	−0.112503
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4.20458e6	1.56477	0.782386	−	0.622794i	$-0.214003\pi$
$373$	4.20458e6	1.56477	0.782386	+	0.622794i	$0.214003\pi$
$374$	0	0
$375$	5.72677e6	2.10296
$376$	0	0
$377$	1.85433e6	0.671943
$378$	0	0
$379$	535586.	0.191527	0.0957637	−	0.995404i	$-0.469471\pi$
$379$	535586.	0.191527	0.0957637	+	0.995404i	$0.469471\pi$
$380$	0	0
$381$	−2.17599e6	−0.767969
$382$	0	0
$383$	3.84697e6	1.34005	0.670027	−	0.742337i	$-0.266283\pi$
$383$	3.84697e6	1.34005	0.670027	+	0.742337i	$0.266283\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.53304e6	−0.859736
$388$	0	0
$389$	−1.69833e6	−0.569045	−0.284523	−	0.958669i	$-0.591835\pi$
$389$	−1.69833e6	−0.569045	−0.284523	+	0.958669i	$0.591835\pi$
$390$	0	0
$391$	−341206.	−0.112869
$392$	0	0
$393$	−3.05533e6	−0.997877
$394$	0	0
$395$	767145.	0.247391
$396$	0	0
$397$	3.62378e6	1.15394	0.576972	−	0.816764i	$-0.304234\pi$
$397$	3.62378e6	1.15394	0.576972	+	0.816764i	$0.304234\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.79801e6	−0.868937	−0.434469	−	0.900687i	$-0.643064\pi$
$401$	−2.79801e6	−0.868937	−0.434469	+	0.900687i	$0.643064\pi$
$402$	0	0
$403$	237543.	0.0728585
$404$	0	0
$405$	−1.28641e6	−0.389710
$406$	0	0
$407$	−636554.	−0.190480
$408$	0	0
$409$	−666281.	−0.196947	−0.0984735	−	0.995140i	$-0.531396\pi$
$409$	−666281.	−0.196947	−0.0984735	+	0.995140i	$0.531396\pi$
$410$	0	0
$411$	2.03103e6	0.593078
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.06595e7	3.03819
$416$	0	0
$417$	2.28006e6	0.642105
$418$	0	0
$419$	−2.89203e6	−0.804762	−0.402381	−	0.915472i	$-0.631817\pi$
$419$	−2.89203e6	−0.804762	−0.402381	+	0.915472i	$0.631817\pi$
$420$	0	0
$421$	6.72027e6	1.84791	0.923956	−	0.382498i	$-0.124936\pi$
$421$	6.72027e6	1.84791	0.923956	+	0.382498i	$0.124936\pi$
$422$	0	0
$423$	−1.80276e6	−0.489876
$424$	0	0
$425$	3.70817e6	0.995835
$426$	0	0
$427$	0	0
$428$	0	0
$429$	665685.	0.174633
$430$	0	0
$431$	−4.72298e6	−1.22468	−0.612341	−	0.790594i	$-0.709772\pi$
$431$	−4.72298e6	−1.22468	−0.612341	+	0.790594i	$0.709772\pi$
$432$	0	0
$433$	−5.78136e6	−1.48187	−0.740935	−	0.671577i	$-0.765617\pi$
$433$	−5.78136e6	−1.48187	−0.740935	+	0.671577i	$0.765617\pi$
$434$	0	0
$435$	3.20708e6	0.812619
$436$	0	0
$437$	827929.	0.207391
$438$	0	0
$439$	−5.36873e6	−1.32957	−0.664784	−	0.747036i	$-0.731476\pi$
$439$	−5.36873e6	−1.32957	−0.664784	+	0.747036i	$0.731476\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.31511e6	−1.04468	−0.522339	−	0.852738i	$-0.674941\pi$
$443$	−4.31511e6	−1.04468	−0.522339	+	0.852738i	$0.674941\pi$
$444$	0	0
$445$	2.12973e6	0.509829
$446$	0	0
$447$	−3.16097e6	−0.748259
$448$	0	0
$449$	4.75292e6	1.11261	0.556307	−	0.830977i	$-0.312218\pi$
$449$	4.75292e6	1.11261	0.556307	+	0.830977i	$0.312218\pi$
$450$	0	0
$451$	216712.	0.0501696
$452$	0	0
$453$	2.20352e6	0.504511
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.13728e6	0.478707	0.239354	−	0.970932i	$-0.423064\pi$
$457$	2.13728e6	0.478707	0.239354	+	0.970932i	$0.423064\pi$
$458$	0	0
$459$	−1.81723e6	−0.402604
$460$	0	0
$461$	−1.72920e6	−0.378959	−0.189479	−	0.981885i	$-0.560680\pi$
$461$	−1.72920e6	−0.378959	−0.189479	+	0.981885i	$0.560680\pi$
$462$	0	0
$463$	6.54367e6	1.41863	0.709315	−	0.704892i	$-0.249004\pi$
$463$	6.54367e6	1.41863	0.709315	+	0.704892i	$0.249004\pi$
$464$	0	0
$465$	410835.	0.0881119
$466$	0	0
$467$	1.79419e6	0.380694	0.190347	−	0.981717i	$-0.439039\pi$
$467$	1.79419e6	0.380694	0.190347	+	0.981717i	$0.439039\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.45194e6	1.13240
$472$	0	0
$473$	1.86549e6	0.383390
$474$	0	0
$475$	−8.99780e6	−1.82979
$476$	0	0
$477$	−3.31690e6	−0.667477
$478$	0	0
$479$	−4.09403e6	−0.815289	−0.407645	−	0.913141i	$-0.633650\pi$
$479$	−4.09403e6	−0.815289	−0.407645	+	0.913141i	$0.633650\pi$
$480$	0	0
$481$	−4.50417e6	−0.887670
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.47448e6	−0.477672
$486$	0	0
$487$	4.45210e6	0.850634	0.425317	−	0.905045i	$-0.360163\pi$
$487$	4.45210e6	0.850634	0.425317	+	0.905045i	$0.360163\pi$
$488$	0	0
$489$	3.06265e6	0.579196
$490$	0	0
$491$	−7.09674e6	−1.32848	−0.664240	−	0.747519i	$-0.731245\pi$
$491$	−7.09674e6	−1.32848	−0.664240	+	0.747519i	$0.731245\pi$
$492$	0	0
$493$	1.27875e6	0.236957
$494$	0	0
$495$	−1.25778e6	−0.230723
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.41903e6	−1.69338	−0.846691	−	0.532085i	$-0.821409\pi$
$499$	−9.41903e6	−1.69338	−0.846691	+	0.532085i	$0.821409\pi$
$500$	0	0
$501$	141663.	0.0252152
$502$	0	0
$503$	−957258.	−0.168698	−0.0843489	−	0.996436i	$-0.526881\pi$
$503$	−957258.	−0.168698	−0.0843489	+	0.996436i	$0.526881\pi$
$504$	0	0
$505$	−7.31854e6	−1.27702
$506$	0	0
$507$	709093.	0.122513
$508$	0	0
$509$	−4.01453e6	−0.686816	−0.343408	−	0.939186i	$-0.611581\pi$
$509$	−4.01453e6	−0.686816	−0.343408	+	0.939186i	$0.611581\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.40947e6	0.739764
$514$	0	0
$515$	−1.20669e7	−2.00483
$516$	0	0
$517$	1.32766e6	0.218455
$518$	0	0
$519$	554478.	0.0903578
$520$	0	0
$521$	−4.31820e6	−0.696960	−0.348480	−	0.937316i	$-0.613302\pi$
$521$	−4.31820e6	−0.696960	−0.348480	+	0.937316i	$0.613302\pi$
$522$	0	0
$523$	−1.34501e6	−0.215017	−0.107508	−	0.994204i	$-0.534287\pi$
$523$	−1.34501e6	−0.215017	−0.107508	+	0.994204i	$0.534287\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	163811.	0.0256931
$528$	0	0
$529$	−5.87626e6	−0.912980
$530$	0	0
$531$	622212.	0.0957640
$532$	0	0
$533$	1.53342e6	0.233799
$534$	0	0
$535$	1.28692e7	1.94388
$536$	0	0
$537$	5.36378e6	0.802667
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.06006e6	−1.03709	−0.518543	−	0.855051i	$-0.673526\pi$
$541$	−7.06006e6	−1.03709	−0.518543	+	0.855051i	$0.673526\pi$
$542$	0	0
$543$	−2.44754e6	−0.356229
$544$	0	0
$545$	−6.51211e6	−0.939140
$546$	0	0
$547$	1.23520e7	1.76510	0.882549	−	0.470221i	$-0.155826\pi$
$547$	1.23520e7	1.76510	0.882549	+	0.470221i	$0.155826\pi$
$548$	0	0
$549$	1.67509e6	0.237196
$550$	0	0
$551$	−3.10287e6	−0.435396
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.79002e6	−1.07351
$556$	0	0
$557$	−1.22251e7	−1.66961	−0.834803	−	0.550549i	$-0.814418\pi$
$557$	−1.22251e7	−1.66961	−0.834803	+	0.550549i	$0.814418\pi$
$558$	0	0
$559$	1.32000e7	1.78666
$560$	0	0
$561$	459060.	0.0615832
$562$	0	0
$563$	812527.	0.108036	0.0540178	−	0.998540i	$-0.482797\pi$
$563$	812527.	0.108036	0.0540178	+	0.998540i	$0.482797\pi$
$564$	0	0
$565$	−2.57772e7	−3.39715
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.43881e6	−1.09270	−0.546350	−	0.837557i	$-0.683983\pi$
$569$	−8.43881e6	−1.09270	−0.546350	+	0.837557i	$0.683983\pi$
$570$	0	0
$571$	−3.71511e6	−0.476849	−0.238425	−	0.971161i	$-0.576631\pi$
$571$	−3.71511e6	−0.476849	−0.238425	+	0.971161i	$0.576631\pi$
$572$	0	0
$573$	−5.41813e6	−0.689387
$574$	0	0
$575$	−6.08695e6	−0.767768
$576$	0	0
$577$	1.30375e7	1.63025	0.815126	−	0.579284i	$-0.196668\pi$
$577$	1.30375e7	1.63025	0.815126	+	0.579284i	$0.196668\pi$
$578$	0	0
$579$	1.51217e6	0.187458
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.44278e6	0.297654
$584$	0	0
$585$	−8.89984e6	−1.07521
$586$	0	0
$587$	−1.15227e7	−1.38026	−0.690128	−	0.723687i	$-0.742446\pi$
$587$	−1.15227e7	−1.38026	−0.690128	+	0.723687i	$0.742446\pi$
$588$	0	0
$589$	−397485.	−0.0472098
$590$	0	0
$591$	4.08417e6	0.480989
$592$	0	0
$593$	−1.20103e7	−1.40254	−0.701271	−	0.712895i	$-0.747384\pi$
$593$	−1.20103e7	−1.40254	−0.701271	+	0.712895i	$0.747384\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.96815e6	0.570504
$598$	0	0
$599$	1.17460e6	0.133759	0.0668794	−	0.997761i	$-0.478696\pi$
$599$	1.17460e6	0.133759	0.0668794	+	0.997761i	$0.478696\pi$
$600$	0	0
$601$	−1.62934e7	−1.84003	−0.920014	−	0.391886i	$-0.871823\pi$
$601$	−1.62934e7	−1.84003	−0.920014	+	0.391886i	$0.871823\pi$
$602$	0	0
$603$	−7.56879e6	−0.847683
$604$	0	0
$605$	−1.61621e7	−1.79519
$606$	0	0
$607$	1.82629e6	0.201186	0.100593	−	0.994928i	$-0.467926\pi$
$607$	1.82629e6	0.201186	0.100593	+	0.994928i	$0.467926\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.39436e6	1.01804
$612$	0	0
$613$	−5.10271e6	−0.548466	−0.274233	−	0.961663i	$-0.588424\pi$
$613$	−5.10271e6	−0.548466	−0.274233	+	0.961663i	$0.588424\pi$
$614$	0	0
$615$	2.65207e6	0.282747
$616$	0	0
$617$	−8.03920e6	−0.850158	−0.425079	−	0.905156i	$-0.639754\pi$
$617$	−8.03920e6	−0.850158	−0.425079	+	0.905156i	$0.639754\pi$
$618$	0	0
$619$	−8.60132e6	−0.902274	−0.451137	−	0.892455i	$-0.648981\pi$
$619$	−8.60132e6	−0.902274	−0.451137	+	0.892455i	$0.648981\pi$
$620$	0	0
$621$	2.98298e6	0.310399
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.09695e7	3.17128
$626$	0	0
$627$	−1.11390e6	−0.113156
$628$	0	0
$629$	−3.10610e6	−0.313032
$630$	0	0
$631$	−6.21151e6	−0.621046	−0.310523	−	0.950566i	$-0.600504\pi$
$631$	−6.21151e6	−0.621046	−0.310523	+	0.950566i	$0.600504\pi$
$632$	0	0
$633$	−2.17833e6	−0.216080
$634$	0	0
$635$	−2.14250e7	−2.10856
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.12995e6	0.109473
$640$	0	0
$641$	4.45143e6	0.427912	0.213956	−	0.976843i	$-0.431365\pi$
$641$	4.45143e6	0.427912	0.213956	+	0.976843i	$0.431365\pi$
$642$	0	0
$643$	−1.58708e7	−1.51381	−0.756907	−	0.653523i	$-0.773290\pi$
$643$	−1.58708e7	−1.51381	−0.756907	+	0.653523i	$0.773290\pi$
$644$	0	0
$645$	2.28295e7	2.16071
$646$	0	0
$647$	3.65619e6	0.343375	0.171687	−	0.985151i	$-0.445078\pi$
$647$	3.65619e6	0.343375	0.171687	+	0.985151i	$0.445078\pi$
$648$	0	0
$649$	−458237.	−0.0427049
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.15039e6	−0.380896	−0.190448	−	0.981697i	$-0.560994\pi$
$653$	−4.15039e6	−0.380896	−0.190448	+	0.981697i	$0.560994\pi$
$654$	0	0
$655$	−3.00832e7	−2.73981
$656$	0	0
$657$	1.32967e6	0.120179
$658$	0	0
$659$	1.33739e7	1.19962	0.599810	−	0.800143i	$-0.295243\pi$
$659$	1.33739e7	1.19962	0.599810	+	0.800143i	$0.295243\pi$
$660$	0	0
$661$	−9.82641e6	−0.874764	−0.437382	−	0.899276i	$-0.644094\pi$
$661$	−9.82641e6	−0.874764	−0.437382	+	0.899276i	$0.644094\pi$
$662$	0	0
$663$	3.24824e6	0.286989
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.09907e6	−0.182689
$668$	0	0
$669$	−7.48248e6	−0.646369
$670$	0	0
$671$	−1.23365e6	−0.105775
$672$	0	0
$673$	401148.	0.0341403	0.0170702	−	0.999854i	$-0.494566\pi$
$673$	401148.	0.0341403	0.0170702	+	0.999854i	$0.494566\pi$
$674$	0	0
$675$	−3.24185e7	−2.73863
$676$	0	0
$677$	2.07302e7	1.73833	0.869164	−	0.494524i	$-0.164657\pi$
$677$	2.07302e7	1.73833	0.869164	+	0.494524i	$0.164657\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.36534e7	−1.12817
$682$	0	0
$683$	1.91996e7	1.57485	0.787427	−	0.616409i	$-0.211413\pi$
$683$	1.91996e7	1.57485	0.787427	+	0.616409i	$0.211413\pi$
$684$	0	0
$685$	1.99978e7	1.62838
$686$	0	0
$687$	9.22972e6	0.746100
$688$	0	0
$689$	1.72847e7	1.38712
$690$	0	0
$691$	1.13093e7	0.901033	0.450517	−	0.892768i	$-0.351240\pi$
$691$	1.13093e7	0.901033	0.450517	+	0.892768i	$0.351240\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.24497e7	1.76299
$696$	0	0
$697$	1.05746e6	0.0824480
$698$	0	0
$699$	−3.06435e6	−0.237217
$700$	0	0
$701$	−1.92159e7	−1.47695	−0.738473	−	0.674283i	$-0.764453\pi$
$701$	−1.92159e7	−1.47695	−0.738473	+	0.674283i	$0.764453\pi$
$702$	0	0
$703$	7.53689e6	0.575180
$704$	0	0
$705$	1.62477e7	1.23117
$706$	0	0
$707$	0	0
$708$	0	0
$709$	5.64480e6	0.421729	0.210864	−	0.977515i	$-0.432372\pi$
$709$	5.64480e6	0.421729	0.210864	+	0.977515i	$0.432372\pi$
$710$	0	0
$711$	−917268.	−0.0680491
$712$	0	0
$713$	−268896.	−0.0198089
$714$	0	0
$715$	6.55441e6	0.479478
$716$	0	0
$717$	4.61166e6	0.335011
$718$	0	0
$719$	−4.42077e6	−0.318916	−0.159458	−	0.987205i	$-0.550975\pi$
$719$	−4.42077e6	−0.318916	−0.159458	+	0.987205i	$0.550975\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	8.72436e6	0.620709
$724$	0	0
$725$	2.28124e7	1.61185
$726$	0	0
$727$	−1.92503e7	−1.35083	−0.675416	−	0.737437i	$-0.736036\pi$
$727$	−1.92503e7	−1.35083	−0.675416	+	0.737437i	$0.736036\pi$
$728$	0	0
$729$	1.19758e7	0.834611
$730$	0	0
$731$	9.10277e6	0.630057
$732$	0	0
$733$	1.30187e7	0.894966	0.447483	−	0.894292i	$-0.352320\pi$
$733$	1.30187e7	0.894966	0.447483	+	0.894292i	$0.352320\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.57414e6	0.378015
$738$	0	0
$739$	1.70955e7	1.15152	0.575758	−	0.817620i	$-0.304707\pi$
$739$	1.70955e7	1.15152	0.575758	+	0.817620i	$0.304707\pi$
$740$	0	0
$741$	−7.88180e6	−0.527327
$742$	0	0
$743$	−2.36546e7	−1.57197	−0.785984	−	0.618247i	$-0.787843\pi$
$743$	−2.36546e7	−1.57197	−0.785984	+	0.618247i	$0.787843\pi$
$744$	0	0
$745$	−3.11233e7	−2.05445
$746$	0	0
$747$	−1.27454e7	−0.835705
$748$	0	0
$749$	0	0
$750$	0	0
$751$	166564.	0.0107766	0.00538828	−	0.999985i	$-0.498285\pi$
$751$	166564.	0.0107766	0.00538828	+	0.999985i	$0.498285\pi$
$752$	0	0
$753$	1.36111e7	0.874794
$754$	0	0
$755$	2.16961e7	1.38520
$756$	0	0
$757$	−4.63842e6	−0.294192	−0.147096	−	0.989122i	$-0.546993\pi$
$757$	−4.63842e6	−0.294192	−0.147096	+	0.989122i	$0.546993\pi$
$758$	0	0
$759$	−753546.	−0.0474794
$760$	0	0
$761$	−6.96999e6	−0.436285	−0.218143	−	0.975917i	$-0.570000\pi$
$761$	−6.96999e6	−0.436285	−0.218143	+	0.975917i	$0.570000\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.13738e6	−0.379166
$766$	0	0
$767$	−3.24242e6	−0.199013
$768$	0	0
$769$	3.02590e6	0.184518	0.0922590	−	0.995735i	$-0.470591\pi$
$769$	3.02590e6	0.184518	0.0922590	+	0.995735i	$0.470591\pi$
$770$	0	0
$771$	604967.	0.0366518
$772$	0	0
$773$	2.78539e7	1.67663	0.838314	−	0.545188i	$-0.183542\pi$
$773$	2.78539e7	1.67663	0.838314	+	0.545188i	$0.183542\pi$
$774$	0	0
$775$	2.92232e6	0.174772
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.56589e6	−0.151494
$780$	0	0
$781$	−832170.	−0.0488185
$782$	0	0
$783$	−1.11795e7	−0.651653
$784$	0	0
$785$	5.36804e7	3.10915
$786$	0	0
$787$	−2.87913e7	−1.65701	−0.828503	−	0.559985i	$-0.810807\pi$
$787$	−2.87913e7	−1.65701	−0.828503	+	0.559985i	$0.810807\pi$
$788$	0	0
$789$	−1.66253e7	−0.950773
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.72910e6	−0.492931
$794$	0	0
$795$	2.98942e7	1.67753
$796$	0	0
$797$	−3.67489e6	−0.204927	−0.102463	−	0.994737i	$-0.532672\pi$
$797$	−3.67489e6	−0.204927	−0.102463	+	0.994737i	$0.532672\pi$
$798$	0	0
$799$	6.47841e6	0.359006
$800$	0	0
$801$	−2.54650e6	−0.140237
$802$	0	0
$803$	−979251.	−0.0535926
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.29894e7	−0.702108
$808$	0	0
$809$	−1.04012e7	−0.558744	−0.279372	−	0.960183i	$-0.590126\pi$
$809$	−1.04012e7	−0.558744	−0.279372	+	0.960183i	$0.590126\pi$
$810$	0	0
$811$	1.53749e7	0.820844	0.410422	−	0.911896i	$-0.365381\pi$
$811$	1.53749e7	0.820844	0.410422	+	0.911896i	$0.365381\pi$
$812$	0	0
$813$	−1.90191e7	−1.00917
$814$	0	0
$815$	3.01552e7	1.59026
$816$	0	0
$817$	−2.20877e7	−1.15770
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.91243e7	0.990209	0.495104	−	0.868834i	$-0.335130\pi$
$821$	1.91243e7	0.990209	0.495104	+	0.868834i	$0.335130\pi$
$822$	0	0
$823$	−2.88895e6	−0.148676	−0.0743381	−	0.997233i	$-0.523684\pi$
$823$	−2.88895e6	−0.148676	−0.0743381	+	0.997233i	$0.523684\pi$
$824$	0	0
$825$	8.18942e6	0.418908
$826$	0	0
$827$	1.99201e7	1.01281	0.506406	−	0.862295i	$-0.330974\pi$
$827$	1.99201e7	1.01281	0.506406	+	0.862295i	$0.330974\pi$
$828$	0	0
$829$	−1.19726e7	−0.605065	−0.302532	−	0.953139i	$-0.597832\pi$
$829$	−1.19726e7	−0.605065	−0.302532	+	0.953139i	$0.597832\pi$
$830$	0	0
$831$	1.59834e7	0.802909
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.39483e6	0.0692316
$836$	0	0
$837$	−1.43211e6	−0.0706584
$838$	0	0
$839$	1.43453e7	0.703568	0.351784	−	0.936081i	$-0.385575\pi$
$839$	1.43453e7	0.703568	0.351784	+	0.936081i	$0.385575\pi$
$840$	0	0
$841$	−1.26444e7	−0.616463
$842$	0	0
$843$	−1.36657e7	−0.662313
$844$	0	0
$845$	6.98181e6	0.336377
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.55888e7	0.742239
$850$	0	0
$851$	5.09865e6	0.241341
$852$	0	0
$853$	−1.41859e7	−0.667551	−0.333776	−	0.942653i	$-0.608323\pi$
$853$	−1.41859e7	−0.667551	−0.333776	+	0.942653i	$0.608323\pi$
$854$	0	0
$855$	1.48922e7	0.696698
$856$	0	0
$857$	2.41394e7	1.12273	0.561363	−	0.827570i	$-0.310277\pi$
$857$	2.41394e7	1.12273	0.561363	+	0.827570i	$0.310277\pi$
$858$	0	0
$859$	−1.37343e7	−0.635071	−0.317536	−	0.948246i	$-0.602855\pi$
$859$	−1.37343e7	−0.635071	−0.317536	+	0.948246i	$0.602855\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.74905e7	−0.799419	−0.399709	−	0.916642i	$-0.630889\pi$
$863$	−1.74905e7	−0.799419	−0.399709	+	0.916642i	$0.630889\pi$
$864$	0	0
$865$	5.45945e6	0.248090
$866$	0	0
$867$	−1.30609e7	−0.590101
$868$	0	0
$869$	675534.	0.0303458
$870$	0	0
$871$	3.94418e7	1.76162
$872$	0	0
$873$	2.95872e6	0.131392
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.51221e7	−1.54199	−0.770995	−	0.636842i	$-0.780241\pi$
$877$	−3.51221e7	−1.54199	−0.770995	+	0.636842i	$0.780241\pi$
$878$	0	0
$879$	3.95510e6	0.172658
$880$	0	0
$881$	−8.78528e6	−0.381343	−0.190672	−	0.981654i	$-0.561067\pi$
$881$	−8.78528e6	−0.381343	−0.190672	+	0.981654i	$0.561067\pi$
$882$	0	0
$883$	−1.87501e7	−0.809287	−0.404643	−	0.914475i	$-0.632604\pi$
$883$	−1.87501e7	−0.809287	−0.404643	+	0.914475i	$0.632604\pi$
$884$	0	0
$885$	−5.60781e6	−0.240677
$886$	0	0
$887$	2.72434e7	1.16266	0.581330	−	0.813668i	$-0.302532\pi$
$887$	2.72434e7	1.16266	0.581330	+	0.813668i	$0.302532\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.13279e6	−0.0478030
$892$	0	0
$893$	−1.57197e7	−0.659654
$894$	0	0
$895$	5.28124e7	2.20383
$896$	0	0
$897$	−5.33198e6	−0.221262
$898$	0	0
$899$	1.00775e6	0.0415868
$900$	0	0
$901$	1.19197e7	0.489161
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.40987e7	−0.978076
$906$	0	0
$907$	−2.90420e7	−1.17222	−0.586110	−	0.810232i	$-0.699341\pi$
$907$	−2.90420e7	−1.17222	−0.586110	+	0.810232i	$0.699341\pi$
$908$	0	0
$909$	8.75071e6	0.351264
$910$	0	0
$911$	−4.12057e7	−1.64498	−0.822491	−	0.568778i	$-0.807417\pi$
$911$	−4.12057e7	−1.64498	−0.822491	+	0.568778i	$0.807417\pi$
$912$	0	0
$913$	9.38655e6	0.372674
$914$	0	0
$915$	−1.50971e7	−0.596130
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3.04768e7	−1.19037	−0.595184	−	0.803589i	$-0.702921\pi$
$919$	−3.04768e7	−1.19037	−0.595184	+	0.803589i	$0.702921\pi$
$920$	0	0
$921$	1.41473e6	0.0549574
$922$	0	0
$923$	−5.88831e6	−0.227503
$924$	0	0
$925$	−5.54114e7	−2.12934
$926$	0	0
$927$	1.44283e7	0.551460
$928$	0	0
$929$	−3.60640e7	−1.37099	−0.685495	−	0.728077i	$-0.740414\pi$
$929$	−3.60640e7	−1.37099	−0.685495	+	0.728077i	$0.740414\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.51598e7	0.946243
$934$	0	0
$935$	4.51996e6	0.169085
$936$	0	0
$937$	4.80602e7	1.78828	0.894141	−	0.447785i	$-0.147787\pi$
$937$	4.80602e7	1.78828	0.894141	+	0.447785i	$0.147787\pi$
$938$	0	0
$939$	−8.32800e6	−0.308231
$940$	0	0
$941$	−4.28577e7	−1.57781	−0.788906	−	0.614514i	$-0.789352\pi$
$941$	−4.28577e7	−1.57781	−0.788906	+	0.614514i	$0.789352\pi$
$942$	0	0
$943$	−1.73581e6	−0.0635657
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.32432e7	−1.56690	−0.783452	−	0.621452i	$-0.786543\pi$
$947$	−4.32432e7	−1.56690	−0.783452	+	0.621452i	$0.786543\pi$
$948$	0	0
$949$	−6.92904e6	−0.249751
$950$	0	0
$951$	5.65045e6	0.202596
$952$	0	0
$953$	−2.81115e7	−1.00265	−0.501327	−	0.865258i	$-0.667155\pi$
$953$	−2.81115e7	−1.00265	−0.501327	+	0.865258i	$0.667155\pi$
$954$	0	0
$955$	−5.33476e7	−1.89281
$956$	0	0
$957$	2.82410e6	0.0996783
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.85001e7	−0.995491
$962$	0	0
$963$	−1.53876e7	−0.534695
$964$	0	0
$965$	1.48890e7	0.514691
$966$	0	0
$967$	−3.70437e7	−1.27394	−0.636968	−	0.770890i	$-0.719812\pi$
$967$	−3.70437e7	−1.27394	−0.636968	+	0.770890i	$0.719812\pi$
$968$	0	0
$969$	−5.43533e6	−0.185959
$970$	0	0
$971$	4.58140e7	1.55937	0.779686	−	0.626170i	$-0.215379\pi$
$971$	4.58140e7	1.55937	0.779686	+	0.626170i	$0.215379\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	5.79471e7	1.95218
$976$	0	0
$977$	1.17497e6	0.0393812	0.0196906	−	0.999806i	$-0.493732\pi$
$977$	1.17497e6	0.0393812	0.0196906	+	0.999806i	$0.493732\pi$
$978$	0	0
$979$	1.87540e6	0.0625372
$980$	0	0
$981$	7.78647e6	0.258326
$982$	0	0
$983$	2.63520e7	0.869822	0.434911	−	0.900474i	$-0.356780\pi$
$983$	2.63520e7	0.869822	0.434911	+	0.900474i	$0.356780\pi$
$984$	0	0
$985$	4.02132e7	1.32062
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.49422e7	−0.485761
$990$	0	0
$991$	−3.82424e7	−1.23697	−0.618487	−	0.785795i	$-0.712254\pi$
$991$	−3.82424e7	−1.23697	−0.618487	+	0.785795i	$0.712254\pi$
$992$	0	0
$993$	−6.80674e6	−0.219062
$994$	0	0
$995$	4.89170e7	1.56640
$996$	0	0
$997$	4.36919e7	1.39208	0.696038	−	0.718005i	$-0.254945\pi$
$997$	4.36919e7	1.39208	0.696038	+	0.718005i	$0.254945\pi$
$998$	0	0
$999$	2.71549e7	0.860865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.s.1.2		2
4.3	odd	2		98.6.a.h.1.1		2
7.2	even	3		112.6.i.d.81.1		4
7.4	even	3		112.6.i.d.65.1		4
7.6	odd	2		784.6.a.bb.1.1		2
12.11	even	2		882.6.a.ba.1.1		2
28.3	even	6		98.6.c.e.79.1		4
28.11	odd	6		14.6.c.a.9.2	✓	4
28.19	even	6		98.6.c.e.67.1		4
28.23	odd	6		14.6.c.a.11.2	yes	4
28.27	even	2		98.6.a.g.1.2		2
84.11	even	6		126.6.g.j.37.2		4
84.23	even	6		126.6.g.j.109.2		4
84.83	odd	2		882.6.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.a.9.2	✓	4	28.11	odd	6
14.6.c.a.11.2	yes	4	28.23	odd	6
98.6.a.g.1.2		2	28.27	even	2
98.6.a.h.1.1		2	4.3	odd	2
98.6.c.e.67.1		4	28.19	even	6
98.6.c.e.79.1		4	28.3	even	6
112.6.i.d.65.1		4	7.4	even	3
112.6.i.d.81.1		4	7.2	even	3
126.6.g.j.37.2		4	84.11	even	6
126.6.g.j.109.2		4	84.23	even	6
784.6.a.s.1.2		2	1.1	even	1	trivial
784.6.a.bb.1.1		2	7.6	odd	2
882.6.a.ba.1.1		2	12.11	even	2
882.6.a.bi.1.2		2	84.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+10.7764 q^{3} +106.106 q^{5} -126.869 q^{9} +93.4347 q^{11} +661.131 q^{13} +1143.43 q^{15} +455.919 q^{17} -1106.28 q^{19} -748.390 q^{23} +8133.39 q^{25} -3985.86 q^{27} +2804.78 q^{29} +359.299 q^{31} +1006.89 q^{33} -6812.82 q^{37} +7124.60 q^{39} +2319.39 q^{41} +19965.7 q^{43} -13461.6 q^{45} +14209.5 q^{47} +4913.17 q^{51} +26144.2 q^{53} +9913.94 q^{55} -11921.7 q^{57} -4904.35 q^{59} -13203.3 q^{61} +70149.6 q^{65} +59658.1 q^{67} -8064.94 q^{69} -8906.43 q^{71} -10480.6 q^{73} +87648.6 q^{75} +7230.02 q^{79} -12123.9 q^{81} +100461. q^{83} +48375.6 q^{85} +30225.4 q^{87} +20071.8 q^{89} +3871.94 q^{93} -117382. q^{95} -23320.9 q^{97} -11854.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} + 70 q^{5} + 244 q^{9} - 62 q^{11} + 1820 q^{13} + 2038 q^{15} + 1694 q^{17} - 826 q^{19} + 2734 q^{23} + 6312 q^{25} - 7154 q^{27} - 2852 q^{29} + 2674 q^{31} + 4858 q^{33} - 9146 q^{37}+ \cdots - 69500 q^{99}+O(q^{100}) \) 2 * q - 14 * q^3 + 70 * q^5 + 244 * q^9 - 62 * q^11 + 1820 * q^13 + 2038 * q^15 + 1694 * q^17 - 826 * q^19 + 2734 * q^23 + 6312 * q^25 - 7154 * q^27 - 2852 * q^29 + 2674 * q^31 + 4858 * q^33 - 9146 * q^37 - 21588 * q^39 + 6132 * q^41 + 16040 * q^43 - 26852 * q^45 + 25326 * q^47 - 25762 * q^51 + 14958 * q^53 + 15526 * q^55 - 18866 * q^57 + 1106 * q^59 - 28042 * q^61 + 28308 * q^65 + 102642 * q^67 - 94346 * q^69 + 11056 * q^71 + 35070 * q^73 + 132776 * q^75 - 101762 * q^79 - 23750 * q^81 + 44632 * q^83 + 3674 * q^85 + 170380 * q^87 - 75474 * q^89 - 53478 * q^93 - 127502 * q^95 - 8316 * q^97 - 69500 * q^99

Introduction

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