LMFDB - Embedded newform 784.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.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:	$125.740914733$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-19.0000 q^{3} -19.0000 q^{5} +118.000 q^{9} +O(q^{10})$

$q-19.0000 q^{3} -19.0000 q^{5} +118.000 q^{9} +559.000 q^{11} -282.000 q^{13} +361.000 q^{15} -1259.00 q^{17} -1957.00 q^{19} +2977.00 q^{23} -2764.00 q^{25} +2375.00 q^{27} -62.0000 q^{29} +2037.00 q^{31} -10621.0 q^{33} +6023.00 q^{37} +5358.00 q^{39} +2178.00 q^{41} -23180.0 q^{43} -2242.00 q^{45} +26235.0 q^{47} +23921.0 q^{51} +30267.0 q^{53} -10621.0 q^{55} +37183.0 q^{57} +44965.0 q^{59} -27639.0 q^{61} +5358.00 q^{65} +58667.0 q^{67} -56563.0 q^{69} +9520.00 q^{71} +6785.00 q^{73} +52516.0 q^{75} +16929.0 q^{79} -73799.0 q^{81} -59572.0 q^{83} +23921.0 q^{85} +1178.00 q^{87} +51873.0 q^{89} -38703.0 q^{93} +37183.0 q^{95} -134110. q^{97} +65962.0 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−19.0000	−1.21885	−0.609425	−	0.792844i	$-0.708600\pi$
$3$	−19.0000	−1.21885	−0.609425	+	0.792844i	$0.708600\pi$
$4$	0	0
$5$	−19.0000	−0.339882	−0.169941	−	0.985454i	$-0.554358\pi$
$5$	−19.0000	−0.339882	−0.169941	+	0.985454i	$0.554358\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	118.000	0.485597
$10$	0	0
$11$	559.000	1.39293	0.696466	−	0.717590i	$-0.254755\pi$
$11$	559.000	1.39293	0.696466	+	0.717590i	$0.254755\pi$
$12$	0	0
$13$	−282.000	−0.462797	−0.231399	−	0.972859i	$-0.574330\pi$
$13$	−282.000	−0.462797	−0.231399	+	0.972859i	$0.574330\pi$
$14$	0	0
$15$	361.000	0.414266
$16$	0	0
$17$	−1259.00	−1.05658	−0.528291	−	0.849063i	$-0.677167\pi$
$17$	−1259.00	−1.05658	−0.528291	+	0.849063i	$0.677167\pi$
$18$	0	0
$19$	−1957.00	−1.24367	−0.621837	−	0.783146i	$-0.713614\pi$
$19$	−1957.00	−1.24367	−0.621837	+	0.783146i	$0.713614\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2977.00	1.17344	0.586718	−	0.809791i	$-0.300420\pi$
$23$	2977.00	1.17344	0.586718	+	0.809791i	$0.300420\pi$
$24$	0	0
$25$	−2764.00	−0.884480
$26$	0	0
$27$	2375.00	0.626981
$28$	0	0
$29$	−62.0000	−0.0136898	−0.00684489	−	0.999977i	$-0.502179\pi$
$29$	−62.0000	−0.0136898	−0.00684489	+	0.999977i	$0.502179\pi$
$30$	0	0
$31$	2037.00	0.380703	0.190352	−	0.981716i	$-0.439037\pi$
$31$	2037.00	0.380703	0.190352	+	0.981716i	$0.439037\pi$
$32$	0	0
$33$	−10621.0	−1.69778
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6023.00	0.723283	0.361642	−	0.932317i	$-0.382216\pi$
$37$	6023.00	0.723283	0.361642	+	0.932317i	$0.382216\pi$
$38$	0	0
$39$	5358.00	0.564081
$40$	0	0
$41$	2178.00	0.202348	0.101174	−	0.994869i	$-0.467740\pi$
$41$	2178.00	0.202348	0.101174	+	0.994869i	$0.467740\pi$
$42$	0	0
$43$	−23180.0	−1.91180	−0.955900	−	0.293694i	$-0.905115\pi$
$43$	−23180.0	−1.91180	−0.955900	+	0.293694i	$0.905115\pi$
$44$	0	0
$45$	−2242.00	−0.165046
$46$	0	0
$47$	26235.0	1.73235	0.866177	−	0.499738i	$-0.166570\pi$
$47$	26235.0	1.73235	0.866177	+	0.499738i	$0.166570\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	23921.0	1.28782
$52$	0	0
$53$	30267.0	1.48006	0.740031	−	0.672573i	$-0.234811\pi$
$53$	30267.0	1.48006	0.740031	+	0.672573i	$0.234811\pi$
$54$	0	0
$55$	−10621.0	−0.473433
$56$	0	0
$57$	37183.0	1.51585
$58$	0	0
$59$	44965.0	1.68168	0.840842	−	0.541280i	$-0.182060\pi$
$59$	44965.0	1.68168	0.840842	+	0.541280i	$0.182060\pi$
$60$	0	0
$61$	−27639.0	−0.951038	−0.475519	−	0.879706i	$-0.657740\pi$
$61$	−27639.0	−0.951038	−0.475519	+	0.879706i	$0.657740\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5358.00	0.157297
$66$	0	0
$67$	58667.0	1.59664	0.798320	−	0.602234i	$-0.205723\pi$
$67$	58667.0	1.59664	0.798320	+	0.602234i	$0.205723\pi$
$68$	0	0
$69$	−56563.0	−1.43024
$70$	0	0
$71$	9520.00	0.224125	0.112063	−	0.993701i	$-0.464254\pi$
$71$	9520.00	0.224125	0.112063	+	0.993701i	$0.464254\pi$
$72$	0	0
$73$	6785.00	0.149019	0.0745097	−	0.997220i	$-0.476261\pi$
$73$	6785.00	0.149019	0.0745097	+	0.997220i	$0.476261\pi$
$74$	0	0
$75$	52516.0	1.07805
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16929.0	0.305185	0.152593	−	0.988289i	$-0.451238\pi$
$79$	16929.0	0.305185	0.152593	+	0.988289i	$0.451238\pi$
$80$	0	0
$81$	−73799.0	−1.24979
$82$	0	0
$83$	−59572.0	−0.949176	−0.474588	−	0.880208i	$-0.657403\pi$
$83$	−59572.0	−0.949176	−0.474588	+	0.880208i	$0.657403\pi$
$84$	0	0
$85$	23921.0	0.359114
$86$	0	0
$87$	1178.00	0.0166858
$88$	0	0
$89$	51873.0	0.694171	0.347085	−	0.937834i	$-0.387171\pi$
$89$	51873.0	0.694171	0.347085	+	0.937834i	$0.387171\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−38703.0	−0.464021
$94$	0	0
$95$	37183.0	0.422703
$96$	0	0
$97$	−134110.	−1.44721	−0.723605	−	0.690214i	$-0.757516\pi$
$97$	−134110.	−1.44721	−0.723605	+	0.690214i	$0.757516\pi$
$98$	0	0
$99$	65962.0	0.676403
$100$	0	0
$101$	−122047.	−1.19048	−0.595242	−	0.803546i	$-0.702944\pi$
$101$	−122047.	−1.19048	−0.595242	+	0.803546i	$0.702944\pi$
$102$	0	0
$103$	−80617.0	−0.748744	−0.374372	−	0.927279i	$-0.622142\pi$
$103$	−80617.0	−0.748744	−0.374372	+	0.927279i	$0.622142\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1467.00	−0.0123871	−0.00619356	−	0.999981i	$-0.501971\pi$
$107$	−1467.00	−0.0123871	−0.00619356	+	0.999981i	$0.501971\pi$
$108$	0	0
$109$	200527.	1.61662	0.808308	−	0.588761i	$-0.200384\pi$
$109$	200527.	1.61662	0.808308	+	0.588761i	$0.200384\pi$
$110$	0	0
$111$	−114437.	−0.881574
$112$	0	0
$113$	−722.000	−0.00531914	−0.00265957	−	0.999996i	$-0.500847\pi$
$113$	−722.000	−0.00531914	−0.00265957	+	0.999996i	$0.500847\pi$
$114$	0	0
$115$	−56563.0	−0.398830
$116$	0	0
$117$	−33276.0	−0.224733
$118$	0	0
$119$	0	0
$120$	0	0
$121$	151430.	0.940261
$122$	0	0
$123$	−41382.0	−0.246632
$124$	0	0
$125$	111891.	0.640501
$126$	0	0
$127$	−147288.	−0.810323	−0.405161	−	0.914245i	$-0.632785\pi$
$127$	−147288.	−0.810323	−0.405161	+	0.914245i	$0.632785\pi$
$128$	0	0
$129$	440420.	2.33020
$130$	0	0
$131$	−53993.0	−0.274890	−0.137445	−	0.990509i	$-0.543889\pi$
$131$	−53993.0	−0.274890	−0.137445	+	0.990509i	$0.543889\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−45125.0	−0.213100
$136$	0	0
$137$	122563.	0.557902	0.278951	−	0.960305i	$-0.410013\pi$
$137$	122563.	0.557902	0.278951	+	0.960305i	$0.410013\pi$
$138$	0	0
$139$	−128108.	−0.562392	−0.281196	−	0.959650i	$-0.590731\pi$
$139$	−128108.	−0.562392	−0.281196	+	0.959650i	$0.590731\pi$
$140$	0	0
$141$	−498465.	−2.11148
$142$	0	0
$143$	−157638.	−0.644645
$144$	0	0
$145$	1178.00	0.00465292
$146$	0	0
$147$	0	0
$148$	0	0
$149$	131955.	0.486923	0.243461	−	0.969911i	$-0.421717\pi$
$149$	131955.	0.486923	0.243461	+	0.969911i	$0.421717\pi$
$150$	0	0
$151$	−140125.	−0.500119	−0.250059	−	0.968230i	$-0.580450\pi$
$151$	−140125.	−0.500119	−0.250059	+	0.968230i	$0.580450\pi$
$152$	0	0
$153$	−148562.	−0.513073
$154$	0	0
$155$	−38703.0	−0.129394
$156$	0	0
$157$	−323339.	−1.04691	−0.523455	−	0.852054i	$-0.675357\pi$
$157$	−323339.	−1.04691	−0.523455	+	0.852054i	$0.675357\pi$
$158$	0	0
$159$	−575073.	−1.80397
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−122159.	−0.360128	−0.180064	−	0.983655i	$-0.557630\pi$
$163$	−122159.	−0.360128	−0.180064	+	0.983655i	$0.557630\pi$
$164$	0	0
$165$	201799.	0.577044
$166$	0	0
$167$	185404.	0.514432	0.257216	−	0.966354i	$-0.417195\pi$
$167$	185404.	0.514432	0.257216	+	0.966354i	$0.417195\pi$
$168$	0	0
$169$	−291769.	−0.785819
$170$	0	0
$171$	−230926.	−0.603924
$172$	0	0
$173$	358625.	0.911015	0.455507	−	0.890232i	$-0.349458\pi$
$173$	358625.	0.911015	0.455507	+	0.890232i	$0.349458\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−854335.	−2.04972
$178$	0	0
$179$	460551.	1.07435	0.537174	−	0.843471i	$-0.319492\pi$
$179$	460551.	1.07435	0.537174	+	0.843471i	$0.319492\pi$
$180$	0	0
$181$	332538.	0.754475	0.377237	−	0.926117i	$-0.376874\pi$
$181$	332538.	0.754475	0.377237	+	0.926117i	$0.376874\pi$
$182$	0	0
$183$	525141.	1.15917
$184$	0	0
$185$	−114437.	−0.245831
$186$	0	0
$187$	−703781.	−1.47175
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−987867.	−1.95936	−0.979682	−	0.200558i	$-0.935725\pi$
$191$	−987867.	−1.95936	−0.979682	+	0.200558i	$0.935725\pi$
$192$	0	0
$193$	−344413.	−0.665559	−0.332779	−	0.943005i	$-0.607986\pi$
$193$	−344413.	−0.665559	−0.332779	+	0.943005i	$0.607986\pi$
$194$	0	0
$195$	−101802.	−0.191721
$196$	0	0
$197$	582362.	1.06912	0.534561	−	0.845130i	$-0.320477\pi$
$197$	582362.	1.06912	0.534561	+	0.845130i	$0.320477\pi$
$198$	0	0
$199$	−150955.	−0.270218	−0.135109	−	0.990831i	$-0.543139\pi$
$199$	−150955.	−0.270218	−0.135109	+	0.990831i	$0.543139\pi$
$200$	0	0
$201$	−1.11467e6	−1.94606
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−41382.0	−0.0687744
$206$	0	0
$207$	351286.	0.569816
$208$	0	0
$209$	−1.09396e6	−1.73236
$210$	0	0
$211$	−272156.	−0.420835	−0.210417	−	0.977612i	$-0.567482\pi$
$211$	−272156.	−0.420835	−0.210417	+	0.977612i	$0.567482\pi$
$212$	0	0
$213$	−180880.	−0.273175
$214$	0	0
$215$	440420.	0.649787
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−128915.	−0.181632
$220$	0	0
$221$	355038.	0.488983
$222$	0	0
$223$	−939112.	−1.26461	−0.632303	−	0.774721i	$-0.717890\pi$
$223$	−939112.	−1.26461	−0.632303	+	0.774721i	$0.717890\pi$
$224$	0	0
$225$	−326152.	−0.429501
$226$	0	0
$227$	481713.	0.620474	0.310237	−	0.950659i	$-0.399592\pi$
$227$	481713.	0.620474	0.310237	+	0.950659i	$0.399592\pi$
$228$	0	0
$229$	−523147.	−0.659227	−0.329614	−	0.944116i	$-0.606918\pi$
$229$	−523147.	−0.659227	−0.329614	+	0.944116i	$0.606918\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.22603e6	−1.47949	−0.739743	−	0.672889i	$-0.765053\pi$
$233$	−1.22603e6	−1.47949	−0.739743	+	0.672889i	$0.765053\pi$
$234$	0	0
$235$	−498465.	−0.588796
$236$	0	0
$237$	−321651.	−0.371975
$238$	0	0
$239$	−1.32568e6	−1.50121	−0.750607	−	0.660749i	$-0.770239\pi$
$239$	−1.32568e6	−1.50121	−0.750607	+	0.660749i	$0.770239\pi$
$240$	0	0
$241$	−271871.	−0.301523	−0.150761	−	0.988570i	$-0.548173\pi$
$241$	−271871.	−0.301523	−0.150761	+	0.988570i	$0.548173\pi$
$242$	0	0
$243$	825056.	0.896330
$244$	0	0
$245$	0	0
$246$	0	0
$247$	551874.	0.575569
$248$	0	0
$249$	1.13187e6	1.15690
$250$	0	0
$251$	−781368.	−0.782837	−0.391418	−	0.920213i	$-0.628015\pi$
$251$	−781368.	−0.782837	−0.391418	+	0.920213i	$0.628015\pi$
$252$	0	0
$253$	1.66414e6	1.63452
$254$	0	0
$255$	−454499.	−0.437706
$256$	0	0
$257$	−1.00337e6	−0.947608	−0.473804	−	0.880630i	$-0.657119\pi$
$257$	−1.00337e6	−0.947608	−0.473804	+	0.880630i	$0.657119\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−7316.00	−0.00664772
$262$	0	0
$263$	1.10958e6	0.989169	0.494584	−	0.869130i	$-0.335320\pi$
$263$	1.10958e6	0.989169	0.494584	+	0.869130i	$0.335320\pi$
$264$	0	0
$265$	−575073.	−0.503047
$266$	0	0
$267$	−985587.	−0.846090
$268$	0	0
$269$	−1.72416e6	−1.45277	−0.726383	−	0.687290i	$-0.758800\pi$
$269$	−1.72416e6	−1.45277	−0.726383	+	0.687290i	$0.758800\pi$
$270$	0	0
$271$	831399.	0.687680	0.343840	−	0.939028i	$-0.388272\pi$
$271$	831399.	0.687680	0.343840	+	0.939028i	$0.388272\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.54508e6	−1.23202
$276$	0	0
$277$	−696757.	−0.545609	−0.272805	−	0.962069i	$-0.587951\pi$
$277$	−696757.	−0.545609	−0.272805	+	0.962069i	$0.587951\pi$
$278$	0	0
$279$	240366.	0.184868
$280$	0	0
$281$	−2.26355e6	−1.71011	−0.855054	−	0.518539i	$-0.826476\pi$
$281$	−2.26355e6	−1.71011	−0.855054	+	0.518539i	$0.826476\pi$
$282$	0	0
$283$	423985.	0.314691	0.157346	−	0.987544i	$-0.449706\pi$
$283$	423985.	0.314691	0.157346	+	0.987544i	$0.449706\pi$
$284$	0	0
$285$	−706477.	−0.515212
$286$	0	0
$287$	0	0
$288$	0	0
$289$	165224.	0.116367
$290$	0	0
$291$	2.54809e6	1.76393
$292$	0	0
$293$	−1.03724e6	−0.705845	−0.352923	−	0.935653i	$-0.614812\pi$
$293$	−1.03724e6	−0.705845	−0.352923	+	0.935653i	$0.614812\pi$
$294$	0	0
$295$	−854335.	−0.571575
$296$	0	0
$297$	1.32762e6	0.873342
$298$	0	0
$299$	−839514.	−0.543063
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.31889e6	1.45102
$304$	0	0
$305$	525141.	0.323241
$306$	0	0
$307$	−152684.	−0.0924587	−0.0462293	−	0.998931i	$-0.514720\pi$
$307$	−152684.	−0.0924587	−0.0462293	+	0.998931i	$0.514720\pi$
$308$	0	0
$309$	1.53172e6	0.912608
$310$	0	0
$311$	1.66634e6	0.976931	0.488466	−	0.872583i	$-0.337557\pi$
$311$	1.66634e6	0.976931	0.488466	+	0.872583i	$0.337557\pi$
$312$	0	0
$313$	−64471.0	−0.0371966	−0.0185983	−	0.999827i	$-0.505920\pi$
$313$	−64471.0	−0.0371966	−0.0185983	+	0.999827i	$0.505920\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.28934e6	0.720643	0.360322	−	0.932828i	$-0.382667\pi$
$317$	1.28934e6	0.720643	0.360322	+	0.932828i	$0.382667\pi$
$318$	0	0
$319$	−34658.0	−0.0190690
$320$	0	0
$321$	27873.0	0.0150981
$322$	0	0
$323$	2.46386e6	1.31405
$324$	0	0
$325$	779448.	0.409335
$326$	0	0
$327$	−3.81001e6	−1.97041
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.64238e6	1.32564	0.662819	−	0.748780i	$-0.269360\pi$
$331$	2.64238e6	1.32564	0.662819	+	0.748780i	$0.269360\pi$
$332$	0	0
$333$	710714.	0.351224
$334$	0	0
$335$	−1.11467e6	−0.542670
$336$	0	0
$337$	−3.00561e6	−1.44164	−0.720822	−	0.693120i	$-0.756235\pi$
$337$	−3.00561e6	−1.44164	−0.720822	+	0.693120i	$0.756235\pi$
$338$	0	0
$339$	13718.0	0.00648323
$340$	0	0
$341$	1.13868e6	0.530294
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.07470e6	0.486114
$346$	0	0
$347$	185307.	0.0826168	0.0413084	−	0.999146i	$-0.486847\pi$
$347$	185307.	0.0826168	0.0413084	+	0.999146i	$0.486847\pi$
$348$	0	0
$349$	−2.82147e6	−1.23997	−0.619987	−	0.784612i	$-0.712862\pi$
$349$	−2.82147e6	−1.23997	−0.619987	+	0.784612i	$0.712862\pi$
$350$	0	0
$351$	−669750.	−0.290165
$352$	0	0
$353$	−1.30752e6	−0.558486	−0.279243	−	0.960220i	$-0.590084\pi$
$353$	−1.30752e6	−0.558486	−0.279243	+	0.960220i	$0.590084\pi$
$354$	0	0
$355$	−180880.	−0.0761763
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.89605e6	−1.18596	−0.592980	−	0.805217i	$-0.702049\pi$
$359$	−2.89605e6	−1.18596	−0.592980	+	0.805217i	$0.702049\pi$
$360$	0	0
$361$	1.35375e6	0.546727
$362$	0	0
$363$	−2.87717e6	−1.14604
$364$	0	0
$365$	−128915.	−0.0506490
$366$	0	0
$367$	−4.71893e6	−1.82885	−0.914427	−	0.404752i	$-0.867358\pi$
$367$	−4.71893e6	−1.82885	−0.914427	+	0.404752i	$0.867358\pi$
$368$	0	0
$369$	257004.	0.0982594
$370$	0	0
$371$	0	0
$372$	0	0
$373$	438079.	0.163035	0.0815174	−	0.996672i	$-0.474023\pi$
$373$	438079.	0.163035	0.0815174	+	0.996672i	$0.474023\pi$
$374$	0	0
$375$	−2.12593e6	−0.780676
$376$	0	0
$377$	17484.0	0.00633560
$378$	0	0
$379$	549632.	0.196550	0.0982752	−	0.995159i	$-0.468667\pi$
$379$	549632.	0.196550	0.0982752	+	0.995159i	$0.468667\pi$
$380$	0	0
$381$	2.79847e6	0.987662
$382$	0	0
$383$	3.53000e6	1.22964	0.614820	−	0.788667i	$-0.289229\pi$
$383$	3.53000e6	1.22964	0.614820	+	0.788667i	$0.289229\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.73524e6	−0.928363
$388$	0	0
$389$	3.13395e6	1.05007	0.525035	−	0.851081i	$-0.324052\pi$
$389$	3.13395e6	1.05007	0.525035	+	0.851081i	$0.324052\pi$
$390$	0	0
$391$	−3.74804e6	−1.23983
$392$	0	0
$393$	1.02587e6	0.335050
$394$	0	0
$395$	−321651.	−0.103727
$396$	0	0
$397$	2.55391e6	0.813260	0.406630	−	0.913593i	$-0.366704\pi$
$397$	2.55391e6	0.813260	0.406630	+	0.913593i	$0.366704\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−702837.	−0.218270	−0.109135	−	0.994027i	$-0.534808\pi$
$401$	−702837.	−0.218270	−0.109135	+	0.994027i	$0.534808\pi$
$402$	0	0
$403$	−574434.	−0.176188
$404$	0	0
$405$	1.40218e6	0.424782
$406$	0	0
$407$	3.36686e6	1.00749
$408$	0	0
$409$	5.47715e6	1.61900	0.809499	−	0.587121i	$-0.199739\pi$
$409$	5.47715e6	1.61900	0.809499	+	0.587121i	$0.199739\pi$
$410$	0	0
$411$	−2.32870e6	−0.679999
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.13187e6	0.322608
$416$	0	0
$417$	2.43405e6	0.685472
$418$	0	0
$419$	5.80976e6	1.61668	0.808339	−	0.588718i	$-0.200367\pi$
$419$	5.80976e6	1.61668	0.808339	+	0.588718i	$0.200367\pi$
$420$	0	0
$421$	1.69370e6	0.465726	0.232863	−	0.972510i	$-0.425191\pi$
$421$	1.69370e6	0.465726	0.232863	+	0.972510i	$0.425191\pi$
$422$	0	0
$423$	3.09573e6	0.841225
$424$	0	0
$425$	3.47988e6	0.934526
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.99512e6	0.785726
$430$	0	0
$431$	1.46468e6	0.379794	0.189897	−	0.981804i	$-0.439185\pi$
$431$	1.46468e6	0.379794	0.189897	+	0.981804i	$0.439185\pi$
$432$	0	0
$433$	3.23418e6	0.828980	0.414490	−	0.910054i	$-0.363960\pi$
$433$	3.23418e6	0.828980	0.414490	+	0.910054i	$0.363960\pi$
$434$	0	0
$435$	−22382.0	−0.00567121
$436$	0	0
$437$	−5.82599e6	−1.45937
$438$	0	0
$439$	−2.54217e6	−0.629570	−0.314785	−	0.949163i	$-0.601932\pi$
$439$	−2.54217e6	−0.629570	−0.314785	+	0.949163i	$0.601932\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−65939.0	−0.0159637	−0.00798184	−	0.999968i	$-0.502541\pi$
$443$	−65939.0	−0.0159637	−0.00798184	+	0.999968i	$0.502541\pi$
$444$	0	0
$445$	−985587.	−0.235936
$446$	0	0
$447$	−2.50714e6	−0.593486
$448$	0	0
$449$	−5.32399e6	−1.24630	−0.623149	−	0.782103i	$-0.714147\pi$
$449$	−5.32399e6	−1.24630	−0.623149	+	0.782103i	$0.714147\pi$
$450$	0	0
$451$	1.21750e6	0.281857
$452$	0	0
$453$	2.66238e6	0.609570
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.05825e6	−0.908967	−0.454484	−	0.890755i	$-0.650176\pi$
$457$	−4.05825e6	−0.908967	−0.454484	+	0.890755i	$0.650176\pi$
$458$	0	0
$459$	−2.99012e6	−0.662457
$460$	0	0
$461$	−3.73021e6	−0.817487	−0.408744	−	0.912649i	$-0.634033\pi$
$461$	−3.73021e6	−0.817487	−0.408744	+	0.912649i	$0.634033\pi$
$462$	0	0
$463$	3.45186e6	0.748342	0.374171	−	0.927360i	$-0.377927\pi$
$463$	3.45186e6	0.748342	0.374171	+	0.927360i	$0.377927\pi$
$464$	0	0
$465$	735357.	0.157712
$466$	0	0
$467$	3.92062e6	0.831884	0.415942	−	0.909391i	$-0.363452\pi$
$467$	3.92062e6	0.831884	0.415942	+	0.909391i	$0.363452\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6.14344e6	1.27603
$472$	0	0
$473$	−1.29576e7	−2.66301
$474$	0	0
$475$	5.40915e6	1.10001
$476$	0	0
$477$	3.57151e6	0.718713
$478$	0	0
$479$	−5.98674e6	−1.19221	−0.596103	−	0.802908i	$-0.703285\pi$
$479$	−5.98674e6	−1.19221	−0.596103	+	0.802908i	$0.703285\pi$
$480$	0	0
$481$	−1.69849e6	−0.334734
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.54809e6	0.491881
$486$	0	0
$487$	−71873.0	−0.0137323	−0.00686615	−	0.999976i	$-0.502186\pi$
$487$	−71873.0	−0.0137323	−0.00686615	+	0.999976i	$0.502186\pi$
$488$	0	0
$489$	2.32102e6	0.438942
$490$	0	0
$491$	−1.01122e6	−0.189295	−0.0946477	−	0.995511i	$-0.530172\pi$
$491$	−1.01122e6	−0.189295	−0.0946477	+	0.995511i	$0.530172\pi$
$492$	0	0
$493$	78058.0	0.0144644
$494$	0	0
$495$	−1.25328e6	−0.229898
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.80469e6	−1.40315	−0.701575	−	0.712596i	$-0.747519\pi$
$499$	−7.80469e6	−1.40315	−0.701575	+	0.712596i	$0.747519\pi$
$500$	0	0
$501$	−3.52268e6	−0.627016
$502$	0	0
$503$	7.89298e6	1.39098	0.695490	−	0.718536i	$-0.255187\pi$
$503$	7.89298e6	1.39098	0.695490	+	0.718536i	$0.255187\pi$
$504$	0	0
$505$	2.31889e6	0.404625
$506$	0	0
$507$	5.54361e6	0.957796
$508$	0	0
$509$	−6.62650e6	−1.13368	−0.566839	−	0.823829i	$-0.691834\pi$
$509$	−6.62650e6	−1.13368	−0.566839	+	0.823829i	$0.691834\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.64788e6	−0.779760
$514$	0	0
$515$	1.53172e6	0.254485
$516$	0	0
$517$	1.46654e7	2.41305
$518$	0	0
$519$	−6.81387e6	−1.11039
$520$	0	0
$521$	−7.27862e6	−1.17477	−0.587387	−	0.809306i	$-0.699843\pi$
$521$	−7.27862e6	−1.17477	−0.587387	+	0.809306i	$0.699843\pi$
$522$	0	0
$523$	4.75278e6	0.759790	0.379895	−	0.925030i	$-0.375960\pi$
$523$	4.75278e6	0.759790	0.379895	+	0.925030i	$0.375960\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.56458e6	−0.402245
$528$	0	0
$529$	2.42619e6	0.376951
$530$	0	0
$531$	5.30587e6	0.816621
$532$	0	0
$533$	−614196.	−0.0936459
$534$	0	0
$535$	27873.0	0.00421017
$536$	0	0
$537$	−8.75047e6	−1.30947
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9.25161e6	−1.35902	−0.679508	−	0.733668i	$-0.737807\pi$
$541$	−9.25161e6	−1.35902	−0.679508	+	0.733668i	$0.737807\pi$
$542$	0	0
$543$	−6.31822e6	−0.919592
$544$	0	0
$545$	−3.81001e6	−0.549459
$546$	0	0
$547$	−4.66834e6	−0.667104	−0.333552	−	0.942732i	$-0.608247\pi$
$547$	−4.66834e6	−0.667104	−0.333552	+	0.942732i	$0.608247\pi$
$548$	0	0
$549$	−3.26140e6	−0.461821
$550$	0	0
$551$	121334.	0.0170256
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.17430e6	0.299632
$556$	0	0
$557$	−6.50094e6	−0.887848	−0.443924	−	0.896065i	$-0.646414\pi$
$557$	−6.50094e6	−0.887848	−0.443924	+	0.896065i	$0.646414\pi$
$558$	0	0
$559$	6.53676e6	0.884775
$560$	0	0
$561$	1.33718e7	1.79384
$562$	0	0
$563$	1.06422e7	1.41501	0.707505	−	0.706708i	$-0.249821\pi$
$563$	1.06422e7	1.41501	0.707505	+	0.706708i	$0.249821\pi$
$564$	0	0
$565$	13718.0	0.00180788
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.71001e6	1.25730	0.628650	−	0.777688i	$-0.283608\pi$
$569$	9.71001e6	1.25730	0.628650	+	0.777688i	$0.283608\pi$
$570$	0	0
$571$	−1.15693e7	−1.48497	−0.742485	−	0.669862i	$-0.766353\pi$
$571$	−1.15693e7	−1.48497	−0.742485	+	0.669862i	$0.766353\pi$
$572$	0	0
$573$	1.87695e7	2.38817
$574$	0	0
$575$	−8.22843e6	−1.03788
$576$	0	0
$577$	5.72550e6	0.715936	0.357968	−	0.933734i	$-0.383470\pi$
$577$	5.72550e6	0.715936	0.357968	+	0.933734i	$0.383470\pi$
$578$	0	0
$579$	6.54385e6	0.811216
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.69193e7	2.06163
$584$	0	0
$585$	632244.	0.0763827
$586$	0	0
$587$	2.35929e6	0.282609	0.141305	−	0.989966i	$-0.454870\pi$
$587$	2.35929e6	0.282609	0.141305	+	0.989966i	$0.454870\pi$
$588$	0	0
$589$	−3.98641e6	−0.473471
$590$	0	0
$591$	−1.10649e7	−1.30310
$592$	0	0
$593$	−2.91024e6	−0.339854	−0.169927	−	0.985457i	$-0.554353\pi$
$593$	−2.91024e6	−0.339854	−0.169927	+	0.985457i	$0.554353\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.86814e6	0.329356
$598$	0	0
$599$	−9.99466e6	−1.13815	−0.569077	−	0.822284i	$-0.692699\pi$
$599$	−9.99466e6	−1.13815	−0.569077	+	0.822284i	$0.692699\pi$
$600$	0	0
$601$	−6.12412e6	−0.691604	−0.345802	−	0.938308i	$-0.612393\pi$
$601$	−6.12412e6	−0.691604	−0.345802	+	0.938308i	$0.612393\pi$
$602$	0	0
$603$	6.92271e6	0.775323
$604$	0	0
$605$	−2.87717e6	−0.319578
$606$	0	0
$607$	1.77047e7	1.95037	0.975185	−	0.221391i	$-0.0710599\pi$
$607$	1.77047e7	1.95037	0.975185	+	0.221391i	$0.0710599\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.39827e6	−0.801728
$612$	0	0
$613$	−8.83215e6	−0.949326	−0.474663	−	0.880168i	$-0.657430\pi$
$613$	−8.83215e6	−0.949326	−0.474663	+	0.880168i	$0.657430\pi$
$614$	0	0
$615$	786258.	0.0838257
$616$	0	0
$617$	1.07392e6	0.113569	0.0567843	−	0.998386i	$-0.481915\pi$
$617$	1.07392e6	0.113569	0.0567843	+	0.998386i	$0.481915\pi$
$618$	0	0
$619$	−352211.	−0.0369468	−0.0184734	−	0.999829i	$-0.505881\pi$
$619$	−352211.	−0.0369468	−0.0184734	+	0.999829i	$0.505881\pi$
$620$	0	0
$621$	7.07038e6	0.735722
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.51157e6	0.666785
$626$	0	0
$627$	2.07853e7	2.11148
$628$	0	0
$629$	−7.58296e6	−0.764209
$630$	0	0
$631$	−775808.	−0.0775677	−0.0387838	−	0.999248i	$-0.512348\pi$
$631$	−775808.	−0.0775677	−0.0387838	+	0.999248i	$0.512348\pi$
$632$	0	0
$633$	5.17096e6	0.512935
$634$	0	0
$635$	2.79847e6	0.275414
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.12336e6	0.108835
$640$	0	0
$641$	−1.18421e7	−1.13837	−0.569187	−	0.822208i	$-0.692742\pi$
$641$	−1.18421e7	−1.13837	−0.569187	+	0.822208i	$0.692742\pi$
$642$	0	0
$643$	307460.	0.0293266	0.0146633	−	0.999892i	$-0.495332\pi$
$643$	307460.	0.0293266	0.0146633	+	0.999892i	$0.495332\pi$
$644$	0	0
$645$	−8.36798e6	−0.791993
$646$	0	0
$647$	1.30708e7	1.22756	0.613780	−	0.789477i	$-0.289648\pi$
$647$	1.30708e7	1.22756	0.613780	+	0.789477i	$0.289648\pi$
$648$	0	0
$649$	2.51354e7	2.34247
$650$	0	0
$651$	0	0
$652$	0	0
$653$	927663.	0.0851348	0.0425674	−	0.999094i	$-0.486446\pi$
$653$	927663.	0.0851348	0.0425674	+	0.999094i	$0.486446\pi$
$654$	0	0
$655$	1.02587e6	0.0934303
$656$	0	0
$657$	800630.	0.0723633
$658$	0	0
$659$	1.90355e7	1.70746	0.853731	−	0.520715i	$-0.174334\pi$
$659$	1.90355e7	1.70746	0.853731	+	0.520715i	$0.174334\pi$
$660$	0	0
$661$	2.17236e6	0.193388	0.0966939	−	0.995314i	$-0.469173\pi$
$661$	2.17236e6	0.193388	0.0966939	+	0.995314i	$0.469173\pi$
$662$	0	0
$663$	−6.74572e6	−0.595998
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−184574.	−0.0160641
$668$	0	0
$669$	1.78431e7	1.54137
$670$	0	0
$671$	−1.54502e7	−1.32473
$672$	0	0
$673$	1.32268e7	1.12569	0.562844	−	0.826563i	$-0.309707\pi$
$673$	1.32268e7	1.12569	0.562844	+	0.826563i	$0.309707\pi$
$674$	0	0
$675$	−6.56450e6	−0.554552
$676$	0	0
$677$	1.59662e7	1.33884	0.669422	−	0.742882i	$-0.266542\pi$
$677$	1.59662e7	1.33884	0.669422	+	0.742882i	$0.266542\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−9.15255e6	−0.756265
$682$	0	0
$683$	−1.40029e7	−1.14859	−0.574297	−	0.818647i	$-0.694724\pi$
$683$	−1.40029e7	−1.14859	−0.574297	+	0.818647i	$0.694724\pi$
$684$	0	0
$685$	−2.32870e6	−0.189621
$686$	0	0
$687$	9.93979e6	0.803499
$688$	0	0
$689$	−8.53529e6	−0.684968
$690$	0	0
$691$	−1.05246e7	−0.838514	−0.419257	−	0.907868i	$-0.637709\pi$
$691$	−1.05246e7	−0.838514	−0.419257	+	0.907868i	$0.637709\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.43405e6	0.191147
$696$	0	0
$697$	−2.74210e6	−0.213797
$698$	0	0
$699$	2.32946e7	1.80327
$700$	0	0
$701$	−1.30811e7	−1.00542	−0.502710	−	0.864455i	$-0.667664\pi$
$701$	−1.30811e7	−1.00542	−0.502710	+	0.864455i	$0.667664\pi$
$702$	0	0
$703$	−1.17870e7	−0.899529
$704$	0	0
$705$	9.47084e6	0.717655
$706$	0	0
$707$	0	0
$708$	0	0
$709$	488519.	0.0364977	0.0182489	−	0.999833i	$-0.494191\pi$
$709$	488519.	0.0364977	0.0182489	+	0.999833i	$0.494191\pi$
$710$	0	0
$711$	1.99762e6	0.148197
$712$	0	0
$713$	6.06415e6	0.446731
$714$	0	0
$715$	2.99512e6	0.219104
$716$	0	0
$717$	2.51878e7	1.82976
$718$	0	0
$719$	−8.40850e6	−0.606592	−0.303296	−	0.952896i	$-0.598087\pi$
$719$	−8.40850e6	−0.606592	−0.303296	+	0.952896i	$0.598087\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	5.16555e6	0.367511
$724$	0	0
$725$	171368.	0.0121083
$726$	0	0
$727$	−2.44145e7	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$727$	−2.44145e7	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$728$	0	0
$729$	2.25709e6	0.157301
$730$	0	0
$731$	2.91836e7	2.01997
$732$	0	0
$733$	−1.13916e7	−0.783111	−0.391556	−	0.920154i	$-0.628063\pi$
$733$	−1.13916e7	−0.783111	−0.391556	+	0.920154i	$0.628063\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.27949e7	2.22401
$738$	0	0
$739$	2.18963e6	0.147489	0.0737445	−	0.997277i	$-0.476505\pi$
$739$	2.18963e6	0.147489	0.0737445	+	0.997277i	$0.476505\pi$
$740$	0	0
$741$	−1.04856e7	−0.701533
$742$	0	0
$743$	7.06982e6	0.469825	0.234913	−	0.972016i	$-0.424520\pi$
$743$	7.06982e6	0.469825	0.234913	+	0.972016i	$0.424520\pi$
$744$	0	0
$745$	−2.50714e6	−0.165496
$746$	0	0
$747$	−7.02950e6	−0.460917
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−6.55880e6	−0.424350	−0.212175	−	0.977232i	$-0.568055\pi$
$751$	−6.55880e6	−0.424350	−0.212175	+	0.977232i	$0.568055\pi$
$752$	0	0
$753$	1.48460e7	0.954161
$754$	0	0
$755$	2.66238e6	0.169982
$756$	0	0
$757$	2.17588e6	0.138005	0.0690026	−	0.997616i	$-0.478018\pi$
$757$	2.17588e6	0.138005	0.0690026	+	0.997616i	$0.478018\pi$
$758$	0	0
$759$	−3.16187e7	−1.99223
$760$	0	0
$761$	−1.58632e7	−0.992954	−0.496477	−	0.868050i	$-0.665373\pi$
$761$	−1.58632e7	−0.992954	−0.496477	+	0.868050i	$0.665373\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.82268e6	0.174384
$766$	0	0
$767$	−1.26801e7	−0.778279
$768$	0	0
$769$	1.86634e7	1.13809	0.569044	−	0.822307i	$-0.307313\pi$
$769$	1.86634e7	1.13809	0.569044	+	0.822307i	$0.307313\pi$
$770$	0	0
$771$	1.90640e7	1.15499
$772$	0	0
$773$	2.26397e7	1.36277	0.681384	−	0.731926i	$-0.261378\pi$
$773$	2.26397e7	1.36277	0.681384	+	0.731926i	$0.261378\pi$
$774$	0	0
$775$	−5.63027e6	−0.336725
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.26235e6	−0.251655
$780$	0	0
$781$	5.32168e6	0.312192
$782$	0	0
$783$	−147250.	−0.00858323
$784$	0	0
$785$	6.14344e6	0.355826
$786$	0	0
$787$	−9.66053e6	−0.555986	−0.277993	−	0.960583i	$-0.589669\pi$
$787$	−9.66053e6	−0.555986	−0.277993	+	0.960583i	$0.589669\pi$
$788$	0	0
$789$	−2.10821e7	−1.20565
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.79420e6	0.440138
$794$	0	0
$795$	1.09264e7	0.613139
$796$	0	0
$797$	−1.15546e7	−0.644330	−0.322165	−	0.946684i	$-0.604411\pi$
$797$	−1.15546e7	−0.644330	−0.322165	+	0.946684i	$0.604411\pi$
$798$	0	0
$799$	−3.30299e7	−1.83037
$800$	0	0
$801$	6.12101e6	0.337087
$802$	0	0
$803$	3.79282e6	0.207574
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.27589e7	1.77070
$808$	0	0
$809$	1.85307e7	0.995453	0.497727	−	0.867334i	$-0.334168\pi$
$809$	1.85307e7	0.995453	0.497727	+	0.867334i	$0.334168\pi$
$810$	0	0
$811$	−1.31357e7	−0.701298	−0.350649	−	0.936507i	$-0.614039\pi$
$811$	−1.31357e7	−0.701298	−0.350649	+	0.936507i	$0.614039\pi$
$812$	0	0
$813$	−1.57966e7	−0.838179
$814$	0	0
$815$	2.32102e6	0.122401
$816$	0	0
$817$	4.53633e7	2.37766
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.20868e6	−0.217915	−0.108958	−	0.994046i	$-0.534751\pi$
$821$	−4.20868e6	−0.217915	−0.108958	+	0.994046i	$0.534751\pi$
$822$	0	0
$823$	−1.10437e7	−0.568350	−0.284175	−	0.958772i	$-0.591720\pi$
$823$	−1.10437e7	−0.568350	−0.284175	+	0.958772i	$0.591720\pi$
$824$	0	0
$825$	2.93564e7	1.50165
$826$	0	0
$827$	−1.74824e7	−0.888867	−0.444433	−	0.895812i	$-0.646595\pi$
$827$	−1.74824e7	−0.888867	−0.444433	+	0.895812i	$0.646595\pi$
$828$	0	0
$829$	−2.35418e7	−1.18974	−0.594871	−	0.803821i	$-0.702797\pi$
$829$	−2.35418e7	−1.18974	−0.594871	+	0.803821i	$0.702797\pi$
$830$	0	0
$831$	1.32384e7	0.665016
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3.52268e6	−0.174846
$836$	0	0
$837$	4.83788e6	0.238694
$838$	0	0
$839$	2.64326e7	1.29639	0.648195	−	0.761474i	$-0.275524\pi$
$839$	2.64326e7	1.29639	0.648195	+	0.761474i	$0.275524\pi$
$840$	0	0
$841$	−2.05073e7	−0.999813
$842$	0	0
$843$	4.30074e7	2.08437
$844$	0	0
$845$	5.54361e6	0.267086
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.05572e6	−0.383561
$850$	0	0
$851$	1.79305e7	0.848727
$852$	0	0
$853$	−3.86472e7	−1.81863	−0.909317	−	0.416103i	$-0.863395\pi$
$853$	−3.86472e7	−1.81863	−0.909317	+	0.416103i	$0.863395\pi$
$854$	0	0
$855$	4.38759e6	0.205263
$856$	0	0
$857$	−2.46708e7	−1.14744	−0.573721	−	0.819051i	$-0.694501\pi$
$857$	−2.46708e7	−1.14744	−0.573721	+	0.819051i	$0.694501\pi$
$858$	0	0
$859$	2.74324e7	1.26847	0.634235	−	0.773140i	$-0.281315\pi$
$859$	2.74324e7	1.26847	0.634235	+	0.773140i	$0.281315\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.10970e6	0.279250	0.139625	−	0.990204i	$-0.455410\pi$
$863$	6.10970e6	0.279250	0.139625	+	0.990204i	$0.455410\pi$
$864$	0	0
$865$	−6.81388e6	−0.309638
$866$	0	0
$867$	−3.13926e6	−0.141834
$868$	0	0
$869$	9.46331e6	0.425103
$870$	0	0
$871$	−1.65441e7	−0.738920
$872$	0	0
$873$	−1.58250e7	−0.702761
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.62612e6	−0.334815	−0.167408	−	0.985888i	$-0.553540\pi$
$877$	−7.62612e6	−0.334815	−0.167408	+	0.985888i	$0.553540\pi$
$878$	0	0
$879$	1.97075e7	0.860320
$880$	0	0
$881$	3.22357e6	0.139925	0.0699627	−	0.997550i	$-0.477712\pi$
$881$	3.22357e6	0.139925	0.0699627	+	0.997550i	$0.477712\pi$
$882$	0	0
$883$	−7.31409e6	−0.315688	−0.157844	−	0.987464i	$-0.550454\pi$
$883$	−7.31409e6	−0.315688	−0.157844	+	0.987464i	$0.550454\pi$
$884$	0	0
$885$	1.62324e7	0.696664
$886$	0	0
$887$	−4.32134e7	−1.84421	−0.922103	−	0.386944i	$-0.873531\pi$
$887$	−4.32134e7	−1.84421	−0.922103	+	0.386944i	$0.873531\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.12536e7	−1.74088
$892$	0	0
$893$	−5.13419e7	−2.15448
$894$	0	0
$895$	−8.75047e6	−0.365152
$896$	0	0
$897$	1.59508e7	0.661912
$898$	0	0
$899$	−126294.	−0.00521175
$900$	0	0
$901$	−3.81062e7	−1.56381
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.31822e6	−0.256433
$906$	0	0
$907$	1.48978e7	0.601317	0.300659	−	0.953732i	$-0.402794\pi$
$907$	1.48978e7	0.601317	0.300659	+	0.953732i	$0.402794\pi$
$908$	0	0
$909$	−1.44015e7	−0.578095
$910$	0	0
$911$	−1.65823e7	−0.661987	−0.330993	−	0.943633i	$-0.607384\pi$
$911$	−1.65823e7	−0.661987	−0.330993	+	0.943633i	$0.607384\pi$
$912$	0	0
$913$	−3.33007e7	−1.32214
$914$	0	0
$915$	−9.97768e6	−0.393982
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.94331e7	0.759021	0.379510	−	0.925187i	$-0.376092\pi$
$919$	1.94331e7	0.759021	0.379510	+	0.925187i	$0.376092\pi$
$920$	0	0
$921$	2.90100e6	0.112693
$922$	0	0
$923$	−2.68464e6	−0.103725
$924$	0	0
$925$	−1.66476e7	−0.639730
$926$	0	0
$927$	−9.51281e6	−0.363588
$928$	0	0
$929$	−4.61166e7	−1.75315	−0.876573	−	0.481269i	$-0.840176\pi$
$929$	−4.61166e7	−1.75315	−0.876573	+	0.481269i	$0.840176\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.16606e7	−1.19073
$934$	0	0
$935$	1.33718e7	0.500221
$936$	0	0
$937$	−1.59157e7	−0.592210	−0.296105	−	0.955155i	$-0.595688\pi$
$937$	−1.59157e7	−0.592210	−0.296105	+	0.955155i	$0.595688\pi$
$938$	0	0
$939$	1.22495e6	0.0453371
$940$	0	0
$941$	1.07301e7	0.395031	0.197516	−	0.980300i	$-0.436713\pi$
$941$	1.07301e7	0.395031	0.197516	+	0.980300i	$0.436713\pi$
$942$	0	0
$943$	6.48391e6	0.237442
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.27632e7	−0.824820	−0.412410	−	0.910998i	$-0.635313\pi$
$947$	−2.27632e7	−0.824820	−0.412410	+	0.910998i	$0.635313\pi$
$948$	0	0
$949$	−1.91337e6	−0.0689657
$950$	0	0
$951$	−2.44975e7	−0.878356
$952$	0	0
$953$	2.69718e7	0.962005	0.481002	−	0.876719i	$-0.340273\pi$
$953$	2.69718e7	0.962005	0.481002	+	0.876719i	$0.340273\pi$
$954$	0	0
$955$	1.87695e7	0.665953
$956$	0	0
$957$	658502.	0.0232422
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.44798e7	−0.855065
$962$	0	0
$963$	−173106.	−0.00601515
$964$	0	0
$965$	6.54385e6	0.226212
$966$	0	0
$967$	−1.02880e7	−0.353805	−0.176902	−	0.984228i	$-0.556608\pi$
$967$	−1.02880e7	−0.353805	−0.176902	+	0.984228i	$0.556608\pi$
$968$	0	0
$969$	−4.68134e7	−1.60162
$970$	0	0
$971$	2.85803e7	0.972791	0.486395	−	0.873739i	$-0.338312\pi$
$971$	2.85803e7	0.972791	0.486395	+	0.873739i	$0.338312\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.48095e7	−0.498918
$976$	0	0
$977$	−2.84299e6	−0.0952881	−0.0476441	−	0.998864i	$-0.515171\pi$
$977$	−2.84299e6	−0.0952881	−0.0476441	+	0.998864i	$0.515171\pi$
$978$	0	0
$979$	2.89970e7	0.966933
$980$	0	0
$981$	2.36622e7	0.785023
$982$	0	0
$983$	−5.31666e7	−1.75491	−0.877455	−	0.479659i	$-0.840760\pi$
$983$	−5.31666e7	−1.75491	−0.877455	+	0.479659i	$0.840760\pi$
$984$	0	0
$985$	−1.10649e7	−0.363376
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.90069e7	−2.24337
$990$	0	0
$991$	−5.41874e7	−1.75273	−0.876363	−	0.481652i	$-0.840037\pi$
$991$	−5.41874e7	−1.75273	−0.876363	+	0.481652i	$0.840037\pi$
$992$	0	0
$993$	−5.02052e7	−1.61575
$994$	0	0
$995$	2.86814e6	0.0918424
$996$	0	0
$997$	−6.55175e6	−0.208747	−0.104373	−	0.994538i	$-0.533284\pi$
$997$	−6.55175e6	−0.208747	−0.104373	+	0.994538i	$0.533284\pi$
$998$	0	0
$999$	1.43046e7	0.453485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.a.1.1		1
4.3	odd	2		196.6.a.g.1.1		1
7.3	odd	6		112.6.i.a.65.1		2
7.5	odd	6		112.6.i.a.81.1		2
7.6	odd	2		784.6.a.k.1.1		1
28.3	even	6		28.6.e.a.9.1	✓	2
28.11	odd	6		196.6.e.b.177.1		2
28.19	even	6		28.6.e.a.25.1	yes	2
28.23	odd	6		196.6.e.b.165.1		2
28.27	even	2		196.6.a.b.1.1		1
84.47	odd	6		252.6.k.c.109.1		2
84.59	odd	6		252.6.k.c.37.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.e.a.9.1	✓	2	28.3	even	6
28.6.e.a.25.1	yes	2	28.19	even	6
112.6.i.a.65.1		2	7.3	odd	6
112.6.i.a.81.1		2	7.5	odd	6
196.6.a.b.1.1		1	28.27	even	2
196.6.a.g.1.1		1	4.3	odd	2
196.6.e.b.165.1		2	28.23	odd	6
196.6.e.b.177.1		2	28.11	odd	6
252.6.k.c.37.1		2	84.59	odd	6
252.6.k.c.109.1		2	84.47	odd	6
784.6.a.a.1.1		1	1.1	even	1	trivial
784.6.a.k.1.1		1	7.6	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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