LMFDB - Embedded newform 1100.6.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,Mod(1,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, 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("1100.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1100.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:	$176.422201794$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 1111x^{6} + 2926x^{5} + 349410x^{4} - 822682x^{3} - 25635603x^{2} + 144346581x - 191266515 $ x^8 - 3x^7 - 1111x^6 + 2926x^5 + 349410x^4 - 822682x^3 - 25635603x^2 + 144346581*x - 191266515
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$6.06056$ of defining polynomial
Character	$\chi$	$=$	1100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.06056 q^{3} +199.386 q^{7} -160.906 q^{9} +121.000 q^{11} -1002.21 q^{13} -501.456 q^{17} -2445.56 q^{19} +1806.55 q^{21} +3138.18 q^{23} -3659.62 q^{27} +704.793 q^{29} +2401.02 q^{31} +1096.33 q^{33} -12368.9 q^{37} -9080.55 q^{39} +13239.2 q^{41} +7456.12 q^{43} +17704.1 q^{47} +22947.9 q^{49} -4543.47 q^{51} +10713.1 q^{53} -22158.2 q^{57} +47877.9 q^{59} +43316.6 q^{61} -32082.5 q^{63} +34008.0 q^{67} +28433.6 q^{69} +68435.7 q^{71} -13319.7 q^{73} +24125.8 q^{77} -59077.7 q^{79} +5942.09 q^{81} -39735.5 q^{83} +6385.82 q^{87} +16098.6 q^{89} -199826. q^{91} +21754.6 q^{93} +23637.5 q^{97} -19469.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 27 q^{3} + 95 q^{7} + 377 q^{9} + 968 q^{11} + 294 q^{13} + 515 q^{17} - 354 q^{19} - 588 q^{21} + 3073 q^{23} + 8502 q^{27} + 2743 q^{29} + 3768 q^{31} + 3267 q^{33} + 9252 q^{37} - 10027 q^{39}+ \cdots + 45617 q^{99}+O(q^{100}) $ 8 * q + 27 * q^3 + 95 * q^7 + 377 * q^9 + 968 * q^11 + 294 * q^13 + 515 * q^17 - 354 * q^19 - 588 * q^21 + 3073 * q^23 + 8502 * q^27 + 2743 * q^29 + 3768 * q^31 + 3267 * q^33 + 9252 * q^37 - 10027 * q^39 + 8756 * q^41 + 15290 * q^43 + 8516 * q^47 + 27341 * q^49 + 4707 * q^51 + 30475 * q^53 + 1903 * q^57 + 20400 * q^59 + 10511 * q^61 - 63122 * q^63 - 616 * q^67 + 72452 * q^69 + 31862 * q^71 - 37377 * q^73 + 11495 * q^77 - 71679 * q^79 - 7228 * q^81 - 61543 * q^83 + 30217 * q^87 - 14961 * q^89 + 132641 * q^91 - 95881 * q^93 + 186419 * q^97 + 45617 * 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
$3$	9.06056	0.581235	0.290617	−	0.956839i	$-0.406139\pi$
$3$	9.06056	0.581235	0.290617	+	0.956839i	$0.406139\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	199.386	1.53798	0.768990	−	0.639261i	$-0.220760\pi$
$7$	199.386	1.53798	0.768990	+	0.639261i	$0.220760\pi$
$8$	0	0
$9$	−160.906	−0.662166
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−1002.21	−1.64475	−0.822373	−	0.568948i	$-0.807351\pi$
$13$	−1002.21	−1.64475	−0.822373	+	0.568948i	$0.807351\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−501.456	−0.420834	−0.210417	−	0.977612i	$-0.567482\pi$
$17$	−501.456	−0.420834	−0.210417	+	0.977612i	$0.567482\pi$
$18$	0	0
$19$	−2445.56	−1.55416	−0.777078	−	0.629404i	$-0.783299\pi$
$19$	−2445.56	−1.55416	−0.777078	+	0.629404i	$0.783299\pi$
$20$	0	0
$21$	1806.55	0.893927
$22$	0	0
$23$	3138.18	1.23697	0.618483	−	0.785798i	$-0.287747\pi$
$23$	3138.18	1.23697	0.618483	+	0.785798i	$0.287747\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3659.62	−0.966109
$28$	0	0
$29$	704.793	0.155620	0.0778102	−	0.996968i	$-0.475207\pi$
$29$	704.793	0.155620	0.0778102	+	0.996968i	$0.475207\pi$
$30$	0	0
$31$	2401.02	0.448737	0.224369	−	0.974504i	$-0.427968\pi$
$31$	2401.02	0.448737	0.224369	+	0.974504i	$0.427968\pi$
$32$	0	0
$33$	1096.33	0.175249
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−12368.9	−1.48534	−0.742669	−	0.669659i	$-0.766440\pi$
$37$	−12368.9	−1.48534	−0.742669	+	0.669659i	$0.766440\pi$
$38$	0	0
$39$	−9080.55	−0.955984
$40$	0	0
$41$	13239.2	1.22999	0.614994	−	0.788532i	$-0.289158\pi$
$41$	13239.2	1.22999	0.614994	+	0.788532i	$0.289158\pi$
$42$	0	0
$43$	7456.12	0.614952	0.307476	−	0.951556i	$-0.400516\pi$
$43$	7456.12	0.614952	0.307476	+	0.951556i	$0.400516\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	17704.1	1.16904	0.584520	−	0.811380i	$-0.301283\pi$
$47$	17704.1	1.16904	0.584520	+	0.811380i	$0.301283\pi$
$48$	0	0
$49$	22947.9	1.36538
$50$	0	0
$51$	−4543.47	−0.244603
$52$	0	0
$53$	10713.1	0.523873	0.261936	−	0.965085i	$-0.415639\pi$
$53$	10713.1	0.523873	0.261936	+	0.965085i	$0.415639\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−22158.2	−0.903330
$58$	0	0
$59$	47877.9	1.79063	0.895314	−	0.445435i	$-0.146951\pi$
$59$	47877.9	1.79063	0.895314	+	0.445435i	$0.146951\pi$
$60$	0	0
$61$	43316.6	1.49049	0.745246	−	0.666790i	$-0.232332\pi$
$61$	43316.6	1.49049	0.745246	+	0.666790i	$0.232332\pi$
$62$	0	0
$63$	−32082.5	−1.01840
$64$	0	0
$65$	0	0
$66$	0	0
$67$	34008.0	0.925537	0.462769	−	0.886479i	$-0.346856\pi$
$67$	34008.0	0.925537	0.462769	+	0.886479i	$0.346856\pi$
$68$	0	0
$69$	28433.6	0.718968
$70$	0	0
$71$	68435.7	1.61115	0.805576	−	0.592492i	$-0.201856\pi$
$71$	68435.7	1.61115	0.805576	+	0.592492i	$0.201856\pi$
$72$	0	0
$73$	−13319.7	−0.292541	−0.146271	−	0.989245i	$-0.546727\pi$
$73$	−13319.7	−0.292541	−0.146271	+	0.989245i	$0.546727\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	24125.8	0.463718
$78$	0	0
$79$	−59077.7	−1.06502	−0.532508	−	0.846425i	$-0.678750\pi$
$79$	−59077.7	−1.06502	−0.532508	+	0.846425i	$0.678750\pi$
$80$	0	0
$81$	5942.09	0.100630
$82$	0	0
$83$	−39735.5	−0.633116	−0.316558	−	0.948573i	$-0.602527\pi$
$83$	−39735.5	−0.633116	−0.316558	+	0.948573i	$0.602527\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6385.82	0.0904520
$88$	0	0
$89$	16098.6	0.215433	0.107717	−	0.994182i	$-0.465646\pi$
$89$	16098.6	0.215433	0.107717	+	0.994182i	$0.465646\pi$
$90$	0	0
$91$	−199826.	−2.52959
$92$	0	0
$93$	21754.6	0.260822
$94$	0	0
$95$	0	0
$96$	0	0
$97$	23637.5	0.255078	0.127539	−	0.991834i	$-0.459292\pi$
$97$	23637.5	0.255078	0.127539	+	0.991834i	$0.459292\pi$
$98$	0	0
$99$	−19469.7	−0.199651
$100$	0	0
$101$	12791.8	0.124775	0.0623877	−	0.998052i	$-0.480128\pi$
$101$	12791.8	0.124775	0.0623877	+	0.998052i	$0.480128\pi$
$102$	0	0
$103$	43957.7	0.408265	0.204132	−	0.978943i	$-0.434563\pi$
$103$	43957.7	0.408265	0.204132	+	0.978943i	$0.434563\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10538.7	−0.0889874	−0.0444937	−	0.999010i	$-0.514167\pi$
$107$	−10538.7	−0.0889874	−0.0444937	+	0.999010i	$0.514167\pi$
$108$	0	0
$109$	69951.3	0.563936	0.281968	−	0.959424i	$-0.409013\pi$
$109$	69951.3	0.563936	0.281968	+	0.959424i	$0.409013\pi$
$110$	0	0
$111$	−112069.	−0.863330
$112$	0	0
$113$	96277.7	0.709299	0.354650	−	0.934999i	$-0.384600\pi$
$113$	96277.7	0.709299	0.354650	+	0.934999i	$0.384600\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	161261.	1.08910
$118$	0	0
$119$	−99983.6	−0.647234
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	119954.	0.714912
$124$	0	0
$125$	0	0
$126$	0	0
$127$	113021.	0.621799	0.310900	−	0.950443i	$-0.399370\pi$
$127$	113021.	0.621799	0.310900	+	0.950443i	$0.399370\pi$
$128$	0	0
$129$	67556.6	0.357432
$130$	0	0
$131$	−41264.6	−0.210087	−0.105044	−	0.994468i	$-0.533498\pi$
$131$	−41264.6	−0.210087	−0.105044	+	0.994468i	$0.533498\pi$
$132$	0	0
$133$	−487612.	−2.39026
$134$	0	0
$135$	0	0
$136$	0	0
$137$	301461.	1.37224	0.686120	−	0.727489i	$-0.259313\pi$
$137$	301461.	1.37224	0.686120	+	0.727489i	$0.259313\pi$
$138$	0	0
$139$	−256904.	−1.12781	−0.563903	−	0.825841i	$-0.690701\pi$
$139$	−256904.	−1.12781	−0.563903	+	0.825841i	$0.690701\pi$
$140$	0	0
$141$	160409.	0.679486
$142$	0	0
$143$	−121267.	−0.495910
$144$	0	0
$145$	0	0
$146$	0	0
$147$	207921.	0.793607
$148$	0	0
$149$	25156.4	0.0928287	0.0464143	−	0.998922i	$-0.485221\pi$
$149$	25156.4	0.0928287	0.0464143	+	0.998922i	$0.485221\pi$
$150$	0	0
$151$	19083.7	0.0681114	0.0340557	−	0.999420i	$-0.489158\pi$
$151$	19083.7	0.0681114	0.0340557	+	0.999420i	$0.489158\pi$
$152$	0	0
$153$	80687.5	0.278662
$154$	0	0
$155$	0	0
$156$	0	0
$157$	110374.	0.357369	0.178685	−	0.983906i	$-0.442816\pi$
$157$	110374.	0.357369	0.178685	+	0.983906i	$0.442816\pi$
$158$	0	0
$159$	97066.7	0.304493
$160$	0	0
$161$	625710.	1.90243
$162$	0	0
$163$	−569442.	−1.67873	−0.839364	−	0.543570i	$-0.817072\pi$
$163$	−569442.	−1.67873	−0.839364	+	0.543570i	$0.817072\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−553462.	−1.53567	−0.767833	−	0.640651i	$-0.778665\pi$
$167$	−553462.	−1.53567	−0.767833	+	0.640651i	$0.778665\pi$
$168$	0	0
$169$	633126.	1.70519
$170$	0	0
$171$	393507.	1.02911
$172$	0	0
$173$	693097.	1.76067	0.880337	−	0.474349i	$-0.157316\pi$
$173$	693097.	1.76067	0.880337	+	0.474349i	$0.157316\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	433801.	1.04078
$178$	0	0
$179$	−655189.	−1.52839	−0.764195	−	0.644985i	$-0.776863\pi$
$179$	−655189.	−1.52839	−0.764195	+	0.644985i	$0.776863\pi$
$180$	0	0
$181$	−281145.	−0.637872	−0.318936	−	0.947776i	$-0.603325\pi$
$181$	−281145.	−0.637872	−0.318936	+	0.947776i	$0.603325\pi$
$182$	0	0
$183$	392472.	0.866326
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−60676.2	−0.126886
$188$	0	0
$189$	−729678.	−1.48586
$190$	0	0
$191$	526382.	1.04404	0.522020	−	0.852933i	$-0.325179\pi$
$191$	526382.	1.04404	0.522020	+	0.852933i	$0.325179\pi$
$192$	0	0
$193$	−630283.	−1.21799	−0.608993	−	0.793175i	$-0.708426\pi$
$193$	−630283.	−1.21799	−0.608993	+	0.793175i	$0.708426\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	676686.	1.24229	0.621143	−	0.783697i	$-0.286669\pi$
$197$	676686.	1.24229	0.621143	+	0.783697i	$0.286669\pi$
$198$	0	0
$199$	941332.	1.68504	0.842520	−	0.538665i	$-0.181071\pi$
$199$	941332.	1.68504	0.842520	+	0.538665i	$0.181071\pi$
$200$	0	0
$201$	308131.	0.537954
$202$	0	0
$203$	140526.	0.239341
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−504953.	−0.819077
$208$	0	0
$209$	−295913.	−0.468596
$210$	0	0
$211$	−905690.	−1.40047	−0.700235	−	0.713913i	$-0.746921\pi$
$211$	−905690.	−1.40047	−0.700235	+	0.713913i	$0.746921\pi$
$212$	0	0
$213$	620065.	0.936458
$214$	0	0
$215$	0	0
$216$	0	0
$217$	478732.	0.690149
$218$	0	0
$219$	−120684.	−0.170035
$220$	0	0
$221$	502563.	0.692165
$222$	0	0
$223$	1.22389e6	1.64809	0.824043	−	0.566527i	$-0.191713\pi$
$223$	1.22389e6	1.64809	0.824043	+	0.566527i	$0.191713\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−691757.	−0.891023	−0.445512	−	0.895276i	$-0.646978\pi$
$227$	−691757.	−0.891023	−0.445512	+	0.895276i	$0.646978\pi$
$228$	0	0
$229$	−130819.	−0.164848	−0.0824240	−	0.996597i	$-0.526266\pi$
$229$	−130819.	−0.164848	−0.0824240	+	0.996597i	$0.526266\pi$
$230$	0	0
$231$	218593.	0.269529
$232$	0	0
$233$	1.12315e6	1.35534	0.677669	−	0.735367i	$-0.262990\pi$
$233$	1.12315e6	1.35534	0.677669	+	0.735367i	$0.262990\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−535277.	−0.619024
$238$	0	0
$239$	−367632.	−0.416312	−0.208156	−	0.978096i	$-0.566746\pi$
$239$	−367632.	−0.416312	−0.208156	+	0.978096i	$0.566746\pi$
$240$	0	0
$241$	1.13015e6	1.25341	0.626703	−	0.779258i	$-0.284404\pi$
$241$	1.13015e6	1.25341	0.626703	+	0.779258i	$0.284404\pi$
$242$	0	0
$243$	943125.	1.02460
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.45096e6	2.55619
$248$	0	0
$249$	−360026.	−0.367989
$250$	0	0
$251$	1.00121e6	1.00309	0.501547	−	0.865130i	$-0.332764\pi$
$251$	1.00121e6	1.00309	0.501547	+	0.865130i	$0.332764\pi$
$252$	0	0
$253$	379720.	0.372960
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−392720.	−0.370895	−0.185447	−	0.982654i	$-0.559373\pi$
$257$	−392720.	−0.370895	−0.185447	+	0.982654i	$0.559373\pi$
$258$	0	0
$259$	−2.46618e6	−2.28442
$260$	0	0
$261$	−113406.	−0.103047
$262$	0	0
$263$	1.11910e6	0.997651	0.498825	−	0.866702i	$-0.333765\pi$
$263$	1.11910e6	0.997651	0.498825	+	0.866702i	$0.333765\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	145862.	0.125217
$268$	0	0
$269$	−582159.	−0.490524	−0.245262	−	0.969457i	$-0.578874\pi$
$269$	−582159.	−0.490524	−0.245262	+	0.969457i	$0.578874\pi$
$270$	0	0
$271$	366743.	0.303346	0.151673	−	0.988431i	$-0.451534\pi$
$271$	366743.	0.303346	0.151673	+	0.988431i	$0.451534\pi$
$272$	0	0
$273$	−1.81054e6	−1.47028
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.24613e6	0.975804	0.487902	−	0.872898i	$-0.337762\pi$
$277$	1.24613e6	0.975804	0.487902	+	0.872898i	$0.337762\pi$
$278$	0	0
$279$	−386340.	−0.297139
$280$	0	0
$281$	−1.72805e6	−1.30554	−0.652771	−	0.757555i	$-0.726394\pi$
$281$	−1.72805e6	−1.30554	−0.652771	+	0.757555i	$0.726394\pi$
$282$	0	0
$283$	−868098.	−0.644321	−0.322161	−	0.946685i	$-0.604409\pi$
$283$	−868098.	−0.644321	−0.322161	+	0.946685i	$0.604409\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.63971e6	1.89170
$288$	0	0
$289$	−1.16840e6	−0.822899
$290$	0	0
$291$	214169.	0.148260
$292$	0	0
$293$	−319096.	−0.217146	−0.108573	−	0.994088i	$-0.534628\pi$
$293$	−319096.	−0.217146	−0.108573	+	0.994088i	$0.534628\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−442813.	−0.291293
$298$	0	0
$299$	−3.14510e6	−2.03450
$300$	0	0
$301$	1.48665e6	0.945784
$302$	0	0
$303$	115901.	0.0725239
$304$	0	0
$305$	0	0
$306$	0	0
$307$	430535.	0.260713	0.130357	−	0.991467i	$-0.458388\pi$
$307$	430535.	0.260713	0.130357	+	0.991467i	$0.458388\pi$
$308$	0	0
$309$	398281.	0.237298
$310$	0	0
$311$	−2.80826e6	−1.64640	−0.823201	−	0.567750i	$-0.807814\pi$
$311$	−2.80826e6	−1.64640	−0.823201	+	0.567750i	$0.807814\pi$
$312$	0	0
$313$	−2.63653e6	−1.52115	−0.760575	−	0.649250i	$-0.775083\pi$
$313$	−2.63653e6	−1.52115	−0.760575	+	0.649250i	$0.775083\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.05195e6	0.587961	0.293980	−	0.955811i	$-0.405020\pi$
$317$	1.05195e6	0.587961	0.293980	+	0.955811i	$0.405020\pi$
$318$	0	0
$319$	85279.9	0.0469213
$320$	0	0
$321$	−95486.7	−0.0517226
$322$	0	0
$323$	1.22634e6	0.654042
$324$	0	0
$325$	0	0
$326$	0	0
$327$	633798.	0.327779
$328$	0	0
$329$	3.52996e6	1.79796
$330$	0	0
$331$	−911806.	−0.457438	−0.228719	−	0.973492i	$-0.573454\pi$
$331$	−911806.	−0.457438	−0.228719	+	0.973492i	$0.573454\pi$
$332$	0	0
$333$	1.99023e6	0.983540
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.68243e6	1.76628	0.883140	−	0.469110i	$-0.155425\pi$
$337$	3.68243e6	1.76628	0.883140	+	0.469110i	$0.155425\pi$
$338$	0	0
$339$	872329.	0.412269
$340$	0	0
$341$	290524.	0.135299
$342$	0	0
$343$	1.22442e6	0.561947
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.28081e6	1.90855	0.954273	−	0.298938i	$-0.0966323\pi$
$347$	4.28081e6	1.90855	0.954273	+	0.298938i	$0.0966323\pi$
$348$	0	0
$349$	−3.47042e6	−1.52517	−0.762585	−	0.646888i	$-0.776070\pi$
$349$	−3.47042e6	−1.52517	−0.762585	+	0.646888i	$0.776070\pi$
$350$	0	0
$351$	3.66769e6	1.58900
$352$	0	0
$353$	−2.79223e6	−1.19266	−0.596328	−	0.802741i	$-0.703374\pi$
$353$	−2.79223e6	−1.19266	−0.596328	+	0.802741i	$0.703374\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−905906.	−0.376195
$358$	0	0
$359$	−524819.	−0.214918	−0.107459	−	0.994210i	$-0.534271\pi$
$359$	−524819.	−0.214918	−0.107459	+	0.994210i	$0.534271\pi$
$360$	0	0
$361$	3.50468e6	1.41540
$362$	0	0
$363$	132656.	0.0528395
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5579.12	0.00216222	0.00108111	−	0.999999i	$-0.499656\pi$
$367$	5579.12	0.00216222	0.00108111	+	0.999999i	$0.499656\pi$
$368$	0	0
$369$	−2.13026e6	−0.814456
$370$	0	0
$371$	2.13605e6	0.805705
$372$	0	0
$373$	3.05618e6	1.13738	0.568692	−	0.822551i	$-0.307450\pi$
$373$	3.05618e6	1.13738	0.568692	+	0.822551i	$0.307450\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−706348.	−0.255956
$378$	0	0
$379$	1.36598e6	0.488481	0.244241	−	0.969715i	$-0.421461\pi$
$379$	1.36598e6	0.488481	0.244241	+	0.969715i	$0.421461\pi$
$380$	0	0
$381$	1.02403e6	0.361411
$382$	0	0
$383$	2.23789e6	0.779546	0.389773	−	0.920911i	$-0.372553\pi$
$383$	2.23789e6	0.779546	0.389773	+	0.920911i	$0.372553\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.19974e6	−0.407201
$388$	0	0
$389$	−4.46369e6	−1.49562	−0.747809	−	0.663914i	$-0.768894\pi$
$389$	−4.46369e6	−1.49562	−0.747809	+	0.663914i	$0.768894\pi$
$390$	0	0
$391$	−1.57366e6	−0.520558
$392$	0	0
$393$	−373881.	−0.122110
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.05861e6	0.973975	0.486988	−	0.873409i	$-0.338096\pi$
$397$	3.05861e6	0.973975	0.486988	+	0.873409i	$0.338096\pi$
$398$	0	0
$399$	−4.41804e6	−1.38930
$400$	0	0
$401$	−2.65046e6	−0.823114	−0.411557	−	0.911384i	$-0.635015\pi$
$401$	−2.65046e6	−0.823114	−0.411557	+	0.911384i	$0.635015\pi$
$402$	0	0
$403$	−2.40632e6	−0.738059
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.49663e6	−0.447846
$408$	0	0
$409$	2.00854e6	0.593707	0.296853	−	0.954923i	$-0.404063\pi$
$409$	2.00854e6	0.593707	0.296853	+	0.954923i	$0.404063\pi$
$410$	0	0
$411$	2.73140e6	0.797593
$412$	0	0
$413$	9.54621e6	2.75395
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.32770e6	−0.655520
$418$	0	0
$419$	−4.20496e6	−1.17011	−0.585055	−	0.810993i	$-0.698927\pi$
$419$	−4.20496e6	−1.17011	−0.585055	+	0.810993i	$0.698927\pi$
$420$	0	0
$421$	4.13287e6	1.13644	0.568220	−	0.822877i	$-0.307632\pi$
$421$	4.13287e6	1.13644	0.568220	+	0.822877i	$0.307632\pi$
$422$	0	0
$423$	−2.84870e6	−0.774098
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.63674e6	2.29235
$428$	0	0
$429$	−1.09875e6	−0.288240
$430$	0	0
$431$	2.31472e6	0.600212	0.300106	−	0.953906i	$-0.402978\pi$
$431$	2.31472e6	0.600212	0.300106	+	0.953906i	$0.402978\pi$
$432$	0	0
$433$	5.73720e6	1.47055	0.735276	−	0.677768i	$-0.237053\pi$
$433$	5.73720e6	1.47055	0.735276	+	0.677768i	$0.237053\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.67461e6	−1.92244
$438$	0	0
$439$	6.49052e6	1.60738	0.803689	−	0.595050i	$-0.202868\pi$
$439$	6.49052e6	1.60738	0.803689	+	0.595050i	$0.202868\pi$
$440$	0	0
$441$	−3.69247e6	−0.904108
$442$	0	0
$443$	−7.66321e6	−1.85525	−0.927623	−	0.373518i	$-0.878152\pi$
$443$	−7.66321e6	−1.85525	−0.927623	+	0.373518i	$0.878152\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	227931.	0.0539553
$448$	0	0
$449$	−251448.	−0.0588616	−0.0294308	−	0.999567i	$-0.509369\pi$
$449$	−251448.	−0.0588616	−0.0294308	+	0.999567i	$0.509369\pi$
$450$	0	0
$451$	1.60194e6	0.370855
$452$	0	0
$453$	172909.	0.0395887
$454$	0	0
$455$	0	0
$456$	0	0
$457$	63954.8	0.0143246	0.00716230	−	0.999974i	$-0.497720\pi$
$457$	63954.8	0.0143246	0.00716230	+	0.999974i	$0.497720\pi$
$458$	0	0
$459$	1.83514e6	0.406571
$460$	0	0
$461$	8.95377e6	1.96225	0.981124	−	0.193381i	$-0.0619453\pi$
$461$	8.95377e6	1.96225	0.981124	+	0.193381i	$0.0619453\pi$
$462$	0	0
$463$	−739148.	−0.160243	−0.0801215	−	0.996785i	$-0.525531\pi$
$463$	−739148.	−0.160243	−0.0801215	+	0.996785i	$0.525531\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.53852e6	1.38735	0.693677	−	0.720287i	$-0.255990\pi$
$467$	6.53852e6	1.38735	0.693677	+	0.720287i	$0.255990\pi$
$468$	0	0
$469$	6.78073e6	1.42346
$470$	0	0
$471$	1.00005e6	0.207716
$472$	0	0
$473$	902190.	0.185415
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.72381e6	−0.346891
$478$	0	0
$479$	5.34320e6	1.06405	0.532026	−	0.846728i	$-0.321431\pi$
$479$	5.34320e6	1.06405	0.532026	+	0.846728i	$0.321431\pi$
$480$	0	0
$481$	1.23962e7	2.44300
$482$	0	0
$483$	5.66928e6	1.10576
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.81268e6	0.728463	0.364232	−	0.931308i	$-0.381332\pi$
$487$	3.81268e6	0.728463	0.364232	+	0.931308i	$0.381332\pi$
$488$	0	0
$489$	−5.15946e6	−0.975735
$490$	0	0
$491$	−7.35416e6	−1.37667	−0.688334	−	0.725394i	$-0.741658\pi$
$491$	−7.35416e6	−1.37667	−0.688334	+	0.725394i	$0.741658\pi$
$492$	0	0
$493$	−353423.	−0.0654903
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.36451e7	2.47792
$498$	0	0
$499$	7.62209e6	1.37032	0.685161	−	0.728391i	$-0.259732\pi$
$499$	7.62209e6	1.37032	0.685161	+	0.728391i	$0.259732\pi$
$500$	0	0
$501$	−5.01467e6	−0.892582
$502$	0	0
$503$	2.03906e6	0.359344	0.179672	−	0.983727i	$-0.442496\pi$
$503$	2.03906e6	0.359344	0.179672	+	0.983727i	$0.442496\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5.73647e6	0.991117
$508$	0	0
$509$	−4.80695e6	−0.822386	−0.411193	−	0.911548i	$-0.634888\pi$
$509$	−4.80695e6	−0.822386	−0.411193	+	0.911548i	$0.634888\pi$
$510$	0	0
$511$	−2.65576e6	−0.449922
$512$	0	0
$513$	8.94982e6	1.50148
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.14220e6	0.352479
$518$	0	0
$519$	6.27985e6	1.02337
$520$	0	0
$521$	1.10303e7	1.78030	0.890148	−	0.455672i	$-0.150601\pi$
$521$	1.10303e7	1.78030	0.890148	+	0.455672i	$0.150601\pi$
$522$	0	0
$523$	−6.12941e6	−0.979861	−0.489930	−	0.871762i	$-0.662978\pi$
$523$	−6.12941e6	−0.979861	−0.489930	+	0.871762i	$0.662978\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.20401e6	−0.188844
$528$	0	0
$529$	3.41182e6	0.530087
$530$	0	0
$531$	−7.70386e6	−1.18569
$532$	0	0
$533$	−1.32684e7	−2.02302
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.93638e6	−0.888353
$538$	0	0
$539$	2.77670e6	0.411678
$540$	0	0
$541$	−6.14186e6	−0.902207	−0.451104	−	0.892472i	$-0.648970\pi$
$541$	−6.14186e6	−0.902207	−0.451104	+	0.892472i	$0.648970\pi$
$542$	0	0
$543$	−2.54733e6	−0.370753
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.46173e6	0.494681	0.247340	−	0.968929i	$-0.420443\pi$
$547$	3.46173e6	0.494681	0.247340	+	0.968929i	$0.420443\pi$
$548$	0	0
$549$	−6.96991e6	−0.986953
$550$	0	0
$551$	−1.72362e6	−0.241859
$552$	0	0
$553$	−1.17793e7	−1.63797
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.10120e7	−1.50394	−0.751969	−	0.659198i	$-0.770896\pi$
$557$	−1.10120e7	−1.50394	−0.751969	+	0.659198i	$0.770896\pi$
$558$	0	0
$559$	−7.47257e6	−1.01144
$560$	0	0
$561$	−549760.	−0.0737507
$562$	0	0
$563$	−440697.	−0.0585961	−0.0292980	−	0.999571i	$-0.509327\pi$
$563$	−440697.	−0.0585961	−0.0292980	+	0.999571i	$0.509327\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.18477e6	0.154767
$568$	0	0
$569$	−5.18526e6	−0.671413	−0.335707	−	0.941967i	$-0.608975\pi$
$569$	−5.18526e6	−0.671413	−0.335707	+	0.941967i	$0.608975\pi$
$570$	0	0
$571$	−217383.	−0.0279019	−0.0139510	−	0.999903i	$-0.504441\pi$
$571$	−217383.	−0.0279019	−0.0139510	+	0.999903i	$0.504441\pi$
$572$	0	0
$573$	4.76931e6	0.606833
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.48074e6	0.435244	0.217622	−	0.976033i	$-0.430170\pi$
$577$	3.48074e6	0.435244	0.217622	+	0.976033i	$0.430170\pi$
$578$	0	0
$579$	−5.71072e6	−0.707936
$580$	0	0
$581$	−7.92272e6	−0.973719
$582$	0	0
$583$	1.29629e6	0.157954
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.74685e6	−0.209248	−0.104624	−	0.994512i	$-0.533364\pi$
$587$	−1.74685e6	−0.209248	−0.104624	+	0.994512i	$0.533364\pi$
$588$	0	0
$589$	−5.87186e6	−0.697408
$590$	0	0
$591$	6.13115e6	0.722060
$592$	0	0
$593$	−606452.	−0.0708207	−0.0354103	−	0.999373i	$-0.511274\pi$
$593$	−606452.	−0.0708207	−0.0354103	+	0.999373i	$0.511274\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.52899e6	0.979404
$598$	0	0
$599$	−8.74412e6	−0.995748	−0.497874	−	0.867249i	$-0.665886\pi$
$599$	−8.74412e6	−0.995748	−0.497874	+	0.867249i	$0.665886\pi$
$600$	0	0
$601$	9.29180e6	1.04933	0.524667	−	0.851308i	$-0.324190\pi$
$601$	9.29180e6	1.04933	0.524667	+	0.851308i	$0.324190\pi$
$602$	0	0
$603$	−5.47210e6	−0.612859
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.84393e6	−0.974257	−0.487128	−	0.873330i	$-0.661956\pi$
$607$	−8.84393e6	−0.974257	−0.487128	+	0.873330i	$0.661956\pi$
$608$	0	0
$609$	1.27324e6	0.139113
$610$	0	0
$611$	−1.77432e7	−1.92277
$612$	0	0
$613$	2.79542e6	0.300467	0.150233	−	0.988651i	$-0.451998\pi$
$613$	2.79542e6	0.300467	0.150233	+	0.988651i	$0.451998\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.79224e6	−0.718290	−0.359145	−	0.933282i	$-0.616932\pi$
$617$	−6.79224e6	−0.718290	−0.359145	+	0.933282i	$0.616932\pi$
$618$	0	0
$619$	5.21853e6	0.547421	0.273711	−	0.961812i	$-0.411749\pi$
$619$	5.21853e6	0.547421	0.273711	+	0.961812i	$0.411749\pi$
$620$	0	0
$621$	−1.14845e7	−1.19504
$622$	0	0
$623$	3.20984e6	0.331332
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.68114e6	−0.272364
$628$	0	0
$629$	6.20244e6	0.625080
$630$	0	0
$631$	4.10205e6	0.410135	0.205068	−	0.978748i	$-0.434259\pi$
$631$	4.10205e6	0.410135	0.205068	+	0.978748i	$0.434259\pi$
$632$	0	0
$633$	−8.20606e6	−0.814001
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.29986e7	−2.24570
$638$	0	0
$639$	−1.10117e7	−1.06685
$640$	0	0
$641$	−3.74532e6	−0.360034	−0.180017	−	0.983663i	$-0.557615\pi$
$641$	−3.74532e6	−0.360034	−0.180017	+	0.983663i	$0.557615\pi$
$642$	0	0
$643$	−1.60830e6	−0.153405	−0.0767023	−	0.997054i	$-0.524439\pi$
$643$	−1.60830e6	−0.153405	−0.0767023	+	0.997054i	$0.524439\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.17131e7	−1.10005	−0.550023	−	0.835149i	$-0.685381\pi$
$647$	−1.17131e7	−1.10005	−0.550023	+	0.835149i	$0.685381\pi$
$648$	0	0
$649$	5.79323e6	0.539895
$650$	0	0
$651$	4.33757e6	0.401139
$652$	0	0
$653$	−4.80648e6	−0.441107	−0.220554	−	0.975375i	$-0.570786\pi$
$653$	−4.80648e6	−0.441107	−0.220554	+	0.975375i	$0.570786\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.14322e6	0.193711
$658$	0	0
$659$	4.25550e6	0.381713	0.190856	−	0.981618i	$-0.438873\pi$
$659$	4.25550e6	0.381713	0.190856	+	0.981618i	$0.438873\pi$
$660$	0	0
$661$	−1.37432e7	−1.22344	−0.611722	−	0.791073i	$-0.709523\pi$
$661$	−1.37432e7	−1.22344	−0.611722	+	0.791073i	$0.709523\pi$
$662$	0	0
$663$	4.55350e6	0.402311
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.21177e6	0.192497
$668$	0	0
$669$	1.10891e7	0.957925
$670$	0	0
$671$	5.24131e6	0.449400
$672$	0	0
$673$	−1.42154e7	−1.20982	−0.604909	−	0.796295i	$-0.706790\pi$
$673$	−1.42154e7	−1.20982	−0.604909	+	0.796295i	$0.706790\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.03422e7	1.70579	0.852894	−	0.522083i	$-0.174845\pi$
$677$	2.03422e7	1.70579	0.852894	+	0.522083i	$0.174845\pi$
$678$	0	0
$679$	4.71301e6	0.392305
$680$	0	0
$681$	−6.26770e6	−0.517894
$682$	0	0
$683$	1.66619e7	1.36670	0.683351	−	0.730090i	$-0.260522\pi$
$683$	1.66619e7	1.36670	0.683351	+	0.730090i	$0.260522\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.18530e6	−0.0958154
$688$	0	0
$689$	−1.07368e7	−0.861638
$690$	0	0
$691$	−5.50835e6	−0.438861	−0.219430	−	0.975628i	$-0.570420\pi$
$691$	−5.50835e6	−0.438861	−0.219430	+	0.975628i	$0.570420\pi$
$692$	0	0
$693$	−3.88199e6	−0.307058
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.63886e6	−0.517621
$698$	0	0
$699$	1.01764e7	0.787770
$700$	0	0
$701$	−1.44472e7	−1.11042	−0.555211	−	0.831709i	$-0.687363\pi$
$701$	−1.44472e7	−1.11042	−0.555211	+	0.831709i	$0.687363\pi$
$702$	0	0
$703$	3.02488e7	2.30845
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.55052e6	0.191902
$708$	0	0
$709$	1.23363e6	0.0921658	0.0460829	−	0.998938i	$-0.485326\pi$
$709$	1.23363e6	0.0921658	0.0460829	+	0.998938i	$0.485326\pi$
$710$	0	0
$711$	9.50598e6	0.705217
$712$	0	0
$713$	7.53484e6	0.555073
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.33095e6	−0.241975
$718$	0	0
$719$	1.53932e7	1.11047	0.555235	−	0.831693i	$-0.312628\pi$
$719$	1.53932e7	1.11047	0.555235	+	0.831693i	$0.312628\pi$
$720$	0	0
$721$	8.76457e6	0.627903
$722$	0	0
$723$	1.02397e7	0.728523
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.78530e7	−1.25278	−0.626390	−	0.779510i	$-0.715468\pi$
$727$	−1.78530e7	−1.25278	−0.626390	+	0.779510i	$0.715468\pi$
$728$	0	0
$729$	7.10131e6	0.494902
$730$	0	0
$731$	−3.73892e6	−0.258793
$732$	0	0
$733$	−2.42891e6	−0.166975	−0.0834876	−	0.996509i	$-0.526606\pi$
$733$	−2.42891e6	−0.166975	−0.0834876	+	0.996509i	$0.526606\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.11497e6	0.279060
$738$	0	0
$739$	2.40946e7	1.62296	0.811482	−	0.584377i	$-0.198661\pi$
$739$	2.40946e7	1.62296	0.811482	+	0.584377i	$0.198661\pi$
$740$	0	0
$741$	2.22071e7	1.48575
$742$	0	0
$743$	−2.31721e7	−1.53990	−0.769952	−	0.638102i	$-0.779720\pi$
$743$	−2.31721e7	−1.53990	−0.769952	+	0.638102i	$0.779720\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.39369e6	0.419228
$748$	0	0
$749$	−2.10128e6	−0.136861
$750$	0	0
$751$	−1.96991e7	−1.27452	−0.637259	−	0.770650i	$-0.719932\pi$
$751$	−1.96991e7	−1.27452	−0.637259	+	0.770650i	$0.719932\pi$
$752$	0	0
$753$	9.07154e6	0.583033
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.90721e7	−1.20965	−0.604825	−	0.796359i	$-0.706757\pi$
$757$	−1.90721e7	−1.20965	−0.604825	+	0.796359i	$0.706757\pi$
$758$	0	0
$759$	3.44047e6	0.216777
$760$	0	0
$761$	1.77948e7	1.11387	0.556933	−	0.830558i	$-0.311978\pi$
$761$	1.77948e7	1.11387	0.556933	+	0.830558i	$0.311978\pi$
$762$	0	0
$763$	1.39473e7	0.867322
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.79836e7	−2.94513
$768$	0	0
$769$	−9.84067e6	−0.600079	−0.300040	−	0.953927i	$-0.597000\pi$
$769$	−9.84067e6	−0.600079	−0.300040	+	0.953927i	$0.597000\pi$
$770$	0	0
$771$	−3.55826e6	−0.215577
$772$	0	0
$773$	1.26324e6	0.0760390	0.0380195	−	0.999277i	$-0.487895\pi$
$773$	1.26324e6	0.0760390	0.0380195	+	0.999277i	$0.487895\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.23450e7	−1.32778
$778$	0	0
$779$	−3.23772e7	−1.91159
$780$	0	0
$781$	8.28072e6	0.485781
$782$	0	0
$783$	−2.57927e6	−0.150346
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.26634e7	−0.728806	−0.364403	−	0.931241i	$-0.618727\pi$
$787$	−1.26634e7	−0.728806	−0.364403	+	0.931241i	$0.618727\pi$
$788$	0	0
$789$	1.01396e7	0.579869
$790$	0	0
$791$	1.91965e7	1.09089
$792$	0	0
$793$	−4.34122e7	−2.45148
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.85646e7	1.03524	0.517619	−	0.855611i	$-0.326819\pi$
$797$	1.85646e7	1.03524	0.517619	+	0.855611i	$0.326819\pi$
$798$	0	0
$799$	−8.87783e6	−0.491971
$800$	0	0
$801$	−2.59037e6	−0.142653
$802$	0	0
$803$	−1.61168e6	−0.0882044
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.27468e6	−0.285110
$808$	0	0
$809$	−3.81470e6	−0.204922	−0.102461	−	0.994737i	$-0.532672\pi$
$809$	−3.81470e6	−0.204922	−0.102461	+	0.994737i	$0.532672\pi$
$810$	0	0
$811$	−2.99995e7	−1.60163	−0.800815	−	0.598912i	$-0.795600\pi$
$811$	−2.99995e7	−1.60163	−0.800815	+	0.598912i	$0.795600\pi$
$812$	0	0
$813$	3.32289e6	0.176315
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.82344e7	−0.955733
$818$	0	0
$819$	3.21533e7	1.67501
$820$	0	0
$821$	−1.26661e7	−0.655819	−0.327910	−	0.944709i	$-0.606344\pi$
$821$	−1.26661e7	−0.655819	−0.327910	+	0.944709i	$0.606344\pi$
$822$	0	0
$823$	5.69298e6	0.292981	0.146491	−	0.989212i	$-0.453202\pi$
$823$	5.69298e6	0.292981	0.146491	+	0.989212i	$0.453202\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.73955e6	−0.444350	−0.222175	−	0.975007i	$-0.571316\pi$
$827$	−8.73955e6	−0.444350	−0.222175	+	0.975007i	$0.571316\pi$
$828$	0	0
$829$	2.74793e7	1.38873	0.694367	−	0.719621i	$-0.255684\pi$
$829$	2.74793e7	1.38873	0.694367	+	0.719621i	$0.255684\pi$
$830$	0	0
$831$	1.12906e7	0.567171
$832$	0	0
$833$	−1.15074e7	−0.574598
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.78682e6	−0.433529
$838$	0	0
$839$	2.26415e7	1.11045	0.555226	−	0.831700i	$-0.312632\pi$
$839$	2.26415e7	1.11045	0.555226	+	0.831700i	$0.312632\pi$
$840$	0	0
$841$	−2.00144e7	−0.975782
$842$	0	0
$843$	−1.56571e7	−0.758827
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.91922e6	0.139816
$848$	0	0
$849$	−7.86545e6	−0.374502
$850$	0	0
$851$	−3.88157e7	−1.83731
$852$	0	0
$853$	9.69760e6	0.456343	0.228172	−	0.973621i	$-0.426725\pi$
$853$	9.69760e6	0.456343	0.228172	+	0.973621i	$0.426725\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.91364e7	−1.35514	−0.677569	−	0.735459i	$-0.736967\pi$
$857$	−2.91364e7	−1.35514	−0.677569	+	0.735459i	$0.736967\pi$
$858$	0	0
$859$	−2.79828e7	−1.29392	−0.646961	−	0.762523i	$-0.723960\pi$
$859$	−2.79828e7	−1.29392	−0.646961	+	0.762523i	$0.723960\pi$
$860$	0	0
$861$	2.39172e7	1.09952
$862$	0	0
$863$	1.59819e7	0.730470	0.365235	−	0.930915i	$-0.380989\pi$
$863$	1.59819e7	0.730470	0.365235	+	0.930915i	$0.380989\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.05863e7	−0.478298
$868$	0	0
$869$	−7.14840e6	−0.321114
$870$	0	0
$871$	−3.40830e7	−1.52227
$872$	0	0
$873$	−3.80343e6	−0.168904
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.11079e7	0.926716	0.463358	−	0.886171i	$-0.346644\pi$
$877$	2.11079e7	0.926716	0.463358	+	0.886171i	$0.346644\pi$
$878$	0	0
$879$	−2.89119e6	−0.126213
$880$	0	0
$881$	−2.03421e7	−0.882992	−0.441496	−	0.897263i	$-0.645552\pi$
$881$	−2.03421e7	−0.882992	−0.441496	+	0.897263i	$0.645552\pi$
$882$	0	0
$883$	2.48733e7	1.07357	0.536787	−	0.843718i	$-0.319638\pi$
$883$	2.48733e7	1.07357	0.536787	+	0.843718i	$0.319638\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.16789e7	0.925185	0.462592	−	0.886571i	$-0.346919\pi$
$887$	2.16789e7	0.925185	0.462592	+	0.886571i	$0.346919\pi$
$888$	0	0
$889$	2.25349e7	0.956314
$890$	0	0
$891$	718993.	0.0303410
$892$	0	0
$893$	−4.32965e7	−1.81687
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.84964e7	−1.18252
$898$	0	0
$899$	1.69222e6	0.0698327
$900$	0	0
$901$	−5.37215e6	−0.220463
$902$	0	0
$903$	1.34699e7	0.549723
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.63074e7	−0.658214	−0.329107	−	0.944293i	$-0.606748\pi$
$907$	−1.63074e7	−0.658214	−0.329107	+	0.944293i	$0.606748\pi$
$908$	0	0
$909$	−2.05829e6	−0.0826221
$910$	0	0
$911$	−6.35223e6	−0.253589	−0.126795	−	0.991929i	$-0.540469\pi$
$911$	−6.35223e6	−0.253589	−0.126795	+	0.991929i	$0.540469\pi$
$912$	0	0
$913$	−4.80799e6	−0.190892
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.22761e6	−0.323110
$918$	0	0
$919$	−4.97426e7	−1.94285	−0.971427	−	0.237340i	$-0.923725\pi$
$919$	−4.97426e7	−1.94285	−0.971427	+	0.237340i	$0.923725\pi$
$920$	0	0
$921$	3.90089e6	0.151536
$922$	0	0
$923$	−6.85867e7	−2.64994
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−7.07308e6	−0.270339
$928$	0	0
$929$	4.86417e6	0.184914	0.0924569	−	0.995717i	$-0.470528\pi$
$929$	4.86417e6	0.184914	0.0924569	+	0.995717i	$0.470528\pi$
$930$	0	0
$931$	−5.61206e7	−2.12201
$932$	0	0
$933$	−2.54444e7	−0.956946
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.02553e7	1.49787	0.748935	−	0.662644i	$-0.230566\pi$
$937$	4.02553e7	1.49787	0.748935	+	0.662644i	$0.230566\pi$
$938$	0	0
$939$	−2.38884e7	−0.884146
$940$	0	0
$941$	4.27399e7	1.57347	0.786737	−	0.617288i	$-0.211769\pi$
$941$	4.27399e7	1.57347	0.786737	+	0.617288i	$0.211769\pi$
$942$	0	0
$943$	4.15469e7	1.52145
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.86980e7	0.677519	0.338759	−	0.940873i	$-0.389993\pi$
$947$	1.86980e7	0.677519	0.338759	+	0.940873i	$0.389993\pi$
$948$	0	0
$949$	1.33491e7	0.481156
$950$	0	0
$951$	9.53128e6	0.341743
$952$	0	0
$953$	1.20455e7	0.429627	0.214813	−	0.976655i	$-0.431086\pi$
$953$	1.20455e7	0.429627	0.214813	+	0.976655i	$0.431086\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	772684.	0.0272723
$958$	0	0
$959$	6.01072e7	2.11048
$960$	0	0
$961$	−2.28642e7	−0.798635
$962$	0	0
$963$	1.69575e6	0.0589244
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.61477e6	0.124312	0.0621562	−	0.998066i	$-0.480202\pi$
$967$	3.61477e6	0.124312	0.0621562	+	0.998066i	$0.480202\pi$
$968$	0	0
$969$	1.11113e7	0.380152
$970$	0	0
$971$	−4.95026e7	−1.68492	−0.842461	−	0.538758i	$-0.818894\pi$
$971$	−4.95026e7	−1.68492	−0.842461	+	0.538758i	$0.818894\pi$
$972$	0	0
$973$	−5.12232e7	−1.73454
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.08480e7	−0.363591	−0.181796	−	0.983336i	$-0.558191\pi$
$977$	−1.08480e7	−0.363591	−0.181796	+	0.983336i	$0.558191\pi$
$978$	0	0
$979$	1.94793e6	0.0649556
$980$	0	0
$981$	−1.12556e7	−0.373419
$982$	0	0
$983$	5.19390e6	0.171439	0.0857195	−	0.996319i	$-0.472681\pi$
$983$	5.19390e6	0.171439	0.0857195	+	0.996319i	$0.472681\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.19834e7	1.04504
$988$	0	0
$989$	2.33986e7	0.760676
$990$	0	0
$991$	2.55274e7	0.825700	0.412850	−	0.910799i	$-0.364533\pi$
$991$	2.55274e7	0.825700	0.412850	+	0.910799i	$0.364533\pi$
$992$	0	0
$993$	−8.26147e6	−0.265879
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.00857e7	−0.321341	−0.160671	−	0.987008i	$-0.551366\pi$
$997$	−1.00857e7	−0.321341	−0.160671	+	0.987008i	$0.551366\pi$
$998$	0	0
$999$	4.52652e7	1.43500

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.a.k.1.6	yes	8
5.2	odd	4		1100.6.b.i.749.6		16
5.3	odd	4		1100.6.b.i.749.11		16
5.4	even	2		1100.6.a.h.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.6.a.h.1.3	✓	8	5.4	even	2
1100.6.a.k.1.6	yes	8	1.1	even	1	trivial
1100.6.b.i.749.6		16	5.2	odd	4
1100.6.b.i.749.11		16	5.3	odd	4

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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)

\(f(q)\)	\(=\)	\(q+9.06056 q^{3} +199.386 q^{7} -160.906 q^{9} +121.000 q^{11} -1002.21 q^{13} -501.456 q^{17} -2445.56 q^{19} +1806.55 q^{21} +3138.18 q^{23} -3659.62 q^{27} +704.793 q^{29} +2401.02 q^{31} +1096.33 q^{33} -12368.9 q^{37} -9080.55 q^{39} +13239.2 q^{41} +7456.12 q^{43} +17704.1 q^{47} +22947.9 q^{49} -4543.47 q^{51} +10713.1 q^{53} -22158.2 q^{57} +47877.9 q^{59} +43316.6 q^{61} -32082.5 q^{63} +34008.0 q^{67} +28433.6 q^{69} +68435.7 q^{71} -13319.7 q^{73} +24125.8 q^{77} -59077.7 q^{79} +5942.09 q^{81} -39735.5 q^{83} +6385.82 q^{87} +16098.6 q^{89} -199826. q^{91} +21754.6 q^{93} +23637.5 q^{97} -19469.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 27 q^{3} + 95 q^{7} + 377 q^{9} + 968 q^{11} + 294 q^{13} + 515 q^{17} - 354 q^{19} - 588 q^{21} + 3073 q^{23} + 8502 q^{27} + 2743 q^{29} + 3768 q^{31} + 3267 q^{33} + 9252 q^{37} - 10027 q^{39}+ \cdots + 45617 q^{99}+O(q^{100}) \) 8 * q + 27 * q^3 + 95 * q^7 + 377 * q^9 + 968 * q^11 + 294 * q^13 + 515 * q^17 - 354 * q^19 - 588 * q^21 + 3073 * q^23 + 8502 * q^27 + 2743 * q^29 + 3768 * q^31 + 3267 * q^33 + 9252 * q^37 - 10027 * q^39 + 8756 * q^41 + 15290 * q^43 + 8516 * q^47 + 27341 * q^49 + 4707 * q^51 + 30475 * q^53 + 1903 * q^57 + 20400 * q^59 + 10511 * q^61 - 63122 * q^63 - 616 * q^67 + 72452 * q^69 + 31862 * q^71 - 37377 * q^73 + 11495 * q^77 - 71679 * q^79 - 7228 * q^81 - 61543 * q^83 + 30217 * q^87 - 14961 * q^89 + 132641 * q^91 - 95881 * q^93 + 186419 * q^97 + 45617 * q^99

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