LMFDB - Embedded newform 441.6.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{249}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 62 $ x^2 - x - 62
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.38987$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.38987 q^{2} +8.83040 q^{4} -38.7291 q^{5} +148.051 q^{8} +247.474 q^{10} +576.390 q^{11} +391.491 q^{13} -1228.60 q^{16} +1329.70 q^{17} +942.474 q^{19} -341.993 q^{20} -3683.05 q^{22} +1632.08 q^{23} -1625.06 q^{25} -2501.58 q^{26} +1463.54 q^{29} -3912.42 q^{31} +3112.95 q^{32} -8496.61 q^{34} -16300.3 q^{37} -6022.28 q^{38} -5733.86 q^{40} +13103.8 q^{41} +14733.5 q^{43} +5089.75 q^{44} -10428.8 q^{46} +6814.52 q^{47} +10383.9 q^{50} +3457.02 q^{52} +2011.34 q^{53} -22323.0 q^{55} -9351.85 q^{58} -51453.1 q^{59} +41097.8 q^{61} +24999.8 q^{62} +19423.8 q^{64} -15162.1 q^{65} +50578.2 q^{67} +11741.8 q^{68} -39970.6 q^{71} -55686.6 q^{73} +104157. q^{74} +8322.42 q^{76} -63151.4 q^{79} +47582.4 q^{80} -83731.7 q^{82} -45572.4 q^{83} -51498.1 q^{85} -94145.1 q^{86} +85334.9 q^{88} -15686.7 q^{89} +14411.9 q^{92} -43543.9 q^{94} -36501.1 q^{95} +3128.49 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 65 q^{4} + 33 q^{5} + 375 q^{8} + 921 q^{10} + 1137 q^{11} + 925 q^{13} - 895 q^{16} + 324 q^{17} + 2311 q^{19} + 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} + 2508 q^{26} + 2217 q^{29}+ \cdots - 9791 q^{97}+O(q^{100}) $ 2 * q + 3 * q^2 + 65 * q^4 + 33 * q^5 + 375 * q^8 + 921 * q^10 + 1137 * q^11 + 925 * q^13 - 895 * q^16 + 324 * q^17 + 2311 * q^19 + 3687 * q^20 + 1581 * q^22 - 1596 * q^23 + 395 * q^25 + 2508 * q^26 + 2217 * q^29 + 4294 * q^31 - 1017 * q^32 - 17940 * q^34 - 19109 * q^37 + 6828 * q^38 + 10545 * q^40 + 12858 * q^41 - 2771 * q^43 + 36579 * q^44 - 40740 * q^46 + 23160 * q^47 + 29352 * q^50 + 33424 * q^52 + 31653 * q^53 + 17889 * q^55 - 2277 * q^58 - 41097 * q^59 + 42052 * q^61 + 102057 * q^62 - 30031 * q^64 + 23106 * q^65 + 30763 * q^67 - 44748 * q^68 - 102096 * q^71 - 28577 * q^73 + 77784 * q^74 + 85192 * q^76 - 18464 * q^79 + 71511 * q^80 - 86040 * q^82 - 61179 * q^83 - 123636 * q^85 - 258510 * q^86 + 212565 * q^88 - 29322 * q^89 - 166908 * q^92 + 109938 * q^94 + 61662 * q^95 - 9791 * q^97

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 $

$2$	−6.38987	−1.12958	−0.564790	−	0.825235i	$-0.691043\pi$
$2$	−6.38987	−1.12958	−0.564790	+	0.825235i	$0.691043\pi$
$3$	0	0
$3$	0	0
$4$	8.83040	0.275950
$5$	−38.7291	−0.692807	−0.346403	−	0.938086i	$-0.612597\pi$
$5$	−38.7291	−0.692807	−0.346403	+	0.938086i	$0.612597\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	148.051	0.817872
$9$	0	0
$10$	247.474	0.782580
$11$	576.390	1.43627	0.718133	−	0.695906i	$-0.244997\pi$
$11$	576.390	1.43627	0.718133	+	0.695906i	$0.244997\pi$
$12$	0	0
$13$	391.491	0.642486	0.321243	−	0.946997i	$-0.395899\pi$
$13$	391.491	0.642486	0.321243	+	0.946997i	$0.395899\pi$
$14$	0	0
$15$	0	0
$16$	−1228.60	−1.19980
$17$	1329.70	1.11592	0.557958	−	0.829869i	$-0.311585\pi$
$17$	1329.70	1.11592	0.557958	+	0.829869i	$0.311585\pi$
$18$	0	0
$19$	942.474	0.598943	0.299471	−	0.954105i	$-0.403190\pi$
$19$	942.474	0.598943	0.299471	+	0.954105i	$0.403190\pi$
$20$	−341.993	−0.191180
$21$	0	0
$22$	−3683.05	−1.62238
$23$	1632.08	0.643312	0.321656	−	0.946857i	$-0.395761\pi$
$23$	1632.08	0.643312	0.321656	+	0.946857i	$0.395761\pi$
$24$	0	0
$25$	−1625.06	−0.520019
$26$	−2501.58	−0.725739
$27$	0	0
$28$	0	0
$29$	1463.54	0.323155	0.161577	−	0.986860i	$-0.448342\pi$
$29$	1463.54	0.323155	0.161577	+	0.986860i	$0.448342\pi$
$30$	0	0
$31$	−3912.42	−0.731208	−0.365604	−	0.930770i	$-0.619138\pi$
$31$	−3912.42	−0.731208	−0.365604	+	0.930770i	$0.619138\pi$
$32$	3112.95	0.537399
$33$	0	0
$34$	−8496.61	−1.26052
$35$	0	0
$36$	0	0
$37$	−16300.3	−1.95746	−0.978729	−	0.205160i	$-0.934229\pi$
$37$	−16300.3	−1.95746	−0.978729	+	0.205160i	$0.934229\pi$
$38$	−6022.28	−0.676553
$39$	0	0
$40$	−5733.86	−0.566627
$41$	13103.8	1.21741	0.608707	−	0.793395i	$-0.291688\pi$
$41$	13103.8	1.21741	0.608707	+	0.793395i	$0.291688\pi$
$42$	0	0
$43$	14733.5	1.21516	0.607582	−	0.794257i	$-0.292140\pi$
$43$	14733.5	1.21516	0.607582	+	0.794257i	$0.292140\pi$
$44$	5089.75	0.396337
$45$	0	0
$46$	−10428.8	−0.726672
$47$	6814.52	0.449977	0.224989	−	0.974361i	$-0.427765\pi$
$47$	6814.52	0.449977	0.224989	+	0.974361i	$0.427765\pi$
$48$	0	0
$49$	0	0
$50$	10383.9	0.587403
$51$	0	0
$52$	3457.02	0.177294
$53$	2011.34	0.0983550	0.0491775	−	0.998790i	$-0.484340\pi$
$53$	2011.34	0.0983550	0.0491775	+	0.998790i	$0.484340\pi$
$54$	0	0
$55$	−22323.0	−0.995054
$56$	0	0
$57$	0	0
$58$	−9351.85	−0.365029
$59$	−51453.1	−1.92434	−0.962170	−	0.272451i	$-0.912166\pi$
$59$	−51453.1	−1.92434	−0.962170	+	0.272451i	$0.912166\pi$
$60$	0	0
$61$	41097.8	1.41415	0.707073	−	0.707141i	$-0.250015\pi$
$61$	41097.8	1.41415	0.707073	+	0.707141i	$0.250015\pi$
$62$	24999.8	0.825958
$63$	0	0
$64$	19423.8	0.592766
$65$	−15162.1	−0.445119
$66$	0	0
$67$	50578.2	1.37650	0.688250	−	0.725473i	$-0.258379\pi$
$67$	50578.2	1.37650	0.688250	+	0.725473i	$0.258379\pi$
$68$	11741.8	0.307937
$69$	0	0
$70$	0	0
$71$	−39970.6	−0.941012	−0.470506	−	0.882397i	$-0.655929\pi$
$71$	−39970.6	−0.941012	−0.470506	+	0.882397i	$0.655929\pi$
$72$	0	0
$73$	−55686.6	−1.22305	−0.611524	−	0.791226i	$-0.709443\pi$
$73$	−55686.6	−1.22305	−0.611524	+	0.791226i	$0.709443\pi$
$74$	104157.	2.21110
$75$	0	0
$76$	8322.42	0.165278
$77$	0	0
$78$	0	0
$79$	−63151.4	−1.13845	−0.569226	−	0.822181i	$-0.692757\pi$
$79$	−63151.4	−1.13845	−0.569226	+	0.822181i	$0.692757\pi$
$80$	47582.4	0.831231
$81$	0	0
$82$	−83731.7	−1.37517
$83$	−45572.4	−0.726116	−0.363058	−	0.931766i	$-0.618267\pi$
$83$	−45572.4	−0.726116	−0.363058	+	0.931766i	$0.618267\pi$
$84$	0	0
$85$	−51498.1	−0.773114
$86$	−94145.1	−1.37262
$87$	0	0
$88$	85334.9	1.17468
$89$	−15686.7	−0.209921	−0.104961	−	0.994476i	$-0.533472\pi$
$89$	−15686.7	−0.209921	−0.104961	+	0.994476i	$0.533472\pi$
$90$	0	0
$91$	0	0
$92$	14411.9	0.177522
$93$	0	0
$94$	−43543.9	−0.508285
$95$	−36501.1	−0.414951
$96$	0	0
$97$	3128.49	0.0337603	0.0168801	−	0.999858i	$-0.494627\pi$
$97$	3128.49	0.0337603	0.0168801	+	0.999858i	$0.494627\pi$
$98$	0	0
$99$	0	0
$100$	−14349.9	−0.143499
$101$	169011.	1.64858	0.824292	−	0.566164i	$-0.191573\pi$
$101$	169011.	1.64858	0.824292	+	0.566164i	$0.191573\pi$
$102$	0	0
$103$	112820.	1.04784	0.523918	−	0.851769i	$-0.324470\pi$
$103$	112820.	1.04784	0.523918	+	0.851769i	$0.324470\pi$
$104$	57960.5	0.525471
$105$	0	0
$106$	−12852.2	−0.111100
$107$	22309.5	0.188378	0.0941890	−	0.995554i	$-0.469974\pi$
$107$	22309.5	0.188378	0.0941890	+	0.995554i	$0.469974\pi$
$108$	0	0
$109$	−83819.5	−0.675739	−0.337869	−	0.941193i	$-0.609706\pi$
$109$	−83819.5	−0.675739	−0.337869	+	0.941193i	$0.609706\pi$
$110$	142641.	1.12399
$111$	0	0
$112$	0	0
$113$	40928.4	0.301529	0.150764	−	0.988570i	$-0.451826\pi$
$113$	40928.4	0.301529	0.150764	+	0.988570i	$0.451826\pi$
$114$	0	0
$115$	−63208.9	−0.445691
$116$	12923.7	0.0891746
$117$	0	0
$118$	328779.	2.17369
$119$	0	0
$120$	0	0
$121$	171174.	1.06286
$122$	−262610.	−1.59739
$123$	0	0
$124$	−34548.2	−0.201777
$125$	183965.	1.05308
$126$	0	0
$127$	83270.1	0.458120	0.229060	−	0.973412i	$-0.426435\pi$
$127$	83270.1	0.458120	0.229060	+	0.973412i	$0.426435\pi$
$128$	−223730.	−1.20698
$129$	0	0
$130$	96883.7	0.502797
$131$	166875.	0.849596	0.424798	−	0.905288i	$-0.360345\pi$
$131$	166875.	0.849596	0.424798	+	0.905288i	$0.360345\pi$
$132$	0	0
$133$	0	0
$134$	−323188.	−1.55487
$135$	0	0
$136$	196863.	0.912676
$137$	−38223.9	−0.173994	−0.0869969	−	0.996209i	$-0.527727\pi$
$137$	−38223.9	−0.173994	−0.0869969	+	0.996209i	$0.527727\pi$
$138$	0	0
$139$	106263.	0.466492	0.233246	−	0.972418i	$-0.425065\pi$
$139$	106263.	0.466492	0.233246	+	0.972418i	$0.425065\pi$
$140$	0	0
$141$	0	0
$142$	255407.	1.06295
$143$	225652.	0.922780
$144$	0	0
$145$	−56681.7	−0.223884
$146$	355830.	1.38153
$147$	0	0
$148$	−143938.	−0.540160
$149$	−192556.	−0.710543	−0.355271	−	0.934763i	$-0.615612\pi$
$149$	−192556.	−0.710543	−0.355271	+	0.934763i	$0.615612\pi$
$150$	0	0
$151$	141699.	0.505735	0.252868	−	0.967501i	$-0.418626\pi$
$151$	141699.	0.505735	0.252868	+	0.967501i	$0.418626\pi$
$152$	139534.	0.489858
$153$	0	0
$154$	0	0
$155$	151524.	0.506586
$156$	0	0
$157$	565771.	1.83186	0.915928	−	0.401342i	$-0.131456\pi$
$157$	565771.	1.83186	0.915928	+	0.401342i	$0.131456\pi$
$158$	403529.	1.28597
$159$	0	0
$160$	−120562.	−0.372314
$161$	0	0
$162$	0	0
$163$	−430201.	−1.26824	−0.634121	−	0.773233i	$-0.718638\pi$
$163$	−430201.	−1.26824	−0.634121	+	0.773233i	$0.718638\pi$
$164$	115712.	0.335946
$165$	0	0
$166$	291201.	0.820206
$167$	240265.	0.666653	0.333327	−	0.942811i	$-0.391829\pi$
$167$	240265.	0.666653	0.333327	+	0.942811i	$0.391829\pi$
$168$	0	0
$169$	−218028.	−0.587212
$170$	329066.	0.873294
$171$	0	0
$172$	130103.	0.335324
$173$	179300.	0.455476	0.227738	−	0.973722i	$-0.426867\pi$
$173$	179300.	0.455476	0.227738	+	0.973722i	$0.426867\pi$
$174$	0	0
$175$	0	0
$176$	−708151.	−1.72323
$177$	0	0
$178$	100236.	0.237123
$179$	575559.	1.34263	0.671317	−	0.741170i	$-0.265729\pi$
$179$	575559.	1.34263	0.671317	+	0.741170i	$0.265729\pi$
$180$	0	0
$181$	581006.	1.31821	0.659105	−	0.752051i	$-0.270935\pi$
$181$	581006.	1.31821	0.659105	+	0.752051i	$0.270935\pi$
$182$	0	0
$183$	0	0
$184$	241630.	0.526147
$185$	631297.	1.35614
$186$	0	0
$187$	766426.	1.60275
$188$	60174.9	0.124171
$189$	0	0
$190$	233237.	0.468721
$191$	660560.	1.31017	0.655087	−	0.755554i	$-0.272632\pi$
$191$	660560.	1.31017	0.655087	+	0.755554i	$0.272632\pi$
$192$	0	0
$193$	−557310.	−1.07697	−0.538485	−	0.842635i	$-0.681003\pi$
$193$	−557310.	−1.07697	−0.538485	+	0.842635i	$0.681003\pi$
$194$	−19990.7	−0.0381349
$195$	0	0
$196$	0	0
$197$	761400.	1.39781	0.698904	−	0.715216i	$-0.253672\pi$
$197$	761400.	1.39781	0.698904	+	0.715216i	$0.253672\pi$
$198$	0	0
$199$	−135860.	−0.243197	−0.121598	−	0.992579i	$-0.538802\pi$
$199$	−135860.	−0.243197	−0.121598	+	0.992579i	$0.538802\pi$
$200$	−240591.	−0.425309
$201$	0	0
$202$	−1.07996e6	−1.86221
$203$	0	0
$204$	0	0
$205$	−507499.	−0.843433
$206$	−720906.	−1.18361
$207$	0	0
$208$	−480985.	−0.770856
$209$	543232.	0.860240
$210$	0	0
$211$	−991157.	−1.53263	−0.766313	−	0.642467i	$-0.777911\pi$
$211$	−991157.	−1.53263	−0.766313	+	0.642467i	$0.777911\pi$
$212$	17761.0	0.0271411
$213$	0	0
$214$	−142555.	−0.212788
$215$	−570615.	−0.841873
$216$	0	0
$217$	0	0
$218$	535596.	0.763301
$219$	0	0
$220$	−197121.	−0.274585
$221$	520566.	0.716960
$222$	0	0
$223$	543344.	0.731666	0.365833	−	0.930681i	$-0.380784\pi$
$223$	543344.	0.731666	0.365833	+	0.930681i	$0.380784\pi$
$224$	0	0
$225$	0	0
$226$	−261527.	−0.340601
$227$	−16.3341	−2.10393e−5 0	−1.05196e−5	−	1.00000i	$-0.500003\pi$
$227$	−16.3341	−2.10393e−5 0	−1.05196e−5		1.00000i	$0.500003\pi$
$228$	0	0
$229$	−77379.0	−0.0975067	−0.0487534	−	0.998811i	$-0.515525\pi$
$229$	−77379.0	−0.0975067	−0.0487534	+	0.998811i	$0.515525\pi$
$230$	403896.	0.503443
$231$	0	0
$232$	216679.	0.264299
$233$	103657.	0.125086	0.0625432	−	0.998042i	$-0.480079\pi$
$233$	103657.	0.125086	0.0625432	+	0.998042i	$0.480079\pi$
$234$	0	0
$235$	−263920.	−0.311747
$236$	−454351.	−0.531021
$237$	0	0
$238$	0	0
$239$	−689109.	−0.780356	−0.390178	−	0.920739i	$-0.627587\pi$
$239$	−689109.	−0.780356	−0.390178	+	0.920739i	$0.627587\pi$
$240$	0	0
$241$	220296.	0.244323	0.122161	−	0.992510i	$-0.461017\pi$
$241$	220296.	0.244323	0.122161	+	0.992510i	$0.461017\pi$
$242$	−1.09378e6	−1.20058
$243$	0	0
$244$	362910.	0.390234
$245$	0	0
$246$	0	0
$247$	368970.	0.384812
$248$	−579236.	−0.598035
$249$	0	0
$250$	−1.17551e6	−1.18954
$251$	−1.43641e6	−1.43912	−0.719558	−	0.694433i	$-0.755655\pi$
$251$	−1.43641e6	−1.43912	−0.719558	+	0.694433i	$0.755655\pi$
$252$	0	0
$253$	940714.	0.923966
$254$	−532085.	−0.517483
$255$	0	0
$256$	808042.	0.770609
$257$	909197.	0.858668	0.429334	−	0.903146i	$-0.358748\pi$
$257$	909197.	0.858668	0.429334	+	0.903146i	$0.358748\pi$
$258$	0	0
$259$	0	0
$260$	−133887.	−0.122830
$261$	0	0
$262$	−1.06631e6	−0.959687
$263$	−749057.	−0.667768	−0.333884	−	0.942614i	$-0.608359\pi$
$263$	−749057.	−0.667768	−0.333884	+	0.942614i	$0.608359\pi$
$264$	0	0
$265$	−77897.4	−0.0681410
$266$	0	0
$267$	0	0
$268$	446626.	0.379845
$269$	−669945.	−0.564493	−0.282246	−	0.959342i	$-0.591080\pi$
$269$	−669945.	−0.564493	−0.282246	+	0.959342i	$0.591080\pi$
$270$	0	0
$271$	540659.	0.447199	0.223599	−	0.974681i	$-0.428219\pi$
$271$	540659.	0.447199	0.223599	+	0.974681i	$0.428219\pi$
$272$	−1.63367e6	−1.33888
$273$	0	0
$274$	244246.	0.196540
$275$	−936668.	−0.746885
$276$	0	0
$277$	−401910.	−0.314723	−0.157362	−	0.987541i	$-0.550299\pi$
$277$	−401910.	−0.314723	−0.157362	+	0.987541i	$0.550299\pi$
$278$	−679005.	−0.526940
$279$	0	0
$280$	0	0
$281$	429139.	0.324214	0.162107	−	0.986773i	$-0.448171\pi$
$281$	429139.	0.324214	0.162107	+	0.986773i	$0.448171\pi$
$282$	0	0
$283$	−340927.	−0.253044	−0.126522	−	0.991964i	$-0.540381\pi$
$283$	−340927.	−0.253044	−0.126522	+	0.991964i	$0.540381\pi$
$284$	−352957.	−0.259672
$285$	0	0
$286$	−1.44188e6	−1.04235
$287$	0	0
$288$	0	0
$289$	348246.	0.245268
$290$	362188.	0.252895
$291$	0	0
$292$	−491735.	−0.337500
$293$	388847.	0.264612	0.132306	−	0.991209i	$-0.457762\pi$
$293$	388847.	0.264612	0.132306	+	0.991209i	$0.457762\pi$
$294$	0	0
$295$	1.99273e6	1.33319
$296$	−2.41328e6	−1.60095
$297$	0	0
$298$	1.23040e6	0.802615
$299$	638945.	0.413319
$300$	0	0
$301$	0	0
$302$	−905435.	−0.571268
$303$	0	0
$304$	−1.15792e6	−0.718612
$305$	−1.59168e6	−0.979730
$306$	0	0
$307$	−2.35747e6	−1.42758	−0.713789	−	0.700361i	$-0.753022\pi$
$307$	−2.35747e6	−1.42758	−0.713789	+	0.700361i	$0.753022\pi$
$308$	0	0
$309$	0	0
$310$	−968220.	−0.572229
$311$	1.43663e6	0.842254	0.421127	−	0.907002i	$-0.361635\pi$
$311$	1.43663e6	0.842254	0.421127	+	0.907002i	$0.361635\pi$
$312$	0	0
$313$	−822800.	−0.474715	−0.237358	−	0.971422i	$-0.576281\pi$
$313$	−822800.	−0.474715	−0.237358	+	0.971422i	$0.576281\pi$
$314$	−3.61520e6	−2.06923
$315$	0	0
$316$	−557652.	−0.314156
$317$	1.76693e6	0.987580	0.493790	−	0.869581i	$-0.335611\pi$
$317$	1.76693e6	0.987580	0.493790	+	0.869581i	$0.335611\pi$
$318$	0	0
$319$	843572.	0.464136
$320$	−752264.	−0.410672
$321$	0	0
$322$	0	0
$323$	1.25321e6	0.668370
$324$	0	0
$325$	−636196.	−0.334105
$326$	2.74893e6	1.43258
$327$	0	0
$328$	1.94003e6	0.995689
$329$	0	0
$330$	0	0
$331$	−3052.27	−0.00153128	−0.000765638	−	1.00000i	$-0.500244\pi$
$331$	−3052.27	−0.00153128	−0.000765638		1.00000i	$0.500244\pi$
$332$	−402422.	−0.200372
$333$	0	0
$334$	−1.53526e6	−0.753038
$335$	−1.95885e6	−0.953649
$336$	0	0
$337$	2.02939e6	0.973398	0.486699	−	0.873570i	$-0.338201\pi$
$337$	2.02939e6	0.973398	0.486699	+	0.873570i	$0.338201\pi$
$338$	1.39317e6	0.663302
$339$	0	0
$340$	−454748.	−0.213341
$341$	−2.25508e6	−1.05021
$342$	0	0
$343$	0	0
$344$	2.18130e6	0.993848
$345$	0	0
$346$	−1.14570e6	−0.514496
$347$	−3.78218e6	−1.68624	−0.843119	−	0.537727i	$-0.819283\pi$
$347$	−3.78218e6	−1.68624	−0.843119	+	0.537727i	$0.819283\pi$
$348$	0	0
$349$	291147.	0.127953	0.0639763	−	0.997951i	$-0.479622\pi$
$349$	291147.	0.127953	0.0639763	+	0.997951i	$0.479622\pi$
$350$	0	0
$351$	0	0
$352$	1.79427e6	0.771848
$353$	−385076.	−0.164479	−0.0822394	−	0.996613i	$-0.526207\pi$
$353$	−385076.	−0.164479	−0.0822394	+	0.996613i	$0.526207\pi$
$354$	0	0
$355$	1.54803e6	0.651939
$356$	−138520.	−0.0579277
$357$	0	0
$358$	−3.67775e6	−1.51661
$359$	3.23014e6	1.32277	0.661385	−	0.750046i	$-0.269969\pi$
$359$	3.23014e6	1.32277	0.661385	+	0.750046i	$0.269969\pi$
$360$	0	0
$361$	−1.58784e6	−0.641268
$362$	−3.71255e6	−1.48902
$363$	0	0
$364$	0	0
$365$	2.15669e6	0.847336
$366$	0	0
$367$	−479559.	−0.185856	−0.0929280	−	0.995673i	$-0.529623\pi$
$367$	−479559.	−0.185856	−0.0929280	+	0.995673i	$0.529623\pi$
$368$	−2.00517e6	−0.771847
$369$	0	0
$370$	−4.03390e6	−1.53187
$371$	0	0
$372$	0	0
$373$	872666.	0.324770	0.162385	−	0.986727i	$-0.448081\pi$
$373$	872666.	0.324770	0.162385	+	0.986727i	$0.448081\pi$
$374$	−4.89736e6	−1.81043
$375$	0	0
$376$	1.00889e6	0.368024
$377$	572965.	0.207622
$378$	0	0
$379$	−2.43493e6	−0.870742	−0.435371	−	0.900251i	$-0.643383\pi$
$379$	−2.43493e6	−0.870742	−0.435371	+	0.900251i	$0.643383\pi$
$380$	−322320.	−0.114506
$381$	0	0
$382$	−4.22089e6	−1.47995
$383$	3.61169e6	1.25809	0.629047	−	0.777367i	$-0.283445\pi$
$383$	3.61169e6	1.25809	0.629047	+	0.777367i	$0.283445\pi$
$384$	0	0
$385$	0	0
$386$	3.56114e6	1.21652
$387$	0	0
$388$	27625.9	0.00931615
$389$	175232.	0.0587138	0.0293569	−	0.999569i	$-0.490654\pi$
$389$	175232.	0.0587138	0.0293569	+	0.999569i	$0.490654\pi$
$390$	0	0
$391$	2.17018e6	0.717882
$392$	0	0
$393$	0	0
$394$	−4.86525e6	−1.57893
$395$	2.44579e6	0.788727
$396$	0	0
$397$	1.88515e6	0.600303	0.300152	−	0.953892i	$-0.402963\pi$
$397$	1.88515e6	0.600303	0.300152	+	0.953892i	$0.402963\pi$
$398$	868126.	0.274710
$399$	0	0
$400$	1.99654e6	0.623920
$401$	−1.33983e6	−0.416091	−0.208046	−	0.978119i	$-0.566710\pi$
$401$	−1.33983e6	−0.416091	−0.208046	+	0.978119i	$0.566710\pi$
$402$	0	0
$403$	−1.53168e6	−0.469791
$404$	1.49243e6	0.454927
$405$	0	0
$406$	0	0
$407$	−9.39535e6	−2.81143
$408$	0	0
$409$	6.58628e6	1.94685	0.973423	−	0.229013i	$-0.0735499\pi$
$409$	6.58628e6	1.94685	0.973423	+	0.229013i	$0.0735499\pi$
$410$	3.24285e6	0.952725
$411$	0	0
$412$	996247.	0.289150
$413$	0	0
$414$	0	0
$415$	1.76497e6	0.503058
$416$	1.21869e6	0.345271
$417$	0	0
$418$	−3.47118e6	−0.971710
$419$	6.96869e6	1.93917	0.969585	−	0.244754i	$-0.0787071\pi$
$419$	6.96869e6	1.93917	0.969585	+	0.244754i	$0.0787071\pi$
$420$	0	0
$421$	3.84041e6	1.05602	0.528010	−	0.849238i	$-0.322938\pi$
$421$	3.84041e6	1.05602	0.528010	+	0.849238i	$0.322938\pi$
$422$	6.33336e6	1.73122
$423$	0	0
$424$	297781.	0.0804418
$425$	−2.16084e6	−0.580297
$426$	0	0
$427$	0	0
$428$	197002.	0.0519829
$429$	0	0
$430$	3.64615e6	0.950963
$431$	−3.03636e6	−0.787337	−0.393668	−	0.919253i	$-0.628794\pi$
$431$	−3.03636e6	−0.787337	−0.393668	+	0.919253i	$0.628794\pi$
$432$	0	0
$433$	−941529.	−0.241332	−0.120666	−	0.992693i	$-0.538503\pi$
$433$	−941529.	−0.241332	−0.120666	+	0.992693i	$0.538503\pi$
$434$	0	0
$435$	0	0
$436$	−740160.	−0.186470
$437$	1.53819e6	0.385307
$438$	0	0
$439$	1.34109e6	0.332122	0.166061	−	0.986116i	$-0.446895\pi$
$439$	1.34109e6	0.332122	0.166061	+	0.986116i	$0.446895\pi$
$440$	−3.30494e6	−0.813827
$441$	0	0
$442$	−3.32635e6	−0.809864
$443$	772341.	0.186982	0.0934910	−	0.995620i	$-0.470197\pi$
$443$	772341.	0.186982	0.0934910	+	0.995620i	$0.470197\pi$
$444$	0	0
$445$	607531.	0.145435
$446$	−3.47190e6	−0.826475
$447$	0	0
$448$	0	0
$449$	−2.25684e6	−0.528304	−0.264152	−	0.964481i	$-0.585092\pi$
$449$	−2.25684e6	−0.528304	−0.264152	+	0.964481i	$0.585092\pi$
$450$	0	0
$451$	7.55291e6	1.74853
$452$	361414.	0.0832069
$453$	0	0
$454$	104.373	2.37655e−5 0
$455$	0	0
$456$	0	0
$457$	4.28470e6	0.959688	0.479844	−	0.877354i	$-0.340693\pi$
$457$	4.28470e6	0.959688	0.479844	+	0.877354i	$0.340693\pi$
$458$	494442.	0.110142
$459$	0	0
$460$	−558160.	−0.122988
$461$	3.10462e6	0.680387	0.340193	−	0.940355i	$-0.389507\pi$
$461$	3.10462e6	0.680387	0.340193	+	0.940355i	$0.389507\pi$
$462$	0	0
$463$	−3.53386e6	−0.766121	−0.383060	−	0.923723i	$-0.625130\pi$
$463$	−3.53386e6	−0.766121	−0.383060	+	0.923723i	$0.625130\pi$
$464$	−1.79811e6	−0.387722
$465$	0	0
$466$	−662356.	−0.141295
$467$	−2.72459e6	−0.578109	−0.289054	−	0.957313i	$-0.593341\pi$
$467$	−2.72459e6	−0.578109	−0.289054	+	0.957313i	$0.593341\pi$
$468$	0	0
$469$	0	0
$470$	1.68641e6	0.352143
$471$	0	0
$472$	−7.61767e6	−1.57386
$473$	8.49224e6	1.74530
$474$	0	0
$475$	−1.53158e6	−0.311461
$476$	0	0
$477$	0	0
$478$	4.40331e6	0.881475
$479$	−978685.	−0.194896	−0.0974482	−	0.995241i	$-0.531068\pi$
$479$	−978685.	−0.194896	−0.0974482	+	0.995241i	$0.531068\pi$
$480$	0	0
$481$	−6.38144e6	−1.25764
$482$	−1.40766e6	−0.275982
$483$	0	0
$484$	1.51154e6	0.293296
$485$	−121164.	−0.0233893
$486$	0	0
$487$	3.92744e6	0.750390	0.375195	−	0.926946i	$-0.377576\pi$
$487$	3.92744e6	0.750390	0.375195	+	0.926946i	$0.377576\pi$
$488$	6.08456e6	1.15659
$489$	0	0
$490$	0	0
$491$	−2.63241e6	−0.492777	−0.246388	−	0.969171i	$-0.579244\pi$
$491$	−2.63241e6	−0.492777	−0.246388	+	0.969171i	$0.579244\pi$
$492$	0	0
$493$	1.94607e6	0.360614
$494$	−2.35767e6	−0.434676
$495$	0	0
$496$	4.80678e6	0.877305
$497$	0	0
$498$	0	0
$499$	2.12544e6	0.382118	0.191059	−	0.981579i	$-0.438808\pi$
$499$	2.12544e6	0.382118	0.191059	+	0.981579i	$0.438808\pi$
$500$	1.62449e6	0.290597
$501$	0	0
$502$	9.17850e6	1.62560
$503$	−2.60929e6	−0.459835	−0.229917	−	0.973210i	$-0.573846\pi$
$503$	−2.60929e6	−0.459835	−0.229917	+	0.973210i	$0.573846\pi$
$504$	0	0
$505$	−6.54564e6	−1.14215
$506$	−6.01104e6	−1.04369
$507$	0	0
$508$	735308.	0.126418
$509$	1.00182e7	1.71394	0.856970	−	0.515366i	$-0.172344\pi$
$509$	1.00182e7	1.71394	0.856970	+	0.515366i	$0.172344\pi$
$510$	0	0
$511$	0	0
$512$	1.99607e6	0.336512
$513$	0	0
$514$	−5.80965e6	−0.969933
$515$	−4.36942e6	−0.725948
$516$	0	0
$517$	3.92782e6	0.646287
$518$	0	0
$519$	0	0
$520$	−2.24476e6	−0.364050
$521$	4.17941e6	0.674560	0.337280	−	0.941404i	$-0.390493\pi$
$521$	4.17941e6	0.674560	0.337280	+	0.941404i	$0.390493\pi$
$522$	0	0
$523$	3.60525e6	0.576343	0.288172	−	0.957579i	$-0.406953\pi$
$523$	3.60525e6	0.576343	0.288172	+	0.957579i	$0.406953\pi$
$524$	1.47357e6	0.234446
$525$	0	0
$526$	4.78637e6	0.754297
$527$	−5.20234e6	−0.815967
$528$	0	0
$529$	−3.77266e6	−0.586150
$530$	497754.	0.0769707
$531$	0	0
$532$	0	0
$533$	5.13003e6	0.782172
$534$	0	0
$535$	−864025.	−0.130509
$536$	7.48814e6	1.12580
$537$	0	0
$538$	4.28086e6	0.637640
$539$	0	0
$540$	0	0
$541$	−6.68166e6	−0.981501	−0.490751	−	0.871300i	$-0.663277\pi$
$541$	−6.68166e6	−0.981501	−0.490751	+	0.871300i	$0.663277\pi$
$542$	−3.45474e6	−0.505146
$543$	0	0
$544$	4.13929e6	0.599692
$545$	3.24625e6	0.468156
$546$	0	0
$547$	8.69076e6	1.24191	0.620954	−	0.783847i	$-0.286745\pi$
$547$	8.69076e6	1.24191	0.620954	+	0.783847i	$0.286745\pi$
$548$	−337532.	−0.0480136
$549$	0	0
$550$	5.98518e6	0.843666
$551$	1.37935e6	0.193551
$552$	0	0
$553$	0	0
$554$	2.56815e6	0.355505
$555$	0	0
$556$	938342.	0.128728
$557$	−6.24742e6	−0.853223	−0.426612	−	0.904435i	$-0.640293\pi$
$557$	−6.24742e6	−0.853223	−0.426612	+	0.904435i	$0.640293\pi$
$558$	0	0
$559$	5.76803e6	0.780725
$560$	0	0
$561$	0	0
$562$	−2.74214e6	−0.366226
$563$	1.19045e7	1.58285	0.791423	−	0.611269i	$-0.209341\pi$
$563$	1.19045e7	1.58285	0.791423	+	0.611269i	$0.209341\pi$
$564$	0	0
$565$	−1.58512e6	−0.208901
$566$	2.17848e6	0.285833
$567$	0	0
$568$	−5.91768e6	−0.769627
$569$	−21414.4	−0.00277284	−0.00138642	−	0.999999i	$-0.500441\pi$
$569$	−21414.4	−0.00277284	−0.00138642	+	0.999999i	$0.500441\pi$
$570$	0	0
$571$	7.11647e6	0.913428	0.456714	−	0.889614i	$-0.349026\pi$
$571$	7.11647e6	0.913428	0.456714	+	0.889614i	$0.349026\pi$
$572$	1.99259e6	0.254641
$573$	0	0
$574$	0	0
$575$	−2.65223e6	−0.334534
$576$	0	0
$577$	−1.06652e7	−1.33361	−0.666805	−	0.745232i	$-0.732339\pi$
$577$	−1.06652e7	−1.33361	−0.666805	+	0.745232i	$0.732339\pi$
$578$	−2.22524e6	−0.277050
$579$	0	0
$580$	−500522.	−0.0617808
$581$	0	0
$582$	0	0
$583$	1.15932e6	0.141264
$584$	−8.24444e6	−1.00030
$585$	0	0
$586$	−2.48468e6	−0.298901
$587$	1.30101e7	1.55843	0.779213	−	0.626759i	$-0.215619\pi$
$587$	1.30101e7	1.55843	0.779213	+	0.626759i	$0.215619\pi$
$588$	0	0
$589$	−3.68735e6	−0.437952
$590$	−1.27333e7	−1.50595
$591$	0	0
$592$	2.00265e7	2.34856
$593$	−4.26086e6	−0.497578	−0.248789	−	0.968558i	$-0.580033\pi$
$593$	−4.26086e6	−0.497578	−0.248789	+	0.968558i	$0.580033\pi$
$594$	0	0
$595$	0	0
$596$	−1.70034e6	−0.196074
$597$	0	0
$598$	−4.08277e6	−0.466877
$599$	1.37958e7	1.57101	0.785507	−	0.618853i	$-0.212402\pi$
$599$	1.37958e7	1.57101	0.785507	+	0.618853i	$0.212402\pi$
$600$	0	0
$601$	4.99695e6	0.564311	0.282155	−	0.959369i	$-0.408951\pi$
$601$	4.99695e6	0.564311	0.282155	+	0.959369i	$0.408951\pi$
$602$	0	0
$603$	0	0
$604$	1.25126e6	0.139558
$605$	−6.62942e6	−0.736355
$606$	0	0
$607$	−3.04946e6	−0.335932	−0.167966	−	0.985793i	$-0.553720\pi$
$607$	−3.04946e6	−0.335932	−0.167966	+	0.985793i	$0.553720\pi$
$608$	2.93387e6	0.321871
$609$	0	0
$610$	1.01706e7	1.10668
$611$	2.66782e6	0.289104
$612$	0	0
$613$	−7.03625e6	−0.756293	−0.378147	−	0.925746i	$-0.623438\pi$
$613$	−7.03625e6	−0.756293	−0.378147	+	0.925746i	$0.623438\pi$
$614$	1.50639e7	1.61256
$615$	0	0
$616$	0	0
$617$	−1.00066e7	−1.05822	−0.529108	−	0.848554i	$-0.677473\pi$
$617$	−1.00066e7	−1.05822	−0.529108	+	0.848554i	$0.677473\pi$
$618$	0	0
$619$	−6.55067e6	−0.687161	−0.343581	−	0.939123i	$-0.611640\pi$
$619$	−6.55067e6	−0.687161	−0.343581	+	0.939123i	$0.611640\pi$
$620$	1.33802e6	0.139792
$621$	0	0
$622$	−9.17986e6	−0.951393
$623$	0	0
$624$	0	0
$625$	−2.04650e6	−0.209561
$626$	5.25758e6	0.536229
$627$	0	0
$628$	4.99598e6	0.505501
$629$	−2.16746e7	−2.18436
$630$	0	0
$631$	2.22672e6	0.222635	0.111317	−	0.993785i	$-0.464493\pi$
$631$	2.22672e6	0.222635	0.111317	+	0.993785i	$0.464493\pi$
$632$	−9.34960e6	−0.931109
$633$	0	0
$634$	−1.12905e7	−1.11555
$635$	−3.22497e6	−0.317389
$636$	0	0
$637$	0	0
$638$	−5.39031e6	−0.524279
$639$	0	0
$640$	8.66484e6	0.836201
$641$	1.59340e7	1.53172	0.765859	−	0.643008i	$-0.222314\pi$
$641$	1.59340e7	1.53172	0.765859	+	0.643008i	$0.222314\pi$
$642$	0	0
$643$	−1.49933e7	−1.43011	−0.715056	−	0.699067i	$-0.753599\pi$
$643$	−1.49933e7	−1.43011	−0.715056	+	0.699067i	$0.753599\pi$
$644$	0	0
$645$	0	0
$646$	−8.00783e6	−0.754977
$647$	−1.29805e7	−1.21908	−0.609540	−	0.792755i	$-0.708646\pi$
$647$	−1.29805e7	−1.21908	−0.609540	+	0.792755i	$0.708646\pi$
$648$	0	0
$649$	−2.96571e7	−2.76386
$650$	4.06521e6	0.377398
$651$	0	0
$652$	−3.79885e6	−0.349972
$653$	1.65492e7	1.51878	0.759389	−	0.650637i	$-0.225498\pi$
$653$	1.65492e7	1.51878	0.759389	+	0.650637i	$0.225498\pi$
$654$	0	0
$655$	−6.46291e6	−0.588606
$656$	−1.60993e7	−1.46066
$657$	0	0
$658$	0	0
$659$	−5.86879e6	−0.526423	−0.263212	−	0.964738i	$-0.584782\pi$
$659$	−5.86879e6	−0.526423	−0.263212	+	0.964738i	$0.584782\pi$
$660$	0	0
$661$	7.27687e6	0.647800	0.323900	−	0.946091i	$-0.395006\pi$
$661$	7.27687e6	0.647800	0.323900	+	0.946091i	$0.395006\pi$
$662$	19503.6	0.00172970
$663$	0	0
$664$	−6.74702e6	−0.593870
$665$	0	0
$666$	0	0
$667$	2.38862e6	0.207889
$668$	2.12164e6	0.183963
$669$	0	0
$670$	1.25168e7	1.07722
$671$	2.36884e7	2.03109
$672$	0	0
$673$	−1.82417e7	−1.55248	−0.776241	−	0.630437i	$-0.782876\pi$
$673$	−1.82417e7	−1.55248	−0.776241	+	0.630437i	$0.782876\pi$
$674$	−1.29675e7	−1.09953
$675$	0	0
$676$	−1.92527e6	−0.162041
$677$	7.76406e6	0.651054	0.325527	−	0.945533i	$-0.394458\pi$
$677$	7.76406e6	0.651054	0.325527	+	0.945533i	$0.394458\pi$
$678$	0	0
$679$	0	0
$680$	−7.62432e6	−0.632308
$681$	0	0
$682$	1.44096e7	1.18629
$683$	−8.56324e6	−0.702403	−0.351201	−	0.936300i	$-0.614227\pi$
$683$	−8.56324e6	−0.702403	−0.351201	+	0.936300i	$0.614227\pi$
$684$	0	0
$685$	1.48038e6	0.120544
$686$	0	0
$687$	0	0
$688$	−1.81015e7	−1.45796
$689$	787423.	0.0631917
$690$	0	0
$691$	1.64509e7	1.31068	0.655338	−	0.755336i	$-0.272526\pi$
$691$	1.64509e7	1.31068	0.655338	+	0.755336i	$0.272526\pi$
$692$	1.58329e6	0.125689
$693$	0	0
$694$	2.41677e7	1.90474
$695$	−4.11546e6	−0.323189
$696$	0	0
$697$	1.74242e7	1.35853
$698$	−1.86039e6	−0.144533
$699$	0	0
$700$	0	0
$701$	1.66928e7	1.28302	0.641512	−	0.767113i	$-0.278307\pi$
$701$	1.66928e7	1.28302	0.641512	+	0.767113i	$0.278307\pi$
$702$	0	0
$703$	−1.53626e7	−1.17240
$704$	1.11957e7	0.851370
$705$	0	0
$706$	2.46059e6	0.185792
$707$	0	0
$708$	0	0
$709$	5.61779e6	0.419711	0.209855	−	0.977732i	$-0.432701\pi$
$709$	5.61779e6	0.419711	0.209855	+	0.977732i	$0.432701\pi$
$710$	−9.89167e6	−0.736417
$711$	0	0
$712$	−2.32242e6	−0.171689
$713$	−6.38537e6	−0.470395
$714$	0	0
$715$	−8.73927e6	−0.639308
$716$	5.08242e6	0.370500
$717$	0	0
$718$	−2.06401e7	−1.49417
$719$	−1.03718e7	−0.748227	−0.374113	−	0.927383i	$-0.622053\pi$
$719$	−1.03718e7	−0.748227	−0.374113	+	0.927383i	$0.622053\pi$
$720$	0	0
$721$	0	0
$722$	1.01461e7	0.724363
$723$	0	0
$724$	5.13052e6	0.363760
$725$	−2.37835e6	−0.168047
$726$	0	0
$727$	1.15369e7	0.809565	0.404783	−	0.914413i	$-0.367347\pi$
$727$	1.15369e7	0.809565	0.404783	+	0.914413i	$0.367347\pi$
$728$	0	0
$729$	0	0
$730$	−1.37810e7	−0.957134
$731$	1.95911e7	1.35602
$732$	0	0
$733$	1.49470e7	1.02753	0.513764	−	0.857932i	$-0.328251\pi$
$733$	1.49470e7	1.02753	0.513764	+	0.857932i	$0.328251\pi$
$734$	3.06432e6	0.209939
$735$	0	0
$736$	5.08058e6	0.345715
$737$	2.91528e7	1.97702
$738$	0	0
$739$	9.01364e6	0.607140	0.303570	−	0.952809i	$-0.401821\pi$
$739$	9.01364e6	0.607140	0.303570	+	0.952809i	$0.401821\pi$
$740$	5.57460e6	0.374227
$741$	0	0
$742$	0	0
$743$	2.10239e7	1.39714	0.698571	−	0.715541i	$-0.253820\pi$
$743$	2.10239e7	1.39714	0.698571	+	0.715541i	$0.253820\pi$
$744$	0	0
$745$	7.45750e6	0.492269
$746$	−5.57622e6	−0.366854
$747$	0	0
$748$	6.76785e6	0.442279
$749$	0	0
$750$	0	0
$751$	4.04219e6	0.261527	0.130764	−	0.991414i	$-0.458257\pi$
$751$	4.04219e6	0.261527	0.130764	+	0.991414i	$0.458257\pi$
$752$	−8.37230e6	−0.539884
$753$	0	0
$754$	−3.66117e6	−0.234526
$755$	−5.48786e6	−0.350377
$756$	0	0
$757$	−1.82059e7	−1.15471	−0.577353	−	0.816495i	$-0.695914\pi$
$757$	−1.82059e7	−1.15471	−0.577353	+	0.816495i	$0.695914\pi$
$758$	1.55589e7	0.983572
$759$	0	0
$760$	−5.40402e6	−0.339377
$761$	1.89200e7	1.18429	0.592146	−	0.805831i	$-0.298281\pi$
$761$	1.89200e7	1.18429	0.592146	+	0.805831i	$0.298281\pi$
$762$	0	0
$763$	0	0
$764$	5.83301e6	0.361542
$765$	0	0
$766$	−2.30782e7	−1.42112
$767$	−2.01434e7	−1.23636
$768$	0	0
$769$	−1.12831e7	−0.688037	−0.344019	−	0.938963i	$-0.611788\pi$
$769$	−1.12831e7	−0.688037	−0.344019	+	0.938963i	$0.611788\pi$
$770$	0	0
$771$	0	0
$772$	−4.92127e6	−0.297190
$773$	3.68565e6	0.221853	0.110926	−	0.993829i	$-0.464618\pi$
$773$	3.68565e6	0.221853	0.110926	+	0.993829i	$0.464618\pi$
$774$	0	0
$775$	6.35791e6	0.380242
$776$	463176.	0.0276116
$777$	0	0
$778$	−1.11971e6	−0.0663219
$779$	1.23500e7	0.729161
$780$	0	0
$781$	−2.30387e7	−1.35154
$782$	−1.38671e7	−0.810905
$783$	0	0
$784$	0	0
$785$	−2.19118e7	−1.26912
$786$	0	0
$787$	−1.40748e7	−0.810039	−0.405019	−	0.914308i	$-0.632735\pi$
$787$	−1.40748e7	−0.810039	−0.405019	+	0.914308i	$0.632735\pi$
$788$	6.72347e6	0.385725
$789$	0	0
$790$	−1.56283e7	−0.890930
$791$	0	0
$792$	0	0
$793$	1.60894e7	0.908569
$794$	−1.20459e7	−0.678090
$795$	0	0
$796$	−1.19970e6	−0.0671102
$797$	1.75191e7	0.976937	0.488469	−	0.872582i	$-0.337556\pi$
$797$	1.75191e7	0.976937	0.488469	+	0.872582i	$0.337556\pi$
$798$	0	0
$799$	9.06127e6	0.502137
$800$	−5.05873e6	−0.279458
$801$	0	0
$802$	8.56134e6	0.470008
$803$	−3.20972e7	−1.75662
$804$	0	0
$805$	0	0
$806$	9.78721e6	0.530666
$807$	0	0
$808$	2.50222e7	1.34833
$809$	−814521.	−0.0437553	−0.0218777	−	0.999761i	$-0.506964\pi$
$809$	−814521.	−0.0437553	−0.0218777	+	0.999761i	$0.506964\pi$
$810$	0	0
$811$	1.26533e7	0.675540	0.337770	−	0.941229i	$-0.390327\pi$
$811$	1.26533e7	0.675540	0.337770	+	0.941229i	$0.390327\pi$
$812$	0	0
$813$	0	0
$814$	6.00350e7	3.17573
$815$	1.66613e7	0.878647
$816$	0	0
$817$	1.38859e7	0.727813
$818$	−4.20854e7	−2.19912
$819$	0	0
$820$	−4.48142e6	−0.232745
$821$	−4.27393e6	−0.221294	−0.110647	−	0.993860i	$-0.535292\pi$
$821$	−4.27393e6	−0.221294	−0.110647	+	0.993860i	$0.535292\pi$
$822$	0	0
$823$	1.70463e7	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$823$	1.70463e7	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$824$	1.67031e7	0.856996
$825$	0	0
$826$	0	0
$827$	−2.60828e6	−0.132614	−0.0663071	−	0.997799i	$-0.521122\pi$
$827$	−2.60828e6	−0.132614	−0.0663071	+	0.997799i	$0.521122\pi$
$828$	0	0
$829$	2.13865e7	1.08082	0.540409	−	0.841403i	$-0.318270\pi$
$829$	2.13865e7	1.08082	0.540409	+	0.841403i	$0.318270\pi$
$830$	−1.12780e7	−0.568244
$831$	0	0
$832$	7.60423e6	0.380844
$833$	0	0
$834$	0	0
$835$	−9.30525e6	−0.461862
$836$	4.79696e6	0.237383
$837$	0	0
$838$	−4.45290e7	−2.19045
$839$	−771393.	−0.0378330	−0.0189165	−	0.999821i	$-0.506022\pi$
$839$	−771393.	−0.0378330	−0.0189165	+	0.999821i	$0.506022\pi$
$840$	0	0
$841$	−1.83692e7	−0.895571
$842$	−2.45397e7	−1.19286
$843$	0	0
$844$	−8.75231e6	−0.422928
$845$	8.44401e6	0.406824
$846$	0	0
$847$	0	0
$848$	−2.47113e6	−0.118006
$849$	0	0
$850$	1.38075e7	0.655492
$851$	−2.66034e7	−1.25926
$852$	0	0
$853$	2.94032e7	1.38364	0.691818	−	0.722072i	$-0.256810\pi$
$853$	2.94032e7	1.38364	0.691818	+	0.722072i	$0.256810\pi$
$854$	0	0
$855$	0	0
$856$	3.30293e6	0.154069
$857$	2.65010e7	1.23257	0.616283	−	0.787525i	$-0.288638\pi$
$857$	2.65010e7	1.23257	0.616283	+	0.787525i	$0.288638\pi$
$858$	0	0
$859$	−3.67519e7	−1.69940	−0.849702	−	0.527264i	$-0.823218\pi$
$859$	−3.67519e7	−1.69940	−0.849702	+	0.527264i	$0.823218\pi$
$860$	−5.03876e6	−0.232315
$861$	0	0
$862$	1.94020e7	0.889360
$863$	−1.18189e7	−0.540196	−0.270098	−	0.962833i	$-0.587056\pi$
$863$	−1.18189e7	−0.540196	−0.270098	+	0.962833i	$0.587056\pi$
$864$	0	0
$865$	−6.94413e6	−0.315557
$866$	6.01625e6	0.272603
$867$	0	0
$868$	0	0
$869$	−3.63998e7	−1.63512
$870$	0	0
$871$	1.98009e7	0.884382
$872$	−1.24095e7	−0.552668
$873$	0	0
$874$	−9.82884e6	−0.435235
$875$	0	0
$876$	0	0
$877$	7.93509e6	0.348380	0.174190	−	0.984712i	$-0.444269\pi$
$877$	7.93509e6	0.348380	0.174190	+	0.984712i	$0.444269\pi$
$878$	−8.56939e6	−0.375158
$879$	0	0
$880$	2.74260e7	1.19387
$881$	−4.20152e7	−1.82375	−0.911877	−	0.410464i	$-0.865367\pi$
$881$	−4.20152e7	−1.82375	−0.911877	+	0.410464i	$0.865367\pi$
$882$	0	0
$883$	2.12461e7	0.917016	0.458508	−	0.888690i	$-0.348384\pi$
$883$	2.12461e7	0.917016	0.458508	+	0.888690i	$0.348384\pi$
$884$	4.59681e6	0.197845
$885$	0	0
$886$	−4.93516e6	−0.211211
$887$	−1.88490e7	−0.804415	−0.402208	−	0.915548i	$-0.631757\pi$
$887$	−1.88490e7	−0.804415	−0.402208	+	0.915548i	$0.631757\pi$
$888$	0	0
$889$	0	0
$890$	−3.88204e6	−0.164280
$891$	0	0
$892$	4.79795e6	0.201903
$893$	6.42251e6	0.269511
$894$	0	0
$895$	−2.22909e7	−0.930186
$896$	0	0
$897$	0	0
$898$	1.44209e7	0.596762
$899$	−5.72600e6	−0.236294
$900$	0	0
$901$	2.67448e6	0.109756
$902$	−4.82621e7	−1.97510
$903$	0	0
$904$	6.05948e6	0.246612
$905$	−2.25018e7	−0.913264
$906$	0	0
$907$	−6.19446e6	−0.250026	−0.125013	−	0.992155i	$-0.539897\pi$
$907$	−6.19446e6	−0.250026	−0.125013	+	0.992155i	$0.539897\pi$
$908$	−144.237	−5.80578e−6 0
$909$	0	0
$910$	0	0
$911$	2.50171e7	0.998712	0.499356	−	0.866397i	$-0.333570\pi$
$911$	2.50171e7	0.998712	0.499356	+	0.866397i	$0.333570\pi$
$912$	0	0
$913$	−2.62674e7	−1.04290
$914$	−2.73787e7	−1.08404
$915$	0	0
$916$	−683288.	−0.0269070
$917$	0	0
$918$	0	0
$919$	1.54992e7	0.605370	0.302685	−	0.953091i	$-0.402117\pi$
$919$	1.54992e7	0.605370	0.302685	+	0.953091i	$0.402117\pi$
$920$	−9.35812e6	−0.364518
$921$	0	0
$922$	−1.98381e7	−0.768551
$923$	−1.56481e7	−0.604587
$924$	0	0
$925$	2.64890e7	1.01791
$926$	2.25809e7	0.865394
$927$	0	0
$928$	4.55594e6	0.173663
$929$	−3.75285e7	−1.42667	−0.713333	−	0.700826i	$-0.752815\pi$
$929$	−3.75285e7	−1.42667	−0.713333	+	0.700826i	$0.752815\pi$
$930$	0	0
$931$	0	0
$932$	915335.	0.0345176
$933$	0	0
$934$	1.74098e7	0.653020
$935$	−2.96830e7	−1.11040
$936$	0	0
$937$	−1.08298e7	−0.402969	−0.201485	−	0.979492i	$-0.564577\pi$
$937$	−1.08298e7	−0.402969	−0.201485	+	0.979492i	$0.564577\pi$
$938$	0	0
$939$	0	0
$940$	−2.33052e6	−0.0860267
$941$	−3.24591e7	−1.19498	−0.597492	−	0.801875i	$-0.703836\pi$
$941$	−3.24591e7	−1.19498	−0.597492	+	0.801875i	$0.703836\pi$
$942$	0	0
$943$	2.13865e7	0.783177
$944$	6.32151e7	2.30883
$945$	0	0
$946$	−5.42643e7	−1.97145
$947$	−7.53877e6	−0.273165	−0.136583	−	0.990629i	$-0.543612\pi$
$947$	−7.53877e6	−0.273165	−0.136583	+	0.990629i	$0.543612\pi$
$948$	0	0
$949$	−2.18008e7	−0.785792
$950$	9.78656e6	0.351821
$951$	0	0
$952$	0	0
$953$	3.01356e7	1.07485	0.537424	−	0.843312i	$-0.319398\pi$
$953$	3.01356e7	1.07485	0.537424	+	0.843312i	$0.319398\pi$
$954$	0	0
$955$	−2.55829e7	−0.907697
$956$	−6.08510e6	−0.215339
$957$	0	0
$958$	6.25366e6	0.220151
$959$	0	0
$960$	0	0
$961$	−1.33221e7	−0.465335
$962$	4.07765e7	1.42060
$963$	0	0
$964$	1.94530e6	0.0674208
$965$	2.15841e7	0.746132
$966$	0	0
$967$	2.88021e6	0.0990509	0.0495255	−	0.998773i	$-0.484229\pi$
$967$	2.88021e6	0.0990509	0.0495255	+	0.998773i	$0.484229\pi$
$968$	2.53425e7	0.869282
$969$	0	0
$970$	774220.	0.0264201
$971$	2.74490e7	0.934283	0.467142	−	0.884183i	$-0.345284\pi$
$971$	2.74490e7	0.934283	0.467142	+	0.884183i	$0.345284\pi$
$972$	0	0
$973$	0	0
$974$	−2.50958e7	−0.847626
$975$	0	0
$976$	−5.04927e7	−1.69669
$977$	5.88524e7	1.97255	0.986274	−	0.165118i	$-0.0528004\pi$
$977$	5.88524e7	1.97255	0.986274	+	0.165118i	$0.0528004\pi$
$978$	0	0
$979$	−9.04164e6	−0.301502
$980$	0	0
$981$	0	0
$982$	1.68208e7	0.556631
$983$	−1.86077e7	−0.614198	−0.307099	−	0.951678i	$-0.599358\pi$
$983$	−1.86077e7	−0.614198	−0.307099	+	0.951678i	$0.599358\pi$
$984$	0	0
$985$	−2.94883e7	−0.968410
$986$	−1.24352e7	−0.407342
$987$	0	0
$988$	3.25815e6	0.106189
$989$	2.40462e7	0.781729
$990$	0	0
$991$	2.47154e7	0.799435	0.399718	−	0.916638i	$-0.369108\pi$
$991$	2.47154e7	0.799435	0.399718	+	0.916638i	$0.369108\pi$
$992$	−1.21792e7	−0.392951
$993$	0	0
$994$	0	0
$995$	5.26172e6	0.168488
$996$	0	0
$997$	−1.79581e7	−0.572167	−0.286084	−	0.958205i	$-0.592354\pi$
$997$	−1.79581e7	−0.572167	−0.286084	+	0.958205i	$0.592354\pi$
$998$	−1.35813e7	−0.431633
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.t.1.1		2
3.2	odd	2		147.6.a.i.1.2		2
7.2	even	3		63.6.e.c.46.2		4
7.4	even	3		63.6.e.c.37.2		4
7.6	odd	2		441.6.a.s.1.1		2
21.2	odd	6		21.6.e.b.4.1	✓	4
21.5	even	6		147.6.e.l.67.1		4
21.11	odd	6		21.6.e.b.16.1	yes	4
21.17	even	6		147.6.e.l.79.1		4
21.20	even	2		147.6.a.k.1.2		2
84.11	even	6		336.6.q.e.289.1		4
84.23	even	6		336.6.q.e.193.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.b.4.1	✓	4	21.2	odd	6
21.6.e.b.16.1	yes	4	21.11	odd	6
63.6.e.c.37.2		4	7.4	even	3
63.6.e.c.46.2		4	7.2	even	3
147.6.a.i.1.2		2	3.2	odd	2
147.6.a.k.1.2		2	21.20	even	2
147.6.e.l.67.1		4	21.5	even	6
147.6.e.l.79.1		4	21.17	even	6
336.6.q.e.193.1		4	84.23	even	6
336.6.q.e.289.1		4	84.11	even	6
441.6.a.s.1.1		2	7.6	odd	2
441.6.a.t.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-6.38987 q^{2} +8.83040 q^{4} -38.7291 q^{5} +148.051 q^{8} +247.474 q^{10} +576.390 q^{11} +391.491 q^{13} -1228.60 q^{16} +1329.70 q^{17} +942.474 q^{19} -341.993 q^{20} -3683.05 q^{22} +1632.08 q^{23} -1625.06 q^{25} -2501.58 q^{26} +1463.54 q^{29} -3912.42 q^{31} +3112.95 q^{32} -8496.61 q^{34} -16300.3 q^{37} -6022.28 q^{38} -5733.86 q^{40} +13103.8 q^{41} +14733.5 q^{43} +5089.75 q^{44} -10428.8 q^{46} +6814.52 q^{47} +10383.9 q^{50} +3457.02 q^{52} +2011.34 q^{53} -22323.0 q^{55} -9351.85 q^{58} -51453.1 q^{59} +41097.8 q^{61} +24999.8 q^{62} +19423.8 q^{64} -15162.1 q^{65} +50578.2 q^{67} +11741.8 q^{68} -39970.6 q^{71} -55686.6 q^{73} +104157. q^{74} +8322.42 q^{76} -63151.4 q^{79} +47582.4 q^{80} -83731.7 q^{82} -45572.4 q^{83} -51498.1 q^{85} -94145.1 q^{86} +85334.9 q^{88} -15686.7 q^{89} +14411.9 q^{92} -43543.9 q^{94} -36501.1 q^{95} +3128.49 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 65 q^{4} + 33 q^{5} + 375 q^{8} + 921 q^{10} + 1137 q^{11} + 925 q^{13} - 895 q^{16} + 324 q^{17} + 2311 q^{19} + 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} + 2508 q^{26} + 2217 q^{29}+ \cdots - 9791 q^{97}+O(q^{100}) \) 2 * q + 3 * q^2 + 65 * q^4 + 33 * q^5 + 375 * q^8 + 921 * q^10 + 1137 * q^11 + 925 * q^13 - 895 * q^16 + 324 * q^17 + 2311 * q^19 + 3687 * q^20 + 1581 * q^22 - 1596 * q^23 + 395 * q^25 + 2508 * q^26 + 2217 * q^29 + 4294 * q^31 - 1017 * q^32 - 17940 * q^34 - 19109 * q^37 + 6828 * q^38 + 10545 * q^40 + 12858 * q^41 - 2771 * q^43 + 36579 * q^44 - 40740 * q^46 + 23160 * q^47 + 29352 * q^50 + 33424 * q^52 + 31653 * q^53 + 17889 * q^55 - 2277 * q^58 - 41097 * q^59 + 42052 * q^61 + 102057 * q^62 - 30031 * q^64 + 23106 * q^65 + 30763 * q^67 - 44748 * q^68 - 102096 * q^71 - 28577 * q^73 + 77784 * q^74 + 85192 * q^76 - 18464 * q^79 + 71511 * q^80 - 86040 * q^82 - 61179 * q^83 - 123636 * q^85 - 258510 * q^86 + 212565 * q^88 - 29322 * q^89 - 166908 * q^92 + 109938 * q^94 + 61662 * q^95 - 9791 * q^97

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