LMFDB - Embedded newform 441.8.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,8,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, 8, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
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:	$137.761796238$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 102x^{2} + 240x + 900 $ x^4 - x^3 - 102x^2 + 240x + 900
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-10.3192$ 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)

$f(q)$	$=$	$q+20.8241 q^{2} +305.642 q^{4} +193.403 q^{5} +3699.23 q^{8} +O(q^{10})$	$q+20.8241 q^{2} +305.642 q^{4} +193.403 q^{5} +3699.23 q^{8} +4027.44 q^{10} -303.944 q^{11} +9891.04 q^{13} +37910.9 q^{16} +12484.5 q^{17} -1123.67 q^{19} +59112.1 q^{20} -6329.35 q^{22} +44719.0 q^{23} -40720.2 q^{25} +205972. q^{26} -51216.6 q^{29} -25633.0 q^{31} +315958. q^{32} +259978. q^{34} -42109.0 q^{37} -23399.5 q^{38} +715443. q^{40} +787100. q^{41} -629737. q^{43} -92898.1 q^{44} +931232. q^{46} +626687. q^{47} -847961. q^{50} +3.02312e6 q^{52} +536611. q^{53} -58783.7 q^{55} -1.06654e6 q^{58} +2.12009e6 q^{59} -2.71468e6 q^{61} -533785. q^{62} +1.72694e6 q^{64} +1.91296e6 q^{65} +3.77405e6 q^{67} +3.81579e6 q^{68} -4.31646e6 q^{71} +710037. q^{73} -876880. q^{74} -343442. q^{76} +2.79211e6 q^{79} +7.33209e6 q^{80} +1.63906e7 q^{82} +538991. q^{83} +2.41454e6 q^{85} -1.31137e7 q^{86} -1.12436e6 q^{88} -2.19854e6 q^{89} +1.36680e7 q^{92} +1.30502e7 q^{94} -217322. q^{95} +1.16799e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{2} + 348 q^{4} + 252 q^{5} + 984 q^{8}+O(q^{10}) $ 4 * q + 6 * q^2 + 348 * q^4 + 252 * q^5 + 984 * q^8	$ 4 q + 6 q^{2} + 348 q^{4} + 252 q^{5} + 984 q^{8} - 4774 q^{10} + 3972 q^{11} + 1176 q^{13} + 57264 q^{16} + 56364 q^{17} - 41748 q^{19} + 162372 q^{20} - 152954 q^{22} - 131748 q^{23} - 68016 q^{25} + 113652 q^{26} - 34056 q^{29} - 401212 q^{31} + 453408 q^{32} - 82110 q^{34} - 5396 q^{37} + 31794 q^{38} + 443688 q^{40} + 410424 q^{41} + 46544 q^{43} + 1465836 q^{44} + 1379202 q^{46} + 1470084 q^{47} - 1395528 q^{50} + 5269768 q^{52} + 642372 q^{53} + 3063340 q^{55} - 1972220 q^{58} + 752220 q^{59} - 1325772 q^{61} - 3016314 q^{62} - 865856 q^{64} + 4868808 q^{65} - 290916 q^{67} + 2453556 q^{68} - 3377760 q^{71} - 6706588 q^{73} - 3616878 q^{74} - 2923004 q^{76} + 3946244 q^{79} + 10695888 q^{80} + 17563252 q^{82} + 9542064 q^{83} + 7006068 q^{85} - 23237832 q^{86} - 8043336 q^{88} + 16165212 q^{89} + 19442628 q^{92} - 5720442 q^{94} + 3268452 q^{95} - 1533112 q^{97}+O(q^{100}) $ 4 * q + 6 * q^2 + 348 * q^4 + 252 * q^5 + 984 * q^8 - 4774 * q^10 + 3972 * q^11 + 1176 * q^13 + 57264 * q^16 + 56364 * q^17 - 41748 * q^19 + 162372 * q^20 - 152954 * q^22 - 131748 * q^23 - 68016 * q^25 + 113652 * q^26 - 34056 * q^29 - 401212 * q^31 + 453408 * q^32 - 82110 * q^34 - 5396 * q^37 + 31794 * q^38 + 443688 * q^40 + 410424 * q^41 + 46544 * q^43 + 1465836 * q^44 + 1379202 * q^46 + 1470084 * q^47 - 1395528 * q^50 + 5269768 * q^52 + 642372 * q^53 + 3063340 * q^55 - 1972220 * q^58 + 752220 * q^59 - 1325772 * q^61 - 3016314 * q^62 - 865856 * q^64 + 4868808 * q^65 - 290916 * q^67 + 2453556 * q^68 - 3377760 * q^71 - 6706588 * q^73 - 3616878 * q^74 - 2923004 * q^76 + 3946244 * q^79 + 10695888 * q^80 + 17563252 * q^82 + 9542064 * q^83 + 7006068 * q^85 - 23237832 * q^86 - 8043336 * q^88 + 16165212 * q^89 + 19442628 * q^92 - 5720442 * q^94 + 3268452 * q^95 - 1533112 * q^97

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 $

$2$	20.8241	1.84061	0.920303	−	0.391207i	$-0.127942\pi$
$2$	20.8241	1.84061	0.920303	+	0.391207i	$0.127942\pi$
$3$	0	0
$3$	0	0
$4$	305.642	2.38783
$5$	193.403	0.691940	0.345970	−	0.938246i	$-0.387550\pi$
$5$	193.403	0.691940	0.345970	+	0.938246i	$0.387550\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3699.23	2.55445
$9$	0	0
$10$	4027.44	1.27359
$11$	−303.944	−0.0688524	−0.0344262	−	0.999407i	$-0.510960\pi$
$11$	−303.944	−0.0688524	−0.0344262	+	0.999407i	$0.510960\pi$
$12$	0	0
$13$	9891.04	1.24865	0.624324	−	0.781165i	$-0.285374\pi$
$13$	9891.04	1.24865	0.624324	+	0.781165i	$0.285374\pi$
$14$	0	0
$15$	0	0
$16$	37910.9	2.31390
$17$	12484.5	0.616312	0.308156	−	0.951336i	$-0.400288\pi$
$17$	12484.5	0.616312	0.308156	+	0.951336i	$0.400288\pi$
$18$	0	0
$19$	−1123.67	−0.0375840	−0.0187920	−	0.999823i	$-0.505982\pi$
$19$	−1123.67	−0.0375840	−0.0187920	+	0.999823i	$0.505982\pi$
$20$	59112.1	1.65223
$21$	0	0
$22$	−6329.35	−0.126730
$23$	44719.0	0.766381	0.383190	−	0.923669i	$-0.374825\pi$
$23$	44719.0	0.766381	0.383190	+	0.923669i	$0.374825\pi$
$24$	0	0
$25$	−40720.2	−0.521219
$26$	205972.	2.29827
$27$	0	0
$28$	0	0
$29$	−51216.6	−0.389958	−0.194979	−	0.980807i	$-0.562464\pi$
$29$	−51216.6	−0.389958	−0.194979	+	0.980807i	$0.562464\pi$
$30$	0	0
$31$	−25633.0	−0.154538	−0.0772688	−	0.997010i	$-0.524620\pi$
$31$	−25633.0	−0.154538	−0.0772688	+	0.997010i	$0.524620\pi$
$32$	315958.	1.70453
$33$	0	0
$34$	259978.	1.13439
$35$	0	0
$36$	0	0
$37$	−42109.0	−0.136669	−0.0683343	−	0.997662i	$-0.521768\pi$
$37$	−42109.0	−0.136669	−0.0683343	+	0.997662i	$0.521768\pi$
$38$	−23399.5	−0.0691773
$39$	0	0
$40$	715443.	1.76752
$41$	787100.	1.78356	0.891778	−	0.452474i	$-0.149458\pi$
$41$	787100.	1.78356	0.891778	+	0.452474i	$0.149458\pi$
$42$	0	0
$43$	−629737.	−1.20787	−0.603934	−	0.797034i	$-0.706401\pi$
$43$	−629737.	−1.20787	−0.603934	+	0.797034i	$0.706401\pi$
$44$	−92898.1	−0.164408
$45$	0	0
$46$	931232.	1.41060
$47$	626687.	0.880457	0.440229	−	0.897886i	$-0.354897\pi$
$47$	626687.	0.880457	0.440229	+	0.897886i	$0.354897\pi$
$48$	0	0
$49$	0	0
$50$	−847961.	−0.959359
$51$	0	0
$52$	3.02312e6	2.98156
$53$	536611.	0.495101	0.247551	−	0.968875i	$-0.420374\pi$
$53$	536611.	0.495101	0.247551	+	0.968875i	$0.420374\pi$
$54$	0	0
$55$	−58783.7	−0.0476417
$56$	0	0
$57$	0	0
$58$	−1.06654e6	−0.717759
$59$	2.12009e6	1.34392	0.671958	−	0.740589i	$-0.265454\pi$
$59$	2.12009e6	1.34392	0.671958	+	0.740589i	$0.265454\pi$
$60$	0	0
$61$	−2.71468e6	−1.53131	−0.765656	−	0.643250i	$-0.777586\pi$
$61$	−2.71468e6	−1.53131	−0.765656	+	0.643250i	$0.777586\pi$
$62$	−533785.	−0.284443
$63$	0	0
$64$	1.72694e6	0.823470
$65$	1.91296e6	0.863990
$66$	0	0
$67$	3.77405e6	1.53301	0.766506	−	0.642238i	$-0.221994\pi$
$67$	3.77405e6	1.53301	0.766506	+	0.642238i	$0.221994\pi$
$68$	3.81579e6	1.47165
$69$	0	0
$70$	0	0
$71$	−4.31646e6	−1.43128	−0.715638	−	0.698472i	$-0.753864\pi$
$71$	−4.31646e6	−1.43128	−0.715638	+	0.698472i	$0.753864\pi$
$72$	0	0
$73$	710037.	0.213624	0.106812	−	0.994279i	$-0.465936\pi$
$73$	710037.	0.213624	0.106812	+	0.994279i	$0.465936\pi$
$74$	−876880.	−0.251553
$75$	0	0
$76$	−343442.	−0.0897442
$77$	0	0
$78$	0	0
$79$	2.79211e6	0.637144	0.318572	−	0.947899i	$-0.396797\pi$
$79$	2.79211e6	0.637144	0.318572	+	0.947899i	$0.396797\pi$
$80$	7.33209e6	1.60108
$81$	0	0
$82$	1.63906e7	3.28282
$83$	538991.	0.103468	0.0517342	−	0.998661i	$-0.483525\pi$
$83$	538991.	0.103468	0.0517342	+	0.998661i	$0.483525\pi$
$84$	0	0
$85$	2.41454e6	0.426451
$86$	−1.31137e7	−2.22321
$87$	0	0
$88$	−1.12436e6	−0.175880
$89$	−2.19854e6	−0.330575	−0.165288	−	0.986245i	$-0.552855\pi$
$89$	−2.19854e6	−0.330575	−0.165288	+	0.986245i	$0.552855\pi$
$90$	0	0
$91$	0	0
$92$	1.36680e7	1.82999
$93$	0	0
$94$	1.30502e7	1.62057
$95$	−217322.	−0.0260059
$96$	0	0
$97$	1.16799e7	1.29938	0.649691	−	0.760198i	$-0.274898\pi$
$97$	1.16799e7	1.29938	0.649691	+	0.760198i	$0.274898\pi$
$98$	0	0
$99$	0	0
$100$	−1.24458e7	−1.24458
$101$	1.15878e7	1.11912	0.559558	−	0.828791i	$-0.310971\pi$
$101$	1.15878e7	1.11912	0.559558	+	0.828791i	$0.310971\pi$
$102$	0	0
$103$	−8.40069e6	−0.757503	−0.378752	−	0.925498i	$-0.623647\pi$
$103$	−8.40069e6	−0.757503	−0.378752	+	0.925498i	$0.623647\pi$
$104$	3.65893e7	3.18961
$105$	0	0
$106$	1.11744e7	0.911286
$107$	−2.90218e6	−0.229024	−0.114512	−	0.993422i	$-0.536530\pi$
$107$	−2.90218e6	−0.229024	−0.114512	+	0.993422i	$0.536530\pi$
$108$	0	0
$109$	−1.98597e7	−1.46886	−0.734428	−	0.678687i	$-0.762549\pi$
$109$	−1.98597e7	−1.46886	−0.734428	+	0.678687i	$0.762549\pi$
$110$	−1.22412e6	−0.0876896
$111$	0	0
$112$	0	0
$113$	−7.27288e6	−0.474167	−0.237084	−	0.971489i	$-0.576192\pi$
$113$	−7.27288e6	−0.474167	−0.237084	+	0.971489i	$0.576192\pi$
$114$	0	0
$115$	8.64879e6	0.530290
$116$	−1.56540e7	−0.931153
$117$	0	0
$118$	4.41489e7	2.47362
$119$	0	0
$120$	0	0
$121$	−1.93948e7	−0.995259
$122$	−5.65307e7	−2.81854
$123$	0	0
$124$	−7.83454e6	−0.369010
$125$	−2.29850e7	−1.05259
$126$	0	0
$127$	2.81015e7	1.21735	0.608676	−	0.793419i	$-0.291701\pi$
$127$	2.81015e7	1.21735	0.608676	+	0.793419i	$0.291701\pi$
$128$	−4.48072e6	−0.188848
$129$	0	0
$130$	3.98356e7	1.59026
$131$	−1.91894e7	−0.745783	−0.372892	−	0.927875i	$-0.621634\pi$
$131$	−1.91894e7	−0.745783	−0.372892	+	0.927875i	$0.621634\pi$
$132$	0	0
$133$	0	0
$134$	7.85910e7	2.82167
$135$	0	0
$136$	4.61831e7	1.57434
$137$	−1.72092e7	−0.571793	−0.285897	−	0.958260i	$-0.592291\pi$
$137$	−1.72092e7	−0.571793	−0.285897	+	0.958260i	$0.592291\pi$
$138$	0	0
$139$	−4.17445e7	−1.31840	−0.659200	−	0.751968i	$-0.729105\pi$
$139$	−4.17445e7	−1.31840	−0.659200	+	0.751968i	$0.729105\pi$
$140$	0	0
$141$	0	0
$142$	−8.98863e7	−2.63441
$143$	−3.00632e6	−0.0859724
$144$	0	0
$145$	−9.90545e6	−0.269827
$146$	1.47859e7	0.393198
$147$	0	0
$148$	−1.28703e7	−0.326341
$149$	−3.99778e6	−0.0990072	−0.0495036	−	0.998774i	$-0.515764\pi$
$149$	−3.99778e6	−0.0990072	−0.0495036	+	0.998774i	$0.515764\pi$
$150$	0	0
$151$	5.72914e7	1.35416	0.677081	−	0.735909i	$-0.263245\pi$
$151$	5.72914e7	1.35416	0.677081	+	0.735909i	$0.263245\pi$
$152$	−4.15674e6	−0.0960063
$153$	0	0
$154$	0	0
$155$	−4.95751e6	−0.106931
$156$	0	0
$157$	2.85227e7	0.588223	0.294112	−	0.955771i	$-0.404976\pi$
$157$	2.85227e7	0.588223	0.294112	+	0.955771i	$0.404976\pi$
$158$	5.81431e7	1.17273
$159$	0	0
$160$	6.11073e7	1.17943
$161$	0	0
$162$	0	0
$163$	3.22706e7	0.583648	0.291824	−	0.956472i	$-0.405738\pi$
$163$	3.22706e7	0.583648	0.291824	+	0.956472i	$0.405738\pi$
$164$	2.40571e8	4.25883
$165$	0	0
$166$	1.12240e7	0.190445
$167$	1.88088e7	0.312502	0.156251	−	0.987717i	$-0.450059\pi$
$167$	1.88088e7	0.312502	0.156251	+	0.987717i	$0.450059\pi$
$168$	0	0
$169$	3.50841e7	0.559123
$170$	5.02807e7	0.784928
$171$	0	0
$172$	−1.92474e8	−2.88418
$173$	9.34458e7	1.37214	0.686070	−	0.727536i	$-0.259334\pi$
$173$	9.34458e7	1.37214	0.686070	+	0.727536i	$0.259334\pi$
$174$	0	0
$175$	0	0
$176$	−1.15228e7	−0.159318
$177$	0	0
$178$	−4.57827e7	−0.608459
$179$	−8.83162e7	−1.15095	−0.575473	−	0.817821i	$-0.695182\pi$
$179$	−8.83162e7	−1.15095	−0.575473	+	0.817821i	$0.695182\pi$
$180$	0	0
$181$	1.09260e7	0.136958	0.0684789	−	0.997653i	$-0.478185\pi$
$181$	1.09260e7	0.136958	0.0684789	+	0.997653i	$0.478185\pi$
$182$	0	0
$183$	0	0
$184$	1.65426e8	1.95768
$185$	−8.14401e6	−0.0945664
$186$	0	0
$187$	−3.79459e6	−0.0424345
$188$	1.91542e8	2.10238
$189$	0	0
$190$	−4.52553e6	−0.0478666
$191$	1.04957e8	1.08992	0.544961	−	0.838461i	$-0.316544\pi$
$191$	1.04957e8	1.08992	0.544961	+	0.838461i	$0.316544\pi$
$192$	0	0
$193$	1.58561e7	0.158762	0.0793809	−	0.996844i	$-0.474706\pi$
$193$	1.58561e7	0.158762	0.0793809	+	0.996844i	$0.474706\pi$
$194$	2.43222e8	2.39165
$195$	0	0
$196$	0	0
$197$	−4.99805e6	−0.0465767	−0.0232883	−	0.999729i	$-0.507414\pi$
$197$	−4.99805e6	−0.0465767	−0.0232883	+	0.999729i	$0.507414\pi$
$198$	0	0
$199$	−5.50267e7	−0.494980	−0.247490	−	0.968890i	$-0.579606\pi$
$199$	−5.50267e7	−0.494980	−0.247490	+	0.968890i	$0.579606\pi$
$200$	−1.50634e8	−1.33143
$201$	0	0
$202$	2.41305e8	2.05985
$203$	0	0
$204$	0	0
$205$	1.52228e8	1.23411
$206$	−1.74937e8	−1.39426
$207$	0	0
$208$	3.74979e8	2.88925
$209$	341534.	0.00258775
$210$	0	0
$211$	−5.55200e7	−0.406875	−0.203437	−	0.979088i	$-0.565211\pi$
$211$	−5.55200e7	−0.406875	−0.203437	+	0.979088i	$0.565211\pi$
$212$	1.64011e8	1.18222
$213$	0	0
$214$	−6.04353e7	−0.421543
$215$	−1.21793e8	−0.835772
$216$	0	0
$217$	0	0
$218$	−4.13559e8	−2.70358
$219$	0	0
$220$	−1.79668e7	−0.113760
$221$	1.23485e8	0.769557
$222$	0	0
$223$	3.31811e7	0.200366	0.100183	−	0.994969i	$-0.468057\pi$
$223$	3.31811e7	0.200366	0.100183	+	0.994969i	$0.468057\pi$
$224$	0	0
$225$	0	0
$226$	−1.51451e8	−0.872755
$227$	−9.76170e6	−0.0553905	−0.0276952	−	0.999616i	$-0.508817\pi$
$227$	−9.76170e6	−0.0553905	−0.0276952	+	0.999616i	$0.508817\pi$
$228$	0	0
$229$	1.37112e7	0.0754484	0.0377242	−	0.999288i	$-0.487989\pi$
$229$	1.37112e7	0.0754484	0.0377242	+	0.999288i	$0.487989\pi$
$230$	1.80103e8	0.976054
$231$	0	0
$232$	−1.89462e8	−0.996127
$233$	−2.29532e8	−1.18877	−0.594385	−	0.804181i	$-0.702604\pi$
$233$	−2.29532e8	−1.18877	−0.594385	+	0.804181i	$0.702604\pi$
$234$	0	0
$235$	1.21203e8	0.609224
$236$	6.47988e8	3.20904
$237$	0	0
$238$	0	0
$239$	1.18140e8	0.559764	0.279882	−	0.960034i	$-0.409705\pi$
$239$	1.18140e8	0.559764	0.279882	+	0.960034i	$0.409705\pi$
$240$	0	0
$241$	−2.21734e8	−1.02040	−0.510202	−	0.860055i	$-0.670429\pi$
$241$	−2.21734e8	−1.02040	−0.510202	+	0.860055i	$0.670429\pi$
$242$	−4.03879e8	−1.83188
$243$	0	0
$244$	−8.29720e8	−3.65651
$245$	0	0
$246$	0	0
$247$	−1.11143e7	−0.0469292
$248$	−9.48226e7	−0.394758
$249$	0	0
$250$	−4.78642e8	−1.93741
$251$	−1.95230e8	−0.779273	−0.389636	−	0.920969i	$-0.627399\pi$
$251$	−1.95230e8	−0.779273	−0.389636	+	0.920969i	$0.627399\pi$
$252$	0	0
$253$	−1.35921e7	−0.0527671
$254$	5.85187e8	2.24067
$255$	0	0
$256$	−3.14355e8	−1.17106
$257$	−5.18254e8	−1.90448	−0.952240	−	0.305350i	$-0.901226\pi$
$257$	−5.18254e8	−1.90448	−0.952240	+	0.305350i	$0.901226\pi$
$258$	0	0
$259$	0	0
$260$	5.84681e8	2.06306
$261$	0	0
$262$	−3.99602e8	−1.37269
$263$	3.90982e8	1.32529	0.662646	−	0.748933i	$-0.269433\pi$
$263$	3.90982e8	1.32529	0.662646	+	0.748933i	$0.269433\pi$
$264$	0	0
$265$	1.03782e8	0.342580
$266$	0	0
$267$	0	0
$268$	1.15351e9	3.66057
$269$	−3.58718e8	−1.12362	−0.561810	−	0.827266i	$-0.689895\pi$
$269$	−3.58718e8	−1.12362	−0.561810	+	0.827266i	$0.689895\pi$
$270$	0	0
$271$	−1.97030e8	−0.601367	−0.300684	−	0.953724i	$-0.597215\pi$
$271$	−1.97030e8	−0.601367	−0.300684	+	0.953724i	$0.597215\pi$
$272$	4.73300e8	1.42608
$273$	0	0
$274$	−3.58366e8	−1.05245
$275$	1.23767e7	0.0358872
$276$	0	0
$277$	−4.01991e8	−1.13641	−0.568207	−	0.822886i	$-0.692363\pi$
$277$	−4.01991e8	−1.13641	−0.568207	+	0.822886i	$0.692363\pi$
$278$	−8.69290e8	−2.42666
$279$	0	0
$280$	0	0
$281$	−2.80445e7	−0.0754008	−0.0377004	−	0.999289i	$-0.512003\pi$
$281$	−2.80445e7	−0.0754008	−0.0377004	+	0.999289i	$0.512003\pi$
$282$	0	0
$283$	−2.06404e6	−0.00541334	−0.00270667	−	0.999996i	$-0.500862\pi$
$283$	−2.06404e6	−0.00541334	−0.00270667	+	0.999996i	$0.500862\pi$
$284$	−1.31929e9	−3.41764
$285$	0	0
$286$	−6.26038e7	−0.158241
$287$	0	0
$288$	0	0
$289$	−2.54476e8	−0.620160
$290$	−2.06272e8	−0.496646
$291$	0	0
$292$	2.17017e8	0.510099
$293$	−3.77433e8	−0.876604	−0.438302	−	0.898828i	$-0.644420\pi$
$293$	−3.77433e8	−0.876604	−0.438302	+	0.898828i	$0.644420\pi$
$294$	0	0
$295$	4.10032e8	0.929909
$296$	−1.55771e8	−0.349113
$297$	0	0
$298$	−8.32501e7	−0.182233
$299$	4.42317e8	0.956940
$300$	0	0
$301$	0	0
$302$	1.19304e9	2.49248
$303$	0	0
$304$	−4.25996e7	−0.0869656
$305$	−5.25027e8	−1.05958
$306$	0	0
$307$	−5.18232e8	−1.02221	−0.511105	−	0.859519i	$-0.670764\pi$
$307$	−5.18232e8	−1.02221	−0.511105	+	0.859519i	$0.670764\pi$
$308$	0	0
$309$	0	0
$310$	−1.03236e8	−0.196817
$311$	2.83833e8	0.535058	0.267529	−	0.963550i	$-0.413793\pi$
$311$	2.83833e8	0.535058	0.267529	+	0.963550i	$0.413793\pi$
$312$	0	0
$313$	3.30100e7	0.0608472	0.0304236	−	0.999537i	$-0.490314\pi$
$313$	3.30100e7	0.0608472	0.0304236	+	0.999537i	$0.490314\pi$
$314$	5.93959e8	1.08269
$315$	0	0
$316$	8.53387e8	1.52139
$317$	−1.09318e9	−1.92745	−0.963726	−	0.266893i	$-0.914003\pi$
$317$	−1.09318e9	−1.92745	−0.963726	+	0.266893i	$0.914003\pi$
$318$	0	0
$319$	1.55670e7	0.0268495
$320$	3.33996e8	0.569792
$321$	0	0
$322$	0	0
$323$	−1.40285e7	−0.0231635
$324$	0	0
$325$	−4.02765e8	−0.650819
$326$	6.72006e8	1.07427
$327$	0	0
$328$	2.91167e9	4.55600
$329$	0	0
$330$	0	0
$331$	−3.21913e8	−0.487911	−0.243955	−	0.969786i	$-0.578445\pi$
$331$	−3.21913e8	−0.487911	−0.243955	+	0.969786i	$0.578445\pi$
$332$	1.64738e8	0.247065
$333$	0	0
$334$	3.91675e8	0.575193
$335$	7.29912e8	1.06075
$336$	0	0
$337$	1.31007e9	1.86461	0.932307	−	0.361668i	$-0.117793\pi$
$337$	1.31007e9	1.86461	0.932307	+	0.361668i	$0.117793\pi$
$338$	7.30595e8	1.02912
$339$	0	0
$340$	7.37986e8	1.01829
$341$	7.79101e6	0.0106403
$342$	0	0
$343$	0	0
$344$	−2.32954e9	−3.08543
$345$	0	0
$346$	1.94592e9	2.52557
$347$	−1.66682e8	−0.214159	−0.107079	−	0.994250i	$-0.534150\pi$
$347$	−1.66682e8	−0.214159	−0.107079	+	0.994250i	$0.534150\pi$
$348$	0	0
$349$	−7.47430e8	−0.941199	−0.470599	−	0.882347i	$-0.655962\pi$
$349$	−7.47430e8	−0.941199	−0.470599	+	0.882347i	$0.655962\pi$
$350$	0	0
$351$	0	0
$352$	−9.60336e7	−0.117361
$353$	6.17912e8	0.747678	0.373839	−	0.927494i	$-0.378041\pi$
$353$	6.17912e8	0.747678	0.373839	+	0.927494i	$0.378041\pi$
$354$	0	0
$355$	−8.34816e8	−0.990357
$356$	−6.71968e8	−0.789357
$357$	0	0
$358$	−1.83910e9	−2.11844
$359$	1.01667e9	1.15971	0.579857	−	0.814718i	$-0.303108\pi$
$359$	1.01667e9	1.15971	0.579857	+	0.814718i	$0.303108\pi$
$360$	0	0
$361$	−8.92609e8	−0.998587
$362$	2.27524e8	0.252085
$363$	0	0
$364$	0	0
$365$	1.37323e8	0.147815
$366$	0	0
$367$	−1.47700e9	−1.55973	−0.779864	−	0.625950i	$-0.784712\pi$
$367$	−1.47700e9	−1.55973	−0.779864	+	0.625950i	$0.784712\pi$
$368$	1.69534e9	1.77333
$369$	0	0
$370$	−1.69591e8	−0.174060
$371$	0	0
$372$	0	0
$373$	−1.63194e9	−1.62826	−0.814130	−	0.580683i	$-0.802786\pi$
$373$	−1.63194e9	−1.62826	−0.814130	+	0.580683i	$0.802786\pi$
$374$	−7.90189e7	−0.0781052
$375$	0	0
$376$	2.31826e9	2.24908
$377$	−5.06585e8	−0.486920
$378$	0	0
$379$	−3.41392e8	−0.322118	−0.161059	−	0.986945i	$-0.551491\pi$
$379$	−3.41392e8	−0.322118	−0.161059	+	0.986945i	$0.551491\pi$
$380$	−6.64228e7	−0.0620976
$381$	0	0
$382$	2.18564e9	2.00612
$383$	−3.71090e8	−0.337508	−0.168754	−	0.985658i	$-0.553974\pi$
$383$	−3.71090e8	−0.337508	−0.168754	+	0.985658i	$0.553974\pi$
$384$	0	0
$385$	0	0
$386$	3.30189e8	0.292218
$387$	0	0
$388$	3.56986e9	3.10270
$389$	9.49022e8	0.817434	0.408717	−	0.912661i	$-0.365976\pi$
$389$	9.49022e8	0.817434	0.408717	+	0.912661i	$0.365976\pi$
$390$	0	0
$391$	5.58295e8	0.472329
$392$	0	0
$393$	0	0
$394$	−1.04080e8	−0.0857293
$395$	5.40003e8	0.440866
$396$	0	0
$397$	−7.63784e8	−0.612637	−0.306319	−	0.951929i	$-0.599097\pi$
$397$	−7.63784e8	−0.612637	−0.306319	+	0.951929i	$0.599097\pi$
$398$	−1.14588e9	−0.911063
$399$	0	0
$400$	−1.54374e9	−1.20605
$401$	−9.20850e8	−0.713154	−0.356577	−	0.934266i	$-0.616056\pi$
$401$	−9.20850e8	−0.713154	−0.356577	+	0.934266i	$0.616056\pi$
$402$	0	0
$403$	−2.53537e8	−0.192963
$404$	3.54171e9	2.67226
$405$	0	0
$406$	0	0
$407$	1.27988e7	0.00940996
$408$	0	0
$409$	8.09329e8	0.584916	0.292458	−	0.956278i	$-0.405527\pi$
$409$	8.09329e8	0.584916	0.292458	+	0.956278i	$0.405527\pi$
$410$	3.17000e9	2.27152
$411$	0	0
$412$	−2.56760e9	−1.80879
$413$	0	0
$414$	0	0
$415$	1.04242e8	0.0715940
$416$	3.12516e9	2.12836
$417$	0	0
$418$	7.11213e6	0.00476302
$419$	2.33725e9	1.55223	0.776115	−	0.630592i	$-0.217188\pi$
$419$	2.33725e9	1.55223	0.776115	+	0.630592i	$0.217188\pi$
$420$	0	0
$421$	6.85194e8	0.447534	0.223767	−	0.974643i	$-0.428164\pi$
$421$	6.85194e8	0.447534	0.223767	+	0.974643i	$0.428164\pi$
$422$	−1.15615e9	−0.748896
$423$	0	0
$424$	1.98505e9	1.26471
$425$	−5.08372e8	−0.321233
$426$	0	0
$427$	0	0
$428$	−8.87029e8	−0.546871
$429$	0	0
$430$	−2.53623e9	−1.53833
$431$	−1.59539e9	−0.959833	−0.479917	−	0.877314i	$-0.659333\pi$
$431$	−1.59539e9	−0.959833	−0.479917	+	0.877314i	$0.659333\pi$
$432$	0	0
$433$	3.20663e9	1.89820	0.949098	−	0.314981i	$-0.101998\pi$
$433$	3.20663e9	1.89820	0.949098	+	0.314981i	$0.101998\pi$
$434$	0	0
$435$	0	0
$436$	−6.06995e9	−3.50738
$437$	−5.02496e7	−0.0288037
$438$	0	0
$439$	−1.86371e9	−1.05136	−0.525681	−	0.850682i	$-0.676190\pi$
$439$	−1.86371e9	−1.05136	−0.525681	+	0.850682i	$0.676190\pi$
$440$	−2.17455e8	−0.121698
$441$	0	0
$442$	2.57146e9	1.41645
$443$	−7.98264e8	−0.436248	−0.218124	−	0.975921i	$-0.569994\pi$
$443$	−7.98264e8	−0.436248	−0.218124	+	0.975921i	$0.569994\pi$
$444$	0	0
$445$	−4.25205e8	−0.228738
$446$	6.90966e8	0.368795
$447$	0	0
$448$	0	0
$449$	4.99626e8	0.260485	0.130242	−	0.991482i	$-0.458424\pi$
$449$	4.99626e8	0.260485	0.130242	+	0.991482i	$0.458424\pi$
$450$	0	0
$451$	−2.39234e8	−0.122802
$452$	−2.22290e9	−1.13223
$453$	0	0
$454$	−2.03278e8	−0.101952
$455$	0	0
$456$	0	0
$457$	2.67411e9	1.31061	0.655305	−	0.755365i	$-0.272540\pi$
$457$	2.67411e9	1.31061	0.655305	+	0.755365i	$0.272540\pi$
$458$	2.85522e8	0.138871
$459$	0	0
$460$	2.64344e9	1.26624
$461$	−2.67032e9	−1.26943	−0.634716	−	0.772746i	$-0.718883\pi$
$461$	−2.67032e9	−1.26943	−0.634716	+	0.772746i	$0.718883\pi$
$462$	0	0
$463$	1.72345e9	0.806983	0.403492	−	0.914983i	$-0.367796\pi$
$463$	1.72345e9	0.806983	0.403492	+	0.914983i	$0.367796\pi$
$464$	−1.94167e9	−0.902324
$465$	0	0
$466$	−4.77979e9	−2.18806
$467$	−2.82708e9	−1.28448	−0.642242	−	0.766502i	$-0.721996\pi$
$467$	−2.82708e9	−1.28448	−0.642242	+	0.766502i	$0.721996\pi$
$468$	0	0
$469$	0	0
$470$	2.52395e9	1.12134
$471$	0	0
$472$	7.84270e9	3.43296
$473$	1.91405e8	0.0831646
$474$	0	0
$475$	4.57563e7	0.0195895
$476$	0	0
$477$	0	0
$478$	2.46016e9	1.03031
$479$	−3.09421e9	−1.28640	−0.643200	−	0.765699i	$-0.722393\pi$
$479$	−3.09421e9	−1.28640	−0.643200	+	0.765699i	$0.722393\pi$
$480$	0	0
$481$	−4.16501e8	−0.170651
$482$	−4.61740e9	−1.87816
$483$	0	0
$484$	−5.92787e9	−2.37651
$485$	2.25892e9	0.899094
$486$	0	0
$487$	−4.23090e8	−0.165990	−0.0829949	−	0.996550i	$-0.526449\pi$
$487$	−4.23090e8	−0.165990	−0.0829949	+	0.996550i	$0.526449\pi$
$488$	−1.00422e10	−3.91166
$489$	0	0
$490$	0	0
$491$	9.85934e8	0.375892	0.187946	−	0.982179i	$-0.439817\pi$
$491$	9.85934e8	0.375892	0.187946	+	0.982179i	$0.439817\pi$
$492$	0	0
$493$	−6.39414e8	−0.240336
$494$	−2.31445e8	−0.0863782
$495$	0	0
$496$	−9.71773e8	−0.357585
$497$	0	0
$498$	0	0
$499$	−5.24739e8	−0.189056	−0.0945282	−	0.995522i	$-0.530134\pi$
$499$	−5.24739e8	−0.189056	−0.0945282	+	0.995522i	$0.530134\pi$
$500$	−7.02520e9	−2.51341
$501$	0	0
$502$	−4.06549e9	−1.43433
$503$	4.23277e9	1.48298	0.741492	−	0.670962i	$-0.234119\pi$
$503$	4.23277e9	1.48298	0.741492	+	0.670962i	$0.234119\pi$
$504$	0	0
$505$	2.24111e9	0.774362
$506$	−2.83042e8	−0.0971235
$507$	0	0
$508$	8.58900e9	2.90683
$509$	−2.32023e9	−0.779863	−0.389931	−	0.920844i	$-0.627501\pi$
$509$	−2.32023e9	−0.779863	−0.389931	+	0.920844i	$0.627501\pi$
$510$	0	0
$511$	0	0
$512$	−5.97263e9	−1.96662
$513$	0	0
$514$	−1.07922e10	−3.50540
$515$	−1.62472e9	−0.524147
$516$	0	0
$517$	−1.90478e8	−0.0606216
$518$	0	0
$519$	0	0
$520$	7.07648e9	2.20702
$521$	2.75969e9	0.854926	0.427463	−	0.904033i	$-0.359407\pi$
$521$	2.75969e9	0.854926	0.427463	+	0.904033i	$0.359407\pi$
$522$	0	0
$523$	−4.44560e9	−1.35886	−0.679430	−	0.733740i	$-0.737773\pi$
$523$	−4.44560e9	−1.35886	−0.679430	+	0.733740i	$0.737773\pi$
$524$	−5.86510e9	−1.78080
$525$	0	0
$526$	8.14184e9	2.43934
$527$	−3.20016e8	−0.0952434
$528$	0	0
$529$	−1.40504e9	−0.412660
$530$	2.16117e9	0.630556
$531$	0	0
$532$	0	0
$533$	7.78524e9	2.22703
$534$	0	0
$535$	−5.61291e8	−0.158471
$536$	1.39611e10	3.91600
$537$	0	0
$538$	−7.46997e9	−2.06814
$539$	0	0
$540$	0	0
$541$	−5.35663e9	−1.45446	−0.727228	−	0.686396i	$-0.759192\pi$
$541$	−5.35663e9	−1.45446	−0.727228	+	0.686396i	$0.759192\pi$
$542$	−4.10297e9	−1.10688
$543$	0	0
$544$	3.94459e9	1.05052
$545$	−3.84092e9	−1.01636
$546$	0	0
$547$	1.36408e9	0.356357	0.178178	−	0.983998i	$-0.442980\pi$
$547$	1.36408e9	0.356357	0.178178	+	0.983998i	$0.442980\pi$
$548$	−5.25986e9	−1.36534
$549$	0	0
$550$	2.57733e8	0.0660541
$551$	5.75508e7	0.0146562
$552$	0	0
$553$	0	0
$554$	−8.37108e9	−2.09169
$555$	0	0
$556$	−1.27589e10	−3.14812
$557$	−3.08958e9	−0.757542	−0.378771	−	0.925490i	$-0.623653\pi$
$557$	−3.08958e9	−0.757542	−0.378771	+	0.925490i	$0.623653\pi$
$558$	0	0
$559$	−6.22875e9	−1.50820
$560$	0	0
$561$	0	0
$562$	−5.84001e8	−0.138783
$563$	−1.89692e9	−0.447991	−0.223995	−	0.974590i	$-0.571910\pi$
$563$	−1.89692e9	−0.447991	−0.223995	+	0.974590i	$0.571910\pi$
$564$	0	0
$565$	−1.40660e9	−0.328095
$566$	−4.29817e7	−0.00996383
$567$	0	0
$568$	−1.59676e10	−3.65612
$569$	−8.35708e9	−1.90179	−0.950893	−	0.309521i	$-0.899831\pi$
$569$	−8.35708e9	−1.90179	−0.950893	+	0.309521i	$0.899831\pi$
$570$	0	0
$571$	4.67543e9	1.05098	0.525491	−	0.850799i	$-0.323882\pi$
$571$	4.67543e9	1.05098	0.525491	+	0.850799i	$0.323882\pi$
$572$	−9.18858e8	−0.205287
$573$	0	0
$574$	0	0
$575$	−1.82097e9	−0.399452
$576$	0	0
$577$	7.76726e8	0.168326	0.0841632	−	0.996452i	$-0.473178\pi$
$577$	7.76726e8	0.168326	0.0841632	+	0.996452i	$0.473178\pi$
$578$	−5.29922e9	−1.14147
$579$	0	0
$580$	−3.02752e9	−0.644302
$581$	0	0
$582$	0	0
$583$	−1.63100e8	−0.0340889
$584$	2.62659e9	0.545692
$585$	0	0
$586$	−7.85970e9	−1.61348
$587$	1.89326e9	0.386347	0.193174	−	0.981165i	$-0.438122\pi$
$587$	1.89326e9	0.386347	0.193174	+	0.981165i	$0.438122\pi$
$588$	0	0
$589$	2.88032e7	0.00580814
$590$	8.53853e9	1.71160
$591$	0	0
$592$	−1.59639e9	−0.316237
$593$	4.44306e9	0.874965	0.437483	−	0.899227i	$-0.355870\pi$
$593$	4.44306e9	0.874965	0.437483	+	0.899227i	$0.355870\pi$
$594$	0	0
$595$	0	0
$596$	−1.22189e9	−0.236412
$597$	0	0
$598$	9.21085e9	1.76135
$599$	−2.22048e9	−0.422136	−0.211068	−	0.977471i	$-0.567694\pi$
$599$	−2.22048e9	−0.422136	−0.211068	+	0.977471i	$0.567694\pi$
$600$	0	0
$601$	−5.04814e9	−0.948574	−0.474287	−	0.880370i	$-0.657294\pi$
$601$	−5.04814e9	−0.948574	−0.474287	+	0.880370i	$0.657294\pi$
$602$	0	0
$603$	0	0
$604$	1.75107e10	3.23351
$605$	−3.75101e9	−0.688660
$606$	0	0
$607$	2.82879e9	0.513381	0.256691	−	0.966494i	$-0.417368\pi$
$607$	2.82879e9	0.513381	0.256691	+	0.966494i	$0.417368\pi$
$608$	−3.55034e8	−0.0640631
$609$	0	0
$610$	−1.09332e10	−1.95026
$611$	6.19859e9	1.09938
$612$	0	0
$613$	−3.21028e9	−0.562900	−0.281450	−	0.959576i	$-0.590815\pi$
$613$	−3.21028e9	−0.562900	−0.281450	+	0.959576i	$0.590815\pi$
$614$	−1.07917e10	−1.88148
$615$	0	0
$616$	0	0
$617$	−2.94679e9	−0.505070	−0.252535	−	0.967588i	$-0.581264\pi$
$617$	−2.94679e9	−0.505070	−0.252535	+	0.967588i	$0.581264\pi$
$618$	0	0
$619$	−6.40987e9	−1.08626	−0.543128	−	0.839650i	$-0.682760\pi$
$619$	−6.40987e9	−1.08626	−0.543128	+	0.839650i	$0.682760\pi$
$620$	−1.51522e9	−0.255332
$621$	0	0
$622$	5.91055e9	0.984831
$623$	0	0
$624$	0	0
$625$	−1.26411e9	−0.207112
$626$	6.87403e8	0.111996
$627$	0	0
$628$	8.71775e9	1.40458
$629$	−5.25710e8	−0.0842304
$630$	0	0
$631$	−1.37351e9	−0.217636	−0.108818	−	0.994062i	$-0.534707\pi$
$631$	−1.37351e9	−0.217636	−0.108818	+	0.994062i	$0.534707\pi$
$632$	1.03287e10	1.62755
$633$	0	0
$634$	−2.27644e10	−3.54768
$635$	5.43491e9	0.842335
$636$	0	0
$637$	0	0
$638$	3.24168e8	0.0494194
$639$	0	0
$640$	−8.66586e8	−0.130672
$641$	4.79105e9	0.718502	0.359251	−	0.933241i	$-0.383032\pi$
$641$	4.79105e9	0.718502	0.359251	+	0.933241i	$0.383032\pi$
$642$	0	0
$643$	9.77303e9	1.44974	0.724871	−	0.688884i	$-0.241899\pi$
$643$	9.77303e9	1.44974	0.724871	+	0.688884i	$0.241899\pi$
$644$	0	0
$645$	0	0
$646$	−2.92131e8	−0.0426348
$647$	1.09805e10	1.59389	0.796943	−	0.604055i	$-0.206449\pi$
$647$	1.09805e10	1.59389	0.796943	+	0.604055i	$0.206449\pi$
$648$	0	0
$649$	−6.44388e8	−0.0925318
$650$	−8.38722e9	−1.19790
$651$	0	0
$652$	9.86326e9	1.39365
$653$	−4.22737e9	−0.594120	−0.297060	−	0.954859i	$-0.596006\pi$
$653$	−4.22737e9	−0.594120	−0.297060	+	0.954859i	$0.596006\pi$
$654$	0	0
$655$	−3.71130e9	−0.516037
$656$	2.98397e10	4.12697
$657$	0	0
$658$	0	0
$659$	−1.09220e9	−0.148663	−0.0743314	−	0.997234i	$-0.523682\pi$
$659$	−1.09220e9	−0.148663	−0.0743314	+	0.997234i	$0.523682\pi$
$660$	0	0
$661$	1.14809e10	1.54622	0.773111	−	0.634270i	$-0.218699\pi$
$661$	1.14809e10	1.54622	0.773111	+	0.634270i	$0.218699\pi$
$662$	−6.70354e9	−0.898051
$663$	0	0
$664$	1.99385e9	0.264305
$665$	0	0
$666$	0	0
$667$	−2.29035e9	−0.298856
$668$	5.74876e9	0.746202
$669$	0	0
$670$	1.51997e10	1.95243
$671$	8.25110e8	0.105435
$672$	0	0
$673$	7.00125e9	0.885366	0.442683	−	0.896678i	$-0.354027\pi$
$673$	7.00125e9	0.885366	0.442683	+	0.896678i	$0.354027\pi$
$674$	2.72809e10	3.43202
$675$	0	0
$676$	1.07232e10	1.33509
$677$	1.23271e10	1.52686	0.763430	−	0.645890i	$-0.223514\pi$
$677$	1.23271e10	1.52686	0.763430	+	0.645890i	$0.223514\pi$
$678$	0	0
$679$	0	0
$680$	8.93196e9	1.08935
$681$	0	0
$682$	1.62241e8	0.0195846
$683$	1.26376e10	1.51772	0.758860	−	0.651254i	$-0.225757\pi$
$683$	1.26376e10	1.51772	0.758860	+	0.651254i	$0.225757\pi$
$684$	0	0
$685$	−3.32831e9	−0.395647
$686$	0	0
$687$	0	0
$688$	−2.38739e10	−2.79489
$689$	5.30764e9	0.618208
$690$	0	0
$691$	4.89154e8	0.0563991	0.0281995	−	0.999602i	$-0.491023\pi$
$691$	4.89154e8	0.0563991	0.0281995	+	0.999602i	$0.491023\pi$
$692$	2.85610e10	3.27644
$693$	0	0
$694$	−3.47100e9	−0.394182
$695$	−8.07351e9	−0.912254
$696$	0	0
$697$	9.82657e9	1.09923
$698$	−1.55645e10	−1.73238
$699$	0	0
$700$	0	0
$701$	1.30414e9	0.142992	0.0714959	−	0.997441i	$-0.477223\pi$
$701$	1.30414e9	0.142992	0.0714959	+	0.997441i	$0.477223\pi$
$702$	0	0
$703$	4.73168e7	0.00513655
$704$	−5.24893e8	−0.0566978
$705$	0	0
$706$	1.28674e10	1.37618
$707$	0	0
$708$	0	0
$709$	8.84454e9	0.931994	0.465997	−	0.884786i	$-0.345696\pi$
$709$	8.84454e9	0.931994	0.465997	+	0.884786i	$0.345696\pi$
$710$	−1.73843e10	−1.82286
$711$	0	0
$712$	−8.13293e9	−0.844437
$713$	−1.14628e9	−0.118435
$714$	0	0
$715$	−5.81432e8	−0.0594878
$716$	−2.69932e10	−2.74826
$717$	0	0
$718$	2.11713e10	2.13458
$719$	−1.50714e10	−1.51218	−0.756088	−	0.654470i	$-0.772892\pi$
$719$	−1.50714e10	−1.51218	−0.756088	+	0.654470i	$0.772892\pi$
$720$	0	0
$721$	0	0
$722$	−1.85878e10	−1.83801
$723$	0	0
$724$	3.33945e9	0.327032
$725$	2.08555e9	0.203253
$726$	0	0
$727$	4.87633e9	0.470677	0.235338	−	0.971914i	$-0.424380\pi$
$727$	4.87633e9	0.470677	0.235338	+	0.971914i	$0.424380\pi$
$728$	0	0
$729$	0	0
$730$	2.85963e9	0.272070
$731$	−7.86196e9	−0.744423
$732$	0	0
$733$	−5.11146e9	−0.479381	−0.239691	−	0.970849i	$-0.577046\pi$
$733$	−5.11146e9	−0.479381	−0.239691	+	0.970849i	$0.577046\pi$
$734$	−3.07571e10	−2.87084
$735$	0	0
$736$	1.41293e10	1.30632
$737$	−1.14710e9	−0.105551
$738$	0	0
$739$	−1.14790e10	−1.04628	−0.523140	−	0.852247i	$-0.675240\pi$
$739$	−1.14790e10	−1.04628	−0.523140	+	0.852247i	$0.675240\pi$
$740$	−2.48915e9	−0.225809
$741$	0	0
$742$	0	0
$743$	3.64003e9	0.325570	0.162785	−	0.986662i	$-0.447952\pi$
$743$	3.64003e9	0.325570	0.162785	+	0.986662i	$0.447952\pi$
$744$	0	0
$745$	−7.73183e8	−0.0685071
$746$	−3.39837e10	−2.99698
$747$	0	0
$748$	−1.15979e9	−0.101326
$749$	0	0
$750$	0	0
$751$	−3.72398e9	−0.320825	−0.160412	−	0.987050i	$-0.551282\pi$
$751$	−3.72398e9	−0.320825	−0.160412	+	0.987050i	$0.551282\pi$
$752$	2.37583e10	2.03729
$753$	0	0
$754$	−1.05492e10	−0.896228
$755$	1.10803e10	0.936998
$756$	0	0
$757$	−1.47428e10	−1.23522	−0.617611	−	0.786484i	$-0.711899\pi$
$757$	−1.47428e10	−1.23522	−0.617611	+	0.786484i	$0.711899\pi$
$758$	−7.10917e9	−0.592893
$759$	0	0
$760$	−8.03926e8	−0.0664306
$761$	−7.34596e9	−0.604230	−0.302115	−	0.953272i	$-0.597693\pi$
$761$	−7.34596e9	−0.604230	−0.302115	+	0.953272i	$0.597693\pi$
$762$	0	0
$763$	0	0
$764$	3.20793e10	2.60255
$765$	0	0
$766$	−7.72762e9	−0.621219
$767$	2.09699e10	1.67808
$768$	0	0
$769$	−9.27906e9	−0.735803	−0.367902	−	0.929865i	$-0.619924\pi$
$769$	−9.27906e9	−0.735803	−0.367902	+	0.929865i	$0.619924\pi$
$770$	0	0
$771$	0	0
$772$	4.84630e9	0.379096
$773$	9.68019e9	0.753799	0.376900	−	0.926254i	$-0.376990\pi$
$773$	9.68019e9	0.753799	0.376900	+	0.926254i	$0.376990\pi$
$774$	0	0
$775$	1.04378e9	0.0805480
$776$	4.32066e10	3.31920
$777$	0	0
$778$	1.97625e10	1.50457
$779$	−8.84445e8	−0.0670332
$780$	0	0
$781$	1.31196e9	0.0985467
$782$	1.16260e10	0.869372
$783$	0	0
$784$	0	0
$785$	5.51638e9	0.407015
$786$	0	0
$787$	−1.61026e9	−0.117756	−0.0588781	−	0.998265i	$-0.518752\pi$
$787$	−1.61026e9	−0.117756	−0.0588781	+	0.998265i	$0.518752\pi$
$788$	−1.52761e9	−0.111217
$789$	0	0
$790$	1.12451e10	0.811460
$791$	0	0
$792$	0	0
$793$	−2.68510e10	−1.91207
$794$	−1.59051e10	−1.12762
$795$	0	0
$796$	−1.68185e10	−1.18193
$797$	1.04405e10	0.730497	0.365249	−	0.930910i	$-0.380984\pi$
$797$	1.04405e10	0.730497	0.365249	+	0.930910i	$0.380984\pi$
$798$	0	0
$799$	7.82388e9	0.542636
$800$	−1.28659e10	−0.888434
$801$	0	0
$802$	−1.91758e10	−1.31264
$803$	−2.15811e8	−0.0147085
$804$	0	0
$805$	0	0
$806$	−5.27968e9	−0.355169
$807$	0	0
$808$	4.28659e10	2.85872
$809$	−1.96370e10	−1.30394	−0.651968	−	0.758247i	$-0.726056\pi$
$809$	−1.96370e10	−1.30394	−0.651968	+	0.758247i	$0.726056\pi$
$810$	0	0
$811$	2.66078e10	1.75161	0.875803	−	0.482669i	$-0.160332\pi$
$811$	2.66078e10	1.75161	0.875803	+	0.482669i	$0.160332\pi$
$812$	0	0
$813$	0	0
$814$	2.66522e8	0.0173200
$815$	6.24124e9	0.403849
$816$	0	0
$817$	7.07619e8	0.0453965
$818$	1.68535e10	1.07660
$819$	0	0
$820$	4.65272e10	2.94685
$821$	7.14495e9	0.450607	0.225304	−	0.974289i	$-0.427663\pi$
$821$	7.14495e9	0.450607	0.225304	+	0.974289i	$0.427663\pi$
$822$	0	0
$823$	8.34387e9	0.521757	0.260878	−	0.965372i	$-0.415988\pi$
$823$	8.34387e9	0.521757	0.260878	+	0.965372i	$0.415988\pi$
$824$	−3.10761e10	−1.93500
$825$	0	0
$826$	0	0
$827$	2.92874e10	1.80058	0.900288	−	0.435295i	$-0.143356\pi$
$827$	2.92874e10	1.80058	0.900288	+	0.435295i	$0.143356\pi$
$828$	0	0
$829$	8.02734e9	0.489362	0.244681	−	0.969604i	$-0.421317\pi$
$829$	8.02734e9	0.489362	0.244681	+	0.969604i	$0.421317\pi$
$830$	2.17075e9	0.131776
$831$	0	0
$832$	1.70812e10	1.02822
$833$	0	0
$834$	0	0
$835$	3.63768e9	0.216233
$836$	1.04387e8	0.00617910
$837$	0	0
$838$	4.86711e10	2.85704
$839$	5.19771e9	0.303840	0.151920	−	0.988393i	$-0.451454\pi$
$839$	5.19771e9	0.303840	0.151920	+	0.988393i	$0.451454\pi$
$840$	0	0
$841$	−1.46267e10	−0.847933
$842$	1.42685e10	0.823734
$843$	0	0
$844$	−1.69692e10	−0.971547
$845$	6.78538e9	0.386879
$846$	0	0
$847$	0	0
$848$	2.03434e10	1.14562
$849$	0	0
$850$	−1.05864e10	−0.591264
$851$	−1.88307e9	−0.104740
$852$	0	0
$853$	−5.41443e9	−0.298697	−0.149349	−	0.988785i	$-0.547718\pi$
$853$	−5.41443e9	−0.298697	−0.149349	+	0.988785i	$0.547718\pi$
$854$	0	0
$855$	0	0
$856$	−1.07359e10	−0.585030
$857$	1.90836e10	1.03569	0.517843	−	0.855476i	$-0.326735\pi$
$857$	1.90836e10	1.03569	0.517843	+	0.855476i	$0.326735\pi$
$858$	0	0
$859$	3.19443e10	1.71956	0.859781	−	0.510663i	$-0.170600\pi$
$859$	3.19443e10	1.71956	0.859781	+	0.510663i	$0.170600\pi$
$860$	−3.72251e10	−1.99568
$861$	0	0
$862$	−3.32225e10	−1.76667
$863$	−1.12017e10	−0.593262	−0.296631	−	0.954992i	$-0.595863\pi$
$863$	−1.12017e10	−0.593262	−0.296631	+	0.954992i	$0.595863\pi$
$864$	0	0
$865$	1.80727e10	0.949438
$866$	6.67751e10	3.49383
$867$	0	0
$868$	0	0
$869$	−8.48645e8	−0.0438689
$870$	0	0
$871$	3.73292e10	1.91419
$872$	−7.34656e10	−3.75211
$873$	0	0
$874$	−1.04640e9	−0.0530162
$875$	0	0
$876$	0	0
$877$	2.73091e9	0.136713	0.0683563	−	0.997661i	$-0.478225\pi$
$877$	2.73091e9	0.136713	0.0683563	+	0.997661i	$0.478225\pi$
$878$	−3.88100e10	−1.93514
$879$	0	0
$880$	−2.22855e9	−0.110238
$881$	2.77466e10	1.36708	0.683539	−	0.729914i	$-0.260440\pi$
$881$	2.77466e10	1.36708	0.683539	+	0.729914i	$0.260440\pi$
$882$	0	0
$883$	1.21160e10	0.592240	0.296120	−	0.955151i	$-0.404307\pi$
$883$	1.21160e10	0.592240	0.296120	+	0.955151i	$0.404307\pi$
$884$	3.77422e10	1.83757
$885$	0	0
$886$	−1.66231e10	−0.802961
$887$	6.57070e9	0.316140	0.158070	−	0.987428i	$-0.449473\pi$
$887$	6.57070e9	0.316140	0.158070	+	0.987428i	$0.449473\pi$
$888$	0	0
$889$	0	0
$890$	−8.85451e9	−0.421017
$891$	0	0
$892$	1.01415e10	0.478440
$893$	−7.04193e8	−0.0330911
$894$	0	0
$895$	−1.70806e10	−0.796385
$896$	0	0
$897$	0	0
$898$	1.04042e10	0.479450
$899$	1.31284e9	0.0602632
$900$	0	0
$901$	6.69933e9	0.305137
$902$	−4.98183e9	−0.226030
$903$	0	0
$904$	−2.69041e10	−1.21124
$905$	2.11313e9	0.0947666
$906$	0	0
$907$	2.43742e9	0.108469	0.0542344	−	0.998528i	$-0.482728\pi$
$907$	2.43742e9	0.108469	0.0542344	+	0.998528i	$0.482728\pi$
$908$	−2.98359e9	−0.132263
$909$	0	0
$910$	0	0
$911$	−2.15307e10	−0.943502	−0.471751	−	0.881732i	$-0.656378\pi$
$911$	−2.15307e10	−0.943502	−0.471751	+	0.881732i	$0.656378\pi$
$912$	0	0
$913$	−1.63823e8	−0.00712405
$914$	5.56859e10	2.41231
$915$	0	0
$916$	4.19071e9	0.180158
$917$	0	0
$918$	0	0
$919$	3.28396e10	1.39571	0.697853	−	0.716241i	$-0.254139\pi$
$919$	3.28396e10	1.39571	0.697853	+	0.716241i	$0.254139\pi$
$920$	3.19939e10	1.35460
$921$	0	0
$922$	−5.56069e10	−2.33652
$923$	−4.26943e10	−1.78716
$924$	0	0
$925$	1.71469e9	0.0712342
$926$	3.58892e10	1.48534
$927$	0	0
$928$	−1.61823e10	−0.664695
$929$	1.28438e10	0.525579	0.262789	−	0.964853i	$-0.415358\pi$
$929$	1.28438e10	0.525579	0.262789	+	0.964853i	$0.415358\pi$
$930$	0	0
$931$	0	0
$932$	−7.01547e10	−2.83858
$933$	0	0
$934$	−5.88713e10	−2.36423
$935$	−7.33886e8	−0.0293622
$936$	0	0
$937$	2.05391e10	0.815628	0.407814	−	0.913065i	$-0.366291\pi$
$937$	2.05391e10	0.815628	0.407814	+	0.913065i	$0.366291\pi$
$938$	0	0
$939$	0	0
$940$	3.70448e10	1.45472
$941$	1.61326e10	0.631160	0.315580	−	0.948899i	$-0.397801\pi$
$941$	1.61326e10	0.631160	0.315580	+	0.948899i	$0.397801\pi$
$942$	0	0
$943$	3.51983e10	1.36688
$944$	8.03746e10	3.10969
$945$	0	0
$946$	3.98583e9	0.153073
$947$	−3.26671e9	−0.124993	−0.0624965	−	0.998045i	$-0.519906\pi$
$947$	−3.26671e9	−0.124993	−0.0624965	+	0.998045i	$0.519906\pi$
$948$	0	0
$949$	7.02300e9	0.266742
$950$	9.52833e8	0.0360565
$951$	0	0
$952$	0	0
$953$	−2.56668e10	−0.960609	−0.480304	−	0.877102i	$-0.659474\pi$
$953$	−2.56668e10	−0.960609	−0.480304	+	0.877102i	$0.659474\pi$
$954$	0	0
$955$	2.02990e10	0.754161
$956$	3.61086e10	1.33662
$957$	0	0
$958$	−6.44341e10	−2.36775
$959$	0	0
$960$	0	0
$961$	−2.68556e10	−0.976118
$962$	−8.67326e9	−0.314101
$963$	0	0
$964$	−6.77711e10	−2.43655
$965$	3.06662e9	0.109854
$966$	0	0
$967$	−4.14546e10	−1.47428	−0.737141	−	0.675739i	$-0.763825\pi$
$967$	−4.14546e10	−1.47428	−0.737141	+	0.675739i	$0.763825\pi$
$968$	−7.17459e10	−2.54234
$969$	0	0
$970$	4.70400e10	1.65488
$971$	1.40608e10	0.492881	0.246440	−	0.969158i	$-0.420739\pi$
$971$	1.40608e10	0.492881	0.246440	+	0.969158i	$0.420739\pi$
$972$	0	0
$973$	0	0
$974$	−8.81046e9	−0.305522
$975$	0	0
$976$	−1.02916e11	−3.54331
$977$	2.44917e10	0.840209	0.420104	−	0.907476i	$-0.361993\pi$
$977$	2.44917e10	0.840209	0.420104	+	0.907476i	$0.361993\pi$
$978$	0	0
$979$	6.68234e8	0.0227609
$980$	0	0
$981$	0	0
$982$	2.05312e10	0.691868
$983$	5.74323e10	1.92850	0.964249	−	0.264997i	$-0.0853711\pi$
$983$	5.74323e10	1.92850	0.964249	+	0.264997i	$0.0853711\pi$
$984$	0	0
$985$	−9.66638e8	−0.0322283
$986$	−1.33152e10	−0.442363
$987$	0	0
$988$	−3.39700e9	−0.112059
$989$	−2.81612e10	−0.925687
$990$	0	0
$991$	−2.45597e10	−0.801613	−0.400806	−	0.916163i	$-0.631270\pi$
$991$	−2.45597e10	−0.801613	−0.400806	+	0.916163i	$0.631270\pi$
$992$	−8.09898e9	−0.263414
$993$	0	0
$994$	0	0
$995$	−1.06423e10	−0.342497
$996$	0	0
$997$	1.65865e10	0.530056	0.265028	−	0.964241i	$-0.414619\pi$
$997$	1.65865e10	0.530056	0.265028	+	0.964241i	$0.414619\pi$
$998$	−1.09272e10	−0.347978
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.8.a.t.1.4		4
3.2	odd	2		49.8.a.e.1.1		4
7.3	odd	6		63.8.e.b.37.1		8
7.5	odd	6		63.8.e.b.46.1		8
7.6	odd	2		441.8.a.s.1.4		4
21.2	odd	6		49.8.c.g.18.4		8
21.5	even	6		7.8.c.a.4.4	yes	8
21.11	odd	6		49.8.c.g.30.4		8
21.17	even	6		7.8.c.a.2.4	✓	8
21.20	even	2		49.8.a.f.1.1		4
84.47	odd	6		112.8.i.c.81.1		8
84.59	odd	6		112.8.i.c.65.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.8.c.a.2.4	✓	8	21.17	even	6
7.8.c.a.4.4	yes	8	21.5	even	6
49.8.a.e.1.1		4	3.2	odd	2
49.8.a.f.1.1		4	21.20	even	2
49.8.c.g.18.4		8	21.2	odd	6
49.8.c.g.30.4		8	21.11	odd	6
63.8.e.b.37.1		8	7.3	odd	6
63.8.e.b.46.1		8	7.5	odd	6
112.8.i.c.65.1		8	84.59	odd	6
112.8.i.c.81.1		8	84.47	odd	6
441.8.a.s.1.4		4	7.6	odd	2
441.8.a.t.1.4		4	1.1	even	1	trivial

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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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+20.8241 q^{2} +305.642 q^{4} +193.403 q^{5} +3699.23 q^{8} +O(q^{10})\)	\(q+20.8241 q^{2} +305.642 q^{4} +193.403 q^{5} +3699.23 q^{8} +4027.44 q^{10} -303.944 q^{11} +9891.04 q^{13} +37910.9 q^{16} +12484.5 q^{17} -1123.67 q^{19} +59112.1 q^{20} -6329.35 q^{22} +44719.0 q^{23} -40720.2 q^{25} +205972. q^{26} -51216.6 q^{29} -25633.0 q^{31} +315958. q^{32} +259978. q^{34} -42109.0 q^{37} -23399.5 q^{38} +715443. q^{40} +787100. q^{41} -629737. q^{43} -92898.1 q^{44} +931232. q^{46} +626687. q^{47} -847961. q^{50} +3.02312e6 q^{52} +536611. q^{53} -58783.7 q^{55} -1.06654e6 q^{58} +2.12009e6 q^{59} -2.71468e6 q^{61} -533785. q^{62} +1.72694e6 q^{64} +1.91296e6 q^{65} +3.77405e6 q^{67} +3.81579e6 q^{68} -4.31646e6 q^{71} +710037. q^{73} -876880. q^{74} -343442. q^{76} +2.79211e6 q^{79} +7.33209e6 q^{80} +1.63906e7 q^{82} +538991. q^{83} +2.41454e6 q^{85} -1.31137e7 q^{86} -1.12436e6 q^{88} -2.19854e6 q^{89} +1.36680e7 q^{92} +1.30502e7 q^{94} -217322. q^{95} +1.16799e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{2} + 348 q^{4} + 252 q^{5} + 984 q^{8}+O(q^{10}) \) 4 * q + 6 * q^2 + 348 * q^4 + 252 * q^5 + 984 * q^8	\( 4 q + 6 q^{2} + 348 q^{4} + 252 q^{5} + 984 q^{8} - 4774 q^{10} + 3972 q^{11} + 1176 q^{13} + 57264 q^{16} + 56364 q^{17} - 41748 q^{19} + 162372 q^{20} - 152954 q^{22} - 131748 q^{23} - 68016 q^{25} + 113652 q^{26} - 34056 q^{29} - 401212 q^{31} + 453408 q^{32} - 82110 q^{34} - 5396 q^{37} + 31794 q^{38} + 443688 q^{40} + 410424 q^{41} + 46544 q^{43} + 1465836 q^{44} + 1379202 q^{46} + 1470084 q^{47} - 1395528 q^{50} + 5269768 q^{52} + 642372 q^{53} + 3063340 q^{55} - 1972220 q^{58} + 752220 q^{59} - 1325772 q^{61} - 3016314 q^{62} - 865856 q^{64} + 4868808 q^{65} - 290916 q^{67} + 2453556 q^{68} - 3377760 q^{71} - 6706588 q^{73} - 3616878 q^{74} - 2923004 q^{76} + 3946244 q^{79} + 10695888 q^{80} + 17563252 q^{82} + 9542064 q^{83} + 7006068 q^{85} - 23237832 q^{86} - 8043336 q^{88} + 16165212 q^{89} + 19442628 q^{92} - 5720442 q^{94} + 3268452 q^{95} - 1533112 q^{97}+O(q^{100}) \) 4 * q + 6 * q^2 + 348 * q^4 + 252 * q^5 + 984 * q^8 - 4774 * q^10 + 3972 * q^11 + 1176 * q^13 + 57264 * q^16 + 56364 * q^17 - 41748 * q^19 + 162372 * q^20 - 152954 * q^22 - 131748 * q^23 - 68016 * q^25 + 113652 * q^26 - 34056 * q^29 - 401212 * q^31 + 453408 * q^32 - 82110 * q^34 - 5396 * q^37 + 31794 * q^38 + 443688 * q^40 + 410424 * q^41 + 46544 * q^43 + 1465836 * q^44 + 1379202 * q^46 + 1470084 * q^47 - 1395528 * q^50 + 5269768 * q^52 + 642372 * q^53 + 3063340 * q^55 - 1972220 * q^58 + 752220 * q^59 - 1325772 * q^61 - 3016314 * q^62 - 865856 * q^64 + 4868808 * q^65 - 290916 * q^67 + 2453556 * q^68 - 3377760 * q^71 - 6706588 * q^73 - 3616878 * q^74 - 2923004 * q^76 + 3946244 * q^79 + 10695888 * q^80 + 17563252 * q^82 + 9542064 * q^83 + 7006068 * q^85 - 23237832 * q^86 - 8043336 * q^88 + 16165212 * q^89 + 19442628 * q^92 - 5720442 * q^94 + 3268452 * q^95 - 1533112 * q^97

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