LMFDB - Embedded newform 272.10.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,10,Mod(1,272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("272.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	272.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:	$140.089747437$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 2986x^{5} + 8252x^{4} + 2252056x^{3} - 10388768x^{2} - 243559296x - 675998208 $ x^7 - x^6 - 2986x^5 + 8252x^4 + 2252056x^3 - 10388768x^2 - 243559296*x - 675998208
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-34.1532$ of defining polynomial
Character	$\chi$	$=$	272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-169.801 q^{3} +195.287 q^{5} +356.628 q^{7} +9149.54 q^{9} +O(q^{10})$	$q-169.801 q^{3} +195.287 q^{5} +356.628 q^{7} +9149.54 q^{9} +21467.8 q^{11} -6206.76 q^{13} -33160.0 q^{15} +83521.0 q^{17} -907187. q^{19} -60556.0 q^{21} +1.23486e6 q^{23} -1.91499e6 q^{25} +1.78860e6 q^{27} -3.01596e6 q^{29} +334277. q^{31} -3.64527e6 q^{33} +69644.7 q^{35} +2.06102e7 q^{37} +1.05392e6 q^{39} +1.47571e7 q^{41} +7.75953e6 q^{43} +1.78679e6 q^{45} -3.19993e7 q^{47} -4.02264e7 q^{49} -1.41820e7 q^{51} +9.46750e7 q^{53} +4.19238e6 q^{55} +1.54042e8 q^{57} -6.01771e7 q^{59} +6.05254e7 q^{61} +3.26298e6 q^{63} -1.21210e6 q^{65} +1.26316e8 q^{67} -2.09681e8 q^{69} -3.33917e7 q^{71} +2.85706e8 q^{73} +3.25168e8 q^{75} +7.65603e6 q^{77} +7.60575e7 q^{79} -4.83797e8 q^{81} +1.73620e7 q^{83} +1.63105e7 q^{85} +5.12115e8 q^{87} +3.96876e8 q^{89} -2.21350e6 q^{91} -5.67608e7 q^{93} -1.77162e8 q^{95} +1.13120e9 q^{97} +1.96421e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9}+O(q^{10}) $ 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9	$ 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9} - 135536 q^{11} + 166122 q^{13} - 159048 q^{15} + 584647 q^{17} - 777172 q^{19} - 3412104 q^{21} - 1357764 q^{23} + 1065785 q^{25} + 4519064 q^{27} + 967002 q^{29} - 3546740 q^{31} + 11928016 q^{33} + 530736 q^{35} + 18296498 q^{37} - 86306872 q^{39} + 10285686 q^{41} - 21913204 q^{43} + 108916410 q^{45} - 56639800 q^{47} + 27010351 q^{49} - 7349848 q^{51} + 121813562 q^{53} - 40793128 q^{55} + 153612960 q^{57} - 29222388 q^{59} - 49915846 q^{61} + 2185356 q^{63} - 122633668 q^{65} - 301863420 q^{67} + 379683432 q^{69} - 652473940 q^{71} + 306656342 q^{73} - 919071912 q^{75} - 102442536 q^{77} - 959147884 q^{79} - 374486977 q^{81} + 1512945268 q^{83} + 113755602 q^{85} + 1612550856 q^{87} - 1971327114 q^{89} + 1061062864 q^{91} - 798598936 q^{93} + 3249631512 q^{95} + 2006526254 q^{97} + 2579159272 q^{99}+O(q^{100}) $ 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9 - 135536 * q^11 + 166122 * q^13 - 159048 * q^15 + 584647 * q^17 - 777172 * q^19 - 3412104 * q^21 - 1357764 * q^23 + 1065785 * q^25 + 4519064 * q^27 + 967002 * q^29 - 3546740 * q^31 + 11928016 * q^33 + 530736 * q^35 + 18296498 * q^37 - 86306872 * q^39 + 10285686 * q^41 - 21913204 * q^43 + 108916410 * q^45 - 56639800 * q^47 + 27010351 * q^49 - 7349848 * q^51 + 121813562 * q^53 - 40793128 * q^55 + 153612960 * q^57 - 29222388 * q^59 - 49915846 * q^61 + 2185356 * q^63 - 122633668 * q^65 - 301863420 * q^67 + 379683432 * q^69 - 652473940 * q^71 + 306656342 * q^73 - 919071912 * q^75 - 102442536 * q^77 - 959147884 * q^79 - 374486977 * q^81 + 1512945268 * q^83 + 113755602 * q^85 + 1612550856 * q^87 - 1971327114 * q^89 + 1061062864 * q^91 - 798598936 * q^93 + 3249631512 * q^95 + 2006526254 * q^97 + 2579159272 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−169.801	−1.21031	−0.605154	−	0.796109i	$-0.706888\pi$
$3$	−169.801	−1.21031	−0.605154	+	0.796109i	$0.706888\pi$
$4$	0	0
$5$	195.287	0.139736	0.0698679	−	0.997556i	$-0.477742\pi$
$5$	195.287	0.139736	0.0698679	+	0.997556i	$0.477742\pi$
$6$	0	0
$7$	356.628	0.0561402	0.0280701	−	0.999606i	$-0.491064\pi$
$7$	356.628	0.0561402	0.0280701	+	0.999606i	$0.491064\pi$
$8$	0	0
$9$	9149.54	0.464845
$10$	0	0
$11$	21467.8	0.442101	0.221050	−	0.975262i	$-0.429051\pi$
$11$	21467.8	0.442101	0.221050	+	0.975262i	$0.429051\pi$
$12$	0	0
$13$	−6206.76	−0.0602726	−0.0301363	−	0.999546i	$-0.509594\pi$
$13$	−6206.76	−0.0602726	−0.0301363	+	0.999546i	$0.509594\pi$
$14$	0	0
$15$	−33160.0	−0.169123
$16$	0	0
$17$	83521.0	0.242536
$17$	83521.0	0.242536
$18$	0	0
$19$	−907187.	−1.59700	−0.798501	−	0.601993i	$-0.794373\pi$
$19$	−907187.	−1.59700	−0.798501	+	0.601993i	$0.794373\pi$
$20$	0	0
$21$	−60556.0	−0.0679470
$22$	0	0
$23$	1.23486e6	0.920115	0.460058	−	0.887889i	$-0.347829\pi$
$23$	1.23486e6	0.920115	0.460058	+	0.887889i	$0.347829\pi$
$24$	0	0
$25$	−1.91499e6	−0.980474
$26$	0	0
$27$	1.78860e6	0.647702
$28$	0	0
$29$	−3.01596e6	−0.791835	−0.395917	−	0.918286i	$-0.629573\pi$
$29$	−3.01596e6	−0.791835	−0.395917	+	0.918286i	$0.629573\pi$
$30$	0	0
$31$	334277.	0.0650099	0.0325049	−	0.999472i	$-0.489652\pi$
$31$	334277.	0.0650099	0.0325049	+	0.999472i	$0.489652\pi$
$32$	0	0
$33$	−3.64527e6	−0.535078
$34$	0	0
$35$	69644.7	0.00784481
$36$	0	0
$37$	2.06102e7	1.80790	0.903949	−	0.427640i	$-0.140655\pi$
$37$	2.06102e7	1.80790	0.903949	+	0.427640i	$0.140655\pi$
$38$	0	0
$39$	1.05392e6	0.0729484
$40$	0	0
$41$	1.47571e7	0.815593	0.407796	−	0.913073i	$-0.366297\pi$
$41$	1.47571e7	0.815593	0.407796	+	0.913073i	$0.366297\pi$
$42$	0	0
$43$	7.75953e6	0.346120	0.173060	−	0.984911i	$-0.444634\pi$
$43$	7.75953e6	0.346120	0.173060	+	0.984911i	$0.444634\pi$
$44$	0	0
$45$	1.78679e6	0.0649555
$46$	0	0
$47$	−3.19993e7	−0.956534	−0.478267	−	0.878214i	$-0.658735\pi$
$47$	−3.19993e7	−0.956534	−0.478267	+	0.878214i	$0.658735\pi$
$48$	0	0
$49$	−4.02264e7	−0.996848
$50$	0	0
$51$	−1.41820e7	−0.293543
$52$	0	0
$53$	9.46750e7	1.64814	0.824070	−	0.566488i	$-0.191698\pi$
$53$	9.46750e7	1.64814	0.824070	+	0.566488i	$0.191698\pi$
$54$	0	0
$55$	4.19238e6	0.0617773
$56$	0	0
$57$	1.54042e8	1.93286
$58$	0	0
$59$	−6.01771e7	−0.646542	−0.323271	−	0.946306i	$-0.604783\pi$
$59$	−6.01771e7	−0.646542	−0.323271	+	0.946306i	$0.604783\pi$
$60$	0	0
$61$	6.05254e7	0.559698	0.279849	−	0.960044i	$-0.409716\pi$
$61$	6.05254e7	0.559698	0.279849	+	0.960044i	$0.409716\pi$
$62$	0	0
$63$	3.26298e6	0.0260965
$64$	0	0
$65$	−1.21210e6	−0.00842224
$66$	0	0
$67$	1.26316e8	0.765813	0.382907	−	0.923787i	$-0.374923\pi$
$67$	1.26316e8	0.765813	0.382907	+	0.923787i	$0.374923\pi$
$68$	0	0
$69$	−2.09681e8	−1.11362
$70$	0	0
$71$	−3.33917e7	−0.155947	−0.0779734	−	0.996955i	$-0.524845\pi$
$71$	−3.33917e7	−0.155947	−0.0779734	+	0.996955i	$0.524845\pi$
$72$	0	0
$73$	2.85706e8	1.17752	0.588758	−	0.808309i	$-0.299617\pi$
$73$	2.85706e8	1.17752	0.588758	+	0.808309i	$0.299617\pi$
$74$	0	0
$75$	3.25168e8	1.18668
$76$	0	0
$77$	7.65603e6	0.0248196
$78$	0	0
$79$	7.60575e7	0.219695	0.109848	−	0.993948i	$-0.464964\pi$
$79$	7.60575e7	0.219695	0.109848	+	0.993948i	$0.464964\pi$
$80$	0	0
$81$	−4.83797e8	−1.24876
$82$	0	0
$83$	1.73620e7	0.0401558	0.0200779	−	0.999798i	$-0.493609\pi$
$83$	1.73620e7	0.0401558	0.0200779	+	0.999798i	$0.493609\pi$
$84$	0	0
$85$	1.63105e7	0.0338909
$86$	0	0
$87$	5.12115e8	0.958364
$88$	0	0
$89$	3.96876e8	0.670501	0.335251	−	0.942129i	$-0.391179\pi$
$89$	3.96876e8	0.670501	0.335251	+	0.942129i	$0.391179\pi$
$90$	0	0
$91$	−2.21350e6	−0.00338372
$92$	0	0
$93$	−5.67608e7	−0.0786819
$94$	0	0
$95$	−1.77162e8	−0.223159
$96$	0	0
$97$	1.13120e9	1.29737	0.648687	−	0.761055i	$-0.275318\pi$
$97$	1.13120e9	1.29737	0.648687	+	0.761055i	$0.275318\pi$
$98$	0	0
$99$	1.96421e8	0.205508
$100$	0	0
$101$	−1.58638e9	−1.51692	−0.758458	−	0.651722i	$-0.774047\pi$
$101$	−1.58638e9	−1.51692	−0.758458	+	0.651722i	$0.774047\pi$
$102$	0	0
$103$	9.77937e8	0.856137	0.428068	−	0.903746i	$-0.359194\pi$
$103$	9.77937e8	0.856137	0.428068	+	0.903746i	$0.359194\pi$
$104$	0	0
$105$	−1.18258e7	−0.00949463
$106$	0	0
$107$	1.71253e8	0.126302	0.0631511	−	0.998004i	$-0.479885\pi$
$107$	1.71253e8	0.126302	0.0631511	+	0.998004i	$0.479885\pi$
$108$	0	0
$109$	1.66962e9	1.13292	0.566461	−	0.824089i	$-0.308312\pi$
$109$	1.66962e9	1.13292	0.566461	+	0.824089i	$0.308312\pi$
$110$	0	0
$111$	−3.49964e9	−2.18811
$112$	0	0
$113$	−2.55660e9	−1.47506	−0.737531	−	0.675313i	$-0.764008\pi$
$113$	−2.55660e9	−1.47506	−0.737531	+	0.675313i	$0.764008\pi$
$114$	0	0
$115$	2.41152e8	0.128573
$116$	0	0
$117$	−5.67890e7	−0.0280174
$118$	0	0
$119$	2.97859e7	0.0136160
$120$	0	0
$121$	−1.89708e9	−0.804547
$122$	0	0
$123$	−2.50578e9	−0.987118
$124$	0	0
$125$	−7.55391e8	−0.276743
$126$	0	0
$127$	1.57297e9	0.536542	0.268271	−	0.963344i	$-0.413548\pi$
$127$	1.57297e9	0.536542	0.268271	+	0.963344i	$0.413548\pi$
$128$	0	0
$129$	−1.31758e9	−0.418912
$130$	0	0
$131$	−1.54185e9	−0.457426	−0.228713	−	0.973494i	$-0.573452\pi$
$131$	−1.54185e9	−0.457426	−0.228713	+	0.973494i	$0.573452\pi$
$132$	0	0
$133$	−3.23528e8	−0.0896561
$134$	0	0
$135$	3.49289e8	0.0905072
$136$	0	0
$137$	−7.48495e9	−1.81529	−0.907646	−	0.419736i	$-0.862123\pi$
$137$	−7.48495e9	−1.81529	−0.907646	+	0.419736i	$0.862123\pi$
$138$	0	0
$139$	−4.97076e9	−1.12942	−0.564711	−	0.825289i	$-0.691012\pi$
$139$	−4.97076e9	−1.12942	−0.564711	+	0.825289i	$0.691012\pi$
$140$	0	0
$141$	5.43353e9	1.15770
$142$	0	0
$143$	−1.33246e8	−0.0266465
$144$	0	0
$145$	−5.88977e8	−0.110648
$146$	0	0
$147$	6.83051e9	1.20649
$148$	0	0
$149$	−2.17167e9	−0.360957	−0.180479	−	0.983579i	$-0.557765\pi$
$149$	−2.17167e9	−0.360957	−0.180479	+	0.983579i	$0.557765\pi$
$150$	0	0
$151$	−5.05353e9	−0.791040	−0.395520	−	0.918457i	$-0.629436\pi$
$151$	−5.05353e9	−0.791040	−0.395520	+	0.918457i	$0.629436\pi$
$152$	0	0
$153$	7.64179e8	0.112741
$154$	0	0
$155$	6.52799e7	0.00908421
$156$	0	0
$157$	−1.14585e10	−1.50515	−0.752577	−	0.658504i	$-0.771190\pi$
$157$	−1.14585e10	−1.50515	−0.752577	+	0.658504i	$0.771190\pi$
$158$	0	0
$159$	−1.60760e10	−1.99476
$160$	0	0
$161$	4.40386e8	0.0516555
$162$	0	0
$163$	−1.58612e10	−1.75992	−0.879960	−	0.475048i	$-0.842431\pi$
$163$	−1.58612e10	−1.75992	−0.879960	+	0.475048i	$0.842431\pi$
$164$	0	0
$165$	−7.11873e8	−0.0747696
$166$	0	0
$167$	7.94993e9	0.790932	0.395466	−	0.918481i	$-0.370583\pi$
$167$	7.94993e9	0.790932	0.395466	+	0.918481i	$0.370583\pi$
$168$	0	0
$169$	−1.05660e10	−0.996367
$170$	0	0
$171$	−8.30035e9	−0.742359
$172$	0	0
$173$	4.82389e9	0.409440	0.204720	−	0.978821i	$-0.434372\pi$
$173$	4.82389e9	0.409440	0.204720	+	0.978821i	$0.434372\pi$
$174$	0	0
$175$	−6.82938e8	−0.0550440
$176$	0	0
$177$	1.02182e10	0.782515
$178$	0	0
$179$	6.50866e9	0.473863	0.236932	−	0.971526i	$-0.423858\pi$
$179$	6.50866e9	0.473863	0.236932	+	0.971526i	$0.423858\pi$
$180$	0	0
$181$	9.87513e9	0.683895	0.341947	−	0.939719i	$-0.388914\pi$
$181$	9.87513e9	0.683895	0.341947	+	0.939719i	$0.388914\pi$
$182$	0	0
$183$	−1.02773e10	−0.677407
$184$	0	0
$185$	4.02490e9	0.252628
$186$	0	0
$187$	1.79301e9	0.107225
$188$	0	0
$189$	6.37864e8	0.0363622
$190$	0	0
$191$	−1.73272e10	−0.942060	−0.471030	−	0.882117i	$-0.656118\pi$
$191$	−1.73272e10	−0.942060	−0.471030	+	0.882117i	$0.656118\pi$
$192$	0	0
$193$	−5.92773e9	−0.307525	−0.153763	−	0.988108i	$-0.549139\pi$
$193$	−5.92773e9	−0.307525	−0.153763	+	0.988108i	$0.549139\pi$
$194$	0	0
$195$	2.05816e8	0.0101935
$196$	0	0
$197$	−1.61285e10	−0.762948	−0.381474	−	0.924380i	$-0.624583\pi$
$197$	−1.61285e10	−0.762948	−0.381474	+	0.924380i	$0.624583\pi$
$198$	0	0
$199$	4.38224e10	1.98088	0.990438	−	0.137958i	$-0.0440539\pi$
$199$	4.38224e10	1.98088	0.990438	+	0.137958i	$0.0440539\pi$
$200$	0	0
$201$	−2.14487e10	−0.926870
$202$	0	0
$203$	−1.07558e9	−0.0444538
$204$	0	0
$205$	2.88186e9	0.113968
$206$	0	0
$207$	1.12984e10	0.427711
$208$	0	0
$209$	−1.94753e10	−0.706036
$210$	0	0
$211$	−4.85013e10	−1.68454	−0.842272	−	0.539053i	$-0.818782\pi$
$211$	−4.85013e10	−1.68454	−0.842272	+	0.539053i	$0.818782\pi$
$212$	0	0
$213$	5.66997e9	0.188744
$214$	0	0
$215$	1.51533e9	0.0483654
$216$	0	0
$217$	1.19213e8	0.00364967
$218$	0	0
$219$	−4.85133e10	−1.42516
$220$	0	0
$221$	−5.18395e8	−0.0146182
$222$	0	0
$223$	1.32607e9	0.0359082	0.0179541	−	0.999839i	$-0.494285\pi$
$223$	1.32607e9	0.0359082	0.0179541	+	0.999839i	$0.494285\pi$
$224$	0	0
$225$	−1.75213e10	−0.455768
$226$	0	0
$227$	6.30032e10	1.57487	0.787437	−	0.616395i	$-0.211407\pi$
$227$	6.30032e10	1.57487	0.787437	+	0.616395i	$0.211407\pi$
$228$	0	0
$229$	3.36520e10	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$229$	3.36520e10	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$230$	0	0
$231$	−1.30001e9	−0.0300394
$232$	0	0
$233$	−3.25593e10	−0.723724	−0.361862	−	0.932232i	$-0.617859\pi$
$233$	−3.25593e10	−0.723724	−0.361862	+	0.932232i	$0.617859\pi$
$234$	0	0
$235$	−6.24905e9	−0.133662
$236$	0	0
$237$	−1.29147e10	−0.265899
$238$	0	0
$239$	−4.25086e10	−0.842725	−0.421363	−	0.906892i	$-0.638448\pi$
$239$	−4.25086e10	−0.842725	−0.421363	+	0.906892i	$0.638448\pi$
$240$	0	0
$241$	−6.46314e10	−1.23415	−0.617074	−	0.786905i	$-0.711682\pi$
$241$	−6.46314e10	−1.23415	−0.617074	+	0.786905i	$0.711682\pi$
$242$	0	0
$243$	4.69445e10	0.863687
$244$	0	0
$245$	−7.85569e9	−0.139295
$246$	0	0
$247$	5.63069e9	0.0962555
$248$	0	0
$249$	−2.94809e9	−0.0486008
$250$	0	0
$251$	1.00795e11	1.60290	0.801452	−	0.598060i	$-0.204061\pi$
$251$	1.00795e11	1.60290	0.801452	+	0.598060i	$0.204061\pi$
$252$	0	0
$253$	2.65098e10	0.406784
$254$	0	0
$255$	−2.76956e9	−0.0410185
$256$	0	0
$257$	−7.98430e10	−1.14166	−0.570831	−	0.821067i	$-0.693379\pi$
$257$	−7.98430e10	−1.14166	−0.570831	+	0.821067i	$0.693379\pi$
$258$	0	0
$259$	7.35017e9	0.101496
$260$	0	0
$261$	−2.75947e10	−0.368080
$262$	0	0
$263$	−9.26392e10	−1.19397	−0.596986	−	0.802252i	$-0.703635\pi$
$263$	−9.26392e10	−1.19397	−0.596986	+	0.802252i	$0.703635\pi$
$264$	0	0
$265$	1.84888e10	0.230304
$266$	0	0
$267$	−6.73901e10	−0.811513
$268$	0	0
$269$	−6.38400e10	−0.743375	−0.371687	−	0.928358i	$-0.621221\pi$
$269$	−6.38400e10	−0.743375	−0.371687	+	0.928358i	$0.621221\pi$
$270$	0	0
$271$	−1.39281e11	−1.56866	−0.784331	−	0.620343i	$-0.786993\pi$
$271$	−1.39281e11	−1.56866	−0.784331	+	0.620343i	$0.786993\pi$
$272$	0	0
$273$	3.75856e8	0.00409534
$274$	0	0
$275$	−4.11106e10	−0.433468
$276$	0	0
$277$	1.33421e11	1.36165	0.680826	−	0.732445i	$-0.261621\pi$
$277$	1.33421e11	1.36165	0.680826	+	0.732445i	$0.261621\pi$
$278$	0	0
$279$	3.05848e9	0.0302195
$280$	0	0
$281$	−1.22205e11	−1.16926	−0.584629	−	0.811301i	$-0.698760\pi$
$281$	−1.22205e11	−1.16926	−0.584629	+	0.811301i	$0.698760\pi$
$282$	0	0
$283$	−1.48138e11	−1.37286	−0.686430	−	0.727196i	$-0.740823\pi$
$283$	−1.48138e11	−1.37286	−0.686430	+	0.727196i	$0.740823\pi$
$284$	0	0
$285$	3.00823e10	0.270090
$286$	0	0
$287$	5.26279e9	0.0457876
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	−1.92079e11	−1.57022
$292$	0	0
$293$	1.17196e11	0.928986	0.464493	−	0.885577i	$-0.346237\pi$
$293$	1.17196e11	0.928986	0.464493	+	0.885577i	$0.346237\pi$
$294$	0	0
$295$	−1.17518e10	−0.0903452
$296$	0	0
$297$	3.83973e10	0.286350
$298$	0	0
$299$	−7.66448e9	−0.0554577
$300$	0	0
$301$	2.76726e9	0.0194313
$302$	0	0
$303$	2.69370e11	1.83594
$304$	0	0
$305$	1.18198e10	0.0782099
$306$	0	0
$307$	4.33130e10	0.278288	0.139144	−	0.990272i	$-0.455565\pi$
$307$	4.33130e10	0.278288	0.139144	+	0.990272i	$0.455565\pi$
$308$	0	0
$309$	−1.66055e11	−1.03619
$310$	0	0
$311$	−1.79166e11	−1.08601	−0.543006	−	0.839729i	$-0.682714\pi$
$311$	−1.79166e11	−1.08601	−0.543006	+	0.839729i	$0.682714\pi$
$312$	0	0
$313$	−6.30182e10	−0.371122	−0.185561	−	0.982633i	$-0.559410\pi$
$313$	−6.30182e10	−0.371122	−0.185561	+	0.982633i	$0.559410\pi$
$314$	0	0
$315$	6.37218e8	0.00364662
$316$	0	0
$317$	4.25331e10	0.236570	0.118285	−	0.992980i	$-0.462260\pi$
$317$	4.25331e10	0.236570	0.118285	+	0.992980i	$0.462260\pi$
$318$	0	0
$319$	−6.47461e10	−0.350071
$320$	0	0
$321$	−2.90790e10	−0.152865
$322$	0	0
$323$	−7.57692e10	−0.387330
$324$	0	0
$325$	1.18859e10	0.0590957
$326$	0	0
$327$	−2.83505e11	−1.37118
$328$	0	0
$329$	−1.14119e10	−0.0537001
$330$	0	0
$331$	1.36038e11	0.622921	0.311461	−	0.950259i	$-0.399182\pi$
$331$	1.36038e11	0.622921	0.311461	+	0.950259i	$0.399182\pi$
$332$	0	0
$333$	1.88574e11	0.840393
$334$	0	0
$335$	2.46679e10	0.107012
$336$	0	0
$337$	−4.52614e11	−1.91158	−0.955792	−	0.294042i	$-0.904999\pi$
$337$	−4.52614e11	−1.91158	−0.955792	+	0.294042i	$0.904999\pi$
$338$	0	0
$339$	4.34115e11	1.78528
$340$	0	0
$341$	7.17621e9	0.0287409
$342$	0	0
$343$	−2.87371e10	−0.112104
$344$	0	0
$345$	−4.09479e10	−0.155613
$346$	0	0
$347$	2.00653e11	0.742955	0.371478	−	0.928442i	$-0.378851\pi$
$347$	2.00653e11	0.742955	0.371478	+	0.928442i	$0.378851\pi$
$348$	0	0
$349$	1.48945e11	0.537416	0.268708	−	0.963222i	$-0.413403\pi$
$349$	1.48945e11	0.537416	0.268708	+	0.963222i	$0.413403\pi$
$350$	0	0
$351$	−1.11014e10	−0.0390387
$352$	0	0
$353$	−3.46998e11	−1.18944	−0.594718	−	0.803934i	$-0.702736\pi$
$353$	−3.46998e11	−1.18944	−0.594718	+	0.803934i	$0.702736\pi$
$354$	0	0
$355$	−6.52097e9	−0.0217914
$356$	0	0
$357$	−5.05769e9	−0.0164796
$358$	0	0
$359$	5.87619e11	1.86711	0.933556	−	0.358431i	$-0.116688\pi$
$359$	5.87619e11	1.86711	0.933556	+	0.358431i	$0.116688\pi$
$360$	0	0
$361$	5.00300e11	1.55042
$362$	0	0
$363$	3.22127e11	0.973750
$364$	0	0
$365$	5.57947e10	0.164541
$366$	0	0
$367$	8.17833e10	0.235325	0.117662	−	0.993054i	$-0.462460\pi$
$367$	8.17833e10	0.235325	0.117662	+	0.993054i	$0.462460\pi$
$368$	0	0
$369$	1.35021e11	0.379124
$370$	0	0
$371$	3.37638e10	0.0925269
$372$	0	0
$373$	3.70187e11	0.990218	0.495109	−	0.868831i	$-0.335128\pi$
$373$	3.70187e11	0.990218	0.495109	+	0.868831i	$0.335128\pi$
$374$	0	0
$375$	1.28267e11	0.334945
$376$	0	0
$377$	1.87193e10	0.0477259
$378$	0	0
$379$	−5.17440e11	−1.28820	−0.644100	−	0.764941i	$-0.722768\pi$
$379$	−5.17440e11	−1.28820	−0.644100	+	0.764941i	$0.722768\pi$
$380$	0	0
$381$	−2.67092e11	−0.649380
$382$	0	0
$383$	−2.03437e11	−0.483099	−0.241549	−	0.970389i	$-0.577656\pi$
$383$	−2.03437e11	−0.483099	−0.241549	+	0.970389i	$0.577656\pi$
$384$	0	0
$385$	1.49512e9	0.00346819
$386$	0	0
$387$	7.09961e10	0.160892
$388$	0	0
$389$	7.76405e11	1.71916	0.859578	−	0.511005i	$-0.170727\pi$
$389$	7.76405e11	1.71916	0.859578	+	0.511005i	$0.170727\pi$
$390$	0	0
$391$	1.03137e11	0.223161
$392$	0	0
$393$	2.61808e11	0.553626
$394$	0	0
$395$	1.48530e10	0.0306993
$396$	0	0
$397$	4.33956e11	0.876776	0.438388	−	0.898786i	$-0.355550\pi$
$397$	4.33956e11	0.876776	0.438388	+	0.898786i	$0.355550\pi$
$398$	0	0
$399$	5.49356e10	0.108511
$400$	0	0
$401$	3.07899e11	0.594645	0.297323	−	0.954777i	$-0.403906\pi$
$401$	3.07899e11	0.594645	0.297323	+	0.954777i	$0.403906\pi$
$402$	0	0
$403$	−2.07478e9	−0.00391831
$404$	0	0
$405$	−9.44791e10	−0.174497
$406$	0	0
$407$	4.42456e11	0.799273
$408$	0	0
$409$	−5.32344e11	−0.940671	−0.470335	−	0.882488i	$-0.655867\pi$
$409$	−5.32344e11	−0.940671	−0.470335	+	0.882488i	$0.655867\pi$
$410$	0	0
$411$	1.27096e12	2.19706
$412$	0	0
$413$	−2.14608e10	−0.0362970
$414$	0	0
$415$	3.39057e9	0.00561120
$416$	0	0
$417$	8.44042e11	1.36695
$418$	0	0
$419$	1.78965e11	0.283665	0.141833	−	0.989891i	$-0.454701\pi$
$419$	1.78965e11	0.283665	0.141833	+	0.989891i	$0.454701\pi$
$420$	0	0
$421$	1.88185e11	0.291954	0.145977	−	0.989288i	$-0.453367\pi$
$421$	1.88185e11	0.291954	0.145977	+	0.989288i	$0.453367\pi$
$422$	0	0
$423$	−2.92779e11	−0.444640
$424$	0	0
$425$	−1.59942e11	−0.237800
$426$	0	0
$427$	2.15851e10	0.0314216
$428$	0	0
$429$	2.26253e10	0.0322505
$430$	0	0
$431$	6.39015e11	0.891997	0.445998	−	0.895034i	$-0.352849\pi$
$431$	6.39015e11	0.891997	0.445998	+	0.895034i	$0.352849\pi$
$432$	0	0
$433$	3.97400e11	0.543291	0.271645	−	0.962397i	$-0.412432\pi$
$433$	3.97400e11	0.543291	0.271645	+	0.962397i	$0.412432\pi$
$434$	0	0
$435$	1.00009e11	0.133918
$436$	0	0
$437$	−1.12025e12	−1.46943
$438$	0	0
$439$	1.41931e11	0.182384	0.0911918	−	0.995833i	$-0.470932\pi$
$439$	1.41931e11	0.182384	0.0911918	+	0.995833i	$0.470932\pi$
$440$	0	0
$441$	−3.68053e11	−0.463380
$442$	0	0
$443$	−1.16614e12	−1.43858	−0.719289	−	0.694711i	$-0.755532\pi$
$443$	−1.16614e12	−1.43858	−0.719289	+	0.694711i	$0.755532\pi$
$444$	0	0
$445$	7.75046e10	0.0936931
$446$	0	0
$447$	3.68753e11	0.436870
$448$	0	0
$449$	9.04654e11	1.05045	0.525223	−	0.850964i	$-0.323982\pi$
$449$	9.04654e11	1.05045	0.525223	+	0.850964i	$0.323982\pi$
$450$	0	0
$451$	3.16803e11	0.360574
$452$	0	0
$453$	8.58097e11	0.957402
$454$	0	0
$455$	−4.32268e8	−0.000472827 0
$456$	0	0
$457$	−2.89019e10	−0.0309959	−0.0154979	−	0.999880i	$-0.504933\pi$
$457$	−2.89019e10	−0.0309959	−0.0154979	+	0.999880i	$0.504933\pi$
$458$	0	0
$459$	1.49385e11	0.157091
$460$	0	0
$461$	−3.33362e11	−0.343765	−0.171883	−	0.985117i	$-0.554985\pi$
$461$	−3.33362e11	−0.343765	−0.171883	+	0.985117i	$0.554985\pi$
$462$	0	0
$463$	−9.65145e11	−0.976064	−0.488032	−	0.872826i	$-0.662285\pi$
$463$	−9.65145e11	−0.976064	−0.488032	+	0.872826i	$0.662285\pi$
$464$	0	0
$465$	−1.10846e10	−0.0109947
$466$	0	0
$467$	7.21341e11	0.701802	0.350901	−	0.936413i	$-0.385875\pi$
$467$	7.21341e11	0.701802	0.350901	+	0.936413i	$0.385875\pi$
$468$	0	0
$469$	4.50479e10	0.0429929
$470$	0	0
$471$	1.94568e12	1.82170
$472$	0	0
$473$	1.66580e11	0.153020
$474$	0	0
$475$	1.73725e12	1.56582
$476$	0	0
$477$	8.66233e11	0.766129
$478$	0	0
$479$	−1.57016e12	−1.36281	−0.681403	−	0.731909i	$-0.738630\pi$
$479$	−1.57016e12	−1.36281	−0.681403	+	0.731909i	$0.738630\pi$
$480$	0	0
$481$	−1.27922e11	−0.108967
$482$	0	0
$483$	−7.47781e10	−0.0625191
$484$	0	0
$485$	2.20908e11	0.181290
$486$	0	0
$487$	−1.48730e12	−1.19817	−0.599084	−	0.800686i	$-0.704468\pi$
$487$	−1.48730e12	−1.19817	−0.599084	+	0.800686i	$0.704468\pi$
$488$	0	0
$489$	2.69326e12	2.13004
$490$	0	0
$491$	−9.71236e11	−0.754150	−0.377075	−	0.926183i	$-0.623070\pi$
$491$	−9.71236e11	−0.754150	−0.377075	+	0.926183i	$0.623070\pi$
$492$	0	0
$493$	−2.51896e11	−0.192048
$494$	0	0
$495$	3.83584e10	0.0287169
$496$	0	0
$497$	−1.19084e10	−0.00875489
$498$	0	0
$499$	−1.21621e12	−0.878127	−0.439064	−	0.898456i	$-0.644690\pi$
$499$	−1.21621e12	−0.878127	−0.439064	+	0.898456i	$0.644690\pi$
$500$	0	0
$501$	−1.34991e12	−0.957272
$502$	0	0
$503$	−2.43415e12	−1.69548	−0.847739	−	0.530414i	$-0.822036\pi$
$503$	−2.43415e12	−1.69548	−0.847739	+	0.530414i	$0.822036\pi$
$504$	0	0
$505$	−3.09799e11	−0.211968
$506$	0	0
$507$	1.79412e12	1.20591
$508$	0	0
$509$	−2.50151e12	−1.65186	−0.825930	−	0.563773i	$-0.809349\pi$
$509$	−2.50151e12	−1.65186	−0.825930	+	0.563773i	$0.809349\pi$
$510$	0	0
$511$	1.01891e11	0.0661060
$512$	0	0
$513$	−1.62259e12	−1.03438
$514$	0	0
$515$	1.90978e11	0.119633
$516$	0	0
$517$	−6.86956e11	−0.422884
$518$	0	0
$519$	−8.19103e11	−0.495548
$520$	0	0
$521$	2.28180e12	1.35678	0.678388	−	0.734704i	$-0.262679\pi$
$521$	2.28180e12	1.35678	0.678388	+	0.734704i	$0.262679\pi$
$522$	0	0
$523$	−1.67259e12	−0.977537	−0.488768	−	0.872414i	$-0.662554\pi$
$523$	−1.67259e12	−0.977537	−0.488768	+	0.872414i	$0.662554\pi$
$524$	0	0
$525$	1.15964e11	0.0666202
$526$	0	0
$527$	2.79192e10	0.0157672
$528$	0	0
$529$	−2.76274e11	−0.153388
$530$	0	0
$531$	−5.50593e11	−0.300542
$532$	0	0
$533$	−9.15937e10	−0.0491579
$534$	0	0
$535$	3.34434e10	0.0176489
$536$	0	0
$537$	−1.10518e12	−0.573520
$538$	0	0
$539$	−8.63574e11	−0.440707
$540$	0	0
$541$	−8.84734e11	−0.444043	−0.222021	−	0.975042i	$-0.571265\pi$
$541$	−8.84734e11	−0.444043	−0.222021	+	0.975042i	$0.571265\pi$
$542$	0	0
$543$	−1.67681e12	−0.827723
$544$	0	0
$545$	3.26056e11	0.158310
$546$	0	0
$547$	2.72146e12	1.29975	0.649874	−	0.760042i	$-0.274822\pi$
$547$	2.72146e12	1.29975	0.649874	+	0.760042i	$0.274822\pi$
$548$	0	0
$549$	5.53780e11	0.260173
$550$	0	0
$551$	2.73604e12	1.26456
$552$	0	0
$553$	2.71242e10	0.0123337
$554$	0	0
$555$	−6.83433e11	−0.305758
$556$	0	0
$557$	−2.68557e12	−1.18219	−0.591096	−	0.806601i	$-0.701305\pi$
$557$	−2.68557e12	−1.18219	−0.591096	+	0.806601i	$0.701305\pi$
$558$	0	0
$559$	−4.81615e10	−0.0208616
$560$	0	0
$561$	−3.04457e11	−0.129775
$562$	0	0
$563$	2.81923e12	1.18261	0.591307	−	0.806446i	$-0.298612\pi$
$563$	2.81923e12	1.18261	0.591307	+	0.806446i	$0.298612\pi$
$564$	0	0
$565$	−4.99271e11	−0.206119
$566$	0	0
$567$	−1.72535e11	−0.0701059
$568$	0	0
$569$	3.57453e11	0.142960	0.0714799	−	0.997442i	$-0.477228\pi$
$569$	3.57453e11	0.142960	0.0714799	+	0.997442i	$0.477228\pi$
$570$	0	0
$571$	3.25953e11	0.128319	0.0641596	−	0.997940i	$-0.479563\pi$
$571$	3.25953e11	0.128319	0.0641596	+	0.997940i	$0.479563\pi$
$572$	0	0
$573$	2.94219e12	1.14018
$574$	0	0
$575$	−2.36474e12	−0.902149
$576$	0	0
$577$	−2.50107e11	−0.0939365	−0.0469683	−	0.998896i	$-0.514956\pi$
$577$	−2.50107e11	−0.0939365	−0.0469683	+	0.998896i	$0.514956\pi$
$578$	0	0
$579$	1.00654e12	0.372200
$580$	0	0
$581$	6.19177e9	0.00225435
$582$	0	0
$583$	2.03247e12	0.728643
$584$	0	0
$585$	−1.10901e10	−0.00391504
$586$	0	0
$587$	−9.14514e10	−0.0317921	−0.0158960	−	0.999874i	$-0.505060\pi$
$587$	−9.14514e10	−0.0317921	−0.0158960	+	0.999874i	$0.505060\pi$
$588$	0	0
$589$	−3.03252e11	−0.103821
$590$	0	0
$591$	2.73864e12	0.923402
$592$	0	0
$593$	−2.60815e12	−0.866138	−0.433069	−	0.901361i	$-0.642569\pi$
$593$	−2.60815e12	−0.866138	−0.433069	+	0.901361i	$0.642569\pi$
$594$	0	0
$595$	5.81680e9	0.00190264
$596$	0	0
$597$	−7.44111e12	−2.39747
$598$	0	0
$599$	−1.25671e12	−0.398853	−0.199427	−	0.979913i	$-0.563908\pi$
$599$	−1.25671e12	−0.398853	−0.199427	+	0.979913i	$0.563908\pi$
$600$	0	0
$601$	−9.01991e11	−0.282012	−0.141006	−	0.990009i	$-0.545034\pi$
$601$	−9.01991e11	−0.282012	−0.141006	+	0.990009i	$0.545034\pi$
$602$	0	0
$603$	1.15574e12	0.355984
$604$	0	0
$605$	−3.70475e11	−0.112424
$606$	0	0
$607$	−4.29234e12	−1.28335	−0.641675	−	0.766977i	$-0.721760\pi$
$607$	−4.29234e12	−1.28335	−0.641675	+	0.766977i	$0.721760\pi$
$608$	0	0
$609$	1.82634e11	0.0538028
$610$	0	0
$611$	1.98612e11	0.0576528
$612$	0	0
$613$	3.09808e12	0.886178	0.443089	−	0.896478i	$-0.353883\pi$
$613$	3.09808e12	0.886178	0.443089	+	0.896478i	$0.353883\pi$
$614$	0	0
$615$	−4.89345e11	−0.137936
$616$	0	0
$617$	−2.97483e12	−0.826378	−0.413189	−	0.910645i	$-0.635585\pi$
$617$	−2.97483e12	−0.826378	−0.413189	+	0.910645i	$0.635585\pi$
$618$	0	0
$619$	3.10484e10	0.00850025	0.00425012	−	0.999991i	$-0.498647\pi$
$619$	3.10484e10	0.00850025	0.00425012	+	0.999991i	$0.498647\pi$
$620$	0	0
$621$	2.20867e12	0.595961
$622$	0	0
$623$	1.41537e11	0.0376421
$624$	0	0
$625$	3.59269e12	0.941803
$626$	0	0
$627$	3.30694e12	0.854521
$628$	0	0
$629$	1.72138e12	0.438480
$630$	0	0
$631$	−5.20764e12	−1.30770	−0.653851	−	0.756623i	$-0.726848\pi$
$631$	−5.20764e12	−1.30770	−0.653851	+	0.756623i	$0.726848\pi$
$632$	0	0
$633$	8.23559e12	2.03882
$634$	0	0
$635$	3.07180e11	0.0749741
$636$	0	0
$637$	2.49676e11	0.0600826
$638$	0	0
$639$	−3.05519e11	−0.0724911
$640$	0	0
$641$	2.80533e12	0.656330	0.328165	−	0.944620i	$-0.393570\pi$
$641$	2.80533e12	0.656330	0.328165	+	0.944620i	$0.393570\pi$
$642$	0	0
$643$	−2.80520e12	−0.647164	−0.323582	−	0.946200i	$-0.604887\pi$
$643$	−2.80520e12	−0.647164	−0.323582	+	0.946200i	$0.604887\pi$
$644$	0	0
$645$	−2.57306e11	−0.0585371
$646$	0	0
$647$	−4.72530e12	−1.06013	−0.530066	−	0.847956i	$-0.677833\pi$
$647$	−4.72530e12	−1.06013	−0.530066	+	0.847956i	$0.677833\pi$
$648$	0	0
$649$	−1.29187e12	−0.285837
$650$	0	0
$651$	−2.02425e10	−0.00441722
$652$	0	0
$653$	−7.04467e12	−1.51618	−0.758091	−	0.652149i	$-0.773868\pi$
$653$	−7.04467e12	−1.51618	−0.758091	+	0.652149i	$0.773868\pi$
$654$	0	0
$655$	−3.01103e11	−0.0639188
$656$	0	0
$657$	2.61408e12	0.547362
$658$	0	0
$659$	2.93823e12	0.606879	0.303439	−	0.952851i	$-0.401865\pi$
$659$	2.93823e12	0.606879	0.303439	+	0.952851i	$0.401865\pi$
$660$	0	0
$661$	−7.72453e11	−0.157386	−0.0786929	−	0.996899i	$-0.525075\pi$
$661$	−7.72453e11	−0.157386	−0.0786929	+	0.996899i	$0.525075\pi$
$662$	0	0
$663$	8.80242e10	0.0176926
$664$	0	0
$665$	−6.31808e10	−0.0125282
$666$	0	0
$667$	−3.72429e12	−0.728580
$668$	0	0
$669$	−2.25168e11	−0.0434599
$670$	0	0
$671$	1.29935e12	0.247443
$672$	0	0
$673$	2.83634e12	0.532954	0.266477	−	0.963841i	$-0.414140\pi$
$673$	2.83634e12	0.532954	0.266477	+	0.963841i	$0.414140\pi$
$674$	0	0
$675$	−3.42514e12	−0.635055
$676$	0	0
$677$	−2.65986e12	−0.486642	−0.243321	−	0.969946i	$-0.578237\pi$
$677$	−2.65986e12	−0.486642	−0.243321	+	0.969946i	$0.578237\pi$
$678$	0	0
$679$	4.03416e11	0.0728349
$680$	0	0
$681$	−1.06980e13	−1.90608
$682$	0	0
$683$	2.44022e12	0.429078	0.214539	−	0.976715i	$-0.431175\pi$
$683$	2.44022e12	0.429078	0.214539	+	0.976715i	$0.431175\pi$
$684$	0	0
$685$	−1.46171e12	−0.253661
$686$	0	0
$687$	−5.71415e12	−0.978693
$688$	0	0
$689$	−5.87625e11	−0.0993376
$690$	0	0
$691$	−5.13446e12	−0.856730	−0.428365	−	0.903606i	$-0.640910\pi$
$691$	−5.13446e12	−0.856730	−0.428365	+	0.903606i	$0.640910\pi$
$692$	0	0
$693$	7.00492e10	0.0115373
$694$	0	0
$695$	−9.70724e11	−0.157821
$696$	0	0
$697$	1.23253e12	0.197810
$698$	0	0
$699$	5.52861e12	0.875929
$700$	0	0
$701$	−4.00510e12	−0.626444	−0.313222	−	0.949680i	$-0.601408\pi$
$701$	−4.00510e12	−0.626444	−0.313222	+	0.949680i	$0.601408\pi$
$702$	0	0
$703$	−1.86973e13	−2.88722
$704$	0	0
$705$	1.06110e12	0.161772
$706$	0	0
$707$	−5.65748e11	−0.0851600
$708$	0	0
$709$	−7.18171e12	−1.06738	−0.533691	−	0.845680i	$-0.679195\pi$
$709$	−7.18171e12	−1.06738	−0.533691	+	0.845680i	$0.679195\pi$
$710$	0	0
$711$	6.95892e11	0.102124
$712$	0	0
$713$	4.12785e11	0.0598166
$714$	0	0
$715$	−2.60211e10	−0.00372348
$716$	0	0
$717$	7.21802e12	1.01996
$718$	0	0
$719$	−2.19372e10	−0.00306127	−0.00153063	−	0.999999i	$-0.500487\pi$
$719$	−2.19372e10	−0.00306127	−0.00153063	+	0.999999i	$0.500487\pi$
$720$	0	0
$721$	3.48760e11	0.0480637
$722$	0	0
$723$	1.09745e13	1.49370
$724$	0	0
$725$	5.77553e12	0.776373
$726$	0	0
$727$	−3.47360e11	−0.0461185	−0.0230593	−	0.999734i	$-0.507341\pi$
$727$	−3.47360e11	−0.0461185	−0.0230593	+	0.999734i	$0.507341\pi$
$728$	0	0
$729$	1.55133e12	0.203437
$730$	0	0
$731$	6.48083e11	0.0839465
$732$	0	0
$733$	3.58285e11	0.0458417	0.0229208	−	0.999737i	$-0.492703\pi$
$733$	3.58285e11	0.0458417	0.0229208	+	0.999737i	$0.492703\pi$
$734$	0	0
$735$	1.33391e12	0.168590
$736$	0	0
$737$	2.71174e12	0.338566
$738$	0	0
$739$	2.16274e12	0.266750	0.133375	−	0.991066i	$-0.457419\pi$
$739$	2.16274e12	0.266750	0.133375	+	0.991066i	$0.457419\pi$
$740$	0	0
$741$	−9.56100e11	−0.116499
$742$	0	0
$743$	−6.42105e12	−0.772959	−0.386479	−	0.922298i	$-0.626309\pi$
$743$	−6.42105e12	−0.772959	−0.386479	+	0.922298i	$0.626309\pi$
$744$	0	0
$745$	−4.24099e11	−0.0504387
$746$	0	0
$747$	1.58854e11	0.0186662
$748$	0	0
$749$	6.10736e10	0.00709064
$750$	0	0
$751$	1.15069e13	1.32002	0.660010	−	0.751257i	$-0.270552\pi$
$751$	1.15069e13	1.32002	0.660010	+	0.751257i	$0.270552\pi$
$752$	0	0
$753$	−1.71151e13	−1.94001
$754$	0	0
$755$	−9.86888e11	−0.110537
$756$	0	0
$757$	−7.80850e12	−0.864243	−0.432122	−	0.901815i	$-0.642235\pi$
$757$	−7.80850e12	−0.864243	−0.432122	+	0.901815i	$0.642235\pi$
$758$	0	0
$759$	−4.50140e12	−0.492333
$760$	0	0
$761$	−6.58284e12	−0.711512	−0.355756	−	0.934579i	$-0.615777\pi$
$761$	−6.58284e12	−0.711512	−0.355756	+	0.934579i	$0.615777\pi$
$762$	0	0
$763$	5.95435e11	0.0636025
$764$	0	0
$765$	1.49234e11	0.0157540
$766$	0	0
$767$	3.73505e11	0.0389688
$768$	0	0
$769$	−1.27459e13	−1.31433	−0.657163	−	0.753748i	$-0.728244\pi$
$769$	−1.27459e13	−1.31433	−0.657163	+	0.753748i	$0.728244\pi$
$770$	0	0
$771$	1.35575e13	1.38176
$772$	0	0
$773$	−1.07389e13	−1.08181	−0.540907	−	0.841083i	$-0.681919\pi$
$773$	−1.07389e13	−1.08181	−0.540907	+	0.841083i	$0.681919\pi$
$774$	0	0
$775$	−6.40137e11	−0.0637405
$776$	0	0
$777$	−1.24807e12	−0.122841
$778$	0	0
$779$	−1.33874e13	−1.30250
$780$	0	0
$781$	−7.16848e11	−0.0689442
$782$	0	0
$783$	−5.39434e12	−0.512873
$784$	0	0
$785$	−2.23770e12	−0.210324
$786$	0	0
$787$	−3.76105e12	−0.349481	−0.174740	−	0.984615i	$-0.555909\pi$
$787$	−3.76105e12	−0.349481	−0.174740	+	0.984615i	$0.555909\pi$
$788$	0	0
$789$	1.57303e13	1.44507
$790$	0	0
$791$	−9.11756e11	−0.0828103
$792$	0	0
$793$	−3.75667e11	−0.0337344
$794$	0	0
$795$	−3.13942e12	−0.278739
$796$	0	0
$797$	2.18200e13	1.91555	0.957773	−	0.287524i	$-0.0928322\pi$
$797$	2.18200e13	1.91555	0.957773	+	0.287524i	$0.0928322\pi$
$798$	0	0
$799$	−2.67262e12	−0.231994
$800$	0	0
$801$	3.63123e12	0.311679
$802$	0	0
$803$	6.13349e12	0.520581
$804$	0	0
$805$	8.60015e10	0.00721813
$806$	0	0
$807$	1.08401e13	0.899712
$808$	0	0
$809$	−1.90714e13	−1.56536	−0.782681	−	0.622423i	$-0.786148\pi$
$809$	−1.90714e13	−1.56536	−0.782681	+	0.622423i	$0.786148\pi$
$810$	0	0
$811$	1.86624e13	1.51486	0.757430	−	0.652916i	$-0.226454\pi$
$811$	1.86624e13	1.51486	0.757430	+	0.652916i	$0.226454\pi$
$812$	0	0
$813$	2.36501e13	1.89856
$814$	0	0
$815$	−3.09749e12	−0.245924
$816$	0	0
$817$	−7.03934e12	−0.552755
$818$	0	0
$819$	−2.02526e10	−0.00157290
$820$	0	0
$821$	−4.79906e12	−0.368648	−0.184324	−	0.982866i	$-0.559010\pi$
$821$	−4.79906e12	−0.368648	−0.184324	+	0.982866i	$0.559010\pi$
$822$	0	0
$823$	−2.14502e13	−1.62979	−0.814895	−	0.579609i	$-0.803205\pi$
$823$	−2.14502e13	−1.62979	−0.814895	+	0.579609i	$0.803205\pi$
$824$	0	0
$825$	6.98065e12	0.524630
$826$	0	0
$827$	1.32157e13	0.982463	0.491231	−	0.871029i	$-0.336547\pi$
$827$	1.32157e13	0.982463	0.491231	+	0.871029i	$0.336547\pi$
$828$	0	0
$829$	−7.78893e12	−0.572773	−0.286386	−	0.958114i	$-0.592454\pi$
$829$	−7.78893e12	−0.572773	−0.286386	+	0.958114i	$0.592454\pi$
$830$	0	0
$831$	−2.26551e13	−1.64802
$832$	0	0
$833$	−3.35975e12	−0.241771
$834$	0	0
$835$	1.55252e12	0.110522
$836$	0	0
$837$	5.97887e11	0.0421070
$838$	0	0
$839$	9.65976e12	0.673035	0.336517	−	0.941677i	$-0.390751\pi$
$839$	9.65976e12	0.673035	0.336517	+	0.941677i	$0.390751\pi$
$840$	0	0
$841$	−5.41113e12	−0.372997
$842$	0	0
$843$	2.07506e13	1.41516
$844$	0	0
$845$	−2.06340e12	−0.139228
$846$	0	0
$847$	−6.76552e11	−0.0451675
$848$	0	0
$849$	2.51540e13	1.66158
$850$	0	0
$851$	2.54507e13	1.66348
$852$	0	0
$853$	2.43508e13	1.57486	0.787430	−	0.616404i	$-0.211411\pi$
$853$	2.43508e13	1.57486	0.787430	+	0.616404i	$0.211411\pi$
$854$	0	0
$855$	−1.62095e12	−0.103734
$856$	0	0
$857$	2.45424e13	1.55419	0.777095	−	0.629383i	$-0.216692\pi$
$857$	2.45424e13	1.55419	0.777095	+	0.629383i	$0.216692\pi$
$858$	0	0
$859$	2.12084e13	1.32904	0.664521	−	0.747270i	$-0.268636\pi$
$859$	2.12084e13	1.32904	0.664521	+	0.747270i	$0.268636\pi$
$860$	0	0
$861$	−8.93630e11	−0.0554170
$862$	0	0
$863$	−1.64190e13	−1.00762	−0.503811	−	0.863814i	$-0.668069\pi$
$863$	−1.64190e13	−1.00762	−0.503811	+	0.863814i	$0.668069\pi$
$864$	0	0
$865$	9.42042e11	0.0572134
$866$	0	0
$867$	−1.18449e12	−0.0711946
$868$	0	0
$869$	1.63279e12	0.0971273
$870$	0	0
$871$	−7.84015e11	−0.0461575
$872$	0	0
$873$	1.03499e13	0.603078
$874$	0	0
$875$	−2.69394e11	−0.0155364
$876$	0	0
$877$	7.74986e12	0.442380	0.221190	−	0.975231i	$-0.429006\pi$
$877$	7.74986e12	0.442380	0.221190	+	0.975231i	$0.429006\pi$
$878$	0	0
$879$	−1.99001e13	−1.12436
$880$	0	0
$881$	1.47006e12	0.0822135	0.0411068	−	0.999155i	$-0.486912\pi$
$881$	1.47006e12	0.0822135	0.0411068	+	0.999155i	$0.486912\pi$
$882$	0	0
$883$	2.27229e13	1.25788	0.628941	−	0.777453i	$-0.283489\pi$
$883$	2.27229e13	1.25788	0.628941	+	0.777453i	$0.283489\pi$
$884$	0	0
$885$	1.99547e12	0.109345
$886$	0	0
$887$	2.54193e13	1.37882	0.689411	−	0.724371i	$-0.257870\pi$
$887$	2.54193e13	1.37882	0.689411	+	0.724371i	$0.257870\pi$
$888$	0	0
$889$	5.60965e11	0.0301216
$890$	0	0
$891$	−1.03861e13	−0.552079
$892$	0	0
$893$	2.90294e13	1.52759
$894$	0	0
$895$	1.27106e12	0.0662157
$896$	0	0
$897$	1.30144e12	0.0671209
$898$	0	0
$899$	−1.00817e12	−0.0514771
$900$	0	0
$901$	7.90735e12	0.399733
$902$	0	0
$903$	−4.69886e11	−0.0235178
$904$	0	0
$905$	1.92848e12	0.0955646
$906$	0	0
$907$	−2.33059e13	−1.14349	−0.571745	−	0.820431i	$-0.693733\pi$
$907$	−2.33059e13	−1.14349	−0.571745	+	0.820431i	$0.693733\pi$
$908$	0	0
$909$	−1.45147e13	−0.705131
$910$	0	0
$911$	6.19635e12	0.298060	0.149030	−	0.988833i	$-0.452385\pi$
$911$	6.19635e12	0.298060	0.149030	+	0.988833i	$0.452385\pi$
$912$	0	0
$913$	3.72724e11	0.0177529
$914$	0	0
$915$	−2.00702e12	−0.0946580
$916$	0	0
$917$	−5.49866e11	−0.0256800
$918$	0	0
$919$	2.57602e13	1.19132	0.595661	−	0.803236i	$-0.296890\pi$
$919$	2.57602e13	1.19132	0.595661	+	0.803236i	$0.296890\pi$
$920$	0	0
$921$	−7.35460e12	−0.336815
$922$	0	0
$923$	2.07255e11	0.00939932
$924$	0	0
$925$	−3.94682e13	−1.77260
$926$	0	0
$927$	8.94768e12	0.397971
$928$	0	0
$929$	1.87578e13	0.826251	0.413126	−	0.910674i	$-0.364437\pi$
$929$	1.87578e13	0.826251	0.413126	+	0.910674i	$0.364437\pi$
$930$	0	0
$931$	3.64929e13	1.59197
$932$	0	0
$933$	3.04227e13	1.31441
$934$	0	0
$935$	3.50152e11	0.0149832
$936$	0	0
$937$	5.64054e12	0.239052	0.119526	−	0.992831i	$-0.461863\pi$
$937$	5.64054e12	0.239052	0.119526	+	0.992831i	$0.461863\pi$
$938$	0	0
$939$	1.07006e13	0.449172
$940$	0	0
$941$	2.26889e13	0.943321	0.471660	−	0.881780i	$-0.343655\pi$
$941$	2.26889e13	0.943321	0.471660	+	0.881780i	$0.343655\pi$
$942$	0	0
$943$	1.82229e13	0.750439
$944$	0	0
$945$	1.24566e11	0.00508110
$946$	0	0
$947$	1.49881e13	0.605582	0.302791	−	0.953057i	$-0.402082\pi$
$947$	1.49881e13	0.605582	0.302791	+	0.953057i	$0.402082\pi$
$948$	0	0
$949$	−1.77331e12	−0.0709719
$950$	0	0
$951$	−7.22218e12	−0.286323
$952$	0	0
$953$	−4.04272e13	−1.58765	−0.793827	−	0.608143i	$-0.791915\pi$
$953$	−4.04272e13	−1.58765	−0.793827	+	0.608143i	$0.791915\pi$
$954$	0	0
$955$	−3.38377e12	−0.131640
$956$	0	0
$957$	1.09940e13	0.423693
$958$	0	0
$959$	−2.66934e12	−0.101911
$960$	0	0
$961$	−2.63279e13	−0.995774
$962$	0	0
$963$	1.56689e12	0.0587109
$964$	0	0
$965$	−1.15761e12	−0.0429723
$966$	0	0
$967$	7.15269e12	0.263057	0.131529	−	0.991312i	$-0.458011\pi$
$967$	7.15269e12	0.263057	0.131529	+	0.991312i	$0.458011\pi$
$968$	0	0
$969$	1.28657e13	0.468788
$970$	0	0
$971$	−2.34500e13	−0.846556	−0.423278	−	0.906000i	$-0.639121\pi$
$971$	−2.34500e13	−0.846556	−0.423278	+	0.906000i	$0.639121\pi$
$972$	0	0
$973$	−1.77271e12	−0.0634060
$974$	0	0
$975$	−2.01824e12	−0.0715240
$976$	0	0
$977$	3.75171e13	1.31736	0.658678	−	0.752425i	$-0.271116\pi$
$977$	3.75171e13	1.31736	0.658678	+	0.752425i	$0.271116\pi$
$978$	0	0
$979$	8.52006e12	0.296429
$980$	0	0
$981$	1.52763e13	0.526633
$982$	0	0
$983$	−5.31777e13	−1.81651	−0.908256	−	0.418414i	$-0.862586\pi$
$983$	−5.31777e13	−1.81651	−0.908256	+	0.418414i	$0.862586\pi$
$984$	0	0
$985$	−3.14968e12	−0.106611
$986$	0	0
$987$	1.93775e12	0.0649936
$988$	0	0
$989$	9.58193e12	0.318471
$990$	0	0
$991$	−3.41823e13	−1.12582	−0.562912	−	0.826517i	$-0.690319\pi$
$991$	−3.41823e13	−1.12582	−0.562912	+	0.826517i	$0.690319\pi$
$992$	0	0
$993$	−2.30994e13	−0.753926
$994$	0	0
$995$	8.55794e12	0.276799
$996$	0	0
$997$	4.83269e13	1.54903	0.774517	−	0.632553i	$-0.217993\pi$
$997$	4.83269e13	1.54903	0.774517	+	0.632553i	$0.217993\pi$
$998$	0	0
$999$	3.68633e13	1.17098

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.g.1.2		7
4.3	odd	2		17.10.a.b.1.6	✓	7
12.11	even	2		153.10.a.f.1.2		7
68.67	odd	2		289.10.a.b.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.b.1.6	✓	7	4.3	odd	2
153.10.a.f.1.2		7	12.11	even	2
272.10.a.g.1.2		7	1.1	even	1	trivial
289.10.a.b.1.6		7	68.67	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	272.a (trivial)

\(f(q)\)	\(=\)	\(q-169.801 q^{3} +195.287 q^{5} +356.628 q^{7} +9149.54 q^{9} +O(q^{10})\)	\(q-169.801 q^{3} +195.287 q^{5} +356.628 q^{7} +9149.54 q^{9} +21467.8 q^{11} -6206.76 q^{13} -33160.0 q^{15} +83521.0 q^{17} -907187. q^{19} -60556.0 q^{21} +1.23486e6 q^{23} -1.91499e6 q^{25} +1.78860e6 q^{27} -3.01596e6 q^{29} +334277. q^{31} -3.64527e6 q^{33} +69644.7 q^{35} +2.06102e7 q^{37} +1.05392e6 q^{39} +1.47571e7 q^{41} +7.75953e6 q^{43} +1.78679e6 q^{45} -3.19993e7 q^{47} -4.02264e7 q^{49} -1.41820e7 q^{51} +9.46750e7 q^{53} +4.19238e6 q^{55} +1.54042e8 q^{57} -6.01771e7 q^{59} +6.05254e7 q^{61} +3.26298e6 q^{63} -1.21210e6 q^{65} +1.26316e8 q^{67} -2.09681e8 q^{69} -3.33917e7 q^{71} +2.85706e8 q^{73} +3.25168e8 q^{75} +7.65603e6 q^{77} +7.60575e7 q^{79} -4.83797e8 q^{81} +1.73620e7 q^{83} +1.63105e7 q^{85} +5.12115e8 q^{87} +3.96876e8 q^{89} -2.21350e6 q^{91} -5.67608e7 q^{93} -1.77162e8 q^{95} +1.13120e9 q^{97} +1.96421e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9}+O(q^{10}) \) 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9	\( 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9} - 135536 q^{11} + 166122 q^{13} - 159048 q^{15} + 584647 q^{17} - 777172 q^{19} - 3412104 q^{21} - 1357764 q^{23} + 1065785 q^{25} + 4519064 q^{27} + 967002 q^{29} - 3546740 q^{31} + 11928016 q^{33} + 530736 q^{35} + 18296498 q^{37} - 86306872 q^{39} + 10285686 q^{41} - 21913204 q^{43} + 108916410 q^{45} - 56639800 q^{47} + 27010351 q^{49} - 7349848 q^{51} + 121813562 q^{53} - 40793128 q^{55} + 153612960 q^{57} - 29222388 q^{59} - 49915846 q^{61} + 2185356 q^{63} - 122633668 q^{65} - 301863420 q^{67} + 379683432 q^{69} - 652473940 q^{71} + 306656342 q^{73} - 919071912 q^{75} - 102442536 q^{77} - 959147884 q^{79} - 374486977 q^{81} + 1512945268 q^{83} + 113755602 q^{85} + 1612550856 q^{87} - 1971327114 q^{89} + 1061062864 q^{91} - 798598936 q^{93} + 3249631512 q^{95} + 2006526254 q^{97} + 2579159272 q^{99}+O(q^{100}) \) 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9 - 135536 * q^11 + 166122 * q^13 - 159048 * q^15 + 584647 * q^17 - 777172 * q^19 - 3412104 * q^21 - 1357764 * q^23 + 1065785 * q^25 + 4519064 * q^27 + 967002 * q^29 - 3546740 * q^31 + 11928016 * q^33 + 530736 * q^35 + 18296498 * q^37 - 86306872 * q^39 + 10285686 * q^41 - 21913204 * q^43 + 108916410 * q^45 - 56639800 * q^47 + 27010351 * q^49 - 7349848 * q^51 + 121813562 * q^53 - 40793128 * q^55 + 153612960 * q^57 - 29222388 * q^59 - 49915846 * q^61 + 2185356 * q^63 - 122633668 * q^65 - 301863420 * q^67 + 379683432 * q^69 - 652473940 * q^71 + 306656342 * q^73 - 919071912 * q^75 - 102442536 * q^77 - 959147884 * q^79 - 374486977 * q^81 + 1512945268 * q^83 + 113755602 * q^85 + 1612550856 * q^87 - 1971327114 * q^89 + 1061062864 * q^91 - 798598936 * q^93 + 3249631512 * q^95 + 2006526254 * q^97 + 2579159272 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more