LMFDB - Embedded newform 272.10.a.g.1.1

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			1.1
Root			$28.6400$ 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)

$f(q)$	$=$	$q-243.971 q^{3} +1776.79 q^{5} +9598.61 q^{7} +39838.7 q^{9} +O(q^{10})$	$q-243.971 q^{3} +1776.79 q^{5} +9598.61 q^{7} +39838.7 q^{9} -18658.0 q^{11} +118081. q^{13} -433484. q^{15} +83521.0 q^{17} -365652. q^{19} -2.34178e6 q^{21} -1.22678e6 q^{23} +1.20385e6 q^{25} -4.91741e6 q^{27} -2.55483e6 q^{29} -8.56431e6 q^{31} +4.55201e6 q^{33} +1.70547e7 q^{35} -7.97315e6 q^{37} -2.88083e7 q^{39} -3.21617e7 q^{41} +2.38617e7 q^{43} +7.07850e7 q^{45} +1.94506e7 q^{47} +5.17797e7 q^{49} -2.03767e7 q^{51} -1.96101e7 q^{53} -3.31513e7 q^{55} +8.92083e7 q^{57} -1.58423e7 q^{59} -1.09913e8 q^{61} +3.82396e8 q^{63} +2.09805e8 q^{65} +8.19198e7 q^{67} +2.99299e8 q^{69} -1.93682e8 q^{71} -1.53405e7 q^{73} -2.93705e8 q^{75} -1.79091e8 q^{77} -1.36544e8 q^{79} +4.15558e8 q^{81} +6.51497e8 q^{83} +1.48399e8 q^{85} +6.23303e8 q^{87} -6.97006e8 q^{89} +1.13341e9 q^{91} +2.08944e9 q^{93} -6.49686e8 q^{95} -4.50260e8 q^{97} -7.43311e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−243.971	−1.73897	−0.869485	−	0.493959i	$-0.835549\pi$
$3$	−243.971	−1.73897	−0.869485	+	0.493959i	$0.835549\pi$
$4$	0	0
$5$	1776.79	1.27137	0.635683	−	0.771950i	$-0.280718\pi$
$5$	1776.79	1.27137	0.635683	+	0.771950i	$0.280718\pi$
$6$	0	0
$7$	9598.61	1.51101	0.755505	−	0.655143i	$-0.227392\pi$
$7$	9598.61	1.51101	0.755505	+	0.655143i	$0.227392\pi$
$8$	0	0
$9$	39838.7	2.02402
$10$	0	0
$11$	−18658.0	−0.384236	−0.192118	−	0.981372i	$-0.561536\pi$
$11$	−18658.0	−0.384236	−0.192118	+	0.981372i	$0.561536\pi$
$12$	0	0
$13$	118081.	1.14666	0.573329	−	0.819325i	$-0.305652\pi$
$13$	118081.	1.14666	0.573329	+	0.819325i	$0.305652\pi$
$14$	0	0
$15$	−433484.	−2.21087
$16$	0	0
$17$	83521.0	0.242536
$17$	83521.0	0.242536
$18$	0	0
$19$	−365652.	−0.643690	−0.321845	−	0.946792i	$-0.604303\pi$
$19$	−365652.	−0.643690	−0.321845	+	0.946792i	$0.604303\pi$
$20$	0	0
$21$	−2.34178e6	−2.62760
$22$	0	0
$23$	−1.22678e6	−0.914098	−0.457049	−	0.889441i	$-0.651094\pi$
$23$	−1.22678e6	−0.914098	−0.457049	+	0.889441i	$0.651094\pi$
$24$	0	0
$25$	1.20385e6	0.616373
$26$	0	0
$27$	−4.91741e6	−1.78073
$28$	0	0
$29$	−2.55483e6	−0.670765	−0.335382	−	0.942082i	$-0.608866\pi$
$29$	−2.55483e6	−0.670765	−0.335382	+	0.942082i	$0.608866\pi$
$30$	0	0
$31$	−8.56431e6	−1.66558	−0.832789	−	0.553591i	$-0.813257\pi$
$31$	−8.56431e6	−1.66558	−0.832789	+	0.553591i	$0.813257\pi$
$32$	0	0
$33$	4.55201e6	0.668175
$34$	0	0
$35$	1.70547e7	1.92105
$36$	0	0
$37$	−7.97315e6	−0.699394	−0.349697	−	0.936863i	$-0.613715\pi$
$37$	−7.97315e6	−0.699394	−0.349697	+	0.936863i	$0.613715\pi$
$38$	0	0
$39$	−2.88083e7	−1.99400
$40$	0	0
$41$	−3.21617e7	−1.77751	−0.888754	−	0.458384i	$-0.848428\pi$
$41$	−3.21617e7	−1.77751	−0.888754	+	0.458384i	$0.848428\pi$
$42$	0	0
$43$	2.38617e7	1.06437	0.532186	−	0.846628i	$-0.321371\pi$
$43$	2.38617e7	1.06437	0.532186	+	0.846628i	$0.321371\pi$
$44$	0	0
$45$	7.07850e7	2.57327
$46$	0	0
$47$	1.94506e7	0.581423	0.290711	−	0.956811i	$-0.406108\pi$
$47$	1.94506e7	0.581423	0.290711	+	0.956811i	$0.406108\pi$
$48$	0	0
$49$	5.17797e7	1.28315
$50$	0	0
$51$	−2.03767e7	−0.421762
$52$	0	0
$53$	−1.96101e7	−0.341380	−0.170690	−	0.985325i	$-0.554600\pi$
$53$	−1.96101e7	−0.341380	−0.170690	+	0.985325i	$0.554600\pi$
$54$	0	0
$55$	−3.31513e7	−0.488505
$56$	0	0
$57$	8.92083e7	1.11936
$58$	0	0
$59$	−1.58423e7	−0.170210	−0.0851051	−	0.996372i	$-0.527123\pi$
$59$	−1.58423e7	−0.170210	−0.0851051	+	0.996372i	$0.527123\pi$
$60$	0	0
$61$	−1.09913e8	−1.01640	−0.508201	−	0.861238i	$-0.669689\pi$
$61$	−1.09913e8	−1.01640	−0.508201	+	0.861238i	$0.669689\pi$
$62$	0	0
$63$	3.82396e8	3.05831
$64$	0	0
$65$	2.09805e8	1.45782
$66$	0	0
$67$	8.19198e7	0.496652	0.248326	−	0.968677i	$-0.420120\pi$
$67$	8.19198e7	0.496652	0.248326	+	0.968677i	$0.420120\pi$
$68$	0	0
$69$	2.99299e8	1.58959
$70$	0	0
$71$	−1.93682e8	−0.904537	−0.452268	−	0.891882i	$-0.649385\pi$
$71$	−1.93682e8	−0.904537	−0.452268	+	0.891882i	$0.649385\pi$
$72$	0	0
$73$	−1.53405e7	−0.0632245	−0.0316123	−	0.999500i	$-0.510064\pi$
$73$	−1.53405e7	−0.0632245	−0.0316123	+	0.999500i	$0.510064\pi$
$74$	0	0
$75$	−2.93705e8	−1.07185
$76$	0	0
$77$	−1.79091e8	−0.580584
$78$	0	0
$79$	−1.36544e8	−0.394413	−0.197206	−	0.980362i	$-0.563187\pi$
$79$	−1.36544e8	−0.394413	−0.197206	+	0.980362i	$0.563187\pi$
$80$	0	0
$81$	4.15558e8	1.07263
$82$	0	0
$83$	6.51497e8	1.50682	0.753409	−	0.657552i	$-0.228408\pi$
$83$	6.51497e8	1.50682	0.753409	+	0.657552i	$0.228408\pi$
$84$	0	0
$85$	1.48399e8	0.308352
$86$	0	0
$87$	6.23303e8	1.16644
$88$	0	0
$89$	−6.97006e8	−1.17756	−0.588778	−	0.808295i	$-0.700391\pi$
$89$	−6.97006e8	−1.17756	−0.588778	+	0.808295i	$0.700391\pi$
$90$	0	0
$91$	1.13341e9	1.73261
$92$	0	0
$93$	2.08944e9	2.89639
$94$	0	0
$95$	−6.49686e8	−0.818365
$96$	0	0
$97$	−4.50260e8	−0.516405	−0.258203	−	0.966091i	$-0.583130\pi$
$97$	−4.50260e8	−0.516405	−0.258203	+	0.966091i	$0.583130\pi$
$98$	0	0
$99$	−7.43311e8	−0.777700
$100$	0	0
$101$	5.43766e8	0.519955	0.259978	−	0.965615i	$-0.416285\pi$
$101$	5.43766e8	0.519955	0.259978	+	0.965615i	$0.416285\pi$
$102$	0	0
$103$	−9.99479e8	−0.874996	−0.437498	−	0.899219i	$-0.644135\pi$
$103$	−9.99479e8	−0.874996	−0.437498	+	0.899219i	$0.644135\pi$
$104$	0	0
$105$	−4.16085e9	−3.34064
$106$	0	0
$107$	−2.11598e7	−0.0156058	−0.00780289	−	0.999970i	$-0.502484\pi$
$107$	−2.11598e7	−0.0156058	−0.00780289	+	0.999970i	$0.502484\pi$
$108$	0	0
$109$	−2.38161e9	−1.61604	−0.808019	−	0.589156i	$-0.799460\pi$
$109$	−2.38161e9	−1.61604	−0.808019	+	0.589156i	$0.799460\pi$
$110$	0	0
$111$	1.94521e9	1.21623
$112$	0	0
$113$	1.14225e9	0.659035	0.329517	−	0.944150i	$-0.393114\pi$
$113$	1.14225e9	0.659035	0.329517	+	0.944150i	$0.393114\pi$
$114$	0	0
$115$	−2.17974e9	−1.16215
$116$	0	0
$117$	4.70419e9	2.32086
$118$	0	0
$119$	8.01686e8	0.366474
$120$	0	0
$121$	−2.00983e9	−0.852363
$122$	0	0
$123$	7.84651e9	3.09103
$124$	0	0
$125$	−1.33130e9	−0.487731
$126$	0	0
$127$	−5.42933e9	−1.85195	−0.925975	−	0.377584i	$-0.876755\pi$
$127$	−5.42933e9	−1.85195	−0.925975	+	0.377584i	$0.876755\pi$
$128$	0	0
$129$	−5.82156e9	−1.85091
$130$	0	0
$131$	−2.35819e9	−0.699613	−0.349806	−	0.936822i	$-0.613753\pi$
$131$	−2.35819e9	−0.699613	−0.349806	+	0.936822i	$0.613753\pi$
$132$	0	0
$133$	−3.50975e9	−0.972621
$134$	0	0
$135$	−8.73719e9	−2.26397
$136$	0	0
$137$	−9.19603e8	−0.223027	−0.111514	−	0.993763i	$-0.535570\pi$
$137$	−9.19603e8	−0.223027	−0.111514	+	0.993763i	$0.535570\pi$
$138$	0	0
$139$	−1.27752e9	−0.290269	−0.145135	−	0.989412i	$-0.546362\pi$
$139$	−1.27752e9	−0.290269	−0.145135	+	0.989412i	$0.546362\pi$
$140$	0	0
$141$	−4.74537e9	−1.01108
$142$	0	0
$143$	−2.20315e9	−0.440588
$144$	0	0
$145$	−4.53939e9	−0.852788
$146$	0	0
$147$	−1.26327e10	−2.23136
$148$	0	0
$149$	−1.13853e9	−0.189237	−0.0946183	−	0.995514i	$-0.530163\pi$
$149$	−1.13853e9	−0.189237	−0.0946183	+	0.995514i	$0.530163\pi$
$150$	0	0
$151$	−1.89477e9	−0.296593	−0.148296	−	0.988943i	$-0.547379\pi$
$151$	−1.89477e9	−0.296593	−0.148296	+	0.988943i	$0.547379\pi$
$152$	0	0
$153$	3.32737e9	0.490896
$154$	0	0
$155$	−1.52170e10	−2.11756
$156$	0	0
$157$	6.87991e9	0.903721	0.451860	−	0.892089i	$-0.350760\pi$
$157$	6.87991e9	0.903721	0.451860	+	0.892089i	$0.350760\pi$
$158$	0	0
$159$	4.78429e9	0.593650
$160$	0	0
$161$	−1.17754e10	−1.38121
$162$	0	0
$163$	−1.18995e10	−1.32034	−0.660171	−	0.751116i	$-0.729516\pi$
$163$	−1.18995e10	−1.32034	−0.660171	+	0.751116i	$0.729516\pi$
$164$	0	0
$165$	8.08795e9	0.849495
$166$	0	0
$167$	1.62243e10	1.61415	0.807073	−	0.590451i	$-0.201050\pi$
$167$	1.62243e10	1.61415	0.807073	+	0.590451i	$0.201050\pi$
$168$	0	0
$169$	3.33857e9	0.314826
$170$	0	0
$171$	−1.45671e10	−1.30284
$172$	0	0
$173$	9.77316e9	0.829521	0.414761	−	0.909931i	$-0.363865\pi$
$173$	9.77316e9	0.829521	0.414761	+	0.909931i	$0.363865\pi$
$174$	0	0
$175$	1.15553e10	0.931345
$176$	0	0
$177$	3.86507e9	0.295990
$178$	0	0
$179$	1.22507e10	0.891915	0.445958	−	0.895054i	$-0.352863\pi$
$179$	1.22507e10	0.891915	0.445958	+	0.895054i	$0.352863\pi$
$180$	0	0
$181$	1.21812e10	0.843597	0.421798	−	0.906690i	$-0.361399\pi$
$181$	1.21812e10	0.843597	0.421798	+	0.906690i	$0.361399\pi$
$182$	0	0
$183$	2.68156e10	1.76749
$184$	0	0
$185$	−1.41666e10	−0.889186
$186$	0	0
$187$	−1.55834e9	−0.0931909
$188$	0	0
$189$	−4.72003e10	−2.69071
$190$	0	0
$191$	1.62713e10	0.884651	0.442325	−	0.896855i	$-0.354154\pi$
$191$	1.62713e10	0.884651	0.442325	+	0.896855i	$0.354154\pi$
$192$	0	0
$193$	1.78626e10	0.926696	0.463348	−	0.886176i	$-0.346648\pi$
$193$	1.78626e10	0.926696	0.463348	+	0.886176i	$0.346648\pi$
$194$	0	0
$195$	−5.11862e10	−2.53511
$196$	0	0
$197$	1.24709e10	0.589927	0.294963	−	0.955509i	$-0.404693\pi$
$197$	1.24709e10	0.589927	0.294963	+	0.955509i	$0.404693\pi$
$198$	0	0
$199$	1.76972e10	0.799957	0.399978	−	0.916525i	$-0.369018\pi$
$199$	1.76972e10	0.799957	0.399978	+	0.916525i	$0.369018\pi$
$200$	0	0
$201$	−1.99860e10	−0.863663
$202$	0	0
$203$	−2.45228e10	−1.01353
$204$	0	0
$205$	−5.71445e10	−2.25986
$206$	0	0
$207$	−4.88735e10	−1.85015
$208$	0	0
$209$	6.82233e9	0.247329
$210$	0	0
$211$	3.49985e10	1.21556	0.607782	−	0.794104i	$-0.292059\pi$
$211$	3.49985e10	1.21556	0.607782	+	0.794104i	$0.292059\pi$
$212$	0	0
$213$	4.72527e10	1.57296
$214$	0	0
$215$	4.23972e10	1.35321
$216$	0	0
$217$	−8.22055e10	−2.51670
$218$	0	0
$219$	3.74262e9	0.109946
$220$	0	0
$221$	9.86223e9	0.278106
$222$	0	0
$223$	−4.75965e10	−1.28885	−0.644426	−	0.764667i	$-0.722904\pi$
$223$	−4.75965e10	−1.28885	−0.644426	+	0.764667i	$0.722904\pi$
$224$	0	0
$225$	4.79600e10	1.24755
$226$	0	0
$227$	−1.17164e10	−0.292873	−0.146436	−	0.989220i	$-0.546780\pi$
$227$	−1.17164e10	−0.292873	−0.146436	+	0.989220i	$0.546780\pi$
$228$	0	0
$229$	2.48997e10	0.598321	0.299161	−	0.954203i	$-0.403293\pi$
$229$	2.48997e10	0.598321	0.299161	+	0.954203i	$0.403293\pi$
$230$	0	0
$231$	4.36929e10	1.00962
$232$	0	0
$233$	6.50866e10	1.44674	0.723369	−	0.690462i	$-0.242593\pi$
$233$	6.50866e10	1.44674	0.723369	+	0.690462i	$0.242593\pi$
$234$	0	0
$235$	3.45596e10	0.739201
$236$	0	0
$237$	3.33128e10	0.685872
$238$	0	0
$239$	−4.29823e10	−0.852117	−0.426058	−	0.904696i	$-0.640098\pi$
$239$	−4.29823e10	−0.852117	−0.426058	+	0.904696i	$0.640098\pi$
$240$	0	0
$241$	−7.83992e10	−1.49705	−0.748523	−	0.663109i	$-0.769236\pi$
$241$	−7.83992e10	−1.49705	−0.748523	+	0.663109i	$0.769236\pi$
$242$	0	0
$243$	−4.59462e9	−0.0845320
$244$	0	0
$245$	9.20017e10	1.63135
$246$	0	0
$247$	−4.31764e10	−0.738092
$248$	0	0
$249$	−1.58946e11	−2.62031
$250$	0	0
$251$	−2.92884e10	−0.465762	−0.232881	−	0.972505i	$-0.574815\pi$
$251$	−2.92884e10	−0.465762	−0.232881	+	0.972505i	$0.574815\pi$
$252$	0	0
$253$	2.28893e10	0.351230
$254$	0	0
$255$	−3.62051e10	−0.536214
$256$	0	0
$257$	3.01696e10	0.431390	0.215695	−	0.976461i	$-0.430798\pi$
$257$	3.01696e10	0.431390	0.215695	+	0.976461i	$0.430798\pi$
$258$	0	0
$259$	−7.65311e10	−1.05679
$260$	0	0
$261$	−1.01781e11	−1.35764
$262$	0	0
$263$	9.10348e10	1.17329	0.586647	−	0.809843i	$-0.300448\pi$
$263$	9.10348e10	1.17329	0.586647	+	0.809843i	$0.300448\pi$
$264$	0	0
$265$	−3.48430e10	−0.434019
$266$	0	0
$267$	1.70049e11	2.04774
$268$	0	0
$269$	−6.95622e10	−0.810005	−0.405003	−	0.914316i	$-0.632729\pi$
$269$	−6.95622e10	−0.810005	−0.405003	+	0.914316i	$0.632729\pi$
$270$	0	0
$271$	9.57591e10	1.07849	0.539247	−	0.842147i	$-0.318709\pi$
$271$	9.57591e10	1.07849	0.539247	+	0.842147i	$0.318709\pi$
$272$	0	0
$273$	−2.76519e11	−3.01296
$274$	0	0
$275$	−2.24615e10	−0.236833
$276$	0	0
$277$	−8.55452e10	−0.873045	−0.436523	−	0.899693i	$-0.643790\pi$
$277$	−8.55452e10	−0.873045	−0.436523	+	0.899693i	$0.643790\pi$
$278$	0	0
$279$	−3.41191e11	−3.37116
$280$	0	0
$281$	−4.39159e10	−0.420188	−0.210094	−	0.977681i	$-0.567377\pi$
$281$	−4.39159e10	−0.420188	−0.210094	+	0.977681i	$0.567377\pi$
$282$	0	0
$283$	−8.04249e10	−0.745335	−0.372668	−	0.927965i	$-0.621557\pi$
$283$	−8.04249e10	−0.745335	−0.372668	+	0.927965i	$0.621557\pi$
$284$	0	0
$285$	1.58504e11	1.42311
$286$	0	0
$287$	−3.08708e11	−2.68583
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	1.09850e11	0.898013
$292$	0	0
$293$	−2.21747e10	−0.175774	−0.0878869	−	0.996130i	$-0.528011\pi$
$293$	−2.21747e10	−0.175774	−0.0878869	+	0.996130i	$0.528011\pi$
$294$	0	0
$295$	−2.81485e10	−0.216399
$296$	0	0
$297$	9.17490e10	0.684222
$298$	0	0
$299$	−1.44860e11	−1.04816
$300$	0	0
$301$	2.29039e11	1.60828
$302$	0	0
$303$	−1.32663e11	−0.904187
$304$	0	0
$305$	−1.95293e11	−1.29222
$306$	0	0
$307$	−1.36198e11	−0.875081	−0.437540	−	0.899199i	$-0.644150\pi$
$307$	−1.36198e11	−0.875081	−0.437540	+	0.899199i	$0.644150\pi$
$308$	0	0
$309$	2.43844e11	1.52159
$310$	0	0
$311$	−2.63133e11	−1.59497	−0.797486	−	0.603337i	$-0.793837\pi$
$311$	−2.63133e11	−1.59497	−0.797486	+	0.603337i	$0.793837\pi$
$312$	0	0
$313$	1.85944e11	1.09504	0.547522	−	0.836791i	$-0.315571\pi$
$313$	1.85944e11	1.09504	0.547522	+	0.836791i	$0.315571\pi$
$314$	0	0
$315$	6.79438e11	3.88823
$316$	0	0
$317$	−3.15409e11	−1.75431	−0.877157	−	0.480203i	$-0.840563\pi$
$317$	−3.15409e11	−1.75431	−0.877157	+	0.480203i	$0.840563\pi$
$318$	0	0
$319$	4.76680e10	0.257732
$320$	0	0
$321$	5.16238e9	0.0271380
$322$	0	0
$323$	−3.05396e10	−0.156118
$324$	0	0
$325$	1.42152e11	0.706769
$326$	0	0
$327$	5.81043e11	2.81024
$328$	0	0
$329$	1.86698e11	0.878535
$330$	0	0
$331$	1.33188e10	0.0609871	0.0304936	−	0.999535i	$-0.490292\pi$
$331$	1.33188e10	0.0609871	0.0304936	+	0.999535i	$0.490292\pi$
$332$	0	0
$333$	−3.17640e11	−1.41559
$334$	0	0
$335$	1.45554e11	0.631426
$336$	0	0
$337$	−8.96864e10	−0.378785	−0.189392	−	0.981902i	$-0.560652\pi$
$337$	−8.96864e10	−0.378785	−0.189392	+	0.981902i	$0.560652\pi$
$338$	0	0
$339$	−2.78676e11	−1.14604
$340$	0	0
$341$	1.59793e11	0.639975
$342$	0	0
$343$	1.09675e11	0.427843
$344$	0	0
$345$	5.31792e11	2.02095
$346$	0	0
$347$	3.71901e11	1.37703	0.688516	−	0.725221i	$-0.258262\pi$
$347$	3.71901e11	1.37703	0.688516	+	0.725221i	$0.258262\pi$
$348$	0	0
$349$	2.40556e11	0.867963	0.433982	−	0.900922i	$-0.357108\pi$
$349$	2.40556e11	0.867963	0.433982	+	0.900922i	$0.357108\pi$
$350$	0	0
$351$	−5.80651e11	−2.04189
$352$	0	0
$353$	−2.31047e11	−0.791980	−0.395990	−	0.918255i	$-0.629599\pi$
$353$	−2.31047e11	−0.791980	−0.395990	+	0.918255i	$0.629599\pi$
$354$	0	0
$355$	−3.44132e11	−1.15000
$356$	0	0
$357$	−1.95588e11	−0.637287
$358$	0	0
$359$	−2.17507e11	−0.691110	−0.345555	−	0.938398i	$-0.612309\pi$
$359$	−2.17507e11	−0.691110	−0.345555	+	0.938398i	$0.612309\pi$
$360$	0	0
$361$	−1.88986e11	−0.585664
$362$	0	0
$363$	4.90339e11	1.48223
$364$	0	0
$365$	−2.72568e10	−0.0803815
$366$	0	0
$367$	−1.24703e11	−0.358823	−0.179412	−	0.983774i	$-0.557419\pi$
$367$	−1.24703e11	−0.358823	−0.179412	+	0.983774i	$0.557419\pi$
$368$	0	0
$369$	−1.28128e12	−3.59771
$370$	0	0
$371$	−1.88230e11	−0.515829
$372$	0	0
$373$	6.71444e11	1.79606	0.898028	−	0.439938i	$-0.144999\pi$
$373$	6.71444e11	1.79606	0.898028	+	0.439938i	$0.144999\pi$
$374$	0	0
$375$	3.24798e11	0.848149
$376$	0	0
$377$	−3.01676e11	−0.769138
$378$	0	0
$379$	2.57132e11	0.640147	0.320073	−	0.947393i	$-0.396292\pi$
$379$	2.57132e11	0.640147	0.320073	+	0.947393i	$0.396292\pi$
$380$	0	0
$381$	1.32460e12	3.22049
$382$	0	0
$383$	1.54303e11	0.366420	0.183210	−	0.983074i	$-0.441351\pi$
$383$	1.54303e11	0.366420	0.183210	+	0.983074i	$0.441351\pi$
$384$	0	0
$385$	−3.18207e11	−0.738135
$386$	0	0
$387$	9.50620e11	2.15431
$388$	0	0
$389$	1.53781e11	0.340510	0.170255	−	0.985400i	$-0.445541\pi$
$389$	1.53781e11	0.340510	0.170255	+	0.985400i	$0.445541\pi$
$390$	0	0
$391$	−1.02462e11	−0.221701
$392$	0	0
$393$	5.75329e11	1.21661
$394$	0	0
$395$	−2.42610e11	−0.501443
$396$	0	0
$397$	−5.53284e10	−0.111787	−0.0558934	−	0.998437i	$-0.517801\pi$
$397$	−5.53284e10	−0.111787	−0.0558934	+	0.998437i	$0.517801\pi$
$398$	0	0
$399$	8.56276e11	1.69136
$400$	0	0
$401$	−5.98775e11	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$401$	−5.98775e11	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$402$	0	0
$403$	−1.01128e12	−1.90985
$404$	0	0
$405$	7.38358e11	1.36370
$406$	0	0
$407$	1.48763e11	0.268732
$408$	0	0
$409$	−8.67423e11	−1.53277	−0.766384	−	0.642383i	$-0.777946\pi$
$409$	−8.67423e11	−1.53277	−0.766384	+	0.642383i	$0.777946\pi$
$410$	0	0
$411$	2.24356e11	0.387838
$412$	0	0
$413$	−1.52065e11	−0.257189
$414$	0	0
$415$	1.15757e12	1.91572
$416$	0	0
$417$	3.11678e11	0.504770
$418$	0	0
$419$	6.41732e11	1.01716	0.508581	−	0.861014i	$-0.330170\pi$
$419$	6.41732e11	1.01716	0.508581	+	0.861014i	$0.330170\pi$
$420$	0	0
$421$	4.57623e11	0.709967	0.354983	−	0.934873i	$-0.384487\pi$
$421$	4.57623e11	0.709967	0.354983	+	0.934873i	$0.384487\pi$
$422$	0	0
$423$	7.74886e11	1.17681
$424$	0	0
$425$	1.00547e11	0.149492
$426$	0	0
$427$	−1.05501e12	−1.53579
$428$	0	0
$429$	5.37505e11	0.766169
$430$	0	0
$431$	−7.38734e10	−0.103119	−0.0515597	−	0.998670i	$-0.516419\pi$
$431$	−7.38734e10	−0.103119	−0.0515597	+	0.998670i	$0.516419\pi$
$432$	0	0
$433$	1.03868e12	1.41999	0.709995	−	0.704206i	$-0.248697\pi$
$433$	1.03868e12	1.41999	0.709995	+	0.704206i	$0.248697\pi$
$434$	0	0
$435$	1.10748e12	1.48297
$436$	0	0
$437$	4.48576e11	0.588396
$438$	0	0
$439$	−9.61297e11	−1.23529	−0.617643	−	0.786459i	$-0.711912\pi$
$439$	−9.61297e11	−1.23529	−0.617643	+	0.786459i	$0.711912\pi$
$440$	0	0
$441$	2.06284e12	2.59712
$442$	0	0
$443$	−2.39240e11	−0.295132	−0.147566	−	0.989052i	$-0.547144\pi$
$443$	−2.39240e11	−0.295132	−0.147566	+	0.989052i	$0.547144\pi$
$444$	0	0
$445$	−1.23843e12	−1.49711
$446$	0	0
$447$	2.77767e11	0.329077
$448$	0	0
$449$	−5.56831e11	−0.646569	−0.323285	−	0.946302i	$-0.604787\pi$
$449$	−5.56831e11	−0.646569	−0.323285	+	0.946302i	$0.604787\pi$
$450$	0	0
$451$	6.00073e11	0.682983
$452$	0	0
$453$	4.62269e11	0.515766
$454$	0	0
$455$	2.01383e12	2.20279
$456$	0	0
$457$	6.83226e11	0.732726	0.366363	−	0.930472i	$-0.380603\pi$
$457$	6.83226e11	0.732726	0.366363	+	0.930472i	$0.380603\pi$
$458$	0	0
$459$	−4.10707e11	−0.431892
$460$	0	0
$461$	−9.09123e11	−0.937494	−0.468747	−	0.883333i	$-0.655294\pi$
$461$	−9.09123e11	−0.937494	−0.468747	+	0.883333i	$0.655294\pi$
$462$	0	0
$463$	−3.46507e11	−0.350427	−0.175213	−	0.984530i	$-0.556062\pi$
$463$	−3.46507e11	−0.350427	−0.175213	+	0.984530i	$0.556062\pi$
$464$	0	0
$465$	3.71250e12	3.68237
$466$	0	0
$467$	−3.54586e11	−0.344982	−0.172491	−	0.985011i	$-0.555182\pi$
$467$	−3.54586e11	−0.344982	−0.172491	+	0.985011i	$0.555182\pi$
$468$	0	0
$469$	7.86316e11	0.750446
$470$	0	0
$471$	−1.67850e12	−1.57154
$472$	0	0
$473$	−4.45212e11	−0.408970
$474$	0	0
$475$	−4.40191e11	−0.396753
$476$	0	0
$477$	−7.81241e11	−0.690959
$478$	0	0
$479$	−1.13858e12	−0.988222	−0.494111	−	0.869399i	$-0.664506\pi$
$479$	−1.13858e12	−0.988222	−0.494111	+	0.869399i	$0.664506\pi$
$480$	0	0
$481$	−9.41475e11	−0.801966
$482$	0	0
$483$	2.87286e12	2.40189
$484$	0	0
$485$	−8.00017e11	−0.656540
$486$	0	0
$487$	−1.19842e10	−0.00965450	−0.00482725	−	0.999988i	$-0.501537\pi$
$487$	−1.19842e10	−0.00965450	−0.00482725	+	0.999988i	$0.501537\pi$
$488$	0	0
$489$	2.90314e12	2.29603
$490$	0	0
$491$	1.60620e11	0.124719	0.0623597	−	0.998054i	$-0.480137\pi$
$491$	1.60620e11	0.124719	0.0623597	+	0.998054i	$0.480137\pi$
$492$	0	0
$493$	−2.13382e11	−0.162684
$494$	0	0
$495$	−1.32071e12	−0.988742
$496$	0	0
$497$	−1.85908e12	−1.36676
$498$	0	0
$499$	1.44413e12	1.04269	0.521345	−	0.853346i	$-0.325431\pi$
$499$	1.44413e12	1.04269	0.521345	+	0.853346i	$0.325431\pi$
$500$	0	0
$501$	−3.95826e12	−2.80695
$502$	0	0
$503$	−1.19095e12	−0.829539	−0.414769	−	0.909927i	$-0.636138\pi$
$503$	−1.19095e12	−0.829539	−0.414769	+	0.909927i	$0.636138\pi$
$504$	0	0
$505$	9.66158e11	0.661054
$506$	0	0
$507$	−8.14514e11	−0.547473
$508$	0	0
$509$	−2.34412e12	−1.54793	−0.773963	−	0.633231i	$-0.781728\pi$
$509$	−2.34412e12	−1.54793	−0.773963	+	0.633231i	$0.781728\pi$
$510$	0	0
$511$	−1.47247e11	−0.0955328
$512$	0	0
$513$	1.79806e12	1.14624
$514$	0	0
$515$	−1.77586e12	−1.11244
$516$	0	0
$517$	−3.62909e11	−0.223404
$518$	0	0
$519$	−2.38437e12	−1.44251
$520$	0	0
$521$	1.55789e12	0.926333	0.463166	−	0.886271i	$-0.346713\pi$
$521$	1.55789e12	0.926333	0.463166	+	0.886271i	$0.346713\pi$
$522$	0	0
$523$	−5.98253e10	−0.0349645	−0.0174823	−	0.999847i	$-0.505565\pi$
$523$	−5.98253e10	−0.0349645	−0.0174823	+	0.999847i	$0.505565\pi$
$524$	0	0
$525$	−2.81916e12	−1.61958
$526$	0	0
$527$	−7.15300e11	−0.403962
$528$	0	0
$529$	−2.96153e11	−0.164424
$530$	0	0
$531$	−6.31139e11	−0.344508
$532$	0	0
$533$	−3.79768e12	−2.03820
$534$	0	0
$535$	−3.75966e10	−0.0198407
$536$	0	0
$537$	−2.98882e12	−1.55101
$538$	0	0
$539$	−9.66107e11	−0.493033
$540$	0	0
$541$	2.35116e12	1.18004	0.590018	−	0.807390i	$-0.299121\pi$
$541$	2.35116e12	1.18004	0.590018	+	0.807390i	$0.299121\pi$
$542$	0	0
$543$	−2.97185e12	−1.46699
$544$	0	0
$545$	−4.23162e12	−2.05458
$546$	0	0
$547$	−3.16333e12	−1.51078	−0.755391	−	0.655274i	$-0.772553\pi$
$547$	−3.16333e12	−1.51078	−0.755391	+	0.655274i	$0.772553\pi$
$548$	0	0
$549$	−4.37880e12	−2.05722
$550$	0	0
$551$	9.34177e11	0.431764
$552$	0	0
$553$	−1.31063e12	−0.595961
$554$	0	0
$555$	3.45624e12	1.54627
$556$	0	0
$557$	3.53542e12	1.55630	0.778148	−	0.628081i	$-0.216159\pi$
$557$	3.53542e12	1.55630	0.778148	+	0.628081i	$0.216159\pi$
$558$	0	0
$559$	2.81761e12	1.22047
$560$	0	0
$561$	3.80188e11	0.162056
$562$	0	0
$563$	1.75567e12	0.736472	0.368236	−	0.929732i	$-0.379962\pi$
$563$	1.75567e12	0.736472	0.368236	+	0.929732i	$0.379962\pi$
$564$	0	0
$565$	2.02954e12	0.837875
$566$	0	0
$567$	3.98878e12	1.62075
$568$	0	0
$569$	−2.32211e9	−0.000928706 0	−0.000464353	−	1.00000i	$-0.500148\pi$
$569$	−2.32211e9	−0.000928706 0	−0.000464353		1.00000i	$0.500148\pi$
$570$	0	0
$571$	2.57323e12	1.01302	0.506509	−	0.862235i	$-0.330936\pi$
$571$	2.57323e12	1.01302	0.506509	+	0.862235i	$0.330936\pi$
$572$	0	0
$573$	−3.96972e12	−1.53838
$574$	0	0
$575$	−1.47687e12	−0.563425
$576$	0	0
$577$	−1.59532e12	−0.599178	−0.299589	−	0.954068i	$-0.596850\pi$
$577$	−1.59532e12	−0.599178	−0.299589	+	0.954068i	$0.596850\pi$
$578$	0	0
$579$	−4.35796e12	−1.61150
$580$	0	0
$581$	6.25346e12	2.27682
$582$	0	0
$583$	3.65885e11	0.131171
$584$	0	0
$585$	8.35835e12	2.95066
$586$	0	0
$587$	2.68434e12	0.933182	0.466591	−	0.884473i	$-0.345482\pi$
$587$	2.68434e12	0.933182	0.466591	+	0.884473i	$0.345482\pi$
$588$	0	0
$589$	3.13156e12	1.07212
$590$	0	0
$591$	−3.04252e12	−1.02586
$592$	0	0
$593$	4.50719e12	1.49679	0.748393	−	0.663256i	$-0.230826\pi$
$593$	4.50719e12	1.49679	0.748393	+	0.663256i	$0.230826\pi$
$594$	0	0
$595$	1.42443e12	0.465922
$596$	0	0
$597$	−4.31761e12	−1.39110
$598$	0	0
$599$	4.19976e12	1.33292	0.666459	−	0.745542i	$-0.267809\pi$
$599$	4.19976e12	1.33292	0.666459	+	0.745542i	$0.267809\pi$
$600$	0	0
$601$	4.96222e12	1.55146	0.775730	−	0.631065i	$-0.217382\pi$
$601$	4.96222e12	1.55146	0.775730	+	0.631065i	$0.217382\pi$
$602$	0	0
$603$	3.26358e12	1.00523
$604$	0	0
$605$	−3.57104e12	−1.08367
$606$	0	0
$607$	−4.06493e12	−1.21536	−0.607679	−	0.794183i	$-0.707899\pi$
$607$	−4.06493e12	−1.21536	−0.607679	+	0.794183i	$0.707899\pi$
$608$	0	0
$609$	5.98284e12	1.76250
$610$	0	0
$611$	2.29674e12	0.666693
$612$	0	0
$613$	−3.12747e12	−0.894583	−0.447291	−	0.894388i	$-0.647611\pi$
$613$	−3.12747e12	−0.894583	−0.447291	+	0.894388i	$0.647611\pi$
$614$	0	0
$615$	1.39416e13	3.92984
$616$	0	0
$617$	1.99199e12	0.553354	0.276677	−	0.960963i	$-0.410767\pi$
$617$	1.99199e12	0.553354	0.276677	+	0.960963i	$0.410767\pi$
$618$	0	0
$619$	−5.76812e12	−1.57916	−0.789581	−	0.613647i	$-0.789702\pi$
$619$	−5.76812e12	−1.57916	−0.789581	+	0.613647i	$0.789702\pi$
$620$	0	0
$621$	6.03260e12	1.62777
$622$	0	0
$623$	−6.69029e12	−1.77930
$624$	0	0
$625$	−4.71671e12	−1.23646
$626$	0	0
$627$	−1.66445e12	−0.430097
$628$	0	0
$629$	−6.65925e11	−0.169628
$630$	0	0
$631$	2.73184e12	0.685999	0.342999	−	0.939336i	$-0.388557\pi$
$631$	2.73184e12	0.685999	0.342999	+	0.939336i	$0.388557\pi$
$632$	0	0
$633$	−8.53860e12	−2.11383
$634$	0	0
$635$	−9.64677e12	−2.35451
$636$	0	0
$637$	6.11419e12	1.47134
$638$	0	0
$639$	−7.71604e12	−1.83080
$640$	0	0
$641$	−3.77475e12	−0.883135	−0.441567	−	0.897228i	$-0.645577\pi$
$641$	−3.77475e12	−0.883135	−0.441567	+	0.897228i	$0.645577\pi$
$642$	0	0
$643$	−1.26222e12	−0.291197	−0.145598	−	0.989344i	$-0.546511\pi$
$643$	−1.26222e12	−0.291197	−0.145598	+	0.989344i	$0.546511\pi$
$644$	0	0
$645$	−1.03437e13	−2.35319
$646$	0	0
$647$	−5.94830e12	−1.33451	−0.667257	−	0.744827i	$-0.732532\pi$
$647$	−5.94830e12	−1.33451	−0.667257	+	0.744827i	$0.732532\pi$
$648$	0	0
$649$	2.95587e11	0.0654009
$650$	0	0
$651$	2.00557e13	4.37647
$652$	0	0
$653$	−1.12364e12	−0.241835	−0.120918	−	0.992663i	$-0.538584\pi$
$653$	−1.12364e12	−0.241835	−0.120918	+	0.992663i	$0.538584\pi$
$654$	0	0
$655$	−4.19000e12	−0.889464
$656$	0	0
$657$	−6.11144e11	−0.127967
$658$	0	0
$659$	8.58385e11	0.177296	0.0886478	−	0.996063i	$-0.471745\pi$
$659$	8.58385e11	0.177296	0.0886478	+	0.996063i	$0.471745\pi$
$660$	0	0
$661$	−2.45857e12	−0.500929	−0.250465	−	0.968126i	$-0.580583\pi$
$661$	−2.45857e12	−0.500929	−0.250465	+	0.968126i	$0.580583\pi$
$662$	0	0
$663$	−2.40609e12	−0.483617
$664$	0	0
$665$	−6.23608e12	−1.23656
$666$	0	0
$667$	3.13422e12	0.613145
$668$	0	0
$669$	1.16121e13	2.24127
$670$	0	0
$671$	2.05076e12	0.390538
$672$	0	0
$673$	2.80053e12	0.526225	0.263113	−	0.964765i	$-0.415251\pi$
$673$	2.80053e12	0.526225	0.263113	+	0.964765i	$0.415251\pi$
$674$	0	0
$675$	−5.91984e12	−1.09760
$676$	0	0
$677$	−3.74996e12	−0.686085	−0.343042	−	0.939320i	$-0.611457\pi$
$677$	−3.74996e12	−0.686085	−0.343042	+	0.939320i	$0.611457\pi$
$678$	0	0
$679$	−4.32187e12	−0.780293
$680$	0	0
$681$	2.85846e12	0.509297
$682$	0	0
$683$	4.38021e12	0.770197	0.385099	−	0.922875i	$-0.374167\pi$
$683$	4.38021e12	0.770197	0.385099	+	0.922875i	$0.374167\pi$
$684$	0	0
$685$	−1.63394e12	−0.283549
$686$	0	0
$687$	−6.07480e12	−1.04046
$688$	0	0
$689$	−2.31558e12	−0.391447
$690$	0	0
$691$	−1.91905e11	−0.0320211	−0.0160105	−	0.999872i	$-0.505097\pi$
$691$	−1.91905e11	−0.0320211	−0.0160105	+	0.999872i	$0.505097\pi$
$692$	0	0
$693$	−7.13475e12	−1.17511
$694$	0	0
$695$	−2.26988e12	−0.369039
$696$	0	0
$697$	−2.68618e12	−0.431109
$698$	0	0
$699$	−1.58792e13	−2.51583
$700$	0	0
$701$	4.79909e12	0.750633	0.375316	−	0.926897i	$-0.377534\pi$
$701$	4.79909e12	0.750633	0.375316	+	0.926897i	$0.377534\pi$
$702$	0	0
$703$	2.91540e12	0.450193
$704$	0	0
$705$	−8.43152e12	−1.28545
$706$	0	0
$707$	5.21940e12	0.785658
$708$	0	0
$709$	−7.09288e12	−1.05418	−0.527090	−	0.849810i	$-0.676717\pi$
$709$	−7.09288e12	−1.05418	−0.527090	+	0.849810i	$0.676717\pi$
$710$	0	0
$711$	−5.43974e12	−0.798298
$712$	0	0
$713$	1.05066e13	1.52250
$714$	0	0
$715$	−3.91454e12	−0.560148
$716$	0	0
$717$	1.04864e13	1.48181
$718$	0	0
$719$	−8.16373e12	−1.13922	−0.569611	−	0.821914i	$-0.692906\pi$
$719$	−8.16373e12	−1.13922	−0.569611	+	0.821914i	$0.692906\pi$
$720$	0	0
$721$	−9.59361e12	−1.32213
$722$	0	0
$723$	1.91271e13	2.60332
$724$	0	0
$725$	−3.07564e12	−0.413441
$726$	0	0
$727$	−1.27003e13	−1.68619	−0.843097	−	0.537762i	$-0.819270\pi$
$727$	−1.27003e13	−1.68619	−0.843097	+	0.537762i	$0.819270\pi$
$728$	0	0
$729$	−7.05847e12	−0.925629
$730$	0	0
$731$	1.99295e12	0.258148
$732$	0	0
$733$	−9.65924e12	−1.23588	−0.617938	−	0.786227i	$-0.712032\pi$
$733$	−9.65924e12	−1.23588	−0.617938	+	0.786227i	$0.712032\pi$
$734$	0	0
$735$	−2.24457e13	−2.83688
$736$	0	0
$737$	−1.52846e12	−0.190832
$738$	0	0
$739$	−5.87320e11	−0.0724394	−0.0362197	−	0.999344i	$-0.511532\pi$
$739$	−5.87320e11	−0.0724394	−0.0362197	+	0.999344i	$0.511532\pi$
$740$	0	0
$741$	1.05338e13	1.28352
$742$	0	0
$743$	6.32905e12	0.761884	0.380942	−	0.924599i	$-0.375600\pi$
$743$	6.32905e12	0.761884	0.380942	+	0.924599i	$0.375600\pi$
$744$	0	0
$745$	−2.02292e12	−0.240589
$746$	0	0
$747$	2.59548e13	3.04982
$748$	0	0
$749$	−2.03105e11	−0.0235805
$750$	0	0
$751$	5.48219e12	0.628889	0.314445	−	0.949276i	$-0.398182\pi$
$751$	5.48219e12	0.628889	0.314445	+	0.949276i	$0.398182\pi$
$752$	0	0
$753$	7.14551e12	0.809946
$754$	0	0
$755$	−3.36661e12	−0.377078
$756$	0	0
$757$	6.94071e12	0.768197	0.384099	−	0.923292i	$-0.374512\pi$
$757$	6.94071e12	0.768197	0.384099	+	0.923292i	$0.374512\pi$
$758$	0	0
$759$	−5.58433e12	−0.610778
$760$	0	0
$761$	−1.18446e13	−1.28023	−0.640117	−	0.768277i	$-0.721114\pi$
$761$	−1.18446e13	−1.28023	−0.640117	+	0.768277i	$0.721114\pi$
$762$	0	0
$763$	−2.28602e13	−2.44185
$764$	0	0
$765$	5.91203e12	0.624109
$766$	0	0
$767$	−1.87068e12	−0.195173
$768$	0	0
$769$	−1.22311e12	−0.126123	−0.0630617	−	0.998010i	$-0.520086\pi$
$769$	−1.22311e12	−0.126123	−0.0630617	+	0.998010i	$0.520086\pi$
$770$	0	0
$771$	−7.36049e12	−0.750174
$772$	0	0
$773$	2.56478e12	0.258370	0.129185	−	0.991621i	$-0.458764\pi$
$773$	2.56478e12	0.258370	0.129185	+	0.991621i	$0.458764\pi$
$774$	0	0
$775$	−1.03102e13	−1.02662
$776$	0	0
$777$	1.86714e13	1.83773
$778$	0	0
$779$	1.17600e13	1.14416
$780$	0	0
$781$	3.61372e12	0.347556
$782$	0	0
$783$	1.25631e13	1.19445
$784$	0	0
$785$	1.22241e13	1.14896
$786$	0	0
$787$	7.44338e12	0.691646	0.345823	−	0.938300i	$-0.387600\pi$
$787$	7.44338e12	0.691646	0.345823	+	0.938300i	$0.387600\pi$
$788$	0	0
$789$	−2.22098e13	−2.04032
$790$	0	0
$791$	1.09640e13	0.995808
$792$	0	0
$793$	−1.29786e13	−1.16547
$794$	0	0
$795$	8.50067e12	0.754747
$796$	0	0
$797$	1.00963e13	0.886337	0.443168	−	0.896438i	$-0.353854\pi$
$797$	1.00963e13	0.886337	0.443168	+	0.896438i	$0.353854\pi$
$798$	0	0
$799$	1.62453e12	0.141016
$800$	0	0
$801$	−2.77678e13	−2.38339
$802$	0	0
$803$	2.86222e11	0.0242931
$804$	0	0
$805$	−2.09224e13	−1.75603
$806$	0	0
$807$	1.69711e13	1.40857
$808$	0	0
$809$	−8.02349e12	−0.658560	−0.329280	−	0.944232i	$-0.606806\pi$
$809$	−8.02349e12	−0.658560	−0.329280	+	0.944232i	$0.606806\pi$
$810$	0	0
$811$	−2.03065e13	−1.64832	−0.824160	−	0.566357i	$-0.808352\pi$
$811$	−2.03065e13	−1.64832	−0.824160	+	0.566357i	$0.808352\pi$
$812$	0	0
$813$	−2.33624e13	−1.87547
$814$	0	0
$815$	−2.11430e13	−1.67864
$816$	0	0
$817$	−8.72507e12	−0.685125
$818$	0	0
$819$	4.51537e13	3.50684
$820$	0	0
$821$	2.20512e13	1.69390	0.846951	−	0.531671i	$-0.178436\pi$
$821$	2.20512e13	1.69390	0.846951	+	0.531671i	$0.178436\pi$
$822$	0	0
$823$	−8.31984e12	−0.632143	−0.316072	−	0.948735i	$-0.602364\pi$
$823$	−8.31984e12	−0.632143	−0.316072	+	0.948735i	$0.602364\pi$
$824$	0	0
$825$	5.47995e12	0.411845
$826$	0	0
$827$	−6.70382e12	−0.498365	−0.249182	−	0.968457i	$-0.580162\pi$
$827$	−6.70382e12	−0.498365	−0.249182	+	0.968457i	$0.580162\pi$
$828$	0	0
$829$	−1.86852e13	−1.37405	−0.687025	−	0.726634i	$-0.741084\pi$
$829$	−1.86852e13	−1.37405	−0.687025	+	0.726634i	$0.741084\pi$
$830$	0	0
$831$	2.08705e13	1.51820
$832$	0	0
$833$	4.32470e12	0.311210
$834$	0	0
$835$	2.88272e13	2.05217
$836$	0	0
$837$	4.21142e13	2.96595
$838$	0	0
$839$	−2.31979e13	−1.61629	−0.808147	−	0.588980i	$-0.799530\pi$
$839$	−2.31979e13	−1.61629	−0.808147	+	0.588980i	$0.799530\pi$
$840$	0	0
$841$	−7.98001e12	−0.550074
$842$	0	0
$843$	1.07142e13	0.730694
$844$	0	0
$845$	5.93194e12	0.400259
$846$	0	0
$847$	−1.92915e13	−1.28793
$848$	0	0
$849$	1.96213e13	1.29612
$850$	0	0
$851$	9.78133e12	0.639315
$852$	0	0
$853$	−6.15810e12	−0.398269	−0.199134	−	0.979972i	$-0.563813\pi$
$853$	−6.15810e12	−0.398269	−0.199134	+	0.979972i	$0.563813\pi$
$854$	0	0
$855$	−2.58827e13	−1.65639
$856$	0	0
$857$	8.90805e12	0.564117	0.282058	−	0.959397i	$-0.408983\pi$
$857$	8.90805e12	0.564117	0.282058	+	0.959397i	$0.408983\pi$
$858$	0	0
$859$	1.06938e13	0.670135	0.335068	−	0.942194i	$-0.391241\pi$
$859$	1.06938e13	0.670135	0.335068	+	0.942194i	$0.391241\pi$
$860$	0	0
$861$	7.53156e13	4.67058
$862$	0	0
$863$	6.21448e12	0.381379	0.190689	−	0.981650i	$-0.438928\pi$
$863$	6.21448e12	0.381379	0.190689	+	0.981650i	$0.438928\pi$
$864$	0	0
$865$	1.73648e13	1.05463
$866$	0	0
$867$	−1.70188e12	−0.102292
$868$	0	0
$869$	2.54764e12	0.151548
$870$	0	0
$871$	9.67315e12	0.569490
$872$	0	0
$873$	−1.79378e13	−1.04521
$874$	0	0
$875$	−1.27786e13	−0.736966
$876$	0	0
$877$	1.74120e13	0.993916	0.496958	−	0.867775i	$-0.334450\pi$
$877$	1.74120e13	0.993916	0.496958	+	0.867775i	$0.334450\pi$
$878$	0	0
$879$	5.40999e12	0.305665
$880$	0	0
$881$	1.85361e13	1.03664	0.518318	−	0.855188i	$-0.326558\pi$
$881$	1.85361e13	1.03664	0.518318	+	0.855188i	$0.326558\pi$
$882$	0	0
$883$	5.46626e12	0.302599	0.151299	−	0.988488i	$-0.451654\pi$
$883$	5.46626e12	0.302599	0.151299	+	0.988488i	$0.451654\pi$
$884$	0	0
$885$	6.86741e12	0.376312
$886$	0	0
$887$	3.61417e13	1.96043	0.980216	−	0.197930i	$-0.0634219\pi$
$887$	3.61417e13	1.96043	0.980216	+	0.197930i	$0.0634219\pi$
$888$	0	0
$889$	−5.21140e13	−2.79832
$890$	0	0
$891$	−7.75348e12	−0.412142
$892$	0	0
$893$	−7.11214e12	−0.374256
$894$	0	0
$895$	2.17670e13	1.13395
$896$	0	0
$897$	3.53415e13	1.82272
$898$	0	0
$899$	2.18803e13	1.11721
$900$	0	0
$901$	−1.63785e12	−0.0827968
$902$	0	0
$903$	−5.58789e13	−2.79674
$904$	0	0
$905$	2.16433e13	1.07252
$906$	0	0
$907$	3.39738e13	1.66691	0.833454	−	0.552589i	$-0.186360\pi$
$907$	3.39738e13	1.66691	0.833454	+	0.552589i	$0.186360\pi$
$908$	0	0
$909$	2.16630e13	1.05240
$910$	0	0
$911$	−2.01195e12	−0.0967797	−0.0483898	−	0.998829i	$-0.515409\pi$
$911$	−2.01195e12	−0.0967797	−0.0483898	+	0.998829i	$0.515409\pi$
$912$	0	0
$913$	−1.21556e13	−0.578974
$914$	0	0
$915$	4.76457e13	2.24713
$916$	0	0
$917$	−2.26353e13	−1.05712
$918$	0	0
$919$	−1.59408e11	−0.00737207	−0.00368604	−	0.999993i	$-0.501173\pi$
$919$	−1.59408e11	−0.00737207	−0.00368604	+	0.999993i	$0.501173\pi$
$920$	0	0
$921$	3.32283e13	1.52174
$922$	0	0
$923$	−2.28701e13	−1.03720
$924$	0	0
$925$	−9.59850e12	−0.431088
$926$	0	0
$927$	−3.98180e13	−1.77101
$928$	0	0
$929$	4.18504e13	1.84344	0.921721	−	0.387854i	$-0.126784\pi$
$929$	4.18504e13	1.84344	0.921721	+	0.387854i	$0.126784\pi$
$930$	0	0
$931$	−1.89334e13	−0.825950
$932$	0	0
$933$	6.41967e13	2.77361
$934$	0	0
$935$	−2.76883e12	−0.118480
$936$	0	0
$937$	1.52091e13	0.644576	0.322288	−	0.946642i	$-0.395548\pi$
$937$	1.52091e13	0.644576	0.322288	+	0.946642i	$0.395548\pi$
$938$	0	0
$939$	−4.53648e13	−1.90425
$940$	0	0
$941$	−1.93636e13	−0.805067	−0.402534	−	0.915405i	$-0.631870\pi$
$941$	−1.93636e13	−0.805067	−0.402534	+	0.915405i	$0.631870\pi$
$942$	0	0
$943$	3.94555e13	1.62482
$944$	0	0
$945$	−8.38649e13	−3.42087
$946$	0	0
$947$	3.99237e13	1.61308	0.806540	−	0.591179i	$-0.201337\pi$
$947$	3.99237e13	1.61308	0.806540	+	0.591179i	$0.201337\pi$
$948$	0	0
$949$	−1.81141e12	−0.0724969
$950$	0	0
$951$	7.69506e13	3.05070
$952$	0	0
$953$	−3.60339e13	−1.41512	−0.707559	−	0.706654i	$-0.750204\pi$
$953$	−3.60339e13	−1.41512	−0.707559	+	0.706654i	$0.750204\pi$
$954$	0	0
$955$	2.89106e13	1.12472
$956$	0	0
$957$	−1.16296e13	−0.448188
$958$	0	0
$959$	−8.82691e12	−0.336996
$960$	0	0
$961$	4.69079e13	1.77415
$962$	0	0
$963$	−8.42981e11	−0.0315864
$964$	0	0
$965$	3.17381e13	1.17817
$966$	0	0
$967$	−2.93298e13	−1.07867	−0.539337	−	0.842090i	$-0.681325\pi$
$967$	−2.93298e13	−1.07867	−0.539337	+	0.842090i	$0.681325\pi$
$968$	0	0
$969$	7.45077e12	0.271484
$970$	0	0
$971$	−4.26932e13	−1.54125	−0.770623	−	0.637291i	$-0.780055\pi$
$971$	−4.26932e13	−1.54125	−0.770623	+	0.637291i	$0.780055\pi$
$972$	0	0
$973$	−1.22624e13	−0.438600
$974$	0	0
$975$	−3.46809e13	−1.22905
$976$	0	0
$977$	−3.65451e12	−0.128323	−0.0641613	−	0.997940i	$-0.520437\pi$
$977$	−3.65451e12	−0.128323	−0.0641613	+	0.997940i	$0.520437\pi$
$978$	0	0
$979$	1.30047e13	0.452460
$980$	0	0
$981$	−9.48803e13	−3.27089
$982$	0	0
$983$	5.51397e13	1.88353	0.941767	−	0.336265i	$-0.109164\pi$
$983$	5.51397e13	1.88353	0.941767	+	0.336265i	$0.109164\pi$
$984$	0	0
$985$	2.21581e13	0.750013
$986$	0	0
$987$	−4.55490e13	−1.52775
$988$	0	0
$989$	−2.92732e13	−0.972941
$990$	0	0
$991$	−4.65397e13	−1.53282	−0.766412	−	0.642350i	$-0.777960\pi$
$991$	−4.65397e13	−1.53282	−0.766412	+	0.642350i	$0.777960\pi$
$992$	0	0
$993$	−3.24939e12	−0.106055
$994$	0	0
$995$	3.14442e13	1.01704
$996$	0	0
$997$	4.29398e13	1.37636	0.688179	−	0.725541i	$-0.258410\pi$
$997$	4.29398e13	1.37636	0.688179	+	0.725541i	$0.258410\pi$
$998$	0	0
$999$	3.92072e13	1.24544

Twists

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

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

\(f(q)\)	\(=\)	\(q-243.971 q^{3} +1776.79 q^{5} +9598.61 q^{7} +39838.7 q^{9} +O(q^{10})\)	\(q-243.971 q^{3} +1776.79 q^{5} +9598.61 q^{7} +39838.7 q^{9} -18658.0 q^{11} +118081. q^{13} -433484. q^{15} +83521.0 q^{17} -365652. q^{19} -2.34178e6 q^{21} -1.22678e6 q^{23} +1.20385e6 q^{25} -4.91741e6 q^{27} -2.55483e6 q^{29} -8.56431e6 q^{31} +4.55201e6 q^{33} +1.70547e7 q^{35} -7.97315e6 q^{37} -2.88083e7 q^{39} -3.21617e7 q^{41} +2.38617e7 q^{43} +7.07850e7 q^{45} +1.94506e7 q^{47} +5.17797e7 q^{49} -2.03767e7 q^{51} -1.96101e7 q^{53} -3.31513e7 q^{55} +8.92083e7 q^{57} -1.58423e7 q^{59} -1.09913e8 q^{61} +3.82396e8 q^{63} +2.09805e8 q^{65} +8.19198e7 q^{67} +2.99299e8 q^{69} -1.93682e8 q^{71} -1.53405e7 q^{73} -2.93705e8 q^{75} -1.79091e8 q^{77} -1.36544e8 q^{79} +4.15558e8 q^{81} +6.51497e8 q^{83} +1.48399e8 q^{85} +6.23303e8 q^{87} -6.97006e8 q^{89} +1.13341e9 q^{91} +2.08944e9 q^{93} -6.49686e8 q^{95} -4.50260e8 q^{97} -7.43311e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9}+O(q^{10}) \) 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

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