LMFDB - Embedded newform 272.10.a.f.1.5

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 1596x^{3} + 5754x^{2} + 488987x - 2711704 $ x^5 - 2x^4 - 1596x^3 + 5754x^2 + 488987x - 2711704
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-35.0613$ 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)

$f(q)$	$=$	$q+256.773 q^{3} +1407.25 q^{5} +5138.64 q^{7} +46249.6 q^{9} +O(q^{10})$	$q+256.773 q^{3} +1407.25 q^{5} +5138.64 q^{7} +46249.6 q^{9} -26560.4 q^{11} +71402.0 q^{13} +361344. q^{15} -83521.0 q^{17} +548715. q^{19} +1.31947e6 q^{21} -1.15988e6 q^{23} +27218.8 q^{25} +6.82160e6 q^{27} +1.44199e6 q^{29} +6.05781e6 q^{31} -6.82000e6 q^{33} +7.23134e6 q^{35} -9.50334e6 q^{37} +1.83341e7 q^{39} +1.75776e7 q^{41} +2.06503e7 q^{43} +6.50846e7 q^{45} -3.15997e7 q^{47} -1.39480e7 q^{49} -2.14460e7 q^{51} -1.02011e8 q^{53} -3.73770e7 q^{55} +1.40895e8 q^{57} +5.95270e7 q^{59} +5.34058e7 q^{61} +2.37660e8 q^{63} +1.00480e8 q^{65} +6.45107e7 q^{67} -2.97827e8 q^{69} +2.71132e8 q^{71} +2.75443e8 q^{73} +6.98908e6 q^{75} -1.36484e8 q^{77} +4.33372e8 q^{79} +8.41275e8 q^{81} -1.23703e8 q^{83} -1.17535e8 q^{85} +3.70264e8 q^{87} -8.87168e8 q^{89} +3.66909e8 q^{91} +1.55548e9 q^{93} +7.72177e8 q^{95} -4.41697e8 q^{97} -1.22841e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9}+O(q^{10}) $ 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9	$ 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9} + 68036 q^{11} - 158862 q^{13} + 687324 q^{15} - 417605 q^{17} + 370992 q^{19} + 1783880 q^{21} - 1645870 q^{23} + 3270239 q^{25} + 2998268 q^{27} + 3668616 q^{29} + 7262362 q^{31} - 11334900 q^{33} + 26503988 q^{35} - 31420708 q^{37} + 42449884 q^{39} - 7996938 q^{41} + 56908268 q^{43} + 12799536 q^{45} + 16903336 q^{47} - 11784059 q^{49} - 19710956 q^{51} - 83362982 q^{53} - 6363364 q^{55} + 136615904 q^{57} + 37946604 q^{59} - 77685452 q^{61} + 191945278 q^{63} - 40321288 q^{65} + 304503600 q^{67} - 333409272 q^{69} + 476602922 q^{71} - 289980486 q^{73} + 153685772 q^{75} - 143385648 q^{77} + 828240610 q^{79} + 891328609 q^{81} - 194681148 q^{83} - 123611080 q^{85} - 158149884 q^{87} + 376848106 q^{89} - 194543664 q^{91} + 3494835920 q^{93} - 1498679864 q^{95} + 692035246 q^{97} - 2027106408 q^{99}+O(q^{100}) $ 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9 + 68036 * q^11 - 158862 * q^13 + 687324 * q^15 - 417605 * q^17 + 370992 * q^19 + 1783880 * q^21 - 1645870 * q^23 + 3270239 * q^25 + 2998268 * q^27 + 3668616 * q^29 + 7262362 * q^31 - 11334900 * q^33 + 26503988 * q^35 - 31420708 * q^37 + 42449884 * q^39 - 7996938 * q^41 + 56908268 * q^43 + 12799536 * q^45 + 16903336 * q^47 - 11784059 * q^49 - 19710956 * q^51 - 83362982 * q^53 - 6363364 * q^55 + 136615904 * q^57 + 37946604 * q^59 - 77685452 * q^61 + 191945278 * q^63 - 40321288 * q^65 + 304503600 * q^67 - 333409272 * q^69 + 476602922 * q^71 - 289980486 * q^73 + 153685772 * q^75 - 143385648 * q^77 + 828240610 * q^79 + 891328609 * q^81 - 194681148 * q^83 - 123611080 * q^85 - 158149884 * q^87 + 376848106 * q^89 - 194543664 * q^91 + 3494835920 * q^93 - 1498679864 * q^95 + 692035246 * q^97 - 2027106408 * 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
$3$	256.773	1.83022	0.915112	−	0.403199i	$-0.132102\pi$
$3$	256.773	1.83022	0.915112	+	0.403199i	$0.132102\pi$
$4$	0	0
$5$	1407.25	1.00694	0.503472	−	0.864012i	$-0.332056\pi$
$5$	1407.25	1.00694	0.503472	+	0.864012i	$0.332056\pi$
$6$	0	0
$7$	5138.64	0.808923	0.404461	−	0.914555i	$-0.367459\pi$
$7$	5138.64	0.808923	0.404461	+	0.914555i	$0.367459\pi$
$8$	0	0
$9$	46249.6	2.34972
$10$	0	0
$11$	−26560.4	−0.546975	−0.273487	−	0.961876i	$-0.588177\pi$
$11$	−26560.4	−0.546975	−0.273487	+	0.961876i	$0.588177\pi$
$12$	0	0
$13$	71402.0	0.693371	0.346685	−	0.937981i	$-0.387307\pi$
$13$	71402.0	0.693371	0.346685	+	0.937981i	$0.387307\pi$
$14$	0	0
$15$	361344.	1.84293
$16$	0	0
$17$	−83521.0	−0.242536
$17$	−83521.0	−0.242536
$18$	0	0
$19$	548715.	0.965951	0.482976	−	0.875634i	$-0.339556\pi$
$19$	548715.	0.965951	0.482976	+	0.875634i	$0.339556\pi$
$20$	0	0
$21$	1.31947e6	1.48051
$22$	0	0
$23$	−1.15988e6	−0.864247	−0.432124	−	0.901814i	$-0.642236\pi$
$23$	−1.15988e6	−0.864247	−0.432124	+	0.901814i	$0.642236\pi$
$24$	0	0
$25$	27218.8	0.0139360
$26$	0	0
$27$	6.82160e6	2.47030
$28$	0	0
$29$	1.44199e6	0.378592	0.189296	−	0.981920i	$-0.439380\pi$
$29$	1.44199e6	0.378592	0.189296	+	0.981920i	$0.439380\pi$
$30$	0	0
$31$	6.05781e6	1.17812	0.589058	−	0.808091i	$-0.299499\pi$
$31$	6.05781e6	1.17812	0.589058	+	0.808091i	$0.299499\pi$
$32$	0	0
$33$	−6.82000e6	−1.00109
$34$	0	0
$35$	7.23134e6	0.814540
$36$	0	0
$37$	−9.50334e6	−0.833621	−0.416810	−	0.908993i	$-0.636852\pi$
$37$	−9.50334e6	−0.833621	−0.416810	+	0.908993i	$0.636852\pi$
$38$	0	0
$39$	1.83341e7	1.26902
$40$	0	0
$41$	1.75776e7	0.971476	0.485738	−	0.874104i	$-0.338551\pi$
$41$	1.75776e7	0.971476	0.485738	+	0.874104i	$0.338551\pi$
$42$	0	0
$43$	2.06503e7	0.921125	0.460562	−	0.887627i	$-0.347648\pi$
$43$	2.06503e7	0.921125	0.460562	+	0.887627i	$0.347648\pi$
$44$	0	0
$45$	6.50846e7	2.36604
$46$	0	0
$47$	−3.15997e7	−0.944587	−0.472294	−	0.881441i	$-0.656574\pi$
$47$	−3.15997e7	−0.944587	−0.472294	+	0.881441i	$0.656574\pi$
$48$	0	0
$49$	−1.39480e7	−0.345644
$50$	0	0
$51$	−2.14460e7	−0.443895
$52$	0	0
$53$	−1.02011e8	−1.77585	−0.887924	−	0.459990i	$-0.847853\pi$
$53$	−1.02011e8	−1.77585	−0.887924	+	0.459990i	$0.847853\pi$
$54$	0	0
$55$	−3.73770e7	−0.550773
$56$	0	0
$57$	1.40895e8	1.76791
$58$	0	0
$59$	5.95270e7	0.639558	0.319779	−	0.947492i	$-0.396391\pi$
$59$	5.95270e7	0.639558	0.319779	+	0.947492i	$0.396391\pi$
$60$	0	0
$61$	5.34058e7	0.493860	0.246930	−	0.969033i	$-0.420578\pi$
$61$	5.34058e7	0.493860	0.246930	+	0.969033i	$0.420578\pi$
$62$	0	0
$63$	2.37660e8	1.90074
$64$	0	0
$65$	1.00480e8	0.698185
$66$	0	0
$67$	6.45107e7	0.391107	0.195553	−	0.980693i	$-0.437350\pi$
$67$	6.45107e7	0.391107	0.195553	+	0.980693i	$0.437350\pi$
$68$	0	0
$69$	−2.97827e8	−1.58177
$70$	0	0
$71$	2.71132e8	1.26625	0.633123	−	0.774051i	$-0.281773\pi$
$71$	2.71132e8	1.26625	0.633123	+	0.774051i	$0.281773\pi$
$72$	0	0
$73$	2.75443e8	1.13522	0.567609	−	0.823298i	$-0.307868\pi$
$73$	2.75443e8	1.13522	0.567609	+	0.823298i	$0.307868\pi$
$74$	0	0
$75$	6.98908e6	0.0255061
$76$	0	0
$77$	−1.36484e8	−0.442460
$78$	0	0
$79$	4.33372e8	1.25181	0.625906	−	0.779898i	$-0.284729\pi$
$79$	4.33372e8	1.25181	0.625906	+	0.779898i	$0.284729\pi$
$80$	0	0
$81$	8.41275e8	2.17148
$82$	0	0
$83$	−1.23703e8	−0.286106	−0.143053	−	0.989715i	$-0.545692\pi$
$83$	−1.23703e8	−0.286106	−0.143053	+	0.989715i	$0.545692\pi$
$84$	0	0
$85$	−1.17535e8	−0.244220
$86$	0	0
$87$	3.70264e8	0.692908
$88$	0	0
$89$	−8.87168e8	−1.49882	−0.749412	−	0.662104i	$-0.769664\pi$
$89$	−8.87168e8	−1.49882	−0.749412	+	0.662104i	$0.769664\pi$
$90$	0	0
$91$	3.66909e8	0.560883
$92$	0	0
$93$	1.55548e9	2.15622
$94$	0	0
$95$	7.72177e8	0.972659
$96$	0	0
$97$	−4.41697e8	−0.506585	−0.253292	−	0.967390i	$-0.581513\pi$
$97$	−4.41697e8	−0.506585	−0.253292	+	0.967390i	$0.581513\pi$
$98$	0	0
$99$	−1.22841e9	−1.28524
$100$	0	0
$101$	−1.14036e9	−1.09042	−0.545212	−	0.838298i	$-0.683551\pi$
$101$	−1.14036e9	−1.09042	−0.545212	+	0.838298i	$0.683551\pi$
$102$	0	0
$103$	−8.49153e8	−0.743393	−0.371696	−	0.928354i	$-0.621224\pi$
$103$	−8.49153e8	−0.743393	−0.371696	+	0.928354i	$0.621224\pi$
$104$	0	0
$105$	1.85682e9	1.49079
$106$	0	0
$107$	−2.23234e9	−1.64639	−0.823196	−	0.567758i	$-0.807811\pi$
$107$	−2.23234e9	−1.64639	−0.823196	+	0.567758i	$0.807811\pi$
$108$	0	0
$109$	1.34014e9	0.909352	0.454676	−	0.890657i	$-0.349755\pi$
$109$	1.34014e9	0.909352	0.454676	+	0.890657i	$0.349755\pi$
$110$	0	0
$111$	−2.44021e9	−1.52571
$112$	0	0
$113$	−2.00028e9	−1.15408	−0.577042	−	0.816715i	$-0.695793\pi$
$113$	−2.00028e9	−1.15408	−0.577042	+	0.816715i	$0.695793\pi$
$114$	0	0
$115$	−1.63224e9	−0.870248
$116$	0	0
$117$	3.30232e9	1.62923
$118$	0	0
$119$	−4.29184e8	−0.196193
$120$	0	0
$121$	−1.65249e9	−0.700819
$122$	0	0
$123$	4.51346e9	1.77802
$124$	0	0
$125$	−2.71023e9	−0.992911
$126$	0	0
$127$	3.35970e9	1.14600	0.572999	−	0.819556i	$-0.305780\pi$
$127$	3.35970e9	1.14600	0.572999	+	0.819556i	$0.305780\pi$
$128$	0	0
$129$	5.30245e9	1.68587
$130$	0	0
$131$	2.27481e8	0.0674876	0.0337438	−	0.999431i	$-0.489257\pi$
$131$	2.27481e8	0.0674876	0.0337438	+	0.999431i	$0.489257\pi$
$132$	0	0
$133$	2.81965e9	0.781380
$134$	0	0
$135$	9.59967e9	2.48745
$136$	0	0
$137$	−3.72862e9	−0.904285	−0.452142	−	0.891946i	$-0.649340\pi$
$137$	−3.72862e9	−0.904285	−0.452142	+	0.891946i	$0.649340\pi$
$138$	0	0
$139$	8.72554e8	0.198256	0.0991279	−	0.995075i	$-0.468395\pi$
$139$	8.72554e8	0.198256	0.0991279	+	0.995075i	$0.468395\pi$
$140$	0	0
$141$	−8.11395e9	−1.72881
$142$	0	0
$143$	−1.89647e9	−0.379256
$144$	0	0
$145$	2.02923e9	0.381220
$146$	0	0
$147$	−3.58147e9	−0.632606
$148$	0	0
$149$	5.86193e9	0.974323	0.487161	−	0.873312i	$-0.338032\pi$
$149$	5.86193e9	0.974323	0.487161	+	0.873312i	$0.338032\pi$
$150$	0	0
$151$	−1.01489e10	−1.58864	−0.794318	−	0.607502i	$-0.792172\pi$
$151$	−1.01489e10	−1.58864	−0.794318	+	0.607502i	$0.792172\pi$
$152$	0	0
$153$	−3.86281e9	−0.569892
$154$	0	0
$155$	8.52483e9	1.18630
$156$	0	0
$157$	−5.31929e9	−0.698723	−0.349362	−	0.936988i	$-0.613601\pi$
$157$	−5.31929e9	−0.698723	−0.349362	+	0.936988i	$0.613601\pi$
$158$	0	0
$159$	−2.61937e10	−3.25020
$160$	0	0
$161$	−5.96021e9	−0.699109
$162$	0	0
$163$	1.08994e10	1.20937	0.604685	−	0.796465i	$-0.293299\pi$
$163$	1.08994e10	1.20937	0.604685	+	0.796465i	$0.293299\pi$
$164$	0	0
$165$	−9.59743e9	−1.00804
$166$	0	0
$167$	5.90585e9	0.587569	0.293784	−	0.955872i	$-0.405085\pi$
$167$	5.90585e9	0.587569	0.293784	+	0.955872i	$0.405085\pi$
$168$	0	0
$169$	−5.50625e9	−0.519237
$170$	0	0
$171$	2.53778e10	2.26972
$172$	0	0
$173$	1.33412e10	1.13237	0.566183	−	0.824280i	$-0.308420\pi$
$173$	1.33412e10	1.13237	0.566183	+	0.824280i	$0.308420\pi$
$174$	0	0
$175$	1.39868e8	0.0112732
$176$	0	0
$177$	1.52850e10	1.17054
$178$	0	0
$179$	−1.62530e10	−1.18330	−0.591651	−	0.806194i	$-0.701524\pi$
$179$	−1.62530e10	−1.18330	−0.591651	+	0.806194i	$0.701524\pi$
$180$	0	0
$181$	−4.03320e9	−0.279316	−0.139658	−	0.990200i	$-0.544600\pi$
$181$	−4.03320e9	−0.279316	−0.139658	+	0.990200i	$0.544600\pi$
$182$	0	0
$183$	1.37132e10	0.903875
$184$	0	0
$185$	−1.33735e10	−0.839409
$186$	0	0
$187$	2.21835e9	0.132661
$188$	0	0
$189$	3.50537e10	1.99828
$190$	0	0
$191$	1.09015e10	0.592700	0.296350	−	0.955079i	$-0.404231\pi$
$191$	1.09015e10	0.592700	0.296350	+	0.955079i	$0.404231\pi$
$192$	0	0
$193$	1.29615e10	0.672428	0.336214	−	0.941786i	$-0.390853\pi$
$193$	1.29615e10	0.672428	0.336214	+	0.941786i	$0.390853\pi$
$194$	0	0
$195$	2.58007e10	1.27784
$196$	0	0
$197$	7.15882e9	0.338644	0.169322	−	0.985561i	$-0.445842\pi$
$197$	7.15882e9	0.338644	0.169322	+	0.985561i	$0.445842\pi$
$198$	0	0
$199$	−3.03762e9	−0.137307	−0.0686537	−	0.997641i	$-0.521870\pi$
$199$	−3.03762e9	−0.137307	−0.0686537	+	0.997641i	$0.521870\pi$
$200$	0	0
$201$	1.65646e10	0.715813
$202$	0	0
$203$	7.40986e9	0.306251
$204$	0	0
$205$	2.47360e10	0.978222
$206$	0	0
$207$	−5.36440e10	−2.03074
$208$	0	0
$209$	−1.45741e10	−0.528351
$210$	0	0
$211$	3.30011e10	1.14619	0.573097	−	0.819488i	$-0.305742\pi$
$211$	3.30011e10	1.14619	0.573097	+	0.819488i	$0.305742\pi$
$212$	0	0
$213$	6.96195e10	2.31752
$214$	0	0
$215$	2.90601e10	0.927521
$216$	0	0
$217$	3.11289e10	0.953004
$218$	0	0
$219$	7.07266e10	2.07771
$220$	0	0
$221$	−5.96357e9	−0.168167
$222$	0	0
$223$	3.03057e10	0.820640	0.410320	−	0.911942i	$-0.365417\pi$
$223$	3.03057e10	0.820640	0.410320	+	0.911942i	$0.365417\pi$
$224$	0	0
$225$	1.25886e9	0.0327459
$226$	0	0
$227$	4.49193e10	1.12284	0.561418	−	0.827532i	$-0.310256\pi$
$227$	4.49193e10	1.12284	0.561418	+	0.827532i	$0.310256\pi$
$228$	0	0
$229$	4.68022e10	1.12462	0.562311	−	0.826926i	$-0.309913\pi$
$229$	4.68022e10	1.12462	0.562311	+	0.826926i	$0.309913\pi$
$230$	0	0
$231$	−3.50455e10	−0.809802
$232$	0	0
$233$	−2.08143e10	−0.462658	−0.231329	−	0.972876i	$-0.574307\pi$
$233$	−2.08143e10	−0.462658	−0.231329	+	0.972876i	$0.574307\pi$
$234$	0	0
$235$	−4.44685e10	−0.951146
$236$	0	0
$237$	1.11279e11	2.29110
$238$	0	0
$239$	−1.23779e9	−0.0245390	−0.0122695	−	0.999925i	$-0.503906\pi$
$239$	−1.23779e9	−0.0245390	−0.0122695	+	0.999925i	$0.503906\pi$
$240$	0	0
$241$	−3.29491e10	−0.629167	−0.314584	−	0.949230i	$-0.601865\pi$
$241$	−3.29491e10	−0.629167	−0.314584	+	0.949230i	$0.601865\pi$
$242$	0	0
$243$	8.17474e10	1.50399
$244$	0	0
$245$	−1.96283e10	−0.348044
$246$	0	0
$247$	3.91793e10	0.669762
$248$	0	0
$249$	−3.17635e10	−0.523638
$250$	0	0
$251$	3.15140e10	0.501154	0.250577	−	0.968097i	$-0.419380\pi$
$251$	3.15140e10	0.501154	0.250577	+	0.968097i	$0.419380\pi$
$252$	0	0
$253$	3.08069e10	0.472721
$254$	0	0
$255$	−3.01798e10	−0.446977
$256$	0	0
$257$	−7.36840e10	−1.05360	−0.526798	−	0.849991i	$-0.676607\pi$
$257$	−7.36840e10	−1.05360	−0.526798	+	0.849991i	$0.676607\pi$
$258$	0	0
$259$	−4.88342e10	−0.674335
$260$	0	0
$261$	6.66914e10	0.889585
$262$	0	0
$263$	5.81832e10	0.749888	0.374944	−	0.927047i	$-0.377662\pi$
$263$	5.81832e10	0.749888	0.374944	+	0.927047i	$0.377662\pi$
$264$	0	0
$265$	−1.43555e11	−1.78818
$266$	0	0
$267$	−2.27801e11	−2.74318
$268$	0	0
$269$	1.32703e11	1.54523	0.772617	−	0.634872i	$-0.218947\pi$
$269$	1.32703e11	1.54523	0.772617	+	0.634872i	$0.218947\pi$
$270$	0	0
$271$	1.25505e11	1.41351	0.706757	−	0.707456i	$-0.250157\pi$
$271$	1.25505e11	1.41351	0.706757	+	0.707456i	$0.250157\pi$
$272$	0	0
$273$	9.42126e10	1.02654
$274$	0	0
$275$	−7.22943e8	−0.00762267
$276$	0	0
$277$	−1.08560e11	−1.10792	−0.553961	−	0.832542i	$-0.686884\pi$
$277$	−1.08560e11	−1.10792	−0.553961	+	0.832542i	$0.686884\pi$
$278$	0	0
$279$	2.80171e11	2.76825
$280$	0	0
$281$	−1.21647e11	−1.16392	−0.581961	−	0.813216i	$-0.697714\pi$
$281$	−1.21647e11	−1.16392	−0.581961	+	0.813216i	$0.697714\pi$
$282$	0	0
$283$	−1.97694e11	−1.83212	−0.916060	−	0.401041i	$-0.868648\pi$
$283$	−1.97694e11	−1.83212	−0.916060	+	0.401041i	$0.868648\pi$
$284$	0	0
$285$	1.98275e11	1.78018
$286$	0	0
$287$	9.03249e10	0.785849
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	−1.13416e11	−0.927164
$292$	0	0
$293$	−5.29206e10	−0.419489	−0.209745	−	0.977756i	$-0.567263\pi$
$293$	−5.29206e10	−0.419489	−0.209745	+	0.977756i	$0.567263\pi$
$294$	0	0
$295$	8.37692e10	0.643999
$296$	0	0
$297$	−1.81184e11	−1.35119
$298$	0	0
$299$	−8.28178e10	−0.599244
$300$	0	0
$301$	1.06115e11	0.745119
$302$	0	0
$303$	−2.92814e11	−1.99572
$304$	0	0
$305$	7.51551e10	0.497290
$306$	0	0
$307$	−6.66906e10	−0.428491	−0.214246	−	0.976780i	$-0.568729\pi$
$307$	−6.66906e10	−0.428491	−0.214246	+	0.976780i	$0.568729\pi$
$308$	0	0
$309$	−2.18040e11	−1.36058
$310$	0	0
$311$	−2.16895e11	−1.31470	−0.657352	−	0.753584i	$-0.728324\pi$
$311$	−2.16895e11	−1.31470	−0.657352	+	0.753584i	$0.728324\pi$
$312$	0	0
$313$	2.03438e11	1.19807	0.599034	−	0.800723i	$-0.295551\pi$
$313$	2.03438e11	1.19807	0.599034	+	0.800723i	$0.295551\pi$
$314$	0	0
$315$	3.34446e11	1.91394
$316$	0	0
$317$	2.27635e11	1.26611	0.633056	−	0.774106i	$-0.281800\pi$
$317$	2.27635e11	1.26611	0.633056	+	0.774106i	$0.281800\pi$
$318$	0	0
$319$	−3.82998e10	−0.207080
$320$	0	0
$321$	−5.73205e11	−3.01327
$322$	0	0
$323$	−4.58292e10	−0.234278
$324$	0	0
$325$	1.94348e9	0.00966285
$326$	0	0
$327$	3.44113e11	1.66432
$328$	0	0
$329$	−1.62379e11	−0.764098
$330$	0	0
$331$	−1.97602e11	−0.904828	−0.452414	−	0.891808i	$-0.649437\pi$
$331$	−1.97602e11	−0.904828	−0.452414	+	0.891808i	$0.649437\pi$
$332$	0	0
$333$	−4.39526e11	−1.95878
$334$	0	0
$335$	9.07825e10	0.393822
$336$	0	0
$337$	4.61396e9	0.0194867	0.00974337	−	0.999953i	$-0.496899\pi$
$337$	4.61396e9	0.0194867	0.00974337	+	0.999953i	$0.496899\pi$
$338$	0	0
$339$	−5.13618e11	−2.11223
$340$	0	0
$341$	−1.60898e11	−0.644399
$342$	0	0
$343$	−2.79036e11	−1.08852
$344$	0	0
$345$	−4.19115e11	−1.59275
$346$	0	0
$347$	3.34855e11	1.23987	0.619933	−	0.784655i	$-0.287160\pi$
$347$	3.34855e11	1.23987	0.619933	+	0.784655i	$0.287160\pi$
$348$	0	0
$349$	−1.38047e11	−0.498095	−0.249048	−	0.968491i	$-0.580118\pi$
$349$	−1.38047e11	−0.498095	−0.249048	+	0.968491i	$0.580118\pi$
$350$	0	0
$351$	4.87076e11	1.71283
$352$	0	0
$353$	3.22524e11	1.10554	0.552771	−	0.833333i	$-0.313570\pi$
$353$	3.22524e11	1.10554	0.552771	+	0.833333i	$0.313570\pi$
$354$	0	0
$355$	3.81550e11	1.27504
$356$	0	0
$357$	−1.10203e11	−0.359077
$358$	0	0
$359$	−1.09669e11	−0.348465	−0.174232	−	0.984705i	$-0.555744\pi$
$359$	−1.09669e11	−0.348465	−0.174232	+	0.984705i	$0.555744\pi$
$360$	0	0
$361$	−2.16000e10	−0.0669378
$362$	0	0
$363$	−4.24316e11	−1.28266
$364$	0	0
$365$	3.87617e11	1.14310
$366$	0	0
$367$	−1.91239e11	−0.550273	−0.275137	−	0.961405i	$-0.588723\pi$
$367$	−1.91239e11	−0.550273	−0.275137	+	0.961405i	$0.588723\pi$
$368$	0	0
$369$	8.12957e11	2.28270
$370$	0	0
$371$	−5.24198e11	−1.43652
$372$	0	0
$373$	1.18077e11	0.315846	0.157923	−	0.987451i	$-0.449520\pi$
$373$	1.18077e11	0.315846	0.157923	+	0.987451i	$0.449520\pi$
$374$	0	0
$375$	−6.95914e11	−1.81725
$376$	0	0
$377$	1.02961e11	0.262504
$378$	0	0
$379$	1.76959e11	0.440551	0.220276	−	0.975438i	$-0.429304\pi$
$379$	1.76959e11	0.440551	0.220276	+	0.975438i	$0.429304\pi$
$380$	0	0
$381$	8.62682e11	2.09743
$382$	0	0
$383$	−1.29043e11	−0.306436	−0.153218	−	0.988192i	$-0.548964\pi$
$383$	−1.29043e11	−0.306436	−0.153218	+	0.988192i	$0.548964\pi$
$384$	0	0
$385$	−1.92067e11	−0.445533
$386$	0	0
$387$	9.55069e11	2.16439
$388$	0	0
$389$	6.86269e10	0.151957	0.0759785	−	0.997109i	$-0.475792\pi$
$389$	6.86269e10	0.151957	0.0759785	+	0.997109i	$0.475792\pi$
$390$	0	0
$391$	9.68744e10	0.209611
$392$	0	0
$393$	5.84110e10	0.123517
$394$	0	0
$395$	6.09862e11	1.26051
$396$	0	0
$397$	7.83412e11	1.58283	0.791413	−	0.611282i	$-0.209346\pi$
$397$	7.83412e11	1.58283	0.791413	+	0.611282i	$0.209346\pi$
$398$	0	0
$399$	7.24011e11	1.43010
$400$	0	0
$401$	1.06747e11	0.206161	0.103080	−	0.994673i	$-0.467130\pi$
$401$	1.06747e11	0.206161	0.103080	+	0.994673i	$0.467130\pi$
$402$	0	0
$403$	4.32540e11	0.816871
$404$	0	0
$405$	1.18388e12	2.18656
$406$	0	0
$407$	2.52412e11	0.455969
$408$	0	0
$409$	−1.08641e12	−1.91973	−0.959863	−	0.280469i	$-0.909510\pi$
$409$	−1.08641e12	−1.91973	−0.959863	+	0.280469i	$0.909510\pi$
$410$	0	0
$411$	−9.57410e11	−1.65504
$412$	0	0
$413$	3.05888e11	0.517353
$414$	0	0
$415$	−1.74080e11	−0.288093
$416$	0	0
$417$	2.24049e11	0.362853
$418$	0	0
$419$	−6.48628e11	−1.02809	−0.514047	−	0.857762i	$-0.671854\pi$
$419$	−6.48628e11	−1.02809	−0.514047	+	0.857762i	$0.671854\pi$
$420$	0	0
$421$	−8.74376e11	−1.35653	−0.678264	−	0.734818i	$-0.737267\pi$
$421$	−8.74376e11	−1.35653	−0.678264	+	0.734818i	$0.737267\pi$
$422$	0	0
$423$	−1.46147e12	−2.21952
$424$	0	0
$425$	−2.27334e9	−0.00337999
$426$	0	0
$427$	2.74433e11	0.399495
$428$	0	0
$429$	−4.86962e11	−0.694124
$430$	0	0
$431$	−2.93756e10	−0.0410052	−0.0205026	−	0.999790i	$-0.506527\pi$
$431$	−2.93756e10	−0.0410052	−0.0205026	+	0.999790i	$0.506527\pi$
$432$	0	0
$433$	−1.33717e12	−1.82806	−0.914029	−	0.405648i	$-0.867046\pi$
$433$	−1.33717e12	−1.82806	−0.914029	+	0.405648i	$0.867046\pi$
$434$	0	0
$435$	5.21053e11	0.697719
$436$	0	0
$437$	−6.36443e11	−0.834821
$438$	0	0
$439$	−7.90604e11	−1.01594	−0.507971	−	0.861374i	$-0.669604\pi$
$439$	−7.90604e11	−1.01594	−0.507971	+	0.861374i	$0.669604\pi$
$440$	0	0
$441$	−6.45089e11	−0.812168
$442$	0	0
$443$	−1.18811e12	−1.46569	−0.732843	−	0.680398i	$-0.761807\pi$
$443$	−1.18811e12	−1.46569	−0.732843	+	0.680398i	$0.761807\pi$
$444$	0	0
$445$	−1.24846e12	−1.50923
$446$	0	0
$447$	1.50519e12	1.78323
$448$	0	0
$449$	6.04411e11	0.701817	0.350909	−	0.936410i	$-0.385873\pi$
$449$	6.04411e11	0.701817	0.350909	+	0.936410i	$0.385873\pi$
$450$	0	0
$451$	−4.66868e11	−0.531373
$452$	0	0
$453$	−2.60598e12	−2.90756
$454$	0	0
$455$	5.16332e11	0.564778
$456$	0	0
$457$	−9.65513e11	−1.03546	−0.517732	−	0.855543i	$-0.673224\pi$
$457$	−9.65513e11	−1.03546	−0.517732	+	0.855543i	$0.673224\pi$
$458$	0	0
$459$	−5.69747e11	−0.599135
$460$	0	0
$461$	5.15082e10	0.0531156	0.0265578	−	0.999647i	$-0.491545\pi$
$461$	5.15082e10	0.0531156	0.0265578	+	0.999647i	$0.491545\pi$
$462$	0	0
$463$	7.66043e11	0.774709	0.387355	−	0.921931i	$-0.373389\pi$
$463$	7.66043e11	0.774709	0.387355	+	0.921931i	$0.373389\pi$
$464$	0	0
$465$	2.18895e12	2.17119
$466$	0	0
$467$	1.55842e12	1.51621	0.758104	−	0.652133i	$-0.226126\pi$
$467$	1.55842e12	1.51621	0.758104	+	0.652133i	$0.226126\pi$
$468$	0	0
$469$	3.31497e11	0.316375
$470$	0	0
$471$	−1.36585e12	−1.27882
$472$	0	0
$473$	−5.48480e11	−0.503832
$474$	0	0
$475$	1.49354e10	0.0134615
$476$	0	0
$477$	−4.71797e12	−4.17275
$478$	0	0
$479$	−1.36399e12	−1.18386	−0.591931	−	0.805989i	$-0.701634\pi$
$479$	−1.36399e12	−1.18386	−0.591931	+	0.805989i	$0.701634\pi$
$480$	0	0
$481$	−6.78558e11	−0.578008
$482$	0	0
$483$	−1.53042e12	−1.27953
$484$	0	0
$485$	−6.21577e11	−0.510102
$486$	0	0
$487$	9.00584e11	0.725511	0.362755	−	0.931884i	$-0.381836\pi$
$487$	9.00584e11	0.725511	0.362755	+	0.931884i	$0.381836\pi$
$488$	0	0
$489$	2.79868e12	2.21342
$490$	0	0
$491$	−2.39477e12	−1.85950	−0.929751	−	0.368189i	$-0.879978\pi$
$491$	−2.39477e12	−1.85950	−0.929751	+	0.368189i	$0.879978\pi$
$492$	0	0
$493$	−1.20436e11	−0.0918219
$494$	0	0
$495$	−1.72867e12	−1.29416
$496$	0	0
$497$	1.39325e12	1.02430
$498$	0	0
$499$	1.19891e12	0.865637	0.432819	−	0.901481i	$-0.357519\pi$
$499$	1.19891e12	0.865637	0.432819	+	0.901481i	$0.357519\pi$
$500$	0	0
$501$	1.51647e12	1.07538
$502$	0	0
$503$	3.61742e11	0.251967	0.125983	−	0.992032i	$-0.459791\pi$
$503$	3.61742e11	0.251967	0.125983	+	0.992032i	$0.459791\pi$
$504$	0	0
$505$	−1.60477e12	−1.09800
$506$	0	0
$507$	−1.41386e12	−0.950321
$508$	0	0
$509$	1.61404e12	1.06582	0.532910	−	0.846172i	$-0.321098\pi$
$509$	1.61404e12	1.06582	0.532910	+	0.846172i	$0.321098\pi$
$510$	0	0
$511$	1.41541e12	0.918304
$512$	0	0
$513$	3.74311e12	2.38619
$514$	0	0
$515$	−1.19497e12	−0.748555
$516$	0	0
$517$	8.39299e11	0.516665
$518$	0	0
$519$	3.42566e12	2.07248
$520$	0	0
$521$	−2.86199e12	−1.70176	−0.850881	−	0.525358i	$-0.823931\pi$
$521$	−2.86199e12	−1.70176	−0.850881	+	0.525358i	$0.823931\pi$
$522$	0	0
$523$	−1.01799e12	−0.594957	−0.297479	−	0.954728i	$-0.596146\pi$
$523$	−1.01799e12	−0.594957	−0.297479	+	0.954728i	$0.596146\pi$
$524$	0	0
$525$	3.59143e10	0.0206325
$526$	0	0
$527$	−5.05954e11	−0.285735
$528$	0	0
$529$	−4.55830e11	−0.253077
$530$	0	0
$531$	2.75310e12	1.50278
$532$	0	0
$533$	1.25508e12	0.673593
$534$	0	0
$535$	−3.14145e12	−1.65782
$536$	0	0
$537$	−4.17335e12	−2.16571
$538$	0	0
$539$	3.70464e11	0.189058
$540$	0	0
$541$	−1.10637e12	−0.555283	−0.277642	−	0.960685i	$-0.589553\pi$
$541$	−1.10637e12	−0.555283	−0.277642	+	0.960685i	$0.589553\pi$
$542$	0	0
$543$	−1.03562e12	−0.511212
$544$	0	0
$545$	1.88591e12	0.915666
$546$	0	0
$547$	1.37273e12	0.655605	0.327803	−	0.944746i	$-0.393692\pi$
$547$	1.37273e12	0.655605	0.327803	+	0.944746i	$0.393692\pi$
$548$	0	0
$549$	2.47000e12	1.16044
$550$	0	0
$551$	7.91240e11	0.365701
$552$	0	0
$553$	2.22695e12	1.01262
$554$	0	0
$555$	−3.43397e12	−1.53631
$556$	0	0
$557$	−2.29710e12	−1.01119	−0.505594	−	0.862771i	$-0.668727\pi$
$557$	−2.29710e12	−1.01119	−0.505594	+	0.862771i	$0.668727\pi$
$558$	0	0
$559$	1.47447e12	0.638681
$560$	0	0
$561$	5.69613e11	0.242799
$562$	0	0
$563$	1.84812e11	0.0775250	0.0387625	−	0.999248i	$-0.487658\pi$
$563$	1.84812e11	0.0775250	0.0387625	+	0.999248i	$0.487658\pi$
$564$	0	0
$565$	−2.81488e12	−1.16210
$566$	0	0
$567$	4.32301e12	1.75656
$568$	0	0
$569$	−2.10912e11	−0.0843523	−0.0421761	−	0.999110i	$-0.513429\pi$
$569$	−2.10912e11	−0.0843523	−0.0421761	+	0.999110i	$0.513429\pi$
$570$	0	0
$571$	−2.37391e12	−0.934548	−0.467274	−	0.884112i	$-0.654764\pi$
$571$	−2.37391e12	−0.934548	−0.467274	+	0.884112i	$0.654764\pi$
$572$	0	0
$573$	2.79921e12	1.08477
$574$	0	0
$575$	−3.15706e10	−0.0120442
$576$	0	0
$577$	1.71792e12	0.645225	0.322613	−	0.946531i	$-0.395439\pi$
$577$	1.71792e12	0.645225	0.322613	+	0.946531i	$0.395439\pi$
$578$	0	0
$579$	3.32816e12	1.23070
$580$	0	0
$581$	−6.35663e11	−0.231438
$582$	0	0
$583$	2.70945e12	0.971344
$584$	0	0
$585$	4.64717e12	1.64054
$586$	0	0
$587$	−2.32455e12	−0.808103	−0.404052	−	0.914736i	$-0.632398\pi$
$587$	−2.32455e12	−0.808103	−0.404052	+	0.914736i	$0.632398\pi$
$588$	0	0
$589$	3.32401e12	1.13800
$590$	0	0
$591$	1.83820e12	0.619795
$592$	0	0
$593$	2.36534e12	0.785503	0.392751	−	0.919645i	$-0.371523\pi$
$593$	2.36534e12	0.785503	0.392751	+	0.919645i	$0.371523\pi$
$594$	0	0
$595$	−6.03968e11	−0.197555
$596$	0	0
$597$	−7.79979e11	−0.251303
$598$	0	0
$599$	−1.14807e12	−0.364375	−0.182188	−	0.983264i	$-0.558318\pi$
$599$	−1.14807e12	−0.364375	−0.182188	+	0.983264i	$0.558318\pi$
$600$	0	0
$601$	−3.38041e12	−1.05690	−0.528451	−	0.848964i	$-0.677227\pi$
$601$	−3.38041e12	−1.05690	−0.528451	+	0.848964i	$0.677227\pi$
$602$	0	0
$603$	2.98359e12	0.918992
$604$	0	0
$605$	−2.32547e12	−0.705685
$606$	0	0
$607$	−2.57549e12	−0.770036	−0.385018	−	0.922909i	$-0.625805\pi$
$607$	−2.57549e12	−0.770036	−0.385018	+	0.922909i	$0.625805\pi$
$608$	0	0
$609$	1.90266e12	0.560509
$610$	0	0
$611$	−2.25628e12	−0.654949
$612$	0	0
$613$	−1.91477e12	−0.547701	−0.273851	−	0.961772i	$-0.588297\pi$
$613$	−1.91477e12	−0.547701	−0.273851	+	0.961772i	$0.588297\pi$
$614$	0	0
$615$	6.35155e12	1.79037
$616$	0	0
$617$	5.24069e12	1.45581	0.727907	−	0.685676i	$-0.240493\pi$
$617$	5.24069e12	1.45581	0.727907	+	0.685676i	$0.240493\pi$
$618$	0	0
$619$	5.64576e10	0.0154566	0.00772831	−	0.999970i	$-0.497540\pi$
$619$	5.64576e10	0.0154566	0.00772831	+	0.999970i	$0.497540\pi$
$620$	0	0
$621$	−7.91224e12	−2.13495
$622$	0	0
$623$	−4.55884e12	−1.21243
$624$	0	0
$625$	−3.86712e12	−1.01374
$626$	0	0
$627$	−3.74223e12	−0.967001
$628$	0	0
$629$	7.93728e11	0.202183
$630$	0	0
$631$	−3.44629e12	−0.865405	−0.432703	−	0.901537i	$-0.642440\pi$
$631$	−3.44629e12	−0.865405	−0.432703	+	0.901537i	$0.642440\pi$
$632$	0	0
$633$	8.47382e12	2.09779
$634$	0	0
$635$	4.72793e12	1.15396
$636$	0	0
$637$	−9.95914e11	−0.239659
$638$	0	0
$639$	1.25398e13	2.97533
$640$	0	0
$641$	−5.50971e12	−1.28904	−0.644522	−	0.764586i	$-0.722943\pi$
$641$	−5.50971e12	−1.28904	−0.644522	+	0.764586i	$0.722943\pi$
$642$	0	0
$643$	−5.98753e11	−0.138133	−0.0690667	−	0.997612i	$-0.522002\pi$
$643$	−5.98753e11	−0.138133	−0.0690667	+	0.997612i	$0.522002\pi$
$644$	0	0
$645$	7.46186e12	1.69757
$646$	0	0
$647$	2.84687e12	0.638701	0.319351	−	0.947637i	$-0.396535\pi$
$647$	2.84687e12	0.638701	0.319351	+	0.947637i	$0.396535\pi$
$648$	0	0
$649$	−1.58106e12	−0.349822
$650$	0	0
$651$	7.99307e12	1.74421
$652$	0	0
$653$	−1.82215e12	−0.392170	−0.196085	−	0.980587i	$-0.562823\pi$
$653$	−1.82215e12	−0.392170	−0.196085	+	0.980587i	$0.562823\pi$
$654$	0	0
$655$	3.20122e11	0.0679562
$656$	0	0
$657$	1.27392e13	2.66745
$658$	0	0
$659$	3.55378e12	0.734017	0.367008	−	0.930218i	$-0.380382\pi$
$659$	3.55378e12	0.734017	0.367008	+	0.930218i	$0.380382\pi$
$660$	0	0
$661$	4.80541e12	0.979093	0.489546	−	0.871977i	$-0.337162\pi$
$661$	4.80541e12	0.979093	0.489546	+	0.871977i	$0.337162\pi$
$662$	0	0
$663$	−1.53129e12	−0.307784
$664$	0	0
$665$	3.96794e12	0.786806
$666$	0	0
$667$	−1.67253e12	−0.327197
$668$	0	0
$669$	7.78170e12	1.50196
$670$	0	0
$671$	−1.41848e12	−0.270129
$672$	0	0
$673$	5.10315e12	0.958894	0.479447	−	0.877571i	$-0.340837\pi$
$673$	5.10315e12	0.958894	0.479447	+	0.877571i	$0.340837\pi$
$674$	0	0
$675$	1.85676e11	0.0344262
$676$	0	0
$677$	2.66255e12	0.487134	0.243567	−	0.969884i	$-0.421682\pi$
$677$	2.66255e12	0.487134	0.243567	+	0.969884i	$0.421682\pi$
$678$	0	0
$679$	−2.26972e12	−0.409788
$680$	0	0
$681$	1.15341e13	2.05504
$682$	0	0
$683$	3.15981e12	0.555607	0.277804	−	0.960638i	$-0.410394\pi$
$683$	3.15981e12	0.555607	0.277804	+	0.960638i	$0.410394\pi$
$684$	0	0
$685$	−5.24708e12	−0.910564
$686$	0	0
$687$	1.20176e13	2.05831
$688$	0	0
$689$	−7.28380e12	−1.23132
$690$	0	0
$691$	7.41599e12	1.23742	0.618711	−	0.785619i	$-0.287655\pi$
$691$	7.41599e12	1.23742	0.618711	+	0.785619i	$0.287655\pi$
$692$	0	0
$693$	−6.31234e12	−1.03966
$694$	0	0
$695$	1.22790e12	0.199632
$696$	0	0
$697$	−1.46810e12	−0.235618
$698$	0	0
$699$	−5.34456e12	−0.846768
$700$	0	0
$701$	−9.39002e12	−1.46871	−0.734354	−	0.678767i	$-0.762515\pi$
$701$	−9.39002e12	−1.46871	−0.734354	+	0.678767i	$0.762515\pi$
$702$	0	0
$703$	−5.21462e12	−0.805237
$704$	0	0
$705$	−1.14183e13	−1.74081
$706$	0	0
$707$	−5.85989e12	−0.882069
$708$	0	0
$709$	4.58730e11	0.0681788	0.0340894	−	0.999419i	$-0.489147\pi$
$709$	4.58730e11	0.0681788	0.0340894	+	0.999419i	$0.489147\pi$
$710$	0	0
$711$	2.00433e13	2.94141
$712$	0	0
$713$	−7.02633e12	−1.01818
$714$	0	0
$715$	−2.66880e12	−0.381890
$716$	0	0
$717$	−3.17832e11	−0.0449119
$718$	0	0
$719$	−4.10035e12	−0.572191	−0.286096	−	0.958201i	$-0.592357\pi$
$719$	−4.10035e12	−0.572191	−0.286096	+	0.958201i	$0.592357\pi$
$720$	0	0
$721$	−4.36349e12	−0.601347
$722$	0	0
$723$	−8.46044e12	−1.15152
$724$	0	0
$725$	3.92493e10	0.00527607
$726$	0	0
$727$	−3.64967e12	−0.484561	−0.242280	−	0.970206i	$-0.577895\pi$
$727$	−3.64967e12	−0.484561	−0.242280	+	0.970206i	$0.577895\pi$
$728$	0	0
$729$	4.43176e12	0.581169
$730$	0	0
$731$	−1.72473e12	−0.223406
$732$	0	0
$733$	−2.77480e11	−0.0355029	−0.0177514	−	0.999842i	$-0.505651\pi$
$733$	−2.77480e11	−0.0355029	−0.0177514	+	0.999842i	$0.505651\pi$
$734$	0	0
$735$	−5.04001e12	−0.636999
$736$	0	0
$737$	−1.71343e12	−0.213925
$738$	0	0
$739$	−1.15877e13	−1.42922	−0.714608	−	0.699525i	$-0.753395\pi$
$739$	−1.15877e13	−1.42922	−0.714608	+	0.699525i	$0.753395\pi$
$740$	0	0
$741$	1.00602e13	1.22582
$742$	0	0
$743$	−4.79681e12	−0.577434	−0.288717	−	0.957414i	$-0.593229\pi$
$743$	−4.79681e12	−0.577434	−0.288717	+	0.957414i	$0.593229\pi$
$744$	0	0
$745$	8.24919e12	0.981088
$746$	0	0
$747$	−5.72119e12	−0.672270
$748$	0	0
$749$	−1.14712e13	−1.33180
$750$	0	0
$751$	2.32776e12	0.267029	0.133515	−	0.991047i	$-0.457374\pi$
$751$	2.32776e12	0.267029	0.133515	+	0.991047i	$0.457374\pi$
$752$	0	0
$753$	8.09196e12	0.917225
$754$	0	0
$755$	−1.42821e13	−1.59967
$756$	0	0
$757$	−6.01494e12	−0.665732	−0.332866	−	0.942974i	$-0.608016\pi$
$757$	−6.01494e12	−0.665732	−0.332866	+	0.942974i	$0.608016\pi$
$758$	0	0
$759$	7.91039e12	0.865187
$760$	0	0
$761$	−1.12427e13	−1.21518	−0.607589	−	0.794252i	$-0.707863\pi$
$761$	−1.12427e13	−1.21518	−0.607589	+	0.794252i	$0.707863\pi$
$762$	0	0
$763$	6.88651e12	0.735595
$764$	0	0
$765$	−5.43593e12	−0.573849
$766$	0	0
$767$	4.25035e12	0.443451
$768$	0	0
$769$	1.35036e13	1.39246	0.696230	−	0.717819i	$-0.254860\pi$
$769$	1.35036e13	1.39246	0.696230	+	0.717819i	$0.254860\pi$
$770$	0	0
$771$	−1.89201e13	−1.92832
$772$	0	0
$773$	2.43055e12	0.244848	0.122424	−	0.992478i	$-0.460933\pi$
$773$	2.43055e12	0.244848	0.122424	+	0.992478i	$0.460933\pi$
$774$	0	0
$775$	1.64886e11	0.0164183
$776$	0	0
$777$	−1.25393e13	−1.23418
$778$	0	0
$779$	9.64508e12	0.938399
$780$	0	0
$781$	−7.20137e12	−0.692605
$782$	0	0
$783$	9.83667e12	0.935234
$784$	0	0
$785$	−7.48555e12	−0.703575
$786$	0	0
$787$	3.78714e12	0.351905	0.175953	−	0.984399i	$-0.443699\pi$
$787$	3.78714e12	0.351905	0.175953	+	0.984399i	$0.443699\pi$
$788$	0	0
$789$	1.49399e13	1.37246
$790$	0	0
$791$	−1.02787e13	−0.933565
$792$	0	0
$793$	3.81328e12	0.342428
$794$	0	0
$795$	−3.68611e13	−3.27277
$796$	0	0
$797$	2.23878e12	0.196539	0.0982694	−	0.995160i	$-0.468669\pi$
$797$	2.23878e12	0.196539	0.0982694	+	0.995160i	$0.468669\pi$
$798$	0	0
$799$	2.63924e12	0.229096
$800$	0	0
$801$	−4.10311e13	−3.52182
$802$	0	0
$803$	−7.31588e12	−0.620936
$804$	0	0
$805$	−8.38749e12	−0.703964
$806$	0	0
$807$	3.40745e13	2.82813
$808$	0	0
$809$	−6.77107e12	−0.555762	−0.277881	−	0.960615i	$-0.589632\pi$
$809$	−6.77107e12	−0.555762	−0.277881	+	0.960615i	$0.589632\pi$
$810$	0	0
$811$	−1.26099e13	−1.02357	−0.511785	−	0.859113i	$-0.671016\pi$
$811$	−1.26099e13	−1.02357	−0.511785	+	0.859113i	$0.671016\pi$
$812$	0	0
$813$	3.22264e13	2.58705
$814$	0	0
$815$	1.53382e13	1.21777
$816$	0	0
$817$	1.13311e13	0.889762
$818$	0	0
$819$	1.69694e13	1.31792
$820$	0	0
$821$	1.40858e13	1.08203	0.541013	−	0.841014i	$-0.318041\pi$
$821$	1.40858e13	1.08203	0.541013	+	0.841014i	$0.318041\pi$
$822$	0	0
$823$	1.62081e13	1.23150	0.615748	−	0.787943i	$-0.288854\pi$
$823$	1.62081e13	1.23150	0.615748	+	0.787943i	$0.288854\pi$
$824$	0	0
$825$	−1.85633e11	−0.0139512
$826$	0	0
$827$	−8.84797e12	−0.657762	−0.328881	−	0.944371i	$-0.606671\pi$
$827$	−8.84797e12	−0.657762	−0.328881	+	0.944371i	$0.606671\pi$
$828$	0	0
$829$	9.64554e12	0.709302	0.354651	−	0.934999i	$-0.384600\pi$
$829$	9.64554e12	0.709302	0.354651	+	0.934999i	$0.384600\pi$
$830$	0	0
$831$	−2.78752e13	−2.02775
$832$	0	0
$833$	1.16495e12	0.0838310
$834$	0	0
$835$	8.31099e12	0.591649
$836$	0	0
$837$	4.13239e13	2.91030
$838$	0	0
$839$	−1.77288e13	−1.23524	−0.617618	−	0.786478i	$-0.711902\pi$
$839$	−1.77288e13	−1.23524	−0.617618	+	0.786478i	$0.711902\pi$
$840$	0	0
$841$	−1.24278e13	−0.856668
$842$	0	0
$843$	−3.12358e13	−2.13024
$844$	0	0
$845$	−7.74865e12	−0.522843
$846$	0	0
$847$	−8.49157e12	−0.566908
$848$	0	0
$849$	−5.07625e13	−3.35319
$850$	0	0
$851$	1.10227e13	0.720454
$852$	0	0
$853$	1.82317e13	1.17911	0.589557	−	0.807727i	$-0.299303\pi$
$853$	1.82317e13	1.17911	0.589557	+	0.807727i	$0.299303\pi$
$854$	0	0
$855$	3.57129e13	2.28548
$856$	0	0
$857$	−4.46174e12	−0.282547	−0.141273	−	0.989971i	$-0.545120\pi$
$857$	−4.46174e12	−0.282547	−0.141273	+	0.989971i	$0.545120\pi$
$858$	0	0
$859$	−2.49571e13	−1.56395	−0.781977	−	0.623307i	$-0.785789\pi$
$859$	−2.49571e13	−1.56395	−0.781977	+	0.623307i	$0.785789\pi$
$860$	0	0
$861$	2.31930e13	1.43828
$862$	0	0
$863$	4.92247e12	0.302089	0.151044	−	0.988527i	$-0.451736\pi$
$863$	4.92247e12	0.302089	0.151044	+	0.988527i	$0.451736\pi$
$864$	0	0
$865$	1.87743e13	1.14023
$866$	0	0
$867$	1.79119e12	0.107660
$868$	0	0
$869$	−1.15105e13	−0.684710
$870$	0	0
$871$	4.60619e12	0.271182
$872$	0	0
$873$	−2.04283e13	−1.19033
$874$	0	0
$875$	−1.39269e13	−0.803188
$876$	0	0
$877$	−1.68831e13	−0.963725	−0.481863	−	0.876247i	$-0.660040\pi$
$877$	−1.68831e13	−0.963725	−0.481863	+	0.876247i	$0.660040\pi$
$878$	0	0
$879$	−1.35886e13	−0.767760
$880$	0	0
$881$	−1.26880e13	−0.709583	−0.354792	−	0.934945i	$-0.615448\pi$
$881$	−1.26880e13	−0.709583	−0.354792	+	0.934945i	$0.615448\pi$
$882$	0	0
$883$	2.65648e13	1.47056	0.735281	−	0.677762i	$-0.237050\pi$
$883$	2.65648e13	1.47056	0.735281	+	0.677762i	$0.237050\pi$
$884$	0	0
$885$	2.15097e13	1.17866
$886$	0	0
$887$	5.71289e11	0.0309885	0.0154942	−	0.999880i	$-0.495068\pi$
$887$	5.71289e11	0.0309885	0.0154942	+	0.999880i	$0.495068\pi$
$888$	0	0
$889$	1.72643e13	0.927023
$890$	0	0
$891$	−2.23446e13	−1.18774
$892$	0	0
$893$	−1.73392e13	−0.912425
$894$	0	0
$895$	−2.28720e13	−1.19152
$896$	0	0
$897$	−2.12654e13	−1.09675
$898$	0	0
$899$	8.73529e12	0.446025
$900$	0	0
$901$	8.52007e12	0.430706
$902$	0	0
$903$	2.72474e13	1.36374
$904$	0	0
$905$	−5.67571e12	−0.281256
$906$	0	0
$907$	−3.29210e12	−0.161525	−0.0807625	−	0.996733i	$-0.525736\pi$
$907$	−3.29210e12	−0.161525	−0.0807625	+	0.996733i	$0.525736\pi$
$908$	0	0
$909$	−5.27411e13	−2.56219
$910$	0	0
$911$	1.27945e13	0.615445	0.307723	−	0.951476i	$-0.400433\pi$
$911$	1.27945e13	0.615445	0.307723	+	0.951476i	$0.400433\pi$
$912$	0	0
$913$	3.28559e12	0.156493
$914$	0	0
$915$	1.92978e13	0.910152
$916$	0	0
$917$	1.16894e12	0.0545922
$918$	0	0
$919$	3.16626e13	1.46429	0.732146	−	0.681148i	$-0.238519\pi$
$919$	3.16626e13	1.46429	0.732146	+	0.681148i	$0.238519\pi$
$920$	0	0
$921$	−1.71244e13	−0.784235
$922$	0	0
$923$	1.93594e13	0.877979
$924$	0	0
$925$	−2.58670e11	−0.0116174
$926$	0	0
$927$	−3.92730e13	−1.74677
$928$	0	0
$929$	−1.31826e13	−0.580673	−0.290336	−	0.956925i	$-0.593767\pi$
$929$	−1.31826e13	−0.580673	−0.290336	+	0.956925i	$0.593767\pi$
$930$	0	0
$931$	−7.65346e12	−0.333875
$932$	0	0
$933$	−5.56929e13	−2.40620
$934$	0	0
$935$	3.12177e12	0.133582
$936$	0	0
$937$	−2.06204e13	−0.873916	−0.436958	−	0.899482i	$-0.643944\pi$
$937$	−2.06204e13	−0.873916	−0.436958	+	0.899482i	$0.643944\pi$
$938$	0	0
$939$	5.22374e13	2.19273
$940$	0	0
$941$	2.43975e13	1.01436	0.507180	−	0.861840i	$-0.330688\pi$
$941$	2.43975e13	1.01436	0.507180	+	0.861840i	$0.330688\pi$
$942$	0	0
$943$	−2.03879e13	−0.839596
$944$	0	0
$945$	4.93293e13	2.01216
$946$	0	0
$947$	4.10278e13	1.65769	0.828845	−	0.559478i	$-0.188998\pi$
$947$	4.10278e13	1.65769	0.828845	+	0.559478i	$0.188998\pi$
$948$	0	0
$949$	1.96672e13	0.787128
$950$	0	0
$951$	5.84506e13	2.31727
$952$	0	0
$953$	4.09880e13	1.60968	0.804839	−	0.593494i	$-0.202252\pi$
$953$	4.09880e13	1.60968	0.804839	+	0.593494i	$0.202252\pi$
$954$	0	0
$955$	1.53410e13	0.596815
$956$	0	0
$957$	−9.83437e12	−0.379003
$958$	0	0
$959$	−1.91600e13	−0.731496
$960$	0	0
$961$	1.02574e13	0.387956
$962$	0	0
$963$	−1.03245e14	−3.86856
$964$	0	0
$965$	1.82400e13	0.677098
$966$	0	0
$967$	4.45707e12	0.163919	0.0819597	−	0.996636i	$-0.473882\pi$
$967$	4.45707e12	0.163919	0.0819597	+	0.996636i	$0.473882\pi$
$968$	0	0
$969$	−1.17677e13	−0.428781
$970$	0	0
$971$	−3.69975e13	−1.33563	−0.667815	−	0.744327i	$-0.732770\pi$
$971$	−3.69975e13	−1.33563	−0.667815	+	0.744327i	$0.732770\pi$
$972$	0	0
$973$	4.48374e12	0.160374
$974$	0	0
$975$	4.99034e11	0.0176852
$976$	0	0
$977$	4.72608e13	1.65949	0.829746	−	0.558141i	$-0.188485\pi$
$977$	4.72608e13	1.65949	0.829746	+	0.558141i	$0.188485\pi$
$978$	0	0
$979$	2.35635e13	0.819819
$980$	0	0
$981$	6.19811e13	2.13673
$982$	0	0
$983$	1.37629e13	0.470130	0.235065	−	0.971980i	$-0.424470\pi$
$983$	1.37629e13	0.470130	0.235065	+	0.971980i	$0.424470\pi$
$984$	0	0
$985$	1.00742e13	0.340996
$986$	0	0
$987$	−4.16947e13	−1.39847
$988$	0	0
$989$	−2.39519e13	−0.796080
$990$	0	0
$991$	3.34745e13	1.10251	0.551255	−	0.834337i	$-0.314149\pi$
$991$	3.34745e13	1.10251	0.551255	+	0.834337i	$0.314149\pi$
$992$	0	0
$993$	−5.07390e13	−1.65604
$994$	0	0
$995$	−4.27468e12	−0.138261
$996$	0	0
$997$	−4.78709e13	−1.53442	−0.767209	−	0.641398i	$-0.778355\pi$
$997$	−4.78709e13	−1.53442	−0.767209	+	0.641398i	$0.778355\pi$
$998$	0	0
$999$	−6.48280e13	−2.05929

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.f.1.5		5
4.3	odd	2		17.10.a.a.1.1	✓	5
12.11	even	2		153.10.a.c.1.5		5
68.67	odd	2		289.10.a.a.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.a.1.1	✓	5	4.3	odd	2
153.10.a.c.1.5		5	12.11	even	2
272.10.a.f.1.5		5	1.1	even	1	trivial
289.10.a.a.1.1		5	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)

\(f(q)\)	\(=\)	\(q+256.773 q^{3} +1407.25 q^{5} +5138.64 q^{7} +46249.6 q^{9} +O(q^{10})\)	\(q+256.773 q^{3} +1407.25 q^{5} +5138.64 q^{7} +46249.6 q^{9} -26560.4 q^{11} +71402.0 q^{13} +361344. q^{15} -83521.0 q^{17} +548715. q^{19} +1.31947e6 q^{21} -1.15988e6 q^{23} +27218.8 q^{25} +6.82160e6 q^{27} +1.44199e6 q^{29} +6.05781e6 q^{31} -6.82000e6 q^{33} +7.23134e6 q^{35} -9.50334e6 q^{37} +1.83341e7 q^{39} +1.75776e7 q^{41} +2.06503e7 q^{43} +6.50846e7 q^{45} -3.15997e7 q^{47} -1.39480e7 q^{49} -2.14460e7 q^{51} -1.02011e8 q^{53} -3.73770e7 q^{55} +1.40895e8 q^{57} +5.95270e7 q^{59} +5.34058e7 q^{61} +2.37660e8 q^{63} +1.00480e8 q^{65} +6.45107e7 q^{67} -2.97827e8 q^{69} +2.71132e8 q^{71} +2.75443e8 q^{73} +6.98908e6 q^{75} -1.36484e8 q^{77} +4.33372e8 q^{79} +8.41275e8 q^{81} -1.23703e8 q^{83} -1.17535e8 q^{85} +3.70264e8 q^{87} -8.87168e8 q^{89} +3.66909e8 q^{91} +1.55548e9 q^{93} +7.72177e8 q^{95} -4.41697e8 q^{97} -1.22841e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9}+O(q^{10}) \) 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9	\( 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9} + 68036 q^{11} - 158862 q^{13} + 687324 q^{15} - 417605 q^{17} + 370992 q^{19} + 1783880 q^{21} - 1645870 q^{23} + 3270239 q^{25} + 2998268 q^{27} + 3668616 q^{29} + 7262362 q^{31} - 11334900 q^{33} + 26503988 q^{35} - 31420708 q^{37} + 42449884 q^{39} - 7996938 q^{41} + 56908268 q^{43} + 12799536 q^{45} + 16903336 q^{47} - 11784059 q^{49} - 19710956 q^{51} - 83362982 q^{53} - 6363364 q^{55} + 136615904 q^{57} + 37946604 q^{59} - 77685452 q^{61} + 191945278 q^{63} - 40321288 q^{65} + 304503600 q^{67} - 333409272 q^{69} + 476602922 q^{71} - 289980486 q^{73} + 153685772 q^{75} - 143385648 q^{77} + 828240610 q^{79} + 891328609 q^{81} - 194681148 q^{83} - 123611080 q^{85} - 158149884 q^{87} + 376848106 q^{89} - 194543664 q^{91} + 3494835920 q^{93} - 1498679864 q^{95} + 692035246 q^{97} - 2027106408 q^{99}+O(q^{100}) \) 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9 + 68036 * q^11 - 158862 * q^13 + 687324 * q^15 - 417605 * q^17 + 370992 * q^19 + 1783880 * q^21 - 1645870 * q^23 + 3270239 * q^25 + 2998268 * q^27 + 3668616 * q^29 + 7262362 * q^31 - 11334900 * q^33 + 26503988 * q^35 - 31420708 * q^37 + 42449884 * q^39 - 7996938 * q^41 + 56908268 * q^43 + 12799536 * q^45 + 16903336 * q^47 - 11784059 * q^49 - 19710956 * q^51 - 83362982 * q^53 - 6363364 * q^55 + 136615904 * q^57 + 37946604 * q^59 - 77685452 * q^61 + 191945278 * q^63 - 40321288 * q^65 + 304503600 * q^67 - 333409272 * q^69 + 476602922 * q^71 - 289980486 * q^73 + 153685772 * q^75 - 143385648 * q^77 + 828240610 * q^79 + 891328609 * q^81 - 194681148 * q^83 - 123611080 * q^85 - 158149884 * q^87 + 376848106 * q^89 - 194543664 * q^91 + 3494835920 * q^93 - 1498679864 * q^95 + 692035246 * q^97 - 2027106408 * q^99

Introduction

L-functions

Modular forms

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Database

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