LMFDB - Embedded newform 272.10.a.f.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:	$0$
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			1.2
Root			$-21.1654$ 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)

$f(q)$	$=$	$q+3.02373 q^{3} +762.851 q^{5} -5573.11 q^{7} -19673.9 q^{9} +O(q^{10})$	$q+3.02373 q^{3} +762.851 q^{5} -5573.11 q^{7} -19673.9 q^{9} +47641.7 q^{11} -92260.9 q^{13} +2306.65 q^{15} -83521.0 q^{17} +8373.93 q^{19} -16851.6 q^{21} -364592. q^{23} -1.37118e6 q^{25} -119004. q^{27} -3.50595e6 q^{29} +5.20629e6 q^{31} +144055. q^{33} -4.25145e6 q^{35} -499530. q^{37} -278972. q^{39} -5.43648e6 q^{41} +3.54411e7 q^{43} -1.50082e7 q^{45} -1.21753e7 q^{47} -9.29403e6 q^{49} -252545. q^{51} +1.04471e8 q^{53} +3.63435e7 q^{55} +25320.5 q^{57} -4.16714e7 q^{59} +5.67537e7 q^{61} +1.09645e8 q^{63} -7.03813e7 q^{65} +1.74621e8 q^{67} -1.10243e6 q^{69} -3.46330e7 q^{71} -3.93220e8 q^{73} -4.14609e6 q^{75} -2.65512e8 q^{77} -1.85772e8 q^{79} +3.86881e8 q^{81} -3.62239e8 q^{83} -6.37141e7 q^{85} -1.06010e7 q^{87} -5.04798e7 q^{89} +5.14180e8 q^{91} +1.57424e7 q^{93} +6.38806e6 q^{95} -9.67620e8 q^{97} -9.37295e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.02373	0.0215525	0.0107762	−	0.999942i	$-0.496570\pi$
$3$	3.02373	0.0215525	0.0107762	+	0.999942i	$0.496570\pi$
$4$	0	0
$5$	762.851	0.545852	0.272926	−	0.962035i	$-0.412009\pi$
$5$	762.851	0.545852	0.272926	+	0.962035i	$0.412009\pi$
$6$	0	0
$7$	−5573.11	−0.877317	−0.438659	−	0.898654i	$-0.644546\pi$
$7$	−5573.11	−0.877317	−0.438659	+	0.898654i	$0.644546\pi$
$8$	0	0
$9$	−19673.9	−0.999535
$10$	0	0
$11$	47641.7	0.981115	0.490558	−	0.871409i	$-0.336793\pi$
$11$	47641.7	0.981115	0.490558	+	0.871409i	$0.336793\pi$
$12$	0	0
$13$	−92260.9	−0.895927	−0.447963	−	0.894052i	$-0.647851\pi$
$13$	−92260.9	−0.895927	−0.447963	+	0.894052i	$0.647851\pi$
$14$	0	0
$15$	2306.65	0.0117645
$16$	0	0
$17$	−83521.0	−0.242536
$17$	−83521.0	−0.242536
$18$	0	0
$19$	8373.93	0.0147414	0.00737069	−	0.999973i	$-0.497654\pi$
$19$	8373.93	0.0147414	0.00737069	+	0.999973i	$0.497654\pi$
$20$	0	0
$21$	−16851.6	−0.0189084
$22$	0	0
$23$	−364592.	−0.271664	−0.135832	−	0.990732i	$-0.543371\pi$
$23$	−364592.	−0.271664	−0.135832	+	0.990732i	$0.543371\pi$
$24$	0	0
$25$	−1.37118e6	−0.702046
$26$	0	0
$27$	−119004.	−0.0430949
$28$	0	0
$29$	−3.50595e6	−0.920482	−0.460241	−	0.887794i	$-0.652237\pi$
$29$	−3.50595e6	−0.920482	−0.460241	+	0.887794i	$0.652237\pi$
$30$	0	0
$31$	5.20629e6	1.01251	0.506257	−	0.862383i	$-0.331029\pi$
$31$	5.20629e6	1.01251	0.506257	+	0.862383i	$0.331029\pi$
$32$	0	0
$33$	144055.	0.0211455
$34$	0	0
$35$	−4.25145e6	−0.478885
$36$	0	0
$37$	−499530.	−0.0438182	−0.0219091	−	0.999760i	$-0.506974\pi$
$37$	−499530.	−0.0438182	−0.0219091	+	0.999760i	$0.506974\pi$
$38$	0	0
$39$	−278972.	−0.0193094
$40$	0	0
$41$	−5.43648e6	−0.300463	−0.150231	−	0.988651i	$-0.548002\pi$
$41$	−5.43648e6	−0.300463	−0.150231	+	0.988651i	$0.548002\pi$
$42$	0	0
$43$	3.54411e7	1.58088	0.790440	−	0.612539i	$-0.209852\pi$
$43$	3.54411e7	1.58088	0.790440	+	0.612539i	$0.209852\pi$
$44$	0	0
$45$	−1.50082e7	−0.545598
$46$	0	0
$47$	−1.21753e7	−0.363947	−0.181973	−	0.983303i	$-0.558248\pi$
$47$	−1.21753e7	−0.363947	−0.181973	+	0.983303i	$0.558248\pi$
$48$	0	0
$49$	−9.29403e6	−0.230315
$50$	0	0
$51$	−252545.	−0.00522724
$52$	0	0
$53$	1.04471e8	1.81867	0.909336	−	0.416063i	$-0.136590\pi$
$53$	1.04471e8	1.81867	0.909336	+	0.416063i	$0.136590\pi$
$54$	0	0
$55$	3.63435e7	0.535543
$56$	0	0
$57$	25320.5	0.000317713 0
$58$	0	0
$59$	−4.16714e7	−0.447718	−0.223859	−	0.974622i	$-0.571865\pi$
$59$	−4.16714e7	−0.447718	−0.223859	+	0.974622i	$0.571865\pi$
$60$	0	0
$61$	5.67537e7	0.524819	0.262410	−	0.964957i	$-0.415483\pi$
$61$	5.67537e7	0.524819	0.262410	+	0.964957i	$0.415483\pi$
$62$	0	0
$63$	1.09645e8	0.876910
$64$	0	0
$65$	−7.03813e7	−0.489043
$66$	0	0
$67$	1.74621e8	1.05867	0.529333	−	0.848414i	$-0.322442\pi$
$67$	1.74621e8	1.05867	0.529333	+	0.848414i	$0.322442\pi$
$68$	0	0
$69$	−1.10243e6	−0.00585503
$70$	0	0
$71$	−3.46330e7	−0.161744	−0.0808719	−	0.996725i	$-0.525770\pi$
$71$	−3.46330e7	−0.161744	−0.0808719	+	0.996725i	$0.525770\pi$
$72$	0	0
$73$	−3.93220e8	−1.62063	−0.810314	−	0.585996i	$-0.800703\pi$
$73$	−3.93220e8	−1.62063	−0.810314	+	0.585996i	$0.800703\pi$
$74$	0	0
$75$	−4.14609e6	−0.0151308
$76$	0	0
$77$	−2.65512e8	−0.860749
$78$	0	0
$79$	−1.85772e8	−0.536609	−0.268305	−	0.963334i	$-0.586463\pi$
$79$	−1.85772e8	−0.536609	−0.268305	+	0.963334i	$0.586463\pi$
$80$	0	0
$81$	3.86881e8	0.998607
$82$	0	0
$83$	−3.62239e8	−0.837807	−0.418903	−	0.908031i	$-0.637585\pi$
$83$	−3.62239e8	−0.837807	−0.418903	+	0.908031i	$0.637585\pi$
$84$	0	0
$85$	−6.37141e7	−0.132388
$86$	0	0
$87$	−1.06010e7	−0.0198387
$88$	0	0
$89$	−5.04798e7	−0.0852831	−0.0426416	−	0.999090i	$-0.513577\pi$
$89$	−5.04798e7	−0.0852831	−0.0426416	+	0.999090i	$0.513577\pi$
$90$	0	0
$91$	5.14180e8	0.786012
$92$	0	0
$93$	1.57424e7	0.0218222
$94$	0	0
$95$	6.38806e6	0.00804661
$96$	0	0
$97$	−9.67620e8	−1.10977	−0.554884	−	0.831928i	$-0.687237\pi$
$97$	−9.67620e8	−1.10977	−0.554884	+	0.831928i	$0.687237\pi$
$98$	0	0
$99$	−9.37295e8	−0.980659
$100$	0	0
$101$	1.60604e9	1.53572	0.767858	−	0.640621i	$-0.221323\pi$
$101$	1.60604e9	1.53572	0.767858	+	0.640621i	$0.221323\pi$
$102$	0	0
$103$	−1.76819e9	−1.54796	−0.773982	−	0.633208i	$-0.781738\pi$
$103$	−1.76819e9	−1.54796	−0.773982	+	0.633208i	$0.781738\pi$
$104$	0	0
$105$	−1.28552e7	−0.0103212
$106$	0	0
$107$	2.59627e9	1.91479	0.957397	−	0.288774i	$-0.0932477\pi$
$107$	2.59627e9	1.91479	0.957397	+	0.288774i	$0.0932477\pi$
$108$	0	0
$109$	2.64333e9	1.79363	0.896814	−	0.442408i	$-0.145876\pi$
$109$	2.64333e9	1.79363	0.896814	+	0.442408i	$0.145876\pi$
$110$	0	0
$111$	−1.51044e6	−0.000944390 0
$112$	0	0
$113$	9.53189e8	0.549954	0.274977	−	0.961451i	$-0.411330\pi$
$113$	9.53189e8	0.549954	0.274977	+	0.961451i	$0.411330\pi$
$114$	0	0
$115$	−2.78129e8	−0.148288
$116$	0	0
$117$	1.81513e9	0.895511
$118$	0	0
$119$	4.65472e8	0.212781
$120$	0	0
$121$	−8.82187e7	−0.0374133
$122$	0	0
$123$	−1.64384e7	−0.00647571
$124$	0	0
$125$	−2.53595e9	−0.929065
$126$	0	0
$127$	−9.64764e8	−0.329082	−0.164541	−	0.986370i	$-0.552614\pi$
$127$	−9.64764e8	−0.329082	−0.164541	+	0.986370i	$0.552614\pi$
$128$	0	0
$129$	1.07164e8	0.0340719
$130$	0	0
$131$	5.24089e9	1.55483	0.777417	−	0.628985i	$-0.216529\pi$
$131$	5.24089e9	1.55483	0.777417	+	0.628985i	$0.216529\pi$
$132$	0	0
$133$	−4.66689e7	−0.0129329
$134$	0	0
$135$	−9.07826e7	−0.0235234
$136$	0	0
$137$	3.09452e9	0.750500	0.375250	−	0.926924i	$-0.377557\pi$
$137$	3.09452e9	0.750500	0.375250	+	0.926924i	$0.377557\pi$
$138$	0	0
$139$	6.10981e8	0.138823	0.0694115	−	0.997588i	$-0.477888\pi$
$139$	6.10981e8	0.138823	0.0694115	+	0.997588i	$0.477888\pi$
$140$	0	0
$141$	−3.68147e7	−0.00784395
$142$	0	0
$143$	−4.39546e9	−0.879007
$144$	0	0
$145$	−2.67452e9	−0.502446
$146$	0	0
$147$	−2.81026e7	−0.00496385
$148$	0	0
$149$	2.02582e9	0.336716	0.168358	−	0.985726i	$-0.446154\pi$
$149$	2.02582e9	0.336716	0.168358	+	0.985726i	$0.446154\pi$
$150$	0	0
$151$	7.09269e9	1.11024	0.555118	−	0.831772i	$-0.312673\pi$
$151$	7.09269e9	1.11024	0.555118	+	0.831772i	$0.312673\pi$
$152$	0	0
$153$	1.64318e9	0.242423
$154$	0	0
$155$	3.97162e9	0.552682
$156$	0	0
$157$	−8.25715e9	−1.08463	−0.542315	−	0.840175i	$-0.682452\pi$
$157$	−8.25715e9	−1.08463	−0.542315	+	0.840175i	$0.682452\pi$
$158$	0	0
$159$	3.15892e8	0.0391969
$160$	0	0
$161$	2.03191e9	0.238335
$162$	0	0
$163$	6.13273e9	0.680471	0.340235	−	0.940340i	$-0.389493\pi$
$163$	6.13273e9	0.680471	0.340235	+	0.940340i	$0.389493\pi$
$164$	0	0
$165$	1.09893e8	0.0115423
$166$	0	0
$167$	1.53224e10	1.52442	0.762208	−	0.647332i	$-0.224115\pi$
$167$	1.53224e10	1.52442	0.762208	+	0.647332i	$0.224115\pi$
$168$	0	0
$169$	−2.09243e9	−0.197315
$170$	0	0
$171$	−1.64748e8	−0.0147345
$172$	0	0
$173$	1.45473e10	1.23474	0.617370	−	0.786673i	$-0.288198\pi$
$173$	1.45473e10	1.23474	0.617370	+	0.786673i	$0.288198\pi$
$174$	0	0
$175$	7.64176e9	0.615917
$176$	0	0
$177$	−1.26003e8	−0.00964942
$178$	0	0
$179$	−4.64898e9	−0.338469	−0.169235	−	0.985576i	$-0.554130\pi$
$179$	−4.64898e9	−0.338469	−0.169235	+	0.985576i	$0.554130\pi$
$180$	0	0
$181$	1.08585e9	0.0751997	0.0375998	−	0.999293i	$-0.488029\pi$
$181$	1.08585e9	0.0751997	0.0375998	+	0.999293i	$0.488029\pi$
$182$	0	0
$183$	1.71608e8	0.0113112
$184$	0	0
$185$	−3.81067e8	−0.0239182
$186$	0	0
$187$	−3.97908e9	−0.237955
$188$	0	0
$189$	6.63225e8	0.0378079
$190$	0	0
$191$	3.78328e9	0.205693	0.102846	−	0.994697i	$-0.467205\pi$
$191$	3.78328e9	0.205693	0.102846	+	0.994697i	$0.467205\pi$
$192$	0	0
$193$	3.16352e10	1.64121	0.820603	−	0.571499i	$-0.193638\pi$
$193$	3.16352e10	1.64121	0.820603	+	0.571499i	$0.193638\pi$
$194$	0	0
$195$	−2.12814e8	−0.0105401
$196$	0	0
$197$	1.24212e10	0.587576	0.293788	−	0.955871i	$-0.405084\pi$
$197$	1.24212e10	0.587576	0.293788	+	0.955871i	$0.405084\pi$
$198$	0	0
$199$	−4.01627e10	−1.81545	−0.907724	−	0.419568i	$-0.862182\pi$
$199$	−4.01627e10	−1.81545	−0.907724	+	0.419568i	$0.862182\pi$
$200$	0	0
$201$	5.28005e8	0.0228169
$202$	0	0
$203$	1.95391e10	0.807554
$204$	0	0
$205$	−4.14722e9	−0.164008
$206$	0	0
$207$	7.17293e9	0.271538
$208$	0	0
$209$	3.98948e8	0.0144630
$210$	0	0
$211$	−1.34292e10	−0.466422	−0.233211	−	0.972426i	$-0.574923\pi$
$211$	−1.34292e10	−0.466422	−0.233211	+	0.972426i	$0.574923\pi$
$212$	0	0
$213$	−1.04721e8	−0.00348598
$214$	0	0
$215$	2.70363e10	0.862926
$216$	0	0
$217$	−2.90153e10	−0.888296
$218$	0	0
$219$	−1.18899e9	−0.0349285
$220$	0	0
$221$	7.70572e9	0.217294
$222$	0	0
$223$	4.41977e10	1.19682	0.598409	−	0.801191i	$-0.295800\pi$
$223$	4.41977e10	1.19682	0.598409	+	0.801191i	$0.295800\pi$
$224$	0	0
$225$	2.69765e10	0.701720
$226$	0	0
$227$	3.42932e10	0.857218	0.428609	−	0.903490i	$-0.359004\pi$
$227$	3.42932e10	0.857218	0.428609	+	0.903490i	$0.359004\pi$
$228$	0	0
$229$	5.87161e10	1.41090	0.705452	−	0.708757i	$-0.250744\pi$
$229$	5.87161e10	1.41090	0.705452	+	0.708757i	$0.250744\pi$
$230$	0	0
$231$	−8.02837e8	−0.0185513
$232$	0	0
$233$	8.16442e10	1.81478	0.907390	−	0.420290i	$-0.138072\pi$
$233$	8.16442e10	1.81478	0.907390	+	0.420290i	$0.138072\pi$
$234$	0	0
$235$	−9.28790e9	−0.198661
$236$	0	0
$237$	−5.61724e8	−0.0115653
$238$	0	0
$239$	6.99278e10	1.38631	0.693153	−	0.720790i	$-0.256221\pi$
$239$	6.99278e10	1.38631	0.693153	+	0.720790i	$0.256221\pi$
$240$	0	0
$241$	2.20186e10	0.420448	0.210224	−	0.977653i	$-0.432581\pi$
$241$	2.20186e10	0.420448	0.210224	+	0.977653i	$0.432581\pi$
$242$	0	0
$243$	3.51219e9	0.0646174
$244$	0	0
$245$	−7.08996e9	−0.125718
$246$	0	0
$247$	−7.72586e8	−0.0132072
$248$	0	0
$249$	−1.09531e9	−0.0180568
$250$	0	0
$251$	7.51995e10	1.19587	0.597934	−	0.801545i	$-0.295988\pi$
$251$	7.51995e10	1.19587	0.597934	+	0.801545i	$0.295988\pi$
$252$	0	0
$253$	−1.73698e10	−0.266534
$254$	0	0
$255$	−1.92654e8	−0.00285330
$256$	0	0
$257$	−4.88179e10	−0.698039	−0.349020	−	0.937115i	$-0.613485\pi$
$257$	−4.88179e10	−0.698039	−0.349020	+	0.937115i	$0.613485\pi$
$258$	0	0
$259$	2.78394e9	0.0384424
$260$	0	0
$261$	6.89756e10	0.920054
$262$	0	0
$263$	−1.51056e10	−0.194687	−0.0973437	−	0.995251i	$-0.531035\pi$
$263$	−1.51056e10	−0.194687	−0.0973437	+	0.995251i	$0.531035\pi$
$264$	0	0
$265$	7.96958e10	0.992725
$266$	0	0
$267$	−1.52637e8	−0.00183806
$268$	0	0
$269$	7.34701e9	0.0855511	0.0427755	−	0.999085i	$-0.486380\pi$
$269$	7.34701e9	0.0855511	0.0427755	+	0.999085i	$0.486380\pi$
$270$	0	0
$271$	−9.26149e10	−1.04308	−0.521542	−	0.853226i	$-0.674643\pi$
$271$	−9.26149e10	−1.04308	−0.521542	+	0.853226i	$0.674643\pi$
$272$	0	0
$273$	1.55474e9	0.0169405
$274$	0	0
$275$	−6.53255e10	−0.688788
$276$	0	0
$277$	2.09446e10	0.213753	0.106877	−	0.994272i	$-0.465915\pi$
$277$	2.09446e10	0.213753	0.106877	+	0.994272i	$0.465915\pi$
$278$	0	0
$279$	−1.02428e11	−1.01204
$280$	0	0
$281$	−7.54010e10	−0.721438	−0.360719	−	0.932675i	$-0.617469\pi$
$281$	−7.54010e10	−0.721438	−0.360719	+	0.932675i	$0.617469\pi$
$282$	0	0
$283$	1.56859e11	1.45368	0.726842	−	0.686805i	$-0.240987\pi$
$283$	1.56859e11	1.45368	0.726842	+	0.686805i	$0.240987\pi$
$284$	0	0
$285$	1.93158e7	0.000173424 0
$286$	0	0
$287$	3.02981e10	0.263601
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	−2.92582e9	−0.0239182
$292$	0	0
$293$	−1.90997e11	−1.51399	−0.756994	−	0.653422i	$-0.773333\pi$
$293$	−1.90997e11	−1.51399	−0.756994	+	0.653422i	$0.773333\pi$
$294$	0	0
$295$	−3.17891e10	−0.244387
$296$	0	0
$297$	−5.66957e9	−0.0422811
$298$	0	0
$299$	3.36376e10	0.243391
$300$	0	0
$301$	−1.97517e11	−1.38693
$302$	0	0
$303$	4.85623e9	0.0330985
$304$	0	0
$305$	4.32946e10	0.286473
$306$	0	0
$307$	−7.83989e10	−0.503718	−0.251859	−	0.967764i	$-0.581042\pi$
$307$	−7.83989e10	−0.503718	−0.251859	+	0.967764i	$0.581042\pi$
$308$	0	0
$309$	−5.34652e9	−0.0333624
$310$	0	0
$311$	2.81000e11	1.70328	0.851639	−	0.524130i	$-0.175609\pi$
$311$	2.81000e11	1.70328	0.851639	+	0.524130i	$0.175609\pi$
$312$	0	0
$313$	−8.83831e10	−0.520499	−0.260249	−	0.965541i	$-0.583805\pi$
$313$	−8.83831e10	−0.520499	−0.260249	+	0.965541i	$0.583805\pi$
$314$	0	0
$315$	8.36425e10	0.478662
$316$	0	0
$317$	2.32664e11	1.29409	0.647043	−	0.762454i	$-0.276006\pi$
$317$	2.32664e11	1.29409	0.647043	+	0.762454i	$0.276006\pi$
$318$	0	0
$319$	−1.67029e11	−0.903098
$320$	0	0
$321$	7.85040e9	0.0412686
$322$	0	0
$323$	−6.99399e8	−0.00357531
$324$	0	0
$325$	1.26507e11	0.628982
$326$	0	0
$327$	7.99271e9	0.0386571
$328$	0	0
$329$	6.78540e10	0.319297
$330$	0	0
$331$	1.08788e11	0.498144	0.249072	−	0.968485i	$-0.419874\pi$
$331$	1.08788e11	0.498144	0.249072	+	0.968485i	$0.419874\pi$
$332$	0	0
$333$	9.82769e9	0.0437978
$334$	0	0
$335$	1.33209e11	0.577874
$336$	0	0
$337$	1.67127e11	0.705849	0.352925	−	0.935652i	$-0.385187\pi$
$337$	1.67127e11	0.705849	0.352925	+	0.935652i	$0.385187\pi$
$338$	0	0
$339$	2.88218e9	0.0118529
$340$	0	0
$341$	2.48037e11	0.993393
$342$	0	0
$343$	2.76692e11	1.07938
$344$	0	0
$345$	−8.40988e8	−0.00319598
$346$	0	0
$347$	−4.77989e11	−1.76985	−0.884923	−	0.465737i	$-0.845789\pi$
$347$	−4.77989e11	−1.76985	−0.884923	+	0.465737i	$0.845789\pi$
$348$	0	0
$349$	−4.80098e11	−1.73227	−0.866134	−	0.499811i	$-0.833403\pi$
$349$	−4.80098e11	−1.73227	−0.866134	+	0.499811i	$0.833403\pi$
$350$	0	0
$351$	1.09795e10	0.0386099
$352$	0	0
$353$	−2.98926e11	−1.02466	−0.512328	−	0.858790i	$-0.671217\pi$
$353$	−2.98926e11	−1.02466	−0.512328	+	0.858790i	$0.671217\pi$
$354$	0	0
$355$	−2.64198e10	−0.0882881
$356$	0	0
$357$	1.40746e9	0.00458595
$358$	0	0
$359$	−3.68878e11	−1.17208	−0.586041	−	0.810282i	$-0.699314\pi$
$359$	−3.68878e11	−1.17208	−0.586041	+	0.810282i	$0.699314\pi$
$360$	0	0
$361$	−3.22618e11	−0.999783
$362$	0	0
$363$	−2.66749e8	−0.000806350 0
$364$	0	0
$365$	−2.99969e11	−0.884622
$366$	0	0
$367$	6.01112e11	1.72965	0.864825	−	0.502074i	$-0.167429\pi$
$367$	6.01112e11	1.72965	0.864825	+	0.502074i	$0.167429\pi$
$368$	0	0
$369$	1.06957e11	0.300323
$370$	0	0
$371$	−5.82229e11	−1.59555
$372$	0	0
$373$	3.28495e11	0.878697	0.439349	−	0.898317i	$-0.355209\pi$
$373$	3.28495e11	0.878697	0.439349	+	0.898317i	$0.355209\pi$
$374$	0	0
$375$	−7.66803e9	−0.0200236
$376$	0	0
$377$	3.23462e11	0.824684
$378$	0	0
$379$	−1.10170e11	−0.274276	−0.137138	−	0.990552i	$-0.543790\pi$
$379$	−1.10170e11	−0.274276	−0.137138	+	0.990552i	$0.543790\pi$
$380$	0	0
$381$	−2.91718e9	−0.00709253
$382$	0	0
$383$	−4.46966e11	−1.06140	−0.530702	−	0.847559i	$-0.678072\pi$
$383$	−4.46966e11	−1.06140	−0.530702	+	0.847559i	$0.678072\pi$
$384$	0	0
$385$	−2.02546e11	−0.469841
$386$	0	0
$387$	−6.97263e11	−1.58015
$388$	0	0
$389$	5.35885e11	1.18658	0.593292	−	0.804987i	$-0.297828\pi$
$389$	5.35885e11	1.18658	0.593292	+	0.804987i	$0.297828\pi$
$390$	0	0
$391$	3.04511e10	0.0658882
$392$	0	0
$393$	1.58470e10	0.0335105
$394$	0	0
$395$	−1.41716e11	−0.292909
$396$	0	0
$397$	−5.15293e11	−1.04111	−0.520555	−	0.853828i	$-0.674275\pi$
$397$	−5.15293e11	−1.04111	−0.520555	+	0.853828i	$0.674275\pi$
$398$	0	0
$399$	−1.41114e8	−0.000278735 0
$400$	0	0
$401$	−4.93193e11	−0.952504	−0.476252	−	0.879309i	$-0.658005\pi$
$401$	−4.93193e11	−0.952504	−0.476252	+	0.879309i	$0.658005\pi$
$402$	0	0
$403$	−4.80337e11	−0.907138
$404$	0	0
$405$	2.95132e11	0.545091
$406$	0	0
$407$	−2.37985e10	−0.0429907
$408$	0	0
$409$	−2.86274e11	−0.505857	−0.252928	−	0.967485i	$-0.581394\pi$
$409$	−2.86274e11	−0.505857	−0.252928	+	0.967485i	$0.581394\pi$
$410$	0	0
$411$	9.35699e9	0.0161751
$412$	0	0
$413$	2.32239e11	0.392790
$414$	0	0
$415$	−2.76334e11	−0.457318
$416$	0	0
$417$	1.84744e9	0.00299198
$418$	0	0
$419$	1.66697e11	0.264219	0.132110	−	0.991235i	$-0.457825\pi$
$419$	1.66697e11	0.264219	0.132110	+	0.991235i	$0.457825\pi$
$420$	0	0
$421$	−5.15187e11	−0.799274	−0.399637	−	0.916674i	$-0.630864\pi$
$421$	−5.15187e11	−0.799274	−0.399637	+	0.916674i	$0.630864\pi$
$422$	0	0
$423$	2.39534e11	0.363777
$424$	0	0
$425$	1.14523e11	0.170271
$426$	0	0
$427$	−3.16295e11	−0.460433
$428$	0	0
$429$	−1.32907e10	−0.0189448
$430$	0	0
$431$	−5.42449e10	−0.0757202	−0.0378601	−	0.999283i	$-0.512054\pi$
$431$	−5.42449e10	−0.0757202	−0.0378601	+	0.999283i	$0.512054\pi$
$432$	0	0
$433$	−2.49025e11	−0.340445	−0.170222	−	0.985406i	$-0.554449\pi$
$433$	−2.49025e11	−0.340445	−0.170222	+	0.985406i	$0.554449\pi$
$434$	0	0
$435$	−8.08702e9	−0.0108290
$436$	0	0
$437$	−3.05307e9	−0.00400470
$438$	0	0
$439$	−2.46288e11	−0.316485	−0.158242	−	0.987400i	$-0.550583\pi$
$439$	−2.46288e11	−0.316485	−0.158242	+	0.987400i	$0.550583\pi$
$440$	0	0
$441$	1.82849e11	0.230208
$442$	0	0
$443$	1.20603e12	1.48779	0.743895	−	0.668297i	$-0.232976\pi$
$443$	1.20603e12	1.48779	0.743895	+	0.668297i	$0.232976\pi$
$444$	0	0
$445$	−3.85086e10	−0.0465519
$446$	0	0
$447$	6.12554e9	0.00725706
$448$	0	0
$449$	−1.43303e12	−1.66398	−0.831988	−	0.554794i	$-0.812797\pi$
$449$	−1.43303e12	−1.66398	−0.831988	+	0.554794i	$0.812797\pi$
$450$	0	0
$451$	−2.59003e11	−0.294788
$452$	0	0
$453$	2.14464e10	0.0239283
$454$	0	0
$455$	3.92243e11	0.429046
$456$	0	0
$457$	−1.45072e12	−1.55582	−0.777911	−	0.628375i	$-0.783721\pi$
$457$	−1.45072e12	−1.55582	−0.777911	+	0.628375i	$0.783721\pi$
$458$	0	0
$459$	9.93937e9	0.0104521
$460$	0	0
$461$	4.79650e11	0.494619	0.247309	−	0.968937i	$-0.420454\pi$
$461$	4.79650e11	0.494619	0.247309	+	0.968937i	$0.420454\pi$
$462$	0	0
$463$	1.02183e11	0.103339	0.0516697	−	0.998664i	$-0.483546\pi$
$463$	1.02183e11	0.103339	0.0516697	+	0.998664i	$0.483546\pi$
$464$	0	0
$465$	1.20091e10	0.0119117
$466$	0	0
$467$	9.48806e11	0.923106	0.461553	−	0.887113i	$-0.347292\pi$
$467$	9.48806e11	0.923106	0.461553	+	0.887113i	$0.347292\pi$
$468$	0	0
$469$	−9.73180e11	−0.928785
$470$	0	0
$471$	−2.49674e10	−0.0233765
$472$	0	0
$473$	1.68847e12	1.55103
$474$	0	0
$475$	−1.14822e10	−0.0103491
$476$	0	0
$477$	−2.05535e12	−1.81783
$478$	0	0
$479$	−1.54645e12	−1.34223	−0.671114	−	0.741354i	$-0.734184\pi$
$479$	−1.54645e12	−1.34223	−0.671114	+	0.741354i	$0.734184\pi$
$480$	0	0
$481$	4.60871e10	0.0392579
$482$	0	0
$483$	6.14395e9	0.00513672
$484$	0	0
$485$	−7.38150e11	−0.605768
$486$	0	0
$487$	8.16740e11	0.657966	0.328983	−	0.944336i	$-0.393294\pi$
$487$	8.16740e11	0.657966	0.328983	+	0.944336i	$0.393294\pi$
$488$	0	0
$489$	1.85437e10	0.0146658
$490$	0	0
$491$	−2.36995e11	−0.184023	−0.0920116	−	0.995758i	$-0.529330\pi$
$491$	−2.36995e11	−0.184023	−0.0920116	+	0.995758i	$0.529330\pi$
$492$	0	0
$493$	2.92821e11	0.223250
$494$	0	0
$495$	−7.15017e11	−0.535294
$496$	0	0
$497$	1.93014e11	0.141901
$498$	0	0
$499$	−1.95695e12	−1.41295	−0.706477	−	0.707736i	$-0.749717\pi$
$499$	−1.95695e12	−1.41295	−0.706477	+	0.707736i	$0.749717\pi$
$500$	0	0
$501$	4.63309e10	0.0328549
$502$	0	0
$503$	−1.25952e12	−0.877302	−0.438651	−	0.898658i	$-0.644544\pi$
$503$	−1.25952e12	−0.877302	−0.438651	+	0.898658i	$0.644544\pi$
$504$	0	0
$505$	1.22517e12	0.838273
$506$	0	0
$507$	−6.32693e9	−0.00425263
$508$	0	0
$509$	8.63543e11	0.570235	0.285118	−	0.958493i	$-0.407967\pi$
$509$	8.63543e11	0.570235	0.285118	+	0.958493i	$0.407967\pi$
$510$	0	0
$511$	2.19146e12	1.42180
$512$	0	0
$513$	−9.96535e8	−0.000635279 0
$514$	0	0
$515$	−1.34886e12	−0.844958
$516$	0	0
$517$	−5.80049e11	−0.357073
$518$	0	0
$519$	4.39872e10	0.0266117
$520$	0	0
$521$	8.44577e11	0.502192	0.251096	−	0.967962i	$-0.419209\pi$
$521$	8.44577e11	0.502192	0.251096	+	0.967962i	$0.419209\pi$
$522$	0	0
$523$	−9.06283e11	−0.529671	−0.264835	−	0.964294i	$-0.585318\pi$
$523$	−9.06283e11	−0.529671	−0.264835	+	0.964294i	$0.585318\pi$
$524$	0	0
$525$	2.31066e10	0.0132745
$526$	0	0
$527$	−4.34835e11	−0.245571
$528$	0	0
$529$	−1.66823e12	−0.926199
$530$	0	0
$531$	8.19837e11	0.447510
$532$	0	0
$533$	5.01575e11	0.269193
$534$	0	0
$535$	1.98056e12	1.04519
$536$	0	0
$537$	−1.40573e10	−0.00729485
$538$	0	0
$539$	−4.42783e11	−0.225965
$540$	0	0
$541$	1.55099e12	0.778435	0.389217	−	0.921146i	$-0.372745\pi$
$541$	1.55099e12	0.778435	0.389217	+	0.921146i	$0.372745\pi$
$542$	0	0
$543$	3.28331e9	0.00162074
$544$	0	0
$545$	2.01647e12	0.979054
$546$	0	0
$547$	2.93476e12	1.40162	0.700808	−	0.713350i	$-0.252823\pi$
$547$	2.93476e12	1.40162	0.700808	+	0.713350i	$0.252823\pi$
$548$	0	0
$549$	−1.11656e12	−0.524576
$550$	0	0
$551$	−2.93586e10	−0.0135692
$552$	0	0
$553$	1.03533e12	0.470776
$554$	0	0
$555$	−1.15224e9	−0.000515497 0
$556$	0	0
$557$	−2.35783e12	−1.03792	−0.518961	−	0.854798i	$-0.673681\pi$
$557$	−2.35783e12	−1.03792	−0.518961	+	0.854798i	$0.673681\pi$
$558$	0	0
$559$	−3.26983e12	−1.41635
$560$	0	0
$561$	−1.20317e10	−0.00512853
$562$	0	0
$563$	−2.31243e12	−0.970020	−0.485010	−	0.874509i	$-0.661184\pi$
$563$	−2.31243e12	−0.970020	−0.485010	+	0.874509i	$0.661184\pi$
$564$	0	0
$565$	7.27141e11	0.300193
$566$	0	0
$567$	−2.15613e12	−0.876095
$568$	0	0
$569$	−8.35674e11	−0.334220	−0.167110	−	0.985938i	$-0.553443\pi$
$569$	−8.35674e11	−0.334220	−0.167110	+	0.985938i	$0.553443\pi$
$570$	0	0
$571$	2.79849e12	1.10169	0.550846	−	0.834607i	$-0.314305\pi$
$571$	2.79849e12	1.10169	0.550846	+	0.834607i	$0.314305\pi$
$572$	0	0
$573$	1.14396e10	0.00443318
$574$	0	0
$575$	4.99923e11	0.190721
$576$	0	0
$577$	4.23010e12	1.58876	0.794382	−	0.607419i	$-0.207795\pi$
$577$	4.23010e12	1.58876	0.794382	+	0.607419i	$0.207795\pi$
$578$	0	0
$579$	9.56563e10	0.0353720
$580$	0	0
$581$	2.01880e12	0.735022
$582$	0	0
$583$	4.97717e12	1.78433
$584$	0	0
$585$	1.38467e12	0.488816
$586$	0	0
$587$	9.29095e11	0.322990	0.161495	−	0.986874i	$-0.448368\pi$
$587$	9.29095e11	0.322990	0.161495	+	0.986874i	$0.448368\pi$
$588$	0	0
$589$	4.35971e10	0.0149259
$590$	0	0
$591$	3.75582e10	0.0126637
$592$	0	0
$593$	3.56436e12	1.18368	0.591841	−	0.806055i	$-0.298401\pi$
$593$	3.56436e12	1.18368	0.591841	+	0.806055i	$0.298401\pi$
$594$	0	0
$595$	3.55086e11	0.116147
$596$	0	0
$597$	−1.21441e11	−0.0391274
$598$	0	0
$599$	5.69493e12	1.80746	0.903728	−	0.428106i	$-0.140819\pi$
$599$	5.69493e12	1.80746	0.903728	+	0.428106i	$0.140819\pi$
$600$	0	0
$601$	−3.23003e12	−1.00989	−0.504943	−	0.863153i	$-0.668486\pi$
$601$	−3.23003e12	−1.00989	−0.504943	+	0.863153i	$0.668486\pi$
$602$	0	0
$603$	−3.43546e12	−1.05817
$604$	0	0
$605$	−6.72977e10	−0.0204221
$606$	0	0
$607$	1.62163e12	0.484846	0.242423	−	0.970171i	$-0.422058\pi$
$607$	1.62163e12	0.484846	0.242423	+	0.970171i	$0.422058\pi$
$608$	0	0
$609$	5.90808e10	0.0174048
$610$	0	0
$611$	1.12330e12	0.326070
$612$	0	0
$613$	3.22227e12	0.921700	0.460850	−	0.887478i	$-0.347545\pi$
$613$	3.22227e12	0.921700	0.460850	+	0.887478i	$0.347545\pi$
$614$	0	0
$615$	−1.25401e10	−0.00353478
$616$	0	0
$617$	−3.76562e12	−1.04605	−0.523026	−	0.852317i	$-0.675197\pi$
$617$	−3.76562e12	−1.04605	−0.523026	+	0.852317i	$0.675197\pi$
$618$	0	0
$619$	−3.75883e12	−1.02907	−0.514535	−	0.857470i	$-0.672035\pi$
$619$	−3.75883e12	−1.02907	−0.514535	+	0.857470i	$0.672035\pi$
$620$	0	0
$621$	4.33881e10	0.0117073
$622$	0	0
$623$	2.81330e11	0.0748203
$624$	0	0
$625$	7.43541e11	0.194915
$626$	0	0
$627$	1.20631e9	0.000311713 0
$628$	0	0
$629$	4.17213e10	0.0106275
$630$	0	0
$631$	5.72141e12	1.43672	0.718358	−	0.695673i	$-0.244894\pi$
$631$	5.72141e12	1.43672	0.718358	+	0.695673i	$0.244894\pi$
$632$	0	0
$633$	−4.06062e10	−0.0100525
$634$	0	0
$635$	−7.35971e11	−0.179630
$636$	0	0
$637$	8.57475e11	0.206345
$638$	0	0
$639$	6.81365e11	0.161669
$640$	0	0
$641$	−2.45043e12	−0.573299	−0.286650	−	0.958035i	$-0.592542\pi$
$641$	−2.45043e12	−0.573299	−0.286650	+	0.958035i	$0.592542\pi$
$642$	0	0
$643$	−5.20042e12	−1.19975	−0.599873	−	0.800095i	$-0.704782\pi$
$643$	−5.20042e12	−1.19975	−0.599873	+	0.800095i	$0.704782\pi$
$644$	0	0
$645$	8.17503e10	0.0185982
$646$	0	0
$647$	−9.06811e11	−0.203445	−0.101723	−	0.994813i	$-0.532435\pi$
$647$	−9.06811e11	−0.203445	−0.101723	+	0.994813i	$0.532435\pi$
$648$	0	0
$649$	−1.98530e12	−0.439262
$650$	0	0
$651$	−8.77342e10	−0.0191450
$652$	0	0
$653$	5.74944e12	1.23742	0.618708	−	0.785621i	$-0.287656\pi$
$653$	5.74944e12	1.23742	0.618708	+	0.785621i	$0.287656\pi$
$654$	0	0
$655$	3.99802e12	0.848709
$656$	0	0
$657$	7.73616e12	1.61987
$658$	0	0
$659$	1.14504e12	0.236503	0.118252	−	0.992984i	$-0.462271\pi$
$659$	1.14504e12	0.236503	0.118252	+	0.992984i	$0.462271\pi$
$660$	0	0
$661$	−2.18103e12	−0.444380	−0.222190	−	0.975003i	$-0.571321\pi$
$661$	−2.18103e12	−0.444380	−0.222190	+	0.975003i	$0.571321\pi$
$662$	0	0
$663$	2.33000e10	0.00468323
$664$	0	0
$665$	−3.56014e10	−0.00705943
$666$	0	0
$667$	1.27824e12	0.250062
$668$	0	0
$669$	1.33642e11	0.0257944
$670$	0	0
$671$	2.70384e12	0.514908
$672$	0	0
$673$	−5.45288e12	−1.02461	−0.512304	−	0.858804i	$-0.671208\pi$
$673$	−5.45288e12	−1.02461	−0.512304	+	0.858804i	$0.671208\pi$
$674$	0	0
$675$	1.63177e11	0.0302546
$676$	0	0
$677$	−4.84898e12	−0.887159	−0.443579	−	0.896235i	$-0.646292\pi$
$677$	−4.84898e12	−0.887159	−0.443579	+	0.896235i	$0.646292\pi$
$678$	0	0
$679$	5.39266e12	0.973618
$680$	0	0
$681$	1.03693e11	0.0184752
$682$	0	0
$683$	1.30756e11	0.0229916	0.0114958	−	0.999934i	$-0.496341\pi$
$683$	1.30756e11	0.0229916	0.0114958	+	0.999934i	$0.496341\pi$
$684$	0	0
$685$	2.36066e12	0.409662
$686$	0	0
$687$	1.77542e11	0.0304085
$688$	0	0
$689$	−9.63859e12	−1.62940
$690$	0	0
$691$	−3.22565e12	−0.538227	−0.269113	−	0.963108i	$-0.586731\pi$
$691$	−3.22565e12	−0.538227	−0.269113	+	0.963108i	$0.586731\pi$
$692$	0	0
$693$	5.22365e12	0.860349
$694$	0	0
$695$	4.66088e11	0.0757767
$696$	0	0
$697$	4.54060e11	0.0728729
$698$	0	0
$699$	2.46870e11	0.0391130
$700$	0	0
$701$	8.43820e12	1.31983	0.659916	−	0.751339i	$-0.270592\pi$
$701$	8.43820e12	1.31983	0.659916	+	0.751339i	$0.270592\pi$
$702$	0	0
$703$	−4.18303e9	−0.000645940 0
$704$	0	0
$705$	−2.80841e10	−0.00428163
$706$	0	0
$707$	−8.95065e12	−1.34731
$708$	0	0
$709$	5.72297e12	0.850576	0.425288	−	0.905058i	$-0.360173\pi$
$709$	5.72297e12	0.850576	0.425288	+	0.905058i	$0.360173\pi$
$710$	0	0
$711$	3.65485e12	0.536360
$712$	0	0
$713$	−1.89817e12	−0.275064
$714$	0	0
$715$	−3.35308e12	−0.479808
$716$	0	0
$717$	2.11443e11	0.0298783
$718$	0	0
$719$	6.14814e12	0.857954	0.428977	−	0.903316i	$-0.358874\pi$
$719$	6.14814e12	0.857954	0.428977	+	0.903316i	$0.358874\pi$
$720$	0	0
$721$	9.85431e12	1.35805
$722$	0	0
$723$	6.65782e10	0.00906170
$724$	0	0
$725$	4.80731e12	0.646220
$726$	0	0
$727$	9.21269e12	1.22316	0.611578	−	0.791184i	$-0.290535\pi$
$727$	9.21269e12	1.22316	0.611578	+	0.791184i	$0.290535\pi$
$728$	0	0
$729$	−7.60435e12	−0.997214
$730$	0	0
$731$	−2.96008e12	−0.383420
$732$	0	0
$733$	−8.69962e12	−1.11310	−0.556548	−	0.830816i	$-0.687874\pi$
$733$	−8.69962e12	−1.11310	−0.556548	+	0.830816i	$0.687874\pi$
$734$	0	0
$735$	−2.14381e10	−0.00270953
$736$	0	0
$737$	8.31921e12	1.03867
$738$	0	0
$739$	8.74888e12	1.07908	0.539539	−	0.841961i	$-0.318599\pi$
$739$	8.74888e12	1.07908	0.539539	+	0.841961i	$0.318599\pi$
$740$	0	0
$741$	−2.33609e9	−0.000284648 0
$742$	0	0
$743$	4.65215e11	0.0560021	0.0280010	−	0.999608i	$-0.491086\pi$
$743$	4.65215e11	0.0560021	0.0280010	+	0.999608i	$0.491086\pi$
$744$	0	0
$745$	1.54540e12	0.183797
$746$	0	0
$747$	7.12664e12	0.837418
$748$	0	0
$749$	−1.44693e13	−1.67988
$750$	0	0
$751$	8.93867e12	1.02540	0.512700	−	0.858568i	$-0.328645\pi$
$751$	8.93867e12	1.02540	0.512700	+	0.858568i	$0.328645\pi$
$752$	0	0
$753$	2.27383e11	0.0257739
$754$	0	0
$755$	5.41067e12	0.606024
$756$	0	0
$757$	6.47580e12	0.716741	0.358371	−	0.933579i	$-0.383332\pi$
$757$	6.47580e12	0.716741	0.358371	+	0.933579i	$0.383332\pi$
$758$	0	0
$759$	−5.25215e10	−0.00574446
$760$	0	0
$761$	1.50384e12	0.162544	0.0812720	−	0.996692i	$-0.474102\pi$
$761$	1.50384e12	0.162544	0.0812720	+	0.996692i	$0.474102\pi$
$762$	0	0
$763$	−1.47316e13	−1.57358
$764$	0	0
$765$	1.25350e12	0.132327
$766$	0	0
$767$	3.84464e12	0.401122
$768$	0	0
$769$	1.49083e13	1.53731	0.768653	−	0.639666i	$-0.220927\pi$
$769$	1.49083e13	1.53731	0.768653	+	0.639666i	$0.220927\pi$
$770$	0	0
$771$	−1.47612e11	−0.0150445
$772$	0	0
$773$	−2.45775e11	−0.0247588	−0.0123794	−	0.999923i	$-0.503941\pi$
$773$	−2.45775e11	−0.0247588	−0.0123794	+	0.999923i	$0.503941\pi$
$774$	0	0
$775$	−7.13878e12	−0.710831
$776$	0	0
$777$	8.41787e9	0.000828529 0
$778$	0	0
$779$	−4.55247e10	−0.00442924
$780$	0	0
$781$	−1.64997e12	−0.158689
$782$	0	0
$783$	4.17224e11	0.0396681
$784$	0	0
$785$	−6.29898e12	−0.592047
$786$	0	0
$787$	−8.04995e12	−0.748009	−0.374004	−	0.927427i	$-0.622016\pi$
$787$	−8.04995e12	−0.748009	−0.374004	+	0.927427i	$0.622016\pi$
$788$	0	0
$789$	−4.56753e10	−0.00419600
$790$	0	0
$791$	−5.31223e12	−0.482484
$792$	0	0
$793$	−5.23615e12	−0.470200
$794$	0	0
$795$	2.40978e11	0.0213957
$796$	0	0
$797$	2.27278e12	0.199524	0.0997618	−	0.995011i	$-0.468192\pi$
$797$	2.27278e12	0.199524	0.0997618	+	0.995011i	$0.468192\pi$
$798$	0	0
$799$	1.01689e12	0.0882700
$800$	0	0
$801$	9.93133e11	0.0852435
$802$	0	0
$803$	−1.87337e13	−1.59002
$804$	0	0
$805$	1.55005e12	0.130096
$806$	0	0
$807$	2.22154e10	0.00184384
$808$	0	0
$809$	2.47519e12	0.203161	0.101581	−	0.994827i	$-0.467610\pi$
$809$	2.47519e12	0.203161	0.101581	+	0.994827i	$0.467610\pi$
$810$	0	0
$811$	−2.45293e13	−1.99110	−0.995548	−	0.0942611i	$-0.969951\pi$
$811$	−2.45293e13	−1.99110	−0.995548	+	0.0942611i	$0.969951\pi$
$812$	0	0
$813$	−2.80042e11	−0.0224810
$814$	0	0
$815$	4.67836e12	0.371436
$816$	0	0
$817$	2.96781e11	0.0233044
$818$	0	0
$819$	−1.01159e13	−0.785647
$820$	0	0
$821$	−8.21852e12	−0.631319	−0.315660	−	0.948872i	$-0.602226\pi$
$821$	−8.21852e12	−0.631319	−0.315660	+	0.948872i	$0.602226\pi$
$822$	0	0
$823$	−1.55798e12	−0.118376	−0.0591878	−	0.998247i	$-0.518851\pi$
$823$	−1.55798e12	−0.118376	−0.0591878	+	0.998247i	$0.518851\pi$
$824$	0	0
$825$	−1.97527e11	−0.0148451
$826$	0	0
$827$	−1.18962e13	−0.884367	−0.442184	−	0.896924i	$-0.645796\pi$
$827$	−1.18962e13	−0.884367	−0.442184	+	0.896924i	$0.645796\pi$
$828$	0	0
$829$	−8.80333e12	−0.647368	−0.323684	−	0.946165i	$-0.604922\pi$
$829$	−8.80333e12	−0.647368	−0.323684	+	0.946165i	$0.604922\pi$
$830$	0	0
$831$	6.33308e10	0.00460691
$832$	0	0
$833$	7.76247e11	0.0558595
$834$	0	0
$835$	1.16887e13	0.832105
$836$	0	0
$837$	−6.19572e11	−0.0436342
$838$	0	0
$839$	6.42832e12	0.447887	0.223944	−	0.974602i	$-0.428107\pi$
$839$	6.42832e12	0.447887	0.223944	+	0.974602i	$0.428107\pi$
$840$	0	0
$841$	−2.21544e12	−0.152714
$842$	0	0
$843$	−2.27992e11	−0.0155488
$844$	0	0
$845$	−1.59621e12	−0.107705
$846$	0	0
$847$	4.91653e11	0.0328234
$848$	0	0
$849$	4.74298e11	0.0313305
$850$	0	0
$851$	1.82125e11	0.0119038
$852$	0	0
$853$	−1.34906e13	−0.872492	−0.436246	−	0.899827i	$-0.643692\pi$
$853$	−1.34906e13	−0.872492	−0.436246	+	0.899827i	$0.643692\pi$
$854$	0	0
$855$	−1.25678e11	−0.00804287
$856$	0	0
$857$	−2.46136e13	−1.55870	−0.779348	−	0.626592i	$-0.784449\pi$
$857$	−2.46136e13	−1.55870	−0.779348	+	0.626592i	$0.784449\pi$
$858$	0	0
$859$	−2.45934e13	−1.54117	−0.770583	−	0.637340i	$-0.780035\pi$
$859$	−2.45934e13	−1.54117	−0.770583	+	0.637340i	$0.780035\pi$
$860$	0	0
$861$	9.16133e10	0.00568125
$862$	0	0
$863$	−1.57068e12	−0.0963918	−0.0481959	−	0.998838i	$-0.515347\pi$
$863$	−1.57068e12	−0.0963918	−0.0481959	+	0.998838i	$0.515347\pi$
$864$	0	0
$865$	1.10974e13	0.673985
$866$	0	0
$867$	2.10928e10	0.00126779
$868$	0	0
$869$	−8.85048e12	−0.526475
$870$	0	0
$871$	−1.61106e13	−0.948487
$872$	0	0
$873$	1.90368e13	1.10925
$874$	0	0
$875$	1.41331e13	0.815084
$876$	0	0
$877$	−2.26638e12	−0.129370	−0.0646852	−	0.997906i	$-0.520604\pi$
$877$	−2.26638e12	−0.129370	−0.0646852	+	0.997906i	$0.520604\pi$
$878$	0	0
$879$	−5.77523e11	−0.0326302
$880$	0	0
$881$	−2.20342e13	−1.23227	−0.616134	−	0.787641i	$-0.711302\pi$
$881$	−2.20342e13	−1.23227	−0.616134	+	0.787641i	$0.711302\pi$
$882$	0	0
$883$	−7.71424e12	−0.427041	−0.213521	−	0.976939i	$-0.568493\pi$
$883$	−7.71424e12	−0.427041	−0.213521	+	0.976939i	$0.568493\pi$
$884$	0	0
$885$	−9.61215e10	−0.00526715
$886$	0	0
$887$	3.10912e13	1.68648	0.843239	−	0.537539i	$-0.180646\pi$
$887$	3.10912e13	1.68648	0.843239	+	0.537539i	$0.180646\pi$
$888$	0	0
$889$	5.37674e12	0.288709
$890$	0	0
$891$	1.84316e13	0.979748
$892$	0	0
$893$	−1.01955e11	−0.00536508
$894$	0	0
$895$	−3.54648e12	−0.184754
$896$	0	0
$897$	1.01711e11	0.00524568
$898$	0	0
$899$	−1.82530e13	−0.932000
$900$	0	0
$901$	−8.72552e12	−0.441093
$902$	0	0
$903$	−5.97238e11	−0.0298918
$904$	0	0
$905$	8.28341e11	0.0410479
$906$	0	0
$907$	9.61906e12	0.471954	0.235977	−	0.971759i	$-0.424171\pi$
$907$	9.61906e12	0.471954	0.235977	+	0.971759i	$0.424171\pi$
$908$	0	0
$909$	−3.15970e13	−1.53500
$910$	0	0
$911$	1.52076e13	0.731525	0.365762	−	0.930708i	$-0.380808\pi$
$911$	1.52076e13	0.731525	0.365762	+	0.930708i	$0.380808\pi$
$912$	0	0
$913$	−1.72577e13	−0.821985
$914$	0	0
$915$	1.30911e11	0.00617421
$916$	0	0
$917$	−2.92081e13	−1.36408
$918$	0	0
$919$	−1.96701e13	−0.909675	−0.454838	−	0.890574i	$-0.650303\pi$
$919$	−1.96701e13	−0.909675	−0.454838	+	0.890574i	$0.650303\pi$
$920$	0	0
$921$	−2.37057e11	−0.0108564
$922$	0	0
$923$	3.19527e12	0.144911
$924$	0	0
$925$	6.84948e11	0.0307624
$926$	0	0
$927$	3.47871e13	1.54724
$928$	0	0
$929$	3.18903e13	1.40471	0.702357	−	0.711825i	$-0.252131\pi$
$929$	3.18903e13	1.40471	0.702357	+	0.711825i	$0.252131\pi$
$930$	0	0
$931$	−7.78276e10	−0.00339516
$932$	0	0
$933$	8.49669e11	0.0367098
$934$	0	0
$935$	−3.03544e12	−0.129888
$936$	0	0
$937$	−2.61645e13	−1.10888	−0.554439	−	0.832224i	$-0.687067\pi$
$937$	−2.61645e13	−1.10888	−0.554439	+	0.832224i	$0.687067\pi$
$938$	0	0
$939$	−2.67246e11	−0.0112180
$940$	0	0
$941$	3.93723e13	1.63696	0.818479	−	0.574536i	$-0.194817\pi$
$941$	3.93723e13	1.63696	0.818479	+	0.574536i	$0.194817\pi$
$942$	0	0
$943$	1.98210e12	0.0816249
$944$	0	0
$945$	5.05942e11	0.0206375
$946$	0	0
$947$	2.26031e13	0.913258	0.456629	−	0.889657i	$-0.349057\pi$
$947$	2.26031e13	0.913258	0.456629	+	0.889657i	$0.349057\pi$
$948$	0	0
$949$	3.62789e13	1.45196
$950$	0	0
$951$	7.03513e11	0.0278907
$952$	0	0
$953$	−3.27556e13	−1.28637	−0.643187	−	0.765709i	$-0.722388\pi$
$953$	−3.27556e13	−1.28637	−0.643187	+	0.765709i	$0.722388\pi$
$954$	0	0
$955$	2.88608e12	0.112278
$956$	0	0
$957$	−5.05052e11	−0.0194640
$958$	0	0
$959$	−1.72461e13	−0.658427
$960$	0	0
$961$	6.65864e11	0.0251843
$962$	0	0
$963$	−5.10786e13	−1.91390
$964$	0	0
$965$	2.41329e13	0.895854
$966$	0	0
$967$	−2.07251e13	−0.762217	−0.381108	−	0.924530i	$-0.624457\pi$
$967$	−2.07251e13	−0.762217	−0.381108	+	0.924530i	$0.624457\pi$
$968$	0	0
$969$	−2.11479e9	−7.70568e−5 0
$970$	0	0
$971$	−5.19068e13	−1.87386	−0.936932	−	0.349512i	$-0.886348\pi$
$971$	−5.19068e13	−1.87386	−0.936932	+	0.349512i	$0.886348\pi$
$972$	0	0
$973$	−3.40507e12	−0.121792
$974$	0	0
$975$	3.82522e11	0.0135561
$976$	0	0
$977$	2.18772e12	0.0768185	0.0384093	−	0.999262i	$-0.487771\pi$
$977$	2.18772e12	0.0768185	0.0384093	+	0.999262i	$0.487771\pi$
$978$	0	0
$979$	−2.40494e12	−0.0836725
$980$	0	0
$981$	−5.20045e13	−1.79279
$982$	0	0
$983$	1.84743e13	0.631068	0.315534	−	0.948914i	$-0.397816\pi$
$983$	1.84743e13	0.631068	0.315534	+	0.948914i	$0.397816\pi$
$984$	0	0
$985$	9.47549e12	0.320729
$986$	0	0
$987$	2.05172e11	0.00688163
$988$	0	0
$989$	−1.29215e13	−0.429468
$990$	0	0
$991$	3.55303e13	1.17022	0.585110	−	0.810954i	$-0.301051\pi$
$991$	3.55303e13	1.17022	0.585110	+	0.810954i	$0.301051\pi$
$992$	0	0
$993$	3.28945e11	0.0107362
$994$	0	0
$995$	−3.06381e13	−0.990965
$996$	0	0
$997$	5.54350e13	1.77687	0.888435	−	0.459003i	$-0.151793\pi$
$997$	5.54350e13	1.77687	0.888435	+	0.459003i	$0.151793\pi$
$998$	0	0
$999$	5.94463e10	0.00188834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.f.1.2		5
4.3	odd	2		17.10.a.a.1.2	✓	5
12.11	even	2		153.10.a.c.1.4		5
68.67	odd	2		289.10.a.a.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.a.1.2	✓	5	4.3	odd	2
153.10.a.c.1.4		5	12.11	even	2
272.10.a.f.1.2		5	1.1	even	1	trivial
289.10.a.a.1.2		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)

\(f(q)\)	\(=\)	\(q+3.02373 q^{3} +762.851 q^{5} -5573.11 q^{7} -19673.9 q^{9} +O(q^{10})\)	\(q+3.02373 q^{3} +762.851 q^{5} -5573.11 q^{7} -19673.9 q^{9} +47641.7 q^{11} -92260.9 q^{13} +2306.65 q^{15} -83521.0 q^{17} +8373.93 q^{19} -16851.6 q^{21} -364592. q^{23} -1.37118e6 q^{25} -119004. q^{27} -3.50595e6 q^{29} +5.20629e6 q^{31} +144055. q^{33} -4.25145e6 q^{35} -499530. q^{37} -278972. q^{39} -5.43648e6 q^{41} +3.54411e7 q^{43} -1.50082e7 q^{45} -1.21753e7 q^{47} -9.29403e6 q^{49} -252545. q^{51} +1.04471e8 q^{53} +3.63435e7 q^{55} +25320.5 q^{57} -4.16714e7 q^{59} +5.67537e7 q^{61} +1.09645e8 q^{63} -7.03813e7 q^{65} +1.74621e8 q^{67} -1.10243e6 q^{69} -3.46330e7 q^{71} -3.93220e8 q^{73} -4.14609e6 q^{75} -2.65512e8 q^{77} -1.85772e8 q^{79} +3.86881e8 q^{81} -3.62239e8 q^{83} -6.37141e7 q^{85} -1.06010e7 q^{87} -5.04798e7 q^{89} +5.14180e8 q^{91} +1.57424e7 q^{93} +6.38806e6 q^{95} -9.67620e8 q^{97} -9.37295e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more