LMFDB - Embedded newform 272.10.a.g.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:	$1$
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			1.5
Root			$16.8116$ 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)

$f(q)$	$=$	$q+116.887 q^{3} -1103.40 q^{5} +5164.29 q^{7} -6020.47 q^{9} +O(q^{10})$	$q+116.887 q^{3} -1103.40 q^{5} +5164.29 q^{7} -6020.47 q^{9} -44537.1 q^{11} +69656.8 q^{13} -128973. q^{15} +83521.0 q^{17} +170217. q^{19} +603638. q^{21} +2.00737e6 q^{23} -735637. q^{25} -3.00440e6 q^{27} +155688. q^{29} -4.27490e6 q^{31} -5.20580e6 q^{33} -5.69827e6 q^{35} +1.51022e7 q^{37} +8.14197e6 q^{39} +1.59320e7 q^{41} -1.49261e7 q^{43} +6.64298e6 q^{45} -3.36137e7 q^{47} -1.36837e7 q^{49} +9.76250e6 q^{51} -5.50379e7 q^{53} +4.91421e7 q^{55} +1.98961e7 q^{57} +7.94611e7 q^{59} +1.27852e7 q^{61} -3.10915e7 q^{63} -7.68593e7 q^{65} -2.73169e8 q^{67} +2.34635e8 q^{69} -3.88392e6 q^{71} +2.32369e8 q^{73} -8.59862e7 q^{75} -2.30002e8 q^{77} -3.38948e8 q^{79} -2.32673e8 q^{81} -5.74624e8 q^{83} -9.21569e7 q^{85} +1.81978e7 q^{87} -9.82429e8 q^{89} +3.59728e8 q^{91} -4.99680e8 q^{93} -1.87817e8 q^{95} -1.03147e9 q^{97} +2.68134e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	116.887	0.833144	0.416572	−	0.909103i	$-0.363231\pi$
$3$	116.887	0.833144	0.416572	+	0.909103i	$0.363231\pi$
$4$	0	0
$5$	−1103.40	−0.789528	−0.394764	−	0.918783i	$-0.629174\pi$
$5$	−1103.40	−0.789528	−0.394764	+	0.918783i	$0.629174\pi$
$6$	0	0
$7$	5164.29	0.812961	0.406480	−	0.913660i	$-0.366756\pi$
$7$	5164.29	0.812961	0.406480	+	0.913660i	$0.366756\pi$
$8$	0	0
$9$	−6020.47	−0.305872
$10$	0	0
$11$	−44537.1	−0.917180	−0.458590	−	0.888648i	$-0.651645\pi$
$11$	−44537.1	−0.917180	−0.458590	+	0.888648i	$0.651645\pi$
$12$	0	0
$13$	69656.8	0.676424	0.338212	−	0.941070i	$-0.390178\pi$
$13$	69656.8	0.676424	0.338212	+	0.941070i	$0.390178\pi$
$14$	0	0
$15$	−128973.	−0.657790
$16$	0	0
$17$	83521.0	0.242536
$17$	83521.0	0.242536
$18$	0	0
$19$	170217.	0.299648	0.149824	−	0.988713i	$-0.452129\pi$
$19$	170217.	0.299648	0.149824	+	0.988713i	$0.452129\pi$
$20$	0	0
$21$	603638.	0.677313
$22$	0	0
$23$	2.00737e6	1.49572	0.747861	−	0.663855i	$-0.231081\pi$
$23$	2.00737e6	1.49572	0.747861	+	0.663855i	$0.231081\pi$
$24$	0	0
$25$	−735637.	−0.376646
$26$	0	0
$27$	−3.00440e6	−1.08798
$28$	0	0
$29$	155688.	0.0408755	0.0204378	−	0.999791i	$-0.493494\pi$
$29$	155688.	0.0408755	0.0204378	+	0.999791i	$0.493494\pi$
$30$	0	0
$31$	−4.27490e6	−0.831379	−0.415689	−	0.909507i	$-0.636460\pi$
$31$	−4.27490e6	−0.831379	−0.415689	+	0.909507i	$0.636460\pi$
$32$	0	0
$33$	−5.20580e6	−0.764143
$34$	0	0
$35$	−5.69827e6	−0.641855
$36$	0	0
$37$	1.51022e7	1.32474	0.662371	−	0.749176i	$-0.269550\pi$
$37$	1.51022e7	1.32474	0.662371	+	0.749176i	$0.269550\pi$
$38$	0	0
$39$	8.14197e6	0.563558
$40$	0	0
$41$	1.59320e7	0.880526	0.440263	−	0.897869i	$-0.354885\pi$
$41$	1.59320e7	0.880526	0.440263	+	0.897869i	$0.354885\pi$
$42$	0	0
$43$	−1.49261e7	−0.665790	−0.332895	−	0.942964i	$-0.608025\pi$
$43$	−1.49261e7	−0.665790	−0.332895	+	0.942964i	$0.608025\pi$
$44$	0	0
$45$	6.64298e6	0.241494
$46$	0	0
$47$	−3.36137e7	−1.00479	−0.502396	−	0.864638i	$-0.667548\pi$
$47$	−3.36137e7	−1.00479	−0.502396	+	0.864638i	$0.667548\pi$
$48$	0	0
$49$	−1.36837e7	−0.339095
$50$	0	0
$51$	9.76250e6	0.202067
$52$	0	0
$53$	−5.50379e7	−0.958121	−0.479060	−	0.877782i	$-0.659023\pi$
$53$	−5.50379e7	−0.958121	−0.479060	+	0.877782i	$0.659023\pi$
$54$	0	0
$55$	4.91421e7	0.724139
$56$	0	0
$57$	1.98961e7	0.249650
$58$	0	0
$59$	7.94611e7	0.853730	0.426865	−	0.904315i	$-0.359618\pi$
$59$	7.94611e7	0.853730	0.426865	+	0.904315i	$0.359618\pi$
$60$	0	0
$61$	1.27852e7	0.118229	0.0591146	−	0.998251i	$-0.481172\pi$
$61$	1.27852e7	0.118229	0.0591146	+	0.998251i	$0.481172\pi$
$62$	0	0
$63$	−3.10915e7	−0.248662
$64$	0	0
$65$	−7.68593e7	−0.534055
$66$	0	0
$67$	−2.73169e8	−1.65613	−0.828066	−	0.560631i	$-0.810559\pi$
$67$	−2.73169e8	−1.65613	−0.828066	+	0.560631i	$0.810559\pi$
$68$	0	0
$69$	2.34635e8	1.24615
$70$	0	0
$71$	−3.88392e6	−0.0181388	−0.00906938	−	0.999959i	$-0.502887\pi$
$71$	−3.88392e6	−0.0181388	−0.00906938	+	0.999959i	$0.502887\pi$
$72$	0	0
$73$	2.32369e8	0.957690	0.478845	−	0.877900i	$-0.341056\pi$
$73$	2.32369e8	0.957690	0.478845	+	0.877900i	$0.341056\pi$
$74$	0	0
$75$	−8.59862e7	−0.313800
$76$	0	0
$77$	−2.30002e8	−0.745631
$78$	0	0
$79$	−3.38948e8	−0.979064	−0.489532	−	0.871985i	$-0.662832\pi$
$79$	−3.38948e8	−0.979064	−0.489532	+	0.871985i	$0.662832\pi$
$80$	0	0
$81$	−2.32673e8	−0.600571
$82$	0	0
$83$	−5.74624e8	−1.32902	−0.664511	−	0.747278i	$-0.731360\pi$
$83$	−5.74624e8	−1.32902	−0.664511	+	0.747278i	$0.731360\pi$
$84$	0	0
$85$	−9.21569e7	−0.191489
$86$	0	0
$87$	1.81978e7	0.0340552
$88$	0	0
$89$	−9.82429e8	−1.65976	−0.829882	−	0.557940i	$-0.811592\pi$
$89$	−9.82429e8	−1.65976	−0.829882	+	0.557940i	$0.811592\pi$
$90$	0	0
$91$	3.59728e8	0.549906
$92$	0	0
$93$	−4.99680e8	−0.692658
$94$	0	0
$95$	−1.87817e8	−0.236581
$96$	0	0
$97$	−1.03147e9	−1.18300	−0.591500	−	0.806305i	$-0.701464\pi$
$97$	−1.03147e9	−1.18300	−0.591500	+	0.806305i	$0.701464\pi$
$98$	0	0
$99$	2.68134e8	0.280539
$100$	0	0
$101$	−5.23482e7	−0.0500559	−0.0250280	−	0.999687i	$-0.507967\pi$
$101$	−5.23482e7	−0.0500559	−0.0250280	+	0.999687i	$0.507967\pi$
$102$	0	0
$103$	−4.65791e8	−0.407778	−0.203889	−	0.978994i	$-0.565358\pi$
$103$	−4.65791e8	−0.407778	−0.203889	+	0.978994i	$0.565358\pi$
$104$	0	0
$105$	−6.66053e8	−0.534757
$106$	0	0
$107$	1.44228e9	1.06371	0.531854	−	0.846836i	$-0.321496\pi$
$107$	1.44228e9	1.06371	0.531854	+	0.846836i	$0.321496\pi$
$108$	0	0
$109$	2.37882e9	1.61414	0.807072	−	0.590453i	$-0.201051\pi$
$109$	2.37882e9	1.61414	0.807072	+	0.590453i	$0.201051\pi$
$110$	0	0
$111$	1.76524e9	1.10370
$112$	0	0
$113$	−5.54552e8	−0.319955	−0.159978	−	0.987121i	$-0.551142\pi$
$113$	−5.54552e8	−0.319955	−0.159978	+	0.987121i	$0.551142\pi$
$114$	0	0
$115$	−2.21492e9	−1.18091
$116$	0	0
$117$	−4.19367e8	−0.206899
$118$	0	0
$119$	4.31327e8	0.197172
$120$	0	0
$121$	−3.74397e8	−0.158781
$122$	0	0
$123$	1.86224e9	0.733605
$124$	0	0
$125$	2.96678e9	1.08690
$126$	0	0
$127$	−3.25473e9	−1.11019	−0.555097	−	0.831786i	$-0.687319\pi$
$127$	−3.25473e9	−1.11019	−0.555097	+	0.831786i	$0.687319\pi$
$128$	0	0
$129$	−1.74466e9	−0.554699
$130$	0	0
$131$	2.29117e9	0.679731	0.339865	−	0.940474i	$-0.389618\pi$
$131$	2.29117e9	0.679731	0.339865	+	0.940474i	$0.389618\pi$
$132$	0	0
$133$	8.79051e8	0.243602
$134$	0	0
$135$	3.31505e9	0.858989
$136$	0	0
$137$	7.07071e9	1.71483	0.857413	−	0.514628i	$-0.172070\pi$
$137$	7.07071e9	1.71483	0.857413	+	0.514628i	$0.172070\pi$
$138$	0	0
$139$	2.13323e9	0.484697	0.242349	−	0.970189i	$-0.422082\pi$
$139$	2.13323e9	0.484697	0.242349	+	0.970189i	$0.422082\pi$
$140$	0	0
$141$	−3.92900e9	−0.837136
$142$	0	0
$143$	−3.10231e9	−0.620402
$144$	0	0
$145$	−1.71786e8	−0.0322724
$146$	0	0
$147$	−1.59944e9	−0.282515
$148$	0	0
$149$	−8.98904e9	−1.49408	−0.747042	−	0.664777i	$-0.768527\pi$
$149$	−8.98904e9	−1.49408	−0.747042	+	0.664777i	$0.768527\pi$
$150$	0	0
$151$	−4.53054e9	−0.709175	−0.354588	−	0.935023i	$-0.615379\pi$
$151$	−4.53054e9	−0.709175	−0.354588	+	0.935023i	$0.615379\pi$
$152$	0	0
$153$	−5.02836e8	−0.0741848
$154$	0	0
$155$	4.71692e9	0.656396
$156$	0	0
$157$	−3.69961e9	−0.485967	−0.242984	−	0.970030i	$-0.578126\pi$
$157$	−3.69961e9	−0.485967	−0.242984	+	0.970030i	$0.578126\pi$
$158$	0	0
$159$	−6.43320e9	−0.798252
$160$	0	0
$161$	1.03666e10	1.21596
$162$	0	0
$163$	−1.71522e10	−1.90317	−0.951583	−	0.307393i	$-0.900544\pi$
$163$	−1.71522e10	−1.90317	−0.951583	+	0.307393i	$0.900544\pi$
$164$	0	0
$165$	5.74407e9	0.603312
$166$	0	0
$167$	−1.39036e10	−1.38326	−0.691628	−	0.722254i	$-0.743106\pi$
$167$	−1.39036e10	−1.38326	−0.691628	+	0.722254i	$0.743106\pi$
$168$	0	0
$169$	−5.75242e9	−0.542451
$170$	0	0
$171$	−1.02479e9	−0.0916540
$172$	0	0
$173$	1.51288e9	0.128409	0.0642047	−	0.997937i	$-0.479549\pi$
$173$	1.51288e9	0.128409	0.0642047	+	0.997937i	$0.479549\pi$
$174$	0	0
$175$	−3.79904e9	−0.306198
$176$	0	0
$177$	9.28796e9	0.711280
$178$	0	0
$179$	2.09830e10	1.52767	0.763835	−	0.645411i	$-0.223314\pi$
$179$	2.09830e10	1.52767	0.763835	+	0.645411i	$0.223314\pi$
$180$	0	0
$181$	−1.75141e10	−1.21293	−0.606464	−	0.795111i	$-0.707413\pi$
$181$	−1.75141e10	−1.21293	−0.606464	+	0.795111i	$0.707413\pi$
$182$	0	0
$183$	1.49443e9	0.0985018
$184$	0	0
$185$	−1.66637e10	−1.04592
$186$	0	0
$187$	−3.71978e9	−0.222449
$188$	0	0
$189$	−1.55156e10	−0.884484
$190$	0	0
$191$	−1.64385e10	−0.893741	−0.446871	−	0.894599i	$-0.647462\pi$
$191$	−1.64385e10	−0.893741	−0.446871	+	0.894599i	$0.647462\pi$
$192$	0	0
$193$	−9.12709e9	−0.473505	−0.236753	−	0.971570i	$-0.576083\pi$
$193$	−9.12709e9	−0.473505	−0.236753	+	0.971570i	$0.576083\pi$
$194$	0	0
$195$	−8.98383e9	−0.444945
$196$	0	0
$197$	1.53151e9	0.0724472	0.0362236	−	0.999344i	$-0.488467\pi$
$197$	1.53151e9	0.0724472	0.0362236	+	0.999344i	$0.488467\pi$
$198$	0	0
$199$	−1.77679e10	−0.803152	−0.401576	−	0.915826i	$-0.631537\pi$
$199$	−1.77679e10	−0.803152	−0.401576	+	0.915826i	$0.631537\pi$
$200$	0	0
$201$	−3.19299e10	−1.37980
$202$	0	0
$203$	8.04017e8	0.0332302
$204$	0	0
$205$	−1.75793e10	−0.695200
$206$	0	0
$207$	−1.20853e10	−0.457499
$208$	0	0
$209$	−7.58097e9	−0.274832
$210$	0	0
$211$	−5.19005e10	−1.80260	−0.901302	−	0.433192i	$-0.857387\pi$
$211$	−5.19005e10	−1.80260	−0.901302	+	0.433192i	$0.857387\pi$
$212$	0	0
$213$	−4.53979e8	−0.0151122
$214$	0	0
$215$	1.64694e10	0.525660
$216$	0	0
$217$	−2.20769e10	−0.675878
$218$	0	0
$219$	2.71608e10	0.797893
$220$	0	0
$221$	5.81781e9	0.164057
$222$	0	0
$223$	6.09533e10	1.65054	0.825269	−	0.564740i	$-0.191024\pi$
$223$	6.09533e10	1.65054	0.825269	+	0.564740i	$0.191024\pi$
$224$	0	0
$225$	4.42888e9	0.115205
$226$	0	0
$227$	−6.35858e10	−1.58944	−0.794720	−	0.606977i	$-0.792382\pi$
$227$	−6.35858e10	−1.58944	−0.794720	+	0.606977i	$0.792382\pi$
$228$	0	0
$229$	−6.22903e10	−1.49679	−0.748395	−	0.663254i	$-0.769175\pi$
$229$	−6.22903e10	−1.49679	−0.748395	+	0.663254i	$0.769175\pi$
$230$	0	0
$231$	−2.68842e10	−0.621218
$232$	0	0
$233$	8.69026e10	1.93166	0.965831	−	0.259172i	$-0.0834496\pi$
$233$	8.69026e10	1.93166	0.965831	+	0.259172i	$0.0834496\pi$
$234$	0	0
$235$	3.70893e10	0.793311
$236$	0	0
$237$	−3.96186e10	−0.815701
$238$	0	0
$239$	−7.04686e10	−1.39703	−0.698514	−	0.715597i	$-0.746155\pi$
$239$	−7.04686e10	−1.39703	−0.698514	+	0.715597i	$0.746155\pi$
$240$	0	0
$241$	6.25058e10	1.19356	0.596779	−	0.802405i	$-0.296447\pi$
$241$	6.25058e10	1.19356	0.596779	+	0.802405i	$0.296447\pi$
$242$	0	0
$243$	3.19391e10	0.587617
$244$	0	0
$245$	1.50986e10	0.267725
$246$	0	0
$247$	1.18568e10	0.202689
$248$	0	0
$249$	−6.71660e10	−1.10727
$250$	0	0
$251$	7.26518e9	0.115535	0.0577676	−	0.998330i	$-0.481602\pi$
$251$	7.26518e9	0.115535	0.0577676	+	0.998330i	$0.481602\pi$
$252$	0	0
$253$	−8.94022e10	−1.37185
$254$	0	0
$255$	−1.07719e10	−0.159538
$256$	0	0
$257$	−8.23779e10	−1.17791	−0.588955	−	0.808166i	$-0.700460\pi$
$257$	−8.23779e10	−1.17791	−0.588955	+	0.808166i	$0.700460\pi$
$258$	0	0
$259$	7.79920e10	1.07696
$260$	0	0
$261$	−9.37314e8	−0.0125027
$262$	0	0
$263$	−8.05229e10	−1.03781	−0.518906	−	0.854831i	$-0.673660\pi$
$263$	−8.05229e10	−1.03781	−0.518906	+	0.854831i	$0.673660\pi$
$264$	0	0
$265$	6.07287e10	0.756463
$266$	0	0
$267$	−1.14833e11	−1.38282
$268$	0	0
$269$	3.37804e10	0.393350	0.196675	−	0.980469i	$-0.436986\pi$
$269$	3.37804e10	0.393350	0.196675	+	0.980469i	$0.436986\pi$
$270$	0	0
$271$	−5.31396e10	−0.598489	−0.299245	−	0.954176i	$-0.596735\pi$
$271$	−5.31396e10	−0.598489	−0.299245	+	0.954176i	$0.596735\pi$
$272$	0	0
$273$	4.20475e10	0.458150
$274$	0	0
$275$	3.27631e10	0.345452
$276$	0	0
$277$	−1.33887e11	−1.36640	−0.683201	−	0.730231i	$-0.739413\pi$
$277$	−1.33887e11	−1.36640	−0.683201	+	0.730231i	$0.739413\pi$
$278$	0	0
$279$	2.57369e10	0.254295
$280$	0	0
$281$	−6.71427e10	−0.642422	−0.321211	−	0.947008i	$-0.604090\pi$
$281$	−6.71427e10	−0.642422	−0.321211	+	0.947008i	$0.604090\pi$
$282$	0	0
$283$	−1.11880e11	−1.03684	−0.518420	−	0.855126i	$-0.673480\pi$
$283$	−1.11880e11	−1.03684	−0.518420	+	0.855126i	$0.673480\pi$
$284$	0	0
$285$	−2.19534e10	−0.197106
$286$	0	0
$287$	8.22774e10	0.715833
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	−1.20566e11	−0.985609
$292$	0	0
$293$	1.51192e11	1.19846	0.599230	−	0.800577i	$-0.295474\pi$
$293$	1.51192e11	1.19846	0.599230	+	0.800577i	$0.295474\pi$
$294$	0	0
$295$	−8.76773e10	−0.674044
$296$	0	0
$297$	1.33807e11	0.997872
$298$	0	0
$299$	1.39827e11	1.01174
$300$	0	0
$301$	−7.70825e10	−0.541261
$302$	0	0
$303$	−6.11881e9	−0.0417038
$304$	0	0
$305$	−1.41072e10	−0.0933452
$306$	0	0
$307$	−6.62355e10	−0.425567	−0.212784	−	0.977099i	$-0.568253\pi$
$307$	−6.62355e10	−0.425567	−0.212784	+	0.977099i	$0.568253\pi$
$308$	0	0
$309$	−5.44449e10	−0.339738
$310$	0	0
$311$	9.27102e10	0.561960	0.280980	−	0.959714i	$-0.409340\pi$
$311$	9.27102e10	0.561960	0.280980	+	0.959714i	$0.409340\pi$
$312$	0	0
$313$	2.58675e11	1.52337	0.761684	−	0.647949i	$-0.224373\pi$
$313$	2.58675e11	1.52337	0.761684	+	0.647949i	$0.224373\pi$
$314$	0	0
$315$	3.43063e10	0.196325
$316$	0	0
$317$	−4.47169e10	−0.248717	−0.124358	−	0.992237i	$-0.539687\pi$
$317$	−4.47169e10	−0.248717	−0.124358	+	0.992237i	$0.539687\pi$
$318$	0	0
$319$	−6.93388e9	−0.0374902
$320$	0	0
$321$	1.68583e11	0.886221
$322$	0	0
$323$	1.42167e10	0.0726754
$324$	0	0
$325$	−5.12421e10	−0.254772
$326$	0	0
$327$	2.78053e11	1.34481
$328$	0	0
$329$	−1.73591e11	−0.816856
$330$	0	0
$331$	−4.78159e10	−0.218951	−0.109475	−	0.993990i	$-0.534917\pi$
$331$	−4.78159e10	−0.218951	−0.109475	+	0.993990i	$0.534917\pi$
$332$	0	0
$333$	−9.09221e10	−0.405201
$334$	0	0
$335$	3.01414e11	1.30756
$336$	0	0
$337$	3.76342e11	1.58945	0.794727	−	0.606968i	$-0.207614\pi$
$337$	3.76342e11	1.58945	0.794727	+	0.606968i	$0.207614\pi$
$338$	0	0
$339$	−6.48198e10	−0.266569
$340$	0	0
$341$	1.90392e11	0.762524
$342$	0	0
$343$	−2.79064e11	−1.08863
$344$	0	0
$345$	−2.58895e11	−0.983872
$346$	0	0
$347$	1.77078e11	0.655665	0.327833	−	0.944736i	$-0.393682\pi$
$347$	1.77078e11	0.655665	0.327833	+	0.944736i	$0.393682\pi$
$348$	0	0
$349$	2.43883e11	0.879967	0.439984	−	0.898006i	$-0.354984\pi$
$349$	2.43883e11	0.879967	0.439984	+	0.898006i	$0.354984\pi$
$350$	0	0
$351$	−2.09277e11	−0.735934
$352$	0	0
$353$	1.91758e11	0.657304	0.328652	−	0.944451i	$-0.393406\pi$
$353$	1.91758e11	0.657304	0.328652	+	0.944451i	$0.393406\pi$
$354$	0	0
$355$	4.28551e9	0.0143211
$356$	0	0
$357$	5.04164e10	0.164273
$358$	0	0
$359$	6.43139e10	0.204352	0.102176	−	0.994766i	$-0.467419\pi$
$359$	6.43139e10	0.204352	0.102176	+	0.994766i	$0.467419\pi$
$360$	0	0
$361$	−2.93714e11	−0.910211
$362$	0	0
$363$	−4.37621e10	−0.132287
$364$	0	0
$365$	−2.56395e11	−0.756123
$366$	0	0
$367$	−7.24947e10	−0.208597	−0.104299	−	0.994546i	$-0.533260\pi$
$367$	−7.24947e10	−0.208597	−0.104299	+	0.994546i	$0.533260\pi$
$368$	0	0
$369$	−9.59180e10	−0.269328
$370$	0	0
$371$	−2.84232e11	−0.778914
$372$	0	0
$373$	−5.44526e10	−0.145656	−0.0728281	−	0.997345i	$-0.523202\pi$
$373$	−5.44526e10	−0.145656	−0.0728281	+	0.997345i	$0.523202\pi$
$374$	0	0
$375$	3.46777e11	0.905544
$376$	0	0
$377$	1.08447e10	0.0276492
$378$	0	0
$379$	2.15005e11	0.535270	0.267635	−	0.963520i	$-0.413758\pi$
$379$	2.15005e11	0.535270	0.267635	+	0.963520i	$0.413758\pi$
$380$	0	0
$381$	−3.80435e11	−0.924951
$382$	0	0
$383$	4.29189e11	1.01919	0.509594	−	0.860415i	$-0.329796\pi$
$383$	4.29189e11	1.01919	0.509594	+	0.860415i	$0.329796\pi$
$384$	0	0
$385$	2.53784e11	0.588696
$386$	0	0
$387$	8.98619e10	0.203646
$388$	0	0
$389$	−2.02217e11	−0.447760	−0.223880	−	0.974617i	$-0.571872\pi$
$389$	−2.02217e11	−0.447760	−0.223880	+	0.974617i	$0.571872\pi$
$390$	0	0
$391$	1.67657e11	0.362766
$392$	0	0
$393$	2.67808e11	0.566313
$394$	0	0
$395$	3.73995e11	0.772998
$396$	0	0
$397$	1.81106e11	0.365911	0.182955	−	0.983121i	$-0.441434\pi$
$397$	1.81106e11	0.365911	0.182955	+	0.983121i	$0.441434\pi$
$398$	0	0
$399$	1.02749e11	0.202956
$400$	0	0
$401$	−1.40602e11	−0.271545	−0.135772	−	0.990740i	$-0.543352\pi$
$401$	−1.40602e11	−0.271545	−0.135772	+	0.990740i	$0.543352\pi$
$402$	0	0
$403$	−2.97776e11	−0.562364
$404$	0	0
$405$	2.56732e11	0.474167
$406$	0	0
$407$	−6.72606e11	−1.21503
$408$	0	0
$409$	−6.11843e11	−1.08115	−0.540574	−	0.841297i	$-0.681793\pi$
$409$	−6.11843e11	−1.08115	−0.540574	+	0.841297i	$0.681793\pi$
$410$	0	0
$411$	8.26472e11	1.42870
$412$	0	0
$413$	4.10360e11	0.694049
$414$	0	0
$415$	6.34039e11	1.04930
$416$	0	0
$417$	2.49346e11	0.403822
$418$	0	0
$419$	−2.89071e10	−0.0458186	−0.0229093	−	0.999738i	$-0.507293\pi$
$419$	−2.89071e10	−0.0458186	−0.0229093	+	0.999738i	$0.507293\pi$
$420$	0	0
$421$	1.06260e12	1.64854	0.824272	−	0.566193i	$-0.191584\pi$
$421$	1.06260e12	1.64854	0.824272	+	0.566193i	$0.191584\pi$
$422$	0	0
$423$	2.02370e11	0.307337
$424$	0	0
$425$	−6.14411e10	−0.0913501
$426$	0	0
$427$	6.60267e10	0.0961156
$428$	0	0
$429$	−3.62619e11	−0.516884
$430$	0	0
$431$	5.09485e10	0.0711187	0.0355594	−	0.999368i	$-0.488679\pi$
$431$	5.09485e10	0.0711187	0.0355594	+	0.999368i	$0.488679\pi$
$432$	0	0
$433$	−1.02428e12	−1.40030	−0.700152	−	0.713994i	$-0.746885\pi$
$433$	−1.02428e12	−1.40030	−0.700152	+	0.713994i	$0.746885\pi$
$434$	0	0
$435$	−2.00795e10	−0.0268875
$436$	0	0
$437$	3.41688e11	0.448191
$438$	0	0
$439$	−3.70271e11	−0.475805	−0.237903	−	0.971289i	$-0.576460\pi$
$439$	−3.70271e11	−0.475805	−0.237903	+	0.971289i	$0.576460\pi$
$440$	0	0
$441$	8.23824e10	0.103720
$442$	0	0
$443$	1.17113e12	1.44473	0.722366	−	0.691511i	$-0.243054\pi$
$443$	1.17113e12	1.44473	0.722366	+	0.691511i	$0.243054\pi$
$444$	0	0
$445$	1.08401e12	1.31043
$446$	0	0
$447$	−1.05070e12	−1.24479
$448$	0	0
$449$	6.79800e10	0.0789355	0.0394678	−	0.999221i	$-0.487434\pi$
$449$	6.79800e10	0.0789355	0.0394678	+	0.999221i	$0.487434\pi$
$450$	0	0
$451$	−7.09563e11	−0.807601
$452$	0	0
$453$	−5.29560e11	−0.590845
$454$	0	0
$455$	−3.96924e11	−0.434166
$456$	0	0
$457$	1.72167e12	1.84640	0.923202	−	0.384315i	$-0.125562\pi$
$457$	1.72167e12	1.84640	0.923202	+	0.384315i	$0.125562\pi$
$458$	0	0
$459$	−2.50930e11	−0.263874
$460$	0	0
$461$	1.42332e12	1.46774	0.733868	−	0.679292i	$-0.237713\pi$
$461$	1.42332e12	1.46774	0.733868	+	0.679292i	$0.237713\pi$
$462$	0	0
$463$	−1.62089e12	−1.63922	−0.819611	−	0.572920i	$-0.805811\pi$
$463$	−1.62089e12	−1.63922	−0.819611	+	0.572920i	$0.805811\pi$
$464$	0	0
$465$	5.51346e11	0.546873
$466$	0	0
$467$	1.13569e12	1.10493	0.552465	−	0.833536i	$-0.313687\pi$
$467$	1.13569e12	1.10493	0.552465	+	0.833536i	$0.313687\pi$
$468$	0	0
$469$	−1.41072e12	−1.34637
$470$	0	0
$471$	−4.32435e11	−0.404881
$472$	0	0
$473$	6.64763e11	0.610649
$474$	0	0
$475$	−1.25218e11	−0.112861
$476$	0	0
$477$	3.31354e11	0.293062
$478$	0	0
$479$	−1.54480e11	−0.134080	−0.0670398	−	0.997750i	$-0.521355\pi$
$479$	−1.54480e11	−0.134080	−0.0670398	+	0.997750i	$0.521355\pi$
$480$	0	0
$481$	1.05197e12	0.896087
$482$	0	0
$483$	1.21172e12	1.01307
$484$	0	0
$485$	1.13813e12	0.934011
$486$	0	0
$487$	9.40063e11	0.757315	0.378657	−	0.925537i	$-0.376386\pi$
$487$	9.40063e11	0.757315	0.378657	+	0.925537i	$0.376386\pi$
$488$	0	0
$489$	−2.00487e12	−1.58561
$490$	0	0
$491$	5.38393e11	0.418055	0.209027	−	0.977910i	$-0.432970\pi$
$491$	5.38393e11	0.418055	0.209027	+	0.977910i	$0.432970\pi$
$492$	0	0
$493$	1.30032e10	0.00991377
$494$	0	0
$495$	−2.95859e11	−0.221494
$496$	0	0
$497$	−2.00577e10	−0.0147461
$498$	0	0
$499$	6.12934e11	0.442549	0.221274	−	0.975212i	$-0.428978\pi$
$499$	6.12934e11	0.442549	0.221274	+	0.975212i	$0.428978\pi$
$500$	0	0
$501$	−1.62514e12	−1.15245
$502$	0	0
$503$	2.31510e12	1.61255	0.806277	−	0.591538i	$-0.201479\pi$
$503$	2.31510e12	1.61255	0.806277	+	0.591538i	$0.201479\pi$
$504$	0	0
$505$	5.77609e10	0.0395205
$506$	0	0
$507$	−6.72382e11	−0.451940
$508$	0	0
$509$	−1.48597e12	−0.981249	−0.490625	−	0.871371i	$-0.663231\pi$
$509$	−1.48597e12	−0.981249	−0.490625	+	0.871371i	$0.663231\pi$
$510$	0	0
$511$	1.20002e12	0.778564
$512$	0	0
$513$	−5.11400e11	−0.326011
$514$	0	0
$515$	5.13954e11	0.321952
$516$	0	0
$517$	1.49706e12	0.921575
$518$	0	0
$519$	1.76836e11	0.106984
$520$	0	0
$521$	−2.26648e11	−0.134766	−0.0673832	−	0.997727i	$-0.521465\pi$
$521$	−2.26648e11	−0.134766	−0.0673832	+	0.997727i	$0.521465\pi$
$522$	0	0
$523$	1.52624e12	0.891998	0.445999	−	0.895033i	$-0.352848\pi$
$523$	1.52624e12	0.891998	0.445999	+	0.895033i	$0.352848\pi$
$524$	0	0
$525$	−4.44058e11	−0.255107
$526$	0	0
$527$	−3.57044e11	−0.201639
$528$	0	0
$529$	2.22836e12	1.23719
$530$	0	0
$531$	−4.78393e11	−0.261132
$532$	0	0
$533$	1.10977e12	0.595609
$534$	0	0
$535$	−1.59141e12	−0.839827
$536$	0	0
$537$	2.45264e12	1.27277
$538$	0	0
$539$	6.09432e11	0.311011
$540$	0	0
$541$	8.24488e11	0.413806	0.206903	−	0.978361i	$-0.433662\pi$
$541$	8.24488e11	0.413806	0.206903	+	0.978361i	$0.433662\pi$
$542$	0	0
$543$	−2.04717e12	−1.01054
$544$	0	0
$545$	−2.62479e12	−1.27441
$546$	0	0
$547$	3.27510e11	0.156416	0.0782082	−	0.996937i	$-0.475080\pi$
$547$	3.27510e11	0.156416	0.0782082	+	0.996937i	$0.475080\pi$
$548$	0	0
$549$	−7.69731e10	−0.0361629
$550$	0	0
$551$	2.65007e10	0.0122483
$552$	0	0
$553$	−1.75043e12	−0.795941
$554$	0	0
$555$	−1.94777e12	−0.871402
$556$	0	0
$557$	6.54333e11	0.288038	0.144019	−	0.989575i	$-0.453997\pi$
$557$	6.54333e11	0.288038	0.144019	+	0.989575i	$0.453997\pi$
$558$	0	0
$559$	−1.03970e12	−0.450356
$560$	0	0
$561$	−4.34793e11	−0.185332
$562$	0	0
$563$	2.05130e12	0.860483	0.430242	−	0.902714i	$-0.358428\pi$
$563$	2.05130e12	0.860483	0.430242	+	0.902714i	$0.358428\pi$
$564$	0	0
$565$	6.11892e11	0.252614
$566$	0	0
$567$	−1.20159e12	−0.488241
$568$	0	0
$569$	9.21470e11	0.368533	0.184266	−	0.982876i	$-0.441009\pi$
$569$	9.21470e11	0.368533	0.184266	+	0.982876i	$0.441009\pi$
$570$	0	0
$571$	1.70877e12	0.672701	0.336351	−	0.941737i	$-0.390807\pi$
$571$	1.70877e12	0.672701	0.336351	+	0.941737i	$0.390807\pi$
$572$	0	0
$573$	−1.92144e12	−0.744615
$574$	0	0
$575$	−1.47669e12	−0.563358
$576$	0	0
$577$	−1.50384e12	−0.564820	−0.282410	−	0.959294i	$-0.591134\pi$
$577$	−1.50384e12	−0.564820	−0.282410	+	0.959294i	$0.591134\pi$
$578$	0	0
$579$	−1.06684e12	−0.394498
$580$	0	0
$581$	−2.96753e12	−1.08044
$582$	0	0
$583$	2.45123e12	0.878769
$584$	0	0
$585$	4.62729e11	0.163352
$586$	0	0
$587$	3.06312e12	1.06486	0.532430	−	0.846474i	$-0.321279\pi$
$587$	3.06312e12	1.06486	0.532430	+	0.846474i	$0.321279\pi$
$588$	0	0
$589$	−7.27662e11	−0.249121
$590$	0	0
$591$	1.79013e11	0.0603589
$592$	0	0
$593$	−7.03492e11	−0.233622	−0.116811	−	0.993154i	$-0.537267\pi$
$593$	−7.03492e11	−0.233622	−0.116811	+	0.993154i	$0.537267\pi$
$594$	0	0
$595$	−4.75925e11	−0.155673
$596$	0	0
$597$	−2.07683e12	−0.669141
$598$	0	0
$599$	−2.22780e12	−0.707060	−0.353530	−	0.935423i	$-0.615019\pi$
$599$	−2.22780e12	−0.707060	−0.353530	+	0.935423i	$0.615019\pi$
$600$	0	0
$601$	−3.12659e12	−0.977541	−0.488771	−	0.872412i	$-0.662555\pi$
$601$	−3.12659e12	−0.977541	−0.488771	+	0.872412i	$0.662555\pi$
$602$	0	0
$603$	1.64461e12	0.506564
$604$	0	0
$605$	4.13109e11	0.125362
$606$	0	0
$607$	−8.02028e11	−0.239795	−0.119898	−	0.992786i	$-0.538257\pi$
$607$	−8.02028e11	−0.239795	−0.119898	+	0.992786i	$0.538257\pi$
$608$	0	0
$609$	9.39790e10	0.0276855
$610$	0	0
$611$	−2.34143e12	−0.679665
$612$	0	0
$613$	4.27823e12	1.22375	0.611874	−	0.790955i	$-0.290416\pi$
$613$	4.27823e12	1.22375	0.611874	+	0.790955i	$0.290416\pi$
$614$	0	0
$615$	−2.05479e12	−0.579201
$616$	0	0
$617$	−2.33959e12	−0.649914	−0.324957	−	0.945729i	$-0.605350\pi$
$617$	−2.33959e12	−0.649914	−0.324957	+	0.945729i	$0.605350\pi$
$618$	0	0
$619$	−7.17485e12	−1.96429	−0.982144	−	0.188131i	$-0.939757\pi$
$619$	−7.17485e12	−1.96429	−0.982144	+	0.188131i	$0.939757\pi$
$620$	0	0
$621$	−6.03092e12	−1.62731
$622$	0	0
$623$	−5.07355e12	−1.34932
$624$	0	0
$625$	−1.83675e12	−0.481492
$626$	0	0
$627$	−8.86116e11	−0.228974
$628$	0	0
$629$	1.26135e12	0.321297
$630$	0	0
$631$	−2.21641e12	−0.556567	−0.278283	−	0.960499i	$-0.589765\pi$
$631$	−2.21641e12	−0.556567	−0.278283	+	0.960499i	$0.589765\pi$
$632$	0	0
$633$	−6.06648e12	−1.50183
$634$	0	0
$635$	3.59127e12	0.876528
$636$	0	0
$637$	−9.53164e11	−0.229372
$638$	0	0
$639$	2.33830e10	0.00554813
$640$	0	0
$641$	5.69709e10	0.0133288	0.00666442	−	0.999978i	$-0.497879\pi$
$641$	5.69709e10	0.0133288	0.00666442	+	0.999978i	$0.497879\pi$
$642$	0	0
$643$	3.26271e12	0.752713	0.376356	−	0.926475i	$-0.377177\pi$
$643$	3.26271e12	0.752713	0.376356	+	0.926475i	$0.377177\pi$
$644$	0	0
$645$	1.92506e12	0.437950
$646$	0	0
$647$	−4.42290e12	−0.992289	−0.496144	−	0.868240i	$-0.665251\pi$
$647$	−4.42290e12	−0.992289	−0.496144	+	0.868240i	$0.665251\pi$
$648$	0	0
$649$	−3.53897e12	−0.783024
$650$	0	0
$651$	−2.58049e12	−0.563104
$652$	0	0
$653$	4.99365e12	1.07475	0.537376	−	0.843343i	$-0.319416\pi$
$653$	4.99365e12	1.07475	0.537376	+	0.843343i	$0.319416\pi$
$654$	0	0
$655$	−2.52808e12	−0.536666
$656$	0	0
$657$	−1.39897e12	−0.292930
$658$	0	0
$659$	−1.68196e12	−0.347401	−0.173701	−	0.984799i	$-0.555572\pi$
$659$	−1.68196e12	−0.347401	−0.173701	+	0.984799i	$0.555572\pi$
$660$	0	0
$661$	8.83788e12	1.80070	0.900351	−	0.435165i	$-0.143310\pi$
$661$	8.83788e12	1.80070	0.900351	+	0.435165i	$0.143310\pi$
$662$	0	0
$663$	6.80025e11	0.136683
$664$	0	0
$665$	−9.69943e11	−0.192331
$666$	0	0
$667$	3.12522e11	0.0611385
$668$	0	0
$669$	7.12464e12	1.37514
$670$	0	0
$671$	−5.69417e11	−0.108437
$672$	0	0
$673$	4.41749e12	0.830057	0.415029	−	0.909808i	$-0.363772\pi$
$673$	4.41749e12	0.830057	0.415029	+	0.909808i	$0.363772\pi$
$674$	0	0
$675$	2.21014e12	0.409783
$676$	0	0
$677$	−8.50941e12	−1.55686	−0.778431	−	0.627730i	$-0.783984\pi$
$677$	−8.50941e12	−1.55686	−0.778431	+	0.627730i	$0.783984\pi$
$678$	0	0
$679$	−5.32682e12	−0.961733
$680$	0	0
$681$	−7.43235e12	−1.32423
$682$	0	0
$683$	−3.49030e12	−0.613719	−0.306859	−	0.951755i	$-0.599278\pi$
$683$	−3.49030e12	−0.613719	−0.306859	+	0.951755i	$0.599278\pi$
$684$	0	0
$685$	−7.80181e12	−1.35390
$686$	0	0
$687$	−7.28092e12	−1.24704
$688$	0	0
$689$	−3.83376e12	−0.648095
$690$	0	0
$691$	−3.30831e12	−0.552019	−0.276010	−	0.961155i	$-0.589012\pi$
$691$	−3.30831e12	−0.552019	−0.276010	+	0.961155i	$0.589012\pi$
$692$	0	0
$693$	1.38472e12	0.228067
$694$	0	0
$695$	−2.35380e12	−0.382682
$696$	0	0
$697$	1.33065e12	0.213559
$698$	0	0
$699$	1.01578e13	1.60935
$700$	0	0
$701$	1.72300e12	0.269497	0.134749	−	0.990880i	$-0.456977\pi$
$701$	1.72300e12	0.269497	0.134749	+	0.990880i	$0.456977\pi$
$702$	0	0
$703$	2.57065e12	0.396957
$704$	0	0
$705$	4.33525e12	0.660942
$706$	0	0
$707$	−2.70341e11	−0.0406935
$708$	0	0
$709$	8.11666e12	1.20634	0.603169	−	0.797613i	$-0.293904\pi$
$709$	8.11666e12	1.20634	0.603169	+	0.797613i	$0.293904\pi$
$710$	0	0
$711$	2.04063e12	0.299468
$712$	0	0
$713$	−8.58130e12	−1.24351
$714$	0	0
$715$	3.42309e12	0.489825
$716$	0	0
$717$	−8.23685e12	−1.16392
$718$	0	0
$719$	3.41170e12	0.476092	0.238046	−	0.971254i	$-0.423493\pi$
$719$	3.41170e12	0.476092	0.238046	+	0.971254i	$0.423493\pi$
$720$	0	0
$721$	−2.40548e12	−0.331508
$722$	0	0
$723$	7.30611e12	0.994406
$724$	0	0
$725$	−1.14530e11	−0.0153956
$726$	0	0
$727$	6.77588e12	0.899624	0.449812	−	0.893123i	$-0.351491\pi$
$727$	6.77588e12	0.899624	0.449812	+	0.893123i	$0.351491\pi$
$728$	0	0
$729$	8.31297e12	1.09014
$730$	0	0
$731$	−1.24664e12	−0.161478
$732$	0	0
$733$	−1.34191e13	−1.71694	−0.858469	−	0.512866i	$-0.828584\pi$
$733$	−1.34191e13	−1.71694	−0.858469	+	0.512866i	$0.828584\pi$
$734$	0	0
$735$	1.76483e12	0.223053
$736$	0	0
$737$	1.21661e13	1.51897
$738$	0	0
$739$	−9.85851e12	−1.21594	−0.607969	−	0.793961i	$-0.708016\pi$
$739$	−9.85851e12	−1.21594	−0.607969	+	0.793961i	$0.708016\pi$
$740$	0	0
$741$	1.38590e12	0.168869
$742$	0	0
$743$	−6.66107e12	−0.801853	−0.400926	−	0.916110i	$-0.631312\pi$
$743$	−6.66107e12	−0.801853	−0.400926	+	0.916110i	$0.631312\pi$
$744$	0	0
$745$	9.91849e12	1.17962
$746$	0	0
$747$	3.45951e12	0.406510
$748$	0	0
$749$	7.44835e12	0.864752
$750$	0	0
$751$	−2.09306e12	−0.240106	−0.120053	−	0.992767i	$-0.538306\pi$
$751$	−2.09306e12	−0.240106	−0.120053	+	0.992767i	$0.538306\pi$
$752$	0	0
$753$	8.49203e11	0.0962575
$754$	0	0
$755$	4.99899e12	0.559913
$756$	0	0
$757$	1.26189e13	1.39666	0.698330	−	0.715776i	$-0.253927\pi$
$757$	1.26189e13	1.39666	0.698330	+	0.715776i	$0.253927\pi$
$758$	0	0
$759$	−1.04499e13	−1.14295
$760$	0	0
$761$	−6.62893e12	−0.716494	−0.358247	−	0.933627i	$-0.616625\pi$
$761$	−6.62893e12	−0.716494	−0.358247	+	0.933627i	$0.616625\pi$
$762$	0	0
$763$	1.22849e13	1.31224
$764$	0	0
$765$	5.54828e11	0.0585709
$766$	0	0
$767$	5.53501e12	0.577483
$768$	0	0
$769$	3.47382e12	0.358210	0.179105	−	0.983830i	$-0.442680\pi$
$769$	3.47382e12	0.358210	0.179105	+	0.983830i	$0.442680\pi$
$770$	0	0
$771$	−9.62889e12	−0.981368
$772$	0	0
$773$	−1.16131e13	−1.16988	−0.584941	−	0.811076i	$-0.698882\pi$
$773$	−1.16131e13	−1.16988	−0.584941	+	0.811076i	$0.698882\pi$
$774$	0	0
$775$	3.14478e12	0.313135
$776$	0	0
$777$	9.11623e12	0.897265
$778$	0	0
$779$	2.71190e12	0.263848
$780$	0	0
$781$	1.72978e11	0.0166365
$782$	0	0
$783$	−4.67748e11	−0.0444717
$784$	0	0
$785$	4.08214e12	0.383685
$786$	0	0
$787$	1.50665e13	1.39999	0.699997	−	0.714146i	$-0.253185\pi$
$787$	1.50665e13	1.39999	0.699997	+	0.714146i	$0.253185\pi$
$788$	0	0
$789$	−9.41207e12	−0.864646
$790$	0	0
$791$	−2.86387e12	−0.260111
$792$	0	0
$793$	8.90579e11	0.0799730
$794$	0	0
$795$	7.09839e12	0.630242
$796$	0	0
$797$	−7.67142e12	−0.673463	−0.336731	−	0.941601i	$-0.609321\pi$
$797$	−7.67142e12	−0.673463	−0.336731	+	0.941601i	$0.609321\pi$
$798$	0	0
$799$	−2.80745e12	−0.243698
$800$	0	0
$801$	5.91468e12	0.507674
$802$	0	0
$803$	−1.03490e13	−0.878374
$804$	0	0
$805$	−1.14385e13	−0.960037
$806$	0	0
$807$	3.94848e12	0.327717
$808$	0	0
$809$	−6.00907e12	−0.493218	−0.246609	−	0.969115i	$-0.579316\pi$
$809$	−6.00907e12	−0.493218	−0.246609	+	0.969115i	$0.579316\pi$
$810$	0	0
$811$	1.51786e13	1.23208	0.616038	−	0.787717i	$-0.288737\pi$
$811$	1.51786e13	1.23208	0.616038	+	0.787717i	$0.288737\pi$
$812$	0	0
$813$	−6.21132e12	−0.498627
$814$	0	0
$815$	1.89258e13	1.50260
$816$	0	0
$817$	−2.54067e12	−0.199503
$818$	0	0
$819$	−2.16573e12	−0.168201
$820$	0	0
$821$	1.51618e13	1.16468	0.582339	−	0.812946i	$-0.302138\pi$
$821$	1.51618e13	1.16468	0.582339	+	0.812946i	$0.302138\pi$
$822$	0	0
$823$	8.88314e12	0.674943	0.337472	−	0.941336i	$-0.390428\pi$
$823$	8.88314e12	0.674943	0.337472	+	0.941336i	$0.390428\pi$
$824$	0	0
$825$	3.82958e12	0.287811
$826$	0	0
$827$	−9.71703e9	−0.000722368 0	−0.000361184	−	1.00000i	$-0.500115\pi$
$827$	−9.71703e9	−0.000722368 0	−0.000361184		1.00000i	$0.500115\pi$
$828$	0	0
$829$	−6.15249e12	−0.452434	−0.226217	−	0.974077i	$-0.572636\pi$
$829$	−6.15249e12	−0.452434	−0.226217	+	0.974077i	$0.572636\pi$
$830$	0	0
$831$	−1.56496e13	−1.13841
$832$	0	0
$833$	−1.14288e12	−0.0822426
$834$	0	0
$835$	1.53412e13	1.09212
$836$	0	0
$837$	1.28435e13	0.904522
$838$	0	0
$839$	−2.01899e13	−1.40671	−0.703355	−	0.710839i	$-0.748316\pi$
$839$	−2.01899e13	−1.40671	−0.703355	+	0.710839i	$0.748316\pi$
$840$	0	0
$841$	−1.44829e13	−0.998329
$842$	0	0
$843$	−7.84810e12	−0.535230
$844$	0	0
$845$	6.34722e12	0.428280
$846$	0	0
$847$	−1.93350e12	−0.129083
$848$	0	0
$849$	−1.30773e13	−0.863837
$850$	0	0
$851$	3.03156e13	1.98145
$852$	0	0
$853$	−2.71140e13	−1.75357	−0.876784	−	0.480885i	$-0.840315\pi$
$853$	−2.71140e13	−1.75357	−0.876784	+	0.480885i	$0.840315\pi$
$854$	0	0
$855$	1.13075e12	0.0723633
$856$	0	0
$857$	2.66987e13	1.69074	0.845371	−	0.534180i	$-0.179380\pi$
$857$	2.66987e13	1.69074	0.845371	+	0.534180i	$0.179380\pi$
$858$	0	0
$859$	−2.62702e13	−1.64624	−0.823122	−	0.567864i	$-0.807770\pi$
$859$	−2.62702e13	−1.64624	−0.823122	+	0.567864i	$0.807770\pi$
$860$	0	0
$861$	9.61714e12	0.596392
$862$	0	0
$863$	−2.34256e13	−1.43761	−0.718806	−	0.695211i	$-0.755311\pi$
$863$	−2.34256e13	−1.43761	−0.718806	+	0.695211i	$0.755311\pi$
$864$	0	0
$865$	−1.66931e12	−0.101383
$866$	0	0
$867$	8.15374e11	0.0490085
$868$	0	0
$869$	1.50958e13	0.897978
$870$	0	0
$871$	−1.90281e13	−1.12025
$872$	0	0
$873$	6.20995e12	0.361846
$874$	0	0
$875$	1.53213e13	0.883607
$876$	0	0
$877$	−4.84217e12	−0.276402	−0.138201	−	0.990404i	$-0.544132\pi$
$877$	−4.84217e12	−0.276402	−0.138201	+	0.990404i	$0.544132\pi$
$878$	0	0
$879$	1.76723e13	0.998489
$880$	0	0
$881$	1.59269e13	0.890718	0.445359	−	0.895352i	$-0.353076\pi$
$881$	1.59269e13	0.890718	0.445359	+	0.895352i	$0.353076\pi$
$882$	0	0
$883$	−4.71830e12	−0.261194	−0.130597	−	0.991436i	$-0.541689\pi$
$883$	−4.71830e12	−0.261194	−0.130597	+	0.991436i	$0.541689\pi$
$884$	0	0
$885$	−1.02483e13	−0.561575
$886$	0	0
$887$	−1.26007e13	−0.683502	−0.341751	−	0.939790i	$-0.611020\pi$
$887$	−1.26007e13	−0.683502	−0.341751	+	0.939790i	$0.611020\pi$
$888$	0	0
$889$	−1.68084e13	−0.902544
$890$	0	0
$891$	1.03626e13	0.550832
$892$	0	0
$893$	−5.72163e12	−0.301084
$894$	0	0
$895$	−2.31527e13	−1.20614
$896$	0	0
$897$	1.63439e13	0.842927
$898$	0	0
$899$	−6.65550e11	−0.0339830
$900$	0	0
$901$	−4.59682e12	−0.232378
$902$	0	0
$903$	−9.00993e12	−0.450948
$904$	0	0
$905$	1.93251e13	0.957640
$906$	0	0
$907$	2.04459e13	1.00317	0.501584	−	0.865109i	$-0.332751\pi$
$907$	2.04459e13	1.00317	0.501584	+	0.865109i	$0.332751\pi$
$908$	0	0
$909$	3.15161e11	0.0153107
$910$	0	0
$911$	−6.21126e12	−0.298777	−0.149388	−	0.988779i	$-0.547730\pi$
$911$	−6.21126e12	−0.298777	−0.149388	+	0.988779i	$0.547730\pi$
$912$	0	0
$913$	2.55921e13	1.21895
$914$	0	0
$915$	−1.64895e12	−0.0777699
$916$	0	0
$917$	1.18323e13	0.552594
$918$	0	0
$919$	9.32576e12	0.431285	0.215642	−	0.976472i	$-0.430815\pi$
$919$	9.32576e12	0.431285	0.215642	+	0.976472i	$0.430815\pi$
$920$	0	0
$921$	−7.74206e12	−0.354559
$922$	0	0
$923$	−2.70542e11	−0.0122695
$924$	0	0
$925$	−1.11097e13	−0.498959
$926$	0	0
$927$	2.80428e12	0.124728
$928$	0	0
$929$	2.57119e13	1.13256	0.566282	−	0.824211i	$-0.308381\pi$
$929$	2.57119e13	1.13256	0.566282	+	0.824211i	$0.308381\pi$
$930$	0	0
$931$	−2.32920e12	−0.101609
$932$	0	0
$933$	1.08366e13	0.468194
$934$	0	0
$935$	4.10440e12	0.175629
$936$	0	0
$937$	7.29891e12	0.309336	0.154668	−	0.987967i	$-0.450569\pi$
$937$	7.29891e12	0.309336	0.154668	+	0.987967i	$0.450569\pi$
$938$	0	0
$939$	3.02357e13	1.26918
$940$	0	0
$941$	−1.10493e13	−0.459389	−0.229695	−	0.973263i	$-0.573773\pi$
$941$	−1.10493e13	−0.459389	−0.229695	+	0.973263i	$0.573773\pi$
$942$	0	0
$943$	3.19813e13	1.31702
$944$	0	0
$945$	1.71199e13	0.698324
$946$	0	0
$947$	9.10292e12	0.367795	0.183898	−	0.982945i	$-0.441129\pi$
$947$	9.10292e12	0.367795	0.183898	+	0.982945i	$0.441129\pi$
$948$	0	0
$949$	1.61861e13	0.647804
$950$	0	0
$951$	−5.22681e12	−0.207217
$952$	0	0
$953$	4.09267e13	1.60727	0.803635	−	0.595123i	$-0.202896\pi$
$953$	4.09267e13	1.60727	0.803635	+	0.595123i	$0.202896\pi$
$954$	0	0
$955$	1.81382e13	0.705633
$956$	0	0
$957$	−8.10479e11	−0.0312347
$958$	0	0
$959$	3.65152e13	1.39409
$960$	0	0
$961$	−8.16481e12	−0.308810
$962$	0	0
$963$	−8.68320e12	−0.325358
$964$	0	0
$965$	1.00708e13	0.373845
$966$	0	0
$967$	−5.16443e13	−1.89934	−0.949671	−	0.313248i	$-0.898583\pi$
$967$	−5.16443e13	−1.89934	−0.949671	+	0.313248i	$0.898583\pi$
$968$	0	0
$969$	1.66175e12	0.0605491
$970$	0	0
$971$	4.69234e13	1.69396	0.846979	−	0.531627i	$-0.178419\pi$
$971$	4.69234e13	1.69396	0.846979	+	0.531627i	$0.178419\pi$
$972$	0	0
$973$	1.10166e13	0.394040
$974$	0	0
$975$	−5.98953e12	−0.212262
$976$	0	0
$977$	−4.71530e13	−1.65571	−0.827854	−	0.560944i	$-0.810438\pi$
$977$	−4.71530e13	−1.65571	−0.827854	+	0.560944i	$0.810438\pi$
$978$	0	0
$979$	4.37545e13	1.52230
$980$	0	0
$981$	−1.43216e13	−0.493721
$982$	0	0
$983$	−4.10494e13	−1.40222	−0.701110	−	0.713053i	$-0.747312\pi$
$983$	−4.10494e13	−1.40222	−0.701110	+	0.713053i	$0.747312\pi$
$984$	0	0
$985$	−1.68986e12	−0.0571990
$986$	0	0
$987$	−2.02905e13	−0.680559
$988$	0	0
$989$	−2.99621e13	−0.995837
$990$	0	0
$991$	−8.40949e12	−0.276973	−0.138487	−	0.990364i	$-0.544224\pi$
$991$	−8.40949e12	−0.276973	−0.138487	+	0.990364i	$0.544224\pi$
$992$	0	0
$993$	−5.58905e12	−0.182417
$994$	0	0
$995$	1.96051e13	0.634110
$996$	0	0
$997$	2.54531e13	0.815854	0.407927	−	0.913015i	$-0.366252\pi$
$997$	2.54531e13	0.815854	0.407927	+	0.913015i	$0.366252\pi$
$998$	0	0
$999$	−4.53729e13	−1.44129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.g.1.5		7
4.3	odd	2		17.10.a.b.1.3	✓	7
12.11	even	2		153.10.a.f.1.5		7
68.67	odd	2		289.10.a.b.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.b.1.3	✓	7	4.3	odd	2
153.10.a.f.1.5		7	12.11	even	2
272.10.a.g.1.5		7	1.1	even	1	trivial
289.10.a.b.1.3		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)

\(f(q)\)	\(=\)	\(q+116.887 q^{3} -1103.40 q^{5} +5164.29 q^{7} -6020.47 q^{9} +O(q^{10})\)	\(q+116.887 q^{3} -1103.40 q^{5} +5164.29 q^{7} -6020.47 q^{9} -44537.1 q^{11} +69656.8 q^{13} -128973. q^{15} +83521.0 q^{17} +170217. q^{19} +603638. q^{21} +2.00737e6 q^{23} -735637. q^{25} -3.00440e6 q^{27} +155688. q^{29} -4.27490e6 q^{31} -5.20580e6 q^{33} -5.69827e6 q^{35} +1.51022e7 q^{37} +8.14197e6 q^{39} +1.59320e7 q^{41} -1.49261e7 q^{43} +6.64298e6 q^{45} -3.36137e7 q^{47} -1.36837e7 q^{49} +9.76250e6 q^{51} -5.50379e7 q^{53} +4.91421e7 q^{55} +1.98961e7 q^{57} +7.94611e7 q^{59} +1.27852e7 q^{61} -3.10915e7 q^{63} -7.68593e7 q^{65} -2.73169e8 q^{67} +2.34635e8 q^{69} -3.88392e6 q^{71} +2.32369e8 q^{73} -8.59862e7 q^{75} -2.30002e8 q^{77} -3.38948e8 q^{79} -2.32673e8 q^{81} -5.74624e8 q^{83} -9.21569e7 q^{85} +1.81978e7 q^{87} -9.82429e8 q^{89} +3.59728e8 q^{91} -4.99680e8 q^{93} -1.87817e8 q^{95} -1.03147e9 q^{97} +2.68134e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9}+O(q^{10}) \) 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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