LMFDB - Embedded newform 16.42.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,42,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 42, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 42);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 42 $
Character orbit:	$[\chi]$	$=$	16.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:	$170.354672730$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2784108376x + 1945534874860 $ x^3 - x^2 - 2784108376*x + 1945534874860
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{3}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$698.922$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.82217e9 q^{3} +9.10518e13 q^{5} +1.69829e16 q^{7} -2.18640e19 q^{9} +O(q^{10})$	$q-3.82217e9 q^{3} +9.10518e13 q^{5} +1.69829e16 q^{7} -2.18640e19 q^{9} +3.48555e21 q^{11} -1.03131e23 q^{13} -3.48015e23 q^{15} +1.01608e25 q^{17} +1.72697e26 q^{19} -6.49115e25 q^{21} -1.09881e28 q^{23} -3.71843e28 q^{25} +2.22974e29 q^{27} +1.22079e30 q^{29} -5.57694e29 q^{31} -1.33224e31 q^{33} +1.54633e30 q^{35} +1.14515e32 q^{37} +3.94185e32 q^{39} -6.17804e32 q^{41} +1.51443e32 q^{43} -1.99076e33 q^{45} +1.94739e34 q^{47} -4.42792e34 q^{49} -3.88362e34 q^{51} -1.38685e35 q^{53} +3.17366e35 q^{55} -6.60078e35 q^{57} -2.19974e36 q^{59} +2.83478e36 q^{61} -3.71315e35 q^{63} -9.39028e36 q^{65} -2.71602e37 q^{67} +4.19982e37 q^{69} -4.91576e37 q^{71} -2.29426e38 q^{73} +1.42125e38 q^{75} +5.91949e37 q^{77} -1.69414e38 q^{79} -5.47960e37 q^{81} +3.52307e39 q^{83} +9.25158e38 q^{85} -4.66608e39 q^{87} +6.37675e39 q^{89} -1.75147e39 q^{91} +2.13160e39 q^{93} +1.57244e40 q^{95} +3.79303e40 q^{97} -7.62083e40 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 10820953044 q^{3} - 212302350281550 q^{5} - 57\!\cdots\!92 q^{7}+ \cdots + 13\!\cdots\!19 q^{9}+O(q^{10}) $ 3 * q + 10820953044 * q^3 - 212302350281550 * q^5 - 57878416258239192 * q^7 + 13277004110931878919 * q^9	$ 3 q + 10820953044 q^{3} - 212302350281550 q^{5} - 57\!\cdots\!92 q^{7}+ \cdots - 45\!\cdots\!28 q^{99}+O(q^{100}) $ 3 * q + 10820953044 * q^3 - 212302350281550 * q^5 - 57878416258239192 * q^7 + 13277004110931878919 * q^9 + 3064080929502798535164 * q^11 - 98515763053518962936694 * q^13 - 2322231962154213782968200 * q^15 + 35582277130790298867301302 * q^17 + 233456308095243376865478180 * q^19 - 784315067894306487816760224 * q^21 - 2812708670812770779627026056 * q^23 - 12630087813294771903109831875 * q^25 - 40981593702376529051426180280 * q^27 - 12736322664295346653179017670 * q^29 + 55977552834365071134409017504 * q^31 - 13829226714693448959900097020528 * q^33 - 35330944415319146254058128592400 * q^35 + 4945769903781582545269841455122 * q^37 + 546470556552953849463965686898712 * q^39 - 3123151500646359746292805811918274 * q^41 - 1491980858362212236675453498061156 * q^43 - 3706804708974790460103758635956150 * q^45 + 63038106682044077762675687404413168 * q^47 - 97136113131667199274890334287198229 * q^49 + 157319866621422817701623275797858024 * q^51 + 79896749344137562310547533743021506 * q^53 + 1107430856215374719020572615529578600 * q^55 - 120534128028213908789981458568129040 * q^57 - 192512048683375799719343226678403860 * q^59 + 8740556115036354715092732710435455386 * q^61 - 4195060549596204507618981258341049336 * q^63 + 23272939413910136331858458170260680700 * q^65 - 11266872945514675454702655007279407852 * q^67 + 110718129232411678126981084073658946848 * q^69 + 140970915469820223119231433051520161384 * q^71 + 45825493210252135256288871746499029406 * q^73 + 247029389196459510298088488221820147500 * q^75 - 429017620126390531124795910645506217696 * q^77 + 520184835554750143118446358634240850320 * q^79 - 3191652631710460747876906046824191542997 * q^81 + 619777491338381742927488037142663372644 * q^83 - 106299433069331216483654652405294143100 * q^85 - 13958176049210326842816445061163645008040 * q^87 - 142389067128374093799211866144944441010 * q^89 - 25080596935320871627133801126628392473296 * q^91 + 3477058704647981071307340824432458334592 * q^93 + 33170437179755379504247502944935502503000 * q^95 + 116011691364103339137369463947331771078182 * q^97 - 45849669342635847189814810549498192693428 * 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
$3$	−3.82217e9	−0.632884	−0.316442	−	0.948612i	$-0.602488\pi$
$3$	−3.82217e9	−0.632884	−0.316442	+	0.948612i	$0.602488\pi$
$4$	0	0
$5$	9.10518e13	0.426976	0.213488	−	0.976946i	$-0.431518\pi$
$5$	9.10518e13	0.426976	0.213488	+	0.976946i	$0.431518\pi$
$6$	0	0
$7$	1.69829e16	0.0804457	0.0402228	−	0.999191i	$-0.487193\pi$
$7$	1.69829e16	0.0804457	0.0402228	+	0.999191i	$0.487193\pi$
$8$	0	0
$9$	−2.18640e19	−0.599458
$10$	0	0
$11$	3.48555e21	1.56215	0.781073	−	0.624440i	$-0.214673\pi$
$11$	3.48555e21	1.56215	0.781073	+	0.624440i	$0.214673\pi$
$12$	0	0
$13$	−1.03131e23	−1.50505	−0.752526	−	0.658563i	$-0.771165\pi$
$13$	−1.03131e23	−1.50505	−0.752526	+	0.658563i	$0.771165\pi$
$14$	0	0
$15$	−3.48015e23	−0.270226
$16$	0	0
$17$	1.01608e25	0.606352	0.303176	−	0.952935i	$-0.401953\pi$
$17$	1.01608e25	0.606352	0.303176	+	0.952935i	$0.401953\pi$
$18$	0	0
$19$	1.72697e26	1.05399	0.526995	−	0.849868i	$-0.323319\pi$
$19$	1.72697e26	1.05399	0.526995	+	0.849868i	$0.323319\pi$
$20$	0	0
$21$	−6.49115e25	−0.0509127
$22$	0	0
$23$	−1.09881e28	−1.33506	−0.667530	−	0.744583i	$-0.732648\pi$
$23$	−1.09881e28	−1.33506	−0.667530	+	0.744583i	$0.732648\pi$
$24$	0	0
$25$	−3.71843e28	−0.817692
$26$	0	0
$27$	2.22974e29	1.01227
$28$	0	0
$29$	1.22079e30	1.28081	0.640403	−	0.768039i	$-0.278767\pi$
$29$	1.22079e30	1.28081	0.640403	+	0.768039i	$0.278767\pi$
$30$	0	0
$31$	−5.57694e29	−0.149101	−0.0745506	−	0.997217i	$-0.523752\pi$
$31$	−5.57694e29	−0.149101	−0.0745506	+	0.997217i	$0.523752\pi$
$32$	0	0
$33$	−1.33224e31	−0.988657
$34$	0	0
$35$	1.54633e30	0.0343484
$36$	0	0
$37$	1.14515e32	0.814194	0.407097	−	0.913385i	$-0.366541\pi$
$37$	1.14515e32	0.814194	0.407097	+	0.913385i	$0.366541\pi$
$38$	0	0
$39$	3.94185e32	0.952522
$40$	0	0
$41$	−6.17804e32	−0.535527	−0.267764	−	0.963485i	$-0.586285\pi$
$41$	−6.17804e32	−0.535527	−0.267764	+	0.963485i	$0.586285\pi$
$42$	0	0
$43$	1.51443e32	0.0494478	0.0247239	−	0.999694i	$-0.492129\pi$
$43$	1.51443e32	0.0494478	0.0247239	+	0.999694i	$0.492129\pi$
$44$	0	0
$45$	−1.99076e33	−0.255954
$46$	0	0
$47$	1.94739e34	1.02671	0.513354	−	0.858177i	$-0.328403\pi$
$47$	1.94739e34	1.02671	0.513354	+	0.858177i	$0.328403\pi$
$48$	0	0
$49$	−4.42792e34	−0.993528
$50$	0	0
$51$	−3.88362e34	−0.383750
$52$	0	0
$53$	−1.38685e35	−0.622837	−0.311418	−	0.950273i	$-0.600804\pi$
$53$	−1.38685e35	−0.622837	−0.311418	+	0.950273i	$0.600804\pi$
$54$	0	0
$55$	3.17366e35	0.666999
$56$	0	0
$57$	−6.60078e35	−0.667053
$58$	0	0
$59$	−2.19974e36	−1.09624	−0.548119	−	0.836400i	$-0.684656\pi$
$59$	−2.19974e36	−1.09624	−0.548119	+	0.836400i	$0.684656\pi$
$60$	0	0
$61$	2.83478e36	0.713277	0.356639	−	0.934242i	$-0.383923\pi$
$61$	2.83478e36	0.713277	0.356639	+	0.934242i	$0.383923\pi$
$62$	0	0
$63$	−3.71315e35	−0.0482238
$64$	0	0
$65$	−9.39028e36	−0.642621
$66$	0	0
$67$	−2.71602e37	−0.998617	−0.499308	−	0.866424i	$-0.666413\pi$
$67$	−2.71602e37	−0.998617	−0.499308	+	0.866424i	$0.666413\pi$
$68$	0	0
$69$	4.19982e37	0.844937
$70$	0	0
$71$	−4.91576e37	−0.550548	−0.275274	−	0.961366i	$-0.588769\pi$
$71$	−4.91576e37	−0.550548	−0.275274	+	0.961366i	$0.588769\pi$
$72$	0	0
$73$	−2.29426e38	−1.45386	−0.726932	−	0.686709i	$-0.759055\pi$
$73$	−2.29426e38	−1.45386	−0.726932	+	0.686709i	$0.759055\pi$
$74$	0	0
$75$	1.42125e38	0.517504
$76$	0	0
$77$	5.91949e37	0.125668
$78$	0	0
$79$	−1.69414e38	−0.212615	−0.106307	−	0.994333i	$-0.533903\pi$
$79$	−1.69414e38	−0.212615	−0.106307	+	0.994333i	$0.533903\pi$
$80$	0	0
$81$	−5.47960e37	−0.0411913
$82$	0	0
$83$	3.52307e39	1.60628	0.803138	−	0.595793i	$-0.203162\pi$
$83$	3.52307e39	1.60628	0.803138	+	0.595793i	$0.203162\pi$
$84$	0	0
$85$	9.25158e38	0.258898
$86$	0	0
$87$	−4.66608e39	−0.810601
$88$	0	0
$89$	6.37675e39	0.695192	0.347596	−	0.937644i	$-0.386998\pi$
$89$	6.37675e39	0.695192	0.347596	+	0.937644i	$0.386998\pi$
$90$	0	0
$91$	−1.75147e39	−0.121075
$92$	0	0
$93$	2.13160e39	0.0943638
$94$	0	0
$95$	1.57244e40	0.450029
$96$	0	0
$97$	3.79303e40	0.708216	0.354108	−	0.935205i	$-0.384785\pi$
$97$	3.79303e40	0.708216	0.354108	+	0.935205i	$0.384785\pi$
$98$	0	0
$99$	−7.62083e40	−0.936441
$100$	0	0
$101$	−2.06559e41	−1.68444	−0.842222	−	0.539131i	$-0.818753\pi$
$101$	−2.06559e41	−1.68444	−0.842222	+	0.539131i	$0.818753\pi$
$102$	0	0
$103$	2.70863e40	0.147770	0.0738851	−	0.997267i	$-0.476460\pi$
$103$	2.70863e40	0.147770	0.0738851	+	0.997267i	$0.476460\pi$
$104$	0	0
$105$	−5.91031e39	−0.0217385
$106$	0	0
$107$	3.86524e41	0.965626	0.482813	−	0.875723i	$-0.339615\pi$
$107$	3.86524e41	0.965626	0.482813	+	0.875723i	$0.339615\pi$
$108$	0	0
$109$	5.41460e41	0.925387	0.462693	−	0.886518i	$-0.346883\pi$
$109$	5.41460e41	0.925387	0.462693	+	0.886518i	$0.346883\pi$
$110$	0	0
$111$	−4.37696e41	−0.515290
$112$	0	0
$113$	−1.99438e42	−1.62817	−0.814086	−	0.580744i	$-0.802762\pi$
$113$	−1.99438e42	−1.62817	−0.814086	+	0.580744i	$0.802762\pi$
$114$	0	0
$115$	−1.00048e42	−0.570038
$116$	0	0
$117$	2.25487e42	0.902216
$118$	0	0
$119$	1.72560e41	0.0487784
$120$	0	0
$121$	7.17056e42	1.44030
$122$	0	0
$123$	2.36135e42	0.338926
$124$	0	0
$125$	−7.52625e42	−0.776111
$126$	0	0
$127$	7.87891e42	0.586800	0.293400	−	0.955990i	$-0.405213\pi$
$127$	7.87891e42	0.586800	0.293400	+	0.955990i	$0.405213\pi$
$128$	0	0
$129$	−5.78842e41	−0.0312947
$130$	0	0
$131$	−8.75549e42	−0.345318	−0.172659	−	0.984982i	$-0.555236\pi$
$131$	−8.75549e42	−0.345318	−0.172659	+	0.984982i	$0.555236\pi$
$132$	0	0
$133$	2.93290e42	0.0847889
$134$	0	0
$135$	2.03022e43	0.432215
$136$	0	0
$137$	−1.30585e43	−0.205647	−0.102824	−	0.994700i	$-0.532788\pi$
$137$	−1.30585e43	−0.205647	−0.102824	+	0.994700i	$0.532788\pi$
$138$	0	0
$139$	7.47721e43	0.874855	0.437428	−	0.899254i	$-0.355890\pi$
$139$	7.47721e43	0.874855	0.437428	+	0.899254i	$0.355890\pi$
$140$	0	0
$141$	−7.44325e43	−0.649787
$142$	0	0
$143$	−3.59469e44	−2.35111
$144$	0	0
$145$	1.11155e44	0.546873
$146$	0	0
$147$	1.69243e44	0.628788
$148$	0	0
$149$	3.41118e44	0.960696	0.480348	−	0.877078i	$-0.340510\pi$
$149$	3.41118e44	0.960696	0.480348	+	0.877078i	$0.340510\pi$
$150$	0	0
$151$	4.86675e44	1.04282	0.521412	−	0.853305i	$-0.325405\pi$
$151$	4.86675e44	1.04282	0.521412	+	0.853305i	$0.325405\pi$
$152$	0	0
$153$	−2.22156e44	−0.363483
$154$	0	0
$155$	−5.07791e43	−0.0636627
$156$	0	0
$157$	1.02260e45	0.985744	0.492872	−	0.870102i	$-0.335947\pi$
$157$	1.02260e45	0.985744	0.492872	+	0.870102i	$0.335947\pi$
$158$	0	0
$159$	5.30077e44	0.394183
$160$	0	0
$161$	−1.86609e44	−0.107400
$162$	0	0
$163$	2.57464e45	1.15046	0.575228	−	0.817993i	$-0.304913\pi$
$163$	2.57464e45	1.15046	0.575228	+	0.817993i	$0.304913\pi$
$164$	0	0
$165$	−1.21302e45	−0.422133
$166$	0	0
$167$	1.90182e45	0.516991	0.258495	−	0.966013i	$-0.416773\pi$
$167$	1.90182e45	0.516991	0.258495	+	0.966013i	$0.416773\pi$
$168$	0	0
$169$	5.94059e45	1.26518
$170$	0	0
$171$	−3.77586e45	−0.631823
$172$	0	0
$173$	−1.22944e45	−0.162092	−0.0810462	−	0.996710i	$-0.525826\pi$
$173$	−1.22944e45	−0.162092	−0.0810462	+	0.996710i	$0.525826\pi$
$174$	0	0
$175$	−6.31498e44	−0.0657797
$176$	0	0
$177$	8.40776e45	0.693791
$178$	0	0
$179$	2.04820e46	1.34241	0.671206	−	0.741271i	$-0.265777\pi$
$179$	2.04820e46	1.34241	0.671206	+	0.741271i	$0.265777\pi$
$180$	0	0
$181$	1.96476e46	1.02541	0.512706	−	0.858564i	$-0.328643\pi$
$181$	1.96476e46	1.02541	0.512706	+	0.858564i	$0.328643\pi$
$182$	0	0
$183$	−1.08350e46	−0.451421
$184$	0	0
$185$	1.04268e46	0.347641
$186$	0	0
$187$	3.54160e46	0.947210
$188$	0	0
$189$	3.78675e45	0.0814328
$190$	0	0
$191$	8.46016e46	1.46620	0.733102	−	0.680118i	$-0.238072\pi$
$191$	8.46016e46	1.46620	0.733102	+	0.680118i	$0.238072\pi$
$192$	0	0
$193$	1.25564e46	0.175768	0.0878840	−	0.996131i	$-0.471990\pi$
$193$	1.25564e46	0.175768	0.0878840	+	0.996131i	$0.471990\pi$
$194$	0	0
$195$	3.58912e46	0.406704
$196$	0	0
$197$	1.78208e47	1.63821	0.819103	−	0.573647i	$-0.194472\pi$
$197$	1.78208e47	1.63821	0.819103	+	0.573647i	$0.194472\pi$
$198$	0	0
$199$	1.36946e47	1.02343	0.511717	−	0.859154i	$-0.329010\pi$
$199$	1.36946e47	1.02343	0.511717	+	0.859154i	$0.329010\pi$
$200$	0	0
$201$	1.03811e47	0.632008
$202$	0	0
$203$	2.07326e46	0.103035
$204$	0	0
$205$	−5.62521e46	−0.228657
$206$	0	0
$207$	2.40243e47	0.800312
$208$	0	0
$209$	6.01945e47	1.64649
$210$	0	0
$211$	−1.52351e47	−0.342812	−0.171406	−	0.985200i	$-0.554831\pi$
$211$	−1.52351e47	−0.342812	−0.171406	+	0.985200i	$0.554831\pi$
$212$	0	0
$213$	1.87889e47	0.348433
$214$	0	0
$215$	1.37892e46	0.0211130
$216$	0	0
$217$	−9.47128e45	−0.0119946
$218$	0	0
$219$	8.76904e47	0.920127
$220$	0	0
$221$	−1.04789e48	−0.912591
$222$	0	0
$223$	5.14471e47	0.372487	0.186243	−	0.982504i	$-0.440369\pi$
$223$	5.14471e47	0.372487	0.186243	+	0.982504i	$0.440369\pi$
$224$	0	0
$225$	8.12999e47	0.490172
$226$	0	0
$227$	−1.86061e47	−0.0935677	−0.0467838	−	0.998905i	$-0.514897\pi$
$227$	−1.86061e47	−0.0935677	−0.0467838	+	0.998905i	$0.514897\pi$
$228$	0	0
$229$	1.08478e48	0.455738	0.227869	−	0.973692i	$-0.426824\pi$
$229$	1.08478e48	0.455738	0.227869	+	0.973692i	$0.426824\pi$
$230$	0	0
$231$	−2.26253e47	−0.0795331
$232$	0	0
$233$	−1.31896e48	−0.388541	−0.194270	−	0.980948i	$-0.562234\pi$
$233$	−1.31896e48	−0.388541	−0.194270	+	0.980948i	$0.562234\pi$
$234$	0	0
$235$	1.77313e48	0.438380
$236$	0	0
$237$	6.47528e47	0.134560
$238$	0	0
$239$	2.46866e48	0.431821	0.215911	−	0.976413i	$-0.430728\pi$
$239$	2.46866e48	0.431821	0.215911	+	0.976413i	$0.430728\pi$
$240$	0	0
$241$	7.05413e48	1.04015	0.520073	−	0.854122i	$-0.325905\pi$
$241$	7.05413e48	1.04015	0.520073	+	0.854122i	$0.325905\pi$
$242$	0	0
$243$	−7.92309e48	−0.986202
$244$	0	0
$245$	−4.03170e48	−0.424213
$246$	0	0
$247$	−1.78105e49	−1.58631
$248$	0	0
$249$	−1.34657e49	−1.01659
$250$	0	0
$251$	1.10391e48	0.0707330	0.0353665	−	0.999374i	$-0.488740\pi$
$251$	1.10391e48	0.0707330	0.0353665	+	0.999374i	$0.488740\pi$
$252$	0	0
$253$	−3.82995e49	−2.08556
$254$	0	0
$255$	−3.53611e48	−0.163852
$256$	0	0
$257$	1.99173e49	0.786324	0.393162	−	0.919469i	$-0.371381\pi$
$257$	1.99173e49	0.786324	0.393162	+	0.919469i	$0.371381\pi$
$258$	0	0
$259$	1.94480e48	0.0654984
$260$	0	0
$261$	−2.66915e49	−0.767790
$262$	0	0
$263$	−3.95658e49	−0.973255	−0.486627	−	0.873610i	$-0.661773\pi$
$263$	−3.95658e49	−0.973255	−0.486627	+	0.873610i	$0.661773\pi$
$264$	0	0
$265$	−1.26275e49	−0.265936
$266$	0	0
$267$	−2.43730e49	−0.439975
$268$	0	0
$269$	−7.39690e49	−1.14585	−0.572926	−	0.819607i	$-0.694192\pi$
$269$	−7.39690e49	−1.14585	−0.572926	+	0.819607i	$0.694192\pi$
$270$	0	0
$271$	2.66721e49	0.354966	0.177483	−	0.984124i	$-0.443205\pi$
$271$	2.66721e49	0.354966	0.177483	+	0.984124i	$0.443205\pi$
$272$	0	0
$273$	6.69441e48	0.0766263
$274$	0	0
$275$	−1.29608e50	−1.27735
$276$	0	0
$277$	1.32709e50	1.12737	0.563684	−	0.825991i	$-0.309384\pi$
$277$	1.32709e50	1.12737	0.563684	+	0.825991i	$0.309384\pi$
$278$	0	0
$279$	1.21935e49	0.0893800
$280$	0	0
$281$	1.25230e49	0.0792918	0.0396459	−	0.999214i	$-0.487377\pi$
$281$	1.25230e49	0.0792918	0.0396459	+	0.999214i	$0.487377\pi$
$282$	0	0
$283$	9.97157e49	0.545936	0.272968	−	0.962023i	$-0.411995\pi$
$283$	9.97157e49	0.545936	0.272968	+	0.962023i	$0.411995\pi$
$284$	0	0
$285$	−6.01013e49	−0.284816
$286$	0	0
$287$	−1.04921e49	−0.0430808
$288$	0	0
$289$	−1.77564e50	−0.632337
$290$	0	0
$291$	−1.44976e50	−0.448218
$292$	0	0
$293$	−3.65149e50	−0.981035	−0.490517	−	0.871431i	$-0.663192\pi$
$293$	−3.65149e50	−0.981035	−0.490517	+	0.871431i	$0.663192\pi$
$294$	0	0
$295$	−2.00290e50	−0.468067
$296$	0	0
$297$	7.77187e50	1.58131
$298$	0	0
$299$	1.13321e51	2.00933
$300$	0	0
$301$	2.57195e48	0.00397786
$302$	0	0
$303$	7.89504e50	1.06606
$304$	0	0
$305$	2.58112e50	0.304552
$306$	0	0
$307$	−1.30418e51	−1.34586	−0.672932	−	0.739705i	$-0.734965\pi$
$307$	−1.30418e51	−1.34586	−0.672932	+	0.739705i	$0.734965\pi$
$308$	0	0
$309$	−1.03528e50	−0.0935214
$310$	0	0
$311$	−3.10188e50	−0.245492	−0.122746	−	0.992438i	$-0.539170\pi$
$311$	−3.10188e50	−0.245492	−0.122746	+	0.992438i	$0.539170\pi$
$312$	0	0
$313$	9.82549e50	0.681860	0.340930	−	0.940089i	$-0.389258\pi$
$313$	9.82549e50	0.681860	0.340930	+	0.940089i	$0.389258\pi$
$314$	0	0
$315$	−3.38089e49	−0.0205904
$316$	0	0
$317$	2.98911e51	1.59892	0.799462	−	0.600717i	$-0.205118\pi$
$317$	2.98911e51	1.59892	0.799462	+	0.600717i	$0.205118\pi$
$318$	0	0
$319$	4.25514e51	2.00081
$320$	0	0
$321$	−1.47736e51	−0.611129
$322$	0	0
$323$	1.75474e51	0.639089
$324$	0	0
$325$	3.83486e51	1.23067
$326$	0	0
$327$	−2.06955e51	−0.585662
$328$	0	0
$329$	3.30724e50	0.0825942
$330$	0	0
$331$	3.54686e51	0.782294	0.391147	−	0.920328i	$-0.372078\pi$
$331$	3.54686e51	0.782294	0.391147	+	0.920328i	$0.372078\pi$
$332$	0	0
$333$	−2.50377e51	−0.488076
$334$	0	0
$335$	−2.47298e51	−0.426385
$336$	0	0
$337$	−3.30444e51	−0.504295	−0.252147	−	0.967689i	$-0.581137\pi$
$337$	−3.30444e51	−0.504295	−0.252147	+	0.967689i	$0.581137\pi$
$338$	0	0
$339$	7.62286e51	1.03044
$340$	0	0
$341$	−1.94387e51	−0.232918
$342$	0	0
$343$	−1.50888e51	−0.160371
$344$	0	0
$345$	3.82401e51	0.360768
$346$	0	0
$347$	−1.33458e52	−1.11838	−0.559189	−	0.829040i	$-0.688887\pi$
$347$	−1.33458e52	−1.11838	−0.559189	+	0.829040i	$0.688887\pi$
$348$	0	0
$349$	5.90464e51	0.439817	0.219908	−	0.975521i	$-0.429424\pi$
$349$	5.90464e51	0.439817	0.219908	+	0.975521i	$0.429424\pi$
$350$	0	0
$351$	−2.29956e52	−1.52352
$352$	0	0
$353$	−2.48492e52	−1.46531	−0.732657	−	0.680598i	$-0.761720\pi$
$353$	−2.48492e52	−1.46531	−0.732657	+	0.680598i	$0.761720\pi$
$354$	0	0
$355$	−4.47589e51	−0.235071
$356$	0	0
$357$	−6.59553e50	−0.0308710
$358$	0	0
$359$	2.54547e52	1.06251	0.531254	−	0.847213i	$-0.321721\pi$
$359$	2.54547e52	1.06251	0.531254	+	0.847213i	$0.321721\pi$
$360$	0	0
$361$	2.97723e51	0.110896
$362$	0	0
$363$	−2.74071e52	−0.911542
$364$	0	0
$365$	−2.08897e52	−0.620765
$366$	0	0
$367$	−8.12057e51	−0.215741	−0.107870	−	0.994165i	$-0.534403\pi$
$367$	−8.12057e51	−0.215741	−0.107870	+	0.994165i	$0.534403\pi$
$368$	0	0
$369$	1.35077e52	0.321026
$370$	0	0
$371$	−2.35528e51	−0.0501045
$372$	0	0
$373$	9.33073e51	0.177781	0.0888903	−	0.996041i	$-0.471668\pi$
$373$	9.33073e51	0.177781	0.0888903	+	0.996041i	$0.471668\pi$
$374$	0	0
$375$	2.87666e52	0.491188
$376$	0	0
$377$	−1.25902e53	−1.92768
$378$	0	0
$379$	−2.43584e52	−0.334614	−0.167307	−	0.985905i	$-0.553507\pi$
$379$	−2.43584e52	−0.334614	−0.167307	+	0.985905i	$0.553507\pi$
$380$	0	0
$381$	−3.01145e52	−0.371376
$382$	0	0
$383$	−5.83695e52	−0.646564	−0.323282	−	0.946303i	$-0.604786\pi$
$383$	−5.83695e52	−0.646564	−0.323282	+	0.946303i	$0.604786\pi$
$384$	0	0
$385$	5.38980e51	0.0536571
$386$	0	0
$387$	−3.31116e51	−0.0296419
$388$	0	0
$389$	1.42025e53	1.14393	0.571964	−	0.820279i	$-0.306182\pi$
$389$	1.42025e53	1.14393	0.571964	+	0.820279i	$0.306182\pi$
$390$	0	0
$391$	−1.11647e53	−0.809516
$392$	0	0
$393$	3.34649e52	0.218546
$394$	0	0
$395$	−1.54254e52	−0.0907814
$396$	0	0
$397$	−2.90471e53	−1.54133	−0.770667	−	0.637238i	$-0.780077\pi$
$397$	−2.90471e53	−1.54133	−0.770667	+	0.637238i	$0.780077\pi$
$398$	0	0
$399$	−1.12100e52	−0.0536615
$400$	0	0
$401$	2.01288e53	0.869677	0.434839	−	0.900508i	$-0.356805\pi$
$401$	2.01288e53	0.869677	0.434839	+	0.900508i	$0.356805\pi$
$402$	0	0
$403$	5.75157e52	0.224405
$404$	0	0
$405$	−4.98927e51	−0.0175877
$406$	0	0
$407$	3.99149e53	1.27189
$408$	0	0
$409$	6.23107e53	1.79570	0.897852	−	0.440298i	$-0.145127\pi$
$409$	6.23107e53	1.79570	0.897852	+	0.440298i	$0.145127\pi$
$410$	0	0
$411$	4.99119e52	0.130151
$412$	0	0
$413$	−3.73580e52	−0.0881876
$414$	0	0
$415$	3.20782e53	0.685841
$416$	0	0
$417$	−2.85791e53	−0.553682
$418$	0	0
$419$	7.69430e53	1.35139	0.675696	−	0.737180i	$-0.263843\pi$
$419$	7.69430e53	1.35139	0.675696	+	0.737180i	$0.263843\pi$
$420$	0	0
$421$	3.14569e53	0.501110	0.250555	−	0.968102i	$-0.419387\pi$
$421$	3.14569e53	0.501110	0.250555	+	0.968102i	$0.419387\pi$
$422$	0	0
$423$	−4.25779e53	−0.615469
$424$	0	0
$425$	−3.77822e53	−0.495809
$426$	0	0
$427$	4.81429e52	0.0573800
$428$	0	0
$429$	1.37395e54	1.48798
$430$	0	0
$431$	5.35565e53	0.527264	0.263632	−	0.964623i	$-0.415079\pi$
$431$	5.35565e53	0.527264	0.263632	+	0.964623i	$0.415079\pi$
$432$	0	0
$433$	1.35139e54	1.20999	0.604993	−	0.796231i	$-0.293176\pi$
$433$	1.35139e54	1.20999	0.604993	+	0.796231i	$0.293176\pi$
$434$	0	0
$435$	−4.24855e53	−0.346107
$436$	0	0
$437$	−1.89761e54	−1.40714
$438$	0	0
$439$	−2.63178e52	−0.0177716	−0.00888582	−	0.999961i	$-0.502828\pi$
$439$	−2.63178e52	−0.0177716	−0.00888582	+	0.999961i	$0.502828\pi$
$440$	0	0
$441$	9.68123e53	0.595579
$442$	0	0
$443$	9.66050e53	0.541656	0.270828	−	0.962628i	$-0.412703\pi$
$443$	9.66050e53	0.541656	0.270828	+	0.962628i	$0.412703\pi$
$444$	0	0
$445$	5.80615e53	0.296830
$446$	0	0
$447$	−1.30381e54	−0.608009
$448$	0	0
$449$	−5.56332e53	−0.236747	−0.118373	−	0.992969i	$-0.537768\pi$
$449$	−5.56332e53	−0.236747	−0.118373	+	0.992969i	$0.537768\pi$
$450$	0	0
$451$	−2.15339e54	−0.836571
$452$	0	0
$453$	−1.86015e54	−0.659987
$454$	0	0
$455$	−1.59474e53	−0.0516961
$456$	0	0
$457$	−2.92141e54	−0.865587	−0.432794	−	0.901493i	$-0.642472\pi$
$457$	−2.92141e54	−0.865587	−0.432794	+	0.901493i	$0.642472\pi$
$458$	0	0
$459$	2.26559e54	0.613792
$460$	0	0
$461$	−1.66774e54	−0.413295	−0.206648	−	0.978415i	$-0.566255\pi$
$461$	−1.66774e54	−0.413295	−0.206648	+	0.978415i	$0.566255\pi$
$462$	0	0
$463$	2.72429e54	0.617793	0.308896	−	0.951096i	$-0.400040\pi$
$463$	2.72429e54	0.617793	0.308896	+	0.951096i	$0.400040\pi$
$464$	0	0
$465$	1.94086e53	0.0402911
$466$	0	0
$467$	−4.71337e54	−0.896057	−0.448028	−	0.894019i	$-0.647874\pi$
$467$	−4.71337e54	−0.896057	−0.448028	+	0.894019i	$0.647874\pi$
$468$	0	0
$469$	−4.61259e53	−0.0803344
$470$	0	0
$471$	−3.90856e54	−0.623861
$472$	0	0
$473$	5.27864e53	0.0772447
$474$	0	0
$475$	−6.42163e54	−0.861839
$476$	0	0
$477$	3.03222e54	0.373365
$478$	0	0
$479$	6.42979e54	0.726639	0.363320	−	0.931665i	$-0.381643\pi$
$479$	6.42979e54	0.726639	0.363320	+	0.931665i	$0.381643\pi$
$480$	0	0
$481$	−1.18101e55	−1.22540
$482$	0	0
$483$	7.13252e53	0.0679715
$484$	0	0
$485$	3.45362e54	0.302391
$486$	0	0
$487$	−2.20621e55	−1.77543	−0.887714	−	0.460396i	$-0.847708\pi$
$487$	−2.20621e55	−1.77543	−0.887714	+	0.460396i	$0.847708\pi$
$488$	0	0
$489$	−9.84068e54	−0.728104
$490$	0	0
$491$	2.26714e54	0.154279	0.0771395	−	0.997020i	$-0.475421\pi$
$491$	2.26714e54	0.154279	0.0771395	+	0.997020i	$0.475421\pi$
$492$	0	0
$493$	1.24042e55	0.776619
$494$	0	0
$495$	−6.93890e54	−0.399838
$496$	0	0
$497$	−8.34840e53	−0.0442892
$498$	0	0
$499$	2.08851e54	0.102041	0.0510205	−	0.998698i	$-0.483753\pi$
$499$	2.08851e54	0.102041	0.0510205	+	0.998698i	$0.483753\pi$
$500$	0	0
$501$	−7.26909e54	−0.327195
$502$	0	0
$503$	4.18071e55	1.73423	0.867115	−	0.498107i	$-0.165971\pi$
$503$	4.18071e55	1.73423	0.867115	+	0.498107i	$0.165971\pi$
$504$	0	0
$505$	−1.88076e55	−0.719217
$506$	0	0
$507$	−2.27059e55	−0.800712
$508$	0	0
$509$	−4.54501e55	−1.47850	−0.739248	−	0.673433i	$-0.764819\pi$
$509$	−4.54501e55	−1.47850	−0.739248	+	0.673433i	$0.764819\pi$
$510$	0	0
$511$	−3.89632e54	−0.116957
$512$	0	0
$513$	3.85070e55	1.06692
$514$	0	0
$515$	2.46626e54	0.0630943
$516$	0	0
$517$	6.78774e55	1.60387
$518$	0	0
$519$	4.69913e54	0.102586
$520$	0	0
$521$	4.52665e55	0.913278	0.456639	−	0.889652i	$-0.349053\pi$
$521$	4.52665e55	0.913278	0.456639	+	0.889652i	$0.349053\pi$
$522$	0	0
$523$	1.00124e56	1.86748	0.933738	−	0.357957i	$-0.116527\pi$
$523$	1.00124e56	1.86748	0.933738	+	0.357957i	$0.116527\pi$
$524$	0	0
$525$	2.41369e54	0.0416309
$526$	0	0
$527$	−5.66662e54	−0.0904078
$528$	0	0
$529$	5.29981e55	0.782383
$530$	0	0
$531$	4.80952e55	0.657149
$532$	0	0
$533$	6.37148e55	0.805996
$534$	0	0
$535$	3.51937e55	0.412299
$536$	0	0
$537$	−7.82857e55	−0.849590
$538$	0	0
$539$	−1.54338e56	−1.55204
$540$	0	0
$541$	1.94080e55	0.180899	0.0904495	−	0.995901i	$-0.471170\pi$
$541$	1.94080e55	0.180899	0.0904495	+	0.995901i	$0.471170\pi$
$542$	0	0
$543$	−7.50963e55	−0.648966
$544$	0	0
$545$	4.93009e55	0.395118
$546$	0	0
$547$	1.89248e56	1.40699	0.703495	−	0.710700i	$-0.251622\pi$
$547$	1.89248e56	1.40699	0.703495	+	0.710700i	$0.251622\pi$
$548$	0	0
$549$	−6.19798e55	−0.427580
$550$	0	0
$551$	2.10828e56	1.34996
$552$	0	0
$553$	−2.87714e54	−0.0171039
$554$	0	0
$555$	−3.98530e55	−0.220017
$556$	0	0
$557$	−4.31584e55	−0.221327	−0.110663	−	0.993858i	$-0.535298\pi$
$557$	−4.31584e55	−0.221327	−0.110663	+	0.993858i	$0.535298\pi$
$558$	0	0
$559$	−1.56185e55	−0.0744215
$560$	0	0
$561$	−1.35366e56	−0.599474
$562$	0	0
$563$	−3.50637e56	−1.44356	−0.721781	−	0.692121i	$-0.756676\pi$
$563$	−3.50637e56	−1.44356	−0.721781	+	0.692121i	$0.756676\pi$
$564$	0	0
$565$	−1.81592e56	−0.695191
$566$	0	0
$567$	−9.30596e53	−0.00331366
$568$	0	0
$569$	3.92278e56	1.29955	0.649775	−	0.760127i	$-0.274863\pi$
$569$	3.92278e56	1.29955	0.649775	+	0.760127i	$0.274863\pi$
$570$	0	0
$571$	−3.12164e56	−0.962370	−0.481185	−	0.876619i	$-0.659794\pi$
$571$	−3.12164e56	−0.962370	−0.481185	+	0.876619i	$0.659794\pi$
$572$	0	0
$573$	−3.23361e56	−0.927937
$574$	0	0
$575$	4.08584e56	1.09167
$576$	0	0
$577$	5.15710e55	0.128322	0.0641611	−	0.997940i	$-0.479563\pi$
$577$	5.15710e55	0.128322	0.0641611	+	0.997940i	$0.479563\pi$
$578$	0	0
$579$	−4.79926e55	−0.111241
$580$	0	0
$581$	5.98320e55	0.129218
$582$	0	0
$583$	−4.83394e56	−0.972962
$584$	0	0
$585$	2.05310e56	0.385224
$586$	0	0
$587$	1.62999e56	0.285170	0.142585	−	0.989783i	$-0.454459\pi$
$587$	1.62999e56	0.285170	0.142585	+	0.989783i	$0.454459\pi$
$588$	0	0
$589$	−9.63123e55	−0.157151
$590$	0	0
$591$	−6.81139e56	−1.03679
$592$	0	0
$593$	−6.76525e56	−0.960866	−0.480433	−	0.877031i	$-0.659520\pi$
$593$	−6.76525e56	−0.960866	−0.480433	+	0.877031i	$0.659520\pi$
$594$	0	0
$595$	1.57119e55	0.0208272
$596$	0	0
$597$	−5.23429e56	−0.647715
$598$	0	0
$599$	−8.22842e56	−0.950751	−0.475376	−	0.879783i	$-0.657688\pi$
$599$	−8.22842e56	−0.950751	−0.475376	+	0.879783i	$0.657688\pi$
$600$	0	0
$601$	−1.40310e57	−1.51412	−0.757062	−	0.653343i	$-0.773366\pi$
$601$	−1.40310e57	−1.51412	−0.757062	+	0.653343i	$0.773366\pi$
$602$	0	0
$603$	5.93831e56	0.598629
$604$	0	0
$605$	6.52892e56	0.614973
$606$	0	0
$607$	1.85452e57	1.63254	0.816271	−	0.577669i	$-0.196038\pi$
$607$	1.85452e57	1.63254	0.816271	+	0.577669i	$0.196038\pi$
$608$	0	0
$609$	−7.92436e55	−0.0652093
$610$	0	0
$611$	−2.00837e57	−1.54525
$612$	0	0
$613$	1.52593e57	1.09798	0.548992	−	0.835828i	$-0.315012\pi$
$613$	1.52593e57	1.09798	0.548992	+	0.835828i	$0.315012\pi$
$614$	0	0
$615$	2.15005e56	0.144713
$616$	0	0
$617$	2.09012e56	0.131621	0.0658105	−	0.997832i	$-0.479037\pi$
$617$	2.09012e56	0.131621	0.0658105	+	0.997832i	$0.479037\pi$
$618$	0	0
$619$	5.03746e56	0.296861	0.148431	−	0.988923i	$-0.452578\pi$
$619$	5.03746e56	0.296861	0.148431	+	0.988923i	$0.452578\pi$
$620$	0	0
$621$	−2.45005e57	−1.35144
$622$	0	0
$623$	1.08296e56	0.0559251
$624$	0	0
$625$	1.00567e57	0.486311
$626$	0	0
$627$	−2.30074e57	−1.04203
$628$	0	0
$629$	1.16357e57	0.493688
$630$	0	0
$631$	2.05980e57	0.818887	0.409444	−	0.912335i	$-0.365723\pi$
$631$	2.05980e57	0.818887	0.409444	+	0.912335i	$0.365723\pi$
$632$	0	0
$633$	5.82312e56	0.216960
$634$	0	0
$635$	7.17389e56	0.250549
$636$	0	0
$637$	4.56657e57	1.49531
$638$	0	0
$639$	1.07478e57	0.330030
$640$	0	0
$641$	4.32782e57	1.24646	0.623232	−	0.782037i	$-0.285819\pi$
$641$	4.32782e57	1.24646	0.623232	+	0.782037i	$0.285819\pi$
$642$	0	0
$643$	2.48467e57	0.671341	0.335670	−	0.941980i	$-0.391037\pi$
$643$	2.48467e57	0.671341	0.335670	+	0.941980i	$0.391037\pi$
$644$	0	0
$645$	−5.27046e55	−0.0133621
$646$	0	0
$647$	4.23499e57	1.00766	0.503831	−	0.863802i	$-0.331923\pi$
$647$	4.23499e57	1.00766	0.503831	+	0.863802i	$0.331923\pi$
$648$	0	0
$649$	−7.66730e57	−1.71248
$650$	0	0
$651$	3.62008e55	0.00759115
$652$	0	0
$653$	−1.80067e57	−0.354580	−0.177290	−	0.984159i	$-0.556733\pi$
$653$	−1.80067e57	−0.354580	−0.177290	+	0.984159i	$0.556733\pi$
$654$	0	0
$655$	−7.97203e56	−0.147442
$656$	0	0
$657$	5.01618e57	0.871532
$658$	0	0
$659$	6.74103e57	1.10046	0.550231	−	0.835012i	$-0.314540\pi$
$659$	6.74103e57	1.10046	0.550231	+	0.835012i	$0.314540\pi$
$660$	0	0
$661$	9.42263e57	1.44558	0.722790	−	0.691068i	$-0.242859\pi$
$661$	9.42263e57	1.44558	0.722790	+	0.691068i	$0.242859\pi$
$662$	0	0
$663$	4.00523e57	0.577564
$664$	0	0
$665$	2.67046e56	0.0362028
$666$	0	0
$667$	−1.34142e58	−1.70995
$668$	0	0
$669$	−1.96639e57	−0.235741
$670$	0	0
$671$	9.88079e57	1.11424
$672$	0	0
$673$	−7.22539e57	−0.766571	−0.383285	−	0.923630i	$-0.625207\pi$
$673$	−7.22539e57	−0.766571	−0.383285	+	0.923630i	$0.625207\pi$
$674$	0	0
$675$	−8.29113e57	−0.827725
$676$	0	0
$677$	−9.31845e57	−0.875540	−0.437770	−	0.899087i	$-0.644232\pi$
$677$	−9.31845e57	−0.875540	−0.437770	+	0.899087i	$0.644232\pi$
$678$	0	0
$679$	6.44167e56	0.0569729
$680$	0	0
$681$	7.11157e56	0.0592175
$682$	0	0
$683$	−1.54469e58	−1.21121	−0.605603	−	0.795767i	$-0.707068\pi$
$683$	−1.54469e58	−1.21121	−0.605603	+	0.795767i	$0.707068\pi$
$684$	0	0
$685$	−1.18900e57	−0.0878064
$686$	0	0
$687$	−4.14622e57	−0.288429
$688$	0	0
$689$	1.43028e58	0.937401
$690$	0	0
$691$	−1.40721e58	−0.869078	−0.434539	−	0.900653i	$-0.643089\pi$
$691$	−1.40721e58	−0.869078	−0.434539	+	0.900653i	$0.643089\pi$
$692$	0	0
$693$	−1.29424e57	−0.0753326
$694$	0	0
$695$	6.80814e57	0.373542
$696$	0	0
$697$	−6.27738e57	−0.324718
$698$	0	0
$699$	5.04130e57	0.245901
$700$	0	0
$701$	−2.16291e58	−0.994991	−0.497495	−	0.867467i	$-0.665747\pi$
$701$	−2.16291e58	−0.994991	−0.497495	+	0.867467i	$0.665747\pi$
$702$	0	0
$703$	1.97765e58	0.858153
$704$	0	0
$705$	−6.77722e57	−0.277443
$706$	0	0
$707$	−3.50798e57	−0.135506
$708$	0	0
$709$	−1.27581e58	−0.465090	−0.232545	−	0.972586i	$-0.574705\pi$
$709$	−1.27581e58	−0.465090	−0.232545	+	0.972586i	$0.574705\pi$
$710$	0	0
$711$	3.70407e57	0.127454
$712$	0	0
$713$	6.12798e57	0.199059
$714$	0	0
$715$	−3.27303e58	−1.00387
$716$	0	0
$717$	−9.43563e57	−0.273293
$718$	0	0
$719$	−2.41677e58	−0.661138	−0.330569	−	0.943782i	$-0.607241\pi$
$719$	−2.41677e58	−0.661138	−0.330569	+	0.943782i	$0.607241\pi$
$720$	0	0
$721$	4.60005e56	0.0118875
$722$	0	0
$723$	−2.69621e58	−0.658291
$724$	0	0
$725$	−4.53944e58	−1.04730
$726$	0	0
$727$	4.92924e58	1.07479	0.537396	−	0.843330i	$-0.319408\pi$
$727$	4.92924e58	1.07479	0.537396	+	0.843330i	$0.319408\pi$
$728$	0	0
$729$	3.22819e58	0.665342
$730$	0	0
$731$	1.53878e57	0.0299828
$732$	0	0
$733$	4.81159e58	0.886456	0.443228	−	0.896409i	$-0.353833\pi$
$733$	4.81159e58	0.886456	0.443228	+	0.896409i	$0.353833\pi$
$734$	0	0
$735$	1.54098e58	0.268477
$736$	0	0
$737$	−9.46682e58	−1.55999
$738$	0	0
$739$	−9.13074e58	−1.42330	−0.711648	−	0.702537i	$-0.752051\pi$
$739$	−9.13074e58	−1.42330	−0.711648	+	0.702537i	$0.752051\pi$
$740$	0	0
$741$	6.80746e58	1.00395
$742$	0	0
$743$	−1.50391e57	−0.0209870	−0.0104935	−	0.999945i	$-0.503340\pi$
$743$	−1.50391e57	−0.0209870	−0.0104935	+	0.999945i	$0.503340\pi$
$744$	0	0
$745$	3.10594e58	0.410194
$746$	0	0
$747$	−7.70285e58	−0.962895
$748$	0	0
$749$	6.56431e57	0.0776804
$750$	0	0
$751$	1.25195e59	1.40272	0.701358	−	0.712809i	$-0.252577\pi$
$751$	1.25195e59	1.40272	0.701358	+	0.712809i	$0.252577\pi$
$752$	0	0
$753$	−4.21933e57	−0.0447658
$754$	0	0
$755$	4.43126e58	0.445261
$756$	0	0
$757$	1.29607e58	0.123357	0.0616784	−	0.998096i	$-0.480355\pi$
$757$	1.29607e58	0.123357	0.0616784	+	0.998096i	$0.480355\pi$
$758$	0	0
$759$	1.46387e59	1.31991
$760$	0	0
$761$	6.14300e58	0.524801	0.262401	−	0.964959i	$-0.415486\pi$
$761$	6.14300e58	0.524801	0.262401	+	0.964959i	$0.415486\pi$
$762$	0	0
$763$	9.19557e57	0.0744433
$764$	0	0
$765$	−2.02277e58	−0.155198
$766$	0	0
$767$	2.26862e59	1.64989
$768$	0	0
$769$	1.38280e59	0.953388	0.476694	−	0.879069i	$-0.341835\pi$
$769$	1.38280e59	0.953388	0.476694	+	0.879069i	$0.341835\pi$
$770$	0	0
$771$	−7.61272e58	−0.497651
$772$	0	0
$773$	1.19232e59	0.739119	0.369560	−	0.929207i	$-0.379509\pi$
$773$	1.19232e59	0.739119	0.369560	+	0.929207i	$0.379509\pi$
$774$	0	0
$775$	2.07375e58	0.121919
$776$	0	0
$777$	−7.43336e57	−0.0414529
$778$	0	0
$779$	−1.06693e59	−0.564440
$780$	0	0
$781$	−1.71342e59	−0.860036
$782$	0	0
$783$	2.72205e59	1.29652
$784$	0	0
$785$	9.31098e58	0.420889
$786$	0	0
$787$	−2.56981e59	−1.10260	−0.551301	−	0.834307i	$-0.685868\pi$
$787$	−2.56981e59	−1.10260	−0.551301	+	0.834307i	$0.685868\pi$
$788$	0	0
$789$	1.51227e59	0.615957
$790$	0	0
$791$	−3.38704e58	−0.130979
$792$	0	0
$793$	−2.92355e59	−1.07352
$794$	0	0
$795$	4.82645e58	0.168307
$796$	0	0
$797$	5.39123e59	1.78563	0.892816	−	0.450421i	$-0.148726\pi$
$797$	5.39123e59	1.78563	0.892816	+	0.450421i	$0.148726\pi$
$798$	0	0
$799$	1.97870e59	0.622547
$800$	0	0
$801$	−1.39422e59	−0.416738
$802$	0	0
$803$	−7.99677e59	−2.27115
$804$	0	0
$805$	−1.69911e58	−0.0458571
$806$	0	0
$807$	2.82722e59	0.725190
$808$	0	0
$809$	−4.68209e59	−1.14155	−0.570775	−	0.821106i	$-0.693357\pi$
$809$	−4.68209e59	−1.14155	−0.570775	+	0.821106i	$0.693357\pi$
$810$	0	0
$811$	−5.03719e58	−0.116751	−0.0583756	−	0.998295i	$-0.518592\pi$
$811$	−5.03719e58	−0.116751	−0.0583756	+	0.998295i	$0.518592\pi$
$812$	0	0
$813$	−1.01945e59	−0.224652
$814$	0	0
$815$	2.34425e59	0.491217
$816$	0	0
$817$	2.61539e58	0.0521175
$818$	0	0
$819$	3.82942e58	0.0725793
$820$	0	0
$821$	−7.92919e59	−1.42953	−0.714766	−	0.699363i	$-0.753467\pi$
$821$	−7.92919e59	−1.42953	−0.714766	+	0.699363i	$0.753467\pi$
$822$	0	0
$823$	−5.91534e58	−0.101457	−0.0507287	−	0.998712i	$-0.516154\pi$
$823$	−5.91534e58	−0.101457	−0.0507287	+	0.998712i	$0.516154\pi$
$824$	0	0
$825$	4.95383e59	0.808416
$826$	0	0
$827$	−9.18303e59	−1.42601	−0.713006	−	0.701158i	$-0.752667\pi$
$827$	−9.18303e59	−1.42601	−0.713006	+	0.701158i	$0.752667\pi$
$828$	0	0
$829$	−6.10924e59	−0.902858	−0.451429	−	0.892307i	$-0.649086\pi$
$829$	−6.10924e59	−0.902858	−0.451429	+	0.892307i	$0.649086\pi$
$830$	0	0
$831$	−5.07236e59	−0.713493
$832$	0	0
$833$	−4.49912e59	−0.602428
$834$	0	0
$835$	1.73164e59	0.220743
$836$	0	0
$837$	−1.24351e59	−0.150931
$838$	0	0
$839$	1.07495e60	1.24242	0.621210	−	0.783644i	$-0.286641\pi$
$839$	1.07495e60	1.24242	0.621210	+	0.783644i	$0.286641\pi$
$840$	0	0
$841$	5.81851e59	0.640463
$842$	0	0
$843$	−4.78649e58	−0.0501825
$844$	0	0
$845$	5.40902e59	0.540202
$846$	0	0
$847$	1.21777e59	0.115866
$848$	0	0
$849$	−3.81130e59	−0.345514
$850$	0	0
$851$	−1.25830e60	−1.08700
$852$	0	0
$853$	−7.04560e59	−0.580047	−0.290023	−	0.957020i	$-0.593663\pi$
$853$	−7.04560e59	−0.580047	−0.290023	+	0.957020i	$0.593663\pi$
$854$	0	0
$855$	−3.43799e59	−0.269773
$856$	0	0
$857$	1.18884e60	0.889234	0.444617	−	0.895721i	$-0.353340\pi$
$857$	1.18884e60	0.889234	0.444617	+	0.895721i	$0.353340\pi$
$858$	0	0
$859$	−2.54870e60	−1.81744	−0.908718	−	0.417410i	$-0.862938\pi$
$859$	−2.54870e60	−1.81744	−0.908718	+	0.417410i	$0.862938\pi$
$860$	0	0
$861$	4.01026e58	0.0272651
$862$	0	0
$863$	1.01594e60	0.658640	0.329320	−	0.944218i	$-0.393181\pi$
$863$	1.01594e60	0.658640	0.329320	+	0.944218i	$0.393181\pi$
$864$	0	0
$865$	−1.11943e59	−0.0692095
$866$	0	0
$867$	6.78679e59	0.400196
$868$	0	0
$869$	−5.90501e59	−0.332135
$870$	0	0
$871$	2.80106e60	1.50297
$872$	0	0
$873$	−8.29309e59	−0.424546
$874$	0	0
$875$	−1.27818e59	−0.0624347
$876$	0	0
$877$	−3.07649e59	−0.143405	−0.0717023	−	0.997426i	$-0.522843\pi$
$877$	−3.07649e59	−0.143405	−0.0717023	+	0.997426i	$0.522843\pi$
$878$	0	0
$879$	1.39566e60	0.620881
$880$	0	0
$881$	1.76559e60	0.749694	0.374847	−	0.927087i	$-0.377695\pi$
$881$	1.76559e60	0.749694	0.374847	+	0.927087i	$0.377695\pi$
$882$	0	0
$883$	−1.02366e60	−0.414918	−0.207459	−	0.978244i	$-0.566519\pi$
$883$	−1.02366e60	−0.414918	−0.207459	+	0.978244i	$0.566519\pi$
$884$	0	0
$885$	7.65542e59	0.296232
$886$	0	0
$887$	4.52674e59	0.167244	0.0836221	−	0.996498i	$-0.473351\pi$
$887$	4.52674e59	0.167244	0.0836221	+	0.996498i	$0.473351\pi$
$888$	0	0
$889$	1.33807e59	0.0472055
$890$	0	0
$891$	−1.90994e59	−0.0643468
$892$	0	0
$893$	3.36309e60	1.08214
$894$	0	0
$895$	1.86493e60	0.573178
$896$	0	0
$897$	−4.33133e60	−1.27167
$898$	0	0
$899$	−6.80829e59	−0.190970
$900$	0	0
$901$	−1.40915e60	−0.377658
$902$	0	0
$903$	−9.83043e57	−0.00251752
$904$	0	0
$905$	1.78895e60	0.437826
$906$	0	0
$907$	6.39985e60	1.49699	0.748497	−	0.663138i	$-0.230776\pi$
$907$	6.39985e60	1.49699	0.748497	+	0.663138i	$0.230776\pi$
$908$	0	0
$909$	4.51622e60	1.00975
$910$	0	0
$911$	3.60826e60	0.771208	0.385604	−	0.922664i	$-0.373993\pi$
$911$	3.60826e60	0.771208	0.385604	+	0.922664i	$0.373993\pi$
$912$	0	0
$913$	1.22798e61	2.50924
$914$	0	0
$915$	−9.86547e59	−0.192746
$916$	0	0
$917$	−1.48694e59	−0.0277793
$918$	0	0
$919$	−6.99835e60	−1.25034	−0.625170	−	0.780489i	$-0.714970\pi$
$919$	−6.99835e60	−1.25034	−0.625170	+	0.780489i	$0.714970\pi$
$920$	0	0
$921$	4.98479e60	0.851775
$922$	0	0
$923$	5.06969e60	0.828603
$924$	0	0
$925$	−4.25817e60	−0.665760
$926$	0	0
$927$	−5.92217e59	−0.0885821
$928$	0	0
$929$	−1.28627e61	−1.84081	−0.920406	−	0.390963i	$-0.872142\pi$
$929$	−1.28627e61	−1.84081	−0.920406	+	0.390963i	$0.872142\pi$
$930$	0	0
$931$	−7.64690e60	−1.04717
$932$	0	0
$933$	1.18559e60	0.155368
$934$	0	0
$935$	3.22469e60	0.404436
$936$	0	0
$937$	−1.05021e61	−1.26071	−0.630354	−	0.776308i	$-0.717090\pi$
$937$	−1.05021e61	−1.26071	−0.630354	+	0.776308i	$0.717090\pi$
$938$	0	0
$939$	−3.75546e60	−0.431538
$940$	0	0
$941$	−7.41023e60	−0.815163	−0.407581	−	0.913169i	$-0.633628\pi$
$941$	−7.41023e60	−0.815163	−0.407581	+	0.913169i	$0.633628\pi$
$942$	0	0
$943$	6.78847e60	0.714960
$944$	0	0
$945$	3.44790e59	0.0347698
$946$	0	0
$947$	−5.73646e60	−0.553949	−0.276974	−	0.960877i	$-0.589332\pi$
$947$	−5.73646e60	−0.553949	−0.276974	+	0.960877i	$0.589332\pi$
$948$	0	0
$949$	2.36610e61	2.18814
$950$	0	0
$951$	−1.14249e61	−1.01193
$952$	0	0
$953$	−1.08849e61	−0.923467	−0.461733	−	0.887019i	$-0.652772\pi$
$953$	−1.08849e61	−0.923467	−0.461733	+	0.887019i	$0.652772\pi$
$954$	0	0
$955$	7.70313e60	0.626034
$956$	0	0
$957$	−1.62639e61	−1.26628
$958$	0	0
$959$	−2.21772e59	−0.0165434
$960$	0	0
$961$	−1.36794e61	−0.977769
$962$	0	0
$963$	−8.45098e60	−0.578853
$964$	0	0
$965$	1.14328e60	0.0750487
$966$	0	0
$967$	−2.37432e61	−1.49381	−0.746906	−	0.664930i	$-0.768461\pi$
$967$	−2.37432e61	−1.49381	−0.746906	+	0.664930i	$0.768461\pi$
$968$	0	0
$969$	−6.70691e60	−0.404469
$970$	0	0
$971$	1.71671e61	0.992436	0.496218	−	0.868198i	$-0.334722\pi$
$971$	1.71671e61	0.992436	0.496218	+	0.868198i	$0.334722\pi$
$972$	0	0
$973$	1.26985e60	0.0703783
$974$	0	0
$975$	−1.46575e61	−0.778870
$976$	0	0
$977$	3.29474e61	1.67873	0.839367	−	0.543565i	$-0.182926\pi$
$977$	3.29474e61	1.67873	0.839367	+	0.543565i	$0.182926\pi$
$978$	0	0
$979$	2.22265e61	1.08599
$980$	0	0
$981$	−1.18385e61	−0.554731
$982$	0	0
$983$	−3.50912e61	−1.57707	−0.788535	−	0.614990i	$-0.789160\pi$
$983$	−3.50912e61	−1.57707	−0.788535	+	0.614990i	$0.789160\pi$
$984$	0	0
$985$	1.62261e61	0.699474
$986$	0	0
$987$	−1.26408e60	−0.0522725
$988$	0	0
$989$	−1.66407e60	−0.0660157
$990$	0	0
$991$	1.41075e61	0.536959	0.268480	−	0.963285i	$-0.413479\pi$
$991$	1.41075e61	0.536959	0.268480	+	0.963285i	$0.413479\pi$
$992$	0	0
$993$	−1.35567e61	−0.495101
$994$	0	0
$995$	1.24691e61	0.436982
$996$	0	0
$997$	−1.88767e61	−0.634857	−0.317429	−	0.948282i	$-0.602819\pi$
$997$	−1.88767e61	−0.634857	−0.317429	+	0.948282i	$0.602819\pi$
$998$	0	0
$999$	2.55339e61	0.824185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.42.a.c.1.1		3
4.3	odd	2		1.42.a.a.1.2	✓	3
12.11	even	2		9.42.a.b.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.42.a.a.1.2	✓	3	4.3	odd	2
9.42.a.b.1.2		3	12.11	even	2
16.42.a.c.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 42 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-3.82217e9 q^{3} +9.10518e13 q^{5} +1.69829e16 q^{7} -2.18640e19 q^{9} +O(q^{10})\)	\(q-3.82217e9 q^{3} +9.10518e13 q^{5} +1.69829e16 q^{7} -2.18640e19 q^{9} +3.48555e21 q^{11} -1.03131e23 q^{13} -3.48015e23 q^{15} +1.01608e25 q^{17} +1.72697e26 q^{19} -6.49115e25 q^{21} -1.09881e28 q^{23} -3.71843e28 q^{25} +2.22974e29 q^{27} +1.22079e30 q^{29} -5.57694e29 q^{31} -1.33224e31 q^{33} +1.54633e30 q^{35} +1.14515e32 q^{37} +3.94185e32 q^{39} -6.17804e32 q^{41} +1.51443e32 q^{43} -1.99076e33 q^{45} +1.94739e34 q^{47} -4.42792e34 q^{49} -3.88362e34 q^{51} -1.38685e35 q^{53} +3.17366e35 q^{55} -6.60078e35 q^{57} -2.19974e36 q^{59} +2.83478e36 q^{61} -3.71315e35 q^{63} -9.39028e36 q^{65} -2.71602e37 q^{67} +4.19982e37 q^{69} -4.91576e37 q^{71} -2.29426e38 q^{73} +1.42125e38 q^{75} +5.91949e37 q^{77} -1.69414e38 q^{79} -5.47960e37 q^{81} +3.52307e39 q^{83} +9.25158e38 q^{85} -4.66608e39 q^{87} +6.37675e39 q^{89} -1.75147e39 q^{91} +2.13160e39 q^{93} +1.57244e40 q^{95} +3.79303e40 q^{97} -7.62083e40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 10820953044 q^{3} - 212302350281550 q^{5} - 57\!\cdots\!92 q^{7}+ \cdots + 13\!\cdots\!19 q^{9}+O(q^{10}) \) 3 * q + 10820953044 * q^3 - 212302350281550 * q^5 - 57878416258239192 * q^7 + 13277004110931878919 * q^9	\( 3 q + 10820953044 q^{3} - 212302350281550 q^{5} - 57\!\cdots\!92 q^{7}+ \cdots - 45\!\cdots\!28 q^{99}+O(q^{100}) \) 3 * q + 10820953044 * q^3 - 212302350281550 * q^5 - 57878416258239192 * q^7 + 13277004110931878919 * q^9 + 3064080929502798535164 * q^11 - 98515763053518962936694 * q^13 - 2322231962154213782968200 * q^15 + 35582277130790298867301302 * q^17 + 233456308095243376865478180 * q^19 - 784315067894306487816760224 * q^21 - 2812708670812770779627026056 * q^23 - 12630087813294771903109831875 * q^25 - 40981593702376529051426180280 * q^27 - 12736322664295346653179017670 * q^29 + 55977552834365071134409017504 * q^31 - 13829226714693448959900097020528 * q^33 - 35330944415319146254058128592400 * q^35 + 4945769903781582545269841455122 * q^37 + 546470556552953849463965686898712 * q^39 - 3123151500646359746292805811918274 * q^41 - 1491980858362212236675453498061156 * q^43 - 3706804708974790460103758635956150 * q^45 + 63038106682044077762675687404413168 * q^47 - 97136113131667199274890334287198229 * q^49 + 157319866621422817701623275797858024 * q^51 + 79896749344137562310547533743021506 * q^53 + 1107430856215374719020572615529578600 * q^55 - 120534128028213908789981458568129040 * q^57 - 192512048683375799719343226678403860 * q^59 + 8740556115036354715092732710435455386 * q^61 - 4195060549596204507618981258341049336 * q^63 + 23272939413910136331858458170260680700 * q^65 - 11266872945514675454702655007279407852 * q^67 + 110718129232411678126981084073658946848 * q^69 + 140970915469820223119231433051520161384 * q^71 + 45825493210252135256288871746499029406 * q^73 + 247029389196459510298088488221820147500 * q^75 - 429017620126390531124795910645506217696 * q^77 + 520184835554750143118446358634240850320 * q^79 - 3191652631710460747876906046824191542997 * q^81 + 619777491338381742927488037142663372644 * q^83 - 106299433069331216483654652405294143100 * q^85 - 13958176049210326842816445061163645008040 * q^87 - 142389067128374093799211866144944441010 * q^89 - 25080596935320871627133801126628392473296 * q^91 + 3477058704647981071307340824432458334592 * q^93 + 33170437179755379504247502944935502503000 * q^95 + 116011691364103339137369463947331771078182 * q^97 - 45849669342635847189814810549498192693428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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