LMFDB - Embedded newform 16.38.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,38,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, 38, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 38);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 38 $
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:	$138.742460999$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 15934380 $ x^2 - x - 15934380
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3991.29$ 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)

$f(q)$	$=$	$q+2.05955e7 q^{3} -8.29715e12 q^{5} -1.97377e15 q^{7} -4.49860e17 q^{9} +O(q^{10})$	$q+2.05955e7 q^{3} -8.29715e12 q^{5} -1.97377e15 q^{7} -4.49860e17 q^{9} +2.57012e19 q^{11} +5.42906e20 q^{13} -1.70884e20 q^{15} -3.52797e22 q^{17} -6.82122e23 q^{19} -4.06509e22 q^{21} +8.19547e22 q^{23} -3.91685e24 q^{25} -1.85389e25 q^{27} -1.51991e27 q^{29} -2.60785e27 q^{31} +5.29330e26 q^{33} +1.63767e28 q^{35} -1.30205e29 q^{37} +1.11814e28 q^{39} -4.07079e29 q^{41} +2.92424e30 q^{43} +3.73255e30 q^{45} -3.58323e30 q^{47} -1.46663e31 q^{49} -7.26605e29 q^{51} +3.56714e31 q^{53} -2.13247e32 q^{55} -1.40487e31 q^{57} +3.03666e32 q^{59} +1.16214e33 q^{61} +8.87921e32 q^{63} -4.50457e33 q^{65} +2.44324e33 q^{67} +1.68790e30 q^{69} -6.30224e33 q^{71} +1.02725e34 q^{73} -8.06697e31 q^{75} -5.07283e34 q^{77} -1.20547e35 q^{79} +2.02183e35 q^{81} +3.26699e35 q^{83} +2.92721e35 q^{85} -3.13034e34 q^{87} -1.56115e36 q^{89} -1.07157e36 q^{91} -5.37100e34 q^{93} +5.65967e36 q^{95} -1.07155e36 q^{97} -1.15619e37 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 13991400 q^{3} + 5529584385900 q^{5} + 34\!\cdots\!00 q^{7}+ \cdots - 89\!\cdots\!14 q^{9}+O(q^{10}) $ 2 * q - 13991400 * q^3 + 5529584385900 * q^5 + 3448443953486000 * q^7 - 898947378401769414 * q^9	$ 2 q - 13991400 q^{3} + 5529584385900 q^{5} + 34\!\cdots\!00 q^{7}+ \cdots - 12\!\cdots\!92 q^{99}+O(q^{100}) $ 2 * q - 13991400 * q^3 + 5529584385900 * q^5 + 3448443953486000 * q^7 - 898947378401769414 * q^9 + 26734036354848538056 * q^11 + 530581741653933178300 * q^13 - 649108943683113884400 * q^15 - 89439073881931767332700 * q^17 - 373276466572513713706120 * q^19 - 228188865811378281195456 * q^21 + 26241180149933881945462800 * q^23 + 114502199615774855754278750 * q^25 + 12567565757525316335881200 * q^27 - 1270673827125882417951629220 * q^29 - 261420819791418895481545024 * q^31 + 493606999076023001294056800 * q^33 + 91348258395203301140779687200 * q^35 - 68025363793820786588733238100 * q^37 + 11607709470242600846801346768 * q^39 - 1260483295466373133974684841836 * q^41 + 2570241023157918831169581605000 * q^43 - 2476861984515775772921684271300 * q^45 - 4252206875568934025158407583200 * q^47 - 3828013658149768338307153838286 * q^49 + 1146602989088714802084517292976 * q^51 + 159799736258590810071505678893900 * q^53 - 198965756266726963295068969294800 * q^55 - 24730693798747893221767420706400 * q^57 + 237962459606090128758899369700840 * q^59 + 101798841373038700200106255199644 * q^61 - 1547129676521036250657950915523600 * q^63 - 4674979790566994364848214315456600 * q^65 + 10891923280981358108643809352546200 * q^67 - 903079825405531082422221799280448 * q^69 + 746133089793832492689962158868016 * q^71 + 19639851676496426023909382507487700 * q^73 - 4176423071329708963282786324335000 * q^75 - 45127994201424729507574115685000000 * q^77 - 272401638034095839035647945144674080 * q^79 + 403323837550038341003235827350619442 * q^81 + 470460275390909929308746217601073400 * q^83 - 456126475708535200141442602834276200 * q^85 - 39923812909186495628003917423124400 * q^87 - 1332868921711238117914343978794942860 * q^89 - 1138398786311531726477856383010207584 * q^91 - 134865729229255425577378938731155200 * q^93 + 9929993151899196579173099213128626000 * q^95 + 6002061888473229973039130090661237700 * q^97 - 12025768918384639461946415841980309592 * 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
$3$	2.05955e7	0.0306923	0.0153462	−	0.999882i	$-0.495115\pi$
$3$	2.05955e7	0.0306923	0.0153462	+	0.999882i	$0.495115\pi$
$4$	0	0
$5$	−8.29715e12	−0.972711	−0.486356	−	0.873761i	$-0.661674\pi$
$5$	−8.29715e12	−0.972711	−0.486356	+	0.873761i	$0.661674\pi$
$6$	0	0
$7$	−1.97377e15	−0.458124	−0.229062	−	0.973412i	$-0.573566\pi$
$7$	−1.97377e15	−0.458124	−0.229062	+	0.973412i	$0.573566\pi$
$8$	0	0
$9$	−4.49860e17	−0.999058
$10$	0	0
$11$	2.57012e19	1.39376	0.696881	−	0.717187i	$-0.254571\pi$
$11$	2.57012e19	1.39376	0.696881	+	0.717187i	$0.254571\pi$
$12$	0	0
$13$	5.42906e20	1.33898	0.669488	−	0.742823i	$-0.266514\pi$
$13$	5.42906e20	1.33898	0.669488	+	0.742823i	$0.266514\pi$
$14$	0	0
$15$	−1.70884e20	−0.0298548
$16$	0	0
$17$	−3.52797e22	−0.608443	−0.304221	−	0.952601i	$-0.598396\pi$
$17$	−3.52797e22	−0.608443	−0.304221	+	0.952601i	$0.598396\pi$
$18$	0	0
$19$	−6.82122e23	−1.50287	−0.751433	−	0.659809i	$-0.770637\pi$
$19$	−6.82122e23	−1.50287	−0.751433	+	0.659809i	$0.770637\pi$
$20$	0	0
$21$	−4.06509e22	−0.0140609
$22$	0	0
$23$	8.19547e22	0.00526755	0.00263378	−	0.999997i	$-0.499162\pi$
$23$	8.19547e22	0.00526755	0.00263378	+	0.999997i	$0.499162\pi$
$24$	0	0
$25$	−3.91685e24	−0.0538328
$26$	0	0
$27$	−1.85389e25	−0.0613558
$28$	0	0
$29$	−1.51991e27	−1.34108	−0.670541	−	0.741872i	$-0.733938\pi$
$29$	−1.51991e27	−1.34108	−0.670541	+	0.741872i	$0.733938\pi$
$30$	0	0
$31$	−2.60785e27	−0.670025	−0.335012	−	0.942214i	$-0.608740\pi$
$31$	−2.60785e27	−0.670025	−0.335012	+	0.942214i	$0.608740\pi$
$32$	0	0
$33$	5.29330e26	0.0427778
$34$	0	0
$35$	1.63767e28	0.445623
$36$	0	0
$37$	−1.30205e29	−1.26734	−0.633672	−	0.773602i	$-0.718453\pi$
$37$	−1.30205e29	−1.26734	−0.633672	+	0.773602i	$0.718453\pi$
$38$	0	0
$39$	1.11814e28	0.0410963
$40$	0	0
$41$	−4.07079e29	−0.593168	−0.296584	−	0.955007i	$-0.595847\pi$
$41$	−4.07079e29	−0.593168	−0.296584	+	0.955007i	$0.595847\pi$
$42$	0	0
$43$	2.92424e30	1.76541	0.882706	−	0.469926i	$-0.155719\pi$
$43$	2.92424e30	1.76541	0.882706	+	0.469926i	$0.155719\pi$
$44$	0	0
$45$	3.73255e30	0.971795
$46$	0	0
$47$	−3.58323e30	−0.417315	−0.208658	−	0.977989i	$-0.566909\pi$
$47$	−3.58323e30	−0.417315	−0.208658	+	0.977989i	$0.566909\pi$
$48$	0	0
$49$	−1.46663e31	−0.790122
$50$	0	0
$51$	−7.26605e29	−0.0186745
$52$	0	0
$53$	3.56714e31	0.450005	0.225002	−	0.974358i	$-0.427761\pi$
$53$	3.56714e31	0.450005	0.225002	+	0.974358i	$0.427761\pi$
$54$	0	0
$55$	−2.13247e32	−1.35573
$56$	0	0
$57$	−1.40487e31	−0.0461265
$58$	0	0
$59$	3.03666e32	0.526785	0.263393	−	0.964689i	$-0.415159\pi$
$59$	3.03666e32	0.526785	0.263393	+	0.964689i	$0.415159\pi$
$60$	0	0
$61$	1.16214e33	1.08807	0.544034	−	0.839063i	$-0.316896\pi$
$61$	1.16214e33	1.08807	0.544034	+	0.839063i	$0.316896\pi$
$62$	0	0
$63$	8.87921e32	0.457693
$64$	0	0
$65$	−4.50457e33	−1.30244
$66$	0	0
$67$	2.44324e33	0.403257	0.201628	−	0.979462i	$-0.435377\pi$
$67$	2.44324e33	0.403257	0.201628	+	0.979462i	$0.435377\pi$
$68$	0	0
$69$	1.68790e30	0.000161673 0
$70$	0	0
$71$	−6.30224e33	−0.355808	−0.177904	−	0.984048i	$-0.556932\pi$
$71$	−6.30224e33	−0.355808	−0.177904	+	0.984048i	$0.556932\pi$
$72$	0	0
$73$	1.02725e34	0.346899	0.173450	−	0.984843i	$-0.444509\pi$
$73$	1.02725e34	0.346899	0.173450	+	0.984843i	$0.444509\pi$
$74$	0	0
$75$	−8.06697e31	−0.00165226
$76$	0	0
$77$	−5.07283e34	−0.638516
$78$	0	0
$79$	−1.20547e35	−0.944182	−0.472091	−	0.881550i	$-0.656501\pi$
$79$	−1.20547e35	−0.944182	−0.472091	+	0.881550i	$0.656501\pi$
$80$	0	0
$81$	2.02183e35	0.997175
$82$	0	0
$83$	3.26699e35	1.02613	0.513066	−	0.858349i	$-0.328510\pi$
$83$	3.26699e35	1.02613	0.513066	+	0.858349i	$0.328510\pi$
$84$	0	0
$85$	2.92721e35	0.591839
$86$	0	0
$87$	−3.13034e34	−0.0411610
$88$	0	0
$89$	−1.56115e36	−1.34812	−0.674062	−	0.738674i	$-0.735452\pi$
$89$	−1.56115e36	−1.34812	−0.674062	+	0.738674i	$0.735452\pi$
$90$	0	0
$91$	−1.07157e36	−0.613418
$92$	0	0
$93$	−5.37100e34	−0.0205646
$94$	0	0
$95$	5.65967e36	1.46186
$96$	0	0
$97$	−1.07155e36	−0.188251	−0.0941254	−	0.995560i	$-0.530005\pi$
$97$	−1.07155e36	−0.188251	−0.0941254	+	0.995560i	$0.530005\pi$
$98$	0	0
$99$	−1.15619e37	−1.39245
$100$	0	0
$101$	1.29521e37	1.07744	0.538721	−	0.842484i	$-0.318908\pi$
$101$	1.29521e37	1.07744	0.538721	+	0.842484i	$0.318908\pi$
$102$	0	0
$103$	3.39568e36	0.196534	0.0982672	−	0.995160i	$-0.468670\pi$
$103$	3.39568e36	0.196534	0.0982672	+	0.995160i	$0.468670\pi$
$104$	0	0
$105$	3.37287e35	0.0136772
$106$	0	0
$107$	1.58608e37	0.453653	0.226827	−	0.973935i	$-0.427165\pi$
$107$	1.58608e37	0.453653	0.226827	+	0.973935i	$0.427165\pi$
$108$	0	0
$109$	−1.06858e36	−0.0216978	−0.0108489	−	0.999941i	$-0.503453\pi$
$109$	−1.06858e36	−0.0216978	−0.0108489	+	0.999941i	$0.503453\pi$
$110$	0	0
$111$	−2.68163e36	−0.0388977
$112$	0	0
$113$	1.64170e37	0.171136	0.0855682	−	0.996332i	$-0.472729\pi$
$113$	1.64170e37	0.171136	0.0855682	+	0.996332i	$0.472729\pi$
$114$	0	0
$115$	−6.79990e35	−0.00512381
$116$	0	0
$117$	−2.44232e38	−1.33771
$118$	0	0
$119$	6.96341e37	0.278743
$120$	0	0
$121$	3.20512e38	0.942572
$122$	0	0
$123$	−8.38402e36	−0.0182057
$124$	0	0
$125$	6.36196e38	1.02508
$126$	0	0
$127$	1.42456e39	1.71124	0.855619	−	0.517606i	$-0.173177\pi$
$127$	1.42456e39	1.71124	0.855619	+	0.517606i	$0.173177\pi$
$128$	0	0
$129$	6.02262e37	0.0541846
$130$	0	0
$131$	−2.12044e39	−1.43518	−0.717590	−	0.696466i	$-0.754755\pi$
$131$	−2.12044e39	−1.43518	−0.717590	+	0.696466i	$0.754755\pi$
$132$	0	0
$133$	1.34635e39	0.688500
$134$	0	0
$135$	1.53820e38	0.0596815
$136$	0	0
$137$	−1.25855e39	−0.371999	−0.185999	−	0.982550i	$-0.559552\pi$
$137$	−1.25855e39	−0.371999	−0.185999	+	0.982550i	$0.559552\pi$
$138$	0	0
$139$	3.12966e39	0.707494	0.353747	−	0.935341i	$-0.384907\pi$
$139$	3.12966e39	0.707494	0.353747	+	0.935341i	$0.384907\pi$
$140$	0	0
$141$	−7.37986e37	−0.0128084
$142$	0	0
$143$	1.39533e40	1.86621
$144$	0	0
$145$	1.26109e40	1.30449
$146$	0	0
$147$	−3.02061e38	−0.0242507
$148$	0	0
$149$	2.99066e39	0.186991	0.0934956	−	0.995620i	$-0.470196\pi$
$149$	2.99066e39	0.186991	0.0934956	+	0.995620i	$0.470196\pi$
$150$	0	0
$151$	7.06984e39	0.345411	0.172706	−	0.984973i	$-0.444749\pi$
$151$	7.06984e39	0.345411	0.172706	+	0.984973i	$0.444749\pi$
$152$	0	0
$153$	1.58709e40	0.607870
$154$	0	0
$155$	2.16377e40	0.651740
$156$	0	0
$157$	7.04676e40	1.67435	0.837174	−	0.546937i	$-0.184206\pi$
$157$	7.04676e40	1.67435	0.837174	+	0.546937i	$0.184206\pi$
$158$	0	0
$159$	7.34673e38	0.0138117
$160$	0	0
$161$	−1.61760e38	−0.00241319
$162$	0	0
$163$	8.16698e40	0.969596	0.484798	−	0.874626i	$-0.338893\pi$
$163$	8.16698e40	0.969596	0.484798	+	0.874626i	$0.338893\pi$
$164$	0	0
$165$	−4.39193e39	−0.0416105
$166$	0	0
$167$	3.94031e40	0.298728	0.149364	−	0.988782i	$-0.452277\pi$
$167$	3.94031e40	0.298728	0.149364	+	0.988782i	$0.452277\pi$
$168$	0	0
$169$	1.30346e41	0.792856
$170$	0	0
$171$	3.06859e41	1.50145
$172$	0	0
$173$	2.61566e41	1.03212	0.516058	−	0.856554i	$-0.327399\pi$
$173$	2.61566e41	1.03212	0.516058	+	0.856554i	$0.327399\pi$
$174$	0	0
$175$	7.73098e39	0.0246621
$176$	0	0
$177$	6.25416e39	0.0161683
$178$	0	0
$179$	4.21879e41	0.885946	0.442973	−	0.896535i	$-0.353924\pi$
$179$	4.21879e41	0.885946	0.442973	+	0.896535i	$0.353924\pi$
$180$	0	0
$181$	−9.51260e40	−0.162647	−0.0813234	−	0.996688i	$-0.525915\pi$
$181$	−9.51260e40	−0.162647	−0.0813234	+	0.996688i	$0.525915\pi$
$182$	0	0
$183$	2.39349e40	0.0333954
$184$	0	0
$185$	1.08033e42	1.23276
$186$	0	0
$187$	−9.06730e41	−0.848024
$188$	0	0
$189$	3.65917e40	0.0281086
$190$	0	0
$191$	2.61439e41	0.165292	0.0826462	−	0.996579i	$-0.473663\pi$
$191$	2.61439e41	0.165292	0.0826462	+	0.996579i	$0.473663\pi$
$192$	0	0
$193$	−6.27072e41	−0.326969	−0.163485	−	0.986546i	$-0.552273\pi$
$193$	−6.27072e41	−0.326969	−0.163485	+	0.986546i	$0.552273\pi$
$194$	0	0
$195$	−9.27742e40	−0.0399748
$196$	0	0
$197$	−1.81081e42	−0.646021	−0.323010	−	0.946395i	$-0.604695\pi$
$197$	−1.81081e42	−0.646021	−0.323010	+	0.946395i	$0.604695\pi$
$198$	0	0
$199$	3.78123e41	0.111906	0.0559528	−	0.998433i	$-0.482180\pi$
$199$	3.78123e41	0.111906	0.0559528	+	0.998433i	$0.482180\pi$
$200$	0	0
$201$	5.03198e40	0.0123769
$202$	0	0
$203$	2.99996e42	0.614383
$204$	0	0
$205$	3.37760e42	0.576981
$206$	0	0
$207$	−3.68681e40	−0.00526259
$208$	0	0
$209$	−1.75313e43	−2.09464
$210$	0	0
$211$	−1.66783e43	−1.67081	−0.835404	−	0.549637i	$-0.814766\pi$
$211$	−1.66783e43	−1.67081	−0.835404	+	0.549637i	$0.814766\pi$
$212$	0	0
$213$	−1.29798e41	−0.0109206
$214$	0	0
$215$	−2.42628e43	−1.71724
$216$	0	0
$217$	5.14730e42	0.306955
$218$	0	0
$219$	2.11568e41	0.0106472
$220$	0	0
$221$	−1.91536e43	−0.814690
$222$	0	0
$223$	3.60083e43	1.29647	0.648236	−	0.761439i	$-0.275507\pi$
$223$	3.60083e43	1.29647	0.648236	+	0.761439i	$0.275507\pi$
$224$	0	0
$225$	1.76203e42	0.0537821
$226$	0	0
$227$	−6.76307e43	−1.75253	−0.876265	−	0.481830i	$-0.839972\pi$
$227$	−6.76307e43	−1.75253	−0.876265	+	0.481830i	$0.839972\pi$
$228$	0	0
$229$	−5.09302e43	−1.12207	−0.561033	−	0.827793i	$-0.689596\pi$
$229$	−5.09302e43	−1.12207	−0.561033	+	0.827793i	$0.689596\pi$
$230$	0	0
$231$	−1.04478e42	−0.0195976
$232$	0	0
$233$	1.79375e43	0.286865	0.143432	−	0.989660i	$-0.454186\pi$
$233$	1.79375e43	0.286865	0.143432	+	0.989660i	$0.454186\pi$
$234$	0	0
$235$	2.97306e43	0.405927
$236$	0	0
$237$	−2.48273e42	−0.0289792
$238$	0	0
$239$	2.86950e43	0.286711	0.143356	−	0.989671i	$-0.454211\pi$
$239$	2.86950e43	0.286711	0.143356	+	0.989671i	$0.454211\pi$
$240$	0	0
$241$	−1.12255e44	−0.961372	−0.480686	−	0.876893i	$-0.659612\pi$
$241$	−1.12255e44	−0.961372	−0.480686	+	0.876893i	$0.659612\pi$
$242$	0	0
$243$	1.25119e43	0.0919614
$244$	0	0
$245$	1.21689e44	0.768561
$246$	0	0
$247$	−3.70328e44	−2.01230
$248$	0	0
$249$	6.72854e42	0.0314944
$250$	0	0
$251$	4.02278e44	1.62391	0.811955	−	0.583720i	$-0.198403\pi$
$251$	4.02278e44	1.62391	0.811955	+	0.583720i	$0.198403\pi$
$252$	0	0
$253$	2.10633e42	0.00734171
$254$	0	0
$255$	6.02875e42	0.0181649
$256$	0	0
$257$	5.42969e44	1.41583	0.707917	−	0.706296i	$-0.249635\pi$
$257$	5.42969e44	1.41583	0.707917	+	0.706296i	$0.249635\pi$
$258$	0	0
$259$	2.56994e44	0.580601
$260$	0	0
$261$	6.83747e44	1.33982
$262$	0	0
$263$	7.98181e44	1.35806	0.679031	−	0.734109i	$-0.262400\pi$
$263$	7.98181e44	1.35806	0.679031	+	0.734109i	$0.262400\pi$
$264$	0	0
$265$	−2.95971e44	−0.437725
$266$	0	0
$267$	−3.21528e43	−0.0413771
$268$	0	0
$269$	−1.02189e45	−1.14548	−0.572739	−	0.819738i	$-0.694119\pi$
$269$	−1.02189e45	−1.14548	−0.572739	+	0.819738i	$0.694119\pi$
$270$	0	0
$271$	2.63101e44	0.257152	0.128576	−	0.991700i	$-0.458959\pi$
$271$	2.63101e44	0.257152	0.128576	+	0.991700i	$0.458959\pi$
$272$	0	0
$273$	−2.20696e43	−0.0188272
$274$	0	0
$275$	−1.00668e44	−0.0750301
$276$	0	0
$277$	1.60882e45	1.04865	0.524327	−	0.851517i	$-0.324317\pi$
$277$	1.60882e45	1.04865	0.524327	+	0.851517i	$0.324317\pi$
$278$	0	0
$279$	1.17317e45	0.669393
$280$	0	0
$281$	2.86236e45	1.43106	0.715529	−	0.698583i	$-0.246186\pi$
$281$	2.86236e45	1.43106	0.715529	+	0.698583i	$0.246186\pi$
$282$	0	0
$283$	−7.97480e44	−0.349680	−0.174840	−	0.984597i	$-0.555941\pi$
$283$	−7.97480e44	−0.349680	−0.174840	+	0.984597i	$0.555941\pi$
$284$	0	0
$285$	1.16564e44	0.0448678
$286$	0	0
$287$	8.03482e44	0.271745
$288$	0	0
$289$	−2.11744e45	−0.629797
$290$	0	0
$291$	−2.20692e43	−0.00577786
$292$	0	0
$293$	5.17198e44	0.119290	0.0596452	−	0.998220i	$-0.481003\pi$
$293$	5.17198e44	0.119290	0.0596452	+	0.998220i	$0.481003\pi$
$294$	0	0
$295$	−2.51956e45	−0.512410
$296$	0	0
$297$	−4.76473e44	−0.0855153
$298$	0	0
$299$	4.44937e43	0.00705312
$300$	0	0
$301$	−5.77178e45	−0.808778
$302$	0	0
$303$	2.66755e44	0.0330692
$304$	0	0
$305$	−9.64245e45	−1.05838
$306$	0	0
$307$	−9.59117e45	−0.932850	−0.466425	−	0.884561i	$-0.654458\pi$
$307$	−9.59117e45	−0.932850	−0.466425	+	0.884561i	$0.654458\pi$
$308$	0	0
$309$	6.99360e43	0.00603210
$310$	0	0
$311$	5.05857e45	0.387222	0.193611	−	0.981078i	$-0.437980\pi$
$311$	5.05857e45	0.387222	0.193611	+	0.981078i	$0.437980\pi$
$312$	0	0
$313$	5.12462e45	0.348410	0.174205	−	0.984709i	$-0.444264\pi$
$313$	5.12462e45	0.348410	0.174205	+	0.984709i	$0.444264\pi$
$314$	0	0
$315$	−7.36722e45	−0.445203
$316$	0	0
$317$	−2.14514e46	−1.15308	−0.576541	−	0.817068i	$-0.695598\pi$
$317$	−2.14514e46	−1.15308	−0.576541	+	0.817068i	$0.695598\pi$
$318$	0	0
$319$	−3.90635e46	−1.86915
$320$	0	0
$321$	3.26661e44	0.0139237
$322$	0	0
$323$	2.40651e46	0.914408
$324$	0	0
$325$	−2.12648e45	−0.0720808
$326$	0	0
$327$	−2.20080e43	−0.000665958 0
$328$	0	0
$329$	7.07248e45	0.191182
$330$	0	0
$331$	−1.85276e46	−0.447714	−0.223857	−	0.974622i	$-0.571865\pi$
$331$	−1.85276e46	−0.447714	−0.223857	+	0.974622i	$0.571865\pi$
$332$	0	0
$333$	5.85738e46	1.26615
$334$	0	0
$335$	−2.02719e46	−0.392253
$336$	0	0
$337$	−9.22947e46	−1.59965	−0.799823	−	0.600236i	$-0.795073\pi$
$337$	−9.22947e46	−1.59965	−0.799823	+	0.600236i	$0.795073\pi$
$338$	0	0
$339$	3.38117e44	0.00525258
$340$	0	0
$341$	−6.70248e46	−0.933855
$342$	0	0
$343$	6.55854e46	0.820099
$344$	0	0
$345$	−1.40048e43	−0.000157262 0
$346$	0	0
$347$	5.27249e46	0.532011	0.266005	−	0.963972i	$-0.414296\pi$
$347$	5.27249e46	0.532011	0.266005	+	0.963972i	$0.414296\pi$
$348$	0	0
$349$	9.71277e46	0.881196	0.440598	−	0.897704i	$-0.354766\pi$
$349$	9.71277e46	0.881196	0.440598	+	0.897704i	$0.354766\pi$
$350$	0	0
$351$	−1.00649e46	−0.0821539
$352$	0	0
$353$	1.73375e47	1.27396	0.636979	−	0.770881i	$-0.280184\pi$
$353$	1.73375e47	1.27396	0.636979	+	0.770881i	$0.280184\pi$
$354$	0	0
$355$	5.22906e46	0.346098
$356$	0	0
$357$	1.43415e45	0.00855526
$358$	0	0
$359$	1.13075e47	0.608301	0.304150	−	0.952624i	$-0.401627\pi$
$359$	1.13075e47	0.608301	0.304150	+	0.952624i	$0.401627\pi$
$360$	0	0
$361$	2.59283e47	1.25861
$362$	0	0
$363$	6.60111e45	0.0289297
$364$	0	0
$365$	−8.52325e46	−0.337433
$366$	0	0
$367$	8.74068e46	0.312768	0.156384	−	0.987696i	$-0.450016\pi$
$367$	8.74068e46	0.312768	0.156384	+	0.987696i	$0.450016\pi$
$368$	0	0
$369$	1.83129e47	0.592609
$370$	0	0
$371$	−7.04073e46	−0.206158
$372$	0	0
$373$	2.70122e47	0.716056	0.358028	−	0.933711i	$-0.383449\pi$
$373$	2.70122e47	0.716056	0.358028	+	0.933711i	$0.383449\pi$
$374$	0	0
$375$	1.31028e46	0.0314620
$376$	0	0
$377$	−8.25170e47	−1.79568
$378$	0	0
$379$	4.13508e47	0.815940	0.407970	−	0.912995i	$-0.366237\pi$
$379$	4.13508e47	0.815940	0.407970	+	0.912995i	$0.366237\pi$
$380$	0	0
$381$	2.93395e46	0.0525219
$382$	0	0
$383$	6.64131e47	1.07914	0.539569	−	0.841942i	$-0.318587\pi$
$383$	6.64131e47	1.07914	0.539569	+	0.841942i	$0.318587\pi$
$384$	0	0
$385$	4.20901e47	0.621092
$386$	0	0
$387$	−1.31550e48	−1.76375
$388$	0	0
$389$	4.74132e46	0.0577872	0.0288936	−	0.999582i	$-0.490802\pi$
$389$	4.74132e46	0.0577872	0.0288936	+	0.999582i	$0.490802\pi$
$390$	0	0
$391$	−2.89134e45	−0.00320500
$392$	0	0
$393$	−4.36716e46	−0.0440491
$394$	0	0
$395$	1.00020e48	0.918417
$396$	0	0
$397$	6.00594e47	0.502292	0.251146	−	0.967949i	$-0.419193\pi$
$397$	6.00594e47	0.502292	0.251146	+	0.967949i	$0.419193\pi$
$398$	0	0
$399$	2.77289e46	0.0211317
$400$	0	0
$401$	−9.30169e47	−0.646235	−0.323118	−	0.946359i	$-0.604731\pi$
$401$	−9.30169e47	−0.646235	−0.323118	+	0.946359i	$0.604731\pi$
$402$	0	0
$403$	−1.41582e48	−0.897147
$404$	0	0
$405$	−1.67754e48	−0.969963
$406$	0	0
$407$	−3.34641e48	−1.76637
$408$	0	0
$409$	−2.60436e47	−0.125551	−0.0627755	−	0.998028i	$-0.519995\pi$
$409$	−2.60436e47	−0.125551	−0.0627755	+	0.998028i	$0.519995\pi$
$410$	0	0
$411$	−2.59206e46	−0.0114175
$412$	0	0
$413$	−5.99367e47	−0.241333
$414$	0	0
$415$	−2.71067e48	−0.998130
$416$	0	0
$417$	6.44570e46	0.0217147
$418$	0	0
$419$	3.92255e48	1.20951	0.604755	−	0.796412i	$-0.293271\pi$
$419$	3.92255e48	1.20951	0.604755	+	0.796412i	$0.293271\pi$
$420$	0	0
$421$	−3.86983e48	−1.09263	−0.546315	−	0.837580i	$-0.683970\pi$
$421$	−3.86983e48	−1.09263	−0.546315	+	0.837580i	$0.683970\pi$
$422$	0	0
$423$	1.61195e48	0.416922
$424$	0	0
$425$	1.38185e47	0.0327542
$426$	0	0
$427$	−2.29380e48	−0.498471
$428$	0	0
$429$	2.87377e47	0.0572785
$430$	0	0
$431$	−2.97992e48	−0.544973	−0.272487	−	0.962160i	$-0.587846\pi$
$431$	−2.97992e48	−0.544973	−0.272487	+	0.962160i	$0.587846\pi$
$432$	0	0
$433$	−2.20884e48	−0.370798	−0.185399	−	0.982663i	$-0.559358\pi$
$433$	−2.20884e48	−0.370798	−0.185399	+	0.982663i	$0.559358\pi$
$434$	0	0
$435$	2.59729e47	0.0400377
$436$	0	0
$437$	−5.59031e46	−0.00791643
$438$	0	0
$439$	1.14225e49	1.48650	0.743252	−	0.669011i	$-0.233282\pi$
$439$	1.14225e49	1.48650	0.743252	+	0.669011i	$0.233282\pi$
$440$	0	0
$441$	6.59779e48	0.789378
$442$	0	0
$443$	−8.78568e48	−0.966733	−0.483366	−	0.875418i	$-0.660586\pi$
$443$	−8.78568e48	−0.966733	−0.483366	+	0.875418i	$0.660586\pi$
$444$	0	0
$445$	1.29531e49	1.31134
$446$	0	0
$447$	6.15943e46	0.00573920
$448$	0	0
$449$	−8.94932e48	−0.767773	−0.383886	−	0.923380i	$-0.625415\pi$
$449$	−8.94932e48	−0.767773	−0.383886	+	0.923380i	$0.625415\pi$
$450$	0	0
$451$	−1.04624e49	−0.826735
$452$	0	0
$453$	1.45607e47	0.0106015
$454$	0	0
$455$	8.89101e48	0.596678
$456$	0	0
$457$	3.77717e48	0.233731	0.116866	−	0.993148i	$-0.462715\pi$
$457$	3.77717e48	0.233731	0.116866	+	0.993148i	$0.462715\pi$
$458$	0	0
$459$	6.54048e47	0.0373315
$460$	0	0
$461$	8.89831e48	0.468641	0.234320	−	0.972159i	$-0.424714\pi$
$461$	8.89831e48	0.468641	0.234320	+	0.972159i	$0.424714\pi$
$462$	0	0
$463$	3.35423e49	1.63059	0.815294	−	0.579047i	$-0.196575\pi$
$463$	3.35423e49	1.63059	0.815294	+	0.579047i	$0.196575\pi$
$464$	0	0
$465$	4.45640e47	0.0200034
$466$	0	0
$467$	3.67520e49	1.52377	0.761884	−	0.647714i	$-0.224275\pi$
$467$	3.67520e49	1.52377	0.761884	+	0.647714i	$0.224275\pi$
$468$	0	0
$469$	−4.82240e48	−0.184742
$470$	0	0
$471$	1.45132e48	0.0513897
$472$	0	0
$473$	7.51563e49	2.46056
$474$	0	0
$475$	2.67177e48	0.0809035
$476$	0	0
$477$	−1.60471e49	−0.449581
$478$	0	0
$479$	−4.73384e49	−1.22746	−0.613730	−	0.789516i	$-0.710331\pi$
$479$	−4.73384e49	−1.22746	−0.613730	+	0.789516i	$0.710331\pi$
$480$	0	0
$481$	−7.06889e49	−1.69694
$482$	0	0
$483$	−3.33153e45	−7.40666e−5 0
$484$	0	0
$485$	8.89084e48	0.183114
$486$	0	0
$487$	−7.97298e48	−0.152172	−0.0760860	−	0.997101i	$-0.524242\pi$
$487$	−7.97298e48	−0.152172	−0.0760860	+	0.997101i	$0.524242\pi$
$488$	0	0
$489$	1.68203e48	0.0297592
$490$	0	0
$491$	−3.59851e49	−0.590358	−0.295179	−	0.955442i	$-0.595379\pi$
$491$	−3.59851e49	−0.590358	−0.295179	+	0.955442i	$0.595379\pi$
$492$	0	0
$493$	5.36220e49	0.815972
$494$	0	0
$495$	9.59311e49	1.35445
$496$	0	0
$497$	1.24392e49	0.163004
$498$	0	0
$499$	−7.98372e49	−0.971282	−0.485641	−	0.874158i	$-0.661414\pi$
$499$	−7.98372e49	−0.971282	−0.485641	+	0.874158i	$0.661414\pi$
$500$	0	0
$501$	8.11529e47	0.00916866
$502$	0	0
$503$	−6.32137e49	−0.663443	−0.331721	−	0.943377i	$-0.607629\pi$
$503$	−6.32137e49	−0.663443	−0.331721	+	0.943377i	$0.607629\pi$
$504$	0	0
$505$	−1.07465e50	−1.04804
$506$	0	0
$507$	2.68455e48	0.0243346
$508$	0	0
$509$	−1.42477e50	−1.20078	−0.600392	−	0.799706i	$-0.704989\pi$
$509$	−1.42477e50	−1.20078	−0.600392	+	0.799706i	$0.704989\pi$
$510$	0	0
$511$	−2.02756e49	−0.158923
$512$	0	0
$513$	1.26458e49	0.0922095
$514$	0	0
$515$	−2.81745e49	−0.191171
$516$	0	0
$517$	−9.20932e49	−0.581638
$518$	0	0
$519$	5.38710e48	0.0316781
$520$	0	0
$521$	1.18735e50	0.650251	0.325125	−	0.945671i	$-0.394593\pi$
$521$	1.18735e50	0.650251	0.325125	+	0.945671i	$0.394593\pi$
$522$	0	0
$523$	−6.66586e49	−0.340075	−0.170038	−	0.985438i	$-0.554389\pi$
$523$	−6.66586e49	−0.340075	−0.170038	+	0.985438i	$0.554389\pi$
$524$	0	0
$525$	1.59224e47	0.000756938 0
$526$	0	0
$527$	9.20040e49	0.407672
$528$	0	0
$529$	−2.42057e50	−0.999972
$530$	0	0
$531$	−1.36607e50	−0.526289
$532$	0	0
$533$	−2.21006e50	−0.794238
$534$	0	0
$535$	−1.31599e50	−0.441274
$536$	0	0
$537$	8.68884e48	0.0271918
$538$	0	0
$539$	−3.76942e50	−1.10124
$540$	0	0
$541$	7.31652e50	1.99597	0.997987	−	0.0634127i	$-0.0201984\pi$
$541$	7.31652e50	1.99597	0.997987	+	0.0634127i	$0.0201984\pi$
$542$	0	0
$543$	−1.95917e48	−0.00499201
$544$	0	0
$545$	8.86616e48	0.0211057
$546$	0	0
$547$	−1.90668e50	−0.424143	−0.212072	−	0.977254i	$-0.568021\pi$
$547$	−1.90668e50	−0.424143	−0.212072	+	0.977254i	$0.568021\pi$
$548$	0	0
$549$	−5.22800e50	−1.08704
$550$	0	0
$551$	1.03677e51	2.01547
$552$	0	0
$553$	2.37933e50	0.432553
$554$	0	0
$555$	2.22499e49	0.0378363
$556$	0	0
$557$	6.78617e50	1.07970	0.539851	−	0.841761i	$-0.318481\pi$
$557$	6.78617e50	1.07970	0.539851	+	0.841761i	$0.318481\pi$
$558$	0	0
$559$	1.58759e51	2.36384
$560$	0	0
$561$	−1.86746e49	−0.0260279
$562$	0	0
$563$	−6.45010e50	−0.841708	−0.420854	−	0.907128i	$-0.638270\pi$
$563$	−6.45010e50	−0.841708	−0.420854	+	0.907128i	$0.638270\pi$
$564$	0	0
$565$	−1.36214e50	−0.166466
$566$	0	0
$567$	−3.99063e50	−0.456830
$568$	0	0
$569$	2.68168e50	0.287627	0.143814	−	0.989605i	$-0.454063\pi$
$569$	2.68168e50	0.287627	0.143814	+	0.989605i	$0.454063\pi$
$570$	0	0
$571$	3.44946e50	0.346724	0.173362	−	0.984858i	$-0.444537\pi$
$571$	3.44946e50	0.346724	0.173362	+	0.984858i	$0.444537\pi$
$572$	0	0
$573$	5.38448e48	0.00507321
$574$	0	0
$575$	−3.21004e47	−0.000283567 0
$576$	0	0
$577$	2.52648e50	0.209297	0.104648	−	0.994509i	$-0.466628\pi$
$577$	2.52648e50	0.209297	0.104648	+	0.994509i	$0.466628\pi$
$578$	0	0
$579$	−1.29149e49	−0.0100355
$580$	0	0
$581$	−6.44830e50	−0.470096
$582$	0	0
$583$	9.16799e50	0.627199
$584$	0	0
$585$	2.02643e51	1.30121
$586$	0	0
$587$	−1.13653e51	−0.685134	−0.342567	−	0.939493i	$-0.611296\pi$
$587$	−1.13653e51	−0.685134	−0.342567	+	0.939493i	$0.611296\pi$
$588$	0	0
$589$	1.77887e51	1.00696
$590$	0	0
$591$	−3.72945e49	−0.0198279
$592$	0	0
$593$	−1.23614e51	−0.617383	−0.308692	−	0.951162i	$-0.599891\pi$
$593$	−1.23614e51	−0.617383	−0.308692	+	0.951162i	$0.599891\pi$
$594$	0	0
$595$	−5.77765e50	−0.271136
$596$	0	0
$597$	7.78766e48	0.00343464
$598$	0	0
$599$	4.59427e51	1.90467	0.952336	−	0.305052i	$-0.0986737\pi$
$599$	4.59427e51	1.90467	0.952336	+	0.305052i	$0.0986737\pi$
$600$	0	0
$601$	−1.48401e49	−0.00578442	−0.00289221	−	0.999996i	$-0.500921\pi$
$601$	−1.48401e49	−0.00578442	−0.00289221	+	0.999996i	$0.500921\pi$
$602$	0	0
$603$	−1.09911e51	−0.402877
$604$	0	0
$605$	−2.65933e51	−0.916850
$606$	0	0
$607$	−2.59968e51	−0.843198	−0.421599	−	0.906782i	$-0.638531\pi$
$607$	−2.59968e51	−0.843198	−0.421599	+	0.906782i	$0.638531\pi$
$608$	0	0
$609$	6.17859e49	0.0188568
$610$	0	0
$611$	−1.94536e51	−0.558775
$612$	0	0
$613$	1.49376e50	0.0403890	0.0201945	−	0.999796i	$-0.493571\pi$
$613$	1.49376e50	0.0403890	0.0201945	+	0.999796i	$0.493571\pi$
$614$	0	0
$615$	6.95635e49	0.0177089
$616$	0	0
$617$	−6.30081e51	−1.51050	−0.755250	−	0.655437i	$-0.772484\pi$
$617$	−6.30081e51	−1.51050	−0.755250	+	0.655437i	$0.772484\pi$
$618$	0	0
$619$	−5.97852e51	−1.34995	−0.674974	−	0.737842i	$-0.735845\pi$
$619$	−5.97852e51	−1.34995	−0.674974	+	0.737842i	$0.735845\pi$
$620$	0	0
$621$	−1.51935e48	−0.000323195 0
$622$	0	0
$623$	3.08136e51	0.617609
$624$	0	0
$625$	−4.99363e51	−0.943269
$626$	0	0
$627$	−3.61068e50	−0.0642893
$628$	0	0
$629$	4.59358e51	0.771106
$630$	0	0
$631$	3.17011e51	0.501800	0.250900	−	0.968013i	$-0.419273\pi$
$631$	3.17011e51	0.501800	0.250900	+	0.968013i	$0.419273\pi$
$632$	0	0
$633$	−3.43499e50	−0.0512810
$634$	0	0
$635$	−1.18198e52	−1.66454
$636$	0	0
$637$	−7.96244e51	−1.05795
$638$	0	0
$639$	2.83512e51	0.355473
$640$	0	0
$641$	1.15243e52	1.36377	0.681885	−	0.731459i	$-0.261160\pi$
$641$	1.15243e52	1.36377	0.681885	+	0.731459i	$0.261160\pi$
$642$	0	0
$643$	−1.66832e52	−1.86370	−0.931851	−	0.362842i	$-0.881807\pi$
$643$	−1.66832e52	−1.86370	−0.931851	+	0.362842i	$0.881807\pi$
$644$	0	0
$645$	−4.99706e50	−0.0527060
$646$	0	0
$647$	1.67531e52	1.66865	0.834327	−	0.551270i	$-0.185856\pi$
$647$	1.67531e52	1.66865	0.834327	+	0.551270i	$0.185856\pi$
$648$	0	0
$649$	7.80457e51	0.734213
$650$	0	0
$651$	1.06011e50	0.00942116
$652$	0	0
$653$	−9.01096e51	−0.756619	−0.378309	−	0.925679i	$-0.623494\pi$
$653$	−9.01096e51	−0.756619	−0.378309	+	0.925679i	$0.623494\pi$
$654$	0	0
$655$	1.75936e52	1.39602
$656$	0	0
$657$	−4.62118e51	−0.346572
$658$	0	0
$659$	−1.92183e52	−1.36249	−0.681247	−	0.732054i	$-0.738562\pi$
$659$	−1.92183e52	−1.36249	−0.681247	+	0.732054i	$0.738562\pi$
$660$	0	0
$661$	−1.12368e51	−0.0753213	−0.0376606	−	0.999291i	$-0.511991\pi$
$661$	−1.12368e51	−0.0753213	−0.0376606	+	0.999291i	$0.511991\pi$
$662$	0	0
$663$	−3.94478e50	−0.0250048
$664$	0	0
$665$	−1.11709e52	−0.669712
$666$	0	0
$667$	−1.24564e50	−0.00706422
$668$	0	0
$669$	7.41611e50	0.0397918
$670$	0	0
$671$	2.98684e52	1.51651
$672$	0	0
$673$	6.07438e51	0.291893	0.145946	−	0.989292i	$-0.453377\pi$
$673$	6.07438e51	0.291893	0.145946	+	0.989292i	$0.453377\pi$
$674$	0	0
$675$	7.26143e49	0.00330295
$676$	0	0
$677$	3.94010e52	1.69675	0.848373	−	0.529398i	$-0.177582\pi$
$677$	3.94010e52	1.69675	0.848373	+	0.529398i	$0.177582\pi$
$678$	0	0
$679$	2.11500e51	0.0862423
$680$	0	0
$681$	−1.39289e51	−0.0537892
$682$	0	0
$683$	2.05041e52	0.749993	0.374996	−	0.927026i	$-0.377644\pi$
$683$	2.05041e52	0.749993	0.374996	+	0.927026i	$0.377644\pi$
$684$	0	0
$685$	1.04424e52	0.361847
$686$	0	0
$687$	−1.04893e51	−0.0344389
$688$	0	0
$689$	1.93662e52	0.602545
$690$	0	0
$691$	−4.63109e51	−0.136565	−0.0682825	−	0.997666i	$-0.521752\pi$
$691$	−4.63109e51	−0.136565	−0.0682825	+	0.997666i	$0.521752\pi$
$692$	0	0
$693$	2.28206e52	0.637915
$694$	0	0
$695$	−2.59672e52	−0.688188
$696$	0	0
$697$	1.43616e52	0.360909
$698$	0	0
$699$	3.69433e50	0.00880456
$700$	0	0
$701$	−6.34322e51	−0.143392	−0.0716962	−	0.997427i	$-0.522841\pi$
$701$	−6.34322e51	−0.143392	−0.0716962	+	0.997427i	$0.522841\pi$
$702$	0	0
$703$	8.88154e52	1.90465
$704$	0	0
$705$	6.12318e50	0.0124589
$706$	0	0
$707$	−2.55645e52	−0.493602
$708$	0	0
$709$	−7.70939e52	−1.41275	−0.706374	−	0.707839i	$-0.749670\pi$
$709$	−7.70939e52	−1.41275	−0.706374	+	0.707839i	$0.749670\pi$
$710$	0	0
$711$	5.42293e52	0.943293
$712$	0	0
$713$	−2.13725e50	−0.00352939
$714$	0	0
$715$	−1.15773e53	−1.81529
$716$	0	0
$717$	5.90989e50	0.00879985
$718$	0	0
$719$	−5.22428e52	−0.738826	−0.369413	−	0.929265i	$-0.620441\pi$
$719$	−5.22428e52	−0.738826	−0.369413	+	0.929265i	$0.620441\pi$
$720$	0	0
$721$	−6.70231e51	−0.0900372
$722$	0	0
$723$	−2.31196e51	−0.0295068
$724$	0	0
$725$	5.95327e51	0.0721942
$726$	0	0
$727$	1.24495e53	1.43471	0.717354	−	0.696708i	$-0.245353\pi$
$727$	1.24495e53	1.43471	0.717354	+	0.696708i	$0.245353\pi$
$728$	0	0
$729$	−9.07820e52	−0.994352
$730$	0	0
$731$	−1.03166e53	−1.07415
$732$	0	0
$733$	−1.12627e52	−0.111485	−0.0557427	−	0.998445i	$-0.517753\pi$
$733$	−1.12627e52	−0.111485	−0.0557427	+	0.998445i	$0.517753\pi$
$734$	0	0
$735$	2.50625e51	0.0235889
$736$	0	0
$737$	6.27941e52	0.562044
$738$	0	0
$739$	2.92067e52	0.248634	0.124317	−	0.992243i	$-0.460326\pi$
$739$	2.92067e52	0.248634	0.124317	+	0.992243i	$0.460326\pi$
$740$	0	0
$741$	−7.62711e51	−0.0617623
$742$	0	0
$743$	−3.47917e52	−0.268029	−0.134015	−	0.990979i	$-0.542787\pi$
$743$	−3.47917e52	−0.268029	−0.134015	+	0.990979i	$0.542787\pi$
$744$	0	0
$745$	−2.48140e52	−0.181889
$746$	0	0
$747$	−1.46969e53	−1.02516
$748$	0	0
$749$	−3.13055e52	−0.207830
$750$	0	0
$751$	−1.55213e53	−0.980818	−0.490409	−	0.871492i	$-0.663153\pi$
$751$	−1.55213e53	−0.980818	−0.490409	+	0.871492i	$0.663153\pi$
$752$	0	0
$753$	8.28513e51	0.0498416
$754$	0	0
$755$	−5.86596e52	−0.335985
$756$	0	0
$757$	−1.00018e51	−0.00545510	−0.00272755	−	0.999996i	$-0.500868\pi$
$757$	−1.00018e51	−0.00545510	−0.00272755	+	0.999996i	$0.500868\pi$
$758$	0	0
$759$	4.33811e49	0.000225334 0
$760$	0	0
$761$	−3.93324e53	−1.94596	−0.972981	−	0.230886i	$-0.925838\pi$
$761$	−3.93324e53	−1.94596	−0.972981	+	0.230886i	$0.925838\pi$
$762$	0	0
$763$	2.10913e51	0.00994031
$764$	0	0
$765$	−1.31683e53	−0.591282
$766$	0	0
$767$	1.64862e53	0.705353
$768$	0	0
$769$	4.31322e53	1.75859	0.879295	−	0.476278i	$-0.158015\pi$
$769$	4.31322e53	1.75859	0.879295	+	0.476278i	$0.158015\pi$
$770$	0	0
$771$	1.11827e52	0.0434552
$772$	0	0
$773$	1.93734e53	0.717604	0.358802	−	0.933414i	$-0.383185\pi$
$773$	1.93734e53	0.717604	0.358802	+	0.933414i	$0.383185\pi$
$774$	0	0
$775$	1.02145e52	0.0360693
$776$	0	0
$777$	5.29294e51	0.0178200
$778$	0	0
$779$	2.77678e53	0.891452
$780$	0	0
$781$	−1.61975e53	−0.495911
$782$	0	0
$783$	2.81776e52	0.0822832
$784$	0	0
$785$	−5.84680e53	−1.62866
$786$	0	0
$787$	1.53439e52	0.0407759	0.0203880	−	0.999792i	$-0.493510\pi$
$787$	1.53439e52	0.0407759	0.0203880	+	0.999792i	$0.493510\pi$
$788$	0	0
$789$	1.64390e52	0.0416821
$790$	0	0
$791$	−3.24034e52	−0.0784018
$792$	0	0
$793$	6.30933e53	1.45690
$794$	0	0
$795$	−6.09569e51	−0.0134348
$796$	0	0
$797$	5.82231e53	1.22494	0.612472	−	0.790493i	$-0.290175\pi$
$797$	5.82231e53	1.22494	0.612472	+	0.790493i	$0.290175\pi$
$798$	0	0
$799$	1.26415e53	0.253912
$800$	0	0
$801$	7.02299e53	1.34685
$802$	0	0
$803$	2.64015e53	0.483495
$804$	0	0
$805$	1.34215e51	0.00234734
$806$	0	0
$807$	−2.10464e52	−0.0351574
$808$	0	0
$809$	7.29683e53	1.16436	0.582179	−	0.813061i	$-0.302200\pi$
$809$	7.29683e53	1.16436	0.582179	+	0.813061i	$0.302200\pi$
$810$	0	0
$811$	8.04109e53	1.22583	0.612914	−	0.790150i	$-0.289997\pi$
$811$	8.04109e53	1.22583	0.612914	+	0.790150i	$0.289997\pi$
$812$	0	0
$813$	5.41870e51	0.00789261
$814$	0	0
$815$	−6.77627e53	−0.943137
$816$	0	0
$817$	−1.99469e54	−2.65318
$818$	0	0
$819$	4.82058e53	0.612840
$820$	0	0
$821$	−1.06873e52	−0.0129873	−0.00649364	−	0.999979i	$-0.502067\pi$
$821$	−1.06873e52	−0.0129873	−0.00649364	+	0.999979i	$0.502067\pi$
$822$	0	0
$823$	−1.55439e54	−1.80578	−0.902889	−	0.429874i	$-0.858558\pi$
$823$	−1.55439e54	−1.80578	−0.902889	+	0.429874i	$0.858558\pi$
$824$	0	0
$825$	−2.07331e51	−0.00230285
$826$	0	0
$827$	1.52865e54	1.62352	0.811759	−	0.583992i	$-0.198510\pi$
$827$	1.52865e54	1.62352	0.811759	+	0.583992i	$0.198510\pi$
$828$	0	0
$829$	4.48559e53	0.455576	0.227788	−	0.973711i	$-0.426851\pi$
$829$	4.48559e53	0.455576	0.227788	+	0.973711i	$0.426851\pi$
$830$	0	0
$831$	3.31346e52	0.0321857
$832$	0	0
$833$	5.17424e53	0.480744
$834$	0	0
$835$	−3.26934e53	−0.290576
$836$	0	0
$837$	4.83467e52	0.0411099
$838$	0	0
$839$	1.70786e54	1.38949	0.694747	−	0.719254i	$-0.255516\pi$
$839$	1.70786e54	1.38949	0.694747	+	0.719254i	$0.255516\pi$
$840$	0	0
$841$	1.02566e54	0.798502
$842$	0	0
$843$	5.89519e52	0.0439226
$844$	0	0
$845$	−1.08150e54	−0.771220
$846$	0	0
$847$	−6.32617e53	−0.431815
$848$	0	0
$849$	−1.64245e52	−0.0107325
$850$	0	0
$851$	−1.06709e52	−0.00667580
$852$	0	0
$853$	−1.02278e54	−0.612667	−0.306334	−	0.951924i	$-0.599102\pi$
$853$	−1.02278e54	−0.612667	−0.306334	+	0.951924i	$0.599102\pi$
$854$	0	0
$855$	−2.54606e54	−1.46048
$856$	0	0
$857$	7.33586e52	0.0403001	0.0201500	−	0.999797i	$-0.493586\pi$
$857$	7.33586e52	0.0403001	0.0201500	+	0.999797i	$0.493586\pi$
$858$	0	0
$859$	−1.46329e54	−0.769938	−0.384969	−	0.922929i	$-0.625788\pi$
$859$	−1.46329e54	−0.769938	−0.384969	+	0.922929i	$0.625788\pi$
$860$	0	0
$861$	1.65482e52	0.00834049
$862$	0	0
$863$	−2.49462e54	−1.20449	−0.602247	−	0.798310i	$-0.705728\pi$
$863$	−2.49462e54	−1.20449	−0.602247	+	0.798310i	$0.705728\pi$
$864$	0	0
$865$	−2.17025e54	−1.00395
$866$	0	0
$867$	−4.36098e52	−0.0193300
$868$	0	0
$869$	−3.09821e54	−1.31597
$870$	0	0
$871$	1.32645e54	0.539951
$872$	0	0
$873$	4.82049e53	0.188073
$874$	0	0
$875$	−1.25571e54	−0.469612
$876$	0	0
$877$	4.03070e54	1.44507	0.722534	−	0.691335i	$-0.242977\pi$
$877$	4.03070e54	1.44507	0.722534	+	0.691335i	$0.242977\pi$
$878$	0	0
$879$	1.06520e52	0.00366130
$880$	0	0
$881$	−2.32935e54	−0.767679	−0.383840	−	0.923400i	$-0.625398\pi$
$881$	−2.32935e54	−0.767679	−0.383840	+	0.923400i	$0.625398\pi$
$882$	0	0
$883$	2.99176e54	0.945482	0.472741	−	0.881202i	$-0.343265\pi$
$883$	2.99176e54	0.945482	0.472741	+	0.881202i	$0.343265\pi$
$884$	0	0
$885$	−5.18917e52	−0.0157271
$886$	0	0
$887$	−5.34771e54	−1.55447	−0.777233	−	0.629213i	$-0.783377\pi$
$887$	−5.34771e54	−1.55447	−0.777233	+	0.629213i	$0.783377\pi$
$888$	0	0
$889$	−2.81175e54	−0.783960
$890$	0	0
$891$	5.19634e54	1.38982
$892$	0	0
$893$	2.44420e54	0.627169
$894$	0	0
$895$	−3.50040e54	−0.861770
$896$	0	0
$897$	9.16372e50	0.000216477 0
$898$	0	0
$899$	3.96370e54	0.898558
$900$	0	0
$901$	−1.25848e54	−0.273802
$902$	0	0
$903$	−1.18873e53	−0.0248233
$904$	0	0
$905$	7.89275e53	0.158208
$906$	0	0
$907$	6.45044e54	1.24124	0.620618	−	0.784113i	$-0.286882\pi$
$907$	6.45044e54	1.24124	0.620618	+	0.784113i	$0.286882\pi$
$908$	0	0
$909$	−5.82662e54	−1.07643
$910$	0	0
$911$	−2.81261e54	−0.498907	−0.249453	−	0.968387i	$-0.580251\pi$
$911$	−2.81261e54	−0.498907	−0.249453	+	0.968387i	$0.580251\pi$
$912$	0	0
$913$	8.39655e54	1.43018
$914$	0	0
$915$	−1.98592e53	−0.0324841
$916$	0	0
$917$	4.18526e54	0.657491
$918$	0	0
$919$	−5.50843e54	−0.831171	−0.415586	−	0.909554i	$-0.636423\pi$
$919$	−5.50843e54	−0.831171	−0.415586	+	0.909554i	$0.636423\pi$
$920$	0	0
$921$	−1.97535e53	−0.0286314
$922$	0	0
$923$	−3.42152e54	−0.476418
$924$	0	0
$925$	5.09992e53	0.0682246
$926$	0	0
$927$	−1.52758e54	−0.196349
$928$	0	0
$929$	9.70568e54	1.19877	0.599384	−	0.800461i	$-0.295412\pi$
$929$	9.70568e54	1.19877	0.599384	+	0.800461i	$0.295412\pi$
$930$	0	0
$931$	1.00042e55	1.18745
$932$	0	0
$933$	1.04184e53	0.0118848
$934$	0	0
$935$	7.52328e54	0.824883
$936$	0	0
$937$	−4.05525e54	−0.427401	−0.213701	−	0.976899i	$-0.568552\pi$
$937$	−4.05525e54	−0.427401	−0.213701	+	0.976899i	$0.568552\pi$
$938$	0	0
$939$	1.05544e53	0.0106935
$940$	0	0
$941$	4.56539e54	0.444703	0.222351	−	0.974967i	$-0.428627\pi$
$941$	4.56539e54	0.444703	0.222351	+	0.974967i	$0.428627\pi$
$942$	0	0
$943$	−3.33621e52	−0.00312454
$944$	0	0
$945$	−3.03607e53	−0.0273415
$946$	0	0
$947$	1.72468e55	1.49360	0.746798	−	0.665051i	$-0.231590\pi$
$947$	1.72468e55	1.49360	0.746798	+	0.665051i	$0.231590\pi$
$948$	0	0
$949$	5.57700e54	0.464490
$950$	0	0
$951$	−4.41804e53	−0.0353908
$952$	0	0
$953$	6.30605e52	0.00485891	0.00242945	−	0.999997i	$-0.499227\pi$
$953$	6.30605e52	0.00485891	0.00242945	+	0.999997i	$0.499227\pi$
$954$	0	0
$955$	−2.16920e54	−0.160782
$956$	0	0
$957$	−8.04535e53	−0.0573686
$958$	0	0
$959$	2.48410e54	0.170422
$960$	0	0
$961$	−8.34809e54	−0.551067
$962$	0	0
$963$	−7.13512e54	−0.453226
$964$	0	0
$965$	5.20291e54	0.318047
$966$	0	0
$967$	1.73337e55	1.01977	0.509883	−	0.860244i	$-0.329689\pi$
$967$	1.73337e55	1.01977	0.509883	+	0.860244i	$0.329689\pi$
$968$	0	0
$969$	4.95633e53	0.0280653
$970$	0	0
$971$	−1.04588e55	−0.570067	−0.285033	−	0.958518i	$-0.592005\pi$
$971$	−1.04588e55	−0.570067	−0.285033	+	0.958518i	$0.592005\pi$
$972$	0	0
$973$	−6.17723e54	−0.324121
$974$	0	0
$975$	−4.37961e52	−0.00221233
$976$	0	0
$977$	−1.65065e55	−0.802798	−0.401399	−	0.915903i	$-0.631476\pi$
$977$	−1.65065e55	−0.802798	−0.401399	+	0.915903i	$0.631476\pi$
$978$	0	0
$979$	−4.01235e55	−1.87896
$980$	0	0
$981$	4.80711e53	0.0216774
$982$	0	0
$983$	−3.82232e55	−1.65992	−0.829960	−	0.557823i	$-0.811637\pi$
$983$	−3.82232e55	−1.65992	−0.829960	+	0.557823i	$0.811637\pi$
$984$	0	0
$985$	1.50245e55	0.628392
$986$	0	0
$987$	1.45662e53	0.00586783
$988$	0	0
$989$	2.39655e53	0.00929939
$990$	0	0
$991$	4.27417e55	1.59768	0.798838	−	0.601546i	$-0.205448\pi$
$991$	4.27417e55	1.59768	0.798838	+	0.601546i	$0.205448\pi$
$992$	0	0
$993$	−3.81585e53	−0.0137414
$994$	0	0
$995$	−3.13735e54	−0.108852
$996$	0	0
$997$	−3.39750e55	−1.13579	−0.567896	−	0.823100i	$-0.692243\pi$
$997$	−3.39750e55	−1.13579	−0.567896	+	0.823100i	$0.692243\pi$
$998$	0	0
$999$	2.41386e54	0.0777588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.38.a.b.1.2		2
4.3	odd	2		1.38.a.a.1.2	✓	2
12.11	even	2		9.38.a.a.1.1		2
20.3	even	4		25.38.b.a.24.2		4
20.7	even	4		25.38.b.a.24.3		4
20.19	odd	2		25.38.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.38.a.a.1.2	✓	2	4.3	odd	2
9.38.a.a.1.1		2	12.11	even	2
16.38.a.b.1.2		2	1.1	even	1	trivial
25.38.a.a.1.1		2	20.19	odd	2
25.38.b.a.24.2		4	20.3	even	4
25.38.b.a.24.3		4	20.7	even	4

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:	\( k \)	\(=\)	\( 38 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+2.05955e7 q^{3} -8.29715e12 q^{5} -1.97377e15 q^{7} -4.49860e17 q^{9} +O(q^{10})\)	\(q+2.05955e7 q^{3} -8.29715e12 q^{5} -1.97377e15 q^{7} -4.49860e17 q^{9} +2.57012e19 q^{11} +5.42906e20 q^{13} -1.70884e20 q^{15} -3.52797e22 q^{17} -6.82122e23 q^{19} -4.06509e22 q^{21} +8.19547e22 q^{23} -3.91685e24 q^{25} -1.85389e25 q^{27} -1.51991e27 q^{29} -2.60785e27 q^{31} +5.29330e26 q^{33} +1.63767e28 q^{35} -1.30205e29 q^{37} +1.11814e28 q^{39} -4.07079e29 q^{41} +2.92424e30 q^{43} +3.73255e30 q^{45} -3.58323e30 q^{47} -1.46663e31 q^{49} -7.26605e29 q^{51} +3.56714e31 q^{53} -2.13247e32 q^{55} -1.40487e31 q^{57} +3.03666e32 q^{59} +1.16214e33 q^{61} +8.87921e32 q^{63} -4.50457e33 q^{65} +2.44324e33 q^{67} +1.68790e30 q^{69} -6.30224e33 q^{71} +1.02725e34 q^{73} -8.06697e31 q^{75} -5.07283e34 q^{77} -1.20547e35 q^{79} +2.02183e35 q^{81} +3.26699e35 q^{83} +2.92721e35 q^{85} -3.13034e34 q^{87} -1.56115e36 q^{89} -1.07157e36 q^{91} -5.37100e34 q^{93} +5.65967e36 q^{95} -1.07155e36 q^{97} -1.15619e37 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 13991400 q^{3} + 5529584385900 q^{5} + 34\!\cdots\!00 q^{7}+ \cdots - 89\!\cdots\!14 q^{9}+O(q^{10}) \) 2 * q - 13991400 * q^3 + 5529584385900 * q^5 + 3448443953486000 * q^7 - 898947378401769414 * q^9	\( 2 q - 13991400 q^{3} + 5529584385900 q^{5} + 34\!\cdots\!00 q^{7}+ \cdots - 12\!\cdots\!92 q^{99}+O(q^{100}) \) 2 * q - 13991400 * q^3 + 5529584385900 * q^5 + 3448443953486000 * q^7 - 898947378401769414 * q^9 + 26734036354848538056 * q^11 + 530581741653933178300 * q^13 - 649108943683113884400 * q^15 - 89439073881931767332700 * q^17 - 373276466572513713706120 * q^19 - 228188865811378281195456 * q^21 + 26241180149933881945462800 * q^23 + 114502199615774855754278750 * q^25 + 12567565757525316335881200 * q^27 - 1270673827125882417951629220 * q^29 - 261420819791418895481545024 * q^31 + 493606999076023001294056800 * q^33 + 91348258395203301140779687200 * q^35 - 68025363793820786588733238100 * q^37 + 11607709470242600846801346768 * q^39 - 1260483295466373133974684841836 * q^41 + 2570241023157918831169581605000 * q^43 - 2476861984515775772921684271300 * q^45 - 4252206875568934025158407583200 * q^47 - 3828013658149768338307153838286 * q^49 + 1146602989088714802084517292976 * q^51 + 159799736258590810071505678893900 * q^53 - 198965756266726963295068969294800 * q^55 - 24730693798747893221767420706400 * q^57 + 237962459606090128758899369700840 * q^59 + 101798841373038700200106255199644 * q^61 - 1547129676521036250657950915523600 * q^63 - 4674979790566994364848214315456600 * q^65 + 10891923280981358108643809352546200 * q^67 - 903079825405531082422221799280448 * q^69 + 746133089793832492689962158868016 * q^71 + 19639851676496426023909382507487700 * q^73 - 4176423071329708963282786324335000 * q^75 - 45127994201424729507574115685000000 * q^77 - 272401638034095839035647945144674080 * q^79 + 403323837550038341003235827350619442 * q^81 + 470460275390909929308746217601073400 * q^83 - 456126475708535200141442602834276200 * q^85 - 39923812909186495628003917423124400 * q^87 - 1332868921711238117914343978794942860 * q^89 - 1138398786311531726477856383010207584 * q^91 - 134865729229255425577378938731155200 * q^93 + 9929993151899196579173099213128626000 * q^95 + 6002061888473229973039130090661237700 * q^97 - 12025768918384639461946415841980309592 * q^99

Introduction

L-functions

Modular forms

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Database

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