LMFDB - Embedded newform 16.38.a.b.1.1

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			1.1
Root			$3992.29$ 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)

$f(q)$	$=$	$q-3.45869e7 q^{3} +1.38267e13 q^{5} +5.42222e15 q^{7} -4.49088e17 q^{9} +O(q^{10})$	$q-3.45869e7 q^{3} +1.38267e13 q^{5} +5.42222e15 q^{7} -4.49088e17 q^{9} +1.03285e18 q^{11} -1.23244e19 q^{13} -4.78225e20 q^{15} -5.41594e22 q^{17} +3.08845e23 q^{19} -1.87538e23 q^{21} +2.61592e25 q^{23} +1.18419e26 q^{25} +3.11065e25 q^{27} +2.49238e26 q^{29} +2.34643e27 q^{31} -3.57230e25 q^{33} +7.49716e28 q^{35} +6.21792e28 q^{37} +4.26262e26 q^{39} -8.53404e29 q^{41} -3.53994e29 q^{43} -6.20942e30 q^{45} -6.68978e29 q^{47} +1.08383e31 q^{49} +1.87321e30 q^{51} +1.24128e32 q^{53} +1.42809e31 q^{55} -1.06820e31 q^{57} -6.57033e31 q^{59} -1.06034e33 q^{61} -2.43505e33 q^{63} -1.70406e32 q^{65} +8.44869e33 q^{67} -9.04768e32 q^{69} +7.04837e33 q^{71} +9.36735e33 q^{73} -4.09575e33 q^{75} +5.60032e33 q^{77} -1.51854e35 q^{79} +2.01141e35 q^{81} +1.43761e35 q^{83} -7.48847e35 q^{85} -8.62039e33 q^{87} +2.28283e35 q^{89} -6.68253e34 q^{91} -8.11557e34 q^{93} +4.27032e36 q^{95} +7.07361e36 q^{97} -4.63839e35 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.45869e7	−0.0515429	−0.0257715	−	0.999668i	$-0.508204\pi$
$3$	−3.45869e7	−0.0515429	−0.0257715	+	0.999668i	$0.508204\pi$
$4$	0	0
$5$	1.38267e13	1.62097	0.810484	−	0.585760i	$-0.199204\pi$
$5$	1.38267e13	1.62097	0.810484	+	0.585760i	$0.199204\pi$
$6$	0	0
$7$	5.42222e15	1.25853	0.629264	−	0.777191i	$-0.283356\pi$
$7$	5.42222e15	1.25853	0.629264	+	0.777191i	$0.283356\pi$
$8$	0	0
$9$	−4.49088e17	−0.997343
$10$	0	0
$11$	1.03285e18	0.0560108	0.0280054	−	0.999608i	$-0.491084\pi$
$11$	1.03285e18	0.0560108	0.0280054	+	0.999608i	$0.491084\pi$
$12$	0	0
$13$	−1.23244e19	−0.0303957	−0.0151979	−	0.999885i	$-0.504838\pi$
$13$	−1.23244e19	−0.0303957	−0.0151979	+	0.999885i	$0.504838\pi$
$14$	0	0
$15$	−4.78225e20	−0.0835495
$16$	0	0
$17$	−5.41594e22	−0.934047	−0.467023	−	0.884245i	$-0.654674\pi$
$17$	−5.41594e22	−0.934047	−0.467023	+	0.884245i	$0.654674\pi$
$18$	0	0
$19$	3.08845e23	0.680455	0.340228	−	0.940343i	$-0.389496\pi$
$19$	3.08845e23	0.680455	0.340228	+	0.940343i	$0.389496\pi$
$20$	0	0
$21$	−1.87538e23	−0.0648682
$22$	0	0
$23$	2.61592e25	1.68136	0.840678	−	0.541535i	$-0.182156\pi$
$23$	2.61592e25	1.68136	0.840678	+	0.541535i	$0.182156\pi$
$24$	0	0
$25$	1.18419e26	1.62754
$26$	0	0
$27$	3.11065e25	0.102949
$28$	0	0
$29$	2.49238e26	0.219913	0.109957	−	0.993936i	$-0.464929\pi$
$29$	2.49238e26	0.219913	0.109957	+	0.993936i	$0.464929\pi$
$30$	0	0
$31$	2.34643e27	0.602859	0.301429	−	0.953489i	$-0.402536\pi$
$31$	2.34643e27	0.602859	0.301429	+	0.953489i	$0.402536\pi$
$32$	0	0
$33$	−3.57230e25	−0.00288696
$34$	0	0
$35$	7.49716e28	2.04004
$36$	0	0
$37$	6.21792e28	0.605220	0.302610	−	0.953114i	$-0.402142\pi$
$37$	6.21792e28	0.605220	0.302610	+	0.953114i	$0.402142\pi$
$38$	0	0
$39$	4.26262e26	0.00156668
$40$	0	0
$41$	−8.53404e29	−1.24352	−0.621761	−	0.783207i	$-0.713582\pi$
$41$	−8.53404e29	−1.24352	−0.621761	+	0.783207i	$0.713582\pi$
$42$	0	0
$43$	−3.53994e29	−0.213713	−0.106856	−	0.994274i	$-0.534078\pi$
$43$	−3.53994e29	−0.213713	−0.106856	+	0.994274i	$0.534078\pi$
$44$	0	0
$45$	−6.20942e30	−1.61666
$46$	0	0
$47$	−6.68978e29	−0.0779115	−0.0389557	−	0.999241i	$-0.512403\pi$
$47$	−6.68978e29	−0.0779115	−0.0389557	+	0.999241i	$0.512403\pi$
$48$	0	0
$49$	1.08383e31	0.583895
$50$	0	0
$51$	1.87321e30	0.0481435
$52$	0	0
$53$	1.24128e32	1.56591	0.782956	−	0.622077i	$-0.213711\pi$
$53$	1.24128e32	1.56591	0.782956	+	0.622077i	$0.213711\pi$
$54$	0	0
$55$	1.42809e31	0.0907917
$56$	0	0
$57$	−1.06820e31	−0.0350726
$58$	0	0
$59$	−6.57033e31	−0.113979	−0.0569895	−	0.998375i	$-0.518150\pi$
$59$	−6.57033e31	−0.113979	−0.0569895	+	0.998375i	$0.518150\pi$
$60$	0	0
$61$	−1.06034e33	−0.992758	−0.496379	−	0.868106i	$-0.665337\pi$
$61$	−1.06034e33	−0.992758	−0.496379	+	0.868106i	$0.665337\pi$
$62$	0	0
$63$	−2.43505e33	−1.25519
$64$	0	0
$65$	−1.70406e32	−0.0492705
$66$	0	0
$67$	8.44869e33	1.39446	0.697229	−	0.716849i	$-0.254416\pi$
$67$	8.44869e33	1.39446	0.697229	+	0.716849i	$0.254416\pi$
$68$	0	0
$69$	−9.04768e32	−0.0866620
$70$	0	0
$71$	7.04837e33	0.397932	0.198966	−	0.980006i	$-0.436242\pi$
$71$	7.04837e33	0.397932	0.198966	+	0.980006i	$0.436242\pi$
$72$	0	0
$73$	9.36735e33	0.316333	0.158166	−	0.987412i	$-0.449442\pi$
$73$	9.36735e33	0.316333	0.158166	+	0.987412i	$0.449442\pi$
$74$	0	0
$75$	−4.09575e33	−0.0838881
$76$	0	0
$77$	5.60032e33	0.0704911
$78$	0	0
$79$	−1.51854e35	−1.18940	−0.594698	−	0.803949i	$-0.702728\pi$
$79$	−1.51854e35	−1.18940	−0.594698	+	0.803949i	$0.702728\pi$
$80$	0	0
$81$	2.01141e35	0.992037
$82$	0	0
$83$	1.43761e35	0.451541	0.225771	−	0.974180i	$-0.427510\pi$
$83$	1.43761e35	0.451541	0.225771	+	0.974180i	$0.427510\pi$
$84$	0	0
$85$	−7.48847e35	−1.51406
$86$	0	0
$87$	−8.62039e33	−0.0113350
$88$	0	0
$89$	2.28283e35	0.197133	0.0985664	−	0.995130i	$-0.468574\pi$
$89$	2.28283e35	0.197133	0.0985664	+	0.995130i	$0.468574\pi$
$90$	0	0
$91$	−6.68253e34	−0.0382539
$92$	0	0
$93$	−8.11557e34	−0.0310731
$94$	0	0
$95$	4.27032e36	1.10300
$96$	0	0
$97$	7.07361e36	1.24269	0.621347	−	0.783535i	$-0.286586\pi$
$97$	7.07361e36	1.24269	0.621347	+	0.783535i	$0.286586\pi$
$98$	0	0
$99$	−4.63839e35	−0.0558620
$100$	0	0
$101$	−1.31355e37	−1.09270	−0.546350	−	0.837557i	$-0.683983\pi$
$101$	−1.31355e37	−1.09270	−0.546350	+	0.837557i	$0.683983\pi$
$102$	0	0
$103$	−2.33181e37	−1.34960	−0.674800	−	0.738000i	$-0.735770\pi$
$103$	−2.33181e37	−1.34960	−0.674800	+	0.738000i	$0.735770\pi$
$104$	0	0
$105$	−2.59304e36	−0.105149
$106$	0	0
$107$	−3.01693e36	−0.0862909	−0.0431454	−	0.999069i	$-0.513738\pi$
$107$	−3.01693e36	−0.0862909	−0.0431454	+	0.999069i	$0.513738\pi$
$108$	0	0
$109$	3.05531e37	0.620391	0.310195	−	0.950673i	$-0.399606\pi$
$109$	3.05531e37	0.620391	0.310195	+	0.950673i	$0.399606\pi$
$110$	0	0
$111$	−2.15059e36	−0.0311948
$112$	0	0
$113$	4.21983e37	0.439890	0.219945	−	0.975512i	$-0.429412\pi$
$113$	4.21983e37	0.439890	0.219945	+	0.975512i	$0.429412\pi$
$114$	0	0
$115$	3.61697e38	2.72543
$116$	0	0
$117$	5.53472e36	0.0303150
$118$	0	0
$119$	−2.93664e38	−1.17552
$120$	0	0
$121$	−3.38973e38	−0.996863
$122$	0	0
$123$	2.95166e37	0.0640947
$124$	0	0
$125$	6.31322e38	1.01722
$126$	0	0
$127$	−1.09656e39	−1.31724	−0.658619	−	0.752476i	$-0.728859\pi$
$127$	−1.09656e39	−1.31724	−0.658619	+	0.752476i	$0.728859\pi$
$128$	0	0
$129$	1.22436e37	0.0110154
$130$	0	0
$131$	2.36928e39	1.60360	0.801802	−	0.597590i	$-0.203875\pi$
$131$	2.36928e39	1.60360	0.801802	+	0.597590i	$0.203875\pi$
$132$	0	0
$133$	1.67463e39	0.856373
$134$	0	0
$135$	4.30102e38	0.166877
$136$	0	0
$137$	2.55274e39	0.754529	0.377265	−	0.926105i	$-0.376865\pi$
$137$	2.55274e39	0.754529	0.377265	+	0.926105i	$0.376865\pi$
$138$	0	0
$139$	6.38958e39	1.44444	0.722219	−	0.691665i	$-0.243122\pi$
$139$	6.38958e39	1.44444	0.722219	+	0.691665i	$0.243122\pi$
$140$	0	0
$141$	2.31379e37	0.00401578
$142$	0	0
$143$	−1.27292e37	−0.00170249
$144$	0	0
$145$	3.44615e39	0.356473
$146$	0	0
$147$	−3.74864e38	−0.0300956
$148$	0	0
$149$	−3.49035e39	−0.218234	−0.109117	−	0.994029i	$-0.534802\pi$
$149$	−3.49035e39	−0.218234	−0.109117	+	0.994029i	$0.534802\pi$
$150$	0	0
$151$	1.43880e40	0.702953	0.351476	−	0.936197i	$-0.385680\pi$
$151$	1.43880e40	0.702953	0.351476	+	0.936197i	$0.385680\pi$
$152$	0	0
$153$	2.43223e40	0.931565
$154$	0	0
$155$	3.24434e40	0.977215
$156$	0	0
$157$	3.81842e40	0.907277	0.453639	−	0.891186i	$-0.350126\pi$
$157$	3.81842e40	0.907277	0.453639	+	0.891186i	$0.350126\pi$
$158$	0	0
$159$	−4.29322e39	−0.0807116
$160$	0	0
$161$	1.41841e41	2.11604
$162$	0	0
$163$	7.49639e40	0.889982	0.444991	−	0.895535i	$-0.353207\pi$
$163$	7.49639e40	0.889982	0.444991	+	0.895535i	$0.353207\pi$
$164$	0	0
$165$	−4.93933e38	−0.00467967
$166$	0	0
$167$	1.87857e41	1.42421	0.712104	−	0.702074i	$-0.247743\pi$
$167$	1.87857e41	1.42421	0.712104	+	0.702074i	$0.247743\pi$
$168$	0	0
$169$	−1.64249e41	−0.999076
$170$	0	0
$171$	−1.38699e41	−0.678648
$172$	0	0
$173$	2.98432e41	1.17759	0.588793	−	0.808284i	$-0.299603\pi$
$173$	2.98432e41	1.17759	0.588793	+	0.808284i	$0.299603\pi$
$174$	0	0
$175$	6.42094e41	2.04830
$176$	0	0
$177$	2.27248e39	0.00587481
$178$	0	0
$179$	−3.46258e41	−0.727141	−0.363570	−	0.931567i	$-0.618442\pi$
$179$	−3.46258e41	−0.727141	−0.363570	+	0.931567i	$0.618442\pi$
$180$	0	0
$181$	4.05668e41	0.693613	0.346806	−	0.937937i	$-0.387266\pi$
$181$	4.05668e41	0.693613	0.346806	+	0.937937i	$0.387266\pi$
$182$	0	0
$183$	3.66740e40	0.0511697
$184$	0	0
$185$	8.59735e41	0.981043
$186$	0	0
$187$	−5.59384e40	−0.0523167
$188$	0	0
$189$	1.68666e41	0.129564
$190$	0	0
$191$	1.41675e42	0.895726	0.447863	−	0.894102i	$-0.352185\pi$
$191$	1.41675e42	0.895726	0.447863	+	0.894102i	$0.352185\pi$
$192$	0	0
$193$	−1.28639e42	−0.670751	−0.335375	−	0.942085i	$-0.608863\pi$
$193$	−1.28639e42	−0.670751	−0.335375	+	0.942085i	$0.608863\pi$
$194$	0	0
$195$	5.89381e39	0.00253954
$196$	0	0
$197$	3.68135e41	0.131336	0.0656678	−	0.997842i	$-0.479082\pi$
$197$	3.68135e41	0.131336	0.0656678	+	0.997842i	$0.479082\pi$
$198$	0	0
$199$	−3.12066e42	−0.923557	−0.461779	−	0.886995i	$-0.652789\pi$
$199$	−3.12066e42	−0.923557	−0.461779	+	0.886995i	$0.652789\pi$
$200$	0	0
$201$	−2.92214e41	−0.0718744
$202$	0	0
$203$	1.35142e42	0.276767
$204$	0	0
$205$	−1.17998e43	−2.01571
$206$	0	0
$207$	−1.17478e43	−1.67689
$208$	0	0
$209$	3.18990e41	0.0381128
$210$	0	0
$211$	7.58037e42	0.759391	0.379695	−	0.925112i	$-0.376029\pi$
$211$	7.58037e42	0.759391	0.379695	+	0.925112i	$0.376029\pi$
$212$	0	0
$213$	−2.43782e41	−0.0205106
$214$	0	0
$215$	−4.89459e42	−0.346421
$216$	0	0
$217$	1.27228e43	0.758715
$218$	0	0
$219$	−3.23988e41	−0.0163047
$220$	0	0
$221$	6.67479e41	0.0283910
$222$	0	0
$223$	−4.00362e43	−1.44149	−0.720747	−	0.693198i	$-0.756201\pi$
$223$	−4.00362e43	−1.44149	−0.720747	+	0.693198i	$0.756201\pi$
$224$	0	0
$225$	−5.31805e43	−1.62322
$226$	0	0
$227$	−3.89322e43	−1.00886	−0.504430	−	0.863453i	$-0.668297\pi$
$227$	−3.89322e43	−1.00886	−0.504430	+	0.863453i	$0.668297\pi$
$228$	0	0
$229$	−3.32118e43	−0.731704	−0.365852	−	0.930673i	$-0.619222\pi$
$229$	−3.32118e43	−0.731704	−0.365852	+	0.930673i	$0.619222\pi$
$230$	0	0
$231$	−1.93698e41	−0.00363332
$232$	0	0
$233$	−7.95672e43	−1.27248	−0.636238	−	0.771493i	$-0.719510\pi$
$233$	−7.95672e43	−1.27248	−0.636238	+	0.771493i	$0.719510\pi$
$234$	0	0
$235$	−9.24978e42	−0.126292
$236$	0	0
$237$	5.25218e42	0.0613049
$238$	0	0
$239$	1.42241e44	1.42123	0.710615	−	0.703581i	$-0.248417\pi$
$239$	1.42241e44	1.42123	0.710615	+	0.703581i	$0.248417\pi$
$240$	0	0
$241$	1.48641e44	1.27299	0.636494	−	0.771282i	$-0.280384\pi$
$241$	1.48641e44	1.27299	0.636494	+	0.771282i	$0.280384\pi$
$242$	0	0
$243$	−2.09636e43	−0.154081
$244$	0	0
$245$	1.49859e44	0.946475
$246$	0	0
$247$	−3.80632e42	−0.0206829
$248$	0	0
$249$	−4.97227e42	−0.0232738
$250$	0	0
$251$	−1.84398e44	−0.744377	−0.372189	−	0.928157i	$-0.621393\pi$
$251$	−1.84398e44	−0.744377	−0.372189	+	0.928157i	$0.621393\pi$
$252$	0	0
$253$	2.70185e43	0.0941741
$254$	0	0
$255$	2.59003e43	0.0780391
$256$	0	0
$257$	−2.92391e44	−0.762433	−0.381216	−	0.924486i	$-0.624495\pi$
$257$	−2.92391e44	−0.762433	−0.381216	+	0.924486i	$0.624495\pi$
$258$	0	0
$259$	3.37149e44	0.761687
$260$	0	0
$261$	−1.11930e44	−0.219329
$262$	0	0
$263$	4.07317e44	0.693029	0.346514	−	0.938045i	$-0.387365\pi$
$263$	4.07317e44	0.693029	0.346514	+	0.938045i	$0.387365\pi$
$264$	0	0
$265$	1.71629e45	2.53829
$266$	0	0
$267$	−7.89562e42	−0.0101608
$268$	0	0
$269$	−6.06370e44	−0.679705	−0.339853	−	0.940479i	$-0.610377\pi$
$269$	−6.06370e44	−0.679705	−0.339853	+	0.940479i	$0.610377\pi$
$270$	0	0
$271$	1.82672e45	1.78542	0.892711	−	0.450630i	$-0.148800\pi$
$271$	1.82672e45	1.78542	0.892711	+	0.450630i	$0.148800\pi$
$272$	0	0
$273$	2.31128e42	0.00197172
$274$	0	0
$275$	1.22309e44	0.0911597
$276$	0	0
$277$	2.33400e45	1.52134	0.760668	−	0.649141i	$-0.224872\pi$
$277$	2.33400e45	1.52134	0.760668	+	0.649141i	$0.224872\pi$
$278$	0	0
$279$	−1.05375e45	−0.601257
$280$	0	0
$281$	−1.00137e45	−0.500643	−0.250322	−	0.968163i	$-0.580536\pi$
$281$	−1.00137e45	−0.500643	−0.250322	+	0.968163i	$0.580536\pi$
$282$	0	0
$283$	2.73703e45	1.20014	0.600069	−	0.799948i	$-0.295140\pi$
$283$	2.73703e45	1.20014	0.600069	+	0.799948i	$0.295140\pi$
$284$	0	0
$285$	−1.47697e44	−0.0568517
$286$	0	0
$287$	−4.62734e45	−1.56501
$288$	0	0
$289$	−4.28857e44	−0.127557
$290$	0	0
$291$	−2.44655e44	−0.0640521
$292$	0	0
$293$	1.07452e45	0.247836	0.123918	−	0.992292i	$-0.460454\pi$
$293$	1.07452e45	0.247836	0.123918	+	0.992292i	$0.460454\pi$
$294$	0	0
$295$	−9.08462e44	−0.184756
$296$	0	0
$297$	3.21283e43	0.00576625
$298$	0	0
$299$	−3.22396e44	−0.0511060
$300$	0	0
$301$	−1.91943e45	−0.268963
$302$	0	0
$303$	4.54316e44	0.0563209
$304$	0	0
$305$	−1.46611e46	−1.60923
$306$	0	0
$307$	1.41267e46	1.37398	0.686989	−	0.726668i	$-0.258932\pi$
$307$	1.41267e46	1.37398	0.686989	+	0.726668i	$0.258932\pi$
$308$	0	0
$309$	8.06503e44	0.0695624
$310$	0	0
$311$	−1.21153e46	−0.927399	−0.463700	−	0.885992i	$-0.653478\pi$
$311$	−1.21153e46	−0.927399	−0.463700	+	0.885992i	$0.653478\pi$
$312$	0	0
$313$	8.67864e45	0.590040	0.295020	−	0.955491i	$-0.404674\pi$
$313$	8.67864e45	0.590040	0.295020	+	0.955491i	$0.404674\pi$
$314$	0	0
$315$	−3.36688e46	−2.03462
$316$	0	0
$317$	−2.34417e46	−1.26006	−0.630031	−	0.776570i	$-0.716958\pi$
$317$	−2.34417e46	−1.26006	−0.630031	+	0.776570i	$0.716958\pi$
$318$	0	0
$319$	2.57425e44	0.0123175
$320$	0	0
$321$	1.04346e44	0.00444768
$322$	0	0
$323$	−1.67269e46	−0.635577
$324$	0	0
$325$	−1.45944e45	−0.0494702
$326$	0	0
$327$	−1.05674e45	−0.0319768
$328$	0	0
$329$	−3.62734e45	−0.0980538
$330$	0	0
$331$	−5.11187e46	−1.23527	−0.617635	−	0.786465i	$-0.711909\pi$
$331$	−5.11187e46	−1.23527	−0.617635	+	0.786465i	$0.711909\pi$
$332$	0	0
$333$	−2.79239e46	−0.603612
$334$	0	0
$335$	1.16818e47	2.26037
$336$	0	0
$337$	2.29819e46	0.398320	0.199160	−	0.979967i	$-0.436179\pi$
$337$	2.29819e46	0.398320	0.199160	+	0.979967i	$0.436179\pi$
$338$	0	0
$339$	−1.45951e45	−0.0226732
$340$	0	0
$341$	2.42350e45	0.0337666
$342$	0	0
$343$	−4.18801e46	−0.523680
$344$	0	0
$345$	−1.25100e46	−0.140476
$346$	0	0
$347$	8.38076e46	0.845645	0.422822	−	0.906213i	$-0.361039\pi$
$347$	8.38076e46	0.845645	0.422822	+	0.906213i	$0.361039\pi$
$348$	0	0
$349$	1.33845e47	1.21432	0.607158	−	0.794581i	$-0.292310\pi$
$349$	1.33845e47	1.21432	0.607158	+	0.794581i	$0.292310\pi$
$350$	0	0
$351$	−3.83368e44	−0.00312920
$352$	0	0
$353$	1.44127e46	0.105904	0.0529520	−	0.998597i	$-0.483137\pi$
$353$	1.44127e46	0.105904	0.0529520	+	0.998597i	$0.483137\pi$
$354$	0	0
$355$	9.74560e46	0.645036
$356$	0	0
$357$	1.01569e46	0.0605900
$358$	0	0
$359$	−6.57303e46	−0.353605	−0.176802	−	0.984246i	$-0.556575\pi$
$359$	−6.57303e46	−0.353605	−0.176802	+	0.984246i	$0.556575\pi$
$360$	0	0
$361$	−1.10622e47	−0.536981
$362$	0	0
$363$	1.17240e46	0.0513812
$364$	0	0
$365$	1.29520e47	0.512765
$366$	0	0
$367$	−2.16797e47	−0.775768	−0.387884	−	0.921708i	$-0.626794\pi$
$367$	−2.16797e47	−0.775768	−0.387884	+	0.921708i	$0.626794\pi$
$368$	0	0
$369$	3.83253e47	1.24022
$370$	0	0
$371$	6.73051e47	1.97074
$372$	0	0
$373$	2.47528e47	0.656163	0.328081	−	0.944649i	$-0.393598\pi$
$373$	2.47528e47	0.656163	0.328081	+	0.944649i	$0.393598\pi$
$374$	0	0
$375$	−2.18355e46	−0.0524305
$376$	0	0
$377$	−3.07170e45	−0.00668442
$378$	0	0
$379$	−1.02737e47	−0.202722	−0.101361	−	0.994850i	$-0.532320\pi$
$379$	−1.02737e47	−0.202722	−0.101361	+	0.994850i	$0.532320\pi$
$380$	0	0
$381$	3.79267e46	0.0678943
$382$	0	0
$383$	1.22188e48	1.98541	0.992706	−	0.120559i	$-0.0384687\pi$
$383$	1.22188e48	1.98541	0.992706	+	0.120559i	$0.0384687\pi$
$384$	0	0
$385$	7.74342e46	0.114264
$386$	0	0
$387$	1.58975e47	0.213145
$388$	0	0
$389$	−1.16416e48	−1.41888	−0.709440	−	0.704766i	$-0.751052\pi$
$389$	−1.16416e48	−1.41888	−0.709440	+	0.704766i	$0.751052\pi$
$390$	0	0
$391$	−1.41677e48	−1.57047
$392$	0	0
$393$	−8.19461e46	−0.0826544
$394$	0	0
$395$	−2.09965e48	−1.92797
$396$	0	0
$397$	−1.33628e48	−1.11756	−0.558781	−	0.829315i	$-0.688731\pi$
$397$	−1.33628e48	−1.11756	−0.558781	+	0.829315i	$0.688731\pi$
$398$	0	0
$399$	−5.79202e46	−0.0441399
$400$	0	0
$401$	2.54007e47	0.176472	0.0882358	−	0.996100i	$-0.471877\pi$
$401$	2.54007e47	0.176472	0.0882358	+	0.996100i	$0.471877\pi$
$402$	0	0
$403$	−2.89182e46	−0.0183243
$404$	0	0
$405$	2.78112e48	1.60806
$406$	0	0
$407$	6.42216e46	0.0338988
$408$	0	0
$409$	−1.33788e48	−0.644967	−0.322484	−	0.946575i	$-0.604518\pi$
$409$	−1.33788e48	−0.644967	−0.322484	+	0.946575i	$0.604518\pi$
$410$	0	0
$411$	−8.82915e46	−0.0388906
$412$	0	0
$413$	−3.56257e47	−0.143446
$414$	0	0
$415$	1.98775e48	0.731934
$416$	0	0
$417$	−2.20996e47	−0.0744505
$418$	0	0
$419$	−2.08239e48	−0.642101	−0.321050	−	0.947062i	$-0.604036\pi$
$419$	−2.08239e48	−0.642101	−0.321050	+	0.947062i	$0.604036\pi$
$420$	0	0
$421$	−4.42178e48	−1.24847	−0.624236	−	0.781236i	$-0.714590\pi$
$421$	−4.42178e48	−1.24847	−0.624236	+	0.781236i	$0.714590\pi$
$422$	0	0
$423$	3.00430e47	0.0777045
$424$	0	0
$425$	−6.41350e48	−1.52020
$426$	0	0
$427$	−5.74940e48	−1.24941
$428$	0	0
$429$	4.40263e44	8.77511e−5 0
$430$	0	0
$431$	−3.37175e48	−0.616631	−0.308315	−	0.951284i	$-0.599765\pi$
$431$	−3.37175e48	−0.616631	−0.308315	+	0.951284i	$0.599765\pi$
$432$	0	0
$433$	2.37577e48	0.398822	0.199411	−	0.979916i	$-0.436097\pi$
$433$	2.37577e48	0.398822	0.199411	+	0.979916i	$0.436097\pi$
$434$	0	0
$435$	−1.19192e47	−0.0183736
$436$	0	0
$437$	8.07916e48	1.14409
$438$	0	0
$439$	2.31976e48	0.301890	0.150945	−	0.988542i	$-0.451768\pi$
$439$	2.31976e48	0.301890	0.150945	+	0.988542i	$0.451768\pi$
$440$	0	0
$441$	−4.86736e48	−0.582344
$442$	0	0
$443$	1.52174e49	1.67444	0.837222	−	0.546864i	$-0.184178\pi$
$443$	1.52174e49	1.67444	0.837222	+	0.546864i	$0.184178\pi$
$444$	0	0
$445$	3.15641e48	0.319546
$446$	0	0
$447$	1.20720e47	0.0112484
$448$	0	0
$449$	1.92781e49	1.65390	0.826948	−	0.562279i	$-0.190075\pi$
$449$	1.92781e49	1.65390	0.826948	+	0.562279i	$0.190075\pi$
$450$	0	0
$451$	−8.81436e47	−0.0696506
$452$	0	0
$453$	−4.97636e47	−0.0362322
$454$	0	0
$455$	−9.23976e47	−0.0620083
$456$	0	0
$457$	1.19646e49	0.740367	0.370183	−	0.928959i	$-0.379295\pi$
$457$	1.19646e49	0.740367	0.370183	+	0.928959i	$0.379295\pi$
$458$	0	0
$459$	−1.68471e48	−0.0961591
$460$	0	0
$461$	−2.78549e49	−1.46702	−0.733508	−	0.679680i	$-0.762119\pi$
$461$	−2.78549e49	−1.46702	−0.733508	+	0.679680i	$0.762119\pi$
$462$	0	0
$463$	1.11920e49	0.544075	0.272038	−	0.962287i	$-0.412303\pi$
$463$	1.11920e49	0.544075	0.272038	+	0.962287i	$0.412303\pi$
$464$	0	0
$465$	−1.12212e48	−0.0503685
$466$	0	0
$467$	1.63248e49	0.676840	0.338420	−	0.940995i	$-0.390108\pi$
$467$	1.63248e49	0.676840	0.338420	+	0.940995i	$0.390108\pi$
$468$	0	0
$469$	4.58106e49	1.75497
$470$	0	0
$471$	−1.32067e48	−0.0467637
$472$	0	0
$473$	−3.65622e47	−0.0119702
$474$	0	0
$475$	3.65732e49	1.10747
$476$	0	0
$477$	−5.57445e49	−1.56175
$478$	0	0
$479$	−2.63654e49	−0.683639	−0.341820	−	0.939766i	$-0.611043\pi$
$479$	−2.63654e49	−0.683639	−0.341820	+	0.939766i	$0.611043\pi$
$480$	0	0
$481$	−7.66319e47	−0.0183961
$482$	0	0
$483$	−4.90585e48	−0.109067
$484$	0	0
$485$	9.78050e49	2.01437
$486$	0	0
$487$	7.78447e49	1.48574	0.742871	−	0.669435i	$-0.233464\pi$
$487$	7.78447e49	1.48574	0.742871	+	0.669435i	$0.233464\pi$
$488$	0	0
$489$	−2.59277e48	−0.0458723
$490$	0	0
$491$	−9.85282e49	−1.61642	−0.808208	−	0.588897i	$-0.799562\pi$
$491$	−9.85282e49	−1.61642	−0.808208	+	0.588897i	$0.799562\pi$
$492$	0	0
$493$	−1.34986e49	−0.205409
$494$	0	0
$495$	−6.41338e48	−0.0905505
$496$	0	0
$497$	3.82178e49	0.500809
$498$	0	0
$499$	−1.25338e50	−1.52483	−0.762415	−	0.647088i	$-0.775987\pi$
$499$	−1.25338e50	−1.52483	−0.762415	+	0.647088i	$0.775987\pi$
$500$	0	0
$501$	−6.49741e48	−0.0734078
$502$	0	0
$503$	1.11020e50	1.16518	0.582589	−	0.812767i	$-0.302040\pi$
$503$	1.11020e50	1.16518	0.582589	+	0.812767i	$0.302040\pi$
$504$	0	0
$505$	−1.81621e50	−1.77123
$506$	0	0
$507$	5.68087e48	0.0514953
$508$	0	0
$509$	−1.38813e50	−1.16991	−0.584955	−	0.811066i	$-0.698888\pi$
$509$	−1.38813e50	−1.16991	−0.584955	+	0.811066i	$0.698888\pi$
$510$	0	0
$511$	5.07918e49	0.398114
$512$	0	0
$513$	9.60711e48	0.0700521
$514$	0	0
$515$	−3.22414e50	−2.18766
$516$	0	0
$517$	−6.90952e47	−0.00436388
$518$	0	0
$519$	−1.03219e49	−0.0606962
$520$	0	0
$521$	−2.33871e50	−1.28079	−0.640396	−	0.768045i	$-0.721230\pi$
$521$	−2.33871e50	−1.28079	−0.640396	+	0.768045i	$0.721230\pi$
$522$	0	0
$523$	1.30285e50	0.664679	0.332340	−	0.943160i	$-0.392162\pi$
$523$	1.30285e50	0.664679	0.332340	+	0.943160i	$0.392162\pi$
$524$	0	0
$525$	−2.22081e49	−0.105576
$526$	0	0
$527$	−1.27081e50	−0.563098
$528$	0	0
$529$	4.42241e50	1.82696
$530$	0	0
$531$	2.95065e49	0.113676
$532$	0	0
$533$	1.05177e49	0.0377977
$534$	0	0
$535$	−4.17143e49	−0.139875
$536$	0	0
$537$	1.19760e49	0.0374789
$538$	0	0
$539$	1.11943e49	0.0327044
$540$	0	0
$541$	−5.31273e50	−1.44933	−0.724667	−	0.689099i	$-0.758007\pi$
$541$	−5.31273e50	−1.44933	−0.724667	+	0.689099i	$0.758007\pi$
$542$	0	0
$543$	−1.40308e49	−0.0357508
$544$	0	0
$545$	4.22450e50	1.00563
$546$	0	0
$547$	7.47732e49	0.166334	0.0831669	−	0.996536i	$-0.473497\pi$
$547$	7.47732e49	0.166334	0.0831669	+	0.996536i	$0.473497\pi$
$548$	0	0
$549$	4.76186e50	0.990121
$550$	0	0
$551$	7.69761e49	0.149641
$552$	0	0
$553$	−8.23388e50	−1.49689
$554$	0	0
$555$	−2.97356e49	−0.0505658
$556$	0	0
$557$	3.60496e50	0.573560	0.286780	−	0.957996i	$-0.407415\pi$
$557$	3.60496e50	0.573560	0.286780	+	0.957996i	$0.407415\pi$
$558$	0	0
$559$	4.36275e48	0.00649594
$560$	0	0
$561$	1.93474e48	0.00269655
$562$	0	0
$563$	3.94055e50	0.514223	0.257112	−	0.966382i	$-0.417229\pi$
$563$	3.94055e50	0.514223	0.257112	+	0.966382i	$0.417229\pi$
$564$	0	0
$565$	5.83465e50	0.713048
$566$	0	0
$567$	1.09063e51	1.24851
$568$	0	0
$569$	−1.14686e50	−0.123008	−0.0615040	−	0.998107i	$-0.519590\pi$
$569$	−1.14686e50	−0.123008	−0.0615040	+	0.998107i	$0.519590\pi$
$570$	0	0
$571$	−1.48184e51	−1.48948	−0.744739	−	0.667356i	$-0.767426\pi$
$571$	−1.48184e51	−1.48948	−0.744739	+	0.667356i	$0.767426\pi$
$572$	0	0
$573$	−4.90010e49	−0.0461683
$574$	0	0
$575$	3.09775e51	2.73647
$576$	0	0
$577$	1.51306e51	1.25344	0.626722	−	0.779243i	$-0.284396\pi$
$577$	1.51306e51	1.25344	0.626722	+	0.779243i	$0.284396\pi$
$578$	0	0
$579$	4.44922e49	0.0345725
$580$	0	0
$581$	7.79505e50	0.568278
$582$	0	0
$583$	1.28206e50	0.0877079
$584$	0	0
$585$	7.65271e49	0.0491396
$586$	0	0
$587$	−8.72797e50	−0.526148	−0.263074	−	0.964776i	$-0.584736\pi$
$587$	−8.72797e50	−0.526148	−0.263074	+	0.964776i	$0.584736\pi$
$588$	0	0
$589$	7.24683e50	0.410218
$590$	0	0
$591$	−1.27327e49	−0.00676942
$592$	0	0
$593$	4.58679e49	0.0229085	0.0114543	−	0.999934i	$-0.496354\pi$
$593$	4.58679e49	0.0229085	0.0114543	+	0.999934i	$0.496354\pi$
$594$	0	0
$595$	−4.06041e51	−1.90549
$596$	0	0
$597$	1.07934e50	0.0476028
$598$	0	0
$599$	2.10919e51	0.874420	0.437210	−	0.899360i	$-0.355967\pi$
$599$	2.10919e51	0.874420	0.437210	+	0.899360i	$0.355967\pi$
$600$	0	0
$601$	−2.46594e51	−0.961179	−0.480589	−	0.876946i	$-0.659577\pi$
$601$	−2.46594e51	−0.961179	−0.480589	+	0.876946i	$0.659577\pi$
$602$	0	0
$603$	−3.79420e51	−1.39075
$604$	0	0
$605$	−4.68689e51	−1.61588
$606$	0	0
$607$	4.33541e51	1.40618	0.703088	−	0.711103i	$-0.251804\pi$
$607$	4.33541e51	1.40618	0.703088	+	0.711103i	$0.251804\pi$
$608$	0	0
$609$	−4.67416e49	−0.0142654
$610$	0	0
$611$	8.24472e48	0.00236817
$612$	0	0
$613$	−1.36136e51	−0.368091	−0.184046	−	0.982918i	$-0.558919\pi$
$613$	−1.36136e51	−0.368091	−0.184046	+	0.982918i	$0.558919\pi$
$614$	0	0
$615$	4.08119e50	0.103896
$616$	0	0
$617$	3.77980e51	0.906136	0.453068	−	0.891476i	$-0.350329\pi$
$617$	3.77980e51	0.906136	0.453068	+	0.891476i	$0.350329\pi$
$618$	0	0
$619$	−6.56261e50	−0.148183	−0.0740917	−	0.997251i	$-0.523606\pi$
$619$	−6.56261e50	−0.148183	−0.0740917	+	0.997251i	$0.523606\pi$
$620$	0	0
$621$	8.13722e50	0.173094
$622$	0	0
$623$	1.23780e51	0.248097
$624$	0	0
$625$	1.12996e50	0.0213443
$626$	0	0
$627$	−1.10329e49	−0.00196445
$628$	0	0
$629$	−3.36759e51	−0.565304
$630$	0	0
$631$	−3.19436e51	−0.505640	−0.252820	−	0.967513i	$-0.581358\pi$
$631$	−3.19436e51	−0.505640	−0.252820	+	0.967513i	$0.581358\pi$
$632$	0	0
$633$	−2.62182e50	−0.0391412
$634$	0	0
$635$	−1.51619e52	−2.13520
$636$	0	0
$637$	−1.33575e50	−0.0177479
$638$	0	0
$639$	−3.16534e51	−0.396875
$640$	0	0
$641$	−1.48064e52	−1.75217	−0.876085	−	0.482157i	$-0.839853\pi$
$641$	−1.48064e52	−1.75217	−0.876085	+	0.482157i	$0.839853\pi$
$642$	0	0
$643$	−6.82003e51	−0.761874	−0.380937	−	0.924601i	$-0.624399\pi$
$643$	−6.82003e51	−0.761874	−0.380937	+	0.924601i	$0.624399\pi$
$644$	0	0
$645$	1.69289e50	0.0178556
$646$	0	0
$647$	2.43296e51	0.242329	0.121165	−	0.992632i	$-0.461337\pi$
$647$	2.43296e51	0.242329	0.121165	+	0.992632i	$0.461337\pi$
$648$	0	0
$649$	−6.78614e49	−0.00638405
$650$	0	0
$651$	−4.40044e50	−0.0391064
$652$	0	0
$653$	5.92349e51	0.497375	0.248687	−	0.968584i	$-0.420001\pi$
$653$	5.92349e51	0.497375	0.248687	+	0.968584i	$0.420001\pi$
$654$	0	0
$655$	3.27594e52	2.59939
$656$	0	0
$657$	−4.20676e51	−0.315492
$658$	0	0
$659$	−2.77432e51	−0.196687	−0.0983437	−	0.995153i	$-0.531354\pi$
$659$	−2.77432e51	−0.196687	−0.0983437	+	0.995153i	$0.531354\pi$
$660$	0	0
$661$	1.53847e52	1.03124	0.515622	−	0.856816i	$-0.327561\pi$
$661$	1.53847e52	1.03124	0.515622	+	0.856816i	$0.327561\pi$
$662$	0	0
$663$	−2.30861e49	−0.00146336
$664$	0	0
$665$	2.31546e52	1.38815
$666$	0	0
$667$	6.51988e51	0.369753
$668$	0	0
$669$	1.38473e51	0.0742988
$670$	0	0
$671$	−1.09517e51	−0.0556051
$672$	0	0
$673$	−8.54768e50	−0.0410742	−0.0205371	−	0.999789i	$-0.506538\pi$
$673$	−8.54768e50	−0.0410742	−0.0205371	+	0.999789i	$0.506538\pi$
$674$	0	0
$675$	3.68360e51	0.167553
$676$	0	0
$677$	−3.33629e52	−1.43673	−0.718363	−	0.695669i	$-0.755108\pi$
$677$	−3.33629e52	−1.43673	−0.718363	+	0.695669i	$0.755108\pi$
$678$	0	0
$679$	3.83547e52	1.56397
$680$	0	0
$681$	1.34655e51	0.0519996
$682$	0	0
$683$	−3.41377e52	−1.24868	−0.624340	−	0.781153i	$-0.714632\pi$
$683$	−3.41377e52	−1.24868	−0.624340	+	0.781153i	$0.714632\pi$
$684$	0	0
$685$	3.52961e52	1.22307
$686$	0	0
$687$	1.14869e51	0.0377142
$688$	0	0
$689$	−1.52980e51	−0.0475970
$690$	0	0
$691$	−5.92716e52	−1.74784	−0.873922	−	0.486066i	$-0.838431\pi$
$691$	−5.92716e52	−1.74784	−0.873922	+	0.486066i	$0.838431\pi$
$692$	0	0
$693$	−2.51504e51	−0.0703039
$694$	0	0
$695$	8.83471e52	2.34139
$696$	0	0
$697$	4.62198e52	1.16151
$698$	0	0
$699$	2.75199e51	0.0655871
$700$	0	0
$701$	4.09077e52	0.924744	0.462372	−	0.886686i	$-0.346998\pi$
$701$	4.09077e52	0.924744	0.462372	+	0.886686i	$0.346998\pi$
$702$	0	0
$703$	1.92038e52	0.411825
$704$	0	0
$705$	3.19922e50	0.00650946
$706$	0	0
$707$	−7.12235e52	−1.37519
$708$	0	0
$709$	−8.44115e52	−1.54684	−0.773422	−	0.633891i	$-0.781457\pi$
$709$	−8.44115e52	−1.54684	−0.773422	+	0.633891i	$0.781457\pi$
$710$	0	0
$711$	6.81960e52	1.18624
$712$	0	0
$713$	6.13807e52	1.01362
$714$	0	0
$715$	−1.76003e50	−0.00275968
$716$	0	0
$717$	−4.91969e51	−0.0732544
$718$	0	0
$719$	2.81946e52	0.398733	0.199367	−	0.979925i	$-0.436112\pi$
$719$	2.81946e52	0.398733	0.199367	+	0.979925i	$0.436112\pi$
$720$	0	0
$721$	−1.26436e53	−1.69851
$722$	0	0
$723$	−5.14105e51	−0.0656135
$724$	0	0
$725$	2.95146e52	0.357918
$726$	0	0
$727$	−6.98427e52	−0.804886	−0.402443	−	0.915445i	$-0.631839\pi$
$727$	−6.98427e52	−0.804886	−0.402443	+	0.915445i	$0.631839\pi$
$728$	0	0
$729$	−8.98455e52	−0.984095
$730$	0	0
$731$	1.91721e52	0.199618
$732$	0	0
$733$	3.54513e52	0.350921	0.175460	−	0.984487i	$-0.443859\pi$
$733$	3.54513e52	0.350921	0.175460	+	0.984487i	$0.443859\pi$
$734$	0	0
$735$	−5.18315e51	−0.0487841
$736$	0	0
$737$	8.72620e51	0.0781046
$738$	0	0
$739$	−1.51689e53	−1.29131	−0.645655	−	0.763629i	$-0.723416\pi$
$739$	−1.51689e53	−1.29131	−0.645655	+	0.763629i	$0.723416\pi$
$740$	0	0
$741$	1.31649e50	0.00106606
$742$	0	0
$743$	−1.52901e53	−1.17793	−0.588964	−	0.808159i	$-0.700464\pi$
$743$	−1.52901e53	−1.17793	−0.588964	+	0.808159i	$0.700464\pi$
$744$	0	0
$745$	−4.82601e52	−0.353750
$746$	0	0
$747$	−6.45615e52	−0.450342
$748$	0	0
$749$	−1.63584e52	−0.108600
$750$	0	0
$751$	−1.42568e53	−0.900916	−0.450458	−	0.892798i	$-0.648739\pi$
$751$	−1.42568e53	−0.900916	−0.450458	+	0.892798i	$0.648739\pi$
$752$	0	0
$753$	6.37778e51	0.0383674
$754$	0	0
$755$	1.98939e53	1.13946
$756$	0	0
$757$	−2.82784e53	−1.54234	−0.771171	−	0.636628i	$-0.780329\pi$
$757$	−2.82784e53	−1.54234	−0.771171	+	0.636628i	$0.780329\pi$
$758$	0	0
$759$	−9.34487e50	−0.00485401
$760$	0	0
$761$	3.71288e52	0.183694	0.0918470	−	0.995773i	$-0.470723\pi$
$761$	3.71288e52	0.183694	0.0918470	+	0.995773i	$0.470723\pi$
$762$	0	0
$763$	1.65666e53	0.780780
$764$	0	0
$765$	3.36298e53	1.51004
$766$	0	0
$767$	8.09750e50	0.00346447
$768$	0	0
$769$	−6.15158e52	−0.250813	−0.125406	−	0.992105i	$-0.540023\pi$
$769$	−6.15158e52	−0.250813	−0.125406	+	0.992105i	$0.540023\pi$
$770$	0	0
$771$	1.01129e52	0.0392980
$772$	0	0
$773$	−2.28656e53	−0.846960	−0.423480	−	0.905905i	$-0.639192\pi$
$773$	−2.28656e53	−0.846960	−0.423480	+	0.905905i	$0.639192\pi$
$774$	0	0
$775$	2.77862e53	0.981176
$776$	0	0
$777$	−1.16610e52	−0.0392596
$778$	0	0
$779$	−2.63570e53	−0.846161
$780$	0	0
$781$	7.27989e51	0.0222885
$782$	0	0
$783$	7.75294e51	0.0226399
$784$	0	0
$785$	5.27963e53	1.47067
$786$	0	0
$787$	−7.05791e52	−0.187561	−0.0937807	−	0.995593i	$-0.529895\pi$
$787$	−7.05791e52	−0.187561	−0.0937807	+	0.995593i	$0.529895\pi$
$788$	0	0
$789$	−1.40879e52	−0.0357207
$790$	0	0
$791$	2.28809e53	0.553614
$792$	0	0
$793$	1.30680e52	0.0301756
$794$	0	0
$795$	−5.93612e52	−0.130831
$796$	0	0
$797$	−8.88567e53	−1.86944	−0.934719	−	0.355387i	$-0.884349\pi$
$797$	−8.88567e53	−1.86944	−0.934719	+	0.355387i	$0.884349\pi$
$798$	0	0
$799$	3.62314e52	0.0727729
$800$	0	0
$801$	−1.02519e53	−0.196609
$802$	0	0
$803$	9.67504e51	0.0177180
$804$	0	0
$805$	1.96120e54	3.43003
$806$	0	0
$807$	2.09725e52	0.0350340
$808$	0	0
$809$	3.72232e53	0.593972	0.296986	−	0.954882i	$-0.404019\pi$
$809$	3.72232e53	0.593972	0.296986	+	0.954882i	$0.404019\pi$
$810$	0	0
$811$	5.96718e53	0.909668	0.454834	−	0.890576i	$-0.349699\pi$
$811$	5.96718e53	0.909668	0.454834	+	0.890576i	$0.349699\pi$
$812$	0	0
$813$	−6.31808e52	−0.0920259
$814$	0	0
$815$	1.03651e54	1.44263
$816$	0	0
$817$	−1.09330e53	−0.145422
$818$	0	0
$819$	3.00104e52	0.0381522
$820$	0	0
$821$	8.74697e52	0.106294	0.0531471	−	0.998587i	$-0.483075\pi$
$821$	8.74697e52	0.106294	0.0531471	+	0.998587i	$0.483075\pi$
$822$	0	0
$823$	−1.05135e54	−1.22138	−0.610692	−	0.791869i	$-0.709108\pi$
$823$	−1.05135e54	−1.22138	−0.610692	+	0.791869i	$0.709108\pi$
$824$	0	0
$825$	−4.23029e51	−0.00469864
$826$	0	0
$827$	1.54483e53	0.164069	0.0820347	−	0.996629i	$-0.473858\pi$
$827$	1.54483e53	0.164069	0.0820347	+	0.996629i	$0.473858\pi$
$828$	0	0
$829$	2.69934e53	0.274156	0.137078	−	0.990560i	$-0.456229\pi$
$829$	2.69934e53	0.274156	0.137078	+	0.990560i	$0.456229\pi$
$830$	0	0
$831$	−8.07259e52	−0.0784141
$832$	0	0
$833$	−5.86997e53	−0.545385
$834$	0	0
$835$	2.59745e54	2.30860
$836$	0	0
$837$	7.29891e52	0.0620636
$838$	0	0
$839$	−2.43290e54	−1.97937	−0.989686	−	0.143252i	$-0.954244\pi$
$839$	−2.43290e54	−1.97937	−0.989686	+	0.143252i	$0.954244\pi$
$840$	0	0
$841$	−1.22236e54	−0.951638
$842$	0	0
$843$	3.46344e52	0.0258046
$844$	0	0
$845$	−2.27103e54	−1.61947
$846$	0	0
$847$	−1.83798e54	−1.25458
$848$	0	0
$849$	−9.46656e52	−0.0618586
$850$	0	0
$851$	1.62656e54	1.01759
$852$	0	0
$853$	9.59321e53	0.574654	0.287327	−	0.957832i	$-0.407233\pi$
$853$	9.59321e53	0.574654	0.287327	+	0.957832i	$0.407233\pi$
$854$	0	0
$855$	−1.91775e54	−1.10007
$856$	0	0
$857$	3.59352e52	0.0197413	0.00987063	−	0.999951i	$-0.496858\pi$
$857$	3.59352e52	0.0197413	0.00987063	+	0.999951i	$0.496858\pi$
$858$	0	0
$859$	−8.55580e53	−0.450181	−0.225091	−	0.974338i	$-0.572268\pi$
$859$	−8.55580e53	−0.450181	−0.225091	+	0.974338i	$0.572268\pi$
$860$	0	0
$861$	1.60046e53	0.0806651
$862$	0	0
$863$	−1.56057e54	−0.753502	−0.376751	−	0.926315i	$-0.622959\pi$
$863$	−1.56057e54	−0.753502	−0.376751	+	0.926315i	$0.622959\pi$
$864$	0	0
$865$	4.12634e54	1.90883
$866$	0	0
$867$	1.48329e52	0.00657464
$868$	0	0
$869$	−1.56842e53	−0.0666190
$870$	0	0
$871$	−1.04125e53	−0.0423855
$872$	0	0
$873$	−3.17667e54	−1.23939
$874$	0	0
$875$	3.42316e54	1.28020
$876$	0	0
$877$	−3.37270e54	−1.20916	−0.604581	−	0.796544i	$-0.706659\pi$
$877$	−3.37270e54	−1.20916	−0.604581	+	0.796544i	$0.706659\pi$
$878$	0	0
$879$	−3.71644e52	−0.0127742
$880$	0	0
$881$	1.67513e54	0.552069	0.276034	−	0.961148i	$-0.410980\pi$
$881$	1.67513e54	0.552069	0.276034	+	0.961148i	$0.410980\pi$
$882$	0	0
$883$	−1.42002e54	−0.448767	−0.224383	−	0.974501i	$-0.572037\pi$
$883$	−1.42002e54	−0.448767	−0.224383	+	0.974501i	$0.572037\pi$
$884$	0	0
$885$	3.14209e52	0.00952288
$886$	0	0
$887$	−1.98874e54	−0.578085	−0.289042	−	0.957316i	$-0.593337\pi$
$887$	−1.98874e54	−0.578085	−0.289042	+	0.957316i	$0.593337\pi$
$888$	0	0
$889$	−5.94580e54	−1.65778
$890$	0	0
$891$	2.07748e53	0.0555647
$892$	0	0
$893$	−2.06611e53	−0.0530153
$894$	0	0
$895$	−4.78761e54	−1.17867
$896$	0	0
$897$	1.11507e52	0.00263415
$898$	0	0
$899$	5.84819e53	0.132577
$900$	0	0
$901$	−6.72271e54	−1.46263
$902$	0	0
$903$	6.63874e52	0.0138632
$904$	0	0
$905$	5.60906e54	1.12432
$906$	0	0
$907$	6.36503e54	1.22480	0.612401	−	0.790547i	$-0.290204\pi$
$907$	6.36503e54	1.22480	0.612401	+	0.790547i	$0.290204\pi$
$908$	0	0
$909$	5.89899e54	1.08980
$910$	0	0
$911$	−2.93922e54	−0.521365	−0.260683	−	0.965425i	$-0.583948\pi$
$911$	−2.93922e54	−0.521365	−0.260683	+	0.965425i	$0.583948\pi$
$912$	0	0
$913$	1.48484e53	0.0252912
$914$	0	0
$915$	5.07081e53	0.0829444
$916$	0	0
$917$	1.28467e55	2.01818
$918$	0	0
$919$	7.57823e54	1.14349	0.571743	−	0.820433i	$-0.306267\pi$
$919$	7.57823e54	1.14349	0.571743	+	0.820433i	$0.306267\pi$
$920$	0	0
$921$	−4.88598e53	−0.0708189
$922$	0	0
$923$	−8.68666e52	−0.0120954
$924$	0	0
$925$	7.36320e54	0.985019
$926$	0	0
$927$	1.04719e55	1.34602
$928$	0	0
$929$	6.55047e53	0.0809062	0.0404531	−	0.999181i	$-0.487120\pi$
$929$	6.55047e53	0.0809062	0.0404531	+	0.999181i	$0.487120\pi$
$930$	0	0
$931$	3.34737e54	0.397314
$932$	0	0
$933$	4.19031e53	0.0478009
$934$	0	0
$935$	−7.73445e53	−0.0848037
$936$	0	0
$937$	−3.87596e54	−0.408505	−0.204253	−	0.978918i	$-0.565476\pi$
$937$	−3.87596e54	−0.408505	−0.204253	+	0.978918i	$0.565476\pi$
$938$	0	0
$939$	−3.00168e53	−0.0304124
$940$	0	0
$941$	−1.38620e55	−1.35027	−0.675133	−	0.737696i	$-0.735914\pi$
$941$	−1.38620e55	−1.35027	−0.675133	+	0.737696i	$0.735914\pi$
$942$	0	0
$943$	−2.23244e55	−2.09080
$944$	0	0
$945$	2.33210e54	0.210019
$946$	0	0
$947$	8.86598e53	0.0767807	0.0383903	−	0.999263i	$-0.487777\pi$
$947$	8.86598e53	0.0767807	0.0383903	+	0.999263i	$0.487777\pi$
$948$	0	0
$949$	−1.15447e53	−0.00961515
$950$	0	0
$951$	8.10776e53	0.0649473
$952$	0	0
$953$	1.94425e55	1.49807	0.749036	−	0.662530i	$-0.230517\pi$
$953$	1.94425e55	1.49807	0.749036	+	0.662530i	$0.230517\pi$
$954$	0	0
$955$	1.95890e55	1.45194
$956$	0	0
$957$	−8.90355e51	−0.000634881 0
$958$	0	0
$959$	1.38415e55	0.949597
$960$	0	0
$961$	−9.64324e54	−0.636561
$962$	0	0
$963$	1.35486e54	0.0860616
$964$	0	0
$965$	−1.77865e55	−1.08727
$966$	0	0
$967$	−6.67380e54	−0.392630	−0.196315	−	0.980541i	$-0.562897\pi$
$967$	−6.67380e54	−0.392630	−0.196315	+	0.980541i	$0.562897\pi$
$968$	0	0
$969$	5.78532e53	0.0327595
$970$	0	0
$971$	3.90466e52	0.00212827	0.00106414	−	0.999999i	$-0.499661\pi$
$971$	3.90466e52	0.00212827	0.00106414	+	0.999999i	$0.499661\pi$
$972$	0	0
$973$	3.46457e55	1.81787
$974$	0	0
$975$	5.04775e52	0.00254984
$976$	0	0
$977$	−6.49911e54	−0.316085	−0.158043	−	0.987432i	$-0.550518\pi$
$977$	−6.49911e54	−0.316085	−0.158043	+	0.987432i	$0.550518\pi$
$978$	0	0
$979$	2.35782e53	0.0110416
$980$	0	0
$981$	−1.37210e55	−0.618743
$982$	0	0
$983$	−1.39076e55	−0.603967	−0.301984	−	0.953313i	$-0.597649\pi$
$983$	−1.39076e55	−0.603967	−0.301984	+	0.953313i	$0.597649\pi$
$984$	0	0
$985$	5.09011e54	0.212891
$986$	0	0
$987$	1.25459e53	0.00505398
$988$	0	0
$989$	−9.26022e54	−0.359327
$990$	0	0
$991$	−4.28319e55	−1.60105	−0.800525	−	0.599300i	$-0.795446\pi$
$991$	−4.28319e55	−1.60105	−0.800525	+	0.599300i	$0.795446\pi$
$992$	0	0
$993$	1.76804e54	0.0636695
$994$	0	0
$995$	−4.31485e55	−1.49706
$996$	0	0
$997$	−3.24811e55	−1.08585	−0.542925	−	0.839781i	$-0.682683\pi$
$997$	−3.24811e55	−1.08585	−0.542925	+	0.839781i	$0.682683\pi$
$998$	0	0
$999$	1.93418e54	0.0623067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.38.a.b.1.1		2
4.3	odd	2		1.38.a.a.1.1	✓	2
12.11	even	2		9.38.a.a.1.2		2
20.3	even	4		25.38.b.a.24.4		4
20.7	even	4		25.38.b.a.24.1		4
20.19	odd	2		25.38.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.38.a.a.1.1	✓	2	4.3	odd	2
9.38.a.a.1.2		2	12.11	even	2
16.38.a.b.1.1		2	1.1	even	1	trivial
25.38.a.a.1.2		2	20.19	odd	2
25.38.b.a.24.1		4	20.7	even	4
25.38.b.a.24.4		4	20.3	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)

\(f(q)\)	\(=\)	\(q-3.45869e7 q^{3} +1.38267e13 q^{5} +5.42222e15 q^{7} -4.49088e17 q^{9} +O(q^{10})\)	\(q-3.45869e7 q^{3} +1.38267e13 q^{5} +5.42222e15 q^{7} -4.49088e17 q^{9} +1.03285e18 q^{11} -1.23244e19 q^{13} -4.78225e20 q^{15} -5.41594e22 q^{17} +3.08845e23 q^{19} -1.87538e23 q^{21} +2.61592e25 q^{23} +1.18419e26 q^{25} +3.11065e25 q^{27} +2.49238e26 q^{29} +2.34643e27 q^{31} -3.57230e25 q^{33} +7.49716e28 q^{35} +6.21792e28 q^{37} +4.26262e26 q^{39} -8.53404e29 q^{41} -3.53994e29 q^{43} -6.20942e30 q^{45} -6.68978e29 q^{47} +1.08383e31 q^{49} +1.87321e30 q^{51} +1.24128e32 q^{53} +1.42809e31 q^{55} -1.06820e31 q^{57} -6.57033e31 q^{59} -1.06034e33 q^{61} -2.43505e33 q^{63} -1.70406e32 q^{65} +8.44869e33 q^{67} -9.04768e32 q^{69} +7.04837e33 q^{71} +9.36735e33 q^{73} -4.09575e33 q^{75} +5.60032e33 q^{77} -1.51854e35 q^{79} +2.01141e35 q^{81} +1.43761e35 q^{83} -7.48847e35 q^{85} -8.62039e33 q^{87} +2.28283e35 q^{89} -6.68253e34 q^{91} -8.11557e34 q^{93} +4.27032e36 q^{95} +7.07361e36 q^{97} -4.63839e35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 13991400 q^{3} + 5529584385900 q^{5} + 34\!\cdots\!00 q^{7}+ \cdots - 89\!\cdots\!14 q^{9}+O(q^{10}) \) 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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