LMFDB - Embedded newform 16.44.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,44,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, 44, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 44);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 44 $
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:	$187.376632553$
Analytic rank:	$1$
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} - 11258260111x - 264759545317170 $ x^3 - 11258260111*x - 264759545317170
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{4}\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			$-24885.9$ 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)

$f(q)$	$=$	$q-3.22027e10 q^{3} +5.54482e14 q^{5} +5.57604e17 q^{7} +7.08758e20 q^{9} +O(q^{10})$	$q-3.22027e10 q^{3} +5.54482e14 q^{5} +5.57604e17 q^{7} +7.08758e20 q^{9} +1.13059e22 q^{11} +5.07130e23 q^{13} -1.78558e25 q^{15} +8.73610e25 q^{17} -5.55222e27 q^{19} -1.79564e28 q^{21} -1.85137e29 q^{23} -8.29418e29 q^{25} -1.22532e31 q^{27} +4.19782e31 q^{29} +9.30312e31 q^{31} -3.64082e32 q^{33} +3.09181e32 q^{35} -3.20928e33 q^{37} -1.63309e34 q^{39} +6.48857e33 q^{41} -3.51189e33 q^{43} +3.92993e35 q^{45} -6.21410e35 q^{47} -1.87289e36 q^{49} -2.81326e36 q^{51} +8.28552e36 q^{53} +6.26895e36 q^{55} +1.78797e38 q^{57} +7.48073e36 q^{59} -2.24824e38 q^{61} +3.95206e38 q^{63} +2.81194e38 q^{65} +1.89786e39 q^{67} +5.96191e39 q^{69} +9.53945e39 q^{71} +2.48260e39 q^{73} +2.67095e40 q^{75} +6.30424e39 q^{77} +3.50244e39 q^{79} +1.61930e41 q^{81} +2.23114e41 q^{83} +4.84401e40 q^{85} -1.35181e42 q^{87} -4.38099e41 q^{89} +2.82777e41 q^{91} -2.99586e42 q^{93} -3.07861e42 q^{95} -6.11565e42 q^{97} +8.01318e42 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 24401437812 q^{3} + 535205380774170 q^{5} - 30\!\cdots\!56 q^{7}+ \cdots + 47\!\cdots\!31 q^{9}+O(q^{10}) $ 3 * q - 24401437812 * q^3 + 535205380774170 * q^5 - 301971425665478856 * q^7 + 477482125171066743231 * q^9	$ 3 q - 24401437812 q^{3} + 535205380774170 q^{5} - 30\!\cdots\!56 q^{7}+ \cdots + 14\!\cdots\!08 q^{99}+O(q^{100}) $ 3 * q - 24401437812 * q^3 + 535205380774170 * q^5 - 301971425665478856 * q^7 + 477482125171066743231 * q^9 - 26665053285822824598396 * q^11 + 2675160683143344673595682 * q^13 - 35619426649405115420082840 * q^15 - 408654222932442443796841194 * q^17 - 1577331357975519701176464900 * q^19 + 39305230121617849136195725536 * q^21 + 1215339635928561142484811048 * q^23 - 2310484082897201607166539950475 * q^25 - 12636088863973677226324883265480 * q^27 + 57118989018136587596191009705650 * q^29 + 255565972342397832812135282963424 * q^31 - 274896238871162964163796672607216 * q^33 - 2398213530857285851552147980593520 * q^35 - 2320645576165979428293572256842694 * q^37 + 3966989860384498209469308423541128 * q^39 + 25742376604739002042016577514061166 * q^41 - 240714339689128182512406300123419292 * q^43 + 257230496455952342677786043056557090 * q^45 - 308713553347399687962182796433367856 * q^47 + 3434980292250886821160584750829904379 * q^49 - 13352566627975131351422398848669478056 * q^51 + 15000497558784669132889458695112796362 * q^53 - 3996571389110251928625895283213506440 * q^55 + 194498189404429212580032012092952587760 * q^57 - 226996261182733939538282061249836505900 * q^59 - 9393429368478033721028929889113591854 * q^61 + 967475285671148246145956691206790781848 * q^63 - 270218921102297342477494789001032451460 * q^65 + 733015616887151283726086660162187354444 * q^67 + 1967042253746083561325124578559972227232 * q^69 + 18523878066158899993345833814569898287864 * q^71 - 20251106025908798058937016734369224029298 * q^73 + 21273380043530396090723723962888745519700 * q^75 + 59055613160181778232413942942559747685792 * q^77 - 15233568018599764739559826813006999294800 * q^79 + 73114835779397486920654645241710822062363 * q^81 + 89206610196719670053085531955706819434428 * q^83 + 438725373228516358990025164547060204838420 * q^85 - 1720273148596604243726575204332689696886840 * q^87 + 203691422331055117225246388044508249665950 * q^89 + 1168954967602248602369466051364648836229584 * q^91 - 4949189234312030019905814954548842124935296 * q^93 - 3125746305909018830165692627580442029979000 * q^95 - 3881062021784775072625368160679539917597594 * q^97 + 14255275025472043026078170466415579823927508 * 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
$3$	−3.22027e10	−1.77740	−0.888701	−	0.458488i	$-0.848391\pi$
$3$	−3.22027e10	−1.77740	−0.888701	+	0.458488i	$0.848391\pi$
$4$	0	0
$5$	5.54482e14	0.520035	0.260017	−	0.965604i	$-0.416272\pi$
$5$	5.54482e14	0.520035	0.260017	+	0.965604i	$0.416272\pi$
$6$	0	0
$7$	5.57604e17	0.377327	0.188663	−	0.982042i	$-0.439584\pi$
$7$	5.57604e17	0.377327	0.188663	+	0.982042i	$0.439584\pi$
$8$	0	0
$9$	7.08758e20	2.15915
$10$	0	0
$11$	1.13059e22	0.460643	0.230321	−	0.973115i	$-0.426022\pi$
$11$	1.13059e22	0.460643	0.230321	+	0.973115i	$0.426022\pi$
$12$	0	0
$13$	5.07130e23	0.569294	0.284647	−	0.958632i	$-0.408124\pi$
$13$	5.07130e23	0.569294	0.284647	+	0.958632i	$0.408124\pi$
$14$	0	0
$15$	−1.78558e25	−0.924311
$16$	0	0
$17$	8.73610e25	0.306666	0.153333	−	0.988175i	$-0.450999\pi$
$17$	8.73610e25	0.306666	0.153333	+	0.988175i	$0.450999\pi$
$18$	0	0
$19$	−5.55222e27	−1.78346	−0.891732	−	0.452564i	$-0.850509\pi$
$19$	−5.55222e27	−1.78346	−0.891732	+	0.452564i	$0.850509\pi$
$20$	0	0
$21$	−1.79564e28	−0.670661
$22$	0	0
$23$	−1.85137e29	−0.978012	−0.489006	−	0.872280i	$-0.662640\pi$
$23$	−1.85137e29	−0.978012	−0.489006	+	0.872280i	$0.662640\pi$
$24$	0	0
$25$	−8.29418e29	−0.729564
$26$	0	0
$27$	−1.22532e31	−2.06028
$28$	0	0
$29$	4.19782e31	1.51868	0.759340	−	0.650694i	$-0.225522\pi$
$29$	4.19782e31	1.51868	0.759340	+	0.650694i	$0.225522\pi$
$30$	0	0
$31$	9.30312e31	0.802329	0.401164	−	0.916006i	$-0.368606\pi$
$31$	9.30312e31	0.802329	0.401164	+	0.916006i	$0.368606\pi$
$32$	0	0
$33$	−3.64082e32	−0.818747
$34$	0	0
$35$	3.09181e32	0.196223
$36$	0	0
$37$	−3.20928e33	−0.616695	−0.308347	−	0.951274i	$-0.599776\pi$
$37$	−3.20928e33	−0.616695	−0.308347	+	0.951274i	$0.599776\pi$
$38$	0	0
$39$	−1.63309e34	−1.01186
$40$	0	0
$41$	6.48857e33	0.137182	0.0685908	−	0.997645i	$-0.478150\pi$
$41$	6.48857e33	0.137182	0.0685908	+	0.997645i	$0.478150\pi$
$42$	0	0
$43$	−3.51189e33	−0.0266667	−0.0133333	−	0.999911i	$-0.504244\pi$
$43$	−3.51189e33	−0.0266667	−0.0133333	+	0.999911i	$0.504244\pi$
$44$	0	0
$45$	3.92993e35	1.12284
$46$	0	0
$47$	−6.21410e35	−0.697067	−0.348533	−	0.937296i	$-0.613320\pi$
$47$	−6.21410e35	−0.697067	−0.348533	+	0.937296i	$0.613320\pi$
$48$	0	0
$49$	−1.87289e36	−0.857624
$50$	0	0
$51$	−2.81326e36	−0.545069
$52$	0	0
$53$	8.28552e36	0.702083	0.351042	−	0.936360i	$-0.385828\pi$
$53$	8.28552e36	0.702083	0.351042	+	0.936360i	$0.385828\pi$
$54$	0	0
$55$	6.26895e36	0.239550
$56$	0	0
$57$	1.78797e38	3.16993
$58$	0	0
$59$	7.48073e36	0.0631867	0.0315933	−	0.999501i	$-0.489942\pi$
$59$	7.48073e36	0.0631867	0.0315933	+	0.999501i	$0.489942\pi$
$60$	0	0
$61$	−2.24824e38	−0.927366	−0.463683	−	0.886001i	$-0.653472\pi$
$61$	−2.24824e38	−0.927366	−0.463683	+	0.886001i	$0.653472\pi$
$62$	0	0
$63$	3.95206e38	0.814707
$64$	0	0
$65$	2.81194e38	0.296053
$66$	0	0
$67$	1.89786e39	1.04149	0.520745	−	0.853712i	$-0.325654\pi$
$67$	1.89786e39	1.04149	0.520745	+	0.853712i	$0.325654\pi$
$68$	0	0
$69$	5.96191e39	1.73832
$70$	0	0
$71$	9.53945e39	1.50476	0.752382	−	0.658727i	$-0.228905\pi$
$71$	9.53945e39	1.50476	0.752382	+	0.658727i	$0.228905\pi$
$72$	0	0
$73$	2.48260e39	0.215509	0.107755	−	0.994178i	$-0.465634\pi$
$73$	2.48260e39	0.215509	0.107755	+	0.994178i	$0.465634\pi$
$74$	0	0
$75$	2.67095e40	1.29673
$76$	0	0
$77$	6.30424e39	0.173813
$78$	0	0
$79$	3.50244e39	0.0556402	0.0278201	−	0.999613i	$-0.491143\pi$
$79$	3.50244e39	0.0556402	0.0278201	+	0.999613i	$0.491143\pi$
$80$	0	0
$81$	1.61930e41	1.50280
$82$	0	0
$83$	2.23114e41	1.22560	0.612799	−	0.790239i	$-0.290043\pi$
$83$	2.23114e41	1.22560	0.612799	+	0.790239i	$0.290043\pi$
$84$	0	0
$85$	4.84401e40	0.159477
$86$	0	0
$87$	−1.35181e42	−2.69930
$88$	0	0
$89$	−4.38099e41	−0.536645	−0.268323	−	0.963329i	$-0.586469\pi$
$89$	−4.38099e41	−0.536645	−0.268323	+	0.963329i	$0.586469\pi$
$90$	0	0
$91$	2.82777e41	0.214810
$92$	0	0
$93$	−2.99586e42	−1.42606
$94$	0	0
$95$	−3.07861e42	−0.927463
$96$	0	0
$97$	−6.11565e42	−1.17720	−0.588600	−	0.808424i	$-0.700321\pi$
$97$	−6.11565e42	−1.17720	−0.588600	+	0.808424i	$0.700321\pi$
$98$	0	0
$99$	8.01318e42	0.994599
$100$	0	0
$101$	8.20815e42	0.662728	0.331364	−	0.943503i	$-0.392491\pi$
$101$	8.20815e42	0.662728	0.331364	+	0.943503i	$0.392491\pi$
$102$	0	0
$103$	−2.72199e43	−1.44174	−0.720868	−	0.693072i	$-0.756257\pi$
$103$	−2.72199e43	−1.44174	−0.720868	+	0.693072i	$0.756257\pi$
$104$	0	0
$105$	−9.95648e42	−0.348767
$106$	0	0
$107$	4.26383e43	0.995517	0.497758	−	0.867316i	$-0.334157\pi$
$107$	4.26383e43	0.995517	0.497758	+	0.867316i	$0.334157\pi$
$108$	0	0
$109$	2.42680e43	0.380508	0.190254	−	0.981735i	$-0.439069\pi$
$109$	2.42680e43	0.380508	0.190254	+	0.981735i	$0.439069\pi$
$110$	0	0
$111$	1.03347e44	1.09611
$112$	0	0
$113$	1.73831e44	1.25586	0.627931	−	0.778269i	$-0.283902\pi$
$113$	1.73831e44	1.25586	0.627931	+	0.778269i	$0.283902\pi$
$114$	0	0
$115$	−1.02655e44	−0.508600
$116$	0	0
$117$	3.59432e44	1.22919
$118$	0	0
$119$	4.87128e43	0.115713
$120$	0	0
$121$	−4.74576e44	−0.787808
$122$	0	0
$123$	−2.08949e44	−0.243827
$124$	0	0
$125$	−1.09027e45	−0.899433
$126$	0	0
$127$	2.81930e45	1.65334	0.826669	−	0.562688i	$-0.190233\pi$
$127$	2.81930e45	1.65334	0.826669	+	0.562688i	$0.190233\pi$
$128$	0	0
$129$	1.13092e44	0.0473974
$130$	0	0
$131$	−5.22187e45	−1.57215	−0.786073	−	0.618133i	$-0.787889\pi$
$131$	−5.22187e45	−1.57215	−0.786073	+	0.618133i	$0.787889\pi$
$132$	0	0
$133$	−3.09594e45	−0.672949
$134$	0	0
$135$	−6.79416e45	−1.07142
$136$	0	0
$137$	−1.17929e46	−1.35559	−0.677794	−	0.735252i	$-0.737064\pi$
$137$	−1.17929e46	−1.35559	−0.677794	+	0.735252i	$0.737064\pi$
$138$	0	0
$139$	−7.89914e45	−0.664908	−0.332454	−	0.943119i	$-0.607877\pi$
$139$	−7.89914e45	−0.664908	−0.332454	+	0.943119i	$0.607877\pi$
$140$	0	0
$141$	2.00111e46	1.23897
$142$	0	0
$143$	5.73358e45	0.262241
$144$	0	0
$145$	2.32761e46	0.789767
$146$	0	0
$147$	6.03122e46	1.52434
$148$	0	0
$149$	5.77438e45	0.109144	0.0545721	−	0.998510i	$-0.482621\pi$
$149$	5.77438e45	0.109144	0.0545721	+	0.998510i	$0.482621\pi$
$150$	0	0
$151$	1.97310e46	0.279992	0.139996	−	0.990152i	$-0.455291\pi$
$151$	1.97310e46	0.279992	0.139996	+	0.990152i	$0.455291\pi$
$152$	0	0
$153$	6.19178e46	0.662140
$154$	0	0
$155$	5.15842e46	0.417239
$156$	0	0
$157$	−1.33604e47	−0.820306	−0.410153	−	0.912017i	$-0.634525\pi$
$157$	−1.33604e47	−0.820306	−0.410153	+	0.912017i	$0.634525\pi$
$158$	0	0
$159$	−2.66816e47	−1.24788
$160$	0	0
$161$	−1.03233e47	−0.369030
$162$	0	0
$163$	1.04926e47	0.287641	0.143820	−	0.989604i	$-0.454061\pi$
$163$	1.04926e47	0.287641	0.143820	+	0.989604i	$0.454061\pi$
$164$	0	0
$165$	−2.01877e47	−0.425777
$166$	0	0
$167$	6.74904e47	1.09860	0.549298	−	0.835627i	$-0.314895\pi$
$167$	6.74904e47	1.09860	0.549298	+	0.835627i	$0.314895\pi$
$168$	0	0
$169$	−5.36351e47	−0.675904
$170$	0	0
$171$	−3.93518e48	−3.85077
$172$	0	0
$173$	−4.15452e47	−0.316614	−0.158307	−	0.987390i	$-0.550603\pi$
$173$	−4.15452e47	−0.316614	−0.158307	+	0.987390i	$0.550603\pi$
$174$	0	0
$175$	−4.62486e47	−0.275284
$176$	0	0
$177$	−2.40900e47	−0.112308
$178$	0	0
$179$	6.34938e47	0.232483	0.116241	−	0.993221i	$-0.462915\pi$
$179$	6.34938e47	0.232483	0.116241	+	0.993221i	$0.462915\pi$
$180$	0	0
$181$	5.38506e47	0.155275	0.0776375	−	0.996982i	$-0.475262\pi$
$181$	5.38506e47	0.155275	0.0776375	+	0.996982i	$0.475262\pi$
$182$	0	0
$183$	7.23994e48	1.64830
$184$	0	0
$185$	−1.77949e48	−0.320703
$186$	0	0
$187$	9.87699e47	0.141263
$188$	0	0
$189$	−6.83240e48	−0.777400
$190$	0	0
$191$	−1.22017e49	−1.10714	−0.553572	−	0.832802i	$-0.686735\pi$
$191$	−1.22017e49	−1.10714	−0.553572	+	0.832802i	$0.686735\pi$
$192$	0	0
$193$	8.36996e46	0.00607073	0.00303537	−	0.999995i	$-0.499034\pi$
$193$	8.36996e46	0.00607073	0.00303537	+	0.999995i	$0.499034\pi$
$194$	0	0
$195$	−9.05522e48	−0.526205
$196$	0	0
$197$	1.24344e49	0.580229	0.290115	−	0.956992i	$-0.406307\pi$
$197$	1.24344e49	0.580229	0.290115	+	0.956992i	$0.406307\pi$
$198$	0	0
$199$	−2.54032e49	−0.953997	−0.476999	−	0.878904i	$-0.658275\pi$
$199$	−2.54032e49	−0.953997	−0.476999	+	0.878904i	$0.658275\pi$
$200$	0	0
$201$	−6.11161e49	−1.85115
$202$	0	0
$203$	2.34072e49	0.573039
$204$	0	0
$205$	3.59779e48	0.0713392
$206$	0	0
$207$	−1.31217e50	−2.11168
$208$	0	0
$209$	−6.27731e49	−0.821540
$210$	0	0
$211$	−1.72327e50	−1.83772	−0.918861	−	0.394582i	$-0.870890\pi$
$211$	−1.72327e50	−1.83772	−0.918861	+	0.394582i	$0.870890\pi$
$212$	0	0
$213$	−3.07196e50	−2.67457
$214$	0	0
$215$	−1.94728e48	−0.0138676
$216$	0	0
$217$	5.18746e49	0.302740
$218$	0	0
$219$	−7.99465e49	−0.383046
$220$	0	0
$221$	4.43033e49	0.174583
$222$	0	0
$223$	1.43740e50	0.466683	0.233342	−	0.972395i	$-0.425034\pi$
$223$	1.43740e50	0.466683	0.233342	+	0.972395i	$0.425034\pi$
$224$	0	0
$225$	−5.87856e50	−1.57524
$226$	0	0
$227$	7.21848e48	0.0159915	0.00799576	−	0.999968i	$-0.497455\pi$
$227$	7.21848e48	0.0159915	0.00799576	+	0.999968i	$0.497455\pi$
$228$	0	0
$229$	8.60814e50	1.57923	0.789617	−	0.613600i	$-0.210279\pi$
$229$	8.60814e50	1.57923	0.789617	+	0.613600i	$0.210279\pi$
$230$	0	0
$231$	−2.03014e50	−0.308935
$232$	0	0
$233$	−8.15789e50	−1.03140	−0.515698	−	0.856770i	$-0.672467\pi$
$233$	−8.15789e50	−1.03140	−0.515698	+	0.856770i	$0.672467\pi$
$234$	0	0
$235$	−3.44561e50	−0.362499
$236$	0	0
$237$	−1.12788e50	−0.0988950
$238$	0	0
$239$	−1.19865e51	−0.877276	−0.438638	−	0.898664i	$-0.644539\pi$
$239$	−1.19865e51	−0.877276	−0.438638	+	0.898664i	$0.644539\pi$
$240$	0	0
$241$	1.15117e51	0.704326	0.352163	−	0.935939i	$-0.385446\pi$
$241$	1.15117e51	0.704326	0.352163	+	0.935939i	$0.385446\pi$
$242$	0	0
$243$	−1.19241e51	−0.610786
$244$	0	0
$245$	−1.03849e51	−0.445995
$246$	0	0
$247$	−2.81570e51	−1.01532
$248$	0	0
$249$	−7.18488e51	−2.17838
$250$	0	0
$251$	3.52535e51	0.899948	0.449974	−	0.893042i	$-0.351433\pi$
$251$	3.52535e51	0.899948	0.449974	+	0.893042i	$0.351433\pi$
$252$	0	0
$253$	−2.09315e51	−0.450514
$254$	0	0
$255$	−1.55990e51	−0.283455
$256$	0	0
$257$	−9.00901e51	−1.38393	−0.691967	−	0.721930i	$-0.743255\pi$
$257$	−9.00901e51	−1.38393	−0.691967	+	0.721930i	$0.743255\pi$
$258$	0	0
$259$	−1.78951e51	−0.232696
$260$	0	0
$261$	2.97523e52	3.27907
$262$	0	0
$263$	8.70831e51	0.814489	0.407245	−	0.913319i	$-0.366490\pi$
$263$	8.70831e51	0.814489	0.407245	+	0.913319i	$0.366490\pi$
$264$	0	0
$265$	4.59417e51	0.365108
$266$	0	0
$267$	1.41080e52	0.953834
$268$	0	0
$269$	2.29589e52	1.32214	0.661071	−	0.750324i	$-0.270102\pi$
$269$	2.29589e52	1.32214	0.661071	+	0.750324i	$0.270102\pi$
$270$	0	0
$271$	2.04437e52	1.00397	0.501983	−	0.864877i	$-0.332604\pi$
$271$	2.04437e52	1.00397	0.501983	+	0.864877i	$0.332604\pi$
$272$	0	0
$273$	−9.10620e51	−0.381804
$274$	0	0
$275$	−9.37735e51	−0.336068
$276$	0	0
$277$	3.67190e52	1.12610	0.563049	−	0.826424i	$-0.309628\pi$
$277$	3.67190e52	1.12610	0.563049	+	0.826424i	$0.309628\pi$
$278$	0	0
$279$	6.59366e52	1.73235
$280$	0	0
$281$	−3.21289e52	−0.723952	−0.361976	−	0.932187i	$-0.617898\pi$
$281$	−3.21289e52	−0.723952	−0.361976	+	0.932187i	$0.617898\pi$
$282$	0	0
$283$	−7.76513e52	−1.50224	−0.751121	−	0.660164i	$-0.770487\pi$
$283$	−7.76513e52	−1.50224	−0.751121	+	0.660164i	$0.770487\pi$
$284$	0	0
$285$	9.91395e52	1.64847
$286$	0	0
$287$	3.61805e51	0.0517623
$288$	0	0
$289$	−7.35209e52	−0.905956
$290$	0	0
$291$	1.96940e53	2.09236
$292$	0	0
$293$	7.07016e52	0.648299	0.324150	−	0.946006i	$-0.394922\pi$
$293$	7.07016e52	0.648299	0.324150	+	0.946006i	$0.394922\pi$
$294$	0	0
$295$	4.14793e51	0.0328593
$296$	0	0
$297$	−1.38533e53	−0.949054
$298$	0	0
$299$	−9.38883e52	−0.556777
$300$	0	0
$301$	−1.95824e51	−0.0100621
$302$	0	0
$303$	−2.64325e53	−1.17793
$304$	0	0
$305$	−1.24661e53	−0.482263
$306$	0	0
$307$	4.17987e53	1.40504	0.702518	−	0.711666i	$-0.252059\pi$
$307$	4.17987e53	1.40504	0.702518	+	0.711666i	$0.252059\pi$
$308$	0	0
$309$	8.76554e53	2.56255
$310$	0	0
$311$	−1.08737e53	−0.276714	−0.138357	−	0.990382i	$-0.544182\pi$
$311$	−1.08737e53	−0.276714	−0.138357	+	0.990382i	$0.544182\pi$
$312$	0	0
$313$	5.09671e52	0.113002	0.0565011	−	0.998403i	$-0.482006\pi$
$313$	5.09671e52	0.113002	0.0565011	+	0.998403i	$0.482006\pi$
$314$	0	0
$315$	2.19135e53	0.423676
$316$	0	0
$317$	5.92996e53	1.00064	0.500320	−	0.865841i	$-0.333216\pi$
$317$	5.92996e53	1.00064	0.500320	+	0.865841i	$0.333216\pi$
$318$	0	0
$319$	4.74603e53	0.699569
$320$	0	0
$321$	−1.37307e54	−1.76943
$322$	0	0
$323$	−4.85047e53	−0.546928
$324$	0	0
$325$	−4.20622e53	−0.415336
$326$	0	0
$327$	−7.81494e53	−0.676315
$328$	0	0
$329$	−3.46501e53	−0.263022
$330$	0	0
$331$	−4.62094e53	−0.307913	−0.153957	−	0.988078i	$-0.549202\pi$
$331$	−4.62094e53	−0.307913	−0.153957	+	0.988078i	$0.549202\pi$
$332$	0	0
$333$	−2.27460e54	−1.33154
$334$	0	0
$335$	1.05233e54	0.541611
$336$	0	0
$337$	−2.44988e54	−1.10943	−0.554717	−	0.832039i	$-0.687173\pi$
$337$	−2.44988e54	−1.10943	−0.554717	+	0.832039i	$0.687173\pi$
$338$	0	0
$339$	−5.59784e54	−2.23217
$340$	0	0
$341$	1.05181e54	0.369587
$342$	0	0
$343$	−2.26203e54	−0.700932
$344$	0	0
$345$	3.30577e54	0.903987
$346$	0	0
$347$	−2.06865e54	−0.499579	−0.249789	−	0.968300i	$-0.580361\pi$
$347$	−2.06865e54	−0.499579	−0.249789	+	0.968300i	$0.580361\pi$
$348$	0	0
$349$	−9.24540e54	−1.97324	−0.986618	−	0.163050i	$-0.947867\pi$
$349$	−9.24540e54	−1.97324	−0.986618	+	0.163050i	$0.947867\pi$
$350$	0	0
$351$	−6.21394e54	−1.17291
$352$	0	0
$353$	−3.55103e54	−0.593196	−0.296598	−	0.955002i	$-0.595852\pi$
$353$	−3.55103e54	−0.593196	−0.296598	+	0.955002i	$0.595852\pi$
$354$	0	0
$355$	5.28945e54	0.782530
$356$	0	0
$357$	−1.56868e54	−0.205669
$358$	0	0
$359$	6.37557e54	0.741291	0.370645	−	0.928774i	$-0.379137\pi$
$359$	6.37557e54	0.741291	0.370645	+	0.928774i	$0.379137\pi$
$360$	0	0
$361$	2.11353e55	2.18074
$362$	0	0
$363$	1.52826e55	1.40025
$364$	0	0
$365$	1.37656e54	0.112072
$366$	0	0
$367$	−1.19284e55	−0.863502	−0.431751	−	0.901993i	$-0.642104\pi$
$367$	−1.19284e55	−0.863502	−0.431751	+	0.901993i	$0.642104\pi$
$368$	0	0
$369$	4.59882e54	0.296196
$370$	0	0
$371$	4.62004e54	0.264915
$372$	0	0
$373$	−6.58590e54	−0.336415	−0.168208	−	0.985752i	$-0.553798\pi$
$373$	−6.58590e54	−0.336415	−0.168208	+	0.985752i	$0.553798\pi$
$374$	0	0
$375$	3.51097e55	1.59865
$376$	0	0
$377$	2.12884e55	0.864576
$378$	0	0
$379$	−4.25975e55	−1.54398	−0.771989	−	0.635636i	$-0.780738\pi$
$379$	−4.25975e55	−1.54398	−0.771989	+	0.635636i	$0.780738\pi$
$380$	0	0
$381$	−9.07892e55	−2.93865
$382$	0	0
$383$	1.85965e53	0.00537846	0.00268923	−	0.999996i	$-0.499144\pi$
$383$	1.85965e53	0.00537846	0.00268923	+	0.999996i	$0.499144\pi$
$384$	0	0
$385$	3.49559e54	0.0903888
$386$	0	0
$387$	−2.48908e54	−0.0575775
$388$	0	0
$389$	7.73460e55	1.60148	0.800740	−	0.599012i	$-0.204440\pi$
$389$	7.73460e55	1.60148	0.800740	+	0.599012i	$0.204440\pi$
$390$	0	0
$391$	−1.61737e55	−0.299923
$392$	0	0
$393$	1.68158e56	2.79434
$394$	0	0
$395$	1.94204e54	0.0289349
$396$	0	0
$397$	−5.40998e55	−0.723102	−0.361551	−	0.932352i	$-0.617753\pi$
$397$	−5.40998e55	−0.723102	−0.361551	+	0.932352i	$0.617753\pi$
$398$	0	0
$399$	9.96976e55	1.19610
$400$	0	0
$401$	−1.27603e56	−1.37486	−0.687429	−	0.726252i	$-0.741261\pi$
$401$	−1.27603e56	−1.37486	−0.687429	+	0.726252i	$0.741261\pi$
$402$	0	0
$403$	4.71789e55	0.456761
$404$	0	0
$405$	8.97874e55	0.781506
$406$	0	0
$407$	−3.62839e55	−0.284076
$408$	0	0
$409$	1.15025e56	0.810481	0.405240	−	0.914210i	$-0.367188\pi$
$409$	1.15025e56	0.810481	0.405240	+	0.914210i	$0.367188\pi$
$410$	0	0
$411$	3.79763e56	2.40942
$412$	0	0
$413$	4.17128e54	0.0238420
$414$	0	0
$415$	1.23713e56	0.637354
$416$	0	0
$417$	2.54374e56	1.18181
$418$	0	0
$419$	3.32264e55	0.139278	0.0696389	−	0.997572i	$-0.477815\pi$
$419$	3.32264e55	0.139278	0.0696389	+	0.997572i	$0.477815\pi$
$420$	0	0
$421$	−3.83623e55	−0.145158	−0.0725788	−	0.997363i	$-0.523123\pi$
$421$	−3.83623e55	−0.145158	−0.0725788	+	0.997363i	$0.523123\pi$
$422$	0	0
$423$	−4.40429e56	−1.50507
$424$	0	0
$425$	−7.24588e55	−0.223732
$426$	0	0
$427$	−1.25363e56	−0.349920
$428$	0	0
$429$	−1.84637e56	−0.466108
$430$	0	0
$431$	−2.91528e56	−0.665916	−0.332958	−	0.942942i	$-0.608047\pi$
$431$	−2.91528e56	−0.665916	−0.332958	+	0.942942i	$0.608047\pi$
$432$	0	0
$433$	−8.91932e56	−1.84434	−0.922172	−	0.386780i	$-0.873587\pi$
$433$	−8.91932e56	−1.84434	−0.922172	+	0.386780i	$0.873587\pi$
$434$	0	0
$435$	−7.49555e56	−1.40373
$436$	0	0
$437$	1.02792e57	1.74425
$438$	0	0
$439$	4.59768e56	0.707216	0.353608	−	0.935394i	$-0.384955\pi$
$439$	4.59768e56	0.707216	0.353608	+	0.935394i	$0.384955\pi$
$440$	0	0
$441$	−1.32743e57	−1.85174
$442$	0	0
$443$	−7.17753e56	−0.908437	−0.454219	−	0.890890i	$-0.650082\pi$
$443$	−7.17753e56	−0.908437	−0.454219	+	0.890890i	$0.650082\pi$
$444$	0	0
$445$	−2.42918e56	−0.279074
$446$	0	0
$447$	−1.85951e56	−0.193993
$448$	0	0
$449$	−2.56945e56	−0.243526	−0.121763	−	0.992559i	$-0.538855\pi$
$449$	−2.56945e56	−0.243526	−0.121763	+	0.992559i	$0.538855\pi$
$450$	0	0
$451$	7.33594e55	0.0631917
$452$	0	0
$453$	−6.35392e56	−0.497657
$454$	0	0
$455$	1.56795e56	0.111709
$456$	0	0
$457$	−1.11475e57	−0.722733	−0.361367	−	0.932424i	$-0.617690\pi$
$457$	−1.11475e57	−0.722733	−0.361367	+	0.932424i	$0.617690\pi$
$458$	0	0
$459$	−1.07045e57	−0.631819
$460$	0	0
$461$	2.02024e57	1.08601	0.543003	−	0.839731i	$-0.317287\pi$
$461$	2.02024e57	1.08601	0.543003	+	0.839731i	$0.317287\pi$
$462$	0	0
$463$	2.86426e56	0.140288	0.0701442	−	0.997537i	$-0.477654\pi$
$463$	2.86426e56	0.140288	0.0701442	+	0.997537i	$0.477654\pi$
$464$	0	0
$465$	−1.66115e57	−0.741601
$466$	0	0
$467$	−1.63037e57	−0.663701	−0.331850	−	0.943332i	$-0.607673\pi$
$467$	−1.63037e57	−0.663701	−0.331850	+	0.943332i	$0.607673\pi$
$468$	0	0
$469$	1.05825e57	0.392982
$470$	0	0
$471$	4.30240e57	1.45801
$472$	0	0
$473$	−3.97052e55	−0.0122838
$474$	0	0
$475$	4.60511e57	1.30115
$476$	0	0
$477$	5.87243e57	1.51591
$478$	0	0
$479$	−7.76358e57	−1.83168	−0.915838	−	0.401548i	$-0.868472\pi$
$479$	−7.76358e57	−1.83168	−0.915838	+	0.401548i	$0.868472\pi$
$480$	0	0
$481$	−1.62752e57	−0.351081
$482$	0	0
$483$	3.32438e57	0.655915
$484$	0	0
$485$	−3.39102e57	−0.612185
$486$	0	0
$487$	−1.02143e58	−1.68786	−0.843932	−	0.536451i	$-0.819765\pi$
$487$	−1.02143e58	−1.68786	−0.843932	+	0.536451i	$0.819765\pi$
$488$	0	0
$489$	−3.37891e57	−0.511253
$490$	0	0
$491$	−4.75337e57	−0.658794	−0.329397	−	0.944192i	$-0.606845\pi$
$491$	−4.75337e57	−0.658794	−0.329397	+	0.944192i	$0.606845\pi$
$492$	0	0
$493$	3.66725e57	0.465728
$494$	0	0
$495$	4.44316e57	0.517226
$496$	0	0
$497$	5.31923e57	0.567788
$498$	0	0
$499$	6.28971e57	0.615841	0.307921	−	0.951412i	$-0.400367\pi$
$499$	6.28971e57	0.615841	0.307921	+	0.951412i	$0.400367\pi$
$500$	0	0
$501$	−2.17337e58	−1.95264
$502$	0	0
$503$	2.08189e58	1.71690	0.858451	−	0.512896i	$-0.171427\pi$
$503$	2.08189e58	1.71690	0.858451	+	0.512896i	$0.171427\pi$
$504$	0	0
$505$	4.55127e57	0.344642
$506$	0	0
$507$	1.72720e58	1.20135
$508$	0	0
$509$	4.65765e57	0.297669	0.148835	−	0.988862i	$-0.452448\pi$
$509$	4.65765e57	0.297669	0.148835	+	0.988862i	$0.452448\pi$
$510$	0	0
$511$	1.38431e57	0.0813174
$512$	0	0
$513$	6.80322e58	3.67444
$514$	0	0
$515$	−1.50929e58	−0.749754
$516$	0	0
$517$	−7.02563e57	−0.321099
$518$	0	0
$519$	1.33787e58	0.562749
$520$	0	0
$521$	4.15022e57	0.160716	0.0803582	−	0.996766i	$-0.474394\pi$
$521$	4.15022e57	0.160716	0.0803582	+	0.996766i	$0.474394\pi$
$522$	0	0
$523$	3.61441e57	0.128899	0.0644497	−	0.997921i	$-0.479471\pi$
$523$	3.61441e57	0.128899	0.0644497	+	0.997921i	$0.479471\pi$
$524$	0	0
$525$	1.48933e58	0.489290
$526$	0	0
$527$	8.12730e57	0.246047
$528$	0	0
$529$	−1.55852e57	−0.0434925
$530$	0	0
$531$	5.30202e57	0.136430
$532$	0	0
$533$	3.29054e57	0.0780967
$534$	0	0
$535$	2.36422e58	0.517703
$536$	0	0
$537$	−2.04467e58	−0.413215
$538$	0	0
$539$	−2.11748e58	−0.395058
$540$	0	0
$541$	−4.79722e58	−0.826511	−0.413256	−	0.910615i	$-0.635608\pi$
$541$	−4.79722e58	−0.826511	−0.413256	+	0.910615i	$0.635608\pi$
$542$	0	0
$543$	−1.73414e58	−0.275986
$544$	0	0
$545$	1.34562e58	0.197877
$546$	0	0
$547$	−3.67046e58	−0.498877	−0.249439	−	0.968391i	$-0.580246\pi$
$547$	−3.67046e58	−0.498877	−0.249439	+	0.968391i	$0.580246\pi$
$548$	0	0
$549$	−1.59346e59	−2.00233
$550$	0	0
$551$	−2.33072e59	−2.70851
$552$	0	0
$553$	1.95298e57	0.0209946
$554$	0	0
$555$	5.73043e58	0.570018
$556$	0	0
$557$	−1.44377e59	−1.32926	−0.664632	−	0.747171i	$-0.731412\pi$
$557$	−1.44377e59	−1.32926	−0.664632	+	0.747171i	$0.731412\pi$
$558$	0	0
$559$	−1.78098e57	−0.0151812
$560$	0	0
$561$	−3.18066e58	−0.251082
$562$	0	0
$563$	1.85030e58	0.135304	0.0676521	−	0.997709i	$-0.478449\pi$
$563$	1.85030e58	0.135304	0.0676521	+	0.997709i	$0.478449\pi$
$564$	0	0
$565$	9.63864e58	0.653092
$566$	0	0
$567$	9.02928e58	0.567045
$568$	0	0
$569$	7.55839e58	0.440064	0.220032	−	0.975493i	$-0.429384\pi$
$569$	7.55839e58	0.440064	0.220032	+	0.975493i	$0.429384\pi$
$570$	0	0
$571$	−1.65354e59	−0.892767	−0.446384	−	0.894842i	$-0.647288\pi$
$571$	−1.65354e59	−0.892767	−0.446384	+	0.894842i	$0.647288\pi$
$572$	0	0
$573$	3.92929e59	1.96784
$574$	0	0
$575$	1.53556e59	0.713522
$576$	0	0
$577$	−4.04625e59	−1.74491	−0.872453	−	0.488697i	$-0.837472\pi$
$577$	−4.04625e59	−1.74491	−0.872453	+	0.488697i	$0.837472\pi$
$578$	0	0
$579$	−2.69535e57	−0.0107901
$580$	0	0
$581$	1.24409e59	0.462451
$582$	0	0
$583$	9.36757e58	0.323410
$584$	0	0
$585$	1.99299e59	0.639224
$586$	0	0
$587$	−3.79894e59	−1.13225	−0.566126	−	0.824319i	$-0.691558\pi$
$587$	−3.79894e59	−1.13225	−0.566126	+	0.824319i	$0.691558\pi$
$588$	0	0
$589$	−5.16530e59	−1.43092
$590$	0	0
$591$	−4.00420e59	−1.03130
$592$	0	0
$593$	−4.79450e59	−1.14833	−0.574166	−	0.818739i	$-0.694674\pi$
$593$	−4.79450e59	−1.14833	−0.574166	+	0.818739i	$0.694674\pi$
$594$	0	0
$595$	2.70104e58	0.0601750
$596$	0	0
$597$	8.18052e59	1.69564
$598$	0	0
$599$	4.48279e59	0.864713	0.432356	−	0.901703i	$-0.357682\pi$
$599$	4.48279e59	0.864713	0.432356	+	0.901703i	$0.357682\pi$
$600$	0	0
$601$	−1.41250e59	−0.253621	−0.126811	−	0.991927i	$-0.540474\pi$
$601$	−1.41250e59	−0.253621	−0.126811	+	0.991927i	$0.540474\pi$
$602$	0	0
$603$	1.34512e60	2.24874
$604$	0	0
$605$	−2.63144e59	−0.409688
$606$	0	0
$607$	−7.01925e59	−1.01797	−0.508984	−	0.860776i	$-0.669979\pi$
$607$	−7.01925e59	−1.01797	−0.508984	+	0.860776i	$0.669979\pi$
$608$	0	0
$609$	−7.53775e59	−1.01852
$610$	0	0
$611$	−3.15135e59	−0.396836
$612$	0	0
$613$	1.03814e60	1.21858	0.609292	−	0.792946i	$-0.291454\pi$
$613$	1.03814e60	1.21858	0.609292	+	0.792946i	$0.291454\pi$
$614$	0	0
$615$	−1.15859e59	−0.126798
$616$	0	0
$617$	−1.00931e60	−1.03014	−0.515069	−	0.857149i	$-0.672234\pi$
$617$	−1.00931e60	−1.03014	−0.515069	+	0.857149i	$0.672234\pi$
$618$	0	0
$619$	−1.22876e59	−0.116981	−0.0584907	−	0.998288i	$-0.518629\pi$
$619$	−1.22876e59	−0.116981	−0.0584907	+	0.998288i	$0.518629\pi$
$620$	0	0
$621$	2.26851e60	2.01498
$622$	0	0
$623$	−2.44286e59	−0.202491
$624$	0	0
$625$	3.38403e59	0.261827
$626$	0	0
$627$	2.02146e60	1.46021
$628$	0	0
$629$	−2.80366e59	−0.189119
$630$	0	0
$631$	−2.36287e60	−1.48870	−0.744352	−	0.667787i	$-0.767242\pi$
$631$	−2.36287e60	−1.48870	−0.744352	+	0.667787i	$0.767242\pi$
$632$	0	0
$633$	5.54939e60	3.26637
$634$	0	0
$635$	1.56325e60	0.859794
$636$	0	0
$637$	−9.49799e59	−0.488241
$638$	0	0
$639$	6.76116e60	3.24902
$640$	0	0
$641$	−3.06701e60	−1.37806	−0.689029	−	0.724734i	$-0.741963\pi$
$641$	−3.06701e60	−1.37806	−0.689029	+	0.724734i	$0.741963\pi$
$642$	0	0
$643$	−8.63761e59	−0.362959	−0.181479	−	0.983395i	$-0.558089\pi$
$643$	−8.63761e59	−0.362959	−0.181479	+	0.983395i	$0.558089\pi$
$644$	0	0
$645$	6.27077e58	0.0246483
$646$	0	0
$647$	1.57232e60	0.578228	0.289114	−	0.957295i	$-0.406639\pi$
$647$	1.57232e60	0.578228	0.289114	+	0.957295i	$0.406639\pi$
$648$	0	0
$649$	8.45767e58	0.0291065
$650$	0	0
$651$	−1.67050e60	−0.538091
$652$	0	0
$653$	1.42220e60	0.428870	0.214435	−	0.976738i	$-0.431209\pi$
$653$	1.42220e60	0.428870	0.214435	+	0.976738i	$0.431209\pi$
$654$	0	0
$655$	−2.89543e60	−0.817571
$656$	0	0
$657$	1.75956e60	0.465317
$658$	0	0
$659$	−2.37773e60	−0.589015	−0.294507	−	0.955649i	$-0.595156\pi$
$659$	−2.37773e60	−0.589015	−0.294507	+	0.955649i	$0.595156\pi$
$660$	0	0
$661$	−7.09142e60	−1.64589	−0.822947	−	0.568119i	$-0.807671\pi$
$661$	−7.09142e60	−1.64589	−0.822947	+	0.568119i	$0.807671\pi$
$662$	0	0
$663$	−1.42669e60	−0.310304
$664$	0	0
$665$	−1.71664e60	−0.349957
$666$	0	0
$667$	−7.77170e60	−1.48529
$668$	0	0
$669$	−4.62881e60	−0.829483
$670$	0	0
$671$	−2.54185e60	−0.427184
$672$	0	0
$673$	7.62590e60	1.20217	0.601086	−	0.799184i	$-0.294735\pi$
$673$	7.62590e60	1.20217	0.601086	+	0.799184i	$0.294735\pi$
$674$	0	0
$675$	1.01630e61	1.50311
$676$	0	0
$677$	−1.33549e61	−1.85347	−0.926735	−	0.375717i	$-0.877397\pi$
$677$	−1.33549e61	−1.85347	−0.926735	+	0.375717i	$0.877397\pi$
$678$	0	0
$679$	−3.41011e60	−0.444189
$680$	0	0
$681$	−2.32455e59	−0.0284233
$682$	0	0
$683$	−7.17925e60	−0.824202	−0.412101	−	0.911138i	$-0.635205\pi$
$683$	−7.17925e60	−0.824202	−0.412101	+	0.911138i	$0.635205\pi$
$684$	0	0
$685$	−6.53894e60	−0.704953
$686$	0	0
$687$	−2.77206e61	−2.80693
$688$	0	0
$689$	4.20183e60	0.399692
$690$	0	0
$691$	1.17437e61	1.04961	0.524803	−	0.851224i	$-0.324139\pi$
$691$	1.17437e61	1.04961	0.524803	+	0.851224i	$0.324139\pi$
$692$	0	0
$693$	4.46818e60	0.375289
$694$	0	0
$695$	−4.37993e60	−0.345776
$696$	0	0
$697$	5.66848e59	0.0420689
$698$	0	0
$699$	2.62706e61	1.83320
$700$	0	0
$701$	1.27675e61	0.837854	0.418927	−	0.908020i	$-0.362406\pi$
$701$	1.27675e61	0.837854	0.418927	+	0.908020i	$0.362406\pi$
$702$	0	0
$703$	1.78186e61	1.09985
$704$	0	0
$705$	1.10958e61	0.644306
$706$	0	0
$707$	4.57689e60	0.250065
$708$	0	0
$709$	−2.73809e61	−1.40784	−0.703921	−	0.710278i	$-0.748569\pi$
$709$	−2.73809e61	−1.40784	−0.703921	+	0.710278i	$0.748569\pi$
$710$	0	0
$711$	2.48238e60	0.120136
$712$	0	0
$713$	−1.72235e61	−0.784687
$714$	0	0
$715$	3.17917e60	0.136375
$716$	0	0
$717$	3.85997e61	1.55927
$718$	0	0
$719$	−2.16831e61	−0.824993	−0.412496	−	0.910959i	$-0.635343\pi$
$719$	−2.16831e61	−0.824993	−0.412496	+	0.910959i	$0.635343\pi$
$720$	0	0
$721$	−1.51779e61	−0.544006
$722$	0	0
$723$	−3.70709e61	−1.25187
$724$	0	0
$725$	−3.48174e61	−1.10797
$726$	0	0
$727$	−1.57652e61	−0.472835	−0.236418	−	0.971652i	$-0.575973\pi$
$727$	−1.57652e61	−0.472835	−0.236418	+	0.971652i	$0.575973\pi$
$728$	0	0
$729$	−1.47560e61	−0.417183
$730$	0	0
$731$	−3.06802e59	−0.00817777
$732$	0	0
$733$	2.88722e61	0.725679	0.362840	−	0.931852i	$-0.381807\pi$
$733$	2.88722e61	0.725679	0.362840	+	0.931852i	$0.381807\pi$
$734$	0	0
$735$	3.34420e61	0.792711
$736$	0	0
$737$	2.14571e61	0.479755
$738$	0	0
$739$	−7.66659e60	−0.161714	−0.0808568	−	0.996726i	$-0.525766\pi$
$739$	−7.66659e60	−0.161714	−0.0808568	+	0.996726i	$0.525766\pi$
$740$	0	0
$741$	9.06730e61	1.80462
$742$	0	0
$743$	−5.63267e61	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$743$	−5.63267e61	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$744$	0	0
$745$	3.20179e60	0.0567588
$746$	0	0
$747$	1.58134e62	2.64626
$748$	0	0
$749$	2.37753e61	0.375635
$750$	0	0
$751$	5.65647e60	0.0843892	0.0421946	−	0.999109i	$-0.486565\pi$
$751$	5.65647e60	0.0843892	0.0421946	+	0.999109i	$0.486565\pi$
$752$	0	0
$753$	−1.13526e62	−1.59957
$754$	0	0
$755$	1.09405e61	0.145605
$756$	0	0
$757$	4.86809e61	0.612065	0.306032	−	0.952021i	$-0.400998\pi$
$757$	4.86809e61	0.612065	0.306032	+	0.952021i	$0.400998\pi$
$758$	0	0
$759$	6.74050e61	0.800744
$760$	0	0
$761$	5.98651e61	0.672053	0.336026	−	0.941853i	$-0.390917\pi$
$761$	5.98651e61	0.672053	0.336026	+	0.941853i	$0.390917\pi$
$762$	0	0
$763$	1.35319e61	0.143576
$764$	0	0
$765$	3.43323e61	0.344336
$766$	0	0
$767$	3.79370e60	0.0359718
$768$	0	0
$769$	−2.53056e61	−0.226882	−0.113441	−	0.993545i	$-0.536187\pi$
$769$	−2.53056e61	−0.226882	−0.113441	+	0.993545i	$0.536187\pi$
$770$	0	0
$771$	2.90115e62	2.45980
$772$	0	0
$773$	1.22577e62	0.982992	0.491496	−	0.870880i	$-0.336450\pi$
$773$	1.22577e62	0.982992	0.491496	+	0.870880i	$0.336450\pi$
$774$	0	0
$775$	−7.71618e61	−0.585350
$776$	0	0
$777$	5.76269e61	0.413593
$778$	0	0
$779$	−3.60260e61	−0.244658
$780$	0	0
$781$	1.07852e62	0.693159
$782$	0	0
$783$	−5.14365e62	−3.12891
$784$	0	0
$785$	−7.40809e61	−0.426588
$786$	0	0
$787$	5.88188e61	0.320671	0.160335	−	0.987063i	$-0.448742\pi$
$787$	5.88188e61	0.320671	0.160335	+	0.987063i	$0.448742\pi$
$788$	0	0
$789$	−2.80431e62	−1.44767
$790$	0	0
$791$	9.69290e61	0.473870
$792$	0	0
$793$	−1.14015e62	−0.527944
$794$	0	0
$795$	−1.47945e62	−0.648943
$796$	0	0
$797$	−1.91183e62	−0.794504	−0.397252	−	0.917710i	$-0.630036\pi$
$797$	−1.91183e62	−0.794504	−0.397252	+	0.917710i	$0.630036\pi$
$798$	0	0
$799$	−5.42870e61	−0.213767
$800$	0	0
$801$	−3.10506e62	−1.15870
$802$	0	0
$803$	2.80682e61	0.0992727
$804$	0	0
$805$	−5.72408e61	−0.191909
$806$	0	0
$807$	−7.39340e62	−2.34998
$808$	0	0
$809$	3.32609e62	1.00240	0.501200	−	0.865332i	$-0.332892\pi$
$809$	3.32609e62	1.00240	0.501200	+	0.865332i	$0.332892\pi$
$810$	0	0
$811$	−2.53671e61	−0.0724974	−0.0362487	−	0.999343i	$-0.511541\pi$
$811$	−2.53671e61	−0.0724974	−0.0362487	+	0.999343i	$0.511541\pi$
$812$	0	0
$813$	−6.58342e62	−1.78445
$814$	0	0
$815$	5.81797e61	0.149583
$816$	0	0
$817$	1.94988e61	0.0475591
$818$	0	0
$819$	2.00421e62	0.463808
$820$	0	0
$821$	8.50693e62	1.86808	0.934039	−	0.357170i	$-0.116258\pi$
$821$	8.50693e62	1.86808	0.934039	+	0.357170i	$0.116258\pi$
$822$	0	0
$823$	5.42465e62	1.13051	0.565256	−	0.824915i	$-0.308777\pi$
$823$	5.42465e62	1.13051	0.565256	+	0.824915i	$0.308777\pi$
$824$	0	0
$825$	3.01976e62	0.597328
$826$	0	0
$827$	7.20567e62	1.35303	0.676513	−	0.736431i	$-0.263490\pi$
$827$	7.20567e62	1.35303	0.676513	+	0.736431i	$0.263490\pi$
$828$	0	0
$829$	−8.50026e62	−1.51534	−0.757669	−	0.652639i	$-0.773662\pi$
$829$	−8.50026e62	−1.51534	−0.757669	+	0.652639i	$0.773662\pi$
$830$	0	0
$831$	−1.18245e63	−2.00153
$832$	0	0
$833$	−1.63618e62	−0.263004
$834$	0	0
$835$	3.74222e62	0.571308
$836$	0	0
$837$	−1.13993e63	−1.65302
$838$	0	0
$839$	−5.84952e62	−0.805820	−0.402910	−	0.915240i	$-0.632001\pi$
$839$	−5.84952e62	−0.805820	−0.402910	+	0.915240i	$0.632001\pi$
$840$	0	0
$841$	9.98130e62	1.30639
$842$	0	0
$843$	1.03464e63	1.28675
$844$	0	0
$845$	−2.97397e62	−0.351494
$846$	0	0
$847$	−2.64625e62	−0.297261
$848$	0	0
$849$	2.50058e63	2.67009
$850$	0	0
$851$	5.94155e62	0.603135
$852$	0	0
$853$	6.11541e62	0.590230	0.295115	−	0.955462i	$-0.404642\pi$
$853$	6.11541e62	0.590230	0.295115	+	0.955462i	$0.404642\pi$
$854$	0	0
$855$	−2.18199e63	−2.00254
$856$	0	0
$857$	−3.04840e62	−0.266063	−0.133032	−	0.991112i	$-0.542471\pi$
$857$	−3.04840e62	−0.266063	−0.133032	+	0.991112i	$0.542471\pi$
$858$	0	0
$859$	4.98228e62	0.413595	0.206798	−	0.978384i	$-0.433696\pi$
$859$	4.98228e62	0.413595	0.206798	+	0.978384i	$0.433696\pi$
$860$	0	0
$861$	−1.16511e62	−0.0920024
$862$	0	0
$863$	−2.55705e61	−0.0192091	−0.00960454	−	0.999954i	$-0.503057\pi$
$863$	−2.55705e61	−0.0192091	−0.00960454	+	0.999954i	$0.503057\pi$
$864$	0	0
$865$	−2.30361e62	−0.164650
$866$	0	0
$867$	2.36757e63	1.61025
$868$	0	0
$869$	3.95985e61	0.0256303
$870$	0	0
$871$	9.62459e62	0.592914
$872$	0	0
$873$	−4.33451e63	−2.54176
$874$	0	0
$875$	−6.07939e62	−0.339380
$876$	0	0
$877$	2.16419e63	1.15028	0.575141	−	0.818054i	$-0.304947\pi$
$877$	2.16419e63	1.15028	0.575141	+	0.818054i	$0.304947\pi$
$878$	0	0
$879$	−2.27678e63	−1.15229
$880$	0	0
$881$	−8.01449e62	−0.386273	−0.193136	−	0.981172i	$-0.561866\pi$
$881$	−8.01449e62	−0.386273	−0.193136	+	0.981172i	$0.561866\pi$
$882$	0	0
$883$	1.13427e62	0.0520667	0.0260334	−	0.999661i	$-0.491712\pi$
$883$	1.13427e62	0.0520667	0.0260334	+	0.999661i	$0.491712\pi$
$884$	0	0
$885$	−1.33575e62	−0.0584041
$886$	0	0
$887$	2.15665e63	0.898299	0.449150	−	0.893457i	$-0.351727\pi$
$887$	2.15665e63	0.898299	0.449150	+	0.893457i	$0.351727\pi$
$888$	0	0
$889$	1.57205e63	0.623849
$890$	0	0
$891$	1.83077e63	0.692252
$892$	0	0
$893$	3.45021e63	1.24319
$894$	0	0
$895$	3.52062e62	0.120899
$896$	0	0
$897$	3.02346e63	0.989616
$898$	0	0
$899$	3.90528e63	1.21848
$900$	0	0
$901$	7.23832e62	0.215305
$902$	0	0
$903$	6.30607e61	0.0178843
$904$	0	0
$905$	2.98592e62	0.0807484
$906$	0	0
$907$	−7.76445e62	−0.200241	−0.100121	−	0.994975i	$-0.531923\pi$
$907$	−7.76445e62	−0.200241	−0.100121	+	0.994975i	$0.531923\pi$
$908$	0	0
$909$	5.81759e63	1.43093
$910$	0	0
$911$	2.02334e62	0.0474704	0.0237352	−	0.999718i	$-0.492444\pi$
$911$	2.02334e62	0.0474704	0.0237352	+	0.999718i	$0.492444\pi$
$912$	0	0
$913$	2.52252e63	0.564563
$914$	0	0
$915$	4.01442e63	0.857174
$916$	0	0
$917$	−2.91173e63	−0.593213
$918$	0	0
$919$	−9.59396e63	−1.86515	−0.932577	−	0.360972i	$-0.882445\pi$
$919$	−9.59396e63	−1.86515	−0.932577	+	0.360972i	$0.882445\pi$
$920$	0	0
$921$	−1.34603e64	−2.49731
$922$	0	0
$923$	4.83773e63	0.856654
$924$	0	0
$925$	2.66183e63	0.449918
$926$	0	0
$927$	−1.92923e64	−3.11293
$928$	0	0
$929$	−7.48220e63	−1.15263	−0.576317	−	0.817226i	$-0.695511\pi$
$929$	−7.48220e63	−1.15263	−0.576317	+	0.817226i	$0.695511\pi$
$930$	0	0
$931$	1.03987e64	1.52954
$932$	0	0
$933$	3.50164e63	0.491831
$934$	0	0
$935$	5.47661e62	0.0734619
$936$	0	0
$937$	−5.90450e63	−0.756453	−0.378227	−	0.925713i	$-0.623466\pi$
$937$	−5.90450e63	−0.756453	−0.378227	+	0.925713i	$0.623466\pi$
$938$	0	0
$939$	−1.64128e63	−0.200850
$940$	0	0
$941$	−1.10650e64	−1.29352	−0.646759	−	0.762694i	$-0.723876\pi$
$941$	−1.10650e64	−1.29352	−0.646759	+	0.762694i	$0.723876\pi$
$942$	0	0
$943$	−1.20127e63	−0.134165
$944$	0	0
$945$	−3.78845e63	−0.404275
$946$	0	0
$947$	−7.68969e63	−0.784123	−0.392062	−	0.919939i	$-0.628238\pi$
$947$	−7.68969e63	−0.784123	−0.392062	+	0.919939i	$0.628238\pi$
$948$	0	0
$949$	1.25900e63	0.122688
$950$	0	0
$951$	−1.90961e64	−1.77854
$952$	0	0
$953$	5.87345e63	0.522873	0.261436	−	0.965221i	$-0.415804\pi$
$953$	5.87345e63	0.522873	0.261436	+	0.965221i	$0.415804\pi$
$954$	0	0
$955$	−6.76564e63	−0.575753
$956$	0	0
$957$	−1.52835e64	−1.24341
$958$	0	0
$959$	−6.57576e63	−0.511500
$960$	0	0
$961$	−4.78994e63	−0.356268
$962$	0	0
$963$	3.02202e64	2.14947
$964$	0	0
$965$	4.64099e61	0.00315699
$966$	0	0
$967$	1.95511e64	1.27205	0.636023	−	0.771670i	$-0.280578\pi$
$967$	1.95511e64	1.27205	0.636023	+	0.771670i	$0.280578\pi$
$968$	0	0
$969$	1.56198e64	0.972110
$970$	0	0
$971$	−2.50785e64	−1.49310	−0.746549	−	0.665330i	$-0.768291\pi$
$971$	−2.50785e64	−1.49310	−0.746549	+	0.665330i	$0.768291\pi$
$972$	0	0
$973$	−4.40459e63	−0.250888
$974$	0	0
$975$	1.35452e64	0.738220
$976$	0	0
$977$	3.39735e64	1.77177	0.885884	−	0.463906i	$-0.153553\pi$
$977$	3.39735e64	1.77177	0.885884	+	0.463906i	$0.153553\pi$
$978$	0	0
$979$	−4.95312e63	−0.247202
$980$	0	0
$981$	1.72001e64	0.821575
$982$	0	0
$983$	2.21328e64	1.01189	0.505947	−	0.862565i	$-0.331143\pi$
$983$	2.21328e64	1.01189	0.505947	+	0.862565i	$0.331143\pi$
$984$	0	0
$985$	6.89463e63	0.301739
$986$	0	0
$987$	1.11583e64	0.467496
$988$	0	0
$989$	6.50180e62	0.0260803
$990$	0	0
$991$	−2.88177e64	−1.10682	−0.553408	−	0.832910i	$-0.686673\pi$
$991$	−2.88177e64	−1.10682	−0.553408	+	0.832910i	$0.686673\pi$
$992$	0	0
$993$	1.48807e64	0.547285
$994$	0	0
$995$	−1.40856e64	−0.496112
$996$	0	0
$997$	−2.97127e64	−1.00230	−0.501148	−	0.865361i	$-0.667089\pi$
$997$	−2.97127e64	−1.00230	−0.501148	+	0.865361i	$0.667089\pi$
$998$	0	0
$999$	3.93238e64	1.27057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.44.a.c.1.1		3
4.3	odd	2		1.44.a.a.1.1	✓	3
12.11	even	2		9.44.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.44.a.a.1.1	✓	3	4.3	odd	2
9.44.a.b.1.3		3	12.11	even	2
16.44.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:	\( k \)	\(=\)	\( 44 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-3.22027e10 q^{3} +5.54482e14 q^{5} +5.57604e17 q^{7} +7.08758e20 q^{9} +O(q^{10})\)	\(q-3.22027e10 q^{3} +5.54482e14 q^{5} +5.57604e17 q^{7} +7.08758e20 q^{9} +1.13059e22 q^{11} +5.07130e23 q^{13} -1.78558e25 q^{15} +8.73610e25 q^{17} -5.55222e27 q^{19} -1.79564e28 q^{21} -1.85137e29 q^{23} -8.29418e29 q^{25} -1.22532e31 q^{27} +4.19782e31 q^{29} +9.30312e31 q^{31} -3.64082e32 q^{33} +3.09181e32 q^{35} -3.20928e33 q^{37} -1.63309e34 q^{39} +6.48857e33 q^{41} -3.51189e33 q^{43} +3.92993e35 q^{45} -6.21410e35 q^{47} -1.87289e36 q^{49} -2.81326e36 q^{51} +8.28552e36 q^{53} +6.26895e36 q^{55} +1.78797e38 q^{57} +7.48073e36 q^{59} -2.24824e38 q^{61} +3.95206e38 q^{63} +2.81194e38 q^{65} +1.89786e39 q^{67} +5.96191e39 q^{69} +9.53945e39 q^{71} +2.48260e39 q^{73} +2.67095e40 q^{75} +6.30424e39 q^{77} +3.50244e39 q^{79} +1.61930e41 q^{81} +2.23114e41 q^{83} +4.84401e40 q^{85} -1.35181e42 q^{87} -4.38099e41 q^{89} +2.82777e41 q^{91} -2.99586e42 q^{93} -3.07861e42 q^{95} -6.11565e42 q^{97} +8.01318e42 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 24401437812 q^{3} + 535205380774170 q^{5} - 30\!\cdots\!56 q^{7}+ \cdots + 47\!\cdots\!31 q^{9}+O(q^{10}) \) 3 * q - 24401437812 * q^3 + 535205380774170 * q^5 - 301971425665478856 * q^7 + 477482125171066743231 * q^9	\( 3 q - 24401437812 q^{3} + 535205380774170 q^{5} - 30\!\cdots\!56 q^{7}+ \cdots + 14\!\cdots\!08 q^{99}+O(q^{100}) \) 3 * q - 24401437812 * q^3 + 535205380774170 * q^5 - 301971425665478856 * q^7 + 477482125171066743231 * q^9 - 26665053285822824598396 * q^11 + 2675160683143344673595682 * q^13 - 35619426649405115420082840 * q^15 - 408654222932442443796841194 * q^17 - 1577331357975519701176464900 * q^19 + 39305230121617849136195725536 * q^21 + 1215339635928561142484811048 * q^23 - 2310484082897201607166539950475 * q^25 - 12636088863973677226324883265480 * q^27 + 57118989018136587596191009705650 * q^29 + 255565972342397832812135282963424 * q^31 - 274896238871162964163796672607216 * q^33 - 2398213530857285851552147980593520 * q^35 - 2320645576165979428293572256842694 * q^37 + 3966989860384498209469308423541128 * q^39 + 25742376604739002042016577514061166 * q^41 - 240714339689128182512406300123419292 * q^43 + 257230496455952342677786043056557090 * q^45 - 308713553347399687962182796433367856 * q^47 + 3434980292250886821160584750829904379 * q^49 - 13352566627975131351422398848669478056 * q^51 + 15000497558784669132889458695112796362 * q^53 - 3996571389110251928625895283213506440 * q^55 + 194498189404429212580032012092952587760 * q^57 - 226996261182733939538282061249836505900 * q^59 - 9393429368478033721028929889113591854 * q^61 + 967475285671148246145956691206790781848 * q^63 - 270218921102297342477494789001032451460 * q^65 + 733015616887151283726086660162187354444 * q^67 + 1967042253746083561325124578559972227232 * q^69 + 18523878066158899993345833814569898287864 * q^71 - 20251106025908798058937016734369224029298 * q^73 + 21273380043530396090723723962888745519700 * q^75 + 59055613160181778232413942942559747685792 * q^77 - 15233568018599764739559826813006999294800 * q^79 + 73114835779397486920654645241710822062363 * q^81 + 89206610196719670053085531955706819434428 * q^83 + 438725373228516358990025164547060204838420 * q^85 - 1720273148596604243726575204332689696886840 * q^87 + 203691422331055117225246388044508249665950 * q^89 + 1168954967602248602369466051364648836229584 * q^91 - 4949189234312030019905814954548842124935296 * q^93 - 3125746305909018830165692627580442029979000 * q^95 - 3881062021784775072625368160679539917597594 * q^97 + 14255275025472043026078170466415579823927508 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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