LMFDB - Embedded newform 16.44.a.c.1.3

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			1.3
Root			$116336.$ 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)

$f(q)$	$=$	$q+1.79507e10 q^{3} -6.39116e14 q^{5} +1.72730e18 q^{7} -6.02939e18 q^{9} +O(q^{10})$	$q+1.79507e10 q^{3} -6.39116e14 q^{5} +1.72730e18 q^{7} -6.02939e18 q^{9} -1.05408e22 q^{11} +1.50541e24 q^{13} -1.14726e25 q^{15} -5.54217e26 q^{17} +1.99446e27 q^{19} +3.10063e28 q^{21} -7.48572e28 q^{23} -7.28399e29 q^{25} -6.00067e30 q^{27} -7.64381e30 q^{29} -1.08077e31 q^{31} -1.89215e32 q^{33} -1.10395e33 q^{35} +2.93069e32 q^{37} +2.70232e34 q^{39} +7.49273e34 q^{41} -1.23211e35 q^{43} +3.85348e33 q^{45} +5.61634e35 q^{47} +7.99756e35 q^{49} -9.94859e36 q^{51} +2.77981e36 q^{53} +6.73680e36 q^{55} +3.58019e37 q^{57} -1.73860e38 q^{59} -1.16569e38 q^{61} -1.04146e37 q^{63} -9.62132e38 q^{65} -9.18410e38 q^{67} -1.34374e39 q^{69} +3.76161e39 q^{71} -9.44690e39 q^{73} -1.30753e40 q^{75} -1.82072e40 q^{77} -3.86685e40 q^{79} -1.05737e41 q^{81} -1.39471e41 q^{83} +3.54209e41 q^{85} -1.37212e41 q^{87} +9.05928e41 q^{89} +2.60030e42 q^{91} -1.94005e41 q^{93} -1.27469e42 q^{95} -1.23492e42 q^{97} +6.35546e40 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.79507e10	0.990773	0.495387	−	0.868673i	$-0.335026\pi$
$3$	1.79507e10	0.990773	0.495387	+	0.868673i	$0.335026\pi$
$4$	0	0
$5$	−6.39116e14	−0.599411	−0.299706	−	0.954032i	$-0.596888\pi$
$5$	−6.39116e14	−0.599411	−0.299706	+	0.954032i	$0.596888\pi$
$6$	0	0
$7$	1.72730e18	1.16885	0.584427	−	0.811446i	$-0.301319\pi$
$7$	1.72730e18	1.16885	0.584427	+	0.811446i	$0.301319\pi$
$8$	0	0
$9$	−6.02939e18	−0.0183679
$10$	0	0
$11$	−1.05408e22	−0.429468	−0.214734	−	0.976673i	$-0.568889\pi$
$11$	−1.05408e22	−0.429468	−0.214734	+	0.976673i	$0.568889\pi$
$12$	0	0
$13$	1.50541e24	1.68995	0.844973	−	0.534809i	$-0.179617\pi$
$13$	1.50541e24	1.68995	0.844973	+	0.534809i	$0.179617\pi$
$14$	0	0
$15$	−1.14726e25	−0.593881
$16$	0	0
$17$	−5.54217e26	−1.94549	−0.972743	−	0.231884i	$-0.925511\pi$
$17$	−5.54217e26	−1.94549	−0.972743	+	0.231884i	$0.925511\pi$
$18$	0	0
$19$	1.99446e27	0.640652	0.320326	−	0.947307i	$-0.396208\pi$
$19$	1.99446e27	0.640652	0.320326	+	0.947307i	$0.396208\pi$
$20$	0	0
$21$	3.10063e28	1.15807
$22$	0	0
$23$	−7.48572e28	−0.395444	−0.197722	−	0.980258i	$-0.563354\pi$
$23$	−7.48572e28	−0.395444	−0.197722	+	0.980258i	$0.563354\pi$
$24$	0	0
$25$	−7.28399e29	−0.640706
$26$	0	0
$27$	−6.00067e30	−1.00897
$28$	0	0
$29$	−7.64381e30	−0.276537	−0.138268	−	0.990395i	$-0.544154\pi$
$29$	−7.64381e30	−0.276537	−0.138268	+	0.990395i	$0.544154\pi$
$30$	0	0
$31$	−1.08077e31	−0.0932085	−0.0466043	−	0.998913i	$-0.514840\pi$
$31$	−1.08077e31	−0.0932085	−0.0466043	+	0.998913i	$0.514840\pi$
$32$	0	0
$33$	−1.89215e32	−0.425506
$34$	0	0
$35$	−1.10395e33	−0.700624
$36$	0	0
$37$	2.93069e32	0.0563161	0.0281580	−	0.999603i	$-0.491036\pi$
$37$	2.93069e32	0.0563161	0.0281580	+	0.999603i	$0.491036\pi$
$38$	0	0
$39$	2.70232e34	1.67435
$40$	0	0
$41$	7.49273e34	1.58412	0.792058	−	0.610446i	$-0.209010\pi$
$41$	7.49273e34	1.58412	0.792058	+	0.610446i	$0.209010\pi$
$42$	0	0
$43$	−1.23211e35	−0.935572	−0.467786	−	0.883842i	$-0.654948\pi$
$43$	−1.23211e35	−0.935572	−0.467786	+	0.883842i	$0.654948\pi$
$44$	0	0
$45$	3.85348e33	0.0110099
$46$	0	0
$47$	5.61634e35	0.630013	0.315006	−	0.949090i	$-0.397993\pi$
$47$	5.61634e35	0.630013	0.315006	+	0.949090i	$0.397993\pi$
$48$	0	0
$49$	7.99756e35	0.366220
$50$	0	0
$51$	−9.94859e36	−1.92754
$52$	0	0
$53$	2.77981e36	0.235551	0.117775	−	0.993040i	$-0.462424\pi$
$53$	2.77981e36	0.235551	0.117775	+	0.993040i	$0.462424\pi$
$54$	0	0
$55$	6.73680e36	0.257428
$56$	0	0
$57$	3.58019e37	0.634741
$58$	0	0
$59$	−1.73860e38	−1.46852	−0.734262	−	0.678866i	$-0.762472\pi$
$59$	−1.73860e38	−1.46852	−0.734262	+	0.678866i	$0.762472\pi$
$60$	0	0
$61$	−1.16569e38	−0.480829	−0.240414	−	0.970670i	$-0.577283\pi$
$61$	−1.16569e38	−0.480829	−0.240414	+	0.970670i	$0.577283\pi$
$62$	0	0
$63$	−1.04146e37	−0.0214694
$64$	0	0
$65$	−9.62132e38	−1.01297
$66$	0	0
$67$	−9.18410e38	−0.503998	−0.251999	−	0.967728i	$-0.581088\pi$
$67$	−9.18410e38	−0.503998	−0.251999	+	0.967728i	$0.581088\pi$
$68$	0	0
$69$	−1.34374e39	−0.391796
$70$	0	0
$71$	3.76161e39	0.593361	0.296681	−	0.954977i	$-0.404120\pi$
$71$	3.76161e39	0.593361	0.296681	+	0.954977i	$0.404120\pi$
$72$	0	0
$73$	−9.44690e39	−0.820065	−0.410032	−	0.912071i	$-0.634483\pi$
$73$	−9.44690e39	−0.820065	−0.410032	+	0.912071i	$0.634483\pi$
$74$	0	0
$75$	−1.30753e40	−0.634795
$76$	0	0
$77$	−1.82072e40	−0.501986
$78$	0	0
$79$	−3.86685e40	−0.614293	−0.307146	−	0.951662i	$-0.599374\pi$
$79$	−3.86685e40	−0.614293	−0.307146	+	0.951662i	$0.599374\pi$
$80$	0	0
$81$	−1.05737e41	−0.981295
$82$	0	0
$83$	−1.39471e41	−0.766137	−0.383068	−	0.923720i	$-0.625133\pi$
$83$	−1.39471e41	−0.766137	−0.383068	+	0.923720i	$0.625133\pi$
$84$	0	0
$85$	3.54209e41	1.16615
$86$	0	0
$87$	−1.37212e41	−0.273985
$88$	0	0
$89$	9.05928e41	1.10971	0.554854	−	0.831948i	$-0.312774\pi$
$89$	9.05928e41	1.10971	0.554854	+	0.831948i	$0.312774\pi$
$90$	0	0
$91$	2.60030e42	1.97530
$92$	0	0
$93$	−1.94005e41	−0.0923486
$94$	0	0
$95$	−1.27469e42	−0.384014
$96$	0	0
$97$	−1.23492e42	−0.237709	−0.118855	−	0.992912i	$-0.537922\pi$
$97$	−1.23492e42	−0.237709	−0.118855	+	0.992912i	$0.537922\pi$
$98$	0	0
$99$	6.35546e40	0.00788842
$100$	0	0
$101$	9.34212e41	0.0754286	0.0377143	−	0.999289i	$-0.487992\pi$
$101$	9.34212e41	0.0754286	0.0377143	+	0.999289i	$0.487992\pi$
$102$	0	0
$103$	2.71649e43	1.43882	0.719412	−	0.694584i	$-0.244411\pi$
$103$	2.71649e43	1.43882	0.719412	+	0.694584i	$0.244411\pi$
$104$	0	0
$105$	−1.98166e43	−0.694160
$106$	0	0
$107$	−7.12131e43	−1.66268	−0.831340	−	0.555764i	$-0.812426\pi$
$107$	−7.12131e43	−1.66268	−0.831340	+	0.555764i	$0.812426\pi$
$108$	0	0
$109$	5.74242e43	0.900379	0.450190	−	0.892933i	$-0.351356\pi$
$109$	5.74242e43	0.900379	0.450190	+	0.892933i	$0.351356\pi$
$110$	0	0
$111$	5.26079e42	0.0557965
$112$	0	0
$113$	−6.84403e43	−0.494454	−0.247227	−	0.968958i	$-0.579519\pi$
$113$	−6.84403e43	−0.494454	−0.247227	+	0.968958i	$0.579519\pi$
$114$	0	0
$115$	4.78425e43	0.237034
$116$	0	0
$117$	−9.07670e42	−0.0310407
$118$	0	0
$119$	−9.57300e44	−2.27399
$120$	0	0
$121$	−4.91292e44	−0.815557
$122$	0	0
$123$	1.34500e45	1.56950
$124$	0	0
$125$	1.19212e45	0.983458
$126$	0	0
$127$	2.23889e45	1.31297	0.656483	−	0.754341i	$-0.272043\pi$
$127$	2.23889e45	1.31297	0.656483	+	0.754341i	$0.272043\pi$
$128$	0	0
$129$	−2.21172e45	−0.926940
$130$	0	0
$131$	−2.93570e45	−0.883850	−0.441925	−	0.897052i	$-0.645704\pi$
$131$	−2.93570e45	−0.883850	−0.441925	+	0.897052i	$0.645704\pi$
$132$	0	0
$133$	3.44503e45	0.748829
$134$	0	0
$135$	3.83513e45	0.604789
$136$	0	0
$137$	1.51920e45	0.174631	0.0873157	−	0.996181i	$-0.472171\pi$
$137$	1.51920e45	0.174631	0.0873157	+	0.996181i	$0.472171\pi$
$138$	0	0
$139$	−6.55245e44	−0.0551551	−0.0275775	−	0.999620i	$-0.508779\pi$
$139$	−6.55245e44	−0.0551551	−0.0275775	+	0.999620i	$0.508779\pi$
$140$	0	0
$141$	1.00817e46	0.624200
$142$	0	0
$143$	−1.58682e46	−0.725778
$144$	0	0
$145$	4.88528e45	0.165759
$146$	0	0
$147$	1.43562e46	0.362841
$148$	0	0
$149$	−8.85478e46	−1.67368	−0.836841	−	0.547446i	$-0.815600\pi$
$149$	−8.85478e46	−1.67368	−0.836841	+	0.547446i	$0.815600\pi$
$150$	0	0
$151$	−8.54056e46	−1.21194	−0.605971	−	0.795486i	$-0.707215\pi$
$151$	−8.54056e46	−1.21194	−0.605971	+	0.795486i	$0.707215\pi$
$152$	0	0
$153$	3.34159e45	0.0357345
$154$	0	0
$155$	6.90736e45	0.0558702
$156$	0	0
$157$	−6.33166e46	−0.388754	−0.194377	−	0.980927i	$-0.562269\pi$
$157$	−6.33166e46	−0.388754	−0.194377	+	0.980927i	$0.562269\pi$
$158$	0	0
$159$	4.98996e46	0.233377
$160$	0	0
$161$	−1.29301e47	−0.462217
$162$	0	0
$163$	2.42368e47	0.664419	0.332210	−	0.943206i	$-0.392206\pi$
$163$	2.42368e47	0.664419	0.332210	+	0.943206i	$0.392206\pi$
$164$	0	0
$165$	1.20930e47	0.255053
$166$	0	0
$167$	−2.12976e47	−0.346677	−0.173339	−	0.984862i	$-0.555456\pi$
$167$	−2.12976e47	−0.346677	−0.173339	+	0.984862i	$0.555456\pi$
$168$	0	0
$169$	1.47273e48	1.85592
$170$	0	0
$171$	−1.20253e46	−0.0117674
$172$	0	0
$173$	−9.17738e47	−0.699403	−0.349702	−	0.936861i	$-0.613717\pi$
$173$	−9.17738e47	−0.699403	−0.349702	+	0.936861i	$0.613717\pi$
$174$	0	0
$175$	−1.25816e48	−0.748892
$176$	0	0
$177$	−3.12091e48	−1.45497
$178$	0	0
$179$	−1.88488e48	−0.690150	−0.345075	−	0.938575i	$-0.612147\pi$
$179$	−1.88488e48	−0.690150	−0.345075	+	0.938575i	$0.612147\pi$
$180$	0	0
$181$	−1.01885e47	−0.0293779	−0.0146889	−	0.999892i	$-0.504676\pi$
$181$	−1.01885e47	−0.0293779	−0.0146889	+	0.999892i	$0.504676\pi$
$182$	0	0
$183$	−2.09249e48	−0.476392
$184$	0	0
$185$	−1.87305e47	−0.0337565
$186$	0	0
$187$	5.84190e48	0.835525
$188$	0	0
$189$	−1.03650e49	−1.17934
$190$	0	0
$191$	−8.44008e48	−0.765824	−0.382912	−	0.923785i	$-0.625079\pi$
$191$	−8.44008e48	−0.765824	−0.382912	+	0.923785i	$0.625079\pi$
$192$	0	0
$193$	−2.13716e49	−1.55008	−0.775042	−	0.631909i	$-0.782272\pi$
$193$	−2.13716e49	−1.55008	−0.775042	+	0.631909i	$0.782272\pi$
$194$	0	0
$195$	−1.72709e49	−1.00363
$196$	0	0
$197$	1.92220e49	0.896963	0.448482	−	0.893792i	$-0.351965\pi$
$197$	1.92220e49	0.896963	0.448482	+	0.893792i	$0.351965\pi$
$198$	0	0
$199$	−3.32032e49	−1.24692	−0.623460	−	0.781855i	$-0.714274\pi$
$199$	−3.32032e49	−1.24692	−0.623460	+	0.781855i	$0.714274\pi$
$200$	0	0
$201$	−1.64861e49	−0.499347
$202$	0	0
$203$	−1.32032e49	−0.323231
$204$	0	0
$205$	−4.78872e49	−0.949537
$206$	0	0
$207$	4.51343e47	0.00726347
$208$	0	0
$209$	−2.10232e49	−0.275140
$210$	0	0
$211$	−7.61623e49	−0.812208	−0.406104	−	0.913827i	$-0.633113\pi$
$211$	−7.61623e49	−0.812208	−0.406104	+	0.913827i	$0.633113\pi$
$212$	0	0
$213$	6.75235e49	0.587887
$214$	0	0
$215$	7.87460e49	0.560792
$216$	0	0
$217$	−1.86681e49	−0.108947
$218$	0	0
$219$	−1.69579e50	−0.812498
$220$	0	0
$221$	−8.34324e50	−3.28777
$222$	0	0
$223$	−2.81313e50	−0.913344	−0.456672	−	0.889635i	$-0.650959\pi$
$223$	−2.81313e50	−0.913344	−0.456672	+	0.889635i	$0.650959\pi$
$224$	0	0
$225$	4.39180e48	0.0117684
$226$	0	0
$227$	−4.74569e50	−1.05134	−0.525670	−	0.850689i	$-0.676185\pi$
$227$	−4.74569e50	−1.05134	−0.525670	+	0.850689i	$0.676185\pi$
$228$	0	0
$229$	−1.73705e50	−0.318676	−0.159338	−	0.987224i	$-0.550936\pi$
$229$	−1.73705e50	−0.318676	−0.159338	+	0.987224i	$0.550936\pi$
$230$	0	0
$231$	−3.26831e50	−0.497354
$232$	0	0
$233$	−3.86377e50	−0.488494	−0.244247	−	0.969713i	$-0.578541\pi$
$233$	−3.86377e50	−0.488494	−0.244247	+	0.969713i	$0.578541\pi$
$234$	0	0
$235$	−3.58949e50	−0.377637
$236$	0	0
$237$	−6.94127e50	−0.608625
$238$	0	0
$239$	−1.53965e51	−1.12685	−0.563427	−	0.826166i	$-0.690517\pi$
$239$	−1.53965e51	−1.12685	−0.563427	+	0.826166i	$0.690517\pi$
$240$	0	0
$241$	1.69305e51	1.03586	0.517932	−	0.855422i	$-0.326702\pi$
$241$	1.69305e51	1.03586	0.517932	+	0.855422i	$0.326702\pi$
$242$	0	0
$243$	7.17082e49	0.0367311
$244$	0	0
$245$	−5.11137e50	−0.219516
$246$	0	0
$247$	3.00247e51	1.08267
$248$	0	0
$249$	−2.50361e51	−0.759068
$250$	0	0
$251$	−1.12233e51	−0.286506	−0.143253	−	0.989686i	$-0.545756\pi$
$251$	−1.12233e51	−0.286506	−0.143253	+	0.989686i	$0.545756\pi$
$252$	0	0
$253$	7.89055e50	0.169831
$254$	0	0
$255$	6.35830e51	1.15539
$256$	0	0
$257$	5.15162e51	0.791374	0.395687	−	0.918385i	$-0.370506\pi$
$257$	5.15162e51	0.791374	0.395687	+	0.918385i	$0.370506\pi$
$258$	0	0
$259$	5.06218e50	0.0658252
$260$	0	0
$261$	4.60875e49	0.00507939
$262$	0	0
$263$	−9.78033e51	−0.914755	−0.457377	−	0.889273i	$-0.651211\pi$
$263$	−9.78033e51	−0.914755	−0.457377	+	0.889273i	$0.651211\pi$
$264$	0	0
$265$	−1.77662e51	−0.141192
$266$	0	0
$267$	1.62620e52	1.09947
$268$	0	0
$269$	−1.17046e52	−0.674037	−0.337019	−	0.941498i	$-0.609419\pi$
$269$	−1.17046e52	−0.674037	−0.337019	+	0.941498i	$0.609419\pi$
$270$	0	0
$271$	−1.90564e52	−0.935839	−0.467919	−	0.883771i	$-0.654996\pi$
$271$	−1.90564e52	−0.935839	−0.467919	+	0.883771i	$0.654996\pi$
$272$	0	0
$273$	4.66771e52	1.95707
$274$	0	0
$275$	7.67791e51	0.275163
$276$	0	0
$277$	3.73097e51	0.114421	0.0572106	−	0.998362i	$-0.481779\pi$
$277$	3.73097e51	0.114421	0.0572106	+	0.998362i	$0.481779\pi$
$278$	0	0
$279$	6.51636e49	0.00171204
$280$	0	0
$281$	3.03658e51	0.0684225	0.0342112	−	0.999415i	$-0.489108\pi$
$281$	3.03658e51	0.0684225	0.0342112	+	0.999415i	$0.489108\pi$
$282$	0	0
$283$	6.91493e52	1.33776	0.668882	−	0.743369i	$-0.266773\pi$
$283$	6.91493e52	1.33776	0.668882	+	0.743369i	$0.266773\pi$
$284$	0	0
$285$	−2.28816e52	−0.380471
$286$	0	0
$287$	1.29422e53	1.85160
$288$	0	0
$289$	2.26004e53	2.78492
$290$	0	0
$291$	−2.21676e52	−0.235516
$292$	0	0
$293$	−7.58660e52	−0.695654	−0.347827	−	0.937559i	$-0.613080\pi$
$293$	−7.58660e52	−0.695654	−0.347827	+	0.937559i	$0.613080\pi$
$294$	0	0
$295$	1.11117e53	0.880250
$296$	0	0
$297$	6.32519e52	0.433321
$298$	0	0
$299$	−1.12691e53	−0.668279
$300$	0	0
$301$	−2.12822e53	−1.09355
$302$	0	0
$303$	1.67698e52	0.0747326
$304$	0	0
$305$	7.45010e52	0.288214
$306$	0	0
$307$	3.17728e53	1.06802	0.534011	−	0.845478i	$-0.320684\pi$
$307$	3.17728e53	1.06802	0.534011	+	0.845478i	$0.320684\pi$
$308$	0	0
$309$	4.87629e53	1.42555
$310$	0	0
$311$	−6.14862e53	−1.56469	−0.782347	−	0.622843i	$-0.785978\pi$
$311$	−6.14862e53	−1.56469	−0.782347	+	0.622843i	$0.785978\pi$
$312$	0	0
$313$	3.42329e53	0.758997	0.379498	−	0.925192i	$-0.376097\pi$
$313$	3.42329e53	0.758997	0.379498	+	0.925192i	$0.376097\pi$
$314$	0	0
$315$	6.65612e51	0.0128690
$316$	0	0
$317$	−5.98968e53	−1.01072	−0.505358	−	0.862909i	$-0.668640\pi$
$317$	−5.98968e53	−1.01072	−0.505358	+	0.862909i	$0.668640\pi$
$318$	0	0
$319$	8.05719e52	0.118764
$320$	0	0
$321$	−1.27833e54	−1.64734
$322$	0	0
$323$	−1.10536e54	−1.24638
$324$	0	0
$325$	−1.09654e54	−1.08276
$326$	0	0
$327$	1.03081e54	0.892072
$328$	0	0
$329$	9.70111e53	0.736393
$330$	0	0
$331$	−1.33462e54	−0.889317	−0.444659	−	0.895700i	$-0.646675\pi$
$331$	−1.33462e54	−0.889317	−0.444659	+	0.895700i	$0.646675\pi$
$332$	0	0
$333$	−1.76702e51	−0.00103441
$334$	0	0
$335$	5.86971e53	0.302102
$336$	0	0
$337$	−2.37074e54	−1.07359	−0.536797	−	0.843711i	$-0.680366\pi$
$337$	−2.37074e54	−1.07359	−0.536797	+	0.843711i	$0.680366\pi$
$338$	0	0
$339$	−1.22855e54	−0.489891
$340$	0	0
$341$	1.13922e53	0.0400301
$342$	0	0
$343$	−2.39069e54	−0.740796
$344$	0	0
$345$	8.58806e53	0.234847
$346$	0	0
$347$	5.56591e54	1.34417	0.672083	−	0.740476i	$-0.265400\pi$
$347$	5.56591e54	1.34417	0.672083	+	0.740476i	$0.265400\pi$
$348$	0	0
$349$	5.55165e54	1.18488	0.592441	−	0.805614i	$-0.298164\pi$
$349$	5.55165e54	1.18488	0.592441	+	0.805614i	$0.298164\pi$
$350$	0	0
$351$	−9.03347e54	−1.70511
$352$	0	0
$353$	−2.79315e54	−0.466592	−0.233296	−	0.972406i	$-0.574951\pi$
$353$	−2.79315e54	−0.466592	−0.233296	+	0.972406i	$0.574951\pi$
$354$	0	0
$355$	−2.40411e54	−0.355667
$356$	0	0
$357$	−1.71842e55	−2.25301
$358$	0	0
$359$	−4.56336e54	−0.530584	−0.265292	−	0.964168i	$-0.585468\pi$
$359$	−4.56336e54	−0.530584	−0.265292	+	0.964168i	$0.585468\pi$
$360$	0	0
$361$	−5.71395e54	−0.589565
$362$	0	0
$363$	−8.81904e54	−0.808032
$364$	0	0
$365$	6.03767e54	0.491556
$366$	0	0
$367$	9.24553e54	0.669286	0.334643	−	0.942345i	$-0.391384\pi$
$367$	9.24553e54	0.669286	0.334643	+	0.942345i	$0.391384\pi$
$368$	0	0
$369$	−4.51765e53	−0.0290969
$370$	0	0
$371$	4.80157e54	0.275324
$372$	0	0
$373$	9.10042e54	0.464859	0.232430	−	0.972613i	$-0.425332\pi$
$373$	9.10042e54	0.464859	0.232430	+	0.972613i	$0.425332\pi$
$374$	0	0
$375$	2.13994e55	0.974384
$376$	0	0
$377$	−1.15071e55	−0.467332
$378$	0	0
$379$	2.67644e55	0.970096	0.485048	−	0.874488i	$-0.338802\pi$
$379$	2.67644e55	0.970096	0.485048	+	0.874488i	$0.338802\pi$
$380$	0	0
$381$	4.01897e55	1.30085
$382$	0	0
$383$	−2.36692e55	−0.684556	−0.342278	−	0.939599i	$-0.611199\pi$
$383$	−2.36692e55	−0.684556	−0.342278	+	0.939599i	$0.611199\pi$
$384$	0	0
$385$	1.16365e55	0.300896
$386$	0	0
$387$	7.42886e53	0.0171845
$388$	0	0
$389$	9.04290e55	1.87237	0.936184	−	0.351509i	$-0.114331\pi$
$389$	9.04290e55	1.87237	0.936184	+	0.351509i	$0.114331\pi$
$390$	0	0
$391$	4.14872e55	0.769331
$392$	0	0
$393$	−5.26978e55	−0.875695
$394$	0	0
$395$	2.47137e55	0.368214
$396$	0	0
$397$	5.22770e55	0.698738	0.349369	−	0.936985i	$-0.386396\pi$
$397$	5.22770e55	0.698738	0.349369	+	0.936985i	$0.386396\pi$
$398$	0	0
$399$	6.18407e55	0.741920
$400$	0	0
$401$	2.74708e55	0.295983	0.147992	−	0.988989i	$-0.452719\pi$
$401$	2.74708e55	0.295983	0.147992	+	0.988989i	$0.452719\pi$
$402$	0	0
$403$	−1.62700e55	−0.157517
$404$	0	0
$405$	6.75783e55	0.588199
$406$	0	0
$407$	−3.08918e54	−0.0241860
$408$	0	0
$409$	−2.29499e55	−0.161708	−0.0808538	−	0.996726i	$-0.525765\pi$
$409$	−2.29499e55	−0.161708	−0.0808538	+	0.996726i	$0.525765\pi$
$410$	0	0
$411$	2.72707e55	0.173020
$412$	0	0
$413$	−3.00308e56	−1.71649
$414$	0	0
$415$	8.91385e55	0.459231
$416$	0	0
$417$	−1.17621e55	−0.0546462
$418$	0	0
$419$	3.63801e54	0.0152497	0.00762487	−	0.999971i	$-0.497573\pi$
$419$	3.63801e54	0.0152497	0.00762487	+	0.999971i	$0.497573\pi$
$420$	0	0
$421$	1.16447e56	0.440620	0.220310	−	0.975430i	$-0.429293\pi$
$421$	1.16447e56	0.440620	0.220310	+	0.975430i	$0.429293\pi$
$422$	0	0
$423$	−3.38631e54	−0.0115720
$424$	0	0
$425$	4.03691e56	1.24649
$426$	0	0
$427$	−2.01349e56	−0.562019
$428$	0	0
$429$	−2.84846e56	−0.719082
$430$	0	0
$431$	−5.90924e56	−1.34980	−0.674901	−	0.737908i	$-0.735814\pi$
$431$	−5.90924e56	−1.34980	−0.674901	+	0.737908i	$0.735814\pi$
$432$	0	0
$433$	1.91269e56	0.395507	0.197754	−	0.980252i	$-0.436635\pi$
$433$	1.91269e56	0.395507	0.197754	+	0.980252i	$0.436635\pi$
$434$	0	0
$435$	8.76942e55	0.164230
$436$	0	0
$437$	−1.49299e56	−0.253342
$438$	0	0
$439$	1.21770e57	1.87307	0.936537	−	0.350569i	$-0.114012\pi$
$439$	1.21770e57	1.87307	0.936537	+	0.350569i	$0.114012\pi$
$440$	0	0
$441$	−4.82204e54	−0.00672669
$442$	0	0
$443$	−2.89543e56	−0.366465	−0.183233	−	0.983070i	$-0.558656\pi$
$443$	−2.89543e56	−0.366465	−0.183233	+	0.983070i	$0.558656\pi$
$444$	0	0
$445$	−5.78993e56	−0.665171
$446$	0	0
$447$	−1.58949e57	−1.65824
$448$	0	0
$449$	−7.37718e55	−0.0699188	−0.0349594	−	0.999389i	$-0.511130\pi$
$449$	−7.37718e55	−0.0699188	−0.0349594	+	0.999389i	$0.511130\pi$
$450$	0	0
$451$	−7.89794e56	−0.680327
$452$	0	0
$453$	−1.53309e57	−1.20076
$454$	0	0
$455$	−1.66189e57	−1.18402
$456$	0	0
$457$	−1.57216e57	−1.01929	−0.509646	−	0.860384i	$-0.670224\pi$
$457$	−1.57216e57	−1.01929	−0.509646	+	0.860384i	$0.670224\pi$
$458$	0	0
$459$	3.32568e57	1.96294
$460$	0	0
$461$	4.69628e56	0.252455	0.126228	−	0.992001i	$-0.459713\pi$
$461$	4.69628e56	0.252455	0.126228	+	0.992001i	$0.459713\pi$
$462$	0	0
$463$	3.45809e56	0.169373	0.0846866	−	0.996408i	$-0.473011\pi$
$463$	3.45809e56	0.169373	0.0846866	+	0.996408i	$0.473011\pi$
$464$	0	0
$465$	1.23992e56	0.0553547
$466$	0	0
$467$	−3.48051e57	−1.41687	−0.708436	−	0.705775i	$-0.750599\pi$
$467$	−3.48051e57	−1.41687	−0.708436	+	0.705775i	$0.750599\pi$
$468$	0	0
$469$	−1.58637e57	−0.589100
$470$	0	0
$471$	−1.13658e57	−0.385167
$472$	0	0
$473$	1.29874e57	0.401798
$474$	0	0
$475$	−1.45276e57	−0.410470
$476$	0	0
$477$	−1.67606e55	−0.00432657
$478$	0	0
$479$	−3.13467e56	−0.0739567	−0.0369784	−	0.999316i	$-0.511773\pi$
$479$	−3.13467e56	−0.0739567	−0.0369784	+	0.999316i	$0.511773\pi$
$480$	0	0
$481$	4.41188e56	0.0951711
$482$	0	0
$483$	−2.32104e57	−0.457952
$484$	0	0
$485$	7.89256e56	0.142485
$486$	0	0
$487$	8.38372e57	1.38536	0.692682	−	0.721243i	$-0.256429\pi$
$487$	8.38372e57	1.38536	0.692682	+	0.721243i	$0.256429\pi$
$488$	0	0
$489$	4.35068e57	0.658289
$490$	0	0
$491$	5.79408e57	0.803031	0.401516	−	0.915852i	$-0.368484\pi$
$491$	5.79408e57	0.803031	0.401516	+	0.915852i	$0.368484\pi$
$492$	0	0
$493$	4.23633e57	0.537998
$494$	0	0
$495$	−4.06188e55	−0.00472841
$496$	0	0
$497$	6.49743e57	0.693553
$498$	0	0
$499$	−6.67553e56	−0.0653618	−0.0326809	−	0.999466i	$-0.510405\pi$
$499$	−6.67553e56	−0.0653618	−0.0326809	+	0.999466i	$0.510405\pi$
$500$	0	0
$501$	−3.82306e57	−0.343479
$502$	0	0
$503$	2.00370e58	1.65242	0.826212	−	0.563359i	$-0.190491\pi$
$503$	2.00370e58	1.65242	0.826212	+	0.563359i	$0.190491\pi$
$504$	0	0
$505$	−5.97070e56	−0.0452127
$506$	0	0
$507$	2.64365e58	1.83879
$508$	0	0
$509$	2.35405e58	1.50447	0.752233	−	0.658897i	$-0.228977\pi$
$509$	2.35405e58	1.50447	0.752233	+	0.658897i	$0.228977\pi$
$510$	0	0
$511$	−1.63177e58	−0.958536
$512$	0	0
$513$	−1.19681e58	−0.646400
$514$	0	0
$515$	−1.73615e58	−0.862447
$516$	0	0
$517$	−5.92007e57	−0.270571
$518$	0	0
$519$	−1.64740e58	−0.692950
$520$	0	0
$521$	1.49380e58	0.578471	0.289236	−	0.957258i	$-0.406599\pi$
$521$	1.49380e58	0.578471	0.289236	+	0.957258i	$0.406599\pi$
$522$	0	0
$523$	−1.11209e58	−0.396602	−0.198301	−	0.980141i	$-0.563542\pi$
$523$	−1.11209e58	−0.396602	−0.198301	+	0.980141i	$0.563542\pi$
$524$	0	0
$525$	−2.25849e58	−0.741983
$526$	0	0
$527$	5.98980e57	0.181336
$528$	0	0
$529$	−3.02305e58	−0.843624
$530$	0	0
$531$	1.04827e57	0.0269737
$532$	0	0
$533$	1.12796e59	2.67707
$534$	0	0
$535$	4.55135e58	0.996629
$536$	0	0
$537$	−3.38349e58	−0.683782
$538$	0	0
$539$	−8.43008e57	−0.157280
$540$	0	0
$541$	8.34052e57	0.143698	0.0718492	−	0.997416i	$-0.477110\pi$
$541$	8.34052e57	0.143698	0.0718492	+	0.997416i	$0.477110\pi$
$542$	0	0
$543$	−1.82891e57	−0.0291068
$544$	0	0
$545$	−3.67008e58	−0.539697
$546$	0	0
$547$	6.38951e58	0.868440	0.434220	−	0.900807i	$-0.357024\pi$
$547$	6.38951e58	0.868440	0.434220	+	0.900807i	$0.357024\pi$
$548$	0	0
$549$	7.02838e56	0.00883181
$550$	0	0
$551$	−1.52452e58	−0.177164
$552$	0	0
$553$	−6.67922e58	−0.718019
$554$	0	0
$555$	−3.36225e57	−0.0334450
$556$	0	0
$557$	−1.47103e59	−1.35436	−0.677181	−	0.735816i	$-0.736799\pi$
$557$	−1.47103e59	−1.35436	−0.677181	+	0.735816i	$0.736799\pi$
$558$	0	0
$559$	−1.85483e59	−1.58107
$560$	0	0
$561$	1.04866e59	0.827816
$562$	0	0
$563$	−9.38359e58	−0.686181	−0.343090	−	0.939302i	$-0.611474\pi$
$563$	−9.38359e58	−0.686181	−0.343090	+	0.939302i	$0.611474\pi$
$564$	0	0
$565$	4.37413e58	0.296381
$566$	0	0
$567$	−1.82640e59	−1.14699
$568$	0	0
$569$	−2.42417e59	−1.41140	−0.705698	−	0.708513i	$-0.749366\pi$
$569$	−2.42417e59	−1.41140	−0.705698	+	0.708513i	$0.749366\pi$
$570$	0	0
$571$	2.26275e59	1.22169	0.610844	−	0.791751i	$-0.290830\pi$
$571$	2.26275e59	1.22169	0.610844	+	0.791751i	$0.290830\pi$
$572$	0	0
$573$	−1.51505e59	−0.758758
$574$	0	0
$575$	5.45259e58	0.253364
$576$	0	0
$577$	2.80486e59	1.20957	0.604786	−	0.796388i	$-0.293259\pi$
$577$	2.80486e59	1.20957	0.604786	+	0.796388i	$0.293259\pi$
$578$	0	0
$579$	−3.83636e59	−1.53578
$580$	0	0
$581$	−2.40909e59	−0.895502
$582$	0	0
$583$	−2.93015e58	−0.101162
$584$	0	0
$585$	5.80107e57	0.0186062
$586$	0	0
$587$	−6.13483e59	−1.82845	−0.914226	−	0.405204i	$-0.867200\pi$
$587$	−6.13483e59	−1.82845	−0.914226	+	0.405204i	$0.867200\pi$
$588$	0	0
$589$	−2.15554e58	−0.0597142
$590$	0	0
$591$	3.45048e59	0.888687
$592$	0	0
$593$	3.38684e59	0.811183	0.405591	−	0.914055i	$-0.367066\pi$
$593$	3.38684e59	0.811183	0.405591	+	0.914055i	$0.367066\pi$
$594$	0	0
$595$	6.11826e59	1.36305
$596$	0	0
$597$	−5.96021e59	−1.23542
$598$	0	0
$599$	8.69352e59	1.67695	0.838473	−	0.544944i	$-0.183449\pi$
$599$	8.69352e59	1.67695	0.838473	+	0.544944i	$0.183449\pi$
$600$	0	0
$601$	2.05552e59	0.369080	0.184540	−	0.982825i	$-0.440920\pi$
$601$	2.05552e59	0.369080	0.184540	+	0.982825i	$0.440920\pi$
$602$	0	0
$603$	5.53745e57	0.00925737
$604$	0	0
$605$	3.13993e59	0.488854
$606$	0	0
$607$	−7.51891e59	−1.09043	−0.545215	−	0.838296i	$-0.683552\pi$
$607$	−7.51891e59	−1.09043	−0.545215	+	0.838296i	$0.683552\pi$
$608$	0	0
$609$	−2.37006e59	−0.320249
$610$	0	0
$611$	8.45489e59	1.06469
$612$	0	0
$613$	6.96117e59	0.817112	0.408556	−	0.912733i	$-0.366032\pi$
$613$	6.96117e59	0.817112	0.408556	+	0.912733i	$0.366032\pi$
$614$	0	0
$615$	−8.59609e59	−0.940776
$616$	0	0
$617$	1.68634e60	1.72113	0.860567	−	0.509337i	$-0.170109\pi$
$617$	1.68634e60	1.72113	0.860567	+	0.509337i	$0.170109\pi$
$618$	0	0
$619$	1.32244e60	1.25901	0.629503	−	0.776998i	$-0.283258\pi$
$619$	1.32244e60	1.25901	0.629503	+	0.776998i	$0.283258\pi$
$620$	0	0
$621$	4.49194e59	0.398992
$622$	0	0
$623$	1.56481e60	1.29709
$624$	0	0
$625$	6.61886e58	0.0512110
$626$	0	0
$627$	−3.77381e59	−0.272601
$628$	0	0
$629$	−1.62424e59	−0.109562
$630$	0	0
$631$	−1.50474e60	−0.948048	−0.474024	−	0.880512i	$-0.657199\pi$
$631$	−1.50474e60	−0.948048	−0.474024	+	0.880512i	$0.657199\pi$
$632$	0	0
$633$	−1.36717e60	−0.804714
$634$	0	0
$635$	−1.43091e60	−0.787007
$636$	0	0
$637$	1.20396e60	0.618892
$638$	0	0
$639$	−2.26802e58	−0.0108988
$640$	0	0
$641$	−1.27833e60	−0.574373	−0.287187	−	0.957875i	$-0.592720\pi$
$641$	−1.27833e60	−0.574373	−0.287187	+	0.957875i	$0.592720\pi$
$642$	0	0
$643$	1.28540e60	0.540134	0.270067	−	0.962842i	$-0.412954\pi$
$643$	1.28540e60	0.540134	0.270067	+	0.962842i	$0.412954\pi$
$644$	0	0
$645$	1.41355e60	0.555618
$646$	0	0
$647$	3.49078e60	1.28375	0.641875	−	0.766809i	$-0.278157\pi$
$647$	3.49078e60	1.28375	0.641875	+	0.766809i	$0.278157\pi$
$648$	0	0
$649$	1.83262e60	0.630684
$650$	0	0
$651$	−3.35106e59	−0.107942
$652$	0	0
$653$	−4.91548e60	−1.48229	−0.741143	−	0.671347i	$-0.765716\pi$
$653$	−4.91548e60	−1.48229	−0.741143	+	0.671347i	$0.765716\pi$
$654$	0	0
$655$	1.87625e60	0.529790
$656$	0	0
$657$	5.69590e58	0.0150629
$658$	0	0
$659$	4.43095e60	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$659$	4.43095e60	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$660$	0	0
$661$	3.53419e60	0.820274	0.410137	−	0.912024i	$-0.365481\pi$
$661$	3.53419e60	0.820274	0.410137	+	0.912024i	$0.365481\pi$
$662$	0	0
$663$	−1.49767e61	−3.25743
$664$	0	0
$665$	−2.20177e60	−0.448856
$666$	0	0
$667$	5.72194e59	0.109355
$668$	0	0
$669$	−5.04976e60	−0.904917
$670$	0	0
$671$	1.22873e60	0.206501
$672$	0	0
$673$	3.42060e60	0.539236	0.269618	−	0.962967i	$-0.413103\pi$
$673$	3.42060e60	0.539236	0.269618	+	0.962967i	$0.413103\pi$
$674$	0	0
$675$	4.37088e60	0.646455
$676$	0	0
$677$	6.63114e60	0.920305	0.460152	−	0.887840i	$-0.347795\pi$
$677$	6.63114e60	0.920305	0.460152	+	0.887840i	$0.347795\pi$
$678$	0	0
$679$	−2.13307e60	−0.277847
$680$	0	0
$681$	−8.51884e60	−1.04164
$682$	0	0
$683$	−4.07501e60	−0.467825	−0.233913	−	0.972258i	$-0.575153\pi$
$683$	−4.07501e60	−0.467825	−0.233913	+	0.972258i	$0.575153\pi$
$684$	0	0
$685$	−9.70945e59	−0.104676
$686$	0	0
$687$	−3.11813e60	−0.315736
$688$	0	0
$689$	4.18476e60	0.398068
$690$	0	0
$691$	−1.23638e61	−1.10503	−0.552514	−	0.833504i	$-0.686331\pi$
$691$	−1.23638e61	−1.10503	−0.552514	+	0.833504i	$0.686331\pi$
$692$	0	0
$693$	1.09778e59	0.00922042
$694$	0	0
$695$	4.18778e59	0.0330606
$696$	0	0
$697$	−4.15260e61	−3.08188
$698$	0	0
$699$	−6.93574e60	−0.483987
$700$	0	0
$701$	1.41956e61	0.931574	0.465787	−	0.884897i	$-0.345771\pi$
$701$	1.41956e61	0.931574	0.465787	+	0.884897i	$0.345771\pi$
$702$	0	0
$703$	5.84513e59	0.0360790
$704$	0	0
$705$	−6.44339e60	−0.374152
$706$	0	0
$707$	1.61367e60	0.0881650
$708$	0	0
$709$	−6.77303e60	−0.348248	−0.174124	−	0.984724i	$-0.555709\pi$
$709$	−6.77303e60	−0.348248	−0.174124	+	0.984724i	$0.555709\pi$
$710$	0	0
$711$	2.33148e59	0.0112833
$712$	0	0
$713$	8.09032e59	0.0368588
$714$	0	0
$715$	1.01416e61	0.435039
$716$	0	0
$717$	−2.76378e61	−1.11646
$718$	0	0
$719$	−3.64283e61	−1.38601	−0.693006	−	0.720931i	$-0.743714\pi$
$719$	−3.64283e61	−1.38601	−0.693006	+	0.720931i	$0.743714\pi$
$720$	0	0
$721$	4.69219e61	1.68177
$722$	0	0
$723$	3.03914e61	1.02631
$724$	0	0
$725$	5.56774e60	0.177179
$726$	0	0
$727$	1.87172e61	0.561372	0.280686	−	0.959800i	$-0.409438\pi$
$727$	1.87172e61	0.561372	0.280686	+	0.959800i	$0.409438\pi$
$728$	0	0
$729$	3.59962e61	1.01769
$730$	0	0
$731$	6.82856e61	1.82014
$732$	0	0
$733$	4.58324e61	1.15196	0.575980	−	0.817464i	$-0.304621\pi$
$733$	4.58324e61	1.15196	0.575980	+	0.817464i	$0.304621\pi$
$734$	0	0
$735$	−9.17527e60	−0.217491
$736$	0	0
$737$	9.68078e60	0.216451
$738$	0	0
$739$	6.45765e59	0.0136213	0.00681066	−	0.999977i	$-0.497832\pi$
$739$	6.45765e59	0.0136213	0.00681066	+	0.999977i	$0.497832\pi$
$740$	0	0
$741$	5.38965e61	1.07268
$742$	0	0
$743$	−3.46765e61	−0.651291	−0.325646	−	0.945492i	$-0.605582\pi$
$743$	−3.46765e61	−0.651291	−0.325646	+	0.945492i	$0.605582\pi$
$744$	0	0
$745$	5.65923e61	1.00322
$746$	0	0
$747$	8.40927e59	0.0140723
$748$	0	0
$749$	−1.23007e62	−1.94343
$750$	0	0
$751$	3.57168e61	0.532861	0.266430	−	0.963854i	$-0.414156\pi$
$751$	3.57168e61	0.532861	0.266430	+	0.963854i	$0.414156\pi$
$752$	0	0
$753$	−2.01465e61	−0.283863
$754$	0	0
$755$	5.45841e61	0.726452
$756$	0	0
$757$	−1.06737e62	−1.34200	−0.671002	−	0.741456i	$-0.734136\pi$
$757$	−1.06737e62	−1.34200	−0.671002	+	0.741456i	$0.734136\pi$
$758$	0	0
$759$	1.41641e61	0.168264
$760$	0	0
$761$	9.90270e61	1.11169	0.555845	−	0.831286i	$-0.312395\pi$
$761$	9.90270e61	1.11169	0.555845	+	0.831286i	$0.312395\pi$
$762$	0	0
$763$	9.91890e61	1.05241
$764$	0	0
$765$	−2.13566e60	−0.0214196
$766$	0	0
$767$	−2.61730e62	−2.48173
$768$	0	0
$769$	−2.55003e61	−0.228628	−0.114314	−	0.993445i	$-0.536467\pi$
$769$	−2.55003e61	−0.228628	−0.114314	+	0.993445i	$0.536467\pi$
$770$	0	0
$771$	9.24752e61	0.784073
$772$	0	0
$773$	1.50600e62	1.20772	0.603861	−	0.797090i	$-0.293628\pi$
$773$	1.50600e62	1.20772	0.603861	+	0.797090i	$0.293628\pi$
$774$	0	0
$775$	7.87229e60	0.0597193
$776$	0	0
$777$	9.08696e60	0.0652179
$778$	0	0
$779$	1.49439e62	1.01487
$780$	0	0
$781$	−3.96504e61	−0.254830
$782$	0	0
$783$	4.58680e61	0.279018
$784$	0	0
$785$	4.04667e61	0.233024
$786$	0	0
$787$	−1.74985e62	−0.953989	−0.476994	−	0.878906i	$-0.658274\pi$
$787$	−1.74985e62	−0.953989	−0.476994	+	0.878906i	$0.658274\pi$
$788$	0	0
$789$	−1.75564e62	−0.906315
$790$	0	0
$791$	−1.18217e62	−0.577944
$792$	0	0
$793$	−1.75484e62	−0.812575
$794$	0	0
$795$	−3.18916e61	−0.139889
$796$	0	0
$797$	3.53459e62	1.46888	0.734438	−	0.678676i	$-0.237446\pi$
$797$	3.53459e62	1.46888	0.734438	+	0.678676i	$0.237446\pi$
$798$	0	0
$799$	−3.11267e62	−1.22568
$800$	0	0
$801$	−5.46219e60	−0.0203830
$802$	0	0
$803$	9.95780e61	0.352192
$804$	0	0
$805$	8.26384e61	0.277058
$806$	0	0
$807$	−2.10106e62	−0.667818
$808$	0	0
$809$	−4.31937e62	−1.30175	−0.650875	−	0.759185i	$-0.725598\pi$
$809$	−4.31937e62	−1.30175	−0.650875	+	0.759185i	$0.725598\pi$
$810$	0	0
$811$	5.78809e62	1.65419	0.827097	−	0.562059i	$-0.189990\pi$
$811$	5.78809e62	1.65419	0.827097	+	0.562059i	$0.189990\pi$
$812$	0	0
$813$	−3.42076e62	−0.927204
$814$	0	0
$815$	−1.54901e62	−0.398260
$816$	0	0
$817$	−2.45739e62	−0.599376
$818$	0	0
$819$	−1.56782e61	−0.0362821
$820$	0	0
$821$	−1.86128e62	−0.408727	−0.204364	−	0.978895i	$-0.565512\pi$
$821$	−1.86128e62	−0.408727	−0.204364	+	0.978895i	$0.565512\pi$
$822$	0	0
$823$	−6.65210e62	−1.38632	−0.693158	−	0.720785i	$-0.743781\pi$
$823$	−6.65210e62	−1.38632	−0.693158	+	0.720785i	$0.743781\pi$
$824$	0	0
$825$	1.37824e62	0.272624
$826$	0	0
$827$	6.49406e61	0.121940	0.0609702	−	0.998140i	$-0.480581\pi$
$827$	6.49406e61	0.121940	0.0609702	+	0.998140i	$0.480581\pi$
$828$	0	0
$829$	4.83956e62	0.862746	0.431373	−	0.902174i	$-0.358029\pi$
$829$	4.83956e62	0.862746	0.431373	+	0.902174i	$0.358029\pi$
$830$	0	0
$831$	6.69735e61	0.113365
$832$	0	0
$833$	−4.43239e62	−0.712476
$834$	0	0
$835$	1.36116e62	0.207802
$836$	0	0
$837$	6.48533e61	0.0940448
$838$	0	0
$839$	−5.22756e62	−0.720140	−0.360070	−	0.932925i	$-0.617247\pi$
$839$	−5.22756e62	−0.720140	−0.360070	+	0.932925i	$0.617247\pi$
$840$	0	0
$841$	−7.05608e62	−0.923527
$842$	0	0
$843$	5.45087e61	0.0677912
$844$	0	0
$845$	−9.41244e62	−1.11246
$846$	0	0
$847$	−8.48610e62	−0.953267
$848$	0	0
$849$	1.24128e63	1.32542
$850$	0	0
$851$	−2.19383e61	−0.0222699
$852$	0	0
$853$	3.48905e62	0.336747	0.168373	−	0.985723i	$-0.446149\pi$
$853$	3.48905e62	0.336747	0.168373	+	0.985723i	$0.446149\pi$
$854$	0	0
$855$	7.68560e60	0.00705352
$856$	0	0
$857$	5.09487e62	0.444678	0.222339	−	0.974969i	$-0.428631\pi$
$857$	5.09487e62	0.444678	0.222339	+	0.974969i	$0.428631\pi$
$858$	0	0
$859$	1.66852e63	1.38509	0.692546	−	0.721374i	$-0.256489\pi$
$859$	1.66852e63	1.38509	0.692546	+	0.721374i	$0.256489\pi$
$860$	0	0
$861$	2.32321e63	1.83452
$862$	0	0
$863$	−6.32017e62	−0.474785	−0.237392	−	0.971414i	$-0.576293\pi$
$863$	−6.32017e62	−0.474785	−0.237392	+	0.971414i	$0.576293\pi$
$864$	0	0
$865$	5.86541e62	0.419230
$866$	0	0
$867$	4.05693e63	2.75922
$868$	0	0
$869$	4.07598e62	0.263819
$870$	0	0
$871$	−1.38258e63	−0.851728
$872$	0	0
$873$	7.44579e60	0.00436621
$874$	0	0
$875$	2.05916e63	1.14952
$876$	0	0
$877$	−2.16093e63	−1.14855	−0.574274	−	0.818664i	$-0.694715\pi$
$877$	−2.16093e63	−1.14855	−0.574274	+	0.818664i	$0.694715\pi$
$878$	0	0
$879$	−1.36185e63	−0.689236
$880$	0	0
$881$	−1.77705e63	−0.856481	−0.428241	−	0.903665i	$-0.640866\pi$
$881$	−1.77705e63	−0.856481	−0.428241	+	0.903665i	$0.640866\pi$
$882$	0	0
$883$	2.16218e63	0.992515	0.496258	−	0.868175i	$-0.334707\pi$
$883$	2.16218e63	0.992515	0.496258	+	0.868175i	$0.334707\pi$
$884$	0	0
$885$	1.99462e63	0.872128
$886$	0	0
$887$	−9.41993e62	−0.392365	−0.196182	−	0.980567i	$-0.562854\pi$
$887$	−9.41993e62	−0.392365	−0.196182	+	0.980567i	$0.562854\pi$
$888$	0	0
$889$	3.86725e63	1.53467
$890$	0	0
$891$	1.11455e63	0.421435
$892$	0	0
$893$	1.12015e63	0.403619
$894$	0	0
$895$	1.20466e63	0.413684
$896$	0	0
$897$	−2.02288e63	−0.662113
$898$	0	0
$899$	8.26118e61	0.0257756
$900$	0	0
$901$	−1.54062e63	−0.458261
$902$	0	0
$903$	−3.82031e63	−1.08346
$904$	0	0
$905$	6.51163e61	0.0176094
$906$	0	0
$907$	−4.24432e63	−1.09459	−0.547295	−	0.836940i	$-0.684342\pi$
$907$	−4.24432e63	−1.09459	−0.547295	+	0.836940i	$0.684342\pi$
$908$	0	0
$909$	−5.63272e60	−0.00138546
$910$	0	0
$911$	1.73895e63	0.407983	0.203991	−	0.978973i	$-0.434609\pi$
$911$	1.73895e63	0.407983	0.203991	+	0.978973i	$0.434609\pi$
$912$	0	0
$913$	1.47014e63	0.329031
$914$	0	0
$915$	1.33734e63	0.285555
$916$	0	0
$917$	−5.07084e63	−1.03309
$918$	0	0
$919$	−8.49800e63	−1.65209	−0.826044	−	0.563606i	$-0.809414\pi$
$919$	−8.49800e63	−1.65209	−0.826044	+	0.563606i	$0.809414\pi$
$920$	0	0
$921$	5.70343e63	1.05817
$922$	0	0
$923$	5.66276e63	1.00275
$924$	0	0
$925$	−2.13471e62	−0.0360821
$926$	0	0
$927$	−1.63788e62	−0.0264281
$928$	0	0
$929$	−6.41656e63	−0.988472	−0.494236	−	0.869328i	$-0.664552\pi$
$929$	−6.41656e63	−0.988472	−0.494236	+	0.869328i	$0.664552\pi$
$930$	0	0
$931$	1.59508e63	0.234619
$932$	0	0
$933$	−1.10372e64	−1.55026
$934$	0	0
$935$	−3.73365e63	−0.500823
$936$	0	0
$937$	4.22960e63	0.541874	0.270937	−	0.962597i	$-0.412666\pi$
$937$	4.22960e63	0.541874	0.270937	+	0.962597i	$0.412666\pi$
$938$	0	0
$939$	6.14504e63	0.751994
$940$	0	0
$941$	5.97856e63	0.698907	0.349454	−	0.936954i	$-0.386367\pi$
$941$	5.97856e63	0.698907	0.349454	+	0.936954i	$0.386367\pi$
$942$	0	0
$943$	−5.60885e63	−0.626429
$944$	0	0
$945$	6.62442e63	0.706910
$946$	0	0
$947$	3.09116e63	0.315208	0.157604	−	0.987502i	$-0.449623\pi$
$947$	3.09116e63	0.315208	0.157604	+	0.987502i	$0.449623\pi$
$948$	0	0
$949$	−1.42215e64	−1.38586
$950$	0	0
$951$	−1.07519e64	−1.00139
$952$	0	0
$953$	−8.78118e62	−0.0781727	−0.0390864	−	0.999236i	$-0.512445\pi$
$953$	−8.78118e62	−0.0781727	−0.0390864	+	0.999236i	$0.512445\pi$
$954$	0	0
$955$	5.39419e63	0.459044
$956$	0	0
$957$	1.44632e63	0.117668
$958$	0	0
$959$	2.62411e63	0.204119
$960$	0	0
$961$	−1.33279e64	−0.991312
$962$	0	0
$963$	4.29371e62	0.0305399
$964$	0	0
$965$	1.36590e64	0.929138
$966$	0	0
$967$	−1.67955e63	−0.109275	−0.0546377	−	0.998506i	$-0.517400\pi$
$967$	−1.67955e63	−0.109275	−0.0546377	+	0.998506i	$0.517400\pi$
$968$	0	0
$969$	−1.98420e64	−1.23488
$970$	0	0
$971$	−2.39350e64	−1.42502	−0.712508	−	0.701664i	$-0.752441\pi$
$971$	−2.39350e64	−1.42502	−0.712508	+	0.701664i	$0.752441\pi$
$972$	0	0
$973$	−1.13181e63	−0.0644682
$974$	0	0
$975$	−1.96836e64	−1.07277
$976$	0	0
$977$	1.55784e64	0.812438	0.406219	−	0.913776i	$-0.366847\pi$
$977$	1.55784e64	0.812438	0.406219	+	0.913776i	$0.366847\pi$
$978$	0	0
$979$	−9.54921e63	−0.476584
$980$	0	0
$981$	−3.46233e62	−0.0165381
$982$	0	0
$983$	−8.28128e63	−0.378614	−0.189307	−	0.981918i	$-0.560624\pi$
$983$	−8.28128e63	−0.378614	−0.189307	+	0.981918i	$0.560624\pi$
$984$	0	0
$985$	−1.22851e64	−0.537650
$986$	0	0
$987$	1.74142e64	0.729599
$988$	0	0
$989$	9.22322e63	0.369966
$990$	0	0
$991$	1.98538e64	0.762537	0.381269	−	0.924464i	$-0.375487\pi$
$991$	1.98538e64	0.762537	0.381269	+	0.924464i	$0.375487\pi$
$992$	0	0
$993$	−2.39574e64	−0.881112
$994$	0	0
$995$	2.12207e64	0.747418
$996$	0	0
$997$	5.32681e64	1.79689	0.898444	−	0.439088i	$-0.144698\pi$
$997$	5.32681e64	1.79689	0.898444	+	0.439088i	$0.144698\pi$
$998$	0	0
$999$	−1.75861e63	−0.0568213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.44.a.c.1.3		3
4.3	odd	2		1.44.a.a.1.2	✓	3
12.11	even	2		9.44.a.b.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.44.a.a.1.2	✓	3	4.3	odd	2
9.44.a.b.1.2		3	12.11	even	2
16.44.a.c.1.3		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)

\(f(q)\)	\(=\)	\(q+1.79507e10 q^{3} -6.39116e14 q^{5} +1.72730e18 q^{7} -6.02939e18 q^{9} +O(q^{10})\)	\(q+1.79507e10 q^{3} -6.39116e14 q^{5} +1.72730e18 q^{7} -6.02939e18 q^{9} -1.05408e22 q^{11} +1.50541e24 q^{13} -1.14726e25 q^{15} -5.54217e26 q^{17} +1.99446e27 q^{19} +3.10063e28 q^{21} -7.48572e28 q^{23} -7.28399e29 q^{25} -6.00067e30 q^{27} -7.64381e30 q^{29} -1.08077e31 q^{31} -1.89215e32 q^{33} -1.10395e33 q^{35} +2.93069e32 q^{37} +2.70232e34 q^{39} +7.49273e34 q^{41} -1.23211e35 q^{43} +3.85348e33 q^{45} +5.61634e35 q^{47} +7.99756e35 q^{49} -9.94859e36 q^{51} +2.77981e36 q^{53} +6.73680e36 q^{55} +3.58019e37 q^{57} -1.73860e38 q^{59} -1.16569e38 q^{61} -1.04146e37 q^{63} -9.62132e38 q^{65} -9.18410e38 q^{67} -1.34374e39 q^{69} +3.76161e39 q^{71} -9.44690e39 q^{73} -1.30753e40 q^{75} -1.82072e40 q^{77} -3.86685e40 q^{79} -1.05737e41 q^{81} -1.39471e41 q^{83} +3.54209e41 q^{85} -1.37212e41 q^{87} +9.05928e41 q^{89} +2.60030e42 q^{91} -1.94005e41 q^{93} -1.27469e42 q^{95} -1.23492e42 q^{97} +6.35546e40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 24401437812 q^{3} + 535205380774170 q^{5} - 30\!\cdots\!56 q^{7}+ \cdots + 47\!\cdots\!31 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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