LMFDB - Embedded newform 144.4.i.a.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,4,Mod(49,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("144.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	144.i (of order $3$, degree $2$, not minimal)

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:	no
Analytic conductor:	$8.49627504083$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	144.49
Dual form			144.4.i.a.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615i q^{3} +(4.50000 - 7.79423i) q^{5} +(-15.5000 - 26.8468i) q^{7} -27.0000 q^{9} +O(q^{10})$	\(q+5.19615i q^{3} +(4.50000 - 7.79423i) q^{5} +(-15.5000 - 26.8468i) q^{7} -27.0000 q^{9} +(-7.50000 - 12.9904i) q^{11} +(18.5000 - 32.0429i) q^{13} +(40.5000 + 23.3827i) q^{15} -42.0000 q^{17} +28.0000 q^{19} +(139.500 - 80.5404i) q^{21} +(97.5000 - 168.875i) q^{23} +(22.0000 + 38.1051i) q^{25} -140.296i q^{27} +(-55.5000 - 96.1288i) q^{29} +(-102.500 + 177.535i) q^{31} +(67.5000 - 38.9711i) q^{33} -279.000 q^{35} -166.000 q^{37} +(166.500 + 96.1288i) q^{39} +(130.500 - 226.033i) q^{41} +(-21.5000 - 37.2391i) q^{43} +(-121.500 + 210.444i) q^{45} +(88.5000 + 153.286i) q^{47} +(-309.000 + 535.204i) q^{49} -218.238i q^{51} +114.000 q^{53} -135.000 q^{55} +145.492i q^{57} +(79.5000 - 137.698i) q^{59} +(-95.5000 - 165.411i) q^{61} +(418.500 + 724.863i) q^{63} +(-166.500 - 288.386i) q^{65} +(-210.500 + 364.597i) q^{67} +(877.500 + 506.625i) q^{69} -156.000 q^{71} +182.000 q^{73} +(-198.000 + 114.315i) q^{75} +(-232.500 + 402.702i) q^{77} +(566.500 + 981.207i) q^{79} +729.000 q^{81} +(-541.500 - 937.906i) q^{83} +(-189.000 + 327.358i) q^{85} +(499.500 - 288.386i) q^{87} -1050.00 q^{89} -1147.00 q^{91} +(-922.500 - 532.606i) q^{93} +(126.000 - 218.238i) q^{95} +(450.500 + 780.289i) q^{97} +(202.500 + 350.740i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{5} - 31 q^{7} - 54 q^{9}+O(q^{10}) $ 2 * q + 9 * q^5 - 31 * q^7 - 54 * q^9	$ 2 q + 9 q^{5} - 31 q^{7} - 54 q^{9} - 15 q^{11} + 37 q^{13} + 81 q^{15} - 84 q^{17} + 56 q^{19} + 279 q^{21} + 195 q^{23} + 44 q^{25} - 111 q^{29} - 205 q^{31} + 135 q^{33} - 558 q^{35} - 332 q^{37} + 333 q^{39} + 261 q^{41} - 43 q^{43} - 243 q^{45} + 177 q^{47} - 618 q^{49} + 228 q^{53} - 270 q^{55} + 159 q^{59} - 191 q^{61} + 837 q^{63} - 333 q^{65} - 421 q^{67} + 1755 q^{69} - 312 q^{71} + 364 q^{73} - 396 q^{75} - 465 q^{77} + 1133 q^{79} + 1458 q^{81} - 1083 q^{83} - 378 q^{85} + 999 q^{87} - 2100 q^{89} - 2294 q^{91} - 1845 q^{93} + 252 q^{95} + 901 q^{97} + 405 q^{99}+O(q^{100}) $ 2 * q + 9 * q^5 - 31 * q^7 - 54 * q^9 - 15 * q^11 + 37 * q^13 + 81 * q^15 - 84 * q^17 + 56 * q^19 + 279 * q^21 + 195 * q^23 + 44 * q^25 - 111 * q^29 - 205 * q^31 + 135 * q^33 - 558 * q^35 - 332 * q^37 + 333 * q^39 + 261 * q^41 - 43 * q^43 - 243 * q^45 + 177 * q^47 - 618 * q^49 + 228 * q^53 - 270 * q^55 + 159 * q^59 - 191 * q^61 + 837 * q^63 - 333 * q^65 - 421 * q^67 + 1755 * q^69 - 312 * q^71 + 364 * q^73 - 396 * q^75 - 465 * q^77 + 1133 * q^79 + 1458 * q^81 - 1083 * q^83 - 378 * q^85 + 999 * q^87 - 2100 * q^89 - 2294 * q^91 - 1845 * q^93 + 252 * q^95 + 901 * q^97 + 405 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$			5.19615i			1.00000i
$3$			5.19615i			1.00000i
$4$		0			0
$5$	4.50000	−	7.79423i	0.402492	−	0.697137i	−0.591534	−	0.806280i	$-0.701477\pi$
$5$	4.50000	−	7.79423i	0.402492	−	0.697137i	0.994026	+	0.109143i	$0.0348107\pi$
$6$		0			0
$7$	−15.5000	−	26.8468i	−0.836921	−	1.44959i	−0.892456	−	0.451134i	$-0.851020\pi$
$7$	−15.5000	−	26.8468i	−0.836921	−	1.44959i	0.0555351	−	0.998457i	$-0.482314\pi$
$8$		0			0
$9$	−27.0000			−1.00000
$10$		0			0
$11$	−7.50000	−	12.9904i	−0.205576	−	0.356068i	0.744740	−	0.667355i	$-0.232573\pi$
$11$	−7.50000	−	12.9904i	−0.205576	−	0.356068i	−0.950316	+	0.311287i	$0.899240\pi$
$12$		0			0
$13$	18.5000	−	32.0429i	0.394691	−	0.683624i	−0.598371	−	0.801219i	$-0.704185\pi$
$13$	18.5000	−	32.0429i	0.394691	−	0.683624i	0.993062	+	0.117595i	$0.0375185\pi$
$14$		0			0
$15$	40.5000	+	23.3827i	0.697137	+	0.402492i
$16$		0			0
$17$	−42.0000			−0.599206			−0.299603	−	0.954064i	$-0.596854\pi$
$17$	−42.0000			−0.599206			−0.299603	+	0.954064i	$0.596854\pi$
$18$		0			0
$19$	28.0000			0.338086			0.169043	−	0.985609i	$-0.445932\pi$
$19$	28.0000			0.338086			0.169043	+	0.985609i	$0.445932\pi$
$20$		0			0
$21$	139.500	−	80.5404i	1.44959	−	0.836921i
$22$		0			0
$23$	97.5000	−	168.875i	0.883920	−	1.53099i	0.0369731	−	0.999316i	$-0.488228\pi$
$23$	97.5000	−	168.875i	0.883920	−	1.53099i	0.846947	−	0.531678i	$-0.178438\pi$
$24$		0			0
$25$	22.0000	+	38.1051i	0.176000	+	0.304841i
$26$		0			0
$27$		−	140.296i		−	1.00000i
$28$		0			0
$29$	−55.5000	−	96.1288i	−0.355382	−	0.615540i	0.631801	−	0.775131i	$-0.282316\pi$
$29$	−55.5000	−	96.1288i	−0.355382	−	0.615540i	−0.987183	+	0.159590i	$0.948983\pi$
$30$		0			0
$31$	−102.500	+	177.535i	−0.593856	+	1.02859i	0.399851	+	0.916580i	$0.369062\pi$
$31$	−102.500	+	177.535i	−0.593856	+	1.02859i	−0.993707	+	0.112009i	$0.964271\pi$
$32$		0			0
$33$	67.5000	−	38.9711i	0.356068	−	0.205576i
$34$		0			0
$35$	−279.000			−1.34742
$36$		0			0
$37$	−166.000			−0.737574			−0.368787	−	0.929514i	$-0.620227\pi$
$37$	−166.000			−0.737574			−0.368787	+	0.929514i	$0.620227\pi$
$38$		0			0
$39$	166.500	+	96.1288i	0.683624	+	0.394691i
$40$		0			0
$41$	130.500	−	226.033i	0.497090	−	0.860985i	−0.502905	−	0.864342i	$-0.667735\pi$
$41$	130.500	−	226.033i	0.497090	−	0.860985i	0.999994	+	0.00335732i	$0.00106867\pi$
$42$		0			0
$43$	−21.5000	−	37.2391i	−0.0762493	−	0.132068i	0.825380	−	0.564578i	$-0.190961\pi$
$43$	−21.5000	−	37.2391i	−0.0762493	−	0.132068i	−0.901629	+	0.432511i	$0.857628\pi$
$44$		0			0
$45$	−121.500	+	210.444i	−0.402492	+	0.697137i
$46$		0			0
$47$	88.5000	+	153.286i	0.274661	+	0.475726i	0.970049	−	0.242907i	$-0.0781011\pi$
$47$	88.5000	+	153.286i	0.274661	+	0.475726i	−0.695389	+	0.718634i	$0.744768\pi$
$48$		0			0
$49$	−309.000	+	535.204i	−0.900875	+	1.56036i
$50$		0			0
$51$		−	218.238i		−	0.599206i
$52$		0			0
$53$	114.000			0.295455			0.147727	−	0.989028i	$-0.452804\pi$
$53$	114.000			0.295455			0.147727	+	0.989028i	$0.452804\pi$
$54$		0			0
$55$	−135.000			−0.330971
$56$		0			0
$57$			145.492i			0.338086i
$58$		0			0
$59$	79.5000	−	137.698i	0.175424	−	0.303843i	−0.764884	−	0.644168i	$-0.777204\pi$
$59$	79.5000	−	137.698i	0.175424	−	0.303843i	0.940308	+	0.340325i	$0.110537\pi$
$60$		0			0
$61$	−95.5000	−	165.411i	−0.200451	−	0.347192i	0.748223	−	0.663448i	$-0.230907\pi$
$61$	−95.5000	−	165.411i	−0.200451	−	0.347192i	−0.948674	+	0.316256i	$0.897574\pi$
$62$		0			0
$63$	418.500	+	724.863i	0.836921	+	1.44959i
$64$		0			0
$65$	−166.500	−	288.386i	−0.317720	−	0.550307i
$66$		0			0
$67$	−210.500	+	364.597i	−0.383831	+	0.664815i	−0.991606	−	0.129294i	$-0.958729\pi$
$67$	−210.500	+	364.597i	−0.383831	+	0.664815i	0.607775	+	0.794109i	$0.292062\pi$
$68$		0			0
$69$	877.500	+	506.625i	1.53099	+	0.883920i
$70$		0			0
$71$	−156.000			−0.260758			−0.130379	−	0.991464i	$-0.541619\pi$
$71$	−156.000			−0.260758			−0.130379	+	0.991464i	$0.541619\pi$
$72$		0			0
$73$	182.000			0.291801			0.145901	−	0.989299i	$-0.453392\pi$
$73$	182.000			0.291801			0.145901	+	0.989299i	$0.453392\pi$
$74$		0			0
$75$	−198.000	+	114.315i	−0.304841	+	0.176000i
$76$		0			0
$77$	−232.500	+	402.702i	−0.344102	+	0.596002i
$78$		0			0
$79$	566.500	+	981.207i	0.806788	+	1.39740i	0.915078	+	0.403278i	$0.132129\pi$
$79$	566.500	+	981.207i	0.806788	+	1.39740i	−0.108290	+	0.994119i	$0.534537\pi$
$80$		0			0
$81$	729.000			1.00000
$82$		0			0
$83$	−541.500	−	937.906i	−0.716113	−	1.24034i	−0.962529	−	0.271179i	$-0.912586\pi$
$83$	−541.500	−	937.906i	−0.716113	−	1.24034i	0.246416	−	0.969164i	$-0.420747\pi$
$84$		0			0
$85$	−189.000	+	327.358i	−0.241176	+	0.417728i
$86$		0			0
$87$	499.500	−	288.386i	0.615540	−	0.355382i
$88$		0			0
$89$	−1050.00			−1.25056			−0.625280	−	0.780401i	$-0.715015\pi$
$89$	−1050.00			−1.25056			−0.625280	+	0.780401i	$0.715015\pi$
$90$		0			0
$91$	−1147.00			−1.32130
$92$		0			0
$93$	−922.500	−	532.606i	−1.02859	−	0.593856i
$94$		0			0
$95$	126.000	−	218.238i	0.136077	−	0.235693i
$96$		0			0
$97$	450.500	+	780.289i	0.471560	+	0.816766i	0.999471	−	0.0325338i	$-0.0103576\pi$
$97$	450.500	+	780.289i	0.471560	+	0.816766i	−0.527910	+	0.849300i	$0.677024\pi$
$98$		0			0
$99$	202.500	+	350.740i	0.205576	+	0.356068i
$100$		0			0
$101$	−193.500	−	335.152i	−0.190633	−	0.330187i	0.754827	−	0.655924i	$-0.227721\pi$
$101$	−193.500	−	335.152i	−0.190633	−	0.330187i	−0.945460	+	0.325737i	$0.894387\pi$
$102$		0			0
$103$	275.500	−	477.180i	0.263552	−	0.456485i	−0.703631	−	0.710565i	$-0.748439\pi$
$103$	275.500	−	477.180i	0.263552	−	0.456485i	0.967183	+	0.254080i	$0.0817727\pi$
$104$		0			0
$105$		−	1449.73i		−	1.34742i
$106$		0			0
$107$	12.0000			0.0108419			0.00542095	−	0.999985i	$-0.498274\pi$
$107$	12.0000			0.0108419			0.00542095	+	0.999985i	$0.498274\pi$
$108$		0			0
$109$	−502.000			−0.441127			−0.220564	−	0.975373i	$-0.570790\pi$
$109$	−502.000			−0.441127			−0.220564	+	0.975373i	$0.570790\pi$
$110$		0			0
$111$		−	862.561i		−	0.737574i
$112$		0			0
$113$	700.500	−	1213.30i	0.583164	−	1.01007i	−0.411938	−	0.911212i	$-0.635148\pi$
$113$	700.500	−	1213.30i	0.583164	−	1.01007i	0.995102	−	0.0988572i	$-0.0315187\pi$
$114$		0			0
$115$	−877.500	−	1519.87i	−0.711542	−	1.23243i
$116$		0			0
$117$	−499.500	+	865.159i	−0.394691	+	0.683624i
$118$		0			0
$119$	651.000	+	1127.57i	0.501488	+	0.868603i
$120$		0			0
$121$	553.000	−	957.824i	0.415477	−	0.719627i
$122$		0			0
$123$	1174.50	+	678.098i	0.860985	+	0.497090i
$124$		0			0
$125$	1521.00			1.08834
$126$		0			0
$127$	880.000			0.614861			0.307431	−	0.951571i	$-0.400531\pi$
$127$	880.000			0.614861			0.307431	+	0.951571i	$0.400531\pi$
$128$		0			0
$129$	193.500	−	111.717i	0.132068	−	0.0762493i
$130$		0			0
$131$	751.500	−	1301.64i	0.501213	−	0.868126i	−0.498786	−	0.866725i	$-0.666221\pi$
$131$	751.500	−	1301.64i	0.501213	−	0.868126i	0.999999	−	0.00140084i	$-0.000445901\pi$
$132$		0			0
$133$	−434.000	−	751.710i	−0.282952	−	0.490087i
$134$		0			0
$135$	−1093.50	−	631.333i	−0.697137	−	0.402492i
$136$		0			0
$137$	1330.50	+	2304.49i	0.829725	+	1.43713i	0.898254	+	0.439476i	$0.144836\pi$
$137$	1330.50	+	2304.49i	0.829725	+	1.43713i	−0.0685295	+	0.997649i	$0.521831\pi$
$138$		0			0
$139$	−60.5000	+	104.789i	−0.0369176	+	0.0639431i	−0.883894	−	0.467688i	$-0.845087\pi$
$139$	−60.5000	+	104.789i	−0.0369176	+	0.0639431i	0.846976	+	0.531631i	$0.178421\pi$
$140$		0			0
$141$	−796.500	+	459.859i	−0.475726	+	0.274661i
$142$		0			0
$143$	−555.000			−0.324555
$144$		0			0
$145$	−999.000			−0.572155
$146$		0			0
$147$	−2781.00	−	1605.61i	−1.56036	−	0.900875i
$148$		0			0
$149$	1414.50	−	2449.99i	0.777721	−	1.34705i	−0.155532	−	0.987831i	$-0.549709\pi$
$149$	1414.50	−	2449.99i	0.777721	−	1.34705i	0.933253	−	0.359221i	$-0.116957\pi$
$150$		0			0
$151$	230.500	+	399.238i	0.124224	+	0.215162i	0.921429	−	0.388546i	$-0.127023\pi$
$151$	230.500	+	399.238i	0.124224	+	0.215162i	−0.797205	+	0.603708i	$0.793689\pi$
$152$		0			0
$153$	1134.00			0.599206
$154$		0			0
$155$	922.500	+	1597.82i	0.478045	+	0.827998i
$156$		0			0
$157$	1488.50	−	2578.16i	0.756658	−	1.31057i	−0.187889	−	0.982190i	$-0.560164\pi$
$157$	1488.50	−	2578.16i	0.756658	−	1.31057i	0.944546	−	0.328379i	$-0.106502\pi$
$158$		0			0
$159$			592.361i			0.295455i
$160$		0			0
$161$	−6045.00			−2.95909
$162$		0			0
$163$	3316.00			1.59343			0.796715	−	0.604355i	$-0.206569\pi$
$163$	3316.00			1.59343			0.796715	+	0.604355i	$0.206569\pi$
$164$		0			0
$165$		−	701.481i		−	0.330971i
$166$		0			0
$167$	−340.500	+	589.763i	−0.157777	+	0.273277i	−0.934067	−	0.357099i	$-0.883766\pi$
$167$	−340.500	+	589.763i	−0.157777	+	0.273277i	0.776290	+	0.630376i	$0.217099\pi$
$168$		0			0
$169$	414.000	+	717.069i	0.188439	+	0.326386i
$170$		0			0
$171$	−756.000			−0.338086
$172$		0			0
$173$	1990.50	+	3447.65i	0.874768	+	1.51514i	0.857009	+	0.515301i	$0.172320\pi$
$173$	1990.50	+	3447.65i	0.874768	+	1.51514i	0.0177589	+	0.999842i	$0.494347\pi$
$174$		0			0
$175$	682.000	−	1181.26i	0.294596	−	0.510256i
$176$		0			0
$177$	715.500	+	413.094i	0.303843	+	0.175424i
$178$		0			0
$179$	−2004.00			−0.836793			−0.418397	−	0.908264i	$-0.637408\pi$
$179$	−2004.00			−0.836793			−0.418397	+	0.908264i	$0.637408\pi$
$180$		0			0
$181$	1274.00			0.523181			0.261590	−	0.965179i	$-0.415753\pi$
$181$	1274.00			0.523181			0.261590	+	0.965179i	$0.415753\pi$
$182$		0			0
$183$	859.500	−	496.233i	0.347192	−	0.200451i
$184$		0			0
$185$	−747.000	+	1293.84i	−0.296868	+	0.514190i
$186$		0			0
$187$	315.000	+	545.596i	0.123182	+	0.213358i
$188$		0			0
$189$	−3766.50	+	2174.59i	−1.44959	+	0.836921i
$190$		0			0
$191$	580.500	+	1005.46i	0.219914	+	0.380902i	0.954781	−	0.297309i	$-0.0960891\pi$
$191$	580.500	+	1005.46i	0.219914	+	0.380902i	−0.734868	+	0.678210i	$0.762756\pi$
$192$		0			0
$193$	−1805.50	+	3127.22i	−0.673382	+	1.16633i	0.303557	+	0.952813i	$0.401826\pi$
$193$	−1805.50	+	3127.22i	−0.673382	+	1.16633i	−0.976939	+	0.213519i	$0.931508\pi$
$194$		0			0
$195$	1498.50	−	865.159i	0.550307	−	0.317720i
$196$		0			0
$197$	2046.00			0.739957			0.369978	−	0.929040i	$-0.379365\pi$
$197$	2046.00			0.739957			0.369978	+	0.929040i	$0.379365\pi$
$198$		0			0
$199$	−2996.00			−1.06724			−0.533620	−	0.845724i	$-0.679169\pi$
$199$	−2996.00			−1.06724			−0.533620	+	0.845724i	$0.679169\pi$
$200$		0			0
$201$	−1894.50	−	1093.79i	−0.664815	−	0.383831i
$202$		0			0
$203$	−1720.50	+	2979.99i	−0.594854	+	1.03032i
$204$		0			0
$205$	−1174.50	−	2034.29i	−0.400149	−	0.693079i
$206$		0			0
$207$	−2632.50	+	4559.62i	−0.883920	+	1.53099i
$208$		0			0
$209$	−210.000	−	363.731i	−0.0695024	−	0.120382i
$210$		0			0
$211$	377.500	−	653.849i	0.123167	−	0.213331i	−0.797848	−	0.602858i	$-0.794028\pi$
$211$	377.500	−	653.849i	0.123167	−	0.213331i	0.921015	+	0.389528i	$0.127362\pi$
$212$		0			0
$213$		−	810.600i		−	0.260758i
$214$		0			0
$215$	−387.000			−0.122759
$216$		0			0
$217$	6355.00			1.98804
$218$		0			0
$219$			945.700i			0.291801i
$220$		0			0
$221$	−777.000	+	1345.80i	−0.236501	+	0.409631i
$222$		0			0
$223$	−1731.50	−	2999.05i	−0.519954	−	0.900587i	−0.999731	−	0.0231966i	$-0.992616\pi$
$223$	−1731.50	−	2999.05i	−0.519954	−	0.900587i	0.479777	−	0.877391i	$-0.340718\pi$
$224$		0			0
$225$	−594.000	−	1028.84i	−0.176000	−	0.304841i
$226$		0			0
$227$	3112.50	+	5391.01i	0.910061	+	1.57627i	0.813976	+	0.580899i	$0.197299\pi$
$227$	3112.50	+	5391.01i	0.910061	+	1.57627i	0.0960856	+	0.995373i	$0.469368\pi$
$228$		0			0
$229$	732.500	−	1268.73i	0.211375	−	0.366113i	−0.740770	−	0.671759i	$-0.765539\pi$
$229$	732.500	−	1268.73i	0.211375	−	0.366113i	0.952145	+	0.305646i	$0.0988724\pi$
$230$		0			0
$231$	−2092.50	−	1208.11i	−0.596002	−	0.344102i
$232$		0			0
$233$	2634.00			0.740597			0.370298	−	0.928913i	$-0.379255\pi$
$233$	2634.00			0.740597			0.370298	+	0.928913i	$0.379255\pi$
$234$		0			0
$235$	1593.00			0.442195
$236$		0			0
$237$	−5098.50	+	2943.62i	−1.39740	+	0.806788i
$238$		0			0
$239$	3457.50	−	5988.57i	0.935762	−	1.62079i	0.162492	−	0.986710i	$-0.448047\pi$
$239$	3457.50	−	5988.57i	0.935762	−	1.62079i	0.773270	−	0.634077i	$-0.218620\pi$
$240$		0			0
$241$	744.500	+	1289.51i	0.198994	+	0.344667i	0.948202	−	0.317667i	$-0.102899\pi$
$241$	744.500	+	1289.51i	0.198994	+	0.344667i	−0.749209	+	0.662334i	$0.769566\pi$
$242$		0			0
$243$			3788.00i			1.00000i
$244$		0			0
$245$	2781.00	+	4816.83i	0.725190	+	1.25607i
$246$		0			0
$247$	518.000	−	897.202i	0.133439	−	0.231124i
$248$		0			0
$249$	4873.50	−	2813.72i	1.24034	−	0.716113i
$250$		0			0
$251$	4620.00			1.16180			0.580900	−	0.813975i	$-0.302701\pi$
$251$	4620.00			1.16180			0.580900	+	0.813975i	$0.302701\pi$
$252$		0			0
$253$	−2925.00			−0.726850
$254$		0			0
$255$	−1701.00	−	982.073i	−0.417728	−	0.241176i
$256$		0			0
$257$	−1675.50	+	2902.05i	−0.406672	+	0.704377i	−0.994515	−	0.104598i	$-0.966644\pi$
$257$	−1675.50	+	2902.05i	−0.406672	+	0.704377i	0.587842	+	0.808976i	$0.299978\pi$
$258$		0			0
$259$	2573.00	+	4456.57i	0.617291	+	1.06918i
$260$		0			0
$261$	1498.50	+	2595.48i	0.355382	+	0.615540i
$262$		0			0
$263$	−301.500	−	522.213i	−0.0706893	−	0.122437i	0.828514	−	0.559968i	$-0.189187\pi$
$263$	−301.500	−	522.213i	−0.0706893	−	0.122437i	−0.899204	+	0.437530i	$0.855853\pi$
$264$		0			0
$265$	513.000	−	888.542i	0.118918	−	0.205972i
$266$		0			0
$267$		−	5455.96i		−	1.25056i
$268$		0			0
$269$	−1470.00			−0.333188			−0.166594	−	0.986026i	$-0.553277\pi$
$269$	−1470.00			−0.333188			−0.166594	+	0.986026i	$0.553277\pi$
$270$		0			0
$271$	−2072.00			−0.464447			−0.232223	−	0.972662i	$-0.574600\pi$
$271$	−2072.00			−0.464447			−0.232223	+	0.972662i	$0.574600\pi$
$272$		0			0
$273$		−	5959.99i		−	1.32130i
$274$		0			0
$275$	330.000	−	571.577i	0.0723627	−	0.125336i
$276$		0			0
$277$	−3569.50	−	6182.56i	−0.774262	−	1.34106i	−0.935209	−	0.354097i	$-0.884788\pi$
$277$	−3569.50	−	6182.56i	−0.774262	−	1.34106i	0.160947	−	0.986963i	$-0.448545\pi$
$278$		0			0
$279$	2767.50	−	4793.45i	0.593856	−	1.02859i
$280$		0			0
$281$	−4213.50	−	7298.00i	−0.894507	−	1.54933i	−0.834414	−	0.551138i	$-0.814194\pi$
$281$	−4213.50	−	7298.00i	−0.894507	−	1.54933i	−0.0600924	−	0.998193i	$-0.519140\pi$
$282$		0			0
$283$	−228.500	+	395.774i	−0.0479962	+	0.0831318i	−0.889025	−	0.457858i	$-0.848617\pi$
$283$	−228.500	+	395.774i	−0.0479962	+	0.0831318i	0.841029	+	0.540990i	$0.181950\pi$
$284$		0			0
$285$	1134.00	+	654.715i	0.235693	+	0.136077i
$286$		0			0
$287$	−8091.00			−1.66410
$288$		0			0
$289$	−3149.00			−0.640953
$290$		0			0
$291$	−4054.50	+	2340.87i	−0.816766	+	0.471560i
$292$		0			0
$293$	2944.50	−	5100.02i	0.587097	−	1.01688i	−0.407513	−	0.913199i	$-0.633604\pi$
$293$	2944.50	−	5100.02i	0.587097	−	1.01688i	0.994610	−	0.103683i	$-0.0330628\pi$
$294$		0			0
$295$	−715.500	−	1239.28i	−0.141214	−	0.244589i
$296$		0			0
$297$	−1822.50	+	1052.22i	−0.356068	+	0.205576i
$298$		0			0
$299$	−3607.50	−	6248.37i	−0.697750	−	1.20854i
$300$		0			0
$301$	−666.500	+	1154.41i	−0.127629	+	0.221060i
$302$		0			0
$303$	1741.50	−	1005.46i	0.330187	−	0.190633i
$304$		0			0
$305$	−1719.00			−0.322720
$306$		0			0
$307$	1204.00			0.223830			0.111915	−	0.993718i	$-0.464302\pi$
$307$	1204.00			0.223830			0.111915	+	0.993718i	$0.464302\pi$
$308$		0			0
$309$	2479.50	+	1431.54i	0.456485	+	0.263552i
$310$		0			0
$311$	−1642.50	+	2844.89i	−0.299478	+	0.518711i	−0.976017	−	0.217696i	$-0.930146\pi$
$311$	−1642.50	+	2844.89i	−0.299478	+	0.518711i	0.676539	+	0.736407i	$0.263479\pi$
$312$		0			0
$313$	5028.50	+	8709.62i	0.908075	+	1.57283i	0.816735	+	0.577013i	$0.195782\pi$
$313$	5028.50	+	8709.62i	0.908075	+	1.57283i	0.0913406	+	0.995820i	$0.470885\pi$
$314$		0			0
$315$	7533.00			1.34742
$316$		0			0
$317$	−1147.50	−	1987.53i	−0.203312	−	0.352147i	0.746281	−	0.665631i	$-0.231837\pi$
$317$	−1147.50	−	1987.53i	−0.203312	−	0.352147i	−0.949594	+	0.313483i	$0.898504\pi$
$318$		0			0
$319$	−832.500	+	1441.93i	−0.146116	+	0.253081i
$320$		0			0
$321$			62.3538i			0.0108419i
$322$		0			0
$323$	−1176.00			−0.202583
$324$		0			0
$325$	1628.00			0.277862
$326$		0			0
$327$		−	2608.47i		−	0.441127i
$328$		0			0
$329$	2743.50	−	4751.88i	0.459739	−	0.796291i
$330$		0			0
$331$	−3339.50	−	5784.18i	−0.554548	−	0.960506i	−0.997939	−	0.0641773i	$-0.979558\pi$
$331$	−3339.50	−	5784.18i	−0.554548	−	0.960506i	0.443390	−	0.896329i	$-0.353776\pi$
$332$		0			0
$333$	4482.00			0.737574
$334$		0			0
$335$	1894.50	+	3281.37i	0.308978	+	0.535165i
$336$		0			0
$337$	−1091.50	+	1890.53i	−0.176433	+	0.305590i	−0.940656	−	0.339361i	$-0.889789\pi$
$337$	−1091.50	+	1890.53i	−0.176433	+	0.305590i	0.764224	+	0.644951i	$0.223122\pi$
$338$		0			0
$339$	6304.50	+	3639.90i	1.01007	+	0.583164i
$340$		0			0
$341$	3075.00			0.488330
$342$		0			0
$343$	8525.00			1.34200
$344$		0			0
$345$	7897.50	−	4559.62i	1.23243	−	0.711542i
$346$		0			0
$347$	1945.50	−	3369.70i	0.300980	−	0.521312i	−0.675379	−	0.737471i	$-0.736020\pi$
$347$	1945.50	−	3369.70i	0.300980	−	0.521312i	0.976358	+	0.216159i	$0.0693531\pi$
$348$		0			0
$349$	−1397.50	−	2420.54i	−0.214345	−	0.371257i	0.738725	−	0.674007i	$-0.235428\pi$
$349$	−1397.50	−	2420.54i	−0.214345	−	0.371257i	−0.953070	+	0.302751i	$0.902095\pi$
$350$		0			0
$351$	−4495.50	−	2595.48i	−0.683624	−	0.394691i
$352$		0			0
$353$	−2377.50	−	4117.95i	−0.358475	−	0.620896i	0.629232	−	0.777218i	$-0.283370\pi$
$353$	−2377.50	−	4117.95i	−0.358475	−	0.620896i	−0.987706	+	0.156322i	$0.950036\pi$
$354$		0			0
$355$	−702.000	+	1215.90i	−0.104953	+	0.181784i
$356$		0			0
$357$	−5859.00	+	3382.70i	−0.868603	+	0.501488i
$358$		0			0
$359$	−4608.00			−0.677440			−0.338720	−	0.940887i	$-0.609994\pi$
$359$	−4608.00			−0.677440			−0.338720	+	0.940887i	$0.609994\pi$
$360$		0			0
$361$	−6075.00			−0.885698
$362$		0			0
$363$	4977.00	+	2873.47i	0.719627	+	0.415477i
$364$		0			0
$365$	819.000	−	1418.55i	0.117448	−	0.203425i
$366$		0			0
$367$	1922.50	+	3329.87i	0.273443	+	0.473618i	0.969741	−	0.244135i	$-0.0785041\pi$
$367$	1922.50	+	3329.87i	0.273443	+	0.473618i	−0.696298	+	0.717753i	$0.745171\pi$
$368$		0			0
$369$	−3523.50	+	6102.88i	−0.497090	+	0.860985i
$370$		0			0
$371$	−1767.00	−	3060.53i	−0.247272	−	0.428288i
$372$		0			0
$373$	4158.50	−	7202.73i	0.577263	−	0.999848i	−0.418529	−	0.908203i	$-0.637454\pi$
$373$	4158.50	−	7202.73i	0.577263	−	0.999848i	0.995792	−	0.0916449i	$-0.0292125\pi$
$374$		0			0
$375$			7903.35i			1.08834i
$376$		0			0
$377$	−4107.00			−0.561064
$378$		0			0
$379$	−12560.0			−1.70228			−0.851140	−	0.524939i	$-0.824088\pi$
$379$	−12560.0			−1.70228			−0.851140	+	0.524939i	$0.824088\pi$
$380$		0			0
$381$			4572.61i			0.614861i
$382$		0			0
$383$	6043.50	−	10467.6i	0.806288	−	1.39653i	−0.109130	−	0.994028i	$-0.534806\pi$
$383$	6043.50	−	10467.6i	0.806288	−	1.39653i	0.915418	−	0.402505i	$-0.131860\pi$
$384$		0			0
$385$	2092.50	+	3624.32i	0.276997	+	0.479772i
$386$		0			0
$387$	580.500	+	1005.46i	0.0762493	+	0.132068i
$388$		0			0
$389$	4270.50	+	7396.72i	0.556614	+	0.964084i	0.997776	+	0.0666565i	$0.0212332\pi$
$389$	4270.50	+	7396.72i	0.556614	+	0.964084i	−0.441162	+	0.897428i	$0.645433\pi$
$390$		0			0
$391$	−4095.00	+	7092.75i	−0.529650	+	0.917380i
$392$		0			0
$393$	6763.50	+	3904.91i	0.868126	+	0.501213i
$394$		0			0
$395$	10197.0			1.29890
$396$		0			0
$397$	−13174.0			−1.66545			−0.832726	−	0.553686i	$-0.813221\pi$
$397$	−13174.0			−1.66545			−0.832726	+	0.553686i	$0.813221\pi$
$398$		0			0
$399$	3906.00	−	2255.13i	0.490087	−	0.282952i
$400$		0			0
$401$	−4801.50	+	8316.44i	−0.597944	+	1.03567i	0.395180	+	0.918604i	$0.370682\pi$
$401$	−4801.50	+	8316.44i	−0.597944	+	1.03567i	−0.993124	+	0.117066i	$0.962651\pi$
$402$		0			0
$403$	3792.50	+	6568.80i	0.468779	+	0.811949i
$404$		0			0
$405$	3280.50	−	5681.99i	0.402492	−	0.697137i
$406$		0			0
$407$	1245.00	+	2156.40i	0.151627	+	0.262626i
$408$		0			0
$409$	−5735.50	+	9934.18i	−0.693404	+	1.20101i	0.277312	+	0.960780i	$0.410557\pi$
$409$	−5735.50	+	9934.18i	−0.693404	+	1.20101i	−0.970716	+	0.240231i	$0.922777\pi$
$410$		0			0
$411$	−11974.5	+	6913.48i	−1.43713	+	0.829725i
$412$		0			0
$413$	−4929.00			−0.587264
$414$		0			0
$415$	−9747.00			−1.15292
$416$		0			0
$417$	−544.500	−	314.367i	−0.0639431	−	0.0369176i
$418$		0			0
$419$	−2986.50	+	5172.77i	−0.348210	+	0.603118i	−0.985932	−	0.167149i	$-0.946544\pi$
$419$	−2986.50	+	5172.77i	−0.348210	+	0.603118i	0.637721	+	0.770267i	$0.279877\pi$
$420$		0			0
$421$	4452.50	+	7711.96i	0.515443	+	0.892774i	0.999839	+	0.0179250i	$0.00570601\pi$
$421$	4452.50	+	7711.96i	0.515443	+	0.892774i	−0.484396	+	0.874849i	$0.660961\pi$
$422$		0			0
$423$	−2389.50	−	4138.74i	−0.274661	−	0.475726i
$424$		0			0
$425$	−924.000	−	1600.41i	−0.105460	−	0.182662i
$426$		0			0
$427$	−2960.50	+	5127.74i	−0.335524	+	0.581144i
$428$		0			0
$429$		−	2883.86i		−	0.324555i
$430$		0			0
$431$	−1416.00			−0.158251			−0.0791257	−	0.996865i	$-0.525213\pi$
$431$	−1416.00			−0.158251			−0.0791257	+	0.996865i	$0.525213\pi$
$432$		0			0
$433$	10766.0			1.19488			0.597438	−	0.801915i	$-0.296186\pi$
$433$	10766.0			1.19488			0.597438	+	0.801915i	$0.296186\pi$
$434$		0			0
$435$		−	5190.96i		−	0.572155i
$436$		0			0
$437$	2730.00	−	4728.50i	0.298841	−	0.517608i
$438$		0			0
$439$	2174.50	+	3766.34i	0.236408	+	0.409471i	0.959681	−	0.281091i	$-0.0906964\pi$
$439$	2174.50	+	3766.34i	0.236408	+	0.409471i	−0.723273	+	0.690562i	$0.757363\pi$
$440$		0			0
$441$	8343.00	−	14450.5i	0.900875	−	1.56036i
$442$		0			0
$443$	−7273.50	−	12598.1i	−0.780078	−	1.35113i	−0.931896	−	0.362726i	$-0.881846\pi$
$443$	−7273.50	−	12598.1i	−0.780078	−	1.35113i	0.151818	−	0.988408i	$-0.451487\pi$
$444$		0			0
$445$	−4725.00	+	8183.94i	−0.503340	+	0.871811i
$446$		0			0
$447$	12730.5	+	7349.96i	1.34705	+	0.777721i
$448$		0			0
$449$	−3330.00			−0.350005			−0.175003	−	0.984568i	$-0.555993\pi$
$449$	−3330.00			−0.350005			−0.175003	+	0.984568i	$0.555993\pi$
$450$		0			0
$451$	−3915.00			−0.408759
$452$		0			0
$453$	−2074.50	+	1197.71i	−0.215162	+	0.124224i
$454$		0			0
$455$	−5161.50	+	8939.98i	−0.531813	+	0.921127i
$456$		0			0
$457$	−4073.50	−	7055.51i	−0.416959	−	0.722194i	0.578673	−	0.815560i	$-0.303571\pi$
$457$	−4073.50	−	7055.51i	−0.416959	−	0.722194i	−0.995632	+	0.0933655i	$0.970237\pi$
$458$		0			0
$459$			5892.44i			0.599206i
$460$		0			0
$461$	−4015.50	−	6955.05i	−0.405684	−	0.702666i	0.588717	−	0.808340i	$-0.299633\pi$
$461$	−4015.50	−	6955.05i	−0.405684	−	0.702666i	−0.994401	+	0.105674i	$0.966300\pi$
$462$		0			0
$463$	2141.50	−	3709.19i	0.214955	−	0.372312i	−0.738304	−	0.674468i	$-0.764373\pi$
$463$	2141.50	−	3709.19i	0.214955	−	0.372312i	0.953259	+	0.302156i	$0.0977063\pi$
$464$		0			0
$465$	−8302.50	+	4793.45i	−0.827998	+	0.478045i
$466$		0			0
$467$	−5460.00			−0.541025			−0.270512	−	0.962716i	$-0.587193\pi$
$467$	−5460.00			−0.541025			−0.270512	+	0.962716i	$0.587193\pi$
$468$		0			0
$469$	13051.0			1.28494
$470$		0			0
$471$	13396.5	+	7734.47i	1.31057	+	0.756658i
$472$		0			0
$473$	−322.500	+	558.586i	−0.0313500	+	0.0542999i
$474$		0			0
$475$	616.000	+	1066.94i	0.0595032	+	0.103063i
$476$		0			0
$477$	−3078.00			−0.295455
$478$		0			0
$479$	214.500	+	371.525i	0.0204609	+	0.0354393i	0.876075	−	0.482176i	$-0.160153\pi$
$479$	214.500	+	371.525i	0.0204609	+	0.0354393i	−0.855614	+	0.517615i	$0.826820\pi$
$480$		0			0
$481$	−3071.00	+	5319.13i	−0.291113	+	0.504223i
$482$		0			0
$483$		−	31410.7i		−	2.95909i
$484$		0			0
$485$	8109.00			0.759197
$486$		0			0
$487$	11296.0			1.05107			0.525535	−	0.850772i	$-0.323865\pi$
$487$	11296.0			1.05107			0.525535	+	0.850772i	$0.323865\pi$
$488$		0			0
$489$			17230.4i			1.59343i
$490$		0			0
$491$	−7336.50	+	12707.2i	−0.674321	+	1.16796i	0.302346	+	0.953198i	$0.402230\pi$
$491$	−7336.50	+	12707.2i	−0.674321	+	1.16796i	−0.976667	+	0.214760i	$0.931103\pi$
$492$		0			0
$493$	2331.00	+	4037.41i	0.212947	+	0.368835i
$494$		0			0
$495$	3645.00			0.330971
$496$		0			0
$497$	2418.00	+	4188.10i	0.218234	+	0.377992i
$498$		0			0
$499$	6719.50	−	11638.5i	0.602818	−	1.04411i	−0.389574	−	0.920995i	$-0.627378\pi$
$499$	6719.50	−	11638.5i	0.602818	−	1.04411i	0.992392	−	0.123116i	$-0.0392888\pi$
$500$		0			0
$501$	−3064.50	−	1769.29i	−0.273277	−	0.157777i
$502$		0			0
$503$	17388.0			1.54134			0.770669	−	0.637236i	$-0.219922\pi$
$503$	17388.0			1.54134			0.770669	+	0.637236i	$0.219922\pi$
$504$		0			0
$505$	−3483.00			−0.306914
$506$		0			0
$507$	−3726.00	+	2151.21i	−0.326386	+	0.188439i
$508$		0			0
$509$	1894.50	−	3281.37i	0.164975	−	0.285745i	−0.771671	−	0.636021i	$-0.780579\pi$
$509$	1894.50	−	3281.37i	0.164975	−	0.285745i	0.936646	+	0.350276i	$0.113912\pi$
$510$		0			0
$511$	−2821.00	−	4886.12i	−0.244215	−	0.422992i
$512$		0			0
$513$		−	3928.29i		−	0.338086i
$514$		0			0
$515$	−2479.50	−	4294.62i	−0.212155	−	0.367463i
$516$		0			0
$517$	1327.50	−	2299.30i	0.112927	−	0.195596i
$518$		0			0
$519$	−17914.5	+	10342.9i	−1.51514	+	0.874768i
$520$		0			0
$521$	9786.00			0.822903			0.411451	−	0.911432i	$-0.365022\pi$
$521$	9786.00			0.822903			0.411451	+	0.911432i	$0.365022\pi$
$522$		0			0
$523$	8008.00			0.669532			0.334766	−	0.942301i	$-0.391343\pi$
$523$	8008.00			0.669532			0.334766	+	0.942301i	$0.391343\pi$
$524$		0			0
$525$	6138.00	+	3543.78i	0.510256	+	0.294596i
$526$		0			0
$527$	4305.00	−	7456.48i	0.355842	−	0.616336i
$528$		0			0
$529$	−12929.0	−	22393.7i	−1.06263	−	1.84053i
$530$		0			0
$531$	−2146.50	+	3717.85i	−0.175424	+	0.303843i
$532$		0			0
$533$	−4828.50	−	8363.21i	−0.392393	−	0.679645i
$534$		0			0
$535$	54.0000	−	93.5307i	0.00436378	−	0.00755829i
$536$		0			0
$537$		−	10413.1i		−	0.836793i
$538$		0			0
$539$	9270.00			0.740793
$540$		0			0
$541$	−2938.00			−0.233483			−0.116742	−	0.993162i	$-0.537245\pi$
$541$	−2938.00			−0.233483			−0.116742	+	0.993162i	$0.537245\pi$
$542$		0			0
$543$			6619.90i			0.523181i
$544$		0			0
$545$	−2259.00	+	3912.70i	−0.177550	+	0.307526i
$546$		0			0
$547$	−5187.50	−	8985.01i	−0.405487	−	0.702324i	0.588891	−	0.808213i	$-0.299565\pi$
$547$	−5187.50	−	8985.01i	−0.405487	−	0.702324i	−0.994378	+	0.105888i	$0.966231\pi$
$548$		0			0
$549$	2578.50	+	4466.09i	0.200451	+	0.347192i
$550$		0			0
$551$	−1554.00	−	2691.61i	−0.120150	−	0.208106i
$552$		0			0
$553$	17561.5	−	30417.4i	1.35044	−	2.33902i
$554$		0			0
$555$	−6723.00	−	3881.53i	−0.514190	−	0.296868i
$556$		0			0
$557$	3306.00			0.251490			0.125745	−	0.992063i	$-0.459868\pi$
$557$	3306.00			0.251490			0.125745	+	0.992063i	$0.459868\pi$
$558$		0			0
$559$	−1591.00			−0.120379
$560$		0			0
$561$	−2835.00	+	1636.79i	−0.213358	+	0.123182i
$562$		0			0
$563$	−10546.5	+	18267.1i	−0.789488	+	1.36743i	0.136792	+	0.990600i	$0.456321\pi$
$563$	−10546.5	+	18267.1i	−0.789488	+	1.36743i	−0.926281	+	0.376834i	$0.877013\pi$
$564$		0			0
$565$	−6304.50	−	10919.7i	−0.469438	−	0.813090i
$566$		0			0
$567$	−11299.5	−	19571.3i	−0.836921	−	1.44959i
$568$		0			0
$569$	−643.500	−	1114.57i	−0.0474111	−	0.0821185i	0.841346	−	0.540497i	$-0.181764\pi$
$569$	−643.500	−	1114.57i	−0.0474111	−	0.0821185i	−0.888757	+	0.458379i	$0.848430\pi$
$570$		0			0
$571$	7517.50	−	13020.7i	0.550959	−	0.954289i	−0.447247	−	0.894411i	$-0.647595\pi$
$571$	7517.50	−	13020.7i	0.550959	−	0.954289i	0.998206	−	0.0598783i	$-0.0190713\pi$
$572$		0			0
$573$	−5224.50	+	3016.37i	−0.380902	+	0.219914i
$574$		0			0
$575$	8580.00			0.622280
$576$		0			0
$577$	1190.00			0.0858585			0.0429292	−	0.999078i	$-0.486331\pi$
$577$	1190.00			0.0858585			0.0429292	+	0.999078i	$0.486331\pi$
$578$		0			0
$579$	−16249.5	−	9381.65i	−1.16633	−	0.673382i
$580$		0			0
$581$	−16786.5	+	29075.1i	−1.19866	+	2.07614i
$582$		0			0
$583$	−855.000	−	1480.90i	−0.0607384	−	0.105202i
$584$		0			0
$585$	4495.50	+	7786.43i	0.317720	+	0.550307i
$586$		0			0
$587$	−8941.50	−	15487.1i	−0.628714	−	1.08896i	−0.987810	−	0.155664i	$-0.950248\pi$
$587$	−8941.50	−	15487.1i	−0.628714	−	1.08896i	0.359096	−	0.933301i	$-0.383085\pi$
$588$		0			0
$589$	−2870.00	+	4970.99i	−0.200775	+	0.347752i
$590$		0			0
$591$			10631.3i			0.739957i
$592$		0			0
$593$	20118.0			1.39317			0.696583	−	0.717476i	$-0.254703\pi$
$593$	20118.0			1.39317			0.696583	+	0.717476i	$0.254703\pi$
$594$		0			0
$595$	11718.0			0.807380
$596$		0			0
$597$		−	15567.7i		−	1.06724i
$598$		0			0
$599$	−532.500	+	922.317i	−0.0363228	+	0.0629129i	−0.883615	−	0.468214i	$-0.844898\pi$
$599$	−532.500	+	922.317i	−0.0363228	+	0.0629129i	0.847293	+	0.531127i	$0.178231\pi$
$600$		0			0
$601$	10362.5	+	17948.4i	0.703320	+	1.21819i	0.967294	+	0.253656i	$0.0816331\pi$
$601$	10362.5	+	17948.4i	0.703320	+	1.21819i	−0.263975	+	0.964530i	$0.585034\pi$
$602$		0			0
$603$	5683.50	−	9844.11i	0.383831	−	0.664815i
$604$		0			0
$605$	−4977.00	−	8620.42i	−0.334453	−	0.579289i
$606$		0			0
$607$	−7872.50	+	13635.6i	−0.526417	+	0.911780i	0.473110	+	0.881004i	$0.343131\pi$
$607$	−7872.50	+	13635.6i	−0.526417	+	0.911780i	−0.999526	+	0.0307768i	$0.990202\pi$
$608$		0			0
$609$	−15484.5	−	8939.98i	−1.03032	−	0.594854i
$610$		0			0
$611$	6549.00			0.433624
$612$		0			0
$613$	5042.00			0.332210			0.166105	−	0.986108i	$-0.446881\pi$
$613$	5042.00			0.332210			0.166105	+	0.986108i	$0.446881\pi$
$614$		0			0
$615$	10570.5	−	6102.88i	0.693079	−	0.400149i
$616$		0			0
$617$	5026.50	−	8706.15i	0.327973	−	0.568066i	−0.654137	−	0.756376i	$-0.726968\pi$
$617$	5026.50	−	8706.15i	0.327973	−	0.568066i	0.982110	+	0.188311i	$0.0603012\pi$
$618$		0			0
$619$	−2991.50	−	5181.43i	−0.194246	−	0.336445i	0.752407	−	0.658699i	$-0.228893\pi$
$619$	−2991.50	−	5181.43i	−0.194246	−	0.336445i	−0.946653	+	0.322254i	$0.895559\pi$
$620$		0			0
$621$	−23692.5	−	13678.9i	−1.53099	−	0.883920i
$622$		0			0
$623$	16275.0	+	28189.1i	1.04662	+	1.81280i
$624$		0			0
$625$	4094.50	−	7091.88i	0.262048	−	0.453880i
$626$		0			0
$627$	1890.00	−	1091.19i	0.120382	−	0.0695024i
$628$		0			0
$629$	6972.00			0.441958
$630$		0			0
$631$	19696.0			1.24261			0.621304	−	0.783570i	$-0.286603\pi$
$631$	19696.0			1.24261			0.621304	+	0.783570i	$0.286603\pi$
$632$		0			0
$633$	3397.50	+	1961.55i	0.213331	+	0.123167i
$634$		0			0
$635$	3960.00	−	6858.92i	0.247477	−	0.428642i
$636$		0			0
$637$	11433.0	+	19802.5i	0.711133	+	1.23172i
$638$		0			0
$639$	4212.00			0.260758
$640$		0			0
$641$	5488.50	+	9506.36i	0.338195	+	0.585770i	0.984093	−	0.177653i	$-0.0568505\pi$
$641$	5488.50	+	9506.36i	0.338195	+	0.585770i	−0.645899	+	0.763423i	$0.723517\pi$
$642$		0			0
$643$	−7914.50	+	13708.3i	−0.485408	+	0.840752i	−0.999859	−	0.0167681i	$-0.994662\pi$
$643$	−7914.50	+	13708.3i	−0.485408	+	0.840752i	0.514451	+	0.857520i	$0.327996\pi$
$644$		0			0
$645$		−	2010.91i		−	0.122759i
$646$		0			0
$647$	−28224.0			−1.71499			−0.857496	−	0.514490i	$-0.827981\pi$
$647$	−28224.0			−1.71499			−0.857496	+	0.514490i	$0.827981\pi$
$648$		0			0
$649$	−2385.00			−0.144252
$650$		0			0
$651$			33021.5i			1.98804i
$652$		0			0
$653$	−14083.5	+	24393.3i	−0.843997	+	1.46185i	0.0424927	+	0.999097i	$0.486470\pi$
$653$	−14083.5	+	24393.3i	−0.843997	+	1.46185i	−0.886490	+	0.462749i	$0.846863\pi$
$654$		0			0
$655$	−6763.50	−	11714.7i	−0.403468	−	0.698828i
$656$		0			0
$657$	−4914.00			−0.291801
$658$		0			0
$659$	5368.50	+	9298.51i	0.317340	+	0.549649i	0.979932	−	0.199331i	$-0.0638770\pi$
$659$	5368.50	+	9298.51i	0.317340	+	0.549649i	−0.662592	+	0.748980i	$0.730544\pi$
$660$		0			0
$661$	−5063.50	+	8770.24i	−0.297954	+	0.516071i	−0.975668	−	0.219255i	$-0.929637\pi$
$661$	−5063.50	+	8770.24i	−0.297954	+	0.516071i	0.677714	+	0.735326i	$0.262971\pi$
$662$		0			0
$663$	−6993.00	−	4037.41i	−0.409631	−	0.236501i
$664$		0			0
$665$	−7812.00			−0.455543
$666$		0			0
$667$	−21645.0			−1.25652
$668$		0			0
$669$	15583.5	−	8997.14i	0.900587	−	0.519954i
$670$		0			0
$671$	−1432.50	+	2481.16i	−0.0824159	+	0.142748i
$672$		0			0
$673$	−125.500	−	217.372i	−0.00718822	−	0.0124504i	0.862409	−	0.506212i	$-0.168955\pi$
$673$	−125.500	−	217.372i	−0.00718822	−	0.0124504i	−0.869597	+	0.493762i	$0.835621\pi$
$674$		0			0
$675$	5346.00	−	3086.51i	0.304841	−	0.176000i
$676$		0			0
$677$	−4225.50	−	7318.78i	−0.239881	−	0.415485i	0.720799	−	0.693144i	$-0.243775\pi$
$677$	−4225.50	−	7318.78i	−0.239881	−	0.415485i	−0.960680	+	0.277659i	$0.910442\pi$
$678$		0			0
$679$	13965.5	−	24189.0i	0.789318	−	1.36714i
$680$		0			0
$681$	−28012.5	+	16173.0i	−1.57627	+	0.910061i
$682$		0			0
$683$	25884.0			1.45011			0.725054	−	0.688692i	$-0.241815\pi$
$683$	25884.0			1.45011			0.725054	+	0.688692i	$0.241815\pi$
$684$		0			0
$685$	23949.0			1.33583
$686$		0			0
$687$	6592.50	+	3806.18i	0.366113	+	0.211375i
$688$		0			0
$689$	2109.00	−	3652.90i	0.116613	−	0.201980i
$690$		0			0
$691$	3182.50	+	5512.25i	0.175207	+	0.303467i	0.940233	−	0.340532i	$-0.110607\pi$
$691$	3182.50	+	5512.25i	0.175207	+	0.303467i	−0.765026	+	0.643999i	$0.777274\pi$
$692$		0			0
$693$	6277.50	−	10872.9i	0.344102	−	0.596002i
$694$		0			0
$695$	544.500	+	943.102i	0.0297181	+	0.0514732i
$696$		0			0
$697$	−5481.00	+	9493.37i	−0.297859	+	0.515907i
$698$		0			0
$699$			13686.7i			0.740597i
$700$		0			0
$701$	1122.00			0.0604527			0.0302264	−	0.999543i	$-0.490377\pi$
$701$	1122.00			0.0604527			0.0302264	+	0.999543i	$0.490377\pi$
$702$		0			0
$703$	−4648.00			−0.249364
$704$		0			0
$705$			8277.47i			0.442195i
$706$		0			0
$707$	−5998.50	+	10389.7i	−0.319090	+	0.552681i
$708$		0			0
$709$	−2141.50	−	3709.19i	−0.113435	−	0.196476i	0.803718	−	0.595011i	$-0.202852\pi$
$709$	−2141.50	−	3709.19i	−0.113435	−	0.196476i	−0.917153	+	0.398535i	$0.869519\pi$
$710$		0			0
$711$	−15295.5	−	26492.6i	−0.806788	−	1.39740i
$712$		0			0
$713$	19987.5	+	34619.4i	1.04984	+	1.81838i
$714$		0			0
$715$	−2497.50	+	4325.80i	−0.130631	+	0.226260i
$716$		0			0
$717$	31117.5	+	17965.7i	1.62079	+	0.935762i
$718$		0			0
$719$	−4032.00			−0.209135			−0.104568	−	0.994518i	$-0.533346\pi$
$719$	−4032.00			−0.209135			−0.104568	+	0.994518i	$0.533346\pi$
$720$		0			0
$721$	−17081.0			−0.882288
$722$		0			0
$723$	−6700.50	+	3868.54i	−0.344667	+	0.198994i
$724$		0			0
$725$	2442.00	−	4229.67i	0.125095	−	0.216670i
$726$		0			0
$727$	12002.5	+	20788.9i	0.612308	+	1.06055i	0.990850	+	0.134965i	$0.0430922\pi$
$727$	12002.5	+	20788.9i	0.612308	+	1.06055i	−0.378542	+	0.925584i	$0.623574\pi$
$728$		0			0
$729$	−19683.0			−1.00000
$730$		0			0
$731$	903.000	+	1564.04i	0.0456890	+	0.0791357i
$732$		0			0
$733$	18750.5	−	32476.8i	0.944837	−	1.63651i	0.188760	−	0.982023i	$-0.439553\pi$
$733$	18750.5	−	32476.8i	0.944837	−	1.63651i	0.756077	−	0.654482i	$-0.227113\pi$
$734$		0			0
$735$	−25029.0	+	14450.5i	−1.25607	+	0.725190i
$736$		0			0
$737$	6315.00			0.315626
$738$		0			0
$739$	880.000			0.0438042			0.0219021	−	0.999760i	$-0.493028\pi$
$739$	880.000			0.0438042			0.0219021	+	0.999760i	$0.493028\pi$
$740$		0			0
$741$	4662.00	+	2691.61i	0.231124	+	0.133439i
$742$		0			0
$743$	811.500	−	1405.56i	0.0400687	−	0.0694010i	−0.845296	−	0.534299i	$-0.820576\pi$
$743$	811.500	−	1405.56i	0.0400687	−	0.0694010i	0.885364	+	0.464898i	$0.153909\pi$
$744$		0			0
$745$	−12730.5	−	22049.9i	−0.626053	−	1.08436i
$746$		0			0
$747$	14620.5	+	25323.4i	0.716113	+	1.24034i
$748$		0			0
$749$	−186.000	−	322.161i	−0.00907382	−	0.0157163i
$750$		0			0
$751$	−3444.50	+	5966.05i	−0.167366	+	0.289886i	−0.937493	−	0.348005i	$-0.886859\pi$
$751$	−3444.50	+	5966.05i	−0.167366	+	0.289886i	0.770127	+	0.637890i	$0.220193\pi$
$752$		0			0
$753$			24006.2i			1.16180i
$754$		0			0
$755$	4149.00			0.199997
$756$		0			0
$757$	−12850.0			−0.616963			−0.308482	−	0.951230i	$-0.599821\pi$
$757$	−12850.0			−0.616963			−0.308482	+	0.951230i	$0.599821\pi$
$758$		0			0
$759$		−	15198.7i		−	0.726850i
$760$		0			0
$761$	−2305.50	+	3993.24i	−0.109822	+	0.190217i	−0.915698	−	0.401867i	$-0.868361\pi$
$761$	−2305.50	+	3993.24i	−0.109822	+	0.190217i	0.805876	+	0.592084i	$0.201695\pi$
$762$		0			0
$763$	7781.00	+	13477.1i	0.369189	+	0.639454i
$764$		0			0
$765$	5103.00	−	8838.66i	0.241176	−	0.417728i
$766$		0			0
$767$	−2941.50	−	5094.83i	−0.138476	−	0.239848i
$768$		0			0
$769$	1152.50	−	1996.19i	0.0540445	−	0.0936078i	−0.837737	−	0.546073i	$-0.816122\pi$
$769$	1152.50	−	1996.19i	0.0540445	−	0.0936078i	0.891782	+	0.452465i	$0.149455\pi$
$770$		0			0
$771$	−15079.5	−	8706.15i	−0.704377	−	0.406672i
$772$		0			0
$773$	−34902.0			−1.62398			−0.811991	−	0.583670i	$-0.801616\pi$
$773$	−34902.0			−1.62398			−0.811991	+	0.583670i	$0.801616\pi$
$774$		0			0
$775$	−9020.00			−0.418075
$776$		0			0
$777$	−23157.0	+	13369.7i	−1.06918	+	0.617291i
$778$		0			0
$779$	3654.00	−	6328.91i	0.168059	−	0.291087i
$780$		0			0
$781$	1170.00	+	2026.50i	0.0536055	+	0.0928474i
$782$		0			0
$783$	−13486.5	+	7786.43i	−0.615540	+	0.355382i
$784$		0			0
$785$	−13396.5	−	23203.4i	−0.609098	−	1.05499i
$786$

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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	144.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615i q^{3} +(4.50000 - 7.79423i) q^{5} +(-15.5000 - 26.8468i) q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q+5.19615i q^{3} +(4.50000 - 7.79423i) q^{5} +(-15.5000 - 26.8468i) q^{7} -27.0000 q^{9} +(-7.50000 - 12.9904i) q^{11} +(18.5000 - 32.0429i) q^{13} +(40.5000 + 23.3827i) q^{15} -42.0000 q^{17} +28.0000 q^{19} +(139.500 - 80.5404i) q^{21} +(97.5000 - 168.875i) q^{23} +(22.0000 + 38.1051i) q^{25} -140.296i q^{27} +(-55.5000 - 96.1288i) q^{29} +(-102.500 + 177.535i) q^{31} +(67.5000 - 38.9711i) q^{33} -279.000 q^{35} -166.000 q^{37} +(166.500 + 96.1288i) q^{39} +(130.500 - 226.033i) q^{41} +(-21.5000 - 37.2391i) q^{43} +(-121.500 + 210.444i) q^{45} +(88.5000 + 153.286i) q^{47} +(-309.000 + 535.204i) q^{49} -218.238i q^{51} +114.000 q^{53} -135.000 q^{55} +145.492i q^{57} +(79.5000 - 137.698i) q^{59} +(-95.5000 - 165.411i) q^{61} +(418.500 + 724.863i) q^{63} +(-166.500 - 288.386i) q^{65} +(-210.500 + 364.597i) q^{67} +(877.500 + 506.625i) q^{69} -156.000 q^{71} +182.000 q^{73} +(-198.000 + 114.315i) q^{75} +(-232.500 + 402.702i) q^{77} +(566.500 + 981.207i) q^{79} +729.000 q^{81} +(-541.500 - 937.906i) q^{83} +(-189.000 + 327.358i) q^{85} +(499.500 - 288.386i) q^{87} -1050.00 q^{89} -1147.00 q^{91} +(-922.500 - 532.606i) q^{93} +(126.000 - 218.238i) q^{95} +(450.500 + 780.289i) q^{97} +(202.500 + 350.740i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{5} - 31 q^{7} - 54 q^{9}+O(q^{10}) \) 2 * q + 9 * q^5 - 31 * q^7 - 54 * q^9	\( 2 q + 9 q^{5} - 31 q^{7} - 54 q^{9} - 15 q^{11} + 37 q^{13} + 81 q^{15} - 84 q^{17} + 56 q^{19} + 279 q^{21} + 195 q^{23} + 44 q^{25} - 111 q^{29} - 205 q^{31} + 135 q^{33} - 558 q^{35} - 332 q^{37} + 333 q^{39} + 261 q^{41} - 43 q^{43} - 243 q^{45} + 177 q^{47} - 618 q^{49} + 228 q^{53} - 270 q^{55} + 159 q^{59} - 191 q^{61} + 837 q^{63} - 333 q^{65} - 421 q^{67} + 1755 q^{69} - 312 q^{71} + 364 q^{73} - 396 q^{75} - 465 q^{77} + 1133 q^{79} + 1458 q^{81} - 1083 q^{83} - 378 q^{85} + 999 q^{87} - 2100 q^{89} - 2294 q^{91} - 1845 q^{93} + 252 q^{95} + 901 q^{97} + 405 q^{99}+O(q^{100}) \) 2 * q + 9 * q^5 - 31 * q^7 - 54 * q^9 - 15 * q^11 + 37 * q^13 + 81 * q^15 - 84 * q^17 + 56 * q^19 + 279 * q^21 + 195 * q^23 + 44 * q^25 - 111 * q^29 - 205 * q^31 + 135 * q^33 - 558 * q^35 - 332 * q^37 + 333 * q^39 + 261 * q^41 - 43 * q^43 - 243 * q^45 + 177 * q^47 - 618 * q^49 + 228 * q^53 - 270 * q^55 + 159 * q^59 - 191 * q^61 + 837 * q^63 - 333 * q^65 - 421 * q^67 + 1755 * q^69 - 312 * q^71 + 364 * q^73 - 396 * q^75 - 465 * q^77 + 1133 * q^79 + 1458 * q^81 - 1083 * q^83 - 378 * q^85 + 999 * q^87 - 2100 * q^89 - 2294 * q^91 - 1845 * q^93 + 252 * q^95 + 901 * q^97 + 405 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more