LMFDB - Embedded newform 432.4.i.a.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,4,Mod(145,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, 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("432.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	432.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:	$25.4888251225$
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			289.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	432.289
Dual form			432.4.i.a.145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 - 7.79423i) q^{5} +(-15.5000 + 26.8468i) q^{7} +O(q^{10})$	$q+(-4.50000 - 7.79423i) q^{5} +(-15.5000 + 26.8468i) q^{7} +(7.50000 - 12.9904i) q^{11} +(18.5000 + 32.0429i) q^{13} +42.0000 q^{17} +28.0000 q^{19} +(-97.5000 - 168.875i) q^{23} +(22.0000 - 38.1051i) q^{25} +(55.5000 - 96.1288i) q^{29} +(-102.500 - 177.535i) q^{31} +279.000 q^{35} -166.000 q^{37} +(-130.500 - 226.033i) q^{41} +(-21.5000 + 37.2391i) q^{43} +(-88.5000 + 153.286i) q^{47} +(-309.000 - 535.204i) q^{49} -114.000 q^{53} -135.000 q^{55} +(-79.5000 - 137.698i) q^{59} +(-95.5000 + 165.411i) q^{61} +(166.500 - 288.386i) q^{65} +(-210.500 - 364.597i) q^{67} +156.000 q^{71} +182.000 q^{73} +(232.500 + 402.702i) q^{77} +(566.500 - 981.207i) q^{79} +(541.500 - 937.906i) q^{83} +(-189.000 - 327.358i) q^{85} +1050.00 q^{89} -1147.00 q^{91} +(-126.000 - 218.238i) q^{95} +(450.500 - 780.289i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{5} - 31 q^{7}+O(q^{10}) $ 2 * q - 9 * q^5 - 31 * q^7	$ 2 q - 9 q^{5} - 31 q^{7} + 15 q^{11} + 37 q^{13} + 84 q^{17} + 56 q^{19} - 195 q^{23} + 44 q^{25} + 111 q^{29} - 205 q^{31} + 558 q^{35} - 332 q^{37} - 261 q^{41} - 43 q^{43} - 177 q^{47} - 618 q^{49} - 228 q^{53} - 270 q^{55} - 159 q^{59} - 191 q^{61} + 333 q^{65} - 421 q^{67} + 312 q^{71} + 364 q^{73} + 465 q^{77} + 1133 q^{79} + 1083 q^{83} - 378 q^{85} + 2100 q^{89} - 2294 q^{91} - 252 q^{95} + 901 q^{97}+O(q^{100}) $ 2 * q - 9 * q^5 - 31 * q^7 + 15 * q^11 + 37 * q^13 + 84 * q^17 + 56 * q^19 - 195 * q^23 + 44 * q^25 + 111 * q^29 - 205 * q^31 + 558 * q^35 - 332 * q^37 - 261 * q^41 - 43 * q^43 - 177 * q^47 - 618 * q^49 - 228 * q^53 - 270 * q^55 - 159 * q^59 - 191 * q^61 + 333 * q^65 - 421 * q^67 + 312 * q^71 + 364 * q^73 + 465 * q^77 + 1133 * q^79 + 1083 * q^83 - 378 * q^85 + 2100 * q^89 - 2294 * q^91 - 252 * q^95 + 901 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−4.50000	−	7.79423i	−0.402492	−	0.697137i	0.591534	−	0.806280i	$-0.298523\pi$
$5$	−4.50000	−	7.79423i	−0.402492	−	0.697137i	−0.994026	+	0.109143i	$0.965189\pi$
$6$		0			0
$7$	−15.5000	+	26.8468i	−0.836921	+	1.44959i	0.0555351	+	0.998457i	$0.482314\pi$
$7$	−15.5000	+	26.8468i	−0.836921	+	1.44959i	−0.892456	+	0.451134i	$0.851020\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.50000	−	12.9904i	0.205576	−	0.356068i	−0.744740	−	0.667355i	$-0.767427\pi$
$11$	7.50000	−	12.9904i	0.205576	−	0.356068i	0.950316	+	0.311287i	$0.100760\pi$
$12$		0			0
$13$	18.5000	+	32.0429i	0.394691	+	0.683624i	0.993062	−	0.117595i	$-0.0375185\pi$
$13$	18.5000	+	32.0429i	0.394691	+	0.683624i	−0.598371	+	0.801219i	$0.704185\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	42.0000			0.599206			0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000			0.599206			0.299603	+	0.954064i	$0.403146\pi$
$18$		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
$21$		0			0
$22$		0			0
$23$	−97.5000	−	168.875i	−0.883920	−	1.53099i	−0.846947	−	0.531678i	$-0.821562\pi$
$23$	−97.5000	−	168.875i	−0.883920	−	1.53099i	−0.0369731	−	0.999316i	$-0.511772\pi$
$24$		0			0
$25$	22.0000	−	38.1051i	0.176000	−	0.304841i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	55.5000	−	96.1288i	0.355382	−	0.615540i	−0.631801	−	0.775131i	$-0.717684\pi$
$29$	55.5000	−	96.1288i	0.355382	−	0.615540i	0.987183	+	0.159590i	$0.0510173\pi$
$30$		0			0
$31$	−102.500	−	177.535i	−0.593856	−	1.02859i	−0.993707	−	0.112009i	$-0.964271\pi$
$31$	−102.500	−	177.535i	−0.593856	−	1.02859i	0.399851	−	0.916580i	$-0.369062\pi$
$32$		0			0
$33$		0			0
$34$		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
$39$		0			0
$40$		0			0
$41$	−130.500	−	226.033i	−0.497090	−	0.860985i	0.502905	−	0.864342i	$-0.332265\pi$
$41$	−130.500	−	226.033i	−0.497090	−	0.860985i	−0.999994	+	0.00335732i	$0.998931\pi$
$42$		0			0
$43$	−21.5000	+	37.2391i	−0.0762493	+	0.132068i	−0.901629	−	0.432511i	$-0.857628\pi$
$43$	−21.5000	+	37.2391i	−0.0762493	+	0.132068i	0.825380	+	0.564578i	$0.190961\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−88.5000	+	153.286i	−0.274661	+	0.475726i	−0.970049	−	0.242907i	$-0.921899\pi$
$47$	−88.5000	+	153.286i	−0.274661	+	0.475726i	0.695389	+	0.718634i	$0.255232\pi$
$48$		0			0
$49$	−309.000	−	535.204i	−0.900875	−	1.56036i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−114.000			−0.295455			−0.147727	−	0.989028i	$-0.547196\pi$
$53$	−114.000			−0.295455			−0.147727	+	0.989028i	$0.547196\pi$
$54$		0			0
$55$	−135.000			−0.330971
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−79.5000	−	137.698i	−0.175424	−	0.303843i	0.764884	−	0.644168i	$-0.222796\pi$
$59$	−79.5000	−	137.698i	−0.175424	−	0.303843i	−0.940308	+	0.340325i	$0.889463\pi$
$60$		0			0
$61$	−95.5000	+	165.411i	−0.200451	+	0.347192i	−0.948674	−	0.316256i	$-0.897574\pi$
$61$	−95.5000	+	165.411i	−0.200451	+	0.347192i	0.748223	+	0.663448i	$0.230907\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	166.500	−	288.386i	0.317720	−	0.550307i
$66$		0			0
$67$	−210.500	−	364.597i	−0.383831	−	0.664815i	0.607775	−	0.794109i	$-0.292062\pi$
$67$	−210.500	−	364.597i	−0.383831	−	0.664815i	−0.991606	+	0.129294i	$0.958729\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	156.000			0.260758			0.130379	−	0.991464i	$-0.458381\pi$
$71$	156.000			0.260758			0.130379	+	0.991464i	$0.458381\pi$
$72$		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
$75$		0			0
$76$		0			0
$77$	232.500	+	402.702i	0.344102	+	0.596002i
$78$		0			0
$79$	566.500	−	981.207i	0.806788	−	1.39740i	−0.108290	−	0.994119i	$-0.534537\pi$
$79$	566.500	−	981.207i	0.806788	−	1.39740i	0.915078	−	0.403278i	$-0.132129\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	541.500	−	937.906i	0.716113	−	1.24034i	−0.246416	−	0.969164i	$-0.579253\pi$
$83$	541.500	−	937.906i	0.716113	−	1.24034i	0.962529	−	0.271179i	$-0.0874136\pi$
$84$		0			0
$85$	−189.000	−	327.358i	−0.241176	−	0.417728i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1050.00			1.25056			0.625280	−	0.780401i	$-0.284985\pi$
$89$	1050.00			1.25056			0.625280	+	0.780401i	$0.284985\pi$
$90$		0			0
$91$	−1147.00			−1.32130
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−126.000	−	218.238i	−0.136077	−	0.235693i
$96$		0			0
$97$	450.500	−	780.289i	0.471560	−	0.816766i	−0.527910	−	0.849300i	$-0.677024\pi$
$97$	450.500	−	780.289i	0.471560	−	0.816766i	0.999471	+	0.0325338i	$0.0103576\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	193.500	−	335.152i	0.190633	−	0.330187i	−0.754827	−	0.655924i	$-0.772279\pi$
$101$	193.500	−	335.152i	0.190633	−	0.330187i	0.945460	+	0.325737i	$0.105613\pi$
$102$		0			0
$103$	275.500	+	477.180i	0.263552	+	0.456485i	0.967183	−	0.254080i	$-0.0817727\pi$
$103$	275.500	+	477.180i	0.263552	+	0.456485i	−0.703631	+	0.710565i	$0.748439\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−0.0108419			−0.00542095	−	0.999985i	$-0.501726\pi$
$107$	−12.0000			−0.0108419			−0.00542095	+	0.999985i	$0.501726\pi$
$108$		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
$111$		0			0
$112$		0			0
$113$	−700.500	−	1213.30i	−0.583164	−	1.01007i	−0.995102	−	0.0988572i	$-0.968481\pi$
$113$	−700.500	−	1213.30i	−0.583164	−	1.01007i	0.411938	−	0.911212i	$-0.364852\pi$
$114$		0			0
$115$	−877.500	+	1519.87i	−0.711542	+	1.23243i
$116$		0			0
$117$		0			0
$118$		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
$123$		0			0
$124$		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
$129$		0			0
$130$		0			0
$131$	−751.500	−	1301.64i	−0.501213	−	0.868126i	−0.999999	−	0.00140084i	$-0.999554\pi$
$131$	−751.500	−	1301.64i	−0.501213	−	0.868126i	0.498786	−	0.866725i	$-0.333779\pi$
$132$		0			0
$133$	−434.000	+	751.710i	−0.282952	+	0.490087i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1330.50	+	2304.49i	−0.829725	+	1.43713i	0.0685295	+	0.997649i	$0.478169\pi$
$137$	−1330.50	+	2304.49i	−0.829725	+	1.43713i	−0.898254	+	0.439476i	$0.855164\pi$
$138$		0			0
$139$	−60.5000	−	104.789i	−0.0369176	−	0.0639431i	0.846976	−	0.531631i	$-0.178421\pi$
$139$	−60.5000	−	104.789i	−0.0369176	−	0.0639431i	−0.883894	+	0.467688i	$0.845087\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	555.000			0.324555
$144$		0			0
$145$	−999.000			−0.572155
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1414.50	−	2449.99i	−0.777721	−	1.34705i	−0.933253	−	0.359221i	$-0.883043\pi$
$149$	−1414.50	−	2449.99i	−0.777721	−	1.34705i	0.155532	−	0.987831i	$-0.450291\pi$
$150$		0			0
$151$	230.500	−	399.238i	0.124224	−	0.215162i	−0.797205	−	0.603708i	$-0.793689\pi$
$151$	230.500	−	399.238i	0.124224	−	0.215162i	0.921429	+	0.388546i	$0.127023\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−922.500	+	1597.82i	−0.478045	+	0.827998i
$156$		0			0
$157$	1488.50	+	2578.16i	0.756658	+	1.31057i	0.944546	+	0.328379i	$0.106502\pi$
$157$	1488.50	+	2578.16i	0.756658	+	1.31057i	−0.187889	+	0.982190i	$0.560164\pi$
$158$		0			0
$159$		0			0
$160$		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
$165$		0			0
$166$		0			0
$167$	340.500	+	589.763i	0.157777	+	0.273277i	0.934067	−	0.357099i	$-0.116234\pi$
$167$	340.500	+	589.763i	0.157777	+	0.273277i	−0.776290	+	0.630376i	$0.782901\pi$
$168$		0			0
$169$	414.000	−	717.069i	0.188439	−	0.326386i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1990.50	+	3447.65i	−0.874768	+	1.51514i	−0.0177589	+	0.999842i	$0.505653\pi$
$173$	−1990.50	+	3447.65i	−0.874768	+	1.51514i	−0.857009	+	0.515301i	$0.827680\pi$
$174$		0			0
$175$	682.000	+	1181.26i	0.294596	+	0.510256i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2004.00			0.836793			0.418397	−	0.908264i	$-0.362592\pi$
$179$	2004.00			0.836793			0.418397	+	0.908264i	$0.362592\pi$
$180$		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
$183$		0			0
$184$		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
$189$		0			0
$190$		0			0
$191$	−580.500	+	1005.46i	−0.219914	+	0.380902i	−0.954781	−	0.297309i	$-0.903911\pi$
$191$	−580.500	+	1005.46i	−0.219914	+	0.380902i	0.734868	+	0.678210i	$0.237244\pi$
$192$		0			0
$193$	−1805.50	−	3127.22i	−0.673382	−	1.16633i	−0.976939	−	0.213519i	$-0.931508\pi$
$193$	−1805.50	−	3127.22i	−0.673382	−	1.16633i	0.303557	−	0.952813i	$-0.401826\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2046.00			−0.739957			−0.369978	−	0.929040i	$-0.620635\pi$
$197$	−2046.00			−0.739957			−0.369978	+	0.929040i	$0.620635\pi$
$198$		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
$201$		0			0
$202$		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
$207$		0			0
$208$		0			0
$209$	210.000	−	363.731i	0.0695024	−	0.120382i
$210$		0			0
$211$	377.500	+	653.849i	0.123167	+	0.213331i	0.921015	−	0.389528i	$-0.127362\pi$
$211$	377.500	+	653.849i	0.123167	+	0.213331i	−0.797848	+	0.602858i	$0.794028\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	387.000			0.122759
$216$		0			0
$217$	6355.00			1.98804
$218$		0			0
$219$		0			0
$220$		0			0
$221$	777.000	+	1345.80i	0.236501	+	0.409631i
$222$		0			0
$223$	−1731.50	+	2999.05i	−0.519954	+	0.900587i	0.479777	+	0.877391i	$0.340718\pi$
$223$	−1731.50	+	2999.05i	−0.519954	+	0.900587i	−0.999731	+	0.0231966i	$0.992616\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3112.50	+	5391.01i	−0.910061	+	1.57627i	−0.0960856	+	0.995373i	$0.530632\pi$
$227$	−3112.50	+	5391.01i	−0.910061	+	1.57627i	−0.813976	+	0.580899i	$0.802701\pi$
$228$		0			0
$229$	732.500	+	1268.73i	0.211375	+	0.366113i	0.952145	−	0.305646i	$-0.0988724\pi$
$229$	732.500	+	1268.73i	0.211375	+	0.366113i	−0.740770	+	0.671759i	$0.765539\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2634.00			−0.740597			−0.370298	−	0.928913i	$-0.620745\pi$
$233$	−2634.00			−0.740597			−0.370298	+	0.928913i	$0.620745\pi$
$234$		0			0
$235$	1593.00			0.442195
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3457.50	−	5988.57i	−0.935762	−	1.62079i	−0.773270	−	0.634077i	$-0.781380\pi$
$239$	−3457.50	−	5988.57i	−0.935762	−	1.62079i	−0.162492	−	0.986710i	$-0.551953\pi$
$240$		0			0
$241$	744.500	−	1289.51i	0.198994	−	0.344667i	−0.749209	−	0.662334i	$-0.769566\pi$
$241$	744.500	−	1289.51i	0.198994	−	0.344667i	0.948202	+	0.317667i	$0.102899\pi$
$242$		0			0
$243$		0			0
$244$		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
$249$		0			0
$250$		0			0
$251$	−4620.00			−1.16180			−0.580900	−	0.813975i	$-0.697299\pi$
$251$	−4620.00			−1.16180			−0.580900	+	0.813975i	$0.697299\pi$
$252$		0			0
$253$	−2925.00			−0.726850
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1675.50	+	2902.05i	0.406672	+	0.704377i	0.994515	−	0.104598i	$-0.0333557\pi$
$257$	1675.50	+	2902.05i	0.406672	+	0.704377i	−0.587842	+	0.808976i	$0.700022\pi$
$258$		0			0
$259$	2573.00	−	4456.57i	0.617291	−	1.06918i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	301.500	−	522.213i	0.0706893	−	0.122437i	−0.828514	−	0.559968i	$-0.810813\pi$
$263$	301.500	−	522.213i	0.0706893	−	0.122437i	0.899204	+	0.437530i	$0.144147\pi$
$264$		0			0
$265$	513.000	+	888.542i	0.118918	+	0.205972i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1470.00			0.333188			0.166594	−	0.986026i	$-0.446723\pi$
$269$	1470.00			0.333188			0.166594	+	0.986026i	$0.446723\pi$
$270$		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
$273$		0			0
$274$		0			0
$275$	−330.000	−	571.577i	−0.0723627	−	0.125336i
$276$		0			0
$277$	−3569.50	+	6182.56i	−0.774262	+	1.34106i	0.160947	+	0.986963i	$0.448545\pi$
$277$	−3569.50	+	6182.56i	−0.774262	+	1.34106i	−0.935209	+	0.354097i	$0.884788\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4213.50	−	7298.00i	0.894507	−	1.54933i	0.0600924	−	0.998193i	$-0.480860\pi$
$281$	4213.50	−	7298.00i	0.894507	−	1.54933i	0.834414	−	0.551138i	$-0.185806\pi$
$282$		0			0
$283$	−228.500	−	395.774i	−0.0479962	−	0.0831318i	0.841029	−	0.540990i	$-0.181950\pi$
$283$	−228.500	−	395.774i	−0.0479962	−	0.0831318i	−0.889025	+	0.457858i	$0.848617\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8091.00			1.66410
$288$		0			0
$289$	−3149.00			−0.640953
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2944.50	−	5100.02i	−0.587097	−	1.01688i	−0.994610	−	0.103683i	$-0.966937\pi$
$293$	−2944.50	−	5100.02i	−0.587097	−	1.01688i	0.407513	−	0.913199i	$-0.366396\pi$
$294$		0			0
$295$	−715.500	+	1239.28i	−0.141214	+	0.244589i
$296$		0			0
$297$		0			0
$298$		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
$303$		0			0
$304$		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
$309$		0			0
$310$		0			0
$311$	1642.50	+	2844.89i	0.299478	+	0.518711i	0.976017	−	0.217696i	$-0.0698542\pi$
$311$	1642.50	+	2844.89i	0.299478	+	0.518711i	−0.676539	+	0.736407i	$0.736521\pi$
$312$		0			0
$313$	5028.50	−	8709.62i	0.908075	−	1.57283i	0.0913406	−	0.995820i	$-0.470885\pi$
$313$	5028.50	−	8709.62i	0.908075	−	1.57283i	0.816735	−	0.577013i	$-0.195782\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1147.50	−	1987.53i	0.203312	−	0.352147i	−0.746281	−	0.665631i	$-0.768163\pi$
$317$	1147.50	−	1987.53i	0.203312	−	0.352147i	0.949594	+	0.313483i	$0.101496\pi$
$318$		0			0
$319$	−832.500	−	1441.93i	−0.146116	−	0.253081i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1176.00			0.202583
$324$		0			0
$325$	1628.00			0.277862
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2743.50	−	4751.88i	−0.459739	−	0.796291i
$330$		0			0
$331$	−3339.50	+	5784.18i	−0.554548	+	0.960506i	0.443390	+	0.896329i	$0.353776\pi$
$331$	−3339.50	+	5784.18i	−0.554548	+	0.960506i	−0.997939	+	0.0641773i	$0.979558\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1894.50	+	3281.37i	−0.308978	+	0.535165i
$336$		0			0
$337$	−1091.50	−	1890.53i	−0.176433	−	0.305590i	0.764224	−	0.644951i	$-0.223122\pi$
$337$	−1091.50	−	1890.53i	−0.176433	−	0.305590i	−0.940656	+	0.339361i	$0.889789\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3075.00			−0.488330
$342$		0			0
$343$	8525.00			1.34200
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1945.50	−	3369.70i	−0.300980	−	0.521312i	0.675379	−	0.737471i	$-0.263980\pi$
$347$	−1945.50	−	3369.70i	−0.300980	−	0.521312i	−0.976358	+	0.216159i	$0.930647\pi$
$348$		0			0
$349$	−1397.50	+	2420.54i	−0.214345	+	0.371257i	−0.953070	−	0.302751i	$-0.902095\pi$
$349$	−1397.50	+	2420.54i	−0.214345	+	0.371257i	0.738725	+	0.674007i	$0.235428\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2377.50	−	4117.95i	0.358475	−	0.620896i	−0.629232	−	0.777218i	$-0.716630\pi$
$353$	2377.50	−	4117.95i	0.358475	−	0.620896i	0.987706	+	0.156322i	$0.0499636\pi$
$354$		0			0
$355$	−702.000	−	1215.90i	−0.104953	−	0.181784i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4608.00			0.677440			0.338720	−	0.940887i	$-0.390006\pi$
$359$	4608.00			0.677440			0.338720	+	0.940887i	$0.390006\pi$
$360$		0			0
$361$	−6075.00			−0.885698
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−819.000	−	1418.55i	−0.117448	−	0.203425i
$366$		0			0
$367$	1922.50	−	3329.87i	0.273443	−	0.473618i	−0.696298	−	0.717753i	$-0.745171\pi$
$367$	1922.50	−	3329.87i	0.273443	−	0.473618i	0.969741	+	0.244135i	$0.0785041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1767.00	−	3060.53i	0.247272	−	0.428288i
$372$		0			0
$373$	4158.50	+	7202.73i	0.577263	+	0.999848i	0.995792	+	0.0916449i	$0.0292125\pi$
$373$	4158.50	+	7202.73i	0.577263	+	0.999848i	−0.418529	+	0.908203i	$0.637454\pi$
$374$		0			0
$375$		0			0
$376$		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
$381$		0			0
$382$		0			0
$383$	−6043.50	−	10467.6i	−0.806288	−	1.39653i	−0.915418	−	0.402505i	$-0.868140\pi$
$383$	−6043.50	−	10467.6i	−0.806288	−	1.39653i	0.109130	−	0.994028i	$-0.465194\pi$
$384$		0			0
$385$	2092.50	−	3624.32i	0.276997	−	0.479772i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4270.50	+	7396.72i	−0.556614	+	0.964084i	0.441162	+	0.897428i	$0.354567\pi$
$389$	−4270.50	+	7396.72i	−0.556614	+	0.964084i	−0.997776	+	0.0666565i	$0.978767\pi$
$390$		0			0
$391$	−4095.00	−	7092.75i	−0.529650	−	0.917380i
$392$		0			0
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$	4801.50	+	8316.44i	0.597944	+	1.03567i	0.993124	+	0.117066i	$0.0373488\pi$
$401$	4801.50	+	8316.44i	0.597944	+	1.03567i	−0.395180	+	0.918604i	$0.629318\pi$
$402$		0			0
$403$	3792.50	−	6568.80i	0.468779	−	0.811949i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1245.00	+	2156.40i	−0.151627	+	0.262626i
$408$		0			0
$409$	−5735.50	−	9934.18i	−0.693404	−	1.20101i	−0.970716	−	0.240231i	$-0.922777\pi$
$409$	−5735.50	−	9934.18i	−0.693404	−	1.20101i	0.277312	−	0.960780i	$-0.410557\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4929.00			0.587264
$414$		0			0
$415$	−9747.00			−1.15292
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2986.50	+	5172.77i	0.348210	+	0.603118i	0.985932	−	0.167149i	$-0.0534562\pi$
$419$	2986.50	+	5172.77i	0.348210	+	0.603118i	−0.637721	+	0.770267i	$0.720123\pi$
$420$		0			0
$421$	4452.50	−	7711.96i	0.515443	−	0.892774i	−0.484396	−	0.874849i	$-0.660961\pi$
$421$	4452.50	−	7711.96i	0.515443	−	0.892774i	0.999839	−	0.0179250i	$-0.00570601\pi$
$422$		0			0
$423$		0			0
$424$		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
$429$		0			0
$430$		0			0
$431$	1416.00			0.158251			0.0791257	−	0.996865i	$-0.474787\pi$
$431$	1416.00			0.158251			0.0791257	+	0.996865i	$0.474787\pi$
$432$		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
$435$		0			0
$436$		0			0
$437$	−2730.00	−	4728.50i	−0.298841	−	0.517608i
$438$		0			0
$439$	2174.50	−	3766.34i	0.236408	−	0.409471i	−0.723273	−	0.690562i	$-0.757363\pi$
$439$	2174.50	−	3766.34i	0.236408	−	0.409471i	0.959681	+	0.281091i	$0.0906964\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7273.50	−	12598.1i	0.780078	−	1.35113i	−0.151818	−	0.988408i	$-0.548513\pi$
$443$	7273.50	−	12598.1i	0.780078	−	1.35113i	0.931896	−	0.362726i	$-0.118154\pi$
$444$		0			0
$445$	−4725.00	−	8183.94i	−0.503340	−	0.871811i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3330.00			0.350005			0.175003	−	0.984568i	$-0.444007\pi$
$449$	3330.00			0.350005			0.175003	+	0.984568i	$0.444007\pi$
$450$		0			0
$451$	−3915.00			−0.408759
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5161.50	+	8939.98i	0.531813	+	0.921127i
$456$		0			0
$457$	−4073.50	+	7055.51i	−0.416959	+	0.722194i	−0.995632	−	0.0933655i	$-0.970237\pi$
$457$	−4073.50	+	7055.51i	−0.416959	+	0.722194i	0.578673	+	0.815560i	$0.303571\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4015.50	−	6955.05i	0.405684	−	0.702666i	−0.588717	−	0.808340i	$-0.700367\pi$
$461$	4015.50	−	6955.05i	0.405684	−	0.702666i	0.994401	+	0.105674i	$0.0336999\pi$
$462$		0			0
$463$	2141.50	+	3709.19i	0.214955	+	0.372312i	0.953259	−	0.302156i	$-0.0977063\pi$
$463$	2141.50	+	3709.19i	0.214955	+	0.372312i	−0.738304	+	0.674468i	$0.764373\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5460.00			0.541025			0.270512	−	0.962716i	$-0.412807\pi$
$467$	5460.00			0.541025			0.270512	+	0.962716i	$0.412807\pi$
$468$		0			0
$469$	13051.0			1.28494
$470$		0			0
$471$		0			0
$472$		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
$477$		0			0
$478$		0			0
$479$	−214.500	+	371.525i	−0.0204609	+	0.0354393i	−0.876075	−	0.482176i	$-0.839847\pi$
$479$	−214.500	+	371.525i	−0.0204609	+	0.0354393i	0.855614	+	0.517615i	$0.173180\pi$
$480$		0			0
$481$	−3071.00	−	5319.13i	−0.291113	−	0.504223i
$482$		0			0
$483$		0			0
$484$		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
$489$		0			0
$490$		0			0
$491$	7336.50	+	12707.2i	0.674321	+	1.16796i	0.976667	+	0.214760i	$0.0688969\pi$
$491$	7336.50	+	12707.2i	0.674321	+	1.16796i	−0.302346	+	0.953198i	$0.597770\pi$
$492$		0			0
$493$	2331.00	−	4037.41i	0.212947	−	0.368835i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2418.00	+	4188.10i	−0.218234	+	0.377992i
$498$		0			0
$499$	6719.50	+	11638.5i	0.602818	+	1.04411i	0.992392	+	0.123116i	$0.0392888\pi$
$499$	6719.50	+	11638.5i	0.602818	+	1.04411i	−0.389574	+	0.920995i	$0.627378\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17388.0			−1.54134			−0.770669	−	0.637236i	$-0.780078\pi$
$503$	−17388.0			−1.54134			−0.770669	+	0.637236i	$0.780078\pi$
$504$		0			0
$505$	−3483.00			−0.306914
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1894.50	−	3281.37i	−0.164975	−	0.285745i	0.771671	−	0.636021i	$-0.219421\pi$
$509$	−1894.50	−	3281.37i	−0.164975	−	0.285745i	−0.936646	+	0.350276i	$0.886088\pi$
$510$		0			0
$511$	−2821.00	+	4886.12i	−0.244215	+	0.422992i
$512$		0			0
$513$		0			0
$514$		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
$519$		0			0
$520$		0			0
$521$	−9786.00			−0.822903			−0.411451	−	0.911432i	$-0.634978\pi$
$521$	−9786.00			−0.822903			−0.411451	+	0.911432i	$0.634978\pi$
$522$		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
$525$		0			0
$526$		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
$531$		0			0
$532$		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
$537$		0			0
$538$		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
$543$		0			0
$544$		0			0
$545$	2259.00	+	3912.70i	0.177550	+	0.307526i
$546$		0			0
$547$	−5187.50	+	8985.01i	−0.405487	+	0.702324i	−0.994378	−	0.105888i	$-0.966231\pi$
$547$	−5187.50	+	8985.01i	−0.405487	+	0.702324i	0.588891	+	0.808213i	$0.299565\pi$
$548$		0			0
$549$		0			0
$550$		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
$555$		0			0
$556$		0			0
$557$	−3306.00			−0.251490			−0.125745	−	0.992063i	$-0.540132\pi$
$557$	−3306.00			−0.251490			−0.125745	+	0.992063i	$0.540132\pi$
$558$		0			0
$559$	−1591.00			−0.120379
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10546.5	+	18267.1i	0.789488	+	1.36743i	0.926281	+	0.376834i	$0.122987\pi$
$563$	10546.5	+	18267.1i	0.789488	+	1.36743i	−0.136792	+	0.990600i	$0.543679\pi$
$564$		0			0
$565$	−6304.50	+	10919.7i	−0.469438	+	0.813090i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	643.500	−	1114.57i	0.0474111	−	0.0821185i	−0.841346	−	0.540497i	$-0.818236\pi$
$569$	643.500	−	1114.57i	0.0474111	−	0.0821185i	0.888757	+	0.458379i	$0.151570\pi$
$570$		0			0
$571$	7517.50	+	13020.7i	0.550959	+	0.954289i	0.998206	+	0.0598783i	$0.0190713\pi$
$571$	7517.50	+	13020.7i	0.550959	+	0.954289i	−0.447247	+	0.894411i	$0.647595\pi$
$572$		0			0
$573$		0			0
$574$		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
$579$		0			0
$580$		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
$585$		0			0
$586$		0			0
$587$	8941.50	−	15487.1i	0.628714	−	1.08896i	−0.359096	−	0.933301i	$-0.616915\pi$
$587$	8941.50	−	15487.1i	0.628714	−	1.08896i	0.987810	−	0.155664i	$-0.0497518\pi$
$588$		0			0
$589$	−2870.00	−	4970.99i	−0.200775	−	0.347752i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−20118.0			−1.39317			−0.696583	−	0.717476i	$-0.745297\pi$
$593$	−20118.0			−1.39317			−0.696583	+	0.717476i	$0.745297\pi$
$594$		0			0
$595$	11718.0			0.807380
$596$		0			0
$597$		0			0
$598$		0			0
$599$	532.500	+	922.317i	0.0363228	+	0.0629129i	0.883615	−	0.468214i	$-0.155102\pi$
$599$	532.500	+	922.317i	0.0363228	+	0.0629129i	−0.847293	+	0.531127i	$0.821769\pi$
$600$		0			0
$601$	10362.5	−	17948.4i	0.703320	−	1.21819i	−0.263975	−	0.964530i	$-0.585034\pi$
$601$	10362.5	−	17948.4i	0.703320	−	1.21819i	0.967294	−	0.253656i	$-0.0816331\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4977.00	−	8620.42i	0.334453	−	0.579289i
$606$		0			0
$607$	−7872.50	−	13635.6i	−0.526417	−	0.911780i	−0.999526	−	0.0307768i	$-0.990202\pi$
$607$	−7872.50	−	13635.6i	−0.526417	−	0.911780i	0.473110	−	0.881004i	$-0.343131\pi$
$608$		0			0
$609$		0			0
$610$		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
$615$		0			0
$616$		0			0
$617$	−5026.50	−	8706.15i	−0.327973	−	0.568066i	0.654137	−	0.756376i	$-0.273032\pi$
$617$	−5026.50	−	8706.15i	−0.327973	−	0.568066i	−0.982110	+	0.188311i	$0.939699\pi$
$618$		0			0
$619$	−2991.50	+	5181.43i	−0.194246	+	0.336445i	−0.946653	−	0.322254i	$-0.895559\pi$
$619$	−2991.50	+	5181.43i	−0.194246	+	0.336445i	0.752407	+	0.658699i	$0.228893\pi$
$620$		0			0
$621$		0			0
$622$		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
$627$		0			0
$628$		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
$633$		0			0
$634$		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
$639$		0			0
$640$		0			0
$641$	−5488.50	+	9506.36i	−0.338195	+	0.585770i	−0.984093	−	0.177653i	$-0.943150\pi$
$641$	−5488.50	+	9506.36i	−0.338195	+	0.585770i	0.645899	+	0.763423i	$0.276483\pi$
$642$		0			0
$643$	−7914.50	−	13708.3i	−0.485408	−	0.840752i	0.514451	−	0.857520i	$-0.327996\pi$
$643$	−7914.50	−	13708.3i	−0.485408	−	0.840752i	−0.999859	+	0.0167681i	$0.994662\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	28224.0			1.71499			0.857496	−	0.514490i	$-0.172019\pi$
$647$	28224.0			1.71499			0.857496	+	0.514490i	$0.172019\pi$
$648$		0			0
$649$	−2385.00			−0.144252
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14083.5	+	24393.3i	0.843997	+	1.46185i	0.886490	+	0.462749i	$0.153137\pi$
$653$	14083.5	+	24393.3i	0.843997	+	1.46185i	−0.0424927	+	0.999097i	$0.513530\pi$
$654$		0			0
$655$	−6763.50	+	11714.7i	−0.403468	+	0.698828i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5368.50	+	9298.51i	−0.317340	+	0.549649i	−0.979932	−	0.199331i	$-0.936123\pi$
$659$	−5368.50	+	9298.51i	−0.317340	+	0.549649i	0.662592	+	0.748980i	$0.269456\pi$
$660$		0			0
$661$	−5063.50	−	8770.24i	−0.297954	−	0.516071i	0.677714	−	0.735326i	$-0.262971\pi$
$661$	−5063.50	−	8770.24i	−0.297954	−	0.516071i	−0.975668	+	0.219255i	$0.929637\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7812.00			0.455543
$666$		0			0
$667$	−21645.0			−1.25652
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1432.50	+	2481.16i	0.0824159	+	0.142748i
$672$		0			0
$673$	−125.500	+	217.372i	−0.00718822	+	0.0124504i	−0.869597	−	0.493762i	$-0.835621\pi$
$673$	−125.500	+	217.372i	−0.00718822	+	0.0124504i	0.862409	+	0.506212i	$0.168955\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4225.50	−	7318.78i	0.239881	−	0.415485i	−0.720799	−	0.693144i	$-0.756225\pi$
$677$	4225.50	−	7318.78i	0.239881	−	0.415485i	0.960680	+	0.277659i	$0.0895584\pi$
$678$		0			0
$679$	13965.5	+	24189.0i	0.789318	+	1.36714i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−25884.0			−1.45011			−0.725054	−	0.688692i	$-0.758185\pi$
$683$	−25884.0			−1.45011			−0.725054	+	0.688692i	$0.758185\pi$
$684$		0			0
$685$	23949.0			1.33583
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2109.00	−	3652.90i	−0.116613	−	0.201980i
$690$		0			0
$691$	3182.50	−	5512.25i	0.175207	−	0.303467i	−0.765026	−	0.643999i	$-0.777274\pi$
$691$	3182.50	−	5512.25i	0.175207	−	0.303467i	0.940233	+	0.340532i	$0.110607\pi$
$692$		0			0
$693$		0			0
$694$		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
$699$		0			0
$700$		0			0
$701$	−1122.00			−0.0604527			−0.0302264	−	0.999543i	$-0.509623\pi$
$701$	−1122.00			−0.0604527			−0.0302264	+	0.999543i	$0.509623\pi$
$702$		0			0
$703$	−4648.00			−0.249364
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5998.50	+	10389.7i	0.319090	+	0.552681i
$708$		0			0
$709$	−2141.50	+	3709.19i	−0.113435	+	0.196476i	−0.917153	−	0.398535i	$-0.869519\pi$
$709$	−2141.50	+	3709.19i	−0.113435	+	0.196476i	0.803718	+	0.595011i	$0.202852\pi$
$710$		0			0
$711$		0			0
$712$		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
$717$		0			0
$718$		0			0
$719$	4032.00			0.209135			0.104568	−	0.994518i	$-0.466654\pi$
$719$	4032.00			0.209135			0.104568	+	0.994518i	$0.466654\pi$
$720$		0			0
$721$	−17081.0			−0.882288
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2442.00	−	4229.67i	−0.125095	−	0.216670i
$726$		0			0
$727$	12002.5	−	20788.9i	0.612308	−	1.06055i	−0.378542	−	0.925584i	$-0.623574\pi$
$727$	12002.5	−	20788.9i	0.612308	−	1.06055i	0.990850	−	0.134965i	$-0.0430922\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−903.000	+	1564.04i	−0.0456890	+	0.0791357i
$732$		0			0
$733$	18750.5	+	32476.8i	0.944837	+	1.63651i	0.756077	+	0.654482i	$0.227113\pi$
$733$	18750.5	+	32476.8i	0.944837	+	1.63651i	0.188760	+	0.982023i	$0.439553\pi$
$734$		0			0
$735$		0			0
$736$		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
$741$		0			0
$742$		0			0
$743$	−811.500	−	1405.56i	−0.0400687	−	0.0694010i	0.845296	−	0.534299i	$-0.179424\pi$
$743$	−811.500	−	1405.56i	−0.0400687	−	0.0694010i	−0.885364	+	0.464898i	$0.846091\pi$
$744$		0			0
$745$	−12730.5	+	22049.9i	−0.626053	+	1.08436i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	186.000	−	322.161i	0.00907382	−	0.0157163i
$750$		0			0
$751$	−3444.50	−	5966.05i	−0.167366	−	0.289886i	0.770127	−	0.637890i	$-0.220193\pi$
$751$	−3444.50	−	5966.05i	−0.167366	−	0.289886i	−0.937493	+	0.348005i	$0.886859\pi$
$752$		0			0
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$	2305.50	+	3993.24i	0.109822	+	0.190217i	0.915698	−	0.401867i	$-0.131639\pi$
$761$	2305.50	+	3993.24i	0.109822	+	0.190217i	−0.805876	+	0.592084i	$0.798305\pi$
$762$		0			0
$763$	7781.00	−	13477.1i	0.369189	−	0.639454i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2941.50	−	5094.83i	0.138476	−	0.239848i
$768$		0			0
$769$	1152.50	+	1996.19i	0.0540445	+	0.0936078i	0.891782	−	0.452465i	$-0.149455\pi$
$769$	1152.50	+	1996.19i	0.0540445	+	0.0936078i	−0.837737	+	0.546073i	$0.816122\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	34902.0			1.62398			0.811991	−	0.583670i	$-0.198384\pi$
$773$	34902.0			1.62398			0.811991	+	0.583670i	$0.198384\pi$
$774$		0			0
$775$	−9020.00			−0.418075
$776$		0			0
$777$		0			0
$778$		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
$783$		0			0
$784$		0			0
$785$	13396.5	−	23203.4i	0.609098	−	1.05499i
$786$		0			0
$787$	13127.5	+	22737.5i	0.594593	+	1.02987i	0.993604	+	0.112919i	$0.0360202\pi$
$787$	13127.5	+	22737.5i	0.594593	+	1.02987i	−0.399011	+	0.916946i	$0.630646\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	43431.0			1.95225
$792$		0			0

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	432.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 7.79423i) q^{5} +(-15.5000 + 26.8468i) q^{7} +O(q^{10})\)	\(q+(-4.50000 - 7.79423i) q^{5} +(-15.5000 + 26.8468i) q^{7} +(7.50000 - 12.9904i) q^{11} +(18.5000 + 32.0429i) q^{13} +42.0000 q^{17} +28.0000 q^{19} +(-97.5000 - 168.875i) q^{23} +(22.0000 - 38.1051i) q^{25} +(55.5000 - 96.1288i) q^{29} +(-102.500 - 177.535i) q^{31} +279.000 q^{35} -166.000 q^{37} +(-130.500 - 226.033i) q^{41} +(-21.5000 + 37.2391i) q^{43} +(-88.5000 + 153.286i) q^{47} +(-309.000 - 535.204i) q^{49} -114.000 q^{53} -135.000 q^{55} +(-79.5000 - 137.698i) q^{59} +(-95.5000 + 165.411i) q^{61} +(166.500 - 288.386i) q^{65} +(-210.500 - 364.597i) q^{67} +156.000 q^{71} +182.000 q^{73} +(232.500 + 402.702i) q^{77} +(566.500 - 981.207i) q^{79} +(541.500 - 937.906i) q^{83} +(-189.000 - 327.358i) q^{85} +1050.00 q^{89} -1147.00 q^{91} +(-126.000 - 218.238i) q^{95} +(450.500 - 780.289i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{5} - 31 q^{7}+O(q^{10}) \) 2 * q - 9 * q^5 - 31 * q^7	\( 2 q - 9 q^{5} - 31 q^{7} + 15 q^{11} + 37 q^{13} + 84 q^{17} + 56 q^{19} - 195 q^{23} + 44 q^{25} + 111 q^{29} - 205 q^{31} + 558 q^{35} - 332 q^{37} - 261 q^{41} - 43 q^{43} - 177 q^{47} - 618 q^{49} - 228 q^{53} - 270 q^{55} - 159 q^{59} - 191 q^{61} + 333 q^{65} - 421 q^{67} + 312 q^{71} + 364 q^{73} + 465 q^{77} + 1133 q^{79} + 1083 q^{83} - 378 q^{85} + 2100 q^{89} - 2294 q^{91} - 252 q^{95} + 901 q^{97}+O(q^{100}) \) 2 * q - 9 * q^5 - 31 * q^7 + 15 * q^11 + 37 * q^13 + 84 * q^17 + 56 * q^19 - 195 * q^23 + 44 * q^25 + 111 * q^29 - 205 * q^31 + 558 * q^35 - 332 * q^37 - 261 * q^41 - 43 * q^43 - 177 * q^47 - 618 * q^49 - 228 * q^53 - 270 * q^55 - 159 * q^59 - 191 * q^61 + 333 * q^65 - 421 * q^67 + 312 * q^71 + 364 * q^73 + 465 * q^77 + 1133 * q^79 + 1083 * q^83 - 378 * q^85 + 2100 * q^89 - 2294 * q^91 - 252 * q^95 + 901 * q^97

Introduction

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