LMFDB - Embedded newform 432.4.i.a.145.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,4,Mod(145,432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			145.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	432.145
Dual form			432.4.i.a.289.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 7.79423i) q^{5} +(-15.5000 - 26.8468i) q^{7} +(7.50000 + 12.9904i) q^{11} +(18.5000 - 32.0429i) q^{13} +42.0000 q^{17} +28.0000 q^{19} +(-97.5000 + 168.875i) q^{23} +(22.0000 + 38.1051i) q^{25} +(55.5000 + 96.1288i) q^{29} +(-102.500 + 177.535i) q^{31} +279.000 q^{35} -166.000 q^{37} +(-130.500 + 226.033i) q^{41} +(-21.5000 - 37.2391i) q^{43} +(-88.5000 - 153.286i) q^{47} +(-309.000 + 535.204i) q^{49} -114.000 q^{53} -135.000 q^{55} +(-79.5000 + 137.698i) q^{59} +(-95.5000 - 165.411i) q^{61} +(166.500 + 288.386i) q^{65} +(-210.500 + 364.597i) q^{67} +156.000 q^{71} +182.000 q^{73} +(232.500 - 402.702i) q^{77} +(566.500 + 981.207i) q^{79} +(541.500 + 937.906i) q^{83} +(-189.000 + 327.358i) q^{85} +1050.00 q^{89} -1147.00 q^{91} +(-126.000 + 218.238i) q^{95} +(450.500 + 780.289i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{5} - 31 q^{7} + 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}+ \cdots + 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

Character values

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

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

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	−0.994026	−	0.109143i	$-0.965189\pi$
$5$	−4.50000	+	7.79423i	−0.402492	+	0.697137i	0.591534	+	0.806280i	$0.298523\pi$
$6$		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
$9$		0			0
$10$		0			0
$11$	7.50000	+	12.9904i	0.205576	+	0.356068i	0.950316	−	0.311287i	$-0.100760\pi$
$11$	7.50000	+	12.9904i	0.205576	+	0.356068i	−0.744740	+	0.667355i	$0.767427\pi$
$12$		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
$15$		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	−0.0369731	+	0.999316i	$0.511772\pi$
$23$	−97.5000	+	168.875i	−0.883920	+	1.53099i	−0.846947	+	0.531678i	$0.821562\pi$
$24$		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	0.987183	−	0.159590i	$-0.0510173\pi$
$29$	55.5000	+	96.1288i	0.355382	+	0.615540i	−0.631801	+	0.775131i	$0.717684\pi$
$30$		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
$33$		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	−0.999994	−	0.00335732i	$-0.998931\pi$
$41$	−130.500	+	226.033i	−0.497090	+	0.860985i	0.502905	+	0.864342i	$0.332265\pi$
$42$		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
$45$		0			0
$46$		0			0
$47$	−88.5000	−	153.286i	−0.274661	−	0.475726i	0.695389	−	0.718634i	$-0.255232\pi$
$47$	−88.5000	−	153.286i	−0.274661	−	0.475726i	−0.970049	+	0.242907i	$0.921899\pi$
$48$		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	−0.940308	−	0.340325i	$-0.889463\pi$
$59$	−79.5000	+	137.698i	−0.175424	+	0.303843i	0.764884	+	0.644168i	$0.222796\pi$
$60$		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
$63$		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	−0.991606	−	0.129294i	$-0.958729\pi$
$67$	−210.500	+	364.597i	−0.383831	+	0.664815i	0.607775	+	0.794109i	$0.292062\pi$
$68$		0			0
$69$		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	0.915078	+	0.403278i	$0.132129\pi$
$79$	566.500	+	981.207i	0.806788	+	1.39740i	−0.108290	+	0.994119i	$0.534537\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	541.500	+	937.906i	0.716113	+	1.24034i	0.962529	+	0.271179i	$0.0874136\pi$
$83$	541.500	+	937.906i	0.716113	+	1.24034i	−0.246416	+	0.969164i	$0.579253\pi$
$84$		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	0.999471	−	0.0325338i	$-0.0103576\pi$
$97$	450.500	+	780.289i	0.471560	+	0.816766i	−0.527910	+	0.849300i	$0.677024\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	193.500	+	335.152i	0.190633	+	0.330187i	0.945460	−	0.325737i	$-0.105613\pi$
$101$	193.500	+	335.152i	0.190633	+	0.330187i	−0.754827	+	0.655924i	$0.772279\pi$
$102$		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
$105$		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	0.411938	+	0.911212i	$0.364852\pi$
$113$	−700.500	+	1213.30i	−0.583164	+	1.01007i	−0.995102	+	0.0988572i	$0.968481\pi$
$114$		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	0.498786	+	0.866725i	$0.333779\pi$
$131$	−751.500	+	1301.64i	−0.501213	+	0.868126i	−0.999999	+	0.00140084i	$0.999554\pi$
$132$		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	−0.898254	−	0.439476i	$-0.855164\pi$
$137$	−1330.50	−	2304.49i	−0.829725	−	1.43713i	0.0685295	−	0.997649i	$-0.478169\pi$
$138$		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
$141$		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	0.155532	+	0.987831i	$0.450291\pi$
$149$	−1414.50	+	2449.99i	−0.777721	+	1.34705i	−0.933253	+	0.359221i	$0.883043\pi$
$150$		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
$153$		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	−0.187889	−	0.982190i	$-0.560164\pi$
$157$	1488.50	−	2578.16i	0.756658	−	1.31057i	0.944546	−	0.328379i	$-0.106502\pi$
$158$		0			0
$159$		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	−0.776290	−	0.630376i	$-0.782901\pi$
$167$	340.500	−	589.763i	0.157777	−	0.273277i	0.934067	+	0.357099i	$0.116234\pi$
$168$		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	−0.857009	−	0.515301i	$-0.827680\pi$
$173$	−1990.50	−	3447.65i	−0.874768	−	1.51514i	−0.0177589	−	0.999842i	$-0.505653\pi$
$174$		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	0.734868	−	0.678210i	$-0.237244\pi$
$191$	−580.500	−	1005.46i	−0.219914	−	0.380902i	−0.954781	+	0.297309i	$0.903911\pi$
$192$		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
$195$		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	−0.797848	−	0.602858i	$-0.794028\pi$
$211$	377.500	−	653.849i	0.123167	−	0.213331i	0.921015	+	0.389528i	$0.127362\pi$
$212$		0			0
$213$		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	−0.999731	−	0.0231966i	$-0.992616\pi$
$223$	−1731.50	−	2999.05i	−0.519954	−	0.900587i	0.479777	−	0.877391i	$-0.340718\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3112.50	−	5391.01i	−0.910061	−	1.57627i	−0.813976	−	0.580899i	$-0.802701\pi$
$227$	−3112.50	−	5391.01i	−0.910061	−	1.57627i	−0.0960856	−	0.995373i	$-0.530632\pi$
$228$		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
$231$		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	−0.162492	+	0.986710i	$0.551953\pi$
$239$	−3457.50	+	5988.57i	−0.935762	+	1.62079i	−0.773270	+	0.634077i	$0.781380\pi$
$240$		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
$243$		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	−0.587842	−	0.808976i	$-0.700022\pi$
$257$	1675.50	−	2902.05i	0.406672	−	0.704377i	0.994515	+	0.104598i	$0.0333557\pi$
$258$		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	0.899204	−	0.437530i	$-0.144147\pi$
$263$	301.500	+	522.213i	0.0706893	+	0.122437i	−0.828514	+	0.559968i	$0.810813\pi$
$264$		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	−0.935209	−	0.354097i	$-0.884788\pi$
$277$	−3569.50	−	6182.56i	−0.774262	−	1.34106i	0.160947	−	0.986963i	$-0.448545\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4213.50	+	7298.00i	0.894507	+	1.54933i	0.834414	+	0.551138i	$0.185806\pi$
$281$	4213.50	+	7298.00i	0.894507	+	1.54933i	0.0600924	+	0.998193i	$0.480860\pi$
$282$		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
$285$		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	0.407513	+	0.913199i	$0.366396\pi$
$293$	−2944.50	+	5100.02i	−0.587097	+	1.01688i	−0.994610	+	0.103683i	$0.966937\pi$
$294$		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	−0.676539	−	0.736407i	$-0.736521\pi$
$311$	1642.50	−	2844.89i	0.299478	−	0.518711i	0.976017	+	0.217696i	$0.0698542\pi$
$312$		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
$315$		0			0
$316$		0			0
$317$	1147.50	+	1987.53i	0.203312	+	0.352147i	0.949594	−	0.313483i	$-0.101496\pi$
$317$	1147.50	+	1987.53i	0.203312	+	0.352147i	−0.746281	+	0.665631i	$0.768163\pi$
$318$		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	−0.997939	−	0.0641773i	$-0.979558\pi$
$331$	−3339.50	−	5784.18i	−0.554548	−	0.960506i	0.443390	−	0.896329i	$-0.353776\pi$
$332$		0			0
$333$		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	−0.940656	−	0.339361i	$-0.889789\pi$
$337$	−1091.50	+	1890.53i	−0.176433	+	0.305590i	0.764224	+	0.644951i	$0.223122\pi$
$338$		0			0
$339$		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	−0.976358	−	0.216159i	$-0.930647\pi$
$347$	−1945.50	+	3369.70i	−0.300980	+	0.521312i	0.675379	+	0.737471i	$0.263980\pi$
$348$		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
$351$		0			0
$352$		0			0
$353$	2377.50	+	4117.95i	0.358475	+	0.620896i	0.987706	−	0.156322i	$-0.0499636\pi$
$353$	2377.50	+	4117.95i	0.358475	+	0.620896i	−0.629232	+	0.777218i	$0.716630\pi$
$354$		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	0.969741	−	0.244135i	$-0.0785041\pi$
$367$	1922.50	+	3329.87i	0.273443	+	0.473618i	−0.696298	+	0.717753i	$0.745171\pi$
$368$		0			0
$369$		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	−0.418529	−	0.908203i	$-0.637454\pi$
$373$	4158.50	−	7202.73i	0.577263	−	0.999848i	0.995792	−	0.0916449i	$-0.0292125\pi$
$374$		0			0
$375$		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	0.109130	+	0.994028i	$0.465194\pi$
$383$	−6043.50	+	10467.6i	−0.806288	+	1.39653i	−0.915418	+	0.402505i	$0.868140\pi$
$384$		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	−0.997776	−	0.0666565i	$-0.978767\pi$
$389$	−4270.50	−	7396.72i	−0.556614	−	0.964084i	0.441162	−	0.897428i	$-0.354567\pi$
$390$		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	−0.395180	−	0.918604i	$-0.629318\pi$
$401$	4801.50	−	8316.44i	0.597944	−	1.03567i	0.993124	−	0.117066i	$-0.0373488\pi$
$402$		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	0.277312	+	0.960780i	$0.410557\pi$
$409$	−5735.50	+	9934.18i	−0.693404	+	1.20101i	−0.970716	+	0.240231i	$0.922777\pi$
$410$		0			0
$411$		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	−0.637721	−	0.770267i	$-0.720123\pi$
$419$	2986.50	−	5172.77i	0.348210	−	0.603118i	0.985932	+	0.167149i	$0.0534562\pi$
$420$		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
$423$		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	0.959681	−	0.281091i	$-0.0906964\pi$
$439$	2174.50	+	3766.34i	0.236408	+	0.409471i	−0.723273	+	0.690562i	$0.757363\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7273.50	+	12598.1i	0.780078	+	1.35113i	0.931896	+	0.362726i	$0.118154\pi$
$443$	7273.50	+	12598.1i	0.780078	+	1.35113i	−0.151818	+	0.988408i	$0.548513\pi$
$444$		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	0.578673	−	0.815560i	$-0.303571\pi$
$457$	−4073.50	−	7055.51i	−0.416959	−	0.722194i	−0.995632	+	0.0933655i	$0.970237\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4015.50	+	6955.05i	0.405684	+	0.702666i	0.994401	−	0.105674i	$-0.0336999\pi$
$461$	4015.50	+	6955.05i	0.405684	+	0.702666i	−0.588717	+	0.808340i	$0.700367\pi$
$462$		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
$465$		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	0.855614	−	0.517615i	$-0.173180\pi$
$479$	−214.500	−	371.525i	−0.0204609	−	0.0354393i	−0.876075	+	0.482176i	$0.839847\pi$
$480$		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	−0.302346	−	0.953198i	$-0.597770\pi$
$491$	7336.50	−	12707.2i	0.674321	−	1.16796i	0.976667	−	0.214760i	$-0.0688969\pi$
$492$		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	−0.389574	−	0.920995i	$-0.627378\pi$
$499$	6719.50	−	11638.5i	0.602818	−	1.04411i	0.992392	−	0.123116i	$-0.0392888\pi$
$500$		0			0
$501$		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	−0.936646	−	0.350276i	$-0.886088\pi$
$509$	−1894.50	+	3281.37i	−0.164975	+	0.285745i	0.771671	+	0.636021i	$0.219421\pi$
$510$		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	0.588891	−	0.808213i	$-0.299565\pi$
$547$	−5187.50	−	8985.01i	−0.405487	−	0.702324i	−0.994378	+	0.105888i	$0.966231\pi$
$548$		0			0
$549$		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	−0.136792	−	0.990600i	$-0.543679\pi$
$563$	10546.5	−	18267.1i	0.789488	−	1.36743i	0.926281	−	0.376834i	$-0.122987\pi$
$564$		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	0.888757	−	0.458379i	$-0.151570\pi$
$569$	643.500	+	1114.57i	0.0474111	+	0.0821185i	−0.841346	+	0.540497i	$0.818236\pi$
$570$		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
$573$		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	0.987810	+	0.155664i	$0.0497518\pi$
$587$	8941.50	+	15487.1i	0.628714	+	1.08896i	−0.359096	+	0.933301i	$0.616915\pi$
$588$		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	−0.847293	−	0.531127i	$-0.821769\pi$
$599$	532.500	−	922.317i	0.0363228	−	0.0629129i	0.883615	+	0.468214i	$0.155102\pi$
$600$		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
$603$		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	0.473110	+	0.881004i	$0.343131\pi$
$607$	−7872.50	+	13635.6i	−0.526417	+	0.911780i	−0.999526	+	0.0307768i	$0.990202\pi$
$608$		0			0
$609$		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	−0.982110	−	0.188311i	$-0.939699\pi$
$617$	−5026.50	+	8706.15i	−0.327973	+	0.568066i	0.654137	+	0.756376i	$0.273032\pi$
$618$		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
$621$		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	0.645899	−	0.763423i	$-0.276483\pi$
$641$	−5488.50	−	9506.36i	−0.338195	−	0.585770i	−0.984093	+	0.177653i	$0.943150\pi$
$642$		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
$645$		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	−0.0424927	−	0.999097i	$-0.513530\pi$
$653$	14083.5	−	24393.3i	0.843997	−	1.46185i	0.886490	−	0.462749i	$-0.153137\pi$
$654$		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	0.662592	−	0.748980i	$-0.269456\pi$
$659$	−5368.50	−	9298.51i	−0.317340	−	0.549649i	−0.979932	+	0.199331i	$0.936123\pi$
$660$		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
$663$		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	0.862409	−	0.506212i	$-0.168955\pi$
$673$	−125.500	−	217.372i	−0.00718822	−	0.0124504i	−0.869597	+	0.493762i	$0.835621\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4225.50	+	7318.78i	0.239881	+	0.415485i	0.960680	−	0.277659i	$-0.0895584\pi$
$677$	4225.50	+	7318.78i	0.239881	+	0.415485i	−0.720799	+	0.693144i	$0.756225\pi$
$678$		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	0.940233	−	0.340532i	$-0.110607\pi$
$691$	3182.50	+	5512.25i	0.175207	+	0.303467i	−0.765026	+	0.643999i	$0.777274\pi$
$692$		0			0
$693$		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	0.803718	−	0.595011i	$-0.202852\pi$
$709$	−2141.50	−	3709.19i	−0.113435	−	0.196476i	−0.917153	+	0.398535i	$0.869519\pi$
$710$		0			0
$711$		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	0.990850	+	0.134965i	$0.0430922\pi$
$727$	12002.5	+	20788.9i	0.612308	+	1.06055i	−0.378542	+	0.925584i	$0.623574\pi$
$728$		0			0
$729$		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	0.188760	−	0.982023i	$-0.439553\pi$
$733$	18750.5	−	32476.8i	0.944837	−	1.63651i	0.756077	−	0.654482i	$-0.227113\pi$
$734$		0			0
$735$		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	−0.885364	−	0.464898i	$-0.846091\pi$
$743$	−811.500	+	1405.56i	−0.0400687	+	0.0694010i	0.845296	+	0.534299i	$0.179424\pi$
$744$		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	−0.937493	−	0.348005i	$-0.886859\pi$
$751$	−3444.50	+	5966.05i	−0.167366	+	0.289886i	0.770127	+	0.637890i	$0.220193\pi$
$752$		0			0
$753$		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	−0.805876	−	0.592084i	$-0.798305\pi$
$761$	2305.50	−	3993.24i	0.109822	−	0.190217i	0.915698	+	0.401867i	$0.131639\pi$
$762$		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	−0.837737	−	0.546073i	$-0.816122\pi$
$769$	1152.50	−	1996.19i	0.0540445	−	0.0936078i	0.891782	+	0.452465i	$0.149455\pi$
$770$		0			0
$771$		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	−0.399011	−	0.916946i	$-0.630646\pi$
$787$	13127.5	−	22737.5i	0.594593	−	1.02987i	0.993604	−	0.112919i	$-0.0360202\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	43431.0			1.95225
$792$		0			0
$793$	−7067.00			−0.316465
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−718.500	+	1244.48i	−0.0319330	+	0.0553095i	−0.881550	−	0.472090i	$-0.843500\pi$
$797$	−718.500	+	1244.48i	−0.0319330	+	0.0553095i	0.849617	+	0.527400i	$0.176833\pi$
$798$		0			0
$799$	−3717.00	−	6438.03i	−0.164578	−	0.285058i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1365.00	+	2364.25i	0.0599873	+	0.103901i
$804$		0			0
$805$	−27202.5	+	47116.1i	−1.19101	+	2.06289i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−25734.0			−1.11837			−0.559184	−	0.829044i	$-0.688885\pi$
$809$	−25734.0			−1.11837			−0.559184	+	0.829044i	$0.688885\pi$
$810$		0			0
$811$	3220.00			0.139420			0.0697099	−	0.997567i	$-0.477793\pi$
$811$	3220.00			0.139420			0.0697099	+	0.997567i	$0.477793\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−14922.0	+	25845.7i	−0.641343	+	1.11084i
$816$		0			0
$817$	−602.000	−	1042.69i	−0.0257788	−	0.0446503i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22315.5	+	38651.6i	0.948619	+	1.64306i	0.748337	+	0.663318i	$0.230852\pi$
$821$	22315.5	+	38651.6i	0.948619	+	1.64306i	0.200282	+	0.979738i	$0.435814\pi$
$822$		0			0
$823$	12473.5	−	21604.7i	0.528310	−	0.915059i	−0.471146	−	0.882055i	$-0.656159\pi$
$823$	12473.5	−	21604.7i	0.528310	−	0.915059i	0.999455	−	0.0330037i	$-0.0105073\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−14964.0			−0.629201			−0.314601	−	0.949224i	$-0.601871\pi$
$827$	−14964.0			−0.629201			−0.314601	+	0.949224i	$0.601871\pi$
$828$		0			0
$829$	−7462.00			−0.312625			−0.156312	−	0.987708i	$-0.549961\pi$
$829$	−7462.00			−0.312625			−0.156312	+	0.987708i	$0.549961\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12978.0	+	22478.6i	−0.539809	+	0.934977i
$834$		0			0
$835$	3064.50	+	5307.87i	0.127008	+	0.219984i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1348.50	−	2335.67i	−0.0554891	−	0.0961100i	0.836947	−	0.547285i	$-0.184338\pi$
$839$	−1348.50	−	2335.67i	−0.0554891	−	0.0961100i	−0.892436	+	0.451175i	$0.851005\pi$
$840$		0			0
$841$	6034.00	−	10451.2i	0.247407	−	0.428521i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7452.00			−0.303381
$846$		0			0
$847$	−34286.0			−1.39089
$848$		0			0
$849$		0			0
$850$		0			0
$851$	16185.0	−	28033.2i	0.651956	−	1.12922i
$852$		0			0
$853$	19812.5	+	34316.3i	0.795272	+	1.37745i	0.922666	+	0.385599i	$0.126005\pi$
$853$	19812.5	+	34316.3i	0.795272	+	1.37745i	−0.127394	+	0.991852i	$0.540661\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1486.50	−	2574.69i	−0.0592507	−	0.102625i	0.834879	−	0.550434i	$-0.185538\pi$
$857$	−1486.50	−	2574.69i	−0.0592507	−	0.102625i	−0.894129	+	0.447809i	$0.852204\pi$
$858$		0			0
$859$	−22614.5	+	39169.5i	−0.898250	+	1.55581i	−0.0685200	+	0.997650i	$0.521828\pi$
$859$	−22614.5	+	39169.5i	−0.898250	+	1.55581i	−0.829730	+	0.558165i	$0.811506\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1416.00			0.0558531			0.0279265	−	0.999610i	$-0.491110\pi$
$863$	1416.00			0.0558531			0.0279265	+	0.999610i	$0.491110\pi$
$864$		0			0
$865$	35829.0			1.40835
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8497.50	+	14718.1i	−0.331712	+	0.574542i
$870$		0			0
$871$	7788.50	+	13490.1i	0.302989	+	0.524792i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	23575.5	+	40834.0i	0.910854	+	1.57765i
$876$		0			0
$877$	−8777.50	+	15203.1i	−0.337965	+	0.585372i	−0.984050	−	0.177893i	$-0.943072\pi$
$877$	−8777.50	+	15203.1i	−0.337965	+	0.585372i	0.646085	+	0.763265i	$0.276405\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30030.0			−1.14840			−0.574198	−	0.818717i	$-0.694686\pi$
$881$	−30030.0			−1.14840			−0.574198	+	0.818717i	$0.694686\pi$
$882$		0			0
$883$	35740.0			1.36211			0.681057	−	0.732230i	$-0.261521\pi$
$883$	35740.0			1.36211			0.681057	+	0.732230i	$0.261521\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−373.500	+	646.921i	−0.0141386	+	0.0244887i	−0.873008	−	0.487706i	$-0.837834\pi$
$887$	−373.500	+	646.921i	−0.0141386	+	0.0244887i	0.858870	+	0.512194i	$0.171167\pi$
$888$		0			0
$889$	−13640.0	−	23625.2i	−0.514590	−	0.891297i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2478.00	−	4292.02i	−0.0928590	−	0.160837i
$894$		0			0
$895$	−9018.00	+	15619.6i	−0.336803	+	0.583360i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22755.0			−0.844184
$900$		0			0
$901$	−4788.00			−0.177038
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5733.00	+	9929.85i	−0.210576	+	0.364728i
$906$		0			0
$907$	−10269.5	−	17787.3i	−0.375957	−	0.651177i	0.614513	−	0.788907i	$-0.289353\pi$
$907$	−10269.5	−	17787.3i	−0.375957	−	0.651177i	−0.990470	+	0.137730i	$0.956019\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12061.5	+	20891.1i	0.438656	+	0.759774i	0.997586	−	0.0694406i	$-0.0221214\pi$
$911$	12061.5	+	20891.1i	0.438656	+	0.759774i	−0.558930	+	0.829215i	$0.688788\pi$
$912$		0			0
$913$	−8122.50	+	14068.6i	−0.294431	+	0.509969i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	46593.0			1.67790
$918$		0			0
$919$	−23312.0			−0.836770			−0.418385	−	0.908270i	$-0.637404\pi$
$919$	−23312.0			−0.836770			−0.418385	+	0.908270i	$0.637404\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2886.00	−	4998.70i	0.102919	−	0.178260i
$924$		0			0
$925$	−3652.00	−	6325.45i	−0.129813	−	0.224843i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9349.50	+	16193.8i	0.330191	+	0.571907i	0.982549	−	0.186004i	$-0.0595536\pi$
$929$	9349.50	+	16193.8i	0.330191	+	0.571907i	−0.652358	+	0.757911i	$0.726220\pi$
$930$		0			0
$931$	−8652.00	+	14985.7i	−0.304573	+	0.527537i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5670.00			−0.198320
$936$		0			0
$937$	−31234.0			−1.08898			−0.544488	−	0.838769i	$-0.683276\pi$
$937$	−31234.0			−1.08898			−0.544488	+	0.838769i	$0.683276\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5875.50	−	10176.7i	0.203545	−	0.352550i	−0.746123	−	0.665808i	$-0.768087\pi$
$941$	5875.50	−	10176.7i	0.203545	−	0.352550i	0.949668	+	0.313258i	$0.101420\pi$
$942$		0			0
$943$	−25447.5	−	44076.4i	−0.878775	−	1.52208i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12943.5	+	22418.8i	0.444147	+	0.769285i	0.997992	−	0.0633343i	$-0.0201735\pi$
$947$	12943.5	+	22418.8i	0.444147	+	0.769285i	−0.553845	+	0.832620i	$0.686840\pi$
$948$		0			0
$949$	3367.00	−	5831.82i	0.115171	−	0.199482i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	41130.0			1.39804			0.699020	−	0.715102i	$-0.253620\pi$
$953$	41130.0			1.39804			0.699020	+	0.715102i	$0.253620\pi$
$954$		0			0
$955$	10449.0			0.354054
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−41245.5	+	71439.3i	−1.38883	+	2.40552i
$960$		0			0
$961$	−6117.00	−	10595.0i	−0.205330	−	0.355643i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16249.5	−	28145.0i	−0.542062	−	0.938879i
$966$		0			0
$967$	−14322.5	+	24807.3i	−0.476298	+	0.824973i	−0.999631	−	0.0271553i	$-0.991355\pi$
$967$	−14322.5	+	24807.3i	−0.476298	+	0.824973i	0.523333	+	0.852128i	$0.324688\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−32256.0			−1.06606			−0.533030	−	0.846096i	$-0.678947\pi$
$971$	−32256.0			−1.06606			−0.533030	+	0.846096i	$0.678947\pi$
$972$		0			0
$973$	3751.00			0.123588
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4507.50	−	7807.22i	0.147603	−	0.255655i	−0.782738	−	0.622351i	$-0.786178\pi$
$977$	4507.50	−	7807.22i	0.147603	−	0.255655i	0.930341	+	0.366696i	$0.119511\pi$
$978$		0			0
$979$	7875.00	+	13639.9i	0.257085	+	0.445284i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7942.50	−	13756.8i	−0.257707	−	0.446362i	0.707920	−	0.706293i	$-0.249634\pi$
$983$	−7942.50	−	13756.8i	−0.257707	−	0.446362i	−0.965627	+	0.259930i	$0.916300\pi$
$984$		0			0
$985$	9207.00	−	15947.0i	0.297827	−	0.515851i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8385.00			0.269593
$990$		0			0
$991$	−34904.0			−1.11883			−0.559416	−	0.828887i	$-0.688974\pi$
$991$	−34904.0			−1.11883			−0.559416	+	0.828887i	$0.688974\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	13482.0	−	23351.5i	0.429556	−	0.744013i
$996$		0			0
$997$	17670.5	+	30606.2i	0.561314	+	0.972225i	0.997382	+	0.0723111i	$0.0230374\pi$
$997$	17670.5	+	30606.2i	0.561314	+	0.972225i	−0.436068	+	0.899914i	$0.643629\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.4.i.a.145.1		2
3.2	odd	2		144.4.i.a.49.1		2
4.3	odd	2		54.4.c.a.37.1		2
9.2	odd	6		144.4.i.a.97.1		2
9.4	even	3		1296.4.a.g.1.1		1
9.5	odd	6		1296.4.a.b.1.1		1
9.7	even	3	inner	432.4.i.a.289.1		2
12.11	even	2		18.4.c.a.13.1	yes	2
36.7	odd	6		54.4.c.a.19.1		2
36.11	even	6		18.4.c.a.7.1	✓	2
36.23	even	6		162.4.a.d.1.1		1
36.31	odd	6		162.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.a.7.1	✓	2	36.11	even	6
18.4.c.a.13.1	yes	2	12.11	even	2
54.4.c.a.19.1		2	36.7	odd	6
54.4.c.a.37.1		2	4.3	odd	2
144.4.i.a.49.1		2	3.2	odd	2
144.4.i.a.97.1		2	9.2	odd	6
162.4.a.a.1.1		1	36.31	odd	6
162.4.a.d.1.1		1	36.23	even	6
432.4.i.a.145.1		2	1.1	even	1	trivial
432.4.i.a.289.1		2	9.7	even	3	inner
1296.4.a.b.1.1		1	9.5	odd	6
1296.4.a.g.1.1		1	9.4	even	3

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 $\F_{q}$
Belyi maps

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

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

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