LMFDB - Embedded newform 1728.3.o.d.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(127,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1728.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.o (of order $6$, 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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.121550625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 $ x^8 - x^7 - 4x^6 - 9x^5 + 23x^4 + 18x^3 - 16x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			127.3
Root			$-1.44918 + 1.77086i$ of defining polynomial
Character	$\chi$	$=$	1728.127
Dual form			1728.3.o.d.1279.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.81174 + 3.13802i) q^{5} +(-1.59422 - 0.920424i) q^{7} +O(q^{10})$	$q+(1.81174 + 3.13802i) q^{5} +(-1.59422 - 0.920424i) q^{7} +(10.0623 + 5.80948i) q^{11} +(-6.43521 - 11.1461i) q^{13} -12.6235 q^{17} -25.6526i q^{19} +(-25.9012 + 14.9541i) q^{23} +(5.93521 - 10.2801i) q^{25} +(-10.8117 + 18.7265i) q^{29} +(-52.4027 + 30.2547i) q^{31} -6.67027i q^{35} +25.7409 q^{37} +(33.3704 + 57.7993i) q^{41} +(-14.2447 - 8.22418i) q^{43} +(-66.1505 - 38.1920i) q^{47} +(-22.8056 - 39.5005i) q^{49} +14.2591 q^{53} +42.1010i q^{55} +(50.3115 - 29.0474i) q^{59} +(9.43521 - 16.3423i) q^{61} +(23.3178 - 40.3877i) q^{65} +(-20.6216 + 11.9059i) q^{67} -46.4758i q^{71} +49.3521 q^{73} +(-10.6944 - 18.5232i) q^{77} +(-52.4027 - 30.2547i) q^{79} +(-86.2751 - 49.8109i) q^{83} +(-22.8704 - 39.6127i) q^{85} -154.988 q^{89} +23.6925i q^{91} +(80.4984 - 46.4758i) q^{95} +(-21.1113 + 36.5658i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5}+O(q^{10}) $ 8 * q - 6 * q^5	$ 8 q - 6 q^{5} + 10 q^{13} - 60 q^{17} - 14 q^{25} - 66 q^{29} - 40 q^{37} + 144 q^{41} + 2 q^{49} + 360 q^{53} + 14 q^{61} + 330 q^{65} - 220 q^{73} - 270 q^{77} - 60 q^{85} - 912 q^{89} + 200 q^{97}+O(q^{100}) $ 8 * q - 6 * q^5 + 10 * q^13 - 60 * q^17 - 14 * q^25 - 66 * q^29 - 40 * q^37 + 144 * q^41 + 2 * q^49 + 360 * q^53 + 14 * q^61 + 330 * q^65 - 220 * q^73 - 270 * q^77 - 60 * q^85 - 912 * q^89 + 200 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{2}{3}\right)$

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
$5$	1.81174	+	3.13802i	0.362348	+	0.627604i	0.988347	−	0.152219i	$-0.0486420\pi$
$5$	1.81174	+	3.13802i	0.362348	+	0.627604i	−0.625999	+	0.779824i	$0.715309\pi$
$6$		0			0
$7$	−1.59422	−	0.920424i	−0.227746	−	0.131489i	0.381786	−	0.924251i	$-0.375309\pi$
$7$	−1.59422	−	0.920424i	−0.227746	−	0.131489i	−0.609532	+	0.792762i	$0.708643\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.0623	+	5.80948i	0.914755	+	0.528134i	0.881958	−	0.471328i	$-0.156225\pi$
$11$	10.0623	+	5.80948i	0.914755	+	0.528134i	0.0327970	+	0.999462i	$0.489559\pi$
$12$		0			0
$13$	−6.43521	−	11.1461i	−0.495016	−	0.857394i	0.504967	−	0.863139i	$-0.331505\pi$
$13$	−6.43521	−	11.1461i	−0.495016	−	0.857394i	−0.999983	+	0.00574505i	$0.998171\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.6235			−0.742557			−0.371279	−	0.928521i	$-0.621081\pi$
$17$	−12.6235			−0.742557			−0.371279	+	0.928521i	$0.621081\pi$
$18$		0			0
$19$		−	25.6526i		−	1.35014i	−0.737755	−	0.675069i	$-0.764114\pi$
$19$		−	25.6526i		−	1.35014i	0.737755	−	0.675069i	$-0.235886\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−25.9012	+	14.9541i	−1.12614	+	0.650178i	−0.942961	−	0.332902i	$-0.891972\pi$
$23$	−25.9012	+	14.9541i	−1.12614	+	0.650178i	−0.183179	+	0.983080i	$0.558639\pi$
$24$		0			0
$25$	5.93521	−	10.2801i	0.237409	−	0.411204i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−10.8117	+	18.7265i	−0.372819	+	0.645741i	−0.989998	−	0.141082i	$-0.954942\pi$
$29$	−10.8117	+	18.7265i	−0.372819	+	0.645741i	0.617179	+	0.786822i	$0.288275\pi$
$30$		0			0
$31$	−52.4027	+	30.2547i	−1.69041	+	0.975959i	−0.736228	+	0.676733i	$0.763395\pi$
$31$	−52.4027	+	30.2547i	−1.69041	+	0.975959i	−0.954182	+	0.299226i	$0.903272\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	6.67027i		−	0.190579i
$36$		0			0
$37$	25.7409			0.695699			0.347849	−	0.937550i	$-0.386912\pi$
$37$	25.7409			0.695699			0.347849	+	0.937550i	$0.386912\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	33.3704	+	57.7993i	0.813913	+	1.40974i	0.910106	+	0.414376i	$0.136000\pi$
$41$	33.3704	+	57.7993i	0.813913	+	1.40974i	−0.0961931	+	0.995363i	$0.530667\pi$
$42$		0			0
$43$	−14.2447	−	8.22418i	−0.331272	−	0.191260i	0.325134	−	0.945668i	$-0.394591\pi$
$43$	−14.2447	−	8.22418i	−0.331272	−	0.191260i	−0.656406	+	0.754408i	$0.727924\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−66.1505	−	38.1920i	−1.40746	−	0.812595i	−0.412314	−	0.911042i	$-0.635279\pi$
$47$	−66.1505	−	38.1920i	−1.40746	−	0.812595i	−0.995142	+	0.0984463i	$0.968613\pi$
$48$		0			0
$49$	−22.8056	−	39.5005i	−0.465421	−	0.806133i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	14.2591			0.269041			0.134520	−	0.990911i	$-0.457051\pi$
$53$	14.2591			0.269041			0.134520	+	0.990911i	$0.457051\pi$
$54$		0			0
$55$			42.1010i			0.765472i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	50.3115	−	29.0474i	0.852738	−	0.492328i	−0.00883587	−	0.999961i	$-0.502813\pi$
$59$	50.3115	−	29.0474i	0.852738	−	0.492328i	0.861574	+	0.507633i	$0.169479\pi$
$60$		0			0
$61$	9.43521	−	16.3423i	0.154676	−	0.267906i	−0.778265	−	0.627936i	$-0.783900\pi$
$61$	9.43521	−	16.3423i	0.154676	−	0.267906i	0.932941	+	0.360030i	$0.117233\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	23.3178	−	40.3877i	0.358736	−	0.621349i
$66$		0			0
$67$	−20.6216	+	11.9059i	−0.307785	+	0.177700i	−0.645935	−	0.763393i	$-0.723532\pi$
$67$	−20.6216	+	11.9059i	−0.307785	+	0.177700i	0.338150	+	0.941092i	$0.390199\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	46.4758i		−	0.654589i	−0.944922	−	0.327294i	$-0.893863\pi$
$71$		−	46.4758i		−	0.654589i	0.944922	−	0.327294i	$-0.106137\pi$
$72$		0			0
$73$	49.3521			0.676057			0.338028	−	0.941136i	$-0.390240\pi$
$73$	49.3521			0.676057			0.338028	+	0.941136i	$0.390240\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.6944	−	18.5232i	−0.138888	−	0.240561i
$78$		0			0
$79$	−52.4027	−	30.2547i	−0.663326	−	0.382971i	0.130217	−	0.991485i	$-0.458433\pi$
$79$	−52.4027	−	30.2547i	−0.663326	−	0.382971i	−0.793543	+	0.608514i	$0.791766\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−86.2751	−	49.8109i	−1.03946	−	0.600132i	−0.119779	−	0.992801i	$-0.538219\pi$
$83$	−86.2751	−	49.8109i	−1.03946	−	0.600132i	−0.919680	+	0.392669i	$0.871552\pi$
$84$		0			0
$85$	−22.8704	−	39.6127i	−0.269064	−	0.466032i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−154.988			−1.74144			−0.870718	−	0.491783i	$-0.836345\pi$
$89$	−154.988			−1.74144			−0.870718	+	0.491783i	$0.836345\pi$
$90$		0			0
$91$			23.6925i			0.260357i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	80.4984	−	46.4758i	0.847352	−	0.489219i
$96$		0			0
$97$	−21.1113	+	36.5658i	−0.217642	+	0.376967i	−0.954087	−	0.299531i	$-0.903170\pi$
$97$	−21.1113	+	36.5658i	−0.217642	+	0.376967i	0.736445	+	0.676498i	$0.236503\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−63.1761	+	109.424i	−0.625506	+	1.08341i	0.362937	+	0.931814i	$0.381774\pi$
$101$	−63.1761	+	109.424i	−0.625506	+	1.08341i	−0.988443	+	0.151594i	$0.951559\pi$
$102$		0			0
$103$	23.9133	−	13.8064i	0.232168	−	0.134042i	−0.379404	−	0.925231i	$-0.623871\pi$
$103$	23.9133	−	13.8064i	0.232168	−	0.134042i	0.611572	+	0.791189i	$0.290538\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	121.138i		−	1.13213i	−0.824360	−	0.566066i	$-0.808465\pi$
$107$		−	121.138i		−	1.13213i	0.824360	−	0.566066i	$-0.191535\pi$
$108$		0			0
$109$	−61.7409			−0.566430			−0.283215	−	0.959056i	$-0.591401\pi$
$109$	−61.7409			−0.566430			−0.283215	+	0.959056i	$0.591401\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	43.0709	+	74.6010i	0.381158	+	0.660186i	0.991228	−	0.132162i	$-0.0421920\pi$
$113$	43.0709	+	74.6010i	0.381158	+	0.660186i	−0.610070	+	0.792348i	$0.708859\pi$
$114$		0			0
$115$	−93.8525	−	54.1858i	−0.816109	−	0.471180i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	20.1246	+	11.6190i	0.169114	+	0.0976382i
$120$		0			0
$121$	7.00000	+	12.1244i	0.0578512	+	0.100201i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	133.599			1.06879
$126$		0			0
$127$		−	183.369i		−	1.44385i	−0.691970	−	0.721926i	$-0.743257\pi$
$127$		−	183.369i		−	1.44385i	0.691970	−	0.721926i	$-0.256743\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−149.631	+	86.3894i	−1.14222	+	0.659461i	−0.946979	−	0.321295i	$-0.895882\pi$
$131$	−149.631	+	86.3894i	−1.14222	+	0.659461i	−0.195240	+	0.980755i	$0.562549\pi$
$132$		0			0
$133$	−23.6113	+	40.8959i	−0.177528	+	0.307488i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	80.1235	−	138.778i	0.584843	−	1.01298i	−0.410052	−	0.912062i	$-0.634489\pi$
$137$	80.1235	−	138.778i	0.584843	−	1.01298i	0.994895	−	0.100915i	$-0.0321772\pi$
$138$		0			0
$139$	−191.971	+	110.835i	−1.38109	+	0.797371i	−0.992288	−	0.123951i	$-0.960443\pi$
$139$	−191.971	+	110.835i	−1.38109	+	0.797371i	−0.388799	+	0.921322i	$0.627110\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	149.541i		−	1.04574i
$144$		0			0
$145$	−78.3521			−0.540360
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−94.0343	−	162.872i	−0.631103	−	1.09310i	−0.987327	−	0.158701i	$-0.949269\pi$
$149$	−94.0343	−	162.872i	−0.631103	−	1.09310i	0.356224	−	0.934401i	$-0.384064\pi$
$150$		0			0
$151$	−96.8344	−	55.9073i	−0.641287	−	0.370247i	0.143823	−	0.989603i	$-0.454060\pi$
$151$	−96.8344	−	55.9073i	−0.641287	−	0.370247i	−0.785110	+	0.619356i	$0.787394\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−189.880	−	109.627i	−1.22503	−	0.707273i
$156$		0			0
$157$	51.4352	+	89.0884i	0.327613	+	0.567442i	0.982038	−	0.188685i	$-0.0604225\pi$
$157$	51.4352	+	89.0884i	0.327613	+	0.567442i	−0.654425	+	0.756127i	$0.727089\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	55.0564			0.341965
$162$		0			0
$163$			205.221i			1.25902i	0.776991	+	0.629512i	$0.216745\pi$
$163$			205.221i			1.25902i	−0.776991	+	0.629512i	$0.783255\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	46.0258	−	26.5730i	0.275604	−	0.159120i	−0.355828	−	0.934552i	$-0.615801\pi$
$167$	46.0258	−	26.5730i	0.275604	−	0.159120i	0.631432	+	0.775432i	$0.282468\pi$
$168$		0			0
$169$	1.67607	−	2.90303i	0.00991755	−	0.0171777i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	94.1517	−	163.075i	0.544229	−	0.942633i	−0.454426	−	0.890785i	$-0.650155\pi$
$173$	94.1517	−	163.075i	0.544229	−	0.942633i	0.998655	−	0.0518482i	$-0.0165112\pi$
$174$		0			0
$175$	−18.9241	+	10.9258i	−0.108138	+	0.0624333i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	92.9516i		−	0.519283i	−0.965705	−	0.259641i	$-0.916396\pi$
$179$		−	92.9516i		−	0.519283i	0.965705	−	0.259641i	$-0.0836043\pi$
$180$		0			0
$181$	240.445			1.32843			0.664213	−	0.747543i	$-0.268767\pi$
$181$	240.445			1.32843			0.664213	+	0.747543i	$0.268767\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	46.6357	+	80.7754i	0.252085	+	0.436624i
$186$		0			0
$187$	−127.021	−	73.3358i	−0.679258	−	0.392170i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−49.0077	−	28.2946i	−0.256585	−	0.148139i	0.366191	−	0.930540i	$-0.380662\pi$
$191$	−49.0077	−	28.2946i	−0.256585	−	0.148139i	−0.622776	+	0.782400i	$0.713995\pi$
$192$		0			0
$193$	50.3704	+	87.2441i	0.260987	+	0.452042i	0.966504	−	0.256650i	$-0.0826188\pi$
$193$	50.3704	+	87.2441i	0.260987	+	0.452042i	−0.705518	+	0.708692i	$0.749285\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−115.247			−0.585010			−0.292505	−	0.956264i	$-0.594489\pi$
$197$	−115.247			−0.585010			−0.292505	+	0.956264i	$0.594489\pi$
$198$		0			0
$199$			190.733i			0.958455i	0.877691	+	0.479228i	$0.159083\pi$
$199$			190.733i			0.958455i	−0.877691	+	0.479228i	$0.840917\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	34.4726	−	19.9028i	0.169816	−	0.0980432i
$204$		0			0
$205$	−120.917	+	209.434i	−0.589839	+	1.02163i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	149.028	−	258.124i	0.713054	−	1.23505i
$210$		0			0
$211$	−68.3449	+	39.4590i	−0.323910	+	0.187009i	−0.653134	−	0.757242i	$-0.726546\pi$
$211$	−68.3449	+	39.4590i	−0.323910	+	0.187009i	0.329224	+	0.944252i	$0.393213\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	59.6003i		−	0.277211i
$216$		0			0
$217$	111.389			0.513312
$218$		0			0
$219$		0			0
$220$		0			0
$221$	81.2348	+	140.703i	0.367578	+	0.636664i
$222$		0			0
$223$	−192.074	−	110.894i	−0.861321	−	0.497284i	0.00313373	−	0.999995i	$-0.499003\pi$
$223$	−192.074	−	110.894i	−0.861321	−	0.497284i	−0.864454	+	0.502711i	$0.832336\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.0623	+	5.80948i	0.0443273	+	0.0255924i	0.522000	−	0.852946i	$-0.325186\pi$
$227$	10.0623	+	5.80948i	0.0443273	+	0.0255924i	−0.477673	+	0.878538i	$0.658519\pi$
$228$		0			0
$229$	−24.4352	−	42.3230i	−0.106704	−	0.184817i	0.807729	−	0.589554i	$-0.200696\pi$
$229$	−24.4352	−	42.3230i	−0.106704	−	0.184817i	−0.914433	+	0.404737i	$0.867363\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−384.785			−1.65144			−0.825719	−	0.564082i	$-0.809230\pi$
$233$	−384.785			−1.65144			−0.825719	+	0.564082i	$0.809230\pi$
$234$		0			0
$235$		−	276.775i		−	1.17777i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−169.755	+	98.0083i	−0.710274	+	0.410077i	−0.811162	−	0.584821i	$-0.801165\pi$
$239$	−169.755	+	98.0083i	−0.710274	+	0.410077i	0.100889	+	0.994898i	$0.467831\pi$
$240$		0			0
$241$	−130.593	+	226.194i	−0.541880	+	0.938563i	0.456917	+	0.889510i	$0.348954\pi$
$241$	−130.593	+	226.194i	−0.541880	+	0.938563i	−0.998796	+	0.0490534i	$0.984380\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	82.6357	−	143.129i	0.337288	−	0.584201i
$246$		0			0
$247$	−285.927	+	165.080i	−1.15760	+	0.668340i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	137.922i		−	0.549490i	−0.961517	−	0.274745i	$-0.911407\pi$
$251$		−	137.922i		−	0.549490i	0.961517	−	0.274745i	$-0.0885934\pi$
$252$		0			0
$253$	−347.502			−1.37352
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13.7348	+	23.7893i	0.0534426	+	0.0925653i	0.891509	−	0.453003i	$-0.149647\pi$
$257$	13.7348	+	23.7893i	0.0534426	+	0.0925653i	−0.838066	+	0.545568i	$0.816314\pi$
$258$		0			0
$259$	−41.0366	−	23.6925i	−0.158443	−	0.0914768i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−250.254	−	144.484i	−0.951535	−	0.549369i	−0.0579779	−	0.998318i	$-0.518465\pi$
$263$	−250.254	−	144.484i	−0.951535	−	0.549369i	−0.893558	+	0.448949i	$0.851799\pi$
$264$		0			0
$265$	25.8338	+	44.7455i	0.0974862	+	0.168851i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−180.235			−0.670018			−0.335009	−	0.942215i	$-0.608739\pi$
$269$	−180.235			−0.670018			−0.335009	+	0.942215i	$0.608739\pi$
$270$		0			0
$271$			28.9765i			0.106924i	0.998570	+	0.0534622i	$0.0170257\pi$
$271$			28.9765i			0.106924i	−0.998570	+	0.0534622i	$0.982974\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	119.444	−	68.9609i	0.434341	−	0.250767i
$276$		0			0
$277$	146.528	−	253.794i	0.528983	−	0.916225i	−0.470446	−	0.882429i	$-0.655907\pi$
$277$	146.528	−	253.794i	0.528983	−	0.916225i	0.999429	−	0.0337961i	$-0.0107597\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−14.6700	+	25.4091i	−0.0522063	+	0.0904239i	−0.890948	−	0.454106i	$-0.849959\pi$
$281$	−14.6700	+	25.4091i	−0.0522063	+	0.0904239i	0.838741	+	0.544530i	$0.183292\pi$
$282$		0			0
$283$	179.527	−	103.650i	0.634372	−	0.366255i	−0.148071	−	0.988977i	$-0.547307\pi$
$283$	179.527	−	103.650i	0.634372	−	0.366255i	0.782443	+	0.622722i	$0.213973\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	122.860i		−	0.428083i
$288$		0			0
$289$	−129.648			−0.448609
$290$		0			0
$291$		0			0
$292$		0			0
$293$	248.905	+	431.116i	0.849504	+	1.47138i	0.881651	+	0.471901i	$0.156432\pi$
$293$	248.905	+	431.116i	0.849504	+	1.47138i	−0.0321473	+	0.999483i	$0.510235\pi$
$294$		0			0
$295$	182.303	+	105.252i	0.617975	+	0.356788i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	333.360	+	192.465i	1.11492	+	0.643697i
$300$		0			0
$301$	15.1395	+	26.2223i	0.0502973	+	0.0871174i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	68.3765			0.224185
$306$		0			0
$307$		−	32.7775i		−	0.106767i	−0.998574	−	0.0533835i	$-0.982999\pi$
$307$		−	32.7775i		−	0.106767i	0.998574	−	0.0533835i	$-0.0170006\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−330.752	+	190.960i	−1.06351	+	0.614019i	−0.926402	−	0.376537i	$-0.877115\pi$
$311$	−330.752	+	190.960i	−1.06351	+	0.614019i	−0.137110	+	0.990556i	$0.543782\pi$
$312$		0			0
$313$	247.593	−	428.844i	0.791032	−	1.37011i	−0.134297	−	0.990941i	$-0.542878\pi$
$313$	247.593	−	428.844i	0.791032	−	1.37011i	0.925329	−	0.379166i	$-0.123789\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	140.200	−	242.834i	0.442273	−	0.766039i	−0.555585	−	0.831460i	$-0.687506\pi$
$317$	140.200	−	242.834i	0.442273	−	0.766039i	0.997858	+	0.0654209i	$0.0208390\pi$
$318$		0			0
$319$	−217.582	+	125.621i	−0.682075	+	0.393796i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			323.825i			1.00255i
$324$		0			0
$325$	−152.777			−0.470084
$326$		0			0
$327$		0			0
$328$		0			0
$329$	70.3056	+	121.773i	0.213695	+	0.370131i
$330$		0			0
$331$	−4.57609	−	2.64201i	−0.0138251	−	0.00798190i	0.493072	−	0.869989i	$-0.335874\pi$
$331$	−4.57609	−	2.64201i	−0.0138251	−	0.00798190i	−0.506897	+	0.862007i	$0.669207\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−74.7218	−	43.1407i	−0.223050	−	0.128778i
$336$		0			0
$337$	−121.945	−	211.215i	−0.361855	−	0.626751i	0.626411	−	0.779493i	$-0.284523\pi$
$337$	−121.945	−	211.215i	−0.361855	−	0.626751i	−0.988266	+	0.152742i	$0.951190\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−703.056			−2.06175
$342$		0			0
$343$			174.165i			0.507770i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	168.452	−	97.2556i	0.485451	−	0.280275i	−0.237234	−	0.971452i	$-0.576241\pi$
$347$	168.452	−	97.2556i	0.485451	−	0.280275i	0.722685	+	0.691177i	$0.242908\pi$
$348$		0			0
$349$	202.139	−	350.116i	0.579196	−	1.00320i	−0.416376	−	0.909193i	$-0.636700\pi$
$349$	202.139	−	350.116i	0.579196	−	1.00320i	0.995572	−	0.0940045i	$-0.0299668\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−145.463	+	251.950i	−0.412078	+	0.713739i	−0.995117	−	0.0987047i	$-0.968530\pi$
$353$	−145.463	+	251.950i	−0.412078	+	0.713739i	0.583039	+	0.812444i	$0.301863\pi$
$354$		0			0
$355$	145.842	−	84.2020i	0.410823	−	0.237189i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			554.699i			1.54512i	0.634941	+	0.772561i	$0.281024\pi$
$359$			554.699i			1.54512i	−0.634941	+	0.772561i	$0.718976\pi$
$360$		0			0
$361$	−297.056			−0.822871
$362$		0			0
$363$		0			0
$364$		0			0
$365$	89.4131	+	154.868i	0.244967	+	0.424296i
$366$		0			0
$367$	11.3661	+	6.56223i	0.0309703	+	0.0178807i	0.515405	−	0.856947i	$-0.327641\pi$
$367$	11.3661	+	6.56223i	0.0309703	+	0.0178807i	−0.484435	+	0.874827i	$0.660975\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−22.7322	−	13.1245i	−0.0612729	−	0.0353759i
$372$		0			0
$373$	209.306	+	362.528i	0.561141	+	0.971925i	0.997397	+	0.0721024i	$0.0229708\pi$
$373$	209.306	+	362.528i	0.561141	+	0.971925i	−0.436256	+	0.899823i	$0.643696\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	278.303			0.738205
$378$		0			0
$379$		−	546.545i		−	1.44207i	−0.692898	−	0.721036i	$-0.743666\pi$
$379$		−	546.545i		−	1.44207i	0.692898	−	0.721036i	$-0.256334\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	511.874	−	295.530i	1.33649	−	0.771620i	0.350201	−	0.936675i	$-0.386113\pi$
$383$	511.874	−	295.530i	1.33649	−	0.771620i	0.986284	+	0.165055i	$0.0527801\pi$
$384$		0			0
$385$	38.7508	−	67.1183i	0.100651	−	0.174333i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	172.694	−	299.115i	0.443944	−	0.768934i	−0.554034	−	0.832494i	$-0.686912\pi$
$389$	172.694	−	299.115i	0.443944	−	0.768934i	0.997978	+	0.0635601i	$0.0202454\pi$
$390$		0			0
$391$	326.964	−	188.773i	0.836224	−	0.482794i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	219.255i		−	0.555075i
$396$		0			0
$397$	385.741			0.971639			0.485820	−	0.874059i	$-0.338521\pi$
$397$	385.741			0.971639			0.485820	+	0.874059i	$0.338521\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	59.0869	+	102.341i	0.147349	+	0.255216i	0.930247	−	0.366934i	$-0.119593\pi$
$401$	59.0869	+	102.341i	0.147349	+	0.255216i	−0.782898	+	0.622150i	$0.786259\pi$
$402$		0			0
$403$	674.445	+	389.391i	1.67356	+	0.966231i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	259.012	+	149.541i	0.636394	+	0.367422i
$408$		0			0
$409$	167.852	+	290.728i	0.410396	+	0.710827i	0.994933	−	0.100540i	$-0.0320570\pi$
$409$	167.852	+	290.728i	0.410396	+	0.710827i	−0.584537	+	0.811367i	$0.698724\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−106.944			−0.258943
$414$		0			0
$415$		−	360.977i		−	0.869825i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28.8831	+	16.6757i	−0.0689334	+	0.0397987i	−0.534071	−	0.845440i	$-0.679338\pi$
$419$	−28.8831	+	16.6757i	−0.0689334	+	0.0397987i	0.465137	+	0.885239i	$0.346005\pi$
$420$		0			0
$421$	−138.824	+	240.450i	−0.329748	+	0.571140i	−0.982462	−	0.186464i	$-0.940297\pi$
$421$	−138.824	+	240.450i	−0.329748	+	0.571140i	0.652714	+	0.757605i	$0.273630\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−74.9230	+	129.770i	−0.176289	+	0.305342i
$426$		0			0
$427$	−30.0836	+	17.3688i	−0.0704535	+	0.0406763i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	368.795i		−	0.855674i	−0.903856	−	0.427837i	$-0.859276\pi$
$431$		−	368.795i		−	0.855674i	0.903856	−	0.427837i	$-0.140724\pi$
$432$		0			0
$433$	585.093			1.35125			0.675627	−	0.737244i	$-0.263873\pi$
$433$	585.093			1.35125			0.675627	+	0.737244i	$0.263873\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	383.611	+	664.434i	0.877829	+	1.52044i
$438$		0			0
$439$	569.640	+	328.882i	1.29759	+	0.749161i	0.979987	−	0.199064i	$-0.0637901\pi$
$439$	569.640	+	328.882i	1.29759	+	0.749161i	0.317599	+	0.948225i	$0.397123\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−96.1502	−	55.5124i	−0.217043	−	0.125310i	0.387537	−	0.921854i	$-0.373326\pi$
$443$	−96.1502	−	55.5124i	−0.217043	−	0.125310i	−0.604581	+	0.796544i	$0.706659\pi$
$444$		0			0
$445$	−280.797	−	486.355i	−0.631005	−	1.09293i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−136.056			−0.303021			−0.151510	−	0.988456i	$-0.548414\pi$
$449$	−136.056			−0.303021			−0.151510	+	0.988456i	$0.548414\pi$
$450$		0			0
$451$			775.459i			1.71942i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−74.3476	+	42.9246i	−0.163401	+	0.0943398i
$456$		0			0
$457$	29.0747	−	50.3588i	0.0636208	−	0.110194i	−0.832461	−	0.554084i	$-0.813069\pi$
$457$	29.0747	−	50.3588i	0.0636208	−	0.110194i	0.896081	+	0.443890i	$0.146402\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−51.4916	+	89.1861i	−0.111695	+	0.193462i	−0.916454	−	0.400140i	$-0.868961\pi$
$461$	−51.4916	+	89.1861i	−0.111695	+	0.193462i	0.804759	+	0.593602i	$0.202295\pi$
$462$		0			0
$463$	672.083	−	388.027i	1.45158	−	0.838072i	0.453012	−	0.891504i	$-0.350349\pi$
$463$	672.083	−	388.027i	1.45158	−	0.838072i	0.998571	+	0.0534320i	$0.0170160\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			692.620i			1.48313i	0.670882	+	0.741564i	$0.265915\pi$
$467$			692.620i			1.48313i	−0.670882	+	0.741564i	$0.734085\pi$
$468$		0			0
$469$	43.8338			0.0934623
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−95.5564	−	165.509i	−0.202022	−	0.349912i
$474$		0			0
$475$	−263.711	−	152.254i	−0.555181	−	0.320534i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	630.014	+	363.739i	1.31527	+	0.759371i	0.982963	−	0.183801i	$-0.0588402\pi$
$479$	630.014	+	363.739i	1.31527	+	0.759371i	0.332305	+	0.943172i	$0.392174\pi$
$480$		0			0
$481$	−165.648	−	286.911i	−0.344382	−	0.596488i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−152.992			−0.315448
$486$		0			0
$487$		−	285.503i		−	0.586248i	−0.956074	−	0.293124i	$-0.905305\pi$
$487$		−	285.503i		−	0.586248i	0.956074	−	0.293124i	$-0.0946948\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−188.576	+	108.875i	−0.384066	+	0.221740i	−0.679586	−	0.733596i	$-0.737840\pi$
$491$	−188.576	+	108.875i	−0.384066	+	0.221740i	0.295520	+	0.955337i	$0.404507\pi$
$492$		0			0
$493$	136.482	−	236.393i	0.276839	−	0.479499i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−42.7774	+	74.0927i	−0.0860713	+	0.149080i
$498$		0			0
$499$	30.1869	−	17.4284i	0.0604948	−	0.0349267i	−0.469448	−	0.882960i	$-0.655547\pi$
$499$	30.1869	−	17.4284i	0.0604948	−	0.0349267i	0.529942	+	0.848034i	$0.322214\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			345.125i			0.686134i	0.939311	+	0.343067i	$0.111466\pi$
$503$			345.125i			0.686134i	−0.939311	+	0.343067i	$0.888534\pi$
$504$		0			0
$505$	−457.834			−0.906602
$506$		0			0
$507$		0			0
$508$		0			0
$509$	189.528	+	328.272i	0.372354	+	0.644936i	0.989927	−	0.141577i	$-0.0452174\pi$
$509$	189.528	+	328.272i	0.372354	+	0.644936i	−0.617573	+	0.786513i	$0.711884\pi$
$510$		0			0
$511$	−78.6782	−	45.4249i	−0.153969	−	0.0888941i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	86.6493	+	50.0270i	0.168251	+	0.0971398i
$516$		0			0
$517$	−443.751	−	768.599i	−0.858319	−	1.48665i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−40.2028			−0.0771646			−0.0385823	−	0.999255i	$-0.512284\pi$
$521$	−40.2028			−0.0771646			−0.0385823	+	0.999255i	$0.512284\pi$
$522$		0			0
$523$			623.980i			1.19308i	0.802584	+	0.596539i	$0.203458\pi$
$523$			623.980i			1.19308i	−0.802584	+	0.596539i	$0.796542\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	661.505	−	381.920i	1.25523	−	0.724706i
$528$		0			0
$529$	182.749	−	316.531i	0.345462	−	0.598357i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	429.492	−	743.901i	0.805800	−	1.39569i
$534$		0			0
$535$	380.134	−	219.471i	0.710531	−	0.410225i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	529.955i		−	0.983219i
$540$		0			0
$541$	−529.741			−0.979188			−0.489594	−	0.871950i	$-0.662855\pi$
$541$	−529.741			−0.979188			−0.489594	+	0.871950i	$0.662855\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−111.858	−	193.744i	−0.205244	−	0.355494i
$546$		0			0
$547$	512.558	+	295.925i	0.937035	+	0.540997i	0.889029	−	0.457850i	$-0.151380\pi$
$547$	512.558	+	295.925i	0.937035	+	0.540997i	0.0480051	+	0.998847i	$0.484714\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	480.383	+	277.349i	0.871839	+	0.503356i
$552$		0			0
$553$	55.6944	+	96.4655i	0.100713	+	0.174440i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−682.841			−1.22593			−0.612964	−	0.790111i	$-0.710023\pi$
$557$	−682.841			−1.22593			−0.612964	+	0.790111i	$0.710023\pi$
$558$		0			0
$559$			211.698i			0.378708i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−889.956	+	513.816i	−1.58074	+	0.912640i	−0.585987	+	0.810321i	$0.699293\pi$
$563$	−889.956	+	513.816i	−1.58074	+	0.912640i	−0.994752	+	0.102319i	$0.967374\pi$
$564$		0			0
$565$	−156.066	+	270.315i	−0.276224	+	0.478433i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	165.913	−	287.370i	0.291587	−	0.505044i	−0.682598	−	0.730794i	$-0.739150\pi$
$569$	165.913	−	287.370i	0.291587	−	0.505044i	0.974185	+	0.225750i	$0.0724833\pi$
$570$		0			0
$571$	289.987	−	167.424i	0.507858	−	0.293212i	−0.224095	−	0.974567i	$-0.571943\pi$
$571$	289.987	−	167.424i	0.507858	−	0.293212i	0.731953	+	0.681356i	$0.238609\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			355.023i			0.617431i
$576$		0			0
$577$	−87.6845			−0.151966			−0.0759831	−	0.997109i	$-0.524209\pi$
$577$	−87.6845			−0.151966			−0.0759831	+	0.997109i	$0.524209\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	91.6944	+	158.819i	0.157822	+	0.273355i
$582$		0			0
$583$	143.480	+	82.8382i	0.246106	+	0.142089i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	116.275	+	67.1313i	0.198083	+	0.114363i	0.595761	−	0.803162i	$-0.296851\pi$
$587$	116.275	+	67.1313i	0.198083	+	0.114363i	−0.397678	+	0.917525i	$0.630184\pi$
$588$		0			0
$589$	776.113	+	1344.27i	1.31768	+	2.28229i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−256.915			−0.433246			−0.216623	−	0.976255i	$-0.569504\pi$
$593$	−256.915			−0.433246			−0.216623	+	0.976255i	$0.569504\pi$
$594$		0			0
$595$			84.2020i			0.141516i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−851.385	+	491.547i	−1.42134	+	0.820613i	−0.996414	−	0.0846136i	$-0.973034\pi$
$599$	−851.385	+	491.547i	−1.42134	+	0.820613i	−0.424929	+	0.905227i	$0.639701\pi$
$600$		0			0
$601$	−558.909	+	968.058i	−0.929964	+	1.61075i	−0.146588	+	0.989198i	$0.546829\pi$
$601$	−558.909	+	968.058i	−0.929964	+	1.61075i	−0.783376	+	0.621548i	$0.786504\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−25.3643	+	43.9323i	−0.0419245	+	0.0726154i
$606$		0			0
$607$	259.238	−	149.671i	0.427081	−	0.246575i	−0.271021	−	0.962573i	$-0.587361\pi$
$607$	259.238	−	149.671i	0.427081	−	0.246575i	0.698102	+	0.715998i	$0.254028\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			983.094i			1.60899i
$612$		0			0
$613$	468.332			0.764001			0.382000	−	0.924162i	$-0.375235\pi$
$613$	468.332			0.764001			0.382000	+	0.924162i	$0.375235\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	396.884	+	687.423i	0.643248	+	1.11414i	0.984703	+	0.174240i	$0.0557470\pi$
$617$	396.884	+	687.423i	0.643248	+	1.11414i	−0.341455	+	0.939898i	$0.610920\pi$
$618$		0			0
$619$	−782.130	−	451.563i	−1.26354	−	0.729504i	−0.289780	−	0.957093i	$-0.593582\pi$
$619$	−782.130	−	451.563i	−1.26354	−	0.729504i	−0.973757	+	0.227590i	$0.926915\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	247.085	+	142.654i	0.396605	+	0.228980i
$624$		0			0
$625$	93.6662	+	162.235i	0.149866	+	0.259575i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−324.939			−0.516596
$630$		0			0
$631$			484.076i			0.767156i	0.923508	+	0.383578i	$0.125308\pi$
$631$			484.076i			0.767156i	−0.923508	+	0.383578i	$0.874692\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	575.417	−	332.217i	0.906168	−	0.523176i
$636$		0			0
$637$	−293.518	+	508.389i	−0.460782	+	0.798098i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	173.864	−	301.142i	0.271239	−	0.469800i	−0.697940	−	0.716156i	$-0.745900\pi$
$641$	173.864	−	301.142i	0.271239	−	0.469800i	0.969179	+	0.246356i	$0.0792332\pi$
$642$		0			0
$643$	162.862	−	94.0285i	0.253285	−	0.146234i	−0.367983	−	0.929833i	$-0.619951\pi$
$643$	162.862	−	94.0285i	0.253285	−	0.146234i	0.621267	+	0.783599i	$0.286618\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			853.348i			1.31893i	0.751735	+	0.659465i	$0.229217\pi$
$647$			853.348i			1.31893i	−0.751735	+	0.659465i	$0.770783\pi$
$648$		0			0
$649$	675.000			1.04006
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−599.678	−	1038.67i	−0.918342	−	1.59062i	−0.801933	−	0.597414i	$-0.796195\pi$
$653$	−599.678	−	1038.67i	−0.918342	−	1.59062i	−0.116410	−	0.993201i	$-0.537139\pi$
$654$		0			0
$655$	−542.183	−	313.030i	−0.827761	−	0.477908i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	534.606	+	308.655i	0.811238	+	0.468369i	0.847386	−	0.530978i	$-0.178175\pi$
$659$	534.606	+	308.655i	0.811238	+	0.468369i	−0.0361474	+	0.999346i	$0.511509\pi$
$660$		0			0
$661$	109.880	+	190.318i	0.166233	+	0.287925i	0.937093	−	0.349081i	$-0.113506\pi$
$661$	109.880	+	190.318i	0.166233	+	0.287925i	−0.770859	+	0.637006i	$0.780173\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−171.110			−0.257308
$666$		0			0
$667$		−	646.719i		−	0.969593i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	189.880	−	109.627i	0.282981	−	0.163379i
$672$		0			0
$673$	−289.492	+	501.414i	−0.430151	+	0.745043i	−0.996886	−	0.0788568i	$-0.974873\pi$
$673$	−289.492	+	501.414i	−0.430151	+	0.745043i	0.566735	+	0.823900i	$0.308206\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	351.528	−	608.865i	0.519244	−	0.899357i	−0.480506	−	0.876992i	$-0.659547\pi$
$677$	351.528	−	608.865i	0.519244	−	0.899357i	0.999750	−	0.0223655i	$-0.00711975\pi$
$678$		0			0
$679$	67.3121	−	38.8627i	0.0991342	−	0.0572351i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			881.535i			1.29068i	0.763895	+	0.645340i	$0.223284\pi$
$683$			881.535i			1.29068i	−0.763895	+	0.645340i	$0.776716\pi$
$684$		0			0
$685$	580.651			0.847666
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−91.7607	−	158.934i	−0.133179	−	0.230674i
$690$		0			0
$691$	890.846	+	514.330i	1.28921	+	0.744328i	0.978514	−	0.206181i	$-0.0661034\pi$
$691$	890.846	+	514.330i	1.28921	+	0.744328i	0.310699	+	0.950508i	$0.399437\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−695.603	−	401.607i	−1.00087	−	0.577851i
$696$		0			0
$697$	−421.251	−	729.628i	−0.604377	−	1.04681i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	813.287			1.16018			0.580090	−	0.814552i	$-0.303017\pi$
$701$	813.287			1.16018			0.580090	+	0.814552i	$0.303017\pi$
$702$		0			0
$703$		−	660.320i		−	0.939289i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	201.433	−	116.298i	0.284913	−	0.164494i
$708$		0			0
$709$	−82.4916	+	142.880i	−0.116349	+	0.201523i	−0.918318	−	0.395843i	$-0.870453\pi$
$709$	−82.4916	+	142.880i	−0.116349	+	0.201523i	0.801969	+	0.597366i	$0.203786\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	904.864	−	1567.27i	1.26909	−	2.19813i
$714$		0			0
$715$	469.262	−	270.929i	0.656311	−	0.378921i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			179.881i			0.250182i	0.992145	+	0.125091i	$0.0399223\pi$
$719$			179.881i			0.250182i	−0.992145	+	0.125091i	$0.960078\pi$
$720$		0			0
$721$	−50.8308			−0.0705004
$722$		0			0
$723$		0			0
$724$		0			0
$725$	128.340	+	222.291i	0.177021	+	0.306609i
$726$		0			0
$727$	664.674	+	383.749i	0.914269	+	0.527853i	0.881802	−	0.471619i	$-0.156330\pi$
$727$	664.674	+	383.749i	0.914269	+	0.527853i	0.0324668	+	0.999473i	$0.489664\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	179.818	+	103.818i	0.245989	+	0.142022i
$732$		0			0
$733$	−505.139	−	874.927i	−0.689140	−	1.19363i	−0.972117	−	0.234498i	$-0.924655\pi$
$733$	−505.139	−	874.927i	−0.689140	−	1.19363i	0.282977	−	0.959127i	$-0.408678\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−276.668			−0.375397
$738$		0			0
$739$		−	941.306i		−	1.27376i	−0.770964	−	0.636878i	$-0.780225\pi$
$739$		−	941.306i		−	1.27376i	0.770964	−	0.636878i	$-0.219775\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	94.8464	−	54.7596i	0.127653	−	0.0737007i	−0.434814	−	0.900521i	$-0.643186\pi$
$743$	94.8464	−	54.7596i	0.127653	−	0.0737007i	0.562467	+	0.826820i	$0.309852\pi$
$744$		0			0
$745$	340.731	−	590.163i	0.457357	−	0.792166i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−111.498	+	193.121i	−0.148863	+	0.257839i
$750$		0			0
$751$	−706.330	+	407.800i	−0.940519	+	0.543009i	−0.890123	−	0.455720i	$-0.849382\pi$
$751$	−706.330	+	407.800i	−0.940519	+	0.543009i	−0.0503961	+	0.998729i	$0.516048\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	405.158i		−	0.536633i
$756$		0			0
$757$	−1253.34			−1.65566			−0.827830	−	0.560978i	$-0.810425\pi$
$757$	−1253.34			−1.65566			−0.827830	+	0.560978i	$0.810425\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	183.095	+	317.130i	0.240598	+	0.416728i	0.960885	−	0.276948i	$-0.0893230\pi$
$761$	183.095	+	317.130i	0.240598	+	0.416728i	−0.720287	+	0.693677i	$0.755990\pi$
$762$		0			0
$763$	98.4286	+	56.8278i	0.129002	+	0.0744794i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−647.531	−	373.852i	−0.844238	−	0.487421i
$768$		0			0
$769$	539.066	+	933.690i	0.700996	+	1.21416i	0.968117	+	0.250498i	$0.0805945\pi$
$769$	539.066	+	933.690i	0.700996	+	1.21416i	−0.267121	+	0.963663i	$0.586072\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	283.085			0.366217			0.183108	−	0.983093i	$-0.441384\pi$
$773$	283.085			0.366217			0.183108	+	0.983093i	$0.441384\pi$
$774$		0			0
$775$			718.273i			0.926804i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1482.70	−	856.039i	1.90334	−	1.09889i
$780$		0			0
$781$	270.000	−	467.654i	0.345711	−	0.598788i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−186.374	+	322.810i	−0.237419	+	0.411222i
$786$		0			0
$787$	255.224	−	147.353i	0.324299	−	0.187234i	−0.329008	−	0.944327i	$-0.606714\pi$
$787$	255.224	−	147.353i	0.324299	−	0.187234i	0.653307	+	0.757093i	$0.273381\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	158.574i		−	0.200473i
$792$		0			0
$793$	−242.870			−0.306268
$794$		0			0
$795$		0			0
$796$		0			0
$797$	396.653	+	687.023i	0.497683	+	0.862012i	0.999996	−	0.00267363i	$-0.000851044\pi$
$797$	396.653	+	687.023i	0.497683	+	0.862012i	−0.502314	+	0.864685i	$0.667518\pi$
$798$		0			0
$799$	835.049	+	482.116i	1.04512	+	0.603399i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	496.596	+	286.710i	0.618426	+	0.357049i
$804$		0			0
$805$	99.7477	+	172.768i	0.123910	+	0.214619i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1215.12			−1.50200			−0.751000	−	0.660303i	$-0.770428\pi$
$809$	−1215.12			−1.50200			−0.751000	+	0.660303i	$0.770428\pi$
$810$		0			0
$811$		−	429.447i		−	0.529527i	−0.964313	−	0.264764i	$-0.914706\pi$
$811$		−	429.447i		−	0.529527i	0.964313	−	0.264764i	$-0.0852939\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−643.988	+	371.806i	−0.790169	+	0.456204i
$816$		0			0
$817$	−210.972	+	365.414i	−0.258227	+	0.447263i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.68216	−	6.37769i	0.00448497	−	0.00776820i	−0.863774	−	0.503879i	$-0.831906\pi$
$821$	3.68216	−	6.37769i	0.00448497	−	0.00776820i	0.868259	+	0.496111i	$0.165239\pi$
$822$		0			0
$823$	−1169.36	+	675.132i	−1.42085	+	0.820331i	−0.996372	−	0.0851049i	$-0.972877\pi$
$823$	−1169.36	+	675.132i	−1.42085	+	0.820331i	−0.424483	+	0.905436i	$0.639544\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	571.482i		−	0.691031i	−0.938413	−	0.345515i	$-0.887704\pi$
$827$		−	571.482i		−	0.691031i	0.938413	−	0.345515i	$-0.112296\pi$
$828$		0			0
$829$	888.445			1.07171			0.535854	−	0.844311i	$-0.319990\pi$
$829$	888.445			1.07171			0.535854	+	0.844311i	$0.319990\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	287.886	+	498.634i	0.345602	+	0.598600i
$834$		0			0
$835$	166.774	+	96.2867i	0.199729	+	0.115313i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	506.659	+	292.519i	0.603884	+	0.348652i	0.770568	−	0.637358i	$-0.219973\pi$
$839$	506.659	+	292.519i	0.603884	+	0.348652i	−0.166684	+	0.986010i	$0.553306\pi$
$840$		0			0
$841$	186.713	+	323.396i	0.222013	+	0.384537i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.1464			0.0143744
$846$		0			0
$847$		−	25.7719i		−	0.0304272i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−666.720	+	384.931i	−0.783455	+	0.452328i
$852$		0			0
$853$	−437.917	+	758.494i	−0.513384	+	0.889208i	0.486495	+	0.873683i	$0.338275\pi$
$853$	−437.917	+	758.494i	−0.513384	+	0.889208i	−0.999879	+	0.0155246i	$0.995058\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	186.836	−	323.610i	0.218012	−	0.377608i	−0.736188	−	0.676777i	$-0.763376\pi$
$857$	186.836	−	323.610i	0.218012	−	0.377608i	0.954200	+	0.299169i	$0.0967095\pi$
$858$		0			0
$859$	253.378	−	146.288i	0.294968	−	0.170300i	−0.345212	−	0.938525i	$-0.612193\pi$
$859$	253.378	−	146.288i	0.294968	−	0.170300i	0.640180	+	0.768225i	$0.278860\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1461.41i		−	1.69341i	−0.532066	−	0.846703i	$-0.678584\pi$
$863$		−	1461.41i		−	1.69341i	0.532066	−	0.846703i	$-0.321416\pi$
$864$		0			0
$865$	682.313			0.788801
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−351.528	−	608.865i	−0.404520	−	0.700650i
$870$		0			0
$871$	265.409	+	153.234i	0.304717	+	0.175929i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−212.986	−	122.968i	−0.243413	−	0.140535i
$876$		0			0
$877$	−38.5648	−	66.7962i	−0.0439735	−	0.0761644i	0.843201	−	0.537599i	$-0.180668\pi$
$877$	−38.5648	−	66.7962i	−0.0439735	−	0.0761644i	−0.887174	+	0.461434i	$0.847335\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−371.927			−0.422164			−0.211082	−	0.977468i	$-0.567699\pi$
$881$	−371.927			−0.422164			−0.211082	+	0.977468i	$0.567699\pi$
$882$		0			0
$883$			875.990i			0.992061i	0.868305	+	0.496030i	$0.165210\pi$
$883$			875.990i			0.992061i	−0.868305	+	0.496030i	$0.834790\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−247.646	+	142.979i	−0.279195	+	0.161193i	−0.633059	−	0.774104i	$-0.718201\pi$
$887$	−247.646	+	142.979i	−0.279195	+	0.161193i	0.353864	+	0.935297i	$0.384868\pi$
$888$		0			0
$889$	−168.777	+	292.331i	−0.189851	+	0.328831i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−979.724	+	1696.93i	−1.09712	+	1.90026i
$894$		0			0
$895$	291.684	−	168.404i	0.325904	−	0.188161i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1308.42i		−	1.45542i
$900$		0			0
$901$	−180.000			−0.199778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	435.623	+	754.522i	0.481352	+	0.833726i
$906$		0			0
$907$	615.208	+	355.190i	0.678289	+	0.391610i	0.799210	−	0.601052i	$-0.205252\pi$
$907$	615.208	+	355.190i	0.678289	+	0.391610i	−0.120921	+	0.992662i	$0.538585\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−167.148	−	96.5028i	−0.183477	−	0.105931i	0.405448	−	0.914118i	$-0.367115\pi$
$911$	−167.148	−	96.5028i	−0.183477	−	0.105931i	−0.588925	+	0.808187i	$0.700449\pi$
$912$		0			0
$913$	−578.751	−	1002.43i	−0.633900	−	1.09795i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	318.059			0.346848
$918$		0			0
$919$			1106.39i			1.20390i	0.798533	+	0.601951i	$0.205610\pi$
$919$			1106.39i			1.20390i	−0.798533	+	0.601951i	$0.794390\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−518.025	+	299.082i	−0.561240	+	0.324032i
$924$		0			0
$925$	152.777	−	264.618i	0.165165	−	0.286074i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−54.4108	+	94.2423i	−0.0585692	+	0.101445i	−0.893823	−	0.448419i	$-0.851987\pi$
$929$	−54.4108	+	94.2423i	−0.0585692	+	0.101445i	0.835254	+	0.549864i	$0.185321\pi$
$930$		0			0
$931$	−1013.29	+	585.024i	−1.08839	+	0.628383i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	531.461i		−	0.568407i
$936$		0			0
$937$	−1530.74			−1.63366			−0.816832	−	0.576875i	$-0.804272\pi$
$937$	−1530.74			−1.63366			−0.816832	+	0.576875i	$0.804272\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−730.350	−	1265.00i	−0.776142	−	1.34432i	−0.934150	−	0.356880i	$-0.883841\pi$
$941$	−730.350	−	1265.00i	−0.776142	−	1.34432i	0.158008	−	0.987438i	$-0.449493\pi$
$942$		0			0
$943$	−1728.67	−	998.048i	−1.83316	−	1.05838i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−292.181	−	168.691i	−0.308533	−	0.178132i	0.337737	−	0.941241i	$-0.390339\pi$
$947$	−292.181	−	168.691i	−0.308533	−	0.178132i	−0.646270	+	0.763109i	$0.723672\pi$
$948$		0			0
$949$	−317.591	−	550.085i	−0.334659	−	0.579647i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.6997			0.0122767			0.00613834	−	0.999981i	$-0.498046\pi$
$953$	11.6997			0.0122767			0.00613834	+	0.999981i	$0.498046\pi$
$954$		0			0
$955$		−	205.050i		−	0.214712i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−255.469	+	147.495i	−0.266391	+	0.153801i
$960$		0			0
$961$	1350.20	−	2338.61i	1.40499	−	2.43352i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−182.516	+	316.127i	−0.189136	+	0.327593i
$966$		0			0
$967$	1072.38	−	619.139i	1.10898	−	0.640268i	0.170413	−	0.985373i	$-0.445490\pi$
$967$	1072.38	−	619.139i	1.10898	−	0.640268i	0.938564	+	0.345104i	$0.112156\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	52.4978i		−	0.0540658i	−0.999635	−	0.0270329i	$-0.991394\pi$
$971$		−	52.4978i		−	0.0540658i	0.999635	−	0.0270329i	$-0.00860588\pi$
$972$		0			0
$973$	408.059			0.419383
$974$		0			0
$975$		0			0
$976$		0			0
$977$	52.7789	+	91.4158i	0.0540214	+	0.0935679i	0.891772	−	0.452486i	$-0.149463\pi$
$977$	52.7789	+	91.4158i	0.0540214	+	0.0935679i	−0.837750	+	0.546054i	$0.816129\pi$
$978$		0			0
$979$	−1559.53	−	900.398i	−1.59299	−	0.919712i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	5.77662	+	3.33513i	0.00587652	+	0.00339281i	0.502935	−	0.864324i	$-0.332253\pi$
$983$	5.77662	+	3.33513i	0.00587652	+	0.00339281i	−0.497059	+	0.867717i	$0.665587\pi$
$984$		0			0
$985$	−208.797	−	361.647i	−0.211977	−	0.367155i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	491.941			0.497412
$990$		0			0
$991$		−	1394.27i		−	1.40694i	−0.710727	−	0.703468i	$-0.751634\pi$
$991$		−	1394.27i		−	1.40694i	0.710727	−	0.703468i	$-0.248366\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−598.523	+	345.557i	−0.601531	+	0.347294i
$996$		0			0
$997$	634.771	−	1099.45i	0.636681	−	1.10276i	−0.349476	−	0.936945i	$-0.613640\pi$
$997$	634.771	−	1099.45i	0.636681	−	1.10276i	0.986156	−	0.165818i	$-0.0530263\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.o.d.127.3		8
3.2	odd	2		576.3.o.e.511.4		8
4.3	odd	2	inner	1728.3.o.d.127.4		8
8.3	odd	2		432.3.o.c.127.2		8
8.5	even	2		432.3.o.c.127.1		8
9.4	even	3	inner	1728.3.o.d.1279.4		8
9.5	odd	6		576.3.o.e.319.1		8
12.11	even	2		576.3.o.e.511.1		8
24.5	odd	2		144.3.o.b.79.1	yes	8
24.11	even	2		144.3.o.b.79.4	yes	8
36.23	even	6		576.3.o.e.319.4		8
36.31	odd	6	inner	1728.3.o.d.1279.3		8
72.5	odd	6		144.3.o.b.31.4	yes	8
72.11	even	6		1296.3.g.h.1135.1		4
72.13	even	6		432.3.o.c.415.2		8
72.29	odd	6		1296.3.g.h.1135.2		4
72.43	odd	6		1296.3.g.d.1135.3		4
72.59	even	6		144.3.o.b.31.1	✓	8
72.61	even	6		1296.3.g.d.1135.4		4
72.67	odd	6		432.3.o.c.415.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.o.b.31.1	✓	8	72.59	even	6
144.3.o.b.31.4	yes	8	72.5	odd	6
144.3.o.b.79.1	yes	8	24.5	odd	2
144.3.o.b.79.4	yes	8	24.11	even	2
432.3.o.c.127.1		8	8.5	even	2
432.3.o.c.127.2		8	8.3	odd	2
432.3.o.c.415.1		8	72.67	odd	6
432.3.o.c.415.2		8	72.13	even	6
576.3.o.e.319.1		8	9.5	odd	6
576.3.o.e.319.4		8	36.23	even	6
576.3.o.e.511.1		8	12.11	even	2
576.3.o.e.511.4		8	3.2	odd	2
1296.3.g.d.1135.3		4	72.43	odd	6
1296.3.g.d.1135.4		4	72.61	even	6
1296.3.g.h.1135.1		4	72.11	even	6
1296.3.g.h.1135.2		4	72.29	odd	6
1728.3.o.d.127.3		8	1.1	even	1	trivial
1728.3.o.d.127.4		8	4.3	odd	2	inner
1728.3.o.d.1279.3		8	36.31	odd	6	inner
1728.3.o.d.1279.4		8	9.4	even	3	inner

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.81174 + 3.13802i) q^{5} +(-1.59422 - 0.920424i) q^{7} +O(q^{10})\)	\(q+(1.81174 + 3.13802i) q^{5} +(-1.59422 - 0.920424i) q^{7} +(10.0623 + 5.80948i) q^{11} +(-6.43521 - 11.1461i) q^{13} -12.6235 q^{17} -25.6526i q^{19} +(-25.9012 + 14.9541i) q^{23} +(5.93521 - 10.2801i) q^{25} +(-10.8117 + 18.7265i) q^{29} +(-52.4027 + 30.2547i) q^{31} -6.67027i q^{35} +25.7409 q^{37} +(33.3704 + 57.7993i) q^{41} +(-14.2447 - 8.22418i) q^{43} +(-66.1505 - 38.1920i) q^{47} +(-22.8056 - 39.5005i) q^{49} +14.2591 q^{53} +42.1010i q^{55} +(50.3115 - 29.0474i) q^{59} +(9.43521 - 16.3423i) q^{61} +(23.3178 - 40.3877i) q^{65} +(-20.6216 + 11.9059i) q^{67} -46.4758i q^{71} +49.3521 q^{73} +(-10.6944 - 18.5232i) q^{77} +(-52.4027 - 30.2547i) q^{79} +(-86.2751 - 49.8109i) q^{83} +(-22.8704 - 39.6127i) q^{85} -154.988 q^{89} +23.6925i q^{91} +(80.4984 - 46.4758i) q^{95} +(-21.1113 + 36.5658i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5}+O(q^{10}) \) 8 * q - 6 * q^5	\( 8 q - 6 q^{5} + 10 q^{13} - 60 q^{17} - 14 q^{25} - 66 q^{29} - 40 q^{37} + 144 q^{41} + 2 q^{49} + 360 q^{53} + 14 q^{61} + 330 q^{65} - 220 q^{73} - 270 q^{77} - 60 q^{85} - 912 q^{89} + 200 q^{97}+O(q^{100}) \) 8 * q - 6 * q^5 + 10 * q^13 - 60 * q^17 - 14 * q^25 - 66 * q^29 - 40 * q^37 + 144 * q^41 + 2 * q^49 + 360 * q^53 + 14 * q^61 + 330 * q^65 - 220 * q^73 - 270 * q^77 - 60 * q^85 - 912 * q^89 + 200 * q^97

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