LMFDB - Embedded newform 1728.3.q.l.449.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(449,1728)]

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([0, 0, 5]))

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

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

chi := DirichletCharacter("1728.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.q (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:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.11
Character	$\chi$	$=$	1728.449
Dual form			1728.3.q.l.1601.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.92504 + 3.99817i) q^{5} +(-6.10647 - 10.5767i) q^{7} +O(q^{10})$	$q+(6.92504 + 3.99817i) q^{5} +(-6.10647 - 10.5767i) q^{7} +(-2.04017 + 1.17789i) q^{11} +(-8.01829 + 13.8881i) q^{13} -9.26058i q^{17} -4.43352 q^{19} +(-10.9115 - 6.29976i) q^{23} +(19.4708 + 33.7243i) q^{25} +(21.3763 - 12.3416i) q^{29} +(17.9645 - 31.1154i) q^{31} -97.6589i q^{35} +62.3141 q^{37} +(-19.9813 - 11.5362i) q^{41} +(-18.4520 - 31.9598i) q^{43} +(18.4657 - 10.6612i) q^{47} +(-50.0781 + 86.7377i) q^{49} -94.9767i q^{53} -18.8377 q^{55} +(46.4263 + 26.8043i) q^{59} +(-53.0732 - 91.9255i) q^{61} +(-111.054 + 64.1170i) q^{65} +(5.50175 - 9.52931i) q^{67} -108.498i q^{71} +26.0269 q^{73} +(24.9165 + 14.3855i) q^{77} +(26.7447 + 46.3231i) q^{79} +(-111.198 + 64.2000i) q^{83} +(37.0254 - 64.1299i) q^{85} -39.8009i q^{89} +195.854 q^{91} +(-30.7023 - 17.7260i) q^{95} +(-29.4518 - 51.0121i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97}+O(q^{100}) $ 24 * q + 60 * q^25 + 72 * q^29 + 36 * q^41 - 132 * q^49 - 96 * q^61 - 576 * q^65 + 24 * q^73 + 432 * q^77 + 96 * q^85 + 252 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	6.92504	+	3.99817i	1.38501	+	0.799634i	0.992747	−	0.120220i	$-0.0383600\pi$
$5$	6.92504	+	3.99817i	1.38501	+	0.799634i	0.392260	+	0.919854i	$0.371693\pi$
$6$		0			0
$7$	−6.10647	−	10.5767i	−0.872354	−	1.51096i	−0.859555	−	0.511042i	$-0.829259\pi$
$7$	−6.10647	−	10.5767i	−0.872354	−	1.51096i	−0.0127981	−	0.999918i	$-0.504074\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.04017	+	1.17789i	−0.185470	+	0.107081i	−0.589860	−	0.807505i	$-0.700817\pi$
$11$	−2.04017	+	1.17789i	−0.185470	+	0.107081i	0.404390	+	0.914587i	$0.367484\pi$
$12$		0			0
$13$	−8.01829	+	13.8881i	−0.616792	+	1.06831i	0.373276	+	0.927720i	$0.378235\pi$
$13$	−8.01829	+	13.8881i	−0.616792	+	1.06831i	−0.990067	+	0.140594i	$0.955099\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	9.26058i		−	0.544740i	−0.962193	−	0.272370i	$-0.912193\pi$
$17$		−	9.26058i		−	0.544740i	0.962193	−	0.272370i	$-0.0878075\pi$
$18$		0			0
$19$	−4.43352			−0.233343			−0.116672	−	0.993171i	$-0.537222\pi$
$19$	−4.43352			−0.233343			−0.116672	+	0.993171i	$0.537222\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−10.9115	−	6.29976i	−0.474414	−	0.273903i	0.243672	−	0.969858i	$-0.421648\pi$
$23$	−10.9115	−	6.29976i	−0.474414	−	0.273903i	−0.718085	+	0.695955i	$0.754981\pi$
$24$		0			0
$25$	19.4708	+	33.7243i	0.778830	+	1.34897i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	21.3763	−	12.3416i	0.737114	−	0.425573i	−0.0839051	−	0.996474i	$-0.526739\pi$
$29$	21.3763	−	12.3416i	0.737114	−	0.425573i	0.821019	+	0.570901i	$0.193406\pi$
$30$		0			0
$31$	17.9645	−	31.1154i	0.579500	−	1.00372i	−0.416036	−	0.909348i	$-0.636581\pi$
$31$	17.9645	−	31.1154i	0.579500	−	1.00372i	0.995537	−	0.0943759i	$-0.0300855\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	97.6589i		−	2.79026i
$36$		0			0
$37$	62.3141			1.68416			0.842082	−	0.539349i	$-0.181330\pi$
$37$	62.3141			1.68416			0.842082	+	0.539349i	$0.181330\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−19.9813	−	11.5362i	−0.487348	−	0.281371i	0.236125	−	0.971723i	$-0.424122\pi$
$41$	−19.9813	−	11.5362i	−0.487348	−	0.281371i	−0.723474	+	0.690352i	$0.757456\pi$
$42$		0			0
$43$	−18.4520	−	31.9598i	−0.429116	−	0.743251i	0.567679	−	0.823250i	$-0.307841\pi$
$43$	−18.4520	−	31.9598i	−0.429116	−	0.743251i	−0.996795	+	0.0799994i	$0.974508\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	18.4657	−	10.6612i	0.392888	−	0.226834i	−0.290523	−	0.956868i	$-0.593829\pi$
$47$	18.4657	−	10.6612i	0.392888	−	0.226834i	0.683411	+	0.730034i	$0.260496\pi$
$48$		0			0
$49$	−50.0781	+	86.7377i	−1.02200	+	1.77016i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	94.9767i		−	1.79201i	−0.444041	−	0.896007i	$-0.646455\pi$
$53$		−	94.9767i		−	1.79201i	0.444041	−	0.896007i	$-0.353545\pi$
$54$		0			0
$55$	−18.8377			−0.342503
$56$		0			0
$57$		0			0
$58$		0			0
$59$	46.4263	+	26.8043i	0.786887	+	0.454309i	0.838865	−	0.544339i	$-0.183219\pi$
$59$	46.4263	+	26.8043i	0.786887	+	0.454309i	−0.0519785	+	0.998648i	$0.516553\pi$
$60$		0			0
$61$	−53.0732	−	91.9255i	−0.870053	−	1.50698i	−0.861941	−	0.507009i	$-0.830751\pi$
$61$	−53.0732	−	91.9255i	−0.870053	−	1.50698i	−0.00811229	−	0.999967i	$-0.502582\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−111.054	+	64.1170i	−1.70852	+	0.986416i
$66$		0			0
$67$	5.50175	−	9.52931i	0.0821157	−	0.142229i	−0.822043	−	0.569426i	$-0.807166\pi$
$67$	5.50175	−	9.52931i	0.0821157	−	0.142229i	0.904159	+	0.427197i	$0.140499\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	108.498i		−	1.52814i	−0.645135	−	0.764068i	$-0.723199\pi$
$71$		−	108.498i		−	1.52814i	0.645135	−	0.764068i	$-0.276801\pi$
$72$		0			0
$73$	26.0269			0.356533			0.178267	−	0.983982i	$-0.442951\pi$
$73$	26.0269			0.356533			0.178267	+	0.983982i	$0.442951\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	24.9165	+	14.3855i	0.323591	+	0.186825i
$78$		0			0
$79$	26.7447	+	46.3231i	0.338540	+	0.586369i	0.984158	−	0.177292i	$-0.0567336\pi$
$79$	26.7447	+	46.3231i	0.338540	+	0.586369i	−0.645618	+	0.763660i	$0.723400\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−111.198	+	64.2000i	−1.33973	+	0.773494i	−0.986767	−	0.162143i	$-0.948159\pi$
$83$	−111.198	+	64.2000i	−1.33973	+	0.773494i	−0.352963	+	0.935637i	$0.614826\pi$
$84$		0			0
$85$	37.0254	−	64.1299i	0.435593	−	0.754469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	39.8009i		−	0.447202i	−0.974681	−	0.223601i	$-0.928219\pi$
$89$		−	39.8009i		−	0.447202i	0.974681	−	0.223601i	$-0.0717812\pi$
$90$		0			0
$91$	195.854			2.15224
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−30.7023	−	17.7260i	−0.323182	−	0.186589i
$96$		0			0
$97$	−29.4518	−	51.0121i	−0.303627	−	0.525898i	0.673328	−	0.739344i	$-0.264864\pi$
$97$	−29.4518	−	51.0121i	−0.303627	−	0.525898i	−0.976955	+	0.213447i	$0.931531\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	34.3380	−	19.8251i	0.339980	−	0.196288i	−0.320283	−	0.947322i	$-0.603778\pi$
$101$	34.3380	−	19.8251i	0.339980	−	0.196288i	0.660263	+	0.751034i	$0.270445\pi$
$102$		0			0
$103$	42.9542	−	74.3989i	0.417031	−	0.722319i	−0.578608	−	0.815606i	$-0.696404\pi$
$103$	42.9542	−	74.3989i	0.417031	−	0.722319i	0.995639	+	0.0932864i	$0.0297372\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			58.9252i			0.550702i	0.961344	+	0.275351i	$0.0887941\pi$
$107$			58.9252i			0.550702i	−0.961344	+	0.275351i	$0.911206\pi$
$108$		0			0
$109$	−41.4153			−0.379957			−0.189978	−	0.981788i	$-0.560842\pi$
$109$	−41.4153			−0.379957			−0.189978	+	0.981788i	$0.560842\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−113.242	−	65.3804i	−1.00214	−	0.578588i	−0.0932619	−	0.995642i	$-0.529729\pi$
$113$	−113.242	−	65.3804i	−1.00214	−	0.578588i	−0.908882	+	0.417054i	$0.863063\pi$
$114$		0			0
$115$	−50.3751	−	87.2522i	−0.438044	−	0.758715i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−97.9466	+	56.5495i	−0.823081	+	0.475206i
$120$		0			0
$121$	−57.7251	+	99.9829i	−0.477067	+	0.826305i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			111.481i			0.891850i
$126$		0			0
$127$	132.557			1.04375			0.521877	−	0.853021i	$-0.325232\pi$
$127$	132.557			1.04375			0.521877	+	0.853021i	$0.325232\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−164.641	−	95.0556i	−1.25680	−	0.725615i	−0.284351	−	0.958720i	$-0.591778\pi$
$131$	−164.641	−	95.0556i	−1.25680	−	0.725615i	−0.972451	+	0.233105i	$0.925111\pi$
$132$		0			0
$133$	27.0732	+	46.8921i	0.203558	+	0.352572i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	40.0008	−	23.0945i	0.291976	−	0.168573i	−0.346856	−	0.937918i	$-0.612751\pi$
$137$	40.0008	−	23.0945i	0.291976	−	0.168573i	0.638833	+	0.769346i	$0.279418\pi$
$138$		0			0
$139$	65.6376	−	113.688i	0.472213	−	0.817896i	−0.527282	−	0.849691i	$-0.676789\pi$
$139$	65.6376	−	113.688i	0.472213	−	0.817896i	0.999494	+	0.0317941i	$0.0101221\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	37.7787i		−	0.264187i
$144$		0			0
$145$	197.376			1.36121
$146$		0			0
$147$		0			0
$148$		0			0
$149$	107.105	+	61.8372i	0.718827	+	0.415015i	0.814321	−	0.580415i	$-0.197110\pi$
$149$	107.105	+	61.8372i	0.718827	+	0.415015i	−0.0954937	+	0.995430i	$0.530443\pi$
$150$		0			0
$151$	84.1852	+	145.813i	0.557518	+	0.965649i	0.997703	+	0.0677420i	$0.0215795\pi$
$151$	84.1852	+	145.813i	0.557518	+	0.965649i	−0.440185	+	0.897907i	$0.645087\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	248.810	−	143.650i	1.60522	−	0.926777i
$156$		0			0
$157$	−25.4285	+	44.0434i	−0.161965	+	0.280531i	−0.935573	−	0.353133i	$-0.885116\pi$
$157$	−25.4285	+	44.0434i	−0.161965	+	0.280531i	0.773608	+	0.633664i	$0.218450\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			153.877i			0.955760i
$162$		0			0
$163$	94.1471			0.577589			0.288795	−	0.957391i	$-0.406746\pi$
$163$	94.1471			0.577589			0.288795	+	0.957391i	$0.406746\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14.1942	+	8.19501i	0.0849951	+	0.0490719i	0.541895	−	0.840446i	$-0.317707\pi$
$167$	14.1942	+	8.19501i	0.0849951	+	0.0490719i	−0.456900	+	0.889518i	$0.651040\pi$
$168$		0			0
$169$	−44.0860	−	76.3591i	−0.260864	−	0.451829i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	41.6348	−	24.0378i	0.240663	−	0.138947i	−0.374818	−	0.927098i	$-0.622295\pi$
$173$	41.6348	−	24.0378i	0.240663	−	0.138947i	0.615482	+	0.788151i	$0.288962\pi$
$174$		0			0
$175$	237.795	−	411.874i	1.35883	−	2.35356i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	178.722i		−	0.998446i	−0.866474	−	0.499223i	$-0.833619\pi$
$179$		−	178.722i		−	0.998446i	0.866474	−	0.499223i	$-0.166381\pi$
$180$		0			0
$181$	65.0626			0.359462			0.179731	−	0.983716i	$-0.442477\pi$
$181$	65.0626			0.359462			0.179731	+	0.983716i	$0.442477\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	431.527	+	249.142i	2.33258	+	1.34672i
$186$		0			0
$187$	10.9080	+	18.8931i	0.0583314	+	0.101033i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	58.1864	−	33.5939i	0.304641	−	0.175884i	−0.339885	−	0.940467i	$-0.610388\pi$
$191$	58.1864	−	33.5939i	0.304641	−	0.175884i	0.644526	+	0.764583i	$0.277055\pi$
$192$		0			0
$193$	33.7306	−	58.4231i	0.174770	−	0.302710i	−0.765312	−	0.643660i	$-0.777415\pi$
$193$	33.7306	−	58.4231i	0.174770	−	0.302710i	0.940082	+	0.340949i	$0.110749\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	89.4955i		−	0.454292i	−0.973861	−	0.227146i	$-0.927061\pi$
$197$		−	89.4955i		−	0.454292i	0.973861	−	0.227146i	$-0.0729395\pi$
$198$		0			0
$199$	−291.423			−1.46444			−0.732220	−	0.681068i	$-0.761516\pi$
$199$	−291.423			−1.46444			−0.732220	+	0.681068i	$0.761516\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−261.068	−	150.728i	−1.28605	−	0.742500i
$204$		0			0
$205$	−92.2474	−	159.777i	−0.449987	−	0.779401i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.04512	−	5.22220i	0.0432781	−	0.0249866i
$210$		0			0
$211$	−116.263	+	201.374i	−0.551011	+	0.954379i	0.447191	+	0.894439i	$0.352424\pi$
$211$	−116.263	+	201.374i	−0.551011	+	0.954379i	−0.998202	+	0.0599405i	$0.980909\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	295.097i		−	1.37254i
$216$		0			0
$217$	−438.799			−2.02212
$218$		0			0
$219$		0			0
$220$		0			0
$221$	128.612	+	74.2540i	0.581954	+	0.335991i
$222$		0			0
$223$	−156.139	−	270.441i	−0.700176	−	1.21274i	−0.968405	−	0.249385i	$-0.919772\pi$
$223$	−156.139	−	270.441i	−0.700176	−	1.21274i	0.268229	−	0.963355i	$-0.413562\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	138.685	−	80.0701i	0.610949	−	0.352732i	−0.162388	−	0.986727i	$-0.551920\pi$
$227$	138.685	−	80.0701i	0.610949	−	0.352732i	0.773337	+	0.633995i	$0.218586\pi$
$228$		0			0
$229$	−9.54914	+	16.5396i	−0.0416993	+	0.0722253i	−0.886122	−	0.463452i	$-0.846610\pi$
$229$	−9.54914	+	16.5396i	−0.0416993	+	0.0722253i	0.844422	+	0.535678i	$0.179944\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	194.029i		−	0.832741i	−0.909195	−	0.416371i	$-0.863302\pi$
$233$		−	194.029i		−	0.832741i	0.909195	−	0.416371i	$-0.136698\pi$
$234$		0			0
$235$	170.501			0.725537
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−333.871	−	192.760i	−1.39695	−	0.806529i	−0.402877	−	0.915254i	$-0.631990\pi$
$239$	−333.871	−	192.760i	−1.39695	−	0.806529i	−0.994072	+	0.108725i	$0.965323\pi$
$240$		0			0
$241$	−20.3136	−	35.1841i	−0.0842886	−	0.145992i	0.820799	−	0.571217i	$-0.193528\pi$
$241$	−20.3136	−	35.1841i	−0.0842886	−	0.145992i	−0.905088	+	0.425225i	$0.860195\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−693.585	+	400.441i	−2.83096	+	1.63445i
$246$		0			0
$247$	35.5492	−	61.5731i	0.143924	−	0.249284i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	33.7966i		−	0.134648i	−0.997731	−	0.0673238i	$-0.978554\pi$
$251$		−	33.7966i		−	0.134648i	0.997731	−	0.0673238i	$-0.0214461\pi$
$252$		0			0
$253$	29.6818			0.117319
$254$		0			0
$255$		0			0
$256$		0			0
$257$	186.282	+	107.550i	0.724832	+	0.418482i	0.816528	−	0.577305i	$-0.195896\pi$
$257$	186.282	+	107.550i	0.724832	+	0.418482i	−0.0916967	+	0.995787i	$0.529229\pi$
$258$		0			0
$259$	−380.519	−	659.079i	−1.46919	−	2.54471i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	291.375	−	168.225i	1.10789	−	0.639640i	0.169607	−	0.985512i	$-0.445750\pi$
$263$	291.375	−	168.225i	1.10789	−	0.639640i	0.938282	+	0.345872i	$0.112417\pi$
$264$		0			0
$265$	379.733	−	657.717i	1.43296	−	2.48195i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	195.385i		−	0.726338i	−0.931723	−	0.363169i	$-0.881695\pi$
$269$		−	195.385i		−	0.726338i	0.931723	−	0.363169i	$-0.118305\pi$
$270$		0			0
$271$	−256.273			−0.945657			−0.472828	−	0.881154i	$-0.656767\pi$
$271$	−256.273			−0.945657			−0.472828	+	0.881154i	$0.656767\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−79.4473	−	45.8689i	−0.288899	−	0.166796i
$276$		0			0
$277$	161.046	+	278.940i	0.581394	+	1.00700i	0.995315	+	0.0966903i	$0.0308256\pi$
$277$	161.046	+	278.940i	0.581394	+	1.00700i	−0.413921	+	0.910313i	$0.635841\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−231.774	+	133.815i	−0.824817	+	0.476208i	−0.852075	−	0.523420i	$-0.824656\pi$
$281$	−231.774	+	133.815i	−0.824817	+	0.476208i	0.0272577	+	0.999628i	$0.491323\pi$
$282$		0			0
$283$	−143.305	+	248.212i	−0.506379	+	0.877075i	0.493594	+	0.869693i	$0.335683\pi$
$283$	−143.305	+	248.212i	−0.506379	+	0.877075i	−0.999973	+	0.00738184i	$0.997650\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			281.782i			0.981819i
$288$		0			0
$289$	203.242			0.703258
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−223.818	−	129.222i	−0.763885	−	0.441029i	0.0668040	−	0.997766i	$-0.478720\pi$
$293$	−223.818	−	129.222i	−0.763885	−	0.441029i	−0.830689	+	0.556737i	$0.812053\pi$
$294$		0			0
$295$	214.336	+	371.241i	0.726563	+	1.25844i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	174.983	−	101.027i	0.585229	−	0.337882i
$300$		0			0
$301$	−225.353	+	390.323i	−0.748682	+	1.29675i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	848.784i		−	2.78290i
$306$		0			0
$307$	185.607			0.604583			0.302292	−	0.953216i	$-0.402248\pi$
$307$	185.607			0.604583			0.302292	+	0.953216i	$0.402248\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	70.5350	+	40.7234i	0.226801	+	0.130943i	0.609095	−	0.793097i	$-0.291533\pi$
$311$	70.5350	+	40.7234i	0.226801	+	0.130943i	−0.382295	+	0.924040i	$0.624866\pi$
$312$		0			0
$313$	7.26701	+	12.5868i	0.0232173	+	0.0402135i	0.877401	−	0.479758i	$-0.159276\pi$
$313$	7.26701	+	12.5868i	0.0232173	+	0.0402135i	−0.854183	+	0.519972i	$0.825942\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−155.242	+	89.6289i	−0.489722	+	0.282741i	−0.724459	−	0.689318i	$-0.757910\pi$
$317$	−155.242	+	89.6289i	−0.489722	+	0.282741i	0.234737	+	0.972059i	$0.424577\pi$
$318$		0			0
$319$	−29.0742	+	50.3579i	−0.0911416	+	0.157862i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			41.0569i			0.127111i
$324$		0			0
$325$	−624.489			−1.92150
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−225.521	−	130.205i	−0.685474	−	0.395759i
$330$		0			0
$331$	122.621	+	212.385i	0.370455	+	0.641648i	0.989636	−	0.143601i	$-0.0458683\pi$
$331$	122.621	+	212.385i	0.370455	+	0.641648i	−0.619180	+	0.785249i	$0.712535\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	76.1997	−	43.9939i	0.227462	−	0.131325i
$336$		0			0
$337$	157.997	−	273.658i	0.468832	−	0.812042i	−0.530533	−	0.847664i	$-0.678008\pi$
$337$	157.997	−	273.658i	0.468832	−	0.812042i	0.999365	+	0.0356228i	$0.0113415\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			84.6410i			0.248214i
$342$		0			0
$343$	624.767			1.82148
$344$		0			0
$345$		0			0
$346$		0			0
$347$	346.789	+	200.218i	0.999391	+	0.576998i	0.908068	−	0.418823i	$-0.137557\pi$
$347$	346.789	+	200.218i	0.999391	+	0.576998i	0.0913228	+	0.995821i	$0.470891\pi$
$348$		0			0
$349$	200.146	+	346.663i	0.573485	+	0.993305i	0.996204	+	0.0870444i	$0.0277422\pi$
$349$	200.146	+	346.663i	0.573485	+	0.993305i	−0.422720	+	0.906261i	$0.638924\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	175.941	−	101.579i	0.498416	−	0.287760i	−0.229643	−	0.973275i	$-0.573756\pi$
$353$	175.941	−	101.579i	0.498416	−	0.287760i	0.728059	+	0.685514i	$0.240423\pi$
$354$		0			0
$355$	433.792	−	751.351i	1.22195	−	2.11648i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			345.898i			0.963505i	0.876307	+	0.481753i	$0.160000\pi$
$359$			345.898i			0.963505i	−0.876307	+	0.481753i	$0.840000\pi$
$360$		0			0
$361$	−341.344			−0.945551
$362$		0			0
$363$		0			0
$364$		0			0
$365$	180.237	+	104.060i	0.493801	+	0.285096i
$366$		0			0
$367$	174.321	+	301.933i	0.474990	+	0.822706i	0.999590	−	0.0286425i	$-0.00911843\pi$
$367$	174.321	+	301.933i	0.474990	+	0.822706i	−0.524600	+	0.851349i	$0.675785\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1004.54	+	579.973i	−2.70766	+	1.56327i
$372$		0			0
$373$	226.324	−	392.004i	0.606766	−	1.05095i	−0.385003	−	0.922915i	$-0.625800\pi$
$373$	226.324	−	392.004i	0.606766	−	1.05095i	0.991770	−	0.128035i	$-0.0408670\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			395.835i			1.04996i
$378$		0			0
$379$	−39.1096			−0.103192			−0.0515958	−	0.998668i	$-0.516431\pi$
$379$	−39.1096			−0.103192			−0.0515958	+	0.998668i	$0.516431\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−422.286	−	243.807i	−1.10257	−	0.636572i	−0.165679	−	0.986180i	$-0.552981\pi$
$383$	−422.286	−	243.807i	−1.10257	−	0.636572i	−0.936896	+	0.349608i	$0.886315\pi$
$384$		0			0
$385$	115.032	+	199.241i	0.298784	+	0.517508i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	46.2319	−	26.6920i	0.118848	−	0.0686170i	−0.439398	−	0.898293i	$-0.644808\pi$
$389$	46.2319	−	26.6920i	0.118848	−	0.0686170i	0.558246	+	0.829676i	$0.311475\pi$
$390$		0			0
$391$	−58.3395	+	101.047i	−0.149206	+	0.258432i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			427.719i			1.08283i
$396$		0			0
$397$	266.942			0.672399			0.336199	−	0.941791i	$-0.390858\pi$
$397$	266.942			0.672399			0.336199	+	0.941791i	$0.390858\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	501.937	+	289.793i	1.25171	+	0.722677i	0.971450	−	0.237246i	$-0.0762449\pi$
$401$	501.937	+	289.793i	1.25171	+	0.722677i	0.280263	+	0.959923i	$0.409578\pi$
$402$		0			0
$403$	288.089	+	498.985i	0.714862	+	1.23818i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−127.131	+	73.3992i	−0.312362	+	0.180342i
$408$		0			0
$409$	−357.583	+	619.352i	−0.874286	+	1.51431i	−0.0167650	+	0.999859i	$0.505337\pi$
$409$	−357.583	+	619.352i	−0.874286	+	1.51431i	−0.857521	+	0.514449i	$0.827997\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	654.718i		−	1.58527i
$414$		0			0
$415$	−1026.73			−2.47405
$416$		0			0
$417$		0			0
$418$		0			0
$419$	678.912	+	391.970i	1.62032	+	0.935489i	0.986835	+	0.161730i	$0.0517075\pi$
$419$	678.912	+	391.970i	1.62032	+	0.935489i	0.633480	+	0.773759i	$0.281626\pi$
$420$		0			0
$421$	346.253	+	599.727i	0.822453	+	1.42453i	0.903850	+	0.427849i	$0.140728\pi$
$421$	346.253	+	599.727i	0.822453	+	1.42453i	−0.0813974	+	0.996682i	$0.525938\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	312.307	−	180.311i	0.734840	−	0.424260i
$426$		0			0
$427$	−648.181	+	1122.68i	−1.51799	+	2.62923i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			35.9087i			0.0833148i	0.999132	+	0.0416574i	$0.0132638\pi$
$431$			35.9087i			0.0833148i	−0.999132	+	0.0416574i	$0.986736\pi$
$432$		0			0
$433$	292.547			0.675629			0.337814	−	0.941213i	$-0.390312\pi$
$433$	292.547			0.675629			0.337814	+	0.941213i	$0.390312\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	48.3764	+	27.9301i	0.110701	+	0.0639133i
$438$		0			0
$439$	43.2227	+	74.8639i	0.0984572	+	0.170533i	0.911046	−	0.412304i	$-0.135276\pi$
$439$	43.2227	+	74.8639i	0.0984572	+	0.170533i	−0.812589	+	0.582837i	$0.801943\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−43.3778	+	25.0442i	−0.0979183	+	0.0565332i	−0.548160	−	0.836374i	$-0.684671\pi$
$443$	−43.3778	+	25.0442i	−0.0979183	+	0.0565332i	0.450241	+	0.892907i	$0.351338\pi$
$444$		0			0
$445$	159.131	−	275.623i	0.357598	−	0.619377i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	181.014i		−	0.403150i	−0.979473	−	0.201575i	$-0.935394\pi$
$449$		−	181.014i		−	0.403150i	0.979473	−	0.201575i	$-0.0646060\pi$
$450$		0			0
$451$	54.3536			0.120518
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1356.30	+	783.058i	2.98087	+	1.72101i
$456$		0			0
$457$	302.620	+	524.153i	0.662188	+	1.14694i	0.980040	+	0.198802i	$0.0637051\pi$
$457$	302.620	+	524.153i	0.662188	+	1.14694i	−0.317852	+	0.948140i	$0.602962\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	156.278	−	90.2273i	0.338999	−	0.195721i	−0.320831	−	0.947137i	$-0.603962\pi$
$461$	156.278	−	90.2273i	0.338999	−	0.195721i	0.659829	+	0.751416i	$0.270629\pi$
$462$		0			0
$463$	−126.203	+	218.591i	−0.272577	+	0.472118i	−0.969521	−	0.245008i	$-0.921209\pi$
$463$	−126.203	+	218.591i	−0.272577	+	0.472118i	0.696944	+	0.717126i	$0.254543\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	131.155i		−	0.280845i	−0.990092	−	0.140422i	$-0.955154\pi$
$467$		−	131.155i		−	0.280845i	0.990092	−	0.140422i	$-0.0448461\pi$
$468$		0			0
$469$	−134.385			−0.286536
$470$		0			0
$471$		0			0
$472$		0			0
$473$	75.2903	+	43.4689i	0.159176	+	0.0919004i
$474$		0			0
$475$	−86.3240	−	149.517i	−0.181735	−	0.314774i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−612.191	+	353.449i	−1.27806	+	0.737889i	−0.976492	−	0.215556i	$-0.930844\pi$
$479$	−612.191	+	353.449i	−1.27806	+	0.737889i	−0.301569	+	0.953444i	$0.597510\pi$
$480$		0			0
$481$	−499.652	+	865.423i	−1.03878	+	1.79922i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	471.014i		−	0.971163i
$486$		0			0
$487$	181.895			0.373502			0.186751	−	0.982407i	$-0.440204\pi$
$487$	181.895			0.373502			0.186751	+	0.982407i	$0.440204\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	47.2326	+	27.2697i	0.0961967	+	0.0555392i	0.547327	−	0.836919i	$-0.315646\pi$
$491$	47.2326	+	27.2697i	0.0961967	+	0.0555392i	−0.451130	+	0.892458i	$0.648979\pi$
$492$		0			0
$493$	−114.291	−	197.957i	−0.231827	−	0.401536i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1147.55	+	662.538i	−2.30895	+	1.33308i
$498$		0			0
$499$	−347.353	+	601.632i	−0.696097	+	1.20568i	0.273712	+	0.961812i	$0.411749\pi$
$499$	−347.353	+	601.632i	−0.696097	+	1.20568i	−0.969809	+	0.243864i	$0.921585\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			521.419i			1.03662i	0.855193	+	0.518309i	$0.173438\pi$
$503$			521.419i			1.03662i	−0.855193	+	0.518309i	$0.826562\pi$
$504$		0			0
$505$	317.056			0.627833
$506$		0			0
$507$		0			0
$508$		0			0
$509$	216.489	+	124.990i	0.425322	+	0.245560i	0.697352	−	0.716729i	$-0.254362\pi$
$509$	216.489	+	124.990i	0.425322	+	0.245560i	−0.272030	+	0.962289i	$0.587695\pi$
$510$		0			0
$511$	−158.933	−	275.280i	−0.311023	−	0.538708i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	594.919	−	343.477i	1.15518	−	0.666945i
$516$		0			0
$517$	−25.1155	+	43.5013i	−0.0485792	+	0.0841417i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	487.492i		−	0.935685i	−0.883812	−	0.467842i	$-0.845031\pi$
$521$		−	487.492i		−	0.935685i	0.883812	−	0.467842i	$-0.154969\pi$
$522$		0			0
$523$	−505.871			−0.967248			−0.483624	−	0.875276i	$-0.660680\pi$
$523$	−505.871			−0.967248			−0.483624	+	0.875276i	$0.660680\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−288.147	−	166.362i	−0.546769	−	0.315677i
$528$		0			0
$529$	−185.126	−	320.648i	−0.349955	−	0.606139i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	320.431	−	185.001i	0.601185	−	0.347094i
$534$		0			0
$535$	−235.593	+	408.059i	−0.440361	+	0.762727i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	235.946i		−	0.437748i
$540$		0			0
$541$	384.098			0.709977			0.354989	−	0.934871i	$-0.384485\pi$
$541$	384.098			0.709977			0.354989	+	0.934871i	$0.384485\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−286.802	−	165.585i	−0.526243	−	0.303826i
$546$		0			0
$547$	−117.073	−	202.777i	−0.214028	−	0.370707i	0.738944	−	0.673767i	$-0.235325\pi$
$547$	−117.073	−	202.777i	−0.214028	−	0.370707i	−0.952971	+	0.303060i	$0.901992\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−94.7722	+	54.7168i	−0.172000	+	0.0993045i
$552$		0			0
$553$	326.631	−	565.742i	0.590653	−	1.02304i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			461.176i			0.827964i	0.910285	+	0.413982i	$0.135862\pi$
$557$			461.176i			0.827964i	−0.910285	+	0.413982i	$0.864138\pi$
$558$		0			0
$559$	591.814			1.05870
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−621.735	−	358.959i	−1.10433	−	0.637582i	−0.166972	−	0.985962i	$-0.553399\pi$
$563$	−621.735	−	358.959i	−1.10433	−	0.637582i	−0.937354	+	0.348379i	$0.886732\pi$
$564$		0			0
$565$	−522.805	−	905.524i	−0.925318	−	1.60270i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−351.827	+	203.128i	−0.618326	+	0.356990i	−0.776217	−	0.630466i	$-0.782864\pi$
$569$	−351.827	+	203.128i	−0.618326	+	0.356990i	0.157891	+	0.987457i	$0.449530\pi$
$570$		0			0
$571$	247.658	−	428.955i	0.433726	−	0.751235i	−0.563465	−	0.826140i	$-0.690532\pi$
$571$	247.658	−	428.955i	0.433726	−	0.751235i	0.997191	+	0.0749047i	$0.0238652\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	490.645i		−	0.853295i
$576$		0			0
$577$	−254.103			−0.440387			−0.220194	−	0.975456i	$-0.570669\pi$
$577$	−254.103			−0.440387			−0.220194	+	0.975456i	$0.570669\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1358.05	+	784.071i	2.33744	+	1.34952i
$582$		0			0
$583$	111.872	+	193.768i	0.191891	+	0.332364i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	329.669	−	190.335i	0.561617	−	0.324250i	−0.192177	−	0.981360i	$-0.561555\pi$
$587$	329.669	−	190.335i	0.561617	−	0.324250i	0.753794	+	0.657110i	$0.228222\pi$
$588$		0			0
$589$	−79.6460	+	137.951i	−0.135222	+	0.234212i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			76.6170i			0.129202i	0.997911	+	0.0646012i	$0.0205775\pi$
$593$			76.6170i			0.129202i	−0.997911	+	0.0646012i	$0.979422\pi$
$594$		0			0
$595$	−904.379			−1.51996
$596$		0			0
$597$		0			0
$598$		0			0
$599$	905.152	+	522.590i	1.51110	+	0.872437i	0.999916	+	0.0129680i	$0.00412796\pi$
$599$	905.152	+	522.590i	1.51110	+	0.872437i	0.511189	+	0.859469i	$0.329205\pi$
$600$		0			0
$601$	7.76050	+	13.4416i	0.0129127	+	0.0223654i	0.872410	−	0.488776i	$-0.162556\pi$
$601$	7.76050	+	13.4416i	0.0129127	+	0.0223654i	−0.859497	+	0.511141i	$0.829223\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−799.498	+	461.590i	−1.32148	+	0.762959i
$606$		0			0
$607$	−16.9276	+	29.3195i	−0.0278874	+	0.0483023i	−0.879632	−	0.475654i	$-0.842211\pi$
$607$	−16.9276	+	29.3195i	−0.0278874	+	0.0483023i	0.851745	+	0.523957i	$0.175545\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			341.938i			0.559637i
$612$		0			0
$613$	−78.0809			−0.127375			−0.0636875	−	0.997970i	$-0.520286\pi$
$613$	−78.0809			−0.127375			−0.0636875	+	0.997970i	$0.520286\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−619.489	−	357.662i	−1.00403	−	0.579679i	−0.0945946	−	0.995516i	$-0.530155\pi$
$617$	−619.489	−	357.662i	−1.00403	−	0.579679i	−0.909439	+	0.415837i	$0.863489\pi$
$618$		0			0
$619$	569.364	+	986.167i	0.919812	+	1.59316i	0.799699	+	0.600402i	$0.204993\pi$
$619$	569.364	+	986.167i	0.919812	+	1.59316i	0.120114	+	0.992760i	$0.461674\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−420.964	+	243.043i	−0.675704	+	0.390118i
$624$		0			0
$625$	41.0480	−	71.0972i	0.0656768	−	0.113756i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	577.065i		−	0.917432i
$630$		0			0
$631$	−1051.70			−1.66672			−0.833359	−	0.552732i	$-0.813585\pi$
$631$	−1051.70			−1.66672			−0.833359	+	0.552732i	$0.813585\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	917.961	+	529.985i	1.44561	+	0.834622i
$636$		0			0
$637$	−803.081	−	1390.98i	−1.26072	−	2.18364i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−476.586	+	275.157i	−0.743504	+	0.429262i	−0.823342	−	0.567546i	$-0.807893\pi$
$641$	−476.586	+	275.157i	−0.743504	+	0.429262i	0.0798379	+	0.996808i	$0.474560\pi$
$642$		0			0
$643$	225.621	−	390.788i	0.350889	−	0.607757i	−0.635517	−	0.772087i	$-0.719213\pi$
$643$	225.621	−	390.788i	0.350889	−	0.607757i	0.986405	+	0.164330i	$0.0525462\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	428.912i		−	0.662924i	−0.943469	−	0.331462i	$-0.892458\pi$
$647$		−	428.912i		−	0.662924i	0.943469	−	0.331462i	$-0.107542\pi$
$648$		0			0
$649$	−126.290			−0.194592
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−763.269	−	440.673i	−1.16886	−	0.674844i	−0.215452	−	0.976514i	$-0.569123\pi$
$653$	−763.269	−	440.673i	−1.16886	−	0.674844i	−0.953412	+	0.301670i	$0.902456\pi$
$654$		0			0
$655$	−760.097	−	1316.53i	−1.16045	−	2.00996i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−426.907	+	246.475i	−0.647810	+	0.374013i	−0.787617	−	0.616166i	$-0.788685\pi$
$659$	−426.907	+	246.475i	−0.647810	+	0.374013i	0.139807	+	0.990179i	$0.455352\pi$
$660$		0			0
$661$	372.280	−	644.808i	0.563207	−	0.975504i	−0.434007	−	0.900910i	$-0.642901\pi$
$661$	372.280	−	644.808i	0.563207	−	0.975504i	0.997214	−	0.0745941i	$-0.0237661\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			432.973i			0.651087i
$666$		0			0
$667$	−310.997			−0.466262
$668$		0			0
$669$		0			0
$670$		0			0
$671$	216.557	+	125.029i	0.322737	+	0.186332i
$672$		0			0
$673$	−501.342	−	868.349i	−0.744935	−	1.29027i	−0.950225	−	0.311565i	$-0.899147\pi$
$673$	−501.342	−	868.349i	−0.744935	−	1.29027i	0.205290	−	0.978701i	$-0.434186\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	361.329	−	208.613i	0.533720	−	0.308144i	−0.208810	−	0.977956i	$-0.566959\pi$
$677$	361.329	−	208.613i	0.533720	−	0.308144i	0.742530	+	0.669813i	$0.233626\pi$
$678$		0			0
$679$	−359.694	+	623.008i	−0.529741	+	0.917538i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			760.675i			1.11373i	0.830604	+	0.556863i	$0.187995\pi$
$683$			760.675i			1.11373i	−0.830604	+	0.556863i	$0.812005\pi$
$684$		0			0
$685$	369.342			0.539186
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1319.04	+	761.551i	1.91443	+	1.10530i
$690$		0			0
$691$	−289.301	−	501.085i	−0.418671	−	0.725159i	0.577135	−	0.816648i	$-0.304170\pi$
$691$	−289.301	−	501.085i	−0.418671	−	0.725159i	−0.995806	+	0.0914898i	$0.970837\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	909.085	−	524.861i	1.30804	−	0.755195i
$696$		0			0
$697$	−106.832	+	185.038i	−0.153274	+	0.265478i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			790.696i			1.12795i	0.825791	+	0.563977i	$0.190729\pi$
$701$			790.696i			1.12795i	−0.825791	+	0.563977i	$0.809271\pi$
$702$		0			0
$703$	−276.271			−0.392988
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−419.368	−	242.122i	−0.593166	−	0.342464i
$708$		0			0
$709$	−264.420	−	457.990i	−0.372948	−	0.645966i	0.617069	−	0.786909i	$-0.288320\pi$
$709$	−264.420	−	457.990i	−0.372948	−	0.645966i	−0.990018	+	0.140943i	$0.954987\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−392.040	+	226.344i	−0.549846	+	0.317453i
$714$		0			0
$715$	151.046	−	261.619i	0.211253	−	0.365901i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	614.949i		−	0.855284i	−0.903948	−	0.427642i	$-0.859344\pi$
$719$		−	614.949i		−	0.855284i	0.903948	−	0.427642i	$-0.140656\pi$
$720$		0			0
$721$	−1049.20			−1.45519
$722$		0			0
$723$		0			0
$724$		0			0
$725$	832.426	+	480.601i	1.14817	+	0.662898i
$726$		0			0
$727$	393.521	+	681.599i	0.541295	+	0.937550i	0.998830	+	0.0483589i	$0.0153991\pi$
$727$	393.521	+	681.599i	0.541295	+	0.937550i	−0.457535	+	0.889192i	$0.651268\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−295.966	+	170.876i	−0.404878	+	0.233757i
$732$		0			0
$733$	14.2526	−	24.6863i	0.0194442	−	0.0336784i	−0.856140	−	0.516745i	$-0.827144\pi$
$733$	14.2526	−	24.6863i	0.0194442	−	0.0336784i	0.875584	+	0.483066i	$0.160477\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			25.9219i			0.0351721i
$738$		0			0
$739$	−641.231			−0.867701			−0.433851	−	0.900985i	$-0.642845\pi$
$739$	−641.231			−0.867701			−0.433851	+	0.900985i	$0.642845\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1136.16	−	655.960i	−1.52915	−	0.882853i	−0.999398	−	0.0346987i	$-0.988953\pi$
$743$	−1136.16	−	655.960i	−1.52915	−	0.882853i	−0.529749	−	0.848155i	$-0.677714\pi$
$744$		0			0
$745$	494.472	+	856.450i	0.663721	+	1.14960i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	623.235	−	359.825i	0.832090	−	0.480407i
$750$		0			0
$751$	−489.224	+	847.360i	−0.651430	+	1.12831i	0.331346	+	0.943509i	$0.392497\pi$
$751$	−489.224	+	847.360i	−0.651430	+	1.12831i	−0.982776	+	0.184800i	$0.940836\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1346.35i			1.78324i
$756$		0			0
$757$	1205.99			1.59312			0.796559	−	0.604561i	$-0.206651\pi$
$757$	1205.99			1.59312			0.796559	+	0.604561i	$0.206651\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−248.111	−	143.247i	−0.326033	−	0.188235i	0.328045	−	0.944662i	$-0.393610\pi$
$761$	−248.111	−	143.247i	−0.326033	−	0.188235i	−0.654079	+	0.756427i	$0.726943\pi$
$762$		0			0
$763$	252.901	+	438.038i	0.331457	+	0.574100i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−744.520	+	429.849i	−0.970690	+	0.560428i
$768$		0			0
$769$	−431.839	+	747.967i	−0.561559	+	0.972649i	0.435802	+	0.900043i	$0.356465\pi$
$769$	−431.839	+	747.967i	−0.561559	+	0.972649i	−0.997361	+	0.0726059i	$0.976868\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1326.95i		−	1.71663i	−0.513126	−	0.858313i	$-0.671513\pi$
$773$		−	1326.95i		−	1.71663i	0.513126	−	0.858313i	$-0.328487\pi$
$774$		0			0
$775$	1399.13			1.80533
$776$		0			0
$777$		0			0
$778$		0			0
$779$	88.5874	+	51.1459i	0.113719	+	0.0656559i
$780$		0			0
$781$	127.799	+	221.354i	0.163634	+	0.283423i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−352.187	+	203.335i	−0.448645	+	0.259025i
$786$		0			0
$787$	271.162	−	469.666i	0.344551	−	0.596781i	−0.640721	−	0.767774i	$-0.721364\pi$
$787$	271.162	−	469.666i	0.344551	−	0.596781i	0.985272	+	0.170993i	$0.0546977\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1596.98i			2.01893i
$792$		0			0
$793$	1702.23			2.14657
$794$		0			0
$795$		0			0
$796$		0			0
$797$	100.953	+	58.2853i	0.126666	+	0.0731308i	0.561994	−	0.827141i	$-0.310034\pi$
$797$	100.953	+	58.2853i	0.126666	+	0.0731308i	−0.435328	+	0.900272i	$0.643368\pi$
$798$		0			0
$799$	−98.7288	−	171.003i	−0.123565	−	0.214022i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−53.0993	+	30.6569i	−0.0661262	+	0.0381780i
$804$		0			0
$805$	−615.228	+	1065.61i	−0.764259	+	1.32374i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			40.7031i			0.0503128i	0.999684	+	0.0251564i	$0.00800838\pi$
$809$			40.7031i			0.0503128i	−0.999684	+	0.0251564i	$0.991992\pi$
$810$		0			0
$811$	366.179			0.451515			0.225758	−	0.974183i	$-0.427514\pi$
$811$	366.179			0.451515			0.225758	+	0.974183i	$0.427514\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	651.972	+	376.416i	0.799966	+	0.461860i
$816$		0			0
$817$	81.8072	+	141.694i	0.100131	+	0.173432i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1331.20	+	768.571i	−1.62144	+	0.936140i	−0.634907	+	0.772588i	$0.718962\pi$
$821$	−1331.20	+	768.571i	−1.62144	+	0.936140i	−0.986535	+	0.163552i	$0.947705\pi$
$822$		0			0
$823$	−597.445	+	1034.81i	−0.725936	+	1.25736i	0.232652	+	0.972560i	$0.425260\pi$
$823$	−597.445	+	1034.81i	−0.725936	+	1.25736i	−0.958588	+	0.284798i	$0.908074\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1025.51i		−	1.24004i	−0.784587	−	0.620018i	$-0.787125\pi$
$827$		−	1025.51i		−	1.24004i	0.784587	−	0.620018i	$-0.212875\pi$
$828$		0			0
$829$	−151.730			−0.183027			−0.0915136	−	0.995804i	$-0.529171\pi$
$829$	−151.730			−0.183027			−0.0915136	+	0.995804i	$0.529171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	803.242	+	463.752i	0.964276	+	0.556725i
$834$		0			0
$835$	65.5301	+	113.502i	0.0784792	+	0.135930i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	712.702	−	411.479i	0.849466	−	0.490440i	−0.0110045	−	0.999939i	$-0.503503\pi$
$839$	712.702	−	411.479i	0.849466	−	0.490440i	0.860471	+	0.509500i	$0.170170\pi$
$840$		0			0
$841$	−115.869	+	200.691i	−0.137775	+	0.238634i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	705.053i		−	0.834382i
$846$		0			0
$847$	1409.99			1.66469
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−679.941	−	392.564i	−0.798990	−	0.461297i
$852$		0			0
$853$	409.497	+	709.270i	0.480067	+	0.831501i	0.999739	−	0.0228656i	$-0.00727898\pi$
$853$	409.497	+	709.270i	0.480067	+	0.831501i	−0.519671	+	0.854366i	$0.673946\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1163.67	−	671.846i	1.35784	−	0.783951i	0.368510	−	0.929624i	$-0.379868\pi$
$857$	1163.67	−	671.846i	1.35784	−	0.783951i	0.989333	+	0.145673i	$0.0465349\pi$
$858$		0			0
$859$	−281.268	+	487.170i	−0.327436	+	0.567136i	−0.982002	−	0.188869i	$-0.939518\pi$
$859$	−281.268	+	487.170i	−0.327436	+	0.567136i	0.654566	+	0.756005i	$0.272851\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1326.03i			1.53654i	0.640126	+	0.768270i	$0.278882\pi$
$863$			1326.03i			1.53654i	−0.640126	+	0.768270i	$0.721118\pi$
$864$		0			0
$865$	384.430			0.444427
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−109.127	−	63.0047i	−0.125578	−	0.0725025i
$870$		0			0
$871$	88.2293	+	152.818i	0.101297	+	0.175451i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1179.11	−	680.757i	1.34755	−	0.778008i
$876$		0			0
$877$	−127.627	+	221.056i	−0.145527	+	0.252060i	−0.929569	−	0.368647i	$-0.879821\pi$
$877$	−127.627	+	221.056i	−0.145527	+	0.252060i	0.784043	+	0.620707i	$0.213154\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	358.375i		−	0.406782i	−0.979098	−	0.203391i	$-0.934804\pi$
$881$		−	358.375i		−	0.406782i	0.979098	−	0.203391i	$-0.0651963\pi$
$882$		0			0
$883$	112.147			0.127007			0.0635033	−	0.997982i	$-0.479773\pi$
$883$	112.147			0.127007			0.0635033	+	0.997982i	$0.479773\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1013.97	−	585.419i	−1.14315	−	0.659998i	−0.195942	−	0.980616i	$-0.562776\pi$
$887$	−1013.97	−	585.419i	−1.14315	−	0.659998i	−0.947209	+	0.320617i	$0.896110\pi$
$888$		0			0
$889$	−809.455	−	1402.02i	−0.910523	−	1.57707i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−81.8681	+	47.2666i	−0.0916776	+	0.0529301i
$894$		0			0
$895$	714.560	−	1237.66i	0.798392	−	1.38285i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	886.844i		−	0.986478i
$900$		0			0
$901$	−879.539			−0.976181
$902$		0			0
$903$		0			0
$904$		0			0
$905$	450.561	+	260.132i	0.497858	+	0.287438i
$906$		0			0
$907$	765.705	+	1326.24i	0.844218	+	1.46223i	0.886299	+	0.463114i	$0.153268\pi$
$907$	765.705	+	1326.24i	0.844218	+	1.46223i	−0.0420812	+	0.999114i	$0.513399\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1268.74	−	732.505i	1.39269	−	0.804067i	0.399074	−	0.916919i	$-0.369332\pi$
$911$	1268.74	−	732.505i	1.39269	−	0.804067i	0.993612	+	0.112851i	$0.0359984\pi$
$912$		0			0
$913$	151.241	−	261.958i	0.165653	−	0.286920i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2321.82i			2.53197i
$918$		0			0
$919$	−17.7049			−0.0192654			−0.00963269	−	0.999954i	$-0.503066\pi$
$919$	−17.7049			−0.0192654			−0.00963269	+	0.999954i	$0.503066\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1506.83	+	869.966i	1.63253	+	0.942542i
$924$		0			0
$925$	1213.30	+	2101.50i	1.31168	+	2.27189i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1256.59	−	725.491i	1.35262	−	0.780937i	0.364007	−	0.931396i	$-0.381408\pi$
$929$	1256.59	−	725.491i	1.35262	−	0.780937i	0.988616	+	0.150459i	$0.0480751\pi$
$930$		0			0
$931$	222.022	−	384.553i	0.238477	−	0.413054i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			174.448i			0.186575i
$936$		0			0
$937$	471.414			0.503110			0.251555	−	0.967843i	$-0.419058\pi$
$937$	471.414			0.503110			0.251555	+	0.967843i	$0.419058\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	126.626	+	73.1076i	0.134565	+	0.0776914i	0.565771	−	0.824562i	$-0.308578\pi$
$941$	126.626	+	73.1076i	0.134565	+	0.0776914i	−0.431206	+	0.902253i	$0.641912\pi$
$942$		0			0
$943$	145.351	+	251.755i	0.154136	+	0.266972i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−152.698	+	88.1600i	−0.161244	+	0.0930940i	−0.578450	−	0.815718i	$-0.696342\pi$
$947$	−152.698	+	88.1600i	−0.161244	+	0.0930940i	0.417207	+	0.908812i	$0.363009\pi$
$948$		0			0
$949$	−208.691	+	361.464i	−0.219907	+	0.380890i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1111.46i		−	1.16628i	−0.812373	−	0.583138i	$-0.801825\pi$
$953$		−	1111.46i		−	1.16628i	0.812373	−	0.583138i	$-0.198175\pi$
$954$		0			0
$955$	537.257			0.562573
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−488.527	−	282.051i	−0.509413	−	0.294110i
$960$		0			0
$961$	−164.947	−	285.697i	−0.171641	−	0.297291i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	467.171	−	269.721i	0.484115	−	0.279504i
$966$		0			0
$967$	−32.9276	+	57.0323i	−0.0340513	+	0.0589786i	−0.882549	−	0.470221i	$-0.844174\pi$
$967$	−32.9276	+	57.0323i	−0.0340513	+	0.0589786i	0.848498	+	0.529199i	$0.177508\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1269.23i			1.30714i	0.756867	+	0.653569i	$0.226729\pi$
$971$			1269.23i			1.30714i	−0.756867	+	0.653569i	$0.773271\pi$
$972$		0			0
$973$	−1603.26			−1.64775
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−764.333	−	441.288i	−0.782327	−	0.451677i	0.0549275	−	0.998490i	$-0.482507\pi$
$977$	−764.333	−	441.288i	−0.782327	−	0.451677i	−0.837254	+	0.546814i	$0.815841\pi$
$978$		0			0
$979$	46.8812	+	81.2006i	0.0478868	+	0.0829424i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	59.1795	−	34.1673i	0.0602029	−	0.0347582i	−0.469596	−	0.882881i	$-0.655601\pi$
$983$	59.1795	−	34.1673i	0.0602029	−	0.0347582i	0.529799	+	0.848123i	$0.322267\pi$
$984$		0			0
$985$	357.819	−	619.760i	0.363268	−	0.629198i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			464.973i			0.470144i
$990$		0			0
$991$	−418.216			−0.422014			−0.211007	−	0.977485i	$-0.567674\pi$
$991$	−418.216			−0.422014			−0.211007	+	0.977485i	$0.567674\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2018.12	−	1165.16i	−2.02826	−	1.17102i
$996$		0			0
$997$	−255.039	−	441.741i	−0.255807	−	0.443070i	0.709308	−	0.704899i	$-0.249008\pi$
$997$	−255.039	−	441.741i	−0.255807	−	0.443070i	−0.965114	+	0.261829i	$0.915674\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.q.l.449.11		24
3.2	odd	2		576.3.q.k.257.4		24
4.3	odd	2	inner	1728.3.q.l.449.12		24
8.3	odd	2		864.3.q.b.449.1		24
8.5	even	2		864.3.q.b.449.2		24
9.2	odd	6	inner	1728.3.q.l.1601.11		24
9.7	even	3		576.3.q.k.65.4		24
12.11	even	2		576.3.q.k.257.9		24
24.5	odd	2		288.3.q.a.257.9	yes	24
24.11	even	2		288.3.q.a.257.4	yes	24
36.7	odd	6		576.3.q.k.65.9		24
36.11	even	6	inner	1728.3.q.l.1601.12		24
72.5	odd	6		2592.3.e.j.161.23		24
72.11	even	6		864.3.q.b.737.1		24
72.13	even	6		2592.3.e.j.161.24		24
72.29	odd	6		864.3.q.b.737.2		24
72.43	odd	6		288.3.q.a.65.4	✓	24
72.59	even	6		2592.3.e.j.161.1		24
72.61	even	6		288.3.q.a.65.9	yes	24
72.67	odd	6		2592.3.e.j.161.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.a.65.4	✓	24	72.43	odd	6
288.3.q.a.65.9	yes	24	72.61	even	6
288.3.q.a.257.4	yes	24	24.11	even	2
288.3.q.a.257.9	yes	24	24.5	odd	2
576.3.q.k.65.4		24	9.7	even	3
576.3.q.k.65.9		24	36.7	odd	6
576.3.q.k.257.4		24	3.2	odd	2
576.3.q.k.257.9		24	12.11	even	2
864.3.q.b.449.1		24	8.3	odd	2
864.3.q.b.449.2		24	8.5	even	2
864.3.q.b.737.1		24	72.11	even	6
864.3.q.b.737.2		24	72.29	odd	6
1728.3.q.l.449.11		24	1.1	even	1	trivial
1728.3.q.l.449.12		24	4.3	odd	2	inner
1728.3.q.l.1601.11		24	9.2	odd	6	inner
1728.3.q.l.1601.12		24	36.11	even	6	inner
2592.3.e.j.161.1		24	72.59	even	6
2592.3.e.j.161.2		24	72.67	odd	6
2592.3.e.j.161.23		24	72.5	odd	6
2592.3.e.j.161.24		24	72.13	even	6

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.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.92504 + 3.99817i) q^{5} +(-6.10647 - 10.5767i) q^{7} +O(q^{10})\)	\(q+(6.92504 + 3.99817i) q^{5} +(-6.10647 - 10.5767i) q^{7} +(-2.04017 + 1.17789i) q^{11} +(-8.01829 + 13.8881i) q^{13} -9.26058i q^{17} -4.43352 q^{19} +(-10.9115 - 6.29976i) q^{23} +(19.4708 + 33.7243i) q^{25} +(21.3763 - 12.3416i) q^{29} +(17.9645 - 31.1154i) q^{31} -97.6589i q^{35} +62.3141 q^{37} +(-19.9813 - 11.5362i) q^{41} +(-18.4520 - 31.9598i) q^{43} +(18.4657 - 10.6612i) q^{47} +(-50.0781 + 86.7377i) q^{49} -94.9767i q^{53} -18.8377 q^{55} +(46.4263 + 26.8043i) q^{59} +(-53.0732 - 91.9255i) q^{61} +(-111.054 + 64.1170i) q^{65} +(5.50175 - 9.52931i) q^{67} -108.498i q^{71} +26.0269 q^{73} +(24.9165 + 14.3855i) q^{77} +(26.7447 + 46.3231i) q^{79} +(-111.198 + 64.2000i) q^{83} +(37.0254 - 64.1299i) q^{85} -39.8009i q^{89} +195.854 q^{91} +(-30.7023 - 17.7260i) q^{95} +(-29.4518 - 51.0121i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97}+O(q^{100}) \) 24 * q + 60 * q^25 + 72 * q^29 + 36 * q^41 - 132 * q^49 - 96 * q^61 - 576 * q^65 + 24 * q^73 + 432 * q^77 + 96 * q^85 + 252 * q^97

Introduction

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