LMFDB - Embedded newform 2052.3.bl.a.145.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(145,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 5]))

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

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

chi := DirichletCharacter("2052.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.bl (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:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.4
Character	$\chi$	$=$	2052.145
Dual form			2052.3.bl.a.1585.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.87469 q^{5} +(2.97107 + 5.14604i) q^{7} +O(q^{10})$	$q-7.87469 q^{5} +(2.97107 + 5.14604i) q^{7} +(-2.93551 - 5.08444i) q^{11} +(8.80601 - 5.08415i) q^{13} +(2.78184 + 4.81829i) q^{17} +(-11.6693 + 14.9942i) q^{19} +(22.7998 + 39.4904i) q^{23} +37.0108 q^{25} -39.4060i q^{29} +(-18.3346 - 10.5855i) q^{31} +(-23.3963 - 40.5235i) q^{35} -32.9592i q^{37} +50.5716i q^{41} +(12.0530 - 20.8765i) q^{43} +59.3968 q^{47} +(6.84550 - 11.8568i) q^{49} +(9.43881 + 5.44950i) q^{53} +(23.1162 + 40.0384i) q^{55} +36.1350i q^{59} +16.4438 q^{61} +(-69.3446 + 40.0361i) q^{65} +(-106.274 + 61.3572i) q^{67} +(-111.397 + 64.3151i) q^{71} +(35.7968 + 62.0019i) q^{73} +(17.4432 - 30.2125i) q^{77} +(-26.5147 - 15.3083i) q^{79} +(-61.1539 - 105.922i) q^{83} +(-21.9061 - 37.9426i) q^{85} +(-51.6517 - 29.8211i) q^{89} +(52.3265 + 30.2107i) q^{91} +(91.8925 - 118.075i) q^{95} +(79.5266 + 45.9147i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - q^{7}+O(q^{10}) $ 80 * q - q^7	$ 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) $ 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−7.87469			−1.57494			−0.787469	−	0.616354i	$-0.788609\pi$
$5$	−7.87469			−1.57494			−0.787469	+	0.616354i	$0.788609\pi$
$6$		0			0
$7$	2.97107	+	5.14604i	0.424438	+	0.735149i	0.996368	−	0.0851540i	$-0.0271382\pi$
$7$	2.97107	+	5.14604i	0.424438	+	0.735149i	−0.571929	+	0.820303i	$0.693805\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.93551	−	5.08444i	−0.266864	−	0.462222i	0.701186	−	0.712978i	$-0.252654\pi$
$11$	−2.93551	−	5.08444i	−0.266864	−	0.462222i	−0.968050	+	0.250756i	$0.919321\pi$
$12$		0			0
$13$	8.80601	−	5.08415i	0.677385	−	0.391089i	−0.121484	−	0.992593i	$-0.538765\pi$
$13$	8.80601	−	5.08415i	0.677385	−	0.391089i	0.798869	+	0.601505i	$0.205432\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.78184	+	4.81829i	0.163638	+	0.283429i	0.936171	−	0.351546i	$-0.114344\pi$
$17$	2.78184	+	4.81829i	0.163638	+	0.283429i	−0.772533	+	0.634975i	$0.781011\pi$
$18$		0			0
$19$	−11.6693	+	14.9942i	−0.614176	+	0.789169i
$19$	−11.6693	+	14.9942i	−0.614176	+	0.789169i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	22.7998	+	39.4904i	0.991294	+	1.71697i	0.609672	+	0.792654i	$0.291301\pi$
$23$	22.7998	+	39.4904i	0.991294	+	1.71697i	0.381622	+	0.924318i	$0.375365\pi$
$24$		0			0
$25$	37.0108			1.48043
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	39.4060i		−	1.35883i	−0.733756	−	0.679413i	$-0.762235\pi$
$29$		−	39.4060i		−	1.35883i	0.733756	−	0.679413i	$-0.237765\pi$
$30$		0			0
$31$	−18.3346	−	10.5855i	−0.591438	−	0.341467i	0.174228	−	0.984705i	$-0.444257\pi$
$31$	−18.3346	−	10.5855i	−0.591438	−	0.341467i	−0.765666	+	0.643238i	$0.777590\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−23.3963	−	40.5235i	−0.668464	−	1.15781i
$36$		0			0
$37$		−	32.9592i		−	0.890789i	−0.895334	−	0.445394i	$-0.853063\pi$
$37$		−	32.9592i		−	0.890789i	0.895334	−	0.445394i	$-0.146937\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			50.5716i			1.23345i	0.787177	+	0.616727i	$0.211542\pi$
$41$			50.5716i			1.23345i	−0.787177	+	0.616727i	$0.788458\pi$
$42$		0			0
$43$	12.0530	−	20.8765i	0.280303	−	0.485499i	−0.691156	−	0.722705i	$-0.742898\pi$
$43$	12.0530	−	20.8765i	0.280303	−	0.485499i	0.971459	+	0.237206i	$0.0762317\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	59.3968			1.26376			0.631880	−	0.775066i	$-0.282283\pi$
$47$	59.3968			1.26376			0.631880	+	0.775066i	$0.282283\pi$
$48$		0			0
$49$	6.84550	−	11.8568i	0.139704	−	0.241975i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.43881	+	5.44950i	0.178091	+	0.102821i	0.586395	−	0.810025i	$-0.300547\pi$
$53$	9.43881	+	5.44950i	0.178091	+	0.102821i	−0.408305	+	0.912846i	$0.633880\pi$
$54$		0			0
$55$	23.1162	+	40.0384i	0.420295	+	0.727972i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			36.1350i			0.612457i	0.951958	+	0.306229i	$0.0990672\pi$
$59$			36.1350i			0.612457i	−0.951958	+	0.306229i	$0.900933\pi$
$60$		0			0
$61$	16.4438			0.269570			0.134785	−	0.990875i	$-0.456966\pi$
$61$	16.4438			0.269570			0.134785	+	0.990875i	$0.456966\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−69.3446	+	40.0361i	−1.06684	+	0.615941i
$66$		0			0
$67$	−106.274	+	61.3572i	−1.58618	+	0.915779i	−0.592246	+	0.805757i	$0.701759\pi$
$67$	−106.274	+	61.3572i	−1.58618	+	0.915779i	−0.993929	+	0.110021i	$0.964908\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−111.397	+	64.3151i	−1.56897	+	0.905847i	−0.572684	+	0.819776i	$0.694098\pi$
$71$	−111.397	+	64.3151i	−1.56897	+	0.905847i	−0.996289	+	0.0860708i	$0.972569\pi$
$72$		0			0
$73$	35.7968	+	62.0019i	0.490368	+	0.849342i	0.999939	−	0.0110870i	$-0.00352918\pi$
$73$	35.7968	+	62.0019i	0.490368	+	0.849342i	−0.509571	+	0.860429i	$0.670196\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	17.4432	−	30.2125i	0.226535	−	0.392370i
$78$		0			0
$79$	−26.5147	−	15.3083i	−0.335629	−	0.193775i	0.322708	−	0.946498i	$-0.395407\pi$
$79$	−26.5147	−	15.3083i	−0.335629	−	0.193775i	−0.658337	+	0.752723i	$0.728740\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−61.1539	−	105.922i	−0.736794	−	1.27616i	−0.953932	−	0.300023i	$-0.903005\pi$
$83$	−61.1539	−	105.922i	−0.736794	−	1.27616i	0.217138	−	0.976141i	$-0.430328\pi$
$84$		0			0
$85$	−21.9061	−	37.9426i	−0.257719	−	0.446383i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−51.6517	−	29.8211i	−0.580356	−	0.335069i	0.180919	−	0.983498i	$-0.442093\pi$
$89$	−51.6517	−	29.8211i	−0.580356	−	0.335069i	−0.761275	+	0.648429i	$0.775426\pi$
$90$		0			0
$91$	52.3265	+	30.2107i	0.575017	+	0.331986i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	91.8925	−	118.075i	0.967289	−	1.24289i
$96$		0			0
$97$	79.5266	+	45.9147i	0.819862	+	0.473347i	0.850369	−	0.526187i	$-0.176379\pi$
$97$	79.5266	+	45.9147i	0.819862	+	0.473347i	−0.0305069	+	0.999535i	$0.509712\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−21.2266			−0.210164			−0.105082	−	0.994464i	$-0.533511\pi$
$101$	−21.2266			−0.210164			−0.105082	+	0.994464i	$0.533511\pi$
$102$		0			0
$103$	−7.07481	−	4.08464i	−0.0686875	−	0.0396567i	0.465263	−	0.885173i	$-0.345960\pi$
$103$	−7.07481	−	4.08464i	−0.0686875	−	0.0396567i	−0.533950	+	0.845516i	$0.679293\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	28.0913i		−	0.262536i	−0.991347	−	0.131268i	$-0.958095\pi$
$107$		−	28.0913i		−	0.262536i	0.991347	−	0.131268i	$-0.0419048\pi$
$108$		0			0
$109$	−115.785	+	66.8483i	−1.06225	+	0.613288i	−0.926052	−	0.377395i	$-0.876820\pi$
$109$	−115.785	+	66.8483i	−1.06225	+	0.613288i	−0.136193	+	0.990682i	$0.543487\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.57918	+	1.48909i	0.0228246	+	0.0131778i	0.511369	−	0.859361i	$-0.329139\pi$
$113$	2.57918	+	1.48909i	0.0228246	+	0.0131778i	−0.488544	+	0.872539i	$0.662472\pi$
$114$		0			0
$115$	−179.541	−	310.974i	−1.56123	−	2.70413i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−16.5301	+	28.6309i	−0.138908	+	0.240596i
$120$		0			0
$121$	43.2656	−	74.9382i	0.357567	−	0.619324i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−94.5813			−0.756651
$126$		0			0
$127$	−83.0708	−	47.9610i	−0.654101	−	0.377645i	0.135925	−	0.990719i	$-0.456599\pi$
$127$	−83.0708	−	47.9610i	−0.654101	−	0.377645i	−0.790026	+	0.613074i	$0.789933\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−203.065			−1.55011			−0.775057	−	0.631892i	$-0.782279\pi$
$131$	−203.065			−1.55011			−0.775057	+	0.631892i	$0.782279\pi$
$132$		0			0
$133$	−111.831	−	15.5021i	−0.840837	−	0.116557i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−134.366			−0.980773			−0.490386	−	0.871505i	$-0.663144\pi$
$137$	−134.366			−0.980773			−0.490386	+	0.871505i	$0.663144\pi$
$138$		0			0
$139$	−90.9735	−	157.571i	−0.654485	−	1.13360i	−0.982023	−	0.188764i	$-0.939552\pi$
$139$	−90.9735	−	157.571i	−0.654485	−	1.13360i	0.327537	−	0.944838i	$-0.393781\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−51.7002	−	29.8491i	−0.361540	−	0.208735i
$144$		0			0
$145$			310.310i			2.14007i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−191.808			−1.28730			−0.643651	−	0.765319i	$-0.722581\pi$
$149$	−191.808			−1.28730			−0.643651	+	0.765319i	$0.722581\pi$
$150$		0			0
$151$	−95.6299	+	55.2120i	−0.633311	+	0.365642i	−0.782033	−	0.623237i	$-0.785817\pi$
$151$	−95.6299	+	55.2120i	−0.633311	+	0.365642i	0.148722	+	0.988879i	$0.452484\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	144.379	+	83.3574i	0.931479	+	0.537789i
$156$		0			0
$157$	−26.4117			−0.168228			−0.0841138	−	0.996456i	$-0.526806\pi$
$157$	−26.4117			−0.168228			−0.0841138	+	0.996456i	$0.526806\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−135.479	+	234.657i	−0.841487	+	1.45750i
$162$		0			0
$163$	−86.6847			−0.531808			−0.265904	−	0.964000i	$-0.585670\pi$
$163$	−86.6847			−0.531808			−0.265904	+	0.964000i	$0.585670\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−227.524	+	131.361i	−1.36242	+	0.786592i	−0.989945	−	0.141451i	$-0.954823\pi$
$167$	−227.524	+	131.361i	−1.36242	+	0.786592i	−0.372472	+	0.928043i	$0.621490\pi$
$168$		0			0
$169$	−32.8028	+	56.8161i	−0.194099	+	0.336190i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	258.789	+	149.412i	1.49589	+	0.863652i	0.999989	−	0.00472721i	$-0.00150472\pi$
$173$	258.789	+	149.412i	1.49589	+	0.863652i	0.495901	+	0.868379i	$0.334838\pi$
$174$		0			0
$175$	109.962	+	190.459i	0.628352	+	1.08834i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			29.5036i			0.164825i	0.996598	+	0.0824124i	$0.0262625\pi$
$179$			29.5036i			0.164825i	−0.996598	+	0.0824124i	$0.973738\pi$
$180$		0			0
$181$	−158.150	−	91.3082i	−0.873759	−	0.504465i	−0.00516334	−	0.999987i	$-0.501644\pi$
$181$	−158.150	−	91.3082i	−0.873759	−	0.504465i	−0.868596	+	0.495522i	$0.834977\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			259.544i			1.40294i
$186$		0			0
$187$	16.3322	−	28.2882i	0.0873381	−	0.151274i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−70.1739	−	121.545i	−0.367403	−	0.636360i	0.621756	−	0.783211i	$-0.286420\pi$
$191$	−70.1739	−	121.545i	−0.367403	−	0.636360i	−0.989159	+	0.146851i	$0.953086\pi$
$192$		0			0
$193$		−	332.739i		−	1.72404i	−0.506878	−	0.862018i	$-0.669200\pi$
$193$		−	332.739i		−	1.72404i	0.506878	−	0.862018i	$-0.330800\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	52.4559			0.266274			0.133137	−	0.991098i	$-0.457495\pi$
$197$	52.4559			0.266274			0.133137	+	0.991098i	$0.457495\pi$
$198$		0			0
$199$	−108.793	+	188.435i	−0.546698	+	0.946909i	0.451800	+	0.892119i	$0.350782\pi$
$199$	−108.793	+	188.435i	−0.546698	+	0.946909i	−0.998498	+	0.0547893i	$0.982551\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	202.785	−	117.078i	0.998939	−	0.576738i
$204$		0			0
$205$		−	398.236i		−	1.94262i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	110.493	+	15.3165i	0.528673	+	0.0732848i
$210$		0			0
$211$			343.428i			1.62762i	0.581130	+	0.813811i	$0.302611\pi$
$211$			343.428i			1.62762i	−0.581130	+	0.813811i	$0.697389\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−94.9139	+	164.396i	−0.441460	+	0.764631i
$216$		0			0
$217$		−	125.801i		−	0.579727i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	48.9939	+	28.2866i	0.221692	+	0.127994i
$222$		0			0
$223$	−62.1329	−	35.8725i	−0.278623	−	0.160863i	0.354177	−	0.935178i	$-0.384761\pi$
$223$	−62.1329	−	35.8725i	−0.278623	−	0.160863i	−0.632800	+	0.774315i	$0.718094\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	226.307	−	130.659i	0.996949	−	0.575589i	0.0896049	−	0.995977i	$-0.471440\pi$
$227$	226.307	−	130.659i	0.996949	−	0.575589i	0.907344	+	0.420389i	$0.138106\pi$
$228$		0			0
$229$	7.31766	−	12.6746i	0.0319548	−	0.0553474i	−0.849606	−	0.527418i	$-0.823160\pi$
$229$	7.31766	−	12.6746i	0.0319548	−	0.0553474i	0.881561	+	0.472071i	$0.156493\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	51.4515	+	89.1166i	0.220822	+	0.382475i	0.955058	−	0.296420i	$-0.0957928\pi$
$233$	51.4515	+	89.1166i	0.220822	+	0.382475i	−0.734236	+	0.678894i	$0.762459\pi$
$234$		0			0
$235$	−467.731			−1.99035
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−60.9446	+	105.559i	−0.254998	+	0.441670i	−0.964895	−	0.262636i	$-0.915408\pi$
$239$	−60.9446	+	105.559i	−0.254998	+	0.441670i	0.709897	+	0.704306i	$0.248742\pi$
$240$		0			0
$241$		−	187.951i		−	0.779881i	−0.920840	−	0.389940i	$-0.872496\pi$
$241$		−	187.951i		−	0.779881i	0.920840	−	0.389940i	$-0.127504\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−53.9062	+	93.3684i	−0.220025	+	0.381095i
$246$		0			0
$247$	−26.5275	+	191.368i	−0.107399	+	0.774769i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.6291	−	21.8742i	0.0503150	−	0.0871482i	−0.839771	−	0.542941i	$-0.817311\pi$
$251$	12.6291	−	21.8742i	0.0503150	−	0.0871482i	0.890086	+	0.455793i	$0.150644\pi$
$252$		0			0
$253$	133.858	−	231.848i	0.529082	−	0.916397i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	171.492	−	99.0107i	0.667282	−	0.385256i	−0.127764	−	0.991805i	$-0.540780\pi$
$257$	171.492	−	99.0107i	0.667282	−	0.385256i	0.795046	+	0.606549i	$0.207447\pi$
$258$		0			0
$259$	169.609	−	97.9240i	0.654862	−	0.378085i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−81.4534	+	141.081i	−0.309709	+	0.536431i	−0.978299	−	0.207200i	$-0.933565\pi$
$263$	−81.4534	+	141.081i	−0.309709	+	0.536431i	0.668590	+	0.743631i	$0.266898\pi$
$264$		0			0
$265$	−74.3277	−	42.9131i	−0.280482	−	0.161936i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−341.865	+	197.376i	−1.27088	+	0.733740i	−0.975153	−	0.221534i	$-0.928894\pi$
$269$	−341.865	+	197.376i	−1.27088	+	0.733740i	−0.295722	+	0.955274i	$0.595560\pi$
$270$		0			0
$271$	−179.436	−	310.792i	−0.662125	−	1.14683i	−0.980056	−	0.198721i	$-0.936321\pi$
$271$	−179.436	−	310.792i	−0.662125	−	1.14683i	0.317931	−	0.948114i	$-0.397012\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−108.645	−	188.179i	−0.395074	−	0.684288i
$276$		0			0
$277$	54.4726	+	94.3494i	0.196652	+	0.340611i	0.947441	−	0.319931i	$-0.103660\pi$
$277$	54.4726	+	94.3494i	0.196652	+	0.340611i	−0.750789	+	0.660542i	$0.770326\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			440.659i			1.56818i	0.620646	+	0.784091i	$0.286871\pi$
$281$			440.659i			1.56818i	−0.620646	+	0.784091i	$0.713129\pi$
$282$		0			0
$283$	43.7500			0.154594			0.0772969	−	0.997008i	$-0.475371\pi$
$283$	43.7500			0.154594			0.0772969	+	0.997008i	$0.475371\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−260.244	+	150.252i	−0.906773	+	0.523525i
$288$		0			0
$289$	129.023	−	223.474i	0.446445	−	0.773266i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.3446	−	8.85919i	−0.0523706	−	0.0302362i	0.473586	−	0.880748i	$-0.342959\pi$
$293$	−15.3446	−	8.85919i	−0.0523706	−	0.0302362i	−0.525957	+	0.850511i	$0.676293\pi$
$294$		0			0
$295$		−	284.552i		−	0.964582i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	401.550	+	231.835i	1.34298	+	0.775368i
$300$		0			0
$301$	143.242			0.475885
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−129.490			−0.424557
$306$		0			0
$307$	−124.766	+	72.0337i	−0.406404	+	0.234638i	−0.689244	−	0.724530i	$-0.742057\pi$
$307$	−124.766	+	72.0337i	−0.406404	+	0.234638i	0.282839	+	0.959167i	$0.408724\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	100.334	−	173.784i	0.322617	−	0.558790i	−0.658410	−	0.752660i	$-0.728771\pi$
$311$	100.334	−	173.784i	0.322617	−	0.558790i	0.981027	+	0.193870i	$0.0621040\pi$
$312$		0			0
$313$	−346.971			−1.10853			−0.554267	−	0.832339i	$-0.687001\pi$
$313$	−346.971			−1.10853			−0.554267	+	0.832339i	$0.687001\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	35.8229i		−	0.113006i	−0.998402	−	0.0565031i	$-0.982005\pi$
$317$		−	35.8229i		−	0.113006i	0.998402	−	0.0565031i	$-0.0179951\pi$
$318$		0			0
$319$	−200.357	+	115.676i	−0.628080	+	0.362622i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−104.709	−	14.5147i	−0.324176	−	0.0449373i
$324$		0			0
$325$	325.917	−	188.169i	1.00282	−	0.578980i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	176.472	+	305.658i	0.536388	+	0.929052i
$330$		0			0
$331$	−524.816	+	303.003i	−1.58555	+	0.915416i	−0.591519	+	0.806291i	$0.701471\pi$
$331$	−524.816	+	303.003i	−1.58555	+	0.915416i	−0.994028	+	0.109125i	$0.965195\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	836.873	−	483.169i	2.49813	−	1.44230i
$336$		0			0
$337$		−	351.512i		−	1.04306i	−0.853232	−	0.521532i	$-0.825361\pi$
$337$		−	351.512i		−	1.04306i	0.853232	−	0.521532i	$-0.174639\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			124.295i			0.364501i
$342$		0			0
$343$	372.519			1.08606
$344$		0			0
$345$		0			0
$346$		0			0
$347$	336.496			0.969729			0.484865	−	0.874589i	$-0.338869\pi$
$347$	336.496			0.969729			0.484865	+	0.874589i	$0.338869\pi$
$348$		0			0
$349$	147.329	+	255.181i	0.422146	+	0.731178i	0.996149	−	0.0876751i	$-0.0279437\pi$
$349$	147.329	+	255.181i	0.422146	+	0.731178i	−0.574003	+	0.818853i	$0.694610\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−147.760	−	255.928i	−0.418584	−	0.725009i	0.577213	−	0.816594i	$-0.304140\pi$
$353$	−147.760	−	255.928i	−0.418584	−	0.725009i	−0.995797	+	0.0915844i	$0.970807\pi$
$354$		0			0
$355$	877.218	−	506.462i	2.47104	−	1.42665i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	36.1082	+	62.5412i	0.100580	+	0.174210i	0.911924	−	0.410360i	$-0.134597\pi$
$359$	36.1082	+	62.5412i	0.100580	+	0.174210i	−0.811344	+	0.584569i	$0.801264\pi$
$360$		0			0
$361$	−88.6530	−	349.945i	−0.245576	−	0.969377i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−281.889	−	488.246i	−0.772299	−	1.33766i
$366$		0			0
$367$	651.057			1.77400			0.886998	−	0.461773i	$-0.152786\pi$
$367$	651.057			1.77400			0.886998	+	0.461773i	$0.152786\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			64.7633i			0.174564i
$372$		0			0
$373$	25.4034	+	14.6667i	0.0681057	+	0.0393209i	0.533666	−	0.845695i	$-0.320814\pi$
$373$	25.4034	+	14.6667i	0.0681057	+	0.0393209i	−0.465560	+	0.885016i	$0.654147\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−200.346	−	347.009i	−0.531421	−	0.920449i
$378$		0			0
$379$			680.418i			1.79530i	0.440712	+	0.897648i	$0.354726\pi$
$379$			680.418i			1.79530i	−0.440712	+	0.897648i	$0.645274\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			154.629i			0.403730i	0.979413	+	0.201865i	$0.0647003\pi$
$383$			154.629i			0.403730i	−0.979413	+	0.201865i	$0.935300\pi$
$384$		0			0
$385$	−137.360	+	237.914i	−0.356778	+	0.617958i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−466.183			−1.19841			−0.599207	−	0.800594i	$-0.704517\pi$
$389$	−466.183			−1.19841			−0.599207	+	0.800594i	$0.704517\pi$
$390$		0			0
$391$	−126.851	+	219.712i	−0.324426	+	0.561923i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	208.795	+	120.548i	0.528595	+	0.305184i
$396$		0			0
$397$	−185.889	−	321.969i	−0.468233	−	0.811004i	0.531107	−	0.847304i	$-0.321776\pi$
$397$	−185.889	−	321.969i	−0.468233	−	0.811004i	−0.999341	+	0.0363003i	$0.988443\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	437.412i		−	1.09080i	−0.838175	−	0.545401i	$-0.816377\pi$
$401$		−	437.412i		−	1.09080i	0.838175	−	0.545401i	$-0.183623\pi$
$402$		0			0
$403$	−215.273			−0.534175
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−167.579	+	96.7519i	−0.411742	+	0.237720i
$408$		0			0
$409$	−243.379	+	140.515i	−0.595059	+	0.343558i	−0.767095	−	0.641533i	$-0.778299\pi$
$409$	−243.379	+	140.515i	−0.595059	+	0.343558i	0.172036	+	0.985091i	$0.444965\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−185.952	+	107.359i	−0.450247	+	0.259950i
$414$		0			0
$415$	481.568	+	834.100i	1.16041	+	2.00988i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	308.348	−	534.075i	0.735914	−	1.27464i	−0.218406	−	0.975858i	$-0.570086\pi$
$419$	308.348	−	534.075i	0.735914	−	1.27464i	0.954321	−	0.298783i	$-0.0965808\pi$
$420$		0			0
$421$	−200.316	−	115.653i	−0.475810	−	0.274709i	0.242859	−	0.970062i	$-0.421915\pi$
$421$	−200.316	−	115.653i	−0.475810	−	0.274709i	−0.718669	+	0.695353i	$0.755248\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	102.958	+	178.329i	0.242255	+	0.419597i
$426$		0			0
$427$	48.8556	+	84.6204i	0.114416	+	0.198174i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−229.645	−	132.586i	−0.532819	−	0.307623i	0.209344	−	0.977842i	$-0.432867\pi$
$431$	−229.645	−	132.586i	−0.532819	−	0.307623i	−0.742164	+	0.670219i	$0.766200\pi$
$432$		0			0
$433$	−273.676	−	158.007i	−0.632047	−	0.364913i	0.149497	−	0.988762i	$-0.452234\pi$
$433$	−273.676	−	158.007i	−0.632047	−	0.364913i	−0.781545	+	0.623849i	$0.785568\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−858.185	−	118.962i	−1.96381	−	0.272224i
$438$		0			0
$439$	284.047	+	163.995i	0.647032	+	0.373564i	0.787318	−	0.616547i	$-0.211469\pi$
$439$	284.047	+	163.995i	0.647032	+	0.373564i	−0.140286	+	0.990111i	$0.544802\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	491.688			1.10990			0.554952	−	0.831882i	$-0.312737\pi$
$443$	491.688			1.10990			0.554952	+	0.831882i	$0.312737\pi$
$444$		0			0
$445$	406.741	+	234.832i	0.914025	+	0.527713i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			270.439i			0.602315i	0.953574	+	0.301157i	$0.0973730\pi$
$449$			270.439i			0.602315i	−0.953574	+	0.301157i	$0.902627\pi$
$450$		0			0
$451$	257.129	−	148.453i	0.570130	−	0.329165i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−412.055	−	237.900i	−0.905616	−	0.522858i
$456$		0			0
$457$	419.072	+	725.854i	0.917007	+	1.58830i	0.803936	+	0.594716i	$0.202735\pi$
$457$	419.072	+	725.854i	0.917007	+	1.58830i	0.113071	+	0.993587i	$0.463931\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	236.446	−	409.537i	0.512899	−	0.888366i	−0.486990	−	0.873408i	$-0.661905\pi$
$461$	236.446	−	409.537i	0.512899	−	0.888366i	0.999888	−	0.0149586i	$-0.00476166\pi$
$462$		0			0
$463$	−380.427	+	658.918i	−0.821656	+	1.42315i	0.0827930	+	0.996567i	$0.473616\pi$
$463$	−380.427	+	658.918i	−0.821656	+	1.42315i	−0.904449	+	0.426583i	$0.859717\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	428.831			0.918268			0.459134	−	0.888367i	$-0.348160\pi$
$467$	428.831			0.918268			0.459134	+	0.888367i	$0.348160\pi$
$468$		0			0
$469$	−631.493	−	364.593i	−1.34647	−	0.777383i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−141.527			−0.299211
$474$		0			0
$475$	−431.892	+	554.948i	−0.909245	+	1.16831i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−546.631			−1.14119			−0.570596	−	0.821231i	$-0.693288\pi$
$479$	−546.631			−1.14119			−0.570596	+	0.821231i	$0.693288\pi$
$480$		0			0
$481$	−167.570	−	290.239i	−0.348377	−	0.603408i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−626.248	−	361.564i	−1.29123	−	0.745493i
$486$		0			0
$487$			22.0584i			0.0452945i	0.999744	+	0.0226472i	$0.00720946\pi$
$487$			22.0584i			0.0452945i	−0.999744	+	0.0226472i	$0.992791\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−915.493			−1.86455			−0.932274	−	0.361754i	$-0.882178\pi$
$491$	−915.493			−1.86455			−0.932274	+	0.361754i	$0.882178\pi$
$492$		0			0
$493$	189.869	−	109.621i	0.385131	−	0.222355i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−661.937	−	382.169i	−1.33186	−	0.768952i
$498$		0			0
$499$	821.232			1.64576			0.822878	−	0.568218i	$-0.192367\pi$
$499$	821.232			1.64576			0.822878	+	0.568218i	$0.192367\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−227.039	+	393.244i	−0.451371	+	0.781797i	−0.998471	−	0.0552698i	$-0.982398\pi$
$503$	−227.039	+	393.244i	−0.451371	+	0.781797i	0.547101	+	0.837067i	$0.315731\pi$
$504$		0			0
$505$	167.153			0.330996
$506$		0			0
$507$		0			0
$508$		0			0
$509$	296.836	−	171.379i	0.583175	−	0.336696i	−0.179219	−	0.983809i	$-0.557357\pi$
$509$	296.836	−	171.379i	0.583175	−	0.336696i	0.762394	+	0.647113i	$0.224024\pi$
$510$		0			0
$511$	−212.710	+	368.424i	−0.416262	+	0.720986i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	55.7120	+	32.1653i	0.108179	+	0.0624569i
$516$		0			0
$517$	−174.359	−	301.999i	−0.337252	−	0.584138i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	346.949i		−	0.665929i	−0.942939	−	0.332965i	$-0.891951\pi$
$521$		−	346.949i		−	0.665929i	0.942939	−	0.332965i	$-0.108049\pi$
$522$		0			0
$523$	652.430	+	376.681i	1.24748	+	0.720231i	0.970605	−	0.240676i	$-0.0773693\pi$
$523$	652.430	+	376.681i	1.24748	+	0.720231i	0.276871	+	0.960907i	$0.410703\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	117.788i		−	0.223507i
$528$		0			0
$529$	−775.159	+	1342.61i	−1.46533	+	2.53802i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	257.114	+	445.334i	0.482390	+	0.835524i
$534$		0			0
$535$			221.211i			0.413478i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−80.3800			−0.149128
$540$		0			0
$541$	131.685	−	228.086i	0.243411	−	0.421600i	−0.718273	−	0.695762i	$-0.755067\pi$
$541$	131.685	−	228.086i	0.243411	−	0.421600i	0.961684	+	0.274162i	$0.0884003\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	911.769	−	526.410i	1.67297	−	0.965890i
$546$		0			0
$547$		−	130.371i		−	0.238338i	−0.992874	−	0.119169i	$-0.961977\pi$
$547$		−	130.371i		−	0.238338i	0.992874	−	0.119169i	$-0.0380230\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	590.861	+	459.841i	1.07234	+	0.834558i
$552$		0			0
$553$		−	181.928i		−	0.328983i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−85.9690	+	148.903i	−0.154343	+	0.267330i	−0.932820	−	0.360344i	$-0.882659\pi$
$557$	−85.9690	+	148.903i	−0.154343	+	0.267330i	0.778477	+	0.627674i	$0.215993\pi$
$558$		0			0
$559$		−	245.118i		−	0.438493i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	327.977	+	189.358i	0.582553	+	0.336337i	0.762147	−	0.647404i	$-0.224145\pi$
$563$	327.977	+	189.358i	0.582553	+	0.336337i	−0.179595	+	0.983741i	$0.557479\pi$
$564$		0			0
$565$	−20.3103	−	11.7261i	−0.0359474	−	0.0207542i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	279.072	−	161.122i	0.490460	−	0.283167i	−0.234305	−	0.972163i	$-0.575282\pi$
$569$	279.072	−	161.122i	0.490460	−	0.283167i	0.724765	+	0.688996i	$0.241948\pi$
$570$		0			0
$571$	24.9340	−	43.1869i	0.0436672	−	0.0756339i	−0.843366	−	0.537340i	$-0.819429\pi$
$571$	24.9340	−	43.1869i	0.0436672	−	0.0756339i	0.887033	+	0.461706i	$0.152763\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	843.838	+	1461.57i	1.46754	+	2.54186i
$576$		0			0
$577$	−219.364			−0.380181			−0.190090	−	0.981767i	$-0.560878\pi$
$577$	−219.364			−0.380181			−0.190090	+	0.981767i	$0.560878\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	363.385	−	629.401i	0.625447	−	1.08331i
$582$		0			0
$583$		−	63.9881i		−	0.109757i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	354.197	−	613.488i	0.603403	−	1.04512i	−0.388899	−	0.921280i	$-0.627144\pi$
$587$	354.197	−	613.488i	0.603403	−	1.04512i	0.992302	−	0.123844i	$-0.0395222\pi$
$588$		0			0
$589$	372.673	−	151.387i	0.632722	−	0.257024i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	461.768	−	799.805i	0.778698	−	1.34874i	−0.153995	−	0.988072i	$-0.549214\pi$
$593$	461.768	−	799.805i	0.778698	−	1.34874i	0.932693	−	0.360672i	$-0.117453\pi$
$594$		0			0
$595$	130.169	−	225.460i	0.218772	−	0.378924i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	151.086	−	87.2294i	0.252230	−	0.145625i	−0.368555	−	0.929606i	$-0.620147\pi$
$599$	151.086	−	87.2294i	0.252230	−	0.145625i	0.620785	+	0.783981i	$0.286814\pi$
$600$		0			0
$601$	749.076	−	432.479i	1.24638	−	0.719599i	0.275997	−	0.961159i	$-0.410992\pi$
$601$	749.076	−	432.479i	1.24638	−	0.719599i	0.970386	+	0.241559i	$0.0776588\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−340.703	+	590.116i	−0.563146	+	0.975398i
$606$		0			0
$607$	−78.0604	−	45.0682i	−0.128600	−	0.0742475i	0.434320	−	0.900759i	$-0.356989\pi$
$607$	−78.0604	−	45.0682i	−0.128600	−	0.0742475i	−0.562920	+	0.826511i	$0.690322\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	523.048	−	301.982i	0.856053	−	0.494243i
$612$		0			0
$613$	−272.257	−	471.563i	−0.444139	−	0.769271i	0.553853	−	0.832615i	$-0.313157\pi$
$613$	−272.257	−	471.563i	−0.444139	−	0.769271i	−0.997992	+	0.0633433i	$0.979824\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−522.262	−	904.584i	−0.846453	−	1.46610i	−0.884353	−	0.466819i	$-0.845400\pi$
$617$	−522.262	−	904.584i	−0.846453	−	1.46610i	0.0378998	−	0.999282i	$-0.487933\pi$
$618$		0			0
$619$	−234.817	−	406.714i	−0.379348	−	0.657051i	0.611619	−	0.791152i	$-0.290518\pi$
$619$	−234.817	−	406.714i	−0.379348	−	0.657051i	−0.990968	+	0.134102i	$0.957185\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	354.402i		−	0.568864i
$624$		0			0
$625$	−180.471			−0.288753
$626$		0			0
$627$		0			0
$628$		0			0
$629$	158.807	−	91.6872i	0.252475	−	0.145767i
$630$		0			0
$631$	−357.598	+	619.379i	−0.566717	+	0.981582i	0.430171	+	0.902748i	$0.358453\pi$
$631$	−357.598	+	619.379i	−0.566717	+	0.981582i	−0.996888	+	0.0788350i	$0.974880\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	654.157	+	377.678i	1.03017	+	0.594768i
$636$		0			0
$637$		−	139.214i		−	0.218547i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	802.683	+	463.429i	1.25224	+	0.722979i	0.971553	−	0.236822i	$-0.0761058\pi$
$641$	802.683	+	463.429i	1.25224	+	0.722979i	0.280683	+	0.959801i	$0.409439\pi$
$642$		0			0
$643$	−975.729			−1.51746			−0.758732	−	0.651403i	$-0.774181\pi$
$643$	−975.729			−1.51746			−0.758732	+	0.651403i	$0.774181\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−709.164			−1.09608			−0.548040	−	0.836452i	$-0.684626\pi$
$647$	−709.164			−1.09608			−0.548040	+	0.836452i	$0.684626\pi$
$648$		0			0
$649$	183.726	−	106.074i	0.283091	−	0.163443i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−230.160	+	398.649i	−0.352466	+	0.610489i	−0.986681	−	0.162668i	$-0.947990\pi$
$653$	−230.160	+	398.649i	−0.352466	+	0.610489i	0.634215	+	0.773157i	$0.281323\pi$
$654$		0			0
$655$	1599.07			2.44133
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	59.4528i		−	0.0902167i	−0.998982	−	0.0451083i	$-0.985637\pi$
$659$		−	59.4528i		−	0.0902167i	0.998982	−	0.0451083i	$-0.0143633\pi$
$660$		0			0
$661$	−663.511	+	383.078i	−1.00380	+	0.579544i	−0.909370	−	0.415988i	$-0.863436\pi$
$661$	−663.511	+	383.078i	−1.00380	+	0.579544i	−0.0944292	+	0.995532i	$0.530103\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	880.637	+	122.074i	1.32427	+	0.183570i
$666$		0			0
$667$	1556.16	−	898.447i	2.33307	−	1.34700i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−48.2708	−	83.6076i	−0.0719387	−	0.124601i
$672$		0			0
$673$	−235.641	+	136.048i	−0.350136	+	0.202151i	−0.664745	−	0.747070i	$-0.731460\pi$
$673$	−235.641	+	136.048i	−0.350136	+	0.202151i	0.314609	+	0.949221i	$0.398127\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−156.365	+	90.2772i	−0.230967	+	0.133349i	−0.611018	−	0.791617i	$-0.709240\pi$
$677$	−156.365	+	90.2772i	−0.230967	+	0.133349i	0.380051	+	0.924966i	$0.375906\pi$
$678$		0			0
$679$			545.663i			0.803627i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			581.525i			0.851427i	0.904858	+	0.425714i	$0.139977\pi$
$683$			581.525i			0.851427i	−0.904858	+	0.425714i	$0.860023\pi$
$684$		0			0
$685$	1058.09			1.54466
$686$		0			0
$687$		0			0
$688$		0			0
$689$	110.824			0.160848
$690$		0			0
$691$	−533.790	−	924.552i	−0.772489	−	1.33799i	−0.936195	−	0.351482i	$-0.885678\pi$
$691$	−533.790	−	924.552i	−0.772489	−	1.33799i	0.163705	−	0.986509i	$-0.447655\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	716.388	+	1240.82i	1.03077	+	1.78535i
$696$		0			0
$697$	−243.669	+	140.682i	−0.349597	+	0.201840i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−155.364	−	269.099i	−0.221632	−	0.383878i	0.733671	−	0.679504i	$-0.237805\pi$
$701$	−155.364	−	269.099i	−0.221632	−	0.383878i	−0.955304	+	0.295626i	$0.904472\pi$
$702$		0			0
$703$	494.197	+	384.612i	0.702983	+	0.547101i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−63.0657	−	109.233i	−0.0892019	−	0.154502i
$708$		0			0
$709$	961.035			1.35548			0.677739	−	0.735302i	$-0.262960\pi$
$709$	961.035			1.35548			0.677739	+	0.735302i	$0.262960\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	965.385i		−	1.35398i
$714$		0			0
$715$	407.123	+	235.053i	0.569403	+	0.328745i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−103.698	−	179.611i	−0.144226	−	0.249807i	0.784858	−	0.619676i	$-0.212736\pi$
$719$	−103.698	−	179.611i	−0.144226	−	0.249807i	−0.929084	+	0.369869i	$0.879403\pi$
$720$		0			0
$721$		−	48.5430i		−	0.0673274i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1458.45i		−	2.01165i
$726$		0			0
$727$	279.503	−	484.113i	0.384460	−	0.665905i	−0.607234	−	0.794523i	$-0.707721\pi$
$727$	279.503	−	484.113i	0.384460	−	0.665905i	0.991694	+	0.128618i	$0.0410542\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	134.118			0.183473
$732$		0			0
$733$	−474.704	+	822.211i	−0.647618	+	1.12171i	0.336072	+	0.941836i	$0.390901\pi$
$733$	−474.704	+	822.211i	−0.647618	+	1.12171i	−0.983690	+	0.179871i	$0.942432\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	623.934	+	360.229i	0.846586	+	0.488777i
$738$		0			0
$739$	216.376	+	374.774i	0.292796	+	0.507137i	0.974470	−	0.224519i	$-0.0720811\pi$
$739$	216.376	+	374.774i	0.292796	+	0.507137i	−0.681674	+	0.731656i	$0.738748\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1464.58i			1.97117i	0.169191	+	0.985583i	$0.445885\pi$
$743$			1464.58i			1.97117i	−0.169191	+	0.985583i	$0.554115\pi$
$744$		0			0
$745$	1510.43			2.02742
$746$		0			0
$747$		0			0
$748$		0			0
$749$	144.559	−	83.4613i	0.193003	−	0.111430i
$750$		0			0
$751$	324.606	−	187.411i	0.432231	−	0.249549i	−0.268066	−	0.963401i	$-0.586384\pi$
$751$	324.606	−	187.411i	0.432231	−	0.249549i	0.700297	+	0.713852i	$0.253051\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	753.056	−	434.777i	0.997426	−	0.575864i
$756$		0			0
$757$	744.294	+	1289.15i	0.983215	+	1.70298i	0.649615	+	0.760263i	$0.274930\pi$
$757$	744.294	+	1289.15i	0.983215	+	1.70298i	0.333600	+	0.942715i	$0.391737\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	380.900	−	659.738i	0.500526	−	0.866936i	−0.499474	−	0.866329i	$-0.666473\pi$
$761$	380.900	−	659.738i	0.500526	−	0.866936i	1.00000		0.000607148i	$-0.000193261\pi$
$762$		0			0
$763$	−688.009	−	397.222i	−0.901715	−	0.520605i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	183.716	+	318.205i	0.239525	+	0.414870i
$768$		0			0
$769$	−169.141	−	292.961i	−0.219949	−	0.380963i	0.734843	−	0.678237i	$-0.237256\pi$
$769$	−169.141	−	292.961i	−0.219949	−	0.380963i	−0.954792	+	0.297274i	$0.903923\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−119.263	−	68.8563i	−0.154285	−	0.0890767i	0.420870	−	0.907121i	$-0.361725\pi$
$773$	−119.263	−	68.8563i	−0.154285	−	0.0890767i	−0.575155	+	0.818044i	$0.695058\pi$
$774$		0			0
$775$	−678.577	−	391.777i	−0.875584	−	0.505518i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−758.282	−	590.138i	−0.973404	−	0.757558i
$780$		0			0
$781$	654.013	+	377.595i	0.837405	+	0.483476i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	207.984			0.264948
$786$		0			0
$787$	463.682	+	267.707i	0.589177	+	0.340161i	0.764772	−	0.644301i	$-0.222852\pi$
$787$	463.682	+	267.707i	0.589177	+	0.340161i	−0.175595	+	0.984462i	$0.556185\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			17.6968i			0.0223726i
$792$		0			0
$793$	144.804	−	83.6028i	0.182603	−	0.105426i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−652.486	−	376.713i	−0.818678	−	0.472664i	0.0312825	−	0.999511i	$-0.490041\pi$
$797$	−652.486	−	376.713i	−0.818678	−	0.472664i	−0.849960	+	0.526847i	$0.823374\pi$
$798$		0			0
$799$	165.232	+	286.191i	0.206799	+	0.358186i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	210.164	−	364.014i	0.261723	−	0.453318i
$804$		0			0
$805$	1066.86	−	1847.85i	1.32529	−	2.29547i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1183.21			−1.46256			−0.731279	−	0.682078i	$-0.761076\pi$
$809$	−1183.21			−1.46256			−0.731279	+	0.682078i	$0.761076\pi$
$810$		0			0
$811$	−457.899	−	264.368i	−0.564610	−	0.325978i	0.190384	−	0.981710i	$-0.439027\pi$
$811$	−457.899	−	264.368i	−0.564610	−	0.325978i	−0.754994	+	0.655732i	$0.772360\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	682.615			0.837565
$816$		0			0
$817$	172.375	+	424.340i	0.210986	+	0.519388i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−524.249			−0.638549			−0.319275	−	0.947662i	$-0.603439\pi$
$821$	−524.249			−0.638549			−0.319275	+	0.947662i	$0.603439\pi$
$822$		0			0
$823$	−451.891	−	782.698i	−0.549078	−	0.951030i	−0.998338	−	0.0576294i	$-0.981646\pi$
$823$	−451.891	−	782.698i	−0.549078	−	0.951030i	0.449260	−	0.893401i	$-0.351688\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	269.472	+	155.580i	0.325843	+	0.188126i	0.653994	−	0.756500i	$-0.273092\pi$
$827$	269.472	+	155.580i	0.325843	+	0.188126i	−0.328151	+	0.944625i	$0.606425\pi$
$828$		0			0
$829$			243.583i			0.293827i	0.989149	+	0.146914i	$0.0469339\pi$
$829$			243.583i			0.293827i	−0.989149	+	0.146914i	$0.953066\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	76.1724			0.0914435
$834$		0			0
$835$	1791.68	−	1034.43i	2.14572	−	1.23883i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	641.374	+	370.298i	0.764451	+	0.441356i	0.830891	−	0.556435i	$-0.187831\pi$
$839$	641.374	+	370.298i	0.764451	+	0.441356i	−0.0664407	+	0.997790i	$0.521164\pi$
$840$		0			0
$841$	−711.829			−0.846408
$842$		0			0
$843$		0			0
$844$		0			0
$845$	258.312	−	447.409i	0.305694	−	0.529478i
$846$		0			0
$847$	514.180			0.607061
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1301.57	−	751.462i	1.52946	−	0.883034i
$852$		0			0
$853$	−125.559	+	217.475i	−0.147198	+	0.254954i	−0.930191	−	0.367077i	$-0.880359\pi$
$853$	−125.559	+	217.475i	−0.147198	+	0.254954i	0.782993	+	0.622030i	$0.213692\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	852.690	+	492.301i	0.994971	+	0.574447i	0.906757	−	0.421655i	$-0.138551\pi$
$857$	852.690	+	492.301i	0.994971	+	0.574447i	0.0882147	+	0.996101i	$0.471884\pi$
$858$		0			0
$859$	−432.773	−	749.585i	−0.503811	−	0.872626i	−0.999990	−	0.00440567i	$-0.998598\pi$
$859$	−432.773	−	749.585i	−0.503811	−	0.872626i	0.496180	−	0.868220i	$-0.334736\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			809.822i			0.938380i	0.883097	+	0.469190i	$0.155454\pi$
$863$			809.822i			0.938380i	−0.883097	+	0.469190i	$0.844546\pi$
$864$		0			0
$865$	−2037.88	−	1176.57i	−2.35593	−	1.36020i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			179.750i			0.206847i
$870$		0			0
$871$	−623.898	+	1080.62i	−0.716301	+	1.24067i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−281.008	−	486.720i	−0.321152	−	0.556251i
$876$		0			0
$877$			141.039i			0.160820i	0.996762	+	0.0804101i	$0.0256230\pi$
$877$			141.039i			0.160820i	−0.996762	+	0.0804101i	$0.974377\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1590.79			1.80567			0.902833	−	0.429991i	$-0.141483\pi$
$881$	1590.79			1.80567			0.902833	+	0.429991i	$0.141483\pi$
$882$		0			0
$883$	−323.546	+	560.398i	−0.366417	+	0.634652i	−0.989002	−	0.147899i	$-0.952749\pi$
$883$	−323.546	+	560.398i	−0.366417	+	0.634652i	0.622586	+	0.782552i	$0.286082\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−403.790	+	233.128i	−0.455231	+	0.262828i	−0.710037	−	0.704165i	$-0.751322\pi$
$887$	−403.790	+	233.128i	−0.455231	+	0.262828i	0.254806	+	0.966992i	$0.417988\pi$
$888$		0			0
$889$		−	569.981i		−	0.641149i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−693.121	+	890.608i	−0.776171	+	0.997321i
$894$		0			0
$895$		−	232.332i		−	0.259589i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−417.131	+	722.491i	−0.463994	+	0.803661i
$900$		0			0
$901$			60.6386i			0.0673014i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1245.39	+	719.024i	1.37612	+	0.794501i
$906$		0			0
$907$	−41.8546	−	24.1647i	−0.0461462	−	0.0266425i	0.476749	−	0.879039i	$-0.341815\pi$
$907$	−41.8546	−	24.1647i	−0.0461462	−	0.0266425i	−0.522896	+	0.852397i	$0.675148\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−820.395	+	473.655i	−0.900543	+	0.519929i	−0.877376	−	0.479803i	$-0.840708\pi$
$911$	−820.395	+	473.655i	−0.900543	+	0.519929i	−0.0231666	+	0.999732i	$0.507375\pi$
$912$		0			0
$913$	−359.035	+	621.867i	−0.393248	+	0.681125i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−603.320	−	1044.98i	−0.657928	−	1.13956i
$918$		0			0
$919$	−227.469			−0.247518			−0.123759	−	0.992312i	$-0.539495\pi$
$919$	−227.469			−0.247518			−0.123759	+	0.992312i	$0.539495\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−653.976	+	1132.72i	−0.708533	+	1.22722i
$924$		0			0
$925$		−	1219.85i		−	1.31875i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−171.224	+	296.568i	−0.184310	+	0.319234i	−0.943344	−	0.331817i	$-0.892338\pi$
$929$	−171.224	+	296.568i	−0.184310	+	0.319234i	0.759034	+	0.651051i	$0.225672\pi$
$930$		0			0
$931$	97.9003	+	241.004i	0.105156	+	0.258865i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−128.611	+	222.761i	−0.137552	+	0.238247i
$936$		0			0
$937$	−190.923	+	330.688i	−0.203760	+	0.352922i	−0.949737	−	0.313049i	$-0.898649\pi$
$937$	−190.923	+	330.688i	−0.203760	+	0.352922i	0.745977	+	0.665972i	$0.231983\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−356.597	+	205.881i	−0.378955	+	0.218790i	−0.677364	−	0.735648i	$-0.736878\pi$
$941$	−356.597	+	205.881i	−0.378955	+	0.218790i	0.298408	+	0.954438i	$0.403544\pi$
$942$		0			0
$943$	−1997.09	+	1153.02i	−2.11781	+	1.22272i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	549.842	−	952.354i	0.580614	−	1.00565i	−0.414792	−	0.909916i	$-0.636146\pi$
$947$	549.842	−	952.354i	0.580614	−	1.00565i	0.995407	−	0.0957374i	$-0.0305209\pi$
$948$		0			0
$949$	630.455	+	363.993i	0.664336	+	0.383554i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−311.556	+	179.877i	−0.326921	+	0.188748i	−0.654473	−	0.756085i	$-0.727110\pi$
$953$	−311.556	+	179.877i	−0.326921	+	0.188748i	0.327552	+	0.944833i	$0.393776\pi$
$954$		0			0
$955$	552.598	+	957.128i	0.578637	+	1.00223i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−399.210	−	691.452i	−0.416278	−	0.721014i
$960$		0			0
$961$	−256.396	−	444.090i	−0.266801	−	0.462112i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2620.22i			2.71525i
$966$		0			0
$967$	−343.440			−0.355160			−0.177580	−	0.984106i	$-0.556827\pi$
$967$	−343.440			−0.355160			−0.177580	+	0.984106i	$0.556827\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−657.424	+	379.564i	−0.677059	+	0.390900i	−0.798746	−	0.601668i	$-0.794503\pi$
$971$	−657.424	+	379.564i	−0.677059	+	0.390900i	0.121687	+	0.992569i	$0.461170\pi$
$972$		0			0
$973$	540.577	−	936.306i	0.555577	−	0.962288i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−984.799	−	568.574i	−1.00798	−	0.581959i	−0.0973823	−	0.995247i	$-0.531047\pi$
$977$	−984.799	−	568.574i	−1.00798	−	0.581959i	−0.910600	+	0.413288i	$0.864380\pi$
$978$		0			0
$979$			350.160i			0.357671i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1080.85	+	624.029i	1.09954	+	0.634821i	0.936101	−	0.351731i	$-0.114407\pi$
$983$	1080.85	+	624.029i	1.09954	+	0.634821i	0.163442	+	0.986553i	$0.447740\pi$
$984$		0			0
$985$	−413.074			−0.419365
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1099.23			1.11145
$990$		0			0
$991$	−657.089	+	379.370i	−0.663056	+	0.382816i	−0.793441	−	0.608648i	$-0.791712\pi$
$991$	−657.089	+	379.370i	−0.663056	+	0.382816i	0.130384	+	0.991464i	$0.458379\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	856.711	−	1483.87i	0.861016	−	1.49132i
$996$		0			0
$997$	−1445.15			−1.44950			−0.724750	−	0.689012i	$-0.758045\pi$
$997$	−1445.15			−1.44950			−0.724750	+	0.689012i	$0.758045\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.bl.a.145.4		80
3.2	odd	2		684.3.bl.a.373.34	yes	80
9.2	odd	6		684.3.s.a.601.21	yes	80
9.7	even	3		2052.3.s.a.829.37		80
19.8	odd	6		2052.3.s.a.901.37		80
57.8	even	6		684.3.s.a.445.21	✓	80
171.65	even	6		684.3.bl.a.673.34	yes	80
171.160	odd	6	inner	2052.3.bl.a.1585.4		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.21	✓	80	57.8	even	6
684.3.s.a.601.21	yes	80	9.2	odd	6
684.3.bl.a.373.34	yes	80	3.2	odd	2
684.3.bl.a.673.34	yes	80	171.65	even	6
2052.3.s.a.829.37		80	9.7	even	3
2052.3.s.a.901.37		80	19.8	odd	6
2052.3.bl.a.145.4		80	1.1	even	1	trivial
2052.3.bl.a.1585.4		80	171.160	odd	6	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:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-7.87469 q^{5} +(2.97107 + 5.14604i) q^{7} +O(q^{10})\)	\(q-7.87469 q^{5} +(2.97107 + 5.14604i) q^{7} +(-2.93551 - 5.08444i) q^{11} +(8.80601 - 5.08415i) q^{13} +(2.78184 + 4.81829i) q^{17} +(-11.6693 + 14.9942i) q^{19} +(22.7998 + 39.4904i) q^{23} +37.0108 q^{25} -39.4060i q^{29} +(-18.3346 - 10.5855i) q^{31} +(-23.3963 - 40.5235i) q^{35} -32.9592i q^{37} +50.5716i q^{41} +(12.0530 - 20.8765i) q^{43} +59.3968 q^{47} +(6.84550 - 11.8568i) q^{49} +(9.43881 + 5.44950i) q^{53} +(23.1162 + 40.0384i) q^{55} +36.1350i q^{59} +16.4438 q^{61} +(-69.3446 + 40.0361i) q^{65} +(-106.274 + 61.3572i) q^{67} +(-111.397 + 64.3151i) q^{71} +(35.7968 + 62.0019i) q^{73} +(17.4432 - 30.2125i) q^{77} +(-26.5147 - 15.3083i) q^{79} +(-61.1539 - 105.922i) q^{83} +(-21.9061 - 37.9426i) q^{85} +(-51.6517 - 29.8211i) q^{89} +(52.3265 + 30.2107i) q^{91} +(91.8925 - 118.075i) q^{95} +(79.5266 + 45.9147i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7}+O(q^{10}) \) 80 * q - q^7	\( 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) \) 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

Introduction

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