LMFDB - Embedded newform 2052.3.m.a.881.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(881,2052)]

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, 5, 2]))

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

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

chi := DirichletCharacter("2052.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.m (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			881.13
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.96092i q^{5} +(-6.03683 - 10.4561i) q^{7} +O(q^{10})$	$q-3.96092i q^{5} +(-6.03683 - 10.4561i) q^{7} +(-13.9578 + 8.05857i) q^{11} +(-11.3972 - 19.7406i) q^{13} +(-7.80805 + 4.50798i) q^{17} +(7.69436 - 17.3723i) q^{19} +(-3.57360 + 2.06322i) q^{23} +9.31110 q^{25} -32.5606i q^{29} +(17.6133 - 30.5071i) q^{31} +(-41.4158 + 23.9114i) q^{35} -12.1006 q^{37} +17.7572i q^{41} +(-0.131207 + 0.227258i) q^{43} +62.9065i q^{47} +(-48.3866 + 83.8080i) q^{49} +(-70.5478 - 40.7308i) q^{53} +(31.9193 + 55.2859i) q^{55} -17.2013i q^{59} -68.4804 q^{61} +(-78.1909 + 45.1435i) q^{65} +(37.2470 + 64.5137i) q^{67} +(104.803 - 60.5078i) q^{71} +(22.6715 + 39.2682i) q^{73} +(168.522 + 97.2963i) q^{77} +(37.4287 - 64.8283i) q^{79} +(-44.5212 + 25.7043i) q^{83} +(17.8558 + 30.9271i) q^{85} +(20.7932 + 12.0050i) q^{89} +(-137.606 + 238.341i) q^{91} +(-68.8103 - 30.4767i) q^{95} +(67.9519 - 117.696i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$		−	3.96092i		−	0.792184i	−0.918211	−	0.396092i	$-0.870366\pi$
$5$		−	3.96092i		−	0.792184i	0.918211	−	0.396092i	$-0.129634\pi$
$6$		0			0
$7$	−6.03683	−	10.4561i	−0.862404	−	1.49373i	−0.869602	−	0.493753i	$-0.835625\pi$
$7$	−6.03683	−	10.4561i	−0.862404	−	1.49373i	0.00719814	−	0.999974i	$-0.497709\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.9578	+	8.05857i	−1.26890	+	0.732597i	−0.974779	−	0.223173i	$-0.928359\pi$
$11$	−13.9578	+	8.05857i	−1.26890	+	0.732597i	−0.294116	+	0.955770i	$0.595025\pi$
$12$		0			0
$13$	−11.3972	−	19.7406i	−0.876710	−	1.51851i	−0.854930	−	0.518743i	$-0.826400\pi$
$13$	−11.3972	−	19.7406i	−0.876710	−	1.51851i	−0.0217797	−	0.999763i	$-0.506933\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.80805	+	4.50798i	−0.459297	+	0.265175i	−0.711749	−	0.702434i	$-0.752096\pi$
$17$	−7.80805	+	4.50798i	−0.459297	+	0.265175i	0.252452	+	0.967610i	$0.418763\pi$
$18$		0			0
$19$	7.69436	−	17.3723i	0.404966	−	0.914332i
$19$	7.69436	−	17.3723i	0.404966	−	0.914332i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.57360	+	2.06322i	−0.155374	+	0.0897053i	−0.575671	−	0.817681i	$-0.695259\pi$
$23$	−3.57360	+	2.06322i	−0.155374	+	0.0897053i	0.420297	+	0.907387i	$0.361926\pi$
$24$		0			0
$25$	9.31110			0.372444
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	32.5606i		−	1.12278i	−0.827551	−	0.561390i	$-0.810267\pi$
$29$		−	32.5606i		−	1.12278i	0.827551	−	0.561390i	$-0.189733\pi$
$30$		0			0
$31$	17.6133	−	30.5071i	0.568171	−	0.984101i	−0.428576	−	0.903506i	$-0.640985\pi$
$31$	17.6133	−	30.5071i	0.568171	−	0.984101i	0.996747	−	0.0805952i	$-0.0256821\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−41.4158	+	23.9114i	−1.18331	+	0.683183i
$36$		0			0
$37$	−12.1006			−0.327042			−0.163521	−	0.986540i	$-0.552285\pi$
$37$	−12.1006			−0.327042			−0.163521	+	0.986540i	$0.552285\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			17.7572i			0.433101i	0.976271	+	0.216551i	$0.0694807\pi$
$41$			17.7572i			0.433101i	−0.976271	+	0.216551i	$0.930519\pi$
$42$		0			0
$43$	−0.131207	+	0.227258i	−0.00305133	+	0.00528506i	−0.867547	−	0.497355i	$-0.834305\pi$
$43$	−0.131207	+	0.227258i	−0.00305133	+	0.00528506i	0.864496	+	0.502640i	$0.167638\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			62.9065i			1.33844i	0.743066	+	0.669218i	$0.233371\pi$
$47$			62.9065i			1.33844i	−0.743066	+	0.669218i	$0.766629\pi$
$48$		0			0
$49$	−48.3866	+	83.8080i	−0.987481	+	1.71037i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−70.5478	−	40.7308i	−1.33109	−	0.768506i	−0.345624	−	0.938373i	$-0.612333\pi$
$53$	−70.5478	−	40.7308i	−1.33109	−	0.768506i	−0.985467	+	0.169867i	$0.945666\pi$
$54$		0			0
$55$	31.9193	+	55.2859i	0.580352	+	1.00520i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	17.2013i		−	0.291547i	−0.989318	−	0.145774i	$-0.953433\pi$
$59$		−	17.2013i		−	0.291547i	0.989318	−	0.145774i	$-0.0465671\pi$
$60$		0			0
$61$	−68.4804			−1.12263			−0.561315	−	0.827602i	$-0.689704\pi$
$61$	−68.4804			−1.12263			−0.561315	+	0.827602i	$0.689704\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−78.1909	+	45.1435i	−1.20294	+	0.694516i
$66$		0			0
$67$	37.2470	+	64.5137i	0.555925	+	0.962891i	0.997831	+	0.0658294i	$0.0209693\pi$
$67$	37.2470	+	64.5137i	0.555925	+	0.962891i	−0.441906	+	0.897062i	$0.645697\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	104.803	−	60.5078i	1.47609	−	0.852222i	0.476456	−	0.879198i	$-0.341921\pi$
$71$	104.803	−	60.5078i	1.47609	−	0.852222i	0.999636	+	0.0269762i	$0.00858782\pi$
$72$		0			0
$73$	22.6715	+	39.2682i	0.310568	+	0.537920i	0.978486	−	0.206315i	$-0.0661473\pi$
$73$	22.6715	+	39.2682i	0.310568	+	0.537920i	−0.667917	+	0.744236i	$0.732814\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	168.522	+	97.2963i	2.18860	+	1.26359i
$78$		0			0
$79$	37.4287	−	64.8283i	0.473780	−	0.820612i	−0.525769	−	0.850627i	$-0.676222\pi$
$79$	37.4287	−	64.8283i	0.473780	−	0.820612i	0.999549	+	0.0300155i	$0.00955568\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−44.5212	+	25.7043i	−0.536400	+	0.309690i	−0.743618	−	0.668604i	$-0.766892\pi$
$83$	−44.5212	+	25.7043i	−0.536400	+	0.309690i	0.207219	+	0.978295i	$0.433559\pi$
$84$		0			0
$85$	17.8558	+	30.9271i	0.210068	+	0.363848i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	20.7932	+	12.0050i	0.233632	+	0.134887i	0.612246	−	0.790667i	$-0.290266\pi$
$89$	20.7932	+	12.0050i	0.233632	+	0.134887i	−0.378614	+	0.925554i	$0.623599\pi$
$90$		0			0
$91$	−137.606	+	238.341i	−1.51216	+	2.61913i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−68.8103	−	30.4767i	−0.724319	−	0.320808i
$96$		0			0
$97$	67.9519	−	117.696i	0.700535	−	1.21336i	−0.267744	−	0.963490i	$-0.586278\pi$
$97$	67.9519	−	117.696i	0.700535	−	1.21336i	0.968279	−	0.249872i	$-0.0803887\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			155.767i			1.54225i	0.636683	+	0.771125i	$0.280306\pi$
$101$			155.767i			1.54225i	−0.636683	+	0.771125i	$0.719694\pi$
$102$		0			0
$103$	65.0924	−	112.743i	0.631965	−	1.09459i	−0.355185	−	0.934796i	$-0.615582\pi$
$103$	65.0924	−	112.743i	0.631965	−	1.09459i	0.987150	−	0.159799i	$-0.0510845\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			68.7768i			0.642774i	0.946948	+	0.321387i	$0.104149\pi$
$107$			68.7768i			0.642774i	−0.946948	+	0.321387i	$0.895851\pi$
$108$		0			0
$109$	−27.4720	−	47.5829i	−0.252037	−	0.436540i	0.712050	−	0.702129i	$-0.247767\pi$
$109$	−27.4720	−	47.5829i	−0.252037	−	0.436540i	−0.964086	+	0.265589i	$0.914434\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.96136	+	3.44179i	0.0527554	+	0.0304584i	0.526146	−	0.850394i	$-0.323637\pi$
$113$	5.96136	+	3.44179i	0.0527554	+	0.0304584i	−0.473390	+	0.880853i	$0.656970\pi$
$114$		0			0
$115$	8.17226	+	14.1548i	0.0710631	+	0.123085i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	94.2717	+	54.4278i	0.792199	+	0.457376i
$120$		0			0
$121$	69.3810	−	120.171i	0.573397	−	0.993152i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	135.904i		−	1.08723i
$126$		0			0
$127$	10.9101	−	18.8968i	0.0859059	−	0.148793i	−0.819871	−	0.572548i	$-0.805955\pi$
$127$	10.9101	−	18.8968i	0.0859059	−	0.148793i	0.905777	+	0.423755i	$0.139288\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	33.8889i		−	0.258694i	−0.991599	−	0.129347i	$-0.958712\pi$
$131$		−	33.8889i		−	0.258694i	0.991599	−	0.129347i	$-0.0412881\pi$
$132$		0			0
$133$	−228.096	+	24.4207i	−1.71501	+	0.183614i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			181.387i			1.32399i	0.749508	+	0.661996i	$0.230290\pi$
$137$			181.387i			1.32399i	−0.749508	+	0.661996i	$0.769710\pi$
$138$		0			0
$139$	23.2236	+	40.2244i	0.167076	+	0.289384i	0.937391	−	0.348280i	$-0.113234\pi$
$139$	23.2236	+	40.2244i	0.167076	+	0.289384i	−0.770314	+	0.637664i	$0.779901\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	318.162	+	183.691i	2.22491	+	1.28455i
$144$		0			0
$145$	−128.970			−0.889449
$146$		0			0
$147$		0			0
$148$		0			0
$149$			270.026i			1.81226i	0.423001	+	0.906129i	$0.360977\pi$
$149$			270.026i			1.81226i	−0.423001	+	0.906129i	$0.639023\pi$
$150$		0			0
$151$	88.8801	+	153.945i	0.588610	+	1.01950i	0.994415	+	0.105543i	$0.0336580\pi$
$151$	88.8801	+	153.945i	0.588610	+	1.01950i	−0.405805	+	0.913960i	$0.633009\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−120.836	−	69.7649i	−0.779589	−	0.450096i
$156$		0			0
$157$	−16.5176			−0.105208			−0.0526038	−	0.998615i	$-0.516752\pi$
$157$	−16.5176			−0.105208			−0.0526038	+	0.998615i	$0.516752\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	43.1465	+	24.9106i	0.267990	+	0.154724i
$162$		0			0
$163$	197.434			1.21125			0.605626	−	0.795749i	$-0.292923\pi$
$163$	197.434			1.21125			0.605626	+	0.795749i	$0.292923\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	118.671	−	68.5146i	0.710603	−	0.410267i	−0.100681	−	0.994919i	$-0.532102\pi$
$167$	118.671	−	68.5146i	0.710603	−	0.410267i	0.811284	+	0.584652i	$0.198769\pi$
$168$		0			0
$169$	−175.294	+	303.617i	−1.03724	+	1.79655i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−65.4756	−	37.8024i	−0.378472	−	0.218511i	0.298681	−	0.954353i	$-0.403453\pi$
$173$	−65.4756	−	37.8024i	−0.378472	−	0.218511i	−0.677153	+	0.735842i	$0.736787\pi$
$174$		0			0
$175$	−56.2095	−	97.3578i	−0.321197	−	0.556330i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	230.214i		−	1.28611i	−0.765820	−	0.643055i	$-0.777666\pi$
$179$		−	230.214i		−	1.28611i	0.765820	−	0.643055i	$-0.222334\pi$
$180$		0			0
$181$	−77.5967	+	134.401i	−0.428711	+	0.742549i	−0.996759	−	0.0804464i	$-0.974365\pi$
$181$	−77.5967	+	134.401i	−0.428711	+	0.742549i	0.568048	+	0.822995i	$0.307699\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			47.9294i			0.259078i
$186$		0			0
$187$	72.6557	−	125.843i	0.388533	−	0.672959i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	56.6731	−	32.7202i	0.296718	−	0.171310i	−0.344250	−	0.938878i	$-0.611867\pi$
$191$	56.6731	−	32.7202i	0.296718	−	0.171310i	0.640967	+	0.767568i	$0.278533\pi$
$192$		0			0
$193$	−67.3588			−0.349009			−0.174505	−	0.984656i	$-0.555832\pi$
$193$	−67.3588			−0.349009			−0.174505	+	0.984656i	$0.555832\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	247.679i		−	1.25726i	−0.777706	−	0.628628i	$-0.783617\pi$
$197$		−	247.679i		−	1.25726i	0.777706	−	0.628628i	$-0.216383\pi$
$198$		0			0
$199$	−134.396	+	232.781i	−0.675357	+	1.16975i	0.301007	+	0.953622i	$0.402677\pi$
$199$	−134.396	+	232.781i	−0.675357	+	1.16975i	−0.976364	+	0.216131i	$0.930656\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−340.457	+	196.563i	−1.67713	+	0.968290i
$204$		0			0
$205$	70.3347			0.343096
$206$		0			0
$207$		0			0
$208$		0			0
$209$	32.5992	+	304.485i	0.155977	+	1.45687i
$210$		0			0
$211$	−151.874			−0.719782			−0.359891	−	0.932994i	$-0.617186\pi$
$211$	−151.874			−0.719782			−0.359891	+	0.932994i	$0.617186\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.900149	+	0.519701i	0.00418674	+	0.00241722i
$216$		0			0
$217$	−425.314			−1.95997
$218$		0			0
$219$		0			0
$220$		0			0
$221$	177.980	+	102.757i	0.805341	+	0.464964i
$222$		0			0
$223$	−45.4208	+	78.6712i	−0.203681	+	0.352786i	−0.949712	−	0.313126i	$-0.898624\pi$
$223$	−45.4208	+	78.6712i	−0.203681	+	0.352786i	0.746031	+	0.665912i	$0.231957\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−167.766	+	96.8596i	−0.739056	+	0.426694i	−0.821726	−	0.569883i	$-0.806989\pi$
$227$	−167.766	+	96.8596i	−0.739056	+	0.426694i	0.0826697	+	0.996577i	$0.473655\pi$
$228$		0			0
$229$	114.878	−	198.974i	0.501648	−	0.868881i	−0.498350	−	0.866976i	$-0.666060\pi$
$229$	114.878	−	198.974i	0.501648	−	0.868881i	0.999998	−	0.00190456i	$-0.000606241\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−165.357	+	95.4689i	−0.709686	+	0.409738i	−0.810945	−	0.585123i	$-0.801046\pi$
$233$	−165.357	+	95.4689i	−0.709686	+	0.409738i	0.101259	+	0.994860i	$0.467713\pi$
$234$		0			0
$235$	249.168			1.06029
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−236.368	−	136.467i	−0.988986	−	0.570991i	−0.0840147	−	0.996465i	$-0.526774\pi$
$239$	−236.368	−	136.467i	−0.988986	−	0.570991i	−0.904971	+	0.425473i	$0.860108\pi$
$240$		0			0
$241$	−448.904			−1.86267			−0.931337	−	0.364158i	$-0.881357\pi$
$241$	−448.904			−1.86267			−0.931337	+	0.364158i	$0.881357\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	331.957	+	191.655i	1.35493	+	0.782267i
$246$		0			0
$247$	−430.634	+	46.1050i	−1.74346	+	0.186660i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−59.5359	−	34.3731i	−0.237195	−	0.136945i	0.376692	−	0.926339i	$-0.377062\pi$
$251$	−59.5359	−	34.3731i	−0.237195	−	0.136945i	−0.613887	+	0.789394i	$0.710395\pi$
$252$		0			0
$253$	33.2532	−	57.5962i	0.131436	−	0.227653i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	256.450	−	148.061i	0.997860	−	0.576115i	0.0902453	−	0.995920i	$-0.471235\pi$
$257$	256.450	−	148.061i	0.997860	−	0.576115i	0.907614	+	0.419805i	$0.137902\pi$
$258$		0			0
$259$	73.0490	+	126.525i	0.282042	+	0.488512i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−134.418	−	77.6060i	−0.511093	−	0.295080i	0.222190	−	0.975003i	$-0.428680\pi$
$263$	−134.418	−	77.6060i	−0.511093	−	0.295080i	−0.733283	+	0.679924i	$0.762013\pi$
$264$		0			0
$265$	−161.332	+	279.434i	−0.608798	+	1.05447i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	375.343	−	216.704i	1.39533	−	0.805592i	0.401428	−	0.915891i	$-0.368514\pi$
$269$	375.343	−	216.704i	1.39533	−	0.805592i	0.993899	+	0.110298i	$0.0351806\pi$
$270$		0			0
$271$	−11.2795	−	19.5367i	−0.0416217	−	0.0720910i	0.844464	−	0.535612i	$-0.179919\pi$
$271$	−11.2795	−	19.5367i	−0.0416217	−	0.0720910i	−0.886086	+	0.463521i	$0.846586\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−129.963	+	75.0341i	−0.472593	+	0.272851i
$276$		0			0
$277$	−72.4225	−	125.439i	−0.261453	−	0.452850i	0.705175	−	0.709033i	$-0.250868\pi$
$277$	−72.4225	−	125.439i	−0.261453	−	0.452850i	−0.966628	+	0.256183i	$0.917535\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			33.4205i			0.118934i	0.998230	+	0.0594671i	$0.0189401\pi$
$281$			33.4205i			0.118934i	−0.998230	+	0.0594671i	$0.981060\pi$
$282$		0			0
$283$	39.2274			0.138613			0.0693063	−	0.997595i	$-0.477921\pi$
$283$	39.2274			0.138613			0.0693063	+	0.997595i	$0.477921\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	185.670	−	107.197i	0.646935	−	0.373508i
$288$		0			0
$289$	−103.856	+	179.884i	−0.359364	+	0.622437i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−287.114	−	165.766i	−0.979912	−	0.565753i	−0.0776687	−	0.996979i	$-0.524748\pi$
$293$	−287.114	−	165.766i	−0.979912	−	0.565753i	−0.902244	+	0.431227i	$0.858081\pi$
$294$		0			0
$295$	−68.1330			−0.230959
$296$		0			0
$297$		0			0
$298$		0			0
$299$	81.4583	+	47.0300i	0.272436	+	0.157291i
$300$		0			0
$301$	3.16830			0.0105259
$302$		0			0
$303$		0			0
$304$		0			0
$305$			271.245i			0.889329i
$306$		0			0
$307$	89.4260	+	154.890i	0.291290	+	0.504529i	0.974115	−	0.226053i	$-0.0725823\pi$
$307$	89.4260	+	154.890i	0.291290	+	0.504529i	−0.682825	+	0.730582i	$0.739249\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−278.477	−	160.779i	−0.895424	−	0.516973i	−0.0197111	−	0.999806i	$-0.506275\pi$
$311$	−278.477	−	160.779i	−0.895424	−	0.516973i	−0.875713	+	0.482833i	$0.839608\pi$
$312$		0			0
$313$	−216.484			−0.691641			−0.345820	−	0.938301i	$-0.612399\pi$
$313$	−216.484			−0.691641			−0.345820	+	0.938301i	$0.612399\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	322.150i		−	1.01625i	−0.861284	−	0.508123i	$-0.830339\pi$
$317$		−	322.150i		−	1.01625i	0.861284	−	0.508123i	$-0.169661\pi$
$318$		0			0
$319$	262.392	+	454.476i	0.822545	+	1.42469i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	18.2361	+	170.330i	0.0564584	+	0.527337i
$324$		0			0
$325$	−106.121	−	183.807i	−0.326525	−	0.565559i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	657.756	−	379.755i	1.99926	−	1.15427i
$330$		0			0
$331$	155.064	+	268.579i	0.468471	+	0.811416i	0.999351	−	0.0360315i	$-0.0114717\pi$
$331$	155.064	+	268.579i	0.468471	+	0.811416i	−0.530879	+	0.847447i	$0.678138\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	255.534	−	147.532i	0.762787	−	0.440395i
$336$		0			0
$337$	463.800			1.37626			0.688131	−	0.725586i	$-0.258431\pi$
$337$	463.800			1.37626			0.688131	+	0.725586i	$0.258431\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			567.752i			1.66496i
$342$		0			0
$343$	576.796			1.68162
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	567.679i		−	1.63596i	−0.575245	−	0.817981i	$-0.695093\pi$
$347$		−	567.679i		−	1.63596i	0.575245	−	0.817981i	$-0.304907\pi$
$348$		0			0
$349$	110.715	+	191.763i	0.317234	+	0.549465i	0.979910	−	0.199441i	$-0.0639127\pi$
$349$	110.715	+	191.763i	0.317234	+	0.549465i	−0.662676	+	0.748906i	$0.730579\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−330.736	+	190.950i	−0.936928	+	0.540936i	−0.888996	−	0.457915i	$-0.848596\pi$
$353$	−330.736	+	190.950i	−0.936928	+	0.540936i	−0.0479322	+	0.998851i	$0.515263\pi$
$354$		0			0
$355$	−239.667	−	415.115i	−0.675117	−	1.16934i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	553.318	−	319.458i	1.54128	−	0.889856i	0.542518	−	0.840044i	$-0.317471\pi$
$359$	553.318	−	319.458i	1.54128	−	0.889856i	0.998759	−	0.0498123i	$-0.0158623\pi$
$360$		0			0
$361$	−242.594	−	267.337i	−0.672005	−	0.740547i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	155.538	−	89.8000i	0.426132	−	0.246027i
$366$		0			0
$367$	−34.4305			−0.0938161			−0.0469080	−	0.998899i	$-0.514937\pi$
$367$	−34.4305			−0.0938161			−0.0469080	+	0.998899i	$0.514937\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			983.539i			2.65105i
$372$		0			0
$373$	−179.525	+	310.946i	−0.481299	+	0.833634i	−0.999770	−	0.0214610i	$-0.993168\pi$
$373$	−179.525	+	310.946i	−0.481299	+	0.833634i	0.518471	+	0.855095i	$0.326502\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−642.766	+	371.101i	−1.70495	+	0.984352i
$378$		0			0
$379$	−285.289			−0.752740			−0.376370	−	0.926469i	$-0.622828\pi$
$379$	−285.289			−0.752740			−0.376370	+	0.926469i	$0.622828\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			432.538i			1.12934i	0.825316	+	0.564671i	$0.190997\pi$
$383$			432.538i			1.12934i	−0.825316	+	0.564671i	$0.809003\pi$
$384$		0			0
$385$	385.383	−	667.503i	1.00100	−	1.73377i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	106.924i		−	0.274869i	−0.990511	−	0.137434i	$-0.956114\pi$
$389$		−	106.924i		−	0.274869i	0.990511	−	0.137434i	$-0.0438856\pi$
$390$		0			0
$391$	18.6019	−	32.2195i	0.0475752	−	0.0824027i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−256.780	−	148.252i	−0.650076	−	0.375321i
$396$		0			0
$397$	213.672	+	370.091i	0.538217	+	0.932220i	0.999000	+	0.0447068i	$0.0142354\pi$
$397$	213.672	+	370.091i	0.538217	+	0.932220i	−0.460783	+	0.887513i	$0.652431\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			258.464i			0.644548i	0.946646	+	0.322274i	$0.104447\pi$
$401$			258.464i			0.644548i	−0.946646	+	0.322274i	$0.895553\pi$
$402$		0			0
$403$	−802.971			−1.99248
$404$		0			0
$405$		0			0
$406$		0			0
$407$	168.898	−	97.5132i	0.414982	−	0.239590i
$408$		0			0
$409$	219.754	+	380.625i	0.537295	+	0.930622i	0.999048	+	0.0436140i	$0.0138872\pi$
$409$	219.754	+	380.625i	0.537295	+	0.930622i	−0.461753	+	0.887008i	$0.652779\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−179.858	+	103.841i	−0.435492	+	0.251432i
$414$		0			0
$415$	101.813	+	176.345i	0.245332	+	0.424927i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	602.592	+	347.907i	1.43817	+	0.830326i	0.997722	−	0.0674535i	$-0.0214874\pi$
$419$	602.592	+	347.907i	1.43817	+	0.830326i	0.440445	+	0.897780i	$0.354821\pi$
$420$		0			0
$421$	143.187	−	248.008i	0.340112	−	0.589092i	−0.644341	−	0.764738i	$-0.722868\pi$
$421$	143.187	−	248.008i	0.340112	−	0.589092i	0.984453	+	0.175646i	$0.0562015\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−72.7016	+	41.9743i	−0.171063	+	0.0987630i
$426$		0			0
$427$	413.404	+	716.037i	0.968160	+	1.67690i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−316.602	−	182.790i	−0.734575	−	0.424107i	0.0855187	−	0.996337i	$-0.472745\pi$
$431$	−316.602	−	182.790i	−0.734575	−	0.424107i	−0.820093	+	0.572230i	$0.806079\pi$
$432$		0			0
$433$	356.861	−	618.102i	0.824160	−	1.42749i	−0.0784001	−	0.996922i	$-0.524981\pi$
$433$	356.861	−	618.102i	0.824160	−	1.42749i	0.902560	−	0.430564i	$-0.141685\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.34631	+	77.9569i	0.0190991	+	0.178391i
$438$		0			0
$439$	169.668	−	293.874i	0.386488	−	0.669417i	−0.605486	−	0.795856i	$-0.707021\pi$
$439$	169.668	−	293.874i	0.386488	−	0.669417i	0.991974	+	0.126439i	$0.0403547\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	362.351i		−	0.817948i	−0.912546	−	0.408974i	$-0.865887\pi$
$443$		−	362.351i		−	0.817948i	0.912546	−	0.408974i	$-0.134113\pi$
$444$		0			0
$445$	47.5508	−	82.3604i	0.106856	−	0.185080i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	528.543i		−	1.17716i	−0.808440	−	0.588578i	$-0.799688\pi$
$449$		−	528.543i		−	1.17716i	0.808440	−	0.588578i	$-0.200312\pi$
$450$		0			0
$451$	−143.097	−	247.852i	−0.317289	−	0.549560i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	944.050	+	545.047i	2.07483	+	1.19791i
$456$		0			0
$457$	−192.523	−	333.460i	−0.421276	−	0.729671i	0.574789	−	0.818302i	$-0.305084\pi$
$457$	−192.523	−	333.460i	−0.421276	−	0.729671i	−0.996065	+	0.0886308i	$0.971751\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−300.661	−	173.587i	−0.652193	−	0.376544i	0.137103	−	0.990557i	$-0.456221\pi$
$461$	−300.661	−	173.587i	−0.652193	−	0.376544i	−0.789296	+	0.614013i	$0.789554\pi$
$462$		0			0
$463$	351.439	−	608.710i	0.759047	−	1.31471i	−0.184289	−	0.982872i	$-0.558998\pi$
$463$	351.439	−	608.710i	0.759047	−	1.31471i	0.943337	−	0.331837i	$-0.107668\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			79.7158i			0.170698i	0.996351	+	0.0853488i	$0.0272005\pi$
$467$			79.7158i			0.170698i	−0.996351	+	0.0853488i	$0.972800\pi$
$468$		0			0
$469$	449.707	−	778.916i	0.958864	−	1.66080i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	4.22937i		−	0.00894158i
$474$		0			0
$475$	71.6430	−	161.755i	0.150827	−	0.340537i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			836.060i			1.74543i	0.488232	+	0.872714i	$0.337642\pi$
$479$			836.060i			1.74543i	−0.488232	+	0.872714i	$0.662358\pi$
$480$		0			0
$481$	137.913	+	238.872i	0.286721	+	0.496616i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−466.185	−	269.152i	−0.961207	−	0.554953i
$486$		0			0
$487$	103.687			0.212909			0.106455	−	0.994318i	$-0.466050\pi$
$487$	103.687			0.212909			0.106455	+	0.994318i	$0.466050\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			587.375i			1.19628i	0.801390	+	0.598142i	$0.204094\pi$
$491$			587.375i			1.19628i	−0.801390	+	0.598142i	$0.795906\pi$
$492$		0			0
$493$	146.783	+	254.235i	0.297734	+	0.515690i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1265.35	−	730.550i	−2.54598	−	1.46992i
$498$		0			0
$499$	848.727			1.70086			0.850428	−	0.526091i	$-0.176343\pi$
$499$	848.727			1.70086			0.850428	+	0.526091i	$0.176343\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−557.422	−	321.828i	−1.10820	−	0.639817i	−0.169834	−	0.985473i	$-0.554323\pi$
$503$	−557.422	−	321.828i	−1.10820	−	0.639817i	−0.938361	+	0.345656i	$0.887656\pi$
$504$		0			0
$505$	616.982			1.22175
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−644.052	+	371.844i	−1.26533	+	0.730538i	−0.974100	−	0.226116i	$-0.927397\pi$
$509$	−644.052	+	371.844i	−1.26533	+	0.730538i	−0.291228	+	0.956654i	$0.594064\pi$
$510$		0			0
$511$	273.728	−	474.111i	0.535671	−	0.927809i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−446.567	−	257.826i	−0.867121	−	0.500632i
$516$		0			0
$517$	−506.936	−	878.039i	−0.980534	−	1.69833i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	289.315i		−	0.555307i	−0.960681	−	0.277654i	$-0.910443\pi$
$521$		−	289.315i		−	0.555307i	0.960681	−	0.277654i	$-0.0895568\pi$
$522$		0			0
$523$	−332.610	+	576.097i	−0.635965	+	1.10152i	0.350345	+	0.936621i	$0.386064\pi$
$523$	−332.610	+	576.097i	−0.635965	+	1.10152i	−0.986310	+	0.164903i	$0.947269\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			317.602i			0.602660i
$528$		0			0
$529$	−255.986	+	443.381i	−0.483906	+	0.838150i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	350.537	−	202.382i	0.657667	−	0.379704i
$534$		0			0
$535$	272.420			0.509195
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1559.71i		−	2.89370i
$540$		0			0
$541$	108.091	−	187.219i	0.199799	−	0.346062i	−0.748664	−	0.662949i	$-0.769304\pi$
$541$	108.091	−	187.219i	0.199799	−	0.346062i	0.948463	+	0.316888i	$0.102638\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−188.472	+	108.814i	−0.345820	+	0.199660i
$546$		0			0
$547$	−404.904			−0.740226			−0.370113	−	0.928987i	$-0.620681\pi$
$547$	−404.904			−0.740226			−0.370113	+	0.928987i	$0.620681\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−565.653	−	250.533i	−1.02659	−	0.454688i
$552$		0			0
$553$	−903.801			−1.63436
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−301.106	−	173.843i	−0.540585	−	0.312107i	0.204731	−	0.978818i	$-0.434368\pi$
$557$	−301.106	−	173.843i	−0.540585	−	0.312107i	−0.745316	+	0.666712i	$0.767701\pi$
$558$		0			0
$559$	5.98159			0.0107005
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−272.302	−	157.214i	−0.483663	−	0.279243i	0.238279	−	0.971197i	$-0.423417\pi$
$563$	−272.302	−	157.214i	−0.483663	−	0.279243i	−0.721942	+	0.691954i	$0.756750\pi$
$564$		0			0
$565$	13.6327	−	23.6125i	0.0241286	−	0.0417920i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−377.181	+	217.766i	−0.662884	+	0.382716i	−0.793375	−	0.608733i	$-0.791678\pi$
$569$	−377.181	+	217.766i	−0.662884	+	0.382716i	0.130491	+	0.991450i	$0.458345\pi$
$570$		0			0
$571$	238.902	−	413.791i	0.418393	−	0.724678i	−0.577385	−	0.816472i	$-0.695927\pi$
$571$	238.902	−	413.791i	0.418393	−	0.724678i	0.995778	+	0.0917941i	$0.0292602\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−33.2742	+	19.2109i	−0.0578682	+	0.0334102i
$576$		0			0
$577$	−1012.34			−1.75448			−0.877242	−	0.480049i	$-0.840619\pi$
$577$	−1012.34			−1.75448			−0.877242	+	0.480049i	$0.840619\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	537.533	+	310.345i	0.925186	+	0.534156i
$582$		0			0
$583$	1312.93			2.25202
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−293.061	−	169.199i	−0.499252	−	0.288243i	0.229153	−	0.973390i	$-0.426404\pi$
$587$	−293.061	−	169.199i	−0.499252	−	0.288243i	−0.728405	+	0.685147i	$0.759738\pi$
$588$		0			0
$589$	−394.456	−	540.716i	−0.669705	−	0.918024i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−619.042	−	357.404i	−1.04392	−	0.602705i	−0.122975	−	0.992410i	$-0.539244\pi$
$593$	−619.042	−	357.404i	−1.04392	−	0.602705i	−0.920940	+	0.389705i	$0.872577\pi$
$594$		0			0
$595$	215.584	−	373.403i	0.362326	−	0.627568i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	788.749	−	455.384i	1.31678	−	0.760241i	0.333567	−	0.942726i	$-0.391748\pi$
$599$	788.749	−	455.384i	1.31678	−	0.760241i	0.983209	+	0.182485i	$0.0584142\pi$
$600$		0			0
$601$	518.851	+	898.676i	0.863312	+	1.49530i	0.868713	+	0.495316i	$0.164948\pi$
$601$	518.851	+	898.676i	0.863312	+	1.49530i	−0.00540074	+	0.999985i	$0.501719\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−475.989	−	274.813i	−0.786759	−	0.454236i
$606$		0			0
$607$	−245.610	+	425.410i	−0.404630	+	0.700840i	−0.994278	−	0.106820i	$-0.965933\pi$
$607$	−245.610	+	425.410i	−0.404630	+	0.700840i	0.589648	+	0.807660i	$0.299266\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1241.81	−	716.959i	2.03242	−	1.17342i
$612$		0			0
$613$	−426.072	−	737.979i	−0.695061	−	1.20388i	−0.970160	−	0.242465i	$-0.922044\pi$
$613$	−426.072	−	737.979i	−0.695061	−	1.20388i	0.275100	−	0.961416i	$-0.411289\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	474.863	−	274.163i	0.769633	−	0.444348i	−0.0631109	−	0.998007i	$-0.520102\pi$
$617$	474.863	−	274.163i	0.769633	−	0.444348i	0.832744	+	0.553659i	$0.186769\pi$
$618$		0			0
$619$	−354.141	−	613.391i	−0.572118	−	0.990938i	−0.996348	−	0.0853837i	$-0.972788\pi$
$619$	−354.141	−	613.391i	−0.572118	−	0.990938i	0.424230	−	0.905555i	$-0.360545\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	289.888i		−	0.465310i
$624$		0			0
$625$	−305.526			−0.488841
$626$		0			0
$627$		0			0
$628$		0			0
$629$	94.4818	−	54.5491i	0.150210	−	0.0867235i
$630$		0			0
$631$	471.188	−	816.121i	0.746732	−	1.29338i	−0.202650	−	0.979251i	$-0.564955\pi$
$631$	471.188	−	816.121i	0.746732	−	1.29338i	0.949381	−	0.314126i	$-0.101711\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−74.8486	−	43.2139i	−0.117872	−	0.0680533i
$636$		0			0
$637$	2205.89			3.46294
$638$		0			0
$639$		0			0
$640$		0			0
$641$	369.500	+	213.331i	0.576442	+	0.332809i	0.759718	−	0.650252i	$-0.225337\pi$
$641$	369.500	+	213.331i	0.576442	+	0.332809i	−0.183276	+	0.983062i	$0.558670\pi$
$642$		0			0
$643$	590.704			0.918668			0.459334	−	0.888264i	$-0.348088\pi$
$643$	590.704			0.918668			0.459334	+	0.888264i	$0.348088\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	595.345i		−	0.920162i	−0.887877	−	0.460081i	$-0.847820\pi$
$647$		−	595.345i		−	0.920162i	0.887877	−	0.460081i	$-0.152180\pi$
$648$		0			0
$649$	138.618	+	240.093i	0.213587	+	0.369943i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	906.009	+	523.084i	1.38746	+	0.801048i	0.993028	−	0.117879i	$-0.0376094\pi$
$653$	906.009	+	523.084i	1.38746	+	0.801048i	0.394428	+	0.918927i	$0.370943\pi$
$654$		0			0
$655$	−134.231			−0.204933
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	323.660i		−	0.491138i	−0.969379	−	0.245569i	$-0.921025\pi$
$659$		−	323.660i		−	0.491138i	0.969379	−	0.245569i	$-0.0789748\pi$
$660$		0			0
$661$	114.328	+	198.021i	0.172961	+	0.299578i	0.939454	−	0.342675i	$-0.111333\pi$
$661$	114.328	+	198.021i	0.172961	+	0.299578i	−0.766492	+	0.642253i	$0.778000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	96.7284	+	903.470i	0.145456	+	1.35860i
$666$		0			0
$667$	67.1798	+	116.359i	0.100719	+	0.174451i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	955.839	−	551.854i	1.42450	−	0.822435i
$672$		0			0
$673$	−52.3224	−	90.6251i	−0.0777451	−	0.134658i	0.824532	−	0.565816i	$-0.191439\pi$
$673$	−52.3224	−	90.6251i	−0.0777451	−	0.134658i	−0.902277	+	0.431157i	$0.858105\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−893.825	+	516.050i	−1.32027	+	0.762260i	−0.983772	−	0.179423i	$-0.942577\pi$
$677$	−893.825	+	516.050i	−1.32027	+	0.762260i	−0.336501	+	0.941683i	$0.609244\pi$
$678$		0			0
$679$	−1640.86			−2.41658
$680$		0			0
$681$		0			0
$682$		0			0
$683$			40.8914i			0.0598703i	0.999552	+	0.0299351i	$0.00953007\pi$
$683$			40.8914i			0.0598703i	−0.999552	+	0.0299351i	$0.990470\pi$
$684$		0			0
$685$	718.459			1.04884
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1856.87i			2.69503i
$690$		0			0
$691$	359.412	+	622.520i	0.520133	+	0.900897i	0.999726	+	0.0234062i	$0.00745110\pi$
$691$	359.412	+	622.520i	0.520133	+	0.900897i	−0.479593	+	0.877491i	$0.659216\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	159.326	−	91.9868i	0.229246	−	0.132355i
$696$		0			0
$697$	−80.0489	−	138.649i	−0.114848	−	0.198922i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	614.261	−	354.644i	0.876263	−	0.505911i	0.00683877	−	0.999977i	$-0.497823\pi$
$701$	614.261	−	354.644i	0.876263	−	0.505911i	0.869425	+	0.494066i	$0.164490\pi$
$702$		0			0
$703$	−93.1061	+	210.215i	−0.132441	+	0.299025i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1628.72	−	940.341i	2.30370	−	1.33004i
$708$		0			0
$709$	−582.136			−0.821067			−0.410533	−	0.911846i	$-0.634657\pi$
$709$	−582.136			−0.821067			−0.410533	+	0.911846i	$0.634657\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			145.361i			0.203872i
$714$		0			0
$715$	727.584	−	1260.21i	1.01760	−	1.76254i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−445.208	+	257.041i	−0.619204	+	0.357498i	−0.776559	−	0.630044i	$-0.783037\pi$
$719$	−445.208	+	257.041i	−0.619204	+	0.357498i	0.157355	+	0.987542i	$0.449703\pi$
$720$		0			0
$721$	−1571.81			−2.18004
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	303.175i		−	0.418173i
$726$		0			0
$727$	−141.230	+	244.618i	−0.194265	+	0.336476i	−0.946659	−	0.322237i	$-0.895565\pi$
$727$	−141.230	+	244.618i	−0.194265	+	0.336476i	0.752395	+	0.658713i	$0.228899\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	2.36592i		−	0.00323655i
$732$		0			0
$733$	−363.972	+	630.418i	−0.496551	+	0.860052i	−0.999992	−	0.00397748i	$-0.998734\pi$
$733$	−363.972	+	630.418i	−0.496551	+	0.860052i	0.503441	+	0.864030i	$0.332067\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1039.78	−	600.315i	−1.41082	−	0.814538i
$738$		0			0
$739$	452.621	+	783.962i	0.612477	+	1.06084i	0.990822	+	0.135176i	$0.0431601\pi$
$739$	452.621	+	783.962i	0.612477	+	1.06084i	−0.378345	+	0.925665i	$0.623507\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1108.03i		−	1.49129i	−0.666341	−	0.745647i	$-0.732140\pi$
$743$		−	1108.03i		−	1.49129i	0.666341	−	0.745647i	$-0.267860\pi$
$744$		0			0
$745$	1069.55			1.43564
$746$		0			0
$747$		0			0
$748$		0			0
$749$	719.137	−	415.194i	0.960129	−	0.554331i
$750$		0			0
$751$	617.846	+	1070.14i	0.822697	+	1.42495i	0.903666	+	0.428237i	$0.140865\pi$
$751$	617.846	+	1070.14i	0.822697	+	1.42495i	−0.0809692	+	0.996717i	$0.525802\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	609.764	−	352.047i	0.807634	−	0.466288i
$756$		0			0
$757$	−562.964	−	975.082i	−0.743678	−	1.28809i	−0.950810	−	0.309775i	$-0.899746\pi$
$757$	−562.964	−	975.082i	−0.743678	−	1.28809i	0.207132	−	0.978313i	$-0.433587\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−751.713	−	434.002i	−0.987796	−	0.570304i	−0.0831813	−	0.996534i	$-0.526508\pi$
$761$	−751.713	−	434.002i	−0.987796	−	0.570304i	−0.904615	+	0.426230i	$0.859841\pi$
$762$		0			0
$763$	−331.688	+	574.500i	−0.434715	+	0.752948i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−339.563	+	196.047i	−0.442716	+	0.255602i
$768$		0			0
$769$	252.639	+	437.583i	0.328529	+	0.569029i	0.982220	−	0.187733i	$-0.0601138\pi$
$769$	252.639	+	437.583i	0.328529	+	0.569029i	−0.653691	+	0.756761i	$0.726780\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	106.301	+	61.3729i	0.137518	+	0.0793958i	0.567180	−	0.823594i	$-0.308034\pi$
$773$	106.301	+	61.3729i	0.137518	+	0.0793958i	−0.429663	+	0.902989i	$0.641368\pi$
$774$		0			0
$775$	163.999	−	284.055i	0.211612	−	0.366523i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	308.483	+	136.630i	0.395998	+	0.175391i
$780$		0			0
$781$	−975.212	+	1689.12i	−1.24867	+	2.16276i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			65.4248i			0.0833437i
$786$		0			0
$787$	99.1669	−	171.762i	0.126006	−	0.218249i	−0.796120	−	0.605139i	$-0.793117\pi$
$787$	99.1669	−	171.762i	0.126006	−	0.218249i	0.922126	+	0.386890i	$0.126451\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	83.1101i		−	0.105070i
$792$		0			0
$793$	780.487	+	1351.84i	0.984220	+	1.70472i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1105.31	−	638.151i	−1.38684	−	0.800691i	−0.393881	−	0.919162i	$-0.628868\pi$
$797$	−1105.31	−	638.151i	−1.38684	−	0.800691i	−0.992958	+	0.118470i	$0.962201\pi$
$798$		0			0
$799$	−283.581	−	491.177i	−0.354920	−	0.614739i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−632.891	−	365.400i	−0.788158	−	0.455043i
$804$		0			0
$805$	98.6690	−	170.900i	0.122570	−	0.212298i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1049.44i			1.29720i	0.761128	+	0.648602i	$0.224646\pi$
$809$			1049.44i			1.29720i	−0.761128	+	0.648602i	$0.775354\pi$
$810$		0			0
$811$	247.735	−	429.090i	0.305469	−	0.529088i	−0.671897	−	0.740645i	$-0.734520\pi$
$811$	247.735	−	429.090i	0.305469	−	0.529088i	0.977366	+	0.211557i	$0.0678534\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	782.021i		−	0.959535i
$816$		0			0
$817$	2.93843	+	4.02797i	0.00359661	+	0.00493020i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	283.927i		−	0.345831i	−0.984937	−	0.172915i	$-0.944681\pi$
$821$		−	283.927i		−	0.345831i	0.984937	−	0.172915i	$-0.0553187\pi$
$822$		0			0
$823$	−398.100	−	689.530i	−0.483718	−	0.837824i	0.516107	−	0.856524i	$-0.327381\pi$
$823$	−398.100	−	689.530i	−0.483718	−	0.837824i	−0.999825	+	0.0186997i	$0.994047\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−921.590	−	532.080i	−1.11438	−	0.643386i	−0.174417	−	0.984672i	$-0.555804\pi$
$827$	−921.590	−	532.080i	−1.11438	−	0.643386i	−0.939959	+	0.341286i	$0.889138\pi$
$828$		0			0
$829$	30.3330			0.0365899			0.0182949	−	0.999833i	$-0.494176\pi$
$829$	30.3330			0.0365899			0.0182949	+	0.999833i	$0.494176\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	872.503i		−	1.04742i
$834$		0			0
$835$	−271.381	−	470.045i	−0.325007	−	0.562929i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1054.44	+	608.778i	1.25678	+	0.725600i	0.972446	−	0.233126i	$-0.0748956\pi$
$839$	1054.44	+	608.778i	1.25678	+	0.725600i	0.284330	+	0.958727i	$0.408229\pi$
$840$		0			0
$841$	−219.194			−0.260635
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1202.60	+	694.324i	1.42320	+	0.821685i
$846$		0			0
$847$	−1675.36			−1.97800
$848$		0			0
$849$		0			0
$850$		0			0
$851$	43.2426	−	24.9661i	0.0508139	−	0.0293374i
$852$		0			0
$853$	−282.564	+	489.416i	−0.331260	+	0.573758i	−0.982759	−	0.184891i	$-0.940807\pi$
$853$	−282.564	+	489.416i	−0.331260	+	0.573758i	0.651499	+	0.758649i	$0.274140\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−112.874	−	65.1679i	−0.131708	−	0.0760419i	0.432698	−	0.901539i	$-0.357562\pi$
$857$	−112.874	−	65.1679i	−0.131708	−	0.0760419i	−0.564407	+	0.825497i	$0.690895\pi$
$858$		0			0
$859$	−158.988	−	275.376i	−0.185085	−	0.320577i	0.758520	−	0.651650i	$-0.225923\pi$
$859$	−158.988	−	275.376i	−0.185085	−	0.320577i	−0.943605	+	0.331073i	$0.892589\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			449.132i			0.520431i	0.965551	+	0.260216i	$0.0837936\pi$
$863$			449.132i			0.520431i	−0.965551	+	0.260216i	$0.916206\pi$
$864$		0			0
$865$	−149.732	+	259.344i	−0.173101	+	0.299819i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1206.49i			1.38836i
$870$		0			0
$871$	849.025	−	1470.55i	0.974770	−	1.68835i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1421.02	+	820.426i	−1.62402	+	0.937630i
$876$		0			0
$877$	−1420.74			−1.62000			−0.810000	−	0.586430i	$-0.800533\pi$
$877$	−1420.74			−1.62000			−0.810000	+	0.586430i	$0.800533\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1534.41i		−	1.74166i	−0.491580	−	0.870832i	$-0.663581\pi$
$881$		−	1534.41i		−	1.74166i	0.491580	−	0.870832i	$-0.336419\pi$
$882$		0			0
$883$	255.107	−	441.859i	0.288910	−	0.500406i	−0.684640	−	0.728881i	$-0.740041\pi$
$883$	255.107	−	441.859i	0.288910	−	0.500406i	0.973550	+	0.228475i	$0.0733739\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−128.442	+	74.1563i	−0.144805	+	0.0836034i	−0.570652	−	0.821192i	$-0.693310\pi$
$887$	−128.442	+	74.1563i	−0.144805	+	0.0836034i	0.425847	+	0.904795i	$0.359976\pi$
$888$		0			0
$889$	−263.448			−0.296342
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1092.83	+	484.025i	1.22377	+	0.542021i
$894$		0			0
$895$	−911.859			−1.01884
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−993.331	−	573.500i	−1.10493	−	0.637931i
$900$		0			0
$901$	734.455			0.815155
$902$		0			0
$903$		0			0
$904$		0			0
$905$	532.353	+	307.354i	0.588236	+	0.339618i
$906$		0			0
$907$	668.842	−	1158.47i	0.737422	−	1.27725i	−0.216230	−	0.976342i	$-0.569376\pi$
$907$	668.842	−	1158.47i	0.737422	−	1.27725i	0.953652	−	0.300911i	$-0.0972906\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	728.807	−	420.777i	0.800008	−	0.461885i	−0.0434662	−	0.999055i	$-0.513840\pi$
$911$	728.807	−	420.777i	0.800008	−	0.461885i	0.843474	+	0.537170i	$0.180507\pi$
$912$		0			0
$913$	414.280	−	717.553i	0.453757	−	0.785929i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−354.345	+	204.581i	−0.386418	+	0.223099i
$918$		0			0
$919$	1282.29			1.39531			0.697657	−	0.716431i	$-0.254226\pi$
$919$	1282.29			1.39531			0.697657	+	0.716431i	$0.254226\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2388.92	−	1379.24i	−2.58821	−	1.49430i
$924$		0			0
$925$	−112.670			−0.121805
$926$		0			0
$927$		0			0
$928$		0			0
$929$	563.774	+	325.495i	0.606861	+	0.350371i	0.771736	−	0.635943i	$-0.219389\pi$
$929$	563.774	+	325.495i	0.606861	+	0.350371i	−0.164875	+	0.986315i	$0.552722\pi$
$930$		0			0
$931$	1083.63	+	1485.43i	1.16395	+	1.59553i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−498.456	−	287.784i	−0.533108	−	0.307790i
$936$		0			0
$937$	252.334	−	437.056i	0.269300	−	0.466441i	−0.699381	−	0.714749i	$-0.746541\pi$
$937$	252.334	−	437.056i	0.269300	−	0.466441i	0.968681	+	0.248308i	$0.0798743\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1241.70	+	716.898i	−1.31956	+	0.761847i	−0.983657	−	0.180050i	$-0.942374\pi$
$941$	−1241.70	+	716.898i	−1.31956	+	0.761847i	−0.335900	+	0.941897i	$0.609041\pi$
$942$		0			0
$943$	−36.6369	−	63.4570i	−0.0388515	−	0.0672927i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	695.809	+	401.725i	0.734751	+	0.424208i	0.820158	−	0.572138i	$-0.193886\pi$
$947$	695.809	+	401.725i	0.734751	+	0.424208i	−0.0854070	+	0.996346i	$0.527219\pi$
$948$		0			0
$949$	516.784	−	895.097i	0.544557	−	0.943200i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−796.306	+	459.747i	−0.835578	+	0.482421i	−0.855759	−	0.517375i	$-0.826909\pi$
$953$	−796.306	+	459.747i	−0.835578	+	0.482421i	0.0201809	+	0.999796i	$0.493576\pi$
$954$		0			0
$955$	−129.602	−	224.477i	−0.135709	−	0.235055i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1896.60	−	1095.00i	1.97768	−	1.14182i
$960$		0			0
$961$	−139.957	−	242.412i	−0.145636	−	0.252250i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			266.803i			0.276480i
$966$		0			0
$967$	−603.095			−0.623676			−0.311838	−	0.950135i	$-0.600945\pi$
$967$	−603.095			−0.623676			−0.311838	+	0.950135i	$0.600945\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−562.056	+	324.503i	−0.578842	+	0.334195i	−0.760673	−	0.649135i	$-0.775131\pi$
$971$	−562.056	+	324.503i	−0.578842	+	0.334195i	0.181831	+	0.983330i	$0.441798\pi$
$972$		0			0
$973$	280.394	−	485.656i	0.288174	−	0.499132i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	45.8246	+	26.4568i	0.0469033	+	0.0270796i	0.523268	−	0.852168i	$-0.324713\pi$
$977$	45.8246	+	26.4568i	0.0469033	+	0.0270796i	−0.476365	+	0.879248i	$0.658046\pi$
$978$		0			0
$979$	−386.972			−0.395273
$980$		0			0
$981$		0			0
$982$		0			0
$983$	779.865	+	450.255i	0.793352	+	0.458042i	0.841141	−	0.540815i	$-0.181884\pi$
$983$	779.865	+	450.255i	0.793352	+	0.458042i	−0.0477892	+	0.998857i	$0.515218\pi$
$984$		0			0
$985$	−981.039			−0.995978
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1.08284i		−	0.00109488i
$990$		0			0
$991$	456.255	+	790.257i	0.460399	+	0.797434i	0.998981	−	0.0451394i	$-0.0143732\pi$
$991$	456.255	+	790.257i	0.460399	+	0.797434i	−0.538582	+	0.842573i	$0.681040\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	922.026	+	532.332i	0.926660	+	0.535007i
$996$		0			0
$997$	1284.31			1.28818			0.644088	−	0.764951i	$-0.277237\pi$
$997$	1284.31			1.28818			0.644088	+	0.764951i	$0.277237\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.m.a.881.13		80
3.2	odd	2		684.3.m.a.653.30	yes	80
9.2	odd	6		2052.3.be.a.197.13		80
9.7	even	3		684.3.be.a.425.24	yes	80
19.11	even	3		2052.3.be.a.125.13		80
57.11	odd	6		684.3.be.a.581.24	yes	80
171.11	odd	6	inner	2052.3.m.a.1493.28		80
171.106	even	3		684.3.m.a.353.30	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.30	✓	80	171.106	even	3
684.3.m.a.653.30	yes	80	3.2	odd	2
684.3.be.a.425.24	yes	80	9.7	even	3
684.3.be.a.581.24	yes	80	57.11	odd	6
2052.3.m.a.881.13		80	1.1	even	1	trivial
2052.3.m.a.1493.28		80	171.11	odd	6	inner
2052.3.be.a.125.13		80	19.11	even	3
2052.3.be.a.197.13		80	9.2	odd	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:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-3.96092i q^{5} +(-6.03683 - 10.4561i) q^{7} +O(q^{10})\)	\(q-3.96092i q^{5} +(-6.03683 - 10.4561i) q^{7} +(-13.9578 + 8.05857i) q^{11} +(-11.3972 - 19.7406i) q^{13} +(-7.80805 + 4.50798i) q^{17} +(7.69436 - 17.3723i) q^{19} +(-3.57360 + 2.06322i) q^{23} +9.31110 q^{25} -32.5606i q^{29} +(17.6133 - 30.5071i) q^{31} +(-41.4158 + 23.9114i) q^{35} -12.1006 q^{37} +17.7572i q^{41} +(-0.131207 + 0.227258i) q^{43} +62.9065i q^{47} +(-48.3866 + 83.8080i) q^{49} +(-70.5478 - 40.7308i) q^{53} +(31.9193 + 55.2859i) q^{55} -17.2013i q^{59} -68.4804 q^{61} +(-78.1909 + 45.1435i) q^{65} +(37.2470 + 64.5137i) q^{67} +(104.803 - 60.5078i) q^{71} +(22.6715 + 39.2682i) q^{73} +(168.522 + 97.2963i) q^{77} +(37.4287 - 64.8283i) q^{79} +(-44.5212 + 25.7043i) q^{83} +(17.8558 + 30.9271i) q^{85} +(20.7932 + 12.0050i) q^{89} +(-137.606 + 238.341i) q^{91} +(-68.8103 - 30.4767i) q^{95} +(67.9519 - 117.696i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

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