LMFDB - Embedded newform 2052.3.m.a.881.5

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			881.5
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.36

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.78017i q^{5} +(-0.841737 - 1.45793i) q^{7} +O(q^{10})$	$q-6.78017i q^{5} +(-0.841737 - 1.45793i) q^{7} +(-1.95022 + 1.12596i) q^{11} +(7.98882 + 13.8370i) q^{13} +(-3.45027 + 1.99201i) q^{17} +(17.2374 + 7.99203i) q^{19} +(36.2471 - 20.9273i) q^{23} -20.9708 q^{25} +25.4531i q^{29} +(-27.0551 + 46.8607i) q^{31} +(-9.88503 + 5.70713i) q^{35} -6.30115 q^{37} -4.87714i q^{41} +(6.55967 - 11.3617i) q^{43} +74.4130i q^{47} +(23.0830 - 39.9809i) q^{49} +(20.7509 + 11.9805i) q^{53} +(7.63421 + 13.2228i) q^{55} +72.4427i q^{59} +19.5370 q^{61} +(93.8175 - 54.1656i) q^{65} +(-21.4612 - 37.1718i) q^{67} +(67.4688 - 38.9531i) q^{71} +(47.8585 + 82.8933i) q^{73} +(3.28315 + 1.89553i) q^{77} +(22.7089 - 39.3329i) q^{79} +(17.1623 - 9.90864i) q^{83} +(13.5062 + 23.3934i) q^{85} +(-35.7852 - 20.6606i) q^{89} +(13.4490 - 23.2943i) q^{91} +(54.1874 - 116.872i) q^{95} +(-52.1590 + 90.3421i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	6.78017i		−	1.35603i	−0.735046	−	0.678017i	$-0.762839\pi$
$5$		−	6.78017i		−	1.35603i	0.735046	−	0.678017i	$-0.237161\pi$
$6$		0			0
$7$	−0.841737	−	1.45793i	−0.120248	−	0.208276i	0.799617	−	0.600510i	$-0.205036\pi$
$7$	−0.841737	−	1.45793i	−0.120248	−	0.208276i	−0.919866	+	0.392234i	$0.871702\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.95022	+	1.12596i	−0.177293	+	0.102360i	−0.586020	−	0.810296i	$-0.699306\pi$
$11$	−1.95022	+	1.12596i	−0.177293	+	0.102360i	0.408727	+	0.912657i	$0.365973\pi$
$12$		0			0
$13$	7.98882	+	13.8370i	0.614524	+	1.06439i	0.990468	+	0.137745i	$0.0439854\pi$
$13$	7.98882	+	13.8370i	0.614524	+	1.06439i	−0.375943	+	0.926643i	$0.622681\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.45027	+	1.99201i	−0.202957	+	0.117177i	−0.598034	−	0.801471i	$-0.704051\pi$
$17$	−3.45027	+	1.99201i	−0.202957	+	0.117177i	0.395077	+	0.918648i	$0.370718\pi$
$18$		0			0
$19$	17.2374	+	7.99203i	0.907231	+	0.420633i
$19$	17.2374	+	7.99203i	0.907231	+	0.420633i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	36.2471	−	20.9273i	1.57596	−	0.909881i	0.580545	−	0.814228i	$-0.302839\pi$
$23$	36.2471	−	20.9273i	1.57596	−	0.909881i	0.995415	−	0.0956529i	$-0.0304939\pi$
$24$		0			0
$25$	−20.9708			−0.838830
$26$		0			0
$27$		0			0
$28$		0			0
$29$			25.4531i			0.877694i	0.898562	+	0.438847i	$0.144613\pi$
$29$			25.4531i			0.877694i	−0.898562	+	0.438847i	$0.855387\pi$
$30$		0			0
$31$	−27.0551	+	46.8607i	−0.872744	+	1.51164i	−0.0135973	+	0.999908i	$0.504328\pi$
$31$	−27.0551	+	46.8607i	−0.872744	+	1.51164i	−0.859147	+	0.511729i	$0.829005\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.88503	+	5.70713i	−0.282429	+	0.163061i
$36$		0			0
$37$	−6.30115			−0.170301			−0.0851507	−	0.996368i	$-0.527137\pi$
$37$	−6.30115			−0.170301			−0.0851507	+	0.996368i	$0.527137\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.87714i		−	0.118955i	−0.998230	−	0.0594773i	$-0.981057\pi$
$41$		−	4.87714i		−	0.118955i	0.998230	−	0.0594773i	$-0.0189434\pi$
$42$		0			0
$43$	6.55967	−	11.3617i	0.152551	−	0.264225i	−0.779614	−	0.626260i	$-0.784585\pi$
$43$	6.55967	−	11.3617i	0.152551	−	0.264225i	0.932164	+	0.362035i	$0.117918\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			74.4130i			1.58326i	0.611004	+	0.791628i	$0.290766\pi$
$47$			74.4130i			1.58326i	−0.611004	+	0.791628i	$0.709234\pi$
$48$		0			0
$49$	23.0830	−	39.9809i	0.471081	−	0.815936i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	20.7509	+	11.9805i	0.391526	+	0.226048i	0.682821	−	0.730585i	$-0.260753\pi$
$53$	20.7509	+	11.9805i	0.391526	+	0.226048i	−0.291295	+	0.956633i	$0.594086\pi$
$54$		0			0
$55$	7.63421	+	13.2228i	0.138804	+	0.240415i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			72.4427i			1.22784i	0.789367	+	0.613921i	$0.210409\pi$
$59$			72.4427i			1.22784i	−0.789367	+	0.613921i	$0.789591\pi$
$60$		0			0
$61$	19.5370			0.320279			0.160140	−	0.987094i	$-0.448806\pi$
$61$	19.5370			0.320279			0.160140	+	0.987094i	$0.448806\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	93.8175	−	54.1656i	1.44335	−	0.833316i
$66$		0			0
$67$	−21.4612	−	37.1718i	−0.320316	−	0.554803i	0.660237	−	0.751057i	$-0.270456\pi$
$67$	−21.4612	−	37.1718i	−0.320316	−	0.554803i	−0.980553	+	0.196254i	$0.937122\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	67.4688	−	38.9531i	0.950264	−	0.548635i	0.0571013	−	0.998368i	$-0.481814\pi$
$71$	67.4688	−	38.9531i	0.950264	−	0.548635i	0.893163	+	0.449733i	$0.148481\pi$
$72$		0			0
$73$	47.8585	+	82.8933i	0.655596	+	1.13553i	0.981744	+	0.190206i	$0.0609157\pi$
$73$	47.8585	+	82.8933i	0.655596	+	1.13553i	−0.326148	+	0.945319i	$0.605751\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.28315	+	1.89553i	0.0426383	+	0.0246172i
$78$		0			0
$79$	22.7089	−	39.3329i	0.287454	−	0.497885i	−0.685747	−	0.727840i	$-0.740525\pi$
$79$	22.7089	−	39.3329i	0.287454	−	0.497885i	0.973201	+	0.229955i	$0.0738579\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	17.1623	−	9.90864i	0.206774	−	0.119381i	−0.393037	−	0.919523i	$-0.628576\pi$
$83$	17.1623	−	9.90864i	0.206774	−	0.119381i	0.599811	+	0.800141i	$0.295242\pi$
$84$		0			0
$85$	13.5062	+	23.3934i	0.158897	+	0.275217i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−35.7852	−	20.6606i	−0.402081	−	0.232142i	0.285300	−	0.958438i	$-0.407907\pi$
$89$	−35.7852	−	20.6606i	−0.402081	−	0.232142i	−0.687382	+	0.726296i	$0.741240\pi$
$90$		0			0
$91$	13.4490	−	23.2943i	0.147791	−	0.255981i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	54.1874	−	116.872i	0.570393	−	1.23024i
$96$		0			0
$97$	−52.1590	+	90.3421i	−0.537722	+	0.931362i	0.461304	+	0.887242i	$0.347382\pi$
$97$	−52.1590	+	90.3421i	−0.537722	+	0.931362i	−0.999026	+	0.0441198i	$0.985952\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	157.680i		−	1.56119i	−0.625037	−	0.780595i	$-0.714916\pi$
$101$		−	157.680i		−	1.56119i	0.625037	−	0.780595i	$-0.285084\pi$
$102$		0			0
$103$	54.0333	−	93.5884i	0.524595	−	0.908625i	−0.474995	−	0.879989i	$-0.657550\pi$
$103$	54.0333	−	93.5884i	0.524595	−	0.908625i	0.999590	−	0.0286368i	$-0.00911661\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	105.210i		−	0.983267i	−0.870802	−	0.491633i	$-0.836400\pi$
$107$		−	105.210i		−	0.983267i	0.870802	−	0.491633i	$-0.163600\pi$
$108$		0			0
$109$	71.6115	+	124.035i	0.656986	+	1.13793i	0.981392	+	0.192015i	$0.0615023\pi$
$109$	71.6115	+	124.035i	0.656986	+	1.13793i	−0.324406	+	0.945918i	$0.605164\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	39.2823	+	22.6796i	0.347631	+	0.200705i	0.663641	−	0.748051i	$-0.269010\pi$
$113$	39.2823	+	22.6796i	0.347631	+	0.200705i	−0.316010	+	0.948756i	$0.602343\pi$
$114$		0			0
$115$	−141.890	−	245.761i	−1.23383	−	2.13706i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.80844	+	3.35351i	0.0488104	+	0.0281807i
$120$		0			0
$121$	−57.9644	+	100.397i	−0.479045	+	0.829730i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	27.3190i		−	0.218552i
$126$		0			0
$127$	22.6095	−	39.1608i	0.178028	−	0.308353i	−0.763177	−	0.646189i	$-0.776362\pi$
$127$	22.6095	−	39.1608i	0.178028	−	0.308353i	0.941205	+	0.337836i	$0.109695\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	150.920i		−	1.15206i	−0.817427	−	0.576032i	$-0.804600\pi$
$131$		−	150.920i		−	1.15206i	0.817427	−	0.576032i	$-0.195400\pi$
$132$		0			0
$133$	−2.85751	−	31.8581i	−0.0214851	−	0.239535i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	83.1959i		−	0.607269i	−0.952789	−	0.303635i	$-0.901800\pi$
$137$		−	83.1959i		−	0.607269i	0.952789	−	0.303635i	$-0.0982003\pi$
$138$		0			0
$139$	−83.9534	−	145.412i	−0.603981	−	1.04613i	−0.992212	−	0.124564i	$-0.960247\pi$
$139$	−83.9534	−	145.412i	−0.603981	−	1.04613i	0.388230	−	0.921562i	$-0.373086\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−31.1599	−	17.9902i	−0.217902	−	0.125806i
$144$		0			0
$145$	172.577			1.19018
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	58.5286i		−	0.392810i	−0.980523	−	0.196405i	$-0.937073\pi$
$149$		−	58.5286i		−	0.392810i	0.980523	−	0.196405i	$-0.0629267\pi$
$150$		0			0
$151$	98.4954	+	170.599i	0.652287	+	1.12979i	0.982567	+	0.185911i	$0.0595237\pi$
$151$	98.4954	+	170.599i	0.652287	+	1.12979i	−0.330279	+	0.943883i	$0.607143\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	317.724	+	183.438i	2.04983	+	1.18347i
$156$		0			0
$157$	104.655			0.666593			0.333297	−	0.942822i	$-0.391839\pi$
$157$	104.655			0.666593			0.333297	+	0.942822i	$0.391839\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−61.0210	−	35.2305i	−0.379013	−	0.218823i
$162$		0			0
$163$	322.358			1.97765			0.988827	−	0.149066i	$-0.0476265\pi$
$163$	322.358			1.97765			0.988827	+	0.149066i	$0.0476265\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−175.361	+	101.245i	−1.05006	+	0.606255i	−0.922668	−	0.385596i	$-0.873996\pi$
$167$	−175.361	+	101.245i	−1.05006	+	0.606255i	−0.127397	+	0.991852i	$0.540662\pi$
$168$		0			0
$169$	−43.1424	+	74.7248i	−0.255280	+	0.442158i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	192.308	+	111.029i	1.11161	+	0.641788i	0.939246	−	0.343245i	$-0.111526\pi$
$173$	192.308	+	111.029i	1.11161	+	0.641788i	0.172364	+	0.985033i	$0.444860\pi$
$174$		0			0
$175$	17.6519	+	30.5739i	0.100868	+	0.174708i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	168.340i		−	0.940448i	−0.882547	−	0.470224i	$-0.844173\pi$
$179$		−	168.340i		−	0.940448i	0.882547	−	0.470224i	$-0.155827\pi$
$180$		0			0
$181$	−130.553	+	226.124i	−0.721286	+	1.24930i	0.239199	+	0.970971i	$0.423115\pi$
$181$	−130.553	+	226.124i	−0.721286	+	1.24930i	−0.960485	+	0.278333i	$0.910218\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			42.7229i			0.230934i
$186$		0			0
$187$	4.48586	−	7.76974i	0.0239886	−	0.0415494i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	191.894	−	110.790i	1.00468	−	0.580053i	0.0950513	−	0.995472i	$-0.469698\pi$
$191$	191.894	−	110.790i	1.00468	−	0.580053i	0.909630	+	0.415419i	$0.136365\pi$
$192$		0			0
$193$	73.3218			0.379906			0.189953	−	0.981793i	$-0.439166\pi$
$193$	73.3218			0.379906			0.189953	+	0.981793i	$0.439166\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	102.260i		−	0.519087i	−0.965731	−	0.259544i	$-0.916428\pi$
$197$		−	102.260i		−	0.519087i	0.965731	−	0.259544i	$-0.0835721\pi$
$198$		0			0
$199$	95.6170	−	165.613i	0.480487	−	0.832228i	−0.519262	−	0.854615i	$-0.673793\pi$
$199$	95.6170	−	165.613i	0.480487	−	0.832228i	0.999749	+	0.0223867i	$0.00712652\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	37.1089	−	21.4248i	0.182803	−	0.105541i
$204$		0			0
$205$	−33.0678			−0.161307
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−42.6154	+	3.82239i	−0.203902	+	0.0182890i
$210$		0			0
$211$	−48.5051			−0.229882			−0.114941	−	0.993372i	$-0.536668\pi$
$211$	−48.5051			−0.229882			−0.114941	+	0.993372i	$0.536668\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−77.0342	−	44.4757i	−0.358299	−	0.206864i
$216$		0			0
$217$	91.0930			0.419784
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−55.1272	−	31.8277i	−0.249444	−	0.144017i
$222$		0			0
$223$	−79.9794	+	138.528i	−0.358652	+	0.621204i	−0.987736	−	0.156134i	$-0.950097\pi$
$223$	−79.9794	+	138.528i	−0.358652	+	0.621204i	0.629084	+	0.777337i	$0.283430\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−243.852	+	140.788i	−1.07424	+	0.620212i	−0.929336	−	0.369234i	$-0.879620\pi$
$227$	−243.852	+	140.788i	−1.07424	+	0.620212i	−0.144902	+	0.989446i	$0.546287\pi$
$228$		0			0
$229$	33.4297	−	57.9020i	0.145981	−	0.252847i	−0.783757	−	0.621067i	$-0.786699\pi$
$229$	33.4297	−	57.9020i	0.145981	−	0.252847i	0.929739	+	0.368220i	$0.120033\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	158.308	−	91.3991i	0.679433	−	0.392271i	−0.120208	−	0.992749i	$-0.538356\pi$
$233$	158.308	−	91.3991i	0.679433	−	0.392271i	0.799641	+	0.600478i	$0.205023\pi$
$234$		0			0
$235$	504.533			2.14695
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−276.398	−	159.578i	−1.15648	−	0.667692i	−0.206019	−	0.978548i	$-0.566051\pi$
$239$	−276.398	−	159.578i	−1.15648	−	0.667692i	−0.950457	+	0.310856i	$0.899384\pi$
$240$		0			0
$241$	372.519			1.54572			0.772861	−	0.634576i	$-0.218825\pi$
$241$	372.519			1.54572			0.772861	+	0.634576i	$0.218825\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−271.077	−	156.506i	−1.10644	−	0.638802i
$246$		0			0
$247$	27.1203	+	302.361i	0.109799	+	1.22413i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.6863	+	6.16971i	0.0425747	+	0.0245805i	0.521136	−	0.853473i	$-0.325508\pi$
$251$	10.6863	+	6.16971i	0.0425747	+	0.0245805i	−0.478562	+	0.878054i	$0.658842\pi$
$252$		0			0
$253$	−47.1266	+	81.6256i	−0.186271	+	0.322631i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	90.2167	−	52.0866i	0.351038	−	0.202672i	−0.314105	−	0.949388i	$-0.601704\pi$
$257$	90.2167	−	52.0866i	0.351038	−	0.202672i	0.665142	+	0.746717i	$0.268371\pi$
$258$		0			0
$259$	5.30391	+	9.18665i	0.0204784	+	0.0354697i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	343.118	+	198.099i	1.30463	+	0.753228i	0.981194	−	0.193022i	$-0.0618288\pi$
$263$	343.118	+	198.099i	1.30463	+	0.753228i	0.323436	+	0.946250i	$0.395162\pi$
$264$		0			0
$265$	81.2301	−	140.695i	0.306529	−	0.530923i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−336.491	+	194.273i	−1.25090	+	0.722205i	−0.971288	−	0.237909i	$-0.923538\pi$
$269$	−336.491	+	194.273i	−1.25090	+	0.722205i	−0.279609	+	0.960114i	$0.590205\pi$
$270$		0			0
$271$	−188.040	−	325.694i	−0.693873	−	1.20182i	−0.970559	−	0.240864i	$-0.922569\pi$
$271$	−188.040	−	325.694i	−0.693873	−	1.20182i	0.276685	−	0.960961i	$-0.410764\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	40.8976	−	23.6123i	0.148719	−	0.0858627i
$276$		0			0
$277$	170.595	+	295.480i	0.615868	+	1.06671i	0.990232	+	0.139432i	$0.0445278\pi$
$277$	170.595	+	295.480i	0.615868	+	1.06671i	−0.374364	+	0.927282i	$0.622139\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			301.495i			1.07293i	0.843921	+	0.536467i	$0.180242\pi$
$281$			301.495i			1.07293i	−0.843921	+	0.536467i	$0.819758\pi$
$282$		0			0
$283$	199.815			0.706059			0.353029	−	0.935612i	$-0.385152\pi$
$283$	199.815			0.706059			0.353029	+	0.935612i	$0.385152\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.11054	+	4.10527i	−0.0247754	+	0.0143041i
$288$		0			0
$289$	−136.564	+	236.535i	−0.472539	+	0.818461i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−146.982	−	84.8600i	−0.501644	−	0.289624i	0.227748	−	0.973720i	$-0.426864\pi$
$293$	−146.982	−	84.8600i	−0.501644	−	0.289624i	−0.729392	+	0.684096i	$0.760197\pi$
$294$		0			0
$295$	491.174			1.66500
$296$		0			0
$297$		0			0
$298$		0			0
$299$	579.143	+	334.368i	1.93693	+	1.11829i
$300$		0			0
$301$	−22.0861			−0.0733757
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	132.464i		−	0.434310i
$306$		0			0
$307$	−10.2784	−	17.8027i	−0.0334802	−	0.0579894i	0.848800	−	0.528714i	$-0.177326\pi$
$307$	−10.2784	−	17.8027i	−0.0334802	−	0.0579894i	−0.882280	+	0.470725i	$0.843992\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−56.8944	−	32.8480i	−0.182940	−	0.105621i	0.405733	−	0.913992i	$-0.367016\pi$
$311$	−56.8944	−	32.8480i	−0.182940	−	0.105621i	−0.588673	+	0.808371i	$0.700350\pi$
$312$		0			0
$313$	300.999			0.961657			0.480829	−	0.876815i	$-0.340336\pi$
$313$	300.999			0.961657			0.480829	+	0.876815i	$0.340336\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			102.797i			0.324280i	0.986768	+	0.162140i	$0.0518396\pi$
$317$			102.797i			0.324280i	−0.986768	+	0.162140i	$0.948160\pi$
$318$		0			0
$319$	−28.6592	−	49.6392i	−0.0898408	−	0.155609i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−75.3939	+	6.76245i	−0.233418	+	0.0209364i
$324$		0			0
$325$	−167.532	−	290.173i	−0.515482	−	0.892840i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	108.489	−	62.6362i	0.329754	−	0.190384i
$330$		0			0
$331$	−2.49535	−	4.32207i	−0.00753881	−	0.0130576i	0.862231	−	0.506515i	$-0.169066\pi$
$331$	−2.49535	−	4.32207i	−0.00753881	−	0.0130576i	−0.869770	+	0.493457i	$0.835733\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−252.031	+	145.510i	−0.752332	+	0.434359i
$336$		0			0
$337$	346.729			1.02887			0.514434	−	0.857530i	$-0.328002\pi$
$337$	346.729			1.02887			0.514434	+	0.857530i	$0.328002\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	121.852i		−	0.357337i
$342$		0			0
$343$	−160.209			−0.467083
$344$		0			0
$345$		0			0
$346$		0			0
$347$			479.146i			1.38082i	0.723416	+	0.690412i	$0.242571\pi$
$347$			479.146i			1.38082i	−0.723416	+	0.690412i	$0.757429\pi$
$348$		0			0
$349$	95.5350	+	165.472i	0.273739	+	0.474131i	0.969816	−	0.243837i	$-0.0784061\pi$
$349$	95.5350	+	165.472i	0.273739	+	0.474131i	−0.696077	+	0.717967i	$0.745073\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−157.395	+	90.8723i	−0.445879	+	0.257429i	−0.706088	−	0.708124i	$-0.749542\pi$
$353$	−157.395	+	90.8723i	−0.445879	+	0.257429i	0.260209	+	0.965552i	$0.416209\pi$
$354$		0			0
$355$	−264.109	−	457.450i	−0.743969	−	1.28859i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	238.668	−	137.795i	0.664815	−	0.383831i	−0.129294	−	0.991606i	$-0.541271\pi$
$359$	238.668	−	137.795i	0.664815	−	0.383831i	0.794109	+	0.607775i	$0.207938\pi$
$360$		0			0
$361$	233.255	+	275.523i	0.646135	+	0.763223i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	562.031	−	324.489i	1.53981	−	0.889011i
$366$		0			0
$367$	−514.613			−1.40221			−0.701107	−	0.713056i	$-0.747311\pi$
$367$	−514.613			−1.40221			−0.701107	+	0.713056i	$0.747311\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	40.3379i		−	0.108727i
$372$		0			0
$373$	200.091	−	346.568i	0.536438	−	0.929137i	−0.462654	−	0.886539i	$-0.653103\pi$
$373$	200.091	−	346.568i	0.536438	−	0.929137i	0.999092	−	0.0425988i	$-0.0135637\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−352.196	+	203.340i	−0.934206	+	0.539364i
$378$		0			0
$379$	−447.617			−1.18105			−0.590524	−	0.807020i	$-0.701079\pi$
$379$	−447.617			−1.18105			−0.590524	+	0.807020i	$0.701079\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	649.759i		−	1.69650i	−0.529597	−	0.848250i	$-0.677657\pi$
$383$		−	649.759i		−	1.69650i	0.529597	−	0.848250i	$-0.322343\pi$
$384$		0			0
$385$	12.8520	−	22.2603i	0.0333818	−	0.0578190i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	100.871i		−	0.259308i	−0.991559	−	0.129654i	$-0.958613\pi$
$389$		−	100.871i		−	0.259308i	0.991559	−	0.129654i	$-0.0413866\pi$
$390$		0			0
$391$	−83.3748	+	144.409i	−0.213235	+	0.369334i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−266.684	−	153.970i	−0.675149	−	0.389797i
$396$		0			0
$397$	11.4121	+	19.7663i	0.0287457	+	0.0497891i	0.880040	−	0.474899i	$-0.157515\pi$
$397$	11.4121	+	19.7663i	0.0287457	+	0.0497891i	−0.851295	+	0.524688i	$0.824182\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	273.448i		−	0.681916i	−0.940079	−	0.340958i	$-0.889249\pi$
$401$		−	273.448i		−	0.681916i	0.940079	−	0.340958i	$-0.110751\pi$
$402$		0			0
$403$	−864.552			−2.14529
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.2886	−	7.09485i	0.0301932	−	0.0174321i
$408$		0			0
$409$	181.068	+	313.618i	0.442708	+	0.766793i	0.997889	−	0.0649363i	$-0.0206844\pi$
$409$	181.068	+	313.618i	0.442708	+	0.766793i	−0.555181	+	0.831729i	$0.687351\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	105.617	−	60.9777i	0.255730	−	0.147646i
$414$		0			0
$415$	−67.1823	−	116.363i	−0.161885	−	0.280393i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	277.629	+	160.289i	0.662599	+	0.382551i	0.793266	−	0.608875i	$-0.208379\pi$
$419$	277.629	+	160.289i	0.662599	+	0.382551i	−0.130668	+	0.991426i	$0.541712\pi$
$420$		0			0
$421$	−408.525	+	707.585i	−0.970367	+	1.68073i	−0.275921	+	0.961180i	$0.588983\pi$
$421$	−408.525	+	707.585i	−0.970367	+	1.68073i	−0.694446	+	0.719545i	$0.744351\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	72.3548	−	41.7741i	0.170247	−	0.0982919i
$426$		0			0
$427$	−16.4450	−	28.4837i	−0.0385130	−	0.0667064i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−600.247	−	346.553i	−1.39268	−	0.804067i	−0.399072	−	0.916919i	$-0.630668\pi$
$431$	−600.247	−	346.553i	−1.39268	−	0.804067i	−0.993612	+	0.112853i	$0.964001\pi$
$432$		0			0
$433$	−114.105	+	197.636i	−0.263522	+	0.456434i	−0.967175	−	0.254110i	$-0.918218\pi$
$433$	−114.105	+	197.636i	−0.263522	+	0.456434i	0.703653	+	0.710544i	$0.251551\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	792.056	−	71.0435i	1.81249	−	0.162571i
$438$		0			0
$439$	−96.4899	+	167.125i	−0.219795	+	0.380696i	−0.954745	−	0.297425i	$-0.903872\pi$
$439$	−96.4899	+	167.125i	−0.219795	+	0.380696i	0.734950	+	0.678121i	$0.237205\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	82.7959i		−	0.186898i	−0.995624	−	0.0934492i	$-0.970211\pi$
$443$		−	82.7959i		−	0.186898i	0.995624	−	0.0934492i	$-0.0297893\pi$
$444$		0			0
$445$	−140.083	+	242.630i	−0.314792	+	0.545236i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			423.667i			0.943579i	0.881711	+	0.471790i	$0.156392\pi$
$449$			423.667i			0.943579i	−0.881711	+	0.471790i	$0.843608\pi$
$450$		0			0
$451$	5.49147	+	9.51150i	0.0121762	+	0.0210898i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−157.939	−	91.1864i	−0.347120	−	0.200410i
$456$		0			0
$457$	−310.956	−	538.592i	−0.680429	−	1.17854i	−0.974850	−	0.222862i	$-0.928460\pi$
$457$	−310.956	−	538.592i	−0.680429	−	1.17854i	0.294421	−	0.955676i	$-0.404873\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	640.993	+	370.077i	1.39044	+	0.802771i	0.993364	−	0.115014i	$-0.0366914\pi$
$461$	640.993	+	370.077i	1.39044	+	0.802771i	0.397077	+	0.917786i	$0.370025\pi$
$462$		0			0
$463$	122.352	−	211.919i	0.264258	−	0.457708i	−0.703111	−	0.711080i	$-0.748206\pi$
$463$	122.352	−	211.919i	0.264258	−	0.457708i	0.967369	+	0.253372i	$0.0815396\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			770.000i			1.64882i	0.565991	+	0.824412i	$0.308494\pi$
$467$			770.000i			1.64882i	−0.565991	+	0.824412i	$0.691506\pi$
$468$		0			0
$469$	−36.1293	+	62.5778i	−0.0770348	+	0.133428i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			29.5437i			0.0624603i
$474$		0			0
$475$	−361.481	−	167.599i	−0.761013	−	0.352840i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	822.107i		−	1.71630i	−0.513401	−	0.858149i	$-0.671615\pi$
$479$		−	822.107i		−	1.71630i	0.513401	−	0.858149i	$-0.328385\pi$
$480$		0			0
$481$	−50.3387	−	87.1892i	−0.104654	−	0.181267i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	612.535	+	353.647i	1.26296	+	0.729170i
$486$		0			0
$487$	−589.573			−1.21062			−0.605311	−	0.795989i	$-0.706951\pi$
$487$	−589.573			−1.21062			−0.605311	+	0.795989i	$0.706951\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			226.040i			0.460367i	0.973147	+	0.230184i	$0.0739327\pi$
$491$			226.040i			0.460367i	−0.973147	+	0.230184i	$0.926067\pi$
$492$		0			0
$493$	−50.7030	−	87.8201i	−0.102846	−	0.178134i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−113.582	−	65.5766i	−0.228535	−	0.131945i
$498$		0			0
$499$	619.012			1.24050			0.620252	−	0.784402i	$-0.287030\pi$
$499$	619.012			1.24050			0.620252	+	0.784402i	$0.287030\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−558.159	−	322.253i	−1.10966	−	0.640663i	−0.170919	−	0.985285i	$-0.554674\pi$
$503$	−558.159	−	322.253i	−1.10966	−	0.640663i	−0.938741	+	0.344622i	$0.888007\pi$
$504$		0			0
$505$	−1069.10			−2.11703
$506$		0			0
$507$		0			0
$508$		0			0
$509$	121.920	−	70.3908i	0.239529	−	0.138292i	−0.375431	−	0.926850i	$-0.622505\pi$
$509$	121.920	−	70.3908i	0.239529	−	0.138292i	0.614960	+	0.788558i	$0.289172\pi$
$510$		0			0
$511$	80.5686	−	139.549i	0.157668	−	0.273090i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−634.546	−	366.355i	−1.23213	−	0.711369i
$516$		0			0
$517$	−83.7861	−	145.122i	−0.162062	−	0.280700i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			522.469i			1.00282i	0.865210	+	0.501410i	$0.167185\pi$
$521$			522.469i			1.00282i	−0.865210	+	0.501410i	$0.832815\pi$
$522$		0			0
$523$	247.654	−	428.950i	0.473526	−	0.820172i	−0.526014	−	0.850476i	$-0.676314\pi$
$523$	247.654	−	428.950i	0.473526	−	0.820172i	0.999541	+	0.0303040i	$0.00964754\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	215.576i		−	0.409063i
$528$		0			0
$529$	611.401	−	1058.98i	1.15577	−	2.00185i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	67.4851	−	38.9626i	0.126614	−	0.0731005i
$534$		0			0
$535$	−713.339			−1.33334
$536$		0			0
$537$		0			0
$538$		0			0
$539$			103.962i			0.192879i
$540$		0			0
$541$	134.964	−	233.764i	0.249471	−	0.432096i	−0.713908	−	0.700239i	$-0.753077\pi$
$541$	134.964	−	233.764i	0.249471	−	0.432096i	0.963379	+	0.268143i	$0.0864100\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	840.977	−	485.538i	1.54308	−	0.890896i
$546$		0			0
$547$	−211.641			−0.386913			−0.193456	−	0.981109i	$-0.561970\pi$
$547$	−211.641			−0.386913			−0.193456	+	0.981109i	$0.561970\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−203.422	+	438.745i	−0.369187	+	0.796271i
$552$		0			0
$553$	−76.4596			−0.138263
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−273.449	−	157.876i	−0.490932	−	0.283440i	0.234029	−	0.972230i	$-0.424809\pi$
$557$	−273.449	−	157.876i	−0.490932	−	0.283440i	−0.724961	+	0.688790i	$0.758142\pi$
$558$		0			0
$559$	209.616			0.374984
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−389.868	−	225.090i	−0.692483	−	0.399805i	0.112058	−	0.993702i	$-0.464256\pi$
$563$	−389.868	−	225.090i	−0.692483	−	0.399805i	−0.804542	+	0.593896i	$0.797589\pi$
$564$		0			0
$565$	153.772	−	266.341i	0.272163	−	0.471400i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−135.883	+	78.4518i	−0.238809	+	0.137877i	−0.614629	−	0.788816i	$-0.710694\pi$
$569$	−135.883	+	78.4518i	−0.238809	+	0.137877i	0.375820	+	0.926693i	$0.377361\pi$
$570$		0			0
$571$	225.433	−	390.462i	0.394804	−	0.683821i	−0.598272	−	0.801293i	$-0.704146\pi$
$571$	225.433	−	390.462i	0.394804	−	0.683821i	0.993076	+	0.117472i	$0.0374792\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−760.129	+	438.860i	−1.32196	+	0.763236i
$576$		0			0
$577$	−684.430			−1.18619			−0.593094	−	0.805133i	$-0.702094\pi$
$577$	−684.430			−1.18619			−0.593094	+	0.805133i	$0.702094\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−28.8923	−	16.6810i	−0.0497285	−	0.0287108i
$582$		0			0
$583$	−53.9585			−0.0925531
$584$		0			0
$585$		0			0
$586$		0			0
$587$	815.845	+	471.028i	1.38985	+	0.802433i	0.993299	−	0.115577i	$-0.0368718\pi$
$587$	815.845	+	471.028i	1.38985	+	0.802433i	0.396556	+	0.918010i	$0.370205\pi$
$588$		0			0
$589$	−840.871	+	591.532i	−1.42762	+	1.00430i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	118.019	+	68.1386i	0.199021	+	0.114905i	0.596199	−	0.802837i	$-0.296677\pi$
$593$	118.019	+	68.1386i	0.199021	+	0.114905i	−0.397178	+	0.917742i	$0.630010\pi$
$594$		0			0
$595$	22.7374	−	39.3823i	0.0382140	−	0.0661887i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−418.632	+	241.697i	−0.698885	+	0.403501i	−0.806932	−	0.590644i	$-0.798874\pi$
$599$	−418.632	+	241.697i	−0.698885	+	0.403501i	0.108047	+	0.994146i	$0.465540\pi$
$600$		0			0
$601$	453.178	+	784.928i	0.754040	+	1.30604i	0.945850	+	0.324604i	$0.105231\pi$
$601$	453.178	+	784.928i	0.754040	+	1.30604i	−0.191809	+	0.981432i	$0.561436\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	680.711	+	393.009i	1.12514	+	0.649601i
$606$		0			0
$607$	161.082	−	279.002i	0.265374	−	0.459641i	−0.702288	−	0.711893i	$-0.747838\pi$
$607$	161.082	−	279.002i	0.265374	−	0.459641i	0.967662	+	0.252252i	$0.0811712\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1029.66	+	594.472i	−1.68520	+	0.972949i
$612$		0			0
$613$	−491.796	−	851.816i	−0.802277	−	1.38959i	−0.918114	−	0.396317i	$-0.870288\pi$
$613$	−491.796	−	851.816i	−0.802277	−	1.38959i	0.115836	−	0.993268i	$-0.463045\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	173.604	−	100.230i	0.281368	−	0.162448i	−0.352675	−	0.935746i	$-0.614728\pi$
$617$	173.604	−	100.230i	0.281368	−	0.162448i	0.634042	+	0.773298i	$0.281394\pi$
$618$		0			0
$619$	131.387	+	227.569i	0.212257	+	0.367640i	0.952421	−	0.304787i	$-0.0985852\pi$
$619$	131.387	+	227.569i	0.212257	+	0.367640i	−0.740163	+	0.672427i	$0.765252\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			69.5632i			0.111658i
$624$		0			0
$625$	−709.496			−1.13519
$626$		0			0
$627$		0			0
$628$		0			0
$629$	21.7407	−	12.5520i	0.0345639	−	0.0199555i
$630$		0			0
$631$	11.4806	−	19.8851i	0.0181944	−	0.0315136i	−0.856785	−	0.515674i	$-0.827542\pi$
$631$	11.4806	−	19.8851i	0.0181944	−	0.0315136i	0.874979	+	0.484160i	$0.160875\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−265.517	−	153.297i	−0.418138	−	0.241412i
$636$		0			0
$637$	737.622			1.15796
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−424.315	−	244.978i	−0.661958	−	0.382181i	0.131065	−	0.991374i	$-0.458160\pi$
$641$	−424.315	−	244.978i	−0.661958	−	0.382181i	−0.793023	+	0.609192i	$0.791494\pi$
$642$		0			0
$643$	−272.087			−0.423152			−0.211576	−	0.977362i	$-0.567860\pi$
$643$	−272.087			−0.423152			−0.211576	+	0.977362i	$0.567860\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			216.446i			0.334538i	0.985911	+	0.167269i	$0.0534949\pi$
$647$			216.446i			0.334538i	−0.985911	+	0.167269i	$0.946505\pi$
$648$		0			0
$649$	−81.5676	−	141.279i	−0.125682	−	0.217688i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−518.737	−	299.493i	−0.794391	−	0.458642i	0.0471153	−	0.998889i	$-0.484997\pi$
$653$	−518.737	−	299.493i	−0.794391	−	0.458642i	−0.841506	+	0.540248i	$0.818331\pi$
$654$		0			0
$655$	−1023.27			−1.56224
$656$		0			0
$657$		0			0
$658$		0			0
$659$			48.5561i			0.0736815i	0.999321	+	0.0368407i	$0.0117294\pi$
$659$			48.5561i			0.0736815i	−0.999321	+	0.0368407i	$0.988271\pi$
$660$		0			0
$661$	288.238	+	499.243i	0.436063	+	0.755284i	0.997382	−	0.0723161i	$-0.0230390\pi$
$661$	288.238	+	499.243i	0.436063	+	0.755284i	−0.561318	+	0.827600i	$0.689706\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−216.004	+	19.3744i	−0.324817	+	0.0291345i
$666$		0			0
$667$	532.664	+	922.601i	0.798597	+	1.38321i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−38.1015	+	21.9979i	−0.0567832	+	0.0327838i
$672$		0			0
$673$	−31.3885	−	54.3665i	−0.0466397	−	0.0807823i	0.841763	−	0.539847i	$-0.181518\pi$
$673$	−31.3885	−	54.3665i	−0.0466397	−	0.0807823i	−0.888403	+	0.459065i	$0.848185\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.0440	+	7.53096i	−0.0192674	+	0.0111240i	−0.509603	−	0.860410i	$-0.670208\pi$
$677$	−13.0440	+	7.53096i	−0.0192674	+	0.0111240i	0.490335	+	0.871534i	$0.336874\pi$
$678$		0			0
$679$	175.617			0.258640
$680$		0			0
$681$		0			0
$682$		0			0
$683$			952.885i			1.39515i	0.716514	+	0.697573i	$0.245737\pi$
$683$			952.885i			1.39515i	−0.716514	+	0.697573i	$0.754263\pi$
$684$		0			0
$685$	−564.083			−0.823479
$686$		0			0
$687$		0			0
$688$		0			0
$689$			382.841i			0.555648i
$690$		0			0
$691$	−190.389	−	329.764i	−0.275527	−	0.477227i	0.694741	−	0.719260i	$-0.255519\pi$
$691$	−190.389	−	329.764i	−0.275527	−	0.477227i	−0.970268	+	0.242033i	$0.922186\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−985.916	+	569.219i	−1.41858	+	0.819020i
$696$		0			0
$697$	9.71533	+	16.8274i	0.0139388	+	0.0241427i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−99.7822	+	57.6093i	−0.142343	+	0.0821816i	−0.569480	−	0.822005i	$-0.692855\pi$
$701$	−99.7822	+	57.6093i	−0.142343	+	0.0821816i	0.427137	+	0.904187i	$0.359522\pi$
$702$		0			0
$703$	−108.615	−	50.3590i	−0.154503	−	0.0716344i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−229.887	+	132.725i	−0.325158	+	0.187730i
$708$		0			0
$709$	1142.71			1.61171			0.805857	−	0.592110i	$-0.201705\pi$
$709$	1142.71			1.61171			0.805857	+	0.592110i	$0.201705\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			2264.75i			3.17637i
$714$		0			0
$715$	−121.977	+	211.270i	−0.170597	+	0.295482i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−452.298	+	261.135i	−0.629066	+	0.363191i	−0.780390	−	0.625293i	$-0.784980\pi$
$719$	−452.298	+	261.135i	−0.629066	+	0.363191i	0.151324	+	0.988484i	$0.451646\pi$
$720$		0			0
$721$	−181.927			−0.252326
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	533.771i		−	0.736236i
$726$		0			0
$727$	582.132	−	1008.28i	0.800732	−	1.38691i	−0.118404	−	0.992966i	$-0.537778\pi$
$727$	582.132	−	1008.28i	0.800732	−	1.38691i	0.919135	−	0.393942i	$-0.128889\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			52.2678i			0.0715018i
$732$		0			0
$733$	−731.629	+	1267.22i	−0.998129	+	1.72881i	−0.446114	+	0.894976i	$0.647192\pi$
$733$	−731.629	+	1267.22i	−0.998129	+	1.72881i	−0.552015	+	0.833834i	$0.686141\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	83.7080	+	48.3288i	0.113579	+	0.0655751i
$738$		0			0
$739$	587.645	+	1017.83i	0.795189	+	1.37731i	0.922719	+	0.385474i	$0.125962\pi$
$739$	587.645	+	1017.83i	0.795189	+	1.37731i	−0.127529	+	0.991835i	$0.540705\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	414.414i		−	0.557758i	−0.960326	−	0.278879i	$-0.910037\pi$
$743$		−	414.414i		−	0.557758i	0.960326	−	0.278879i	$-0.0899628\pi$
$744$		0			0
$745$	−396.834			−0.532663
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−153.388	+	88.5588i	−0.204791	+	0.118236i
$750$		0			0
$751$	672.614	+	1165.00i	0.895624	+	1.55127i	0.833030	+	0.553228i	$0.186604\pi$
$751$	672.614	+	1165.00i	0.895624	+	1.55127i	0.0625944	+	0.998039i	$0.480063\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1156.69	−	667.816i	1.53204	−	0.884524i
$756$		0			0
$757$	−138.046	−	239.103i	−0.182360	−	0.315856i	0.760324	−	0.649544i	$-0.225040\pi$
$757$	−138.046	−	239.103i	−0.182360	−	0.315856i	−0.942684	+	0.333688i	$0.891707\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	489.422	+	282.568i	0.643130	+	0.371311i	0.785819	−	0.618456i	$-0.212242\pi$
$761$	489.422	+	282.568i	0.643130	+	0.371311i	−0.142689	+	0.989768i	$0.545575\pi$
$762$		0			0
$763$	120.556	−	208.809i	0.158003	−	0.273669i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1002.39	+	578.731i	−1.30690	+	0.754539i
$768$		0			0
$769$	−245.139	−	424.593i	−0.318776	−	0.552136i	0.661457	−	0.749983i	$-0.269938\pi$
$769$	−245.139	−	424.593i	−0.318776	−	0.552136i	−0.980233	+	0.197847i	$0.936605\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−971.685	−	561.002i	−1.25703	−	0.725747i	−0.284534	−	0.958666i	$-0.591839\pi$
$773$	−971.685	−	561.002i	−1.25703	−	0.725747i	−0.972496	+	0.232919i	$0.925172\pi$
$774$		0			0
$775$	567.365	−	982.705i	0.732084	−	1.26801i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	38.9782	−	84.0691i	0.0500363	−	0.107919i
$780$		0			0
$781$	−87.7194	+	151.934i	−0.112317	+	0.194538i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	709.580i		−	0.903924i
$786$		0			0
$787$	−331.125	+	573.525i	−0.420743	+	0.728749i	−0.996012	−	0.0892155i	$-0.971564\pi$
$787$	−331.125	+	573.525i	−0.420743	+	0.728749i	0.575269	+	0.817964i	$0.304897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	76.3612i		−	0.0965376i
$792$		0			0
$793$	156.078	+	270.334i	0.196819	+	0.340901i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	235.847	+	136.166i	0.295918	+	0.170848i	0.640608	−	0.767868i	$-0.278683\pi$
$797$	235.847	+	136.166i	0.295918	+	0.170848i	−0.344690	+	0.938717i	$0.612016\pi$
$798$		0			0
$799$	−148.232	−	256.745i	−0.185522	−	0.321333i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−186.669	−	107.774i	−0.232465	−	0.134214i
$804$		0			0
$805$	−238.869	+	413.733i	−0.296732	+	0.513954i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1210.05i			1.49573i	0.663851	+	0.747865i	$0.268921\pi$
$809$			1210.05i			1.49573i	−0.663851	+	0.747865i	$0.731079\pi$
$810$		0			0
$811$	40.7638	−	70.6050i	0.0502636	−	0.0870591i	−0.839799	−	0.542898i	$-0.817327\pi$
$811$	40.7638	−	70.6050i	0.0502636	−	0.0870591i	0.890063	+	0.455838i	$0.150661\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	2185.64i		−	2.68177i
$816$		0			0
$817$	203.875	−	143.421i	0.249540	−	0.175545i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			30.7672i			0.0374753i	0.999824	+	0.0187376i	$0.00596473\pi$
$821$			30.7672i			0.0374753i	−0.999824	+	0.0187376i	$0.994035\pi$
$822$		0			0
$823$	−516.641	−	894.849i	−0.627754	−	1.08730i	−0.988001	−	0.154445i	$-0.950641\pi$
$823$	−516.641	−	894.849i	−0.627754	−	1.08730i	0.360248	−	0.932857i	$-0.382692\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1062.64	−	613.513i	−1.28493	−	0.741854i	−0.307184	−	0.951650i	$-0.599387\pi$
$827$	−1062.64	−	613.513i	−1.28493	−	0.741854i	−0.977745	+	0.209796i	$0.932720\pi$
$828$		0			0
$829$	453.167			0.546643			0.273322	−	0.961923i	$-0.411878\pi$
$829$	453.167			0.546643			0.273322	+	0.961923i	$0.411878\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			183.926i			0.220800i
$834$		0			0
$835$	686.456	+	1188.98i	0.822103	+	1.42392i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−343.027	−	198.047i	−0.408853	−	0.236051i	0.281444	−	0.959578i	$-0.409187\pi$
$839$	−343.027	−	198.047i	−0.408853	−	0.236051i	−0.690297	+	0.723526i	$0.742520\pi$
$840$		0			0
$841$	193.139			0.229654
$842$		0			0
$843$		0			0
$844$		0			0
$845$	506.647	+	292.513i	0.599582	+	0.346169i
$846$		0			0
$847$	195.163			0.230417
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−228.398	+	131.866i	−0.268388	+	0.154954i
$852$		0			0
$853$	−451.552	+	782.112i	−0.529370	+	0.916895i	0.470043	+	0.882643i	$0.344238\pi$
$853$	−451.552	+	782.112i	−0.529370	+	0.916895i	−0.999413	+	0.0342521i	$0.989095\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−820.248	−	473.570i	−0.957115	−	0.552591i	−0.0618312	−	0.998087i	$-0.519694\pi$
$857$	−820.248	−	473.570i	−0.957115	−	0.552591i	−0.895284	+	0.445496i	$0.853027\pi$
$858$		0			0
$859$	−190.223	−	329.476i	−0.221447	−	0.383557i	0.733801	−	0.679365i	$-0.237745\pi$
$859$	−190.223	−	329.476i	−0.221447	−	0.383557i	−0.955248	+	0.295808i	$0.904411\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	946.505i		−	1.09676i	−0.836229	−	0.548381i	$-0.815245\pi$
$863$		−	946.505i		−	1.09676i	0.836229	−	0.548381i	$-0.184755\pi$
$864$		0			0
$865$	752.798	−	1303.88i	0.870287	−	1.50738i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			102.277i			0.117695i
$870$		0			0
$871$	342.898	−	593.918i	0.393684	−	0.681880i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−39.8292	+	22.9954i	−0.0455191	+	0.0262805i
$876$		0			0
$877$	1262.26			1.43929			0.719645	−	0.694342i	$-0.244305\pi$
$877$	1262.26			1.43929			0.719645	+	0.694342i	$0.244305\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			360.879i			0.409625i	0.978801	+	0.204812i	$0.0656584\pi$
$881$			360.879i			0.409625i	−0.978801	+	0.204812i	$0.934342\pi$
$882$		0			0
$883$	−527.235	+	913.198i	−0.597095	+	1.03420i	0.396153	+	0.918185i	$0.370345\pi$
$883$	−527.235	+	913.198i	−0.597095	+	1.03420i	−0.993248	+	0.116014i	$0.962988\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1306.74	+	754.445i	−1.47321	+	0.850558i	−0.999545	−	0.0301472i	$-0.990402\pi$
$887$	−1306.74	+	754.445i	−1.47321	+	0.850558i	−0.473664	+	0.880705i	$0.657069\pi$
$888$		0			0
$889$	−76.1251			−0.0856301
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−594.711	+	1282.69i	−0.665970	+	1.43638i
$894$		0			0
$895$	−1141.38			−1.27528
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1192.75	−	688.636i	−1.32675	−	0.766002i
$900$		0			0
$901$	−95.4616			−0.105951
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1533.16	+	885.170i	1.69410	+	0.978089i
$906$		0			0
$907$	658.551	−	1140.64i	0.726077	−	1.25760i	−0.232453	−	0.972608i	$-0.574675\pi$
$907$	658.551	−	1140.64i	0.726077	−	1.25760i	0.958529	−	0.284994i	$-0.0919915\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1102.10	−	636.299i	1.20977	−	0.698463i	0.247063	−	0.968999i	$-0.420535\pi$
$911$	1102.10	−	636.299i	1.20977	−	0.698463i	0.962710	+	0.270537i	$0.0872012\pi$
$912$		0			0
$913$	−22.3135	+	38.6481i	−0.0244397	+	0.0423309i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−220.032	+	127.035i	−0.239947	+	0.138534i
$918$		0			0
$919$	390.752			0.425193			0.212596	−	0.977140i	$-0.431808\pi$
$919$	390.752			0.425193			0.212596	+	0.977140i	$0.431808\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1077.99	+	622.378i	1.16792	+	0.674300i
$924$		0			0
$925$	132.140			0.142854
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−374.794	−	216.388i	−0.403439	−	0.232925i	0.284528	−	0.958668i	$-0.408163\pi$
$929$	−374.794	−	216.388i	−0.403439	−	0.232925i	−0.687967	+	0.725742i	$0.741496\pi$
$930$		0			0
$931$	717.418	−	504.686i	0.770589	−	0.542090i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−52.6802	−	30.4149i	−0.0563424	−	0.0325293i
$936$		0			0
$937$	242.942	−	420.788i	0.259276	−	0.449080i	−0.706772	−	0.707441i	$-0.749849\pi$
$937$	242.942	−	420.788i	0.259276	−	0.449080i	0.966048	+	0.258362i	$0.0831827\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1602.51	+	925.212i	−1.70299	+	0.983222i	−0.760288	+	0.649586i	$0.774942\pi$
$941$	−1602.51	+	925.212i	−1.70299	+	0.983222i	−0.942702	+	0.333636i	$0.891725\pi$
$942$		0			0
$943$	−102.065	−	176.782i	−0.108235	−	0.187468i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1023.83	+	591.106i	1.08113	+	0.624188i	0.931200	−	0.364509i	$-0.118763\pi$
$947$	1023.83	+	591.106i	1.08113	+	0.624188i	0.149926	+	0.988697i	$0.452097\pi$
$948$		0			0
$949$	−764.665	+	1324.44i	−0.805759	+	1.39562i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	861.089	−	497.150i	0.903556	−	0.521668i	0.0252039	−	0.999682i	$-0.491977\pi$
$953$	861.089	−	497.150i	0.903556	−	0.521668i	0.878352	+	0.478014i	$0.158643\pi$
$954$		0			0
$955$	−751.176	−	1301.08i	−0.786572	−	1.36238i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−121.294	+	70.0291i	−0.126480	+	0.0730231i
$960$		0			0
$961$	−983.453	−	1703.39i	−1.02336	−	1.77252i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	497.135i		−	0.515166i
$966$		0			0
$967$	−312.208			−0.322863			−0.161431	−	0.986884i	$-0.551611\pi$
$967$	−312.208			−0.322863			−0.161431	+	0.986884i	$0.551611\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−522.246	+	301.519i	−0.537844	+	0.310524i	−0.744205	−	0.667952i	$-0.767171\pi$
$971$	−522.246	+	301.519i	−0.537844	+	0.310524i	0.206361	+	0.978476i	$0.433838\pi$
$972$		0			0
$973$	−141.333	+	244.797i	−0.145255	+	0.251590i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	531.713	+	306.984i	0.544230	+	0.314211i	0.746791	−	0.665058i	$-0.231593\pi$
$977$	531.713	+	306.984i	0.544230	+	0.314211i	−0.202562	+	0.979270i	$0.564927\pi$
$978$		0			0
$979$	93.0522			0.0950482
$980$		0			0
$981$		0			0
$982$		0			0
$983$	719.685	+	415.510i	0.732131	+	0.422696i	0.819201	−	0.573506i	$-0.194417\pi$
$983$	719.685	+	415.510i	0.732131	+	0.422696i	−0.0870700	+	0.996202i	$0.527750\pi$
$984$		0			0
$985$	−693.342			−0.703900
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	549.104i		−	0.555211i
$990$		0			0
$991$	−339.423	−	587.898i	−0.342505	−	0.593237i	0.642392	−	0.766376i	$-0.277942\pi$
$991$	−339.423	−	587.898i	−0.342505	−	0.593237i	−0.984897	+	0.173139i	$0.944609\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1122.89	−	648.300i	−1.12853	−	0.651557i
$996$		0			0
$997$	−552.180			−0.553841			−0.276921	−	0.960893i	$-0.589314\pi$
$997$	−552.180			−0.553841			−0.276921	+	0.960893i	$0.589314\pi$
$998$		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.881.5		80
3.2	odd	2		684.3.m.a.653.24	yes	80
9.2	odd	6		2052.3.be.a.197.5		80
9.7	even	3		684.3.be.a.425.30	yes	80
19.11	even	3		2052.3.be.a.125.5		80
57.11	odd	6		684.3.be.a.581.30	yes	80
171.11	odd	6	inner	2052.3.m.a.1493.36		80
171.106	even	3		684.3.m.a.353.24	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.24	✓	80	171.106	even	3
684.3.m.a.653.24	yes	80	3.2	odd	2
684.3.be.a.425.30	yes	80	9.7	even	3
684.3.be.a.581.30	yes	80	57.11	odd	6
2052.3.m.a.881.5		80	1.1	even	1	trivial
2052.3.m.a.1493.36		80	171.11	odd	6	inner
2052.3.be.a.125.5		80	19.11	even	3
2052.3.be.a.197.5		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)

\(f(q)\)	\(=\)	\(q-6.78017i q^{5} +(-0.841737 - 1.45793i) q^{7} +O(q^{10})\)	\(q-6.78017i q^{5} +(-0.841737 - 1.45793i) q^{7} +(-1.95022 + 1.12596i) q^{11} +(7.98882 + 13.8370i) q^{13} +(-3.45027 + 1.99201i) q^{17} +(17.2374 + 7.99203i) q^{19} +(36.2471 - 20.9273i) q^{23} -20.9708 q^{25} +25.4531i q^{29} +(-27.0551 + 46.8607i) q^{31} +(-9.88503 + 5.70713i) q^{35} -6.30115 q^{37} -4.87714i q^{41} +(6.55967 - 11.3617i) q^{43} +74.4130i q^{47} +(23.0830 - 39.9809i) q^{49} +(20.7509 + 11.9805i) q^{53} +(7.63421 + 13.2228i) q^{55} +72.4427i q^{59} +19.5370 q^{61} +(93.8175 - 54.1656i) q^{65} +(-21.4612 - 37.1718i) q^{67} +(67.4688 - 38.9531i) q^{71} +(47.8585 + 82.8933i) q^{73} +(3.28315 + 1.89553i) q^{77} +(22.7089 - 39.3329i) q^{79} +(17.1623 - 9.90864i) q^{83} +(13.5062 + 23.3934i) q^{85} +(-35.7852 - 20.6606i) q^{89} +(13.4490 - 23.2943i) q^{91} +(54.1874 - 116.872i) q^{95} +(-52.1590 + 90.3421i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 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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