LMFDB - Embedded newform 2052.3.be.a.125.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(125,2052)]

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, 4]))

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

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

chi := DirichletCharacter("2052.125");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.be (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			125.4
Character	$\chi$	$=$	2052.125
Dual form			2052.3.be.a.197.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.09418 + 3.51848i) q^{5} +(2.95479 + 5.11784i) q^{7} +O(q^{10})$	$q+(-6.09418 + 3.51848i) q^{5} +(2.95479 + 5.11784i) q^{7} +(-14.8621 + 8.58065i) q^{11} -1.69095 q^{13} +(-22.0157 - 12.7108i) q^{17} +(9.01588 + 16.7247i) q^{19} -24.5609i q^{23} +(12.2594 - 21.2339i) q^{25} +(-4.33936 - 2.50533i) q^{29} +(-25.4929 + 44.1550i) q^{31} +(-36.0140 - 20.7927i) q^{35} -30.6535 q^{37} +(-61.7782 + 35.6677i) q^{41} +5.36109 q^{43} +(33.2249 + 19.1824i) q^{47} +(7.03844 - 12.1909i) q^{49} +(12.6088 - 7.27967i) q^{53} +(60.3817 - 104.584i) q^{55} +(40.9541 - 23.6449i) q^{59} +(47.3668 - 82.0417i) q^{61} +(10.3050 - 5.94958i) q^{65} +80.5590 q^{67} +(116.450 + 67.2326i) q^{71} +(-0.432590 + 0.749268i) q^{73} +(-87.8289 - 50.7080i) q^{77} -67.3609 q^{79} +(-85.0365 + 49.0958i) q^{83} +178.890 q^{85} +(-28.9575 + 16.7186i) q^{89} +(-4.99641 - 8.65404i) q^{91} +(-113.790 - 70.2009i) q^{95} +7.99364 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−6.09418	+	3.51848i	−1.21884	+	0.703696i	−0.964669	−	0.263465i	$-0.915135\pi$
$5$	−6.09418	+	3.51848i	−1.21884	+	0.703696i	−0.254167	+	0.967160i	$0.581801\pi$
$6$		0			0
$7$	2.95479	+	5.11784i	0.422113	+	0.731121i	0.996146	−	0.0877114i	$-0.0279553\pi$
$7$	2.95479	+	5.11784i	0.422113	+	0.731121i	−0.574033	+	0.818832i	$0.694622\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−14.8621	+	8.58065i	−1.35110	+	0.780059i	−0.988404	−	0.151848i	$-0.951478\pi$
$11$	−14.8621	+	8.58065i	−1.35110	+	0.780059i	−0.362698	+	0.931907i	$0.618144\pi$
$12$		0			0
$13$	−1.69095			−0.130073			−0.0650367	−	0.997883i	$-0.520716\pi$
$13$	−1.69095			−0.130073			−0.0650367	+	0.997883i	$0.520716\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−22.0157	−	12.7108i	−1.29504	−	0.747692i	−0.315497	−	0.948927i	$-0.602171\pi$
$17$	−22.0157	−	12.7108i	−1.29504	−	0.747692i	−0.979543	+	0.201235i	$0.935505\pi$
$18$		0			0
$19$	9.01588	+	16.7247i	0.474520	+	0.880245i
$19$	9.01588	+	16.7247i	0.474520	+	0.880245i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	24.5609i		−	1.06787i	−0.845526	−	0.533934i	$-0.820713\pi$
$23$		−	24.5609i		−	1.06787i	0.845526	−	0.533934i	$-0.179287\pi$
$24$		0			0
$25$	12.2594	−	21.2339i	0.490375	−	0.849354i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.33936	−	2.50533i	−0.149633	−	0.0863907i	0.423314	−	0.905983i	$-0.360867\pi$
$29$	−4.33936	−	2.50533i	−0.149633	−	0.0863907i	−0.572947	+	0.819592i	$0.694200\pi$
$30$		0			0
$31$	−25.4929	+	44.1550i	−0.822352	+	1.42436i	0.0815735	+	0.996667i	$0.474005\pi$
$31$	−25.4929	+	44.1550i	−0.822352	+	1.42436i	−0.903926	+	0.427689i	$0.859328\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−36.0140	−	20.7927i	−1.02897	−	0.594078i
$36$		0			0
$37$	−30.6535			−0.828473			−0.414237	−	0.910169i	$-0.635951\pi$
$37$	−30.6535			−0.828473			−0.414237	+	0.910169i	$0.635951\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−61.7782	+	35.6677i	−1.50679	+	0.869943i	−0.506816	+	0.862054i	$0.669178\pi$
$41$	−61.7782	+	35.6677i	−1.50679	+	0.869943i	−0.999969	+	0.00788886i	$0.997489\pi$
$42$		0			0
$43$	5.36109			0.124677			0.0623383	−	0.998055i	$-0.480144\pi$
$43$	5.36109			0.124677			0.0623383	+	0.998055i	$0.480144\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	33.2249	+	19.1824i	0.706912	+	0.408136i	0.809917	−	0.586545i	$-0.199512\pi$
$47$	33.2249	+	19.1824i	0.706912	+	0.408136i	−0.103005	+	0.994681i	$0.532846\pi$
$48$		0			0
$49$	7.03844	−	12.1909i	0.143642	−	0.248795i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.6088	−	7.27967i	0.237901	−	0.137352i	−0.376311	−	0.926494i	$-0.622807\pi$
$53$	12.6088	−	7.27967i	0.237901	−	0.137352i	0.614212	+	0.789141i	$0.289474\pi$
$54$		0			0
$55$	60.3817	−	104.584i	1.09785	−	1.90153i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	40.9541	−	23.6449i	0.694137	−	0.400760i	−0.111023	−	0.993818i	$-0.535413\pi$
$59$	40.9541	−	23.6449i	0.694137	−	0.400760i	0.805160	+	0.593057i	$0.202079\pi$
$60$		0			0
$61$	47.3668	−	82.0417i	0.776505	−	1.34495i	−0.157440	−	0.987529i	$-0.550324\pi$
$61$	47.3668	−	82.0417i	0.776505	−	1.34495i	0.933945	−	0.357417i	$-0.116343\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.3050	−	5.94958i	0.158538	−	0.0915321i
$66$		0			0
$67$	80.5590			1.20237			0.601186	−	0.799109i	$-0.294695\pi$
$67$	80.5590			1.20237			0.601186	+	0.799109i	$0.294695\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	116.450	+	67.2326i	1.64014	+	0.946938i	0.980780	+	0.195114i	$0.0625078\pi$
$71$	116.450	+	67.2326i	1.64014	+	0.946938i	0.659364	+	0.751824i	$0.270825\pi$
$72$		0			0
$73$	−0.432590	+	0.749268i	−0.00592589	+	0.0102639i	−0.868973	−	0.494859i	$-0.835220\pi$
$73$	−0.432590	+	0.749268i	−0.00592589	+	0.0102639i	0.863047	+	0.505123i	$0.168553\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−87.8289	−	50.7080i	−1.14063	−	0.658546i
$78$		0			0
$79$	−67.3609			−0.852669			−0.426335	−	0.904566i	$-0.640195\pi$
$79$	−67.3609			−0.852669			−0.426335	+	0.904566i	$0.640195\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−85.0365	+	49.0958i	−1.02454	+	0.591516i	−0.915414	−	0.402512i	$-0.868137\pi$
$83$	−85.0365	+	49.0958i	−1.02454	+	0.591516i	−0.109121	+	0.994028i	$0.534804\pi$
$84$		0			0
$85$	178.890			2.10459
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−28.9575	+	16.7186i	−0.325365	+	0.187849i	−0.653781	−	0.756684i	$-0.726818\pi$
$89$	−28.9575	+	16.7186i	−0.325365	+	0.187849i	0.328417	+	0.944533i	$0.393485\pi$
$90$		0			0
$91$	−4.99641	−	8.65404i	−0.0549056	−	0.0950994i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−113.790	−	70.2009i	−1.19779	−	0.738957i
$96$		0			0
$97$	7.99364			0.0824086			0.0412043	−	0.999151i	$-0.486881\pi$
$97$	7.99364			0.0824086			0.0412043	+	0.999151i	$0.486881\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−28.8327	−	16.6466i	−0.285473	−	0.164818i	0.350426	−	0.936591i	$-0.386037\pi$
$101$	−28.8327	−	16.6466i	−0.285473	−	0.164818i	−0.635898	+	0.771773i	$0.719370\pi$
$102$		0			0
$103$	−67.4528	+	116.832i	−0.654882	+	1.13429i	0.327042	+	0.945010i	$0.393948\pi$
$103$	−67.4528	+	116.832i	−0.654882	+	1.13429i	−0.981923	+	0.189278i	$0.939385\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			19.5810i			0.183000i	0.995805	+	0.0915002i	$0.0291662\pi$
$107$			19.5810i			0.183000i	−0.995805	+	0.0915002i	$0.970834\pi$
$108$		0			0
$109$	86.2205	−	149.338i	0.791014	−	1.37008i	−0.134326	−	0.990937i	$-0.542887\pi$
$109$	86.2205	−	149.338i	0.791014	−	1.37008i	0.925340	−	0.379139i	$-0.123780\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	116.759	+	67.4111i	1.03327	+	0.596559i	0.917920	−	0.396766i	$-0.129868\pi$
$113$	116.759	+	67.4111i	1.03327	+	0.596559i	0.115350	+	0.993325i	$0.463201\pi$
$114$		0			0
$115$	86.4171	+	149.679i	0.751453	+	1.30156i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	150.230i		−	1.26244i
$120$		0			0
$121$	86.7551	−	150.264i	0.716985	−	1.24185i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	3.38662i		−	0.0270929i
$126$		0			0
$127$	−73.3589	−	127.061i	−0.577629	−	1.00048i	−0.995751	−	0.0920915i	$-0.970645\pi$
$127$	−73.3589	−	127.061i	−0.577629	−	1.00048i	0.418122	−	0.908391i	$-0.362689\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	161.973	−	93.5151i	1.23643	−	0.713855i	0.268070	−	0.963399i	$-0.413614\pi$
$131$	161.973	−	93.5151i	1.23643	−	0.713855i	0.968363	+	0.249544i	$0.0802807\pi$
$132$		0			0
$133$	−58.9542	+	95.5597i	−0.443264	+	0.718494i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−41.2192	−	23.7979i	−0.300870	−	0.173707i	0.341964	−	0.939713i	$-0.388908\pi$
$137$	−41.2192	−	23.7979i	−0.300870	−	0.173707i	−0.642834	+	0.766006i	$0.722241\pi$
$138$		0			0
$139$	129.892			0.934476			0.467238	−	0.884132i	$-0.345249\pi$
$139$	129.892			0.934476			0.467238	+	0.884132i	$0.345249\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	25.1312	−	14.5095i	0.175742	−	0.101465i
$144$		0			0
$145$	35.2598			0.243171
$146$		0			0
$147$		0			0
$148$		0			0
$149$	222.082	−	128.219i	1.49048	−	0.860530i	0.490541	−	0.871418i	$-0.336799\pi$
$149$	222.082	−	128.219i	1.49048	−	0.860530i	0.999941	+	0.0108886i	$0.00346601\pi$
$150$		0			0
$151$	−37.8395	−	65.5400i	−0.250593	−	0.434040i	0.713096	−	0.701066i	$-0.247292\pi$
$151$	−37.8395	−	65.5400i	−0.250593	−	0.434040i	−0.963689	+	0.267026i	$0.913959\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	358.785i		−	2.31474i
$156$		0			0
$157$	−68.6526	−	118.910i	−0.437278	−	0.757387i	0.560201	−	0.828357i	$-0.310724\pi$
$157$	−68.6526	−	118.910i	−0.437278	−	0.757387i	−0.997478	+	0.0709695i	$0.977391\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	125.699	−	72.5724i	0.780740	−	0.450760i
$162$		0			0
$163$	−204.632			−1.25541			−0.627707	−	0.778450i	$-0.716006\pi$
$163$	−204.632			−1.25541			−0.627707	+	0.778450i	$0.716006\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	48.9598i		−	0.293172i	−0.989198	−	0.146586i	$-0.953171\pi$
$167$		−	48.9598i		−	0.293172i	0.989198	−	0.146586i	$-0.0468286\pi$
$168$		0			0
$169$	−166.141			−0.983081
$170$		0			0
$171$		0			0
$172$		0			0
$173$			235.500i			1.36127i	0.732622	+	0.680635i	$0.238296\pi$
$173$			235.500i			1.36127i	−0.732622	+	0.680635i	$0.761704\pi$
$174$		0			0
$175$	144.895			0.827974
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	295.150i		−	1.64888i	−0.565949	−	0.824440i	$-0.691490\pi$
$179$		−	295.150i		−	1.64888i	0.565949	−	0.824440i	$-0.308510\pi$
$180$		0			0
$181$	17.0737	+	29.5724i	0.0943296	+	0.163384i	0.909329	−	0.416079i	$-0.136596\pi$
$181$	17.0737	+	29.5724i	0.0943296	+	0.163384i	−0.814999	+	0.579462i	$0.803263\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	186.808	−	107.854i	1.00977	−	0.582993i
$186$		0			0
$187$	436.266			2.33297
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−282.384	+	163.034i	−1.47845	+	0.853584i	−0.999703	−	0.0243694i	$-0.992242\pi$
$191$	−282.384	+	163.034i	−1.47845	+	0.853584i	−0.478747	+	0.877953i	$0.658909\pi$
$192$		0			0
$193$	54.2566	+	93.9752i	0.281122	+	0.486918i	0.971661	−	0.236377i	$-0.0759600\pi$
$193$	54.2566	+	93.9752i	0.281122	+	0.486918i	−0.690539	+	0.723295i	$0.742627\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			62.3757i			0.316628i	0.987389	+	0.158314i	$0.0506058\pi$
$197$			62.3757i			0.316628i	−0.987389	+	0.158314i	$0.949394\pi$
$198$		0			0
$199$	−66.6982	−	115.525i	−0.335167	−	0.580526i	0.648350	−	0.761342i	$-0.275459\pi$
$199$	−66.6982	−	115.525i	−0.335167	−	0.580526i	−0.983517	+	0.180817i	$0.942126\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	29.6109i		−	0.145866i
$204$		0			0
$205$	250.992	−	434.730i	1.22435	−	2.12064i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−277.503	−	171.202i	−1.32777	−	0.819147i
$210$		0			0
$211$	79.6230	+	137.911i	0.377360	+	0.653607i	0.990677	−	0.136230i	$-0.0434985\pi$
$211$	79.6230	+	137.911i	0.377360	+	0.653607i	−0.613317	+	0.789837i	$0.710165\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−32.6715	+	18.8629i	−0.151960	+	0.0877343i
$216$		0			0
$217$	−301.305			−1.38850
$218$		0			0
$219$		0			0
$220$		0			0
$221$	37.2275	+	21.4933i	0.168450	+	0.0972548i
$222$		0			0
$223$	−42.1356			−0.188949			−0.0944744	−	0.995527i	$-0.530117\pi$
$223$	−42.1356			−0.188949			−0.0944744	+	0.995527i	$0.530117\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−121.312	+	70.0396i	−0.534415	+	0.308545i	−0.742812	−	0.669500i	$-0.766509\pi$
$227$	−121.312	+	70.0396i	−0.534415	+	0.308545i	0.208398	+	0.978044i	$0.433175\pi$
$228$		0			0
$229$	95.4470	−	165.319i	0.416799	−	0.721917i	−0.578816	−	0.815458i	$-0.696485\pi$
$229$	95.4470	−	165.319i	0.416799	−	0.721917i	0.995615	+	0.0935406i	$0.0298185\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−84.4012	−	48.7290i	−0.362237	−	0.209138i	0.307825	−	0.951443i	$-0.400399\pi$
$233$	−84.4012	−	48.7290i	−0.362237	−	0.209138i	−0.670062	+	0.742306i	$0.733732\pi$
$234$		0			0
$235$	−269.971			−1.14881
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.7618	+	6.79066i	0.0492124	+	0.0284128i	0.524404	−	0.851469i	$-0.324288\pi$
$239$	11.7618	+	6.79066i	0.0492124	+	0.0284128i	−0.475192	+	0.879882i	$0.657621\pi$
$240$		0			0
$241$	12.5763	−	21.7829i	0.0521840	−	0.0903853i	−0.838753	−	0.544511i	$-0.816715\pi$
$241$	12.5763	−	21.7829i	0.0521840	−	0.0903853i	0.890937	+	0.454126i	$0.150048\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			99.0584i			0.404320i
$246$		0			0
$247$	−15.2454	−	28.2806i	−0.0617224	−	0.114496i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	284.518	−	164.266i	1.13354	−	0.654447i	0.188714	−	0.982032i	$-0.439568\pi$
$251$	284.518	−	164.266i	1.13354	−	0.654447i	0.944822	+	0.327585i	$0.106235\pi$
$252$		0			0
$253$	210.749	+	365.028i	0.833000	+	1.44280i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	497.762i		−	1.93682i	−0.249371	−	0.968408i	$-0.580224\pi$
$257$		−	497.762i		−	1.93682i	0.249371	−	0.968408i	$-0.419776\pi$
$258$		0			0
$259$	−90.5747	−	156.880i	−0.349709	−	0.605714i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	443.885i		−	1.68777i	−0.536521	−	0.843887i	$-0.680262\pi$
$263$		−	443.885i		−	1.68777i	0.536521	−	0.843887i	$-0.319738\pi$
$264$		0			0
$265$	−51.2267	+	88.7272i	−0.193308	+	0.334820i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	67.1951	+	38.7951i	0.249796	+	0.144220i	0.619671	−	0.784862i	$-0.287266\pi$
$269$	67.1951	+	38.7951i	0.249796	+	0.144220i	−0.369875	+	0.929082i	$0.620599\pi$
$270$		0			0
$271$	−14.1771	+	24.5554i	−0.0523140	+	0.0906105i	−0.890997	−	0.454010i	$-0.849993\pi$
$271$	−14.1771	+	24.5554i	−0.0523140	+	0.0906105i	0.838683	+	0.544621i	$0.183326\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			420.773i			1.53009i
$276$		0			0
$277$	−80.2383	−	138.977i	−0.289669	−	0.501721i	0.684062	−	0.729424i	$-0.260212\pi$
$277$	−80.2383	−	138.977i	−0.289669	−	0.501721i	−0.973731	+	0.227703i	$0.926878\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	166.952	+	96.3898i	0.594135	+	0.343024i	0.766731	−	0.641969i	$-0.221882\pi$
$281$	166.952	+	96.3898i	0.594135	+	0.343024i	−0.172596	+	0.984993i	$0.555215\pi$
$282$		0			0
$283$	140.393	+	243.168i	0.496088	+	0.859250i	0.999990	−	0.00451135i	$-0.00143601\pi$
$283$	140.393	+	243.168i	0.496088	+	0.859250i	−0.503902	+	0.863761i	$0.668103\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−365.083	−	210.781i	−1.27207	−	0.734428i
$288$		0			0
$289$	178.627	+	309.391i	0.618086	+	1.07056i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	17.7905	+	10.2714i	0.0607185	+	0.0350559i	0.530052	−	0.847965i	$-0.322172\pi$
$293$	17.7905	+	10.2714i	0.0607185	+	0.0350559i	−0.469333	+	0.883021i	$0.655506\pi$
$294$		0			0
$295$	−166.388	+	288.192i	−0.564027	+	0.976923i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			41.5314i			0.138901i
$300$		0			0
$301$	15.8409	+	27.4372i	0.0526275	+	0.0911536i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			666.636i			2.18569i
$306$		0			0
$307$	−43.0811	+	74.6187i	−0.140329	+	0.243058i	−0.927621	−	0.373524i	$-0.878149\pi$
$307$	−43.0811	+	74.6187i	−0.140329	+	0.243058i	0.787291	+	0.616581i	$0.211483\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	519.182	+	299.750i	1.66940	+	0.963826i	0.967964	+	0.251090i	$0.0807892\pi$
$311$	519.182	+	299.750i	1.66940	+	0.963826i	0.701433	+	0.712736i	$0.252544\pi$
$312$		0			0
$313$	81.4152	−	141.015i	0.260112	−	0.450528i	−0.706159	−	0.708053i	$-0.749574\pi$
$313$	81.4152	−	141.015i	0.260112	−	0.450528i	0.966272	+	0.257525i	$0.0829071\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−185.646	−	107.183i	−0.585634	−	0.338116i	0.177735	−	0.984078i	$-0.443123\pi$
$317$	−185.646	−	107.183i	−0.585634	−	0.338116i	−0.763369	+	0.645962i	$0.776456\pi$
$318$		0			0
$319$	85.9894			0.269559
$320$		0			0
$321$		0			0
$322$		0			0
$323$	14.0924	−	482.803i	0.0436296	−	1.49475i
$324$		0			0
$325$	−20.7300	+	35.9055i	−0.0637847	+	0.110478i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			226.720i			0.689117i
$330$		0			0
$331$	−233.185	−	403.888i	−0.704487	−	1.22021i	−0.966877	−	0.255245i	$-0.917844\pi$
$331$	−233.185	−	403.888i	−0.704487	−	1.22021i	0.262390	−	0.964962i	$-0.415489\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−490.941	+	283.445i	−1.46550	+	0.846104i
$336$		0			0
$337$	5.52445	+	9.56863i	0.0163930	+	0.0283936i	0.874106	−	0.485736i	$-0.161448\pi$
$337$	5.52445	+	9.56863i	0.0163930	+	0.0283936i	−0.857713	+	0.514130i	$0.828115\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	874.984i		−	2.56593i
$342$		0			0
$343$	372.758			1.08676
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−57.5958	+	33.2529i	−0.165982	+	0.0958298i	−0.580690	−	0.814125i	$-0.697217\pi$
$347$	−57.5958	+	33.2529i	−0.165982	+	0.0958298i	0.414708	+	0.909955i	$0.363884\pi$
$348$		0			0
$349$	−74.8996	−	129.730i	−0.214612	−	0.371719i	0.738540	−	0.674209i	$-0.235515\pi$
$349$	−74.8996	−	129.730i	−0.214612	−	0.371719i	−0.953152	+	0.302490i	$0.902182\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−402.700	+	232.499i	−1.14079	+	0.658638i	−0.946627	−	0.322331i	$-0.895534\pi$
$353$	−402.700	+	232.499i	−1.14079	+	0.658638i	−0.194167	+	0.980968i	$0.562200\pi$
$354$		0			0
$355$	−946.226			−2.66542
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−62.8015	−	36.2585i	−0.174935	−	0.100999i	0.409976	−	0.912096i	$-0.365537\pi$
$359$	−62.8015	−	36.2585i	−0.174935	−	0.100999i	−0.584911	+	0.811098i	$0.698870\pi$
$360$		0			0
$361$	−198.428	+	301.575i	−0.549662	+	0.835387i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	6.08823i		−	0.0166801i
$366$		0			0
$367$	−43.6168	+	75.5466i	−0.118847	+	0.205849i	−0.919311	−	0.393532i	$-0.871253\pi$
$367$	−43.6168	+	75.5466i	−0.118847	+	0.205849i	0.800464	+	0.599381i	$0.204586\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	74.5124	+	43.0198i	0.200842	+	0.115956i
$372$		0			0
$373$	22.9854	−	39.8119i	0.0616231	−	0.106734i	−0.833568	−	0.552417i	$-0.813706\pi$
$373$	22.9854	−	39.8119i	0.0616231	−	0.106734i	0.895191	+	0.445683i	$0.147039\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.33766	+	4.23640i	0.0194633	+	0.0112371i
$378$		0			0
$379$	619.505			1.63458			0.817288	−	0.576229i	$-0.195476\pi$
$379$	619.505			1.63458			0.817288	+	0.576229i	$0.195476\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−478.580	+	276.308i	−1.24956	+	0.721432i	−0.971021	−	0.238994i	$-0.923182\pi$
$383$	−478.580	+	276.308i	−1.24956	+	0.721432i	−0.278535	+	0.960426i	$0.589849\pi$
$384$		0			0
$385$	713.660			1.85366
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−57.9442	−	33.4541i	−0.148957	−	0.0860003i	0.423669	−	0.905817i	$-0.360742\pi$
$389$	−57.9442	−	33.4541i	−0.148957	−	0.0860003i	−0.572626	+	0.819817i	$0.694075\pi$
$390$		0			0
$391$	−312.188	+	540.726i	−0.798435	+	1.38293i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	410.509	−	237.008i	1.03926	−	0.600020i
$396$		0			0
$397$	−380.194	+	658.515i	−0.957667	+	1.65873i	−0.229523	+	0.973303i	$0.573717\pi$
$397$	−380.194	+	658.515i	−0.957667	+	1.65873i	−0.728144	+	0.685424i	$0.759617\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−229.471	+	132.485i	−0.572247	+	0.330387i	−0.758046	−	0.652201i	$-0.773846\pi$
$401$	−229.471	+	132.485i	−0.572247	+	0.330387i	0.185799	+	0.982588i	$0.440513\pi$
$402$		0			0
$403$	43.1074	−	74.6642i	0.106966	−	0.185271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	455.576	−	263.027i	1.11935	−	0.646258i
$408$		0			0
$409$	−446.236			−1.09104			−0.545521	−	0.838097i	$-0.683668\pi$
$409$	−446.236			−1.09104			−0.545521	+	0.838097i	$0.683668\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	242.022	+	139.731i	0.586008	+	0.338332i
$414$		0			0
$415$	345.485	−	598.398i	0.832494	−	1.44192i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	314.327	+	181.477i	0.750183	+	0.433119i	0.825760	−	0.564021i	$-0.190746\pi$
$419$	314.327	+	181.477i	0.750183	+	0.433119i	−0.0755768	+	0.997140i	$0.524080\pi$
$420$		0			0
$421$	−170.301			−0.404515			−0.202257	−	0.979332i	$-0.564828\pi$
$421$	−170.301			−0.404515			−0.202257	+	0.979332i	$0.564828\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−539.797	+	311.652i	−1.27011	+	0.733298i
$426$		0			0
$427$	559.835			1.31109
$428$		0			0
$429$		0			0
$430$		0			0
$431$	183.959	−	106.209i	0.426818	−	0.246424i	−0.271172	−	0.962531i	$-0.587411\pi$
$431$	183.959	−	106.209i	0.426818	−	0.246424i	0.697990	+	0.716107i	$0.254078\pi$
$432$		0			0
$433$	−178.510	−	309.188i	−0.412263	−	0.714061i	0.582874	−	0.812563i	$-0.301928\pi$
$433$	−178.510	−	309.188i	−0.412263	−	0.714061i	−0.995137	+	0.0985019i	$0.968595\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	410.773	−	221.438i	0.939985	−	0.506724i
$438$		0			0
$439$	−129.965			−0.296048			−0.148024	−	0.988984i	$-0.547291\pi$
$439$	−129.965			−0.296048			−0.148024	+	0.988984i	$0.547291\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	310.039	+	179.001i	0.699863	+	0.404066i	0.807296	−	0.590146i	$-0.200930\pi$
$443$	310.039	+	179.001i	0.699863	+	0.404066i	−0.107434	+	0.994212i	$0.534263\pi$
$444$		0			0
$445$	117.648	−	203.772i	0.264378	−	0.457915i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	146.426i		−	0.326116i	−0.986616	−	0.163058i	$-0.947864\pi$
$449$		−	146.426i		−	0.326116i	0.986616	−	0.163058i	$-0.0521358\pi$
$450$		0			0
$451$	612.103	−	1060.19i	1.35721	−	2.35076i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	60.8981	+	35.1595i	0.133842	+	0.0772737i
$456$		0			0
$457$	424.977	+	736.082i	0.929928	+	1.61068i	0.783437	+	0.621471i	$0.213465\pi$
$457$	424.977	+	736.082i	0.929928	+	1.61068i	0.146491	+	0.989212i	$0.453202\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			611.058i			1.32551i	0.748838	+	0.662753i	$0.230612\pi$
$461$			611.058i			1.32551i	−0.748838	+	0.662753i	$0.769388\pi$
$462$		0			0
$463$	295.664	−	512.105i	0.638583	−	1.10606i	−0.347161	−	0.937806i	$-0.612854\pi$
$463$	295.664	−	512.105i	0.638583	−	1.10606i	0.985744	−	0.168252i	$-0.0538124\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	288.535i		−	0.617848i	−0.951087	−	0.308924i	$-0.900031\pi$
$467$		−	288.535i		−	0.617848i	0.951087	−	0.308924i	$-0.0999688\pi$
$468$		0			0
$469$	238.035	+	412.288i	0.507537	+	0.879080i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−79.6772	+	46.0016i	−0.168451	+	0.0972551i
$474$		0			0
$475$	465.658	+	13.5919i	0.980332	+	0.0286145i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.66681	+	5.58113i	0.0201812	+	0.0116516i	0.510057	−	0.860141i	$-0.329624\pi$
$479$	9.66681	+	5.58113i	0.0201812	+	0.0116516i	−0.489875	+	0.871792i	$0.662958\pi$
$480$		0			0
$481$	51.8337			0.107762
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−48.7147	+	28.1254i	−0.100443	+	0.0579906i
$486$		0			0
$487$	372.703			0.765304			0.382652	−	0.923892i	$-0.375011\pi$
$487$	372.703			0.765304			0.382652	+	0.923892i	$0.375011\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−219.366	+	126.651i	−0.446774	+	0.257945i	−0.706467	−	0.707746i	$-0.749712\pi$
$491$	−219.366	+	126.651i	−0.446774	+	0.257945i	0.259693	+	0.965691i	$0.416379\pi$
$492$		0			0
$493$	63.6893	+	110.313i	0.129187	+	0.223759i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			794.633i			1.59886i
$498$		0			0
$499$	−426.697	−	739.062i	−0.855105	−	1.48109i	−0.876547	−	0.481315i	$-0.840159\pi$
$499$	−426.697	−	739.062i	−0.855105	−	1.48109i	0.0214423	−	0.999770i	$-0.493174\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−362.114	+	209.067i	−0.719908	+	0.415639i	−0.814719	−	0.579856i	$-0.803109\pi$
$503$	−362.114	+	209.067i	−0.719908	+	0.415639i	0.0948105	+	0.995495i	$0.469775\pi$
$504$		0			0
$505$	234.283			0.463926
$506$		0			0
$507$		0			0
$508$		0			0
$509$			183.285i			0.360088i	0.983659	+	0.180044i	$0.0576240\pi$
$509$			183.285i			0.360088i	−0.983659	+	0.180044i	$0.942376\pi$
$510$		0			0
$511$	−5.11285			−0.0100056
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	949.325i		−	1.84335i
$516$		0			0
$517$	−658.389			−1.27348
$518$		0			0
$519$		0			0
$520$		0			0
$521$			434.260i			0.833513i	0.909018	+	0.416757i	$0.136833\pi$
$521$			434.260i			0.833513i	−0.909018	+	0.416757i	$0.863167\pi$
$522$		0			0
$523$	−389.570	−	674.755i	−0.744875	−	1.29016i	−0.950253	−	0.311479i	$-0.899176\pi$
$523$	−389.570	−	674.755i	−0.744875	−	1.29016i	0.205378	−	0.978683i	$-0.434158\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1122.49	−	648.069i	2.12996	−	1.22973i
$528$		0			0
$529$	−74.2401			−0.140340
$530$		0			0
$531$		0			0
$532$		0			0
$533$	104.464	−	60.3124i	0.195993	−	0.113156i
$534$		0			0
$535$	−68.8955	−	119.330i	−0.128777	−	0.223048i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			241.578i			0.448196i
$540$		0			0
$541$	−191.863	−	332.316i	−0.354644	−	0.614262i	0.632413	−	0.774632i	$-0.282065\pi$
$541$	−191.863	−	332.316i	−0.354644	−	0.614262i	−0.987057	+	0.160370i	$0.948731\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1213.46i			2.22653i
$546$		0			0
$547$	262.281	−	454.284i	0.479489	−	0.830500i	−0.520234	−	0.854024i	$-0.674155\pi$
$547$	262.281	−	454.284i	0.479489	−	0.830500i	0.999723	+	0.0235238i	$0.00748855\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.77765	−	95.1620i	0.00504110	−	0.172708i
$552$		0			0
$553$	−199.037	−	344.742i	−0.359923	−	0.623404i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−648.964	+	374.680i	−1.16511	+	0.672674i	−0.952522	−	0.304469i	$-0.901521\pi$
$557$	−648.964	+	374.680i	−1.16511	+	0.672674i	−0.212583	+	0.977143i	$0.568188\pi$
$558$		0			0
$559$	−9.06536			−0.0162171
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−384.547	−	222.018i	−0.683032	−	0.394348i	0.117965	−	0.993018i	$-0.462363\pi$
$563$	−384.547	−	222.018i	−0.683032	−	0.394348i	−0.800996	+	0.598669i	$0.795696\pi$
$564$		0			0
$565$	−948.738			−1.67918
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−142.613	+	82.3379i	−0.250639	+	0.144706i	−0.620057	−	0.784557i	$-0.712890\pi$
$569$	−142.613	+	82.3379i	−0.250639	+	0.144706i	0.369418	+	0.929263i	$0.379557\pi$
$570$		0			0
$571$	264.801	−	458.649i	0.463750	−	0.803238i	−0.535394	−	0.844602i	$-0.679837\pi$
$571$	264.801	−	458.649i	0.463750	−	0.803238i	0.999144	+	0.0413641i	$0.0131703\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−521.523	−	301.102i	−0.906997	−	0.523655i
$576$		0			0
$577$	−907.310			−1.57246			−0.786230	−	0.617933i	$-0.787970\pi$
$577$	−907.310			−1.57246			−0.786230	+	0.617933i	$0.787970\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−502.530	−	290.136i	−0.864939	−	0.499373i
$582$		0			0
$583$	−124.929	+	216.383i	−0.214286	+	0.371154i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	60.8421i		−	0.103649i	−0.998656	−	0.0518246i	$-0.983496\pi$
$587$		−	60.8421i		−	0.103649i	0.998656	−	0.0518246i	$-0.0165037\pi$
$588$		0			0
$589$	−968.319	−	28.2639i	−1.64400	−	0.0479862i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	431.969	−	249.397i	0.728447	−	0.420569i	−0.0894067	−	0.995995i	$-0.528497\pi$
$593$	431.969	−	249.397i	0.728447	−	0.420569i	0.817854	+	0.575426i	$0.195164\pi$
$594$		0			0
$595$	528.582	+	915.532i	0.888374	+	1.53871i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			903.010i			1.50753i	0.657145	+	0.753764i	$0.271764\pi$
$599$			903.010i			1.50753i	−0.657145	+	0.753764i	$0.728236\pi$
$600$		0			0
$601$	98.9079	+	171.314i	0.164572	+	0.285047i	0.936503	−	0.350659i	$-0.114042\pi$
$601$	98.9079	+	171.314i	0.164572	+	0.285047i	−0.771931	+	0.635706i	$0.780709\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			1220.98i			2.01816i
$606$		0			0
$607$	−262.991	+	455.514i	−0.433264	+	0.750435i	−0.997152	−	0.0754162i	$-0.975971\pi$
$607$	−262.991	+	455.514i	−0.433264	+	0.750435i	0.563888	+	0.825851i	$0.309305\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−56.1817	−	32.4365i	−0.0919505	−	0.0530876i
$612$		0			0
$613$	−246.020	+	426.118i	−0.401337	+	0.695136i	−0.993888	−	0.110397i	$-0.964788\pi$
$613$	−246.020	+	426.118i	−0.401337	+	0.695136i	0.592551	+	0.805533i	$0.298121\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			656.509i			1.06403i	0.846734	+	0.532017i	$0.178566\pi$
$617$			656.509i			1.06403i	−0.846734	+	0.532017i	$0.821434\pi$
$618$		0			0
$619$	−483.420	−	837.308i	−0.780969	−	1.35268i	−0.931378	−	0.364054i	$-0.881392\pi$
$619$	−483.420	−	837.308i	−0.780969	−	1.35268i	0.150409	−	0.988624i	$-0.451941\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−171.126	−	98.7999i	−0.274681	−	0.158587i
$624$		0			0
$625$	318.400	+	551.485i	0.509440	+	0.882376i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	674.858	+	389.629i	1.07291	+	0.619443i
$630$		0			0
$631$	340.653	+	590.028i	0.539862	+	0.935068i	0.998911	+	0.0466568i	$0.0148567\pi$
$631$	340.653	+	590.028i	0.539862	+	0.935068i	−0.459049	+	0.888411i	$0.651810\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	894.125	+	516.223i	1.40807	+	0.812950i
$636$		0			0
$637$	−11.9017	+	20.6143i	−0.0186840	+	0.0323616i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	514.323i		−	0.802377i	−0.915996	−	0.401188i	$-0.868597\pi$
$641$		−	514.323i		−	0.802377i	0.915996	−	0.401188i	$-0.131403\pi$
$642$		0			0
$643$	105.365	+	182.498i	0.163865	+	0.283823i	0.936252	−	0.351330i	$-0.114270\pi$
$643$	105.365	+	182.498i	0.163865	+	0.283823i	−0.772387	+	0.635153i	$0.780937\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	547.085i		−	0.845572i	−0.906230	−	0.422786i	$-0.861052\pi$
$647$		−	547.085i		−	0.845572i	0.906230	−	0.422786i	$-0.138948\pi$
$648$		0			0
$649$	−405.777	+	702.826i	−0.625234	+	1.08294i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−713.415	−	411.890i	−1.09252	−	0.630766i	−0.158273	−	0.987395i	$-0.550593\pi$
$653$	−713.415	−	411.890i	−1.09252	−	0.630766i	−0.934246	+	0.356629i	$0.883926\pi$
$654$		0			0
$655$	−658.061	+	1139.80i	−1.00467	+	1.74015i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−34.8216	−	20.1043i	−0.0528401	−	0.0305073i	0.473347	−	0.880876i	$-0.343046\pi$
$659$	−34.8216	−	20.1043i	−0.0528401	−	0.0305073i	−0.526187	+	0.850369i	$0.676379\pi$
$660$		0			0
$661$	−786.041			−1.18917			−0.594585	−	0.804033i	$-0.702684\pi$
$661$	−786.041			−1.18917			−0.594585	+	0.804033i	$0.702684\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	23.0528	−	789.787i	0.0346659	−	1.18765i
$666$		0			0
$667$	−61.5333	+	106.579i	−0.0922538	+	0.159788i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1625.75i			2.42288i
$672$		0			0
$673$	512.431	+	887.556i	0.761413	+	1.31881i	0.942122	+	0.335269i	$0.108827\pi$
$673$	512.431	+	887.556i	0.761413	+	1.31881i	−0.180710	+	0.983536i	$0.557840\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	202.004	−	116.627i	0.298380	−	0.172270i	−0.343335	−	0.939213i	$-0.611556\pi$
$677$	202.004	−	116.627i	0.298380	−	0.172270i	0.641715	+	0.766943i	$0.278223\pi$
$678$		0			0
$679$	23.6195	+	40.9102i	0.0347857	+	0.0602507i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	669.562i		−	0.980326i	−0.871631	−	0.490163i	$-0.836937\pi$
$683$		−	669.562i		−	0.980326i	0.871631	−	0.490163i	$-0.163063\pi$
$684$		0			0
$685$	334.929			0.488948
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.3208	+	12.3096i	−0.0309446	+	0.0178659i
$690$		0			0
$691$	495.592	+	858.390i	0.717210	+	1.24224i	0.962101	+	0.272693i	$0.0879143\pi$
$691$	495.592	+	858.390i	0.717210	+	1.24224i	−0.244891	+	0.969551i	$0.578752\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−791.586	+	457.023i	−1.13897	+	0.657586i
$696$		0			0
$697$	1813.45			2.60180
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−225.129	−	129.978i	−0.321154	−	0.185418i	0.330753	−	0.943717i	$-0.392697\pi$
$701$	−225.129	−	129.978i	−0.321154	−	0.185418i	−0.651907	+	0.758299i	$0.726031\pi$
$702$		0			0
$703$	−276.368	−	512.669i	−0.393127	−	0.729259i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	196.749i		−	0.278287i
$708$		0			0
$709$	−119.700	+	207.326i	−0.168829	+	0.292420i	−0.938008	−	0.346613i	$-0.887332\pi$
$709$	−119.700	+	207.326i	−0.168829	+	0.292420i	0.769180	+	0.639033i	$0.220665\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1084.49	+	626.130i	1.52102	+	0.878163i
$714$		0			0
$715$	−102.103	+	176.847i	−0.142801	+	0.247338i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−369.672	−	213.430i	−0.514147	−	0.296843i	0.220390	−	0.975412i	$-0.429267\pi$
$719$	−369.672	−	213.430i	−0.514147	−	0.296843i	−0.734537	+	0.678569i	$0.762600\pi$
$720$		0			0
$721$	−797.235			−1.10574
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−106.396	+	61.4275i	−0.146753	+	0.0847276i
$726$		0			0
$727$	−1034.99			−1.42365			−0.711824	−	0.702358i	$-0.752131\pi$
$727$	−1034.99			−1.42365			−0.711824	+	0.702358i	$0.752131\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−118.028	−	68.1435i	−0.161461	−	0.0932196i
$732$		0			0
$733$	−327.088	+	566.533i	−0.446232	+	0.772896i	−0.998137	−	0.0610108i	$-0.980568\pi$
$733$	−327.088	+	566.533i	−0.446232	+	0.772896i	0.551905	+	0.833907i	$0.313901\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1197.28	+	691.248i	−1.62453	+	0.937922i
$738$		0			0
$739$	−683.303	+	1183.51i	−0.924631	+	1.60151i	−0.132478	+	0.991186i	$0.542293\pi$
$739$	−683.303	+	1183.51i	−0.924631	+	1.60151i	−0.792153	+	0.610322i	$0.791040\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−280.625	+	162.019i	−0.377692	+	0.218061i	−0.676814	−	0.736154i	$-0.736640\pi$
$743$	−280.625	+	162.019i	−0.377692	+	0.218061i	0.299122	+	0.954215i	$0.403306\pi$
$744$		0			0
$745$	−902.271	+	1562.78i	−1.21110	+	2.09769i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−100.213	+	57.8579i	−0.133795	+	0.0772468i
$750$		0			0
$751$	363.697			0.484284			0.242142	−	0.970241i	$-0.422150\pi$
$751$	363.697			0.484284			0.242142	+	0.970241i	$0.422150\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	461.202	+	266.275i	0.610864	+	0.352682i
$756$		0			0
$757$	538.057	−	931.942i	0.710775	−	1.23110i	−0.253791	−	0.967259i	$-0.581678\pi$
$757$	538.057	−	931.942i	0.710775	−	1.23110i	0.964566	−	0.263840i	$-0.0849891\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	659.022	+	380.487i	0.865995	+	0.499983i	0.866015	−	0.500017i	$-0.166673\pi$
$761$	659.022	+	380.487i	0.865995	+	0.499983i	−2.00331e−5		1.00000i	$0.500006\pi$
$762$		0			0
$763$	1019.05			1.33559
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−69.2515	+	39.9824i	−0.0902888	+	0.0521283i
$768$		0			0
$769$	665.877			0.865900			0.432950	−	0.901418i	$-0.357473\pi$
$769$	665.877			0.865900			0.432950	+	0.901418i	$0.357473\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	182.002	−	105.079i	0.235449	−	0.135936i	−0.377634	−	0.925955i	$-0.623262\pi$
$773$	182.002	−	105.079i	0.235449	−	0.135936i	0.613083	+	0.790018i	$0.289929\pi$
$774$		0			0
$775$	625.054	+	1082.63i	0.806522	+	1.39694i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1153.51	−	711.644i	−1.48076	−	0.913535i
$780$		0			0
$781$	−2307.60			−2.95467
$782$		0			0
$783$		0			0
$784$		0			0
$785$	836.763	+	483.105i	1.06594	+	0.615421i
$786$		0			0
$787$	643.922	−	1115.30i	0.818198	−	1.41716i	−0.0888113	−	0.996048i	$-0.528307\pi$
$787$	643.922	−	1115.30i	0.818198	−	1.41716i	0.907009	−	0.421111i	$-0.138360\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			796.743i			1.00726i
$792$		0			0
$793$	−80.0951	+	138.729i	−0.101003	+	0.174942i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	353.437	+	204.057i	0.443459	+	0.256031i	0.705064	−	0.709144i	$-0.250918\pi$
$797$	353.437	+	204.057i	0.443459	+	0.256031i	−0.261605	+	0.965175i	$0.584252\pi$
$798$		0			0
$799$	−487.645	−	844.626i	−0.610320	−	1.05710i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	14.8476i		−	0.0184902i
$804$		0			0
$805$	−510.689	+	884.539i	−0.634396	+	1.09881i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1173.72i		−	1.45082i	−0.688315	−	0.725412i	$-0.741649\pi$
$809$		−	1173.72i		−	1.45082i	0.688315	−	0.725412i	$-0.258351\pi$
$810$		0			0
$811$	−695.637	−	1204.88i	−0.857752	−	1.48567i	−0.874068	−	0.485804i	$-0.838527\pi$
$811$	−695.637	−	1204.88i	−0.857752	−	1.48567i	0.0163156	−	0.999867i	$-0.494806\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1247.07	−	719.994i	1.53014	−	0.883429i
$816$		0			0
$817$	48.3349	+	89.6624i	0.0591615	+	0.109746i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	125.681	+	72.5620i	0.153083	+	0.0883824i	0.574585	−	0.818445i	$-0.305164\pi$
$821$	125.681	+	72.5620i	0.153083	+	0.0883824i	−0.421502	+	0.906828i	$0.638497\pi$
$822$		0			0
$823$	−1165.56			−1.41623			−0.708117	−	0.706096i	$-0.750455\pi$
$823$	−1165.56			−1.41623			−0.708117	+	0.706096i	$0.750455\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1350.34	−	779.619i	1.63282	−	0.942707i	0.649597	−	0.760278i	$-0.274937\pi$
$827$	1350.34	−	779.619i	1.63282	−	0.942707i	0.983219	−	0.182429i	$-0.0583959\pi$
$828$		0			0
$829$	−1328.51			−1.60255			−0.801275	−	0.598297i	$-0.795844\pi$
$829$	−1328.51			−1.60255			−0.801275	+	0.598297i	$0.795844\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−309.912	+	178.928i	−0.372044	+	0.214799i
$834$		0			0
$835$	172.264	+	298.370i	0.206304	+	0.357329i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			140.127i			0.167017i	0.996507	+	0.0835084i	$0.0266125\pi$
$839$			140.127i			0.167017i	−0.996507	+	0.0835084i	$0.973387\pi$
$840$		0			0
$841$	−407.947	−	706.584i	−0.485073	−	0.840172i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1012.49	−	584.562i	1.19821	−	0.691790i
$846$		0			0
$847$	1025.37			1.21059
$848$		0			0
$849$		0			0
$850$		0			0
$851$			752.879i			0.884699i
$852$		0			0
$853$	−372.208			−0.436351			−0.218176	−	0.975910i	$-0.570011\pi$
$853$	−372.208			−0.436351			−0.218176	+	0.975910i	$0.570011\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1289.23i		−	1.50435i	−0.658964	−	0.752175i	$-0.729005\pi$
$857$		−	1289.23i		−	1.50435i	0.658964	−	0.752175i	$-0.270995\pi$
$858$		0			0
$859$	−346.837			−0.403769			−0.201884	−	0.979409i	$-0.564707\pi$
$859$	−346.837			−0.403769			−0.201884	+	0.979409i	$0.564707\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	450.329i		−	0.521818i	−0.965363	−	0.260909i	$-0.915978\pi$
$863$		−	450.329i		−	0.521818i	0.965363	−	0.260909i	$-0.0840222\pi$
$864$		0			0
$865$	−828.601	−	1435.18i	−0.957920	−	1.65917i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1001.13	−	578.000i	1.15204	−	0.665132i
$870$		0			0
$871$	−136.222			−0.156397
$872$		0			0
$873$		0			0
$874$		0			0
$875$	17.3322	−	10.0067i	0.0198082	−	0.0114363i
$876$		0			0
$877$	752.685	+	1303.69i	0.858250	+	1.48653i	0.873597	+	0.486650i	$0.161781\pi$
$877$	752.685	+	1303.69i	0.858250	+	1.48653i	−0.0153476	+	0.999882i	$0.504885\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	42.9592i		−	0.0487619i	−0.999703	−	0.0243809i	$-0.992239\pi$
$881$		−	42.9592i		−	0.0487619i	0.999703	−	0.0243809i	$-0.00776147\pi$
$882$		0			0
$883$	603.628	+	1045.51i	0.683610	+	1.18405i	0.973871	+	0.227100i	$0.0729243\pi$
$883$	603.628	+	1045.51i	0.683610	+	1.18405i	−0.290262	+	0.956947i	$0.593742\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			970.655i			1.09431i	0.837031	+	0.547156i	$0.184290\pi$
$887$			970.655i			1.09431i	−0.837031	+	0.547156i	$0.815710\pi$
$888$		0			0
$889$	433.520	−	750.879i	0.487649	−	0.844633i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−21.2674	+	728.620i	−0.0238157	+	0.815924i
$894$		0			0
$895$	1038.48	+	1798.70i	1.16031	+	2.00972i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	221.246	−	127.736i	0.246102	−	0.142087i
$900$		0			0
$901$	−370.120			−0.410788
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−208.100	−	120.147i	−0.229945	−	0.132759i
$906$		0			0
$907$	67.9690			0.0749383			0.0374692	−	0.999298i	$-0.488070\pi$
$907$	67.9690			0.0749383			0.0374692	+	0.999298i	$0.488070\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−583.794	+	337.054i	−0.640828	+	0.369982i	−0.784933	−	0.619580i	$-0.787303\pi$
$911$	−583.794	+	337.054i	−0.640828	+	0.369982i	0.144106	+	0.989562i	$0.453969\pi$
$912$		0			0
$913$	842.548	−	1459.34i	0.922835	−	1.59840i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	957.191	+	552.635i	1.04383	+	0.602655i
$918$		0			0
$919$	−1229.42			−1.33778			−0.668892	−	0.743360i	$-0.733231\pi$
$919$	−1229.42			−1.33778			−0.668892	+	0.743360i	$0.733231\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−196.912	−	113.687i	−0.213339	−	0.123171i
$924$		0			0
$925$	−375.793	+	650.892i	−0.406262	+	0.703667i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	230.413i		−	0.248023i	−0.992281	−	0.124012i	$-0.960424\pi$
$929$		−	230.413i		−	0.248023i	0.992281	−	0.124012i	$-0.0395760\pi$
$930$		0			0
$931$	267.347	+	7.80349i	0.287161	+	0.00838184i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2658.69	+	1534.99i	−2.84351	+	1.64170i
$936$		0			0
$937$	−78.2109	−	135.465i	−0.0834695	−	0.144573i	0.821269	−	0.570542i	$-0.193267\pi$
$937$	−78.2109	−	135.465i	−0.0834695	−	0.144573i	−0.904738	+	0.425968i	$0.859933\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	105.417i		−	0.112027i	−0.998430	−	0.0560133i	$-0.982161\pi$
$941$		−	105.417i		−	0.112027i	0.998430	−	0.0560133i	$-0.0178389\pi$
$942$		0			0
$943$	876.031	+	1517.33i	0.928984	+	1.60905i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	428.358i		−	0.452331i	−0.974089	−	0.226166i	$-0.927381\pi$
$947$		−	428.358i		−	0.452331i	0.974089	−	0.226166i	$-0.0726190\pi$
$948$		0			0
$949$	0.731490	−	1.26698i	0.000770801	−	0.00133507i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1015.22	−	586.140i	−1.06529	−	0.615047i	−0.138401	−	0.990376i	$-0.544196\pi$
$953$	−1015.22	−	586.140i	−1.06529	−	0.615047i	−0.926891	+	0.375330i	$0.877530\pi$
$954$		0			0
$955$	1147.27	−	1987.12i	1.20133	−	2.08076i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	281.271i		−	0.293296i
$960$		0			0
$961$	−819.279	−	1419.03i	−0.852527	−	1.47662i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−661.300	−	381.801i	−0.685285	−	0.395649i
$966$		0			0
$967$	−322.838	−	559.172i	−0.333856	−	0.578255i	0.649409	−	0.760440i	$-0.275016\pi$
$967$	−322.838	−	559.172i	−0.333856	−	0.578255i	−0.983264	+	0.182185i	$0.941683\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−213.035	−	122.996i	−0.219398	−	0.126669i	0.386274	−	0.922384i	$-0.373762\pi$
$971$	−213.035	−	122.996i	−0.219398	−	0.126669i	−0.605671	+	0.795715i	$0.707095\pi$
$972$		0			0
$973$	383.804	+	664.768i	0.394454	+	0.683215i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1330.58	+	768.213i	1.36191	+	0.786297i	0.989878	−	0.141923i	$-0.0453287\pi$
$977$	1330.58	+	768.213i	1.36191	+	0.786297i	0.372030	+	0.928221i	$0.378662\pi$
$978$		0			0
$979$	286.913	−	496.948i	0.293067	−	0.507608i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1091.10i		−	1.10997i	−0.831860	−	0.554985i	$-0.812724\pi$
$983$		−	1091.10i		−	1.10997i	0.831860	−	0.554985i	$-0.187276\pi$
$984$		0			0
$985$	−219.468	−	380.129i	−0.222810	−	0.385918i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	131.673i		−	0.133138i
$990$		0			0
$991$	56.2143	−	97.3661i	0.0567248	−	0.0982503i	−0.836269	−	0.548320i	$-0.815268\pi$
$991$	56.2143	−	97.3661i	0.0567248	−	0.0982503i	0.892993	+	0.450070i	$0.148601\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	812.942	+	469.352i	0.817027	+	0.471711i
$996$		0			0
$997$	−144.320	+	249.970i	−0.144754	+	0.250722i	−0.929281	−	0.369373i	$-0.879573\pi$
$997$	−144.320	+	249.970i	−0.144754	+	0.250722i	0.784527	+	0.620095i	$0.212906\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.be.a.125.4		80
3.2	odd	2		684.3.be.a.581.13	yes	80
9.2	odd	6		2052.3.m.a.1493.37		80
9.7	even	3		684.3.m.a.353.40	✓	80
19.7	even	3		2052.3.m.a.881.4		80
57.26	odd	6		684.3.m.a.653.40	yes	80
171.7	even	3		684.3.be.a.425.13	yes	80
171.83	odd	6	inner	2052.3.be.a.197.4		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.40	✓	80	9.7	even	3
684.3.m.a.653.40	yes	80	57.26	odd	6
684.3.be.a.425.13	yes	80	171.7	even	3
684.3.be.a.581.13	yes	80	3.2	odd	2
2052.3.m.a.881.4		80	19.7	even	3
2052.3.m.a.1493.37		80	9.2	odd	6
2052.3.be.a.125.4		80	1.1	even	1	trivial
2052.3.be.a.197.4		80	171.83	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.be (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.09418 + 3.51848i) q^{5} +(2.95479 + 5.11784i) q^{7} +O(q^{10})\)	\(q+(-6.09418 + 3.51848i) q^{5} +(2.95479 + 5.11784i) q^{7} +(-14.8621 + 8.58065i) q^{11} -1.69095 q^{13} +(-22.0157 - 12.7108i) q^{17} +(9.01588 + 16.7247i) q^{19} -24.5609i q^{23} +(12.2594 - 21.2339i) q^{25} +(-4.33936 - 2.50533i) q^{29} +(-25.4929 + 44.1550i) q^{31} +(-36.0140 - 20.7927i) q^{35} -30.6535 q^{37} +(-61.7782 + 35.6677i) q^{41} +5.36109 q^{43} +(33.2249 + 19.1824i) q^{47} +(7.03844 - 12.1909i) q^{49} +(12.6088 - 7.27967i) q^{53} +(60.3817 - 104.584i) q^{55} +(40.9541 - 23.6449i) q^{59} +(47.3668 - 82.0417i) q^{61} +(10.3050 - 5.94958i) q^{65} +80.5590 q^{67} +(116.450 + 67.2326i) q^{71} +(-0.432590 + 0.749268i) q^{73} +(-87.8289 - 50.7080i) q^{77} -67.3609 q^{79} +(-85.0365 + 49.0958i) q^{83} +178.890 q^{85} +(-28.9575 + 16.7186i) q^{89} +(-4.99641 - 8.65404i) q^{91} +(-113.790 - 70.2009i) q^{95} +7.99364 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Introduction

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