LMFDB - Embedded newform 2052.3.m.a.881.3

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			881.3
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.26979i q^{5} +(3.75776 + 6.50863i) q^{7} +O(q^{10})$	$q-7.26979i q^{5} +(3.75776 + 6.50863i) q^{7} +(14.6898 - 8.48117i) q^{11} +(10.1476 + 17.5762i) q^{13} +(-12.9313 + 7.46590i) q^{17} +(-17.9208 + 6.31229i) q^{19} +(-28.8029 + 16.6293i) q^{23} -27.8498 q^{25} +30.3086i q^{29} +(-23.8575 + 41.3223i) q^{31} +(47.3163 - 27.3181i) q^{35} +25.1099 q^{37} -5.04512i q^{41} +(-24.8411 + 43.0261i) q^{43} +62.4682i q^{47} +(-3.74149 + 6.48044i) q^{49} +(-47.5227 - 27.4372i) q^{53} +(-61.6563 - 106.792i) q^{55} -73.6284i q^{59} -95.3512 q^{61} +(127.775 - 73.7709i) q^{65} +(0.588665 + 1.01960i) q^{67} +(12.3658 - 7.13938i) q^{71} +(-59.5390 - 103.125i) q^{73} +(110.401 + 63.7403i) q^{77} +(0.0750630 - 0.130013i) q^{79} +(41.9662 - 24.2292i) q^{83} +(54.2755 + 94.0079i) q^{85} +(-67.4096 - 38.9190i) q^{89} +(-76.2645 + 132.094i) q^{91} +(45.8890 + 130.280i) q^{95} +(-47.2818 + 81.8945i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	7.26979i		−	1.45396i	−0.686660	−	0.726979i	$-0.740924\pi$
$5$		−	7.26979i		−	1.45396i	0.686660	−	0.726979i	$-0.259076\pi$
$6$		0			0
$7$	3.75776	+	6.50863i	0.536823	+	0.929804i	0.999073	+	0.0430544i	$0.0137089\pi$
$7$	3.75776	+	6.50863i	0.536823	+	0.929804i	−0.462250	+	0.886750i	$0.652958\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	14.6898	−	8.48117i	1.33544	−	0.771015i	0.349310	−	0.937007i	$-0.386416\pi$
$11$	14.6898	−	8.48117i	1.33544	−	0.771015i	0.986127	+	0.165992i	$0.0530826\pi$
$12$		0			0
$13$	10.1476	+	17.5762i	0.780585	+	1.35201i	0.931601	+	0.363482i	$0.118412\pi$
$13$	10.1476	+	17.5762i	0.780585	+	1.35201i	−0.151016	+	0.988531i	$0.548255\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.9313	+	7.46590i	−0.760666	+	0.439170i	−0.829535	−	0.558455i	$-0.811394\pi$
$17$	−12.9313	+	7.46590i	−0.760666	+	0.439170i	0.0688691	+	0.997626i	$0.478061\pi$
$18$		0			0
$19$	−17.9208	+	6.31229i	−0.943200	+	0.332226i
$19$	−17.9208	+	6.31229i	−0.943200	+	0.332226i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−28.8029	+	16.6293i	−1.25230	+	0.723015i	−0.971565	−	0.236771i	$-0.923911\pi$
$23$	−28.8029	+	16.6293i	−1.25230	+	0.723015i	−0.280733	+	0.959786i	$0.590578\pi$
$24$		0			0
$25$	−27.8498			−1.11399
$26$		0			0
$27$		0			0
$28$		0			0
$29$			30.3086i			1.04512i	0.852601	+	0.522562i	$0.175024\pi$
$29$			30.3086i			1.04512i	−0.852601	+	0.522562i	$0.824976\pi$
$30$		0			0
$31$	−23.8575	+	41.3223i	−0.769596	+	1.33298i	0.168187	+	0.985755i	$0.446209\pi$
$31$	−23.8575	+	41.3223i	−0.769596	+	1.33298i	−0.937782	+	0.347224i	$0.887124\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	47.3163	−	27.3181i	1.35190	−	0.780517i
$36$		0			0
$37$	25.1099			0.678647			0.339324	−	0.940670i	$-0.389802\pi$
$37$	25.1099			0.678647			0.339324	+	0.940670i	$0.389802\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.04512i		−	0.123052i	−0.998105	−	0.0615258i	$-0.980403\pi$
$41$		−	5.04512i		−	0.123052i	0.998105	−	0.0615258i	$-0.0195967\pi$
$42$		0			0
$43$	−24.8411	+	43.0261i	−0.577701	+	1.00061i	0.418042	+	0.908428i	$0.362717\pi$
$43$	−24.8411	+	43.0261i	−0.577701	+	1.00061i	−0.995742	+	0.0921791i	$0.970617\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			62.4682i			1.32911i	0.747239	+	0.664555i	$0.231379\pi$
$47$			62.4682i			1.32911i	−0.747239	+	0.664555i	$0.768621\pi$
$48$		0			0
$49$	−3.74149	+	6.48044i	−0.0763569	+	0.132254i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−47.5227	−	27.4372i	−0.896654	−	0.517684i	−0.0205412	−	0.999789i	$-0.506539\pi$
$53$	−47.5227	−	27.4372i	−0.896654	−	0.517684i	−0.876113	+	0.482105i	$0.839872\pi$
$54$		0			0
$55$	−61.6563	−	106.792i	−1.12102	−	1.94167i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	73.6284i		−	1.24794i	−0.781449	−	0.623969i	$-0.785519\pi$
$59$		−	73.6284i		−	1.24794i	0.781449	−	0.623969i	$-0.214481\pi$
$60$		0			0
$61$	−95.3512			−1.56314			−0.781568	−	0.623821i	$-0.785580\pi$
$61$	−95.3512			−1.56314			−0.781568	+	0.623821i	$0.785580\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	127.775	−	73.7709i	1.96577	−	1.13494i
$66$		0			0
$67$	0.588665	+	1.01960i	0.00878605	+	0.0152179i	0.870385	−	0.492372i	$-0.163870\pi$
$67$	0.588665	+	1.01960i	0.00878605	+	0.0152179i	−0.861599	+	0.507590i	$0.830537\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.3658	−	7.13938i	0.174166	−	0.100555i	−0.410383	−	0.911913i	$-0.634605\pi$
$71$	12.3658	−	7.13938i	0.174166	−	0.100555i	0.584549	+	0.811359i	$0.301272\pi$
$72$		0			0
$73$	−59.5390	−	103.125i	−0.815603	−	1.41267i	−0.908894	−	0.417027i	$-0.863072\pi$
$73$	−59.5390	−	103.125i	−0.815603	−	1.41267i	0.0932911	−	0.995639i	$-0.470261\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	110.401	+	63.7403i	1.43379	+	0.827796i
$78$		0			0
$79$	0.0750630	−	0.130013i	0.000950165	−	0.00164573i	−0.865550	−	0.500823i	$-0.833031\pi$
$79$	0.0750630	−	0.130013i	0.000950165	−	0.00164573i	0.866500	+	0.499177i	$0.166364\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	41.9662	−	24.2292i	0.505617	−	0.291918i	−0.225413	−	0.974263i	$-0.572373\pi$
$83$	41.9662	−	24.2292i	0.505617	−	0.291918i	0.731030	+	0.682345i	$0.239040\pi$
$84$		0			0
$85$	54.2755	+	94.0079i	0.638535	+	1.10598i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−67.4096	−	38.9190i	−0.757412	−	0.437292i	0.0709540	−	0.997480i	$-0.477396\pi$
$89$	−67.4096	−	38.9190i	−0.757412	−	0.437292i	−0.828366	+	0.560188i	$0.810729\pi$
$90$		0			0
$91$	−76.2645	+	132.094i	−0.838071	+	1.45158i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	45.8890	+	130.280i	0.483042	+	1.37137i
$96$		0			0
$97$	−47.2818	+	81.8945i	−0.487441	+	0.844273i	−0.999896	−	0.0144414i	$-0.995403\pi$
$97$	−47.2818	+	81.8945i	−0.487441	+	0.844273i	0.512455	+	0.858714i	$0.328736\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			84.7721i			0.839328i	0.907680	+	0.419664i	$0.137852\pi$
$101$			84.7721i			0.839328i	−0.907680	+	0.419664i	$0.862148\pi$
$102$		0			0
$103$	−3.50170	+	6.06512i	−0.0339971	+	0.0588846i	−0.882523	−	0.470269i	$-0.844157\pi$
$103$	−3.50170	+	6.06512i	−0.0339971	+	0.0588846i	0.848526	+	0.529153i	$0.177490\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			97.9295i			0.915229i	0.889151	+	0.457614i	$0.151296\pi$
$107$			97.9295i			0.915229i	−0.889151	+	0.457614i	$0.848704\pi$
$108$		0			0
$109$	75.3829	+	130.567i	0.691587	+	1.19786i	0.971318	+	0.237785i	$0.0764213\pi$
$109$	75.3829	+	130.567i	0.691587	+	1.19786i	−0.279731	+	0.960078i	$0.590245\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−39.6676	−	22.9021i	−0.351041	−	0.202674i	0.314103	−	0.949389i	$-0.398296\pi$
$113$	−39.6676	−	22.9021i	−0.351041	−	0.202674i	−0.665144	+	0.746715i	$0.731630\pi$
$114$		0			0
$115$	120.892	+	209.391i	1.05123	+	1.82079i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−97.1855	−	56.1101i	−0.816685	−	0.471513i
$120$		0			0
$121$	83.3603	−	144.384i	0.688928	−	1.19326i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			20.7173i			0.165739i
$126$		0			0
$127$	−14.0912	+	24.4067i	−0.110955	+	0.192179i	−0.916155	−	0.400823i	$-0.868724\pi$
$127$	−14.0912	+	24.4067i	−0.110955	+	0.192179i	0.805201	+	0.593002i	$0.202058\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			252.087i			1.92433i	0.272466	+	0.962166i	$0.412161\pi$
$131$			252.087i			1.92433i	−0.272466	+	0.962166i	$0.587839\pi$
$132$		0			0
$133$	−108.426	−	92.9197i	−0.815236	−	0.698645i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	86.3432i		−	0.630242i	−0.949051	−	0.315121i	$-0.897955\pi$
$137$		−	86.3432i		−	0.630242i	0.949051	−	0.315121i	$-0.102045\pi$
$138$		0			0
$139$	−82.7671	−	143.357i	−0.595447	−	1.03134i	−0.993484	−	0.113974i	$-0.963642\pi$
$139$	−82.7671	−	143.357i	−0.595447	−	1.03134i	0.398037	−	0.917369i	$-0.369692\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	298.133	+	172.127i	2.08484	+	1.20369i
$144$		0			0
$145$	220.337			1.51957
$146$		0			0
$147$		0			0
$148$		0			0
$149$			196.064i			1.31587i	0.753076	+	0.657934i	$0.228569\pi$
$149$			196.064i			1.31587i	−0.753076	+	0.657934i	$0.771431\pi$
$150$		0			0
$151$	0.412875	+	0.715121i	0.00273427	+	0.00473590i	0.867389	−	0.497630i	$-0.165796\pi$
$151$	0.412875	+	0.715121i	0.00273427	+	0.00473590i	−0.864655	+	0.502366i	$0.832463\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	300.405	+	173.439i	1.93809	+	1.11896i
$156$		0			0
$157$	−128.511			−0.818541			−0.409271	−	0.912413i	$-0.634217\pi$
$157$	−128.511			−0.818541			−0.409271	+	0.912413i	$0.634217\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−216.468	−	124.978i	−1.34452	−	0.776261i
$162$		0			0
$163$	236.259			1.44944			0.724722	−	0.689041i	$-0.241968\pi$
$163$	236.259			1.44944			0.724722	+	0.689041i	$0.241968\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−144.149	+	83.2244i	−0.863167	+	0.498350i	−0.865072	−	0.501648i	$-0.832727\pi$
$167$	−144.149	+	83.2244i	−0.863167	+	0.498350i	0.00190452	+	0.999998i	$0.499394\pi$
$168$		0			0
$169$	−121.448	+	210.354i	−0.718626	+	1.24470i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	58.6776	+	33.8776i	0.339177	+	0.195824i	0.659908	−	0.751346i	$-0.270595\pi$
$173$	58.6776	+	33.8776i	0.339177	+	0.195824i	−0.320731	+	0.947170i	$0.603929\pi$
$174$		0			0
$175$	−104.653	−	181.264i	−0.598016	−	1.03579i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	60.5311i		−	0.338163i	−0.985602	−	0.169081i	$-0.945920\pi$
$179$		−	60.5311i		−	0.338163i	0.985602	−	0.169081i	$-0.0540801\pi$
$180$		0			0
$181$	83.4816	−	144.594i	0.461224	−	0.798864i	−0.537798	−	0.843074i	$-0.680744\pi$
$181$	83.4816	−	144.594i	0.461224	−	0.798864i	0.999022	+	0.0442096i	$0.0140769\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	182.544i		−	0.986724i
$186$		0			0
$187$	−126.639	+	219.345i	−0.677214	+	1.17297i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−133.030	+	76.8050i	−0.696493	+	0.402120i	−0.806040	−	0.591861i	$-0.798393\pi$
$191$	−133.030	+	76.8050i	−0.696493	+	0.402120i	0.109547	+	0.993982i	$0.465060\pi$
$192$		0			0
$193$	238.302			1.23473			0.617364	−	0.786678i	$-0.288201\pi$
$193$	238.302			1.23473			0.617364	+	0.786678i	$0.288201\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	109.602i		−	0.556357i	−0.960529	−	0.278178i	$-0.910269\pi$
$197$		−	109.602i		−	0.556357i	0.960529	−	0.278178i	$-0.0897306\pi$
$198$		0			0
$199$	−126.692	+	219.437i	−0.636643	+	1.10270i	0.349522	+	0.936928i	$0.386344\pi$
$199$	−126.692	+	219.437i	−0.636643	+	1.10270i	−0.986165	+	0.165770i	$0.946989\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−197.267	+	113.892i	−0.971760	+	0.561046i
$204$		0			0
$205$	−36.6769			−0.178912
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−209.718	+	244.716i	−1.00343	+	1.17089i
$210$		0			0
$211$	394.462			1.86949			0.934745	−	0.355320i	$-0.115628\pi$
$211$	394.462			1.86949			0.934745	+	0.355320i	$0.115628\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	312.791	+	180.590i	1.45484	+	0.839952i
$216$		0			0
$217$	−358.602			−1.65255
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−262.444	−	151.522i	−1.18753	−	0.685620i
$222$		0			0
$223$	97.6081	−	169.062i	0.437705	−	0.758127i	−0.559807	−	0.828623i	$-0.689125\pi$
$223$	97.6081	−	169.062i	0.437705	−	0.758127i	0.997512	+	0.0704962i	$0.0224583\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	253.263	−	146.222i	1.11570	−	0.644148i	0.175398	−	0.984498i	$-0.443879\pi$
$227$	253.263	−	146.222i	1.11570	−	0.644148i	0.940299	+	0.340349i	$0.110545\pi$
$228$		0			0
$229$	−70.4763	+	122.069i	−0.307757	+	0.533050i	−0.977871	−	0.209207i	$-0.932912\pi$
$229$	−70.4763	+	122.069i	−0.307757	+	0.533050i	0.670114	+	0.742258i	$0.266245\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	198.955	−	114.867i	0.853885	−	0.492991i	−0.00807462	−	0.999967i	$-0.502570\pi$
$233$	198.955	−	114.867i	0.853885	−	0.492991i	0.861960	+	0.506977i	$0.169237\pi$
$234$		0			0
$235$	454.130			1.93247
$236$		0			0
$237$		0			0
$238$		0			0
$239$	203.641	+	117.572i	0.852053	+	0.491933i	0.861343	−	0.508024i	$-0.169624\pi$
$239$	203.641	+	117.572i	0.852053	+	0.491933i	−0.00929033	+	0.999957i	$0.502957\pi$
$240$		0			0
$241$	69.1883			0.287089			0.143544	−	0.989644i	$-0.454150\pi$
$241$	69.1883			0.287089			0.143544	+	0.989644i	$0.454150\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	47.1114	+	27.1998i	0.192292	+	0.111020i
$246$		0			0
$247$	−292.799	−	250.924i	−1.18542	−	1.01589i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	18.6371	+	10.7602i	0.0742515	+	0.0428692i	0.536666	−	0.843795i	$-0.319683\pi$
$251$	18.6371	+	10.7602i	0.0742515	+	0.0428692i	−0.462415	+	0.886664i	$0.653017\pi$
$252$		0			0
$253$	−282.072	+	488.564i	−1.11491	+	1.93108i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	252.333	−	145.684i	0.981839	−	0.566865i	0.0790144	−	0.996873i	$-0.474823\pi$
$257$	252.333	−	145.684i	0.981839	−	0.566865i	0.902825	+	0.430008i	$0.141489\pi$
$258$		0			0
$259$	94.3571	+	163.431i	0.364313	+	0.631009i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	140.817	+	81.3007i	0.535426	+	0.309128i	0.743223	−	0.669044i	$-0.233296\pi$
$263$	140.817	+	81.3007i	0.535426	+	0.309128i	−0.207797	+	0.978172i	$0.566629\pi$
$264$		0			0
$265$	−199.463	+	345.480i	−0.752690	+	1.30370i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−29.2134	+	16.8664i	−0.108600	+	0.0627003i	−0.553316	−	0.832971i	$-0.686638\pi$
$269$	−29.2134	+	16.8664i	−0.108600	+	0.0627003i	0.444716	+	0.895672i	$0.353305\pi$
$270$		0			0
$271$	−88.5330	−	153.344i	−0.326690	−	0.565844i	0.655163	−	0.755488i	$-0.272600\pi$
$271$	−88.5330	−	153.344i	−0.326690	−	0.565844i	−0.981853	+	0.189644i	$0.939267\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−409.108	+	236.199i	−1.48767	+	0.858904i
$276$		0			0
$277$	88.8575	+	153.906i	0.320785	+	0.555616i	0.980650	−	0.195768i	$-0.0627199\pi$
$277$	88.8575	+	153.906i	0.320785	+	0.555616i	−0.659865	+	0.751384i	$0.729387\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	336.283i		−	1.19674i	−0.801221	−	0.598369i	$-0.795816\pi$
$281$		−	336.283i		−	1.19674i	0.801221	−	0.598369i	$-0.204184\pi$
$282$		0			0
$283$	31.1149			0.109947			0.0549733	−	0.998488i	$-0.482493\pi$
$283$	31.1149			0.109947			0.0549733	+	0.998488i	$0.482493\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	32.8368	−	18.9583i	0.114414	−	0.0660569i
$288$		0			0
$289$	−33.0207	+	57.1936i	−0.114259	+	0.197902i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−180.338	−	104.118i	−0.615487	−	0.355351i	0.159623	−	0.987178i	$-0.448972\pi$
$293$	−180.338	−	104.118i	−0.615487	−	0.355351i	−0.775110	+	0.631827i	$0.782305\pi$
$294$		0			0
$295$	−535.263			−1.81445
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−584.560	−	337.496i	−1.95505	−	1.12875i
$300$		0			0
$301$	−373.388			−1.24049
$302$		0			0
$303$		0			0
$304$		0			0
$305$			693.183i			2.27273i
$306$		0			0
$307$	251.455	+	435.532i	0.819071	+	1.41867i	0.906368	+	0.422489i	$0.138844\pi$
$307$	251.455	+	435.532i	0.819071	+	1.41867i	−0.0872974	+	0.996182i	$0.527823\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−241.751	−	139.575i	−0.777334	−	0.448794i	0.0581507	−	0.998308i	$-0.481480\pi$
$311$	−241.751	−	139.575i	−0.777334	−	0.448794i	−0.835485	+	0.549514i	$0.814813\pi$
$312$		0			0
$313$	−113.259			−0.361850			−0.180925	−	0.983497i	$-0.557909\pi$
$313$	−113.259			−0.361850			−0.180925	+	0.983497i	$0.557909\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	109.393i		−	0.345089i	−0.985002	−	0.172544i	$-0.944801\pi$
$317$		−	109.393i		−	0.345089i	0.985002	−	0.172544i	$-0.0551988\pi$
$318$		0			0
$319$	257.052	+	445.227i	0.805806	+	1.39570i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	184.613	−	215.421i	0.571556	−	0.666938i
$324$		0			0
$325$	−282.609	−	489.492i	−0.869565	−	1.50613i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−406.582	+	234.740i	−1.23581	+	0.713497i
$330$		0			0
$331$	−50.5317	−	87.5235i	−0.152664	−	0.264422i	0.779542	−	0.626350i	$-0.215452\pi$
$331$	−50.5317	−	87.5235i	−0.152664	−	0.264422i	−0.932206	+	0.361928i	$0.882119\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.41226	−	4.27947i	0.0221262	−	0.0127745i
$336$		0			0
$337$	−560.509			−1.66323			−0.831616	−	0.555351i	$-0.812584\pi$
$337$	−560.509			−1.66323			−0.831616	+	0.555351i	$0.812584\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			809.357i			2.37348i
$342$		0			0
$343$	312.022			0.909685
$344$		0			0
$345$		0			0
$346$		0			0
$347$			106.009i			0.305501i	0.988265	+	0.152751i	$0.0488132\pi$
$347$			106.009i			0.305501i	−0.988265	+	0.152751i	$0.951187\pi$
$348$		0			0
$349$	120.332	+	208.421i	0.344791	+	0.597196i	0.985316	−	0.170742i	$-0.0546164\pi$
$349$	120.332	+	208.421i	0.344791	+	0.597196i	−0.640525	+	0.767938i	$0.721283\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	438.326	−	253.068i	1.24172	−	0.716906i	0.272274	−	0.962220i	$-0.412224\pi$
$353$	438.326	−	253.068i	1.24172	−	0.716906i	0.969444	+	0.245314i	$0.0788909\pi$
$354$		0			0
$355$	−51.9018	−	89.8965i	−0.146202	−	0.253229i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	481.722	−	278.122i	1.34184	−	0.774714i	0.354766	−	0.934955i	$-0.384561\pi$
$359$	481.722	−	278.122i	1.34184	−	0.774714i	0.987078	+	0.160242i	$0.0512274\pi$
$360$		0			0
$361$	281.310	−	226.243i	0.779252	−	0.626711i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−749.694	+	432.836i	−2.05396	+	1.18585i
$366$		0			0
$367$	608.527			1.65811			0.829056	−	0.559165i	$-0.188878\pi$
$367$	608.527			1.65811			0.829056	+	0.559165i	$0.188878\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	412.410i		−	1.11162i
$372$		0			0
$373$	61.6756	−	106.825i	0.165350	−	0.286395i	−0.771429	−	0.636315i	$-0.780458\pi$
$373$	61.6756	−	106.825i	0.165350	−	0.286395i	0.936780	+	0.349920i	$0.113791\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−532.709	+	307.560i	−1.41302	+	0.815808i
$378$		0			0
$379$	−358.690			−0.946412			−0.473206	−	0.880952i	$-0.656903\pi$
$379$	−358.690			−0.946412			−0.473206	+	0.880952i	$0.656903\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	366.840i		−	0.957806i	−0.877868	−	0.478903i	$-0.841035\pi$
$383$		−	366.840i		−	0.957806i	0.877868	−	0.478903i	$-0.158965\pi$
$384$		0			0
$385$	463.379	−	802.595i	1.20358	−	2.08466i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			190.120i			0.488740i	0.969682	+	0.244370i	$0.0785812\pi$
$389$			190.120i			0.488740i	−0.969682	+	0.244370i	$0.921419\pi$
$390$		0			0
$391$	248.306	−	430.078i	0.635054	−	1.09995i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.945166	−	0.545692i	−0.00239283	−	0.00138150i
$396$		0			0
$397$	115.434	+	199.937i	0.290765	+	0.503619i	0.973991	−	0.226588i	$-0.0727570\pi$
$397$	115.434	+	199.937i	0.290765	+	0.503619i	−0.683226	+	0.730207i	$0.739424\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			439.408i			1.09578i	0.836551	+	0.547890i	$0.184569\pi$
$401$			439.408i			1.09578i	−0.836551	+	0.547890i	$0.815431\pi$
$402$		0			0
$403$	−968.385			−2.40294
$404$		0			0
$405$		0			0
$406$		0			0
$407$	368.860	−	212.962i	0.906291	−	0.523247i
$408$		0			0
$409$	131.307	+	227.431i	0.321045	+	0.556065i	0.980704	−	0.195499i	$-0.0626328\pi$
$409$	131.307	+	227.431i	0.321045	+	0.556065i	−0.659659	+	0.751565i	$0.729299\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	479.220	−	276.678i	1.16034	−	0.669922i
$414$		0			0
$415$	−176.141	−	305.086i	−0.424437	−	0.735146i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−490.534	−	283.210i	−1.17072	−	0.675918i	−0.216874	−	0.976200i	$-0.569586\pi$
$419$	−490.534	−	283.210i	−1.17072	−	0.675918i	−0.953851	+	0.300281i	$0.902919\pi$
$420$		0			0
$421$	239.616	−	415.027i	0.569158	−	0.985811i	−0.427491	−	0.904020i	$-0.640602\pi$
$421$	239.616	−	415.027i	0.569158	−	0.985811i	0.996649	−	0.0817918i	$-0.0260642\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	360.134	−	207.924i	0.847375	−	0.489232i
$426$		0			0
$427$	−358.307	−	620.606i	−0.839126	−	1.45341i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−480.030	−	277.145i	−1.11376	−	0.643029i	−0.173958	−	0.984753i	$-0.555656\pi$
$431$	−480.030	−	277.145i	−1.11376	−	0.643029i	−0.939800	+	0.341724i	$0.888989\pi$
$432$		0			0
$433$	−116.826	+	202.348i	−0.269805	+	0.467316i	−0.968811	−	0.247799i	$-0.920293\pi$
$433$	−116.826	+	202.348i	−0.269805	+	0.467316i	0.699006	+	0.715116i	$0.253626\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	411.201	−	479.823i	0.940964	−	1.09799i
$438$		0			0
$439$	−100.290	+	173.707i	−0.228451	+	0.395688i	−0.957349	−	0.288934i	$-0.906699\pi$
$439$	−100.290	+	173.707i	−0.228451	+	0.395688i	0.728898	+	0.684622i	$0.240033\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	296.907i		−	0.670219i	−0.942179	−	0.335110i	$-0.891227\pi$
$443$		−	296.907i		−	0.670219i	0.942179	−	0.335110i	$-0.108773\pi$
$444$		0			0
$445$	−282.933	+	490.054i	−0.635804	+	1.10124i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			392.134i			0.873349i	0.899620	+	0.436675i	$0.143844\pi$
$449$			392.134i			0.873349i	−0.899620	+	0.436675i	$0.856156\pi$
$450$		0			0
$451$	−42.7885	−	74.1118i	−0.0948747	−	0.164328i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	960.295	+	554.426i	2.11054	+	1.21852i
$456$		0			0
$457$	101.699	+	176.148i	0.222537	+	0.385445i	0.955578	−	0.294740i	$-0.0952330\pi$
$457$	101.699	+	176.148i	0.222537	+	0.385445i	−0.733041	+	0.680184i	$0.761900\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	121.048	+	69.8873i	0.262578	+	0.151599i	0.625510	−	0.780216i	$-0.284891\pi$
$461$	121.048	+	69.8873i	0.262578	+	0.151599i	−0.362932	+	0.931816i	$0.618224\pi$
$462$		0			0
$463$	94.0240	−	162.854i	0.203076	−	0.351737i	−0.746442	−	0.665450i	$-0.768240\pi$
$463$	94.0240	−	162.854i	0.203076	−	0.351737i	0.949518	+	0.313713i	$0.101573\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	370.749i		−	0.793896i	−0.917841	−	0.396948i	$-0.870069\pi$
$467$		−	370.749i		−	0.793896i	0.917841	−	0.396948i	$-0.129931\pi$
$468$		0			0
$469$	−4.42412	+	7.66281i	−0.00943310	+	0.0163386i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			842.727i			1.78166i
$474$		0			0
$475$	499.090	−	175.796i	1.05072	−	0.370097i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	330.120i		−	0.689185i	−0.938752	−	0.344592i	$-0.888017\pi$
$479$		−	330.120i		−	0.689185i	0.938752	−	0.344592i	$-0.111983\pi$
$480$		0			0
$481$	254.806	+	441.337i	0.529742	+	0.917540i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	595.355	+	343.729i	1.22754	+	0.708719i
$486$		0			0
$487$	709.032			1.45592			0.727959	−	0.685621i	$-0.240469\pi$
$487$	709.032			1.45592			0.727959	+	0.685621i	$0.240469\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	363.330i		−	0.739980i	−0.929036	−	0.369990i	$-0.879361\pi$
$491$		−	363.330i		−	0.739980i	0.929036	−	0.369990i	$-0.120639\pi$
$492$		0			0
$493$	−226.281	−	391.930i	−0.458988	−	0.794990i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	92.9351	+	53.6561i	0.186992	+	0.107960i
$498$		0			0
$499$	−743.071			−1.48912			−0.744560	−	0.667556i	$-0.767341\pi$
$499$	−743.071			−1.48912			−0.744560	+	0.667556i	$0.767341\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	319.580	+	184.509i	0.635348	+	0.366818i	0.782820	−	0.622248i	$-0.213780\pi$
$503$	319.580	+	184.509i	0.635348	+	0.366818i	−0.147473	+	0.989066i	$0.547114\pi$
$504$		0			0
$505$	616.275			1.22035
$506$		0			0
$507$		0			0
$508$		0			0
$509$	106.451	−	61.4595i	0.209137	−	0.120746i	−0.391773	−	0.920062i	$-0.628138\pi$
$509$	106.451	−	61.4595i	0.209137	−	0.120746i	0.600910	+	0.799316i	$0.294805\pi$
$510$		0			0
$511$	447.466	−	775.035i	0.875668	−	1.51670i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	44.0921	+	25.4566i	0.0856157	+	0.0494303i
$516$		0			0
$517$	529.803	+	917.646i	1.02476	+	1.77494i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	883.174i		−	1.69515i	−0.530674	−	0.847576i	$-0.678061\pi$
$521$		−	883.174i		−	1.69515i	0.530674	−	0.847576i	$-0.321939\pi$
$522$		0			0
$523$	82.8317	−	143.469i	0.158378	−	0.274319i	−0.775906	−	0.630849i	$-0.782707\pi$
$523$	82.8317	−	143.469i	0.158378	−	0.274319i	0.934284	+	0.356530i	$0.116040\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	712.470i		−	1.35193i
$528$		0			0
$529$	288.570	−	499.818i	0.545501	−	0.944835i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	88.6738	−	51.1959i	0.166367	−	0.0960523i
$534$		0			0
$535$	711.926			1.33070
$536$		0			0
$537$		0			0
$538$		0			0
$539$			126.929i			0.235489i
$540$		0			0
$541$	−239.705	+	415.182i	−0.443079	+	0.767435i	−0.997916	−	0.0645241i	$-0.979447\pi$
$541$	−239.705	+	415.182i	−0.443079	+	0.767435i	0.554838	+	0.831959i	$0.312780\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	949.195	−	548.018i	1.74164	−	1.00554i
$546$		0			0
$547$	323.732			0.591831			0.295916	−	0.955214i	$-0.404375\pi$
$547$	323.732			0.591831			0.295916	+	0.955214i	$0.404375\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−191.317	−	543.154i	−0.347217	−	0.985761i
$552$		0			0
$553$	1.12827			0.00204028
$554$		0			0
$555$		0			0
$556$		0			0
$557$	139.331	+	80.4425i	0.250145	+	0.144421i	0.619830	−	0.784736i	$-0.287201\pi$
$557$	139.331	+	80.4425i	0.250145	+	0.144421i	−0.369686	+	0.929157i	$0.620535\pi$
$558$		0			0
$559$	−1008.31			−1.80378
$560$		0			0
$561$		0			0
$562$		0			0
$563$	351.195	+	202.763i	0.623793	+	0.360147i	0.778344	−	0.627838i	$-0.216060\pi$
$563$	351.195	+	202.763i	0.623793	+	0.360147i	−0.154551	+	0.987985i	$0.549393\pi$
$564$		0			0
$565$	−166.493	+	288.375i	−0.294679	+	0.510398i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−937.203	+	541.095i	−1.64711	+	0.950957i	−0.668892	+	0.743360i	$0.733231\pi$
$569$	−937.203	+	541.095i	−1.64711	+	0.950957i	−0.978214	+	0.207597i	$0.933436\pi$
$570$		0			0
$571$	−275.915	+	477.899i	−0.483214	+	0.836951i	−0.999814	−	0.0192758i	$-0.993864\pi$
$571$	−275.915	+	477.899i	−0.483214	+	0.836951i	0.516600	+	0.856227i	$0.327197\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	802.154	−	463.124i	1.39505	−	0.805432i
$576$		0			0
$577$	−7.66081			−0.0132770			−0.00663848	−	0.999978i	$-0.502113\pi$
$577$	−7.66081			−0.0132770			−0.00663848	+	0.999978i	$0.502113\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	315.398	+	182.095i	0.542854	+	0.313417i
$582$		0			0
$583$	−930.799			−1.59657
$584$		0			0
$585$		0			0
$586$		0			0
$587$	686.241	+	396.201i	1.16906	+	0.674960i	0.953459	−	0.301521i	$-0.0974944\pi$
$587$	686.241	+	396.201i	1.16906	+	0.674960i	0.215605	+	0.976481i	$0.430828\pi$
$588$		0			0
$589$	166.706	−	891.125i	0.283033	−	1.51295i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	541.981	+	312.913i	0.913965	+	0.527678i	0.881705	−	0.471802i	$-0.156396\pi$
$593$	541.981	+	312.913i	0.913965	+	0.527678i	0.0322602	+	0.999480i	$0.489729\pi$
$594$		0			0
$595$	−407.908	+	706.518i	−0.685560	+	1.18742i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	150.471	−	86.8742i	0.251203	−	0.145032i	−0.369112	−	0.929385i	$-0.620338\pi$
$599$	150.471	−	86.8742i	0.251203	−	0.145032i	0.620315	+	0.784353i	$0.287005\pi$
$600$		0			0
$601$	−138.816	−	240.436i	−0.230975	−	0.400060i	0.727120	−	0.686510i	$-0.240858\pi$
$601$	−138.816	−	240.436i	−0.230975	−	0.400060i	−0.958095	+	0.286450i	$0.907525\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1049.64	−	606.012i	−1.73495	−	1.00167i
$606$		0			0
$607$	236.591	−	409.787i	0.389770	−	0.675102i	−0.602648	−	0.798007i	$-0.705888\pi$
$607$	236.591	−	409.787i	0.389770	−	0.675102i	0.992418	+	0.122905i	$0.0392211\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1097.95	+	633.903i	−1.79697	+	1.03748i
$612$		0			0
$613$	412.386	+	714.274i	0.672735	+	1.16521i	0.977126	+	0.212663i	$0.0682137\pi$
$613$	412.386	+	714.274i	0.672735	+	1.16521i	−0.304391	+	0.952547i	$0.598453\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−449.661	+	259.612i	−0.728786	+	0.420765i	−0.817978	−	0.575250i	$-0.804905\pi$
$617$	−449.661	+	259.612i	−0.728786	+	0.420765i	0.0891918	+	0.996014i	$0.471572\pi$
$618$		0			0
$619$	515.153	+	892.272i	0.832235	+	1.44147i	0.896262	+	0.443525i	$0.146272\pi$
$619$	515.153	+	892.272i	0.832235	+	1.44147i	−0.0640275	+	0.997948i	$0.520395\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	584.992i		−	0.938992i
$624$		0			0
$625$	−545.634			−0.873015
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−324.705	+	187.468i	−0.516224	+	0.298042i
$630$		0			0
$631$	282.848	−	489.908i	0.448254	−	0.776399i	−0.550018	−	0.835153i	$-0.685379\pi$
$631$	282.848	−	489.908i	0.448254	−	0.776399i	0.998273	+	0.0587536i	$0.0187126\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	177.432	+	102.440i	0.279420	+	0.161323i
$636$		0			0
$637$	−151.868			−0.238412
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−427.649	−	246.903i	−0.667159	−	0.385184i	0.127840	−	0.991795i	$-0.459195\pi$
$641$	−427.649	−	246.903i	−0.667159	−	0.385184i	−0.794999	+	0.606610i	$0.792529\pi$
$642$		0			0
$643$	921.506			1.43314			0.716568	−	0.697518i	$-0.245712\pi$
$643$	921.506			1.43314			0.716568	+	0.697518i	$0.245712\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	838.370i		−	1.29578i	−0.761734	−	0.647890i	$-0.775652\pi$
$647$		−	838.370i		−	1.29578i	0.761734	−	0.647890i	$-0.224348\pi$
$648$		0			0
$649$	−624.454	−	1081.59i	−0.962179	−	1.66654i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−671.238	−	387.539i	−1.02793	−	0.593475i	−0.111539	−	0.993760i	$-0.535578\pi$
$653$	−671.238	−	387.539i	−1.02793	−	0.593475i	−0.916391	+	0.400285i	$0.868911\pi$
$654$		0			0
$655$	1832.62			2.79789
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	82.4435i		−	0.125104i	−0.998042	−	0.0625520i	$-0.980076\pi$
$659$		−	82.4435i		−	0.125104i	0.998042	−	0.0625520i	$-0.0199239\pi$
$660$		0			0
$661$	−434.594	−	752.739i	−0.657479	−	1.13879i	−0.981266	−	0.192657i	$-0.938289\pi$
$661$	−434.594	−	752.739i	−0.657479	−	1.13879i	0.323787	−	0.946130i	$-0.395044\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−675.507	+	788.236i	−1.01580	+	1.18532i
$666$		0			0
$667$	−504.012	−	872.974i	−0.755640	−	1.30881i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1400.69	+	808.690i	−2.08747	+	1.20520i
$672$		0			0
$673$	−379.173	−	656.747i	−0.563407	−	0.975850i	−0.997196	−	0.0748351i	$-0.976157\pi$
$673$	−379.173	−	656.747i	−0.563407	−	0.975850i	0.433789	−	0.901015i	$-0.357176\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−910.405	+	525.622i	−1.34476	+	0.776400i	−0.987502	−	0.157605i	$-0.949623\pi$
$677$	−910.405	+	525.622i	−1.34476	+	0.776400i	−0.357261	+	0.934005i	$0.616289\pi$
$678$		0			0
$679$	−710.694			−1.04668
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	834.766i		−	1.22221i	−0.791551	−	0.611103i	$-0.790726\pi$
$683$		−	834.766i		−	1.22221i	0.791551	−	0.611103i	$-0.209274\pi$
$684$		0			0
$685$	−627.697			−0.916345
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1113.69i		−	1.61638i
$690$		0			0
$691$	41.2273	+	71.4077i	0.0596632	+	0.103340i	0.894314	−	0.447439i	$-0.147664\pi$
$691$	41.2273	+	71.4077i	0.0596632	+	0.103340i	−0.834651	+	0.550779i	$0.814331\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1042.17	+	601.699i	−1.49953	+	0.865754i
$696$		0			0
$697$	37.6663	+	65.2400i	0.0540407	+	0.0936012i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	44.1386	−	25.4834i	0.0629652	−	0.0363530i	−0.468187	−	0.883629i	$-0.655093\pi$
$701$	44.1386	−	25.4834i	0.0629652	−	0.0363530i	0.531152	+	0.847276i	$0.321759\pi$
$702$		0			0
$703$	−449.990	+	158.501i	−0.640100	+	0.225464i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−551.750	+	318.553i	−0.780410	+	0.450570i
$708$		0			0
$709$	−210.795			−0.297313			−0.148657	−	0.988889i	$-0.547495\pi$
$709$	−210.795			−0.297313			−0.148657	+	0.988889i	$0.547495\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1586.94i		−	2.22572i
$714$		0			0
$715$	1251.33	−	2167.36i	1.75011	−	3.03127i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−399.243	+	230.503i	−0.555276	+	0.320589i	−0.751247	−	0.660021i	$-0.770547\pi$
$719$	−399.243	+	230.503i	−0.555276	+	0.320589i	0.195971	+	0.980610i	$0.437214\pi$
$720$		0			0
$721$	−52.6341			−0.0730015
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	844.088i		−	1.16426i
$726$		0			0
$727$	−267.029	+	462.507i	−0.367302	+	0.636186i	−0.989143	−	0.146958i	$-0.953052\pi$
$727$	−267.029	+	462.507i	−0.367302	+	0.636186i	0.621841	+	0.783144i	$0.286385\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	741.845i		−	1.01484i
$732$		0			0
$733$	262.174	−	454.098i	0.357672	−	0.619506i	−0.629899	−	0.776677i	$-0.716904\pi$
$733$	262.174	−	454.098i	0.357672	−	0.619506i	0.987571	+	0.157171i	$0.0502372\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	17.2948	+	9.98514i	0.0234664	+	0.0135484i
$738$		0			0
$739$	−16.8402	−	29.1681i	−0.0227878	−	0.0394696i	0.854407	−	0.519605i	$-0.173921\pi$
$739$	−16.8402	−	29.1681i	−0.0227878	−	0.0394696i	−0.877194	+	0.480135i	$0.840588\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			12.7053i			0.0171000i	0.999963	+	0.00855001i	$0.00272159\pi$
$743$			12.7053i			0.0171000i	−0.999963	+	0.00855001i	$0.997278\pi$
$744$		0			0
$745$	1425.35			1.91321
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−637.387	+	367.995i	−0.850983	+	0.491316i
$750$		0			0
$751$	734.945	+	1272.96i	0.978622	+	1.69502i	0.667424	+	0.744678i	$0.267397\pi$
$751$	734.945	+	1272.96i	0.978622	+	1.69502i	0.311197	+	0.950345i	$0.399270\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.19877	−	3.00151i	0.00688579	−	0.00397551i
$756$		0			0
$757$	−141.145	−	244.470i	−0.186453	−	0.322945i	0.757612	−	0.652705i	$-0.226366\pi$
$757$	−141.145	−	244.470i	−0.186453	−	0.322945i	−0.944065	+	0.329759i	$0.893032\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−917.495	−	529.716i	−1.20564	−	0.696079i	−0.243839	−	0.969816i	$-0.578407\pi$
$761$	−917.495	−	529.716i	−1.20564	−	0.696079i	−0.961805	+	0.273737i	$0.911740\pi$
$762$		0			0
$763$	−566.542	+	981.279i	−0.742519	+	1.28608i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1294.10	−	747.152i	1.68723	−	0.974122i
$768$		0			0
$769$	548.055	+	949.259i	0.712685	+	1.23441i	0.963845	+	0.266462i	$0.0858546\pi$
$769$	548.055	+	949.259i	0.712685	+	1.23441i	−0.251160	+	0.967946i	$0.580812\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	428.849	+	247.596i	0.554786	+	0.320306i	0.751050	−	0.660245i	$-0.229548\pi$
$773$	428.849	+	247.596i	0.554786	+	0.320306i	−0.196264	+	0.980551i	$0.562881\pi$
$774$		0			0
$775$	664.425	−	1150.82i	0.857323	−	1.48493i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	31.8462	+	90.4125i	0.0408809	+	0.116062i
$780$		0			0
$781$	121.101	−	209.752i	0.155058	−	0.268569i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			934.247i			1.19012i
$786$		0			0
$787$	−134.383	+	232.757i	−0.170753	+	0.295753i	−0.938683	−	0.344781i	$-0.887953\pi$
$787$	−134.383	+	232.757i	−0.170753	+	0.295753i	0.767930	+	0.640533i	$0.221287\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	344.242i		−	0.435199i
$792$		0			0
$793$	−967.587	−	1675.91i	−1.22016	−	2.11338i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1058.28	−	610.999i	−1.32783	−	0.766624i	−0.342868	−	0.939384i	$-0.611398\pi$
$797$	−1058.28	−	610.999i	−1.32783	−	0.766624i	−0.984964	+	0.172760i	$0.944732\pi$
$798$		0			0
$799$	−466.381	−	807.796i	−0.583706	−	1.01101i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1749.23	−	1009.92i	−2.17837	−	1.25768i
$804$		0			0
$805$	−908.564	+	1573.68i	−1.12865	+	1.95488i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1065.41i			1.31695i	0.752605	+	0.658473i	$0.228797\pi$
$809$			1065.41i			1.31695i	−0.752605	+	0.658473i	$0.771203\pi$
$810$		0			0
$811$	270.922	−	469.251i	0.334059	−	0.578608i	−0.649244	−	0.760580i	$-0.724915\pi$
$811$	270.922	−	469.251i	0.334059	−	0.578608i	0.983304	+	0.181972i	$0.0582481\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	1717.56i		−	2.10743i
$816$		0			0
$817$	173.580	−	927.866i	0.212460	−	1.13570i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			340.662i			0.414936i	0.978242	+	0.207468i	$0.0665222\pi$
$821$			340.662i			0.414936i	−0.978242	+	0.207468i	$0.933478\pi$
$822$		0			0
$823$	532.112	+	921.644i	0.646551	+	1.11986i	0.983941	+	0.178494i	$0.0571227\pi$
$823$	532.112	+	921.644i	0.646551	+	1.11986i	−0.337390	+	0.941365i	$0.609544\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	885.426	+	511.201i	1.07065	+	0.618139i	0.928358	−	0.371686i	$-0.121220\pi$
$827$	885.426	+	511.201i	1.07065	+	0.618139i	0.142289	+	0.989825i	$0.454554\pi$
$828$		0			0
$829$	−671.808			−0.810384			−0.405192	−	0.914232i	$-0.632795\pi$
$829$	−671.808			−0.810384			−0.405192	+	0.914232i	$0.632795\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	111.734i		−	0.134135i
$834$		0			0
$835$	605.024	+	1047.93i	0.724579	+	1.25501i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	824.312	+	475.917i	0.982493	+	0.567243i	0.903022	−	0.429595i	$-0.141343\pi$
$839$	824.312	+	475.917i	0.982493	+	0.567243i	0.0794712	+	0.996837i	$0.474677\pi$
$840$		0			0
$841$	−77.6109			−0.0922840
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1529.23	+	882.899i	1.80974	+	1.04485i
$846$		0			0
$847$	1252.99			1.47933
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−723.238	+	417.562i	−0.849869	+	0.490672i
$852$		0			0
$853$	−18.4431	+	31.9445i	−0.0216215	+	0.0374495i	−0.876634	−	0.481158i	$-0.840216\pi$
$853$	−18.4431	+	31.9445i	−0.0216215	+	0.0374495i	0.855012	+	0.518608i	$0.173550\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	393.745	+	227.329i	0.459446	+	0.265261i	0.711811	−	0.702371i	$-0.247875\pi$
$857$	393.745	+	227.329i	0.459446	+	0.265261i	−0.252365	+	0.967632i	$0.581208\pi$
$858$		0			0
$859$	−261.644	−	453.180i	−0.304591	−	0.527567i	0.672579	−	0.740025i	$-0.265187\pi$
$859$	−261.644	−	453.180i	−0.304591	−	0.527567i	−0.977170	+	0.212458i	$0.931853\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			537.201i			0.622481i	0.950331	+	0.311240i	$0.100744\pi$
$863$			537.201i			0.622481i	−0.950331	+	0.311240i	$0.899256\pi$
$864$		0			0
$865$	246.283	−	426.574i	0.284720	−	0.493149i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	2.54649i		−	0.00293037i
$870$		0			0
$871$	−11.9471	+	20.6930i	−0.0137165	+	0.0237577i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−134.841	+	77.8507i	−0.154104	+	0.0889722i
$876$		0			0
$877$	731.890			0.834539			0.417269	−	0.908783i	$-0.362987\pi$
$877$	731.890			0.834539			0.417269	+	0.908783i	$0.362987\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			520.652i			0.590979i	0.955346	+	0.295489i	$0.0954827\pi$
$881$			520.652i			0.590979i	−0.955346	+	0.295489i	$0.904517\pi$
$882$		0			0
$883$	−492.648	+	853.291i	−0.557925	+	0.966354i	0.439745	+	0.898123i	$0.355069\pi$
$883$	−492.648	+	853.291i	−0.557925	+	0.966354i	−0.997670	+	0.0682312i	$0.978264\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−742.547	+	428.710i	−0.837144	+	0.483325i	−0.856292	−	0.516491i	$-0.827238\pi$
$887$	−742.547	+	428.710i	−0.837144	+	0.483325i	0.0191484	+	0.999817i	$0.493904\pi$
$888$		0			0
$889$	−211.806			−0.238252
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−394.317	−	1119.48i	−0.441565	−	1.25362i
$894$		0			0
$895$	−440.048			−0.491674
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1252.42	−	723.086i	−1.39313	−	0.804323i
$900$		0			0
$901$	819.374			0.909406
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1051.17	−	606.894i	−1.16151	−	0.670601i
$906$		0			0
$907$	620.439	−	1074.63i	0.684057	−	1.18482i	−0.289676	−	0.957125i	$-0.593547\pi$
$907$	620.439	−	1074.63i	0.684057	−	1.18482i	0.973732	−	0.227696i	$-0.0731192\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	692.109	−	399.589i	0.759725	−	0.438627i	−0.0694723	−	0.997584i	$-0.522132\pi$
$911$	692.109	−	399.589i	0.759725	−	0.438627i	0.829197	+	0.558957i	$0.188798\pi$
$912$		0			0
$913$	410.984	−	711.845i	0.450147	−	0.779677i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1640.74	+	947.283i	−1.78925	+	1.03302i
$918$		0			0
$919$	1507.71			1.64060			0.820299	−	0.571934i	$-0.193807\pi$
$919$	1507.71			1.64060			0.820299	+	0.571934i	$0.193807\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	250.966	+	144.895i	0.271902	+	0.156983i
$924$		0			0
$925$	−699.307			−0.756007
$926$		0			0
$927$		0			0
$928$		0			0
$929$	871.341	+	503.069i	0.937934	+	0.541516i	0.889312	−	0.457301i	$-0.151184\pi$
$929$	871.341	+	503.069i	0.937934	+	0.541516i	0.0486219	+	0.998817i	$0.484517\pi$
$930$		0			0
$931$	26.1440	−	139.752i	0.0280816	−	0.150110i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1594.59	+	920.639i	1.70545	+	0.984640i
$936$		0			0
$937$	−622.963	+	1079.00i	−0.664848	+	1.15155i	0.314478	+	0.949265i	$0.398170\pi$
$937$	−622.963	+	1079.00i	−0.664848	+	1.15155i	−0.979326	+	0.202286i	$0.935163\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	952.114	−	549.703i	1.01181	−	0.584169i	0.100090	−	0.994978i	$-0.468087\pi$
$941$	952.114	−	549.703i	1.01181	−	0.584169i	0.911721	+	0.410809i	$0.134754\pi$
$942$		0			0
$943$	83.8970	+	145.314i	0.0889682	+	0.154097i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−533.426	−	307.974i	−0.563280	−	0.325210i	0.191181	−	0.981555i	$-0.438768\pi$
$947$	−533.426	−	307.974i	−0.563280	−	0.325210i	−0.754461	+	0.656345i	$0.772102\pi$
$948$		0			0
$949$	1208.36	−	2092.94i	1.27329	−	2.20541i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−447.017	+	258.085i	−0.469063	+	0.270814i	−0.715847	−	0.698257i	$-0.753959\pi$
$953$	−447.017	+	258.085i	−0.469063	+	0.270814i	0.246784	+	0.969070i	$0.420626\pi$
$954$		0			0
$955$	558.356	+	967.101i	0.584666	+	1.01267i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	561.976	−	324.457i	0.586002	−	0.338328i
$960$		0			0
$961$	−657.858	−	1139.44i	−0.684555	−	1.18568i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1732.41i		−	1.79524i
$966$		0			0
$967$	864.526			0.894029			0.447014	−	0.894527i	$-0.352487\pi$
$967$	864.526			0.894029			0.447014	+	0.894527i	$0.352487\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1640.42	−	947.099i	1.68942	−	0.975386i	0.734456	−	0.678657i	$-0.237438\pi$
$971$	1640.42	−	947.099i	1.68942	−	0.975386i	0.954962	−	0.296729i	$-0.0958957\pi$
$972$		0			0
$973$	622.037	−	1077.40i	0.639298	−	1.10730i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1561.64	+	901.613i	1.59840	+	0.922838i	0.991795	+	0.127837i	$0.0408034\pi$
$977$	1561.64	+	901.613i	1.59840	+	0.922838i	0.606608	+	0.795001i	$0.292530\pi$
$978$		0			0
$979$	−1320.31			−1.34863
$980$		0			0
$981$		0			0
$982$		0			0
$983$	345.988	+	199.756i	0.351972	+	0.203211i	0.665553	−	0.746350i	$-0.268196\pi$
$983$	345.988	+	199.756i	0.351972	+	0.203211i	−0.313582	+	0.949561i	$0.601529\pi$
$984$		0			0
$985$	−796.785			−0.808919
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1652.37i		−	1.67074i
$990$		0			0
$991$	479.340	+	830.240i	0.483693	+	0.837780i	0.999825	−	0.0187287i	$-0.00596188\pi$
$991$	479.340	+	830.240i	0.483693	+	0.837780i	−0.516132	+	0.856509i	$0.672629\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1595.26	+	921.023i	1.60328	+	0.925651i
$996$		0			0
$997$	−1169.13			−1.17265			−0.586326	−	0.810075i	$-0.699426\pi$
$997$	−1169.13			−1.17265			−0.586326	+	0.810075i	$0.699426\pi$
$998$		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.881.3		80
3.2	odd	2		684.3.m.a.653.3	yes	80
9.2	odd	6		2052.3.be.a.197.3		80
9.7	even	3		684.3.be.a.425.23	yes	80
19.11	even	3		2052.3.be.a.125.3		80
57.11	odd	6		684.3.be.a.581.23	yes	80
171.11	odd	6	inner	2052.3.m.a.1493.38		80
171.106	even	3		684.3.m.a.353.3	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.3	✓	80	171.106	even	3
684.3.m.a.653.3	yes	80	3.2	odd	2
684.3.be.a.425.23	yes	80	9.7	even	3
684.3.be.a.581.23	yes	80	57.11	odd	6
2052.3.m.a.881.3		80	1.1	even	1	trivial
2052.3.m.a.1493.38		80	171.11	odd	6	inner
2052.3.be.a.125.3		80	19.11	even	3
2052.3.be.a.197.3		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)

\(f(q)\)	\(=\)	\(q-7.26979i q^{5} +(3.75776 + 6.50863i) q^{7} +O(q^{10})\)	\(q-7.26979i q^{5} +(3.75776 + 6.50863i) q^{7} +(14.6898 - 8.48117i) q^{11} +(10.1476 + 17.5762i) q^{13} +(-12.9313 + 7.46590i) q^{17} +(-17.9208 + 6.31229i) q^{19} +(-28.8029 + 16.6293i) q^{23} -27.8498 q^{25} +30.3086i q^{29} +(-23.8575 + 41.3223i) q^{31} +(47.3163 - 27.3181i) q^{35} +25.1099 q^{37} -5.04512i q^{41} +(-24.8411 + 43.0261i) q^{43} +62.4682i q^{47} +(-3.74149 + 6.48044i) q^{49} +(-47.5227 - 27.4372i) q^{53} +(-61.6563 - 106.792i) q^{55} -73.6284i q^{59} -95.3512 q^{61} +(127.775 - 73.7709i) q^{65} +(0.588665 + 1.01960i) q^{67} +(12.3658 - 7.13938i) q^{71} +(-59.5390 - 103.125i) q^{73} +(110.401 + 63.7403i) q^{77} +(0.0750630 - 0.130013i) q^{79} +(41.9662 - 24.2292i) q^{83} +(54.2755 + 94.0079i) q^{85} +(-67.4096 - 38.9190i) q^{89} +(-76.2645 + 132.094i) q^{91} +(45.8890 + 130.280i) q^{95} +(-47.2818 + 81.8945i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 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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