LMFDB - Embedded newform 4140.3.d.c.2161.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(2161,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("4140.2161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.d (of order $2$, degree $1$, 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:	$112.806829445$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.13
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.c.2161.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +9.24833i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +9.24833i q^{7} -16.5611i q^{11} -0.981118 q^{13} -32.3578i q^{17} -22.7691i q^{19} +(-13.0254 + 18.9563i) q^{23} -5.00000 q^{25} -18.2073 q^{29} -51.1444 q^{31} +20.6799 q^{35} -35.8500i q^{37} +76.1329 q^{41} +27.4468i q^{43} +25.9596 q^{47} -36.5316 q^{49} +93.6068i q^{53} -37.0317 q^{55} -91.5180 q^{59} +81.3134i q^{61} +2.19385i q^{65} -9.90151i q^{67} +49.8518 q^{71} -1.04840 q^{73} +153.162 q^{77} +103.967i q^{79} -101.150i q^{83} -72.3543 q^{85} +153.631i q^{89} -9.07370i q^{91} -50.9132 q^{95} +36.0497i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) $ 32 * q + 24 * q^13 - 64 * q^23 - 160 * q^25 + 60 * q^29 - 4 * q^31 + 60 * q^35 + 108 * q^41 - 136 * q^47 - 428 * q^49 + 120 * q^55 + 84 * q^59 - 188 * q^71 + 472 * q^73 + 120 * q^77 + 60 * q^85 - 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			9.24833i			1.32119i	0.750743	+	0.660595i	$0.229696\pi$
$7$			9.24833i			1.32119i	−0.750743	+	0.660595i	$0.770304\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	16.5611i		−	1.50555i	−0.658276	−	0.752777i	$-0.728714\pi$
$11$		−	16.5611i		−	1.50555i	0.658276	−	0.752777i	$-0.271286\pi$
$12$		0			0
$13$	−0.981118			−0.0754706			−0.0377353	−	0.999288i	$-0.512014\pi$
$13$	−0.981118			−0.0754706			−0.0377353	+	0.999288i	$0.512014\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	32.3578i		−	1.90340i	−0.307028	−	0.951700i	$-0.599335\pi$
$17$		−	32.3578i		−	1.90340i	0.307028	−	0.951700i	$-0.400665\pi$
$18$		0			0
$19$		−	22.7691i		−	1.19837i	−0.800609	−	0.599187i	$-0.795491\pi$
$19$		−	22.7691i		−	1.19837i	0.800609	−	0.599187i	$-0.204509\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−13.0254	+	18.9563i	−0.566320	+	0.824186i
$23$	−13.0254	+	18.9563i	−0.566320	+	0.824186i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−18.2073			−0.627838			−0.313919	−	0.949450i	$-0.601642\pi$
$29$	−18.2073			−0.627838			−0.313919	+	0.949450i	$0.601642\pi$
$30$		0			0
$31$	−51.1444			−1.64982			−0.824909	−	0.565265i	$-0.808774\pi$
$31$	−51.1444			−1.64982			−0.824909	+	0.565265i	$0.808774\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	20.6799			0.590854
$36$		0			0
$37$		−	35.8500i		−	0.968918i	−0.874814	−	0.484459i	$-0.839016\pi$
$37$		−	35.8500i		−	0.968918i	0.874814	−	0.484459i	$-0.160984\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	76.1329			1.85690			0.928450	−	0.371458i	$-0.121142\pi$
$41$	76.1329			1.85690			0.928450	+	0.371458i	$0.121142\pi$
$42$		0			0
$43$			27.4468i			0.638297i	0.947705	+	0.319149i	$0.103397\pi$
$43$			27.4468i			0.638297i	−0.947705	+	0.319149i	$0.896603\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	25.9596			0.552332			0.276166	−	0.961110i	$-0.410936\pi$
$47$	25.9596			0.552332			0.276166	+	0.961110i	$0.410936\pi$
$48$		0			0
$49$	−36.5316			−0.745542
$50$		0			0
$51$		0			0
$52$		0			0
$53$			93.6068i			1.76617i	0.469216	+	0.883083i	$0.344536\pi$
$53$			93.6068i			1.76617i	−0.469216	+	0.883083i	$0.655464\pi$
$54$		0			0
$55$	−37.0317			−0.673304
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−91.5180			−1.55115			−0.775576	−	0.631254i	$-0.782541\pi$
$59$	−91.5180			−1.55115			−0.775576	+	0.631254i	$0.782541\pi$
$60$		0			0
$61$			81.3134i			1.33301i	0.745502	+	0.666503i	$0.232210\pi$
$61$			81.3134i			1.33301i	−0.745502	+	0.666503i	$0.767790\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			2.19385i			0.0337515i
$66$		0			0
$67$		−	9.90151i		−	0.147784i	−0.997266	−	0.0738919i	$-0.976458\pi$
$67$		−	9.90151i		−	0.147784i	0.997266	−	0.0738919i	$-0.0235420\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	49.8518			0.702138			0.351069	−	0.936350i	$-0.385818\pi$
$71$	49.8518			0.702138			0.351069	+	0.936350i	$0.385818\pi$
$72$		0			0
$73$	−1.04840			−0.0143616			−0.00718082	−	0.999974i	$-0.502286\pi$
$73$	−1.04840			−0.0143616			−0.00718082	+	0.999974i	$0.502286\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	153.162			1.98912
$78$		0			0
$79$			103.967i			1.31604i	0.753001	+	0.658019i	$0.228605\pi$
$79$			103.967i			1.31604i	−0.753001	+	0.658019i	$0.771395\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	101.150i		−	1.21868i	−0.792909	−	0.609340i	$-0.791435\pi$
$83$		−	101.150i		−	1.21868i	0.792909	−	0.609340i	$-0.208565\pi$
$84$		0			0
$85$	−72.3543			−0.851227
$86$		0			0
$87$		0			0
$88$		0			0
$89$			153.631i			1.72619i	0.505043	+	0.863094i	$0.331477\pi$
$89$			153.631i			1.72619i	−0.505043	+	0.863094i	$0.668523\pi$
$90$		0			0
$91$		−	9.07370i		−	0.0997110i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−50.9132			−0.535929
$96$		0			0
$97$			36.0497i			0.371646i	0.982583	+	0.185823i	$0.0594951\pi$
$97$			36.0497i			0.371646i	−0.982583	+	0.185823i	$0.940505\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−22.1270			−0.219080			−0.109540	−	0.993982i	$-0.534938\pi$
$101$	−22.1270			−0.219080			−0.109540	+	0.993982i	$0.534938\pi$
$102$		0			0
$103$		−	2.88248i		−	0.0279852i	−0.999902	−	0.0139926i	$-0.995546\pi$
$103$		−	2.88248i		−	0.0279852i	0.999902	−	0.0139926i	$-0.00445413\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	20.3688i		−	0.190362i	−0.995460	−	0.0951812i	$-0.969657\pi$
$107$		−	20.3688i		−	0.190362i	0.995460	−	0.0951812i	$-0.0303431\pi$
$108$		0			0
$109$			52.5105i			0.481748i	0.970556	+	0.240874i	$0.0774340\pi$
$109$			52.5105i			0.481748i	−0.970556	+	0.240874i	$0.922566\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	81.4603i		−	0.720887i	−0.932781	−	0.360444i	$-0.882625\pi$
$113$		−	81.4603i		−	0.720887i	0.932781	−	0.360444i	$-0.117375\pi$
$114$		0			0
$115$	42.3875	+	29.1256i	0.368587	+	0.253266i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	299.256			2.51475
$120$		0			0
$121$	−153.270			−1.26669
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	111.720			0.879683			0.439842	−	0.898075i	$-0.355035\pi$
$127$	111.720			0.879683			0.439842	+	0.898075i	$0.355035\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−231.398			−1.76640			−0.883199	−	0.468998i	$-0.844615\pi$
$131$	−231.398			−1.76640			−0.883199	+	0.468998i	$0.844615\pi$
$132$		0			0
$133$	210.576			1.58328
$134$		0			0
$135$		0			0
$136$		0			0
$137$			33.3860i			0.243693i	0.992549	+	0.121847i	$0.0388816\pi$
$137$			33.3860i			0.243693i	−0.992549	+	0.121847i	$0.961118\pi$
$138$		0			0
$139$	−24.1023			−0.173398			−0.0866988	−	0.996235i	$-0.527632\pi$
$139$	−24.1023			−0.173398			−0.0866988	+	0.996235i	$0.527632\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			16.2484i			0.113625i
$144$		0			0
$145$			40.7127i			0.280777i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	264.767i		−	1.77696i	−0.458912	−	0.888482i	$-0.651761\pi$
$149$		−	264.767i		−	1.77696i	0.458912	−	0.888482i	$-0.348239\pi$
$150$		0			0
$151$	32.8905			0.217818			0.108909	−	0.994052i	$-0.465264\pi$
$151$	32.8905			0.217818			0.108909	+	0.994052i	$0.465264\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			114.362i			0.737821i
$156$		0			0
$157$			19.7098i			0.125540i	0.998028	+	0.0627700i	$0.0199935\pi$
$157$			19.7098i			0.125540i	−0.998028	+	0.0627700i	$0.980007\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−175.314	−	120.463i	−1.08891	−	0.748216i
$162$		0			0
$163$	−257.790			−1.58153			−0.790767	−	0.612118i	$-0.790318\pi$
$163$	−257.790			−1.58153			−0.790767	+	0.612118i	$0.790318\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−308.644			−1.84817			−0.924084	−	0.382190i	$-0.875170\pi$
$167$	−308.644			−1.84817			−0.924084	+	0.382190i	$0.875170\pi$
$168$		0			0
$169$	−168.037			−0.994304
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−222.684			−1.28719			−0.643595	−	0.765367i	$-0.722558\pi$
$173$	−222.684			−1.28719			−0.643595	+	0.765367i	$0.722558\pi$
$174$		0			0
$175$		−	46.2416i		−	0.264238i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	91.3844			0.510527			0.255264	−	0.966871i	$-0.417838\pi$
$179$	91.3844			0.510527			0.255264	+	0.966871i	$0.417838\pi$
$180$		0			0
$181$			289.376i			1.59876i	0.600825	+	0.799380i	$0.294839\pi$
$181$			289.376i			1.59876i	−0.600825	+	0.799380i	$0.705161\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−80.1630			−0.433313
$186$		0			0
$187$	−535.881			−2.86567
$188$		0			0
$189$		0			0
$190$		0			0
$191$			157.455i			0.824371i	0.911100	+	0.412186i	$0.135234\pi$
$191$			157.455i			0.824371i	−0.911100	+	0.412186i	$0.864766\pi$
$192$		0			0
$193$	−97.8648			−0.507072			−0.253536	−	0.967326i	$-0.581594\pi$
$193$	−97.8648			−0.507072			−0.253536	+	0.967326i	$0.581594\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	98.3572			0.499275			0.249637	−	0.968339i	$-0.419689\pi$
$197$	98.3572			0.499275			0.249637	+	0.968339i	$0.419689\pi$
$198$		0			0
$199$		−	141.254i		−	0.709818i	−0.934901	−	0.354909i	$-0.884512\pi$
$199$		−	141.254i		−	0.709818i	0.934901	−	0.354909i	$-0.115488\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	168.387i		−	0.829493i
$204$		0			0
$205$		−	170.238i		−	0.830431i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−377.081			−1.80422
$210$		0			0
$211$	−290.334			−1.37599			−0.687994	−	0.725716i	$-0.741509\pi$
$211$	−290.334			−1.37599			−0.687994	+	0.725716i	$0.741509\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	61.3729			0.285455
$216$		0			0
$217$		−	473.000i		−	2.17972i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			31.7468i			0.143651i
$222$		0			0
$223$	346.556			1.55406			0.777032	−	0.629462i	$-0.216724\pi$
$223$	346.556			1.55406			0.777032	+	0.629462i	$0.216724\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	51.9414i		−	0.228817i	−0.993434	−	0.114408i	$-0.963503\pi$
$227$		−	51.9414i		−	0.228817i	0.993434	−	0.114408i	$-0.0364972\pi$
$228$		0			0
$229$			17.8456i			0.0779285i	0.999241	+	0.0389643i	$0.0124059\pi$
$229$			17.8456i			0.0779285i	−0.999241	+	0.0389643i	$0.987594\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	239.497			1.02789			0.513943	−	0.857825i	$-0.328184\pi$
$233$	239.497			1.02789			0.513943	+	0.857825i	$0.328184\pi$
$234$		0			0
$235$		−	58.0475i		−	0.247011i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−130.103			−0.544362			−0.272181	−	0.962246i	$-0.587745\pi$
$239$	−130.103			−0.544362			−0.272181	+	0.962246i	$0.587745\pi$
$240$		0			0
$241$			398.401i			1.65311i	0.562853	+	0.826557i	$0.309704\pi$
$241$			398.401i			1.65311i	−0.562853	+	0.826557i	$0.690296\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			81.6871i			0.333417i
$246$		0			0
$247$			22.3392i			0.0904420i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			99.2266i			0.395325i	0.980270	+	0.197663i	$0.0633350\pi$
$251$			99.2266i			0.395325i	−0.980270	+	0.197663i	$0.936665\pi$
$252$		0			0
$253$	313.937	+	215.714i	1.24086	+	0.852625i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	329.639			1.28264			0.641320	−	0.767273i	$-0.278387\pi$
$257$	329.639			1.28264			0.641320	+	0.767273i	$0.278387\pi$
$258$		0			0
$259$	331.552			1.28013
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	320.965i		−	1.22040i	−0.792248	−	0.610199i	$-0.791089\pi$
$263$		−	320.965i		−	1.22040i	0.792248	−	0.610199i	$-0.208911\pi$
$264$		0			0
$265$	209.311			0.789854
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−35.5356			−0.132103			−0.0660513	−	0.997816i	$-0.521040\pi$
$269$	−35.5356			−0.132103			−0.0660513	+	0.997816i	$0.521040\pi$
$270$		0			0
$271$	41.5410			0.153288			0.0766439	−	0.997059i	$-0.475580\pi$
$271$	41.5410			0.153288			0.0766439	+	0.997059i	$0.475580\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			82.8055i			0.301111i
$276$		0			0
$277$	−91.5111			−0.330365			−0.165182	−	0.986263i	$-0.552821\pi$
$277$	−91.5111			−0.330365			−0.165182	+	0.986263i	$0.552821\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			109.708i			0.390421i	0.980761	+	0.195210i	$0.0625390\pi$
$281$			109.708i			0.390421i	−0.980761	+	0.195210i	$0.937461\pi$
$282$		0			0
$283$		−	197.784i		−	0.698882i	−0.936958	−	0.349441i	$-0.886371\pi$
$283$		−	197.784i		−	0.698882i	0.936958	−	0.349441i	$-0.113629\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			704.102i			2.45332i
$288$		0			0
$289$	−758.028			−2.62293
$290$		0			0
$291$		0			0
$292$		0			0
$293$			84.7674i			0.289309i	0.989482	+	0.144654i	$0.0462070\pi$
$293$			84.7674i			0.289309i	−0.989482	+	0.144654i	$0.953793\pi$
$294$		0			0
$295$			204.640i			0.693696i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.7794	−	18.5983i	0.0427405	−	0.0622018i
$300$		0			0
$301$	−253.837			−0.843312
$302$		0			0
$303$		0			0
$304$		0			0
$305$	181.822			0.596138
$306$		0			0
$307$	−37.2050			−0.121189			−0.0605945	−	0.998162i	$-0.519300\pi$
$307$	−37.2050			−0.121189			−0.0605945	+	0.998162i	$0.519300\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−342.730			−1.10203			−0.551013	−	0.834497i	$-0.685758\pi$
$311$	−342.730			−1.10203			−0.551013	+	0.834497i	$0.685758\pi$
$312$		0			0
$313$			268.986i			0.859380i	0.902976	+	0.429690i	$0.141377\pi$
$313$			268.986i			0.859380i	−0.902976	+	0.429690i	$0.858623\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	64.4934			0.203449			0.101725	−	0.994813i	$-0.467564\pi$
$317$	64.4934			0.203449			0.101725	+	0.994813i	$0.467564\pi$
$318$		0			0
$319$			301.533i			0.945243i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−736.758			−2.28098
$324$		0			0
$325$	4.90559			0.0150941
$326$		0			0
$327$		0			0
$328$		0			0
$329$			240.083i			0.729736i
$330$		0			0
$331$	174.273			0.526505			0.263252	−	0.964727i	$-0.415205\pi$
$331$	174.273			0.526505			0.263252	+	0.964727i	$0.415205\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−22.1405			−0.0660909
$336$		0			0
$337$			263.858i			0.782962i	0.920186	+	0.391481i	$0.128037\pi$
$337$			263.858i			0.782962i	−0.920186	+	0.391481i	$0.871963\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			847.007i			2.48389i
$342$		0			0
$343$			115.312i			0.336187i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−460.628			−1.32746			−0.663730	−	0.747973i	$-0.731027\pi$
$347$	−460.628			−1.32746			−0.663730	+	0.747973i	$0.731027\pi$
$348$		0			0
$349$	351.887			1.00827			0.504136	−	0.863624i	$-0.331811\pi$
$349$	351.887			1.00827			0.504136	+	0.863624i	$0.331811\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	419.316			1.18786			0.593932	−	0.804515i	$-0.297575\pi$
$353$	419.316			1.18786			0.593932	+	0.804515i	$0.297575\pi$
$354$		0			0
$355$		−	111.472i		−	0.314006i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	413.133i		−	1.15079i	−0.817876	−	0.575394i	$-0.804849\pi$
$359$		−	413.133i		−	1.15079i	0.817876	−	0.575394i	$-0.195151\pi$
$360$		0			0
$361$	−157.432			−0.436099
$362$		0			0
$363$		0			0
$364$		0			0
$365$			2.34429i			0.00642272i
$366$		0			0
$367$		−	172.694i		−	0.470555i	−0.971928	−	0.235278i	$-0.924400\pi$
$367$		−	172.694i		−	0.470555i	0.971928	−	0.235278i	$-0.0756000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−865.707			−2.33344
$372$		0			0
$373$		−	597.195i		−	1.60106i	−0.599294	−	0.800529i	$-0.704552\pi$
$373$		−	597.195i		−	1.60106i	0.599294	−	0.800529i	$-0.295448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	17.8635			0.0473833
$378$		0			0
$379$			127.052i			0.335229i	0.985853	+	0.167615i	$0.0536064\pi$
$379$			127.052i			0.335229i	−0.985853	+	0.167615i	$0.946394\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			652.032i			1.70243i	0.524814	+	0.851217i	$0.324135\pi$
$383$			652.032i			1.70243i	−0.524814	+	0.851217i	$0.675865\pi$
$384$		0			0
$385$		−	342.482i		−	0.889563i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			89.3298i			0.229640i	0.993386	+	0.114820i	$0.0366291\pi$
$389$			89.3298i			0.229640i	−0.993386	+	0.114820i	$0.963371\pi$
$390$		0			0
$391$	613.383	+	421.472i	1.56876	+	1.07793i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	232.477			0.588550
$396$		0			0
$397$	−509.232			−1.28270			−0.641350	−	0.767249i	$-0.721625\pi$
$397$	−509.232			−1.28270			−0.641350	+	0.767249i	$0.721625\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			293.326i			0.731486i	0.930716	+	0.365743i	$0.119185\pi$
$401$			293.326i			0.731486i	−0.930716	+	0.365743i	$0.880815\pi$
$402$		0			0
$403$	50.1787			0.124513
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−593.715			−1.45876
$408$		0			0
$409$	710.069			1.73611			0.868055	−	0.496468i	$-0.165370\pi$
$409$	710.069			1.73611			0.868055	+	0.496468i	$0.165370\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	846.388i		−	2.04937i
$414$		0			0
$415$	−226.179			−0.545010
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	177.329i		−	0.423220i	−0.977354	−	0.211610i	$-0.932129\pi$
$419$		−	177.329i		−	0.423220i	0.977354	−	0.211610i	$-0.0678706\pi$
$420$		0			0
$421$			745.499i			1.77078i	0.464847	+	0.885391i	$0.346109\pi$
$421$			745.499i			1.77078i	−0.464847	+	0.885391i	$0.653891\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			161.789i			0.380680i
$426$		0			0
$427$	−752.013			−1.76115
$428$		0			0
$429$		0			0
$430$		0			0
$431$			300.853i			0.698035i	0.937116	+	0.349018i	$0.113485\pi$
$431$			300.853i			0.698035i	−0.937116	+	0.349018i	$0.886515\pi$
$432$		0			0
$433$			269.505i			0.622413i	0.950342	+	0.311206i	$0.100733\pi$
$433$			269.505i			0.622413i	−0.950342	+	0.311206i	$0.899267\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	431.617	+	296.576i	0.987682	+	0.678663i
$438$		0			0
$439$	412.629			0.939929			0.469965	−	0.882685i	$-0.344267\pi$
$439$	412.629			0.939929			0.469965	+	0.882685i	$0.344267\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−252.557			−0.570106			−0.285053	−	0.958512i	$-0.592011\pi$
$443$	−252.557			−0.570106			−0.285053	+	0.958512i	$0.592011\pi$
$444$		0			0
$445$	343.529			0.771975
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−829.502			−1.84744			−0.923721	−	0.383065i	$-0.874868\pi$
$449$	−829.502			−1.84744			−0.923721	+	0.383065i	$0.874868\pi$
$450$		0			0
$451$		−	1260.84i		−	2.79566i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−20.2894			−0.0445921
$456$		0			0
$457$		−	92.5377i		−	0.202490i	−0.994862	−	0.101245i	$-0.967717\pi$
$457$		−	92.5377i		−	0.202490i	0.994862	−	0.101245i	$-0.0322825\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−509.434			−1.10506			−0.552532	−	0.833492i	$-0.686338\pi$
$461$	−509.434			−1.10506			−0.552532	+	0.833492i	$0.686338\pi$
$462$		0			0
$463$	318.186			0.687226			0.343613	−	0.939111i	$-0.388349\pi$
$463$	318.186			0.687226			0.343613	+	0.939111i	$0.388349\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			22.0560i			0.0472291i	0.999721	+	0.0236145i	$0.00751744\pi$
$467$			22.0560i			0.0472291i	−0.999721	+	0.0236145i	$0.992483\pi$
$468$		0			0
$469$	91.5724			0.195250
$470$		0			0
$471$		0			0
$472$		0			0
$473$	454.549			0.960991
$474$		0			0
$475$			113.845i			0.239675i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	40.1343i		−	0.0837877i	−0.999122	−	0.0418939i	$-0.986661\pi$
$479$		−	40.1343i		−	0.0837877i	0.999122	−	0.0418939i	$-0.0133391\pi$
$480$		0			0
$481$			35.1731i			0.0731249i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	80.6095			0.166205
$486$		0			0
$487$	325.889			0.669178			0.334589	−	0.942364i	$-0.391403\pi$
$487$	325.889			0.669178			0.334589	+	0.942364i	$0.391403\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−100.091			−0.203852			−0.101926	−	0.994792i	$-0.532501\pi$
$491$	−100.091			−0.203852			−0.101926	+	0.994792i	$0.532501\pi$
$492$		0			0
$493$			589.148i			1.19503i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			461.046i			0.927658i
$498$		0			0
$499$	162.286			0.325222			0.162611	−	0.986690i	$-0.448008\pi$
$499$	162.286			0.325222			0.162611	+	0.986690i	$0.448008\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			670.221i			1.33245i	0.745752	+	0.666224i	$0.232090\pi$
$503$			670.221i			1.33245i	−0.745752	+	0.666224i	$0.767910\pi$
$504$		0			0
$505$			49.4776i			0.0979754i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−514.522			−1.01085			−0.505424	−	0.862871i	$-0.668664\pi$
$509$	−514.522			−1.01085			−0.505424	+	0.862871i	$0.668664\pi$
$510$		0			0
$511$		−	9.69594i		−	0.0189744i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.44542			−0.0125154
$516$		0			0
$517$		−	429.920i		−	0.831566i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	790.254i		−	1.51680i	−0.651788	−	0.758401i	$-0.725981\pi$
$521$		−	790.254i		−	1.51680i	0.651788	−	0.758401i	$-0.274019\pi$
$522$		0			0
$523$		−	382.026i		−	0.730452i	−0.930919	−	0.365226i	$-0.880992\pi$
$523$		−	382.026i		−	0.730452i	0.930919	−	0.365226i	$-0.119008\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1654.92i			3.14027i
$528$		0			0
$529$	−189.680	−	493.824i	−0.358564	−	0.933505i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−74.6953			−0.140141
$534$		0			0
$535$	−45.5460			−0.0851327
$536$		0			0
$537$		0			0
$538$		0			0
$539$			605.003i			1.12245i
$540$		0			0
$541$	195.666			0.361675			0.180837	−	0.983513i	$-0.442119\pi$
$541$	195.666			0.361675			0.180837	+	0.983513i	$0.442119\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	117.417			0.215444
$546$		0			0
$547$	−877.854			−1.60485			−0.802426	−	0.596752i	$-0.796458\pi$
$547$	−877.854			−1.60485			−0.802426	+	0.596752i	$0.796458\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			414.563i			0.752384i
$552$		0			0
$553$	−961.521			−1.73874
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	429.433i		−	0.770975i	−0.922713	−	0.385487i	$-0.874033\pi$
$557$		−	429.433i		−	0.770975i	0.922713	−	0.385487i	$-0.125967\pi$
$558$		0			0
$559$		−	26.9285i		−	0.0481727i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			232.993i			0.413842i	0.978358	+	0.206921i	$0.0663442\pi$
$563$			232.993i			0.413842i	−0.978358	+	0.206921i	$0.933656\pi$
$564$		0			0
$565$	−182.151			−0.322391
$566$		0			0
$567$		0			0
$568$		0			0
$569$			211.168i			0.371121i	0.982633	+	0.185561i	$0.0594101\pi$
$569$			211.168i			0.371121i	−0.982633	+	0.185561i	$0.940590\pi$
$570$		0			0
$571$		−	305.687i		−	0.535354i	−0.963509	−	0.267677i	$-0.913744\pi$
$571$		−	305.687i		−	0.535354i	0.963509	−	0.267677i	$-0.0862561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	65.1268	−	94.7813i	0.113264	−	0.164837i
$576$		0			0
$577$	475.332			0.823798			0.411899	−	0.911229i	$-0.364866\pi$
$577$	475.332			0.823798			0.411899	+	0.911229i	$0.364866\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	935.472			1.61011
$582$		0			0
$583$	1550.23			2.65906
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−288.293			−0.491130			−0.245565	−	0.969380i	$-0.578973\pi$
$587$	−288.293			−0.491130			−0.245565	+	0.969380i	$0.578973\pi$
$588$		0			0
$589$			1164.51i			1.97710i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	59.2270			0.0998768			0.0499384	−	0.998752i	$-0.484097\pi$
$593$	59.2270			0.0998768			0.0499384	+	0.998752i	$0.484097\pi$
$594$		0			0
$595$		−	669.156i		−	1.12463i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−775.933			−1.29538			−0.647690	−	0.761904i	$-0.724265\pi$
$599$	−775.933			−1.29538			−0.647690	+	0.761904i	$0.724265\pi$
$600$		0			0
$601$	133.544			0.222204			0.111102	−	0.993809i	$-0.464562\pi$
$601$	133.544			0.222204			0.111102	+	0.993809i	$0.464562\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			342.722i			0.566482i
$606$		0			0
$607$	−604.134			−0.995278			−0.497639	−	0.867384i	$-0.665800\pi$
$607$	−604.134			−0.995278			−0.497639	+	0.867384i	$0.665800\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−25.4695			−0.0416849
$612$		0			0
$613$		−	896.881i		−	1.46310i	−0.681787	−	0.731550i	$-0.738797\pi$
$613$		−	896.881i		−	1.46310i	0.681787	−	0.731550i	$-0.261203\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			140.577i			0.227839i	0.993490	+	0.113920i	$0.0363406\pi$
$617$			140.577i			0.227839i	−0.993490	+	0.113920i	$0.963659\pi$
$618$		0			0
$619$		−	217.160i		−	0.350824i	−0.984495	−	0.175412i	$-0.943874\pi$
$619$		−	217.160i		−	0.350824i	0.984495	−	0.175412i	$-0.0561257\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1420.83			−2.28062
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1160.03			−1.84424
$630$		0			0
$631$			231.996i			0.367663i	0.982958	+	0.183832i	$0.0588501\pi$
$631$			231.996i			0.367663i	−0.982958	+	0.183832i	$0.941150\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	249.813i		−	0.393406i
$636$		0			0
$637$	35.8418			0.0562665
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	968.715i		−	1.51126i	−0.655002	−	0.755628i	$-0.727332\pi$
$641$		−	968.715i		−	1.51126i	0.655002	−	0.755628i	$-0.272668\pi$
$642$		0			0
$643$			127.161i			0.197763i	0.995099	+	0.0988813i	$0.0315264\pi$
$643$			127.161i			0.197763i	−0.995099	+	0.0988813i	$0.968474\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−522.649			−0.807804			−0.403902	−	0.914802i	$-0.632346\pi$
$647$	−522.649			−0.807804			−0.403902	+	0.914802i	$0.632346\pi$
$648$		0			0
$649$			1515.64i			2.33534i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−46.4519			−0.0711361			−0.0355681	−	0.999367i	$-0.511324\pi$
$653$	−46.4519			−0.0711361			−0.0355681	+	0.999367i	$0.511324\pi$
$654$		0			0
$655$			517.422i			0.789958i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1154.05i		−	1.75121i	−0.483028	−	0.875605i	$-0.660463\pi$
$659$		−	1154.05i		−	1.75121i	0.483028	−	0.875605i	$-0.339537\pi$
$660$		0			0
$661$			1077.57i			1.63021i	0.579313	+	0.815105i	$0.303321\pi$
$661$			1077.57i			1.63021i	−0.579313	+	0.815105i	$0.696679\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	470.862i		−	0.708064i
$666$		0			0
$667$	237.156	−	345.142i	0.355557	−	0.517455i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1346.64			2.00691
$672$		0			0
$673$	−87.3694			−0.129821			−0.0649104	−	0.997891i	$-0.520676\pi$
$673$	−87.3694			−0.129821			−0.0649104	+	0.997891i	$0.520676\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	347.722i		−	0.513622i	−0.966462	−	0.256811i	$-0.917328\pi$
$677$		−	347.722i		−	0.513622i	0.966462	−	0.256811i	$-0.0826719\pi$
$678$		0			0
$679$	−333.399			−0.491015
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−959.572			−1.40494			−0.702468	−	0.711715i	$-0.747919\pi$
$683$	−959.572			−1.40494			−0.702468	+	0.711715i	$0.747919\pi$
$684$		0			0
$685$	74.6533			0.108983
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	91.8394i		−	0.133294i
$690$		0			0
$691$	768.435			1.11206			0.556031	−	0.831162i	$-0.312324\pi$
$691$	768.435			1.11206			0.556031	+	0.831162i	$0.312324\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			53.8943i			0.0775458i
$696$		0			0
$697$		−	2463.49i		−	3.53442i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1068.35i		−	1.52404i	−0.647551	−	0.762022i	$-0.724207\pi$
$701$		−	1068.35i		−	1.52404i	0.647551	−	0.762022i	$-0.275793\pi$
$702$		0			0
$703$	−816.272			−1.16113
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	204.638i		−	0.289446i
$708$		0			0
$709$		−	598.457i		−	0.844085i	−0.906576	−	0.422043i	$-0.861313\pi$
$709$		−	598.457i		−	0.844085i	0.906576	−	0.422043i	$-0.138687\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	666.174	−	969.506i	0.934325	−	1.35976i
$714$		0			0
$715$	36.3325			0.0508147
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−664.339			−0.923976			−0.461988	−	0.886886i	$-0.652864\pi$
$719$	−664.339			−0.923976			−0.461988	+	0.886886i	$0.652864\pi$
$720$		0			0
$721$	26.6581			0.0369738
$722$		0			0
$723$		0			0
$724$		0			0
$725$	91.0364			0.125568
$726$		0			0
$727$		−	397.701i		−	0.547044i	−0.961866	−	0.273522i	$-0.911811\pi$
$727$		−	397.701i		−	0.547044i	0.961866	−	0.273522i	$-0.0881887\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	888.118			1.21494
$732$		0			0
$733$			693.029i			0.945470i	0.881205	+	0.472735i	$0.156733\pi$
$733$			693.029i			0.945470i	−0.881205	+	0.472735i	$0.843267\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−163.980			−0.222496
$738$		0			0
$739$	−1053.65			−1.42578			−0.712889	−	0.701276i	$-0.752614\pi$
$739$	−1053.65			−1.42578			−0.712889	+	0.701276i	$0.752614\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	224.302i		−	0.301887i	−0.988542	−	0.150943i	$-0.951769\pi$
$743$		−	224.302i		−	0.301887i	0.988542	−	0.150943i	$-0.0482311\pi$
$744$		0			0
$745$	−592.038			−0.794682
$746$		0			0
$747$		0			0
$748$		0			0
$749$	188.377			0.251505
$750$		0			0
$751$			1296.78i			1.72674i	0.504574	+	0.863368i	$0.331650\pi$
$751$			1296.78i			1.72674i	−0.504574	+	0.863368i	$0.668350\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	73.5454i		−	0.0974111i
$756$		0			0
$757$		−	988.661i		−	1.30603i	−0.757347	−	0.653013i	$-0.773505\pi$
$757$		−	988.661i		−	1.30603i	0.757347	−	0.653013i	$-0.226495\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	166.442			0.218715			0.109358	−	0.994002i	$-0.465121\pi$
$761$	166.442			0.218715			0.109358	+	0.994002i	$0.465121\pi$
$762$		0			0
$763$	−485.634			−0.636480
$764$		0			0
$765$		0			0
$766$		0			0
$767$	89.7899			0.117066
$768$		0			0
$769$			176.465i			0.229473i	0.993396	+	0.114736i	$0.0366024\pi$
$769$			176.465i			0.229473i	−0.993396	+	0.114736i	$0.963398\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			923.487i			1.19468i	0.801988	+	0.597340i	$0.203776\pi$
$773$			923.487i			1.19468i	−0.801988	+	0.597340i	$0.796224\pi$
$774$		0			0
$775$	255.722			0.329964
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1733.48i		−	2.22526i
$780$		0			0
$781$		−	825.601i		−	1.05711i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	44.0724			0.0561432
$786$		0			0
$787$		−	515.459i		−	0.654967i	−0.944857	−	0.327484i	$-0.893799\pi$
$787$		−	515.459i		−	0.654967i	0.944857	−	0.327484i	$-0.106201\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	753.371			0.952429
$792$		0			0
$793$		−	79.7780i		−	0.100603i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	2.43902i		−	0.00306025i	−0.999999	−	0.00153012i	$-0.999513\pi$
$797$		−	2.43902i		−	0.00306025i	0.999999	−	0.00153012i	$-0.000487054\pi$
$798$		0			0
$799$		−	839.997i		−	1.05131i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			17.3626i			0.0216222i
$804$		0			0
$805$	−269.363	+	392.014i	−0.334612	+	0.486973i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−638.985			−0.789846			−0.394923	−	0.918714i	$-0.629229\pi$
$809$	−638.985			−0.789846			−0.394923	+	0.918714i	$0.629229\pi$
$810$		0			0
$811$	−1425.33			−1.75750			−0.878749	−	0.477285i	$-0.841621\pi$
$811$	−1425.33			−1.75750			−0.878749	+	0.477285i	$0.841621\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			576.436i			0.707283i
$816$		0			0
$817$	624.938			0.764919
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1046.36			1.27450			0.637250	−	0.770657i	$-0.280072\pi$
$821$	1046.36			1.27450			0.637250	+	0.770657i	$0.280072\pi$
$822$		0			0
$823$	−677.830			−0.823609			−0.411804	−	0.911272i	$-0.635101\pi$
$823$	−677.830			−0.823609			−0.411804	+	0.911272i	$0.635101\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			990.322i			1.19749i	0.800941	+	0.598744i	$0.204333\pi$
$827$			990.322i			1.19749i	−0.800941	+	0.598744i	$0.795667\pi$
$828$		0			0
$829$	−1188.46			−1.43360			−0.716801	−	0.697278i	$-0.754394\pi$
$829$	−1188.46			−1.43360			−0.716801	+	0.697278i	$0.754394\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1182.08i			1.41907i
$834$		0			0
$835$			690.149i			0.826526i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	589.853i		−	0.703042i	−0.936180	−	0.351521i	$-0.885665\pi$
$839$		−	589.853i		−	0.703042i	0.936180	−	0.351521i	$-0.114335\pi$
$840$		0			0
$841$	−509.495			−0.605820
$842$		0			0
$843$		0			0
$844$		0			0
$845$			375.743i			0.444666i
$846$		0			0
$847$		−	1417.49i		−	1.67354i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	679.582	+	466.959i	0.798569	+	0.548718i
$852$		0			0
$853$	−1284.62			−1.50600			−0.753000	−	0.658021i	$-0.771394\pi$
$853$	−1284.62			−1.50600			−0.753000	+	0.658021i	$0.771394\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	505.528			0.589881			0.294940	−	0.955516i	$-0.404700\pi$
$857$	505.528			0.589881			0.294940	+	0.955516i	$0.404700\pi$
$858$		0			0
$859$	478.855			0.557457			0.278728	−	0.960370i	$-0.410087\pi$
$859$	478.855			0.557457			0.278728	+	0.960370i	$0.410087\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−54.1974			−0.0628012			−0.0314006	−	0.999507i	$-0.509997\pi$
$863$	−54.1974			−0.0628012			−0.0314006	+	0.999507i	$0.509997\pi$
$864$		0			0
$865$			497.936i			0.575648i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1721.81			1.98137
$870$		0			0
$871$			9.71455i			0.0111533i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−103.399			−0.118171
$876$		0			0
$877$	−1199.26			−1.36746			−0.683728	−	0.729737i	$-0.739642\pi$
$877$	−1199.26			−1.36746			−0.683728	+	0.729737i	$0.739642\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1429.88i		−	1.62301i	−0.584343	−	0.811507i	$-0.698648\pi$
$881$		−	1429.88i		−	1.62301i	0.584343	−	0.811507i	$-0.301352\pi$
$882$		0			0
$883$	1346.12			1.52448			0.762240	−	0.647295i	$-0.224100\pi$
$883$	1346.12			1.52448			0.762240	+	0.647295i	$0.224100\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1030.52			−1.16181			−0.580903	−	0.813973i	$-0.697301\pi$
$887$	−1030.52			−1.16181			−0.580903	+	0.813973i	$0.697301\pi$
$888$		0			0
$889$			1033.22i			1.16223i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	591.077i		−	0.661900i
$894$		0			0
$895$		−	204.342i		−	0.228315i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	931.200			1.03582
$900$		0			0
$901$	3028.91			3.36172
$902$		0			0
$903$		0			0
$904$		0			0
$905$	647.064			0.714988
$906$		0			0
$907$			1125.12i			1.24048i	0.784411	+	0.620241i	$0.212965\pi$
$907$			1125.12i			1.24048i	−0.784411	+	0.620241i	$0.787035\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1517.18i			1.66540i	0.553723	+	0.832701i	$0.313207\pi$
$911$			1517.18i			1.66540i	−0.553723	+	0.832701i	$0.686793\pi$
$912$		0			0
$913$	−1675.16			−1.83479
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2140.05i		−	2.33375i
$918$		0			0
$919$			1362.42i			1.48250i	0.671226	+	0.741252i	$0.265768\pi$
$919$			1362.42i			1.48250i	−0.671226	+	0.741252i	$0.734232\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−48.9105			−0.0529908
$924$		0			0
$925$			179.250i			0.193784i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1443.42			1.55374			0.776868	−	0.629664i	$-0.216807\pi$
$929$	1443.42			1.55374			0.776868	+	0.629664i	$0.216807\pi$
$930$		0			0
$931$			831.791i			0.893438i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			1198.27i			1.28157i
$936$		0			0
$937$			101.121i			0.107920i	0.998543	+	0.0539598i	$0.0171843\pi$
$937$			101.121i			0.107920i	−0.998543	+	0.0539598i	$0.982816\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	941.259i		−	1.00028i	−0.865946	−	0.500138i	$-0.833283\pi$
$941$		−	941.259i		−	1.00028i	0.865946	−	0.500138i	$-0.166717\pi$
$942$		0			0
$943$	−991.658	+	1443.19i	−1.05160	+	1.53043i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	403.480			0.426061			0.213031	−	0.977046i	$-0.431667\pi$
$947$	403.480			0.426061			0.213031	+	0.977046i	$0.431667\pi$
$948$		0			0
$949$	1.02860			0.00108388
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1812.26i		−	1.90164i	−0.309743	−	0.950820i	$-0.600243\pi$
$953$		−	1812.26i		−	1.90164i	0.309743	−	0.950820i	$-0.399757\pi$
$954$		0			0
$955$	352.080			0.368670
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−308.764			−0.321965
$960$		0			0
$961$	1654.75			1.72190
$962$		0			0
$963$		0			0
$964$		0			0
$965$			218.832i			0.226769i
$966$		0			0
$967$	−897.419			−0.928044			−0.464022	−	0.885824i	$-0.653594\pi$
$967$	−897.419			−0.928044			−0.464022	+	0.885824i	$0.653594\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			991.478i			1.02109i	0.859851	+	0.510545i	$0.170556\pi$
$971$			991.478i			1.02109i	−0.859851	+	0.510545i	$0.829444\pi$
$972$		0			0
$973$		−	222.906i		−	0.229091i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			147.071i			0.150533i	0.997163	+	0.0752665i	$0.0239808\pi$
$977$			147.071i			0.150533i	−0.997163	+	0.0752665i	$0.976019\pi$
$978$		0			0
$979$	2544.29			2.59887
$980$		0			0
$981$		0			0
$982$		0			0
$983$			456.715i			0.464613i	0.972643	+	0.232306i	$0.0746272\pi$
$983$			456.715i			0.464613i	−0.972643	+	0.232306i	$0.925373\pi$
$984$		0			0
$985$		−	219.933i		−	0.223283i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−520.289	−	357.504i	−0.526076	−	0.361481i
$990$		0			0
$991$	668.858			0.674932			0.337466	−	0.941338i	$-0.390430\pi$
$991$	668.858			0.674932			0.337466	+	0.941338i	$0.390430\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−315.853			−0.317440
$996$		0			0
$997$	−289.844			−0.290716			−0.145358	−	0.989379i	$-0.546433\pi$
$997$	−289.844			−0.290716			−0.145358	+	0.989379i	$0.546433\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.c.2161.13		32
3.2	odd	2		1380.3.d.a.781.30	yes	32
23.22	odd	2	inner	4140.3.d.c.2161.20		32
69.68	even	2		1380.3.d.a.781.19	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.3.d.a.781.19	✓	32	69.68	even	2
1380.3.d.a.781.30	yes	32	3.2	odd	2
4140.3.d.c.2161.13		32	1.1	even	1	trivial
4140.3.d.c.2161.20		32	23.22	odd	2	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +9.24833i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +9.24833i q^{7} -16.5611i q^{11} -0.981118 q^{13} -32.3578i q^{17} -22.7691i q^{19} +(-13.0254 + 18.9563i) q^{23} -5.00000 q^{25} -18.2073 q^{29} -51.1444 q^{31} +20.6799 q^{35} -35.8500i q^{37} +76.1329 q^{41} +27.4468i q^{43} +25.9596 q^{47} -36.5316 q^{49} +93.6068i q^{53} -37.0317 q^{55} -91.5180 q^{59} +81.3134i q^{61} +2.19385i q^{65} -9.90151i q^{67} +49.8518 q^{71} -1.04840 q^{73} +153.162 q^{77} +103.967i q^{79} -101.150i q^{83} -72.3543 q^{85} +153.631i q^{89} -9.07370i q^{91} -50.9132 q^{95} +36.0497i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) \) 32 * q + 24 * q^13 - 64 * q^23 - 160 * q^25 + 60 * q^29 - 4 * q^31 + 60 * q^35 + 108 * q^41 - 136 * q^47 - 428 * q^49 + 120 * q^55 + 84 * q^59 - 188 * q^71 + 472 * q^73 + 120 * q^77 + 60 * q^85 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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