LMFDB - Embedded newform 1452.3.e.l.485.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,3,Mod(485,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.485"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1452.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,-4,0,14,0,0,0,-2,0,-8,0,0,0,-2,0,1,0,0,0,-70,0, -54,0,0,0,-50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.5641343851$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7 x^{14} - 18 x^{13} + 41 x^{12} + 216 x^{11} + 199 x^{10} - 1278 x^{9} - 468 x^{8} + \cdots + 43046721 $ x^16 - 7x^14 - 18x^13 + 41x^12 + 216x^11 + 199x^10 - 1278x^9 - 468x^8 - 11502x^7 + 16119x^6 + 157464x^5 + 269001x^4 - 1062882x^3 - 3720087*x^2 + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.2
Root			$2.93080 + 0.640644i$ of defining polynomial
Character	$\chi$	$=$	1452.485
Dual form			1452.3.e.l.485.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.93080 + 0.640644i) q^{3} -6.71318i q^{5} -3.02703 q^{7} +(8.17915 - 3.75519i) q^{9} +0.160975 q^{13} +(4.30075 + 19.6750i) q^{15} -24.4165i q^{17} -0.622290 q^{19} +(8.87160 - 1.93924i) q^{21} -6.09109i q^{23} -20.0667 q^{25} +(-21.5657 + 16.2456i) q^{27} -52.2104i q^{29} +44.6273 q^{31} +20.3210i q^{35} +30.7814 q^{37} +(-0.471786 + 0.103128i) q^{39} -1.89383i q^{41} +79.5435 q^{43} +(-25.2093 - 54.9081i) q^{45} -14.7756i q^{47} -39.8371 q^{49} +(15.6423 + 71.5598i) q^{51} +47.7201i q^{53} +(1.82381 - 0.398666i) q^{57} +21.9283i q^{59} -61.6667 q^{61} +(-24.7585 + 11.3671i) q^{63} -1.08065i q^{65} -56.3242 q^{67} +(3.90222 + 17.8518i) q^{69} -90.1606i q^{71} -70.4497 q^{73} +(58.8116 - 12.8556i) q^{75} -98.7362 q^{79} +(52.7970 - 61.4286i) q^{81} +103.496i q^{83} -163.912 q^{85} +(33.4483 + 153.018i) q^{87} +165.487i q^{89} -0.487276 q^{91} +(-130.793 + 28.5902i) q^{93} +4.17755i q^{95} +97.8555 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7} + 14 q^{9} - 2 q^{13} - 8 q^{15} - 2 q^{19} + q^{21} - 70 q^{25} - 54 q^{27} - 50 q^{31} + 30 q^{37} + 68 q^{39} - 50 q^{43} + 17 q^{45} + 232 q^{49} + 218 q^{51} + 205 q^{57} - 80 q^{61}+ \cdots - 416 q^{97}+O(q^{100}) $ 16 * q - 4 * q^7 + 14 * q^9 - 2 * q^13 - 8 * q^15 - 2 * q^19 + q^21 - 70 * q^25 - 54 * q^27 - 50 * q^31 + 30 * q^37 + 68 * q^39 - 50 * q^43 + 17 * q^45 + 232 * q^49 + 218 * q^51 + 205 * q^57 - 80 * q^61 + 250 * q^63 + 44 * q^67 + 80 * q^69 - 450 * q^73 + 171 * q^75 - 104 * q^79 - 66 * q^81 - 130 * q^85 + 177 * q^87 - 312 * q^91 - 426 * q^93 - 416 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$1333$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	−2.93080	+	0.640644i	−0.976933	+	0.213548i
$3$	−2.93080	+	0.640644i	−0.976933	+	0.213548i
$4$		0			0
$5$		−	6.71318i		−	1.34264i	−0.741170	−	0.671318i	$-0.765729\pi$
$5$		−	6.71318i		−	1.34264i	0.741170	−	0.671318i	$-0.234271\pi$
$6$		0			0
$7$	−3.02703			−0.432432			−0.216216	−	0.976346i	$-0.569372\pi$
$7$	−3.02703			−0.432432			−0.216216	+	0.976346i	$0.569372\pi$
$8$		0			0
$9$	8.17915	−	3.75519i	0.908795	−	0.417244i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$	0.160975			0.0123827			0.00619135	−	0.999981i	$-0.498029\pi$
$13$	0.160975			0.0123827			0.00619135	+	0.999981i	$0.498029\pi$
$14$		0			0
$15$	4.30075	+	19.6750i	0.286717	+	1.31166i
$16$		0			0
$17$		−	24.4165i		−	1.43626i	−0.695906	−	0.718132i	$-0.744997\pi$
$17$		−	24.4165i		−	1.43626i	0.695906	−	0.718132i	$-0.255003\pi$
$18$		0			0
$19$	−0.622290			−0.0327521			−0.0163761	−	0.999866i	$-0.505213\pi$
$19$	−0.622290			−0.0327521			−0.0163761	+	0.999866i	$0.505213\pi$
$20$		0			0
$21$	8.87160	−	1.93924i	0.422457	−	0.0923450i
$22$		0			0
$23$		−	6.09109i		−	0.264830i	−0.991194	−	0.132415i	$-0.957727\pi$
$23$		−	6.09109i		−	0.264830i	0.991194	−	0.132415i	$-0.0422732\pi$
$24$		0			0
$25$	−20.0667			−0.802670
$26$		0			0
$27$	−21.5657	+	16.2456i	−0.798730	+	0.601690i
$28$		0			0
$29$		−	52.2104i		−	1.80036i	−0.435518	−	0.900180i	$-0.643435\pi$
$29$		−	52.2104i		−	1.80036i	0.435518	−	0.900180i	$-0.356565\pi$
$30$		0			0
$31$	44.6273			1.43959			0.719795	−	0.694187i	$-0.244236\pi$
$31$	44.6273			1.43959			0.719795	+	0.694187i	$0.244236\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			20.3210i			0.580599i
$36$		0			0
$37$	30.7814			0.831929			0.415964	−	0.909381i	$-0.363444\pi$
$37$	30.7814			0.831929			0.415964	+	0.909381i	$0.363444\pi$
$38$		0			0
$39$	−0.471786	+	0.103128i	−0.0120971	+	0.00264430i
$40$		0			0
$41$		−	1.89383i		−	0.0461909i	−0.999733	−	0.0230955i	$-0.992648\pi$
$41$		−	1.89383i		−	0.0461909i	0.999733	−	0.0230955i	$-0.00735217\pi$
$42$		0			0
$43$	79.5435			1.84985			0.924924	−	0.380152i	$-0.124128\pi$
$43$	79.5435			1.84985			0.924924	+	0.380152i	$0.124128\pi$
$44$		0			0
$45$	−25.2093	−	54.9081i	−0.560206	−	1.22018i
$46$		0			0
$47$		−	14.7756i		−	0.314375i	−0.987569	−	0.157187i	$-0.949757\pi$
$47$		−	14.7756i		−	0.314375i	0.987569	−	0.157187i	$-0.0502426\pi$
$48$		0			0
$49$	−39.8371			−0.813002
$50$		0			0
$51$	15.6423	+	71.5598i	0.306711	+	1.40313i
$52$		0			0
$53$			47.7201i			0.900379i	0.892933	+	0.450190i	$0.148644\pi$
$53$			47.7201i			0.900379i	−0.892933	+	0.450190i	$0.851356\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.82381	−	0.398666i	0.0319966	−	0.00699415i
$58$		0			0
$59$			21.9283i			0.371666i	0.982581	+	0.185833i	$0.0594983\pi$
$59$			21.9283i			0.371666i	−0.982581	+	0.185833i	$0.940502\pi$
$60$		0			0
$61$	−61.6667			−1.01093			−0.505465	−	0.862847i	$-0.668679\pi$
$61$	−61.6667			−1.01093			−0.505465	+	0.862847i	$0.668679\pi$
$62$		0			0
$63$	−24.7585	+	11.3671i	−0.392992	+	0.180430i
$64$		0			0
$65$		−	1.08065i		−	0.0166255i
$66$		0			0
$67$	−56.3242			−0.840659			−0.420330	−	0.907371i	$-0.638086\pi$
$67$	−56.3242			−0.840659			−0.420330	+	0.907371i	$0.638086\pi$
$68$		0			0
$69$	3.90222	+	17.8518i	0.0565539	+	0.258721i
$70$		0			0
$71$		−	90.1606i		−	1.26987i	−0.772566	−	0.634934i	$-0.781027\pi$
$71$		−	90.1606i		−	1.26987i	0.772566	−	0.634934i	$-0.218973\pi$
$72$		0			0
$73$	−70.4497			−0.965064			−0.482532	−	0.875878i	$-0.660283\pi$
$73$	−70.4497			−0.965064			−0.482532	+	0.875878i	$0.660283\pi$
$74$		0			0
$75$	58.8116	−	12.8556i	0.784154	−	0.171408i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−98.7362			−1.24983			−0.624913	−	0.780695i	$-0.714865\pi$
$79$	−98.7362			−1.24983			−0.624913	+	0.780695i	$0.714865\pi$
$80$		0			0
$81$	52.7970	−	61.4286i	0.651815	−	0.758378i
$82$		0			0
$83$			103.496i			1.24694i	0.781846	+	0.623472i	$0.214278\pi$
$83$			103.496i			1.24694i	−0.781846	+	0.623472i	$0.785722\pi$
$84$		0			0
$85$	−163.912			−1.92838
$86$		0			0
$87$	33.4483	+	153.018i	0.384463	+	1.75883i
$88$		0			0
$89$			165.487i			1.85940i	0.368312	+	0.929702i	$0.379936\pi$
$89$			165.487i			1.85940i	−0.368312	+	0.929702i	$0.620064\pi$
$90$		0			0
$91$	−0.487276			−0.00535468
$92$		0			0
$93$	−130.793	+	28.5902i	−1.40638	+	0.307421i
$94$		0			0
$95$			4.17755i			0.0439742i
$96$		0			0
$97$	97.8555			1.00882			0.504410	−	0.863464i	$-0.331710\pi$
$97$	97.8555			1.00882			0.504410	+	0.863464i	$0.331710\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	150.869i		−	1.49375i	−0.664965	−	0.746874i	$-0.731554\pi$
$101$		−	150.869i		−	1.49375i	0.664965	−	0.746874i	$-0.268446\pi$
$102$		0			0
$103$	21.3127			0.206919			0.103460	−	0.994634i	$-0.467009\pi$
$103$	21.3127			0.206919			0.103460	+	0.994634i	$0.467009\pi$
$104$		0			0
$105$	−13.0185	−	59.5566i	−0.123986	−	0.567206i
$106$		0			0
$107$		−	117.968i		−	1.10251i	−0.834338	−	0.551253i	$-0.814150\pi$
$107$		−	117.968i		−	1.10251i	0.834338	−	0.551253i	$-0.185850\pi$
$108$		0			0
$109$	−101.381			−0.930102			−0.465051	−	0.885284i	$-0.653964\pi$
$109$	−101.381			−0.930102			−0.465051	+	0.885284i	$0.653964\pi$
$110$		0			0
$111$	−90.2140	+	19.7199i	−0.812738	+	0.177657i
$112$		0			0
$113$		−	149.367i		−	1.32183i	−0.750459	−	0.660917i	$-0.770168\pi$
$113$		−	149.367i		−	1.32183i	0.750459	−	0.660917i	$-0.229832\pi$
$114$		0			0
$115$	−40.8906			−0.355570
$116$		0			0
$117$	1.31664	−	0.604493i	0.0112533	−	0.00516661i
$118$		0			0
$119$			73.9094i			0.621087i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	1.21327	+	5.55042i	0.00986397	+	0.0451254i
$124$		0			0
$125$		−	33.1179i		−	0.264943i
$126$		0			0
$127$	−195.108			−1.53628			−0.768141	−	0.640280i	$-0.778818\pi$
$127$	−195.108			−1.53628			−0.768141	+	0.640280i	$0.778818\pi$
$128$		0			0
$129$	−233.126	+	50.9590i	−1.80718	+	0.395031i
$130$		0			0
$131$		−	132.420i		−	1.01084i	−0.862873	−	0.505420i	$-0.831337\pi$
$131$		−	132.420i		−	1.01084i	0.862873	−	0.505420i	$-0.168663\pi$
$132$		0			0
$133$	1.88369			0.0141631
$134$		0			0
$135$	109.060	+	144.774i	0.807850	+	1.07240i
$136$		0			0
$137$			77.1905i			0.563434i	0.959498	+	0.281717i	$0.0909040\pi$
$137$			77.1905i			0.563434i	−0.959498	+	0.281717i	$0.909096\pi$
$138$		0			0
$139$	14.9947			0.107875			0.0539377	−	0.998544i	$-0.482823\pi$
$139$	14.9947			0.107875			0.0539377	+	0.998544i	$0.482823\pi$
$140$		0			0
$141$	9.46590	+	43.3043i	0.0671340	+	0.307123i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−350.498			−2.41723
$146$		0			0
$147$	116.755	−	25.5214i	0.794248	−	0.173615i
$148$		0			0
$149$			5.13835i			0.0344855i	0.999851	+	0.0172428i	$0.00548882\pi$
$149$			5.13835i			0.0344855i	−0.999851	+	0.0172428i	$0.994511\pi$
$150$		0			0
$151$	−169.710			−1.12391			−0.561954	−	0.827168i	$-0.689950\pi$
$151$	−169.710			−1.12391			−0.561954	+	0.827168i	$0.689950\pi$
$152$		0			0
$153$	−91.6887	−	199.706i	−0.599273	−	1.30527i
$154$		0			0
$155$		−	299.591i		−	1.93284i
$156$		0			0
$157$	51.9190			0.330694			0.165347	−	0.986235i	$-0.447126\pi$
$157$	51.9190			0.330694			0.165347	+	0.986235i	$0.447126\pi$
$158$		0			0
$159$	−30.5716	−	139.858i	−0.192274	−	0.879610i
$160$		0			0
$161$			18.4379i			0.114521i
$162$		0			0
$163$	97.3027			0.596949			0.298475	−	0.954418i	$-0.403522\pi$
$163$	97.3027			0.596949			0.298475	+	0.954418i	$0.403522\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			162.320i			0.971974i	0.873966	+	0.485987i	$0.161540\pi$
$167$			162.320i			0.971974i	−0.873966	+	0.485987i	$0.838460\pi$
$168$		0			0
$169$	−168.974			−0.999847
$170$		0			0
$171$	−5.08981	+	2.33682i	−0.0297650	+	0.0136656i
$172$		0			0
$173$		−	198.267i		−	1.14605i	−0.819537	−	0.573026i	$-0.805770\pi$
$173$		−	198.267i		−	1.14605i	0.819537	−	0.573026i	$-0.194230\pi$
$174$		0			0
$175$	60.7425			0.347100
$176$		0			0
$177$	−14.0482	−	64.2673i	−0.0793684	−	0.363092i
$178$		0			0
$179$			195.002i			1.08940i	0.838631	+	0.544699i	$0.183356\pi$
$179$			195.002i			1.08940i	−0.838631	+	0.544699i	$0.816644\pi$
$180$		0			0
$181$	23.7912			0.131443			0.0657215	−	0.997838i	$-0.479065\pi$
$181$	23.7912			0.131443			0.0657215	+	0.997838i	$0.479065\pi$
$182$		0			0
$183$	180.733	−	39.5064i	0.987610	−	0.215882i
$184$		0			0
$185$		−	206.641i		−	1.11698i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	65.2799	−	49.1760i	0.345396	−	0.260190i
$190$		0			0
$191$			152.246i			0.797101i	0.917146	+	0.398550i	$0.130487\pi$
$191$			152.246i			0.797101i	−0.917146	+	0.398550i	$0.869513\pi$
$192$		0			0
$193$	−128.743			−0.667063			−0.333531	−	0.942739i	$-0.608240\pi$
$193$	−128.743			−0.667063			−0.333531	+	0.942739i	$0.608240\pi$
$194$		0			0
$195$	0.692315	+	3.16718i	0.00355033	+	0.0162420i
$196$		0			0
$197$			141.406i			0.717798i	0.933376	+	0.358899i	$0.116848\pi$
$197$			141.406i			0.717798i	−0.933376	+	0.358899i	$0.883152\pi$
$198$		0			0
$199$	−162.261			−0.815381			−0.407690	−	0.913120i	$-0.633666\pi$
$199$	−162.261			−0.815381			−0.407690	+	0.913120i	$0.633666\pi$
$200$		0			0
$201$	165.075	−	36.0837i	0.821267	−	0.179521i
$202$		0			0
$203$			158.042i			0.778534i
$204$		0			0
$205$	−12.7136			−0.0620175
$206$		0			0
$207$	−22.8732	−	49.8199i	−0.110499	−	0.240676i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−250.547			−1.18742			−0.593712	−	0.804677i	$-0.702338\pi$
$211$	−250.547			−1.18742			−0.593712	+	0.804677i	$0.702338\pi$
$212$		0			0
$213$	57.7608	+	264.243i	0.271178	+	1.24058i
$214$		0			0
$215$		−	533.989i		−	2.48367i
$216$		0			0
$217$	−135.088			−0.622525
$218$		0			0
$219$	206.474	−	45.1332i	0.942803	−	0.206087i
$220$		0			0
$221$		−	3.93045i		−	0.0177848i
$222$		0			0
$223$	−73.8810			−0.331305			−0.165652	−	0.986184i	$-0.552973\pi$
$223$	−73.8810			−0.331305			−0.165652	+	0.986184i	$0.552973\pi$
$224$		0			0
$225$	−164.129	+	75.3545i	−0.729462	+	0.334909i
$226$		0			0
$227$		−	27.4064i		−	0.120733i	−0.998176	−	0.0603666i	$-0.980773\pi$
$227$		−	27.4064i		−	0.120733i	0.998176	−	0.0603666i	$-0.0192270\pi$
$228$		0			0
$229$	9.00423			0.0393198			0.0196599	−	0.999807i	$-0.493742\pi$
$229$	9.00423			0.0393198			0.0196599	+	0.999807i	$0.493742\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	119.019i		−	0.510813i	−0.966834	−	0.255407i	$-0.917791\pi$
$233$		−	119.019i		−	0.510813i	0.966834	−	0.255407i	$-0.0822093\pi$
$234$		0			0
$235$	−99.1913			−0.422090
$236$		0			0
$237$	289.376	−	63.2547i	1.22099	−	0.266897i
$238$		0			0
$239$			367.492i			1.53762i	0.639475	+	0.768812i	$0.279152\pi$
$239$			367.492i			1.53762i	−0.639475	+	0.768812i	$0.720848\pi$
$240$		0			0
$241$	−237.557			−0.985713			−0.492856	−	0.870111i	$-0.664047\pi$
$241$	−237.557			−0.985713			−0.492856	+	0.870111i	$0.664047\pi$
$242$		0			0
$243$	−115.384	+	213.859i	−0.474830	+	0.880078i
$244$		0			0
$245$			267.434i			1.09157i
$246$		0			0
$247$	−0.100173			−0.000405560
$248$		0			0
$249$	−66.3043	−	303.327i	−0.266282	−	1.21818i
$250$		0			0
$251$		−	62.6930i		−	0.249773i	−0.992171	−	0.124886i	$-0.960143\pi$
$251$		−	62.6930i		−	0.249773i	0.992171	−	0.124886i	$-0.0398567\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	480.394	−	105.009i	1.88390	−	0.411801i
$256$		0			0
$257$		−	210.876i		−	0.820530i	−0.911966	−	0.410265i	$-0.865436\pi$
$257$		−	210.876i		−	0.820530i	0.911966	−	0.410265i	$-0.134564\pi$
$258$		0			0
$259$	−93.1760			−0.359753
$260$		0			0
$261$	−196.060	−	427.037i	−0.751189	−	1.63616i
$262$		0			0
$263$			113.017i			0.429723i	0.976645	+	0.214861i	$0.0689300\pi$
$263$			113.017i			0.429723i	−0.976645	+	0.214861i	$0.931070\pi$
$264$		0			0
$265$	320.353			1.20888
$266$		0			0
$267$	−106.018	−	485.009i	−0.397072	−	1.81651i
$268$		0			0
$269$		−	475.033i		−	1.76592i	−0.469446	−	0.882961i	$-0.655546\pi$
$269$		−	475.033i		−	1.76592i	0.469446	−	0.882961i	$-0.344454\pi$
$270$		0			0
$271$	−265.303			−0.978980			−0.489490	−	0.872009i	$-0.662817\pi$
$271$	−265.303			−0.978980			−0.489490	+	0.872009i	$0.662817\pi$
$272$		0			0
$273$	1.42811	−	0.312170i	0.00523116	−	0.00114348i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−151.472			−0.546832			−0.273416	−	0.961896i	$-0.588153\pi$
$277$	−151.472			−0.546832			−0.273416	+	0.961896i	$0.588153\pi$
$278$		0			0
$279$	365.013	−	167.584i	1.30829	−	0.600660i
$280$		0			0
$281$			383.206i			1.36372i	0.731482	+	0.681861i	$0.238829\pi$
$281$			383.206i			1.36372i	−0.731482	+	0.681861i	$0.761171\pi$
$282$		0			0
$283$	414.272			1.46386			0.731929	−	0.681381i	$-0.238620\pi$
$283$	414.272			1.46386			0.731929	+	0.681381i	$0.238620\pi$
$284$		0			0
$285$	−2.67632	−	12.2435i	−0.00939059	−	0.0429598i
$286$		0			0
$287$			5.73266i			0.0199744i
$288$		0			0
$289$	−307.166			−1.06286
$290$		0			0
$291$	−286.795	+	62.6905i	−0.985549	+	0.215431i
$292$		0			0
$293$			347.883i			1.18732i	0.804718	+	0.593658i	$0.202317\pi$
$293$			347.883i			1.18732i	−0.804718	+	0.593658i	$0.797683\pi$
$294$		0			0
$295$	147.208			0.499012
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	0.980514i		−	0.00327931i
$300$		0			0
$301$	−240.780			−0.799934
$302$		0			0
$303$	96.6530	+	442.165i	0.318987	+	1.45929i
$304$		0			0
$305$			413.979i			1.35731i
$306$		0			0
$307$	507.760			1.65394			0.826971	−	0.562245i	$-0.190062\pi$
$307$	507.760			1.65394			0.826971	+	0.562245i	$0.190062\pi$
$308$		0			0
$309$	−62.4631	+	13.6538i	−0.202146	+	0.0441871i
$310$		0			0
$311$		−	91.3825i		−	0.293834i	−0.989149	−	0.146917i	$-0.953065\pi$
$311$		−	91.3825i		−	0.293834i	0.989149	−	0.146917i	$-0.0469351\pi$
$312$		0			0
$313$	478.683			1.52934			0.764669	−	0.644423i	$-0.222903\pi$
$313$	478.683			1.52934			0.764669	+	0.644423i	$0.222903\pi$
$314$		0			0
$315$	76.3091	+	166.208i	0.242251	+	0.527645i
$316$		0			0
$317$			503.112i			1.58710i	0.608503	+	0.793552i	$0.291770\pi$
$317$			503.112i			1.58710i	−0.608503	+	0.793552i	$0.708230\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	75.5756	+	345.741i	0.235438	+	1.07707i
$322$		0			0
$323$			15.1942i			0.0470407i
$324$		0			0
$325$	−3.23025			−0.00993922
$326$		0			0
$327$	297.128	−	64.9492i	0.908647	−	0.198621i
$328$		0			0
$329$			44.7261i			0.135946i
$330$		0			0
$331$	154.468			0.466670			0.233335	−	0.972396i	$-0.425036\pi$
$331$	154.468			0.466670			0.233335	+	0.972396i	$0.425036\pi$
$332$		0			0
$333$	251.765	−	115.590i	0.756053	−	0.347117i
$334$		0			0
$335$			378.114i			1.12870i
$336$		0			0
$337$	−303.331			−0.900093			−0.450046	−	0.893005i	$-0.648593\pi$
$337$	−303.331			−0.900093			−0.450046	+	0.893005i	$0.648593\pi$
$338$		0			0
$339$	95.6911	+	437.765i	0.282275	+	1.29134i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	268.912			0.784001
$344$		0			0
$345$	119.842	−	26.1963i	0.347368	−	0.0759312i
$346$		0			0
$347$			323.307i			0.931720i	0.884858	+	0.465860i	$0.154255\pi$
$347$			323.307i			0.931720i	−0.884858	+	0.465860i	$0.845745\pi$
$348$		0			0
$349$	557.743			1.59812			0.799059	−	0.601253i	$-0.205332\pi$
$349$	557.743			1.59812			0.799059	+	0.601253i	$0.205332\pi$
$350$		0			0
$351$	−3.47154	+	2.61514i	−0.00989043	+	0.00745055i
$352$		0			0
$353$			31.0942i			0.0880857i	0.999030	+	0.0440428i	$0.0140238\pi$
$353$			31.0942i			0.0880857i	−0.999030	+	0.0440428i	$0.985976\pi$
$354$		0			0
$355$	−605.264			−1.70497
$356$		0			0
$357$	−47.3496	−	216.613i	−0.132632	−	0.606760i
$358$		0			0
$359$		−	104.701i		−	0.291646i	−0.989311	−	0.145823i	$-0.953417\pi$
$359$		−	104.701i		−	0.291646i	0.989311	−	0.145823i	$-0.0465829\pi$
$360$		0			0
$361$	−360.613			−0.998927
$362$		0			0
$363$		0			0
$364$		0			0
$365$			472.941i			1.29573i
$366$		0			0
$367$	135.588			0.369448			0.184724	−	0.982790i	$-0.440861\pi$
$367$	135.588			0.369448			0.184724	+	0.982790i	$0.440861\pi$
$368$		0			0
$369$	−7.11169	−	15.4899i	−0.0192729	−	0.0419780i
$370$		0			0
$371$		−	144.450i		−	0.389353i
$372$		0			0
$373$	−275.618			−0.738921			−0.369461	−	0.929246i	$-0.620457\pi$
$373$	−275.618			−0.738921			−0.369461	+	0.929246i	$0.620457\pi$
$374$		0			0
$375$	21.2167	+	97.0617i	0.0565780	+	0.258831i
$376$		0			0
$377$		−	8.40458i		−	0.0222933i
$378$		0			0
$379$	281.262			0.742116			0.371058	−	0.928610i	$-0.378995\pi$
$379$	281.262			0.742116			0.371058	+	0.928610i	$0.378995\pi$
$380$		0			0
$381$	571.822	−	124.995i	1.50084	−	0.328070i
$382$		0			0
$383$		−	396.904i		−	1.03630i	−0.855289	−	0.518151i	$-0.826621\pi$
$383$		−	396.904i		−	1.03630i	0.855289	−	0.518151i	$-0.173379\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	650.598	−	298.701i	1.68113	−	0.771837i
$388$		0			0
$389$		−	83.2177i		−	0.213927i	−0.994263	−	0.106964i	$-0.965887\pi$
$389$		−	83.2177i		−	0.213927i	0.994263	−	0.106964i	$-0.0341128\pi$
$390$		0			0
$391$	−148.723			−0.380366
$392$		0			0
$393$	84.8341	+	388.097i	0.215863	+	0.987523i
$394$		0			0
$395$			662.833i			1.67806i
$396$		0			0
$397$	173.738			0.437627			0.218813	−	0.975767i	$-0.429781\pi$
$397$	173.738			0.437627			0.218813	+	0.975767i	$0.429781\pi$
$398$		0			0
$399$	−5.52071	+	1.20677i	−0.0138364	+	0.00302449i
$400$		0			0
$401$			553.248i			1.37967i	0.723966	+	0.689835i	$0.242317\pi$
$401$			553.248i			1.37967i	−0.723966	+	0.689835i	$0.757683\pi$
$402$		0			0
$403$	7.18388			0.0178260
$404$		0			0
$405$	−412.381	−	354.436i	−1.01822	−	0.875150i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−148.662			−0.363476			−0.181738	−	0.983347i	$-0.558172\pi$
$409$	−148.662			−0.363476			−0.181738	+	0.983347i	$0.558172\pi$
$410$		0			0
$411$	−49.4516	−	226.230i	−0.120320	−	0.550437i
$412$		0			0
$413$		−	66.3775i		−	0.160720i
$414$		0			0
$415$	694.789			1.67419
$416$		0			0
$417$	−43.9464	+	9.60625i	−0.105387	+	0.0230366i
$418$		0			0
$419$			326.983i			0.780389i	0.920732	+	0.390195i	$0.127592\pi$
$419$			326.983i			0.780389i	−0.920732	+	0.390195i	$0.872408\pi$
$420$		0			0
$421$	471.641			1.12029			0.560143	−	0.828396i	$-0.310746\pi$
$421$	471.641			1.12029			0.560143	+	0.828396i	$0.310746\pi$
$422$		0			0
$423$	−55.4853	−	120.852i	−0.131171	−	0.285702i
$424$		0			0
$425$			489.960i			1.15285i
$426$		0			0
$427$	186.667			0.437158
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	565.829i		−	1.31283i	−0.754401	−	0.656414i	$-0.772072\pi$
$431$		−	565.829i		−	1.31283i	0.754401	−	0.656414i	$-0.227928\pi$
$432$		0			0
$433$	−573.754			−1.32507			−0.662534	−	0.749032i	$-0.730519\pi$
$433$	−573.754			−1.32507			−0.662534	+	0.749032i	$0.730519\pi$
$434$		0			0
$435$	1027.24	−	224.544i	2.36147	−	0.516194i
$436$		0			0
$437$			3.79043i			0.00867374i
$438$		0			0
$439$	262.112			0.597066			0.298533	−	0.954399i	$-0.403503\pi$
$439$	262.112			0.597066			0.298533	+	0.954399i	$0.403503\pi$
$440$		0			0
$441$	−325.834	+	149.596i	−0.738852	+	0.339220i
$442$		0			0
$443$		−	409.726i		−	0.924889i	−0.886648	−	0.462444i	$-0.846972\pi$
$443$		−	409.726i		−	0.924889i	0.886648	−	0.462444i	$-0.153028\pi$
$444$		0			0
$445$	1110.94			2.49650
$446$		0			0
$447$	−3.29185	−	15.0595i	−0.00736432	−	0.0336901i
$448$		0			0
$449$		−	210.760i		−	0.469398i	−0.972068	−	0.234699i	$-0.924590\pi$
$449$		−	210.760i		−	0.469398i	0.972068	−	0.234699i	$-0.0754104\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	497.386	−	108.724i	1.09798	−	0.240008i
$454$		0			0
$455$			3.27117i			0.00718939i
$456$		0			0
$457$	−96.5269			−0.211219			−0.105609	−	0.994408i	$-0.533679\pi$
$457$	−96.5269			−0.211219			−0.105609	+	0.994408i	$0.533679\pi$
$458$		0			0
$459$	396.662	+	526.559i	0.864186	+	1.14719i
$460$		0			0
$461$		−	542.354i		−	1.17647i	−0.808689	−	0.588237i	$-0.799822\pi$
$461$		−	542.354i		−	1.17647i	0.808689	−	0.588237i	$-0.200178\pi$
$462$		0			0
$463$	488.061			1.05413			0.527064	−	0.849826i	$-0.323293\pi$
$463$	488.061			1.05413			0.527064	+	0.849826i	$0.323293\pi$
$464$		0			0
$465$	191.931	+	878.040i	0.412755	+	1.88826i
$466$		0			0
$467$		−	535.757i		−	1.14723i	−0.819124	−	0.573616i	$-0.805540\pi$
$467$		−	535.757i		−	1.14723i	0.819124	−	0.573616i	$-0.194460\pi$
$468$		0			0
$469$	170.495			0.363528
$470$		0			0
$471$	−152.164	+	33.2616i	−0.323066	+	0.0706191i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	12.4873			0.0262891
$476$		0			0
$477$	179.198	+	390.310i	0.375678	+	0.818260i
$478$		0			0
$479$		−	556.175i		−	1.16112i	−0.814219	−	0.580558i	$-0.802834\pi$
$479$		−	556.175i		−	1.16112i	0.814219	−	0.580558i	$-0.197166\pi$
$480$		0			0
$481$	4.95504			0.0103015
$482$		0			0
$483$	−11.8121	−	54.0377i	−0.0244557	−	0.111879i
$484$		0			0
$485$		−	656.921i		−	1.35448i
$486$		0			0
$487$	−488.920			−1.00394			−0.501971	−	0.864884i	$-0.667392\pi$
$487$	−488.920			−1.00394			−0.501971	+	0.864884i	$0.667392\pi$
$488$		0			0
$489$	−285.175	+	62.3363i	−0.583179	+	0.127477i
$490$		0			0
$491$		−	421.420i		−	0.858289i	−0.903236	−	0.429145i	$-0.858815\pi$
$491$		−	421.420i		−	0.858289i	0.903236	−	0.429145i	$-0.141185\pi$
$492$		0			0
$493$	−1274.80			−2.58579
$494$		0			0
$495$		0			0
$496$		0			0
$497$			272.919i			0.549132i
$498$		0			0
$499$	−541.359			−1.08489			−0.542444	−	0.840092i	$-0.682501\pi$
$499$	−541.359			−1.08489			−0.542444	+	0.840092i	$0.682501\pi$
$500$		0			0
$501$	−103.989	−	475.726i	−0.207563	−	0.949553i
$502$		0			0
$503$		−	476.681i		−	0.947675i	−0.880612	−	0.473838i	$-0.842868\pi$
$503$		−	476.681i		−	0.947675i	0.880612	−	0.473838i	$-0.157132\pi$
$504$		0			0
$505$	−1012.81			−2.00556
$506$		0			0
$507$	495.229	−	108.252i	0.976783	−	0.213515i
$508$		0			0
$509$		−	59.1682i		−	0.116244i	−0.998309	−	0.0581220i	$-0.981489\pi$
$509$		−	59.1682i		−	0.116244i	0.998309	−	0.0581220i	$-0.0185112\pi$
$510$		0			0
$511$	213.253			0.417325
$512$		0			0
$513$	13.4201	−	10.1095i	0.0261601	−	0.0197066i
$514$		0			0
$515$		−	143.076i		−	0.277817i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	127.018	+	581.080i	0.244737	+	1.11961i
$520$		0			0
$521$			347.098i			0.666214i	0.942889	+	0.333107i	$0.108097\pi$
$521$			347.098i			0.666214i	−0.942889	+	0.333107i	$0.891903\pi$
$522$		0			0
$523$	768.286			1.46900			0.734499	−	0.678609i	$-0.237417\pi$
$523$	768.286			1.46900			0.734499	+	0.678609i	$0.237417\pi$
$524$		0			0
$525$	−178.024	+	38.9143i	−0.339094	+	0.0741225i
$526$		0			0
$527$		−	1089.64i		−	2.06763i
$528$		0			0
$529$	491.899			0.929865
$530$		0			0
$531$	82.3449	+	179.355i	0.155075	+	0.337768i
$532$		0			0
$533$		−	0.304859i		−	0.000571968i
$534$		0			0
$535$	−791.942			−1.48026
$536$		0			0
$537$	−124.927	−	571.512i	−0.232639	−	1.06427i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−477.003			−0.881705			−0.440853	−	0.897579i	$-0.645324\pi$
$541$	−477.003			−0.881705			−0.440853	+	0.897579i	$0.645324\pi$
$542$		0			0
$543$	−69.7272	+	15.2417i	−0.128411	+	0.0280694i
$544$		0			0
$545$			680.590i			1.24879i
$546$		0			0
$547$	32.6495			0.0596884			0.0298442	−	0.999555i	$-0.490499\pi$
$547$	32.6495			0.0596884			0.0298442	+	0.999555i	$0.490499\pi$
$548$		0			0
$549$	−504.381	+	231.570i	−0.918727	+	0.421804i
$550$		0			0
$551$			32.4901i			0.0589656i
$552$		0			0
$553$	298.877			0.540465
$554$		0			0
$555$	132.383	+	605.622i	0.238528	+	1.09121i
$556$		0			0
$557$		−	791.760i		−	1.42147i	−0.703459	−	0.710736i	$-0.748362\pi$
$557$		−	791.760i		−	1.42147i	0.703459	−	0.710736i	$-0.251638\pi$
$558$		0			0
$559$	12.8045			0.0229061
$560$		0			0
$561$		0			0
$562$		0			0
$563$			526.375i			0.934946i	0.884007	+	0.467473i	$0.154835\pi$
$563$			526.375i			0.934946i	−0.884007	+	0.467473i	$0.845165\pi$
$564$		0			0
$565$	−1002.73			−1.77474
$566$		0			0
$567$	−159.818	+	185.946i	−0.281866	+	0.327947i
$568$		0			0
$569$		−	725.934i		−	1.27581i	−0.770117	−	0.637903i	$-0.779802\pi$
$569$		−	725.934i		−	1.27581i	0.770117	−	0.637903i	$-0.220198\pi$
$570$		0			0
$571$	−227.235			−0.397959			−0.198980	−	0.980004i	$-0.563763\pi$
$571$	−227.235			−0.397959			−0.198980	+	0.980004i	$0.563763\pi$
$572$		0			0
$573$	−97.5356	−	446.203i	−0.170219	−	0.778714i
$574$		0			0
$575$			122.228i			0.212571i
$576$		0			0
$577$	497.327			0.861919			0.430960	−	0.902371i	$-0.358175\pi$
$577$	497.327			0.861919			0.430960	+	0.902371i	$0.358175\pi$
$578$		0			0
$579$	377.320	−	82.4784i	0.651675	−	0.142450i
$580$		0			0
$581$		−	313.286i		−	0.539219i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−4.05807	−	8.83884i	−0.00693687	−	0.0151091i
$586$		0			0
$587$			225.177i			0.383607i	0.981433	+	0.191803i	$0.0614336\pi$
$587$			225.177i			0.383607i	−0.981433	+	0.191803i	$0.938566\pi$
$588$		0			0
$589$	−27.7711			−0.0471496
$590$		0			0
$591$	−90.5910	−	414.433i	−0.153284	−	0.701240i
$592$		0			0
$593$		−	418.371i		−	0.705516i	−0.935715	−	0.352758i	$-0.885244\pi$
$593$		−	418.371i		−	0.705516i	0.935715	−	0.352758i	$-0.114756\pi$
$594$		0			0
$595$	496.167			0.833894
$596$		0			0
$597$	475.554	−	103.951i	0.796572	−	0.174123i
$598$		0			0
$599$			426.796i			0.712514i	0.934388	+	0.356257i	$0.115947\pi$
$599$			426.796i			0.712514i	−0.934388	+	0.356257i	$0.884053\pi$
$600$		0			0
$601$	624.158			1.03853			0.519267	−	0.854612i	$-0.326205\pi$
$601$	624.158			1.03853			0.519267	+	0.854612i	$0.326205\pi$
$602$		0			0
$603$	−460.684	+	211.508i	−0.763987	+	0.350760i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−88.8697			−0.146408			−0.0732041	−	0.997317i	$-0.523322\pi$
$607$	−88.8697			−0.146408			−0.0732041	+	0.997317i	$0.523322\pi$
$608$		0			0
$609$	−101.249	−	463.190i	−0.166254	−	0.760575i
$610$		0			0
$611$		−	2.37851i		−	0.00389281i
$612$		0			0
$613$	750.203			1.22382			0.611911	−	0.790927i	$-0.290401\pi$
$613$	750.203			1.22382			0.611911	+	0.790927i	$0.290401\pi$
$614$		0			0
$615$	37.2610	−	8.14488i	0.0605870	−	0.0132437i
$616$		0			0
$617$			996.446i			1.61499i	0.589878	+	0.807493i	$0.299176\pi$
$617$			996.446i			1.61499i	−0.589878	+	0.807493i	$0.700824\pi$
$618$		0			0
$619$	−276.055			−0.445970			−0.222985	−	0.974822i	$-0.571580\pi$
$619$	−276.055			−0.445970			−0.222985	+	0.974822i	$0.571580\pi$
$620$		0			0
$621$	98.9536	+	131.359i	0.159346	+	0.211528i
$622$		0			0
$623$		−	500.933i		−	0.804066i
$624$		0			0
$625$	−723.994			−1.15839
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	751.573i		−	1.19487i
$630$		0			0
$631$	569.065			0.901846			0.450923	−	0.892563i	$-0.351095\pi$
$631$	569.065			0.901846			0.450923	+	0.892563i	$0.351095\pi$
$632$		0			0
$633$	734.302	−	160.511i	1.16003	−	0.253572i
$634$		0			0
$635$			1309.79i			2.06267i
$636$		0			0
$637$	−6.41279			−0.0100672
$638$		0			0
$639$	−338.571	−	737.438i	−0.529845	−	1.15405i
$640$		0			0
$641$		−	244.812i		−	0.381921i	−0.981598	−	0.190961i	$-0.938840\pi$
$641$		−	244.812i		−	0.381921i	0.981598	−	0.190961i	$-0.0611603\pi$
$642$		0			0
$643$	−305.759			−0.475519			−0.237759	−	0.971324i	$-0.576413\pi$
$643$	−305.759			−0.475519			−0.237759	+	0.971324i	$0.576413\pi$
$644$		0			0
$645$	342.097	+	1565.01i	0.530383	+	2.42638i
$646$		0			0
$647$			1017.96i			1.57335i	0.617365	+	0.786677i	$0.288200\pi$
$647$			1017.96i			1.57335i	−0.617365	+	0.786677i	$0.711800\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	395.915	−	86.5432i	0.608165	−	0.132939i
$652$		0			0
$653$		−	637.028i		−	0.975540i	−0.872972	−	0.487770i	$-0.837811\pi$
$653$		−	637.028i		−	0.975540i	0.872972	−	0.487770i	$-0.162189\pi$
$654$		0			0
$655$	−888.960			−1.35719
$656$		0			0
$657$	−576.219	+	264.552i	−0.877045	+	0.402667i
$658$		0			0
$659$			163.018i			0.247373i	0.992321	+	0.123686i	$0.0394716\pi$
$659$			163.018i			0.247373i	−0.992321	+	0.123686i	$0.960528\pi$
$660$		0			0
$661$	−361.095			−0.546285			−0.273143	−	0.961974i	$-0.588063\pi$
$661$	−361.095			−0.546285			−0.273143	+	0.961974i	$0.588063\pi$
$662$		0			0
$663$	2.51802	+	11.5194i	0.00379792	+	0.0173746i
$664$		0			0
$665$		−	12.6455i		−	0.0190158i
$666$		0			0
$667$	−318.018			−0.476789
$668$		0			0
$669$	216.530	−	47.3314i	0.323662	−	0.0707494i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	415.802			0.617833			0.308917	−	0.951089i	$-0.400034\pi$
$673$	415.802			0.617833			0.308917	+	0.951089i	$0.400034\pi$
$674$		0			0
$675$	432.753	−	325.997i	0.641116	−	0.482958i
$676$		0			0
$677$		−	448.680i		−	0.662747i	−0.943500	−	0.331373i	$-0.892488\pi$
$677$		−	448.680i		−	0.662747i	0.943500	−	0.331373i	$-0.107512\pi$
$678$		0			0
$679$	−296.211			−0.436246
$680$		0			0
$681$	17.5578	+	80.3227i	0.0257823	+	0.117948i
$682$		0			0
$683$			759.557i			1.11209i	0.831152	+	0.556045i	$0.187682\pi$
$683$			759.557i			1.11209i	−0.831152	+	0.556045i	$0.812318\pi$
$684$		0			0
$685$	518.193			0.756487
$686$		0			0
$687$	−26.3896	+	5.76850i	−0.0384128	+	0.00839665i
$688$		0			0
$689$			7.68175i			0.0111491i
$690$		0			0
$691$	1127.86			1.63221			0.816104	−	0.577905i	$-0.196129\pi$
$691$	1127.86			1.63221			0.816104	+	0.577905i	$0.196129\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	100.662i		−	0.144837i
$696$		0			0
$697$	−46.2406			−0.0663424
$698$		0			0
$699$	76.2490	+	348.822i	0.109083	+	0.499030i
$700$		0			0
$701$		−	274.523i		−	0.391616i	−0.980642	−	0.195808i	$-0.937267\pi$
$701$		−	274.523i		−	0.391616i	0.980642	−	0.195808i	$-0.0627329\pi$
$702$		0			0
$703$	−19.1550			−0.0272474
$704$		0			0
$705$	290.710	−	63.5462i	0.412354	−	0.0901365i
$706$		0			0
$707$			456.683i			0.645945i
$708$		0			0
$709$	820.824			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$709$	820.824			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$710$		0			0
$711$	−807.578	+	370.773i	−1.13583	+	0.521482i
$712$		0			0
$713$		−	271.829i		−	0.381246i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−235.431	−	1077.04i	−0.328356	−	1.50215i
$718$		0			0
$719$		−	88.6310i		−	0.123270i	−0.998099	−	0.0616349i	$-0.980369\pi$
$719$		−	88.6310i		−	0.123270i	0.998099	−	0.0616349i	$-0.0196314\pi$
$720$		0			0
$721$	−64.5140			−0.0894785
$722$		0			0
$723$	696.231	−	152.189i	0.962975	−	0.210497i
$724$		0			0
$725$			1047.69i			1.44509i
$726$		0			0
$727$	−1060.77			−1.45910			−0.729552	−	0.683925i	$-0.760272\pi$
$727$	−1060.77			−1.45910			−0.729552	+	0.683925i	$0.760272\pi$
$728$		0			0
$729$	201.159	−	700.697i	0.275938	−	0.961175i
$730$		0			0
$731$		−	1942.17i		−	2.65687i
$732$		0			0
$733$	899.505			1.22716			0.613578	−	0.789634i	$-0.289730\pi$
$733$	899.505			1.22716			0.613578	+	0.789634i	$0.289730\pi$
$734$		0			0
$735$	−171.330	−	783.794i	−0.233102	−	1.06639i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−960.819			−1.30016			−0.650080	−	0.759865i	$-0.725265\pi$
$739$	−960.819			−1.30016			−0.650080	+	0.759865i	$0.725265\pi$
$740$		0			0
$741$	0.293588	−	0.0641754i	0.000396205	−	8.66065e-5i
$742$		0			0
$743$			953.907i			1.28386i	0.766764	+	0.641930i	$0.221866\pi$
$743$			953.907i			1.28386i	−0.766764	+	0.641930i	$0.778134\pi$
$744$		0			0
$745$	34.4946			0.0463015
$746$		0			0
$747$	388.649	+	846.512i	0.520279	+	1.13322i
$748$		0			0
$749$			357.093i			0.476760i
$750$		0			0
$751$	−372.940			−0.496592			−0.248296	−	0.968684i	$-0.579871\pi$
$751$	−372.940			−0.496592			−0.248296	+	0.968684i	$0.579871\pi$
$752$		0			0
$753$	40.1639	+	183.740i	0.0533385	+	0.244011i
$754$		0			0
$755$			1139.29i			1.50900i
$756$		0			0
$757$	−23.1928			−0.0306377			−0.0153189	−	0.999883i	$-0.504876\pi$
$757$	−23.1928			−0.0306377			−0.0153189	+	0.999883i	$0.504876\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	207.287i		−	0.272387i	−0.990682	−	0.136194i	$-0.956513\pi$
$761$		−	207.287i		−	0.272387i	0.990682	−	0.136194i	$-0.0434869\pi$
$762$		0			0
$763$	306.883			0.402206
$764$		0			0
$765$	−1340.66	+	615.522i	−1.75250	+	0.804604i
$766$		0			0
$767$			3.52991i			0.00460223i
$768$		0			0
$769$	−140.439			−0.182626			−0.0913130	−	0.995822i	$-0.529106\pi$
$769$	−140.439			−0.182626			−0.0913130	+	0.995822i	$0.529106\pi$
$770$		0			0
$771$	135.096	+	618.035i	0.175222	+	0.801602i
$772$		0			0
$773$			251.302i			0.325099i	0.986700	+	0.162550i	$0.0519717\pi$
$773$			251.302i			0.325099i	−0.986700	+	0.162550i	$0.948028\pi$
$774$		0			0
$775$	−895.524			−1.15551
$776$		0			0
$777$	273.080	−	59.6926i	0.351454	−	0.0768245i
$778$		0			0
$779$			1.17851i			0.00151285i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	848.192	+	1125.95i	1.08326	+	1.43800i
$784$		0			0
$785$		−	348.542i		−	0.444002i
$786$		0			0
$787$	−186.027			−0.236374			−0.118187	−	0.992991i	$-0.537708\pi$
$787$	−186.027			−0.236374			−0.118187	+	0.992991i	$0.537708\pi$
$788$		0			0
$789$	−72.4037	−	331.230i	−0.0917664	−	0.419810i
$790$		0			0
$791$			452.138i			0.571604i
$792$		0			0
$793$	−9.92680			−0.0125180
$794$		0			0
$795$	−938.891	+	205.232i	−1.18100	+	0.258154i
$796$		0			0
$797$			787.425i			0.987986i	0.869466	+	0.493993i	$0.164463\pi$
$797$			787.425i			0.987986i	−0.869466	+	0.493993i	$0.835537\pi$
$798$		0			0
$799$	−360.769			−0.451525
$800$		0			0
$801$	621.436	+	1353.54i	0.775825	+	1.68982i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	123.777			0.153760
$806$		0			0
$807$	304.327	+	1392.23i	0.377109	+	1.72519i
$808$		0			0
$809$		−	348.517i		−	0.430800i	−0.976526	−	0.215400i	$-0.930894\pi$
$809$		−	348.517i		−	0.430800i	0.976526	−	0.215400i	$-0.0691055\pi$
$810$		0			0
$811$	−1047.61			−1.29176			−0.645878	−	0.763441i	$-0.723508\pi$
$811$	−1047.61			−1.29176			−0.645878	+	0.763441i	$0.723508\pi$
$812$		0			0
$813$	777.551	−	169.965i	0.956397	−	0.209059i
$814$		0			0
$815$		−	653.210i		−	0.801485i
$816$		0			0
$817$	−49.4991			−0.0605864
$818$		0			0
$819$	−3.98551	+	1.82982i	−0.00486631	+	0.00223421i
$820$		0			0
$821$		−	352.171i		−	0.428954i	−0.976729	−	0.214477i	$-0.931195\pi$
$821$		−	352.171i		−	0.428954i	0.976729	−	0.214477i	$-0.0688047\pi$
$822$		0			0
$823$	949.556			1.15377			0.576887	−	0.816824i	$-0.304267\pi$
$823$	949.556			1.15377			0.576887	+	0.816824i	$0.304267\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			830.269i			1.00395i	0.864881	+	0.501976i	$0.167394\pi$
$827$			830.269i			1.00395i	−0.864881	+	0.501976i	$0.832606\pi$
$828$		0			0
$829$	−50.2704			−0.0606398			−0.0303199	−	0.999540i	$-0.509653\pi$
$829$	−50.2704			−0.0606398			−0.0303199	+	0.999540i	$0.509653\pi$
$830$		0			0
$831$	443.935	−	97.0398i	0.534218	−	0.116775i
$832$		0			0
$833$			972.683i			1.16769i
$834$		0			0
$835$	1089.68			1.30501
$836$		0			0
$837$	−962.418	+	724.998i	−1.14984	+	0.866187i
$838$		0			0
$839$		−	1052.41i		−	1.25436i	−0.778876	−	0.627178i	$-0.784210\pi$
$839$		−	1052.41i		−	1.25436i	0.778876	−	0.627178i	$-0.215790\pi$
$840$		0			0
$841$	−1884.93			−2.24130
$842$		0			0
$843$	−245.498	−	1123.10i	−0.291220	−	1.33226i
$844$		0			0
$845$			1134.35i			1.34243i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1214.15	+	265.401i	−1.43009	+	0.312604i
$850$		0			0
$851$		−	187.492i		−	0.220320i
$852$		0			0
$853$	610.830			0.716096			0.358048	−	0.933703i	$-0.383442\pi$
$853$	610.830			0.716096			0.358048	+	0.933703i	$0.383442\pi$
$854$		0			0
$855$	15.6875	+	34.1688i	0.0183479	+	0.0399635i
$856$		0			0
$857$		−	125.921i		−	0.146932i	−0.997298	−	0.0734661i	$-0.976594\pi$
$857$		−	125.921i		−	0.146932i	0.997298	−	0.0734661i	$-0.0234061\pi$
$858$		0			0
$859$	382.660			0.445472			0.222736	−	0.974879i	$-0.428501\pi$
$859$	382.660			0.445472			0.222736	+	0.974879i	$0.428501\pi$
$860$		0			0
$861$	−3.67259	−	16.8013i	−0.00426550	−	0.0195137i
$862$		0			0
$863$			350.666i			0.406334i	0.979144	+	0.203167i	$0.0651234\pi$
$863$			350.666i			0.406334i	−0.979144	+	0.203167i	$0.934877\pi$
$864$		0			0
$865$	−1331.00			−1.53873
$866$		0			0
$867$	900.240	−	196.784i	1.03834	−	0.226971i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−9.06679			−0.0104096
$872$		0			0
$873$	800.375	−	367.466i	0.916810	−	0.420924i
$874$		0			0
$875$			100.249i			0.114570i
$876$		0			0
$877$	−530.525			−0.604931			−0.302466	−	0.953160i	$-0.597810\pi$
$877$	−530.525			−0.604931			−0.302466	+	0.953160i	$0.597810\pi$
$878$		0			0
$879$	−222.869	−	1019.58i	−0.253549	−	1.15993i
$880$		0			0
$881$		−	1455.22i		−	1.65179i	−0.563826	−	0.825893i	$-0.690671\pi$
$881$		−	1455.22i		−	1.65179i	0.563826	−	0.825893i	$-0.309329\pi$
$882$		0			0
$883$	−735.414			−0.832859			−0.416429	−	0.909168i	$-0.636719\pi$
$883$	−735.414			−0.832859			−0.416429	+	0.909168i	$0.636719\pi$
$884$		0			0
$885$	−431.438	+	94.3081i	−0.487501	+	0.106563i
$886$		0			0
$887$		−	248.096i		−	0.279703i	−0.990173	−	0.139851i	$-0.955338\pi$
$887$		−	248.096i		−	0.279703i	0.990173	−	0.139851i	$-0.0446625\pi$
$888$		0			0
$889$	590.597			0.664338
$890$		0			0
$891$		0			0
$892$		0			0
$893$			9.19472i			0.0102964i
$894$		0			0
$895$	1309.08			1.46266
$896$		0			0
$897$	0.628160	+	2.87369i	0.000700290	+	0.00320367i
$898$		0			0
$899$		−	2330.01i		−	2.59178i
$900$		0			0
$901$	1165.16			1.29318
$902$		0			0
$903$	705.678	−	154.254i	0.781481	−	0.170824i
$904$		0			0
$905$		−	159.714i		−	0.176480i
$906$		0			0
$907$	−969.523			−1.06893			−0.534467	−	0.845190i	$-0.679488\pi$
$907$	−969.523			−1.06893			−0.534467	+	0.845190i	$0.679488\pi$
$908$		0			0
$909$	−566.541	−	1233.98i	−0.623257	−	1.35751i
$910$		0			0
$911$		−	253.147i		−	0.277879i	−0.990301	−	0.138939i	$-0.955631\pi$
$911$		−	253.147i		−	0.277879i	0.990301	−	0.138939i	$-0.0443693\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−265.213	−	1213.29i	−0.289850	−	1.32600i
$916$		0			0
$917$			400.839i			0.437120i
$918$		0			0
$919$	−1207.57			−1.31401			−0.657005	−	0.753887i	$-0.728177\pi$
$919$	−1207.57			−1.31401			−0.657005	+	0.753887i	$0.728177\pi$
$920$		0			0
$921$	−1488.14	+	325.293i	−1.61579	+	0.353196i
$922$		0			0
$923$		−	14.5136i		−	0.0157244i
$924$		0			0
$925$	−617.682			−0.667764
$926$		0			0
$927$	174.319	−	80.0332i	0.188047	−	0.0863357i
$928$		0			0
$929$		−	53.6487i		−	0.0577488i	−0.999583	−	0.0288744i	$-0.990808\pi$
$929$		−	53.6487i		−	0.0577488i	0.999583	−	0.0288744i	$-0.00919229\pi$
$930$		0			0
$931$	24.7903			0.0266276
$932$		0			0
$933$	58.5436	+	267.824i	0.0627477	+	0.287056i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	383.183			0.408947			0.204473	−	0.978872i	$-0.434452\pi$
$937$	383.183			0.408947			0.204473	+	0.978872i	$0.434452\pi$
$938$		0			0
$939$	−1402.92	+	306.665i	−1.49406	+	0.326587i
$940$		0			0
$941$		−	1223.65i		−	1.30037i	−0.759774	−	0.650187i	$-0.774691\pi$
$941$		−	1223.65i		−	1.30037i	0.759774	−	0.650187i	$-0.225309\pi$
$942$		0			0
$943$	−11.5355			−0.0122327
$944$		0			0
$945$	−330.127	−	438.236i	−0.349341	−	0.463741i
$946$		0			0
$947$		−	1190.44i		−	1.25707i	−0.777782	−	0.628534i	$-0.783655\pi$
$947$		−	1190.44i		−	1.25707i	0.777782	−	0.628534i	$-0.216345\pi$
$948$		0			0
$949$	−11.3407			−0.0119501
$950$		0			0
$951$	−322.315	−	1474.52i	−0.338923	−	1.55049i
$952$		0			0
$953$		−	117.498i		−	0.123293i	−0.998098	−	0.0616465i	$-0.980365\pi$
$953$		−	117.498i		−	0.123293i	0.998098	−	0.0616465i	$-0.0196351\pi$
$954$		0			0
$955$	1022.06			1.07022
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	233.658i		−	0.243647i
$960$		0			0
$961$	1030.59			1.07242
$962$		0			0
$963$	−442.994	−	964.880i	−0.460014	−	1.00195i
$964$		0			0
$965$			864.275i			0.895622i
$966$		0			0
$967$	488.396			0.505063			0.252531	−	0.967589i	$-0.418737\pi$
$967$	488.396			0.505063			0.252531	+	0.967589i	$0.418737\pi$
$968$		0			0
$969$	−9.73404	−	44.5310i	−0.0100454	−	0.0459556i
$970$		0			0
$971$			131.573i			0.135503i	0.997702	+	0.0677514i	$0.0215825\pi$
$971$			131.573i			0.135503i	−0.997702	+	0.0677514i	$0.978418\pi$
$972$		0			0
$973$	−45.3893			−0.0466488
$974$		0			0
$975$	9.46720	−	2.06944i	0.00970995	−	0.00212250i
$976$		0			0
$977$		−	1248.05i		−	1.27743i	−0.769442	−	0.638716i	$-0.779466\pi$
$977$		−	1248.05i		−	1.27743i	0.769442	−	0.638716i	$-0.220534\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−829.212	+	380.706i	−0.845272	+	0.388079i
$982$		0			0
$983$			831.804i			0.846189i	0.906085	+	0.423095i	$0.139056\pi$
$983$			831.804i			0.846189i	−0.906085	+	0.423095i	$0.860944\pi$
$984$		0			0
$985$	949.285			0.963741
$986$		0			0
$987$	−28.6535	−	131.083i	−0.0290309	−	0.132810i
$988$		0			0
$989$		−	484.506i		−	0.489895i
$990$		0			0
$991$	−1450.80			−1.46397			−0.731986	−	0.681319i	$-0.761407\pi$
$991$	−1450.80			−1.46397			−0.731986	+	0.681319i	$0.761407\pi$
$992$		0			0
$993$	−452.714	+	98.9587i	−0.455905	+	0.0996563i
$994$		0			0
$995$			1089.29i			1.09476i
$996$		0			0
$997$	1328.69			1.33269			0.666346	−	0.745643i	$-0.267857\pi$
$997$	1328.69			1.33269			0.666346	+	0.745643i	$0.267857\pi$
$998$		0			0
$999$	−663.822	+	500.063i	−0.664486	+	0.500563i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.3.e.l.485.2		16
3.2	odd	2	inner	1452.3.e.l.485.1		16
11.5	even	5		132.3.m.a.113.3	✓	32
11.9	even	5		132.3.m.a.125.4	yes	32
11.10	odd	2		1452.3.e.m.485.2		16
33.5	odd	10		132.3.m.a.113.4	yes	32
33.20	odd	10		132.3.m.a.125.3	yes	32
33.32	even	2		1452.3.e.m.485.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.3.m.a.113.3	✓	32	11.5	even	5
132.3.m.a.113.4	yes	32	33.5	odd	10
132.3.m.a.125.3	yes	32	33.20	odd	10
132.3.m.a.125.4	yes	32	11.9	even	5
1452.3.e.l.485.1		16	3.2	odd	2	inner
1452.3.e.l.485.2		16	1.1	even	1	trivial
1452.3.e.m.485.1		16	33.32	even	2
1452.3.e.m.485.2		16	11.10	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1452.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.93080 + 0.640644i) q^{3} -6.71318i q^{5} -3.02703 q^{7} +(8.17915 - 3.75519i) q^{9} +0.160975 q^{13} +(4.30075 + 19.6750i) q^{15} -24.4165i q^{17} -0.622290 q^{19} +(8.87160 - 1.93924i) q^{21} -6.09109i q^{23} -20.0667 q^{25} +(-21.5657 + 16.2456i) q^{27} -52.2104i q^{29} +44.6273 q^{31} +20.3210i q^{35} +30.7814 q^{37} +(-0.471786 + 0.103128i) q^{39} -1.89383i q^{41} +79.5435 q^{43} +(-25.2093 - 54.9081i) q^{45} -14.7756i q^{47} -39.8371 q^{49} +(15.6423 + 71.5598i) q^{51} +47.7201i q^{53} +(1.82381 - 0.398666i) q^{57} +21.9283i q^{59} -61.6667 q^{61} +(-24.7585 + 11.3671i) q^{63} -1.08065i q^{65} -56.3242 q^{67} +(3.90222 + 17.8518i) q^{69} -90.1606i q^{71} -70.4497 q^{73} +(58.8116 - 12.8556i) q^{75} -98.7362 q^{79} +(52.7970 - 61.4286i) q^{81} +103.496i q^{83} -163.912 q^{85} +(33.4483 + 153.018i) q^{87} +165.487i q^{89} -0.487276 q^{91} +(-130.793 + 28.5902i) q^{93} +4.17755i q^{95} +97.8555 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} + 14 q^{9} - 2 q^{13} - 8 q^{15} - 2 q^{19} + q^{21} - 70 q^{25} - 54 q^{27} - 50 q^{31} + 30 q^{37} + 68 q^{39} - 50 q^{43} + 17 q^{45} + 232 q^{49} + 218 q^{51} + 205 q^{57} - 80 q^{61}+ \cdots - 416 q^{97}+O(q^{100}) \) 16 * q - 4 * q^7 + 14 * q^9 - 2 * q^13 - 8 * q^15 - 2 * q^19 + q^21 - 70 * q^25 - 54 * q^27 - 50 * q^31 + 30 * q^37 + 68 * q^39 - 50 * q^43 + 17 * q^45 + 232 * q^49 + 218 * q^51 + 205 * q^57 - 80 * q^61 + 250 * q^63 + 44 * q^67 + 80 * q^69 - 450 * q^73 + 171 * q^75 - 104 * q^79 - 66 * q^81 - 130 * q^85 + 177 * q^87 - 312 * q^91 - 426 * q^93 - 416 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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