LMFDB - Embedded newform 1452.3.e.l.485.9

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			485.9
Root			$-1.08635 - 2.79640i$ of defining polynomial
Character	$\chi$	$=$	1452.485
Dual form			1452.3.e.l.485.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.08635 - 2.79640i) q^{3} -3.92851i q^{5} -4.86735 q^{7} +(-6.63968 - 6.07575i) q^{9} -23.4682 q^{13} +(-10.9857 - 4.26774i) q^{15} +1.24388i q^{17} -8.80998 q^{19} +(-5.28766 + 13.6111i) q^{21} +2.99618i q^{23} +9.56683 q^{25} +(-24.2032 + 11.9668i) q^{27} +51.7070i q^{29} +38.5697 q^{31} +19.1214i q^{35} +38.1187 q^{37} +(-25.4948 + 65.6265i) q^{39} -19.6699i q^{41} +8.37836 q^{43} +(-23.8686 + 26.0840i) q^{45} +52.5197i q^{47} -25.3089 q^{49} +(3.47838 + 1.35129i) q^{51} -66.7993i q^{53} +(-9.57074 + 24.6362i) q^{57} +38.1323i q^{59} +48.4584 q^{61} +(32.3177 + 29.5728i) q^{63} +92.1951i q^{65} -27.6614 q^{67} +(8.37850 + 3.25490i) q^{69} -111.327i q^{71} -72.2982 q^{73} +(10.3929 - 26.7526i) q^{75} +19.0712 q^{79} +(7.17062 + 80.6820i) q^{81} +73.2064i q^{83} +4.88659 q^{85} +(144.593 + 56.1720i) q^{87} +99.5693i q^{89} +114.228 q^{91} +(41.9003 - 107.856i) q^{93} +34.6101i q^{95} -139.199 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.08635	−	2.79640i	0.362117	−	0.932132i
$3$	1.08635	−	2.79640i	0.362117	−	0.932132i
$4$		0			0
$5$		−	3.92851i		−	0.785702i	−0.919602	−	0.392851i	$-0.871489\pi$
$5$		−	3.92851i		−	0.785702i	0.919602	−	0.392851i	$-0.128511\pi$
$6$		0			0
$7$	−4.86735			−0.695336			−0.347668	−	0.937618i	$-0.613026\pi$
$7$	−4.86735			−0.695336			−0.347668	+	0.937618i	$0.613026\pi$
$8$		0			0
$9$	−6.63968	−	6.07575i	−0.737742	−	0.675083i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$	−23.4682			−1.80525			−0.902624	−	0.430429i	$-0.858362\pi$
$13$	−23.4682			−1.80525			−0.902624	+	0.430429i	$0.858362\pi$
$14$		0			0
$15$	−10.9857	−	4.26774i	−0.732378	−	0.284516i
$16$		0			0
$17$			1.24388i			0.0731694i	0.999331	+	0.0365847i	$0.0116479\pi$
$17$			1.24388i			0.0731694i	−0.999331	+	0.0365847i	$0.988352\pi$
$18$		0			0
$19$	−8.80998			−0.463683			−0.231842	−	0.972754i	$-0.574475\pi$
$19$	−8.80998			−0.463683			−0.231842	+	0.972754i	$0.574475\pi$
$20$		0			0
$21$	−5.28766	+	13.6111i	−0.251793	+	0.648146i
$22$		0			0
$23$			2.99618i			0.130269i	0.997877	+	0.0651343i	$0.0207476\pi$
$23$			2.99618i			0.130269i	−0.997877	+	0.0651343i	$0.979252\pi$
$24$		0			0
$25$	9.56683			0.382673
$26$		0			0
$27$	−24.2032	+	11.9668i	−0.896416	+	0.443214i
$28$		0			0
$29$			51.7070i			1.78300i	0.453021	+	0.891500i	$0.350346\pi$
$29$			51.7070i			1.78300i	−0.453021	+	0.891500i	$0.649654\pi$
$30$		0			0
$31$	38.5697			1.24418			0.622092	−	0.782944i	$-0.286283\pi$
$31$	38.5697			1.24418			0.622092	+	0.782944i	$0.286283\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			19.1214i			0.546327i
$36$		0			0
$37$	38.1187			1.03024			0.515118	−	0.857119i	$-0.327748\pi$
$37$	38.1187			1.03024			0.515118	+	0.857119i	$0.327748\pi$
$38$		0			0
$39$	−25.4948	+	65.6265i	−0.653712	+	1.68273i
$40$		0			0
$41$		−	19.6699i		−	0.479755i	−0.970803	−	0.239877i	$-0.922893\pi$
$41$		−	19.6699i		−	0.479755i	0.970803	−	0.239877i	$-0.0771072\pi$
$42$		0			0
$43$	8.37836			0.194845			0.0974227	−	0.995243i	$-0.468940\pi$
$43$	8.37836			0.194845			0.0974227	+	0.995243i	$0.468940\pi$
$44$		0			0
$45$	−23.8686	+	26.0840i	−0.530414	+	0.579645i
$46$		0			0
$47$			52.5197i			1.11744i	0.829356	+	0.558720i	$0.188707\pi$
$47$			52.5197i			1.11744i	−0.829356	+	0.558720i	$0.811293\pi$
$48$		0			0
$49$	−25.3089			−0.516507
$50$		0			0
$51$	3.47838	+	1.35129i	0.0682036	+	0.0264959i
$52$		0			0
$53$		−	66.7993i		−	1.26036i	−0.776448	−	0.630182i	$-0.782980\pi$
$53$		−	66.7993i		−	1.26036i	0.776448	−	0.630182i	$-0.217020\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−9.57074	+	24.6362i	−0.167908	+	0.432214i
$58$		0			0
$59$			38.1323i			0.646310i	0.946346	+	0.323155i	$0.104744\pi$
$59$			38.1323i			0.646310i	−0.946346	+	0.323155i	$0.895256\pi$
$60$		0			0
$61$	48.4584			0.794400			0.397200	−	0.917732i	$-0.369982\pi$
$61$	48.4584			0.794400			0.397200	+	0.917732i	$0.369982\pi$
$62$		0			0
$63$	32.3177	+	29.5728i	0.512979	+	0.469410i
$64$		0			0
$65$			92.1951i			1.41839i
$66$		0			0
$67$	−27.6614			−0.412856			−0.206428	−	0.978462i	$-0.566184\pi$
$67$	−27.6614			−0.412856			−0.206428	+	0.978462i	$0.566184\pi$
$68$		0			0
$69$	8.37850	+	3.25490i	0.121428	+	0.0471725i
$70$		0			0
$71$		−	111.327i		−	1.56799i	−0.620766	−	0.783996i	$-0.713178\pi$
$71$		−	111.327i		−	1.56799i	0.620766	−	0.783996i	$-0.286822\pi$
$72$		0			0
$73$	−72.2982			−0.990386			−0.495193	−	0.868783i	$-0.664903\pi$
$73$	−72.2982			−0.990386			−0.495193	+	0.868783i	$0.664903\pi$
$74$		0			0
$75$	10.3929	−	26.7526i	0.138573	−	0.356702i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	19.0712			0.241408			0.120704	−	0.992689i	$-0.461485\pi$
$79$	19.0712			0.241408			0.120704	+	0.992689i	$0.461485\pi$
$80$		0			0
$81$	7.17062	+	80.6820i	0.0885262	+	0.996074i
$82$		0			0
$83$			73.2064i			0.882005i	0.897506	+	0.441002i	$0.145377\pi$
$83$			73.2064i			0.882005i	−0.897506	+	0.441002i	$0.854623\pi$
$84$		0			0
$85$	4.88659			0.0574893
$86$		0			0
$87$	144.593	+	56.1720i	1.66199	+	0.645655i
$88$		0			0
$89$			99.5693i			1.11876i	0.828912	+	0.559378i	$0.188960\pi$
$89$			99.5693i			1.11876i	−0.828912	+	0.559378i	$0.811040\pi$
$90$		0			0
$91$	114.228			1.25525
$92$		0			0
$93$	41.9003	−	107.856i	0.450541	−	1.15974i
$94$		0			0
$95$			34.6101i			0.364317i
$96$		0			0
$97$	−139.199			−1.43504			−0.717519	−	0.696539i	$-0.754723\pi$
$97$	−139.199			−1.43504			−0.717519	+	0.696539i	$0.754723\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	112.642i		−	1.11527i	−0.830087	−	0.557634i	$-0.811709\pi$
$101$		−	112.642i		−	1.11527i	0.830087	−	0.557634i	$-0.188291\pi$
$102$		0			0
$103$	−96.5798			−0.937668			−0.468834	−	0.883286i	$-0.655326\pi$
$103$	−96.5798			−0.937668			−0.468834	+	0.883286i	$0.655326\pi$
$104$		0			0
$105$	53.4711	+	20.7726i	0.509249	+	0.197834i
$106$		0			0
$107$		−	76.5598i		−	0.715512i	−0.933815	−	0.357756i	$-0.883542\pi$
$107$		−	76.5598i		−	0.715512i	0.933815	−	0.357756i	$-0.116458\pi$
$108$		0			0
$109$	0.480728			0.00441035			0.00220518	−	0.999998i	$-0.499298\pi$
$109$	0.480728			0.00441035			0.00220518	+	0.999998i	$0.499298\pi$
$110$		0			0
$111$	41.4104	−	106.595i	0.373066	−	0.960316i
$112$		0			0
$113$			218.286i			1.93174i	0.259035	+	0.965868i	$0.416595\pi$
$113$			218.286i			1.93174i	−0.259035	+	0.965868i	$0.583405\pi$
$114$		0			0
$115$	11.7705			0.102352
$116$		0			0
$117$	155.821	+	142.587i	1.33181	+	1.21869i
$118$		0			0
$119$		−	6.05440i		−	0.0508773i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	−55.0050	−	21.3685i	−0.447195	−	0.173727i
$124$		0			0
$125$		−	135.796i		−	1.08637i
$126$		0			0
$127$	1.07327			0.00845093			0.00422547	−	0.999991i	$-0.498655\pi$
$127$	1.07327			0.00845093			0.00422547	+	0.999991i	$0.498655\pi$
$128$		0			0
$129$	9.10185	−	23.4292i	0.0705569	−	0.181622i
$130$		0			0
$131$			199.521i			1.52306i	0.648128	+	0.761531i	$0.275552\pi$
$131$			199.521i			1.52306i	−0.648128	+	0.761531i	$0.724448\pi$
$132$		0			0
$133$	42.8813			0.322416
$134$		0			0
$135$	47.0116	+	95.0826i	0.348234	+	0.704315i
$136$		0			0
$137$		−	27.8147i		−	0.203027i	−0.994834	−	0.101514i	$-0.967631\pi$
$137$		−	27.8147i		−	0.203027i	0.994834	−	0.101514i	$-0.0323686\pi$
$138$		0			0
$139$	163.034			1.17290			0.586452	−	0.809984i	$-0.300524\pi$
$139$	163.034			1.17290			0.586452	+	0.809984i	$0.300524\pi$
$140$		0			0
$141$	146.866	+	57.0549i	1.04160	+	0.404644i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	203.131			1.40091
$146$		0			0
$147$	−27.4943	+	70.7737i	−0.187036	+	0.481453i
$148$		0			0
$149$			188.752i			1.26680i	0.773827	+	0.633398i	$0.218340\pi$
$149$			188.752i			1.26680i	−0.773827	+	0.633398i	$0.781660\pi$
$150$		0			0
$151$	−193.942			−1.28438			−0.642191	−	0.766545i	$-0.721974\pi$
$151$	−193.942			−1.28438			−0.642191	+	0.766545i	$0.721974\pi$
$152$		0			0
$153$	7.55750	−	8.25896i	0.0493954	−	0.0539801i
$154$		0			0
$155$		−	151.521i		−	0.977557i
$156$		0			0
$157$	−164.580			−1.04828			−0.524141	−	0.851631i	$-0.675614\pi$
$157$	−164.580			−1.04828			−0.524141	+	0.851631i	$0.675614\pi$
$158$		0			0
$159$	−186.797	−	72.5676i	−1.17483	−	0.456400i
$160$		0			0
$161$		−	14.5835i		−	0.0905804i
$162$		0			0
$163$	−239.237			−1.46771			−0.733855	−	0.679306i	$-0.762281\pi$
$163$	−239.237			−1.46771			−0.733855	+	0.679306i	$0.762281\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	172.264i		−	1.03152i	−0.856733	−	0.515760i	$-0.827509\pi$
$167$		−	172.264i		−	1.03152i	0.856733	−	0.515760i	$-0.172491\pi$
$168$		0			0
$169$	381.758			2.25892
$170$		0			0
$171$	58.4954	+	53.5272i	0.342078	+	0.313025i
$172$		0			0
$173$			57.7934i			0.334066i	0.985951	+	0.167033i	$0.0534187\pi$
$173$			57.7934i			0.334066i	−0.985951	+	0.167033i	$0.946581\pi$
$174$		0			0
$175$	−46.5651			−0.266086
$176$		0			0
$177$	106.633	+	41.4251i	0.602447	+	0.234040i
$178$		0			0
$179$		−	18.5040i		−	0.103375i	−0.998663	−	0.0516873i	$-0.983540\pi$
$179$		−	18.5040i		−	0.103375i	0.998663	−	0.0516873i	$-0.0164599\pi$
$180$		0			0
$181$	−32.9019			−0.181779			−0.0908893	−	0.995861i	$-0.528971\pi$
$181$	−32.9019			−0.181779			−0.0908893	+	0.995861i	$0.528971\pi$
$182$		0			0
$183$	52.6429	−	135.509i	0.287666	−	0.740486i
$184$		0			0
$185$		−	149.750i		−	0.809458i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	117.806	−	58.2465i	0.623310	−	0.308183i
$190$		0			0
$191$			221.269i			1.15848i	0.815158	+	0.579238i	$0.196650\pi$
$191$			221.269i			1.15848i	−0.815158	+	0.579238i	$0.803350\pi$
$192$		0			0
$193$	264.763			1.37183			0.685913	−	0.727683i	$-0.259403\pi$
$193$	264.763			1.37183			0.685913	+	0.727683i	$0.259403\pi$
$194$		0			0
$195$	257.814	+	100.156i	1.32212	+	0.513623i
$196$		0			0
$197$			230.752i			1.17133i	0.810553	+	0.585665i	$0.199167\pi$
$197$			230.752i			1.17133i	−0.810553	+	0.585665i	$0.800833\pi$
$198$		0			0
$199$	34.0846			0.171280			0.0856398	−	0.996326i	$-0.472707\pi$
$199$	34.0846			0.171280			0.0856398	+	0.996326i	$0.472707\pi$
$200$		0			0
$201$	−30.0500	+	77.3522i	−0.149503	+	0.384837i
$202$		0			0
$203$		−	251.676i		−	1.23978i
$204$		0			0
$205$	−77.2735			−0.376944
$206$		0			0
$207$	18.2040	−	19.8936i	0.0879421	−	0.0961046i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−392.819			−1.86170			−0.930850	−	0.365402i	$-0.880932\pi$
$211$	−392.819			−1.86170			−0.930850	+	0.365402i	$0.880932\pi$
$212$		0			0
$213$	−311.316	−	120.941i	−1.46158	−	0.567797i
$214$		0			0
$215$		−	32.9144i		−	0.153090i
$216$		0			0
$217$	−187.732			−0.865126
$218$		0			0
$219$	−78.5413	+	202.174i	−0.358636	+	0.923171i
$220$		0			0
$221$		−	29.1917i		−	0.132089i
$222$		0			0
$223$	−13.2044			−0.0592125			−0.0296063	−	0.999562i	$-0.509425\pi$
$223$	−13.2044			−0.0592125			−0.0296063	+	0.999562i	$0.509425\pi$
$224$		0			0
$225$	−63.5206	−	58.1256i	−0.282314	−	0.258336i
$226$		0			0
$227$		−	22.5130i		−	0.0991764i	−0.998770	−	0.0495882i	$-0.984209\pi$
$227$		−	22.5130i		−	0.0991764i	0.998770	−	0.0495882i	$-0.0157909\pi$
$228$		0			0
$229$	−142.304			−0.621413			−0.310706	−	0.950506i	$-0.600566\pi$
$229$	−142.304			−0.621413			−0.310706	+	0.950506i	$0.600566\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			416.553i			1.78778i	0.448284	+	0.893891i	$0.352035\pi$
$233$			416.553i			1.78778i	−0.448284	+	0.893891i	$0.647965\pi$
$234$		0			0
$235$	206.324			0.877974
$236$		0			0
$237$	20.7180	−	53.3306i	0.0874179	−	0.225024i
$238$		0			0
$239$			170.568i			0.713675i	0.934166	+	0.356837i	$0.116145\pi$
$239$			170.568i			0.713675i	−0.934166	+	0.356837i	$0.883855\pi$
$240$		0			0
$241$	151.910			0.630330			0.315165	−	0.949037i	$-0.397940\pi$
$241$	151.910			0.630330			0.315165	+	0.949037i	$0.397940\pi$
$242$		0			0
$243$	233.409	+	67.5971i	0.960530	+	0.278178i
$244$		0			0
$245$			99.4261i			0.405821i
$246$		0			0
$247$	206.755			0.837063
$248$		0			0
$249$	204.714	+	79.5279i	0.822145	+	0.319389i
$250$		0			0
$251$			388.488i			1.54776i	0.633332	+	0.773880i	$0.281687\pi$
$251$			388.488i			1.54776i	−0.633332	+	0.773880i	$0.718313\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	5.30856	−	13.6649i	0.0208179	−	0.0535877i
$256$		0			0
$257$			192.084i			0.747409i	0.927548	+	0.373705i	$0.121913\pi$
$257$			192.084i			0.747409i	−0.927548	+	0.373705i	$0.878087\pi$
$258$		0			0
$259$	−185.537			−0.716360
$260$		0			0
$261$	314.158	−	343.318i	1.20367	−	1.31539i
$262$		0			0
$263$		−	402.957i		−	1.53216i	−0.642747	−	0.766078i	$-0.722205\pi$
$263$		−	402.957i		−	1.53216i	0.642747	−	0.766078i	$-0.277795\pi$
$264$		0			0
$265$	−262.422			−0.990270
$266$		0			0
$267$	278.435	+	108.167i	1.04283	+	0.405121i
$268$		0			0
$269$			152.284i			0.566111i	0.959104	+	0.283055i	$0.0913481\pi$
$269$			152.284i			0.566111i	−0.959104	+	0.283055i	$0.908652\pi$
$270$		0			0
$271$	−521.761			−1.92532			−0.962658	−	0.270721i	$-0.912738\pi$
$271$	−521.761			−1.92532			−0.962658	+	0.270721i	$0.912738\pi$
$272$		0			0
$273$	124.092	−	319.427i	0.454550	−	1.17006i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	65.3160			0.235798			0.117899	−	0.993026i	$-0.462384\pi$
$277$	65.3160			0.235798			0.117899	+	0.993026i	$0.462384\pi$
$278$		0			0
$279$	−256.090	−	234.340i	−0.917887	−	0.839927i
$280$		0			0
$281$			45.9893i			0.163663i	0.996646	+	0.0818315i	$0.0260769\pi$
$281$			45.9893i			0.163663i	−0.996646	+	0.0818315i	$0.973923\pi$
$282$		0			0
$283$	−408.174			−1.44231			−0.721155	−	0.692774i	$-0.756388\pi$
$283$	−408.174			−1.44231			−0.721155	+	0.692774i	$0.756388\pi$
$284$		0			0
$285$	96.7835	+	37.5987i	0.339591	+	0.131925i
$286$		0			0
$287$			95.7405i			0.333591i
$288$		0			0
$289$	287.453			0.994646
$290$		0			0
$291$	−151.219	+	389.255i	−0.519653	+	1.33765i
$292$		0			0
$293$			458.355i			1.56435i	0.623059	+	0.782175i	$0.285890\pi$
$293$			458.355i			1.56435i	−0.623059	+	0.782175i	$0.714110\pi$
$294$		0			0
$295$	149.803			0.507807
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	70.3150i		−	0.235167i
$300$		0			0
$301$	−40.7804			−0.135483
$302$		0			0
$303$	−314.992	−	122.369i	−1.03958	−	0.403858i
$304$		0			0
$305$		−	190.369i		−	0.624161i
$306$		0			0
$307$	−292.014			−0.951185			−0.475592	−	0.879666i	$-0.657766\pi$
$307$	−292.014			−0.951185			−0.475592	+	0.879666i	$0.657766\pi$
$308$		0			0
$309$	−104.920	+	270.075i	−0.339546	+	0.874030i
$310$		0			0
$311$		−	128.356i		−	0.412720i	−0.978476	−	0.206360i	$-0.933838\pi$
$311$		−	128.356i		−	0.412720i	0.978476	−	0.206360i	$-0.0661618\pi$
$312$		0			0
$313$	153.622			0.490805			0.245403	−	0.969421i	$-0.421080\pi$
$313$	153.622			0.490805			0.245403	+	0.969421i	$0.421080\pi$
$314$		0			0
$315$	116.177	−	126.960i	0.368816	−	0.403048i
$316$		0			0
$317$		−	335.277i		−	1.05766i	−0.848729	−	0.528828i	$-0.822631\pi$
$317$		−	335.277i		−	1.05766i	0.848729	−	0.528828i	$-0.177369\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−214.092	−	83.1709i	−0.666952	−	0.259099i
$322$		0			0
$323$		−	10.9586i		−	0.0339274i
$324$		0			0
$325$	−224.516			−0.690820
$326$		0			0
$327$	0.522240	−	1.34431i	0.00159707	−	0.00411103i
$328$		0			0
$329$		−	255.632i		−	0.776996i
$330$		0			0
$331$	−438.407			−1.32449			−0.662246	−	0.749287i	$-0.730396\pi$
$331$	−438.407			−1.32449			−0.662246	+	0.749287i	$0.730396\pi$
$332$		0			0
$333$	−253.096	−	231.600i	−0.760048	−	0.695495i
$334$		0			0
$335$			108.668i			0.324382i
$336$		0			0
$337$	−76.1881			−0.226078			−0.113039	−	0.993591i	$-0.536058\pi$
$337$	−76.1881			−0.226078			−0.113039	+	0.993591i	$0.536058\pi$
$338$		0			0
$339$	610.415	+	237.136i	1.80063	+	0.699515i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	361.688			1.05448
$344$		0			0
$345$	12.7869	−	32.9150i	0.0370635	−	0.0954058i
$346$		0			0
$347$		−	45.5255i		−	0.131197i	−0.997846	−	0.0655987i	$-0.979104\pi$
$347$		−	45.5255i		−	0.131197i	0.997846	−	0.0655987i	$-0.0208957\pi$
$348$		0			0
$349$	44.8532			0.128519			0.0642595	−	0.997933i	$-0.479531\pi$
$349$	44.8532			0.128519			0.0642595	+	0.997933i	$0.479531\pi$
$350$		0			0
$351$	568.007	−	280.839i	1.61825	−	0.800111i
$352$		0			0
$353$			30.9900i			0.0877903i	0.999036	+	0.0438952i	$0.0139768\pi$
$353$			30.9900i			0.0877903i	−0.999036	+	0.0438952i	$0.986023\pi$
$354$		0			0
$355$	−437.351			−1.23197
$356$		0			0
$357$	−16.9305	−	6.57722i	−0.0474244	−	0.0184236i
$358$		0			0
$359$		−	442.020i		−	1.23125i	−0.788038	−	0.615627i	$-0.788903\pi$
$359$		−	442.020i		−	1.23125i	0.788038	−	0.615627i	$-0.211097\pi$
$360$		0			0
$361$	−283.384			−0.784998
$362$		0			0
$363$		0			0
$364$		0			0
$365$			284.024i			0.778148i
$366$		0			0
$367$	163.998			0.446860			0.223430	−	0.974720i	$-0.428275\pi$
$367$	163.998			0.446860			0.223430	+	0.974720i	$0.428275\pi$
$368$		0			0
$369$	−119.510	+	130.602i	−0.323874	+	0.353935i
$370$		0			0
$371$			325.136i			0.876377i
$372$		0			0
$373$	−494.457			−1.32562			−0.662810	−	0.748787i	$-0.730637\pi$
$373$	−494.457			−1.32562			−0.662810	+	0.748787i	$0.730637\pi$
$374$		0			0
$375$	−379.740	−	147.522i	−1.01264	−	0.393393i
$376$		0			0
$377$		−	1213.47i		−	3.21876i
$378$		0			0
$379$	−33.4644			−0.0882967			−0.0441483	−	0.999025i	$-0.514057\pi$
$379$	−33.4644			−0.0882967			−0.0441483	+	0.999025i	$0.514057\pi$
$380$		0			0
$381$	1.16595	−	3.00128i	0.00306023	−	0.00787739i
$382$		0			0
$383$		−	43.0569i		−	0.112420i	−0.998419	−	0.0562100i	$-0.982098\pi$
$383$		−	43.0569i		−	0.112420i	0.998419	−	0.0562100i	$-0.0179016\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−55.6296	−	50.9048i	−0.143746	−	0.131537i
$388$		0			0
$389$		−	205.336i		−	0.527857i	−0.964542	−	0.263929i	$-0.914982\pi$
$389$		−	205.336i		−	0.527857i	0.964542	−	0.263929i	$-0.0850183\pi$
$390$		0			0
$391$	−3.72688			−0.00953167
$392$		0			0
$393$	557.940	+	216.750i	1.41970	+	0.551527i
$394$		0			0
$395$		−	74.9213i		−	0.189674i
$396$		0			0
$397$	305.317			0.769061			0.384530	−	0.923112i	$-0.374363\pi$
$397$	305.317			0.769061			0.384530	+	0.923112i	$0.374363\pi$
$398$		0			0
$399$	46.5842	−	119.913i	0.116752	−	0.300534i
$400$		0			0
$401$			375.495i			0.936396i	0.883624	+	0.468198i	$0.155097\pi$
$401$			375.495i			0.936396i	−0.883624	+	0.468198i	$0.844903\pi$
$402$		0			0
$403$	−905.163			−2.24606
$404$		0			0
$405$	316.960	−	28.1699i	0.782617	−	0.0695552i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−33.0529			−0.0808140			−0.0404070	−	0.999183i	$-0.512865\pi$
$409$	−33.0529			−0.0808140			−0.0404070	+	0.999183i	$0.512865\pi$
$410$		0			0
$411$	−77.7811	−	30.2166i	−0.189248	−	0.0735197i
$412$		0			0
$413$		−	185.603i		−	0.449403i
$414$		0			0
$415$	287.592			0.692993
$416$		0			0
$417$	177.112	−	455.907i	0.424729	−	1.09330i
$418$		0			0
$419$		−	225.453i		−	0.538073i	−0.963130	−	0.269037i	$-0.913295\pi$
$419$		−	225.453i		−	0.538073i	0.963130	−	0.269037i	$-0.0867053\pi$
$420$		0			0
$421$	149.943			0.356159			0.178080	−	0.984016i	$-0.443012\pi$
$421$	149.943			0.356159			0.178080	+	0.984016i	$0.443012\pi$
$422$		0			0
$423$	319.096	−	348.714i	0.754365	−	0.824382i
$424$		0			0
$425$			11.9000i			0.0280000i
$426$		0			0
$427$	−235.864			−0.552375
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	407.178i		−	0.944730i	−0.881403	−	0.472365i	$-0.843401\pi$
$431$		−	407.178i		−	0.944730i	0.881403	−	0.472365i	$-0.156599\pi$
$432$		0			0
$433$	−193.788			−0.447547			−0.223774	−	0.974641i	$-0.571838\pi$
$433$	−193.788			−0.447547			−0.223774	+	0.974641i	$0.571838\pi$
$434$		0			0
$435$	220.672	−	568.036i	0.507292	−	1.30583i
$436$		0			0
$437$		−	26.3963i		−	0.0604033i
$438$		0			0
$439$	−511.496			−1.16514			−0.582569	−	0.812781i	$-0.697953\pi$
$439$	−511.496			−1.16514			−0.582569	+	0.812781i	$0.697953\pi$
$440$		0			0
$441$	168.043	+	153.770i	0.381049	+	0.348685i
$442$		0			0
$443$			198.177i			0.447353i	0.974663	+	0.223677i	$0.0718060\pi$
$443$			198.177i			0.447353i	−0.974663	+	0.223677i	$0.928194\pi$
$444$		0			0
$445$	391.159			0.879009
$446$		0			0
$447$	527.827	+	205.052i	1.18082	+	0.458729i
$448$		0			0
$449$			528.711i			1.17753i	0.808305	+	0.588765i	$0.200385\pi$
$449$			528.711i			1.17753i	−0.808305	+	0.588765i	$0.799615\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−210.689	+	542.338i	−0.465097	+	1.19721i
$454$		0			0
$455$		−	448.746i		−	0.986256i
$456$		0			0
$457$	494.121			1.08123			0.540614	−	0.841271i	$-0.318192\pi$
$457$	494.121			1.08123			0.540614	+	0.841271i	$0.318192\pi$
$458$		0			0
$459$	−14.8852	−	30.1059i	−0.0324297	−	0.0655902i
$460$		0			0
$461$		−	630.172i		−	1.36697i	−0.729966	−	0.683484i	$-0.760464\pi$
$461$		−	630.172i		−	1.36697i	0.729966	−	0.683484i	$-0.239536\pi$
$462$		0			0
$463$	−140.450			−0.303347			−0.151674	−	0.988431i	$-0.548466\pi$
$463$	−140.450			−0.303347			−0.151674	+	0.988431i	$0.548466\pi$
$464$		0			0
$465$	−423.714	−	164.606i	−0.911213	−	0.353991i
$466$		0			0
$467$			732.037i			1.56753i	0.621057	+	0.783766i	$0.286704\pi$
$467$			732.037i			1.56753i	−0.621057	+	0.783766i	$0.713296\pi$
$468$		0			0
$469$	134.638			0.287074
$470$		0			0
$471$	−178.792	+	460.232i	−0.379601	+	0.977138i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−84.2835			−0.177439
$476$		0			0
$477$	−405.855	+	443.526i	−0.850850	+	0.929823i
$478$		0			0
$479$		−	693.824i		−	1.44848i	−0.689545	−	0.724242i	$-0.742190\pi$
$479$		−	693.824i		−	1.44848i	0.689545	−	0.724242i	$-0.257810\pi$
$480$		0			0
$481$	−894.579			−1.85983
$482$		0			0
$483$	−40.7811	−	15.8428i	−0.0844330	−	0.0328008i
$484$		0			0
$485$			546.843i			1.12751i
$486$		0			0
$487$	−132.947			−0.272991			−0.136496	−	0.990641i	$-0.543584\pi$
$487$	−132.947			−0.272991			−0.136496	+	0.990641i	$0.543584\pi$
$488$		0			0
$489$	−259.895	+	669.001i	−0.531483	+	1.36810i
$490$		0			0
$491$		−	951.503i		−	1.93789i	−0.247280	−	0.968944i	$-0.579537\pi$
$491$		−	951.503i		−	1.93789i	0.247280	−	0.968944i	$-0.420463\pi$
$492$		0			0
$493$	−64.3173			−0.130461
$494$		0			0
$495$		0			0
$496$		0			0
$497$			541.870i			1.09028i
$498$		0			0
$499$	392.408			0.786389			0.393195	−	0.919455i	$-0.371370\pi$
$499$	392.408			0.786389			0.393195	+	0.919455i	$0.371370\pi$
$500$		0			0
$501$	−481.718	−	187.139i	−0.961514	−	0.373532i
$502$		0			0
$503$		−	548.215i		−	1.08989i	−0.838472	−	0.544945i	$-0.816550\pi$
$503$		−	548.215i		−	1.08989i	0.838472	−	0.544945i	$-0.183450\pi$
$504$		0			0
$505$	−442.515			−0.876268
$506$		0			0
$507$	414.724	−	1067.55i	0.817995	−	2.10562i
$508$		0			0
$509$			469.090i			0.921591i	0.887506	+	0.460795i	$0.152436\pi$
$509$			469.090i			0.921591i	−0.887506	+	0.460795i	$0.847564\pi$
$510$		0			0
$511$	351.901			0.688651
$512$		0			0
$513$	213.230	−	105.427i	0.415653	−	0.205511i
$514$		0			0
$515$			379.414i			0.736727i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	161.613	+	62.7840i	0.311394	+	0.120971i
$520$		0			0
$521$		−	77.8258i		−	0.149378i	−0.997207	−	0.0746888i	$-0.976204\pi$
$521$		−	77.8258i		−	0.149378i	0.997207	−	0.0746888i	$-0.0237964\pi$
$522$		0			0
$523$	−373.600			−0.714341			−0.357170	−	0.934039i	$-0.616258\pi$
$523$	−373.600			−0.714341			−0.357170	+	0.934039i	$0.616258\pi$
$524$		0			0
$525$	−50.5861	+	130.215i	−0.0963545	+	0.248028i
$526$		0			0
$527$			47.9761i			0.0910362i
$528$		0			0
$529$	520.023			0.983030
$530$		0			0
$531$	231.682	−	253.186i	0.436313	−	0.476810i
$532$		0			0
$533$			461.619i			0.866076i
$534$		0			0
$535$	−300.766			−0.562179
$536$		0			0
$537$	−51.7447	−	20.1019i	−0.0963588	−	0.0374337i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	88.0993			0.162845			0.0814226	−	0.996680i	$-0.474054\pi$
$541$	88.0993			0.162845			0.0814226	+	0.996680i	$0.474054\pi$
$542$		0			0
$543$	−35.7431	+	92.0069i	−0.0658252	+	0.169442i
$544$		0			0
$545$		−	1.88855i		−	0.00346522i
$546$		0			0
$547$	−354.731			−0.648502			−0.324251	−	0.945971i	$-0.605112\pi$
$547$	−354.731			−0.648502			−0.324251	+	0.945971i	$0.605112\pi$
$548$		0			0
$549$	−321.748	−	294.421i	−0.586062	−	0.536286i
$550$		0			0
$551$		−	455.537i		−	0.826747i
$552$		0			0
$553$	−92.8263			−0.167859
$554$		0			0
$555$	−418.760	−	162.681i	−0.754522	−	0.293119i
$556$		0			0
$557$		−	441.848i		−	0.793264i	−0.917978	−	0.396632i	$-0.870179\pi$
$557$		−	441.848i		−	0.793264i	0.917978	−	0.396632i	$-0.129821\pi$
$558$		0			0
$559$	−196.625			−0.351745
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	732.386i		−	1.30086i	−0.759565	−	0.650432i	$-0.774588\pi$
$563$		−	732.386i		−	1.30086i	0.759565	−	0.650432i	$-0.225412\pi$
$564$		0			0
$565$	857.539			1.51777
$566$		0			0
$567$	−34.9020	−	392.708i	−0.0615555	−	0.692606i
$568$		0			0
$569$			460.340i			0.809034i	0.914531	+	0.404517i	$0.132560\pi$
$569$			460.340i			0.809034i	−0.914531	+	0.404517i	$0.867440\pi$
$570$		0			0
$571$	−64.8107			−0.113504			−0.0567519	−	0.998388i	$-0.518074\pi$
$571$	−64.8107			−0.113504			−0.0567519	+	0.998388i	$0.518074\pi$
$572$		0			0
$573$	618.756	+	240.376i	1.07985	+	0.419505i
$574$		0			0
$575$			28.6639i			0.0498503i
$576$		0			0
$577$	−222.871			−0.386258			−0.193129	−	0.981173i	$-0.561864\pi$
$577$	−222.871			−0.386258			−0.193129	+	0.981173i	$0.561864\pi$
$578$		0			0
$579$	287.625	−	740.381i	0.496762	−	1.27872i
$580$		0			0
$581$		−	356.321i		−	0.613290i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	560.154	−	612.146i	0.957529	−	1.04640i
$586$		0			0
$587$		−	783.407i		−	1.33459i	−0.744792	−	0.667297i	$-0.767451\pi$
$587$		−	783.407i		−	1.33459i	0.744792	−	0.667297i	$-0.232549\pi$
$588$		0			0
$589$	−339.798			−0.576907
$590$		0			0
$591$	645.274	+	250.678i	1.09183	+	0.424159i
$592$		0			0
$593$		−	258.436i		−	0.435812i	−0.975970	−	0.217906i	$-0.930077\pi$
$593$		−	258.436i		−	0.435812i	0.975970	−	0.217906i	$-0.0699226\pi$
$594$		0			0
$595$	−23.7848			−0.0399744
$596$		0			0
$597$	37.0279	−	95.3142i	0.0620233	−	0.159655i
$598$		0			0
$599$		−	708.581i		−	1.18294i	−0.806327	−	0.591470i	$-0.798548\pi$
$599$		−	708.581i		−	1.18294i	0.806327	−	0.591470i	$-0.201452\pi$
$600$		0			0
$601$	720.547			1.19891			0.599457	−	0.800407i	$-0.295383\pi$
$601$	720.547			1.19891			0.599457	+	0.800407i	$0.295383\pi$
$602$		0			0
$603$	183.663	+	168.064i	0.304581	+	0.278712i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	107.231			0.176657			0.0883287	−	0.996091i	$-0.471847\pi$
$607$	107.231			0.176657			0.0883287	+	0.996091i	$0.471847\pi$
$608$		0			0
$609$	−703.787	−	273.409i	−1.15564	−	0.448947i
$610$		0			0
$611$		−	1232.54i		−	2.01726i
$612$		0			0
$613$	−828.985			−1.35234			−0.676170	−	0.736746i	$-0.736362\pi$
$613$	−828.985			−1.35234			−0.676170	+	0.736746i	$0.736362\pi$
$614$		0			0
$615$	−83.9462	+	216.087i	−0.136498	+	0.351362i
$616$		0			0
$617$		−	44.6300i		−	0.0723338i	−0.999346	−	0.0361669i	$-0.988485\pi$
$617$		−	44.6300i		−	0.0723338i	0.999346	−	0.0361669i	$-0.0115148\pi$
$618$		0			0
$619$	−960.665			−1.55196			−0.775981	−	0.630756i	$-0.782745\pi$
$619$	−960.665			−1.55196			−0.775981	+	0.630756i	$0.782745\pi$
$620$		0			0
$621$	−35.8546	−	72.5172i	−0.0577368	−	0.116775i
$622$		0			0
$623$		−	484.639i		−	0.777912i
$624$		0			0
$625$	−294.305			−0.470888
$626$		0			0
$627$		0			0
$628$		0			0
$629$			47.4151i			0.0753818i
$630$		0			0
$631$	−86.3415			−0.136833			−0.0684164	−	0.997657i	$-0.521795\pi$
$631$	−86.3415			−0.136833			−0.0684164	+	0.997657i	$0.521795\pi$
$632$		0			0
$633$	−426.739	+	1098.48i	−0.674154	+	1.73535i
$634$		0			0
$635$		−	4.21634i		−	0.00663991i
$636$		0			0
$637$	593.954			0.932424
$638$		0			0
$639$	−676.397	+	739.178i	−1.05852	+	1.15677i
$640$		0			0
$641$		−	956.235i		−	1.49179i	−0.666066	−	0.745893i	$-0.732023\pi$
$641$		−	956.235i		−	1.49179i	0.666066	−	0.745893i	$-0.267977\pi$
$642$		0			0
$643$	345.210			0.536874			0.268437	−	0.963297i	$-0.413493\pi$
$643$	345.210			0.536874			0.268437	+	0.963297i	$0.413493\pi$
$644$		0			0
$645$	−92.0418	−	35.7567i	−0.142701	−	0.0554367i
$646$		0			0
$647$			305.845i			0.472712i	0.971666	+	0.236356i	$0.0759532\pi$
$647$			305.845i			0.472712i	−0.971666	+	0.236356i	$0.924047\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−203.944	+	524.974i	−0.313277	+	0.806412i
$652$		0			0
$653$			654.227i			1.00188i	0.865482	+	0.500939i	$0.167012\pi$
$653$			654.227i			1.00188i	−0.865482	+	0.500939i	$0.832988\pi$
$654$		0			0
$655$	783.820			1.19667
$656$		0			0
$657$	480.036	+	439.265i	0.730649	+	0.668592i
$658$		0			0
$659$		−	111.907i		−	0.169813i	−0.996389	−	0.0849063i	$-0.972941\pi$
$659$		−	111.907i		−	0.169813i	0.996389	−	0.0849063i	$-0.0270591\pi$
$660$		0			0
$661$	−86.0243			−0.130143			−0.0650713	−	0.997881i	$-0.520727\pi$
$661$	−86.0243			−0.130143			−0.0650713	+	0.997881i	$0.520727\pi$
$662$		0			0
$663$	−81.6315	−	31.7124i	−0.123124	−	0.0478317i
$664$		0			0
$665$		−	168.459i		−	0.253323i
$666$		0			0
$667$	−154.923			−0.232269
$668$		0			0
$669$	−14.3446	+	36.9247i	−0.0214419	+	0.0551939i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	182.608			0.271334			0.135667	−	0.990754i	$-0.456682\pi$
$673$	182.608			0.271334			0.135667	+	0.990754i	$0.456682\pi$
$674$		0			0
$675$	−231.548	+	114.484i	−0.343034	+	0.169606i
$676$		0			0
$677$		−	340.463i		−	0.502899i	−0.967870	−	0.251450i	$-0.919093\pi$
$677$		−	340.463i		−	0.502899i	0.967870	−	0.251450i	$-0.0809073\pi$
$678$		0			0
$679$	677.530			0.997834
$680$		0			0
$681$	−62.9554	−	24.4571i	−0.0924455	−	0.0359135i
$682$		0			0
$683$		−	746.115i		−	1.09241i	−0.837652	−	0.546204i	$-0.816072\pi$
$683$		−	746.115i		−	1.09241i	0.837652	−	0.546204i	$-0.183928\pi$
$684$		0			0
$685$	−109.270			−0.159519
$686$		0			0
$687$	−154.592	+	397.937i	−0.225024	+	0.579239i
$688$		0			0
$689$			1567.66i			2.27527i
$690$		0			0
$691$	364.621			0.527671			0.263836	−	0.964568i	$-0.415012\pi$
$691$	364.621			0.527671			0.263836	+	0.964568i	$0.415012\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	640.479i		−	0.921553i
$696$		0			0
$697$	24.4670			0.0351034
$698$		0			0
$699$	1164.85	+	452.524i	1.66645	+	0.647387i
$700$		0			0
$701$		−	88.0985i		−	0.125675i	−0.998024	−	0.0628377i	$-0.979985\pi$
$701$		−	88.0985i		−	0.125675i	0.998024	−	0.0628377i	$-0.0200151\pi$
$702$		0			0
$703$	−335.825			−0.477703
$704$		0			0
$705$	224.141	−	576.964i	0.317930	−	0.818388i
$706$		0			0
$707$			548.269i			0.775486i
$708$		0			0
$709$	691.437			0.975228			0.487614	−	0.873059i	$-0.337867\pi$
$709$	691.437			0.975228			0.487614	+	0.873059i	$0.337867\pi$
$710$		0			0
$711$	−126.627	−	115.872i	−0.178096	−	0.162970i
$712$		0			0
$713$			115.562i			0.162078i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	476.977	+	185.297i	0.665240	+	0.258434i
$718$		0			0
$719$			139.223i			0.193634i	0.995302	+	0.0968171i	$0.0308662\pi$
$719$			139.223i			0.193634i	−0.995302	+	0.0968171i	$0.969134\pi$
$720$		0			0
$721$	470.088			0.651994
$722$		0			0
$723$	165.027	−	424.800i	0.228254	−	0.587551i
$724$		0			0
$725$			494.672i			0.682306i
$726$		0			0
$727$	−822.403			−1.13123			−0.565614	−	0.824670i	$-0.691361\pi$
$727$	−822.403			−1.13123			−0.565614	+	0.824670i	$0.691361\pi$
$728$		0			0
$729$	442.593	−	579.269i	0.607123	−	0.794608i
$730$		0			0
$731$			10.4217i			0.0142567i
$732$		0			0
$733$	−569.900			−0.777490			−0.388745	−	0.921345i	$-0.627091\pi$
$733$	−569.900			−0.777490			−0.388745	+	0.921345i	$0.627091\pi$
$734$		0			0
$735$	278.035	+	108.012i	0.378279	+	0.146955i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−240.719			−0.325735			−0.162868	−	0.986648i	$-0.552074\pi$
$739$	−240.719			−0.325735			−0.162868	+	0.986648i	$0.552074\pi$
$740$		0			0
$741$	224.608	−	578.168i	0.303115	−	0.780254i
$742$		0			0
$743$			843.654i			1.13547i	0.823211	+	0.567735i	$0.192180\pi$
$743$			843.654i			1.13547i	−0.823211	+	0.567735i	$0.807820\pi$
$744$		0			0
$745$	741.516			0.995323
$746$		0			0
$747$	444.783	−	486.067i	0.595426	−	0.650692i
$748$		0			0
$749$			372.644i			0.497522i
$750$		0			0
$751$	1321.29			1.75938			0.879689	−	0.475549i	$-0.157750\pi$
$751$	1321.29			1.75938			0.879689	+	0.475549i	$0.157750\pi$
$752$		0			0
$753$	1086.37	+	422.035i	1.44272	+	0.560471i
$754$		0			0
$755$			761.901i			1.00914i
$756$		0			0
$757$	−1138.10			−1.50343			−0.751717	−	0.659485i	$-0.770774\pi$
$757$	−1138.10			−1.50343			−0.751717	+	0.659485i	$0.770774\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			854.619i			1.12302i	0.827469	+	0.561511i	$0.189780\pi$
$761$			854.619i			1.12302i	−0.827469	+	0.561511i	$0.810220\pi$
$762$		0			0
$763$	−2.33988			−0.00306668
$764$		0			0
$765$	−32.4454	−	29.6897i	−0.0424123	−	0.0388101i
$766$		0			0
$767$		−	894.898i		−	1.16675i
$768$		0			0
$769$	1278.12			1.66205			0.831026	−	0.556233i	$-0.187754\pi$
$769$	1278.12			1.66205			0.831026	+	0.556233i	$0.187754\pi$
$770$		0			0
$771$	537.144	+	208.671i	0.696684	+	0.270650i
$772$		0			0
$773$			670.898i			0.867915i	0.900933	+	0.433957i	$0.142883\pi$
$773$			670.898i			0.867915i	−0.900933	+	0.433957i	$0.857117\pi$
$774$		0			0
$775$	368.990			0.476116
$776$		0			0
$777$	−201.559	+	518.836i	−0.259407	+	0.667743i
$778$		0			0
$779$			173.292i			0.222454i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−618.766	−	1251.48i	−0.790250	−	1.59831i
$784$		0			0
$785$			646.555i			0.823637i
$786$		0			0
$787$	−1402.76			−1.78241			−0.891206	−	0.453600i	$-0.850140\pi$
$787$	−1402.76			−1.78241			−0.891206	+	0.453600i	$0.850140\pi$
$788$		0			0
$789$	−1126.83	−	437.753i	−1.42817	−	0.554821i
$790$		0			0
$791$		−	1062.48i		−	1.34321i
$792$		0			0
$793$	−1137.23			−1.43409
$794$		0			0
$795$	−285.082	+	733.835i	−0.358594	+	0.923063i
$796$		0			0
$797$		−	566.092i		−	0.710278i	−0.934814	−	0.355139i	$-0.884434\pi$
$797$		−	566.092i		−	0.710278i	0.934814	−	0.355139i	$-0.115566\pi$
$798$		0			0
$799$	−65.3282			−0.0817624
$800$		0			0
$801$	604.958	−	661.108i	0.755253	−	0.825354i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−57.2912			−0.0711692
$806$		0			0
$807$	425.846	+	165.434i	0.527690	+	0.204999i
$808$		0			0
$809$			989.854i			1.22355i	0.791031	+	0.611776i	$0.209545\pi$
$809$			989.854i			1.22355i	−0.791031	+	0.611776i	$0.790455\pi$
$810$		0			0
$811$	−167.646			−0.206716			−0.103358	−	0.994644i	$-0.532959\pi$
$811$	−167.646			−0.206716			−0.103358	+	0.994644i	$0.532959\pi$
$812$		0			0
$813$	−566.816	+	1459.05i	−0.697190	+	1.79465i
$814$		0			0
$815$			939.843i			1.15318i
$816$		0			0
$817$	−73.8131			−0.0903466
$818$		0			0
$819$	−758.438	−	694.021i	−0.926054	−	0.847401i
$820$		0			0
$821$			194.351i			0.236724i	0.992971	+	0.118362i	$0.0377643\pi$
$821$			194.351i			0.236724i	−0.992971	+	0.118362i	$0.962236\pi$
$822$		0			0
$823$	653.508			0.794056			0.397028	−	0.917806i	$-0.370042\pi$
$823$	653.508			0.794056			0.397028	+	0.917806i	$0.370042\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			595.831i			0.720473i	0.932861	+	0.360236i	$0.117304\pi$
$827$			595.831i			0.720473i	−0.932861	+	0.360236i	$0.882696\pi$
$828$		0			0
$829$	−1235.22			−1.49002			−0.745008	−	0.667056i	$-0.767554\pi$
$829$	−1235.22			−1.49002			−0.745008	+	0.667056i	$0.767554\pi$
$830$		0			0
$831$	70.9562	−	182.649i	0.0853865	−	0.219795i
$832$		0			0
$833$		−	31.4812i		−	0.0377925i
$834$		0			0
$835$	−676.740			−0.810467
$836$		0			0
$837$	−933.511	+	461.555i	−1.11531	+	0.551440i
$838$		0			0
$839$			223.414i			0.266286i	0.991097	+	0.133143i	$0.0425070\pi$
$839$			223.414i			0.266286i	−0.991097	+	0.133143i	$0.957493\pi$
$840$		0			0
$841$	−1832.61			−2.17909
$842$		0			0
$843$	128.604	+	49.9606i	0.152556	+	0.0592653i
$844$		0			0
$845$		−	1499.74i		−	1.77484i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−443.420	+	1141.42i	−0.522285	+	1.34442i
$850$		0			0
$851$			114.210i			0.134207i
$852$		0			0
$853$	−113.818			−0.133433			−0.0667165	−	0.997772i	$-0.521252\pi$
$853$	−113.818			−0.133433			−0.0667165	+	0.997772i	$0.521252\pi$
$854$		0			0
$855$	210.282	−	229.800i	0.245944	−	0.268772i
$856$		0			0
$857$			953.012i			1.11203i	0.831171	+	0.556017i	$0.187671\pi$
$857$			953.012i			1.11203i	−0.831171	+	0.556017i	$0.812329\pi$
$858$		0			0
$859$	1139.47			1.32651			0.663254	−	0.748394i	$-0.269175\pi$
$859$	1139.47			1.32651			0.663254	+	0.748394i	$0.269175\pi$
$860$		0			0
$861$	267.729	+	104.008i	0.310951	+	0.120799i
$862$		0			0
$863$		−	403.944i		−	0.468069i	−0.972228	−	0.234035i	$-0.924807\pi$
$863$		−	403.944i		−	0.468069i	0.972228	−	0.234035i	$-0.0751929\pi$
$864$		0			0
$865$	227.042			0.262476
$866$		0			0
$867$	312.275	−	803.832i	0.360179	−	0.927142i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	649.164			0.745309
$872$		0			0
$873$	924.235	+	845.736i	1.05869	+	0.968770i
$874$		0			0
$875$			660.967i			0.755391i
$876$		0			0
$877$	−125.630			−0.143250			−0.0716249	−	0.997432i	$-0.522818\pi$
$877$	−125.630			−0.143250			−0.0716249	+	0.997432i	$0.522818\pi$
$878$		0			0
$879$	1281.74	+	497.934i	1.45818	+	0.566478i
$880$		0			0
$881$			151.533i			0.172001i	0.996295	+	0.0860004i	$0.0274086\pi$
$881$			151.533i			0.172001i	−0.996295	+	0.0860004i	$0.972591\pi$
$882$		0			0
$883$	−181.979			−0.206092			−0.103046	−	0.994677i	$-0.532859\pi$
$883$	−181.979			−0.206092			−0.103046	+	0.994677i	$0.532859\pi$
$884$		0			0
$885$	162.739	−	418.909i	0.183886	−	0.473343i
$886$		0			0
$887$		−	391.155i		−	0.440987i	−0.975388	−	0.220493i	$-0.929233\pi$
$887$		−	391.155i		−	0.440987i	0.975388	−	0.220493i	$-0.0707668\pi$
$888$		0			0
$889$	−5.22398			−0.00587624
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	462.697i		−	0.518138i
$894$		0			0
$895$	−72.6933			−0.0812215
$896$		0			0
$897$	−196.629	−	76.3868i	−0.219207	−	0.0851581i
$898$		0			0
$899$			1994.32i			2.21838i
$900$		0			0
$901$	83.0903			0.0922201
$902$		0			0
$903$	−44.3019	+	114.038i	−0.0490608	+	0.126288i
$904$		0			0
$905$			129.256i			0.142824i
$906$		0			0
$907$	−1265.28			−1.39501			−0.697507	−	0.716578i	$-0.745708\pi$
$907$	−1265.28			−1.39501			−0.697507	+	0.716578i	$0.745708\pi$
$908$		0			0
$909$	−684.384	+	747.907i	−0.752898	+	0.822780i
$910$		0			0
$911$		−	401.672i		−	0.440913i	−0.975397	−	0.220457i	$-0.929245\pi$
$911$		−	401.672i		−	0.440913i	0.975397	−	0.220457i	$-0.0707547\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−532.348	−	206.808i	−0.581801	−	0.226020i
$916$		0			0
$917$		−	971.140i		−	1.05904i
$918$		0			0
$919$	−798.609			−0.868998			−0.434499	−	0.900672i	$-0.643075\pi$
$919$	−798.609			−0.868998			−0.434499	+	0.900672i	$0.643075\pi$
$920$		0			0
$921$	−317.230	+	816.586i	−0.344441	+	0.886630i
$922$		0			0
$923$			2612.66i			2.83062i
$924$		0			0
$925$	364.675			0.394243
$926$		0			0
$927$	641.258	+	586.794i	0.691757	+	0.633003i
$928$		0			0
$929$		−	371.191i		−	0.399560i	−0.979841	−	0.199780i	$-0.935977\pi$
$929$		−	371.191i		−	0.399560i	0.979841	−	0.199780i	$-0.0640227\pi$
$930$		0			0
$931$	222.971			0.239496
$932$		0			0
$933$	−358.934	−	139.440i	−0.384710	−	0.149453i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1185.70			−1.26542			−0.632710	−	0.774389i	$-0.718058\pi$
$937$	−1185.70			−1.26542			−0.632710	+	0.774389i	$0.718058\pi$
$938$		0			0
$939$	166.888	−	429.588i	0.177729	−	0.457495i
$940$		0			0
$941$			1098.26i			1.16712i	0.812071	+	0.583558i	$0.198340\pi$
$941$			1098.26i			1.16712i	−0.812071	+	0.583558i	$0.801660\pi$
$942$		0			0
$943$	58.9346			0.0624969
$944$		0			0
$945$	−228.822	−	462.801i	−0.242140	−	0.489736i
$946$		0			0
$947$			544.356i			0.574821i	0.957807	+	0.287411i	$0.0927944\pi$
$947$			544.356i			0.574821i	−0.957807	+	0.287411i	$0.907206\pi$
$948$		0			0
$949$	1696.71			1.78789
$950$		0			0
$951$	−937.568	−	364.229i	−0.985876	−	0.382996i
$952$		0			0
$953$		−	140.712i		−	0.147651i	−0.997271	−	0.0738257i	$-0.976479\pi$
$953$		−	140.712i		−	0.147651i	0.997271	−	0.0738257i	$-0.0235209\pi$
$954$		0			0
$955$	869.257			0.910217
$956$		0			0
$957$		0			0
$958$		0			0
$959$			135.384i			0.141172i
$960$		0			0
$961$	526.622			0.547994
$962$		0			0
$963$	−465.158	+	508.332i	−0.483030	+	0.527863i
$964$		0			0
$965$		−	1040.12i		−	1.07785i
$966$		0			0
$967$	487.128			0.503751			0.251876	−	0.967760i	$-0.418953\pi$
$967$	487.128			0.503751			0.251876	+	0.967760i	$0.418953\pi$
$968$		0			0
$969$	−30.6445	−	11.9049i	−0.0316249	−	0.0122857i
$970$		0			0
$971$			750.766i			0.773188i	0.922250	+	0.386594i	$0.126349\pi$
$971$			750.766i			0.773188i	−0.922250	+	0.386594i	$0.873651\pi$
$972$		0			0
$973$	−793.543			−0.815563
$974$		0			0
$975$	−243.904	+	627.837i	−0.250158	+	0.643936i
$976$		0			0
$977$		−	190.661i		−	0.195149i	−0.995228	−	0.0975747i	$-0.968892\pi$
$977$		−	190.661i		−	0.195149i	0.995228	−	0.0975747i	$-0.0311085\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−3.19188	−	2.92078i	−0.00325370	−	0.00297735i
$982$		0			0
$983$			1611.38i			1.63925i	0.572902	+	0.819624i	$0.305818\pi$
$983$			1611.38i			1.63925i	−0.572902	+	0.819624i	$0.694182\pi$
$984$		0			0
$985$	906.511			0.920316
$986$		0			0
$987$	−714.848	−	277.706i	−0.724264	−	0.281364i
$988$		0			0
$989$			25.1030i			0.0253822i
$990$		0			0
$991$	188.780			0.190495			0.0952473	−	0.995454i	$-0.469636\pi$
$991$	188.780			0.190495			0.0952473	+	0.995454i	$0.469636\pi$
$992$		0			0
$993$	−476.264	+	1225.96i	−0.479621	+	1.23460i
$994$		0			0
$995$		−	133.902i		−	0.134575i
$996$		0			0
$997$	545.159			0.546799			0.273400	−	0.961901i	$-0.411852\pi$
$997$	545.159			0.546799			0.273400	+	0.961901i	$0.411852\pi$
$998$		0			0
$999$	−922.596	+	456.158i	−0.923520	+	0.456615i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.3.e.l.485.9		16
3.2	odd	2	inner	1452.3.e.l.485.10		16
11.3	even	5		132.3.m.a.53.5	yes	32
11.4	even	5		132.3.m.a.5.2	✓	32
11.10	odd	2		1452.3.e.m.485.9		16
33.14	odd	10		132.3.m.a.53.2	yes	32
33.26	odd	10		132.3.m.a.5.5	yes	32
33.32	even	2		1452.3.e.m.485.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.3.m.a.5.2	✓	32	11.4	even	5
132.3.m.a.5.5	yes	32	33.26	odd	10
132.3.m.a.53.2	yes	32	33.14	odd	10
132.3.m.a.53.5	yes	32	11.3	even	5
1452.3.e.l.485.9		16	1.1	even	1	trivial
1452.3.e.l.485.10		16	3.2	odd	2	inner
1452.3.e.m.485.9		16	11.10	odd	2
1452.3.e.m.485.10		16	33.32	even	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)

\(f(q)\)	\(=\)	\(q+(1.08635 - 2.79640i) q^{3} -3.92851i q^{5} -4.86735 q^{7} +(-6.63968 - 6.07575i) q^{9} -23.4682 q^{13} +(-10.9857 - 4.26774i) q^{15} +1.24388i q^{17} -8.80998 q^{19} +(-5.28766 + 13.6111i) q^{21} +2.99618i q^{23} +9.56683 q^{25} +(-24.2032 + 11.9668i) q^{27} +51.7070i q^{29} +38.5697 q^{31} +19.1214i q^{35} +38.1187 q^{37} +(-25.4948 + 65.6265i) q^{39} -19.6699i q^{41} +8.37836 q^{43} +(-23.8686 + 26.0840i) q^{45} +52.5197i q^{47} -25.3089 q^{49} +(3.47838 + 1.35129i) q^{51} -66.7993i q^{53} +(-9.57074 + 24.6362i) q^{57} +38.1323i q^{59} +48.4584 q^{61} +(32.3177 + 29.5728i) q^{63} +92.1951i q^{65} -27.6614 q^{67} +(8.37850 + 3.25490i) q^{69} -111.327i q^{71} -72.2982 q^{73} +(10.3929 - 26.7526i) q^{75} +19.0712 q^{79} +(7.17062 + 80.6820i) q^{81} +73.2064i q^{83} +4.88659 q^{85} +(144.593 + 56.1720i) q^{87} +99.5693i q^{89} +114.228 q^{91} +(41.9003 - 107.856i) q^{93} +34.6101i q^{95} -139.199 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} + 14 q^{9} - 2 q^{13} - 8 q^{15} - 2 q^{19} + q^{21} - 70 q^{25} - 54 q^{27} - 50 q^{31} + 30 q^{37} + 68 q^{39} - 50 q^{43} + 17 q^{45} + 232 q^{49} + 218 q^{51} + 205 q^{57} - 80 q^{61}+ \cdots - 416 q^{97}+O(q^{100}) \) 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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