LMFDB - Embedded newform 1100.3.j.b.1057.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,3,Mod(793,1100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1100.793"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1100.j (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$29.9728290796$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 6 x^{17} + 1579 x^{16} - 3420 x^{15} + 3700 x^{14} - 2060 x^{13} + \cdots + 30250000 $ x^20 - 2x^19 + 2x^18 - 6x^17 + 1579x^16 - 3420x^15 + 3700x^14 - 2060x^13 + 650315x^12 - 1522610x^11 + 1762850x^10 + 2919770x^9 + 55614409x^8 - 104629208x^7 + 100253328x^6 + 76100096x^5 + 730390456x^4 - 1123974720x^3 + 837223200x^2 - 225060000*x + 30250000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1057.1
Root			$-3.95222 - 3.95222i$ of defining polynomial
Character	$\chi$	$=$	1100.1057
Dual form			1100.3.j.b.793.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.95222 + 3.95222i) q^{3} +(7.19082 + 7.19082i) q^{7} -22.2400i q^{9} +3.31662 q^{11} +(9.00000 - 9.00000i) q^{13} +(-12.3662 - 12.3662i) q^{17} +16.8538i q^{19} -56.8393 q^{21} +(-24.1259 + 24.1259i) q^{23} +(52.3274 + 52.3274i) q^{27} +13.5506i q^{29} -38.7747 q^{31} +(-13.1080 + 13.1080i) q^{33} +(-17.2266 - 17.2266i) q^{37} +71.1399i q^{39} +18.8877 q^{41} +(-46.8590 + 46.8590i) q^{43} +(-24.1412 - 24.1412i) q^{47} +54.4157i q^{49} +97.7481 q^{51} +(42.8165 - 42.8165i) q^{53} +(-66.6098 - 66.6098i) q^{57} +5.26637i q^{59} -95.5197 q^{61} +(159.924 - 159.924i) q^{63} +(-41.5998 - 41.5998i) q^{67} -190.702i q^{69} -9.80142 q^{71} +(-57.7709 + 57.7709i) q^{73} +(23.8492 + 23.8492i) q^{77} +42.6126i q^{79} -213.458 q^{81} +(82.6490 - 82.6490i) q^{83} +(-53.5550 - 53.5550i) q^{87} +24.6536i q^{89} +129.435 q^{91} +(153.246 - 153.246i) q^{93} +(19.7682 + 19.7682i) q^{97} -73.7618i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3} + 28 q^{13} - 24 q^{17} - 106 q^{23} + 50 q^{27} + 88 q^{31} + 22 q^{33} - 2 q^{37} + 72 q^{41} - 168 q^{43} - 108 q^{47} + 112 q^{51} + 164 q^{53} - 48 q^{57} - 280 q^{61} + 348 q^{63} - 110 q^{67}+ \cdots + 542 q^{97}+O(q^{100}) $ 20 * q + 2 * q^3 + 28 * q^13 - 24 * q^17 - 106 * q^23 + 50 * q^27 + 88 * q^31 + 22 * q^33 - 2 * q^37 + 72 * q^41 - 168 * q^43 - 108 * q^47 + 112 * q^51 + 164 * q^53 - 48 * q^57 - 280 * q^61 + 348 * q^63 - 110 * q^67 - 140 * q^71 - 28 * q^73 - 552 * q^81 - 16 * q^83 - 208 * q^87 + 480 * q^91 + 374 * q^93 + 542 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	−3.95222	+	3.95222i	−1.31741	+	1.31741i	−0.401582	+	0.915823i	$0.631540\pi$
$3$	−3.95222	+	3.95222i	−1.31741	+	1.31741i	−0.915823	+	0.401582i	$0.868460\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.19082	+	7.19082i	1.02726	+	1.02726i	0.999618	+	0.0276416i	$0.00879973\pi$
$7$	7.19082	+	7.19082i	1.02726	+	1.02726i	0.0276416	+	0.999618i	$0.491200\pi$
$8$		0			0
$9$		−	22.2400i		−	2.47111i
$10$		0			0
$11$	3.31662			0.301511
$11$	3.31662			0.301511
$12$		0			0
$13$	9.00000	−	9.00000i	0.692308	−	0.692308i	−0.270431	−	0.962739i	$-0.587166\pi$
$13$	9.00000	−	9.00000i	0.692308	−	0.692308i	0.962739	+	0.270431i	$0.0871663\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.3662	−	12.3662i	−0.727426	−	0.727426i	0.242680	−	0.970106i	$-0.421973\pi$
$17$	−12.3662	−	12.3662i	−0.727426	−	0.727426i	−0.970106	+	0.242680i	$0.921973\pi$
$18$		0			0
$19$			16.8538i			0.887041i	0.896264	+	0.443521i	$0.146271\pi$
$19$			16.8538i			0.887041i	−0.896264	+	0.443521i	$0.853729\pi$
$20$		0			0
$21$	−56.8393			−2.70663
$22$		0			0
$23$	−24.1259	+	24.1259i	−1.04895	+	1.04895i	−0.0502150	+	0.998738i	$0.515991\pi$
$23$	−24.1259	+	24.1259i	−1.04895	+	1.04895i	−0.998738	+	0.0502150i	$0.984009\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	52.3274	+	52.3274i	1.93805	+	1.93805i
$28$		0			0
$29$			13.5506i			0.467263i	0.972325	+	0.233631i	$0.0750609\pi$
$29$			13.5506i			0.467263i	−0.972325	+	0.233631i	$0.924939\pi$
$30$		0			0
$31$	−38.7747			−1.25080			−0.625399	−	0.780305i	$-0.715064\pi$
$31$	−38.7747			−1.25080			−0.625399	+	0.780305i	$0.715064\pi$
$32$		0			0
$33$	−13.1080	+	13.1080i	−0.397213	+	0.397213i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−17.2266	−	17.2266i	−0.465583	−	0.465583i	0.434897	−	0.900480i	$-0.356785\pi$
$37$	−17.2266	−	17.2266i	−0.465583	−	0.465583i	−0.900480	+	0.434897i	$0.856785\pi$
$38$		0			0
$39$			71.1399i			1.82410i
$40$		0			0
$41$	18.8877			0.460677			0.230338	−	0.973111i	$-0.426017\pi$
$41$	18.8877			0.460677			0.230338	+	0.973111i	$0.426017\pi$
$42$		0			0
$43$	−46.8590	+	46.8590i	−1.08975	+	1.08975i	−0.0941909	+	0.995554i	$0.530026\pi$
$43$	−46.8590	+	46.8590i	−1.08975	+	1.08975i	−0.995554	+	0.0941909i	$0.969974\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−24.1412	−	24.1412i	−0.513644	−	0.513644i	0.401997	−	0.915641i	$-0.368316\pi$
$47$	−24.1412	−	24.1412i	−0.513644	−	0.513644i	−0.915641	+	0.401997i	$0.868316\pi$
$48$		0			0
$49$			54.4157i			1.11052i
$50$		0			0
$51$	97.7481			1.91663
$52$		0			0
$53$	42.8165	−	42.8165i	0.807858	−	0.807858i	−0.176451	−	0.984309i	$-0.556462\pi$
$53$	42.8165	−	42.8165i	0.807858	−	0.807858i	0.984309	+	0.176451i	$0.0564618\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−66.6098	−	66.6098i	−1.16859	−	1.16859i
$58$		0			0
$59$			5.26637i			0.0892606i	0.999004	+	0.0446303i	$0.0142110\pi$
$59$			5.26637i			0.0892606i	−0.999004	+	0.0446303i	$0.985789\pi$
$60$		0			0
$61$	−95.5197			−1.56590			−0.782948	−	0.622087i	$-0.786285\pi$
$61$	−95.5197			−1.56590			−0.782948	+	0.622087i	$0.786285\pi$
$62$		0			0
$63$	159.924	−	159.924i	2.53847	−	2.53847i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−41.5998	−	41.5998i	−0.620892	−	0.620892i	0.324868	−	0.945759i	$-0.394680\pi$
$67$	−41.5998	−	41.5998i	−0.620892	−	0.620892i	−0.945759	+	0.324868i	$0.894680\pi$
$68$		0			0
$69$		−	190.702i		−	2.76379i
$70$		0			0
$71$	−9.80142			−0.138048			−0.0690241	−	0.997615i	$-0.521989\pi$
$71$	−9.80142			−0.138048			−0.0690241	+	0.997615i	$0.521989\pi$
$72$		0			0
$73$	−57.7709	+	57.7709i	−0.791383	+	0.791383i	−0.981719	−	0.190336i	$-0.939042\pi$
$73$	−57.7709	+	57.7709i	−0.791383	+	0.791383i	0.190336	+	0.981719i	$0.439042\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	23.8492	+	23.8492i	0.309730	+	0.309730i
$78$		0			0
$79$			42.6126i			0.539400i	0.962944	+	0.269700i	$0.0869245\pi$
$79$			42.6126i			0.539400i	−0.962944	+	0.269700i	$0.913075\pi$
$80$		0			0
$81$	−213.458			−2.63529
$82$		0			0
$83$	82.6490	−	82.6490i	0.995771	−	0.995771i	−0.00422058	−	0.999991i	$-0.501343\pi$
$83$	82.6490	−	82.6490i	0.995771	−	0.995771i	0.999991	+	0.00422058i	$0.00134346\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−53.5550	−	53.5550i	−0.615574	−	0.615574i
$88$		0			0
$89$			24.6536i			0.277006i	0.990362	+	0.138503i	$0.0442291\pi$
$89$			24.6536i			0.277006i	−0.990362	+	0.138503i	$0.955771\pi$
$90$		0			0
$91$	129.435			1.42236
$92$		0			0
$93$	153.246	−	153.246i	1.64781	−	1.64781i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	19.7682	+	19.7682i	0.203796	+	0.203796i	0.801624	−	0.597828i	$-0.203970\pi$
$97$	19.7682	+	19.7682i	0.203796	+	0.203796i	−0.597828	+	0.801624i	$0.703970\pi$
$98$		0			0
$99$		−	73.7618i		−	0.745069i
$100$		0			0
$101$	−52.8786			−0.523550			−0.261775	−	0.965129i	$-0.584308\pi$
$101$	−52.8786			−0.523550			−0.261775	+	0.965129i	$0.584308\pi$
$102$		0			0
$103$	−29.0656	+	29.0656i	−0.282190	+	0.282190i	−0.833982	−	0.551792i	$-0.813944\pi$
$103$	−29.0656	+	29.0656i	−0.282190	+	0.282190i	0.551792	+	0.833982i	$0.313944\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−43.8578	−	43.8578i	−0.409886	−	0.409886i	0.471813	−	0.881699i	$-0.343600\pi$
$107$	−43.8578	−	43.8578i	−0.409886	−	0.409886i	−0.881699	+	0.471813i	$0.843600\pi$
$108$		0			0
$109$			35.4173i			0.324929i	0.986714	+	0.162464i	$0.0519443\pi$
$109$			35.4173i			0.324929i	−0.986714	+	0.162464i	$0.948056\pi$
$110$		0			0
$111$	136.166			1.22672
$112$		0			0
$113$	−10.7512	+	10.7512i	−0.0951437	+	0.0951437i	−0.753077	−	0.657933i	$-0.771431\pi$
$113$	−10.7512	+	10.7512i	−0.0951437	+	0.0951437i	0.657933	+	0.753077i	$0.271431\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−200.160	−	200.160i	−1.71077	−	1.71077i
$118$		0			0
$119$		−	177.847i		−	1.49451i
$120$		0			0
$121$	11.0000			0.0909091
$122$		0			0
$123$	−74.6485	+	74.6485i	−0.606898	+	0.606898i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−108.059	−	108.059i	−0.850856	−	0.850856i	0.139382	−	0.990239i	$-0.455488\pi$
$127$	−108.059	−	108.059i	−0.850856	−	0.850856i	−0.990239	+	0.139382i	$0.955488\pi$
$128$		0			0
$129$		−	370.394i		−	2.87127i
$130$		0			0
$131$	−189.077			−1.44333			−0.721666	−	0.692241i	$-0.756623\pi$
$131$	−189.077			−1.44333			−0.721666	+	0.692241i	$0.756623\pi$
$132$		0			0
$133$	−121.192	+	121.192i	−0.911221	+	0.911221i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.0927	+	12.0927i	0.0882678	+	0.0882678i	0.749862	−	0.661594i	$-0.230120\pi$
$137$	12.0927	+	12.0927i	0.0882678	+	0.0882678i	−0.661594	+	0.749862i	$0.730120\pi$
$138$		0			0
$139$		−	165.100i		−	1.18777i	−0.804550	−	0.593885i	$-0.797593\pi$
$139$		−	165.100i		−	1.18777i	0.804550	−	0.593885i	$-0.202407\pi$
$140$		0			0
$141$	190.823			1.35335
$142$		0			0
$143$	29.8496	−	29.8496i	0.208739	−	0.208739i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−215.063	−	215.063i	−1.46301	−	1.46301i
$148$		0			0
$149$		−	210.631i		−	1.41363i	−0.707396	−	0.706817i	$-0.750130\pi$
$149$		−	210.631i		−	1.41363i	0.707396	−	0.706817i	$-0.249870\pi$
$150$		0			0
$151$	−39.1302			−0.259140			−0.129570	−	0.991570i	$-0.541360\pi$
$151$	−39.1302			−0.259140			−0.129570	+	0.991570i	$0.541360\pi$
$152$		0			0
$153$	−275.025	+	275.025i	−1.79755	+	1.79755i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	150.537	+	150.537i	0.958833	+	0.958833i	0.999185	−	0.0403528i	$-0.0128482\pi$
$157$	150.537	+	150.537i	0.958833	+	0.958833i	−0.0403528	+	0.999185i	$0.512848\pi$
$158$		0			0
$159$			338.440i			2.12855i
$160$		0			0
$161$	−346.970			−2.15509
$162$		0			0
$163$	146.432	−	146.432i	0.898355	−	0.898355i	−0.0969357	−	0.995291i	$-0.530904\pi$
$163$	146.432	−	146.432i	0.898355	−	0.898355i	0.995291	+	0.0969357i	$0.0309041\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	169.812	+	169.812i	1.01684	+	1.01684i	0.999856	+	0.0169844i	$0.00540656\pi$
$167$	169.812	+	169.812i	1.01684	+	1.01684i	0.0169844	+	0.999856i	$0.494593\pi$
$168$		0			0
$169$			6.99993i			0.0414197i
$170$		0			0
$171$	374.828			2.19198
$172$		0			0
$173$	81.6275	−	81.6275i	0.471836	−	0.471836i	−0.430673	−	0.902508i	$-0.641724\pi$
$173$	81.6275	−	81.6275i	0.471836	−	0.471836i	0.902508	+	0.430673i	$0.141724\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−20.8138	−	20.8138i	−0.117592	−	0.117592i
$178$		0			0
$179$		−	232.603i		−	1.29946i	−0.760166	−	0.649729i	$-0.774882\pi$
$179$		−	232.603i		−	1.29946i	0.760166	−	0.649729i	$-0.225118\pi$
$180$		0			0
$181$	−44.3335			−0.244936			−0.122468	−	0.992472i	$-0.539081\pi$
$181$	−44.3335			−0.244936			−0.122468	+	0.992472i	$0.539081\pi$
$182$		0			0
$183$	377.514	−	377.514i	2.06292	−	2.06292i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−41.0142	−	41.0142i	−0.219327	−	0.219327i
$188$		0			0
$189$			752.554i			3.98176i
$190$		0			0
$191$	81.1677			0.424962			0.212481	−	0.977165i	$-0.431846\pi$
$191$	81.1677			0.424962			0.212481	+	0.977165i	$0.431846\pi$
$192$		0			0
$193$	−84.8626	+	84.8626i	−0.439702	+	0.439702i	−0.891912	−	0.452209i	$-0.850636\pi$
$193$	−84.8626	+	84.8626i	−0.439702	+	0.439702i	0.452209	+	0.891912i	$0.350636\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	37.0287	+	37.0287i	0.187963	+	0.187963i	0.794815	−	0.606852i	$-0.207568\pi$
$197$	37.0287	+	37.0287i	0.187963	+	0.187963i	−0.606852	+	0.794815i	$0.707568\pi$
$198$		0			0
$199$			106.137i			0.533350i	0.963786	+	0.266675i	$0.0859251\pi$
$199$			106.137i			0.533350i	−0.963786	+	0.266675i	$0.914075\pi$
$200$		0			0
$201$	328.822			1.63593
$202$		0			0
$203$	−97.4400	+	97.4400i	−0.480000	+	0.480000i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	536.561	+	536.561i	2.59208	+	2.59208i
$208$		0			0
$209$			55.8977i			0.267453i
$210$		0			0
$211$	−306.354			−1.45191			−0.725957	−	0.687740i	$-0.758603\pi$
$211$	−306.354			−1.45191			−0.725957	+	0.687740i	$0.758603\pi$
$212$		0			0
$213$	38.7373	−	38.7373i	0.181865	−	0.181865i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−278.822	−	278.822i	−1.28489	−	1.28489i
$218$		0			0
$219$		−	456.646i		−	2.08514i
$220$		0			0
$221$	−222.592			−1.00721
$222$		0			0
$223$	51.4818	−	51.4818i	0.230860	−	0.230860i	−0.582192	−	0.813052i	$-0.697805\pi$
$223$	51.4818	−	51.4818i	0.230860	−	0.230860i	0.813052	+	0.582192i	$0.197805\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	97.2323	+	97.2323i	0.428336	+	0.428336i	0.888061	−	0.459725i	$-0.152052\pi$
$227$	97.2323	+	97.2323i	0.428336	+	0.428336i	−0.459725	+	0.888061i	$0.652052\pi$
$228$		0			0
$229$			102.149i			0.446067i	0.974811	+	0.223034i	$0.0715960\pi$
$229$			102.149i			0.446067i	−0.974811	+	0.223034i	$0.928404\pi$
$230$		0			0
$231$	−188.515			−0.816081
$232$		0			0
$233$	−126.126	+	126.126i	−0.541311	+	0.541311i	−0.923913	−	0.382602i	$-0.875028\pi$
$233$	−126.126	+	126.126i	−0.541311	+	0.541311i	0.382602	+	0.923913i	$0.375028\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−168.414	−	168.414i	−0.710608	−	0.710608i
$238$		0			0
$239$		−	68.4687i		−	0.286480i	−0.989688	−	0.143240i	$-0.954248\pi$
$239$		−	68.4687i		−	0.286480i	0.989688	−	0.143240i	$-0.0457521\pi$
$240$		0			0
$241$	−104.075			−0.431847			−0.215924	−	0.976410i	$-0.569276\pi$
$241$	−104.075			−0.431847			−0.215924	+	0.976410i	$0.569276\pi$
$242$		0			0
$243$	372.686	−	372.686i	1.53369	−	1.53369i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	151.684	+	151.684i	0.614105	+	0.614105i
$248$		0			0
$249$			653.293i			2.62367i
$250$		0			0
$251$	79.1882			0.315491			0.157746	−	0.987480i	$-0.449577\pi$
$251$	79.1882			0.315491			0.157746	+	0.987480i	$0.449577\pi$
$252$		0			0
$253$	−80.0167	+	80.0167i	−0.316271	+	0.316271i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	218.723	+	218.723i	0.851064	+	0.851064i	0.990264	−	0.139201i	$-0.0444533\pi$
$257$	218.723	+	218.723i	0.851064	+	0.851064i	−0.139201	+	0.990264i	$0.544453\pi$
$258$		0			0
$259$		−	247.746i		−	0.956550i
$260$		0			0
$261$	301.366			1.15466
$262$		0			0
$263$	−318.361	+	318.361i	−1.21050	+	1.21050i	−0.239637	+	0.970863i	$0.577028\pi$
$263$	−318.361	+	318.361i	−1.21050	+	1.21050i	−0.970863	+	0.239637i	$0.922972\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−97.4362	−	97.4362i	−0.364930	−	0.364930i
$268$		0			0
$269$			200.013i			0.743544i	0.928324	+	0.371772i	$0.121250\pi$
$269$			200.013i			0.743544i	−0.928324	+	0.371772i	$0.878750\pi$
$270$		0			0
$271$	394.460			1.45557			0.727785	−	0.685805i	$-0.240550\pi$
$271$	394.460			1.45557			0.727785	+	0.685805i	$0.240550\pi$
$272$		0			0
$273$	−511.554	+	511.554i	−1.87382	+	1.87382i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	98.3770	+	98.3770i	0.355151	+	0.355151i	0.862022	−	0.506871i	$-0.169198\pi$
$277$	98.3770	+	98.3770i	0.355151	+	0.355151i	−0.506871	+	0.862022i	$0.669198\pi$
$278$		0			0
$279$			862.351i			3.09086i
$280$		0			0
$281$	−51.3994			−0.182916			−0.0914580	−	0.995809i	$-0.529153\pi$
$281$	−51.3994			−0.182916			−0.0914580	+	0.995809i	$0.529153\pi$
$282$		0			0
$283$	−333.612	+	333.612i	−1.17884	+	1.17884i	−0.198801	+	0.980040i	$0.563705\pi$
$283$	−333.612	+	333.612i	−1.17884	+	1.17884i	−0.980040	+	0.198801i	$0.936295\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	135.818	+	135.818i	0.473235	+	0.473235i
$288$		0			0
$289$			16.8480i			0.0582975i
$290$		0			0
$291$	−156.256			−0.536963
$292$		0			0
$293$	306.323	−	306.323i	1.04547	−	1.04547i	0.0465563	−	0.998916i	$-0.485175\pi$
$293$	306.323	−	306.323i	1.04547	−	1.04547i	0.998916	−	0.0465563i	$-0.0148247\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	173.550	+	173.550i	0.584345	+	0.584345i
$298$		0			0
$299$			434.267i			1.45240i
$300$		0			0
$301$	−673.910			−2.23890
$302$		0			0
$303$	208.987	−	208.987i	0.689728	−	0.689728i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	110.822	+	110.822i	0.360985	+	0.360985i	0.864175	−	0.503191i	$-0.167841\pi$
$307$	110.822	+	110.822i	0.360985	+	0.360985i	−0.503191	+	0.864175i	$0.667841\pi$
$308$		0			0
$309$		−	229.747i		−	0.743518i
$310$		0			0
$311$	−530.500			−1.70579			−0.852894	−	0.522085i	$-0.825154\pi$
$311$	−530.500			−1.70579			−0.852894	+	0.522085i	$0.825154\pi$
$312$		0			0
$313$	−385.274	+	385.274i	−1.23091	+	1.23091i	−0.267293	+	0.963615i	$0.586129\pi$
$313$	−385.274	+	385.274i	−1.23091	+	1.23091i	−0.963615	+	0.267293i	$0.913871\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−369.972	−	369.972i	−1.16710	−	1.16710i	−0.982885	−	0.184218i	$-0.941025\pi$
$317$	−369.972	−	369.972i	−1.16710	−	1.16710i	−0.184218	−	0.982885i	$-0.558975\pi$
$318$		0			0
$319$			44.9423i			0.140885i
$320$		0			0
$321$	346.671			1.07997
$322$		0			0
$323$	208.418	−	208.418i	0.645257	−	0.645257i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−139.977	−	139.977i	−0.428063	−	0.428063i
$328$		0			0
$329$		−	347.191i		−	1.05529i
$330$		0			0
$331$	−434.871			−1.31381			−0.656904	−	0.753974i	$-0.728134\pi$
$331$	−434.871			−1.31381			−0.656904	+	0.753974i	$0.728134\pi$
$332$		0			0
$333$	−383.119	+	383.119i	−1.15051	+	1.15051i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−380.700	−	380.700i	−1.12967	−	1.12967i	−0.990230	−	0.139442i	$-0.955469\pi$
$337$	−380.700	−	380.700i	−1.12967	−	1.12967i	−0.139442	−	0.990230i	$-0.544531\pi$
$338$		0			0
$339$		−	84.9825i		−	0.250686i
$340$		0			0
$341$	−128.601			−0.377130
$342$		0			0
$343$	−38.9432	+	38.9432i	−0.113537	+	0.113537i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.29216	−	7.29216i	−0.0210149	−	0.0210149i	0.696521	−	0.717536i	$-0.254730\pi$
$347$	−7.29216	−	7.29216i	−0.0210149	−	0.0210149i	−0.717536	+	0.696521i	$0.754730\pi$
$348$		0			0
$349$			390.203i			1.11806i	0.829147	+	0.559030i	$0.188826\pi$
$349$			390.203i			1.11806i	−0.829147	+	0.559030i	$0.811174\pi$
$350$		0			0
$351$	941.893			2.68346
$352$		0			0
$353$	302.231	−	302.231i	0.856178	−	0.856178i	−0.134708	−	0.990885i	$-0.543010\pi$
$353$	302.231	−	302.231i	0.856178	−	0.856178i	0.990885	+	0.134708i	$0.0430096\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	702.889	+	702.889i	1.96888	+	1.96888i
$358$		0			0
$359$		−	184.439i		−	0.513758i	−0.966444	−	0.256879i	$-0.917306\pi$
$359$		−	184.439i		−	0.513758i	0.966444	−	0.256879i	$-0.0826942\pi$
$360$		0			0
$361$	76.9501			0.213158
$362$		0			0
$363$	−43.4744	+	43.4744i	−0.119764	+	0.119764i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	121.109	+	121.109i	0.329998	+	0.329998i	0.852586	−	0.522588i	$-0.175033\pi$
$367$	121.109	+	121.109i	0.329998	+	0.329998i	−0.522588	+	0.852586i	$0.675033\pi$
$368$		0			0
$369$		−	420.064i		−	1.13838i
$370$		0			0
$371$	615.771			1.65976
$372$		0			0
$373$	57.2108	−	57.2108i	0.153380	−	0.153380i	−0.626246	−	0.779626i	$-0.715409\pi$
$373$	57.2108	−	57.2108i	0.153380	−	0.153380i	0.779626	+	0.626246i	$0.215409\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	121.956	+	121.956i	0.323490	+	0.323490i
$378$		0			0
$379$			502.705i			1.32640i	0.748443	+	0.663199i	$0.230802\pi$
$379$			502.705i			1.32640i	−0.748443	+	0.663199i	$0.769198\pi$
$380$		0			0
$381$	854.143			2.24185
$382$		0			0
$383$	−341.372	+	341.372i	−0.891310	+	0.891310i	−0.994646	−	0.103336i	$-0.967048\pi$
$383$	−341.372	+	341.372i	−0.891310	+	0.891310i	0.103336	+	0.994646i	$0.467048\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1042.15	+	1042.15i	2.69288	+	2.69288i
$388$		0			0
$389$		−	495.697i		−	1.27429i	−0.770746	−	0.637143i	$-0.780116\pi$
$389$		−	495.697i		−	1.27429i	0.770746	−	0.637143i	$-0.219884\pi$
$390$		0			0
$391$	596.694			1.52607
$392$		0			0
$393$	747.271	−	747.271i	1.90145	−	1.90145i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−321.828	−	321.828i	−0.810651	−	0.810651i	0.174081	−	0.984731i	$-0.444305\pi$
$397$	−321.828	−	321.828i	−0.810651	−	0.810651i	−0.984731	+	0.174081i	$0.944305\pi$
$398$		0			0
$399$		−	957.957i		−	2.40090i
$400$		0			0
$401$	−148.698			−0.370819			−0.185410	−	0.982661i	$-0.559361\pi$
$401$	−148.698			−0.370819			−0.185410	+	0.982661i	$0.559361\pi$
$402$		0			0
$403$	−348.973	+	348.973i	−0.865937	+	0.865937i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−57.1341	−	57.1341i	−0.140379	−	0.140379i
$408$		0			0
$409$		−	414.463i		−	1.01336i	−0.862135	−	0.506678i	$-0.830873\pi$
$409$		−	414.463i		−	1.01336i	0.862135	−	0.506678i	$-0.169127\pi$
$410$		0			0
$411$	−95.5858			−0.232569
$412$		0			0
$413$	−37.8695	+	37.8695i	−0.0916938	+	0.0916938i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	652.511	+	652.511i	1.56477	+	1.56477i
$418$		0			0
$419$		−	463.547i		−	1.10632i	−0.833076	−	0.553159i	$-0.813422\pi$
$419$		−	463.547i		−	1.10632i	0.833076	−	0.553159i	$-0.186578\pi$
$420$		0			0
$421$	801.861			1.90466			0.952329	−	0.305074i	$-0.0986813\pi$
$421$	801.861			1.90466			0.952329	+	0.305074i	$0.0986813\pi$
$422$		0			0
$423$	−536.902	+	536.902i	−1.26927	+	1.26927i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−686.865	−	686.865i	−1.60858	−	1.60858i
$428$		0			0
$429$			235.944i			0.549987i
$430$		0			0
$431$	−163.062			−0.378335			−0.189167	−	0.981945i	$-0.560579\pi$
$431$	−163.062			−0.378335			−0.189167	+	0.981945i	$0.560579\pi$
$432$		0			0
$433$	65.2169	−	65.2169i	0.150616	−	0.150616i	−0.627777	−	0.778393i	$-0.716035\pi$
$433$	65.2169	−	65.2169i	0.150616	−	0.150616i	0.778393	+	0.627777i	$0.216035\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−406.613	−	406.613i	−0.930465	−	0.930465i
$438$		0			0
$439$		−	104.810i		−	0.238748i	−0.992849	−	0.119374i	$-0.961911\pi$
$439$		−	104.810i		−	0.238748i	0.992849	−	0.119374i	$-0.0380887\pi$
$440$		0			0
$441$	1210.21			2.74423
$442$		0			0
$443$	307.200	−	307.200i	0.693454	−	0.693454i	−0.269536	−	0.962990i	$-0.586870\pi$
$443$	307.200	−	307.200i	0.693454	−	0.693454i	0.962990	+	0.269536i	$0.0868703\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	832.461	+	832.461i	1.86233	+	1.86233i
$448$		0			0
$449$			716.991i			1.59686i	0.602087	+	0.798430i	$0.294336\pi$
$449$			716.991i			1.59686i	−0.602087	+	0.798430i	$0.705664\pi$
$450$		0			0
$451$	62.6436			0.138899
$452$		0			0
$453$	154.651	−	154.651i	0.341393	−	0.341393i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	81.7651	+	81.7651i	0.178917	+	0.178917i	0.790884	−	0.611967i	$-0.209621\pi$
$457$	81.7651	+	81.7651i	0.178917	+	0.178917i	−0.611967	+	0.790884i	$0.709621\pi$
$458$		0			0
$459$		−	1294.19i		−	2.81958i
$460$		0			0
$461$	577.600			1.25293			0.626464	−	0.779450i	$-0.284502\pi$
$461$	577.600			1.25293			0.626464	+	0.779450i	$0.284502\pi$
$462$		0			0
$463$	−275.856	+	275.856i	−0.595802	+	0.595802i	−0.939193	−	0.343391i	$-0.888424\pi$
$463$	−275.856	+	275.856i	−0.595802	+	0.595802i	0.343391	+	0.939193i	$0.388424\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−521.011	−	521.011i	−1.11566	−	1.11566i	−0.992371	−	0.123284i	$-0.960657\pi$
$467$	−521.011	−	521.011i	−1.11566	−	1.11566i	−0.123284	−	0.992371i	$-0.539343\pi$
$468$		0			0
$469$		−	598.272i		−	1.27563i
$470$		0			0
$471$	−1189.91			−2.52634
$472$		0			0
$473$	−155.414	+	155.414i	−0.328570	+	0.328570i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−952.239	−	952.239i	−1.99631	−	1.99631i
$478$		0			0
$479$		−	226.368i		−	0.472585i	−0.971682	−	0.236292i	$-0.924068\pi$
$479$		−	226.368i		−	0.472585i	0.971682	−	0.236292i	$-0.0759323\pi$
$480$		0			0
$481$	−310.078			−0.644654
$482$		0			0
$483$	1371.30	−	1371.30i	2.83913	−	2.83913i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−273.568	−	273.568i	−0.561740	−	0.561740i	0.368061	−	0.929802i	$-0.380022\pi$
$487$	−273.568	−	273.568i	−0.561740	−	0.561740i	−0.929802	+	0.368061i	$0.880022\pi$
$488$		0			0
$489$			1157.46i			2.36699i
$490$		0			0
$491$	−125.180			−0.254948			−0.127474	−	0.991842i	$-0.540687\pi$
$491$	−125.180			−0.254948			−0.127474	+	0.991842i	$0.540687\pi$
$492$		0			0
$493$	167.570	−	167.570i	0.339899	−	0.339899i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−70.4802	−	70.4802i	−0.141811	−	0.141811i
$498$		0			0
$499$			17.3246i			0.0347186i	0.999849	+	0.0173593i	$0.00552591\pi$
$499$			17.3246i			0.0347186i	−0.999849	+	0.0173593i	$0.994474\pi$
$500$		0			0
$501$	−1342.27			−2.67918
$502$		0			0
$503$	−440.004	+	440.004i	−0.874759	+	0.874759i	−0.992987	−	0.118227i	$-0.962279\pi$
$503$	−440.004	+	440.004i	−0.874759	+	0.874759i	0.118227	+	0.992987i	$0.462279\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−27.6652	−	27.6652i	−0.0545665	−	0.0545665i
$508$		0			0
$509$			71.8708i			0.141200i	0.997505	+	0.0706000i	$0.0224914\pi$
$509$			71.8708i			0.141200i	−0.997505	+	0.0706000i	$0.977509\pi$
$510$		0			0
$511$	−830.840			−1.62591
$512$		0			0
$513$	−881.914	+	881.914i	−1.71913	+	1.71913i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−80.0675	−	80.0675i	−0.154869	−	0.154869i
$518$		0			0
$519$			645.219i			1.24320i
$520$		0			0
$521$	531.546			1.02024			0.510121	−	0.860103i	$-0.329601\pi$
$521$	531.546			1.02024			0.510121	+	0.860103i	$0.329601\pi$
$522$		0			0
$523$	541.418	−	541.418i	1.03522	−	1.03522i	0.0358591	−	0.999357i	$-0.488583\pi$
$523$	541.418	−	541.418i	1.03522	−	1.03522i	0.999357	−	0.0358591i	$-0.0114168\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	479.498	+	479.498i	0.909863	+	0.909863i
$528$		0			0
$529$		−	635.121i		−	1.20061i
$530$		0			0
$531$	117.124			0.220573
$532$		0			0
$533$	169.990	−	169.990i	0.318930	−	0.318930i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	919.298	+	919.298i	1.71191	+	1.71191i
$538$		0			0
$539$			180.476i			0.334836i
$540$		0			0
$541$	−623.541			−1.15257			−0.576286	−	0.817248i	$-0.695498\pi$
$541$	−623.541			−1.15257			−0.576286	+	0.817248i	$0.695498\pi$
$542$		0			0
$543$	175.215	−	175.215i	0.322680	−	0.322680i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−450.982	−	450.982i	−0.824464	−	0.824464i	0.162281	−	0.986745i	$-0.448115\pi$
$547$	−450.982	−	450.982i	−0.824464	−	0.824464i	−0.986745	+	0.162281i	$0.948115\pi$
$548$		0			0
$549$			2124.36i			3.86951i
$550$		0			0
$551$	−228.379			−0.414481
$552$		0			0
$553$	−306.419	+	306.419i	−0.554104	+	0.554104i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−313.715	−	313.715i	−0.563222	−	0.563222i	0.366999	−	0.930221i	$-0.380385\pi$
$557$	−313.715	−	313.715i	−0.563222	−	0.563222i	−0.930221	+	0.366999i	$0.880385\pi$
$558$		0			0
$559$			843.463i			1.50888i
$560$		0			0
$561$	324.194			0.577886
$562$		0			0
$563$	−461.442	+	461.442i	−0.819613	+	0.819613i	−0.986052	−	0.166439i	$-0.946773\pi$
$563$	−461.442	+	461.442i	−0.819613	+	0.819613i	0.166439	+	0.986052i	$0.446773\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1534.94	−	1534.94i	−2.70712	−	2.70712i
$568$		0			0
$569$			929.437i			1.63346i	0.577022	+	0.816729i	$0.304215\pi$
$569$			929.437i			1.63346i	−0.577022	+	0.816729i	$0.695785\pi$
$570$		0			0
$571$	207.841			0.363994			0.181997	−	0.983299i	$-0.441744\pi$
$571$	207.841			0.363994			0.181997	+	0.983299i	$0.441744\pi$
$572$		0			0
$573$	−320.792	+	320.792i	−0.559847	+	0.559847i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	183.868	+	183.868i	0.318663	+	0.318663i	0.848253	−	0.529591i	$-0.177654\pi$
$577$	183.868	+	183.868i	0.318663	+	0.318663i	−0.529591	+	0.848253i	$0.677654\pi$
$578$		0			0
$579$		−	670.790i		−	1.15853i
$580$		0			0
$581$	1188.63			2.04583
$582$		0			0
$583$	142.006	−	142.006i	0.243578	−	0.243578i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	721.923	+	721.923i	1.22985	+	1.22985i	0.964019	+	0.265833i	$0.0856469\pi$
$587$	721.923	+	721.923i	1.22985	+	1.22985i	0.265833	+	0.964019i	$0.414353\pi$
$588$		0			0
$589$		−	653.501i		−	1.10951i
$590$		0			0
$591$	−292.691			−0.495247
$592$		0			0
$593$	57.9020	−	57.9020i	0.0976425	−	0.0976425i	−0.656598	−	0.754241i	$-0.728005\pi$
$593$	57.9020	−	57.9020i	0.0976425	−	0.0976425i	0.754241	+	0.656598i	$0.228005\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−419.475	−	419.475i	−0.702639	−	0.702639i
$598$		0			0
$599$		−	649.630i		−	1.08452i	−0.840210	−	0.542262i	$-0.817568\pi$
$599$		−	649.630i		−	1.08452i	0.840210	−	0.542262i	$-0.182432\pi$
$600$		0			0
$601$	−123.798			−0.205986			−0.102993	−	0.994682i	$-0.532842\pi$
$601$	−123.798			−0.205986			−0.102993	+	0.994682i	$0.532842\pi$
$602$		0			0
$603$	−925.179	+	925.179i	−1.53429	+	1.53429i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	238.329	+	238.329i	0.392635	+	0.392635i	0.875626	−	0.482991i	$-0.160449\pi$
$607$	238.329	+	238.329i	0.392635	+	0.392635i	−0.482991	+	0.875626i	$0.660449\pi$
$608$		0			0
$609$		−	770.208i		−	1.26471i
$610$		0			0
$611$	−434.543			−0.711199
$612$		0			0
$613$	−265.136	+	265.136i	−0.432521	+	0.432521i	−0.889485	−	0.456964i	$-0.848937\pi$
$613$	−265.136	+	265.136i	−0.432521	+	0.432521i	0.456964	+	0.889485i	$0.348937\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−296.733	−	296.733i	−0.480929	−	0.480929i	0.424499	−	0.905428i	$-0.360450\pi$
$617$	−296.733	−	296.733i	−0.480929	−	0.480929i	−0.905428	+	0.424499i	$0.860450\pi$
$618$		0			0
$619$			263.457i			0.425617i	0.977094	+	0.212809i	$0.0682611\pi$
$619$			263.457i			0.425617i	−0.977094	+	0.212809i	$0.931739\pi$
$620$		0			0
$621$	−2524.89			−4.06585
$622$		0			0
$623$	−177.279	+	177.279i	−0.284557	+	0.284557i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−220.920	−	220.920i	−0.352344	−	0.352344i
$628$		0			0
$629$			426.056i			0.677355i
$630$		0			0
$631$	−415.483			−0.658451			−0.329226	−	0.944251i	$-0.606788\pi$
$631$	−415.483			−0.658451			−0.329226	+	0.944251i	$0.606788\pi$
$632$		0			0
$633$	1210.78	−	1210.78i	1.91276	−	1.91276i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	489.741	+	489.741i	0.768825	+	0.768825i
$638$		0			0
$639$			217.984i			0.341133i
$640$		0			0
$641$	136.257			0.212569			0.106284	−	0.994336i	$-0.466105\pi$
$641$	136.257			0.212569			0.106284	+	0.994336i	$0.466105\pi$
$642$		0			0
$643$	388.304	−	388.304i	0.603895	−	0.603895i	−0.337449	−	0.941344i	$-0.609564\pi$
$643$	388.304	−	388.304i	0.603895	−	0.603895i	0.941344	+	0.337449i	$0.109564\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	325.747	+	325.747i	0.503474	+	0.503474i	0.912516	−	0.409042i	$-0.134137\pi$
$647$	325.747	+	325.747i	0.503474	+	0.503474i	−0.409042	+	0.912516i	$0.634137\pi$
$648$		0			0
$649$			17.4666i			0.0269131i
$650$		0			0
$651$	2203.93			3.38545
$652$		0			0
$653$	352.232	−	352.232i	0.539406	−	0.539406i	−0.383948	−	0.923355i	$-0.625436\pi$
$653$	352.232	−	352.232i	0.539406	−	0.539406i	0.923355	+	0.383948i	$0.125436\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1284.83	+	1284.83i	1.95560	+	1.95560i
$658$		0			0
$659$			9.72092i			0.0147510i	0.999973	+	0.00737551i	$0.00234772\pi$
$659$			9.72092i			0.0147510i	−0.999973	+	0.00737551i	$0.997652\pi$
$660$		0			0
$661$	−489.500			−0.740544			−0.370272	−	0.928923i	$-0.620736\pi$
$661$	−489.500			−0.740544			−0.370272	+	0.928923i	$0.620736\pi$
$662$		0			0
$663$	879.733	−	879.733i	1.32690	−	1.32690i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−326.921	−	326.921i	−0.490137	−	0.490137i
$668$		0			0
$669$			406.934i			0.608272i
$670$		0			0
$671$	−316.803			−0.472136
$672$		0			0
$673$	−714.517	+	714.517i	−1.06169	+	1.06169i	−0.0637224	+	0.997968i	$0.520297\pi$
$673$	−714.517	+	714.517i	−1.06169	+	1.06169i	−0.997968	+	0.0637224i	$0.979703\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	498.782	+	498.782i	0.736753	+	0.736753i	0.971948	−	0.235195i	$-0.0755730\pi$
$677$	498.782	+	498.782i	0.736753	+	0.736753i	−0.235195	+	0.971948i	$0.575573\pi$
$678$		0			0
$679$			284.299i			0.418702i
$680$		0			0
$681$	−768.566			−1.12858
$682$		0			0
$683$	40.5043	−	40.5043i	0.0593035	−	0.0593035i	−0.676833	−	0.736137i	$-0.736648\pi$
$683$	40.5043	−	40.5043i	0.0593035	−	0.0593035i	0.736137	+	0.676833i	$0.236648\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−403.716	−	403.716i	−0.587651	−	0.587651i
$688$		0			0
$689$		−	770.697i		−	1.11857i
$690$		0			0
$691$	10.5277			0.0152354			0.00761771	−	0.999971i	$-0.497575\pi$
$691$	10.5277			0.0152354			0.00761771	+	0.999971i	$0.497575\pi$
$692$		0			0
$693$	530.408	−	530.408i	0.765379	−	0.765379i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−233.571	−	233.571i	−0.335108	−	0.335108i
$698$		0			0
$699$		−	996.951i		−	1.42625i
$700$		0			0
$701$	411.838			0.587500			0.293750	−	0.955882i	$-0.405097\pi$
$701$	411.838			0.587500			0.293750	+	0.955882i	$0.405097\pi$
$702$		0			0
$703$	290.333	−	290.333i	0.412991	−	0.412991i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−380.240	−	380.240i	−0.537822	−	0.537822i
$708$		0			0
$709$			632.010i			0.891411i	0.895180	+	0.445705i	$0.147047\pi$
$709$			632.010i			0.891411i	−0.895180	+	0.445705i	$0.852953\pi$
$710$		0			0
$711$	947.705			1.33292
$712$		0			0
$713$	935.476	−	935.476i	1.31203	−	1.31203i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	270.603	+	270.603i	0.377410	+	0.377410i
$718$		0			0
$719$			339.623i			0.472354i	0.971710	+	0.236177i	$0.0758945\pi$
$719$			339.623i			0.472354i	−0.971710	+	0.236177i	$0.924105\pi$
$720$		0			0
$721$	−418.011			−0.579765
$722$		0			0
$723$	411.328	−	411.328i	0.568918	−	0.568918i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	889.248	+	889.248i	1.22318	+	1.22318i	0.966497	+	0.256678i	$0.0826281\pi$
$727$	889.248	+	889.248i	1.22318	+	1.22318i	0.256678	+	0.966497i	$0.417372\pi$
$728$		0			0
$729$			1024.75i			1.40569i
$730$		0			0
$731$	1158.94			1.58542
$732$		0			0
$733$	198.708	−	198.708i	0.271089	−	0.271089i	−0.558450	−	0.829538i	$-0.688604\pi$
$733$	198.708	−	198.708i	0.271089	−	0.271089i	0.829538	+	0.558450i	$0.188604\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−137.971	−	137.971i	−0.187206	−	0.187206i
$738$		0			0
$739$			513.878i			0.695370i	0.937611	+	0.347685i	$0.113032\pi$
$739$			513.878i			0.695370i	−0.937611	+	0.347685i	$0.886968\pi$
$740$		0			0
$741$	−1198.98			−1.61805
$742$		0			0
$743$	−272.464	+	272.464i	−0.366708	+	0.366708i	−0.866275	−	0.499567i	$-0.833492\pi$
$743$	−272.464	+	272.464i	−0.366708	+	0.366708i	0.499567	+	0.866275i	$0.333492\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1838.11	−	1838.11i	−2.46066	−	2.46066i
$748$		0			0
$749$		−	630.747i		−	0.842118i
$750$		0			0
$751$	−904.450			−1.20433			−0.602164	−	0.798373i	$-0.705695\pi$
$751$	−904.450			−1.20433			−0.602164	+	0.798373i	$0.705695\pi$
$752$		0			0
$753$	−312.969	+	312.969i	−0.415630	+	0.415630i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	821.722	+	821.722i	1.08550	+	1.08550i	0.995986	+	0.0895122i	$0.0285308\pi$
$757$	821.722	+	821.722i	1.08550	+	1.08550i	0.0895122	+	0.995986i	$0.471469\pi$
$758$		0			0
$759$		−	632.486i		−	0.833315i
$760$		0			0
$761$	−279.035			−0.366668			−0.183334	−	0.983051i	$-0.558689\pi$
$761$	−279.035			−0.366668			−0.183334	+	0.983051i	$0.558689\pi$
$762$		0			0
$763$	−254.679	+	254.679i	−0.333786	+	0.333786i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	47.3974	+	47.3974i	0.0617958	+	0.0617958i
$768$		0			0
$769$		−	1.86178i		−	0.00242104i	−0.999999	−	0.00121052i	$-0.999615\pi$
$769$		−	1.86178i		−	0.00242104i	0.999999	−	0.00121052i	$-0.000385321\pi$
$770$		0			0
$771$	−1728.88			−2.24239
$772$		0			0
$773$	−410.232	+	410.232i	−0.530701	+	0.530701i	−0.920781	−	0.390080i	$-0.872447\pi$
$773$	−410.232	+	410.232i	−0.530701	+	0.530701i	0.390080	+	0.920781i	$0.372447\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	979.147	+	979.147i	1.26016	+	1.26016i
$778$		0			0
$779$			318.330i			0.408639i
$780$		0			0
$781$	−32.5076			−0.0416231
$782$		0			0
$783$	−709.069	+	709.069i	−0.905580	+	0.905580i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−362.180	−	362.180i	−0.460204	−	0.460204i	0.438519	−	0.898722i	$-0.355503\pi$
$787$	−362.180	−	362.180i	−0.460204	−	0.460204i	−0.898722	+	0.438519i	$0.855503\pi$
$788$		0			0
$789$		−	2516.47i		−	3.18944i
$790$		0			0
$791$	−154.620			−0.195475
$792$		0			0
$793$	−859.677	+	859.677i	−1.08408	+	1.08408i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−530.618	−	530.618i	−0.665769	−	0.665769i	0.290965	−	0.956734i	$-0.406024\pi$
$797$	−530.618	−	530.618i	−0.665769	−	0.665769i	−0.956734	+	0.290965i	$0.906024\pi$
$798$		0			0
$799$			597.073i			0.747275i
$800$		0			0
$801$	548.296			0.684514
$802$		0			0
$803$	−191.604	+	191.604i	−0.238611	+	0.238611i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−790.496	−	790.496i	−0.979549	−	0.979549i
$808$		0			0
$809$			767.584i			0.948806i	0.880308	+	0.474403i	$0.157336\pi$
$809$			767.584i			0.948806i	−0.880308	+	0.474403i	$0.842664\pi$
$810$		0			0
$811$	1160.22			1.43060			0.715302	−	0.698816i	$-0.246289\pi$
$811$	1160.22			1.43060			0.715302	+	0.698816i	$0.246289\pi$
$812$		0			0
$813$	−1558.99	+	1558.99i	−1.91758	+	1.91758i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−789.752	−	789.752i	−0.966649	−	0.966649i
$818$		0			0
$819$		−	2878.63i		−	3.51481i
$820$		0			0
$821$	337.531			0.411121			0.205561	−	0.978644i	$-0.434098\pi$
$821$	337.531			0.411121			0.205561	+	0.978644i	$0.434098\pi$
$822$		0			0
$823$	−376.069	+	376.069i	−0.456949	+	0.456949i	−0.897653	−	0.440704i	$-0.854729\pi$
$823$	−376.069	+	376.069i	−0.456949	+	0.456949i	0.440704	+	0.897653i	$0.354729\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1048.91	+	1048.91i	1.26833	+	1.26833i	0.946951	+	0.321378i	$0.104146\pi$
$827$	1048.91	+	1048.91i	1.26833	+	1.26833i	0.321378	+	0.946951i	$0.395854\pi$
$828$		0			0
$829$			100.841i			0.121642i	0.998149	+	0.0608208i	$0.0193718\pi$
$829$			100.841i			0.121642i	−0.998149	+	0.0608208i	$0.980628\pi$
$830$		0			0
$831$	−777.614			−0.935757
$832$		0			0
$833$	672.918	−	672.918i	0.807824	−	0.807824i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−2028.98	−	2028.98i	−2.42411	−	2.42411i
$838$		0			0
$839$		−	13.0660i		−	0.0155734i	−0.999970	−	0.00778668i	$-0.997521\pi$
$839$		−	13.0660i		−	0.0155734i	0.999970	−	0.00778668i	$-0.00247860\pi$
$840$		0			0
$841$	657.381			0.781666
$842$		0			0
$843$	203.141	−	203.141i	0.240974	−	0.240974i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	79.0990	+	79.0990i	0.0933872	+	0.0933872i
$848$		0			0
$849$		−	2637.01i		−	3.10602i
$850$		0			0
$851$	831.214			0.976750
$852$		0			0
$853$	−904.060	+	904.060i	−1.05986	+	1.05986i	−0.0617686	+	0.998090i	$0.519674\pi$
$853$	−904.060	+	904.060i	−1.05986	+	1.05986i	−0.998090	+	0.0617686i	$0.980326\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−515.656	−	515.656i	−0.601699	−	0.601699i	0.339064	−	0.940763i	$-0.389890\pi$
$857$	−515.656	−	515.656i	−0.601699	−	0.601699i	−0.940763	+	0.339064i	$0.889890\pi$
$858$		0			0
$859$			1663.47i			1.93652i	0.249939	+	0.968261i	$0.419589\pi$
$859$			1663.47i			1.93652i	−0.249939	+	0.968261i	$0.580411\pi$
$860$		0			0
$861$	−1073.57			−1.24688
$862$		0			0
$863$	−28.4642	+	28.4642i	−0.0329828	+	0.0329828i	−0.723406	−	0.690423i	$-0.757424\pi$
$863$	−28.4642	+	28.4642i	−0.0329828	+	0.0329828i	0.690423	+	0.723406i	$0.257424\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−66.5869	−	66.5869i	−0.0768015	−	0.0768015i
$868$		0			0
$869$			141.330i			0.162635i
$870$		0			0
$871$	−748.796			−0.859697
$872$		0			0
$873$	439.645	−	439.645i	0.503602	−	0.503602i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1127.16	−	1127.16i	−1.28524	−	1.28524i	−0.937644	−	0.347596i	$-0.886998\pi$
$877$	−1127.16	−	1127.16i	−1.28524	−	1.28524i	−0.347596	−	0.937644i	$-0.613002\pi$
$878$		0			0
$879$			2421.31i			2.75462i
$880$		0			0
$881$	−355.371			−0.403372			−0.201686	−	0.979450i	$-0.564642\pi$
$881$	−355.371			−0.403372			−0.201686	+	0.979450i	$0.564642\pi$
$882$		0			0
$883$	264.850	−	264.850i	0.299943	−	0.299943i	−0.541048	−	0.840991i	$-0.681972\pi$
$883$	264.850	−	264.850i	0.299943	−	0.299943i	0.840991	+	0.541048i	$0.181972\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	851.097	+	851.097i	0.959523	+	0.959523i	0.999212	−	0.0396890i	$-0.0126367\pi$
$887$	851.097	+	851.097i	0.959523	+	0.959523i	−0.0396890	+	0.999212i	$0.512637\pi$
$888$		0			0
$889$		−	1554.06i		−	1.74810i
$890$		0			0
$891$	−707.961			−0.794569
$892$		0			0
$893$	406.871	−	406.871i	0.455623	−	0.455623i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1716.32	−	1716.32i	−1.91340	−	1.91340i
$898$		0			0
$899$		−	525.422i		−	0.584451i
$900$		0			0
$901$	−1058.96			−1.17531
$902$		0			0
$903$	2663.44	−	2663.44i	2.94954	−	2.94954i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	590.259	+	590.259i	0.650781	+	0.650781i	0.953181	−	0.302400i	$-0.0977877\pi$
$907$	590.259	+	590.259i	0.650781	+	0.650781i	−0.302400	+	0.953181i	$0.597788\pi$
$908$		0			0
$909$			1176.02i			1.29375i
$910$		0			0
$911$	617.826			0.678185			0.339092	−	0.940753i	$-0.389880\pi$
$911$	617.826			0.678185			0.339092	+	0.940753i	$0.389880\pi$
$912$		0			0
$913$	274.116	−	274.116i	0.300236	−	0.300236i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1359.61	−	1359.61i	−1.48268	−	1.48268i
$918$		0			0
$919$			1001.25i			1.08950i	0.838599	+	0.544750i	$0.183375\pi$
$919$			1001.25i			1.08950i	−0.838599	+	0.544750i	$0.816625\pi$
$920$		0			0
$921$	−875.988			−0.951127
$922$		0			0
$923$	−88.2128	+	88.2128i	−0.0955718	+	0.0955718i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	646.419	+	646.419i	0.697324	+	0.697324i
$928$		0			0
$929$		−	1140.11i		−	1.22724i	−0.789601	−	0.613621i	$-0.789712\pi$
$929$		−	1140.11i		−	1.22724i	0.789601	−	0.613621i	$-0.210288\pi$
$930$		0			0
$931$	−917.110			−0.985081
$932$		0			0
$933$	2096.65	−	2096.65i	2.24721	−	2.24721i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	55.8420	+	55.8420i	0.0595966	+	0.0595966i	0.736277	−	0.676680i	$-0.236582\pi$
$937$	55.8420	+	55.8420i	0.0595966	+	0.0595966i	−0.676680	+	0.736277i	$0.736582\pi$
$938$		0			0
$939$		−	3045.37i		−	3.24321i
$940$		0			0
$941$	557.134			0.592065			0.296033	−	0.955178i	$-0.404336\pi$
$941$	557.134			0.592065			0.296033	+	0.955178i	$0.404336\pi$
$942$		0			0
$943$	−455.685	+	455.685i	−0.483229	+	0.483229i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−920.019	−	920.019i	−0.971509	−	0.971509i	0.0280967	−	0.999605i	$-0.491055\pi$
$947$	−920.019	−	920.019i	−0.971509	−	0.971509i	−0.999605	+	0.0280967i	$0.991055\pi$
$948$		0			0
$949$			1039.88i			1.09576i
$950$		0			0
$951$	2924.42			3.07510
$952$		0			0
$953$	29.6394	−	29.6394i	0.0311012	−	0.0311012i	−0.691385	−	0.722486i	$-0.742999\pi$
$953$	29.6394	−	29.6394i	0.0311012	−	0.0311012i	0.722486	+	0.691385i	$0.242999\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−177.622	−	177.622i	−0.185603	−	0.185603i
$958$		0			0
$959$			173.913i			0.181348i
$960$		0			0
$961$	542.479			0.564495
$962$		0			0
$963$	−975.398	+	975.398i	−1.01287	+	1.01287i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	501.846	+	501.846i	0.518972	+	0.518972i	0.917260	−	0.398288i	$-0.130396\pi$
$967$	501.846	+	501.846i	0.518972	+	0.518972i	−0.398288	+	0.917260i	$0.630396\pi$
$968$		0			0
$969$			1647.43i			1.70013i
$970$		0			0
$971$	−281.520			−0.289927			−0.144964	−	0.989437i	$-0.546307\pi$
$971$	−281.520			−0.289927			−0.144964	+	0.989437i	$0.546307\pi$
$972$		0			0
$973$	1187.20	−	1187.20i	1.22015	−	1.22015i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−503.792	−	503.792i	−0.515652	−	0.515652i	0.400601	−	0.916253i	$-0.368801\pi$
$977$	−503.792	−	503.792i	−0.515652	−	0.515652i	−0.916253	+	0.400601i	$0.868801\pi$
$978$		0			0
$979$			81.7666i			0.0835206i
$980$		0			0
$981$	787.680			0.802936
$982$		0			0
$983$	385.480	−	385.480i	0.392147	−	0.392147i	−0.483305	−	0.875452i	$-0.660564\pi$
$983$	385.480	−	385.480i	0.392147	−	0.392147i	0.875452	+	0.483305i	$0.160564\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1372.17	+	1372.17i	1.39025	+	1.39025i
$988$		0			0
$989$		−	2261.04i		−	2.28618i
$990$		0			0
$991$	−435.164			−0.439116			−0.219558	−	0.975599i	$-0.570462\pi$
$991$	−435.164			−0.439116			−0.219558	+	0.975599i	$0.570462\pi$
$992$		0			0
$993$	1718.70	−	1718.70i	1.73082	−	1.73082i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	812.819	+	812.819i	0.815264	+	0.815264i	0.985418	−	0.170153i	$-0.0544263\pi$
$997$	812.819	+	812.819i	0.815264	+	0.815264i	−0.170153	+	0.985418i	$0.554426\pi$
$998$		0			0
$999$		−	1802.84i		−	1.80465i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.3.j.b.1057.1		20
5.2	odd	4		220.3.j.a.133.10	✓	20
5.3	odd	4	inner	1100.3.j.b.793.1		20
5.4	even	2		220.3.j.a.177.10	yes	20
15.2	even	4		1980.3.s.a.793.1		20
15.14	odd	2		1980.3.s.a.397.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.j.a.133.10	✓	20	5.2	odd	4
220.3.j.a.177.10	yes	20	5.4	even	2
1100.3.j.b.793.1		20	5.3	odd	4	inner
1100.3.j.b.1057.1		20	1.1	even	1	trivial
1980.3.s.a.397.1		20	15.14	odd	2
1980.3.s.a.793.1		20	15.2	even	4

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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1100.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.95222 + 3.95222i) q^{3} +(7.19082 + 7.19082i) q^{7} -22.2400i q^{9} +3.31662 q^{11} +(9.00000 - 9.00000i) q^{13} +(-12.3662 - 12.3662i) q^{17} +16.8538i q^{19} -56.8393 q^{21} +(-24.1259 + 24.1259i) q^{23} +(52.3274 + 52.3274i) q^{27} +13.5506i q^{29} -38.7747 q^{31} +(-13.1080 + 13.1080i) q^{33} +(-17.2266 - 17.2266i) q^{37} +71.1399i q^{39} +18.8877 q^{41} +(-46.8590 + 46.8590i) q^{43} +(-24.1412 - 24.1412i) q^{47} +54.4157i q^{49} +97.7481 q^{51} +(42.8165 - 42.8165i) q^{53} +(-66.6098 - 66.6098i) q^{57} +5.26637i q^{59} -95.5197 q^{61} +(159.924 - 159.924i) q^{63} +(-41.5998 - 41.5998i) q^{67} -190.702i q^{69} -9.80142 q^{71} +(-57.7709 + 57.7709i) q^{73} +(23.8492 + 23.8492i) q^{77} +42.6126i q^{79} -213.458 q^{81} +(82.6490 - 82.6490i) q^{83} +(-53.5550 - 53.5550i) q^{87} +24.6536i q^{89} +129.435 q^{91} +(153.246 - 153.246i) q^{93} +(19.7682 + 19.7682i) q^{97} -73.7618i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} + 28 q^{13} - 24 q^{17} - 106 q^{23} + 50 q^{27} + 88 q^{31} + 22 q^{33} - 2 q^{37} + 72 q^{41} - 168 q^{43} - 108 q^{47} + 112 q^{51} + 164 q^{53} - 48 q^{57} - 280 q^{61} + 348 q^{63} - 110 q^{67}+ \cdots + 542 q^{97}+O(q^{100}) \) 20 * q + 2 * q^3 + 28 * q^13 - 24 * q^17 - 106 * q^23 + 50 * q^27 + 88 * q^31 + 22 * q^33 - 2 * q^37 + 72 * q^41 - 168 * q^43 - 108 * q^47 + 112 * q^51 + 164 * q^53 - 48 * q^57 - 280 * q^61 + 348 * q^63 - 110 * q^67 - 140 * q^71 - 28 * q^73 - 552 * q^81 - 16 * q^83 - 208 * q^87 + 480 * q^91 + 374 * q^93 + 542 * q^97

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