LMFDB - Embedded newform 1008.3.dc.e.305.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(305,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.305"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.dc (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,2,0,0,0,0,0,-52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 307x^{8} - 1824x^{6} + 8289x^{4} - 1674x^{2} + 324 $ x^12 - 20x^10 + 307x^8 - 1824x^6 + 8289x^4 - 1674*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			305.6
Root			$0.389477 + 0.224865i$ of defining polynomial
Character	$\chi$	$=$	1008.305
Dual form			1008.3.dc.e.737.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.49023 - 4.32449i) q^{5} +(-6.25176 - 3.14888i) q^{7} +(6.95899 + 4.01778i) q^{11} -10.8897 q^{13} +(-10.2166 - 5.89854i) q^{17} +(-5.44485 - 9.43076i) q^{19} +(20.2398 - 11.6854i) q^{23} +(24.9024 - 43.1322i) q^{25} -38.3888i q^{29} +(10.9575 - 18.9790i) q^{31} +(-60.4445 + 3.44983i) q^{35} +(3.94838 + 6.83879i) q^{37} +35.1994i q^{41} -28.1174 q^{43} +(-46.1528 + 26.6463i) q^{47} +(29.1691 + 39.3721i) q^{49} +(-61.0817 - 35.2655i) q^{53} +69.4993 q^{55} +(-23.6821 - 13.6729i) q^{59} +(-17.5678 - 30.4284i) q^{61} +(-81.5664 + 47.0924i) q^{65} +(15.9024 - 27.5438i) q^{67} -127.309i q^{71} +(14.5969 - 25.2826i) q^{73} +(-30.8545 - 47.0312i) q^{77} +(35.5164 + 61.5161i) q^{79} -1.94121i q^{83} -102.033 q^{85} +(-66.4198 + 38.3475i) q^{89} +(68.0798 + 34.2904i) q^{91} +(-81.5664 - 47.0924i) q^{95} +169.973 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{7} - 52 q^{13} - 26 q^{19} + 106 q^{25} - 22 q^{31} - 146 q^{37} - 108 q^{43} + 114 q^{49} - 16 q^{55} - 136 q^{61} - 2 q^{67} - 482 q^{73} - 42 q^{79} - 288 q^{85} - 222 q^{91} + 568 q^{97}+O(q^{100}) $ 12 * q + 2 * q^7 - 52 * q^13 - 26 * q^19 + 106 * q^25 - 22 * q^31 - 146 * q^37 - 108 * q^43 + 114 * q^49 - 16 * q^55 - 136 * q^61 - 2 * q^67 - 482 * q^73 - 42 * q^79 - 288 * q^85 - 222 * q^91 + 568 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	7.49023	−	4.32449i	1.49805	−	0.864898i	0.498049	−	0.867149i	$-0.334050\pi$
$5$	7.49023	−	4.32449i	1.49805	−	0.864898i	0.999997	+	0.00225107i	$0.000716537\pi$
$6$		0			0
$7$	−6.25176	−	3.14888i	−0.893109	−	0.449840i
$7$	−6.25176	−	3.14888i	−0.893109	−	0.449840i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.95899	+	4.01778i	0.632636	+	0.365252i	0.781772	−	0.623564i	$-0.214316\pi$
$11$	6.95899	+	4.01778i	0.632636	+	0.365252i	−0.149136	+	0.988817i	$0.547649\pi$
$12$		0			0
$13$	−10.8897			−0.837669			−0.418835	−	0.908063i	$-0.637561\pi$
$13$	−10.8897			−0.837669			−0.418835	+	0.908063i	$0.637561\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−10.2166	−	5.89854i	−0.600975	−	0.346973i	0.168450	−	0.985710i	$-0.446124\pi$
$17$	−10.2166	−	5.89854i	−0.600975	−	0.346973i	−0.769425	+	0.638737i	$0.779457\pi$
$18$		0			0
$19$	−5.44485	−	9.43076i	−0.286571	−	0.496356i	0.686418	−	0.727207i	$-0.259182\pi$
$19$	−5.44485	−	9.43076i	−0.286571	−	0.496356i	−0.972989	+	0.230852i	$0.925849\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	20.2398	−	11.6854i	0.879991	−	0.508063i	0.00933533	−	0.999956i	$-0.497028\pi$
$23$	20.2398	−	11.6854i	0.879991	−	0.508063i	0.870655	+	0.491894i	$0.163695\pi$
$24$		0			0
$25$	24.9024	−	43.1322i	0.996096	−	1.72529i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	38.3888i		−	1.32375i	−0.749613	−	0.661876i	$-0.769760\pi$
$29$		−	38.3888i		−	1.32375i	0.749613	−	0.661876i	$-0.230240\pi$
$30$		0			0
$31$	10.9575	−	18.9790i	0.353469	−	0.612227i	−0.633385	−	0.773836i	$-0.718335\pi$
$31$	10.9575	−	18.9790i	0.353469	−	0.612227i	0.986855	+	0.161610i	$0.0516685\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−60.4445	+	3.44983i	−1.72698	+	0.0985666i
$36$		0			0
$37$	3.94838	+	6.83879i	0.106713	+	0.184832i	0.914437	−	0.404729i	$-0.132634\pi$
$37$	3.94838	+	6.83879i	0.106713	+	0.184832i	−0.807724	+	0.589561i	$0.799301\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			35.1994i			0.858521i	0.903181	+	0.429260i	$0.141226\pi$
$41$			35.1994i			0.858521i	−0.903181	+	0.429260i	$0.858774\pi$
$42$		0			0
$43$	−28.1174			−0.653892			−0.326946	−	0.945043i	$-0.606020\pi$
$43$	−28.1174			−0.653892			−0.326946	+	0.945043i	$0.606020\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−46.1528	+	26.6463i	−0.981974	+	0.566943i	−0.902866	−	0.429923i	$-0.858541\pi$
$47$	−46.1528	+	26.6463i	−0.981974	+	0.566943i	−0.0791087	+	0.996866i	$0.525207\pi$
$48$		0			0
$49$	29.1691	+	39.3721i	0.595288	+	0.803513i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−61.0817	−	35.2655i	−1.15248	−	0.665387i	−0.202993	−	0.979180i	$-0.565067\pi$
$53$	−61.0817	−	35.2655i	−1.15248	−	0.665387i	−0.949491	+	0.313793i	$0.898400\pi$
$54$		0			0
$55$	69.4993			1.26362
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−23.6821	−	13.6729i	−0.401391	−	0.231743i	0.285693	−	0.958321i	$-0.407776\pi$
$59$	−23.6821	−	13.6729i	−0.401391	−	0.231743i	−0.687084	+	0.726578i	$0.741110\pi$
$60$		0			0
$61$	−17.5678	−	30.4284i	−0.287997	−	0.498826i	0.685334	−	0.728229i	$-0.259656\pi$
$61$	−17.5678	−	30.4284i	−0.287997	−	0.498826i	−0.973332	+	0.229402i	$0.926323\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−81.5664	+	47.0924i	−1.25487	+	0.724498i
$66$		0			0
$67$	15.9024	−	27.5438i	0.237349	−	0.411101i	−0.722604	−	0.691263i	$-0.757055\pi$
$67$	15.9024	−	27.5438i	0.237349	−	0.411101i	0.959953	+	0.280162i	$0.0903881\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	127.309i		−	1.79309i	−0.442956	−	0.896543i	$-0.646070\pi$
$71$		−	127.309i		−	1.79309i	0.442956	−	0.896543i	$-0.353930\pi$
$72$		0			0
$73$	14.5969	−	25.2826i	0.199958	−	0.346337i	−0.748557	−	0.663071i	$-0.769253\pi$
$73$	14.5969	−	25.2826i	0.199958	−	0.346337i	0.948514	+	0.316734i	$0.102586\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−30.8545	−	47.0312i	−0.400708	−	0.610795i
$78$		0			0
$79$	35.5164	+	61.5161i	0.449574	+	0.778685i	0.998358	−	0.0572789i	$-0.0182424\pi$
$79$	35.5164	+	61.5161i	0.449574	+	0.778685i	−0.548784	+	0.835964i	$0.684909\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	1.94121i		−	0.0233881i	−0.999932	−	0.0116941i	$-0.996278\pi$
$83$		−	1.94121i		−	0.0233881i	0.999932	−	0.0116941i	$-0.00372242\pi$
$84$		0			0
$85$	−102.033			−1.20038
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−66.4198	+	38.3475i	−0.746289	+	0.430870i	−0.824352	−	0.566078i	$-0.808460\pi$
$89$	−66.4198	+	38.3475i	−0.746289	+	0.430870i	0.0780622	+	0.996948i	$0.475127\pi$
$90$		0			0
$91$	68.0798	+	34.2904i	0.748130	+	0.376817i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−81.5664	−	47.0924i	−0.858594	−	0.495709i
$96$		0			0
$97$	169.973			1.75230			0.876151	−	0.482038i	$-0.160103\pi$
$97$	169.973			1.75230			0.876151	+	0.482038i	$0.160103\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	121.412	+	70.0970i	1.20209	+	0.694030i	0.961020	−	0.276477i	$-0.0891670\pi$
$101$	121.412	+	70.0970i	1.20209	+	0.694030i	0.241074	+	0.970507i	$0.422500\pi$
$102$		0			0
$103$	−48.6818	−	84.3194i	−0.472639	−	0.818635i	0.526871	−	0.849945i	$-0.323365\pi$
$103$	−48.6818	−	84.3194i	−0.472639	−	0.818635i	−0.999510	+	0.0313108i	$0.990032\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−83.6756	+	48.3101i	−0.782015	+	0.451496i	−0.837144	−	0.546983i	$-0.815776\pi$
$107$	−83.6756	+	48.3101i	−0.782015	+	0.451496i	0.0551290	+	0.998479i	$0.482443\pi$
$108$		0			0
$109$	7.41721	−	12.8470i	0.0680478	−	0.117862i	−0.829994	−	0.557772i	$-0.811656\pi$
$109$	7.41721	−	12.8470i	0.0680478	−	0.117862i	0.898042	+	0.439910i	$0.144990\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	40.2522i		−	0.356214i	−0.984011	−	0.178107i	$-0.943003\pi$
$113$		−	40.2522i		−	0.356214i	0.984011	−	0.178107i	$-0.0569974\pi$
$114$		0			0
$115$	101.067	−	175.053i	0.878845	−	1.52220i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	45.2978	+	69.0471i	0.380654	+	0.580227i
$120$		0			0
$121$	−28.2150	−	48.8697i	−0.233181	−	0.403882i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	214.536i		−	1.71629i
$126$		0			0
$127$	−88.6423			−0.697971			−0.348985	−	0.937128i	$-0.613474\pi$
$127$	−88.6423			−0.697971			−0.348985	+	0.937128i	$0.613474\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	141.626	−	81.7676i	1.08111	−	0.624180i	0.149916	−	0.988699i	$-0.452100\pi$
$131$	141.626	−	81.7676i	1.08111	−	0.624180i	0.931196	+	0.364519i	$0.118766\pi$
$132$		0			0
$133$	4.34359	+	76.1041i	0.0326586	+	0.572211i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−20.8199	−	12.0204i	−0.151970	−	0.0877399i	0.422087	−	0.906555i	$-0.361298\pi$
$137$	−20.8199	−	12.0204i	−0.151970	−	0.0877399i	−0.574057	+	0.818816i	$0.694631\pi$
$138$		0			0
$139$	−167.635			−1.20601			−0.603004	−	0.797738i	$-0.706030\pi$
$139$	−167.635			−1.20601			−0.603004	+	0.797738i	$0.706030\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−75.7813	−	43.7524i	−0.529939	−	0.305961i
$144$		0			0
$145$	−166.012	−	287.541i	−1.14491	−	1.98304i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.5974	−	7.85046i	0.0912576	−	0.0526876i	−0.453677	−	0.891166i	$-0.649888\pi$
$149$	13.5974	−	7.85046i	0.0912576	−	0.0526876i	0.544934	+	0.838479i	$0.316555\pi$
$150$		0			0
$151$	96.3155	−	166.823i	0.637851	−	1.10479i	−0.348052	−	0.937475i	$-0.613157\pi$
$151$	96.3155	−	166.823i	0.637851	−	1.10479i	0.985903	−	0.167315i	$-0.0535098\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	189.543i		−	1.22286i
$156$		0			0
$157$	58.4739	−	101.280i	0.372445	−	0.645094i	−0.617496	−	0.786574i	$-0.711853\pi$
$157$	58.4739	−	101.280i	0.372445	−	0.645094i	0.989941	+	0.141480i	$0.0451861\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−163.330	+	9.32198i	−1.01447	+	0.0579005i
$162$		0			0
$163$	54.5932	+	94.5583i	0.334928	+	0.580112i	0.983471	−	0.181066i	$-0.0579547\pi$
$163$	54.5932	+	94.5583i	0.334928	+	0.580112i	−0.648543	+	0.761178i	$0.724621\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			217.273i			1.30103i	0.759492	+	0.650517i	$0.225448\pi$
$167$			217.273i			1.30103i	−0.759492	+	0.650517i	$0.774552\pi$
$168$		0			0
$169$	−50.4144			−0.298310
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.58828	+	4.95845i	−0.0496433	+	0.0286615i	−0.524616	−	0.851339i	$-0.675791\pi$
$173$	−8.58828	+	4.95845i	−0.0496433	+	0.0286615i	0.474973	+	0.880000i	$0.342458\pi$
$174$		0			0
$175$	−291.502	+	191.238i	−1.66573	+	1.09279i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−37.5658	−	21.6886i	−0.209865	−	0.121165i	0.391384	−	0.920228i	$-0.371996\pi$
$179$	−37.5658	−	21.6886i	−0.209865	−	0.121165i	−0.601248	+	0.799062i	$0.705330\pi$
$180$		0			0
$181$	183.194			1.01212			0.506060	−	0.862498i	$-0.331101\pi$
$181$	183.194			1.01212			0.506060	+	0.862498i	$0.331101\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	59.1486	+	34.1494i	0.319722	+	0.184592i
$186$		0			0
$187$	−47.3980	−	82.0958i	−0.253465	−	0.439015i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	32.5684	−	18.8034i	0.170515	−	0.0984471i	−0.412313	−	0.911042i	$-0.635279\pi$
$191$	32.5684	−	18.8034i	0.170515	−	0.0984471i	0.582829	+	0.812595i	$0.301946\pi$
$192$		0			0
$193$	16.8013	−	29.1007i	0.0870532	−	0.150781i	−0.819211	−	0.573492i	$-0.805588\pi$
$193$	16.8013	−	29.1007i	0.0870532	−	0.150781i	0.906264	+	0.422712i	$0.138922\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	172.788i		−	0.877095i	−0.898708	−	0.438548i	$-0.855493\pi$
$197$		−	172.788i		−	0.877095i	0.898708	−	0.438548i	$-0.144507\pi$
$198$		0			0
$199$	89.4485	−	154.929i	0.449490	−	0.778540i	−0.548863	−	0.835912i	$-0.684939\pi$
$199$	89.4485	−	154.929i	0.449490	−	0.778540i	0.998353	+	0.0573729i	$0.0182724\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−120.882	+	239.998i	−0.595477	+	1.18226i
$204$		0			0
$205$	152.219	+	263.651i	0.742533	+	1.28610i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	87.5048i		−	0.418683i
$210$		0			0
$211$	293.408			1.39056			0.695279	−	0.718740i	$-0.255281\pi$
$211$	293.408			1.39056			0.695279	+	0.718740i	$0.255281\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−210.606	+	121.593i	−0.979561	+	0.565550i
$216$		0			0
$217$	−128.267	+	84.1484i	−0.591091	+	0.387781i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	111.255	+	64.2334i	0.503418	+	0.290649i
$222$		0			0
$223$	102.718			0.460621			0.230311	−	0.973117i	$-0.426026\pi$
$223$	102.718			0.460621			0.230311	+	0.973117i	$0.426026\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	192.733	+	111.274i	0.849042	+	0.490195i	0.860328	−	0.509742i	$-0.170259\pi$
$227$	192.733	+	111.274i	0.849042	+	0.490195i	−0.0112855	+	0.999936i	$0.503592\pi$
$228$		0			0
$229$	186.758	+	323.474i	0.815537	+	1.41255i	0.908942	+	0.416923i	$0.136892\pi$
$229$	186.758	+	323.474i	0.815537	+	1.41255i	−0.0934045	+	0.995628i	$0.529775\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−257.051	+	148.408i	−1.10322	+	0.636945i	−0.937065	−	0.349155i	$-0.886469\pi$
$233$	−257.051	+	148.408i	−1.10322	+	0.636945i	−0.166156	+	0.986099i	$0.553136\pi$
$234$		0			0
$235$	−230.463	+	399.174i	−0.980696	+	1.69861i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	73.8411i		−	0.308959i	−0.987996	−	0.154479i	$-0.950630\pi$
$239$		−	73.8411i		−	0.308959i	0.987996	−	0.154479i	$-0.0493700\pi$
$240$		0			0
$241$	−90.1564	+	156.155i	−0.374093	+	0.647948i	−0.990191	−	0.139722i	$-0.955379\pi$
$241$	−90.1564	+	156.155i	−0.374093	+	0.647948i	0.616098	+	0.787670i	$0.288713\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	388.748	+	168.765i	1.58673	+	0.688836i
$246$		0			0
$247$	59.2928	+	102.698i	0.240052	+	0.415782i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	141.050i		−	0.561954i	−0.959715	−	0.280977i	$-0.909342\pi$
$251$		−	141.050i		−	0.561954i	0.959715	−	0.280977i	$-0.0906584\pi$
$252$		0			0
$253$	187.798			0.742285
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−222.864	+	128.671i	−0.867177	+	0.500665i	−0.866409	−	0.499335i	$-0.833578\pi$
$257$	−222.864	+	128.671i	−0.867177	+	0.500665i	−0.000767847		1.00000i	$0.500244\pi$
$258$		0			0
$259$	−3.14979	−	55.1875i	−0.0121614	−	0.213079i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	377.586	+	217.999i	1.43569	+	0.828895i	0.997546	−	0.0700101i	$-0.0223031\pi$
$263$	377.586	+	217.999i	1.43569	+	0.828895i	0.438143	+	0.898905i	$0.355636\pi$
$264$		0			0
$265$	−610.021			−2.30197
$266$		0			0
$267$		0			0
$268$		0			0
$269$	438.287	+	253.045i	1.62932	+	0.940688i	0.984296	+	0.176524i	$0.0564853\pi$
$269$	438.287	+	253.045i	1.62932	+	0.940688i	0.645022	+	0.764164i	$0.276848\pi$
$270$		0			0
$271$	183.222	+	317.350i	0.676097	+	1.17103i	0.976147	+	0.217110i	$0.0696631\pi$
$271$	183.222	+	317.350i	0.676097	+	1.17103i	−0.300051	+	0.953923i	$0.597004\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	346.591	−	200.105i	1.26033	−	0.727653i
$276$		0			0
$277$	14.7159	−	25.4887i	0.0531260	−	0.0920170i	−0.838239	−	0.545302i	$-0.816415\pi$
$277$	14.7159	−	25.4887i	0.0531260	−	0.0920170i	0.891365	+	0.453285i	$0.149748\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			67.1419i			0.238939i	0.992838	+	0.119470i	$0.0381194\pi$
$281$			67.1419i			0.238939i	−0.992838	+	0.119470i	$0.961881\pi$
$282$		0			0
$283$	−222.216	+	384.889i	−0.785214	+	1.36003i	0.143657	+	0.989627i	$0.454114\pi$
$283$	−222.216	+	384.889i	−0.785214	+	1.36003i	−0.928871	+	0.370403i	$0.879220\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	110.839	−	220.058i	0.386197	−	0.766753i
$288$		0			0
$289$	−74.9144	−	129.756i	−0.259219	−	0.448981i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			334.732i			1.14243i	0.820800	+	0.571215i	$0.193528\pi$
$293$			334.732i			1.14243i	−0.820800	+	0.571215i	$0.806472\pi$
$294$		0			0
$295$	−236.513			−0.801737
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−220.405	+	127.251i	−0.737141	+	0.425589i
$300$		0			0
$301$	175.783	+	88.5382i	0.583997	+	0.294147i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−263.175	−	151.944i	−0.862867	−	0.498177i
$306$		0			0
$307$	258.213			0.841086			0.420543	−	0.907273i	$-0.361840\pi$
$307$	258.213			0.841086			0.420543	+	0.907273i	$0.361840\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	183.756	+	106.092i	0.590856	+	0.341131i	0.765436	−	0.643512i	$-0.222523\pi$
$311$	183.756	+	106.092i	0.590856	+	0.341131i	−0.174580	+	0.984643i	$0.555857\pi$
$312$		0			0
$313$	−109.144	−	189.042i	−0.348702	−	0.603969i	0.637317	−	0.770602i	$-0.280044\pi$
$313$	−109.144	−	189.042i	−0.348702	−	0.603969i	−0.986019	+	0.166632i	$0.946711\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	282.647	−	163.186i	0.891631	−	0.514783i	0.0171555	−	0.999853i	$-0.494539\pi$
$317$	282.647	−	163.186i	0.891631	−	0.514783i	0.874476	+	0.485069i	$0.161206\pi$
$318$		0			0
$319$	154.238	−	267.147i	0.483504	−	0.837453i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			128.467i			0.397730i
$324$		0			0
$325$	−271.180	+	469.697i	−0.834399	+	1.44522i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	372.442	−	21.2569i	1.13204	−	0.0646107i
$330$		0			0
$331$	1.83301	+	3.17486i	0.00553779	+	0.00959174i	0.868781	−	0.495196i	$-0.164904\pi$
$331$	1.83301	+	3.17486i	0.00553779	+	0.00959174i	−0.863243	+	0.504788i	$0.831571\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	275.079i		−	0.821131i
$336$		0			0
$337$	−117.056			−0.347347			−0.173673	−	0.984803i	$-0.555564\pi$
$337$	−117.056			−0.347347			−0.173673	+	0.984803i	$0.555564\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	152.507	−	88.0499i	0.447235	−	0.258211i
$342$		0			0
$343$	−58.3803	−	337.995i	−0.170205	−	0.985409i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−296.287	−	171.061i	−0.853853	−	0.492972i	0.00809576	−	0.999967i	$-0.497423\pi$
$347$	−296.287	−	171.061i	−0.853853	−	0.492972i	−0.861949	+	0.506995i	$0.830756\pi$
$348$		0			0
$349$	161.765			0.463509			0.231755	−	0.972774i	$-0.425553\pi$
$349$	161.765			0.463509			0.231755	+	0.972774i	$0.425553\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	41.8270	+	24.1488i	0.118490	+	0.0684103i	0.558074	−	0.829791i	$-0.311541\pi$
$353$	41.8270	+	24.1488i	0.118490	+	0.0684103i	−0.439584	+	0.898202i	$0.644874\pi$
$354$		0			0
$355$	−550.547	−	953.575i	−1.55084	−	2.68613i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	176.022	−	101.626i	0.490312	−	0.283082i	−0.234392	−	0.972142i	$-0.575310\pi$
$359$	176.022	−	101.626i	0.490312	−	0.283082i	0.724704	+	0.689060i	$0.241977\pi$
$360$		0			0
$361$	121.207	−	209.937i	0.335754	−	0.581543i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	252.497i		−	0.691771i
$366$		0			0
$367$	−350.904	+	607.783i	−0.956142	+	1.65609i	−0.224408	+	0.974495i	$0.572045\pi$
$367$	−350.904	+	607.783i	−0.956142	+	1.65609i	−0.731734	+	0.681590i	$0.761289\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	270.821	+	412.810i	0.729976	+	1.11270i
$372$		0			0
$373$	−174.154	−	301.644i	−0.466902	−	0.808697i	0.532383	−	0.846503i	$-0.321296\pi$
$373$	−174.154	−	301.644i	−0.466902	−	0.808697i	−0.999285	+	0.0378060i	$0.987963\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			418.043i			1.10887i
$378$		0			0
$379$	464.851			1.22652			0.613260	−	0.789881i	$-0.289858\pi$
$379$	464.851			1.22652			0.613260	+	0.789881i	$0.289858\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−173.981	+	100.448i	−0.454257	+	0.262266i	−0.709627	−	0.704578i	$-0.751136\pi$
$383$	−173.981	+	100.448i	−0.454257	+	0.262266i	0.255369	+	0.966844i	$0.417803\pi$
$384$		0			0
$385$	−434.493	−	218.845i	−1.12855	−	0.568429i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.5433	+	6.08718i	0.0271036	+	0.0156483i	0.513491	−	0.858095i	$-0.328352\pi$
$389$	10.5433	+	6.08718i	0.0271036	+	0.0156483i	−0.486387	+	0.873744i	$0.661685\pi$
$390$		0			0
$391$	−275.708			−0.705137
$392$		0			0
$393$		0			0
$394$		0			0
$395$	532.052	+	307.180i	1.34697	+	0.777671i
$396$		0			0
$397$	−47.9960	−	83.1315i	−0.120897	−	0.209399i	0.799225	−	0.601032i	$-0.205244\pi$
$397$	−47.9960	−	83.1315i	−0.120897	−	0.209399i	−0.920122	+	0.391633i	$0.871910\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	449.101	−	259.289i	1.11995	−	0.646606i	0.178565	−	0.983928i	$-0.442855\pi$
$401$	449.101	−	259.289i	1.11995	−	0.646606i	0.941389	+	0.337323i	$0.109521\pi$
$402$		0			0
$403$	−119.324	+	206.676i	−0.296090	+	0.512844i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			63.4548i			0.155909i
$408$		0			0
$409$	−76.8817	+	133.163i	−0.187975	+	0.325582i	−0.944575	−	0.328296i	$-0.893526\pi$
$409$	−76.8817	+	133.163i	−0.187975	+	0.325582i	0.756600	+	0.653878i	$0.226859\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	105.001	+	160.052i	0.254239	+	0.387534i
$414$		0			0
$415$	−8.39476	−	14.5402i	−0.0202283	−	0.0350365i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	76.4753i		−	0.182519i	−0.995827	−	0.0912593i	$-0.970911\pi$
$419$		−	76.4753i		−	0.182519i	0.995827	−	0.0912593i	$-0.0290892\pi$
$420$		0			0
$421$	639.803			1.51972			0.759861	−	0.650086i	$-0.225267\pi$
$421$	639.803			1.51972			0.759861	+	0.650086i	$0.225267\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−508.834	+	293.776i	−1.19726	+	0.691237i
$426$		0			0
$427$	14.0146	+	245.550i	0.0328211	+	0.575059i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	126.587	+	73.0849i	0.293705	+	0.169570i	0.639611	−	0.768698i	$-0.279095\pi$
$431$	126.587	+	73.0849i	0.293705	+	0.169570i	−0.345907	+	0.938269i	$0.612429\pi$
$432$		0			0
$433$	426.699			0.985448			0.492724	−	0.870186i	$-0.336001\pi$
$433$	426.699			0.985448			0.492724	+	0.870186i	$0.336001\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−220.405	−	127.251i	−0.504360	−	0.291192i
$438$		0			0
$439$	356.170	+	616.904i	0.811321	+	1.40525i	0.911940	+	0.410324i	$0.134584\pi$
$439$	356.170	+	616.904i	0.811321	+	1.40525i	−0.100619	+	0.994925i	$0.532082\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	49.7225	−	28.7073i	0.112240	−	0.0648020i	−0.442829	−	0.896606i	$-0.646025\pi$
$443$	49.7225	−	28.7073i	0.112240	−	0.0648020i	0.555069	+	0.831804i	$0.312692\pi$
$444$		0			0
$445$	−331.666	+	574.463i	−0.745318	+	1.29093i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			405.746i			0.903666i	0.892103	+	0.451833i	$0.149230\pi$
$449$			405.746i			0.903666i	−0.892103	+	0.451833i	$0.850770\pi$
$450$		0			0
$451$	−141.423	+	244.952i	−0.313577	+	0.543131i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	658.222	−	37.5676i	1.44664	−	0.0825662i
$456$		0			0
$457$	124.387	+	215.445i	0.272182	+	0.471433i	0.969420	−	0.245406i	$-0.0789214\pi$
$457$	124.387	+	215.445i	0.272182	+	0.471433i	−0.697238	+	0.716839i	$0.745588\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	835.580i		−	1.81254i	−0.422702	−	0.906269i	$-0.638918\pi$
$461$		−	835.580i		−	1.81254i	0.422702	−	0.906269i	$-0.361082\pi$
$462$		0			0
$463$	323.497			0.698698			0.349349	−	0.936993i	$-0.386403\pi$
$463$	323.497			0.698698			0.349349	+	0.936993i	$0.386403\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−212.171	+	122.497i	−0.454327	+	0.262306i	−0.709656	−	0.704549i	$-0.751149\pi$
$467$	−212.171	+	122.497i	−0.454327	+	0.262306i	0.255329	+	0.966854i	$0.417816\pi$
$468$		0			0
$469$	−186.150	+	122.122i	−0.396908	+	0.260389i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−195.668	−	112.969i	−0.413675	−	0.238836i
$474$		0			0
$475$	−542.359			−1.14181
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−613.130	−	353.991i	−1.28002	−	0.739020i	−0.303169	−	0.952937i	$-0.598045\pi$
$479$	−613.130	−	353.991i	−1.28002	−	0.739020i	−0.976852	+	0.213917i	$0.931378\pi$
$480$		0			0
$481$	−42.9967	−	74.4724i	−0.0893902	−	0.154828i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1273.14	−	735.047i	2.62503	−	1.51556i
$486$		0			0
$487$	401.598	−	695.589i	0.824637	−	1.42831i	−0.0775585	−	0.996988i	$-0.524712\pi$
$487$	401.598	−	695.589i	0.824637	−	1.42831i	0.902196	−	0.431326i	$-0.141954\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			389.098i			0.792461i	0.918151	+	0.396230i	$0.129682\pi$
$491$			389.098i			0.792461i	−0.918151	+	0.396230i	$0.870318\pi$
$492$		0			0
$493$	−226.438	+	392.202i	−0.459306	+	0.795542i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−400.881	+	795.907i	−0.806602	+	1.60142i
$498$		0			0
$499$	−155.635	−	269.567i	−0.311893	−	0.540215i	0.666879	−	0.745166i	$-0.267630\pi$
$499$	−155.635	−	269.567i	−0.311893	−	0.540215i	−0.978772	+	0.204951i	$0.934296\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			800.342i			1.59114i	0.605864	+	0.795568i	$0.292827\pi$
$503$			800.342i			1.59114i	−0.605864	+	0.795568i	$0.707173\pi$
$504$		0			0
$505$	1212.53			2.40106
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−485.974	+	280.577i	−0.954762	+	0.551232i	−0.894557	−	0.446954i	$-0.852509\pi$
$509$	−485.974	+	280.577i	−0.954762	+	0.551232i	−0.0602049	+	0.998186i	$0.519175\pi$
$510$		0			0
$511$	−170.868	+	112.097i	−0.334380	+	0.219367i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−729.276	−	421.048i	−1.41607	−	0.817569i
$516$		0			0
$517$	−428.236			−0.828309
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−406.597	−	234.749i	−0.780416	−	0.450573i	0.0561618	−	0.998422i	$-0.482114\pi$
$521$	−406.597	−	234.749i	−0.780416	−	0.450573i	−0.836578	+	0.547848i	$0.815447\pi$
$522$		0			0
$523$	−68.4449	−	118.550i	−0.130870	−	0.226673i	0.793142	−	0.609036i	$-0.208444\pi$
$523$	−68.4449	−	118.550i	−0.130870	−	0.226673i	−0.924012	+	0.382363i	$0.875110\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−223.897	+	129.267i	−0.424852	+	0.245289i
$528$		0			0
$529$	8.59927	−	14.8944i	0.0162557	−	0.0281557i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	383.310i		−	0.719157i
$534$		0			0
$535$	−417.833	+	723.708i	−0.780997	+	1.35273i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	44.7992	+	391.185i	0.0831155	+	0.725761i
$540$		0			0
$541$	326.316	+	565.195i	0.603171	+	1.04472i	0.992338	+	0.123556i	$0.0394298\pi$
$541$	326.316	+	565.195i	0.603171	+	1.04472i	−0.389166	+	0.921168i	$0.627237\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	128.303i		−	0.235418i
$546$		0			0
$547$	994.957			1.81894			0.909468	−	0.415775i	$-0.136490\pi$
$547$	994.957			1.81894			0.909468	+	0.415775i	$0.136490\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−362.036	+	209.021i	−0.657052	+	0.379349i
$552$		0			0
$553$	−28.3329	−	496.421i	−0.0512350	−	0.897687i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	557.400	+	321.815i	1.00072	+	0.577764i	0.908460	−	0.417971i	$-0.137259\pi$
$557$	557.400	+	321.815i	1.00072	+	0.577764i	0.0922569	+	0.995735i	$0.470592\pi$
$558$		0			0
$559$	306.190			0.547745
$560$		0			0
$561$		0			0
$562$		0			0
$563$	45.3507	+	26.1832i	0.0805518	+	0.0465066i	0.539735	−	0.841835i	$-0.318524\pi$
$563$	45.3507	+	26.1832i	0.0805518	+	0.0465066i	−0.459183	+	0.888342i	$0.651858\pi$
$564$		0			0
$565$	−174.070	−	301.499i	−0.308089	−	0.533626i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−682.939	+	394.295i	−1.20024	+	0.692961i	−0.960610	−	0.277902i	$-0.910361\pi$
$569$	−682.939	+	394.295i	−1.20024	+	0.692961i	−0.239635	+	0.970863i	$0.577028\pi$
$570$		0			0
$571$	−356.934	+	618.228i	−0.625103	+	1.08271i	0.363417	+	0.931626i	$0.381610\pi$
$571$	−356.934	+	618.228i	−0.625103	+	1.08271i	−0.988521	+	0.151084i	$0.951723\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1163.98i		−	2.02432i
$576$		0			0
$577$	184.770	−	320.031i	0.320225	−	0.554646i	−0.660309	−	0.750994i	$-0.729575\pi$
$577$	184.770	−	320.031i	0.320225	−	0.554646i	0.980534	+	0.196347i	$0.0629081\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.11265	+	12.1360i	−0.0105209	+	0.0208882i
$582$		0			0
$583$	−283.378	−	490.825i	−0.486068	−	0.841895i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1125.62i		−	1.91758i	−0.284113	−	0.958791i	$-0.591699\pi$
$587$		−	1125.62i		−	1.91758i	0.284113	−	0.958791i	$-0.408301\pi$
$588$		0			0
$589$	−238.649			−0.405176
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−777.102	+	448.660i	−1.31046	+	0.756594i	−0.982172	−	0.187984i	$-0.939805\pi$
$593$	−777.102	+	448.660i	−1.31046	+	0.756594i	−0.328287	+	0.944578i	$0.606471\pi$
$594$		0			0
$595$	637.884	+	321.289i	1.07207	+	0.539981i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−788.076	−	454.996i	−1.31565	−	0.759593i	−0.332627	−	0.943058i	$-0.607935\pi$
$599$	−788.076	−	454.996i	−1.31565	−	0.759593i	−0.983026	+	0.183466i	$0.941268\pi$
$600$		0			0
$601$	−62.2700			−0.103611			−0.0518053	−	0.998657i	$-0.516498\pi$
$601$	−62.2700			−0.103611			−0.0518053	+	0.998657i	$0.516498\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−422.673	−	244.030i	−0.698633	−	0.403356i
$606$		0			0
$607$	−121.764	−	210.901i	−0.200599	−	0.347448i	0.748122	−	0.663561i	$-0.230956\pi$
$607$	−121.764	−	210.901i	−0.200599	−	0.347448i	−0.948722	+	0.316113i	$0.897622\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	502.590	−	290.171i	0.822570	−	0.474911i
$612$		0			0
$613$	258.552	−	447.825i	0.421781	−	0.730546i	−0.574333	−	0.818622i	$-0.694738\pi$
$613$	258.552	−	447.825i	0.421781	−	0.730546i	0.996114	+	0.0880757i	$0.0280717\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	543.011i		−	0.880083i	−0.897977	−	0.440042i	$-0.854964\pi$
$617$		−	543.011i		−	0.880083i	0.897977	−	0.440042i	$-0.145036\pi$
$618$		0			0
$619$	443.162	−	767.579i	0.715932	−	1.24003i	−0.246667	−	0.969100i	$-0.579335\pi$
$619$	443.162	−	767.579i	0.715932	−	1.24003i	0.962599	−	0.270930i	$-0.0873313\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	535.992	−	30.5914i	0.860341	−	0.0491034i
$624$		0			0
$625$	−305.199	−	528.620i	−0.488318	−	0.845792i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	93.1587i		−	0.148106i
$630$		0			0
$631$	26.8549			0.0425593			0.0212797	−	0.999774i	$-0.493226\pi$
$631$	26.8549			0.0425593			0.0212797	+	0.999774i	$0.493226\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−663.952	+	383.333i	−1.04559	+	0.603673i
$636$		0			0
$637$	−317.643	−	428.751i	−0.498654	−	0.673078i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−21.0385	−	12.1466i	−0.0328214	−	0.0189494i	0.483500	−	0.875345i	$-0.339365\pi$
$641$	−21.0385	−	12.1466i	−0.0328214	−	0.0189494i	−0.516321	+	0.856395i	$0.672699\pi$
$642$		0			0
$643$	670.076			1.04211			0.521055	−	0.853523i	$-0.325539\pi$
$643$	670.076			1.04211			0.521055	+	0.853523i	$0.325539\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	867.091	+	500.615i	1.34017	+	0.773748i	0.986832	−	0.161747i	$-0.0517128\pi$
$647$	867.091	+	500.615i	1.34017	+	0.773748i	0.353339	+	0.935495i	$0.385046\pi$
$648$		0			0
$649$	−109.869	−	190.299i	−0.169290	−	0.293218i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−499.826	+	288.575i	−0.765431	+	0.441922i	−0.831242	−	0.555910i	$-0.812370\pi$
$653$	−499.826	+	288.575i	−0.765431	+	0.441922i	0.0658113	+	0.997832i	$0.479036\pi$
$654$		0			0
$655$	707.206	−	1224.92i	1.07970	−	1.87010i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	188.823i		−	0.286530i	−0.989684	−	0.143265i	$-0.954240\pi$
$659$		−	188.823i		−	0.286530i	0.989684	−	0.143265i	$-0.0457601\pi$
$660$		0			0
$661$	55.7396	−	96.5438i	0.0843261	−	0.146057i	−0.820778	−	0.571248i	$-0.806460\pi$
$661$	55.7396	−	96.5438i	0.0843261	−	0.146057i	0.905104	+	0.425191i	$0.139793\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	361.646	+	551.253i	0.543828	+	0.828952i
$666$		0			0
$667$	−448.590	−	776.981i	−0.672549	−	1.16489i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	282.335i		−	0.420767i
$672$		0			0
$673$	−92.6835			−0.137717			−0.0688585	−	0.997626i	$-0.521936\pi$
$673$	−92.6835			−0.137717			−0.0688585	+	0.997626i	$0.521936\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	164.532	−	94.9929i	0.243032	−	0.140314i	−0.373538	−	0.927615i	$-0.621855\pi$
$677$	164.532	−	94.9929i	0.243032	−	0.140314i	0.616569	+	0.787301i	$0.288522\pi$
$678$		0			0
$679$	−1062.63	−	535.225i	−1.56500	−	0.788255i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−29.9576	−	17.2960i	−0.0438618	−	0.0253236i	0.477909	−	0.878409i	$-0.341395\pi$
$683$	−29.9576	−	17.2960i	−0.0438618	−	0.0253236i	−0.521771	+	0.853086i	$0.674728\pi$
$684$		0			0
$685$	−207.928			−0.303544
$686$		0			0
$687$		0			0
$688$		0			0
$689$	665.161	+	384.031i	0.965401	+	0.557374i
$690$		0			0
$691$	−81.1658	−	140.583i	−0.117461	−	0.203449i	0.801300	−	0.598263i	$-0.204142\pi$
$691$	−81.1658	−	140.583i	−0.117461	−	0.203449i	−0.918761	+	0.394814i	$0.870809\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1255.63	+	724.936i	−1.80666	+	1.04307i
$696$		0			0
$697$	207.625	−	359.617i	0.297884	−	0.515950i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	873.813i		−	1.24652i	−0.782013	−	0.623262i	$-0.785807\pi$
$701$		−	873.813i		−	1.24652i	0.782013	−	0.623262i	$-0.214193\pi$
$702$		0			0
$703$	42.9967	−	74.4724i	0.0611617	−	0.105935i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−538.309	−	820.540i	−0.761399	−	1.16059i
$708$		0			0
$709$	−14.2034	−	24.6011i	−0.0200330	−	0.0346982i	0.855835	−	0.517249i	$-0.173044\pi$
$709$	−14.2034	−	24.6011i	−0.0200330	−	0.0346982i	−0.875868	+	0.482551i	$0.839710\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	512.175i		−	0.718339i
$714$		0			0
$715$	−756.827			−1.05850
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−762.868	+	440.442i	−1.06101	+	0.612576i	−0.925712	−	0.378228i	$-0.876533\pi$
$719$	−762.868	+	440.442i	−1.06101	+	0.612576i	−0.135301	+	0.990805i	$0.543200\pi$
$720$		0			0
$721$	38.8356	+	680.438i	0.0538635	+	0.943742i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1655.79	−	955.974i	−2.28386	−	1.31858i
$726$		0			0
$727$	614.261			0.844926			0.422463	−	0.906380i	$-0.361166\pi$
$727$	614.261			0.844926			0.422463	+	0.906380i	$0.361166\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	287.263	+	165.851i	0.392973	+	0.226883i
$732$		0			0
$733$	−491.390	−	851.112i	−0.670382	−	1.16114i	−0.977796	−	0.209559i	$-0.932797\pi$
$733$	−491.390	−	851.112i	−0.670382	−	1.16114i	0.307414	−	0.951576i	$-0.400536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	221.329	−	127.785i	0.300311	−	0.173385i
$738$		0			0
$739$	−156.790	+	271.568i	−0.212165	+	0.367481i	−0.952392	−	0.304876i	$-0.901385\pi$
$739$	−156.790	+	271.568i	−0.212165	+	0.367481i	0.740227	+	0.672357i	$0.234718\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	969.364i		−	1.30466i	−0.757934	−	0.652331i	$-0.773791\pi$
$743$		−	969.364i		−	1.30466i	0.757934	−	0.652331i	$-0.226209\pi$
$744$		0			0
$745$	67.8984	−	117.604i	0.0911388	−	0.157857i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	675.243	−	38.5391i	0.901526	−	0.0514540i
$750$		0			0
$751$	359.002	+	621.810i	0.478033	+	0.827977i	0.999683	−	0.0251827i	$-0.00801676\pi$
$751$	359.002	+	621.810i	0.478033	+	0.827977i	−0.521650	+	0.853159i	$0.674683\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1666.06i		−	2.20670i
$756$		0			0
$757$	941.339			1.24351			0.621757	−	0.783211i	$-0.286419\pi$
$757$	941.339			1.24351			0.621757	+	0.783211i	$0.286419\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	198.359	−	114.523i	0.260656	−	0.150490i	−0.363978	−	0.931408i	$-0.618582\pi$
$761$	198.359	−	114.523i	0.260656	−	0.150490i	0.624634	+	0.780918i	$0.285248\pi$
$762$		0			0
$763$	−86.8243	+	56.9604i	−0.113793	+	0.0746532i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	257.891	+	148.893i	0.336233	+	0.194124i
$768$		0			0
$769$	144.825			0.188328			0.0941642	−	0.995557i	$-0.469982\pi$
$769$	144.825			0.188328			0.0941642	+	0.995557i	$0.469982\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	321.268	+	185.484i	0.415612	+	0.239954i	0.693198	−	0.720747i	$-0.256201\pi$
$773$	321.268	+	185.484i	0.415612	+	0.239954i	−0.277586	+	0.960701i	$0.589534\pi$
$774$		0			0
$775$	−545.739	−	945.247i	−0.704179	−	1.21967i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	331.957	−	191.655i	0.426132	−	0.246027i
$780$		0			0
$781$	511.500	−	885.943i	0.654929	−	1.13437i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	1011.48i		−	1.28851i
$786$		0			0
$787$	−122.010	+	211.328i	−0.155032	+	0.268524i	−0.933071	−	0.359693i	$-0.882882\pi$
$787$	−122.010	+	211.328i	−0.155032	+	0.268524i	0.778038	+	0.628217i	$0.216215\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−126.749	+	251.647i	−0.160239	+	0.318138i
$792$		0			0
$793$	191.309	+	331.356i	0.241247	+	0.417851i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1229.06i			1.54211i	0.636767	+	0.771056i	$0.280271\pi$
$797$			1229.06i			1.54211i	−0.636767	+	0.771056i	$0.719729\pi$
$798$		0			0
$799$	628.698			0.786856
$800$		0			0
$801$		0			0
$802$		0			0
$803$	203.159	−	117.294i	0.253001	−	0.146070i
$804$		0			0
$805$	−1183.07	+	776.144i	−1.46965	+	0.964155i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−498.214	−	287.644i	−0.615839	−	0.355555i	0.159408	−	0.987213i	$-0.449041\pi$
$809$	−498.214	−	287.644i	−0.615839	−	0.355555i	−0.775247	+	0.631658i	$0.782375\pi$
$810$		0			0
$811$	597.338			0.736545			0.368273	−	0.929718i	$-0.379949\pi$
$811$	597.338			0.736545			0.368273	+	0.929718i	$0.379949\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	817.832	+	472.176i	1.00348	+	0.579357i
$816$		0			0
$817$	153.095	+	265.168i	0.187387	+	0.324563i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−910.909	+	525.913i	−1.10951	+	0.640577i	−0.938703	−	0.344728i	$-0.887971\pi$
$821$	−910.909	+	525.913i	−1.10951	+	0.640577i	−0.170809	+	0.985304i	$0.554638\pi$
$822$		0			0
$823$	−365.162	+	632.480i	−0.443697	+	0.768505i	−0.997960	−	0.0638357i	$-0.979667\pi$
$823$	−365.162	+	632.480i	−0.443697	+	0.768505i	0.554264	+	0.832341i	$0.313000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1604.57i			1.94023i	0.242639	+	0.970117i	$0.421987\pi$
$827$			1604.57i			1.94023i	−0.242639	+	0.970117i	$0.578013\pi$
$828$		0			0
$829$	208.909	−	361.840i	0.252001	−	0.436478i	−0.712076	−	0.702103i	$-0.752245\pi$
$829$	208.909	−	361.840i	0.252001	−	0.436478i	0.964077	+	0.265624i	$0.0855781\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−65.7703	−	574.303i	−0.0789559	−	0.689440i
$834$		0			0
$835$	939.593	+	1627.42i	1.12526	+	1.94901i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			526.847i			0.627946i	0.949432	+	0.313973i	$0.101660\pi$
$839$			526.847i			0.627946i	−0.949432	+	0.313973i	$0.898340\pi$
$840$		0			0
$841$	−632.701			−0.752320
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−377.616	+	218.016i	−0.446882	+	0.258008i
$846$		0			0
$847$	22.5083	+	394.368i	0.0265741	+	0.465605i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	159.829	+	92.2771i	0.187813	+	0.108434i
$852$		0			0
$853$	1298.77			1.52259			0.761297	−	0.648403i	$-0.224563\pi$
$853$	1298.77			1.52259			0.761297	+	0.648403i	$0.224563\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−776.026	−	448.039i	−0.905515	−	0.522799i	−0.0265297	−	0.999648i	$-0.508446\pi$
$857$	−776.026	−	448.039i	−0.905515	−	0.522799i	−0.878985	+	0.476849i	$0.841779\pi$
$858$		0			0
$859$	−699.841	−	1212.16i	−0.814716	−	1.41113i	−0.909532	−	0.415634i	$-0.863560\pi$
$859$	−699.841	−	1212.16i	−0.814716	−	1.41113i	0.0948158	−	0.995495i	$-0.469774\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	287.391	−	165.925i	0.333013	−	0.192265i	−0.324165	−	0.946001i	$-0.605083\pi$
$863$	287.391	−	165.925i	0.333013	−	0.192265i	0.657178	+	0.753735i	$0.271750\pi$
$864$		0			0
$865$	−42.8855	+	74.2799i	−0.0495786	+	0.0858727i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			570.787i			0.656832i
$870$		0			0
$871$	−173.172	+	299.943i	−0.198820	+	0.344367i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−675.549	+	1341.23i	−0.772056	+	1.53283i
$876$		0			0
$877$	494.521	+	856.536i	0.563878	+	0.976666i	0.997153	+	0.0754040i	$0.0240246\pi$
$877$	494.521	+	856.536i	0.563878	+	0.976666i	−0.433275	+	0.901262i	$0.642642\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			676.723i			0.768131i	0.923306	+	0.384065i	$0.125476\pi$
$881$			676.723i			0.768131i	−0.923306	+	0.384065i	$0.874524\pi$
$882$		0			0
$883$	−230.145			−0.260640			−0.130320	−	0.991472i	$-0.541601\pi$
$883$	−230.145			−0.260640			−0.130320	+	0.991472i	$0.541601\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−928.185	+	535.888i	−1.04643	+	0.604157i	−0.921648	−	0.388027i	$-0.873157\pi$
$887$	−928.185	+	535.888i	−1.04643	+	0.604157i	−0.124783	+	0.992184i	$0.539824\pi$
$888$		0			0
$889$	554.171	+	279.124i	0.623364	+	0.313975i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	502.590	+	290.171i	0.562811	+	0.324939i
$894$		0			0
$895$	−375.168			−0.419183
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−728.583	−	420.647i	−0.810437	−	0.467906i
$900$		0			0
$901$	416.030	+	720.586i	0.461743	+	0.799762i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1372.16	−	792.219i	1.51620	−	0.875381i
$906$		0			0
$907$	312.161	−	540.679i	0.344169	−	0.596118i	−0.641034	−	0.767513i	$-0.721494\pi$
$907$	312.161	−	540.679i	0.344169	−	0.596118i	0.985202	+	0.171395i	$0.0548274\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1652.45i		−	1.81389i	−0.421253	−	0.906943i	$-0.638409\pi$
$911$		−	1652.45i		−	1.81389i	0.421253	−	0.906943i	$-0.361591\pi$
$912$		0			0
$913$	7.79937	−	13.5089i	0.00854257	−	0.0147962i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1142.89	+	65.2296i	−1.24633	+	0.0711336i
$918$		0			0
$919$	592.158	+	1025.65i	0.644351	+	1.11605i	0.984451	+	0.175659i	$0.0562056\pi$
$919$	592.158	+	1025.65i	0.644351	+	1.11605i	−0.340101	+	0.940389i	$0.610461\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1386.36i			1.50201i
$924$		0			0
$925$	393.296			0.425185
$926$		0			0
$927$		0			0
$928$		0			0
$929$	78.6575	−	45.4129i	0.0846690	−	0.0488837i	−0.457068	−	0.889432i	$-0.651100\pi$
$929$	78.6575	−	45.4129i	0.0846690	−	0.0488837i	0.541737	+	0.840548i	$0.317767\pi$
$930$		0			0
$931$	212.487	−	489.462i	0.228236	−	0.525738i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−710.045	−	409.945i	−0.759406	−	0.438443i
$936$		0			0
$937$	−435.637			−0.464928			−0.232464	−	0.972605i	$-0.574679\pi$
$937$	−435.637			−0.464928			−0.232464	+	0.972605i	$0.574679\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	999.310	+	576.952i	1.06197	+	0.613126i	0.925975	−	0.377584i	$-0.123245\pi$
$941$	999.310	+	576.952i	1.06197	+	0.613126i	0.135990	+	0.990710i	$0.456578\pi$
$942$		0			0
$943$	411.320	+	712.427i	0.436183	+	0.755490i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−568.610	+	328.287i	−0.600433	+	0.346660i	−0.769212	−	0.638994i	$-0.779351\pi$
$947$	−568.610	+	328.287i	−0.600433	+	0.346660i	0.168779	+	0.985654i	$0.446018\pi$
$948$		0			0
$949$	−158.956	+	275.320i	−0.167498	+	0.290116i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1559.33i		−	1.63623i	−0.575055	−	0.818114i	$-0.695020\pi$
$953$		−	1559.33i		−	1.63623i	0.575055	−	0.818114i	$-0.304980\pi$
$954$		0			0
$955$	162.630	−	281.684i	0.170293	−	0.294957i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	92.3103	+	140.708i	0.0962568	+	0.146723i
$960$		0			0
$961$	240.364	+	416.323i	0.250119	+	0.433219i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	290.628i		−	0.301169i
$966$		0			0
$967$	−1677.36			−1.73460			−0.867299	−	0.497788i	$-0.834146\pi$
$967$	−1677.36			−1.73460			−0.867299	+	0.497788i	$0.834146\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−118.068	+	68.1669i	−0.121595	+	0.0702028i	−0.559564	−	0.828787i	$-0.689031\pi$
$971$	−118.068	+	68.1669i	−0.121595	+	0.0702028i	0.437969	+	0.898990i	$0.355698\pi$
$972$		0			0
$973$	1048.01	+	527.863i	1.07710	+	0.542510i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−472.748	−	272.941i	−0.483877	−	0.279367i	0.238154	−	0.971227i	$-0.423458\pi$
$977$	−472.748	−	272.941i	−0.483877	−	0.279367i	−0.722031	+	0.691861i	$0.756791\pi$
$978$		0			0
$979$	−616.286			−0.629506
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−515.332	−	297.527i	−0.524245	−	0.302673i	0.214425	−	0.976740i	$-0.431212\pi$
$983$	−515.332	−	297.527i	−0.524245	−	0.302673i	−0.738670	+	0.674068i	$0.764546\pi$
$984$		0			0
$985$	−747.219	−	1294.22i	−0.758598	−	1.31393i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−569.089	+	328.564i	−0.575419	+	0.332218i
$990$		0			0
$991$	865.832	−	1499.67i	0.873696	−	1.51329i	0.0155496	−	0.999879i	$-0.495050\pi$
$991$	865.832	−	1499.67i	0.873696	−	1.51329i	0.858146	−	0.513406i	$-0.171616\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1547.28i		−	1.55505i
$996$		0			0
$997$	851.605	−	1475.02i	0.854168	−	1.47946i	−0.0232474	−	0.999730i	$-0.507401\pi$
$997$	851.605	−	1475.02i	0.854168	−	1.47946i	0.877415	−	0.479732i	$-0.159266\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.dc.e.305.6		12
3.2	odd	2	inner	1008.3.dc.e.305.1		12
4.3	odd	2		63.3.q.a.53.4	yes	12
7.2	even	3	inner	1008.3.dc.e.737.1		12
12.11	even	2		63.3.q.a.53.3	yes	12
21.2	odd	6	inner	1008.3.dc.e.737.6		12
28.3	even	6		441.3.b.c.197.4		6
28.11	odd	6		441.3.b.d.197.4		6
28.19	even	6		441.3.q.d.422.3		12
28.23	odd	6		63.3.q.a.44.3	✓	12
28.27	even	2		441.3.q.d.116.4		12
84.11	even	6		441.3.b.d.197.3		6
84.23	even	6		63.3.q.a.44.4	yes	12
84.47	odd	6		441.3.q.d.422.4		12
84.59	odd	6		441.3.b.c.197.3		6
84.83	odd	2		441.3.q.d.116.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.3.q.a.44.3	✓	12	28.23	odd	6
63.3.q.a.44.4	yes	12	84.23	even	6
63.3.q.a.53.3	yes	12	12.11	even	2
63.3.q.a.53.4	yes	12	4.3	odd	2
441.3.b.c.197.3		6	84.59	odd	6
441.3.b.c.197.4		6	28.3	even	6
441.3.b.d.197.3		6	84.11	even	6
441.3.b.d.197.4		6	28.11	odd	6
441.3.q.d.116.3		12	84.83	odd	2
441.3.q.d.116.4		12	28.27	even	2
441.3.q.d.422.3		12	28.19	even	6
441.3.q.d.422.4		12	84.47	odd	6
1008.3.dc.e.305.1		12	3.2	odd	2	inner
1008.3.dc.e.305.6		12	1.1	even	1	trivial
1008.3.dc.e.737.1		12	7.2	even	3	inner
1008.3.dc.e.737.6		12	21.2	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(7.49023 - 4.32449i) q^{5} +(-6.25176 - 3.14888i) q^{7} +(6.95899 + 4.01778i) q^{11} -10.8897 q^{13} +(-10.2166 - 5.89854i) q^{17} +(-5.44485 - 9.43076i) q^{19} +(20.2398 - 11.6854i) q^{23} +(24.9024 - 43.1322i) q^{25} -38.3888i q^{29} +(10.9575 - 18.9790i) q^{31} +(-60.4445 + 3.44983i) q^{35} +(3.94838 + 6.83879i) q^{37} +35.1994i q^{41} -28.1174 q^{43} +(-46.1528 + 26.6463i) q^{47} +(29.1691 + 39.3721i) q^{49} +(-61.0817 - 35.2655i) q^{53} +69.4993 q^{55} +(-23.6821 - 13.6729i) q^{59} +(-17.5678 - 30.4284i) q^{61} +(-81.5664 + 47.0924i) q^{65} +(15.9024 - 27.5438i) q^{67} -127.309i q^{71} +(14.5969 - 25.2826i) q^{73} +(-30.8545 - 47.0312i) q^{77} +(35.5164 + 61.5161i) q^{79} -1.94121i q^{83} -102.033 q^{85} +(-66.4198 + 38.3475i) q^{89} +(68.0798 + 34.2904i) q^{91} +(-81.5664 - 47.0924i) q^{95} +169.973 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{7} - 52 q^{13} - 26 q^{19} + 106 q^{25} - 22 q^{31} - 146 q^{37} - 108 q^{43} + 114 q^{49} - 16 q^{55} - 136 q^{61} - 2 q^{67} - 482 q^{73} - 42 q^{79} - 288 q^{85} - 222 q^{91} + 568 q^{97}+O(q^{100}) \) 12 * q + 2 * q^7 - 52 * q^13 - 26 * q^19 + 106 * q^25 - 22 * q^31 - 146 * q^37 - 108 * q^43 + 114 * q^49 - 16 * q^55 - 136 * q^61 - 2 * q^67 - 482 * q^73 - 42 * q^79 - 288 * q^85 - 222 * q^91 + 568 * q^97

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