LMFDB - Embedded newform 1008.3.dc.d.305.1

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 = [8,0,0,0,0,0,-12,0,0,0,0,0,-112] 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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.92844527616.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 13x^{6} - 2x^{5} + 239x^{4} - 200x^{3} - 50x^{2} - 288x + 324 $ x^8 - 2x^7 - 13x^6 - 2x^5 + 239x^4 - 200x^3 - 50x^2 - 288*x + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			305.1
Root			$3.55140 - 1.47305i$ of defining polynomial
Character	$\chi$	$=$	1008.305
Dual form			1008.3.dc.d.737.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.60280 + 3.81213i) q^{5} +(-4.89116 + 5.00764i) q^{7} +(-4.41990 - 2.55183i) q^{11} -20.7823 q^{13} +(-22.7370 - 13.1272i) q^{17} +(11.1735 + 19.3531i) q^{19} +(34.4512 - 19.8904i) q^{23} +(16.5647 - 28.6908i) q^{25} -7.62426i q^{29} +(-5.89116 + 10.2038i) q^{31} +(13.2056 - 51.7102i) q^{35} +(30.1735 + 52.2620i) q^{37} -11.8044i q^{41} +30.3470 q^{43} +(-33.0681 + 19.0919i) q^{47} +(-1.15301 - 48.9864i) q^{49} +(5.11147 + 2.95111i) q^{53} +38.9116 q^{55} +(38.2878 + 22.1055i) q^{59} +(24.7382 + 42.8477i) q^{61} +(137.222 - 79.2249i) q^{65} +(31.5647 - 54.6716i) q^{67} +41.6279i q^{71} +(2.08835 - 3.61713i) q^{73} +(34.3971 - 9.65187i) q^{77} +(-20.8912 - 36.1846i) q^{79} -145.784i q^{83} +200.170 q^{85} +(38.0173 - 21.9493i) q^{89} +(101.650 - 104.071i) q^{91} +(-147.553 - 85.1896i) q^{95} +39.5173 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{7} - 112 q^{13} + 8 q^{19} + 24 q^{25} - 20 q^{31} + 160 q^{37} + 80 q^{43} - 172 q^{49} + 40 q^{55} + 8 q^{61} + 144 q^{67} + 288 q^{73} - 140 q^{79} + 896 q^{85} + 352 q^{91} - 552 q^{97}+O(q^{100}) $ 8 * q - 12 * q^7 - 112 * q^13 + 8 * q^19 + 24 * q^25 - 20 * q^31 + 160 * q^37 + 80 * q^43 - 172 * q^49 + 40 * q^55 + 8 * q^61 + 144 * q^67 + 288 * q^73 - 140 * q^79 + 896 * q^85 + 352 * q^91 - 552 * 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
$5$	−6.60280	+	3.81213i	−1.32056	+	0.762426i	−0.983818	−	0.179171i	$-0.942658\pi$
$5$	−6.60280	+	3.81213i	−1.32056	+	0.762426i	−0.336742	+	0.941597i	$0.609325\pi$
$6$		0			0
$7$	−4.89116	+	5.00764i	−0.698738	+	0.715378i
$7$	−4.89116	+	5.00764i	−0.698738	+	0.715378i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.41990	−	2.55183i	−0.401809	−	0.231985i	0.285455	−	0.958392i	$-0.407855\pi$
$11$	−4.41990	−	2.55183i	−0.401809	−	0.231985i	−0.687264	+	0.726407i	$0.741189\pi$
$12$		0			0
$13$	−20.7823			−1.59864			−0.799320	−	0.600905i	$-0.794807\pi$
$13$	−20.7823			−1.59864			−0.799320	+	0.600905i	$0.794807\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−22.7370	−	13.1272i	−1.33747	−	0.772188i	−0.351037	−	0.936361i	$-0.614171\pi$
$17$	−22.7370	−	13.1272i	−1.33747	−	0.772188i	−0.986432	+	0.164173i	$0.947504\pi$
$18$		0			0
$19$	11.1735	+	19.3531i	0.588079	+	1.01858i	0.994484	+	0.104889i	$0.0334488\pi$
$19$	11.1735	+	19.3531i	0.588079	+	1.01858i	−0.406405	+	0.913693i	$0.633218\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	34.4512	−	19.8904i	1.49788	−	0.864801i	0.497883	−	0.867244i	$-0.334111\pi$
$23$	34.4512	−	19.8904i	1.49788	−	0.864801i	0.999997	+	0.00244278i	$0.000777562\pi$
$24$		0			0
$25$	16.5647	−	28.6908i	0.662586	−	1.14763i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	7.62426i		−	0.262905i	−0.991322	−	0.131453i	$-0.958036\pi$
$29$		−	7.62426i		−	0.262905i	0.991322	−	0.131453i	$-0.0419642\pi$
$30$		0			0
$31$	−5.89116	+	10.2038i	−0.190038	+	0.329155i	−0.945262	−	0.326311i	$-0.894194\pi$
$31$	−5.89116	+	10.2038i	−0.190038	+	0.329155i	0.755225	+	0.655466i	$0.227528\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	13.2056	−	51.7102i	0.377303	−	1.47744i
$36$		0			0
$37$	30.1735	+	52.2620i	0.815500	+	1.41249i	0.908968	+	0.416865i	$0.136871\pi$
$37$	30.1735	+	52.2620i	0.815500	+	1.41249i	−0.0934686	+	0.995622i	$0.529795\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.8044i		−	0.287913i	−0.989584	−	0.143956i	$-0.954017\pi$
$41$		−	11.8044i		−	0.287913i	0.989584	−	0.143956i	$-0.0459825\pi$
$42$		0			0
$43$	30.3470			0.705744			0.352872	−	0.935672i	$-0.385205\pi$
$43$	30.3470			0.705744			0.352872	+	0.935672i	$0.385205\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−33.0681	+	19.0919i	−0.703577	+	0.406210i	−0.808678	−	0.588251i	$-0.799817\pi$
$47$	−33.0681	+	19.0919i	−0.703577	+	0.406210i	0.105101	+	0.994462i	$0.466483\pi$
$48$		0			0
$49$	−1.15301	−	48.9864i	−0.0235308	−	0.999723i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.11147	+	2.95111i	0.0964428	+	0.0556813i	0.547446	−	0.836841i	$-0.315600\pi$
$53$	5.11147	+	2.95111i	0.0964428	+	0.0556813i	−0.451003	+	0.892522i	$0.648934\pi$
$54$		0			0
$55$	38.9116			0.707485
$56$		0			0
$57$		0			0
$58$		0			0
$59$	38.2878	+	22.1055i	0.648945	+	0.374669i	0.788052	−	0.615609i	$-0.211090\pi$
$59$	38.2878	+	22.1055i	0.648945	+	0.374669i	−0.139107	+	0.990277i	$0.544423\pi$
$60$		0			0
$61$	24.7382	+	42.8477i	0.405544	+	0.702422i	0.994385	−	0.105827i	$-0.0337489\pi$
$61$	24.7382	+	42.8477i	0.405544	+	0.702422i	−0.588841	+	0.808249i	$0.700416\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	137.222	−	79.2249i	2.11110	−	1.21885i
$66$		0			0
$67$	31.5647	−	54.6716i	0.471114	−	0.815994i	−0.528340	−	0.849033i	$-0.677185\pi$
$67$	31.5647	−	54.6716i	0.471114	−	0.815994i	0.999454	+	0.0330391i	$0.0105186\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			41.6279i			0.586308i	0.956065	+	0.293154i	$0.0947048\pi$
$71$			41.6279i			0.586308i	−0.956065	+	0.293154i	$0.905295\pi$
$72$		0			0
$73$	2.08835	−	3.61713i	0.0286075	−	0.0495497i	−0.851367	−	0.524570i	$-0.824226\pi$
$73$	2.08835	−	3.61713i	0.0286075	−	0.0495497i	0.879975	+	0.475021i	$0.157559\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	34.3971	−	9.65187i	0.446716	−	0.125349i
$78$		0			0
$79$	−20.8912	−	36.1846i	−0.264445	−	0.458032i	0.702973	−	0.711217i	$-0.251855\pi$
$79$	−20.8912	−	36.1846i	−0.264445	−	0.458032i	−0.967418	+	0.253184i	$0.918522\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	145.784i		−	1.75644i	−0.478258	−	0.878219i	$-0.658732\pi$
$83$		−	145.784i		−	1.75644i	0.478258	−	0.878219i	$-0.341268\pi$
$84$		0			0
$85$	200.170			2.35494
$86$		0			0
$87$		0			0
$88$		0			0
$89$	38.0173	−	21.9493i	0.427160	−	0.246621i	−0.270976	−	0.962586i	$-0.587346\pi$
$89$	38.0173	−	21.9493i	0.427160	−	0.246621i	0.698136	+	0.715965i	$0.254013\pi$
$90$		0			0
$91$	101.650	−	104.071i	1.11703	−	1.14363i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−147.553	−	85.1896i	−1.55319	−	0.896733i
$96$		0			0
$97$	39.5173			0.407395			0.203697	−	0.979034i	$-0.434704\pi$
$97$	39.5173			0.407395			0.203697	+	0.979034i	$0.434704\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−80.9414	−	46.7315i	−0.801400	−	0.462688i	0.0425607	−	0.999094i	$-0.486448\pi$
$101$	−80.9414	−	46.7315i	−0.801400	−	0.462688i	−0.843960	+	0.536406i	$0.819782\pi$
$102$		0			0
$103$	−29.0000	−	50.2295i	−0.281553	−	0.487665i	0.690214	−	0.723605i	$-0.257516\pi$
$103$	−29.0000	−	50.2295i	−0.281553	−	0.487665i	−0.971768	+	0.235940i	$0.924183\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−148.407	+	85.6826i	−1.38698	+	0.800772i	−0.992974	−	0.118337i	$-0.962244\pi$
$107$	−148.407	+	85.6826i	−1.38698	+	0.800772i	−0.394004	+	0.919109i	$0.628910\pi$
$108$		0			0
$109$	−98.6056	+	170.790i	−0.904639	+	1.56688i	−0.0832375	+	0.996530i	$0.526526\pi$
$109$	−98.6056	+	170.790i	−0.904639	+	1.56688i	−0.821401	+	0.570351i	$0.806807\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	135.640i		−	1.20035i	−0.799869	−	0.600175i	$-0.795098\pi$
$113$		−	135.640i		−	1.20035i	0.799869	−	0.600175i	$-0.204902\pi$
$114$		0			0
$115$	−151.650	+	262.665i	−1.31869	+	2.28404i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	176.947	−	49.6514i	1.48695	−	0.417239i
$120$		0			0
$121$	−47.4763	−	82.2314i	−0.392366	−	0.679598i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			61.9800i			0.495840i
$126$		0			0
$127$	186.558			1.46896			0.734481	−	0.678629i	$-0.237426\pi$
$127$	186.558			1.46896			0.734481	+	0.678629i	$0.237426\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	43.8203	−	25.2997i	0.334506	−	0.193127i	−0.323334	−	0.946285i	$-0.604804\pi$
$131$	43.8203	−	25.2997i	0.334506	−	0.193127i	0.657840	+	0.753158i	$0.271470\pi$
$132$		0			0
$133$	−151.565	−	38.7061i	−1.13958	−	0.291023i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−137.805	−	79.5617i	−1.00588	−	0.580743i	−0.0958942	−	0.995392i	$-0.530571\pi$
$137$	−137.805	−	79.5617i	−1.00588	−	0.580743i	−0.909981	+	0.414649i	$0.863904\pi$
$138$		0			0
$139$	−1.82330			−0.0131173			−0.00655863	−	0.999978i	$-0.502088\pi$
$139$	−1.82330			−0.0131173			−0.00655863	+	0.999978i	$0.502088\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	91.8559	+	53.0330i	0.642349	+	0.370860i
$144$		0			0
$145$	29.0647	+	50.3415i	0.200446	+	0.347183i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	146.969	−	84.8528i	0.986372	−	0.569482i	0.0821839	−	0.996617i	$-0.473811\pi$
$149$	146.969	−	84.8528i	0.986372	−	0.569482i	0.904188	+	0.427135i	$0.140477\pi$
$150$		0			0
$151$	−64.4968	+	111.712i	−0.427131	+	0.739813i	−0.996617	−	0.0821885i	$-0.973809\pi$
$151$	−64.4968	+	111.712i	−0.427131	+	0.739813i	0.569486	+	0.822001i	$0.307142\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	89.8315i		−	0.579558i
$156$		0			0
$157$	28.3470	−	49.0984i	0.180554	−	0.312729i	−0.761515	−	0.648147i	$-0.775544\pi$
$157$	28.3470	−	49.0984i	0.180554	−	0.312729i	0.942069	+	0.335418i	$0.108878\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−68.9025	+	269.807i	−0.427966	+	1.67582i
$162$		0			0
$163$	−67.4795	−	116.878i	−0.413985	−	0.717043i	0.581337	−	0.813663i	$-0.302530\pi$
$163$	−67.4795	−	116.878i	−0.413985	−	0.717043i	−0.995321	+	0.0966206i	$0.969197\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	322.191i		−	1.92929i	−0.263560	−	0.964643i	$-0.584897\pi$
$167$		−	322.191i		−	1.92929i	0.263560	−	0.964643i	$-0.415103\pi$
$168$		0			0
$169$	262.905			1.55565
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−49.7315	+	28.7125i	−0.287466	+	0.165968i	−0.636798	−	0.771030i	$-0.719742\pi$
$173$	−49.7315	+	28.7125i	−0.287466	+	0.165968i	0.349333	+	0.936999i	$0.386408\pi$
$174$		0			0
$175$	62.6530	+	223.282i	0.358017	+	1.27589i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	164.216	+	94.8102i	0.917409	+	0.529666i	0.882807	−	0.469735i	$-0.155651\pi$
$179$	164.216	+	94.8102i	0.917409	+	0.529666i	0.0346013	+	0.999401i	$0.488984\pi$
$180$		0			0
$181$	−220.599			−1.21878			−0.609390	−	0.792871i	$-0.708586\pi$
$181$	−220.599			−1.21878			−0.609390	+	0.792871i	$0.708586\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−398.459	−	230.051i	−2.15383	−	1.24352i
$186$		0			0
$187$	66.9968	+	116.042i	0.358272	+	0.620545i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	127.474	−	73.5970i	0.667402	−	0.385325i	−0.127689	−	0.991814i	$-0.540756\pi$
$191$	127.474	−	73.5970i	0.667402	−	0.385325i	0.795092	+	0.606489i	$0.207423\pi$
$192$		0			0
$193$	60.2349	−	104.330i	0.312098	−	0.540570i	−0.666718	−	0.745310i	$-0.732302\pi$
$193$	60.2349	−	104.330i	0.312098	−	0.540570i	0.978816	+	0.204740i	$0.0656349\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.6369i			0.109832i	0.998491	+	0.0549160i	$0.0174891\pi$
$197$			21.6369i			0.109832i	−0.998491	+	0.0549160i	$0.982511\pi$
$198$		0			0
$199$	−9.95903	+	17.2495i	−0.0500454	+	0.0866811i	−0.889963	−	0.456033i	$-0.849270\pi$
$199$	−9.95903	+	17.2495i	−0.0500454	+	0.0866811i	0.839918	+	0.542714i	$0.182603\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	38.1796	+	37.2915i	0.188077	+	0.183702i
$204$		0			0
$205$	45.0000	+	77.9423i	0.219512	+	0.380206i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	114.052i		−	0.545701i
$210$		0			0
$211$	31.6530			0.150014			0.0750071	−	0.997183i	$-0.476102\pi$
$211$	31.6530			0.150014			0.0750071	+	0.997183i	$0.476102\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−200.375	+	115.687i	−0.931977	+	0.538077i
$216$		0			0
$217$	−22.2823	−	79.4093i	−0.102684	−	0.365942i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	472.527	+	272.814i	2.13813	+	1.23445i
$222$		0			0
$223$	250.653			1.12400			0.562002	−	0.827136i	$-0.310031\pi$
$223$	250.653			1.12400			0.562002	+	0.827136i	$0.310031\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	196.971	+	113.722i	0.867716	+	0.500976i	0.866588	−	0.499024i	$-0.166308\pi$
$227$	196.971	+	113.722i	0.867716	+	0.500976i	0.00112715	+	0.999999i	$0.499641\pi$
$228$		0			0
$229$	−112.565	−	194.968i	−0.491549	−	0.851387i	0.508404	−	0.861119i	$-0.330236\pi$
$229$	−112.565	−	194.968i	−0.491549	−	0.851387i	−0.999953	+	0.00973134i	$0.996902\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−45.9491	+	26.5287i	−0.197206	+	0.113857i	−0.595352	−	0.803465i	$-0.702987\pi$
$233$	−45.9491	+	26.5287i	−0.197206	+	0.113857i	0.398145	+	0.917322i	$0.369654\pi$
$234$		0			0
$235$	145.561	−	252.120i	0.619410	−	1.07285i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			451.366i			1.88856i	0.329143	+	0.944280i	$0.393240\pi$
$239$			451.366i			1.88856i	−0.329143	+	0.944280i	$0.606760\pi$
$240$		0			0
$241$	44.1940	−	76.5462i	0.183378	−	0.317619i	−0.759651	−	0.650331i	$-0.774630\pi$
$241$	44.1940	−	76.5462i	0.183378	−	0.317619i	0.943029	+	0.332712i	$0.107964\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	194.356	+	319.052i	0.793289	+	1.30225i
$246$		0			0
$247$	−232.211	−	402.202i	−0.940127	−	1.62835i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			233.706i			0.931102i	0.885021	+	0.465551i	$0.154144\pi$
$251$			233.706i			0.931102i	−0.885021	+	0.465551i	$0.845856\pi$
$252$		0			0
$253$	−203.028			−0.802483
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−324.608	+	187.412i	−1.26306	+	0.729231i	−0.973666	−	0.227978i	$-0.926789\pi$
$257$	−324.608	+	187.412i	−1.26306	+	0.729231i	−0.289398	+	0.957209i	$0.593455\pi$
$258$		0			0
$259$	−409.293	−	104.524i	−1.58028	−	0.403568i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−347.128	−	200.415i	−1.31988	−	0.762032i	−0.336170	−	0.941801i	$-0.609132\pi$
$263$	−347.128	−	200.415i	−1.31988	−	0.762032i	−0.983709	+	0.179769i	$0.942465\pi$
$264$		0			0
$265$	−45.0000			−0.169811
$266$		0			0
$267$		0			0
$268$		0			0
$269$	292.544	+	168.900i	1.08752	+	0.627882i	0.932916	−	0.360094i	$-0.117255\pi$
$269$	292.544	+	168.900i	1.08752	+	0.627882i	0.154608	+	0.987976i	$0.450589\pi$
$270$		0			0
$271$	−177.279	−	307.056i	−0.654167	−	1.13305i	−0.982102	−	0.188350i	$-0.939686\pi$
$271$	−177.279	−	307.056i	−0.654167	−	1.13305i	0.327936	−	0.944700i	$-0.393647\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−146.428	+	84.5404i	−0.532467	+	0.307420i
$276$		0			0
$277$	137.653	−	238.422i	0.496942	−	0.860729i	−0.503052	−	0.864256i	$-0.667789\pi$
$277$	137.653	−	238.422i	0.496942	−	0.860729i	0.999994	+	0.00352717i	$0.00112273\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	190.120i		−	0.676585i	−0.941041	−	0.338292i	$-0.890151\pi$
$281$		−	190.120i		−	0.676585i	0.941041	−	0.338292i	$-0.109849\pi$
$282$		0			0
$283$	−72.9149	+	126.292i	−0.257650	+	0.446262i	−0.965612	−	0.259988i	$-0.916281\pi$
$283$	−72.9149	+	126.292i	−0.257650	+	0.446262i	0.707962	+	0.706250i	$0.249615\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	59.1124	+	57.7374i	0.205966	+	0.201176i
$288$		0			0
$289$	200.147	+	346.664i	0.692549	+	1.19953i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			340.870i			1.16338i	0.813411	+	0.581689i	$0.197608\pi$
$293$			340.870i			1.16338i	−0.813411	+	0.581689i	$0.802392\pi$
$294$		0			0
$295$	−337.076			−1.14263
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−715.977	+	413.370i	−2.39457	+	1.38251i
$300$		0			0
$301$	−148.432	+	151.967i	−0.493130	+	0.504874i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−326.682	−	188.610i	−1.07109	−	0.618394i
$306$		0			0
$307$	115.300			0.375569			0.187784	−	0.982210i	$-0.439869\pi$
$307$	115.300			0.375569			0.187784	+	0.982210i	$0.439869\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	117.684	+	67.9447i	0.378404	+	0.218472i	0.677124	−	0.735869i	$-0.263226\pi$
$311$	117.684	+	67.9447i	0.378404	+	0.218472i	−0.298720	+	0.954341i	$0.596560\pi$
$312$		0			0
$313$	9.58835	+	16.6075i	0.0306337	+	0.0530591i	0.880936	−	0.473236i	$-0.156914\pi$
$313$	9.58835	+	16.6075i	0.0306337	+	0.0530591i	−0.850302	+	0.526295i	$0.823581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	231.314	−	133.549i	0.729699	−	0.421292i	−0.0886133	−	0.996066i	$-0.528244\pi$
$317$	231.314	−	133.549i	0.729699	−	0.421292i	0.818312	+	0.574774i	$0.194910\pi$
$318$		0			0
$319$	−19.4558	+	33.6985i	−0.0609900	+	0.105638i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	586.707i		−	1.81643i
$324$		0			0
$325$	−344.252	+	596.262i	−1.05924	+	1.83465i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	66.1362	−	258.975i	0.201022	−	0.787158i
$330$		0			0
$331$	240.347	+	416.293i	0.726124	+	1.25768i	0.958510	+	0.285060i	$0.0920134\pi$
$331$	240.347	+	416.293i	0.726124	+	1.25768i	−0.232386	+	0.972624i	$0.574653\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			481.314i			1.43676i
$336$		0			0
$337$	−234.823			−0.696805			−0.348403	−	0.937345i	$-0.613276\pi$
$337$	−234.823			−0.696805			−0.348403	+	0.937345i	$0.613276\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	52.0767	−	30.0665i	0.152718	−	0.0881716i
$342$		0			0
$343$	250.946	+	233.827i	0.731622	+	0.681711i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	53.0388	+	30.6220i	0.152850	+	0.0882478i	0.574474	−	0.818523i	$-0.305207\pi$
$347$	53.0388	+	30.6220i	0.152850	+	0.0882478i	−0.421624	+	0.906771i	$0.638540\pi$
$348$		0			0
$349$	84.5173			0.242170			0.121085	−	0.992642i	$-0.461363\pi$
$349$	84.5173			0.242170			0.121085	+	0.992642i	$0.461363\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	477.650	+	275.772i	1.35312	+	0.781223i	0.988685	−	0.150007i	$-0.0479296\pi$
$353$	477.650	+	275.772i	1.35312	+	0.781223i	0.364433	+	0.931230i	$0.381263\pi$
$354$		0			0
$355$	−158.691	−	274.860i	−0.447016	−	0.774255i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−247.383	+	142.827i	−0.689088	+	0.397845i	−0.803270	−	0.595614i	$-0.796909\pi$
$359$	−247.383	+	142.827i	−0.689088	+	0.397845i	0.114182	+	0.993460i	$0.463575\pi$
$360$		0			0
$361$	−69.1940	+	119.847i	−0.191673	+	0.331988i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			31.8442i			0.0872445i
$366$		0			0
$367$	5.63253	−	9.75582i	0.0153475	−	0.0265826i	−0.858250	−	0.513232i	$-0.828448\pi$
$367$	5.63253	−	9.75582i	0.0153475	−	0.0265826i	0.873597	+	0.486650i	$0.161781\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−39.7791	+	11.1621i	−0.107221	+	0.0300864i
$372$		0			0
$373$	−79.6876	−	138.023i	−0.213640	−	0.370035i	0.739211	−	0.673474i	$-0.235198\pi$
$373$	−79.6876	−	138.023i	−0.213640	−	0.370035i	−0.952851	+	0.303439i	$0.901865\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			158.450i			0.420291i
$378$		0			0
$379$	−315.375			−0.832124			−0.416062	−	0.909336i	$-0.636590\pi$
$379$	−315.375			−0.832124			−0.416062	+	0.909336i	$0.636590\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	432.369	−	249.629i	1.12890	−	0.651772i	0.185243	−	0.982693i	$-0.440693\pi$
$383$	432.369	−	249.629i	1.12890	−	0.651772i	0.943659	+	0.330921i	$0.107359\pi$
$384$		0			0
$385$	−190.323	+	194.856i	−0.494346	+	0.506119i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−472.960	−	273.064i	−1.21584	−	0.701963i	−0.251811	−	0.967776i	$-0.581026\pi$
$389$	−472.960	−	273.064i	−1.21584	−	0.701963i	−0.964025	+	0.265813i	$0.914360\pi$
$390$		0			0
$391$	−1044.42			−2.67116
$392$		0			0
$393$		0			0
$394$		0			0
$395$	275.880	+	159.280i	0.698431	+	0.403240i
$396$		0			0
$397$	−261.823	−	453.491i	−0.659505	−	1.14230i	−0.980744	−	0.195297i	$-0.937433\pi$
$397$	−261.823	−	453.491i	−0.659505	−	1.14230i	0.321240	−	0.946998i	$-0.395901\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−473.802	+	273.550i	−1.18155	+	0.682169i	−0.956373	−	0.292148i	$-0.905630\pi$
$401$	−473.802	+	273.550i	−1.18155	+	0.682169i	−0.225179	+	0.974318i	$0.572297\pi$
$402$		0			0
$403$	122.432	−	212.059i	0.303802	−	0.526200i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	307.991i		−	0.756734i
$408$		0			0
$409$	322.399	−	558.411i	0.788261	−	1.36531i	−0.138770	−	0.990325i	$-0.544315\pi$
$409$	322.399	−	558.411i	0.788261	−	1.36531i	0.927031	−	0.374984i	$-0.122352\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−297.968	+	83.6102i	−0.721473	+	0.202446i
$414$		0			0
$415$	555.749	+	962.586i	1.33915	+	2.31948i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	64.9379i		−	0.154983i	−0.996993	−	0.0774916i	$-0.975309\pi$
$419$		−	64.9379i		−	0.154983i	0.996993	−	0.0774916i	$-0.0246911\pi$
$420$		0			0
$421$	−283.722			−0.673924			−0.336962	−	0.941518i	$-0.609399\pi$
$421$	−283.722			−0.673924			−0.336962	+	0.941518i	$0.609399\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−753.260	+	434.895i	−1.77238	+	1.02328i
$426$		0			0
$427$	−335.565	−	85.6955i	−0.785866	−	0.200692i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	31.6191	+	18.2553i	0.0733622	+	0.0423557i	0.536232	−	0.844070i	$-0.319847\pi$
$431$	31.6191	+	18.2553i	0.0733622	+	0.0423557i	−0.462870	+	0.886426i	$0.653180\pi$
$432$		0			0
$433$	382.082			0.882406			0.441203	−	0.897407i	$-0.354552\pi$
$433$	382.082			0.882406			0.441203	+	0.897407i	$0.354552\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	769.882	+	444.491i	1.76174	+	1.01714i
$438$		0			0
$439$	−289.973	−	502.248i	−0.660531	−	1.14407i	−0.980476	−	0.196637i	$-0.936998\pi$
$439$	−289.973	−	502.248i	−0.660531	−	1.14407i	0.319945	−	0.947436i	$-0.396335\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	610.819	−	352.657i	1.37882	−	0.796065i	0.386806	−	0.922161i	$-0.373578\pi$
$443$	610.819	−	352.657i	1.37882	−	0.796065i	0.992018	+	0.126096i	$0.0402449\pi$
$444$		0			0
$445$	−167.347	+	289.853i	−0.376061	+	0.651356i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	532.400i		−	1.18575i	−0.805296	−	0.592873i	$-0.797994\pi$
$449$		−	532.400i		−	1.18575i	0.805296	−	0.592873i	$-0.202006\pi$
$450$		0			0
$451$	−30.1229	+	52.1744i	−0.0667914	+	0.115686i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−274.443	+	1074.66i	−0.603172	+	2.36189i
$456$		0			0
$457$	−222.453	−	385.299i	−0.486767	−	0.843106i	0.513117	−	0.858319i	$-0.328491\pi$
$457$	−222.453	−	385.299i	−0.486767	−	0.843106i	−0.999884	+	0.0152131i	$0.995157\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			380.463i			0.825300i	0.910890	+	0.412650i	$0.135397\pi$
$461$			380.463i			0.825300i	−0.910890	+	0.412650i	$0.864603\pi$
$462$		0			0
$463$	847.035			1.82945			0.914724	−	0.404079i	$-0.132408\pi$
$463$	847.035			1.82945			0.914724	+	0.404079i	$0.132408\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−151.876	+	87.6858i	−0.325217	+	0.187764i	−0.653715	−	0.756740i	$-0.726791\pi$
$467$	−151.876	+	87.6858i	−0.325217	+	0.187764i	0.328499	+	0.944504i	$0.393457\pi$
$468$		0			0
$469$	119.388	+	425.472i	0.254559	+	0.907191i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−134.131	−	77.4404i	−0.283574	−	0.163722i
$474$		0			0
$475$	740.341			1.55861
$476$		0			0
$477$		0			0
$478$		0			0
$479$	186.629	+	107.750i	0.389621	+	0.224948i	0.681996	−	0.731356i	$-0.261112\pi$
$479$	186.629	+	107.750i	0.389621	+	0.224948i	−0.292375	+	0.956304i	$0.594445\pi$
$480$		0			0
$481$	−627.076	−	1086.13i	−1.30369	−	2.25806i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−260.925	+	150.645i	−0.537989	+	0.310608i
$486$		0			0
$487$	−196.143	+	339.730i	−0.402759	+	0.697598i	−0.994058	−	0.108854i	$-0.965282\pi$
$487$	−196.143	+	339.730i	−0.402759	+	0.697598i	0.591299	+	0.806452i	$0.298615\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	491.584i		−	1.00119i	−0.865682	−	0.500595i	$-0.833115\pi$
$491$		−	491.584i		−	1.00119i	0.865682	−	0.500595i	$-0.166885\pi$
$492$		0			0
$493$	−100.085	+	173.353i	−0.203012	+	0.351628i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−208.458	−	203.609i	−0.419432	−	0.409675i
$498$		0			0
$499$	62.4827	+	108.223i	0.125216	+	0.216880i	0.921817	−	0.387625i	$-0.126704\pi$
$499$	62.4827	+	108.223i	0.125216	+	0.216880i	−0.796601	+	0.604505i	$0.793371\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	21.9357i		−	0.0436097i	−0.999762	−	0.0218049i	$-0.993059\pi$
$503$		−	21.9357i		−	0.0436097i	0.999762	−	0.0218049i	$-0.00694125\pi$
$504$		0			0
$505$	712.586			1.41106
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−252.734	+	145.916i	−0.496531	+	0.286672i	−0.727280	−	0.686341i	$-0.759216\pi$
$509$	−252.734	+	145.916i	−0.496531	+	0.286672i	0.230749	+	0.973013i	$0.425882\pi$
$510$		0			0
$511$	7.89883	+	28.1497i	0.0154576	+	0.0550875i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	382.962	+	221.104i	0.743616	+	0.429327i
$516$		0			0
$517$	194.877			0.376938
$518$		0			0
$519$		0			0
$520$		0			0
$521$	455.130	+	262.769i	0.873570	+	0.504356i	0.868533	−	0.495631i	$-0.165063\pi$
$521$	455.130	+	262.769i	0.873570	+	0.504356i	0.00503702	+	0.999987i	$0.498397\pi$
$522$		0			0
$523$	120.337	+	208.430i	0.230091	+	0.398529i	0.957835	−	0.287320i	$-0.0927644\pi$
$523$	120.337	+	208.430i	0.230091	+	0.398529i	−0.727744	+	0.685849i	$0.759431\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	267.895	−	154.669i	0.508339	−	0.293489i
$528$		0			0
$529$	526.759	−	912.373i	0.995763	−	1.72471i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			245.323i			0.460269i
$534$		0			0
$535$	653.266	−	1131.49i	1.22106	−	2.11494i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−119.909	+	219.458i	−0.222466	+	0.407157i
$540$		0			0
$541$	171.208	+	296.541i	0.316466	+	0.548135i	0.979748	−	0.200234i	$-0.0641704\pi$
$541$	171.208	+	296.541i	0.316466	+	0.548135i	−0.663282	+	0.748369i	$0.730837\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1503.59i		−	2.75888i
$546$		0			0
$547$	−94.5237			−0.172804			−0.0864019	−	0.996260i	$-0.527537\pi$
$547$	−94.5237			−0.172804			−0.0864019	+	0.996260i	$0.527537\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	147.553	−	85.1896i	0.267791	−	0.154609i
$552$		0			0
$553$	283.382	+	72.3691i	0.512444	+	0.130866i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	58.6678	+	33.8719i	0.105328	+	0.0608112i	0.551739	−	0.834017i	$-0.313965\pi$
$557$	58.6678	+	33.8719i	0.105328	+	0.0608112i	−0.446411	+	0.894828i	$0.647298\pi$
$558$		0			0
$559$	−630.681			−1.12823
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−447.878	−	258.582i	−0.795520	−	0.459294i	0.0463821	−	0.998924i	$-0.485231\pi$
$563$	−447.878	−	258.582i	−0.795520	−	0.459294i	−0.841902	+	0.539630i	$0.818564\pi$
$564$		0			0
$565$	517.076	+	895.601i	0.915178	+	1.58513i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	186.087	−	107.438i	0.327043	−	0.188818i	−0.327484	−	0.944857i	$-0.606201\pi$
$569$	186.087	−	107.438i	0.327043	−	0.188818i	0.654528	+	0.756038i	$0.272868\pi$
$570$		0			0
$571$	217.467	−	376.663i	0.380852	−	0.659656i	−0.610332	−	0.792146i	$-0.708964\pi$
$571$	217.467	−	376.663i	0.380852	−	0.659656i	0.991184	+	0.132490i	$0.0422973\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1317.91i		−	2.29202i
$576$		0			0
$577$	−153.983	+	266.706i	−0.266868	+	0.462229i	−0.968051	−	0.250752i	$-0.919322\pi$
$577$	−153.983	+	266.706i	−0.266868	+	0.462229i	0.701183	+	0.712981i	$0.252655\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	730.037	+	713.056i	1.25652	+	1.22729i
$582$		0			0
$583$	−15.0615	−	26.0872i	−0.0258344	−	0.0447465i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			342.668i			0.583761i	0.956455	+	0.291881i	$0.0942810\pi$
$587$			342.668i			0.583761i	−0.956455	+	0.291881i	$0.905719\pi$
$588$		0			0
$589$	−263.300			−0.447028
$590$		0			0
$591$		0			0
$592$		0			0
$593$	502.396	−	290.059i	0.847211	−	0.489138i	−0.0124977	−	0.999922i	$-0.503978\pi$
$593$	502.396	−	290.059i	0.847211	−	0.489138i	0.859709	+	0.510784i	$0.170645\pi$
$594$		0			0
$595$	−979.066	+	1002.38i	−1.64549	+	1.68468i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−461.655	−	266.537i	−0.770710	−	0.444969i	0.0624181	−	0.998050i	$-0.480119\pi$
$599$	−461.655	−	266.537i	−0.770710	−	0.444969i	−0.833128	+	0.553081i	$0.813452\pi$
$600$		0			0
$601$	−244.388			−0.406636			−0.203318	−	0.979113i	$-0.565172\pi$
$601$	−244.388			−0.406636			−0.203318	+	0.979113i	$0.565172\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	626.953	+	361.972i	1.03629	+	0.598300i
$606$		0			0
$607$	−244.708	−	423.847i	−0.403143	−	0.698265i	0.590960	−	0.806701i	$-0.298749\pi$
$607$	−244.708	−	423.847i	−0.403143	−	0.698265i	−0.994103	+	0.108436i	$0.965416\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	687.232	−	396.774i	1.12477	−	0.649384i
$612$		0			0
$613$	199.779	−	346.028i	0.325904	−	0.564482i	−0.655791	−	0.754943i	$-0.727665\pi$
$613$	199.779	−	346.028i	0.325904	−	0.564482i	0.981695	+	0.190460i	$0.0609981\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	800.485i		−	1.29738i	−0.761052	−	0.648691i	$-0.775317\pi$
$617$		−	800.485i		−	1.29738i	0.761052	−	0.648691i	$-0.224683\pi$
$618$		0			0
$619$	368.596	−	638.427i	0.595470	−	1.03138i	−0.398010	−	0.917381i	$-0.630299\pi$
$619$	368.596	−	638.427i	0.595470	−	1.03138i	0.993480	−	0.114004i	$-0.0363675\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−76.0345	+	297.735i	−0.122046	+	0.477905i
$624$		0			0
$625$	177.841	+	308.029i	0.284545	+	0.492846i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1584.37i		−	2.51888i
$630$		0			0
$631$	1046.82			1.65898			0.829490	−	0.558521i	$-0.188631\pi$
$631$	1046.82			1.65898			0.829490	+	0.558521i	$0.188631\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1231.81	+	711.184i	−1.93985	+	1.11998i
$636$		0			0
$637$	23.9622	+	1018.05i	0.0376173	+	1.59820i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−152.784	−	88.2100i	−0.238353	−	0.137613i	0.376067	−	0.926593i	$-0.377276\pi$
$641$	−152.784	−	88.2100i	−0.238353	−	0.137613i	−0.614419	+	0.788980i	$0.710610\pi$
$642$		0			0
$643$	865.349			1.34580			0.672900	−	0.739733i	$-0.265048\pi$
$643$	865.349			1.34580			0.672900	+	0.739733i	$0.265048\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−471.727	−	272.352i	−0.729100	−	0.420946i	0.0889930	−	0.996032i	$-0.471635\pi$
$647$	−471.727	−	272.352i	−0.729100	−	0.420946i	−0.818093	+	0.575086i	$0.804968\pi$
$648$		0			0
$649$	−112.819	−	195.408i	−0.173835	−	0.301091i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−560.396	+	323.545i	−0.858187	+	0.495474i	−0.863405	−	0.504512i	$-0.831672\pi$
$653$	−560.396	+	323.545i	−0.858187	+	0.495474i	0.00521781	+	0.999986i	$0.498339\pi$
$654$		0			0
$655$	−192.891	+	334.097i	−0.294490	+	0.510072i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			265.738i			0.403245i	0.979463	+	0.201622i	$0.0646213\pi$
$659$			265.738i			0.403245i	−0.979463	+	0.201622i	$0.935379\pi$
$660$		0			0
$661$	−432.779	+	749.595i	−0.654734	+	1.13403i	0.327227	+	0.944946i	$0.393886\pi$
$661$	−432.779	+	749.595i	−0.654734	+	1.13403i	−0.981960	+	0.189086i	$0.939447\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1148.30	−	322.215i	1.72677	−	0.484534i
$666$		0			0
$667$	−151.650	−	262.665i	−0.227361	−	0.393801i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	252.510i		−	0.376320i
$672$		0			0
$673$	−344.022			−0.511176			−0.255588	−	0.966786i	$-0.582269\pi$
$673$	−344.022			−0.511176			−0.255588	+	0.966786i	$0.582269\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−86.9185	+	50.1824i	−0.128388	+	0.0741247i	−0.562818	−	0.826581i	$-0.690283\pi$
$677$	−86.9185	+	50.1824i	−0.128388	+	0.0741247i	0.434431	+	0.900705i	$0.356950\pi$
$678$		0			0
$679$	−193.286	+	197.888i	−0.284662	+	0.291441i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−231.465	−	133.636i	−0.338895	−	0.195661i	0.320889	−	0.947117i	$-0.396019\pi$
$683$	−231.465	−	133.636i	−0.338895	−	0.195661i	−0.659783	+	0.751456i	$0.729352\pi$
$684$		0			0
$685$	1213.20			1.77109
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−106.228	−	61.3309i	−0.154177	−	0.0890143i
$690$		0			0
$691$	269.940	+	467.549i	0.390651	+	0.676627i	0.992536	−	0.121956i	$-0.0389166\pi$
$691$	269.940	+	467.549i	0.390651	+	0.676627i	−0.601885	+	0.798583i	$0.705583\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12.0389	−	6.95065i	0.0173221	−	0.0100009i
$696$		0			0
$697$	−154.959	+	268.397i	−0.222323	+	0.385074i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1083.03i		−	1.54498i	−0.635025	−	0.772492i	$-0.719010\pi$
$701$		−	1083.03i		−	1.54498i	0.635025	−	0.772492i	$-0.280990\pi$
$702$		0			0
$703$	−674.287	+	1167.90i	−0.959156	+	1.66131i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	629.912	−	176.754i	0.890965	−	0.250006i
$708$		0			0
$709$	286.981	+	497.065i	0.404768	+	0.701079i	0.994294	−	0.106670i	$-0.0340188\pi$
$709$	286.981	+	497.065i	0.404768	+	0.701079i	−0.589526	+	0.807749i	$0.700685\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			468.711i			0.657379i
$714$		0			0
$715$	−808.675			−1.13101
$716$		0			0
$717$		0			0
$718$		0			0
$719$	582.754	−	336.453i	0.810507	−	0.467946i	−0.0366251	−	0.999329i	$-0.511661\pi$
$719$	582.754	−	336.453i	0.810507	−	0.467946i	0.847132	+	0.531383i	$0.178327\pi$
$720$		0			0
$721$	393.375	+	100.459i	0.545597	+	0.139333i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−218.746	−	126.293i	−0.301719	−	0.174198i
$726$		0			0
$727$	924.627			1.27184			0.635920	−	0.771755i	$-0.280621\pi$
$727$	924.627			1.27184			0.635920	+	0.771755i	$0.280621\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−689.999	−	398.371i	−0.943911	−	0.544967i
$732$		0			0
$733$	−491.120	−	850.644i	−0.670013	−	1.16050i	−0.977900	−	0.209074i	$-0.932955\pi$
$733$	−491.120	−	850.644i	−0.670013	−	1.16050i	0.307887	−	0.951423i	$-0.400378\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−279.025	+	161.095i	−0.378596	+	0.218583i
$738$		0			0
$739$	−52.1165	+	90.2684i	−0.0705230	+	0.122149i	−0.899131	−	0.437680i	$-0.855800\pi$
$739$	−52.1165	+	90.2684i	−0.0705230	+	0.122149i	0.828608	+	0.559830i	$0.189133\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1381.60i		−	1.85949i	−0.368203	−	0.929745i	$-0.620027\pi$
$743$		−	1381.60i		−	1.85949i	0.368203	−	0.929745i	$-0.379973\pi$
$744$		0			0
$745$	−646.940	+	1120.53i	−0.868376	+	1.50407i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	296.813	−	1162.26i	0.396279	−	1.55174i
$750$		0			0
$751$	−159.156	−	275.667i	−0.211926	−	0.367066i	0.740391	−	0.672176i	$-0.234640\pi$
$751$	−159.156	−	275.667i	−0.211926	−	0.367066i	−0.952317	+	0.305110i	$0.901307\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	983.480i		−	1.30262i
$756$		0			0
$757$	541.097			0.714792			0.357396	−	0.933953i	$-0.383665\pi$
$757$	541.097			0.714792			0.357396	+	0.933953i	$0.383665\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−594.210	+	343.067i	−0.780828	+	0.450811i	−0.836724	−	0.547626i	$-0.815532\pi$
$761$	−594.210	+	343.067i	−0.780828	+	0.450811i	0.0558959	+	0.998437i	$0.482199\pi$
$762$		0			0
$763$	−372.959	−	1329.14i	−0.488806	−	1.74200i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−795.709	−	459.403i	−1.03743	−	0.598961i
$768$		0			0
$769$	1233.76			1.60437			0.802187	−	0.597073i	$-0.203670\pi$
$769$	1233.76			1.60437			0.802187	+	0.597073i	$0.203670\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−307.770	−	177.691i	−0.398150	−	0.229872i	0.287535	−	0.957770i	$-0.407164\pi$
$773$	−307.770	−	177.691i	−0.398150	−	0.229872i	−0.685685	+	0.727898i	$0.740497\pi$
$774$		0			0
$775$	195.170	+	338.045i	0.251833	+	0.436187i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	228.452	−	131.897i	0.293263	−	0.169315i
$780$		0			0
$781$	106.227	−	183.991i	0.136014	−	0.235584i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			432.250i			0.550636i
$786$		0			0
$787$	−31.3912	+	54.3711i	−0.0398871	+	0.0690865i	−0.885280	−	0.465059i	$-0.846033\pi$
$787$	−31.3912	+	54.3711i	−0.0398871	+	0.0690865i	0.845393	+	0.534145i	$0.179367\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	679.235	+	663.435i	0.858704	+	0.838730i
$792$		0			0
$793$	−514.116	−	890.476i	−0.648318	−	1.12292i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	294.027i		−	0.368917i	−0.982840	−	0.184459i	$-0.940947\pi$
$797$		−	294.027i		−	0.368917i	0.982840	−	0.184459i	$-0.0590531\pi$
$798$		0			0
$799$	1002.49			1.25468
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−18.4606	+	10.6582i	−0.0229895	+	0.0132730i
$804$		0			0
$805$	−573.590	−	2044.15i	−0.712534	−	2.53931i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−149.303	−	86.2000i	−0.184552	−	0.106551i	0.404877	−	0.914371i	$-0.367314\pi$
$809$	−149.303	−	86.2000i	−0.184552	−	0.106551i	−0.589430	+	0.807820i	$0.700648\pi$
$810$		0			0
$811$	−28.1511			−0.0347115			−0.0173558	−	0.999849i	$-0.505525\pi$
$811$	−28.1511			−0.0347115			−0.0173558	+	0.999849i	$0.505525\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	891.108	+	514.481i	1.09338	+	0.631265i
$816$		0			0
$817$	339.082	+	587.307i	0.415033	+	0.718858i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	775.318	−	447.630i	0.944358	−	0.545225i	0.0530340	−	0.998593i	$-0.483111\pi$
$821$	775.318	−	447.630i	0.944358	−	0.545225i	0.891324	+	0.453368i	$0.149778\pi$
$822$		0			0
$823$	668.994	−	1158.73i	0.812872	−	1.40794i	−0.0979740	−	0.995189i	$-0.531236\pi$
$823$	668.994	−	1158.73i	0.812872	−	1.40794i	0.910846	−	0.412746i	$-0.135430\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			330.739i			0.399926i	0.979803	+	0.199963i	$0.0640821\pi$
$827$			330.739i			0.399926i	−0.979803	+	0.199963i	$0.935918\pi$
$828$		0			0
$829$	535.284	−	927.138i	0.645698	−	1.11838i	−0.338442	−	0.940987i	$-0.609900\pi$
$829$	535.284	−	927.138i	0.645698	−	1.11838i	0.984140	−	0.177394i	$-0.0567668\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−616.839	+	1128.94i	−0.740503	+	1.35527i
$834$		0			0
$835$	1228.23	+	2127.36i	1.47094	+	2.54774i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			818.205i			0.975214i	0.873063	+	0.487607i	$0.162130\pi$
$839$			818.205i			0.975214i	−0.873063	+	0.487607i	$0.837870\pi$
$840$		0			0
$841$	782.871			0.930881
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1735.91	+	1002.23i	−2.05433	+	1.18607i
$846$		0			0
$847$	644.000	+	164.463i	0.760331	+	0.194171i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2079.03	+	1200.33i	2.44304	+	1.41049i
$852$		0			0
$853$	−1455.47			−1.70630			−0.853148	−	0.521670i	$-0.825309\pi$
$853$	−1455.47			−1.70630			−0.853148	+	0.521670i	$0.825309\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−26.9946	−	15.5853i	−0.0314989	−	0.0181859i	0.484168	−	0.874975i	$-0.339122\pi$
$857$	−26.9946	−	15.5853i	−0.0314989	−	0.0181859i	−0.515667	+	0.856789i	$0.672456\pi$
$858$		0			0
$859$	−589.337	−	1020.76i	−0.686074	−	1.18831i	−0.973098	−	0.230391i	$-0.925999\pi$
$859$	−589.337	−	1020.76i	−0.686074	−	1.18831i	0.287024	−	0.957923i	$-0.407334\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	468.246	−	270.342i	0.542580	−	0.313258i	−0.203544	−	0.979066i	$-0.565246\pi$
$863$	468.246	−	270.342i	0.542580	−	0.313258i	0.746124	+	0.665807i	$0.231913\pi$
$864$		0			0
$865$	218.912	−	379.166i	0.253077	−	0.438342i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			213.243i			0.245389i
$870$		0			0
$871$	−655.987	+	1136.20i	−0.753143	+	1.30448i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−310.374	−	303.155i	−0.354713	−	0.346462i
$876$		0			0
$877$	−737.974	−	1278.21i	−0.841476	−	1.45748i	−0.888647	−	0.458592i	$-0.848354\pi$
$877$	−737.974	−	1278.21i	−0.841476	−	1.45748i	0.0471710	−	0.998887i	$-0.484979\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			919.702i			1.04393i	0.852967	+	0.521965i	$0.174801\pi$
$881$			919.702i			1.04393i	−0.852967	+	0.521965i	$0.825199\pi$
$882$		0			0
$883$	938.164			1.06247			0.531237	−	0.847223i	$-0.321728\pi$
$883$	938.164			1.06247			0.531237	+	0.847223i	$0.321728\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−655.439	+	378.418i	−0.738939	+	0.426627i	−0.821684	−	0.569944i	$-0.806965\pi$
$887$	−655.439	+	378.418i	−0.738939	+	0.426627i	0.0827442	+	0.996571i	$0.473632\pi$
$888$		0			0
$889$	−912.487	+	934.217i	−1.02642	+	1.05086i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−738.973	−	426.646i	−0.827517	−	0.477767i
$894$		0			0
$895$	−1445.72			−1.61532
$896$		0			0
$897$		0			0
$898$		0			0
$899$	77.7964	+	44.9158i	0.0865366	+	0.0499619i
$900$		0			0
$901$	−77.4795	−	134.198i	−0.0859928	−	0.148944i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1456.57	−	840.953i	1.60947	−	0.929230i
$906$		0			0
$907$	−191.839	+	332.275i	−0.211510	+	0.366346i	−0.952187	−	0.305515i	$-0.901171\pi$
$907$	−191.839	+	332.275i	−0.211510	+	0.366346i	0.740677	+	0.671861i	$0.234505\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			25.2548i			0.0277221i	0.999904	+	0.0138610i	$0.00441225\pi$
$911$			25.2548i			0.0277221i	−0.999904	+	0.0138610i	$0.995588\pi$
$912$		0			0
$913$	−372.017	+	644.353i	−0.407467	+	0.705753i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−87.6406	+	343.181i	−0.0955732	+	0.374244i
$918$		0			0
$919$	−406.558	−	704.180i	−0.442392	−	0.766245i	0.555474	−	0.831534i	$-0.312537\pi$
$919$	−406.558	−	704.180i	−0.442392	−	0.766245i	−0.997866	+	0.0652882i	$0.979203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	865.124i		−	0.937296i
$924$		0			0
$925$	1999.25			2.16136
$926$		0			0
$927$		0			0
$928$		0			0
$929$	279.717	−	161.495i	0.301095	−	0.173837i	−0.341840	−	0.939758i	$-0.611050\pi$
$929$	279.717	−	161.495i	0.301095	−	0.173837i	0.642935	+	0.765921i	$0.277717\pi$
$930$		0			0
$931$	935.154	−	569.664i	1.00446	−	0.611884i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−884.733	−	510.801i	−0.946239	−	0.546311i
$936$		0			0
$937$	649.858			0.693552			0.346776	−	0.937948i	$-0.387276\pi$
$937$	649.858			0.693552			0.346776	+	0.937948i	$0.387276\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	576.584	+	332.891i	0.612736	+	0.353763i	0.774035	−	0.633142i	$-0.218235\pi$
$941$	576.584	+	332.891i	0.612736	+	0.353763i	−0.161300	+	0.986905i	$0.551569\pi$
$942$		0			0
$943$	−234.795	−	406.677i	−0.248987	−	0.431259i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1256.46	−	725.415i	1.32678	−	0.766014i	0.341976	−	0.939709i	$-0.388904\pi$
$947$	1256.46	−	725.415i	1.32678	−	0.766014i	0.984800	+	0.173694i	$0.0555705\pi$
$948$		0			0
$949$	−43.4008	+	75.1724i	−0.0457332	+	0.0792122i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			562.272i			0.590002i	0.955497	+	0.295001i	$0.0953200\pi$
$953$			562.272i			0.590002i	−0.955497	+	0.295001i	$0.904680\pi$
$954$		0			0
$955$	−561.123	+	971.893i	−0.587563	+	1.01769i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1072.44	−	300.929i	1.11829	−	0.313794i
$960$		0			0
$961$	411.088	+	712.026i	0.427771	+	0.740922i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			918.494i			0.951807i
$966$		0			0
$967$	1238.53			1.28080			0.640399	−	0.768042i	$-0.278769\pi$
$967$	1238.53			1.28080			0.640399	+	0.768042i	$0.278769\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−380.984	+	219.961i	−0.392363	+	0.226531i	−0.683183	−	0.730247i	$-0.739405\pi$
$971$	−380.984	+	219.961i	−0.392363	+	0.226531i	0.290821	+	0.956778i	$0.406072\pi$
$972$		0			0
$973$	8.91806	−	9.13044i	0.00916553	−	0.00938380i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−671.736	−	387.827i	−0.687549	−	0.396957i	0.115144	−	0.993349i	$-0.463267\pi$
$977$	−671.736	−	387.827i	−0.687549	−	0.396957i	−0.802693	+	0.596392i	$0.796600\pi$
$978$		0			0
$979$	−224.043			−0.228849
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1171.93	+	676.614i	1.19220	+	0.688316i	0.958804	−	0.284067i	$-0.0916837\pi$
$983$	1171.93	+	676.614i	1.19220	+	0.688316i	0.233393	+	0.972382i	$0.425017\pi$
$984$		0			0
$985$	−82.4827	−	142.864i	−0.0837388	−	0.145040i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1045.49	−	603.615i	1.05712	−	0.610328i
$990$		0			0
$991$	−107.307	+	185.862i	−0.108282	+	0.187550i	−0.915074	−	0.403285i	$-0.867868\pi$
$991$	−107.307	+	185.862i	−0.108282	+	0.187550i	0.806792	+	0.590835i	$0.201202\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	151.860i		−	0.152624i
$996$		0			0
$997$	519.306	−	899.464i	0.520869	−	0.902171i	−0.478837	−	0.877904i	$-0.658941\pi$
$997$	519.306	−	899.464i	0.520869	−	0.902171i	0.999706	−	0.0242670i	$-0.00772520\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.dc.d.305.1		8
3.2	odd	2	inner	1008.3.dc.d.305.4		8
4.3	odd	2		252.3.bk.b.53.1	✓	8
7.2	even	3	inner	1008.3.dc.d.737.4		8
12.11	even	2		252.3.bk.b.53.4	yes	8
21.2	odd	6	inner	1008.3.dc.d.737.1		8
28.3	even	6		1764.3.c.g.197.4		4
28.11	odd	6		1764.3.c.e.197.1		4
28.19	even	6		1764.3.bk.f.1745.1		8
28.23	odd	6		252.3.bk.b.233.4	yes	8
28.27	even	2		1764.3.bk.f.557.4		8
84.11	even	6		1764.3.c.e.197.4		4
84.23	even	6		252.3.bk.b.233.1	yes	8
84.47	odd	6		1764.3.bk.f.1745.4		8
84.59	odd	6		1764.3.c.g.197.1		4
84.83	odd	2		1764.3.bk.f.557.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.bk.b.53.1	✓	8	4.3	odd	2
252.3.bk.b.53.4	yes	8	12.11	even	2
252.3.bk.b.233.1	yes	8	84.23	even	6
252.3.bk.b.233.4	yes	8	28.23	odd	6
1008.3.dc.d.305.1		8	1.1	even	1	trivial
1008.3.dc.d.305.4		8	3.2	odd	2	inner
1008.3.dc.d.737.1		8	21.2	odd	6	inner
1008.3.dc.d.737.4		8	7.2	even	3	inner
1764.3.c.e.197.1		4	28.11	odd	6
1764.3.c.e.197.4		4	84.11	even	6
1764.3.c.g.197.1		4	84.59	odd	6
1764.3.c.g.197.4		4	28.3	even	6
1764.3.bk.f.557.1		8	84.83	odd	2
1764.3.bk.f.557.4		8	28.27	even	2
1764.3.bk.f.1745.1		8	28.19	even	6
1764.3.bk.f.1745.4		8	84.47	odd	6

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)

\(f(q)\)	\(=\)	\(q+(-6.60280 + 3.81213i) q^{5} +(-4.89116 + 5.00764i) q^{7} +(-4.41990 - 2.55183i) q^{11} -20.7823 q^{13} +(-22.7370 - 13.1272i) q^{17} +(11.1735 + 19.3531i) q^{19} +(34.4512 - 19.8904i) q^{23} +(16.5647 - 28.6908i) q^{25} -7.62426i q^{29} +(-5.89116 + 10.2038i) q^{31} +(13.2056 - 51.7102i) q^{35} +(30.1735 + 52.2620i) q^{37} -11.8044i q^{41} +30.3470 q^{43} +(-33.0681 + 19.0919i) q^{47} +(-1.15301 - 48.9864i) q^{49} +(5.11147 + 2.95111i) q^{53} +38.9116 q^{55} +(38.2878 + 22.1055i) q^{59} +(24.7382 + 42.8477i) q^{61} +(137.222 - 79.2249i) q^{65} +(31.5647 - 54.6716i) q^{67} +41.6279i q^{71} +(2.08835 - 3.61713i) q^{73} +(34.3971 - 9.65187i) q^{77} +(-20.8912 - 36.1846i) q^{79} -145.784i q^{83} +200.170 q^{85} +(38.0173 - 21.9493i) q^{89} +(101.650 - 104.071i) q^{91} +(-147.553 - 85.1896i) q^{95} +39.5173 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{7} - 112 q^{13} + 8 q^{19} + 24 q^{25} - 20 q^{31} + 160 q^{37} + 80 q^{43} - 172 q^{49} + 40 q^{55} + 8 q^{61} + 144 q^{67} + 288 q^{73} - 140 q^{79} + 896 q^{85} + 352 q^{91} - 552 q^{97}+O(q^{100}) \) 8 * q - 12 * q^7 - 112 * q^13 + 8 * q^19 + 24 * q^25 - 20 * q^31 + 160 * q^37 + 80 * q^43 - 172 * q^49 + 40 * q^55 + 8 * q^61 + 144 * q^67 + 288 * q^73 - 140 * q^79 + 896 * q^85 + 352 * q^91 - 552 * q^97

Introduction

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