LMFDB - Embedded newform 1512.2.bl.c.593.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(593,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1512.593"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.bl (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 26x^{14} + 521x^{12} + 3668x^{10} + 19270x^{8} + 25507x^{6} + 25166x^{4} + 8869x^{2} + 2401 $ x^16 + 26x^14 + 521x^12 + 3668x^10 + 19270x^8 + 25507x^6 + 25166x^4 + 8869*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.1
Root			$-0.314872 + 0.545374i$ of defining polynomial
Character	$\chi$	$=$	1512.593
Dual form			1512.2.bl.c.1025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.69652 + 2.93845i) q^{5} +(-2.25634 + 1.38164i) q^{7} +(-0.669013 + 0.386255i) q^{11} -0.401495i q^{13} +(1.38164 + 2.39308i) q^{17} +(-3.68797 - 2.12925i) q^{19} +(-0.0785903 - 0.0453741i) q^{23} +(-3.25634 - 5.64014i) q^{25} +5.51267i q^{29} +(3.34770 - 1.93280i) q^{31} +(-0.231984 - 8.97412i) q^{35} +(3.39308 - 5.87698i) q^{37} -7.88837 q^{41} -12.0713 q^{43} +(1.84991 - 3.20414i) q^{47} +(3.18212 - 6.23491i) q^{49} +(11.7927 - 6.80854i) q^{53} -2.62115i q^{55} +(2.67667 + 4.63612i) q^{59} +(-8.61025 - 4.97113i) q^{61} +(1.17977 + 0.681143i) q^{65} +(-5.69479 - 9.86367i) q^{67} -12.6748i q^{71} +(-2.39211 + 1.38108i) q^{73} +(0.975853 - 1.79586i) q^{77} +(4.46145 - 7.72746i) q^{79} -10.3438 q^{83} -9.37593 q^{85} +(-1.41344 + 2.44816i) q^{89} +(0.554723 + 0.905908i) q^{91} +(12.5134 - 7.22461i) q^{95} +6.45676i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{7} - 18 q^{19} - 12 q^{25} + 24 q^{31} + 16 q^{37} - 52 q^{43} + 44 q^{49} + 6 q^{61} - 4 q^{67} - 12 q^{73} + 34 q^{79} - 68 q^{85} + 102 q^{91}+O(q^{100}) $ 16 * q + 4 * q^7 - 18 * q^19 - 12 * q^25 + 24 * q^31 + 16 * q^37 - 52 * q^43 + 44 * q^49 + 6 * q^61 - 4 * q^67 - 12 * q^73 + 34 * q^79 - 68 * q^85 + 102 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.69652	+	2.93845i	−0.758705	+	1.31412i	0.184806	+	0.982775i	$0.440834\pi$
$5$	−1.69652	+	2.93845i	−0.758705	+	1.31412i	−0.943511	+	0.331341i	$0.892499\pi$
$6$		0			0
$7$	−2.25634	+	1.38164i	−0.852815	+	0.522213i
$7$	−2.25634	+	1.38164i	−0.852815	+	0.522213i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.669013	+	0.386255i	−0.201715	+	0.116460i	−0.597455	−	0.801902i	$-0.703821\pi$
$11$	−0.669013	+	0.386255i	−0.201715	+	0.116460i	0.395740	+	0.918363i	$0.370488\pi$
$12$		0			0
$13$		−	0.401495i		−	0.111355i	−0.998449	−	0.0556773i	$-0.982268\pi$
$13$		−	0.401495i		−	0.111355i	0.998449	−	0.0556773i	$-0.0177318\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.38164	+	2.39308i	0.335098	+	0.580407i	0.983504	−	0.180888i	$-0.0578972\pi$
$17$	1.38164	+	2.39308i	0.335098	+	0.580407i	−0.648406	+	0.761295i	$0.724564\pi$
$18$		0			0
$19$	−3.68797	−	2.12925i	−0.846078	−	0.488483i	0.0132479	−	0.999912i	$-0.495783\pi$
$19$	−3.68797	−	2.12925i	−0.846078	−	0.488483i	−0.859325	+	0.511429i	$0.829116\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.0785903	−	0.0453741i	−0.0163872	−	0.00946116i	0.491784	−	0.870717i	$-0.336345\pi$
$23$	−0.0785903	−	0.0453741i	−0.0163872	−	0.00946116i	−0.508171	+	0.861256i	$0.669678\pi$
$24$		0			0
$25$	−3.25634	−	5.64014i	−0.651267	−	1.12803i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.51267i			1.02368i	0.859081	+	0.511839i	$0.171036\pi$
$29$			5.51267i			1.02368i	−0.859081	+	0.511839i	$0.828964\pi$
$30$		0			0
$31$	3.34770	−	1.93280i	0.601266	−	0.347141i	−0.168274	−	0.985740i	$-0.553819\pi$
$31$	3.34770	−	1.93280i	0.601266	−	0.347141i	0.769539	+	0.638599i	$0.220486\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.231984	−	8.97412i	−0.0392125	−	1.51690i
$36$		0			0
$37$	3.39308	−	5.87698i	0.557819	−	0.966170i	−0.439860	−	0.898067i	$-0.644972\pi$
$37$	3.39308	−	5.87698i	0.557819	−	0.966170i	0.997678	−	0.0681038i	$-0.0216949\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.88837			−1.23196			−0.615978	−	0.787763i	$-0.711239\pi$
$41$	−7.88837			−1.23196			−0.615978	+	0.787763i	$0.711239\pi$
$42$		0			0
$43$	−12.0713			−1.84086			−0.920431	−	0.390905i	$-0.872162\pi$
$43$	−12.0713			−1.84086			−0.920431	+	0.390905i	$0.872162\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.84991	−	3.20414i	0.269837	−	0.467372i	−0.698982	−	0.715139i	$-0.746363\pi$
$47$	1.84991	−	3.20414i	0.269837	−	0.467372i	0.968820	+	0.247767i	$0.0796968\pi$
$48$		0			0
$49$	3.18212	−	6.23491i	0.454588	−	0.890702i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	11.7927	−	6.80854i	1.61986	−	0.935224i	0.632900	−	0.774234i	$-0.281864\pi$
$53$	11.7927	−	6.80854i	1.61986	−	0.935224i	0.986956	−	0.160991i	$-0.0514689\pi$
$54$		0			0
$55$		−	2.62115i		−	0.353436i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.67667	+	4.63612i	0.348472	+	0.603572i	0.985978	−	0.166874i	$-0.0533672\pi$
$59$	2.67667	+	4.63612i	0.348472	+	0.603572i	−0.637506	+	0.770445i	$0.720034\pi$
$60$		0			0
$61$	−8.61025	−	4.97113i	−1.10243	−	0.636488i	−0.165571	−	0.986198i	$-0.552947\pi$
$61$	−8.61025	−	4.97113i	−1.10243	−	0.636488i	−0.936858	+	0.349710i	$0.886280\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.17977	+	0.681143i	0.146333	+	0.0844854i
$66$		0			0
$67$	−5.69479	−	9.86367i	−0.695729	−	1.20504i	−0.969934	−	0.243367i	$-0.921748\pi$
$67$	−5.69479	−	9.86367i	−0.695729	−	1.20504i	0.274205	−	0.961671i	$-0.411585\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.6748i		−	1.50422i	−0.659039	−	0.752109i	$-0.729037\pi$
$71$		−	12.6748i		−	1.50422i	0.659039	−	0.752109i	$-0.270963\pi$
$72$		0			0
$73$	−2.39211	+	1.38108i	−0.279975	+	0.161644i	−0.633412	−	0.773815i	$-0.718346\pi$
$73$	−2.39211	+	1.38108i	−0.279975	+	0.161644i	0.353437	+	0.935458i	$0.385013\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.975853	−	1.79586i	0.111209	−	0.204657i
$78$		0			0
$79$	4.46145	−	7.72746i	0.501952	−	0.869407i	−0.498045	−	0.867151i	$-0.665949\pi$
$79$	4.46145	−	7.72746i	0.501952	−	0.869407i	0.999997	−	0.00225579i	$-0.000718042\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.3438			−1.13537			−0.567687	−	0.823244i	$-0.692162\pi$
$83$	−10.3438			−1.13537			−0.567687	+	0.823244i	$0.692162\pi$
$84$		0			0
$85$	−9.37593			−1.01696
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.41344	+	2.44816i	−0.149825	+	0.259504i	−0.931163	−	0.364604i	$-0.881204\pi$
$89$	−1.41344	+	2.44816i	−0.149825	+	0.259504i	0.781338	+	0.624108i	$0.214538\pi$
$90$		0			0
$91$	0.554723	+	0.905908i	0.0581508	+	0.0949649i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.5134	−	7.22461i	1.28385	−	0.741229i
$96$		0			0
$97$			6.45676i			0.655585i	0.944750	+	0.327793i	$0.106305\pi$
$97$			6.45676i			0.655585i	−0.944750	+	0.327793i	$0.893695\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.91518	+	11.9774i	0.688087	+	1.19180i	0.972456	+	0.233086i	$0.0748824\pi$
$101$	6.91518	+	11.9774i	0.688087	+	1.19180i	−0.284370	+	0.958715i	$0.591784\pi$
$102$		0			0
$103$	10.1112	+	5.83772i	0.996288	+	0.575207i	0.907148	−	0.420812i	$-0.138255\pi$
$103$	10.1112	+	5.83772i	0.996288	+	0.575207i	0.0891402	+	0.996019i	$0.471588\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.02875	−	4.05805i	−0.679495	−	0.392306i	0.120170	−	0.992753i	$-0.461656\pi$
$107$	−7.02875	−	4.05805i	−0.679495	−	0.392306i	−0.799665	+	0.600447i	$0.794989\pi$
$108$		0			0
$109$	−0.772510	−	1.33803i	−0.0739931	−	0.128160i	0.826655	−	0.562709i	$-0.190241\pi$
$109$	−0.772510	−	1.33803i	−0.0739931	−	0.128160i	−0.900648	+	0.434550i	$0.856908\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			8.85750i			0.833244i	0.909080	+	0.416622i	$0.136786\pi$
$113$			8.85750i			0.833244i	−0.909080	+	0.416622i	$0.863214\pi$
$114$		0			0
$115$	0.266659	−	0.153956i	0.0248661	−	0.0143565i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.42384	−	3.49065i	−0.588873	−	0.319987i
$120$		0			0
$121$	−5.20161	+	9.00946i	−0.472874	+	0.819042i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	5.13255			0.459070
$126$		0			0
$127$	−9.27224			−0.822779			−0.411389	−	0.911460i	$-0.634956\pi$
$127$	−9.27224			−0.822779			−0.411389	+	0.911460i	$0.634956\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.27835	+	5.67826i	−0.286431	+	0.496112i	−0.972955	−	0.230994i	$-0.925802\pi$
$131$	−3.27835	+	5.67826i	−0.286431	+	0.496112i	0.686525	+	0.727107i	$0.259135\pi$
$132$		0			0
$133$	11.2632	−	0.291157i	0.976640	−	0.0252465i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.78508	+	5.64942i	−0.835995	+	0.482662i	−0.855901	−	0.517140i	$-0.826997\pi$
$137$	−9.78508	+	5.64942i	−0.835995	+	0.482662i	0.0199056	+	0.999802i	$0.493663\pi$
$138$		0			0
$139$		−	13.4999i		−	1.14505i	−0.819887	−	0.572525i	$-0.805964\pi$
$139$		−	13.4999i		−	1.14505i	0.819887	−	0.572525i	$-0.194036\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.155079	+	0.268605i	0.0129684	+	0.0224619i
$144$		0			0
$145$	−16.1987	−	9.35234i	−1.34523	−	0.776670i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.44769	−	5.45463i	−0.773985	−	0.446860i	0.0603094	−	0.998180i	$-0.480791\pi$
$149$	−9.44769	−	5.45463i	−0.773985	−	0.446860i	−0.834294	+	0.551319i	$0.814125\pi$
$150$		0			0
$151$	3.11960	+	5.40330i	0.253869	+	0.439714i	0.964588	−	0.263762i	$-0.0849634\pi$
$151$	3.11960	+	5.40330i	0.253869	+	0.439714i	−0.710719	+	0.703476i	$0.751630\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			13.1161i			1.05351i
$156$		0			0
$157$	−13.7441	+	7.93517i	−1.09690	+	0.633295i	−0.935405	−	0.353579i	$-0.884965\pi$
$157$	−13.7441	+	7.93517i	−1.09690	+	0.633295i	−0.161494	+	0.986874i	$0.551631\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.240017	−	0.00620453i	0.0189160	−	0.000488985i
$162$		0			0
$163$	7.14295	−	12.3719i	0.559479	−	0.969046i	−0.438061	−	0.898945i	$-0.644335\pi$
$163$	7.14295	−	12.3719i	0.559479	−	0.969046i	0.997540	−	0.0701004i	$-0.0223320\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.06261			−0.236992			−0.118496	−	0.992955i	$-0.537807\pi$
$167$	−3.06261			−0.236992			−0.118496	+	0.992955i	$0.537807\pi$
$168$		0			0
$169$	12.8388			0.987600
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.2685	+	21.2496i	−0.932754	+	1.61558i	−0.154164	+	0.988045i	$0.549268\pi$
$173$	−12.2685	+	21.2496i	−0.932754	+	1.61558i	−0.778591	+	0.627532i	$0.784065\pi$
$174$		0			0
$175$	15.1401	+	8.22696i	1.14448	+	0.621900i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.69553	−	2.71096i	0.350960	−	0.202627i	−0.314148	−	0.949374i	$-0.601719\pi$
$179$	4.69553	−	2.71096i	0.350960	−	0.202627i	0.665108	+	0.746747i	$0.268385\pi$
$180$		0			0
$181$			25.6404i			1.90583i	0.303233	+	0.952917i	$0.401934\pi$
$181$			25.6404i			1.90583i	−0.303233	+	0.952917i	$0.598066\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.5128	+	19.9408i	0.846440	+	1.46608i
$186$		0			0
$187$	−1.84868	−	1.06733i	−0.135189	−	0.0780512i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.65321	+	3.84123i	0.481410	+	0.277942i	0.721004	−	0.692931i	$-0.243681\pi$
$191$	6.65321	+	3.84123i	0.481410	+	0.277942i	−0.239594	+	0.970873i	$0.577014\pi$
$192$		0			0
$193$	−9.19576	−	15.9275i	−0.661926	−	1.14649i	−0.980109	−	0.198460i	$-0.936406\pi$
$193$	−9.19576	−	15.9275i	−0.661926	−	1.14649i	0.318184	−	0.948029i	$-0.396927\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	0.966947i		−	0.0688921i	−0.999407	−	0.0344460i	$-0.989033\pi$
$197$		−	0.966947i		−	0.0688921i	0.999407	−	0.0344460i	$-0.0109667\pi$
$198$		0			0
$199$	15.8436	−	9.14730i	1.12312	−	0.648435i	0.180926	−	0.983497i	$-0.442091\pi$
$199$	15.8436	−	9.14730i	1.12312	−	0.648435i	0.942196	+	0.335062i	$0.108757\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.61656	−	12.4385i	−0.534578	−	0.873008i
$204$		0			0
$205$	13.3828	−	23.1796i	0.934692	−	1.61893i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.28973			0.227556
$210$		0			0
$211$	−25.2411			−1.73767			−0.868836	−	0.495100i	$-0.835131\pi$
$211$	−25.2411			−1.73767			−0.868836	+	0.495100i	$0.835131\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	20.4792	−	35.4711i	1.39667	−	2.41911i
$216$		0			0
$217$	−4.88311	+	8.98638i	−0.331487	+	0.610035i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.960809	−	0.554723i	0.0646310	−	0.0373147i
$222$		0			0
$223$			19.4754i			1.30417i	0.758145	+	0.652086i	$0.226106\pi$
$223$			19.4754i			1.30417i	−0.758145	+	0.652086i	$0.773894\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.03859	−	13.9233i	−0.533540	−	0.924119i	−0.999232	−	0.0391720i	$-0.987528\pi$
$227$	−8.03859	−	13.9233i	−0.533540	−	0.924119i	0.465692	−	0.884947i	$-0.345805\pi$
$228$		0			0
$229$	3.56487	+	2.05818i	0.235573	+	0.136008i	0.613141	−	0.789974i	$-0.289906\pi$
$229$	3.56487	+	2.05818i	0.235573	+	0.136008i	−0.377567	+	0.925982i	$0.623239\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.32621	−	0.765686i	−0.0868828	−	0.0501618i	0.455929	−	0.890016i	$-0.349307\pi$
$233$	−1.32621	−	0.765686i	−0.0868828	−	0.0501618i	−0.542812	+	0.839854i	$0.682640\pi$
$234$		0			0
$235$	6.27681	+	10.8718i	0.409454	+	0.709195i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			4.36423i			0.282299i	0.989988	+	0.141149i	$0.0450798\pi$
$239$			4.36423i			0.282299i	−0.989988	+	0.141149i	$0.954920\pi$
$240$		0			0
$241$	25.8242	−	14.9096i	1.66348	−	0.960412i	0.692448	−	0.721467i	$-0.256532\pi$
$241$	25.8242	−	14.9096i	1.66348	−	0.960412i	0.971033	−	0.238944i	$-0.0768013\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.9225	+	19.9281i	0.825587	+	1.27316i
$246$		0			0
$247$	−0.854882	+	1.48070i	−0.0543949	+	0.0942147i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.8466			−0.810868			−0.405434	−	0.914124i	$-0.632880\pi$
$251$	−12.8466			−0.810868			−0.405434	+	0.914124i	$0.632880\pi$
$252$		0			0
$253$	0.0701039			0.00440740
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.14450	+	10.6426i	−0.383284	+	0.663867i	−0.991529	−	0.129882i	$-0.958540\pi$
$257$	−6.14450	+	10.6426i	−0.383284	+	0.663867i	0.608246	+	0.793749i	$0.291873\pi$
$258$		0			0
$259$	0.463975	+	17.9485i	0.0288300	+	1.11526i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.20071	+	5.31203i	−0.567340	+	0.327554i	−0.756086	−	0.654472i	$-0.772891\pi$
$263$	−9.20071	+	5.31203i	−0.567340	+	0.327554i	0.188746	+	0.982026i	$0.439558\pi$
$264$		0			0
$265$			46.2032i			2.83824i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.2212	−	22.8997i	−0.806109	−	1.39622i	−0.915540	−	0.402227i	$-0.868236\pi$
$269$	−13.2212	−	22.8997i	−0.806109	−	1.39622i	0.109431	−	0.993994i	$-0.465097\pi$
$270$		0			0
$271$	−0.935128	−	0.539896i	−0.0568050	−	0.0327964i	0.471329	−	0.881958i	$-0.343775\pi$
$271$	−0.935128	−	0.539896i	−0.0568050	−	0.0327964i	−0.528134	+	0.849161i	$0.677108\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.35707	+	2.51555i	0.262741	+	0.151694i
$276$		0			0
$277$	8.04932	+	13.9418i	0.483637	+	0.837683i	0.999823	−	0.0187928i	$-0.00598227\pi$
$277$	8.04932	+	13.9418i	0.483637	+	0.837683i	−0.516187	+	0.856476i	$0.672649\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	23.6494i		−	1.41081i	−0.708807	−	0.705403i	$-0.750766\pi$
$281$		−	23.6494i		−	1.41081i	0.708807	−	0.705403i	$-0.249234\pi$
$282$		0			0
$283$	6.07360	−	3.50660i	0.361038	−	0.208445i	−0.308498	−	0.951225i	$-0.599826\pi$
$283$	6.07360	−	3.50660i	0.361038	−	0.208445i	0.669536	+	0.742780i	$0.266493\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	17.7988	−	10.8989i	1.05063	−	0.643343i
$288$		0			0
$289$	4.68212	−	8.10966i	0.275419	−	0.477039i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.88725			0.460778			0.230389	−	0.973099i	$-0.426000\pi$
$293$	7.88725			0.460778			0.230389	+	0.973099i	$0.426000\pi$
$294$		0			0
$295$	−18.1640			−1.05755
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.0182175	+	0.0315536i	−0.00105354	+	0.00182479i
$300$		0			0
$301$	27.2370	−	16.6783i	1.56992	−	0.961322i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	29.2149	−	16.8672i	1.67284	−	0.965813i
$306$		0			0
$307$			27.9636i			1.59597i	0.602680	+	0.797983i	$0.294099\pi$
$307$			27.9636i			1.59597i	−0.602680	+	0.797983i	$0.705901\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.58046	−	13.1297i	−0.429849	−	0.744520i	0.567011	−	0.823710i	$-0.308100\pi$
$311$	−7.58046	−	13.1297i	−0.429849	−	0.744520i	−0.996859	+	0.0791907i	$0.974766\pi$
$312$		0			0
$313$	18.7334	+	10.8158i	1.05888	+	0.611342i	0.925121	−	0.379672i	$-0.123963\pi$
$313$	18.7334	+	10.8158i	1.05888	+	0.611342i	0.133755	+	0.991014i	$0.457296\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.71998	−	3.87979i	−0.377432	−	0.217910i	0.299268	−	0.954169i	$-0.403257\pi$
$317$	−6.71998	−	3.87979i	−0.377432	−	0.217910i	−0.676700	+	0.736259i	$0.736591\pi$
$318$		0			0
$319$	−2.12930	−	3.68805i	−0.119218	−	0.206491i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	11.7675i		−	0.654759i
$324$		0			0
$325$	−2.26449	+	1.30740i	−0.125611	+	0.0725217i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.252960	+	9.78554i	0.0139461	+	0.539494i
$330$		0			0
$331$	−6.50297	+	11.2635i	−0.357436	+	0.619097i	−0.987532	−	0.157421i	$-0.949682\pi$
$331$	−6.50297	+	11.2635i	−0.357436	+	0.619097i	0.630096	+	0.776517i	$0.283016\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	38.6452			2.11141
$336$		0			0
$337$	−4.31948			−0.235297			−0.117648	−	0.993055i	$-0.537536\pi$
$337$	−4.31948			−0.235297			−0.117648	+	0.993055i	$0.537536\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.49311	+	2.58614i	−0.0808562	+	0.140047i
$342$		0			0
$343$	1.43451	+	18.4646i	0.0774562	+	0.996996i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.2331	+	17.4551i	−1.62300	+	0.937037i	−0.636884	+	0.770960i	$0.719777\pi$
$347$	−30.2331	+	17.4551i	−1.62300	+	0.937037i	−0.986113	+	0.166077i	$0.946890\pi$
$348$		0			0
$349$		−	2.48536i		−	0.133038i	−0.997785	−	0.0665192i	$-0.978811\pi$
$349$		−	2.48536i		−	0.133038i	0.997785	−	0.0665192i	$-0.0211894\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.9429	−	20.6857i	−0.635656	−	1.10099i	−0.986376	−	0.164508i	$-0.947396\pi$
$353$	−11.9429	−	20.6857i	−0.635656	−	1.10099i	0.350719	−	0.936481i	$-0.385937\pi$
$354$		0			0
$355$	37.2442	+	21.5029i	1.97672	+	1.14126i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	19.3512	+	11.1724i	1.02132	+	0.589657i	0.914485	−	0.404620i	$-0.132596\pi$
$359$	19.3512	+	11.1724i	1.02132	+	0.589657i	0.106832	+	0.994277i	$0.465929\pi$
$360$		0			0
$361$	−0.432602	−	0.749289i	−0.0227685	−	0.0394363i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	9.37212i		−	0.490559i
$366$		0			0
$367$	−23.7863	+	13.7330i	−1.24163	+	0.716857i	−0.969426	−	0.245382i	$-0.921086\pi$
$367$	−23.7863	+	13.7330i	−1.24163	+	0.716857i	−0.272206	+	0.962239i	$0.587753\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−17.2014	+	31.6557i	−0.893052	+	1.64348i
$372$		0			0
$373$	−12.9023	+	22.3474i	−0.668053	+	1.15710i	0.310394	+	0.950608i	$0.399539\pi$
$373$	−12.9023	+	22.3474i	−0.668053	+	1.15710i	−0.978448	+	0.206495i	$0.933794\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.21331			0.113991
$378$		0			0
$379$	15.6424			0.803497			0.401749	−	0.915750i	$-0.368403\pi$
$379$	15.6424			0.803497			0.401749	+	0.915750i	$0.368403\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.99471	−	5.18699i	0.153023	−	0.265043i	−0.779315	−	0.626633i	$-0.784433\pi$
$383$	2.99471	−	5.18699i	0.153023	−	0.265043i	0.932337	+	0.361590i	$0.117766\pi$
$384$		0			0
$385$	3.62150	+	5.91420i	0.184569	+	0.301416i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−29.8059	+	17.2084i	−1.51122	+	0.872502i	−0.511304	+	0.859400i	$0.670838\pi$
$389$	−29.8059	+	17.2084i	−1.51122	+	0.872502i	−0.999914	+	0.0131027i	$0.995829\pi$
$390$		0			0
$391$		−	0.250764i		−	0.0126817i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.1378	+	26.2195i	0.761668	+	1.31925i
$396$		0			0
$397$	3.89760	+	2.25028i	0.195615	+	0.112938i	0.594609	−	0.804015i	$-0.297307\pi$
$397$	3.89760	+	2.25028i	0.195615	+	0.112938i	−0.398993	+	0.916954i	$0.630640\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.5634	+	7.25346i	0.627384	+	0.362220i	0.779738	−	0.626106i	$-0.215352\pi$
$401$	12.5634	+	7.25346i	0.627384	+	0.362220i	−0.152354	+	0.988326i	$0.548685\pi$
$402$		0			0
$403$	−0.776009	−	1.34409i	−0.0386557	−	0.0669537i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5.24238i			0.259855i
$408$		0			0
$409$	15.8510	−	9.15160i	0.783783	−	0.452517i	−0.0539863	−	0.998542i	$-0.517193\pi$
$409$	15.8510	−	9.15160i	0.783783	−	0.452517i	0.837769	+	0.546024i	$0.183859\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.4449	−	6.76245i	−0.612375	−	0.332759i
$414$		0			0
$415$	17.5483	−	30.3946i	0.861414	−	1.49201i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−27.2777			−1.33260			−0.666302	−	0.745682i	$-0.732124\pi$
$419$	−27.2777			−1.33260			−0.666302	+	0.745682i	$0.732124\pi$
$420$		0			0
$421$	−8.33694			−0.406317			−0.203159	−	0.979146i	$-0.565121\pi$
$421$	−8.33694			−0.406317			−0.203159	+	0.979146i	$0.565121\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.99820	−	15.5853i	0.436477	−	0.756000i
$426$		0			0
$427$	26.2960	−	0.679760i	1.27255	−	0.0328959i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−28.4948	+	16.4515i	−1.37255	+	0.792440i	−0.991248	−	0.132012i	$-0.957856\pi$
$431$	−28.4948	+	16.4515i	−1.37255	+	0.792440i	−0.381298	+	0.924452i	$0.624523\pi$
$432$		0			0
$433$			11.1943i			0.537962i	0.963145	+	0.268981i	$0.0866870\pi$
$433$			11.1943i			0.537962i	−0.963145	+	0.268981i	$0.913313\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.193226	+	0.334676i	0.00924323	+	0.0160097i
$438$		0			0
$439$	−25.8047	−	14.8983i	−1.23159	−	0.711059i	−0.264229	−	0.964460i	$-0.585117\pi$
$439$	−25.8047	−	14.8983i	−1.23159	−	0.711059i	−0.967361	+	0.253401i	$0.918451\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.7877	+	10.8471i	0.892631	+	0.515361i	0.874802	−	0.484480i	$-0.160991\pi$
$443$	18.7877	+	10.8471i	0.892631	+	0.515361i	0.0178286	+	0.999841i	$0.494325\pi$
$444$		0			0
$445$	−4.79586	−	8.30667i	−0.227346	−	0.393774i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			9.73564i			0.459453i	0.973255	+	0.229727i	$0.0737832\pi$
$449$			9.73564i			0.459453i	−0.973255	+	0.229727i	$0.926217\pi$
$450$		0			0
$451$	5.27743	−	3.04692i	0.248504	−	0.143474i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.60306	+	0.0931405i	−0.168914	+	0.00436650i
$456$		0			0
$457$	7.14197	−	12.3703i	0.334087	−	0.578656i	−0.649222	−	0.760599i	$-0.724905\pi$
$457$	7.14197	−	12.3703i	0.334087	−	0.578656i	0.983309	+	0.181943i	$0.0582386\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	20.3229			0.946531			0.473266	−	0.880920i	$-0.343075\pi$
$461$	20.3229			0.946531			0.473266	+	0.880920i	$0.343075\pi$
$462$		0			0
$463$	−19.5606			−0.909059			−0.454529	−	0.890732i	$-0.650193\pi$
$463$	−19.5606			−0.909059			−0.454529	+	0.890732i	$0.650193\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.49744	+	9.52185i	−0.254391	+	0.440619i	−0.964730	−	0.263241i	$-0.915208\pi$
$467$	−5.49744	+	9.52185i	−0.254391	+	0.440619i	0.710339	+	0.703860i	$0.248542\pi$
$468$		0			0
$469$	26.4774	+	14.3876i	1.22262	+	0.664357i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.07589	−	4.66262i	0.371330	−	0.214387i
$474$		0			0
$475$			27.7342i			1.27253i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.5145	−	18.2117i	−0.480420	−	0.832112i	0.519327	−	0.854575i	$-0.326183\pi$
$479$	−10.5145	−	18.2117i	−0.480420	−	0.832112i	−0.999748	+	0.0224629i	$0.992849\pi$
$480$		0			0
$481$	−2.35958	−	1.36230i	−0.107588	−	0.0621157i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.9729	−	10.9540i	−0.861515	−	0.497396i
$486$		0			0
$487$	−3.25049	−	5.63001i	−0.147294	−	0.255120i	0.782933	−	0.622106i	$-0.213723\pi$
$487$	−3.25049	−	5.63001i	−0.147294	−	0.255120i	−0.930226	+	0.366986i	$0.880390\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			30.0184i			1.35471i	0.735657	+	0.677355i	$0.236874\pi$
$491$			30.0184i			1.35471i	−0.735657	+	0.677355i	$0.763126\pi$
$492$		0			0
$493$	−13.1923	+	7.61656i	−0.594150	+	0.343033i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	17.5120	+	28.5985i	0.785522	+	1.28282i
$498$		0			0
$499$	15.6919	−	27.1792i	0.702466	−	1.21671i	−0.265132	−	0.964212i	$-0.585415\pi$
$499$	15.6919	−	27.1792i	0.702466	−	1.21671i	0.967598	−	0.252495i	$-0.0812512\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	14.1671			0.631682			0.315841	−	0.948812i	$-0.397713\pi$
$503$	14.1671			0.631682			0.315841	+	0.948812i	$0.397713\pi$
$504$		0			0
$505$	−46.9269			−2.08822
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.16409	−	2.01627i	0.0515974	−	0.0893694i	−0.839073	−	0.544019i	$-0.816902\pi$
$509$	1.16409	−	2.01627i	0.0515974	−	0.0893694i	0.890671	+	0.454649i	$0.150235\pi$
$510$		0			0
$511$	3.48923	−	6.42123i	0.154355	−	0.284058i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−34.3077	+	19.8076i	−1.51178	+	0.872825i
$516$		0			0
$517$			2.85815i			0.125701i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−19.3887	−	33.5823i	−0.849436	−	1.47127i	−0.881712	−	0.471788i	$-0.843609\pi$
$521$	−19.3887	−	33.5823i	−0.849436	−	1.47127i	0.0322757	−	0.999479i	$-0.489725\pi$
$522$		0			0
$523$	−14.4817	−	8.36099i	−0.633239	−	0.365600i	0.148767	−	0.988872i	$-0.452470\pi$
$523$	−14.4817	−	8.36099i	−0.633239	−	0.365600i	−0.782005	+	0.623272i	$0.785803\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.25068	+	5.34088i	0.402966	+	0.232652i
$528$		0			0
$529$	−11.4959	−	19.9115i	−0.499821	−	0.865715i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			3.16714i			0.137184i
$534$		0			0
$535$	23.8488	−	13.7691i	1.03107	−	0.595290i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.279389	+	5.40035i	0.0120341	+	0.232609i
$540$		0			0
$541$	7.71132	−	13.3564i	0.331535	−	0.574236i	−0.651278	−	0.758839i	$-0.725767\pi$
$541$	7.71132	−	13.3564i	0.331535	−	0.574236i	0.982813	+	0.184603i	$0.0591000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.24231			0.224556
$546$		0			0
$547$	−5.64942			−0.241552			−0.120776	−	0.992680i	$-0.538538\pi$
$547$	−5.64942			−0.241552			−0.120776	+	0.992680i	$0.538538\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	11.7379	−	20.3306i	0.500049	−	0.866111i
$552$		0			0
$553$	0.610066	+	23.5999i	0.0259426	+	1.00357i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.39858	−	2.53952i	0.186374	−	0.107603i	−0.403910	−	0.914799i	$-0.632349\pi$
$557$	4.39858	−	2.53952i	0.186374	−	0.107603i	0.590284	+	0.807196i	$0.299016\pi$
$558$		0			0
$559$			4.84658i			0.204989i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.79684	+	15.2366i	0.370743	+	0.642145i	0.989680	−	0.143295i	$-0.0457698\pi$
$563$	8.79684	+	15.2366i	0.370743	+	0.642145i	−0.618937	+	0.785440i	$0.712437\pi$
$564$		0			0
$565$	−26.0273	−	15.0269i	−1.09498	−	0.632186i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−27.2399	−	15.7270i	−1.14196	−	0.659309i	−0.195042	−	0.980795i	$-0.562484\pi$
$569$	−27.2399	−	15.7270i	−1.14196	−	0.659309i	−0.946914	+	0.321486i	$0.895818\pi$
$570$		0			0
$571$	0.00225988	+	0.00391423i	9.45732e−5	+	0.000163806i	0.866073	−	0.499918i	$-0.166637\pi$
$571$	0.00225988	+	0.00391423i	9.45732e−5	+	0.000163806i	−0.865978	+	0.500082i	$0.833303\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			0.591014i			0.0246470i
$576$		0			0
$577$	11.2450	−	6.49230i	0.468135	−	0.270278i	−0.247324	−	0.968933i	$-0.579551\pi$
$577$	11.2450	−	6.49230i	0.468135	−	0.270278i	0.715459	+	0.698655i	$0.246218\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	23.3390	−	14.2914i	0.968265	−	0.592907i
$582$		0			0
$583$	−5.25966	+	9.11000i	−0.217833	+	0.377298i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17.3258			0.715114			0.357557	−	0.933891i	$-0.383610\pi$
$587$	17.3258			0.715114			0.357557	+	0.933891i	$0.383610\pi$
$588$		0			0
$589$	−16.4616			−0.678290
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.2777	+	17.8014i	−0.422053	+	0.731018i	−0.996140	−	0.0877766i	$-0.972024\pi$
$593$	−10.2777	+	17.8014i	−0.422053	+	0.731018i	0.574087	+	0.818794i	$0.305357\pi$
$594$		0			0
$595$	21.1553	−	12.9542i	0.867281	−	0.531071i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	36.5063	−	21.0769i	1.49161	−	0.861180i	0.491654	−	0.870791i	$-0.336393\pi$
$599$	36.5063	−	21.0769i	1.49161	−	0.861180i	0.999954	+	0.00961097i	$0.00305931\pi$
$600$		0			0
$601$			20.8943i			0.852295i	0.904654	+	0.426147i	$0.140129\pi$
$601$			20.8943i			0.852295i	−0.904654	+	0.426147i	$0.859871\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17.6492	−	30.5694i	−0.717544	−	1.24282i
$606$		0			0
$607$	0.451389	+	0.260609i	0.0183213	+	0.0105778i	0.509133	−	0.860688i	$-0.329966\pi$
$607$	0.451389	+	0.260609i	0.0183213	+	0.0105778i	−0.490811	+	0.871266i	$0.663300\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.28645	−	0.742730i	−0.0520440	−	0.0300476i
$612$		0			0
$613$	22.7406	+	39.3879i	0.918485	+	1.59086i	0.801718	+	0.597703i	$0.203920\pi$
$613$	22.7406	+	39.3879i	0.918485	+	1.59086i	0.116767	+	0.993159i	$0.462747\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	39.7150i		−	1.59887i	−0.600755	−	0.799433i	$-0.705133\pi$
$617$		−	39.7150i		−	1.59887i	0.600755	−	0.799433i	$-0.294867\pi$
$618$		0			0
$619$	19.4589	−	11.2346i	0.782120	−	0.451557i	−0.0550610	−	0.998483i	$-0.517535\pi$
$619$	19.4589	−	11.2346i	0.782120	−	0.451557i	0.837181	+	0.546926i	$0.184202\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.193277	−	7.47674i	−0.00774346	−	0.299549i
$624$		0			0
$625$	7.57422	−	13.1189i	0.302969	−	0.524757i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18.7521			0.747696
$630$		0			0
$631$	−7.67982			−0.305729			−0.152864	−	0.988247i	$-0.548850\pi$
$631$	−7.67982			−0.305729			−0.152864	+	0.988247i	$0.548850\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.7305	−	27.2461i	0.624247	−	1.08123i
$636$		0			0
$637$	−2.50329	−	1.27760i	−0.0991838	−	0.0506205i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.32345	−	2.49615i	0.170766	−	0.0985919i	−0.412181	−	0.911102i	$-0.635233\pi$
$641$	4.32345	−	2.49615i	0.170766	−	0.0985919i	0.582947	+	0.812510i	$0.301899\pi$
$642$		0			0
$643$			6.24243i			0.246178i	0.992396	+	0.123089i	$0.0392800\pi$
$643$			6.24243i			0.246178i	−0.992396	+	0.123089i	$0.960720\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.5158	+	28.6061i	0.649301	+	1.12462i	0.983290	+	0.182046i	$0.0582719\pi$
$647$	16.5158	+	28.6061i	0.649301	+	1.12462i	−0.333989	+	0.942577i	$0.608395\pi$
$648$		0			0
$649$	−3.58145	−	2.06775i	−0.140584	−	0.0811664i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−36.1543	−	20.8737i	−1.41483	−	0.816850i	−0.418988	−	0.907992i	$-0.637615\pi$
$653$	−36.1543	−	20.8737i	−1.41483	−	0.816850i	−0.995838	+	0.0911420i	$0.970948\pi$
$654$		0			0
$655$	−11.1235	−	19.2665i	−0.434633	−	0.752806i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			6.11610i			0.238249i	0.992879	+	0.119125i	$0.0380088\pi$
$659$			6.11610i			0.238249i	−0.992879	+	0.119125i	$0.961991\pi$
$660$		0			0
$661$	−41.3405	+	23.8680i	−1.60796	+	0.928356i	−0.618135	+	0.786072i	$0.712111\pi$
$661$	−41.3405	+	23.8680i	−1.60796	+	0.928356i	−0.989826	+	0.142284i	$0.954555\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−18.2526	+	33.5902i	−0.707805	+	1.30257i
$666$		0			0
$667$	0.250133	−	0.433243i	0.00968518	−	0.0167752i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.68049			0.296502
$672$		0			0
$673$	−16.1051			−0.620806			−0.310403	−	0.950605i	$-0.600464\pi$
$673$	−16.1051			−0.620806			−0.310403	+	0.950605i	$0.600464\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.7342	−	34.1807i	0.758449	−	1.31367i	−0.185193	−	0.982702i	$-0.559291\pi$
$677$	19.7342	−	34.1807i	0.758449	−	1.31367i	0.943642	−	0.330969i	$-0.107376\pi$
$678$		0			0
$679$	−8.92095	−	14.5686i	−0.342355	−	0.559093i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−42.6561	+	24.6275i	−1.63219	+	0.942345i	−0.648772	+	0.760983i	$0.724717\pi$
$683$	−42.6561	+	24.6275i	−1.63219	+	0.942345i	−0.983416	+	0.181362i	$0.941949\pi$
$684$		0			0
$685$		−	38.3373i		−	1.46479i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.73359	−	4.73472i	−0.104142	−	0.180378i
$690$		0			0
$691$	−7.31536	−	4.22352i	−0.278289	−	0.160670i	0.354359	−	0.935109i	$-0.384699\pi$
$691$	−7.31536	−	4.22352i	−0.278289	−	0.160670i	−0.632649	+	0.774439i	$0.718032\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	39.6689	+	22.9029i	1.50473	+	0.868756i
$696$		0			0
$697$	−10.8989	−	18.8775i	−0.412826	−	0.715036i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1.40969i			0.0532434i	0.999646	+	0.0266217i	$0.00847495\pi$
$701$			1.40969i			0.0532434i	−0.999646	+	0.0266217i	$0.991525\pi$
$702$		0			0
$703$	−25.0271	+	14.4494i	−0.943916	+	0.544970i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−32.1516	−	17.4708i	−1.20918	−	0.657059i
$708$		0			0
$709$	23.2574	−	40.2830i	0.873450	−	1.51286i	0.0150454	−	0.999887i	$-0.495211\pi$
$709$	23.2574	−	40.2830i	0.873450	−	1.51286i	0.858405	−	0.512973i	$-0.171456\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.350796			−0.0131374
$714$		0			0
$715$	−1.05238			−0.0393568
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.99888	+	3.46216i	−0.0745456	+	0.129117i	−0.900889	−	0.434050i	$-0.857084\pi$
$719$	−1.99888	+	3.46216i	−0.0745456	+	0.129117i	0.826343	+	0.563167i	$0.190417\pi$
$720$		0			0
$721$	−30.8800	+	0.798259i	−1.15003	+	0.0297287i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	31.0923	−	17.9511i	1.15474	−	0.666688i
$726$		0			0
$727$			13.3117i			0.493702i	0.969053	+	0.246851i	$0.0793958\pi$
$727$			13.3117i			0.493702i	−0.969053	+	0.246851i	$0.920604\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−16.6783	−	28.8877i	−0.616869	−	1.06845i
$732$		0			0
$733$	−34.4428	−	19.8856i	−1.27218	−	0.734491i	−0.296778	−	0.954946i	$-0.595912\pi$
$733$	−34.4428	−	19.8856i	−1.27218	−	0.734491i	−0.975397	+	0.220456i	$0.929246\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.61978	+	4.39928i	0.280678	+	0.162050i
$738$		0			0
$739$	−20.6810	−	35.8205i	−0.760762	−	1.31768i	−0.942458	−	0.334323i	$-0.891492\pi$
$739$	−20.6810	−	35.8205i	−0.760762	−	1.31768i	0.181697	−	0.983355i	$-0.441841\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7.98724i		−	0.293024i	−0.989209	−	0.146512i	$-0.953195\pi$
$743$		−	7.98724i		−	0.293024i	0.989209	−	0.146512i	$-0.0468046\pi$
$744$		0			0
$745$	32.0563	−	18.5077i	1.17445	−	0.678071i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	21.4660	−	0.554904i	0.784351	−	0.0202758i
$750$		0			0
$751$	4.12965	−	7.15277i	0.150693	−	0.261008i	−0.780789	−	0.624794i	$-0.785183\pi$
$751$	4.12965	−	7.15277i	0.150693	−	0.261008i	0.931482	+	0.363786i	$0.118516\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.1698			−0.770447
$756$		0			0
$757$	−24.1678			−0.878395			−0.439198	−	0.898390i	$-0.644737\pi$
$757$	−24.1678			−0.878395			−0.439198	+	0.898390i	$0.644737\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.0652	+	22.6296i	−0.473613	+	0.820321i	−0.999544	−	0.0302061i	$-0.990384\pi$
$761$	−13.0652	+	22.6296i	−0.473613	+	0.820321i	0.525931	+	0.850527i	$0.323717\pi$
$762$		0			0
$763$	3.59172	+	1.95171i	0.130029	+	0.0706565i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.86138	−	1.07467i	0.0672105	−	0.0388040i
$768$		0			0
$769$			2.48254i			0.0895227i	0.998998	+	0.0447613i	$0.0142527\pi$
$769$			2.48254i			0.0895227i	−0.998998	+	0.0447613i	$0.985747\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	14.5225	+	25.1538i	0.522339	+	0.904718i	0.999662	+	0.0259902i	$0.00827386\pi$
$773$	14.5225	+	25.1538i	0.522339	+	0.904718i	−0.477323	+	0.878728i	$0.658393\pi$
$774$		0			0
$775$	−21.8025	−	12.5877i	−0.783169	−	0.452163i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	29.0921	+	16.7963i	1.04233	+	0.601790i
$780$		0			0
$781$	4.89569	+	8.47959i	0.175182	+	0.303424i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	53.8486i		−	1.92194i
$786$		0			0
$787$	−1.81957	+	1.05053i	−0.0648606	+	0.0374473i	−0.532080	−	0.846694i	$-0.678589\pi$
$787$	−1.81957	+	1.05053i	−0.0648606	+	0.0374473i	0.467219	+	0.884142i	$0.345256\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.2379	−	19.9855i	−0.435130	−	0.710603i
$792$		0			0
$793$	−1.99588	+	3.45697i	−0.0708759	+	0.122761i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−30.4617			−1.07901			−0.539505	−	0.841982i	$-0.681389\pi$
$797$	−30.4617			−1.07901			−0.539505	+	0.841982i	$0.681389\pi$
$798$		0			0
$799$	10.2237			0.361688
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.06690	−	1.84793i	0.0376501	−	0.0652119i
$804$		0			0
$805$	−0.388961	+	0.715805i	−0.0137091	+	0.0252288i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22.2427	−	12.8419i	0.782013	−	0.451496i	−0.0551301	−	0.998479i	$-0.517557\pi$
$809$	22.2427	−	12.8419i	0.782013	−	0.451496i	0.837143	+	0.546984i	$0.184224\pi$
$810$		0			0
$811$			21.7048i			0.762159i	0.924542	+	0.381080i	$0.124448\pi$
$811$			21.7048i			0.762159i	−0.924542	+	0.381080i	$0.875552\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.2363	+	41.9784i	0.848959	+	1.47044i
$816$		0			0
$817$	44.5187	+	25.7029i	1.55751	+	0.899230i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.19178	+	2.99747i	0.181194	+	0.104613i	0.587854	−	0.808967i	$-0.299973\pi$
$821$	5.19178	+	2.99747i	0.181194	+	0.104613i	−0.406659	+	0.913580i	$0.633306\pi$
$822$		0			0
$823$	−17.7234	−	30.6978i	−0.617798	−	1.07006i	−0.989887	−	0.141860i	$-0.954692\pi$
$823$	−17.7234	−	30.6978i	−0.617798	−	1.07006i	0.372089	−	0.928197i	$-0.378642\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			44.0703i			1.53247i	0.642558	+	0.766237i	$0.277873\pi$
$827$			44.0703i			1.53247i	−0.642558	+	0.766237i	$0.722127\pi$
$828$		0			0
$829$	30.8006	−	17.7827i	1.06975	−	0.617619i	0.141635	−	0.989919i	$-0.454764\pi$
$829$	30.8006	−	17.7827i	1.06975	−	0.617619i	0.928113	+	0.372300i	$0.121431\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	19.3172	−	0.999381i	0.669301	−	0.0346265i
$834$		0			0
$835$	5.19576	−	8.99933i	0.179807	−	0.311435i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21.3775			−0.738032			−0.369016	−	0.929423i	$-0.620305\pi$
$839$	−21.3775			−0.738032			−0.369016	+	0.929423i	$0.620305\pi$
$840$		0			0
$841$	−1.38958			−0.0479166
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−21.7812	+	37.7262i	−0.749297	+	1.29782i
$846$		0			0
$847$	−0.711277	−	27.5152i	−0.0244398	−	0.945432i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.533326	+	0.307916i	−0.0182822	+	0.0105552i
$852$		0			0
$853$		−	43.8662i		−	1.50195i	−0.660330	−	0.750976i	$-0.729583\pi$
$853$		−	43.8662i		−	1.50195i	0.660330	−	0.750976i	$-0.270417\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.32987	+	2.30340i	0.0454274	+	0.0786825i	0.887845	−	0.460143i	$-0.152202\pi$
$857$	1.32987	+	2.30340i	0.0454274	+	0.0786825i	−0.842418	+	0.538825i	$0.818868\pi$
$858$		0			0
$859$	−45.6614	−	26.3626i	−1.55795	−	0.899481i	−0.997453	−	0.0713218i	$-0.977278\pi$
$859$	−45.6614	−	26.3626i	−1.55795	−	0.899481i	−0.560493	−	0.828159i	$-0.689388\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7.94811	+	4.58884i	0.270557	+	0.156206i	0.629141	−	0.777292i	$-0.283407\pi$
$863$	7.94811	+	4.58884i	0.270557	+	0.156206i	−0.358584	+	0.933498i	$0.616740\pi$
$864$		0			0
$865$	−41.6273	−	72.1006i	−1.41537	−	2.45149i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			6.89303i			0.233830i
$870$		0			0
$871$	−3.96021	+	2.28643i	−0.134187	+	0.0774727i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−11.5808	+	7.09137i	−0.391502	+	0.239732i
$876$		0			0
$877$	12.3455	−	21.3831i	0.416879	−	0.722056i	−0.578745	−	0.815509i	$-0.696457\pi$
$877$	12.3455	−	21.3831i	0.416879	−	0.722056i	0.995624	+	0.0934532i	$0.0297905\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−36.9178			−1.24379			−0.621897	−	0.783099i	$-0.713638\pi$
$881$	−36.9178			−1.24379			−0.621897	+	0.783099i	$0.713638\pi$
$882$		0			0
$883$	−0.325420			−0.0109512			−0.00547562	−	0.999985i	$-0.501743\pi$
$883$	−0.325420			−0.0109512			−0.00547562	+	0.999985i	$0.501743\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.9652	+	29.3845i	−0.569635	+	0.986636i	0.426967	+	0.904267i	$0.359582\pi$
$887$	−16.9652	+	29.3845i	−0.569635	+	0.986636i	−0.996602	+	0.0823689i	$0.973751\pi$
$888$		0			0
$889$	20.9213	−	12.8109i	0.701678	−	0.429665i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.6448	+	7.87784i	−0.456606	+	0.263622i
$894$		0			0
$895$			18.3968i			0.614936i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10.6549	+	18.4548i	0.355360	+	0.615502i
$900$		0			0
$901$	32.5867	+	18.8140i	1.08562	+	0.626784i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−75.3430	−	43.4993i	−2.50449	−	1.44597i
$906$		0			0
$907$	−13.4226	−	23.2487i	−0.445691	−	0.771960i	0.552409	−	0.833573i	$-0.313709\pi$
$907$	−13.4226	−	23.2487i	−0.445691	−	0.771960i	−0.998100	+	0.0616134i	$0.980375\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			26.4947i			0.877808i	0.898534	+	0.438904i	$0.144633\pi$
$911$			26.4947i			0.877808i	−0.898534	+	0.438904i	$0.855367\pi$
$912$		0			0
$913$	6.92011	−	3.99533i	0.229022	−	0.132226i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.448286	−	17.3416i	−0.0148037	−	0.572670i
$918$		0			0
$919$	−14.0127	+	24.2707i	−0.462235	+	0.800615i	−0.999072	−	0.0430709i	$-0.986286\pi$
$919$	−14.0127	+	24.2707i	−0.462235	+	0.800615i	0.536837	+	0.843686i	$0.319619\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.08885			−0.167502
$924$		0			0
$925$	−44.1960			−1.45316
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.54656	+	9.60692i	−0.181977	+	0.315193i	−0.942554	−	0.334055i	$-0.891583\pi$
$929$	−5.54656	+	9.60692i	−0.181977	+	0.315193i	0.760577	+	0.649248i	$0.224916\pi$
$930$		0			0
$931$	−25.0112	+	16.2186i	−0.819709	+	0.531544i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.27263	−	3.62150i	0.205137	−	0.118436i
$936$		0			0
$937$		−	2.93878i		−	0.0960058i	−0.998847	−	0.0480029i	$-0.984714\pi$
$937$		−	2.93878i		−	0.0960058i	0.998847	−	0.0480029i	$-0.0152857\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.94394	+	15.4914i	0.291564	+	0.505004i	0.974180	−	0.225774i	$-0.0724910\pi$
$941$	8.94394	+	15.4914i	0.291564	+	0.505004i	−0.682616	+	0.730778i	$0.739158\pi$
$942$		0			0
$943$	0.619949	+	0.357928i	0.0201883	+	0.0116557i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.78474	+	5.07187i	0.285466	+	0.164814i	0.635895	−	0.771775i	$-0.280631\pi$
$947$	8.78474	+	5.07187i	0.285466	+	0.164814i	−0.350430	+	0.936589i	$0.613964\pi$
$948$		0			0
$949$	0.554498	+	0.960418i	0.0179998	+	0.0311765i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	3.84180i		−	0.124448i	−0.998062	−	0.0622241i	$-0.980181\pi$
$953$		−	3.84180i		−	0.124448i	0.998062	−	0.0622241i	$-0.0198193\pi$
$954$		0			0
$955$	−22.5746	+	13.0334i	−0.730496	+	0.421752i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.2729	−	26.2665i	0.460897	−	0.848189i
$960$		0			0
$961$	−8.02858	+	13.9059i	−0.258987	+	0.448578i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	62.4031			2.00883
$966$		0			0
$967$	−11.2131			−0.360590			−0.180295	−	0.983613i	$-0.557705\pi$
$967$	−11.2131			−0.360590			−0.180295	+	0.983613i	$0.557705\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.4029	+	42.2670i	−0.783126	+	1.35641i	0.146987	+	0.989138i	$0.453043\pi$
$971$	−24.4029	+	42.2670i	−0.783126	+	1.35641i	−0.930112	+	0.367275i	$0.880291\pi$
$972$		0			0
$973$	18.6521	+	30.4604i	0.597960	+	0.976516i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	40.7751	−	23.5415i	1.30451	−	0.753160i	0.323337	−	0.946284i	$-0.395195\pi$
$977$	40.7751	−	23.5415i	1.30451	−	0.753160i	0.981174	+	0.193124i	$0.0618619\pi$
$978$		0			0
$979$		−	2.18380i		−	0.0697945i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.9560	−	32.8328i	−0.604604	−	1.04720i	−0.992114	−	0.125339i	$-0.959998\pi$
$983$	−18.9560	−	32.8328i	−0.604604	−	1.04720i	0.387510	−	0.921865i	$-0.373335\pi$
$984$		0			0
$985$	2.84133	+	1.64044i	0.0905322	+	0.0522688i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.948690	+	0.547726i	0.0301666	+	0.0174167i
$990$		0			0
$991$	−3.40637	−	5.90001i	−0.108207	−	0.187420i	0.806837	−	0.590774i	$-0.201178\pi$
$991$	−3.40637	−	5.90001i	−0.108207	−	0.187420i	−0.915044	+	0.403354i	$0.867844\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			62.0742i			1.96788i
$996$		0			0
$997$	9.33662	−	5.39050i	0.295694	−	0.170719i	−0.344813	−	0.938671i	$-0.612058\pi$
$997$	9.33662	−	5.39050i	0.295694	−	0.170719i	0.640507	+	0.767953i	$0.278724\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.bl.c.593.1	✓	16
3.2	odd	2	inner	1512.2.bl.c.593.8	yes	16
7.3	odd	6	inner	1512.2.bl.c.1025.8	yes	16
21.17	even	6	inner	1512.2.bl.c.1025.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.bl.c.593.1	✓	16	1.1	even	1	trivial
1512.2.bl.c.593.8	yes	16	3.2	odd	2	inner
1512.2.bl.c.1025.1	yes	16	21.17	even	6	inner
1512.2.bl.c.1025.8	yes	16	7.3	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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.69652 + 2.93845i) q^{5} +(-2.25634 + 1.38164i) q^{7} +(-0.669013 + 0.386255i) q^{11} -0.401495i q^{13} +(1.38164 + 2.39308i) q^{17} +(-3.68797 - 2.12925i) q^{19} +(-0.0785903 - 0.0453741i) q^{23} +(-3.25634 - 5.64014i) q^{25} +5.51267i q^{29} +(3.34770 - 1.93280i) q^{31} +(-0.231984 - 8.97412i) q^{35} +(3.39308 - 5.87698i) q^{37} -7.88837 q^{41} -12.0713 q^{43} +(1.84991 - 3.20414i) q^{47} +(3.18212 - 6.23491i) q^{49} +(11.7927 - 6.80854i) q^{53} -2.62115i q^{55} +(2.67667 + 4.63612i) q^{59} +(-8.61025 - 4.97113i) q^{61} +(1.17977 + 0.681143i) q^{65} +(-5.69479 - 9.86367i) q^{67} -12.6748i q^{71} +(-2.39211 + 1.38108i) q^{73} +(0.975853 - 1.79586i) q^{77} +(4.46145 - 7.72746i) q^{79} -10.3438 q^{83} -9.37593 q^{85} +(-1.41344 + 2.44816i) q^{89} +(0.554723 + 0.905908i) q^{91} +(12.5134 - 7.22461i) q^{95} +6.45676i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{7} - 18 q^{19} - 12 q^{25} + 24 q^{31} + 16 q^{37} - 52 q^{43} + 44 q^{49} + 6 q^{61} - 4 q^{67} - 12 q^{73} + 34 q^{79} - 68 q^{85} + 102 q^{91}+O(q^{100}) \) 16 * q + 4 * q^7 - 18 * q^19 - 12 * q^25 + 24 * q^31 + 16 * q^37 - 52 * q^43 + 44 * q^49 + 6 * q^61 - 4 * q^67 - 12 * q^73 + 34 * q^79 - 68 * q^85 + 102 * q^91

Introduction

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