LMFDB - Embedded newform 855.2.k.b.406.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(406,855)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("855.406");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.k (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.82720937282$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			406.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	855.406
Dual form			855.2.k.b.676.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{4} +(-0.500000 - 0.866025i) q^{5} -4.00000 q^{7} +O(q^{10})$	\(q+(1.00000 - 1.73205i) q^{4} +(-0.500000 - 0.866025i) q^{5} -4.00000 q^{7} -3.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(3.00000 + 5.19615i) q^{17} +(-3.50000 + 2.59808i) q^{19} -2.00000 q^{20} +(-0.500000 + 0.866025i) q^{25} +(-4.00000 + 6.92820i) q^{28} +(-1.50000 + 2.59808i) q^{29} -7.00000 q^{31} +(2.00000 + 3.46410i) q^{35} +8.00000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +(-3.00000 + 5.19615i) q^{44} +(3.00000 - 5.19615i) q^{47} +9.00000 q^{49} +(2.00000 + 3.46410i) q^{52} +(-3.00000 + 5.19615i) q^{53} +(1.50000 + 2.59808i) q^{55} +(-7.50000 - 12.9904i) q^{59} +(-2.50000 + 4.33013i) q^{61} -8.00000 q^{64} +2.00000 q^{65} +(-1.00000 + 1.73205i) q^{67} +12.0000 q^{68} +(-1.50000 - 2.59808i) q^{71} +(-4.00000 - 6.92820i) q^{73} +(1.00000 + 8.66025i) q^{76} +12.0000 q^{77} +(-2.50000 - 4.33013i) q^{79} +(-2.00000 + 3.46410i) q^{80} -12.0000 q^{83} +(3.00000 - 5.19615i) q^{85} +(-7.50000 + 12.9904i) q^{89} +(4.00000 - 6.92820i) q^{91} +(4.00000 + 1.73205i) q^{95} +(-4.00000 - 6.92820i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - q^{5} - 8 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 - q^5 - 8 * q^7	$ 2 q + 2 q^{4} - q^{5} - 8 q^{7} - 6 q^{11} - 2 q^{13} - 4 q^{16} + 6 q^{17} - 7 q^{19} - 4 q^{20} - q^{25} - 8 q^{28} - 3 q^{29} - 14 q^{31} + 4 q^{35} + 16 q^{37} - 6 q^{41} + 4 q^{43} - 6 q^{44} + 6 q^{47} + 18 q^{49} + 4 q^{52} - 6 q^{53} + 3 q^{55} - 15 q^{59} - 5 q^{61} - 16 q^{64} + 4 q^{65} - 2 q^{67} + 24 q^{68} - 3 q^{71} - 8 q^{73} + 2 q^{76} + 24 q^{77} - 5 q^{79} - 4 q^{80} - 24 q^{83} + 6 q^{85} - 15 q^{89} + 8 q^{91} + 8 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - q^5 - 8 * q^7 - 6 * q^11 - 2 * q^13 - 4 * q^16 + 6 * q^17 - 7 * q^19 - 4 * q^20 - q^25 - 8 * q^28 - 3 * q^29 - 14 * q^31 + 4 * q^35 + 16 * q^37 - 6 * q^41 + 4 * q^43 - 6 * q^44 + 6 * q^47 + 18 * q^49 + 4 * q^52 - 6 * q^53 + 3 * q^55 - 15 * q^59 - 5 * q^61 - 16 * q^64 + 4 * q^65 - 2 * q^67 + 24 * q^68 - 3 * q^71 - 8 * q^73 + 2 * q^76 + 24 * q^77 - 5 * q^79 - 4 * q^80 - 24 * q^83 + 6 * q^85 - 15 * q^89 + 8 * q^91 + 8 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$	−4.00000			−1.51186			−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000			−1.51186			−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	$0.0927008\pi$
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	−0.230285	+	0.973123i	$0.573966\pi$
$18$		0			0
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$20$	−2.00000			−0.447214
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$	−4.00000	+	6.92820i	−0.755929	+	1.30931i
$29$	−1.50000	+	2.59808i	−0.278543	+	0.482451i	−0.971023	−	0.238987i	$-0.923185\pi$
$29$	−1.50000	+	2.59808i	−0.278543	+	0.482451i	0.692480	+	0.721437i	$0.256518\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000	+	3.46410i	0.338062	+	0.585540i
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	0.530831	−	0.847477i	$-0.321880\pi$
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	0.977261	−	0.212041i	$-0.0680112\pi$
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	−0.672264	+	0.740312i	$0.734678\pi$
$44$	−3.00000	+	5.19615i	−0.452267	+	0.783349i
$45$		0			0
$46$		0			0
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	$-0.689164\pi$
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	0.997503	+	0.0706177i	$0.0224970\pi$
$48$		0			0
$49$	9.00000			1.28571
$50$		0			0
$51$		0			0
$52$	2.00000	+	3.46410i	0.277350	+	0.480384i
$53$	−3.00000	+	5.19615i	−0.412082	+	0.713746i	−0.995117	−	0.0987002i	$-0.968532\pi$
$53$	−3.00000	+	5.19615i	−0.412082	+	0.713746i	0.583036	+	0.812447i	$0.301865\pi$
$54$		0			0
$55$	1.50000	+	2.59808i	0.202260	+	0.350325i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.50000	−	12.9904i	−0.976417	−	1.69120i	−0.675178	−	0.737655i	$-0.735933\pi$
$59$	−7.50000	−	12.9904i	−0.976417	−	1.69120i	−0.301239	−	0.953549i	$-0.597400\pi$
$60$		0			0
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	$-0.937047\pi$
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	0.660415	+	0.750901i	$0.270381\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	2.00000			0.248069
$66$		0			0
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$	12.0000			1.45521
$69$		0			0
$70$		0			0
$71$	−1.50000	−	2.59808i	−0.178017	−	0.308335i	0.763184	−	0.646181i	$-0.223635\pi$
$71$	−1.50000	−	2.59808i	−0.178017	−	0.308335i	−0.941201	+	0.337846i	$0.890302\pi$
$72$		0			0
$73$	−4.00000	−	6.92820i	−0.468165	−	0.810885i	0.531174	−	0.847263i	$-0.321751\pi$
$73$	−4.00000	−	6.92820i	−0.468165	−	0.810885i	−0.999338	+	0.0363782i	$0.988418\pi$
$74$		0			0
$75$		0			0
$76$	1.00000	+	8.66025i	0.114708	+	0.993399i
$77$	12.0000			1.36753
$78$		0			0
$79$	−2.50000	−	4.33013i	−0.281272	−	0.487177i	0.690426	−	0.723403i	$-0.257423\pi$
$79$	−2.50000	−	4.33013i	−0.281272	−	0.487177i	−0.971698	+	0.236225i	$0.924090\pi$
$80$	−2.00000	+	3.46410i	−0.223607	+	0.387298i
$81$		0			0
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	3.00000	−	5.19615i	0.325396	−	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.50000	+	12.9904i	−0.794998	+	1.37698i	0.127842	+	0.991795i	$0.459195\pi$
$89$	−7.50000	+	12.9904i	−0.794998	+	1.37698i	−0.922840	+	0.385183i	$0.874138\pi$
$90$		0			0
$91$	4.00000	−	6.92820i	0.419314	−	0.726273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	+	1.73205i	0.410391	+	0.177705i
$96$		0			0
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	0.588315	−	0.808632i	$-0.299792\pi$
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	−0.994453	+	0.105180i	$0.966458\pi$
$98$		0			0
$99$		0			0
$100$	1.00000	+	1.73205i	0.100000	+	0.173205i
$101$	7.50000	−	12.9904i	0.746278	−	1.29259i	−0.203317	−	0.979113i	$-0.565172\pi$
$101$	7.50000	−	12.9904i	0.746278	−	1.29259i	0.949595	−	0.313478i	$-0.101494\pi$
$102$		0			0
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	−0.999512	−	0.0312328i	$-0.990057\pi$
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	0.472708	−	0.881219i	$-0.343277\pi$
$110$		0			0
$111$		0			0
$112$	8.00000	+	13.8564i	0.755929	+	1.30931i
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$	3.00000	+	5.19615i	0.278543	+	0.482451i
$117$		0			0
$118$		0			0
$119$	−12.0000	−	20.7846i	−1.10004	−	1.90532i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$	−7.00000	+	12.1244i	−0.628619	+	1.08880i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−1.00000	+	1.73205i	−0.0887357	+	0.153695i	−0.906977	−	0.421180i	$-0.861616\pi$
$127$	−1.00000	+	1.73205i	−0.0887357	+	0.153695i	0.818241	+	0.574875i	$0.194949\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$	14.0000	−	10.3923i	1.21395	−	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	0.487278	+	0.873247i	$0.337990\pi$
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	−0.999893	+	0.0146279i	$0.995344\pi$
$138$		0			0
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	0.975417	−	0.220366i	$-0.0707252\pi$
$140$	8.00000			0.676123
$141$		0			0
$142$		0			0
$143$	3.00000	−	5.19615i	0.250873	−	0.434524i
$144$		0			0
$145$	3.00000			0.249136
$146$		0			0
$147$		0			0
$148$	8.00000	−	13.8564i	0.657596	−	1.13899i
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	0.920904	−	0.389789i	$-0.127452\pi$
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	−0.798019	+	0.602632i	$0.794119\pi$
$150$		0			0
$151$	17.0000			1.38344			0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000			1.38344			0.691720	+	0.722166i	$0.256853\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.50000	+	6.06218i	0.281127	+	0.486926i
$156$		0			0
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	0.823359	−	0.567521i	$-0.192098\pi$
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	−0.903167	+	0.429289i	$0.858764\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.0000			−0.783260			−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000			−0.783260			−0.391630	+	0.920123i	$0.628089\pi$
$164$	−12.0000			−0.937043
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$		0			0
$171$		0			0
$172$	8.00000			0.609994
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	2.00000	−	3.46410i	0.151186	−	0.261861i
$176$	6.00000	+	10.3923i	0.452267	+	0.783349i
$177$		0			0
$178$		0			0
$179$	9.00000			0.672692			0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000			0.672692			0.336346	+	0.941739i	$0.390809\pi$
$180$		0			0
$181$	−1.00000	+	1.73205i	−0.0743294	+	0.128742i	−0.900794	−	0.434246i	$-0.857015\pi$
$181$	−1.00000	+	1.73205i	−0.0743294	+	0.128742i	0.826465	+	0.562988i	$0.190348\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00000	−	6.92820i	−0.294086	−	0.509372i
$186$		0			0
$187$	−9.00000	−	15.5885i	−0.658145	−	1.13994i
$188$	−6.00000	−	10.3923i	−0.437595	−	0.757937i
$189$		0			0
$190$		0			0
$191$	15.0000			1.08536			0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000			1.08536			0.542681	+	0.839939i	$0.317409\pi$
$192$		0			0
$193$	8.00000	+	13.8564i	0.575853	+	0.997406i	0.995948	+	0.0899262i	$0.0286631\pi$
$193$	8.00000	+	13.8564i	0.575853	+	0.997406i	−0.420096	+	0.907480i	$0.638004\pi$
$194$		0			0
$195$		0			0
$196$	9.00000	−	15.5885i	0.642857	−	1.11346i
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	9.50000	−	16.4545i	0.673437	−	1.16643i	−0.303486	−	0.952836i	$-0.598151\pi$
$199$	9.50000	−	16.4545i	0.673437	−	1.16643i	0.976923	−	0.213591i	$-0.0685161\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.00000	−	10.3923i	0.421117	−	0.729397i
$204$		0			0
$205$	−3.00000	+	5.19615i	−0.209529	+	0.362915i
$206$		0			0
$207$		0			0
$208$	8.00000			0.554700
$209$	10.5000	−	7.79423i	0.726300	−	0.539138i
$210$		0			0
$211$	3.50000	+	6.06218i	0.240950	+	0.417338i	0.960985	−	0.276600i	$-0.0892077\pi$
$211$	3.50000	+	6.06218i	0.240950	+	0.417338i	−0.720035	+	0.693938i	$0.755874\pi$
$212$	6.00000	+	10.3923i	0.412082	+	0.713746i
$213$		0			0
$214$		0			0
$215$	2.00000	−	3.46410i	0.136399	−	0.236250i
$216$		0			0
$217$	28.0000			1.90076
$218$		0			0
$219$		0			0
$220$	6.00000			0.404520
$221$	−12.0000			−0.807207
$222$		0			0
$223$	2.00000	+	3.46410i	0.133930	+	0.231973i	0.925188	−	0.379509i	$-0.123907\pi$
$223$	2.00000	+	3.46410i	0.133930	+	0.231973i	−0.791258	+	0.611482i	$0.790574\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	24.0000			1.59294			0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000			1.59294			0.796468	+	0.604681i	$0.206699\pi$
$228$		0			0
$229$	−7.00000			−0.462573			−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000			−0.462573			−0.231287	+	0.972886i	$0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	0.947403	−	0.320043i	$-0.103697\pi$
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	−0.750867	+	0.660454i	$0.770364\pi$
$234$		0			0
$235$	−6.00000			−0.391397
$236$	−30.0000			−1.95283
$237$		0			0
$238$		0			0
$239$	3.00000			0.194054			0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000			0.194054			0.0970269	+	0.995282i	$0.469067\pi$
$240$		0			0
$241$	−2.50000	+	4.33013i	−0.161039	+	0.278928i	−0.935242	−	0.354010i	$-0.884818\pi$
$241$	−2.50000	+	4.33013i	−0.161039	+	0.278928i	0.774202	+	0.632938i	$0.218151\pi$
$242$		0			0
$243$		0			0
$244$	5.00000	+	8.66025i	0.320092	+	0.554416i
$245$	−4.50000	−	7.79423i	−0.287494	−	0.497955i
$246$		0			0
$247$	−1.00000	−	8.66025i	−0.0636285	−	0.551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.50000	+	12.9904i	−0.473396	+	0.819946i	−0.999536	−	0.0304521i	$-0.990305\pi$
$251$	−7.50000	+	12.9904i	−0.473396	+	0.819946i	0.526140	+	0.850398i	$0.323639\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	0.435970	+	0.899961i	$0.356405\pi$
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	−0.997374	+	0.0724199i	$0.976928\pi$
$258$		0			0
$259$	−32.0000			−1.98838
$260$	2.00000	−	3.46410i	0.124035	−	0.214834i
$261$		0			0
$262$		0			0
$263$	−9.00000	−	15.5885i	−0.554964	−	0.961225i	−0.997906	−	0.0646755i	$-0.979399\pi$
$263$	−9.00000	−	15.5885i	−0.554964	−	0.961225i	0.442943	−	0.896550i	$-0.353935\pi$
$264$		0			0
$265$	6.00000			0.368577
$266$		0			0
$267$		0			0
$268$	2.00000	+	3.46410i	0.122169	+	0.211604i
$269$	−10.5000	−	18.1865i	−0.640196	−	1.10885i	−0.985389	−	0.170321i	$-0.945520\pi$
$269$	−10.5000	−	18.1865i	−0.640196	−	1.10885i	0.345192	−	0.938532i	$-0.387814\pi$
$270$		0			0
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	0.649211	−	0.760609i	$-0.275099\pi$
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	−0.983312	+	0.181928i	$0.941766\pi$
$272$	12.0000	−	20.7846i	0.727607	−	1.26025i
$273$		0			0
$274$		0			0
$275$	1.50000	−	2.59808i	0.0904534	−	0.156670i
$276$		0			0
$277$	8.00000			0.480673			0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000			0.480673			0.240337	+	0.970690i	$0.422742\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	−0.762561	−	0.646916i	$-0.776058\pi$
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	0.941526	+	0.336939i	$0.109392\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$	−6.00000			−0.356034
$285$		0			0
$286$		0			0
$287$	12.0000	+	20.7846i	0.708338	+	1.22688i
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$		0			0
$292$	−16.0000			−0.936329
$293$	−24.0000			−1.40209			−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000			−1.40209			−0.701047	+	0.713115i	$0.747284\pi$
$294$		0			0
$295$	−7.50000	+	12.9904i	−0.436667	+	0.756329i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−8.00000	−	13.8564i	−0.461112	−	0.798670i
$302$		0			0
$303$		0			0
$304$	16.0000	+	6.92820i	0.917663	+	0.397360i
$305$	5.00000			0.286299
$306$		0			0
$307$	17.0000	+	29.4449i	0.970241	+	1.68051i	0.694820	+	0.719183i	$0.255484\pi$
$307$	17.0000	+	29.4449i	0.970241	+	1.68051i	0.275421	+	0.961324i	$0.411183\pi$
$308$	12.0000	−	20.7846i	0.683763	−	1.18431i
$309$		0			0
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	−0.689412	−	0.724370i	$-0.742131\pi$
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	0.972028	+	0.234863i	$0.0754642\pi$
$314$		0			0
$315$		0			0
$316$	−10.0000			−0.562544
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	−0.302777	−	0.953062i	$-0.597914\pi$
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	0.976764	−	0.214318i	$-0.0687530\pi$
$318$		0			0
$319$	4.50000	−	7.79423i	0.251952	−	0.436393i
$320$	4.00000	+	6.92820i	0.223607	+	0.387298i
$321$		0			0
$322$		0			0
$323$	−24.0000	−	10.3923i	−1.33540	−	0.578243i
$324$		0			0
$325$	−1.00000	−	1.73205i	−0.0554700	−	0.0960769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.0000	+	20.7846i	−0.661581	+	1.14589i
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$	−12.0000	+	20.7846i	−0.658586	+	1.14070i
$333$		0			0
$334$		0			0
$335$	2.00000			0.109272
$336$		0			0
$337$	8.00000	+	13.8564i	0.435788	+	0.754807i	0.997360	−	0.0726214i	$-0.0231365\pi$
$337$	8.00000	+	13.8564i	0.435788	+	0.754807i	−0.561572	+	0.827428i	$0.689803\pi$
$338$		0			0
$339$		0			0
$340$	−6.00000	−	10.3923i	−0.325396	−	0.563602i
$341$	21.0000			1.13721
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.0000			−0.638696			−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000			−0.638696			−0.319348	+	0.947638i	$0.603464\pi$
$354$		0			0
$355$	−1.50000	+	2.59808i	−0.0796117	+	0.137892i
$356$	15.0000	+	25.9808i	0.794998	+	1.37698i
$357$		0			0
$358$		0			0
$359$	12.0000	+	20.7846i	0.633336	+	1.09697i	0.986865	+	0.161546i	$0.0516481\pi$
$359$	12.0000	+	20.7846i	0.633336	+	1.09697i	−0.353529	+	0.935423i	$0.615019\pi$
$360$		0			0
$361$	5.50000	−	18.1865i	0.289474	−	0.957186i
$362$		0			0
$363$		0			0
$364$	−8.00000	−	13.8564i	−0.419314	−	0.726273i
$365$	−4.00000	+	6.92820i	−0.209370	+	0.362639i
$366$		0			0
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	0.913493	+	0.406855i	$0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.0000	−	20.7846i	0.623009	−	1.07908i
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.00000	−	5.19615i	−0.154508	−	0.267615i
$378$		0			0
$379$	−37.0000			−1.90056			−0.950281	−	0.311393i	$-0.899204\pi$
$379$	−37.0000			−1.90056			−0.950281	+	0.311393i	$0.899204\pi$
$380$	7.00000	−	5.19615i	0.359092	−	0.266557i
$381$		0			0
$382$		0			0
$383$	15.0000	+	25.9808i	0.766464	+	1.32755i	0.939469	+	0.342634i	$0.111319\pi$
$383$	15.0000	+	25.9808i	0.766464	+	1.32755i	−0.173005	+	0.984921i	$0.555348\pi$
$384$		0			0
$385$	−6.00000	−	10.3923i	−0.305788	−	0.529641i
$386$		0			0
$387$		0			0
$388$	−16.0000			−0.812277
$389$	7.50000	−	12.9904i	0.380265	−	0.658638i	−0.610835	−	0.791758i	$-0.709166\pi$
$389$	7.50000	−	12.9904i	0.380265	−	0.658638i	0.991100	+	0.133120i	$0.0424994\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.50000	+	4.33013i	−0.125789	+	0.217872i
$396$		0			0
$397$	−4.00000	−	6.92820i	−0.200754	−	0.347717i	0.748017	−	0.663679i	$-0.231006\pi$
$397$	−4.00000	−	6.92820i	−0.200754	−	0.347717i	−0.948772	+	0.315963i	$0.897673\pi$
$398$		0			0
$399$		0			0
$400$	4.00000			0.200000
$401$	−16.5000	−	28.5788i	−0.823971	−	1.42716i	−0.902703	−	0.430263i	$-0.858421\pi$
$401$	−16.5000	−	28.5788i	−0.823971	−	1.42716i	0.0787327	−	0.996896i	$-0.474913\pi$
$402$		0			0
$403$	7.00000	−	12.1244i	0.348695	−	0.603957i
$404$	−15.0000	−	25.9808i	−0.746278	−	1.29259i
$405$		0			0
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$	−11.5000	+	19.9186i	−0.568638	+	0.984911i	0.428063	+	0.903749i	$0.359196\pi$
$409$	−11.5000	+	19.9186i	−0.568638	+	0.984911i	−0.996701	+	0.0811615i	$0.974137\pi$
$410$		0			0
$411$		0			0
$412$	−16.0000	+	27.7128i	−0.788263	+	1.36531i
$413$	30.0000	+	51.9615i	1.47620	+	2.55686i
$414$		0			0
$415$	6.00000	+	10.3923i	0.294528	+	0.510138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.00000			0.146560			0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000			0.146560			0.0732798	+	0.997311i	$0.476653\pi$
$420$		0			0
$421$	9.50000	+	16.4545i	0.463002	+	0.801942i	0.999109	−	0.0422075i	$-0.0134391\pi$
$421$	9.50000	+	16.4545i	0.463002	+	0.801942i	−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.00000			−0.291043
$426$		0			0
$427$	10.0000	−	17.3205i	0.483934	−	0.838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.50000	+	12.9904i	−0.361262	+	0.625725i	−0.988169	−	0.153370i	$-0.950987\pi$
$431$	−7.50000	+	12.9904i	−0.361262	+	0.625725i	0.626907	+	0.779094i	$0.284321\pi$
$432$		0			0
$433$	−4.00000	+	6.92820i	−0.192228	+	0.332948i	−0.945988	−	0.324201i	$-0.894905\pi$
$433$	−4.00000	+	6.92820i	−0.192228	+	0.332948i	0.753760	+	0.657149i	$0.228238\pi$
$434$		0			0
$435$		0			0
$436$	−22.0000			−1.05361
$437$		0			0
$438$		0			0
$439$	6.50000	+	11.2583i	0.310228	+	0.537331i	0.978412	−	0.206666i	$-0.0662612\pi$
$439$	6.50000	+	11.2583i	0.310228	+	0.537331i	−0.668184	+	0.743996i	$0.732928\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	−0.972626	−	0.232377i	$-0.925350\pi$
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	0.687557	+	0.726130i	$0.258683\pi$
$444$		0			0
$445$	15.0000			0.711068
$446$		0			0
$447$		0			0
$448$	32.0000			1.51186
$449$	9.00000			0.424736			0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000			0.424736			0.212368	+	0.977190i	$0.431882\pi$
$450$		0			0
$451$	9.00000	+	15.5885i	0.423793	+	0.734032i
$452$	−6.00000	+	10.3923i	−0.282216	+	0.488813i
$453$		0			0
$454$		0			0
$455$	−8.00000			−0.375046
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.50000	−	7.79423i	−0.209586	−	0.363013i	0.741998	−	0.670402i	$-0.233878\pi$
$461$	−4.50000	−	7.79423i	−0.209586	−	0.363013i	−0.951584	+	0.307388i	$0.900545\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$	12.0000			0.557086
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	4.00000	−	6.92820i	0.184703	−	0.319915i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.00000	−	10.3923i	−0.275880	−	0.477839i
$474$		0			0
$475$	−0.500000	−	4.33013i	−0.0229416	−	0.198680i
$476$	−48.0000			−2.20008
$477$		0			0
$478$		0			0
$479$	7.50000	−	12.9904i	0.342684	−	0.593546i	−0.642246	−	0.766498i	$-0.721997\pi$
$479$	7.50000	−	12.9904i	0.342684	−	0.593546i	0.984930	+	0.172953i	$0.0553307\pi$
$480$		0			0
$481$	−8.00000	+	13.8564i	−0.364769	+	0.631798i
$482$		0			0
$483$		0			0
$484$	−2.00000	+	3.46410i	−0.0909091	+	0.157459i
$485$	−4.00000	+	6.92820i	−0.181631	+	0.314594i
$486$		0			0
$487$	2.00000			0.0906287			0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000			0.0906287			0.0453143	+	0.998973i	$0.485571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7.50000	−	12.9904i	−0.338470	−	0.586248i	0.645675	−	0.763612i	$-0.276576\pi$
$491$	−7.50000	−	12.9904i	−0.338470	−	0.586248i	−0.984145	+	0.177365i	$0.943243\pi$
$492$		0			0
$493$	−18.0000			−0.810679
$494$		0			0
$495$		0			0
$496$	14.0000	+	24.2487i	0.628619	+	1.08880i
$497$	6.00000	+	10.3923i	0.269137	+	0.466159i
$498$		0			0
$499$	−10.0000	−	17.3205i	−0.447661	−	0.775372i	0.550572	−	0.834788i	$-0.314410\pi$
$499$	−10.0000	−	17.3205i	−0.447661	−	0.775372i	−0.998233	+	0.0594153i	$0.981076\pi$
$500$	1.00000	−	1.73205i	0.0447214	−	0.0774597i
$501$		0			0
$502$		0			0
$503$	−21.0000	+	36.3731i	−0.936344	+	1.62179i	−0.164124	+	0.986440i	$0.552480\pi$
$503$	−21.0000	+	36.3731i	−0.936344	+	1.62179i	−0.772220	+	0.635355i	$0.780854\pi$
$504$		0			0
$505$	−15.0000			−0.667491
$506$		0			0
$507$		0			0
$508$	2.00000	+	3.46410i	0.0887357	+	0.153695i
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	−0.993593	−	0.113020i	$-0.963948\pi$
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	0.594675	+	0.803966i	$0.297281\pi$
$510$		0			0
$511$	16.0000	+	27.7128i	0.707798	+	1.22594i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.00000	+	13.8564i	0.352522	+	0.610586i
$516$		0			0
$517$	−9.00000	+	15.5885i	−0.395820	+	0.685580i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	45.0000			1.97149			0.985743	−	0.168259i	$-0.0538144\pi$
$521$	45.0000			1.97149			0.985743	+	0.168259i	$0.0538144\pi$
$522$		0			0
$523$	−13.0000	+	22.5167i	−0.568450	+	0.984585i	0.428269	+	0.903651i	$0.359124\pi$
$523$	−13.0000	+	22.5167i	−0.568450	+	0.984585i	−0.996719	+	0.0809336i	$0.974210\pi$
$524$	24.0000			1.04844
$525$		0			0
$526$		0			0
$527$	−21.0000	−	36.3731i	−0.914774	−	1.58444i
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−4.00000	−	34.6410i	−0.173422	−	1.50188i
$533$	12.0000			0.519778
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−27.0000			−1.16297
$540$		0			0
$541$	18.5000	−	32.0429i	0.795377	−	1.37763i	−0.127222	−	0.991874i	$-0.540606\pi$
$541$	18.5000	−	32.0429i	0.795377	−	1.37763i	0.922599	−	0.385759i	$-0.126061\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.50000	+	9.52628i	−0.235594	+	0.408061i
$546$		0			0
$547$	2.00000	−	3.46410i	0.0855138	−	0.148114i	−0.820096	−	0.572226i	$-0.806080\pi$
$547$	2.00000	−	3.46410i	0.0855138	−	0.148114i	0.905610	+	0.424111i	$0.139413\pi$
$548$	12.0000	+	20.7846i	0.512615	+	0.887875i
$549$		0			0
$550$		0			0
$551$	−1.50000	−	12.9904i	−0.0639021	−	0.553409i
$552$		0			0
$553$	10.0000	+	17.3205i	0.425243	+	0.736543i
$554$		0			0
$555$		0			0
$556$	−16.0000	−	27.7128i	−0.678551	−	1.17529i
$557$	6.00000	−	10.3923i	0.254228	−	0.440336i	−0.710457	−	0.703740i	$-0.751512\pi$
$557$	6.00000	−	10.3923i	0.254228	−	0.440336i	0.964686	+	0.263404i	$0.0848453\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$	8.00000	−	13.8564i	0.338062	−	0.585540i
$561$		0			0
$562$		0			0
$563$	−6.00000			−0.252870			−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000			−0.252870			−0.126435	+	0.991975i	$0.540353\pi$
$564$		0			0
$565$	3.00000	+	5.19615i	0.126211	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−39.0000			−1.63497			−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000			−1.63497			−0.817483	+	0.575953i	$0.804631\pi$
$570$		0			0
$571$	−7.00000			−0.292941			−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000			−0.292941			−0.146470	+	0.989215i	$0.546791\pi$
$572$	−6.00000	−	10.3923i	−0.250873	−	0.434524i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−16.0000			−0.666089			−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000			−0.666089			−0.333044	+	0.942911i	$0.608076\pi$
$578$		0			0
$579$		0			0
$580$	3.00000	−	5.19615i	0.124568	−	0.215758i
$581$	48.0000			1.99138
$582$		0			0
$583$	9.00000	−	15.5885i	0.372742	−	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.0000	−	31.1769i	−0.742940	−	1.28681i	−0.951151	−	0.308725i	$-0.900098\pi$
$587$	−18.0000	−	31.1769i	−0.742940	−	1.28681i	0.208212	−	0.978084i	$-0.433236\pi$
$588$		0			0
$589$	24.5000	−	18.1865i	1.00950	−	0.749363i
$590$		0			0
$591$		0			0
$592$	−16.0000	−	27.7128i	−0.657596	−	1.13899i
$593$	6.00000	−	10.3923i	0.246390	−	0.426761i	−0.716131	−	0.697966i	$-0.754089\pi$
$593$	6.00000	−	10.3923i	0.246390	−	0.426761i	0.962522	+	0.271205i	$0.0874221\pi$
$594$		0			0
$595$	−12.0000	+	20.7846i	−0.491952	+	0.852086i
$596$	6.00000			0.245770
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−31.0000			−1.26452			−0.632258	−	0.774758i	$-0.717872\pi$
$601$	−31.0000			−1.26452			−0.632258	+	0.774758i	$0.717872\pi$
$602$		0			0
$603$		0			0
$604$	17.0000	−	29.4449i	0.691720	−	1.19809i
$605$	1.00000	+	1.73205i	0.0406558	+	0.0704179i
$606$		0			0
$607$	2.00000			0.0811775			0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000			0.0811775			0.0405887	+	0.999176i	$0.487077\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.00000	+	10.3923i	0.242734	+	0.420428i
$612$		0			0
$613$	−1.00000	−	1.73205i	−0.0403896	−	0.0699569i	0.845124	−	0.534570i	$-0.179527\pi$
$613$	−1.00000	−	1.73205i	−0.0403896	−	0.0699569i	−0.885514	+	0.464614i	$0.846193\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.0000	−	31.1769i	0.724653	−	1.25514i	−0.234464	−	0.972125i	$-0.575334\pi$
$617$	18.0000	−	31.1769i	0.724653	−	1.25514i	0.959117	−	0.283011i	$-0.0913331\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$	14.0000			0.562254
$621$		0			0
$622$		0			0
$623$	30.0000	−	51.9615i	1.20192	−	2.08179i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$	−4.00000			−0.159617
$629$	24.0000	+	41.5692i	0.956943	+	1.65747i
$630$		0			0
$631$	9.50000	−	16.4545i	0.378189	−	0.655043i	−0.612610	−	0.790386i	$-0.709880\pi$
$631$	9.50000	−	16.4545i	0.378189	−	0.655043i	0.990799	+	0.135343i	$0.0432136\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.00000			0.0793676
$636$		0			0
$637$	−9.00000	+	15.5885i	−0.356593	+	0.617637i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.50000	+	7.79423i	0.177739	+	0.307854i	0.941106	−	0.338112i	$-0.109788\pi$
$641$	4.50000	+	7.79423i	0.177739	+	0.307854i	−0.763367	+	0.645966i	$0.776455\pi$
$642$		0			0
$643$	17.0000	+	29.4449i	0.670415	+	1.16119i	0.977787	+	0.209603i	$0.0672170\pi$
$643$	17.0000	+	29.4449i	0.670415	+	1.16119i	−0.307372	+	0.951589i	$0.599450\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.0000			−0.707653			−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000			−0.707653			−0.353827	+	0.935311i	$0.615120\pi$
$648$		0			0
$649$	22.5000	+	38.9711i	0.883202	+	1.52975i
$650$		0			0
$651$		0			0
$652$	−10.0000	+	17.3205i	−0.391630	+	0.678323i
$653$	−24.0000			−0.939193			−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000			−0.939193			−0.469596	+	0.882881i	$0.655601\pi$
$654$		0			0
$655$	6.00000	−	10.3923i	0.234439	−	0.406061i
$656$	−12.0000	+	20.7846i	−0.468521	+	0.811503i
$657$		0			0
$658$		0			0
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	−0.958902	−	0.283738i	$-0.908425\pi$
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	0.725175	+	0.688565i	$0.241759\pi$
$660$		0			0
$661$	−2.50000	+	4.33013i	−0.0972387	+	0.168422i	−0.910541	−	0.413419i	$-0.864334\pi$
$661$	−2.50000	+	4.33013i	−0.0972387	+	0.168422i	0.813302	+	0.581842i	$0.197668\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−16.0000	−	6.92820i	−0.620453	−	0.268664i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.50000	−	12.9904i	0.289534	−	0.501488i
$672$		0			0
$673$	−4.00000			−0.154189			−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000			−0.154189			−0.0770943	+	0.997024i	$0.524564\pi$
$674$		0			0
$675$		0			0
$676$	18.0000			0.692308
$677$	−48.0000			−1.84479			−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000			−1.84479			−0.922395	+	0.386248i	$0.873771\pi$
$678$		0			0
$679$	16.0000	+	27.7128i	0.614024	+	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	18.0000			0.688751			0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000			0.688751			0.344375	+	0.938832i	$0.388091\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$		0			0
$688$	8.00000	−	13.8564i	0.304997	−	0.528271i
$689$	−6.00000	−	10.3923i	−0.228582	−	0.395915i
$690$		0			0
$691$	−37.0000			−1.40755			−0.703773	−	0.710425i	$-0.748503\pi$
$691$	−37.0000			−1.40755			−0.703773	+	0.710425i	$0.748503\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−16.0000			−0.606915
$696$		0			0
$697$	18.0000	−	31.1769i	0.681799	−	1.18091i
$698$		0			0
$699$		0			0
$700$	−4.00000	−	6.92820i	−0.151186	−	0.261861i
$701$	3.00000	+	5.19615i	0.113308	+	0.196256i	0.917102	−	0.398652i	$-0.130522\pi$
$701$	3.00000	+	5.19615i	0.113308	+	0.196256i	−0.803794	+	0.594908i	$0.797189\pi$
$702$		0			0
$703$	−28.0000	+	20.7846i	−1.05604	+	0.783906i
$704$	24.0000			0.904534
$705$		0			0
$706$		0			0
$707$	−30.0000	+	51.9615i	−1.12827	+	1.95421i
$708$		0			0
$709$	15.5000	−	26.8468i	0.582115	−	1.00825i	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	15.5000	−	26.8468i	0.582115	−	1.00825i	0.995228	−	0.0975728i	$-0.0311079\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−6.00000			−0.224387
$716$	9.00000	−	15.5885i	0.336346	−	0.582568i
$717$		0			0
$718$		0			0
$719$	7.50000	+	12.9904i	0.279703	+	0.484459i	0.971311	−	0.237814i	$-0.0764307\pi$
$719$	7.50000	+	12.9904i	0.279703	+	0.484459i	−0.691608	+	0.722273i	$0.743097\pi$
$720$		0			0
$721$	64.0000			2.38348
$722$		0			0
$723$		0			0
$724$	2.00000	+	3.46410i	0.0743294	+	0.128742i
$725$	−1.50000	−	2.59808i	−0.0557086	−	0.0964901i
$726$		0			0
$727$	−7.00000	−	12.1244i	−0.259616	−	0.449667i	0.706523	−	0.707690i	$-0.250263\pi$
$727$	−7.00000	−	12.1244i	−0.259616	−	0.449667i	−0.966139	+	0.258022i	$0.916929\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000	+	20.7846i	−0.443836	+	0.768747i
$732$		0			0
$733$	32.0000			1.18195			0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000			1.18195			0.590973	+	0.806691i	$0.298744\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.00000	−	5.19615i	0.110506	−	0.191403i
$738$		0			0
$739$	−17.5000	−	30.3109i	−0.643748	−	1.11500i	−0.984589	−	0.174883i	$-0.944045\pi$
$739$	−17.5000	−	30.3109i	−0.643748	−	1.11500i	0.340841	−	0.940121i	$-0.389288\pi$
$740$	−16.0000			−0.588172
$741$		0			0
$742$		0			0
$743$	−12.0000	−	20.7846i	−0.440237	−	0.762513i	0.557470	−	0.830197i	$-0.311772\pi$
$743$	−12.0000	−	20.7846i	−0.440237	−	0.762513i	−0.997707	+	0.0676840i	$0.978439\pi$
$744$		0			0
$745$	1.50000	−	2.59808i	0.0549557	−	0.0951861i
$746$		0			0
$747$		0			0
$748$	−36.0000			−1.31629
$749$		0			0
$750$		0			0
$751$	3.50000	−	6.06218i	0.127717	−	0.221212i	−0.795075	−	0.606511i	$-0.792568\pi$
$751$	3.50000	−	6.06218i	0.127717	−	0.221212i	0.922792	+	0.385299i	$0.125902\pi$
$752$	−24.0000			−0.875190
$753$		0			0
$754$		0			0
$755$	−8.50000	−	14.7224i	−0.309347	−	0.535804i
$756$		0			0
$757$	5.00000	+	8.66025i	0.181728	+	0.314762i	0.942469	−	0.334293i	$-0.108498\pi$
$757$	5.00000	+	8.66025i	0.181728	+	0.314762i	−0.760741	+	0.649056i	$0.775164\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$	22.0000	+	38.1051i	0.796453	+	1.37950i
$764$	15.0000	−	25.9808i	0.542681	−	0.939951i
$765$		0			0
$766$		0			0
$767$	30.0000			1.08324
$768$		0			0
$769$	−14.5000	+	25.1147i	−0.522883	+	0.905661i	0.476762	+	0.879032i	$0.341810\pi$
$769$	−14.5000	+	25.1147i	−0.522883	+	0.905661i	−0.999645	+	0.0266282i	$0.991523\pi$
$770$		0			0
$771$		0			0
$772$	32.0000			1.15171
$773$	15.0000	−	25.9808i	0.539513	−	0.934463i	−0.459418	−	0.888220i	$-0.651942\pi$
$773$	15.0000	−	25.9808i	0.539513	−	0.934463i	0.998930	−	0.0462427i	$-0.0147248\pi$
$774$		0			0
$775$	3.50000	−	6.06218i	0.125724	−	0.217760i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.0000	+	10.3923i	0.859889	+	0.372343i
$780$		0			0
$781$	4.50000	+	7.79423i	0.161023	+	0.278899i
$782$		0			0
$783$		0			0
$784$	−18.0000	−	31.1769i	−0.642857	−	1.11346i
$785$	−1.00000	+	1.73205i	−0.0356915	+	0.0618195i
$786$		0			0
$787$	−34.0000			−1.21197			−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000			−1.21197			−0.605985	+	0.795476i	$0.707221\pi$
$788$	18.0000	−	31.1769i	0.641223	−	1.11063i
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$	−5.00000	−	8.66025i	−0.177555	−	0.307535i

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{4} +(-0.500000 - 0.866025i) q^{5} -4.00000 q^{7} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{4} +(-0.500000 - 0.866025i) q^{5} -4.00000 q^{7} -3.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(3.00000 + 5.19615i) q^{17} +(-3.50000 + 2.59808i) q^{19} -2.00000 q^{20} +(-0.500000 + 0.866025i) q^{25} +(-4.00000 + 6.92820i) q^{28} +(-1.50000 + 2.59808i) q^{29} -7.00000 q^{31} +(2.00000 + 3.46410i) q^{35} +8.00000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +(-3.00000 + 5.19615i) q^{44} +(3.00000 - 5.19615i) q^{47} +9.00000 q^{49} +(2.00000 + 3.46410i) q^{52} +(-3.00000 + 5.19615i) q^{53} +(1.50000 + 2.59808i) q^{55} +(-7.50000 - 12.9904i) q^{59} +(-2.50000 + 4.33013i) q^{61} -8.00000 q^{64} +2.00000 q^{65} +(-1.00000 + 1.73205i) q^{67} +12.0000 q^{68} +(-1.50000 - 2.59808i) q^{71} +(-4.00000 - 6.92820i) q^{73} +(1.00000 + 8.66025i) q^{76} +12.0000 q^{77} +(-2.50000 - 4.33013i) q^{79} +(-2.00000 + 3.46410i) q^{80} -12.0000 q^{83} +(3.00000 - 5.19615i) q^{85} +(-7.50000 + 12.9904i) q^{89} +(4.00000 - 6.92820i) q^{91} +(4.00000 + 1.73205i) q^{95} +(-4.00000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - q^{5} - 8 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 - q^5 - 8 * q^7	\( 2 q + 2 q^{4} - q^{5} - 8 q^{7} - 6 q^{11} - 2 q^{13} - 4 q^{16} + 6 q^{17} - 7 q^{19} - 4 q^{20} - q^{25} - 8 q^{28} - 3 q^{29} - 14 q^{31} + 4 q^{35} + 16 q^{37} - 6 q^{41} + 4 q^{43} - 6 q^{44} + 6 q^{47} + 18 q^{49} + 4 q^{52} - 6 q^{53} + 3 q^{55} - 15 q^{59} - 5 q^{61} - 16 q^{64} + 4 q^{65} - 2 q^{67} + 24 q^{68} - 3 q^{71} - 8 q^{73} + 2 q^{76} + 24 q^{77} - 5 q^{79} - 4 q^{80} - 24 q^{83} + 6 q^{85} - 15 q^{89} + 8 q^{91} + 8 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - q^5 - 8 * q^7 - 6 * q^11 - 2 * q^13 - 4 * q^16 + 6 * q^17 - 7 * q^19 - 4 * q^20 - q^25 - 8 * q^28 - 3 * q^29 - 14 * q^31 + 4 * q^35 + 16 * q^37 - 6 * q^41 + 4 * q^43 - 6 * q^44 + 6 * q^47 + 18 * q^49 + 4 * q^52 - 6 * q^53 + 3 * q^55 - 15 * q^59 - 5 * q^61 - 16 * q^64 + 4 * q^65 - 2 * q^67 + 24 * q^68 - 3 * q^71 - 8 * q^73 + 2 * q^76 + 24 * q^77 - 5 * q^79 - 4 * q^80 - 24 * q^83 + 6 * q^85 - 15 * q^89 + 8 * q^91 + 8 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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