LMFDB - Embedded newform 1805.2.b.a.1084.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1084,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("1805.1084");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1805 = 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1805.b (of order $2$, degree $1$, 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:	$14.4129975648$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1084.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1805.1084
Dual form			1805.2.b.a.1084.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -2.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -4.00000i q^{7} +3.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} -2.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -4.00000i q^{7} +3.00000 q^{9} +(-4.00000 + 2.00000i) q^{10} -1.00000 q^{11} -2.00000i q^{13} -8.00000 q^{14} -4.00000 q^{16} -2.00000i q^{17} -6.00000i q^{18} +(2.00000 + 4.00000i) q^{20} +2.00000i q^{22} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000 q^{26} +8.00000i q^{28} +9.00000 q^{29} +7.00000 q^{31} +8.00000i q^{32} -4.00000 q^{34} +(-8.00000 + 4.00000i) q^{35} -6.00000 q^{36} -2.00000i q^{37} -2.00000 q^{41} -2.00000i q^{43} +2.00000 q^{44} +(-3.00000 - 6.00000i) q^{45} -12.0000 q^{46} +6.00000i q^{47} -9.00000 q^{49} +(8.00000 + 6.00000i) q^{50} +4.00000i q^{52} +4.00000i q^{53} +(1.00000 + 2.00000i) q^{55} -18.0000i q^{58} +9.00000 q^{59} -7.00000 q^{61} -14.0000i q^{62} -12.0000i q^{63} +8.00000 q^{64} +(-4.00000 + 2.00000i) q^{65} +10.0000i q^{67} +4.00000i q^{68} +(8.00000 + 16.0000i) q^{70} -1.00000 q^{71} -10.0000i q^{73} -4.00000 q^{74} +4.00000i q^{77} +1.00000 q^{79} +(4.00000 + 8.00000i) q^{80} +9.00000 q^{81} +4.00000i q^{82} +6.00000i q^{83} +(-4.00000 + 2.00000i) q^{85} -4.00000 q^{86} -11.0000 q^{89} +(-12.0000 + 6.00000i) q^{90} -8.00000 q^{91} +12.0000i q^{92} +12.0000 q^{94} +6.00000i q^{97} +18.0000i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 2 q^{5} + 6 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 - 2 * q^5 + 6 * q^9	$ 2 q - 4 q^{4} - 2 q^{5} + 6 q^{9} - 8 q^{10} - 2 q^{11} - 16 q^{14} - 8 q^{16} + 4 q^{20} - 6 q^{25} - 8 q^{26} + 18 q^{29} + 14 q^{31} - 8 q^{34} - 16 q^{35} - 12 q^{36} - 4 q^{41} + 4 q^{44} - 6 q^{45} - 24 q^{46} - 18 q^{49} + 16 q^{50} + 2 q^{55} + 18 q^{59} - 14 q^{61} + 16 q^{64} - 8 q^{65} + 16 q^{70} - 2 q^{71} - 8 q^{74} + 2 q^{79} + 8 q^{80} + 18 q^{81} - 8 q^{85} - 8 q^{86} - 22 q^{89} - 24 q^{90} - 16 q^{91} + 24 q^{94} - 6 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 - 2 * q^5 + 6 * q^9 - 8 * q^10 - 2 * q^11 - 16 * q^14 - 8 * q^16 + 4 * q^20 - 6 * q^25 - 8 * q^26 + 18 * q^29 + 14 * q^31 - 8 * q^34 - 16 * q^35 - 12 * q^36 - 4 * q^41 + 4 * q^44 - 6 * q^45 - 24 * q^46 - 18 * q^49 + 16 * q^50 + 2 * q^55 + 18 * q^59 - 14 * q^61 + 16 * q^64 - 8 * q^65 + 16 * q^70 - 2 * q^71 - 8 * q^74 + 2 * q^79 + 8 * q^80 + 18 * q^81 - 8 * q^85 - 8 * q^86 - 22 * q^89 - 24 * q^90 - 16 * q^91 + 24 * q^94 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$.

$n$	$362$	$1446$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	2.00000i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$		−	2.00000i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	−2.00000			−1.00000
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$	−4.00000	+	2.00000i	−1.26491	+	0.632456i
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−8.00000			−2.13809
$15$		0			0
$16$	−4.00000			−1.00000
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		−	6.00000i		−	1.41421i
$19$		0			0
$19$		0			0
$20$	2.00000	+	4.00000i	0.447214	+	0.894427i
$21$		0			0
$22$			2.00000i			0.426401i
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$	−4.00000			−0.784465
$27$		0			0
$28$			8.00000i			1.51186i
$29$	9.00000			1.67126			0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000			1.67126			0.835629	+	0.549294i	$0.185103\pi$
$30$		0			0
$31$	7.00000			1.25724			0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000			1.25724			0.628619	+	0.777714i	$0.283621\pi$
$32$			8.00000i			1.41421i
$33$		0			0
$34$	−4.00000			−0.685994
$35$	−8.00000	+	4.00000i	−1.35225	+	0.676123i
$36$	−6.00000			−1.00000
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$		−	2.00000i		−	0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$		−	2.00000i		−	0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	2.00000			0.301511
$45$	−3.00000	−	6.00000i	−0.447214	−	0.894427i
$46$	−12.0000			−1.76930
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$	8.00000	+	6.00000i	1.13137	+	0.848528i
$51$		0			0
$52$			4.00000i			0.554700i
$53$			4.00000i			0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$			4.00000i			0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$	1.00000	+	2.00000i	0.134840	+	0.269680i
$56$		0			0
$57$		0			0
$58$		−	18.0000i		−	2.36352i
$59$	9.00000			1.17170			0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000			1.17170			0.585850	+	0.810419i	$0.300761\pi$
$60$		0			0
$61$	−7.00000			−0.896258			−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000			−0.896258			−0.448129	+	0.893969i	$0.647910\pi$
$62$		−	14.0000i		−	1.77800i
$63$		−	12.0000i		−	1.51186i
$64$	8.00000			1.00000
$65$	−4.00000	+	2.00000i	−0.496139	+	0.248069i
$66$		0			0
$67$			10.0000i			1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$			10.0000i			1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$			4.00000i			0.485071i
$69$		0			0
$70$	8.00000	+	16.0000i	0.956183	+	1.91237i
$71$	−1.00000			−0.118678			−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000			−0.118678			−0.0593391	+	0.998238i	$0.518899\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	−4.00000			−0.464991
$75$		0			0
$76$		0			0
$77$			4.00000i			0.455842i
$78$		0			0
$79$	1.00000			0.112509			0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000			0.112509			0.0562544	+	0.998416i	$0.482084\pi$
$80$	4.00000	+	8.00000i	0.447214	+	0.894427i
$81$	9.00000			1.00000
$82$			4.00000i			0.441726i
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		0			0
$85$	−4.00000	+	2.00000i	−0.433861	+	0.216930i
$86$	−4.00000			−0.431331
$87$		0			0
$88$		0			0
$89$	−11.0000			−1.16600			−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000			−1.16600			−0.582999	+	0.812473i	$0.698121\pi$
$90$	−12.0000	+	6.00000i	−1.26491	+	0.632456i
$91$	−8.00000			−0.838628
$92$			12.0000i			1.25109i
$93$		0			0
$94$	12.0000			1.23771
$95$		0			0
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$			18.0000i			1.81827i
$99$	−3.00000			−0.301511
$100$	6.00000	−	8.00000i	0.600000	−	0.800000i
$101$	15.0000			1.49256			0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000			1.49256			0.746278	+	0.665635i	$0.231839\pi$
$102$		0			0
$103$			16.0000i			1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$			16.0000i			1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$		0			0
$105$		0			0
$106$	8.00000			0.777029
$107$			10.0000i			0.966736i	0.875417	+	0.483368i	$0.160587\pi$
$107$			10.0000i			0.966736i	−0.875417	+	0.483368i	$0.839413\pi$
$108$		0			0
$109$	15.0000			1.43674			0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000			1.43674			0.718370	+	0.695662i	$0.244889\pi$
$110$	4.00000	−	2.00000i	0.381385	−	0.190693i
$111$		0			0
$112$			16.0000i			1.51186i
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$		0			0
$115$	−12.0000	+	6.00000i	−1.11901	+	0.559503i
$116$	−18.0000			−1.67126
$117$		−	6.00000i		−	0.554700i
$118$		−	18.0000i		−	1.65703i
$119$	−8.00000			−0.733359
$120$		0			0
$121$	−10.0000			−0.909091
$122$			14.0000i			1.26750i
$123$		0			0
$124$	−14.0000			−1.25724
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$	−24.0000			−2.13809
$127$			6.00000i			0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$			6.00000i			0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$		0			0
$129$		0			0
$130$	4.00000	+	8.00000i	0.350823	+	0.701646i
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$		0			0
$134$	20.0000			1.72774
$135$		0			0
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$	16.0000	−	8.00000i	1.35225	−	0.676123i
$141$		0			0
$142$			2.00000i			0.167836i
$143$			2.00000i			0.167248i
$144$	−12.0000			−1.00000
$145$	−9.00000	−	18.0000i	−0.747409	−	1.49482i
$146$	−20.0000			−1.65521
$147$		0			0
$148$			4.00000i			0.328798i
$149$	1.00000			0.0819232			0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000			0.0819232			0.0409616	+	0.999161i	$0.486958\pi$
$150$		0			0
$151$	−9.00000			−0.732410			−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000			−0.732410			−0.366205	+	0.930534i	$0.619343\pi$
$152$		0			0
$153$		−	6.00000i		−	0.485071i
$154$	8.00000			0.644658
$155$	−7.00000	−	14.0000i	−0.562254	−	1.12451i
$156$		0			0
$157$			4.00000i			0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$			4.00000i			0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$		−	2.00000i		−	0.159111i
$159$		0			0
$160$	16.0000	−	8.00000i	1.26491	−	0.632456i
$161$	−24.0000			−1.89146
$162$		−	18.0000i		−	1.41421i
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	4.00000			0.312348
$165$		0			0
$166$	12.0000			0.931381
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	4.00000	+	8.00000i	0.306786	+	0.613572i
$171$		0			0
$172$			4.00000i			0.304997i
$173$		−	24.0000i		−	1.82469i	−0.409426	−	0.912343i	$-0.634271\pi$
$173$		−	24.0000i		−	1.82469i	0.409426	−	0.912343i	$-0.365729\pi$
$174$		0			0
$175$	16.0000	+	12.0000i	1.20949	+	0.907115i
$176$	4.00000			0.301511
$177$		0			0
$178$			22.0000i			1.64897i
$179$	15.0000			1.12115			0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000			1.12115			0.560576	+	0.828103i	$0.310580\pi$
$180$	6.00000	+	12.0000i	0.447214	+	0.894427i
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$			16.0000i			1.18600i
$183$		0			0
$184$		0			0
$185$	−4.00000	+	2.00000i	−0.294086	+	0.147043i
$186$		0			0
$187$			2.00000i			0.146254i
$188$		−	12.0000i		−	0.875190i
$189$		0			0
$190$		0			0
$191$	−3.00000			−0.217072			−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000			−0.217072			−0.108536	+	0.994092i	$0.534616\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	12.0000			0.861550
$195$		0			0
$196$	18.0000			1.28571
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$			6.00000i			0.426401i
$199$	−13.0000			−0.921546			−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000			−0.921546			−0.460773	+	0.887518i	$0.652428\pi$
$200$		0			0
$201$		0			0
$202$		−	30.0000i		−	2.11079i
$203$		−	36.0000i		−	2.52670i
$204$		0			0
$205$	2.00000	+	4.00000i	0.139686	+	0.279372i
$206$	32.0000			2.22955
$207$		−	18.0000i		−	1.25109i
$208$			8.00000i			0.554700i
$209$		0			0
$210$		0			0
$211$	−5.00000			−0.344214			−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000			−0.344214			−0.172107	+	0.985078i	$0.555058\pi$
$212$		−	8.00000i		−	0.549442i
$213$		0			0
$214$	20.0000			1.36717
$215$	−4.00000	+	2.00000i	−0.272798	+	0.136399i
$216$		0			0
$217$		−	28.0000i		−	1.90076i
$218$		−	30.0000i		−	2.03186i
$219$		0			0
$220$	−2.00000	−	4.00000i	−0.134840	−	0.269680i
$221$	−4.00000			−0.269069
$222$		0			0
$223$		−	2.00000i		−	0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$		−	2.00000i		−	0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	32.0000			2.13809
$225$	−9.00000	+	12.0000i	−0.600000	+	0.800000i
$226$	−24.0000			−1.59646
$227$		−	14.0000i		−	0.929213i	−0.885517	−	0.464606i	$-0.846196\pi$
$227$		−	14.0000i		−	0.929213i	0.885517	−	0.464606i	$-0.153804\pi$
$228$		0			0
$229$	−17.0000			−1.12339			−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000			−1.12339			−0.561696	+	0.827344i	$0.689851\pi$
$230$	12.0000	+	24.0000i	0.791257	+	1.58251i
$231$		0			0
$232$		0			0
$233$			8.00000i			0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$			8.00000i			0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$	−12.0000			−0.784465
$235$	12.0000	−	6.00000i	0.782794	−	0.391397i
$236$	−18.0000			−1.17170
$237$		0			0
$238$			16.0000i			1.03713i
$239$	19.0000			1.22901			0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000			1.22901			0.614504	+	0.788914i	$0.289356\pi$
$240$		0			0
$241$	−1.00000			−0.0644157			−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000			−0.0644157			−0.0322078	+	0.999481i	$0.510254\pi$
$242$			20.0000i			1.28565i
$243$		0			0
$244$	14.0000			0.896258
$245$	9.00000	+	18.0000i	0.574989	+	1.14998i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	4.00000	−	22.0000i	0.252982	−	1.39140i
$251$	13.0000			0.820553			0.410276	−	0.911961i	$-0.365432\pi$
$251$	13.0000			0.820553			0.410276	+	0.911961i	$0.365432\pi$
$252$			24.0000i			1.51186i
$253$			6.00000i			0.377217i
$254$	12.0000			0.752947
$255$		0			0
$256$	16.0000			1.00000
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	−8.00000			−0.497096
$260$	8.00000	−	4.00000i	0.496139	−	0.248069i
$261$	27.0000			1.67126
$262$		−	24.0000i		−	1.48272i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	8.00000	−	4.00000i	0.491436	−	0.245718i
$266$		0			0
$267$		0			0
$268$		−	20.0000i		−	1.22169i
$269$	3.00000			0.182913			0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000			0.182913			0.0914566	+	0.995809i	$0.470848\pi$
$270$		0			0
$271$	3.00000			0.182237			0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000			0.182237			0.0911185	+	0.995840i	$0.470956\pi$
$272$			8.00000i			0.485071i
$273$		0			0
$274$	24.0000			1.44989
$275$	3.00000	−	4.00000i	0.180907	−	0.241209i
$276$		0			0
$277$		−	28.0000i		−	1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$		−	28.0000i		−	1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$			40.0000i			2.39904i
$279$	21.0000			1.25724
$280$		0			0
$281$	−10.0000			−0.596550			−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000			−0.596550			−0.298275	+	0.954480i	$0.596411\pi$
$282$		0			0
$283$		−	14.0000i		−	0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$		−	14.0000i		−	0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$	4.00000			0.236525
$287$			8.00000i			0.472225i
$288$			24.0000i			1.41421i
$289$	13.0000			0.764706
$290$	−36.0000	+	18.0000i	−2.11399	+	1.05700i
$291$		0			0
$292$			20.0000i			1.17041i
$293$			4.00000i			0.233682i	0.993151	+	0.116841i	$0.0372769\pi$
$293$			4.00000i			0.233682i	−0.993151	+	0.116841i	$0.962723\pi$
$294$		0			0
$295$	−9.00000	−	18.0000i	−0.524000	−	1.04800i
$296$		0			0
$297$		0			0
$298$		−	2.00000i		−	0.115857i
$299$	−12.0000			−0.693978
$300$		0			0
$301$	−8.00000			−0.461112
$302$			18.0000i			1.03578i
$303$		0			0
$304$		0			0
$305$	7.00000	+	14.0000i	0.400819	+	0.801638i
$306$	−12.0000			−0.685994
$307$		−	16.0000i		−	0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$		−	16.0000i		−	0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$		−	8.00000i		−	0.455842i
$309$		0			0
$310$	−28.0000	+	14.0000i	−1.59029	+	0.795147i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	30.0000i		−	1.69570i	−0.530236	−	0.847850i	$-0.677897\pi$
$313$		−	30.0000i		−	1.69570i	0.530236	−	0.847850i	$-0.322103\pi$
$314$	8.00000			0.451466
$315$	−24.0000	+	12.0000i	−1.35225	+	0.676123i
$316$	−2.00000			−0.112509
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$		0			0
$319$	−9.00000			−0.503903
$320$	−8.00000	−	16.0000i	−0.447214	−	0.894427i
$321$		0			0
$322$			48.0000i			2.67494i
$323$		0			0
$324$	−18.0000			−1.00000
$325$	8.00000	+	6.00000i	0.443760	+	0.332820i
$326$	−8.00000			−0.443079
$327$		0			0
$328$		0			0
$329$	24.0000			1.32316
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		−	12.0000i		−	0.658586i
$333$		−	6.00000i		−	0.328798i
$334$	24.0000			1.31322
$335$	20.0000	−	10.0000i	1.09272	−	0.546358i
$336$		0			0
$337$		−	34.0000i		−	1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$		−	34.0000i		−	1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$		−	18.0000i		−	0.979071i
$339$		0			0
$340$	8.00000	−	4.00000i	0.433861	−	0.216930i
$341$	−7.00000			−0.379071
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$		0			0
$346$	−48.0000			−2.58050
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$	24.0000	−	32.0000i	1.28285	−	1.71047i
$351$		0			0
$352$		−	8.00000i		−	0.426401i
$353$			8.00000i			0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$			8.00000i			0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$		0			0
$355$	1.00000	+	2.00000i	0.0530745	+	0.106149i
$356$	22.0000			1.16600
$357$		0			0
$358$		−	30.0000i		−	1.58555i
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$		0			0
$362$		−	12.0000i		−	0.630706i
$363$		0			0
$364$	16.0000			0.838628
$365$	−20.0000	+	10.0000i	−1.04685	+	0.523424i
$366$		0			0
$367$		−	16.0000i		−	0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$		−	16.0000i		−	0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$			24.0000i			1.25109i
$369$	−6.00000			−0.312348
$370$	4.00000	+	8.00000i	0.207950	+	0.415900i
$371$	16.0000			0.830679
$372$		0			0
$373$		−	12.0000i		−	0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$		−	12.0000i		−	0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	4.00000			0.206835
$375$		0			0
$376$		0			0
$377$		−	18.0000i		−	0.927047i
$378$		0			0
$379$	−29.0000			−1.48963			−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000			−1.48963			−0.744815	+	0.667271i	$0.767462\pi$
$380$		0			0
$381$		0			0
$382$			6.00000i			0.306987i
$383$			6.00000i			0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$			6.00000i			0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$		0			0
$385$	8.00000	−	4.00000i	0.407718	−	0.203859i
$386$	32.0000			1.62876
$387$		−	6.00000i		−	0.304997i
$388$		−	12.0000i		−	0.609208i
$389$	33.0000			1.67317			0.836583	−	0.547840i	$-0.184550\pi$
$389$	33.0000			1.67317			0.836583	+	0.547840i	$0.184550\pi$
$390$		0			0
$391$	−12.0000			−0.606866
$392$		0			0
$393$		0			0
$394$	−36.0000			−1.81365
$395$	−1.00000	−	2.00000i	−0.0503155	−	0.100631i
$396$	6.00000			0.301511
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$			26.0000i			1.30326i
$399$		0			0
$400$	12.0000	−	16.0000i	0.600000	−	0.800000i
$401$	−3.00000			−0.149813			−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000			−0.149813			−0.0749064	+	0.997191i	$0.523866\pi$
$402$		0			0
$403$		−	14.0000i		−	0.697390i
$404$	−30.0000			−1.49256
$405$	−9.00000	−	18.0000i	−0.447214	−	0.894427i
$406$	−72.0000			−3.57330
$407$			2.00000i			0.0991363i
$408$		0			0
$409$	−5.00000			−0.247234			−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000			−0.247234			−0.123617	+	0.992330i	$0.539449\pi$
$410$	8.00000	−	4.00000i	0.395092	−	0.197546i
$411$		0			0
$412$		−	32.0000i		−	1.57653i
$413$		−	36.0000i		−	1.77144i
$414$	−36.0000			−1.76930
$415$	12.0000	−	6.00000i	0.589057	−	0.294528i
$416$	16.0000			0.784465
$417$		0			0
$418$		0			0
$419$	−9.00000			−0.439679			−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000			−0.439679			−0.219839	+	0.975536i	$0.570553\pi$
$420$		0			0
$421$	15.0000			0.731055			0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000			0.731055			0.365528	+	0.930800i	$0.380889\pi$
$422$			10.0000i			0.486792i
$423$			18.0000i			0.875190i
$424$		0			0
$425$	8.00000	+	6.00000i	0.388057	+	0.291043i
$426$		0			0
$427$			28.0000i			1.35501i
$428$		−	20.0000i		−	0.966736i
$429$		0			0
$430$	4.00000	+	8.00000i	0.192897	+	0.385794i
$431$	−21.0000			−1.01153			−0.505767	−	0.862670i	$-0.668791\pi$
$431$	−21.0000			−1.01153			−0.505767	+	0.862670i	$0.668791\pi$
$432$		0			0
$433$		−	4.00000i		−	0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$		−	4.00000i		−	0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	−56.0000			−2.68809
$435$		0			0
$436$	−30.0000			−1.43674
$437$		0			0
$438$		0			0
$439$	−13.0000			−0.620456			−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000			−0.620456			−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$	−27.0000			−1.28571
$442$			8.00000i			0.380521i
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$	11.0000	+	22.0000i	0.521450	+	1.04290i
$446$	−4.00000			−0.189405
$447$		0			0
$448$		−	32.0000i		−	1.51186i
$449$	31.0000			1.46298			0.731490	−	0.681852i	$-0.238825\pi$
$449$	31.0000			1.46298			0.731490	+	0.681852i	$0.238825\pi$
$450$	24.0000	+	18.0000i	1.13137	+	0.848528i
$451$	2.00000			0.0941763
$452$			24.0000i			1.12887i
$453$		0			0
$454$	−28.0000			−1.31411
$455$	8.00000	+	16.0000i	0.375046	+	0.750092i
$456$		0			0
$457$			34.0000i			1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$			34.0000i			1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$			34.0000i			1.58872i
$459$		0			0
$460$	24.0000	−	12.0000i	1.11901	−	0.559503i
$461$	−25.0000			−1.16437			−0.582183	−	0.813058i	$-0.697801\pi$
$461$	−25.0000			−1.16437			−0.582183	+	0.813058i	$0.697801\pi$
$462$		0			0
$463$			4.00000i			0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$			4.00000i			0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−36.0000			−1.67126
$465$		0			0
$466$	16.0000			0.741186
$467$			10.0000i			0.462745i	0.972865	+	0.231372i	$0.0743216\pi$
$467$			10.0000i			0.462745i	−0.972865	+	0.231372i	$0.925678\pi$
$468$			12.0000i			0.554700i
$469$	40.0000			1.84703
$470$	−12.0000	−	24.0000i	−0.553519	−	1.10704i
$471$		0			0
$472$		0			0
$473$			2.00000i			0.0919601i
$474$		0			0
$475$		0			0
$476$	16.0000			0.733359
$477$			12.0000i			0.549442i
$478$		−	38.0000i		−	1.73808i
$479$	−15.0000			−0.685367			−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000			−0.685367			−0.342684	+	0.939451i	$0.611336\pi$
$480$		0			0
$481$	−4.00000			−0.182384
$482$			2.00000i			0.0910975i
$483$		0			0
$484$	20.0000			0.909091
$485$	12.0000	−	6.00000i	0.544892	−	0.272446i
$486$		0			0
$487$			38.0000i			1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$			38.0000i			1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$		0			0
$489$		0			0
$490$	36.0000	−	18.0000i	1.62631	−	0.813157i
$491$	13.0000			0.586682			0.293341	−	0.956008i	$-0.405233\pi$
$491$	13.0000			0.586682			0.293341	+	0.956008i	$0.405233\pi$
$492$		0			0
$493$		−	18.0000i		−	0.810679i
$494$		0			0
$495$	3.00000	+	6.00000i	0.134840	+	0.269680i
$496$	−28.0000			−1.25724
$497$			4.00000i			0.179425i
$498$		0			0
$499$	−28.0000			−1.25345			−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000			−1.25345			−0.626726	+	0.779240i	$0.715605\pi$
$500$	−22.0000	−	4.00000i	−0.983870	−	0.178885i
$501$		0			0
$502$		−	26.0000i		−	1.16044i
$503$			26.0000i			1.15928i	0.814872	+	0.579641i	$0.196807\pi$
$503$			26.0000i			1.15928i	−0.814872	+	0.579641i	$0.803193\pi$
$504$		0			0
$505$	−15.0000	−	30.0000i	−0.667491	−	1.33498i
$506$	12.0000			0.533465
$507$		0			0
$508$		−	12.0000i		−	0.532414i
$509$	18.0000			0.797836			0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000			0.797836			0.398918	+	0.916987i	$0.369386\pi$
$510$		0			0
$511$	−40.0000			−1.76950
$512$		−	32.0000i		−	1.41421i
$513$		0			0
$514$	12.0000			0.529297
$515$	32.0000	−	16.0000i	1.41009	−	0.705044i
$516$		0			0
$517$		−	6.00000i		−	0.263880i
$518$			16.0000i			0.703000i
$519$		0			0
$520$		0			0
$521$	−15.0000			−0.657162			−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000			−0.657162			−0.328581	+	0.944476i	$0.606570\pi$
$522$		−	54.0000i		−	2.36352i
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	−24.0000			−1.04844
$525$		0			0
$526$		0			0
$527$		−	14.0000i		−	0.609850i
$528$		0			0
$529$	−13.0000			−0.565217
$530$	−8.00000	−	16.0000i	−0.347498	−	0.694996i
$531$	27.0000			1.17170
$532$		0			0
$533$			4.00000i			0.173259i
$534$		0			0
$535$	20.0000	−	10.0000i	0.864675	−	0.432338i
$536$		0			0
$537$		0			0
$538$		−	6.00000i		−	0.258678i
$539$	9.00000			0.387657
$540$		0			0
$541$	23.0000			0.988847			0.494424	−	0.869221i	$-0.335379\pi$
$541$	23.0000			0.988847			0.494424	+	0.869221i	$0.335379\pi$
$542$		−	6.00000i		−	0.257722i
$543$		0			0
$544$	16.0000			0.685994
$545$	−15.0000	−	30.0000i	−0.642529	−	1.28506i
$546$		0			0
$547$		−	22.0000i		−	0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$547$		−	22.0000i		−	0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$548$		−	24.0000i		−	1.02523i
$549$	−21.0000			−0.896258
$550$	−8.00000	−	6.00000i	−0.341121	−	0.255841i
$551$		0			0
$552$		0			0
$553$		−	4.00000i		−	0.170097i
$554$	−56.0000			−2.37921
$555$		0			0
$556$	40.0000			1.69638
$557$			14.0000i			0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$			14.0000i			0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$		−	42.0000i		−	1.77800i
$559$	−4.00000			−0.169182
$560$	32.0000	−	16.0000i	1.35225	−	0.676123i
$561$		0			0
$562$			20.0000i			0.843649i
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	−24.0000	+	12.0000i	−1.00969	+	0.504844i
$566$	−28.0000			−1.17693
$567$		−	36.0000i		−	1.51186i
$568$		0			0
$569$	15.0000			0.628833			0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000			0.628833			0.314416	+	0.949285i	$0.398191\pi$
$570$		0			0
$571$	13.0000			0.544033			0.272017	−	0.962293i	$-0.412309\pi$
$571$	13.0000			0.544033			0.272017	+	0.962293i	$0.412309\pi$
$572$		−	4.00000i		−	0.167248i
$573$		0			0
$574$	16.0000			0.667827
$575$	24.0000	+	18.0000i	1.00087	+	0.750652i
$576$	24.0000			1.00000
$577$			6.00000i			0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$			6.00000i			0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$		−	26.0000i		−	1.08146i
$579$		0			0
$580$	18.0000	+	36.0000i	0.747409	+	1.49482i
$581$	24.0000			0.995688
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$	−12.0000	+	6.00000i	−0.496139	+	0.248069i
$586$	8.00000			0.330477
$587$		−	24.0000i		−	0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$		−	24.0000i		−	0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$		0			0
$589$		0			0
$590$	−36.0000	+	18.0000i	−1.48210	+	0.741048i
$591$		0			0
$592$			8.00000i			0.328798i
$593$			32.0000i			1.31408i	0.753855	+	0.657041i	$0.228192\pi$
$593$			32.0000i			1.31408i	−0.753855	+	0.657041i	$0.771808\pi$
$594$		0			0
$595$	8.00000	+	16.0000i	0.327968	+	0.655936i
$596$	−2.00000			−0.0819232
$597$		0			0
$598$			24.0000i			0.981433i
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$		0			0
$601$	−17.0000			−0.693444			−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000			−0.693444			−0.346722	+	0.937968i	$0.612705\pi$
$602$			16.0000i			0.652111i
$603$			30.0000i			1.22169i
$604$	18.0000			0.732410
$605$	10.0000	+	20.0000i	0.406558	+	0.813116i
$606$		0			0
$607$			14.0000i			0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$			14.0000i			0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$		0			0
$609$		0			0
$610$	28.0000	−	14.0000i	1.13369	−	0.566843i
$611$	12.0000			0.485468
$612$			12.0000i			0.485071i
$613$		−	24.0000i		−	0.969351i	−0.874694	−	0.484675i	$-0.838938\pi$
$613$		−	24.0000i		−	0.969351i	0.874694	−	0.484675i	$-0.161062\pi$
$614$	−32.0000			−1.29141
$615$		0			0
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		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	+	28.0000i	0.562254	+	1.12451i
$621$		0			0
$622$		0			0
$623$			44.0000i			1.76282i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−60.0000			−2.39808
$627$		0			0
$628$		−	8.00000i		−	0.319235i
$629$	−4.00000			−0.159490
$630$	24.0000	+	48.0000i	0.956183	+	1.91237i
$631$	1.00000			0.0398094			0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000			0.0398094			0.0199047	+	0.999802i	$0.493664\pi$
$632$		0			0
$633$		0			0
$634$	4.00000			0.158860
$635$	12.0000	−	6.00000i	0.476205	−	0.238103i
$636$		0			0
$637$			18.0000i			0.713186i
$638$			18.0000i			0.712627i
$639$	−3.00000			−0.118678
$640$		0			0
$641$	−21.0000			−0.829450			−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000			−0.829450			−0.414725	+	0.909947i	$0.636122\pi$
$642$		0			0
$643$			46.0000i			1.81406i	0.421063	+	0.907031i	$0.361657\pi$
$643$			46.0000i			1.81406i	−0.421063	+	0.907031i	$0.638343\pi$
$644$	48.0000			1.89146
$645$		0			0
$646$		0			0
$647$		−	6.00000i		−	0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$		−	6.00000i		−	0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$		0			0
$649$	−9.00000			−0.353281
$650$	12.0000	−	16.0000i	0.470679	−	0.627572i
$651$		0			0
$652$			8.00000i			0.313304i
$653$		−	10.0000i		−	0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$		−	10.0000i		−	0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$		0			0
$655$	−12.0000	−	24.0000i	−0.468879	−	0.937758i
$656$	8.00000			0.312348
$657$		−	30.0000i		−	1.17041i
$658$		−	48.0000i		−	1.87123i
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	15.0000			0.583432			0.291716	−	0.956505i	$-0.405774\pi$
$661$	15.0000			0.583432			0.291716	+	0.956505i	$0.405774\pi$
$662$			40.0000i			1.55464i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−12.0000			−0.464991
$667$		−	54.0000i		−	2.09089i
$668$		−	24.0000i		−	0.928588i
$669$		0			0
$670$	−20.0000	−	40.0000i	−0.772667	−	1.54533i
$671$	7.00000			0.270232
$672$		0			0
$673$		−	20.0000i		−	0.770943i	−0.922720	−	0.385472i	$-0.874039\pi$
$673$		−	20.0000i		−	0.770943i	0.922720	−	0.385472i	$-0.125961\pi$
$674$	−68.0000			−2.61926
$675$		0			0
$676$	−18.0000			−0.692308
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	24.0000			0.921035
$680$		0			0
$681$		0			0
$682$			14.0000i			0.536088i
$683$			30.0000i			1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$			30.0000i			1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$		0			0
$685$	24.0000	−	12.0000i	0.916993	−	0.458496i
$686$	16.0000			0.610883
$687$		0			0
$688$			8.00000i			0.304997i
$689$	8.00000			0.304776
$690$		0			0
$691$	3.00000			0.114125			0.0570627	−	0.998371i	$-0.481827\pi$
$691$	3.00000			0.114125			0.0570627	+	0.998371i	$0.481827\pi$
$692$			48.0000i			1.82469i
$693$			12.0000i			0.455842i
$694$	−24.0000			−0.911028
$695$	20.0000	+	40.0000i	0.758643	+	1.51729i
$696$		0			0
$697$			4.00000i			0.151511i
$698$			28.0000i			1.05982i
$699$		0			0
$700$	−32.0000	−	24.0000i	−1.20949	−	0.907115i
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$		0			0
$704$	−8.00000			−0.301511
$705$		0			0
$706$	16.0000			0.602168
$707$		−	60.0000i		−	2.25653i
$708$		0			0
$709$	−25.0000			−0.938895			−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000			−0.938895			−0.469447	+	0.882960i	$0.655547\pi$
$710$	4.00000	−	2.00000i	0.150117	−	0.0750587i
$711$	3.00000			0.112509
$712$		0			0
$713$		−	42.0000i		−	1.57291i
$714$		0			0
$715$	4.00000	−	2.00000i	0.149592	−	0.0747958i
$716$	−30.0000			−1.12115
$717$		0			0
$718$		−	40.0000i		−	1.49279i
$719$	21.0000			0.783168			0.391584	−	0.920142i	$-0.371927\pi$
$719$	21.0000			0.783168			0.391584	+	0.920142i	$0.371927\pi$
$720$	12.0000	+	24.0000i	0.447214	+	0.894427i
$721$	64.0000			2.38348
$722$		0			0
$723$		0			0
$724$	−12.0000			−0.445976
$725$	−27.0000	+	36.0000i	−1.00275	+	1.33701i
$726$		0			0
$727$			8.00000i			0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$			8.00000i			0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$	20.0000	+	40.0000i	0.740233	+	1.48047i
$731$	−4.00000			−0.147945
$732$		0			0
$733$			6.00000i			0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$			6.00000i			0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	−32.0000			−1.18114
$735$		0			0
$736$	48.0000			1.76930
$737$		−	10.0000i		−	0.368355i
$738$			12.0000i			0.441726i
$739$	29.0000			1.06678			0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000			1.06678			0.533391	+	0.845869i	$0.320917\pi$
$740$	8.00000	−	4.00000i	0.294086	−	0.147043i
$741$		0			0
$742$		−	32.0000i		−	1.17476i
$743$			50.0000i			1.83432i	0.398517	+	0.917161i	$0.369525\pi$
$743$			50.0000i			1.83432i	−0.398517	+	0.917161i	$0.630475\pi$
$744$		0			0
$745$	−1.00000	−	2.00000i	−0.0366372	−	0.0732743i
$746$	−24.0000			−0.878702
$747$			18.0000i			0.658586i
$748$		−	4.00000i		−	0.146254i
$749$	40.0000			1.46157
$750$		0			0
$751$	−41.0000			−1.49611			−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−41.0000			−1.49611			−0.748056	+	0.663636i	$0.769012\pi$
$752$		−	24.0000i		−	0.875190i
$753$		0			0
$754$	−36.0000			−1.31104
$755$	9.00000	+	18.0000i	0.327544	+	0.655087i
$756$		0			0
$757$			26.0000i			0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$			26.0000i			0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$			58.0000i			2.10665i
$759$		0			0
$760$		0			0
$761$	−18.0000			−0.652499			−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000			−0.652499			−0.326250	+	0.945284i	$0.605785\pi$
$762$		0			0
$763$		−	60.0000i		−	2.17215i
$764$	6.00000			0.217072
$765$	−12.0000	+	6.00000i	−0.433861	+	0.216930i
$766$	12.0000			0.433578
$767$		−	18.0000i		−	0.649942i
$768$		0			0
$769$	−5.00000			−0.180305			−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000			−0.180305			−0.0901523	+	0.995928i	$0.528735\pi$
$770$	−8.00000	−	16.0000i	−0.288300	−	0.576600i
$771$		0			0
$772$		−	32.0000i		−	1.15171i
$773$			34.0000i			1.22290i	0.791285	+	0.611448i	$0.209412\pi$
$773$			34.0000i			1.22290i	−0.791285	+	0.611448i	$0.790588\pi$
$774$	−12.0000			−0.431331
$775$	−21.0000	+	28.0000i	−0.754342	+	1.00579i
$776$		0			0
$777$		0			0
$778$		−	66.0000i		−	2.36621i
$779$		0			0
$780$		0			0
$781$	1.00000			0.0357828
$782$			24.0000i			0.858238i
$783$		0			0
$784$	36.0000			1.28571
$785$	8.00000	−	4.00000i	0.285532	−	0.142766i
$786$		0			0
$787$		−	22.0000i		−	0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$		−	22.0000i		−	0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$			36.0000i			1.28245i
$789$		0			0
$790$

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:	\( N \)	\(=\)	\( 1805 = 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1805.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -2.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -4.00000i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} -2.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -4.00000i q^{7} +3.00000 q^{9} +(-4.00000 + 2.00000i) q^{10} -1.00000 q^{11} -2.00000i q^{13} -8.00000 q^{14} -4.00000 q^{16} -2.00000i q^{17} -6.00000i q^{18} +(2.00000 + 4.00000i) q^{20} +2.00000i q^{22} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000 q^{26} +8.00000i q^{28} +9.00000 q^{29} +7.00000 q^{31} +8.00000i q^{32} -4.00000 q^{34} +(-8.00000 + 4.00000i) q^{35} -6.00000 q^{36} -2.00000i q^{37} -2.00000 q^{41} -2.00000i q^{43} +2.00000 q^{44} +(-3.00000 - 6.00000i) q^{45} -12.0000 q^{46} +6.00000i q^{47} -9.00000 q^{49} +(8.00000 + 6.00000i) q^{50} +4.00000i q^{52} +4.00000i q^{53} +(1.00000 + 2.00000i) q^{55} -18.0000i q^{58} +9.00000 q^{59} -7.00000 q^{61} -14.0000i q^{62} -12.0000i q^{63} +8.00000 q^{64} +(-4.00000 + 2.00000i) q^{65} +10.0000i q^{67} +4.00000i q^{68} +(8.00000 + 16.0000i) q^{70} -1.00000 q^{71} -10.0000i q^{73} -4.00000 q^{74} +4.00000i q^{77} +1.00000 q^{79} +(4.00000 + 8.00000i) q^{80} +9.00000 q^{81} +4.00000i q^{82} +6.00000i q^{83} +(-4.00000 + 2.00000i) q^{85} -4.00000 q^{86} -11.0000 q^{89} +(-12.0000 + 6.00000i) q^{90} -8.00000 q^{91} +12.0000i q^{92} +12.0000 q^{94} +6.00000i q^{97} +18.0000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 - 2 * q^5 + 6 * q^9	\( 2 q - 4 q^{4} - 2 q^{5} + 6 q^{9} - 8 q^{10} - 2 q^{11} - 16 q^{14} - 8 q^{16} + 4 q^{20} - 6 q^{25} - 8 q^{26} + 18 q^{29} + 14 q^{31} - 8 q^{34} - 16 q^{35} - 12 q^{36} - 4 q^{41} + 4 q^{44} - 6 q^{45} - 24 q^{46} - 18 q^{49} + 16 q^{50} + 2 q^{55} + 18 q^{59} - 14 q^{61} + 16 q^{64} - 8 q^{65} + 16 q^{70} - 2 q^{71} - 8 q^{74} + 2 q^{79} + 8 q^{80} + 18 q^{81} - 8 q^{85} - 8 q^{86} - 22 q^{89} - 24 q^{90} - 16 q^{91} + 24 q^{94} - 6 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 - 2 * q^5 + 6 * q^9 - 8 * q^10 - 2 * q^11 - 16 * q^14 - 8 * q^16 + 4 * q^20 - 6 * q^25 - 8 * q^26 + 18 * q^29 + 14 * q^31 - 8 * q^34 - 16 * q^35 - 12 * q^36 - 4 * q^41 + 4 * q^44 - 6 * q^45 - 24 * q^46 - 18 * q^49 + 16 * q^50 + 2 * q^55 + 18 * q^59 - 14 * q^61 + 16 * q^64 - 8 * q^65 + 16 * q^70 - 2 * q^71 - 8 * q^74 + 2 * q^79 + 8 * q^80 + 18 * q^81 - 8 * q^85 - 8 * q^86 - 22 * q^89 - 24 * q^90 - 16 * q^91 + 24 * q^94 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more