LMFDB - Embedded newform 1960.2.q.e.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(361,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("1960.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.q (of order $3$, degree $2$, not 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:	$15.6506787962$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1960.961
Dual form			1960.2.q.e.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +(2.50000 + 4.33013i) q^{11} -1.00000 q^{13} +1.00000 q^{15} +(1.50000 + 2.59808i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(3.00000 - 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} -5.00000 q^{27} -9.00000 q^{29} +(2.50000 - 4.33013i) q^{33} +(-3.00000 + 5.19615i) q^{37} +(0.500000 + 0.866025i) q^{39} -8.00000 q^{41} +6.00000 q^{43} +(1.00000 + 1.73205i) q^{45} +(1.50000 - 2.59808i) q^{47} +(1.50000 - 2.59808i) q^{51} +(6.00000 + 10.3923i) q^{53} -5.00000 q^{55} +6.00000 q^{57} +(4.00000 + 6.92820i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(0.500000 - 0.866025i) q^{65} +(2.00000 + 3.46410i) q^{67} -6.00000 q^{69} +8.00000 q^{71} +(5.00000 + 8.66025i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(1.50000 - 2.59808i) q^{79} +(-0.500000 - 0.866025i) q^{81} +12.0000 q^{83} -3.00000 q^{85} +(4.50000 + 7.79423i) q^{87} +(-8.00000 + 13.8564i) q^{89} +(-3.00000 - 5.19615i) q^{95} -7.00000 q^{97} +10.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - q^3 - q^5 + 2 * q^9	$ 2 q - q^{3} - q^{5} + 2 q^{9} + 5 q^{11} - 2 q^{13} + 2 q^{15} + 3 q^{17} - 6 q^{19} + 6 q^{23} - q^{25} - 10 q^{27} - 18 q^{29} + 5 q^{33} - 6 q^{37} + q^{39} - 16 q^{41} + 12 q^{43} + 2 q^{45} + 3 q^{47} + 3 q^{51} + 12 q^{53} - 10 q^{55} + 12 q^{57} + 8 q^{59} - 4 q^{61} + q^{65} + 4 q^{67} - 12 q^{69} + 16 q^{71} + 10 q^{73} - q^{75} + 3 q^{79} - q^{81} + 24 q^{83} - 6 q^{85} + 9 q^{87} - 16 q^{89} - 6 q^{95} - 14 q^{97} + 20 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 + 2 * q^9 + 5 * q^11 - 2 * q^13 + 2 * q^15 + 3 * q^17 - 6 * q^19 + 6 * q^23 - q^25 - 10 * q^27 - 18 * q^29 + 5 * q^33 - 6 * q^37 + q^39 - 16 * q^41 + 12 * q^43 + 2 * q^45 + 3 * q^47 + 3 * q^51 + 12 * q^53 - 10 * q^55 + 12 * q^57 + 8 * q^59 - 4 * q^61 + q^65 + 4 * q^67 - 12 * q^69 + 16 * q^71 + 10 * q^73 - q^75 + 3 * q^79 - q^81 + 24 * q^83 - 6 * q^85 + 9 * q^87 - 16 * q^89 - 6 * q^95 - 14 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$.

$n$	$981$	$1081$	$1177$	$1471$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i	0.684819	−	0.728714i	$-0.259881\pi$
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i	−0.973494	+	0.228714i	$0.926548\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000	−	1.73205i	0.333333	−	0.577350i
$10$		0			0
$11$	2.50000	+	4.33013i	0.753778	+	1.30558i	0.945979	+	0.324227i	$0.105104\pi$
$11$	2.50000	+	4.33013i	0.753778	+	1.30558i	−0.192201	+	0.981356i	$0.561563\pi$
$12$		0			0
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	0.284157	+	0.958778i	$0.408286\pi$
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	−0.972404	+	0.233301i	$0.925047\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$		0			0
$33$	2.50000	−	4.33013i	0.435194	−	0.753778i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.00000	+	5.19615i	−0.493197	+	0.854242i	−0.999969	−	0.00783774i	$-0.997505\pi$
$37$	−3.00000	+	5.19615i	−0.493197	+	0.854242i	0.506772	+	0.862080i	$0.330838\pi$
$38$		0			0
$39$	0.500000	+	0.866025i	0.0800641	+	0.138675i
$40$		0			0
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$		0			0
$45$	1.00000	+	1.73205i	0.149071	+	0.258199i
$46$		0			0
$47$	1.50000	−	2.59808i	0.218797	−	0.378968i	−0.735643	−	0.677369i	$-0.763120\pi$
$47$	1.50000	−	2.59808i	0.218797	−	0.378968i	0.954441	+	0.298401i	$0.0964533\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.50000	−	2.59808i	0.210042	−	0.363803i
$52$		0			0
$53$	6.00000	+	10.3923i	0.824163	+	1.42749i	0.902557	+	0.430570i	$0.141688\pi$
$53$	6.00000	+	10.3923i	0.824163	+	1.42749i	−0.0783936	+	0.996922i	$0.524979\pi$
$54$		0			0
$55$	−5.00000			−0.674200
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$	4.00000	+	6.92820i	0.520756	+	0.901975i	0.999709	+	0.0241347i	$0.00768307\pi$
$59$	4.00000	+	6.92820i	0.520756	+	0.901975i	−0.478953	+	0.877841i	$0.658984\pi$
$60$		0			0
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	−0.965187	−	0.261562i	$-0.915762\pi$
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	0.709113	+	0.705095i	$0.249096\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.500000	−	0.866025i	0.0620174	−	0.107417i
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	5.00000	+	8.66025i	0.585206	+	1.01361i	0.994850	+	0.101361i	$0.0323196\pi$
$73$	5.00000	+	8.66025i	0.585206	+	1.01361i	−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$	−0.500000	+	0.866025i	−0.0577350	+	0.100000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.50000	−	2.59808i	0.168763	−	0.292306i	−0.769222	−	0.638982i	$-0.779356\pi$
$79$	1.50000	−	2.59808i	0.168763	−	0.292306i	0.937985	+	0.346675i	$0.112689\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$	−3.00000			−0.325396
$86$		0			0
$87$	4.50000	+	7.79423i	0.482451	+	0.835629i
$88$		0			0
$89$	−8.00000	+	13.8564i	−0.847998	+	1.46878i	0.0349934	+	0.999388i	$0.488859\pi$
$89$	−8.00000	+	13.8564i	−0.847998	+	1.46878i	−0.882992	+	0.469389i	$0.844474\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.00000	−	5.19615i	−0.307794	−	0.533114i
$96$		0			0
$97$	−7.00000			−0.710742			−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000			−0.710742			−0.355371	+	0.934725i	$0.615646\pi$
$98$		0			0
$99$	10.0000			1.00504
$100$		0			0
$101$	−7.00000	−	12.1244i	−0.696526	−	1.20642i	−0.969664	−	0.244443i	$-0.921395\pi$
$101$	−7.00000	−	12.1244i	−0.696526	−	1.20642i	0.273138	−	0.961975i	$-0.411939\pi$
$102$		0			0
$103$	−4.50000	+	7.79423i	−0.443398	+	0.767988i	−0.997939	−	0.0641683i	$-0.979561\pi$
$103$	−4.50000	+	7.79423i	−0.443398	+	0.767988i	0.554541	+	0.832156i	$0.312894\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.00000	+	1.73205i	−0.0966736	+	0.167444i	−0.910306	−	0.413936i	$-0.864154\pi$
$107$	−1.00000	+	1.73205i	−0.0966736	+	0.167444i	0.813632	+	0.581380i	$0.197487\pi$
$108$		0			0
$109$	5.50000	+	9.52628i	0.526804	+	0.912452i	0.999512	+	0.0312328i	$0.00994332\pi$
$109$	5.50000	+	9.52628i	0.526804	+	0.912452i	−0.472708	+	0.881219i	$0.656723\pi$
$110$		0			0
$111$	6.00000			0.569495
$112$		0			0
$113$	−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
$115$	3.00000	+	5.19615i	0.279751	+	0.484544i
$116$		0			0
$117$	−1.00000	+	1.73205i	−0.0924500	+	0.160128i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.00000	+	12.1244i	−0.636364	+	1.10221i
$122$		0			0
$123$	4.00000	+	6.92820i	0.360668	+	0.624695i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	−3.00000	−	5.19615i	−0.264135	−	0.457496i
$130$		0			0
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	−0.966803	−	0.255524i	$-0.917752\pi$
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	0.704692	+	0.709514i	$0.251085\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.50000	−	4.33013i	0.215166	−	0.372678i
$136$		0			0
$137$	8.00000	+	13.8564i	0.683486	+	1.18383i	0.973910	+	0.226935i	$0.0728704\pi$
$137$	8.00000	+	13.8564i	0.683486	+	1.18383i	−0.290424	+	0.956898i	$0.593796\pi$
$138$		0			0
$139$	−18.0000			−1.52674			−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000			−1.52674			−0.763370	+	0.645961i	$0.776457\pi$
$140$		0			0
$141$	−3.00000			−0.252646
$142$		0			0
$143$	−2.50000	−	4.33013i	−0.209061	−	0.362103i
$144$		0			0
$145$	4.50000	−	7.79423i	0.373705	−	0.647275i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.00000	−	12.1244i	0.573462	−	0.993266i	−0.422744	−	0.906249i	$-0.638933\pi$
$149$	7.00000	−	12.1244i	0.573462	−	0.993266i	0.996207	−	0.0870170i	$-0.0277334\pi$
$150$		0			0
$151$	9.50000	+	16.4545i	0.773099	+	1.33905i	0.935857	+	0.352381i	$0.114628\pi$
$151$	9.50000	+	16.4545i	0.773099	+	1.33905i	−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$		0			0
$156$		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
$159$	6.00000	−	10.3923i	0.475831	−	0.824163i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3.00000	+	5.19615i	−0.234978	+	0.406994i	−0.959266	−	0.282503i	$-0.908835\pi$
$163$	−3.00000	+	5.19615i	−0.234978	+	0.406994i	0.724288	+	0.689497i	$0.242169\pi$
$164$		0			0
$165$	2.50000	+	4.33013i	0.194625	+	0.337100i
$166$		0			0
$167$	9.00000			0.696441			0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000			0.696441			0.348220	+	0.937413i	$0.386786\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$	6.00000	+	10.3923i	0.458831	+	0.794719i
$172$		0			0
$173$	9.50000	−	16.4545i	0.722272	−	1.25101i	−0.237816	−	0.971310i	$-0.576431\pi$
$173$	9.50000	−	16.4545i	0.722272	−	1.25101i	0.960087	−	0.279701i	$-0.0902353\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.00000	−	6.92820i	0.300658	−	0.520756i
$178$		0			0
$179$	−2.00000	−	3.46410i	−0.149487	−	0.258919i	0.781551	−	0.623841i	$-0.214429\pi$
$179$	−2.00000	−	3.46410i	−0.149487	−	0.258919i	−0.931038	+	0.364922i	$0.881096\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$	4.00000			0.295689
$184$		0			0
$185$	−3.00000	−	5.19615i	−0.220564	−	0.382029i
$186$		0			0
$187$	−7.50000	+	12.9904i	−0.548454	+	0.949951i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.50000	+	9.52628i	−0.397966	+	0.689297i	−0.993475	−	0.114051i	$-0.963617\pi$
$191$	−5.50000	+	9.52628i	−0.397966	+	0.689297i	0.595509	+	0.803349i	$0.296950\pi$
$192$		0			0
$193$	−4.00000	−	6.92820i	−0.287926	−	0.498703i	0.685388	−	0.728178i	$-0.259632\pi$
$193$	−4.00000	−	6.92820i	−0.287926	−	0.498703i	−0.973315	+	0.229475i	$0.926299\pi$
$194$		0			0
$195$	−1.00000			−0.0716115
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$		0			0
$201$	2.00000	−	3.46410i	0.141069	−	0.244339i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.00000	−	6.92820i	0.279372	−	0.483887i
$206$		0			0
$207$	−6.00000	−	10.3923i	−0.417029	−	0.722315i
$208$		0			0
$209$	−30.0000			−2.07514
$210$		0			0
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$	−4.00000	−	6.92820i	−0.274075	−	0.474713i
$214$		0			0
$215$	−3.00000	+	5.19615i	−0.204598	+	0.354375i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	5.00000	−	8.66025i	0.337869	−	0.585206i
$220$		0			0
$221$	−1.50000	−	2.59808i	−0.100901	−	0.174766i
$222$		0			0
$223$	25.0000			1.67412			0.837062	−	0.547108i	$-0.184271\pi$
$223$	25.0000			1.67412			0.837062	+	0.547108i	$0.184271\pi$
$224$		0			0
$225$	−2.00000			−0.133333
$226$		0			0
$227$	−6.50000	−	11.2583i	−0.431420	−	0.747242i	0.565576	−	0.824696i	$-0.308654\pi$
$227$	−6.50000	−	11.2583i	−0.431420	−	0.747242i	−0.996996	+	0.0774548i	$0.975321\pi$
$228$		0			0
$229$	8.00000	−	13.8564i	0.528655	−	0.915657i	−0.470787	−	0.882247i	$-0.656030\pi$
$229$	8.00000	−	13.8564i	0.528655	−	0.915657i	0.999442	−	0.0334101i	$-0.0106368\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.00000	+	6.92820i	−0.262049	+	0.453882i	−0.966786	−	0.255586i	$-0.917731\pi$
$233$	−4.00000	+	6.92820i	−0.262049	+	0.453882i	0.704737	+	0.709468i	$0.251065\pi$
$234$		0			0
$235$	1.50000	+	2.59808i	0.0978492	+	0.169480i
$236$		0			0
$237$	−3.00000			−0.194871
$238$		0			0
$239$	−7.00000			−0.452792			−0.226396	−	0.974035i	$-0.572694\pi$
$239$	−7.00000			−0.452792			−0.226396	+	0.974035i	$0.572694\pi$
$240$		0			0
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	0.995509	+	0.0946700i	$0.0301796\pi$
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	−0.415768	+	0.909471i	$0.636487\pi$
$242$		0			0
$243$	−8.00000	+	13.8564i	−0.513200	+	0.888889i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.00000	−	5.19615i	0.190885	−	0.330623i
$248$		0			0
$249$	−6.00000	−	10.3923i	−0.380235	−	0.658586i
$250$		0			0
$251$	−14.0000			−0.883672			−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000			−0.883672			−0.441836	+	0.897096i	$0.645673\pi$
$252$		0			0
$253$	30.0000			1.88608
$254$		0			0
$255$	1.50000	+	2.59808i	0.0939336	+	0.162698i
$256$		0			0
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	−0.757159	−	0.653231i	$-0.773413\pi$
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	0.944294	+	0.329104i	$0.106747\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−9.00000	+	15.5885i	−0.557086	+	0.964901i
$262$		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
$265$	−12.0000			−0.737154
$266$		0			0
$267$	16.0000			0.979184
$268$		0			0
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	0.977229	−	0.212187i	$-0.0680585\pi$
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	−0.672374	+	0.740212i	$0.734725\pi$
$270$		0			0
$271$	2.00000	−	3.46410i	0.121491	−	0.210429i	−0.798865	−	0.601511i	$-0.794566\pi$
$271$	2.00000	−	3.46410i	0.121491	−	0.210429i	0.920356	+	0.391082i	$0.127899\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.50000	−	4.33013i	0.150756	−	0.261116i
$276$		0			0
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	0.575408	−	0.817867i	$-0.304843\pi$
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	−0.995997	+	0.0893846i	$0.971510\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.0000			−0.775515			−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000			−0.775515			−0.387757	+	0.921761i	$0.626750\pi$
$282$		0			0
$283$	14.5000	+	25.1147i	0.861936	+	1.49292i	0.870058	+	0.492949i	$0.164081\pi$
$283$	14.5000	+	25.1147i	0.861936	+	1.49292i	−0.00812260	+	0.999967i	$0.502586\pi$
$284$		0			0
$285$	−3.00000	+	5.19615i	−0.177705	+	0.307794i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$	3.50000	+	6.06218i	0.205174	+	0.355371i
$292$		0			0
$293$	1.00000			0.0584206			0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000			0.0584206			0.0292103	+	0.999573i	$0.490701\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$		0			0
$297$	−12.5000	−	21.6506i	−0.725324	−	1.25630i
$298$		0			0
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−7.00000	+	12.1244i	−0.402139	+	0.696526i
$304$		0			0
$305$	−2.00000	−	3.46410i	−0.114520	−	0.198354i
$306$		0			0
$307$	27.0000			1.54097			0.770486	−	0.637457i	$-0.220014\pi$
$307$	27.0000			1.54097			0.770486	+	0.637457i	$0.220014\pi$
$308$		0			0
$309$	9.00000			0.511992
$310$		0			0
$311$	−7.00000	−	12.1244i	−0.396934	−	0.687509i	0.596412	−	0.802678i	$-0.296592\pi$
$311$	−7.00000	−	12.1244i	−0.396934	−	0.687509i	−0.993346	+	0.115169i	$0.963259\pi$
$312$		0			0
$313$	14.5000	−	25.1147i	0.819588	−	1.41957i	−0.0863973	−	0.996261i	$-0.527535\pi$
$313$	14.5000	−	25.1147i	0.819588	−	1.41957i	0.905986	−	0.423308i	$-0.139131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.0000	+	25.9808i	−0.842484	+	1.45922i	0.0453045	+	0.998973i	$0.485574\pi$
$317$	−15.0000	+	25.9808i	−0.842484	+	1.45922i	−0.887788	+	0.460252i	$0.847759\pi$
$318$		0			0
$319$	−22.5000	−	38.9711i	−1.25976	−	2.18197i
$320$		0			0
$321$	2.00000			0.111629
$322$		0			0
$323$	−18.0000			−1.00155
$324$		0			0
$325$	0.500000	+	0.866025i	0.0277350	+	0.0480384i
$326$		0			0
$327$	5.50000	−	9.52628i	0.304151	−	0.526804i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	−0.448649	−	0.893708i	$-0.648095\pi$
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	0.998298	−	0.0583130i	$-0.0185721\pi$
$332$		0			0
$333$	6.00000	+	10.3923i	0.328798	+	0.569495i
$334$		0			0
$335$	−4.00000			−0.218543
$336$		0			0
$337$	−26.0000			−1.41631			−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000			−1.41631			−0.708155	+	0.706057i	$0.750472\pi$
$338$		0			0
$339$	3.00000	+	5.19615i	0.162938	+	0.282216i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	3.00000	−	5.19615i	0.161515	−	0.279751i
$346$		0			0
$347$	5.00000	+	8.66025i	0.268414	+	0.464907i	0.968452	−	0.249198i	$-0.0801671\pi$
$347$	5.00000	+	8.66025i	0.268414	+	0.464907i	−0.700038	+	0.714105i	$0.746834\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	5.00000			0.266880
$352$		0			0
$353$	−16.5000	−	28.5788i	−0.878206	−	1.52110i	−0.853307	−	0.521408i	$-0.825407\pi$
$353$	−16.5000	−	28.5788i	−0.878206	−	1.52110i	−0.0248989	−	0.999690i	$-0.507926\pi$
$354$		0			0
$355$	−4.00000	+	6.92820i	−0.212298	+	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	−8.50000	−	14.7224i	−0.447368	−	0.774865i
$362$		0			0
$363$	14.0000			0.734809
$364$		0			0
$365$	−10.0000			−0.523424
$366$		0			0
$367$	−14.5000	−	25.1147i	−0.756894	−	1.31098i	−0.944427	−	0.328720i	$-0.893383\pi$
$367$	−14.5000	−	25.1147i	−0.756894	−	1.31098i	0.187533	−	0.982258i	$-0.439951\pi$
$368$		0			0
$369$	−8.00000	+	13.8564i	−0.416463	+	0.721336i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$373$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$374$		0			0
$375$	−0.500000	−	0.866025i	−0.0258199	−	0.0447214i
$376$		0			0
$377$	9.00000			0.463524
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$	−4.00000	−	6.92820i	−0.204926	−	0.354943i
$382$		0			0
$383$	6.00000	−	10.3923i	0.306586	−	0.531022i	−0.671027	−	0.741433i	$-0.734147\pi$
$383$	6.00000	−	10.3923i	0.306586	−	0.531022i	0.977613	+	0.210411i	$0.0674801\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	6.00000	−	10.3923i	0.304997	−	0.528271i
$388$		0			0
$389$	−12.5000	−	21.6506i	−0.633775	−	1.09773i	−0.986773	−	0.162107i	$-0.948171\pi$
$389$	−12.5000	−	21.6506i	−0.633775	−	1.09773i	0.352998	−	0.935624i	$-0.385162\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$	6.00000			0.302660
$394$		0			0
$395$	1.50000	+	2.59808i	0.0754732	+	0.130723i
$396$		0			0
$397$	−14.5000	+	25.1147i	−0.727734	+	1.26047i	0.230105	+	0.973166i	$0.426093\pi$
$397$	−14.5000	+	25.1147i	−0.727734	+	1.26047i	−0.957839	+	0.287307i	$0.907240\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.50000	+	7.79423i	−0.224719	+	0.389225i	−0.956235	−	0.292599i	$-0.905480\pi$
$401$	−4.50000	+	7.79423i	−0.224719	+	0.389225i	0.731516	+	0.681824i	$0.238813\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	−30.0000			−1.48704
$408$		0			0
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	0.985558	−	0.169338i	$-0.0541630\pi$
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	−0.639430	+	0.768849i	$0.720830\pi$
$410$		0			0
$411$	8.00000	−	13.8564i	0.394611	−	0.683486i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−6.00000	+	10.3923i	−0.294528	+	0.510138i
$416$		0			0
$417$	9.00000	+	15.5885i	0.440732	+	0.763370i
$418$		0			0
$419$	−36.0000			−1.75872			−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000			−1.75872			−0.879358	+	0.476162i	$0.842028\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$	−3.00000	−	5.19615i	−0.145865	−	0.252646i
$424$		0			0
$425$	1.50000	−	2.59808i	0.0727607	−	0.126025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−2.50000	+	4.33013i	−0.120701	+	0.209061i
$430$		0			0
$431$	−11.5000	−	19.9186i	−0.553936	−	0.959444i	−0.997985	−	0.0634424i	$-0.979792\pi$
$431$	−11.5000	−	19.9186i	−0.553936	−	0.959444i	0.444050	−	0.896002i	$-0.353541\pi$
$432$		0			0
$433$	30.0000			1.44171			0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000			1.44171			0.720854	+	0.693087i	$0.243750\pi$
$434$		0			0
$435$	−9.00000			−0.431517
$436$		0			0
$437$	18.0000	+	31.1769i	0.861057	+	1.49139i
$438$		0			0
$439$	−17.0000	+	29.4449i	−0.811366	+	1.40533i	0.100543	+	0.994933i	$0.467942\pi$
$439$	−17.0000	+	29.4449i	−0.811366	+	1.40533i	−0.911908	+	0.410394i	$0.865391\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−15.0000	+	25.9808i	−0.712672	+	1.23438i	0.251179	+	0.967941i	$0.419182\pi$
$443$	−15.0000	+	25.9808i	−0.712672	+	1.23438i	−0.963851	+	0.266443i	$0.914152\pi$
$444$		0			0
$445$	−8.00000	−	13.8564i	−0.379236	−	0.656857i
$446$		0			0
$447$	−14.0000			−0.662177
$448$		0			0
$449$	−33.0000			−1.55737			−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000			−1.55737			−0.778683	+	0.627417i	$0.784112\pi$
$450$		0			0
$451$	−20.0000	−	34.6410i	−0.941763	−	1.63118i
$452$		0			0
$453$	9.50000	−	16.4545i	0.446349	−	0.773099i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.0000	−	24.2487i	0.654892	−	1.13431i	−0.327028	−	0.945015i	$-0.606047\pi$
$457$	14.0000	−	24.2487i	0.654892	−	1.13431i	0.981921	−	0.189292i	$-0.0606194\pi$
$458$		0			0
$459$	−7.50000	−	12.9904i	−0.350070	−	0.606339i
$460$		0			0
$461$	24.0000			1.11779			0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000			1.11779			0.558896	+	0.829238i	$0.311225\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.5000	−	28.5788i	0.763529	−	1.32247i	−0.177492	−	0.984122i	$-0.556798\pi$
$467$	16.5000	−	28.5788i	0.763529	−	1.32247i	0.941021	−	0.338349i	$-0.109868\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.00000	+	1.73205i	−0.0460776	+	0.0798087i
$472$		0			0
$473$	15.0000	+	25.9808i	0.689701	+	1.19460i
$474$		0			0
$475$	6.00000			0.275299
$476$		0			0
$477$	24.0000			1.09888
$478$		0			0
$479$	−3.00000	−	5.19615i	−0.137073	−	0.237418i	0.789314	−	0.613990i	$-0.210436\pi$
$479$	−3.00000	−	5.19615i	−0.137073	−	0.237418i	−0.926388	+	0.376571i	$0.877103\pi$
$480$		0			0
$481$	3.00000	−	5.19615i	0.136788	−	0.236924i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.50000	−	6.06218i	0.158927	−	0.275269i
$486$		0			0
$487$	−13.0000	−	22.5167i	−0.589086	−	1.02033i	−0.994352	−	0.106129i	$-0.966154\pi$
$487$	−13.0000	−	22.5167i	−0.589086	−	1.02033i	0.405266	−	0.914199i	$-0.367179\pi$
$488$		0			0
$489$	6.00000			0.271329
$490$		0			0
$491$	33.0000			1.48927			0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000			1.48927			0.744635	+	0.667472i	$0.232624\pi$
$492$		0			0
$493$	−13.5000	−	23.3827i	−0.608009	−	1.05310i
$494$		0			0
$495$	−5.00000	+	8.66025i	−0.224733	+	0.389249i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.5000	−	21.6506i	0.559577	−	0.969216i	−0.437955	−	0.898997i	$-0.644297\pi$
$499$	12.5000	−	21.6506i	0.559577	−	0.969216i	0.997532	−	0.0702185i	$-0.0223697\pi$
$500$		0			0
$501$	−4.50000	−	7.79423i	−0.201045	−	0.348220i
$502$		0			0
$503$	31.0000			1.38222			0.691111	−	0.722749i	$-0.257122\pi$
$503$	31.0000			1.38222			0.691111	+	0.722749i	$0.257122\pi$
$504$		0			0
$505$	14.0000			0.622992
$506$		0			0
$507$	6.00000	+	10.3923i	0.266469	+	0.461538i
$508$		0			0
$509$	9.00000	−	15.5885i	0.398918	−	0.690946i	−0.594675	−	0.803966i	$-0.702719\pi$
$509$	9.00000	−	15.5885i	0.398918	−	0.690946i	0.993593	+	0.113020i	$0.0360525\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	15.0000	−	25.9808i	0.662266	−	1.14708i
$514$		0			0
$515$	−4.50000	−	7.79423i	−0.198294	−	0.343455i
$516$		0			0
$517$	15.0000			0.659699
$518$		0			0
$519$	−19.0000			−0.834007
$520$		0			0
$521$	−3.00000	−	5.19615i	−0.131432	−	0.227648i	0.792797	−	0.609486i	$-0.208624\pi$
$521$	−3.00000	−	5.19615i	−0.131432	−	0.227648i	−0.924229	+	0.381839i	$0.875291\pi$
$522$		0			0
$523$	2.00000	−	3.46410i	0.0874539	−	0.151475i	−0.818980	−	0.573822i	$-0.805460\pi$
$523$	2.00000	−	3.46410i	0.0874539	−	0.151475i	0.906434	+	0.422347i	$0.138794\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$	16.0000			0.694341
$532$		0			0
$533$	8.00000			0.346518
$534$		0			0
$535$	−1.00000	−	1.73205i	−0.0432338	−	0.0748831i
$536$		0			0
$537$	−2.00000	+	3.46410i	−0.0863064	+	0.149487i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4.50000	−	7.79423i	0.193470	−	0.335100i	−0.752928	−	0.658103i	$-0.771359\pi$
$541$	4.50000	−	7.79423i	0.193470	−	0.335100i	0.946398	+	0.323003i	$0.104692\pi$
$542$		0			0
$543$	−10.0000	−	17.3205i	−0.429141	−	0.743294i
$544$		0			0
$545$	−11.0000			−0.471188
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$		0			0
$549$	4.00000	+	6.92820i	0.170716	+	0.295689i
$550$		0			0
$551$	27.0000	−	46.7654i	1.15024	−	1.99227i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−3.00000	+	5.19615i	−0.127343	+	0.220564i
$556$		0			0
$557$	−12.0000	−	20.7846i	−0.508456	−	0.880672i	−0.999952	−	0.00979220i	$-0.996883\pi$
$557$	−12.0000	−	20.7846i	−0.508456	−	0.880672i	0.491496	−	0.870880i	$-0.336450\pi$
$558$		0			0
$559$	−6.00000			−0.253773
$560$		0			0
$561$	15.0000			0.633300
$562$		0			0
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	0.905088	−	0.425223i	$-0.139804\pi$
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	−0.820798	+	0.571218i	$0.806471\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17.0000	+	29.4449i	−0.712677	+	1.23439i	0.251172	+	0.967943i	$0.419184\pi$
$569$	−17.0000	+	29.4449i	−0.712677	+	1.23439i	−0.963849	+	0.266450i	$0.914149\pi$
$570$		0			0
$571$	18.0000	+	31.1769i	0.753277	+	1.30471i	0.946227	+	0.323505i	$0.104861\pi$
$571$	18.0000	+	31.1769i	0.753277	+	1.30471i	−0.192950	+	0.981209i	$0.561806\pi$
$572$		0			0
$573$	11.0000			0.459532
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	8.50000	+	14.7224i	0.353860	+	0.612903i	0.986922	−	0.161198i	$-0.0515357\pi$
$577$	8.50000	+	14.7224i	0.353860	+	0.612903i	−0.633062	+	0.774101i	$0.718202\pi$
$578$		0			0
$579$	−4.00000	+	6.92820i	−0.166234	+	0.287926i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−30.0000	+	51.9615i	−1.24247	+	2.15203i
$584$		0			0
$585$	−1.00000	−	1.73205i	−0.0413449	−	0.0716115i
$586$		0			0
$587$	−8.00000			−0.330195			−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000			−0.330195			−0.165098	+	0.986277i	$0.552794\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	6.00000	+	10.3923i	0.246807	+	0.427482i
$592$		0			0
$593$	−3.50000	+	6.06218i	−0.143728	+	0.248944i	−0.928898	−	0.370337i	$-0.879242\pi$
$593$	−3.50000	+	6.06218i	−0.143728	+	0.248944i	0.785170	+	0.619281i	$0.212576\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.5000	+	28.5788i	0.674172	+	1.16770i	0.976710	+	0.214563i	$0.0688326\pi$
$599$	16.5000	+	28.5788i	0.674172	+	1.16770i	−0.302539	+	0.953137i	$0.597834\pi$
$600$		0			0
$601$	34.0000			1.38689			0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000			1.38689			0.693444	+	0.720510i	$0.256092\pi$
$602$		0			0
$603$	8.00000			0.325785
$604$		0			0
$605$	−7.00000	−	12.1244i	−0.284590	−	0.492925i
$606$		0			0
$607$	8.50000	−	14.7224i	0.345004	−	0.597565i	−0.640350	−	0.768083i	$-0.721211\pi$
$607$	8.50000	−	14.7224i	0.345004	−	0.597565i	0.985355	+	0.170518i	$0.0545441\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.50000	+	2.59808i	−0.0606835	+	0.105107i
$612$		0			0
$613$	11.0000	+	19.0526i	0.444286	+	0.769526i	0.998002	−	0.0631797i	$-0.0201241\pi$
$613$	11.0000	+	19.0526i	0.444286	+	0.769526i	−0.553716	+	0.832705i	$0.686791\pi$
$614$		0			0
$615$	−8.00000			−0.322591
$616$		0			0
$617$	10.0000			0.402585			0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000			0.402585			0.201292	+	0.979531i	$0.435486\pi$
$618$		0			0
$619$	1.00000	+	1.73205i	0.0401934	+	0.0696170i	0.885422	−	0.464787i	$-0.153869\pi$
$619$	1.00000	+	1.73205i	0.0401934	+	0.0696170i	−0.845229	+	0.534404i	$0.820536\pi$
$620$		0			0
$621$	−15.0000	+	25.9808i	−0.601929	+	1.04257i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	15.0000	+	25.9808i	0.599042	+	1.03757i
$628$		0			0
$629$	−18.0000			−0.717707
$630$		0			0
$631$	−9.00000			−0.358284			−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000			−0.358284			−0.179142	+	0.983823i	$0.557332\pi$
$632$		0			0
$633$	−6.50000	−	11.2583i	−0.258352	−	0.447478i
$634$		0			0
$635$	−4.00000	+	6.92820i	−0.158735	+	0.274937i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	8.00000	−	13.8564i	0.316475	−	0.548151i
$640$		0			0
$641$	−17.0000	−	29.4449i	−0.671460	−	1.16300i	−0.977490	−	0.210981i	$-0.932334\pi$
$641$	−17.0000	−	29.4449i	−0.671460	−	1.16300i	0.306031	−	0.952022i	$-0.400999\pi$
$642$		0			0
$643$	−47.0000			−1.85350			−0.926750	−	0.375680i	$-0.877409\pi$
$643$	−47.0000			−1.85350			−0.926750	+	0.375680i	$0.877409\pi$
$644$		0			0
$645$	6.00000			0.236250
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$	−20.0000	+	34.6410i	−0.785069	+	1.35978i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.0000	+	25.9808i	−0.586995	+	1.01671i	0.407628	+	0.913148i	$0.366356\pi$
$653$	−15.0000	+	25.9808i	−0.586995	+	1.01671i	−0.994623	+	0.103558i	$0.966977\pi$
$654$		0			0
$655$	−3.00000	−	5.19615i	−0.117220	−	0.203030i
$656$		0			0
$657$	20.0000			0.780274
$658$		0			0
$659$	−25.0000			−0.973862			−0.486931	−	0.873441i	$-0.661884\pi$
$659$	−25.0000			−0.973862			−0.486931	+	0.873441i	$0.661884\pi$
$660$		0			0
$661$	4.00000	+	6.92820i	0.155582	+	0.269476i	0.933271	−	0.359174i	$-0.116941\pi$
$661$	4.00000	+	6.92820i	0.155582	+	0.269476i	−0.777689	+	0.628649i	$0.783608\pi$
$662$		0			0
$663$	−1.50000	+	2.59808i	−0.0582552	+	0.100901i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−27.0000	+	46.7654i	−1.04544	+	1.81076i
$668$		0			0
$669$	−12.5000	−	21.6506i	−0.483278	−	0.837062i
$670$		0			0
$671$	−20.0000			−0.772091
$672$		0			0
$673$	32.0000			1.23351			0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000			1.23351			0.616755	+	0.787155i	$0.288447\pi$
$674$		0			0
$675$	2.50000	+	4.33013i	0.0962250	+	0.166667i
$676$		0			0
$677$	16.5000	−	28.5788i	0.634147	−	1.09837i	−0.352549	−	0.935793i	$-0.614685\pi$
$677$	16.5000	−	28.5788i	0.634147	−	1.09837i	0.986695	−	0.162581i	$-0.0519817\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−6.50000	+	11.2583i	−0.249081	+	0.431420i
$682$		0			0
$683$	10.0000	+	17.3205i	0.382639	+	0.662751i	0.991439	−	0.130573i	$-0.0416818\pi$
$683$	10.0000	+	17.3205i	0.382639	+	0.662751i	−0.608799	+	0.793324i	$0.708349\pi$
$684$		0			0
$685$	−16.0000			−0.611329
$686$		0			0
$687$	−16.0000			−0.610438
$688$		0			0
$689$	−6.00000	−	10.3923i	−0.228582	−	0.395915i
$690$		0			0
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	−0.610704	−	0.791859i	$-0.709113\pi$
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	0.991122	+	0.132956i	$0.0424468\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.00000	−	15.5885i	0.341389	−	0.591304i
$696$		0			0
$697$	−12.0000	−	20.7846i	−0.454532	−	0.787273i
$698$		0			0
$699$	8.00000			0.302588
$700$		0			0
$701$	−21.0000			−0.793159			−0.396580	−	0.918000i	$-0.629803\pi$
$701$	−21.0000			−0.793159			−0.396580	+	0.918000i	$0.629803\pi$
$702$		0			0
$703$	−18.0000	−	31.1769i	−0.678883	−	1.17586i
$704$		0			0
$705$	1.50000	−	2.59808i	0.0564933	−	0.0978492i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	20.5000	−	35.5070i	0.769894	−	1.33349i	−0.167727	−	0.985834i	$-0.553643\pi$
$709$	20.5000	−	35.5070i	0.769894	−	1.33349i	0.937620	−	0.347661i	$-0.113024\pi$
$710$		0			0
$711$	−3.00000	−	5.19615i	−0.112509	−	0.194871i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	5.00000			0.186989
$716$		0			0
$717$	3.50000	+	6.06218i	0.130710	+	0.226396i
$718$		0			0
$719$	25.0000	−	43.3013i	0.932343	−	1.61486i	0.153037	−	0.988220i	$-0.451094\pi$
$719$	25.0000	−	43.3013i	0.932343	−	1.61486i	0.779305	−	0.626644i	$-0.215572\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	9.00000	−	15.5885i	0.334714	−	0.579741i
$724$		0			0
$725$	4.50000	+	7.79423i	0.167126	+	0.289470i
$726$		0			0
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	9.00000	+	15.5885i	0.332877	+	0.576560i
$732$		0			0
$733$	2.50000	−	4.33013i	0.0923396	−	0.159937i	−0.816156	−	0.577832i	$-0.803899\pi$
$733$	2.50000	−	4.33013i	0.0923396	−	0.159937i	0.908495	+	0.417895i	$0.137232\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10.0000	+	17.3205i	−0.368355	+	0.638009i
$738$		0			0
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	0.974818	+	0.223001i	$0.0715853\pi$
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$	−6.00000			−0.220416
$742$		0			0
$743$	−36.0000			−1.32071			−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000			−1.32071			−0.660356	+	0.750953i	$0.729595\pi$
$744$		0			0
$745$	7.00000	+	12.1244i	0.256460	+	0.444202i
$746$		0			0
$747$	12.0000	−	20.7846i	0.439057	−	0.760469i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	17.5000	−	30.3109i	0.638584	−	1.10606i	−0.347160	−	0.937806i	$-0.612854\pi$
$751$	17.5000	−	30.3109i	0.638584	−	1.10606i	0.985744	−	0.168254i	$-0.0538129\pi$
$752$		0			0
$753$	7.00000	+	12.1244i	0.255094	+	0.441836i
$754$		0			0
$755$	−19.0000			−0.691481
$756$		0			0
$757$	8.00000			0.290765			0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000			0.290765			0.145382	+	0.989376i	$0.453559\pi$
$758$		0			0
$759$	−15.0000	−	25.9808i	−0.544466	−	0.943042i
$760$		0			0
$761$	−23.0000	+	39.8372i	−0.833749	+	1.44410i	0.0612953	+	0.998120i	$0.480477\pi$
$761$	−23.0000	+	39.8372i	−0.833749	+	1.44410i	−0.895045	+	0.445977i	$0.852856\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−3.00000	+	5.19615i	−0.108465	+	0.187867i
$766$		0			0
$767$	−4.00000	−	6.92820i	−0.144432	−	0.250163i
$768$		0			0
$769$	10.0000			0.360609			0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000			0.360609			0.180305	+	0.983611i	$0.442292\pi$
$770$		0			0
$771$	−6.00000			−0.216085
$772$		0			0
$773$	0.500000	+	0.866025i	0.0179838	+	0.0311488i	0.874877	−	0.484345i	$-0.160942\pi$
$773$	0.500000	+	0.866025i	0.0179838	+	0.0311488i	−0.856893	+	0.515494i	$0.827609\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.0000	−	41.5692i	0.859889	−	1.48937i
$780$		0			0
$781$	20.0000	+	34.6410i	0.715656	+	1.23955i
$782$		0			0
$783$	45.0000			1.60817
$784$		0			0
$785$	2.00000			0.0713831
$786$		0			0
$787$	5.50000	+	9.52628i	0.196054	+	0.339575i	0.947245	−	0.320509i	$-0.103854\pi$
$787$	5.50000	+	9.52628i	0.196054	+	0.339575i	−0.751192	+	0.660084i	$0.770521\pi$
$788$		0			0
$789$	−9.00000	+	15.5885i	−0.320408	+	0.554964i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.00000	−	3.46410i	0.0710221	−	0.123014i

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:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +(2.50000 + 4.33013i) q^{11} -1.00000 q^{13} +1.00000 q^{15} +(1.50000 + 2.59808i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(3.00000 - 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} -5.00000 q^{27} -9.00000 q^{29} +(2.50000 - 4.33013i) q^{33} +(-3.00000 + 5.19615i) q^{37} +(0.500000 + 0.866025i) q^{39} -8.00000 q^{41} +6.00000 q^{43} +(1.00000 + 1.73205i) q^{45} +(1.50000 - 2.59808i) q^{47} +(1.50000 - 2.59808i) q^{51} +(6.00000 + 10.3923i) q^{53} -5.00000 q^{55} +6.00000 q^{57} +(4.00000 + 6.92820i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(0.500000 - 0.866025i) q^{65} +(2.00000 + 3.46410i) q^{67} -6.00000 q^{69} +8.00000 q^{71} +(5.00000 + 8.66025i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(1.50000 - 2.59808i) q^{79} +(-0.500000 - 0.866025i) q^{81} +12.0000 q^{83} -3.00000 q^{85} +(4.50000 + 7.79423i) q^{87} +(-8.00000 + 13.8564i) q^{89} +(-3.00000 - 5.19615i) q^{95} -7.00000 q^{97} +10.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - q^3 - q^5 + 2 * q^9	\( 2 q - q^{3} - q^{5} + 2 q^{9} + 5 q^{11} - 2 q^{13} + 2 q^{15} + 3 q^{17} - 6 q^{19} + 6 q^{23} - q^{25} - 10 q^{27} - 18 q^{29} + 5 q^{33} - 6 q^{37} + q^{39} - 16 q^{41} + 12 q^{43} + 2 q^{45} + 3 q^{47} + 3 q^{51} + 12 q^{53} - 10 q^{55} + 12 q^{57} + 8 q^{59} - 4 q^{61} + q^{65} + 4 q^{67} - 12 q^{69} + 16 q^{71} + 10 q^{73} - q^{75} + 3 q^{79} - q^{81} + 24 q^{83} - 6 q^{85} + 9 q^{87} - 16 q^{89} - 6 q^{95} - 14 q^{97} + 20 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 + 2 * q^9 + 5 * q^11 - 2 * q^13 + 2 * q^15 + 3 * q^17 - 6 * q^19 + 6 * q^23 - q^25 - 10 * q^27 - 18 * q^29 + 5 * q^33 - 6 * q^37 + q^39 - 16 * q^41 + 12 * q^43 + 2 * q^45 + 3 * q^47 + 3 * q^51 + 12 * q^53 - 10 * q^55 + 12 * q^57 + 8 * q^59 - 4 * q^61 + q^65 + 4 * q^67 - 12 * q^69 + 16 * q^71 + 10 * q^73 - q^75 + 3 * q^79 - q^81 + 24 * q^83 - 6 * q^85 + 9 * q^87 - 16 * q^89 - 6 * q^95 - 14 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more