LMFDB - Embedded newform 975.2.bc.c.901.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(751,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("975.751");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.bc (of order $6$, 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:	$7.78541419707$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			901.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	975.901
Dual form			975.2.bc.c.751.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(1.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(1.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} +2.00000 q^{12} +(-3.50000 + 0.866025i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(3.00000 - 1.73205i) q^{19} +1.73205i q^{21} +(-3.00000 + 5.19615i) q^{23} +1.00000 q^{27} +(-3.00000 - 1.73205i) q^{28} +(-3.00000 + 5.19615i) q^{29} +1.73205i q^{31} +(3.00000 - 1.73205i) q^{33} +(-1.00000 + 1.73205i) q^{36} +(1.00000 - 3.46410i) q^{39} +(-6.00000 - 3.46410i) q^{41} +(-0.500000 - 0.866025i) q^{43} +6.92820i q^{44} -3.46410i q^{47} +(-2.00000 - 3.46410i) q^{48} +(-2.00000 + 3.46410i) q^{49} +(5.00000 + 5.19615i) q^{52} -12.0000 q^{53} +3.46410i q^{57} +(-3.00000 + 1.73205i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(-1.50000 - 0.866025i) q^{63} +8.00000 q^{64} +(-7.50000 - 4.33013i) q^{67} +(-3.00000 - 5.19615i) q^{69} +(-9.00000 + 5.19615i) q^{71} -1.73205i q^{73} +(-6.00000 - 3.46410i) q^{76} -6.00000 q^{77} -11.0000 q^{79} +(-0.500000 + 0.866025i) q^{81} +13.8564i q^{83} +(3.00000 - 1.73205i) q^{84} +(-3.00000 - 5.19615i) q^{87} +(6.00000 + 3.46410i) q^{89} +(-4.50000 + 4.33013i) q^{91} +12.0000 q^{92} +(-1.50000 - 0.866025i) q^{93} +(4.50000 - 2.59808i) q^{97} +3.46410i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{4} + 3 q^{7} - q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^4 + 3 * q^7 - q^9	$ 2 q - q^{3} - 2 q^{4} + 3 q^{7} - q^{9} - 6 q^{11} + 4 q^{12} - 7 q^{13} - 4 q^{16} + 6 q^{19} - 6 q^{23} + 2 q^{27} - 6 q^{28} - 6 q^{29} + 6 q^{33} - 2 q^{36} + 2 q^{39} - 12 q^{41} - q^{43} - 4 q^{48} - 4 q^{49} + 10 q^{52} - 24 q^{53} - 6 q^{59} - q^{61} - 3 q^{63} + 16 q^{64} - 15 q^{67} - 6 q^{69} - 18 q^{71} - 12 q^{76} - 12 q^{77} - 22 q^{79} - q^{81} + 6 q^{84} - 6 q^{87} + 12 q^{89} - 9 q^{91} + 24 q^{92} - 3 q^{93} + 9 q^{97}+O(q^{100}) $ 2 * q - q^3 - 2 * q^4 + 3 * q^7 - q^9 - 6 * q^11 + 4 * q^12 - 7 * q^13 - 4 * q^16 + 6 * q^19 - 6 * q^23 + 2 * q^27 - 6 * q^28 - 6 * q^29 + 6 * q^33 - 2 * q^36 + 2 * q^39 - 12 * q^41 - q^43 - 4 * q^48 - 4 * q^49 + 10 * q^52 - 24 * q^53 - 6 * q^59 - q^61 - 3 * q^63 + 16 * q^64 - 15 * q^67 - 6 * q^69 - 18 * q^71 - 12 * q^76 - 12 * q^77 - 22 * q^79 - q^81 + 6 * q^84 - 6 * q^87 + 12 * q^89 - 9 * q^91 + 24 * q^92 - 3 * q^93 + 9 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.50000	−	0.866025i	0.566947	−	0.327327i	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	1.50000	−	0.866025i	0.566947	−	0.327327i	0.755929	+	0.654654i	$0.227186\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.0258656	−	0.999665i	$-0.508234\pi$
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.878668	+	0.477432i	$0.841568\pi$
$12$	2.00000			0.577350
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	0.802955	+	0.596040i	$0.203260\pi$
$20$		0			0
$21$			1.73205i			0.377964i
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.00000			0.192450
$28$	−3.00000	−	1.73205i	−0.566947	−	0.327327i
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$		0			0
$31$			1.73205i			0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$			1.73205i			0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$	3.00000	−	1.73205i	0.522233	−	0.301511i
$34$		0			0
$35$		0			0
$36$	−1.00000	+	1.73205i	−0.166667	+	0.288675i
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	1.00000	−	3.46410i	0.160128	−	0.554700i
$40$		0			0
$41$	−6.00000	−	3.46410i	−0.937043	−	0.541002i	−0.0480106	−	0.998847i	$-0.515288\pi$
$41$	−6.00000	−	3.46410i	−0.937043	−	0.541002i	−0.889032	+	0.457845i	$0.848621\pi$
$42$		0			0
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	0.825380	−	0.564578i	$-0.190961\pi$
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	−0.901629	+	0.432511i	$0.857628\pi$
$44$			6.92820i			1.04447i
$45$		0			0
$46$		0			0
$47$		−	3.46410i		−	0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$		−	3.46410i		−	0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$	−2.00000	−	3.46410i	−0.288675	−	0.500000i
$49$	−2.00000	+	3.46410i	−0.285714	+	0.494872i
$50$		0			0
$51$		0			0
$52$	5.00000	+	5.19615i	0.693375	+	0.720577i
$53$	−12.0000			−1.64833			−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000			−1.64833			−0.824163	+	0.566352i	$0.808354\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			3.46410i			0.458831i
$58$		0			0
$59$	−3.00000	+	1.73205i	−0.390567	+	0.225494i	−0.682406	−	0.730974i	$-0.739066\pi$
$59$	−3.00000	+	1.73205i	−0.390567	+	0.225494i	0.291839	+	0.956467i	$0.405733\pi$
$60$		0			0
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	0.832240	−	0.554416i	$-0.187058\pi$
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	−1.50000	−	0.866025i	−0.188982	−	0.109109i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−7.50000	−	4.33013i	−0.916271	−	0.529009i	−0.0338274	−	0.999428i	$-0.510770\pi$
$67$	−7.50000	−	4.33013i	−0.916271	−	0.529009i	−0.882443	+	0.470418i	$0.844103\pi$
$68$		0			0
$69$	−3.00000	−	5.19615i	−0.361158	−	0.625543i
$70$		0			0
$71$	−9.00000	+	5.19615i	−1.06810	+	0.616670i	−0.927663	−	0.373419i	$-0.878185\pi$
$71$	−9.00000	+	5.19615i	−1.06810	+	0.616670i	−0.140441	+	0.990089i	$0.544852\pi$
$72$		0			0
$73$		−	1.73205i		−	0.202721i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$		−	1.73205i		−	0.202721i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$		0			0
$76$	−6.00000	−	3.46410i	−0.688247	−	0.397360i
$77$	−6.00000			−0.683763
$78$		0			0
$79$	−11.0000			−1.23760			−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000			−1.23760			−0.618798	+	0.785550i	$0.712380\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$			13.8564i			1.52094i	0.649374	+	0.760469i	$0.275031\pi$
$83$			13.8564i			1.52094i	−0.649374	+	0.760469i	$0.724969\pi$
$84$	3.00000	−	1.73205i	0.327327	−	0.188982i
$85$		0			0
$86$		0			0
$87$	−3.00000	−	5.19615i	−0.321634	−	0.557086i
$88$		0			0
$89$	6.00000	+	3.46410i	0.635999	+	0.367194i	0.783072	−	0.621932i	$-0.213652\pi$
$89$	6.00000	+	3.46410i	0.635999	+	0.367194i	−0.147073	+	0.989126i	$0.546985\pi$
$90$		0			0
$91$	−4.50000	+	4.33013i	−0.471728	+	0.453921i
$92$	12.0000			1.25109
$93$	−1.50000	−	0.866025i	−0.155543	−	0.0898027i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.50000	−	2.59808i	0.456906	−	0.263795i	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	4.50000	−	2.59808i	0.456906	−	0.263795i	0.710742	+	0.703452i	$0.248359\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.0623905	−	0.998052i	$-0.480128\pi$
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.833143	−	0.553058i	$-0.186539\pi$
$102$		0			0
$103$	1.00000			0.0985329			0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000			0.0985329			0.0492665	+	0.998786i	$0.484312\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	−0.973814	−	0.227345i	$-0.926996\pi$
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	0.683793	+	0.729676i	$0.260329\pi$
$108$	−1.00000	−	1.73205i	−0.0962250	−	0.166667i
$109$		−	15.5885i		−	1.49310i	−0.665327	−	0.746552i	$-0.731708\pi$
$109$		−	15.5885i		−	1.49310i	0.665327	−	0.746552i	$-0.268292\pi$
$110$		0			0
$111$		0			0
$112$			6.92820i			0.654654i
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$		0			0
$116$	12.0000			1.11417
$117$	2.50000	+	2.59808i	0.231125	+	0.240192i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$		0			0
$123$	6.00000	−	3.46410i	0.541002	−	0.312348i
$124$	3.00000	−	1.73205i	0.269408	−	0.155543i
$125$		0			0
$126$		0			0
$127$	6.50000	−	11.2583i	0.576782	−	0.999015i	−0.419064	−	0.907957i	$-0.637642\pi$
$127$	6.50000	−	11.2583i	0.576782	−	0.999015i	0.995846	−	0.0910585i	$-0.0290250\pi$
$128$		0			0
$129$	1.00000			0.0880451
$130$		0			0
$131$	6.00000			0.524222			0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000			0.524222			0.262111	+	0.965038i	$0.415581\pi$
$132$	−6.00000	−	3.46410i	−0.522233	−	0.301511i
$133$	3.00000	−	5.19615i	0.260133	−	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	0.740308	−	0.672268i	$-0.234680\pi$
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	−0.952355	+	0.304991i	$0.901346\pi$
$140$		0			0
$141$	3.00000	+	1.73205i	0.252646	+	0.145865i
$142$		0			0
$143$	12.0000	+	3.46410i	1.00349	+	0.289683i
$144$	4.00000			0.333333
$145$		0			0
$146$		0			0
$147$	−2.00000	−	3.46410i	−0.164957	−	0.285714i
$148$		0			0
$149$	−6.00000	+	3.46410i	−0.491539	+	0.283790i	−0.725213	−	0.688525i	$-0.758259\pi$
$149$	−6.00000	+	3.46410i	−0.491539	+	0.283790i	0.233674	+	0.972315i	$0.424925\pi$
$150$		0			0
$151$		−	3.46410i		−	0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$		−	3.46410i		−	0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−7.00000	+	1.73205i	−0.560449	+	0.138675i
$157$	−11.0000			−0.877896			−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000			−0.877896			−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$	6.00000	−	10.3923i	0.475831	−	0.824163i
$160$		0			0
$161$			10.3923i			0.819028i
$162$		0			0
$163$	16.5000	−	9.52628i	1.29238	−	0.746156i	0.313304	−	0.949653i	$-0.398564\pi$
$163$	16.5000	−	9.52628i	1.29238	−	0.746156i	0.979076	+	0.203497i	$0.0652307\pi$
$164$			13.8564i			1.08200i
$165$		0			0
$166$		0			0
$167$	6.00000	+	3.46410i	0.464294	+	0.268060i	0.713848	−	0.700301i	$-0.246951\pi$
$167$	6.00000	+	3.46410i	0.464294	+	0.268060i	−0.249554	+	0.968361i	$0.580284\pi$
$168$		0			0
$169$	11.5000	−	6.06218i	0.884615	−	0.466321i
$170$		0			0
$171$	−3.00000	−	1.73205i	−0.229416	−	0.132453i
$172$	−1.00000	+	1.73205i	−0.0762493	+	0.132068i
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	0.729155	−	0.684349i	$-0.239913\pi$
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	−0.957241	+	0.289292i	$0.906580\pi$
$174$		0			0
$175$		0			0
$176$	12.0000	−	6.92820i	0.904534	−	0.522233i
$177$		−	3.46410i		−	0.260378i
$178$		0			0
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	−0.998286	−	0.0585225i	$-0.981361\pi$
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	0.549825	+	0.835280i	$0.314694\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$	1.00000			0.0739221
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−6.00000	+	3.46410i	−0.437595	+	0.252646i
$189$	1.50000	−	0.866025i	0.109109	−	0.0629941i
$190$		0			0
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	0.982828	+	0.184525i	$0.0590746\pi$
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	−0.331611	+	0.943416i	$0.607592\pi$
$192$	−4.00000	+	6.92820i	−0.288675	+	0.500000i
$193$	13.5000	+	7.79423i	0.971751	+	0.561041i	0.899770	−	0.436365i	$-0.143734\pi$
$193$	13.5000	+	7.79423i	0.971751	+	0.561041i	0.0719816	+	0.997406i	$0.477068\pi$
$194$		0			0
$195$		0			0
$196$	8.00000			0.571429
$197$	−12.0000	−	6.92820i	−0.854965	−	0.493614i	0.00735824	−	0.999973i	$-0.497658\pi$
$197$	−12.0000	−	6.92820i	−0.854965	−	0.493614i	−0.862323	+	0.506359i	$0.830991\pi$
$198$		0			0
$199$	−3.50000	−	6.06218i	−0.248108	−	0.429736i	0.714893	−	0.699234i	$-0.246476\pi$
$199$	−3.50000	−	6.06218i	−0.248108	−	0.429736i	−0.963001	+	0.269498i	$0.913142\pi$
$200$		0			0
$201$	7.50000	−	4.33013i	0.529009	−	0.305424i
$202$		0			0
$203$			10.3923i			0.729397i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	6.00000			0.417029
$208$	4.00000	−	13.8564i	0.277350	−	0.960769i
$209$	−12.0000			−0.830057
$210$		0			0
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	0.550743	+	0.834675i	$0.314345\pi$
$212$	12.0000	+	20.7846i	0.824163	+	1.42749i
$213$		−	10.3923i		−	0.712069i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.50000	+	2.59808i	0.101827	+	0.176369i
$218$		0			0
$219$	1.50000	+	0.866025i	0.101361	+	0.0585206i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−15.0000	−	8.66025i	−1.00447	−	0.579934i	−0.0949052	−	0.995486i	$-0.530255\pi$
$223$	−15.0000	−	8.66025i	−1.00447	−	0.579934i	−0.909569	+	0.415553i	$0.863588\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.0000	+	10.3923i	−1.19470	+	0.689761i	−0.959369	−	0.282153i	$-0.908951\pi$
$227$	−18.0000	+	10.3923i	−1.19470	+	0.689761i	−0.235333	+	0.971915i	$0.575618\pi$
$228$	6.00000	−	3.46410i	0.397360	−	0.229416i
$229$			27.7128i			1.83131i	0.401960	+	0.915657i	$0.368329\pi$
$229$			27.7128i			1.83131i	−0.401960	+	0.915657i	$0.631671\pi$
$230$		0			0
$231$	3.00000	−	5.19615i	0.197386	−	0.341882i
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$		0			0
$236$	6.00000	+	3.46410i	0.390567	+	0.225494i
$237$	5.50000	−	9.52628i	0.357263	−	0.618798i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	18.0000	−	10.3923i	1.15948	−	0.669427i	0.208302	−	0.978065i	$-0.433206\pi$
$241$	18.0000	−	10.3923i	1.15948	−	0.669427i	0.951180	+	0.308637i	$0.0998729\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$	−1.00000	+	1.73205i	−0.0640184	+	0.110883i
$245$		0			0
$246$		0			0
$247$	−9.00000	+	8.66025i	−0.572656	+	0.551039i
$248$		0			0
$249$	−12.0000	−	6.92820i	−0.760469	−	0.439057i
$250$		0			0
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	0.612159	−	0.790735i	$-0.290301\pi$
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	−0.990876	+	0.134778i	$0.956968\pi$
$252$			3.46410i			0.218218i
$253$	18.0000	−	10.3923i	1.13165	−	0.653359i
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−6.00000	+	3.46410i	−0.367194	+	0.212000i
$268$			17.3205i			1.05802i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−4.50000	−	2.59808i	−0.273356	−	0.157822i	0.357056	−	0.934083i	$-0.383781\pi$
$271$	−4.50000	−	2.59808i	−0.273356	−	0.157822i	−0.630412	+	0.776261i	$0.717114\pi$
$272$		0			0
$273$	−1.50000	−	6.06218i	−0.0907841	−	0.366900i
$274$		0			0
$275$		0			0
$276$	−6.00000	+	10.3923i	−0.361158	+	0.625543i
$277$	−5.00000	−	8.66025i	−0.300421	−	0.520344i	0.675810	−	0.737075i	$-0.263794\pi$
$277$	−5.00000	−	8.66025i	−0.300421	−	0.520344i	−0.976231	+	0.216731i	$0.930460\pi$
$278$		0			0
$279$	1.50000	−	0.866025i	0.0898027	−	0.0518476i
$280$		0			0
$281$		−	24.2487i		−	1.44656i	−0.690557	−	0.723278i	$-0.742634\pi$
$281$		−	24.2487i		−	1.44656i	0.690557	−	0.723278i	$-0.257366\pi$
$282$		0			0
$283$	−5.50000	+	9.52628i	−0.326941	+	0.566279i	−0.981903	−	0.189383i	$-0.939351\pi$
$283$	−5.50000	+	9.52628i	−0.326941	+	0.566279i	0.654962	+	0.755662i	$0.272685\pi$
$284$	18.0000	+	10.3923i	1.06810	+	0.616670i
$285$		0			0
$286$		0			0
$287$	−12.0000			−0.708338
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$			5.19615i			0.304604i
$292$	−3.00000	+	1.73205i	−0.175562	+	0.101361i
$293$	15.0000	−	8.66025i	0.876309	−	0.505937i	0.00686959	−	0.999976i	$-0.497813\pi$
$293$	15.0000	−	8.66025i	0.876309	−	0.505937i	0.869440	+	0.494039i	$0.164480\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.00000	−	1.73205i	−0.174078	−	0.100504i
$298$		0			0
$299$	6.00000	−	20.7846i	0.346989	−	1.20201i
$300$		0			0
$301$	−1.50000	−	0.866025i	−0.0864586	−	0.0499169i
$302$		0			0
$303$	9.00000	+	15.5885i	0.517036	+	0.895533i
$304$			13.8564i			0.794719i
$305$		0			0
$306$		0			0
$307$			1.73205i			0.0988534i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$			1.73205i			0.0988534i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$	6.00000	+	10.3923i	0.341882	+	0.592157i
$309$	−0.500000	+	0.866025i	−0.0284440	+	0.0492665i
$310$		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
$313$	−13.0000			−0.734803			−0.367402	−	0.930062i	$-0.619753\pi$
$313$	−13.0000			−0.734803			−0.367402	+	0.930062i	$0.619753\pi$
$314$		0			0
$315$		0			0
$316$	11.0000	+	19.0526i	0.618798	+	1.07179i
$317$		−	6.92820i		−	0.389127i	−0.980890	−	0.194563i	$-0.937671\pi$
$317$		−	6.92820i		−	0.389127i	0.980890	−	0.194563i	$-0.0623290\pi$
$318$		0			0
$319$	18.0000	−	10.3923i	1.00781	−	0.581857i
$320$		0			0
$321$	−3.00000	−	5.19615i	−0.167444	−	0.290021i
$322$		0			0
$323$		0			0
$324$	2.00000			0.111111
$325$		0			0
$326$		0			0
$327$	13.5000	+	7.79423i	0.746552	+	0.431022i
$328$		0			0
$329$	−3.00000	−	5.19615i	−0.165395	−	0.286473i
$330$		0			0
$331$	4.50000	−	2.59808i	0.247342	−	0.142803i	−0.371204	−	0.928551i	$-0.621055\pi$
$331$	4.50000	−	2.59808i	0.247342	−	0.142803i	0.618547	+	0.785748i	$0.287722\pi$
$332$	24.0000	−	13.8564i	1.31717	−	0.760469i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−6.00000	−	3.46410i	−0.327327	−	0.188982i
$337$	5.00000			0.272367			0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000			0.272367			0.136184	+	0.990684i	$0.456516\pi$
$338$		0			0
$339$	−6.00000			−0.325875
$340$		0			0
$341$	3.00000	−	5.19615i	0.162459	−	0.281387i
$342$		0			0
$343$			19.0526i			1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000	−	20.7846i	−0.644194	−	1.11578i	−0.984487	−	0.175457i	$-0.943860\pi$
$347$	−12.0000	−	20.7846i	−0.644194	−	1.11578i	0.340293	−	0.940319i	$-0.389474\pi$
$348$	−6.00000	+	10.3923i	−0.321634	+	0.557086i
$349$	−16.5000	−	9.52628i	−0.883225	−	0.509930i	−0.0115044	−	0.999934i	$-0.503662\pi$
$349$	−16.5000	−	9.52628i	−0.883225	−	0.509930i	−0.871720	+	0.490004i	$0.836995\pi$
$350$		0			0
$351$	−3.50000	+	0.866025i	−0.186816	+	0.0462250i
$352$		0			0
$353$	−9.00000	−	5.19615i	−0.479022	−	0.276563i	0.240987	−	0.970528i	$-0.422529\pi$
$353$	−9.00000	−	5.19615i	−0.479022	−	0.276563i	−0.720009	+	0.693965i	$0.755862\pi$
$354$		0			0
$355$		0			0
$356$		−	13.8564i		−	0.734388i
$357$		0			0
$358$		0			0
$359$			6.92820i			0.365657i	0.983145	+	0.182828i	$0.0585252\pi$
$359$			6.92820i			0.365657i	−0.983145	+	0.182828i	$0.941475\pi$
$360$		0			0
$361$	−3.50000	+	6.06218i	−0.184211	+	0.319062i
$362$		0			0
$363$	−1.00000			−0.0524864
$364$	12.0000	+	3.46410i	0.628971	+	0.181568i
$365$		0			0
$366$		0			0
$367$	8.50000	−	14.7224i	0.443696	−	0.768505i	−0.554264	−	0.832341i	$-0.687000\pi$
$367$	8.50000	−	14.7224i	0.443696	−	0.768505i	0.997960	+	0.0638362i	$0.0203335\pi$
$368$	−12.0000	−	20.7846i	−0.625543	−	1.08347i
$369$			6.92820i			0.360668i
$370$		0			0
$371$	−18.0000	+	10.3923i	−0.934513	+	0.539542i
$372$			3.46410i			0.179605i
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	0.687776	−	0.725923i	$-0.258587\pi$
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	−0.972556	+	0.232671i	$0.925254\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.00000	−	20.7846i	0.309016	−	1.07046i
$378$		0			0
$379$	−19.5000	−	11.2583i	−1.00165	−	0.578302i	−0.0929123	−	0.995674i	$-0.529618\pi$
$379$	−19.5000	−	11.2583i	−1.00165	−	0.578302i	−0.908735	+	0.417373i	$0.862951\pi$
$380$		0			0
$381$	6.50000	+	11.2583i	0.333005	+	0.576782i
$382$		0			0
$383$	−24.0000	+	13.8564i	−1.22634	+	0.708029i	−0.966263	−	0.257558i	$-0.917082\pi$
$383$	−24.0000	+	13.8564i	−1.22634	+	0.708029i	−0.260080	+	0.965587i	$0.583749\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.500000	+	0.866025i	−0.0254164	+	0.0440225i
$388$	−9.00000	−	5.19615i	−0.456906	−	0.263795i
$389$	12.0000			0.608424			0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000			0.608424			0.304212	+	0.952604i	$0.401607\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−3.00000	+	5.19615i	−0.151330	+	0.262111i
$394$		0			0
$395$		0			0
$396$	6.00000	−	3.46410i	0.301511	−	0.174078i
$397$	−13.5000	+	7.79423i	−0.677546	+	0.391181i	−0.798930	−	0.601424i	$-0.794600\pi$
$397$	−13.5000	+	7.79423i	−0.677546	+	0.391181i	0.121384	+	0.992606i	$0.461267\pi$
$398$		0			0
$399$	3.00000	+	5.19615i	0.150188	+	0.260133i
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−1.50000	−	6.06218i	−0.0747203	−	0.301979i
$404$	−36.0000			−1.79107
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	−0.302979	−	0.952997i	$-0.597981\pi$
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	0.673830	+	0.738886i	$0.264648\pi$
$410$		0			0
$411$		0			0
$412$	−1.00000	−	1.73205i	−0.0492665	−	0.0853320i
$413$	−3.00000	+	5.19615i	−0.147620	+	0.255686i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	5.00000			0.244851
$418$		0			0
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	−0.681426	−	0.731887i	$-0.738640\pi$
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	0.974546	+	0.224189i	$0.0719734\pi$
$420$		0			0
$421$		−	12.1244i		−	0.590905i	−0.955357	−	0.295452i	$-0.904530\pi$
$421$		−	12.1244i		−	0.590905i	0.955357	−	0.295452i	$-0.0954704\pi$
$422$		0			0
$423$	−3.00000	+	1.73205i	−0.145865	+	0.0842152i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.50000	−	0.866025i	−0.0725901	−	0.0419099i
$428$	12.0000			0.580042
$429$	−9.00000	+	8.66025i	−0.434524	+	0.418121i
$430$		0			0
$431$	18.0000	+	10.3923i	0.867029	+	0.500580i	0.866360	−	0.499420i	$-0.166454\pi$
$431$	18.0000	+	10.3923i	0.867029	+	0.500580i	0.000669521		1.00000i	$0.499787\pi$
$432$	−2.00000	+	3.46410i	−0.0962250	+	0.166667i
$433$	−11.5000	−	19.9186i	−0.552655	−	0.957226i	−0.998082	−	0.0619079i	$-0.980282\pi$
$433$	−11.5000	−	19.9186i	−0.552655	−	0.957226i	0.445427	−	0.895318i	$-0.353052\pi$
$434$		0			0
$435$		0			0
$436$	−27.0000	+	15.5885i	−1.29307	+	0.746552i
$437$			20.7846i			0.994263i
$438$		0			0
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	0.0586141	+	0.998281i	$0.481332\pi$
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	−0.893843	+	0.448379i	$0.852001\pi$
$440$		0			0
$441$	4.00000			0.190476
$442$		0			0
$443$	−24.0000			−1.14027			−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000			−1.14027			−0.570137	+	0.821549i	$0.693110\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	6.92820i		−	0.327693i
$448$	12.0000	−	6.92820i	0.566947	−	0.327327i
$449$	33.0000	−	19.0526i	1.55737	−	0.899146i	0.559859	−	0.828588i	$-0.310855\pi$
$449$	33.0000	−	19.0526i	1.55737	−	0.899146i	0.997508	−	0.0705577i	$-0.0224779\pi$
$450$		0			0
$451$	12.0000	+	20.7846i	0.565058	+	0.978709i
$452$	6.00000	−	10.3923i	0.282216	−	0.488813i
$453$	3.00000	+	1.73205i	0.140952	+	0.0813788i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	31.5000	+	18.1865i	1.47351	+	0.850730i	0.999555	−	0.0298202i	$-0.00949348\pi$
$457$	31.5000	+	18.1865i	1.47351	+	0.850730i	0.473953	+	0.880550i	$0.342827\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−36.0000	+	20.7846i	−1.67669	+	0.968036i	−0.712938	+	0.701228i	$0.752636\pi$
$461$	−36.0000	+	20.7846i	−1.67669	+	0.968036i	−0.963750	+	0.266808i	$0.914031\pi$
$462$		0			0
$463$			36.3731i			1.69040i	0.534450	+	0.845200i	$0.320519\pi$
$463$			36.3731i			1.69040i	−0.534450	+	0.845200i	$0.679481\pi$
$464$	−12.0000	−	20.7846i	−0.557086	−	0.964901i
$465$		0			0
$466$		0			0
$467$	−6.00000			−0.277647			−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000			−0.277647			−0.138823	+	0.990317i	$0.544332\pi$
$468$	2.00000	−	6.92820i	0.0924500	−	0.320256i
$469$	−15.0000			−0.692636
$470$		0			0
$471$	5.50000	−	9.52628i	0.253427	−	0.438948i
$472$		0			0
$473$			3.46410i			0.159280i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	6.00000	+	10.3923i	0.274721	+	0.475831i
$478$		0			0
$479$	30.0000	+	17.3205i	1.37073	+	0.791394i	0.991021	−	0.133710i	$-0.0426889\pi$
$479$	30.0000	+	17.3205i	1.37073	+	0.791394i	0.379714	+	0.925104i	$0.376022\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−9.00000	−	5.19615i	−0.409514	−	0.236433i
$484$	1.00000	−	1.73205i	0.0454545	−	0.0787296i
$485$		0			0
$486$		0			0
$487$	21.0000	−	12.1244i	0.951601	−	0.549407i	0.0580230	−	0.998315i	$-0.481520\pi$
$487$	21.0000	−	12.1244i	0.951601	−	0.549407i	0.893578	+	0.448908i	$0.148187\pi$
$488$		0			0
$489$			19.0526i			0.861586i
$490$		0			0
$491$	−3.00000	+	5.19615i	−0.135388	+	0.234499i	−0.925746	−	0.378147i	$-0.876561\pi$
$491$	−3.00000	+	5.19615i	−0.135388	+	0.234499i	0.790358	+	0.612646i	$0.209895\pi$
$492$	−12.0000	−	6.92820i	−0.541002	−	0.312348i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−6.00000	−	3.46410i	−0.269408	−	0.155543i
$497$	−9.00000	+	15.5885i	−0.403705	+	0.699238i
$498$		0			0
$499$			31.1769i			1.39567i	0.716258	+	0.697835i	$0.245853\pi$
$499$			31.1769i			1.39567i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$	−6.00000	+	3.46410i	−0.268060	+	0.154765i
$502$		0			0
$503$	−15.0000	−	25.9808i	−0.668817	−	1.15842i	−0.978235	−	0.207499i	$-0.933468\pi$
$503$	−15.0000	−	25.9808i	−0.668817	−	1.15842i	0.309418	−	0.950926i	$-0.399866\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.500000	+	12.9904i	−0.0222058	+	0.576923i
$508$	−26.0000			−1.15356
$509$	−15.0000	−	8.66025i	−0.664863	−	0.383859i	0.129264	−	0.991610i	$-0.458738\pi$
$509$	−15.0000	−	8.66025i	−0.664863	−	0.383859i	−0.794128	+	0.607751i	$0.792072\pi$
$510$		0			0
$511$	−1.50000	−	2.59808i	−0.0663561	−	0.114932i
$512$		0			0
$513$	3.00000	−	1.73205i	0.132453	−	0.0764719i
$514$		0			0
$515$		0			0
$516$	−1.00000	−	1.73205i	−0.0440225	−	0.0762493i
$517$	−6.00000	+	10.3923i	−0.263880	+	0.457053i
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$	−24.0000			−1.05146			−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000			−1.05146			−0.525730	+	0.850652i	$0.676208\pi$
$522$		0			0
$523$	−14.0000	+	24.2487i	−0.612177	+	1.06032i	0.378695	+	0.925521i	$0.376373\pi$
$523$	−14.0000	+	24.2487i	−0.612177	+	1.06032i	−0.990873	+	0.134801i	$0.956961\pi$
$524$	−6.00000	−	10.3923i	−0.262111	−	0.453990i
$525$		0			0
$526$		0			0
$527$		0			0
$528$			13.8564i			0.603023i
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$	3.00000	+	1.73205i	0.130189	+	0.0751646i
$532$	−12.0000			−0.520266
$533$	24.0000	+	6.92820i	1.03956	+	0.300094i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−6.00000	−	10.3923i	−0.258919	−	0.448461i
$538$		0			0
$539$	12.0000	−	6.92820i	0.516877	−	0.298419i
$540$		0			0
$541$		−	29.4449i		−	1.26593i	−0.774179	−	0.632967i	$-0.781837\pi$
$541$		−	29.4449i		−	1.26593i	0.774179	−	0.632967i	$-0.218163\pi$
$542$		0			0
$543$	−7.00000	+	12.1244i	−0.300399	+	0.520306i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	19.0000			0.812381			0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000			0.812381			0.406191	+	0.913788i	$0.366857\pi$
$548$		0			0
$549$	−0.500000	+	0.866025i	−0.0213395	+	0.0369611i
$550$		0			0
$551$			20.7846i			0.885454i
$552$		0			0
$553$	−16.5000	+	9.52628i	−0.701651	+	0.405099i
$554$		0			0
$555$		0			0
$556$	−5.00000	+	8.66025i	−0.212047	+	0.367277i
$557$	24.0000	+	13.8564i	1.01691	+	0.587115i	0.913208	−	0.407493i	$-0.133597\pi$
$557$	24.0000	+	13.8564i	1.01691	+	0.587115i	0.103704	+	0.994608i	$0.466930\pi$
$558$		0			0
$559$	2.50000	+	2.59808i	0.105739	+	0.109887i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.0000	−	36.3731i	−0.885044	−	1.53294i	−0.845663	−	0.533718i	$-0.820794\pi$
$563$	−21.0000	−	36.3731i	−0.885044	−	1.53294i	−0.0393818	−	0.999224i	$-0.512539\pi$
$564$		−	6.92820i		−	0.291730i
$565$		0			0
$566$		0			0
$567$			1.73205i			0.0727393i
$568$		0			0
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	−44.0000			−1.84134			−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000			−1.84134			−0.920671	+	0.390339i	$0.872358\pi$
$572$	−6.00000	−	24.2487i	−0.250873	−	1.01389i
$573$	−18.0000			−0.751961
$574$		0			0
$575$		0			0
$576$	−4.00000	−	6.92820i	−0.166667	−	0.288675i
$577$			34.6410i			1.44212i	0.692870	+	0.721062i	$0.256346\pi$
$577$			34.6410i			1.44212i	−0.692870	+	0.721062i	$0.743654\pi$
$578$		0			0
$579$	−13.5000	+	7.79423i	−0.561041	+	0.323917i
$580$		0			0
$581$	12.0000	+	20.7846i	0.497844	+	0.862291i
$582$		0			0
$583$	36.0000	+	20.7846i	1.49097	+	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.0000	+	15.5885i	1.11441	+	0.643404i	0.939968	−	0.341263i	$-0.110855\pi$
$587$	27.0000	+	15.5885i	1.11441	+	0.643404i	0.174441	+	0.984668i	$0.444188\pi$
$588$	−4.00000	+	6.92820i	−0.164957	+	0.285714i
$589$	3.00000	+	5.19615i	0.123613	+	0.214104i
$590$		0			0
$591$	12.0000	−	6.92820i	0.493614	−	0.284988i
$592$		0			0
$593$		−	3.46410i		−	0.142254i	−0.997467	−	0.0711268i	$-0.977341\pi$
$593$		−	3.46410i		−	0.142254i	0.997467	−	0.0711268i	$-0.0226595\pi$
$594$		0			0
$595$		0			0
$596$	12.0000	+	6.92820i	0.491539	+	0.283790i
$597$	7.00000			0.286491
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	0.999376	−	0.0353259i	$-0.0112469\pi$
$602$		0			0
$603$			8.66025i			0.352673i
$604$	−6.00000	+	3.46410i	−0.244137	+	0.140952i
$605$		0			0
$606$		0			0
$607$	4.00000	+	6.92820i	0.162355	+	0.281207i	0.935713	−	0.352763i	$-0.114758\pi$
$607$	4.00000	+	6.92820i	0.162355	+	0.281207i	−0.773358	+	0.633970i	$0.781424\pi$
$608$		0			0
$609$	−9.00000	−	5.19615i	−0.364698	−	0.210559i
$610$		0			0
$611$	3.00000	+	12.1244i	0.121367	+	0.490499i
$612$		0			0
$613$	−7.50000	−	4.33013i	−0.302922	−	0.174892i	0.340833	−	0.940124i	$-0.389291\pi$
$613$	−7.50000	−	4.33013i	−0.302922	−	0.174892i	−0.643755	+	0.765232i	$0.722624\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.00000	−	5.19615i	0.362326	−	0.209189i	−0.307774	−	0.951459i	$-0.599584\pi$
$617$	9.00000	−	5.19615i	0.362326	−	0.209189i	0.670101	+	0.742270i	$0.266251\pi$
$618$		0			0
$619$			25.9808i			1.04425i	0.852867	+	0.522127i	$0.174861\pi$
$619$			25.9808i			1.04425i	−0.852867	+	0.522127i	$0.825139\pi$
$620$		0			0
$621$	−3.00000	+	5.19615i	−0.120386	+	0.208514i
$622$		0			0
$623$	12.0000			0.480770
$624$	10.0000	+	10.3923i	0.400320	+	0.416025i
$625$		0			0
$626$		0			0
$627$	6.00000	−	10.3923i	0.239617	−	0.415029i
$628$	11.0000	+	19.0526i	0.438948	+	0.760280i
$629$		0			0
$630$		0			0
$631$	1.50000	−	0.866025i	0.0597141	−	0.0344759i	−0.469846	−	0.882749i	$-0.655690\pi$
$631$	1.50000	−	0.866025i	0.0597141	−	0.0344759i	0.529560	+	0.848273i	$0.322357\pi$
$632$		0			0
$633$	−6.50000	−	11.2583i	−0.258352	−	0.447478i
$634$		0			0
$635$		0			0
$636$	−24.0000			−0.951662
$637$	4.00000	−	13.8564i	0.158486	−	0.549011i
$638$		0			0
$639$	9.00000	+	5.19615i	0.356034	+	0.205557i
$640$		0			0
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	0.631721	−	0.775196i	$-0.282349\pi$
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	−0.987200	+	0.159489i	$0.949015\pi$
$642$		0			0
$643$	−16.5000	+	9.52628i	−0.650696	+	0.375680i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−16.5000	+	9.52628i	−0.650696	+	0.375680i	0.138027	+	0.990429i	$0.455924\pi$
$644$	18.0000	−	10.3923i	0.709299	−	0.409514i
$645$		0			0
$646$		0			0
$647$	−9.00000	+	15.5885i	−0.353827	+	0.612845i	−0.986916	−	0.161233i	$-0.948453\pi$
$647$	−9.00000	+	15.5885i	−0.353827	+	0.612845i	0.633090	+	0.774078i	$0.281786\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$	−3.00000			−0.117579
$652$	−33.0000	−	19.0526i	−1.29238	−	0.746156i
$653$	18.0000	−	31.1769i	0.704394	−	1.22005i	−0.262515	−	0.964928i	$-0.584552\pi$
$653$	18.0000	−	31.1769i	0.704394	−	1.22005i	0.966910	−	0.255119i	$-0.0821147\pi$
$654$		0			0
$655$		0			0
$656$	24.0000	−	13.8564i	0.937043	−	0.541002i
$657$	−1.50000	+	0.866025i	−0.0585206	+	0.0337869i
$658$		0			0
$659$	−24.0000	−	41.5692i	−0.934907	−	1.61931i	−0.774799	−	0.632207i	$-0.782149\pi$
$659$	−24.0000	−	41.5692i	−0.934907	−	1.61931i	−0.160108	−	0.987099i	$-0.551184\pi$
$660$		0			0
$661$	−22.5000	−	12.9904i	−0.875149	−	0.505267i	−0.00609283	−	0.999981i	$-0.501939\pi$
$661$	−22.5000	−	12.9904i	−0.875149	−	0.505267i	−0.869056	+	0.494714i	$0.835273\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−18.0000	−	31.1769i	−0.696963	−	1.20717i
$668$		−	13.8564i		−	0.536120i
$669$	15.0000	−	8.66025i	0.579934	−	0.334825i
$670$		0			0
$671$			3.46410i			0.133730i
$672$		0			0
$673$	−0.500000	+	0.866025i	−0.0192736	+	0.0333828i	−0.875501	−	0.483216i	$-0.839469\pi$
$673$	−0.500000	+	0.866025i	−0.0192736	+	0.0333828i	0.856228	+	0.516599i	$0.172802\pi$
$674$		0			0
$675$		0			0
$676$	−22.0000	−	13.8564i	−0.846154	−	0.532939i
$677$	−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
$679$	4.50000	−	7.79423i	0.172694	−	0.299115i
$680$		0			0
$681$		−	20.7846i		−	0.796468i
$682$		0			0
$683$	−21.0000	+	12.1244i	−0.803543	+	0.463926i	−0.844708	−	0.535227i	$-0.820226\pi$
$683$	−21.0000	+	12.1244i	−0.803543	+	0.463926i	0.0411658	+	0.999152i	$0.486893\pi$
$684$			6.92820i			0.264906i
$685$		0			0
$686$		0			0
$687$	−24.0000	−	13.8564i	−0.915657	−	0.528655i
$688$	4.00000			0.152499
$689$	42.0000	−	10.3923i	1.60007	−	0.395915i
$690$		0			0
$691$	37.5000	+	21.6506i	1.42657	+	0.823629i	0.996848	−	0.0793336i	$-0.0252792\pi$
$691$	37.5000	+	21.6506i	1.42657	+	0.823629i	0.429719	+	0.902963i	$0.358613\pi$
$692$	−6.00000	+	10.3923i	−0.228086	+	0.395056i
$693$	3.00000	+	5.19615i	0.113961	+	0.197386i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−9.00000	+	15.5885i	−0.340411	+	0.589610i
$700$		0			0
$701$	−12.0000			−0.453234			−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000			−0.453234			−0.226617	+	0.973984i	$0.572767\pi$
$702$		0			0
$703$		0			0
$704$	−24.0000	−	13.8564i	−0.904534	−	0.522233i
$705$		0			0
$706$		0			0
$707$		−	31.1769i		−	1.17253i
$708$	−6.00000	+	3.46410i	−0.225494	+	0.130189i
$709$	−16.5000	+	9.52628i	−0.619671	+	0.357767i	−0.776741	−	0.629821i	$-0.783128\pi$
$709$	−16.5000	+	9.52628i	−0.619671	+	0.357767i	0.157070	+	0.987587i	$0.449795\pi$
$710$		0			0
$711$	5.50000	+	9.52628i	0.206266	+	0.357263i
$712$		0			0
$713$	−9.00000	−	5.19615i	−0.337053	−	0.194597i
$714$		0			0
$715$		0			0
$716$	24.0000			0.896922
$717$		0			0
$718$		0			0
$719$	3.00000	+	5.19615i	0.111881	+	0.193784i	0.916529	−	0.399969i	$-0.130979\pi$
$719$	3.00000	+	5.19615i	0.111881	+	0.193784i	−0.804648	+	0.593753i	$0.797646\pi$
$720$		0			0
$721$	1.50000	−	0.866025i	0.0558629	−	0.0322525i
$722$		0			0
$723$			20.7846i			0.772988i
$724$	−14.0000	−	24.2487i	−0.520306	−	0.901196i
$725$		0			0
$726$		0			0
$727$	35.0000			1.29808			0.649039	−	0.760755i	$-0.275171\pi$
$727$	35.0000			1.29808			0.649039	+	0.760755i	$0.275171\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$	−1.00000	−	1.73205i	−0.0369611	−	0.0640184i
$733$		−	39.8372i		−	1.47142i	−0.677297	−	0.735710i	$-0.736849\pi$
$733$		−	39.8372i		−	1.47142i	0.677297	−	0.735710i	$-0.263151\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.0000	+	25.9808i	0.552532	+	0.957014i
$738$		0			0
$739$	39.0000	+	22.5167i	1.43464	+	0.828289i	0.997470	−	0.0710909i	$-0.0226481\pi$
$739$	39.0000	+	22.5167i	1.43464	+	0.828289i	0.437168	+	0.899380i	$0.355981\pi$
$740$		0			0
$741$	−3.00000	−	12.1244i	−0.110208	−	0.445399i
$742$		0			0
$743$	9.00000	+	5.19615i	0.330178	+	0.190628i	0.655920	−	0.754830i	$-0.272281\pi$
$743$	9.00000	+	5.19615i	0.330178	+	0.190628i	−0.325742	+	0.945459i	$0.605614\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	12.0000	−	6.92820i	0.439057	−	0.253490i
$748$		0			0
$749$			10.3923i			0.379727i
$750$		0			0
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	−0.929731	−	0.368238i	$-0.879961\pi$
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	0.783769	+	0.621052i	$0.213294\pi$
$752$	12.0000	+	6.92820i	0.437595	+	0.252646i
$753$	12.0000			0.437304
$754$		0			0
$755$		0			0
$756$	−3.00000	−	1.73205i	−0.109109	−	0.0629941i
$757$	17.0000	−	29.4449i	0.617876	−	1.07019i	−0.371997	−	0.928234i	$-0.621327\pi$
$757$	17.0000	−	29.4449i	0.617876	−	1.07019i	0.989873	−	0.141958i	$-0.0453398\pi$
$758$		0			0
$759$			20.7846i			0.754434i
$760$		0			0
$761$	−18.0000	+	10.3923i	−0.652499	+	0.376721i	−0.789413	−	0.613862i	$-0.789615\pi$
$761$	−18.0000	+	10.3923i	−0.652499	+	0.376721i	0.136914	+	0.990583i	$0.456282\pi$
$762$		0			0
$763$	−13.5000	−	23.3827i	−0.488733	−	0.846510i
$764$	18.0000	−	31.1769i	0.651217	−	1.12794i
$765$		0			0
$766$		0			0
$767$	9.00000	−	8.66025i	0.324971	−	0.312704i
$768$	16.0000			0.577350
$769$	−6.00000	−	3.46410i	−0.216366	−	0.124919i	0.387901	−	0.921701i	$-0.373200\pi$
$769$	−6.00000	−	3.46410i	−0.216366	−	0.124919i	−0.604266	+	0.796782i	$0.706534\pi$
$770$		0			0
$771$		0			0
$772$		−	31.1769i		−	1.12208i
$773$	45.0000	−	25.9808i	1.61854	−	0.934463i	0.631239	−	0.775589i	$-0.282547\pi$
$773$	45.0000	−	25.9808i	1.61854	−	0.934463i	0.987299	−	0.158874i	$-0.0507865\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000			−0.859889
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$	−3.00000	+	5.19615i	−0.107211	+	0.185695i
$784$	−8.00000	−	13.8564i	−0.285714	−	0.494872i
$785$		0			0
$786$		0			0
$787$	−28.5000	+	16.4545i	−1.01592	+	0.586539i	−0.912918	−	0.408143i	$-0.866177\pi$
$787$	−28.5000	+	16.4545i	−1.01592	+	0.586539i	−0.102997	+	0.994682i	$0.532843\pi$
$788$			27.7128i			0.987228i
$789$	6.00000	+

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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.bc (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(1.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(1.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} +2.00000 q^{12} +(-3.50000 + 0.866025i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(3.00000 - 1.73205i) q^{19} +1.73205i q^{21} +(-3.00000 + 5.19615i) q^{23} +1.00000 q^{27} +(-3.00000 - 1.73205i) q^{28} +(-3.00000 + 5.19615i) q^{29} +1.73205i q^{31} +(3.00000 - 1.73205i) q^{33} +(-1.00000 + 1.73205i) q^{36} +(1.00000 - 3.46410i) q^{39} +(-6.00000 - 3.46410i) q^{41} +(-0.500000 - 0.866025i) q^{43} +6.92820i q^{44} -3.46410i q^{47} +(-2.00000 - 3.46410i) q^{48} +(-2.00000 + 3.46410i) q^{49} +(5.00000 + 5.19615i) q^{52} -12.0000 q^{53} +3.46410i q^{57} +(-3.00000 + 1.73205i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(-1.50000 - 0.866025i) q^{63} +8.00000 q^{64} +(-7.50000 - 4.33013i) q^{67} +(-3.00000 - 5.19615i) q^{69} +(-9.00000 + 5.19615i) q^{71} -1.73205i q^{73} +(-6.00000 - 3.46410i) q^{76} -6.00000 q^{77} -11.0000 q^{79} +(-0.500000 + 0.866025i) q^{81} +13.8564i q^{83} +(3.00000 - 1.73205i) q^{84} +(-3.00000 - 5.19615i) q^{87} +(6.00000 + 3.46410i) q^{89} +(-4.50000 + 4.33013i) q^{91} +12.0000 q^{92} +(-1.50000 - 0.866025i) q^{93} +(4.50000 - 2.59808i) q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{4} + 3 q^{7} - q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^4 + 3 * q^7 - q^9	\( 2 q - q^{3} - 2 q^{4} + 3 q^{7} - q^{9} - 6 q^{11} + 4 q^{12} - 7 q^{13} - 4 q^{16} + 6 q^{19} - 6 q^{23} + 2 q^{27} - 6 q^{28} - 6 q^{29} + 6 q^{33} - 2 q^{36} + 2 q^{39} - 12 q^{41} - q^{43} - 4 q^{48} - 4 q^{49} + 10 q^{52} - 24 q^{53} - 6 q^{59} - q^{61} - 3 q^{63} + 16 q^{64} - 15 q^{67} - 6 q^{69} - 18 q^{71} - 12 q^{76} - 12 q^{77} - 22 q^{79} - q^{81} + 6 q^{84} - 6 q^{87} + 12 q^{89} - 9 q^{91} + 24 q^{92} - 3 q^{93} + 9 q^{97}+O(q^{100}) \) 2 * q - q^3 - 2 * q^4 + 3 * q^7 - q^9 - 6 * q^11 + 4 * q^12 - 7 * q^13 - 4 * q^16 + 6 * q^19 - 6 * q^23 + 2 * q^27 - 6 * q^28 - 6 * q^29 + 6 * q^33 - 2 * q^36 + 2 * q^39 - 12 * q^41 - q^43 - 4 * q^48 - 4 * q^49 + 10 * q^52 - 24 * q^53 - 6 * q^59 - q^61 - 3 * q^63 + 16 * q^64 - 15 * q^67 - 6 * q^69 - 18 * q^71 - 12 * q^76 - 12 * q^77 - 22 * q^79 - q^81 + 6 * q^84 - 6 * q^87 + 12 * q^89 - 9 * q^91 + 24 * q^92 - 3 * q^93 + 9 * q^97

Introduction

L-functions

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