LMFDB - Embedded newform 880.2.b.a.529.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,2,Mod(529,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.529"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-4,0,0,0,4,0,-2,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$7.02683537787$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	880.529
Dual form			880.2.b.a.529.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +3.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +4.00000i q^{13} +(1.00000 - 2.00000i) q^{15} -3.00000i q^{17} -5.00000 q^{19} -3.00000 q^{21} -4.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +5.00000i q^{27} -5.00000 q^{29} -7.00000 q^{31} -1.00000i q^{33} +(3.00000 - 6.00000i) q^{35} +7.00000i q^{37} -4.00000 q^{39} -8.00000 q^{41} +6.00000i q^{43} +(-4.00000 - 2.00000i) q^{45} +8.00000i q^{47} -2.00000 q^{49} +3.00000 q^{51} +9.00000i q^{53} +(2.00000 + 1.00000i) q^{55} -5.00000i q^{57} -13.0000 q^{61} +6.00000i q^{63} +(4.00000 - 8.00000i) q^{65} -12.0000i q^{67} +4.00000 q^{69} +3.00000 q^{71} -6.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -3.00000i q^{77} +1.00000 q^{81} -4.00000i q^{83} +(-3.00000 + 6.00000i) q^{85} -5.00000i q^{87} +15.0000 q^{89} -12.0000 q^{91} -7.00000i q^{93} +(10.0000 + 5.00000i) q^{95} +12.0000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 4 q^{9} - 2 q^{11} + 2 q^{15} - 10 q^{19} - 6 q^{21} + 6 q^{25} - 10 q^{29} - 14 q^{31} + 6 q^{35} - 8 q^{39} - 16 q^{41} - 8 q^{45} - 4 q^{49} + 6 q^{51} + 4 q^{55} - 26 q^{61} + 8 q^{65}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 4 * q^5 + 4 * q^9 - 2 * q^11 + 2 * q^15 - 10 * q^19 - 6 * q^21 + 6 * q^25 - 10 * q^29 - 14 * q^31 + 6 * q^35 - 8 * q^39 - 16 * q^41 - 8 * q^45 - 4 * q^49 + 6 * q^51 + 4 * q^55 - 26 * q^61 + 8 * q^65 + 8 * q^69 + 6 * q^71 - 8 * q^75 + 2 * q^81 - 6 * q^85 + 30 * q^89 - 24 * q^91 + 20 * q^95 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$			1.00000i			0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$		0			0
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$6$		0			0
$7$			3.00000i			1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$			3.00000i			1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	1.00000	−	2.00000i	0.258199	−	0.516398i
$16$		0			0
$17$		−	3.00000i		−	0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$		−	3.00000i		−	0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$		0			0
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$	−3.00000			−0.654654
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	3.00000	−	6.00000i	0.507093	−	1.01419i
$36$		0			0
$37$			7.00000i			1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$			7.00000i			1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$		0			0
$39$	−4.00000			−0.640513
$40$		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
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$		0			0
$45$	−4.00000	−	2.00000i	−0.596285	−	0.298142i
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$		0			0
$51$	3.00000			0.420084
$52$		0			0
$53$			9.00000i			1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$			9.00000i			1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$		0			0
$55$	2.00000	+	1.00000i	0.269680	+	0.134840i
$56$		0			0
$57$		−	5.00000i		−	0.662266i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$			6.00000i			0.755929i
$64$		0			0
$65$	4.00000	−	8.00000i	0.496139	−	0.992278i
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$	3.00000			0.356034			0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000			0.356034			0.178017	+	0.984027i	$0.443032\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$		0			0
$77$		−	3.00000i		−	0.341882i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$	−3.00000	+	6.00000i	−0.325396	+	0.650791i
$86$		0			0
$87$		−	5.00000i		−	0.536056i
$88$		0			0
$89$	15.0000			1.59000			0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000			1.59000			0.794998	+	0.606612i	$0.207472\pi$
$90$		0			0
$91$	−12.0000			−1.25794
$92$		0			0
$93$		−	7.00000i		−	0.725866i
$94$		0			0
$95$	10.0000	+	5.00000i	1.02598	+	0.512989i
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$	−2.00000			−0.201008
$100$		0			0
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		0			0
$105$	6.00000	+	3.00000i	0.585540	+	0.292770i
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$	−7.00000			−0.664411
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$	−4.00000	+	8.00000i	−0.373002	+	0.746004i
$116$		0			0
$117$			8.00000i			0.739600i
$118$		0			0
$119$	9.00000			0.825029
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	8.00000i		−	0.721336i
$124$		0			0
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		0			0
$129$	−6.00000			−0.528271
$130$		0			0
$131$	−7.00000			−0.611593			−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000			−0.611593			−0.305796	+	0.952097i	$0.598923\pi$
$132$		0			0
$133$		−	15.0000i		−	1.30066i
$134$		0			0
$135$	5.00000	−	10.0000i	0.430331	−	0.860663i
$136$		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	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$		−	4.00000i		−	0.334497i
$144$		0			0
$145$	10.0000	+	5.00000i	0.830455	+	0.415227i
$146$		0			0
$147$		−	2.00000i		−	0.164957i
$148$		0			0
$149$	5.00000			0.409616			0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000			0.409616			0.204808	+	0.978802i	$0.434343\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		−	6.00000i		−	0.485071i
$154$		0			0
$155$	14.0000	+	7.00000i	1.12451	+	0.562254i
$156$		0			0
$157$		−	13.0000i		−	1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$		−	13.0000i		−	1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$		0			0
$159$	−9.00000			−0.713746
$160$		0			0
$161$	12.0000			0.945732
$162$		0			0
$163$			21.0000i			1.64485i	0.568876	+	0.822423i	$0.307379\pi$
$163$			21.0000i			1.64485i	−0.568876	+	0.822423i	$0.692621\pi$
$164$		0			0
$165$	−1.00000	+	2.00000i	−0.0778499	+	0.155700i
$166$		0			0
$167$			23.0000i			1.77979i	0.456162	+	0.889897i	$0.349224\pi$
$167$			23.0000i			1.77979i	−0.456162	+	0.889897i	$0.650776\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	−10.0000			−0.764719
$172$		0			0
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		0			0
$175$	−12.0000	+	9.00000i	−0.907115	+	0.680336i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		−	13.0000i		−	0.960988i
$184$		0			0
$185$	7.00000	−	14.0000i	0.514650	−	1.02930i
$186$		0			0
$187$			3.00000i			0.219382i
$188$		0			0
$189$	−15.0000			−1.09109
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	1.00000i		−	0.0719816i	−0.999352	−	0.0359908i	$-0.988541\pi$
$193$		−	1.00000i		−	0.0719816i	0.999352	−	0.0359908i	$-0.0114587\pi$
$194$		0			0
$195$	8.00000	+	4.00000i	0.572892	+	0.286446i
$196$		0			0
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		0			0
$199$	25.0000			1.77220			0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000			1.77220			0.886102	+	0.463491i	$0.153403\pi$
$200$		0			0
$201$	12.0000			0.846415
$202$		0			0
$203$		−	15.0000i		−	1.05279i
$204$		0			0
$205$	16.0000	+	8.00000i	1.11749	+	0.558744i
$206$		0			0
$207$		−	8.00000i		−	0.556038i
$208$		0			0
$209$	5.00000			0.345857
$210$		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
$213$			3.00000i			0.205557i
$214$		0			0
$215$	6.00000	−	12.0000i	0.409197	−	0.818393i
$216$		0			0
$217$		−	21.0000i		−	1.42557i
$218$		0			0
$219$	6.00000			0.405442
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$		0			0
$225$	6.00000	+	8.00000i	0.400000	+	0.533333i
$226$		0			0
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$	3.00000			0.197386
$232$		0			0
$233$			19.0000i			1.24473i	0.782727	+	0.622366i	$0.213828\pi$
$233$			19.0000i			1.24473i	−0.782727	+	0.622366i	$0.786172\pi$
$234$		0			0
$235$	8.00000	−	16.0000i	0.521862	−	1.04372i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$		0			0
$243$			16.0000i			1.02640i
$244$		0			0
$245$	4.00000	+	2.00000i	0.255551	+	0.127775i
$246$		0			0
$247$		−	20.0000i		−	1.27257i
$248$		0			0
$249$	4.00000			0.253490
$250$		0			0
$251$	−2.00000			−0.126239			−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000			−0.126239			−0.0631194	+	0.998006i	$0.520105\pi$
$252$		0			0
$253$			4.00000i			0.251478i
$254$		0			0
$255$	−6.00000	−	3.00000i	−0.375735	−	0.187867i
$256$		0			0
$257$		−	28.0000i		−	1.74659i	−0.487190	−	0.873296i	$-0.661978\pi$
$257$		−	28.0000i		−	1.74659i	0.487190	−	0.873296i	$-0.338022\pi$
$258$		0			0
$259$	−21.0000			−1.30488
$260$		0			0
$261$	−10.0000			−0.618984
$262$		0			0
$263$		−	9.00000i		−	0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$		−	9.00000i		−	0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$		0			0
$265$	9.00000	−	18.0000i	0.552866	−	1.10573i
$266$		0			0
$267$			15.0000i			0.917985i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−22.0000			−1.33640			−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000			−1.33640			−0.668202	+	0.743980i	$0.732936\pi$
$272$		0			0
$273$		−	12.0000i		−	0.726273i
$274$		0			0
$275$	−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	$-0.318138\pi$
$278$		0			0
$279$	−14.0000			−0.838158
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$		0			0
$285$	−5.00000	+	10.0000i	−0.296174	+	0.592349i
$286$		0			0
$287$		−	24.0000i		−	1.41668i
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$	−12.0000			−0.703452
$292$		0			0
$293$		−	16.0000i		−	0.934730i	−0.884064	−	0.467365i	$-0.845203\pi$
$293$		−	16.0000i		−	0.934730i	0.884064	−	0.467365i	$-0.154797\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	5.00000i		−	0.290129i
$298$		0			0
$299$	16.0000			0.925304
$300$		0			0
$301$	−18.0000			−1.03750
$302$		0			0
$303$			2.00000i			0.114897i
$304$		0			0
$305$	26.0000	+	13.0000i	1.48876	+	0.744378i
$306$		0			0
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$		0			0
$309$	4.00000			0.227552
$310$		0			0
$311$	13.0000			0.737162			0.368581	−	0.929596i	$-0.379844\pi$
$311$	13.0000			0.737162			0.368581	+	0.929596i	$0.379844\pi$
$312$		0			0
$313$			14.0000i			0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$			14.0000i			0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$		0			0
$315$	6.00000	−	12.0000i	0.338062	−	0.676123i
$316$		0			0
$317$		−	23.0000i		−	1.29181i	−0.763418	−	0.645904i	$-0.776480\pi$
$317$		−	23.0000i		−	1.29181i	0.763418	−	0.645904i	$-0.223520\pi$
$318$		0			0
$319$	5.00000			0.279946
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$			15.0000i			0.834622i
$324$		0			0
$325$	−16.0000	+	12.0000i	−0.887520	+	0.665640i
$326$		0			0
$327$			10.0000i			0.553001i
$328$		0			0
$329$	−24.0000			−1.32316
$330$		0			0
$331$	18.0000			0.989369			0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000			0.989369			0.494685	+	0.869072i	$0.335284\pi$
$332$		0			0
$333$			14.0000i			0.767195i
$334$		0			0
$335$	−12.0000	+	24.0000i	−0.655630	+	1.31126i
$336$		0			0
$337$			7.00000i			0.381314i	0.981657	+	0.190657i	$0.0610619\pi$
$337$			7.00000i			0.381314i	−0.981657	+	0.190657i	$0.938938\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$	7.00000			0.379071
$342$		0			0
$343$			15.0000i			0.809924i
$344$		0			0
$345$	−8.00000	−	4.00000i	−0.430706	−	0.215353i
$346$		0			0
$347$			18.0000i			0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$			18.0000i			0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	−20.0000			−1.06752
$352$		0			0
$353$			14.0000i			0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$			14.0000i			0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$		0			0
$355$	−6.00000	−	3.00000i	−0.318447	−	0.159223i
$356$		0			0
$357$			9.00000i			0.476331i
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$			1.00000i			0.0524864i
$364$		0			0
$365$	−6.00000	+	12.0000i	−0.314054	+	0.628109i
$366$		0			0
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		0			0
$369$	−16.0000			−0.832927
$370$		0			0
$371$	−27.0000			−1.40177
$372$		0			0
$373$			4.00000i			0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$			4.00000i			0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$		0			0
$375$	11.0000	−	2.00000i	0.568038	−	0.103280i
$376$		0			0
$377$		−	20.0000i		−	1.03005i
$378$		0			0
$379$	−10.0000			−0.513665			−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000			−0.513665			−0.256833	+	0.966456i	$0.582679\pi$
$380$		0			0
$381$	−8.00000			−0.409852
$382$		0			0
$383$			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
$385$	−3.00000	+	6.00000i	−0.152894	+	0.305788i
$386$		0			0
$387$			12.0000i			0.609994i
$388$		0			0
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−12.0000			−0.606866
$392$		0			0
$393$		−	7.00000i		−	0.353103i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			22.0000i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$			22.0000i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$		0			0
$399$	15.0000			0.750939
$400$		0			0
$401$	27.0000			1.34832			0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000			1.34832			0.674158	+	0.738587i	$0.264507\pi$
$402$		0			0
$403$		−	28.0000i		−	1.39478i
$404$		0			0
$405$	−2.00000	−	1.00000i	−0.0993808	−	0.0496904i
$406$		0			0
$407$		−	7.00000i		−	0.346977i
$408$		0			0
$409$	−20.0000			−0.988936			−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000			−0.988936			−0.494468	+	0.869196i	$0.664637\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.00000	+	8.00000i	−0.196352	+	0.392705i
$416$		0			0
$417$			20.0000i			0.979404i
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$			16.0000i			0.777947i
$424$		0			0
$425$	12.0000	−	9.00000i	0.582086	−	0.436564i
$426$		0			0
$427$		−	39.0000i		−	1.88734i
$428$		0			0
$429$	4.00000			0.193122
$430$		0			0
$431$	−32.0000			−1.54139			−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000			−1.54139			−0.770693	+	0.637207i	$0.780090\pi$
$432$		0			0
$433$			14.0000i			0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$			14.0000i			0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$		0			0
$435$	−5.00000	+	10.0000i	−0.239732	+	0.479463i
$436$		0			0
$437$			20.0000i			0.956730i
$438$		0			0
$439$	10.0000			0.477274			0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000			0.477274			0.238637	+	0.971109i	$0.423299\pi$
$440$		0			0
$441$	−4.00000			−0.190476
$442$		0			0
$443$		−	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
$445$	−30.0000	−	15.0000i	−1.42214	−	0.711068i
$446$		0			0
$447$			5.00000i			0.236492i
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	8.00000			0.376705
$452$		0			0
$453$		−	2.00000i		−	0.0939682i
$454$		0			0
$455$	24.0000	+	12.0000i	1.12514	+	0.562569i
$456$		0			0
$457$			27.0000i			1.26301i	0.775373	+	0.631503i	$0.217562\pi$
$457$			27.0000i			1.26301i	−0.775373	+	0.631503i	$0.782438\pi$
$458$		0			0
$459$	15.0000			0.700140
$460$		0			0
$461$	−13.0000			−0.605470			−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000			−0.605470			−0.302735	+	0.953075i	$0.597900\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$		0			0
$465$	−7.00000	+	14.0000i	−0.324617	+	0.649234i
$466$		0			0
$467$			3.00000i			0.138823i	0.997588	+	0.0694117i	$0.0221122\pi$
$467$			3.00000i			0.138823i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$		0			0
$469$	36.0000			1.66233
$470$		0			0
$471$	13.0000			0.599008
$472$		0			0
$473$		−	6.00000i		−	0.275880i
$474$		0			0
$475$	−15.0000	−	20.0000i	−0.688247	−	0.917663i
$476$		0			0
$477$			18.0000i			0.824163i
$478$		0			0
$479$	−10.0000			−0.456912			−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000			−0.456912			−0.228456	+	0.973554i	$0.573368\pi$
$480$		0			0
$481$	−28.0000			−1.27669
$482$		0			0
$483$			12.0000i			0.546019i
$484$		0			0
$485$	12.0000	−	24.0000i	0.544892	−	1.08978i
$486$		0			0
$487$		−	22.0000i		−	0.996915i	−0.866914	−	0.498458i	$-0.833900\pi$
$487$		−	22.0000i		−	0.996915i	0.866914	−	0.498458i	$-0.166100\pi$
$488$		0			0
$489$	−21.0000			−0.949653
$490$		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
$493$			15.0000i			0.675566i
$494$		0			0
$495$	4.00000	+	2.00000i	0.179787	+	0.0898933i
$496$		0			0
$497$			9.00000i			0.403705i
$498$		0			0
$499$	−40.0000			−1.79065			−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000			−1.79065			−0.895323	+	0.445418i	$0.853055\pi$
$500$		0			0
$501$	−23.0000			−1.02756
$502$		0			0
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$		0			0
$505$	−4.00000	−	2.00000i	−0.177998	−	0.0889988i
$506$		0			0
$507$		−	3.00000i		−	0.133235i
$508$		0			0
$509$	20.0000			0.886484			0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000			0.886484			0.443242	+	0.896402i	$0.353828\pi$
$510$		0			0
$511$	18.0000			0.796273
$512$		0			0
$513$		−	25.0000i		−	1.10378i
$514$		0			0
$515$	−4.00000	+	8.00000i	−0.176261	+	0.352522i
$516$		0			0
$517$		−	8.00000i		−	0.351840i
$518$		0			0
$519$	−14.0000			−0.614532
$520$		0			0
$521$	22.0000			0.963837			0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000			0.963837			0.481919	+	0.876216i	$0.339940\pi$
$522$		0			0
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$		0			0
$525$	−9.00000	−	12.0000i	−0.392792	−	0.523723i
$526$		0			0
$527$			21.0000i			0.914774i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	32.0000i		−	1.38607i
$534$		0			0
$535$	−12.0000	+	24.0000i	−0.518805	+	1.03761i
$536$		0			0
$537$			10.0000i			0.431532i
$538$		0			0
$539$	2.00000			0.0861461
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$		−	18.0000i		−	0.772454i
$544$		0			0
$545$	−20.0000	−	10.0000i	−0.856706	−	0.428353i
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$		0			0
$549$	−26.0000			−1.10965
$550$		0			0
$551$	25.0000			1.06504
$552$		0			0
$553$		0			0
$554$		0			0
$555$	14.0000	+	7.00000i	0.594267	+	0.297133i
$556$		0			0
$557$		−	28.0000i		−	1.18640i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$		−	28.0000i		−	1.18640i	0.805056	−	0.593199i	$-0.202135\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$	−3.00000			−0.126660
$562$		0			0
$563$		−	4.00000i		−	0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$		−	4.00000i		−	0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$		0			0
$565$	−6.00000	+	12.0000i	−0.252422	+	0.504844i
$566$		0			0
$567$			3.00000i			0.125988i
$568$		0			0
$569$	−20.0000			−0.838444			−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000			−0.838444			−0.419222	+	0.907884i	$0.637697\pi$
$570$		0			0
$571$	3.00000			0.125546			0.0627730	−	0.998028i	$-0.480006\pi$
$571$	3.00000			0.125546			0.0627730	+	0.998028i	$0.480006\pi$
$572$		0			0
$573$			8.00000i			0.334205i
$574$		0			0
$575$	16.0000	−	12.0000i	0.667246	−	0.500435i
$576$		0			0
$577$		−	8.00000i		−	0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$		−	8.00000i		−	0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$		0			0
$579$	1.00000			0.0415586
$580$		0			0
$581$	12.0000			0.497844
$582$		0			0
$583$		−	9.00000i		−	0.372742i
$584$		0			0
$585$	8.00000	−	16.0000i	0.330759	−	0.661519i
$586$		0			0
$587$		−	7.00000i		−	0.288921i	−0.989511	−	0.144460i	$-0.953855\pi$
$587$		−	7.00000i		−	0.288921i	0.989511	−	0.144460i	$-0.0461446\pi$
$588$		0			0
$589$	35.0000			1.44215
$590$		0			0
$591$	−22.0000			−0.904959
$592$		0			0
$593$		−	6.00000i		−	0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$		−	6.00000i		−	0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$		0			0
$595$	−18.0000	−	9.00000i	−0.737928	−	0.368964i
$596$		0			0
$597$			25.0000i			1.02318i
$598$		0			0
$599$	−5.00000			−0.204294			−0.102147	−	0.994769i	$-0.532571\pi$
$599$	−5.00000			−0.204294			−0.102147	+	0.994769i	$0.532571\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$		−	24.0000i		−	0.977356i
$604$		0			0
$605$	−2.00000	−	1.00000i	−0.0813116	−	0.0406558i
$606$		0			0
$607$		−	7.00000i		−	0.284121i	−0.989858	−	0.142061i	$-0.954627\pi$
$607$		−	7.00000i		−	0.284121i	0.989858	−	0.142061i	$-0.0453728\pi$
$608$		0			0
$609$	15.0000			0.607831
$610$		0			0
$611$	−32.0000			−1.29458
$612$		0			0
$613$		−	6.00000i		−	0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$		−	6.00000i		−	0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$		0			0
$615$	−8.00000	+	16.0000i	−0.322591	+	0.645182i
$616$		0			0
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$	20.0000			0.802572
$622$		0			0
$623$			45.0000i			1.80289i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$			5.00000i			0.199681i
$628$		0			0
$629$	21.0000			0.837325
$630$		0			0
$631$	43.0000			1.71180			0.855901	−	0.517139i	$-0.173003\pi$
$631$	43.0000			1.71180			0.855901	+	0.517139i	$0.173003\pi$
$632$		0			0
$633$			13.0000i			0.516704i
$634$		0			0
$635$	8.00000	−	16.0000i	0.317470	−	0.634941i
$636$		0			0
$637$		−	8.00000i		−	0.316972i
$638$		0			0
$639$	6.00000			0.237356
$640$		0			0
$641$	−33.0000			−1.30342			−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000			−1.30342			−0.651711	+	0.758468i	$0.725948\pi$
$642$		0			0
$643$			1.00000i			0.0394362i	0.999806	+	0.0197181i	$0.00627687\pi$
$643$			1.00000i			0.0394362i	−0.999806	+	0.0197181i	$0.993723\pi$
$644$		0			0
$645$	12.0000	+	6.00000i	0.472500	+	0.236250i
$646$		0			0
$647$		−	12.0000i		−	0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$		−	12.0000i		−	0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	21.0000			0.823055
$652$		0			0
$653$		−	21.0000i		−	0.821794i	−0.911682	−	0.410897i	$-0.865216\pi$
$653$		−	21.0000i		−	0.821794i	0.911682	−	0.410897i	$-0.134784\pi$
$654$		0			0
$655$	14.0000	+	7.00000i	0.547025	+	0.273513i
$656$		0			0
$657$		−	12.0000i		−	0.468165i
$658$		0			0
$659$	−35.0000			−1.36341			−0.681703	−	0.731629i	$-0.738760\pi$
$659$	−35.0000			−1.36341			−0.681703	+	0.731629i	$0.738760\pi$
$660$		0			0
$661$	12.0000			0.466746			0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000			0.466746			0.233373	+	0.972387i	$0.425024\pi$
$662$		0			0
$663$			12.0000i			0.466041i
$664$		0			0
$665$	−15.0000	+	30.0000i	−0.581675	+	1.16335i
$666$		0			0
$667$			20.0000i			0.774403i
$668$		0			0
$669$	14.0000			0.541271
$670$		0			0
$671$	13.0000			0.501859
$672$		0			0
$673$		−	1.00000i		−	0.0385472i	−0.999814	−	0.0192736i	$-0.993865\pi$
$673$		−	1.00000i		−	0.0385472i	0.999814	−	0.0192736i	$-0.00613535\pi$
$674$		0			0
$675$	−20.0000	+	15.0000i	−0.769800	+	0.577350i
$676$		0			0
$677$		−	8.00000i		−	0.307465i	−0.988113	−	0.153732i	$-0.950871\pi$
$677$		−	8.00000i		−	0.307465i	0.988113	−	0.153732i	$-0.0491294\pi$
$678$		0			0
$679$	−36.0000			−1.38155
$680$		0			0
$681$	12.0000			0.459841
$682$		0			0
$683$		−	49.0000i		−	1.87493i	−0.348076	−	0.937466i	$-0.613165\pi$
$683$		−	49.0000i		−	1.87493i	0.348076	−	0.937466i	$-0.386835\pi$
$684$		0			0
$685$	12.0000	−	24.0000i	0.458496	−	0.916993i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−36.0000			−1.37149
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$		−	6.00000i		−	0.227921i
$694$		0			0
$695$	−40.0000	−	20.0000i	−1.51729	−	0.758643i
$696$		0			0
$697$			24.0000i			0.909065i
$698$		0			0
$699$	−19.0000			−0.718646
$700$		0			0
$701$	27.0000			1.01978			0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000			1.01978			0.509888	+	0.860241i	$0.329687\pi$
$702$		0			0
$703$		−	35.0000i		−	1.32005i
$704$		0			0
$705$	16.0000	+	8.00000i	0.602595	+	0.301297i
$706$		0			0
$707$			6.00000i			0.225653i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			28.0000i			1.04861i
$714$		0			0
$715$	−4.00000	+	8.00000i	−0.149592	+	0.299183i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.0000			0.559406			0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000			0.559406			0.279703	+	0.960087i	$0.409764\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$		0			0
$723$		−	8.00000i		−	0.297523i
$724$		0			0
$725$	−15.0000	−	20.0000i	−0.557086	−	0.742781i
$726$		0			0
$727$			18.0000i			0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$			18.0000i			0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	18.0000			0.665754
$732$		0			0
$733$			14.0000i			0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$			14.0000i			0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$		0			0
$735$	−2.00000	+	4.00000i	−0.0737711	+	0.147542i
$736$		0			0
$737$			12.0000i			0.442026i
$738$		0			0
$739$	40.0000			1.47142			0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000			1.47142			0.735712	+	0.677295i	$0.236848\pi$
$740$		0			0
$741$	20.0000			0.734718
$742$		0			0
$743$			1.00000i			0.0366864i	0.999832	+	0.0183432i	$0.00583916\pi$
$743$			1.00000i			0.0366864i	−0.999832	+	0.0183432i	$0.994161\pi$
$744$		0			0
$745$	−10.0000	−	5.00000i	−0.366372	−	0.183186i
$746$		0			0
$747$		−	8.00000i		−	0.292705i
$748$		0			0
$749$	36.0000			1.31541
$750$		0			0
$751$	−27.0000			−0.985244			−0.492622	−	0.870243i	$-0.663961\pi$
$751$	−27.0000			−0.985244			−0.492622	+	0.870243i	$0.663961\pi$
$752$		0			0
$753$		−	2.00000i		−	0.0728841i
$754$		0			0
$755$	4.00000	+	2.00000i	0.145575	+	0.0727875i
$756$		0			0
$757$		−	18.0000i		−	0.654221i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$		−	18.0000i		−	0.654221i	0.944986	−	0.327111i	$-0.106075\pi$
$758$		0			0
$759$	−4.00000			−0.145191
$760$		0			0
$761$	12.0000			0.435000			0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000			0.435000			0.217500	+	0.976060i	$0.430210\pi$
$762$		0			0
$763$			30.0000i			1.08607i
$764$		0			0
$765$	−6.00000	+	12.0000i	−0.216930	+	0.433861i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$	28.0000			1.00840
$772$		0			0
$773$		−	11.0000i		−	0.395643i	−0.980238	−	0.197821i	$-0.936613\pi$
$773$		−	11.0000i		−	0.395643i	0.980238	−	0.197821i	$-0.0633866\pi$
$774$		0			0
$775$	−21.0000	−	28.0000i	−0.754342	−	1.00579i
$776$		0			0
$777$		−	21.0000i		−	0.753371i
$778$		0			0
$779$	40.0000			1.43315
$780$		0			0
$781$	−3.00000			−0.107348
$782$		0			0
$783$		−	25.0000i		−	0.893427i
$784$		0			0
$785$	−13.0000	+	26.0000i	−0.463990	+	0.927980i
$786$		0			0
$787$			38.0000i			1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$			38.0000i			1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$		0			0
$789$	9.00000			0.320408
$790$		0			0
$791$	18.0000			0.640006
$792$		0			0
$793$		−	52.0000i		−	1.84657i
$794$		0			0
$795$	18.0000	+	9.00000i	0.638394	+	0.319197i
$796$		0			0
$797$		−	18.0000i		−	0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$		−	18.0000i		−	0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$		0			0
$801$	30.0000			1.06000
$802$		0			0
$803$			6.00000i			0.211735i
$804$		0			0
$805$	−24.0000	−	12.0000i	−0.845889	−	0.422944i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10.0000			−0.351581			−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000			−0.351581			−0.175791	+	0.984428i	$0.556248\pi$
$810$		0			0
$811$	−37.0000			−1.29925			−0.649623	−	0.760257i	$-0.725073\pi$
$811$	−37.0000			−1.29925			−0.649623	+	0.760257i	$0.725073\pi$
$812$		0			0
$813$		−	22.0000i		−	0.771574i
$814$		0			0
$815$	21.0000	−	42.0000i	0.735598	−	1.47120i
$816$		0			0
$817$		−	30.0000i		−	1.04957i
$818$		0			0
$819$	−24.0000			−0.838628
$820$		0			0
$821$	22.0000			0.767805			0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000			0.767805			0.383903	+	0.923374i	$0.374580\pi$
$822$		0			0
$823$			26.0000i			0.906303i	0.891434	+	0.453152i	$0.149700\pi$
$823$			26.0000i			0.906303i	−0.891434	+	0.453152i	$0.850300\pi$
$824$		0			0
$825$	4.00000	−	3.00000i	0.139262	−	0.104447i
$826$		0			0
$827$		−	22.0000i		−	0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$		−	22.0000i		−	0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$		0			0
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$	28.0000			0.971309
$832$		0			0
$833$			6.00000i			0.207888i
$834$		0			0
$835$	23.0000	−	46.0000i	0.795948	−	1.59190i
$836$		0			0
$837$		−	35.0000i		−	1.20978i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$		−	18.0000i		−	0.619953i
$844$		0			0
$845$	6.00000	+	3.00000i	0.206406	+	0.103203i
$846$		0			0
$847$			3.00000i			0.103081i
$848$		0			0
$849$	−16.0000			−0.549119
$850$		0			0
$851$	28.0000			0.959828
$852$		0			0
$853$			44.0000i			1.50653i	0.657716	+	0.753266i	$0.271523\pi$
$853$			44.0000i			1.50653i	−0.657716	+	0.753266i	$0.728477\pi$
$854$		0			0
$855$	20.0000	+	10.0000i	0.683986	+	0.341993i
$856$		0			0
$857$			17.0000i			0.580709i	0.956919	+	0.290354i	$0.0937732\pi$
$857$			17.0000i			0.580709i	−0.956919	+	0.290354i	$0.906227\pi$
$858$		0			0
$859$	−10.0000			−0.341196			−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000			−0.341196			−0.170598	+	0.985341i	$0.554570\pi$
$860$		0			0
$861$	24.0000			0.817918
$862$		0			0
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$	14.0000	−	28.0000i	0.476014	−	0.952029i
$866$		0			0
$867$			8.00000i			0.271694i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	48.0000			1.62642
$872$		0			0
$873$			24.0000i			0.812277i
$874$		0			0
$875$	33.0000	−	6.00000i	1.11560	−	0.202837i
$876$		0			0
$877$			2.00000i			0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$			2.00000i			0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$		0			0
$879$	16.0000			0.539667
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$		−	9.00000i		−	0.302874i	−0.988467	−	0.151437i	$-0.951610\pi$
$883$		−	9.00000i		−	0.302874i	0.988467	−	0.151437i	$-0.0483901\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	32.0000i		−	1.07445i	−0.843437	−	0.537227i	$-0.819472\pi$
$887$		−	32.0000i		−	1.07445i	0.843437	−	0.537227i	$-0.180528\pi$
$888$		0			0
$889$	−24.0000			−0.804934
$890$		0			0
$891$	−1.00000			−0.0335013
$892$		0			0
$893$		−	40.0000i		−	1.33855i
$894$		0			0
$895$	−20.0000	−	10.0000i	−0.668526	−	0.334263i
$896$		0			0
$897$			16.0000i			0.534224i
$898$		0			0
$899$	35.0000			1.16732
$900$		0			0
$901$	27.0000			0.899500
$902$		0			0
$903$		−	18.0000i		−	0.599002i
$904$		0			0
$905$	36.0000	+	18.0000i	1.19668	+	0.598340i
$906$		0			0
$907$			23.0000i			0.763702i	0.924224	+	0.381851i	$0.124713\pi$
$907$			23.0000i			0.763702i	−0.924224	+	0.381851i	$0.875287\pi$
$908$		0			0
$909$	4.00000			0.132672
$910$		0			0
$911$	−37.0000			−1.22586			−0.612932	−	0.790135i	$-0.710010\pi$
$911$	−37.0000			−1.22586			−0.612932	+	0.790135i	$0.710010\pi$
$912$		0			0
$913$			4.00000i			0.132381i
$914$		0			0
$915$	−13.0000	+	26.0000i	−0.429767	+	0.859533i
$916$		0			0
$917$		−	21.0000i		−	0.693481i
$918$		0			0
$919$	50.0000			1.64935			0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000			1.64935			0.824674	+	0.565608i	$0.191359\pi$
$920$		0			0
$921$	−8.00000			−0.263609
$922$		0			0
$923$			12.0000i			0.394985i
$924$		0			0
$925$	−28.0000	+	21.0000i	−0.920634	+	0.690476i
$926$		0			0
$927$		−	8.00000i		−	0.262754i
$928$		0			0
$929$	−15.0000			−0.492134			−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000			−0.492134			−0.246067	+	0.969253i	$0.579138\pi$
$930$		0			0
$931$	10.0000			0.327737
$932$		0			0
$933$			13.0000i			0.425601i
$934$		0			0
$935$	3.00000	−	6.00000i	0.0981105	−	0.196221i
$936$		0			0
$937$			42.0000i			1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$			42.0000i			1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$		0			0
$939$	−14.0000			−0.456873
$940$		0			0
$941$	17.0000			0.554184			0.277092	−	0.960843i	$-0.410629\pi$
$941$	17.0000			0.554184			0.277092	+	0.960843i	$0.410629\pi$
$942$		0			0
$943$			32.0000i			1.04206i
$944$		0			0
$945$	30.0000	+	15.0000i	0.975900	+	0.487950i
$946$		0			0
$947$			53.0000i			1.72227i	0.508378	+	0.861134i	$0.330245\pi$
$947$			53.0000i			1.72227i	−0.508378	+	0.861134i	$0.669755\pi$
$948$		0			0
$949$	24.0000			0.779073
$950$		0			0
$951$	23.0000			0.745826
$952$		0			0
$953$			29.0000i			0.939402i	0.882826	+	0.469701i	$0.155638\pi$
$953$			29.0000i			0.939402i	−0.882826	+	0.469701i	$0.844362\pi$
$954$		0			0
$955$	−16.0000	−	8.00000i	−0.517748	−	0.258874i
$956$		0			0
$957$			5.00000i			0.161627i
$958$		0			0
$959$	−36.0000			−1.16250
$960$		0			0
$961$	18.0000			0.580645
$962$		0			0
$963$		−	24.0000i		−	0.773389i
$964$		0			0
$965$	−1.00000	+	2.00000i	−0.0321911	+	0.0643823i
$966$		0			0
$967$			13.0000i			0.418052i	0.977910	+	0.209026i	$0.0670293\pi$
$967$			13.0000i			0.418052i	−0.977910	+	0.209026i	$0.932971\pi$
$968$		0			0
$969$	−15.0000			−0.481869
$970$		0			0
$971$	28.0000			0.898563			0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000			0.898563			0.449281	+	0.893390i	$0.351680\pi$
$972$		0			0
$973$			60.0000i			1.92351i
$974$		0			0
$975$	−12.0000	−	16.0000i	−0.384308	−	0.512410i
$976$		0			0
$977$		−	28.0000i		−	0.895799i	−0.894084	−	0.447900i	$-0.852172\pi$
$977$		−	28.0000i		−	0.895799i	0.894084	−	0.447900i	$-0.147828\pi$
$978$		0			0
$979$	−15.0000			−0.479402
$980$		0			0
$981$	20.0000			0.638551
$982$		0			0
$983$		−	14.0000i		−	0.446531i	−0.974758	−	0.223265i	$-0.928328\pi$
$983$		−	14.0000i		−	0.446531i	0.974758	−	0.223265i	$-0.0716716\pi$
$984$		0			0
$985$	22.0000	−	44.0000i	0.700978	−	1.40196i
$986$		0			0
$987$		−	24.0000i		−	0.763928i
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$	8.00000			0.254128			0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000			0.254128			0.127064	+	0.991894i	$0.459445\pi$
$992$		0			0
$993$			18.0000i			0.571213i
$994$		0			0
$995$	−50.0000	−	25.0000i	−1.58511	−	0.792553i
$996$		0			0
$997$			2.00000i			0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$			2.00000i			0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$		0			0
$999$	−35.0000			−1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.b.a.529.2		2
4.3	odd	2		110.2.b.a.89.1	✓	2
5.2	odd	4		4400.2.a.s.1.1		1
5.3	odd	4		4400.2.a.k.1.1		1
5.4	even	2	inner	880.2.b.a.529.1		2
12.11	even	2		990.2.c.d.199.2		2
20.3	even	4		550.2.a.e.1.1		1
20.7	even	4		550.2.a.j.1.1		1
20.19	odd	2		110.2.b.a.89.2	yes	2
44.43	even	2		1210.2.b.a.969.2		2
60.23	odd	4		4950.2.a.ba.1.1		1
60.47	odd	4		4950.2.a.q.1.1		1
60.59	even	2		990.2.c.d.199.1		2
220.43	odd	4		6050.2.a.bk.1.1		1
220.87	odd	4		6050.2.a.f.1.1		1
220.219	even	2		1210.2.b.a.969.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.b.a.89.1	✓	2	4.3	odd	2
110.2.b.a.89.2	yes	2	20.19	odd	2
550.2.a.e.1.1		1	20.3	even	4
550.2.a.j.1.1		1	20.7	even	4
880.2.b.a.529.1		2	5.4	even	2	inner
880.2.b.a.529.2		2	1.1	even	1	trivial
990.2.c.d.199.1		2	60.59	even	2
990.2.c.d.199.2		2	12.11	even	2
1210.2.b.a.969.1		2	220.219	even	2
1210.2.b.a.969.2		2	44.43	even	2
4400.2.a.k.1.1		1	5.3	odd	4
4400.2.a.s.1.1		1	5.2	odd	4
4950.2.a.q.1.1		1	60.47	odd	4
4950.2.a.ba.1.1		1	60.23	odd	4
6050.2.a.f.1.1		1	220.87	odd	4
6050.2.a.bk.1.1		1	220.43	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +3.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +4.00000i q^{13} +(1.00000 - 2.00000i) q^{15} -3.00000i q^{17} -5.00000 q^{19} -3.00000 q^{21} -4.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +5.00000i q^{27} -5.00000 q^{29} -7.00000 q^{31} -1.00000i q^{33} +(3.00000 - 6.00000i) q^{35} +7.00000i q^{37} -4.00000 q^{39} -8.00000 q^{41} +6.00000i q^{43} +(-4.00000 - 2.00000i) q^{45} +8.00000i q^{47} -2.00000 q^{49} +3.00000 q^{51} +9.00000i q^{53} +(2.00000 + 1.00000i) q^{55} -5.00000i q^{57} -13.0000 q^{61} +6.00000i q^{63} +(4.00000 - 8.00000i) q^{65} -12.0000i q^{67} +4.00000 q^{69} +3.00000 q^{71} -6.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -3.00000i q^{77} +1.00000 q^{81} -4.00000i q^{83} +(-3.00000 + 6.00000i) q^{85} -5.00000i q^{87} +15.0000 q^{89} -12.0000 q^{91} -7.00000i q^{93} +(10.0000 + 5.00000i) q^{95} +12.0000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 4 q^{9} - 2 q^{11} + 2 q^{15} - 10 q^{19} - 6 q^{21} + 6 q^{25} - 10 q^{29} - 14 q^{31} + 6 q^{35} - 8 q^{39} - 16 q^{41} - 8 q^{45} - 4 q^{49} + 6 q^{51} + 4 q^{55} - 26 q^{61} + 8 q^{65}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 4 * q^5 + 4 * q^9 - 2 * q^11 + 2 * q^15 - 10 * q^19 - 6 * q^21 + 6 * q^25 - 10 * q^29 - 14 * q^31 + 6 * q^35 - 8 * q^39 - 16 * q^41 - 8 * q^45 - 4 * q^49 + 6 * q^51 + 4 * q^55 - 26 * q^61 + 8 * q^65 + 8 * q^69 + 6 * q^71 - 8 * q^75 + 2 * q^81 - 6 * q^85 + 30 * q^89 - 24 * q^91 + 20 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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