LMFDB - Embedded newform 440.2.b.d.89.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [440,2,Mod(89,440)] 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("440.89"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$3.51341768894$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.47985531136.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 $ x^8 + 19x^6 + 91x^4 + 45*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.1
Root			$-3.36007i$ of defining polynomial
Character	$\chi$	$=$	440.89
Dual form			440.2.b.d.89.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.36007i q^{3} +(-0.256321 - 2.22133i) q^{5} +1.08258i q^{7} -8.29009 q^{9} +1.00000 q^{11} -4.00000i q^{13} +(-7.46382 + 0.861256i) q^{15} -0.107866i q^{17} +6.61228 q^{19} +3.63756 q^{21} +5.97235i q^{23} +(-4.86860 + 1.13874i) q^{25} +17.7751i q^{27} -7.80273 q^{29} +1.12492 q^{31} -3.36007i q^{33} +(2.40477 - 0.277489i) q^{35} -7.05494i q^{37} -13.4403 q^{39} +5.19045 q^{41} -5.52969i q^{43} +(2.12492 + 18.4150i) q^{45} -7.69486i q^{47} +5.82801 q^{49} -0.362439 q^{51} +4.77745i q^{53} +(-0.256321 - 2.22133i) q^{55} -22.2177i q^{57} +0.677809 q^{59} +0.197271 q^{61} -8.97472i q^{63} +(-8.88531 + 1.02528i) q^{65} -1.41737i q^{67} +20.0675 q^{69} -6.15020 q^{71} -6.16517i q^{73} +(3.82626 + 16.3588i) q^{75} +1.08258i q^{77} +9.35562 q^{79} +34.8553 q^{81} -13.7454i q^{83} +(-0.239607 + 0.0276484i) q^{85} +26.2177i q^{87} +1.29009 q^{89} +4.33034 q^{91} -3.77981i q^{93} +(-1.69486 - 14.6880i) q^{95} -6.60782i q^{97} -8.29009 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{9} + 8 q^{11} - 12 q^{15} + 18 q^{19} - 14 q^{21} - 2 q^{25} - 6 q^{29} - 30 q^{31} + 30 q^{35} - 8 q^{39} + 20 q^{41} - 22 q^{45} - 18 q^{49} - 46 q^{51} - 12 q^{59} + 58 q^{61} - 8 q^{65}+ \cdots - 14 q^{99}+O(q^{100}) $ 8 * q - 14 * q^9 + 8 * q^11 - 12 * q^15 + 18 * q^19 - 14 * q^21 - 2 * q^25 - 6 * q^29 - 30 * q^31 + 30 * q^35 - 8 * q^39 + 20 * q^41 - 22 * q^45 - 18 * q^49 - 46 * q^51 - 12 * q^59 + 58 * q^61 - 8 * q^65 + 60 * q^69 - 2 * q^71 + 26 * q^75 + 40 * q^79 + 88 * q^81 - 26 * q^85 - 42 * q^89 + 8 * q^91 + 28 * q^95 - 14 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$221$	$321$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	3.36007i		−	1.93994i	−0.243227	−	0.969969i	$-0.578206\pi$
$3$		−	3.36007i		−	1.93994i	0.243227	−	0.969969i	$-0.421794\pi$
$4$		0			0
$5$	−0.256321	−	2.22133i	−0.114630	−	0.993408i
$5$	−0.256321	−	2.22133i	−0.114630	−	0.993408i
$6$		0			0
$7$			1.08258i			0.409178i	0.978848	+	0.204589i	$0.0655858\pi$
$7$			1.08258i			0.409178i	−0.978848	+	0.204589i	$0.934414\pi$
$8$		0			0
$9$	−8.29009			−2.76336
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$	−7.46382	+	0.861256i	−1.92715	+	0.222375i
$16$		0			0
$17$		−	0.107866i		−	0.0261614i	−0.999914	−	0.0130807i	$-0.995836\pi$
$17$		−	0.107866i		−	0.0261614i	0.999914	−	0.0130807i	$-0.00416384\pi$
$18$		0			0
$19$	6.61228			1.51696			0.758480	−	0.651696i	$-0.225942\pi$
$19$	6.61228			1.51696			0.758480	+	0.651696i	$0.225942\pi$
$20$		0			0
$21$	3.63756			0.793781
$22$		0			0
$23$			5.97235i			1.24532i	0.782492	+	0.622661i	$0.213948\pi$
$23$			5.97235i			1.24532i	−0.782492	+	0.622661i	$0.786052\pi$
$24$		0			0
$25$	−4.86860	+	1.13874i	−0.973720	+	0.227749i
$26$		0			0
$27$			17.7751i			3.42082i
$28$		0			0
$29$	−7.80273			−1.44893			−0.724465	−	0.689311i	$-0.757913\pi$
$29$	−7.80273			−1.44893			−0.724465	+	0.689311i	$0.757913\pi$
$30$		0			0
$31$	1.12492			0.202042			0.101021	−	0.994884i	$-0.467789\pi$
$31$	1.12492			0.202042			0.101021	+	0.994884i	$0.467789\pi$
$32$		0			0
$33$		−	3.36007i		−	0.584914i
$34$		0			0
$35$	2.40477	−	0.277489i	0.406481	−	0.0469041i
$36$		0			0
$37$		−	7.05494i		−	1.15982i	−0.814679	−	0.579912i	$-0.803087\pi$
$37$		−	7.05494i		−	1.15982i	0.814679	−	0.579912i	$-0.196913\pi$
$38$		0			0
$39$	−13.4403			−2.15217
$40$		0			0
$41$	5.19045			0.810612			0.405306	−	0.914181i	$-0.367165\pi$
$41$	5.19045			0.810612			0.405306	+	0.914181i	$0.367165\pi$
$42$		0			0
$43$		−	5.52969i		−	0.843271i	−0.906766	−	0.421635i	$-0.861456\pi$
$43$		−	5.52969i		−	0.843271i	0.906766	−	0.421635i	$-0.138544\pi$
$44$		0			0
$45$	2.12492	+	18.4150i	0.316764	+	2.74515i
$46$		0			0
$47$		−	7.69486i		−	1.12241i	−0.827676	−	0.561206i	$-0.810338\pi$
$47$		−	7.69486i		−	1.12241i	0.827676	−	0.561206i	$-0.189662\pi$
$48$		0			0
$49$	5.82801			0.832573
$50$		0			0
$51$	−0.362439			−0.0507516
$52$		0			0
$53$			4.77745i			0.656233i	0.944637	+	0.328116i	$0.106414\pi$
$53$			4.77745i			0.656233i	−0.944637	+	0.328116i	$0.893586\pi$
$54$		0			0
$55$	−0.256321	−	2.22133i	−0.0345623	−	0.299524i
$56$		0			0
$57$		−	22.2177i		−	2.94281i
$58$		0			0
$59$	0.677809			0.0882433			0.0441216	−	0.999026i	$-0.485951\pi$
$59$	0.677809			0.0882433			0.0441216	+	0.999026i	$0.485951\pi$
$60$		0			0
$61$	0.197271			0.0252579			0.0126290	−	0.999920i	$-0.495980\pi$
$61$	0.197271			0.0252579			0.0126290	+	0.999920i	$0.495980\pi$
$62$		0			0
$63$		−	8.97472i		−	1.13071i
$64$		0			0
$65$	−8.88531	+	1.02528i	−1.10209	+	0.127171i
$66$		0			0
$67$		−	1.41737i		−	0.173160i	−0.996245	−	0.0865799i	$-0.972406\pi$
$67$		−	1.41737i		−	0.173160i	0.996245	−	0.0865799i	$-0.0275938\pi$
$68$		0			0
$69$	20.0675			2.41585
$70$		0			0
$71$	−6.15020			−0.729895			−0.364947	−	0.931028i	$-0.618913\pi$
$71$	−6.15020			−0.729895			−0.364947	+	0.931028i	$0.618913\pi$
$72$		0			0
$73$		−	6.16517i		−	0.721578i	−0.932647	−	0.360789i	$-0.882507\pi$
$73$		−	6.16517i		−	0.721578i	0.932647	−	0.360789i	$-0.117493\pi$
$74$		0			0
$75$	3.82626	+	16.3588i	0.441819	+	1.88896i
$76$		0			0
$77$			1.08258i			0.123372i
$78$		0			0
$79$	9.35562			1.05259			0.526295	−	0.850302i	$-0.323581\pi$
$79$	9.35562			1.05259			0.526295	+	0.850302i	$0.323581\pi$
$80$		0			0
$81$	34.8553			3.87281
$82$		0			0
$83$		−	13.7454i		−	1.50876i	−0.656440	−	0.754378i	$-0.727938\pi$
$83$		−	13.7454i		−	1.50876i	0.656440	−	0.754378i	$-0.272062\pi$
$84$		0			0
$85$	−0.239607	+	0.0276484i	−0.0259890	+	0.00299889i
$86$		0			0
$87$			26.2177i			2.81084i
$88$		0			0
$89$	1.29009			0.136749			0.0683745	−	0.997660i	$-0.478219\pi$
$89$	1.29009			0.136749			0.0683745	+	0.997660i	$0.478219\pi$
$90$		0			0
$91$	4.33034			0.453943
$92$		0			0
$93$		−	3.77981i		−	0.391948i
$94$		0			0
$95$	−1.69486	−	14.6880i	−0.173889	−	1.50696i
$96$		0			0
$97$		−	6.60782i		−	0.670923i	−0.942054	−	0.335461i	$-0.891108\pi$
$97$		−	6.60782i		−	0.670923i	0.942054	−	0.335461i	$-0.108892\pi$
$98$		0			0
$99$	−8.29009			−0.833185
$100$		0			0
$101$	−15.4403			−1.53637			−0.768183	−	0.640230i	$-0.778839\pi$
$101$	−15.4403			−1.53637			−0.768183	+	0.640230i	$0.778839\pi$
$102$		0			0
$103$			5.74543i			0.566114i	0.959103	+	0.283057i	$0.0913485\pi$
$103$			5.74543i			0.566114i	−0.959103	+	0.283057i	$0.908651\pi$
$104$		0			0
$105$	−0.932382	−	8.08022i	−0.0909911	−	0.788549i
$106$		0			0
$107$			4.30514i			0.416193i	0.978108	+	0.208097i	$0.0667269\pi$
$107$			4.30514i			0.416193i	−0.978108	+	0.208097i	$0.933273\pi$
$108$		0			0
$109$	2.33034			0.223206			0.111603	−	0.993753i	$-0.464402\pi$
$109$	2.33034			0.223206			0.111603	+	0.993753i	$0.464402\pi$
$110$		0			0
$111$	−23.7051			−2.24999
$112$		0			0
$113$		−	10.9471i		−	1.02981i	−0.857246	−	0.514907i	$-0.827827\pi$
$113$		−	10.9471i		−	1.02981i	0.857246	−	0.514907i	$-0.172173\pi$
$114$		0			0
$115$	13.2666	−	1.53084i	1.23711	−	0.142751i
$116$		0			0
$117$			33.1604i			3.06568i
$118$		0			0
$119$	0.116774			0.0107047
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	17.4403i		−	1.57254i
$124$		0			0
$125$	3.77745	+	10.5229i	0.337865	+	0.941195i
$126$		0			0
$127$			12.5802i			1.11631i	0.829737	+	0.558155i	$0.188491\pi$
$127$			12.5802i			1.11631i	−0.829737	+	0.558155i	$0.811509\pi$
$128$		0			0
$129$	−18.5802			−1.63589
$130$		0			0
$131$	15.2430			1.33179			0.665894	−	0.746046i	$-0.268050\pi$
$131$	15.2430			1.33179			0.665894	+	0.746046i	$0.268050\pi$
$132$		0			0
$133$			7.15835i			0.620707i
$134$		0			0
$135$	39.4843	−	4.55612i	3.39827	−	0.392128i
$136$		0			0
$137$			14.6078i			1.24803i	0.781412	+	0.624015i	$0.214500\pi$
$137$			14.6078i			1.24803i	−0.781412	+	0.624015i	$0.785500\pi$
$138$		0			0
$139$	20.4150			1.73158			0.865789	−	0.500409i	$-0.166817\pi$
$139$	20.4150			1.73158			0.865789	+	0.500409i	$0.166817\pi$
$140$		0			0
$141$	−25.8553			−2.17741
$142$		0			0
$143$		−	4.00000i		−	0.334497i
$144$		0			0
$145$	2.00000	+	17.3324i	0.166091	+	1.43938i
$146$		0			0
$147$		−	19.5825i		−	1.61514i
$148$		0			0
$149$	15.8874			1.30155			0.650773	−	0.759272i	$-0.274445\pi$
$149$	15.8874			1.30155			0.650773	+	0.759272i	$0.274445\pi$
$150$		0			0
$151$	6.58018			0.535487			0.267744	−	0.963490i	$-0.413722\pi$
$151$	6.58018			0.535487			0.267744	+	0.963490i	$0.413722\pi$
$152$		0			0
$153$			0.894222i			0.0722935i
$154$		0			0
$155$	−0.288340	−	2.49882i	−0.0231600	−	0.200710i
$156$		0			0
$157$			6.38535i			0.509607i	0.966993	+	0.254803i	$0.0820108\pi$
$157$			6.38535i			0.509607i	−0.966993	+	0.254803i	$0.917989\pi$
$158$		0			0
$159$	16.0526			1.27305
$160$		0			0
$161$	−6.46557			−0.509559
$162$		0			0
$163$			18.3071i			1.43393i	0.697111	+	0.716963i	$0.254468\pi$
$163$			18.3071i			1.43393i	−0.697111	+	0.716963i	$0.745532\pi$
$164$		0			0
$165$	−7.46382	+	0.861256i	−0.581058	+	0.0670487i
$166$		0			0
$167$			0.197271i			0.0152653i	0.999971	+	0.00763263i	$0.00242956\pi$
$167$			0.197271i			0.0152653i	−0.999971	+	0.00763263i	$0.997570\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	−54.8164			−4.19191
$172$		0			0
$173$		−	14.7201i		−	1.11915i	−0.828779	−	0.559576i	$-0.810964\pi$
$173$		−	14.7201i		−	1.11915i	0.828779	−	0.559576i	$-0.189036\pi$
$174$		0			0
$175$	−1.23279	−	5.27067i	−0.0931899	−	0.398425i
$176$		0			0
$177$		−	2.27749i		−	0.171187i
$178$		0			0
$179$	−11.8518			−0.885845			−0.442923	−	0.896560i	$-0.646058\pi$
$179$	−11.8518			−0.885845			−0.442923	+	0.896560i	$0.646058\pi$
$180$		0			0
$181$	−10.7625			−0.799969			−0.399984	−	0.916522i	$-0.630984\pi$
$181$	−10.7625			−0.799969			−0.399984	+	0.916522i	$0.630984\pi$
$182$		0			0
$183$		−	0.662843i		−	0.0489988i
$184$		0			0
$185$	−15.6713	+	1.80833i	−1.15218	+	0.132951i
$186$		0			0
$187$		−	0.107866i		−	0.00788797i
$188$		0			0
$189$	−19.2430			−1.39972
$190$		0			0
$191$	1.70309			0.123231			0.0616157	−	0.998100i	$-0.480375\pi$
$191$	1.70309			0.123231			0.0616157	+	0.998100i	$0.480375\pi$
$192$		0			0
$193$		−	10.8280i		−	0.779417i	−0.920938	−	0.389709i	$-0.872576\pi$
$193$		−	10.8280i		−	0.779417i	0.920938	−	0.389709i	$-0.127424\pi$
$194$		0			0
$195$	3.44502	+	29.8553i	0.246703	+	2.13798i
$196$		0			0
$197$		−	6.72015i		−	0.478791i	−0.970922	−	0.239395i	$-0.923051\pi$
$197$		−	6.72015i		−	0.478791i	0.970922	−	0.239395i	$-0.0769492\pi$
$198$		0			0
$199$	10.8280			0.767577			0.383789	−	0.923421i	$-0.374619\pi$
$199$	10.8280			0.767577			0.383789	+	0.923421i	$0.374619\pi$
$200$		0			0
$201$	−4.76248			−0.335920
$202$		0			0
$203$		−	8.44711i		−	0.592871i
$204$		0			0
$205$	−1.33042	−	11.5297i	−0.0929205	−	0.805269i
$206$		0			0
$207$		−	49.5113i		−	3.44127i
$208$		0			0
$209$	6.61228			0.457381
$210$		0			0
$211$	0.447111			0.0307804			0.0153902	−	0.999882i	$-0.495101\pi$
$211$	0.447111			0.0307804			0.0153902	+	0.999882i	$0.495101\pi$
$212$		0			0
$213$			20.6651i			1.41595i
$214$		0			0
$215$	−12.2833	+	1.41737i	−0.837712	+	0.0966641i
$216$		0			0
$217$			1.21782i			0.0826710i
$218$		0			0
$219$	−20.7154			−1.39982
$220$		0			0
$221$	−0.431465			−0.0290235
$222$		0			0
$223$			17.8577i			1.19584i	0.801555	+	0.597922i	$0.204007\pi$
$223$			17.8577i			1.19584i	−0.801555	+	0.597922i	$0.795993\pi$
$224$		0			0
$225$	40.3611	−	9.44029i	2.69074	−	0.629353i
$226$		0			0
$227$			1.31396i			0.0872107i	0.999049	+	0.0436054i	$0.0138844\pi$
$227$			1.31396i			0.0872107i	−0.999049	+	0.0436054i	$0.986116\pi$
$228$		0			0
$229$	−4.84298			−0.320033			−0.160016	−	0.987114i	$-0.551155\pi$
$229$	−4.84298			−0.320033			−0.160016	+	0.987114i	$0.551155\pi$
$230$		0			0
$231$	3.63756			0.239334
$232$		0			0
$233$			20.3829i			1.33533i	0.744462	+	0.667664i	$0.232706\pi$
$233$			20.3829i			1.33533i	−0.744462	+	0.667664i	$0.767294\pi$
$234$		0			0
$235$	−17.0928	+	1.97235i	−1.11501	+	0.128662i
$236$		0			0
$237$		−	31.4356i		−	2.04196i
$238$		0			0
$239$	9.35562			0.605165			0.302582	−	0.953123i	$-0.402151\pi$
$239$	9.35562			0.605165			0.302582	+	0.953123i	$0.402151\pi$
$240$		0			0
$241$	22.3004			1.43650			0.718248	−	0.695788i	$-0.244944\pi$
$241$	22.3004			1.43650			0.718248	+	0.695788i	$0.244944\pi$
$242$		0			0
$243$		−	63.7911i		−	4.09220i
$244$		0			0
$245$	−1.49384	−	12.9459i	−0.0954379	−	0.827085i
$246$		0			0
$247$		−	26.4491i		−	1.68292i
$248$		0			0
$249$	−46.1856			−2.92690
$250$		0			0
$251$	−10.9276			−0.689747			−0.344874	−	0.938649i	$-0.612078\pi$
$251$	−10.9276			−0.689747			−0.344874	+	0.938649i	$0.612078\pi$
$252$		0			0
$253$			5.97235i			0.375479i
$254$		0			0
$255$	0.0929005	+	0.805096i	0.00581766	+	0.0504170i
$256$		0			0
$257$			21.4403i			1.33741i	0.743528	+	0.668704i	$0.233151\pi$
$257$			21.4403i			1.33741i	−0.743528	+	0.668704i	$0.766849\pi$
$258$		0			0
$259$	7.63756			0.474575
$260$		0			0
$261$	64.6853			4.00392
$262$		0			0
$263$			6.52287i			0.402218i	0.979569	+	0.201109i	$0.0644545\pi$
$263$			6.52287i			0.402218i	−0.979569	+	0.201109i	$0.935546\pi$
$264$		0			0
$265$	10.6123	−	1.22456i	0.651907	−	0.0752240i
$266$		0			0
$267$		−	4.33479i		−	0.265285i
$268$		0			0
$269$	−5.60546			−0.341771			−0.170885	−	0.985291i	$-0.554663\pi$
$269$	−5.60546			−0.341771			−0.170885	+	0.985291i	$0.554663\pi$
$270$		0			0
$271$	−28.9446			−1.75826			−0.879130	−	0.476582i	$-0.841876\pi$
$271$	−28.9446			−1.75826			−0.879130	+	0.476582i	$0.841876\pi$
$272$		0			0
$273$		−	14.5502i		−	0.880621i
$274$		0			0
$275$	−4.86860	+	1.13874i	−0.293588	+	0.0686689i
$276$		0			0
$277$			25.3850i			1.52524i	0.646849	+	0.762618i	$0.276086\pi$
$277$			25.3850i			1.52524i	−0.646849	+	0.762618i	$0.723914\pi$
$278$		0			0
$279$	−9.32569			−0.558314
$280$		0			0
$281$	−27.1604			−1.62025			−0.810125	−	0.586257i	$-0.800601\pi$
$281$	−27.1604			−1.62025			−0.810125	+	0.586257i	$0.800601\pi$
$282$		0			0
$283$			22.0205i			1.30898i	0.756070	+	0.654490i	$0.227117\pi$
$283$			22.0205i			1.30898i	−0.756070	+	0.654490i	$0.772883\pi$
$284$		0			0
$285$	−49.3529	+	5.69486i	−2.92341	+	0.337335i
$286$		0			0
$287$			5.61910i			0.331685i
$288$		0			0
$289$	16.9884			0.999316
$290$		0			0
$291$	−22.2028			−1.30155
$292$		0			0
$293$		−	1.44029i		−	0.0841427i	−0.999115	−	0.0420713i	$-0.986604\pi$
$293$		−	1.44029i		−	0.0841427i	0.999115	−	0.0420713i	$-0.0133957\pi$
$294$		0			0
$295$	−0.173736	−	1.50564i	−0.0101153	−	0.0876616i
$296$		0			0
$297$			17.7751i			1.03141i
$298$		0			0
$299$	23.8894			1.38156
$300$		0			0
$301$	5.98636			0.345048
$302$		0			0
$303$			51.8805i			2.98046i
$304$		0			0
$305$	−0.0505645	−	0.438203i	−0.00289532	−	0.0250914i
$306$		0			0
$307$		−	2.80955i		−	0.160349i	−0.996781	−	0.0801747i	$-0.974452\pi$
$307$		−	2.80955i		−	0.160349i	0.996781	−	0.0801747i	$-0.0255478\pi$
$308$		0			0
$309$	19.3051			1.09823
$310$		0			0
$311$	−11.5529			−0.655104			−0.327552	−	0.944833i	$-0.606224\pi$
$311$	−11.5529			−0.655104			−0.327552	+	0.944833i	$0.606224\pi$
$312$		0			0
$313$		−	26.1580i		−	1.47854i	−0.673411	−	0.739268i	$-0.735171\pi$
$313$		−	26.1580i		−	1.47854i	0.673411	−	0.739268i	$-0.264829\pi$
$314$		0			0
$315$	−19.9358	+	2.30040i	−1.12325	+	0.129613i
$316$		0			0
$317$			29.3805i			1.65018i	0.565005	+	0.825088i	$0.308874\pi$
$317$			29.3805i			1.65018i	−0.565005	+	0.825088i	$0.691126\pi$
$318$		0			0
$319$	−7.80273			−0.436869
$320$		0			0
$321$	14.4656			0.807390
$322$		0			0
$323$		−	0.713242i		−	0.0396859i
$324$		0			0
$325$	4.55498	+	19.4744i	0.252665	+	1.08025i
$326$		0			0
$327$		−	7.83010i		−	0.433005i
$328$		0			0
$329$	8.33034			0.459266
$330$		0			0
$331$	12.2833			0.675149			0.337575	−	0.941299i	$-0.390393\pi$
$331$	12.2833			0.675149			0.337575	+	0.941299i	$0.390393\pi$
$332$		0			0
$333$			58.4860i			3.20502i
$334$		0			0
$335$	−3.14845	+	0.363302i	−0.172018	+	0.0198493i
$336$		0			0
$337$		−	21.3324i		−	1.16205i	−0.813885	−	0.581026i	$-0.802652\pi$
$337$		−	21.3324i		−	1.16205i	0.813885	−	0.581026i	$-0.197348\pi$
$338$		0			0
$339$	−36.7829			−1.99778
$340$		0			0
$341$	1.12492			0.0609178
$342$		0			0
$343$			13.8874i			0.749849i
$344$		0			0
$345$	−5.14372	−	44.5766i	−0.276929	−	2.39992i
$346$		0			0
$347$			10.4150i			0.559107i	0.960130	+	0.279553i	$0.0901864\pi$
$347$			10.4150i			0.559107i	−0.960130	+	0.279553i	$0.909814\pi$
$348$		0			0
$349$	13.4909			0.722149			0.361074	−	0.932537i	$-0.382410\pi$
$349$	13.4909			0.722149			0.361074	+	0.932537i	$0.382410\pi$
$350$		0			0
$351$	71.1003			3.79505
$352$		0			0
$353$			12.7177i			0.676895i	0.940985	+	0.338447i	$0.109902\pi$
$353$			12.7177i			0.676895i	−0.940985	+	0.338447i	$0.890098\pi$
$354$		0			0
$355$	1.57642	+	13.6616i	0.0836679	+	0.725083i
$356$		0			0
$357$		−	0.392370i		−	0.0207664i
$358$		0			0
$359$	19.8212			1.04612			0.523061	−	0.852295i	$-0.324790\pi$
$359$	19.8212			1.04612			0.523061	+	0.852295i	$0.324790\pi$
$360$		0			0
$361$	24.7222			1.30117
$362$		0			0
$363$		−	3.36007i		−	0.176358i
$364$		0			0
$365$	−13.6949	+	1.58026i	−0.716822	+	0.0827146i
$366$		0			0
$367$			10.3027i			0.537796i	0.963169	+	0.268898i	$0.0866594\pi$
$367$			10.3027i			0.537796i	−0.963169	+	0.268898i	$0.913341\pi$
$368$		0			0
$369$	−43.0293			−2.24002
$370$		0			0
$371$	−5.17199			−0.268516
$372$		0			0
$373$		−	23.3344i		−	1.20821i	−0.796904	−	0.604105i	$-0.793531\pi$
$373$		−	23.3344i		−	1.20821i	0.796904	−	0.604105i	$-0.206469\pi$
$374$		0			0
$375$	35.3576	−	12.6925i	1.82586	−	0.655438i
$376$		0			0
$377$			31.2109i			1.60744i
$378$		0			0
$379$	21.2074			1.08935			0.544676	−	0.838647i	$-0.316653\pi$
$379$	21.2074			1.08935			0.544676	+	0.838647i	$0.316653\pi$
$380$		0			0
$381$	42.2703			2.16557
$382$		0			0
$383$			0.972434i			0.0496891i	0.999691	+	0.0248445i	$0.00790908\pi$
$383$			0.972434i			0.0496891i	−0.999691	+	0.0248445i	$0.992091\pi$
$384$		0			0
$385$	2.40477	−	0.277489i	0.122559	−	0.0141421i
$386$		0			0
$387$			45.8417i			2.33026i
$388$		0			0
$389$	−2.76248			−0.140063			−0.0700317	−	0.997545i	$-0.522310\pi$
$389$	−2.76248			−0.140063			−0.0700317	+	0.997545i	$0.522310\pi$
$390$		0			0
$391$	0.644216			0.0325794
$392$		0			0
$393$		−	51.2177i		−	2.58359i
$394$		0			0
$395$	−2.39804	−	20.7819i	−0.120658	−	1.04565i
$396$		0			0
$397$		−	11.2751i		−	0.565882i	−0.959137	−	0.282941i	$-0.908690\pi$
$397$		−	11.2751i		−	0.565882i	0.959137	−	0.282941i	$-0.0913101\pi$
$398$		0			0
$399$	24.0526			1.20413
$400$		0			0
$401$	−10.4471			−0.521704			−0.260852	−	0.965379i	$-0.584003\pi$
$401$	−10.4471			−0.521704			−0.260852	+	0.965379i	$0.584003\pi$
$402$		0			0
$403$		−	4.49968i		−	0.224145i
$404$		0			0
$405$	−8.93413	−	77.4251i	−0.443940	−	3.84728i
$406$		0			0
$407$		−	7.05494i		−	0.349700i
$408$		0			0
$409$	1.27512			0.0630507			0.0315254	−	0.999503i	$-0.489963\pi$
$409$	1.27512			0.0630507			0.0315254	+	0.999503i	$0.489963\pi$
$410$		0			0
$411$	49.0834			2.42110
$412$		0			0
$413$			0.733786i			0.0361072i
$414$		0			0
$415$	−30.5331	+	3.52324i	−1.49881	+	0.172949i
$416$		0			0
$417$		−	68.5959i		−	3.35916i
$418$		0			0
$419$	−18.7795			−0.917436			−0.458718	−	0.888582i	$-0.651691\pi$
$419$	−18.7795			−0.917436			−0.458718	+	0.888582i	$0.651691\pi$
$420$		0			0
$421$	−1.27512			−0.0621457			−0.0310728	−	0.999517i	$-0.509892\pi$
$421$	−1.27512			−0.0621457			−0.0310728	+	0.999517i	$0.509892\pi$
$422$		0			0
$423$			63.7911i			3.10163i
$424$		0			0
$425$	0.122832	+	0.525158i	0.00595824	+	0.0254739i
$426$		0			0
$427$			0.213562i			0.0103350i
$428$		0			0
$429$	−13.4403			−0.648903
$430$		0			0
$431$	1.63556			0.0787820			0.0393910	−	0.999224i	$-0.487458\pi$
$431$	1.63556			0.0787820			0.0393910	+	0.999224i	$0.487458\pi$
$432$		0			0
$433$			26.4932i			1.27318i	0.771201	+	0.636591i	$0.219656\pi$
$433$			26.4932i			1.27318i	−0.771201	+	0.636591i	$0.780344\pi$
$434$		0			0
$435$	58.2382	−	6.72015i	2.79231	−	0.322206i
$436$		0			0
$437$			39.4909i			1.88910i
$438$		0			0
$439$	−31.9358			−1.52421			−0.762106	−	0.647452i	$-0.775835\pi$
$439$	−31.9358			−1.52421			−0.762106	+	0.647452i	$0.775835\pi$
$440$		0			0
$441$	−48.3147			−2.30070
$442$		0			0
$443$			2.02765i			0.0963365i	0.998839	+	0.0481682i	$0.0153384\pi$
$443$			2.02765i			0.0963365i	−0.998839	+	0.0481682i	$0.984662\pi$
$444$		0			0
$445$	−0.330676	−	2.86571i	−0.0156756	−	0.135848i
$446$		0			0
$447$		−	53.3828i		−	2.52492i
$448$		0			0
$449$	10.0675			0.475116			0.237558	−	0.971373i	$-0.423653\pi$
$449$	10.0675			0.475116			0.237558	+	0.971373i	$0.423653\pi$
$450$		0			0
$451$	5.19045			0.244409
$452$		0			0
$453$		−	22.1099i		−	1.03881i
$454$		0			0
$455$	−1.10995	−	9.61910i	−0.0520355	−	0.450950i
$456$		0			0
$457$		−	6.04839i		−	0.282932i	−0.989943	−	0.141466i	$-0.954818\pi$
$457$		−	6.04839i		−	0.282932i	0.989943	−	0.141466i	$-0.0451816\pi$
$458$		0			0
$459$	1.91733			0.0894934
$460$		0			0
$461$	31.8397			1.48292			0.741460	−	0.670997i	$-0.234134\pi$
$461$	31.8397			1.48292			0.741460	+	0.670997i	$0.234134\pi$
$462$		0			0
$463$		−	30.2474i		−	1.40572i	−0.711330	−	0.702858i	$-0.751907\pi$
$463$		−	30.2474i		−	1.40572i	0.711330	−	0.702858i	$-0.248093\pi$
$464$		0			0
$465$	−8.39621	+	0.968844i	−0.389365	+	0.0449291i
$466$		0			0
$467$		−	31.3417i		−	1.45032i	−0.688580	−	0.725160i	$-0.741766\pi$
$467$		−	31.3417i		−	1.45032i	0.688580	−	0.725160i	$-0.258234\pi$
$468$		0			0
$469$	1.53443			0.0708533
$470$		0			0
$471$	21.4553			0.988606
$472$		0			0
$473$		−	5.52969i		−	0.254256i
$474$		0			0
$475$	−32.1925	+	7.52969i	−1.47709	+	0.345486i
$476$		0			0
$477$		−	39.6055i		−	1.81341i
$478$		0			0
$479$	−6.59663			−0.301408			−0.150704	−	0.988579i	$-0.548154\pi$
$479$	−6.59663			−0.301408			−0.150704	+	0.988579i	$0.548154\pi$
$480$		0			0
$481$	−28.2197			−1.28671
$482$		0			0
$483$			21.7248i			0.988512i
$484$		0			0
$485$	−14.6781	+	1.69372i	−0.666500	+	0.0769079i
$486$		0			0
$487$		−	1.13343i		−	0.0513605i	−0.999670	−	0.0256802i	$-0.991825\pi$
$487$		−	1.13343i		−	0.0513605i	0.999670	−	0.0256802i	$-0.00817517\pi$
$488$		0			0
$489$	61.5133			2.78173
$490$		0			0
$491$	43.5569			1.96570			0.982848	−	0.184419i	$-0.0590404\pi$
$491$	43.5569			1.96570			0.982848	+	0.184419i	$0.0590404\pi$
$492$		0			0
$493$			0.841652i			0.0379061i
$494$		0			0
$495$	2.12492	+	18.4150i	0.0955081	+	0.827693i
$496$		0			0
$497$		−	6.65811i		−	0.298657i
$498$		0			0
$499$	34.5502			1.54668			0.773341	−	0.633991i	$-0.218584\pi$
$499$	34.5502			1.54668			0.773341	+	0.633991i	$0.218584\pi$
$500$		0			0
$501$	0.662843			0.0296137
$502$		0			0
$503$		−	5.92424i		−	0.264149i	−0.991240	−	0.132074i	$-0.957836\pi$
$503$		−	5.92424i		−	0.264149i	0.991240	−	0.132074i	$-0.0421638\pi$
$504$		0			0
$505$	3.95766	+	34.2980i	0.176114	+	1.52624i
$506$		0			0
$507$			10.0802i			0.447678i
$508$		0			0
$509$	42.3679			1.87793			0.938963	−	0.344018i	$-0.111788\pi$
$509$	42.3679			1.87793			0.938963	+	0.344018i	$0.111788\pi$
$510$		0			0
$511$	6.67431			0.295254
$512$		0			0
$513$			117.534i			5.18924i
$514$		0			0
$515$	12.7625	−	1.47267i	0.562382	−	0.0648936i
$516$		0			0
$517$		−	7.69486i		−	0.338420i
$518$		0			0
$519$	−49.4608			−2.17109
$520$		0			0
$521$	21.2116			0.929297			0.464648	−	0.885495i	$-0.346181\pi$
$521$	21.2116			0.929297			0.464648	+	0.885495i	$0.346181\pi$
$522$		0			0
$523$			5.19045i			0.226963i	0.993540	+	0.113481i	$0.0362002\pi$
$523$			5.19045i			0.226963i	−0.993540	+	0.113481i	$0.963800\pi$
$524$		0			0
$525$	−17.7098	+	4.14225i	−0.772920	+	0.180783i
$526$		0			0
$527$		−	0.121341i		−	0.00528570i
$528$		0			0
$529$	−12.6690			−0.550825
$530$		0			0
$531$	−5.61910			−0.243848
$532$		0			0
$533$		−	20.7618i		−	0.899293i
$534$		0			0
$535$	9.56312	−	1.10350i	0.413450	−	0.0477083i
$536$		0			0
$537$			39.8229i			1.71849i
$538$		0			0
$539$	5.82801			0.251030
$540$		0			0
$541$	−24.0185			−1.03263			−0.516317	−	0.856397i	$-0.672697\pi$
$541$	−24.0185			−1.03263			−0.516317	+	0.856397i	$0.672697\pi$
$542$		0			0
$543$			36.1627i			1.55189i
$544$		0			0
$545$	−0.597313	−	5.17644i	−0.0255861	−	0.221734i
$546$		0			0
$547$		−	12.8601i		−	0.549859i	−0.961464	−	0.274929i	$-0.911346\pi$
$547$		−	12.8601i		−	0.549859i	0.961464	−	0.274929i	$-0.0886545\pi$
$548$		0			0
$549$	−1.63539			−0.0697968
$550$		0			0
$551$	−51.5938			−2.19797
$552$		0			0
$553$			10.1282i			0.430697i
$554$		0			0
$555$	6.07610	+	52.6568i	0.257916	+	2.23516i
$556$		0			0
$557$		−	31.3344i		−	1.32768i	−0.747874	−	0.663841i	$-0.768925\pi$
$557$		−	31.3344i		−	1.32768i	0.747874	−	0.663841i	$-0.231075\pi$
$558$		0			0
$559$	−22.1188			−0.935525
$560$		0			0
$561$	−0.362439			−0.0153022
$562$		0			0
$563$			17.6901i			0.745550i	0.927922	+	0.372775i	$0.121594\pi$
$563$			17.6901i			0.745550i	−0.927922	+	0.372775i	$0.878406\pi$
$564$		0			0
$565$	−24.3170	+	2.80596i	−1.02303	+	0.118048i
$566$		0			0
$567$			37.7338i			1.58467i
$568$		0			0
$569$	−26.3004			−1.10257			−0.551285	−	0.834317i	$-0.685862\pi$
$569$	−26.3004			−1.10257			−0.551285	+	0.834317i	$0.685862\pi$
$570$		0			0
$571$	−35.9884			−1.50607			−0.753033	−	0.657983i	$-0.771410\pi$
$571$	−35.9884			−1.50607			−0.753033	+	0.657983i	$0.771410\pi$
$572$		0			0
$573$		−	5.72251i		−	0.239061i
$574$		0			0
$575$	−6.80098	−	29.0770i	−0.283621	−	1.21259i
$576$		0			0
$577$		−	32.6031i		−	1.35728i	−0.734469	−	0.678642i	$-0.762569\pi$
$577$		−	32.6031i		−	1.35728i	0.734469	−	0.678642i	$-0.237431\pi$
$578$		0			0
$579$	−36.3829			−1.51202
$580$		0			0
$581$	14.8806			0.617351
$582$		0			0
$583$			4.77745i			0.197862i
$584$		0			0
$585$	73.6600	−	8.49968i	3.04547	−	0.351419i
$586$		0			0
$587$			17.6287i			0.727612i	0.931475	+	0.363806i	$0.118523\pi$
$587$			17.6287i			0.727612i	−0.931475	+	0.363806i	$0.881477\pi$
$588$		0			0
$589$	7.43829			0.306489
$590$		0			0
$591$	−22.5802			−0.928824
$592$		0			0
$593$		−	13.2246i		−	0.543068i	−0.962429	−	0.271534i	$-0.912469\pi$
$593$		−	13.2246i		−	0.543068i	0.962429	−	0.271534i	$-0.0875309\pi$
$594$		0			0
$595$	−0.0299317	−	0.259394i	−0.00122708	−	0.0106341i
$596$		0			0
$597$		−	36.3829i		−	1.48905i
$598$		0			0
$599$	−37.2771			−1.52310			−0.761551	−	0.648105i	$-0.775562\pi$
$599$	−37.2771			−1.52310			−0.761551	+	0.648105i	$0.775562\pi$
$600$		0			0
$601$	17.0594			0.695867			0.347934	−	0.937519i	$-0.386883\pi$
$601$	17.0594			0.695867			0.347934	+	0.937519i	$0.386883\pi$
$602$		0			0
$603$			11.7502i			0.478503i
$604$		0			0
$605$	−0.256321	−	2.22133i	−0.0104209	−	0.0903098i
$606$		0			0
$607$			7.92624i			0.321716i	0.986978	+	0.160858i	$0.0514262\pi$
$607$			7.92624i			0.321716i	−0.986978	+	0.160858i	$0.948574\pi$
$608$		0			0
$609$	−28.3829			−1.15013
$610$		0			0
$611$	−30.7795			−1.24520
$612$		0			0
$613$		−	9.93596i		−	0.401310i	−0.979662	−	0.200655i	$-0.935693\pi$
$613$		−	9.93596i		−	0.401310i	0.979662	−	0.200655i	$-0.0643070\pi$
$614$		0			0
$615$	−38.7406	+	4.47031i	−1.56217	+	0.180260i
$616$		0			0
$617$			38.1010i			1.53389i	0.641715	+	0.766944i	$0.278223\pi$
$617$			38.1010i			1.53389i	−0.641715	+	0.766944i	$0.721777\pi$
$618$		0			0
$619$	−5.70309			−0.229227			−0.114613	−	0.993410i	$-0.536563\pi$
$619$	−5.70309			−0.229227			−0.114613	+	0.993410i	$0.536563\pi$
$620$		0			0
$621$	−106.159			−4.26002
$622$		0			0
$623$			1.39663i			0.0559548i
$624$		0			0
$625$	22.4065	−	11.0882i	0.896261	−	0.443527i
$626$		0			0
$627$		−	22.2177i		−	0.887291i
$628$		0			0
$629$	−0.760990			−0.0303427
$630$		0			0
$631$	17.3242			0.689665			0.344833	−	0.938664i	$-0.387936\pi$
$631$	17.3242			0.689665			0.344833	+	0.938664i	$0.387936\pi$
$632$		0			0
$633$		−	1.50232i		−	0.0597120i
$634$		0			0
$635$	27.9447	−	3.22456i	1.10895	−	0.127963i
$636$		0			0
$637$		−	23.3120i		−	0.923657i
$638$		0			0
$639$	50.9857			2.01696
$640$		0			0
$641$	−22.3523			−0.882863			−0.441431	−	0.897295i	$-0.645529\pi$
$641$	−22.3523			−0.882863			−0.441431	+	0.897295i	$0.645529\pi$
$642$		0			0
$643$			25.2911i			0.997385i	0.866779	+	0.498693i	$0.166186\pi$
$643$			25.2911i			0.997385i	−0.866779	+	0.498693i	$0.833814\pi$
$644$		0			0
$645$	4.76248	+	41.2727i	0.187523	+	1.62511i
$646$		0			0
$647$			22.4262i			0.881665i	0.897589	+	0.440832i	$0.145317\pi$
$647$			22.4262i			0.881665i	−0.897589	+	0.440832i	$0.854683\pi$
$648$		0			0
$649$	0.677809			0.0266063
$650$		0			0
$651$	4.09197			0.160377
$652$		0			0
$653$		−	11.8391i		−	0.463301i	−0.972799	−	0.231650i	$-0.925587\pi$
$653$		−	11.8391i		−	0.463301i	0.972799	−	0.231650i	$-0.0744125\pi$
$654$		0			0
$655$	−3.90710	−	33.8598i	−0.152663	−	1.32301i
$656$		0			0
$657$			51.1098i			1.99398i
$658$		0			0
$659$	15.7522			0.613617			0.306809	−	0.951771i	$-0.400739\pi$
$659$	15.7522			0.613617			0.306809	+	0.951771i	$0.400739\pi$
$660$		0			0
$661$	−7.15702			−0.278376			−0.139188	−	0.990266i	$-0.544449\pi$
$661$	−7.15702			−0.278376			−0.139188	+	0.990266i	$0.544449\pi$
$662$		0			0
$663$			1.44976i			0.0563038i
$664$		0			0
$665$	15.9010	−	1.83483i	0.616616	−	0.0711517i
$666$		0			0
$667$		−	46.6006i		−	1.80438i
$668$		0			0
$669$	60.0033			2.31986
$670$		0			0
$671$	0.197271			0.00761555
$672$		0			0
$673$			11.4976i			0.443200i	0.975138	+	0.221600i	$0.0711279\pi$
$673$			11.4976i			0.443200i	−0.975138	+	0.221600i	$0.928872\pi$
$674$		0			0
$675$	−20.2413	−	86.5398i	−0.779087	−	3.33092i
$676$		0			0
$677$		−	21.9953i		−	0.845347i	−0.906282	−	0.422673i	$-0.861092\pi$
$677$		−	21.9953i		−	0.845347i	0.906282	−	0.422673i	$-0.138908\pi$
$678$		0			0
$679$	7.15353			0.274527
$680$		0			0
$681$	4.41501			0.169183
$682$		0			0
$683$		−	0.468300i		−	0.0179190i	−0.999960	−	0.00895950i	$-0.997148\pi$
$683$		−	0.468300i		−	0.0179190i	0.999960	−	0.00895950i	$-0.00285194\pi$
$684$		0			0
$685$	32.4488	−	3.74429i	1.23980	−	0.143062i
$686$		0			0
$687$			16.2728i			0.620844i
$688$		0			0
$689$	19.1098			0.728025
$690$		0			0
$691$	−36.9140			−1.40428			−0.702138	−	0.712041i	$-0.747771\pi$
$691$	−36.9140			−1.40428			−0.702138	+	0.712041i	$0.747771\pi$
$692$		0			0
$693$		−	8.97472i		−	0.340921i
$694$		0			0
$695$	−5.23279	−	45.3484i	−0.198491	−	1.72016i
$696$		0			0
$697$		−	0.559875i		−	0.0212068i
$698$		0			0
$699$	68.4880			2.59046
$700$		0			0
$701$	18.6628			0.704886			0.352443	−	0.935833i	$-0.385351\pi$
$701$	18.6628			0.704886			0.352443	+	0.935833i	$0.385351\pi$
$702$		0			0
$703$		−	46.6492i		−	1.75941i
$704$		0			0
$705$	6.62724	+	57.4331i	0.249596	+	2.16306i
$706$		0			0
$707$		−	16.7154i		−	0.628648i
$708$		0			0
$709$	−6.66135			−0.250172			−0.125086	−	0.992146i	$-0.539921\pi$
$709$	−6.66135			−0.250172			−0.125086	+	0.992146i	$0.539921\pi$
$710$		0			0
$711$	−77.5589			−2.90869
$712$		0			0
$713$			6.71842i			0.251607i
$714$		0			0
$715$	−8.88531	+	1.02528i	−0.332292	+	0.0383434i
$716$		0			0
$717$		−	31.4356i		−	1.17398i
$718$		0			0
$719$	3.11128			0.116031			0.0580156	−	0.998316i	$-0.481523\pi$
$719$	3.11128			0.116031			0.0580156	+	0.998316i	$0.481523\pi$
$720$		0			0
$721$	−6.21991			−0.231641
$722$		0			0
$723$		−	74.9310i		−	2.78671i
$724$		0			0
$725$	37.9884	−	8.88531i	1.41085	−	0.329992i
$726$		0			0
$727$			33.9540i			1.25928i	0.776886	+	0.629642i	$0.216798\pi$
$727$			33.9540i			1.25928i	−0.776886	+	0.629642i	$0.783202\pi$
$728$		0			0
$729$	−109.777			−4.06581
$730$		0			0
$731$	−0.596468			−0.0220612
$732$		0			0
$733$		−	21.0457i		−	0.777342i	−0.921377	−	0.388671i	$-0.872934\pi$
$733$		−	21.0457i		−	0.777342i	0.921377	−	0.388671i	$-0.127066\pi$
$734$		0			0
$735$	−43.4993	+	5.01941i	−1.60449	+	0.185144i
$736$		0			0
$737$		−	1.41737i		−	0.0522097i
$738$		0			0
$739$	17.0253			0.626285			0.313143	−	0.949706i	$-0.398618\pi$
$739$	17.0253			0.626285			0.313143	+	0.949706i	$0.398618\pi$
$740$		0			0
$741$	−88.8710			−3.26476
$742$		0			0
$743$		−	23.7563i		−	0.871536i	−0.900059	−	0.435768i	$-0.856477\pi$
$743$		−	23.7563i		−	0.871536i	0.900059	−	0.435768i	$-0.143523\pi$
$744$		0			0
$745$	−4.07227	−	35.2911i	−0.149196	−	1.29297i
$746$		0			0
$747$			113.951i			4.16924i
$748$		0			0
$749$	−4.66067			−0.170297
$750$		0			0
$751$	−44.8654			−1.63716			−0.818582	−	0.574390i	$-0.805239\pi$
$751$	−44.8654			−1.63716			−0.818582	+	0.574390i	$0.805239\pi$
$752$		0			0
$753$			36.7177i			1.33807i
$754$		0			0
$755$	−1.68663	−	14.6167i	−0.0613829	−	0.531957i
$756$		0			0
$757$			41.3761i			1.50384i	0.659255	+	0.751920i	$0.270872\pi$
$757$			41.3761i			1.50384i	−0.659255	+	0.751920i	$0.729128\pi$
$758$		0			0
$759$	20.0675			0.728405
$760$		0			0
$761$	16.3002			0.590883			0.295442	−	0.955361i	$-0.404533\pi$
$761$	16.3002			0.590883			0.295442	+	0.955361i	$0.404533\pi$
$762$		0			0
$763$			2.52279i			0.0913310i
$764$		0			0
$765$	1.98636	−	0.229207i	0.0718170	−	0.00828701i
$766$		0			0
$767$		−	2.71124i		−	0.0978971i
$768$		0			0
$769$	−41.1262			−1.48305			−0.741525	−	0.670925i	$-0.765897\pi$
$769$	−41.1262			−1.48305			−0.741525	+	0.670925i	$0.765897\pi$
$770$		0			0
$771$	72.0409			2.59449
$772$		0			0
$773$		−	9.77280i		−	0.351503i	−0.984435	−	0.175752i	$-0.943764\pi$
$773$		−	9.77280i		−	0.351503i	0.984435	−	0.175752i	$-0.0562355\pi$
$774$		0			0
$775$	−5.47679	+	1.28100i	−0.196732	+	0.0460147i
$776$		0			0
$777$		−	25.6628i		−	0.920646i
$778$		0			0
$779$	34.3207			1.22967
$780$		0			0
$781$	−6.15020			−0.220072
$782$		0			0
$783$		−	138.694i		−	4.95652i
$784$		0			0
$785$	14.1840	−	1.63670i	0.506248	−	0.0584162i
$786$		0			0
$787$			5.25466i			0.187308i	0.995605	+	0.0936541i	$0.0298548\pi$
$787$			5.25466i			0.187308i	−0.995605	+	0.0936541i	$0.970145\pi$
$788$		0			0
$789$	21.9173			0.780278
$790$		0			0
$791$	11.8511			0.421377
$792$		0			0
$793$		−	0.789082i		−	0.0280211i
$794$		0			0
$795$	−4.11460	−	35.6580i	−0.145930	−	1.26466i
$796$		0			0
$797$		−	14.2133i		−	0.503460i	−0.967797	−	0.251730i	$-0.919000\pi$
$797$		−	14.2133i		−	0.503460i	0.967797	−	0.251730i	$-0.0809995\pi$
$798$		0			0
$799$	−0.830017			−0.0293639
$800$		0			0
$801$	−10.6949			−0.377887
$802$		0			0
$803$		−	6.16517i		−	0.217564i
$804$		0			0
$805$	1.65726	+	14.3622i	0.0584107	+	0.506200i
$806$		0			0
$807$			18.8347i			0.663015i
$808$		0			0
$809$	12.8641			0.452279			0.226139	−	0.974095i	$-0.427390\pi$
$809$	12.8641			0.452279			0.226139	+	0.974095i	$0.427390\pi$
$810$		0			0
$811$	4.31605			0.151557			0.0757785	−	0.997125i	$-0.475856\pi$
$811$	4.31605			0.151557			0.0757785	+	0.997125i	$0.475856\pi$
$812$		0			0
$813$			97.2560i			3.41092i
$814$		0			0
$815$	40.6662	−	4.69250i	1.42447	−	0.164371i
$816$		0			0
$817$		−	36.5639i		−	1.27921i
$818$		0			0
$819$	−35.8989			−1.25441
$820$		0			0
$821$	−42.9994			−1.50069			−0.750344	−	0.661048i	$-0.770112\pi$
$821$	−42.9994			−1.50069			−0.750344	+	0.661048i	$0.770112\pi$
$822$		0			0
$823$			48.1834i			1.67957i	0.542922	+	0.839783i	$0.317318\pi$
$823$			48.1834i			1.67957i	−0.542922	+	0.839783i	$0.682682\pi$
$824$		0			0
$825$	3.82626	+	16.3588i	0.133213	+	0.569542i
$826$		0			0
$827$			32.2868i			1.12272i	0.827571	+	0.561360i	$0.189722\pi$
$827$			32.2868i			1.12272i	−0.827571	+	0.561360i	$0.810278\pi$
$828$		0			0
$829$	−54.9345			−1.90795			−0.953977	−	0.299881i	$-0.903053\pi$
$829$	−54.9345			−1.90795			−0.953977	+	0.299881i	$0.903053\pi$
$830$		0			0
$831$	85.2954			2.95887
$832$		0			0
$833$		−	0.628646i		−	0.0217813i
$834$		0			0
$835$	0.438203	−	0.0505645i	0.0151646	−	0.00174986i
$836$		0			0
$837$			19.9955i			0.691147i
$838$		0			0
$839$	−11.2281			−0.387635			−0.193818	−	0.981038i	$-0.562087\pi$
$839$	−11.2281			−0.387635			−0.193818	+	0.981038i	$0.562087\pi$
$840$		0			0
$841$	31.8826			1.09940
$842$		0			0
$843$			91.2608i			3.14319i
$844$		0			0
$845$	0.768962	+	6.66399i	0.0264531	+	0.229248i
$846$		0			0
$847$			1.08258i			0.0371980i
$848$		0			0
$849$	73.9904			2.53934
$850$		0			0
$851$	42.1346			1.44435
$852$		0			0
$853$		−	2.40328i		−	0.0822869i	−0.999153	−	0.0411434i	$-0.986900\pi$
$853$		−	2.40328i		−	0.0822869i	0.999153	−	0.0411434i	$-0.0131001\pi$
$854$		0			0
$855$	14.0506	+	121.765i	0.480519	+	4.16428i
$856$		0			0
$857$			34.8280i			1.18970i	0.803836	+	0.594851i	$0.202789\pi$
$857$			34.8280i			1.18970i	−0.803836	+	0.594851i	$0.797211\pi$
$858$		0			0
$859$	−27.0627			−0.923368			−0.461684	−	0.887044i	$-0.652755\pi$
$859$	−27.0627			−0.923368			−0.461684	+	0.887044i	$0.652755\pi$
$860$		0			0
$861$	18.8806			0.643448
$862$		0			0
$863$		−	40.0157i		−	1.36215i	−0.732213	−	0.681076i	$-0.761512\pi$
$863$		−	40.0157i		−	1.36215i	0.732213	−	0.681076i	$-0.238488\pi$
$864$		0			0
$865$	−32.6983	+	3.77308i	−1.11177	+	0.128288i
$866$		0			0
$867$		−	57.0821i		−	1.93861i
$868$		0			0
$869$	9.35562			0.317368
$870$		0			0
$871$	−5.66950			−0.192104
$872$		0			0
$873$			54.7795i			1.85400i
$874$		0			0
$875$	−11.3919	+	4.08940i	−0.385116	+	0.138247i
$876$		0			0
$877$		−	18.3167i		−	0.618511i	−0.950979	−	0.309255i	$-0.899920\pi$
$877$		−	18.3167i		−	0.618511i	0.950979	−	0.309255i	$-0.100080\pi$
$878$		0			0
$879$	−4.83948			−0.163232
$880$		0			0
$881$	11.8560			0.399438			0.199719	−	0.979853i	$-0.435997\pi$
$881$	11.8560			0.399438			0.199719	+	0.979853i	$0.435997\pi$
$882$		0			0
$883$		−	15.2478i		−	0.513128i	−0.966527	−	0.256564i	$-0.917410\pi$
$883$		−	15.2478i		−	0.513128i	0.966527	−	0.256564i	$-0.0825904\pi$
$884$		0			0
$885$	−5.05905	+	0.583767i	−0.170058	+	0.0196231i
$886$		0			0
$887$			3.29631i			0.110679i	0.998468	+	0.0553397i	$0.0176242\pi$
$887$			3.29631i			0.110679i	−0.998468	+	0.0553397i	$0.982376\pi$
$888$		0			0
$889$	−13.6191			−0.456770
$890$		0			0
$891$	34.8553			1.16770
$892$		0			0
$893$		−	50.8806i		−	1.70265i
$894$		0			0
$895$	3.03786	+	26.3267i	0.101544	+	0.880006i
$896$		0			0
$897$		−	80.2701i		−	2.68014i
$898$		0			0
$899$	−8.77745			−0.292744
$900$		0			0
$901$	0.515326			0.0171680
$902$		0			0
$903$		−	20.1146i		−	0.669372i
$904$		0			0
$905$	2.75865	+	23.9070i	0.0917005	+	0.794696i
$906$		0			0
$907$		−	16.3577i		−	0.543149i	−0.962417	−	0.271574i	$-0.912456\pi$
$907$		−	16.3577i		−	0.543149i	0.962417	−	0.271574i	$-0.0875443\pi$
$908$		0			0
$909$	128.001			4.24554
$910$		0			0
$911$	−8.34598			−0.276515			−0.138257	−	0.990396i	$-0.544150\pi$
$911$	−8.34598			−0.276515			−0.138257	+	0.990396i	$0.544150\pi$
$912$		0			0
$913$		−	13.7454i		−	0.454907i
$914$		0			0
$915$	−1.47239	+	0.169900i	−0.0486758	+	0.00561673i
$916$		0			0
$917$			16.5019i			0.544939i
$918$		0			0
$919$	12.9611			0.427546			0.213773	−	0.976883i	$-0.431425\pi$
$919$	12.9611			0.427546			0.213773	+	0.976883i	$0.431425\pi$
$920$		0			0
$921$	−9.44029			−0.311068
$922$		0			0
$923$			24.6008i			0.809746i
$924$		0			0
$925$	8.03377	+	34.3477i	0.264149	+	1.12934i
$926$		0			0
$927$		−	47.6301i		−	1.56438i
$928$		0			0
$929$	−37.2635			−1.22258			−0.611288	−	0.791408i	$-0.709348\pi$
$929$	−37.2635			−1.22258			−0.611288	+	0.791408i	$0.709348\pi$
$930$		0			0
$931$	38.5364			1.26298
$932$		0			0
$933$			38.8185i			1.27086i
$934$		0			0
$935$	−0.239607	+	0.0276484i	−0.00783597	+	0.000904198i
$936$		0			0
$937$		−	4.28395i		−	0.139950i	−0.997549	−	0.0699752i	$-0.977708\pi$
$937$		−	4.28395i		−	0.139950i	0.997549	−	0.0699752i	$-0.0222920\pi$
$938$		0			0
$939$	−87.8927			−2.86827
$940$		0			0
$941$	23.2089			0.756589			0.378294	−	0.925685i	$-0.376511\pi$
$941$	23.2089			0.756589			0.378294	+	0.925685i	$0.376511\pi$
$942$		0			0
$943$			30.9992i			1.00947i
$944$		0			0
$945$	4.93238	+	42.7451i	0.160450	+	1.39050i
$946$		0			0
$947$			20.8646i			0.678007i	0.940785	+	0.339004i	$0.110090\pi$
$947$			20.8646i			0.678007i	−0.940785	+	0.339004i	$0.889910\pi$
$948$		0			0
$949$	−24.6607			−0.800519
$950$		0			0
$951$	98.7207			3.20124
$952$		0			0
$953$		−	45.0061i		−	1.45789i	−0.684572	−	0.728945i	$-0.740011\pi$
$953$		−	45.0061i		−	1.45789i	0.684572	−	0.728945i	$-0.259989\pi$
$954$		0			0
$955$	−0.436537	−	3.78313i	−0.0141260	−	0.122419i
$956$		0			0
$957$			26.2177i			0.847499i
$958$		0			0
$959$	−15.8142			−0.510667
$960$		0			0
$961$	−29.7346			−0.959179
$962$		0			0
$963$		−	35.6900i		−	1.15009i
$964$		0			0
$965$	−24.0526	+	2.77544i	−0.774280	+	0.0893446i
$966$		0			0
$967$		−	21.9679i		−	0.706440i	−0.935540	−	0.353220i	$-0.885087\pi$
$967$		−	21.9679i		−	0.706440i	0.935540	−	0.353220i	$-0.114913\pi$
$968$		0			0
$969$	−2.39655			−0.0769882
$970$		0			0
$971$	−16.5837			−0.532195			−0.266098	−	0.963946i	$-0.585734\pi$
$971$	−16.5837			−0.532195			−0.266098	+	0.963946i	$0.585734\pi$
$972$		0			0
$973$			22.1010i			0.708524i
$974$		0			0
$975$	65.4354	−	15.3051i	2.09561	−	0.490154i
$976$		0			0
$977$			19.8189i			0.634063i	0.948415	+	0.317032i	$0.102686\pi$
$977$			19.8189i			0.634063i	−0.948415	+	0.317032i	$0.897314\pi$
$978$		0			0
$979$	1.29009			0.0412314
$980$		0			0
$981$	−19.3187			−0.616798
$982$		0			0
$983$			13.9634i			0.445365i	0.974891	+	0.222682i	$0.0714813\pi$
$983$			13.9634i			0.445365i	−0.974891	+	0.222682i	$0.928519\pi$
$984$		0			0
$985$	−14.9276	+	1.72251i	−0.475634	+	0.0548838i
$986$		0			0
$987$		−	27.9905i		−	0.890949i
$988$		0			0
$989$	33.0253			1.05014
$990$		0			0
$991$	4.10113			0.130277			0.0651383	−	0.997876i	$-0.479251\pi$
$991$	4.10113			0.130277			0.0651383	+	0.997876i	$0.479251\pi$
$992$		0			0
$993$		−	41.2727i		−	1.30975i
$994$		0			0
$995$	−2.77544	−	24.0526i	−0.0879874	−	0.762518i
$996$		0			0
$997$			33.3208i			1.05528i	0.849468	+	0.527640i	$0.176923\pi$
$997$			33.3208i			1.05528i	−0.849468	+	0.527640i	$0.823077\pi$
$998$		0			0
$999$	125.402			3.96755

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.b.d.89.1	✓	8
3.2	odd	2		3960.2.d.f.3169.6		8
4.3	odd	2		880.2.b.j.529.8		8
5.2	odd	4		2200.2.a.x.1.1		4
5.3	odd	4		2200.2.a.y.1.4		4
5.4	even	2	inner	440.2.b.d.89.8	yes	8
15.14	odd	2		3960.2.d.f.3169.5		8
20.3	even	4		4400.2.a.cb.1.1		4
20.7	even	4		4400.2.a.ce.1.4		4
20.19	odd	2		880.2.b.j.529.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.d.89.1	✓	8	1.1	even	1	trivial
440.2.b.d.89.8	yes	8	5.4	even	2	inner
880.2.b.j.529.1		8	20.19	odd	2
880.2.b.j.529.8		8	4.3	odd	2
2200.2.a.x.1.1		4	5.2	odd	4
2200.2.a.y.1.4		4	5.3	odd	4
3960.2.d.f.3169.5		8	15.14	odd	2
3960.2.d.f.3169.6		8	3.2	odd	2
4400.2.a.cb.1.1		4	20.3	even	4
4400.2.a.ce.1.4		4	20.7	even	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:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.36007i q^{3} +(-0.256321 - 2.22133i) q^{5} +1.08258i q^{7} -8.29009 q^{9} +1.00000 q^{11} -4.00000i q^{13} +(-7.46382 + 0.861256i) q^{15} -0.107866i q^{17} +6.61228 q^{19} +3.63756 q^{21} +5.97235i q^{23} +(-4.86860 + 1.13874i) q^{25} +17.7751i q^{27} -7.80273 q^{29} +1.12492 q^{31} -3.36007i q^{33} +(2.40477 - 0.277489i) q^{35} -7.05494i q^{37} -13.4403 q^{39} +5.19045 q^{41} -5.52969i q^{43} +(2.12492 + 18.4150i) q^{45} -7.69486i q^{47} +5.82801 q^{49} -0.362439 q^{51} +4.77745i q^{53} +(-0.256321 - 2.22133i) q^{55} -22.2177i q^{57} +0.677809 q^{59} +0.197271 q^{61} -8.97472i q^{63} +(-8.88531 + 1.02528i) q^{65} -1.41737i q^{67} +20.0675 q^{69} -6.15020 q^{71} -6.16517i q^{73} +(3.82626 + 16.3588i) q^{75} +1.08258i q^{77} +9.35562 q^{79} +34.8553 q^{81} -13.7454i q^{83} +(-0.239607 + 0.0276484i) q^{85} +26.2177i q^{87} +1.29009 q^{89} +4.33034 q^{91} -3.77981i q^{93} +(-1.69486 - 14.6880i) q^{95} -6.60782i q^{97} -8.29009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{9} + 8 q^{11} - 12 q^{15} + 18 q^{19} - 14 q^{21} - 2 q^{25} - 6 q^{29} - 30 q^{31} + 30 q^{35} - 8 q^{39} + 20 q^{41} - 22 q^{45} - 18 q^{49} - 46 q^{51} - 12 q^{59} + 58 q^{61} - 8 q^{65}+ \cdots - 14 q^{99}+O(q^{100}) \) 8 * q - 14 * q^9 + 8 * q^11 - 12 * q^15 + 18 * q^19 - 14 * q^21 - 2 * q^25 - 6 * q^29 - 30 * q^31 + 30 * q^35 - 8 * q^39 + 20 * q^41 - 22 * q^45 - 18 * q^49 - 46 * q^51 - 12 * q^59 + 58 * q^61 - 8 * q^65 + 60 * q^69 - 2 * q^71 + 26 * q^75 + 40 * q^79 + 88 * q^81 - 26 * q^85 - 42 * q^89 + 8 * q^91 + 28 * q^95 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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