LMFDB - Embedded newform 3360.2.w.a.559.31

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(559,3360)] 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("3360.559"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.w (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.31
Character	$\chi$	$=$	3360.559
Dual form			3360.2.w.a.559.32

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{3} +(0.840958 - 2.07190i) q^{5} +(-2.31981 - 1.27219i) q^{7} +1.00000 q^{9} +1.15873 q^{11} +0.698885i q^{13} +(-0.840958 + 2.07190i) q^{15} -6.87158 q^{17} +4.94979i q^{19} +(2.31981 + 1.27219i) q^{21} +5.30311 q^{23} +(-3.58558 - 3.48477i) q^{25} -1.00000 q^{27} +10.0699i q^{29} +0.109215 q^{31} -1.15873 q^{33} +(-4.58672 + 3.73658i) q^{35} +4.30570 q^{37} -0.698885i q^{39} -2.58861i q^{41} +7.55358i q^{43} +(0.840958 - 2.07190i) q^{45} +3.53225i q^{47} +(3.76308 + 5.90248i) q^{49} +6.87158 q^{51} -1.70909 q^{53} +(0.974447 - 2.40079i) q^{55} -4.94979i q^{57} -12.3942i q^{59} +4.88851 q^{61} +(-2.31981 - 1.27219i) q^{63} +(1.44802 + 0.587733i) q^{65} +8.12528i q^{67} -5.30311 q^{69} -5.80082i q^{71} -3.86018 q^{73} +(3.58558 + 3.48477i) q^{75} +(-2.68805 - 1.47413i) q^{77} +6.30589i q^{79} +1.00000 q^{81} +12.7204 q^{83} +(-5.77872 + 14.2373i) q^{85} -10.0699i q^{87} -0.841441i q^{89} +(0.889113 - 1.62128i) q^{91} -0.109215 q^{93} +(10.2555 + 4.16257i) q^{95} -0.0488298 q^{97} +1.15873 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{3} + 48 q^{9} - 48 q^{27} - 8 q^{35} + 48 q^{81} + 16 q^{91} - 48 q^{97}+O(q^{100}) $ 48 * q - 48 * q^3 + 48 * q^9 - 48 * q^27 - 8 * q^35 + 48 * q^81 + 16 * q^91 - 48 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$-1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	0.840958	−	2.07190i	0.376088	−	0.926584i
$5$	0.840958	−	2.07190i	0.376088	−	0.926584i
$6$		0			0
$7$	−2.31981	−	1.27219i	−0.876807	−	0.480842i
$7$	−2.31981	−	1.27219i	−0.876807	−	0.480842i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	1.15873			0.349372			0.174686	−	0.984624i	$-0.444109\pi$
$11$	1.15873			0.349372			0.174686	+	0.984624i	$0.444109\pi$
$12$		0			0
$13$			0.698885i			0.193836i	0.995292	+	0.0969180i	$0.0308984\pi$
$13$			0.698885i			0.193836i	−0.995292	+	0.0969180i	$0.969102\pi$
$14$		0			0
$15$	−0.840958	+	2.07190i	−0.217135	+	0.534963i
$16$		0			0
$17$	−6.87158			−1.66660			−0.833302	−	0.552818i	$-0.813552\pi$
$17$	−6.87158			−1.66660			−0.833302	+	0.552818i	$0.813552\pi$
$18$		0			0
$19$			4.94979i			1.13556i	0.823180	+	0.567780i	$0.192198\pi$
$19$			4.94979i			1.13556i	−0.823180	+	0.567780i	$0.807802\pi$
$20$		0			0
$21$	2.31981	+	1.27219i	0.506225	+	0.277614i
$22$		0			0
$23$	5.30311			1.10577			0.552887	−	0.833256i	$-0.313526\pi$
$23$	5.30311			1.10577			0.552887	+	0.833256i	$0.313526\pi$
$24$		0			0
$25$	−3.58558	−	3.48477i	−0.717116	−	0.696954i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$			10.0699i			1.86993i	0.354742	+	0.934964i	$0.384569\pi$
$29$			10.0699i			1.86993i	−0.354742	+	0.934964i	$0.615431\pi$
$30$		0			0
$31$	0.109215			0.0196157			0.00980783	−	0.999952i	$-0.496878\pi$
$31$	0.109215			0.0196157			0.00980783	+	0.999952i	$0.496878\pi$
$32$		0			0
$33$	−1.15873			−0.201710
$34$		0			0
$35$	−4.58672	+	3.73658i	−0.775297	+	0.631597i
$36$		0			0
$37$	4.30570			0.707853			0.353926	−	0.935273i	$-0.384846\pi$
$37$	4.30570			0.707853			0.353926	+	0.935273i	$0.384846\pi$
$38$		0			0
$39$		−	0.698885i		−	0.111911i
$40$		0			0
$41$		−	2.58861i		−	0.404273i	−0.979357	−	0.202137i	$-0.935212\pi$
$41$		−	2.58861i		−	0.404273i	0.979357	−	0.202137i	$-0.0647885\pi$
$42$		0			0
$43$			7.55358i			1.15191i	0.817481	+	0.575955i	$0.195370\pi$
$43$			7.55358i			1.15191i	−0.817481	+	0.575955i	$0.804630\pi$
$44$		0			0
$45$	0.840958	−	2.07190i	0.125363	−	0.308861i
$46$		0			0
$47$			3.53225i			0.515231i	0.966247	+	0.257616i	$0.0829368\pi$
$47$			3.53225i			0.515231i	−0.966247	+	0.257616i	$0.917063\pi$
$48$		0			0
$49$	3.76308	+	5.90248i	0.537583	+	0.843211i
$50$		0			0
$51$	6.87158			0.962214
$52$		0			0
$53$	−1.70909			−0.234762			−0.117381	−	0.993087i	$-0.537450\pi$
$53$	−1.70909			−0.234762			−0.117381	+	0.993087i	$0.537450\pi$
$54$		0			0
$55$	0.974447	−	2.40079i	0.131394	−	0.323722i
$56$		0			0
$57$		−	4.94979i		−	0.655616i
$58$		0			0
$59$		−	12.3942i		−	1.61359i	−0.590834	−	0.806793i	$-0.701201\pi$
$59$		−	12.3942i		−	1.61359i	0.590834	−	0.806793i	$-0.298799\pi$
$60$		0			0
$61$	4.88851			0.625909			0.312955	−	0.949768i	$-0.398681\pi$
$61$	4.88851			0.625909			0.312955	+	0.949768i	$0.398681\pi$
$62$		0			0
$63$	−2.31981	−	1.27219i	−0.292269	−	0.160281i
$64$		0			0
$65$	1.44802	+	0.587733i	0.179605	+	0.0728994i
$66$		0			0
$67$			8.12528i			0.992661i	0.868134	+	0.496330i	$0.165320\pi$
$67$			8.12528i			0.992661i	−0.868134	+	0.496330i	$0.834680\pi$
$68$		0			0
$69$	−5.30311			−0.638419
$70$		0			0
$71$		−	5.80082i		−	0.688431i	−0.938891	−	0.344215i	$-0.888145\pi$
$71$		−	5.80082i		−	0.688431i	0.938891	−	0.344215i	$-0.111855\pi$
$72$		0			0
$73$	−3.86018			−0.451799			−0.225900	−	0.974151i	$-0.572532\pi$
$73$	−3.86018			−0.451799			−0.225900	+	0.974151i	$0.572532\pi$
$74$		0			0
$75$	3.58558	+	3.48477i	0.414027	+	0.402387i
$76$		0			0
$77$	−2.68805	−	1.47413i	−0.306332	−	0.167992i
$78$		0			0
$79$			6.30589i			0.709468i	0.934967	+	0.354734i	$0.115428\pi$
$79$			6.30589i			0.709468i	−0.934967	+	0.354734i	$0.884572\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	12.7204			1.39624			0.698121	−	0.715980i	$-0.254020\pi$
$83$	12.7204			1.39624			0.698121	+	0.715980i	$0.254020\pi$
$84$		0			0
$85$	−5.77872	+	14.2373i	−0.626790	+	1.54425i
$86$		0			0
$87$		−	10.0699i		−	1.07960i
$88$		0			0
$89$		−	0.841441i		−	0.0891926i	−0.999005	−	0.0445963i	$-0.985800\pi$
$89$		−	0.841441i		−	0.0891926i	0.999005	−	0.0445963i	$-0.0142002\pi$
$90$		0			0
$91$	0.889113	−	1.62128i	0.0932044	−	0.169957i
$92$		0			0
$93$	−0.109215			−0.0113251
$94$		0			0
$95$	10.2555	+	4.16257i	1.05219	+	0.427071i
$96$		0			0
$97$	−0.0488298			−0.00495792			−0.00247896	−	0.999997i	$-0.500789\pi$
$97$	−0.0488298			−0.00495792			−0.00247896	+	0.999997i	$0.500789\pi$
$98$		0			0
$99$	1.15873			0.116457
$100$		0			0
$101$	15.0202			1.49456			0.747281	−	0.664508i	$-0.231359\pi$
$101$	15.0202			1.49456			0.747281	+	0.664508i	$0.231359\pi$
$102$		0			0
$103$		−	16.0240i		−	1.57890i	−0.613817	−	0.789448i	$-0.710367\pi$
$103$		−	16.0240i		−	1.57890i	0.613817	−	0.789448i	$-0.289633\pi$
$104$		0			0
$105$	4.58672	−	3.73658i	0.447618	−	0.364653i
$106$		0			0
$107$			9.62303i			0.930293i	0.885234	+	0.465146i	$0.153998\pi$
$107$			9.62303i			0.930293i	−0.885234	+	0.465146i	$0.846002\pi$
$108$		0			0
$109$			4.74400i			0.454393i	0.973849	+	0.227196i	$0.0729559\pi$
$109$			4.74400i			0.454393i	−0.973849	+	0.227196i	$0.927044\pi$
$110$		0			0
$111$	−4.30570			−0.408679
$112$		0			0
$113$		−	11.3962i		−	1.07206i	−0.844198	−	0.536031i	$-0.819923\pi$
$113$		−	11.3962i		−	1.07206i	0.844198	−	0.536031i	$-0.180077\pi$
$114$		0			0
$115$	4.45969	−	10.9875i	0.415869	−	1.02459i
$116$		0			0
$117$			0.698885i			0.0646120i
$118$		0			0
$119$	15.9408	+	8.74194i	1.46129	+	0.801373i
$120$		0			0
$121$	−9.65733			−0.877940
$122$		0			0
$123$			2.58861i			0.233407i
$124$		0			0
$125$	−10.2354	+	4.49843i	−0.915485	+	0.402352i
$126$		0			0
$127$	11.4203			1.01339			0.506693	−	0.862126i	$-0.330868\pi$
$127$	11.4203			1.01339			0.506693	+	0.862126i	$0.330868\pi$
$128$		0			0
$129$		−	7.55358i		−	0.665055i
$130$		0			0
$131$		−	1.20053i		−	0.104891i	−0.998624	−	0.0524453i	$-0.983298\pi$
$131$		−	1.20053i		−	0.104891i	0.998624	−	0.0524453i	$-0.0167015\pi$
$132$		0			0
$133$	6.29706	−	11.4826i	0.546025	−	0.995668i
$134$		0			0
$135$	−0.840958	+	2.07190i	−0.0723782	+	0.178321i
$136$		0			0
$137$		−	4.29412i		−	0.366872i	−0.983032	−	0.183436i	$-0.941278\pi$
$137$		−	4.29412i		−	0.366872i	0.983032	−	0.183436i	$-0.0587219\pi$
$138$		0			0
$139$		−	12.0100i		−	1.01867i	−0.860568	−	0.509336i	$-0.829891\pi$
$139$		−	12.0100i		−	1.01867i	0.860568	−	0.509336i	$-0.170109\pi$
$140$		0			0
$141$		−	3.53225i		−	0.297469i
$142$		0			0
$143$			0.809822i			0.0677207i
$144$		0			0
$145$	20.8638	+	8.46834i	1.73265	+	0.703258i
$146$		0			0
$147$	−3.76308	−	5.90248i	−0.310373	−	0.486828i
$148$		0			0
$149$			2.09953i			0.172000i	0.996295	+	0.0860001i	$0.0274085\pi$
$149$			2.09953i			0.172000i	−0.996295	+	0.0860001i	$0.972591\pi$
$150$		0			0
$151$			21.0358i			1.71187i	0.517083	+	0.855935i	$0.327018\pi$
$151$			21.0358i			1.71187i	−0.517083	+	0.855935i	$0.672982\pi$
$152$		0			0
$153$	−6.87158			−0.555535
$154$		0			0
$155$	0.0918456	−	0.226284i	0.00737721	−	0.0181756i
$156$		0			0
$157$		−	5.76627i		−	0.460198i	−0.973167	−	0.230099i	$-0.926095\pi$
$157$		−	5.76627i		−	0.460198i	0.973167	−	0.230099i	$-0.0739051\pi$
$158$		0			0
$159$	1.70909			0.135540
$160$		0			0
$161$	−12.3022	−	6.74655i	−0.969552	−	0.531703i
$162$		0			0
$163$		−	11.4647i		−	0.897984i	−0.893536	−	0.448992i	$-0.851783\pi$
$163$		−	11.4647i		−	0.897984i	0.893536	−	0.448992i	$-0.148217\pi$
$164$		0			0
$165$	−0.974447	+	2.40079i	−0.0758606	+	0.186901i
$166$		0			0
$167$			19.8534i			1.53630i	0.640271	+	0.768149i	$0.278822\pi$
$167$			19.8534i			1.53630i	−0.640271	+	0.768149i	$0.721178\pi$
$168$		0			0
$169$	12.5116			0.962428
$170$		0			0
$171$			4.94979i			0.378520i
$172$		0			0
$173$			16.3132i			1.24027i	0.784495	+	0.620135i	$0.212922\pi$
$173$			16.3132i			1.24027i	−0.784495	+	0.620135i	$0.787078\pi$
$174$		0			0
$175$	3.88459	+	12.6456i	0.293648	+	0.955914i
$176$		0			0
$177$			12.3942i			0.931605i
$178$		0			0
$179$	20.4707			1.53005			0.765025	−	0.644001i	$-0.222727\pi$
$179$	20.4707			1.53005			0.765025	+	0.644001i	$0.222727\pi$
$180$		0			0
$181$	14.5030			1.07800			0.538999	−	0.842306i	$-0.318803\pi$
$181$	14.5030			1.07800			0.538999	+	0.842306i	$0.318803\pi$
$182$		0			0
$183$	−4.88851			−0.361369
$184$		0			0
$185$	3.62091	−	8.92100i	0.266215	−	0.655885i
$186$		0			0
$187$	−7.96234			−0.582264
$188$		0			0
$189$	2.31981	+	1.27219i	0.168742	+	0.0925380i
$190$		0			0
$191$			19.8316i			1.43496i	0.696578	+	0.717481i	$0.254705\pi$
$191$			19.8316i			1.43496i	−0.696578	+	0.717481i	$0.745295\pi$
$192$		0			0
$193$		−	0.938946i		−	0.0675868i	−0.999429	−	0.0337934i	$-0.989241\pi$
$193$		−	0.938946i		−	0.0675868i	0.999429	−	0.0337934i	$-0.0107588\pi$
$194$		0			0
$195$	−1.44802	−	0.587733i	−0.103695	−	0.0420885i
$196$		0			0
$197$	22.2269			1.58360			0.791800	−	0.610780i	$-0.209144\pi$
$197$	22.2269			1.58360			0.791800	+	0.610780i	$0.209144\pi$
$198$		0			0
$199$	7.57854			0.537228			0.268614	−	0.963248i	$-0.413434\pi$
$199$	7.57854			0.537228			0.268614	+	0.963248i	$0.413434\pi$
$200$		0			0
$201$		−	8.12528i		−	0.573113i
$202$		0			0
$203$	12.8108	−	23.3602i	0.899139	−	1.63957i
$204$		0			0
$205$	−5.36335	−	2.17691i	−0.374593	−	0.152042i
$206$		0			0
$207$	5.30311			0.368592
$208$		0			0
$209$			5.73549i			0.396732i
$210$		0			0
$211$	−20.4549			−1.40817			−0.704086	−	0.710115i	$-0.748643\pi$
$211$	−20.4549			−1.40817			−0.704086	+	0.710115i	$0.748643\pi$
$212$		0			0
$213$			5.80082i			0.397466i
$214$		0			0
$215$	15.6503	+	6.35224i	1.06734	+	0.433219i
$216$		0			0
$217$	−0.253359	−	0.138942i	−0.0171992	−	0.00943203i
$218$		0			0
$219$	3.86018			0.260846
$220$		0			0
$221$		−	4.80245i		−	0.323048i
$222$		0			0
$223$			4.53719i			0.303833i	0.988393	+	0.151916i	$0.0485445\pi$
$223$			4.53719i			0.303833i	−0.988393	+	0.151916i	$0.951456\pi$
$224$		0			0
$225$	−3.58558	−	3.48477i	−0.239039	−	0.232318i
$226$		0			0
$227$	−12.4331			−0.825215			−0.412607	−	0.910909i	$-0.635382\pi$
$227$	−12.4331			−0.825215			−0.412607	+	0.910909i	$0.635382\pi$
$228$		0			0
$229$	3.24679			0.214554			0.107277	−	0.994229i	$-0.465787\pi$
$229$	3.24679			0.214554			0.107277	+	0.994229i	$0.465787\pi$
$230$		0			0
$231$	2.68805	+	1.47413i	0.176861	+	0.0969905i
$232$		0			0
$233$			26.6500i			1.74590i	0.487812	+	0.872949i	$0.337795\pi$
$233$			26.6500i			1.74590i	−0.487812	+	0.872949i	$0.662205\pi$
$234$		0			0
$235$	7.31848	+	2.97047i	0.477405	+	0.193772i
$236$		0			0
$237$		−	6.30589i		−	0.409611i
$238$		0			0
$239$		−	10.8623i		−	0.702622i	−0.936259	−	0.351311i	$-0.885736\pi$
$239$		−	10.8623i		−	0.702622i	0.936259	−	0.351311i	$-0.114264\pi$
$240$		0			0
$241$			16.1547i			1.04061i	0.853979	+	0.520307i	$0.174182\pi$
$241$			16.1547i			1.04061i	−0.853979	+	0.520307i	$0.825818\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	15.3940	−	2.83300i	0.983484	−	0.180994i
$246$		0			0
$247$	−3.45934			−0.220112
$248$		0			0
$249$	−12.7204			−0.806121
$250$		0			0
$251$			22.5756i			1.42496i	0.701694	+	0.712478i	$0.252427\pi$
$251$			22.5756i			1.42496i	−0.701694	+	0.712478i	$0.747573\pi$
$252$		0			0
$253$	6.14490			0.386326
$254$		0			0
$255$	5.77872	−	14.2373i	0.361877	−	0.891572i
$256$		0			0
$257$	−12.0533			−0.751863			−0.375931	−	0.926648i	$-0.622677\pi$
$257$	−12.0533			−0.751863			−0.375931	+	0.926648i	$0.622677\pi$
$258$		0			0
$259$	−9.98842	−	5.47766i	−0.620650	−	0.340365i
$260$		0			0
$261$			10.0699i			0.623309i
$262$		0			0
$263$	19.6608			1.21234			0.606169	−	0.795336i	$-0.292705\pi$
$263$	19.6608			1.21234			0.606169	+	0.795336i	$0.292705\pi$
$264$		0			0
$265$	−1.43727	+	3.54107i	−0.0882910	+	0.217526i
$266$		0			0
$267$			0.841441i			0.0514954i
$268$		0			0
$269$	1.22906			0.0749372			0.0374686	−	0.999298i	$-0.488071\pi$
$269$	1.22906			0.0749372			0.0374686	+	0.999298i	$0.488071\pi$
$270$		0			0
$271$	−12.1597			−0.738648			−0.369324	−	0.929301i	$-0.620411\pi$
$271$	−12.1597			−0.738648			−0.369324	+	0.929301i	$0.620411\pi$
$272$		0			0
$273$	−0.889113	+	1.62128i	−0.0538116	+	0.0981246i
$274$		0			0
$275$	−4.15473	−	4.03792i	−0.250540	−	0.243496i
$276$		0			0
$277$	−24.1319			−1.44995			−0.724973	−	0.688778i	$-0.758148\pi$
$277$	−24.1319			−1.44995			−0.724973	+	0.688778i	$0.758148\pi$
$278$		0			0
$279$	0.109215			0.00653855
$280$		0			0
$281$	2.45095			0.146211			0.0731056	−	0.997324i	$-0.476709\pi$
$281$	2.45095			0.146211			0.0731056	+	0.997324i	$0.476709\pi$
$282$		0			0
$283$	0.770452			0.0457986			0.0228993	−	0.999738i	$-0.492710\pi$
$283$	0.770452			0.0457986			0.0228993	+	0.999738i	$0.492710\pi$
$284$		0			0
$285$	−10.2555	−	4.16257i	−0.607483	−	0.246569i
$286$		0			0
$287$	−3.29320	+	6.00510i	−0.194391	+	0.354470i
$288$		0			0
$289$	30.2187			1.77757
$290$		0			0
$291$	0.0488298			0.00286246
$292$		0			0
$293$			14.0583i			0.821295i	0.911794	+	0.410648i	$0.134697\pi$
$293$			14.0583i			0.821295i	−0.911794	+	0.410648i	$0.865303\pi$
$294$		0			0
$295$	−25.6796	−	10.4230i	−1.49512	−	0.606851i
$296$		0			0
$297$	−1.15873			−0.0672366
$298$		0			0
$299$			3.70627i			0.214339i
$300$		0			0
$301$	9.60957	−	17.5229i	0.553886	−	1.01000i
$302$		0			0
$303$	−15.0202			−0.862886
$304$		0			0
$305$	4.11103	−	10.1285i	0.235397	−	0.579957i
$306$		0			0
$307$	−33.8488			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$307$	−33.8488			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$308$		0			0
$309$			16.0240i			0.911576i
$310$		0			0
$311$	14.3608			0.814324			0.407162	−	0.913356i	$-0.366518\pi$
$311$	14.3608			0.814324			0.407162	+	0.913356i	$0.366518\pi$
$312$		0			0
$313$	−16.3655			−0.925033			−0.462516	−	0.886611i	$-0.653053\pi$
$313$	−16.3655			−0.925033			−0.462516	+	0.886611i	$0.653053\pi$
$314$		0			0
$315$	−4.58672	+	3.73658i	−0.258432	+	0.210532i
$316$		0			0
$317$	25.5784			1.43662			0.718312	−	0.695721i	$-0.244915\pi$
$317$	25.5784			1.43662			0.718312	+	0.695721i	$0.244915\pi$
$318$		0			0
$319$			11.6683i			0.653300i
$320$		0			0
$321$		−	9.62303i		−	0.537105i
$322$		0			0
$323$		−	34.0129i		−	1.89253i
$324$		0			0
$325$	2.43546	−	2.50591i	0.135095	−	0.139003i
$326$		0			0
$327$		−	4.74400i		−	0.262344i
$328$		0			0
$329$	4.49368	−	8.19416i	0.247745	−	0.451759i
$330$		0			0
$331$	4.64867			0.255514			0.127757	−	0.991806i	$-0.459222\pi$
$331$	4.64867			0.255514			0.127757	+	0.991806i	$0.459222\pi$
$332$		0			0
$333$	4.30570			0.235951
$334$		0			0
$335$	16.8348	+	6.83302i	0.919784	+	0.373328i
$336$		0			0
$337$			1.71956i			0.0936701i	0.998903	+	0.0468351i	$0.0149135\pi$
$337$			1.71956i			0.0936701i	−0.998903	+	0.0468351i	$0.985086\pi$
$338$		0			0
$339$			11.3962i			0.618955i
$340$		0			0
$341$	0.126552			0.00685315
$342$		0			0
$343$	−1.22058	−	18.4800i	−0.0659053	−	0.997826i
$344$		0			0
$345$	−4.45969	+	10.9875i	−0.240102	+	0.591549i
$346$		0			0
$347$			18.7988i			1.00917i	0.863361	+	0.504587i	$0.168355\pi$
$347$			18.7988i			1.00917i	−0.863361	+	0.504587i	$0.831645\pi$
$348$		0			0
$349$	−27.6131			−1.47809			−0.739047	−	0.673654i	$-0.764724\pi$
$349$	−27.6131			−1.47809			−0.739047	+	0.673654i	$0.764724\pi$
$350$		0			0
$351$		−	0.698885i		−	0.0373037i
$352$		0			0
$353$	20.2417			1.07736			0.538678	−	0.842512i	$-0.318924\pi$
$353$	20.2417			1.07736			0.538678	+	0.842512i	$0.318924\pi$
$354$		0			0
$355$	−12.0187	−	4.87825i	−0.637889	−	0.258911i
$356$		0			0
$357$	−15.9408	−	8.74194i	−0.843677	−	0.462673i
$358$		0			0
$359$		−	23.0400i		−	1.21601i	−0.793935	−	0.608003i	$-0.791971\pi$
$359$		−	23.0400i		−	1.21601i	0.793935	−	0.608003i	$-0.208029\pi$
$360$		0			0
$361$	−5.50045			−0.289497
$362$		0			0
$363$	9.65733			0.506879
$364$		0			0
$365$	−3.24625	+	7.99792i	−0.169916	+	0.418630i
$366$		0			0
$367$			10.9328i			0.570689i	0.958425	+	0.285345i	$0.0921080\pi$
$367$			10.9328i			0.570689i	−0.958425	+	0.285345i	$0.907892\pi$
$368$		0			0
$369$		−	2.58861i		−	0.134758i
$370$		0			0
$371$	3.96477	+	2.17428i	0.205841	+	0.112883i
$372$		0			0
$373$	23.4976			1.21666			0.608330	−	0.793684i	$-0.291840\pi$
$373$	23.4976			1.21666			0.608330	+	0.793684i	$0.291840\pi$
$374$		0			0
$375$	10.2354	−	4.49843i	0.528556	−	0.232298i
$376$		0			0
$377$	−7.03769			−0.362459
$378$		0			0
$379$	2.54308			0.130629			0.0653147	−	0.997865i	$-0.479195\pi$
$379$	2.54308			0.130629			0.0653147	+	0.997865i	$0.479195\pi$
$380$		0			0
$381$	−11.4203			−0.585079
$382$		0			0
$383$			18.1612i			0.927992i	0.885837	+	0.463996i	$0.153585\pi$
$383$			18.1612i			0.927992i	−0.885837	+	0.463996i	$0.846415\pi$
$384$		0			0
$385$	−5.31479	+	4.32970i	−0.270867	+	0.220662i
$386$		0			0
$387$			7.55358i			0.383970i
$388$		0			0
$389$		−	16.7007i		−	0.846760i	−0.905952	−	0.423380i	$-0.860844\pi$
$389$		−	16.7007i		−	0.846760i	0.905952	−	0.423380i	$-0.139156\pi$
$390$		0			0
$391$	−36.4408			−1.84289
$392$		0			0
$393$			1.20053i			0.0605586i
$394$		0			0
$395$	13.0652	+	5.30299i	0.657381	+	0.266822i
$396$		0			0
$397$			11.8222i			0.593339i	0.954980	+	0.296669i	$0.0958759\pi$
$397$			11.8222i			0.593339i	−0.954980	+	0.296669i	$0.904124\pi$
$398$		0			0
$399$	−6.29706	+	11.4826i	−0.315248	+	0.574849i
$400$		0			0
$401$	31.5542			1.57574			0.787871	−	0.615841i	$-0.211184\pi$
$401$	31.5542			1.57574			0.787871	+	0.615841i	$0.211184\pi$
$402$		0			0
$403$			0.0763290i			0.00380222i
$404$		0			0
$405$	0.840958	−	2.07190i	0.0417876	−	0.102954i
$406$		0			0
$407$	4.98916			0.247304
$408$		0			0
$409$		−	20.0561i		−	0.991709i	−0.868406	−	0.495855i	$-0.834855\pi$
$409$		−	20.0561i		−	0.991709i	0.868406	−	0.495855i	$-0.165145\pi$
$410$		0			0
$411$			4.29412i			0.211813i
$412$		0			0
$413$	−15.7677	+	28.7522i	−0.775880	+	1.41480i
$414$		0			0
$415$	10.6973	−	26.3554i	0.525110	−	1.29374i
$416$		0			0
$417$			12.0100i			0.588131i
$418$		0			0
$419$			6.04413i			0.295275i	0.989042	+	0.147638i	$0.0471669\pi$
$419$			6.04413i			0.295275i	−0.989042	+	0.147638i	$0.952833\pi$
$420$		0			0
$421$			20.6123i			1.00458i	0.864699	+	0.502290i	$0.167509\pi$
$421$			20.6123i			1.00458i	−0.864699	+	0.502290i	$0.832491\pi$
$422$		0			0
$423$			3.53225i			0.171744i
$424$		0			0
$425$	24.6386	+	23.9459i	1.19515	+	1.16155i
$426$		0			0
$427$	−11.3404	−	6.21910i	−0.548802	−	0.300963i
$428$		0			0
$429$		−	0.809822i		−	0.0390986i
$430$		0			0
$431$		−	4.31002i		−	0.207606i	−0.994598	−	0.103803i	$-0.966899\pi$
$431$		−	4.31002i		−	0.207606i	0.994598	−	0.103803i	$-0.0331012\pi$
$432$		0			0
$433$	24.8247			1.19300			0.596500	−	0.802613i	$-0.296558\pi$
$433$	24.8247			1.19300			0.596500	+	0.802613i	$0.296558\pi$
$434$		0			0
$435$	−20.8638	−	8.46834i	−1.00034	−	0.406026i
$436$		0			0
$437$			26.2493i			1.25567i
$438$		0			0
$439$	1.81801			0.0867687			0.0433843	−	0.999058i	$-0.486186\pi$
$439$	1.81801			0.0867687			0.0433843	+	0.999058i	$0.486186\pi$
$440$		0			0
$441$	3.76308	+	5.90248i	0.179194	+	0.281070i
$442$		0			0
$443$			37.3237i			1.77330i	0.462439	+	0.886651i	$0.346975\pi$
$443$			37.3237i			1.77330i	−0.462439	+	0.886651i	$0.653025\pi$
$444$		0			0
$445$	−1.74339	−	0.707617i	−0.0826444	−	0.0335443i
$446$		0			0
$447$		−	2.09953i		−	0.0993043i
$448$		0			0
$449$	−22.9898			−1.08496			−0.542478	−	0.840070i	$-0.682514\pi$
$449$	−22.9898			−1.08496			−0.542478	+	0.840070i	$0.682514\pi$
$450$		0			0
$451$		−	2.99951i		−	0.141242i
$452$		0			0
$453$		−	21.0358i		−	0.988349i
$454$		0			0
$455$	−2.61144	−	3.20559i	−0.122426	−	0.150280i
$456$		0			0
$457$			37.1559i			1.73808i	0.494741	+	0.869040i	$0.335263\pi$
$457$			37.1559i			1.73808i	−0.494741	+	0.869040i	$0.664737\pi$
$458$		0			0
$459$	6.87158			0.320738
$460$		0			0
$461$	33.7273			1.57083			0.785417	−	0.618966i	$-0.212448\pi$
$461$	33.7273			1.57083			0.785417	+	0.618966i	$0.212448\pi$
$462$		0			0
$463$	−17.3016			−0.804072			−0.402036	−	0.915624i	$-0.631697\pi$
$463$	−17.3016			−0.804072			−0.402036	+	0.915624i	$0.631697\pi$
$464$		0			0
$465$	−0.0918456	+	0.226284i	−0.00425924	+	0.0104937i
$466$		0			0
$467$	6.46441			0.299137			0.149569	−	0.988751i	$-0.452211\pi$
$467$	6.46441			0.299137			0.149569	+	0.988751i	$0.452211\pi$
$468$		0			0
$469$	10.3369	−	18.8491i	0.477313	−	0.870373i
$470$		0			0
$471$			5.76627i			0.265696i
$472$		0			0
$473$			8.75259i			0.402444i
$474$		0			0
$475$	17.2489	−	17.7479i	0.791434	−	0.814328i
$476$		0			0
$477$	−1.70909			−0.0782539
$478$		0			0
$479$	−29.1194			−1.33050			−0.665250	−	0.746621i	$-0.731675\pi$
$479$	−29.1194			−1.33050			−0.665250	+	0.746621i	$0.731675\pi$
$480$		0			0
$481$			3.00919i			0.137207i
$482$		0			0
$483$	12.3022	+	6.74655i	0.559771	+	0.306979i
$484$		0			0
$485$	−0.0410639	+	0.101171i	−0.00186461	+	0.00459393i
$486$		0			0
$487$	−9.61283			−0.435599			−0.217799	−	0.975994i	$-0.569888\pi$
$487$	−9.61283			−0.435599			−0.217799	+	0.975994i	$0.569888\pi$
$488$		0			0
$489$			11.4647i			0.518452i
$490$		0			0
$491$	3.85331			0.173897			0.0869487	−	0.996213i	$-0.472288\pi$
$491$	3.85331			0.173897			0.0869487	+	0.996213i	$0.472288\pi$
$492$		0			0
$493$		−	69.1960i		−	3.11643i
$494$		0			0
$495$	0.974447	−	2.40079i	0.0437981	−	0.107907i
$496$		0			0
$497$	−7.37973	+	13.4568i	−0.331026	+	0.603621i
$498$		0			0
$499$	−11.4197			−0.511218			−0.255609	−	0.966780i	$-0.582276\pi$
$499$	−11.4197			−0.511218			−0.255609	+	0.966780i	$0.582276\pi$
$500$		0			0
$501$		−	19.8534i		−	0.886982i
$502$		0			0
$503$		−	9.67677i		−	0.431466i	−0.976452	−	0.215733i	$-0.930786\pi$
$503$		−	9.67677i		−	0.431466i	0.976452	−	0.215733i	$-0.0692141\pi$
$504$		0			0
$505$	12.6313	−	31.1203i	0.562087	−	1.38484i
$506$		0			0
$507$	−12.5116			−0.555658
$508$		0			0
$509$	29.4349			1.30468			0.652340	−	0.757927i	$-0.273788\pi$
$509$	29.4349			1.30468			0.652340	+	0.757927i	$0.273788\pi$
$510$		0			0
$511$	8.95489	+	4.91087i	0.396141	+	0.217244i
$512$		0			0
$513$		−	4.94979i		−	0.218539i
$514$		0			0
$515$	−33.2003	−	13.4756i	−1.46298	−	0.593804i
$516$		0			0
$517$			4.09294i			0.180007i
$518$		0			0
$519$		−	16.3132i		−	0.716070i
$520$		0			0
$521$		−	37.2925i		−	1.63381i	−0.576771	−	0.816906i	$-0.695687\pi$
$521$		−	37.2925i		−	1.63381i	0.576771	−	0.816906i	$-0.304313\pi$
$522$		0			0
$523$	28.2787			1.23654			0.618271	−	0.785965i	$-0.287834\pi$
$523$	28.2787			1.23654			0.618271	+	0.785965i	$0.287834\pi$
$524$		0			0
$525$	−3.88459	−	12.6456i	−0.169538	−	0.551897i
$526$		0			0
$527$	−0.750483			−0.0326915
$528$		0			0
$529$	5.12297			0.222738
$530$		0			0
$531$		−	12.3942i		−	0.537862i
$532$		0			0
$533$	1.80914			0.0783626
$534$		0			0
$535$	19.9380	+	8.09256i	0.861995	+	0.349872i
$536$		0			0
$537$	−20.4707			−0.883375
$538$		0			0
$539$	4.36041	+	6.83940i	0.187816	+	0.294594i
$540$		0			0
$541$			11.0367i			0.474505i	0.971448	+	0.237252i	$0.0762468\pi$
$541$			11.0367i			0.474505i	−0.971448	+	0.237252i	$0.923753\pi$
$542$		0			0
$543$	−14.5030			−0.622382
$544$		0			0
$545$	9.82911	+	3.98951i	0.421033	+	0.170892i
$546$		0			0
$547$		−	28.0990i		−	1.20143i	−0.799464	−	0.600714i	$-0.794883\pi$
$547$		−	28.0990i		−	1.20143i	0.799464	−	0.600714i	$-0.205117\pi$
$548$		0			0
$549$	4.88851			0.208636
$550$		0			0
$551$	−49.8438			−2.12342
$552$		0			0
$553$	8.02227	−	14.6285i	0.341142	−	0.622066i
$554$		0			0
$555$	−3.62091	+	8.92100i	−0.153699	+	0.378675i
$556$		0			0
$557$	7.34006			0.311008			0.155504	−	0.987835i	$-0.450300\pi$
$557$	7.34006			0.311008			0.155504	+	0.987835i	$0.450300\pi$
$558$		0			0
$559$	−5.27908			−0.223281
$560$		0			0
$561$	7.96234			0.336170
$562$		0			0
$563$	−36.4524			−1.53628			−0.768142	−	0.640279i	$-0.778819\pi$
$563$	−36.4524			−1.53628			−0.768142	+	0.640279i	$0.778819\pi$
$564$		0			0
$565$	−23.6118	−	9.58371i	−0.993355	−	0.403190i
$566$		0			0
$567$	−2.31981	−	1.27219i	−0.0974230	−	0.0534269i
$568$		0			0
$569$	1.02524			0.0429803			0.0214902	−	0.999769i	$-0.493159\pi$
$569$	1.02524			0.0429803			0.0214902	+	0.999769i	$0.493159\pi$
$570$		0			0
$571$	10.7464			0.449722			0.224861	−	0.974391i	$-0.427807\pi$
$571$	10.7464			0.449722			0.224861	+	0.974391i	$0.427807\pi$
$572$		0			0
$573$		−	19.8316i		−	0.828476i
$574$		0			0
$575$	−19.0147	−	18.4801i	−0.792968	−	0.770674i
$576$		0			0
$577$	−45.2022			−1.88179			−0.940897	−	0.338693i	$-0.890015\pi$
$577$	−45.2022			−1.88179			−0.940897	+	0.338693i	$0.890015\pi$
$578$		0			0
$579$			0.938946i			0.0390213i
$580$		0			0
$581$	−29.5089	−	16.1827i	−1.22424	−	0.671372i
$582$		0			0
$583$	−1.98038			−0.0820190
$584$		0			0
$585$	1.44802	+	0.587733i	0.0598684	+	0.0242998i
$586$		0			0
$587$	−9.41098			−0.388433			−0.194216	−	0.980959i	$-0.562216\pi$
$587$	−9.41098			−0.388433			−0.194216	+	0.980959i	$0.562216\pi$
$588$		0			0
$589$			0.540593i			0.0222748i
$590$		0			0
$591$	−22.2269			−0.914292
$592$		0			0
$593$	28.1625			1.15650			0.578248	−	0.815861i	$-0.303736\pi$
$593$	28.1625			1.15650			0.578248	+	0.815861i	$0.303736\pi$
$594$		0			0
$595$	31.5180	−	25.6762i	1.29211	−	1.05262i
$596$		0			0
$597$	−7.57854			−0.310169
$598$		0			0
$599$		−	11.5112i		−	0.470336i	−0.971955	−	0.235168i	$-0.924436\pi$
$599$		−	11.5112i		−	0.470336i	0.971955	−	0.235168i	$-0.0755640\pi$
$600$		0			0
$601$		−	14.8292i		−	0.604894i	−0.953166	−	0.302447i	$-0.902196\pi$
$601$		−	14.8292i		−	0.604894i	0.953166	−	0.302447i	$-0.0978035\pi$
$602$		0			0
$603$			8.12528i			0.330887i
$604$		0			0
$605$	−8.12142	+	20.0091i	−0.330183	+	0.813485i
$606$		0			0
$607$		−	19.9890i		−	0.811327i	−0.914023	−	0.405663i	$-0.867041\pi$
$607$		−	19.9890i		−	0.811327i	0.914023	−	0.405663i	$-0.132959\pi$
$608$		0			0
$609$	−12.8108	+	23.3602i	−0.519118	+	0.946604i
$610$		0			0
$611$	−2.46864			−0.0998703
$612$		0			0
$613$	−38.5925			−1.55874			−0.779368	−	0.626566i	$-0.784460\pi$
$613$	−38.5925			−1.55874			−0.779368	+	0.626566i	$0.784460\pi$
$614$		0			0
$615$	5.36335	+	2.17691i	0.216271	+	0.0877816i
$616$		0			0
$617$		−	16.2711i		−	0.655048i	−0.944843	−	0.327524i	$-0.893786\pi$
$617$		−	16.2711i		−	0.655048i	0.944843	−	0.327524i	$-0.106214\pi$
$618$		0			0
$619$			5.64262i			0.226796i	0.993550	+	0.113398i	$0.0361735\pi$
$619$			5.64262i			0.226796i	−0.993550	+	0.113398i	$0.963826\pi$
$620$		0			0
$621$	−5.30311			−0.212806
$622$		0			0
$623$	−1.07047	+	1.95199i	−0.0428875	+	0.0782047i
$624$		0			0
$625$	0.712738	+	24.9898i	0.0285095	+	0.999594i
$626$		0			0
$627$		−	5.73549i		−	0.229054i
$628$		0			0
$629$	−29.5870			−1.17971
$630$		0			0
$631$		−	4.18154i		−	0.166465i	−0.996530	−	0.0832323i	$-0.973476\pi$
$631$		−	4.18154i		−	0.166465i	0.996530	−	0.0832323i	$-0.0265244\pi$
$632$		0			0
$633$	20.4549			0.813009
$634$		0			0
$635$	9.60398	−	23.6617i	0.381123	−	0.938988i
$636$		0			0
$637$	−4.12515	+	2.62996i	−0.163445	+	0.104203i
$638$		0			0
$639$		−	5.80082i		−	0.229477i
$640$		0			0
$641$	−23.6502			−0.934127			−0.467064	−	0.884224i	$-0.654688\pi$
$641$	−23.6502			−0.934127			−0.467064	+	0.884224i	$0.654688\pi$
$642$		0			0
$643$	−12.6788			−0.500002			−0.250001	−	0.968246i	$-0.580431\pi$
$643$	−12.6788			−0.500002			−0.250001	+	0.968246i	$0.580431\pi$
$644$		0			0
$645$	−15.6503	−	6.35224i	−0.616230	−	0.250119i
$646$		0			0
$647$			4.87226i			0.191548i	0.995403	+	0.0957741i	$0.0305326\pi$
$647$			4.87226i			0.191548i	−0.995403	+	0.0957741i	$0.969467\pi$
$648$		0			0
$649$		−	14.3616i		−	0.563741i
$650$		0			0
$651$	0.253359	+	0.138942i	0.00992994	+	0.00544558i
$652$		0			0
$653$	−18.1794			−0.711414			−0.355707	−	0.934597i	$-0.615760\pi$
$653$	−18.1794			−0.711414			−0.355707	+	0.934597i	$0.615760\pi$
$654$		0			0
$655$	−2.48738	−	1.00959i	−0.0971899	−	0.0394481i
$656$		0			0
$657$	−3.86018			−0.150600
$658$		0			0
$659$	−38.8661			−1.51401			−0.757005	−	0.653409i	$-0.773338\pi$
$659$	−38.8661			−1.51401			−0.757005	+	0.653409i	$0.773338\pi$
$660$		0			0
$661$	−32.1806			−1.25168			−0.625841	−	0.779951i	$-0.715244\pi$
$661$	−32.1806			−1.25168			−0.625841	+	0.779951i	$0.715244\pi$
$662$		0			0
$663$			4.80245i			0.186512i
$664$		0			0
$665$	−18.4953	−	22.7033i	−0.717216	−	0.880397i
$666$		0			0
$667$			53.4016i			2.06772i
$668$		0			0
$669$		−	4.53719i		−	0.175418i
$670$		0			0
$671$	5.66448			0.218675
$672$		0			0
$673$		−	40.4479i		−	1.55915i	−0.626308	−	0.779576i	$-0.715435\pi$
$673$		−	40.4479i		−	1.55915i	0.626308	−	0.779576i	$-0.284565\pi$
$674$		0			0
$675$	3.58558	+	3.48477i	0.138009	+	0.134129i
$676$		0			0
$677$		−	0.721800i		−	0.0277410i	−0.999904	−	0.0138705i	$-0.995585\pi$
$677$		−	0.721800i		−	0.0277410i	0.999904	−	0.0138705i	$-0.00441527\pi$
$678$		0			0
$679$	0.113276	+	0.0621207i	0.00434714	+	0.00238397i
$680$		0			0
$681$	12.4331			0.476438
$682$		0			0
$683$			19.6600i			0.752270i	0.926565	+	0.376135i	$0.122747\pi$
$683$			19.6600i			0.752270i	−0.926565	+	0.376135i	$0.877253\pi$
$684$		0			0
$685$	−8.89701	−	3.61118i	−0.339937	−	0.137976i
$686$		0			0
$687$	−3.24679			−0.123873
$688$		0			0
$689$		−	1.19446i		−	0.0455052i
$690$		0			0
$691$			39.4005i			1.49886i	0.662081	+	0.749432i	$0.269673\pi$
$691$			39.4005i			1.49886i	−0.662081	+	0.749432i	$0.730327\pi$
$692$		0			0
$693$	−2.68805	−	1.47413i	−0.102111	−	0.0559975i
$694$		0			0
$695$	−24.8835	−	10.0999i	−0.943885	−	0.383110i
$696$		0			0
$697$			17.7879i			0.673763i
$698$		0			0
$699$		−	26.6500i		−	1.00799i
$700$		0			0
$701$		−	5.00623i		−	0.189083i	−0.995521	−	0.0945413i	$-0.969862\pi$
$701$		−	5.00623i		−	0.189083i	0.995521	−	0.0945413i	$-0.0301384\pi$
$702$		0			0
$703$			21.3123i			0.803809i
$704$		0			0
$705$	−7.31848	−	2.97047i	−0.275630	−	0.111875i
$706$		0			0
$707$	−34.8440	−	19.1085i	−1.31044	−	0.718648i
$708$		0			0
$709$		−	33.3192i		−	1.25133i	−0.780093	−	0.625664i	$-0.784828\pi$
$709$		−	33.3192i		−	1.25133i	0.780093	−	0.625664i	$-0.215172\pi$
$710$		0			0
$711$			6.30589i			0.236489i
$712$		0			0
$713$	0.579181			0.0216905
$714$		0			0
$715$	1.67787	+	0.681027i	0.0627490	+	0.0254690i
$716$		0			0
$717$			10.8623i			0.405659i
$718$		0			0
$719$	31.7733			1.18495			0.592473	−	0.805590i	$-0.298152\pi$
$719$	31.7733			1.18495			0.592473	+	0.805590i	$0.298152\pi$
$720$		0			0
$721$	−20.3856	+	37.1728i	−0.759199	+	1.38439i
$722$		0			0
$723$		−	16.1547i		−	0.600798i
$724$		0			0
$725$	35.0912	−	36.1063i	1.30325	−	1.34095i
$726$		0			0
$727$		−	19.6746i		−	0.729689i	−0.931068	−	0.364844i	$-0.881122\pi$
$727$		−	19.6746i		−	0.729689i	0.931068	−	0.364844i	$-0.118878\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	51.9050i		−	1.91978i
$732$		0			0
$733$			29.8147i			1.10123i	0.834758	+	0.550616i	$0.185607\pi$
$733$			29.8147i			1.10123i	−0.834758	+	0.550616i	$0.814393\pi$
$734$		0			0
$735$	−15.3940	+	2.83300i	−0.567815	+	0.104497i
$736$		0			0
$737$			9.41504i			0.346807i
$738$		0			0
$739$	8.15114			0.299845			0.149922	−	0.988698i	$-0.452098\pi$
$739$	8.15114			0.299845			0.149922	+	0.988698i	$0.452098\pi$
$740$		0			0
$741$	3.45934			0.127082
$742$		0			0
$743$	22.4631			0.824092			0.412046	−	0.911163i	$-0.364814\pi$
$743$	22.4631			0.824092			0.412046	+	0.911163i	$0.364814\pi$
$744$		0			0
$745$	4.35002	+	1.76562i	0.159373	+	0.0646872i
$746$		0			0
$747$	12.7204			0.465414
$748$		0			0
$749$	12.2423	−	22.3236i	0.447324	−	0.815688i
$750$		0			0
$751$		−	36.0935i		−	1.31707i	−0.752551	−	0.658534i	$-0.771177\pi$
$751$		−	36.0935i		−	1.31707i	0.752551	−	0.658534i	$-0.228823\pi$
$752$		0			0
$753$		−	22.5756i		−	0.822699i
$754$		0			0
$755$	43.5842	+	17.6902i	1.58619	+	0.643814i
$756$		0			0
$757$	−26.8241			−0.974937			−0.487469	−	0.873141i	$-0.662080\pi$
$757$	−26.8241			−0.974937			−0.487469	+	0.873141i	$0.662080\pi$
$758$		0			0
$759$	−6.14490			−0.223046
$760$		0			0
$761$			49.5586i			1.79650i	0.439488	+	0.898248i	$0.355160\pi$
$761$			49.5586i			1.79650i	−0.439488	+	0.898248i	$0.644840\pi$
$762$		0			0
$763$	6.03526	−	11.0052i	0.218491	−	0.398415i
$764$		0			0
$765$	−5.77872	+	14.2373i	−0.208930	+	0.514749i
$766$		0			0
$767$	8.66212			0.312771
$768$		0			0
$769$		−	10.3517i		−	0.373292i	−0.982427	−	0.186646i	$-0.940238\pi$
$769$		−	10.3517i		−	0.373292i	0.982427	−	0.186646i	$-0.0597618\pi$
$770$		0			0
$771$	12.0533			0.434088
$772$		0			0
$773$			43.7636i			1.57407i	0.616910	+	0.787033i	$0.288384\pi$
$773$			43.7636i			1.57407i	−0.616910	+	0.787033i	$0.711616\pi$
$774$		0			0
$775$	−0.391600	−	0.380591i	−0.0140667	−	0.0136712i
$776$		0			0
$777$	9.98842	+	5.47766i	0.358333	+	0.196510i
$778$		0			0
$779$	12.8131			0.459077
$780$		0			0
$781$		−	6.72161i		−	0.240518i
$782$		0			0
$783$		−	10.0699i		−	0.359868i
$784$		0			0
$785$	−11.9472	−	4.84919i	−0.426412	−	0.173075i
$786$		0			0
$787$	3.23802			0.115423			0.0577114	−	0.998333i	$-0.481620\pi$
$787$	3.23802			0.115423			0.0577114	+	0.998333i	$0.481620\pi$
$788$		0			0
$789$	−19.6608			−0.699944
$790$		0			0
$791$	−14.4981	+	26.4370i	−0.515492	+	0.939992i
$792$		0			0
$793$			3.41651i			0.121324i
$794$		0			0
$795$	1.43727	−	3.54107i	0.0509749	−	0.125589i
$796$		0			0
$797$			24.9995i			0.885529i	0.896638	+	0.442764i	$0.146002\pi$
$797$			24.9995i			0.885529i	−0.896638	+	0.442764i	$0.853998\pi$
$798$		0			0
$799$		−	24.2721i		−	0.858687i
$800$		0			0
$801$		−	0.841441i		−	0.0297309i
$802$		0			0
$803$	−4.47292			−0.157846
$804$		0			0
$805$	−24.3239	+	19.8155i	−0.857304	+	0.698404i
$806$		0			0
$807$	−1.22906			−0.0432650
$808$		0			0
$809$	−15.5950			−0.548290			−0.274145	−	0.961688i	$-0.588395\pi$
$809$	−15.5950			−0.548290			−0.274145	+	0.961688i	$0.588395\pi$
$810$		0			0
$811$			12.4994i			0.438913i	0.975622	+	0.219456i	$0.0704283\pi$
$811$			12.4994i			0.438913i	−0.975622	+	0.219456i	$0.929572\pi$
$812$		0			0
$813$	12.1597			0.426459
$814$		0			0
$815$	−23.7538	−	9.64133i	−0.832058	−	0.337721i
$816$		0			0
$817$	−37.3886			−1.30806
$818$		0			0
$819$	0.889113	−	1.62128i	0.0310681	−	0.0566523i
$820$		0			0
$821$		−	31.9570i		−	1.11531i	−0.830074	−	0.557653i	$-0.811702\pi$
$821$		−	31.9570i		−	1.11531i	0.830074	−	0.557653i	$-0.188298\pi$
$822$		0			0
$823$	−16.0320			−0.558839			−0.279419	−	0.960169i	$-0.590142\pi$
$823$	−16.0320			−0.558839			−0.279419	+	0.960169i	$0.590142\pi$
$824$		0			0
$825$	4.15473	+	4.03792i	0.144649	+	0.140582i
$826$		0			0
$827$			21.7120i			0.755001i	0.926009	+	0.377501i	$0.123216\pi$
$827$			21.7120i			0.755001i	−0.926009	+	0.377501i	$0.876784\pi$
$828$		0			0
$829$	−48.2421			−1.67552			−0.837759	−	0.546040i	$-0.816135\pi$
$829$	−48.2421			−1.67552			−0.837759	+	0.546040i	$0.816135\pi$
$830$		0			0
$831$	24.1319			0.837126
$832$		0			0
$833$	−25.8583	−	40.5594i	−0.895937	−	1.40530i
$834$		0			0
$835$	41.1343	+	16.6958i	1.42351	+	0.577783i
$836$		0			0
$837$	−0.109215			−0.00377504
$838$		0			0
$839$	25.1122			0.866968			0.433484	−	0.901161i	$-0.357284\pi$
$839$	25.1122			0.866968			0.433484	+	0.901161i	$0.357284\pi$
$840$		0			0
$841$	−72.4023			−2.49663
$842$		0			0
$843$	−2.45095			−0.0844151
$844$		0			0
$845$	10.5217	−	25.9228i	0.361958	−	0.891770i
$846$		0			0
$847$	22.4032	+	12.2859i	0.769784	+	0.422150i
$848$		0			0
$849$	−0.770452			−0.0264419
$850$		0			0
$851$	22.8336			0.782726
$852$		0			0
$853$		−	9.94325i		−	0.340450i	−0.985405	−	0.170225i	$-0.945551\pi$
$853$		−	9.94325i		−	0.340450i	0.985405	−	0.170225i	$-0.0544495\pi$
$854$		0			0
$855$	10.2555	+	4.16257i	0.350731	+	0.142357i
$856$		0			0
$857$	−22.3499			−0.763457			−0.381728	−	0.924275i	$-0.624671\pi$
$857$	−22.3499			−0.763457			−0.381728	+	0.924275i	$0.624671\pi$
$858$		0			0
$859$			11.5753i			0.394944i	0.980309	+	0.197472i	$0.0632731\pi$
$859$			11.5753i			0.394944i	−0.980309	+	0.197472i	$0.936727\pi$
$860$		0			0
$861$	3.29320	−	6.00510i	0.112232	−	0.204653i
$862$		0			0
$863$	42.1305			1.43414			0.717070	−	0.697001i	$-0.245483\pi$
$863$	42.1305			1.43414			0.717070	+	0.697001i	$0.245483\pi$
$864$		0			0
$865$	33.7994	+	13.7187i	1.14921	+	0.466451i
$866$		0			0
$867$	−30.2187			−1.02628
$868$		0			0
$869$			7.30685i			0.247868i
$870$		0			0
$871$	−5.67864			−0.192413
$872$		0			0
$873$	−0.0488298			−0.00165264
$874$		0			0
$875$	29.4672	+	2.58588i	0.996172	+	0.0874185i
$876$		0			0
$877$	1.06166			0.0358496			0.0179248	−	0.999839i	$-0.494294\pi$
$877$	1.06166			0.0358496			0.0179248	+	0.999839i	$0.494294\pi$
$878$		0			0
$879$		−	14.0583i		−	0.474175i
$880$		0			0
$881$		−	32.0206i		−	1.07880i	−0.842049	−	0.539401i	$-0.818651\pi$
$881$		−	32.0206i		−	1.07880i	0.842049	−	0.539401i	$-0.181349\pi$
$882$		0			0
$883$		−	28.8735i		−	0.971670i	−0.874050	−	0.485835i	$-0.838516\pi$
$883$		−	28.8735i		−	0.971670i	0.874050	−	0.485835i	$-0.161484\pi$
$884$		0			0
$885$	25.6796	+	10.4230i	0.863210	+	0.350365i
$886$		0			0
$887$			32.6544i			1.09643i	0.836339	+	0.548213i	$0.184692\pi$
$887$			32.6544i			1.09643i	−0.836339	+	0.548213i	$0.815308\pi$
$888$		0			0
$889$	−26.4929	−	14.5287i	−0.888545	−	0.487278i
$890$		0			0
$891$	1.15873			0.0388191
$892$		0			0
$893$	−17.4839			−0.585076
$894$		0			0
$895$	17.2150	−	42.4133i	0.575433	−	1.41772i
$896$		0			0
$897$		−	3.70627i		−	0.123749i
$898$		0			0
$899$			1.09978i			0.0366799i
$900$		0			0
$901$	11.7442			0.391255
$902$		0			0
$903$	−9.60957	+	17.5229i	−0.319786	+	0.583126i
$904$		0			0
$905$	12.1964	−	30.0488i	0.405422	−	0.998856i
$906$		0			0
$907$			36.6780i			1.21787i	0.793218	+	0.608937i	$0.208404\pi$
$907$			36.6780i			1.21787i	−0.793218	+	0.608937i	$0.791596\pi$
$908$		0			0
$909$	15.0202			0.498187
$910$		0			0
$911$		−	38.1450i		−	1.26380i	−0.775049	−	0.631901i	$-0.782275\pi$
$911$		−	38.1450i		−	1.26380i	0.775049	−	0.631901i	$-0.217725\pi$
$912$		0			0
$913$	14.7395			0.487807
$914$		0			0
$915$	−4.11103	+	10.1285i	−0.135906	+	0.334839i
$916$		0			0
$917$	−1.52730	+	2.78500i	−0.0504358	+	0.0919688i
$918$		0			0
$919$		−	10.1269i		−	0.334056i	−0.985952	−	0.167028i	$-0.946583\pi$
$919$		−	10.1269i		−	0.334056i	0.985952	−	0.167028i	$-0.0534170\pi$
$920$		0			0
$921$	33.8488			1.11536
$922$		0			0
$923$	4.05411			0.133443
$924$		0			0
$925$	−15.4384	−	15.0044i	−0.507612	−	0.493341i
$926$		0			0
$927$		−	16.0240i		−	0.526299i
$928$		0			0
$929$			28.8015i			0.944947i	0.881345	+	0.472474i	$0.156639\pi$
$929$			28.8015i			0.944947i	−0.881345	+	0.472474i	$0.843361\pi$
$930$		0			0
$931$	−29.2160	+	18.6265i	−0.957517	+	0.610457i
$932$		0			0
$933$	−14.3608			−0.470150
$934$		0			0
$935$	−6.69600	+	16.4972i	−0.218983	+	0.539516i
$936$		0			0
$937$	−31.8351			−1.04001			−0.520003	−	0.854164i	$-0.674069\pi$
$937$	−31.8351			−1.04001			−0.520003	+	0.854164i	$0.674069\pi$
$938$		0			0
$939$	16.3655			0.534068
$940$		0			0
$941$	15.0378			0.490217			0.245109	−	0.969496i	$-0.421176\pi$
$941$	15.0378			0.490217			0.245109	+	0.969496i	$0.421176\pi$
$942$		0			0
$943$		−	13.7277i		−	0.447035i
$944$		0			0
$945$	4.58672	−	3.73658i	0.149206	−	0.121551i
$946$		0			0
$947$			45.0438i			1.46373i	0.681451	+	0.731864i	$0.261349\pi$
$947$			45.0438i			1.46373i	−0.681451	+	0.731864i	$0.738651\pi$
$948$		0			0
$949$		−	2.69782i		−	0.0875749i
$950$		0			0
$951$	−25.5784			−0.829436
$952$		0			0
$953$		−	28.4561i		−	0.921782i	−0.887457	−	0.460891i	$-0.847530\pi$
$953$		−	28.4561i		−	0.921782i	0.887457	−	0.460891i	$-0.152470\pi$
$954$		0			0
$955$	41.0891	+	16.6775i	1.32961	+	0.539672i
$956$		0			0
$957$		−	11.6683i		−	0.377183i
$958$		0			0
$959$	−5.46293	+	9.96157i	−0.176407	+	0.321676i
$960$		0			0
$961$	−30.9881			−0.999615
$962$		0			0
$963$			9.62303i			0.310098i
$964$		0			0
$965$	−1.94541	−	0.789614i	−0.0626248	−	0.0254186i
$966$		0			0
$967$	−16.7411			−0.538357			−0.269178	−	0.963090i	$-0.586752\pi$
$967$	−16.7411			−0.538357			−0.269178	+	0.963090i	$0.586752\pi$
$968$		0			0
$969$			34.0129i			1.09265i
$970$		0			0
$971$		−	8.17899i		−	0.262476i	−0.991351	−	0.131238i	$-0.958105\pi$
$971$		−	8.17899i		−	0.262476i	0.991351	−	0.131238i	$-0.0418953\pi$
$972$		0			0
$973$	−15.2789	+	27.8609i	−0.489820	+	0.893179i
$974$		0			0
$975$	−2.43546	+	2.50591i	−0.0779970	+	0.0802533i
$976$		0			0
$977$			44.9685i			1.43867i	0.694663	+	0.719335i	$0.255553\pi$
$977$			44.9685i			1.43867i	−0.694663	+	0.719335i	$0.744447\pi$
$978$		0			0
$979$		−	0.975007i		−	0.0311614i
$980$		0			0
$981$			4.74400i			0.151464i
$982$		0			0
$983$			29.0890i			0.927794i	0.885889	+	0.463897i	$0.153549\pi$
$983$			29.0890i			0.927794i	−0.885889	+	0.463897i	$0.846451\pi$
$984$		0			0
$985$	18.6919	−	46.0520i	0.595573	−	1.46734i
$986$		0			0
$987$	−4.49368	+	8.19416i	−0.143035	+	0.260823i
$988$		0			0
$989$			40.0575i			1.27375i
$990$		0			0
$991$		−	0.727753i		−	0.0231178i	−0.999933	−	0.0115589i	$-0.996321\pi$
$991$		−	0.727753i		−	0.0231178i	0.999933	−	0.0115589i	$-0.00367940\pi$
$992$		0			0
$993$	−4.64867			−0.147521
$994$		0			0
$995$	6.37324	−	15.7020i	0.202045	−	0.497787i
$996$		0			0
$997$		−	15.1844i		−	0.480893i	−0.970662	−	0.240447i	$-0.922706\pi$
$997$		−	15.1844i		−	0.480893i	0.970662	−	0.240447i	$-0.0772939\pi$
$998$		0			0
$999$	−4.30570			−0.136226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.w.a.559.31		48
4.3	odd	2		840.2.w.b.139.46	yes	48
5.4	even	2		3360.2.w.b.559.32		48
7.6	odd	2		3360.2.w.b.559.18		48
8.3	odd	2	inner	3360.2.w.a.559.18		48
8.5	even	2		840.2.w.b.139.4	yes	48
20.19	odd	2		840.2.w.a.139.3	✓	48
28.27	even	2		840.2.w.a.139.46	yes	48
35.34	odd	2	inner	3360.2.w.a.559.17		48
40.19	odd	2		3360.2.w.b.559.17		48
40.29	even	2		840.2.w.a.139.45	yes	48
56.13	odd	2		840.2.w.a.139.4	yes	48
56.27	even	2		3360.2.w.b.559.31		48
140.139	even	2		840.2.w.b.139.3	yes	48
280.69	odd	2		840.2.w.b.139.45	yes	48
280.139	even	2	inner	3360.2.w.a.559.32		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.w.a.139.3	✓	48	20.19	odd	2
840.2.w.a.139.4	yes	48	56.13	odd	2
840.2.w.a.139.45	yes	48	40.29	even	2
840.2.w.a.139.46	yes	48	28.27	even	2
840.2.w.b.139.3	yes	48	140.139	even	2
840.2.w.b.139.4	yes	48	8.5	even	2
840.2.w.b.139.45	yes	48	280.69	odd	2
840.2.w.b.139.46	yes	48	4.3	odd	2
3360.2.w.a.559.17		48	35.34	odd	2	inner
3360.2.w.a.559.18		48	8.3	odd	2	inner
3360.2.w.a.559.31		48	1.1	even	1	trivial
3360.2.w.a.559.32		48	280.139	even	2	inner
3360.2.w.b.559.17		48	40.19	odd	2
3360.2.w.b.559.18		48	7.6	odd	2
3360.2.w.b.559.31		48	56.27	even	2
3360.2.w.b.559.32		48	5.4	even	2

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:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.w (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +(0.840958 - 2.07190i) q^{5} +(-2.31981 - 1.27219i) q^{7} +1.00000 q^{9} +1.15873 q^{11} +0.698885i q^{13} +(-0.840958 + 2.07190i) q^{15} -6.87158 q^{17} +4.94979i q^{19} +(2.31981 + 1.27219i) q^{21} +5.30311 q^{23} +(-3.58558 - 3.48477i) q^{25} -1.00000 q^{27} +10.0699i q^{29} +0.109215 q^{31} -1.15873 q^{33} +(-4.58672 + 3.73658i) q^{35} +4.30570 q^{37} -0.698885i q^{39} -2.58861i q^{41} +7.55358i q^{43} +(0.840958 - 2.07190i) q^{45} +3.53225i q^{47} +(3.76308 + 5.90248i) q^{49} +6.87158 q^{51} -1.70909 q^{53} +(0.974447 - 2.40079i) q^{55} -4.94979i q^{57} -12.3942i q^{59} +4.88851 q^{61} +(-2.31981 - 1.27219i) q^{63} +(1.44802 + 0.587733i) q^{65} +8.12528i q^{67} -5.30311 q^{69} -5.80082i q^{71} -3.86018 q^{73} +(3.58558 + 3.48477i) q^{75} +(-2.68805 - 1.47413i) q^{77} +6.30589i q^{79} +1.00000 q^{81} +12.7204 q^{83} +(-5.77872 + 14.2373i) q^{85} -10.0699i q^{87} -0.841441i q^{89} +(0.889113 - 1.62128i) q^{91} -0.109215 q^{93} +(10.2555 + 4.16257i) q^{95} -0.0488298 q^{97} +1.15873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{3} + 48 q^{9} - 48 q^{27} - 8 q^{35} + 48 q^{81} + 16 q^{91} - 48 q^{97}+O(q^{100}) \) 48 * q - 48 * q^3 + 48 * q^9 - 48 * q^27 - 8 * q^35 + 48 * q^81 + 16 * q^91 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more