LMFDB - Embedded newform 3360.2.e.c.911.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(911,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.911"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [44,0,-4,0,-44] 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:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			911.11
Character	$\chi$	$=$	3360.911
Dual form			3360.2.e.c.911.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.39892 - 1.02128i) q^{3} -1.00000 q^{5} +1.00000i q^{7} +(0.913962 + 2.85739i) q^{9} +2.60621i q^{11} -3.64377i q^{13} +(1.39892 + 1.02128i) q^{15} -4.39342i q^{17} -6.32881 q^{19} +(1.02128 - 1.39892i) q^{21} +0.872985 q^{23} +1.00000 q^{25} +(1.63964 - 4.93068i) q^{27} +2.82610 q^{29} +4.51893i q^{31} +(2.66168 - 3.64589i) q^{33} -1.00000i q^{35} +0.410585i q^{37} +(-3.72132 + 5.09735i) q^{39} -1.31574i q^{41} +2.37889 q^{43} +(-0.913962 - 2.85739i) q^{45} -12.8953 q^{47} -1.00000 q^{49} +(-4.48693 + 6.14606i) q^{51} +6.44201 q^{53} -2.60621i q^{55} +(8.85351 + 6.46351i) q^{57} -10.3119i q^{59} +11.4151i q^{61} +(-2.85739 + 0.913962i) q^{63} +3.64377i q^{65} +9.90769 q^{67} +(-1.22124 - 0.891565i) q^{69} -2.67806 q^{71} +14.6519 q^{73} +(-1.39892 - 1.02128i) q^{75} -2.60621 q^{77} +11.3895i q^{79} +(-7.32935 + 5.22309i) q^{81} +15.8054i q^{83} +4.39342i q^{85} +(-3.95349 - 2.88625i) q^{87} -1.07306i q^{89} +3.64377 q^{91} +(4.61510 - 6.32162i) q^{93} +6.32881 q^{95} -11.4590 q^{97} +(-7.44697 + 2.38198i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{3} - 44 q^{5} + 4 q^{9} + 4 q^{15} - 16 q^{19} - 16 q^{23} + 44 q^{25} - 4 q^{27} + 16 q^{29} + 24 q^{33} + 16 q^{39} - 24 q^{43} - 4 q^{45} + 24 q^{47} - 44 q^{49} - 32 q^{53} - 24 q^{67} - 8 q^{69}+ \cdots + 64 q^{99}+O(q^{100}) $ 44 * q - 4 * q^3 - 44 * q^5 + 4 * q^9 + 4 * q^15 - 16 * q^19 - 16 * q^23 + 44 * q^25 - 4 * q^27 + 16 * q^29 + 24 * q^33 + 16 * q^39 - 24 * q^43 - 4 * q^45 + 24 * q^47 - 44 * q^49 - 32 * q^53 - 24 * q^67 - 8 * q^69 - 8 * q^71 + 8 * q^73 - 4 * q^75 + 8 * q^77 + 20 * q^81 + 16 * q^87 + 16 * q^91 + 16 * q^93 + 16 * q^95 + 24 * q^97 + 64 * q^99

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
$3$	−1.39892	−	1.02128i	−0.807668	−	0.589638i
$3$	−1.39892	−	1.02128i	−0.807668	−	0.589638i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	0.913962	+	2.85739i	0.304654	+	0.952463i
$10$		0			0
$11$			2.60621i			0.785803i	0.919581	+	0.392901i	$0.128529\pi$
$11$			2.60621i			0.785803i	−0.919581	+	0.392901i	$0.871471\pi$
$12$		0			0
$13$		−	3.64377i		−	1.01060i	−0.862944	−	0.505300i	$-0.831382\pi$
$13$		−	3.64377i		−	1.01060i	0.862944	−	0.505300i	$-0.168618\pi$
$14$		0			0
$15$	1.39892	+	1.02128i	0.361200	+	0.263694i
$16$		0			0
$17$		−	4.39342i		−	1.06556i	−0.846253	−	0.532781i	$-0.821147\pi$
$17$		−	4.39342i		−	1.06556i	0.846253	−	0.532781i	$-0.178853\pi$
$18$		0			0
$19$	−6.32881			−1.45193			−0.725965	−	0.687732i	$-0.758606\pi$
$19$	−6.32881			−1.45193			−0.725965	+	0.687732i	$0.758606\pi$
$20$		0			0
$21$	1.02128	−	1.39892i	0.222862	−	0.305270i
$22$		0			0
$23$	0.872985			0.182030			0.0910150	−	0.995850i	$-0.470989\pi$
$23$	0.872985			0.182030			0.0910150	+	0.995850i	$0.470989\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.63964	−	4.93068i	0.315549	−	0.948909i
$28$		0			0
$29$	2.82610			0.524793			0.262397	−	0.964960i	$-0.415487\pi$
$29$	2.82610			0.524793			0.262397	+	0.964960i	$0.415487\pi$
$30$		0			0
$31$			4.51893i			0.811623i	0.913957	+	0.405812i	$0.133011\pi$
$31$			4.51893i			0.811623i	−0.913957	+	0.405812i	$0.866989\pi$
$32$		0			0
$33$	2.66168	−	3.64589i	0.463339	−	0.634668i
$34$		0			0
$35$		−	1.00000i		−	0.169031i
$36$		0			0
$37$			0.410585i			0.0674998i	0.999430	+	0.0337499i	$0.0107450\pi$
$37$			0.410585i			0.0674998i	−0.999430	+	0.0337499i	$0.989255\pi$
$38$		0			0
$39$	−3.72132	+	5.09735i	−0.595888	+	0.816229i
$40$		0			0
$41$		−	1.31574i		−	0.205484i	−0.994708	−	0.102742i	$-0.967238\pi$
$41$		−	1.31574i		−	0.205484i	0.994708	−	0.102742i	$-0.0327616\pi$
$42$		0			0
$43$	2.37889			0.362778			0.181389	−	0.983411i	$-0.441941\pi$
$43$	2.37889			0.362778			0.181389	+	0.983411i	$0.441941\pi$
$44$		0			0
$45$	−0.913962	−	2.85739i	−0.136245	−	0.425954i
$46$		0			0
$47$	−12.8953			−1.88097			−0.940483	−	0.339840i	$-0.889627\pi$
$47$	−12.8953			−1.88097			−0.940483	+	0.339840i	$0.889627\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−4.48693	+	6.14606i	−0.628296	+	0.860620i
$52$		0			0
$53$	6.44201			0.884879			0.442439	−	0.896798i	$-0.354113\pi$
$53$	6.44201			0.884879			0.442439	+	0.896798i	$0.354113\pi$
$54$		0			0
$55$		−	2.60621i		−	0.351422i
$56$		0			0
$57$	8.85351	+	6.46351i	1.17268	+	0.856113i
$58$		0			0
$59$		−	10.3119i		−	1.34250i	−0.741232	−	0.671249i	$-0.765758\pi$
$59$		−	10.3119i		−	1.34250i	0.741232	−	0.671249i	$-0.234242\pi$
$60$		0			0
$61$			11.4151i			1.46156i	0.682614	+	0.730779i	$0.260843\pi$
$61$			11.4151i			1.46156i	−0.682614	+	0.730779i	$0.739157\pi$
$62$		0			0
$63$	−2.85739	+	0.913962i	−0.359997	+	0.115148i
$64$		0			0
$65$			3.64377i			0.451954i
$66$		0			0
$67$	9.90769			1.21042			0.605208	−	0.796067i	$-0.293090\pi$
$67$	9.90769			1.21042			0.605208	+	0.796067i	$0.293090\pi$
$68$		0			0
$69$	−1.22124	−	0.891565i	−0.147020	−	0.107332i
$70$		0			0
$71$	−2.67806			−0.317828			−0.158914	−	0.987292i	$-0.550799\pi$
$71$	−2.67806			−0.317828			−0.158914	+	0.987292i	$0.550799\pi$
$72$		0			0
$73$	14.6519			1.71487			0.857435	−	0.514592i	$-0.172056\pi$
$73$	14.6519			1.71487			0.857435	+	0.514592i	$0.172056\pi$
$74$		0			0
$75$	−1.39892	−	1.02128i	−0.161534	−	0.117928i
$76$		0			0
$77$	−2.60621			−0.297006
$78$		0			0
$79$			11.3895i			1.28142i	0.767784	+	0.640709i	$0.221359\pi$
$79$			11.3895i			1.28142i	−0.767784	+	0.640709i	$0.778641\pi$
$80$		0			0
$81$	−7.32935	+	5.22309i	−0.814372	+	0.580344i
$82$		0			0
$83$			15.8054i			1.73487i	0.497551	+	0.867435i	$0.334233\pi$
$83$			15.8054i			1.73487i	−0.497551	+	0.867435i	$0.665767\pi$
$84$		0			0
$85$			4.39342i			0.476534i
$86$		0			0
$87$	−3.95349	−	2.88625i	−0.423858	−	0.309438i
$88$		0			0
$89$		−	1.07306i		−	0.113744i	−0.998381	−	0.0568720i	$-0.981887\pi$
$89$		−	1.07306i		−	0.113744i	0.998381	−	0.0568720i	$-0.0181127\pi$
$90$		0			0
$91$	3.64377			0.381971
$92$		0			0
$93$	4.61510	−	6.32162i	0.478564	−	0.655522i
$94$		0			0
$95$	6.32881			0.649322
$96$		0			0
$97$	−11.4590			−1.16349			−0.581743	−	0.813373i	$-0.697629\pi$
$97$	−11.4590			−1.16349			−0.581743	+	0.813373i	$0.697629\pi$
$98$		0			0
$99$	−7.44697	+	2.38198i	−0.748448	+	0.239398i
$100$		0			0
$101$	−8.86572			−0.882173			−0.441086	−	0.897465i	$-0.645407\pi$
$101$	−8.86572			−0.882173			−0.441086	+	0.897465i	$0.645407\pi$
$102$		0			0
$103$			4.62254i			0.455472i	0.973723	+	0.227736i	$0.0731323\pi$
$103$			4.62254i			0.455472i	−0.973723	+	0.227736i	$0.926868\pi$
$104$		0			0
$105$	−1.02128	+	1.39892i	−0.0996670	+	0.136521i
$106$		0			0
$107$			2.82698i			0.273295i	0.990620	+	0.136647i	$0.0436327\pi$
$107$			2.82698i			0.273295i	−0.990620	+	0.136647i	$0.956367\pi$
$108$		0			0
$109$			1.54341i			0.147832i	0.997264	+	0.0739158i	$0.0235496\pi$
$109$			1.54341i			0.147832i	−0.997264	+	0.0739158i	$0.976450\pi$
$110$		0			0
$111$	0.419324	−	0.574377i	0.0398005	−	0.0545174i
$112$		0			0
$113$		−	6.52668i		−	0.613978i	−0.951713	−	0.306989i	$-0.900678\pi$
$113$		−	6.52668i		−	0.613978i	0.951713	−	0.306989i	$-0.0993215\pi$
$114$		0			0
$115$	−0.872985			−0.0814063
$116$		0			0
$117$	10.4117	−	3.33027i	0.962559	−	0.307884i
$118$		0			0
$119$	4.39342			0.402745
$120$		0			0
$121$	4.20765			0.382514
$122$		0			0
$123$	−1.34374	+	1.84062i	−0.121161	+	0.165963i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			16.4148i			1.45658i	0.685270	+	0.728289i	$0.259684\pi$
$127$			16.4148i			1.45658i	−0.685270	+	0.728289i	$0.740316\pi$
$128$		0			0
$129$	−3.32789	−	2.42952i	−0.293004	−	0.213908i
$130$		0			0
$131$		−	13.1259i		−	1.14681i	−0.819271	−	0.573406i	$-0.805622\pi$
$131$		−	13.1259i		−	1.14681i	0.819271	−	0.573406i	$-0.194378\pi$
$132$		0			0
$133$		−	6.32881i		−	0.548778i
$134$		0			0
$135$	−1.63964	+	4.93068i	−0.141118	+	0.424365i
$136$		0			0
$137$		−	5.14434i		−	0.439510i	−0.975555	−	0.219755i	$-0.929474\pi$
$137$		−	5.14434i		−	0.439510i	0.975555	−	0.219755i	$-0.0705258\pi$
$138$		0			0
$139$	23.0112			1.95179			0.975893	−	0.218251i	$-0.0700351\pi$
$139$	23.0112			1.95179			0.975893	+	0.218251i	$0.0700351\pi$
$140$		0			0
$141$	18.0395	+	13.1697i	1.51920	+	1.10909i
$142$		0			0
$143$	9.49645			0.794133
$144$		0			0
$145$	−2.82610			−0.234695
$146$		0			0
$147$	1.39892	+	1.02128i	0.115381	+	0.0842340i
$148$		0			0
$149$	12.1682			0.996860			0.498430	−	0.866930i	$-0.333910\pi$
$149$	12.1682			0.996860			0.498430	+	0.866930i	$0.333910\pi$
$150$		0			0
$151$		−	0.557777i		−	0.0453912i	−0.999742	−	0.0226956i	$-0.992775\pi$
$151$		−	0.557777i		−	0.0453912i	0.999742	−	0.0226956i	$-0.00722486\pi$
$152$		0			0
$153$	12.5537	−	4.01543i	1.01491	−	0.324628i
$154$		0			0
$155$		−	4.51893i		−	0.362969i
$156$		0			0
$157$			15.7226i			1.25480i	0.778698	+	0.627399i	$0.215880\pi$
$157$			15.7226i			1.25480i	−0.778698	+	0.627399i	$0.784120\pi$
$158$		0			0
$159$	−9.01187	−	6.57912i	−0.714688	−	0.521758i
$160$		0			0
$161$			0.872985i			0.0688009i
$162$		0			0
$163$	15.7464			1.23335			0.616677	−	0.787217i	$-0.288479\pi$
$163$	15.7464			1.23335			0.616677	+	0.787217i	$0.288479\pi$
$164$		0			0
$165$	−2.66168	+	3.64589i	−0.207212	+	0.283832i
$166$		0			0
$167$	24.0522			1.86122			0.930608	−	0.366016i	$-0.119279\pi$
$167$	24.0522			1.86122			0.930608	+	0.366016i	$0.119279\pi$
$168$		0			0
$169$	−0.277070			−0.0213131
$170$		0			0
$171$	−5.78430	−	18.0839i	−0.442336	−	1.38291i
$172$		0			0
$173$	10.8040			0.821412			0.410706	−	0.911768i	$-0.365282\pi$
$173$	10.8040			0.821412			0.410706	+	0.911768i	$0.365282\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$	−10.5314	+	14.4256i	−0.791588	+	1.08429i
$178$		0			0
$179$		−	5.38759i		−	0.402687i	−0.979521	−	0.201344i	$-0.935469\pi$
$179$		−	5.38759i		−	0.402687i	0.979521	−	0.201344i	$-0.0645308\pi$
$180$		0			0
$181$			10.3954i			0.772684i	0.922356	+	0.386342i	$0.126262\pi$
$181$			10.3954i			0.772684i	−0.922356	+	0.386342i	$0.873738\pi$
$182$		0			0
$183$	11.6581	−	15.9689i	0.861790	−	1.18045i
$184$		0			0
$185$		−	0.410585i		−	0.0301868i
$186$		0			0
$187$	11.4502			0.837322
$188$		0			0
$189$	4.93068	+	1.63964i	0.358654	+	0.119266i
$190$		0			0
$191$	22.6073			1.63580			0.817902	−	0.575358i	$-0.195137\pi$
$191$	22.6073			1.63580			0.817902	+	0.575358i	$0.195137\pi$
$192$		0			0
$193$	17.1364			1.23351			0.616753	−	0.787157i	$-0.288448\pi$
$193$	17.1364			1.23351			0.616753	+	0.787157i	$0.288448\pi$
$194$		0			0
$195$	3.72132	−	5.09735i	0.266489	−	0.365029i
$196$		0			0
$197$	9.63654			0.686575			0.343287	−	0.939230i	$-0.388460\pi$
$197$	9.63654			0.686575			0.343287	+	0.939230i	$0.388460\pi$
$198$		0			0
$199$		−	6.03091i		−	0.427520i	−0.976886	−	0.213760i	$-0.931429\pi$
$199$		−	6.03091i		−	0.427520i	0.976886	−	0.213760i	$-0.0685711\pi$
$200$		0			0
$201$	−13.8601	−	10.1186i	−0.977614	−	0.713708i
$202$		0			0
$203$			2.82610i			0.198353i
$204$		0			0
$205$			1.31574i			0.0918952i
$206$		0			0
$207$	0.797876	+	2.49446i	0.0554562	+	0.173377i
$208$		0			0
$209$		−	16.4942i		−	1.14093i
$210$		0			0
$211$	−9.41330			−0.648039			−0.324019	−	0.946050i	$-0.605034\pi$
$211$	−9.41330			−0.648039			−0.324019	+	0.946050i	$0.605034\pi$
$212$		0			0
$213$	3.74640	+	2.73506i	0.256699	+	0.187403i
$214$		0			0
$215$	−2.37889			−0.162239
$216$		0			0
$217$	−4.51893			−0.306765
$218$		0			0
$219$	−20.4968	−	14.9637i	−1.38505	−	1.01115i
$220$		0			0
$221$	−16.0086			−1.07686
$222$		0			0
$223$		−	2.79614i		−	0.187243i	−0.995608	−	0.0936217i	$-0.970156\pi$
$223$		−	2.79614i		−	0.187243i	0.995608	−	0.0936217i	$-0.0298444\pi$
$224$		0			0
$225$	0.913962	+	2.85739i	0.0609308	+	0.190493i
$226$		0			0
$227$		−	23.8776i		−	1.58481i	−0.609996	−	0.792405i	$-0.708829\pi$
$227$		−	23.8776i		−	1.58481i	0.609996	−	0.792405i	$-0.291171\pi$
$228$		0			0
$229$			19.4866i			1.28771i	0.765147	+	0.643856i	$0.222666\pi$
$229$			19.4866i			1.28771i	−0.765147	+	0.643856i	$0.777334\pi$
$230$		0			0
$231$	3.64589	+	2.66168i	0.239882	+	0.175126i
$232$		0			0
$233$			16.8441i			1.10349i	0.834012	+	0.551746i	$0.186038\pi$
$233$			16.8441i			1.10349i	−0.834012	+	0.551746i	$0.813962\pi$
$234$		0			0
$235$	12.8953			0.841194
$236$		0			0
$237$	11.6319	−	15.9330i	0.755573	−	1.03496i
$238$		0			0
$239$	7.03283			0.454916			0.227458	−	0.973788i	$-0.426959\pi$
$239$	7.03283			0.454916			0.227458	+	0.973788i	$0.426959\pi$
$240$		0			0
$241$	4.94865			0.318771			0.159385	−	0.987216i	$-0.449049\pi$
$241$	4.94865			0.318771			0.159385	+	0.987216i	$0.449049\pi$
$242$		0			0
$243$	15.5874	+	0.178639i	0.999934	+	0.0114597i
$244$		0			0
$245$	1.00000			0.0638877
$246$		0			0
$247$			23.0607i			1.46732i
$248$		0			0
$249$	16.1418	−	22.1105i	1.02294	−	1.40120i
$250$		0			0
$251$			2.99335i			0.188939i	0.995528	+	0.0944693i	$0.0301154\pi$
$251$			2.99335i			0.188939i	−0.995528	+	0.0944693i	$0.969885\pi$
$252$		0			0
$253$			2.27519i			0.143040i
$254$		0			0
$255$	4.48693	−	6.14606i	0.280982	−	0.384881i
$256$		0			0
$257$			10.2629i			0.640184i	0.947386	+	0.320092i	$0.103714\pi$
$257$			10.2629i			0.640184i	−0.947386	+	0.320092i	$0.896286\pi$
$258$		0			0
$259$	−0.410585			−0.0255125
$260$		0			0
$261$	2.58295	+	8.07526i	0.159880	+	0.499846i
$262$		0			0
$263$	−18.3429			−1.13107			−0.565536	−	0.824723i	$-0.691331\pi$
$263$	−18.3429			−1.13107			−0.565536	+	0.824723i	$0.691331\pi$
$264$		0			0
$265$	−6.44201			−0.395730
$266$		0			0
$267$	−1.09590	+	1.50112i	−0.0670677	+	0.0918673i
$268$		0			0
$269$	−24.8952			−1.51789			−0.758944	−	0.651156i	$-0.774284\pi$
$269$	−24.8952			−1.51789			−0.758944	+	0.651156i	$0.774284\pi$
$270$		0			0
$271$		−	5.68426i		−	0.345294i	−0.984984	−	0.172647i	$-0.944768\pi$
$271$		−	5.68426i		−	0.345294i	0.984984	−	0.172647i	$-0.0552320\pi$
$272$		0			0
$273$	−5.09735	−	3.72132i	−0.308506	−	0.225225i
$274$		0			0
$275$			2.60621i			0.157161i
$276$		0			0
$277$		−	20.4907i		−	1.23117i	−0.788072	−	0.615583i	$-0.788921\pi$
$277$		−	20.4907i		−	1.23117i	0.788072	−	0.615583i	$-0.211079\pi$
$278$		0			0
$279$	−12.9123	+	4.13013i	−0.773041	+	0.247264i
$280$		0			0
$281$			1.78658i			0.106579i	0.998579	+	0.0532893i	$0.0169706\pi$
$281$			1.78658i			0.106579i	−0.998579	+	0.0532893i	$0.983029\pi$
$282$		0			0
$283$	3.73574			0.222067			0.111033	−	0.993817i	$-0.464584\pi$
$283$	3.73574			0.222067			0.111033	+	0.993817i	$0.464584\pi$
$284$		0			0
$285$	−8.85351	−	6.46351i	−0.524437	−	0.382865i
$286$		0			0
$287$	1.31574			0.0776656
$288$		0			0
$289$	−2.30218			−0.135422
$290$		0			0
$291$	16.0302	+	11.7029i	0.939709	+	0.686035i
$292$		0			0
$293$	5.42748			0.317077			0.158538	−	0.987353i	$-0.449322\pi$
$293$	5.42748			0.317077			0.158538	+	0.987353i	$0.449322\pi$
$294$		0			0
$295$			10.3119i			0.600383i
$296$		0			0
$297$	12.8504	+	4.27325i	0.745656	+	0.247959i
$298$		0			0
$299$		−	3.18096i		−	0.183960i
$300$		0			0
$301$			2.37889i			0.137117i
$302$		0			0
$303$	12.4025	+	9.05441i	0.712502	+	0.520162i
$304$		0			0
$305$		−	11.4151i		−	0.653628i
$306$		0			0
$307$	3.30494			0.188623			0.0943114	−	0.995543i	$-0.469935\pi$
$307$	3.30494			0.188623			0.0943114	+	0.995543i	$0.469935\pi$
$308$		0			0
$309$	4.72092	−	6.46657i	0.268564	−	0.367870i
$310$		0			0
$311$	18.9308			1.07347			0.536734	−	0.843751i	$-0.319658\pi$
$311$	18.9308			1.07347			0.536734	+	0.843751i	$0.319658\pi$
$312$		0			0
$313$	19.4397			1.09879			0.549397	−	0.835561i	$-0.314857\pi$
$313$	19.4397			1.09879			0.549397	+	0.835561i	$0.314857\pi$
$314$		0			0
$315$	2.85739	−	0.913962i	0.160996	−	0.0514960i
$316$		0			0
$317$	−13.4464			−0.755224			−0.377612	−	0.925964i	$-0.623255\pi$
$317$	−13.4464			−0.755224			−0.377612	+	0.925964i	$0.623255\pi$
$318$		0			0
$319$			7.36541i			0.412384i
$320$		0			0
$321$	2.88715	−	3.95472i	0.161145	−	0.220731i
$322$		0			0
$323$			27.8052i			1.54712i
$324$		0			0
$325$		−	3.64377i		−	0.202120i
$326$		0			0
$327$	1.57626	−	2.15911i	0.0871671	−	0.119399i
$328$		0			0
$329$		−	12.8953i		−	0.710939i
$330$		0			0
$331$	−22.3577			−1.22889			−0.614445	−	0.788960i	$-0.710620\pi$
$331$	−22.3577			−1.22889			−0.614445	+	0.788960i	$0.710620\pi$
$332$		0			0
$333$	−1.17320	+	0.375260i	−0.0642911	+	0.0205641i
$334$		0			0
$335$	−9.90769			−0.541315
$336$		0			0
$337$	−25.2446			−1.37516			−0.687581	−	0.726108i	$-0.741327\pi$
$337$	−25.2446			−1.37516			−0.687581	+	0.726108i	$0.741327\pi$
$338$		0			0
$339$	−6.66559	+	9.13031i	−0.362025	+	0.495890i
$340$		0			0
$341$	−11.7773			−0.637776
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	1.22124	+	0.891565i	0.0657492	+	0.0480002i
$346$		0			0
$347$		−	10.8735i		−	0.583721i	−0.956461	−	0.291860i	$-0.905726\pi$
$347$		−	10.8735i		−	0.583721i	0.956461	−	0.291860i	$-0.0942742\pi$
$348$		0			0
$349$			32.5032i			1.73986i	0.493180	+	0.869928i	$0.335835\pi$
$349$			32.5032i			1.73986i	−0.493180	+	0.869928i	$0.664165\pi$
$350$		0			0
$351$	−17.9663	−	5.97448i	−0.958968	−	0.318894i
$352$		0			0
$353$			9.36201i			0.498290i	0.968466	+	0.249145i	$0.0801495\pi$
$353$			9.36201i			0.498290i	−0.968466	+	0.249145i	$0.919850\pi$
$354$		0			0
$355$	2.67806			0.142137
$356$		0			0
$357$	−6.14606	−	4.48693i	−0.325284	−	0.237474i
$358$		0			0
$359$	5.27997			0.278666			0.139333	−	0.990246i	$-0.455504\pi$
$359$	5.27997			0.278666			0.139333	+	0.990246i	$0.455504\pi$
$360$		0			0
$361$	21.0539			1.10810
$362$		0			0
$363$	−5.88617	−	4.29720i	−0.308944	−	0.225545i
$364$		0			0
$365$	−14.6519			−0.766913
$366$		0			0
$367$			27.9027i			1.45651i	0.685306	+	0.728255i	$0.259668\pi$
$367$			27.9027i			1.45651i	−0.685306	+	0.728255i	$0.740332\pi$
$368$		0			0
$369$	3.75958	−	1.20254i	0.195716	−	0.0626015i
$370$		0			0
$371$			6.44201i			0.334453i
$372$		0			0
$373$			28.1209i			1.45604i	0.685554	+	0.728021i	$0.259560\pi$
$373$			28.1209i			1.45604i	−0.685554	+	0.728021i	$0.740440\pi$
$374$		0			0
$375$	1.39892	+	1.02128i	0.0722400	+	0.0527388i
$376$		0			0
$377$		−	10.2977i		−	0.530356i
$378$		0			0
$379$	−26.7110			−1.37205			−0.686027	−	0.727576i	$-0.740647\pi$
$379$	−26.7110			−1.37205			−0.686027	+	0.727576i	$0.740647\pi$
$380$		0			0
$381$	16.7642	−	22.9630i	0.858854	−	1.17643i
$382$		0			0
$383$	29.7598			1.52065			0.760327	−	0.649541i	$-0.225039\pi$
$383$	29.7598			1.52065			0.760327	+	0.649541i	$0.225039\pi$
$384$		0			0
$385$	2.60621			0.132825
$386$		0			0
$387$	2.17422	+	6.79743i	0.110522	+	0.345533i
$388$		0			0
$389$	−7.44233			−0.377341			−0.188670	−	0.982040i	$-0.560418\pi$
$389$	−7.44233			−0.377341			−0.188670	+	0.982040i	$0.560418\pi$
$390$		0			0
$391$		−	3.83540i		−	0.193964i
$392$		0			0
$393$	−13.4052	+	18.3621i	−0.676204	+	0.926244i
$394$		0			0
$395$		−	11.3895i		−	0.573067i
$396$		0			0
$397$			16.3521i			0.820687i	0.911931	+	0.410343i	$0.134591\pi$
$397$			16.3521i			0.820687i	−0.911931	+	0.410343i	$0.865409\pi$
$398$		0			0
$399$	−6.46351	+	8.85351i	−0.323580	+	0.443230i
$400$		0			0
$401$		−	7.07083i		−	0.353100i	−0.984292	−	0.176550i	$-0.943506\pi$
$401$		−	7.07083i		−	0.353100i	0.984292	−	0.176550i	$-0.0564938\pi$
$402$		0			0
$403$	16.4659			0.820227
$404$		0			0
$405$	7.32935	−	5.22309i	0.364198	−	0.259538i
$406$		0			0
$407$	−1.07007			−0.0530416
$408$		0			0
$409$	4.95747			0.245131			0.122566	−	0.992460i	$-0.460888\pi$
$409$	4.95747			0.245131			0.122566	+	0.992460i	$0.460888\pi$
$410$		0			0
$411$	−5.25382	+	7.19652i	−0.259152	+	0.354978i
$412$		0			0
$413$	10.3119			0.507416
$414$		0			0
$415$		−	15.8054i		−	0.775857i
$416$		0			0
$417$	−32.1909	−	23.5010i	−1.57639	−	1.15085i
$418$		0			0
$419$			21.0449i			1.02811i	0.857758	+	0.514054i	$0.171857\pi$
$419$			21.0449i			1.02811i	−0.857758	+	0.514054i	$0.828143\pi$
$420$		0			0
$421$			5.58224i			0.272062i	0.990705	+	0.136031i	$0.0434346\pi$
$421$			5.58224i			0.272062i	−0.990705	+	0.136031i	$0.956565\pi$
$422$		0			0
$423$	−11.7858	−	36.8468i	−0.573044	−	1.79155i
$424$		0			0
$425$		−	4.39342i		−	0.213112i
$426$		0			0
$427$	−11.4151			−0.552417
$428$		0			0
$429$	−13.2848	−	9.69856i	−0.641395	−	0.468251i
$430$		0			0
$431$	22.2746			1.07293			0.536465	−	0.843923i	$-0.319759\pi$
$431$	22.2746			1.07293			0.536465	+	0.843923i	$0.319759\pi$
$432$		0			0
$433$	27.3240			1.31311			0.656553	−	0.754280i	$-0.272014\pi$
$433$	27.3240			1.31311			0.656553	+	0.754280i	$0.272014\pi$
$434$		0			0
$435$	3.95349	+	2.88625i	0.189555	+	0.138385i
$436$		0			0
$437$	−5.52496			−0.264295
$438$		0			0
$439$		−	2.62754i		−	0.125406i	−0.998032	−	0.0627029i	$-0.980028\pi$
$439$		−	2.62754i		−	0.125406i	0.998032	−	0.0627029i	$-0.0199721\pi$
$440$		0			0
$441$	−0.913962	−	2.85739i	−0.0435220	−	0.136066i
$442$		0			0
$443$			27.5339i			1.30818i	0.756418	+	0.654088i	$0.226948\pi$
$443$			27.5339i			1.30818i	−0.756418	+	0.654088i	$0.773052\pi$
$444$		0			0
$445$			1.07306i			0.0508678i
$446$		0			0
$447$	−17.0224	−	12.4272i	−0.805131	−	0.587786i
$448$		0			0
$449$		−	30.5641i		−	1.44241i	−0.692722	−	0.721205i	$-0.743589\pi$
$449$		−	30.5641i		−	1.44241i	0.692722	−	0.721205i	$-0.256411\pi$
$450$		0			0
$451$	3.42910			0.161470
$452$		0			0
$453$	−0.569648	+	0.780286i	−0.0267644	+	0.0366610i
$454$		0			0
$455$	−3.64377			−0.170823
$456$		0			0
$457$	−30.6201			−1.43235			−0.716174	−	0.697922i	$-0.754108\pi$
$457$	−30.6201			−1.43235			−0.716174	+	0.697922i	$0.754108\pi$
$458$		0			0
$459$	−21.6626	−	7.20364i	−1.01112	−	0.336237i
$460$		0			0
$461$	3.21566			0.149768			0.0748841	−	0.997192i	$-0.476141\pi$
$461$	3.21566			0.149768			0.0748841	+	0.997192i	$0.476141\pi$
$462$		0			0
$463$		−	26.6753i		−	1.23971i	−0.784717	−	0.619854i	$-0.787192\pi$
$463$		−	26.6753i		−	1.23971i	0.784717	−	0.619854i	$-0.212808\pi$
$464$		0			0
$465$	−4.61510	+	6.32162i	−0.214020	+	0.293158i
$466$		0			0
$467$			22.6319i			1.04728i	0.851941	+	0.523639i	$0.175426\pi$
$467$			22.6319i			1.04728i	−0.851941	+	0.523639i	$0.824574\pi$
$468$		0			0
$469$			9.90769i			0.457495i
$470$		0			0
$471$	16.0572	−	21.9946i	0.739876	−	1.01346i
$472$		0			0
$473$			6.19991i			0.285072i
$474$		0			0
$475$	−6.32881			−0.290386
$476$		0			0
$477$	5.88776	+	18.4073i	0.269582	+	0.842814i
$478$		0			0
$479$	−12.8958			−0.589226			−0.294613	−	0.955617i	$-0.595191\pi$
$479$	−12.8958			−0.589226			−0.294613	+	0.955617i	$0.595191\pi$
$480$		0			0
$481$	1.49608			0.0682153
$482$		0			0
$483$	0.891565	−	1.22124i	0.0405676	−	0.0555682i
$484$		0			0
$485$	11.4590			0.520326
$486$		0			0
$487$		−	0.438571i		−	0.0198736i	−0.999951	−	0.00993678i	$-0.996837\pi$
$487$		−	0.438571i		−	0.0198736i	0.999951	−	0.00993678i	$-0.00316303\pi$
$488$		0			0
$489$	−22.0280	−	16.0815i	−0.996139	−	0.727232i
$490$		0			0
$491$			22.4073i			1.01123i	0.862760	+	0.505614i	$0.168734\pi$
$491$			22.4073i			1.01123i	−0.862760	+	0.505614i	$0.831266\pi$
$492$		0			0
$493$		−	12.4162i		−	0.559200i
$494$		0			0
$495$	7.44697	−	2.38198i	0.334716	−	0.107062i
$496$		0			0
$497$		−	2.67806i		−	0.120128i
$498$		0			0
$499$	−20.1466			−0.901884			−0.450942	−	0.892553i	$-0.648912\pi$
$499$	−20.1466			−0.901884			−0.450942	+	0.892553i	$0.648912\pi$
$500$		0			0
$501$	−33.6472	−	24.5641i	−1.50324	−	1.09744i
$502$		0			0
$503$	−16.2587			−0.724942			−0.362471	−	0.931995i	$-0.618067\pi$
$503$	−16.2587			−0.724942			−0.362471	+	0.931995i	$0.618067\pi$
$504$		0			0
$505$	8.86572			0.394520
$506$		0			0
$507$	0.387599	+	0.282967i	0.0172139	+	0.0125670i
$508$		0			0
$509$	5.87104			0.260229			0.130115	−	0.991499i	$-0.458465\pi$
$509$	5.87104			0.260229			0.130115	+	0.991499i	$0.458465\pi$
$510$		0			0
$511$			14.6519i			0.648160i
$512$		0			0
$513$	−10.3770	+	31.2053i	−0.458155	+	1.37775i
$514$		0			0
$515$		−	4.62254i		−	0.203693i
$516$		0			0
$517$		−	33.6078i		−	1.47807i
$518$		0			0
$519$	−15.1139	−	11.0339i	−0.663428	−	0.484335i
$520$		0			0
$521$			26.7975i			1.17402i	0.809579	+	0.587010i	$0.199695\pi$
$521$			26.7975i			1.17402i	−0.809579	+	0.587010i	$0.800305\pi$
$522$		0			0
$523$	−20.0527			−0.876844			−0.438422	−	0.898769i	$-0.644462\pi$
$523$	−20.0527			−0.876844			−0.438422	+	0.898769i	$0.644462\pi$
$524$		0			0
$525$	1.02128	−	1.39892i	0.0445724	−	0.0610539i
$526$		0			0
$527$	19.8536			0.864835
$528$		0			0
$529$	−22.2379			−0.966865
$530$		0			0
$531$	29.4652	−	9.42471i	1.27868	−	0.408997i
$532$		0			0
$533$	−4.79425			−0.207662
$534$		0			0
$535$		−	2.82698i		−	0.122221i
$536$		0			0
$537$	−5.50225	+	7.53682i	−0.237440	+	0.325238i
$538$		0			0
$539$		−	2.60621i		−	0.112258i
$540$		0			0
$541$		−	3.77748i		−	0.162406i	−0.996698	−	0.0812032i	$-0.974124\pi$
$541$		−	3.77748i		−	0.162406i	0.996698	−	0.0812032i	$-0.0258763\pi$
$542$		0			0
$543$	10.6166	−	14.5423i	0.455604	−	0.624072i
$544$		0			0
$545$		−	1.54341i		−	0.0661123i
$546$		0			0
$547$	33.2021			1.41962			0.709811	−	0.704393i	$-0.248781\pi$
$547$	33.2021			1.41962			0.709811	+	0.704393i	$0.248781\pi$
$548$		0			0
$549$	−32.6175	+	10.4330i	−1.39208	+	0.445270i
$550$		0			0
$551$	−17.8858			−0.761963
$552$		0			0
$553$	−11.3895			−0.484330
$554$		0			0
$555$	−0.419324	+	0.574377i	−0.0177993	+	0.0243809i
$556$		0			0
$557$	−19.9269			−0.844329			−0.422165	−	0.906519i	$-0.638730\pi$
$557$	−19.9269			−0.844329			−0.422165	+	0.906519i	$0.638730\pi$
$558$		0			0
$559$		−	8.66815i		−	0.366623i
$560$		0			0
$561$	−16.0179	−	11.6939i	−0.676278	−	0.493717i
$562$		0			0
$563$		−	22.0502i		−	0.929304i	−0.885493	−	0.464652i	$-0.846180\pi$
$563$		−	22.0502i		−	0.929304i	0.885493	−	0.464652i	$-0.153820\pi$
$564$		0			0
$565$			6.52668i			0.274579i
$566$		0			0
$567$	−5.22309	−	7.32935i	−0.219349	−	0.307804i
$568$		0			0
$569$			20.5477i			0.861406i	0.902494	+	0.430703i	$0.141734\pi$
$569$			20.5477i			0.861406i	−0.902494	+	0.430703i	$0.858266\pi$
$570$		0			0
$571$	5.11523			0.214066			0.107033	−	0.994255i	$-0.465865\pi$
$571$	5.11523			0.214066			0.107033	+	0.994255i	$0.465865\pi$
$572$		0			0
$573$	−31.6258	−	23.0884i	−1.32119	−	0.964532i
$574$		0			0
$575$	0.872985			0.0364060
$576$		0			0
$577$	42.2812			1.76019			0.880094	−	0.474800i	$-0.157480\pi$
$577$	42.2812			1.76019			0.880094	+	0.474800i	$0.157480\pi$
$578$		0			0
$579$	−23.9725	−	17.5011i	−0.996262	−	0.727321i
$580$		0			0
$581$	−15.8054			−0.655719
$582$		0			0
$583$			16.7893i			0.695340i
$584$		0			0
$585$	−10.4117	+	3.33027i	−0.430470	+	0.137690i
$586$		0			0
$587$		−	38.6250i		−	1.59422i	−0.603833	−	0.797111i	$-0.706360\pi$
$587$		−	38.6250i		−	1.59422i	0.603833	−	0.797111i	$-0.293640\pi$
$588$		0			0
$589$		−	28.5994i		−	1.17842i
$590$		0			0
$591$	−13.4808	−	9.84163i	−0.554524	−	0.404831i
$592$		0			0
$593$			15.3379i			0.629854i	0.949116	+	0.314927i	$0.101980\pi$
$593$			15.3379i			0.629854i	−0.949116	+	0.314927i	$0.898020\pi$
$594$		0			0
$595$	−4.39342			−0.180113
$596$		0			0
$597$	−6.15927	+	8.43677i	−0.252082	+	0.345294i
$598$		0			0
$599$	−29.8200			−1.21841			−0.609207	−	0.793011i	$-0.708512\pi$
$599$	−29.8200			−1.21841			−0.609207	+	0.793011i	$0.708512\pi$
$600$		0			0
$601$	45.2523			1.84588			0.922939	−	0.384946i	$-0.125780\pi$
$601$	45.2523			1.84588			0.922939	+	0.384946i	$0.125780\pi$
$602$		0			0
$603$	9.05525	+	28.3101i	0.368758	+	1.15288i
$604$		0			0
$605$	−4.20765			−0.171065
$606$		0			0
$607$		−	35.2729i		−	1.43168i	−0.698264	−	0.715841i	$-0.746044\pi$
$607$		−	35.2729i		−	1.43168i	0.698264	−	0.715841i	$-0.253956\pi$
$608$		0			0
$609$	2.88625	−	3.95349i	0.116957	−	0.160203i
$610$		0			0
$611$			46.9874i			1.90091i
$612$		0			0
$613$		−	15.4175i		−	0.622709i	−0.950294	−	0.311354i	$-0.899217\pi$
$613$		−	15.4175i		−	0.622709i	0.950294	−	0.311354i	$-0.100783\pi$
$614$		0			0
$615$	1.34374	−	1.84062i	0.0541849	−	0.0742208i
$616$		0			0
$617$			15.6194i			0.628812i	0.949289	+	0.314406i	$0.101805\pi$
$617$			15.6194i			0.628812i	−0.949289	+	0.314406i	$0.898195\pi$
$618$		0			0
$619$	−31.7001			−1.27413			−0.637067	−	0.770808i	$-0.719853\pi$
$619$	−31.7001			−1.27413			−0.637067	+	0.770808i	$0.719853\pi$
$620$		0			0
$621$	1.43138	−	4.30441i	0.0574394	−	0.172730i
$622$		0			0
$623$	1.07306			0.0429912
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−16.8453	+	23.0741i	−0.672736	+	0.921492i
$628$		0			0
$629$	1.80388			0.0719253
$630$		0			0
$631$			36.6634i			1.45955i	0.683688	+	0.729774i	$0.260375\pi$
$631$			36.6634i			1.45955i	−0.683688	+	0.729774i	$0.739625\pi$
$632$		0			0
$633$	13.1685	+	9.61365i	0.523400	+	0.382108i
$634$		0			0
$635$		−	16.4148i		−	0.651402i
$636$		0			0
$637$			3.64377i			0.144371i
$638$		0			0
$639$	−2.44765	−	7.65227i	−0.0968275	−	0.302719i
$640$		0			0
$641$		−	3.14434i		−	0.124194i	−0.998070	−	0.0620969i	$-0.980221\pi$
$641$		−	3.14434i		−	0.124194i	0.998070	−	0.0620969i	$-0.0197788\pi$
$642$		0			0
$643$	32.3300			1.27497			0.637486	−	0.770462i	$-0.279975\pi$
$643$	32.3300			1.27497			0.637486	+	0.770462i	$0.279975\pi$
$644$		0			0
$645$	3.32789	+	2.42952i	0.131035	+	0.0956624i
$646$		0			0
$647$	11.3287			0.445379			0.222689	−	0.974889i	$-0.428516\pi$
$647$	11.3287			0.445379			0.222689	+	0.974889i	$0.428516\pi$
$648$		0			0
$649$	26.8751			1.05494
$650$		0			0
$651$	6.32162	+	4.61510i	0.247764	+	0.180880i
$652$		0			0
$653$	29.7903			1.16579			0.582893	−	0.812549i	$-0.301921\pi$
$653$	29.7903			1.16579			0.582893	+	0.812549i	$0.301921\pi$
$654$		0			0
$655$			13.1259i			0.512870i
$656$		0			0
$657$	13.3913	+	41.8661i	0.522442	+	1.63335i
$658$		0			0
$659$			37.0098i			1.44170i	0.693092	+	0.720849i	$0.256248\pi$
$659$			37.0098i			1.44170i	−0.693092	+	0.720849i	$0.743752\pi$
$660$		0			0
$661$		−	6.31954i		−	0.245802i	−0.992419	−	0.122901i	$-0.960780\pi$
$661$		−	6.31954i		−	0.245802i	0.992419	−	0.122901i	$-0.0392197\pi$
$662$		0			0
$663$	22.3948	+	16.3493i	0.869743	+	0.634956i
$664$		0			0
$665$			6.32881i			0.245421i
$666$		0			0
$667$	2.46714			0.0955281
$668$		0			0
$669$	−2.85565	+	3.91158i	−0.110406	+	0.151230i
$670$		0			0
$671$	−29.7503			−1.14850
$672$		0			0
$673$	−37.4069			−1.44193			−0.720964	−	0.692972i	$-0.756301\pi$
$673$	−37.4069			−1.44193			−0.720964	+	0.692972i	$0.756301\pi$
$674$		0			0
$675$	1.63964	−	4.93068i	0.0631098	−	0.189782i
$676$		0			0
$677$	−24.4385			−0.939248			−0.469624	−	0.882866i	$-0.655611\pi$
$677$	−24.4385			−0.939248			−0.469624	+	0.882866i	$0.655611\pi$
$678$		0			0
$679$		−	11.4590i		−	0.439756i
$680$		0			0
$681$	−24.3857	+	33.4028i	−0.934464	+	1.28000i
$682$		0			0
$683$		−	35.3525i		−	1.35273i	−0.736568	−	0.676363i	$-0.763555\pi$
$683$		−	35.3525i		−	1.35273i	0.736568	−	0.676363i	$-0.236445\pi$
$684$		0			0
$685$			5.14434i			0.196555i
$686$		0			0
$687$	19.9014	−	27.2603i	0.759284	−	1.04004i
$688$		0			0
$689$		−	23.4732i		−	0.894259i
$690$		0			0
$691$	−10.8725			−0.413610			−0.206805	−	0.978382i	$-0.566306\pi$
$691$	−10.8725			−0.413610			−0.206805	+	0.978382i	$0.566306\pi$
$692$		0			0
$693$	−2.38198	−	7.44697i	−0.0904840	−	0.282887i
$694$		0			0
$695$	−23.0112			−0.872865
$696$		0			0
$697$	−5.78060			−0.218956
$698$		0			0
$699$	17.2026	−	23.5636i	0.650661	−	0.891255i
$700$		0			0
$701$	24.9216			0.941276			0.470638	−	0.882326i	$-0.344024\pi$
$701$	24.9216			0.941276			0.470638	+	0.882326i	$0.344024\pi$
$702$		0			0
$703$		−	2.59852i		−	0.0980050i
$704$		0			0
$705$	−18.0395	−	13.1697i	−0.679405	−	0.496000i
$706$		0			0
$707$		−	8.86572i		−	0.333430i
$708$		0			0
$709$			42.2314i			1.58603i	0.609200	+	0.793016i	$0.291490\pi$
$709$			42.2314i			1.58603i	−0.609200	+	0.793016i	$0.708510\pi$
$710$		0			0
$711$	−32.5442	+	10.4096i	−1.22050	+	0.390389i
$712$		0			0
$713$			3.94496i			0.147740i
$714$		0			0
$715$	−9.49645			−0.355147
$716$		0			0
$717$	−9.83838	−	7.18251i	−0.367421	−	0.268236i
$718$		0			0
$719$	−27.6757			−1.03213			−0.516064	−	0.856550i	$-0.672603\pi$
$719$	−27.6757			−1.03213			−0.516064	+	0.856550i	$0.672603\pi$
$720$		0			0
$721$	−4.62254			−0.172152
$722$		0			0
$723$	−6.92277	−	5.05397i	−0.257461	−	0.187959i
$724$		0			0
$725$	2.82610			0.104959
$726$		0			0
$727$		−	38.7201i		−	1.43605i	−0.696018	−	0.718024i	$-0.745047\pi$
$727$		−	38.7201i		−	1.43605i	0.696018	−	0.718024i	$-0.254953\pi$
$728$		0			0
$729$	−21.6232	−	16.1691i	−0.800858	−	0.598855i
$730$		0			0
$731$		−	10.4515i		−	0.386562i
$732$		0			0
$733$		−	30.8836i		−	1.14071i	−0.821397	−	0.570356i	$-0.806805\pi$
$733$		−	30.8836i		−	1.14071i	0.821397	−	0.570356i	$-0.193195\pi$
$734$		0			0
$735$	−1.39892	−	1.02128i	−0.0516000	−	0.0376706i
$736$		0			0
$737$			25.8215i			0.951149i
$738$		0			0
$739$	11.8412			0.435586			0.217793	−	0.975995i	$-0.430114\pi$
$739$	11.8412			0.435586			0.217793	+	0.975995i	$0.430114\pi$
$740$		0			0
$741$	23.5515	−	32.2602i	0.865188	−	1.18511i
$742$		0			0
$743$	−18.3374			−0.672734			−0.336367	−	0.941731i	$-0.609198\pi$
$743$	−18.3374			−0.672734			−0.336367	+	0.941731i	$0.609198\pi$
$744$		0			0
$745$	−12.1682			−0.445809
$746$		0			0
$747$	−45.1622	+	14.4456i	−1.65240	+	0.528535i
$748$		0			0
$749$	−2.82698			−0.103296
$750$		0			0
$751$		−	31.9649i		−	1.16642i	−0.812323	−	0.583208i	$-0.801797\pi$
$751$		−	31.9649i		−	1.16642i	0.812323	−	0.583208i	$-0.198203\pi$
$752$		0			0
$753$	3.05706	−	4.18746i	0.111405	−	0.152600i
$754$		0			0
$755$			0.557777i			0.0202996i
$756$		0			0
$757$		−	9.13341i		−	0.331960i	−0.986129	−	0.165980i	$-0.946921\pi$
$757$		−	9.13341i		−	0.331960i	0.986129	−	0.165980i	$-0.0530787\pi$
$758$		0			0
$759$	2.32361	−	3.18281i	0.0843416	−	0.115529i
$760$		0			0
$761$			51.4586i			1.86537i	0.360687	+	0.932687i	$0.382542\pi$
$761$			51.4586i			1.86537i	−0.360687	+	0.932687i	$0.617458\pi$
$762$		0			0
$763$	−1.54341			−0.0558751
$764$		0			0
$765$	−12.5537	+	4.01543i	−0.453881	+	0.145178i
$766$		0			0
$767$	−37.5743			−1.35673
$768$		0			0
$769$	−4.72688			−0.170456			−0.0852279	−	0.996361i	$-0.527162\pi$
$769$	−4.72688			−0.170456			−0.0852279	+	0.996361i	$0.527162\pi$
$770$		0			0
$771$	10.4814	−	14.3570i	0.377477	−	0.517056i
$772$		0			0
$773$	7.49571			0.269602			0.134801	−	0.990873i	$-0.456960\pi$
$773$	7.49571			0.269602			0.134801	+	0.990873i	$0.456960\pi$
$774$		0			0
$775$			4.51893i			0.162325i
$776$		0			0
$777$	0.574377	+	0.419324i	0.0206057	+	0.0150432i
$778$		0			0
$779$			8.32706i			0.298348i
$780$		0			0
$781$		−	6.97961i		−	0.249750i
$782$		0			0
$783$	4.63379	−	13.9346i	0.165598	−	0.497981i
$784$		0			0
$785$		−	15.7226i		−	0.561163i
$786$		0			0
$787$	15.6092			0.556408			0.278204	−	0.960522i	$-0.410261\pi$
$787$	15.6092			0.556408			0.278204	+	0.960522i	$0.410261\pi$
$788$		0			0
$789$	25.6603	+	18.7333i	0.913531	+	0.666923i
$790$		0			0
$791$	6.52668			0.232062
$792$		0			0
$793$	41.5941			1.47705
$794$		0			0
$795$	9.01187	+	6.57912i	0.319618	+	0.233337i
$796$		0			0
$797$	−31.1859			−1.10466			−0.552331	−	0.833625i	$-0.686262\pi$
$797$	−31.1859			−1.10466			−0.552331	+	0.833625i	$0.686262\pi$
$798$		0			0
$799$			56.6543i			2.00429i
$800$		0			0
$801$	3.06614	−	0.980735i	0.108337	−	0.0346526i
$802$		0			0
$803$			38.1859i			1.34755i
$804$		0			0
$805$		−	0.872985i		−	0.0307687i
$806$		0			0
$807$	34.8264	+	25.4251i	1.22595	+	0.895004i
$808$		0			0
$809$			7.84808i			0.275924i	0.990438	+	0.137962i	$0.0440551\pi$
$809$			7.84808i			0.275924i	−0.990438	+	0.137962i	$0.955945\pi$
$810$		0			0
$811$	−33.3680			−1.17171			−0.585855	−	0.810416i	$-0.699241\pi$
$811$	−33.3680			−1.17171			−0.585855	+	0.810416i	$0.699241\pi$
$812$		0			0
$813$	−5.80524	+	7.95184i	−0.203599	+	0.278883i
$814$		0			0
$815$	−15.7464			−0.551572
$816$		0			0
$817$	−15.0556			−0.526728
$818$		0			0
$819$	3.33027	+	10.4117i	0.116369	+	0.363813i
$820$		0			0
$821$	37.6886			1.31534			0.657670	−	0.753306i	$-0.271542\pi$
$821$	37.6886			1.31534			0.657670	+	0.753306i	$0.271542\pi$
$822$		0			0
$823$		−	16.6410i		−	0.580068i	−0.957016	−	0.290034i	$-0.906333\pi$
$823$		−	16.6410i		−	0.580068i	0.957016	−	0.290034i	$-0.0936667\pi$
$824$		0			0
$825$	2.66168	−	3.64589i	0.0926678	−	0.126934i
$826$		0			0
$827$		−	12.5710i		−	0.437136i	−0.975822	−	0.218568i	$-0.929861\pi$
$827$		−	12.5710i		−	0.437136i	0.975822	−	0.218568i	$-0.0701386\pi$
$828$		0			0
$829$		−	20.9681i		−	0.728251i	−0.931350	−	0.364125i	$-0.881368\pi$
$829$		−	20.9681i		−	0.728251i	0.931350	−	0.364125i	$-0.118632\pi$
$830$		0			0
$831$	−20.9268	+	28.6649i	−0.725942	+	0.994373i
$832$		0			0
$833$			4.39342i			0.152223i
$834$		0			0
$835$	−24.0522			−0.832362
$836$		0			0
$837$	22.2814	+	7.40942i	0.770157	+	0.256107i
$838$		0			0
$839$	−31.7318			−1.09550			−0.547752	−	0.836641i	$-0.684516\pi$
$839$	−31.7318			−1.09550			−0.547752	+	0.836641i	$0.684516\pi$
$840$		0			0
$841$	−21.0132			−0.724592
$842$		0			0
$843$	1.82461	−	2.49929i	0.0628428	−	0.0860801i
$844$		0			0
$845$	0.277070			0.00953149
$846$		0			0
$847$			4.20765i			0.144577i
$848$		0			0
$849$	−5.22601	−	3.81525i	−0.179356	−	0.130939i
$850$		0			0
$851$			0.358435i			0.0122870i
$852$		0			0
$853$		−	38.4646i		−	1.31700i	−0.752580	−	0.658501i	$-0.771191\pi$
$853$		−	38.4646i		−	1.31700i	0.752580	−	0.658501i	$-0.228809\pi$
$854$		0			0
$855$	5.78430	+	18.0839i	0.197819	+	0.618456i
$856$		0			0
$857$			43.6229i			1.49013i	0.666991	+	0.745066i	$0.267582\pi$
$857$			43.6229i			1.49013i	−0.666991	+	0.745066i	$0.732418\pi$
$858$		0			0
$859$	2.10572			0.0718463			0.0359232	−	0.999355i	$-0.488563\pi$
$859$	2.10572			0.0718463			0.0359232	+	0.999355i	$0.488563\pi$
$860$		0			0
$861$	−1.84062	−	1.34374i	−0.0627280	−	0.0457946i
$862$		0			0
$863$	−14.1968			−0.483263			−0.241632	−	0.970368i	$-0.577683\pi$
$863$	−14.1968			−0.483263			−0.241632	+	0.970368i	$0.577683\pi$
$864$		0			0
$865$	−10.8040			−0.367346
$866$		0			0
$867$	3.22057	+	2.35118i	0.109376	+	0.0798502i
$868$		0			0
$869$	−29.6834			−1.00694
$870$		0			0
$871$		−	36.1014i		−	1.22325i
$872$		0			0
$873$	−10.4731	−	32.7428i	−0.354461	−	1.10818i
$874$		0			0
$875$		−	1.00000i		−	0.0338062i
$876$		0			0
$877$		−	38.8947i		−	1.31338i	−0.754161	−	0.656690i	$-0.771956\pi$
$877$		−	38.8947i		−	1.31338i	0.754161	−	0.656690i	$-0.228044\pi$
$878$		0			0
$879$	−7.59261	−	5.54299i	−0.256093	−	0.186960i
$880$		0			0
$881$		−	37.6748i		−	1.26930i	−0.772801	−	0.634649i	$-0.781145\pi$
$881$		−	37.6748i		−	1.26930i	0.772801	−	0.634649i	$-0.218855\pi$
$882$		0			0
$883$	7.31822			0.246278			0.123139	−	0.992389i	$-0.460704\pi$
$883$	7.31822			0.246278			0.123139	+	0.992389i	$0.460704\pi$
$884$		0			0
$885$	10.5314	−	14.4256i	0.354009	−	0.484910i
$886$		0			0
$887$	8.75464			0.293952			0.146976	−	0.989140i	$-0.453046\pi$
$887$	8.75464			0.293952			0.146976	+	0.989140i	$0.453046\pi$
$888$		0			0
$889$	−16.4148			−0.550535
$890$		0			0
$891$	−13.6125	−	19.1018i	−0.456036	−	0.639936i
$892$		0			0
$893$	81.6117			2.73103
$894$		0			0
$895$			5.38759i			0.180087i
$896$		0			0
$897$	−3.24866	+	4.44991i	−0.108470	+	0.148578i
$898$		0			0
$899$			12.7709i			0.425934i
$900$		0			0
$901$		−	28.3025i		−	0.942893i
$902$		0			0
$903$	2.42952	−	3.32789i	0.0808495	−	0.110745i
$904$		0			0
$905$		−	10.3954i		−	0.345555i
$906$		0			0
$907$	50.4547			1.67532			0.837660	−	0.546191i	$-0.183923\pi$
$907$	50.4547			1.67532			0.837660	+	0.546191i	$0.183923\pi$
$908$		0			0
$909$	−8.10294	−	25.3328i	−0.268758	−	0.840237i
$910$		0			0
$911$	−26.7091			−0.884913			−0.442456	−	0.896790i	$-0.645893\pi$
$911$	−26.7091			−0.884913			−0.442456	+	0.896790i	$0.645893\pi$
$912$		0			0
$913$	−41.1923			−1.36327
$914$		0			0
$915$	−11.6581	+	15.9689i	−0.385404	+	0.527915i
$916$		0			0
$917$	13.1259			0.433454
$918$		0			0
$919$			18.4650i			0.609104i	0.952496	+	0.304552i	$0.0985067\pi$
$919$			18.4650i			0.609104i	−0.952496	+	0.304552i	$0.901493\pi$
$920$		0			0
$921$	−4.62335	−	3.37528i	−0.152344	−	0.111219i
$922$		0			0
$923$			9.75825i			0.321197i
$924$		0			0
$925$			0.410585i			0.0135000i
$926$		0			0
$927$	−13.2084	+	4.22483i	−0.433820	+	0.138761i
$928$		0			0
$929$			30.4117i			0.997777i	0.866666	+	0.498889i	$0.166258\pi$
$929$			30.4117i			0.997777i	−0.866666	+	0.498889i	$0.833742\pi$
$930$		0			0
$931$	6.32881			0.207418
$932$		0			0
$933$	−26.4827	−	19.3337i	−0.867006	−	0.632958i
$934$		0			0
$935$	−11.4502			−0.374462
$936$		0			0
$937$	27.8966			0.911343			0.455671	−	0.890148i	$-0.349399\pi$
$937$	27.8966			0.911343			0.455671	+	0.890148i	$0.349399\pi$
$938$		0			0
$939$	−27.1946	−	19.8534i	−0.887461	−	0.647891i
$940$		0			0
$941$	−25.9404			−0.845633			−0.422816	−	0.906215i	$-0.638959\pi$
$941$	−25.9404			−0.845633			−0.422816	+	0.906215i	$0.638959\pi$
$942$		0			0
$943$		−	1.14862i		−	0.0374042i
$944$		0			0
$945$	−4.93068	−	1.63964i	−0.160395	−	0.0533375i
$946$		0			0
$947$		−	22.9262i		−	0.745001i	−0.928032	−	0.372501i	$-0.878500\pi$
$947$		−	22.9262i		−	0.745001i	0.928032	−	0.372501i	$-0.121500\pi$
$948$		0			0
$949$		−	53.3880i		−	1.73305i
$950$		0			0
$951$	18.8104	+	13.7326i	0.609970	+	0.445309i
$952$		0			0
$953$			9.26963i			0.300273i	0.988665	+	0.150136i	$0.0479713\pi$
$953$			9.26963i			0.300273i	−0.988665	+	0.150136i	$0.952029\pi$
$954$		0			0
$955$	−22.6073			−0.731554
$956$		0			0
$957$	7.52217	−	10.3036i	0.243157	−	0.333069i
$958$		0			0
$959$	5.14434			0.166119
$960$		0			0
$961$	10.5793			0.341268
$962$		0			0
$963$	−8.07778	+	2.58375i	−0.260303	+	0.0832603i
$964$		0			0
$965$	−17.1364			−0.551640
$966$		0			0
$967$			37.2682i			1.19846i	0.800575	+	0.599232i	$0.204527\pi$
$967$			37.2682i			1.19846i	−0.800575	+	0.599232i	$0.795473\pi$
$968$		0			0
$969$	28.3969	−	38.8972i	0.912241	−	1.24956i
$970$		0			0
$971$		−	43.3878i		−	1.39238i	−0.717857	−	0.696191i	$-0.754877\pi$
$971$		−	43.3878i		−	1.39238i	0.717857	−	0.696191i	$-0.245123\pi$
$972$		0			0
$973$			23.0112i			0.737706i
$974$		0			0
$975$	−3.72132	+	5.09735i	−0.119178	+	0.163246i
$976$		0			0
$977$		−	48.3279i		−	1.54615i	−0.634318	−	0.773073i	$-0.718719\pi$
$977$		−	48.3279i		−	1.54615i	0.634318	−	0.773073i	$-0.281281\pi$
$978$		0			0
$979$	2.79662			0.0893803
$980$		0			0
$981$	−4.41012	+	1.41062i	−0.140804	+	0.0450375i
$982$		0			0
$983$	−6.14477			−0.195988			−0.0979938	−	0.995187i	$-0.531243\pi$
$983$	−6.14477			−0.195988			−0.0979938	+	0.995187i	$0.531243\pi$
$984$		0			0
$985$	−9.63654			−0.307046
$986$		0			0
$987$	−13.1697	+	18.0395i	−0.419196	+	0.574202i
$988$		0			0
$989$	2.07674			0.0660365
$990$		0			0
$991$			10.6251i			0.337518i	0.985657	+	0.168759i	$0.0539759\pi$
$991$			10.6251i			0.337518i	−0.985657	+	0.168759i	$0.946024\pi$
$992$		0			0
$993$	31.2766	+	22.8335i	0.992534	+	0.724600i
$994$		0			0
$995$			6.03091i			0.191193i
$996$		0			0
$997$			56.9151i			1.80252i	0.433280	+	0.901259i	$0.357356\pi$
$997$			56.9151i			1.80252i	−0.433280	+	0.901259i	$0.642644\pi$
$998$		0			0
$999$	2.02446	+	0.673213i	0.0640512	+	0.0212995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.e.c.911.11		44
3.2	odd	2		3360.2.e.d.911.12		44
4.3	odd	2		840.2.e.d.491.9	yes	44
8.3	odd	2		3360.2.e.d.911.11		44
8.5	even	2		840.2.e.c.491.35	✓	44
12.11	even	2		840.2.e.c.491.36	yes	44
24.5	odd	2		840.2.e.d.491.10	yes	44
24.11	even	2	inner	3360.2.e.c.911.12		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.e.c.491.35	✓	44	8.5	even	2
840.2.e.c.491.36	yes	44	12.11	even	2
840.2.e.d.491.9	yes	44	4.3	odd	2
840.2.e.d.491.10	yes	44	24.5	odd	2
3360.2.e.c.911.11		44	1.1	even	1	trivial
3360.2.e.c.911.12		44	24.11	even	2	inner
3360.2.e.d.911.11		44	8.3	odd	2
3360.2.e.d.911.12		44	3.2	odd	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.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.39892 - 1.02128i) q^{3} -1.00000 q^{5} +1.00000i q^{7} +(0.913962 + 2.85739i) q^{9} +2.60621i q^{11} -3.64377i q^{13} +(1.39892 + 1.02128i) q^{15} -4.39342i q^{17} -6.32881 q^{19} +(1.02128 - 1.39892i) q^{21} +0.872985 q^{23} +1.00000 q^{25} +(1.63964 - 4.93068i) q^{27} +2.82610 q^{29} +4.51893i q^{31} +(2.66168 - 3.64589i) q^{33} -1.00000i q^{35} +0.410585i q^{37} +(-3.72132 + 5.09735i) q^{39} -1.31574i q^{41} +2.37889 q^{43} +(-0.913962 - 2.85739i) q^{45} -12.8953 q^{47} -1.00000 q^{49} +(-4.48693 + 6.14606i) q^{51} +6.44201 q^{53} -2.60621i q^{55} +(8.85351 + 6.46351i) q^{57} -10.3119i q^{59} +11.4151i q^{61} +(-2.85739 + 0.913962i) q^{63} +3.64377i q^{65} +9.90769 q^{67} +(-1.22124 - 0.891565i) q^{69} -2.67806 q^{71} +14.6519 q^{73} +(-1.39892 - 1.02128i) q^{75} -2.60621 q^{77} +11.3895i q^{79} +(-7.32935 + 5.22309i) q^{81} +15.8054i q^{83} +4.39342i q^{85} +(-3.95349 - 2.88625i) q^{87} -1.07306i q^{89} +3.64377 q^{91} +(4.61510 - 6.32162i) q^{93} +6.32881 q^{95} -11.4590 q^{97} +(-7.44697 + 2.38198i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{3} - 44 q^{5} + 4 q^{9} + 4 q^{15} - 16 q^{19} - 16 q^{23} + 44 q^{25} - 4 q^{27} + 16 q^{29} + 24 q^{33} + 16 q^{39} - 24 q^{43} - 4 q^{45} + 24 q^{47} - 44 q^{49} - 32 q^{53} - 24 q^{67} - 8 q^{69}+ \cdots + 64 q^{99}+O(q^{100}) \) 44 * q - 4 * q^3 - 44 * q^5 + 4 * q^9 + 4 * q^15 - 16 * q^19 - 16 * q^23 + 44 * q^25 - 4 * q^27 + 16 * q^29 + 24 * q^33 + 16 * q^39 - 24 * q^43 - 4 * q^45 + 24 * q^47 - 44 * q^49 - 32 * q^53 - 24 * q^67 - 8 * q^69 - 8 * q^71 + 8 * q^73 - 4 * q^75 + 8 * q^77 + 20 * q^81 + 16 * q^87 + 16 * q^91 + 16 * q^93 + 16 * q^95 + 24 * q^97 + 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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