LMFDB - Embedded newform 1120.2.n.b.559.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1120,2,Mod(559,1120)] 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("1120.559"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.n (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.14
Character	$\chi$	$=$	1120.559
Dual form			1120.2.n.b.559.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.19038 q^{3} +(-2.22369 - 0.234978i) q^{5} +(2.11797 - 1.58562i) q^{7} -1.58299 q^{9} +3.65078 q^{11} +5.56044i q^{13} +(2.64704 + 0.279713i) q^{15} -0.808468 q^{17} -4.54845i q^{19} +(-2.52120 + 1.88750i) q^{21} -1.75488 q^{23} +(4.88957 + 1.04503i) q^{25} +5.45551 q^{27} -8.36272i q^{29} -4.73282 q^{31} -4.34582 q^{33} +(-5.08229 + 3.02825i) q^{35} -6.15399 q^{37} -6.61905i q^{39} -7.65085i q^{41} +2.66483i q^{43} +(3.52008 + 0.371968i) q^{45} -4.79391i q^{47} +(1.97161 - 6.71660i) q^{49} +0.962386 q^{51} -2.98963 q^{53} +(-8.11818 - 0.857851i) q^{55} +5.41439i q^{57} -8.92626i q^{59} -4.08673 q^{61} +(-3.35273 + 2.51002i) q^{63} +(1.30658 - 12.3647i) q^{65} -11.7723i q^{67} +2.08897 q^{69} -8.17275i q^{71} +15.1783 q^{73} +(-5.82046 - 1.24399i) q^{75} +(7.73224 - 5.78875i) q^{77} -2.49307i q^{79} -1.74517 q^{81} +5.75885 q^{83} +(1.79778 + 0.189972i) q^{85} +9.95483i q^{87} +4.95764i q^{89} +(8.81676 + 11.7769i) q^{91} +5.63387 q^{93} +(-1.06878 + 10.1143i) q^{95} -11.7195 q^{97} -5.77914 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 40 q^{9} + 16 q^{25} - 16 q^{35} + 8 q^{49} + 32 q^{51} - 24 q^{65} - 72 q^{81} + 128 q^{99}+O(q^{100}) $ 40 * q + 40 * q^9 + 16 * q^25 - 16 * q^35 + 8 * q^49 + 32 * q^51 - 24 * q^65 - 72 * q^81 + 128 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.19038			−0.687268			−0.343634	−	0.939104i	$-0.611658\pi$
$3$	−1.19038			−0.687268			−0.343634	+	0.939104i	$0.611658\pi$
$4$		0			0
$5$	−2.22369	−	0.234978i	−0.994463	−	0.105085i
$5$	−2.22369	−	0.234978i	−0.994463	−	0.105085i
$6$		0			0
$7$	2.11797	−	1.58562i	0.800518	−	0.599309i
$7$	2.11797	−	1.58562i	0.800518	−	0.599309i
$8$		0			0
$9$	−1.58299			−0.527663
$10$		0			0
$11$	3.65078			1.10075			0.550375	−	0.834917i	$-0.314485\pi$
$11$	3.65078			1.10075			0.550375	+	0.834917i	$0.314485\pi$
$12$		0			0
$13$			5.56044i			1.54219i	0.636721	+	0.771095i	$0.280290\pi$
$13$			5.56044i			1.54219i	−0.636721	+	0.771095i	$0.719710\pi$
$14$		0			0
$15$	2.64704	+	0.279713i	0.683462	+	0.0722217i
$16$		0			0
$17$	−0.808468			−0.196082			−0.0980411	−	0.995182i	$-0.531258\pi$
$17$	−0.808468			−0.196082			−0.0980411	+	0.995182i	$0.531258\pi$
$18$		0			0
$19$		−	4.54845i		−	1.04349i	−0.853103	−	0.521743i	$-0.825282\pi$
$19$		−	4.54845i		−	1.04349i	0.853103	−	0.521743i	$-0.174718\pi$
$20$		0			0
$21$	−2.52120	+	1.88750i	−0.550170	+	0.411885i
$22$		0			0
$23$	−1.75488			−0.365917			−0.182959	−	0.983121i	$-0.558567\pi$
$23$	−1.75488			−0.365917			−0.182959	+	0.983121i	$0.558567\pi$
$24$		0			0
$25$	4.88957	+	1.04503i	0.977914	+	0.209007i
$26$		0			0
$27$	5.45551			1.04991
$28$		0			0
$29$		−	8.36272i		−	1.55292i	−0.630168	−	0.776459i	$-0.717014\pi$
$29$		−	8.36272i		−	1.55292i	0.630168	−	0.776459i	$-0.282986\pi$
$30$		0			0
$31$	−4.73282			−0.850040			−0.425020	−	0.905184i	$-0.639733\pi$
$31$	−4.73282			−0.850040			−0.425020	+	0.905184i	$0.639733\pi$
$32$		0			0
$33$	−4.34582			−0.756510
$34$		0			0
$35$	−5.08229	+	3.02825i	−0.859064	+	0.511868i
$36$		0			0
$37$	−6.15399			−1.01171			−0.505855	−	0.862619i	$-0.668823\pi$
$37$	−6.15399			−1.01171			−0.505855	+	0.862619i	$0.668823\pi$
$38$		0			0
$39$		−	6.61905i		−	1.05990i
$40$		0			0
$41$		−	7.65085i		−	1.19486i	−0.801920	−	0.597431i	$-0.796188\pi$
$41$		−	7.65085i		−	1.19486i	0.801920	−	0.597431i	$-0.203812\pi$
$42$		0			0
$43$			2.66483i			0.406382i	0.979139	+	0.203191i	$0.0651312\pi$
$43$			2.66483i			0.406382i	−0.979139	+	0.203191i	$0.934869\pi$
$44$		0			0
$45$	3.52008	+	0.371968i	0.524742	+	0.0554497i
$46$		0			0
$47$		−	4.79391i		−	0.699263i	−0.936887	−	0.349632i	$-0.886307\pi$
$47$		−	4.79391i		−	0.699263i	0.936887	−	0.349632i	$-0.113693\pi$
$48$		0			0
$49$	1.97161	−	6.71660i	0.281658	−	0.959515i
$50$		0			0
$51$	0.962386			0.134761
$52$		0			0
$53$	−2.98963			−0.410658			−0.205329	−	0.978693i	$-0.565826\pi$
$53$	−2.98963			−0.410658			−0.205329	+	0.978693i	$0.565826\pi$
$54$		0			0
$55$	−8.11818	−	0.857851i	−1.09466	−	0.115673i
$56$		0			0
$57$			5.41439i			0.717154i
$58$		0			0
$59$		−	8.92626i		−	1.16210i	−0.813868	−	0.581050i	$-0.802642\pi$
$59$		−	8.92626i		−	1.16210i	0.813868	−	0.581050i	$-0.197358\pi$
$60$		0			0
$61$	−4.08673			−0.523253			−0.261626	−	0.965169i	$-0.584259\pi$
$61$	−4.08673			−0.523253			−0.261626	+	0.965169i	$0.584259\pi$
$62$		0			0
$63$	−3.35273	+	2.51002i	−0.422404	+	0.316233i
$64$		0			0
$65$	1.30658	−	12.3647i	0.162061	−	1.53365i
$66$		0			0
$67$		−	11.7723i		−	1.43821i	−0.694901	−	0.719105i	$-0.744552\pi$
$67$		−	11.7723i		−	1.43821i	0.694901	−	0.719105i	$-0.255448\pi$
$68$		0			0
$69$	2.08897			0.251483
$70$		0			0
$71$		−	8.17275i		−	0.969927i	−0.874534	−	0.484963i	$-0.838833\pi$
$71$		−	8.17275i		−	0.969927i	0.874534	−	0.484963i	$-0.161167\pi$
$72$		0			0
$73$	15.1783			1.77648			0.888241	−	0.459378i	$-0.151928\pi$
$73$	15.1783			1.77648			0.888241	+	0.459378i	$0.151928\pi$
$74$		0			0
$75$	−5.82046	−	1.24399i	−0.672089	−	0.143644i
$76$		0			0
$77$	7.73224	−	5.78875i	0.881170	−	0.659689i
$78$		0			0
$79$		−	2.49307i		−	0.280492i	−0.990117	−	0.140246i	$-0.955211\pi$
$79$		−	2.49307i		−	0.280492i	0.990117	−	0.140246i	$-0.0447893\pi$
$80$		0			0
$81$	−1.74517			−0.193908
$82$		0			0
$83$	5.75885			0.632116			0.316058	−	0.948740i	$-0.397641\pi$
$83$	5.75885			0.632116			0.316058	+	0.948740i	$0.397641\pi$
$84$		0			0
$85$	1.79778	+	0.189972i	0.194997	+	0.0206054i
$86$		0			0
$87$			9.95483i			1.06727i
$88$		0			0
$89$			4.95764i			0.525509i	0.964863	+	0.262755i	$0.0846310\pi$
$89$			4.95764i			0.525509i	−0.964863	+	0.262755i	$0.915369\pi$
$90$		0			0
$91$	8.81676	+	11.7769i	0.924247	+	1.23455i
$92$		0			0
$93$	5.63387			0.584205
$94$		0			0
$95$	−1.06878	+	10.1143i	−0.109655	+	1.03771i
$96$		0			0
$97$	−11.7195			−1.18993			−0.594967	−	0.803750i	$-0.702835\pi$
$97$	−11.7195			−1.18993			−0.594967	+	0.803750i	$0.702835\pi$
$98$		0			0
$99$	−5.77914			−0.580826
$100$		0			0
$101$	−7.10724			−0.707197			−0.353598	−	0.935397i	$-0.615042\pi$
$101$	−7.10724			−0.707197			−0.353598	+	0.935397i	$0.615042\pi$
$102$		0			0
$103$		−	5.06789i		−	0.499354i	−0.968329	−	0.249677i	$-0.919676\pi$
$103$		−	5.06789i		−	0.499354i	0.968329	−	0.249677i	$-0.0803244\pi$
$104$		0			0
$105$	6.04987	−	3.60478i	0.590407	−	0.351790i
$106$		0			0
$107$		−	10.8670i		−	1.05056i	−0.850931	−	0.525278i	$-0.823961\pi$
$107$		−	10.8670i		−	1.05056i	0.850931	−	0.525278i	$-0.176039\pi$
$108$		0			0
$109$			7.15158i			0.684997i	0.939518	+	0.342498i	$0.111273\pi$
$109$			7.15158i			0.684997i	−0.939518	+	0.342498i	$0.888727\pi$
$110$		0			0
$111$	7.32560			0.695315
$112$		0			0
$113$		−	0.0847428i		−	0.00797193i	−0.999992	−	0.00398597i	$-0.998731\pi$
$113$		−	0.0847428i		−	0.00797193i	0.999992	−	0.00398597i	$-0.00126878\pi$
$114$		0			0
$115$	3.90230	+	0.412357i	0.363891	+	0.0384525i
$116$		0			0
$117$		−	8.80212i		−	0.813757i
$118$		0			0
$119$	−1.71231	+	1.28192i	−0.156967	+	0.117514i
$120$		0			0
$121$	2.32816			0.211651
$122$		0			0
$123$			9.10743i			0.821190i
$124$		0			0
$125$	−10.6273	−	3.47277i	−0.950536	−	0.310614i
$126$		0			0
$127$	−5.93446			−0.526598			−0.263299	−	0.964714i	$-0.584811\pi$
$127$	−5.93446			−0.526598			−0.263299	+	0.964714i	$0.584811\pi$
$128$		0			0
$129$		−	3.17216i		−	0.279293i
$130$		0			0
$131$		−	17.2792i		−	1.50969i	−0.655905	−	0.754844i	$-0.727713\pi$
$131$		−	17.2792i		−	1.50969i	0.655905	−	0.754844i	$-0.272287\pi$
$132$		0			0
$133$	−7.21212	−	9.63349i	−0.625370	−	0.835329i
$134$		0			0
$135$	−12.1313	−	1.28192i	−1.04410	−	0.110330i
$136$		0			0
$137$			6.59924i			0.563811i	0.959442	+	0.281906i	$0.0909665\pi$
$137$			6.59924i			0.563811i	−0.959442	+	0.281906i	$0.909033\pi$
$138$		0			0
$139$			6.37544i			0.540757i	0.962754	+	0.270379i	$0.0871489\pi$
$139$			6.37544i			0.540757i	−0.962754	+	0.270379i	$0.912851\pi$
$140$		0			0
$141$			5.70658i			0.480581i
$142$		0			0
$143$			20.2999i			1.69756i
$144$		0			0
$145$	−1.96505	+	18.5961i	−0.163189	+	1.54432i
$146$		0			0
$147$	−2.34697	+	7.99533i	−0.193575	+	0.659443i
$148$		0			0
$149$		−	17.0131i		−	1.39377i	−0.717184	−	0.696883i	$-0.754569\pi$
$149$		−	17.0131i		−	1.39377i	0.717184	−	0.696883i	$-0.245431\pi$
$150$		0			0
$151$			14.7918i			1.20374i	0.798594	+	0.601870i	$0.205577\pi$
$151$			14.7918i			1.20374i	−0.798594	+	0.601870i	$0.794423\pi$
$152$		0			0
$153$	1.27980			0.103465
$154$		0			0
$155$	10.5243	+	1.11211i	0.845333	+	0.0893267i
$156$		0			0
$157$			16.6742i			1.33075i	0.746510	+	0.665374i	$0.231728\pi$
$157$			16.6742i			1.33075i	−0.746510	+	0.665374i	$0.768272\pi$
$158$		0			0
$159$	3.55881			0.282232
$160$		0			0
$161$	−3.71678	+	2.78257i	−0.292923	+	0.219297i
$162$		0			0
$163$			10.8670i			0.851172i	0.904918	+	0.425586i	$0.139932\pi$
$163$			10.8670i			0.851172i	−0.904918	+	0.425586i	$0.860068\pi$
$164$		0			0
$165$	9.66374	+	1.02117i	0.752321	+	0.0794981i
$166$		0			0
$167$			5.06789i			0.392165i	0.980587	+	0.196082i	$0.0628220\pi$
$167$			5.06789i			0.392165i	−0.980587	+	0.196082i	$0.937178\pi$
$168$		0			0
$169$	−17.9185			−1.37835
$170$		0			0
$171$			7.20015i			0.550609i
$172$		0			0
$173$		−	3.71164i		−	0.282191i	−0.989996	−	0.141095i	$-0.954938\pi$
$173$		−	3.71164i		−	0.282191i	0.989996	−	0.141095i	$-0.0450624\pi$
$174$		0			0
$175$	12.0130	−	5.53965i	0.908098	−	0.418759i
$176$		0			0
$177$			10.6257i			0.798674i
$178$		0			0
$179$	−7.26949			−0.543347			−0.271673	−	0.962390i	$-0.587577\pi$
$179$	−7.26949			−0.543347			−0.271673	+	0.962390i	$0.587577\pi$
$180$		0			0
$181$	18.2341			1.35533			0.677666	−	0.735370i	$-0.262992\pi$
$181$	18.2341			1.35533			0.677666	+	0.735370i	$0.262992\pi$
$182$		0			0
$183$	4.86477			0.359614
$184$		0			0
$185$	13.6845	+	1.44605i	1.00611	+	0.106316i
$186$		0			0
$187$	−2.95153			−0.215838
$188$		0			0
$189$	11.5546	−	8.65037i	0.840475	−	0.629222i
$190$		0			0
$191$			2.49307i			0.180392i	0.995924	+	0.0901960i	$0.0287494\pi$
$191$			2.49307i			0.180392i	−0.995924	+	0.0901960i	$0.971251\pi$
$192$		0			0
$193$		−	14.0190i		−	1.00911i	−0.863380	−	0.504554i	$-0.831657\pi$
$193$		−	14.0190i		−	1.00911i	0.863380	−	0.504554i	$-0.168343\pi$
$194$		0			0
$195$	−1.55533	+	14.7187i	−0.111380	+	1.05403i
$196$		0			0
$197$	8.27025			0.589231			0.294615	−	0.955616i	$-0.404808\pi$
$197$	8.27025			0.589231			0.294615	+	0.955616i	$0.404808\pi$
$198$		0			0
$199$	27.1530			1.92483			0.962413	−	0.271592i	$-0.0875500\pi$
$199$	27.1530			1.92483			0.962413	+	0.271592i	$0.0875500\pi$
$200$		0			0
$201$			14.0135i			0.988435i
$202$		0			0
$203$	−13.2601	−	17.7120i	−0.930677	−	1.24314i
$204$		0			0
$205$	−1.79778	+	17.0131i	−0.125562	+	1.18825i
$206$		0			0
$207$	2.77795			0.193081
$208$		0			0
$209$		−	16.6054i		−	1.14862i
$210$		0			0
$211$	−20.1183			−1.38500			−0.692501	−	0.721417i	$-0.743491\pi$
$211$	−20.1183			−1.38500			−0.692501	+	0.721417i	$0.743491\pi$
$212$		0			0
$213$			9.72869i			0.666599i
$214$		0			0
$215$	0.626175	−	5.92574i	0.0427048	−	0.404132i
$216$		0			0
$217$	−10.0240	+	7.50446i	−0.680472	+	0.509436i
$218$		0			0
$219$	−18.0679			−1.22092
$220$		0			0
$221$		−	4.49544i		−	0.302396i
$222$		0			0
$223$		−	15.7322i		−	1.05350i	−0.850019	−	0.526752i	$-0.823410\pi$
$223$		−	15.7322i		−	1.05350i	0.850019	−	0.526752i	$-0.176590\pi$
$224$		0			0
$225$	−7.74014	−	1.65428i	−0.516009	−	0.110285i
$226$		0			0
$227$	17.5252			1.16319			0.581595	−	0.813478i	$-0.302429\pi$
$227$	17.5252			1.16319			0.581595	+	0.813478i	$0.302429\pi$
$228$		0			0
$229$	16.5729			1.09517			0.547583	−	0.836751i	$-0.315548\pi$
$229$	16.5729			1.09517			0.547583	+	0.836751i	$0.315548\pi$
$230$		0			0
$231$	−9.20432	+	6.89082i	−0.605600	+	0.453383i
$232$		0			0
$233$			14.1037i			0.923964i	0.886889	+	0.461982i	$0.152862\pi$
$233$			14.1037i			0.923964i	−0.886889	+	0.461982i	$0.847138\pi$
$234$		0			0
$235$	−1.12646	+	10.6602i	−0.0734823	+	0.695392i
$236$		0			0
$237$			2.96770i			0.192773i
$238$		0			0
$239$		−	22.7217i		−	1.46975i	−0.678205	−	0.734873i	$-0.737242\pi$
$239$		−	22.7217i		−	1.46975i	0.678205	−	0.734873i	$-0.262758\pi$
$240$		0			0
$241$			15.0216i			0.967629i	0.875171	+	0.483814i	$0.160749\pi$
$241$			15.0216i			0.967629i	−0.875171	+	0.483814i	$0.839251\pi$
$242$		0			0
$243$	−14.2891			−0.916647
$244$		0			0
$245$	−5.96249	+	14.4723i	−0.380930	+	0.924604i
$246$		0			0
$247$	25.2914			1.60925
$248$		0			0
$249$	−6.85524			−0.434433
$250$		0			0
$251$			15.4522i			0.975332i	0.873030	+	0.487666i	$0.162152\pi$
$251$			15.4522i			0.975332i	−0.873030	+	0.487666i	$0.837848\pi$
$252$		0			0
$253$	−6.40666			−0.402783
$254$		0			0
$255$	−2.14005	−	0.226139i	−0.134015	−	0.0141614i
$256$		0			0
$257$	−19.9954			−1.24728			−0.623640	−	0.781712i	$-0.714347\pi$
$257$	−19.9954			−1.24728			−0.623640	+	0.781712i	$0.714347\pi$
$258$		0			0
$259$	−13.0340	+	9.75789i	−0.809891	+	0.606326i
$260$		0			0
$261$			13.2381i			0.819418i
$262$		0			0
$263$	−3.10990			−0.191764			−0.0958822	−	0.995393i	$-0.530567\pi$
$263$	−3.10990			−0.191764			−0.0958822	+	0.995393i	$0.530567\pi$
$264$		0			0
$265$	6.64801	+	0.702498i	0.408384	+	0.0431541i
$266$		0			0
$267$		−	5.90149i		−	0.361165i
$268$		0			0
$269$	11.2181			0.683980			0.341990	−	0.939704i	$-0.388899\pi$
$269$	11.2181			0.683980			0.341990	+	0.939704i	$0.388899\pi$
$270$		0			0
$271$	−6.60431			−0.401183			−0.200592	−	0.979675i	$-0.564286\pi$
$271$	−6.60431			−0.401183			−0.200592	+	0.979675i	$0.564286\pi$
$272$		0			0
$273$	−10.4953	−	14.0190i	−0.635205	−	0.848466i
$274$		0			0
$275$	17.8507	+	3.81519i	1.07644	+	0.230064i
$276$		0			0
$277$	−16.0704			−0.965577			−0.482789	−	0.875737i	$-0.660376\pi$
$277$	−16.0704			−0.965577			−0.482789	+	0.875737i	$0.660376\pi$
$278$		0			0
$279$	7.49201			0.448535
$280$		0			0
$281$	21.1540			1.26194			0.630972	−	0.775806i	$-0.282656\pi$
$281$	21.1540			1.26194			0.630972	+	0.775806i	$0.282656\pi$
$282$		0			0
$283$	24.7231			1.46964			0.734819	−	0.678263i	$-0.237267\pi$
$283$	24.7231			1.46964			0.734819	+	0.678263i	$0.237267\pi$
$284$		0			0
$285$	1.27226	−	12.0399i	0.0753623	−	0.713183i
$286$		0			0
$287$	−12.1313	−	16.2043i	−0.716091	−	0.956508i
$288$		0			0
$289$	−16.3464			−0.961552
$290$		0			0
$291$	13.9507			0.817803
$292$		0			0
$293$			11.4011i			0.666060i	0.942916	+	0.333030i	$0.108071\pi$
$293$			11.4011i			0.666060i	−0.942916	+	0.333030i	$0.891929\pi$
$294$		0			0
$295$	−2.09747	+	19.8492i	−0.122120	+	1.15567i
$296$		0			0
$297$	19.9168			1.15569
$298$		0			0
$299$		−	9.75789i		−	0.564314i
$300$		0			0
$301$	4.22540	+	5.64402i	0.243548	+	0.325316i
$302$		0			0
$303$	8.46033			0.486034
$304$		0			0
$305$	9.08762	+	0.960292i	0.520355	+	0.0549861i
$306$		0			0
$307$	0.426554			0.0243447			0.0121723	−	0.999926i	$-0.496125\pi$
$307$	0.426554			0.0243447			0.0121723	+	0.999926i	$0.496125\pi$
$308$		0			0
$309$			6.03272i			0.343190i
$310$		0			0
$311$	−0.586913			−0.0332808			−0.0166404	−	0.999862i	$-0.505297\pi$
$311$	−0.586913			−0.0332808			−0.0166404	+	0.999862i	$0.505297\pi$
$312$		0			0
$313$	6.45704			0.364973			0.182487	−	0.983208i	$-0.441585\pi$
$313$	6.45704			0.364973			0.182487	+	0.983208i	$0.441585\pi$
$314$		0			0
$315$	8.04522	−	4.79369i	0.453297	−	0.270094i
$316$		0			0
$317$	−10.3086			−0.578986			−0.289493	−	0.957180i	$-0.593487\pi$
$317$	−10.3086			−0.578986			−0.289493	+	0.957180i	$0.593487\pi$
$318$		0			0
$319$		−	30.5304i		−	1.70937i
$320$		0			0
$321$			12.9359i			0.722013i
$322$		0			0
$323$			3.67728i			0.204609i
$324$		0			0
$325$	−5.81085	+	27.1882i	−0.322328	+	1.50813i
$326$		0			0
$327$		−	8.51311i		−	0.470776i
$328$		0			0
$329$	−7.60132	−	10.1534i	−0.419074	−	0.559773i
$330$		0			0
$331$	−8.02828			−0.441274			−0.220637	−	0.975356i	$-0.570814\pi$
$331$	−8.02828			−0.441274			−0.220637	+	0.975356i	$0.570814\pi$
$332$		0			0
$333$	9.74170			0.533842
$334$		0			0
$335$	−2.76622	+	26.1778i	−0.151135	+	1.43025i
$336$		0			0
$337$		−	33.3339i		−	1.81582i	−0.419170	−	0.907908i	$-0.637679\pi$
$337$		−	33.3339i		−	1.81582i	0.419170	−	0.907908i	$-0.362321\pi$
$338$		0			0
$339$			0.100876i			0.00547885i
$340$		0			0
$341$	−17.2785			−0.935681
$342$		0			0
$343$	−6.47418	−	17.3518i	−0.349573	−	0.936909i
$344$		0			0
$345$	−4.64523	−	0.490863i	−0.250091	−	0.0264272i
$346$		0			0
$347$			16.7685i			0.900182i	0.892983	+	0.450091i	$0.148608\pi$
$347$			16.7685i			0.900182i	−0.892983	+	0.450091i	$0.851392\pi$
$348$		0			0
$349$	−0.234376			−0.0125458			−0.00627292	−	0.999980i	$-0.501997\pi$
$349$	−0.234376			−0.0125458			−0.00627292	+	0.999980i	$0.501997\pi$
$350$		0			0
$351$			30.3350i			1.61916i
$352$		0			0
$353$	−14.2235			−0.757039			−0.378519	−	0.925593i	$-0.623567\pi$
$353$	−14.2235			−0.757039			−0.378519	+	0.925593i	$0.623567\pi$
$354$		0			0
$355$	−1.92041	+	18.1736i	−0.101925	+	0.964556i
$356$		0			0
$357$	2.03831	−	1.52598i	0.107879	−	0.0807634i
$358$		0			0
$359$			14.2398i			0.751549i	0.926711	+	0.375774i	$0.122623\pi$
$359$			14.2398i			0.751549i	−0.926711	+	0.375774i	$0.877377\pi$
$360$		0			0
$361$	−1.68839			−0.0888626
$362$		0			0
$363$	−2.77140			−0.145461
$364$		0			0
$365$	−33.7517	−	3.56656i	−1.76665	−	0.186682i
$366$		0			0
$367$			5.09891i			0.266161i	0.991105	+	0.133080i	$0.0424868\pi$
$367$			5.09891i			0.266161i	−0.991105	+	0.133080i	$0.957513\pi$
$368$		0			0
$369$			12.1112i			0.630485i
$370$		0			0
$371$	−6.33196	+	4.74043i	−0.328739	+	0.246111i
$372$		0			0
$373$	25.1282			1.30109			0.650545	−	0.759468i	$-0.274541\pi$
$373$	25.1282			1.30109			0.650545	+	0.759468i	$0.274541\pi$
$374$		0			0
$375$	12.6506	+	4.13393i	0.653273	+	0.213475i
$376$		0			0
$377$	46.5004			2.39489
$378$		0			0
$379$	5.77914			0.296855			0.148427	−	0.988923i	$-0.452579\pi$
$379$	5.77914			0.296855			0.148427	+	0.988923i	$0.452579\pi$
$380$		0			0
$381$	7.06428			0.361914
$382$		0			0
$383$		−	24.8199i		−	1.26824i	−0.773237	−	0.634118i	$-0.781364\pi$
$383$		−	24.8199i		−	1.26824i	0.773237	−	0.634118i	$-0.218636\pi$
$384$		0			0
$385$	−18.5543	+	11.0555i	−0.945615	+	0.563438i
$386$		0			0
$387$		−	4.21839i		−	0.214433i
$388$		0			0
$389$			7.34694i			0.372505i	0.982502	+	0.186252i	$0.0596342\pi$
$389$			7.34694i			0.372505i	−0.982502	+	0.186252i	$0.940366\pi$
$390$		0			0
$391$	1.41876			0.0717499
$392$		0			0
$393$			20.5688i			1.03756i
$394$		0			0
$395$	−0.585816	+	5.54380i	−0.0294756	+	0.278939i
$396$		0			0
$397$		−	10.0197i		−	0.502872i	−0.967874	−	0.251436i	$-0.919097\pi$
$397$		−	10.0197i		−	0.502872i	0.967874	−	0.251436i	$-0.0809029\pi$
$398$		0			0
$399$	8.58518	+	11.4675i	0.429796	+	0.574095i
$400$		0			0
$401$	11.7525			0.586893			0.293447	−	0.955975i	$-0.405198\pi$
$401$	11.7525			0.586893			0.293447	+	0.955975i	$0.405198\pi$
$402$		0			0
$403$		−	26.3166i		−	1.31092i
$404$		0			0
$405$	3.88072	+	0.410077i	0.192834	+	0.0203769i
$406$		0			0
$407$	−22.4668			−1.11364
$408$		0			0
$409$			4.67758i			0.231292i	0.993291	+	0.115646i	$0.0368938\pi$
$409$			4.67758i			0.231292i	−0.993291	+	0.115646i	$0.963106\pi$
$410$		0			0
$411$		−	7.85562i		−	0.387489i
$412$		0			0
$413$	−14.1537	−	18.9056i	−0.696457	−	0.930282i
$414$		0			0
$415$	−12.8059	−	1.35320i	−0.628616	−	0.0664261i
$416$		0			0
$417$		−	7.58921i		−	0.371645i
$418$		0			0
$419$		−	21.7400i		−	1.06207i	−0.847351	−	0.531034i	$-0.821804\pi$
$419$		−	21.7400i		−	1.06207i	0.847351	−	0.531034i	$-0.178196\pi$
$420$		0			0
$421$		−	9.76923i		−	0.476123i	−0.971250	−	0.238062i	$-0.923488\pi$
$421$		−	9.76923i		−	0.476123i	0.971250	−	0.238062i	$-0.0765120\pi$
$422$		0			0
$423$			7.58871i			0.368976i
$424$		0			0
$425$	−3.95306	−	0.844877i	−0.191752	−	0.0409826i
$426$		0			0
$427$	−8.65559	+	6.48001i	−0.418873	+	0.313590i
$428$		0			0
$429$		−	24.1647i		−	1.16668i
$430$		0			0
$431$			2.12901i			0.102551i	0.998685	+	0.0512753i	$0.0163286\pi$
$431$			2.12901i			0.102551i	−0.998685	+	0.0512753i	$0.983671\pi$
$432$		0			0
$433$	−1.12265			−0.0539513			−0.0269756	−	0.999636i	$-0.508588\pi$
$433$	−1.12265			−0.0539513			−0.0269756	+	0.999636i	$0.508588\pi$
$434$		0			0
$435$	2.33917	−	22.1364i	0.112154	−	1.06136i
$436$		0			0
$437$			7.98197i			0.381829i
$438$		0			0
$439$	17.9973			0.858964			0.429482	−	0.903075i	$-0.358696\pi$
$439$	17.9973			0.858964			0.429482	+	0.903075i	$0.358696\pi$
$440$		0			0
$441$	−3.12104	+	10.6323i	−0.148621	+	0.506301i
$442$		0			0
$443$			10.1199i			0.480810i	0.970673	+	0.240405i	$0.0772803\pi$
$443$			10.1199i			0.480810i	−0.970673	+	0.240405i	$0.922720\pi$
$444$		0			0
$445$	1.16494	−	11.0243i	0.0552233	−	0.522600i
$446$		0			0
$447$			20.2521i			0.957891i
$448$		0			0
$449$	0.750627			0.0354243			0.0177121	−	0.999843i	$-0.494362\pi$
$449$	0.750627			0.0354243			0.0177121	+	0.999843i	$0.494362\pi$
$450$		0			0
$451$		−	27.9315i		−	1.31524i
$452$		0			0
$453$		−	17.6079i		−	0.827291i
$454$		0			0
$455$	−16.8384	−	28.2598i	−0.789397	−	1.32484i
$456$		0			0
$457$			20.4998i			0.958941i	0.877558	+	0.479471i	$0.159171\pi$
$457$			20.4998i			0.958941i	−0.877558	+	0.479471i	$0.840829\pi$
$458$		0			0
$459$	−4.41060			−0.205869
$460$		0			0
$461$	−14.9599			−0.696752			−0.348376	−	0.937355i	$-0.613267\pi$
$461$	−14.9599			−0.696752			−0.348376	+	0.937355i	$0.613267\pi$
$462$		0			0
$463$	23.1072			1.07388			0.536941	−	0.843619i	$-0.319580\pi$
$463$	23.1072			1.07388			0.536941	+	0.843619i	$0.319580\pi$
$464$		0			0
$465$	−12.5280	−	1.32383i	−0.580970	−	0.0613913i
$466$		0			0
$467$	−22.3760			−1.03544			−0.517720	−	0.855550i	$-0.673219\pi$
$467$	−22.3760			−1.03544			−0.517720	+	0.855550i	$0.673219\pi$
$468$		0			0
$469$	−18.6663	−	24.9333i	−0.861932	−	1.15131i
$470$		0			0
$471$		−	19.8487i		−	0.914580i
$472$		0			0
$473$			9.72868i			0.447325i
$474$		0			0
$475$	4.75329	−	22.2400i	0.218096	−	1.02044i
$476$		0			0
$477$	4.73256			0.216689
$478$		0			0
$479$	−32.6071			−1.48986			−0.744928	−	0.667145i	$-0.767516\pi$
$479$	−32.6071			−1.48986			−0.744928	+	0.667145i	$0.767516\pi$
$480$		0			0
$481$		−	34.2189i		−	1.56025i
$482$		0			0
$483$	4.42439	−	3.31232i	0.201317	−	0.150716i
$484$		0			0
$485$	26.0605	+	2.75382i	1.18335	+	0.125045i
$486$		0			0
$487$	−3.90400			−0.176907			−0.0884536	−	0.996080i	$-0.528192\pi$
$487$	−3.90400			−0.176907			−0.0884536	+	0.996080i	$0.528192\pi$
$488$		0			0
$489$		−	12.9359i		−	0.584983i
$490$		0			0
$491$	20.5527			0.927532			0.463766	−	0.885958i	$-0.346498\pi$
$491$	20.5527			0.927532			0.463766	+	0.885958i	$0.346498\pi$
$492$		0			0
$493$			6.76099i			0.304500i
$494$		0			0
$495$	12.8510	+	1.35797i	0.577610	+	0.0610362i
$496$		0			0
$497$	−12.9589	−	17.3096i	−0.581285	−	0.776444i
$498$		0			0
$499$	25.8654			1.15789			0.578947	−	0.815365i	$-0.303464\pi$
$499$	25.8654			1.15789			0.578947	+	0.815365i	$0.303464\pi$
$500$		0			0
$501$		−	6.03272i		−	0.269522i
$502$		0			0
$503$			31.0150i			1.38289i	0.722428	+	0.691446i	$0.243026\pi$
$503$			31.0150i			1.38289i	−0.722428	+	0.691446i	$0.756974\pi$
$504$		0			0
$505$	15.8043	+	1.67004i	0.703281	+	0.0743160i
$506$		0			0
$507$	21.3299			0.947293
$508$		0			0
$509$	−14.2066			−0.629695			−0.314847	−	0.949142i	$-0.601953\pi$
$509$	−14.2066			−0.629695			−0.314847	+	0.949142i	$0.601953\pi$
$510$		0			0
$511$	32.1471	−	24.0670i	1.42211	−	1.06466i
$512$		0			0
$513$		−	24.8141i		−	1.09557i
$514$		0			0
$515$	−1.19084	+	11.2694i	−0.0524747	+	0.496589i
$516$		0			0
$517$		−	17.5015i		−	0.769714i
$518$		0			0
$519$			4.41827i			0.193940i
$520$		0			0
$521$		−	37.9559i		−	1.66288i	−0.555617	−	0.831438i	$-0.687518\pi$
$521$		−	37.9559i		−	1.66288i	0.555617	−	0.831438i	$-0.312482\pi$
$522$		0			0
$523$	−20.8846			−0.913219			−0.456609	−	0.889667i	$-0.650936\pi$
$523$	−20.8846			−0.913219			−0.456609	+	0.889667i	$0.650936\pi$
$524$		0			0
$525$	−14.3001	+	6.59431i	−0.624106	+	0.287799i
$526$		0			0
$527$	3.82633			0.166678
$528$		0			0
$529$	−19.9204			−0.866105
$530$		0			0
$531$			14.1302i			0.613198i
$532$		0			0
$533$	42.5421			1.84270
$534$		0			0
$535$	−2.55351	+	24.1649i	−0.110398	+	1.04474i
$536$		0			0
$537$	8.65347			0.373425
$538$		0			0
$539$	7.19790	−	24.5208i	0.310036	−	1.05619i
$540$		0			0
$541$		−	1.19268i		−	0.0512773i	−0.999671	−	0.0256387i	$-0.991838\pi$
$541$		−	1.19268i		−	0.0512773i	0.999671	−	0.0256387i	$-0.00816194\pi$
$542$		0			0
$543$	−21.7056			−0.931476
$544$		0			0
$545$	1.68046	−	15.9029i	0.0719831	−	0.681204i
$546$		0			0
$547$			15.5500i			0.664872i	0.943126	+	0.332436i	$0.107871\pi$
$547$			15.5500i			0.664872i	−0.943126	+	0.332436i	$0.892129\pi$
$548$		0			0
$549$	6.46926			0.276101
$550$		0			0
$551$	−38.0374			−1.62045
$552$		0			0
$553$	−3.95306	−	5.28025i	−0.168101	−	0.224539i
$554$		0			0
$555$	−16.2898	−	1.72135i	−0.691465	−	0.0730674i
$556$		0			0
$557$	31.9739			1.35478			0.677390	−	0.735624i	$-0.263111\pi$
$557$	31.9739			1.35478			0.677390	+	0.735624i	$0.263111\pi$
$558$		0			0
$559$	−14.8176			−0.626718
$560$		0			0
$561$	3.51345			0.148338
$562$		0			0
$563$	6.48451			0.273290			0.136645	−	0.990620i	$-0.456368\pi$
$563$	6.48451			0.273290			0.136645	+	0.990620i	$0.456368\pi$
$564$		0			0
$565$	−0.0199127	+	0.188441i	−0.000837733	+	0.00792779i
$566$		0			0
$567$	−3.69622	+	2.76718i	−0.155227	+	0.116211i
$568$		0			0
$569$	36.4610			1.52853			0.764263	−	0.644905i	$-0.223103\pi$
$569$	36.4610			1.52853			0.764263	+	0.644905i	$0.223103\pi$
$570$		0			0
$571$	−24.9527			−1.04424			−0.522119	−	0.852872i	$-0.674858\pi$
$571$	−24.9527			−1.04424			−0.522119	+	0.852872i	$0.674858\pi$
$572$		0			0
$573$		−	2.96770i		−	0.123978i
$574$		0			0
$575$	−8.58060	−	1.83391i	−0.357836	−	0.0764792i
$576$		0			0
$577$	21.7435			0.905192			0.452596	−	0.891716i	$-0.350498\pi$
$577$	21.7435			0.905192			0.452596	+	0.891716i	$0.350498\pi$
$578$		0			0
$579$			16.6879i			0.693526i
$580$		0			0
$581$	12.1971	−	9.13136i	0.506020	−	0.378833i
$582$		0			0
$583$	−10.9145			−0.452032
$584$		0			0
$585$	−2.06830	+	19.5732i	−0.0855139	+	0.809251i
$586$		0			0
$587$	−24.7885			−1.02313			−0.511565	−	0.859245i	$-0.670934\pi$
$587$	−24.7885			−1.02313			−0.511565	+	0.859245i	$0.670934\pi$
$588$		0			0
$589$			21.5270i			0.887004i
$590$		0			0
$591$	−9.84476			−0.404959
$592$		0			0
$593$	−41.5751			−1.70728			−0.853642	−	0.520860i	$-0.825611\pi$
$593$	−41.5751			−1.70728			−0.853642	+	0.520860i	$0.825611\pi$
$594$		0			0
$595$	4.10887	−	2.44824i	0.168447	−	0.100368i
$596$		0			0
$597$	−32.3224			−1.32287
$598$		0			0
$599$		−	38.2863i		−	1.56434i	−0.623068	−	0.782168i	$-0.714114\pi$
$599$		−	38.2863i		−	1.56434i	0.623068	−	0.782168i	$-0.285886\pi$
$600$		0			0
$601$			22.8512i			0.932121i	0.884753	+	0.466060i	$0.154327\pi$
$601$			22.8512i			0.932121i	−0.884753	+	0.466060i	$0.845673\pi$
$602$		0			0
$603$			18.6354i			0.758891i
$604$		0			0
$605$	−5.17710	−	0.547067i	−0.210479	−	0.0222414i
$606$		0			0
$607$			3.04718i			0.123681i	0.998086	+	0.0618407i	$0.0196971\pi$
$607$			3.04718i			0.123681i	−0.998086	+	0.0618407i	$0.980303\pi$
$608$		0			0
$609$	15.7846	+	21.0841i	0.639624	+	0.854369i
$610$		0			0
$611$	26.6562			1.07840
$612$		0			0
$613$	−36.3023			−1.46624			−0.733118	−	0.680101i	$-0.761936\pi$
$613$	−36.3023			−1.46624			−0.733118	+	0.680101i	$0.761936\pi$
$614$		0			0
$615$	2.14005	−	20.2521i	0.0862950	−	0.816643i
$616$		0			0
$617$		−	15.0501i		−	0.605894i	−0.953007	−	0.302947i	$-0.902029\pi$
$617$		−	15.0501i		−	0.605894i	0.953007	−	0.302947i	$-0.0979705\pi$
$618$		0			0
$619$		−	1.73303i		−	0.0696563i	−0.999393	−	0.0348281i	$-0.988912\pi$
$619$		−	1.73303i		−	0.0696563i	0.999393	−	0.0348281i	$-0.0110884\pi$
$620$		0			0
$621$	−9.57375			−0.384181
$622$		0			0
$623$	7.86095	+	10.5002i	0.314942	+	0.420680i
$624$		0			0
$625$	22.8158	+	10.2195i	0.912632	+	0.408782i
$626$		0			0
$627$			19.7667i			0.789407i
$628$		0			0
$629$	4.97530			0.198378
$630$		0			0
$631$		−	19.5984i		−	0.780202i	−0.920772	−	0.390101i	$-0.872440\pi$
$631$		−	19.5984i		−	0.780202i	0.920772	−	0.390101i	$-0.127560\pi$
$632$		0			0
$633$	23.9485			0.951866
$634$		0			0
$635$	13.1964	+	1.39447i	0.523682	+	0.0553377i
$636$		0			0
$637$	37.3473	+	10.9630i	1.47975	+	0.434371i
$638$		0			0
$639$			12.9374i			0.511795i
$640$		0			0
$641$	5.76448			0.227683			0.113842	−	0.993499i	$-0.463684\pi$
$641$	5.76448			0.227683			0.113842	+	0.993499i	$0.463684\pi$
$642$		0			0
$643$	−12.8313			−0.506017			−0.253009	−	0.967464i	$-0.581420\pi$
$643$	−12.8313			−0.506017			−0.253009	+	0.967464i	$0.581420\pi$
$644$		0			0
$645$	−0.745388	+	7.05389i	−0.0293496	+	0.277747i
$646$		0			0
$647$			1.42452i			0.0560037i	0.999608	+	0.0280018i	$0.00891443\pi$
$647$			1.42452i			0.0560037i	−0.999608	+	0.0280018i	$0.991086\pi$
$648$		0			0
$649$		−	32.5878i		−	1.27918i
$650$		0			0
$651$	11.9324	−	8.93318i	0.467666	−	0.350119i
$652$		0			0
$653$	−29.1628			−1.14123			−0.570615	−	0.821218i	$-0.693295\pi$
$653$	−29.1628			−1.14123			−0.570615	+	0.821218i	$0.693295\pi$
$654$		0			0
$655$	−4.06022	+	38.4234i	−0.158646	+	1.50133i
$656$		0			0
$657$	−24.0270			−0.937384
$658$		0			0
$659$	7.06968			0.275396			0.137698	−	0.990474i	$-0.456030\pi$
$659$	7.06968			0.275396			0.137698	+	0.990474i	$0.456030\pi$
$660$		0			0
$661$	3.04463			0.118423			0.0592113	−	0.998245i	$-0.481141\pi$
$661$	3.04463			0.118423			0.0592113	+	0.998245i	$0.481141\pi$
$662$		0			0
$663$			5.35129i			0.207827i
$664$		0			0
$665$	13.7738	+	23.1166i	0.534127	+	0.896421i
$666$		0			0
$667$			14.6755i			0.568239i
$668$		0			0
$669$			18.7273i			0.724038i
$670$		0			0
$671$	−14.9197			−0.575970
$672$		0			0
$673$			45.1287i			1.73958i	0.493420	+	0.869791i	$0.335747\pi$
$673$			45.1287i			1.73958i	−0.493420	+	0.869791i	$0.664253\pi$
$674$		0			0
$675$	26.6751	+	5.70120i	1.02673	+	0.219439i
$676$		0			0
$677$		−	3.41915i		−	0.131408i	−0.997839	−	0.0657042i	$-0.979071\pi$
$677$		−	3.41915i		−	0.131408i	0.997839	−	0.0657042i	$-0.0209294\pi$
$678$		0			0
$679$	−24.8215	+	18.5827i	−0.952563	+	0.713138i
$680$		0			0
$681$	−20.8617			−0.799423
$682$		0			0
$683$		−	52.1925i		−	1.99709i	−0.0539068	−	0.998546i	$-0.517167\pi$
$683$		−	52.1925i		−	1.99709i	0.0539068	−	0.998546i	$-0.482833\pi$
$684$		0			0
$685$	1.55068	−	14.6747i	0.0592483	−	0.560690i
$686$		0			0
$687$	−19.7281			−0.752673
$688$		0			0
$689$		−	16.6237i		−	0.633312i
$690$		0			0
$691$			22.1744i			0.843554i	0.906700	+	0.421777i	$0.138594\pi$
$691$			22.1744i			0.843554i	−0.906700	+	0.421777i	$0.861406\pi$
$692$		0			0
$693$	−12.2401	+	9.16353i	−0.464961	+	0.348094i
$694$		0			0
$695$	1.49809	−	14.1770i	0.0568256	−	0.537763i
$696$		0			0
$697$			6.18546i			0.234291i
$698$		0			0
$699$		−	16.7888i		−	0.635011i
$700$		0			0
$701$			17.3192i			0.654138i	0.945001	+	0.327069i	$0.106061\pi$
$701$			17.3192i			0.654138i	−0.945001	+	0.327069i	$0.893939\pi$
$702$		0			0
$703$			27.9911i			1.05570i
$704$		0			0
$705$	1.34092	−	12.6897i	0.0505020	−	0.477920i
$706$		0			0
$707$	−15.0529	+	11.2694i	−0.566124	+	0.423829i
$708$		0			0
$709$		−	3.57195i		−	0.134147i	−0.997748	−	0.0670737i	$-0.978634\pi$
$709$		−	3.57195i		−	0.134147i	0.997748	−	0.0670737i	$-0.0213663\pi$
$710$		0			0
$711$			3.94650i			0.148005i
$712$		0			0
$713$	8.30552			0.311044
$714$		0			0
$715$	4.77003	−	45.1407i	0.178389	−	1.68817i
$716$		0			0
$717$			27.0475i			1.01011i
$718$		0			0
$719$	−1.20606			−0.0449783			−0.0224891	−	0.999747i	$-0.507159\pi$
$719$	−1.20606			−0.0449783			−0.0224891	+	0.999747i	$0.507159\pi$
$720$		0			0
$721$	−8.03575	−	10.7336i	−0.299267	−	0.399742i
$722$		0			0
$723$		−	17.8815i		−	0.665020i
$724$		0			0
$725$	8.73933	−	40.8901i	0.324571	−	1.51862i
$726$		0			0
$727$		−	28.9323i		−	1.07304i	−0.843888	−	0.536520i	$-0.819739\pi$
$727$		−	28.9323i		−	1.07304i	0.843888	−	0.536520i	$-0.180261\pi$
$728$		0			0
$729$	22.2450			0.823890
$730$		0			0
$731$		−	2.15443i		−	0.0796843i
$732$		0			0
$733$		−	11.5527i		−	0.426710i	−0.976975	−	0.213355i	$-0.931561\pi$
$733$		−	11.5527i		−	0.426710i	0.976975	−	0.213355i	$-0.0684391\pi$
$734$		0			0
$735$	7.09765	−	17.2276i	0.261801	−	0.635450i
$736$		0			0
$737$		−	42.9779i		−	1.58311i
$738$		0			0
$739$	11.3868			0.418869			0.209434	−	0.977823i	$-0.432838\pi$
$739$	11.3868			0.418869			0.209434	+	0.977823i	$0.432838\pi$
$740$		0			0
$741$	−30.1064			−1.10599
$742$		0			0
$743$	48.8376			1.79168			0.895840	−	0.444377i	$-0.146575\pi$
$743$	48.8376			1.79168			0.895840	+	0.444377i	$0.146575\pi$
$744$		0			0
$745$	−3.99770	+	37.8318i	−0.146464	+	1.38605i
$746$		0			0
$747$	−9.11621			−0.333545
$748$		0			0
$749$	−17.2310	−	23.0161i	−0.629607	−	0.840989i
$750$		0			0
$751$			0.434842i			0.0158676i	0.999969	+	0.00793381i	$0.00252544\pi$
$751$			0.434842i			0.0158676i	−0.999969	+	0.00793381i	$0.997475\pi$
$752$		0			0
$753$		−	18.3940i		−	0.670314i
$754$		0			0
$755$	3.47575	−	32.8923i	0.126495	−	1.19707i
$756$		0			0
$757$	−13.5346			−0.491923			−0.245962	−	0.969280i	$-0.579104\pi$
$757$	−13.5346			−0.491923			−0.245962	+	0.969280i	$0.579104\pi$
$758$		0			0
$759$	7.62638			0.276820
$760$		0			0
$761$		−	28.4347i		−	1.03076i	−0.856963	−	0.515378i	$-0.827651\pi$
$761$		−	28.4347i		−	1.03076i	0.856963	−	0.515378i	$-0.172349\pi$
$762$		0			0
$763$	11.3397	+	15.1468i	0.410524	+	0.548352i
$764$		0			0
$765$	−2.84587	−	0.300724i	−0.102893	−	0.0108727i
$766$		0			0
$767$	49.6339			1.79218
$768$		0			0
$769$			37.6894i			1.35912i	0.733622	+	0.679558i	$0.237828\pi$
$769$			37.6894i			1.35912i	−0.733622	+	0.679558i	$0.762172\pi$
$770$		0			0
$771$	23.8022			0.857215
$772$		0			0
$773$			8.24274i			0.296471i	0.988952	+	0.148235i	$0.0473593\pi$
$773$			8.24274i			0.296471i	−0.988952	+	0.148235i	$0.952641\pi$
$774$		0			0
$775$	−23.1415	−	4.94596i	−0.831266	−	0.177664i
$776$		0			0
$777$	15.5154	−	11.6156i	0.556612	−	0.416708i
$778$		0			0
$779$	−34.7995			−1.24682
$780$		0			0
$781$		−	29.8369i		−	1.06765i
$782$		0			0
$783$		−	45.6229i		−	1.63043i
$784$		0			0
$785$	3.91808	−	37.0783i	0.139842	−	1.32338i
$786$		0			0
$787$	−22.9785			−0.819095			−0.409547	−	0.912289i	$-0.634313\pi$
$787$	−22.9785			−0.819095			−0.409547	+	0.912289i	$0.634313\pi$
$788$		0			0
$789$	3.70197			0.131793
$790$		0			0
$791$	−0.134370	−	0.179483i	−0.00477765	−	0.00638167i
$792$		0			0
$793$		−	22.7240i		−	0.806954i
$794$		0			0
$795$	−7.91367	−	0.836241i	−0.280669	−	0.0296584i
$796$		0			0
$797$			33.7838i			1.19668i	0.801241	+	0.598341i	$0.204173\pi$
$797$			33.7838i			1.19668i	−0.801241	+	0.598341i	$0.795827\pi$
$798$		0			0
$799$			3.87572i			0.137113i
$800$		0			0
$801$		−	7.84790i		−	0.277292i
$802$		0			0
$803$	55.4124			1.95546
$804$		0			0
$805$	8.91880	−	5.31421i	0.314346	−	0.187301i
$806$		0			0
$807$	−13.3538			−0.470077
$808$		0			0
$809$	20.4803			0.720049			0.360024	−	0.932943i	$-0.382768\pi$
$809$	20.4803			0.720049			0.360024	+	0.932943i	$0.382768\pi$
$810$		0			0
$811$		−	12.3156i		−	0.432460i	−0.976342	−	0.216230i	$-0.930624\pi$
$811$		−	12.3156i		−	0.432460i	0.976342	−	0.216230i	$-0.0693762\pi$
$812$		0			0
$813$	7.86166			0.275720
$814$		0			0
$815$	2.55351	−	24.1649i	0.0894457	−	0.846459i
$816$		0			0
$817$	12.1208			0.424054
$818$		0			0
$819$	−13.9568	−	18.6427i	−0.487691	−	0.651427i
$820$		0			0
$821$			26.8208i			0.936053i	0.883715	+	0.468026i	$0.155035\pi$
$821$			26.8208i			0.936053i	−0.883715	+	0.468026i	$0.844965\pi$
$822$		0			0
$823$	−26.3764			−0.919425			−0.459712	−	0.888068i	$-0.652047\pi$
$823$	−26.3764			−0.919425			−0.459712	+	0.888068i	$0.652047\pi$
$824$		0			0
$825$	−21.2492	−	4.54153i	−0.739802	−	0.158116i
$826$		0			0
$827$			13.0913i			0.455228i	0.973751	+	0.227614i	$0.0730924\pi$
$827$			13.0913i			0.455228i	−0.973751	+	0.227614i	$0.926908\pi$
$828$		0			0
$829$	−13.7348			−0.477030			−0.238515	−	0.971139i	$-0.576661\pi$
$829$	−13.7348			−0.477030			−0.238515	+	0.971139i	$0.576661\pi$
$830$		0			0
$831$	19.1299			0.663610
$832$		0			0
$833$	−1.59398	+	5.43016i	−0.0552282	+	0.188144i
$834$		0			0
$835$	1.19084	−	11.2694i	0.0412108	−	0.389994i
$836$		0			0
$837$	−25.8200			−0.892468
$838$		0			0
$839$	25.7342			0.888445			0.444222	−	0.895917i	$-0.353480\pi$
$839$	25.7342			0.888445			0.444222	+	0.895917i	$0.353480\pi$
$840$		0			0
$841$	−40.9351			−1.41155
$842$		0			0
$843$	−25.1814			−0.867293
$844$		0			0
$845$	39.8452	+	4.21045i	1.37072	+	0.144844i
$846$		0			0
$847$	4.93098	−	3.69158i	0.169431	−	0.126844i
$848$		0			0
$849$	−29.4300			−1.01003
$850$		0			0
$851$	10.7995			0.370202
$852$		0			0
$853$			8.40578i			0.287808i	0.989592	+	0.143904i	$0.0459657\pi$
$853$			8.40578i			0.287808i	−0.989592	+	0.143904i	$0.954034\pi$
$854$		0			0
$855$	1.69188	−	16.0109i	0.0578609	−	0.547561i
$856$		0			0
$857$	−25.2022			−0.860892			−0.430446	−	0.902616i	$-0.641644\pi$
$857$	−25.2022			−0.860892			−0.430446	+	0.902616i	$0.641644\pi$
$858$		0			0
$859$			44.9663i			1.53423i	0.641510	+	0.767114i	$0.278308\pi$
$859$			44.9663i			1.53423i	−0.641510	+	0.767114i	$0.721692\pi$
$860$		0			0
$861$	14.4409	+	19.2893i	0.492146	+	0.657377i
$862$		0			0
$863$	−21.2327			−0.722768			−0.361384	−	0.932417i	$-0.617696\pi$
$863$	−21.2327			−0.722768			−0.361384	+	0.932417i	$0.617696\pi$
$864$		0			0
$865$	−0.872153	+	8.25352i	−0.0296541	+	0.280628i
$866$		0			0
$867$	19.4584			0.660843
$868$		0			0
$869$		−	9.10163i		−	0.308752i
$870$		0			0
$871$	65.4590			2.21799
$872$		0			0
$873$	18.5518			0.627884
$874$		0			0
$875$	−28.0149	+	9.49567i	−0.947075	+	0.321012i
$876$		0			0
$877$	23.4510			0.791884			0.395942	−	0.918275i	$-0.370418\pi$
$877$	23.4510			0.791884			0.395942	+	0.918275i	$0.370418\pi$
$878$		0			0
$879$		−	13.5717i		−	0.457762i
$880$		0			0
$881$			30.2220i			1.01821i	0.860706	+	0.509103i	$0.170023\pi$
$881$			30.2220i			1.01821i	−0.860706	+	0.509103i	$0.829977\pi$
$882$		0			0
$883$			55.2980i			1.86092i	0.366387	+	0.930462i	$0.380595\pi$
$883$			55.2980i			1.86092i	−0.366387	+	0.930462i	$0.619405\pi$
$884$		0			0
$885$	2.49680	−	23.6281i	0.0839289	−	0.794252i
$886$		0			0
$887$		−	8.28733i		−	0.278261i	−0.990274	−	0.139131i	$-0.955569\pi$
$887$		−	8.28733i		−	0.278261i	0.990274	−	0.139131i	$-0.0444308\pi$
$888$		0			0
$889$	−12.5690	+	9.40981i	−0.421551	+	0.315595i
$890$		0			0
$891$	−6.37123			−0.213444
$892$		0			0
$893$	−21.8048			−0.729671
$894$		0			0
$895$	16.1651	+	1.70817i	0.540338	+	0.0570978i
$896$		0			0
$897$			11.6156i			0.387834i
$898$		0			0
$899$			39.5793i			1.32004i
$900$		0			0
$901$	2.41702			0.0805227
$902$		0			0
$903$	−5.02985	−	6.71855i	−0.167383	−	0.223579i
$904$		0			0
$905$	−40.5470	−	4.28462i	−1.34783	−	0.142425i
$906$		0			0
$907$		−	18.0875i		−	0.600587i	−0.953847	−	0.300293i	$-0.902915\pi$
$907$		−	18.0875i		−	0.600587i	0.953847	−	0.300293i	$-0.0970845\pi$
$908$		0			0
$909$	11.2507			0.373162
$910$		0			0
$911$			6.16833i			0.204366i	0.994766	+	0.102183i	$0.0325827\pi$
$911$			6.16833i			0.204366i	−0.994766	+	0.102183i	$0.967417\pi$
$912$		0			0
$913$	21.0243			0.695802
$914$		0			0
$915$	−10.8177	−	1.14311i	−0.357623	−	0.0377902i
$916$		0			0
$917$	−27.3982	−	36.5968i	−0.904768	−	1.20853i
$918$		0			0
$919$			55.8085i			1.84095i	0.390797	+	0.920477i	$0.372199\pi$
$919$			55.8085i			1.84095i	−0.390797	+	0.920477i	$0.627801\pi$
$920$		0			0
$921$	−0.507762			−0.0167313
$922$		0			0
$923$	45.4441			1.49581
$924$		0			0
$925$	−30.0904	−	6.43113i	−0.989365	−	0.211454i
$926$		0			0
$927$			8.02241i			0.263491i
$928$		0			0
$929$			14.2579i			0.467787i	0.972262	+	0.233894i	$0.0751467\pi$
$929$			14.2579i			0.467787i	−0.972262	+	0.233894i	$0.924853\pi$
$930$		0			0
$931$	−30.5501	−	8.96776i	−1.00124	−	0.293907i
$932$		0			0
$933$	0.698651			0.0228728
$934$		0			0
$935$	6.56329	+	0.693545i	0.214643	+	0.0226814i
$936$		0			0
$937$	−22.5672			−0.737237			−0.368618	−	0.929581i	$-0.620169\pi$
$937$	−22.5672			−0.737237			−0.368618	+	0.929581i	$0.620169\pi$
$938$		0			0
$939$	−7.68634			−0.250834
$940$		0			0
$941$	−12.3410			−0.402307			−0.201153	−	0.979560i	$-0.564469\pi$
$941$	−12.3410			−0.402307			−0.201153	+	0.979560i	$0.564469\pi$
$942$		0			0
$943$			13.4263i			0.437220i
$944$		0			0
$945$	−27.7265	+	16.5206i	−0.901943	+	0.537417i
$946$		0			0
$947$		−	38.9941i		−	1.26714i	−0.773687	−	0.633568i	$-0.781590\pi$
$947$		−	38.9941i		−	1.26714i	0.773687	−	0.633568i	$-0.218410\pi$
$948$		0			0
$949$			84.3979i			2.73967i
$950$		0			0
$951$	12.2711			0.397918
$952$		0			0
$953$			57.1070i			1.84988i	0.380117	+	0.924938i	$0.375884\pi$
$953$			57.1070i			1.84988i	−0.380117	+	0.924938i	$0.624116\pi$
$954$		0			0
$955$	0.585816	−	5.54380i	0.0189565	−	0.179393i
$956$		0			0
$957$			36.3429i			1.17480i
$958$		0			0
$959$	10.4639	+	13.9770i	0.337897	+	0.451341i
$960$		0			0
$961$	−8.60041			−0.277433
$962$		0			0
$963$			17.2024i			0.554340i
$964$		0			0
$965$	−3.29415	+	31.1738i	−0.106042	+	1.00352i
$966$		0			0
$967$	−11.1371			−0.358146			−0.179073	−	0.983836i	$-0.557310\pi$
$967$	−11.1371			−0.358146			−0.179073	+	0.983836i	$0.557310\pi$
$968$		0			0
$969$		−	4.37736i		−	0.140621i
$970$		0			0
$971$			7.38403i			0.236965i	0.992956	+	0.118482i	$0.0378029\pi$
$971$			7.38403i			0.236965i	−0.992956	+	0.118482i	$0.962197\pi$
$972$		0			0
$973$	10.1090	+	13.5030i	0.324081	+	0.432886i
$974$		0			0
$975$	6.91714	−	32.3643i	0.221526	−	1.03649i
$976$		0			0
$977$			23.4951i			0.751675i	0.926686	+	0.375838i	$0.122645\pi$
$977$			23.4951i			0.751675i	−0.926686	+	0.375838i	$0.877355\pi$
$978$		0			0
$979$			18.0992i			0.578454i
$980$		0			0
$981$		−	11.3209i		−	0.361448i
$982$		0			0
$983$		−	60.7158i		−	1.93653i	−0.249918	−	0.968267i	$-0.580404\pi$
$983$		−	60.7158i		−	1.93653i	0.249918	−	0.968267i	$-0.419596\pi$
$984$		0			0
$985$	−18.3905	−	1.94333i	−0.585969	−	0.0619195i
$986$		0			0
$987$	9.04848	+	12.0864i	0.288016	+	0.384714i
$988$		0			0
$989$		−	4.67644i		−	0.148702i
$990$		0			0
$991$		−	8.22655i		−	0.261325i	−0.991427	−	0.130663i	$-0.958290\pi$
$991$		−	8.22655i		−	0.261325i	0.991427	−	0.130663i	$-0.0417104\pi$
$992$		0			0
$993$	9.55672			0.303273
$994$		0			0
$995$	−60.3798	−	6.38035i	−1.91417	−	0.202271i
$996$		0			0
$997$		−	49.5766i		−	1.57011i	−0.619428	−	0.785054i	$-0.712635\pi$
$997$		−	49.5766i		−	1.57011i	0.619428	−	0.785054i	$-0.287365\pi$
$998$		0			0
$999$	−33.5731			−1.06221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.n.b.559.14		40
4.3	odd	2		280.2.n.b.139.12	yes	40
5.4	even	2	inner	1120.2.n.b.559.28		40
7.6	odd	2	inner	1120.2.n.b.559.27		40
8.3	odd	2	inner	1120.2.n.b.559.13		40
8.5	even	2		280.2.n.b.139.32	yes	40
20.19	odd	2		280.2.n.b.139.29	yes	40
28.27	even	2		280.2.n.b.139.11	yes	40
35.34	odd	2	inner	1120.2.n.b.559.16		40
40.19	odd	2	inner	1120.2.n.b.559.26		40
40.29	even	2		280.2.n.b.139.9	✓	40
56.13	odd	2		280.2.n.b.139.31	yes	40
56.27	even	2	inner	1120.2.n.b.559.25		40
140.139	even	2		280.2.n.b.139.30	yes	40
280.69	odd	2		280.2.n.b.139.10	yes	40
280.139	even	2	inner	1120.2.n.b.559.15		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.n.b.139.9	✓	40	40.29	even	2
280.2.n.b.139.10	yes	40	280.69	odd	2
280.2.n.b.139.11	yes	40	28.27	even	2
280.2.n.b.139.12	yes	40	4.3	odd	2
280.2.n.b.139.29	yes	40	20.19	odd	2
280.2.n.b.139.30	yes	40	140.139	even	2
280.2.n.b.139.31	yes	40	56.13	odd	2
280.2.n.b.139.32	yes	40	8.5	even	2
1120.2.n.b.559.13		40	8.3	odd	2	inner
1120.2.n.b.559.14		40	1.1	even	1	trivial
1120.2.n.b.559.15		40	280.139	even	2	inner
1120.2.n.b.559.16		40	35.34	odd	2	inner
1120.2.n.b.559.25		40	56.27	even	2	inner
1120.2.n.b.559.26		40	40.19	odd	2	inner
1120.2.n.b.559.27		40	7.6	odd	2	inner
1120.2.n.b.559.28		40	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.n (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.19038 q^{3} +(-2.22369 - 0.234978i) q^{5} +(2.11797 - 1.58562i) q^{7} -1.58299 q^{9} +3.65078 q^{11} +5.56044i q^{13} +(2.64704 + 0.279713i) q^{15} -0.808468 q^{17} -4.54845i q^{19} +(-2.52120 + 1.88750i) q^{21} -1.75488 q^{23} +(4.88957 + 1.04503i) q^{25} +5.45551 q^{27} -8.36272i q^{29} -4.73282 q^{31} -4.34582 q^{33} +(-5.08229 + 3.02825i) q^{35} -6.15399 q^{37} -6.61905i q^{39} -7.65085i q^{41} +2.66483i q^{43} +(3.52008 + 0.371968i) q^{45} -4.79391i q^{47} +(1.97161 - 6.71660i) q^{49} +0.962386 q^{51} -2.98963 q^{53} +(-8.11818 - 0.857851i) q^{55} +5.41439i q^{57} -8.92626i q^{59} -4.08673 q^{61} +(-3.35273 + 2.51002i) q^{63} +(1.30658 - 12.3647i) q^{65} -11.7723i q^{67} +2.08897 q^{69} -8.17275i q^{71} +15.1783 q^{73} +(-5.82046 - 1.24399i) q^{75} +(7.73224 - 5.78875i) q^{77} -2.49307i q^{79} -1.74517 q^{81} +5.75885 q^{83} +(1.79778 + 0.189972i) q^{85} +9.95483i q^{87} +4.95764i q^{89} +(8.81676 + 11.7769i) q^{91} +5.63387 q^{93} +(-1.06878 + 10.1143i) q^{95} -11.7195 q^{97} -5.77914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{9} + 16 q^{25} - 16 q^{35} + 8 q^{49} + 32 q^{51} - 24 q^{65} - 72 q^{81} + 128 q^{99}+O(q^{100}) \) 40 * q + 40 * q^9 + 16 * q^25 - 16 * q^35 + 8 * q^49 + 32 * q^51 - 24 * q^65 - 72 * q^81 + 128 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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