LMFDB - Embedded newform 1152.2.k.f.289.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,2,Mod(289,1152)] 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("1152.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.k (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.18939904.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 2.10607i$ of defining polynomial
Character	$\chi$	$=$	1152.289
Dual form			1152.2.k.f.865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.68554 + 2.68554i) q^{5} +2.15894i q^{7} +(-1.79793 + 1.79793i) q^{11} +(-1.38372 - 1.38372i) q^{13} +0.224777 q^{17} +(0.158942 + 0.158942i) q^{19} -2.82843i q^{23} -9.42429i q^{25} +(-1.85712 - 1.85712i) q^{29} -1.84106 q^{31} +(-5.79793 - 5.79793i) q^{35} +(3.66949 - 3.66949i) q^{37} -5.88163i q^{41} +(-7.75481 + 7.75481i) q^{43} -2.82843 q^{47} +2.33897 q^{49} +(7.51397 - 7.51397i) q^{53} -9.65685i q^{55} +(-4.00000 + 4.00000i) q^{59} +(-5.98737 - 5.98737i) q^{61} +7.43208 q^{65} +(-10.4243 - 10.4243i) q^{67} +4.31788i q^{71} +5.97474i q^{73} +(-3.88163 - 3.88163i) q^{77} -15.0075 q^{79} +(10.1158 + 10.1158i) q^{83} +(-0.603650 + 0.603650i) q^{85} -1.42847i q^{89} +(2.98737 - 2.98737i) q^{91} -0.853690 q^{95} -16.3990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{11} - 8 q^{19} - 16 q^{29} - 24 q^{31} - 24 q^{35} + 16 q^{37} - 8 q^{43} - 8 q^{49} + 16 q^{53} - 32 q^{59} - 16 q^{61} + 16 q^{65} - 16 q^{67} + 16 q^{77} + 24 q^{79} + 40 q^{83} + 16 q^{85}+ \cdots - 48 q^{95}+O(q^{100}) $ 8 * q + 8 * q^11 - 8 * q^19 - 16 * q^29 - 24 * q^31 - 24 * q^35 + 16 * q^37 - 8 * q^43 - 8 * q^49 + 16 * q^53 - 32 * q^59 - 16 * q^61 + 16 * q^65 - 16 * q^67 + 16 * q^77 + 24 * q^79 + 40 * q^83 + 16 * q^85 - 8 * q^91 - 48 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.68554	+	2.68554i	−1.20101	+	1.20101i	−0.227153	+	0.973859i	$0.572942\pi$
$5$	−2.68554	+	2.68554i	−1.20101	+	1.20101i	−0.973859	+	0.227153i	$0.927058\pi$
$6$		0			0
$7$			2.15894i			0.816003i	0.912981	+	0.408002i	$0.133774\pi$
$7$			2.15894i			0.816003i	−0.912981	+	0.408002i	$0.866226\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.79793	+	1.79793i	−0.542097	+	0.542097i	−0.924143	−	0.382046i	$-0.875220\pi$
$11$	−1.79793	+	1.79793i	−0.542097	+	0.542097i	0.382046	+	0.924143i	$0.375220\pi$
$12$		0			0
$13$	−1.38372	−	1.38372i	−0.383775	−	0.383775i	0.488685	−	0.872460i	$-0.337477\pi$
$13$	−1.38372	−	1.38372i	−0.383775	−	0.383775i	−0.872460	+	0.488685i	$0.837477\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.224777			0.0545165			0.0272583	−	0.999628i	$-0.491322\pi$
$17$	0.224777			0.0545165			0.0272583	+	0.999628i	$0.491322\pi$
$18$		0			0
$19$	0.158942	+	0.158942i	0.0364637	+	0.0364637i	0.725104	−	0.688640i	$-0.241792\pi$
$19$	0.158942	+	0.158942i	0.0364637	+	0.0364637i	−0.688640	+	0.725104i	$0.741792\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$		−	9.42429i		−	1.88486i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.85712	−	1.85712i	−0.344858	−	0.344858i	0.513332	−	0.858190i	$-0.328411\pi$
$29$	−1.85712	−	1.85712i	−0.344858	−	0.344858i	−0.858190	+	0.513332i	$0.828411\pi$
$30$		0			0
$31$	−1.84106			−0.330664			−0.165332	−	0.986238i	$-0.552870\pi$
$31$	−1.84106			−0.330664			−0.165332	+	0.986238i	$0.552870\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.79793	−	5.79793i	−0.980029	−	0.980029i
$36$		0			0
$37$	3.66949	−	3.66949i	0.603260	−	0.603260i	−0.337916	−	0.941176i	$-0.609722\pi$
$37$	3.66949	−	3.66949i	0.603260	−	0.603260i	0.941176	+	0.337916i	$0.109722\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.88163i		−	0.918557i	−0.888292	−	0.459278i	$-0.848108\pi$
$41$		−	5.88163i		−	0.918557i	0.888292	−	0.459278i	$-0.151892\pi$
$42$		0			0
$43$	−7.75481	+	7.75481i	−1.18260	+	1.18260i	−0.203528	+	0.979069i	$0.565241\pi$
$43$	−7.75481	+	7.75481i	−1.18260	+	1.18260i	−0.979069	+	0.203528i	$0.934759\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.82843			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	2.33897			0.334139
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.51397	−	7.51397i	1.03212	−	1.03212i	0.0326567	−	0.999467i	$-0.489603\pi$
$53$	7.51397	−	7.51397i	1.03212	−	1.03212i	0.999467	−	0.0326567i	$-0.0103968\pi$
$54$		0			0
$55$		−	9.65685i		−	1.30213i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000	+	4.00000i	−0.520756	+	0.520756i	−0.917800	−	0.397044i	$-0.870036\pi$
$59$	−4.00000	+	4.00000i	−0.520756	+	0.520756i	0.397044	+	0.917800i	$0.370036\pi$
$60$		0			0
$61$	−5.98737	−	5.98737i	−0.766604	−	0.766604i	0.210903	−	0.977507i	$-0.432360\pi$
$61$	−5.98737	−	5.98737i	−0.766604	−	0.766604i	−0.977507	+	0.210903i	$0.932360\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.43208			0.921836
$66$		0			0
$67$	−10.4243	−	10.4243i	−1.27353	−	1.27353i	−0.944223	−	0.329307i	$-0.893185\pi$
$67$	−10.4243	−	10.4243i	−1.27353	−	1.27353i	−0.329307	−	0.944223i	$-0.606815\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.31788i			0.512438i	0.966619	+	0.256219i	$0.0824769\pi$
$71$			4.31788i			0.512438i	−0.966619	+	0.256219i	$0.917523\pi$
$72$		0			0
$73$			5.97474i			0.699290i	0.936882	+	0.349645i	$0.113698\pi$
$73$			5.97474i			0.699290i	−0.936882	+	0.349645i	$0.886302\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.88163	−	3.88163i	−0.442353	−	0.442353i
$78$		0			0
$79$	−15.0075			−1.68848			−0.844239	−	0.535966i	$-0.819947\pi$
$79$	−15.0075			−1.68848			−0.844239	+	0.535966i	$0.819947\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.1158	+	10.1158i	1.11036	+	1.11036i	0.993102	+	0.117253i	$0.0374088\pi$
$83$	10.1158	+	10.1158i	1.11036	+	1.11036i	0.117253	+	0.993102i	$0.462591\pi$
$84$		0			0
$85$	−0.603650	+	0.603650i	−0.0654750	+	0.0654750i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.42847i		−	0.151417i	−0.997130	−	0.0757086i	$-0.975878\pi$
$89$		−	1.42847i		−	0.151417i	0.997130	−	0.0757086i	$-0.0241219\pi$
$90$		0			0
$91$	2.98737	−	2.98737i	0.313161	−	0.313161i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.853690			−0.0875867
$96$		0			0
$97$	−16.3990			−1.66507			−0.832535	−	0.553973i	$-0.813111\pi$
$97$	−16.3990			−1.66507			−0.832535	+	0.553973i	$0.813111\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.0818942	−	0.0818942i	0.00814878	−	0.00814878i	−0.703021	−	0.711169i	$-0.748166\pi$
$101$	0.0818942	−	0.0818942i	0.00814878	−	0.00814878i	0.711169	+	0.703021i	$0.248166\pi$
$102$		0			0
$103$			13.3507i			1.31548i	0.753245	+	0.657740i	$0.228488\pi$
$103$			13.3507i			1.31548i	−0.753245	+	0.657740i	$0.771512\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.27798	−	7.27798i	0.703589	−	0.703589i	−0.261590	−	0.965179i	$-0.584247\pi$
$107$	7.27798	−	7.27798i	0.703589	−	0.703589i	0.965179	+	0.261590i	$0.0842468\pi$
$108$		0			0
$109$	7.04057	+	7.04057i	0.674365	+	0.674365i	0.958719	−	0.284355i	$-0.0917793\pi$
$109$	7.04057	+	7.04057i	0.674365	+	0.674365i	−0.284355	+	0.958719i	$0.591779\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.8486			−1.77313			−0.886563	−	0.462608i	$-0.846914\pi$
$113$	−18.8486			−1.77313			−0.886563	+	0.462608i	$0.846914\pi$
$114$		0			0
$115$	7.59587	+	7.59587i	0.708318	+	0.708318i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.485281i			0.0444857i
$120$		0			0
$121$			4.53488i			0.412261i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.8816	+	11.8816i	1.06273	+	1.06273i
$126$		0			0
$127$	3.81580			0.338597			0.169299	−	0.985565i	$-0.445850\pi$
$127$	3.81580			0.338597			0.169299	+	0.985565i	$0.445850\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.767438	+	0.767438i	0.0670514	+	0.0670514i	0.739837	−	0.672786i	$-0.234902\pi$
$131$	0.767438	+	0.767438i	0.0670514	+	0.0670514i	−0.672786	+	0.739837i	$0.734902\pi$
$132$		0			0
$133$	−0.343146	+	0.343146i	−0.0297545	+	0.0297545i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			5.31010i			0.453672i	0.973933	+	0.226836i	$0.0728382\pi$
$137$			5.31010i			0.453672i	−0.973933	+	0.226836i	$0.927162\pi$
$138$		0			0
$139$	8.76744	−	8.76744i	0.743644	−	0.743644i	−0.229633	−	0.973277i	$-0.573753\pi$
$139$	8.76744	−	8.76744i	0.743644	−	0.743644i	0.973277	+	0.229633i	$0.0737526\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.97567			0.416086
$144$		0			0
$145$	9.97474			0.828357
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.02869	+	1.02869i	−0.0842735	+	0.0842735i	−0.747987	−	0.663713i	$-0.768979\pi$
$149$	−1.02869	+	1.02869i	−0.0842735	+	0.0842735i	0.663713	+	0.747987i	$0.268979\pi$
$150$		0			0
$151$		−	2.03696i		−	0.165766i	−0.996559	−	0.0828829i	$-0.973587\pi$
$151$		−	2.03696i		−	0.165766i	0.996559	−	0.0828829i	$-0.0264127\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.94424	−	4.94424i	0.397131	−	0.397131i
$156$		0			0
$157$	−6.09378	−	6.09378i	−0.486336	−	0.486336i	0.420812	−	0.907148i	$-0.361745\pi$
$157$	−6.09378	−	6.09378i	−0.486336	−	0.486336i	−0.907148	+	0.420812i	$0.861745\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.10641			0.481252
$162$		0			0
$163$	3.43692	+	3.43692i	0.269201	+	0.269201i	0.828778	−	0.559577i	$-0.189037\pi$
$163$	3.43692	+	3.43692i	0.269201	+	0.269201i	−0.559577	+	0.828778i	$0.689037\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			21.7023i			1.67937i	0.543072	+	0.839686i	$0.317261\pi$
$167$			21.7023i			1.67937i	−0.543072	+	0.839686i	$0.682739\pi$
$168$		0			0
$169$		−	9.17064i		−	0.705434i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.74653	−	8.74653i	−0.664987	−	0.664987i	0.291565	−	0.956551i	$-0.405824\pi$
$173$	−8.74653	−	8.74653i	−0.664987	−	0.664987i	−0.956551	+	0.291565i	$0.905824\pi$
$174$		0			0
$175$	20.3465			1.53805
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.23163	−	8.23163i	−0.615261	−	0.615261i	0.329051	−	0.944312i	$-0.393271\pi$
$179$	−8.23163	−	8.23163i	−0.615261	−	0.615261i	−0.944312	+	0.329051i	$0.893271\pi$
$180$		0			0
$181$	−6.72269	+	6.72269i	−0.499694	+	0.499694i	−0.911343	−	0.411649i	$-0.864953\pi$
$181$	−6.72269	+	6.72269i	−0.499694	+	0.499694i	0.411649	+	0.911343i	$0.364953\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			19.7091i			1.44904i
$186$		0			0
$187$	−0.404135	+	0.404135i	−0.0295533	+	0.0295533i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−20.8032			−1.50526			−0.752632	−	0.658441i	$-0.771216\pi$
$191$	−20.8032			−1.50526			−0.752632	+	0.658441i	$0.771216\pi$
$192$		0			0
$193$	14.1454			1.01821			0.509103	−	0.860705i	$-0.329977\pi$
$193$	14.1454			1.01821			0.509103	+	0.860705i	$0.329977\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.42865	+	2.42865i	−0.173034	+	0.173034i	−0.788311	−	0.615277i	$-0.789044\pi$
$197$	−2.42865	+	2.42865i	−0.173034	+	0.173034i	0.615277	+	0.788311i	$0.289044\pi$
$198$		0			0
$199$			0.306182i			0.0217047i	0.999941	+	0.0108523i	$0.00345447\pi$
$199$			0.306182i			0.0217047i	−0.999941	+	0.0108523i	$0.996546\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.00941	−	4.00941i	0.281405	−	0.281405i
$204$		0			0
$205$	15.7954	+	15.7954i	1.10320	+	1.10320i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.571533			−0.0395337
$210$		0			0
$211$	7.23256	+	7.23256i	0.497910	+	0.497910i	0.910787	−	0.412877i	$-0.135476\pi$
$211$	7.23256	+	7.23256i	0.497910	+	0.497910i	−0.412877	+	0.910787i	$0.635476\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	41.6517i		−	2.84063i
$216$		0			0
$217$		−	3.97474i		−	0.269823i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.311029	−	0.311029i	−0.0209221	−	0.0209221i
$222$		0			0
$223$	1.71908			0.115118			0.0575591	−	0.998342i	$-0.481668\pi$
$223$	1.71908			0.115118			0.0575591	+	0.998342i	$0.481668\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.1158	−	10.1158i	−0.671410	−	0.671410i	0.286631	−	0.958041i	$-0.407465\pi$
$227$	−10.1158	−	10.1158i	−0.671410	−	0.671410i	−0.958041	+	0.286631i	$0.907465\pi$
$228$		0			0
$229$	12.0195	−	12.0195i	0.794270	−	0.794270i	−0.187915	−	0.982185i	$-0.560173\pi$
$229$	12.0195	−	12.0195i	0.794270	−	0.794270i	0.982185	+	0.187915i	$0.0601730\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	13.3779i		−	0.876418i	−0.898873	−	0.438209i	$-0.855613\pi$
$233$		−	13.3779i		−	0.876418i	0.898873	−	0.438209i	$-0.144387\pi$
$234$		0			0
$235$	7.59587	−	7.59587i	0.495500	−	0.495500i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.3675			0.864670			0.432335	−	0.901713i	$-0.357690\pi$
$239$	13.3675			0.864670			0.432335	+	0.901713i	$0.357690\pi$
$240$		0			0
$241$	0.211474			0.0136222			0.00681112	−	0.999977i	$-0.497832\pi$
$241$	0.211474			0.0136222			0.00681112	+	0.999977i	$0.497832\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.28141	+	6.28141i	−0.401305	+	0.401305i
$246$		0			0
$247$		−	0.439861i		−	0.0279877i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.4337	+	10.4337i	−0.658569	+	0.658569i	−0.955041	−	0.296472i	$-0.904190\pi$
$251$	−10.4337	+	10.4337i	−0.658569	+	0.658569i	0.296472	+	0.955041i	$0.404190\pi$
$252$		0			0
$253$	5.08532	+	5.08532i	0.319711	+	0.319711i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.742176			0.0462957			0.0231478	−	0.999732i	$-0.492631\pi$
$257$	0.742176			0.0462957			0.0231478	+	0.999732i	$0.492631\pi$
$258$		0			0
$259$	7.92221	+	7.92221i	0.492262	+	0.492262i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5.48435i		−	0.338180i	−0.985601	−	0.169090i	$-0.945917\pi$
$263$		−	5.48435i		−	0.338180i	0.985601	−	0.169090i	$-0.0540828\pi$
$264$		0			0
$265$			40.3582i			2.47918i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.4741	+	14.4741i	0.882500	+	0.882500i	0.993788	−	0.111289i	$-0.0354978\pi$
$269$	14.4741	+	14.4741i	0.882500	+	0.882500i	−0.111289	+	0.993788i	$0.535498\pi$
$270$		0			0
$271$	14.0370			0.852685			0.426342	−	0.904562i	$-0.359802\pi$
$271$	14.0370			0.852685			0.426342	+	0.904562i	$0.359802\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16.9442	+	16.9442i	1.02178	+	1.02178i
$276$		0			0
$277$	−9.49013	+	9.49013i	−0.570207	+	0.570207i	−0.932186	−	0.361980i	$-0.882101\pi$
$277$	−9.49013	+	9.49013i	−0.570207	+	0.570207i	0.361980	+	0.932186i	$0.382101\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.89359i			0.232272i	0.993233	+	0.116136i	$0.0370509\pi$
$281$			3.89359i			0.232272i	−0.993233	+	0.116136i	$0.962949\pi$
$282$		0			0
$283$	−12.4853	+	12.4853i	−0.742173	+	0.742173i	−0.972996	−	0.230823i	$-0.925858\pi$
$283$	−12.4853	+	12.4853i	−0.742173	+	0.742173i	0.230823	+	0.972996i	$0.425858\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.6981			0.749545
$288$		0			0
$289$	−16.9495			−0.997028
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.1553	+	11.1553i	−0.651697	+	0.651697i	−0.953402	−	0.301704i	$-0.902444\pi$
$293$	−11.1553	+	11.1553i	−0.651697	+	0.651697i	0.301704	+	0.953402i	$0.402444\pi$
$294$		0			0
$295$		−	21.4844i		−	1.25087i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.91375	+	3.91375i	−0.226338	+	0.226338i
$300$		0			0
$301$	−16.7422	−	16.7422i	−0.965003	−	0.965003i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	32.1587			1.84140
$306$		0			0
$307$	−5.40320	−	5.40320i	−0.308377	−	0.308377i	0.535903	−	0.844280i	$-0.319971\pi$
$307$	−5.40320	−	5.40320i	−0.308377	−	0.308377i	−0.844280	+	0.535903i	$0.819971\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			24.1623i			1.37012i	0.728488	+	0.685059i	$0.240224\pi$
$311$			24.1623i			1.37012i	−0.728488	+	0.685059i	$0.759776\pi$
$312$		0			0
$313$			16.6105i			0.938881i	0.882964	+	0.469441i	$0.155544\pi$
$313$			16.6105i			0.938881i	−0.882964	+	0.469441i	$0.844456\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.81170	+	1.81170i	0.101755	+	0.101755i	0.756152	−	0.654397i	$-0.227077\pi$
$317$	1.81170	+	1.81170i	0.101755	+	0.101755i	−0.654397	+	0.756152i	$0.727077\pi$
$318$		0			0
$319$	6.67794			0.373893
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.0357265	+	0.0357265i	0.00198788	+	0.00198788i
$324$		0			0
$325$	−13.0406	+	13.0406i	−0.723361	+	0.723361i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	6.10641i		−	0.336657i
$330$		0			0
$331$	13.5252	−	13.5252i	0.743411	−	0.743411i	−0.229822	−	0.973233i	$-0.573814\pi$
$331$	13.5252	−	13.5252i	0.743411	−	0.743411i	0.973233	+	0.229822i	$0.0738142\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	55.9898			3.05905
$336$		0			0
$337$	−1.12615			−0.0613454			−0.0306727	−	0.999529i	$-0.509765\pi$
$337$	−1.12615			−0.0613454			−0.0306727	+	0.999529i	$0.509765\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.31010	−	3.31010i	0.179252	−	0.179252i
$342$		0			0
$343$			20.1623i			1.08866i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.7938	+	20.7938i	−1.11627	+	1.11627i	−0.123983	+	0.992284i	$0.539567\pi$
$347$	−20.7938	+	20.7938i	−1.11627	+	1.11627i	−0.992284	+	0.123983i	$0.960433\pi$
$348$		0			0
$349$	19.2855	+	19.2855i	1.03233	+	1.03233i	0.999460	+	0.0328700i	$0.0104647\pi$
$349$	19.2855	+	19.2855i	1.03233	+	1.03233i	0.0328700	+	0.999460i	$0.489535\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.5908			−1.36206			−0.681029	−	0.732256i	$-0.738467\pi$
$353$	−25.5908			−1.36206			−0.681029	+	0.732256i	$0.738467\pi$
$354$		0			0
$355$	−11.5959	−	11.5959i	−0.615445	−	0.615445i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	3.77296i		−	0.199129i	−0.995031	−	0.0995645i	$-0.968255\pi$
$359$		−	3.77296i		−	0.199129i	0.995031	−	0.0995645i	$-0.0317450\pi$
$360$		0			0
$361$		−	18.9495i		−	0.997341i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.0454	−	16.0454i	−0.839856	−	0.839856i
$366$		0			0
$367$	27.4474			1.43274			0.716371	−	0.697720i	$-0.245802\pi$
$367$	27.4474			1.43274			0.716371	+	0.697720i	$0.245802\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.2222	+	16.2222i	0.842216	+	0.842216i
$372$		0			0
$373$	−12.6231	+	12.6231i	−0.653601	+	0.653601i	−0.953858	−	0.300257i	$-0.902928\pi$
$373$	−12.6231	+	12.6231i	−0.653601	+	0.653601i	0.300257	+	0.953858i	$0.402928\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			5.13946i			0.264695i
$378$		0			0
$379$	−11.6686	+	11.6686i	−0.599373	+	0.599373i	−0.940146	−	0.340772i	$-0.889311\pi$
$379$	−11.6686	+	11.6686i	−0.599373	+	0.599373i	0.340772	+	0.940146i	$0.389311\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.1885			−0.878291			−0.439145	−	0.898416i	$-0.644719\pi$
$383$	−17.1885			−0.878291			−0.439145	+	0.898416i	$0.644719\pi$
$384$		0			0
$385$	20.8486			1.06254
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.88238	+	1.88238i	−0.0954404	+	0.0954404i	−0.753215	−	0.657774i	$-0.771498\pi$
$389$	−1.88238	+	1.88238i	−0.0954404	+	0.0954404i	0.657774	+	0.753215i	$0.271498\pi$
$390$		0			0
$391$		−	0.635767i		−	0.0321521i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	40.3034	−	40.3034i	2.02788	−	2.02788i
$396$		0			0
$397$	−8.41166	−	8.41166i	−0.422169	−	0.422169i	0.463781	−	0.885950i	$-0.346493\pi$
$397$	−8.41166	−	8.41166i	−0.422169	−	0.422169i	−0.885950	+	0.463781i	$0.846493\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.12389			−0.0561242			−0.0280621	−	0.999606i	$-0.508934\pi$
$401$	−1.12389			−0.0561242			−0.0280621	+	0.999606i	$0.508934\pi$
$402$		0			0
$403$	2.54751	+	2.54751i	0.126900	+	0.126900i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			13.1950i			0.654051i
$408$		0			0
$409$		−	13.7211i		−	0.678464i	−0.940703	−	0.339232i	$-0.889833\pi$
$409$		−	13.7211i		−	0.678464i	0.940703	−	0.339232i	$-0.110167\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.63577	−	8.63577i	−0.424938	−	0.424938i
$414$		0			0
$415$	−54.3329			−2.66710
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.30755	+	9.30755i	0.454703	+	0.454703i	0.896912	−	0.442209i	$-0.145805\pi$
$419$	9.30755	+	9.30755i	0.454703	+	0.454703i	−0.442209	+	0.896912i	$0.645805\pi$
$420$		0			0
$421$	−8.44378	+	8.44378i	−0.411525	+	0.411525i	−0.882269	−	0.470745i	$-0.843985\pi$
$421$	−8.44378	+	8.44378i	−0.411525	+	0.411525i	0.470745	+	0.882269i	$0.343985\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.11837i		−	0.102756i
$426$		0			0
$427$	12.9264	−	12.9264i	0.625551	−	0.625551i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.6054			−1.47421			−0.737105	−	0.675778i	$-0.763808\pi$
$431$	−30.6054			−1.47421			−0.737105	+	0.675778i	$0.763808\pi$
$432$		0			0
$433$	−15.3137			−0.735930			−0.367965	−	0.929840i	$-0.619945\pi$
$433$	−15.3137			−0.735930			−0.367965	+	0.929840i	$0.619945\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.449555	−	0.449555i	0.0215051	−	0.0215051i
$438$		0			0
$439$		−	33.3676i		−	1.59255i	−0.604936	−	0.796274i	$-0.706801\pi$
$439$		−	33.3676i		−	1.59255i	0.604936	−	0.796274i	$-0.293199\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.28832	+	2.28832i	−0.108721	+	0.108721i	−0.759375	−	0.650653i	$-0.774495\pi$
$443$	−2.28832	+	2.28832i	−0.108721	+	0.108721i	0.650653	+	0.759375i	$0.274495\pi$
$444$		0			0
$445$	3.83621	+	3.83621i	0.181854	+	0.181854i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	27.4165			1.29387			0.646933	−	0.762547i	$-0.276052\pi$
$449$	27.4165			1.29387			0.646933	+	0.762547i	$0.276052\pi$
$450$		0			0
$451$	10.5748	+	10.5748i	0.497947	+	0.497947i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			16.0454i			0.752221i
$456$		0			0
$457$		−	10.9147i		−	0.510567i	−0.966866	−	0.255284i	$-0.917831\pi$
$457$		−	10.9147i		−	0.510567i	0.966866	−	0.255284i	$-0.0821688\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.8319	+	17.8319i	0.830512	+	0.830512i	0.987587	−	0.157075i	$-0.0502063\pi$
$461$	17.8319	+	17.8319i	0.830512	+	0.830512i	−0.157075	+	0.987587i	$0.550206\pi$
$462$		0			0
$463$	−22.4937			−1.04537			−0.522686	−	0.852525i	$-0.675070\pi$
$463$	−22.4937			−1.04537			−0.522686	+	0.852525i	$0.675070\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.2171	−	24.2171i	−1.12063	−	1.12063i	−0.991646	−	0.128989i	$-0.958827\pi$
$467$	−24.2171	−	24.2171i	−1.12063	−	1.12063i	−0.128989	−	0.991646i	$-0.541173\pi$
$468$		0			0
$469$	22.5054	−	22.5054i	1.03920	−	1.03920i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	27.8852i		−	1.28216i
$474$		0			0
$475$	1.49791	−	1.49791i	0.0687289	−	0.0687289i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−36.2362			−1.65568			−0.827838	−	0.560968i	$-0.810429\pi$
$479$	−36.2362			−1.65568			−0.827838	+	0.560968i	$0.810429\pi$
$480$		0			0
$481$	−10.1551			−0.463032
$482$		0			0
$483$		0			0
$484$		0			0
$485$	44.0403	−	44.0403i	1.99977	−	1.99977i
$486$		0			0
$487$			16.8200i			0.762186i	0.924537	+	0.381093i	$0.124452\pi$
$487$			16.8200i			0.762186i	−0.924537	+	0.381093i	$0.875548\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.10641	+	6.10641i	−0.275578	+	0.275578i	−0.831341	−	0.555763i	$-0.812426\pi$
$491$	−6.10641	+	6.10641i	−0.275578	+	0.275578i	0.555763	+	0.831341i	$0.312426\pi$
$492$		0			0
$493$	−0.417438	−	0.417438i	−0.0188005	−	0.0188005i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.32206			−0.418151
$498$		0			0
$499$	−19.6770	−	19.6770i	−0.880864	−	0.880864i	0.112758	−	0.993622i	$-0.464031\pi$
$499$	−19.6770	−	19.6770i	−0.880864	−	0.880864i	−0.993622	+	0.112758i	$0.964031\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	25.7308i		−	1.14728i	−0.819108	−	0.573639i	$-0.805531\pi$
$503$		−	25.7308i		−	1.14728i	0.819108	−	0.573639i	$-0.194469\pi$
$504$		0			0
$505$			0.439861i			0.0195736i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.73514	+	1.73514i	0.0769087	+	0.0769087i	0.744515	−	0.667606i	$-0.232681\pi$
$509$	1.73514	+	1.73514i	0.0769087	+	0.0769087i	−0.667606	+	0.744515i	$0.732681\pi$
$510$		0			0
$511$	−12.8991			−0.570623
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−35.8538	−	35.8538i	−1.57991	−	1.57991i
$516$		0			0
$517$	5.08532	−	5.08532i	0.223652	−	0.223652i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			33.5944i			1.47180i	0.677092	+	0.735898i	$0.263240\pi$
$521$			33.5944i			1.47180i	−0.677092	+	0.735898i	$0.736760\pi$
$522$		0			0
$523$	−21.8158	+	21.8158i	−0.953938	+	0.953938i	−0.998985	−	0.0450467i	$-0.985656\pi$
$523$	−21.8158	+	21.8158i	−0.953938	+	0.953938i	0.0450467	+	0.998985i	$0.485656\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.413828			−0.0180266
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−8.13853	+	8.13853i	−0.352519	+	0.352519i
$534$		0			0
$535$			39.0907i			1.69004i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.20531	+	4.20531i	−0.181136	+	0.181136i
$540$		0			0
$541$	−27.2112	−	27.2112i	−1.16990	−	1.16990i	−0.982232	−	0.187669i	$-0.939907\pi$
$541$	−27.2112	−	27.2112i	−1.16990	−	1.16990i	−0.187669	−	0.982232i	$-0.560093\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−37.8155			−1.61984
$546$		0			0
$547$	−6.80116	−	6.80116i	−0.290796	−	0.290796i	0.546598	−	0.837395i	$-0.315922\pi$
$547$	−6.80116	−	6.80116i	−0.290796	−	0.290796i	−0.837395	+	0.546598i	$0.815922\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	0.590346i		−	0.0251496i
$552$		0			0
$553$		−	32.4004i		−	1.37780i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.29337	−	4.29337i	−0.181916	−	0.181916i	0.610274	−	0.792190i	$-0.291059\pi$
$557$	−4.29337	−	4.29337i	−0.181916	−	0.181916i	−0.792190	+	0.610274i	$0.791059\pi$
$558$		0			0
$559$	21.4609			0.907701
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.0801	+	10.0801i	0.424825	+	0.424825i	0.886861	−	0.462036i	$-0.152881\pi$
$563$	10.0801	+	10.0801i	0.424825	+	0.424825i	−0.462036	+	0.886861i	$0.652881\pi$
$564$		0			0
$565$	50.6187	−	50.6187i	2.12954	−	2.12954i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	32.5018i		−	1.36255i	−0.732029	−	0.681274i	$-0.761426\pi$
$569$		−	32.5018i		−	1.36255i	0.732029	−	0.681274i	$-0.238574\pi$
$570$		0			0
$571$	−9.17157	+	9.17157i	−0.383818	+	0.383818i	−0.872476	−	0.488657i	$-0.837487\pi$
$571$	−9.17157	+	9.17157i	−0.383818	+	0.383818i	0.488657	+	0.872476i	$0.337487\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−26.6559			−1.11163
$576$		0			0
$577$	11.7536			0.489308			0.244654	−	0.969611i	$-0.421326\pi$
$577$	11.7536			0.489308			0.244654	+	0.969611i	$0.421326\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−21.8395	+	21.8395i	−0.906053	+	0.906053i
$582$		0			0
$583$			27.0192i			1.11902i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.46002	+	6.46002i	−0.266634	+	0.266634i	−0.827742	−	0.561109i	$-0.810375\pi$
$587$	−6.46002	+	6.46002i	−0.266634	+	0.266634i	0.561109	+	0.827742i	$0.310375\pi$
$588$		0			0
$589$	−0.292621	−	0.292621i	−0.0120572	−	0.0120572i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.49270			−0.225558			−0.112779	−	0.993620i	$-0.535975\pi$
$593$	−5.49270			−0.225558			−0.112779	+	0.993620i	$0.535975\pi$
$594$		0			0
$595$	−1.30324	−	1.30324i	−0.0534278	−	0.0534278i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	36.4348i		−	1.48868i	−0.667799	−	0.744342i	$-0.732763\pi$
$599$		−	36.4348i		−	1.48868i	0.667799	−	0.744342i	$-0.267237\pi$
$600$		0			0
$601$			9.97474i			0.406878i	0.979088	+	0.203439i	$0.0652119\pi$
$601$			9.97474i			0.406878i	−0.979088	+	0.203439i	$0.934788\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.1786	−	12.1786i	−0.495131	−	0.495131i
$606$		0			0
$607$	−4.51900			−0.183421			−0.0917103	−	0.995786i	$-0.529233\pi$
$607$	−4.51900			−0.183421			−0.0917103	+	0.995786i	$0.529233\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.91375	+	3.91375i	0.158333	+	0.158333i
$612$		0			0
$613$	8.43692	−	8.43692i	0.340764	−	0.340764i	−0.515890	−	0.856655i	$-0.672539\pi$
$613$	8.43692	−	8.43692i	0.340764	−	0.340764i	0.856655	+	0.515890i	$0.172539\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			32.1201i			1.29311i	0.762869	+	0.646554i	$0.223790\pi$
$617$			32.1201i			1.29311i	−0.762869	+	0.646554i	$0.776210\pi$
$618$		0			0
$619$	15.0412	−	15.0412i	0.604559	−	0.604559i	−0.336960	−	0.941519i	$-0.609399\pi$
$619$	15.0412	−	15.0412i	0.604559	−	0.604559i	0.941519	+	0.336960i	$0.109399\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.08398			0.123557
$624$		0			0
$625$	−16.6958			−0.667833
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.824818	−	0.824818i	0.0328876	−	0.0328876i
$630$		0			0
$631$		−	36.4685i		−	1.45179i	−0.687807	−	0.725894i	$-0.741426\pi$
$631$		−	36.4685i		−	1.45179i	0.687807	−	0.725894i	$-0.258574\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.2475	+	10.2475i	−0.406659	+	0.406659i
$636$		0			0
$637$	−3.23648	−	3.23648i	−0.128234	−	0.128234i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.0036			−0.553109			−0.276555	−	0.960998i	$-0.589193\pi$
$641$	−14.0036			−0.553109			−0.276555	+	0.960998i	$0.589193\pi$
$642$		0			0
$643$	16.6034	+	16.6034i	0.654774	+	0.654774i	0.954139	−	0.299365i	$-0.0967748\pi$
$643$	16.6034	+	16.6034i	0.654774	+	0.654774i	−0.299365	+	0.954139i	$0.596775\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			12.1908i			0.479270i	0.970863	+	0.239635i	$0.0770277\pi$
$647$			12.1908i			0.479270i	−0.970863	+	0.239635i	$0.922972\pi$
$648$		0			0
$649$		−	14.3835i		−	0.564600i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.983270	+	0.983270i	0.0384783	+	0.0384783i	0.726084	−	0.687606i	$-0.241338\pi$
$653$	0.983270	+	0.983270i	0.0384783	+	0.0384783i	−0.687606	+	0.726084i	$0.741338\pi$
$654$		0			0
$655$	−4.12198			−0.161059
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.0559	+	18.0559i	0.703357	+	0.703357i	0.965130	−	0.261772i	$-0.0843069\pi$
$659$	18.0559	+	18.0559i	0.703357	+	0.703357i	−0.261772	+	0.965130i	$0.584307\pi$
$660$		0			0
$661$	4.55890	−	4.55890i	0.177321	−	0.177321i	−0.612866	−	0.790187i	$-0.709983\pi$
$661$	4.55890	−	4.55890i	0.177321	−	0.177321i	0.790187	+	0.612866i	$0.209983\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.84307i		−	0.0714710i
$666$		0			0
$667$	−5.25272	+	5.25272i	−0.203386	+	0.203386i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21.5298			0.831148
$672$		0			0
$673$	−10.8569			−0.418504			−0.209252	−	0.977862i	$-0.567103\pi$
$673$	−10.8569			−0.418504			−0.209252	+	0.977862i	$0.567103\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	23.7066	−	23.7066i	0.911120	−	0.911120i	−0.0852405	−	0.996360i	$-0.527166\pi$
$677$	23.7066	−	23.7066i	0.911120	−	0.911120i	0.996360	+	0.0852405i	$0.0271659\pi$
$678$		0			0
$679$		−	35.4045i		−	1.35870i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	17.8337	−	17.8337i	0.682386	−	0.682386i	−0.278151	−	0.960537i	$-0.589722\pi$
$683$	17.8337	−	17.8337i	0.682386	−	0.682386i	0.960537	+	0.278151i	$0.0897217\pi$
$684$		0			0
$685$	−14.2605	−	14.2605i	−0.544866	−	0.544866i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−20.7945			−0.792205
$690$		0			0
$691$	10.8557	+	10.8557i	0.412970	+	0.412970i	0.882772	−	0.469802i	$-0.155675\pi$
$691$	10.8557	+	10.8557i	0.412970	+	0.412970i	−0.469802	+	0.882772i	$0.655675\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			47.0907i			1.78625i
$696$		0			0
$697$		−	1.32206i		−	0.0500765i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.08875	−	6.08875i	−0.229969	−	0.229969i	0.582711	−	0.812680i	$-0.301992\pi$
$701$	−6.08875	−	6.08875i	−0.229969	−	0.229969i	−0.812680	+	0.582711i	$0.801992\pi$
$702$		0			0
$703$	1.16647			0.0439942
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.176805	+	0.176805i	0.00664943	+	0.00664943i
$708$		0			0
$709$	−22.8836	+	22.8836i	−0.859413	+	0.859413i	−0.991269	−	0.131856i	$-0.957906\pi$
$709$	−22.8836	+	22.8836i	−0.859413	+	0.859413i	0.131856	+	0.991269i	$0.457906\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			5.20730i			0.195015i
$714$		0			0
$715$	−13.3624	+	13.3624i	−0.499724	+	0.499724i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.46744			−0.0547262			−0.0273631	−	0.999626i	$-0.508711\pi$
$719$	−1.46744			−0.0547262			−0.0273631	+	0.999626i	$0.508711\pi$
$720$		0			0
$721$	−28.8233			−1.07344
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−17.5020	+	17.5020i	−0.650008	+	0.650008i
$726$		0			0
$727$			15.3928i			0.570889i	0.958395	+	0.285445i	$0.0921412\pi$
$727$			15.3928i			0.570889i	−0.958395	+	0.285445i	$0.907859\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.74311	+	1.74311i	−0.0644711	+	0.0644711i
$732$		0			0
$733$	12.4185	+	12.4185i	0.458688	+	0.458688i	0.898225	−	0.439536i	$-0.144857\pi$
$733$	12.4185	+	12.4185i	0.458688	+	0.458688i	−0.439536	+	0.898225i	$0.644857\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	37.4844			1.38075
$738$		0			0
$739$	−14.6559	−	14.6559i	−0.539127	−	0.539127i	0.384146	−	0.923273i	$-0.374496\pi$
$739$	−14.6559	−	14.6559i	−0.539127	−	0.539127i	−0.923273	+	0.384146i	$0.874496\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			31.7821i			1.16597i	0.812482	+	0.582986i	$0.198116\pi$
$743$			31.7821i			1.16597i	−0.812482	+	0.582986i	$0.801884\pi$
$744$		0			0
$745$		−	5.52518i		−	0.202427i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.7127	+	15.7127i	0.574131	+	0.574131i
$750$		0			0
$751$	−29.7594			−1.08594			−0.542968	−	0.839753i	$-0.682699\pi$
$751$	−29.7594			−1.08594			−0.542968	+	0.839753i	$0.682699\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.47036	+	5.47036i	0.199087	+	0.199087i
$756$		0			0
$757$	−15.6355	+	15.6355i	−0.568282	+	0.568282i	−0.931647	−	0.363365i	$-0.881628\pi$
$757$	−15.6355	+	15.6355i	−0.568282	+	0.568282i	0.363365	+	0.931647i	$0.381628\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	4.55957i		−	0.165284i	−0.996579	−	0.0826422i	$-0.973664\pi$
$761$		−	4.55957i		−	0.165284i	0.996579	−	0.0826422i	$-0.0263359\pi$
$762$		0			0
$763$	−15.2002	+	15.2002i	−0.550284	+	0.550284i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11.0698			0.399706
$768$		0			0
$769$	36.5794			1.31909			0.659543	−	0.751667i	$-0.270750\pi$
$769$	36.5794			1.31909			0.659543	+	0.751667i	$0.270750\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18.7108	+	18.7108i	−0.672981	+	0.672981i	−0.958402	−	0.285421i	$-0.907867\pi$
$773$	−18.7108	+	18.7108i	−0.672981	+	0.672981i	0.285421	+	0.958402i	$0.407867\pi$
$774$		0			0
$775$			17.3507i			0.623255i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.934836	−	0.934836i	0.0334940	−	0.0334940i
$780$		0			0
$781$	−7.76326	−	7.76326i	−0.277791	−	0.277791i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	32.7302			1.16819
$786$		0			0
$787$	13.3759	+	13.3759i	0.476801	+	0.476801i	0.904107	−	0.427306i	$-0.140537\pi$
$787$	13.3759	+	13.3759i	0.476801	+	0.476801i	−0.427306	+	0.904107i	$0.640537\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	40.6930i		−	1.44688i
$792$		0			0
$793$			16.5697i			0.588406i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−33.8043	−	33.8043i	−1.19741	−	1.19741i	−0.974939	−	0.222471i	$-0.928588\pi$
$797$	−33.8043	−	33.8043i	−1.19741	−	1.19741i	−0.222471	−	0.974939i	$-0.571412\pi$
$798$		0			0
$799$	−0.635767			−0.0224918
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.7422	−	10.7422i	−0.379083	−	0.379083i
$804$		0			0
$805$	−16.3990	+	16.3990i	−0.577990	+	0.577990i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	29.9862i		−	1.05426i	−0.849785	−	0.527129i	$-0.823268\pi$
$809$		−	29.9862i		−	1.05426i	0.849785	−	0.527129i	$-0.176732\pi$
$810$		0			0
$811$	−8.05388	+	8.05388i	−0.282810	+	0.282810i	−0.834229	−	0.551419i	$-0.814087\pi$
$811$	−8.05388	+	8.05388i	−0.282810	+	0.282810i	0.551419	+	0.834229i	$0.314087\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.4600			−0.646626
$816$		0			0
$817$	−2.46512			−0.0862438
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.9250	−	13.9250i	0.485984	−	0.485984i	−0.421052	−	0.907036i	$-0.638339\pi$
$821$	13.9250	−	13.9250i	0.485984	−	0.485984i	0.907036	+	0.421052i	$0.138339\pi$
$822$		0			0
$823$		−	22.4666i		−	0.783137i	−0.920149	−	0.391568i	$-0.871933\pi$
$823$		−	22.4666i		−	0.783137i	0.920149	−	0.391568i	$-0.128067\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.63159	+	7.63159i	−0.265376	+	0.265376i	−0.827234	−	0.561858i	$-0.810087\pi$
$827$	−7.63159	+	7.63159i	−0.265376	+	0.265376i	0.561858	+	0.827234i	$0.310087\pi$
$828$		0			0
$829$	32.4860	+	32.4860i	1.12828	+	1.12828i	0.990456	+	0.137828i	$0.0440120\pi$
$829$	32.4860	+	32.4860i	1.12828	+	1.12828i	0.137828	+	0.990456i	$0.455988\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.525748			0.0182161
$834$		0			0
$835$	−58.2824	−	58.2824i	−2.01695	−	2.01695i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	22.9142i		−	0.791085i	−0.918448	−	0.395542i	$-0.870557\pi$
$839$		−	22.9142i		−	0.791085i	0.918448	−	0.395542i	$-0.129443\pi$
$840$		0			0
$841$		−	22.1022i		−	0.762146i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	24.6282	+	24.6282i	0.847235	+	0.847235i
$846$		0			0
$847$	−9.79053			−0.336407
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−10.3789	−	10.3789i	−0.355783	−	0.355783i
$852$		0			0
$853$	−38.9424	+	38.9424i	−1.33336	+	1.33336i	−0.431018	+	0.902344i	$0.641845\pi$
$853$	−38.9424	+	38.9424i	−1.33336	+	1.33336i	−0.902344	+	0.431018i	$0.858155\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.79079i		−	0.0611723i	−0.999532	−	0.0305861i	$-0.990263\pi$
$857$		−	1.79079i		−	0.0611723i	0.999532	−	0.0305861i	$-0.00973739\pi$
$858$		0			0
$859$	18.0643	−	18.0643i	0.616347	−	0.616347i	−0.328245	−	0.944593i	$-0.606457\pi$
$859$	18.0643	−	18.0643i	0.616347	−	0.616347i	0.944593	+	0.328245i	$0.106457\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	42.1150			1.43361			0.716806	−	0.697273i	$-0.245603\pi$
$863$	42.1150			1.43361			0.716806	+	0.697273i	$0.245603\pi$
$864$		0			0
$865$	46.9784			1.59731
$866$		0			0
$867$		0			0
$868$		0			0
$869$	26.9825	−	26.9825i	0.915319	−	0.915319i
$870$		0			0
$871$			28.8486i			0.977497i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25.6517	+	25.6517i	−0.867187	+	0.867187i
$876$		0			0
$877$	−0.714491	−	0.714491i	−0.0241267	−	0.0241267i	0.694941	−	0.719067i	$-0.255431\pi$
$877$	−0.714491	−	0.714491i	−0.0241267	−	0.0241267i	−0.719067	+	0.694941i	$0.755431\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	44.3972			1.49578			0.747889	−	0.663823i	$-0.231067\pi$
$881$	44.3972			1.49578			0.747889	+	0.663823i	$0.231067\pi$
$882$		0			0
$883$	1.28968	+	1.28968i	0.0434013	+	0.0434013i	0.728474	−	0.685073i	$-0.240230\pi$
$883$	1.28968	+	1.28968i	0.0434013	+	0.0434013i	−0.685073	+	0.728474i	$0.740230\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4.38532i			0.147245i	0.997286	+	0.0736223i	$0.0234559\pi$
$887$			4.38532i			0.147245i	−0.997286	+	0.0736223i	$0.976544\pi$
$888$		0			0
$889$			8.23808i			0.276296i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.449555	−	0.449555i	−0.0150438	−	0.0150438i
$894$		0			0
$895$	44.2128			1.47787
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.41906	+	3.41906i	0.114032	+	0.114032i
$900$		0			0
$901$	1.68897	−	1.68897i	0.0562678	−	0.0562678i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	36.1082i		−	1.20028i
$906$		0			0
$907$	−2.87261	+	2.87261i	−0.0953835	+	0.0953835i	−0.753188	−	0.657805i	$-0.771485\pi$
$907$	−2.87261	+	2.87261i	−0.0953835	+	0.0953835i	0.657805	+	0.753188i	$0.271485\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	42.3784			1.40406			0.702029	−	0.712149i	$-0.252278\pi$
$911$	42.3784			1.40406			0.702029	+	0.712149i	$0.252278\pi$
$912$		0			0
$913$	−36.3751			−1.20384
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.65685	+	1.65685i	−0.0547141	+	0.0547141i
$918$		0			0
$919$		−	44.8603i		−	1.47980i	−0.672715	−	0.739902i	$-0.734872\pi$
$919$		−	44.8603i		−	1.47980i	0.672715	−	0.739902i	$-0.265128\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.97474	−	5.97474i	0.196661	−	0.196661i
$924$		0			0
$925$	−34.5823	−	34.5823i	−1.13706	−	1.13706i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.96695			0.0973426			0.0486713	−	0.998815i	$-0.484501\pi$
$929$	2.96695			0.0973426			0.0486713	+	0.998815i	$0.484501\pi$
$930$		0			0
$931$	0.371760	+	0.371760i	0.0121839	+	0.0121839i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	2.17064i		−	0.0709876i
$936$		0			0
$937$			54.7669i			1.78916i	0.446910	+	0.894579i	$0.352524\pi$
$937$			54.7669i			1.78916i	−0.446910	+	0.894579i	$0.647476\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.29012	−	6.29012i	−0.205052	−	0.205052i	0.597108	−	0.802161i	$-0.296316\pi$
$941$	−6.29012	−	6.29012i	−0.205052	−	0.205052i	−0.802161	+	0.597108i	$0.796316\pi$
$942$		0			0
$943$	−16.6358			−0.541735
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.2105	+	11.2105i	0.364294	+	0.364294i	0.865391	−	0.501097i	$-0.167070\pi$
$947$	11.2105	+	11.2105i	0.364294	+	0.364294i	−0.501097	+	0.865391i	$0.667070\pi$
$948$		0			0
$949$	8.26736	−	8.26736i	0.268370	−	0.268370i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	30.2807i		−	0.980887i	−0.871473	−	0.490443i	$-0.836835\pi$
$953$		−	30.2807i		−	0.980887i	0.871473	−	0.490443i	$-0.163165\pi$
$954$		0			0
$955$	55.8678	−	55.8678i	1.80784	−	1.80784i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.4642			−0.370198
$960$		0			0
$961$	−27.6105			−0.890661
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−37.9880	+	37.9880i	−1.22288	+	1.22288i
$966$		0			0
$967$		−	10.5273i		−	0.338537i	−0.985570	−	0.169268i	$-0.945860\pi$
$967$		−	10.5273i		−	0.338537i	0.985570	−	0.169268i	$-0.0541405\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−27.3685	+	27.3685i	−0.878298	+	0.878298i	−0.993358	−	0.115060i	$-0.963294\pi$
$971$	−27.3685	+	27.3685i	−0.878298	+	0.878298i	0.115060	+	0.993358i	$0.463294\pi$
$972$		0			0
$973$	18.9284	+	18.9284i	0.606816	+	0.606816i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.9009			0.540706			0.270353	−	0.962761i	$-0.412860\pi$
$977$	16.9009			0.540706			0.270353	+	0.962761i	$0.412860\pi$
$978$		0			0
$979$	2.56829	+	2.56829i	0.0820828	+	0.0820828i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	10.0798i		−	0.321496i	−0.986995	−	0.160748i	$-0.948609\pi$
$983$		−	10.0798i		−	0.321496i	0.986995	−	0.160748i	$-0.0513906\pi$
$984$		0			0
$985$		−	13.0445i		−	0.415632i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	21.9339	+	21.9339i	0.697458	+	0.697458i
$990$		0			0
$991$	42.3446			1.34512			0.672561	−	0.740042i	$-0.265194\pi$
$991$	42.3446			1.34512			0.672561	+	0.740042i	$0.265194\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.822265	−	0.822265i	−0.0260676	−	0.0260676i
$996$		0			0
$997$	33.6676	−	33.6676i	1.06626	−	1.06626i	0.0686216	−	0.997643i	$-0.478140\pi$
$997$	33.6676	−	33.6676i	1.06626	−	1.06626i	0.997643	−	0.0686216i	$-0.0218601\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.k.f.289.1		8
3.2	odd	2		384.2.j.a.289.4		8
4.3	odd	2		1152.2.k.c.289.1		8
8.3	odd	2		144.2.k.b.109.2		8
8.5	even	2		576.2.k.b.145.4		8
12.11	even	2		384.2.j.b.289.2		8
16.3	odd	4		144.2.k.b.37.2		8
16.5	even	4	inner	1152.2.k.f.865.1		8
16.11	odd	4		1152.2.k.c.865.1		8
16.13	even	4		576.2.k.b.433.4		8
24.5	odd	2		192.2.j.a.145.1		8
24.11	even	2		48.2.j.a.13.3	✓	8
32.5	even	8		9216.2.a.x.1.1		4
32.11	odd	8		9216.2.a.bo.1.4		4
32.21	even	8		9216.2.a.bn.1.4		4
32.27	odd	8		9216.2.a.y.1.1		4
48.5	odd	4		384.2.j.a.97.4		8
48.11	even	4		384.2.j.b.97.2		8
48.29	odd	4		192.2.j.a.49.1		8
48.35	even	4		48.2.j.a.37.3	yes	8
96.5	odd	8		3072.2.a.n.1.4		4
96.11	even	8		3072.2.a.i.1.1		4
96.29	odd	8		3072.2.d.i.1537.8		8
96.35	even	8		3072.2.d.f.1537.4		8
96.53	odd	8		3072.2.a.o.1.1		4
96.59	even	8		3072.2.a.t.1.4		4
96.77	odd	8		3072.2.d.i.1537.1		8
96.83	even	8		3072.2.d.f.1537.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.3	✓	8	24.11	even	2
48.2.j.a.37.3	yes	8	48.35	even	4
144.2.k.b.37.2		8	16.3	odd	4
144.2.k.b.109.2		8	8.3	odd	2
192.2.j.a.49.1		8	48.29	odd	4
192.2.j.a.145.1		8	24.5	odd	2
384.2.j.a.97.4		8	48.5	odd	4
384.2.j.a.289.4		8	3.2	odd	2
384.2.j.b.97.2		8	48.11	even	4
384.2.j.b.289.2		8	12.11	even	2
576.2.k.b.145.4		8	8.5	even	2
576.2.k.b.433.4		8	16.13	even	4
1152.2.k.c.289.1		8	4.3	odd	2
1152.2.k.c.865.1		8	16.11	odd	4
1152.2.k.f.289.1		8	1.1	even	1	trivial
1152.2.k.f.865.1		8	16.5	even	4	inner
3072.2.a.i.1.1		4	96.11	even	8
3072.2.a.n.1.4		4	96.5	odd	8
3072.2.a.o.1.1		4	96.53	odd	8
3072.2.a.t.1.4		4	96.59	even	8
3072.2.d.f.1537.4		8	96.35	even	8
3072.2.d.f.1537.5		8	96.83	even	8
3072.2.d.i.1537.1		8	96.77	odd	8
3072.2.d.i.1537.8		8	96.29	odd	8
9216.2.a.x.1.1		4	32.5	even	8
9216.2.a.y.1.1		4	32.27	odd	8
9216.2.a.bn.1.4		4	32.21	even	8
9216.2.a.bo.1.4		4	32.11	odd	8

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.68554 + 2.68554i) q^{5} +2.15894i q^{7} +(-1.79793 + 1.79793i) q^{11} +(-1.38372 - 1.38372i) q^{13} +0.224777 q^{17} +(0.158942 + 0.158942i) q^{19} -2.82843i q^{23} -9.42429i q^{25} +(-1.85712 - 1.85712i) q^{29} -1.84106 q^{31} +(-5.79793 - 5.79793i) q^{35} +(3.66949 - 3.66949i) q^{37} -5.88163i q^{41} +(-7.75481 + 7.75481i) q^{43} -2.82843 q^{47} +2.33897 q^{49} +(7.51397 - 7.51397i) q^{53} -9.65685i q^{55} +(-4.00000 + 4.00000i) q^{59} +(-5.98737 - 5.98737i) q^{61} +7.43208 q^{65} +(-10.4243 - 10.4243i) q^{67} +4.31788i q^{71} +5.97474i q^{73} +(-3.88163 - 3.88163i) q^{77} -15.0075 q^{79} +(10.1158 + 10.1158i) q^{83} +(-0.603650 + 0.603650i) q^{85} -1.42847i q^{89} +(2.98737 - 2.98737i) q^{91} -0.853690 q^{95} -16.3990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{11} - 8 q^{19} - 16 q^{29} - 24 q^{31} - 24 q^{35} + 16 q^{37} - 8 q^{43} - 8 q^{49} + 16 q^{53} - 32 q^{59} - 16 q^{61} + 16 q^{65} - 16 q^{67} + 16 q^{77} + 24 q^{79} + 40 q^{83} + 16 q^{85}+ \cdots - 48 q^{95}+O(q^{100}) \) 8 * q + 8 * q^11 - 8 * q^19 - 16 * q^29 - 24 * q^31 - 24 * q^35 + 16 * q^37 - 8 * q^43 - 8 * q^49 + 16 * q^53 - 32 * q^59 - 16 * q^61 + 16 * q^65 - 16 * q^67 + 16 * q^77 + 24 * q^79 + 40 * q^83 + 16 * q^85 - 8 * q^91 - 48 * q^95

Introduction

L-functions

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