LMFDB - Embedded newform 1152.2.k.c.865.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(289,1152)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			865.1
Root			$0.500000 + 2.10607i$ of defining polynomial
Character	$\chi$	$=$	1152.865
Dual form			1152.2.k.c.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.68554 - 2.68554i) q^{5} +2.15894i q^{7} +O(q^{10})$	$q+(-2.68554 - 2.68554i) q^{5} +2.15894i q^{7} +(1.79793 + 1.79793i) q^{11} +(-1.38372 + 1.38372i) q^{13} +0.224777 q^{17} +(-0.158942 + 0.158942i) q^{19} -2.82843i q^{23} +9.42429i q^{25} +(-1.85712 + 1.85712i) q^{29} +1.84106 q^{31} +(5.79793 - 5.79793i) q^{35} +(3.66949 + 3.66949i) q^{37} +5.88163i q^{41} +(7.75481 + 7.75481i) q^{43} +2.82843 q^{47} +2.33897 q^{49} +(7.51397 + 7.51397i) q^{53} -9.65685i q^{55} +(4.00000 + 4.00000i) q^{59} +(-5.98737 + 5.98737i) q^{61} +7.43208 q^{65} +(10.4243 - 10.4243i) q^{67} +4.31788i q^{71} -5.97474i q^{73} +(-3.88163 + 3.88163i) q^{77} +15.0075 q^{79} +(-10.1158 + 10.1158i) q^{83} +(-0.603650 - 0.603650i) q^{85} +1.42847i q^{89} +(-2.98737 - 2.98737i) q^{91} +0.853690 q^{95} -16.3990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * 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} + 8 q^{91} + 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

Display 100 coefficients 10 coefficients

Character values

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

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

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	−0.973859	−	0.227153i	$-0.927058\pi$
$5$	−2.68554	−	2.68554i	−1.20101	−	1.20101i	−0.227153	−	0.973859i	$-0.572942\pi$
$6$		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	$-0.124780\pi$
$11$	1.79793	+	1.79793i	0.542097	+	0.542097i	−0.382046	+	0.924143i	$0.624780\pi$
$12$		0			0
$13$	−1.38372	+	1.38372i	−0.383775	+	0.383775i	−0.872460	−	0.488685i	$-0.837477\pi$
$13$	−1.38372	+	1.38372i	−0.383775	+	0.383775i	0.488685	+	0.872460i	$0.337477\pi$
$14$		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	$-0.758208\pi$
$19$	−0.158942	+	0.158942i	−0.0364637	+	0.0364637i	0.688640	+	0.725104i	$0.258208\pi$
$20$		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	−0.858190	−	0.513332i	$-0.828411\pi$
$29$	−1.85712	+	1.85712i	−0.344858	+	0.344858i	0.513332	+	0.858190i	$0.328411\pi$
$30$		0			0
$31$	1.84106			0.330664			0.165332	−	0.986238i	$-0.447130\pi$
$31$	1.84106			0.330664			0.165332	+	0.986238i	$0.447130\pi$
$32$		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	0.941176	−	0.337916i	$-0.109722\pi$
$37$	3.66949	+	3.66949i	0.603260	+	0.603260i	−0.337916	+	0.941176i	$0.609722\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.88163i			0.918557i	0.888292	+	0.459278i	$0.151892\pi$
$41$			5.88163i			0.918557i	−0.888292	+	0.459278i	$0.848108\pi$
$42$		0			0
$43$	7.75481	+	7.75481i	1.18260	+	1.18260i	0.979069	+	0.203528i	$0.0652407\pi$
$43$	7.75481	+	7.75481i	1.18260	+	1.18260i	0.203528	+	0.979069i	$0.434759\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.82843			0.412568			0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843			0.412568			0.206284	+	0.978492i	$0.433863\pi$
$48$		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	0.999467	+	0.0326567i	$0.0103968\pi$
$53$	7.51397	+	7.51397i	1.03212	+	1.03212i	0.0326567	+	0.999467i	$0.489603\pi$
$54$		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	$-0.129964\pi$
$59$	4.00000	+	4.00000i	0.520756	+	0.520756i	−0.397044	+	0.917800i	$0.629964\pi$
$60$		0			0
$61$	−5.98737	+	5.98737i	−0.766604	+	0.766604i	−0.977507	−	0.210903i	$-0.932360\pi$
$61$	−5.98737	+	5.98737i	−0.766604	+	0.766604i	0.210903	+	0.977507i	$0.432360\pi$
$62$		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	0.329307	−	0.944223i	$-0.393185\pi$
$67$	10.4243	−	10.4243i	1.27353	−	1.27353i	0.944223	−	0.329307i	$-0.106815\pi$
$68$		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	$-0.886302\pi$
$73$		−	5.97474i		−	0.699290i	0.936882	−	0.349645i	$-0.113698\pi$
$74$		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	$-0.180053\pi$
$79$	15.0075			1.68848			0.844239	+	0.535966i	$0.180053\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.1158	+	10.1158i	−1.11036	+	1.11036i	−0.117253	+	0.993102i	$0.537409\pi$
$83$	−10.1158	+	10.1158i	−1.11036	+	1.11036i	−0.993102	+	0.117253i	$0.962591\pi$
$84$		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	$0.0241219\pi$
$89$			1.42847i			0.151417i	−0.997130	+	0.0757086i	$0.975878\pi$
$90$		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	0.711169	−	0.703021i	$-0.248166\pi$
$101$	0.0818942	+	0.0818942i	0.00814878	+	0.00814878i	−0.703021	+	0.711169i	$0.748166\pi$
$102$		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	$-0.415753\pi$
$107$	−7.27798	−	7.27798i	−0.703589	−	0.703589i	−0.965179	+	0.261590i	$0.915753\pi$
$108$		0			0
$109$	7.04057	−	7.04057i	0.674365	−	0.674365i	−0.284355	−	0.958719i	$-0.591779\pi$
$109$	7.04057	−	7.04057i	0.674365	−	0.674365i	0.958719	+	0.284355i	$0.0917793\pi$
$110$		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	$-0.554150\pi$
$127$	−3.81580			−0.338597			−0.169299	+	0.985565i	$0.554150\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.767438	+	0.767438i	−0.0670514	+	0.0670514i	−0.739837	−	0.672786i	$-0.765098\pi$
$131$	−0.767438	+	0.767438i	−0.0670514	+	0.0670514i	0.672786	+	0.739837i	$0.265098\pi$
$132$		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	$-0.927162\pi$
$137$		−	5.31010i		−	0.453672i	0.973933	−	0.226836i	$-0.0728382\pi$
$138$		0			0
$139$	−8.76744	−	8.76744i	−0.743644	−	0.743644i	0.229633	−	0.973277i	$-0.426247\pi$
$139$	−8.76744	−	8.76744i	−0.743644	−	0.743644i	−0.973277	+	0.229633i	$0.926247\pi$
$140$		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	0.663713	−	0.747987i	$-0.268979\pi$
$149$	−1.02869	−	1.02869i	−0.0842735	−	0.0842735i	−0.747987	+	0.663713i	$0.768979\pi$
$150$		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	−0.907148	−	0.420812i	$-0.861745\pi$
$157$	−6.09378	+	6.09378i	−0.486336	+	0.486336i	0.420812	+	0.907148i	$0.361745\pi$
$158$		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	$-0.810963\pi$
$163$	−3.43692	+	3.43692i	−0.269201	+	0.269201i	0.559577	+	0.828778i	$0.310963\pi$
$164$		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	−0.956551	−	0.291565i	$-0.905824\pi$
$173$	−8.74653	+	8.74653i	−0.664987	+	0.664987i	0.291565	+	0.956551i	$0.405824\pi$
$174$		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	$-0.606729\pi$
$179$	8.23163	−	8.23163i	0.615261	−	0.615261i	0.944312	+	0.329051i	$0.106729\pi$
$180$		0			0
$181$	−6.72269	−	6.72269i	−0.499694	−	0.499694i	0.411649	−	0.911343i	$-0.364953\pi$
$181$	−6.72269	−	6.72269i	−0.499694	−	0.499694i	−0.911343	+	0.411649i	$0.864953\pi$
$182$		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	$-0.228784\pi$
$191$	20.8032			1.50526			0.752632	+	0.658441i	$0.228784\pi$
$192$		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	0.615277	−	0.788311i	$-0.289044\pi$
$197$	−2.42865	−	2.42865i	−0.173034	−	0.173034i	−0.788311	+	0.615277i	$0.789044\pi$
$198$		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	$-0.864524\pi$
$211$	−7.23256	+	7.23256i	−0.497910	+	0.497910i	0.412877	+	0.910787i	$0.364524\pi$
$212$		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	$-0.518332\pi$
$223$	−1.71908			−0.115118			−0.0575591	+	0.998342i	$0.518332\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.1158	−	10.1158i	0.671410	−	0.671410i	−0.286631	−	0.958041i	$-0.592535\pi$
$227$	10.1158	−	10.1158i	0.671410	−	0.671410i	0.958041	+	0.286631i	$0.0925353\pi$
$228$		0			0
$229$	12.0195	+	12.0195i	0.794270	+	0.794270i	0.982185	−	0.187915i	$-0.0601730\pi$
$229$	12.0195	+	12.0195i	0.794270	+	0.794270i	−0.187915	+	0.982185i	$0.560173\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			13.3779i			0.876418i	0.898873	+	0.438209i	$0.144387\pi$
$233$			13.3779i			0.876418i	−0.898873	+	0.438209i	$0.855613\pi$
$234$		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	$-0.642310\pi$
$239$	−13.3675			−0.864670			−0.432335	+	0.901713i	$0.642310\pi$
$240$		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	$-0.0958102\pi$
$251$	10.4337	+	10.4337i	0.658569	+	0.658569i	−0.296472	+	0.955041i	$0.595810\pi$
$252$		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	−0.111289	−	0.993788i	$-0.535498\pi$
$269$	14.4741	−	14.4741i	0.882500	−	0.882500i	0.993788	+	0.111289i	$0.0354978\pi$
$270$		0			0
$271$	−14.0370			−0.852685			−0.426342	−	0.904562i	$-0.640198\pi$
$271$	−14.0370			−0.852685			−0.426342	+	0.904562i	$0.640198\pi$
$272$		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	0.361980	−	0.932186i	$-0.382101\pi$
$277$	−9.49013	−	9.49013i	−0.570207	−	0.570207i	−0.932186	+	0.361980i	$0.882101\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3.89359i		−	0.232272i	−0.993233	−	0.116136i	$-0.962949\pi$
$281$		−	3.89359i		−	0.232272i	0.993233	−	0.116136i	$-0.0370509\pi$
$282$		0			0
$283$	12.4853	+	12.4853i	0.742173	+	0.742173i	0.972996	−	0.230823i	$-0.0741418\pi$
$283$	12.4853	+	12.4853i	0.742173	+	0.742173i	−0.230823	+	0.972996i	$0.574142\pi$
$284$		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	0.301704	−	0.953402i	$-0.402444\pi$
$293$	−11.1553	−	11.1553i	−0.651697	−	0.651697i	−0.953402	+	0.301704i	$0.902444\pi$
$294$		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	$-0.680029\pi$
$307$	5.40320	−	5.40320i	0.308377	−	0.308377i	0.844280	+	0.535903i	$0.180029\pi$
$308$		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	$-0.844456\pi$
$313$		−	16.6105i		−	0.938881i	0.882964	−	0.469441i	$-0.155544\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.81170	−	1.81170i	0.101755	−	0.101755i	−0.654397	−	0.756152i	$-0.727077\pi$
$317$	1.81170	−	1.81170i	0.101755	−	0.101755i	0.756152	+	0.654397i	$0.227077\pi$
$318$		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	$-0.426186\pi$
$331$	−13.5252	−	13.5252i	−0.743411	−	0.743411i	−0.973233	+	0.229822i	$0.926186\pi$
$332$		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	0.992284	+	0.123983i	$0.0395669\pi$
$347$	20.7938	+	20.7938i	1.11627	+	1.11627i	0.123983	+	0.992284i	$0.460433\pi$
$348$		0			0
$349$	19.2855	−	19.2855i	1.03233	−	1.03233i	0.0328700	−	0.999460i	$-0.489535\pi$
$349$	19.2855	−	19.2855i	1.03233	−	1.03233i	0.999460	−	0.0328700i	$-0.0104647\pi$
$350$		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	$-0.754198\pi$
$367$	−27.4474			−1.43274			−0.716371	+	0.697720i	$0.754198\pi$
$368$		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	0.300257	−	0.953858i	$-0.402928\pi$
$373$	−12.6231	−	12.6231i	−0.653601	−	0.653601i	−0.953858	+	0.300257i	$0.902928\pi$
$374$		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	$-0.110689\pi$
$379$	11.6686	+	11.6686i	0.599373	+	0.599373i	−0.340772	+	0.940146i	$0.610689\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.1885			0.878291			0.439145	−	0.898416i	$-0.355281\pi$
$383$	17.1885			0.878291			0.439145	+	0.898416i	$0.355281\pi$
$384$		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	0.657774	−	0.753215i	$-0.271498\pi$
$389$	−1.88238	−	1.88238i	−0.0954404	−	0.0954404i	−0.753215	+	0.657774i	$0.771498\pi$
$390$		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	−0.885950	−	0.463781i	$-0.846493\pi$
$397$	−8.41166	+	8.41166i	−0.422169	+	0.422169i	0.463781	+	0.885950i	$0.346493\pi$
$398$		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	$0.110167\pi$
$409$			13.7211i			0.678464i	−0.940703	+	0.339232i	$0.889833\pi$
$410$		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	$-0.854195\pi$
$419$	−9.30755	+	9.30755i	−0.454703	+	0.454703i	0.442209	+	0.896912i	$0.354195\pi$
$420$		0			0
$421$	−8.44378	−	8.44378i	−0.411525	−	0.411525i	0.470745	−	0.882269i	$-0.343985\pi$
$421$	−8.44378	−	8.44378i	−0.411525	−	0.411525i	−0.882269	+	0.470745i	$0.843985\pi$
$422$		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	$-0.236192\pi$
$431$	30.6054			1.47421			0.737105	+	0.675778i	$0.236192\pi$
$432$		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	$-0.225505\pi$
$443$	2.28832	+	2.28832i	0.108721	+	0.108721i	−0.650653	+	0.759375i	$0.725505\pi$
$444$		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	$0.0821688\pi$
$457$			10.9147i			0.510567i	−0.966866	+	0.255284i	$0.917831\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.8319	−	17.8319i	0.830512	−	0.830512i	−0.157075	−	0.987587i	$-0.550206\pi$
$461$	17.8319	−	17.8319i	0.830512	−	0.830512i	0.987587	+	0.157075i	$0.0502063\pi$
$462$		0			0
$463$	22.4937			1.04537			0.522686	−	0.852525i	$-0.324930\pi$
$463$	22.4937			1.04537			0.522686	+	0.852525i	$0.324930\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.2171	−	24.2171i	1.12063	−	1.12063i	0.128989	−	0.991646i	$-0.458827\pi$
$467$	24.2171	−	24.2171i	1.12063	−	1.12063i	0.991646	−	0.128989i	$-0.0411731\pi$
$468$		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	$-0.189571\pi$
$479$	36.2362			1.65568			0.827838	+	0.560968i	$0.189571\pi$
$480$		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	$-0.187574\pi$
$491$	6.10641	+	6.10641i	0.275578	+	0.275578i	−0.555763	+	0.831341i	$0.687574\pi$
$492$		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	$-0.535969\pi$
$499$	19.6770	−	19.6770i	0.880864	−	0.880864i	0.993622	+	0.112758i	$0.0359686\pi$
$500$		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	−0.667606	−	0.744515i	$-0.732681\pi$
$509$	1.73514	−	1.73514i	0.0769087	−	0.0769087i	0.744515	+	0.667606i	$0.232681\pi$
$510$		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	$-0.736760\pi$
$521$		−	33.5944i		−	1.47180i	0.677092	−	0.735898i	$-0.263240\pi$
$522$		0			0
$523$	21.8158	+	21.8158i	0.953938	+	0.953938i	0.998985	−	0.0450467i	$-0.0143437\pi$
$523$	21.8158	+	21.8158i	0.953938	+	0.953938i	−0.0450467	+	0.998985i	$0.514344\pi$
$524$		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	−0.187669	+	0.982232i	$0.560093\pi$
$541$	−27.2112	+	27.2112i	−1.16990	+	1.16990i	−0.982232	+	0.187669i	$0.939907\pi$
$542$		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	$-0.684078\pi$
$547$	6.80116	−	6.80116i	0.290796	−	0.290796i	0.837395	+	0.546598i	$0.184078\pi$
$548$		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	−0.792190	−	0.610274i	$-0.791059\pi$
$557$	−4.29337	+	4.29337i	−0.181916	+	0.181916i	0.610274	+	0.792190i	$0.291059\pi$
$558$		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	$-0.847119\pi$
$563$	−10.0801	+	10.0801i	−0.424825	+	0.424825i	0.462036	+	0.886861i	$0.347119\pi$
$564$		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	$0.238574\pi$
$569$			32.5018i			1.36255i	−0.732029	+	0.681274i	$0.761426\pi$
$570$		0			0
$571$	9.17157	+	9.17157i	0.383818	+	0.383818i	0.872476	−	0.488657i	$-0.162513\pi$
$571$	9.17157	+	9.17157i	0.383818	+	0.383818i	−0.488657	+	0.872476i	$0.662513\pi$
$572$		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	$-0.189625\pi$
$587$	6.46002	+	6.46002i	0.266634	+	0.266634i	−0.561109	+	0.827742i	$0.689625\pi$
$588$		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	$-0.934788\pi$
$601$		−	9.97474i		−	0.406878i	0.979088	−	0.203439i	$-0.0652119\pi$
$602$		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	$-0.470767\pi$
$607$	4.51900			0.183421			0.0917103	+	0.995786i	$0.470767\pi$
$608$		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	0.856655	−	0.515890i	$-0.172539\pi$
$613$	8.43692	+	8.43692i	0.340764	+	0.340764i	−0.515890	+	0.856655i	$0.672539\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.1201i		−	1.29311i	−0.762869	−	0.646554i	$-0.776210\pi$
$617$		−	32.1201i		−	1.29311i	0.762869	−	0.646554i	$-0.223790\pi$
$618$		0			0
$619$	−15.0412	−	15.0412i	−0.604559	−	0.604559i	0.336960	−	0.941519i	$-0.390601\pi$
$619$	−15.0412	−	15.0412i	−0.604559	−	0.604559i	−0.941519	+	0.336960i	$0.890601\pi$
$620$		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	$-0.903225\pi$
$643$	−16.6034	+	16.6034i	−0.654774	+	0.654774i	0.299365	+	0.954139i	$0.403225\pi$
$644$		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	−0.687606	−	0.726084i	$-0.741338\pi$
$653$	0.983270	−	0.983270i	0.0384783	−	0.0384783i	0.726084	+	0.687606i	$0.241338\pi$
$654$		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	$-0.915693\pi$
$659$	−18.0559	+	18.0559i	−0.703357	+	0.703357i	0.261772	+	0.965130i	$0.415693\pi$
$660$		0			0
$661$	4.55890	+	4.55890i	0.177321	+	0.177321i	0.790187	−	0.612866i	$-0.209983\pi$
$661$	4.55890	+	4.55890i	0.177321	+	0.177321i	−0.612866	+	0.790187i	$0.709983\pi$
$662$		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	0.996360	−	0.0852405i	$-0.0271659\pi$
$677$	23.7066	+	23.7066i	0.911120	+	0.911120i	−0.0852405	+	0.996360i	$0.527166\pi$
$678$		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	$-0.410278\pi$
$683$	−17.8337	−	17.8337i	−0.682386	−	0.682386i	−0.960537	+	0.278151i	$0.910278\pi$
$684$		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	$-0.844325\pi$
$691$	−10.8557	+	10.8557i	−0.412970	+	0.412970i	0.469802	+	0.882772i	$0.344325\pi$
$692$		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	−0.812680	−	0.582711i	$-0.801992\pi$
$701$	−6.08875	+	6.08875i	−0.229969	+	0.229969i	0.582711	+	0.812680i	$0.301992\pi$
$702$		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	0.131856	−	0.991269i	$-0.457906\pi$
$709$	−22.8836	−	22.8836i	−0.859413	−	0.859413i	−0.991269	+	0.131856i	$0.957906\pi$
$710$		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	$-0.491289\pi$
$719$	1.46744			0.0547262			0.0273631	+	0.999626i	$0.491289\pi$
$720$		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	−0.439536	−	0.898225i	$-0.644857\pi$
$733$	12.4185	−	12.4185i	0.458688	−	0.458688i	0.898225	+	0.439536i	$0.144857\pi$
$734$		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	$-0.625504\pi$
$739$	14.6559	−	14.6559i	0.539127	−	0.539127i	0.923273	+	0.384146i	$0.125504\pi$
$740$		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	$-0.317301\pi$
$751$	29.7594			1.08594			0.542968	+	0.839753i	$0.317301\pi$
$752$		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	0.363365	−	0.931647i	$-0.381628\pi$
$757$	−15.6355	−	15.6355i	−0.568282	−	0.568282i	−0.931647	+	0.363365i	$0.881628\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			4.55957i			0.165284i	0.996579	+	0.0826422i	$0.0263359\pi$
$761$			4.55957i			0.165284i	−0.996579	+	0.0826422i	$0.973664\pi$
$762$		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	0.285421	−	0.958402i	$-0.407867\pi$
$773$	−18.7108	−	18.7108i	−0.672981	−	0.672981i	−0.958402	+	0.285421i	$0.907867\pi$
$774$		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	$-0.859463\pi$
$787$	−13.3759	+	13.3759i	−0.476801	+	0.476801i	0.427306	+	0.904107i	$0.359463\pi$
$788$		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	−0.222471	+	0.974939i	$0.571412\pi$
$797$	−33.8043	+	33.8043i	−1.19741	+	1.19741i	−0.974939	+	0.222471i	$0.928588\pi$
$798$		0			0

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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-2.68554 - 2.68554i) q^{5} +2.15894i q^{7} +O(q^{10})\)	\(q+(-2.68554 - 2.68554i) q^{5} +2.15894i q^{7} +(1.79793 + 1.79793i) q^{11} +(-1.38372 + 1.38372i) q^{13} +0.224777 q^{17} +(-0.158942 + 0.158942i) q^{19} -2.82843i q^{23} +9.42429i q^{25} +(-1.85712 + 1.85712i) q^{29} +1.84106 q^{31} +(5.79793 - 5.79793i) q^{35} +(3.66949 + 3.66949i) q^{37} +5.88163i q^{41} +(7.75481 + 7.75481i) q^{43} +2.82843 q^{47} +2.33897 q^{49} +(7.51397 + 7.51397i) q^{53} -9.65685i q^{55} +(4.00000 + 4.00000i) q^{59} +(-5.98737 + 5.98737i) q^{61} +7.43208 q^{65} +(10.4243 - 10.4243i) q^{67} +4.31788i q^{71} -5.97474i q^{73} +(-3.88163 + 3.88163i) q^{77} +15.0075 q^{79} +(-10.1158 + 10.1158i) q^{83} +(-0.603650 - 0.603650i) q^{85} +1.42847i q^{89} +(-2.98737 - 2.98737i) q^{91} +0.853690 q^{95} -16.3990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{11} + 8 q^{19} - 16 q^{29} + 24 q^{31} + 24 q^{35} + 16 q^{37} + 8 q^{43} - 8 q^{49} + 16 q^{53} + 32 q^{59} - 16 q^{61} + 16 q^{65} + 16 q^{67} + 16 q^{77} - 24 q^{79} - 40 q^{83} + 16 q^{85} + 8 q^{91} + 48 q^{95}+O(q^{100}) \) 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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