LMFDB - Embedded newform 8048.2.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.a (trivial)

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:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 $ x^10 - 2x^9 - 9x^8 + 14x^7 + 27x^6 - 27x^5 - 34x^4 + 14x^3 + 17x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.489003$ of defining polynomial
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48900 q^{3} -1.79865 q^{5} -0.552233 q^{7} -0.782869 q^{9} +O(q^{10})$	$q+1.48900 q^{3} -1.79865 q^{5} -0.552233 q^{7} -0.782869 q^{9} -4.33718 q^{11} +2.54873 q^{13} -2.67819 q^{15} +2.52304 q^{17} +5.11893 q^{19} -0.822276 q^{21} +3.78409 q^{23} -1.76486 q^{25} -5.63270 q^{27} +0.907287 q^{29} -0.380820 q^{31} -6.45807 q^{33} +0.993273 q^{35} -5.43266 q^{37} +3.79507 q^{39} +5.72459 q^{41} +9.21553 q^{43} +1.40811 q^{45} +8.81077 q^{47} -6.69504 q^{49} +3.75682 q^{51} -6.46357 q^{53} +7.80106 q^{55} +7.62210 q^{57} -3.40568 q^{59} -1.06109 q^{61} +0.432326 q^{63} -4.58427 q^{65} +0.253929 q^{67} +5.63451 q^{69} -11.7311 q^{71} -16.1271 q^{73} -2.62789 q^{75} +2.39513 q^{77} -7.76447 q^{79} -6.03851 q^{81} +16.3846 q^{83} -4.53807 q^{85} +1.35095 q^{87} -3.09992 q^{89} -1.40749 q^{91} -0.567042 q^{93} -9.20715 q^{95} -4.28605 q^{97} +3.39544 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.48900	0.859676	0.429838	−	0.902906i	$-0.358571\pi$
$3$	1.48900	0.859676	0.429838	+	0.902906i	$0.358571\pi$
$4$	0	0
$5$	−1.79865	−0.804380	−0.402190	−	0.915556i	$-0.631751\pi$
$5$	−1.79865	−0.804380	−0.402190	+	0.915556i	$0.631751\pi$
$6$	0	0
$7$	−0.552233	−0.208724	−0.104362	−	0.994539i	$-0.533280\pi$
$7$	−0.552233	−0.208724	−0.104362	+	0.994539i	$0.533280\pi$
$8$	0	0
$9$	−0.782869	−0.260956
$10$	0	0
$11$	−4.33718	−1.30771	−0.653854	−	0.756620i	$-0.726849\pi$
$11$	−4.33718	−1.30771	−0.653854	+	0.756620i	$0.726849\pi$
$12$	0	0
$13$	2.54873	0.706891	0.353445	−	0.935455i	$-0.385010\pi$
$13$	2.54873	0.706891	0.353445	+	0.935455i	$0.385010\pi$
$14$	0	0
$15$	−2.67819	−0.691507
$16$	0	0
$17$	2.52304	0.611928	0.305964	−	0.952043i	$-0.401021\pi$
$17$	2.52304	0.611928	0.305964	+	0.952043i	$0.401021\pi$
$18$	0	0
$19$	5.11893	1.17436	0.587181	−	0.809455i	$-0.300238\pi$
$19$	5.11893	1.17436	0.587181	+	0.809455i	$0.300238\pi$
$20$	0	0
$21$	−0.822276	−0.179435
$22$	0	0
$23$	3.78409	0.789036	0.394518	−	0.918888i	$-0.370912\pi$
$23$	3.78409	0.789036	0.394518	+	0.918888i	$0.370912\pi$
$24$	0	0
$25$	−1.76486	−0.352972
$26$	0	0
$27$	−5.63270	−1.08401
$28$	0	0
$29$	0.907287	0.168479	0.0842395	−	0.996446i	$-0.473154\pi$
$29$	0.907287	0.168479	0.0842395	+	0.996446i	$0.473154\pi$
$30$	0	0
$31$	−0.380820	−0.0683972	−0.0341986	−	0.999415i	$-0.510888\pi$
$31$	−0.380820	−0.0683972	−0.0341986	+	0.999415i	$0.510888\pi$
$32$	0	0
$33$	−6.45807	−1.12421
$34$	0	0
$35$	0.993273	0.167894
$36$	0	0
$37$	−5.43266	−0.893124	−0.446562	−	0.894753i	$-0.647352\pi$
$37$	−5.43266	−0.893124	−0.446562	+	0.894753i	$0.647352\pi$
$38$	0	0
$39$	3.79507	0.607697
$40$	0	0
$41$	5.72459	0.894031	0.447015	−	0.894526i	$-0.352487\pi$
$41$	5.72459	0.894031	0.447015	+	0.894526i	$0.352487\pi$
$42$	0	0
$43$	9.21553	1.40536	0.702678	−	0.711508i	$-0.251988\pi$
$43$	9.21553	1.40536	0.702678	+	0.711508i	$0.251988\pi$
$44$	0	0
$45$	1.40811	0.209908
$46$	0	0
$47$	8.81077	1.28518	0.642592	−	0.766209i	$-0.277859\pi$
$47$	8.81077	1.28518	0.642592	+	0.766209i	$0.277859\pi$
$48$	0	0
$49$	−6.69504	−0.956434
$50$	0	0
$51$	3.75682	0.526060
$52$	0	0
$53$	−6.46357	−0.887840	−0.443920	−	0.896066i	$-0.646413\pi$
$53$	−6.46357	−0.887840	−0.443920	+	0.896066i	$0.646413\pi$
$54$	0	0
$55$	7.80106	1.05190
$56$	0	0
$57$	7.62210	1.00957
$58$	0	0
$59$	−3.40568	−0.443381	−0.221691	−	0.975117i	$-0.571157\pi$
$59$	−3.40568	−0.443381	−0.221691	+	0.975117i	$0.571157\pi$
$60$	0	0
$61$	−1.06109	−0.135859	−0.0679294	−	0.997690i	$-0.521639\pi$
$61$	−1.06109	−0.135859	−0.0679294	+	0.997690i	$0.521639\pi$
$62$	0	0
$63$	0.432326	0.0544679
$64$	0	0
$65$	−4.58427	−0.568609
$66$	0	0
$67$	0.253929	0.0310223	0.0155112	−	0.999880i	$-0.495062\pi$
$67$	0.253929	0.0310223	0.0155112	+	0.999880i	$0.495062\pi$
$68$	0	0
$69$	5.63451	0.678316
$70$	0	0
$71$	−11.7311	−1.39222	−0.696112	−	0.717933i	$-0.745088\pi$
$71$	−11.7311	−1.39222	−0.696112	+	0.717933i	$0.745088\pi$
$72$	0	0
$73$	−16.1271	−1.88753	−0.943765	−	0.330617i	$-0.892743\pi$
$73$	−16.1271	−1.88753	−0.943765	+	0.330617i	$0.892743\pi$
$74$	0	0
$75$	−2.62789	−0.303442
$76$	0	0
$77$	2.39513	0.272951
$78$	0	0
$79$	−7.76447	−0.873571	−0.436785	−	0.899566i	$-0.643883\pi$
$79$	−7.76447	−0.873571	−0.436785	+	0.899566i	$0.643883\pi$
$80$	0	0
$81$	−6.03851	−0.670945
$82$	0	0
$83$	16.3846	1.79844	0.899222	−	0.437492i	$-0.144133\pi$
$83$	16.3846	1.79844	0.899222	+	0.437492i	$0.144133\pi$
$84$	0	0
$85$	−4.53807	−0.492223
$86$	0	0
$87$	1.35095	0.144837
$88$	0	0
$89$	−3.09992	−0.328590	−0.164295	−	0.986411i	$-0.552535\pi$
$89$	−3.09992	−0.328590	−0.164295	+	0.986411i	$0.552535\pi$
$90$	0	0
$91$	−1.40749	−0.147545
$92$	0	0
$93$	−0.567042	−0.0587995
$94$	0	0
$95$	−9.20715	−0.944634
$96$	0	0
$97$	−4.28605	−0.435183	−0.217591	−	0.976040i	$-0.569820\pi$
$97$	−4.28605	−0.435183	−0.217591	+	0.976040i	$0.569820\pi$
$98$	0	0
$99$	3.39544	0.341255
$100$	0	0
$101$	−3.73446	−0.371592	−0.185796	−	0.982588i	$-0.559486\pi$
$101$	−3.73446	−0.371592	−0.185796	+	0.982588i	$0.559486\pi$
$102$	0	0
$103$	−2.83220	−0.279065	−0.139533	−	0.990217i	$-0.544560\pi$
$103$	−2.83220	−0.279065	−0.139533	+	0.990217i	$0.544560\pi$
$104$	0	0
$105$	1.47899	0.144334
$106$	0	0
$107$	−10.0994	−0.976343	−0.488172	−	0.872748i	$-0.662336\pi$
$107$	−10.0994	−0.976343	−0.488172	+	0.872748i	$0.662336\pi$
$108$	0	0
$109$	−6.86764	−0.657800	−0.328900	−	0.944365i	$-0.606678\pi$
$109$	−6.86764	−0.657800	−0.328900	+	0.944365i	$0.606678\pi$
$110$	0	0
$111$	−8.08925	−0.767798
$112$	0	0
$113$	2.80558	0.263927	0.131963	−	0.991255i	$-0.457872\pi$
$113$	2.80558	0.263927	0.131963	+	0.991255i	$0.457872\pi$
$114$	0	0
$115$	−6.80624	−0.634685
$116$	0	0
$117$	−1.99532	−0.184468
$118$	0	0
$119$	−1.39331	−0.127724
$120$	0	0
$121$	7.81112	0.710102
$122$	0	0
$123$	8.52393	0.768577
$124$	0	0
$125$	12.1676	1.08830
$126$	0	0
$127$	14.9147	1.32347	0.661734	−	0.749739i	$-0.269821\pi$
$127$	14.9147	1.32347	0.661734	+	0.749739i	$0.269821\pi$
$128$	0	0
$129$	13.7220	1.20815
$130$	0	0
$131$	−13.5207	−1.18131	−0.590654	−	0.806925i	$-0.701130\pi$
$131$	−13.5207	−1.18131	−0.590654	+	0.806925i	$0.701130\pi$
$132$	0	0
$133$	−2.82684	−0.245118
$134$	0	0
$135$	10.1313	0.871960
$136$	0	0
$137$	−12.4479	−1.06350	−0.531750	−	0.846901i	$-0.678465\pi$
$137$	−12.4479	−1.06350	−0.531750	+	0.846901i	$0.678465\pi$
$138$	0	0
$139$	−2.15443	−0.182736	−0.0913681	−	0.995817i	$-0.529124\pi$
$139$	−2.15443	−0.182736	−0.0913681	+	0.995817i	$0.529124\pi$
$140$	0	0
$141$	13.1193	1.10484
$142$	0	0
$143$	−11.0543	−0.924407
$144$	0	0
$145$	−1.63189	−0.135521
$146$	0	0
$147$	−9.96894	−0.822224
$148$	0	0
$149$	−8.59253	−0.703927	−0.351964	−	0.936014i	$-0.614486\pi$
$149$	−8.59253	−0.703927	−0.351964	+	0.936014i	$0.614486\pi$
$150$	0	0
$151$	−0.527862	−0.0429568	−0.0214784	−	0.999769i	$-0.506837\pi$
$151$	−0.527862	−0.0429568	−0.0214784	+	0.999769i	$0.506837\pi$
$152$	0	0
$153$	−1.97521	−0.159687
$154$	0	0
$155$	0.684961	0.0550174
$156$	0	0
$157$	11.1711	0.891554	0.445777	−	0.895144i	$-0.352927\pi$
$157$	11.1711	0.891554	0.445777	+	0.895144i	$0.352927\pi$
$158$	0	0
$159$	−9.62428	−0.763255
$160$	0	0
$161$	−2.08970	−0.164691
$162$	0	0
$163$	−20.0109	−1.56738	−0.783689	−	0.621153i	$-0.786664\pi$
$163$	−20.0109	−1.56738	−0.783689	+	0.621153i	$0.786664\pi$
$164$	0	0
$165$	11.6158	0.904289
$166$	0	0
$167$	11.9520	0.924877	0.462439	−	0.886651i	$-0.346975\pi$
$167$	11.9520	0.924877	0.462439	+	0.886651i	$0.346975\pi$
$168$	0	0
$169$	−6.50397	−0.500305
$170$	0	0
$171$	−4.00745	−0.306457
$172$	0	0
$173$	3.14561	0.239156	0.119578	−	0.992825i	$-0.461846\pi$
$173$	3.14561	0.239156	0.119578	+	0.992825i	$0.461846\pi$
$174$	0	0
$175$	0.974615	0.0736739
$176$	0	0
$177$	−5.07106	−0.381164
$178$	0	0
$179$	6.02096	0.450028	0.225014	−	0.974356i	$-0.427757\pi$
$179$	6.02096	0.450028	0.225014	+	0.974356i	$0.427757\pi$
$180$	0	0
$181$	−23.0475	−1.71311	−0.856553	−	0.516059i	$-0.827398\pi$
$181$	−23.0475	−1.71311	−0.856553	+	0.516059i	$0.827398\pi$
$182$	0	0
$183$	−1.57997	−0.116795
$184$	0	0
$185$	9.77145	0.718412
$186$	0	0
$187$	−10.9429	−0.800224
$188$	0	0
$189$	3.11056	0.226260
$190$	0	0
$191$	17.9774	1.30080	0.650398	−	0.759594i	$-0.274602\pi$
$191$	17.9774	1.30080	0.650398	+	0.759594i	$0.274602\pi$
$192$	0	0
$193$	−10.4429	−0.751697	−0.375848	−	0.926681i	$-0.622649\pi$
$193$	−10.4429	−0.751697	−0.375848	+	0.926681i	$0.622649\pi$
$194$	0	0
$195$	−6.82600	−0.488820
$196$	0	0
$197$	−4.36026	−0.310656	−0.155328	−	0.987863i	$-0.549643\pi$
$197$	−4.36026	−0.310656	−0.155328	+	0.987863i	$0.549643\pi$
$198$	0	0
$199$	−14.0205	−0.993888	−0.496944	−	0.867783i	$-0.665545\pi$
$199$	−14.0205	−0.993888	−0.496944	+	0.867783i	$0.665545\pi$
$200$	0	0
$201$	0.378101	0.0266692
$202$	0	0
$203$	−0.501034	−0.0351657
$204$	0	0
$205$	−10.2965	−0.719141
$206$	0	0
$207$	−2.96244	−0.205904
$208$	0	0
$209$	−22.2017	−1.53572
$210$	0	0
$211$	11.0580	0.761265	0.380633	−	0.924726i	$-0.375706\pi$
$211$	11.0580	0.761265	0.380633	+	0.924726i	$0.375706\pi$
$212$	0	0
$213$	−17.4676	−1.19686
$214$	0	0
$215$	−16.5755	−1.13044
$216$	0	0
$217$	0.210301	0.0142762
$218$	0	0
$219$	−24.0133	−1.62267
$220$	0	0
$221$	6.43056	0.432566
$222$	0	0
$223$	15.5502	1.04132	0.520658	−	0.853765i	$-0.325687\pi$
$223$	15.5502	1.04132	0.520658	+	0.853765i	$0.325687\pi$
$224$	0	0
$225$	1.38166	0.0921104
$226$	0	0
$227$	−1.27577	−0.0846759	−0.0423379	−	0.999103i	$-0.513481\pi$
$227$	−1.27577	−0.0846759	−0.0423379	+	0.999103i	$0.513481\pi$
$228$	0	0
$229$	−15.5232	−1.02580	−0.512901	−	0.858448i	$-0.671429\pi$
$229$	−15.5232	−1.02580	−0.512901	+	0.858448i	$0.671429\pi$
$230$	0	0
$231$	3.56636	0.234649
$232$	0	0
$233$	−27.8010	−1.82130	−0.910651	−	0.413177i	$-0.864419\pi$
$233$	−27.8010	−1.82130	−0.910651	+	0.413177i	$0.864419\pi$
$234$	0	0
$235$	−15.8475	−1.03378
$236$	0	0
$237$	−11.5613	−0.750988
$238$	0	0
$239$	−1.20369	−0.0778601	−0.0389301	−	0.999242i	$-0.512395\pi$
$239$	−1.20369	−0.0778601	−0.0389301	+	0.999242i	$0.512395\pi$
$240$	0	0
$241$	−25.6025	−1.64920	−0.824602	−	0.565713i	$-0.808601\pi$
$241$	−25.6025	−1.64920	−0.824602	+	0.565713i	$0.808601\pi$
$242$	0	0
$243$	7.90676	0.507219
$244$	0	0
$245$	12.0420	0.769337
$246$	0	0
$247$	13.0468	0.830146
$248$	0	0
$249$	24.3967	1.54608
$250$	0	0
$251$	−13.2638	−0.837205	−0.418603	−	0.908170i	$-0.637480\pi$
$251$	−13.2638	−0.837205	−0.418603	+	0.908170i	$0.637480\pi$
$252$	0	0
$253$	−16.4123	−1.03183
$254$	0	0
$255$	−6.75720	−0.423152
$256$	0	0
$257$	−16.1006	−1.00433	−0.502163	−	0.864773i	$-0.667462\pi$
$257$	−16.1006	−1.00433	−0.502163	+	0.864773i	$0.667462\pi$
$258$	0	0
$259$	3.00009	0.186417
$260$	0	0
$261$	−0.710288	−0.0439657
$262$	0	0
$263$	−6.77336	−0.417663	−0.208832	−	0.977952i	$-0.566966\pi$
$263$	−6.77336	−0.417663	−0.208832	+	0.977952i	$0.566966\pi$
$264$	0	0
$265$	11.6257	0.714161
$266$	0	0
$267$	−4.61579	−0.282481
$268$	0	0
$269$	23.4853	1.43192	0.715962	−	0.698139i	$-0.245988\pi$
$269$	23.4853	1.43192	0.715962	+	0.698139i	$0.245988\pi$
$270$	0	0
$271$	−17.5886	−1.06843	−0.534216	−	0.845348i	$-0.679393\pi$
$271$	−17.5886	−1.06843	−0.534216	+	0.845348i	$0.679393\pi$
$272$	0	0
$273$	−2.09576	−0.126841
$274$	0	0
$275$	7.65452	0.461585
$276$	0	0
$277$	−11.3448	−0.681645	−0.340823	−	0.940128i	$-0.610706\pi$
$277$	−11.3448	−0.681645	−0.340823	+	0.940128i	$0.610706\pi$
$278$	0	0
$279$	0.298132	0.0178487
$280$	0	0
$281$	−9.36752	−0.558820	−0.279410	−	0.960172i	$-0.590139\pi$
$281$	−9.36752	−0.558820	−0.279410	+	0.960172i	$0.590139\pi$
$282$	0	0
$283$	−26.8789	−1.59778	−0.798892	−	0.601474i	$-0.794580\pi$
$283$	−26.8789	−1.59778	−0.798892	+	0.601474i	$0.794580\pi$
$284$	0	0
$285$	−13.7095	−0.812080
$286$	0	0
$287$	−3.16131	−0.186606
$288$	0	0
$289$	−10.6342	−0.625544
$290$	0	0
$291$	−6.38195	−0.374117
$292$	0	0
$293$	17.2365	1.00697	0.503485	−	0.864004i	$-0.332051\pi$
$293$	17.2365	1.00697	0.503485	+	0.864004i	$0.332051\pi$
$294$	0	0
$295$	6.12561	0.356647
$296$	0	0
$297$	24.4301	1.41758
$298$	0	0
$299$	9.64462	0.557763
$300$	0	0
$301$	−5.08912	−0.293332
$302$	0	0
$303$	−5.56062	−0.319449
$304$	0	0
$305$	1.90853	0.109282
$306$	0	0
$307$	−29.0001	−1.65512	−0.827561	−	0.561376i	$-0.810272\pi$
$307$	−29.0001	−1.65512	−0.827561	+	0.561376i	$0.810272\pi$
$308$	0	0
$309$	−4.21716	−0.239906
$310$	0	0
$311$	−27.2312	−1.54414	−0.772071	−	0.635536i	$-0.780779\pi$
$311$	−27.2312	−1.54414	−0.772071	+	0.635536i	$0.780779\pi$
$312$	0	0
$313$	5.53059	0.312607	0.156304	−	0.987709i	$-0.450042\pi$
$313$	5.53059	0.312607	0.156304	+	0.987709i	$0.450042\pi$
$314$	0	0
$315$	−0.777603	−0.0438129
$316$	0	0
$317$	31.2030	1.75253	0.876267	−	0.481826i	$-0.160026\pi$
$317$	31.2030	1.75253	0.876267	+	0.481826i	$0.160026\pi$
$318$	0	0
$319$	−3.93507	−0.220322
$320$	0	0
$321$	−15.0380	−0.839339
$322$	0	0
$323$	12.9153	0.718626
$324$	0	0
$325$	−4.49816	−0.249513
$326$	0	0
$327$	−10.2259	−0.565495
$328$	0	0
$329$	−4.86560	−0.268249
$330$	0	0
$331$	−16.8025	−0.923551	−0.461775	−	0.886997i	$-0.652787\pi$
$331$	−16.8025	−0.923551	−0.461775	+	0.886997i	$0.652787\pi$
$332$	0	0
$333$	4.25307	0.233067
$334$	0	0
$335$	−0.456729	−0.0249538
$336$	0	0
$337$	7.57943	0.412878	0.206439	−	0.978459i	$-0.433813\pi$
$337$	7.57943	0.412878	0.206439	+	0.978459i	$0.433813\pi$
$338$	0	0
$339$	4.17752	0.226892
$340$	0	0
$341$	1.65168	0.0894437
$342$	0	0
$343$	7.56285	0.408355
$344$	0	0
$345$	−10.1345	−0.545624
$346$	0	0
$347$	28.0855	1.50771	0.753853	−	0.657043i	$-0.228193\pi$
$347$	28.0855	1.50771	0.753853	+	0.657043i	$0.228193\pi$
$348$	0	0
$349$	11.6230	0.622164	0.311082	−	0.950383i	$-0.399309\pi$
$349$	11.6230	0.622164	0.311082	+	0.950383i	$0.399309\pi$
$350$	0	0
$351$	−14.3563	−0.766280
$352$	0	0
$353$	−6.16556	−0.328160	−0.164080	−	0.986447i	$-0.552465\pi$
$353$	−6.16556	−0.328160	−0.164080	+	0.986447i	$0.552465\pi$
$354$	0	0
$355$	21.1001	1.11988
$356$	0	0
$357$	−2.07464	−0.109802
$358$	0	0
$359$	10.3896	0.548344	0.274172	−	0.961681i	$-0.411596\pi$
$359$	10.3896	0.548344	0.274172	+	0.961681i	$0.411596\pi$
$360$	0	0
$361$	7.20342	0.379127
$362$	0	0
$363$	11.6308	0.610458
$364$	0	0
$365$	29.0069	1.51829
$366$	0	0
$367$	−0.571294	−0.0298213	−0.0149107	−	0.999889i	$-0.504746\pi$
$367$	−0.571294	−0.0298213	−0.0149107	+	0.999889i	$0.504746\pi$
$368$	0	0
$369$	−4.48161	−0.233303
$370$	0	0
$371$	3.56940	0.185314
$372$	0	0
$373$	2.95332	0.152917	0.0764586	−	0.997073i	$-0.475639\pi$
$373$	2.95332	0.152917	0.0764586	+	0.997073i	$0.475639\pi$
$374$	0	0
$375$	18.1176	0.935590
$376$	0	0
$377$	2.31243	0.119096
$378$	0	0
$379$	26.2705	1.34942	0.674711	−	0.738082i	$-0.264268\pi$
$379$	26.2705	1.34942	0.674711	+	0.738082i	$0.264268\pi$
$380$	0	0
$381$	22.2081	1.13775
$382$	0	0
$383$	−3.44503	−0.176033	−0.0880164	−	0.996119i	$-0.528053\pi$
$383$	−3.44503	−0.176033	−0.0880164	+	0.996119i	$0.528053\pi$
$384$	0	0
$385$	−4.30800	−0.219556
$386$	0	0
$387$	−7.21456	−0.366736
$388$	0	0
$389$	−7.95150	−0.403157	−0.201578	−	0.979472i	$-0.564607\pi$
$389$	−7.95150	−0.403157	−0.201578	+	0.979472i	$0.564607\pi$
$390$	0	0
$391$	9.54742	0.482834
$392$	0	0
$393$	−20.1323	−1.01554
$394$	0	0
$395$	13.9656	0.702683
$396$	0	0
$397$	21.7255	1.09037	0.545186	−	0.838315i	$-0.316459\pi$
$397$	21.7255	1.09037	0.545186	+	0.838315i	$0.316459\pi$
$398$	0	0
$399$	−4.20917	−0.210722
$400$	0	0
$401$	14.1072	0.704481	0.352241	−	0.935909i	$-0.385420\pi$
$401$	14.1072	0.704481	0.352241	+	0.935909i	$0.385420\pi$
$402$	0	0
$403$	−0.970607	−0.0483494
$404$	0	0
$405$	10.8612	0.539695
$406$	0	0
$407$	23.5624	1.16795
$408$	0	0
$409$	−10.2905	−0.508834	−0.254417	−	0.967095i	$-0.581884\pi$
$409$	−10.2905	−0.508834	−0.254417	+	0.967095i	$0.581884\pi$
$410$	0	0
$411$	−18.5350	−0.914266
$412$	0	0
$413$	1.88072	0.0925444
$414$	0	0
$415$	−29.4702	−1.44663
$416$	0	0
$417$	−3.20795	−0.157094
$418$	0	0
$419$	−19.0965	−0.932925	−0.466463	−	0.884541i	$-0.654472\pi$
$419$	−19.0965	−0.932925	−0.466463	+	0.884541i	$0.654472\pi$
$420$	0	0
$421$	26.3025	1.28191	0.640953	−	0.767580i	$-0.278539\pi$
$421$	26.3025	1.28191	0.640953	+	0.767580i	$0.278539\pi$
$422$	0	0
$423$	−6.89768	−0.335377
$424$	0	0
$425$	−4.45283	−0.215994
$426$	0	0
$427$	0.585969	0.0283570
$428$	0	0
$429$	−16.4599	−0.794691
$430$	0	0
$431$	16.5246	0.795963	0.397982	−	0.917393i	$-0.369711\pi$
$431$	16.5246	0.795963	0.397982	+	0.917393i	$0.369711\pi$
$432$	0	0
$433$	4.52467	0.217442	0.108721	−	0.994072i	$-0.465325\pi$
$433$	4.52467	0.217442	0.108721	+	0.994072i	$0.465325\pi$
$434$	0	0
$435$	−2.42989	−0.116504
$436$	0	0
$437$	19.3705	0.926615
$438$	0	0
$439$	9.06783	0.432784	0.216392	−	0.976307i	$-0.430571\pi$
$439$	9.06783	0.432784	0.216392	+	0.976307i	$0.430571\pi$
$440$	0	0
$441$	5.24134	0.249588
$442$	0	0
$443$	12.6542	0.601219	0.300609	−	0.953747i	$-0.402810\pi$
$443$	12.6542	0.601219	0.300609	+	0.953747i	$0.402810\pi$
$444$	0	0
$445$	5.57566	0.264312
$446$	0	0
$447$	−12.7943	−0.605150
$448$	0	0
$449$	34.8336	1.64390	0.821950	−	0.569560i	$-0.192886\pi$
$449$	34.8336	1.64390	0.821950	+	0.569560i	$0.192886\pi$
$450$	0	0
$451$	−24.8286	−1.16913
$452$	0	0
$453$	−0.785988	−0.0369290
$454$	0	0
$455$	2.53158	0.118683
$456$	0	0
$457$	−14.3718	−0.672283	−0.336141	−	0.941812i	$-0.609122\pi$
$457$	−14.3718	−0.672283	−0.336141	+	0.941812i	$0.609122\pi$
$458$	0	0
$459$	−14.2116	−0.663339
$460$	0	0
$461$	−20.9323	−0.974914	−0.487457	−	0.873147i	$-0.662075\pi$
$461$	−20.9323	−0.974914	−0.487457	+	0.873147i	$0.662075\pi$
$462$	0	0
$463$	−19.4204	−0.902541	−0.451271	−	0.892387i	$-0.649029\pi$
$463$	−19.4204	−0.902541	−0.451271	+	0.892387i	$0.649029\pi$
$464$	0	0
$465$	1.01991	0.0472971
$466$	0	0
$467$	−22.0847	−1.02196	−0.510980	−	0.859593i	$-0.670717\pi$
$467$	−22.0847	−1.02196	−0.510980	+	0.859593i	$0.670717\pi$
$468$	0	0
$469$	−0.140228	−0.00647512
$470$	0	0
$471$	16.6339	0.766448
$472$	0	0
$473$	−39.9694	−1.83780
$474$	0	0
$475$	−9.03420	−0.414518
$476$	0	0
$477$	5.06013	0.231688
$478$	0	0
$479$	−25.4118	−1.16109	−0.580547	−	0.814227i	$-0.697161\pi$
$479$	−25.4118	−1.16109	−0.580547	+	0.814227i	$0.697161\pi$
$480$	0	0
$481$	−13.8464	−0.631342
$482$	0	0
$483$	−3.11156	−0.141581
$484$	0	0
$485$	7.70911	0.350053
$486$	0	0
$487$	36.7671	1.66608	0.833039	−	0.553214i	$-0.186599\pi$
$487$	36.7671	1.66608	0.833039	+	0.553214i	$0.186599\pi$
$488$	0	0
$489$	−29.7964	−1.34744
$490$	0	0
$491$	12.6572	0.571212	0.285606	−	0.958347i	$-0.407805\pi$
$491$	12.6572	0.571212	0.285606	+	0.958347i	$0.407805\pi$
$492$	0	0
$493$	2.28913	0.103097
$494$	0	0
$495$	−6.10721	−0.274499
$496$	0	0
$497$	6.47829	0.290591
$498$	0	0
$499$	−2.39500	−0.107215	−0.0536074	−	0.998562i	$-0.517072\pi$
$499$	−2.39500	−0.107215	−0.0536074	+	0.998562i	$0.517072\pi$
$500$	0	0
$501$	17.7966	0.795095
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	6.71697	0.298901
$506$	0	0
$507$	−9.68443	−0.430101
$508$	0	0
$509$	6.46329	0.286480	0.143240	−	0.989688i	$-0.454248\pi$
$509$	6.46329	0.286480	0.143240	+	0.989688i	$0.454248\pi$
$510$	0	0
$511$	8.90589	0.393973
$512$	0	0
$513$	−28.8334	−1.27303
$514$	0	0
$515$	5.09414	0.224475
$516$	0	0
$517$	−38.2139	−1.68065
$518$	0	0
$519$	4.68382	0.205597
$520$	0	0
$521$	34.3036	1.50287	0.751434	−	0.659808i	$-0.229363\pi$
$521$	34.3036	1.50287	0.751434	+	0.659808i	$0.229363\pi$
$522$	0	0
$523$	−30.3662	−1.32782	−0.663910	−	0.747812i	$-0.731104\pi$
$523$	−30.3662	−1.32782	−0.663910	+	0.747812i	$0.731104\pi$
$524$	0	0
$525$	1.45120	0.0633357
$526$	0	0
$527$	−0.960825	−0.0418542
$528$	0	0
$529$	−8.68070	−0.377422
$530$	0	0
$531$	2.66620	0.115703
$532$	0	0
$533$	14.5904	0.631982
$534$	0	0
$535$	18.1652	0.785351
$536$	0	0
$537$	8.96523	0.386878
$538$	0	0
$539$	29.0376	1.25074
$540$	0	0
$541$	39.1371	1.68263	0.841317	−	0.540542i	$-0.181781\pi$
$541$	39.1371	1.68263	0.841317	+	0.540542i	$0.181781\pi$
$542$	0	0
$543$	−34.3178	−1.47272
$544$	0	0
$545$	12.3525	0.529121
$546$	0	0
$547$	−24.0487	−1.02825	−0.514125	−	0.857715i	$-0.671883\pi$
$547$	−24.0487	−1.02825	−0.514125	+	0.857715i	$0.671883\pi$
$548$	0	0
$549$	0.830696	0.0354532
$550$	0	0
$551$	4.64434	0.197855
$552$	0	0
$553$	4.28779	0.182335
$554$	0	0
$555$	14.5497	0.617602
$556$	0	0
$557$	−17.2987	−0.732971	−0.366486	−	0.930424i	$-0.619439\pi$
$557$	−17.2987	−0.732971	−0.366486	+	0.930424i	$0.619439\pi$
$558$	0	0
$559$	23.4879	0.993433
$560$	0	0
$561$	−16.2940	−0.687934
$562$	0	0
$563$	−5.63944	−0.237674	−0.118837	−	0.992914i	$-0.537917\pi$
$563$	−5.63944	−0.237674	−0.118837	+	0.992914i	$0.537917\pi$
$564$	0	0
$565$	−5.04626	−0.212298
$566$	0	0
$567$	3.33466	0.140043
$568$	0	0
$569$	−12.0350	−0.504535	−0.252268	−	0.967658i	$-0.581176\pi$
$569$	−12.0350	−0.504535	−0.252268	+	0.967658i	$0.581176\pi$
$570$	0	0
$571$	−29.9404	−1.25297	−0.626484	−	0.779434i	$-0.715507\pi$
$571$	−29.9404	−1.25297	−0.626484	+	0.779434i	$0.715507\pi$
$572$	0	0
$573$	26.7683	1.11826
$574$	0	0
$575$	−6.67839	−0.278508
$576$	0	0
$577$	17.6080	0.733032	0.366516	−	0.930412i	$-0.380551\pi$
$577$	17.6080	0.733032	0.366516	+	0.930412i	$0.380551\pi$
$578$	0	0
$579$	−15.5495	−0.646216
$580$	0	0
$581$	−9.04811	−0.375379
$582$	0	0
$583$	28.0337	1.16104
$584$	0	0
$585$	3.58889	0.148382
$586$	0	0
$587$	21.2268	0.876125	0.438062	−	0.898945i	$-0.355665\pi$
$587$	21.2268	0.876125	0.438062	+	0.898945i	$0.355665\pi$
$588$	0	0
$589$	−1.94939	−0.0803231
$590$	0	0
$591$	−6.49244	−0.267063
$592$	0	0
$593$	1.78783	0.0734175	0.0367087	−	0.999326i	$-0.488313\pi$
$593$	1.78783	0.0734175	0.0367087	+	0.999326i	$0.488313\pi$
$594$	0	0
$595$	2.50607	0.102739
$596$	0	0
$597$	−20.8766	−0.854422
$598$	0	0
$599$	4.15199	0.169646	0.0848228	−	0.996396i	$-0.472968\pi$
$599$	4.15199	0.169646	0.0848228	+	0.996396i	$0.472968\pi$
$600$	0	0
$601$	−4.87221	−0.198742	−0.0993708	−	0.995050i	$-0.531683\pi$
$601$	−4.87221	−0.198742	−0.0993708	+	0.995050i	$0.531683\pi$
$602$	0	0
$603$	−0.198793	−0.00809548
$604$	0	0
$605$	−14.0495	−0.571192
$606$	0	0
$607$	22.0579	0.895303	0.447652	−	0.894208i	$-0.352261\pi$
$607$	22.0579	0.895303	0.447652	+	0.894208i	$0.352261\pi$
$608$	0	0
$609$	−0.746041	−0.0302311
$610$	0	0
$611$	22.4563	0.908484
$612$	0	0
$613$	−19.0369	−0.768895	−0.384447	−	0.923147i	$-0.625608\pi$
$613$	−19.0369	−0.768895	−0.384447	+	0.923147i	$0.625608\pi$
$614$	0	0
$615$	−15.3316	−0.618228
$616$	0	0
$617$	−22.0697	−0.888494	−0.444247	−	0.895904i	$-0.646529\pi$
$617$	−22.0697	−0.888494	−0.444247	+	0.895904i	$0.646529\pi$
$618$	0	0
$619$	5.69372	0.228850	0.114425	−	0.993432i	$-0.463497\pi$
$619$	5.69372	0.228850	0.114425	+	0.993432i	$0.463497\pi$
$620$	0	0
$621$	−21.3146	−0.855327
$622$	0	0
$623$	1.71187	0.0685848
$624$	0	0
$625$	−13.0609	−0.522438
$626$	0	0
$627$	−33.0584	−1.32023
$628$	0	0
$629$	−13.7069	−0.546528
$630$	0	0
$631$	−37.6327	−1.49813	−0.749067	−	0.662494i	$-0.769498\pi$
$631$	−37.6327	−1.49813	−0.749067	+	0.662494i	$0.769498\pi$
$632$	0	0
$633$	16.4654	0.654442
$634$	0	0
$635$	−26.8263	−1.06457
$636$	0	0
$637$	−17.0639	−0.676095
$638$	0	0
$639$	9.18391	0.363310
$640$	0	0
$641$	6.63550	0.262086	0.131043	−	0.991377i	$-0.458167\pi$
$641$	6.63550	0.262086	0.131043	+	0.991377i	$0.458167\pi$
$642$	0	0
$643$	29.4970	1.16325	0.581625	−	0.813457i	$-0.302417\pi$
$643$	29.4970	1.16325	0.581625	+	0.813457i	$0.302417\pi$
$644$	0	0
$645$	−24.6810	−0.971813
$646$	0	0
$647$	45.6726	1.79558	0.897788	−	0.440427i	$-0.145173\pi$
$647$	45.6726	1.79558	0.897788	+	0.440427i	$0.145173\pi$
$648$	0	0
$649$	14.7710	0.579813
$650$	0	0
$651$	0.313139	0.0122729
$652$	0	0
$653$	−31.6691	−1.23931	−0.619655	−	0.784875i	$-0.712727\pi$
$653$	−31.6691	−1.23931	−0.619655	+	0.784875i	$0.712727\pi$
$654$	0	0
$655$	24.3190	0.950220
$656$	0	0
$657$	12.6254	0.492563
$658$	0	0
$659$	−9.73186	−0.379099	−0.189550	−	0.981871i	$-0.560703\pi$
$659$	−9.73186	−0.379099	−0.189550	+	0.981871i	$0.560703\pi$
$660$	0	0
$661$	−14.5544	−0.566100	−0.283050	−	0.959105i	$-0.591346\pi$
$661$	−14.5544	−0.566100	−0.283050	+	0.959105i	$0.591346\pi$
$662$	0	0
$663$	9.57513	0.371867
$664$	0	0
$665$	5.08449	0.197168
$666$	0	0
$667$	3.43325	0.132936
$668$	0	0
$669$	23.1542	0.895195
$670$	0	0
$671$	4.60214	0.177664
$672$	0	0
$673$	−46.1914	−1.78055	−0.890273	−	0.455427i	$-0.849486\pi$
$673$	−46.1914	−1.78055	−0.890273	+	0.455427i	$0.849486\pi$
$674$	0	0
$675$	9.94095	0.382627
$676$	0	0
$677$	−10.6087	−0.407725	−0.203863	−	0.978999i	$-0.565350\pi$
$677$	−10.6087	−0.407725	−0.203863	+	0.978999i	$0.565350\pi$
$678$	0	0
$679$	2.36690	0.0908333
$680$	0	0
$681$	−1.89963	−0.0727939
$682$	0	0
$683$	35.9985	1.37745	0.688723	−	0.725025i	$-0.258172\pi$
$683$	35.9985	1.37745	0.688723	+	0.725025i	$0.258172\pi$
$684$	0	0
$685$	22.3895	0.855458
$686$	0	0
$687$	−23.1141	−0.881858
$688$	0	0
$689$	−16.4739	−0.627606
$690$	0	0
$691$	−24.7115	−0.940070	−0.470035	−	0.882648i	$-0.655759\pi$
$691$	−24.7115	−0.940070	−0.470035	+	0.882648i	$0.655759\pi$
$692$	0	0
$693$	−1.87508	−0.0712282
$694$	0	0
$695$	3.87506	0.146989
$696$	0	0
$697$	14.4434	0.547083
$698$	0	0
$699$	−41.3957	−1.56573
$700$	0	0
$701$	28.8712	1.09045	0.545225	−	0.838290i	$-0.316444\pi$
$701$	28.8712	1.09045	0.545225	+	0.838290i	$0.316444\pi$
$702$	0	0
$703$	−27.8094	−1.04885
$704$	0	0
$705$	−23.5970	−0.888713
$706$	0	0
$707$	2.06229	0.0775603
$708$	0	0
$709$	−31.6656	−1.18923	−0.594614	−	0.804012i	$-0.702695\pi$
$709$	−31.6656	−1.18923	−0.594614	+	0.804012i	$0.702695\pi$
$710$	0	0
$711$	6.07856	0.227964
$712$	0	0
$713$	−1.44105	−0.0539679
$714$	0	0
$715$	19.8828	0.743575
$716$	0	0
$717$	−1.79230	−0.0669345
$718$	0	0
$719$	−30.1524	−1.12449	−0.562247	−	0.826969i	$-0.690063\pi$
$719$	−30.1524	−1.12449	−0.562247	+	0.826969i	$0.690063\pi$
$720$	0	0
$721$	1.56403	0.0582477
$722$	0	0
$723$	−38.1222	−1.41778
$724$	0	0
$725$	−1.60124	−0.0594685
$726$	0	0
$727$	−15.1905	−0.563384	−0.281692	−	0.959505i	$-0.590896\pi$
$727$	−15.1905	−0.563384	−0.281692	+	0.959505i	$0.590896\pi$
$728$	0	0
$729$	29.8887	1.10699
$730$	0	0
$731$	23.2512	0.859977
$732$	0	0
$733$	39.8270	1.47104	0.735522	−	0.677501i	$-0.236937\pi$
$733$	39.8270	1.47104	0.735522	+	0.677501i	$0.236937\pi$
$734$	0	0
$735$	17.9306	0.661381
$736$	0	0
$737$	−1.10134	−0.0405682
$738$	0	0
$739$	26.4367	0.972489	0.486245	−	0.873823i	$-0.338366\pi$
$739$	26.4367	0.972489	0.486245	+	0.873823i	$0.338366\pi$
$740$	0	0
$741$	19.4267	0.713657
$742$	0	0
$743$	26.5745	0.974923	0.487462	−	0.873144i	$-0.337923\pi$
$743$	26.5745	0.974923	0.487462	+	0.873144i	$0.337923\pi$
$744$	0	0
$745$	15.4549	0.566225
$746$	0	0
$747$	−12.8270	−0.469316
$748$	0	0
$749$	5.57720	0.203787
$750$	0	0
$751$	30.1977	1.10193	0.550964	−	0.834529i	$-0.314260\pi$
$751$	30.1977	1.10193	0.550964	+	0.834529i	$0.314260\pi$
$752$	0	0
$753$	−19.7499	−0.719725
$754$	0	0
$755$	0.949439	0.0345536
$756$	0	0
$757$	31.3606	1.13982	0.569911	−	0.821706i	$-0.306978\pi$
$757$	31.3606	1.13982	0.569911	+	0.821706i	$0.306978\pi$
$758$	0	0
$759$	−24.4379	−0.887040
$760$	0	0
$761$	40.3426	1.46242	0.731210	−	0.682153i	$-0.238956\pi$
$761$	40.3426	1.46242	0.731210	+	0.682153i	$0.238956\pi$
$762$	0	0
$763$	3.79253	0.137299
$764$	0	0
$765$	3.55272	0.128449
$766$	0	0
$767$	−8.68015	−0.313422
$768$	0	0
$769$	−20.4304	−0.736739	−0.368369	−	0.929680i	$-0.620084\pi$
$769$	−20.4304	−0.736739	−0.368369	+	0.929680i	$0.620084\pi$
$770$	0	0
$771$	−23.9738	−0.863395
$772$	0	0
$773$	45.6859	1.64321	0.821605	−	0.570058i	$-0.193079\pi$
$773$	45.6859	1.64321	0.821605	+	0.570058i	$0.193079\pi$
$774$	0	0
$775$	0.672094	0.0241423
$776$	0	0
$777$	4.46715	0.160258
$778$	0	0
$779$	29.3038	1.04992
$780$	0	0
$781$	50.8798	1.82062
$782$	0	0
$783$	−5.11048	−0.182634
$784$	0	0
$785$	−20.0930	−0.717149
$786$	0	0
$787$	−15.1038	−0.538392	−0.269196	−	0.963085i	$-0.586758\pi$
$787$	−15.1038	−0.538392	−0.269196	+	0.963085i	$0.586758\pi$
$788$	0	0
$789$	−10.0856	−0.359055
$790$	0	0
$791$	−1.54933	−0.0550880
$792$	0	0
$793$	−2.70444	−0.0960373
$794$	0	0
$795$	17.3107	0.613947
$796$	0	0
$797$	13.0697	0.462954	0.231477	−	0.972840i	$-0.425644\pi$
$797$	13.0697	0.462954	0.231477	+	0.972840i	$0.425644\pi$
$798$	0	0
$799$	22.2300	0.786440
$800$	0	0
$801$	2.42683	0.0857478
$802$	0	0
$803$	69.9460	2.46834
$804$	0	0
$805$	3.75863	0.132474
$806$	0	0
$807$	34.9697	1.23099
$808$	0	0
$809$	−19.0282	−0.668996	−0.334498	−	0.942396i	$-0.608567\pi$
$809$	−19.0282	−0.668996	−0.334498	+	0.942396i	$0.608567\pi$
$810$	0	0
$811$	−21.0749	−0.740042	−0.370021	−	0.929023i	$-0.620649\pi$
$811$	−21.0749	−0.740042	−0.370021	+	0.929023i	$0.620649\pi$
$812$	0	0
$813$	−26.1895	−0.918506
$814$	0	0
$815$	35.9927	1.26077
$816$	0	0
$817$	47.1736	1.65040
$818$	0	0
$819$	1.10188	0.0385029
$820$	0	0
$821$	10.5704	0.368909	0.184454	−	0.982841i	$-0.440948\pi$
$821$	10.5704	0.368909	0.184454	+	0.982841i	$0.440948\pi$
$822$	0	0
$823$	−40.5293	−1.41276	−0.706381	−	0.707832i	$-0.749673\pi$
$823$	−40.5293	−1.41276	−0.706381	+	0.707832i	$0.749673\pi$
$824$	0	0
$825$	11.3976	0.396814
$826$	0	0
$827$	47.1057	1.63803	0.819013	−	0.573775i	$-0.194522\pi$
$827$	47.1057	1.63803	0.819013	+	0.573775i	$0.194522\pi$
$828$	0	0
$829$	−9.69487	−0.336717	−0.168358	−	0.985726i	$-0.553847\pi$
$829$	−9.69487	−0.336717	−0.168358	+	0.985726i	$0.553847\pi$
$830$	0	0
$831$	−16.8925	−0.585994
$832$	0	0
$833$	−16.8919	−0.585269
$834$	0	0
$835$	−21.4975	−0.743953
$836$	0	0
$837$	2.14504	0.0741436
$838$	0	0
$839$	−29.2517	−1.00988	−0.504940	−	0.863154i	$-0.668485\pi$
$839$	−29.2517	−1.00988	−0.504940	+	0.863154i	$0.668485\pi$
$840$	0	0
$841$	−28.1768	−0.971615
$842$	0	0
$843$	−13.9483	−0.480404
$844$	0	0
$845$	11.6984	0.402436
$846$	0	0
$847$	−4.31356	−0.148216
$848$	0	0
$849$	−40.0228	−1.37358
$850$	0	0
$851$	−20.5577	−0.704708
$852$	0	0
$853$	−31.0610	−1.06351	−0.531754	−	0.846899i	$-0.678467\pi$
$853$	−31.0610	−1.06351	−0.531754	+	0.846899i	$0.678467\pi$
$854$	0	0
$855$	7.20800	0.246508
$856$	0	0
$857$	−27.3567	−0.934486	−0.467243	−	0.884129i	$-0.654753\pi$
$857$	−27.3567	−0.934486	−0.467243	+	0.884129i	$0.654753\pi$
$858$	0	0
$859$	37.0124	1.26285	0.631424	−	0.775438i	$-0.282471\pi$
$859$	37.0124	1.26285	0.631424	+	0.775438i	$0.282471\pi$
$860$	0	0
$861$	−4.70719	−0.160421
$862$	0	0
$863$	37.6253	1.28078	0.640389	−	0.768051i	$-0.278773\pi$
$863$	37.6253	1.28078	0.640389	+	0.768051i	$0.278773\pi$
$864$	0	0
$865$	−5.65785	−0.192373
$866$	0	0
$867$	−15.8344	−0.537765
$868$	0	0
$869$	33.6759	1.14238
$870$	0	0
$871$	0.647196	0.0219294
$872$	0	0
$873$	3.35542	0.113564
$874$	0	0
$875$	−6.71935	−0.227156
$876$	0	0
$877$	35.6727	1.20458	0.602291	−	0.798277i	$-0.294255\pi$
$877$	35.6727	1.20458	0.602291	+	0.798277i	$0.294255\pi$
$878$	0	0
$879$	25.6653	0.865668
$880$	0	0
$881$	−30.3930	−1.02397	−0.511984	−	0.858995i	$-0.671089\pi$
$881$	−30.3930	−1.02397	−0.511984	+	0.858995i	$0.671089\pi$
$882$	0	0
$883$	−23.6017	−0.794260	−0.397130	−	0.917762i	$-0.629994\pi$
$883$	−23.6017	−0.794260	−0.397130	+	0.917762i	$0.629994\pi$
$884$	0	0
$885$	9.12106	0.306601
$886$	0	0
$887$	18.1804	0.610437	0.305219	−	0.952282i	$-0.401270\pi$
$887$	18.1804	0.610437	0.305219	+	0.952282i	$0.401270\pi$
$888$	0	0
$889$	−8.23639	−0.276240
$890$	0	0
$891$	26.1901	0.877401
$892$	0	0
$893$	45.1017	1.50927
$894$	0	0
$895$	−10.8296	−0.361993
$896$	0	0
$897$	14.3609	0.479495
$898$	0	0
$899$	−0.345513	−0.0115235
$900$	0	0
$901$	−16.3079	−0.543294
$902$	0	0
$903$	−7.57771	−0.252170
$904$	0	0
$905$	41.4543	1.37799
$906$	0	0
$907$	26.8615	0.891921	0.445961	−	0.895053i	$-0.352862\pi$
$907$	26.8615	0.891921	0.445961	+	0.895053i	$0.352862\pi$
$908$	0	0
$909$	2.92359	0.0969694
$910$	0	0
$911$	−42.7019	−1.41478	−0.707388	−	0.706825i	$-0.750127\pi$
$911$	−42.7019	−1.41478	−0.707388	+	0.706825i	$0.750127\pi$
$912$	0	0
$913$	−71.0630	−2.35184
$914$	0	0
$915$	2.84181	0.0939473
$916$	0	0
$917$	7.46656	0.246568
$918$	0	0
$919$	7.59767	0.250624	0.125312	−	0.992117i	$-0.460007\pi$
$919$	7.59767	0.250624	0.125312	+	0.992117i	$0.460007\pi$
$920$	0	0
$921$	−43.1812	−1.42287
$922$	0	0
$923$	−29.8994	−0.984151
$924$	0	0
$925$	9.58790	0.315248
$926$	0	0
$927$	2.21724	0.0728239
$928$	0	0
$929$	−34.5765	−1.13442	−0.567210	−	0.823574i	$-0.691977\pi$
$929$	−34.5765	−1.13442	−0.567210	+	0.823574i	$0.691977\pi$
$930$	0	0
$931$	−34.2714	−1.12320
$932$	0	0
$933$	−40.5474	−1.32746
$934$	0	0
$935$	19.6824	0.643684
$936$	0	0
$937$	40.4353	1.32096	0.660482	−	0.750842i	$-0.270352\pi$
$937$	40.4353	1.32096	0.660482	+	0.750842i	$0.270352\pi$
$938$	0	0
$939$	8.23506	0.268741
$940$	0	0
$941$	−22.3193	−0.727587	−0.363794	−	0.931480i	$-0.618519\pi$
$941$	−22.3193	−0.727587	−0.363794	+	0.931480i	$0.618519\pi$
$942$	0	0
$943$	21.6623	0.705423
$944$	0	0
$945$	−5.59481	−0.181999
$946$	0	0
$947$	40.0277	1.30073	0.650363	−	0.759623i	$-0.274617\pi$
$947$	40.0277	1.30073	0.650363	+	0.759623i	$0.274617\pi$
$948$	0	0
$949$	−41.1036	−1.33428
$950$	0	0
$951$	46.4613	1.50661
$952$	0	0
$953$	−6.33099	−0.205081	−0.102540	−	0.994729i	$-0.532697\pi$
$953$	−6.33099	−0.205081	−0.102540	+	0.994729i	$0.532697\pi$
$954$	0	0
$955$	−32.3349	−1.04633
$956$	0	0
$957$	−5.85933	−0.189405
$958$	0	0
$959$	6.87416	0.221978
$960$	0	0
$961$	−30.8550	−0.995322
$962$	0	0
$963$	7.90649	0.254783
$964$	0	0
$965$	18.7831	0.604650
$966$	0	0
$967$	12.8388	0.412867	0.206433	−	0.978461i	$-0.433814\pi$
$967$	12.8388	0.412867	0.206433	+	0.978461i	$0.433814\pi$
$968$	0	0
$969$	19.2309	0.617785
$970$	0	0
$971$	−22.1505	−0.710842	−0.355421	−	0.934706i	$-0.615662\pi$
$971$	−22.1505	−0.710842	−0.355421	+	0.934706i	$0.615662\pi$
$972$	0	0
$973$	1.18975	0.0381415
$974$	0	0
$975$	−6.69778	−0.214500
$976$	0	0
$977$	41.0297	1.31266	0.656328	−	0.754475i	$-0.272109\pi$
$977$	41.0297	1.31266	0.656328	+	0.754475i	$0.272109\pi$
$978$	0	0
$979$	13.4449	0.429701
$980$	0	0
$981$	5.37646	0.171657
$982$	0	0
$983$	−43.9691	−1.40240	−0.701199	−	0.712966i	$-0.747351\pi$
$983$	−43.9691	−1.40240	−0.701199	+	0.712966i	$0.747351\pi$
$984$	0	0
$985$	7.84258	0.249885
$986$	0	0
$987$	−7.24489	−0.230607
$988$	0	0
$989$	34.8724	1.10888
$990$	0	0
$991$	−20.6965	−0.657446	−0.328723	−	0.944426i	$-0.606618\pi$
$991$	−20.6965	−0.657446	−0.328723	+	0.944426i	$0.606618\pi$
$992$	0	0
$993$	−25.0190	−0.793955
$994$	0	0
$995$	25.2180	0.799464
$996$	0	0
$997$	52.9018	1.67542	0.837709	−	0.546117i	$-0.183895\pi$
$997$	52.9018	1.67542	0.837709	+	0.546117i	$0.183895\pi$
$998$	0	0
$999$	30.6006	0.968160

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.p.1.6		10
4.3	odd	2		503.2.a.e.1.3	✓	10
12.11	even	2		4527.2.a.k.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.3	✓	10	4.3	odd	2
4527.2.a.k.1.8		10	12.11	even	2
8048.2.a.p.1.6		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48900 q^{3} -1.79865 q^{5} -0.552233 q^{7} -0.782869 q^{9} +O(q^{10})\)	\(q+1.48900 q^{3} -1.79865 q^{5} -0.552233 q^{7} -0.782869 q^{9} -4.33718 q^{11} +2.54873 q^{13} -2.67819 q^{15} +2.52304 q^{17} +5.11893 q^{19} -0.822276 q^{21} +3.78409 q^{23} -1.76486 q^{25} -5.63270 q^{27} +0.907287 q^{29} -0.380820 q^{31} -6.45807 q^{33} +0.993273 q^{35} -5.43266 q^{37} +3.79507 q^{39} +5.72459 q^{41} +9.21553 q^{43} +1.40811 q^{45} +8.81077 q^{47} -6.69504 q^{49} +3.75682 q^{51} -6.46357 q^{53} +7.80106 q^{55} +7.62210 q^{57} -3.40568 q^{59} -1.06109 q^{61} +0.432326 q^{63} -4.58427 q^{65} +0.253929 q^{67} +5.63451 q^{69} -11.7311 q^{71} -16.1271 q^{73} -2.62789 q^{75} +2.39513 q^{77} -7.76447 q^{79} -6.03851 q^{81} +16.3846 q^{83} -4.53807 q^{85} +1.35095 q^{87} -3.09992 q^{89} -1.40749 q^{91} -0.567042 q^{93} -9.20715 q^{95} -4.28605 q^{97} +3.39544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * q^99

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