LMFDB - Embedded newform 8048.2.a.y.1.15

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:	$0$
Dimension:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-0.220191 q^{3} +1.20642 q^{5} +3.77303 q^{7} -2.95152 q^{9} +O(q^{10})$	$q-0.220191 q^{3} +1.20642 q^{5} +3.77303 q^{7} -2.95152 q^{9} +3.28775 q^{11} -2.57020 q^{13} -0.265642 q^{15} +2.82949 q^{17} -0.306992 q^{19} -0.830786 q^{21} -8.21155 q^{23} -3.54456 q^{25} +1.31047 q^{27} +1.89079 q^{29} +2.28862 q^{31} -0.723932 q^{33} +4.55185 q^{35} +6.00030 q^{37} +0.565934 q^{39} +1.51588 q^{41} +1.64672 q^{43} -3.56076 q^{45} +6.84269 q^{47} +7.23576 q^{49} -0.623028 q^{51} -5.95625 q^{53} +3.96640 q^{55} +0.0675967 q^{57} -8.38851 q^{59} +12.5491 q^{61} -11.1362 q^{63} -3.10074 q^{65} -4.93868 q^{67} +1.80811 q^{69} +15.0765 q^{71} +15.8834 q^{73} +0.780478 q^{75} +12.4048 q^{77} -3.13062 q^{79} +8.56600 q^{81} -5.57184 q^{83} +3.41355 q^{85} -0.416335 q^{87} +0.0590559 q^{89} -9.69745 q^{91} -0.503933 q^{93} -0.370360 q^{95} +15.7419 q^{97} -9.70385 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * 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
$3$	−0.220191	−0.127127	−0.0635636	−	0.997978i	$-0.520247\pi$
$3$	−0.220191	−0.127127	−0.0635636	+	0.997978i	$0.520247\pi$
$4$	0	0
$5$	1.20642	0.539526	0.269763	−	0.962927i	$-0.413055\pi$
$5$	1.20642	0.539526	0.269763	+	0.962927i	$0.413055\pi$
$6$	0	0
$7$	3.77303	1.42607	0.713036	−	0.701128i	$-0.247320\pi$
$7$	3.77303	1.42607	0.713036	+	0.701128i	$0.247320\pi$
$8$	0	0
$9$	−2.95152	−0.983839
$10$	0	0
$11$	3.28775	0.991294	0.495647	−	0.868524i	$-0.334931\pi$
$11$	3.28775	0.991294	0.495647	+	0.868524i	$0.334931\pi$
$12$	0	0
$13$	−2.57020	−0.712845	−0.356423	−	0.934325i	$-0.616004\pi$
$13$	−2.57020	−0.712845	−0.356423	+	0.934325i	$0.616004\pi$
$14$	0	0
$15$	−0.265642	−0.0685884
$16$	0	0
$17$	2.82949	0.686253	0.343126	−	0.939289i	$-0.388514\pi$
$17$	2.82949	0.686253	0.343126	+	0.939289i	$0.388514\pi$
$18$	0	0
$19$	−0.306992	−0.0704287	−0.0352144	−	0.999380i	$-0.511211\pi$
$19$	−0.306992	−0.0704287	−0.0352144	+	0.999380i	$0.511211\pi$
$20$	0	0
$21$	−0.830786	−0.181292
$22$	0	0
$23$	−8.21155	−1.71223	−0.856114	−	0.516788i	$-0.827128\pi$
$23$	−8.21155	−1.71223	−0.856114	+	0.516788i	$0.827128\pi$
$24$	0	0
$25$	−3.54456	−0.708911
$26$	0	0
$27$	1.31047	0.252200
$28$	0	0
$29$	1.89079	0.351111	0.175556	−	0.984469i	$-0.443828\pi$
$29$	1.89079	0.351111	0.175556	+	0.984469i	$0.443828\pi$
$30$	0	0
$31$	2.28862	0.411048	0.205524	−	0.978652i	$-0.434110\pi$
$31$	2.28862	0.411048	0.205524	+	0.978652i	$0.434110\pi$
$32$	0	0
$33$	−0.723932	−0.126020
$34$	0	0
$35$	4.55185	0.769403
$36$	0	0
$37$	6.00030	0.986443	0.493222	−	0.869904i	$-0.335819\pi$
$37$	6.00030	0.986443	0.493222	+	0.869904i	$0.335819\pi$
$38$	0	0
$39$	0.565934	0.0906220
$40$	0	0
$41$	1.51588	0.236741	0.118370	−	0.992970i	$-0.462233\pi$
$41$	1.51588	0.236741	0.118370	+	0.992970i	$0.462233\pi$
$42$	0	0
$43$	1.64672	0.251122	0.125561	−	0.992086i	$-0.459927\pi$
$43$	1.64672	0.251122	0.125561	+	0.992086i	$0.459927\pi$
$44$	0	0
$45$	−3.56076	−0.530807
$46$	0	0
$47$	6.84269	0.998109	0.499054	−	0.866571i	$-0.333681\pi$
$47$	6.84269	0.998109	0.499054	+	0.866571i	$0.333681\pi$
$48$	0	0
$49$	7.23576	1.03368
$50$	0	0
$51$	−0.623028	−0.0872413
$52$	0	0
$53$	−5.95625	−0.818154	−0.409077	−	0.912500i	$-0.634149\pi$
$53$	−5.95625	−0.818154	−0.409077	+	0.912500i	$0.634149\pi$
$54$	0	0
$55$	3.96640	0.534829
$56$	0	0
$57$	0.0675967	0.00895340
$58$	0	0
$59$	−8.38851	−1.09209	−0.546045	−	0.837756i	$-0.683867\pi$
$59$	−8.38851	−1.09209	−0.546045	+	0.837756i	$0.683867\pi$
$60$	0	0
$61$	12.5491	1.60675	0.803377	−	0.595471i	$-0.203034\pi$
$61$	12.5491	1.60675	0.803377	+	0.595471i	$0.203034\pi$
$62$	0	0
$63$	−11.1362	−1.40302
$64$	0	0
$65$	−3.10074	−0.384599
$66$	0	0
$67$	−4.93868	−0.603355	−0.301678	−	0.953410i	$-0.597547\pi$
$67$	−4.93868	−0.603355	−0.301678	+	0.953410i	$0.597547\pi$
$68$	0	0
$69$	1.80811	0.217671
$70$	0	0
$71$	15.0765	1.78925	0.894626	−	0.446816i	$-0.147442\pi$
$71$	15.0765	1.78925	0.894626	+	0.446816i	$0.147442\pi$
$72$	0	0
$73$	15.8834	1.85901	0.929504	−	0.368812i	$-0.120235\pi$
$73$	15.8834	1.85901	0.929504	+	0.368812i	$0.120235\pi$
$74$	0	0
$75$	0.780478	0.0901219
$76$	0	0
$77$	12.4048	1.41366
$78$	0	0
$79$	−3.13062	−0.352223	−0.176111	−	0.984370i	$-0.556352\pi$
$79$	−3.13062	−0.352223	−0.176111	+	0.984370i	$0.556352\pi$
$80$	0	0
$81$	8.56600	0.951777
$82$	0	0
$83$	−5.57184	−0.611589	−0.305795	−	0.952098i	$-0.598922\pi$
$83$	−5.57184	−0.611589	−0.305795	+	0.952098i	$0.598922\pi$
$84$	0	0
$85$	3.41355	0.370251
$86$	0	0
$87$	−0.416335	−0.0446358
$88$	0	0
$89$	0.0590559	0.00625991	0.00312996	−	0.999995i	$-0.499004\pi$
$89$	0.0590559	0.00625991	0.00312996	+	0.999995i	$0.499004\pi$
$90$	0	0
$91$	−9.69745	−1.01657
$92$	0	0
$93$	−0.503933	−0.0522554
$94$	0	0
$95$	−0.370360	−0.0379982
$96$	0	0
$97$	15.7419	1.59834	0.799172	−	0.601103i	$-0.205272\pi$
$97$	15.7419	1.59834	0.799172	+	0.601103i	$0.205272\pi$
$98$	0	0
$99$	−9.70385	−0.975273
$100$	0	0
$101$	−2.17998	−0.216916	−0.108458	−	0.994101i	$-0.534591\pi$
$101$	−2.17998	−0.216916	−0.108458	+	0.994101i	$0.534591\pi$
$102$	0	0
$103$	−0.0816921	−0.00804936	−0.00402468	−	0.999992i	$-0.501281\pi$
$103$	−0.0816921	−0.00804936	−0.00402468	+	0.999992i	$0.501281\pi$
$104$	0	0
$105$	−1.00227	−0.0978120
$106$	0	0
$107$	9.56409	0.924595	0.462298	−	0.886725i	$-0.347025\pi$
$107$	9.56409	0.924595	0.462298	+	0.886725i	$0.347025\pi$
$108$	0	0
$109$	8.38110	0.802764	0.401382	−	0.915911i	$-0.368530\pi$
$109$	8.38110	0.802764	0.401382	+	0.915911i	$0.368530\pi$
$110$	0	0
$111$	−1.32121	−0.125404
$112$	0	0
$113$	15.5479	1.46262	0.731310	−	0.682045i	$-0.238909\pi$
$113$	15.5479	1.46262	0.731310	+	0.682045i	$0.238909\pi$
$114$	0	0
$115$	−9.90656	−0.923792
$116$	0	0
$117$	7.58599	0.701325
$118$	0	0
$119$	10.6758	0.978646
$120$	0	0
$121$	−0.190701	−0.0173365
$122$	0	0
$123$	−0.333783	−0.0300962
$124$	0	0
$125$	−10.3083	−0.922003
$126$	0	0
$127$	−3.99639	−0.354622	−0.177311	−	0.984155i	$-0.556740\pi$
$127$	−3.99639	−0.354622	−0.177311	+	0.984155i	$0.556740\pi$
$128$	0	0
$129$	−0.362592	−0.0319244
$130$	0	0
$131$	1.40034	0.122348	0.0611742	−	0.998127i	$-0.480515\pi$
$131$	1.40034	0.122348	0.0611742	+	0.998127i	$0.480515\pi$
$132$	0	0
$133$	−1.15829	−0.100436
$134$	0	0
$135$	1.58097	0.136068
$136$	0	0
$137$	10.9701	0.937237	0.468618	−	0.883401i	$-0.344752\pi$
$137$	10.9701	0.937237	0.468618	+	0.883401i	$0.344752\pi$
$138$	0	0
$139$	8.92896	0.757345	0.378672	−	0.925531i	$-0.376381\pi$
$139$	8.92896	0.757345	0.378672	+	0.925531i	$0.376381\pi$
$140$	0	0
$141$	−1.50670	−0.126887
$142$	0	0
$143$	−8.45018	−0.706639
$144$	0	0
$145$	2.28109	0.189434
$146$	0	0
$147$	−1.59325	−0.131409
$148$	0	0
$149$	20.1530	1.65099	0.825497	−	0.564406i	$-0.190895\pi$
$149$	20.1530	1.65099	0.825497	+	0.564406i	$0.190895\pi$
$150$	0	0
$151$	−10.5257	−0.856570	−0.428285	−	0.903644i	$-0.640882\pi$
$151$	−10.5257	−0.856570	−0.428285	+	0.903644i	$0.640882\pi$
$152$	0	0
$153$	−8.35129	−0.675162
$154$	0	0
$155$	2.76103	0.221771
$156$	0	0
$157$	2.63794	0.210530	0.105265	−	0.994444i	$-0.466431\pi$
$157$	2.63794	0.210530	0.105265	+	0.994444i	$0.466431\pi$
$158$	0	0
$159$	1.31151	0.104010
$160$	0	0
$161$	−30.9824	−2.44176
$162$	0	0
$163$	13.0111	1.01911	0.509553	−	0.860439i	$-0.329811\pi$
$163$	13.0111	1.01911	0.509553	+	0.860439i	$0.329811\pi$
$164$	0	0
$165$	−0.873364	−0.0679913
$166$	0	0
$167$	−0.187556	−0.0145135	−0.00725676	−	0.999974i	$-0.502310\pi$
$167$	−0.187556	−0.0145135	−0.00725676	+	0.999974i	$0.502310\pi$
$168$	0	0
$169$	−6.39407	−0.491851
$170$	0	0
$171$	0.906091	0.0692905
$172$	0	0
$173$	−5.38200	−0.409186	−0.204593	−	0.978847i	$-0.565587\pi$
$173$	−5.38200	−0.409186	−0.204593	+	0.978847i	$0.565587\pi$
$174$	0	0
$175$	−13.3737	−1.01096
$176$	0	0
$177$	1.84707	0.138834
$178$	0	0
$179$	−3.12372	−0.233478	−0.116739	−	0.993163i	$-0.537244\pi$
$179$	−3.12372	−0.233478	−0.116739	+	0.993163i	$0.537244\pi$
$180$	0	0
$181$	15.2333	1.13228	0.566141	−	0.824309i	$-0.308436\pi$
$181$	15.2333	1.13228	0.566141	+	0.824309i	$0.308436\pi$
$182$	0	0
$183$	−2.76321	−0.204262
$184$	0	0
$185$	7.23887	0.532212
$186$	0	0
$187$	9.30266	0.680278
$188$	0	0
$189$	4.94444	0.359655
$190$	0	0
$191$	−5.73551	−0.415007	−0.207503	−	0.978234i	$-0.566534\pi$
$191$	−5.73551	−0.415007	−0.207503	+	0.978234i	$0.566534\pi$
$192$	0	0
$193$	−15.7493	−1.13366	−0.566831	−	0.823834i	$-0.691831\pi$
$193$	−15.7493	−1.13366	−0.566831	+	0.823834i	$0.691831\pi$
$194$	0	0
$195$	0.682753	0.0488930
$196$	0	0
$197$	−15.7662	−1.12330	−0.561649	−	0.827376i	$-0.689833\pi$
$197$	−15.7662	−1.12330	−0.561649	+	0.827376i	$0.689833\pi$
$198$	0	0
$199$	−6.08398	−0.431282	−0.215641	−	0.976473i	$-0.569184\pi$
$199$	−6.08398	−0.431282	−0.215641	+	0.976473i	$0.569184\pi$
$200$	0	0
$201$	1.08745	0.0767028
$202$	0	0
$203$	7.13402	0.500710
$204$	0	0
$205$	1.82879	0.127728
$206$	0	0
$207$	24.2365	1.68456
$208$	0	0
$209$	−1.00931	−0.0698156
$210$	0	0
$211$	0.223669	0.0153980	0.00769901	−	0.999970i	$-0.497549\pi$
$211$	0.223669	0.0153980	0.00769901	+	0.999970i	$0.497549\pi$
$212$	0	0
$213$	−3.31970	−0.227462
$214$	0	0
$215$	1.98663	0.135487
$216$	0	0
$217$	8.63503	0.586184
$218$	0	0
$219$	−3.49737	−0.236330
$220$	0	0
$221$	−7.27236	−0.489192
$222$	0	0
$223$	−22.0486	−1.47648	−0.738242	−	0.674536i	$-0.764344\pi$
$223$	−22.0486	−1.47648	−0.738242	+	0.674536i	$0.764344\pi$
$224$	0	0
$225$	10.4618	0.697454
$226$	0	0
$227$	3.63641	0.241357	0.120679	−	0.992692i	$-0.461493\pi$
$227$	3.63641	0.241357	0.120679	+	0.992692i	$0.461493\pi$
$228$	0	0
$229$	−18.7881	−1.24155	−0.620776	−	0.783988i	$-0.713182\pi$
$229$	−18.7881	−1.24155	−0.620776	+	0.783988i	$0.713182\pi$
$230$	0	0
$231$	−2.73142	−0.179714
$232$	0	0
$233$	−15.2250	−0.997421	−0.498710	−	0.866769i	$-0.666193\pi$
$233$	−15.2250	−0.997421	−0.498710	+	0.866769i	$0.666193\pi$
$234$	0	0
$235$	8.25514	0.538506
$236$	0	0
$237$	0.689334	0.0447771
$238$	0	0
$239$	−17.8793	−1.15652	−0.578259	−	0.815854i	$-0.696268\pi$
$239$	−17.8793	−1.15652	−0.578259	+	0.815854i	$0.696268\pi$
$240$	0	0
$241$	−6.42601	−0.413936	−0.206968	−	0.978348i	$-0.566360\pi$
$241$	−6.42601	−0.413936	−0.206968	+	0.978348i	$0.566360\pi$
$242$	0	0
$243$	−5.81756	−0.373196
$244$	0	0
$245$	8.72935	0.557698
$246$	0	0
$247$	0.789030	0.0502048
$248$	0	0
$249$	1.22687	0.0777496
$250$	0	0
$251$	11.6135	0.733039	0.366520	−	0.930410i	$-0.380549\pi$
$251$	11.6135	0.733039	0.366520	+	0.930410i	$0.380549\pi$
$252$	0	0
$253$	−26.9975	−1.69732
$254$	0	0
$255$	−0.751632	−0.0470690
$256$	0	0
$257$	−0.184618	−0.0115161	−0.00575807	−	0.999983i	$-0.501833\pi$
$257$	−0.184618	−0.0115161	−0.00575807	+	0.999983i	$0.501833\pi$
$258$	0	0
$259$	22.6393	1.40674
$260$	0	0
$261$	−5.58071	−0.345437
$262$	0	0
$263$	−11.4618	−0.706766	−0.353383	−	0.935479i	$-0.614969\pi$
$263$	−11.4618	−0.706766	−0.353383	+	0.935479i	$0.614969\pi$
$264$	0	0
$265$	−7.18573	−0.441416
$266$	0	0
$267$	−0.0130036	−0.000795805 0
$268$	0	0
$269$	23.2720	1.41892	0.709458	−	0.704747i	$-0.248940\pi$
$269$	23.2720	1.41892	0.709458	+	0.704747i	$0.248940\pi$
$270$	0	0
$271$	17.5288	1.06480	0.532398	−	0.846494i	$-0.321291\pi$
$271$	17.5288	1.06480	0.532398	+	0.846494i	$0.321291\pi$
$272$	0	0
$273$	2.13529	0.129233
$274$	0	0
$275$	−11.6536	−0.702739
$276$	0	0
$277$	15.5409	0.933760	0.466880	−	0.884321i	$-0.345378\pi$
$277$	15.5409	0.933760	0.466880	+	0.884321i	$0.345378\pi$
$278$	0	0
$279$	−6.75490	−0.404405
$280$	0	0
$281$	21.7911	1.29995	0.649974	−	0.759957i	$-0.274780\pi$
$281$	21.7911	1.29995	0.649974	+	0.759957i	$0.274780\pi$
$282$	0	0
$283$	11.2752	0.670240	0.335120	−	0.942175i	$-0.391223\pi$
$283$	11.2752	0.670240	0.335120	+	0.942175i	$0.391223\pi$
$284$	0	0
$285$	0.0815499	0.00483060
$286$	0	0
$287$	5.71947	0.337610
$288$	0	0
$289$	−8.99397	−0.529057
$290$	0	0
$291$	−3.46621	−0.203193
$292$	0	0
$293$	18.2795	1.06790	0.533950	−	0.845516i	$-0.320707\pi$
$293$	18.2795	1.06790	0.533950	+	0.845516i	$0.320707\pi$
$294$	0	0
$295$	−10.1200	−0.589212
$296$	0	0
$297$	4.30849	0.250004
$298$	0	0
$299$	21.1053	1.22055
$300$	0	0
$301$	6.21312	0.358118
$302$	0	0
$303$	0.480012	0.0275760
$304$	0	0
$305$	15.1395	0.866886
$306$	0	0
$307$	−2.27283	−0.129717	−0.0648585	−	0.997894i	$-0.520660\pi$
$307$	−2.27283	−0.129717	−0.0648585	+	0.997894i	$0.520660\pi$
$308$	0	0
$309$	0.0179878	0.00102329
$310$	0	0
$311$	9.80746	0.556130	0.278065	−	0.960562i	$-0.410307\pi$
$311$	9.80746	0.556130	0.278065	+	0.960562i	$0.410307\pi$
$312$	0	0
$313$	16.5012	0.932704	0.466352	−	0.884599i	$-0.345568\pi$
$313$	16.5012	0.932704	0.466352	+	0.884599i	$0.345568\pi$
$314$	0	0
$315$	−13.4349	−0.756969
$316$	0	0
$317$	8.42278	0.473070	0.236535	−	0.971623i	$-0.423988\pi$
$317$	8.42278	0.473070	0.236535	+	0.971623i	$0.423988\pi$
$318$	0	0
$319$	6.21645	0.348055
$320$	0	0
$321$	−2.10592	−0.117541
$322$	0	0
$323$	−0.868631	−0.0483319
$324$	0	0
$325$	9.11022	0.505344
$326$	0	0
$327$	−1.84544	−0.102053
$328$	0	0
$329$	25.8177	1.42337
$330$	0	0
$331$	−1.54569	−0.0849587	−0.0424794	−	0.999097i	$-0.513526\pi$
$331$	−1.54569	−0.0849587	−0.0424794	+	0.999097i	$0.513526\pi$
$332$	0	0
$333$	−17.7100	−0.970501
$334$	0	0
$335$	−5.95811	−0.325526
$336$	0	0
$337$	18.2291	0.993001	0.496501	−	0.868036i	$-0.334618\pi$
$337$	18.2291	0.993001	0.496501	+	0.868036i	$0.334618\pi$
$338$	0	0
$339$	−3.42349	−0.185939
$340$	0	0
$341$	7.52441	0.407470
$342$	0	0
$343$	0.889532	0.0480302
$344$	0	0
$345$	2.18133	0.117439
$346$	0	0
$347$	30.5835	1.64181	0.820905	−	0.571065i	$-0.193470\pi$
$347$	30.5835	1.64181	0.820905	+	0.571065i	$0.193470\pi$
$348$	0	0
$349$	−27.0717	−1.44912	−0.724558	−	0.689214i	$-0.757956\pi$
$349$	−27.0717	−1.44912	−0.724558	+	0.689214i	$0.757956\pi$
$350$	0	0
$351$	−3.36817	−0.179779
$352$	0	0
$353$	10.2880	0.547575	0.273787	−	0.961790i	$-0.411724\pi$
$353$	10.2880	0.547575	0.273787	+	0.961790i	$0.411724\pi$
$354$	0	0
$355$	18.1886	0.965348
$356$	0	0
$357$	−2.35070	−0.124412
$358$	0	0
$359$	−0.722012	−0.0381064	−0.0190532	−	0.999818i	$-0.506065\pi$
$359$	−0.722012	−0.0381064	−0.0190532	+	0.999818i	$0.506065\pi$
$360$	0	0
$361$	−18.9058	−0.995040
$362$	0	0
$363$	0.0419906	0.00220393
$364$	0	0
$365$	19.1620	1.00298
$366$	0	0
$367$	−20.6605	−1.07847	−0.539235	−	0.842155i	$-0.681287\pi$
$367$	−20.6605	−1.07847	−0.539235	+	0.842155i	$0.681287\pi$
$368$	0	0
$369$	−4.47415	−0.232915
$370$	0	0
$371$	−22.4731	−1.16675
$372$	0	0
$373$	7.56284	0.391589	0.195795	−	0.980645i	$-0.437271\pi$
$373$	7.56284	0.391589	0.195795	+	0.980645i	$0.437271\pi$
$374$	0	0
$375$	2.26979	0.117212
$376$	0	0
$377$	−4.85972	−0.250288
$378$	0	0
$379$	−2.88903	−0.148400	−0.0741998	−	0.997243i	$-0.523640\pi$
$379$	−2.88903	−0.148400	−0.0741998	+	0.997243i	$0.523640\pi$
$380$	0	0
$381$	0.879968	0.0450821
$382$	0	0
$383$	20.7831	1.06196	0.530982	−	0.847383i	$-0.321823\pi$
$383$	20.7831	1.06196	0.530982	+	0.847383i	$0.321823\pi$
$384$	0	0
$385$	14.9653	0.762705
$386$	0	0
$387$	−4.86032	−0.247064
$388$	0	0
$389$	2.61615	0.132644	0.0663219	−	0.997798i	$-0.478874\pi$
$389$	2.61615	0.132644	0.0663219	+	0.997798i	$0.478874\pi$
$390$	0	0
$391$	−23.2345	−1.17502
$392$	0	0
$393$	−0.308342	−0.0155538
$394$	0	0
$395$	−3.77684	−0.190033
$396$	0	0
$397$	27.3012	1.37021	0.685105	−	0.728445i	$-0.259757\pi$
$397$	27.3012	1.37021	0.685105	+	0.728445i	$0.259757\pi$
$398$	0	0
$399$	0.255044	0.0127682
$400$	0	0
$401$	−22.4714	−1.12217	−0.561085	−	0.827758i	$-0.689616\pi$
$401$	−22.4714	−1.12217	−0.561085	+	0.827758i	$0.689616\pi$
$402$	0	0
$403$	−5.88221	−0.293014
$404$	0	0
$405$	10.3342	0.513509
$406$	0	0
$407$	19.7275	0.977855
$408$	0	0
$409$	−3.40540	−0.168386	−0.0841931	−	0.996449i	$-0.526831\pi$
$409$	−3.40540	−0.168386	−0.0841931	+	0.996449i	$0.526831\pi$
$410$	0	0
$411$	−2.41551	−0.119148
$412$	0	0
$413$	−31.6501	−1.55740
$414$	0	0
$415$	−6.72197	−0.329968
$416$	0	0
$417$	−1.96607	−0.0962791
$418$	0	0
$419$	−15.6123	−0.762713	−0.381357	−	0.924428i	$-0.624543\pi$
$419$	−15.6123	−0.762713	−0.381357	+	0.924428i	$0.624543\pi$
$420$	0	0
$421$	7.26856	0.354248	0.177124	−	0.984189i	$-0.443321\pi$
$421$	7.26856	0.354248	0.177124	+	0.984189i	$0.443321\pi$
$422$	0	0
$423$	−20.1963	−0.981978
$424$	0	0
$425$	−10.0293	−0.486492
$426$	0	0
$427$	47.3483	2.29135
$428$	0	0
$429$	1.86065	0.0898330
$430$	0	0
$431$	3.95422	0.190468	0.0952340	−	0.995455i	$-0.469640\pi$
$431$	3.95422	0.190468	0.0952340	+	0.995455i	$0.469640\pi$
$432$	0	0
$433$	−23.2296	−1.11634	−0.558172	−	0.829725i	$-0.688497\pi$
$433$	−23.2296	−1.11634	−0.558172	+	0.829725i	$0.688497\pi$
$434$	0	0
$435$	−0.502274	−0.0240822
$436$	0	0
$437$	2.52088	0.120590
$438$	0	0
$439$	26.3518	1.25770	0.628852	−	0.777525i	$-0.283525\pi$
$439$	26.3518	1.25770	0.628852	+	0.777525i	$0.283525\pi$
$440$	0	0
$441$	−21.3565	−1.01697
$442$	0	0
$443$	9.77231	0.464296	0.232148	−	0.972680i	$-0.425425\pi$
$443$	9.77231	0.464296	0.232148	+	0.972680i	$0.425425\pi$
$444$	0	0
$445$	0.0712461	0.00337739
$446$	0	0
$447$	−4.43749	−0.209886
$448$	0	0
$449$	−4.21439	−0.198889	−0.0994446	−	0.995043i	$-0.531707\pi$
$449$	−4.21439	−0.198889	−0.0994446	+	0.995043i	$0.531707\pi$
$450$	0	0
$451$	4.98384	0.234680
$452$	0	0
$453$	2.31766	0.108893
$454$	0	0
$455$	−11.6992	−0.548466
$456$	0	0
$457$	−0.667430	−0.0312211	−0.0156105	−	0.999878i	$-0.504969\pi$
$457$	−0.667430	−0.0312211	−0.0156105	+	0.999878i	$0.504969\pi$
$458$	0	0
$459$	3.70796	0.173073
$460$	0	0
$461$	17.0573	0.794438	0.397219	−	0.917724i	$-0.369975\pi$
$461$	17.0573	0.794438	0.397219	+	0.917724i	$0.369975\pi$
$462$	0	0
$463$	−25.2670	−1.17426	−0.587128	−	0.809494i	$-0.699741\pi$
$463$	−25.2670	−1.17426	−0.587128	+	0.809494i	$0.699741\pi$
$464$	0	0
$465$	−0.607953	−0.0281932
$466$	0	0
$467$	−23.1378	−1.07069	−0.535345	−	0.844634i	$-0.679818\pi$
$467$	−23.1378	−1.07069	−0.535345	+	0.844634i	$0.679818\pi$
$468$	0	0
$469$	−18.6338	−0.860428
$470$	0	0
$471$	−0.580849	−0.0267641
$472$	0	0
$473$	5.41400	0.248936
$474$	0	0
$475$	1.08815	0.0499277
$476$	0	0
$477$	17.5800	0.804932
$478$	0	0
$479$	−18.9551	−0.866082	−0.433041	−	0.901374i	$-0.642559\pi$
$479$	−18.9551	−0.866082	−0.433041	+	0.901374i	$0.642559\pi$
$480$	0	0
$481$	−15.4220	−0.703182
$482$	0	0
$483$	6.82204	0.310414
$484$	0	0
$485$	18.9913	0.862348
$486$	0	0
$487$	−22.1705	−1.00464	−0.502320	−	0.864682i	$-0.667520\pi$
$487$	−22.1705	−1.00464	−0.502320	+	0.864682i	$0.667520\pi$
$488$	0	0
$489$	−2.86491	−0.129556
$490$	0	0
$491$	−35.8904	−1.61971	−0.809855	−	0.586630i	$-0.800454\pi$
$491$	−35.8904	−1.61971	−0.809855	+	0.586630i	$0.800454\pi$
$492$	0	0
$493$	5.34998	0.240951
$494$	0	0
$495$	−11.7069	−0.526186
$496$	0	0
$497$	56.8841	2.55160
$498$	0	0
$499$	−14.6962	−0.657892	−0.328946	−	0.944349i	$-0.606693\pi$
$499$	−14.6962	−0.657892	−0.328946	+	0.944349i	$0.606693\pi$
$500$	0	0
$501$	0.0412981	0.00184506
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−2.62997	−0.117032
$506$	0	0
$507$	1.40791	0.0625277
$508$	0	0
$509$	−2.73261	−0.121121	−0.0605605	−	0.998165i	$-0.519289\pi$
$509$	−2.73261	−0.121121	−0.0605605	+	0.998165i	$0.519289\pi$
$510$	0	0
$511$	59.9285	2.65108
$512$	0	0
$513$	−0.402303	−0.0177621
$514$	0	0
$515$	−0.0985548	−0.00434284
$516$	0	0
$517$	22.4970	0.989419
$518$	0	0
$519$	1.18507	0.0520186
$520$	0	0
$521$	−10.5586	−0.462579	−0.231290	−	0.972885i	$-0.574295\pi$
$521$	−10.5586	−0.462579	−0.231290	+	0.972885i	$0.574295\pi$
$522$	0	0
$523$	35.7557	1.56349	0.781744	−	0.623600i	$-0.214331\pi$
$523$	35.7557	1.56349	0.781744	+	0.623600i	$0.214331\pi$
$524$	0	0
$525$	2.94477	0.128520
$526$	0	0
$527$	6.47563	0.282083
$528$	0	0
$529$	44.4296	1.93172
$530$	0	0
$531$	24.7588	1.07444
$532$	0	0
$533$	−3.89612	−0.168760
$534$	0	0
$535$	11.5383	0.498843
$536$	0	0
$537$	0.687814	0.0296814
$538$	0	0
$539$	23.7894	1.02468
$540$	0	0
$541$	−4.24525	−0.182517	−0.0912587	−	0.995827i	$-0.529089\pi$
$541$	−4.24525	−0.182517	−0.0912587	+	0.995827i	$0.529089\pi$
$542$	0	0
$543$	−3.35423	−0.143944
$544$	0	0
$545$	10.1111	0.433112
$546$	0	0
$547$	−23.9608	−1.02449	−0.512244	−	0.858840i	$-0.671186\pi$
$547$	−23.9608	−1.02449	−0.512244	+	0.858840i	$0.671186\pi$
$548$	0	0
$549$	−37.0390	−1.58079
$550$	0	0
$551$	−0.580458	−0.0247283
$552$	0	0
$553$	−11.8119	−0.502295
$554$	0	0
$555$	−1.59393	−0.0676586
$556$	0	0
$557$	−13.6650	−0.579003	−0.289502	−	0.957178i	$-0.593490\pi$
$557$	−13.6650	−0.579003	−0.289502	+	0.957178i	$0.593490\pi$
$558$	0	0
$559$	−4.23240	−0.179011
$560$	0	0
$561$	−2.04836	−0.0864818
$562$	0	0
$563$	−26.2997	−1.10840	−0.554200	−	0.832384i	$-0.686976\pi$
$563$	−26.2997	−1.10840	−0.554200	+	0.832384i	$0.686976\pi$
$564$	0	0
$565$	18.7572	0.789122
$566$	0	0
$567$	32.3198	1.35730
$568$	0	0
$569$	−34.5816	−1.44974	−0.724869	−	0.688887i	$-0.758100\pi$
$569$	−34.5816	−1.44974	−0.724869	+	0.688887i	$0.758100\pi$
$570$	0	0
$571$	25.5743	1.07025	0.535126	−	0.844772i	$-0.320264\pi$
$571$	25.5743	1.07025	0.535126	+	0.844772i	$0.320264\pi$
$572$	0	0
$573$	1.26291	0.0527586
$574$	0	0
$575$	29.1063	1.21382
$576$	0	0
$577$	16.4769	0.685944	0.342972	−	0.939346i	$-0.388566\pi$
$577$	16.4769	0.685944	0.342972	+	0.939346i	$0.388566\pi$
$578$	0	0
$579$	3.46786	0.144119
$580$	0	0
$581$	−21.0227	−0.872170
$582$	0	0
$583$	−19.5827	−0.811031
$584$	0	0
$585$	9.15187	0.378383
$586$	0	0
$587$	−12.6565	−0.522391	−0.261195	−	0.965286i	$-0.584117\pi$
$587$	−12.6565	−0.522391	−0.261195	+	0.965286i	$0.584117\pi$
$588$	0	0
$589$	−0.702587	−0.0289496
$590$	0	0
$591$	3.47158	0.142802
$592$	0	0
$593$	20.2200	0.830334	0.415167	−	0.909745i	$-0.363723\pi$
$593$	20.2200	0.830334	0.415167	+	0.909745i	$0.363723\pi$
$594$	0	0
$595$	12.8794	0.528005
$596$	0	0
$597$	1.33964	0.0548277
$598$	0	0
$599$	13.0346	0.532578	0.266289	−	0.963893i	$-0.414202\pi$
$599$	13.0346	0.532578	0.266289	+	0.963893i	$0.414202\pi$
$600$	0	0
$601$	43.6227	1.77941	0.889703	−	0.456541i	$-0.150912\pi$
$601$	43.6227	1.77941	0.889703	+	0.456541i	$0.150912\pi$
$602$	0	0
$603$	14.5766	0.593604
$604$	0	0
$605$	−0.230065	−0.00935347
$606$	0	0
$607$	30.1490	1.22371	0.611856	−	0.790969i	$-0.290423\pi$
$607$	30.1490	1.22371	0.611856	+	0.790969i	$0.290423\pi$
$608$	0	0
$609$	−1.57084	−0.0636538
$610$	0	0
$611$	−17.5871	−0.711497
$612$	0	0
$613$	−17.6171	−0.711547	−0.355773	−	0.934572i	$-0.615782\pi$
$613$	−17.6171	−0.711547	−0.355773	+	0.934572i	$0.615782\pi$
$614$	0	0
$615$	−0.402682	−0.0162377
$616$	0	0
$617$	−23.5913	−0.949752	−0.474876	−	0.880053i	$-0.657507\pi$
$617$	−23.5913	−0.949752	−0.474876	+	0.880053i	$0.657507\pi$
$618$	0	0
$619$	20.4303	0.821164	0.410582	−	0.911824i	$-0.365326\pi$
$619$	20.4303	0.821164	0.410582	+	0.911824i	$0.365326\pi$
$620$	0	0
$621$	−10.7610	−0.431823
$622$	0	0
$623$	0.222820	0.00892709
$624$	0	0
$625$	5.28666	0.211467
$626$	0	0
$627$	0.222241	0.00887545
$628$	0	0
$629$	16.9778	0.676949
$630$	0	0
$631$	1.11790	0.0445029	0.0222515	−	0.999752i	$-0.492917\pi$
$631$	1.11790	0.0445029	0.0222515	+	0.999752i	$0.492917\pi$
$632$	0	0
$633$	−0.0492499	−0.00195751
$634$	0	0
$635$	−4.82132	−0.191328
$636$	0	0
$637$	−18.5974	−0.736854
$638$	0	0
$639$	−44.4985	−1.76033
$640$	0	0
$641$	20.7298	0.818776	0.409388	−	0.912360i	$-0.365742\pi$
$641$	20.7298	0.818776	0.409388	+	0.912360i	$0.365742\pi$
$642$	0	0
$643$	−16.6636	−0.657148	−0.328574	−	0.944478i	$-0.606568\pi$
$643$	−16.6636	−0.657148	−0.328574	+	0.944478i	$0.606568\pi$
$644$	0	0
$645$	−0.437437	−0.0172241
$646$	0	0
$647$	9.98512	0.392556	0.196278	−	0.980548i	$-0.437115\pi$
$647$	9.98512	0.392556	0.196278	+	0.980548i	$0.437115\pi$
$648$	0	0
$649$	−27.5793	−1.08258
$650$	0	0
$651$	−1.90135	−0.0745199
$652$	0	0
$653$	−38.5220	−1.50748	−0.753741	−	0.657172i	$-0.771752\pi$
$653$	−38.5220	−1.50748	−0.753741	+	0.657172i	$0.771752\pi$
$654$	0	0
$655$	1.68940	0.0660102
$656$	0	0
$657$	−46.8800	−1.82896
$658$	0	0
$659$	−26.5650	−1.03482	−0.517412	−	0.855736i	$-0.673104\pi$
$659$	−26.5650	−1.03482	−0.517412	+	0.855736i	$0.673104\pi$
$660$	0	0
$661$	22.1313	0.860808	0.430404	−	0.902636i	$-0.358371\pi$
$661$	22.1313	0.860808	0.430404	+	0.902636i	$0.358371\pi$
$662$	0	0
$663$	1.60131	0.0621896
$664$	0	0
$665$	−1.39738	−0.0541881
$666$	0	0
$667$	−15.5263	−0.601183
$668$	0	0
$669$	4.85490	0.187701
$670$	0	0
$671$	41.2585	1.59277
$672$	0	0
$673$	−13.7480	−0.529947	−0.264973	−	0.964256i	$-0.585363\pi$
$673$	−13.7480	−0.529947	−0.264973	+	0.964256i	$0.585363\pi$
$674$	0	0
$675$	−4.64503	−0.178787
$676$	0	0
$677$	17.9188	0.688675	0.344338	−	0.938846i	$-0.388103\pi$
$677$	17.9188	0.688675	0.344338	+	0.938846i	$0.388103\pi$
$678$	0	0
$679$	59.3945	2.27935
$680$	0	0
$681$	−0.800704	−0.0306830
$682$	0	0
$683$	26.6647	1.02029	0.510147	−	0.860087i	$-0.329591\pi$
$683$	26.6647	1.02029	0.510147	+	0.860087i	$0.329591\pi$
$684$	0	0
$685$	13.2345	0.505664
$686$	0	0
$687$	4.13696	0.157835
$688$	0	0
$689$	15.3088	0.583217
$690$	0	0
$691$	38.8083	1.47634	0.738168	−	0.674617i	$-0.235691\pi$
$691$	38.8083	1.47634	0.738168	+	0.674617i	$0.235691\pi$
$692$	0	0
$693$	−36.6129	−1.39081
$694$	0	0
$695$	10.7721	0.408607
$696$	0	0
$697$	4.28918	0.162464
$698$	0	0
$699$	3.35240	0.126799
$700$	0	0
$701$	37.6129	1.42062	0.710310	−	0.703889i	$-0.248555\pi$
$701$	37.6129	1.42062	0.710310	+	0.703889i	$0.248555\pi$
$702$	0	0
$703$	−1.84204	−0.0694740
$704$	0	0
$705$	−1.81770	−0.0684587
$706$	0	0
$707$	−8.22515	−0.309338
$708$	0	0
$709$	47.3761	1.77925	0.889624	−	0.456694i	$-0.150967\pi$
$709$	47.3761	1.77925	0.889624	+	0.456694i	$0.150967\pi$
$710$	0	0
$711$	9.24009	0.346530
$712$	0	0
$713$	−18.7931	−0.703808
$714$	0	0
$715$	−10.1944	−0.381251
$716$	0	0
$717$	3.93686	0.147025
$718$	0	0
$719$	−33.7267	−1.25779	−0.628896	−	0.777489i	$-0.716493\pi$
$719$	−33.7267	−1.25779	−0.628896	+	0.777489i	$0.716493\pi$
$720$	0	0
$721$	−0.308227	−0.0114790
$722$	0	0
$723$	1.41495	0.0526225
$724$	0	0
$725$	−6.70202	−0.248907
$726$	0	0
$727$	−7.40357	−0.274583	−0.137292	−	0.990531i	$-0.543840\pi$
$727$	−7.40357	−0.274583	−0.137292	+	0.990531i	$0.543840\pi$
$728$	0	0
$729$	−24.4170	−0.904334
$730$	0	0
$731$	4.65938	0.172333
$732$	0	0
$733$	30.3978	1.12277	0.561384	−	0.827555i	$-0.310269\pi$
$733$	30.3978	1.12277	0.561384	+	0.827555i	$0.310269\pi$
$734$	0	0
$735$	−1.92212	−0.0708985
$736$	0	0
$737$	−16.2371	−0.598102
$738$	0	0
$739$	3.50774	0.129034	0.0645171	−	0.997917i	$-0.479449\pi$
$739$	3.50774	0.129034	0.0645171	+	0.997917i	$0.479449\pi$
$740$	0	0
$741$	−0.173737	−0.00638239
$742$	0	0
$743$	49.3864	1.81181	0.905906	−	0.423479i	$-0.139191\pi$
$743$	49.3864	1.81181	0.905906	+	0.423479i	$0.139191\pi$
$744$	0	0
$745$	24.3129	0.890755
$746$	0	0
$747$	16.4454	0.601705
$748$	0	0
$749$	36.0856	1.31854
$750$	0	0
$751$	−34.1016	−1.24438	−0.622192	−	0.782865i	$-0.713758\pi$
$751$	−34.1016	−1.24438	−0.622192	+	0.782865i	$0.713758\pi$
$752$	0	0
$753$	−2.55719	−0.0931892
$754$	0	0
$755$	−12.6984	−0.462142
$756$	0	0
$757$	3.01859	0.109712	0.0548562	−	0.998494i	$-0.482530\pi$
$757$	3.01859	0.109712	0.0548562	+	0.998494i	$0.482530\pi$
$758$	0	0
$759$	5.94460	0.215775
$760$	0	0
$761$	−40.7815	−1.47833	−0.739164	−	0.673526i	$-0.764779\pi$
$761$	−40.7815	−1.47833	−0.739164	+	0.673526i	$0.764779\pi$
$762$	0	0
$763$	31.6222	1.14480
$764$	0	0
$765$	−10.0751	−0.364268
$766$	0	0
$767$	21.5601	0.778492
$768$	0	0
$769$	37.4334	1.34988	0.674941	−	0.737871i	$-0.264169\pi$
$769$	37.4334	1.34988	0.674941	+	0.737871i	$0.264169\pi$
$770$	0	0
$771$	0.0406511	0.00146401
$772$	0	0
$773$	−10.1952	−0.366696	−0.183348	−	0.983048i	$-0.558693\pi$
$773$	−10.1952	−0.366696	−0.183348	+	0.983048i	$0.558693\pi$
$774$	0	0
$775$	−8.11214	−0.291397
$776$	0	0
$777$	−4.98497	−0.178835
$778$	0	0
$779$	−0.465363	−0.0166734
$780$	0	0
$781$	49.5678	1.77367
$782$	0	0
$783$	2.47782	0.0885502
$784$	0	0
$785$	3.18246	0.113587
$786$	0	0
$787$	22.4880	0.801611	0.400806	−	0.916163i	$-0.368730\pi$
$787$	22.4880	0.801611	0.400806	+	0.916163i	$0.368730\pi$
$788$	0	0
$789$	2.52379	0.0898492
$790$	0	0
$791$	58.6626	2.08580
$792$	0	0
$793$	−32.2538	−1.14537
$794$	0	0
$795$	1.58223	0.0561159
$796$	0	0
$797$	6.07922	0.215337	0.107669	−	0.994187i	$-0.465661\pi$
$797$	6.07922	0.215337	0.107669	+	0.994187i	$0.465661\pi$
$798$	0	0
$799$	19.3613	0.684955
$800$	0	0
$801$	−0.174304	−0.00615875
$802$	0	0
$803$	52.2206	1.84282
$804$	0	0
$805$	−37.3778	−1.31739
$806$	0	0
$807$	−5.12427	−0.180383
$808$	0	0
$809$	1.55154	0.0545494	0.0272747	−	0.999628i	$-0.491317\pi$
$809$	1.55154	0.0545494	0.0272747	+	0.999628i	$0.491317\pi$
$810$	0	0
$811$	23.4919	0.824913	0.412456	−	0.910977i	$-0.364671\pi$
$811$	23.4919	0.824913	0.412456	+	0.910977i	$0.364671\pi$
$812$	0	0
$813$	−3.85967	−0.135365
$814$	0	0
$815$	15.6968	0.549834
$816$	0	0
$817$	−0.505529	−0.0176862
$818$	0	0
$819$	28.6222	1.00014
$820$	0	0
$821$	25.8660	0.902731	0.451366	−	0.892339i	$-0.350937\pi$
$821$	25.8660	0.902731	0.451366	+	0.892339i	$0.350937\pi$
$822$	0	0
$823$	43.1775	1.50507	0.752536	−	0.658551i	$-0.228830\pi$
$823$	43.1775	1.50507	0.752536	+	0.658551i	$0.228830\pi$
$824$	0	0
$825$	2.56602	0.0893373
$826$	0	0
$827$	−3.09266	−0.107542	−0.0537712	−	0.998553i	$-0.517124\pi$
$827$	−3.09266	−0.107542	−0.0537712	+	0.998553i	$0.517124\pi$
$828$	0	0
$829$	0.607227	0.0210899	0.0105449	−	0.999944i	$-0.496643\pi$
$829$	0.607227	0.0210899	0.0105449	+	0.999944i	$0.496643\pi$
$830$	0	0
$831$	−3.42195	−0.118706
$832$	0	0
$833$	20.4735	0.709366
$834$	0	0
$835$	−0.226271	−0.00783043
$836$	0	0
$837$	2.99916	0.103666
$838$	0	0
$839$	−29.3015	−1.01160	−0.505800	−	0.862651i	$-0.668803\pi$
$839$	−29.3015	−1.01160	−0.505800	+	0.862651i	$0.668803\pi$
$840$	0	0
$841$	−25.4249	−0.876721
$842$	0	0
$843$	−4.79820	−0.165259
$844$	0	0
$845$	−7.71392	−0.265367
$846$	0	0
$847$	−0.719521	−0.0247230
$848$	0	0
$849$	−2.48269	−0.0852057
$850$	0	0
$851$	−49.2718	−1.68902
$852$	0	0
$853$	1.12789	0.0386182	0.0193091	−	0.999814i	$-0.493853\pi$
$853$	1.12789	0.0386182	0.0193091	+	0.999814i	$0.493853\pi$
$854$	0	0
$855$	1.09312	0.0373841
$856$	0	0
$857$	−46.0217	−1.57207	−0.786035	−	0.618181i	$-0.787870\pi$
$857$	−46.0217	−1.57207	−0.786035	+	0.618181i	$0.787870\pi$
$858$	0	0
$859$	−6.93170	−0.236507	−0.118253	−	0.992983i	$-0.537729\pi$
$859$	−6.93170	−0.236507	−0.118253	+	0.992983i	$0.537729\pi$
$860$	0	0
$861$	−1.25937	−0.0429193
$862$	0	0
$863$	−33.4159	−1.13749	−0.568745	−	0.822514i	$-0.692571\pi$
$863$	−33.4159	−1.13749	−0.568745	+	0.822514i	$0.692571\pi$
$864$	0	0
$865$	−6.49294	−0.220767
$866$	0	0
$867$	1.98039	0.0672575
$868$	0	0
$869$	−10.2927	−0.349156
$870$	0	0
$871$	12.6934	0.430099
$872$	0	0
$873$	−46.4623	−1.57251
$874$	0	0
$875$	−38.8935	−1.31484
$876$	0	0
$877$	18.8804	0.637547	0.318774	−	0.947831i	$-0.396729\pi$
$877$	18.8804	0.637547	0.318774	+	0.947831i	$0.396729\pi$
$878$	0	0
$879$	−4.02498	−0.135759
$880$	0	0
$881$	20.6170	0.694604	0.347302	−	0.937753i	$-0.387098\pi$
$881$	20.6170	0.694604	0.347302	+	0.937753i	$0.387098\pi$
$882$	0	0
$883$	33.7702	1.13646	0.568230	−	0.822870i	$-0.307629\pi$
$883$	33.7702	1.13646	0.568230	+	0.822870i	$0.307629\pi$
$884$	0	0
$885$	2.22834	0.0749048
$886$	0	0
$887$	−19.9222	−0.668921	−0.334460	−	0.942410i	$-0.608554\pi$
$887$	−19.9222	−0.668921	−0.334460	+	0.942410i	$0.608554\pi$
$888$	0	0
$889$	−15.0785	−0.505717
$890$	0	0
$891$	28.1628	0.943491
$892$	0	0
$893$	−2.10065	−0.0702955
$894$	0	0
$895$	−3.76851	−0.125967
$896$	0	0
$897$	−4.64720	−0.155165
$898$	0	0
$899$	4.32731	0.144324
$900$	0	0
$901$	−16.8532	−0.561460
$902$	0	0
$903$	−1.36807	−0.0455265
$904$	0	0
$905$	18.3777	0.610896
$906$	0	0
$907$	−6.49970	−0.215819	−0.107909	−	0.994161i	$-0.534416\pi$
$907$	−6.49970	−0.215819	−0.107909	+	0.994161i	$0.534416\pi$
$908$	0	0
$909$	6.43426	0.213411
$910$	0	0
$911$	−8.30471	−0.275147	−0.137574	−	0.990492i	$-0.543930\pi$
$911$	−8.30471	−0.275147	−0.137574	+	0.990492i	$0.543930\pi$
$912$	0	0
$913$	−18.3188	−0.606265
$914$	0	0
$915$	−3.33358	−0.110205
$916$	0	0
$917$	5.28353	0.174478
$918$	0	0
$919$	−35.2968	−1.16433	−0.582166	−	0.813070i	$-0.697795\pi$
$919$	−35.2968	−1.16433	−0.582166	+	0.813070i	$0.697795\pi$
$920$	0	0
$921$	0.500455	0.0164905
$922$	0	0
$923$	−38.7496	−1.27546
$924$	0	0
$925$	−21.2684	−0.699301
$926$	0	0
$927$	0.241116	0.00791927
$928$	0	0
$929$	−30.2437	−0.992265	−0.496132	−	0.868247i	$-0.665247\pi$
$929$	−30.2437	−0.992265	−0.496132	+	0.868247i	$0.665247\pi$
$930$	0	0
$931$	−2.22132	−0.0728008
$932$	0	0
$933$	−2.15951	−0.0706992
$934$	0	0
$935$	11.2229	0.367028
$936$	0	0
$937$	−20.6550	−0.674771	−0.337385	−	0.941367i	$-0.609543\pi$
$937$	−20.6550	−0.674771	−0.337385	+	0.941367i	$0.609543\pi$
$938$	0	0
$939$	−3.63341	−0.118572
$940$	0	0
$941$	53.9112	1.75746	0.878728	−	0.477323i	$-0.158393\pi$
$941$	53.9112	1.75746	0.878728	+	0.477323i	$0.158393\pi$
$942$	0	0
$943$	−12.4477	−0.405354
$944$	0	0
$945$	5.96506	0.194043
$946$	0	0
$947$	−3.53030	−0.114719	−0.0573597	−	0.998354i	$-0.518268\pi$
$947$	−3.53030	−0.114719	−0.0573597	+	0.998354i	$0.518268\pi$
$948$	0	0
$949$	−40.8235	−1.32519
$950$	0	0
$951$	−1.85462	−0.0601401
$952$	0	0
$953$	−44.7276	−1.44887	−0.724434	−	0.689344i	$-0.757899\pi$
$953$	−44.7276	−1.44887	−0.724434	+	0.689344i	$0.757899\pi$
$954$	0	0
$955$	−6.91942	−0.223907
$956$	0	0
$957$	−1.36881	−0.0442472
$958$	0	0
$959$	41.3904	1.33657
$960$	0	0
$961$	−25.7622	−0.831039
$962$	0	0
$963$	−28.2286	−0.909652
$964$	0	0
$965$	−19.0003	−0.611641
$966$	0	0
$967$	32.9631	1.06002	0.530011	−	0.847991i	$-0.322188\pi$
$967$	32.9631	1.06002	0.530011	+	0.847991i	$0.322188\pi$
$968$	0	0
$969$	0.191264	0.00614430
$970$	0	0
$971$	−54.0561	−1.73474	−0.867371	−	0.497661i	$-0.834192\pi$
$971$	−54.0561	−1.73474	−0.867371	+	0.497661i	$0.834192\pi$
$972$	0	0
$973$	33.6892	1.08003
$974$	0	0
$975$	−2.00599	−0.0642430
$976$	0	0
$977$	−42.1987	−1.35006	−0.675028	−	0.737792i	$-0.735868\pi$
$977$	−42.1987	−1.35006	−0.675028	+	0.737792i	$0.735868\pi$
$978$	0	0
$979$	0.194161	0.00620542
$980$	0	0
$981$	−24.7370	−0.789791
$982$	0	0
$983$	26.0189	0.829873	0.414936	−	0.909850i	$-0.363804\pi$
$983$	26.0189	0.829873	0.414936	+	0.909850i	$0.363804\pi$
$984$	0	0
$985$	−19.0207	−0.606048
$986$	0	0
$987$	−5.68481	−0.180950
$988$	0	0
$989$	−13.5221	−0.429978
$990$	0	0
$991$	17.2574	0.548200	0.274100	−	0.961701i	$-0.411620\pi$
$991$	17.2574	0.548200	0.274100	+	0.961701i	$0.411620\pi$
$992$	0	0
$993$	0.340346	0.0108006
$994$	0	0
$995$	−7.33983	−0.232688
$996$	0	0
$997$	24.7282	0.783151	0.391576	−	0.920146i	$-0.371930\pi$
$997$	24.7282	0.783151	0.391576	+	0.920146i	$0.371930\pi$
$998$	0	0
$999$	7.86320	0.248781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.y.1.15		33
4.3	odd	2		4024.2.a.f.1.19	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.19	✓	33	4.3	odd	2
8048.2.a.y.1.15		33	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)

\(f(q)\)	\(=\)	\(q-0.220191 q^{3} +1.20642 q^{5} +3.77303 q^{7} -2.95152 q^{9} +O(q^{10})\)	\(q-0.220191 q^{3} +1.20642 q^{5} +3.77303 q^{7} -2.95152 q^{9} +3.28775 q^{11} -2.57020 q^{13} -0.265642 q^{15} +2.82949 q^{17} -0.306992 q^{19} -0.830786 q^{21} -8.21155 q^{23} -3.54456 q^{25} +1.31047 q^{27} +1.89079 q^{29} +2.28862 q^{31} -0.723932 q^{33} +4.55185 q^{35} +6.00030 q^{37} +0.565934 q^{39} +1.51588 q^{41} +1.64672 q^{43} -3.56076 q^{45} +6.84269 q^{47} +7.23576 q^{49} -0.623028 q^{51} -5.95625 q^{53} +3.96640 q^{55} +0.0675967 q^{57} -8.38851 q^{59} +12.5491 q^{61} -11.1362 q^{63} -3.10074 q^{65} -4.93868 q^{67} +1.80811 q^{69} +15.0765 q^{71} +15.8834 q^{73} +0.780478 q^{75} +12.4048 q^{77} -3.13062 q^{79} +8.56600 q^{81} -5.57184 q^{83} +3.41355 q^{85} -0.416335 q^{87} +0.0590559 q^{89} -9.69745 q^{91} -0.503933 q^{93} -0.370360 q^{95} +15.7419 q^{97} -9.70385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * q^99

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