LMFDB - Embedded newform 8048.2.a.v.1.14

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:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q-0.659121 q^{3} +0.997881 q^{5} -0.366743 q^{7} -2.56556 q^{9} +O(q^{10})$	$q-0.659121 q^{3} +0.997881 q^{5} -0.366743 q^{7} -2.56556 q^{9} -2.34193 q^{11} +1.58340 q^{13} -0.657724 q^{15} +0.164328 q^{17} +2.94082 q^{19} +0.241728 q^{21} +6.49861 q^{23} -4.00423 q^{25} +3.66838 q^{27} -7.89072 q^{29} +7.02250 q^{31} +1.54361 q^{33} -0.365966 q^{35} -5.39311 q^{37} -1.04365 q^{39} -4.46312 q^{41} -8.37356 q^{43} -2.56012 q^{45} +12.8414 q^{47} -6.86550 q^{49} -0.108312 q^{51} +12.7113 q^{53} -2.33697 q^{55} -1.93836 q^{57} -9.64319 q^{59} +9.98548 q^{61} +0.940901 q^{63} +1.58004 q^{65} +6.58675 q^{67} -4.28337 q^{69} -9.53600 q^{71} +2.32994 q^{73} +2.63927 q^{75} +0.858886 q^{77} -2.51367 q^{79} +5.27878 q^{81} +2.90312 q^{83} +0.163980 q^{85} +5.20093 q^{87} -13.3006 q^{89} -0.580700 q^{91} -4.62868 q^{93} +2.93459 q^{95} +0.261925 q^{97} +6.00836 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	−0.659121	−0.380543	−0.190272	−	0.981731i	$-0.560937\pi$
$3$	−0.659121	−0.380543	−0.190272	+	0.981731i	$0.560937\pi$
$4$	0	0
$5$	0.997881	0.446266	0.223133	−	0.974788i	$-0.428372\pi$
$5$	0.997881	0.446266	0.223133	+	0.974788i	$0.428372\pi$
$6$	0	0
$7$	−0.366743	−0.138616	−0.0693079	−	0.997595i	$-0.522079\pi$
$7$	−0.366743	−0.138616	−0.0693079	+	0.997595i	$0.522079\pi$
$8$	0	0
$9$	−2.56556	−0.855187
$10$	0	0
$11$	−2.34193	−0.706118	−0.353059	−	0.935601i	$-0.614859\pi$
$11$	−2.34193	−0.706118	−0.353059	+	0.935601i	$0.614859\pi$
$12$	0	0
$13$	1.58340	0.439155	0.219578	−	0.975595i	$-0.429532\pi$
$13$	1.58340	0.439155	0.219578	+	0.975595i	$0.429532\pi$
$14$	0	0
$15$	−0.657724	−0.169824
$16$	0	0
$17$	0.164328	0.0398554	0.0199277	−	0.999801i	$-0.493656\pi$
$17$	0.164328	0.0398554	0.0199277	+	0.999801i	$0.493656\pi$
$18$	0	0
$19$	2.94082	0.674671	0.337336	−	0.941384i	$-0.390474\pi$
$19$	2.94082	0.674671	0.337336	+	0.941384i	$0.390474\pi$
$20$	0	0
$21$	0.241728	0.0527493
$22$	0	0
$23$	6.49861	1.35505	0.677527	−	0.735498i	$-0.263052\pi$
$23$	6.49861	1.35505	0.677527	+	0.735498i	$0.263052\pi$
$24$	0	0
$25$	−4.00423	−0.800847
$26$	0	0
$27$	3.66838	0.705979
$28$	0	0
$29$	−7.89072	−1.46527	−0.732635	−	0.680622i	$-0.761710\pi$
$29$	−7.89072	−1.46527	−0.732635	+	0.680622i	$0.761710\pi$
$30$	0	0
$31$	7.02250	1.26128	0.630640	−	0.776076i	$-0.282793\pi$
$31$	7.02250	1.26128	0.630640	+	0.776076i	$0.282793\pi$
$32$	0	0
$33$	1.54361	0.268709
$34$	0	0
$35$	−0.365966	−0.0618595
$36$	0	0
$37$	−5.39311	−0.886622	−0.443311	−	0.896368i	$-0.646196\pi$
$37$	−5.39311	−0.886622	−0.443311	+	0.896368i	$0.646196\pi$
$38$	0	0
$39$	−1.04365	−0.167118
$40$	0	0
$41$	−4.46312	−0.697023	−0.348511	−	0.937305i	$-0.613313\pi$
$41$	−4.46312	−0.697023	−0.348511	+	0.937305i	$0.613313\pi$
$42$	0	0
$43$	−8.37356	−1.27696	−0.638478	−	0.769640i	$-0.720436\pi$
$43$	−8.37356	−1.27696	−0.638478	+	0.769640i	$0.720436\pi$
$44$	0	0
$45$	−2.56012	−0.381641
$46$	0	0
$47$	12.8414	1.87311	0.936555	−	0.350520i	$-0.113995\pi$
$47$	12.8414	1.87311	0.936555	+	0.350520i	$0.113995\pi$
$48$	0	0
$49$	−6.86550	−0.980786
$50$	0	0
$51$	−0.108312	−0.0151667
$52$	0	0
$53$	12.7113	1.74603	0.873013	−	0.487697i	$-0.162163\pi$
$53$	12.7113	1.74603	0.873013	+	0.487697i	$0.162163\pi$
$54$	0	0
$55$	−2.33697	−0.315116
$56$	0	0
$57$	−1.93836	−0.256742
$58$	0	0
$59$	−9.64319	−1.25544	−0.627718	−	0.778440i	$-0.716011\pi$
$59$	−9.64319	−1.25544	−0.627718	+	0.778440i	$0.716011\pi$
$60$	0	0
$61$	9.98548	1.27851	0.639255	−	0.768995i	$-0.279243\pi$
$61$	9.98548	1.27851	0.639255	+	0.768995i	$0.279243\pi$
$62$	0	0
$63$	0.940901	0.118542
$64$	0	0
$65$	1.58004	0.195980
$66$	0	0
$67$	6.58675	0.804700	0.402350	−	0.915486i	$-0.368193\pi$
$67$	6.58675	0.804700	0.402350	+	0.915486i	$0.368193\pi$
$68$	0	0
$69$	−4.28337	−0.515657
$70$	0	0
$71$	−9.53600	−1.13171	−0.565857	−	0.824503i	$-0.691455\pi$
$71$	−9.53600	−1.13171	−0.565857	+	0.824503i	$0.691455\pi$
$72$	0	0
$73$	2.32994	0.272699	0.136350	−	0.990661i	$-0.456463\pi$
$73$	2.32994	0.272699	0.136350	+	0.990661i	$0.456463\pi$
$74$	0	0
$75$	2.63927	0.304757
$76$	0	0
$77$	0.858886	0.0978791
$78$	0	0
$79$	−2.51367	−0.282810	−0.141405	−	0.989952i	$-0.545162\pi$
$79$	−2.51367	−0.282810	−0.141405	+	0.989952i	$0.545162\pi$
$80$	0	0
$81$	5.27878	0.586531
$82$	0	0
$83$	2.90312	0.318659	0.159330	−	0.987225i	$-0.449067\pi$
$83$	2.90312	0.318659	0.159330	+	0.987225i	$0.449067\pi$
$84$	0	0
$85$	0.163980	0.0177861
$86$	0	0
$87$	5.20093	0.557599
$88$	0	0
$89$	−13.3006	−1.40986	−0.704930	−	0.709276i	$-0.749022\pi$
$89$	−13.3006	−1.40986	−0.704930	+	0.709276i	$0.749022\pi$
$90$	0	0
$91$	−0.580700	−0.0608739
$92$	0	0
$93$	−4.62868	−0.479971
$94$	0	0
$95$	2.93459	0.301083
$96$	0	0
$97$	0.261925	0.0265945	0.0132972	−	0.999912i	$-0.495767\pi$
$97$	0.261925	0.0265945	0.0132972	+	0.999912i	$0.495767\pi$
$98$	0	0
$99$	6.00836	0.603863
$100$	0	0
$101$	−4.46200	−0.443985	−0.221993	−	0.975048i	$-0.571256\pi$
$101$	−4.46200	−0.443985	−0.221993	+	0.975048i	$0.571256\pi$
$102$	0	0
$103$	0.734821	0.0724040	0.0362020	−	0.999344i	$-0.488474\pi$
$103$	0.734821	0.0724040	0.0362020	+	0.999344i	$0.488474\pi$
$104$	0	0
$105$	0.241216	0.0235402
$106$	0	0
$107$	2.54784	0.246309	0.123154	−	0.992388i	$-0.460699\pi$
$107$	2.54784	0.246309	0.123154	+	0.992388i	$0.460699\pi$
$108$	0	0
$109$	−4.06803	−0.389647	−0.194823	−	0.980838i	$-0.562413\pi$
$109$	−4.06803	−0.389647	−0.194823	+	0.980838i	$0.562413\pi$
$110$	0	0
$111$	3.55471	0.337398
$112$	0	0
$113$	14.4829	1.36243	0.681216	−	0.732082i	$-0.261451\pi$
$113$	14.4829	1.36243	0.681216	+	0.732082i	$0.261451\pi$
$114$	0	0
$115$	6.48484	0.604714
$116$	0	0
$117$	−4.06230	−0.375560
$118$	0	0
$119$	−0.0602661	−0.00552458
$120$	0	0
$121$	−5.51537	−0.501397
$122$	0	0
$123$	2.94174	0.265248
$124$	0	0
$125$	−8.98515	−0.803656
$126$	0	0
$127$	2.68557	0.238306	0.119153	−	0.992876i	$-0.461982\pi$
$127$	2.68557	0.238306	0.119153	+	0.992876i	$0.461982\pi$
$128$	0	0
$129$	5.51918	0.485937
$130$	0	0
$131$	−9.68434	−0.846125	−0.423062	−	0.906101i	$-0.639045\pi$
$131$	−9.68434	−0.846125	−0.423062	+	0.906101i	$0.639045\pi$
$132$	0	0
$133$	−1.07853	−0.0935201
$134$	0	0
$135$	3.66060	0.315054
$136$	0	0
$137$	−18.8157	−1.60753	−0.803766	−	0.594945i	$-0.797174\pi$
$137$	−18.8157	−1.60753	−0.803766	+	0.594945i	$0.797174\pi$
$138$	0	0
$139$	−16.5214	−1.40132	−0.700662	−	0.713493i	$-0.747112\pi$
$139$	−16.5214	−1.40132	−0.700662	+	0.713493i	$0.747112\pi$
$140$	0	0
$141$	−8.46403	−0.712800
$142$	0	0
$143$	−3.70820	−0.310096
$144$	0	0
$145$	−7.87399	−0.653899
$146$	0	0
$147$	4.52519	0.373232
$148$	0	0
$149$	−10.9148	−0.894173	−0.447086	−	0.894491i	$-0.647538\pi$
$149$	−10.9148	−0.894173	−0.447086	+	0.894491i	$0.647538\pi$
$150$	0	0
$151$	0.258942	0.0210724	0.0105362	−	0.999944i	$-0.496646\pi$
$151$	0.258942	0.0210724	0.0105362	+	0.999944i	$0.496646\pi$
$152$	0	0
$153$	−0.421593	−0.0340838
$154$	0	0
$155$	7.00762	0.562866
$156$	0	0
$157$	3.76206	0.300245	0.150123	−	0.988667i	$-0.452033\pi$
$157$	3.76206	0.300245	0.150123	+	0.988667i	$0.452033\pi$
$158$	0	0
$159$	−8.37826	−0.664439
$160$	0	0
$161$	−2.38332	−0.187832
$162$	0	0
$163$	15.1296	1.18504	0.592520	−	0.805556i	$-0.298133\pi$
$163$	15.1296	1.18504	0.592520	+	0.805556i	$0.298133\pi$
$164$	0	0
$165$	1.54034	0.119915
$166$	0	0
$167$	5.73732	0.443967	0.221983	−	0.975050i	$-0.428747\pi$
$167$	5.73732	0.443967	0.221983	+	0.975050i	$0.428747\pi$
$168$	0	0
$169$	−10.4929	−0.807143
$170$	0	0
$171$	−7.54486	−0.576970
$172$	0	0
$173$	4.95955	0.377068	0.188534	−	0.982067i	$-0.439626\pi$
$173$	4.95955	0.377068	0.188534	+	0.982067i	$0.439626\pi$
$174$	0	0
$175$	1.46852	0.111010
$176$	0	0
$177$	6.35603	0.477748
$178$	0	0
$179$	7.47906	0.559011	0.279506	−	0.960144i	$-0.409829\pi$
$179$	7.47906	0.559011	0.279506	+	0.960144i	$0.409829\pi$
$180$	0	0
$181$	−8.26533	−0.614357	−0.307179	−	0.951652i	$-0.599385\pi$
$181$	−8.26533	−0.614357	−0.307179	+	0.951652i	$0.599385\pi$
$182$	0	0
$183$	−6.58164	−0.486528
$184$	0	0
$185$	−5.38168	−0.395669
$186$	0	0
$187$	−0.384844	−0.0281426
$188$	0	0
$189$	−1.34535	−0.0978599
$190$	0	0
$191$	2.25713	0.163320	0.0816601	−	0.996660i	$-0.473978\pi$
$191$	2.25713	0.163320	0.0816601	+	0.996660i	$0.473978\pi$
$192$	0	0
$193$	−18.4007	−1.32451	−0.662257	−	0.749277i	$-0.730401\pi$
$193$	−18.4007	−1.32451	−0.662257	+	0.749277i	$0.730401\pi$
$194$	0	0
$195$	−1.04144	−0.0745789
$196$	0	0
$197$	−11.6058	−0.826877	−0.413438	−	0.910532i	$-0.635672\pi$
$197$	−11.6058	−0.826877	−0.413438	+	0.910532i	$0.635672\pi$
$198$	0	0
$199$	−16.8781	−1.19646	−0.598228	−	0.801326i	$-0.704128\pi$
$199$	−16.8781	−1.19646	−0.598228	+	0.801326i	$0.704128\pi$
$200$	0	0
$201$	−4.34147	−0.306223
$202$	0	0
$203$	2.89386	0.203109
$204$	0	0
$205$	−4.45367	−0.311057
$206$	0	0
$207$	−16.6726	−1.15882
$208$	0	0
$209$	−6.88720	−0.476397
$210$	0	0
$211$	−8.12192	−0.559136	−0.279568	−	0.960126i	$-0.590191\pi$
$211$	−8.12192	−0.559136	−0.279568	+	0.960126i	$0.590191\pi$
$212$	0	0
$213$	6.28537	0.430667
$214$	0	0
$215$	−8.35581	−0.569862
$216$	0	0
$217$	−2.57545	−0.174833
$218$	0	0
$219$	−1.53571	−0.103774
$220$	0	0
$221$	0.260196	0.0175027
$222$	0	0
$223$	−5.67745	−0.380190	−0.190095	−	0.981766i	$-0.560880\pi$
$223$	−5.67745	−0.380190	−0.190095	+	0.981766i	$0.560880\pi$
$224$	0	0
$225$	10.2731	0.684874
$226$	0	0
$227$	−13.0486	−0.866063	−0.433032	−	0.901379i	$-0.642556\pi$
$227$	−13.0486	−0.866063	−0.433032	+	0.901379i	$0.642556\pi$
$228$	0	0
$229$	0.823786	0.0544373	0.0272187	−	0.999630i	$-0.491335\pi$
$229$	0.823786	0.0544373	0.0272187	+	0.999630i	$0.491335\pi$
$230$	0	0
$231$	−0.566109	−0.0372473
$232$	0	0
$233$	11.0371	0.723065	0.361532	−	0.932360i	$-0.382254\pi$
$233$	11.0371	0.723065	0.361532	+	0.932360i	$0.382254\pi$
$234$	0	0
$235$	12.8142	0.835905
$236$	0	0
$237$	1.65681	0.107621
$238$	0	0
$239$	−24.2559	−1.56898	−0.784491	−	0.620140i	$-0.787076\pi$
$239$	−24.2559	−1.56898	−0.784491	+	0.620140i	$0.787076\pi$
$240$	0	0
$241$	−13.3061	−0.857121	−0.428560	−	0.903513i	$-0.640979\pi$
$241$	−13.3061	−0.857121	−0.428560	+	0.903513i	$0.640979\pi$
$242$	0	0
$243$	−14.4845	−0.929180
$244$	0	0
$245$	−6.85095	−0.437691
$246$	0	0
$247$	4.65649	0.296285
$248$	0	0
$249$	−1.91351	−0.121264
$250$	0	0
$251$	−6.73438	−0.425071	−0.212535	−	0.977153i	$-0.568172\pi$
$251$	−6.73438	−0.425071	−0.212535	+	0.977153i	$0.568172\pi$
$252$	0	0
$253$	−15.2193	−0.956828
$254$	0	0
$255$	−0.108082	−0.00676838
$256$	0	0
$257$	−27.0972	−1.69028	−0.845138	−	0.534549i	$-0.820482\pi$
$257$	−27.0972	−1.69028	−0.845138	+	0.534549i	$0.820482\pi$
$258$	0	0
$259$	1.97789	0.122900
$260$	0	0
$261$	20.2441	1.25308
$262$	0	0
$263$	20.5250	1.26562	0.632811	−	0.774306i	$-0.281901\pi$
$263$	20.5250	1.26562	0.632811	+	0.774306i	$0.281901\pi$
$264$	0	0
$265$	12.6843	0.779192
$266$	0	0
$267$	8.76670	0.536513
$268$	0	0
$269$	22.6469	1.38081	0.690404	−	0.723424i	$-0.257433\pi$
$269$	22.6469	1.38081	0.690404	+	0.723424i	$0.257433\pi$
$270$	0	0
$271$	8.87654	0.539211	0.269606	−	0.962971i	$-0.413107\pi$
$271$	8.87654	0.539211	0.269606	+	0.962971i	$0.413107\pi$
$272$	0	0
$273$	0.382751	0.0231652
$274$	0	0
$275$	9.37763	0.565492
$276$	0	0
$277$	19.6591	1.18120	0.590602	−	0.806963i	$-0.298890\pi$
$277$	19.6591	1.18120	0.590602	+	0.806963i	$0.298890\pi$
$278$	0	0
$279$	−18.0167	−1.07863
$280$	0	0
$281$	15.2756	0.911265	0.455632	−	0.890168i	$-0.349413\pi$
$281$	15.2756	0.911265	0.455632	+	0.890168i	$0.349413\pi$
$282$	0	0
$283$	−21.6629	−1.28773	−0.643864	−	0.765140i	$-0.722670\pi$
$283$	−21.6629	−1.28773	−0.643864	+	0.765140i	$0.722670\pi$
$284$	0	0
$285$	−1.93425	−0.114575
$286$	0	0
$287$	1.63682	0.0966184
$288$	0	0
$289$	−16.9730	−0.998412
$290$	0	0
$291$	−0.172640	−0.0101204
$292$	0	0
$293$	9.59095	0.560310	0.280155	−	0.959955i	$-0.409614\pi$
$293$	9.59095	0.560310	0.280155	+	0.959955i	$0.409614\pi$
$294$	0	0
$295$	−9.62276	−0.560259
$296$	0	0
$297$	−8.59107	−0.498505
$298$	0	0
$299$	10.2899	0.595079
$300$	0	0
$301$	3.07094	0.177006
$302$	0	0
$303$	2.94099	0.168956
$304$	0	0
$305$	9.96432	0.570555
$306$	0	0
$307$	7.78430	0.444273	0.222137	−	0.975016i	$-0.428697\pi$
$307$	7.78430	0.444273	0.222137	+	0.975016i	$0.428697\pi$
$308$	0	0
$309$	−0.484335	−0.0275529
$310$	0	0
$311$	33.5497	1.90243	0.951216	−	0.308527i	$-0.0998360\pi$
$311$	33.5497	1.90243	0.951216	+	0.308527i	$0.0998360\pi$
$312$	0	0
$313$	−6.19902	−0.350390	−0.175195	−	0.984534i	$-0.556056\pi$
$313$	−6.19902	−0.350390	−0.175195	+	0.984534i	$0.556056\pi$
$314$	0	0
$315$	0.938907	0.0529014
$316$	0	0
$317$	−11.4833	−0.644968	−0.322484	−	0.946575i	$-0.604518\pi$
$317$	−11.4833	−0.644968	−0.322484	+	0.946575i	$0.604518\pi$
$318$	0	0
$319$	18.4795	1.03465
$320$	0	0
$321$	−1.67933	−0.0937313
$322$	0	0
$323$	0.483259	0.0268893
$324$	0	0
$325$	−6.34029	−0.351696
$326$	0	0
$327$	2.68132	0.148278
$328$	0	0
$329$	−4.70949	−0.259643
$330$	0	0
$331$	3.29804	0.181277	0.0906383	−	0.995884i	$-0.471109\pi$
$331$	3.29804	0.181277	0.0906383	+	0.995884i	$0.471109\pi$
$332$	0	0
$333$	13.8363	0.758227
$334$	0	0
$335$	6.57280	0.359110
$336$	0	0
$337$	7.62848	0.415550	0.207775	−	0.978177i	$-0.433378\pi$
$337$	7.62848	0.415550	0.207775	+	0.978177i	$0.433378\pi$
$338$	0	0
$339$	−9.54595	−0.518465
$340$	0	0
$341$	−16.4462	−0.890612
$342$	0	0
$343$	5.08508	0.274568
$344$	0	0
$345$	−4.27429	−0.230120
$346$	0	0
$347$	−30.6857	−1.64730	−0.823648	−	0.567102i	$-0.808065\pi$
$347$	−30.6857	−1.64730	−0.823648	+	0.567102i	$0.808065\pi$
$348$	0	0
$349$	−9.46969	−0.506901	−0.253450	−	0.967348i	$-0.581565\pi$
$349$	−9.46969	−0.506901	−0.253450	+	0.967348i	$0.581565\pi$
$350$	0	0
$351$	5.80850	0.310035
$352$	0	0
$353$	−17.2010	−0.915516	−0.457758	−	0.889077i	$-0.651347\pi$
$353$	−17.2010	−0.915516	−0.457758	+	0.889077i	$0.651347\pi$
$354$	0	0
$355$	−9.51579	−0.505046
$356$	0	0
$357$	0.0397226	0.00210234
$358$	0	0
$359$	7.41231	0.391207	0.195603	−	0.980683i	$-0.437333\pi$
$359$	7.41231	0.391207	0.195603	+	0.980683i	$0.437333\pi$
$360$	0	0
$361$	−10.3516	−0.544819
$362$	0	0
$363$	3.63530	0.190804
$364$	0	0
$365$	2.32500	0.121696
$366$	0	0
$367$	−21.7723	−1.13651	−0.568253	−	0.822854i	$-0.692381\pi$
$367$	−21.7723	−1.13651	−0.568253	+	0.822854i	$0.692381\pi$
$368$	0	0
$369$	11.4504	0.596085
$370$	0	0
$371$	−4.66177	−0.242027
$372$	0	0
$373$	−22.7062	−1.17568	−0.587841	−	0.808977i	$-0.700022\pi$
$373$	−22.7062	−1.17568	−0.587841	+	0.808977i	$0.700022\pi$
$374$	0	0
$375$	5.92230	0.305826
$376$	0	0
$377$	−12.4941	−0.643481
$378$	0	0
$379$	−22.5815	−1.15994	−0.579968	−	0.814639i	$-0.696935\pi$
$379$	−22.5815	−1.15994	−0.579968	+	0.814639i	$0.696935\pi$
$380$	0	0
$381$	−1.77011	−0.0906857
$382$	0	0
$383$	29.5809	1.51151	0.755756	−	0.654854i	$-0.227270\pi$
$383$	29.5809	1.51151	0.755756	+	0.654854i	$0.227270\pi$
$384$	0	0
$385$	0.857066	0.0436801
$386$	0	0
$387$	21.4829	1.09204
$388$	0	0
$389$	4.11718	0.208749	0.104375	−	0.994538i	$-0.466716\pi$
$389$	4.11718	0.208749	0.104375	+	0.994538i	$0.466716\pi$
$390$	0	0
$391$	1.06790	0.0540061
$392$	0	0
$393$	6.38315	0.321987
$394$	0	0
$395$	−2.50834	−0.126208
$396$	0	0
$397$	−19.4331	−0.975319	−0.487660	−	0.873034i	$-0.662149\pi$
$397$	−19.4331	−0.975319	−0.487660	+	0.873034i	$0.662149\pi$
$398$	0	0
$399$	0.710879	0.0355885
$400$	0	0
$401$	15.1714	0.757621	0.378811	−	0.925474i	$-0.376333\pi$
$401$	15.1714	0.757621	0.378811	+	0.925474i	$0.376333\pi$
$402$	0	0
$403$	11.1194	0.553898
$404$	0	0
$405$	5.26759	0.261749
$406$	0	0
$407$	12.6303	0.626059
$408$	0	0
$409$	32.6254	1.61322	0.806610	−	0.591083i	$-0.201300\pi$
$409$	32.6254	1.61322	0.806610	+	0.591083i	$0.201300\pi$
$410$	0	0
$411$	12.4018	0.611736
$412$	0	0
$413$	3.53657	0.174023
$414$	0	0
$415$	2.89697	0.142207
$416$	0	0
$417$	10.8896	0.533265
$418$	0	0
$419$	−30.1541	−1.47312	−0.736562	−	0.676370i	$-0.763552\pi$
$419$	−30.1541	−1.47312	−0.736562	+	0.676370i	$0.763552\pi$
$420$	0	0
$421$	−16.0071	−0.780140	−0.390070	−	0.920785i	$-0.627549\pi$
$421$	−16.0071	−0.780140	−0.390070	+	0.920785i	$0.627549\pi$
$422$	0	0
$423$	−32.9454	−1.60186
$424$	0	0
$425$	−0.658007	−0.0319180
$426$	0	0
$427$	−3.66211	−0.177222
$428$	0	0
$429$	2.44415	0.118005
$430$	0	0
$431$	18.3797	0.885321	0.442661	−	0.896689i	$-0.354035\pi$
$431$	18.3797	0.885321	0.442661	+	0.896689i	$0.354035\pi$
$432$	0	0
$433$	−19.9237	−0.957471	−0.478735	−	0.877959i	$-0.658905\pi$
$433$	−19.9237	−0.957471	−0.478735	+	0.877959i	$0.658905\pi$
$434$	0	0
$435$	5.18991	0.248837
$436$	0	0
$437$	19.1113	0.914215
$438$	0	0
$439$	−25.5087	−1.21747	−0.608733	−	0.793375i	$-0.708322\pi$
$439$	−25.5087	−1.21747	−0.608733	+	0.793375i	$0.708322\pi$
$440$	0	0
$441$	17.6139	0.838755
$442$	0	0
$443$	−3.79329	−0.180225	−0.0901124	−	0.995932i	$-0.528723\pi$
$443$	−3.79329	−0.180225	−0.0901124	+	0.995932i	$0.528723\pi$
$444$	0	0
$445$	−13.2724	−0.629173
$446$	0	0
$447$	7.19415	0.340272
$448$	0	0
$449$	2.08159	0.0982363	0.0491182	−	0.998793i	$-0.484359\pi$
$449$	2.08159	0.0982363	0.0491182	+	0.998793i	$0.484359\pi$
$450$	0	0
$451$	10.4523	0.492180
$452$	0	0
$453$	−0.170674	−0.00801895
$454$	0	0
$455$	−0.579469	−0.0271659
$456$	0	0
$457$	34.4508	1.61154	0.805771	−	0.592227i	$-0.201751\pi$
$457$	34.4508	1.61154	0.805771	+	0.592227i	$0.201751\pi$
$458$	0	0
$459$	0.602816	0.0281370
$460$	0	0
$461$	−25.3819	−1.18215	−0.591076	−	0.806616i	$-0.701297\pi$
$461$	−25.3819	−1.18215	−0.591076	+	0.806616i	$0.701297\pi$
$462$	0	0
$463$	−15.2540	−0.708912	−0.354456	−	0.935073i	$-0.615334\pi$
$463$	−15.2540	−0.708912	−0.354456	+	0.935073i	$0.615334\pi$
$464$	0	0
$465$	−4.61887	−0.214195
$466$	0	0
$467$	−6.38856	−0.295627	−0.147814	−	0.989015i	$-0.547224\pi$
$467$	−6.38856	−0.295627	−0.147814	+	0.989015i	$0.547224\pi$
$468$	0	0
$469$	−2.41565	−0.111544
$470$	0	0
$471$	−2.47965	−0.114256
$472$	0	0
$473$	19.6103	0.901681
$474$	0	0
$475$	−11.7757	−0.540308
$476$	0	0
$477$	−32.6115	−1.49318
$478$	0	0
$479$	37.5331	1.71493	0.857464	−	0.514543i	$-0.172038\pi$
$479$	37.5331	1.71493	0.857464	+	0.514543i	$0.172038\pi$
$480$	0	0
$481$	−8.53943	−0.389365
$482$	0	0
$483$	1.57089	0.0714782
$484$	0	0
$485$	0.261370	0.0118682
$486$	0	0
$487$	4.33788	0.196568	0.0982840	−	0.995158i	$-0.468665\pi$
$487$	4.33788	0.196568	0.0982840	+	0.995158i	$0.468665\pi$
$488$	0	0
$489$	−9.97221	−0.450959
$490$	0	0
$491$	−32.0024	−1.44425	−0.722124	−	0.691764i	$-0.756834\pi$
$491$	−32.0024	−1.44425	−0.722124	+	0.691764i	$0.756834\pi$
$492$	0	0
$493$	−1.29666	−0.0583988
$494$	0	0
$495$	5.99562	0.269483
$496$	0	0
$497$	3.49726	0.156874
$498$	0	0
$499$	−3.66389	−0.164018	−0.0820091	−	0.996632i	$-0.526134\pi$
$499$	−3.66389	−0.164018	−0.0820091	+	0.996632i	$0.526134\pi$
$500$	0	0
$501$	−3.78158	−0.168949
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−4.45254	−0.198135
$506$	0	0
$507$	6.91606	0.307153
$508$	0	0
$509$	−34.8313	−1.54387	−0.771936	−	0.635700i	$-0.780711\pi$
$509$	−34.8313	−1.54387	−0.771936	+	0.635700i	$0.780711\pi$
$510$	0	0
$511$	−0.854490	−0.0378004
$512$	0	0
$513$	10.7880	0.476304
$514$	0	0
$515$	0.733263	0.0323114
$516$	0	0
$517$	−30.0736	−1.32264
$518$	0	0
$519$	−3.26894	−0.143491
$520$	0	0
$521$	−24.0129	−1.05202	−0.526012	−	0.850477i	$-0.676313\pi$
$521$	−24.0129	−1.05202	−0.526012	+	0.850477i	$0.676313\pi$
$522$	0	0
$523$	−29.8701	−1.30613	−0.653064	−	0.757303i	$-0.726517\pi$
$523$	−29.8701	−1.30613	−0.653064	+	0.757303i	$0.726517\pi$
$524$	0	0
$525$	−0.967935	−0.0422441
$526$	0	0
$527$	1.15399	0.0502687
$528$	0	0
$529$	19.2319	0.836170
$530$	0	0
$531$	24.7402	1.07363
$532$	0	0
$533$	−7.06690	−0.306101
$534$	0	0
$535$	2.54244	0.109919
$536$	0	0
$537$	−4.92960	−0.212728
$538$	0	0
$539$	16.0785	0.692550
$540$	0	0
$541$	11.2005	0.481547	0.240774	−	0.970581i	$-0.422599\pi$
$541$	11.2005	0.481547	0.240774	+	0.970581i	$0.422599\pi$
$542$	0	0
$543$	5.44785	0.233790
$544$	0	0
$545$	−4.05941	−0.173886
$546$	0	0
$547$	24.3236	1.04000	0.520001	−	0.854166i	$-0.325932\pi$
$547$	24.3236	1.04000	0.520001	+	0.854166i	$0.325932\pi$
$548$	0	0
$549$	−25.6183	−1.09336
$550$	0	0
$551$	−23.2052	−0.988575
$552$	0	0
$553$	0.921870	0.0392019
$554$	0	0
$555$	3.54718	0.150569
$556$	0	0
$557$	−15.5932	−0.660705	−0.330353	−	0.943858i	$-0.607168\pi$
$557$	−15.5932	−0.660705	−0.330353	+	0.943858i	$0.607168\pi$
$558$	0	0
$559$	−13.2587	−0.560782
$560$	0	0
$561$	0.253659	0.0107095
$562$	0	0
$563$	2.04046	0.0859952	0.0429976	−	0.999075i	$-0.486309\pi$
$563$	2.04046	0.0859952	0.0429976	+	0.999075i	$0.486309\pi$
$564$	0	0
$565$	14.4522	0.608007
$566$	0	0
$567$	−1.93596	−0.0813025
$568$	0	0
$569$	29.7908	1.24890	0.624448	−	0.781067i	$-0.285324\pi$
$569$	29.7908	1.24890	0.624448	+	0.781067i	$0.285324\pi$
$570$	0	0
$571$	1.67514	0.0701025	0.0350513	−	0.999386i	$-0.488841\pi$
$571$	1.67514	0.0701025	0.0350513	+	0.999386i	$0.488841\pi$
$572$	0	0
$573$	−1.48772	−0.0621504
$574$	0	0
$575$	−26.0220	−1.08519
$576$	0	0
$577$	3.76042	0.156548	0.0782742	−	0.996932i	$-0.475059\pi$
$577$	3.76042	0.156548	0.0782742	+	0.996932i	$0.475059\pi$
$578$	0	0
$579$	12.1283	0.504035
$580$	0	0
$581$	−1.06470	−0.0441712
$582$	0	0
$583$	−29.7689	−1.23290
$584$	0	0
$585$	−4.05369	−0.167600
$586$	0	0
$587$	25.9499	1.07107	0.535533	−	0.844514i	$-0.320111\pi$
$587$	25.9499	1.07107	0.535533	+	0.844514i	$0.320111\pi$
$588$	0	0
$589$	20.6519	0.850949
$590$	0	0
$591$	7.64960	0.314663
$592$	0	0
$593$	9.45049	0.388085	0.194043	−	0.980993i	$-0.437840\pi$
$593$	9.45049	0.388085	0.194043	+	0.980993i	$0.437840\pi$
$594$	0	0
$595$	−0.0601384	−0.00246543
$596$	0	0
$597$	11.1247	0.455304
$598$	0	0
$599$	−30.7653	−1.25703	−0.628517	−	0.777796i	$-0.716338\pi$
$599$	−30.7653	−1.25703	−0.628517	+	0.777796i	$0.716338\pi$
$600$	0	0
$601$	9.70280	0.395785	0.197893	−	0.980224i	$-0.436590\pi$
$601$	9.70280	0.395785	0.197893	+	0.980224i	$0.436590\pi$
$602$	0	0
$603$	−16.8987	−0.688169
$604$	0	0
$605$	−5.50368	−0.223757
$606$	0	0
$607$	3.66881	0.148912	0.0744562	−	0.997224i	$-0.476278\pi$
$607$	3.66881	0.148912	0.0744562	+	0.997224i	$0.476278\pi$
$608$	0	0
$609$	−1.90741	−0.0772920
$610$	0	0
$611$	20.3330	0.822587
$612$	0	0
$613$	44.3930	1.79302	0.896509	−	0.443025i	$-0.146095\pi$
$613$	44.3930	1.79302	0.896509	+	0.443025i	$0.146095\pi$
$614$	0	0
$615$	2.93550	0.118371
$616$	0	0
$617$	39.0543	1.57227	0.786134	−	0.618057i	$-0.212080\pi$
$617$	39.0543	1.57227	0.786134	+	0.618057i	$0.212080\pi$
$618$	0	0
$619$	−24.2073	−0.972973	−0.486487	−	0.873688i	$-0.661722\pi$
$619$	−24.2073	−0.972973	−0.486487	+	0.873688i	$0.661722\pi$
$620$	0	0
$621$	23.8393	0.956640
$622$	0	0
$623$	4.87790	0.195429
$624$	0	0
$625$	11.0551	0.442202
$626$	0	0
$627$	4.53949	0.181290
$628$	0	0
$629$	−0.886238	−0.0353366
$630$	0	0
$631$	−12.3183	−0.490385	−0.245193	−	0.969474i	$-0.578851\pi$
$631$	−12.3183	−0.490385	−0.245193	+	0.969474i	$0.578851\pi$
$632$	0	0
$633$	5.35332	0.212775
$634$	0	0
$635$	2.67988	0.106348
$636$	0	0
$637$	−10.8708	−0.430717
$638$	0	0
$639$	24.4652	0.967828
$640$	0	0
$641$	−16.9113	−0.667956	−0.333978	−	0.942581i	$-0.608391\pi$
$641$	−16.9113	−0.667956	−0.333978	+	0.942581i	$0.608391\pi$
$642$	0	0
$643$	−26.1189	−1.03003	−0.515015	−	0.857181i	$-0.672213\pi$
$643$	−26.1189	−1.03003	−0.515015	+	0.857181i	$0.672213\pi$
$644$	0	0
$645$	5.50749	0.216857
$646$	0	0
$647$	−8.24455	−0.324127	−0.162063	−	0.986780i	$-0.551815\pi$
$647$	−8.24455	−0.324127	−0.162063	+	0.986780i	$0.551815\pi$
$648$	0	0
$649$	22.5837	0.886486
$650$	0	0
$651$	1.69754	0.0665317
$652$	0	0
$653$	−39.8697	−1.56022	−0.780110	−	0.625642i	$-0.784837\pi$
$653$	−39.8697	−1.56022	−0.780110	+	0.625642i	$0.784837\pi$
$654$	0	0
$655$	−9.66382	−0.377597
$656$	0	0
$657$	−5.97761	−0.233209
$658$	0	0
$659$	−43.2594	−1.68515	−0.842573	−	0.538582i	$-0.818960\pi$
$659$	−43.2594	−1.68515	−0.842573	+	0.538582i	$0.818960\pi$
$660$	0	0
$661$	−36.2663	−1.41059	−0.705297	−	0.708912i	$-0.749187\pi$
$661$	−36.2663	−1.41059	−0.705297	+	0.708912i	$0.749187\pi$
$662$	0	0
$663$	−0.171501	−0.00666054
$664$	0	0
$665$	−1.07624	−0.0417348
$666$	0	0
$667$	−51.2787	−1.98552
$668$	0	0
$669$	3.74212	0.144679
$670$	0	0
$671$	−23.3853	−0.902779
$672$	0	0
$673$	6.83729	0.263558	0.131779	−	0.991279i	$-0.457931\pi$
$673$	6.83729	0.263558	0.131779	+	0.991279i	$0.457931\pi$
$674$	0	0
$675$	−14.6890	−0.565381
$676$	0	0
$677$	−28.3308	−1.08884	−0.544420	−	0.838812i	$-0.683250\pi$
$677$	−28.3308	−1.08884	−0.544420	+	0.838812i	$0.683250\pi$
$678$	0	0
$679$	−0.0960593	−0.00368642
$680$	0	0
$681$	8.60057	0.329575
$682$	0	0
$683$	22.3969	0.856995	0.428497	−	0.903543i	$-0.359043\pi$
$683$	22.3969	0.856995	0.428497	+	0.903543i	$0.359043\pi$
$684$	0	0
$685$	−18.7758	−0.717387
$686$	0	0
$687$	−0.542975	−0.0207158
$688$	0	0
$689$	20.1270	0.766777
$690$	0	0
$691$	−8.86769	−0.337343	−0.168671	−	0.985672i	$-0.553948\pi$
$691$	−8.86769	−0.337343	−0.168671	+	0.985672i	$0.553948\pi$
$692$	0	0
$693$	−2.20352	−0.0837049
$694$	0	0
$695$	−16.4864	−0.625363
$696$	0	0
$697$	−0.733416	−0.0277801
$698$	0	0
$699$	−7.27478	−0.275158
$700$	0	0
$701$	−28.7295	−1.08510	−0.542550	−	0.840024i	$-0.682541\pi$
$701$	−28.7295	−1.08510	−0.542550	+	0.840024i	$0.682541\pi$
$702$	0	0
$703$	−15.8602	−0.598178
$704$	0	0
$705$	−8.44609	−0.318098
$706$	0	0
$707$	1.63641	0.0615434
$708$	0	0
$709$	−16.2925	−0.611878	−0.305939	−	0.952051i	$-0.598970\pi$
$709$	−16.2925	−0.611878	−0.305939	+	0.952051i	$0.598970\pi$
$710$	0	0
$711$	6.44896	0.241855
$712$	0	0
$713$	45.6365	1.70910
$714$	0	0
$715$	−3.70034	−0.138385
$716$	0	0
$717$	15.9876	0.597066
$718$	0	0
$719$	−8.67655	−0.323581	−0.161790	−	0.986825i	$-0.551727\pi$
$719$	−8.67655	−0.323581	−0.161790	+	0.986825i	$0.551727\pi$
$720$	0	0
$721$	−0.269490	−0.0100363
$722$	0	0
$723$	8.77032	0.326172
$724$	0	0
$725$	31.5963	1.17346
$726$	0	0
$727$	38.9559	1.44479	0.722396	−	0.691479i	$-0.243041\pi$
$727$	38.9559	1.44479	0.722396	+	0.691479i	$0.243041\pi$
$728$	0	0
$729$	−6.28932	−0.232938
$730$	0	0
$731$	−1.37601	−0.0508935
$732$	0	0
$733$	−0.822898	−0.0303944	−0.0151972	−	0.999885i	$-0.504838\pi$
$733$	−0.822898	−0.0303944	−0.0151972	+	0.999885i	$0.504838\pi$
$734$	0	0
$735$	4.51560	0.166560
$736$	0	0
$737$	−15.4257	−0.568213
$738$	0	0
$739$	37.6813	1.38613	0.693065	−	0.720875i	$-0.256260\pi$
$739$	37.6813	1.38613	0.693065	+	0.720875i	$0.256260\pi$
$740$	0	0
$741$	−3.06919	−0.112749
$742$	0	0
$743$	−35.6314	−1.30719	−0.653594	−	0.756845i	$-0.726740\pi$
$743$	−35.6314	−1.30719	−0.653594	+	0.756845i	$0.726740\pi$
$744$	0	0
$745$	−10.8916	−0.399039
$746$	0	0
$747$	−7.44814	−0.272513
$748$	0	0
$749$	−0.934403	−0.0341423
$750$	0	0
$751$	−16.1886	−0.590731	−0.295366	−	0.955384i	$-0.595441\pi$
$751$	−16.1886	−0.590731	−0.295366	+	0.955384i	$0.595441\pi$
$752$	0	0
$753$	4.43877	0.161758
$754$	0	0
$755$	0.258393	0.00940388
$756$	0	0
$757$	33.2456	1.20833	0.604165	−	0.796859i	$-0.293507\pi$
$757$	33.2456	1.20833	0.604165	+	0.796859i	$0.293507\pi$
$758$	0	0
$759$	10.0313	0.364114
$760$	0	0
$761$	−54.0682	−1.95997	−0.979985	−	0.199074i	$-0.936207\pi$
$761$	−54.0682	−1.95997	−0.979985	+	0.199074i	$0.936207\pi$
$762$	0	0
$763$	1.49192	0.0540112
$764$	0	0
$765$	−0.420699	−0.0152104
$766$	0	0
$767$	−15.2690	−0.551332
$768$	0	0
$769$	4.35847	0.157171	0.0785853	−	0.996907i	$-0.474960\pi$
$769$	4.35847	0.157171	0.0785853	+	0.996907i	$0.474960\pi$
$770$	0	0
$771$	17.8603	0.643223
$772$	0	0
$773$	−43.6469	−1.56987	−0.784936	−	0.619577i	$-0.787304\pi$
$773$	−43.6469	−1.56987	−0.784936	+	0.619577i	$0.787304\pi$
$774$	0	0
$775$	−28.1198	−1.01009
$776$	0	0
$777$	−1.30366	−0.0467687
$778$	0	0
$779$	−13.1253	−0.470261
$780$	0	0
$781$	22.3326	0.799124
$782$	0	0
$783$	−28.9461	−1.03445
$784$	0	0
$785$	3.75409	0.133989
$786$	0	0
$787$	−39.1261	−1.39469	−0.697347	−	0.716734i	$-0.745636\pi$
$787$	−39.1261	−1.39469	−0.697347	+	0.716734i	$0.745636\pi$
$788$	0	0
$789$	−13.5284	−0.481624
$790$	0	0
$791$	−5.31149	−0.188855
$792$	0	0
$793$	15.8110	0.561464
$794$	0	0
$795$	−8.36050	−0.296516
$796$	0	0
$797$	14.4918	0.513326	0.256663	−	0.966501i	$-0.417377\pi$
$797$	14.4918	0.513326	0.256663	+	0.966501i	$0.417377\pi$
$798$	0	0
$799$	2.11020	0.0746535
$800$	0	0
$801$	34.1235	1.20569
$802$	0	0
$803$	−5.45656	−0.192558
$804$	0	0
$805$	−2.37827	−0.0838230
$806$	0	0
$807$	−14.9271	−0.525458
$808$	0	0
$809$	6.26269	0.220184	0.110092	−	0.993921i	$-0.464885\pi$
$809$	6.26269	0.220184	0.110092	+	0.993921i	$0.464885\pi$
$810$	0	0
$811$	11.8822	0.417240	0.208620	−	0.977997i	$-0.433103\pi$
$811$	11.8822	0.417240	0.208620	+	0.977997i	$0.433103\pi$
$812$	0	0
$813$	−5.85071	−0.205193
$814$	0	0
$815$	15.0975	0.528843
$816$	0	0
$817$	−24.6252	−0.861525
$818$	0	0
$819$	1.48982	0.0520585
$820$	0	0
$821$	−14.4622	−0.504735	−0.252367	−	0.967631i	$-0.581209\pi$
$821$	−14.4622	−0.504735	−0.252367	+	0.967631i	$0.581209\pi$
$822$	0	0
$823$	−18.0941	−0.630719	−0.315360	−	0.948972i	$-0.602125\pi$
$823$	−18.0941	−0.630719	−0.315360	+	0.948972i	$0.602125\pi$
$824$	0	0
$825$	−6.18099	−0.215194
$826$	0	0
$827$	6.61196	0.229920	0.114960	−	0.993370i	$-0.463326\pi$
$827$	6.61196	0.229920	0.114960	+	0.993370i	$0.463326\pi$
$828$	0	0
$829$	−25.1172	−0.872356	−0.436178	−	0.899860i	$-0.643668\pi$
$829$	−25.1172	−0.872356	−0.436178	+	0.899860i	$0.643668\pi$
$830$	0	0
$831$	−12.9577	−0.449499
$832$	0	0
$833$	−1.12819	−0.0390896
$834$	0	0
$835$	5.72516	0.198127
$836$	0	0
$837$	25.7612	0.890437
$838$	0	0
$839$	7.13587	0.246358	0.123179	−	0.992384i	$-0.460691\pi$
$839$	7.13587	0.246358	0.123179	+	0.992384i	$0.460691\pi$
$840$	0	0
$841$	33.2634	1.14701
$842$	0	0
$843$	−10.0684	−0.346776
$844$	0	0
$845$	−10.4706	−0.360200
$846$	0	0
$847$	2.02272	0.0695016
$848$	0	0
$849$	14.2785	0.490036
$850$	0	0
$851$	−35.0477	−1.20142
$852$	0	0
$853$	−10.1428	−0.347281	−0.173641	−	0.984809i	$-0.555553\pi$
$853$	−10.1428	−0.347281	−0.173641	+	0.984809i	$0.555553\pi$
$854$	0	0
$855$	−7.52887	−0.257482
$856$	0	0
$857$	33.1710	1.13310	0.566550	−	0.824027i	$-0.308278\pi$
$857$	33.1710	1.13310	0.566550	+	0.824027i	$0.308278\pi$
$858$	0	0
$859$	15.8176	0.539691	0.269845	−	0.962904i	$-0.413027\pi$
$859$	15.8176	0.539691	0.269845	+	0.962904i	$0.413027\pi$
$860$	0	0
$861$	−1.07886	−0.0367675
$862$	0	0
$863$	−17.9450	−0.610856	−0.305428	−	0.952215i	$-0.598800\pi$
$863$	−17.9450	−0.610856	−0.305428	+	0.952215i	$0.598800\pi$
$864$	0	0
$865$	4.94904	0.168272
$866$	0	0
$867$	11.1873	0.379939
$868$	0	0
$869$	5.88683	0.199697
$870$	0	0
$871$	10.4294	0.353388
$872$	0	0
$873$	−0.671985	−0.0227432
$874$	0	0
$875$	3.29524	0.111400
$876$	0	0
$877$	20.5719	0.694664	0.347332	−	0.937742i	$-0.387088\pi$
$877$	20.5719	0.694664	0.347332	+	0.937742i	$0.387088\pi$
$878$	0	0
$879$	−6.32160	−0.213222
$880$	0	0
$881$	−19.5457	−0.658510	−0.329255	−	0.944241i	$-0.606798\pi$
$881$	−19.5457	−0.658510	−0.329255	+	0.944241i	$0.606798\pi$
$882$	0	0
$883$	21.5650	0.725719	0.362859	−	0.931844i	$-0.381800\pi$
$883$	21.5650	0.725719	0.362859	+	0.931844i	$0.381800\pi$
$884$	0	0
$885$	6.34256	0.213203
$886$	0	0
$887$	50.5182	1.69624	0.848118	−	0.529808i	$-0.177736\pi$
$887$	50.5182	1.69624	0.848118	+	0.529808i	$0.177736\pi$
$888$	0	0
$889$	−0.984914	−0.0330329
$890$	0	0
$891$	−12.3625	−0.414160
$892$	0	0
$893$	37.7643	1.26373
$894$	0	0
$895$	7.46321	0.249468
$896$	0	0
$897$	−6.78227	−0.226453
$898$	0	0
$899$	−55.4126	−1.84811
$900$	0	0
$901$	2.08881	0.0695885
$902$	0	0
$903$	−2.02412	−0.0673586
$904$	0	0
$905$	−8.24781	−0.274167
$906$	0	0
$907$	18.9017	0.627622	0.313811	−	0.949486i	$-0.398394\pi$
$907$	18.9017	0.627622	0.313811	+	0.949486i	$0.398394\pi$
$908$	0	0
$909$	11.4475	0.379690
$910$	0	0
$911$	52.1637	1.72826	0.864130	−	0.503268i	$-0.167869\pi$
$911$	52.1637	1.72826	0.864130	+	0.503268i	$0.167869\pi$
$912$	0	0
$913$	−6.79891	−0.225011
$914$	0	0
$915$	−6.56769	−0.217121
$916$	0	0
$917$	3.55166	0.117286
$918$	0	0
$919$	26.7580	0.882665	0.441332	−	0.897344i	$-0.354506\pi$
$919$	26.7580	0.882665	0.441332	+	0.897344i	$0.354506\pi$
$920$	0	0
$921$	−5.13079	−0.169065
$922$	0	0
$923$	−15.0993	−0.496999
$924$	0	0
$925$	21.5953	0.710048
$926$	0	0
$927$	−1.88523	−0.0619190
$928$	0	0
$929$	31.6778	1.03931	0.519657	−	0.854375i	$-0.326060\pi$
$929$	31.6778	1.03931	0.519657	+	0.854375i	$0.326060\pi$
$930$	0	0
$931$	−20.1902	−0.661708
$932$	0	0
$933$	−22.1133	−0.723958
$934$	0	0
$935$	−0.384028	−0.0125591
$936$	0	0
$937$	−35.1014	−1.14671	−0.573356	−	0.819307i	$-0.694359\pi$
$937$	−35.1014	−1.14671	−0.573356	+	0.819307i	$0.694359\pi$
$938$	0	0
$939$	4.08590	0.133338
$940$	0	0
$941$	27.2279	0.887604	0.443802	−	0.896125i	$-0.353629\pi$
$941$	27.2279	0.887604	0.443802	+	0.896125i	$0.353629\pi$
$942$	0	0
$943$	−29.0041	−0.944503
$944$	0	0
$945$	−1.34250	−0.0436715
$946$	0	0
$947$	24.6119	0.799779	0.399890	−	0.916563i	$-0.369048\pi$
$947$	24.6119	0.799779	0.399890	+	0.916563i	$0.369048\pi$
$948$	0	0
$949$	3.68922	0.119757
$950$	0	0
$951$	7.56890	0.245438
$952$	0	0
$953$	3.01549	0.0976813	0.0488407	−	0.998807i	$-0.484447\pi$
$953$	3.01549	0.0976813	0.0488407	+	0.998807i	$0.484447\pi$
$954$	0	0
$955$	2.25235	0.0728842
$956$	0	0
$957$	−12.1802	−0.393730
$958$	0	0
$959$	6.90052	0.222829
$960$	0	0
$961$	18.3156	0.590825
$962$	0	0
$963$	−6.53664	−0.210640
$964$	0	0
$965$	−18.3617	−0.591085
$966$	0	0
$967$	−6.03746	−0.194152	−0.0970758	−	0.995277i	$-0.530949\pi$
$967$	−6.03746	−0.194152	−0.0970758	+	0.995277i	$0.530949\pi$
$968$	0	0
$969$	−0.318526	−0.0102325
$970$	0	0
$971$	8.23721	0.264345	0.132172	−	0.991227i	$-0.457805\pi$
$971$	8.23721	0.264345	0.132172	+	0.991227i	$0.457805\pi$
$972$	0	0
$973$	6.05910	0.194246
$974$	0	0
$975$	4.17902	0.133836
$976$	0	0
$977$	−36.4331	−1.16560	−0.582799	−	0.812617i	$-0.698042\pi$
$977$	−36.4331	−1.16560	−0.582799	+	0.812617i	$0.698042\pi$
$978$	0	0
$979$	31.1491	0.995528
$980$	0	0
$981$	10.4368	0.333221
$982$	0	0
$983$	−16.4411	−0.524391	−0.262195	−	0.965015i	$-0.584447\pi$
$983$	−16.4411	−0.524391	−0.262195	+	0.965015i	$0.584447\pi$
$984$	0	0
$985$	−11.5812	−0.369007
$986$	0	0
$987$	3.10412	0.0988054
$988$	0	0
$989$	−54.4165	−1.73034
$990$	0	0
$991$	−12.0291	−0.382118	−0.191059	−	0.981579i	$-0.561192\pi$
$991$	−12.0291	−0.382118	−0.191059	+	0.981579i	$0.561192\pi$
$992$	0	0
$993$	−2.17381	−0.0689837
$994$	0	0
$995$	−16.8423	−0.533937
$996$	0	0
$997$	10.9894	0.348037	0.174019	−	0.984742i	$-0.444325\pi$
$997$	10.9894	0.348037	0.174019	+	0.984742i	$0.444325\pi$
$998$	0	0
$999$	−19.7839	−0.625936

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.14		28
4.3	odd	2		4024.2.a.d.1.15	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.15	✓	28	4.3	odd	2
8048.2.a.v.1.14		28	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)

\(f(q)\)	\(=\)	\(q-0.659121 q^{3} +0.997881 q^{5} -0.366743 q^{7} -2.56556 q^{9} +O(q^{10})\)	\(q-0.659121 q^{3} +0.997881 q^{5} -0.366743 q^{7} -2.56556 q^{9} -2.34193 q^{11} +1.58340 q^{13} -0.657724 q^{15} +0.164328 q^{17} +2.94082 q^{19} +0.241728 q^{21} +6.49861 q^{23} -4.00423 q^{25} +3.66838 q^{27} -7.89072 q^{29} +7.02250 q^{31} +1.54361 q^{33} -0.365966 q^{35} -5.39311 q^{37} -1.04365 q^{39} -4.46312 q^{41} -8.37356 q^{43} -2.56012 q^{45} +12.8414 q^{47} -6.86550 q^{49} -0.108312 q^{51} +12.7113 q^{53} -2.33697 q^{55} -1.93836 q^{57} -9.64319 q^{59} +9.98548 q^{61} +0.940901 q^{63} +1.58004 q^{65} +6.58675 q^{67} -4.28337 q^{69} -9.53600 q^{71} +2.32994 q^{73} +2.63927 q^{75} +0.858886 q^{77} -2.51367 q^{79} +5.27878 q^{81} +2.90312 q^{83} +0.163980 q^{85} +5.20093 q^{87} -13.3006 q^{89} -0.580700 q^{91} -4.62868 q^{93} +2.93459 q^{95} +0.261925 q^{97} +6.00836 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

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