LMFDB - Embedded newform 8048.2.a.r.1.3

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 22 x^{10} + 67 x^{9} + 180 x^{8} - 383 x^{7} - 674 x^{6} + 875 x^{5} + 1077 x^{4} + \cdots + 2 $ x^12 - 4x^11 - 22x^10 + 67x^9 + 180x^8 - 383x^7 - 674x^6 + 875x^5 + 1077x^4 - 704x^3 - 467x^2 + 210*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.753640$ 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)

$f(q)$	$=$	$q-2.49671 q^{3} +2.79920 q^{5} -1.31469 q^{7} +3.23355 q^{9} +O(q^{10})$	$q-2.49671 q^{3} +2.79920 q^{5} -1.31469 q^{7} +3.23355 q^{9} -0.214910 q^{11} +4.48880 q^{13} -6.98879 q^{15} +0.531363 q^{17} +7.59936 q^{19} +3.28240 q^{21} +6.80244 q^{23} +2.83553 q^{25} -0.583104 q^{27} +10.5592 q^{29} -3.24171 q^{31} +0.536567 q^{33} -3.68009 q^{35} -5.24012 q^{37} -11.2072 q^{39} +3.41910 q^{41} +7.59335 q^{43} +9.05136 q^{45} +5.87068 q^{47} -5.27159 q^{49} -1.32666 q^{51} +3.35707 q^{53} -0.601576 q^{55} -18.9734 q^{57} +9.93028 q^{59} -10.7166 q^{61} -4.25112 q^{63} +12.5651 q^{65} -13.4958 q^{67} -16.9837 q^{69} -3.31548 q^{71} -2.20600 q^{73} -7.07948 q^{75} +0.282540 q^{77} +3.11263 q^{79} -8.24481 q^{81} +3.88502 q^{83} +1.48739 q^{85} -26.3633 q^{87} -6.38189 q^{89} -5.90139 q^{91} +8.09360 q^{93} +21.2721 q^{95} +12.8118 q^{97} -0.694922 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9	$ 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9} - 18 q^{11} - 4 q^{13} + 2 q^{15} + 12 q^{17} + 7 q^{21} + 9 q^{23} + 25 q^{25} + 18 q^{27} + 34 q^{29} + 11 q^{31} + 4 q^{33} - 21 q^{35} - 22 q^{37} - 13 q^{39} + 32 q^{41} + 8 q^{43} + 13 q^{45} - 24 q^{47} + 36 q^{49} - 16 q^{51} - 2 q^{53} + 12 q^{55} + 26 q^{57} - 26 q^{59} + 12 q^{61} - 5 q^{63} + 66 q^{65} + 21 q^{67} + 20 q^{69} - 50 q^{71} + 17 q^{73} + 14 q^{75} + 25 q^{77} + 9 q^{79} + 48 q^{81} - 25 q^{83} + 24 q^{85} + 10 q^{87} + 21 q^{89} + 9 q^{91} + 31 q^{93} - 22 q^{95} + 18 q^{97} - 102 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9 - 18 * q^11 - 4 * q^13 + 2 * q^15 + 12 * q^17 + 7 * q^21 + 9 * q^23 + 25 * q^25 + 18 * q^27 + 34 * q^29 + 11 * q^31 + 4 * q^33 - 21 * q^35 - 22 * q^37 - 13 * q^39 + 32 * q^41 + 8 * q^43 + 13 * q^45 - 24 * q^47 + 36 * q^49 - 16 * q^51 - 2 * q^53 + 12 * q^55 + 26 * q^57 - 26 * q^59 + 12 * q^61 - 5 * q^63 + 66 * q^65 + 21 * q^67 + 20 * q^69 - 50 * q^71 + 17 * q^73 + 14 * q^75 + 25 * q^77 + 9 * q^79 + 48 * q^81 - 25 * q^83 + 24 * q^85 + 10 * q^87 + 21 * q^89 + 9 * q^91 + 31 * q^93 - 22 * q^95 + 18 * q^97 - 102 * 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
$3$	−2.49671	−1.44147	−0.720737	−	0.693208i	$-0.756197\pi$
$3$	−2.49671	−1.44147	−0.720737	+	0.693208i	$0.756197\pi$
$4$	0	0
$5$	2.79920	1.25184	0.625920	−	0.779887i	$-0.284723\pi$
$5$	2.79920	1.25184	0.625920	+	0.779887i	$0.284723\pi$
$6$	0	0
$7$	−1.31469	−0.496907	−0.248453	−	0.968644i	$-0.579922\pi$
$7$	−1.31469	−0.496907	−0.248453	+	0.968644i	$0.579922\pi$
$8$	0	0
$9$	3.23355	1.07785
$10$	0	0
$11$	−0.214910	−0.0647978	−0.0323989	−	0.999475i	$-0.510315\pi$
$11$	−0.214910	−0.0647978	−0.0323989	+	0.999475i	$0.510315\pi$
$12$	0	0
$13$	4.48880	1.24497	0.622485	−	0.782632i	$-0.286123\pi$
$13$	4.48880	1.24497	0.622485	+	0.782632i	$0.286123\pi$
$14$	0	0
$15$	−6.98879	−1.80450
$16$	0	0
$17$	0.531363	0.128875	0.0644373	−	0.997922i	$-0.479475\pi$
$17$	0.531363	0.128875	0.0644373	+	0.997922i	$0.479475\pi$
$18$	0	0
$19$	7.59936	1.74341	0.871706	−	0.490029i	$-0.163014\pi$
$19$	7.59936	1.74341	0.871706	+	0.490029i	$0.163014\pi$
$20$	0	0
$21$	3.28240	0.716279
$22$	0	0
$23$	6.80244	1.41841	0.709203	−	0.705004i	$-0.249055\pi$
$23$	6.80244	1.41841	0.709203	+	0.705004i	$0.249055\pi$
$24$	0	0
$25$	2.83553	0.567106
$26$	0	0
$27$	−0.583104	−0.112218
$28$	0	0
$29$	10.5592	1.96080	0.980398	−	0.197025i	$-0.0631280\pi$
$29$	10.5592	1.96080	0.980398	+	0.197025i	$0.0631280\pi$
$30$	0	0
$31$	−3.24171	−0.582228	−0.291114	−	0.956688i	$-0.594026\pi$
$31$	−3.24171	−0.582228	−0.291114	+	0.956688i	$0.594026\pi$
$32$	0	0
$33$	0.536567	0.0934044
$34$	0	0
$35$	−3.68009	−0.622048
$36$	0	0
$37$	−5.24012	−0.861471	−0.430736	−	0.902478i	$-0.641746\pi$
$37$	−5.24012	−0.861471	−0.430736	+	0.902478i	$0.641746\pi$
$38$	0	0
$39$	−11.2072	−1.79459
$40$	0	0
$41$	3.41910	0.533973	0.266987	−	0.963700i	$-0.413972\pi$
$41$	3.41910	0.533973	0.266987	+	0.963700i	$0.413972\pi$
$42$	0	0
$43$	7.59335	1.15798	0.578988	−	0.815336i	$-0.303448\pi$
$43$	7.59335	1.15798	0.578988	+	0.815336i	$0.303448\pi$
$44$	0	0
$45$	9.05136	1.34930
$46$	0	0
$47$	5.87068	0.856327	0.428164	−	0.903701i	$-0.359161\pi$
$47$	5.87068	0.856327	0.428164	+	0.903701i	$0.359161\pi$
$48$	0	0
$49$	−5.27159	−0.753084
$50$	0	0
$51$	−1.32666	−0.185769
$52$	0	0
$53$	3.35707	0.461129	0.230564	−	0.973057i	$-0.425943\pi$
$53$	3.35707	0.461129	0.230564	+	0.973057i	$0.425943\pi$
$54$	0	0
$55$	−0.601576	−0.0811165
$56$	0	0
$57$	−18.9734	−2.51308
$58$	0	0
$59$	9.93028	1.29281	0.646406	−	0.762993i	$-0.276271\pi$
$59$	9.93028	1.29281	0.646406	+	0.762993i	$0.276271\pi$
$60$	0	0
$61$	−10.7166	−1.37213	−0.686063	−	0.727542i	$-0.740662\pi$
$61$	−10.7166	−1.37213	−0.686063	+	0.727542i	$0.740662\pi$
$62$	0	0
$63$	−4.25112	−0.535591
$64$	0	0
$65$	12.5651	1.55850
$66$	0	0
$67$	−13.4958	−1.64877	−0.824387	−	0.566027i	$-0.808480\pi$
$67$	−13.4958	−1.64877	−0.824387	+	0.566027i	$0.808480\pi$
$68$	0	0
$69$	−16.9837	−2.04460
$70$	0	0
$71$	−3.31548	−0.393475	−0.196737	−	0.980456i	$-0.563035\pi$
$71$	−3.31548	−0.393475	−0.196737	+	0.980456i	$0.563035\pi$
$72$	0	0
$73$	−2.20600	−0.258193	−0.129096	−	0.991632i	$-0.541208\pi$
$73$	−2.20600	−0.258193	−0.129096	+	0.991632i	$0.541208\pi$
$74$	0	0
$75$	−7.07948	−0.817468
$76$	0	0
$77$	0.282540	0.0321985
$78$	0	0
$79$	3.11263	0.350198	0.175099	−	0.984551i	$-0.443975\pi$
$79$	3.11263	0.350198	0.175099	+	0.984551i	$0.443975\pi$
$80$	0	0
$81$	−8.24481	−0.916090
$82$	0	0
$83$	3.88502	0.426436	0.213218	−	0.977005i	$-0.431606\pi$
$83$	3.88502	0.426436	0.213218	+	0.977005i	$0.431606\pi$
$84$	0	0
$85$	1.48739	0.161330
$86$	0	0
$87$	−26.3633	−2.82644
$88$	0	0
$89$	−6.38189	−0.676479	−0.338240	−	0.941060i	$-0.609831\pi$
$89$	−6.38189	−0.676479	−0.338240	+	0.941060i	$0.609831\pi$
$90$	0	0
$91$	−5.90139	−0.618634
$92$	0	0
$93$	8.09360	0.839268
$94$	0	0
$95$	21.2721	2.18247
$96$	0	0
$97$	12.8118	1.30084	0.650419	−	0.759575i	$-0.274593\pi$
$97$	12.8118	1.30084	0.650419	+	0.759575i	$0.274593\pi$
$98$	0	0
$99$	−0.694922	−0.0698423
$100$	0	0
$101$	12.1509	1.20906	0.604530	−	0.796582i	$-0.293361\pi$
$101$	12.1509	1.20906	0.604530	+	0.796582i	$0.293361\pi$
$102$	0	0
$103$	−3.64846	−0.359494	−0.179747	−	0.983713i	$-0.557528\pi$
$103$	−3.64846	−0.359494	−0.179747	+	0.983713i	$0.557528\pi$
$104$	0	0
$105$	9.18810	0.896667
$106$	0	0
$107$	−6.15421	−0.594950	−0.297475	−	0.954730i	$-0.596144\pi$
$107$	−6.15421	−0.594950	−0.297475	+	0.954730i	$0.596144\pi$
$108$	0	0
$109$	16.0449	1.53683	0.768413	−	0.639954i	$-0.221047\pi$
$109$	16.0449	1.53683	0.768413	+	0.639954i	$0.221047\pi$
$110$	0	0
$111$	13.0831	1.24179
$112$	0	0
$113$	6.87399	0.646651	0.323325	−	0.946288i	$-0.395199\pi$
$113$	6.87399	0.646651	0.323325	+	0.946288i	$0.395199\pi$
$114$	0	0
$115$	19.0414	1.77562
$116$	0	0
$117$	14.5148	1.34189
$118$	0	0
$119$	−0.698579	−0.0640386
$120$	0	0
$121$	−10.9538	−0.995801
$122$	0	0
$123$	−8.53648	−0.769709
$124$	0	0
$125$	−6.05879	−0.541915
$126$	0	0
$127$	−9.45181	−0.838713	−0.419356	−	0.907822i	$-0.637744\pi$
$127$	−9.45181	−0.838713	−0.419356	+	0.907822i	$0.637744\pi$
$128$	0	0
$129$	−18.9584	−1.66919
$130$	0	0
$131$	−9.60429	−0.839131	−0.419566	−	0.907725i	$-0.637818\pi$
$131$	−9.60429	−0.839131	−0.419566	+	0.907725i	$0.637818\pi$
$132$	0	0
$133$	−9.99081	−0.866313
$134$	0	0
$135$	−1.63223	−0.140480
$136$	0	0
$137$	12.2873	1.04977	0.524887	−	0.851172i	$-0.324107\pi$
$137$	12.2873	1.04977	0.524887	+	0.851172i	$0.324107\pi$
$138$	0	0
$139$	15.3864	1.30506	0.652530	−	0.757763i	$-0.273708\pi$
$139$	15.3864	1.30506	0.652530	+	0.757763i	$0.273708\pi$
$140$	0	0
$141$	−14.6574	−1.23437
$142$	0	0
$143$	−0.964689	−0.0806714
$144$	0	0
$145$	29.5574	2.45461
$146$	0	0
$147$	13.1616	1.08555
$148$	0	0
$149$	−12.5606	−1.02901	−0.514503	−	0.857489i	$-0.672024\pi$
$149$	−12.5606	−1.02901	−0.514503	+	0.857489i	$0.672024\pi$
$150$	0	0
$151$	−7.82680	−0.636936	−0.318468	−	0.947934i	$-0.603168\pi$
$151$	−7.82680	−0.636936	−0.318468	+	0.947934i	$0.603168\pi$
$152$	0	0
$153$	1.71819	0.138907
$154$	0	0
$155$	−9.07420	−0.728857
$156$	0	0
$157$	−18.6010	−1.48452	−0.742262	−	0.670109i	$-0.766247\pi$
$157$	−18.6010	−1.48452	−0.742262	+	0.670109i	$0.766247\pi$
$158$	0	0
$159$	−8.38161	−0.664705
$160$	0	0
$161$	−8.94311	−0.704816
$162$	0	0
$163$	7.36672	0.577006	0.288503	−	0.957479i	$-0.406842\pi$
$163$	7.36672	0.577006	0.288503	+	0.957479i	$0.406842\pi$
$164$	0	0
$165$	1.50196	0.116927
$166$	0	0
$167$	−12.6120	−0.975948	−0.487974	−	0.872858i	$-0.662264\pi$
$167$	−12.6120	−0.975948	−0.487974	+	0.872858i	$0.662264\pi$
$168$	0	0
$169$	7.14937	0.549951
$170$	0	0
$171$	24.5729	1.87914
$172$	0	0
$173$	14.5677	1.10756	0.553780	−	0.832663i	$-0.313185\pi$
$173$	14.5677	1.10756	0.553780	+	0.832663i	$0.313185\pi$
$174$	0	0
$175$	−3.72785	−0.281799
$176$	0	0
$177$	−24.7930	−1.86356
$178$	0	0
$179$	−14.8248	−1.10806	−0.554028	−	0.832498i	$-0.686910\pi$
$179$	−14.8248	−1.10806	−0.554028	+	0.832498i	$0.686910\pi$
$180$	0	0
$181$	−1.29653	−0.0963700	−0.0481850	−	0.998838i	$-0.515344\pi$
$181$	−1.29653	−0.0963700	−0.0481850	+	0.998838i	$0.515344\pi$
$182$	0	0
$183$	26.7563	1.97788
$184$	0	0
$185$	−14.6682	−1.07842
$186$	0	0
$187$	−0.114195	−0.00835079
$188$	0	0
$189$	0.766602	0.0557621
$190$	0	0
$191$	6.61849	0.478897	0.239449	−	0.970909i	$-0.423033\pi$
$191$	6.61849	0.478897	0.239449	+	0.970909i	$0.423033\pi$
$192$	0	0
$193$	−15.2763	−1.09961	−0.549806	−	0.835293i	$-0.685298\pi$
$193$	−15.2763	−1.09961	−0.549806	+	0.835293i	$0.685298\pi$
$194$	0	0
$195$	−31.3713	−2.24655
$196$	0	0
$197$	18.2507	1.30031	0.650154	−	0.759803i	$-0.274705\pi$
$197$	18.2507	1.30031	0.650154	+	0.759803i	$0.274705\pi$
$198$	0	0
$199$	3.96715	0.281224	0.140612	−	0.990065i	$-0.455093\pi$
$199$	3.96715	0.281224	0.140612	+	0.990065i	$0.455093\pi$
$200$	0	0
$201$	33.6950	2.37666
$202$	0	0
$203$	−13.8821	−0.974333
$204$	0	0
$205$	9.57074	0.668449
$206$	0	0
$207$	21.9960	1.52883
$208$	0	0
$209$	−1.63318	−0.112969
$210$	0	0
$211$	−13.4060	−0.922908	−0.461454	−	0.887164i	$-0.652672\pi$
$211$	−13.4060	−0.922908	−0.461454	+	0.887164i	$0.652672\pi$
$212$	0	0
$213$	8.27777	0.567184
$214$	0	0
$215$	21.2553	1.44960
$216$	0	0
$217$	4.26185	0.289313
$218$	0	0
$219$	5.50774	0.372178
$220$	0	0
$221$	2.38519	0.160445
$222$	0	0
$223$	24.7289	1.65597	0.827985	−	0.560751i	$-0.189487\pi$
$223$	24.7289	1.65597	0.827985	+	0.560751i	$0.189487\pi$
$224$	0	0
$225$	9.16882	0.611255
$226$	0	0
$227$	−15.5037	−1.02902	−0.514509	−	0.857485i	$-0.672026\pi$
$227$	−15.5037	−1.02902	−0.514509	+	0.857485i	$0.672026\pi$
$228$	0	0
$229$	−10.7101	−0.707745	−0.353873	−	0.935294i	$-0.615135\pi$
$229$	−10.7101	−0.707745	−0.353873	+	0.935294i	$0.615135\pi$
$230$	0	0
$231$	−0.705421	−0.0464133
$232$	0	0
$233$	21.8231	1.42968	0.714840	−	0.699288i	$-0.246500\pi$
$233$	21.8231	1.42968	0.714840	+	0.699288i	$0.246500\pi$
$234$	0	0
$235$	16.4332	1.07199
$236$	0	0
$237$	−7.77133	−0.504802
$238$	0	0
$239$	10.2065	0.660201	0.330101	−	0.943946i	$-0.392917\pi$
$239$	10.2065	0.660201	0.330101	+	0.943946i	$0.392917\pi$
$240$	0	0
$241$	−14.9915	−0.965690	−0.482845	−	0.875706i	$-0.660397\pi$
$241$	−14.9915	−0.965690	−0.482845	+	0.875706i	$0.660397\pi$
$242$	0	0
$243$	22.3342	1.43274
$244$	0	0
$245$	−14.7562	−0.942741
$246$	0	0
$247$	34.1120	2.17050
$248$	0	0
$249$	−9.69975	−0.614696
$250$	0	0
$251$	−19.8887	−1.25536	−0.627682	−	0.778470i	$-0.715996\pi$
$251$	−19.8887	−1.25536	−0.627682	+	0.778470i	$0.715996\pi$
$252$	0	0
$253$	−1.46191	−0.0919096
$254$	0	0
$255$	−3.71359	−0.232554
$256$	0	0
$257$	−30.4500	−1.89942	−0.949709	−	0.313135i	$-0.898621\pi$
$257$	−30.4500	−1.89942	−0.949709	+	0.313135i	$0.898621\pi$
$258$	0	0
$259$	6.88915	0.428071
$260$	0	0
$261$	34.1437	2.11344
$262$	0	0
$263$	4.72105	0.291112	0.145556	−	0.989350i	$-0.453503\pi$
$263$	4.72105	0.291112	0.145556	+	0.989350i	$0.453503\pi$
$264$	0	0
$265$	9.39710	0.577260
$266$	0	0
$267$	15.9337	0.975128
$268$	0	0
$269$	−21.3513	−1.30181	−0.650907	−	0.759158i	$-0.725611\pi$
$269$	−21.3513	−1.30181	−0.650907	+	0.759158i	$0.725611\pi$
$270$	0	0
$271$	−10.0054	−0.607783	−0.303891	−	0.952707i	$-0.598286\pi$
$271$	−10.0054	−0.607783	−0.303891	+	0.952707i	$0.598286\pi$
$272$	0	0
$273$	14.7341	0.891746
$274$	0	0
$275$	−0.609383	−0.0367472
$276$	0	0
$277$	−22.6138	−1.35873	−0.679367	−	0.733799i	$-0.737745\pi$
$277$	−22.6138	−1.35873	−0.679367	+	0.733799i	$0.737745\pi$
$278$	0	0
$279$	−10.4822	−0.627555
$280$	0	0
$281$	22.5518	1.34533	0.672663	−	0.739949i	$-0.265151\pi$
$281$	22.5518	1.34533	0.672663	+	0.739949i	$0.265151\pi$
$282$	0	0
$283$	21.6726	1.28830	0.644152	−	0.764898i	$-0.277211\pi$
$283$	21.6726	1.28830	0.644152	+	0.764898i	$0.277211\pi$
$284$	0	0
$285$	−53.1103	−3.14598
$286$	0	0
$287$	−4.49506	−0.265335
$288$	0	0
$289$	−16.7177	−0.983391
$290$	0	0
$291$	−31.9873	−1.87513
$292$	0	0
$293$	0.0357357	0.00208770	0.00104385	−	0.999999i	$-0.499668\pi$
$293$	0.0357357	0.00208770	0.00104385	+	0.999999i	$0.499668\pi$
$294$	0	0
$295$	27.7969	1.61840
$296$	0	0
$297$	0.125315	0.00727151
$298$	0	0
$299$	30.5348	1.76587
$300$	0	0
$301$	−9.98292	−0.575406
$302$	0	0
$303$	−30.3373	−1.74283
$304$	0	0
$305$	−29.9980	−1.71768
$306$	0	0
$307$	32.1186	1.83311	0.916554	−	0.399911i	$-0.130959\pi$
$307$	32.1186	1.83311	0.916554	+	0.399911i	$0.130959\pi$
$308$	0	0
$309$	9.10914	0.518201
$310$	0	0
$311$	10.2371	0.580492	0.290246	−	0.956952i	$-0.406263\pi$
$311$	10.2371	0.580492	0.290246	+	0.956952i	$0.406263\pi$
$312$	0	0
$313$	−6.28376	−0.355179	−0.177589	−	0.984105i	$-0.556830\pi$
$313$	−6.28376	−0.355179	−0.177589	+	0.984105i	$0.556830\pi$
$314$	0	0
$315$	−11.8997	−0.670474
$316$	0	0
$317$	−11.3267	−0.636173	−0.318087	−	0.948062i	$-0.603040\pi$
$317$	−11.3267	−0.636173	−0.318087	+	0.948062i	$0.603040\pi$
$318$	0	0
$319$	−2.26928	−0.127055
$320$	0	0
$321$	15.3653	0.857605
$322$	0	0
$323$	4.03802	0.224681
$324$	0	0
$325$	12.7281	0.706030
$326$	0	0
$327$	−40.0595	−2.21530
$328$	0	0
$329$	−7.71814	−0.425515
$330$	0	0
$331$	−21.5175	−1.18271	−0.591355	−	0.806411i	$-0.701407\pi$
$331$	−21.5175	−1.18271	−0.591355	+	0.806411i	$0.701407\pi$
$332$	0	0
$333$	−16.9442	−0.928536
$334$	0	0
$335$	−37.7774	−2.06400
$336$	0	0
$337$	6.19687	0.337565	0.168783	−	0.985653i	$-0.446016\pi$
$337$	6.19687	0.337565	0.168783	+	0.985653i	$0.446016\pi$
$338$	0	0
$339$	−17.1623	−0.932131
$340$	0	0
$341$	0.696676	0.0377271
$342$	0	0
$343$	16.1334	0.871119
$344$	0	0
$345$	−47.5408	−2.55951
$346$	0	0
$347$	−19.6683	−1.05585	−0.527924	−	0.849292i	$-0.677029\pi$
$347$	−19.6683	−1.05585	−0.527924	+	0.849292i	$0.677029\pi$
$348$	0	0
$349$	32.0717	1.71676	0.858379	−	0.513015i	$-0.171472\pi$
$349$	32.0717	1.71676	0.858379	+	0.513015i	$0.171472\pi$
$350$	0	0
$351$	−2.61744	−0.139709
$352$	0	0
$353$	−1.66289	−0.0885065	−0.0442532	−	0.999020i	$-0.514091\pi$
$353$	−1.66289	−0.0885065	−0.0442532	+	0.999020i	$0.514091\pi$
$354$	0	0
$355$	−9.28068	−0.492568
$356$	0	0
$357$	1.74415	0.0923101
$358$	0	0
$359$	6.62921	0.349876	0.174938	−	0.984579i	$-0.444027\pi$
$359$	6.62921	0.349876	0.174938	+	0.984579i	$0.444027\pi$
$360$	0	0
$361$	38.7502	2.03949
$362$	0	0
$363$	27.3485	1.43542
$364$	0	0
$365$	−6.17504	−0.323216
$366$	0	0
$367$	−2.90874	−0.151835	−0.0759174	−	0.997114i	$-0.524189\pi$
$367$	−2.90874	−0.151835	−0.0759174	+	0.997114i	$0.524189\pi$
$368$	0	0
$369$	11.0558	0.575543
$370$	0	0
$371$	−4.41351	−0.229138
$372$	0	0
$373$	3.59871	0.186334	0.0931671	−	0.995650i	$-0.470301\pi$
$373$	3.59871	0.186334	0.0931671	+	0.995650i	$0.470301\pi$
$374$	0	0
$375$	15.1270	0.781157
$376$	0	0
$377$	47.3982	2.44113
$378$	0	0
$379$	−16.9376	−0.870026	−0.435013	−	0.900424i	$-0.643256\pi$
$379$	−16.9376	−0.870026	−0.435013	+	0.900424i	$0.643256\pi$
$380$	0	0
$381$	23.5984	1.20898
$382$	0	0
$383$	18.1570	0.927781	0.463891	−	0.885893i	$-0.346453\pi$
$383$	18.1570	0.927781	0.463891	+	0.885893i	$0.346453\pi$
$384$	0	0
$385$	0.790888	0.0403074
$386$	0	0
$387$	24.5535	1.24812
$388$	0	0
$389$	−6.79403	−0.344471	−0.172235	−	0.985056i	$-0.555099\pi$
$389$	−6.79403	−0.344471	−0.172235	+	0.985056i	$0.555099\pi$
$390$	0	0
$391$	3.61457	0.182796
$392$	0	0
$393$	23.9791	1.20959
$394$	0	0
$395$	8.71288	0.438393
$396$	0	0
$397$	−3.28781	−0.165010	−0.0825052	−	0.996591i	$-0.526292\pi$
$397$	−3.28781	−0.165010	−0.0825052	+	0.996591i	$0.526292\pi$
$398$	0	0
$399$	24.9441	1.24877
$400$	0	0
$401$	24.3263	1.21480	0.607399	−	0.794397i	$-0.292213\pi$
$401$	24.3263	1.21480	0.607399	+	0.794397i	$0.292213\pi$
$402$	0	0
$403$	−14.5514	−0.724857
$404$	0	0
$405$	−23.0789	−1.14680
$406$	0	0
$407$	1.12616	0.0558214
$408$	0	0
$409$	16.6999	0.825755	0.412878	−	0.910787i	$-0.364524\pi$
$409$	16.6999	0.825755	0.412878	+	0.910787i	$0.364524\pi$
$410$	0	0
$411$	−30.6778	−1.51322
$412$	0	0
$413$	−13.0553	−0.642407
$414$	0	0
$415$	10.8749	0.533830
$416$	0	0
$417$	−38.4154	−1.88121
$418$	0	0
$419$	−8.61001	−0.420627	−0.210313	−	0.977634i	$-0.567448\pi$
$419$	−8.61001	−0.420627	−0.210313	+	0.977634i	$0.567448\pi$
$420$	0	0
$421$	−7.16303	−0.349105	−0.174552	−	0.984648i	$-0.555848\pi$
$421$	−7.16303	−0.349105	−0.174552	+	0.984648i	$0.555848\pi$
$422$	0	0
$423$	18.9831	0.922992
$424$	0	0
$425$	1.50670	0.0730855
$426$	0	0
$427$	14.0891	0.681818
$428$	0	0
$429$	2.40855	0.116286
$430$	0	0
$431$	−2.82687	−0.136165	−0.0680827	−	0.997680i	$-0.521688\pi$
$431$	−2.82687	−0.136165	−0.0680827	+	0.997680i	$0.521688\pi$
$432$	0	0
$433$	11.8263	0.568336	0.284168	−	0.958775i	$-0.408283\pi$
$433$	11.8263	0.568336	0.284168	+	0.958775i	$0.408283\pi$
$434$	0	0
$435$	−73.7961	−3.53825
$436$	0	0
$437$	51.6942	2.47287
$438$	0	0
$439$	16.1184	0.769287	0.384644	−	0.923065i	$-0.374324\pi$
$439$	16.1184	0.769287	0.384644	+	0.923065i	$0.374324\pi$
$440$	0	0
$441$	−17.0459	−0.811711
$442$	0	0
$443$	33.1314	1.57412	0.787060	−	0.616877i	$-0.211602\pi$
$443$	33.1314	1.57412	0.787060	+	0.616877i	$0.211602\pi$
$444$	0	0
$445$	−17.8642	−0.846844
$446$	0	0
$447$	31.3602	1.48329
$448$	0	0
$449$	−3.39951	−0.160433	−0.0802164	−	0.996777i	$-0.525561\pi$
$449$	−3.39951	−0.160433	−0.0802164	+	0.996777i	$0.525561\pi$
$450$	0	0
$451$	−0.734798	−0.0346003
$452$	0	0
$453$	19.5412	0.918127
$454$	0	0
$455$	−16.5192	−0.774432
$456$	0	0
$457$	−35.5847	−1.66458	−0.832291	−	0.554338i	$-0.812971\pi$
$457$	−35.5847	−1.66458	−0.832291	+	0.554338i	$0.812971\pi$
$458$	0	0
$459$	−0.309840	−0.0144621
$460$	0	0
$461$	34.9030	1.62560	0.812798	−	0.582545i	$-0.197943\pi$
$461$	34.9030	1.62560	0.812798	+	0.582545i	$0.197943\pi$
$462$	0	0
$463$	−14.3855	−0.668550	−0.334275	−	0.942476i	$-0.608491\pi$
$463$	−14.3855	−0.668550	−0.334275	+	0.942476i	$0.608491\pi$
$464$	0	0
$465$	22.6556	1.05063
$466$	0	0
$467$	−14.3321	−0.663213	−0.331606	−	0.943418i	$-0.607591\pi$
$467$	−14.3321	−0.663213	−0.331606	+	0.943418i	$0.607591\pi$
$468$	0	0
$469$	17.7428	0.819286
$470$	0	0
$471$	46.4414	2.13990
$472$	0	0
$473$	−1.63189	−0.0750343
$474$	0	0
$475$	21.5482	0.988699
$476$	0	0
$477$	10.8552	0.497027
$478$	0	0
$479$	36.3487	1.66082	0.830408	−	0.557156i	$-0.188108\pi$
$479$	36.3487	1.66082	0.830408	+	0.557156i	$0.188108\pi$
$480$	0	0
$481$	−23.5219	−1.07251
$482$	0	0
$483$	22.3283	1.01597
$484$	0	0
$485$	35.8627	1.62844
$486$	0	0
$487$	33.5406	1.51987	0.759935	−	0.650000i	$-0.225231\pi$
$487$	33.5406	1.51987	0.759935	+	0.650000i	$0.225231\pi$
$488$	0	0
$489$	−18.3925	−0.831739
$490$	0	0
$491$	−37.3160	−1.68405	−0.842024	−	0.539441i	$-0.818636\pi$
$491$	−37.3160	−1.68405	−0.842024	+	0.539441i	$0.818636\pi$
$492$	0	0
$493$	5.61078	0.252697
$494$	0	0
$495$	−1.94523	−0.0874314
$496$	0	0
$497$	4.35883	0.195520
$498$	0	0
$499$	−3.12726	−0.139995	−0.0699977	−	0.997547i	$-0.522299\pi$
$499$	−3.12726	−0.139995	−0.0699977	+	0.997547i	$0.522299\pi$
$500$	0	0
$501$	31.4886	1.40680
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	34.0128	1.51355
$506$	0	0
$507$	−17.8499	−0.792741
$508$	0	0
$509$	38.0821	1.68796	0.843980	−	0.536375i	$-0.180207\pi$
$509$	38.0821	1.68796	0.843980	+	0.536375i	$0.180207\pi$
$510$	0	0
$511$	2.90021	0.128298
$512$	0	0
$513$	−4.43122	−0.195643
$514$	0	0
$515$	−10.2128	−0.450029
$516$	0	0
$517$	−1.26167	−0.0554881
$518$	0	0
$519$	−36.3713	−1.59652
$520$	0	0
$521$	−23.0831	−1.01129	−0.505645	−	0.862742i	$-0.668745\pi$
$521$	−23.0831	−1.01129	−0.505645	+	0.862742i	$0.668745\pi$
$522$	0	0
$523$	−26.4097	−1.15482	−0.577409	−	0.816455i	$-0.695936\pi$
$523$	−26.4097	−1.15482	−0.577409	+	0.816455i	$0.695936\pi$
$524$	0	0
$525$	9.30734	0.406206
$526$	0	0
$527$	−1.72253	−0.0750344
$528$	0	0
$529$	23.2732	1.01188
$530$	0	0
$531$	32.1100	1.39346
$532$	0	0
$533$	15.3477	0.664781
$534$	0	0
$535$	−17.2269	−0.744783
$536$	0	0
$537$	37.0132	1.59724
$538$	0	0
$539$	1.13292	0.0487982
$540$	0	0
$541$	−5.30510	−0.228084	−0.114042	−	0.993476i	$-0.536380\pi$
$541$	−5.30510	−0.228084	−0.114042	+	0.993476i	$0.536380\pi$
$542$	0	0
$543$	3.23704	0.138915
$544$	0	0
$545$	44.9130	1.92386
$546$	0	0
$547$	9.08255	0.388342	0.194171	−	0.980968i	$-0.437798\pi$
$547$	9.08255	0.388342	0.194171	+	0.980968i	$0.437798\pi$
$548$	0	0
$549$	−34.6528	−1.47895
$550$	0	0
$551$	80.2432	3.41848
$552$	0	0
$553$	−4.09215	−0.174016
$554$	0	0
$555$	36.6221	1.55452
$556$	0	0
$557$	−2.19469	−0.0929922	−0.0464961	−	0.998918i	$-0.514806\pi$
$557$	−2.19469	−0.0929922	−0.0464961	+	0.998918i	$0.514806\pi$
$558$	0	0
$559$	34.0851	1.44165
$560$	0	0
$561$	0.285112	0.0120375
$562$	0	0
$563$	36.1536	1.52369	0.761846	−	0.647758i	$-0.224293\pi$
$563$	36.1536	1.52369	0.761846	+	0.647758i	$0.224293\pi$
$564$	0	0
$565$	19.2417	0.809504
$566$	0	0
$567$	10.8394	0.455211
$568$	0	0
$569$	−16.2612	−0.681706	−0.340853	−	0.940117i	$-0.610716\pi$
$569$	−16.2612	−0.681706	−0.340853	+	0.940117i	$0.610716\pi$
$570$	0	0
$571$	−12.2329	−0.511932	−0.255966	−	0.966686i	$-0.582394\pi$
$571$	−12.2329	−0.511932	−0.255966	+	0.966686i	$0.582394\pi$
$572$	0	0
$573$	−16.5244	−0.690318
$574$	0	0
$575$	19.2885	0.804386
$576$	0	0
$577$	30.4384	1.26717	0.633584	−	0.773674i	$-0.281583\pi$
$577$	30.4384	1.26717	0.633584	+	0.773674i	$0.281583\pi$
$578$	0	0
$579$	38.1404	1.58506
$580$	0	0
$581$	−5.10760	−0.211899
$582$	0	0
$583$	−0.721467	−0.0298801
$584$	0	0
$585$	40.6298	1.67983
$586$	0	0
$587$	−1.33753	−0.0552056	−0.0276028	−	0.999619i	$-0.508787\pi$
$587$	−1.33753	−0.0552056	−0.0276028	+	0.999619i	$0.508787\pi$
$588$	0	0
$589$	−24.6349	−1.01506
$590$	0	0
$591$	−45.5666	−1.87436
$592$	0	0
$593$	39.2257	1.61081	0.805404	−	0.592726i	$-0.201948\pi$
$593$	39.2257	1.61081	0.805404	+	0.592726i	$0.201948\pi$
$594$	0	0
$595$	−1.95546	−0.0801662
$596$	0	0
$597$	−9.90481	−0.405377
$598$	0	0
$599$	−4.25729	−0.173948	−0.0869741	−	0.996211i	$-0.527720\pi$
$599$	−4.25729	−0.173948	−0.0869741	+	0.996211i	$0.527720\pi$
$600$	0	0
$601$	−3.16250	−0.129001	−0.0645005	−	0.997918i	$-0.520545\pi$
$601$	−3.16250	−0.129001	−0.0645005	+	0.997918i	$0.520545\pi$
$602$	0	0
$603$	−43.6393	−1.77713
$604$	0	0
$605$	−30.6619	−1.24658
$606$	0	0
$607$	−27.6650	−1.12289	−0.561445	−	0.827514i	$-0.689754\pi$
$607$	−27.6650	−1.12289	−0.561445	+	0.827514i	$0.689754\pi$
$608$	0	0
$609$	34.6596	1.40448
$610$	0	0
$611$	26.3523	1.06610
$612$	0	0
$613$	36.3191	1.46691	0.733457	−	0.679736i	$-0.237906\pi$
$613$	36.3191	1.46691	0.733457	+	0.679736i	$0.237906\pi$
$614$	0	0
$615$	−23.8953	−0.963553
$616$	0	0
$617$	−3.90613	−0.157255	−0.0786273	−	0.996904i	$-0.525054\pi$
$617$	−3.90613	−0.157255	−0.0786273	+	0.996904i	$0.525054\pi$
$618$	0	0
$619$	−17.7479	−0.713348	−0.356674	−	0.934229i	$-0.616089\pi$
$619$	−17.7479	−0.713348	−0.356674	+	0.934229i	$0.616089\pi$
$620$	0	0
$621$	−3.96653	−0.159171
$622$	0	0
$623$	8.39022	0.336147
$624$	0	0
$625$	−31.1374	−1.24550
$626$	0	0
$627$	4.07757	0.162842
$628$	0	0
$629$	−2.78441	−0.111022
$630$	0	0
$631$	13.6000	0.541407	0.270703	−	0.962663i	$-0.412744\pi$
$631$	13.6000	0.541407	0.270703	+	0.962663i	$0.412744\pi$
$632$	0	0
$633$	33.4709	1.33035
$634$	0	0
$635$	−26.4575	−1.04993
$636$	0	0
$637$	−23.6631	−0.937567
$638$	0	0
$639$	−10.7208	−0.424106
$640$	0	0
$641$	13.9364	0.550456	0.275228	−	0.961379i	$-0.411247\pi$
$641$	13.9364	0.550456	0.275228	+	0.961379i	$0.411247\pi$
$642$	0	0
$643$	−18.5556	−0.731763	−0.365881	−	0.930662i	$-0.619232\pi$
$643$	−18.5556	−0.731763	−0.365881	+	0.930662i	$0.619232\pi$
$644$	0	0
$645$	−53.0683	−2.08956
$646$	0	0
$647$	13.8632	0.545018	0.272509	−	0.962153i	$-0.412146\pi$
$647$	13.8632	0.545018	0.272509	+	0.962153i	$0.412146\pi$
$648$	0	0
$649$	−2.13412	−0.0837714
$650$	0	0
$651$	−10.6406	−0.417038
$652$	0	0
$653$	−26.8331	−1.05006	−0.525031	−	0.851083i	$-0.675946\pi$
$653$	−26.8331	−1.05006	−0.525031	+	0.851083i	$0.675946\pi$
$654$	0	0
$655$	−26.8844	−1.05046
$656$	0	0
$657$	−7.13321	−0.278293
$658$	0	0
$659$	−19.2991	−0.751785	−0.375892	−	0.926663i	$-0.622664\pi$
$659$	−19.2991	−0.751785	−0.375892	+	0.926663i	$0.622664\pi$
$660$	0	0
$661$	12.1669	0.473238	0.236619	−	0.971603i	$-0.423961\pi$
$661$	12.1669	0.473238	0.236619	+	0.971603i	$0.423961\pi$
$662$	0	0
$663$	−5.95511	−0.231277
$664$	0	0
$665$	−27.9663	−1.08449
$666$	0	0
$667$	71.8284	2.78121
$668$	0	0
$669$	−61.7408	−2.38704
$670$	0	0
$671$	2.30311	0.0889107
$672$	0	0
$673$	16.8266	0.648618	0.324309	−	0.945951i	$-0.394868\pi$
$673$	16.8266	0.648618	0.324309	+	0.945951i	$0.394868\pi$
$674$	0	0
$675$	−1.65341	−0.0636397
$676$	0	0
$677$	9.81619	0.377267	0.188633	−	0.982048i	$-0.439594\pi$
$677$	9.81619	0.377267	0.188633	+	0.982048i	$0.439594\pi$
$678$	0	0
$679$	−16.8435	−0.646395
$680$	0	0
$681$	38.7082	1.48330
$682$	0	0
$683$	−43.0568	−1.64752	−0.823761	−	0.566937i	$-0.808128\pi$
$683$	−43.0568	−1.64752	−0.823761	+	0.566937i	$0.808128\pi$
$684$	0	0
$685$	34.3946	1.31415
$686$	0	0
$687$	26.7401	1.02020
$688$	0	0
$689$	15.0692	0.574091
$690$	0	0
$691$	−31.0812	−1.18239	−0.591193	−	0.806530i	$-0.701343\pi$
$691$	−31.0812	−1.18239	−0.591193	+	0.806530i	$0.701343\pi$
$692$	0	0
$693$	0.913608	0.0347051
$694$	0	0
$695$	43.0697	1.63373
$696$	0	0
$697$	1.81678	0.0688155
$698$	0	0
$699$	−54.4859	−2.06085
$700$	0	0
$701$	−14.9328	−0.564003	−0.282002	−	0.959414i	$-0.590998\pi$
$701$	−14.9328	−0.564003	−0.282002	+	0.959414i	$0.590998\pi$
$702$	0	0
$703$	−39.8216	−1.50190
$704$	0	0
$705$	−41.0290	−1.54524
$706$	0	0
$707$	−15.9747	−0.600790
$708$	0	0
$709$	17.7868	0.667998	0.333999	−	0.942573i	$-0.391602\pi$
$709$	17.7868	0.667998	0.333999	+	0.942573i	$0.391602\pi$
$710$	0	0
$711$	10.0648	0.377461
$712$	0	0
$713$	−22.0515	−0.825837
$714$	0	0
$715$	−2.70036	−0.100988
$716$	0	0
$717$	−25.4825	−0.951663
$718$	0	0
$719$	−2.34669	−0.0875169	−0.0437585	−	0.999042i	$-0.513933\pi$
$719$	−2.34669	−0.0875169	−0.0437585	+	0.999042i	$0.513933\pi$
$720$	0	0
$721$	4.79660	0.178635
$722$	0	0
$723$	37.4295	1.39202
$724$	0	0
$725$	29.9409	1.11198
$726$	0	0
$727$	−13.2559	−0.491636	−0.245818	−	0.969316i	$-0.579056\pi$
$727$	−13.2559	−0.491636	−0.245818	+	0.969316i	$0.579056\pi$
$728$	0	0
$729$	−31.0275	−1.14917
$730$	0	0
$731$	4.03483	0.149234
$732$	0	0
$733$	41.9658	1.55004	0.775021	−	0.631936i	$-0.217739\pi$
$733$	41.9658	1.55004	0.775021	+	0.631936i	$0.217739\pi$
$734$	0	0
$735$	36.8420	1.35894
$736$	0	0
$737$	2.90038	0.106837
$738$	0	0
$739$	15.7393	0.578979	0.289489	−	0.957181i	$-0.406514\pi$
$739$	15.7393	0.578979	0.289489	+	0.957181i	$0.406514\pi$
$740$	0	0
$741$	−85.1678	−3.12872
$742$	0	0
$743$	48.6983	1.78657	0.893284	−	0.449494i	$-0.148396\pi$
$743$	48.6983	1.78657	0.893284	+	0.449494i	$0.148396\pi$
$744$	0	0
$745$	−35.1597	−1.28815
$746$	0	0
$747$	12.5624	0.459634
$748$	0	0
$749$	8.09089	0.295635
$750$	0	0
$751$	19.3337	0.705496	0.352748	−	0.935718i	$-0.385247\pi$
$751$	19.3337	0.705496	0.352748	+	0.935718i	$0.385247\pi$
$752$	0	0
$753$	49.6563	1.80958
$754$	0	0
$755$	−21.9088	−0.797343
$756$	0	0
$757$	−49.3985	−1.79542	−0.897710	−	0.440587i	$-0.854770\pi$
$757$	−49.3985	−1.79542	−0.897710	+	0.440587i	$0.854770\pi$
$758$	0	0
$759$	3.64997	0.132485
$760$	0	0
$761$	−10.3490	−0.375153	−0.187576	−	0.982250i	$-0.560063\pi$
$761$	−10.3490	−0.375153	−0.187576	+	0.982250i	$0.560063\pi$
$762$	0	0
$763$	−21.0941	−0.763659
$764$	0	0
$765$	4.80956	0.173890
$766$	0	0
$767$	44.5751	1.60951
$768$	0	0
$769$	15.7765	0.568914	0.284457	−	0.958689i	$-0.408187\pi$
$769$	15.7765	0.568914	0.284457	+	0.958689i	$0.408187\pi$
$770$	0	0
$771$	76.0247	2.73796
$772$	0	0
$773$	−33.7577	−1.21418	−0.607089	−	0.794634i	$-0.707663\pi$
$773$	−33.7577	−1.21418	−0.607089	+	0.794634i	$0.707663\pi$
$774$	0	0
$775$	−9.19196	−0.330185
$776$	0	0
$777$	−17.2002	−0.617053
$778$	0	0
$779$	25.9829	0.930935
$780$	0	0
$781$	0.712529	0.0254963
$782$	0	0
$783$	−6.15712	−0.220038
$784$	0	0
$785$	−52.0681	−1.85839
$786$	0	0
$787$	29.6976	1.05861	0.529303	−	0.848433i	$-0.322453\pi$
$787$	29.6976	1.05861	0.529303	+	0.848433i	$0.322453\pi$
$788$	0	0
$789$	−11.7871	−0.419631
$790$	0	0
$791$	−9.03718	−0.321325
$792$	0	0
$793$	−48.1049	−1.70826
$794$	0	0
$795$	−23.4618	−0.832105
$796$	0	0
$797$	22.0734	0.781882	0.390941	−	0.920416i	$-0.372150\pi$
$797$	22.0734	0.781882	0.390941	+	0.920416i	$0.372150\pi$
$798$	0	0
$799$	3.11947	0.110359
$800$	0	0
$801$	−20.6362	−0.729143
$802$	0	0
$803$	0.474091	0.0167303
$804$	0	0
$805$	−25.0336	−0.882317
$806$	0	0
$807$	53.3080	1.87653
$808$	0	0
$809$	30.0030	1.05485	0.527424	−	0.849602i	$-0.323158\pi$
$809$	30.0030	1.05485	0.527424	+	0.849602i	$0.323158\pi$
$810$	0	0
$811$	19.1169	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$811$	19.1169	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$812$	0	0
$813$	24.9805	0.876104
$814$	0	0
$815$	20.6209	0.722320
$816$	0	0
$817$	57.7046	2.01883
$818$	0	0
$819$	−19.0824	−0.666795
$820$	0	0
$821$	25.6672	0.895792	0.447896	−	0.894086i	$-0.352173\pi$
$821$	25.6672	0.895792	0.447896	+	0.894086i	$0.352173\pi$
$822$	0	0
$823$	−11.1225	−0.387705	−0.193853	−	0.981031i	$-0.562098\pi$
$823$	−11.1225	−0.387705	−0.193853	+	0.981031i	$0.562098\pi$
$824$	0	0
$825$	1.52145	0.0529702
$826$	0	0
$827$	36.5469	1.27086	0.635430	−	0.772158i	$-0.280823\pi$
$827$	36.5469	1.27086	0.635430	+	0.772158i	$0.280823\pi$
$828$	0	0
$829$	−12.9358	−0.449277	−0.224639	−	0.974442i	$-0.572120\pi$
$829$	−12.9358	−0.449277	−0.224639	+	0.974442i	$0.572120\pi$
$830$	0	0
$831$	56.4601	1.95858
$832$	0	0
$833$	−2.80113	−0.0970533
$834$	0	0
$835$	−35.3036	−1.22173
$836$	0	0
$837$	1.89025	0.0653368
$838$	0	0
$839$	12.5439	0.433062	0.216531	−	0.976276i	$-0.430526\pi$
$839$	12.5439	0.433062	0.216531	+	0.976276i	$0.430526\pi$
$840$	0	0
$841$	82.4970	2.84472
$842$	0	0
$843$	−56.3051	−1.93925
$844$	0	0
$845$	20.0125	0.688451
$846$	0	0
$847$	14.4009	0.494820
$848$	0	0
$849$	−54.1102	−1.85706
$850$	0	0
$851$	−35.6456	−1.22192
$852$	0	0
$853$	10.9760	0.375813	0.187906	−	0.982187i	$-0.439830\pi$
$853$	10.9760	0.375813	0.187906	+	0.982187i	$0.439830\pi$
$854$	0	0
$855$	68.7845	2.35238
$856$	0	0
$857$	−36.4080	−1.24367	−0.621837	−	0.783146i	$-0.713614\pi$
$857$	−36.4080	−1.24367	−0.621837	+	0.783146i	$0.713614\pi$
$858$	0	0
$859$	−21.8751	−0.746370	−0.373185	−	0.927757i	$-0.621734\pi$
$859$	−21.8751	−0.746370	−0.373185	+	0.927757i	$0.621734\pi$
$860$	0	0
$861$	11.2228	0.382474
$862$	0	0
$863$	−55.3352	−1.88363	−0.941817	−	0.336127i	$-0.890883\pi$
$863$	−55.3352	−1.88363	−0.941817	+	0.336127i	$0.890883\pi$
$864$	0	0
$865$	40.7779	1.38649
$866$	0	0
$867$	41.7391	1.41753
$868$	0	0
$869$	−0.668936	−0.0226921
$870$	0	0
$871$	−60.5800	−2.05267
$872$	0	0
$873$	41.4275	1.40211
$874$	0	0
$875$	7.96544	0.269281
$876$	0	0
$877$	43.4681	1.46781	0.733907	−	0.679250i	$-0.237695\pi$
$877$	43.4681	1.46781	0.733907	+	0.679250i	$0.237695\pi$
$878$	0	0
$879$	−0.0892216	−0.00300937
$880$	0	0
$881$	−12.7587	−0.429851	−0.214925	−	0.976630i	$-0.568951\pi$
$881$	−12.7587	−0.429851	−0.214925	+	0.976630i	$0.568951\pi$
$882$	0	0
$883$	29.3044	0.986172	0.493086	−	0.869981i	$-0.335869\pi$
$883$	29.3044	0.986172	0.493086	+	0.869981i	$0.335869\pi$
$884$	0	0
$885$	−69.4006	−2.33288
$886$	0	0
$887$	−39.5947	−1.32946	−0.664730	−	0.747084i	$-0.731453\pi$
$887$	−39.5947	−1.32946	−0.664730	+	0.747084i	$0.731453\pi$
$888$	0	0
$889$	12.4262	0.416762
$890$	0	0
$891$	1.77189	0.0593606
$892$	0	0
$893$	44.6134	1.49293
$894$	0	0
$895$	−41.4976	−1.38711
$896$	0	0
$897$	−76.2365	−2.54546
$898$	0	0
$899$	−34.2299	−1.14163
$900$	0	0
$901$	1.78382	0.0594277
$902$	0	0
$903$	24.9244	0.829433
$904$	0	0
$905$	−3.62923	−0.120640
$906$	0	0
$907$	−3.16892	−0.105222	−0.0526111	−	0.998615i	$-0.516754\pi$
$907$	−3.16892	−0.105222	−0.0526111	+	0.998615i	$0.516754\pi$
$908$	0	0
$909$	39.2905	1.30319
$910$	0	0
$911$	5.58375	0.184998	0.0924989	−	0.995713i	$-0.470515\pi$
$911$	5.58375	0.184998	0.0924989	+	0.995713i	$0.470515\pi$
$912$	0	0
$913$	−0.834929	−0.0276321
$914$	0	0
$915$	74.8963	2.47600
$916$	0	0
$917$	12.6267	0.416970
$918$	0	0
$919$	−56.4947	−1.86359	−0.931794	−	0.362988i	$-0.881756\pi$
$919$	−56.4947	−1.86359	−0.931794	+	0.362988i	$0.881756\pi$
$920$	0	0
$921$	−80.1909	−2.64238
$922$	0	0
$923$	−14.8825	−0.489864
$924$	0	0
$925$	−14.8585	−0.488545
$926$	0	0
$927$	−11.7975	−0.387480
$928$	0	0
$929$	−25.8228	−0.847218	−0.423609	−	0.905845i	$-0.639237\pi$
$929$	−25.8228	−0.847218	−0.423609	+	0.905845i	$0.639237\pi$
$930$	0	0
$931$	−40.0607	−1.31294
$932$	0	0
$933$	−25.5590	−0.836765
$934$	0	0
$935$	−0.319656	−0.0104539
$936$	0	0
$937$	−13.6840	−0.447037	−0.223518	−	0.974700i	$-0.571754\pi$
$937$	−13.6840	−0.447037	−0.223518	+	0.974700i	$0.571754\pi$
$938$	0	0
$939$	15.6887	0.511981
$940$	0	0
$941$	27.5733	0.898862	0.449431	−	0.893315i	$-0.351627\pi$
$941$	27.5733	0.898862	0.449431	+	0.893315i	$0.351627\pi$
$942$	0	0
$943$	23.2582	0.757391
$944$	0	0
$945$	2.14587	0.0698053
$946$	0	0
$947$	24.7426	0.804026	0.402013	−	0.915634i	$-0.368311\pi$
$947$	24.7426	0.804026	0.402013	+	0.915634i	$0.368311\pi$
$948$	0	0
$949$	−9.90230	−0.321442
$950$	0	0
$951$	28.2796	0.917028
$952$	0	0
$953$	58.1228	1.88278	0.941391	−	0.337317i	$-0.109520\pi$
$953$	58.1228	1.88278	0.941391	+	0.337317i	$0.109520\pi$
$954$	0	0
$955$	18.5265	0.599503
$956$	0	0
$957$	5.66573	0.183147
$958$	0	0
$959$	−16.1540	−0.521640
$960$	0	0
$961$	−20.4913	−0.661010
$962$	0	0
$963$	−19.8999	−0.641267
$964$	0	0
$965$	−42.7614	−1.37654
$966$	0	0
$967$	7.56870	0.243393	0.121697	−	0.992567i	$-0.461167\pi$
$967$	7.56870	0.243393	0.121697	+	0.992567i	$0.461167\pi$
$968$	0	0
$969$	−10.0818	−0.323873
$970$	0	0
$971$	16.0968	0.516571	0.258286	−	0.966069i	$-0.416842\pi$
$971$	16.0968	0.516571	0.258286	+	0.966069i	$0.416842\pi$
$972$	0	0
$973$	−20.2284	−0.648493
$974$	0	0
$975$	−31.7784	−1.01772
$976$	0	0
$977$	−17.1730	−0.549413	−0.274707	−	0.961528i	$-0.588581\pi$
$977$	−17.1730	−0.549413	−0.274707	+	0.961528i	$0.588581\pi$
$978$	0	0
$979$	1.37153	0.0438344
$980$	0	0
$981$	51.8821	1.65647
$982$	0	0
$983$	−20.0335	−0.638970	−0.319485	−	0.947591i	$-0.603510\pi$
$983$	−20.0335	−0.638970	−0.319485	+	0.947591i	$0.603510\pi$
$984$	0	0
$985$	51.0873	1.62778
$986$	0	0
$987$	19.2699	0.613369
$988$	0	0
$989$	51.6533	1.64248
$990$	0	0
$991$	−32.5514	−1.03403	−0.517014	−	0.855977i	$-0.672957\pi$
$991$	−32.5514	−1.03403	−0.517014	+	0.855977i	$0.672957\pi$
$992$	0	0
$993$	53.7230	1.70485
$994$	0	0
$995$	11.1049	0.352047
$996$	0	0
$997$	20.4199	0.646703	0.323352	−	0.946279i	$-0.395190\pi$
$997$	20.4199	0.646703	0.323352	+	0.946279i	$0.395190\pi$
$998$	0	0
$999$	3.05554	0.0966729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.r.1.3		12
4.3	odd	2		1006.2.a.i.1.10	✓	12
12.11	even	2		9054.2.a.bj.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.i.1.10	✓	12	4.3	odd	2
8048.2.a.r.1.3		12	1.1	even	1	trivial
9054.2.a.bj.1.4		12	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q-2.49671 q^{3} +2.79920 q^{5} -1.31469 q^{7} +3.23355 q^{9} +O(q^{10})\)	\(q-2.49671 q^{3} +2.79920 q^{5} -1.31469 q^{7} +3.23355 q^{9} -0.214910 q^{11} +4.48880 q^{13} -6.98879 q^{15} +0.531363 q^{17} +7.59936 q^{19} +3.28240 q^{21} +6.80244 q^{23} +2.83553 q^{25} -0.583104 q^{27} +10.5592 q^{29} -3.24171 q^{31} +0.536567 q^{33} -3.68009 q^{35} -5.24012 q^{37} -11.2072 q^{39} +3.41910 q^{41} +7.59335 q^{43} +9.05136 q^{45} +5.87068 q^{47} -5.27159 q^{49} -1.32666 q^{51} +3.35707 q^{53} -0.601576 q^{55} -18.9734 q^{57} +9.93028 q^{59} -10.7166 q^{61} -4.25112 q^{63} +12.5651 q^{65} -13.4958 q^{67} -16.9837 q^{69} -3.31548 q^{71} -2.20600 q^{73} -7.07948 q^{75} +0.282540 q^{77} +3.11263 q^{79} -8.24481 q^{81} +3.88502 q^{83} +1.48739 q^{85} -26.3633 q^{87} -6.38189 q^{89} -5.90139 q^{91} +8.09360 q^{93} +21.2721 q^{95} +12.8118 q^{97} -0.694922 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9} - 18 q^{11} - 4 q^{13} + 2 q^{15} + 12 q^{17} + 7 q^{21} + 9 q^{23} + 25 q^{25} + 18 q^{27} + 34 q^{29} + 11 q^{31} + 4 q^{33} - 21 q^{35} - 22 q^{37} - 13 q^{39} + 32 q^{41} + 8 q^{43} + 13 q^{45} - 24 q^{47} + 36 q^{49} - 16 q^{51} - 2 q^{53} + 12 q^{55} + 26 q^{57} - 26 q^{59} + 12 q^{61} - 5 q^{63} + 66 q^{65} + 21 q^{67} + 20 q^{69} - 50 q^{71} + 17 q^{73} + 14 q^{75} + 25 q^{77} + 9 q^{79} + 48 q^{81} - 25 q^{83} + 24 q^{85} + 10 q^{87} + 21 q^{89} + 9 q^{91} + 31 q^{93} - 22 q^{95} + 18 q^{97} - 102 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9 - 18 * q^11 - 4 * q^13 + 2 * q^15 + 12 * q^17 + 7 * q^21 + 9 * q^23 + 25 * q^25 + 18 * q^27 + 34 * q^29 + 11 * q^31 + 4 * q^33 - 21 * q^35 - 22 * q^37 - 13 * q^39 + 32 * q^41 + 8 * q^43 + 13 * q^45 - 24 * q^47 + 36 * q^49 - 16 * q^51 - 2 * q^53 + 12 * q^55 + 26 * q^57 - 26 * q^59 + 12 * q^61 - 5 * q^63 + 66 * q^65 + 21 * q^67 + 20 * q^69 - 50 * q^71 + 17 * q^73 + 14 * q^75 + 25 * q^77 + 9 * q^79 + 48 * q^81 - 25 * q^83 + 24 * q^85 + 10 * q^87 + 21 * q^89 + 9 * q^91 + 31 * q^93 - 22 * q^95 + 18 * q^97 - 102 * q^99

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