LMFDB - Embedded newform 8048.2.a.q.1.2

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:	$12$
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} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$2.49390$ 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)

$f(q)$	$=$	$q-2.49390 q^{3} +1.94778 q^{5} +1.87622 q^{7} +3.21954 q^{9} +O(q^{10})$	$q-2.49390 q^{3} +1.94778 q^{5} +1.87622 q^{7} +3.21954 q^{9} -5.83443 q^{11} -0.298003 q^{13} -4.85757 q^{15} -7.33964 q^{17} +0.888571 q^{19} -4.67910 q^{21} +8.64250 q^{23} -1.20615 q^{25} -0.547512 q^{27} +0.804767 q^{29} +5.43347 q^{31} +14.5505 q^{33} +3.65446 q^{35} +4.16366 q^{37} +0.743190 q^{39} +2.14268 q^{41} -5.11414 q^{43} +6.27096 q^{45} -10.2919 q^{47} -3.47981 q^{49} +18.3043 q^{51} +9.43805 q^{53} -11.3642 q^{55} -2.21601 q^{57} +14.1014 q^{59} +14.1214 q^{61} +6.04056 q^{63} -0.580445 q^{65} +6.24714 q^{67} -21.5535 q^{69} -4.03233 q^{71} -8.07158 q^{73} +3.00801 q^{75} -10.9467 q^{77} -15.7749 q^{79} -8.29318 q^{81} -3.93865 q^{83} -14.2960 q^{85} -2.00701 q^{87} -4.09320 q^{89} -0.559119 q^{91} -13.5505 q^{93} +1.73074 q^{95} -8.46702 q^{97} -18.7842 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * 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
$3$	−2.49390	−1.43985	−0.719927	−	0.694050i	$-0.755825\pi$
$3$	−2.49390	−1.43985	−0.719927	+	0.694050i	$0.755825\pi$
$4$	0	0
$5$	1.94778	0.871075	0.435537	−	0.900171i	$-0.356558\pi$
$5$	1.94778	0.871075	0.435537	+	0.900171i	$0.356558\pi$
$6$	0	0
$7$	1.87622	0.709144	0.354572	−	0.935029i	$-0.384627\pi$
$7$	1.87622	0.709144	0.354572	+	0.935029i	$0.384627\pi$
$8$	0	0
$9$	3.21954	1.07318
$10$	0	0
$11$	−5.83443	−1.75915	−0.879574	−	0.475763i	$-0.842172\pi$
$11$	−5.83443	−1.75915	−0.879574	+	0.475763i	$0.842172\pi$
$12$	0	0
$13$	−0.298003	−0.0826512	−0.0413256	−	0.999146i	$-0.513158\pi$
$13$	−0.298003	−0.0826512	−0.0413256	+	0.999146i	$0.513158\pi$
$14$	0	0
$15$	−4.85757	−1.25422
$16$	0	0
$17$	−7.33964	−1.78012	−0.890062	−	0.455840i	$-0.849339\pi$
$17$	−7.33964	−1.78012	−0.890062	+	0.455840i	$0.849339\pi$
$18$	0	0
$19$	0.888571	0.203852	0.101926	−	0.994792i	$-0.467499\pi$
$19$	0.888571	0.203852	0.101926	+	0.994792i	$0.467499\pi$
$20$	0	0
$21$	−4.67910	−1.02106
$22$	0	0
$23$	8.64250	1.80209	0.901043	−	0.433730i	$-0.142803\pi$
$23$	8.64250	1.80209	0.901043	+	0.433730i	$0.142803\pi$
$24$	0	0
$25$	−1.20615	−0.241229
$26$	0	0
$27$	−0.547512	−0.105369
$28$	0	0
$29$	0.804767	0.149441	0.0747207	−	0.997204i	$-0.476193\pi$
$29$	0.804767	0.149441	0.0747207	+	0.997204i	$0.476193\pi$
$30$	0	0
$31$	5.43347	0.975880	0.487940	−	0.872877i	$-0.337748\pi$
$31$	5.43347	0.975880	0.487940	+	0.872877i	$0.337748\pi$
$32$	0	0
$33$	14.5505	2.53292
$34$	0	0
$35$	3.65446	0.617717
$36$	0	0
$37$	4.16366	0.684501	0.342251	−	0.939609i	$-0.388811\pi$
$37$	4.16366	0.684501	0.342251	+	0.939609i	$0.388811\pi$
$38$	0	0
$39$	0.743190	0.119006
$40$	0	0
$41$	2.14268	0.334630	0.167315	−	0.985903i	$-0.446490\pi$
$41$	2.14268	0.334630	0.167315	+	0.985903i	$0.446490\pi$
$42$	0	0
$43$	−5.11414	−0.779899	−0.389950	−	0.920836i	$-0.627508\pi$
$43$	−5.11414	−0.779899	−0.389950	+	0.920836i	$0.627508\pi$
$44$	0	0
$45$	6.27096	0.934820
$46$	0	0
$47$	−10.2919	−1.50123	−0.750617	−	0.660738i	$-0.770243\pi$
$47$	−10.2919	−1.50123	−0.750617	+	0.660738i	$0.770243\pi$
$48$	0	0
$49$	−3.47981	−0.497115
$50$	0	0
$51$	18.3043	2.56312
$52$	0	0
$53$	9.43805	1.29642	0.648208	−	0.761464i	$-0.275519\pi$
$53$	9.43805	1.29642	0.648208	+	0.761464i	$0.275519\pi$
$54$	0	0
$55$	−11.3642	−1.53235
$56$	0	0
$57$	−2.21601	−0.293517
$58$	0	0
$59$	14.1014	1.83585	0.917924	−	0.396756i	$-0.129864\pi$
$59$	14.1014	1.83585	0.917924	+	0.396756i	$0.129864\pi$
$60$	0	0
$61$	14.1214	1.80806	0.904030	−	0.427469i	$-0.140595\pi$
$61$	14.1214	1.80806	0.904030	+	0.427469i	$0.140595\pi$
$62$	0	0
$63$	6.04056	0.761039
$64$	0	0
$65$	−0.580445	−0.0719953
$66$	0	0
$67$	6.24714	0.763210	0.381605	−	0.924326i	$-0.375371\pi$
$67$	6.24714	0.763210	0.381605	+	0.924326i	$0.375371\pi$
$68$	0	0
$69$	−21.5535	−2.59474
$70$	0	0
$71$	−4.03233	−0.478549	−0.239275	−	0.970952i	$-0.576910\pi$
$71$	−4.03233	−0.478549	−0.239275	+	0.970952i	$0.576910\pi$
$72$	0	0
$73$	−8.07158	−0.944707	−0.472354	−	0.881409i	$-0.656595\pi$
$73$	−8.07158	−0.944707	−0.472354	+	0.881409i	$0.656595\pi$
$74$	0	0
$75$	3.00801	0.347335
$76$	0	0
$77$	−10.9467	−1.24749
$78$	0	0
$79$	−15.7749	−1.77482	−0.887409	−	0.460982i	$-0.847497\pi$
$79$	−15.7749	−1.77482	−0.887409	+	0.460982i	$0.847497\pi$
$80$	0	0
$81$	−8.29318	−0.921465
$82$	0	0
$83$	−3.93865	−0.432323	−0.216162	−	0.976358i	$-0.569354\pi$
$83$	−3.93865	−0.432323	−0.216162	+	0.976358i	$0.569354\pi$
$84$	0	0
$85$	−14.2960	−1.55062
$86$	0	0
$87$	−2.00701	−0.215174
$88$	0	0
$89$	−4.09320	−0.433878	−0.216939	−	0.976185i	$-0.569607\pi$
$89$	−4.09320	−0.433878	−0.216939	+	0.976185i	$0.569607\pi$
$90$	0	0
$91$	−0.559119	−0.0586116
$92$	0	0
$93$	−13.5505	−1.40513
$94$	0	0
$95$	1.73074	0.177570
$96$	0	0
$97$	−8.46702	−0.859696	−0.429848	−	0.902901i	$-0.641433\pi$
$97$	−8.46702	−0.859696	−0.429848	+	0.902901i	$0.641433\pi$
$98$	0	0
$99$	−18.7842	−1.88788
$100$	0	0
$101$	−6.91486	−0.688054	−0.344027	−	0.938960i	$-0.611791\pi$
$101$	−6.91486	−0.688054	−0.344027	+	0.938960i	$0.611791\pi$
$102$	0	0
$103$	−5.31246	−0.523452	−0.261726	−	0.965142i	$-0.584292\pi$
$103$	−5.31246	−0.523452	−0.261726	+	0.965142i	$0.584292\pi$
$104$	0	0
$105$	−9.11387	−0.889423
$106$	0	0
$107$	10.3616	1.00169	0.500846	−	0.865536i	$-0.333022\pi$
$107$	10.3616	1.00169	0.500846	+	0.865536i	$0.333022\pi$
$108$	0	0
$109$	11.0031	1.05390	0.526952	−	0.849895i	$-0.323335\pi$
$109$	11.0031	1.05390	0.526952	+	0.849895i	$0.323335\pi$
$110$	0	0
$111$	−10.3838	−0.985582
$112$	0	0
$113$	−5.93637	−0.558447	−0.279223	−	0.960226i	$-0.590077\pi$
$113$	−5.93637	−0.558447	−0.279223	+	0.960226i	$0.590077\pi$
$114$	0	0
$115$	16.8337	1.56975
$116$	0	0
$117$	−0.959433	−0.0886996
$118$	0	0
$119$	−13.7708	−1.26236
$120$	0	0
$121$	23.0406	2.09460
$122$	0	0
$123$	−5.34363	−0.481819
$124$	0	0
$125$	−12.0882	−1.08120
$126$	0	0
$127$	−4.22816	−0.375188	−0.187594	−	0.982247i	$-0.560069\pi$
$127$	−4.22816	−0.375188	−0.187594	+	0.982247i	$0.560069\pi$
$128$	0	0
$129$	12.7542	1.12294
$130$	0	0
$131$	12.9405	1.13061	0.565307	−	0.824881i	$-0.308758\pi$
$131$	12.9405	1.13061	0.565307	+	0.824881i	$0.308758\pi$
$132$	0	0
$133$	1.66715	0.144560
$134$	0	0
$135$	−1.06643	−0.0917840
$136$	0	0
$137$	3.32249	0.283860	0.141930	−	0.989877i	$-0.454669\pi$
$137$	3.32249	0.283860	0.141930	+	0.989877i	$0.454669\pi$
$138$	0	0
$139$	−17.4440	−1.47958	−0.739792	−	0.672836i	$-0.765076\pi$
$139$	−17.4440	−1.47958	−0.739792	+	0.672836i	$0.765076\pi$
$140$	0	0
$141$	25.6671	2.16156
$142$	0	0
$143$	1.73868	0.145396
$144$	0	0
$145$	1.56751	0.130175
$146$	0	0
$147$	8.67829	0.715773
$148$	0	0
$149$	−2.15279	−0.176363	−0.0881817	−	0.996104i	$-0.528106\pi$
$149$	−2.15279	−0.176363	−0.0881817	+	0.996104i	$0.528106\pi$
$150$	0	0
$151$	−8.86908	−0.721756	−0.360878	−	0.932613i	$-0.617523\pi$
$151$	−8.86908	−0.721756	−0.360878	+	0.932613i	$0.617523\pi$
$152$	0	0
$153$	−23.6303	−1.91039
$154$	0	0
$155$	10.5832	0.850064
$156$	0	0
$157$	−6.74771	−0.538526	−0.269263	−	0.963067i	$-0.586780\pi$
$157$	−6.74771	−0.538526	−0.269263	+	0.963067i	$0.586780\pi$
$158$	0	0
$159$	−23.5375	−1.86665
$160$	0	0
$161$	16.2152	1.27794
$162$	0	0
$163$	−6.34872	−0.497270	−0.248635	−	0.968597i	$-0.579982\pi$
$163$	−6.34872	−0.497270	−0.248635	+	0.968597i	$0.579982\pi$
$164$	0	0
$165$	28.3412	2.20636
$166$	0	0
$167$	22.9525	1.77612	0.888060	−	0.459727i	$-0.152053\pi$
$167$	22.9525	1.77612	0.888060	+	0.459727i	$0.152053\pi$
$168$	0	0
$169$	−12.9112	−0.993169
$170$	0	0
$171$	2.86079	0.218770
$172$	0	0
$173$	24.5497	1.86648	0.933238	−	0.359258i	$-0.116970\pi$
$173$	24.5497	1.86648	0.933238	+	0.359258i	$0.116970\pi$
$174$	0	0
$175$	−2.26299	−0.171066
$176$	0	0
$177$	−35.1675	−2.64335
$178$	0	0
$179$	−9.50528	−0.710458	−0.355229	−	0.934779i	$-0.615597\pi$
$179$	−9.50528	−0.710458	−0.355229	+	0.934779i	$0.615597\pi$
$180$	0	0
$181$	−20.8706	−1.55130	−0.775651	−	0.631162i	$-0.782578\pi$
$181$	−20.8706	−1.55130	−0.775651	+	0.631162i	$0.782578\pi$
$182$	0	0
$183$	−35.2174	−2.60334
$184$	0	0
$185$	8.10990	0.596252
$186$	0	0
$187$	42.8226	3.13150
$188$	0	0
$189$	−1.02725	−0.0747215
$190$	0	0
$191$	−19.8473	−1.43610	−0.718049	−	0.695993i	$-0.754965\pi$
$191$	−19.8473	−1.43610	−0.718049	+	0.695993i	$0.754965\pi$
$192$	0	0
$193$	−14.6490	−1.05446	−0.527228	−	0.849724i	$-0.676769\pi$
$193$	−14.6490	−1.05446	−0.527228	+	0.849724i	$0.676769\pi$
$194$	0	0
$195$	1.44757	0.103663
$196$	0	0
$197$	−9.37280	−0.667785	−0.333892	−	0.942611i	$-0.608362\pi$
$197$	−9.37280	−0.667785	−0.333892	+	0.942611i	$0.608362\pi$
$198$	0	0
$199$	14.6360	1.03752	0.518760	−	0.854920i	$-0.326394\pi$
$199$	14.6360	1.03752	0.518760	+	0.854920i	$0.326394\pi$
$200$	0	0
$201$	−15.5798	−1.09891
$202$	0	0
$203$	1.50992	0.105975
$204$	0	0
$205$	4.17347	0.291488
$206$	0	0
$207$	27.8249	1.93396
$208$	0	0
$209$	−5.18431	−0.358606
$210$	0	0
$211$	−3.94613	−0.271663	−0.135831	−	0.990732i	$-0.543371\pi$
$211$	−3.94613	−0.271663	−0.135831	+	0.990732i	$0.543371\pi$
$212$	0	0
$213$	10.0562	0.689041
$214$	0	0
$215$	−9.96123	−0.679350
$216$	0	0
$217$	10.1944	0.692039
$218$	0	0
$219$	20.1297	1.36024
$220$	0	0
$221$	2.18723	0.147129
$222$	0	0
$223$	4.32975	0.289941	0.144971	−	0.989436i	$-0.453691\pi$
$223$	4.32975	0.289941	0.144971	+	0.989436i	$0.453691\pi$
$224$	0	0
$225$	−3.88324	−0.258882
$226$	0	0
$227$	−1.56410	−0.103813	−0.0519064	−	0.998652i	$-0.516530\pi$
$227$	−1.56410	−0.103813	−0.0519064	+	0.998652i	$0.516530\pi$
$228$	0	0
$229$	−23.6879	−1.56534	−0.782672	−	0.622435i	$-0.786144\pi$
$229$	−23.6879	−1.56534	−0.782672	+	0.622435i	$0.786144\pi$
$230$	0	0
$231$	27.2999	1.79620
$232$	0	0
$233$	19.7694	1.29513	0.647567	−	0.762008i	$-0.275787\pi$
$233$	19.7694	1.29513	0.647567	+	0.762008i	$0.275787\pi$
$234$	0	0
$235$	−20.0464	−1.30769
$236$	0	0
$237$	39.3411	2.55548
$238$	0	0
$239$	2.12172	0.137243	0.0686214	−	0.997643i	$-0.478140\pi$
$239$	2.12172	0.137243	0.0686214	+	0.997643i	$0.478140\pi$
$240$	0	0
$241$	20.8089	1.34042	0.670210	−	0.742172i	$-0.266204\pi$
$241$	20.8089	1.34042	0.670210	+	0.742172i	$0.266204\pi$
$242$	0	0
$243$	22.3249	1.43214
$244$	0	0
$245$	−6.77790	−0.433024
$246$	0	0
$247$	−0.264797	−0.0168486
$248$	0	0
$249$	9.82260	0.622482
$250$	0	0
$251$	−19.6095	−1.23774	−0.618870	−	0.785494i	$-0.712409\pi$
$251$	−19.6095	−1.23774	−0.618870	+	0.785494i	$0.712409\pi$
$252$	0	0
$253$	−50.4241	−3.17013
$254$	0	0
$255$	35.6528	2.23267
$256$	0	0
$257$	4.20578	0.262349	0.131175	−	0.991359i	$-0.458125\pi$
$257$	4.20578	0.262349	0.131175	+	0.991359i	$0.458125\pi$
$258$	0	0
$259$	7.81193	0.485410
$260$	0	0
$261$	2.59098	0.160378
$262$	0	0
$263$	−4.95237	−0.305376	−0.152688	−	0.988274i	$-0.548793\pi$
$263$	−4.95237	−0.305376	−0.152688	+	0.988274i	$0.548793\pi$
$264$	0	0
$265$	18.3833	1.12927
$266$	0	0
$267$	10.2080	0.624721
$268$	0	0
$269$	17.6754	1.07769	0.538843	−	0.842406i	$-0.318862\pi$
$269$	17.6754	1.07769	0.538843	+	0.842406i	$0.318862\pi$
$270$	0	0
$271$	−6.71122	−0.407678	−0.203839	−	0.979004i	$-0.565342\pi$
$271$	−6.71122	−0.407678	−0.203839	+	0.979004i	$0.565342\pi$
$272$	0	0
$273$	1.39439	0.0843921
$274$	0	0
$275$	7.03718	0.424358
$276$	0	0
$277$	−22.4543	−1.34915	−0.674575	−	0.738206i	$-0.735673\pi$
$277$	−22.4543	−1.34915	−0.674575	+	0.738206i	$0.735673\pi$
$278$	0	0
$279$	17.4933	1.04730
$280$	0	0
$281$	20.1623	1.20278	0.601391	−	0.798955i	$-0.294613\pi$
$281$	20.1623	1.20278	0.601391	+	0.798955i	$0.294613\pi$
$282$	0	0
$283$	−26.3840	−1.56837	−0.784183	−	0.620530i	$-0.786918\pi$
$283$	−26.3840	−1.56837	−0.784183	+	0.620530i	$0.786918\pi$
$284$	0	0
$285$	−4.31630	−0.255675
$286$	0	0
$287$	4.02013	0.237301
$288$	0	0
$289$	36.8703	2.16884
$290$	0	0
$291$	21.1159	1.23784
$292$	0	0
$293$	−7.77069	−0.453969	−0.226984	−	0.973898i	$-0.572887\pi$
$293$	−7.77069	−0.453969	−0.226984	+	0.973898i	$0.572887\pi$
$294$	0	0
$295$	27.4665	1.59916
$296$	0	0
$297$	3.19442	0.185359
$298$	0	0
$299$	−2.57549	−0.148945
$300$	0	0
$301$	−9.59524	−0.553061
$302$	0	0
$303$	17.2450	0.990697
$304$	0	0
$305$	27.5054	1.57496
$306$	0	0
$307$	−27.3128	−1.55883	−0.779413	−	0.626510i	$-0.784483\pi$
$307$	−27.3128	−1.55883	−0.779413	+	0.626510i	$0.784483\pi$
$308$	0	0
$309$	13.2488	0.753695
$310$	0	0
$311$	−10.3916	−0.589252	−0.294626	−	0.955613i	$-0.595195\pi$
$311$	−10.3916	−0.589252	−0.294626	+	0.955613i	$0.595195\pi$
$312$	0	0
$313$	−8.62733	−0.487645	−0.243823	−	0.969820i	$-0.578401\pi$
$313$	−8.62733	−0.487645	−0.243823	+	0.969820i	$0.578401\pi$
$314$	0	0
$315$	11.7657	0.662922
$316$	0	0
$317$	23.7917	1.33628	0.668138	−	0.744037i	$-0.267092\pi$
$317$	23.7917	1.33628	0.668138	+	0.744037i	$0.267092\pi$
$318$	0	0
$319$	−4.69536	−0.262890
$320$	0	0
$321$	−25.8408	−1.44229
$322$	0	0
$323$	−6.52179	−0.362882
$324$	0	0
$325$	0.359435	0.0199379
$326$	0	0
$327$	−27.4406	−1.51747
$328$	0	0
$329$	−19.3099	−1.06459
$330$	0	0
$331$	16.6716	0.916353	0.458177	−	0.888861i	$-0.348503\pi$
$331$	16.6716	0.916353	0.458177	+	0.888861i	$0.348503\pi$
$332$	0	0
$333$	13.4051	0.734593
$334$	0	0
$335$	12.1681	0.664813
$336$	0	0
$337$	−11.6612	−0.635225	−0.317612	−	0.948221i	$-0.602881\pi$
$337$	−11.6612	−0.635225	−0.317612	+	0.948221i	$0.602881\pi$
$338$	0	0
$339$	14.8047	0.804082
$340$	0	0
$341$	−31.7012	−1.71672
$342$	0	0
$343$	−19.6624	−1.06167
$344$	0	0
$345$	−41.9816	−2.26021
$346$	0	0
$347$	13.8126	0.741500	0.370750	−	0.928733i	$-0.379101\pi$
$347$	13.8126	0.741500	0.370750	+	0.928733i	$0.379101\pi$
$348$	0	0
$349$	13.7647	0.736807	0.368404	−	0.929666i	$-0.379904\pi$
$349$	13.7647	0.736807	0.368404	+	0.929666i	$0.379904\pi$
$350$	0	0
$351$	0.163160	0.00870885
$352$	0	0
$353$	7.48060	0.398152	0.199076	−	0.979984i	$-0.436206\pi$
$353$	7.48060	0.398152	0.199076	+	0.979984i	$0.436206\pi$
$354$	0	0
$355$	−7.85410	−0.416852
$356$	0	0
$357$	34.3429	1.81762
$358$	0	0
$359$	−13.8737	−0.732227	−0.366113	−	0.930570i	$-0.619312\pi$
$359$	−13.8737	−0.732227	−0.366113	+	0.930570i	$0.619312\pi$
$360$	0	0
$361$	−18.2104	−0.958444
$362$	0	0
$363$	−57.4610	−3.01592
$364$	0	0
$365$	−15.7217	−0.822910
$366$	0	0
$367$	10.7012	0.558599	0.279300	−	0.960204i	$-0.409898\pi$
$367$	10.7012	0.558599	0.279300	+	0.960204i	$0.409898\pi$
$368$	0	0
$369$	6.89844	0.359119
$370$	0	0
$371$	17.7078	0.919345
$372$	0	0
$373$	−27.4619	−1.42192	−0.710962	−	0.703230i	$-0.751740\pi$
$373$	−27.4619	−1.42192	−0.710962	+	0.703230i	$0.751740\pi$
$374$	0	0
$375$	30.1468	1.55677
$376$	0	0
$377$	−0.239823	−0.0123515
$378$	0	0
$379$	17.8711	0.917978	0.458989	−	0.888442i	$-0.348212\pi$
$379$	17.8711	0.917978	0.458989	+	0.888442i	$0.348212\pi$
$380$	0	0
$381$	10.5446	0.540217
$382$	0	0
$383$	−10.8778	−0.555832	−0.277916	−	0.960605i	$-0.589644\pi$
$383$	−10.8778	−0.555832	−0.277916	+	0.960605i	$0.589644\pi$
$384$	0	0
$385$	−21.3217	−1.08666
$386$	0	0
$387$	−16.4652	−0.836972
$388$	0	0
$389$	−4.52471	−0.229412	−0.114706	−	0.993399i	$-0.536593\pi$
$389$	−4.52471	−0.229412	−0.114706	+	0.993399i	$0.536593\pi$
$390$	0	0
$391$	−63.4328	−3.20794
$392$	0	0
$393$	−32.2722	−1.62792
$394$	0	0
$395$	−30.7261	−1.54600
$396$	0	0
$397$	−27.6450	−1.38746	−0.693731	−	0.720234i	$-0.744034\pi$
$397$	−27.6450	−1.38746	−0.693731	+	0.720234i	$0.744034\pi$
$398$	0	0
$399$	−4.15771	−0.208146
$400$	0	0
$401$	−1.76900	−0.0883397	−0.0441699	−	0.999024i	$-0.514064\pi$
$401$	−1.76900	−0.0883397	−0.0441699	+	0.999024i	$0.514064\pi$
$402$	0	0
$403$	−1.61919	−0.0806577
$404$	0	0
$405$	−16.1533	−0.802664
$406$	0	0
$407$	−24.2926	−1.20414
$408$	0	0
$409$	−26.0238	−1.28679	−0.643396	−	0.765533i	$-0.722475\pi$
$409$	−26.0238	−1.28679	−0.643396	+	0.765533i	$0.722475\pi$
$410$	0	0
$411$	−8.28596	−0.408716
$412$	0	0
$413$	26.4573	1.30188
$414$	0	0
$415$	−7.67163	−0.376586
$416$	0	0
$417$	43.5037	2.13038
$418$	0	0
$419$	−4.90699	−0.239722	−0.119861	−	0.992791i	$-0.538245\pi$
$419$	−4.90699	−0.239722	−0.119861	+	0.992791i	$0.538245\pi$
$420$	0	0
$421$	7.00976	0.341635	0.170817	−	0.985303i	$-0.445359\pi$
$421$	7.00976	0.341635	0.170817	+	0.985303i	$0.445359\pi$
$422$	0	0
$423$	−33.1353	−1.61109
$424$	0	0
$425$	8.85267	0.429418
$426$	0	0
$427$	26.4948	1.28217
$428$	0	0
$429$	−4.33609	−0.209348
$430$	0	0
$431$	−16.6838	−0.803628	−0.401814	−	0.915721i	$-0.631620\pi$
$431$	−16.6838	−0.803628	−0.401814	+	0.915721i	$0.631620\pi$
$432$	0	0
$433$	−16.3926	−0.787777	−0.393889	−	0.919158i	$-0.628870\pi$
$433$	−16.3926	−0.787777	−0.393889	+	0.919158i	$0.628870\pi$
$434$	0	0
$435$	−3.90921	−0.187433
$436$	0	0
$437$	7.67947	0.367359
$438$	0	0
$439$	10.9201	0.521188	0.260594	−	0.965448i	$-0.416082\pi$
$439$	10.9201	0.521188	0.260594	+	0.965448i	$0.416082\pi$
$440$	0	0
$441$	−11.2034	−0.533494
$442$	0	0
$443$	8.44935	0.401441	0.200720	−	0.979649i	$-0.435672\pi$
$443$	8.44935	0.401441	0.200720	+	0.979649i	$0.435672\pi$
$444$	0	0
$445$	−7.97266	−0.377940
$446$	0	0
$447$	5.36884	0.253938
$448$	0	0
$449$	12.1220	0.572071	0.286036	−	0.958219i	$-0.407662\pi$
$449$	12.1220	0.572071	0.286036	+	0.958219i	$0.407662\pi$
$450$	0	0
$451$	−12.5013	−0.588664
$452$	0	0
$453$	22.1186	1.03922
$454$	0	0
$455$	−1.08904	−0.0510551
$456$	0	0
$457$	−37.0355	−1.73245	−0.866223	−	0.499657i	$-0.833459\pi$
$457$	−37.0355	−1.73245	−0.866223	+	0.499657i	$0.833459\pi$
$458$	0	0
$459$	4.01854	0.187569
$460$	0	0
$461$	−3.06027	−0.142531	−0.0712656	−	0.997457i	$-0.522704\pi$
$461$	−3.06027	−0.142531	−0.0712656	+	0.997457i	$0.522704\pi$
$462$	0	0
$463$	−7.69735	−0.357726	−0.178863	−	0.983874i	$-0.557242\pi$
$463$	−7.69735	−0.357726	−0.178863	+	0.983874i	$0.557242\pi$
$464$	0	0
$465$	−26.3935	−1.22397
$466$	0	0
$467$	22.8952	1.05946	0.529732	−	0.848165i	$-0.322293\pi$
$467$	22.8952	1.05946	0.529732	+	0.848165i	$0.322293\pi$
$468$	0	0
$469$	11.7210	0.541226
$470$	0	0
$471$	16.8281	0.775399
$472$	0	0
$473$	29.8381	1.37196
$474$	0	0
$475$	−1.07175	−0.0491751
$476$	0	0
$477$	30.3862	1.39129
$478$	0	0
$479$	−25.5092	−1.16555	−0.582773	−	0.812635i	$-0.698032\pi$
$479$	−25.5092	−1.16555	−0.582773	+	0.812635i	$0.698032\pi$
$480$	0	0
$481$	−1.24078	−0.0565749
$482$	0	0
$483$	−40.4391	−1.84004
$484$	0	0
$485$	−16.4919	−0.748859
$486$	0	0
$487$	−27.0841	−1.22730	−0.613648	−	0.789580i	$-0.710299\pi$
$487$	−27.0841	−1.22730	−0.613648	+	0.789580i	$0.710299\pi$
$488$	0	0
$489$	15.8331	0.715997
$490$	0	0
$491$	−36.9457	−1.66734	−0.833669	−	0.552265i	$-0.813764\pi$
$491$	−36.9457	−1.66734	−0.833669	+	0.552265i	$0.813764\pi$
$492$	0	0
$493$	−5.90670	−0.266024
$494$	0	0
$495$	−36.5875	−1.64449
$496$	0	0
$497$	−7.56553	−0.339360
$498$	0	0
$499$	−43.5308	−1.94871	−0.974353	−	0.225027i	$-0.927753\pi$
$499$	−43.5308	−1.94871	−0.974353	+	0.225027i	$0.927753\pi$
$500$	0	0
$501$	−57.2413	−2.55735
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−13.4686	−0.599346
$506$	0	0
$507$	32.1992	1.43002
$508$	0	0
$509$	−8.75597	−0.388101	−0.194051	−	0.980992i	$-0.562163\pi$
$509$	−8.75597	−0.388101	−0.194051	+	0.980992i	$0.562163\pi$
$510$	0	0
$511$	−15.1441	−0.669933
$512$	0	0
$513$	−0.486503	−0.0214796
$514$	0	0
$515$	−10.3475	−0.455966
$516$	0	0
$517$	60.0476	2.64089
$518$	0	0
$519$	−61.2244	−2.68745
$520$	0	0
$521$	23.4843	1.02887	0.514433	−	0.857530i	$-0.328002\pi$
$521$	23.4843	1.02887	0.514433	+	0.857530i	$0.328002\pi$
$522$	0	0
$523$	16.0353	0.701177	0.350588	−	0.936530i	$-0.385982\pi$
$523$	16.0353	0.701177	0.350588	+	0.936530i	$0.385982\pi$
$524$	0	0
$525$	5.64368	0.246310
$526$	0	0
$527$	−39.8797	−1.73719
$528$	0	0
$529$	51.6928	2.24751
$530$	0	0
$531$	45.4001	1.97020
$532$	0	0
$533$	−0.638525	−0.0276576
$534$	0	0
$535$	20.1821	0.872549
$536$	0	0
$537$	23.7052	1.02296
$538$	0	0
$539$	20.3027	0.874499
$540$	0	0
$541$	20.8582	0.896765	0.448382	−	0.893842i	$-0.352000\pi$
$541$	20.8582	0.896765	0.448382	+	0.893842i	$0.352000\pi$
$542$	0	0
$543$	52.0493	2.23365
$544$	0	0
$545$	21.4316	0.918029
$546$	0	0
$547$	−0.852077	−0.0364322	−0.0182161	−	0.999834i	$-0.505799\pi$
$547$	−0.852077	−0.0364322	−0.0182161	+	0.999834i	$0.505799\pi$
$548$	0	0
$549$	45.4644	1.94037
$550$	0	0
$551$	0.715092	0.0304640
$552$	0	0
$553$	−29.5972	−1.25860
$554$	0	0
$555$	−20.2253	−0.858516
$556$	0	0
$557$	−35.0509	−1.48515	−0.742576	−	0.669762i	$-0.766396\pi$
$557$	−35.0509	−1.48515	−0.742576	+	0.669762i	$0.766396\pi$
$558$	0	0
$559$	1.52403	0.0644596
$560$	0	0
$561$	−106.795	−4.50890
$562$	0	0
$563$	−32.9907	−1.39039	−0.695196	−	0.718820i	$-0.744683\pi$
$563$	−32.9907	−1.39039	−0.695196	+	0.718820i	$0.744683\pi$
$564$	0	0
$565$	−11.5628	−0.486449
$566$	0	0
$567$	−15.5598	−0.653451
$568$	0	0
$569$	−18.1590	−0.761263	−0.380632	−	0.924727i	$-0.624293\pi$
$569$	−18.1590	−0.761263	−0.380632	+	0.924727i	$0.624293\pi$
$570$	0	0
$571$	29.0313	1.21492	0.607462	−	0.794349i	$-0.292188\pi$
$571$	29.0313	1.21492	0.607462	+	0.794349i	$0.292188\pi$
$572$	0	0
$573$	49.4971	2.06777
$574$	0	0
$575$	−10.4241	−0.434716
$576$	0	0
$577$	15.0405	0.626145	0.313073	−	0.949729i	$-0.398642\pi$
$577$	15.0405	0.626145	0.313073	+	0.949729i	$0.398642\pi$
$578$	0	0
$579$	36.5331	1.51826
$580$	0	0
$581$	−7.38977	−0.306579
$582$	0	0
$583$	−55.0656	−2.28059
$584$	0	0
$585$	−1.86877	−0.0772640
$586$	0	0
$587$	−27.1709	−1.12146	−0.560732	−	0.827997i	$-0.689480\pi$
$587$	−27.1709	−1.12146	−0.560732	+	0.827997i	$0.689480\pi$
$588$	0	0
$589$	4.82802	0.198935
$590$	0	0
$591$	23.3748	0.961512
$592$	0	0
$593$	12.0238	0.493757	0.246878	−	0.969046i	$-0.420595\pi$
$593$	12.0238	0.493757	0.246878	+	0.969046i	$0.420595\pi$
$594$	0	0
$595$	−26.8224	−1.09961
$596$	0	0
$597$	−36.5008	−1.49388
$598$	0	0
$599$	−15.3739	−0.628160	−0.314080	−	0.949396i	$-0.601696\pi$
$599$	−15.3739	−0.628160	−0.314080	+	0.949396i	$0.601696\pi$
$600$	0	0
$601$	−16.1170	−0.657427	−0.328713	−	0.944430i	$-0.606615\pi$
$601$	−16.1170	−0.657427	−0.328713	+	0.944430i	$0.606615\pi$
$602$	0	0
$603$	20.1129	0.819062
$604$	0	0
$605$	44.8781	1.82455
$606$	0	0
$607$	−42.5003	−1.72503	−0.862517	−	0.506029i	$-0.831113\pi$
$607$	−42.5003	−1.72503	−0.862517	+	0.506029i	$0.831113\pi$
$608$	0	0
$609$	−3.76559	−0.152589
$610$	0	0
$611$	3.06703	0.124079
$612$	0	0
$613$	7.01602	0.283374	0.141687	−	0.989911i	$-0.454747\pi$
$613$	7.01602	0.283374	0.141687	+	0.989911i	$0.454747\pi$
$614$	0	0
$615$	−10.4082	−0.419700
$616$	0	0
$617$	−14.1457	−0.569483	−0.284742	−	0.958604i	$-0.591908\pi$
$617$	−14.1457	−0.569483	−0.284742	+	0.958604i	$0.591908\pi$
$618$	0	0
$619$	−9.37865	−0.376960	−0.188480	−	0.982077i	$-0.560356\pi$
$619$	−9.37865	−0.376960	−0.188480	+	0.982077i	$0.560356\pi$
$620$	0	0
$621$	−4.73187	−0.189883
$622$	0	0
$623$	−7.67973	−0.307682
$624$	0	0
$625$	−17.5145	−0.700579
$626$	0	0
$627$	12.9291	0.516340
$628$	0	0
$629$	−30.5598	−1.21850
$630$	0	0
$631$	8.90810	0.354626	0.177313	−	0.984155i	$-0.443260\pi$
$631$	8.90810	0.354626	0.177313	+	0.984155i	$0.443260\pi$
$632$	0	0
$633$	9.84126	0.391155
$634$	0	0
$635$	−8.23553	−0.326817
$636$	0	0
$637$	1.03699	0.0410871
$638$	0	0
$639$	−12.9822	−0.513570
$640$	0	0
$641$	−31.0208	−1.22525	−0.612625	−	0.790374i	$-0.709886\pi$
$641$	−31.0208	−1.22525	−0.612625	+	0.790374i	$0.709886\pi$
$642$	0	0
$643$	36.4005	1.43550	0.717749	−	0.696302i	$-0.245173\pi$
$643$	36.4005	1.43550	0.717749	+	0.696302i	$0.245173\pi$
$644$	0	0
$645$	24.8423	0.978165
$646$	0	0
$647$	−6.09466	−0.239606	−0.119803	−	0.992798i	$-0.538226\pi$
$647$	−6.09466	−0.239606	−0.119803	+	0.992798i	$0.538226\pi$
$648$	0	0
$649$	−82.2738	−3.22953
$650$	0	0
$651$	−25.4238	−0.996436
$652$	0	0
$653$	−25.9660	−1.01613	−0.508063	−	0.861320i	$-0.669638\pi$
$653$	−25.9660	−1.01613	−0.508063	+	0.861320i	$0.669638\pi$
$654$	0	0
$655$	25.2052	0.984849
$656$	0	0
$657$	−25.9868	−1.01384
$658$	0	0
$659$	39.3845	1.53420	0.767102	−	0.641525i	$-0.221698\pi$
$659$	39.3845	1.53420	0.767102	+	0.641525i	$0.221698\pi$
$660$	0	0
$661$	−3.77429	−0.146803	−0.0734015	−	0.997302i	$-0.523385\pi$
$661$	−3.77429	−0.146803	−0.0734015	+	0.997302i	$0.523385\pi$
$662$	0	0
$663$	−5.45475	−0.211845
$664$	0	0
$665$	3.24725	0.125923
$666$	0	0
$667$	6.95520	0.269306
$668$	0	0
$669$	−10.7980	−0.417473
$670$	0	0
$671$	−82.3904	−3.18064
$672$	0	0
$673$	0.143299	0.00552375	0.00276188	−	0.999996i	$-0.499121\pi$
$673$	0.143299	0.00552375	0.00276188	+	0.999996i	$0.499121\pi$
$674$	0	0
$675$	0.660379	0.0254180
$676$	0	0
$677$	47.2278	1.81511	0.907556	−	0.419931i	$-0.137946\pi$
$677$	47.2278	1.81511	0.907556	+	0.419931i	$0.137946\pi$
$678$	0	0
$679$	−15.8860	−0.609648
$680$	0	0
$681$	3.90070	0.149475
$682$	0	0
$683$	−30.6265	−1.17189	−0.585945	−	0.810351i	$-0.699276\pi$
$683$	−30.6265	−1.17189	−0.585945	+	0.810351i	$0.699276\pi$
$684$	0	0
$685$	6.47149	0.247263
$686$	0	0
$687$	59.0754	2.25387
$688$	0	0
$689$	−2.81257	−0.107150
$690$	0	0
$691$	−16.3359	−0.621448	−0.310724	−	0.950500i	$-0.600571\pi$
$691$	−16.3359	−0.621448	−0.310724	+	0.950500i	$0.600571\pi$
$692$	0	0
$693$	−35.2432	−1.33878
$694$	0	0
$695$	−33.9772	−1.28883
$696$	0	0
$697$	−15.7265	−0.595683
$698$	0	0
$699$	−49.3028	−1.86480
$700$	0	0
$701$	−26.1216	−0.986599	−0.493300	−	0.869859i	$-0.664209\pi$
$701$	−26.1216	−0.986599	−0.493300	+	0.869859i	$0.664209\pi$
$702$	0	0
$703$	3.69971	0.139537
$704$	0	0
$705$	49.9938	1.88288
$706$	0	0
$707$	−12.9738	−0.487929
$708$	0	0
$709$	−8.13805	−0.305631	−0.152815	−	0.988255i	$-0.548834\pi$
$709$	−8.13805	−0.305631	−0.152815	+	0.988255i	$0.548834\pi$
$710$	0	0
$711$	−50.7880	−1.90470
$712$	0	0
$713$	46.9588	1.75862
$714$	0	0
$715$	3.38657	0.126650
$716$	0	0
$717$	−5.29137	−0.197610
$718$	0	0
$719$	5.26201	0.196240	0.0981199	−	0.995175i	$-0.468717\pi$
$719$	5.26201	0.196240	0.0981199	+	0.995175i	$0.468717\pi$
$720$	0	0
$721$	−9.96734	−0.371203
$722$	0	0
$723$	−51.8953	−1.93001
$724$	0	0
$725$	−0.970666	−0.0360496
$726$	0	0
$727$	8.93001	0.331196	0.165598	−	0.986193i	$-0.447045\pi$
$727$	8.93001	0.331196	0.165598	+	0.986193i	$0.447045\pi$
$728$	0	0
$729$	−30.7966	−1.14061
$730$	0	0
$731$	37.5359	1.38832
$732$	0	0
$733$	−22.6911	−0.838114	−0.419057	−	0.907960i	$-0.637639\pi$
$733$	−22.6911	−0.838114	−0.419057	+	0.907960i	$0.637639\pi$
$734$	0	0
$735$	16.9034	0.623492
$736$	0	0
$737$	−36.4485	−1.34260
$738$	0	0
$739$	28.2017	1.03742	0.518709	−	0.854951i	$-0.326413\pi$
$739$	28.2017	1.03742	0.518709	+	0.854951i	$0.326413\pi$
$740$	0	0
$741$	0.660377	0.0242596
$742$	0	0
$743$	9.58989	0.351819	0.175910	−	0.984406i	$-0.443713\pi$
$743$	9.58989	0.351819	0.175910	+	0.984406i	$0.443713\pi$
$744$	0	0
$745$	−4.19316	−0.153626
$746$	0	0
$747$	−12.6806	−0.463961
$748$	0	0
$749$	19.4406	0.710344
$750$	0	0
$751$	6.78458	0.247573	0.123786	−	0.992309i	$-0.460496\pi$
$751$	6.78458	0.247573	0.123786	+	0.992309i	$0.460496\pi$
$752$	0	0
$753$	48.9041	1.78216
$754$	0	0
$755$	−17.2750	−0.628703
$756$	0	0
$757$	13.5670	0.493102	0.246551	−	0.969130i	$-0.420703\pi$
$757$	13.5670	0.493102	0.246551	+	0.969130i	$0.420703\pi$
$758$	0	0
$759$	125.753	4.56453
$760$	0	0
$761$	12.5351	0.454396	0.227198	−	0.973849i	$-0.427043\pi$
$761$	12.5351	0.454396	0.227198	+	0.973849i	$0.427043\pi$
$762$	0	0
$763$	20.6442	0.747369
$764$	0	0
$765$	−46.0266	−1.66409
$766$	0	0
$767$	−4.20227	−0.151735
$768$	0	0
$769$	−7.08694	−0.255562	−0.127781	−	0.991802i	$-0.540785\pi$
$769$	−7.08694	−0.255562	−0.127781	+	0.991802i	$0.540785\pi$
$770$	0	0
$771$	−10.4888	−0.377744
$772$	0	0
$773$	−30.5161	−1.09759	−0.548793	−	0.835958i	$-0.684912\pi$
$773$	−30.5161	−1.09759	−0.548793	+	0.835958i	$0.684912\pi$
$774$	0	0
$775$	−6.55356	−0.235411
$776$	0	0
$777$	−19.4822	−0.698920
$778$	0	0
$779$	1.90392	0.0682151
$780$	0	0
$781$	23.5263	0.841839
$782$	0	0
$783$	−0.440619	−0.0157464
$784$	0	0
$785$	−13.1431	−0.469096
$786$	0	0
$787$	9.83380	0.350537	0.175269	−	0.984521i	$-0.443921\pi$
$787$	9.83380	0.350537	0.175269	+	0.984521i	$0.443921\pi$
$788$	0	0
$789$	12.3507	0.439697
$790$	0	0
$791$	−11.1379	−0.396019
$792$	0	0
$793$	−4.20822	−0.149438
$794$	0	0
$795$	−45.8460	−1.62599
$796$	0	0
$797$	−32.6934	−1.15806	−0.579030	−	0.815307i	$-0.696568\pi$
$797$	−32.6934	−1.15806	−0.579030	+	0.815307i	$0.696568\pi$
$798$	0	0
$799$	75.5391	2.67238
$800$	0	0
$801$	−13.1782	−0.465629
$802$	0	0
$803$	47.0931	1.66188
$804$	0	0
$805$	31.5837	1.11318
$806$	0	0
$807$	−44.0806	−1.55171
$808$	0	0
$809$	23.3454	0.820780	0.410390	−	0.911910i	$-0.365392\pi$
$809$	23.3454	0.820780	0.410390	+	0.911910i	$0.365392\pi$
$810$	0	0
$811$	29.0983	1.02178	0.510889	−	0.859646i	$-0.329316\pi$
$811$	29.0983	1.02178	0.510889	+	0.859646i	$0.329316\pi$
$812$	0	0
$813$	16.7371	0.586997
$814$	0	0
$815$	−12.3659	−0.433159
$816$	0	0
$817$	−4.54428	−0.158984
$818$	0	0
$819$	−1.80011	−0.0629008
$820$	0	0
$821$	−12.7977	−0.446644	−0.223322	−	0.974745i	$-0.571690\pi$
$821$	−12.7977	−0.446644	−0.223322	+	0.974745i	$0.571690\pi$
$822$	0	0
$823$	−17.5980	−0.613428	−0.306714	−	0.951802i	$-0.599229\pi$
$823$	−17.5980	−0.613428	−0.306714	+	0.951802i	$0.599229\pi$
$824$	0	0
$825$	−17.5500	−0.611013
$826$	0	0
$827$	−7.92016	−0.275411	−0.137706	−	0.990473i	$-0.543973\pi$
$827$	−7.92016	−0.275411	−0.137706	+	0.990473i	$0.543973\pi$
$828$	0	0
$829$	8.92241	0.309888	0.154944	−	0.987923i	$-0.450480\pi$
$829$	8.92241	0.309888	0.154944	+	0.987923i	$0.450480\pi$
$830$	0	0
$831$	55.9989	1.94258
$832$	0	0
$833$	25.5405	0.884926
$834$	0	0
$835$	44.7065	1.54713
$836$	0	0
$837$	−2.97489	−0.102827
$838$	0	0
$839$	25.0092	0.863415	0.431707	−	0.902014i	$-0.357911\pi$
$839$	25.0092	0.863415	0.431707	+	0.902014i	$0.357911\pi$
$840$	0	0
$841$	−28.3524	−0.977667
$842$	0	0
$843$	−50.2828	−1.73183
$844$	0	0
$845$	−25.1482	−0.865124
$846$	0	0
$847$	43.2292	1.48537
$848$	0	0
$849$	65.7991	2.25822
$850$	0	0
$851$	35.9844	1.23353
$852$	0	0
$853$	49.3810	1.69077	0.845386	−	0.534156i	$-0.179370\pi$
$853$	49.3810	1.69077	0.845386	+	0.534156i	$0.179370\pi$
$854$	0	0
$855$	5.57219	0.190565
$856$	0	0
$857$	−19.6134	−0.669981	−0.334990	−	0.942222i	$-0.608733\pi$
$857$	−19.6134	−0.669981	−0.334990	+	0.942222i	$0.608733\pi$
$858$	0	0
$859$	−49.7275	−1.69668	−0.848340	−	0.529452i	$-0.822398\pi$
$859$	−49.7275	−1.69668	−0.848340	+	0.529452i	$0.822398\pi$
$860$	0	0
$861$	−10.0258	−0.341679
$862$	0	0
$863$	25.8661	0.880493	0.440247	−	0.897877i	$-0.354891\pi$
$863$	25.8661	0.880493	0.440247	+	0.897877i	$0.354891\pi$
$864$	0	0
$865$	47.8174	1.62584
$866$	0	0
$867$	−91.9508	−3.12281
$868$	0	0
$869$	92.0378	3.12217
$870$	0	0
$871$	−1.86167	−0.0630802
$872$	0	0
$873$	−27.2599	−0.922609
$874$	0	0
$875$	−22.6801	−0.766728
$876$	0	0
$877$	12.2543	0.413798	0.206899	−	0.978362i	$-0.433663\pi$
$877$	12.2543	0.413798	0.206899	+	0.978362i	$0.433663\pi$
$878$	0	0
$879$	19.3793	0.653649
$880$	0	0
$881$	29.4082	0.990787	0.495393	−	0.868669i	$-0.335024\pi$
$881$	29.4082	0.990787	0.495393	+	0.868669i	$0.335024\pi$
$882$	0	0
$883$	−28.0532	−0.944064	−0.472032	−	0.881581i	$-0.656479\pi$
$883$	−28.0532	−0.944064	−0.472032	+	0.881581i	$0.656479\pi$
$884$	0	0
$885$	−68.4987	−2.30256
$886$	0	0
$887$	14.5827	0.489640	0.244820	−	0.969569i	$-0.421271\pi$
$887$	14.5827	0.489640	0.244820	+	0.969569i	$0.421271\pi$
$888$	0	0
$889$	−7.93295	−0.266063
$890$	0	0
$891$	48.3860	1.62099
$892$	0	0
$893$	−9.14511	−0.306030
$894$	0	0
$895$	−18.5142	−0.618862
$896$	0	0
$897$	6.42302	0.214458
$898$	0	0
$899$	4.37268	0.145837
$900$	0	0
$901$	−69.2718	−2.30778
$902$	0	0
$903$	23.9296	0.796327
$904$	0	0
$905$	−40.6514	−1.35130
$906$	0	0
$907$	34.1779	1.13486	0.567430	−	0.823422i	$-0.307938\pi$
$907$	34.1779	1.13486	0.567430	+	0.823422i	$0.307938\pi$
$908$	0	0
$909$	−22.2627	−0.738406
$910$	0	0
$911$	−25.0912	−0.831309	−0.415654	−	0.909523i	$-0.636447\pi$
$911$	−25.0912	−0.831309	−0.415654	+	0.909523i	$0.636447\pi$
$912$	0	0
$913$	22.9798	0.760520
$914$	0	0
$915$	−68.5958	−2.26771
$916$	0	0
$917$	24.2791	0.801768
$918$	0	0
$919$	−16.8082	−0.554452	−0.277226	−	0.960805i	$-0.589415\pi$
$919$	−16.8082	−0.554452	−0.277226	+	0.960805i	$0.589415\pi$
$920$	0	0
$921$	68.1155	2.24448
$922$	0	0
$923$	1.20165	0.0395527
$924$	0	0
$925$	−5.02198	−0.165122
$926$	0	0
$927$	−17.1037	−0.561759
$928$	0	0
$929$	7.62631	0.250211	0.125105	−	0.992143i	$-0.460073\pi$
$929$	7.62631	0.250211	0.125105	+	0.992143i	$0.460073\pi$
$930$	0	0
$931$	−3.09205	−0.101338
$932$	0	0
$933$	25.9155	0.848437
$934$	0	0
$935$	83.4091	2.72777
$936$	0	0
$937$	38.6313	1.26203	0.631015	−	0.775771i	$-0.282639\pi$
$937$	38.6313	1.26203	0.631015	+	0.775771i	$0.282639\pi$
$938$	0	0
$939$	21.5157	0.702138
$940$	0	0
$941$	25.9587	0.846230	0.423115	−	0.906076i	$-0.360937\pi$
$941$	25.9587	0.846230	0.423115	+	0.906076i	$0.360937\pi$
$942$	0	0
$943$	18.5181	0.603032
$944$	0	0
$945$	−2.00086	−0.0650880
$946$	0	0
$947$	42.9918	1.39705	0.698523	−	0.715588i	$-0.253841\pi$
$947$	42.9918	1.39705	0.698523	+	0.715588i	$0.253841\pi$
$948$	0	0
$949$	2.40536	0.0780812
$950$	0	0
$951$	−59.3342	−1.92404
$952$	0	0
$953$	48.4871	1.57065	0.785326	−	0.619083i	$-0.212496\pi$
$953$	48.4871	1.57065	0.785326	+	0.619083i	$0.212496\pi$
$954$	0	0
$955$	−38.6581	−1.25095
$956$	0	0
$957$	11.7098	0.378523
$958$	0	0
$959$	6.23372	0.201297
$960$	0	0
$961$	−1.47740	−0.0476579
$962$	0	0
$963$	33.3596	1.07500
$964$	0	0
$965$	−28.5330	−0.918511
$966$	0	0
$967$	32.6038	1.04847	0.524234	−	0.851574i	$-0.324352\pi$
$967$	32.6038	1.04847	0.524234	+	0.851574i	$0.324352\pi$
$968$	0	0
$969$	16.2647	0.522497
$970$	0	0
$971$	21.5755	0.692390	0.346195	−	0.938163i	$-0.387474\pi$
$971$	21.5755	0.692390	0.346195	+	0.938163i	$0.387474\pi$
$972$	0	0
$973$	−32.7288	−1.04924
$974$	0	0
$975$	−0.896396	−0.0287076
$976$	0	0
$977$	3.84983	0.123167	0.0615835	−	0.998102i	$-0.480385\pi$
$977$	3.84983	0.123167	0.0615835	+	0.998102i	$0.480385\pi$
$978$	0	0
$979$	23.8815	0.763256
$980$	0	0
$981$	35.4249	1.13103
$982$	0	0
$983$	16.1448	0.514939	0.257470	−	0.966286i	$-0.417111\pi$
$983$	16.1448	0.514939	0.257470	+	0.966286i	$0.417111\pi$
$984$	0	0
$985$	−18.2562	−0.581690
$986$	0	0
$987$	48.1570	1.53285
$988$	0	0
$989$	−44.1990	−1.40544
$990$	0	0
$991$	−11.3906	−0.361834	−0.180917	−	0.983498i	$-0.557907\pi$
$991$	−11.3906	−0.361834	−0.180917	+	0.983498i	$0.557907\pi$
$992$	0	0
$993$	−41.5773	−1.31942
$994$	0	0
$995$	28.5078	0.903757
$996$	0	0
$997$	−47.5526	−1.50601	−0.753003	−	0.658017i	$-0.771396\pi$
$997$	−47.5526	−1.50601	−0.753003	+	0.658017i	$0.771396\pi$
$998$	0	0
$999$	−2.27965	−0.0721250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.q.1.2		12
4.3	odd	2		1006.2.a.j.1.11	✓	12
12.11	even	2		9054.2.a.bi.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.11	✓	12	4.3	odd	2
8048.2.a.q.1.2		12	1.1	even	1	trivial
9054.2.a.bi.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)

\(f(q)\)	\(=\)	\(q-2.49390 q^{3} +1.94778 q^{5} +1.87622 q^{7} +3.21954 q^{9} +O(q^{10})\)	\(q-2.49390 q^{3} +1.94778 q^{5} +1.87622 q^{7} +3.21954 q^{9} -5.83443 q^{11} -0.298003 q^{13} -4.85757 q^{15} -7.33964 q^{17} +0.888571 q^{19} -4.67910 q^{21} +8.64250 q^{23} -1.20615 q^{25} -0.547512 q^{27} +0.804767 q^{29} +5.43347 q^{31} +14.5505 q^{33} +3.65446 q^{35} +4.16366 q^{37} +0.743190 q^{39} +2.14268 q^{41} -5.11414 q^{43} +6.27096 q^{45} -10.2919 q^{47} -3.47981 q^{49} +18.3043 q^{51} +9.43805 q^{53} -11.3642 q^{55} -2.21601 q^{57} +14.1014 q^{59} +14.1214 q^{61} +6.04056 q^{63} -0.580445 q^{65} +6.24714 q^{67} -21.5535 q^{69} -4.03233 q^{71} -8.07158 q^{73} +3.00801 q^{75} -10.9467 q^{77} -15.7749 q^{79} -8.29318 q^{81} -3.93865 q^{83} -14.2960 q^{85} -2.00701 q^{87} -4.09320 q^{89} -0.559119 q^{91} -13.5505 q^{93} +1.73074 q^{95} -8.46702 q^{97} -18.7842 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

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