LMFDB - Embedded newform 8048.2.a.r.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$0$
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			1.6
Root			$5.07033$ 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)

$f(q)$	$=$	$q+0.163781 q^{3} -2.50185 q^{5} -0.0804128 q^{7} -2.97318 q^{9} +O(q^{10})$	$q+0.163781 q^{3} -2.50185 q^{5} -0.0804128 q^{7} -2.97318 q^{9} +1.12207 q^{11} -6.34257 q^{13} -0.409754 q^{15} +3.88068 q^{17} -6.17439 q^{19} -0.0131701 q^{21} -5.65328 q^{23} +1.25923 q^{25} -0.978290 q^{27} +7.85851 q^{29} +3.07478 q^{31} +0.183773 q^{33} +0.201180 q^{35} -10.6295 q^{37} -1.03879 q^{39} +10.6207 q^{41} -5.76647 q^{43} +7.43843 q^{45} -2.63309 q^{47} -6.99353 q^{49} +0.635579 q^{51} +1.17248 q^{53} -2.80723 q^{55} -1.01124 q^{57} -8.49723 q^{59} -4.59721 q^{61} +0.239081 q^{63} +15.8681 q^{65} -13.0866 q^{67} -0.925897 q^{69} -6.29798 q^{71} -5.82142 q^{73} +0.206237 q^{75} -0.0902285 q^{77} -0.796284 q^{79} +8.75930 q^{81} +7.05177 q^{83} -9.70885 q^{85} +1.28707 q^{87} +0.612804 q^{89} +0.510024 q^{91} +0.503590 q^{93} +15.4474 q^{95} +12.9327 q^{97} -3.33610 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.163781	0.0945587	0.0472794	−	0.998882i	$-0.484945\pi$
$3$	0.163781	0.0945587	0.0472794	+	0.998882i	$0.484945\pi$
$4$	0	0
$5$	−2.50185	−1.11886	−0.559430	−	0.828878i	$-0.688980\pi$
$5$	−2.50185	−1.11886	−0.559430	+	0.828878i	$0.688980\pi$
$6$	0	0
$7$	−0.0804128	−0.0303932	−0.0151966	−	0.999885i	$-0.504837\pi$
$7$	−0.0804128	−0.0303932	−0.0151966	+	0.999885i	$0.504837\pi$
$8$	0	0
$9$	−2.97318	−0.991059
$10$	0	0
$11$	1.12207	0.338316	0.169158	−	0.985589i	$-0.445895\pi$
$11$	1.12207	0.338316	0.169158	+	0.985589i	$0.445895\pi$
$12$	0	0
$13$	−6.34257	−1.75911	−0.879557	−	0.475794i	$-0.842161\pi$
$13$	−6.34257	−1.75911	−0.879557	+	0.475794i	$0.842161\pi$
$14$	0	0
$15$	−0.409754	−0.105798
$16$	0	0
$17$	3.88068	0.941202	0.470601	−	0.882346i	$-0.344037\pi$
$17$	3.88068	0.941202	0.470601	+	0.882346i	$0.344037\pi$
$18$	0	0
$19$	−6.17439	−1.41650	−0.708251	−	0.705961i	$-0.750516\pi$
$19$	−6.17439	−1.41650	−0.708251	+	0.705961i	$0.750516\pi$
$20$	0	0
$21$	−0.0131701	−0.00287394
$22$	0	0
$23$	−5.65328	−1.17879	−0.589395	−	0.807845i	$-0.700634\pi$
$23$	−5.65328	−1.17879	−0.589395	+	0.807845i	$0.700634\pi$
$24$	0	0
$25$	1.25923	0.251846
$26$	0	0
$27$	−0.978290	−0.188272
$28$	0	0
$29$	7.85851	1.45929	0.729644	−	0.683827i	$-0.239686\pi$
$29$	7.85851	1.45929	0.729644	+	0.683827i	$0.239686\pi$
$30$	0	0
$31$	3.07478	0.552247	0.276124	−	0.961122i	$-0.410950\pi$
$31$	3.07478	0.552247	0.276124	+	0.961122i	$0.410950\pi$
$32$	0	0
$33$	0.183773	0.0319907
$34$	0	0
$35$	0.201180	0.0340057
$36$	0	0
$37$	−10.6295	−1.74747	−0.873737	−	0.486398i	$-0.838310\pi$
$37$	−10.6295	−1.74747	−0.873737	+	0.486398i	$0.838310\pi$
$38$	0	0
$39$	−1.03879	−0.166340
$40$	0	0
$41$	10.6207	1.65867	0.829336	−	0.558751i	$-0.188719\pi$
$41$	10.6207	1.65867	0.829336	+	0.558751i	$0.188719\pi$
$42$	0	0
$43$	−5.76647	−0.879378	−0.439689	−	0.898150i	$-0.644911\pi$
$43$	−5.76647	−0.879378	−0.439689	+	0.898150i	$0.644911\pi$
$44$	0	0
$45$	7.43843	1.10886
$46$	0	0
$47$	−2.63309	−0.384075	−0.192038	−	0.981388i	$-0.561510\pi$
$47$	−2.63309	−0.384075	−0.192038	+	0.981388i	$0.561510\pi$
$48$	0	0
$49$	−6.99353	−0.999076
$50$	0	0
$51$	0.635579	0.0889989
$52$	0	0
$53$	1.17248	0.161052	0.0805261	−	0.996752i	$-0.474340\pi$
$53$	1.17248	0.161052	0.0805261	+	0.996752i	$0.474340\pi$
$54$	0	0
$55$	−2.80723	−0.378527
$56$	0	0
$57$	−1.01124	−0.133943
$58$	0	0
$59$	−8.49723	−1.10624	−0.553122	−	0.833100i	$-0.686564\pi$
$59$	−8.49723	−1.10624	−0.553122	+	0.833100i	$0.686564\pi$
$60$	0	0
$61$	−4.59721	−0.588613	−0.294306	−	0.955711i	$-0.595089\pi$
$61$	−4.59721	−0.588613	−0.294306	+	0.955711i	$0.595089\pi$
$62$	0	0
$63$	0.239081	0.0301214
$64$	0	0
$65$	15.8681	1.96820
$66$	0	0
$67$	−13.0866	−1.59878	−0.799391	−	0.600812i	$-0.794844\pi$
$67$	−13.0866	−1.59878	−0.799391	+	0.600812i	$0.794844\pi$
$68$	0	0
$69$	−0.925897	−0.111465
$70$	0	0
$71$	−6.29798	−0.747432	−0.373716	−	0.927543i	$-0.621917\pi$
$71$	−6.29798	−0.747432	−0.373716	+	0.927543i	$0.621917\pi$
$72$	0	0
$73$	−5.82142	−0.681345	−0.340673	−	0.940182i	$-0.610655\pi$
$73$	−5.82142	−0.681345	−0.340673	+	0.940182i	$0.610655\pi$
$74$	0	0
$75$	0.206237	0.0238142
$76$	0	0
$77$	−0.0902285	−0.0102825
$78$	0	0
$79$	−0.796284	−0.0895890	−0.0447945	−	0.998996i	$-0.514263\pi$
$79$	−0.796284	−0.0895890	−0.0447945	+	0.998996i	$0.514263\pi$
$80$	0	0
$81$	8.75930	0.973256
$82$	0	0
$83$	7.05177	0.774032	0.387016	−	0.922073i	$-0.373506\pi$
$83$	7.05177	0.774032	0.387016	+	0.922073i	$0.373506\pi$
$84$	0	0
$85$	−9.70885	−1.05307
$86$	0	0
$87$	1.28707	0.137988
$88$	0	0
$89$	0.612804	0.0649571	0.0324786	−	0.999472i	$-0.489660\pi$
$89$	0.612804	0.0649571	0.0324786	+	0.999472i	$0.489660\pi$
$90$	0	0
$91$	0.510024	0.0534651
$92$	0	0
$93$	0.503590	0.0522198
$94$	0	0
$95$	15.4474	1.58487
$96$	0	0
$97$	12.9327	1.31312	0.656559	−	0.754275i	$-0.272011\pi$
$97$	12.9327	1.31312	0.656559	+	0.754275i	$0.272011\pi$
$98$	0	0
$99$	−3.33610	−0.335291
$100$	0	0
$101$	−10.9451	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$101$	−10.9451	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$102$	0	0
$103$	−16.6080	−1.63644	−0.818219	−	0.574906i	$-0.805039\pi$
$103$	−16.6080	−1.63644	−0.818219	+	0.574906i	$0.805039\pi$
$104$	0	0
$105$	0.0329494	0.00321554
$106$	0	0
$107$	17.8640	1.72698	0.863491	−	0.504364i	$-0.168273\pi$
$107$	17.8640	1.72698	0.863491	+	0.504364i	$0.168273\pi$
$108$	0	0
$109$	13.8099	1.32275	0.661373	−	0.750057i	$-0.269974\pi$
$109$	13.8099	1.32275	0.661373	+	0.750057i	$0.269974\pi$
$110$	0	0
$111$	−1.74090	−0.165239
$112$	0	0
$113$	16.3483	1.53792	0.768959	−	0.639298i	$-0.220775\pi$
$113$	16.3483	1.53792	0.768959	+	0.639298i	$0.220775\pi$
$114$	0	0
$115$	14.1436	1.31890
$116$	0	0
$117$	18.8576	1.74338
$118$	0	0
$119$	−0.312056	−0.0286061
$120$	0	0
$121$	−9.74097	−0.885543
$122$	0	0
$123$	1.73946	0.156842
$124$	0	0
$125$	9.35883	0.837079
$126$	0	0
$127$	21.8774	1.94131	0.970655	−	0.240478i	$-0.0773043\pi$
$127$	21.8774	1.94131	0.970655	+	0.240478i	$0.0773043\pi$
$128$	0	0
$129$	−0.944435	−0.0831529
$130$	0	0
$131$	−4.90997	−0.428986	−0.214493	−	0.976726i	$-0.568810\pi$
$131$	−4.90997	−0.428986	−0.214493	+	0.976726i	$0.568810\pi$
$132$	0	0
$133$	0.496500	0.0430520
$134$	0	0
$135$	2.44753	0.210650
$136$	0	0
$137$	7.90401	0.675285	0.337643	−	0.941274i	$-0.390370\pi$
$137$	7.90401	0.675285	0.337643	+	0.941274i	$0.390370\pi$
$138$	0	0
$139$	2.00923	0.170421	0.0852103	−	0.996363i	$-0.472844\pi$
$139$	2.00923	0.170421	0.0852103	+	0.996363i	$0.472844\pi$
$140$	0	0
$141$	−0.431249	−0.0363177
$142$	0	0
$143$	−7.11678	−0.595135
$144$	0	0
$145$	−19.6608	−1.63274
$146$	0	0
$147$	−1.14540	−0.0944714
$148$	0	0
$149$	5.86246	0.480271	0.240136	−	0.970739i	$-0.422808\pi$
$149$	5.86246	0.480271	0.240136	+	0.970739i	$0.422808\pi$
$150$	0	0
$151$	−17.0262	−1.38557	−0.692785	−	0.721145i	$-0.743616\pi$
$151$	−17.0262	−1.38557	−0.692785	+	0.721145i	$0.743616\pi$
$152$	0	0
$153$	−11.5379	−0.932787
$154$	0	0
$155$	−7.69263	−0.617887
$156$	0	0
$157$	12.3746	0.987597	0.493799	−	0.869576i	$-0.335608\pi$
$157$	12.3746	0.987597	0.493799	+	0.869576i	$0.335608\pi$
$158$	0	0
$159$	0.192029	0.0152289
$160$	0	0
$161$	0.454596	0.0358272
$162$	0	0
$163$	8.58026	0.672058	0.336029	−	0.941852i	$-0.390916\pi$
$163$	8.58026	0.672058	0.336029	+	0.941852i	$0.390916\pi$
$164$	0	0
$165$	−0.459771	−0.0357931
$166$	0	0
$167$	−14.2531	−1.10294	−0.551469	−	0.834196i	$-0.685932\pi$
$167$	−14.2531	−1.10294	−0.551469	+	0.834196i	$0.685932\pi$
$168$	0	0
$169$	27.2282	2.09448
$170$	0	0
$171$	18.3575	1.40384
$172$	0	0
$173$	−13.7182	−1.04298	−0.521489	−	0.853258i	$-0.674623\pi$
$173$	−13.7182	−1.04298	−0.521489	+	0.853258i	$0.674623\pi$
$174$	0	0
$175$	−0.101258	−0.00765440
$176$	0	0
$177$	−1.39168	−0.104605
$178$	0	0
$179$	−0.854287	−0.0638524	−0.0319262	−	0.999490i	$-0.510164\pi$
$179$	−0.854287	−0.0638524	−0.0319262	+	0.999490i	$0.510164\pi$
$180$	0	0
$181$	−5.48292	−0.407542	−0.203771	−	0.979019i	$-0.565320\pi$
$181$	−5.48292	−0.407542	−0.203771	+	0.979019i	$0.565320\pi$
$182$	0	0
$183$	−0.752934	−0.0556585
$184$	0	0
$185$	26.5933	1.95518
$186$	0	0
$187$	4.35437	0.318423
$188$	0	0
$189$	0.0786671	0.00572219
$190$	0	0
$191$	9.66643	0.699438	0.349719	−	0.936855i	$-0.386277\pi$
$191$	9.66643	0.699438	0.349719	+	0.936855i	$0.386277\pi$
$192$	0	0
$193$	15.4395	1.11136	0.555681	−	0.831395i	$-0.312457\pi$
$193$	15.4395	1.11136	0.555681	+	0.831395i	$0.312457\pi$
$194$	0	0
$195$	2.59889	0.186111
$196$	0	0
$197$	−21.9286	−1.56235	−0.781175	−	0.624312i	$-0.785379\pi$
$197$	−21.9286	−1.56235	−0.781175	+	0.624312i	$0.785379\pi$
$198$	0	0
$199$	19.2096	1.36173	0.680867	−	0.732407i	$-0.261603\pi$
$199$	19.2096	1.36173	0.680867	+	0.732407i	$0.261603\pi$
$200$	0	0
$201$	−2.14333	−0.151179
$202$	0	0
$203$	−0.631925	−0.0443524
$204$	0	0
$205$	−26.5713	−1.85582
$206$	0	0
$207$	16.8082	1.16825
$208$	0	0
$209$	−6.92807	−0.479225
$210$	0	0
$211$	−12.5146	−0.861542	−0.430771	−	0.902461i	$-0.641758\pi$
$211$	−12.5146	−0.861542	−0.430771	+	0.902461i	$0.641758\pi$
$212$	0	0
$213$	−1.03149	−0.0706763
$214$	0	0
$215$	14.4268	0.983900
$216$	0	0
$217$	−0.247252	−0.0167846
$218$	0	0
$219$	−0.953435	−0.0644271
$220$	0	0
$221$	−24.6135	−1.65568
$222$	0	0
$223$	18.1402	1.21476	0.607380	−	0.794412i	$-0.292221\pi$
$223$	18.1402	1.21476	0.607380	+	0.794412i	$0.292221\pi$
$224$	0	0
$225$	−3.74391	−0.249594
$226$	0	0
$227$	23.3926	1.55262	0.776311	−	0.630351i	$-0.217089\pi$
$227$	23.3926	1.55262	0.776311	+	0.630351i	$0.217089\pi$
$228$	0	0
$229$	5.89814	0.389760	0.194880	−	0.980827i	$-0.437568\pi$
$229$	5.89814	0.389760	0.194880	+	0.980827i	$0.437568\pi$
$230$	0	0
$231$	−0.0147777	−0.000972299 0
$232$	0	0
$233$	14.6507	0.959800	0.479900	−	0.877323i	$-0.340673\pi$
$233$	14.6507	0.959800	0.479900	+	0.877323i	$0.340673\pi$
$234$	0	0
$235$	6.58758	0.429726
$236$	0	0
$237$	−0.130416	−0.00847142
$238$	0	0
$239$	−25.7852	−1.66791	−0.833954	−	0.551834i	$-0.813928\pi$
$239$	−25.7852	−1.66791	−0.833954	+	0.551834i	$0.813928\pi$
$240$	0	0
$241$	10.8074	0.696166	0.348083	−	0.937464i	$-0.386833\pi$
$241$	10.8074	0.696166	0.348083	+	0.937464i	$0.386833\pi$
$242$	0	0
$243$	4.36947	0.280302
$244$	0	0
$245$	17.4967	1.11783
$246$	0	0
$247$	39.1615	2.49179
$248$	0	0
$249$	1.15494	0.0731915
$250$	0	0
$251$	−2.00333	−0.126449	−0.0632245	−	0.997999i	$-0.520138\pi$
$251$	−2.00333	−0.126449	−0.0632245	+	0.997999i	$0.520138\pi$
$252$	0	0
$253$	−6.34335	−0.398803
$254$	0	0
$255$	−1.59012	−0.0995773
$256$	0	0
$257$	−17.9776	−1.12141	−0.560707	−	0.828014i	$-0.689471\pi$
$257$	−17.9776	−1.12141	−0.560707	+	0.828014i	$0.689471\pi$
$258$	0	0
$259$	0.854746	0.0531113
$260$	0	0
$261$	−23.3647	−1.44624
$262$	0	0
$263$	12.6417	0.779519	0.389760	−	0.920917i	$-0.372558\pi$
$263$	12.6417	0.779519	0.389760	+	0.920917i	$0.372558\pi$
$264$	0	0
$265$	−2.93336	−0.180195
$266$	0	0
$267$	0.100365	0.00614226
$268$	0	0
$269$	−16.8944	−1.03007	−0.515034	−	0.857170i	$-0.672221\pi$
$269$	−16.8944	−1.03007	−0.515034	+	0.857170i	$0.672221\pi$
$270$	0	0
$271$	−19.8240	−1.20422	−0.602112	−	0.798411i	$-0.705674\pi$
$271$	−19.8240	−1.20422	−0.602112	+	0.798411i	$0.705674\pi$
$272$	0	0
$273$	0.0835320	0.00505559
$274$	0	0
$275$	1.41294	0.0852034
$276$	0	0
$277$	4.63829	0.278688	0.139344	−	0.990244i	$-0.455501\pi$
$277$	4.63829	0.278688	0.139344	+	0.990244i	$0.455501\pi$
$278$	0	0
$279$	−9.14187	−0.547310
$280$	0	0
$281$	−8.59241	−0.512580	−0.256290	−	0.966600i	$-0.582500\pi$
$281$	−8.59241	−0.512580	−0.256290	+	0.966600i	$0.582500\pi$
$282$	0	0
$283$	−5.67891	−0.337576	−0.168788	−	0.985652i	$-0.553985\pi$
$283$	−5.67891	−0.337576	−0.168788	+	0.985652i	$0.553985\pi$
$284$	0	0
$285$	2.52998	0.149863
$286$	0	0
$287$	−0.854038	−0.0504123
$288$	0	0
$289$	−1.94035	−0.114138
$290$	0	0
$291$	2.11813	0.124167
$292$	0	0
$293$	0.185824	0.0108560	0.00542798	−	0.999985i	$-0.498272\pi$
$293$	0.185824	0.0108560	0.00542798	+	0.999985i	$0.498272\pi$
$294$	0	0
$295$	21.2588	1.23773
$296$	0	0
$297$	−1.09771	−0.0636953
$298$	0	0
$299$	35.8563	2.07362
$300$	0	0
$301$	0.463698	0.0267271
$302$	0	0
$303$	−1.79259	−0.102982
$304$	0	0
$305$	11.5015	0.658575
$306$	0	0
$307$	26.3840	1.50581	0.752907	−	0.658127i	$-0.228651\pi$
$307$	26.3840	1.50581	0.752907	+	0.658127i	$0.228651\pi$
$308$	0	0
$309$	−2.72007	−0.154740
$310$	0	0
$311$	−25.9322	−1.47048	−0.735240	−	0.677806i	$-0.762931\pi$
$311$	−25.9322	−1.47048	−0.735240	+	0.677806i	$0.762931\pi$
$312$	0	0
$313$	8.14947	0.460635	0.230318	−	0.973115i	$-0.426023\pi$
$313$	8.14947	0.460635	0.230318	+	0.973115i	$0.426023\pi$
$314$	0	0
$315$	−0.598145	−0.0337016
$316$	0	0
$317$	19.5975	1.10071	0.550353	−	0.834932i	$-0.314493\pi$
$317$	19.5975	1.10071	0.550353	+	0.834932i	$0.314493\pi$
$318$	0	0
$319$	8.81776	0.493700
$320$	0	0
$321$	2.92578	0.163301
$322$	0	0
$323$	−23.9608	−1.33322
$324$	0	0
$325$	−7.98676	−0.443026
$326$	0	0
$327$	2.26179	0.125077
$328$	0	0
$329$	0.211734	0.0116733
$330$	0	0
$331$	−31.5281	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$331$	−31.5281	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$332$	0	0
$333$	31.6033	1.73185
$334$	0	0
$335$	32.7406	1.78881
$336$	0	0
$337$	5.44276	0.296486	0.148243	−	0.988951i	$-0.452638\pi$
$337$	5.44276	0.296486	0.148243	+	0.988951i	$0.452638\pi$
$338$	0	0
$339$	2.67753	0.145424
$340$	0	0
$341$	3.45011	0.186834
$342$	0	0
$343$	1.12526	0.0607583
$344$	0	0
$345$	2.31645	0.124713
$346$	0	0
$347$	20.9996	1.12732	0.563659	−	0.826007i	$-0.309393\pi$
$347$	20.9996	1.12732	0.563659	+	0.826007i	$0.309393\pi$
$348$	0	0
$349$	11.8500	0.634315	0.317158	−	0.948373i	$-0.397272\pi$
$349$	11.8500	0.634315	0.317158	+	0.948373i	$0.397272\pi$
$350$	0	0
$351$	6.20488	0.331192
$352$	0	0
$353$	0.0695152	0.00369992	0.00184996	−	0.999998i	$-0.499411\pi$
$353$	0.0695152	0.00369992	0.00184996	+	0.999998i	$0.499411\pi$
$354$	0	0
$355$	15.7566	0.836271
$356$	0	0
$357$	−0.0511087	−0.00270496
$358$	0	0
$359$	19.6673	1.03800	0.519000	−	0.854774i	$-0.326305\pi$
$359$	19.6673	1.03800	0.519000	+	0.854774i	$0.326305\pi$
$360$	0	0
$361$	19.1231	1.00648
$362$	0	0
$363$	−1.59538	−0.0837358
$364$	0	0
$365$	14.5643	0.762329
$366$	0	0
$367$	−18.5255	−0.967026	−0.483513	−	0.875337i	$-0.660639\pi$
$367$	−18.5255	−0.967026	−0.483513	+	0.875337i	$0.660639\pi$
$368$	0	0
$369$	−31.5771	−1.64384
$370$	0	0
$371$	−0.0942823	−0.00489489
$372$	0	0
$373$	11.9756	0.620075	0.310037	−	0.950724i	$-0.399658\pi$
$373$	11.9756	0.620075	0.310037	+	0.950724i	$0.399658\pi$
$374$	0	0
$375$	1.53279	0.0791531
$376$	0	0
$377$	−49.8432	−2.56705
$378$	0	0
$379$	16.2323	0.833800	0.416900	−	0.908952i	$-0.363117\pi$
$379$	16.2323	0.833800	0.416900	+	0.908952i	$0.363117\pi$
$380$	0	0
$381$	3.58310	0.183568
$382$	0	0
$383$	0.855877	0.0437332	0.0218666	−	0.999761i	$-0.493039\pi$
$383$	0.855877	0.0437332	0.0218666	+	0.999761i	$0.493039\pi$
$384$	0	0
$385$	0.225738	0.0115047
$386$	0	0
$387$	17.1447	0.871515
$388$	0	0
$389$	28.4630	1.44313	0.721565	−	0.692347i	$-0.243423\pi$
$389$	28.4630	1.44313	0.721565	+	0.692347i	$0.243423\pi$
$390$	0	0
$391$	−21.9385	−1.10948
$392$	0	0
$393$	−0.804158	−0.0405644
$394$	0	0
$395$	1.99218	0.100237
$396$	0	0
$397$	−16.5632	−0.831283	−0.415642	−	0.909528i	$-0.636443\pi$
$397$	−16.5632	−0.831283	−0.415642	+	0.909528i	$0.636443\pi$
$398$	0	0
$399$	0.0813170	0.00407094
$400$	0	0
$401$	10.1792	0.508323	0.254162	−	0.967162i	$-0.418200\pi$
$401$	10.1792	0.508323	0.254162	+	0.967162i	$0.418200\pi$
$402$	0	0
$403$	−19.5020	−0.971466
$404$	0	0
$405$	−21.9144	−1.08894
$406$	0	0
$407$	−11.9270	−0.591198
$408$	0	0
$409$	28.1040	1.38965	0.694826	−	0.719178i	$-0.255481\pi$
$409$	28.1040	1.38965	0.694826	+	0.719178i	$0.255481\pi$
$410$	0	0
$411$	1.29452	0.0638541
$412$	0	0
$413$	0.683286	0.0336223
$414$	0	0
$415$	−17.6424	−0.866033
$416$	0	0
$417$	0.329073	0.0161148
$418$	0	0
$419$	−25.9397	−1.26724	−0.633619	−	0.773645i	$-0.718431\pi$
$419$	−25.9397	−1.26724	−0.633619	+	0.773645i	$0.718431\pi$
$420$	0	0
$421$	−9.42831	−0.459508	−0.229754	−	0.973249i	$-0.573792\pi$
$421$	−9.42831	−0.459508	−0.229754	+	0.973249i	$0.573792\pi$
$422$	0	0
$423$	7.82863	0.380641
$424$	0	0
$425$	4.88667	0.237038
$426$	0	0
$427$	0.369675	0.0178898
$428$	0	0
$429$	−1.16559	−0.0562753
$430$	0	0
$431$	−35.7533	−1.72218	−0.861088	−	0.508456i	$-0.830216\pi$
$431$	−35.7533	−1.72218	−0.861088	+	0.508456i	$0.830216\pi$
$432$	0	0
$433$	5.86217	0.281718	0.140859	−	0.990030i	$-0.455014\pi$
$433$	5.86217	0.281718	0.140859	+	0.990030i	$0.455014\pi$
$434$	0	0
$435$	−3.22005	−0.154390
$436$	0	0
$437$	34.9055	1.66976
$438$	0	0
$439$	34.2047	1.63250	0.816251	−	0.577698i	$-0.196049\pi$
$439$	34.2047	1.63250	0.816251	+	0.577698i	$0.196049\pi$
$440$	0	0
$441$	20.7930	0.990143
$442$	0	0
$443$	−32.9124	−1.56371	−0.781857	−	0.623457i	$-0.785727\pi$
$443$	−32.9124	−1.56371	−0.781857	+	0.623457i	$0.785727\pi$
$444$	0	0
$445$	−1.53314	−0.0726779
$446$	0	0
$447$	0.960157	0.0454138
$448$	0	0
$449$	0.135448	0.00639219	0.00319610	−	0.999995i	$-0.498983\pi$
$449$	0.135448	0.00639219	0.00319610	+	0.999995i	$0.498983\pi$
$450$	0	0
$451$	11.9171	0.561154
$452$	0	0
$453$	−2.78855	−0.131018
$454$	0	0
$455$	−1.27600	−0.0598199
$456$	0	0
$457$	14.2149	0.664944	0.332472	−	0.943113i	$-0.392117\pi$
$457$	14.2149	0.664944	0.332472	+	0.943113i	$0.392117\pi$
$458$	0	0
$459$	−3.79643	−0.177202
$460$	0	0
$461$	29.8788	1.39160	0.695798	−	0.718237i	$-0.255051\pi$
$461$	29.8788	1.39160	0.695798	+	0.718237i	$0.255051\pi$
$462$	0	0
$463$	34.9796	1.62564	0.812820	−	0.582515i	$-0.197931\pi$
$463$	34.9796	1.62564	0.812820	+	0.582515i	$0.197931\pi$
$464$	0	0
$465$	−1.25990	−0.0584266
$466$	0	0
$467$	38.7257	1.79201	0.896005	−	0.444043i	$-0.146456\pi$
$467$	38.7257	1.79201	0.896005	+	0.444043i	$0.146456\pi$
$468$	0	0
$469$	1.05233	0.0485920
$470$	0	0
$471$	2.02671	0.0933860
$472$	0	0
$473$	−6.47035	−0.297507
$474$	0	0
$475$	−7.77498	−0.356740
$476$	0	0
$477$	−3.48598	−0.159612
$478$	0	0
$479$	−16.4719	−0.752622	−0.376311	−	0.926493i	$-0.622808\pi$
$479$	−16.4719	−0.752622	−0.376311	+	0.926493i	$0.622808\pi$
$480$	0	0
$481$	67.4182	3.07401
$482$	0	0
$483$	0.0744539	0.00338777
$484$	0	0
$485$	−32.3556	−1.46919
$486$	0	0
$487$	2.13819	0.0968904	0.0484452	−	0.998826i	$-0.484573\pi$
$487$	2.13819	0.0968904	0.0484452	+	0.998826i	$0.484573\pi$
$488$	0	0
$489$	1.40528	0.0635490
$490$	0	0
$491$	2.18448	0.0985844	0.0492922	−	0.998784i	$-0.484303\pi$
$491$	2.18448	0.0985844	0.0492922	+	0.998784i	$0.484303\pi$
$492$	0	0
$493$	30.4963	1.37349
$494$	0	0
$495$	8.34640	0.375143
$496$	0	0
$497$	0.506438	0.0227168
$498$	0	0
$499$	−40.9346	−1.83248	−0.916242	−	0.400625i	$-0.868793\pi$
$499$	−40.9346	−1.83248	−0.916242	+	0.400625i	$0.868793\pi$
$500$	0	0
$501$	−2.33438	−0.104292
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	27.3829	1.21852
$506$	0	0
$507$	4.45945	0.198051
$508$	0	0
$509$	−14.2830	−0.633084	−0.316542	−	0.948578i	$-0.602522\pi$
$509$	−14.2830	−0.633084	−0.316542	+	0.948578i	$0.602522\pi$
$510$	0	0
$511$	0.468116	0.0207082
$512$	0	0
$513$	6.04034	0.266688
$514$	0	0
$515$	41.5507	1.83094
$516$	0	0
$517$	−2.95450	−0.129939
$518$	0	0
$519$	−2.24678	−0.0986228
$520$	0	0
$521$	17.3885	0.761805	0.380902	−	0.924615i	$-0.375613\pi$
$521$	17.3885	0.761805	0.380902	+	0.924615i	$0.375613\pi$
$522$	0	0
$523$	−34.2938	−1.49956	−0.749782	−	0.661685i	$-0.769842\pi$
$523$	−34.2938	−1.49956	−0.749782	+	0.661685i	$0.769842\pi$
$524$	0	0
$525$	−0.0165841	−0.000723791 0
$526$	0	0
$527$	11.9322	0.519777
$528$	0	0
$529$	8.95952	0.389544
$530$	0	0
$531$	25.2638	1.09635
$532$	0	0
$533$	−67.3624	−2.91779
$534$	0	0
$535$	−44.6931	−1.93225
$536$	0	0
$537$	−0.139916	−0.00603780
$538$	0	0
$539$	−7.84720	−0.338003
$540$	0	0
$541$	−27.7109	−1.19138	−0.595692	−	0.803213i	$-0.703122\pi$
$541$	−27.7109	−1.19138	−0.595692	+	0.803213i	$0.703122\pi$
$542$	0	0
$543$	−0.897996	−0.0385367
$544$	0	0
$545$	−34.5502	−1.47997
$546$	0	0
$547$	23.8394	1.01930	0.509649	−	0.860382i	$-0.329775\pi$
$547$	23.8394	1.01930	0.509649	+	0.860382i	$0.329775\pi$
$548$	0	0
$549$	13.6683	0.583350
$550$	0	0
$551$	−48.5215	−2.06708
$552$	0	0
$553$	0.0640315	0.00272289
$554$	0	0
$555$	4.35546	0.184879
$556$	0	0
$557$	17.2842	0.732354	0.366177	−	0.930545i	$-0.380667\pi$
$557$	17.2842	0.732354	0.366177	+	0.930545i	$0.380667\pi$
$558$	0	0
$559$	36.5742	1.54693
$560$	0	0
$561$	0.713162	0.0301097
$562$	0	0
$563$	−4.65238	−0.196074	−0.0980372	−	0.995183i	$-0.531256\pi$
$563$	−4.65238	−0.196074	−0.0980372	+	0.995183i	$0.531256\pi$
$564$	0	0
$565$	−40.9009	−1.72071
$566$	0	0
$567$	−0.704360	−0.0295803
$568$	0	0
$569$	25.8946	1.08556	0.542780	−	0.839875i	$-0.317372\pi$
$569$	25.8946	1.08556	0.542780	+	0.839875i	$0.317372\pi$
$570$	0	0
$571$	−14.2330	−0.595632	−0.297816	−	0.954623i	$-0.596258\pi$
$571$	−14.2330	−0.595632	−0.297816	+	0.954623i	$0.596258\pi$
$572$	0	0
$573$	1.58317	0.0661380
$574$	0	0
$575$	−7.11877	−0.296873
$576$	0	0
$577$	−15.2149	−0.633404	−0.316702	−	0.948525i	$-0.602576\pi$
$577$	−15.2149	−0.633404	−0.316702	+	0.948525i	$0.602576\pi$
$578$	0	0
$579$	2.52870	0.105089
$580$	0	0
$581$	−0.567053	−0.0235253
$582$	0	0
$583$	1.31560	0.0544865
$584$	0	0
$585$	−47.1788	−1.95060
$586$	0	0
$587$	4.10658	0.169497	0.0847484	−	0.996402i	$-0.472991\pi$
$587$	4.10658	0.169497	0.0847484	+	0.996402i	$0.472991\pi$
$588$	0	0
$589$	−18.9849	−0.782260
$590$	0	0
$591$	−3.59148	−0.147734
$592$	0	0
$593$	38.8378	1.59488	0.797439	−	0.603399i	$-0.206188\pi$
$593$	38.8378	1.59488	0.797439	+	0.603399i	$0.206188\pi$
$594$	0	0
$595$	0.780716	0.0320062
$596$	0	0
$597$	3.14616	0.128764
$598$	0	0
$599$	5.00282	0.204410	0.102205	−	0.994763i	$-0.467410\pi$
$599$	5.00282	0.204410	0.102205	+	0.994763i	$0.467410\pi$
$600$	0	0
$601$	−25.5387	−1.04174	−0.520872	−	0.853635i	$-0.674393\pi$
$601$	−25.5387	−1.04174	−0.520872	+	0.853635i	$0.674393\pi$
$602$	0	0
$603$	38.9087	1.58449
$604$	0	0
$605$	24.3704	0.990798
$606$	0	0
$607$	23.9276	0.971191	0.485596	−	0.874183i	$-0.338603\pi$
$607$	23.9276	0.971191	0.485596	+	0.874183i	$0.338603\pi$
$608$	0	0
$609$	−0.103497	−0.00419391
$610$	0	0
$611$	16.7006	0.675632
$612$	0	0
$613$	17.9870	0.726489	0.363244	−	0.931694i	$-0.381669\pi$
$613$	17.9870	0.726489	0.363244	+	0.931694i	$0.381669\pi$
$614$	0	0
$615$	−4.35186	−0.175484
$616$	0	0
$617$	6.00486	0.241747	0.120873	−	0.992668i	$-0.461431\pi$
$617$	6.00486	0.241747	0.120873	+	0.992668i	$0.461431\pi$
$618$	0	0
$619$	26.7179	1.07388	0.536941	−	0.843620i	$-0.319580\pi$
$619$	26.7179	1.07388	0.536941	+	0.843620i	$0.319580\pi$
$620$	0	0
$621$	5.53054	0.221933
$622$	0	0
$623$	−0.0492773	−0.00197425
$624$	0	0
$625$	−29.7105	−1.18842
$626$	0	0
$627$	−1.13468	−0.0453149
$628$	0	0
$629$	−41.2495	−1.64473
$630$	0	0
$631$	−22.3491	−0.889703	−0.444852	−	0.895604i	$-0.646744\pi$
$631$	−22.3491	−0.889703	−0.444852	+	0.895604i	$0.646744\pi$
$632$	0	0
$633$	−2.04965	−0.0814664
$634$	0	0
$635$	−54.7340	−2.17205
$636$	0	0
$637$	44.3570	1.75749
$638$	0	0
$639$	18.7250	0.740749
$640$	0	0
$641$	1.48370	0.0586026	0.0293013	−	0.999571i	$-0.490672\pi$
$641$	1.48370	0.0586026	0.0293013	+	0.999571i	$0.490672\pi$
$642$	0	0
$643$	17.3670	0.684887	0.342444	−	0.939538i	$-0.388745\pi$
$643$	17.3670	0.684887	0.342444	+	0.939538i	$0.388745\pi$
$644$	0	0
$645$	2.36283	0.0930364
$646$	0	0
$647$	−18.9918	−0.746643	−0.373321	−	0.927702i	$-0.621781\pi$
$647$	−18.9918	−0.746643	−0.373321	+	0.927702i	$0.621781\pi$
$648$	0	0
$649$	−9.53445	−0.374260
$650$	0	0
$651$	−0.0404951	−0.00158713
$652$	0	0
$653$	−31.0643	−1.21564	−0.607819	−	0.794075i	$-0.707955\pi$
$653$	−31.0643	−1.21564	−0.607819	+	0.794075i	$0.707955\pi$
$654$	0	0
$655$	12.2840	0.479975
$656$	0	0
$657$	17.3081	0.675253
$658$	0	0
$659$	12.4002	0.483043	0.241522	−	0.970395i	$-0.422354\pi$
$659$	12.4002	0.483043	0.241522	+	0.970395i	$0.422354\pi$
$660$	0	0
$661$	−8.50300	−0.330728	−0.165364	−	0.986233i	$-0.552880\pi$
$661$	−8.50300	−0.330728	−0.165364	+	0.986233i	$0.552880\pi$
$662$	0	0
$663$	−4.03121	−0.156559
$664$	0	0
$665$	−1.24217	−0.0481691
$666$	0	0
$667$	−44.4263	−1.72019
$668$	0	0
$669$	2.97102	0.114866
$670$	0	0
$671$	−5.15838	−0.199137
$672$	0	0
$673$	−8.38548	−0.323237	−0.161618	−	0.986853i	$-0.551671\pi$
$673$	−8.38548	−0.323237	−0.161618	+	0.986853i	$0.551671\pi$
$674$	0	0
$675$	−1.23189	−0.0474156
$676$	0	0
$677$	47.4911	1.82523	0.912615	−	0.408820i	$-0.134060\pi$
$677$	47.4911	1.82523	0.912615	+	0.408820i	$0.134060\pi$
$678$	0	0
$679$	−1.03996	−0.0399098
$680$	0	0
$681$	3.83125	0.146814
$682$	0	0
$683$	−39.2994	−1.50375	−0.751875	−	0.659306i	$-0.770850\pi$
$683$	−39.2994	−1.50375	−0.751875	+	0.659306i	$0.770850\pi$
$684$	0	0
$685$	−19.7746	−0.755549
$686$	0	0
$687$	0.966000	0.0368552
$688$	0	0
$689$	−7.43653	−0.283309
$690$	0	0
$691$	−11.8099	−0.449270	−0.224635	−	0.974443i	$-0.572119\pi$
$691$	−11.8099	−0.449270	−0.224635	+	0.974443i	$0.572119\pi$
$692$	0	0
$693$	0.268265	0.0101905
$694$	0	0
$695$	−5.02678	−0.190677
$696$	0	0
$697$	41.2154	1.56115
$698$	0	0
$699$	2.39950	0.0907574
$700$	0	0
$701$	28.8440	1.08942	0.544711	−	0.838624i	$-0.316639\pi$
$701$	28.8440	1.08942	0.544711	+	0.838624i	$0.316639\pi$
$702$	0	0
$703$	65.6305	2.47530
$704$	0	0
$705$	1.07892	0.0406344
$706$	0	0
$707$	0.880123	0.0331004
$708$	0	0
$709$	−16.1118	−0.605093	−0.302547	−	0.953135i	$-0.597837\pi$
$709$	−16.1118	−0.605093	−0.302547	+	0.953135i	$0.597837\pi$
$710$	0	0
$711$	2.36749	0.0887879
$712$	0	0
$713$	−17.3826	−0.650983
$714$	0	0
$715$	17.8051	0.665873
$716$	0	0
$717$	−4.22312	−0.157715
$718$	0	0
$719$	−9.35622	−0.348928	−0.174464	−	0.984664i	$-0.555819\pi$
$719$	−9.35622	−0.348928	−0.174464	+	0.984664i	$0.555819\pi$
$720$	0	0
$721$	1.33550	0.0497366
$722$	0	0
$723$	1.77004	0.0658285
$724$	0	0
$725$	9.89567	0.367516
$726$	0	0
$727$	49.8831	1.85006	0.925031	−	0.379893i	$-0.124039\pi$
$727$	49.8831	1.85006	0.925031	+	0.379893i	$0.124039\pi$
$728$	0	0
$729$	−25.5623	−0.946751
$730$	0	0
$731$	−22.3778	−0.827673
$732$	0	0
$733$	−28.6410	−1.05788	−0.528939	−	0.848660i	$-0.677410\pi$
$733$	−28.6410	−1.05788	−0.528939	+	0.848660i	$0.677410\pi$
$734$	0	0
$735$	2.86563	0.105700
$736$	0	0
$737$	−14.6840	−0.540893
$738$	0	0
$739$	0.582234	0.0214178	0.0107089	−	0.999943i	$-0.496591\pi$
$739$	0.582234	0.0214178	0.0107089	+	0.999943i	$0.496591\pi$
$740$	0	0
$741$	6.41389	0.235620
$742$	0	0
$743$	−32.1567	−1.17971	−0.589857	−	0.807508i	$-0.700816\pi$
$743$	−32.1567	−1.17971	−0.589857	+	0.807508i	$0.700816\pi$
$744$	0	0
$745$	−14.6670	−0.537356
$746$	0	0
$747$	−20.9662	−0.767112
$748$	0	0
$749$	−1.43650	−0.0524885
$750$	0	0
$751$	18.7747	0.685100	0.342550	−	0.939500i	$-0.388709\pi$
$751$	18.7747	0.685100	0.342550	+	0.939500i	$0.388709\pi$
$752$	0	0
$753$	−0.328107	−0.0119569
$754$	0	0
$755$	42.5968	1.55026
$756$	0	0
$757$	42.1066	1.53039	0.765195	−	0.643799i	$-0.222643\pi$
$757$	42.1066	1.53039	0.765195	+	0.643799i	$0.222643\pi$
$758$	0	0
$759$	−1.03892	−0.0377103
$760$	0	0
$761$	48.7456	1.76703	0.883513	−	0.468406i	$-0.155172\pi$
$761$	48.7456	1.76703	0.883513	+	0.468406i	$0.155172\pi$
$762$	0	0
$763$	−1.11049	−0.0402025
$764$	0	0
$765$	28.8661	1.04366
$766$	0	0
$767$	53.8943	1.94601
$768$	0	0
$769$	−7.14665	−0.257715	−0.128857	−	0.991663i	$-0.541131\pi$
$769$	−7.14665	−0.257715	−0.128857	+	0.991663i	$0.541131\pi$
$770$	0	0
$771$	−2.94439	−0.106039
$772$	0	0
$773$	−7.07242	−0.254377	−0.127189	−	0.991879i	$-0.540595\pi$
$773$	−7.07242	−0.254377	−0.127189	+	0.991879i	$0.540595\pi$
$774$	0	0
$775$	3.87186	0.139081
$776$	0	0
$777$	0.139991	0.00502214
$778$	0	0
$779$	−65.5762	−2.34951
$780$	0	0
$781$	−7.06674	−0.252868
$782$	0	0
$783$	−7.68790	−0.274743
$784$	0	0
$785$	−30.9592	−1.10498
$786$	0	0
$787$	−0.432047	−0.0154008	−0.00770041	−	0.999970i	$-0.502451\pi$
$787$	−0.432047	−0.0154008	−0.00770041	+	0.999970i	$0.502451\pi$
$788$	0	0
$789$	2.07046	0.0737104
$790$	0	0
$791$	−1.31461	−0.0467423
$792$	0	0
$793$	29.1582	1.03544
$794$	0	0
$795$	−0.480427	−0.0170390
$796$	0	0
$797$	−48.7184	−1.72569	−0.862847	−	0.505465i	$-0.831321\pi$
$797$	−48.7184	−1.72569	−0.862847	+	0.505465i	$0.831321\pi$
$798$	0	0
$799$	−10.2182	−0.361493
$800$	0	0
$801$	−1.82197	−0.0643763
$802$	0	0
$803$	−6.53201	−0.230510
$804$	0	0
$805$	−1.13733	−0.0400856
$806$	0	0
$807$	−2.76697	−0.0974019
$808$	0	0
$809$	47.7550	1.67898	0.839488	−	0.543378i	$-0.182855\pi$
$809$	47.7550	1.67898	0.839488	+	0.543378i	$0.182855\pi$
$810$	0	0
$811$	−14.7191	−0.516858	−0.258429	−	0.966030i	$-0.583205\pi$
$811$	−14.7191	−0.516858	−0.258429	+	0.966030i	$0.583205\pi$
$812$	0	0
$813$	−3.24679	−0.113870
$814$	0	0
$815$	−21.4665	−0.751938
$816$	0	0
$817$	35.6044	1.24564
$818$	0	0
$819$	−1.51639	−0.0529870
$820$	0	0
$821$	−6.00425	−0.209550	−0.104775	−	0.994496i	$-0.533412\pi$
$821$	−6.00425	−0.209550	−0.104775	+	0.994496i	$0.533412\pi$
$822$	0	0
$823$	−50.2093	−1.75019	−0.875093	−	0.483955i	$-0.839200\pi$
$823$	−50.2093	−1.75019	−0.875093	+	0.483955i	$0.839200\pi$
$824$	0	0
$825$	0.231412	0.00805673
$826$	0	0
$827$	7.53025	0.261852	0.130926	−	0.991392i	$-0.458205\pi$
$827$	7.53025	0.261852	0.130926	+	0.991392i	$0.458205\pi$
$828$	0	0
$829$	−8.54363	−0.296732	−0.148366	−	0.988932i	$-0.547401\pi$
$829$	−8.54363	−0.296732	−0.148366	+	0.988932i	$0.547401\pi$
$830$	0	0
$831$	0.759662	0.0263524
$832$	0	0
$833$	−27.1396	−0.940333
$834$	0	0
$835$	35.6590	1.23403
$836$	0	0
$837$	−3.00803	−0.103973
$838$	0	0
$839$	9.76695	0.337193	0.168596	−	0.985685i	$-0.446077\pi$
$839$	9.76695	0.337193	0.168596	+	0.985685i	$0.446077\pi$
$840$	0	0
$841$	32.7562	1.12952
$842$	0	0
$843$	−1.40727	−0.0484690
$844$	0	0
$845$	−68.1208	−2.34343
$846$	0	0
$847$	0.783299	0.0269145
$848$	0	0
$849$	−0.930095	−0.0319208
$850$	0	0
$851$	60.0913	2.05990
$852$	0	0
$853$	3.04251	0.104173	0.0520867	−	0.998643i	$-0.483413\pi$
$853$	3.04251	0.104173	0.0520867	+	0.998643i	$0.483413\pi$
$854$	0	0
$855$	−45.9277	−1.57070
$856$	0	0
$857$	−37.7097	−1.28814	−0.644069	−	0.764967i	$-0.722755\pi$
$857$	−37.7097	−1.28814	−0.644069	+	0.764967i	$0.722755\pi$
$858$	0	0
$859$	−22.2886	−0.760478	−0.380239	−	0.924888i	$-0.624158\pi$
$859$	−22.2886	−0.760478	−0.380239	+	0.924888i	$0.624158\pi$
$860$	0	0
$861$	−0.139875	−0.00476692
$862$	0	0
$863$	−37.6077	−1.28018	−0.640090	−	0.768300i	$-0.721103\pi$
$863$	−37.6077	−1.28018	−0.640090	+	0.768300i	$0.721103\pi$
$864$	0	0
$865$	34.3209	1.16695
$866$	0	0
$867$	−0.317791	−0.0107927
$868$	0	0
$869$	−0.893483	−0.0303093
$870$	0	0
$871$	83.0026	2.81244
$872$	0	0
$873$	−38.4512	−1.30138
$874$	0	0
$875$	−0.752570	−0.0254415
$876$	0	0
$877$	−43.6088	−1.47256	−0.736282	−	0.676675i	$-0.763420\pi$
$877$	−43.6088	−1.47256	−0.736282	+	0.676675i	$0.763420\pi$
$878$	0	0
$879$	0.0304344	0.00102653
$880$	0	0
$881$	43.9191	1.47967	0.739835	−	0.672788i	$-0.234903\pi$
$881$	43.9191	1.47967	0.739835	+	0.672788i	$0.234903\pi$
$882$	0	0
$883$	16.6273	0.559553	0.279776	−	0.960065i	$-0.409740\pi$
$883$	16.6273	0.559553	0.279776	+	0.960065i	$0.409740\pi$
$884$	0	0
$885$	3.48177	0.117038
$886$	0	0
$887$	48.0518	1.61342	0.806712	−	0.590945i	$-0.201245\pi$
$887$	48.0518	1.61342	0.806712	+	0.590945i	$0.201245\pi$
$888$	0	0
$889$	−1.75923	−0.0590026
$890$	0	0
$891$	9.82851	0.329268
$892$	0	0
$893$	16.2577	0.544044
$894$	0	0
$895$	2.13729	0.0714419
$896$	0	0
$897$	5.87257	0.196079
$898$	0	0
$899$	24.1632	0.805888
$900$	0	0
$901$	4.55001	0.151583
$902$	0	0
$903$	0.0759447	0.00252728
$904$	0	0
$905$	13.7174	0.455982
$906$	0	0
$907$	−22.8889	−0.760014	−0.380007	−	0.924984i	$-0.624078\pi$
$907$	−22.8889	−0.760014	−0.380007	+	0.924984i	$0.624078\pi$
$908$	0	0
$909$	32.5416	1.07934
$910$	0	0
$911$	−44.4322	−1.47210	−0.736051	−	0.676926i	$-0.763312\pi$
$911$	−44.4322	−1.47210	−0.736051	+	0.676926i	$0.763312\pi$
$912$	0	0
$913$	7.91255	0.261867
$914$	0	0
$915$	1.88372	0.0622740
$916$	0	0
$917$	0.394824	0.0130383
$918$	0	0
$919$	−21.6043	−0.712662	−0.356331	−	0.934360i	$-0.615972\pi$
$919$	−21.6043	−0.712662	−0.356331	+	0.934360i	$0.615972\pi$
$920$	0	0
$921$	4.32118	0.142388
$922$	0	0
$923$	39.9454	1.31482
$924$	0	0
$925$	−13.3850	−0.440094
$926$	0	0
$927$	49.3786	1.62181
$928$	0	0
$929$	−34.2834	−1.12480	−0.562400	−	0.826865i	$-0.690122\pi$
$929$	−34.2834	−1.12480	−0.562400	+	0.826865i	$0.690122\pi$
$930$	0	0
$931$	43.1808	1.41519
$932$	0	0
$933$	−4.24719	−0.139047
$934$	0	0
$935$	−10.8940	−0.356271
$936$	0	0
$937$	−14.3054	−0.467338	−0.233669	−	0.972316i	$-0.575073\pi$
$937$	−14.3054	−0.467338	−0.233669	+	0.972316i	$0.575073\pi$
$938$	0	0
$939$	1.33473	0.0435571
$940$	0	0
$941$	−2.22179	−0.0724283	−0.0362141	−	0.999344i	$-0.511530\pi$
$941$	−2.22179	−0.0724283	−0.0362141	+	0.999344i	$0.511530\pi$
$942$	0	0
$943$	−60.0416	−1.95522
$944$	0	0
$945$	−0.196813	−0.00640232
$946$	0	0
$947$	−7.95289	−0.258434	−0.129217	−	0.991616i	$-0.541246\pi$
$947$	−7.95289	−0.258434	−0.129217	+	0.991616i	$0.541246\pi$
$948$	0	0
$949$	36.9228	1.19856
$950$	0	0
$951$	3.20969	0.104081
$952$	0	0
$953$	43.6719	1.41467	0.707336	−	0.706878i	$-0.249897\pi$
$953$	43.6719	1.41467	0.707336	+	0.706878i	$0.249897\pi$
$954$	0	0
$955$	−24.1839	−0.782573
$956$	0	0
$957$	1.44418	0.0466836
$958$	0	0
$959$	−0.635584	−0.0205241
$960$	0	0
$961$	−21.5457	−0.695023
$962$	0	0
$963$	−53.1129	−1.71154
$964$	0	0
$965$	−38.6274	−1.24346
$966$	0	0
$967$	−38.0106	−1.22234	−0.611170	−	0.791500i	$-0.709301\pi$
$967$	−38.0106	−1.22234	−0.611170	+	0.791500i	$0.709301\pi$
$968$	0	0
$969$	−3.92432	−0.126067
$970$	0	0
$971$	7.56063	0.242632	0.121316	−	0.992614i	$-0.461289\pi$
$971$	7.56063	0.242632	0.121316	+	0.992614i	$0.461289\pi$
$972$	0	0
$973$	−0.161568	−0.00517963
$974$	0	0
$975$	−1.30808	−0.0418920
$976$	0	0
$977$	−42.5271	−1.36056	−0.680282	−	0.732951i	$-0.738143\pi$
$977$	−42.5271	−1.36056	−0.680282	+	0.732951i	$0.738143\pi$
$978$	0	0
$979$	0.687607	0.0219760
$980$	0	0
$981$	−41.0592	−1.31092
$982$	0	0
$983$	−13.2488	−0.422570	−0.211285	−	0.977424i	$-0.567765\pi$
$983$	−13.2488	−0.422570	−0.211285	+	0.977424i	$0.567765\pi$
$984$	0	0
$985$	54.8620	1.74805
$986$	0	0
$987$	0.0346779	0.00110381
$988$	0	0
$989$	32.5994	1.03660
$990$	0	0
$991$	34.3884	1.09238	0.546192	−	0.837660i	$-0.316077\pi$
$991$	34.3884	1.09238	0.546192	+	0.837660i	$0.316077\pi$
$992$	0	0
$993$	−5.16369	−0.163865
$994$	0	0
$995$	−48.0595	−1.52359
$996$	0	0
$997$	6.86481	0.217411	0.108705	−	0.994074i	$-0.465329\pi$
$997$	6.86481	0.217411	0.108705	+	0.994074i	$0.465329\pi$
$998$	0	0
$999$	10.3987	0.329001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.r.1.6		12
4.3	odd	2		1006.2.a.i.1.7	✓	12
12.11	even	2		9054.2.a.bj.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.i.1.7	✓	12	4.3	odd	2
8048.2.a.r.1.6		12	1.1	even	1	trivial
9054.2.a.bj.1.11		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)

\(f(q)\)	\(=\)	\(q+0.163781 q^{3} -2.50185 q^{5} -0.0804128 q^{7} -2.97318 q^{9} +O(q^{10})\)	\(q+0.163781 q^{3} -2.50185 q^{5} -0.0804128 q^{7} -2.97318 q^{9} +1.12207 q^{11} -6.34257 q^{13} -0.409754 q^{15} +3.88068 q^{17} -6.17439 q^{19} -0.0131701 q^{21} -5.65328 q^{23} +1.25923 q^{25} -0.978290 q^{27} +7.85851 q^{29} +3.07478 q^{31} +0.183773 q^{33} +0.201180 q^{35} -10.6295 q^{37} -1.03879 q^{39} +10.6207 q^{41} -5.76647 q^{43} +7.43843 q^{45} -2.63309 q^{47} -6.99353 q^{49} +0.635579 q^{51} +1.17248 q^{53} -2.80723 q^{55} -1.01124 q^{57} -8.49723 q^{59} -4.59721 q^{61} +0.239081 q^{63} +15.8681 q^{65} -13.0866 q^{67} -0.925897 q^{69} -6.29798 q^{71} -5.82142 q^{73} +0.206237 q^{75} -0.0902285 q^{77} -0.796284 q^{79} +8.75930 q^{81} +7.05177 q^{83} -9.70885 q^{85} +1.28707 q^{87} +0.612804 q^{89} +0.510024 q^{91} +0.503590 q^{93} +15.4474 q^{95} +12.9327 q^{97} -3.33610 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) \) 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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