LMFDB - Embedded newform 8048.2.a.w.1.9

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:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.29900 q^{3} +2.09702 q^{5} +3.13973 q^{7} -1.31260 q^{9} +O(q^{10})$	$q-1.29900 q^{3} +2.09702 q^{5} +3.13973 q^{7} -1.31260 q^{9} +0.999009 q^{11} -1.50268 q^{13} -2.72403 q^{15} +4.62336 q^{17} -0.597287 q^{19} -4.07850 q^{21} +8.14401 q^{23} -0.602491 q^{25} +5.60206 q^{27} +8.96563 q^{29} -1.32448 q^{31} -1.29771 q^{33} +6.58408 q^{35} -9.16272 q^{37} +1.95198 q^{39} -5.73178 q^{41} +8.83991 q^{43} -2.75256 q^{45} -7.58177 q^{47} +2.85788 q^{49} -6.00574 q^{51} +3.46835 q^{53} +2.09495 q^{55} +0.775874 q^{57} -7.24363 q^{59} +6.02195 q^{61} -4.12121 q^{63} -3.15116 q^{65} +6.73365 q^{67} -10.5791 q^{69} +12.8497 q^{71} -6.43424 q^{73} +0.782635 q^{75} +3.13661 q^{77} +16.4372 q^{79} -3.33927 q^{81} +2.24931 q^{83} +9.69530 q^{85} -11.6463 q^{87} -1.07636 q^{89} -4.71801 q^{91} +1.72050 q^{93} -1.25252 q^{95} -10.1728 q^{97} -1.31130 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * 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
$3$	−1.29900	−0.749977	−0.374989	−	0.927029i	$-0.622353\pi$
$3$	−1.29900	−0.749977	−0.374989	+	0.927029i	$0.622353\pi$
$4$	0	0
$5$	2.09702	0.937818	0.468909	−	0.883247i	$-0.344647\pi$
$5$	2.09702	0.937818	0.468909	+	0.883247i	$0.344647\pi$
$6$	0	0
$7$	3.13973	1.18670	0.593352	−	0.804943i	$-0.297804\pi$
$7$	3.13973	1.18670	0.593352	+	0.804943i	$0.297804\pi$
$8$	0	0
$9$	−1.31260	−0.437534
$10$	0	0
$11$	0.999009	0.301212	0.150606	−	0.988594i	$-0.451877\pi$
$11$	0.999009	0.301212	0.150606	+	0.988594i	$0.451877\pi$
$12$	0	0
$13$	−1.50268	−0.416769	−0.208385	−	0.978047i	$-0.566821\pi$
$13$	−1.50268	−0.416769	−0.208385	+	0.978047i	$0.566821\pi$
$14$	0	0
$15$	−2.72403	−0.703342
$16$	0	0
$17$	4.62336	1.12133	0.560665	−	0.828043i	$-0.310546\pi$
$17$	4.62336	1.12133	0.560665	+	0.828043i	$0.310546\pi$
$18$	0	0
$19$	−0.597287	−0.137027	−0.0685135	−	0.997650i	$-0.521826\pi$
$19$	−0.597287	−0.137027	−0.0685135	+	0.997650i	$0.521826\pi$
$20$	0	0
$21$	−4.07850	−0.890002
$22$	0	0
$23$	8.14401	1.69814	0.849072	−	0.528278i	$-0.177162\pi$
$23$	8.14401	1.69814	0.849072	+	0.528278i	$0.177162\pi$
$24$	0	0
$25$	−0.602491	−0.120498
$26$	0	0
$27$	5.60206	1.07812
$28$	0	0
$29$	8.96563	1.66488	0.832438	−	0.554119i	$-0.186945\pi$
$29$	8.96563	1.66488	0.832438	+	0.554119i	$0.186945\pi$
$30$	0	0
$31$	−1.32448	−0.237884	−0.118942	−	0.992901i	$-0.537950\pi$
$31$	−1.32448	−0.237884	−0.118942	+	0.992901i	$0.537950\pi$
$32$	0	0
$33$	−1.29771	−0.225902
$34$	0	0
$35$	6.58408	1.11291
$36$	0	0
$37$	−9.16272	−1.50634	−0.753171	−	0.657825i	$-0.771477\pi$
$37$	−9.16272	−1.50634	−0.753171	+	0.657825i	$0.771477\pi$
$38$	0	0
$39$	1.95198	0.312567
$40$	0	0
$41$	−5.73178	−0.895153	−0.447577	−	0.894246i	$-0.647713\pi$
$41$	−5.73178	−0.895153	−0.447577	+	0.894246i	$0.647713\pi$
$42$	0	0
$43$	8.83991	1.34807	0.674037	−	0.738698i	$-0.264559\pi$
$43$	8.83991	1.34807	0.674037	+	0.738698i	$0.264559\pi$
$44$	0	0
$45$	−2.75256	−0.410327
$46$	0	0
$47$	−7.58177	−1.10592	−0.552958	−	0.833209i	$-0.686501\pi$
$47$	−7.58177	−1.10592	−0.552958	+	0.833209i	$0.686501\pi$
$48$	0	0
$49$	2.85788	0.408268
$50$	0	0
$51$	−6.00574	−0.840972
$52$	0	0
$53$	3.46835	0.476414	0.238207	−	0.971214i	$-0.423440\pi$
$53$	3.46835	0.476414	0.238207	+	0.971214i	$0.423440\pi$
$54$	0	0
$55$	2.09495	0.282482
$56$	0	0
$57$	0.775874	0.102767
$58$	0	0
$59$	−7.24363	−0.943040	−0.471520	−	0.881855i	$-0.656295\pi$
$59$	−7.24363	−0.943040	−0.471520	+	0.881855i	$0.656295\pi$
$60$	0	0
$61$	6.02195	0.771031	0.385516	−	0.922701i	$-0.374024\pi$
$61$	6.02195	0.771031	0.385516	+	0.922701i	$0.374024\pi$
$62$	0	0
$63$	−4.12121	−0.519224
$64$	0	0
$65$	−3.15116	−0.390854
$66$	0	0
$67$	6.73365	0.822646	0.411323	−	0.911490i	$-0.365067\pi$
$67$	6.73365	0.822646	0.411323	+	0.911490i	$0.365067\pi$
$68$	0	0
$69$	−10.5791	−1.27357
$70$	0	0
$71$	12.8497	1.52498	0.762489	−	0.647001i	$-0.223977\pi$
$71$	12.8497	1.52498	0.762489	+	0.647001i	$0.223977\pi$
$72$	0	0
$73$	−6.43424	−0.753071	−0.376536	−	0.926402i	$-0.622885\pi$
$73$	−6.43424	−0.753071	−0.376536	+	0.926402i	$0.622885\pi$
$74$	0	0
$75$	0.782635	0.0903709
$76$	0	0
$77$	3.13661	0.357450
$78$	0	0
$79$	16.4372	1.84933	0.924663	−	0.380785i	$-0.124346\pi$
$79$	16.4372	1.84933	0.924663	+	0.380785i	$0.124346\pi$
$80$	0	0
$81$	−3.33927	−0.371030
$82$	0	0
$83$	2.24931	0.246894	0.123447	−	0.992351i	$-0.460605\pi$
$83$	2.24931	0.246894	0.123447	+	0.992351i	$0.460605\pi$
$84$	0	0
$85$	9.69530	1.05160
$86$	0	0
$87$	−11.6463	−1.24862
$88$	0	0
$89$	−1.07636	−0.114094	−0.0570469	−	0.998371i	$-0.518168\pi$
$89$	−1.07636	−0.114094	−0.0570469	+	0.998371i	$0.518168\pi$
$90$	0	0
$91$	−4.71801	−0.494582
$92$	0	0
$93$	1.72050	0.178408
$94$	0	0
$95$	−1.25252	−0.128506
$96$	0	0
$97$	−10.1728	−1.03289	−0.516446	−	0.856320i	$-0.672745\pi$
$97$	−10.1728	−1.03289	−0.516446	+	0.856320i	$0.672745\pi$
$98$	0	0
$99$	−1.31130	−0.131791
$100$	0	0
$101$	−8.35202	−0.831057	−0.415528	−	0.909580i	$-0.636403\pi$
$101$	−8.35202	−0.831057	−0.415528	+	0.909580i	$0.636403\pi$
$102$	0	0
$103$	1.38227	0.136199	0.0680996	−	0.997679i	$-0.478306\pi$
$103$	1.38227	0.136199	0.0680996	+	0.997679i	$0.478306\pi$
$104$	0	0
$105$	−8.55271	−0.834659
$106$	0	0
$107$	7.47171	0.722317	0.361159	−	0.932504i	$-0.382381\pi$
$107$	7.47171	0.722317	0.361159	+	0.932504i	$0.382381\pi$
$108$	0	0
$109$	−5.73532	−0.549344	−0.274672	−	0.961538i	$-0.588569\pi$
$109$	−5.73532	−0.549344	−0.274672	+	0.961538i	$0.588569\pi$
$110$	0	0
$111$	11.9024	1.12972
$112$	0	0
$113$	11.4669	1.07871	0.539357	−	0.842077i	$-0.318667\pi$
$113$	11.4669	1.07871	0.539357	+	0.842077i	$0.318667\pi$
$114$	0	0
$115$	17.0782	1.59255
$116$	0	0
$117$	1.97243	0.182351
$118$	0	0
$119$	14.5161	1.33069
$120$	0	0
$121$	−10.0020	−0.909271
$122$	0	0
$123$	7.44557	0.671345
$124$	0	0
$125$	−11.7486	−1.05082
$126$	0	0
$127$	−19.1702	−1.70108	−0.850538	−	0.525913i	$-0.823724\pi$
$127$	−19.1702	−1.70108	−0.850538	+	0.525913i	$0.823724\pi$
$128$	0	0
$129$	−11.4830	−1.01102
$130$	0	0
$131$	15.8064	1.38101	0.690504	−	0.723328i	$-0.257389\pi$
$131$	15.8064	1.38101	0.690504	+	0.723328i	$0.257389\pi$
$132$	0	0
$133$	−1.87532	−0.162611
$134$	0	0
$135$	11.7477	1.01108
$136$	0	0
$137$	10.0844	0.861570	0.430785	−	0.902455i	$-0.358237\pi$
$137$	10.0844	0.861570	0.430785	+	0.902455i	$0.358237\pi$
$138$	0	0
$139$	4.62705	0.392461	0.196231	−	0.980558i	$-0.437130\pi$
$139$	4.62705	0.392461	0.196231	+	0.980558i	$0.437130\pi$
$140$	0	0
$141$	9.84871	0.829411
$142$	0	0
$143$	−1.50119	−0.125536
$144$	0	0
$145$	18.8011	1.56135
$146$	0	0
$147$	−3.71238	−0.306192
$148$	0	0
$149$	2.06956	0.169545	0.0847724	−	0.996400i	$-0.472984\pi$
$149$	2.06956	0.169545	0.0847724	+	0.996400i	$0.472984\pi$
$150$	0	0
$151$	7.10357	0.578080	0.289040	−	0.957317i	$-0.406664\pi$
$151$	7.10357	0.578080	0.289040	+	0.957317i	$0.406664\pi$
$152$	0	0
$153$	−6.06863	−0.490620
$154$	0	0
$155$	−2.77747	−0.223092
$156$	0	0
$157$	−2.82097	−0.225138	−0.112569	−	0.993644i	$-0.535908\pi$
$157$	−2.82097	−0.225138	−0.112569	+	0.993644i	$0.535908\pi$
$158$	0	0
$159$	−4.50538	−0.357300
$160$	0	0
$161$	25.5700	2.01520
$162$	0	0
$163$	−3.75681	−0.294256	−0.147128	−	0.989117i	$-0.547003\pi$
$163$	−3.75681	−0.294256	−0.147128	+	0.989117i	$0.547003\pi$
$164$	0	0
$165$	−2.72133	−0.211855
$166$	0	0
$167$	−19.5251	−1.51090	−0.755450	−	0.655206i	$-0.772582\pi$
$167$	−19.5251	−1.51090	−0.755450	+	0.655206i	$0.772582\pi$
$168$	0	0
$169$	−10.7419	−0.826303
$170$	0	0
$171$	0.784000	0.0599540
$172$	0	0
$173$	12.3589	0.939629	0.469814	−	0.882765i	$-0.344321\pi$
$173$	12.3589	0.939629	0.469814	+	0.882765i	$0.344321\pi$
$174$	0	0
$175$	−1.89166	−0.142996
$176$	0	0
$177$	9.40947	0.707259
$178$	0	0
$179$	20.1158	1.50353	0.751763	−	0.659433i	$-0.229204\pi$
$179$	20.1158	1.50353	0.751763	+	0.659433i	$0.229204\pi$
$180$	0	0
$181$	1.63052	0.121196	0.0605979	−	0.998162i	$-0.480699\pi$
$181$	1.63052	0.121196	0.0605979	+	0.998162i	$0.480699\pi$
$182$	0	0
$183$	−7.82250	−0.578256
$184$	0	0
$185$	−19.2144	−1.41267
$186$	0	0
$187$	4.61878	0.337758
$188$	0	0
$189$	17.5889	1.27941
$190$	0	0
$191$	−0.0495946	−0.00358854	−0.00179427	−	0.999998i	$-0.500571\pi$
$191$	−0.0495946	−0.00358854	−0.00179427	+	0.999998i	$0.500571\pi$
$192$	0	0
$193$	7.43047	0.534857	0.267428	−	0.963578i	$-0.413826\pi$
$193$	7.43047	0.534857	0.267428	+	0.963578i	$0.413826\pi$
$194$	0	0
$195$	4.09336	0.293131
$196$	0	0
$197$	17.8827	1.27409	0.637045	−	0.770827i	$-0.280157\pi$
$197$	17.8827	1.27409	0.637045	+	0.770827i	$0.280157\pi$
$198$	0	0
$199$	0.389776	0.0276305	0.0138153	−	0.999905i	$-0.495602\pi$
$199$	0.389776	0.0276305	0.0138153	+	0.999905i	$0.495602\pi$
$200$	0	0
$201$	−8.74700	−0.616966
$202$	0	0
$203$	28.1496	1.97572
$204$	0	0
$205$	−12.0197	−0.839491
$206$	0	0
$207$	−10.6898	−0.742996
$208$	0	0
$209$	−0.596694	−0.0412742
$210$	0	0
$211$	1.88497	0.129767	0.0648835	−	0.997893i	$-0.479332\pi$
$211$	1.88497	0.129767	0.0648835	+	0.997893i	$0.479332\pi$
$212$	0	0
$213$	−16.6917	−1.14370
$214$	0	0
$215$	18.5375	1.26425
$216$	0	0
$217$	−4.15851	−0.282298
$218$	0	0
$219$	8.35807	0.564786
$220$	0	0
$221$	−6.94745	−0.467336
$222$	0	0
$223$	−17.3957	−1.16490	−0.582450	−	0.812866i	$-0.697906\pi$
$223$	−17.3957	−1.16490	−0.582450	+	0.812866i	$0.697906\pi$
$224$	0	0
$225$	0.790831	0.0527221
$226$	0	0
$227$	9.11436	0.604942	0.302471	−	0.953159i	$-0.402189\pi$
$227$	9.11436	0.604942	0.302471	+	0.953159i	$0.402189\pi$
$228$	0	0
$229$	−13.1448	−0.868634	−0.434317	−	0.900760i	$-0.643010\pi$
$229$	−13.1448	−0.868634	−0.434317	+	0.900760i	$0.643010\pi$
$230$	0	0
$231$	−4.07446	−0.268080
$232$	0	0
$233$	12.9165	0.846187	0.423094	−	0.906086i	$-0.360944\pi$
$233$	12.9165	0.846187	0.423094	+	0.906086i	$0.360944\pi$
$234$	0	0
$235$	−15.8992	−1.03715
$236$	0	0
$237$	−21.3519	−1.38695
$238$	0	0
$239$	−13.5641	−0.877389	−0.438695	−	0.898636i	$-0.644559\pi$
$239$	−13.5641	−0.877389	−0.438695	+	0.898636i	$0.644559\pi$
$240$	0	0
$241$	−11.8863	−0.765662	−0.382831	−	0.923818i	$-0.625051\pi$
$241$	−11.8863	−0.765662	−0.382831	+	0.923818i	$0.625051\pi$
$242$	0	0
$243$	−12.4685	−0.799854
$244$	0	0
$245$	5.99304	0.382881
$246$	0	0
$247$	0.897532	0.0571086
$248$	0	0
$249$	−2.92185	−0.185165
$250$	0	0
$251$	−19.8160	−1.25077	−0.625386	−	0.780315i	$-0.715059\pi$
$251$	−19.8160	−1.25077	−0.625386	+	0.780315i	$0.715059\pi$
$252$	0	0
$253$	8.13594	0.511502
$254$	0	0
$255$	−12.5942	−0.788678
$256$	0	0
$257$	21.1862	1.32156	0.660780	−	0.750579i	$-0.270225\pi$
$257$	21.1862	1.32156	0.660780	+	0.750579i	$0.270225\pi$
$258$	0	0
$259$	−28.7684	−1.78758
$260$	0	0
$261$	−11.7683	−0.728440
$262$	0	0
$263$	−25.3207	−1.56134	−0.780669	−	0.624944i	$-0.785122\pi$
$263$	−25.3207	−1.56134	−0.780669	+	0.624944i	$0.785122\pi$
$264$	0	0
$265$	7.27321	0.446790
$266$	0	0
$267$	1.39819	0.0855677
$268$	0	0
$269$	−3.77750	−0.230318	−0.115159	−	0.993347i	$-0.536738\pi$
$269$	−3.77750	−0.230318	−0.115159	+	0.993347i	$0.536738\pi$
$270$	0	0
$271$	2.57486	0.156412	0.0782059	−	0.996937i	$-0.475081\pi$
$271$	2.57486	0.156412	0.0782059	+	0.996937i	$0.475081\pi$
$272$	0	0
$273$	6.12869	0.370925
$274$	0	0
$275$	−0.601894	−0.0362955
$276$	0	0
$277$	6.20780	0.372991	0.186495	−	0.982456i	$-0.440287\pi$
$277$	6.20780	0.372991	0.186495	+	0.982456i	$0.440287\pi$
$278$	0	0
$279$	1.73852	0.104082
$280$	0	0
$281$	28.7026	1.71226	0.856128	−	0.516764i	$-0.172864\pi$
$281$	28.7026	1.71226	0.856128	+	0.516764i	$0.172864\pi$
$282$	0	0
$283$	5.19892	0.309044	0.154522	−	0.987989i	$-0.450616\pi$
$283$	5.19892	0.309044	0.154522	+	0.987989i	$0.450616\pi$
$284$	0	0
$285$	1.62703	0.0963768
$286$	0	0
$287$	−17.9962	−1.06228
$288$	0	0
$289$	4.37546	0.257380
$290$	0	0
$291$	13.2145	0.774646
$292$	0	0
$293$	−1.42489	−0.0832430	−0.0416215	−	0.999133i	$-0.513252\pi$
$293$	−1.42489	−0.0832430	−0.0416215	+	0.999133i	$0.513252\pi$
$294$	0	0
$295$	−15.1901	−0.884400
$296$	0	0
$297$	5.59651	0.324743
$298$	0	0
$299$	−12.2379	−0.707734
$300$	0	0
$301$	27.7549	1.59977
$302$	0	0
$303$	10.8493	0.623274
$304$	0	0
$305$	12.6282	0.723087
$306$	0	0
$307$	9.53879	0.544408	0.272204	−	0.962240i	$-0.412248\pi$
$307$	9.53879	0.544408	0.272204	+	0.962240i	$0.412248\pi$
$308$	0	0
$309$	−1.79557	−0.102146
$310$	0	0
$311$	2.98799	0.169434	0.0847168	−	0.996405i	$-0.473001\pi$
$311$	2.98799	0.169434	0.0847168	+	0.996405i	$0.473001\pi$
$312$	0	0
$313$	−19.5815	−1.10681	−0.553405	−	0.832912i	$-0.686672\pi$
$313$	−19.5815	−1.10681	−0.553405	+	0.832912i	$0.686672\pi$
$314$	0	0
$315$	−8.64228	−0.486937
$316$	0	0
$317$	−6.19915	−0.348179	−0.174090	−	0.984730i	$-0.555698\pi$
$317$	−6.19915	−0.348179	−0.174090	+	0.984730i	$0.555698\pi$
$318$	0	0
$319$	8.95674	0.501481
$320$	0	0
$321$	−9.70574	−0.541721
$322$	0	0
$323$	−2.76147	−0.153652
$324$	0	0
$325$	0.905353	0.0502199
$326$	0	0
$327$	7.45017	0.411995
$328$	0	0
$329$	−23.8047	−1.31240
$330$	0	0
$331$	14.4254	0.792890	0.396445	−	0.918058i	$-0.370244\pi$
$331$	14.4254	0.792890	0.396445	+	0.918058i	$0.370244\pi$
$332$	0	0
$333$	12.0270	0.659076
$334$	0	0
$335$	14.1206	0.771492
$336$	0	0
$337$	6.52817	0.355612	0.177806	−	0.984066i	$-0.443100\pi$
$337$	6.52817	0.355612	0.177806	+	0.984066i	$0.443100\pi$
$338$	0	0
$339$	−14.8955	−0.809011
$340$	0	0
$341$	−1.32317	−0.0716536
$342$	0	0
$343$	−13.0051	−0.702211
$344$	0	0
$345$	−22.1845	−1.19438
$346$	0	0
$347$	6.10601	0.327788	0.163894	−	0.986478i	$-0.447595\pi$
$347$	6.10601	0.327788	0.163894	+	0.986478i	$0.447595\pi$
$348$	0	0
$349$	1.42282	0.0761619	0.0380809	−	0.999275i	$-0.487876\pi$
$349$	1.42282	0.0761619	0.0380809	+	0.999275i	$0.487876\pi$
$350$	0	0
$351$	−8.41813	−0.449326
$352$	0	0
$353$	−12.6365	−0.672572	−0.336286	−	0.941760i	$-0.609171\pi$
$353$	−12.6365	−0.672572	−0.336286	+	0.941760i	$0.609171\pi$
$354$	0	0
$355$	26.9461	1.43015
$356$	0	0
$357$	−18.8564	−0.997985
$358$	0	0
$359$	18.8694	0.995891	0.497945	−	0.867208i	$-0.334088\pi$
$359$	18.8694	0.995891	0.497945	+	0.867208i	$0.334088\pi$
$360$	0	0
$361$	−18.6432	−0.981224
$362$	0	0
$363$	12.9926	0.681933
$364$	0	0
$365$	−13.4928	−0.706243
$366$	0	0
$367$	2.40410	0.125493	0.0627465	−	0.998029i	$-0.480014\pi$
$367$	2.40410	0.125493	0.0627465	+	0.998029i	$0.480014\pi$
$368$	0	0
$369$	7.52355	0.391660
$370$	0	0
$371$	10.8897	0.565363
$372$	0	0
$373$	−20.3235	−1.05231	−0.526155	−	0.850389i	$-0.676367\pi$
$373$	−20.3235	−1.05231	−0.526155	+	0.850389i	$0.676367\pi$
$374$	0	0
$375$	15.2614	0.788093
$376$	0	0
$377$	−13.4725	−0.693869
$378$	0	0
$379$	20.1332	1.03418	0.517088	−	0.855933i	$-0.327016\pi$
$379$	20.1332	1.03418	0.517088	+	0.855933i	$0.327016\pi$
$380$	0	0
$381$	24.9020	1.27577
$382$	0	0
$383$	−16.3495	−0.835418	−0.417709	−	0.908581i	$-0.637167\pi$
$383$	−16.3495	−0.835418	−0.417709	+	0.908581i	$0.637167\pi$
$384$	0	0
$385$	6.57755	0.335223
$386$	0	0
$387$	−11.6033	−0.589828
$388$	0	0
$389$	31.1263	1.57817	0.789084	−	0.614286i	$-0.210556\pi$
$389$	31.1263	1.57817	0.789084	+	0.614286i	$0.210556\pi$
$390$	0	0
$391$	37.6527	1.90418
$392$	0	0
$393$	−20.5324	−1.03572
$394$	0	0
$395$	34.4692	1.73433
$396$	0	0
$397$	−4.21767	−0.211679	−0.105839	−	0.994383i	$-0.533753\pi$
$397$	−4.21767	−0.211679	−0.105839	+	0.994383i	$0.533753\pi$
$398$	0	0
$399$	2.43603	0.121954
$400$	0	0
$401$	18.5377	0.925730	0.462865	−	0.886429i	$-0.346821\pi$
$401$	18.5377	0.925730	0.462865	+	0.886429i	$0.346821\pi$
$402$	0	0
$403$	1.99028	0.0991427
$404$	0	0
$405$	−7.00252	−0.347958
$406$	0	0
$407$	−9.15363	−0.453729
$408$	0	0
$409$	4.86058	0.240340	0.120170	−	0.992753i	$-0.461656\pi$
$409$	4.86058	0.240340	0.120170	+	0.992753i	$0.461656\pi$
$410$	0	0
$411$	−13.0996	−0.646158
$412$	0	0
$413$	−22.7430	−1.11911
$414$	0	0
$415$	4.71686	0.231542
$416$	0	0
$417$	−6.01053	−0.294337
$418$	0	0
$419$	−36.8390	−1.79970	−0.899852	−	0.436195i	$-0.856326\pi$
$419$	−36.8390	−1.79970	−0.899852	+	0.436195i	$0.856326\pi$
$420$	0	0
$421$	12.3119	0.600048	0.300024	−	0.953932i	$-0.403005\pi$
$421$	12.3119	0.600048	0.300024	+	0.953932i	$0.403005\pi$
$422$	0	0
$423$	9.95186	0.483876
$424$	0	0
$425$	−2.78553	−0.135118
$426$	0	0
$427$	18.9073	0.914987
$428$	0	0
$429$	1.95005	0.0941492
$430$	0	0
$431$	22.5714	1.08723	0.543614	−	0.839335i	$-0.317056\pi$
$431$	22.5714	1.08723	0.543614	+	0.839335i	$0.317056\pi$
$432$	0	0
$433$	7.09112	0.340778	0.170389	−	0.985377i	$-0.445498\pi$
$433$	7.09112	0.340778	0.170389	+	0.985377i	$0.445498\pi$
$434$	0	0
$435$	−24.4226	−1.17098
$436$	0	0
$437$	−4.86431	−0.232691
$438$	0	0
$439$	17.5923	0.839635	0.419818	−	0.907608i	$-0.362094\pi$
$439$	17.5923	0.839635	0.419818	+	0.907608i	$0.362094\pi$
$440$	0	0
$441$	−3.75126	−0.178631
$442$	0	0
$443$	11.2679	0.535352	0.267676	−	0.963509i	$-0.413744\pi$
$443$	11.2679	0.535352	0.267676	+	0.963509i	$0.413744\pi$
$444$	0	0
$445$	−2.25715	−0.106999
$446$	0	0
$447$	−2.68835	−0.127155
$448$	0	0
$449$	23.0328	1.08698	0.543492	−	0.839414i	$-0.317102\pi$
$449$	23.0328	1.08698	0.543492	+	0.839414i	$0.317102\pi$
$450$	0	0
$451$	−5.72610	−0.269631
$452$	0	0
$453$	−9.22752	−0.433547
$454$	0	0
$455$	−9.89379	−0.463828
$456$	0	0
$457$	3.68915	0.172571	0.0862855	−	0.996270i	$-0.472500\pi$
$457$	3.68915	0.172571	0.0862855	+	0.996270i	$0.472500\pi$
$458$	0	0
$459$	25.9004	1.20893
$460$	0	0
$461$	−3.80363	−0.177153	−0.0885764	−	0.996069i	$-0.528232\pi$
$461$	−3.80363	−0.177153	−0.0885764	+	0.996069i	$0.528232\pi$
$462$	0	0
$463$	8.05725	0.374452	0.187226	−	0.982317i	$-0.440050\pi$
$463$	8.05725	0.374452	0.187226	+	0.982317i	$0.440050\pi$
$464$	0	0
$465$	3.60793	0.167314
$466$	0	0
$467$	11.4971	0.532022	0.266011	−	0.963970i	$-0.414294\pi$
$467$	11.4971	0.532022	0.266011	+	0.963970i	$0.414294\pi$
$468$	0	0
$469$	21.1418	0.976238
$470$	0	0
$471$	3.66444	0.168848
$472$	0	0
$473$	8.83115	0.406057
$474$	0	0
$475$	0.359860	0.0165115
$476$	0	0
$477$	−4.55256	−0.208447
$478$	0	0
$479$	26.0120	1.18852	0.594258	−	0.804274i	$-0.297446\pi$
$479$	26.0120	1.18852	0.594258	+	0.804274i	$0.297446\pi$
$480$	0	0
$481$	13.7687	0.627797
$482$	0	0
$483$	−33.2153	−1.51135
$484$	0	0
$485$	−21.3326	−0.968665
$486$	0	0
$487$	15.1515	0.686581	0.343290	−	0.939229i	$-0.388458\pi$
$487$	15.1515	0.686581	0.343290	+	0.939229i	$0.388458\pi$
$488$	0	0
$489$	4.88009	0.220685
$490$	0	0
$491$	21.3340	0.962788	0.481394	−	0.876504i	$-0.340131\pi$
$491$	21.3340	0.962788	0.481394	+	0.876504i	$0.340131\pi$
$492$	0	0
$493$	41.4513	1.86687
$494$	0	0
$495$	−2.74983	−0.123596
$496$	0	0
$497$	40.3445	1.80970
$498$	0	0
$499$	−24.5851	−1.10058	−0.550291	−	0.834973i	$-0.685483\pi$
$499$	−24.5851	−1.10058	−0.550291	+	0.834973i	$0.685483\pi$
$500$	0	0
$501$	25.3631	1.13314
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−17.5144	−0.779380
$506$	0	0
$507$	13.9538	0.619709
$508$	0	0
$509$	−37.8070	−1.67577	−0.837883	−	0.545850i	$-0.816207\pi$
$509$	−37.8070	−1.67577	−0.837883	+	0.545850i	$0.816207\pi$
$510$	0	0
$511$	−20.2018	−0.893673
$512$	0	0
$513$	−3.34604	−0.147731
$514$	0	0
$515$	2.89866	0.127730
$516$	0	0
$517$	−7.57426	−0.333116
$518$	0	0
$519$	−16.0542	−0.704700
$520$	0	0
$521$	43.4947	1.90554	0.952769	−	0.303697i	$-0.0982211\pi$
$521$	43.4947	1.90554	0.952769	+	0.303697i	$0.0982211\pi$
$522$	0	0
$523$	−12.3904	−0.541793	−0.270896	−	0.962608i	$-0.587320\pi$
$523$	−12.3904	−0.541793	−0.270896	+	0.962608i	$0.587320\pi$
$524$	0	0
$525$	2.45726	0.107244
$526$	0	0
$527$	−6.12356	−0.266746
$528$	0	0
$529$	43.3249	1.88369
$530$	0	0
$531$	9.50801	0.412612
$532$	0	0
$533$	8.61305	0.373072
$534$	0	0
$535$	15.6683	0.677402
$536$	0	0
$537$	−26.1304	−1.12761
$538$	0	0
$539$	2.85505	0.122976
$540$	0	0
$541$	26.7583	1.15043	0.575214	−	0.818003i	$-0.304919\pi$
$541$	26.7583	1.15043	0.575214	+	0.818003i	$0.304919\pi$
$542$	0	0
$543$	−2.11805	−0.0908941
$544$	0	0
$545$	−12.0271	−0.515184
$546$	0	0
$547$	43.3690	1.85433	0.927163	−	0.374658i	$-0.122240\pi$
$547$	43.3690	1.85433	0.927163	+	0.374658i	$0.122240\pi$
$548$	0	0
$549$	−7.90442	−0.337353
$550$	0	0
$551$	−5.35505	−0.228133
$552$	0	0
$553$	51.6082	2.19461
$554$	0	0
$555$	24.9595	1.05947
$556$	0	0
$557$	13.5394	0.573682	0.286841	−	0.957978i	$-0.407395\pi$
$557$	13.5394	0.573682	0.286841	+	0.957978i	$0.407395\pi$
$558$	0	0
$559$	−13.2836	−0.561836
$560$	0	0
$561$	−5.99979	−0.253311
$562$	0	0
$563$	−29.8525	−1.25813	−0.629066	−	0.777352i	$-0.716562\pi$
$563$	−29.8525	−1.25813	−0.629066	+	0.777352i	$0.716562\pi$
$564$	0	0
$565$	24.0463	1.01164
$566$	0	0
$567$	−10.4844	−0.440303
$568$	0	0
$569$	26.2428	1.10015	0.550077	−	0.835114i	$-0.314598\pi$
$569$	26.2428	1.10015	0.550077	+	0.835114i	$0.314598\pi$
$570$	0	0
$571$	−23.0726	−0.965558	−0.482779	−	0.875742i	$-0.660373\pi$
$571$	−23.0726	−0.965558	−0.482779	+	0.875742i	$0.660373\pi$
$572$	0	0
$573$	0.0644234	0.00269132
$574$	0	0
$575$	−4.90669	−0.204623
$576$	0	0
$577$	−0.252004	−0.0104911	−0.00524554	−	0.999986i	$-0.501670\pi$
$577$	−0.252004	−0.0104911	−0.00524554	+	0.999986i	$0.501670\pi$
$578$	0	0
$579$	−9.65217	−0.401130
$580$	0	0
$581$	7.06222	0.292990
$582$	0	0
$583$	3.46491	0.143502
$584$	0	0
$585$	4.13622	0.171012
$586$	0	0
$587$	1.43103	0.0590648	0.0295324	−	0.999564i	$-0.490598\pi$
$587$	1.43103	0.0590648	0.0295324	+	0.999564i	$0.490598\pi$
$588$	0	0
$589$	0.791095	0.0325965
$590$	0	0
$591$	−23.2296	−0.955538
$592$	0	0
$593$	−43.5955	−1.79025	−0.895126	−	0.445813i	$-0.852915\pi$
$593$	−43.5955	−1.79025	−0.895126	+	0.445813i	$0.852915\pi$
$594$	0	0
$595$	30.4406	1.24794
$596$	0	0
$597$	−0.506319	−0.0207223
$598$	0	0
$599$	2.02929	0.0829145	0.0414573	−	0.999140i	$-0.486800\pi$
$599$	2.02929	0.0829145	0.0414573	+	0.999140i	$0.486800\pi$
$600$	0	0
$601$	25.0906	1.02347	0.511734	−	0.859144i	$-0.329003\pi$
$601$	25.0906	1.02347	0.511734	+	0.859144i	$0.329003\pi$
$602$	0	0
$603$	−8.83860	−0.359936
$604$	0	0
$605$	−20.9744	−0.852730
$606$	0	0
$607$	40.2620	1.63418	0.817091	−	0.576508i	$-0.195585\pi$
$607$	40.2620	1.63418	0.817091	+	0.576508i	$0.195585\pi$
$608$	0	0
$609$	−36.5663	−1.48174
$610$	0	0
$611$	11.3930	0.460912
$612$	0	0
$613$	28.3439	1.14480	0.572399	−	0.819975i	$-0.306013\pi$
$613$	28.3439	1.14480	0.572399	+	0.819975i	$0.306013\pi$
$614$	0	0
$615$	15.6135	0.629599
$616$	0	0
$617$	−26.6672	−1.07358	−0.536790	−	0.843716i	$-0.680363\pi$
$617$	−26.6672	−1.07358	−0.536790	+	0.843716i	$0.680363\pi$
$618$	0	0
$619$	33.8016	1.35860	0.679301	−	0.733860i	$-0.262283\pi$
$619$	33.8016	1.35860	0.679301	+	0.733860i	$0.262283\pi$
$620$	0	0
$621$	45.6233	1.83080
$622$	0	0
$623$	−3.37947	−0.135396
$624$	0	0
$625$	−21.6246	−0.864982
$626$	0	0
$627$	0.775105	0.0309547
$628$	0	0
$629$	−42.3625	−1.68911
$630$	0	0
$631$	−35.6614	−1.41966	−0.709829	−	0.704374i	$-0.751228\pi$
$631$	−35.6614	−1.41966	−0.709829	+	0.704374i	$0.751228\pi$
$632$	0	0
$633$	−2.44858	−0.0973223
$634$	0	0
$635$	−40.2003	−1.59530
$636$	0	0
$637$	−4.29449	−0.170154
$638$	0	0
$639$	−16.8665	−0.667230
$640$	0	0
$641$	−12.9665	−0.512147	−0.256073	−	0.966657i	$-0.582429\pi$
$641$	−12.9665	−0.512147	−0.256073	+	0.966657i	$0.582429\pi$
$642$	0	0
$643$	14.5651	0.574391	0.287195	−	0.957872i	$-0.407277\pi$
$643$	14.5651	0.574391	0.287195	+	0.957872i	$0.407277\pi$
$644$	0	0
$645$	−24.0802	−0.948157
$646$	0	0
$647$	−45.4683	−1.78754	−0.893772	−	0.448521i	$-0.851951\pi$
$647$	−45.4683	−1.78754	−0.893772	+	0.448521i	$0.851951\pi$
$648$	0	0
$649$	−7.23645	−0.284055
$650$	0	0
$651$	5.40190	0.211717
$652$	0	0
$653$	−18.0901	−0.707921	−0.353961	−	0.935260i	$-0.615165\pi$
$653$	−18.0901	−0.707921	−0.353961	+	0.935260i	$0.615165\pi$
$654$	0	0
$655$	33.1463	1.29513
$656$	0	0
$657$	8.44560	0.329494
$658$	0	0
$659$	32.1392	1.25197	0.625983	−	0.779837i	$-0.284698\pi$
$659$	32.1392	1.25197	0.625983	+	0.779837i	$0.284698\pi$
$660$	0	0
$661$	−20.5884	−0.800798	−0.400399	−	0.916341i	$-0.631128\pi$
$661$	−20.5884	−0.800798	−0.400399	+	0.916341i	$0.631128\pi$
$662$	0	0
$663$	9.02472	0.350491
$664$	0	0
$665$	−3.93258	−0.152499
$666$	0	0
$667$	73.0162	2.82720
$668$	0	0
$669$	22.5970	0.873649
$670$	0	0
$671$	6.01598	0.232244
$672$	0	0
$673$	−7.04846	−0.271698	−0.135849	−	0.990730i	$-0.543376\pi$
$673$	−7.04846	−0.271698	−0.135849	+	0.990730i	$0.543376\pi$
$674$	0	0
$675$	−3.37519	−0.129911
$676$	0	0
$677$	45.0166	1.73013	0.865064	−	0.501661i	$-0.167278\pi$
$677$	45.0166	1.73013	0.865064	+	0.501661i	$0.167278\pi$
$678$	0	0
$679$	−31.9398	−1.22574
$680$	0	0
$681$	−11.8395	−0.453692
$682$	0	0
$683$	−1.45442	−0.0556517	−0.0278258	−	0.999613i	$-0.508858\pi$
$683$	−1.45442	−0.0556517	−0.0278258	+	0.999613i	$0.508858\pi$
$684$	0	0
$685$	21.1473	0.807995
$686$	0	0
$687$	17.0751	0.651456
$688$	0	0
$689$	−5.21183	−0.198555
$690$	0	0
$691$	−3.63990	−0.138468	−0.0692341	−	0.997600i	$-0.522056\pi$
$691$	−3.63990	−0.138468	−0.0692341	+	0.997600i	$0.522056\pi$
$692$	0	0
$693$	−4.11713	−0.156397
$694$	0	0
$695$	9.70303	0.368057
$696$	0	0
$697$	−26.5001	−1.00376
$698$	0	0
$699$	−16.7785	−0.634621
$700$	0	0
$701$	−39.3894	−1.48772	−0.743858	−	0.668337i	$-0.767006\pi$
$701$	−39.3894	−1.48772	−0.743858	+	0.668337i	$0.767006\pi$
$702$	0	0
$703$	5.47277	0.206409
$704$	0	0
$705$	20.6530	0.777837
$706$	0	0
$707$	−26.2230	−0.986219
$708$	0	0
$709$	27.0621	1.01634	0.508170	−	0.861257i	$-0.330322\pi$
$709$	27.0621	1.01634	0.508170	+	0.861257i	$0.330322\pi$
$710$	0	0
$711$	−21.5755	−0.809144
$712$	0	0
$713$	−10.7866	−0.403961
$714$	0	0
$715$	−3.14804	−0.117730
$716$	0	0
$717$	17.6198	0.658022
$718$	0	0
$719$	51.7507	1.92998	0.964988	−	0.262292i	$-0.0844785\pi$
$719$	51.7507	1.92998	0.964988	+	0.262292i	$0.0844785\pi$
$720$	0	0
$721$	4.33995	0.161628
$722$	0	0
$723$	15.4402	0.574229
$724$	0	0
$725$	−5.40171	−0.200614
$726$	0	0
$727$	1.52135	0.0564239	0.0282119	−	0.999602i	$-0.491019\pi$
$727$	1.52135	0.0564239	0.0282119	+	0.999602i	$0.491019\pi$
$728$	0	0
$729$	26.2144	0.970902
$730$	0	0
$731$	40.8701	1.51163
$732$	0	0
$733$	51.3939	1.89828	0.949138	−	0.314861i	$-0.101958\pi$
$733$	51.3939	1.89828	0.949138	+	0.314861i	$0.101958\pi$
$734$	0	0
$735$	−7.78495	−0.287152
$736$	0	0
$737$	6.72697	0.247791
$738$	0	0
$739$	−1.45500	−0.0535230	−0.0267615	−	0.999642i	$-0.508519\pi$
$739$	−1.45500	−0.0535230	−0.0267615	+	0.999642i	$0.508519\pi$
$740$	0	0
$741$	−1.16589	−0.0428302
$742$	0	0
$743$	−51.6275	−1.89403	−0.947014	−	0.321193i	$-0.895916\pi$
$743$	−51.6275	−1.89403	−0.947014	+	0.321193i	$0.895916\pi$
$744$	0	0
$745$	4.33991	0.159002
$746$	0	0
$747$	−2.95245	−0.108025
$748$	0	0
$749$	23.4591	0.857177
$750$	0	0
$751$	−11.3326	−0.413533	−0.206767	−	0.978390i	$-0.566294\pi$
$751$	−11.3326	−0.413533	−0.206767	+	0.978390i	$0.566294\pi$
$752$	0	0
$753$	25.7409	0.938051
$754$	0	0
$755$	14.8963	0.542134
$756$	0	0
$757$	26.7248	0.971331	0.485666	−	0.874145i	$-0.338577\pi$
$757$	26.7248	0.971331	0.485666	+	0.874145i	$0.338577\pi$
$758$	0	0
$759$	−10.5686	−0.383615
$760$	0	0
$761$	20.8543	0.755970	0.377985	−	0.925812i	$-0.376617\pi$
$761$	20.8543	0.755970	0.377985	+	0.925812i	$0.376617\pi$
$762$	0	0
$763$	−18.0073	−0.651909
$764$	0	0
$765$	−12.7261	−0.460112
$766$	0	0
$767$	10.8849	0.393030
$768$	0	0
$769$	−5.72710	−0.206525	−0.103262	−	0.994654i	$-0.532928\pi$
$769$	−5.72710	−0.206525	−0.103262	+	0.994654i	$0.532928\pi$
$770$	0	0
$771$	−27.5209	−0.991140
$772$	0	0
$773$	23.8549	0.858001	0.429001	−	0.903304i	$-0.358866\pi$
$773$	23.8549	0.858001	0.429001	+	0.903304i	$0.358866\pi$
$774$	0	0
$775$	0.797988	0.0286646
$776$	0	0
$777$	37.3701	1.34065
$778$	0	0
$779$	3.42351	0.122660
$780$	0	0
$781$	12.8370	0.459342
$782$	0	0
$783$	50.2260	1.79493
$784$	0	0
$785$	−5.91564	−0.211138
$786$	0	0
$787$	33.4484	1.19231	0.596154	−	0.802870i	$-0.296695\pi$
$787$	33.4484	1.19231	0.596154	+	0.802870i	$0.296695\pi$
$788$	0	0
$789$	32.8915	1.17097
$790$	0	0
$791$	36.0029	1.28012
$792$	0	0
$793$	−9.04908	−0.321342
$794$	0	0
$795$	−9.44789	−0.335082
$796$	0	0
$797$	−15.2192	−0.539092	−0.269546	−	0.962987i	$-0.586874\pi$
$797$	−15.2192	−0.539092	−0.269546	+	0.962987i	$0.586874\pi$
$798$	0	0
$799$	−35.0533	−1.24010
$800$	0	0
$801$	1.41283	0.0499199
$802$	0	0
$803$	−6.42786	−0.226834
$804$	0	0
$805$	53.6208	1.88989
$806$	0	0
$807$	4.90696	0.172733
$808$	0	0
$809$	−40.4700	−1.42285	−0.711424	−	0.702763i	$-0.751950\pi$
$809$	−40.4700	−1.42285	−0.711424	+	0.702763i	$0.751950\pi$
$810$	0	0
$811$	25.9060	0.909681	0.454841	−	0.890573i	$-0.349696\pi$
$811$	25.9060	0.909681	0.454841	+	0.890573i	$0.349696\pi$
$812$	0	0
$813$	−3.34474	−0.117305
$814$	0	0
$815$	−7.87812	−0.275958
$816$	0	0
$817$	−5.27996	−0.184722
$818$	0	0
$819$	6.19288	0.216397
$820$	0	0
$821$	−32.2745	−1.12639	−0.563194	−	0.826324i	$-0.690428\pi$
$821$	−32.2745	−1.12639	−0.563194	+	0.826324i	$0.690428\pi$
$822$	0	0
$823$	−36.0323	−1.25601	−0.628004	−	0.778210i	$-0.716128\pi$
$823$	−36.0323	−1.25601	−0.628004	+	0.778210i	$0.716128\pi$
$824$	0	0
$825$	0.781859	0.0272208
$826$	0	0
$827$	50.0148	1.73919	0.869593	−	0.493769i	$-0.164381\pi$
$827$	50.0148	1.73919	0.869593	+	0.493769i	$0.164381\pi$
$828$	0	0
$829$	37.9305	1.31738	0.658690	−	0.752414i	$-0.271111\pi$
$829$	37.9305	1.31738	0.658690	+	0.752414i	$0.271111\pi$
$830$	0	0
$831$	−8.06393	−0.279734
$832$	0	0
$833$	13.2130	0.457803
$834$	0	0
$835$	−40.9447	−1.41695
$836$	0	0
$837$	−7.41983	−0.256467
$838$	0	0
$839$	−21.8767	−0.755266	−0.377633	−	0.925955i	$-0.623262\pi$
$839$	−21.8767	−0.755266	−0.377633	+	0.925955i	$0.623262\pi$
$840$	0	0
$841$	51.3825	1.77181
$842$	0	0
$843$	−37.2847	−1.28415
$844$	0	0
$845$	−22.5261	−0.774922
$846$	0	0
$847$	−31.4035	−1.07904
$848$	0	0
$849$	−6.75340	−0.231776
$850$	0	0
$851$	−74.6213	−2.55798
$852$	0	0
$853$	28.8686	0.988444	0.494222	−	0.869336i	$-0.335453\pi$
$853$	28.8686	0.988444	0.494222	+	0.869336i	$0.335453\pi$
$854$	0	0
$855$	1.64407	0.0562259
$856$	0	0
$857$	−40.7775	−1.39293	−0.696467	−	0.717589i	$-0.745246\pi$
$857$	−40.7775	−1.39293	−0.696467	+	0.717589i	$0.745246\pi$
$858$	0	0
$859$	−29.3410	−1.00110	−0.500551	−	0.865707i	$-0.666869\pi$
$859$	−29.3410	−1.00110	−0.500551	+	0.865707i	$0.666869\pi$
$860$	0	0
$861$	23.3771	0.796688
$862$	0	0
$863$	8.62528	0.293608	0.146804	−	0.989166i	$-0.453101\pi$
$863$	8.62528	0.293608	0.146804	+	0.989166i	$0.453101\pi$
$864$	0	0
$865$	25.9169	0.881200
$866$	0	0
$867$	−5.68372	−0.193029
$868$	0	0
$869$	16.4209	0.557040
$870$	0	0
$871$	−10.1185	−0.342854
$872$	0	0
$873$	13.3529	0.451926
$874$	0	0
$875$	−36.8872	−1.24702
$876$	0	0
$877$	−18.3520	−0.619703	−0.309851	−	0.950785i	$-0.600279\pi$
$877$	−18.3520	−0.619703	−0.309851	+	0.950785i	$0.600279\pi$
$878$	0	0
$879$	1.85093	0.0624303
$880$	0	0
$881$	−43.2572	−1.45737	−0.728686	−	0.684848i	$-0.759868\pi$
$881$	−43.2572	−1.45737	−0.728686	+	0.684848i	$0.759868\pi$
$882$	0	0
$883$	−23.2194	−0.781393	−0.390697	−	0.920519i	$-0.627766\pi$
$883$	−23.2194	−0.781393	−0.390697	+	0.920519i	$0.627766\pi$
$884$	0	0
$885$	19.7319	0.663280
$886$	0	0
$887$	27.0078	0.906832	0.453416	−	0.891299i	$-0.350205\pi$
$887$	27.0078	0.906832	0.453416	+	0.891299i	$0.350205\pi$
$888$	0	0
$889$	−60.1890	−2.01868
$890$	0	0
$891$	−3.33596	−0.111759
$892$	0	0
$893$	4.52849	0.151540
$894$	0	0
$895$	42.1833	1.41003
$896$	0	0
$897$	15.8970	0.530784
$898$	0	0
$899$	−11.8748	−0.396047
$900$	0	0
$901$	16.0354	0.534217
$902$	0	0
$903$	−36.0536	−1.19979
$904$	0	0
$905$	3.41925	0.113660
$906$	0	0
$907$	33.0086	1.09603	0.548016	−	0.836468i	$-0.315383\pi$
$907$	33.0086	1.09603	0.548016	+	0.836468i	$0.315383\pi$
$908$	0	0
$909$	10.9629	0.363616
$910$	0	0
$911$	−10.6316	−0.352239	−0.176119	−	0.984369i	$-0.556355\pi$
$911$	−10.6316	−0.352239	−0.176119	+	0.984369i	$0.556355\pi$
$912$	0	0
$913$	2.24708	0.0743675
$914$	0	0
$915$	−16.4040	−0.542299
$916$	0	0
$917$	49.6277	1.63885
$918$	0	0
$919$	14.4408	0.476357	0.238179	−	0.971221i	$-0.423450\pi$
$919$	14.4408	0.476357	0.238179	+	0.971221i	$0.423450\pi$
$920$	0	0
$921$	−12.3909	−0.408293
$922$	0	0
$923$	−19.3090	−0.635564
$924$	0	0
$925$	5.52045	0.181511
$926$	0	0
$927$	−1.81437	−0.0595918
$928$	0	0
$929$	−53.9358	−1.76958	−0.884789	−	0.465993i	$-0.845697\pi$
$929$	−53.9358	−1.76958	−0.884789	+	0.465993i	$0.845697\pi$
$930$	0	0
$931$	−1.70697	−0.0559438
$932$	0	0
$933$	−3.88140	−0.127071
$934$	0	0
$935$	9.68569	0.316756
$936$	0	0
$937$	−17.7141	−0.578694	−0.289347	−	0.957224i	$-0.593438\pi$
$937$	−17.7141	−0.578694	−0.289347	+	0.957224i	$0.593438\pi$
$938$	0	0
$939$	25.4363	0.830083
$940$	0	0
$941$	−23.2403	−0.757612	−0.378806	−	0.925476i	$-0.623665\pi$
$941$	−23.2403	−0.757612	−0.378806	+	0.925476i	$0.623665\pi$
$942$	0	0
$943$	−46.6797	−1.52010
$944$	0	0
$945$	36.8844	1.19985
$946$	0	0
$947$	13.9145	0.452159	0.226080	−	0.974109i	$-0.427409\pi$
$947$	13.9145	0.452159	0.226080	+	0.974109i	$0.427409\pi$
$948$	0	0
$949$	9.66863	0.313857
$950$	0	0
$951$	8.05269	0.261126
$952$	0	0
$953$	9.12371	0.295546	0.147773	−	0.989021i	$-0.452789\pi$
$953$	9.12371	0.295546	0.147773	+	0.989021i	$0.452789\pi$
$954$	0	0
$955$	−0.104001	−0.00336540
$956$	0	0
$957$	−11.6348	−0.376099
$958$	0	0
$959$	31.6623	1.02243
$960$	0	0
$961$	−29.2457	−0.943411
$962$	0	0
$963$	−9.80738	−0.316038
$964$	0	0
$965$	15.5819	0.501598
$966$	0	0
$967$	−19.5870	−0.629875	−0.314937	−	0.949112i	$-0.601984\pi$
$967$	−19.5870	−0.629875	−0.314937	+	0.949112i	$0.601984\pi$
$968$	0	0
$969$	3.58715	0.115236
$970$	0	0
$971$	36.4316	1.16914	0.584572	−	0.811342i	$-0.301262\pi$
$971$	36.4316	1.16914	0.584572	+	0.811342i	$0.301262\pi$
$972$	0	0
$973$	14.5277	0.465736
$974$	0	0
$975$	−1.17605	−0.0376638
$976$	0	0
$977$	35.1687	1.12515	0.562573	−	0.826748i	$-0.309812\pi$
$977$	35.1687	1.12515	0.562573	+	0.826748i	$0.309812\pi$
$978$	0	0
$979$	−1.07529	−0.0343665
$980$	0	0
$981$	7.52819	0.240357
$982$	0	0
$983$	−12.3314	−0.393310	−0.196655	−	0.980473i	$-0.563008\pi$
$983$	−12.3314	−0.393310	−0.196655	+	0.980473i	$0.563008\pi$
$984$	0	0
$985$	37.5004	1.19486
$986$	0	0
$987$	30.9223	0.984267
$988$	0	0
$989$	71.9923	2.28922
$990$	0	0
$991$	−43.5898	−1.38468	−0.692338	−	0.721573i	$-0.743419\pi$
$991$	−43.5898	−1.38468	−0.692338	+	0.721573i	$0.743419\pi$
$992$	0	0
$993$	−18.7385	−0.594650
$994$	0	0
$995$	0.817370	0.0259124
$996$	0	0
$997$	−1.06166	−0.0336230	−0.0168115	−	0.999859i	$-0.505352\pi$
$997$	−1.06166	−0.0336230	−0.0168115	+	0.999859i	$0.505352\pi$
$998$	0	0
$999$	−51.3301	−1.62401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.9		29
4.3	odd	2		4024.2.a.e.1.21	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.21	✓	29	4.3	odd	2
8048.2.a.w.1.9		29	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.29900 q^{3} +2.09702 q^{5} +3.13973 q^{7} -1.31260 q^{9} +O(q^{10})\)	\(q-1.29900 q^{3} +2.09702 q^{5} +3.13973 q^{7} -1.31260 q^{9} +0.999009 q^{11} -1.50268 q^{13} -2.72403 q^{15} +4.62336 q^{17} -0.597287 q^{19} -4.07850 q^{21} +8.14401 q^{23} -0.602491 q^{25} +5.60206 q^{27} +8.96563 q^{29} -1.32448 q^{31} -1.29771 q^{33} +6.58408 q^{35} -9.16272 q^{37} +1.95198 q^{39} -5.73178 q^{41} +8.83991 q^{43} -2.75256 q^{45} -7.58177 q^{47} +2.85788 q^{49} -6.00574 q^{51} +3.46835 q^{53} +2.09495 q^{55} +0.775874 q^{57} -7.24363 q^{59} +6.02195 q^{61} -4.12121 q^{63} -3.15116 q^{65} +6.73365 q^{67} -10.5791 q^{69} +12.8497 q^{71} -6.43424 q^{73} +0.782635 q^{75} +3.13661 q^{77} +16.4372 q^{79} -3.33927 q^{81} +2.24931 q^{83} +9.69530 q^{85} -11.6463 q^{87} -1.07636 q^{89} -4.71801 q^{91} +1.72050 q^{93} -1.25252 q^{95} -10.1728 q^{97} -1.31130 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

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