LMFDB - Embedded newform 8048.2.a.t.1.1

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:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.92891 q^{3} -2.56957 q^{5} +1.38532 q^{7} +5.57849 q^{9} +O(q^{10})$	$q-2.92891 q^{3} -2.56957 q^{5} +1.38532 q^{7} +5.57849 q^{9} +0.939680 q^{11} -7.18815 q^{13} +7.52604 q^{15} -2.97207 q^{17} -4.99618 q^{19} -4.05746 q^{21} +2.14523 q^{23} +1.60270 q^{25} -7.55216 q^{27} -6.66769 q^{29} +3.09554 q^{31} -2.75223 q^{33} -3.55967 q^{35} -2.80386 q^{37} +21.0534 q^{39} -1.51270 q^{41} +1.62370 q^{43} -14.3343 q^{45} -6.67119 q^{47} -5.08090 q^{49} +8.70490 q^{51} +7.26474 q^{53} -2.41458 q^{55} +14.6333 q^{57} -1.63908 q^{59} -7.36509 q^{61} +7.72798 q^{63} +18.4705 q^{65} +7.38323 q^{67} -6.28318 q^{69} +0.0410743 q^{71} +3.58460 q^{73} -4.69417 q^{75} +1.30176 q^{77} -4.57726 q^{79} +5.38409 q^{81} -16.4848 q^{83} +7.63694 q^{85} +19.5290 q^{87} -16.2950 q^{89} -9.95787 q^{91} -9.06655 q^{93} +12.8380 q^{95} -3.86000 q^{97} +5.24200 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * 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
$3$	−2.92891	−1.69100	−0.845502	−	0.533972i	$-0.820699\pi$
$3$	−2.92891	−1.69100	−0.845502	+	0.533972i	$0.820699\pi$
$4$	0	0
$5$	−2.56957	−1.14915	−0.574574	−	0.818453i	$-0.694832\pi$
$5$	−2.56957	−1.14915	−0.574574	+	0.818453i	$0.694832\pi$
$6$	0	0
$7$	1.38532	0.523601	0.261800	−	0.965122i	$-0.415684\pi$
$7$	1.38532	0.523601	0.261800	+	0.965122i	$0.415684\pi$
$8$	0	0
$9$	5.57849	1.85950
$10$	0	0
$11$	0.939680	0.283324	0.141662	−	0.989915i	$-0.454755\pi$
$11$	0.939680	0.283324	0.141662	+	0.989915i	$0.454755\pi$
$12$	0	0
$13$	−7.18815	−1.99363	−0.996817	−	0.0797220i	$-0.974597\pi$
$13$	−7.18815	−1.99363	−0.996817	+	0.0797220i	$0.974597\pi$
$14$	0	0
$15$	7.52604	1.94321
$16$	0	0
$17$	−2.97207	−0.720832	−0.360416	−	0.932792i	$-0.617365\pi$
$17$	−2.97207	−0.720832	−0.360416	+	0.932792i	$0.617365\pi$
$18$	0	0
$19$	−4.99618	−1.14620	−0.573101	−	0.819485i	$-0.694260\pi$
$19$	−4.99618	−1.14620	−0.573101	+	0.819485i	$0.694260\pi$
$20$	0	0
$21$	−4.05746	−0.885411
$22$	0	0
$23$	2.14523	0.447312	0.223656	−	0.974668i	$-0.428201\pi$
$23$	2.14523	0.447312	0.223656	+	0.974668i	$0.428201\pi$
$24$	0	0
$25$	1.60270	0.320541
$26$	0	0
$27$	−7.55216	−1.45341
$28$	0	0
$29$	−6.66769	−1.23816	−0.619080	−	0.785328i	$-0.712494\pi$
$29$	−6.66769	−1.23816	−0.619080	+	0.785328i	$0.712494\pi$
$30$	0	0
$31$	3.09554	0.555976	0.277988	−	0.960585i	$-0.410333\pi$
$31$	3.09554	0.555976	0.277988	+	0.960585i	$0.410333\pi$
$32$	0	0
$33$	−2.75223	−0.479103
$34$	0	0
$35$	−3.55967	−0.601695
$36$	0	0
$37$	−2.80386	−0.460952	−0.230476	−	0.973078i	$-0.574028\pi$
$37$	−2.80386	−0.460952	−0.230476	+	0.973078i	$0.574028\pi$
$38$	0	0
$39$	21.0534	3.37124
$40$	0	0
$41$	−1.51270	−0.236244	−0.118122	−	0.992999i	$-0.537687\pi$
$41$	−1.51270	−0.236244	−0.118122	+	0.992999i	$0.537687\pi$
$42$	0	0
$43$	1.62370	0.247612	0.123806	−	0.992306i	$-0.460490\pi$
$43$	1.62370	0.247612	0.123806	+	0.992306i	$0.460490\pi$
$44$	0	0
$45$	−14.3343	−2.13684
$46$	0	0
$47$	−6.67119	−0.973093	−0.486546	−	0.873655i	$-0.661744\pi$
$47$	−6.67119	−0.973093	−0.486546	+	0.873655i	$0.661744\pi$
$48$	0	0
$49$	−5.08090	−0.725842
$50$	0	0
$51$	8.70490	1.21893
$52$	0	0
$53$	7.26474	0.997889	0.498944	−	0.866634i	$-0.333721\pi$
$53$	7.26474	0.997889	0.498944	+	0.866634i	$0.333721\pi$
$54$	0	0
$55$	−2.41458	−0.325581
$56$	0	0
$57$	14.6333	1.93823
$58$	0	0
$59$	−1.63908	−0.213390	−0.106695	−	0.994292i	$-0.534027\pi$
$59$	−1.63908	−0.213390	−0.106695	+	0.994292i	$0.534027\pi$
$60$	0	0
$61$	−7.36509	−0.943003	−0.471501	−	0.881865i	$-0.656288\pi$
$61$	−7.36509	−0.943003	−0.471501	+	0.881865i	$0.656288\pi$
$62$	0	0
$63$	7.72798	0.973634
$64$	0	0
$65$	18.4705	2.29098
$66$	0	0
$67$	7.38323	0.902005	0.451003	−	0.892523i	$-0.351066\pi$
$67$	7.38323	0.902005	0.451003	+	0.892523i	$0.351066\pi$
$68$	0	0
$69$	−6.28318	−0.756406
$70$	0	0
$71$	0.0410743	0.00487462	0.00243731	−	0.999997i	$-0.499224\pi$
$71$	0.0410743	0.00487462	0.00243731	+	0.999997i	$0.499224\pi$
$72$	0	0
$73$	3.58460	0.419545	0.209773	−	0.977750i	$-0.432728\pi$
$73$	3.58460	0.419545	0.209773	+	0.977750i	$0.432728\pi$
$74$	0	0
$75$	−4.69417	−0.542036
$76$	0	0
$77$	1.30176	0.148349
$78$	0	0
$79$	−4.57726	−0.514982	−0.257491	−	0.966281i	$-0.582896\pi$
$79$	−4.57726	−0.514982	−0.257491	+	0.966281i	$0.582896\pi$
$80$	0	0
$81$	5.38409	0.598232
$82$	0	0
$83$	−16.4848	−1.80944	−0.904718	−	0.426010i	$-0.859919\pi$
$83$	−16.4848	−1.80944	−0.904718	+	0.426010i	$0.859919\pi$
$84$	0	0
$85$	7.63694	0.828343
$86$	0	0
$87$	19.5290	2.09373
$88$	0	0
$89$	−16.2950	−1.72726	−0.863632	−	0.504124i	$-0.831816\pi$
$89$	−16.2950	−1.72726	−0.863632	+	0.504124i	$0.831816\pi$
$90$	0	0
$91$	−9.95787	−1.04387
$92$	0	0
$93$	−9.06655	−0.940157
$94$	0	0
$95$	12.8380	1.31715
$96$	0	0
$97$	−3.86000	−0.391924	−0.195962	−	0.980611i	$-0.562783\pi$
$97$	−3.86000	−0.391924	−0.195962	+	0.980611i	$0.562783\pi$
$98$	0	0
$99$	5.24200	0.526840
$100$	0	0
$101$	4.94381	0.491927	0.245964	−	0.969279i	$-0.420896\pi$
$101$	4.94381	0.491927	0.245964	+	0.969279i	$0.420896\pi$
$102$	0	0
$103$	2.92579	0.288287	0.144144	−	0.989557i	$-0.453957\pi$
$103$	2.92579	0.288287	0.144144	+	0.989557i	$0.453957\pi$
$104$	0	0
$105$	10.4260	1.01747
$106$	0	0
$107$	−3.39490	−0.328197	−0.164098	−	0.986444i	$-0.552471\pi$
$107$	−3.39490	−0.328197	−0.164098	+	0.986444i	$0.552471\pi$
$108$	0	0
$109$	−15.9261	−1.52545	−0.762724	−	0.646724i	$-0.776139\pi$
$109$	−15.9261	−1.52545	−0.762724	+	0.646724i	$0.776139\pi$
$110$	0	0
$111$	8.21225	0.779472
$112$	0	0
$113$	−12.0566	−1.13419	−0.567095	−	0.823652i	$-0.691933\pi$
$113$	−12.0566	−1.13419	−0.567095	+	0.823652i	$0.691933\pi$
$114$	0	0
$115$	−5.51233	−0.514027
$116$	0	0
$117$	−40.0990	−3.70716
$118$	0	0
$119$	−4.11726	−0.377428
$120$	0	0
$121$	−10.1170	−0.919727
$122$	0	0
$123$	4.43055	0.399490
$124$	0	0
$125$	8.72960	0.780799
$126$	0	0
$127$	−17.6245	−1.56392	−0.781959	−	0.623330i	$-0.785779\pi$
$127$	−17.6245	−1.56392	−0.781959	+	0.623330i	$0.785779\pi$
$128$	0	0
$129$	−4.75566	−0.418713
$130$	0	0
$131$	−14.9748	−1.30836	−0.654178	−	0.756340i	$-0.726985\pi$
$131$	−14.9748	−1.30836	−0.654178	+	0.756340i	$0.726985\pi$
$132$	0	0
$133$	−6.92129	−0.600152
$134$	0	0
$135$	19.4058	1.67019
$136$	0	0
$137$	3.68175	0.314553	0.157277	−	0.987555i	$-0.449729\pi$
$137$	3.68175	0.314553	0.157277	+	0.987555i	$0.449729\pi$
$138$	0	0
$139$	1.57636	0.133705	0.0668525	−	0.997763i	$-0.478704\pi$
$139$	1.57636	0.133705	0.0668525	+	0.997763i	$0.478704\pi$
$140$	0	0
$141$	19.5393	1.64550
$142$	0	0
$143$	−6.75456	−0.564845
$144$	0	0
$145$	17.1331	1.42283
$146$	0	0
$147$	14.8815	1.22740
$148$	0	0
$149$	−6.67001	−0.546429	−0.273214	−	0.961953i	$-0.588087\pi$
$149$	−6.67001	−0.546429	−0.273214	+	0.961953i	$0.588087\pi$
$150$	0	0
$151$	−11.2795	−0.917911	−0.458955	−	0.888459i	$-0.651776\pi$
$151$	−11.2795	−0.917911	−0.458955	+	0.888459i	$0.651776\pi$
$152$	0	0
$153$	−16.5796	−1.34039
$154$	0	0
$155$	−7.95422	−0.638898
$156$	0	0
$157$	−0.984677	−0.0785858	−0.0392929	−	0.999228i	$-0.512511\pi$
$157$	−0.984677	−0.0785858	−0.0392929	+	0.999228i	$0.512511\pi$
$158$	0	0
$159$	−21.2777	−1.68743
$160$	0	0
$161$	2.97183	0.234213
$162$	0	0
$163$	14.0506	1.10053	0.550264	−	0.834991i	$-0.314527\pi$
$163$	14.0506	1.10053	0.550264	+	0.834991i	$0.314527\pi$
$164$	0	0
$165$	7.07207	0.550560
$166$	0	0
$167$	3.74816	0.290041	0.145021	−	0.989429i	$-0.453675\pi$
$167$	3.74816	0.290041	0.145021	+	0.989429i	$0.453675\pi$
$168$	0	0
$169$	38.6695	2.97458
$170$	0	0
$171$	−27.8711	−2.13136
$172$	0	0
$173$	−9.99365	−0.759803	−0.379901	−	0.925027i	$-0.624042\pi$
$173$	−9.99365	−0.759803	−0.379901	+	0.925027i	$0.624042\pi$
$174$	0	0
$175$	2.22026	0.167836
$176$	0	0
$177$	4.80071	0.360843
$178$	0	0
$179$	16.9394	1.26611	0.633056	−	0.774106i	$-0.281800\pi$
$179$	16.9394	1.26611	0.633056	+	0.774106i	$0.281800\pi$
$180$	0	0
$181$	−7.18450	−0.534020	−0.267010	−	0.963694i	$-0.586036\pi$
$181$	−7.18450	−0.534020	−0.267010	+	0.963694i	$0.586036\pi$
$182$	0	0
$183$	21.5717	1.59462
$184$	0	0
$185$	7.20473	0.529702
$186$	0	0
$187$	−2.79279	−0.204229
$188$	0	0
$189$	−10.4621	−0.761008
$190$	0	0
$191$	−17.9695	−1.30023	−0.650113	−	0.759838i	$-0.725278\pi$
$191$	−17.9695	−1.30023	−0.650113	+	0.759838i	$0.725278\pi$
$192$	0	0
$193$	12.9578	0.932723	0.466361	−	0.884594i	$-0.345565\pi$
$193$	12.9578	0.932723	0.466361	+	0.884594i	$0.345565\pi$
$194$	0	0
$195$	−54.0983	−3.87406
$196$	0	0
$197$	−8.12366	−0.578786	−0.289393	−	0.957210i	$-0.593454\pi$
$197$	−8.12366	−0.578786	−0.289393	+	0.957210i	$0.593454\pi$
$198$	0	0
$199$	−7.31041	−0.518221	−0.259110	−	0.965848i	$-0.583429\pi$
$199$	−7.31041	−0.518221	−0.259110	+	0.965848i	$0.583429\pi$
$200$	0	0
$201$	−21.6248	−1.52530
$202$	0	0
$203$	−9.23687	−0.648301
$204$	0	0
$205$	3.88699	0.271479
$206$	0	0
$207$	11.9672	0.831775
$208$	0	0
$209$	−4.69481	−0.324747
$210$	0	0
$211$	−24.4311	−1.68190	−0.840951	−	0.541111i	$-0.818004\pi$
$211$	−24.4311	−1.68190	−0.840951	+	0.541111i	$0.818004\pi$
$212$	0	0
$213$	−0.120303	−0.00824300
$214$	0	0
$215$	−4.17221	−0.284542
$216$	0	0
$217$	4.28831	0.291109
$218$	0	0
$219$	−10.4990	−0.709453
$220$	0	0
$221$	21.3637	1.43708
$222$	0	0
$223$	−2.57971	−0.172750	−0.0863751	−	0.996263i	$-0.527528\pi$
$223$	−2.57971	−0.172750	−0.0863751	+	0.996263i	$0.527528\pi$
$224$	0	0
$225$	8.94067	0.596045
$226$	0	0
$227$	−3.58524	−0.237961	−0.118980	−	0.992897i	$-0.537963\pi$
$227$	−3.58524	−0.237961	−0.118980	+	0.992897i	$0.537963\pi$
$228$	0	0
$229$	10.3082	0.681187	0.340594	−	0.940211i	$-0.389372\pi$
$229$	10.3082	0.681187	0.340594	+	0.940211i	$0.389372\pi$
$230$	0	0
$231$	−3.81272	−0.250858
$232$	0	0
$233$	21.7133	1.42249	0.711244	−	0.702945i	$-0.248132\pi$
$233$	21.7133	1.42249	0.711244	+	0.702945i	$0.248132\pi$
$234$	0	0
$235$	17.1421	1.11823
$236$	0	0
$237$	13.4064	0.870838
$238$	0	0
$239$	−16.7538	−1.08371	−0.541856	−	0.840471i	$-0.682278\pi$
$239$	−16.7538	−1.08371	−0.541856	+	0.840471i	$0.682278\pi$
$240$	0	0
$241$	3.91112	0.251937	0.125969	−	0.992034i	$-0.459796\pi$
$241$	3.91112	0.251937	0.125969	+	0.992034i	$0.459796\pi$
$242$	0	0
$243$	6.88699	0.441800
$244$	0	0
$245$	13.0557	0.834100
$246$	0	0
$247$	35.9133	2.28511
$248$	0	0
$249$	48.2823	3.05977
$250$	0	0
$251$	9.53273	0.601701	0.300850	−	0.953671i	$-0.402730\pi$
$251$	9.53273	0.601701	0.300850	+	0.953671i	$0.402730\pi$
$252$	0	0
$253$	2.01583	0.126734
$254$	0	0
$255$	−22.3679	−1.40073
$256$	0	0
$257$	−29.8769	−1.86367	−0.931835	−	0.362882i	$-0.881793\pi$
$257$	−29.8769	−1.86367	−0.931835	+	0.362882i	$0.881793\pi$
$258$	0	0
$259$	−3.88424	−0.241355
$260$	0	0
$261$	−37.1957	−2.30235
$262$	0	0
$263$	−4.08055	−0.251617	−0.125809	−	0.992055i	$-0.540153\pi$
$263$	−4.08055	−0.251617	−0.125809	+	0.992055i	$0.540153\pi$
$264$	0	0
$265$	−18.6673	−1.14672
$266$	0	0
$267$	47.7264	2.92081
$268$	0	0
$269$	−27.5190	−1.67787	−0.838933	−	0.544235i	$-0.816820\pi$
$269$	−27.5190	−1.67787	−0.838933	+	0.544235i	$0.816820\pi$
$270$	0	0
$271$	−1.47867	−0.0898230	−0.0449115	−	0.998991i	$-0.514301\pi$
$271$	−1.47867	−0.0898230	−0.0449115	+	0.998991i	$0.514301\pi$
$272$	0	0
$273$	29.1657	1.76519
$274$	0	0
$275$	1.50603	0.0908170
$276$	0	0
$277$	−18.9790	−1.14034	−0.570168	−	0.821528i	$-0.693122\pi$
$277$	−18.9790	−1.14034	−0.570168	+	0.821528i	$0.693122\pi$
$278$	0	0
$279$	17.2684	1.03383
$280$	0	0
$281$	20.9247	1.24826	0.624132	−	0.781319i	$-0.285453\pi$
$281$	20.9247	1.24826	0.624132	+	0.781319i	$0.285453\pi$
$282$	0	0
$283$	8.68900	0.516507	0.258254	−	0.966077i	$-0.416853\pi$
$283$	8.68900	0.516507	0.258254	+	0.966077i	$0.416853\pi$
$284$	0	0
$285$	−37.6014	−2.22731
$286$	0	0
$287$	−2.09557	−0.123697
$288$	0	0
$289$	−8.16682	−0.480401
$290$	0	0
$291$	11.3056	0.662745
$292$	0	0
$293$	19.7851	1.15586	0.577928	−	0.816088i	$-0.303862\pi$
$293$	19.7851	1.15586	0.577928	+	0.816088i	$0.303862\pi$
$294$	0	0
$295$	4.21173	0.245217
$296$	0	0
$297$	−7.09661	−0.411787
$298$	0	0
$299$	−15.4203	−0.891776
$300$	0	0
$301$	2.24934	0.129650
$302$	0	0
$303$	−14.4800	−0.831851
$304$	0	0
$305$	18.9251	1.08365
$306$	0	0
$307$	31.5095	1.79834	0.899170	−	0.437599i	$-0.144171\pi$
$307$	31.5095	1.79834	0.899170	+	0.437599i	$0.144171\pi$
$308$	0	0
$309$	−8.56938	−0.487495
$310$	0	0
$311$	−1.94679	−0.110392	−0.0551960	−	0.998476i	$-0.517578\pi$
$311$	−1.94679	−0.110392	−0.0551960	+	0.998476i	$0.517578\pi$
$312$	0	0
$313$	−23.1264	−1.30718	−0.653591	−	0.756848i	$-0.726738\pi$
$313$	−23.1264	−1.30718	−0.653591	+	0.756848i	$0.726738\pi$
$314$	0	0
$315$	−19.8576	−1.11885
$316$	0	0
$317$	23.6576	1.32874	0.664372	−	0.747402i	$-0.268699\pi$
$317$	23.6576	1.32874	0.664372	+	0.747402i	$0.268699\pi$
$318$	0	0
$319$	−6.26550	−0.350801
$320$	0	0
$321$	9.94333	0.554983
$322$	0	0
$323$	14.8490	0.826219
$324$	0	0
$325$	−11.5205	−0.639041
$326$	0	0
$327$	46.6462	2.57954
$328$	0	0
$329$	−9.24171	−0.509512
$330$	0	0
$331$	33.4116	1.83647	0.918233	−	0.396040i	$-0.129616\pi$
$331$	33.4116	1.83647	0.918233	+	0.396040i	$0.129616\pi$
$332$	0	0
$333$	−15.6413	−0.857139
$334$	0	0
$335$	−18.9718	−1.03654
$336$	0	0
$337$	−31.5947	−1.72107	−0.860535	−	0.509391i	$-0.829871\pi$
$337$	−31.5947	−1.72107	−0.860535	+	0.509391i	$0.829871\pi$
$338$	0	0
$339$	35.3127	1.91792
$340$	0	0
$341$	2.90882	0.157521
$342$	0	0
$343$	−16.7359	−0.903652
$344$	0	0
$345$	16.1451	0.869223
$346$	0	0
$347$	14.6105	0.784331	0.392165	−	0.919895i	$-0.371726\pi$
$347$	14.6105	0.784331	0.392165	+	0.919895i	$0.371726\pi$
$348$	0	0
$349$	−3.75209	−0.200845	−0.100422	−	0.994945i	$-0.532019\pi$
$349$	−3.75209	−0.200845	−0.100422	+	0.994945i	$0.532019\pi$
$350$	0	0
$351$	54.2860	2.89757
$352$	0	0
$353$	8.76527	0.466528	0.233264	−	0.972413i	$-0.425059\pi$
$353$	8.76527	0.466528	0.233264	+	0.972413i	$0.425059\pi$
$354$	0	0
$355$	−0.105543	−0.00560166
$356$	0	0
$357$	12.0591	0.638233
$358$	0	0
$359$	−14.4056	−0.760297	−0.380149	−	0.924925i	$-0.624127\pi$
$359$	−14.4056	−0.760297	−0.380149	+	0.924925i	$0.624127\pi$
$360$	0	0
$361$	5.96177	0.313777
$362$	0	0
$363$	29.6317	1.55526
$364$	0	0
$365$	−9.21089	−0.482120
$366$	0	0
$367$	2.14865	0.112159	0.0560794	−	0.998426i	$-0.482140\pi$
$367$	2.14865	0.112159	0.0560794	+	0.998426i	$0.482140\pi$
$368$	0	0
$369$	−8.43858	−0.439295
$370$	0	0
$371$	10.0640	0.522495
$372$	0	0
$373$	0.303027	0.0156901	0.00784507	−	0.999969i	$-0.497503\pi$
$373$	0.303027	0.0156901	0.00784507	+	0.999969i	$0.497503\pi$
$374$	0	0
$375$	−25.5682	−1.32033
$376$	0	0
$377$	47.9284	2.46844
$378$	0	0
$379$	17.8880	0.918843	0.459421	−	0.888218i	$-0.348057\pi$
$379$	17.8880	0.918843	0.459421	+	0.888218i	$0.348057\pi$
$380$	0	0
$381$	51.6204	2.64459
$382$	0	0
$383$	31.7042	1.62001	0.810004	−	0.586425i	$-0.199465\pi$
$383$	31.7042	1.62001	0.810004	+	0.586425i	$0.199465\pi$
$384$	0	0
$385$	−3.34495	−0.170475
$386$	0	0
$387$	9.05779	0.460433
$388$	0	0
$389$	−0.757144	−0.0383887	−0.0191944	−	0.999816i	$-0.506110\pi$
$389$	−0.757144	−0.0383887	−0.0191944	+	0.999816i	$0.506110\pi$
$390$	0	0
$391$	−6.37577	−0.322437
$392$	0	0
$393$	43.8599	2.21244
$394$	0	0
$395$	11.7616	0.591791
$396$	0	0
$397$	7.65636	0.384262	0.192131	−	0.981369i	$-0.438460\pi$
$397$	7.65636	0.384262	0.192131	+	0.981369i	$0.438460\pi$
$398$	0	0
$399$	20.2718	1.01486
$400$	0	0
$401$	−4.28035	−0.213751	−0.106875	−	0.994272i	$-0.534085\pi$
$401$	−4.28035	−0.213751	−0.106875	+	0.994272i	$0.534085\pi$
$402$	0	0
$403$	−22.2512	−1.10841
$404$	0	0
$405$	−13.8348	−0.687457
$406$	0	0
$407$	−2.63473	−0.130599
$408$	0	0
$409$	33.8882	1.67566	0.837832	−	0.545928i	$-0.183823\pi$
$409$	33.8882	1.67566	0.837832	+	0.545928i	$0.183823\pi$
$410$	0	0
$411$	−10.7835	−0.531911
$412$	0	0
$413$	−2.27065	−0.111731
$414$	0	0
$415$	42.3588	2.07931
$416$	0	0
$417$	−4.61701	−0.226096
$418$	0	0
$419$	−35.7914	−1.74852	−0.874261	−	0.485456i	$-0.838654\pi$
$419$	−35.7914	−1.74852	−0.874261	+	0.485456i	$0.838654\pi$
$420$	0	0
$421$	−1.98414	−0.0967011	−0.0483505	−	0.998830i	$-0.515396\pi$
$421$	−1.98414	−0.0967011	−0.0483505	+	0.998830i	$0.515396\pi$
$422$	0	0
$423$	−37.2152	−1.80946
$424$	0	0
$425$	−4.76335	−0.231056
$426$	0	0
$427$	−10.2030	−0.493757
$428$	0	0
$429$	19.7835	0.955155
$430$	0	0
$431$	0.828177	0.0398919	0.0199459	−	0.999801i	$-0.493651\pi$
$431$	0.828177	0.0398919	0.0199459	+	0.999801i	$0.493651\pi$
$432$	0	0
$433$	−9.85236	−0.473474	−0.236737	−	0.971574i	$-0.576078\pi$
$433$	−9.85236	−0.473474	−0.236737	+	0.971574i	$0.576078\pi$
$434$	0	0
$435$	−50.1813	−2.40601
$436$	0	0
$437$	−10.7180	−0.512709
$438$	0	0
$439$	17.9324	0.855867	0.427934	−	0.903810i	$-0.359242\pi$
$439$	17.9324	0.855867	0.427934	+	0.903810i	$0.359242\pi$
$440$	0	0
$441$	−28.3437	−1.34970
$442$	0	0
$443$	17.2531	0.819717	0.409859	−	0.912149i	$-0.365578\pi$
$443$	17.2531	0.819717	0.409859	+	0.912149i	$0.365578\pi$
$444$	0	0
$445$	41.8711	1.98488
$446$	0	0
$447$	19.5358	0.924013
$448$	0	0
$449$	25.1868	1.18864	0.594319	−	0.804229i	$-0.297422\pi$
$449$	25.1868	1.18864	0.594319	+	0.804229i	$0.297422\pi$
$450$	0	0
$451$	−1.42145	−0.0669336
$452$	0	0
$453$	33.0365	1.55219
$454$	0	0
$455$	25.5875	1.19956
$456$	0	0
$457$	10.7469	0.502721	0.251361	−	0.967894i	$-0.419122\pi$
$457$	10.7469	0.502721	0.251361	+	0.967894i	$0.419122\pi$
$458$	0	0
$459$	22.4455	1.04767
$460$	0	0
$461$	19.6100	0.913331	0.456666	−	0.889638i	$-0.349044\pi$
$461$	19.6100	0.913331	0.456666	+	0.889638i	$0.349044\pi$
$462$	0	0
$463$	18.6811	0.868186	0.434093	−	0.900868i	$-0.357069\pi$
$463$	18.6811	0.868186	0.434093	+	0.900868i	$0.357069\pi$
$464$	0	0
$465$	23.2972	1.08038
$466$	0	0
$467$	−18.3467	−0.848982	−0.424491	−	0.905432i	$-0.639547\pi$
$467$	−18.3467	−0.848982	−0.424491	+	0.905432i	$0.639547\pi$
$468$	0	0
$469$	10.2281	0.472291
$470$	0	0
$471$	2.88403	0.132889
$472$	0	0
$473$	1.52576	0.0701544
$474$	0	0
$475$	−8.00739	−0.367404
$476$	0	0
$477$	40.5263	1.85557
$478$	0	0
$479$	−33.7880	−1.54382	−0.771908	−	0.635735i	$-0.780697\pi$
$479$	−33.7880	−1.54382	−0.771908	+	0.635735i	$0.780697\pi$
$480$	0	0
$481$	20.1546	0.918970
$482$	0	0
$483$	−8.70420	−0.396055
$484$	0	0
$485$	9.91856	0.450379
$486$	0	0
$487$	25.9193	1.17452	0.587258	−	0.809400i	$-0.300207\pi$
$487$	25.9193	1.17452	0.587258	+	0.809400i	$0.300207\pi$
$488$	0	0
$489$	−41.1529	−1.86100
$490$	0	0
$491$	11.3385	0.511700	0.255850	−	0.966717i	$-0.417645\pi$
$491$	11.3385	0.511700	0.255850	+	0.966717i	$0.417645\pi$
$492$	0	0
$493$	19.8168	0.892505
$494$	0	0
$495$	−13.4697	−0.605418
$496$	0	0
$497$	0.0569009	0.00255235
$498$	0	0
$499$	−38.8854	−1.74075	−0.870374	−	0.492392i	$-0.836123\pi$
$499$	−38.8854	−1.74075	−0.870374	+	0.492392i	$0.836123\pi$
$500$	0	0
$501$	−10.9780	−0.490461
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−12.7035	−0.565297
$506$	0	0
$507$	−113.259	−5.03002
$508$	0	0
$509$	17.9349	0.794950	0.397475	−	0.917613i	$-0.369887\pi$
$509$	17.9349	0.794950	0.397475	+	0.917613i	$0.369887\pi$
$510$	0	0
$511$	4.96581	0.219674
$512$	0	0
$513$	37.7319	1.66590
$514$	0	0
$515$	−7.51804	−0.331284
$516$	0	0
$517$	−6.26878	−0.275701
$518$	0	0
$519$	29.2705	1.28483
$520$	0	0
$521$	−8.32467	−0.364711	−0.182355	−	0.983233i	$-0.558372\pi$
$521$	−8.32467	−0.364711	−0.182355	+	0.983233i	$0.558372\pi$
$522$	0	0
$523$	32.6508	1.42772	0.713860	−	0.700288i	$-0.246945\pi$
$523$	32.6508	1.42772	0.713860	+	0.700288i	$0.246945\pi$
$524$	0	0
$525$	−6.50292	−0.283811
$526$	0	0
$527$	−9.20015	−0.400765
$528$	0	0
$529$	−18.3980	−0.799912
$530$	0	0
$531$	−9.14359	−0.396798
$532$	0	0
$533$	10.8735	0.470984
$534$	0	0
$535$	8.72343	0.377147
$536$	0	0
$537$	−49.6140	−2.14100
$538$	0	0
$539$	−4.77442	−0.205649
$540$	0	0
$541$	−11.7107	−0.503483	−0.251742	−	0.967795i	$-0.581003\pi$
$541$	−11.7107	−0.503483	−0.251742	+	0.967795i	$0.581003\pi$
$542$	0	0
$543$	21.0427	0.903030
$544$	0	0
$545$	40.9234	1.75297
$546$	0	0
$547$	19.0172	0.813117	0.406559	−	0.913625i	$-0.366729\pi$
$547$	19.0172	0.813117	0.406559	+	0.913625i	$0.366729\pi$
$548$	0	0
$549$	−41.0861	−1.75351
$550$	0	0
$551$	33.3130	1.41918
$552$	0	0
$553$	−6.34096	−0.269645
$554$	0	0
$555$	−21.1020	−0.895729
$556$	0	0
$557$	−10.7762	−0.456604	−0.228302	−	0.973590i	$-0.573317\pi$
$557$	−10.7762	−0.456604	−0.228302	+	0.973590i	$0.573317\pi$
$558$	0	0
$559$	−11.6714	−0.493647
$560$	0	0
$561$	8.17982	0.345352
$562$	0	0
$563$	−32.7888	−1.38188	−0.690941	−	0.722911i	$-0.742804\pi$
$563$	−32.7888	−1.38188	−0.690941	+	0.722911i	$0.742804\pi$
$564$	0	0
$565$	30.9803	1.30335
$566$	0	0
$567$	7.45867	0.313235
$568$	0	0
$569$	4.29646	0.180117	0.0900585	−	0.995936i	$-0.471295\pi$
$569$	4.29646	0.180117	0.0900585	+	0.995936i	$0.471295\pi$
$570$	0	0
$571$	16.5047	0.690699	0.345350	−	0.938474i	$-0.387760\pi$
$571$	16.5047	0.690699	0.345350	+	0.938474i	$0.387760\pi$
$572$	0	0
$573$	52.6309	2.19869
$574$	0	0
$575$	3.43817	0.143382
$576$	0	0
$577$	42.0272	1.74962	0.874808	−	0.484470i	$-0.160987\pi$
$577$	42.0272	1.74962	0.874808	+	0.484470i	$0.160987\pi$
$578$	0	0
$579$	−37.9522	−1.57724
$580$	0	0
$581$	−22.8366	−0.947423
$582$	0	0
$583$	6.82653	0.282726
$584$	0	0
$585$	103.037	4.26007
$586$	0	0
$587$	33.3612	1.37696	0.688481	−	0.725254i	$-0.258278\pi$
$587$	33.3612	1.37696	0.688481	+	0.725254i	$0.258278\pi$
$588$	0	0
$589$	−15.4659	−0.637260
$590$	0	0
$591$	23.7934	0.978731
$592$	0	0
$593$	−7.12181	−0.292458	−0.146229	−	0.989251i	$-0.546714\pi$
$593$	−7.12181	−0.292458	−0.146229	+	0.989251i	$0.546714\pi$
$594$	0	0
$595$	10.5796	0.433721
$596$	0	0
$597$	21.4115	0.876314
$598$	0	0
$599$	3.68412	0.150529	0.0752645	−	0.997164i	$-0.476020\pi$
$599$	3.68412	0.150529	0.0752645	+	0.997164i	$0.476020\pi$
$600$	0	0
$601$	10.6368	0.433883	0.216942	−	0.976185i	$-0.430392\pi$
$601$	10.6368	0.433883	0.216942	+	0.976185i	$0.430392\pi$
$602$	0	0
$603$	41.1873	1.67728
$604$	0	0
$605$	25.9964	1.05690
$606$	0	0
$607$	45.0092	1.82687	0.913434	−	0.406987i	$-0.133421\pi$
$607$	45.0092	1.82687	0.913434	+	0.406987i	$0.133421\pi$
$608$	0	0
$609$	27.0539	1.09628
$610$	0	0
$611$	47.9535	1.93999
$612$	0	0
$613$	−21.5302	−0.869598	−0.434799	−	0.900528i	$-0.643181\pi$
$613$	−21.5302	−0.869598	−0.434799	+	0.900528i	$0.643181\pi$
$614$	0	0
$615$	−11.3846	−0.459073
$616$	0	0
$617$	−43.3204	−1.74401	−0.872007	−	0.489493i	$-0.837182\pi$
$617$	−43.3204	−1.74401	−0.872007	+	0.489493i	$0.837182\pi$
$618$	0	0
$619$	−10.3039	−0.414149	−0.207074	−	0.978325i	$-0.566394\pi$
$619$	−10.3039	−0.414149	−0.207074	+	0.978325i	$0.566394\pi$
$620$	0	0
$621$	−16.2011	−0.650129
$622$	0	0
$623$	−22.5737	−0.904396
$624$	0	0
$625$	−30.4449	−1.21779
$626$	0	0
$627$	13.7506	0.549148
$628$	0	0
$629$	8.33327	0.332269
$630$	0	0
$631$	25.5848	1.01851	0.509257	−	0.860614i	$-0.329920\pi$
$631$	25.5848	1.01851	0.509257	+	0.860614i	$0.329920\pi$
$632$	0	0
$633$	71.5563	2.84411
$634$	0	0
$635$	45.2873	1.79717
$636$	0	0
$637$	36.5222	1.44706
$638$	0	0
$639$	0.229132	0.00906434
$640$	0	0
$641$	−35.0264	−1.38346	−0.691729	−	0.722157i	$-0.743151\pi$
$641$	−35.0264	−1.38346	−0.691729	+	0.722157i	$0.743151\pi$
$642$	0	0
$643$	−13.0094	−0.513039	−0.256519	−	0.966539i	$-0.582576\pi$
$643$	−13.0094	−0.513039	−0.256519	+	0.966539i	$0.582576\pi$
$644$	0	0
$645$	12.2200	0.481163
$646$	0	0
$647$	−10.3661	−0.407535	−0.203767	−	0.979019i	$-0.565319\pi$
$647$	−10.3661	−0.407535	−0.203767	+	0.979019i	$0.565319\pi$
$648$	0	0
$649$	−1.54021	−0.0604585
$650$	0	0
$651$	−12.5600	−0.492267
$652$	0	0
$653$	24.4697	0.957574	0.478787	−	0.877931i	$-0.341077\pi$
$653$	24.4697	0.957574	0.478787	+	0.877931i	$0.341077\pi$
$654$	0	0
$655$	38.4789	1.50350
$656$	0	0
$657$	19.9966	0.780143
$658$	0	0
$659$	−24.2422	−0.944340	−0.472170	−	0.881507i	$-0.656529\pi$
$659$	−24.2422	−0.944340	−0.472170	+	0.881507i	$0.656529\pi$
$660$	0	0
$661$	−5.85937	−0.227903	−0.113952	−	0.993486i	$-0.536351\pi$
$661$	−5.85937	−0.227903	−0.113952	+	0.993486i	$0.536351\pi$
$662$	0	0
$663$	−62.5722	−2.43010
$664$	0	0
$665$	17.7848	0.689663
$666$	0	0
$667$	−14.3038	−0.553844
$668$	0	0
$669$	7.55573	0.292122
$670$	0	0
$671$	−6.92083	−0.267176
$672$	0	0
$673$	−2.42226	−0.0933711	−0.0466855	−	0.998910i	$-0.514866\pi$
$673$	−2.42226	−0.0933711	−0.0466855	+	0.998910i	$0.514866\pi$
$674$	0	0
$675$	−12.1039	−0.465879
$676$	0	0
$677$	48.6655	1.87037	0.935183	−	0.354164i	$-0.115234\pi$
$677$	48.6655	1.87037	0.935183	+	0.354164i	$0.115234\pi$
$678$	0	0
$679$	−5.34733	−0.205212
$680$	0	0
$681$	10.5008	0.402392
$682$	0	0
$683$	21.5719	0.825425	0.412712	−	0.910861i	$-0.364581\pi$
$683$	21.5719	0.825425	0.412712	+	0.910861i	$0.364581\pi$
$684$	0	0
$685$	−9.46053	−0.361468
$686$	0	0
$687$	−30.1918	−1.15189
$688$	0	0
$689$	−52.2200	−1.98943
$690$	0	0
$691$	12.4968	0.475400	0.237700	−	0.971339i	$-0.423606\pi$
$691$	12.4968	0.475400	0.237700	+	0.971339i	$0.423606\pi$
$692$	0	0
$693$	7.26183	0.275854
$694$	0	0
$695$	−4.05057	−0.153647
$696$	0	0
$697$	4.49584	0.170292
$698$	0	0
$699$	−63.5963	−2.40543
$700$	0	0
$701$	−10.9032	−0.411808	−0.205904	−	0.978572i	$-0.566013\pi$
$701$	−10.9032	−0.411808	−0.205904	+	0.978572i	$0.566013\pi$
$702$	0	0
$703$	14.0086	0.528344
$704$	0	0
$705$	−50.2076	−1.89093
$706$	0	0
$707$	6.84874	0.257574
$708$	0	0
$709$	26.9420	1.01183	0.505915	−	0.862583i	$-0.331155\pi$
$709$	26.9420	1.01183	0.505915	+	0.862583i	$0.331155\pi$
$710$	0	0
$711$	−25.5342	−0.957608
$712$	0	0
$713$	6.64065	0.248694
$714$	0	0
$715$	17.3563	0.649090
$716$	0	0
$717$	49.0702	1.83256
$718$	0	0
$719$	11.7223	0.437170	0.218585	−	0.975818i	$-0.429856\pi$
$719$	11.7223	0.437170	0.218585	+	0.975818i	$0.429856\pi$
$720$	0	0
$721$	4.05315	0.150947
$722$	0	0
$723$	−11.4553	−0.426027
$724$	0	0
$725$	−10.6863	−0.396881
$726$	0	0
$727$	−31.0428	−1.15132	−0.575658	−	0.817691i	$-0.695254\pi$
$727$	−31.0428	−1.15132	−0.575658	+	0.817691i	$0.695254\pi$
$728$	0	0
$729$	−36.3236	−1.34532
$730$	0	0
$731$	−4.82574	−0.178486
$732$	0	0
$733$	−31.1603	−1.15093	−0.575467	−	0.817825i	$-0.695180\pi$
$733$	−31.1603	−1.15093	−0.575467	+	0.817825i	$0.695180\pi$
$734$	0	0
$735$	−38.2390	−1.41047
$736$	0	0
$737$	6.93788	0.255560
$738$	0	0
$739$	−26.8272	−0.986853	−0.493426	−	0.869788i	$-0.664256\pi$
$739$	−26.8272	−0.986853	−0.493426	+	0.869788i	$0.664256\pi$
$740$	0	0
$741$	−105.187	−3.86413
$742$	0	0
$743$	−7.64243	−0.280374	−0.140187	−	0.990125i	$-0.544770\pi$
$743$	−7.64243	−0.280374	−0.140187	+	0.990125i	$0.544770\pi$
$744$	0	0
$745$	17.1391	0.627927
$746$	0	0
$747$	−91.9600	−3.36464
$748$	0	0
$749$	−4.70301	−0.171844
$750$	0	0
$751$	17.3472	0.633009	0.316505	−	0.948591i	$-0.397491\pi$
$751$	17.3472	0.633009	0.316505	+	0.948591i	$0.397491\pi$
$752$	0	0
$753$	−27.9205	−1.01748
$754$	0	0
$755$	28.9834	1.05482
$756$	0	0
$757$	3.63774	0.132216	0.0661079	−	0.997812i	$-0.478942\pi$
$757$	3.63774	0.132216	0.0661079	+	0.997812i	$0.478942\pi$
$758$	0	0
$759$	−5.90418	−0.214308
$760$	0	0
$761$	−25.3825	−0.920117	−0.460058	−	0.887889i	$-0.652171\pi$
$761$	−25.3825	−0.920117	−0.460058	+	0.887889i	$0.652171\pi$
$762$	0	0
$763$	−22.0628	−0.798726
$764$	0	0
$765$	42.6026	1.54030
$766$	0	0
$767$	11.7819	0.425422
$768$	0	0
$769$	50.9883	1.83868	0.919342	−	0.393459i	$-0.128722\pi$
$769$	50.9883	1.83868	0.919342	+	0.393459i	$0.128722\pi$
$770$	0	0
$771$	87.5067	3.15148
$772$	0	0
$773$	−22.5667	−0.811668	−0.405834	−	0.913947i	$-0.633019\pi$
$773$	−22.5667	−0.811668	−0.405834	+	0.913947i	$0.633019\pi$
$774$	0	0
$775$	4.96124	0.178213
$776$	0	0
$777$	11.3766	0.408132
$778$	0	0
$779$	7.55771	0.270783
$780$	0	0
$781$	0.0385967	0.00138110
$782$	0	0
$783$	50.3555	1.79956
$784$	0	0
$785$	2.53020	0.0903067
$786$	0	0
$787$	34.1194	1.21622	0.608112	−	0.793851i	$-0.291927\pi$
$787$	34.1194	1.21622	0.608112	+	0.793851i	$0.291927\pi$
$788$	0	0
$789$	11.9515	0.425486
$790$	0	0
$791$	−16.7022	−0.593863
$792$	0	0
$793$	52.9414	1.88000
$794$	0	0
$795$	54.6747	1.93911
$796$	0	0
$797$	−38.7464	−1.37247	−0.686233	−	0.727382i	$-0.740737\pi$
$797$	−38.7464	−1.37247	−0.686233	+	0.727382i	$0.740737\pi$
$798$	0	0
$799$	19.8272	0.701437
$800$	0	0
$801$	−90.9013	−3.21184
$802$	0	0
$803$	3.36837	0.118867
$804$	0	0
$805$	−7.63633	−0.269145
$806$	0	0
$807$	80.6007	2.83728
$808$	0	0
$809$	−26.5427	−0.933193	−0.466596	−	0.884470i	$-0.654520\pi$
$809$	−26.5427	−0.933193	−0.466596	+	0.884470i	$0.654520\pi$
$810$	0	0
$811$	−18.3930	−0.645866	−0.322933	−	0.946422i	$-0.604669\pi$
$811$	−18.3930	−0.645866	−0.322933	+	0.946422i	$0.604669\pi$
$812$	0	0
$813$	4.33090	0.151891
$814$	0	0
$815$	−36.1041	−1.26467
$816$	0	0
$817$	−8.11228	−0.283813
$818$	0	0
$819$	−55.5499	−1.94107
$820$	0	0
$821$	−35.3299	−1.23302	−0.616511	−	0.787346i	$-0.711454\pi$
$821$	−35.3299	−1.23302	−0.616511	+	0.787346i	$0.711454\pi$
$822$	0	0
$823$	21.4806	0.748765	0.374383	−	0.927274i	$-0.377855\pi$
$823$	21.4806	0.748765	0.374383	+	0.927274i	$0.377855\pi$
$824$	0	0
$825$	−4.41102	−0.153572
$826$	0	0
$827$	23.5366	0.818448	0.409224	−	0.912434i	$-0.365800\pi$
$827$	23.5366	0.818448	0.409224	+	0.912434i	$0.365800\pi$
$828$	0	0
$829$	−0.0860264	−0.00298782	−0.00149391	−	0.999999i	$-0.500476\pi$
$829$	−0.0860264	−0.00298782	−0.00149391	+	0.999999i	$0.500476\pi$
$830$	0	0
$831$	55.5876	1.92831
$832$	0	0
$833$	15.1008	0.523210
$834$	0	0
$835$	−9.63117	−0.333300
$836$	0	0
$837$	−23.3780	−0.808062
$838$	0	0
$839$	−38.7204	−1.33678	−0.668388	−	0.743813i	$-0.733015\pi$
$839$	−38.7204	−1.33678	−0.668388	+	0.743813i	$0.733015\pi$
$840$	0	0
$841$	15.4581	0.533040
$842$	0	0
$843$	−61.2865	−2.11082
$844$	0	0
$845$	−99.3641	−3.41823
$846$	0	0
$847$	−14.0153	−0.481570
$848$	0	0
$849$	−25.4493	−0.873416
$850$	0	0
$851$	−6.01494	−0.206189
$852$	0	0
$853$	−51.5558	−1.76524	−0.882618	−	0.470091i	$-0.844221\pi$
$853$	−51.5558	−1.76524	−0.882618	+	0.470091i	$0.844221\pi$
$854$	0	0
$855$	71.6169	2.44925
$856$	0	0
$857$	34.0602	1.16347	0.581737	−	0.813377i	$-0.302373\pi$
$857$	34.0602	1.16347	0.581737	+	0.813377i	$0.302373\pi$
$858$	0	0
$859$	−5.30313	−0.180940	−0.0904702	−	0.995899i	$-0.528837\pi$
$859$	−5.30313	−0.180940	−0.0904702	+	0.995899i	$0.528837\pi$
$860$	0	0
$861$	6.13772	0.209173
$862$	0	0
$863$	−31.2088	−1.06236	−0.531180	−	0.847259i	$-0.678251\pi$
$863$	−31.2088	−1.06236	−0.531180	+	0.847259i	$0.678251\pi$
$864$	0	0
$865$	25.6794	0.873126
$866$	0	0
$867$	23.9198	0.812360
$868$	0	0
$869$	−4.30116	−0.145907
$870$	0	0
$871$	−53.0718	−1.79827
$872$	0	0
$873$	−21.5330	−0.728781
$874$	0	0
$875$	12.0933	0.408827
$876$	0	0
$877$	−33.2133	−1.12153	−0.560767	−	0.827973i	$-0.689494\pi$
$877$	−33.2133	−1.12153	−0.560767	+	0.827973i	$0.689494\pi$
$878$	0	0
$879$	−57.9486	−1.95456
$880$	0	0
$881$	5.10162	0.171878	0.0859390	−	0.996300i	$-0.472611\pi$
$881$	5.10162	0.171878	0.0859390	+	0.996300i	$0.472611\pi$
$882$	0	0
$883$	−28.1799	−0.948330	−0.474165	−	0.880436i	$-0.657250\pi$
$883$	−28.1799	−0.948330	−0.474165	+	0.880436i	$0.657250\pi$
$884$	0	0
$885$	−12.3358	−0.414662
$886$	0	0
$887$	22.0217	0.739416	0.369708	−	0.929148i	$-0.379458\pi$
$887$	22.0217	0.739416	0.369708	+	0.929148i	$0.379458\pi$
$888$	0	0
$889$	−24.4155	−0.818868
$890$	0	0
$891$	5.05932	0.169494
$892$	0	0
$893$	33.3304	1.11536
$894$	0	0
$895$	−43.5271	−1.45495
$896$	0	0
$897$	45.1645	1.50800
$898$	0	0
$899$	−20.6401	−0.688387
$900$	0	0
$901$	−21.5913	−0.719310
$902$	0	0
$903$	−6.58810	−0.219238
$904$	0	0
$905$	18.4611	0.613668
$906$	0	0
$907$	57.2117	1.89968	0.949842	−	0.312729i	$-0.101243\pi$
$907$	57.2117	1.89968	0.949842	+	0.312729i	$0.101243\pi$
$908$	0	0
$909$	27.5790	0.914737
$910$	0	0
$911$	−27.9387	−0.925652	−0.462826	−	0.886449i	$-0.653165\pi$
$911$	−27.9387	−0.925652	−0.462826	+	0.886449i	$0.653165\pi$
$912$	0	0
$913$	−15.4904	−0.512657
$914$	0	0
$915$	−55.4299	−1.83246
$916$	0	0
$917$	−20.7449	−0.685057
$918$	0	0
$919$	−56.5853	−1.86658	−0.933288	−	0.359129i	$-0.883074\pi$
$919$	−56.5853	−1.86658	−0.933288	+	0.359129i	$0.883074\pi$
$920$	0	0
$921$	−92.2883	−3.04100
$922$	0	0
$923$	−0.295248	−0.00971821
$924$	0	0
$925$	−4.49376	−0.147754
$926$	0	0
$927$	16.3215	0.536069
$928$	0	0
$929$	0.894163	0.0293365	0.0146683	−	0.999892i	$-0.495331\pi$
$929$	0.894163	0.0293365	0.0146683	+	0.999892i	$0.495331\pi$
$930$	0	0
$931$	25.3850	0.831961
$932$	0	0
$933$	5.70195	0.186674
$934$	0	0
$935$	7.17628	0.234690
$936$	0	0
$937$	59.8947	1.95667	0.978337	−	0.207018i	$-0.0663758\pi$
$937$	59.8947	1.95667	0.978337	+	0.207018i	$0.0663758\pi$
$938$	0	0
$939$	67.7351	2.21045
$940$	0	0
$941$	18.1460	0.591543	0.295771	−	0.955259i	$-0.404423\pi$
$941$	18.1460	0.591543	0.295771	+	0.955259i	$0.404423\pi$
$942$	0	0
$943$	−3.24509	−0.105675
$944$	0	0
$945$	26.8832	0.874511
$946$	0	0
$947$	−16.1792	−0.525755	−0.262877	−	0.964829i	$-0.584671\pi$
$947$	−16.1792	−0.525755	−0.262877	+	0.964829i	$0.584671\pi$
$948$	0	0
$949$	−25.7666	−0.836420
$950$	0	0
$951$	−69.2909	−2.24691
$952$	0	0
$953$	−56.4439	−1.82840	−0.914198	−	0.405267i	$-0.867178\pi$
$953$	−56.4439	−1.82840	−0.914198	+	0.405267i	$0.867178\pi$
$954$	0	0
$955$	46.1739	1.49415
$956$	0	0
$957$	18.3511	0.593205
$958$	0	0
$959$	5.10040	0.164700
$960$	0	0
$961$	−21.4176	−0.690891
$962$	0	0
$963$	−18.9384	−0.610281
$964$	0	0
$965$	−33.2960	−1.07184
$966$	0	0
$967$	40.9960	1.31834	0.659171	−	0.751993i	$-0.270907\pi$
$967$	40.9960	1.31834	0.659171	+	0.751993i	$0.270907\pi$
$968$	0	0
$969$	−43.4912	−1.39714
$970$	0	0
$971$	−40.9574	−1.31439	−0.657193	−	0.753722i	$-0.728256\pi$
$971$	−40.9574	−1.31439	−0.657193	+	0.753722i	$0.728256\pi$
$972$	0	0
$973$	2.18376	0.0700081
$974$	0	0
$975$	33.7424	1.08062
$976$	0	0
$977$	30.9144	0.989038	0.494519	−	0.869167i	$-0.335344\pi$
$977$	30.9144	0.989038	0.494519	+	0.869167i	$0.335344\pi$
$978$	0	0
$979$	−15.3121	−0.489375
$980$	0	0
$981$	−88.8439	−2.83657
$982$	0	0
$983$	−5.54467	−0.176847	−0.0884237	−	0.996083i	$-0.528183\pi$
$983$	−5.54467	−0.176847	−0.0884237	+	0.996083i	$0.528183\pi$
$984$	0	0
$985$	20.8743	0.665111
$986$	0	0
$987$	27.0681	0.861588
$988$	0	0
$989$	3.48321	0.110760
$990$	0	0
$991$	31.8521	1.01181	0.505907	−	0.862588i	$-0.331158\pi$
$991$	31.8521	1.01181	0.505907	+	0.862588i	$0.331158\pi$
$992$	0	0
$993$	−97.8594	−3.10547
$994$	0	0
$995$	18.7846	0.595513
$996$	0	0
$997$	−30.7128	−0.972683	−0.486342	−	0.873769i	$-0.661669\pi$
$997$	−30.7128	−0.972683	−0.486342	+	0.873769i	$0.661669\pi$
$998$	0	0
$999$	21.1752	0.669954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.t.1.1		21
4.3	odd	2		2012.2.a.a.1.21	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.a.1.21	✓	21	4.3	odd	2
8048.2.a.t.1.1		21	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)

\(f(q)\)	\(=\)	\(q-2.92891 q^{3} -2.56957 q^{5} +1.38532 q^{7} +5.57849 q^{9} +O(q^{10})\)	\(q-2.92891 q^{3} -2.56957 q^{5} +1.38532 q^{7} +5.57849 q^{9} +0.939680 q^{11} -7.18815 q^{13} +7.52604 q^{15} -2.97207 q^{17} -4.99618 q^{19} -4.05746 q^{21} +2.14523 q^{23} +1.60270 q^{25} -7.55216 q^{27} -6.66769 q^{29} +3.09554 q^{31} -2.75223 q^{33} -3.55967 q^{35} -2.80386 q^{37} +21.0534 q^{39} -1.51270 q^{41} +1.62370 q^{43} -14.3343 q^{45} -6.67119 q^{47} -5.08090 q^{49} +8.70490 q^{51} +7.26474 q^{53} -2.41458 q^{55} +14.6333 q^{57} -1.63908 q^{59} -7.36509 q^{61} +7.72798 q^{63} +18.4705 q^{65} +7.38323 q^{67} -6.28318 q^{69} +0.0410743 q^{71} +3.58460 q^{73} -4.69417 q^{75} +1.30176 q^{77} -4.57726 q^{79} +5.38409 q^{81} -16.4848 q^{83} +7.63694 q^{85} +19.5290 q^{87} -16.2950 q^{89} -9.95787 q^{91} -9.06655 q^{93} +12.8380 q^{95} -3.86000 q^{97} +5.24200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

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