LMFDB - Embedded newform 8048.2.a.v.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.66428 q^{3} -3.44274 q^{5} -4.43077 q^{7} -0.230172 q^{9} +O(q^{10})$	$q-1.66428 q^{3} -3.44274 q^{5} -4.43077 q^{7} -0.230172 q^{9} -1.18904 q^{11} -3.49702 q^{13} +5.72968 q^{15} -2.17128 q^{17} -5.57445 q^{19} +7.37404 q^{21} +7.37514 q^{23} +6.85244 q^{25} +5.37591 q^{27} -4.21812 q^{29} +7.45326 q^{31} +1.97890 q^{33} +15.2540 q^{35} -5.18844 q^{37} +5.82003 q^{39} -1.01920 q^{41} +0.120646 q^{43} +0.792421 q^{45} -3.07754 q^{47} +12.6317 q^{49} +3.61361 q^{51} -2.80462 q^{53} +4.09355 q^{55} +9.27745 q^{57} -2.74850 q^{59} +10.9165 q^{61} +1.01984 q^{63} +12.0393 q^{65} +4.55806 q^{67} -12.2743 q^{69} +0.710487 q^{71} -3.34683 q^{73} -11.4044 q^{75} +5.26836 q^{77} +6.35019 q^{79} -8.25651 q^{81} +13.4018 q^{83} +7.47514 q^{85} +7.02013 q^{87} +9.71069 q^{89} +15.4945 q^{91} -12.4043 q^{93} +19.1914 q^{95} +7.58484 q^{97} +0.273683 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	−1.66428	−0.960873	−0.480436	−	0.877030i	$-0.659522\pi$
$3$	−1.66428	−0.960873	−0.480436	+	0.877030i	$0.659522\pi$
$4$	0	0
$5$	−3.44274	−1.53964	−0.769819	−	0.638262i	$-0.779654\pi$
$5$	−3.44274	−1.53964	−0.769819	+	0.638262i	$0.779654\pi$
$6$	0	0
$7$	−4.43077	−1.67467	−0.837336	−	0.546689i	$-0.815888\pi$
$7$	−4.43077	−1.67467	−0.837336	+	0.546689i	$0.815888\pi$
$8$	0	0
$9$	−0.230172	−0.0767239
$10$	0	0
$11$	−1.18904	−0.358509	−0.179255	−	0.983803i	$-0.557369\pi$
$11$	−1.18904	−0.358509	−0.179255	+	0.983803i	$0.557369\pi$
$12$	0	0
$13$	−3.49702	−0.969900	−0.484950	−	0.874542i	$-0.661162\pi$
$13$	−3.49702	−0.969900	−0.484950	+	0.874542i	$0.661162\pi$
$14$	0	0
$15$	5.72968	1.47940
$16$	0	0
$17$	−2.17128	−0.526612	−0.263306	−	0.964712i	$-0.584813\pi$
$17$	−2.17128	−0.526612	−0.263306	+	0.964712i	$0.584813\pi$
$18$	0	0
$19$	−5.57445	−1.27887	−0.639433	−	0.768847i	$-0.720831\pi$
$19$	−5.57445	−1.27887	−0.639433	+	0.768847i	$0.720831\pi$
$20$	0	0
$21$	7.37404	1.60915
$22$	0	0
$23$	7.37514	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$23$	7.37514	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$24$	0	0
$25$	6.85244	1.37049
$26$	0	0
$27$	5.37591	1.03459
$28$	0	0
$29$	−4.21812	−0.783284	−0.391642	−	0.920118i	$-0.628093\pi$
$29$	−4.21812	−0.783284	−0.391642	+	0.920118i	$0.628093\pi$
$30$	0	0
$31$	7.45326	1.33864	0.669322	−	0.742972i	$-0.266585\pi$
$31$	7.45326	1.33864	0.669322	+	0.742972i	$0.266585\pi$
$32$	0	0
$33$	1.97890	0.344482
$34$	0	0
$35$	15.2540	2.57839
$36$	0	0
$37$	−5.18844	−0.852975	−0.426488	−	0.904493i	$-0.640249\pi$
$37$	−5.18844	−0.852975	−0.426488	+	0.904493i	$0.640249\pi$
$38$	0	0
$39$	5.82003	0.931950
$40$	0	0
$41$	−1.01920	−0.159172	−0.0795861	−	0.996828i	$-0.525360\pi$
$41$	−1.01920	−0.159172	−0.0795861	+	0.996828i	$0.525360\pi$
$42$	0	0
$43$	0.120646	0.0183983	0.00919914	−	0.999958i	$-0.497072\pi$
$43$	0.120646	0.0183983	0.00919914	+	0.999958i	$0.497072\pi$
$44$	0	0
$45$	0.792421	0.118127
$46$	0	0
$47$	−3.07754	−0.448906	−0.224453	−	0.974485i	$-0.572059\pi$
$47$	−3.07754	−0.448906	−0.224453	+	0.974485i	$0.572059\pi$
$48$	0	0
$49$	12.6317	1.80453
$50$	0	0
$51$	3.61361	0.506007
$52$	0	0
$53$	−2.80462	−0.385245	−0.192622	−	0.981273i	$-0.561699\pi$
$53$	−2.80462	−0.385245	−0.192622	+	0.981273i	$0.561699\pi$
$54$	0	0
$55$	4.09355	0.551975
$56$	0	0
$57$	9.27745	1.22883
$58$	0	0
$59$	−2.74850	−0.357825	−0.178912	−	0.983865i	$-0.557258\pi$
$59$	−2.74850	−0.357825	−0.178912	+	0.983865i	$0.557258\pi$
$60$	0	0
$61$	10.9165	1.39771	0.698855	−	0.715263i	$-0.253693\pi$
$61$	10.9165	1.39771	0.698855	+	0.715263i	$0.253693\pi$
$62$	0	0
$63$	1.01984	0.128487
$64$	0	0
$65$	12.0393	1.49330
$66$	0	0
$67$	4.55806	0.556855	0.278428	−	0.960457i	$-0.410187\pi$
$67$	4.55806	0.556855	0.278428	+	0.960457i	$0.410187\pi$
$68$	0	0
$69$	−12.2743	−1.47765
$70$	0	0
$71$	0.710487	0.0843193	0.0421597	−	0.999111i	$-0.486576\pi$
$71$	0.710487	0.0843193	0.0421597	+	0.999111i	$0.486576\pi$
$72$	0	0
$73$	−3.34683	−0.391717	−0.195859	−	0.980632i	$-0.562749\pi$
$73$	−3.34683	−0.391717	−0.195859	+	0.980632i	$0.562749\pi$
$74$	0	0
$75$	−11.4044	−1.31686
$76$	0	0
$77$	5.26836	0.600385
$78$	0	0
$79$	6.35019	0.714452	0.357226	−	0.934018i	$-0.383723\pi$
$79$	6.35019	0.714452	0.357226	+	0.934018i	$0.383723\pi$
$80$	0	0
$81$	−8.25651	−0.917390
$82$	0	0
$83$	13.4018	1.47104	0.735520	−	0.677503i	$-0.236938\pi$
$83$	13.4018	1.47104	0.735520	+	0.677503i	$0.236938\pi$
$84$	0	0
$85$	7.47514	0.810792
$86$	0	0
$87$	7.02013	0.752636
$88$	0	0
$89$	9.71069	1.02933	0.514665	−	0.857391i	$-0.327916\pi$
$89$	9.71069	1.02933	0.514665	+	0.857391i	$0.327916\pi$
$90$	0	0
$91$	15.4945	1.62426
$92$	0	0
$93$	−12.4043	−1.28627
$94$	0	0
$95$	19.1914	1.96899
$96$	0	0
$97$	7.58484	0.770124	0.385062	−	0.922891i	$-0.374180\pi$
$97$	7.58484	0.770124	0.385062	+	0.922891i	$0.374180\pi$
$98$	0	0
$99$	0.273683	0.0275062
$100$	0	0
$101$	−13.3257	−1.32595	−0.662977	−	0.748640i	$-0.730707\pi$
$101$	−13.3257	−1.32595	−0.662977	+	0.748640i	$0.730707\pi$
$102$	0	0
$103$	−6.24375	−0.615215	−0.307607	−	0.951513i	$-0.599528\pi$
$103$	−6.24375	−0.615215	−0.307607	+	0.951513i	$0.599528\pi$
$104$	0	0
$105$	−25.3869	−2.47750
$106$	0	0
$107$	−3.29616	−0.318652	−0.159326	−	0.987226i	$-0.550932\pi$
$107$	−3.29616	−0.318652	−0.159326	+	0.987226i	$0.550932\pi$
$108$	0	0
$109$	0.636565	0.0609719	0.0304860	−	0.999535i	$-0.490295\pi$
$109$	0.636565	0.0609719	0.0304860	+	0.999535i	$0.490295\pi$
$110$	0	0
$111$	8.63503	0.819600
$112$	0	0
$113$	−8.76089	−0.824156	−0.412078	−	0.911149i	$-0.635197\pi$
$113$	−8.76089	−0.824156	−0.412078	+	0.911149i	$0.635197\pi$
$114$	0	0
$115$	−25.3907	−2.36769
$116$	0	0
$117$	0.804916	0.0744145
$118$	0	0
$119$	9.62042	0.881902
$120$	0	0
$121$	−9.58618	−0.871471
$122$	0	0
$123$	1.69623	0.152944
$124$	0	0
$125$	−6.37746	−0.570418
$126$	0	0
$127$	−10.2731	−0.911589	−0.455795	−	0.890085i	$-0.650645\pi$
$127$	−10.2731	−0.911589	−0.455795	+	0.890085i	$0.650645\pi$
$128$	0	0
$129$	−0.200788	−0.0176784
$130$	0	0
$131$	18.7673	1.63970	0.819852	−	0.572576i	$-0.194056\pi$
$131$	18.7673	1.63970	0.819852	+	0.572576i	$0.194056\pi$
$132$	0	0
$133$	24.6991	2.14168
$134$	0	0
$135$	−18.5078	−1.59290
$136$	0	0
$137$	−7.56580	−0.646390	−0.323195	−	0.946332i	$-0.604757\pi$
$137$	−7.56580	−0.646390	−0.323195	+	0.946332i	$0.604757\pi$
$138$	0	0
$139$	10.6960	0.907222	0.453611	−	0.891200i	$-0.350136\pi$
$139$	10.6960	0.907222	0.453611	+	0.891200i	$0.350136\pi$
$140$	0	0
$141$	5.12189	0.431341
$142$	0	0
$143$	4.15810	0.347718
$144$	0	0
$145$	14.5219	1.20598
$146$	0	0
$147$	−21.0227	−1.73392
$148$	0	0
$149$	−0.548032	−0.0448966	−0.0224483	−	0.999748i	$-0.507146\pi$
$149$	−0.548032	−0.0448966	−0.0224483	+	0.999748i	$0.507146\pi$
$150$	0	0
$151$	12.3768	1.00721	0.503605	−	0.863934i	$-0.332007\pi$
$151$	12.3768	1.00721	0.503605	+	0.863934i	$0.332007\pi$
$152$	0	0
$153$	0.499766	0.0404037
$154$	0	0
$155$	−25.6596	−2.06103
$156$	0	0
$157$	−17.4613	−1.39356	−0.696782	−	0.717283i	$-0.745386\pi$
$157$	−17.4613	−1.39356	−0.696782	+	0.717283i	$0.745386\pi$
$158$	0	0
$159$	4.66768	0.370171
$160$	0	0
$161$	−32.6775	−2.57535
$162$	0	0
$163$	−17.7025	−1.38657	−0.693285	−	0.720664i	$-0.743837\pi$
$163$	−17.7025	−1.38657	−0.693285	+	0.720664i	$0.743837\pi$
$164$	0	0
$165$	−6.81282	−0.530377
$166$	0	0
$167$	8.39101	0.649316	0.324658	−	0.945832i	$-0.394751\pi$
$167$	8.39101	0.649316	0.324658	+	0.945832i	$0.394751\pi$
$168$	0	0
$169$	−0.770825	−0.0592942
$170$	0	0
$171$	1.28308	0.0981196
$172$	0	0
$173$	4.47941	0.340563	0.170282	−	0.985395i	$-0.445532\pi$
$173$	4.47941	0.340563	0.170282	+	0.985395i	$0.445532\pi$
$174$	0	0
$175$	−30.3616	−2.29512
$176$	0	0
$177$	4.57428	0.343824
$178$	0	0
$179$	−10.6765	−0.797997	−0.398999	−	0.916951i	$-0.630642\pi$
$179$	−10.6765	−0.797997	−0.398999	+	0.916951i	$0.630642\pi$
$180$	0	0
$181$	−8.00808	−0.595236	−0.297618	−	0.954685i	$-0.596192\pi$
$181$	−8.00808	−0.595236	−0.297618	+	0.954685i	$0.596192\pi$
$182$	0	0
$183$	−18.1681	−1.34302
$184$	0	0
$185$	17.8625	1.31327
$186$	0	0
$187$	2.58174	0.188795
$188$	0	0
$189$	−23.8194	−1.73261
$190$	0	0
$191$	−0.844203	−0.0610844	−0.0305422	−	0.999533i	$-0.509723\pi$
$191$	−0.844203	−0.0610844	−0.0305422	+	0.999533i	$0.509723\pi$
$192$	0	0
$193$	18.4651	1.32915	0.664573	−	0.747223i	$-0.268614\pi$
$193$	18.4651	1.32915	0.664573	+	0.747223i	$0.268614\pi$
$194$	0	0
$195$	−20.0368	−1.43487
$196$	0	0
$197$	−9.91364	−0.706318	−0.353159	−	0.935563i	$-0.614892\pi$
$197$	−9.91364	−0.706318	−0.353159	+	0.935563i	$0.614892\pi$
$198$	0	0
$199$	17.2198	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$199$	17.2198	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$200$	0	0
$201$	−7.58588	−0.535067
$202$	0	0
$203$	18.6895	1.31174
$204$	0	0
$205$	3.50883	0.245068
$206$	0	0
$207$	−1.69755	−0.117988
$208$	0	0
$209$	6.62824	0.458485
$210$	0	0
$211$	10.4567	0.719867	0.359933	−	0.932978i	$-0.382799\pi$
$211$	10.4567	0.719867	0.359933	+	0.932978i	$0.382799\pi$
$212$	0	0
$213$	−1.18245	−0.0810201
$214$	0	0
$215$	−0.415351	−0.0283267
$216$	0	0
$217$	−33.0236	−2.24179
$218$	0	0
$219$	5.57007	0.376390
$220$	0	0
$221$	7.59301	0.510761
$222$	0	0
$223$	2.52993	0.169417	0.0847083	−	0.996406i	$-0.473004\pi$
$223$	2.52993	0.169417	0.0847083	+	0.996406i	$0.473004\pi$
$224$	0	0
$225$	−1.57724	−0.105149
$226$	0	0
$227$	26.9881	1.79126	0.895632	−	0.444795i	$-0.146723\pi$
$227$	26.9881	1.79126	0.895632	+	0.444795i	$0.146723\pi$
$228$	0	0
$229$	−17.6989	−1.16958	−0.584789	−	0.811185i	$-0.698823\pi$
$229$	−17.6989	−1.16958	−0.584789	+	0.811185i	$0.698823\pi$
$230$	0	0
$231$	−8.76802	−0.576894
$232$	0	0
$233$	1.68990	0.110709	0.0553544	−	0.998467i	$-0.482371\pi$
$233$	1.68990	0.110709	0.0553544	+	0.998467i	$0.482371\pi$
$234$	0	0
$235$	10.5952	0.691152
$236$	0	0
$237$	−10.5685	−0.686498
$238$	0	0
$239$	−25.8826	−1.67421	−0.837103	−	0.547045i	$-0.815753\pi$
$239$	−25.8826	−1.67421	−0.837103	+	0.547045i	$0.815753\pi$
$240$	0	0
$241$	24.9079	1.60446	0.802229	−	0.597016i	$-0.203647\pi$
$241$	24.9079	1.60446	0.802229	+	0.597016i	$0.203647\pi$
$242$	0	0
$243$	−2.38659	−0.153100
$244$	0	0
$245$	−43.4876	−2.77832
$246$	0	0
$247$	19.4940	1.24037
$248$	0	0
$249$	−22.3044	−1.41348
$250$	0	0
$251$	24.5831	1.55167	0.775837	−	0.630933i	$-0.217328\pi$
$251$	24.5831	1.55167	0.775837	+	0.630933i	$0.217328\pi$
$252$	0	0
$253$	−8.76933	−0.551323
$254$	0	0
$255$	−12.4407	−0.779068
$256$	0	0
$257$	10.1488	0.633068	0.316534	−	0.948581i	$-0.397481\pi$
$257$	10.1488	0.633068	0.316534	+	0.948581i	$0.397481\pi$
$258$	0	0
$259$	22.9888	1.42845
$260$	0	0
$261$	0.970891	0.0600966
$262$	0	0
$263$	−11.1732	−0.688972	−0.344486	−	0.938791i	$-0.611947\pi$
$263$	−11.1732	−0.688972	−0.344486	+	0.938791i	$0.611947\pi$
$264$	0	0
$265$	9.65558	0.593138
$266$	0	0
$267$	−16.1613	−0.989056
$268$	0	0
$269$	−19.4053	−1.18316	−0.591581	−	0.806245i	$-0.701496\pi$
$269$	−19.4053	−1.18316	−0.591581	+	0.806245i	$0.701496\pi$
$270$	0	0
$271$	12.6613	0.769117	0.384559	−	0.923101i	$-0.374354\pi$
$271$	12.6613	0.769117	0.384559	+	0.923101i	$0.374354\pi$
$272$	0	0
$273$	−25.7872	−1.56071
$274$	0	0
$275$	−8.14783	−0.491332
$276$	0	0
$277$	−11.6562	−0.700351	−0.350175	−	0.936684i	$-0.613878\pi$
$277$	−11.6562	−0.700351	−0.350175	+	0.936684i	$0.613878\pi$
$278$	0	0
$279$	−1.71553	−0.102706
$280$	0	0
$281$	18.2193	1.08687	0.543437	−	0.839450i	$-0.317123\pi$
$281$	18.2193	1.08687	0.543437	+	0.839450i	$0.317123\pi$
$282$	0	0
$283$	−19.4899	−1.15855	−0.579276	−	0.815132i	$-0.696665\pi$
$283$	−19.4899	−1.15855	−0.579276	+	0.815132i	$0.696665\pi$
$284$	0	0
$285$	−31.9398	−1.89195
$286$	0	0
$287$	4.51583	0.266561
$288$	0	0
$289$	−12.2856	−0.722680
$290$	0	0
$291$	−12.6233	−0.739991
$292$	0	0
$293$	−3.98593	−0.232860	−0.116430	−	0.993199i	$-0.537145\pi$
$293$	−3.98593	−0.232860	−0.116430	+	0.993199i	$0.537145\pi$
$294$	0	0
$295$	9.46237	0.550921
$296$	0	0
$297$	−6.39217	−0.370912
$298$	0	0
$299$	−25.7910	−1.49153
$300$	0	0
$301$	−0.534552	−0.0308111
$302$	0	0
$303$	22.1776	1.27407
$304$	0	0
$305$	−37.5825	−2.15197
$306$	0	0
$307$	2.14719	0.122546	0.0612732	−	0.998121i	$-0.480484\pi$
$307$	2.14719	0.122546	0.0612732	+	0.998121i	$0.480484\pi$
$308$	0	0
$309$	10.3913	0.591143
$310$	0	0
$311$	−1.36490	−0.0773963	−0.0386981	−	0.999251i	$-0.512321\pi$
$311$	−1.36490	−0.0773963	−0.0386981	+	0.999251i	$0.512321\pi$
$312$	0	0
$313$	16.1919	0.915222	0.457611	−	0.889153i	$-0.348705\pi$
$313$	16.1919	0.915222	0.457611	+	0.889153i	$0.348705\pi$
$314$	0	0
$315$	−3.51103	−0.197824
$316$	0	0
$317$	20.3104	1.14075	0.570374	−	0.821385i	$-0.306798\pi$
$317$	20.3104	1.14075	0.570374	+	0.821385i	$0.306798\pi$
$318$	0	0
$319$	5.01551	0.280815
$320$	0	0
$321$	5.48574	0.306184
$322$	0	0
$323$	12.1037	0.673466
$324$	0	0
$325$	−23.9631	−1.32924
$326$	0	0
$327$	−1.05942	−0.0585862
$328$	0	0
$329$	13.6359	0.751770
$330$	0	0
$331$	9.23075	0.507368	0.253684	−	0.967287i	$-0.418358\pi$
$331$	9.23075	0.507368	0.253684	+	0.967287i	$0.418358\pi$
$332$	0	0
$333$	1.19423	0.0654436
$334$	0	0
$335$	−15.6922	−0.857356
$336$	0	0
$337$	17.1199	0.932580	0.466290	−	0.884632i	$-0.345590\pi$
$337$	17.1199	0.932580	0.466290	+	0.884632i	$0.345590\pi$
$338$	0	0
$339$	14.5806	0.791908
$340$	0	0
$341$	−8.86222	−0.479916
$342$	0	0
$343$	−24.9527	−1.34732
$344$	0	0
$345$	42.2572	2.27505
$346$	0	0
$347$	−10.7339	−0.576226	−0.288113	−	0.957596i	$-0.593028\pi$
$347$	−10.7339	−0.576226	−0.288113	+	0.957596i	$0.593028\pi$
$348$	0	0
$349$	14.3070	0.765838	0.382919	−	0.923782i	$-0.374919\pi$
$349$	14.3070	0.765838	0.382919	+	0.923782i	$0.374919\pi$
$350$	0	0
$351$	−18.7997	−1.00345
$352$	0	0
$353$	14.3791	0.765320	0.382660	−	0.923889i	$-0.375008\pi$
$353$	14.3791	0.765320	0.382660	+	0.923889i	$0.375008\pi$
$354$	0	0
$355$	−2.44602	−0.129821
$356$	0	0
$357$	−16.0111	−0.847396
$358$	0	0
$359$	−22.5958	−1.19256	−0.596281	−	0.802776i	$-0.703355\pi$
$359$	−22.5958	−1.19256	−0.596281	+	0.802776i	$0.703355\pi$
$360$	0	0
$361$	12.0745	0.635499
$362$	0	0
$363$	15.9541	0.837373
$364$	0	0
$365$	11.5223	0.603103
$366$	0	0
$367$	−33.8333	−1.76609	−0.883043	−	0.469292i	$-0.844509\pi$
$367$	−33.8333	−1.76609	−0.883043	+	0.469292i	$0.844509\pi$
$368$	0	0
$369$	0.234591	0.0122123
$370$	0	0
$371$	12.4266	0.645159
$372$	0	0
$373$	−10.1865	−0.527438	−0.263719	−	0.964600i	$-0.584949\pi$
$373$	−10.1865	−0.527438	−0.263719	+	0.964600i	$0.584949\pi$
$374$	0	0
$375$	10.6139	0.548099
$376$	0	0
$377$	14.7508	0.759707
$378$	0	0
$379$	10.6614	0.547641	0.273821	−	0.961781i	$-0.411713\pi$
$379$	10.6614	0.547641	0.273821	+	0.961781i	$0.411713\pi$
$380$	0	0
$381$	17.0973	0.875921
$382$	0	0
$383$	29.1930	1.49169	0.745846	−	0.666119i	$-0.232046\pi$
$383$	29.1930	1.49169	0.745846	+	0.666119i	$0.232046\pi$
$384$	0	0
$385$	−18.1376	−0.924376
$386$	0	0
$387$	−0.0277692	−0.00141159
$388$	0	0
$389$	−25.9539	−1.31591	−0.657957	−	0.753056i	$-0.728579\pi$
$389$	−25.9539	−1.31591	−0.657957	+	0.753056i	$0.728579\pi$
$390$	0	0
$391$	−16.0135	−0.809836
$392$	0	0
$393$	−31.2340	−1.57555
$394$	0	0
$395$	−21.8620	−1.10000
$396$	0	0
$397$	2.93633	0.147370	0.0736852	−	0.997282i	$-0.476524\pi$
$397$	2.93633	0.147370	0.0736852	+	0.997282i	$0.476524\pi$
$398$	0	0
$399$	−41.1062	−2.05788
$400$	0	0
$401$	−16.7042	−0.834167	−0.417084	−	0.908868i	$-0.636948\pi$
$401$	−16.7042	−0.834167	−0.417084	+	0.908868i	$0.636948\pi$
$402$	0	0
$403$	−26.0642	−1.29835
$404$	0	0
$405$	28.4250	1.41245
$406$	0	0
$407$	6.16927	0.305799
$408$	0	0
$409$	30.7034	1.51819	0.759093	−	0.650982i	$-0.225643\pi$
$409$	30.7034	1.51819	0.759093	+	0.650982i	$0.225643\pi$
$410$	0	0
$411$	12.5916	0.621099
$412$	0	0
$413$	12.1780	0.599239
$414$	0	0
$415$	−46.1389	−2.26487
$416$	0	0
$417$	−17.8011	−0.871725
$418$	0	0
$419$	1.56186	0.0763018	0.0381509	−	0.999272i	$-0.487853\pi$
$419$	1.56186	0.0763018	0.0381509	+	0.999272i	$0.487853\pi$
$420$	0	0
$421$	8.04875	0.392272	0.196136	−	0.980577i	$-0.437161\pi$
$421$	8.04875	0.392272	0.196136	+	0.980577i	$0.437161\pi$
$422$	0	0
$423$	0.708363	0.0344418
$424$	0	0
$425$	−14.8785	−0.721715
$426$	0	0
$427$	−48.3683	−2.34071
$428$	0	0
$429$	−6.92025	−0.334113
$430$	0	0
$431$	−23.5999	−1.13677	−0.568383	−	0.822764i	$-0.692431\pi$
$431$	−23.5999	−1.13677	−0.568383	+	0.822764i	$0.692431\pi$
$432$	0	0
$433$	9.33138	0.448438	0.224219	−	0.974539i	$-0.428017\pi$
$433$	9.33138	0.448438	0.224219	+	0.974539i	$0.428017\pi$
$434$	0	0
$435$	−24.1684	−1.15879
$436$	0	0
$437$	−41.1123	−1.96667
$438$	0	0
$439$	−24.0237	−1.14659	−0.573295	−	0.819349i	$-0.694335\pi$
$439$	−24.0237	−1.14659	−0.573295	+	0.819349i	$0.694335\pi$
$440$	0	0
$441$	−2.90746	−0.138450
$442$	0	0
$443$	−2.08695	−0.0991539	−0.0495769	−	0.998770i	$-0.515787\pi$
$443$	−2.08695	−0.0991539	−0.0495769	+	0.998770i	$0.515787\pi$
$444$	0	0
$445$	−33.4313	−1.58480
$446$	0	0
$447$	0.912079	0.0431399
$448$	0	0
$449$	5.51869	0.260443	0.130221	−	0.991485i	$-0.458431\pi$
$449$	5.51869	0.260443	0.130221	+	0.991485i	$0.458431\pi$
$450$	0	0
$451$	1.21187	0.0570647
$452$	0	0
$453$	−20.5985	−0.967800
$454$	0	0
$455$	−53.3435	−2.50078
$456$	0	0
$457$	−18.9386	−0.885912	−0.442956	−	0.896543i	$-0.646070\pi$
$457$	−18.9386	−0.885912	−0.442956	+	0.896543i	$0.646070\pi$
$458$	0	0
$459$	−11.6726	−0.544830
$460$	0	0
$461$	25.9924	1.21059	0.605293	−	0.796003i	$-0.293056\pi$
$461$	25.9924	1.21059	0.605293	+	0.796003i	$0.293056\pi$
$462$	0	0
$463$	−35.9190	−1.66930	−0.834650	−	0.550781i	$-0.814330\pi$
$463$	−35.9190	−1.66930	−0.834650	+	0.550781i	$0.814330\pi$
$464$	0	0
$465$	42.7048	1.98039
$466$	0	0
$467$	9.83647	0.455177	0.227589	−	0.973757i	$-0.426916\pi$
$467$	9.83647	0.455177	0.227589	+	0.973757i	$0.426916\pi$
$468$	0	0
$469$	−20.1957	−0.932550
$470$	0	0
$471$	29.0605	1.33904
$472$	0	0
$473$	−0.143452	−0.00659595
$474$	0	0
$475$	−38.1986	−1.75267
$476$	0	0
$477$	0.645545	0.0295575
$478$	0	0
$479$	−12.2528	−0.559843	−0.279922	−	0.960023i	$-0.590308\pi$
$479$	−12.2528	−0.559843	−0.279922	+	0.960023i	$0.590308\pi$
$480$	0	0
$481$	18.1441	0.827300
$482$	0	0
$483$	54.3845	2.47458
$484$	0	0
$485$	−26.1126	−1.18571
$486$	0	0
$487$	31.5386	1.42915	0.714576	−	0.699558i	$-0.246620\pi$
$487$	31.5386	1.42915	0.714576	+	0.699558i	$0.246620\pi$
$488$	0	0
$489$	29.4620	1.33232
$490$	0	0
$491$	30.4460	1.37401	0.687004	−	0.726654i	$-0.258926\pi$
$491$	30.4460	1.37401	0.687004	+	0.726654i	$0.258926\pi$
$492$	0	0
$493$	9.15870	0.412487
$494$	0	0
$495$	−0.942220	−0.0423496
$496$	0	0
$497$	−3.14800	−0.141207
$498$	0	0
$499$	−20.7996	−0.931117	−0.465558	−	0.885017i	$-0.654146\pi$
$499$	−20.7996	−0.931117	−0.465558	+	0.885017i	$0.654146\pi$
$500$	0	0
$501$	−13.9650	−0.623910
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	45.8768	2.04149
$506$	0	0
$507$	1.28287	0.0569742
$508$	0	0
$509$	−23.3203	−1.03365	−0.516826	−	0.856090i	$-0.672887\pi$
$509$	−23.3203	−1.03365	−0.516826	+	0.856090i	$0.672887\pi$
$510$	0	0
$511$	14.8290	0.655998
$512$	0	0
$513$	−29.9677	−1.32311
$514$	0	0
$515$	21.4956	0.947208
$516$	0	0
$517$	3.65932	0.160937
$518$	0	0
$519$	−7.45499	−0.327238
$520$	0	0
$521$	−25.4086	−1.11317	−0.556586	−	0.830790i	$-0.687889\pi$
$521$	−25.4086	−1.11317	−0.556586	+	0.830790i	$0.687889\pi$
$522$	0	0
$523$	30.6210	1.33896	0.669480	−	0.742830i	$-0.266517\pi$
$523$	30.6210	1.33896	0.669480	+	0.742830i	$0.266517\pi$
$524$	0	0
$525$	50.5301	2.20532
$526$	0	0
$527$	−16.1831	−0.704946
$528$	0	0
$529$	31.3926	1.36490
$530$	0	0
$531$	0.632627	0.0274537
$532$	0	0
$533$	3.56416	0.154381
$534$	0	0
$535$	11.3478	0.490609
$536$	0	0
$537$	17.7686	0.766774
$538$	0	0
$539$	−15.0196	−0.646939
$540$	0	0
$541$	12.1009	0.520257	0.260129	−	0.965574i	$-0.416235\pi$
$541$	12.1009	0.520257	0.260129	+	0.965574i	$0.416235\pi$
$542$	0	0
$543$	13.3277	0.571946
$544$	0	0
$545$	−2.19153	−0.0938747
$546$	0	0
$547$	−33.3069	−1.42410	−0.712050	−	0.702129i	$-0.752233\pi$
$547$	−33.3069	−1.42410	−0.712050	+	0.702129i	$0.752233\pi$
$548$	0	0
$549$	−2.51266	−0.107238
$550$	0	0
$551$	23.5137	1.00172
$552$	0	0
$553$	−28.1362	−1.19647
$554$	0	0
$555$	−29.7281	−1.26189
$556$	0	0
$557$	31.7330	1.34457	0.672285	−	0.740292i	$-0.265313\pi$
$557$	31.7330	1.34457	0.672285	+	0.740292i	$0.265313\pi$
$558$	0	0
$559$	−0.421900	−0.0178445
$560$	0	0
$561$	−4.29673	−0.181408
$562$	0	0
$563$	−36.9937	−1.55910	−0.779550	−	0.626341i	$-0.784552\pi$
$563$	−36.9937	−1.55910	−0.779550	+	0.626341i	$0.784552\pi$
$564$	0	0
$565$	30.1615	1.26890
$566$	0	0
$567$	36.5826	1.53633
$568$	0	0
$569$	9.61211	0.402961	0.201480	−	0.979493i	$-0.435425\pi$
$569$	9.61211	0.402961	0.201480	+	0.979493i	$0.435425\pi$
$570$	0	0
$571$	6.49307	0.271727	0.135863	−	0.990728i	$-0.456619\pi$
$571$	6.49307	0.271727	0.135863	+	0.990728i	$0.456619\pi$
$572$	0	0
$573$	1.40499	0.0586943
$574$	0	0
$575$	50.5377	2.10757
$576$	0	0
$577$	47.2064	1.96523	0.982614	−	0.185659i	$-0.0594419\pi$
$577$	47.2064	1.96523	0.982614	+	0.185659i	$0.0594419\pi$
$578$	0	0
$579$	−30.7311	−1.27714
$580$	0	0
$581$	−59.3803	−2.46351
$582$	0	0
$583$	3.33481	0.138114
$584$	0	0
$585$	−2.77111	−0.114571
$586$	0	0
$587$	−5.61084	−0.231584	−0.115792	−	0.993273i	$-0.536941\pi$
$587$	−5.61084	−0.231584	−0.115792	+	0.993273i	$0.536941\pi$
$588$	0	0
$589$	−41.5478	−1.71195
$590$	0	0
$591$	16.4991	0.678681
$592$	0	0
$593$	25.0721	1.02959	0.514793	−	0.857314i	$-0.327869\pi$
$593$	25.0721	1.02959	0.514793	+	0.857314i	$0.327869\pi$
$594$	0	0
$595$	−33.1206	−1.35781
$596$	0	0
$597$	−28.6586	−1.17292
$598$	0	0
$599$	28.1633	1.15072	0.575361	−	0.817900i	$-0.304862\pi$
$599$	28.1633	1.15072	0.575361	+	0.817900i	$0.304862\pi$
$600$	0	0
$601$	−26.8480	−1.09515	−0.547576	−	0.836756i	$-0.684449\pi$
$601$	−26.8480	−1.09515	−0.547576	+	0.836756i	$0.684449\pi$
$602$	0	0
$603$	−1.04914	−0.0427241
$604$	0	0
$605$	33.0027	1.34175
$606$	0	0
$607$	42.2396	1.71445	0.857225	−	0.514942i	$-0.172186\pi$
$607$	42.2396	1.71445	0.857225	+	0.514942i	$0.172186\pi$
$608$	0	0
$609$	−31.1045	−1.26042
$610$	0	0
$611$	10.7622	0.435393
$612$	0	0
$613$	−32.3212	−1.30544	−0.652721	−	0.757598i	$-0.726373\pi$
$613$	−32.3212	−1.30544	−0.652721	+	0.757598i	$0.726373\pi$
$614$	0	0
$615$	−5.83968	−0.235479
$616$	0	0
$617$	21.9580	0.883995	0.441997	−	0.897016i	$-0.354270\pi$
$617$	21.9580	0.883995	0.441997	+	0.897016i	$0.354270\pi$
$618$	0	0
$619$	11.7735	0.473216	0.236608	−	0.971605i	$-0.423964\pi$
$619$	11.7735	0.473216	0.236608	+	0.971605i	$0.423964\pi$
$620$	0	0
$621$	39.6481	1.59102
$622$	0	0
$623$	−43.0258	−1.72379
$624$	0	0
$625$	−12.3063	−0.492251
$626$	0	0
$627$	−11.0313	−0.440546
$628$	0	0
$629$	11.2655	0.449187
$630$	0	0
$631$	−18.3254	−0.729523	−0.364762	−	0.931101i	$-0.618850\pi$
$631$	−18.3254	−0.729523	−0.364762	+	0.931101i	$0.618850\pi$
$632$	0	0
$633$	−17.4028	−0.691700
$634$	0	0
$635$	35.3675	1.40352
$636$	0	0
$637$	−44.1733	−1.75021
$638$	0	0
$639$	−0.163534	−0.00646930
$640$	0	0
$641$	19.9149	0.786592	0.393296	−	0.919412i	$-0.371335\pi$
$641$	19.9149	0.786592	0.393296	+	0.919412i	$0.371335\pi$
$642$	0	0
$643$	−1.73358	−0.0683655	−0.0341828	−	0.999416i	$-0.510883\pi$
$643$	−1.73358	−0.0683655	−0.0341828	+	0.999416i	$0.510883\pi$
$644$	0	0
$645$	0.691260	0.0272183
$646$	0	0
$647$	33.9039	1.33290	0.666449	−	0.745551i	$-0.267813\pi$
$647$	33.9039	1.33290	0.666449	+	0.745551i	$0.267813\pi$
$648$	0	0
$649$	3.26808	0.128283
$650$	0	0
$651$	54.9606	2.15407
$652$	0	0
$653$	−15.6760	−0.613450	−0.306725	−	0.951798i	$-0.599233\pi$
$653$	−15.6760	−0.613450	−0.306725	+	0.951798i	$0.599233\pi$
$654$	0	0
$655$	−64.6108	−2.52455
$656$	0	0
$657$	0.770346	0.0300541
$658$	0	0
$659$	39.2087	1.52736	0.763678	−	0.645598i	$-0.223392\pi$
$659$	39.2087	1.52736	0.763678	+	0.645598i	$0.223392\pi$
$660$	0	0
$661$	0.444082	0.0172728	0.00863640	−	0.999963i	$-0.497251\pi$
$661$	0.444082	0.0172728	0.00863640	+	0.999963i	$0.497251\pi$
$662$	0	0
$663$	−12.6369	−0.490776
$664$	0	0
$665$	−85.0324	−3.29742
$666$	0	0
$667$	−31.1092	−1.20455
$668$	0	0
$669$	−4.21051	−0.162788
$670$	0	0
$671$	−12.9801	−0.501092
$672$	0	0
$673$	−7.69877	−0.296766	−0.148383	−	0.988930i	$-0.547407\pi$
$673$	−7.69877	−0.296766	−0.148383	+	0.988930i	$0.547407\pi$
$674$	0	0
$675$	36.8381	1.41790
$676$	0	0
$677$	−23.7491	−0.912751	−0.456376	−	0.889787i	$-0.650853\pi$
$677$	−23.7491	−0.912751	−0.456376	+	0.889787i	$0.650853\pi$
$678$	0	0
$679$	−33.6067	−1.28971
$680$	0	0
$681$	−44.9158	−1.72118
$682$	0	0
$683$	−50.6013	−1.93620	−0.968102	−	0.250556i	$-0.919387\pi$
$683$	−50.6013	−1.93620	−0.968102	+	0.250556i	$0.919387\pi$
$684$	0	0
$685$	26.0471	0.995207
$686$	0	0
$687$	29.4560	1.12382
$688$	0	0
$689$	9.80784	0.373649
$690$	0	0
$691$	23.7377	0.903025	0.451512	−	0.892265i	$-0.350885\pi$
$691$	23.7377	0.903025	0.451512	+	0.892265i	$0.350885\pi$
$692$	0	0
$693$	−1.21263	−0.0460639
$694$	0	0
$695$	−36.8235	−1.39679
$696$	0	0
$697$	2.21296	0.0838220
$698$	0	0
$699$	−2.81246	−0.106377
$700$	0	0
$701$	38.6053	1.45810	0.729051	−	0.684460i	$-0.239962\pi$
$701$	38.6053	1.45810	0.729051	+	0.684460i	$0.239962\pi$
$702$	0	0
$703$	28.9227	1.09084
$704$	0	0
$705$	−17.6333	−0.664109
$706$	0	0
$707$	59.0429	2.22054
$708$	0	0
$709$	−25.7065	−0.965426	−0.482713	−	0.875779i	$-0.660349\pi$
$709$	−25.7065	−0.965426	−0.482713	+	0.875779i	$0.660349\pi$
$710$	0	0
$711$	−1.46163	−0.0548156
$712$	0	0
$713$	54.9688	2.05860
$714$	0	0
$715$	−14.3153	−0.535360
$716$	0	0
$717$	43.0759	1.60870
$718$	0	0
$719$	32.9169	1.22759	0.613796	−	0.789465i	$-0.289642\pi$
$719$	32.9169	1.22759	0.613796	+	0.789465i	$0.289642\pi$
$720$	0	0
$721$	27.6646	1.03028
$722$	0	0
$723$	−41.4537	−1.54168
$724$	0	0
$725$	−28.9044	−1.07348
$726$	0	0
$727$	−31.5790	−1.17120	−0.585600	−	0.810600i	$-0.699141\pi$
$727$	−31.5790	−1.17120	−0.585600	+	0.810600i	$0.699141\pi$
$728$	0	0
$729$	28.7415	1.06450
$730$	0	0
$731$	−0.261955	−0.00968875
$732$	0	0
$733$	3.80843	0.140668	0.0703338	−	0.997524i	$-0.477594\pi$
$733$	3.80843	0.140668	0.0703338	+	0.997524i	$0.477594\pi$
$734$	0	0
$735$	72.3755	2.66961
$736$	0	0
$737$	−5.41971	−0.199638
$738$	0	0
$739$	1.43646	0.0528411	0.0264205	−	0.999651i	$-0.491589\pi$
$739$	1.43646	0.0528411	0.0264205	+	0.999651i	$0.491589\pi$
$740$	0	0
$741$	−32.4434	−1.19184
$742$	0	0
$743$	26.3477	0.966605	0.483302	−	0.875453i	$-0.339437\pi$
$743$	26.3477	0.966605	0.483302	+	0.875453i	$0.339437\pi$
$744$	0	0
$745$	1.88673	0.0691245
$746$	0	0
$747$	−3.08472	−0.112864
$748$	0	0
$749$	14.6045	0.533638
$750$	0	0
$751$	−4.72181	−0.172301	−0.0861506	−	0.996282i	$-0.527457\pi$
$751$	−4.72181	−0.172301	−0.0861506	+	0.996282i	$0.527457\pi$
$752$	0	0
$753$	−40.9132	−1.49096
$754$	0	0
$755$	−42.6101	−1.55074
$756$	0	0
$757$	−39.9298	−1.45127	−0.725637	−	0.688078i	$-0.758455\pi$
$757$	−39.9298	−1.45127	−0.725637	+	0.688078i	$0.758455\pi$
$758$	0	0
$759$	14.5946	0.529751
$760$	0	0
$761$	39.2738	1.42367	0.711837	−	0.702344i	$-0.247863\pi$
$761$	39.2738	1.42367	0.711837	+	0.702344i	$0.247863\pi$
$762$	0	0
$763$	−2.82047	−0.102108
$764$	0	0
$765$	−1.72056	−0.0622071
$766$	0	0
$767$	9.61158	0.347054
$768$	0	0
$769$	42.4607	1.53117	0.765586	−	0.643334i	$-0.222449\pi$
$769$	42.4607	1.53117	0.765586	+	0.643334i	$0.222449\pi$
$770$	0	0
$771$	−16.8905	−0.608297
$772$	0	0
$773$	44.8437	1.61292	0.806459	−	0.591291i	$-0.201381\pi$
$773$	44.8437	1.61292	0.806459	+	0.591291i	$0.201381\pi$
$774$	0	0
$775$	51.0730	1.83460
$776$	0	0
$777$	−38.2598	−1.37256
$778$	0	0
$779$	5.68147	0.203560
$780$	0	0
$781$	−0.844798	−0.0302292
$782$	0	0
$783$	−22.6762	−0.810382
$784$	0	0
$785$	60.1147	2.14559
$786$	0	0
$787$	−50.7168	−1.80786	−0.903929	−	0.427682i	$-0.859330\pi$
$787$	−50.7168	−1.80786	−0.903929	+	0.427682i	$0.859330\pi$
$788$	0	0
$789$	18.5954	0.662014
$790$	0	0
$791$	38.8175	1.38019
$792$	0	0
$793$	−38.1752	−1.35564
$794$	0	0
$795$	−16.0696	−0.569930
$796$	0	0
$797$	31.2596	1.10727	0.553635	−	0.832759i	$-0.313240\pi$
$797$	31.2596	1.10727	0.553635	+	0.832759i	$0.313240\pi$
$798$	0	0
$799$	6.68219	0.236399
$800$	0	0
$801$	−2.23513	−0.0789743
$802$	0	0
$803$	3.97952	0.140434
$804$	0	0
$805$	112.500	3.96511
$806$	0	0
$807$	32.2959	1.13687
$808$	0	0
$809$	−6.92457	−0.243455	−0.121727	−	0.992564i	$-0.538843\pi$
$809$	−6.92457	−0.243455	−0.121727	+	0.992564i	$0.538843\pi$
$810$	0	0
$811$	−10.0880	−0.354236	−0.177118	−	0.984190i	$-0.556677\pi$
$811$	−10.0880	−0.354236	−0.177118	+	0.984190i	$0.556677\pi$
$812$	0	0
$813$	−21.0719	−0.739024
$814$	0	0
$815$	60.9452	2.13482
$816$	0	0
$817$	−0.672533	−0.0235289
$818$	0	0
$819$	−3.56639	−0.124620
$820$	0	0
$821$	50.2386	1.75334	0.876669	−	0.481093i	$-0.159760\pi$
$821$	50.2386	1.75334	0.876669	+	0.481093i	$0.159760\pi$
$822$	0	0
$823$	−50.8517	−1.77258	−0.886290	−	0.463131i	$-0.846726\pi$
$823$	−50.8517	−1.77258	−0.886290	+	0.463131i	$0.846726\pi$
$824$	0	0
$825$	13.5603	0.472108
$826$	0	0
$827$	7.04513	0.244983	0.122492	−	0.992470i	$-0.460912\pi$
$827$	7.04513	0.244983	0.122492	+	0.992470i	$0.460912\pi$
$828$	0	0
$829$	−25.8119	−0.896483	−0.448241	−	0.893913i	$-0.647949\pi$
$829$	−25.8119	−0.896483	−0.448241	+	0.893913i	$0.647949\pi$
$830$	0	0
$831$	19.3991	0.672948
$832$	0	0
$833$	−27.4269	−0.950285
$834$	0	0
$835$	−28.8880	−0.999712
$836$	0	0
$837$	40.0680	1.38495
$838$	0	0
$839$	−37.6885	−1.30115	−0.650576	−	0.759441i	$-0.725473\pi$
$839$	−37.6885	−1.30115	−0.650576	+	0.759441i	$0.725473\pi$
$840$	0	0
$841$	−11.2075	−0.386466
$842$	0	0
$843$	−30.3220	−1.04435
$844$	0	0
$845$	2.65375	0.0912917
$846$	0	0
$847$	42.4741	1.45943
$848$	0	0
$849$	32.4366	1.11322
$850$	0	0
$851$	−38.2655	−1.31172
$852$	0	0
$853$	−50.3708	−1.72466	−0.862332	−	0.506342i	$-0.830997\pi$
$853$	−50.3708	−1.72466	−0.862332	+	0.506342i	$0.830997\pi$
$854$	0	0
$855$	−4.41731	−0.151069
$856$	0	0
$857$	−49.3295	−1.68506	−0.842532	−	0.538647i	$-0.818936\pi$
$857$	−49.3295	−1.68506	−0.842532	+	0.538647i	$0.818936\pi$
$858$	0	0
$859$	−47.9902	−1.63740	−0.818702	−	0.574219i	$-0.805306\pi$
$859$	−47.9902	−1.63740	−0.818702	+	0.574219i	$0.805306\pi$
$860$	0	0
$861$	−7.51561	−0.256131
$862$	0	0
$863$	45.1553	1.53710	0.768552	−	0.639788i	$-0.220978\pi$
$863$	45.1553	1.53710	0.768552	+	0.639788i	$0.220978\pi$
$864$	0	0
$865$	−15.4214	−0.524344
$866$	0	0
$867$	20.4466	0.694403
$868$	0	0
$869$	−7.55063	−0.256138
$870$	0	0
$871$	−15.9396	−0.540094
$872$	0	0
$873$	−1.74582	−0.0590869
$874$	0	0
$875$	28.2571	0.955263
$876$	0	0
$877$	−9.47313	−0.319885	−0.159942	−	0.987126i	$-0.551131\pi$
$877$	−9.47313	−0.319885	−0.159942	+	0.987126i	$0.551131\pi$
$878$	0	0
$879$	6.63370	0.223749
$880$	0	0
$881$	−29.9715	−1.00976	−0.504882	−	0.863188i	$-0.668464\pi$
$881$	−29.9715	−1.00976	−0.504882	+	0.863188i	$0.668464\pi$
$882$	0	0
$883$	46.0660	1.55024	0.775122	−	0.631811i	$-0.217688\pi$
$883$	46.0660	1.55024	0.775122	+	0.631811i	$0.217688\pi$
$884$	0	0
$885$	−15.7480	−0.529365
$886$	0	0
$887$	−54.4170	−1.82715	−0.913573	−	0.406675i	$-0.866688\pi$
$887$	−54.4170	−1.82715	−0.913573	+	0.406675i	$0.866688\pi$
$888$	0	0
$889$	45.5176	1.52661
$890$	0	0
$891$	9.81732	0.328893
$892$	0	0
$893$	17.1556	0.574090
$894$	0	0
$895$	36.7563	1.22863
$896$	0	0
$897$	42.9235	1.43317
$898$	0	0
$899$	−31.4387	−1.04854
$900$	0	0
$901$	6.08962	0.202875
$902$	0	0
$903$	0.889645	0.0296055
$904$	0	0
$905$	27.5697	0.916449
$906$	0	0
$907$	30.0368	0.997357	0.498678	−	0.866787i	$-0.333819\pi$
$907$	30.0368	0.997357	0.498678	+	0.866787i	$0.333819\pi$
$908$	0	0
$909$	3.06719	0.101732
$910$	0	0
$911$	42.0040	1.39166	0.695828	−	0.718209i	$-0.255038\pi$
$911$	42.0040	1.39166	0.695828	+	0.718209i	$0.255038\pi$
$912$	0	0
$913$	−15.9353	−0.527381
$914$	0	0
$915$	62.5479	2.06777
$916$	0	0
$917$	−83.1534	−2.74597
$918$	0	0
$919$	34.7108	1.14500	0.572502	−	0.819903i	$-0.305973\pi$
$919$	34.7108	1.14500	0.572502	+	0.819903i	$0.305973\pi$
$920$	0	0
$921$	−3.57352	−0.117751
$922$	0	0
$923$	−2.48459	−0.0817813
$924$	0	0
$925$	−35.5535	−1.16899
$926$	0	0
$927$	1.43713	0.0472017
$928$	0	0
$929$	−35.0708	−1.15064	−0.575318	−	0.817930i	$-0.695122\pi$
$929$	−35.0708	−1.15064	−0.575318	+	0.817930i	$0.695122\pi$
$930$	0	0
$931$	−70.4147	−2.30775
$932$	0	0
$933$	2.27157	0.0743680
$934$	0	0
$935$	−8.88824	−0.290676
$936$	0	0
$937$	28.9508	0.945781	0.472890	−	0.881121i	$-0.343211\pi$
$937$	28.9508	0.945781	0.472890	+	0.881121i	$0.343211\pi$
$938$	0	0
$939$	−26.9479	−0.879412
$940$	0	0
$941$	4.91056	0.160080	0.0800398	−	0.996792i	$-0.474495\pi$
$941$	4.91056	0.160080	0.0800398	+	0.996792i	$0.474495\pi$
$942$	0	0
$943$	−7.51673	−0.244778
$944$	0	0
$945$	82.0039	2.66759
$946$	0	0
$947$	−16.2429	−0.527825	−0.263912	−	0.964547i	$-0.585013\pi$
$947$	−16.2429	−0.527825	−0.263912	+	0.964547i	$0.585013\pi$
$948$	0	0
$949$	11.7040	0.379926
$950$	0	0
$951$	−33.8023	−1.09611
$952$	0	0
$953$	11.1190	0.360178	0.180089	−	0.983650i	$-0.442361\pi$
$953$	11.1190	0.360178	0.180089	+	0.983650i	$0.442361\pi$
$954$	0	0
$955$	2.90637	0.0940479
$956$	0	0
$957$	−8.34721	−0.269827
$958$	0	0
$959$	33.5223	1.08249
$960$	0	0
$961$	24.5510	0.791969
$962$	0	0
$963$	0.758683	0.0244482
$964$	0	0
$965$	−63.5704	−2.04640
$966$	0	0
$967$	−24.8298	−0.798473	−0.399236	−	0.916848i	$-0.630725\pi$
$967$	−24.8298	−0.798473	−0.399236	+	0.916848i	$0.630725\pi$
$968$	0	0
$969$	−20.1439	−0.647115
$970$	0	0
$971$	−2.97994	−0.0956309	−0.0478154	−	0.998856i	$-0.515226\pi$
$971$	−2.97994	−0.0956309	−0.0478154	+	0.998856i	$0.515226\pi$
$972$	0	0
$973$	−47.3914	−1.51930
$974$	0	0
$975$	39.8814	1.27723
$976$	0	0
$977$	−47.9009	−1.53249	−0.766243	−	0.642551i	$-0.777876\pi$
$977$	−47.9009	−1.53249	−0.766243	+	0.642551i	$0.777876\pi$
$978$	0	0
$979$	−11.5464	−0.369025
$980$	0	0
$981$	−0.146519	−0.00467800
$982$	0	0
$983$	42.3043	1.34930	0.674649	−	0.738139i	$-0.264295\pi$
$983$	42.3043	1.34930	0.674649	+	0.738139i	$0.264295\pi$
$984$	0	0
$985$	34.1301	1.08747
$986$	0	0
$987$	−22.6939	−0.722355
$988$	0	0
$989$	0.889777	0.0282933
$990$	0	0
$991$	−14.4541	−0.459151	−0.229575	−	0.973291i	$-0.573734\pi$
$991$	−14.4541	−0.459151	−0.229575	+	0.973291i	$0.573734\pi$
$992$	0	0
$993$	−15.3626	−0.487516
$994$	0	0
$995$	−59.2834	−1.87941
$996$	0	0
$997$	−19.7728	−0.626211	−0.313105	−	0.949718i	$-0.601369\pi$
$997$	−19.7728	−0.626211	−0.313105	+	0.949718i	$0.601369\pi$
$998$	0	0
$999$	−27.8926	−0.882483

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.8		28
4.3	odd	2		4024.2.a.d.1.21	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.21	✓	28	4.3	odd	2
8048.2.a.v.1.8		28	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)

\(f(q)\)	\(=\)	\(q-1.66428 q^{3} -3.44274 q^{5} -4.43077 q^{7} -0.230172 q^{9} +O(q^{10})\)	\(q-1.66428 q^{3} -3.44274 q^{5} -4.43077 q^{7} -0.230172 q^{9} -1.18904 q^{11} -3.49702 q^{13} +5.72968 q^{15} -2.17128 q^{17} -5.57445 q^{19} +7.37404 q^{21} +7.37514 q^{23} +6.85244 q^{25} +5.37591 q^{27} -4.21812 q^{29} +7.45326 q^{31} +1.97890 q^{33} +15.2540 q^{35} -5.18844 q^{37} +5.82003 q^{39} -1.01920 q^{41} +0.120646 q^{43} +0.792421 q^{45} -3.07754 q^{47} +12.6317 q^{49} +3.61361 q^{51} -2.80462 q^{53} +4.09355 q^{55} +9.27745 q^{57} -2.74850 q^{59} +10.9165 q^{61} +1.01984 q^{63} +12.0393 q^{65} +4.55806 q^{67} -12.2743 q^{69} +0.710487 q^{71} -3.34683 q^{73} -11.4044 q^{75} +5.26836 q^{77} +6.35019 q^{79} -8.25651 q^{81} +13.4018 q^{83} +7.47514 q^{85} +7.02013 q^{87} +9.71069 q^{89} +15.4945 q^{91} -12.4043 q^{93} +19.1914 q^{95} +7.58484 q^{97} +0.273683 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

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