LMFDB - Embedded newform 4864.2.a.bs.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, 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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 23x^{8} + 44x^{7} + 167x^{6} - 266x^{5} - 491x^{4} + 460x^{3} + 546x^{2} + 56x - 8 $ x^10 - 2x^9 - 23x^8 + 44x^7 + 167x^6 - 266x^5 - 491x^4 + 460x^3 + 546x^2 + 56*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-3.58141$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.15181 q^{3} -1.07587 q^{5} -0.238565 q^{7} -1.67333 q^{9} +O(q^{10})$	$q+1.15181 q^{3} -1.07587 q^{5} -0.238565 q^{7} -1.67333 q^{9} +2.84249 q^{11} +1.31444 q^{13} -1.23921 q^{15} -4.47279 q^{17} -1.00000 q^{19} -0.274782 q^{21} -1.94338 q^{23} -3.84249 q^{25} -5.38280 q^{27} +8.83944 q^{29} +6.64877 q^{31} +3.27402 q^{33} +0.256666 q^{35} -4.06763 q^{37} +1.51399 q^{39} -2.78211 q^{41} +1.21363 q^{43} +1.80029 q^{45} +10.2695 q^{47} -6.94309 q^{49} -5.15181 q^{51} -8.36231 q^{53} -3.05817 q^{55} -1.15181 q^{57} -9.25916 q^{59} -4.66035 q^{61} +0.399197 q^{63} -1.41417 q^{65} -4.31849 q^{67} -2.23841 q^{69} -2.78069 q^{71} -0.134460 q^{73} -4.42583 q^{75} -0.678119 q^{77} -12.7841 q^{79} -1.17998 q^{81} -13.4028 q^{83} +4.81216 q^{85} +10.1814 q^{87} +9.20650 q^{89} -0.313579 q^{91} +7.65814 q^{93} +1.07587 q^{95} -0.247499 q^{97} -4.75643 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{3} + 14 q^{9}+O(q^{10}) $ 10 * q - 4 * q^3 + 14 * q^9	$ 10 q - 4 q^{3} + 14 q^{9} - 20 q^{11} + 4 q^{17} - 10 q^{19} + 10 q^{25} - 28 q^{27} - 8 q^{33} - 36 q^{35} - 12 q^{41} + 4 q^{43} + 26 q^{49} - 36 q^{51} + 4 q^{57} - 52 q^{59} - 24 q^{65} - 12 q^{67} + 12 q^{73} + 12 q^{75} + 34 q^{81} - 16 q^{83} - 20 q^{89} - 60 q^{91} - 28 q^{97} - 60 q^{99}+O(q^{100}) $ 10 * q - 4 * q^3 + 14 * q^9 - 20 * q^11 + 4 * q^17 - 10 * q^19 + 10 * q^25 - 28 * q^27 - 8 * q^33 - 36 * q^35 - 12 * q^41 + 4 * q^43 + 26 * q^49 - 36 * q^51 + 4 * q^57 - 52 * q^59 - 24 * q^65 - 12 * q^67 + 12 * q^73 + 12 * q^75 + 34 * q^81 - 16 * q^83 - 20 * q^89 - 60 * q^91 - 28 * q^97 - 60 * 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
$3$	1.15181	0.664999	0.332500	−	0.943103i	$-0.392108\pi$
$3$	1.15181	0.664999	0.332500	+	0.943103i	$0.392108\pi$
$4$	0	0
$5$	−1.07587	−0.481146	−0.240573	−	0.970631i	$-0.577335\pi$
$5$	−1.07587	−0.481146	−0.240573	+	0.970631i	$0.577335\pi$
$6$	0	0
$7$	−0.238565	−0.0901690	−0.0450845	−	0.998983i	$-0.514356\pi$
$7$	−0.238565	−0.0901690	−0.0450845	+	0.998983i	$0.514356\pi$
$8$	0	0
$9$	−1.67333	−0.557776
$10$	0	0
$11$	2.84249	0.857044	0.428522	−	0.903531i	$-0.359034\pi$
$11$	2.84249	0.857044	0.428522	+	0.903531i	$0.359034\pi$
$12$	0	0
$13$	1.31444	0.364560	0.182280	−	0.983247i	$-0.441652\pi$
$13$	1.31444	0.364560	0.182280	+	0.983247i	$0.441652\pi$
$14$	0	0
$15$	−1.23921	−0.319962
$16$	0	0
$17$	−4.47279	−1.08481	−0.542405	−	0.840117i	$-0.682486\pi$
$17$	−4.47279	−1.08481	−0.542405	+	0.840117i	$0.682486\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.274782	−0.0599623
$22$	0	0
$23$	−1.94338	−0.405222	−0.202611	−	0.979259i	$-0.564943\pi$
$23$	−1.94338	−0.405222	−0.202611	+	0.979259i	$0.564943\pi$
$24$	0	0
$25$	−3.84249	−0.768499
$26$	0	0
$27$	−5.38280	−1.03592
$28$	0	0
$29$	8.83944	1.64144	0.820721	−	0.571329i	$-0.193572\pi$
$29$	8.83944	1.64144	0.820721	+	0.571329i	$0.193572\pi$
$30$	0	0
$31$	6.64877	1.19415	0.597077	−	0.802184i	$-0.296329\pi$
$31$	6.64877	1.19415	0.597077	+	0.802184i	$0.296329\pi$
$32$	0	0
$33$	3.27402	0.569933
$34$	0	0
$35$	0.256666	0.0433845
$36$	0	0
$37$	−4.06763	−0.668715	−0.334357	−	0.942446i	$-0.608519\pi$
$37$	−4.06763	−0.668715	−0.334357	+	0.942446i	$0.608519\pi$
$38$	0	0
$39$	1.51399	0.242432
$40$	0	0
$41$	−2.78211	−0.434492	−0.217246	−	0.976117i	$-0.569707\pi$
$41$	−2.78211	−0.434492	−0.217246	+	0.976117i	$0.569707\pi$
$42$	0	0
$43$	1.21363	0.185077	0.0925386	−	0.995709i	$-0.470502\pi$
$43$	1.21363	0.185077	0.0925386	+	0.995709i	$0.470502\pi$
$44$	0	0
$45$	1.80029	0.268372
$46$	0	0
$47$	10.2695	1.49796	0.748978	−	0.662595i	$-0.230545\pi$
$47$	10.2695	1.49796	0.748978	+	0.662595i	$0.230545\pi$
$48$	0	0
$49$	−6.94309	−0.991870
$50$	0	0
$51$	−5.15181	−0.721398
$52$	0	0
$53$	−8.36231	−1.14865	−0.574326	−	0.818627i	$-0.694736\pi$
$53$	−8.36231	−1.14865	−0.574326	+	0.818627i	$0.694736\pi$
$54$	0	0
$55$	−3.05817	−0.412363
$56$	0	0
$57$	−1.15181	−0.152561
$58$	0	0
$59$	−9.25916	−1.20544	−0.602720	−	0.797953i	$-0.705916\pi$
$59$	−9.25916	−1.20544	−0.602720	+	0.797953i	$0.705916\pi$
$60$	0	0
$61$	−4.66035	−0.596697	−0.298349	−	0.954457i	$-0.596436\pi$
$61$	−4.66035	−0.596697	−0.298349	+	0.954457i	$0.596436\pi$
$62$	0	0
$63$	0.399197	0.0502941
$64$	0	0
$65$	−1.41417	−0.175407
$66$	0	0
$67$	−4.31849	−0.527587	−0.263794	−	0.964579i	$-0.584974\pi$
$67$	−4.31849	−0.527587	−0.263794	+	0.964579i	$0.584974\pi$
$68$	0	0
$69$	−2.23841	−0.269472
$70$	0	0
$71$	−2.78069	−0.330007	−0.165003	−	0.986293i	$-0.552764\pi$
$71$	−2.78069	−0.330007	−0.165003	+	0.986293i	$0.552764\pi$
$72$	0	0
$73$	−0.134460	−0.0157373	−0.00786866	−	0.999969i	$-0.502505\pi$
$73$	−0.134460	−0.0157373	−0.00786866	+	0.999969i	$0.502505\pi$
$74$	0	0
$75$	−4.42583	−0.511051
$76$	0	0
$77$	−0.678119	−0.0772788
$78$	0	0
$79$	−12.7841	−1.43832	−0.719162	−	0.694843i	$-0.755474\pi$
$79$	−12.7841	−1.43832	−0.719162	+	0.694843i	$0.755474\pi$
$80$	0	0
$81$	−1.17998	−0.131109
$82$	0	0
$83$	−13.4028	−1.47115	−0.735573	−	0.677445i	$-0.763087\pi$
$83$	−13.4028	−1.47115	−0.735573	+	0.677445i	$0.763087\pi$
$84$	0	0
$85$	4.81216	0.521952
$86$	0	0
$87$	10.1814	1.09156
$88$	0	0
$89$	9.20650	0.975887	0.487944	−	0.872875i	$-0.337747\pi$
$89$	9.20650	0.975887	0.487944	+	0.872875i	$0.337747\pi$
$90$	0	0
$91$	−0.313579	−0.0328720
$92$	0	0
$93$	7.65814	0.794112
$94$	0	0
$95$	1.07587	0.110382
$96$	0	0
$97$	−0.247499	−0.0251297	−0.0125648	−	0.999921i	$-0.504000\pi$
$97$	−0.247499	−0.0251297	−0.0125648	+	0.999921i	$0.504000\pi$
$98$	0	0
$99$	−4.75643	−0.478039
$100$	0	0
$101$	−16.8026	−1.67193	−0.835963	−	0.548786i	$-0.815090\pi$
$101$	−16.8026	−1.67193	−0.835963	+	0.548786i	$0.815090\pi$
$102$	0	0
$103$	−5.40957	−0.533020	−0.266510	−	0.963832i	$-0.585871\pi$
$103$	−5.40957	−0.533020	−0.266510	+	0.963832i	$0.585871\pi$
$104$	0	0
$105$	0.295631	0.0288506
$106$	0	0
$107$	−6.39113	−0.617853	−0.308927	−	0.951086i	$-0.599970\pi$
$107$	−6.39113	−0.617853	−0.308927	+	0.951086i	$0.599970\pi$
$108$	0	0
$109$	14.8779	1.42505	0.712524	−	0.701648i	$-0.247552\pi$
$109$	14.8779	1.42505	0.712524	+	0.701648i	$0.247552\pi$
$110$	0	0
$111$	−4.68515	−0.444695
$112$	0	0
$113$	8.59073	0.808148	0.404074	−	0.914726i	$-0.367594\pi$
$113$	8.59073	0.808148	0.404074	+	0.914726i	$0.367594\pi$
$114$	0	0
$115$	2.09083	0.194971
$116$	0	0
$117$	−2.19949	−0.203343
$118$	0	0
$119$	1.06705	0.0978163
$120$	0	0
$121$	−2.92023	−0.265476
$122$	0	0
$123$	−3.20446	−0.288937
$124$	0	0
$125$	9.51342	0.850906
$126$	0	0
$127$	1.15983	0.102918	0.0514592	−	0.998675i	$-0.483613\pi$
$127$	1.15983	0.102918	0.0514592	+	0.998675i	$0.483613\pi$
$128$	0	0
$129$	1.39788	0.123076
$130$	0	0
$131$	3.85559	0.336864	0.168432	−	0.985713i	$-0.446130\pi$
$131$	3.85559	0.336864	0.168432	+	0.985713i	$0.446130\pi$
$132$	0	0
$133$	0.238565	0.0206862
$134$	0	0
$135$	5.79122	0.498428
$136$	0	0
$137$	−1.42976	−0.122152	−0.0610761	−	0.998133i	$-0.519453\pi$
$137$	−1.42976	−0.122152	−0.0610761	+	0.998133i	$0.519453\pi$
$138$	0	0
$139$	−3.58190	−0.303813	−0.151907	−	0.988395i	$-0.548541\pi$
$139$	−3.58190	−0.303813	−0.151907	+	0.988395i	$0.548541\pi$
$140$	0	0
$141$	11.8285	0.996139
$142$	0	0
$143$	3.73629	0.312444
$144$	0	0
$145$	−9.51013	−0.789773
$146$	0	0
$147$	−7.99713	−0.659592
$148$	0	0
$149$	−16.2763	−1.33341	−0.666705	−	0.745322i	$-0.732296\pi$
$149$	−16.2763	−1.33341	−0.666705	+	0.745322i	$0.732296\pi$
$150$	0	0
$151$	14.3242	1.16569	0.582845	−	0.812583i	$-0.301939\pi$
$151$	14.3242	1.16569	0.582845	+	0.812583i	$0.301939\pi$
$152$	0	0
$153$	7.48445	0.605082
$154$	0	0
$155$	−7.15325	−0.574563
$156$	0	0
$157$	−3.73495	−0.298081	−0.149041	−	0.988831i	$-0.547619\pi$
$157$	−3.73495	−0.298081	−0.149041	+	0.988831i	$0.547619\pi$
$158$	0	0
$159$	−9.63181	−0.763852
$160$	0	0
$161$	0.463621	0.0365385
$162$	0	0
$163$	3.98861	0.312412	0.156206	−	0.987724i	$-0.450074\pi$
$163$	3.98861	0.312412	0.156206	+	0.987724i	$0.450074\pi$
$164$	0	0
$165$	−3.52243	−0.274221
$166$	0	0
$167$	−8.17158	−0.632336	−0.316168	−	0.948703i	$-0.602396\pi$
$167$	−8.17158	−0.632336	−0.316168	+	0.948703i	$0.602396\pi$
$168$	0	0
$169$	−11.2722	−0.867096
$170$	0	0
$171$	1.67333	0.127963
$172$	0	0
$173$	−0.351356	−0.0267131	−0.0133565	−	0.999911i	$-0.504252\pi$
$173$	−0.351356	−0.0267131	−0.0133565	+	0.999911i	$0.504252\pi$
$174$	0	0
$175$	0.916684	0.0692948
$176$	0	0
$177$	−10.6648	−0.801616
$178$	0	0
$179$	−15.2740	−1.14163	−0.570817	−	0.821077i	$-0.693373\pi$
$179$	−15.2740	−1.14163	−0.570817	+	0.821077i	$0.693373\pi$
$180$	0	0
$181$	−8.94630	−0.664974	−0.332487	−	0.943108i	$-0.607888\pi$
$181$	−8.94630	−0.664974	−0.332487	+	0.943108i	$0.607888\pi$
$182$	0	0
$183$	−5.36785	−0.396803
$184$	0	0
$185$	4.37626	0.321749
$186$	0	0
$187$	−12.7139	−0.929730
$188$	0	0
$189$	1.28415	0.0934079
$190$	0	0
$191$	−18.3946	−1.33098	−0.665492	−	0.746405i	$-0.731778\pi$
$191$	−18.3946	−1.33098	−0.665492	+	0.746405i	$0.731778\pi$
$192$	0	0
$193$	−5.09902	−0.367035	−0.183518	−	0.983016i	$-0.558748\pi$
$193$	−5.09902	−0.367035	−0.183518	+	0.983016i	$0.558748\pi$
$194$	0	0
$195$	−1.62886	−0.116645
$196$	0	0
$197$	20.2916	1.44572	0.722860	−	0.690995i	$-0.242827\pi$
$197$	20.2916	1.44572	0.722860	+	0.690995i	$0.242827\pi$
$198$	0	0
$199$	−0.310999	−0.0220461	−0.0110231	−	0.999939i	$-0.503509\pi$
$199$	−0.310999	−0.0220461	−0.0110231	+	0.999939i	$0.503509\pi$
$200$	0	0
$201$	−4.97408	−0.350845
$202$	0	0
$203$	−2.10878	−0.148007
$204$	0	0
$205$	2.99320	0.209054
$206$	0	0
$207$	3.25191	0.226023
$208$	0	0
$209$	−2.84249	−0.196619
$210$	0	0
$211$	−14.3284	−0.986406	−0.493203	−	0.869914i	$-0.664174\pi$
$211$	−14.3284	−0.986406	−0.493203	+	0.869914i	$0.664174\pi$
$212$	0	0
$213$	−3.20283	−0.219454
$214$	0	0
$215$	−1.30572	−0.0890491
$216$	0	0
$217$	−1.58616	−0.107676
$218$	0	0
$219$	−0.154872	−0.0104653
$220$	0	0
$221$	−5.87921	−0.395479
$222$	0	0
$223$	12.8621	0.861312	0.430656	−	0.902516i	$-0.358282\pi$
$223$	12.8621	0.861312	0.430656	+	0.902516i	$0.358282\pi$
$224$	0	0
$225$	6.42976	0.428650
$226$	0	0
$227$	−18.4491	−1.22451	−0.612256	−	0.790659i	$-0.709738\pi$
$227$	−18.4491	−1.22451	−0.612256	+	0.790659i	$0.709738\pi$
$228$	0	0
$229$	4.73279	0.312751	0.156376	−	0.987698i	$-0.450019\pi$
$229$	4.73279	0.312751	0.156376	+	0.987698i	$0.450019\pi$
$230$	0	0
$231$	−0.781066	−0.0513903
$232$	0	0
$233$	−4.27809	−0.280267	−0.140133	−	0.990133i	$-0.544753\pi$
$233$	−4.27809	−0.280267	−0.140133	+	0.990133i	$0.544753\pi$
$234$	0	0
$235$	−11.0487	−0.720735
$236$	0	0
$237$	−14.7249	−0.956483
$238$	0	0
$239$	−20.3905	−1.31895	−0.659474	−	0.751727i	$-0.729221\pi$
$239$	−20.3905	−1.31895	−0.659474	+	0.751727i	$0.729221\pi$
$240$	0	0
$241$	25.5877	1.64825	0.824123	−	0.566410i	$-0.191668\pi$
$241$	25.5877	1.64825	0.824123	+	0.566410i	$0.191668\pi$
$242$	0	0
$243$	14.7893	0.948732
$244$	0	0
$245$	7.46989	0.477234
$246$	0	0
$247$	−1.31444	−0.0836358
$248$	0	0
$249$	−15.4375	−0.978311
$250$	0	0
$251$	−28.9847	−1.82950	−0.914749	−	0.404024i	$-0.867611\pi$
$251$	−28.9847	−1.82950	−0.914749	+	0.404024i	$0.867611\pi$
$252$	0	0
$253$	−5.52404	−0.347293
$254$	0	0
$255$	5.54270	0.347098
$256$	0	0
$257$	5.10203	0.318256	0.159128	−	0.987258i	$-0.449132\pi$
$257$	5.10203	0.318256	0.159128	+	0.987258i	$0.449132\pi$
$258$	0	0
$259$	0.970394	0.0602974
$260$	0	0
$261$	−14.7913	−0.915558
$262$	0	0
$263$	−11.6953	−0.721160	−0.360580	−	0.932728i	$-0.617421\pi$
$263$	−11.6953	−0.721160	−0.360580	+	0.932728i	$0.617421\pi$
$264$	0	0
$265$	8.99680	0.552669
$266$	0	0
$267$	10.6042	0.648964
$268$	0	0
$269$	−0.163433	−0.00996466	−0.00498233	−	0.999988i	$-0.501586\pi$
$269$	−0.163433	−0.00996466	−0.00498233	+	0.999988i	$0.501586\pi$
$270$	0	0
$271$	−1.01216	−0.0614846	−0.0307423	−	0.999527i	$-0.509787\pi$
$271$	−1.01216	−0.0614846	−0.0307423	+	0.999527i	$0.509787\pi$
$272$	0	0
$273$	−0.361184	−0.0218599
$274$	0	0
$275$	−10.9223	−0.658637
$276$	0	0
$277$	3.14825	0.189160	0.0945800	−	0.995517i	$-0.469849\pi$
$277$	3.14825	0.189160	0.0945800	+	0.995517i	$0.469849\pi$
$278$	0	0
$279$	−11.1256	−0.666071
$280$	0	0
$281$	24.6734	1.47189	0.735946	−	0.677040i	$-0.236738\pi$
$281$	24.6734	1.47189	0.735946	+	0.677040i	$0.236738\pi$
$282$	0	0
$283$	13.8095	0.820889	0.410445	−	0.911886i	$-0.365374\pi$
$283$	13.8095	0.820889	0.410445	+	0.911886i	$0.365374\pi$
$284$	0	0
$285$	1.23921	0.0734042
$286$	0	0
$287$	0.663713	0.0391777
$288$	0	0
$289$	3.00584	0.176814
$290$	0	0
$291$	−0.285072	−0.0167112
$292$	0	0
$293$	8.24682	0.481784	0.240892	−	0.970552i	$-0.422560\pi$
$293$	8.24682	0.481784	0.240892	+	0.970552i	$0.422560\pi$
$294$	0	0
$295$	9.96169	0.579992
$296$	0	0
$297$	−15.3006	−0.887829
$298$	0	0
$299$	−2.55445	−0.147728
$300$	0	0
$301$	−0.289530	−0.0166882
$302$	0	0
$303$	−19.3535	−1.11183
$304$	0	0
$305$	5.01396	0.287098
$306$	0	0
$307$	−17.4540	−0.996153	−0.498076	−	0.867133i	$-0.665960\pi$
$307$	−17.4540	−0.996153	−0.498076	+	0.867133i	$0.665960\pi$
$308$	0	0
$309$	−6.23080	−0.354458
$310$	0	0
$311$	−10.5256	−0.596851	−0.298425	−	0.954433i	$-0.596461\pi$
$311$	−10.5256	−0.596851	−0.298425	+	0.954433i	$0.596461\pi$
$312$	0	0
$313$	14.3498	0.811098	0.405549	−	0.914073i	$-0.367080\pi$
$313$	14.3498	0.811098	0.405549	+	0.914073i	$0.367080\pi$
$314$	0	0
$315$	−0.429486	−0.0241988
$316$	0	0
$317$	16.3470	0.918138	0.459069	−	0.888401i	$-0.348183\pi$
$317$	16.3470	0.918138	0.459069	+	0.888401i	$0.348183\pi$
$318$	0	0
$319$	25.1260	1.40679
$320$	0	0
$321$	−7.36138	−0.410872
$322$	0	0
$323$	4.47279	0.248873
$324$	0	0
$325$	−5.05073	−0.280164
$326$	0	0
$327$	17.1366	0.947656
$328$	0	0
$329$	−2.44993	−0.135069
$330$	0	0
$331$	15.0118	0.825123	0.412562	−	0.910930i	$-0.364634\pi$
$331$	15.0118	0.825123	0.412562	+	0.910930i	$0.364634\pi$
$332$	0	0
$333$	6.80649	0.372993
$334$	0	0
$335$	4.64615	0.253846
$336$	0	0
$337$	0.462187	0.0251769	0.0125885	−	0.999921i	$-0.495993\pi$
$337$	0.462187	0.0251769	0.0125885	+	0.999921i	$0.495993\pi$
$338$	0	0
$339$	9.89491	0.537418
$340$	0	0
$341$	18.8991	1.02344
$342$	0	0
$343$	3.32633	0.179605
$344$	0	0
$345$	2.40824	0.129655
$346$	0	0
$347$	−17.2669	−0.926935	−0.463468	−	0.886114i	$-0.653395\pi$
$347$	−17.2669	−0.926935	−0.463468	+	0.886114i	$0.653395\pi$
$348$	0	0
$349$	−26.1466	−1.39960	−0.699798	−	0.714341i	$-0.746727\pi$
$349$	−26.1466	−1.39960	−0.699798	+	0.714341i	$0.746727\pi$
$350$	0	0
$351$	−7.07536	−0.377655
$352$	0	0
$353$	12.3930	0.659614	0.329807	−	0.944048i	$-0.393016\pi$
$353$	12.3930	0.659614	0.329807	+	0.944048i	$0.393016\pi$
$354$	0	0
$355$	2.99167	0.158781
$356$	0	0
$357$	1.22904	0.0650478
$358$	0	0
$359$	14.1664	0.747673	0.373837	−	0.927495i	$-0.378042\pi$
$359$	14.1664	0.747673	0.373837	+	0.927495i	$0.378042\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.36356	−0.176541
$364$	0	0
$365$	0.144662	0.00757195
$366$	0	0
$367$	−3.00328	−0.156770	−0.0783850	−	0.996923i	$-0.524976\pi$
$367$	−3.00328	−0.156770	−0.0783850	+	0.996923i	$0.524976\pi$
$368$	0	0
$369$	4.65538	0.242349
$370$	0	0
$371$	1.99495	0.103573
$372$	0	0
$373$	38.0432	1.96980	0.984901	−	0.173119i	$-0.0553845\pi$
$373$	38.0432	1.96980	0.984901	+	0.173119i	$0.0553845\pi$
$374$	0	0
$375$	10.9577	0.565852
$376$	0	0
$377$	11.6189	0.598404
$378$	0	0
$379$	−28.6291	−1.47058	−0.735290	−	0.677753i	$-0.762954\pi$
$379$	−28.6291	−1.47058	−0.735290	+	0.677753i	$0.762954\pi$
$380$	0	0
$381$	1.33591	0.0684407
$382$	0	0
$383$	24.1407	1.23353	0.616767	−	0.787146i	$-0.288442\pi$
$383$	24.1407	1.23353	0.616767	+	0.787146i	$0.288442\pi$
$384$	0	0
$385$	0.729571	0.0371824
$386$	0	0
$387$	−2.03081	−0.103232
$388$	0	0
$389$	−16.2953	−0.826206	−0.413103	−	0.910684i	$-0.635555\pi$
$389$	−16.2953	−0.826206	−0.413103	+	0.910684i	$0.635555\pi$
$390$	0	0
$391$	8.69231	0.439589
$392$	0	0
$393$	4.44091	0.224014
$394$	0	0
$395$	13.7541	0.692043
$396$	0	0
$397$	−36.8139	−1.84764	−0.923819	−	0.382829i	$-0.874950\pi$
$397$	−36.8139	−1.84764	−0.923819	+	0.382829i	$0.874950\pi$
$398$	0	0
$399$	0.274782	0.0137563
$400$	0	0
$401$	−23.6899	−1.18302	−0.591508	−	0.806299i	$-0.701467\pi$
$401$	−23.6899	−1.18302	−0.591508	+	0.806299i	$0.701467\pi$
$402$	0	0
$403$	8.73941	0.435341
$404$	0	0
$405$	1.26951	0.0630827
$406$	0	0
$407$	−11.5622	−0.573118
$408$	0	0
$409$	−0.102363	−0.00506154	−0.00253077	−	0.999997i	$-0.500806\pi$
$409$	−0.102363	−0.00506154	−0.00253077	+	0.999997i	$0.500806\pi$
$410$	0	0
$411$	−1.64681	−0.0812311
$412$	0	0
$413$	2.20891	0.108693
$414$	0	0
$415$	14.4197	0.707836
$416$	0	0
$417$	−4.12568	−0.202035
$418$	0	0
$419$	−25.7326	−1.25712	−0.628560	−	0.777761i	$-0.716355\pi$
$419$	−25.7326	−1.25712	−0.628560	+	0.777761i	$0.716355\pi$
$420$	0	0
$421$	18.0014	0.877334	0.438667	−	0.898650i	$-0.355451\pi$
$421$	18.0014	0.877334	0.438667	+	0.898650i	$0.355451\pi$
$422$	0	0
$423$	−17.1842	−0.835524
$424$	0	0
$425$	17.1867	0.833675
$426$	0	0
$427$	1.11180	0.0538036
$428$	0	0
$429$	4.30350	0.207775
$430$	0	0
$431$	27.6392	1.33133	0.665667	−	0.746249i	$-0.268147\pi$
$431$	27.6392	1.33133	0.665667	+	0.746249i	$0.268147\pi$
$432$	0	0
$433$	38.5219	1.85124	0.925621	−	0.378451i	$-0.123543\pi$
$433$	38.5219	1.85124	0.925621	+	0.378451i	$0.123543\pi$
$434$	0	0
$435$	−10.9539	−0.525198
$436$	0	0
$437$	1.94338	0.0929643
$438$	0	0
$439$	−23.1302	−1.10394	−0.551971	−	0.833863i	$-0.686124\pi$
$439$	−23.1302	−1.10394	−0.551971	+	0.833863i	$0.686124\pi$
$440$	0	0
$441$	11.6181	0.553241
$442$	0	0
$443$	−19.2436	−0.914290	−0.457145	−	0.889392i	$-0.651128\pi$
$443$	−19.2436	−0.914290	−0.457145	+	0.889392i	$0.651128\pi$
$444$	0	0
$445$	−9.90505	−0.469544
$446$	0	0
$447$	−18.7473	−0.886716
$448$	0	0
$449$	11.0578	0.521851	0.260926	−	0.965359i	$-0.415972\pi$
$449$	11.0578	0.521851	0.260926	+	0.965359i	$0.415972\pi$
$450$	0	0
$451$	−7.90812	−0.372379
$452$	0	0
$453$	16.4988	0.775183
$454$	0	0
$455$	0.337372	0.0158162
$456$	0	0
$457$	−4.08310	−0.190999	−0.0954996	−	0.995429i	$-0.530445\pi$
$457$	−4.08310	−0.190999	−0.0954996	+	0.995429i	$0.530445\pi$
$458$	0	0
$459$	24.0761	1.12378
$460$	0	0
$461$	−7.59017	−0.353509	−0.176755	−	0.984255i	$-0.556560\pi$
$461$	−7.59017	−0.353509	−0.176755	+	0.984255i	$0.556560\pi$
$462$	0	0
$463$	37.2637	1.73179	0.865894	−	0.500227i	$-0.166750\pi$
$463$	37.2637	1.73179	0.865894	+	0.500227i	$0.166750\pi$
$464$	0	0
$465$	−8.23920	−0.382084
$466$	0	0
$467$	8.92480	0.412991	0.206495	−	0.978448i	$-0.433794\pi$
$467$	8.92480	0.412991	0.206495	+	0.978448i	$0.433794\pi$
$468$	0	0
$469$	1.03024	0.0475720
$470$	0	0
$471$	−4.30196	−0.198224
$472$	0	0
$473$	3.44974	0.158619
$474$	0	0
$475$	3.84249	0.176306
$476$	0	0
$477$	13.9929	0.640690
$478$	0	0
$479$	−19.6241	−0.896650	−0.448325	−	0.893871i	$-0.647979\pi$
$479$	−19.6241	−0.896650	−0.448325	+	0.893871i	$0.647979\pi$
$480$	0	0
$481$	−5.34666	−0.243787
$482$	0	0
$483$	0.534005	0.0242981
$484$	0	0
$485$	0.266277	0.0120910
$486$	0	0
$487$	−13.5448	−0.613775	−0.306887	−	0.951746i	$-0.599288\pi$
$487$	−13.5448	−0.613775	−0.306887	+	0.951746i	$0.599288\pi$
$488$	0	0
$489$	4.59413	0.207754
$490$	0	0
$491$	9.58255	0.432454	0.216227	−	0.976343i	$-0.430625\pi$
$491$	9.58255	0.432454	0.216227	+	0.976343i	$0.430625\pi$
$492$	0	0
$493$	−39.5369	−1.78065
$494$	0	0
$495$	5.11732	0.230006
$496$	0	0
$497$	0.663374	0.0297564
$498$	0	0
$499$	−7.24834	−0.324480	−0.162240	−	0.986751i	$-0.551872\pi$
$499$	−7.24834	−0.324480	−0.162240	+	0.986751i	$0.551872\pi$
$500$	0	0
$501$	−9.41213	−0.420503
$502$	0	0
$503$	2.19200	0.0977367	0.0488683	−	0.998805i	$-0.484439\pi$
$503$	2.19200	0.0977367	0.0488683	+	0.998805i	$0.484439\pi$
$504$	0	0
$505$	18.0775	0.804440
$506$	0	0
$507$	−12.9835	−0.576618
$508$	0	0
$509$	−13.5765	−0.601766	−0.300883	−	0.953661i	$-0.597281\pi$
$509$	−13.5765	−0.601766	−0.300883	+	0.953661i	$0.597281\pi$
$510$	0	0
$511$	0.0320774	0.00141902
$512$	0	0
$513$	5.38280	0.237656
$514$	0	0
$515$	5.82002	0.256461
$516$	0	0
$517$	29.1909	1.28381
$518$	0	0
$519$	−0.404696	−0.0177642
$520$	0	0
$521$	−6.57594	−0.288097	−0.144049	−	0.989571i	$-0.546012\pi$
$521$	−6.57594	−0.288097	−0.144049	+	0.989571i	$0.546012\pi$
$522$	0	0
$523$	22.2161	0.971443	0.485722	−	0.874114i	$-0.338557\pi$
$523$	22.2161	0.971443	0.485722	+	0.874114i	$0.338557\pi$
$524$	0	0
$525$	1.05585	0.0460810
$526$	0	0
$527$	−29.7386	−1.29543
$528$	0	0
$529$	−19.2233	−0.835795
$530$	0	0
$531$	15.4936	0.672366
$532$	0	0
$533$	−3.65691	−0.158398
$534$	0	0
$535$	6.87605	0.297278
$536$	0	0
$537$	−17.5928	−0.759185
$538$	0	0
$539$	−19.7357	−0.850076
$540$	0	0
$541$	−22.5004	−0.967368	−0.483684	−	0.875243i	$-0.660702\pi$
$541$	−22.5004	−0.967368	−0.483684	+	0.875243i	$0.660702\pi$
$542$	0	0
$543$	−10.3045	−0.442207
$544$	0	0
$545$	−16.0068	−0.685656
$546$	0	0
$547$	24.1551	1.03280	0.516399	−	0.856348i	$-0.327272\pi$
$547$	24.1551	1.03280	0.516399	+	0.856348i	$0.327272\pi$
$548$	0	0
$549$	7.79830	0.332823
$550$	0	0
$551$	−8.83944	−0.376573
$552$	0	0
$553$	3.04984	0.129692
$554$	0	0
$555$	5.04063	0.213963
$556$	0	0
$557$	21.4469	0.908734	0.454367	−	0.890815i	$-0.349865\pi$
$557$	21.4469	0.908734	0.454367	+	0.890815i	$0.349865\pi$
$558$	0	0
$559$	1.59525	0.0674717
$560$	0	0
$561$	−14.6440	−0.618270
$562$	0	0
$563$	−15.3594	−0.647322	−0.323661	−	0.946173i	$-0.604914\pi$
$563$	−15.3594	−0.647322	−0.323661	+	0.946173i	$0.604914\pi$
$564$	0	0
$565$	−9.24256	−0.388837
$566$	0	0
$567$	0.281503	0.0118220
$568$	0	0
$569$	2.05613	0.0861973	0.0430986	−	0.999071i	$-0.486277\pi$
$569$	2.05613	0.0861973	0.0430986	+	0.999071i	$0.486277\pi$
$570$	0	0
$571$	−35.4718	−1.48445	−0.742225	−	0.670151i	$-0.766229\pi$
$571$	−35.4718	−1.48445	−0.742225	+	0.670151i	$0.766229\pi$
$572$	0	0
$573$	−21.1871	−0.885103
$574$	0	0
$575$	7.46741	0.311413
$576$	0	0
$577$	14.6910	0.611595	0.305798	−	0.952097i	$-0.401077\pi$
$577$	14.6910	0.611595	0.305798	+	0.952097i	$0.401077\pi$
$578$	0	0
$579$	−5.87311	−0.244078
$580$	0	0
$581$	3.19743	0.132652
$582$	0	0
$583$	−23.7698	−0.984445
$584$	0	0
$585$	2.36638	0.0978376
$586$	0	0
$587$	−1.71339	−0.0707193	−0.0353596	−	0.999375i	$-0.511258\pi$
$587$	−1.71339	−0.0707193	−0.0353596	+	0.999375i	$0.511258\pi$
$588$	0	0
$589$	−6.64877	−0.273958
$590$	0	0
$591$	23.3722	0.961402
$592$	0	0
$593$	20.3947	0.837509	0.418754	−	0.908100i	$-0.362467\pi$
$593$	20.3947	0.837509	0.418754	+	0.908100i	$0.362467\pi$
$594$	0	0
$595$	−1.14801	−0.0470639
$596$	0	0
$597$	−0.358212	−0.0146607
$598$	0	0
$599$	21.9447	0.896637	0.448318	−	0.893874i	$-0.352023\pi$
$599$	21.9447	0.896637	0.448318	+	0.893874i	$0.352023\pi$
$600$	0	0
$601$	−3.41894	−0.139461	−0.0697306	−	0.997566i	$-0.522214\pi$
$601$	−3.41894	−0.139461	−0.0697306	+	0.997566i	$0.522214\pi$
$602$	0	0
$603$	7.22625	0.294276
$604$	0	0
$605$	3.14180	0.127732
$606$	0	0
$607$	−23.1235	−0.938555	−0.469278	−	0.883051i	$-0.655486\pi$
$607$	−23.1235	−0.938555	−0.469278	+	0.883051i	$0.655486\pi$
$608$	0	0
$609$	−2.42892	−0.0984247
$610$	0	0
$611$	13.4986	0.546095
$612$	0	0
$613$	46.3763	1.87312	0.936561	−	0.350504i	$-0.113990\pi$
$613$	46.3763	1.87312	0.936561	+	0.350504i	$0.113990\pi$
$614$	0	0
$615$	3.44760	0.139021
$616$	0	0
$617$	−25.2123	−1.01501	−0.507504	−	0.861649i	$-0.669432\pi$
$617$	−25.2123	−1.01501	−0.507504	+	0.861649i	$0.669432\pi$
$618$	0	0
$619$	37.2031	1.49532	0.747660	−	0.664082i	$-0.231178\pi$
$619$	37.2031	1.49532	0.747660	+	0.664082i	$0.231178\pi$
$620$	0	0
$621$	10.4608	0.419778
$622$	0	0
$623$	−2.19635	−0.0879948
$624$	0	0
$625$	8.97722	0.359089
$626$	0	0
$627$	−3.27402	−0.130752
$628$	0	0
$629$	18.1937	0.725429
$630$	0	0
$631$	40.8412	1.62586	0.812931	−	0.582360i	$-0.197871\pi$
$631$	40.8412	1.62586	0.812931	+	0.582360i	$0.197871\pi$
$632$	0	0
$633$	−16.5036	−0.655959
$634$	0	0
$635$	−1.24783	−0.0495188
$636$	0	0
$637$	−9.12627	−0.361596
$638$	0	0
$639$	4.65300	0.184070
$640$	0	0
$641$	5.87446	0.232027	0.116014	−	0.993248i	$-0.462988\pi$
$641$	5.87446	0.232027	0.116014	+	0.993248i	$0.462988\pi$
$642$	0	0
$643$	21.3344	0.841347	0.420673	−	0.907212i	$-0.361794\pi$
$643$	21.3344	0.841347	0.420673	+	0.907212i	$0.361794\pi$
$644$	0	0
$645$	−1.50394	−0.0592176
$646$	0	0
$647$	48.1625	1.89346	0.946732	−	0.322024i	$-0.104363\pi$
$647$	48.1625	1.89346	0.946732	+	0.322024i	$0.104363\pi$
$648$	0	0
$649$	−26.3191	−1.03311
$650$	0	0
$651$	−1.82696	−0.0716043
$652$	0	0
$653$	−11.3441	−0.443929	−0.221965	−	0.975055i	$-0.571247\pi$
$653$	−11.3441	−0.443929	−0.221965	+	0.975055i	$0.571247\pi$
$654$	0	0
$655$	−4.14813	−0.162081
$656$	0	0
$657$	0.224995	0.00877791
$658$	0	0
$659$	−24.6320	−0.959526	−0.479763	−	0.877398i	$-0.659277\pi$
$659$	−24.6320	−0.959526	−0.479763	+	0.877398i	$0.659277\pi$
$660$	0	0
$661$	47.3409	1.84135	0.920673	−	0.390335i	$-0.127641\pi$
$661$	47.3409	1.84135	0.920673	+	0.390335i	$0.127641\pi$
$662$	0	0
$663$	−6.77175	−0.262993
$664$	0	0
$665$	−0.256666	−0.00995308
$666$	0	0
$667$	−17.1784	−0.665149
$668$	0	0
$669$	14.8148	0.572772
$670$	0	0
$671$	−13.2470	−0.511396
$672$	0	0
$673$	−35.1767	−1.35596	−0.677981	−	0.735079i	$-0.737145\pi$
$673$	−35.1767	−1.35596	−0.677981	+	0.735079i	$0.737145\pi$
$674$	0	0
$675$	20.6834	0.796103
$676$	0	0
$677$	−2.71085	−0.104186	−0.0520932	−	0.998642i	$-0.516589\pi$
$677$	−2.71085	−0.104186	−0.0520932	+	0.998642i	$0.516589\pi$
$678$	0	0
$679$	0.0590444	0.00226592
$680$	0	0
$681$	−21.2499	−0.814300
$682$	0	0
$683$	48.4820	1.85511	0.927557	−	0.373682i	$-0.121905\pi$
$683$	48.4820	1.85511	0.927557	+	0.373682i	$0.121905\pi$
$684$	0	0
$685$	1.53824	0.0587730
$686$	0	0
$687$	5.45128	0.207979
$688$	0	0
$689$	−10.9917	−0.418752
$690$	0	0
$691$	−41.3295	−1.57225	−0.786124	−	0.618068i	$-0.787915\pi$
$691$	−41.3295	−1.57225	−0.786124	+	0.618068i	$0.787915\pi$
$692$	0	0
$693$	1.13472	0.0431043
$694$	0	0
$695$	3.85368	0.146178
$696$	0	0
$697$	12.4438	0.471342
$698$	0	0
$699$	−4.92755	−0.186377
$700$	0	0
$701$	−15.9692	−0.603147	−0.301573	−	0.953443i	$-0.597512\pi$
$701$	−15.9692	−0.603147	−0.301573	+	0.953443i	$0.597512\pi$
$702$	0	0
$703$	4.06763	0.153414
$704$	0	0
$705$	−12.7260	−0.479288
$706$	0	0
$707$	4.00852	0.150756
$708$	0	0
$709$	1.27532	0.0478957	0.0239478	−	0.999713i	$-0.492376\pi$
$709$	1.27532	0.0478957	0.0239478	+	0.999713i	$0.492376\pi$
$710$	0	0
$711$	21.3920	0.802263
$712$	0	0
$713$	−12.9211	−0.483898
$714$	0	0
$715$	−4.01978	−0.150331
$716$	0	0
$717$	−23.4860	−0.877100
$718$	0	0
$719$	38.6245	1.44045	0.720225	−	0.693740i	$-0.244038\pi$
$719$	38.6245	1.44045	0.720225	+	0.693740i	$0.244038\pi$
$720$	0	0
$721$	1.29053	0.0480619
$722$	0	0
$723$	29.4722	1.09608
$724$	0	0
$725$	−33.9655	−1.26145
$726$	0	0
$727$	3.96945	0.147219	0.0736095	−	0.997287i	$-0.476548\pi$
$727$	3.96945	0.147219	0.0736095	+	0.997287i	$0.476548\pi$
$728$	0	0
$729$	20.5744	0.762015
$730$	0	0
$731$	−5.42832	−0.200774
$732$	0	0
$733$	−23.9821	−0.885800	−0.442900	−	0.896571i	$-0.646050\pi$
$733$	−23.9821	−0.885800	−0.442900	+	0.896571i	$0.646050\pi$
$734$	0	0
$735$	8.60391	0.317360
$736$	0	0
$737$	−12.2753	−0.452165
$738$	0	0
$739$	34.2553	1.26010	0.630051	−	0.776554i	$-0.283034\pi$
$739$	34.2553	1.26010	0.630051	+	0.776554i	$0.283034\pi$
$740$	0	0
$741$	−1.51399	−0.0556177
$742$	0	0
$743$	−31.0058	−1.13749	−0.568746	−	0.822513i	$-0.692571\pi$
$743$	−31.0058	−1.13749	−0.568746	+	0.822513i	$0.692571\pi$
$744$	0	0
$745$	17.5113	0.641564
$746$	0	0
$747$	22.4273	0.820571
$748$	0	0
$749$	1.52470	0.0557112
$750$	0	0
$751$	−36.0417	−1.31518	−0.657590	−	0.753376i	$-0.728424\pi$
$751$	−36.0417	−1.31518	−0.657590	+	0.753376i	$0.728424\pi$
$752$	0	0
$753$	−33.3849	−1.21661
$754$	0	0
$755$	−15.4111	−0.560867
$756$	0	0
$757$	−26.6253	−0.967712	−0.483856	−	0.875148i	$-0.660764\pi$
$757$	−26.6253	−0.967712	−0.483856	+	0.875148i	$0.660764\pi$
$758$	0	0
$759$	−6.36265	−0.230950
$760$	0	0
$761$	−40.0064	−1.45023	−0.725115	−	0.688628i	$-0.758213\pi$
$761$	−40.0064	−1.45023	−0.725115	+	0.688628i	$0.758213\pi$
$762$	0	0
$763$	−3.54935	−0.128495
$764$	0	0
$765$	−8.05233	−0.291133
$766$	0	0
$767$	−12.1706	−0.439455
$768$	0	0
$769$	19.1147	0.689295	0.344648	−	0.938732i	$-0.387998\pi$
$769$	19.1147	0.689295	0.344648	+	0.938732i	$0.387998\pi$
$770$	0	0
$771$	5.87658	0.211640
$772$	0	0
$773$	−42.2244	−1.51871	−0.759353	−	0.650679i	$-0.774484\pi$
$773$	−42.2244	−1.51871	−0.759353	+	0.650679i	$0.774484\pi$
$774$	0	0
$775$	−25.5479	−0.917706
$776$	0	0
$777$	1.11771	0.0400977
$778$	0	0
$779$	2.78211	0.0996793
$780$	0	0
$781$	−7.90408	−0.282830
$782$	0	0
$783$	−47.5809	−1.70040
$784$	0	0
$785$	4.01833	0.143421
$786$	0	0
$787$	43.6009	1.55420	0.777102	−	0.629375i	$-0.216689\pi$
$787$	43.6009	1.55420	0.777102	+	0.629375i	$0.216689\pi$
$788$	0	0
$789$	−13.4707	−0.479571
$790$	0	0
$791$	−2.04945	−0.0728699
$792$	0	0
$793$	−6.12575	−0.217532
$794$	0	0
$795$	10.3626	0.367524
$796$	0	0
$797$	14.6901	0.520351	0.260176	−	0.965561i	$-0.416220\pi$
$797$	14.6901	0.520351	0.260176	+	0.965561i	$0.416220\pi$
$798$	0	0
$799$	−45.9332	−1.62500
$800$	0	0
$801$	−15.4055	−0.544327
$802$	0	0
$803$	−0.382201	−0.0134876
$804$	0	0
$805$	−0.498799	−0.0175803
$806$	0	0
$807$	−0.188244	−0.00662649
$808$	0	0
$809$	−4.76004	−0.167354	−0.0836771	−	0.996493i	$-0.526666\pi$
$809$	−4.76004	−0.167354	−0.0836771	+	0.996493i	$0.526666\pi$
$810$	0	0
$811$	14.6172	0.513279	0.256640	−	0.966507i	$-0.417385\pi$
$811$	14.6172	0.513279	0.256640	+	0.966507i	$0.417385\pi$
$812$	0	0
$813$	−1.16582	−0.0408872
$814$	0	0
$815$	−4.29125	−0.150316
$816$	0	0
$817$	−1.21363	−0.0424596
$818$	0	0
$819$	0.524721	0.0183352
$820$	0	0
$821$	32.3079	1.12755	0.563776	−	0.825928i	$-0.309348\pi$
$821$	32.3079	1.12755	0.563776	+	0.825928i	$0.309348\pi$
$822$	0	0
$823$	9.79982	0.341600	0.170800	−	0.985306i	$-0.445365\pi$
$823$	9.79982	0.341600	0.170800	+	0.985306i	$0.445365\pi$
$824$	0	0
$825$	−12.5804	−0.437993
$826$	0	0
$827$	11.2066	0.389691	0.194845	−	0.980834i	$-0.437579\pi$
$827$	11.2066	0.389691	0.194845	+	0.980834i	$0.437579\pi$
$828$	0	0
$829$	45.4323	1.57793	0.788965	−	0.614438i	$-0.210617\pi$
$829$	45.4323	1.57793	0.788965	+	0.614438i	$0.210617\pi$
$830$	0	0
$831$	3.62619	0.125791
$832$	0	0
$833$	31.0550	1.07599
$834$	0	0
$835$	8.79160	0.304246
$836$	0	0
$837$	−35.7890	−1.23705
$838$	0	0
$839$	16.8997	0.583441	0.291721	−	0.956504i	$-0.405772\pi$
$839$	16.8997	0.583441	0.291721	+	0.956504i	$0.405772\pi$
$840$	0	0
$841$	49.1356	1.69433
$842$	0	0
$843$	28.4191	0.978807
$844$	0	0
$845$	12.1275	0.417200
$846$	0	0
$847$	0.696664	0.0239377
$848$	0	0
$849$	15.9059	0.545890
$850$	0	0
$851$	7.90494	0.270978
$852$	0	0
$853$	37.7943	1.29405	0.647027	−	0.762467i	$-0.276012\pi$
$853$	37.7943	1.29405	0.647027	+	0.762467i	$0.276012\pi$
$854$	0	0
$855$	−1.80029	−0.0615687
$856$	0	0
$857$	−38.8619	−1.32750	−0.663749	−	0.747956i	$-0.731036\pi$
$857$	−38.8619	−1.32750	−0.663749	+	0.747956i	$0.731036\pi$
$858$	0	0
$859$	41.1795	1.40503	0.702513	−	0.711671i	$-0.252061\pi$
$859$	41.1795	1.40503	0.702513	+	0.711671i	$0.252061\pi$
$860$	0	0
$861$	0.764473	0.0260532
$862$	0	0
$863$	25.5909	0.871125	0.435562	−	0.900159i	$-0.356550\pi$
$863$	25.5909	0.871125	0.435562	+	0.900159i	$0.356550\pi$
$864$	0	0
$865$	0.378015	0.0128529
$866$	0	0
$867$	3.46216	0.117581
$868$	0	0
$869$	−36.3387	−1.23271
$870$	0	0
$871$	−5.67639	−0.192337
$872$	0	0
$873$	0.414146	0.0140167
$874$	0	0
$875$	−2.26957	−0.0767253
$876$	0	0
$877$	−14.4790	−0.488920	−0.244460	−	0.969659i	$-0.578611\pi$
$877$	−14.4790	−0.488920	−0.244460	+	0.969659i	$0.578611\pi$
$878$	0	0
$879$	9.49879	0.320386
$880$	0	0
$881$	48.3407	1.62864	0.814320	−	0.580416i	$-0.197110\pi$
$881$	48.3407	1.62864	0.814320	+	0.580416i	$0.197110\pi$
$882$	0	0
$883$	17.2669	0.581077	0.290539	−	0.956863i	$-0.406166\pi$
$883$	17.2669	0.581077	0.290539	+	0.956863i	$0.406166\pi$
$884$	0	0
$885$	11.4740	0.385694
$886$	0	0
$887$	−51.0650	−1.71460	−0.857298	−	0.514820i	$-0.827859\pi$
$887$	−51.0650	−1.71460	−0.857298	+	0.514820i	$0.827859\pi$
$888$	0	0
$889$	−0.276695	−0.00928006
$890$	0	0
$891$	−3.35410	−0.112366
$892$	0	0
$893$	−10.2695	−0.343655
$894$	0	0
$895$	16.4329	0.549292
$896$	0	0
$897$	−2.94225	−0.0982388
$898$	0	0
$899$	58.7714	1.96014
$900$	0	0
$901$	37.4028	1.24607
$902$	0	0
$903$	−0.333484	−0.0110977
$904$	0	0
$905$	9.62510	0.319949
$906$	0	0
$907$	30.0974	0.999367	0.499684	−	0.866208i	$-0.333450\pi$
$907$	30.0974	0.999367	0.499684	+	0.866208i	$0.333450\pi$
$908$	0	0
$909$	28.1164	0.932561
$910$	0	0
$911$	9.15413	0.303290	0.151645	−	0.988435i	$-0.451543\pi$
$911$	9.15413	0.303290	0.151645	+	0.988435i	$0.451543\pi$
$912$	0	0
$913$	−38.0973	−1.26084
$914$	0	0
$915$	5.77514	0.190920
$916$	0	0
$917$	−0.919807	−0.0303747
$918$	0	0
$919$	7.49270	0.247161	0.123581	−	0.992335i	$-0.460562\pi$
$919$	7.49270	0.247161	0.123581	+	0.992335i	$0.460562\pi$
$920$	0	0
$921$	−20.1037	−0.662440
$922$	0	0
$923$	−3.65505	−0.120307
$924$	0	0
$925$	15.6299	0.513906
$926$	0	0
$927$	9.05198	0.297306
$928$	0	0
$929$	−12.0792	−0.396306	−0.198153	−	0.980171i	$-0.563494\pi$
$929$	−12.0792	−0.396306	−0.198153	+	0.980171i	$0.563494\pi$
$930$	0	0
$931$	6.94309	0.227550
$932$	0	0
$933$	−12.1235	−0.396905
$934$	0	0
$935$	13.6785	0.447336
$936$	0	0
$937$	16.6732	0.544690	0.272345	−	0.962200i	$-0.412201\pi$
$937$	16.6732	0.544690	0.272345	+	0.962200i	$0.412201\pi$
$938$	0	0
$939$	16.5283	0.539380
$940$	0	0
$941$	29.5263	0.962529	0.481265	−	0.876575i	$-0.340178\pi$
$941$	29.5263	0.962529	0.481265	+	0.876575i	$0.340178\pi$
$942$	0	0
$943$	5.40668	0.176066
$944$	0	0
$945$	−1.38158	−0.0449428
$946$	0	0
$947$	−24.2099	−0.786717	−0.393358	−	0.919385i	$-0.628687\pi$
$947$	−24.2099	−0.786717	−0.393358	+	0.919385i	$0.628687\pi$
$948$	0	0
$949$	−0.176739	−0.00573720
$950$	0	0
$951$	18.8287	0.610561
$952$	0	0
$953$	18.5304	0.600258	0.300129	−	0.953899i	$-0.402970\pi$
$953$	18.5304	0.600258	0.300129	+	0.953899i	$0.402970\pi$
$954$	0	0
$955$	19.7903	0.640398
$956$	0	0
$957$	28.9405	0.935513
$958$	0	0
$959$	0.341089	0.0110143
$960$	0	0
$961$	13.2062	0.426005
$962$	0	0
$963$	10.6945	0.344624
$964$	0	0
$965$	5.48590	0.176597
$966$	0	0
$967$	−54.2821	−1.74559	−0.872797	−	0.488084i	$-0.837696\pi$
$967$	−54.2821	−1.74559	−0.872797	+	0.488084i	$0.837696\pi$
$968$	0	0
$969$	5.15181	0.165500
$970$	0	0
$971$	−7.69284	−0.246875	−0.123437	−	0.992352i	$-0.539392\pi$
$971$	−7.69284	−0.246875	−0.123437	+	0.992352i	$0.539392\pi$
$972$	0	0
$973$	0.854516	0.0273945
$974$	0	0
$975$	−5.81749	−0.186309
$976$	0	0
$977$	−32.8587	−1.05124	−0.525621	−	0.850719i	$-0.676167\pi$
$977$	−32.8587	−1.05124	−0.525621	+	0.850719i	$0.676167\pi$
$978$	0	0
$979$	26.1694	0.836378
$980$	0	0
$981$	−24.8957	−0.794858
$982$	0	0
$983$	10.4144	0.332169	0.166084	−	0.986112i	$-0.446888\pi$
$983$	10.4144	0.332169	0.166084	+	0.986112i	$0.446888\pi$
$984$	0	0
$985$	−21.8313	−0.695602
$986$	0	0
$987$	−2.82186	−0.0898209
$988$	0	0
$989$	−2.35855	−0.0749974
$990$	0	0
$991$	−62.1164	−1.97319	−0.986596	−	0.163183i	$-0.947824\pi$
$991$	−62.1164	−1.97319	−0.986596	+	0.163183i	$0.947824\pi$
$992$	0	0
$993$	17.2908	0.548706
$994$	0	0
$995$	0.334596	0.0106074
$996$	0	0
$997$	−5.60362	−0.177468	−0.0887341	−	0.996055i	$-0.528282\pi$
$997$	−5.60362	−0.177468	−0.0887341	+	0.996055i	$0.528282\pi$
$998$	0	0
$999$	21.8952	0.692735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bs.1.7		10
4.3	odd	2		4864.2.a.bt.1.3		10
8.3	odd	2	inner	4864.2.a.bs.1.8		10
8.5	even	2		4864.2.a.bt.1.4		10
16.3	odd	4		2432.2.c.j.1217.8	yes	20
16.5	even	4		2432.2.c.j.1217.7	✓	20
16.11	odd	4		2432.2.c.j.1217.13	yes	20
16.13	even	4		2432.2.c.j.1217.14	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.j.1217.7	✓	20	16.5	even	4
2432.2.c.j.1217.8	yes	20	16.3	odd	4
2432.2.c.j.1217.13	yes	20	16.11	odd	4
2432.2.c.j.1217.14	yes	20	16.13	even	4
4864.2.a.bs.1.7		10	1.1	even	1	trivial
4864.2.a.bs.1.8		10	8.3	odd	2	inner
4864.2.a.bt.1.3		10	4.3	odd	2
4864.2.a.bt.1.4		10	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.15181 q^{3} -1.07587 q^{5} -0.238565 q^{7} -1.67333 q^{9} +O(q^{10})\)	\(q+1.15181 q^{3} -1.07587 q^{5} -0.238565 q^{7} -1.67333 q^{9} +2.84249 q^{11} +1.31444 q^{13} -1.23921 q^{15} -4.47279 q^{17} -1.00000 q^{19} -0.274782 q^{21} -1.94338 q^{23} -3.84249 q^{25} -5.38280 q^{27} +8.83944 q^{29} +6.64877 q^{31} +3.27402 q^{33} +0.256666 q^{35} -4.06763 q^{37} +1.51399 q^{39} -2.78211 q^{41} +1.21363 q^{43} +1.80029 q^{45} +10.2695 q^{47} -6.94309 q^{49} -5.15181 q^{51} -8.36231 q^{53} -3.05817 q^{55} -1.15181 q^{57} -9.25916 q^{59} -4.66035 q^{61} +0.399197 q^{63} -1.41417 q^{65} -4.31849 q^{67} -2.23841 q^{69} -2.78069 q^{71} -0.134460 q^{73} -4.42583 q^{75} -0.678119 q^{77} -12.7841 q^{79} -1.17998 q^{81} -13.4028 q^{83} +4.81216 q^{85} +10.1814 q^{87} +9.20650 q^{89} -0.313579 q^{91} +7.65814 q^{93} +1.07587 q^{95} -0.247499 q^{97} -4.75643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{3} + 14 q^{9}+O(q^{10}) \) 10 * q - 4 * q^3 + 14 * q^9	\( 10 q - 4 q^{3} + 14 q^{9} - 20 q^{11} + 4 q^{17} - 10 q^{19} + 10 q^{25} - 28 q^{27} - 8 q^{33} - 36 q^{35} - 12 q^{41} + 4 q^{43} + 26 q^{49} - 36 q^{51} + 4 q^{57} - 52 q^{59} - 24 q^{65} - 12 q^{67} + 12 q^{73} + 12 q^{75} + 34 q^{81} - 16 q^{83} - 20 q^{89} - 60 q^{91} - 28 q^{97} - 60 q^{99}+O(q^{100}) \) 10 * q - 4 * q^3 + 14 * q^9 - 20 * q^11 + 4 * q^17 - 10 * q^19 + 10 * q^25 - 28 * q^27 - 8 * q^33 - 36 * q^35 - 12 * q^41 + 4 * q^43 + 26 * q^49 - 36 * q^51 + 4 * q^57 - 52 * q^59 - 24 * q^65 - 12 * q^67 + 12 * q^73 + 12 * q^75 + 34 * q^81 - 16 * q^83 - 20 * q^89 - 60 * q^91 - 28 * q^97 - 60 * q^99

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