LMFDB - Embedded newform 4864.2.a.br.1.5

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:	$0$
Dimension:	$8$
Coefficient field:	8.8.34309996544.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 28x^{5} + 31x^{4} - 36x^{3} - 22x^{2} + 12x + 2 $ x^8 - 4x^7 - 8x^6 + 28x^5 + 31x^4 - 36x^3 - 22x^2 + 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.79159$ 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)

$f(q)$	$=$	$q+1.65222 q^{3} -4.30533 q^{5} -2.73423 q^{7} -0.270160 q^{9} +O(q^{10})$	$q+1.65222 q^{3} -4.30533 q^{5} -2.73423 q^{7} -0.270160 q^{9} +1.72984 q^{11} +0.546394 q^{13} -7.11337 q^{15} -3.82843 q^{17} -1.00000 q^{19} -4.51756 q^{21} +0.546394 q^{23} +13.5359 q^{25} -5.40303 q^{27} +0.0738068 q^{29} -1.49730 q^{31} +2.85808 q^{33} +11.7718 q^{35} -8.56253 q^{37} +0.902764 q^{39} +1.90604 q^{41} -9.07622 q^{43} +1.16313 q^{45} -5.33004 q^{47} +0.476019 q^{49} -6.32541 q^{51} +10.1336 q^{53} -7.44754 q^{55} -1.65222 q^{57} +6.03429 q^{59} +1.31074 q^{61} +0.738679 q^{63} -2.35241 q^{65} +9.02097 q^{67} +0.902764 q^{69} -14.6512 q^{71} +8.50162 q^{73} +22.3643 q^{75} -4.72978 q^{77} +7.68543 q^{79} -8.11654 q^{81} +16.8061 q^{83} +16.4827 q^{85} +0.121945 q^{87} +14.0166 q^{89} -1.49397 q^{91} -2.47387 q^{93} +4.30533 q^{95} +7.30445 q^{97} -0.467333 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 4 * q^9	$ 8 q + 4 q^{3} + 4 q^{9} + 20 q^{11} - 8 q^{17} - 8 q^{19} + 20 q^{25} + 4 q^{27} + 24 q^{33} + 12 q^{35} + 8 q^{41} + 28 q^{43} + 8 q^{49} + 12 q^{51} - 4 q^{57} + 36 q^{59} + 8 q^{65} + 28 q^{67} - 8 q^{73} + 68 q^{75} - 32 q^{81} + 40 q^{83} - 8 q^{89} - 12 q^{91} + 40 q^{97} + 60 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 + 4 * q^9 + 20 * q^11 - 8 * q^17 - 8 * q^19 + 20 * q^25 + 4 * q^27 + 24 * q^33 + 12 * q^35 + 8 * q^41 + 28 * q^43 + 8 * q^49 + 12 * q^51 - 4 * q^57 + 36 * q^59 + 8 * q^65 + 28 * q^67 - 8 * q^73 + 68 * q^75 - 32 * q^81 + 40 * q^83 - 8 * q^89 - 12 * q^91 + 40 * 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
$3$	1.65222	0.953911	0.476956	−	0.878927i	$-0.341740\pi$
$3$	1.65222	0.953911	0.476956	+	0.878927i	$0.341740\pi$
$4$	0	0
$5$	−4.30533	−1.92540	−0.962702	−	0.270564i	$-0.912790\pi$
$5$	−4.30533	−1.92540	−0.962702	+	0.270564i	$0.912790\pi$
$6$	0	0
$7$	−2.73423	−1.03344	−0.516721	−	0.856154i	$-0.672848\pi$
$7$	−2.73423	−1.03344	−0.516721	+	0.856154i	$0.672848\pi$
$8$	0	0
$9$	−0.270160	−0.0900532
$10$	0	0
$11$	1.72984	0.521567	0.260783	−	0.965397i	$-0.416019\pi$
$11$	1.72984	0.521567	0.260783	+	0.965397i	$0.416019\pi$
$12$	0	0
$13$	0.546394	0.151542	0.0757712	−	0.997125i	$-0.475858\pi$
$13$	0.546394	0.151542	0.0757712	+	0.997125i	$0.475858\pi$
$14$	0	0
$15$	−7.11337	−1.83666
$16$	0	0
$17$	−3.82843	−0.928530	−0.464265	−	0.885696i	$-0.653681\pi$
$17$	−3.82843	−0.928530	−0.464265	+	0.885696i	$0.653681\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.51756	−0.985812
$22$	0	0
$23$	0.546394	0.113931	0.0569655	−	0.998376i	$-0.481857\pi$
$23$	0.546394	0.113931	0.0569655	+	0.998376i	$0.481857\pi$
$24$	0	0
$25$	13.5359	2.70718
$26$	0	0
$27$	−5.40303	−1.03981
$28$	0	0
$29$	0.0738068	0.0137056	0.00685279	−	0.999977i	$-0.497819\pi$
$29$	0.0738068	0.0137056	0.00685279	+	0.999977i	$0.497819\pi$
$30$	0	0
$31$	−1.49730	−0.268922	−0.134461	−	0.990919i	$-0.542930\pi$
$31$	−1.49730	−0.268922	−0.134461	+	0.990919i	$0.542930\pi$
$32$	0	0
$33$	2.85808	0.497528
$34$	0	0
$35$	11.7718	1.98979
$36$	0	0
$37$	−8.56253	−1.40767	−0.703836	−	0.710363i	$-0.748531\pi$
$37$	−8.56253	−1.40767	−0.703836	+	0.710363i	$0.748531\pi$
$38$	0	0
$39$	0.902764	0.144558
$40$	0	0
$41$	1.90604	0.297674	0.148837	−	0.988862i	$-0.452447\pi$
$41$	1.90604	0.297674	0.148837	+	0.988862i	$0.452447\pi$
$42$	0	0
$43$	−9.07622	−1.38411	−0.692056	−	0.721844i	$-0.743295\pi$
$43$	−9.07622	−1.38411	−0.692056	+	0.721844i	$0.743295\pi$
$44$	0	0
$45$	1.16313	0.173389
$46$	0	0
$47$	−5.33004	−0.777467	−0.388733	−	0.921350i	$-0.627087\pi$
$47$	−5.33004	−0.777467	−0.388733	+	0.921350i	$0.627087\pi$
$48$	0	0
$49$	0.476019	0.0680027
$50$	0	0
$51$	−6.32541	−0.885735
$52$	0	0
$53$	10.1336	1.39196	0.695981	−	0.718060i	$-0.254970\pi$
$53$	10.1336	1.39196	0.695981	+	0.718060i	$0.254970\pi$
$54$	0	0
$55$	−7.44754	−1.00423
$56$	0	0
$57$	−1.65222	−0.218842
$58$	0	0
$59$	6.03429	0.785597	0.392799	−	0.919625i	$-0.371507\pi$
$59$	6.03429	0.785597	0.392799	+	0.919625i	$0.371507\pi$
$60$	0	0
$61$	1.31074	0.167823	0.0839116	−	0.996473i	$-0.473259\pi$
$61$	1.31074	0.167823	0.0839116	+	0.996473i	$0.473259\pi$
$62$	0	0
$63$	0.738679	0.0930648
$64$	0	0
$65$	−2.35241	−0.291780
$66$	0	0
$67$	9.02097	1.10209	0.551043	−	0.834477i	$-0.314230\pi$
$67$	9.02097	1.10209	0.551043	+	0.834477i	$0.314230\pi$
$68$	0	0
$69$	0.902764	0.108680
$70$	0	0
$71$	−14.6512	−1.73878	−0.869388	−	0.494129i	$-0.835487\pi$
$71$	−14.6512	−1.73878	−0.869388	+	0.494129i	$0.835487\pi$
$72$	0	0
$73$	8.50162	0.995039	0.497520	−	0.867453i	$-0.334244\pi$
$73$	8.50162	0.995039	0.497520	+	0.867453i	$0.334244\pi$
$74$	0	0
$75$	22.3643	2.58241
$76$	0	0
$77$	−4.72978	−0.539009
$78$	0	0
$79$	7.68543	0.864679	0.432339	−	0.901711i	$-0.357688\pi$
$79$	7.68543	0.864679	0.432339	+	0.901711i	$0.357688\pi$
$80$	0	0
$81$	−8.11654	−0.901837
$82$	0	0
$83$	16.8061	1.84471	0.922353	−	0.386349i	$-0.126264\pi$
$83$	16.8061	1.84471	0.922353	+	0.386349i	$0.126264\pi$
$84$	0	0
$85$	16.4827	1.78780
$86$	0	0
$87$	0.121945	0.0130739
$88$	0	0
$89$	14.0166	1.48575	0.742876	−	0.669429i	$-0.233461\pi$
$89$	14.0166	1.48575	0.742876	+	0.669429i	$0.233461\pi$
$90$	0	0
$91$	−1.49397	−0.156610
$92$	0	0
$93$	−2.47387	−0.256528
$94$	0	0
$95$	4.30533	0.441718
$96$	0	0
$97$	7.30445	0.741654	0.370827	−	0.928702i	$-0.379074\pi$
$97$	7.30445	0.741654	0.370827	+	0.928702i	$0.379074\pi$
$98$	0	0
$99$	−0.467333	−0.0469687
$100$	0	0
$101$	9.85107	0.980218	0.490109	−	0.871661i	$-0.336957\pi$
$101$	9.85107	0.980218	0.490109	+	0.871661i	$0.336957\pi$
$102$	0	0
$103$	−12.8457	−1.26572	−0.632860	−	0.774266i	$-0.718119\pi$
$103$	−12.8457	−1.26572	−0.632860	+	0.774266i	$0.718119\pi$
$104$	0	0
$105$	19.4496	1.89809
$106$	0	0
$107$	7.81048	0.755067	0.377534	−	0.925996i	$-0.376772\pi$
$107$	7.81048	0.755067	0.377534	+	0.925996i	$0.376772\pi$
$108$	0	0
$109$	16.4792	1.57842	0.789210	−	0.614123i	$-0.210490\pi$
$109$	16.4792	1.57842	0.789210	+	0.614123i	$0.210490\pi$
$110$	0	0
$111$	−14.1472	−1.34279
$112$	0	0
$113$	−14.8673	−1.39860	−0.699301	−	0.714827i	$-0.746505\pi$
$113$	−14.8673	−1.39860	−0.699301	+	0.714827i	$0.746505\pi$
$114$	0	0
$115$	−2.35241	−0.219363
$116$	0	0
$117$	−0.147614	−0.0136469
$118$	0	0
$119$	10.4678	0.959582
$120$	0	0
$121$	−8.00765	−0.727968
$122$	0	0
$123$	3.14921	0.283955
$124$	0	0
$125$	−36.7499	−3.28701
$126$	0	0
$127$	10.1080	0.896937	0.448468	−	0.893799i	$-0.351970\pi$
$127$	10.1080	0.896937	0.448468	+	0.893799i	$0.351970\pi$
$128$	0	0
$129$	−14.9959	−1.32032
$130$	0	0
$131$	0.356822	0.0311757	0.0155879	−	0.999879i	$-0.495038\pi$
$131$	0.356822	0.0311757	0.0155879	+	0.999879i	$0.495038\pi$
$132$	0	0
$133$	2.73423	0.237088
$134$	0	0
$135$	23.2619	2.00206
$136$	0	0
$137$	−8.68953	−0.742397	−0.371198	−	0.928554i	$-0.621053\pi$
$137$	−8.68953	−0.742397	−0.371198	+	0.928554i	$0.621053\pi$
$138$	0	0
$139$	14.4673	1.22710	0.613552	−	0.789655i	$-0.289740\pi$
$139$	14.4673	1.22710	0.613552	+	0.789655i	$0.289740\pi$
$140$	0	0
$141$	−8.80642	−0.741634
$142$	0	0
$143$	0.945174	0.0790394
$144$	0	0
$145$	−0.317763	−0.0263888
$146$	0	0
$147$	0.786489	0.0648685
$148$	0	0
$149$	9.77380	0.800701	0.400350	−	0.916362i	$-0.368888\pi$
$149$	9.77380	0.800701	0.400350	+	0.916362i	$0.368888\pi$
$150$	0	0
$151$	8.61067	0.700726	0.350363	−	0.936614i	$-0.386058\pi$
$151$	8.61067	0.700726	0.350363	+	0.936614i	$0.386058\pi$
$152$	0	0
$153$	1.03429	0.0836171
$154$	0	0
$155$	6.44636	0.517784
$156$	0	0
$157$	18.8862	1.50728	0.753641	−	0.657286i	$-0.228296\pi$
$157$	18.8862	1.50728	0.753641	+	0.657286i	$0.228296\pi$
$158$	0	0
$159$	16.7430	1.32781
$160$	0	0
$161$	−1.49397	−0.117741
$162$	0	0
$163$	−20.6867	−1.62031	−0.810155	−	0.586216i	$-0.800617\pi$
$163$	−20.6867	−1.62031	−0.810155	+	0.586216i	$0.800617\pi$
$164$	0	0
$165$	−12.3050	−0.957943
$166$	0	0
$167$	20.3835	1.57732	0.788661	−	0.614829i	$-0.210775\pi$
$167$	20.3835	1.57732	0.788661	+	0.614829i	$0.210775\pi$
$168$	0	0
$169$	−12.7015	−0.977035
$170$	0	0
$171$	0.270160	0.0206596
$172$	0	0
$173$	8.96704	0.681751	0.340876	−	0.940108i	$-0.389276\pi$
$173$	8.96704	0.681751	0.340876	+	0.940108i	$0.389276\pi$
$174$	0	0
$175$	−37.0103	−2.79771
$176$	0	0
$177$	9.96999	0.749390
$178$	0	0
$179$	10.6702	0.797526	0.398763	−	0.917054i	$-0.369440\pi$
$179$	10.6702	0.797526	0.398763	+	0.917054i	$0.369440\pi$
$180$	0	0
$181$	−14.4424	−1.07350	−0.536749	−	0.843742i	$-0.680348\pi$
$181$	−14.4424	−1.07350	−0.536749	+	0.843742i	$0.680348\pi$
$182$	0	0
$183$	2.16564	0.160088
$184$	0	0
$185$	36.8646	2.71034
$186$	0	0
$187$	−6.62257	−0.484290
$188$	0	0
$189$	14.7731	1.07459
$190$	0	0
$191$	21.8574	1.58154	0.790772	−	0.612111i	$-0.209679\pi$
$191$	21.8574	1.58154	0.790772	+	0.612111i	$0.209679\pi$
$192$	0	0
$193$	0.601599	0.0433040	0.0216520	−	0.999766i	$-0.493107\pi$
$193$	0.601599	0.0433040	0.0216520	+	0.999766i	$0.493107\pi$
$194$	0	0
$195$	−3.88670	−0.278333
$196$	0	0
$197$	6.28441	0.447746	0.223873	−	0.974618i	$-0.428130\pi$
$197$	6.28441	0.447746	0.223873	+	0.974618i	$0.428130\pi$
$198$	0	0
$199$	−24.0262	−1.70317	−0.851586	−	0.524214i	$-0.824359\pi$
$199$	−24.0262	−1.70317	−0.851586	+	0.524214i	$0.824359\pi$
$200$	0	0
$201$	14.9047	1.05129
$202$	0	0
$203$	−0.201805	−0.0141639
$204$	0	0
$205$	−8.20616	−0.573143
$206$	0	0
$207$	−0.147614	−0.0102598
$208$	0	0
$209$	−1.72984	−0.119656
$210$	0	0
$211$	6.25382	0.430531	0.215265	−	0.976556i	$-0.430938\pi$
$211$	6.25382	0.430531	0.215265	+	0.976556i	$0.430938\pi$
$212$	0	0
$213$	−24.2070	−1.65864
$214$	0	0
$215$	39.0762	2.66497
$216$	0	0
$217$	4.09396	0.277916
$218$	0	0
$219$	14.0466	0.949179
$220$	0	0
$221$	−2.09183	−0.140712
$222$	0	0
$223$	12.3449	0.826674	0.413337	−	0.910578i	$-0.364363\pi$
$223$	12.3449	0.826674	0.413337	+	0.910578i	$0.364363\pi$
$224$	0	0
$225$	−3.65685	−0.243790
$226$	0	0
$227$	2.30886	0.153244	0.0766222	−	0.997060i	$-0.475586\pi$
$227$	2.30886	0.153244	0.0766222	+	0.997060i	$0.475586\pi$
$228$	0	0
$229$	2.11976	0.140078	0.0700388	−	0.997544i	$-0.477688\pi$
$229$	2.11976	0.140078	0.0700388	+	0.997544i	$0.477688\pi$
$230$	0	0
$231$	−7.81466	−0.514167
$232$	0	0
$233$	−25.2134	−1.65178	−0.825891	−	0.563829i	$-0.809328\pi$
$233$	−25.2134	−1.65178	−0.825891	+	0.563829i	$0.809328\pi$
$234$	0	0
$235$	22.9476	1.49694
$236$	0	0
$237$	12.6981	0.824827
$238$	0	0
$239$	22.7927	1.47434	0.737170	−	0.675707i	$-0.236162\pi$
$239$	22.7927	1.47434	0.737170	+	0.675707i	$0.236162\pi$
$240$	0	0
$241$	2.98064	0.192000	0.0960000	−	0.995381i	$-0.469395\pi$
$241$	2.98064	0.192000	0.0960000	+	0.995381i	$0.469395\pi$
$242$	0	0
$243$	2.79877	0.179541
$244$	0	0
$245$	−2.04942	−0.130933
$246$	0	0
$247$	−0.546394	−0.0347662
$248$	0	0
$249$	27.7674	1.75969
$250$	0	0
$251$	8.84638	0.558378	0.279189	−	0.960236i	$-0.409934\pi$
$251$	8.84638	0.558378	0.279189	+	0.960236i	$0.409934\pi$
$252$	0	0
$253$	0.945174	0.0594226
$254$	0	0
$255$	27.2330	1.70540
$256$	0	0
$257$	−6.04194	−0.376886	−0.188443	−	0.982084i	$-0.560344\pi$
$257$	−6.04194	−0.376886	−0.188443	+	0.982084i	$0.560344\pi$
$258$	0	0
$259$	23.4119	1.45475
$260$	0	0
$261$	−0.0199396	−0.00123423
$262$	0	0
$263$	13.1317	0.809735	0.404868	−	0.914375i	$-0.367318\pi$
$263$	13.1317	0.809735	0.404868	+	0.914375i	$0.367318\pi$
$264$	0	0
$265$	−43.6287	−2.68009
$266$	0	0
$267$	23.1585	1.41728
$268$	0	0
$269$	13.4659	0.821028	0.410514	−	0.911854i	$-0.365349\pi$
$269$	13.4659	0.821028	0.410514	+	0.911854i	$0.365349\pi$
$270$	0	0
$271$	−3.02026	−0.183468	−0.0917339	−	0.995784i	$-0.529241\pi$
$271$	−3.02026	−0.183468	−0.0917339	+	0.995784i	$0.529241\pi$
$272$	0	0
$273$	−2.46837	−0.149392
$274$	0	0
$275$	23.4150	1.41197
$276$	0	0
$277$	8.16722	0.490721	0.245360	−	0.969432i	$-0.421094\pi$
$277$	8.16722	0.490721	0.245360	+	0.969432i	$0.421094\pi$
$278$	0	0
$279$	0.404509	0.0242173
$280$	0	0
$281$	−20.4956	−1.22266	−0.611332	−	0.791374i	$-0.709366\pi$
$281$	−20.4956	−1.22266	−0.611332	+	0.791374i	$0.709366\pi$
$282$	0	0
$283$	−19.9690	−1.18703	−0.593515	−	0.804823i	$-0.702260\pi$
$283$	−19.9690	−1.18703	−0.593515	+	0.804823i	$0.702260\pi$
$284$	0	0
$285$	7.11337	0.421360
$286$	0	0
$287$	−5.21157	−0.307629
$288$	0	0
$289$	−2.34315	−0.137832
$290$	0	0
$291$	12.0686	0.707472
$292$	0	0
$293$	−10.8432	−0.633465	−0.316733	−	0.948515i	$-0.602586\pi$
$293$	−10.8432	−0.633465	−0.316733	+	0.948515i	$0.602586\pi$
$294$	0	0
$295$	−25.9796	−1.51259
$296$	0	0
$297$	−9.34638	−0.542332
$298$	0	0
$299$	0.298546	0.0172654
$300$	0	0
$301$	24.8165	1.43040
$302$	0	0
$303$	16.2762	0.935041
$304$	0	0
$305$	−5.64318	−0.323127
$306$	0	0
$307$	−5.36573	−0.306238	−0.153119	−	0.988208i	$-0.548932\pi$
$307$	−5.36573	−0.306238	−0.153119	+	0.988208i	$0.548932\pi$
$308$	0	0
$309$	−21.2239	−1.20739
$310$	0	0
$311$	−10.3271	−0.585598	−0.292799	−	0.956174i	$-0.594587\pi$
$311$	−10.3271	−0.585598	−0.292799	+	0.956174i	$0.594587\pi$
$312$	0	0
$313$	−29.2020	−1.65060	−0.825298	−	0.564698i	$-0.808993\pi$
$313$	−29.2020	−1.65060	−0.825298	+	0.564698i	$0.808993\pi$
$314$	0	0
$315$	−3.18026	−0.179187
$316$	0	0
$317$	−11.5115	−0.646551	−0.323276	−	0.946305i	$-0.604784\pi$
$317$	−11.5115	−0.646551	−0.323276	+	0.946305i	$0.604784\pi$
$318$	0	0
$319$	0.127674	0.00714837
$320$	0	0
$321$	12.9047	0.720267
$322$	0	0
$323$	3.82843	0.213019
$324$	0	0
$325$	7.39594	0.410253
$326$	0	0
$327$	27.2273	1.50567
$328$	0	0
$329$	14.5736	0.803467
$330$	0	0
$331$	27.0629	1.48751	0.743756	−	0.668451i	$-0.233042\pi$
$331$	27.0629	1.48751	0.743756	+	0.668451i	$0.233042\pi$
$332$	0	0
$333$	2.31325	0.126765
$334$	0	0
$335$	−38.8383	−2.12196
$336$	0	0
$337$	−26.1851	−1.42639	−0.713197	−	0.700963i	$-0.752754\pi$
$337$	−26.1851	−1.42639	−0.713197	+	0.700963i	$0.752754\pi$
$338$	0	0
$339$	−24.5642	−1.33414
$340$	0	0
$341$	−2.59008	−0.140261
$342$	0	0
$343$	17.8381	0.963165
$344$	0	0
$345$	−3.88670	−0.209253
$346$	0	0
$347$	27.2734	1.46411	0.732056	−	0.681244i	$-0.238561\pi$
$347$	27.2734	1.46411	0.732056	+	0.681244i	$0.238561\pi$
$348$	0	0
$349$	−6.11780	−0.327478	−0.163739	−	0.986504i	$-0.552356\pi$
$349$	−6.11780	−0.327478	−0.163739	+	0.986504i	$0.552356\pi$
$350$	0	0
$351$	−2.95218	−0.157576
$352$	0	0
$353$	−8.51030	−0.452958	−0.226479	−	0.974016i	$-0.572721\pi$
$353$	−8.51030	−0.452958	−0.226479	+	0.974016i	$0.572721\pi$
$354$	0	0
$355$	63.0783	3.34785
$356$	0	0
$357$	17.2951	0.915356
$358$	0	0
$359$	−13.0097	−0.686628	−0.343314	−	0.939221i	$-0.611550\pi$
$359$	−13.0097	−0.686628	−0.343314	+	0.939221i	$0.611550\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−13.2304	−0.694417
$364$	0	0
$365$	−36.6023	−1.91585
$366$	0	0
$367$	−26.1953	−1.36738	−0.683692	−	0.729771i	$-0.739627\pi$
$367$	−26.1953	−1.36738	−0.683692	+	0.729771i	$0.739627\pi$
$368$	0	0
$369$	−0.514936	−0.0268065
$370$	0	0
$371$	−27.7077	−1.43851
$372$	0	0
$373$	−33.4436	−1.73165	−0.865823	−	0.500351i	$-0.833204\pi$
$373$	−33.4436	−1.73165	−0.865823	+	0.500351i	$0.833204\pi$
$374$	0	0
$375$	−60.7191	−3.13552
$376$	0	0
$377$	0.0403276	0.00207698
$378$	0	0
$379$	−28.6269	−1.47046	−0.735231	−	0.677816i	$-0.762927\pi$
$379$	−28.6269	−1.47046	−0.735231	+	0.677816i	$0.762927\pi$
$380$	0	0
$381$	16.7006	0.855598
$382$	0	0
$383$	20.1066	1.02740	0.513701	−	0.857969i	$-0.328274\pi$
$383$	20.1066	1.02740	0.513701	+	0.857969i	$0.328274\pi$
$384$	0	0
$385$	20.3633	1.03781
$386$	0	0
$387$	2.45203	0.124644
$388$	0	0
$389$	25.6141	1.29868	0.649342	−	0.760496i	$-0.275044\pi$
$389$	25.6141	1.29868	0.649342	+	0.760496i	$0.275044\pi$
$390$	0	0
$391$	−2.09183	−0.105788
$392$	0	0
$393$	0.589550	0.0297389
$394$	0	0
$395$	−33.0884	−1.66486
$396$	0	0
$397$	−4.40161	−0.220911	−0.110455	−	0.993881i	$-0.535231\pi$
$397$	−4.40161	−0.220911	−0.110455	+	0.993881i	$0.535231\pi$
$398$	0	0
$399$	4.51756	0.226161
$400$	0	0
$401$	−9.38031	−0.468430	−0.234215	−	0.972185i	$-0.575252\pi$
$401$	−9.38031	−0.468430	−0.234215	+	0.972185i	$0.575252\pi$
$402$	0	0
$403$	−0.818114	−0.0407532
$404$	0	0
$405$	34.9444	1.73640
$406$	0	0
$407$	−14.8118	−0.734194
$408$	0	0
$409$	−0.216515	−0.0107060	−0.00535299	−	0.999986i	$-0.501704\pi$
$409$	−0.216515	−0.0107060	−0.00535299	+	0.999986i	$0.501704\pi$
$410$	0	0
$411$	−14.3570	−0.708181
$412$	0	0
$413$	−16.4991	−0.811869
$414$	0	0
$415$	−72.3557	−3.55180
$416$	0	0
$417$	23.9033	1.17055
$418$	0	0
$419$	−18.7827	−0.917593	−0.458796	−	0.888541i	$-0.651719\pi$
$419$	−18.7827	−0.917593	−0.458796	+	0.888541i	$0.651719\pi$
$420$	0	0
$421$	2.19130	0.106798	0.0533988	−	0.998573i	$-0.482995\pi$
$421$	2.19130	0.106798	0.0533988	+	0.998573i	$0.482995\pi$
$422$	0	0
$423$	1.43996	0.0700134
$424$	0	0
$425$	−51.8212	−2.51370
$426$	0	0
$427$	−3.58387	−0.173436
$428$	0	0
$429$	1.56164	0.0753966
$430$	0	0
$431$	20.2673	0.976240	0.488120	−	0.872777i	$-0.337683\pi$
$431$	20.2673	0.976240	0.488120	+	0.872777i	$0.337683\pi$
$432$	0	0
$433$	10.8766	0.522696	0.261348	−	0.965245i	$-0.415833\pi$
$433$	10.8766	0.522696	0.261348	+	0.965245i	$0.415833\pi$
$434$	0	0
$435$	−0.525015	−0.0251725
$436$	0	0
$437$	−0.546394	−0.0261376
$438$	0	0
$439$	10.3917	0.495970	0.247985	−	0.968764i	$-0.420232\pi$
$439$	10.3917	0.495970	0.247985	+	0.968764i	$0.420232\pi$
$440$	0	0
$441$	−0.128601	−0.00612386
$442$	0	0
$443$	−26.2254	−1.24601	−0.623004	−	0.782219i	$-0.714088\pi$
$443$	−26.2254	−1.24601	−0.623004	+	0.782219i	$0.714088\pi$
$444$	0	0
$445$	−60.3460	−2.86067
$446$	0	0
$447$	16.1485	0.763797
$448$	0	0
$449$	26.6928	1.25971	0.629855	−	0.776713i	$-0.283114\pi$
$449$	26.6928	1.25971	0.629855	+	0.776713i	$0.283114\pi$
$450$	0	0
$451$	3.29715	0.155257
$452$	0	0
$453$	14.2267	0.668431
$454$	0	0
$455$	6.43203	0.301538
$456$	0	0
$457$	14.1819	0.663401	0.331700	−	0.943385i	$-0.392378\pi$
$457$	14.1819	0.663401	0.331700	+	0.943385i	$0.392378\pi$
$458$	0	0
$459$	20.6851	0.965499
$460$	0	0
$461$	−34.9375	−1.62720	−0.813600	−	0.581425i	$-0.802495\pi$
$461$	−34.9375	−1.62720	−0.813600	+	0.581425i	$0.802495\pi$
$462$	0	0
$463$	36.9587	1.71762	0.858808	−	0.512298i	$-0.171206\pi$
$463$	36.9587	1.71762	0.858808	+	0.512298i	$0.171206\pi$
$464$	0	0
$465$	10.6508	0.493920
$466$	0	0
$467$	5.58065	0.258242	0.129121	−	0.991629i	$-0.458785\pi$
$467$	5.58065	0.258242	0.129121	+	0.991629i	$0.458785\pi$
$468$	0	0
$469$	−24.6654	−1.13894
$470$	0	0
$471$	31.2042	1.43781
$472$	0	0
$473$	−15.7004	−0.721906
$474$	0	0
$475$	−13.5359	−0.621070
$476$	0	0
$477$	−2.73770	−0.125351
$478$	0	0
$479$	−35.0828	−1.60297	−0.801487	−	0.598012i	$-0.795958\pi$
$479$	−35.0828	−1.60297	−0.801487	+	0.598012i	$0.795958\pi$
$480$	0	0
$481$	−4.67851	−0.213322
$482$	0	0
$483$	−2.46837	−0.112315
$484$	0	0
$485$	−31.4481	−1.42798
$486$	0	0
$487$	−30.5808	−1.38575	−0.692874	−	0.721059i	$-0.743656\pi$
$487$	−30.5808	−1.38575	−0.692874	+	0.721059i	$0.743656\pi$
$488$	0	0
$489$	−34.1791	−1.54563
$490$	0	0
$491$	12.7282	0.574417	0.287208	−	0.957868i	$-0.407273\pi$
$491$	12.7282	0.574417	0.287208	+	0.957868i	$0.407273\pi$
$492$	0	0
$493$	−0.282564	−0.0127260
$494$	0	0
$495$	2.01202	0.0904338
$496$	0	0
$497$	40.0597	1.79693
$498$	0	0
$499$	13.4286	0.601148	0.300574	−	0.953758i	$-0.402822\pi$
$499$	13.4286	0.601148	0.300574	+	0.953758i	$0.402822\pi$
$500$	0	0
$501$	33.6781	1.50462
$502$	0	0
$503$	−30.3977	−1.35537	−0.677683	−	0.735354i	$-0.737016\pi$
$503$	−30.3977	−1.35537	−0.677683	+	0.735354i	$0.737016\pi$
$504$	0	0
$505$	−42.4122	−1.88732
$506$	0	0
$507$	−20.9856	−0.932005
$508$	0	0
$509$	21.5175	0.953745	0.476873	−	0.878972i	$-0.341770\pi$
$509$	21.5175	0.953745	0.476873	+	0.878972i	$0.341770\pi$
$510$	0	0
$511$	−23.2454	−1.02832
$512$	0	0
$513$	5.40303	0.238550
$514$	0	0
$515$	55.3049	2.43702
$516$	0	0
$517$	−9.22013	−0.405501
$518$	0	0
$519$	14.8155	0.650330
$520$	0	0
$521$	−14.0851	−0.617081	−0.308540	−	0.951211i	$-0.599840\pi$
$521$	−14.0851	−0.617081	−0.308540	+	0.951211i	$0.599840\pi$
$522$	0	0
$523$	−22.8507	−0.999190	−0.499595	−	0.866259i	$-0.666518\pi$
$523$	−22.8507	−0.999190	−0.499595	+	0.866259i	$0.666518\pi$
$524$	0	0
$525$	−61.1492	−2.66877
$526$	0	0
$527$	5.73229	0.249703
$528$	0	0
$529$	−22.7015	−0.987020
$530$	0	0
$531$	−1.63022	−0.0707455
$532$	0	0
$533$	1.04145	0.0451103
$534$	0	0
$535$	−33.6267	−1.45381
$536$	0	0
$537$	17.6295	0.760769
$538$	0	0
$539$	0.823436	0.0354679
$540$	0	0
$541$	36.2741	1.55955	0.779774	−	0.626062i	$-0.215334\pi$
$541$	36.2741	1.55955	0.779774	+	0.626062i	$0.215334\pi$
$542$	0	0
$543$	−23.8621	−1.02402
$544$	0	0
$545$	−70.9484	−3.03910
$546$	0	0
$547$	−22.1698	−0.947913	−0.473957	−	0.880548i	$-0.657175\pi$
$547$	−22.1698	−0.947913	−0.473957	+	0.880548i	$0.657175\pi$
$548$	0	0
$549$	−0.354109	−0.0151130
$550$	0	0
$551$	−0.0738068	−0.00314427
$552$	0	0
$553$	−21.0138	−0.893596
$554$	0	0
$555$	60.9085	2.58542
$556$	0	0
$557$	15.3385	0.649915	0.324957	−	0.945729i	$-0.394650\pi$
$557$	15.3385	0.649915	0.324957	+	0.945729i	$0.394650\pi$
$558$	0	0
$559$	−4.95919	−0.209752
$560$	0	0
$561$	−10.9420	−0.461970
$562$	0	0
$563$	8.47021	0.356977	0.178488	−	0.983942i	$-0.442879\pi$
$563$	8.47021	0.356977	0.178488	+	0.983942i	$0.442879\pi$
$564$	0	0
$565$	64.0089	2.69287
$566$	0	0
$567$	22.1925	0.931997
$568$	0	0
$569$	25.7940	1.08134	0.540670	−	0.841235i	$-0.318171\pi$
$569$	25.7940	1.08134	0.540670	+	0.841235i	$0.318171\pi$
$570$	0	0
$571$	12.3855	0.518318	0.259159	−	0.965835i	$-0.416555\pi$
$571$	12.3855	0.518318	0.259159	+	0.965835i	$0.416555\pi$
$572$	0	0
$573$	36.1133	1.50865
$574$	0	0
$575$	7.39594	0.308432
$576$	0	0
$577$	26.8555	1.11801	0.559005	−	0.829164i	$-0.311183\pi$
$577$	26.8555	1.11801	0.559005	+	0.829164i	$0.311183\pi$
$578$	0	0
$579$	0.993976	0.0413082
$580$	0	0
$581$	−45.9517	−1.90640
$582$	0	0
$583$	17.5296	0.726001
$584$	0	0
$585$	0.635526	0.0262758
$586$	0	0
$587$	7.02574	0.289983	0.144992	−	0.989433i	$-0.453684\pi$
$587$	7.02574	0.289983	0.144992	+	0.989433i	$0.453684\pi$
$588$	0	0
$589$	1.49730	0.0616950
$590$	0	0
$591$	10.3833	0.427110
$592$	0	0
$593$	12.2750	0.504074	0.252037	−	0.967718i	$-0.418900\pi$
$593$	12.2750	0.504074	0.252037	+	0.967718i	$0.418900\pi$
$594$	0	0
$595$	−45.0674	−1.84758
$596$	0	0
$597$	−39.6967	−1.62468
$598$	0	0
$599$	−1.13112	−0.0462163	−0.0231081	−	0.999733i	$-0.507356\pi$
$599$	−1.13112	−0.0462163	−0.0231081	+	0.999733i	$0.507356\pi$
$600$	0	0
$601$	−15.4182	−0.628921	−0.314460	−	0.949271i	$-0.601824\pi$
$601$	−15.4182	−0.628921	−0.314460	+	0.949271i	$0.601824\pi$
$602$	0	0
$603$	−2.43710	−0.0992464
$604$	0	0
$605$	34.4756	1.40163
$606$	0	0
$607$	−41.2922	−1.67600	−0.838000	−	0.545671i	$-0.816275\pi$
$607$	−41.2922	−1.67600	−0.838000	+	0.545671i	$0.816275\pi$
$608$	0	0
$609$	−0.333426	−0.0135111
$610$	0	0
$611$	−2.91230	−0.117819
$612$	0	0
$613$	19.0972	0.771329	0.385664	−	0.922639i	$-0.373972\pi$
$613$	19.0972	0.771329	0.385664	+	0.922639i	$0.373972\pi$
$614$	0	0
$615$	−13.5584	−0.546728
$616$	0	0
$617$	40.7423	1.64022	0.820112	−	0.572202i	$-0.193911\pi$
$617$	40.7423	1.64022	0.820112	+	0.572202i	$0.193911\pi$
$618$	0	0
$619$	35.8186	1.43967	0.719835	−	0.694145i	$-0.244218\pi$
$619$	35.8186	1.43967	0.719835	+	0.694145i	$0.244218\pi$
$620$	0	0
$621$	−2.95218	−0.118467
$622$	0	0
$623$	−38.3245	−1.53544
$624$	0	0
$625$	90.5412	3.62165
$626$	0	0
$627$	−2.85808	−0.114141
$628$	0	0
$629$	32.7810	1.30707
$630$	0	0
$631$	7.14251	0.284339	0.142169	−	0.989842i	$-0.454592\pi$
$631$	7.14251	0.284339	0.142169	+	0.989842i	$0.454592\pi$
$632$	0	0
$633$	10.3327	0.410688
$634$	0	0
$635$	−43.5182	−1.72697
$636$	0	0
$637$	0.260094	0.0103053
$638$	0	0
$639$	3.95816	0.156582
$640$	0	0
$641$	−9.03591	−0.356897	−0.178449	−	0.983949i	$-0.557108\pi$
$641$	−9.03591	−0.356897	−0.178449	+	0.983949i	$0.557108\pi$
$642$	0	0
$643$	33.5484	1.32302	0.661510	−	0.749936i	$-0.269916\pi$
$643$	33.5484	1.32302	0.661510	+	0.749936i	$0.269916\pi$
$644$	0	0
$645$	64.5626	2.54215
$646$	0	0
$647$	29.9641	1.17801	0.589004	−	0.808130i	$-0.299520\pi$
$647$	29.9641	1.17801	0.589004	+	0.808130i	$0.299520\pi$
$648$	0	0
$649$	10.4384	0.409741
$650$	0	0
$651$	6.76413	0.265107
$652$	0	0
$653$	5.44946	0.213254	0.106627	−	0.994299i	$-0.465995\pi$
$653$	5.44946	0.213254	0.106627	+	0.994299i	$0.465995\pi$
$654$	0	0
$655$	−1.53624	−0.0600258
$656$	0	0
$657$	−2.29679	−0.0896065
$658$	0	0
$659$	12.1018	0.471420	0.235710	−	0.971823i	$-0.424259\pi$
$659$	12.1018	0.471420	0.235710	+	0.971823i	$0.424259\pi$
$660$	0	0
$661$	−33.3474	−1.29706	−0.648531	−	0.761188i	$-0.724616\pi$
$661$	−33.3474	−1.29706	−0.648531	+	0.761188i	$0.724616\pi$
$662$	0	0
$663$	−3.45617	−0.134226
$664$	0	0
$665$	−11.7718	−0.456490
$666$	0	0
$667$	0.0403276	0.00156149
$668$	0	0
$669$	20.3965	0.788574
$670$	0	0
$671$	2.26737	0.0875309
$672$	0	0
$673$	37.9052	1.46114	0.730570	−	0.682838i	$-0.239255\pi$
$673$	37.9052	1.46114	0.730570	+	0.682838i	$0.239255\pi$
$674$	0	0
$675$	−73.1349	−2.81496
$676$	0	0
$677$	−47.5427	−1.82721	−0.913607	−	0.406598i	$-0.866715\pi$
$677$	−47.5427	−1.82721	−0.913607	+	0.406598i	$0.866715\pi$
$678$	0	0
$679$	−19.9720	−0.766457
$680$	0	0
$681$	3.81475	0.146182
$682$	0	0
$683$	29.7347	1.13777	0.568883	−	0.822418i	$-0.307376\pi$
$683$	29.7347	1.13777	0.568883	+	0.822418i	$0.307376\pi$
$684$	0	0
$685$	37.4113	1.42941
$686$	0	0
$687$	3.50231	0.133622
$688$	0	0
$689$	5.53696	0.210941
$690$	0	0
$691$	−2.82857	−0.107604	−0.0538019	−	0.998552i	$-0.517134\pi$
$691$	−2.82857	−0.107604	−0.0538019	+	0.998552i	$0.517134\pi$
$692$	0	0
$693$	1.27780	0.0485395
$694$	0	0
$695$	−62.2867	−2.36267
$696$	0	0
$697$	−7.29715	−0.276399
$698$	0	0
$699$	−41.6581	−1.57565
$700$	0	0
$701$	38.0093	1.43559	0.717796	−	0.696253i	$-0.245151\pi$
$701$	38.0093	1.43559	0.717796	+	0.696253i	$0.245151\pi$
$702$	0	0
$703$	8.56253	0.322942
$704$	0	0
$705$	37.9146	1.42795
$706$	0	0
$707$	−26.9351	−1.01300
$708$	0	0
$709$	0.225499	0.00846877	0.00423439	−	0.999991i	$-0.498652\pi$
$709$	0.225499	0.00846877	0.00423439	+	0.999991i	$0.498652\pi$
$710$	0	0
$711$	−2.07629	−0.0778671
$712$	0	0
$713$	−0.818114	−0.0306386
$714$	0	0
$715$	−4.06929	−0.152183
$716$	0	0
$717$	37.6587	1.40639
$718$	0	0
$719$	26.7607	0.998006	0.499003	−	0.866600i	$-0.333700\pi$
$719$	26.7607	0.998006	0.499003	+	0.866600i	$0.333700\pi$
$720$	0	0
$721$	35.1230	1.30805
$722$	0	0
$723$	4.92468	0.183151
$724$	0	0
$725$	0.999041	0.0371035
$726$	0	0
$727$	10.0434	0.372487	0.186244	−	0.982504i	$-0.440369\pi$
$727$	10.0434	0.372487	0.186244	+	0.982504i	$0.440369\pi$
$728$	0	0
$729$	28.9738	1.07310
$730$	0	0
$731$	34.7477	1.28519
$732$	0	0
$733$	−1.32974	−0.0491152	−0.0245576	−	0.999698i	$-0.507818\pi$
$733$	−1.32974	−0.0491152	−0.0245576	+	0.999698i	$0.507818\pi$
$734$	0	0
$735$	−3.38610	−0.124898
$736$	0	0
$737$	15.6048	0.574812
$738$	0	0
$739$	−13.0730	−0.480898	−0.240449	−	0.970662i	$-0.577295\pi$
$739$	−13.0730	−0.480898	−0.240449	+	0.970662i	$0.577295\pi$
$740$	0	0
$741$	−0.902764	−0.0331639
$742$	0	0
$743$	11.0577	0.405666	0.202833	−	0.979213i	$-0.434985\pi$
$743$	11.0577	0.405666	0.202833	+	0.979213i	$0.434985\pi$
$744$	0	0
$745$	−42.0795	−1.54167
$746$	0	0
$747$	−4.54032	−0.166122
$748$	0	0
$749$	−21.3557	−0.780319
$750$	0	0
$751$	43.0977	1.57266	0.786330	−	0.617807i	$-0.211979\pi$
$751$	43.0977	1.57266	0.786330	+	0.617807i	$0.211979\pi$
$752$	0	0
$753$	14.6162	0.532643
$754$	0	0
$755$	−37.0718	−1.34918
$756$	0	0
$757$	7.53688	0.273933	0.136966	−	0.990576i	$-0.456265\pi$
$757$	7.53688	0.273933	0.136966	+	0.990576i	$0.456265\pi$
$758$	0	0
$759$	1.56164	0.0566839
$760$	0	0
$761$	−32.3702	−1.17342	−0.586710	−	0.809797i	$-0.699577\pi$
$761$	−32.3702	−1.17342	−0.586710	+	0.809797i	$0.699577\pi$
$762$	0	0
$763$	−45.0579	−1.63121
$764$	0	0
$765$	−4.45295	−0.160997
$766$	0	0
$767$	3.29710	0.119051
$768$	0	0
$769$	11.1972	0.403780	0.201890	−	0.979408i	$-0.435292\pi$
$769$	11.1972	0.403780	0.201890	+	0.979408i	$0.435292\pi$
$770$	0	0
$771$	−9.98263	−0.359516
$772$	0	0
$773$	−47.2345	−1.69891	−0.849453	−	0.527664i	$-0.823068\pi$
$773$	−47.2345	−1.69891	−0.849453	+	0.527664i	$0.823068\pi$
$774$	0	0
$775$	−20.2673	−0.728022
$776$	0	0
$777$	38.6817	1.38770
$778$	0	0
$779$	−1.90604	−0.0682911
$780$	0	0
$781$	−25.3442	−0.906888
$782$	0	0
$783$	−0.398780	−0.0142512
$784$	0	0
$785$	−81.3114	−2.90213
$786$	0	0
$787$	−35.6791	−1.27182	−0.635911	−	0.771762i	$-0.719376\pi$
$787$	−35.6791	−1.27182	−0.635911	+	0.771762i	$0.719376\pi$
$788$	0	0
$789$	21.6965	0.772415
$790$	0	0
$791$	40.6508	1.44537
$792$	0	0
$793$	0.716181	0.0254323
$794$	0	0
$795$	−72.0843	−2.55657
$796$	0	0
$797$	12.4881	0.442351	0.221175	−	0.975234i	$-0.429011\pi$
$797$	12.4881	0.442351	0.221175	+	0.975234i	$0.429011\pi$
$798$	0	0
$799$	20.4057	0.721901
$800$	0	0
$801$	−3.78671	−0.133797
$802$	0	0
$803$	14.7064	0.518979
$804$	0	0
$805$	6.43203	0.226699
$806$	0	0
$807$	22.2486	0.783188
$808$	0	0
$809$	25.1660	0.884789	0.442394	−	0.896821i	$-0.354129\pi$
$809$	25.1660	0.884789	0.442394	+	0.896821i	$0.354129\pi$
$810$	0	0
$811$	−2.99454	−0.105152	−0.0525762	−	0.998617i	$-0.516743\pi$
$811$	−2.99454	−0.105152	−0.0525762	+	0.998617i	$0.516743\pi$
$812$	0	0
$813$	−4.99015	−0.175012
$814$	0	0
$815$	89.0633	3.11975
$816$	0	0
$817$	9.07622	0.317537
$818$	0	0
$819$	0.403609	0.0141033
$820$	0	0
$821$	3.63700	0.126932	0.0634660	−	0.997984i	$-0.479785\pi$
$821$	3.63700	0.126932	0.0634660	+	0.997984i	$0.479785\pi$
$822$	0	0
$823$	16.7620	0.584287	0.292144	−	0.956374i	$-0.405631\pi$
$823$	16.7620	0.584287	0.292144	+	0.956374i	$0.405631\pi$
$824$	0	0
$825$	38.6867	1.34690
$826$	0	0
$827$	1.31834	0.0458432	0.0229216	−	0.999737i	$-0.492703\pi$
$827$	1.31834	0.0458432	0.0229216	+	0.999737i	$0.492703\pi$
$828$	0	0
$829$	18.2603	0.634205	0.317102	−	0.948391i	$-0.397290\pi$
$829$	18.2603	0.634205	0.317102	+	0.948391i	$0.397290\pi$
$830$	0	0
$831$	13.4941	0.468104
$832$	0	0
$833$	−1.82240	−0.0631425
$834$	0	0
$835$	−87.7577	−3.03698
$836$	0	0
$837$	8.08994	0.279629
$838$	0	0
$839$	−24.0265	−0.829486	−0.414743	−	0.909939i	$-0.636129\pi$
$839$	−24.0265	−0.829486	−0.414743	+	0.909939i	$0.636129\pi$
$840$	0	0
$841$	−28.9946	−0.999812
$842$	0	0
$843$	−33.8633	−1.16631
$844$	0	0
$845$	54.6840	1.88119
$846$	0	0
$847$	21.8948	0.752313
$848$	0	0
$849$	−32.9932	−1.13232
$850$	0	0
$851$	−4.67851	−0.160377
$852$	0	0
$853$	−11.1739	−0.382586	−0.191293	−	0.981533i	$-0.561268\pi$
$853$	−11.1739	−0.382586	−0.191293	+	0.981533i	$0.561268\pi$
$854$	0	0
$855$	−1.16313	−0.0397781
$856$	0	0
$857$	28.6182	0.977578	0.488789	−	0.872402i	$-0.337439\pi$
$857$	28.6182	0.977578	0.488789	+	0.872402i	$0.337439\pi$
$858$	0	0
$859$	16.1420	0.550758	0.275379	−	0.961336i	$-0.411197\pi$
$859$	16.1420	0.550758	0.275379	+	0.961336i	$0.411197\pi$
$860$	0	0
$861$	−8.61067	−0.293451
$862$	0	0
$863$	3.17360	0.108031	0.0540154	−	0.998540i	$-0.482798\pi$
$863$	3.17360	0.108031	0.0540154	+	0.998540i	$0.482798\pi$
$864$	0	0
$865$	−38.6061	−1.31265
$866$	0	0
$867$	−3.87140	−0.131480
$868$	0	0
$869$	13.2946	0.450988
$870$	0	0
$871$	4.92900	0.167013
$872$	0	0
$873$	−1.97337	−0.0667883
$874$	0	0
$875$	100.483	3.39694
$876$	0	0
$877$	−30.9086	−1.04371	−0.521855	−	0.853034i	$-0.674760\pi$
$877$	−30.9086	−1.04371	−0.521855	+	0.853034i	$0.674760\pi$
$878$	0	0
$879$	−17.9153	−0.604270
$880$	0	0
$881$	−13.2076	−0.444976	−0.222488	−	0.974935i	$-0.571418\pi$
$881$	−13.2076	−0.444976	−0.222488	+	0.974935i	$0.571418\pi$
$882$	0	0
$883$	−2.47616	−0.0833294	−0.0416647	−	0.999132i	$-0.513266\pi$
$883$	−2.47616	−0.0833294	−0.0416647	+	0.999132i	$0.513266\pi$
$884$	0	0
$885$	−42.9241	−1.44288
$886$	0	0
$887$	−27.5413	−0.924745	−0.462372	−	0.886686i	$-0.653002\pi$
$887$	−27.5413	−0.924745	−0.462372	+	0.886686i	$0.653002\pi$
$888$	0	0
$889$	−27.6375	−0.926932
$890$	0	0
$891$	−14.0403	−0.470368
$892$	0	0
$893$	5.33004	0.178363
$894$	0	0
$895$	−45.9387	−1.53556
$896$	0	0
$897$	0.493265	0.0164696
$898$	0	0
$899$	−0.110511	−0.00368574
$900$	0	0
$901$	−38.7959	−1.29248
$902$	0	0
$903$	41.0024	1.36447
$904$	0	0
$905$	62.1795	2.06692
$906$	0	0
$907$	10.7841	0.358079	0.179039	−	0.983842i	$-0.442701\pi$
$907$	10.7841	0.358079	0.179039	+	0.983842i	$0.442701\pi$
$908$	0	0
$909$	−2.66136	−0.0882718
$910$	0	0
$911$	14.3560	0.475634	0.237817	−	0.971310i	$-0.423568\pi$
$911$	14.3560	0.475634	0.237817	+	0.971310i	$0.423568\pi$
$912$	0	0
$913$	29.0718	0.962137
$914$	0	0
$915$	−9.32379	−0.308235
$916$	0	0
$917$	−0.975635	−0.0322183
$918$	0	0
$919$	32.1519	1.06059	0.530297	−	0.847812i	$-0.322081\pi$
$919$	32.1519	1.06059	0.530297	+	0.847812i	$0.322081\pi$
$920$	0	0
$921$	−8.86537	−0.292124
$922$	0	0
$923$	−8.00532	−0.263498
$924$	0	0
$925$	−115.902	−3.81082
$926$	0	0
$927$	3.47038	0.113982
$928$	0	0
$929$	14.8371	0.486790	0.243395	−	0.969927i	$-0.421739\pi$
$929$	14.8371	0.486790	0.243395	+	0.969927i	$0.421739\pi$
$930$	0	0
$931$	−0.476019	−0.0156009
$932$	0	0
$933$	−17.0627	−0.558608
$934$	0	0
$935$	28.5124	0.932454
$936$	0	0
$937$	−18.1301	−0.592284	−0.296142	−	0.955144i	$-0.595700\pi$
$937$	−18.1301	−0.592284	−0.296142	+	0.955144i	$0.595700\pi$
$938$	0	0
$939$	−48.2482	−1.57452
$940$	0	0
$941$	30.2219	0.985206	0.492603	−	0.870254i	$-0.336046\pi$
$941$	30.2219	0.985206	0.492603	+	0.870254i	$0.336046\pi$
$942$	0	0
$943$	1.04145	0.0339143
$944$	0	0
$945$	−63.6033	−2.06902
$946$	0	0
$947$	37.5706	1.22088	0.610441	−	0.792062i	$-0.290992\pi$
$947$	37.5706	1.22088	0.610441	+	0.792062i	$0.290992\pi$
$948$	0	0
$949$	4.64523	0.150791
$950$	0	0
$951$	−19.0196	−0.616752
$952$	0	0
$953$	46.0319	1.49112	0.745559	−	0.666440i	$-0.232183\pi$
$953$	46.0319	1.49112	0.745559	+	0.666440i	$0.232183\pi$
$954$	0	0
$955$	−94.1033	−3.04511
$956$	0	0
$957$	0.210946	0.00681891
$958$	0	0
$959$	23.7592	0.767224
$960$	0	0
$961$	−28.7581	−0.927681
$962$	0	0
$963$	−2.11008	−0.0679962
$964$	0	0
$965$	−2.59008	−0.0833778
$966$	0	0
$967$	55.6384	1.78921	0.894605	−	0.446858i	$-0.147457\pi$
$967$	55.6384	1.78921	0.894605	+	0.446858i	$0.147457\pi$
$968$	0	0
$969$	6.32541	0.203202
$970$	0	0
$971$	−3.24836	−0.104245	−0.0521224	−	0.998641i	$-0.516599\pi$
$971$	−3.24836	−0.104245	−0.0521224	+	0.998641i	$0.516599\pi$
$972$	0	0
$973$	−39.5570	−1.26814
$974$	0	0
$975$	12.2197	0.391345
$976$	0	0
$977$	37.9387	1.21377	0.606884	−	0.794791i	$-0.292419\pi$
$977$	37.9387	1.21377	0.606884	+	0.794791i	$0.292419\pi$
$978$	0	0
$979$	24.2464	0.774918
$980$	0	0
$981$	−4.45201	−0.142142
$982$	0	0
$983$	40.6822	1.29756	0.648780	−	0.760976i	$-0.275280\pi$
$983$	40.6822	1.29756	0.648780	+	0.760976i	$0.275280\pi$
$984$	0	0
$985$	−27.0565	−0.862092
$986$	0	0
$987$	24.0788	0.766436
$988$	0	0
$989$	−4.95919	−0.157693
$990$	0	0
$991$	−38.5615	−1.22494	−0.612472	−	0.790492i	$-0.709825\pi$
$991$	−38.5615	−1.22494	−0.612472	+	0.790492i	$0.709825\pi$
$992$	0	0
$993$	44.7140	1.41895
$994$	0	0
$995$	103.441	3.27930
$996$	0	0
$997$	53.2516	1.68649	0.843247	−	0.537526i	$-0.180641\pi$
$997$	53.2516	1.68649	0.843247	+	0.537526i	$0.180641\pi$
$998$	0	0
$999$	46.2636	1.46372

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.br.1.5		8
4.3	odd	2		4864.2.a.bm.1.3		8
8.3	odd	2	inner	4864.2.a.br.1.6		8
8.5	even	2		4864.2.a.bm.1.4		8
16.3	odd	4		2432.2.c.i.1217.6	yes	16
16.5	even	4		2432.2.c.i.1217.5	✓	16
16.11	odd	4		2432.2.c.i.1217.11	yes	16
16.13	even	4		2432.2.c.i.1217.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.i.1217.5	✓	16	16.5	even	4
2432.2.c.i.1217.6	yes	16	16.3	odd	4
2432.2.c.i.1217.11	yes	16	16.11	odd	4
2432.2.c.i.1217.12	yes	16	16.13	even	4
4864.2.a.bm.1.3		8	4.3	odd	2
4864.2.a.bm.1.4		8	8.5	even	2
4864.2.a.br.1.5		8	1.1	even	1	trivial
4864.2.a.br.1.6		8	8.3	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.65222 q^{3} -4.30533 q^{5} -2.73423 q^{7} -0.270160 q^{9} +O(q^{10})\)	\(q+1.65222 q^{3} -4.30533 q^{5} -2.73423 q^{7} -0.270160 q^{9} +1.72984 q^{11} +0.546394 q^{13} -7.11337 q^{15} -3.82843 q^{17} -1.00000 q^{19} -4.51756 q^{21} +0.546394 q^{23} +13.5359 q^{25} -5.40303 q^{27} +0.0738068 q^{29} -1.49730 q^{31} +2.85808 q^{33} +11.7718 q^{35} -8.56253 q^{37} +0.902764 q^{39} +1.90604 q^{41} -9.07622 q^{43} +1.16313 q^{45} -5.33004 q^{47} +0.476019 q^{49} -6.32541 q^{51} +10.1336 q^{53} -7.44754 q^{55} -1.65222 q^{57} +6.03429 q^{59} +1.31074 q^{61} +0.738679 q^{63} -2.35241 q^{65} +9.02097 q^{67} +0.902764 q^{69} -14.6512 q^{71} +8.50162 q^{73} +22.3643 q^{75} -4.72978 q^{77} +7.68543 q^{79} -8.11654 q^{81} +16.8061 q^{83} +16.4827 q^{85} +0.121945 q^{87} +14.0166 q^{89} -1.49397 q^{91} -2.47387 q^{93} +4.30533 q^{95} +7.30445 q^{97} -0.467333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 4 * q^9	\( 8 q + 4 q^{3} + 4 q^{9} + 20 q^{11} - 8 q^{17} - 8 q^{19} + 20 q^{25} + 4 q^{27} + 24 q^{33} + 12 q^{35} + 8 q^{41} + 28 q^{43} + 8 q^{49} + 12 q^{51} - 4 q^{57} + 36 q^{59} + 8 q^{65} + 28 q^{67} - 8 q^{73} + 68 q^{75} - 32 q^{81} + 40 q^{83} - 8 q^{89} - 12 q^{91} + 40 q^{97} + 60 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 + 4 * q^9 + 20 * q^11 - 8 * q^17 - 8 * q^19 + 20 * q^25 + 4 * q^27 + 24 * q^33 + 12 * q^35 + 8 * q^41 + 28 * q^43 + 8 * q^49 + 12 * q^51 - 4 * q^57 + 36 * q^59 + 8 * q^65 + 28 * q^67 - 8 * q^73 + 68 * q^75 - 32 * q^81 + 40 * q^83 - 8 * q^89 - 12 * q^91 + 40 * q^97 + 60 * q^99

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