LMFDB - Embedded newform 4864.2.a.bm.1.3

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:	$8$
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			1.3
Root			$0.545005$ 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	$-0.658260\pi$
$3$	−1.65222	−0.953911	−0.476956	+	0.878927i	$0.658260\pi$
$4$	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	$-0.327152\pi$
$7$	2.73423	1.03344	0.516721	+	0.856154i	$0.327152\pi$
$8$	0	0
$9$	−0.270160	−0.0900532
$10$	0	0
$11$	−1.72984	−0.521567	−0.260783	−	0.965397i	$-0.583981\pi$
$11$	−1.72984	−0.521567	−0.260783	+	0.965397i	$0.583981\pi$
$12$	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	$-0.518143\pi$
$23$	−0.546394	−0.113931	−0.0569655	+	0.998376i	$0.518143\pi$
$24$	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	$-0.457070\pi$
$31$	1.49730	0.268922	0.134461	+	0.990919i	$0.457070\pi$
$32$	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	$-0.256705\pi$
$43$	9.07622	1.38411	0.692056	+	0.721844i	$0.256705\pi$
$44$	0	0
$45$	1.16313	0.173389
$46$	0	0
$47$	5.33004	0.777467	0.388733	−	0.921350i	$-0.372913\pi$
$47$	5.33004	0.777467	0.388733	+	0.921350i	$0.372913\pi$
$48$	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	$-0.628493\pi$
$59$	−6.03429	−0.785597	−0.392799	+	0.919625i	$0.628493\pi$
$60$	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	$-0.685770\pi$
$67$	−9.02097	−1.10209	−0.551043	+	0.834477i	$0.685770\pi$
$68$	0	0
$69$	0.902764	0.108680
$70$	0	0
$71$	14.6512	1.73878	0.869388	−	0.494129i	$-0.164513\pi$
$71$	14.6512	1.73878	0.869388	+	0.494129i	$0.164513\pi$
$72$	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	$-0.642312\pi$
$79$	−7.68543	−0.864679	−0.432339	+	0.901711i	$0.642312\pi$
$80$	0	0
$81$	−8.11654	−0.901837
$82$	0	0
$83$	−16.8061	−1.84471	−0.922353	−	0.386349i	$-0.873736\pi$
$83$	−16.8061	−1.84471	−0.922353	+	0.386349i	$0.873736\pi$
$84$	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	$-0.281881\pi$
$103$	12.8457	1.26572	0.632860	+	0.774266i	$0.281881\pi$
$104$	0	0
$105$	19.4496	1.89809
$106$	0	0
$107$	−7.81048	−0.755067	−0.377534	−	0.925996i	$-0.623228\pi$
$107$	−7.81048	−0.755067	−0.377534	+	0.925996i	$0.623228\pi$
$108$	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	$-0.648030\pi$
$127$	−10.1080	−0.896937	−0.448468	+	0.893799i	$0.648030\pi$
$128$	0	0
$129$	−14.9959	−1.32032
$130$	0	0
$131$	−0.356822	−0.0311757	−0.0155879	−	0.999879i	$-0.504962\pi$
$131$	−0.356822	−0.0311757	−0.0155879	+	0.999879i	$0.504962\pi$
$132$	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	$-0.710260\pi$
$139$	−14.4673	−1.22710	−0.613552	+	0.789655i	$0.710260\pi$
$140$	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	$-0.613942\pi$
$151$	−8.61067	−0.700726	−0.350363	+	0.936614i	$0.613942\pi$
$152$	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	$-0.199383\pi$
$163$	20.6867	1.62031	0.810155	+	0.586216i	$0.199383\pi$
$164$	0	0
$165$	−12.3050	−0.957943
$166$	0	0
$167$	−20.3835	−1.57732	−0.788661	−	0.614829i	$-0.789225\pi$
$167$	−20.3835	−1.57732	−0.788661	+	0.614829i	$0.789225\pi$
$168$	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	$-0.630560\pi$
$179$	−10.6702	−0.797526	−0.398763	+	0.917054i	$0.630560\pi$
$180$	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	$-0.790321\pi$
$191$	−21.8574	−1.58154	−0.790772	+	0.612111i	$0.790321\pi$
$192$	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	$-0.175641\pi$
$199$	24.0262	1.70317	0.851586	+	0.524214i	$0.175641\pi$
$200$	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	$-0.569062\pi$
$211$	−6.25382	−0.430531	−0.215265	+	0.976556i	$0.569062\pi$
$212$	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	$-0.635637\pi$
$223$	−12.3449	−0.826674	−0.413337	+	0.910578i	$0.635637\pi$
$224$	0	0
$225$	−3.65685	−0.243790
$226$	0	0
$227$	−2.30886	−0.153244	−0.0766222	−	0.997060i	$-0.524414\pi$
$227$	−2.30886	−0.153244	−0.0766222	+	0.997060i	$0.524414\pi$
$228$	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	$-0.763838\pi$
$239$	−22.7927	−1.47434	−0.737170	+	0.675707i	$0.763838\pi$
$240$	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	$-0.590066\pi$
$251$	−8.84638	−0.558378	−0.279189	+	0.960236i	$0.590066\pi$
$252$	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	$-0.632682\pi$
$263$	−13.1317	−0.809735	−0.404868	+	0.914375i	$0.632682\pi$
$264$	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	$-0.470759\pi$
$271$	3.02026	0.183468	0.0917339	+	0.995784i	$0.470759\pi$
$272$	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	$-0.297740\pi$
$283$	19.9690	1.18703	0.593515	+	0.804823i	$0.297740\pi$
$284$	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	$-0.451068\pi$
$307$	5.36573	0.306238	0.153119	+	0.988208i	$0.451068\pi$
$308$	0	0
$309$	−21.2239	−1.20739
$310$	0	0
$311$	10.3271	0.585598	0.292799	−	0.956174i	$-0.405413\pi$
$311$	10.3271	0.585598	0.292799	+	0.956174i	$0.405413\pi$
$312$	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	$-0.766958\pi$
$331$	−27.0629	−1.48751	−0.743756	+	0.668451i	$0.766958\pi$
$332$	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	$-0.761439\pi$
$347$	−27.2734	−1.46411	−0.732056	+	0.681244i	$0.761439\pi$
$348$	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	$-0.388450\pi$
$359$	13.0097	0.686628	0.343314	+	0.939221i	$0.388450\pi$
$360$	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	$-0.260373\pi$
$367$	26.1953	1.36738	0.683692	+	0.729771i	$0.260373\pi$
$368$	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	$-0.237073\pi$
$379$	28.6269	1.47046	0.735231	+	0.677816i	$0.237073\pi$
$380$	0	0
$381$	16.7006	0.855598
$382$	0	0
$383$	−20.1066	−1.02740	−0.513701	−	0.857969i	$-0.671726\pi$
$383$	−20.1066	−1.02740	−0.513701	+	0.857969i	$0.671726\pi$
$384$	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	$-0.348281\pi$
$419$	18.7827	0.917593	0.458796	+	0.888541i	$0.348281\pi$
$420$	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	$-0.662317\pi$
$431$	−20.2673	−0.976240	−0.488120	+	0.872777i	$0.662317\pi$
$432$	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	$-0.579768\pi$
$439$	−10.3917	−0.495970	−0.247985	+	0.968764i	$0.579768\pi$
$440$	0	0
$441$	−0.128601	−0.00612386
$442$	0	0
$443$	26.2254	1.24601	0.623004	−	0.782219i	$-0.285912\pi$
$443$	26.2254	1.24601	0.623004	+	0.782219i	$0.285912\pi$
$444$	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	$-0.828794\pi$
$463$	−36.9587	−1.71762	−0.858808	+	0.512298i	$0.828794\pi$
$464$	0	0
$465$	10.6508	0.493920
$466$	0	0
$467$	−5.58065	−0.258242	−0.129121	−	0.991629i	$-0.541215\pi$
$467$	−5.58065	−0.258242	−0.129121	+	0.991629i	$0.541215\pi$
$468$	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	$-0.204042\pi$
$479$	35.0828	1.60297	0.801487	+	0.598012i	$0.204042\pi$
$480$	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	$-0.256344\pi$
$487$	30.5808	1.38575	0.692874	+	0.721059i	$0.256344\pi$
$488$	0	0
$489$	−34.1791	−1.54563
$490$	0	0
$491$	−12.7282	−0.574417	−0.287208	−	0.957868i	$-0.592727\pi$
$491$	−12.7282	−0.574417	−0.287208	+	0.957868i	$0.592727\pi$
$492$	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	$-0.597178\pi$
$499$	−13.4286	−0.601148	−0.300574	+	0.953758i	$0.597178\pi$
$500$	0	0
$501$	33.6781	1.50462
$502$	0	0
$503$	30.3977	1.35537	0.677683	−	0.735354i	$-0.262984\pi$
$503$	30.3977	1.35537	0.677683	+	0.735354i	$0.262984\pi$
$504$	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	$-0.333482\pi$
$523$	22.8507	0.999190	0.499595	+	0.866259i	$0.333482\pi$
$524$	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	$-0.342825\pi$
$547$	22.1698	0.947913	0.473957	+	0.880548i	$0.342825\pi$
$548$	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	$-0.557121\pi$
$563$	−8.47021	−0.356977	−0.178488	+	0.983942i	$0.557121\pi$
$564$	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	$-0.583445\pi$
$571$	−12.3855	−0.518318	−0.259159	+	0.965835i	$0.583445\pi$
$572$	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	$-0.546316\pi$
$587$	−7.02574	−0.289983	−0.144992	+	0.989433i	$0.546316\pi$
$588$	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	$-0.492644\pi$
$599$	1.13112	0.0462163	0.0231081	+	0.999733i	$0.492644\pi$
$600$	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	$-0.183725\pi$
$607$	41.2922	1.67600	0.838000	+	0.545671i	$0.183725\pi$
$608$	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	$-0.755782\pi$
$619$	−35.8186	−1.43967	−0.719835	+	0.694145i	$0.755782\pi$
$620$	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	$-0.545408\pi$
$631$	−7.14251	−0.284339	−0.142169	+	0.989842i	$0.545408\pi$
$632$	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	$-0.730084\pi$
$643$	−33.5484	−1.32302	−0.661510	+	0.749936i	$0.730084\pi$
$644$	0	0
$645$	64.5626	2.54215
$646$	0	0
$647$	−29.9641	−1.17801	−0.589004	−	0.808130i	$-0.700480\pi$
$647$	−29.9641	−1.17801	−0.589004	+	0.808130i	$0.700480\pi$
$648$	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	$-0.575741\pi$
$659$	−12.1018	−0.471420	−0.235710	+	0.971823i	$0.575741\pi$
$660$	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	$-0.692624\pi$
$683$	−29.7347	−1.13777	−0.568883	+	0.822418i	$0.692624\pi$
$684$	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	$-0.482866\pi$
$691$	2.82857	0.107604	0.0538019	+	0.998552i	$0.482866\pi$
$692$	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	$-0.666300\pi$
$719$	−26.7607	−0.998006	−0.499003	+	0.866600i	$0.666300\pi$
$720$	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	$-0.559631\pi$
$727$	−10.0434	−0.372487	−0.186244	+	0.982504i	$0.559631\pi$
$728$	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	$-0.422705\pi$
$739$	13.0730	0.480898	0.240449	+	0.970662i	$0.422705\pi$
$740$	0	0
$741$	−0.902764	−0.0331639
$742$	0	0
$743$	−11.0577	−0.405666	−0.202833	−	0.979213i	$-0.565015\pi$
$743$	−11.0577	−0.405666	−0.202833	+	0.979213i	$0.565015\pi$
$744$	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	$-0.788021\pi$
$751$	−43.0977	−1.57266	−0.786330	+	0.617807i	$0.788021\pi$
$752$	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	$-0.280624\pi$
$787$	35.6791	1.27182	0.635911	+	0.771762i	$0.280624\pi$
$788$	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	$-0.483257\pi$
$811$	2.99454	0.105152	0.0525762	+	0.998617i	$0.483257\pi$
$812$	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	$-0.594369\pi$
$823$	−16.7620	−0.584287	−0.292144	+	0.956374i	$0.594369\pi$
$824$	0	0
$825$	38.6867	1.34690
$826$	0	0
$827$	−1.31834	−0.0458432	−0.0229216	−	0.999737i	$-0.507297\pi$
$827$	−1.31834	−0.0458432	−0.0229216	+	0.999737i	$0.507297\pi$
$828$	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	$-0.363871\pi$
$839$	24.0265	0.829486	0.414743	+	0.909939i	$0.363871\pi$
$840$	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	$-0.588803\pi$
$859$	−16.1420	−0.550758	−0.275379	+	0.961336i	$0.588803\pi$
$860$	0	0
$861$	−8.61067	−0.293451
$862$	0	0
$863$	−3.17360	−0.108031	−0.0540154	−	0.998540i	$-0.517202\pi$
$863$	−3.17360	−0.108031	−0.0540154	+	0.998540i	$0.517202\pi$
$864$	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	$-0.486734\pi$
$883$	2.47616	0.0833294	0.0416647	+	0.999132i	$0.486734\pi$
$884$	0	0
$885$	−42.9241	−1.44288
$886$	0	0
$887$	27.5413	0.924745	0.462372	−	0.886686i	$-0.346998\pi$
$887$	27.5413	0.924745	0.462372	+	0.886686i	$0.346998\pi$
$888$	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	$-0.557299\pi$
$907$	−10.7841	−0.358079	−0.179039	+	0.983842i	$0.557299\pi$
$908$	0	0
$909$	−2.66136	−0.0882718
$910$	0	0
$911$	−14.3560	−0.475634	−0.237817	−	0.971310i	$-0.576432\pi$
$911$	−14.3560	−0.475634	−0.237817	+	0.971310i	$0.576432\pi$
$912$	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	$-0.677919\pi$
$919$	−32.1519	−1.06059	−0.530297	+	0.847812i	$0.677919\pi$
$920$	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	$-0.709008\pi$
$947$	−37.5706	−1.22088	−0.610441	+	0.792062i	$0.709008\pi$
$948$	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	$-0.852543\pi$
$967$	−55.6384	−1.78921	−0.894605	+	0.446858i	$0.852543\pi$
$968$	0	0
$969$	6.32541	0.203202
$970$	0	0
$971$	3.24836	0.104245	0.0521224	−	0.998641i	$-0.483401\pi$
$971$	3.24836	0.104245	0.0521224	+	0.998641i	$0.483401\pi$
$972$	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	$-0.724720\pi$
$983$	−40.6822	−1.29756	−0.648780	+	0.760976i	$0.724720\pi$
$984$	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	$-0.290175\pi$
$991$	38.5615	1.22494	0.612472	+	0.790492i	$0.290175\pi$
$992$	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.bm.1.3		8
4.3	odd	2		4864.2.a.br.1.5		8
8.3	odd	2	inner	4864.2.a.bm.1.4		8
8.5	even	2		4864.2.a.br.1.6		8
16.3	odd	4		2432.2.c.i.1217.12	yes	16
16.5	even	4		2432.2.c.i.1217.11	yes	16
16.11	odd	4		2432.2.c.i.1217.5	✓	16
16.13	even	4		2432.2.c.i.1217.6	yes	16

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more