LMFDB - Embedded newform 4864.2.a.bg.1.1

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:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.470683$ 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)

$f(q)$	$=$	$q-2.24914 q^{3} -1.47068 q^{5} +1.47068 q^{7} +2.05863 q^{9} +O(q^{10})$	$q-2.24914 q^{3} -1.47068 q^{5} +1.47068 q^{7} +2.05863 q^{9} -4.77846 q^{11} -4.49828 q^{13} +3.30777 q^{15} +1.71982 q^{17} -1.00000 q^{19} -3.30777 q^{21} +4.00000 q^{23} -2.83709 q^{25} +2.11727 q^{27} -7.80605 q^{29} -4.13187 q^{31} +10.7474 q^{33} -2.16291 q^{35} -3.30777 q^{37} +10.1173 q^{39} -9.80605 q^{41} +1.83709 q^{43} -3.02760 q^{45} -5.14486 q^{47} -4.83709 q^{49} -3.86813 q^{51} +12.8647 q^{53} +7.02760 q^{55} +2.24914 q^{57} +9.05863 q^{59} -8.91033 q^{61} +3.02760 q^{63} +6.61555 q^{65} +5.92332 q^{67} -8.99656 q^{69} -6.05520 q^{71} -0.335371 q^{73} +6.38101 q^{75} -7.02760 q^{77} -0.366407 q^{79} -10.9379 q^{81} -18.0552 q^{83} -2.52932 q^{85} +17.5569 q^{87} +2.36641 q^{89} -6.61555 q^{91} +9.29317 q^{93} +1.47068 q^{95} -7.05863 q^{97} -9.83709 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 7 * q^9	$ 3 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 7 q^{9} - 6 q^{11} + 4 q^{13} + 2 q^{15} - 4 q^{17} - 3 q^{19} - 2 q^{21} + 12 q^{23} - q^{25} + 8 q^{27} + 2 q^{29} - 2 q^{31} + 6 q^{33} - 14 q^{35} - 2 q^{37} + 32 q^{39} - 4 q^{41} - 2 q^{43} + 8 q^{45} - 7 q^{49} - 22 q^{51} + 14 q^{53} + 4 q^{55} - 2 q^{57} + 28 q^{59} - 8 q^{61} - 8 q^{63} + 4 q^{65} - 6 q^{67} + 8 q^{69} + 16 q^{71} + 24 q^{73} - 4 q^{77} + 6 q^{79} + 3 q^{81} - 20 q^{83} - 8 q^{85} + 36 q^{87} - 4 q^{91} + 32 q^{93} + 4 q^{95} - 22 q^{97} - 22 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 7 * q^9 - 6 * q^11 + 4 * q^13 + 2 * q^15 - 4 * q^17 - 3 * q^19 - 2 * q^21 + 12 * q^23 - q^25 + 8 * q^27 + 2 * q^29 - 2 * q^31 + 6 * q^33 - 14 * q^35 - 2 * q^37 + 32 * q^39 - 4 * q^41 - 2 * q^43 + 8 * q^45 - 7 * q^49 - 22 * q^51 + 14 * q^53 + 4 * q^55 - 2 * q^57 + 28 * q^59 - 8 * q^61 - 8 * q^63 + 4 * q^65 - 6 * q^67 + 8 * q^69 + 16 * q^71 + 24 * q^73 - 4 * q^77 + 6 * q^79 + 3 * q^81 - 20 * q^83 - 8 * q^85 + 36 * q^87 - 4 * q^91 + 32 * q^93 + 4 * q^95 - 22 * q^97 - 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.24914	−1.29854	−0.649271	−	0.760557i	$-0.724926\pi$
$3$	−2.24914	−1.29854	−0.649271	+	0.760557i	$0.724926\pi$
$4$	0	0
$5$	−1.47068	−0.657710	−0.328855	−	0.944380i	$-0.606663\pi$
$5$	−1.47068	−0.657710	−0.328855	+	0.944380i	$0.606663\pi$
$6$	0	0
$7$	1.47068	0.555866	0.277933	−	0.960600i	$-0.410351\pi$
$7$	1.47068	0.555866	0.277933	+	0.960600i	$0.410351\pi$
$8$	0	0
$9$	2.05863	0.686211
$10$	0	0
$11$	−4.77846	−1.44076	−0.720380	−	0.693580i	$-0.756032\pi$
$11$	−4.77846	−1.44076	−0.720380	+	0.693580i	$0.756032\pi$
$12$	0	0
$13$	−4.49828	−1.24760	−0.623799	−	0.781585i	$-0.714412\pi$
$13$	−4.49828	−1.24760	−0.623799	+	0.781585i	$0.714412\pi$
$14$	0	0
$15$	3.30777	0.854063
$16$	0	0
$17$	1.71982	0.417119	0.208559	−	0.978010i	$-0.433123\pi$
$17$	1.71982	0.417119	0.208559	+	0.978010i	$0.433123\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.30777	−0.721815
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−2.83709	−0.567418
$26$	0	0
$27$	2.11727	0.407468
$28$	0	0
$29$	−7.80605	−1.44955	−0.724774	−	0.688987i	$-0.758056\pi$
$29$	−7.80605	−1.44955	−0.724774	+	0.688987i	$0.758056\pi$
$30$	0	0
$31$	−4.13187	−0.742107	−0.371053	−	0.928612i	$-0.621003\pi$
$31$	−4.13187	−0.742107	−0.371053	+	0.928612i	$0.621003\pi$
$32$	0	0
$33$	10.7474	1.87089
$34$	0	0
$35$	−2.16291	−0.365598
$36$	0	0
$37$	−3.30777	−0.543795	−0.271897	−	0.962326i	$-0.587651\pi$
$37$	−3.30777	−0.543795	−0.271897	+	0.962326i	$0.587651\pi$
$38$	0	0
$39$	10.1173	1.62006
$40$	0	0
$41$	−9.80605	−1.53145	−0.765724	−	0.643169i	$-0.777619\pi$
$41$	−9.80605	−1.53145	−0.765724	+	0.643169i	$0.777619\pi$
$42$	0	0
$43$	1.83709	0.280154	0.140077	−	0.990141i	$-0.455265\pi$
$43$	1.83709	0.280154	0.140077	+	0.990141i	$0.455265\pi$
$44$	0	0
$45$	−3.02760	−0.451328
$46$	0	0
$47$	−5.14486	−0.750456	−0.375228	−	0.926933i	$-0.622436\pi$
$47$	−5.14486	−0.750456	−0.375228	+	0.926933i	$0.622436\pi$
$48$	0	0
$49$	−4.83709	−0.691013
$50$	0	0
$51$	−3.86813	−0.541646
$52$	0	0
$53$	12.8647	1.76710	0.883550	−	0.468336i	$-0.155146\pi$
$53$	12.8647	1.76710	0.883550	+	0.468336i	$0.155146\pi$
$54$	0	0
$55$	7.02760	0.947601
$56$	0	0
$57$	2.24914	0.297906
$58$	0	0
$59$	9.05863	1.17933	0.589667	−	0.807647i	$-0.299259\pi$
$59$	9.05863	1.17933	0.589667	+	0.807647i	$0.299259\pi$
$60$	0	0
$61$	−8.91033	−1.14085	−0.570426	−	0.821349i	$-0.693222\pi$
$61$	−8.91033	−1.14085	−0.570426	+	0.821349i	$0.693222\pi$
$62$	0	0
$63$	3.02760	0.381441
$64$	0	0
$65$	6.61555	0.820558
$66$	0	0
$67$	5.92332	0.723649	0.361824	−	0.932246i	$-0.382154\pi$
$67$	5.92332	0.723649	0.361824	+	0.932246i	$0.382154\pi$
$68$	0	0
$69$	−8.99656	−1.08306
$70$	0	0
$71$	−6.05520	−0.718619	−0.359310	−	0.933218i	$-0.616988\pi$
$71$	−6.05520	−0.718619	−0.359310	+	0.933218i	$0.616988\pi$
$72$	0	0
$73$	−0.335371	−0.0392522	−0.0196261	−	0.999807i	$-0.506248\pi$
$73$	−0.335371	−0.0392522	−0.0196261	+	0.999807i	$0.506248\pi$
$74$	0	0
$75$	6.38101	0.736816
$76$	0	0
$77$	−7.02760	−0.800869
$78$	0	0
$79$	−0.366407	−0.0412240	−0.0206120	−	0.999788i	$-0.506561\pi$
$79$	−0.366407	−0.0412240	−0.0206120	+	0.999788i	$0.506561\pi$
$80$	0	0
$81$	−10.9379	−1.21533
$82$	0	0
$83$	−18.0552	−1.98182	−0.990908	−	0.134544i	$-0.957043\pi$
$83$	−18.0552	−1.98182	−0.990908	+	0.134544i	$0.957043\pi$
$84$	0	0
$85$	−2.52932	−0.274343
$86$	0	0
$87$	17.5569	1.88230
$88$	0	0
$89$	2.36641	0.250839	0.125419	−	0.992104i	$-0.459972\pi$
$89$	2.36641	0.250839	0.125419	+	0.992104i	$0.459972\pi$
$90$	0	0
$91$	−6.61555	−0.693498
$92$	0	0
$93$	9.29317	0.963656
$94$	0	0
$95$	1.47068	0.150889
$96$	0	0
$97$	−7.05863	−0.716696	−0.358348	−	0.933588i	$-0.616660\pi$
$97$	−7.05863	−0.716696	−0.358348	+	0.933588i	$0.616660\pi$
$98$	0	0
$99$	−9.83709	−0.988665
$100$	0	0
$101$	6.94137	0.690692	0.345346	−	0.938475i	$-0.387762\pi$
$101$	6.94137	0.690692	0.345346	+	0.938475i	$0.387762\pi$
$102$	0	0
$103$	0.824101	0.0812010	0.0406005	−	0.999175i	$-0.487073\pi$
$103$	0.824101	0.0812010	0.0406005	+	0.999175i	$0.487073\pi$
$104$	0	0
$105$	4.86469	0.474745
$106$	0	0
$107$	11.4396	1.10591	0.552956	−	0.833210i	$-0.313500\pi$
$107$	11.4396	1.10591	0.552956	+	0.833210i	$0.313500\pi$
$108$	0	0
$109$	4.13187	0.395762	0.197881	−	0.980226i	$-0.436594\pi$
$109$	4.13187	0.395762	0.197881	+	0.980226i	$0.436594\pi$
$110$	0	0
$111$	7.43965	0.706140
$112$	0	0
$113$	−7.92332	−0.745363	−0.372682	−	0.927959i	$-0.621562\pi$
$113$	−7.92332	−0.745363	−0.372682	+	0.927959i	$0.621562\pi$
$114$	0	0
$115$	−5.88273	−0.548568
$116$	0	0
$117$	−9.26031	−0.856116
$118$	0	0
$119$	2.52932	0.231862
$120$	0	0
$121$	11.8337	1.07579
$122$	0	0
$123$	22.0552	1.98865
$124$	0	0
$125$	11.5259	1.03091
$126$	0	0
$127$	1.19051	0.105640	0.0528202	−	0.998604i	$-0.483179\pi$
$127$	1.19051	0.105640	0.0528202	+	0.998604i	$0.483179\pi$
$128$	0	0
$129$	−4.13187	−0.363791
$130$	0	0
$131$	2.39744	0.209466	0.104733	−	0.994500i	$-0.466601\pi$
$131$	2.39744	0.209466	0.104733	+	0.994500i	$0.466601\pi$
$132$	0	0
$133$	−1.47068	−0.127524
$134$	0	0
$135$	−3.11383	−0.267996
$136$	0	0
$137$	−9.39400	−0.802584	−0.401292	−	0.915950i	$-0.631439\pi$
$137$	−9.39400	−0.802584	−0.401292	+	0.915950i	$0.631439\pi$
$138$	0	0
$139$	−12.2181	−1.03633	−0.518163	−	0.855282i	$-0.673384\pi$
$139$	−12.2181	−1.03633	−0.518163	+	0.855282i	$0.673384\pi$
$140$	0	0
$141$	11.5715	0.974498
$142$	0	0
$143$	21.4948	1.79749
$144$	0	0
$145$	11.4802	0.953382
$146$	0	0
$147$	10.8793	0.897309
$148$	0	0
$149$	−16.8190	−1.37787	−0.688935	−	0.724823i	$-0.741921\pi$
$149$	−16.8190	−1.37787	−0.688935	+	0.724823i	$0.741921\pi$
$150$	0	0
$151$	−7.17590	−0.583966	−0.291983	−	0.956423i	$-0.594315\pi$
$151$	−7.17590	−0.583966	−0.291983	+	0.956423i	$0.594315\pi$
$152$	0	0
$153$	3.54049	0.286231
$154$	0	0
$155$	6.07668	0.488091
$156$	0	0
$157$	−15.6742	−1.25094	−0.625468	−	0.780250i	$-0.715092\pi$
$157$	−15.6742	−1.25094	−0.625468	+	0.780250i	$0.715092\pi$
$158$	0	0
$159$	−28.9345	−2.29465
$160$	0	0
$161$	5.88273	0.463624
$162$	0	0
$163$	9.32238	0.730185	0.365093	−	0.930971i	$-0.381037\pi$
$163$	9.32238	0.730185	0.365093	+	0.930971i	$0.381037\pi$
$164$	0	0
$165$	−15.8061	−1.23050
$166$	0	0
$167$	20.3043	1.57120	0.785598	−	0.618737i	$-0.212355\pi$
$167$	20.3043	1.57120	0.785598	+	0.618737i	$0.212355\pi$
$168$	0	0
$169$	7.23453	0.556503
$170$	0	0
$171$	−2.05863	−0.157428
$172$	0	0
$173$	0.560352	0.0426028	0.0213014	−	0.999773i	$-0.493219\pi$
$173$	0.560352	0.0426028	0.0213014	+	0.999773i	$0.493219\pi$
$174$	0	0
$175$	−4.17246	−0.315408
$176$	0	0
$177$	−20.3741	−1.53141
$178$	0	0
$179$	4.36641	0.326361	0.163180	−	0.986596i	$-0.447825\pi$
$179$	4.36641	0.326361	0.163180	+	0.986596i	$0.447825\pi$
$180$	0	0
$181$	−5.05863	−0.376005	−0.188003	−	0.982169i	$-0.560201\pi$
$181$	−5.05863	−0.376005	−0.188003	+	0.982169i	$0.560201\pi$
$182$	0	0
$183$	20.0406	1.48144
$184$	0	0
$185$	4.86469	0.357659
$186$	0	0
$187$	−8.21811	−0.600967
$188$	0	0
$189$	3.11383	0.226498
$190$	0	0
$191$	−2.46725	−0.178524	−0.0892618	−	0.996008i	$-0.528451\pi$
$191$	−2.46725	−0.178524	−0.0892618	+	0.996008i	$0.528451\pi$
$192$	0	0
$193$	12.2491	0.881712	0.440856	−	0.897578i	$-0.354675\pi$
$193$	12.2491	0.881712	0.440856	+	0.897578i	$0.354675\pi$
$194$	0	0
$195$	−14.8793	−1.06553
$196$	0	0
$197$	1.23109	0.0877119	0.0438559	−	0.999038i	$-0.486036\pi$
$197$	1.23109	0.0877119	0.0438559	+	0.999038i	$0.486036\pi$
$198$	0	0
$199$	27.0276	1.91594	0.957968	−	0.286876i	$-0.0926167\pi$
$199$	27.0276	1.91594	0.957968	+	0.286876i	$0.0926167\pi$
$200$	0	0
$201$	−13.3224	−0.939688
$202$	0	0
$203$	−11.4802	−0.805755
$204$	0	0
$205$	14.4216	1.00725
$206$	0	0
$207$	8.23453	0.572340
$208$	0	0
$209$	4.77846	0.330533
$210$	0	0
$211$	0.325819	0.0224303	0.0112152	−	0.999937i	$-0.496430\pi$
$211$	0.325819	0.0224303	0.0112152	+	0.999937i	$0.496430\pi$
$212$	0	0
$213$	13.6190	0.933157
$214$	0	0
$215$	−2.70178	−0.184260
$216$	0	0
$217$	−6.07668	−0.412512
$218$	0	0
$219$	0.754297	0.0509707
$220$	0	0
$221$	−7.73625	−0.520397
$222$	0	0
$223$	23.4182	1.56820	0.784098	−	0.620637i	$-0.213126\pi$
$223$	23.4182	1.56820	0.784098	+	0.620637i	$0.213126\pi$
$224$	0	0
$225$	−5.84053	−0.389369
$226$	0	0
$227$	−23.8466	−1.58276	−0.791379	−	0.611326i	$-0.790636\pi$
$227$	−23.8466	−1.58276	−0.791379	+	0.611326i	$0.790636\pi$
$228$	0	0
$229$	13.9690	0.923095	0.461548	−	0.887115i	$-0.347294\pi$
$229$	13.9690	0.923095	0.461548	+	0.887115i	$0.347294\pi$
$230$	0	0
$231$	15.8061	1.03996
$232$	0	0
$233$	−12.8337	−0.840761	−0.420380	−	0.907348i	$-0.638103\pi$
$233$	−12.8337	−0.840761	−0.420380	+	0.907348i	$0.638103\pi$
$234$	0	0
$235$	7.56647	0.493582
$236$	0	0
$237$	0.824101	0.0535311
$238$	0	0
$239$	24.2587	1.56916	0.784582	−	0.620025i	$-0.212877\pi$
$239$	24.2587	1.56916	0.784582	+	0.620025i	$0.212877\pi$
$240$	0	0
$241$	−26.0698	−1.67930	−0.839652	−	0.543125i	$-0.817241\pi$
$241$	−26.0698	−1.67930	−0.839652	+	0.543125i	$0.817241\pi$
$242$	0	0
$243$	18.2491	1.17068
$244$	0	0
$245$	7.11383	0.454486
$246$	0	0
$247$	4.49828	0.286219
$248$	0	0
$249$	40.6087	2.57347
$250$	0	0
$251$	23.3319	1.47270	0.736349	−	0.676602i	$-0.236548\pi$
$251$	23.3319	1.47270	0.736349	+	0.676602i	$0.236548\pi$
$252$	0	0
$253$	−19.1138	−1.20168
$254$	0	0
$255$	5.68879	0.356246
$256$	0	0
$257$	2.82410	0.176163	0.0880813	−	0.996113i	$-0.471926\pi$
$257$	2.82410	0.176163	0.0880813	+	0.996113i	$0.471926\pi$
$258$	0	0
$259$	−4.86469	−0.302277
$260$	0	0
$261$	−16.0698	−0.994696
$262$	0	0
$263$	7.08967	0.437168	0.218584	−	0.975818i	$-0.429856\pi$
$263$	7.08967	0.437168	0.218584	+	0.975818i	$0.429856\pi$
$264$	0	0
$265$	−18.9199	−1.16224
$266$	0	0
$267$	−5.32238	−0.325724
$268$	0	0
$269$	20.8647	1.27214	0.636071	−	0.771630i	$-0.280558\pi$
$269$	20.8647	1.27214	0.636071	+	0.771630i	$0.280558\pi$
$270$	0	0
$271$	0.0620710	0.00377055	0.00188527	−	0.999998i	$-0.499400\pi$
$271$	0.0620710	0.00377055	0.00188527	+	0.999998i	$0.499400\pi$
$272$	0	0
$273$	14.8793	0.900536
$274$	0	0
$275$	13.5569	0.817513
$276$	0	0
$277$	−7.82248	−0.470007	−0.235004	−	0.971994i	$-0.575510\pi$
$277$	−7.82248	−0.470007	−0.235004	+	0.971994i	$0.575510\pi$
$278$	0	0
$279$	−8.50601	−0.509242
$280$	0	0
$281$	20.6448	1.23156	0.615782	−	0.787917i	$-0.288840\pi$
$281$	20.6448	1.23156	0.615782	+	0.787917i	$0.288840\pi$
$282$	0	0
$283$	21.3871	1.27133	0.635666	−	0.771964i	$-0.280725\pi$
$283$	21.3871	1.27133	0.635666	+	0.771964i	$0.280725\pi$
$284$	0	0
$285$	−3.30777	−0.195936
$286$	0	0
$287$	−14.4216	−0.851280
$288$	0	0
$289$	−14.0422	−0.826012
$290$	0	0
$291$	15.8759	0.930659
$292$	0	0
$293$	24.6087	1.43765	0.718827	−	0.695189i	$-0.244679\pi$
$293$	24.6087	1.43765	0.718827	+	0.695189i	$0.244679\pi$
$294$	0	0
$295$	−13.3224	−0.775659
$296$	0	0
$297$	−10.1173	−0.587063
$298$	0	0
$299$	−17.9931	−1.04057
$300$	0	0
$301$	2.70178	0.155728
$302$	0	0
$303$	−15.6121	−0.896892
$304$	0	0
$305$	13.1043	0.750349
$306$	0	0
$307$	−10.4216	−0.594792	−0.297396	−	0.954754i	$-0.596118\pi$
$307$	−10.4216	−0.594792	−0.297396	+	0.954754i	$0.596118\pi$
$308$	0	0
$309$	−1.85352	−0.105443
$310$	0	0
$311$	7.91377	0.448749	0.224374	−	0.974503i	$-0.427966\pi$
$311$	7.91377	0.448749	0.224374	+	0.974503i	$0.427966\pi$
$312$	0	0
$313$	−17.7846	−1.00524	−0.502622	−	0.864506i	$-0.667631\pi$
$313$	−17.7846	−1.00524	−0.502622	+	0.864506i	$0.667631\pi$
$314$	0	0
$315$	−4.45264	−0.250878
$316$	0	0
$317$	30.0552	1.68807	0.844034	−	0.536290i	$-0.180175\pi$
$317$	30.0552	1.68807	0.844034	+	0.536290i	$0.180175\pi$
$318$	0	0
$319$	37.3009	2.08845
$320$	0	0
$321$	−25.7294	−1.43607
$322$	0	0
$323$	−1.71982	−0.0956936
$324$	0	0
$325$	12.7620	0.707910
$326$	0	0
$327$	−9.29317	−0.513913
$328$	0	0
$329$	−7.56647	−0.417153
$330$	0	0
$331$	−18.8095	−1.03386	−0.516932	−	0.856027i	$-0.672926\pi$
$331$	−18.8095	−1.03386	−0.516932	+	0.856027i	$0.672926\pi$
$332$	0	0
$333$	−6.80949	−0.373158
$334$	0	0
$335$	−8.71133	−0.475951
$336$	0	0
$337$	30.2017	1.64519	0.822595	−	0.568628i	$-0.192525\pi$
$337$	30.2017	1.64519	0.822595	+	0.568628i	$0.192525\pi$
$338$	0	0
$339$	17.8207	0.967886
$340$	0	0
$341$	19.7440	1.06920
$342$	0	0
$343$	−17.4086	−0.939977
$344$	0	0
$345$	13.2311	0.712338
$346$	0	0
$347$	9.36802	0.502902	0.251451	−	0.967870i	$-0.419092\pi$
$347$	9.36802	0.502902	0.251451	+	0.967870i	$0.419092\pi$
$348$	0	0
$349$	21.3173	1.14109	0.570545	−	0.821266i	$-0.306732\pi$
$349$	21.3173	1.14109	0.570545	+	0.821266i	$0.306732\pi$
$350$	0	0
$351$	−9.52406	−0.508357
$352$	0	0
$353$	18.4362	0.981260	0.490630	−	0.871368i	$-0.336767\pi$
$353$	18.4362	0.981260	0.490630	+	0.871368i	$0.336767\pi$
$354$	0	0
$355$	8.90528	0.472643
$356$	0	0
$357$	−5.68879	−0.301083
$358$	0	0
$359$	37.1932	1.96298	0.981491	−	0.191510i	$-0.0613386\pi$
$359$	37.1932	1.96298	0.981491	+	0.191510i	$0.0613386\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−26.6155	−1.39695
$364$	0	0
$365$	0.493225	0.0258166
$366$	0	0
$367$	11.6121	0.606147	0.303074	−	0.952967i	$-0.401987\pi$
$367$	11.6121	0.606147	0.303074	+	0.952967i	$0.401987\pi$
$368$	0	0
$369$	−20.1871	−1.05090
$370$	0	0
$371$	18.9199	0.982271
$372$	0	0
$373$	24.6087	1.27419	0.637094	−	0.770786i	$-0.280136\pi$
$373$	24.6087	1.27419	0.637094	+	0.770786i	$0.280136\pi$
$374$	0	0
$375$	−25.9233	−1.33867
$376$	0	0
$377$	35.1138	1.80845
$378$	0	0
$379$	−1.05863	−0.0543783	−0.0271892	−	0.999630i	$-0.508656\pi$
$379$	−1.05863	−0.0543783	−0.0271892	+	0.999630i	$0.508656\pi$
$380$	0	0
$381$	−2.67762	−0.137179
$382$	0	0
$383$	18.4837	0.944472	0.472236	−	0.881472i	$-0.343447\pi$
$383$	18.4837	0.944472	0.472236	+	0.881472i	$0.343447\pi$
$384$	0	0
$385$	10.3354	0.526739
$386$	0	0
$387$	3.78189	0.192245
$388$	0	0
$389$	2.46725	0.125094	0.0625472	−	0.998042i	$-0.480078\pi$
$389$	2.46725	0.125094	0.0625472	+	0.998042i	$0.480078\pi$
$390$	0	0
$391$	6.87930	0.347901
$392$	0	0
$393$	−5.39218	−0.272000
$394$	0	0
$395$	0.538868	0.0271134
$396$	0	0
$397$	6.91721	0.347165	0.173582	−	0.984819i	$-0.444466\pi$
$397$	6.91721	0.347165	0.173582	+	0.984819i	$0.444466\pi$
$398$	0	0
$399$	3.30777	0.165596
$400$	0	0
$401$	20.8578	1.04159	0.520795	−	0.853682i	$-0.325636\pi$
$401$	20.8578	1.04159	0.520795	+	0.853682i	$0.325636\pi$
$402$	0	0
$403$	18.5863	0.925851
$404$	0	0
$405$	16.0862	0.799331
$406$	0	0
$407$	15.8061	0.783477
$408$	0	0
$409$	−24.9820	−1.23528	−0.617639	−	0.786462i	$-0.711911\pi$
$409$	−24.9820	−1.23528	−0.617639	+	0.786462i	$0.711911\pi$
$410$	0	0
$411$	21.1284	1.04219
$412$	0	0
$413$	13.3224	0.655552
$414$	0	0
$415$	26.5535	1.30346
$416$	0	0
$417$	27.4802	1.34571
$418$	0	0
$419$	−31.6742	−1.54738	−0.773692	−	0.633561i	$-0.781592\pi$
$419$	−31.6742	−1.54738	−0.773692	+	0.633561i	$0.781592\pi$
$420$	0	0
$421$	38.5941	1.88096	0.940480	−	0.339850i	$-0.110376\pi$
$421$	38.5941	1.88096	0.940480	+	0.339850i	$0.110376\pi$
$422$	0	0
$423$	−10.5914	−0.514971
$424$	0	0
$425$	−4.87930	−0.236681
$426$	0	0
$427$	−13.1043	−0.634160
$428$	0	0
$429$	−48.3449	−2.33411
$430$	0	0
$431$	37.4948	1.80606	0.903032	−	0.429574i	$-0.141336\pi$
$431$	37.4948	1.80606	0.903032	+	0.429574i	$0.141336\pi$
$432$	0	0
$433$	13.4182	0.644836	0.322418	−	0.946597i	$-0.395504\pi$
$433$	13.4182	0.644836	0.322418	+	0.946597i	$0.395504\pi$
$434$	0	0
$435$	−25.8207	−1.23801
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−36.3043	−1.73271	−0.866356	−	0.499427i	$-0.833544\pi$
$439$	−36.3043	−1.73271	−0.866356	+	0.499427i	$0.833544\pi$
$440$	0	0
$441$	−9.95779	−0.474181
$442$	0	0
$443$	−27.9215	−1.32659	−0.663295	−	0.748358i	$-0.730843\pi$
$443$	−27.9215	−1.32659	−0.663295	+	0.748358i	$0.730843\pi$
$444$	0	0
$445$	−3.48024	−0.164979
$446$	0	0
$447$	37.8284	1.78922
$448$	0	0
$449$	−12.2086	−0.576157	−0.288079	−	0.957607i	$-0.593016\pi$
$449$	−12.2086	−0.576157	−0.288079	+	0.957607i	$0.593016\pi$
$450$	0	0
$451$	46.8578	2.20645
$452$	0	0
$453$	16.1396	0.758305
$454$	0	0
$455$	9.72938	0.456120
$456$	0	0
$457$	−4.42666	−0.207070	−0.103535	−	0.994626i	$-0.533015\pi$
$457$	−4.42666	−0.207070	−0.103535	+	0.994626i	$0.533015\pi$
$458$	0	0
$459$	3.64133	0.169963
$460$	0	0
$461$	6.96553	0.324417	0.162208	−	0.986757i	$-0.448138\pi$
$461$	6.96553	0.324417	0.162208	+	0.986757i	$0.448138\pi$
$462$	0	0
$463$	−30.7501	−1.42908	−0.714539	−	0.699595i	$-0.753364\pi$
$463$	−30.7501	−1.42908	−0.714539	+	0.699595i	$0.753364\pi$
$464$	0	0
$465$	−13.6673	−0.633806
$466$	0	0
$467$	5.16635	0.239070	0.119535	−	0.992830i	$-0.461860\pi$
$467$	5.16635	0.239070	0.119535	+	0.992830i	$0.461860\pi$
$468$	0	0
$469$	8.71133	0.402252
$470$	0	0
$471$	35.2534	1.62439
$472$	0	0
$473$	−8.77846	−0.403634
$474$	0	0
$475$	2.83709	0.130175
$476$	0	0
$477$	26.4837	1.21260
$478$	0	0
$479$	−23.3776	−1.06815	−0.534074	−	0.845437i	$-0.679340\pi$
$479$	−23.3776	−1.06815	−0.534074	+	0.845437i	$0.679340\pi$
$480$	0	0
$481$	14.8793	0.678437
$482$	0	0
$483$	−13.2311	−0.602036
$484$	0	0
$485$	10.3810	0.471378
$486$	0	0
$487$	−41.9018	−1.89875	−0.949377	−	0.314140i	$-0.898284\pi$
$487$	−41.9018	−1.89875	−0.949377	+	0.314140i	$0.898284\pi$
$488$	0	0
$489$	−20.9673	−0.948176
$490$	0	0
$491$	−4.90528	−0.221372	−0.110686	−	0.993855i	$-0.535305\pi$
$491$	−4.90528	−0.221372	−0.110686	+	0.993855i	$0.535305\pi$
$492$	0	0
$493$	−13.4250	−0.604633
$494$	0	0
$495$	14.4672	0.650254
$496$	0	0
$497$	−8.90528	−0.399456
$498$	0	0
$499$	−5.77502	−0.258525	−0.129263	−	0.991610i	$-0.541261\pi$
$499$	−5.77502	−0.258525	−0.129263	+	0.991610i	$0.541261\pi$
$500$	0	0
$501$	−45.6673	−2.04026
$502$	0	0
$503$	−31.3776	−1.39906	−0.699529	−	0.714605i	$-0.746607\pi$
$503$	−31.3776	−1.39906	−0.699529	+	0.714605i	$0.746607\pi$
$504$	0	0
$505$	−10.2086	−0.454275
$506$	0	0
$507$	−16.2715	−0.722642
$508$	0	0
$509$	−17.2603	−0.765050	−0.382525	−	0.923945i	$-0.624945\pi$
$509$	−17.2603	−0.765050	−0.382525	+	0.923945i	$0.624945\pi$
$510$	0	0
$511$	−0.493225	−0.0218190
$512$	0	0
$513$	−2.11727	−0.0934796
$514$	0	0
$515$	−1.21199	−0.0534067
$516$	0	0
$517$	24.5845	1.08123
$518$	0	0
$519$	−1.26031	−0.0553215
$520$	0	0
$521$	−18.8939	−0.827757	−0.413878	−	0.910332i	$-0.635826\pi$
$521$	−18.8939	−0.827757	−0.413878	+	0.910332i	$0.635826\pi$
$522$	0	0
$523$	−40.2017	−1.75790	−0.878948	−	0.476917i	$-0.841754\pi$
$523$	−40.2017	−1.75790	−0.878948	+	0.476917i	$0.841754\pi$
$524$	0	0
$525$	9.38445	0.409571
$526$	0	0
$527$	−7.10610	−0.309546
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	18.6484	0.809272
$532$	0	0
$533$	44.1104	1.91063
$534$	0	0
$535$	−16.8241	−0.727369
$536$	0	0
$537$	−9.82066	−0.423793
$538$	0	0
$539$	23.1138	0.995583
$540$	0	0
$541$	22.4052	0.963274	0.481637	−	0.876371i	$-0.340042\pi$
$541$	22.4052	0.963274	0.481637	+	0.876371i	$0.340042\pi$
$542$	0	0
$543$	11.3776	0.488259
$544$	0	0
$545$	−6.07668	−0.260296
$546$	0	0
$547$	−37.9931	−1.62447	−0.812234	−	0.583331i	$-0.801749\pi$
$547$	−37.9931	−1.62447	−0.812234	+	0.583331i	$0.801749\pi$
$548$	0	0
$549$	−18.3431	−0.782865
$550$	0	0
$551$	7.80605	0.332549
$552$	0	0
$553$	−0.538868	−0.0229150
$554$	0	0
$555$	−10.9414	−0.464435
$556$	0	0
$557$	−16.1775	−0.685463	−0.342732	−	0.939433i	$-0.611352\pi$
$557$	−16.1775	−0.685463	−0.342732	+	0.939433i	$0.611352\pi$
$558$	0	0
$559$	−8.26375	−0.349519
$560$	0	0
$561$	18.4837	0.780381
$562$	0	0
$563$	−9.59750	−0.404486	−0.202243	−	0.979335i	$-0.564823\pi$
$563$	−9.59750	−0.404486	−0.202243	+	0.979335i	$0.564823\pi$
$564$	0	0
$565$	11.6527	0.490233
$566$	0	0
$567$	−16.0862	−0.675558
$568$	0	0
$569$	23.9018	1.00202	0.501009	−	0.865442i	$-0.332963\pi$
$569$	23.9018	1.00202	0.501009	+	0.865442i	$0.332963\pi$
$570$	0	0
$571$	−36.1725	−1.51377	−0.756885	−	0.653548i	$-0.773280\pi$
$571$	−36.1725	−1.51377	−0.756885	+	0.653548i	$0.773280\pi$
$572$	0	0
$573$	5.54918	0.231820
$574$	0	0
$575$	−11.3484	−0.473259
$576$	0	0
$577$	−12.5699	−0.523292	−0.261646	−	0.965164i	$-0.584265\pi$
$577$	−12.5699	−0.523292	−0.261646	+	0.965164i	$0.584265\pi$
$578$	0	0
$579$	−27.5500	−1.14494
$580$	0	0
$581$	−26.5535	−1.10162
$582$	0	0
$583$	−61.4734	−2.54597
$584$	0	0
$585$	13.6190	0.563076
$586$	0	0
$587$	−0.107714	−0.00444585	−0.00222292	−	0.999998i	$-0.500708\pi$
$587$	−0.107714	−0.00444585	−0.00222292	+	0.999998i	$0.500708\pi$
$588$	0	0
$589$	4.13187	0.170251
$590$	0	0
$591$	−2.76891	−0.113898
$592$	0	0
$593$	25.7846	1.05885	0.529423	−	0.848358i	$-0.322409\pi$
$593$	25.7846	1.05885	0.529423	+	0.848358i	$0.322409\pi$
$594$	0	0
$595$	−3.71982	−0.152498
$596$	0	0
$597$	−60.7889	−2.48792
$598$	0	0
$599$	23.6336	0.965642	0.482821	−	0.875719i	$-0.339612\pi$
$599$	23.6336	0.965642	0.482821	+	0.875719i	$0.339612\pi$
$600$	0	0
$601$	35.3415	1.44161	0.720805	−	0.693138i	$-0.243772\pi$
$601$	35.3415	1.44161	0.720805	+	0.693138i	$0.243772\pi$
$602$	0	0
$603$	12.1939	0.496576
$604$	0	0
$605$	−17.4036	−0.707555
$606$	0	0
$607$	−28.8432	−1.17071	−0.585355	−	0.810777i	$-0.699045\pi$
$607$	−28.8432	−1.17071	−0.585355	+	0.810777i	$0.699045\pi$
$608$	0	0
$609$	25.8207	1.04631
$610$	0	0
$611$	23.1430	0.936267
$612$	0	0
$613$	−27.7604	−1.12123	−0.560616	−	0.828076i	$-0.689436\pi$
$613$	−27.7604	−1.12123	−0.560616	+	0.828076i	$0.689436\pi$
$614$	0	0
$615$	−32.4362	−1.30795
$616$	0	0
$617$	32.5370	1.30989	0.654946	−	0.755676i	$-0.272691\pi$
$617$	32.5370	1.30989	0.654946	+	0.755676i	$0.272691\pi$
$618$	0	0
$619$	−6.20855	−0.249543	−0.124771	−	0.992186i	$-0.539820\pi$
$619$	−6.20855	−0.249543	−0.124771	+	0.992186i	$0.539820\pi$
$620$	0	0
$621$	8.46907	0.339852
$622$	0	0
$623$	3.48024	0.139433
$624$	0	0
$625$	−2.76547	−0.110619
$626$	0	0
$627$	−10.7474	−0.429211
$628$	0	0
$629$	−5.68879	−0.226827
$630$	0	0
$631$	2.67256	0.106393	0.0531965	−	0.998584i	$-0.483059\pi$
$631$	2.67256	0.106393	0.0531965	+	0.998584i	$0.483059\pi$
$632$	0	0
$633$	−0.732814	−0.0291267
$634$	0	0
$635$	−1.75086	−0.0694807
$636$	0	0
$637$	21.7586	0.862107
$638$	0	0
$639$	−12.4654	−0.493125
$640$	0	0
$641$	−24.6639	−0.974164	−0.487082	−	0.873356i	$-0.661939\pi$
$641$	−24.6639	−0.974164	−0.487082	+	0.873356i	$0.661939\pi$
$642$	0	0
$643$	25.5405	1.00722	0.503609	−	0.863932i	$-0.332005\pi$
$643$	25.5405	1.00722	0.503609	+	0.863932i	$0.332005\pi$
$644$	0	0
$645$	6.07668	0.239269
$646$	0	0
$647$	−21.7535	−0.855220	−0.427610	−	0.903963i	$-0.640644\pi$
$647$	−21.7535	−0.855220	−0.427610	+	0.903963i	$0.640644\pi$
$648$	0	0
$649$	−43.2863	−1.69914
$650$	0	0
$651$	13.6673	0.535664
$652$	0	0
$653$	−9.37940	−0.367044	−0.183522	−	0.983016i	$-0.558750\pi$
$653$	−9.37940	−0.367044	−0.183522	+	0.983016i	$0.558750\pi$
$654$	0	0
$655$	−3.52588	−0.137767
$656$	0	0
$657$	−0.690407	−0.0269353
$658$	0	0
$659$	39.8138	1.55092	0.775462	−	0.631394i	$-0.217517\pi$
$659$	39.8138	1.55092	0.775462	+	0.631394i	$0.217517\pi$
$660$	0	0
$661$	−19.1138	−0.743442	−0.371721	−	0.928345i	$-0.621232\pi$
$661$	−19.1138	−0.743442	−0.371721	+	0.928345i	$0.621232\pi$
$662$	0	0
$663$	17.3999	0.675757
$664$	0	0
$665$	2.16291	0.0838740
$666$	0	0
$667$	−31.2242	−1.20901
$668$	0	0
$669$	−52.6707	−2.03637
$670$	0	0
$671$	42.5776	1.64369
$672$	0	0
$673$	−21.1138	−0.813878	−0.406939	−	0.913455i	$-0.633404\pi$
$673$	−21.1138	−0.813878	−0.406939	+	0.913455i	$0.633404\pi$
$674$	0	0
$675$	−6.00688	−0.231205
$676$	0	0
$677$	−36.3855	−1.39841	−0.699204	−	0.714922i	$-0.746462\pi$
$677$	−36.3855	−1.39841	−0.699204	+	0.714922i	$0.746462\pi$
$678$	0	0
$679$	−10.3810	−0.398387
$680$	0	0
$681$	53.6344	2.05528
$682$	0	0
$683$	12.6087	0.482457	0.241229	−	0.970468i	$-0.422450\pi$
$683$	12.6087	0.482457	0.241229	+	0.970468i	$0.422450\pi$
$684$	0	0
$685$	13.8156	0.527867
$686$	0	0
$687$	−31.4182	−1.19868
$688$	0	0
$689$	−57.8690	−2.20463
$690$	0	0
$691$	16.8405	0.640644	0.320322	−	0.947309i	$-0.396209\pi$
$691$	16.8405	0.640644	0.320322	+	0.947309i	$0.396209\pi$
$692$	0	0
$693$	−14.4672	−0.549565
$694$	0	0
$695$	17.9690	0.681602
$696$	0	0
$697$	−16.8647	−0.638796
$698$	0	0
$699$	28.8647	1.09176
$700$	0	0
$701$	33.9311	1.28156	0.640779	−	0.767725i	$-0.278611\pi$
$701$	33.9311	1.28156	0.640779	+	0.767725i	$0.278611\pi$
$702$	0	0
$703$	3.30777	0.124755
$704$	0	0
$705$	−17.0180	−0.640937
$706$	0	0
$707$	10.2086	0.383932
$708$	0	0
$709$	−30.8984	−1.16041	−0.580207	−	0.814469i	$-0.697028\pi$
$709$	−30.8984	−1.16041	−0.580207	+	0.814469i	$0.697028\pi$
$710$	0	0
$711$	−0.754297	−0.0282884
$712$	0	0
$713$	−16.5275	−0.618960
$714$	0	0
$715$	−31.6121	−1.18223
$716$	0	0
$717$	−54.5612	−2.03763
$718$	0	0
$719$	26.8122	0.999925	0.499963	−	0.866047i	$-0.333347\pi$
$719$	26.8122	0.999925	0.499963	+	0.866047i	$0.333347\pi$
$720$	0	0
$721$	1.21199	0.0451369
$722$	0	0
$723$	58.6347	2.18065
$724$	0	0
$725$	22.1465	0.822500
$726$	0	0
$727$	30.1897	1.11968	0.559838	−	0.828602i	$-0.310863\pi$
$727$	30.1897	1.11968	0.559838	+	0.828602i	$0.310863\pi$
$728$	0	0
$729$	−8.23109	−0.304855
$730$	0	0
$731$	3.15947	0.116857
$732$	0	0
$733$	−37.8207	−1.39694	−0.698469	−	0.715640i	$-0.746135\pi$
$733$	−37.8207	−1.39694	−0.698469	+	0.715640i	$0.746135\pi$
$734$	0	0
$735$	−16.0000	−0.590169
$736$	0	0
$737$	−28.3043	−1.04260
$738$	0	0
$739$	−39.3803	−1.44863	−0.724313	−	0.689471i	$-0.757843\pi$
$739$	−39.3803	−1.44863	−0.724313	+	0.689471i	$0.757843\pi$
$740$	0	0
$741$	−10.1173	−0.371667
$742$	0	0
$743$	13.9854	0.513074	0.256537	−	0.966534i	$-0.417418\pi$
$743$	13.9854	0.513074	0.256537	+	0.966534i	$0.417418\pi$
$744$	0	0
$745$	24.7355	0.906238
$746$	0	0
$747$	−37.1690	−1.35994
$748$	0	0
$749$	16.8241	0.614739
$750$	0	0
$751$	25.0777	0.915100	0.457550	−	0.889184i	$-0.348727\pi$
$751$	25.0777	0.915100	0.457550	+	0.889184i	$0.348727\pi$
$752$	0	0
$753$	−52.4768	−1.91236
$754$	0	0
$755$	10.5535	0.384080
$756$	0	0
$757$	−24.6137	−0.894601	−0.447301	−	0.894384i	$-0.647615\pi$
$757$	−24.6137	−0.894601	−0.447301	+	0.894384i	$0.647615\pi$
$758$	0	0
$759$	42.9897	1.56043
$760$	0	0
$761$	−37.8010	−1.37029	−0.685143	−	0.728409i	$-0.740260\pi$
$761$	−37.8010	−1.37029	−0.685143	+	0.728409i	$0.740260\pi$
$762$	0	0
$763$	6.07668	0.219991
$764$	0	0
$765$	−5.20693	−0.188257
$766$	0	0
$767$	−40.7483	−1.47134
$768$	0	0
$769$	9.98357	0.360017	0.180008	−	0.983665i	$-0.442388\pi$
$769$	9.98357	0.360017	0.180008	+	0.983665i	$0.442388\pi$
$770$	0	0
$771$	−6.35180	−0.228754
$772$	0	0
$773$	−12.5896	−0.452815	−0.226408	−	0.974033i	$-0.572698\pi$
$773$	−12.5896	−0.452815	−0.226408	+	0.974033i	$0.572698\pi$
$774$	0	0
$775$	11.7225	0.421085
$776$	0	0
$777$	10.9414	0.392519
$778$	0	0
$779$	9.80605	0.351338
$780$	0	0
$781$	28.9345	1.03536
$782$	0	0
$783$	−16.5275	−0.590645
$784$	0	0
$785$	23.0518	0.822753
$786$	0	0
$787$	29.2603	1.04302	0.521509	−	0.853246i	$-0.325369\pi$
$787$	29.2603	1.04302	0.521509	+	0.853246i	$0.325369\pi$
$788$	0	0
$789$	−15.9457	−0.567681
$790$	0	0
$791$	−11.6527	−0.414322
$792$	0	0
$793$	40.0812	1.42332
$794$	0	0
$795$	42.5535	1.50922
$796$	0	0
$797$	−46.8578	−1.65979	−0.829894	−	0.557920i	$-0.811599\pi$
$797$	−46.8578	−1.65979	−0.829894	+	0.557920i	$0.811599\pi$
$798$	0	0
$799$	−8.84826	−0.313029
$800$	0	0
$801$	4.87156	0.172128
$802$	0	0
$803$	1.60256	0.0565530
$804$	0	0
$805$	−8.65164	−0.304930
$806$	0	0
$807$	−46.9276	−1.65193
$808$	0	0
$809$	−22.6681	−0.796967	−0.398483	−	0.917176i	$-0.630463\pi$
$809$	−22.6681	−0.796967	−0.398483	+	0.917176i	$0.630463\pi$
$810$	0	0
$811$	−32.7620	−1.15043	−0.575215	−	0.818002i	$-0.695082\pi$
$811$	−32.7620	−1.15043	−0.575215	+	0.818002i	$0.695082\pi$
$812$	0	0
$813$	−0.139606	−0.00489621
$814$	0	0
$815$	−13.7103	−0.480250
$816$	0	0
$817$	−1.83709	−0.0642717
$818$	0	0
$819$	−13.6190	−0.475886
$820$	0	0
$821$	−3.20006	−0.111683	−0.0558414	−	0.998440i	$-0.517784\pi$
$821$	−3.20006	−0.111683	−0.0558414	+	0.998440i	$0.517784\pi$
$822$	0	0
$823$	−24.4741	−0.853114	−0.426557	−	0.904461i	$-0.640274\pi$
$823$	−24.4741	−0.853114	−0.426557	+	0.904461i	$0.640274\pi$
$824$	0	0
$825$	−30.4914	−1.06157
$826$	0	0
$827$	24.5681	0.854316	0.427158	−	0.904177i	$-0.359515\pi$
$827$	24.5681	0.854316	0.427158	+	0.904177i	$0.359515\pi$
$828$	0	0
$829$	11.2380	0.390311	0.195155	−	0.980772i	$-0.437479\pi$
$829$	11.2380	0.390311	0.195155	+	0.980772i	$0.437479\pi$
$830$	0	0
$831$	17.5939	0.610324
$832$	0	0
$833$	−8.31894	−0.288234
$834$	0	0
$835$	−29.8613	−1.03339
$836$	0	0
$837$	−8.74828	−0.302385
$838$	0	0
$839$	1.94480	0.0671421	0.0335711	−	0.999436i	$-0.489312\pi$
$839$	1.94480	0.0671421	0.0335711	+	0.999436i	$0.489312\pi$
$840$	0	0
$841$	31.9345	1.10119
$842$	0	0
$843$	−46.4330	−1.59924
$844$	0	0
$845$	−10.6397	−0.366017
$846$	0	0
$847$	17.4036	0.597993
$848$	0	0
$849$	−48.1027	−1.65088
$850$	0	0
$851$	−13.2311	−0.453556
$852$	0	0
$853$	29.8207	1.02104	0.510520	−	0.859866i	$-0.329453\pi$
$853$	29.8207	1.02104	0.510520	+	0.859866i	$0.329453\pi$
$854$	0	0
$855$	3.02760	0.103542
$856$	0	0
$857$	−6.49828	−0.221977	−0.110989	−	0.993822i	$-0.535402\pi$
$857$	−6.49828	−0.221977	−0.110989	+	0.993822i	$0.535402\pi$
$858$	0	0
$859$	21.0422	0.717951	0.358975	−	0.933347i	$-0.383126\pi$
$859$	21.0422	0.717951	0.358975	+	0.933347i	$0.383126\pi$
$860$	0	0
$861$	32.4362	1.10542
$862$	0	0
$863$	7.80605	0.265721	0.132861	−	0.991135i	$-0.457584\pi$
$863$	7.80605	0.265721	0.132861	+	0.991135i	$0.457584\pi$
$864$	0	0
$865$	−0.824101	−0.0280203
$866$	0	0
$867$	31.5829	1.07261
$868$	0	0
$869$	1.75086	0.0593938
$870$	0	0
$871$	−26.6448	−0.902823
$872$	0	0
$873$	−14.5311	−0.491804
$874$	0	0
$875$	16.9509	0.573046
$876$	0	0
$877$	4.32582	0.146073	0.0730363	−	0.997329i	$-0.476731\pi$
$877$	4.32582	0.146073	0.0730363	+	0.997329i	$0.476731\pi$
$878$	0	0
$879$	−55.3484	−1.86685
$880$	0	0
$881$	−5.65775	−0.190615	−0.0953073	−	0.995448i	$-0.530383\pi$
$881$	−5.65775	−0.190615	−0.0953073	+	0.995448i	$0.530383\pi$
$882$	0	0
$883$	19.6578	0.661536	0.330768	−	0.943712i	$-0.392692\pi$
$883$	19.6578	0.661536	0.330768	+	0.943712i	$0.392692\pi$
$884$	0	0
$885$	29.9639	1.00723
$886$	0	0
$887$	−40.7811	−1.36930	−0.684648	−	0.728874i	$-0.740044\pi$
$887$	−40.7811	−1.36930	−0.684648	+	0.728874i	$0.740044\pi$
$888$	0	0
$889$	1.75086	0.0587219
$890$	0	0
$891$	52.2664	1.75099
$892$	0	0
$893$	5.14486	0.172166
$894$	0	0
$895$	−6.42160	−0.214650
$896$	0	0
$897$	40.4691	1.35122
$898$	0	0
$899$	32.2536	1.07572
$900$	0	0
$901$	22.1250	0.737091
$902$	0	0
$903$	−6.07668	−0.202219
$904$	0	0
$905$	7.43965	0.247302
$906$	0	0
$907$	−37.1475	−1.23346	−0.616732	−	0.787173i	$-0.711544\pi$
$907$	−37.1475	−1.23346	−0.616732	+	0.787173i	$0.711544\pi$
$908$	0	0
$909$	14.2897	0.473960
$910$	0	0
$911$	20.4070	0.676114	0.338057	−	0.941126i	$-0.390230\pi$
$911$	20.4070	0.676114	0.338057	+	0.941126i	$0.390230\pi$
$912$	0	0
$913$	86.2760	2.85532
$914$	0	0
$915$	−29.4734	−0.974359
$916$	0	0
$917$	3.52588	0.116435
$918$	0	0
$919$	−52.9536	−1.74678	−0.873389	−	0.487023i	$-0.838083\pi$
$919$	−52.9536	−1.74678	−0.873389	+	0.487023i	$0.838083\pi$
$920$	0	0
$921$	23.4396	0.772363
$922$	0	0
$923$	27.2380	0.896549
$924$	0	0
$925$	9.38445	0.308559
$926$	0	0
$927$	1.69652	0.0557210
$928$	0	0
$929$	56.8984	1.86678	0.933388	−	0.358869i	$-0.116837\pi$
$929$	56.8984	1.86678	0.933388	+	0.358869i	$0.116837\pi$
$930$	0	0
$931$	4.83709	0.158529
$932$	0	0
$933$	−17.7992	−0.582719
$934$	0	0
$935$	12.0862	0.395262
$936$	0	0
$937$	40.9148	1.33663	0.668315	−	0.743879i	$-0.267016\pi$
$937$	40.9148	1.33663	0.668315	+	0.743879i	$0.267016\pi$
$938$	0	0
$939$	40.0000	1.30535
$940$	0	0
$941$	−2.59644	−0.0846416	−0.0423208	−	0.999104i	$-0.513475\pi$
$941$	−2.59644	−0.0846416	−0.0423208	+	0.999104i	$0.513475\pi$
$942$	0	0
$943$	−39.2242	−1.27732
$944$	0	0
$945$	−4.57946	−0.148970
$946$	0	0
$947$	25.7586	0.837042	0.418521	−	0.908207i	$-0.362549\pi$
$947$	25.7586	0.837042	0.418521	+	0.908207i	$0.362549\pi$
$948$	0	0
$949$	1.50859	0.0489711
$950$	0	0
$951$	−67.5984	−2.19203
$952$	0	0
$953$	2.25602	0.0730795	0.0365398	−	0.999332i	$-0.488366\pi$
$953$	2.25602	0.0730795	0.0365398	+	0.999332i	$0.488366\pi$
$954$	0	0
$955$	3.62854	0.117417
$956$	0	0
$957$	−83.8950	−2.71194
$958$	0	0
$959$	−13.8156	−0.446129
$960$	0	0
$961$	−13.9276	−0.449278
$962$	0	0
$963$	23.5500	0.758889
$964$	0	0
$965$	−18.0146	−0.579911
$966$	0	0
$967$	45.0449	1.44855	0.724273	−	0.689513i	$-0.242176\pi$
$967$	45.0449	1.44855	0.724273	+	0.689513i	$0.242176\pi$
$968$	0	0
$969$	3.86813	0.124262
$970$	0	0
$971$	17.5760	0.564041	0.282021	−	0.959408i	$-0.408995\pi$
$971$	17.5760	0.564041	0.282021	+	0.959408i	$0.408995\pi$
$972$	0	0
$973$	−17.9690	−0.576059
$974$	0	0
$975$	−28.7036	−0.919251
$976$	0	0
$977$	−15.9042	−0.508821	−0.254410	−	0.967096i	$-0.581881\pi$
$977$	−15.9042	−0.508821	−0.254410	+	0.967096i	$0.581881\pi$
$978$	0	0
$979$	−11.3078	−0.361398
$980$	0	0
$981$	8.50601	0.271576
$982$	0	0
$983$	41.7294	1.33096	0.665480	−	0.746416i	$-0.268227\pi$
$983$	41.7294	1.33096	0.665480	+	0.746416i	$0.268227\pi$
$984$	0	0
$985$	−1.81055	−0.0576889
$986$	0	0
$987$	17.0180	0.541690
$988$	0	0
$989$	7.34836	0.233664
$990$	0	0
$991$	−0.193945	−0.00616087	−0.00308044	−	0.999995i	$-0.500981\pi$
$991$	−0.193945	−0.00616087	−0.00308044	+	0.999995i	$0.500981\pi$
$992$	0	0
$993$	42.3052	1.34251
$994$	0	0
$995$	−39.7490	−1.26013
$996$	0	0
$997$	−40.6949	−1.28882	−0.644410	−	0.764680i	$-0.722897\pi$
$997$	−40.6949	−1.28882	−0.644410	+	0.764680i	$0.722897\pi$
$998$	0	0
$999$	−7.00344	−0.221579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bg.1.1		3
4.3	odd	2		4864.2.a.ba.1.3		3
8.3	odd	2		4864.2.a.bh.1.1		3
8.5	even	2		4864.2.a.bb.1.3		3
16.3	odd	4		2432.2.c.h.1217.5	yes	6
16.5	even	4		2432.2.c.e.1217.5	yes	6
16.11	odd	4		2432.2.c.h.1217.2	yes	6
16.13	even	4		2432.2.c.e.1217.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.e.1217.2	✓	6	16.13	even	4
2432.2.c.e.1217.5	yes	6	16.5	even	4
2432.2.c.h.1217.2	yes	6	16.11	odd	4
2432.2.c.h.1217.5	yes	6	16.3	odd	4
4864.2.a.ba.1.3		3	4.3	odd	2
4864.2.a.bb.1.3		3	8.5	even	2
4864.2.a.bg.1.1		3	1.1	even	1	trivial
4864.2.a.bh.1.1		3	8.3	odd	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)

\(f(q)\)	\(=\)	\(q-2.24914 q^{3} -1.47068 q^{5} +1.47068 q^{7} +2.05863 q^{9} +O(q^{10})\)	\(q-2.24914 q^{3} -1.47068 q^{5} +1.47068 q^{7} +2.05863 q^{9} -4.77846 q^{11} -4.49828 q^{13} +3.30777 q^{15} +1.71982 q^{17} -1.00000 q^{19} -3.30777 q^{21} +4.00000 q^{23} -2.83709 q^{25} +2.11727 q^{27} -7.80605 q^{29} -4.13187 q^{31} +10.7474 q^{33} -2.16291 q^{35} -3.30777 q^{37} +10.1173 q^{39} -9.80605 q^{41} +1.83709 q^{43} -3.02760 q^{45} -5.14486 q^{47} -4.83709 q^{49} -3.86813 q^{51} +12.8647 q^{53} +7.02760 q^{55} +2.24914 q^{57} +9.05863 q^{59} -8.91033 q^{61} +3.02760 q^{63} +6.61555 q^{65} +5.92332 q^{67} -8.99656 q^{69} -6.05520 q^{71} -0.335371 q^{73} +6.38101 q^{75} -7.02760 q^{77} -0.366407 q^{79} -10.9379 q^{81} -18.0552 q^{83} -2.52932 q^{85} +17.5569 q^{87} +2.36641 q^{89} -6.61555 q^{91} +9.29317 q^{93} +1.47068 q^{95} -7.05863 q^{97} -9.83709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 7 * q^9	\( 3 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 7 q^{9} - 6 q^{11} + 4 q^{13} + 2 q^{15} - 4 q^{17} - 3 q^{19} - 2 q^{21} + 12 q^{23} - q^{25} + 8 q^{27} + 2 q^{29} - 2 q^{31} + 6 q^{33} - 14 q^{35} - 2 q^{37} + 32 q^{39} - 4 q^{41} - 2 q^{43} + 8 q^{45} - 7 q^{49} - 22 q^{51} + 14 q^{53} + 4 q^{55} - 2 q^{57} + 28 q^{59} - 8 q^{61} - 8 q^{63} + 4 q^{65} - 6 q^{67} + 8 q^{69} + 16 q^{71} + 24 q^{73} - 4 q^{77} + 6 q^{79} + 3 q^{81} - 20 q^{83} - 8 q^{85} + 36 q^{87} - 4 q^{91} + 32 q^{93} + 4 q^{95} - 22 q^{97} - 22 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 7 * q^9 - 6 * q^11 + 4 * q^13 + 2 * q^15 - 4 * q^17 - 3 * q^19 - 2 * q^21 + 12 * q^23 - q^25 + 8 * q^27 + 2 * q^29 - 2 * q^31 + 6 * q^33 - 14 * q^35 - 2 * q^37 + 32 * q^39 - 4 * q^41 - 2 * q^43 + 8 * q^45 - 7 * q^49 - 22 * q^51 + 14 * q^53 + 4 * q^55 - 2 * q^57 + 28 * q^59 - 8 * q^61 - 8 * q^63 + 4 * q^65 - 6 * q^67 + 8 * q^69 + 16 * q^71 + 24 * q^73 - 4 * q^77 + 6 * q^79 + 3 * q^81 - 20 * q^83 - 8 * q^85 + 36 * q^87 - 4 * q^91 + 32 * q^93 + 4 * q^95 - 22 * q^97 - 22 * q^99

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