LMFDB - Embedded newform 984.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [984,2,Mod(1,984)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(984, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("984.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 984 = 2^{3} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	984.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:	$7.85727955889$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.91729$ of defining polynomial
Character	$\chi$	$=$	984.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.406728 q^{5} -0.406728 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.406728 q^{5} -0.406728 q^{7} +1.00000 q^{9} -5.91729 q^{11} +5.42784 q^{13} -0.406728 q^{15} -3.91729 q^{17} -2.40673 q^{19} +0.406728 q^{21} -1.59327 q^{23} -4.83457 q^{25} -1.00000 q^{27} +1.91729 q^{29} +7.34513 q^{31} +5.91729 q^{33} -0.165428 q^{35} -9.34513 q^{37} -5.42784 q^{39} -1.00000 q^{41} -10.5317 q^{43} +0.406728 q^{45} -0.0827140 q^{47} -6.83457 q^{49} +3.91729 q^{51} -4.00000 q^{53} -2.40673 q^{55} +2.40673 q^{57} -7.18654 q^{59} +5.34513 q^{61} -0.406728 q^{63} +2.20766 q^{65} +3.83457 q^{67} +1.59327 q^{69} -7.10383 q^{71} +9.34513 q^{73} +4.83457 q^{75} +2.40673 q^{77} -13.6691 q^{79} +1.00000 q^{81} -13.4278 q^{83} -1.59327 q^{85} -1.91729 q^{87} +0.373086 q^{89} -2.20766 q^{91} -7.34513 q^{93} -0.978885 q^{95} +12.2413 q^{97} -5.91729 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{9} - 8 q^{11} - 2 q^{13} - 2 q^{17} - 6 q^{19} - 6 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 6 q^{31} + 8 q^{33} - 20 q^{35} + 2 q^{39} - 3 q^{41} - 6 q^{43} - 10 q^{47} - q^{49} + 2 q^{51} - 12 q^{53} - 6 q^{55} + 6 q^{57} - 24 q^{59} - 12 q^{61} - 8 q^{65} - 8 q^{67} + 6 q^{69} - 14 q^{71} - 5 q^{75} + 6 q^{77} - 2 q^{79} + 3 q^{81} - 22 q^{83} - 6 q^{85} + 4 q^{87} + 6 q^{89} + 8 q^{91} + 6 q^{93} - 20 q^{95} + 16 q^{97} - 8 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^9 - 8 * q^11 - 2 * q^13 - 2 * q^17 - 6 * q^19 - 6 * q^23 + 5 * q^25 - 3 * q^27 - 4 * q^29 - 6 * q^31 + 8 * q^33 - 20 * q^35 + 2 * q^39 - 3 * q^41 - 6 * q^43 - 10 * q^47 - q^49 + 2 * q^51 - 12 * q^53 - 6 * q^55 + 6 * q^57 - 24 * q^59 - 12 * q^61 - 8 * q^65 - 8 * q^67 + 6 * q^69 - 14 * q^71 - 5 * q^75 + 6 * q^77 - 2 * q^79 + 3 * q^81 - 22 * q^83 - 6 * q^85 + 4 * q^87 + 6 * q^89 + 8 * q^91 + 6 * q^93 - 20 * q^95 + 16 * q^97 - 8 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.406728	0.181894	0.0909472	−	0.995856i	$-0.471011\pi$
$5$	0.406728	0.181894	0.0909472	+	0.995856i	$0.471011\pi$
$6$	0	0
$7$	−0.406728	−0.153729	−0.0768644	−	0.997042i	$-0.524491\pi$
$7$	−0.406728	−0.153729	−0.0768644	+	0.997042i	$0.524491\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.91729	−1.78413	−0.892064	−	0.451908i	$-0.850743\pi$
$11$	−5.91729	−1.78413	−0.892064	+	0.451908i	$0.850743\pi$
$12$	0	0
$13$	5.42784	1.50541	0.752706	−	0.658356i	$-0.228748\pi$
$13$	5.42784	1.50541	0.752706	+	0.658356i	$0.228748\pi$
$14$	0	0
$15$	−0.406728	−0.105017
$16$	0	0
$17$	−3.91729	−0.950081	−0.475041	−	0.879964i	$-0.657567\pi$
$17$	−3.91729	−0.950081	−0.475041	+	0.879964i	$0.657567\pi$
$18$	0	0
$19$	−2.40673	−0.552141	−0.276071	−	0.961137i	$-0.589032\pi$
$19$	−2.40673	−0.552141	−0.276071	+	0.961137i	$0.589032\pi$
$20$	0	0
$21$	0.406728	0.0887554
$22$	0	0
$23$	−1.59327	−0.332220	−0.166110	−	0.986107i	$-0.553121\pi$
$23$	−1.59327	−0.332220	−0.166110	+	0.986107i	$0.553121\pi$
$24$	0	0
$25$	−4.83457	−0.966914
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.91729	0.356031	0.178016	−	0.984028i	$-0.443032\pi$
$29$	1.91729	0.356031	0.178016	+	0.984028i	$0.443032\pi$
$30$	0	0
$31$	7.34513	1.31922	0.659612	−	0.751606i	$-0.270721\pi$
$31$	7.34513	1.31922	0.659612	+	0.751606i	$0.270721\pi$
$32$	0	0
$33$	5.91729	1.03007
$34$	0	0
$35$	−0.165428	−0.0279624
$36$	0	0
$37$	−9.34513	−1.53633	−0.768165	−	0.640252i	$-0.778830\pi$
$37$	−9.34513	−1.53633	−0.768165	+	0.640252i	$0.778830\pi$
$38$	0	0
$39$	−5.42784	−0.869151
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−10.5317	−1.60607	−0.803033	−	0.595935i	$-0.796781\pi$
$43$	−10.5317	−1.60607	−0.803033	+	0.595935i	$0.796781\pi$
$44$	0	0
$45$	0.406728	0.0606315
$46$	0	0
$47$	−0.0827140	−0.0120651	−0.00603254	−	0.999982i	$-0.501920\pi$
$47$	−0.0827140	−0.0120651	−0.00603254	+	0.999982i	$0.501920\pi$
$48$	0	0
$49$	−6.83457	−0.976367
$50$	0	0
$51$	3.91729	0.548530
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−2.40673	−0.324523
$56$	0	0
$57$	2.40673	0.318779
$58$	0	0
$59$	−7.18654	−0.935608	−0.467804	−	0.883832i	$-0.654955\pi$
$59$	−7.18654	−0.935608	−0.467804	+	0.883832i	$0.654955\pi$
$60$	0	0
$61$	5.34513	0.684374	0.342187	−	0.939632i	$-0.388832\pi$
$61$	5.34513	0.684374	0.342187	+	0.939632i	$0.388832\pi$
$62$	0	0
$63$	−0.406728	−0.0512430
$64$	0	0
$65$	2.20766	0.273826
$66$	0	0
$67$	3.83457	0.468468	0.234234	−	0.972180i	$-0.424742\pi$
$67$	3.83457	0.468468	0.234234	+	0.972180i	$0.424742\pi$
$68$	0	0
$69$	1.59327	0.191807
$70$	0	0
$71$	−7.10383	−0.843069	−0.421535	−	0.906812i	$-0.638508\pi$
$71$	−7.10383	−0.843069	−0.421535	+	0.906812i	$0.638508\pi$
$72$	0	0
$73$	9.34513	1.09376	0.546882	−	0.837209i	$-0.315814\pi$
$73$	9.34513	1.09376	0.546882	+	0.837209i	$0.315814\pi$
$74$	0	0
$75$	4.83457	0.558248
$76$	0	0
$77$	2.40673	0.274272
$78$	0	0
$79$	−13.6691	−1.53790	−0.768949	−	0.639310i	$-0.779220\pi$
$79$	−13.6691	−1.53790	−0.768949	+	0.639310i	$0.779220\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.4278	−1.47390	−0.736949	−	0.675949i	$-0.763734\pi$
$83$	−13.4278	−1.47390	−0.736949	+	0.675949i	$0.763734\pi$
$84$	0	0
$85$	−1.59327	−0.172815
$86$	0	0
$87$	−1.91729	−0.205555
$88$	0	0
$89$	0.373086	0.0395471	0.0197735	−	0.999804i	$-0.493705\pi$
$89$	0.373086	0.0395471	0.0197735	+	0.999804i	$0.493705\pi$
$90$	0	0
$91$	−2.20766	−0.231425
$92$	0	0
$93$	−7.34513	−0.761654
$94$	0	0
$95$	−0.978885	−0.100431
$96$	0	0
$97$	12.2413	1.24292	0.621458	−	0.783448i	$-0.286541\pi$
$97$	12.2413	1.24292	0.621458	+	0.783448i	$0.286541\pi$
$98$	0	0
$99$	−5.91729	−0.594710
$100$	0	0
$101$	−14.5653	−1.44930	−0.724651	−	0.689116i	$-0.757999\pi$
$101$	−14.5653	−1.44930	−0.724651	+	0.689116i	$0.757999\pi$
$102$	0	0
$103$	5.13747	0.506210	0.253105	−	0.967439i	$-0.418548\pi$
$103$	5.13747	0.506210	0.253105	+	0.967439i	$0.418548\pi$
$104$	0	0
$105$	0.165428	0.0161441
$106$	0	0
$107$	−12.6480	−1.22273	−0.611366	−	0.791348i	$-0.709379\pi$
$107$	−12.6480	−1.22273	−0.611366	+	0.791348i	$0.709379\pi$
$108$	0	0
$109$	−10.2413	−0.980939	−0.490469	−	0.871458i	$-0.663175\pi$
$109$	−10.2413	−0.980939	−0.490469	+	0.871458i	$0.663175\pi$
$110$	0	0
$111$	9.34513	0.887000
$112$	0	0
$113$	8.40673	0.790838	0.395419	−	0.918501i	$-0.370599\pi$
$113$	8.40673	0.790838	0.395419	+	0.918501i	$0.370599\pi$
$114$	0	0
$115$	−0.648029	−0.0604290
$116$	0	0
$117$	5.42784	0.501804
$118$	0	0
$119$	1.59327	0.146055
$120$	0	0
$121$	24.0143	2.18312
$122$	0	0
$123$	1.00000	0.0901670
$124$	0	0
$125$	−4.00000	−0.357771
$126$	0	0
$127$	14.2077	1.26073	0.630363	−	0.776301i	$-0.282906\pi$
$127$	14.2077	1.26073	0.630363	+	0.776301i	$0.282906\pi$
$128$	0	0
$129$	10.5317	0.927262
$130$	0	0
$131$	7.63550	0.667117	0.333558	−	0.942729i	$-0.391751\pi$
$131$	7.63550	0.667117	0.333558	+	0.942729i	$0.391751\pi$
$132$	0	0
$133$	0.978885	0.0848801
$134$	0	0
$135$	−0.406728	−0.0350056
$136$	0	0
$137$	−15.4210	−1.31751	−0.658753	−	0.752360i	$-0.728916\pi$
$137$	−15.4210	−1.31751	−0.658753	+	0.752360i	$0.728916\pi$
$138$	0	0
$139$	17.4615	1.48106	0.740532	−	0.672021i	$-0.234574\pi$
$139$	17.4615	1.48106	0.740532	+	0.672021i	$0.234574\pi$
$140$	0	0
$141$	0.0827140	0.00696578
$142$	0	0
$143$	−32.1181	−2.68585
$144$	0	0
$145$	0.779815	0.0647601
$146$	0	0
$147$	6.83457	0.563706
$148$	0	0
$149$	12.6480	1.03617	0.518083	−	0.855330i	$-0.326646\pi$
$149$	12.6480	1.03617	0.518083	+	0.855330i	$0.326646\pi$
$150$	0	0
$151$	14.6480	1.19204	0.596020	−	0.802970i	$-0.296748\pi$
$151$	14.6480	1.19204	0.596020	+	0.802970i	$0.296748\pi$
$152$	0	0
$153$	−3.91729	−0.316694
$154$	0	0
$155$	2.98747	0.239960
$156$	0	0
$157$	−17.0633	−1.36180	−0.680902	−	0.732375i	$-0.738412\pi$
$157$	−17.0633	−1.36180	−0.680902	+	0.732375i	$0.738412\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	0.648029	0.0510718
$162$	0	0
$163$	14.6971	1.15117	0.575583	−	0.817743i	$-0.304775\pi$
$163$	14.6971	1.15117	0.575583	+	0.817743i	$0.304775\pi$
$164$	0	0
$165$	2.40673	0.187364
$166$	0	0
$167$	4.44037	0.343606	0.171803	−	0.985131i	$-0.445041\pi$
$167$	4.44037	0.343606	0.171803	+	0.985131i	$0.445041\pi$
$168$	0	0
$169$	16.4615	1.26627
$170$	0	0
$171$	−2.40673	−0.184047
$172$	0	0
$173$	16.8893	1.28407	0.642036	−	0.766675i	$-0.278090\pi$
$173$	16.8893	1.28407	0.642036	+	0.766675i	$0.278090\pi$
$174$	0	0
$175$	1.96636	0.148643
$176$	0	0
$177$	7.18654	0.540174
$178$	0	0
$179$	4.93840	0.369113	0.184557	−	0.982822i	$-0.440915\pi$
$179$	4.93840	0.369113	0.184557	+	0.982822i	$0.440915\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	−5.34513	−0.395123
$184$	0	0
$185$	−3.80093	−0.279450
$186$	0	0
$187$	23.1797	1.69507
$188$	0	0
$189$	0.406728	0.0295851
$190$	0	0
$191$	21.5037	1.55595	0.777977	−	0.628293i	$-0.216246\pi$
$191$	21.5037	1.55595	0.777977	+	0.628293i	$0.216246\pi$
$192$	0	0
$193$	−21.3046	−1.53354	−0.766771	−	0.641921i	$-0.778138\pi$
$193$	−21.3046	−1.53354	−0.766771	+	0.641921i	$0.778138\pi$
$194$	0	0
$195$	−2.20766	−0.158094
$196$	0	0
$197$	22.9316	1.63381	0.816903	−	0.576775i	$-0.195689\pi$
$197$	22.9316	1.63381	0.816903	+	0.576775i	$0.195689\pi$
$198$	0	0
$199$	−0.572156	−0.0405591	−0.0202795	−	0.999794i	$-0.506456\pi$
$199$	−0.572156	−0.0405591	−0.0202795	+	0.999794i	$0.506456\pi$
$200$	0	0
$201$	−3.83457	−0.270470
$202$	0	0
$203$	−0.779815	−0.0547323
$204$	0	0
$205$	−0.406728	−0.0284071
$206$	0	0
$207$	−1.59327	−0.110740
$208$	0	0
$209$	14.2413	0.985091
$210$	0	0
$211$	−10.5248	−0.724559	−0.362280	−	0.932069i	$-0.618001\pi$
$211$	−10.5248	−0.724559	−0.362280	+	0.932069i	$0.618001\pi$
$212$	0	0
$213$	7.10383	0.486746
$214$	0	0
$215$	−4.28353	−0.292134
$216$	0	0
$217$	−2.98747	−0.202803
$218$	0	0
$219$	−9.34513	−0.631485
$220$	0	0
$221$	−21.2624	−1.43026
$222$	0	0
$223$	9.62691	0.644666	0.322333	−	0.946626i	$-0.395533\pi$
$223$	9.62691	0.644666	0.322333	+	0.946626i	$0.395533\pi$
$224$	0	0
$225$	−4.83457	−0.322305
$226$	0	0
$227$	−18.2345	−1.21026	−0.605132	−	0.796125i	$-0.706880\pi$
$227$	−18.2345	−1.21026	−0.605132	+	0.796125i	$0.706880\pi$
$228$	0	0
$229$	−5.66056	−0.374060	−0.187030	−	0.982354i	$-0.559886\pi$
$229$	−5.66056	−0.374060	−0.187030	+	0.982354i	$0.559886\pi$
$230$	0	0
$231$	−2.40673	−0.158351
$232$	0	0
$233$	14.4153	0.944379	0.472189	−	0.881497i	$-0.343464\pi$
$233$	14.4153	0.944379	0.472189	+	0.881497i	$0.343464\pi$
$234$	0	0
$235$	−0.0336421	−0.00219457
$236$	0	0
$237$	13.6691	0.887906
$238$	0	0
$239$	−4.20766	−0.272171	−0.136085	−	0.990697i	$-0.543452\pi$
$239$	−4.20766	−0.272171	−0.136085	+	0.990697i	$0.543452\pi$
$240$	0	0
$241$	1.67599	0.107960	0.0539799	−	0.998542i	$-0.482809\pi$
$241$	1.67599	0.107960	0.0539799	+	0.998542i	$0.482809\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.77981	−0.177596
$246$	0	0
$247$	−13.0633	−0.831201
$248$	0	0
$249$	13.4278	0.850955
$250$	0	0
$251$	−6.20766	−0.391824	−0.195912	−	0.980621i	$-0.562767\pi$
$251$	−6.20766	−0.391824	−0.195912	+	0.980621i	$0.562767\pi$
$252$	0	0
$253$	9.42784	0.592723
$254$	0	0
$255$	1.59327	0.0997745
$256$	0	0
$257$	25.3788	1.58308	0.791542	−	0.611115i	$-0.209279\pi$
$257$	25.3788	1.58308	0.791542	+	0.611115i	$0.209279\pi$
$258$	0	0
$259$	3.80093	0.236178
$260$	0	0
$261$	1.91729	0.118677
$262$	0	0
$263$	24.8152	1.53017	0.765085	−	0.643929i	$-0.222697\pi$
$263$	24.8152	1.53017	0.765085	+	0.643929i	$0.222697\pi$
$264$	0	0
$265$	−1.62691	−0.0999405
$266$	0	0
$267$	−0.373086	−0.0228325
$268$	0	0
$269$	2.16543	0.132028	0.0660142	−	0.997819i	$-0.478972\pi$
$269$	2.16543	0.132028	0.0660142	+	0.997819i	$0.478972\pi$
$270$	0	0
$271$	−6.11636	−0.371542	−0.185771	−	0.982593i	$-0.559478\pi$
$271$	−6.11636	−0.371542	−0.185771	+	0.982593i	$0.559478\pi$
$272$	0	0
$273$	2.20766	0.133614
$274$	0	0
$275$	28.6075	1.72510
$276$	0	0
$277$	22.5739	1.35633	0.678167	−	0.734908i	$-0.262775\pi$
$277$	22.5739	1.35633	0.678167	+	0.734908i	$0.262775\pi$
$278$	0	0
$279$	7.34513	0.439741
$280$	0	0
$281$	−27.3537	−1.63179	−0.815893	−	0.578203i	$-0.803754\pi$
$281$	−27.3537	−1.63179	−0.815893	+	0.578203i	$0.803754\pi$
$282$	0	0
$283$	28.4758	1.69271	0.846354	−	0.532621i	$-0.178793\pi$
$283$	28.4758	1.69271	0.846354	+	0.532621i	$0.178793\pi$
$284$	0	0
$285$	0.978885	0.0579841
$286$	0	0
$287$	0.406728	0.0240084
$288$	0	0
$289$	−1.65487	−0.0973453
$290$	0	0
$291$	−12.2413	−0.717598
$292$	0	0
$293$	−11.2133	−0.655091	−0.327545	−	0.944835i	$-0.606221\pi$
$293$	−11.2133	−0.655091	−0.327545	+	0.944835i	$0.606221\pi$
$294$	0	0
$295$	−2.92297	−0.170182
$296$	0	0
$297$	5.91729	0.343356
$298$	0	0
$299$	−8.64803	−0.500128
$300$	0	0
$301$	4.28353	0.246899
$302$	0	0
$303$	14.5653	0.836755
$304$	0	0
$305$	2.17402	0.124484
$306$	0	0
$307$	−9.30290	−0.530945	−0.265472	−	0.964119i	$-0.585528\pi$
$307$	−9.30290	−0.530945	−0.265472	+	0.964119i	$0.585528\pi$
$308$	0	0
$309$	−5.13747	−0.292261
$310$	0	0
$311$	−17.8346	−1.01131	−0.505653	−	0.862737i	$-0.668748\pi$
$311$	−17.8346	−1.01131	−0.505653	+	0.862737i	$0.668748\pi$
$312$	0	0
$313$	3.92413	0.221805	0.110902	−	0.993831i	$-0.464626\pi$
$313$	3.92413	0.221805	0.110902	+	0.993831i	$0.464626\pi$
$314$	0	0
$315$	−0.165428	−0.00932081
$316$	0	0
$317$	−20.3577	−1.14340	−0.571700	−	0.820463i	$-0.693716\pi$
$317$	−20.3577	−1.14340	−0.571700	+	0.820463i	$0.693716\pi$
$318$	0	0
$319$	−11.3451	−0.635205
$320$	0	0
$321$	12.6480	0.705944
$322$	0	0
$323$	9.42784	0.524579
$324$	0	0
$325$	−26.2413	−1.45561
$326$	0	0
$327$	10.2413	0.566345
$328$	0	0
$329$	0.0336421	0.00185475
$330$	0	0
$331$	−6.82204	−0.374974	−0.187487	−	0.982267i	$-0.560034\pi$
$331$	−6.82204	−0.374974	−0.187487	+	0.982267i	$0.560034\pi$
$332$	0	0
$333$	−9.34513	−0.512110
$334$	0	0
$335$	1.55963	0.0852117
$336$	0	0
$337$	−14.1586	−0.771267	−0.385634	−	0.922652i	$-0.626017\pi$
$337$	−14.1586	−0.771267	−0.385634	+	0.922652i	$0.626017\pi$
$338$	0	0
$339$	−8.40673	−0.456591
$340$	0	0
$341$	−43.4632	−2.35367
$342$	0	0
$343$	5.62691	0.303825
$344$	0	0
$345$	0.648029	0.0348887
$346$	0	0
$347$	−28.1922	−1.51344	−0.756719	−	0.653740i	$-0.773199\pi$
$347$	−28.1922	−1.51344	−0.756719	+	0.653740i	$0.773199\pi$
$348$	0	0
$349$	−18.4894	−0.989717	−0.494859	−	0.868974i	$-0.664780\pi$
$349$	−18.4894	−0.989717	−0.494859	+	0.868974i	$0.664780\pi$
$350$	0	0
$351$	−5.42784	−0.289717
$352$	0	0
$353$	19.2961	1.02703	0.513513	−	0.858082i	$-0.328344\pi$
$353$	19.2961	1.02703	0.513513	+	0.858082i	$0.328344\pi$
$354$	0	0
$355$	−2.88933	−0.153350
$356$	0	0
$357$	−1.59327	−0.0843249
$358$	0	0
$359$	20.0336	1.05734	0.528668	−	0.848829i	$-0.322692\pi$
$359$	20.0336	1.05734	0.528668	+	0.848829i	$0.322692\pi$
$360$	0	0
$361$	−13.2077	−0.695140
$362$	0	0
$363$	−24.0143	−1.26042
$364$	0	0
$365$	3.80093	0.198950
$366$	0	0
$367$	−5.23562	−0.273297	−0.136648	−	0.990620i	$-0.543633\pi$
$367$	−5.23562	−0.273297	−0.136648	+	0.990620i	$0.543633\pi$
$368$	0	0
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	1.62691	0.0844652
$372$	0	0
$373$	2.25673	0.116849	0.0584245	−	0.998292i	$-0.481392\pi$
$373$	2.25673	0.116849	0.0584245	+	0.998292i	$0.481392\pi$
$374$	0	0
$375$	4.00000	0.206559
$376$	0	0
$377$	10.4067	0.535974
$378$	0	0
$379$	29.8768	1.53467	0.767334	−	0.641247i	$-0.221583\pi$
$379$	29.8768	1.53467	0.767334	+	0.641247i	$0.221583\pi$
$380$	0	0
$381$	−14.2077	−0.727880
$382$	0	0
$383$	4.56531	0.233277	0.116638	−	0.993174i	$-0.462788\pi$
$383$	4.56531	0.233277	0.116638	+	0.993174i	$0.462788\pi$
$384$	0	0
$385$	0.978885	0.0498886
$386$	0	0
$387$	−10.5317	−0.535355
$388$	0	0
$389$	−13.2202	−0.670290	−0.335145	−	0.942167i	$-0.608785\pi$
$389$	−13.2202	−0.670290	−0.335145	+	0.942167i	$0.608785\pi$
$390$	0	0
$391$	6.24130	0.315636
$392$	0	0
$393$	−7.63550	−0.385160
$394$	0	0
$395$	−5.55963	−0.279735
$396$	0	0
$397$	−22.2413	−1.11626	−0.558129	−	0.829754i	$-0.688481\pi$
$397$	−22.2413	−1.11626	−0.558129	+	0.829754i	$0.688481\pi$
$398$	0	0
$399$	−0.978885	−0.0490055
$400$	0	0
$401$	−32.3594	−1.61595	−0.807976	−	0.589216i	$-0.799437\pi$
$401$	−32.3594	−1.61595	−0.807976	+	0.589216i	$0.799437\pi$
$402$	0	0
$403$	39.8682	1.98598
$404$	0	0
$405$	0.406728	0.0202105
$406$	0	0
$407$	55.2978	2.74101
$408$	0	0
$409$	−3.23562	−0.159991	−0.0799954	−	0.996795i	$-0.525491\pi$
$409$	−3.23562	−0.159991	−0.0799954	+	0.996795i	$0.525491\pi$
$410$	0	0
$411$	15.4210	0.760662
$412$	0	0
$413$	2.92297	0.143830
$414$	0	0
$415$	−5.46149	−0.268094
$416$	0	0
$417$	−17.4615	−0.855093
$418$	0	0
$419$	−2.82204	−0.137866	−0.0689330	−	0.997621i	$-0.521959\pi$
$419$	−2.82204	−0.137866	−0.0689330	+	0.997621i	$0.521959\pi$
$420$	0	0
$421$	1.62691	0.0792909	0.0396455	−	0.999214i	$-0.487377\pi$
$421$	1.62691	0.0792909	0.0396455	+	0.999214i	$0.487377\pi$
$422$	0	0
$423$	−0.0827140	−0.00402169
$424$	0	0
$425$	18.9384	0.918647
$426$	0	0
$427$	−2.17402	−0.105208
$428$	0	0
$429$	32.1181	1.55068
$430$	0	0
$431$	−16.8471	−0.811496	−0.405748	−	0.913985i	$-0.632989\pi$
$431$	−16.8471	−0.811496	−0.405748	+	0.913985i	$0.632989\pi$
$432$	0	0
$433$	9.34513	0.449098	0.224549	−	0.974463i	$-0.427909\pi$
$433$	9.34513	0.449098	0.224549	+	0.974463i	$0.427909\pi$
$434$	0	0
$435$	−0.779815	−0.0373893
$436$	0	0
$437$	3.83457	0.183432
$438$	0	0
$439$	−24.0422	−1.14747	−0.573737	−	0.819040i	$-0.694507\pi$
$439$	−24.0422	−1.14747	−0.573737	+	0.819040i	$0.694507\pi$
$440$	0	0
$441$	−6.83457	−0.325456
$442$	0	0
$443$	−28.9652	−1.37618	−0.688089	−	0.725626i	$-0.741550\pi$
$443$	−28.9652	−1.37618	−0.688089	+	0.725626i	$0.741550\pi$
$444$	0	0
$445$	0.151745	0.00719339
$446$	0	0
$447$	−12.6480	−0.598231
$448$	0	0
$449$	15.7923	0.745287	0.372643	−	0.927975i	$-0.378451\pi$
$449$	15.7923	0.745287	0.372643	+	0.927975i	$0.378451\pi$
$450$	0	0
$451$	5.91729	0.278634
$452$	0	0
$453$	−14.6480	−0.688224
$454$	0	0
$455$	−0.897917	−0.0420950
$456$	0	0
$457$	35.0633	1.64019	0.820097	−	0.572224i	$-0.193919\pi$
$457$	35.0633	1.64019	0.820097	+	0.572224i	$0.193919\pi$
$458$	0	0
$459$	3.91729	0.182843
$460$	0	0
$461$	7.71137	0.359155	0.179577	−	0.983744i	$-0.442527\pi$
$461$	7.71137	0.359155	0.179577	+	0.983744i	$0.442527\pi$
$462$	0	0
$463$	12.0759	0.561213	0.280607	−	0.959823i	$-0.409464\pi$
$463$	12.0759	0.561213	0.280607	+	0.959823i	$0.409464\pi$
$464$	0	0
$465$	−2.98747	−0.138541
$466$	0	0
$467$	17.3942	0.804908	0.402454	−	0.915440i	$-0.368157\pi$
$467$	17.3942	0.804908	0.402454	+	0.915440i	$0.368157\pi$
$468$	0	0
$469$	−1.55963	−0.0720170
$470$	0	0
$471$	17.0633	0.786237
$472$	0	0
$473$	62.3189	2.86543
$474$	0	0
$475$	11.6355	0.533873
$476$	0	0
$477$	−4.00000	−0.183147
$478$	0	0
$479$	−27.4210	−1.25290	−0.626449	−	0.779463i	$-0.715492\pi$
$479$	−27.4210	−1.25290	−0.626449	+	0.779463i	$0.715492\pi$
$480$	0	0
$481$	−50.7239	−2.31281
$482$	0	0
$483$	−0.648029	−0.0294863
$484$	0	0
$485$	4.97888	0.226080
$486$	0	0
$487$	−13.0702	−0.592267	−0.296133	−	0.955147i	$-0.595697\pi$
$487$	−13.0702	−0.592267	−0.296133	+	0.955147i	$0.595697\pi$
$488$	0	0
$489$	−14.6971	−0.664626
$490$	0	0
$491$	−19.8682	−0.896640	−0.448320	−	0.893873i	$-0.647977\pi$
$491$	−19.8682	−0.896640	−0.448320	+	0.893873i	$0.647977\pi$
$492$	0	0
$493$	−7.51056	−0.338259
$494$	0	0
$495$	−2.40673	−0.108174
$496$	0	0
$497$	2.88933	0.129604
$498$	0	0
$499$	−1.22877	−0.0550075	−0.0275037	−	0.999622i	$-0.508756\pi$
$499$	−1.22877	−0.0550075	−0.0275037	+	0.999622i	$0.508756\pi$
$500$	0	0
$501$	−4.44037	−0.198381
$502$	0	0
$503$	29.7941	1.32845	0.664226	−	0.747531i	$-0.268761\pi$
$503$	29.7941	1.32845	0.664226	+	0.747531i	$0.268761\pi$
$504$	0	0
$505$	−5.92413	−0.263620
$506$	0	0
$507$	−16.4615	−0.731080
$508$	0	0
$509$	−21.4210	−0.949469	−0.474735	−	0.880129i	$-0.657456\pi$
$509$	−21.4210	−0.949469	−0.474735	+	0.880129i	$0.657456\pi$
$510$	0	0
$511$	−3.80093	−0.168143
$512$	0	0
$513$	2.40673	0.106260
$514$	0	0
$515$	2.08956	0.0920768
$516$	0	0
$517$	0.489442	0.0215257
$518$	0	0
$519$	−16.8893	−0.741359
$520$	0	0
$521$	−3.33654	−0.146177	−0.0730883	−	0.997325i	$-0.523285\pi$
$521$	−3.33654	−0.146177	−0.0730883	+	0.997325i	$0.523285\pi$
$522$	0	0
$523$	−29.5459	−1.29195	−0.645977	−	0.763357i	$-0.723550\pi$
$523$	−29.5459	−1.29195	−0.645977	+	0.763357i	$0.723550\pi$
$524$	0	0
$525$	−1.96636	−0.0858189
$526$	0	0
$527$	−28.7730	−1.25337
$528$	0	0
$529$	−20.4615	−0.889630
$530$	0	0
$531$	−7.18654	−0.311869
$532$	0	0
$533$	−5.42784	−0.235106
$534$	0	0
$535$	−5.14431	−0.222408
$536$	0	0
$537$	−4.93840	−0.213108
$538$	0	0
$539$	40.4421	1.74197
$540$	0	0
$541$	−41.6691	−1.79150	−0.895748	−	0.444562i	$-0.853359\pi$
$541$	−41.6691	−1.79150	−0.895748	+	0.444562i	$0.853359\pi$
$542$	0	0
$543$	16.0000	0.686626
$544$	0	0
$545$	−4.16543	−0.178427
$546$	0	0
$547$	35.2710	1.50808	0.754040	−	0.656829i	$-0.228103\pi$
$547$	35.2710	1.50808	0.754040	+	0.656829i	$0.228103\pi$
$548$	0	0
$549$	5.34513	0.228125
$550$	0	0
$551$	−4.61439	−0.196579
$552$	0	0
$553$	5.55963	0.236419
$554$	0	0
$555$	3.80093	0.161340
$556$	0	0
$557$	−40.0268	−1.69599	−0.847995	−	0.530004i	$-0.822190\pi$
$557$	−40.0268	−1.69599	−0.847995	+	0.530004i	$0.822190\pi$
$558$	0	0
$559$	−57.1643	−2.41779
$560$	0	0
$561$	−23.1797	−0.978648
$562$	0	0
$563$	21.4210	0.902788	0.451394	−	0.892325i	$-0.350927\pi$
$563$	21.4210	0.902788	0.451394	+	0.892325i	$0.350927\pi$
$564$	0	0
$565$	3.41926	0.143849
$566$	0	0
$567$	−0.406728	−0.0170810
$568$	0	0
$569$	4.90301	0.205545	0.102772	−	0.994705i	$-0.467229\pi$
$569$	4.90301	0.205545	0.102772	+	0.994705i	$0.467229\pi$
$570$	0	0
$571$	−25.2961	−1.05861	−0.529304	−	0.848433i	$-0.677547\pi$
$571$	−25.2961	−1.05861	−0.529304	+	0.848433i	$0.677547\pi$
$572$	0	0
$573$	−21.5037	−0.898331
$574$	0	0
$575$	7.70279	0.321228
$576$	0	0
$577$	−22.5807	−0.940049	−0.470024	−	0.882653i	$-0.655755\pi$
$577$	−22.5807	−0.940049	−0.470024	+	0.882653i	$0.655755\pi$
$578$	0	0
$579$	21.3046	0.885391
$580$	0	0
$581$	5.46149	0.226581
$582$	0	0
$583$	23.6691	0.980276
$584$	0	0
$585$	2.20766	0.0912754
$586$	0	0
$587$	−4.62123	−0.190739	−0.0953693	−	0.995442i	$-0.530403\pi$
$587$	−4.62123	−0.190739	−0.0953693	+	0.995442i	$0.530403\pi$
$588$	0	0
$589$	−17.6777	−0.728398
$590$	0	0
$591$	−22.9316	−0.943278
$592$	0	0
$593$	27.5191	1.13008	0.565038	−	0.825065i	$-0.308862\pi$
$593$	27.5191	1.13008	0.565038	+	0.825065i	$0.308862\pi$
$594$	0	0
$595$	0.648029	0.0265666
$596$	0	0
$597$	0.572156	0.0234168
$598$	0	0
$599$	2.82204	0.115306	0.0576528	−	0.998337i	$-0.481638\pi$
$599$	2.82204	0.115306	0.0576528	+	0.998337i	$0.481638\pi$
$600$	0	0
$601$	−20.1095	−0.820284	−0.410142	−	0.912022i	$-0.634521\pi$
$601$	−20.1095	−0.820284	−0.410142	+	0.912022i	$0.634521\pi$
$602$	0	0
$603$	3.83457	0.156156
$604$	0	0
$605$	9.76729	0.397097
$606$	0	0
$607$	−18.2749	−0.741757	−0.370879	−	0.928681i	$-0.620943\pi$
$607$	−18.2749	−0.741757	−0.370879	+	0.928681i	$0.620943\pi$
$608$	0	0
$609$	0.779815	0.0315997
$610$	0	0
$611$	−0.448959	−0.0181629
$612$	0	0
$613$	−21.1729	−0.855164	−0.427582	−	0.903977i	$-0.640634\pi$
$613$	−21.1729	−0.855164	−0.427582	+	0.903977i	$0.640634\pi$
$614$	0	0
$615$	0.406728	0.0164009
$616$	0	0
$617$	1.15290	0.0464140	0.0232070	−	0.999731i	$-0.492612\pi$
$617$	1.15290	0.0464140	0.0232070	+	0.999731i	$0.492612\pi$
$618$	0	0
$619$	31.3451	1.25987	0.629934	−	0.776649i	$-0.283082\pi$
$619$	31.3451	1.25987	0.629934	+	0.776649i	$0.283082\pi$
$620$	0	0
$621$	1.59327	0.0639358
$622$	0	0
$623$	−0.151745	−0.00607953
$624$	0	0
$625$	22.5459	0.901838
$626$	0	0
$627$	−14.2413	−0.568743
$628$	0	0
$629$	36.6075	1.45964
$630$	0	0
$631$	8.48944	0.337959	0.168980	−	0.985620i	$-0.445953\pi$
$631$	8.48944	0.337959	0.168980	+	0.985620i	$0.445953\pi$
$632$	0	0
$633$	10.5248	0.418324
$634$	0	0
$635$	5.77866	0.229319
$636$	0	0
$637$	−37.0970	−1.46984
$638$	0	0
$639$	−7.10383	−0.281023
$640$	0	0
$641$	6.02680	0.238044	0.119022	−	0.992892i	$-0.462024\pi$
$641$	6.02680	0.238044	0.119022	+	0.992892i	$0.462024\pi$
$642$	0	0
$643$	6.44037	0.253983	0.126992	−	0.991904i	$-0.459468\pi$
$643$	6.44037	0.253983	0.126992	+	0.991904i	$0.459468\pi$
$644$	0	0
$645$	4.28353	0.168664
$646$	0	0
$647$	22.0086	0.865247	0.432623	−	0.901575i	$-0.357588\pi$
$647$	22.0086	0.865247	0.432623	+	0.901575i	$0.357588\pi$
$648$	0	0
$649$	42.5248	1.66925
$650$	0	0
$651$	2.98747	0.117088
$652$	0	0
$653$	−29.6709	−1.16111	−0.580556	−	0.814220i	$-0.697165\pi$
$653$	−29.6709	−1.16111	−0.580556	+	0.814220i	$0.697165\pi$
$654$	0	0
$655$	3.10558	0.121345
$656$	0	0
$657$	9.34513	0.364588
$658$	0	0
$659$	8.06728	0.314257	0.157128	−	0.987578i	$-0.449776\pi$
$659$	8.06728	0.314257	0.157128	+	0.987578i	$0.449776\pi$
$660$	0	0
$661$	0.274943	0.0106940	0.00534701	−	0.999986i	$-0.498298\pi$
$661$	0.274943	0.0106940	0.00534701	+	0.999986i	$0.498298\pi$
$662$	0	0
$663$	21.2624	0.825764
$664$	0	0
$665$	0.398140	0.0154392
$666$	0	0
$667$	−3.05476	−0.118281
$668$	0	0
$669$	−9.62691	−0.372198
$670$	0	0
$671$	−31.6287	−1.22101
$672$	0	0
$673$	−8.44037	−0.325352	−0.162676	−	0.986680i	$-0.552013\pi$
$673$	−8.44037	−0.325352	−0.162676	+	0.986680i	$0.552013\pi$
$674$	0	0
$675$	4.83457	0.186083
$676$	0	0
$677$	−12.7376	−0.489545	−0.244773	−	0.969581i	$-0.578713\pi$
$677$	−12.7376	−0.489545	−0.244773	+	0.969581i	$0.578713\pi$
$678$	0	0
$679$	−4.97888	−0.191072
$680$	0	0
$681$	18.2345	0.698746
$682$	0	0
$683$	5.04617	0.193086	0.0965431	−	0.995329i	$-0.469221\pi$
$683$	5.04617	0.193086	0.0965431	+	0.995329i	$0.469221\pi$
$684$	0	0
$685$	−6.27216	−0.239647
$686$	0	0
$687$	5.66056	0.215964
$688$	0	0
$689$	−21.7114	−0.827137
$690$	0	0
$691$	2.17402	0.0827035	0.0413517	−	0.999145i	$-0.486834\pi$
$691$	2.17402	0.0827035	0.0413517	+	0.999145i	$0.486834\pi$
$692$	0	0
$693$	2.40673	0.0914241
$694$	0	0
$695$	7.10208	0.269397
$696$	0	0
$697$	3.91729	0.148378
$698$	0	0
$699$	−14.4153	−0.545237
$700$	0	0
$701$	12.8557	0.485553	0.242776	−	0.970082i	$-0.421942\pi$
$701$	12.8557	0.485553	0.242776	+	0.970082i	$0.421942\pi$
$702$	0	0
$703$	22.4912	0.848271
$704$	0	0
$705$	0.0336421	0.00126704
$706$	0	0
$707$	5.92413	0.222800
$708$	0	0
$709$	38.7575	1.45557	0.727785	−	0.685805i	$-0.240550\pi$
$709$	38.7575	1.45557	0.727785	+	0.685805i	$0.240550\pi$
$710$	0	0
$711$	−13.6691	−0.512633
$712$	0	0
$713$	−11.7028	−0.438273
$714$	0	0
$715$	−13.0633	−0.488541
$716$	0	0
$717$	4.20766	0.157138
$718$	0	0
$719$	14.8403	0.553448	0.276724	−	0.960949i	$-0.410751\pi$
$719$	14.8403	0.553448	0.276724	+	0.960949i	$0.410751\pi$
$720$	0	0
$721$	−2.08956	−0.0778191
$722$	0	0
$723$	−1.67599	−0.0623306
$724$	0	0
$725$	−9.26926	−0.344252
$726$	0	0
$727$	−8.17402	−0.303157	−0.151579	−	0.988445i	$-0.548436\pi$
$727$	−8.17402	−0.303157	−0.151579	+	0.988445i	$0.548436\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	41.2556	1.52589
$732$	0	0
$733$	22.2567	0.822071	0.411036	−	0.911619i	$-0.365167\pi$
$733$	22.2567	0.822071	0.411036	+	0.911619i	$0.365167\pi$
$734$	0	0
$735$	2.77981	0.102535
$736$	0	0
$737$	−22.6903	−0.835806
$738$	0	0
$739$	2.20082	0.0809583	0.0404792	−	0.999180i	$-0.487112\pi$
$739$	2.20082	0.0809583	0.0404792	+	0.999180i	$0.487112\pi$
$740$	0	0
$741$	13.0633	0.479894
$742$	0	0
$743$	19.3520	0.709955	0.354977	−	0.934875i	$-0.384489\pi$
$743$	19.3520	0.709955	0.354977	+	0.934875i	$0.384489\pi$
$744$	0	0
$745$	5.14431	0.188473
$746$	0	0
$747$	−13.4278	−0.491299
$748$	0	0
$749$	5.14431	0.187969
$750$	0	0
$751$	−49.6691	−1.81245	−0.906226	−	0.422793i	$-0.861050\pi$
$751$	−49.6691	−1.81245	−0.906226	+	0.422793i	$0.861050\pi$
$752$	0	0
$753$	6.20766	0.226220
$754$	0	0
$755$	5.95777	0.216825
$756$	0	0
$757$	19.7028	0.716110	0.358055	−	0.933701i	$-0.383440\pi$
$757$	19.7028	0.716110	0.358055	+	0.933701i	$0.383440\pi$
$758$	0	0
$759$	−9.42784	−0.342209
$760$	0	0
$761$	16.2077	0.587527	0.293764	−	0.955878i	$-0.405092\pi$
$761$	16.2077	0.587527	0.293764	+	0.955878i	$0.405092\pi$
$762$	0	0
$763$	4.16543	0.150799
$764$	0	0
$765$	−1.59327	−0.0576049
$766$	0	0
$767$	−39.0074	−1.40848
$768$	0	0
$769$	−51.2082	−1.84662	−0.923308	−	0.384060i	$-0.874526\pi$
$769$	−51.2082	−1.84662	−0.923308	+	0.384060i	$0.874526\pi$
$770$	0	0
$771$	−25.3788	−0.913994
$772$	0	0
$773$	1.98457	0.0713800	0.0356900	−	0.999363i	$-0.488637\pi$
$773$	1.98457	0.0713800	0.0356900	+	0.999363i	$0.488637\pi$
$774$	0	0
$775$	−35.5106	−1.27558
$776$	0	0
$777$	−3.80093	−0.136358
$778$	0	0
$779$	2.40673	0.0862300
$780$	0	0
$781$	42.0354	1.50414
$782$	0	0
$783$	−1.91729	−0.0685182
$784$	0	0
$785$	−6.94015	−0.247704
$786$	0	0
$787$	33.5528	1.19603	0.598014	−	0.801486i	$-0.295957\pi$
$787$	33.5528	1.19603	0.598014	+	0.801486i	$0.295957\pi$
$788$	0	0
$789$	−24.8152	−0.883445
$790$	0	0
$791$	−3.41926	−0.121575
$792$	0	0
$793$	29.0125	1.03027
$794$	0	0
$795$	1.62691	0.0577007
$796$	0	0
$797$	−3.69420	−0.130855	−0.0654276	−	0.997857i	$-0.520841\pi$
$797$	−3.69420	−0.130855	−0.0654276	+	0.997857i	$0.520841\pi$
$798$	0	0
$799$	0.324014	0.0114628
$800$	0	0
$801$	0.373086	0.0131824
$802$	0	0
$803$	−55.2978	−1.95142
$804$	0	0
$805$	0.263572	0.00928968
$806$	0	0
$807$	−2.16543	−0.0762267
$808$	0	0
$809$	−8.77123	−0.308380	−0.154190	−	0.988041i	$-0.549277\pi$
$809$	−8.77123	−0.308380	−0.154190	+	0.988041i	$0.549277\pi$
$810$	0	0
$811$	−20.0913	−0.705501	−0.352751	−	0.935717i	$-0.614754\pi$
$811$	−20.0913	−0.705501	−0.352751	+	0.935717i	$0.614754\pi$
$812$	0	0
$813$	6.11636	0.214510
$814$	0	0
$815$	5.97773	0.209391
$816$	0	0
$817$	25.3469	0.886775
$818$	0	0
$819$	−2.20766	−0.0771418
$820$	0	0
$821$	17.1392	0.598163	0.299081	−	0.954228i	$-0.403320\pi$
$821$	17.1392	0.598163	0.299081	+	0.954228i	$0.403320\pi$
$822$	0	0
$823$	−17.7364	−0.618253	−0.309127	−	0.951021i	$-0.600037\pi$
$823$	−17.7364	−0.618253	−0.309127	+	0.951021i	$0.600037\pi$
$824$	0	0
$825$	−28.6075	−0.995987
$826$	0	0
$827$	11.1461	0.387586	0.193793	−	0.981042i	$-0.437921\pi$
$827$	11.1461	0.387586	0.193793	+	0.981042i	$0.437921\pi$
$828$	0	0
$829$	9.01427	0.313079	0.156539	−	0.987672i	$-0.449966\pi$
$829$	9.01427	0.313079	0.156539	+	0.987672i	$0.449966\pi$
$830$	0	0
$831$	−22.5739	−0.783080
$832$	0	0
$833$	26.7730	0.927629
$834$	0	0
$835$	1.80602	0.0625001
$836$	0	0
$837$	−7.34513	−0.253885
$838$	0	0
$839$	53.5882	1.85007	0.925035	−	0.379883i	$-0.124036\pi$
$839$	53.5882	1.85007	0.925035	+	0.379883i	$0.124036\pi$
$840$	0	0
$841$	−25.3240	−0.873242
$842$	0	0
$843$	27.3537	0.942112
$844$	0	0
$845$	6.69535	0.230327
$846$	0	0
$847$	−9.76729	−0.335608
$848$	0	0
$849$	−28.4758	−0.977286
$850$	0	0
$851$	14.8893	0.510400
$852$	0	0
$853$	−9.50372	−0.325401	−0.162701	−	0.986675i	$-0.552020\pi$
$853$	−9.50372	−0.325401	−0.162701	+	0.986675i	$0.552020\pi$
$854$	0	0
$855$	−0.978885	−0.0334772
$856$	0	0
$857$	24.2921	0.829803	0.414901	−	0.909866i	$-0.363816\pi$
$857$	24.2921	0.829803	0.414901	+	0.909866i	$0.363816\pi$
$858$	0	0
$859$	−42.5317	−1.45116	−0.725581	−	0.688137i	$-0.758429\pi$
$859$	−42.5317	−1.45116	−0.725581	+	0.688137i	$0.758429\pi$
$860$	0	0
$861$	−0.406728	−0.0138613
$862$	0	0
$863$	49.9949	1.70185	0.850923	−	0.525290i	$-0.176043\pi$
$863$	49.9949	1.70185	0.850923	+	0.525290i	$0.176043\pi$
$864$	0	0
$865$	6.86937	0.233566
$866$	0	0
$867$	1.65487	0.0562023
$868$	0	0
$869$	80.8842	2.74381
$870$	0	0
$871$	20.8135	0.705237
$872$	0	0
$873$	12.2413	0.414305
$874$	0	0
$875$	1.62691	0.0549997
$876$	0	0
$877$	43.8836	1.48185	0.740923	−	0.671590i	$-0.234388\pi$
$877$	43.8836	1.48185	0.740923	+	0.671590i	$0.234388\pi$
$878$	0	0
$879$	11.2133	0.378217
$880$	0	0
$881$	−33.0211	−1.11251	−0.556255	−	0.831012i	$-0.687762\pi$
$881$	−33.0211	−1.11251	−0.556255	+	0.831012i	$0.687762\pi$
$882$	0	0
$883$	−0.0336421	−0.00113215	−0.000566074	−	1.00000i	$-0.500180\pi$
$883$	−0.0336421	−0.00113215	−0.000566074		1.00000i	$0.500180\pi$
$884$	0	0
$885$	2.92297	0.0982546
$886$	0	0
$887$	−8.73074	−0.293150	−0.146575	−	0.989200i	$-0.546825\pi$
$887$	−8.73074	−0.293150	−0.146575	+	0.989200i	$0.546825\pi$
$888$	0	0
$889$	−5.77866	−0.193810
$890$	0	0
$891$	−5.91729	−0.198237
$892$	0	0
$893$	0.199070	0.00666163
$894$	0	0
$895$	2.00859	0.0671397
$896$	0	0
$897$	8.64803	0.288749
$898$	0	0
$899$	14.0827	0.469685
$900$	0	0
$901$	15.6691	0.522015
$902$	0	0
$903$	−4.28353	−0.142547
$904$	0	0
$905$	−6.50765	−0.216322
$906$	0	0
$907$	20.2499	0.672387	0.336193	−	0.941793i	$-0.390860\pi$
$907$	20.2499	0.672387	0.336193	+	0.941793i	$0.390860\pi$
$908$	0	0
$909$	−14.5653	−0.483101
$910$	0	0
$911$	47.7200	1.58103	0.790516	−	0.612441i	$-0.209812\pi$
$911$	47.7200	1.58103	0.790516	+	0.612441i	$0.209812\pi$
$912$	0	0
$913$	79.4564	2.62962
$914$	0	0
$915$	−2.17402	−0.0718708
$916$	0	0
$917$	−3.10558	−0.102555
$918$	0	0
$919$	−5.15290	−0.169979	−0.0849893	−	0.996382i	$-0.527086\pi$
$919$	−5.15290	−0.169979	−0.0849893	+	0.996382i	$0.527086\pi$
$920$	0	0
$921$	9.30290	0.306541
$922$	0	0
$923$	−38.5585	−1.26917
$924$	0	0
$925$	45.1797	1.48550
$926$	0	0
$927$	5.13747	0.168737
$928$	0	0
$929$	−5.72680	−0.187890	−0.0939452	−	0.995577i	$-0.529948\pi$
$929$	−5.72680	−0.187890	−0.0939452	+	0.995577i	$0.529948\pi$
$930$	0	0
$931$	16.4490	0.539093
$932$	0	0
$933$	17.8346	0.583878
$934$	0	0
$935$	9.42784	0.308323
$936$	0	0
$937$	−28.3594	−0.926461	−0.463231	−	0.886238i	$-0.653310\pi$
$937$	−28.3594	−0.926461	−0.463231	+	0.886238i	$0.653310\pi$
$938$	0	0
$939$	−3.92413	−0.128059
$940$	0	0
$941$	50.4998	1.64625	0.823123	−	0.567863i	$-0.192230\pi$
$941$	50.4998	1.64625	0.823123	+	0.567863i	$0.192230\pi$
$942$	0	0
$943$	1.59327	0.0518841
$944$	0	0
$945$	0.165428	0.00538137
$946$	0	0
$947$	37.9441	1.23302	0.616509	−	0.787348i	$-0.288547\pi$
$947$	37.9441	1.23302	0.616509	+	0.787348i	$0.288547\pi$
$948$	0	0
$949$	50.7239	1.64657
$950$	0	0
$951$	20.3577	0.660142
$952$	0	0
$953$	−50.8506	−1.64721	−0.823606	−	0.567162i	$-0.808041\pi$
$953$	−50.8506	−1.64721	−0.823606	+	0.567162i	$0.808041\pi$
$954$	0	0
$955$	8.74617	0.283019
$956$	0	0
$957$	11.3451	0.366736
$958$	0	0
$959$	6.27216	0.202539
$960$	0	0
$961$	22.9509	0.740353
$962$	0	0
$963$	−12.6480	−0.407577
$964$	0	0
$965$	−8.66521	−0.278943
$966$	0	0
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	0	0
$969$	−9.42784	−0.302866
$970$	0	0
$971$	−41.1711	−1.32124	−0.660622	−	0.750719i	$-0.729707\pi$
$971$	−41.1711	−1.32124	−0.660622	+	0.750719i	$0.729707\pi$
$972$	0	0
$973$	−7.10208	−0.227682
$974$	0	0
$975$	26.2413	0.840394
$976$	0	0
$977$	20.0827	0.642503	0.321251	−	0.946994i	$-0.395897\pi$
$977$	20.0827	0.642503	0.321251	+	0.946994i	$0.395897\pi$
$978$	0	0
$979$	−2.20766	−0.0705571
$980$	0	0
$981$	−10.2413	−0.326980
$982$	0	0
$983$	−42.8084	−1.36537	−0.682687	−	0.730711i	$-0.739189\pi$
$983$	−42.8084	−1.36537	−0.682687	+	0.730711i	$0.739189\pi$
$984$	0	0
$985$	9.32692	0.297180
$986$	0	0
$987$	−0.0336421	−0.00107084
$988$	0	0
$989$	16.7798	0.533567
$990$	0	0
$991$	−40.2921	−1.27992	−0.639961	−	0.768408i	$-0.721049\pi$
$991$	−40.2921	−1.27992	−0.639961	+	0.768408i	$0.721049\pi$
$992$	0	0
$993$	6.82204	0.216491
$994$	0	0
$995$	−0.232712	−0.00737748
$996$	0	0
$997$	−35.1392	−1.11287	−0.556435	−	0.830891i	$-0.687831\pi$
$997$	−35.1392	−1.11287	−0.556435	+	0.830891i	$0.687831\pi$
$998$	0	0
$999$	9.34513	0.295667

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	984.2.a.f.1.2	✓	3
3.2	odd	2		2952.2.a.n.1.2		3
4.3	odd	2		1968.2.a.u.1.2		3
8.3	odd	2		7872.2.a.bv.1.2		3
8.5	even	2		7872.2.a.ca.1.2		3
12.11	even	2		5904.2.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.2.a.f.1.2	✓	3	1.1	even	1	trivial
1968.2.a.u.1.2		3	4.3	odd	2
2952.2.a.n.1.2		3	3.2	odd	2
5904.2.a.bj.1.2		3	12.11	even	2
7872.2.a.bv.1.2		3	8.3	odd	2
7872.2.a.ca.1.2		3	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:	\( N \)	\(=\)	\( 984 = 2^{3} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	984.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.406728 q^{5} -0.406728 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.406728 q^{5} -0.406728 q^{7} +1.00000 q^{9} -5.91729 q^{11} +5.42784 q^{13} -0.406728 q^{15} -3.91729 q^{17} -2.40673 q^{19} +0.406728 q^{21} -1.59327 q^{23} -4.83457 q^{25} -1.00000 q^{27} +1.91729 q^{29} +7.34513 q^{31} +5.91729 q^{33} -0.165428 q^{35} -9.34513 q^{37} -5.42784 q^{39} -1.00000 q^{41} -10.5317 q^{43} +0.406728 q^{45} -0.0827140 q^{47} -6.83457 q^{49} +3.91729 q^{51} -4.00000 q^{53} -2.40673 q^{55} +2.40673 q^{57} -7.18654 q^{59} +5.34513 q^{61} -0.406728 q^{63} +2.20766 q^{65} +3.83457 q^{67} +1.59327 q^{69} -7.10383 q^{71} +9.34513 q^{73} +4.83457 q^{75} +2.40673 q^{77} -13.6691 q^{79} +1.00000 q^{81} -13.4278 q^{83} -1.59327 q^{85} -1.91729 q^{87} +0.373086 q^{89} -2.20766 q^{91} -7.34513 q^{93} -0.978885 q^{95} +12.2413 q^{97} -5.91729 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{9} - 8 q^{11} - 2 q^{13} - 2 q^{17} - 6 q^{19} - 6 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 6 q^{31} + 8 q^{33} - 20 q^{35} + 2 q^{39} - 3 q^{41} - 6 q^{43} - 10 q^{47} - q^{49} + 2 q^{51} - 12 q^{53} - 6 q^{55} + 6 q^{57} - 24 q^{59} - 12 q^{61} - 8 q^{65} - 8 q^{67} + 6 q^{69} - 14 q^{71} - 5 q^{75} + 6 q^{77} - 2 q^{79} + 3 q^{81} - 22 q^{83} - 6 q^{85} + 4 q^{87} + 6 q^{89} + 8 q^{91} + 6 q^{93} - 20 q^{95} + 16 q^{97} - 8 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^9 - 8 * q^11 - 2 * q^13 - 2 * q^17 - 6 * q^19 - 6 * q^23 + 5 * q^25 - 3 * q^27 - 4 * q^29 - 6 * q^31 + 8 * q^33 - 20 * q^35 + 2 * q^39 - 3 * q^41 - 6 * q^43 - 10 * q^47 - q^49 + 2 * q^51 - 12 * q^53 - 6 * q^55 + 6 * q^57 - 24 * q^59 - 12 * q^61 - 8 * q^65 - 8 * q^67 + 6 * q^69 - 14 * q^71 - 5 * q^75 + 6 * q^77 - 2 * q^79 + 3 * q^81 - 22 * q^83 - 6 * q^85 + 4 * q^87 + 6 * q^89 + 8 * q^91 + 6 * q^93 - 20 * q^95 + 16 * q^97 - 8 * q^99

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