LMFDB - Embedded newform 384.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.87298$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -19.4919 q^{5} +17.4919 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -19.4919 q^{5} +17.4919 q^{7} +9.00000 q^{9} -50.9839 q^{11} -2.00000 q^{13} +58.4758 q^{15} -78.9516 q^{17} -25.0161 q^{19} -52.4758 q^{21} -120.952 q^{23} +254.935 q^{25} -27.0000 q^{27} +273.395 q^{29} +29.5565 q^{31} +152.952 q^{33} -340.952 q^{35} +320.984 q^{37} +6.00000 q^{39} +263.016 q^{41} +388.823 q^{43} -175.427 q^{45} +325.016 q^{47} -37.0323 q^{49} +236.855 q^{51} +567.492 q^{53} +993.774 q^{55} +75.0484 q^{57} -639.806 q^{59} -176.790 q^{61} +157.427 q^{63} +38.9839 q^{65} -1015.55 q^{67} +362.855 q^{69} +15.0484 q^{71} -623.581 q^{73} -764.806 q^{75} -891.806 q^{77} +305.621 q^{79} +81.0000 q^{81} +636.500 q^{83} +1538.92 q^{85} -820.185 q^{87} +1463.84 q^{89} -34.9839 q^{91} -88.6694 q^{93} +487.613 q^{95} -640.161 q^{97} -458.855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 8 q^{5} + 4 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 8 * q^5 + 4 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} - 8 q^{5} + 4 q^{7} + 18 q^{9} - 40 q^{11} - 4 q^{13} + 24 q^{15} + 28 q^{17} - 112 q^{19} - 12 q^{21} - 56 q^{23} + 262 q^{25} - 54 q^{27} + 144 q^{29} + 276 q^{31} + 120 q^{33} - 496 q^{35} + 580 q^{37} + 12 q^{39} + 588 q^{41} + 96 q^{43} - 72 q^{45} + 712 q^{47} - 198 q^{49} - 84 q^{51} + 1104 q^{53} + 1120 q^{55} + 336 q^{57} - 536 q^{59} + 452 q^{61} + 36 q^{63} + 16 q^{65} - 296 q^{67} + 168 q^{69} + 216 q^{71} + 364 q^{73} - 786 q^{75} - 1040 q^{77} + 1076 q^{79} + 162 q^{81} - 648 q^{83} + 2768 q^{85} - 432 q^{87} + 2308 q^{89} - 8 q^{91} - 828 q^{93} - 512 q^{95} - 1900 q^{97} - 360 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 8 * q^5 + 4 * q^7 + 18 * q^9 - 40 * q^11 - 4 * q^13 + 24 * q^15 + 28 * q^17 - 112 * q^19 - 12 * q^21 - 56 * q^23 + 262 * q^25 - 54 * q^27 + 144 * q^29 + 276 * q^31 + 120 * q^33 - 496 * q^35 + 580 * q^37 + 12 * q^39 + 588 * q^41 + 96 * q^43 - 72 * q^45 + 712 * q^47 - 198 * q^49 - 84 * q^51 + 1104 * q^53 + 1120 * q^55 + 336 * q^57 - 536 * q^59 + 452 * q^61 + 36 * q^63 + 16 * q^65 - 296 * q^67 + 168 * q^69 + 216 * q^71 + 364 * q^73 - 786 * q^75 - 1040 * q^77 + 1076 * q^79 + 162 * q^81 - 648 * q^83 + 2768 * q^85 - 432 * q^87 + 2308 * q^89 - 8 * q^91 - 828 * q^93 - 512 * q^95 - 1900 * q^97 - 360 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−19.4919	−1.74341	−0.871706	−	0.490030i	$-0.836986\pi$
$5$	−19.4919	−1.74341	−0.871706	+	0.490030i	$0.836986\pi$
$6$	0	0
$7$	17.4919	0.944476	0.472238	−	0.881471i	$-0.343446\pi$
$7$	17.4919	0.944476	0.472238	+	0.881471i	$0.343446\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−50.9839	−1.39747	−0.698737	−	0.715379i	$-0.746254\pi$
$11$	−50.9839	−1.39747	−0.698737	+	0.715379i	$0.746254\pi$
$12$	0	0
$13$	−2.00000	−0.0426692	−0.0213346	−	0.999772i	$-0.506792\pi$
$13$	−2.00000	−0.0426692	−0.0213346	+	0.999772i	$0.506792\pi$
$14$	0	0
$15$	58.4758	1.00656
$16$	0	0
$17$	−78.9516	−1.12639	−0.563193	−	0.826325i	$-0.690427\pi$
$17$	−78.9516	−1.12639	−0.563193	+	0.826325i	$0.690427\pi$
$18$	0	0
$19$	−25.0161	−0.302058	−0.151029	−	0.988529i	$-0.548259\pi$
$19$	−25.0161	−0.302058	−0.151029	+	0.988529i	$0.548259\pi$
$20$	0	0
$21$	−52.4758	−0.545293
$22$	0	0
$23$	−120.952	−1.09653	−0.548264	−	0.836305i	$-0.684711\pi$
$23$	−120.952	−1.09653	−0.548264	+	0.836305i	$0.684711\pi$
$24$	0	0
$25$	254.935	2.03948
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	273.395	1.75063	0.875314	−	0.483555i	$-0.160655\pi$
$29$	273.395	1.75063	0.875314	+	0.483555i	$0.160655\pi$
$30$	0	0
$31$	29.5565	0.171242	0.0856209	−	0.996328i	$-0.472713\pi$
$31$	29.5565	0.171242	0.0856209	+	0.996328i	$0.472713\pi$
$32$	0	0
$33$	152.952	0.806832
$34$	0	0
$35$	−340.952	−1.64661
$36$	0	0
$37$	320.984	1.42620	0.713100	−	0.701062i	$-0.247290\pi$
$37$	320.984	1.42620	0.713100	+	0.701062i	$0.247290\pi$
$38$	0	0
$39$	6.00000	0.0246351
$40$	0	0
$41$	263.016	1.00186	0.500929	−	0.865488i	$-0.332992\pi$
$41$	263.016	1.00186	0.500929	+	0.865488i	$0.332992\pi$
$42$	0	0
$43$	388.823	1.37895	0.689475	−	0.724309i	$-0.257841\pi$
$43$	388.823	1.37895	0.689475	+	0.724309i	$0.257841\pi$
$44$	0	0
$45$	−175.427	−0.581137
$46$	0	0
$47$	325.016	1.00869	0.504345	−	0.863502i	$-0.331734\pi$
$47$	325.016	1.00869	0.504345	+	0.863502i	$0.331734\pi$
$48$	0	0
$49$	−37.0323	−0.107966
$50$	0	0
$51$	236.855	0.650320
$52$	0	0
$53$	567.492	1.47077	0.735387	−	0.677647i	$-0.237000\pi$
$53$	567.492	1.47077	0.735387	+	0.677647i	$0.237000\pi$
$54$	0	0
$55$	993.774	2.43637
$56$	0	0
$57$	75.0484	0.174393
$58$	0	0
$59$	−639.806	−1.41179	−0.705896	−	0.708316i	$-0.749455\pi$
$59$	−639.806	−1.41179	−0.705896	+	0.708316i	$0.749455\pi$
$60$	0	0
$61$	−176.790	−0.371077	−0.185538	−	0.982637i	$-0.559403\pi$
$61$	−176.790	−0.371077	−0.185538	+	0.982637i	$0.559403\pi$
$62$	0	0
$63$	157.427	0.314825
$64$	0	0
$65$	38.9839	0.0743901
$66$	0	0
$67$	−1015.55	−1.85178	−0.925888	−	0.377799i	$-0.876681\pi$
$67$	−1015.55	−1.85178	−0.925888	+	0.377799i	$0.876681\pi$
$68$	0	0
$69$	362.855	0.633081
$70$	0	0
$71$	15.0484	0.0251538	0.0125769	−	0.999921i	$-0.495997\pi$
$71$	15.0484	0.0251538	0.0125769	+	0.999921i	$0.495997\pi$
$72$	0	0
$73$	−623.581	−0.999789	−0.499894	−	0.866086i	$-0.666628\pi$
$73$	−623.581	−0.999789	−0.499894	+	0.866086i	$0.666628\pi$
$74$	0	0
$75$	−764.806	−1.17750
$76$	0	0
$77$	−891.806	−1.31988
$78$	0	0
$79$	305.621	0.435254	0.217627	−	0.976032i	$-0.430168\pi$
$79$	305.621	0.435254	0.217627	+	0.976032i	$0.430168\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	636.500	0.841746	0.420873	−	0.907120i	$-0.361724\pi$
$83$	636.500	0.841746	0.420873	+	0.907120i	$0.361724\pi$
$84$	0	0
$85$	1538.92	1.96376
$86$	0	0
$87$	−820.185	−1.01073
$88$	0	0
$89$	1463.84	1.74344	0.871722	−	0.490000i	$-0.163003\pi$
$89$	1463.84	1.74344	0.871722	+	0.490000i	$0.163003\pi$
$90$	0	0
$91$	−34.9839	−0.0403001
$92$	0	0
$93$	−88.6694	−0.0988665
$94$	0	0
$95$	487.613	0.526611
$96$	0	0
$97$	−640.161	−0.670088	−0.335044	−	0.942202i	$-0.608751\pi$
$97$	−640.161	−0.670088	−0.335044	+	0.942202i	$0.608751\pi$
$98$	0	0
$99$	−458.855	−0.465825
$100$	0	0
$101$	−290.411	−0.286109	−0.143054	−	0.989715i	$-0.545692\pi$
$101$	−290.411	−0.286109	−0.143054	+	0.989715i	$0.545692\pi$
$102$	0	0
$103$	652.605	0.624302	0.312151	−	0.950033i	$-0.398951\pi$
$103$	652.605	0.624302	0.312151	+	0.950033i	$0.398951\pi$
$104$	0	0
$105$	1022.85	0.950671
$106$	0	0
$107$	1157.39	1.04569	0.522845	−	0.852428i	$-0.324871\pi$
$107$	1157.39	1.04569	0.522845	+	0.852428i	$0.324871\pi$
$108$	0	0
$109$	1019.32	0.895719	0.447860	−	0.894104i	$-0.352186\pi$
$109$	1019.32	0.895719	0.447860	+	0.894104i	$0.352186\pi$
$110$	0	0
$111$	−962.952	−0.823417
$112$	0	0
$113$	207.710	0.172917	0.0864587	−	0.996255i	$-0.472445\pi$
$113$	207.710	0.172917	0.0864587	+	0.996255i	$0.472445\pi$
$114$	0	0
$115$	2357.58	1.91170
$116$	0	0
$117$	−18.0000	−0.0142231
$118$	0	0
$119$	−1381.02	−1.06384
$120$	0	0
$121$	1268.35	0.952934
$122$	0	0
$123$	−789.048	−0.578424
$124$	0	0
$125$	−2532.69	−1.81225
$126$	0	0
$127$	−2174.94	−1.51965	−0.759823	−	0.650130i	$-0.774714\pi$
$127$	−2174.94	−1.51965	−0.759823	+	0.650130i	$0.774714\pi$
$128$	0	0
$129$	−1166.47	−0.796138
$130$	0	0
$131$	−1283.61	−0.856105	−0.428053	−	0.903754i	$-0.640800\pi$
$131$	−1283.61	−0.856105	−0.428053	+	0.903754i	$0.640800\pi$
$132$	0	0
$133$	−437.581	−0.285286
$134$	0	0
$135$	526.282	0.335520
$136$	0	0
$137$	1796.60	1.12039	0.560196	−	0.828360i	$-0.310726\pi$
$137$	1796.60	1.12039	0.560196	+	0.828360i	$0.310726\pi$
$138$	0	0
$139$	392.323	0.239398	0.119699	−	0.992810i	$-0.461807\pi$
$139$	392.323	0.239398	0.119699	+	0.992810i	$0.461807\pi$
$140$	0	0
$141$	−975.048	−0.582368
$142$	0	0
$143$	101.968	0.0596292
$144$	0	0
$145$	−5329.00	−3.05206
$146$	0	0
$147$	111.097	0.0623341
$148$	0	0
$149$	−279.298	−0.153564	−0.0767819	−	0.997048i	$-0.524465\pi$
$149$	−279.298	−0.153564	−0.0767819	+	0.997048i	$0.524465\pi$
$150$	0	0
$151$	979.395	0.527828	0.263914	−	0.964546i	$-0.414986\pi$
$151$	979.395	0.527828	0.263914	+	0.964546i	$0.414986\pi$
$152$	0	0
$153$	−710.564	−0.375462
$154$	0	0
$155$	−576.113	−0.298545
$156$	0	0
$157$	344.855	0.175302	0.0876510	−	0.996151i	$-0.472064\pi$
$157$	344.855	0.175302	0.0876510	+	0.996151i	$0.472064\pi$
$158$	0	0
$159$	−1702.48	−0.849152
$160$	0	0
$161$	−2115.68	−1.03564
$162$	0	0
$163$	3424.89	1.64575	0.822877	−	0.568220i	$-0.192368\pi$
$163$	3424.89	1.64575	0.822877	+	0.568220i	$0.192368\pi$
$164$	0	0
$165$	−2981.32	−1.40664
$166$	0	0
$167$	−256.774	−0.118981	−0.0594904	−	0.998229i	$-0.518948\pi$
$167$	−256.774	−0.118981	−0.0594904	+	0.998229i	$0.518948\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	0	0
$171$	−225.145	−0.100686
$172$	0	0
$173$	−616.073	−0.270746	−0.135373	−	0.990795i	$-0.543223\pi$
$173$	−616.073	−0.270746	−0.135373	+	0.990795i	$0.543223\pi$
$174$	0	0
$175$	4459.31	1.92624
$176$	0	0
$177$	1919.42	0.815098
$178$	0	0
$179$	−3582.90	−1.49608	−0.748041	−	0.663652i	$-0.769005\pi$
$179$	−3582.90	−1.49608	−0.748041	+	0.663652i	$0.769005\pi$
$180$	0	0
$181$	2615.77	1.07419	0.537097	−	0.843521i	$-0.319521\pi$
$181$	2615.77	1.07419	0.537097	+	0.843521i	$0.319521\pi$
$182$	0	0
$183$	530.371	0.214241
$184$	0	0
$185$	−6256.60	−2.48645
$186$	0	0
$187$	4025.26	1.57410
$188$	0	0
$189$	−472.282	−0.181764
$190$	0	0
$191$	−4250.52	−1.61024	−0.805122	−	0.593110i	$-0.797900\pi$
$191$	−4250.52	−1.61024	−0.805122	+	0.593110i	$0.797900\pi$
$192$	0	0
$193$	−213.065	−0.0794650	−0.0397325	−	0.999210i	$-0.512651\pi$
$193$	−213.065	−0.0794650	−0.0397325	+	0.999210i	$0.512651\pi$
$194$	0	0
$195$	−116.952	−0.0429491
$196$	0	0
$197$	528.234	0.191041	0.0955205	−	0.995427i	$-0.469548\pi$
$197$	528.234	0.191041	0.0955205	+	0.995427i	$0.469548\pi$
$198$	0	0
$199$	−2043.88	−0.728074	−0.364037	−	0.931384i	$-0.618602\pi$
$199$	−2043.88	−0.728074	−0.364037	+	0.931384i	$0.618602\pi$
$200$	0	0
$201$	3046.64	1.06912
$202$	0	0
$203$	4782.21	1.65343
$204$	0	0
$205$	−5126.69	−1.74665
$206$	0	0
$207$	−1088.56	−0.365509
$208$	0	0
$209$	1275.42	0.422118
$210$	0	0
$211$	3955.35	1.29051	0.645256	−	0.763967i	$-0.276751\pi$
$211$	3955.35	1.29051	0.645256	+	0.763967i	$0.276751\pi$
$212$	0	0
$213$	−45.1452	−0.0145225
$214$	0	0
$215$	−7578.90	−2.40408
$216$	0	0
$217$	517.000	0.161734
$218$	0	0
$219$	1870.74	0.577228
$220$	0	0
$221$	157.903	0.0480621
$222$	0	0
$223$	3661.96	1.09965	0.549827	−	0.835278i	$-0.314694\pi$
$223$	3661.96	1.09965	0.549827	+	0.835278i	$0.314694\pi$
$224$	0	0
$225$	2294.42	0.679828
$226$	0	0
$227$	117.371	0.0343179	0.0171589	−	0.999853i	$-0.494538\pi$
$227$	117.371	0.0343179	0.0171589	+	0.999853i	$0.494538\pi$
$228$	0	0
$229$	2292.84	0.661638	0.330819	−	0.943694i	$-0.392675\pi$
$229$	2292.84	0.661638	0.330819	+	0.943694i	$0.392675\pi$
$230$	0	0
$231$	2675.42	0.762033
$232$	0	0
$233$	3186.61	0.895974	0.447987	−	0.894040i	$-0.352141\pi$
$233$	3186.61	0.895974	0.447987	+	0.894040i	$0.352141\pi$
$234$	0	0
$235$	−6335.19	−1.75856
$236$	0	0
$237$	−916.863	−0.251294
$238$	0	0
$239$	2879.00	0.779193	0.389596	−	0.920986i	$-0.372615\pi$
$239$	2879.00	0.779193	0.389596	+	0.920986i	$0.372615\pi$
$240$	0	0
$241$	2781.23	0.743380	0.371690	−	0.928357i	$-0.378778\pi$
$241$	2781.23	0.743380	0.371690	+	0.928357i	$0.378778\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	721.830	0.188229
$246$	0	0
$247$	50.0323	0.0128886
$248$	0	0
$249$	−1909.50	−0.485982
$250$	0	0
$251$	1373.02	0.345275	0.172637	−	0.984985i	$-0.444771\pi$
$251$	1373.02	0.345275	0.172637	+	0.984985i	$0.444771\pi$
$252$	0	0
$253$	6166.58	1.53237
$254$	0	0
$255$	−4616.76	−1.13377
$256$	0	0
$257$	1614.97	0.391980	0.195990	−	0.980606i	$-0.437208\pi$
$257$	1614.97	0.391980	0.195990	+	0.980606i	$0.437208\pi$
$258$	0	0
$259$	5614.63	1.34701
$260$	0	0
$261$	2460.56	0.583543
$262$	0	0
$263$	−3527.23	−0.826989	−0.413494	−	0.910507i	$-0.635692\pi$
$263$	−3527.23	−0.826989	−0.413494	+	0.910507i	$0.635692\pi$
$264$	0	0
$265$	−11061.5	−2.56416
$266$	0	0
$267$	−4391.52	−1.00658
$268$	0	0
$269$	1651.69	0.374368	0.187184	−	0.982325i	$-0.440064\pi$
$269$	1651.69	0.374368	0.187184	+	0.982325i	$0.440064\pi$
$270$	0	0
$271$	286.283	0.0641714	0.0320857	−	0.999485i	$-0.489785\pi$
$271$	286.283	0.0641714	0.0320857	+	0.999485i	$0.489785\pi$
$272$	0	0
$273$	104.952	0.0232673
$274$	0	0
$275$	−12997.6	−2.85013
$276$	0	0
$277$	−6525.77	−1.41551	−0.707754	−	0.706459i	$-0.750292\pi$
$277$	−6525.77	−1.41551	−0.707754	+	0.706459i	$0.750292\pi$
$278$	0	0
$279$	266.008	0.0570806
$280$	0	0
$281$	2392.84	0.507989	0.253994	−	0.967206i	$-0.418256\pi$
$281$	2392.84	0.507989	0.253994	+	0.967206i	$0.418256\pi$
$282$	0	0
$283$	−4151.81	−0.872082	−0.436041	−	0.899927i	$-0.643620\pi$
$283$	−4151.81	−0.872082	−0.436041	+	0.899927i	$0.643620\pi$
$284$	0	0
$285$	−1462.84	−0.304039
$286$	0	0
$287$	4600.66	0.946231
$288$	0	0
$289$	1320.36	0.268747
$290$	0	0
$291$	1920.48	0.386875
$292$	0	0
$293$	−9400.96	−1.87444	−0.937218	−	0.348743i	$-0.886608\pi$
$293$	−9400.96	−1.87444	−0.937218	+	0.348743i	$0.886608\pi$
$294$	0	0
$295$	12471.1	2.46133
$296$	0	0
$297$	1376.56	0.268944
$298$	0	0
$299$	241.903	0.0467880
$300$	0	0
$301$	6801.26	1.30239
$302$	0	0
$303$	871.234	0.165185
$304$	0	0
$305$	3445.98	0.646939
$306$	0	0
$307$	−9748.19	−1.81224	−0.906122	−	0.423017i	$-0.860971\pi$
$307$	−9748.19	−1.81224	−0.906122	+	0.423017i	$0.860971\pi$
$308$	0	0
$309$	−1957.81	−0.360441
$310$	0	0
$311$	9601.71	1.75068	0.875342	−	0.483504i	$-0.160636\pi$
$311$	9601.71	1.75068	0.875342	+	0.483504i	$0.160636\pi$
$312$	0	0
$313$	10055.5	1.81587	0.907936	−	0.419110i	$-0.137658\pi$
$313$	10055.5	1.81587	0.907936	+	0.419110i	$0.137658\pi$
$314$	0	0
$315$	−3068.56	−0.548870
$316$	0	0
$317$	4812.94	0.852750	0.426375	−	0.904546i	$-0.359790\pi$
$317$	4812.94	0.852750	0.426375	+	0.904546i	$0.359790\pi$
$318$	0	0
$319$	−13938.7	−2.44646
$320$	0	0
$321$	−3472.16	−0.603729
$322$	0	0
$323$	1975.06	0.340234
$324$	0	0
$325$	−509.871	−0.0870232
$326$	0	0
$327$	−3057.97	−0.517144
$328$	0	0
$329$	5685.16	0.952684
$330$	0	0
$331$	2638.74	0.438182	0.219091	−	0.975704i	$-0.429691\pi$
$331$	2638.74	0.438182	0.219091	+	0.975704i	$0.429691\pi$
$332$	0	0
$333$	2888.85	0.475400
$334$	0	0
$335$	19795.0	3.22841
$336$	0	0
$337$	2937.19	0.474775	0.237387	−	0.971415i	$-0.423709\pi$
$337$	2937.19	0.474775	0.237387	+	0.971415i	$0.423709\pi$
$338$	0	0
$339$	−623.129	−0.0998340
$340$	0	0
$341$	−1506.90	−0.239306
$342$	0	0
$343$	−6647.50	−1.04645
$344$	0	0
$345$	−7072.74	−1.10372
$346$	0	0
$347$	7594.56	1.17492	0.587460	−	0.809253i	$-0.300128\pi$
$347$	7594.56	1.17492	0.587460	+	0.809253i	$0.300128\pi$
$348$	0	0
$349$	7322.11	1.12305	0.561524	−	0.827460i	$-0.310215\pi$
$349$	7322.11	1.12305	0.561524	+	0.827460i	$0.310215\pi$
$350$	0	0
$351$	54.0000	0.00821170
$352$	0	0
$353$	3699.26	0.557767	0.278883	−	0.960325i	$-0.410036\pi$
$353$	3699.26	0.557767	0.278883	+	0.960325i	$0.410036\pi$
$354$	0	0
$355$	−293.322	−0.0438533
$356$	0	0
$357$	4143.05	0.614211
$358$	0	0
$359$	−3.98361	−0.000585645 0	−0.000292823	−	1.00000i	$-0.500093\pi$
$359$	−3.98361	−0.000585645 0	−0.000292823		1.00000i	$0.500093\pi$
$360$	0	0
$361$	−6233.19	−0.908761
$362$	0	0
$363$	−3805.06	−0.550176
$364$	0	0
$365$	12154.8	1.74304
$366$	0	0
$367$	4813.30	0.684611	0.342305	−	0.939589i	$-0.388792\pi$
$367$	4813.30	0.684611	0.342305	+	0.939589i	$0.388792\pi$
$368$	0	0
$369$	2367.15	0.333953
$370$	0	0
$371$	9926.53	1.38911
$372$	0	0
$373$	3059.21	0.424664	0.212332	−	0.977198i	$-0.431894\pi$
$373$	3059.21	0.424664	0.212332	+	0.977198i	$0.431894\pi$
$374$	0	0
$375$	7598.08	1.04630
$376$	0	0
$377$	−546.790	−0.0746980
$378$	0	0
$379$	−2854.37	−0.386858	−0.193429	−	0.981114i	$-0.561961\pi$
$379$	−2854.37	−0.386858	−0.193429	+	0.981114i	$0.561961\pi$
$380$	0	0
$381$	6524.83	0.877368
$382$	0	0
$383$	−10104.5	−1.34808	−0.674038	−	0.738696i	$-0.735442\pi$
$383$	−10104.5	−1.34808	−0.674038	+	0.738696i	$0.735442\pi$
$384$	0	0
$385$	17383.0	2.30109
$386$	0	0
$387$	3499.40	0.459650
$388$	0	0
$389$	2925.77	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$389$	2925.77	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$390$	0	0
$391$	9549.32	1.23512
$392$	0	0
$393$	3850.84	0.494273
$394$	0	0
$395$	−5957.14	−0.758826
$396$	0	0
$397$	7099.72	0.897544	0.448772	−	0.893646i	$-0.351862\pi$
$397$	7099.72	0.897544	0.448772	+	0.893646i	$0.351862\pi$
$398$	0	0
$399$	1312.74	0.164710
$400$	0	0
$401$	2397.08	0.298515	0.149257	−	0.988798i	$-0.452312\pi$
$401$	2397.08	0.298515	0.149257	+	0.988798i	$0.452312\pi$
$402$	0	0
$403$	−59.1129	−0.00730676
$404$	0	0
$405$	−1578.85	−0.193712
$406$	0	0
$407$	−16365.0	−1.99308
$408$	0	0
$409$	2635.65	0.318641	0.159321	−	0.987227i	$-0.449070\pi$
$409$	2635.65	0.318641	0.159321	+	0.987227i	$0.449070\pi$
$410$	0	0
$411$	−5389.79	−0.646858
$412$	0	0
$413$	−11191.5	−1.33340
$414$	0	0
$415$	−12406.6	−1.46751
$416$	0	0
$417$	−1176.97	−0.138217
$418$	0	0
$419$	−7889.14	−0.919833	−0.459916	−	0.887962i	$-0.652121\pi$
$419$	−7889.14	−0.919833	−0.459916	+	0.887962i	$0.652121\pi$
$420$	0	0
$421$	1532.10	0.177363	0.0886815	−	0.996060i	$-0.471735\pi$
$421$	1532.10	0.177363	0.0886815	+	0.996060i	$0.471735\pi$
$422$	0	0
$423$	2925.15	0.336230
$424$	0	0
$425$	−20127.6	−2.29725
$426$	0	0
$427$	−3092.40	−0.350473
$428$	0	0
$429$	−305.903	−0.0344269
$430$	0	0
$431$	11583.6	1.29457	0.647286	−	0.762247i	$-0.275904\pi$
$431$	11583.6	1.29457	0.647286	+	0.762247i	$0.275904\pi$
$432$	0	0
$433$	−8032.71	−0.891518	−0.445759	−	0.895153i	$-0.647066\pi$
$433$	−8032.71	−0.891518	−0.445759	+	0.895153i	$0.647066\pi$
$434$	0	0
$435$	15987.0	1.76211
$436$	0	0
$437$	3025.74	0.331215
$438$	0	0
$439$	2683.51	0.291747	0.145873	−	0.989303i	$-0.453401\pi$
$439$	2683.51	0.291747	0.145873	+	0.989303i	$0.453401\pi$
$440$	0	0
$441$	−333.290	−0.0359886
$442$	0	0
$443$	−390.854	−0.0419188	−0.0209594	−	0.999780i	$-0.506672\pi$
$443$	−390.854	−0.0419188	−0.0209594	+	0.999780i	$0.506672\pi$
$444$	0	0
$445$	−28533.0	−3.03954
$446$	0	0
$447$	837.895	0.0886601
$448$	0	0
$449$	2456.92	0.258239	0.129119	−	0.991629i	$-0.458785\pi$
$449$	2456.92	0.258239	0.129119	+	0.991629i	$0.458785\pi$
$450$	0	0
$451$	−13409.6	−1.40007
$452$	0	0
$453$	−2938.19	−0.304742
$454$	0	0
$455$	681.903	0.0702596
$456$	0	0
$457$	7544.35	0.772232	0.386116	−	0.922450i	$-0.373816\pi$
$457$	7544.35	0.772232	0.386116	+	0.922450i	$0.373816\pi$
$458$	0	0
$459$	2131.69	0.216773
$460$	0	0
$461$	4702.31	0.475073	0.237536	−	0.971379i	$-0.423660\pi$
$461$	4702.31	0.475073	0.237536	+	0.971379i	$0.423660\pi$
$462$	0	0
$463$	−6643.88	−0.666884	−0.333442	−	0.942771i	$-0.608210\pi$
$463$	−6643.88	−0.666884	−0.333442	+	0.942771i	$0.608210\pi$
$464$	0	0
$465$	1728.34	0.172365
$466$	0	0
$467$	−4279.08	−0.424009	−0.212005	−	0.977269i	$-0.567999\pi$
$467$	−4279.08	−0.424009	−0.212005	+	0.977269i	$0.567999\pi$
$468$	0	0
$469$	−17763.9	−1.74896
$470$	0	0
$471$	−1034.56	−0.101211
$472$	0	0
$473$	−19823.7	−1.92705
$474$	0	0
$475$	−6377.50	−0.616042
$476$	0	0
$477$	5107.43	0.490258
$478$	0	0
$479$	8791.69	0.838628	0.419314	−	0.907841i	$-0.362271\pi$
$479$	8791.69	0.838628	0.419314	+	0.907841i	$0.362271\pi$
$480$	0	0
$481$	−641.968	−0.0608549
$482$	0	0
$483$	6347.03	0.597930
$484$	0	0
$485$	12478.0	1.16824
$486$	0	0
$487$	6764.07	0.629383	0.314691	−	0.949194i	$-0.398099\pi$
$487$	6764.07	0.629383	0.314691	+	0.949194i	$0.398099\pi$
$488$	0	0
$489$	−10274.7	−0.950176
$490$	0	0
$491$	19170.7	1.76204	0.881019	−	0.473081i	$-0.156858\pi$
$491$	19170.7	1.76204	0.881019	+	0.473081i	$0.156858\pi$
$492$	0	0
$493$	−21585.0	−1.97188
$494$	0	0
$495$	8943.97	0.812124
$496$	0	0
$497$	263.226	0.0237571
$498$	0	0
$499$	4885.48	0.438285	0.219143	−	0.975693i	$-0.429674\pi$
$499$	4885.48	0.438285	0.219143	+	0.975693i	$0.429674\pi$
$500$	0	0
$501$	770.323	0.0686936
$502$	0	0
$503$	−16417.2	−1.45528	−0.727641	−	0.685958i	$-0.759383\pi$
$503$	−16417.2	−1.45528	−0.727641	+	0.685958i	$0.759383\pi$
$504$	0	0
$505$	5660.68	0.498806
$506$	0	0
$507$	6579.00	0.576299
$508$	0	0
$509$	3997.88	0.348139	0.174070	−	0.984733i	$-0.444308\pi$
$509$	3997.88	0.348139	0.174070	+	0.984733i	$0.444308\pi$
$510$	0	0
$511$	−10907.6	−0.944276
$512$	0	0
$513$	675.436	0.0581310
$514$	0	0
$515$	−12720.5	−1.08841
$516$	0	0
$517$	−16570.6	−1.40962
$518$	0	0
$519$	1848.22	0.156316
$520$	0	0
$521$	7134.76	0.599961	0.299980	−	0.953945i	$-0.403020\pi$
$521$	7134.76	0.599961	0.299980	+	0.953945i	$0.403020\pi$
$522$	0	0
$523$	7820.66	0.653869	0.326935	−	0.945047i	$-0.393984\pi$
$523$	7820.66	0.653869	0.326935	+	0.945047i	$0.393984\pi$
$524$	0	0
$525$	−13377.9	−1.11212
$526$	0	0
$527$	−2333.53	−0.192885
$528$	0	0
$529$	2462.29	0.202374
$530$	0	0
$531$	−5758.26	−0.470597
$532$	0	0
$533$	−526.032	−0.0427486
$534$	0	0
$535$	−22559.7	−1.82307
$536$	0	0
$537$	10748.7	0.863764
$538$	0	0
$539$	1888.05	0.150879
$540$	0	0
$541$	15615.1	1.24094	0.620468	−	0.784232i	$-0.286943\pi$
$541$	15615.1	1.24094	0.620468	+	0.784232i	$0.286943\pi$
$542$	0	0
$543$	−7847.32	−0.620186
$544$	0	0
$545$	−19868.6	−1.56161
$546$	0	0
$547$	−4572.95	−0.357450	−0.178725	−	0.983899i	$-0.557197\pi$
$547$	−4572.95	−0.357450	−0.178725	+	0.983899i	$0.557197\pi$
$548$	0	0
$549$	−1591.11	−0.123692
$550$	0	0
$551$	−6839.29	−0.528790
$552$	0	0
$553$	5345.90	0.411087
$554$	0	0
$555$	18769.8	1.43556
$556$	0	0
$557$	2443.09	0.185847	0.0929237	−	0.995673i	$-0.470379\pi$
$557$	2443.09	0.185847	0.0929237	+	0.995673i	$0.470379\pi$
$558$	0	0
$559$	−777.645	−0.0588388
$560$	0	0
$561$	−12075.8	−0.908805
$562$	0	0
$563$	18343.8	1.37318	0.686588	−	0.727046i	$-0.259107\pi$
$563$	18343.8	1.37318	0.686588	+	0.727046i	$0.259107\pi$
$564$	0	0
$565$	−4048.66	−0.301466
$566$	0	0
$567$	1416.85	0.104942
$568$	0	0
$569$	12298.0	0.906078	0.453039	−	0.891491i	$-0.350340\pi$
$569$	12298.0	0.906078	0.453039	+	0.891491i	$0.350340\pi$
$570$	0	0
$571$	284.678	0.0208641	0.0104321	−	0.999946i	$-0.496679\pi$
$571$	284.678	0.0208641	0.0104321	+	0.999946i	$0.496679\pi$
$572$	0	0
$573$	12751.5	0.929675
$574$	0	0
$575$	−30834.9	−2.23635
$576$	0	0
$577$	−9023.26	−0.651028	−0.325514	−	0.945537i	$-0.605537\pi$
$577$	−9023.26	−0.651028	−0.325514	+	0.945537i	$0.605537\pi$
$578$	0	0
$579$	639.194	0.0458791
$580$	0	0
$581$	11133.6	0.795009
$582$	0	0
$583$	−28932.9	−2.05537
$584$	0	0
$585$	350.855	0.0247967
$586$	0	0
$587$	285.711	0.0200895	0.0100448	−	0.999950i	$-0.496803\pi$
$587$	285.711	0.0200895	0.0100448	+	0.999950i	$0.496803\pi$
$588$	0	0
$589$	−739.388	−0.0517249
$590$	0	0
$591$	−1584.70	−0.110298
$592$	0	0
$593$	−21701.1	−1.50279	−0.751396	−	0.659851i	$-0.770619\pi$
$593$	−21701.1	−1.50279	−0.751396	+	0.659851i	$0.770619\pi$
$594$	0	0
$595$	26918.7	1.85472
$596$	0	0
$597$	6131.64	0.420354
$598$	0	0
$599$	22389.3	1.52721	0.763607	−	0.645682i	$-0.223427\pi$
$599$	22389.3	1.52721	0.763607	+	0.645682i	$0.223427\pi$
$600$	0	0
$601$	−9713.13	−0.659246	−0.329623	−	0.944113i	$-0.606922\pi$
$601$	−9713.13	−0.659246	−0.329623	+	0.944113i	$0.606922\pi$
$602$	0	0
$603$	−9139.93	−0.617259
$604$	0	0
$605$	−24722.7	−1.66136
$606$	0	0
$607$	−5289.91	−0.353725	−0.176862	−	0.984236i	$-0.556595\pi$
$607$	−5289.91	−0.353725	−0.176862	+	0.984236i	$0.556595\pi$
$608$	0	0
$609$	−14346.6	−0.954605
$610$	0	0
$611$	−650.032	−0.0430401
$612$	0	0
$613$	10731.3	0.707071	0.353536	−	0.935421i	$-0.384979\pi$
$613$	10731.3	0.707071	0.353536	+	0.935421i	$0.384979\pi$
$614$	0	0
$615$	15380.1	1.00843
$616$	0	0
$617$	12202.0	0.796165	0.398083	−	0.917350i	$-0.369676\pi$
$617$	12202.0	0.796165	0.398083	+	0.917350i	$0.369676\pi$
$618$	0	0
$619$	−12340.5	−0.801300	−0.400650	−	0.916231i	$-0.631216\pi$
$619$	−12340.5	−0.801300	−0.400650	+	0.916231i	$0.631216\pi$
$620$	0	0
$621$	3265.69	0.211027
$622$	0	0
$623$	25605.4	1.64664
$624$	0	0
$625$	17500.2	1.12001
$626$	0	0
$627$	−3826.26	−0.243710
$628$	0	0
$629$	−25342.2	−1.60645
$630$	0	0
$631$	1072.01	0.0676323	0.0338162	−	0.999428i	$-0.489234\pi$
$631$	1072.01	0.0676323	0.0338162	+	0.999428i	$0.489234\pi$
$632$	0	0
$633$	−11866.1	−0.745077
$634$	0	0
$635$	42393.9	2.64937
$636$	0	0
$637$	74.0645	0.00460682
$638$	0	0
$639$	135.436	0.00838458
$640$	0	0
$641$	−20062.5	−1.23623	−0.618113	−	0.786089i	$-0.712103\pi$
$641$	−20062.5	−1.23623	−0.618113	+	0.786089i	$0.712103\pi$
$642$	0	0
$643$	−4130.50	−0.253330	−0.126665	−	0.991946i	$-0.540427\pi$
$643$	−4130.50	−0.253330	−0.126665	+	0.991946i	$0.540427\pi$
$644$	0	0
$645$	22736.7	1.38800
$646$	0	0
$647$	21264.3	1.29209	0.646047	−	0.763298i	$-0.276421\pi$
$647$	21264.3	1.29209	0.646047	+	0.763298i	$0.276421\pi$
$648$	0	0
$649$	32619.8	1.97294
$650$	0	0
$651$	−1551.00	−0.0933770
$652$	0	0
$653$	1.76698	0.000105892 0	5.29458e−5	−	1.00000i	$-0.499983\pi$
$653$	1.76698	0.000105892 0	5.29458e−5		1.00000i	$0.499983\pi$
$654$	0	0
$655$	25020.1	1.49254
$656$	0	0
$657$	−5612.22	−0.333263
$658$	0	0
$659$	17347.8	1.02545	0.512726	−	0.858552i	$-0.328636\pi$
$659$	17347.8	1.02545	0.512726	+	0.858552i	$0.328636\pi$
$660$	0	0
$661$	10485.2	0.616984	0.308492	−	0.951227i	$-0.400176\pi$
$661$	10485.2	0.616984	0.308492	+	0.951227i	$0.400176\pi$
$662$	0	0
$663$	−473.710	−0.0277487
$664$	0	0
$665$	8529.29	0.497371
$666$	0	0
$667$	−33067.6	−1.91961
$668$	0	0
$669$	−10985.9	−0.634886
$670$	0	0
$671$	9013.45	0.518570
$672$	0	0
$673$	−30739.2	−1.76064	−0.880319	−	0.474381i	$-0.842672\pi$
$673$	−30739.2	−1.76064	−0.880319	+	0.474381i	$0.842672\pi$
$674$	0	0
$675$	−6883.26	−0.392499
$676$	0	0
$677$	−20148.0	−1.14380	−0.571898	−	0.820325i	$-0.693793\pi$
$677$	−20148.0	−1.14380	−0.571898	+	0.820325i	$0.693793\pi$
$678$	0	0
$679$	−11197.7	−0.632882
$680$	0	0
$681$	−352.112	−0.0198134
$682$	0	0
$683$	−15474.8	−0.866951	−0.433476	−	0.901165i	$-0.642713\pi$
$683$	−15474.8	−0.866951	−0.433476	+	0.901165i	$0.642713\pi$
$684$	0	0
$685$	−35019.1	−1.95330
$686$	0	0
$687$	−6878.52	−0.381997
$688$	0	0
$689$	−1134.98	−0.0627568
$690$	0	0
$691$	−19447.9	−1.07067	−0.535335	−	0.844640i	$-0.679814\pi$
$691$	−19447.9	−1.07067	−0.535335	+	0.844640i	$0.679814\pi$
$692$	0	0
$693$	−8026.26	−0.439960
$694$	0	0
$695$	−7647.13	−0.417370
$696$	0	0
$697$	−20765.5	−1.12848
$698$	0	0
$699$	−9559.84	−0.517291
$700$	0	0
$701$	−19294.0	−1.03955	−0.519774	−	0.854304i	$-0.673984\pi$
$701$	−19294.0	−1.03955	−0.519774	+	0.854304i	$0.673984\pi$
$702$	0	0
$703$	−8029.78	−0.430795
$704$	0	0
$705$	19005.6	1.01531
$706$	0	0
$707$	−5079.85	−0.270223
$708$	0	0
$709$	16102.8	0.852969	0.426484	−	0.904495i	$-0.359752\pi$
$709$	16102.8	0.852969	0.426484	+	0.904495i	$0.359752\pi$
$710$	0	0
$711$	2750.59	0.145085
$712$	0	0
$713$	−3574.90	−0.187772
$714$	0	0
$715$	−1987.55	−0.103958
$716$	0	0
$717$	−8637.00	−0.449867
$718$	0	0
$719$	−540.985	−0.0280603	−0.0140301	−	0.999902i	$-0.504466\pi$
$719$	−540.985	−0.0280603	−0.0140301	+	0.999902i	$0.504466\pi$
$720$	0	0
$721$	11415.3	0.589638
$722$	0	0
$723$	−8343.68	−0.429190
$724$	0	0
$725$	69698.1	3.57038
$726$	0	0
$727$	−5066.54	−0.258470	−0.129235	−	0.991614i	$-0.541252\pi$
$727$	−5066.54	−0.258470	−0.129235	+	0.991614i	$0.541252\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−30698.2	−1.55323
$732$	0	0
$733$	13880.5	0.699435	0.349718	−	0.936855i	$-0.386278\pi$
$733$	13880.5	0.699435	0.349718	+	0.936855i	$0.386278\pi$
$734$	0	0
$735$	−2165.49	−0.108674
$736$	0	0
$737$	51776.6	2.58781
$738$	0	0
$739$	25369.3	1.26282	0.631411	−	0.775448i	$-0.282476\pi$
$739$	25369.3	1.26282	0.631411	+	0.775448i	$0.282476\pi$
$740$	0	0
$741$	−150.097	−0.00744122
$742$	0	0
$743$	26218.7	1.29458	0.647290	−	0.762244i	$-0.275902\pi$
$743$	26218.7	1.29458	0.647290	+	0.762244i	$0.275902\pi$
$744$	0	0
$745$	5444.06	0.267725
$746$	0	0
$747$	5728.50	0.280582
$748$	0	0
$749$	20244.9	0.987629
$750$	0	0
$751$	26990.0	1.31142	0.655712	−	0.755011i	$-0.272369\pi$
$751$	26990.0	1.31142	0.655712	+	0.755011i	$0.272369\pi$
$752$	0	0
$753$	−4119.05	−0.199345
$754$	0	0
$755$	−19090.3	−0.920222
$756$	0	0
$757$	9200.58	0.441745	0.220872	−	0.975303i	$-0.429110\pi$
$757$	9200.58	0.441745	0.220872	+	0.975303i	$0.429110\pi$
$758$	0	0
$759$	−18499.7	−0.884714
$760$	0	0
$761$	32932.6	1.56873	0.784366	−	0.620298i	$-0.212988\pi$
$761$	32932.6	1.56873	0.784366	+	0.620298i	$0.212988\pi$
$762$	0	0
$763$	17829.9	0.845985
$764$	0	0
$765$	13850.3	0.654585
$766$	0	0
$767$	1279.61	0.0602401
$768$	0	0
$769$	4946.36	0.231951	0.115975	−	0.993252i	$-0.463001\pi$
$769$	4946.36	0.231951	0.115975	+	0.993252i	$0.463001\pi$
$770$	0	0
$771$	−4844.90	−0.226310
$772$	0	0
$773$	−27479.4	−1.27861	−0.639305	−	0.768953i	$-0.720778\pi$
$773$	−27479.4	−1.27861	−0.639305	+	0.768953i	$0.720778\pi$
$774$	0	0
$775$	7534.99	0.349245
$776$	0	0
$777$	−16843.9	−0.777698
$778$	0	0
$779$	−6579.65	−0.302619
$780$	0	0
$781$	−767.226	−0.0351517
$782$	0	0
$783$	−7381.67	−0.336908
$784$	0	0
$785$	−6721.89	−0.305623
$786$	0	0
$787$	−33882.6	−1.53467	−0.767334	−	0.641247i	$-0.778417\pi$
$787$	−33882.6	−1.53467	−0.767334	+	0.641247i	$0.778417\pi$
$788$	0	0
$789$	10581.7	0.477462
$790$	0	0
$791$	3633.24	0.163316
$792$	0	0
$793$	353.581	0.0158336
$794$	0	0
$795$	33184.5	1.48042
$796$	0	0
$797$	−10266.1	−0.456267	−0.228134	−	0.973630i	$-0.573262\pi$
$797$	−10266.1	−0.456267	−0.228134	+	0.973630i	$0.573262\pi$
$798$	0	0
$799$	−25660.5	−1.13618
$800$	0	0
$801$	13174.5	0.581148
$802$	0	0
$803$	31792.5	1.39718
$804$	0	0
$805$	41238.6	1.80555
$806$	0	0
$807$	−4955.06	−0.216142
$808$	0	0
$809$	−3494.69	−0.151875	−0.0759375	−	0.997113i	$-0.524195\pi$
$809$	−3494.69	−0.151875	−0.0759375	+	0.997113i	$0.524195\pi$
$810$	0	0
$811$	−4921.82	−0.213106	−0.106553	−	0.994307i	$-0.533981\pi$
$811$	−4921.82	−0.213106	−0.106553	+	0.994307i	$0.533981\pi$
$812$	0	0
$813$	−858.848	−0.0370493
$814$	0	0
$815$	−66757.7	−2.86923
$816$	0	0
$817$	−9726.84	−0.416523
$818$	0	0
$819$	−314.855	−0.0134334
$820$	0	0
$821$	39129.9	1.66339	0.831694	−	0.555234i	$-0.187371\pi$
$821$	39129.9	1.66339	0.831694	+	0.555234i	$0.187371\pi$
$822$	0	0
$823$	40616.5	1.72029	0.860147	−	0.510046i	$-0.170372\pi$
$823$	40616.5	1.72029	0.860147	+	0.510046i	$0.170372\pi$
$824$	0	0
$825$	38992.8	1.64552
$826$	0	0
$827$	−23260.9	−0.978065	−0.489032	−	0.872266i	$-0.662650\pi$
$827$	−23260.9	−0.978065	−0.489032	+	0.872266i	$0.662650\pi$
$828$	0	0
$829$	−21507.4	−0.901064	−0.450532	−	0.892760i	$-0.648766\pi$
$829$	−21507.4	−0.901064	−0.450532	+	0.892760i	$0.648766\pi$
$830$	0	0
$831$	19577.3	0.817244
$832$	0	0
$833$	2923.76	0.121611
$834$	0	0
$835$	5005.03	0.207433
$836$	0	0
$837$	−798.025	−0.0329555
$838$	0	0
$839$	21296.2	0.876312	0.438156	−	0.898899i	$-0.355632\pi$
$839$	21296.2	0.876312	0.438156	+	0.898899i	$0.355632\pi$
$840$	0	0
$841$	50355.9	2.06470
$842$	0	0
$843$	−7178.52	−0.293287
$844$	0	0
$845$	42745.8	1.74024
$846$	0	0
$847$	22186.0	0.900023
$848$	0	0
$849$	12455.4	0.503497
$850$	0	0
$851$	−38823.5	−1.56387
$852$	0	0
$853$	1572.21	0.0631084	0.0315542	−	0.999502i	$-0.489954\pi$
$853$	1572.21	0.0631084	0.0315542	+	0.999502i	$0.489954\pi$
$854$	0	0
$855$	4388.52	0.175537
$856$	0	0
$857$	18909.5	0.753719	0.376860	−	0.926270i	$-0.377004\pi$
$857$	18909.5	0.753719	0.376860	+	0.926270i	$0.377004\pi$
$858$	0	0
$859$	−1741.66	−0.0691790	−0.0345895	−	0.999402i	$-0.511012\pi$
$859$	−1741.66	−0.0691790	−0.0345895	+	0.999402i	$0.511012\pi$
$860$	0	0
$861$	−13802.0	−0.546307
$862$	0	0
$863$	−17993.1	−0.709724	−0.354862	−	0.934919i	$-0.615472\pi$
$863$	−17993.1	−0.709724	−0.354862	+	0.934919i	$0.615472\pi$
$864$	0	0
$865$	12008.4	0.472023
$866$	0	0
$867$	−3961.07	−0.155161
$868$	0	0
$869$	−15581.7	−0.608256
$870$	0	0
$871$	2031.10	0.0790139
$872$	0	0
$873$	−5761.45	−0.223363
$874$	0	0
$875$	−44301.7	−1.71162
$876$	0	0
$877$	6794.95	0.261630	0.130815	−	0.991407i	$-0.458241\pi$
$877$	6794.95	0.261630	0.130815	+	0.991407i	$0.458241\pi$
$878$	0	0
$879$	28202.9	1.08221
$880$	0	0
$881$	18857.0	0.721121	0.360560	−	0.932736i	$-0.382585\pi$
$881$	18857.0	0.721121	0.360560	+	0.932736i	$0.382585\pi$
$882$	0	0
$883$	11347.3	0.432467	0.216234	−	0.976342i	$-0.430623\pi$
$883$	11347.3	0.432467	0.216234	+	0.976342i	$0.430623\pi$
$884$	0	0
$885$	−37413.2	−1.42105
$886$	0	0
$887$	1959.61	0.0741797	0.0370899	−	0.999312i	$-0.488191\pi$
$887$	1959.61	0.0741797	0.0370899	+	0.999312i	$0.488191\pi$
$888$	0	0
$889$	−38044.0	−1.43527
$890$	0	0
$891$	−4129.69	−0.155275
$892$	0	0
$893$	−8130.65	−0.304683
$894$	0	0
$895$	69837.7	2.60829
$896$	0	0
$897$	−725.710	−0.0270131
$898$	0	0
$899$	8080.59	0.299781
$900$	0	0
$901$	−44804.4	−1.65666
$902$	0	0
$903$	−20403.8	−0.751933
$904$	0	0
$905$	−50986.5	−1.87276
$906$	0	0
$907$	29727.4	1.08829	0.544147	−	0.838990i	$-0.316853\pi$
$907$	29727.4	1.08829	0.544147	+	0.838990i	$0.316853\pi$
$908$	0	0
$909$	−2613.70	−0.0953696
$910$	0	0
$911$	29432.7	1.07041	0.535207	−	0.844721i	$-0.320233\pi$
$911$	29432.7	1.07041	0.535207	+	0.844721i	$0.320233\pi$
$912$	0	0
$913$	−32451.2	−1.17632
$914$	0	0
$915$	−10338.0	−0.373511
$916$	0	0
$917$	−22452.9	−0.808570
$918$	0	0
$919$	54250.3	1.94728	0.973641	−	0.228086i	$-0.0732468\pi$
$919$	54250.3	1.94728	0.973641	+	0.228086i	$0.0732468\pi$
$920$	0	0
$921$	29244.6	1.04630
$922$	0	0
$923$	−30.0968	−0.00107329
$924$	0	0
$925$	81830.2	2.90871
$926$	0	0
$927$	5873.44	0.208101
$928$	0	0
$929$	−46933.1	−1.65751	−0.828754	−	0.559613i	$-0.810950\pi$
$929$	−46933.1	−1.65751	−0.828754	+	0.559613i	$0.810950\pi$
$930$	0	0
$931$	926.404	0.0326119
$932$	0	0
$933$	−28805.1	−1.01076
$934$	0	0
$935$	−78460.1	−2.74430
$936$	0	0
$937$	−7143.42	−0.249056	−0.124528	−	0.992216i	$-0.539742\pi$
$937$	−7143.42	−0.249056	−0.124528	+	0.992216i	$0.539742\pi$
$938$	0	0
$939$	−30166.4	−1.04839
$940$	0	0
$941$	11662.2	0.404013	0.202007	−	0.979384i	$-0.435254\pi$
$941$	11662.2	0.404013	0.202007	+	0.979384i	$0.435254\pi$
$942$	0	0
$943$	−31812.2	−1.09857
$944$	0	0
$945$	9205.69	0.316890
$946$	0	0
$947$	20691.3	0.710007	0.355003	−	0.934865i	$-0.384480\pi$
$947$	20691.3	0.710007	0.355003	+	0.934865i	$0.384480\pi$
$948$	0	0
$949$	1247.16	0.0426602
$950$	0	0
$951$	−14438.8	−0.492336
$952$	0	0
$953$	−15086.7	−0.512807	−0.256403	−	0.966570i	$-0.582538\pi$
$953$	−15086.7	−0.512807	−0.256403	+	0.966570i	$0.582538\pi$
$954$	0	0
$955$	82850.8	2.80732
$956$	0	0
$957$	41816.2	1.41246
$958$	0	0
$959$	31425.9	1.05818
$960$	0	0
$961$	−28917.4	−0.970676
$962$	0	0
$963$	10416.5	0.348563
$964$	0	0
$965$	4153.04	0.138540
$966$	0	0
$967$	18123.9	0.602714	0.301357	−	0.953511i	$-0.402560\pi$
$967$	18123.9	0.602714	0.301357	+	0.953511i	$0.402560\pi$
$968$	0	0
$969$	−5925.19	−0.196434
$970$	0	0
$971$	−37808.0	−1.24956	−0.624778	−	0.780803i	$-0.714810\pi$
$971$	−37808.0	−1.24956	−0.624778	+	0.780803i	$0.714810\pi$
$972$	0	0
$973$	6862.48	0.226106
$974$	0	0
$975$	1529.61	0.0502429
$976$	0	0
$977$	−30977.2	−1.01438	−0.507190	−	0.861834i	$-0.669316\pi$
$977$	−30977.2	−1.01438	−0.507190	+	0.861834i	$0.669316\pi$
$978$	0	0
$979$	−74632.2	−2.43642
$980$	0	0
$981$	9173.90	0.298573
$982$	0	0
$983$	3162.90	0.102626	0.0513128	−	0.998683i	$-0.483659\pi$
$983$	3162.90	0.102626	0.0513128	+	0.998683i	$0.483659\pi$
$984$	0	0
$985$	−10296.3	−0.333063
$986$	0	0
$987$	−17055.5	−0.550032
$988$	0	0
$989$	−47028.7	−1.51206
$990$	0	0
$991$	15629.7	0.501002	0.250501	−	0.968116i	$-0.419405\pi$
$991$	15629.7	0.501002	0.250501	+	0.968116i	$0.419405\pi$
$992$	0	0
$993$	−7916.22	−0.252985
$994$	0	0
$995$	39839.1	1.26933
$996$	0	0
$997$	18095.9	0.574826	0.287413	−	0.957807i	$-0.407205\pi$
$997$	18095.9	0.574826	0.287413	+	0.957807i	$0.407205\pi$
$998$	0	0
$999$	−8666.56	−0.274472

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.a.j.1.1	✓	2
3.2	odd	2		1152.4.a.v.1.2		2
4.3	odd	2		384.4.a.n.1.1	yes	2
8.3	odd	2		384.4.a.k.1.2	yes	2
8.5	even	2		384.4.a.o.1.2	yes	2
12.11	even	2		1152.4.a.u.1.2		2
16.3	odd	4		768.4.d.t.385.4		4
16.5	even	4		768.4.d.q.385.3		4
16.11	odd	4		768.4.d.t.385.1		4
16.13	even	4		768.4.d.q.385.2		4
24.5	odd	2		1152.4.a.p.1.1		2
24.11	even	2		1152.4.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.a.j.1.1	✓	2	1.1	even	1	trivial
384.4.a.k.1.2	yes	2	8.3	odd	2
384.4.a.n.1.1	yes	2	4.3	odd	2
384.4.a.o.1.2	yes	2	8.5	even	2
768.4.d.q.385.2		4	16.13	even	4
768.4.d.q.385.3		4	16.5	even	4
768.4.d.t.385.1		4	16.11	odd	4
768.4.d.t.385.4		4	16.3	odd	4
1152.4.a.o.1.1		2	24.11	even	2
1152.4.a.p.1.1		2	24.5	odd	2
1152.4.a.u.1.2		2	12.11	even	2
1152.4.a.v.1.2		2	3.2	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -19.4919 q^{5} +17.4919 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -19.4919 q^{5} +17.4919 q^{7} +9.00000 q^{9} -50.9839 q^{11} -2.00000 q^{13} +58.4758 q^{15} -78.9516 q^{17} -25.0161 q^{19} -52.4758 q^{21} -120.952 q^{23} +254.935 q^{25} -27.0000 q^{27} +273.395 q^{29} +29.5565 q^{31} +152.952 q^{33} -340.952 q^{35} +320.984 q^{37} +6.00000 q^{39} +263.016 q^{41} +388.823 q^{43} -175.427 q^{45} +325.016 q^{47} -37.0323 q^{49} +236.855 q^{51} +567.492 q^{53} +993.774 q^{55} +75.0484 q^{57} -639.806 q^{59} -176.790 q^{61} +157.427 q^{63} +38.9839 q^{65} -1015.55 q^{67} +362.855 q^{69} +15.0484 q^{71} -623.581 q^{73} -764.806 q^{75} -891.806 q^{77} +305.621 q^{79} +81.0000 q^{81} +636.500 q^{83} +1538.92 q^{85} -820.185 q^{87} +1463.84 q^{89} -34.9839 q^{91} -88.6694 q^{93} +487.613 q^{95} -640.161 q^{97} -458.855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 8 q^{5} + 4 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 8 * q^5 + 4 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 8 q^{5} + 4 q^{7} + 18 q^{9} - 40 q^{11} - 4 q^{13} + 24 q^{15} + 28 q^{17} - 112 q^{19} - 12 q^{21} - 56 q^{23} + 262 q^{25} - 54 q^{27} + 144 q^{29} + 276 q^{31} + 120 q^{33} - 496 q^{35} + 580 q^{37} + 12 q^{39} + 588 q^{41} + 96 q^{43} - 72 q^{45} + 712 q^{47} - 198 q^{49} - 84 q^{51} + 1104 q^{53} + 1120 q^{55} + 336 q^{57} - 536 q^{59} + 452 q^{61} + 36 q^{63} + 16 q^{65} - 296 q^{67} + 168 q^{69} + 216 q^{71} + 364 q^{73} - 786 q^{75} - 1040 q^{77} + 1076 q^{79} + 162 q^{81} - 648 q^{83} + 2768 q^{85} - 432 q^{87} + 2308 q^{89} - 8 q^{91} - 828 q^{93} - 512 q^{95} - 1900 q^{97} - 360 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 8 * q^5 + 4 * q^7 + 18 * q^9 - 40 * q^11 - 4 * q^13 + 24 * q^15 + 28 * q^17 - 112 * q^19 - 12 * q^21 - 56 * q^23 + 262 * q^25 - 54 * q^27 + 144 * q^29 + 276 * q^31 + 120 * q^33 - 496 * q^35 + 580 * q^37 + 12 * q^39 + 588 * q^41 + 96 * q^43 - 72 * q^45 + 712 * q^47 - 198 * q^49 - 84 * q^51 + 1104 * q^53 + 1120 * q^55 + 336 * q^57 - 536 * q^59 + 452 * q^61 + 36 * q^63 + 16 * q^65 - 296 * q^67 + 168 * q^69 + 216 * q^71 + 364 * q^73 - 786 * q^75 - 1040 * q^77 + 1076 * q^79 + 162 * q^81 - 648 * q^83 + 2768 * q^85 - 432 * q^87 + 2308 * q^89 - 8 * q^91 - 828 * q^93 - 512 * q^95 - 1900 * q^97 - 360 * q^99

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