LMFDB - Embedded newform 384.4.a.n.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,4,Mod(1,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			1.2
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)

$f(q)$	$=$	$q+3.00000 q^{3} +11.4919 q^{5} +13.4919 q^{7} +9.00000 q^{9} -10.9839 q^{11} -2.00000 q^{13} +34.4758 q^{15} +106.952 q^{17} +86.9839 q^{19} +40.4758 q^{21} -64.9516 q^{23} +7.06453 q^{25} +27.0000 q^{27} -129.395 q^{29} -246.444 q^{31} -32.9516 q^{33} +155.048 q^{35} +259.016 q^{37} -6.00000 q^{39} +324.984 q^{41} +292.823 q^{43} +103.427 q^{45} -386.984 q^{47} -160.968 q^{49} +320.855 q^{51} +536.508 q^{53} -126.226 q^{55} +260.952 q^{57} -103.806 q^{59} +628.790 q^{61} +121.427 q^{63} -22.9839 q^{65} -719.548 q^{67} -194.855 q^{69} -200.952 q^{71} +987.581 q^{73} +21.1936 q^{75} -148.194 q^{77} -770.379 q^{79} +81.0000 q^{81} +1284.50 q^{83} +1229.08 q^{85} -388.185 q^{87} +844.161 q^{89} -26.9839 q^{91} -739.331 q^{93} +999.613 q^{95} -1259.84 q^{97} -98.8548 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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
$5$	11.4919	1.02787	0.513935	−	0.857829i	$-0.328187\pi$
$5$	11.4919	1.02787	0.513935	+	0.857829i	$0.328187\pi$
$6$	0	0
$7$	13.4919	0.728496	0.364248	−	0.931302i	$-0.381326\pi$
$7$	13.4919	0.728496	0.364248	+	0.931302i	$0.381326\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−10.9839	−0.301069	−0.150535	−	0.988605i	$-0.548099\pi$
$11$	−10.9839	−0.301069	−0.150535	+	0.988605i	$0.548099\pi$
$12$	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
$15$	34.4758	0.593441
$16$	0	0
$17$	106.952	1.52586	0.762929	−	0.646483i	$-0.223761\pi$
$17$	106.952	1.52586	0.762929	+	0.646483i	$0.223761\pi$
$18$	0	0
$19$	86.9839	1.05029	0.525144	−	0.851013i	$-0.324011\pi$
$19$	86.9839	1.05029	0.525144	+	0.851013i	$0.324011\pi$
$20$	0	0
$21$	40.4758	0.420597
$22$	0	0
$23$	−64.9516	−0.588841	−0.294421	−	0.955676i	$-0.595127\pi$
$23$	−64.9516	−0.588841	−0.294421	+	0.955676i	$0.595127\pi$
$24$	0	0
$25$	7.06453	0.0565163
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−129.395	−0.828554	−0.414277	−	0.910151i	$-0.635966\pi$
$29$	−129.395	−0.828554	−0.414277	+	0.910151i	$0.635966\pi$
$30$	0	0
$31$	−246.444	−1.42782	−0.713912	−	0.700235i	$-0.753079\pi$
$31$	−246.444	−1.42782	−0.713912	+	0.700235i	$0.753079\pi$
$32$	0	0
$33$	−32.9516	−0.173822
$34$	0	0
$35$	155.048	0.748799
$36$	0	0
$37$	259.016	1.15086	0.575432	−	0.817849i	$-0.304834\pi$
$37$	259.016	1.15086	0.575432	+	0.817849i	$0.304834\pi$
$38$	0	0
$39$	−6.00000	−0.0246351
$40$	0	0
$41$	324.984	1.23790	0.618951	−	0.785430i	$-0.287558\pi$
$41$	324.984	1.23790	0.618951	+	0.785430i	$0.287558\pi$
$42$	0	0
$43$	292.823	1.03849	0.519244	−	0.854626i	$-0.326213\pi$
$43$	292.823	1.03849	0.519244	+	0.854626i	$0.326213\pi$
$44$	0	0
$45$	103.427	0.342623
$46$	0	0
$47$	−386.984	−1.20101	−0.600504	−	0.799622i	$-0.705033\pi$
$47$	−386.984	−1.20101	−0.600504	+	0.799622i	$0.705033\pi$
$48$	0	0
$49$	−160.968	−0.469294
$50$	0	0
$51$	320.855	0.880954
$52$	0	0
$53$	536.508	1.39047	0.695236	−	0.718781i	$-0.255300\pi$
$53$	536.508	1.39047	0.695236	+	0.718781i	$0.255300\pi$
$54$	0	0
$55$	−126.226	−0.309460
$56$	0	0
$57$	260.952	0.606384
$58$	0	0
$59$	−103.806	−0.229058	−0.114529	−	0.993420i	$-0.536536\pi$
$59$	−103.806	−0.229058	−0.114529	+	0.993420i	$0.536536\pi$
$60$	0	0
$61$	628.790	1.31981	0.659904	−	0.751350i	$-0.270597\pi$
$61$	628.790	1.31981	0.659904	+	0.751350i	$0.270597\pi$
$62$	0	0
$63$	121.427	0.242832
$64$	0	0
$65$	−22.9839	−0.0438584
$66$	0	0
$67$	−719.548	−1.31204	−0.656021	−	0.754743i	$-0.727762\pi$
$67$	−719.548	−1.31204	−0.656021	+	0.754743i	$0.727762\pi$
$68$	0	0
$69$	−194.855	−0.339968
$70$	0	0
$71$	−200.952	−0.335895	−0.167948	−	0.985796i	$-0.553714\pi$
$71$	−200.952	−0.335895	−0.167948	+	0.985796i	$0.553714\pi$
$72$	0	0
$73$	987.581	1.58339	0.791696	−	0.610916i	$-0.209199\pi$
$73$	987.581	1.58339	0.791696	+	0.610916i	$0.209199\pi$
$74$	0	0
$75$	21.1936	0.0326297
$76$	0	0
$77$	−148.194	−0.219328
$78$	0	0
$79$	−770.379	−1.09714	−0.548572	−	0.836103i	$-0.684828\pi$
$79$	−770.379	−1.09714	−0.548572	+	0.836103i	$0.684828\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1284.50	1.69870	0.849350	−	0.527829i	$-0.176994\pi$
$83$	1284.50	1.69870	0.849350	+	0.527829i	$0.176994\pi$
$84$	0	0
$85$	1229.08	1.56838
$86$	0	0
$87$	−388.185	−0.478366
$88$	0	0
$89$	844.161	1.00540	0.502702	−	0.864460i	$-0.332339\pi$
$89$	844.161	1.00540	0.502702	+	0.864460i	$0.332339\pi$
$90$	0	0
$91$	−26.9839	−0.0310844
$92$	0	0
$93$	−739.331	−0.824355
$94$	0	0
$95$	999.613	1.07956
$96$	0	0
$97$	−1259.84	−1.31873	−0.659367	−	0.751821i	$-0.729176\pi$
$97$	−1259.84	−1.31873	−0.659367	+	0.751821i	$0.729176\pi$
$98$	0	0
$99$	−98.8548	−0.100356
$100$	0	0
$101$	50.4113	0.0496644	0.0248322	−	0.999692i	$-0.492095\pi$
$101$	50.4113	0.0496644	0.0248322	+	0.999692i	$0.492095\pi$
$102$	0	0
$103$	−1055.40	−1.00962	−0.504812	−	0.863230i	$-0.668438\pi$
$103$	−1055.40	−1.00962	−0.504812	+	0.863230i	$0.668438\pi$
$104$	0	0
$105$	465.145	0.432319
$106$	0	0
$107$	1197.39	1.08183	0.540915	−	0.841077i	$-0.318078\pi$
$107$	1197.39	1.08183	0.540915	+	0.841077i	$0.318078\pi$
$108$	0	0
$109$	−1583.32	−1.39133	−0.695664	−	0.718367i	$-0.744890\pi$
$109$	−1583.32	−1.39133	−0.695664	+	0.718367i	$0.744890\pi$
$110$	0	0
$111$	777.048	0.664452
$112$	0	0
$113$	−907.710	−0.755665	−0.377832	−	0.925874i	$-0.623330\pi$
$113$	−907.710	−0.755665	−0.377832	+	0.925874i	$0.623330\pi$
$114$	0	0
$115$	−746.419	−0.605252
$116$	0	0
$117$	−18.0000	−0.0142231
$118$	0	0
$119$	1442.98	1.11158
$120$	0	0
$121$	−1210.35	−0.909357
$122$	0	0
$123$	974.952	0.714703
$124$	0	0
$125$	−1355.31	−0.969778
$126$	0	0
$127$	37.0566	0.0258917	0.0129458	−	0.999916i	$-0.495879\pi$
$127$	37.0566	0.0258917	0.0129458	+	0.999916i	$0.495879\pi$
$128$	0	0
$129$	878.468	0.599572
$130$	0	0
$131$	−203.613	−0.135799	−0.0678997	−	0.997692i	$-0.521630\pi$
$131$	−203.613	−0.135799	−0.0678997	+	0.997692i	$0.521630\pi$
$132$	0	0
$133$	1173.58	0.765130
$134$	0	0
$135$	310.282	0.197814
$136$	0	0
$137$	247.403	0.154285	0.0771427	−	0.997020i	$-0.475420\pi$
$137$	247.403	0.154285	0.0771427	+	0.997020i	$0.475420\pi$
$138$	0	0
$139$	−1631.68	−0.995662	−0.497831	−	0.867274i	$-0.665870\pi$
$139$	−1631.68	−0.995662	−0.497831	+	0.867274i	$0.665870\pi$
$140$	0	0
$141$	−1160.95	−0.693403
$142$	0	0
$143$	21.9677	0.0128464
$144$	0	0
$145$	−1487.00	−0.851646
$146$	0	0
$147$	−482.903	−0.270947
$148$	0	0
$149$	495.298	0.272325	0.136162	−	0.990687i	$-0.456523\pi$
$149$	495.298	0.272325	0.136162	+	0.990687i	$0.456523\pi$
$150$	0	0
$151$	−576.605	−0.310751	−0.155376	−	0.987855i	$-0.549659\pi$
$151$	−576.605	−0.310751	−0.155376	+	0.987855i	$0.549659\pi$
$152$	0	0
$153$	962.564	0.508619
$154$	0	0
$155$	−2832.11	−1.46762
$156$	0	0
$157$	−212.855	−0.108202	−0.0541008	−	0.998535i	$-0.517229\pi$
$157$	−212.855	−0.108202	−0.0541008	+	0.998535i	$0.517229\pi$
$158$	0	0
$159$	1609.52	0.802790
$160$	0	0
$161$	−876.323	−0.428968
$162$	0	0
$163$	−2991.11	−1.43731	−0.718657	−	0.695365i	$-0.755243\pi$
$163$	−2991.11	−1.43731	−0.718657	+	0.695365i	$0.755243\pi$
$164$	0	0
$165$	−378.678	−0.178667
$166$	0	0
$167$	3231.23	1.49724	0.748622	−	0.662997i	$-0.230716\pi$
$167$	3231.23	1.49724	0.748622	+	0.662997i	$0.230716\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	0	0
$171$	782.855	0.350096
$172$	0	0
$173$	−2815.93	−1.23752	−0.618760	−	0.785580i	$-0.712365\pi$
$173$	−2815.93	−1.23752	−0.618760	+	0.785580i	$0.712365\pi$
$174$	0	0
$175$	95.3142	0.0411719
$176$	0	0
$177$	−311.419	−0.132247
$178$	0	0
$179$	−630.903	−0.263441	−0.131720	−	0.991287i	$-0.542050\pi$
$179$	−630.903	−0.263441	−0.131720	+	0.991287i	$0.542050\pi$
$180$	0	0
$181$	1748.23	0.717926	0.358963	−	0.933352i	$-0.383130\pi$
$181$	1748.23	0.717926	0.358963	+	0.933352i	$0.383130\pi$
$182$	0	0
$183$	1886.37	0.761992
$184$	0	0
$185$	2976.60	1.18294
$186$	0	0
$187$	−1174.74	−0.459389
$188$	0	0
$189$	364.282	0.140199
$190$	0	0
$191$	−1450.52	−0.549506	−0.274753	−	0.961515i	$-0.588596\pi$
$191$	−1450.52	−0.549506	−0.274753	+	0.961515i	$0.588596\pi$
$192$	0	0
$193$	−4302.94	−1.60483	−0.802415	−	0.596767i	$-0.796452\pi$
$193$	−4302.94	−1.60483	−0.802415	+	0.596767i	$0.796452\pi$
$194$	0	0
$195$	−68.9516	−0.0253217
$196$	0	0
$197$	−4336.23	−1.56824	−0.784121	−	0.620607i	$-0.786886\pi$
$197$	−4336.23	−1.56824	−0.784121	+	0.620607i	$0.786886\pi$
$198$	0	0
$199$	−4183.88	−1.49039	−0.745194	−	0.666848i	$-0.767643\pi$
$199$	−4183.88	−1.49039	−0.745194	+	0.666848i	$0.767643\pi$
$200$	0	0
$201$	−2158.64	−0.757508
$202$	0	0
$203$	−1745.79	−0.603598
$204$	0	0
$205$	3734.69	1.27240
$206$	0	0
$207$	−584.564	−0.196280
$208$	0	0
$209$	−955.419	−0.316209
$210$	0	0
$211$	−1476.65	−0.481784	−0.240892	−	0.970552i	$-0.577440\pi$
$211$	−1476.65	−0.481784	−0.240892	+	0.970552i	$0.577440\pi$
$212$	0	0
$213$	−602.855	−0.193929
$214$	0	0
$215$	3365.10	1.06743
$216$	0	0
$217$	−3325.00	−1.04016
$218$	0	0
$219$	2962.74	0.914171
$220$	0	0
$221$	−213.903	−0.0651072
$222$	0	0
$223$	6097.96	1.83116	0.915582	−	0.402131i	$-0.131731\pi$
$223$	6097.96	1.83116	0.915582	+	0.402131i	$0.131731\pi$
$224$	0	0
$225$	63.5808	0.0188388
$226$	0	0
$227$	6141.37	1.79567	0.897835	−	0.440332i	$-0.145139\pi$
$227$	6141.37	1.79567	0.897835	+	0.440332i	$0.145139\pi$
$228$	0	0
$229$	−2168.84	−0.625855	−0.312928	−	0.949777i	$-0.601310\pi$
$229$	−2168.84	−0.625855	−0.312928	+	0.949777i	$0.601310\pi$
$230$	0	0
$231$	−444.581	−0.126629
$232$	0	0
$233$	−2142.61	−0.602434	−0.301217	−	0.953556i	$-0.597393\pi$
$233$	−2142.61	−0.602434	−0.301217	+	0.953556i	$0.597393\pi$
$234$	0	0
$235$	−4447.19	−1.23448
$236$	0	0
$237$	−2311.14	−0.633437
$238$	0	0
$239$	−6721.00	−1.81902	−0.909509	−	0.415684i	$-0.863542\pi$
$239$	−6721.00	−1.81902	−0.909509	+	0.415684i	$0.863542\pi$
$240$	0	0
$241$	−193.226	−0.0516463	−0.0258231	−	0.999667i	$-0.508221\pi$
$241$	−193.226	−0.0516463	−0.0258231	+	0.999667i	$0.508221\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1849.83	−0.482373
$246$	0	0
$247$	−173.968	−0.0448150
$248$	0	0
$249$	3853.50	0.980745
$250$	0	0
$251$	−1434.98	−0.360858	−0.180429	−	0.983588i	$-0.557749\pi$
$251$	−1434.98	−0.360858	−0.180429	+	0.983588i	$0.557749\pi$
$252$	0	0
$253$	713.420	0.177282
$254$	0	0
$255$	3687.24	0.905506
$256$	0	0
$257$	5333.03	1.29442	0.647209	−	0.762313i	$-0.275936\pi$
$257$	5333.03	1.29442	0.647209	+	0.762313i	$0.275936\pi$
$258$	0	0
$259$	3494.63	0.838400
$260$	0	0
$261$	−1164.56	−0.276185
$262$	0	0
$263$	552.774	0.129603	0.0648014	−	0.997898i	$-0.479359\pi$
$263$	552.774	0.129603	0.0648014	+	0.997898i	$0.479359\pi$
$264$	0	0
$265$	6165.51	1.42922
$266$	0	0
$267$	2532.48	0.580470
$268$	0	0
$269$	2364.31	0.535891	0.267946	−	0.963434i	$-0.413655\pi$
$269$	2364.31	0.535891	0.267946	+	0.963434i	$0.413655\pi$
$270$	0	0
$271$	−7133.72	−1.59905	−0.799525	−	0.600633i	$-0.794915\pi$
$271$	−7133.72	−1.59905	−0.799525	+	0.600633i	$0.794915\pi$
$272$	0	0
$273$	−80.9516	−0.0179466
$274$	0	0
$275$	−77.5959	−0.0170153
$276$	0	0
$277$	2025.77	0.439411	0.219706	−	0.975566i	$-0.429490\pi$
$277$	2025.77	0.439411	0.219706	+	0.975566i	$0.429490\pi$
$278$	0	0
$279$	−2217.99	−0.475942
$280$	0	0
$281$	−2068.84	−0.439205	−0.219602	−	0.975589i	$-0.570476\pi$
$281$	−2068.84	−0.439205	−0.219602	+	0.975589i	$0.570476\pi$
$282$	0	0
$283$	3408.19	0.715887	0.357944	−	0.933743i	$-0.383478\pi$
$283$	3408.19	0.715887	0.357944	+	0.933743i	$0.383478\pi$
$284$	0	0
$285$	2998.84	0.623284
$286$	0	0
$287$	4384.66	0.901806
$288$	0	0
$289$	6525.64	1.32824
$290$	0	0
$291$	−3779.52	−0.761372
$292$	0	0
$293$	4200.96	0.837620	0.418810	−	0.908074i	$-0.362447\pi$
$293$	4200.96	0.837620	0.418810	+	0.908074i	$0.362447\pi$
$294$	0	0
$295$	−1192.94	−0.235442
$296$	0	0
$297$	−296.564	−0.0579408
$298$	0	0
$299$	129.903	0.0251254
$300$	0	0
$301$	3950.74	0.756535
$302$	0	0
$303$	151.234	0.0286738
$304$	0	0
$305$	7226.02	1.35659
$306$	0	0
$307$	−4876.19	−0.906511	−0.453256	−	0.891381i	$-0.649738\pi$
$307$	−4876.19	−0.906511	−0.453256	+	0.891381i	$0.649738\pi$
$308$	0	0
$309$	−3166.19	−0.582906
$310$	0	0
$311$	6881.71	1.25475	0.627373	−	0.778719i	$-0.284130\pi$
$311$	6881.71	1.25475	0.627373	+	0.778719i	$0.284130\pi$
$312$	0	0
$313$	−7419.45	−1.33985	−0.669924	−	0.742430i	$-0.733673\pi$
$313$	−7419.45	−1.33985	−0.669924	+	0.742430i	$0.733673\pi$
$314$	0	0
$315$	1395.44	0.249600
$316$	0	0
$317$	2675.06	0.473963	0.236981	−	0.971514i	$-0.423842\pi$
$317$	2675.06	0.473963	0.236981	+	0.971514i	$0.423842\pi$
$318$	0	0
$319$	1421.26	0.249452
$320$	0	0
$321$	3592.16	0.624595
$322$	0	0
$323$	9303.06	1.60259
$324$	0	0
$325$	−14.1291	−0.00241151
$326$	0	0
$327$	−4749.97	−0.803284
$328$	0	0
$329$	−5221.16	−0.874930
$330$	0	0
$331$	9878.74	1.64044	0.820219	−	0.572050i	$-0.193852\pi$
$331$	9878.74	1.64044	0.820219	+	0.572050i	$0.193852\pi$
$332$	0	0
$333$	2331.15	0.383622
$334$	0	0
$335$	−8269.00	−1.34861
$336$	0	0
$337$	7522.81	1.21600	0.608002	−	0.793935i	$-0.291971\pi$
$337$	7522.81	1.21600	0.608002	+	0.793935i	$0.291971\pi$
$338$	0	0
$339$	−2723.13	−0.436283
$340$	0	0
$341$	2706.90	0.429874
$342$	0	0
$343$	−6799.50	−1.07037
$344$	0	0
$345$	−2239.26	−0.349442
$346$	0	0
$347$	1762.56	0.272678	0.136339	−	0.990662i	$-0.456466\pi$
$347$	1762.56	0.272678	0.136339	+	0.990662i	$0.456466\pi$
$348$	0	0
$349$	3913.89	0.600303	0.300151	−	0.953892i	$-0.402963\pi$
$349$	3913.89	0.600303	0.300151	+	0.953892i	$0.402963\pi$
$350$	0	0
$351$	−54.0000	−0.00821170
$352$	0	0
$353$	848.742	0.127972	0.0639858	−	0.997951i	$-0.479619\pi$
$353$	848.742	0.127972	0.0639858	+	0.997951i	$0.479619\pi$
$354$	0	0
$355$	−2309.32	−0.345257
$356$	0	0
$357$	4328.95	0.641771
$358$	0	0
$359$	−3899.98	−0.573352	−0.286676	−	0.958028i	$-0.592550\pi$
$359$	−3899.98	−0.573352	−0.286676	+	0.958028i	$0.592550\pi$
$360$	0	0
$361$	707.193	0.103104
$362$	0	0
$363$	−3631.06	−0.525018
$364$	0	0
$365$	11349.2	1.62752
$366$	0	0
$367$	−4038.70	−0.574437	−0.287219	−	0.957865i	$-0.592731\pi$
$367$	−4038.70	−0.574437	−0.287219	+	0.957865i	$0.592731\pi$
$368$	0	0
$369$	2924.85	0.412634
$370$	0	0
$371$	7238.53	1.01295
$372$	0	0
$373$	−11503.2	−1.59682	−0.798410	−	0.602115i	$-0.794325\pi$
$373$	−11503.2	−1.59682	−0.798410	+	0.602115i	$0.794325\pi$
$374$	0	0
$375$	−4065.92	−0.559902
$376$	0	0
$377$	258.790	0.0353538
$378$	0	0
$379$	8121.63	1.10074	0.550369	−	0.834921i	$-0.314487\pi$
$379$	8121.63	1.10074	0.550369	+	0.834921i	$0.314487\pi$
$380$	0	0
$381$	111.170	0.0149486
$382$	0	0
$383$	−3528.45	−0.470745	−0.235373	−	0.971905i	$-0.575631\pi$
$383$	−3528.45	−0.470745	−0.235373	+	0.971905i	$0.575631\pi$
$384$	0	0
$385$	−1703.03	−0.225440
$386$	0	0
$387$	2635.40	0.346163
$388$	0	0
$389$	106.234	0.0138465	0.00692324	−	0.999976i	$-0.497796\pi$
$389$	106.234	0.0138465	0.00692324	+	0.999976i	$0.497796\pi$
$390$	0	0
$391$	−6946.68	−0.898487
$392$	0	0
$393$	−610.838	−0.0784039
$394$	0	0
$395$	−8853.14	−1.12772
$396$	0	0
$397$	−5479.72	−0.692744	−0.346372	−	0.938097i	$-0.612587\pi$
$397$	−5479.72	−0.692744	−0.346372	+	0.938097i	$0.612587\pi$
$398$	0	0
$399$	3520.74	0.441748
$400$	0	0
$401$	−4977.08	−0.619809	−0.309905	−	0.950768i	$-0.600297\pi$
$401$	−4977.08	−0.619809	−0.309905	+	0.950768i	$0.600297\pi$
$402$	0	0
$403$	492.887	0.0609242
$404$	0	0
$405$	930.847	0.114208
$406$	0	0
$407$	−2845.00	−0.346490
$408$	0	0
$409$	1272.35	0.153824	0.0769119	−	0.997038i	$-0.475494\pi$
$409$	1272.35	0.153824	0.0769119	+	0.997038i	$0.475494\pi$
$410$	0	0
$411$	742.210	0.0890767
$412$	0	0
$413$	−1400.55	−0.166868
$414$	0	0
$415$	14761.4	1.74604
$416$	0	0
$417$	−4895.03	−0.574846
$418$	0	0
$419$	−6921.14	−0.806969	−0.403485	−	0.914986i	$-0.632201\pi$
$419$	−6921.14	−0.806969	−0.403485	+	0.914986i	$0.632201\pi$
$420$	0	0
$421$	1903.90	0.220405	0.110203	−	0.993909i	$-0.464850\pi$
$421$	1903.90	0.220405	0.110203	+	0.993909i	$0.464850\pi$
$422$	0	0
$423$	−3482.85	−0.400336
$424$	0	0
$425$	755.563	0.0862358
$426$	0	0
$427$	8483.60	0.961475
$428$	0	0
$429$	65.9032	0.00741687
$430$	0	0
$431$	−13752.4	−1.53696	−0.768482	−	0.639871i	$-0.778988\pi$
$431$	−13752.4	−1.53696	−0.768482	+	0.639871i	$0.778988\pi$
$432$	0	0
$433$	12292.7	1.36432	0.682159	−	0.731204i	$-0.261041\pi$
$433$	12292.7	1.36432	0.682159	+	0.731204i	$0.261041\pi$
$434$	0	0
$435$	−4461.00	−0.491698
$436$	0	0
$437$	−5649.74	−0.618453
$438$	0	0
$439$	16495.5	1.79337	0.896683	−	0.442673i	$-0.145970\pi$
$439$	16495.5	1.79337	0.896683	+	0.442673i	$0.145970\pi$
$440$	0	0
$441$	−1448.71	−0.156431
$442$	0	0
$443$	−15534.9	−1.66610	−0.833051	−	0.553196i	$-0.813408\pi$
$443$	−15534.9	−1.66610	−0.833051	+	0.553196i	$0.813408\pi$
$444$	0	0
$445$	9701.05	1.03342
$446$	0	0
$447$	1485.90	0.157227
$448$	0	0
$449$	2147.08	0.225673	0.112836	−	0.993614i	$-0.464006\pi$
$449$	2147.08	0.225673	0.112836	+	0.993614i	$0.464006\pi$
$450$	0	0
$451$	−3569.58	−0.372694
$452$	0	0
$453$	−1729.81	−0.179412
$454$	0	0
$455$	−310.097	−0.0319507
$456$	0	0
$457$	8907.65	0.911777	0.455888	−	0.890037i	$-0.349322\pi$
$457$	8907.65	0.911777	0.455888	+	0.890037i	$0.349322\pi$
$458$	0	0
$459$	2887.69	0.293651
$460$	0	0
$461$	11673.7	1.17939	0.589694	−	0.807627i	$-0.299248\pi$
$461$	11673.7	1.17939	0.589694	+	0.807627i	$0.299248\pi$
$462$	0	0
$463$	416.121	0.0417685	0.0208842	−	0.999782i	$-0.493352\pi$
$463$	416.121	0.0417685	0.0208842	+	0.999782i	$0.493352\pi$
$464$	0	0
$465$	−8496.34	−0.847330
$466$	0	0
$467$	12272.9	1.21611	0.608055	−	0.793895i	$-0.291950\pi$
$467$	12272.9	1.21611	0.608055	+	0.793895i	$0.291950\pi$
$468$	0	0
$469$	−9708.10	−0.955817
$470$	0	0
$471$	−638.564	−0.0624703
$472$	0	0
$473$	−3216.32	−0.312657
$474$	0	0
$475$	614.500	0.0593583
$476$	0	0
$477$	4828.57	0.463491
$478$	0	0
$479$	19279.7	1.83906	0.919532	−	0.393015i	$-0.128568\pi$
$479$	19279.7	1.83906	0.919532	+	0.393015i	$0.128568\pi$
$480$	0	0
$481$	−518.032	−0.0491065
$482$	0	0
$483$	−2628.97	−0.247665
$484$	0	0
$485$	−14478.0	−1.35549
$486$	0	0
$487$	14088.1	1.31087	0.655433	−	0.755254i	$-0.272486\pi$
$487$	14088.1	1.31087	0.655433	+	0.755254i	$0.272486\pi$
$488$	0	0
$489$	−8973.34	−0.829833
$490$	0	0
$491$	−6405.32	−0.588734	−0.294367	−	0.955693i	$-0.595109\pi$
$491$	−6405.32	−0.588734	−0.294367	+	0.955693i	$0.595109\pi$
$492$	0	0
$493$	−13839.0	−1.26426
$494$	0	0
$495$	−1136.03	−0.103153
$496$	0	0
$497$	−2711.23	−0.244698
$498$	0	0
$499$	4781.48	0.428955	0.214478	−	0.976729i	$-0.431195\pi$
$499$	4781.48	0.428955	0.214478	+	0.976729i	$0.431195\pi$
$500$	0	0
$501$	9693.68	0.864434
$502$	0	0
$503$	17222.8	1.52669	0.763346	−	0.645990i	$-0.223555\pi$
$503$	17222.8	1.52669	0.763346	+	0.645990i	$0.223555\pi$
$504$	0	0
$505$	579.323	0.0510486
$506$	0	0
$507$	−6579.00	−0.576299
$508$	0	0
$509$	−17597.9	−1.53244	−0.766220	−	0.642578i	$-0.777865\pi$
$509$	−17597.9	−1.53244	−0.766220	+	0.642578i	$0.777865\pi$
$510$	0	0
$511$	13324.4	1.15349
$512$	0	0
$513$	2348.56	0.202128
$514$	0	0
$515$	−12128.5	−1.03776
$516$	0	0
$517$	4250.58	0.361587
$518$	0	0
$519$	−8447.78	−0.714483
$520$	0	0
$521$	6205.24	0.521798	0.260899	−	0.965366i	$-0.415981\pi$
$521$	6205.24	0.521798	0.260899	+	0.965366i	$0.415981\pi$
$522$	0	0
$523$	1164.66	0.0973749	0.0486874	−	0.998814i	$-0.484496\pi$
$523$	1164.66	0.0973749	0.0486874	+	0.998814i	$0.484496\pi$
$524$	0	0
$525$	285.943	0.0237706
$526$	0	0
$527$	−26357.5	−2.17866
$528$	0	0
$529$	−7948.29	−0.653266
$530$	0	0
$531$	−934.258	−0.0763528
$532$	0	0
$533$	−649.968	−0.0528203
$534$	0	0
$535$	13760.3	1.11198
$536$	0	0
$537$	−1892.71	−0.152098
$538$	0	0
$539$	1768.05	0.141290
$540$	0	0
$541$	−18467.1	−1.46759	−0.733793	−	0.679373i	$-0.762252\pi$
$541$	−18467.1	−1.46759	−0.733793	+	0.679373i	$0.762252\pi$
$542$	0	0
$543$	5244.68	0.414495
$544$	0	0
$545$	−18195.4	−1.43010
$546$	0	0
$547$	4387.05	0.342919	0.171459	−	0.985191i	$-0.445152\pi$
$547$	4387.05	0.342919	0.171459	+	0.985191i	$0.445152\pi$
$548$	0	0
$549$	5659.11	0.439936
$550$	0	0
$551$	−11255.3	−0.870220
$552$	0	0
$553$	−10393.9	−0.799265
$554$	0	0
$555$	8929.79	0.682970
$556$	0	0
$557$	−2979.09	−0.226621	−0.113311	−	0.993560i	$-0.536146\pi$
$557$	−2979.09	−0.226621	−0.113311	+	0.993560i	$0.536146\pi$
$558$	0	0
$559$	−585.645	−0.0443115
$560$	0	0
$561$	−3524.23	−0.265228
$562$	0	0
$563$	−13696.2	−1.02527	−0.512634	−	0.858607i	$-0.671330\pi$
$563$	−13696.2	−1.02527	−0.512634	+	0.858607i	$0.671330\pi$
$564$	0	0
$565$	−10431.3	−0.776725
$566$	0	0
$567$	1092.85	0.0809440
$568$	0	0
$569$	−6973.98	−0.513822	−0.256911	−	0.966435i	$-0.582705\pi$
$569$	−6973.98	−0.513822	−0.256911	+	0.966435i	$0.582705\pi$
$570$	0	0
$571$	−10571.3	−0.774774	−0.387387	−	0.921917i	$-0.626622\pi$
$571$	−10571.3	−0.774774	−0.387387	+	0.921917i	$0.626622\pi$
$572$	0	0
$573$	−4351.55	−0.317257
$574$	0	0
$575$	−458.853	−0.0332791
$576$	0	0
$577$	9195.26	0.663438	0.331719	−	0.943378i	$-0.392371\pi$
$577$	9195.26	0.663438	0.331719	+	0.943378i	$0.392371\pi$
$578$	0	0
$579$	−12908.8	−0.926549
$580$	0	0
$581$	17330.4	1.23750
$582$	0	0
$583$	−5892.93	−0.418628
$584$	0	0
$585$	−206.855	−0.0146195
$586$	0	0
$587$	−14538.3	−1.02225	−0.511124	−	0.859507i	$-0.670771\pi$
$587$	−14538.3	−1.02225	−0.511124	+	0.859507i	$0.670771\pi$
$588$	0	0
$589$	−21436.6	−1.49963
$590$	0	0
$591$	−13008.7	−0.905425
$592$	0	0
$593$	−10422.9	−0.721785	−0.360893	−	0.932607i	$-0.617528\pi$
$593$	−10422.9	−0.721785	−0.360893	+	0.932607i	$0.617528\pi$
$594$	0	0
$595$	16582.7	1.14256
$596$	0	0
$597$	−12551.6	−0.860476
$598$	0	0
$599$	7293.27	0.497488	0.248744	−	0.968569i	$-0.419982\pi$
$599$	7293.27	0.497488	0.248744	+	0.968569i	$0.419982\pi$
$600$	0	0
$601$	16685.1	1.13245	0.566223	−	0.824252i	$-0.308404\pi$
$601$	16685.1	1.13245	0.566223	+	0.824252i	$0.308404\pi$
$602$	0	0
$603$	−6475.93	−0.437347
$604$	0	0
$605$	−13909.3	−0.934701
$606$	0	0
$607$	14554.1	0.973200	0.486600	−	0.873625i	$-0.338237\pi$
$607$	14554.1	0.973200	0.486600	+	0.873625i	$0.338237\pi$
$608$	0	0
$609$	−5237.37	−0.348488
$610$	0	0
$611$	773.968	0.0512461
$612$	0	0
$613$	−3335.34	−0.219760	−0.109880	−	0.993945i	$-0.535047\pi$
$613$	−3335.34	−0.219760	−0.109880	+	0.993945i	$0.535047\pi$
$614$	0	0
$615$	11204.1	0.734621
$616$	0	0
$617$	−18534.0	−1.20932	−0.604660	−	0.796484i	$-0.706691\pi$
$617$	−18534.0	−1.20932	−0.604660	+	0.796484i	$0.706691\pi$
$618$	0	0
$619$	−1292.45	−0.0839224	−0.0419612	−	0.999119i	$-0.513361\pi$
$619$	−1292.45	−0.0839224	−0.0419612	+	0.999119i	$0.513361\pi$
$620$	0	0
$621$	−1753.69	−0.113323
$622$	0	0
$623$	11389.4	0.732432
$624$	0	0
$625$	−16458.2	−1.05332
$626$	0	0
$627$	−2866.26	−0.182563
$628$	0	0
$629$	27702.2	1.75606
$630$	0	0
$631$	−10708.0	−0.675560	−0.337780	−	0.941225i	$-0.609676\pi$
$631$	−10708.0	−0.675560	−0.337780	+	0.941225i	$0.609676\pi$
$632$	0	0
$633$	−4429.94	−0.278158
$634$	0	0
$635$	425.852	0.0266133
$636$	0	0
$637$	321.935	0.0200244
$638$	0	0
$639$	−1808.56	−0.111965
$640$	0	0
$641$	−2773.50	−0.170900	−0.0854499	−	0.996342i	$-0.527233\pi$
$641$	−2773.50	−0.170900	−0.0854499	+	0.996342i	$0.527233\pi$
$642$	0	0
$643$	−5474.50	−0.335759	−0.167880	−	0.985808i	$-0.553692\pi$
$643$	−5474.50	−0.335759	−0.167880	+	0.985808i	$0.553692\pi$
$644$	0	0
$645$	10095.3	0.616282
$646$	0	0
$647$	19944.3	1.21189	0.605943	−	0.795508i	$-0.292796\pi$
$647$	19944.3	1.21189	0.605943	+	0.795508i	$0.292796\pi$
$648$	0	0
$649$	1140.20	0.0689624
$650$	0	0
$651$	−9975.00	−0.600539
$652$	0	0
$653$	12550.2	0.752111	0.376056	−	0.926597i	$-0.377280\pi$
$653$	12550.2	0.752111	0.376056	+	0.926597i	$0.377280\pi$
$654$	0	0
$655$	−2339.90	−0.139584
$656$	0	0
$657$	8888.22	0.527797
$658$	0	0
$659$	6571.77	0.388467	0.194234	−	0.980955i	$-0.437778\pi$
$659$	6571.77	0.388467	0.194234	+	0.980955i	$0.437778\pi$
$660$	0	0
$661$	11166.8	0.657094	0.328547	−	0.944488i	$-0.393441\pi$
$661$	11166.8	0.657094	0.328547	+	0.944488i	$0.393441\pi$
$662$	0	0
$663$	−641.710	−0.0375896
$664$	0	0
$665$	13486.7	0.786454
$666$	0	0
$667$	8404.42	0.487887
$668$	0	0
$669$	18293.9	1.05722
$670$	0	0
$671$	−6906.55	−0.397354
$672$	0	0
$673$	−4712.78	−0.269932	−0.134966	−	0.990850i	$-0.543093\pi$
$673$	−4712.78	−0.269932	−0.134966	+	0.990850i	$0.543093\pi$
$674$	0	0
$675$	190.742	0.0108766
$676$	0	0
$677$	16444.0	0.933520	0.466760	−	0.884384i	$-0.345421\pi$
$677$	16444.0	0.933520	0.466760	+	0.884384i	$0.345421\pi$
$678$	0	0
$679$	−16997.7	−0.960693
$680$	0	0
$681$	18424.1	1.03673
$682$	0	0
$683$	7109.18	0.398280	0.199140	−	0.979971i	$-0.436185\pi$
$683$	7109.18	0.398280	0.199140	+	0.979971i	$0.436185\pi$
$684$	0	0
$685$	2843.14	0.158585
$686$	0	0
$687$	−6506.52	−0.361338
$688$	0	0
$689$	−1073.02	−0.0593304
$690$	0	0
$691$	−7879.89	−0.433813	−0.216907	−	0.976192i	$-0.569597\pi$
$691$	−7879.89	−0.433813	−0.216907	+	0.976192i	$0.569597\pi$
$692$	0	0
$693$	−1333.74	−0.0731092
$694$	0	0
$695$	−18751.1	−1.02341
$696$	0	0
$697$	34757.5	1.88886
$698$	0	0
$699$	−6427.84	−0.347816
$700$	0	0
$701$	−17218.0	−0.927698	−0.463849	−	0.885914i	$-0.653532\pi$
$701$	−17218.0	−0.927698	−0.463849	+	0.885914i	$0.653532\pi$
$702$	0	0
$703$	22530.2	1.20874
$704$	0	0
$705$	−13341.6	−0.712728
$706$	0	0
$707$	680.145	0.0361803
$708$	0	0
$709$	3957.16	0.209611	0.104806	−	0.994493i	$-0.466578\pi$
$709$	3957.16	0.209611	0.104806	+	0.994493i	$0.466578\pi$
$710$	0	0
$711$	−6933.41	−0.365715
$712$	0	0
$713$	16006.9	0.840762
$714$	0	0
$715$	252.452	0.0132044
$716$	0	0
$717$	−20163.0	−1.05021
$718$	0	0
$719$	23531.0	1.22053	0.610263	−	0.792199i	$-0.291064\pi$
$719$	23531.0	1.22053	0.610263	+	0.792199i	$0.291064\pi$
$720$	0	0
$721$	−14239.3	−0.735506
$722$	0	0
$723$	−579.677	−0.0298180
$724$	0	0
$725$	−914.116	−0.0468268
$726$	0	0
$727$	−2462.54	−0.125627	−0.0628133	−	0.998025i	$-0.520007\pi$
$727$	−2462.54	−0.125627	−0.0628133	+	0.998025i	$0.520007\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	31317.8	1.58459
$732$	0	0
$733$	−7436.45	−0.374722	−0.187361	−	0.982291i	$-0.559993\pi$
$733$	−7436.45	−0.374722	−0.187361	+	0.982291i	$0.559993\pi$
$734$	0	0
$735$	−5549.49	−0.278498
$736$	0	0
$737$	7903.42	0.395015
$738$	0	0
$739$	7969.32	0.396693	0.198347	−	0.980132i	$-0.436443\pi$
$739$	7969.32	0.396693	0.198347	+	0.980132i	$0.436443\pi$
$740$	0	0
$741$	−521.903	−0.0258739
$742$	0	0
$743$	−13701.3	−0.676515	−0.338257	−	0.941054i	$-0.609837\pi$
$743$	−13701.3	−0.676515	−0.338257	+	0.941054i	$0.609837\pi$
$744$	0	0
$745$	5691.94	0.279915
$746$	0	0
$747$	11560.5	0.566234
$748$	0	0
$749$	16155.1	0.788108
$750$	0	0
$751$	−13574.0	−0.659550	−0.329775	−	0.944060i	$-0.606973\pi$
$751$	−13574.0	−0.659550	−0.329775	+	0.944060i	$0.606973\pi$
$752$	0	0
$753$	−4304.95	−0.208342
$754$	0	0
$755$	−6626.30	−0.319412
$756$	0	0
$757$	19115.4	0.917783	0.458891	−	0.888492i	$-0.348247\pi$
$757$	19115.4	0.917783	0.458891	+	0.888492i	$0.348247\pi$
$758$	0	0
$759$	2140.26	0.102354
$760$	0	0
$761$	647.405	0.0308389	0.0154195	−	0.999881i	$-0.495092\pi$
$761$	647.405	0.0308389	0.0154195	+	0.999881i	$0.495092\pi$
$762$	0	0
$763$	−21362.1	−1.01358
$764$	0	0
$765$	11061.7	0.522794
$766$	0	0
$767$	207.613	0.00977375
$768$	0	0
$769$	13993.6	0.656208	0.328104	−	0.944642i	$-0.393590\pi$
$769$	13993.6	0.656208	0.328104	+	0.944642i	$0.393590\pi$
$770$	0	0
$771$	15999.1	0.747333
$772$	0	0
$773$	−38664.6	−1.79905	−0.899527	−	0.436865i	$-0.856089\pi$
$773$	−38664.6	−1.79905	−0.899527	+	0.436865i	$0.856089\pi$
$774$	0	0
$775$	−1741.01	−0.0806953
$776$	0	0
$777$	10483.9	0.484051
$778$	0	0
$779$	28268.4	1.30015
$780$	0	0
$781$	2207.23	0.101128
$782$	0	0
$783$	−3493.67	−0.159455
$784$	0	0
$785$	−2446.11	−0.111217
$786$	0	0
$787$	16965.4	0.768426	0.384213	−	0.923244i	$-0.374473\pi$
$787$	16965.4	0.768426	0.384213	+	0.923244i	$0.374473\pi$
$788$	0	0
$789$	1658.32	0.0748262
$790$	0	0
$791$	−12246.8	−0.550499
$792$	0	0
$793$	−1257.58	−0.0563153
$794$	0	0
$795$	18496.5	0.825163
$796$	0	0
$797$	10338.1	0.459467	0.229734	−	0.973254i	$-0.426214\pi$
$797$	10338.1	0.459467	0.229734	+	0.973254i	$0.426214\pi$
$798$	0	0
$799$	−41388.5	−1.83257
$800$	0	0
$801$	7597.45	0.335135
$802$	0	0
$803$	−10847.5	−0.476710
$804$	0	0
$805$	−10070.6	−0.440924
$806$	0	0
$807$	7092.94	0.309397
$808$	0	0
$809$	28418.7	1.23504	0.617520	−	0.786555i	$-0.288137\pi$
$809$	28418.7	1.23504	0.617520	+	0.786555i	$0.288137\pi$
$810$	0	0
$811$	15766.2	0.682645	0.341323	−	0.939946i	$-0.389125\pi$
$811$	15766.2	0.682645	0.341323	+	0.939946i	$0.389125\pi$
$812$	0	0
$813$	−21401.2	−0.923212
$814$	0	0
$815$	−34373.7	−1.47737
$816$	0	0
$817$	25470.8	1.09071
$818$	0	0
$819$	−242.855	−0.0103615
$820$	0	0
$821$	32902.1	1.39865	0.699325	−	0.714804i	$-0.253484\pi$
$821$	32902.1	1.39865	0.699325	+	0.714804i	$0.253484\pi$
$822$	0	0
$823$	17044.5	0.721912	0.360956	−	0.932583i	$-0.382450\pi$
$823$	17044.5	0.721912	0.360956	+	0.932583i	$0.382450\pi$
$824$	0	0
$825$	−232.788	−0.00982379
$826$	0	0
$827$	18923.1	0.795673	0.397837	−	0.917456i	$-0.369761\pi$
$827$	18923.1	0.795673	0.397837	+	0.917456i	$0.369761\pi$
$828$	0	0
$829$	42319.4	1.77300	0.886498	−	0.462733i	$-0.153131\pi$
$829$	42319.4	1.77300	0.886498	+	0.462733i	$0.153131\pi$
$830$	0	0
$831$	6077.32	0.253694
$832$	0	0
$833$	−17215.8	−0.716075
$834$	0	0
$835$	37133.0	1.53897
$836$	0	0
$837$	−6653.98	−0.274785
$838$	0	0
$839$	−18135.8	−0.746267	−0.373134	−	0.927778i	$-0.621717\pi$
$839$	−18135.8	−0.746267	−0.373134	+	0.927778i	$0.621717\pi$
$840$	0	0
$841$	−7645.90	−0.313498
$842$	0	0
$843$	−6206.52	−0.253575
$844$	0	0
$845$	−25201.8	−1.02600
$846$	0	0
$847$	−16330.0	−0.662463
$848$	0	0
$849$	10224.6	0.413318
$850$	0	0
$851$	−16823.5	−0.677676
$852$	0	0
$853$	13903.8	0.558097	0.279048	−	0.960277i	$-0.409981\pi$
$853$	13903.8	0.558097	0.279048	+	0.960277i	$0.409981\pi$
$854$	0	0
$855$	8996.52	0.359853
$856$	0	0
$857$	−2097.53	−0.0836059	−0.0418030	−	0.999126i	$-0.513310\pi$
$857$	−2097.53	−0.0836059	−0.0418030	+	0.999126i	$0.513310\pi$
$858$	0	0
$859$	19650.3	0.780513	0.390257	−	0.920706i	$-0.372386\pi$
$859$	19650.3	0.780513	0.390257	+	0.920706i	$0.372386\pi$
$860$	0	0
$861$	13154.0	0.520658
$862$	0	0
$863$	37574.9	1.48211	0.741057	−	0.671442i	$-0.234325\pi$
$863$	37574.9	1.48211	0.741057	+	0.671442i	$0.234325\pi$
$864$	0	0
$865$	−32360.4	−1.27201
$866$	0	0
$867$	19576.9	0.766860
$868$	0	0
$869$	8461.74	0.330316
$870$	0	0
$871$	1439.10	0.0559838
$872$	0	0
$873$	−11338.5	−0.439578
$874$	0	0
$875$	−18285.7	−0.706480
$876$	0	0
$877$	21977.0	0.846194	0.423097	−	0.906084i	$-0.360943\pi$
$877$	21977.0	0.846194	0.423097	+	0.906084i	$0.360943\pi$
$878$	0	0
$879$	12602.9	0.483600
$880$	0	0
$881$	45627.0	1.74485	0.872425	−	0.488747i	$-0.162546\pi$
$881$	45627.0	1.74485	0.872425	+	0.488747i	$0.162546\pi$
$882$	0	0
$883$	−51068.7	−1.94632	−0.973158	−	0.230138i	$-0.926082\pi$
$883$	−51068.7	−1.94632	−0.973158	+	0.230138i	$0.926082\pi$
$884$	0	0
$885$	−3578.81	−0.135933
$886$	0	0
$887$	−31208.4	−1.18137	−0.590685	−	0.806902i	$-0.701142\pi$
$887$	−31208.4	−1.18137	−0.590685	+	0.806902i	$0.701142\pi$
$888$	0	0
$889$	499.965	0.0188620
$890$	0	0
$891$	−889.693	−0.0334521
$892$	0	0
$893$	−33661.4	−1.26140
$894$	0	0
$895$	−7250.29	−0.270783
$896$	0	0
$897$	389.710	0.0145062
$898$	0	0
$899$	31888.6	1.18303
$900$	0	0
$901$	57380.4	2.12166
$902$	0	0
$903$	11852.2	0.436786
$904$	0	0
$905$	20090.5	0.737934
$906$	0	0
$907$	−46768.6	−1.71216	−0.856078	−	0.516847i	$-0.827105\pi$
$907$	−46768.6	−1.71216	−0.856078	+	0.516847i	$0.827105\pi$
$908$	0	0
$909$	453.701	0.0165548
$910$	0	0
$911$	21752.7	0.791107	0.395553	−	0.918443i	$-0.370553\pi$
$911$	21752.7	0.791107	0.395553	+	0.918443i	$0.370553\pi$
$912$	0	0
$913$	−14108.8	−0.511426
$914$	0	0
$915$	21678.0	0.783229
$916$	0	0
$917$	−2747.13	−0.0989294
$918$	0	0
$919$	−3529.71	−0.126697	−0.0633483	−	0.997991i	$-0.520178\pi$
$919$	−3529.71	−0.126697	−0.0633483	+	0.997991i	$0.520178\pi$
$920$	0	0
$921$	−14628.6	−0.523375
$922$	0	0
$923$	401.903	0.0143324
$924$	0	0
$925$	1829.83	0.0650426
$926$	0	0
$927$	−9498.56	−0.336541
$928$	0	0
$929$	29473.1	1.04088	0.520442	−	0.853897i	$-0.325767\pi$
$929$	29473.1	1.04088	0.520442	+	0.853897i	$0.325767\pi$
$930$	0	0
$931$	−14001.6	−0.492893
$932$	0	0
$933$	20645.1	0.724428
$934$	0	0
$935$	−13500.1	−0.472192
$936$	0	0
$937$	−43332.6	−1.51079	−0.755397	−	0.655268i	$-0.772556\pi$
$937$	−43332.6	−1.51079	−0.755397	+	0.655268i	$0.772556\pi$
$938$	0	0
$939$	−22258.4	−0.773561
$940$	0	0
$941$	48873.8	1.69314	0.846568	−	0.532281i	$-0.178665\pi$
$941$	48873.8	1.69314	0.846568	+	0.532281i	$0.178665\pi$
$942$	0	0
$943$	−21108.2	−0.728927
$944$	0	0
$945$	4186.31	0.144106
$946$	0	0
$947$	−2596.71	−0.0891043	−0.0445521	−	0.999007i	$-0.514186\pi$
$947$	−2596.71	−0.0891043	−0.0445521	+	0.999007i	$0.514186\pi$
$948$	0	0
$949$	−1975.16	−0.0675621
$950$	0	0
$951$	8025.17	0.273642
$952$	0	0
$953$	−6101.34	−0.207389	−0.103695	−	0.994609i	$-0.533066\pi$
$953$	−6101.34	−0.207389	−0.103695	+	0.994609i	$0.533066\pi$
$954$	0	0
$955$	−16669.2	−0.564821
$956$	0	0
$957$	4263.78	0.144021
$958$	0	0
$959$	3337.95	0.112396
$960$	0	0
$961$	30943.4	1.03868
$962$	0	0
$963$	10776.5	0.360610
$964$	0	0
$965$	−49449.0	−1.64956
$966$	0	0
$967$	7375.86	0.245286	0.122643	−	0.992451i	$-0.460863\pi$
$967$	7375.86	0.245286	0.122643	+	0.992451i	$0.460863\pi$
$968$	0	0
$969$	27909.2	0.925255
$970$	0	0
$971$	34152.0	1.12872	0.564361	−	0.825528i	$-0.309123\pi$
$971$	34152.0	1.12872	0.564361	+	0.825528i	$0.309123\pi$
$972$	0	0
$973$	−22014.5	−0.725336
$974$	0	0
$975$	−42.3872	−0.00139228
$976$	0	0
$977$	−1170.76	−0.0383377	−0.0191689	−	0.999816i	$-0.506102\pi$
$977$	−1170.76	−0.0383377	−0.0191689	+	0.999816i	$0.506102\pi$
$978$	0	0
$979$	−9272.16	−0.302696
$980$	0	0
$981$	−14249.9	−0.463776
$982$	0	0
$983$	31786.9	1.03138	0.515689	−	0.856776i	$-0.327536\pi$
$983$	31786.9	1.03138	0.515689	+	0.856776i	$0.327536\pi$
$984$	0	0
$985$	−49831.7	−1.61195
$986$	0	0
$987$	−15663.5	−0.505141
$988$	0	0
$989$	−19019.3	−0.611505
$990$	0	0
$991$	−47078.3	−1.50907	−0.754537	−	0.656258i	$-0.772138\pi$
$991$	−47078.3	−1.50907	−0.754537	+	0.656258i	$0.772138\pi$
$992$	0	0
$993$	29636.2	0.947107
$994$	0	0
$995$	−48080.9	−1.53193
$996$	0	0
$997$	−40091.9	−1.27354	−0.636771	−	0.771053i	$-0.719730\pi$
$997$	−40091.9	−1.27354	−0.636771	+	0.771053i	$0.719730\pi$
$998$	0	0
$999$	6993.44	0.221484

Twists

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

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +11.4919 q^{5} +13.4919 q^{7} +9.00000 q^{9} -10.9839 q^{11} -2.00000 q^{13} +34.4758 q^{15} +106.952 q^{17} +86.9839 q^{19} +40.4758 q^{21} -64.9516 q^{23} +7.06453 q^{25} +27.0000 q^{27} -129.395 q^{29} -246.444 q^{31} -32.9516 q^{33} +155.048 q^{35} +259.016 q^{37} -6.00000 q^{39} +324.984 q^{41} +292.823 q^{43} +103.427 q^{45} -386.984 q^{47} -160.968 q^{49} +320.855 q^{51} +536.508 q^{53} -126.226 q^{55} +260.952 q^{57} -103.806 q^{59} +628.790 q^{61} +121.427 q^{63} -22.9839 q^{65} -719.548 q^{67} -194.855 q^{69} -200.952 q^{71} +987.581 q^{73} +21.1936 q^{75} -148.194 q^{77} -770.379 q^{79} +81.0000 q^{81} +1284.50 q^{83} +1229.08 q^{85} -388.185 q^{87} +844.161 q^{89} -26.9839 q^{91} -739.331 q^{93} +999.613 q^{95} -1259.84 q^{97} -98.8548 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 8 q^{5} - 4 q^{7} + 18 q^{9} + 40 q^{11} - 4 q^{13} - 24 q^{15} + 28 q^{17} + 112 q^{19} - 12 q^{21} + 56 q^{23} + 262 q^{25} + 54 q^{27} + 144 q^{29} - 276 q^{31} + 120 q^{33} + 496 q^{35}+ \cdots + 360 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

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Representations

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Database

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