LMFDB - Embedded newform 384.4.a.l.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,-6,0,16] 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{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
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			$2.64575$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +18.5830 q^{5} +29.7490 q^{7} +9.00000 q^{9} +9.16601 q^{11} +80.3320 q^{13} -55.7490 q^{15} -31.1660 q^{17} -89.8301 q^{19} -89.2470 q^{21} -57.1660 q^{23} +220.328 q^{25} -27.0000 q^{27} +167.085 q^{29} -270.915 q^{31} -27.4980 q^{33} +552.826 q^{35} -157.498 q^{37} -240.996 q^{39} -404.154 q^{41} +317.166 q^{43} +167.247 q^{45} +63.1581 q^{47} +542.004 q^{49} +93.4980 q^{51} +616.737 q^{53} +170.332 q^{55} +269.490 q^{57} -137.992 q^{59} +200.170 q^{61} +267.741 q^{63} +1492.81 q^{65} +576.664 q^{67} +171.498 q^{69} -305.166 q^{71} -198.988 q^{73} -660.984 q^{75} +272.680 q^{77} -335.563 q^{79} +81.0000 q^{81} -981.474 q^{83} -579.158 q^{85} -501.255 q^{87} -51.0197 q^{89} +2389.80 q^{91} +812.745 q^{93} -1669.31 q^{95} +678.340 q^{97} +82.4941 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 16 q^{5} - 4 q^{7} + 18 q^{9} - 24 q^{11} + 76 q^{13} - 48 q^{15} - 20 q^{17} + 32 q^{19} + 12 q^{21} - 72 q^{23} + 102 q^{25} - 54 q^{27} + 440 q^{29} - 436 q^{31} + 72 q^{33} + 640 q^{35}+ \cdots - 216 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 16 * q^5 - 4 * q^7 + 18 * q^9 - 24 * q^11 + 76 * q^13 - 48 * q^15 - 20 * q^17 + 32 * q^19 + 12 * q^21 - 72 * q^23 + 102 * q^25 - 54 * q^27 + 440 * q^29 - 436 * q^31 + 72 * q^33 + 640 * q^35 - 188 * q^37 - 228 * q^39 - 4 * q^41 + 592 * q^43 + 144 * q^45 - 424 * q^47 + 1338 * q^49 + 60 * q^51 + 408 * q^53 + 256 * q^55 - 96 * q^57 + 232 * q^59 + 612 * q^61 - 36 * q^63 + 1504 * q^65 + 984 * q^67 + 216 * q^69 - 568 * q^71 + 364 * q^73 - 306 * q^75 + 1392 * q^77 + 620 * q^79 + 162 * q^81 - 312 * q^83 - 608 * q^85 - 1320 * q^87 - 1372 * q^89 + 2536 * q^91 + 1308 * q^93 - 1984 * q^95 + 1780 * q^97 - 216 * 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
$5$	18.5830	1.66211	0.831057	−	0.556187i	$-0.187736\pi$
$5$	18.5830	1.66211	0.831057	+	0.556187i	$0.187736\pi$
$6$	0	0
$7$	29.7490	1.60630	0.803148	−	0.595780i	$-0.203157\pi$
$7$	29.7490	1.60630	0.803148	+	0.595780i	$0.203157\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	9.16601	0.251241	0.125621	−	0.992078i	$-0.459908\pi$
$11$	9.16601	0.251241	0.125621	+	0.992078i	$0.459908\pi$
$12$	0	0
$13$	80.3320	1.71385	0.856927	−	0.515438i	$-0.172371\pi$
$13$	80.3320	1.71385	0.856927	+	0.515438i	$0.172371\pi$
$14$	0	0
$15$	−55.7490	−0.959622
$16$	0	0
$17$	−31.1660	−0.444639	−0.222320	−	0.974974i	$-0.571363\pi$
$17$	−31.1660	−0.444639	−0.222320	+	0.974974i	$0.571363\pi$
$18$	0	0
$19$	−89.8301	−1.08465	−0.542327	−	0.840167i	$-0.682457\pi$
$19$	−89.8301	−1.08465	−0.542327	+	0.840167i	$0.682457\pi$
$20$	0	0
$21$	−89.2470	−0.927395
$22$	0	0
$23$	−57.1660	−0.518258	−0.259129	−	0.965843i	$-0.583436\pi$
$23$	−57.1660	−0.518258	−0.259129	+	0.965843i	$0.583436\pi$
$24$	0	0
$25$	220.328	1.76262
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	167.085	1.06989	0.534947	−	0.844886i	$-0.320332\pi$
$29$	167.085	1.06989	0.534947	+	0.844886i	$0.320332\pi$
$30$	0	0
$31$	−270.915	−1.56961	−0.784803	−	0.619746i	$-0.787236\pi$
$31$	−270.915	−1.56961	−0.784803	+	0.619746i	$0.787236\pi$
$32$	0	0
$33$	−27.4980	−0.145054
$34$	0	0
$35$	552.826	2.66985
$36$	0	0
$37$	−157.498	−0.699798	−0.349899	−	0.936787i	$-0.613784\pi$
$37$	−157.498	−0.699798	−0.349899	+	0.936787i	$0.613784\pi$
$38$	0	0
$39$	−240.996	−0.989494
$40$	0	0
$41$	−404.154	−1.53947	−0.769735	−	0.638363i	$-0.779612\pi$
$41$	−404.154	−1.53947	−0.769735	+	0.638363i	$0.779612\pi$
$42$	0	0
$43$	317.166	1.12482	0.562411	−	0.826858i	$-0.309874\pi$
$43$	317.166	1.12482	0.562411	+	0.826858i	$0.309874\pi$
$44$	0	0
$45$	167.247	0.554038
$46$	0	0
$47$	63.1581	0.196012	0.0980060	−	0.995186i	$-0.468754\pi$
$47$	63.1581	0.196012	0.0980060	+	0.995186i	$0.468754\pi$
$48$	0	0
$49$	542.004	1.58019
$50$	0	0
$51$	93.4980	0.256713
$52$	0	0
$53$	616.737	1.59840	0.799202	−	0.601063i	$-0.205256\pi$
$53$	616.737	1.59840	0.799202	+	0.601063i	$0.205256\pi$
$54$	0	0
$55$	170.332	0.417592
$56$	0	0
$57$	269.490	0.626225
$58$	0	0
$59$	−137.992	−0.304492	−0.152246	−	0.988343i	$-0.548651\pi$
$59$	−137.992	−0.304492	−0.152246	+	0.988343i	$0.548651\pi$
$60$	0	0
$61$	200.170	0.420150	0.210075	−	0.977685i	$-0.432629\pi$
$61$	200.170	0.420150	0.210075	+	0.977685i	$0.432629\pi$
$62$	0	0
$63$	267.741	0.535432
$64$	0	0
$65$	1492.81	2.84862
$66$	0	0
$67$	576.664	1.05150	0.525752	−	0.850638i	$-0.323784\pi$
$67$	576.664	1.05150	0.525752	+	0.850638i	$0.323784\pi$
$68$	0	0
$69$	171.498	0.299216
$70$	0	0
$71$	−305.166	−0.510092	−0.255046	−	0.966929i	$-0.582091\pi$
$71$	−305.166	−0.510092	−0.255046	+	0.966929i	$0.582091\pi$
$72$	0	0
$73$	−198.988	−0.319038	−0.159519	−	0.987195i	$-0.550994\pi$
$73$	−198.988	−0.319038	−0.159519	+	0.987195i	$0.550994\pi$
$74$	0	0
$75$	−660.984	−1.01765
$76$	0	0
$77$	272.680	0.403568
$78$	0	0
$79$	−335.563	−0.477896	−0.238948	−	0.971032i	$-0.576803\pi$
$79$	−335.563	−0.477896	−0.238948	+	0.971032i	$0.576803\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−981.474	−1.29796	−0.648981	−	0.760805i	$-0.724804\pi$
$83$	−981.474	−1.29796	−0.648981	+	0.760805i	$0.724804\pi$
$84$	0	0
$85$	−579.158	−0.739041
$86$	0	0
$87$	−501.255	−0.617703
$88$	0	0
$89$	−51.0197	−0.0607649	−0.0303824	−	0.999538i	$-0.509673\pi$
$89$	−51.0197	−0.0607649	−0.0303824	+	0.999538i	$0.509673\pi$
$90$	0	0
$91$	2389.80	2.75296
$92$	0	0
$93$	812.745	0.906212
$94$	0	0
$95$	−1669.31	−1.80282
$96$	0	0
$97$	678.340	0.710051	0.355026	−	0.934857i	$-0.384472\pi$
$97$	678.340	0.710051	0.355026	+	0.934857i	$0.384472\pi$
$98$	0	0
$99$	82.4941	0.0837472
$100$	0	0
$101$	−1334.91	−1.31513	−0.657565	−	0.753397i	$-0.728414\pi$
$101$	−1334.91	−1.31513	−0.657565	+	0.753397i	$0.728414\pi$
$102$	0	0
$103$	901.231	0.862145	0.431073	−	0.902317i	$-0.358135\pi$
$103$	901.231	0.862145	0.431073	+	0.902317i	$0.358135\pi$
$104$	0	0
$105$	−1658.48	−1.54144
$106$	0	0
$107$	−605.684	−0.547230	−0.273615	−	0.961839i	$-0.588219\pi$
$107$	−605.684	−0.547230	−0.273615	+	0.961839i	$0.588219\pi$
$108$	0	0
$109$	400.656	0.352072	0.176036	−	0.984384i	$-0.443672\pi$
$109$	400.656	0.352072	0.176036	+	0.984384i	$0.443672\pi$
$110$	0	0
$111$	472.494	0.404028
$112$	0	0
$113$	−1442.96	−1.20126	−0.600628	−	0.799529i	$-0.705083\pi$
$113$	−1442.96	−1.20126	−0.600628	+	0.799529i	$0.705083\pi$
$114$	0	0
$115$	−1062.32	−0.861404
$116$	0	0
$117$	722.988	0.571284
$118$	0	0
$119$	−927.158	−0.714222
$120$	0	0
$121$	−1246.98	−0.936878
$122$	0	0
$123$	1212.46	0.888814
$124$	0	0
$125$	1771.48	1.26757
$126$	0	0
$127$	2436.55	1.70243	0.851216	−	0.524815i	$-0.175865\pi$
$127$	2436.55	1.70243	0.851216	+	0.524815i	$0.175865\pi$
$128$	0	0
$129$	−951.498	−0.649417
$130$	0	0
$131$	983.953	0.656247	0.328123	−	0.944635i	$-0.393584\pi$
$131$	983.953	0.656247	0.328123	+	0.944635i	$0.393584\pi$
$132$	0	0
$133$	−2672.36	−1.74228
$134$	0	0
$135$	−501.741	−0.319874
$136$	0	0
$137$	−633.205	−0.394879	−0.197439	−	0.980315i	$-0.563263\pi$
$137$	−633.205	−0.394879	−0.197439	+	0.980315i	$0.563263\pi$
$138$	0	0
$139$	−1360.60	−0.830249	−0.415125	−	0.909765i	$-0.636262\pi$
$139$	−1360.60	−0.830249	−0.415125	+	0.909765i	$0.636262\pi$
$140$	0	0
$141$	−189.474	−0.113168
$142$	0	0
$143$	736.324	0.430591
$144$	0	0
$145$	3104.94	1.77828
$146$	0	0
$147$	−1626.01	−0.912321
$148$	0	0
$149$	−3079.33	−1.69308	−0.846539	−	0.532327i	$-0.821318\pi$
$149$	−3079.33	−1.69308	−0.846539	+	0.532327i	$0.821318\pi$
$150$	0	0
$151$	538.769	0.290360	0.145180	−	0.989405i	$-0.453624\pi$
$151$	538.769	0.290360	0.145180	+	0.989405i	$0.453624\pi$
$152$	0	0
$153$	−280.494	−0.148213
$154$	0	0
$155$	−5034.42	−2.60886
$156$	0	0
$157$	−586.842	−0.298313	−0.149156	−	0.988814i	$-0.547656\pi$
$157$	−586.842	−0.298313	−0.149156	+	0.988814i	$0.547656\pi$
$158$	0	0
$159$	−1850.21	−0.922839
$160$	0	0
$161$	−1700.63	−0.832476
$162$	0	0
$163$	585.830	0.281508	0.140754	−	0.990045i	$-0.455047\pi$
$163$	585.830	0.281508	0.140754	+	0.990045i	$0.455047\pi$
$164$	0	0
$165$	−510.996	−0.241097
$166$	0	0
$167$	2369.36	1.09788	0.548942	−	0.835861i	$-0.315031\pi$
$167$	2369.36	1.09788	0.548942	+	0.835861i	$0.315031\pi$
$168$	0	0
$169$	4256.23	1.93729
$170$	0	0
$171$	−808.470	−0.361551
$172$	0	0
$173$	2329.84	1.02390	0.511949	−	0.859016i	$-0.328924\pi$
$173$	2329.84	1.02390	0.511949	+	0.859016i	$0.328924\pi$
$174$	0	0
$175$	6554.54	2.83130
$176$	0	0
$177$	413.976	0.175799
$178$	0	0
$179$	−317.976	−0.132775	−0.0663873	−	0.997794i	$-0.521147\pi$
$179$	−317.976	−0.132775	−0.0663873	+	0.997794i	$0.521147\pi$
$180$	0	0
$181$	−157.399	−0.0646374	−0.0323187	−	0.999478i	$-0.510289\pi$
$181$	−157.399	−0.0646374	−0.0323187	+	0.999478i	$0.510289\pi$
$182$	0	0
$183$	−600.510	−0.242574
$184$	0	0
$185$	−2926.79	−1.16314
$186$	0	0
$187$	−285.668	−0.111712
$188$	0	0
$189$	−803.223	−0.309132
$190$	0	0
$191$	−636.727	−0.241214	−0.120607	−	0.992700i	$-0.538484\pi$
$191$	−636.727	−0.241214	−0.120607	+	0.992700i	$0.538484\pi$
$192$	0	0
$193$	−1344.93	−0.501608	−0.250804	−	0.968038i	$-0.580695\pi$
$193$	−1344.93	−0.501608	−0.250804	+	0.968038i	$0.580695\pi$
$194$	0	0
$195$	−4478.43	−1.64465
$196$	0	0
$197$	1068.37	0.386385	0.193193	−	0.981161i	$-0.438116\pi$
$197$	1068.37	0.386385	0.193193	+	0.981161i	$0.438116\pi$
$198$	0	0
$199$	1537.60	0.547727	0.273864	−	0.961769i	$-0.411698\pi$
$199$	1537.60	0.547727	0.273864	+	0.961769i	$0.411698\pi$
$200$	0	0
$201$	−1729.99	−0.607086
$202$	0	0
$203$	4970.61	1.71856
$204$	0	0
$205$	−7510.40	−2.55878
$206$	0	0
$207$	−514.494	−0.172753
$208$	0	0
$209$	−823.383	−0.272510
$210$	0	0
$211$	5622.56	1.83447	0.917235	−	0.398347i	$-0.130416\pi$
$211$	5622.56	1.83447	0.917235	+	0.398347i	$0.130416\pi$
$212$	0	0
$213$	915.498	0.294502
$214$	0	0
$215$	5893.90	1.86958
$216$	0	0
$217$	−8059.46	−2.52125
$218$	0	0
$219$	596.965	0.184197
$220$	0	0
$221$	−2503.63	−0.762047
$222$	0	0
$223$	−4587.32	−1.37753	−0.688766	−	0.724983i	$-0.741847\pi$
$223$	−4587.32	−1.37753	−0.688766	+	0.724983i	$0.741847\pi$
$224$	0	0
$225$	1982.95	0.587542
$226$	0	0
$227$	488.873	0.142941	0.0714706	−	0.997443i	$-0.477231\pi$
$227$	488.873	0.142941	0.0714706	+	0.997443i	$0.477231\pi$
$228$	0	0
$229$	−4316.89	−1.24571	−0.622856	−	0.782337i	$-0.714028\pi$
$229$	−4316.89	−1.24571	−0.622856	+	0.782337i	$0.714028\pi$
$230$	0	0
$231$	−818.039	−0.233000
$232$	0	0
$233$	−4591.62	−1.29102	−0.645509	−	0.763753i	$-0.723355\pi$
$233$	−4591.62	−1.29102	−0.645509	+	0.763753i	$0.723355\pi$
$234$	0	0
$235$	1173.67	0.325794
$236$	0	0
$237$	1006.69	0.275914
$238$	0	0
$239$	−6358.19	−1.72082	−0.860412	−	0.509598i	$-0.829794\pi$
$239$	−6358.19	−1.72082	−0.860412	+	0.509598i	$0.829794\pi$
$240$	0	0
$241$	3118.65	0.833568	0.416784	−	0.909006i	$-0.363157\pi$
$241$	3118.65	0.833568	0.416784	+	0.909006i	$0.363157\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	10072.1	2.62645
$246$	0	0
$247$	−7216.23	−1.85894
$248$	0	0
$249$	2944.42	0.749378
$250$	0	0
$251$	−6854.71	−1.72377	−0.861883	−	0.507107i	$-0.830715\pi$
$251$	−6854.71	−1.72377	−0.861883	+	0.507107i	$0.830715\pi$
$252$	0	0
$253$	−523.984	−0.130208
$254$	0	0
$255$	1737.47	0.426686
$256$	0	0
$257$	−234.696	−0.0569646	−0.0284823	−	0.999594i	$-0.509067\pi$
$257$	−234.696	−0.0569646	−0.0284823	+	0.999594i	$0.509067\pi$
$258$	0	0
$259$	−4685.41	−1.12408
$260$	0	0
$261$	1503.76	0.356631
$262$	0	0
$263$	5862.58	1.37453	0.687266	−	0.726406i	$-0.258811\pi$
$263$	5862.58	1.37453	0.687266	+	0.726406i	$0.258811\pi$
$264$	0	0
$265$	11460.8	2.65673
$266$	0	0
$267$	153.059	0.0350826
$268$	0	0
$269$	−1399.79	−0.317273	−0.158636	−	0.987337i	$-0.550710\pi$
$269$	−1399.79	−0.317273	−0.158636	+	0.987337i	$0.550710\pi$
$270$	0	0
$271$	7927.86	1.77706	0.888529	−	0.458820i	$-0.151728\pi$
$271$	7927.86	1.77706	0.888529	+	0.458820i	$0.151728\pi$
$272$	0	0
$273$	−7169.40	−1.58942
$274$	0	0
$275$	2019.53	0.442844
$276$	0	0
$277$	−1209.19	−0.262287	−0.131143	−	0.991363i	$-0.541865\pi$
$277$	−1209.19	−0.262287	−0.131143	+	0.991363i	$0.541865\pi$
$278$	0	0
$279$	−2438.24	−0.523202
$280$	0	0
$281$	8983.25	1.90710	0.953551	−	0.301231i	$-0.0973976\pi$
$281$	8983.25	1.90710	0.953551	+	0.301231i	$0.0973976\pi$
$282$	0	0
$283$	2028.65	0.426115	0.213058	−	0.977040i	$-0.431658\pi$
$283$	2028.65	0.426115	0.213058	+	0.977040i	$0.431658\pi$
$284$	0	0
$285$	5007.94	1.04086
$286$	0	0
$287$	−12023.2	−2.47284
$288$	0	0
$289$	−3941.68	−0.802296
$290$	0	0
$291$	−2035.02	−0.409948
$292$	0	0
$293$	14.8238	0.00295569	0.00147784	−	0.999999i	$-0.499530\pi$
$293$	14.8238	0.00295569	0.00147784	+	0.999999i	$0.499530\pi$
$294$	0	0
$295$	−2564.31	−0.506101
$296$	0	0
$297$	−247.482	−0.0483514
$298$	0	0
$299$	−4592.26	−0.888218
$300$	0	0
$301$	9435.38	1.80680
$302$	0	0
$303$	4004.72	0.759291
$304$	0	0
$305$	3719.76	0.698337
$306$	0	0
$307$	5597.88	1.04068	0.520339	−	0.853960i	$-0.325806\pi$
$307$	5597.88	1.04068	0.520339	+	0.853960i	$0.325806\pi$
$308$	0	0
$309$	−2703.69	−0.497760
$310$	0	0
$311$	−6603.50	−1.20402	−0.602010	−	0.798488i	$-0.705633\pi$
$311$	−6603.50	−1.20402	−0.602010	+	0.798488i	$0.705633\pi$
$312$	0	0
$313$	6034.93	1.08982	0.544911	−	0.838494i	$-0.316564\pi$
$313$	6034.93	1.08982	0.544911	+	0.838494i	$0.316564\pi$
$314$	0	0
$315$	4975.44	0.889949
$316$	0	0
$317$	−4150.88	−0.735446	−0.367723	−	0.929935i	$-0.619863\pi$
$317$	−4150.88	−0.735446	−0.367723	+	0.929935i	$0.619863\pi$
$318$	0	0
$319$	1531.50	0.268802
$320$	0	0
$321$	1817.05	0.315944
$322$	0	0
$323$	2799.64	0.482280
$324$	0	0
$325$	17699.4	3.02088
$326$	0	0
$327$	−1201.97	−0.203269
$328$	0	0
$329$	1878.89	0.314853
$330$	0	0
$331$	5448.40	0.904747	0.452374	−	0.891829i	$-0.350577\pi$
$331$	5448.40	0.904747	0.452374	+	0.891829i	$0.350577\pi$
$332$	0	0
$333$	−1417.48	−0.233266
$334$	0	0
$335$	10716.2	1.74772
$336$	0	0
$337$	−9446.83	−1.52701	−0.763504	−	0.645803i	$-0.776523\pi$
$337$	−9446.83	−1.52701	−0.763504	+	0.645803i	$0.776523\pi$
$338$	0	0
$339$	4328.87	0.693546
$340$	0	0
$341$	−2483.21	−0.394350
$342$	0	0
$343$	5920.17	0.931951
$344$	0	0
$345$	3186.95	0.497332
$346$	0	0
$347$	902.478	0.139618	0.0698092	−	0.997560i	$-0.477761\pi$
$347$	902.478	0.139618	0.0698092	+	0.997560i	$0.477761\pi$
$348$	0	0
$349$	2296.76	0.352271	0.176135	−	0.984366i	$-0.443640\pi$
$349$	2296.76	0.352271	0.176135	+	0.984366i	$0.443640\pi$
$350$	0	0
$351$	−2168.96	−0.329831
$352$	0	0
$353$	−683.699	−0.103087	−0.0515434	−	0.998671i	$-0.516414\pi$
$353$	−683.699	−0.103087	−0.0515434	+	0.998671i	$0.516414\pi$
$354$	0	0
$355$	−5670.90	−0.847832
$356$	0	0
$357$	2781.47	0.412356
$358$	0	0
$359$	−10440.2	−1.53486	−0.767430	−	0.641133i	$-0.778465\pi$
$359$	−10440.2	−1.53486	−0.767430	+	0.641133i	$0.778465\pi$
$360$	0	0
$361$	1210.44	0.176474
$362$	0	0
$363$	3740.95	0.540907
$364$	0	0
$365$	−3697.80	−0.530278
$366$	0	0
$367$	−4398.79	−0.625654	−0.312827	−	0.949810i	$-0.601276\pi$
$367$	−4398.79	−0.625654	−0.312827	+	0.949810i	$0.601276\pi$
$368$	0	0
$369$	−3637.39	−0.513157
$370$	0	0
$371$	18347.3	2.56751
$372$	0	0
$373$	−341.111	−0.0473513	−0.0236757	−	0.999720i	$-0.507537\pi$
$373$	−341.111	−0.0473513	−0.0236757	+	0.999720i	$0.507537\pi$
$374$	0	0
$375$	−5314.45	−0.731832
$376$	0	0
$377$	13422.3	1.83364
$378$	0	0
$379$	11900.1	1.61284	0.806421	−	0.591342i	$-0.201402\pi$
$379$	11900.1	1.61284	0.806421	+	0.591342i	$0.201402\pi$
$380$	0	0
$381$	−7309.65	−0.982900
$382$	0	0
$383$	−13863.1	−1.84953	−0.924767	−	0.380535i	$-0.875740\pi$
$383$	−13863.1	−1.84953	−0.924767	+	0.380535i	$0.875740\pi$
$384$	0	0
$385$	5067.21	0.670776
$386$	0	0
$387$	2854.49	0.374941
$388$	0	0
$389$	−335.006	−0.0436645	−0.0218322	−	0.999762i	$-0.506950\pi$
$389$	−335.006	−0.0436645	−0.0218322	+	0.999762i	$0.506950\pi$
$390$	0	0
$391$	1781.64	0.230438
$392$	0	0
$393$	−2951.86	−0.378884
$394$	0	0
$395$	−6235.77	−0.794319
$396$	0	0
$397$	−8731.50	−1.10383	−0.551916	−	0.833900i	$-0.686103\pi$
$397$	−8731.50	−1.10383	−0.551916	+	0.833900i	$0.686103\pi$
$398$	0	0
$399$	8017.07	1.00590
$400$	0	0
$401$	382.565	0.0476419	0.0238209	−	0.999716i	$-0.492417\pi$
$401$	382.565	0.0476419	0.0238209	+	0.999716i	$0.492417\pi$
$402$	0	0
$403$	−21763.2	−2.69007
$404$	0	0
$405$	1505.22	0.184679
$406$	0	0
$407$	−1443.63	−0.175818
$408$	0	0
$409$	−1370.47	−0.165685	−0.0828425	−	0.996563i	$-0.526400\pi$
$409$	−1370.47	−0.165685	−0.0828425	+	0.996563i	$0.526400\pi$
$410$	0	0
$411$	1899.62	0.227983
$412$	0	0
$413$	−4105.13	−0.489105
$414$	0	0
$415$	−18238.7	−2.15736
$416$	0	0
$417$	4081.80	0.479345
$418$	0	0
$419$	2389.63	0.278619	0.139309	−	0.990249i	$-0.455512\pi$
$419$	2389.63	0.278619	0.139309	+	0.990249i	$0.455512\pi$
$420$	0	0
$421$	−1144.48	−0.132491	−0.0662455	−	0.997803i	$-0.521102\pi$
$421$	−1144.48	−0.132491	−0.0662455	+	0.997803i	$0.521102\pi$
$422$	0	0
$423$	568.423	0.0653373
$424$	0	0
$425$	−6866.75	−0.783732
$426$	0	0
$427$	5954.86	0.674885
$428$	0	0
$429$	−2208.97	−0.248602
$430$	0	0
$431$	9624.10	1.07558	0.537792	−	0.843078i	$-0.319259\pi$
$431$	9624.10	1.07558	0.537792	+	0.843078i	$0.319259\pi$
$432$	0	0
$433$	3276.44	0.363638	0.181819	−	0.983332i	$-0.441801\pi$
$433$	3276.44	0.363638	0.181819	+	0.983332i	$0.441801\pi$
$434$	0	0
$435$	−9314.82	−1.02669
$436$	0	0
$437$	5135.23	0.562131
$438$	0	0
$439$	−12192.7	−1.32557	−0.662787	−	0.748808i	$-0.730627\pi$
$439$	−12192.7	−1.32557	−0.662787	+	0.748808i	$0.730627\pi$
$440$	0	0
$441$	4878.04	0.526729
$442$	0	0
$443$	−101.632	−0.0108999	−0.00544997	−	0.999985i	$-0.501735\pi$
$443$	−101.632	−0.0108999	−0.00544997	+	0.999985i	$0.501735\pi$
$444$	0	0
$445$	−948.099	−0.100998
$446$	0	0
$447$	9237.99	0.977499
$448$	0	0
$449$	−7096.37	−0.745876	−0.372938	−	0.927856i	$-0.621650\pi$
$449$	−7096.37	−0.745876	−0.372938	+	0.927856i	$0.621650\pi$
$450$	0	0
$451$	−3704.48	−0.386779
$452$	0	0
$453$	−1616.31	−0.167639
$454$	0	0
$455$	44409.6	4.57573
$456$	0	0
$457$	−7377.47	−0.755150	−0.377575	−	0.925979i	$-0.623242\pi$
$457$	−7377.47	−0.755150	−0.377575	+	0.925979i	$0.623242\pi$
$458$	0	0
$459$	841.482	0.0855709
$460$	0	0
$461$	13627.7	1.37680	0.688398	−	0.725333i	$-0.258314\pi$
$461$	13627.7	1.37680	0.688398	+	0.725333i	$0.258314\pi$
$462$	0	0
$463$	2160.46	0.216858	0.108429	−	0.994104i	$-0.465418\pi$
$463$	2160.46	0.216858	0.108429	+	0.994104i	$0.465418\pi$
$464$	0	0
$465$	15103.2	1.50623
$466$	0	0
$467$	−2242.53	−0.222209	−0.111105	−	0.993809i	$-0.535439\pi$
$467$	−2242.53	−0.222209	−0.111105	+	0.993809i	$0.535439\pi$
$468$	0	0
$469$	17155.2	1.68903
$470$	0	0
$471$	1760.53	0.172231
$472$	0	0
$473$	2907.15	0.282602
$474$	0	0
$475$	−19792.1	−1.91184
$476$	0	0
$477$	5550.63	0.532801
$478$	0	0
$479$	2155.89	0.205648	0.102824	−	0.994700i	$-0.467212\pi$
$479$	2155.89	0.205648	0.102824	+	0.994700i	$0.467212\pi$
$480$	0	0
$481$	−12652.1	−1.19935
$482$	0	0
$483$	5101.90	0.480630
$484$	0	0
$485$	12605.6	1.18019
$486$	0	0
$487$	−4911.41	−0.456996	−0.228498	−	0.973544i	$-0.573381\pi$
$487$	−4911.41	−0.456996	−0.228498	+	0.973544i	$0.573381\pi$
$488$	0	0
$489$	−1757.49	−0.162529
$490$	0	0
$491$	−3840.06	−0.352952	−0.176476	−	0.984305i	$-0.556470\pi$
$491$	−3840.06	−0.352952	−0.176476	+	0.984305i	$0.556470\pi$
$492$	0	0
$493$	−5207.37	−0.475717
$494$	0	0
$495$	1532.99	0.139197
$496$	0	0
$497$	−9078.39	−0.819359
$498$	0	0
$499$	−10463.4	−0.938687	−0.469343	−	0.883016i	$-0.655509\pi$
$499$	−10463.4	−0.938687	−0.469343	+	0.883016i	$0.655509\pi$
$500$	0	0
$501$	−7108.08	−0.633863
$502$	0	0
$503$	2238.90	0.198464	0.0992321	−	0.995064i	$-0.468361\pi$
$503$	2238.90	0.198464	0.0992321	+	0.995064i	$0.468361\pi$
$504$	0	0
$505$	−24806.6	−2.18590
$506$	0	0
$507$	−12768.7	−1.11850
$508$	0	0
$509$	−7481.43	−0.651490	−0.325745	−	0.945458i	$-0.605615\pi$
$509$	−7481.43	−0.651490	−0.325745	+	0.945458i	$0.605615\pi$
$510$	0	0
$511$	−5919.70	−0.512470
$512$	0	0
$513$	2425.41	0.208742
$514$	0	0
$515$	16747.6	1.43298
$516$	0	0
$517$	578.908	0.0492463
$518$	0	0
$519$	−6989.52	−0.591148
$520$	0	0
$521$	2231.61	0.187656	0.0938278	−	0.995588i	$-0.470090\pi$
$521$	2231.61	0.187656	0.0938278	+	0.995588i	$0.470090\pi$
$522$	0	0
$523$	−11380.5	−0.951499	−0.475749	−	0.879581i	$-0.657823\pi$
$523$	−11380.5	−0.951499	−0.475749	+	0.879581i	$0.657823\pi$
$524$	0	0
$525$	−19663.6	−1.63465
$526$	0	0
$527$	8443.34	0.697908
$528$	0	0
$529$	−8899.05	−0.731409
$530$	0	0
$531$	−1241.93	−0.101497
$532$	0	0
$533$	−32466.5	−2.63843
$534$	0	0
$535$	−11255.4	−0.909560
$536$	0	0
$537$	953.929	0.0766575
$538$	0	0
$539$	4968.01	0.397008
$540$	0	0
$541$	−2751.65	−0.218674	−0.109337	−	0.994005i	$-0.534873\pi$
$541$	−2751.65	−0.218674	−0.109337	+	0.994005i	$0.534873\pi$
$542$	0	0
$543$	472.197	0.0373184
$544$	0	0
$545$	7445.40	0.585185
$546$	0	0
$547$	20617.0	1.61155	0.805776	−	0.592220i	$-0.201748\pi$
$547$	20617.0	1.61155	0.805776	+	0.592220i	$0.201748\pi$
$548$	0	0
$549$	1801.53	0.140050
$550$	0	0
$551$	−15009.3	−1.16046
$552$	0	0
$553$	−9982.68	−0.767643
$554$	0	0
$555$	8780.36	0.671542
$556$	0	0
$557$	12165.3	0.925420	0.462710	−	0.886510i	$-0.346877\pi$
$557$	12165.3	0.925420	0.462710	+	0.886510i	$0.346877\pi$
$558$	0	0
$559$	25478.6	1.92778
$560$	0	0
$561$	857.004	0.0644969
$562$	0	0
$563$	−19095.1	−1.42942	−0.714708	−	0.699423i	$-0.753440\pi$
$563$	−19095.1	−1.42942	−0.714708	+	0.699423i	$0.753440\pi$
$564$	0	0
$565$	−26814.5	−1.99663
$566$	0	0
$567$	2409.67	0.178477
$568$	0	0
$569$	−18354.4	−1.35229	−0.676146	−	0.736767i	$-0.736351\pi$
$569$	−18354.4	−1.35229	−0.676146	+	0.736767i	$0.736351\pi$
$570$	0	0
$571$	−7264.64	−0.532427	−0.266213	−	0.963914i	$-0.585773\pi$
$571$	−7264.64	−0.532427	−0.266213	+	0.963914i	$0.585773\pi$
$572$	0	0
$573$	1910.18	0.139265
$574$	0	0
$575$	−12595.3	−0.913495
$576$	0	0
$577$	−4303.73	−0.310514	−0.155257	−	0.987874i	$-0.549621\pi$
$577$	−4303.73	−0.310514	−0.155257	+	0.987874i	$0.549621\pi$
$578$	0	0
$579$	4034.80	0.289604
$580$	0	0
$581$	−29197.9	−2.08491
$582$	0	0
$583$	5653.02	0.401585
$584$	0	0
$585$	13435.3	0.949540
$586$	0	0
$587$	−23123.2	−1.62589	−0.812943	−	0.582343i	$-0.802136\pi$
$587$	−23123.2	−1.62589	−0.812943	+	0.582343i	$0.802136\pi$
$588$	0	0
$589$	24336.3	1.70248
$590$	0	0
$591$	−3205.10	−0.223080
$592$	0	0
$593$	−15038.9	−1.04144	−0.520721	−	0.853727i	$-0.674337\pi$
$593$	−15038.9	−1.04144	−0.520721	+	0.853727i	$0.674337\pi$
$594$	0	0
$595$	−17229.4	−1.18712
$596$	0	0
$597$	−4612.81	−0.316231
$598$	0	0
$599$	26909.4	1.83554	0.917769	−	0.397114i	$-0.129988\pi$
$599$	26909.4	1.83554	0.917769	+	0.397114i	$0.129988\pi$
$600$	0	0
$601$	9863.37	0.669443	0.334721	−	0.942317i	$-0.391358\pi$
$601$	9863.37	0.669443	0.334721	+	0.942317i	$0.391358\pi$
$602$	0	0
$603$	5189.98	0.350501
$604$	0	0
$605$	−23172.7	−1.55720
$606$	0	0
$607$	−17169.9	−1.14812	−0.574058	−	0.818815i	$-0.694631\pi$
$607$	−17169.9	−1.14812	−0.574058	+	0.818815i	$0.694631\pi$
$608$	0	0
$609$	−14911.8	−0.992214
$610$	0	0
$611$	5073.62	0.335936
$612$	0	0
$613$	−6302.75	−0.415279	−0.207639	−	0.978205i	$-0.566578\pi$
$613$	−6302.75	−0.415279	−0.207639	+	0.978205i	$0.566578\pi$
$614$	0	0
$615$	22531.2	1.47731
$616$	0	0
$617$	24410.8	1.59277	0.796386	−	0.604789i	$-0.206742\pi$
$617$	24410.8	1.59277	0.796386	+	0.604789i	$0.206742\pi$
$618$	0	0
$619$	−16981.9	−1.10268	−0.551340	−	0.834280i	$-0.685883\pi$
$619$	−16981.9	−1.10268	−0.551340	+	0.834280i	$0.685883\pi$
$620$	0	0
$621$	1543.48	0.0997388
$622$	0	0
$623$	−1517.79	−0.0976064
$624$	0	0
$625$	5378.45	0.344221
$626$	0	0
$627$	2470.15	0.157334
$628$	0	0
$629$	4908.59	0.311158
$630$	0	0
$631$	−4791.04	−0.302264	−0.151132	−	0.988514i	$-0.548292\pi$
$631$	−4791.04	−0.302264	−0.151132	+	0.988514i	$0.548292\pi$
$632$	0	0
$633$	−16867.7	−1.05913
$634$	0	0
$635$	45278.4	2.82964
$636$	0	0
$637$	43540.3	2.70821
$638$	0	0
$639$	−2746.49	−0.170031
$640$	0	0
$641$	31537.4	1.94329	0.971647	−	0.236435i	$-0.0759790\pi$
$641$	31537.4	1.94329	0.971647	+	0.236435i	$0.0759790\pi$
$642$	0	0
$643$	−13428.9	−0.823617	−0.411809	−	0.911270i	$-0.635103\pi$
$643$	−13428.9	−0.823617	−0.411809	+	0.911270i	$0.635103\pi$
$644$	0	0
$645$	−17681.7	−1.07940
$646$	0	0
$647$	−1108.75	−0.0673719	−0.0336859	−	0.999432i	$-0.510725\pi$
$647$	−1108.75	−0.0673719	−0.0336859	+	0.999432i	$0.510725\pi$
$648$	0	0
$649$	−1264.84	−0.0765011
$650$	0	0
$651$	24178.4	1.45564
$652$	0	0
$653$	9563.52	0.573123	0.286562	−	0.958062i	$-0.407488\pi$
$653$	9563.52	0.573123	0.286562	+	0.958062i	$0.407488\pi$
$654$	0	0
$655$	18284.8	1.09076
$656$	0	0
$657$	−1790.89	−0.106346
$658$	0	0
$659$	7239.84	0.427957	0.213979	−	0.976838i	$-0.431358\pi$
$659$	7239.84	0.427957	0.213979	+	0.976838i	$0.431358\pi$
$660$	0	0
$661$	3653.77	0.215001	0.107500	−	0.994205i	$-0.465715\pi$
$661$	3653.77	0.215001	0.107500	+	0.994205i	$0.465715\pi$
$662$	0	0
$663$	7510.89	0.439968
$664$	0	0
$665$	−49660.4	−2.89586
$666$	0	0
$667$	−9551.58	−0.554481
$668$	0	0
$669$	13762.0	0.795319
$670$	0	0
$671$	1834.76	0.105559
$672$	0	0
$673$	−2498.60	−0.143111	−0.0715557	−	0.997437i	$-0.522796\pi$
$673$	−2498.60	−0.143111	−0.0715557	+	0.997437i	$0.522796\pi$
$674$	0	0
$675$	−5948.86	−0.339217
$676$	0	0
$677$	−4198.61	−0.238354	−0.119177	−	0.992873i	$-0.538026\pi$
$677$	−4198.61	−0.238354	−0.119177	+	0.992873i	$0.538026\pi$
$678$	0	0
$679$	20179.9	1.14055
$680$	0	0
$681$	−1466.62	−0.0825272
$682$	0	0
$683$	−22374.1	−1.25347	−0.626737	−	0.779231i	$-0.715610\pi$
$683$	−22374.1	−1.25347	−0.626737	+	0.779231i	$0.715610\pi$
$684$	0	0
$685$	−11766.9	−0.656334
$686$	0	0
$687$	12950.7	0.719212
$688$	0	0
$689$	49543.7	2.73943
$690$	0	0
$691$	27217.7	1.49842	0.749212	−	0.662330i	$-0.230432\pi$
$691$	27217.7	1.49842	0.749212	+	0.662330i	$0.230432\pi$
$692$	0	0
$693$	2454.12	0.134523
$694$	0	0
$695$	−25284.1	−1.37997
$696$	0	0
$697$	12595.9	0.684509
$698$	0	0
$699$	13774.9	0.745369
$700$	0	0
$701$	−7516.46	−0.404982	−0.202491	−	0.979284i	$-0.564904\pi$
$701$	−7516.46	−0.404982	−0.202491	+	0.979284i	$0.564904\pi$
$702$	0	0
$703$	14148.1	0.759038
$704$	0	0
$705$	−3521.00	−0.188097
$706$	0	0
$707$	−39712.2	−2.11249
$708$	0	0
$709$	29538.0	1.56463	0.782314	−	0.622884i	$-0.214039\pi$
$709$	29538.0	1.56463	0.782314	+	0.622884i	$0.214039\pi$
$710$	0	0
$711$	−3020.07	−0.159299
$712$	0	0
$713$	15487.1	0.813461
$714$	0	0
$715$	13683.1	0.715692
$716$	0	0
$717$	19074.6	0.993519
$718$	0	0
$719$	1327.71	0.0688669	0.0344334	−	0.999407i	$-0.489037\pi$
$719$	1327.71	0.0688669	0.0344334	+	0.999407i	$0.489037\pi$
$720$	0	0
$721$	26810.7	1.38486
$722$	0	0
$723$	−9355.94	−0.481260
$724$	0	0
$725$	36813.5	1.88582
$726$	0	0
$727$	−13636.7	−0.695678	−0.347839	−	0.937554i	$-0.613084\pi$
$727$	−13636.7	−0.695678	−0.347839	+	0.937554i	$0.613084\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−9884.80	−0.500140
$732$	0	0
$733$	−7587.21	−0.382319	−0.191160	−	0.981559i	$-0.561225\pi$
$733$	−7587.21	−0.382319	−0.191160	+	0.981559i	$0.561225\pi$
$734$	0	0
$735$	−30216.2	−1.51638
$736$	0	0
$737$	5285.71	0.264181
$738$	0	0
$739$	−1550.27	−0.0771686	−0.0385843	−	0.999255i	$-0.512285\pi$
$739$	−1550.27	−0.0771686	−0.0385843	+	0.999255i	$0.512285\pi$
$740$	0	0
$741$	21648.7	1.07326
$742$	0	0
$743$	−6931.44	−0.342247	−0.171124	−	0.985250i	$-0.554740\pi$
$743$	−6931.44	−0.342247	−0.171124	+	0.985250i	$0.554740\pi$
$744$	0	0
$745$	−57223.2	−2.81409
$746$	0	0
$747$	−8833.27	−0.432654
$748$	0	0
$749$	−18018.5	−0.879014
$750$	0	0
$751$	−19674.1	−0.955948	−0.477974	−	0.878374i	$-0.658629\pi$
$751$	−19674.1	−0.955948	−0.477974	+	0.878374i	$0.658629\pi$
$752$	0	0
$753$	20564.1	0.995217
$754$	0	0
$755$	10011.9	0.482612
$756$	0	0
$757$	−20509.6	−0.984722	−0.492361	−	0.870391i	$-0.663866\pi$
$757$	−20509.6	−0.984722	−0.492361	+	0.870391i	$0.663866\pi$
$758$	0	0
$759$	1571.95	0.0751756
$760$	0	0
$761$	10727.9	0.511018	0.255509	−	0.966807i	$-0.417757\pi$
$761$	10727.9	0.511018	0.255509	+	0.966807i	$0.417757\pi$
$762$	0	0
$763$	11919.1	0.565533
$764$	0	0
$765$	−5212.42	−0.246347
$766$	0	0
$767$	−11085.2	−0.521855
$768$	0	0
$769$	13923.9	0.652935	0.326468	−	0.945208i	$-0.394142\pi$
$769$	13923.9	0.652935	0.326468	+	0.945208i	$0.394142\pi$
$770$	0	0
$771$	704.087	0.0328885
$772$	0	0
$773$	−16592.6	−0.772051	−0.386026	−	0.922488i	$-0.626152\pi$
$773$	−16592.6	−0.772051	−0.386026	+	0.922488i	$0.626152\pi$
$774$	0	0
$775$	−59690.2	−2.76663
$776$	0	0
$777$	14056.2	0.648989
$778$	0	0
$779$	36305.2	1.66979
$780$	0	0
$781$	−2797.15	−0.128156
$782$	0	0
$783$	−4511.29	−0.205901
$784$	0	0
$785$	−10905.3	−0.495830
$786$	0	0
$787$	28139.8	1.27455	0.637277	−	0.770635i	$-0.280061\pi$
$787$	28139.8	1.27455	0.637277	+	0.770635i	$0.280061\pi$
$788$	0	0
$789$	−17587.7	−0.793587
$790$	0	0
$791$	−42926.5	−1.92957
$792$	0	0
$793$	16080.1	0.720075
$794$	0	0
$795$	−34382.5	−1.53386
$796$	0	0
$797$	12918.6	0.574156	0.287078	−	0.957907i	$-0.407316\pi$
$797$	12918.6	0.574156	0.287078	+	0.957907i	$0.407316\pi$
$798$	0	0
$799$	−1968.39	−0.0871546
$800$	0	0
$801$	−459.177	−0.0202550
$802$	0	0
$803$	−1823.93	−0.0801557
$804$	0	0
$805$	−31602.9	−1.38367
$806$	0	0
$807$	4199.36	0.183178
$808$	0	0
$809$	−28655.9	−1.24535	−0.622675	−	0.782481i	$-0.713954\pi$
$809$	−28655.9	−1.24535	−0.622675	+	0.782481i	$0.713954\pi$
$810$	0	0
$811$	23687.0	1.02560	0.512802	−	0.858507i	$-0.328608\pi$
$811$	23687.0	1.02560	0.512802	+	0.858507i	$0.328608\pi$
$812$	0	0
$813$	−23783.6	−1.02599
$814$	0	0
$815$	10886.5	0.467898
$816$	0	0
$817$	−28491.0	−1.22004
$818$	0	0
$819$	21508.2	0.917652
$820$	0	0
$821$	26268.3	1.11665	0.558326	−	0.829622i	$-0.311444\pi$
$821$	26268.3	1.11665	0.558326	+	0.829622i	$0.311444\pi$
$822$	0	0
$823$	17964.8	0.760893	0.380447	−	0.924803i	$-0.375770\pi$
$823$	17964.8	0.760893	0.380447	+	0.924803i	$0.375770\pi$
$824$	0	0
$825$	−6058.59	−0.255676
$826$	0	0
$827$	7957.87	0.334610	0.167305	−	0.985905i	$-0.446494\pi$
$827$	7957.87	0.334610	0.167305	+	0.985905i	$0.446494\pi$
$828$	0	0
$829$	31973.7	1.33955	0.669777	−	0.742562i	$-0.266390\pi$
$829$	31973.7	1.33955	0.669777	+	0.742562i	$0.266390\pi$
$830$	0	0
$831$	3627.58	0.151431
$832$	0	0
$833$	−16892.1	−0.702613
$834$	0	0
$835$	44029.8	1.82481
$836$	0	0
$837$	7314.71	0.302071
$838$	0	0
$839$	−13956.4	−0.574289	−0.287144	−	0.957887i	$-0.592706\pi$
$839$	−13956.4	−0.574289	−0.287144	+	0.957887i	$0.592706\pi$
$840$	0	0
$841$	3528.39	0.144671
$842$	0	0
$843$	−26949.7	−1.10107
$844$	0	0
$845$	79093.6	3.22000
$846$	0	0
$847$	−37096.6	−1.50490
$848$	0	0
$849$	−6085.94	−0.246018
$850$	0	0
$851$	9003.53	0.362676
$852$	0	0
$853$	30848.1	1.23824	0.619120	−	0.785296i	$-0.287489\pi$
$853$	30848.1	1.23824	0.619120	+	0.785296i	$0.287489\pi$
$854$	0	0
$855$	−15023.8	−0.600940
$856$	0	0
$857$	14292.3	0.569679	0.284839	−	0.958575i	$-0.408060\pi$
$857$	14292.3	0.569679	0.284839	+	0.958575i	$0.408060\pi$
$858$	0	0
$859$	−11549.7	−0.458753	−0.229377	−	0.973338i	$-0.573669\pi$
$859$	−11549.7	−0.458753	−0.229377	+	0.973338i	$0.573669\pi$
$860$	0	0
$861$	36069.6	1.42770
$862$	0	0
$863$	29851.8	1.17748	0.588742	−	0.808321i	$-0.299623\pi$
$863$	29851.8	1.17748	0.588742	+	0.808321i	$0.299623\pi$
$864$	0	0
$865$	43295.4	1.70184
$866$	0	0
$867$	11825.0	0.463206
$868$	0	0
$869$	−3075.78	−0.120067
$870$	0	0
$871$	46324.6	1.80212
$872$	0	0
$873$	6105.06	0.236684
$874$	0	0
$875$	52699.9	2.03609
$876$	0	0
$877$	−40261.8	−1.55022	−0.775110	−	0.631826i	$-0.782306\pi$
$877$	−40261.8	−1.55022	−0.775110	+	0.631826i	$0.782306\pi$
$878$	0	0
$879$	−44.4715	−0.00170647
$880$	0	0
$881$	2868.85	0.109709	0.0548546	−	0.998494i	$-0.482530\pi$
$881$	2868.85	0.109709	0.0548546	+	0.998494i	$0.482530\pi$
$882$	0	0
$883$	−697.808	−0.0265947	−0.0132973	−	0.999912i	$-0.504233\pi$
$883$	−697.808	−0.0265947	−0.0132973	+	0.999912i	$0.504233\pi$
$884$	0	0
$885$	7692.93	0.292198
$886$	0	0
$887$	43009.9	1.62811	0.814054	−	0.580789i	$-0.197256\pi$
$887$	43009.9	1.62811	0.814054	+	0.580789i	$0.197256\pi$
$888$	0	0
$889$	72485.0	2.73461
$890$	0	0
$891$	742.447	0.0279157
$892$	0	0
$893$	−5673.50	−0.212605
$894$	0	0
$895$	−5908.96	−0.220687
$896$	0	0
$897$	13776.8	0.512813
$898$	0	0
$899$	−45265.8	−1.67931
$900$	0	0
$901$	−19221.2	−0.710713
$902$	0	0
$903$	−28306.1	−1.04316
$904$	0	0
$905$	−2924.95	−0.107435
$906$	0	0
$907$	13107.8	0.479867	0.239933	−	0.970789i	$-0.422874\pi$
$907$	13107.8	0.479867	0.239933	+	0.970789i	$0.422874\pi$
$908$	0	0
$909$	−12014.2	−0.438377
$910$	0	0
$911$	24893.1	0.905320	0.452660	−	0.891683i	$-0.350475\pi$
$911$	24893.1	0.905320	0.452660	+	0.891683i	$0.350475\pi$
$912$	0	0
$913$	−8996.20	−0.326102
$914$	0	0
$915$	−11159.3	−0.403185
$916$	0	0
$917$	29271.6	1.05413
$918$	0	0
$919$	4809.81	0.172645	0.0863225	−	0.996267i	$-0.472488\pi$
$919$	4809.81	0.172645	0.0863225	+	0.996267i	$0.472488\pi$
$920$	0	0
$921$	−16793.6	−0.600835
$922$	0	0
$923$	−24514.6	−0.874223
$924$	0	0
$925$	−34701.2	−1.23348
$926$	0	0
$927$	8111.08	0.287382
$928$	0	0
$929$	−32589.2	−1.15093	−0.575466	−	0.817826i	$-0.695179\pi$
$929$	−32589.2	−1.15093	−0.575466	+	0.817826i	$0.695179\pi$
$930$	0	0
$931$	−48688.2	−1.71396
$932$	0	0
$933$	19810.5	0.695141
$934$	0	0
$935$	−5308.57	−0.185678
$936$	0	0
$937$	38928.7	1.35725	0.678627	−	0.734483i	$-0.262575\pi$
$937$	38928.7	1.35725	0.678627	+	0.734483i	$0.262575\pi$
$938$	0	0
$939$	−18104.8	−0.629209
$940$	0	0
$941$	34383.5	1.19115	0.595574	−	0.803301i	$-0.296925\pi$
$941$	34383.5	1.19115	0.595574	+	0.803301i	$0.296925\pi$
$942$	0	0
$943$	23103.9	0.797843
$944$	0	0
$945$	−14926.3	−0.513812
$946$	0	0
$947$	32211.3	1.10531	0.552653	−	0.833411i	$-0.313615\pi$
$947$	32211.3	1.10531	0.552653	+	0.833411i	$0.313615\pi$
$948$	0	0
$949$	−15985.1	−0.546785
$950$	0	0
$951$	12452.6	0.424610
$952$	0	0
$953$	20860.0	0.709046	0.354523	−	0.935047i	$-0.384643\pi$
$953$	20860.0	0.709046	0.354523	+	0.935047i	$0.384643\pi$
$954$	0	0
$955$	−11832.3	−0.400926
$956$	0	0
$957$	−4594.51	−0.155193
$958$	0	0
$959$	−18837.2	−0.634292
$960$	0	0
$961$	43604.0	1.46366
$962$	0	0
$963$	−5451.15	−0.182410
$964$	0	0
$965$	−24992.9	−0.833730
$966$	0	0
$967$	41953.3	1.39517	0.697584	−	0.716503i	$-0.254259\pi$
$967$	41953.3	1.39517	0.697584	+	0.716503i	$0.254259\pi$
$968$	0	0
$969$	−8398.93	−0.278444
$970$	0	0
$971$	3032.34	0.100219	0.0501093	−	0.998744i	$-0.484043\pi$
$971$	3032.34	0.100219	0.0501093	+	0.998744i	$0.484043\pi$
$972$	0	0
$973$	−40476.5	−1.33363
$974$	0	0
$975$	−53098.2	−1.74411
$976$	0	0
$977$	−19012.9	−0.622598	−0.311299	−	0.950312i	$-0.600764\pi$
$977$	−19012.9	−0.622598	−0.311299	+	0.950312i	$0.600764\pi$
$978$	0	0
$979$	−467.647	−0.0152667
$980$	0	0
$981$	3605.91	0.117357
$982$	0	0
$983$	−22706.7	−0.736756	−0.368378	−	0.929676i	$-0.620087\pi$
$983$	−22706.7	−0.736756	−0.368378	+	0.929676i	$0.620087\pi$
$984$	0	0
$985$	19853.4	0.642217
$986$	0	0
$987$	−5636.68	−0.181781
$988$	0	0
$989$	−18131.1	−0.582948
$990$	0	0
$991$	−9554.24	−0.306257	−0.153128	−	0.988206i	$-0.548935\pi$
$991$	−9554.24	−0.306257	−0.153128	+	0.988206i	$0.548935\pi$
$992$	0	0
$993$	−16345.2	−0.522356
$994$	0	0
$995$	28573.3	0.910386
$996$	0	0
$997$	15445.1	0.490623	0.245312	−	0.969444i	$-0.421110\pi$
$997$	15445.1	0.490623	0.245312	+	0.969444i	$0.421110\pi$
$998$	0	0
$999$	4252.45	0.134676

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.a.l.1.2	yes	2
3.2	odd	2		1152.4.a.m.1.1		2
4.3	odd	2		384.4.a.p.1.2	yes	2
8.3	odd	2		384.4.a.i.1.1	✓	2
8.5	even	2		384.4.a.m.1.1	yes	2
12.11	even	2		1152.4.a.n.1.1		2
16.3	odd	4		768.4.d.r.385.3		4
16.5	even	4		768.4.d.s.385.4		4
16.11	odd	4		768.4.d.r.385.2		4
16.13	even	4		768.4.d.s.385.1		4
24.5	odd	2		1152.4.a.w.1.2		2
24.11	even	2		1152.4.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.a.i.1.1	✓	2	8.3	odd	2
384.4.a.l.1.2	yes	2	1.1	even	1	trivial
384.4.a.m.1.1	yes	2	8.5	even	2
384.4.a.p.1.2	yes	2	4.3	odd	2
768.4.d.r.385.2		4	16.11	odd	4
768.4.d.r.385.3		4	16.3	odd	4
768.4.d.s.385.1		4	16.13	even	4
768.4.d.s.385.4		4	16.5	even	4
1152.4.a.m.1.1		2	3.2	odd	2
1152.4.a.n.1.1		2	12.11	even	2
1152.4.a.w.1.2		2	24.5	odd	2
1152.4.a.x.1.2		2	24.11	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +18.5830 q^{5} +29.7490 q^{7} +9.00000 q^{9} +9.16601 q^{11} +80.3320 q^{13} -55.7490 q^{15} -31.1660 q^{17} -89.8301 q^{19} -89.2470 q^{21} -57.1660 q^{23} +220.328 q^{25} -27.0000 q^{27} +167.085 q^{29} -270.915 q^{31} -27.4980 q^{33} +552.826 q^{35} -157.498 q^{37} -240.996 q^{39} -404.154 q^{41} +317.166 q^{43} +167.247 q^{45} +63.1581 q^{47} +542.004 q^{49} +93.4980 q^{51} +616.737 q^{53} +170.332 q^{55} +269.490 q^{57} -137.992 q^{59} +200.170 q^{61} +267.741 q^{63} +1492.81 q^{65} +576.664 q^{67} +171.498 q^{69} -305.166 q^{71} -198.988 q^{73} -660.984 q^{75} +272.680 q^{77} -335.563 q^{79} +81.0000 q^{81} -981.474 q^{83} -579.158 q^{85} -501.255 q^{87} -51.0197 q^{89} +2389.80 q^{91} +812.745 q^{93} -1669.31 q^{95} +678.340 q^{97} +82.4941 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 16 q^{5} - 4 q^{7} + 18 q^{9} - 24 q^{11} + 76 q^{13} - 48 q^{15} - 20 q^{17} + 32 q^{19} + 12 q^{21} - 72 q^{23} + 102 q^{25} - 54 q^{27} + 440 q^{29} - 436 q^{31} + 72 q^{33} + 640 q^{35}+ \cdots - 216 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 16 * q^5 - 4 * q^7 + 18 * q^9 - 24 * q^11 + 76 * q^13 - 48 * q^15 - 20 * q^17 + 32 * q^19 + 12 * q^21 - 72 * q^23 + 102 * q^25 - 54 * q^27 + 440 * q^29 - 436 * q^31 + 72 * q^33 + 640 * q^35 - 188 * q^37 - 228 * q^39 - 4 * q^41 + 592 * q^43 + 144 * q^45 - 424 * q^47 + 1338 * q^49 + 60 * q^51 + 408 * q^53 + 256 * q^55 - 96 * q^57 + 232 * q^59 + 612 * q^61 - 36 * q^63 + 1504 * q^65 + 984 * q^67 + 216 * q^69 - 568 * q^71 + 364 * q^73 - 306 * q^75 + 1392 * q^77 + 620 * q^79 + 162 * q^81 - 312 * q^83 - 608 * q^85 - 1320 * q^87 - 1372 * q^89 + 2536 * q^91 + 1308 * q^93 - 1984 * q^95 + 1780 * q^97 - 216 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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