LMFDB - Embedded newform 768.4.a.u.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,4,Mod(1,768)] 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("768.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-12,0,0,0,0,0,36,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$45.3134668844$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.9792.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 2x + 7 $ x^4 - 2x^3 - 7x^2 + 2*x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.06909$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -17.4288 q^{5} +2.99032 q^{7} +9.00000 q^{9} +10.6274 q^{11} +43.3156 q^{13} +52.2865 q^{15} -37.8823 q^{17} -79.8823 q^{19} -8.97095 q^{21} +191.204 q^{23} +178.765 q^{25} -27.0000 q^{27} -138.918 q^{29} +212.136 q^{31} -31.8823 q^{33} -52.1177 q^{35} +270.404 q^{37} -129.947 q^{39} +441.411 q^{41} -64.1177 q^{43} -156.860 q^{45} -436.234 q^{47} -334.058 q^{49} +113.647 q^{51} -278.348 q^{53} -185.224 q^{55} +239.647 q^{57} -830.039 q^{59} +724.580 q^{61} +26.9128 q^{63} -754.940 q^{65} -859.529 q^{67} -573.613 q^{69} -681.264 q^{71} +785.058 q^{73} -536.294 q^{75} +31.7793 q^{77} -1018.82 q^{79} +81.0000 q^{81} -467.334 q^{83} +660.244 q^{85} +416.753 q^{87} -510.706 q^{89} +129.527 q^{91} -636.409 q^{93} +1392.26 q^{95} -234.235 q^{97} +95.6468 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 36 q^{9} - 48 q^{11} + 120 q^{17} - 48 q^{19} + 172 q^{25} - 108 q^{27} + 144 q^{33} - 480 q^{35} + 408 q^{41} - 528 q^{43} + 836 q^{49} - 360 q^{51} + 144 q^{57} - 1872 q^{59} - 576 q^{65}+ \cdots - 432 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 36 * q^9 - 48 * q^11 + 120 * q^17 - 48 * q^19 + 172 * q^25 - 108 * q^27 + 144 * q^33 - 480 * q^35 + 408 * q^41 - 528 * q^43 + 836 * q^49 - 360 * q^51 + 144 * q^57 - 1872 * q^59 - 576 * q^65 - 2352 * q^67 + 968 * q^73 - 516 * q^75 + 324 * q^81 - 3408 * q^83 - 3672 * q^89 - 5184 * q^91 - 1480 * q^97 - 432 * 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
$5$	−17.4288	−1.55888	−0.779441	−	0.626475i	$-0.784497\pi$
$5$	−17.4288	−1.55888	−0.779441	+	0.626475i	$0.784497\pi$
$6$	0	0
$7$	2.99032	0.161462	0.0807310	−	0.996736i	$-0.474275\pi$
$7$	2.99032	0.161462	0.0807310	+	0.996736i	$0.474275\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	10.6274	0.291299	0.145649	−	0.989336i	$-0.453473\pi$
$11$	10.6274	0.291299	0.145649	+	0.989336i	$0.453473\pi$
$12$	0	0
$13$	43.3156	0.924121	0.462061	−	0.886848i	$-0.347110\pi$
$13$	43.3156	0.924121	0.462061	+	0.886848i	$0.347110\pi$
$14$	0	0
$15$	52.2865	0.900021
$16$	0	0
$17$	−37.8823	−0.540459	−0.270229	−	0.962796i	$-0.587100\pi$
$17$	−37.8823	−0.540459	−0.270229	+	0.962796i	$0.587100\pi$
$18$	0	0
$19$	−79.8823	−0.964539	−0.482270	−	0.876023i	$-0.660187\pi$
$19$	−79.8823	−0.964539	−0.482270	+	0.876023i	$0.660187\pi$
$20$	0	0
$21$	−8.97095	−0.0932201
$22$	0	0
$23$	191.204	1.73343	0.866714	−	0.498806i	$-0.166228\pi$
$23$	191.204	1.73343	0.866714	+	0.498806i	$0.166228\pi$
$24$	0	0
$25$	178.765	1.43012
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−138.918	−0.889530	−0.444765	−	0.895647i	$-0.646713\pi$
$29$	−138.918	−0.889530	−0.444765	+	0.895647i	$0.646713\pi$
$30$	0	0
$31$	212.136	1.22906	0.614529	−	0.788894i	$-0.289346\pi$
$31$	212.136	1.22906	0.614529	+	0.788894i	$0.289346\pi$
$32$	0	0
$33$	−31.8823	−0.168181
$34$	0	0
$35$	−52.1177	−0.251700
$36$	0	0
$37$	270.404	1.20146	0.600731	−	0.799451i	$-0.294876\pi$
$37$	270.404	1.20146	0.600731	+	0.799451i	$0.294876\pi$
$38$	0	0
$39$	−129.947	−0.533542
$40$	0	0
$41$	441.411	1.68139	0.840693	−	0.541511i	$-0.182148\pi$
$41$	441.411	1.68139	0.840693	+	0.541511i	$0.182148\pi$
$42$	0	0
$43$	−64.1177	−0.227392	−0.113696	−	0.993516i	$-0.536269\pi$
$43$	−64.1177	−0.227392	−0.113696	+	0.993516i	$0.536269\pi$
$44$	0	0
$45$	−156.860	−0.519628
$46$	0	0
$47$	−436.234	−1.35386	−0.676929	−	0.736049i	$-0.736689\pi$
$47$	−436.234	−1.35386	−0.676929	+	0.736049i	$0.736689\pi$
$48$	0	0
$49$	−334.058	−0.973930
$50$	0	0
$51$	113.647	0.312034
$52$	0	0
$53$	−278.348	−0.721398	−0.360699	−	0.932682i	$-0.617462\pi$
$53$	−278.348	−0.721398	−0.360699	+	0.932682i	$0.617462\pi$
$54$	0	0
$55$	−185.224	−0.454101
$56$	0	0
$57$	239.647	0.556877
$58$	0	0
$59$	−830.039	−1.83156	−0.915778	−	0.401684i	$-0.868425\pi$
$59$	−830.039	−1.83156	−0.915778	+	0.401684i	$0.868425\pi$
$60$	0	0
$61$	724.580	1.52087	0.760434	−	0.649416i	$-0.224986\pi$
$61$	724.580	1.52087	0.760434	+	0.649416i	$0.224986\pi$
$62$	0	0
$63$	26.9128	0.0538206
$64$	0	0
$65$	−754.940	−1.44060
$66$	0	0
$67$	−859.529	−1.56729	−0.783643	−	0.621211i	$-0.786641\pi$
$67$	−859.529	−1.56729	−0.783643	+	0.621211i	$0.786641\pi$
$68$	0	0
$69$	−573.613	−1.00079
$70$	0	0
$71$	−681.264	−1.13875	−0.569374	−	0.822078i	$-0.692814\pi$
$71$	−681.264	−1.13875	−0.569374	+	0.822078i	$0.692814\pi$
$72$	0	0
$73$	785.058	1.25869	0.629343	−	0.777128i	$-0.283324\pi$
$73$	785.058	1.25869	0.629343	+	0.777128i	$0.283324\pi$
$74$	0	0
$75$	−536.294	−0.825678
$76$	0	0
$77$	31.7793	0.0470337
$78$	0	0
$79$	−1018.82	−1.45096	−0.725481	−	0.688243i	$-0.758382\pi$
$79$	−1018.82	−1.45096	−0.725481	+	0.688243i	$0.758382\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−467.334	−0.618031	−0.309015	−	0.951057i	$-0.599999\pi$
$83$	−467.334	−0.618031	−0.309015	+	0.951057i	$0.599999\pi$
$84$	0	0
$85$	660.244	0.842512
$86$	0	0
$87$	416.753	0.513570
$88$	0	0
$89$	−510.706	−0.608256	−0.304128	−	0.952631i	$-0.598365\pi$
$89$	−510.706	−0.608256	−0.304128	+	0.952631i	$0.598365\pi$
$90$	0	0
$91$	129.527	0.149210
$92$	0	0
$93$	−636.409	−0.709597
$94$	0	0
$95$	1392.26	1.50360
$96$	0	0
$97$	−234.235	−0.245186	−0.122593	−	0.992457i	$-0.539121\pi$
$97$	−234.235	−0.245186	−0.122593	+	0.992457i	$0.539121\pi$
$98$	0	0
$99$	95.6468	0.0970996
$100$	0	0
$101$	205.555	0.202509	0.101255	−	0.994861i	$-0.467714\pi$
$101$	205.555	0.202509	0.101255	+	0.994861i	$0.467714\pi$
$102$	0	0
$103$	391.379	0.374405	0.187203	−	0.982321i	$-0.440058\pi$
$103$	391.379	0.374405	0.187203	+	0.982321i	$0.440058\pi$
$104$	0	0
$105$	156.353	0.145319
$106$	0	0
$107$	−934.274	−0.844109	−0.422055	−	0.906570i	$-0.638691\pi$
$107$	−934.274	−0.844109	−0.422055	+	0.906570i	$0.638691\pi$
$108$	0	0
$109$	−584.123	−0.513292	−0.256646	−	0.966505i	$-0.582617\pi$
$109$	−584.123	−0.513292	−0.256646	+	0.966505i	$0.582617\pi$
$110$	0	0
$111$	−811.211	−0.693664
$112$	0	0
$113$	−582.706	−0.485101	−0.242551	−	0.970139i	$-0.577984\pi$
$113$	−582.706	−0.485101	−0.242551	+	0.970139i	$0.577984\pi$
$114$	0	0
$115$	−3332.47	−2.70221
$116$	0	0
$117$	389.840	0.308040
$118$	0	0
$119$	−113.280	−0.0872635
$120$	0	0
$121$	−1218.06	−0.915145
$122$	0	0
$123$	−1324.23	−0.970749
$124$	0	0
$125$	−937.053	−0.670501
$126$	0	0
$127$	1461.03	1.02083	0.510416	−	0.859928i	$-0.329491\pi$
$127$	1461.03	1.02083	0.510416	+	0.859928i	$0.329491\pi$
$128$	0	0
$129$	192.353	0.131285
$130$	0	0
$131$	−98.4323	−0.0656494	−0.0328247	−	0.999461i	$-0.510450\pi$
$131$	−98.4323	−0.0656494	−0.0328247	+	0.999461i	$0.510450\pi$
$132$	0	0
$133$	−238.873	−0.155736
$134$	0	0
$135$	470.579	0.300007
$136$	0	0
$137$	−2171.06	−1.35391	−0.676956	−	0.736024i	$-0.736701\pi$
$137$	−2171.06	−1.35391	−0.676956	+	0.736024i	$0.736701\pi$
$138$	0	0
$139$	1624.70	0.991407	0.495703	−	0.868492i	$-0.334910\pi$
$139$	1624.70	0.991407	0.495703	+	0.868492i	$0.334910\pi$
$140$	0	0
$141$	1308.70	0.781650
$142$	0	0
$143$	460.333	0.269195
$144$	0	0
$145$	2421.17	1.38667
$146$	0	0
$147$	1002.17	0.562299
$148$	0	0
$149$	−636.658	−0.350047	−0.175024	−	0.984564i	$-0.556000\pi$
$149$	−636.658	−0.350047	−0.175024	+	0.984564i	$0.556000\pi$
$150$	0	0
$151$	1819.34	0.980503	0.490252	−	0.871581i	$-0.336905\pi$
$151$	1819.34	0.980503	0.490252	+	0.871581i	$0.336905\pi$
$152$	0	0
$153$	−340.940	−0.180153
$154$	0	0
$155$	−3697.29	−1.91596
$156$	0	0
$157$	1656.50	0.842059	0.421029	−	0.907047i	$-0.361669\pi$
$157$	1656.50	0.842059	0.421029	+	0.907047i	$0.361669\pi$
$158$	0	0
$159$	835.045	0.416499
$160$	0	0
$161$	571.761	0.279882
$162$	0	0
$163$	−2228.82	−1.07101	−0.535505	−	0.844532i	$-0.679879\pi$
$163$	−2228.82	−1.07101	−0.535505	+	0.844532i	$0.679879\pi$
$164$	0	0
$165$	555.671	0.262175
$166$	0	0
$167$	1667.01	0.772439	0.386219	−	0.922407i	$-0.373781\pi$
$167$	1667.01	0.772439	0.386219	+	0.922407i	$0.373781\pi$
$168$	0	0
$169$	−320.761	−0.146000
$170$	0	0
$171$	−718.940	−0.321513
$172$	0	0
$173$	2500.53	1.09891	0.549457	−	0.835522i	$-0.314835\pi$
$173$	2500.53	1.09891	0.549457	+	0.835522i	$0.314835\pi$
$174$	0	0
$175$	534.562	0.230909
$176$	0	0
$177$	2490.12	1.05745
$178$	0	0
$179$	378.742	0.158148	0.0790740	−	0.996869i	$-0.474804\pi$
$179$	378.742	0.158148	0.0790740	+	0.996869i	$0.474804\pi$
$180$	0	0
$181$	−3093.88	−1.27053	−0.635265	−	0.772294i	$-0.719109\pi$
$181$	−3093.88	−1.27053	−0.635265	+	0.772294i	$0.719109\pi$
$182$	0	0
$183$	−2173.74	−0.878073
$184$	0	0
$185$	−4712.82	−1.87294
$186$	0	0
$187$	−402.590	−0.157435
$188$	0	0
$189$	−80.7385	−0.0310734
$190$	0	0
$191$	−3656.98	−1.38539	−0.692695	−	0.721230i	$-0.743577\pi$
$191$	−3656.98	−1.38539	−0.692695	+	0.721230i	$0.743577\pi$
$192$	0	0
$193$	2788.12	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$193$	2788.12	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$194$	0	0
$195$	2264.82	0.831729
$196$	0	0
$197$	1147.74	0.415091	0.207546	−	0.978225i	$-0.433452\pi$
$197$	1147.74	0.415091	0.207546	+	0.978225i	$0.433452\pi$
$198$	0	0
$199$	−4842.73	−1.72508	−0.862542	−	0.505986i	$-0.831129\pi$
$199$	−4842.73	−1.72508	−0.862542	+	0.505986i	$0.831129\pi$
$200$	0	0
$201$	2578.59	0.904873
$202$	0	0
$203$	−415.408	−0.143625
$204$	0	0
$205$	−7693.29	−2.62109
$206$	0	0
$207$	1720.84	0.577809
$208$	0	0
$209$	−848.942	−0.280969
$210$	0	0
$211$	−3222.35	−1.05135	−0.525677	−	0.850684i	$-0.676188\pi$
$211$	−3222.35	−1.05135	−0.525677	+	0.850684i	$0.676188\pi$
$212$	0	0
$213$	2043.79	0.657457
$214$	0	0
$215$	1117.50	0.354478
$216$	0	0
$217$	634.355	0.198446
$218$	0	0
$219$	−2355.17	−0.726703
$220$	0	0
$221$	−1640.89	−0.499449
$222$	0	0
$223$	−4932.61	−1.48122	−0.740610	−	0.671935i	$-0.765463\pi$
$223$	−4932.61	−1.48122	−0.740610	+	0.671935i	$0.765463\pi$
$224$	0	0
$225$	1608.88	0.476705
$226$	0	0
$227$	3619.49	1.05830	0.529150	−	0.848529i	$-0.322511\pi$
$227$	3619.49	1.05830	0.529150	+	0.848529i	$0.322511\pi$
$228$	0	0
$229$	305.759	0.0882320	0.0441160	−	0.999026i	$-0.485953\pi$
$229$	305.759	0.0882320	0.0441160	+	0.999026i	$0.485953\pi$
$230$	0	0
$231$	−95.3380	−0.0271549
$232$	0	0
$233$	−639.648	−0.179849	−0.0899244	−	0.995949i	$-0.528663\pi$
$233$	−639.648	−0.179849	−0.0899244	+	0.995949i	$0.528663\pi$
$234$	0	0
$235$	7603.05	2.11050
$236$	0	0
$237$	3056.45	0.837713
$238$	0	0
$239$	−1744.94	−0.472262	−0.236131	−	0.971721i	$-0.575879\pi$
$239$	−1744.94	−0.472262	−0.236131	+	0.971721i	$0.575879\pi$
$240$	0	0
$241$	3357.29	0.897354	0.448677	−	0.893694i	$-0.351895\pi$
$241$	3357.29	0.897354	0.448677	+	0.893694i	$0.351895\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	5822.24	1.51824
$246$	0	0
$247$	−3460.15	−0.891351
$248$	0	0
$249$	1402.00	0.356820
$250$	0	0
$251$	1317.45	0.331301	0.165650	−	0.986185i	$-0.447028\pi$
$251$	1317.45	0.331301	0.165650	+	0.986185i	$0.447028\pi$
$252$	0	0
$253$	2032.01	0.504945
$254$	0	0
$255$	−1980.73	−0.486424
$256$	0	0
$257$	−3036.12	−0.736917	−0.368459	−	0.929644i	$-0.620114\pi$
$257$	−3036.12	−0.736917	−0.368459	+	0.929644i	$0.620114\pi$
$258$	0	0
$259$	808.592	0.193990
$260$	0	0
$261$	−1250.26	−0.296510
$262$	0	0
$263$	−1655.76	−0.388206	−0.194103	−	0.980981i	$-0.562180\pi$
$263$	−1655.76	−0.388206	−0.194103	+	0.980981i	$0.562180\pi$
$264$	0	0
$265$	4851.29	1.12458
$266$	0	0
$267$	1532.12	0.351177
$268$	0	0
$269$	5292.72	1.19964	0.599820	−	0.800135i	$-0.295239\pi$
$269$	5292.72	1.19964	0.599820	+	0.800135i	$0.295239\pi$
$270$	0	0
$271$	8010.52	1.79559	0.897795	−	0.440414i	$-0.145168\pi$
$271$	8010.52	1.79559	0.897795	+	0.440414i	$0.145168\pi$
$272$	0	0
$273$	−388.582	−0.0861467
$274$	0	0
$275$	1899.80	0.416591
$276$	0	0
$277$	−5692.81	−1.23483	−0.617415	−	0.786638i	$-0.711820\pi$
$277$	−5692.81	−1.23483	−0.617415	+	0.786638i	$0.711820\pi$
$278$	0	0
$279$	1909.23	0.409686
$280$	0	0
$281$	−2024.83	−0.429861	−0.214931	−	0.976629i	$-0.568953\pi$
$281$	−2024.83	−0.429861	−0.214931	+	0.976629i	$0.568953\pi$
$282$	0	0
$283$	247.761	0.0520419	0.0260210	−	0.999661i	$-0.491716\pi$
$283$	247.761	0.0520419	0.0260210	+	0.999661i	$0.491716\pi$
$284$	0	0
$285$	−4176.77	−0.868106
$286$	0	0
$287$	1319.96	0.271480
$288$	0	0
$289$	−3477.94	−0.707905
$290$	0	0
$291$	702.706	0.141558
$292$	0	0
$293$	−8133.61	−1.62174	−0.810871	−	0.585225i	$-0.801006\pi$
$293$	−8133.61	−1.62174	−0.810871	+	0.585225i	$0.801006\pi$
$294$	0	0
$295$	14466.6	2.85518
$296$	0	0
$297$	−286.940	−0.0560605
$298$	0	0
$299$	8282.12	1.60190
$300$	0	0
$301$	−191.732	−0.0367152
$302$	0	0
$303$	−616.664	−0.116919
$304$	0	0
$305$	−12628.6	−2.37085
$306$	0	0
$307$	2974.82	0.553035	0.276518	−	0.961009i	$-0.410820\pi$
$307$	2974.82	0.553035	0.276518	+	0.961009i	$0.410820\pi$
$308$	0	0
$309$	−1174.14	−0.216163
$310$	0	0
$311$	−4451.52	−0.811648	−0.405824	−	0.913951i	$-0.633015\pi$
$311$	−4451.52	−0.811648	−0.405824	+	0.913951i	$0.633015\pi$
$312$	0	0
$313$	8273.75	1.49412	0.747061	−	0.664755i	$-0.231464\pi$
$313$	8273.75	1.49412	0.747061	+	0.664755i	$0.231464\pi$
$314$	0	0
$315$	−469.060	−0.0839001
$316$	0	0
$317$	−429.036	−0.0760160	−0.0380080	−	0.999277i	$-0.512101\pi$
$317$	−429.036	−0.0760160	−0.0380080	+	0.999277i	$0.512101\pi$
$318$	0	0
$319$	−1476.34	−0.259119
$320$	0	0
$321$	2802.82	0.487347
$322$	0	0
$323$	3026.12	0.521293
$324$	0	0
$325$	7743.29	1.32160
$326$	0	0
$327$	1752.37	0.296349
$328$	0	0
$329$	−1304.48	−0.218596
$330$	0	0
$331$	8196.71	1.36112	0.680562	−	0.732691i	$-0.261736\pi$
$331$	8196.71	1.36112	0.680562	+	0.732691i	$0.261736\pi$
$332$	0	0
$333$	2433.63	0.400487
$334$	0	0
$335$	14980.6	2.44322
$336$	0	0
$337$	−2000.35	−0.323341	−0.161670	−	0.986845i	$-0.551688\pi$
$337$	−2000.35	−0.323341	−0.161670	+	0.986845i	$0.551688\pi$
$338$	0	0
$339$	1748.12	0.280073
$340$	0	0
$341$	2254.46	0.358023
$342$	0	0
$343$	−2024.62	−0.318715
$344$	0	0
$345$	9997.40	1.56012
$346$	0	0
$347$	−7707.48	−1.19239	−0.596195	−	0.802840i	$-0.703321\pi$
$347$	−7707.48	−1.19239	−0.596195	+	0.802840i	$0.703321\pi$
$348$	0	0
$349$	9681.98	1.48500	0.742499	−	0.669847i	$-0.233640\pi$
$349$	9681.98	1.48500	0.742499	+	0.669847i	$0.233640\pi$
$350$	0	0
$351$	−1169.52	−0.177847
$352$	0	0
$353$	−10540.3	−1.58925	−0.794626	−	0.607099i	$-0.792333\pi$
$353$	−10540.3	−1.58925	−0.794626	+	0.607099i	$0.792333\pi$
$354$	0	0
$355$	11873.6	1.77518
$356$	0	0
$357$	339.840	0.0503816
$358$	0	0
$359$	514.158	0.0755884	0.0377942	−	0.999286i	$-0.487967\pi$
$359$	514.158	0.0755884	0.0377942	+	0.999286i	$0.487967\pi$
$360$	0	0
$361$	−477.826	−0.0696641
$362$	0	0
$363$	3654.17	0.528359
$364$	0	0
$365$	−13682.7	−1.96214
$366$	0	0
$367$	−11272.4	−1.60331	−0.801657	−	0.597785i	$-0.796048\pi$
$367$	−11272.4	−1.60331	−0.801657	+	0.597785i	$0.796048\pi$
$368$	0	0
$369$	3972.70	0.560462
$370$	0	0
$371$	−832.350	−0.116478
$372$	0	0
$373$	6956.92	0.965726	0.482863	−	0.875696i	$-0.339597\pi$
$373$	6956.92	0.965726	0.482863	+	0.875696i	$0.339597\pi$
$374$	0	0
$375$	2811.16	0.387114
$376$	0	0
$377$	−6017.30	−0.822034
$378$	0	0
$379$	−10201.3	−1.38260	−0.691299	−	0.722569i	$-0.742961\pi$
$379$	−10201.3	−1.38260	−0.691299	+	0.722569i	$0.742961\pi$
$380$	0	0
$381$	−4383.10	−0.589378
$382$	0	0
$383$	−2461.56	−0.328406	−0.164203	−	0.986427i	$-0.552505\pi$
$383$	−2461.56	−0.328406	−0.164203	+	0.986427i	$0.552505\pi$
$384$	0	0
$385$	−553.877	−0.0733200
$386$	0	0
$387$	−577.060	−0.0757974
$388$	0	0
$389$	546.451	0.0712240	0.0356120	−	0.999366i	$-0.488662\pi$
$389$	546.451	0.0712240	0.0356120	+	0.999366i	$0.488662\pi$
$390$	0	0
$391$	−7243.25	−0.936846
$392$	0	0
$393$	295.297	0.0379027
$394$	0	0
$395$	17756.8	2.26188
$396$	0	0
$397$	−2084.56	−0.263529	−0.131764	−	0.991281i	$-0.542064\pi$
$397$	−2084.56	−0.263529	−0.131764	+	0.991281i	$0.542064\pi$
$398$	0	0
$399$	716.620	0.0899144
$400$	0	0
$401$	−9710.59	−1.20929	−0.604643	−	0.796497i	$-0.706684\pi$
$401$	−9710.59	−1.20929	−0.604643	+	0.796497i	$0.706684\pi$
$402$	0	0
$403$	9188.81	1.13580
$404$	0	0
$405$	−1411.74	−0.173209
$406$	0	0
$407$	2873.69	0.349984
$408$	0	0
$409$	6659.89	0.805160	0.402580	−	0.915385i	$-0.368114\pi$
$409$	6659.89	0.805160	0.402580	+	0.915385i	$0.368114\pi$
$410$	0	0
$411$	6513.17	0.781681
$412$	0	0
$413$	−2482.08	−0.295727
$414$	0	0
$415$	8145.09	0.963438
$416$	0	0
$417$	−4874.11	−0.572389
$418$	0	0
$419$	10576.7	1.23318	0.616592	−	0.787283i	$-0.288513\pi$
$419$	10576.7	1.23318	0.616592	+	0.787283i	$0.288513\pi$
$420$	0	0
$421$	4871.09	0.563901	0.281951	−	0.959429i	$-0.409019\pi$
$421$	4871.09	0.563901	0.281951	+	0.959429i	$0.409019\pi$
$422$	0	0
$423$	−3926.11	−0.451286
$424$	0	0
$425$	−6772.00	−0.772918
$426$	0	0
$427$	2166.72	0.245562
$428$	0	0
$429$	−1381.00	−0.155420
$430$	0	0
$431$	−16916.7	−1.89060	−0.945302	−	0.326196i	$-0.894233\pi$
$431$	−16916.7	−1.89060	−0.945302	+	0.326196i	$0.894233\pi$
$432$	0	0
$433$	−1163.88	−0.129174	−0.0645870	−	0.997912i	$-0.520573\pi$
$433$	−1163.88	−0.129174	−0.0645870	+	0.997912i	$0.520573\pi$
$434$	0	0
$435$	−7263.52	−0.800596
$436$	0	0
$437$	−15273.8	−1.67196
$438$	0	0
$439$	−1856.28	−0.201812	−0.100906	−	0.994896i	$-0.532174\pi$
$439$	−1856.28	−0.201812	−0.100906	+	0.994896i	$0.532174\pi$
$440$	0	0
$441$	−3006.52	−0.324643
$442$	0	0
$443$	−1472.21	−0.157893	−0.0789465	−	0.996879i	$-0.525156\pi$
$443$	−1472.21	−0.157893	−0.0789465	+	0.996879i	$0.525156\pi$
$444$	0	0
$445$	8901.02	0.948200
$446$	0	0
$447$	1909.97	0.202100
$448$	0	0
$449$	9620.94	1.01123	0.505613	−	0.862761i	$-0.331267\pi$
$449$	9620.94	1.01123	0.505613	+	0.862761i	$0.331267\pi$
$450$	0	0
$451$	4691.06	0.489786
$452$	0	0
$453$	−5458.03	−0.566094
$454$	0	0
$455$	−2257.51	−0.232602
$456$	0	0
$457$	−3613.53	−0.369877	−0.184938	−	0.982750i	$-0.559209\pi$
$457$	−3613.53	−0.369877	−0.184938	+	0.982750i	$0.559209\pi$
$458$	0	0
$459$	1022.82	0.104011
$460$	0	0
$461$	17710.7	1.78931	0.894654	−	0.446759i	$-0.147422\pi$
$461$	17710.7	1.78931	0.894654	+	0.446759i	$0.147422\pi$
$462$	0	0
$463$	1674.57	0.168087	0.0840433	−	0.996462i	$-0.473217\pi$
$463$	1674.57	0.168087	0.0840433	+	0.996462i	$0.473217\pi$
$464$	0	0
$465$	11091.9	1.10618
$466$	0	0
$467$	−15208.3	−1.50697	−0.753484	−	0.657466i	$-0.771628\pi$
$467$	−15208.3	−1.50697	−0.753484	+	0.657466i	$0.771628\pi$
$468$	0	0
$469$	−2570.26	−0.253057
$470$	0	0
$471$	−4969.51	−0.486163
$472$	0	0
$473$	−681.406	−0.0662391
$474$	0	0
$475$	−14280.1	−1.37940
$476$	0	0
$477$	−2505.14	−0.240466
$478$	0	0
$479$	−6458.37	−0.616056	−0.308028	−	0.951377i	$-0.599669\pi$
$479$	−6458.37	−0.616056	−0.308028	+	0.951377i	$0.599669\pi$
$480$	0	0
$481$	11712.7	1.11030
$482$	0	0
$483$	−1715.28	−0.161590
$484$	0	0
$485$	4082.45	0.382216
$486$	0	0
$487$	−11337.7	−1.05495	−0.527474	−	0.849571i	$-0.676861\pi$
$487$	−11337.7	−1.05495	−0.527474	+	0.849571i	$0.676861\pi$
$488$	0	0
$489$	6686.46	0.618348
$490$	0	0
$491$	14946.7	1.37380	0.686898	−	0.726754i	$-0.258972\pi$
$491$	14946.7	1.37380	0.686898	+	0.726754i	$0.258972\pi$
$492$	0	0
$493$	5262.51	0.480754
$494$	0	0
$495$	−1667.01	−0.151367
$496$	0	0
$497$	−2037.19	−0.183865
$498$	0	0
$499$	−2631.77	−0.236101	−0.118051	−	0.993008i	$-0.537664\pi$
$499$	−2631.77	−0.236101	−0.118051	+	0.993008i	$0.537664\pi$
$500$	0	0
$501$	−5001.04	−0.445968
$502$	0	0
$503$	6907.45	0.612302	0.306151	−	0.951983i	$-0.400959\pi$
$503$	6907.45	0.612302	0.306151	+	0.951983i	$0.400959\pi$
$504$	0	0
$505$	−3582.58	−0.315689
$506$	0	0
$507$	962.283	0.0842929
$508$	0	0
$509$	−12020.3	−1.04674	−0.523368	−	0.852107i	$-0.675325\pi$
$509$	−12020.3	−1.04674	−0.523368	+	0.852107i	$0.675325\pi$
$510$	0	0
$511$	2347.57	0.203230
$512$	0	0
$513$	2156.82	0.185626
$514$	0	0
$515$	−6821.29	−0.583654
$516$	0	0
$517$	−4636.04	−0.394377
$518$	0	0
$519$	−7501.60	−0.634458
$520$	0	0
$521$	−15846.1	−1.33249	−0.666247	−	0.745731i	$-0.732101\pi$
$521$	−15846.1	−1.33249	−0.666247	+	0.745731i	$0.732101\pi$
$522$	0	0
$523$	−8891.64	−0.743411	−0.371706	−	0.928351i	$-0.621227\pi$
$523$	−8891.64	−0.743411	−0.371706	+	0.928351i	$0.621227\pi$
$524$	0	0
$525$	−1603.69	−0.133316
$526$	0	0
$527$	−8036.20	−0.664255
$528$	0	0
$529$	24392.0	2.00477
$530$	0	0
$531$	−7470.35	−0.610519
$532$	0	0
$533$	19120.0	1.55381
$534$	0	0
$535$	16283.3	1.31587
$536$	0	0
$537$	−1136.23	−0.0913068
$538$	0	0
$539$	−3550.17	−0.283705
$540$	0	0
$541$	−12833.5	−1.01988	−0.509940	−	0.860210i	$-0.670333\pi$
$541$	−12833.5	−1.01988	−0.509940	+	0.860210i	$0.670333\pi$
$542$	0	0
$543$	9281.63	0.733541
$544$	0	0
$545$	10180.6	0.800162
$546$	0	0
$547$	16257.0	1.27075	0.635375	−	0.772204i	$-0.280845\pi$
$547$	16257.0	1.27075	0.635375	+	0.772204i	$0.280845\pi$
$548$	0	0
$549$	6521.22	0.506956
$550$	0	0
$551$	11097.1	0.857986
$552$	0	0
$553$	−3046.59	−0.234275
$554$	0	0
$555$	14138.5	1.08134
$556$	0	0
$557$	−1558.32	−0.118542	−0.0592712	−	0.998242i	$-0.518878\pi$
$557$	−1558.32	−0.118542	−0.0592712	+	0.998242i	$0.518878\pi$
$558$	0	0
$559$	−2777.30	−0.210138
$560$	0	0
$561$	1207.77	0.0908951
$562$	0	0
$563$	−9782.16	−0.732272	−0.366136	−	0.930561i	$-0.619319\pi$
$563$	−9782.16	−0.732272	−0.366136	+	0.930561i	$0.619319\pi$
$564$	0	0
$565$	10155.9	0.756216
$566$	0	0
$567$	242.216	0.0179402
$568$	0	0
$569$	−7887.05	−0.581094	−0.290547	−	0.956861i	$-0.593837\pi$
$569$	−7887.05	−0.581094	−0.290547	+	0.956861i	$0.593837\pi$
$570$	0	0
$571$	−21819.3	−1.59914	−0.799570	−	0.600573i	$-0.794939\pi$
$571$	−21819.3	−1.59914	−0.799570	+	0.600573i	$0.794939\pi$
$572$	0	0
$573$	10970.9	0.799856
$574$	0	0
$575$	34180.5	2.47900
$576$	0	0
$577$	7190.22	0.518774	0.259387	−	0.965773i	$-0.416479\pi$
$577$	7190.22	0.518774	0.259387	+	0.965773i	$0.416479\pi$
$578$	0	0
$579$	−8364.35	−0.600363
$580$	0	0
$581$	−1397.48	−0.0997884
$582$	0	0
$583$	−2958.12	−0.210142
$584$	0	0
$585$	−6794.46	−0.480199
$586$	0	0
$587$	−13305.6	−0.935569	−0.467785	−	0.883843i	$-0.654948\pi$
$587$	−13305.6	−0.935569	−0.467785	+	0.883843i	$0.654948\pi$
$588$	0	0
$589$	−16945.9	−1.18548
$590$	0	0
$591$	−3443.21	−0.239653
$592$	0	0
$593$	−8062.23	−0.558307	−0.279153	−	0.960246i	$-0.590054\pi$
$593$	−8062.23	−0.558307	−0.279153	+	0.960246i	$0.590054\pi$
$594$	0	0
$595$	1974.34	0.136034
$596$	0	0
$597$	14528.2	0.995978
$598$	0	0
$599$	2185.21	0.149058	0.0745288	−	0.997219i	$-0.476255\pi$
$599$	2185.21	0.149058	0.0745288	+	0.997219i	$0.476255\pi$
$600$	0	0
$601$	3542.25	0.240418	0.120209	−	0.992749i	$-0.461643\pi$
$601$	3542.25	0.240418	0.120209	+	0.992749i	$0.461643\pi$
$602$	0	0
$603$	−7735.76	−0.522429
$604$	0	0
$605$	21229.3	1.42660
$606$	0	0
$607$	6050.64	0.404593	0.202296	−	0.979324i	$-0.435160\pi$
$607$	6050.64	0.404593	0.202296	+	0.979324i	$0.435160\pi$
$608$	0	0
$609$	1246.22	0.0829220
$610$	0	0
$611$	−18895.7	−1.25113
$612$	0	0
$613$	−22514.2	−1.48343	−0.741713	−	0.670717i	$-0.765986\pi$
$613$	−22514.2	−1.48343	−0.741713	+	0.670717i	$0.765986\pi$
$614$	0	0
$615$	23079.9	1.51328
$616$	0	0
$617$	−4255.88	−0.277691	−0.138845	−	0.990314i	$-0.544339\pi$
$617$	−4255.88	−0.277691	−0.138845	+	0.990314i	$0.544339\pi$
$618$	0	0
$619$	228.949	0.0148663	0.00743315	−	0.999972i	$-0.497634\pi$
$619$	228.949	0.0148663	0.00743315	+	0.999972i	$0.497634\pi$
$620$	0	0
$621$	−5162.51	−0.333598
$622$	0	0
$623$	−1527.17	−0.0982102
$624$	0	0
$625$	−6013.82	−0.384884
$626$	0	0
$627$	2546.83	0.162218
$628$	0	0
$629$	−10243.5	−0.649340
$630$	0	0
$631$	11429.2	0.721058	0.360529	−	0.932748i	$-0.382596\pi$
$631$	11429.2	0.721058	0.360529	+	0.932748i	$0.382596\pi$
$632$	0	0
$633$	9667.04	0.606999
$634$	0	0
$635$	−25464.1	−1.59136
$636$	0	0
$637$	−14469.9	−0.900030
$638$	0	0
$639$	−6131.38	−0.379583
$640$	0	0
$641$	29381.4	1.81045	0.905223	−	0.424938i	$-0.139704\pi$
$641$	29381.4	1.81045	0.905223	+	0.424938i	$0.139704\pi$
$642$	0	0
$643$	−249.316	−0.0152909	−0.00764546	−	0.999971i	$-0.502434\pi$
$643$	−249.316	−0.0152909	−0.00764546	+	0.999971i	$0.502434\pi$
$644$	0	0
$645$	−3352.49	−0.204658
$646$	0	0
$647$	−16025.8	−0.973785	−0.486893	−	0.873462i	$-0.661870\pi$
$647$	−16025.8	−0.973785	−0.486893	+	0.873462i	$0.661870\pi$
$648$	0	0
$649$	−8821.17	−0.533530
$650$	0	0
$651$	−1903.06	−0.114573
$652$	0	0
$653$	−14008.6	−0.839511	−0.419755	−	0.907637i	$-0.637884\pi$
$653$	−14008.6	−0.839511	−0.419755	+	0.907637i	$0.637884\pi$
$654$	0	0
$655$	1715.56	0.102340
$656$	0	0
$657$	7065.52	0.419562
$658$	0	0
$659$	1011.91	0.0598152	0.0299076	−	0.999553i	$-0.490479\pi$
$659$	1011.91	0.0598152	0.0299076	+	0.999553i	$0.490479\pi$
$660$	0	0
$661$	−23619.4	−1.38985	−0.694923	−	0.719084i	$-0.744561\pi$
$661$	−23619.4	−1.38985	−0.694923	+	0.719084i	$0.744561\pi$
$662$	0	0
$663$	4922.67	0.288357
$664$	0	0
$665$	4163.28	0.242775
$666$	0	0
$667$	−26561.6	−1.54194
$668$	0	0
$669$	14797.8	0.855183
$670$	0	0
$671$	7700.41	0.443027
$672$	0	0
$673$	−25811.9	−1.47842	−0.739208	−	0.673477i	$-0.764800\pi$
$673$	−25811.9	−1.47842	−0.739208	+	0.673477i	$0.764800\pi$
$674$	0	0
$675$	−4826.64	−0.275226
$676$	0	0
$677$	18255.2	1.03634	0.518172	−	0.855277i	$-0.326613\pi$
$677$	18255.2	1.03634	0.518172	+	0.855277i	$0.326613\pi$
$678$	0	0
$679$	−700.438	−0.0395881
$680$	0	0
$681$	−10858.5	−0.611009
$682$	0	0
$683$	20090.4	1.12553	0.562765	−	0.826617i	$-0.309738\pi$
$683$	20090.4	1.12553	0.562765	+	0.826617i	$0.309738\pi$
$684$	0	0
$685$	37839.0	2.11059
$686$	0	0
$687$	−917.278	−0.0509408
$688$	0	0
$689$	−12056.8	−0.666659
$690$	0	0
$691$	−16521.5	−0.909563	−0.454782	−	0.890603i	$-0.650283\pi$
$691$	−16521.5	−0.909563	−0.454782	+	0.890603i	$0.650283\pi$
$692$	0	0
$693$	286.014	0.0156779
$694$	0	0
$695$	−28316.7	−1.54549
$696$	0	0
$697$	−16721.7	−0.908720
$698$	0	0
$699$	1918.95	0.103836
$700$	0	0
$701$	12431.4	0.669795	0.334897	−	0.942255i	$-0.391298\pi$
$701$	12431.4	0.669795	0.334897	+	0.942255i	$0.391298\pi$
$702$	0	0
$703$	−21600.4	−1.15886
$704$	0	0
$705$	−22809.2	−1.21850
$706$	0	0
$707$	614.674	0.0326976
$708$	0	0
$709$	−980.957	−0.0519614	−0.0259807	−	0.999662i	$-0.508271\pi$
$709$	−980.957	−0.0519614	−0.0259807	+	0.999662i	$0.508271\pi$
$710$	0	0
$711$	−9169.36	−0.483654
$712$	0	0
$713$	40561.4	2.13048
$714$	0	0
$715$	−8023.06	−0.419644
$716$	0	0
$717$	5234.81	0.272660
$718$	0	0
$719$	−4115.73	−0.213478	−0.106739	−	0.994287i	$-0.534041\pi$
$719$	−4115.73	−0.213478	−0.106739	+	0.994287i	$0.534041\pi$
$720$	0	0
$721$	1170.35	0.0604522
$722$	0	0
$723$	−10071.9	−0.518088
$724$	0	0
$725$	−24833.5	−1.27213
$726$	0	0
$727$	−20850.1	−1.06367	−0.531833	−	0.846849i	$-0.678497\pi$
$727$	−20850.1	−1.06367	−0.531833	+	0.846849i	$0.678497\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	2428.92	0.122896
$732$	0	0
$733$	−31517.2	−1.58815	−0.794074	−	0.607821i	$-0.792044\pi$
$733$	−31517.2	−1.58815	−0.794074	+	0.607821i	$0.792044\pi$
$734$	0	0
$735$	−17466.7	−0.876558
$736$	0	0
$737$	−9134.57	−0.456549
$738$	0	0
$739$	−11415.0	−0.568213	−0.284106	−	0.958793i	$-0.591697\pi$
$739$	−11415.0	−0.568213	−0.284106	+	0.958793i	$0.591697\pi$
$740$	0	0
$741$	10380.4	0.514622
$742$	0	0
$743$	−5732.08	−0.283028	−0.141514	−	0.989936i	$-0.545197\pi$
$743$	−5732.08	−0.283028	−0.141514	+	0.989936i	$0.545197\pi$
$744$	0	0
$745$	11096.2	0.545683
$746$	0	0
$747$	−4206.01	−0.206010
$748$	0	0
$749$	−2793.78	−0.136291
$750$	0	0
$751$	−7843.07	−0.381089	−0.190544	−	0.981679i	$-0.561025\pi$
$751$	−7843.07	−0.381089	−0.190544	+	0.981679i	$0.561025\pi$
$752$	0	0
$753$	−3952.34	−0.191277
$754$	0	0
$755$	−31709.0	−1.52849
$756$	0	0
$757$	29125.9	1.39841	0.699206	−	0.714920i	$-0.253537\pi$
$757$	29125.9	1.39841	0.699206	+	0.714920i	$0.253537\pi$
$758$	0	0
$759$	−6096.02	−0.291530
$760$	0	0
$761$	−14228.5	−0.677768	−0.338884	−	0.940828i	$-0.610049\pi$
$761$	−14228.5	−0.677768	−0.338884	+	0.940828i	$0.610049\pi$
$762$	0	0
$763$	−1746.71	−0.0828771
$764$	0	0
$765$	5942.19	0.280837
$766$	0	0
$767$	−35953.6	−1.69258
$768$	0	0
$769$	−28133.7	−1.31928	−0.659641	−	0.751581i	$-0.729292\pi$
$769$	−28133.7	−1.31928	−0.659641	+	0.751581i	$0.729292\pi$
$770$	0	0
$771$	9108.35	0.425459
$772$	0	0
$773$	14686.9	0.683377	0.341689	−	0.939813i	$-0.389001\pi$
$773$	14686.9	0.683377	0.341689	+	0.939813i	$0.389001\pi$
$774$	0	0
$775$	37922.5	1.75770
$776$	0	0
$777$	−2425.78	−0.112000
$778$	0	0
$779$	−35260.9	−1.62176
$780$	0	0
$781$	−7240.08	−0.331716
$782$	0	0
$783$	3750.78	0.171190
$784$	0	0
$785$	−28870.9	−1.31267
$786$	0	0
$787$	−21001.0	−0.951214	−0.475607	−	0.879658i	$-0.657772\pi$
$787$	−21001.0	−0.951214	−0.475607	+	0.879658i	$0.657772\pi$
$788$	0	0
$789$	4967.27	0.224131
$790$	0	0
$791$	−1742.48	−0.0783253
$792$	0	0
$793$	31385.6	1.40547
$794$	0	0
$795$	−14553.9	−0.649274
$796$	0	0
$797$	−13362.7	−0.593892	−0.296946	−	0.954894i	$-0.595968\pi$
$797$	−13362.7	−0.593892	−0.296946	+	0.954894i	$0.595968\pi$
$798$	0	0
$799$	16525.5	0.731704
$800$	0	0
$801$	−4596.36	−0.202752
$802$	0	0
$803$	8343.14	0.366654
$804$	0	0
$805$	−9965.13	−0.436304
$806$	0	0
$807$	−15878.2	−0.692612
$808$	0	0
$809$	39481.6	1.71582	0.857911	−	0.513798i	$-0.171762\pi$
$809$	39481.6	1.71582	0.857911	+	0.513798i	$0.171762\pi$
$810$	0	0
$811$	31157.5	1.34906	0.674531	−	0.738247i	$-0.264346\pi$
$811$	31157.5	1.34906	0.674531	+	0.738247i	$0.264346\pi$
$812$	0	0
$813$	−24031.6	−1.03668
$814$	0	0
$815$	38845.8	1.66958
$816$	0	0
$817$	5121.87	0.219329
$818$	0	0
$819$	1165.75	0.0497368
$820$	0	0
$821$	21229.9	0.902470	0.451235	−	0.892405i	$-0.350984\pi$
$821$	21229.9	0.902470	0.451235	+	0.892405i	$0.350984\pi$
$822$	0	0
$823$	−24603.8	−1.04208	−0.521041	−	0.853532i	$-0.674456\pi$
$823$	−24603.8	−1.04208	−0.521041	+	0.853532i	$0.674456\pi$
$824$	0	0
$825$	−5699.41	−0.240519
$826$	0	0
$827$	13668.1	0.574710	0.287355	−	0.957824i	$-0.407224\pi$
$827$	13668.1	0.574710	0.287355	+	0.957824i	$0.407224\pi$
$828$	0	0
$829$	−27518.8	−1.15291	−0.576457	−	0.817127i	$-0.695565\pi$
$829$	−27518.8	−1.15291	−0.576457	+	0.817127i	$0.695565\pi$
$830$	0	0
$831$	17078.4	0.712929
$832$	0	0
$833$	12654.9	0.526369
$834$	0	0
$835$	−29054.1	−1.20414
$836$	0	0
$837$	−5727.68	−0.236532
$838$	0	0
$839$	29951.5	1.23247	0.616234	−	0.787563i	$-0.288657\pi$
$839$	29951.5	1.23247	0.616234	+	0.787563i	$0.288657\pi$
$840$	0	0
$841$	−5090.88	−0.208737
$842$	0	0
$843$	6074.48	0.248180
$844$	0	0
$845$	5590.49	0.227596
$846$	0	0
$847$	−3642.38	−0.147761
$848$	0	0
$849$	−743.283	−0.0300464
$850$	0	0
$851$	51702.3	2.08265
$852$	0	0
$853$	−5174.61	−0.207708	−0.103854	−	0.994593i	$-0.533118\pi$
$853$	−5174.61	−0.207708	−0.103854	+	0.994593i	$0.533118\pi$
$854$	0	0
$855$	12530.3	0.501201
$856$	0	0
$857$	−9258.34	−0.369030	−0.184515	−	0.982830i	$-0.559071\pi$
$857$	−9258.34	−0.369030	−0.184515	+	0.982830i	$0.559071\pi$
$858$	0	0
$859$	24353.0	0.967304	0.483652	−	0.875260i	$-0.339310\pi$
$859$	24353.0	0.967304	0.483652	+	0.875260i	$0.339310\pi$
$860$	0	0
$861$	−3959.88	−0.156739
$862$	0	0
$863$	−42283.4	−1.66784	−0.833919	−	0.551887i	$-0.813908\pi$
$863$	−42283.4	−1.66784	−0.833919	+	0.551887i	$0.813908\pi$
$864$	0	0
$865$	−43581.4	−1.71308
$866$	0	0
$867$	10433.8	0.408709
$868$	0	0
$869$	−10827.4	−0.422663
$870$	0	0
$871$	−37231.0	−1.44836
$872$	0	0
$873$	−2108.12	−0.0817286
$874$	0	0
$875$	−2802.08	−0.108260
$876$	0	0
$877$	49843.1	1.91914	0.959568	−	0.281476i	$-0.0908239\pi$
$877$	49843.1	1.91914	0.959568	+	0.281476i	$0.0908239\pi$
$878$	0	0
$879$	24400.8	0.936314
$880$	0	0
$881$	8986.94	0.343675	0.171837	−	0.985125i	$-0.445030\pi$
$881$	8986.94	0.343675	0.171837	+	0.985125i	$0.445030\pi$
$882$	0	0
$883$	3693.99	0.140784	0.0703922	−	0.997519i	$-0.477575\pi$
$883$	3693.99	0.140784	0.0703922	+	0.997519i	$0.477575\pi$
$884$	0	0
$885$	−43399.8	−1.64844
$886$	0	0
$887$	51613.0	1.95377	0.976886	−	0.213763i	$-0.0685720\pi$
$887$	51613.0	1.95377	0.976886	+	0.213763i	$0.0685720\pi$
$888$	0	0
$889$	4368.95	0.164825
$890$	0	0
$891$	860.821	0.0323665
$892$	0	0
$893$	34847.4	1.30585
$894$	0	0
$895$	−6601.03	−0.246534
$896$	0	0
$897$	−24846.4	−0.924856
$898$	0	0
$899$	−29469.5	−1.09328
$900$	0	0
$901$	10544.5	0.389886
$902$	0	0
$903$	575.197	0.0211975
$904$	0	0
$905$	53922.7	1.98061
$906$	0	0
$907$	−17016.8	−0.622971	−0.311486	−	0.950251i	$-0.600827\pi$
$907$	−17016.8	−0.622971	−0.311486	+	0.950251i	$0.600827\pi$
$908$	0	0
$909$	1849.99	0.0675032
$910$	0	0
$911$	2991.72	0.108804	0.0544019	−	0.998519i	$-0.482675\pi$
$911$	2991.72	0.108804	0.0544019	+	0.998519i	$0.482675\pi$
$912$	0	0
$913$	−4966.55	−0.180032
$914$	0	0
$915$	37885.7	1.36881
$916$	0	0
$917$	−294.344	−0.0105999
$918$	0	0
$919$	5174.82	0.185747	0.0928736	−	0.995678i	$-0.470395\pi$
$919$	5174.82	0.185747	0.0928736	+	0.995678i	$0.470395\pi$
$920$	0	0
$921$	−8924.46	−0.319295
$922$	0	0
$923$	−29509.3	−1.05234
$924$	0	0
$925$	48338.6	1.71823
$926$	0	0
$927$	3522.41	0.124802
$928$	0	0
$929$	49256.5	1.73956	0.869780	−	0.493439i	$-0.164260\pi$
$929$	49256.5	1.73956	0.869780	+	0.493439i	$0.164260\pi$
$930$	0	0
$931$	26685.3	0.939394
$932$	0	0
$933$	13354.6	0.468605
$934$	0	0
$935$	7016.69	0.245423
$936$	0	0
$937$	−31566.5	−1.10057	−0.550283	−	0.834978i	$-0.685480\pi$
$937$	−31566.5	−1.10057	−0.550283	+	0.834978i	$0.685480\pi$
$938$	0	0
$939$	−24821.3	−0.862632
$940$	0	0
$941$	−37575.6	−1.30173	−0.650866	−	0.759193i	$-0.725594\pi$
$941$	−37575.6	−1.30173	−0.650866	+	0.759193i	$0.725594\pi$
$942$	0	0
$943$	84399.7	2.91456
$944$	0	0
$945$	1407.18	0.0484397
$946$	0	0
$947$	10289.3	0.353070	0.176535	−	0.984294i	$-0.443511\pi$
$947$	10289.3	0.353070	0.176535	+	0.984294i	$0.443511\pi$
$948$	0	0
$949$	34005.2	1.16318
$950$	0	0
$951$	1287.11	0.0438879
$952$	0	0
$953$	36779.7	1.25017	0.625085	−	0.780557i	$-0.285064\pi$
$953$	36779.7	1.25017	0.625085	+	0.780557i	$0.285064\pi$
$954$	0	0
$955$	63736.9	2.15966
$956$	0	0
$957$	4429.01	0.149602
$958$	0	0
$959$	−6492.15	−0.218605
$960$	0	0
$961$	15210.9	0.510586
$962$	0	0
$963$	−8408.47	−0.281370
$964$	0	0
$965$	−48593.6	−1.62102
$966$	0	0
$967$	35228.9	1.17155	0.585773	−	0.810475i	$-0.300791\pi$
$967$	35228.9	1.17155	0.585773	+	0.810475i	$0.300791\pi$
$968$	0	0
$969$	−9078.36	−0.300969
$970$	0	0
$971$	15314.8	0.506155	0.253077	−	0.967446i	$-0.418557\pi$
$971$	15314.8	0.506155	0.253077	+	0.967446i	$0.418557\pi$
$972$	0	0
$973$	4858.38	0.160074
$974$	0	0
$975$	−23229.9	−0.763027
$976$	0	0
$977$	24074.3	0.788337	0.394168	−	0.919038i	$-0.371033\pi$
$977$	24074.3	0.788337	0.394168	+	0.919038i	$0.371033\pi$
$978$	0	0
$979$	−5427.49	−0.177184
$980$	0	0
$981$	−5257.10	−0.171097
$982$	0	0
$983$	−15244.1	−0.494620	−0.247310	−	0.968936i	$-0.579547\pi$
$983$	−15244.1	−0.494620	−0.247310	+	0.968936i	$0.579547\pi$
$984$	0	0
$985$	−20003.7	−0.647079
$986$	0	0
$987$	3913.43	0.126207
$988$	0	0
$989$	−12259.6	−0.394168
$990$	0	0
$991$	−37518.6	−1.20264	−0.601321	−	0.799008i	$-0.705359\pi$
$991$	−37518.6	−1.20264	−0.601321	+	0.799008i	$0.705359\pi$
$992$	0	0
$993$	−24590.1	−0.785845
$994$	0	0
$995$	84403.1	2.68920
$996$	0	0
$997$	1717.01	0.0545420	0.0272710	−	0.999628i	$-0.491318\pi$
$997$	1717.01	0.0545420	0.0272710	+	0.999628i	$0.491318\pi$
$998$	0	0
$999$	−7300.90	−0.231221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.u.1.1		4
3.2	odd	2		2304.4.a.cb.1.4		4
4.3	odd	2		768.4.a.v.1.1		4
8.3	odd	2	inner	768.4.a.u.1.4		4
8.5	even	2		768.4.a.v.1.4		4
12.11	even	2		2304.4.a.by.1.4		4
16.3	odd	4		384.4.d.f.193.8	yes	8
16.5	even	4		384.4.d.f.193.5	yes	8
16.11	odd	4		384.4.d.f.193.1	✓	8
16.13	even	4		384.4.d.f.193.4	yes	8
24.5	odd	2		2304.4.a.by.1.1		4
24.11	even	2		2304.4.a.cb.1.1		4
48.5	odd	4		1152.4.d.p.577.7		8
48.11	even	4		1152.4.d.p.577.8		8
48.29	odd	4		1152.4.d.p.577.1		8
48.35	even	4		1152.4.d.p.577.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.f.193.1	✓	8	16.11	odd	4
384.4.d.f.193.4	yes	8	16.13	even	4
384.4.d.f.193.5	yes	8	16.5	even	4
384.4.d.f.193.8	yes	8	16.3	odd	4
768.4.a.u.1.1		4	1.1	even	1	trivial
768.4.a.u.1.4		4	8.3	odd	2	inner
768.4.a.v.1.1		4	4.3	odd	2
768.4.a.v.1.4		4	8.5	even	2
1152.4.d.p.577.1		8	48.29	odd	4
1152.4.d.p.577.2		8	48.35	even	4
1152.4.d.p.577.7		8	48.5	odd	4
1152.4.d.p.577.8		8	48.11	even	4
2304.4.a.by.1.1		4	24.5	odd	2
2304.4.a.by.1.4		4	12.11	even	2
2304.4.a.cb.1.1		4	24.11	even	2
2304.4.a.cb.1.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -17.4288 q^{5} +2.99032 q^{7} +9.00000 q^{9} +10.6274 q^{11} +43.3156 q^{13} +52.2865 q^{15} -37.8823 q^{17} -79.8823 q^{19} -8.97095 q^{21} +191.204 q^{23} +178.765 q^{25} -27.0000 q^{27} -138.918 q^{29} +212.136 q^{31} -31.8823 q^{33} -52.1177 q^{35} +270.404 q^{37} -129.947 q^{39} +441.411 q^{41} -64.1177 q^{43} -156.860 q^{45} -436.234 q^{47} -334.058 q^{49} +113.647 q^{51} -278.348 q^{53} -185.224 q^{55} +239.647 q^{57} -830.039 q^{59} +724.580 q^{61} +26.9128 q^{63} -754.940 q^{65} -859.529 q^{67} -573.613 q^{69} -681.264 q^{71} +785.058 q^{73} -536.294 q^{75} +31.7793 q^{77} -1018.82 q^{79} +81.0000 q^{81} -467.334 q^{83} +660.244 q^{85} +416.753 q^{87} -510.706 q^{89} +129.527 q^{91} -636.409 q^{93} +1392.26 q^{95} -234.235 q^{97} +95.6468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 36 q^{9} - 48 q^{11} + 120 q^{17} - 48 q^{19} + 172 q^{25} - 108 q^{27} + 144 q^{33} - 480 q^{35} + 408 q^{41} - 528 q^{43} + 836 q^{49} - 360 q^{51} + 144 q^{57} - 1872 q^{59} - 576 q^{65}+ \cdots - 432 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 36 * q^9 - 48 * q^11 + 120 * q^17 - 48 * q^19 + 172 * q^25 - 108 * q^27 + 144 * q^33 - 480 * q^35 + 408 * q^41 - 528 * q^43 + 836 * q^49 - 360 * q^51 + 144 * q^57 - 1872 * q^59 - 576 * q^65 - 2352 * q^67 + 968 * q^73 - 516 * q^75 + 324 * q^81 - 3408 * q^83 - 3672 * q^89 - 5184 * q^91 - 1480 * q^97 - 432 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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