LMFDB - Embedded newform 768.4.a.v.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$45.3134668844$
Analytic rank:	$0$
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			1.4
Root			$-1.48330$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +17.4288 q^{5} +2.99032 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +17.4288 q^{5} +2.99032 q^{7} +9.00000 q^{9} -10.6274 q^{11} -43.3156 q^{13} +52.2865 q^{15} -37.8823 q^{17} +79.8823 q^{19} +8.97095 q^{21} +191.204 q^{23} +178.765 q^{25} +27.0000 q^{27} +138.918 q^{29} +212.136 q^{31} -31.8823 q^{33} +52.1177 q^{35} -270.404 q^{37} -129.947 q^{39} +441.411 q^{41} +64.1177 q^{43} +156.860 q^{45} -436.234 q^{47} -334.058 q^{49} -113.647 q^{51} +278.348 q^{53} -185.224 q^{55} +239.647 q^{57} +830.039 q^{59} -724.580 q^{61} +26.9128 q^{63} -754.940 q^{65} +859.529 q^{67} +573.613 q^{69} -681.264 q^{71} +785.058 q^{73} +536.294 q^{75} -31.7793 q^{77} -1018.82 q^{79} +81.0000 q^{81} +467.334 q^{83} -660.244 q^{85} +416.753 q^{87} -510.706 q^{89} -129.527 q^{91} +636.409 q^{93} +1392.26 q^{95} -234.235 q^{97} -95.6468 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 36 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	17.4288	1.55888	0.779441	−	0.626475i	$-0.215503\pi$
$5$	17.4288	1.55888	0.779441	+	0.626475i	$0.215503\pi$
$6$	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	$-0.546527\pi$
$11$	−10.6274	−0.291299	−0.145649	+	0.989336i	$0.546527\pi$
$12$	0	0
$13$	−43.3156	−0.924121	−0.462061	−	0.886848i	$-0.652890\pi$
$13$	−43.3156	−0.924121	−0.462061	+	0.886848i	$0.652890\pi$
$14$	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	$-0.339813\pi$
$19$	79.8823	0.964539	0.482270	+	0.876023i	$0.339813\pi$
$20$	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	$-0.353287\pi$
$29$	138.918	0.889530	0.444765	+	0.895647i	$0.353287\pi$
$30$	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	$-0.705124\pi$
$37$	−270.404	−1.20146	−0.600731	+	0.799451i	$0.705124\pi$
$38$	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	$-0.463731\pi$
$43$	64.1177	0.227392	0.113696	+	0.993516i	$0.463731\pi$
$44$	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	$-0.382538\pi$
$53$	278.348	0.721398	0.360699	+	0.932682i	$0.382538\pi$
$54$	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	$-0.131575\pi$
$59$	830.039	1.83156	0.915778	+	0.401684i	$0.131575\pi$
$60$	0	0
$61$	−724.580	−1.52087	−0.760434	−	0.649416i	$-0.775014\pi$
$61$	−724.580	−1.52087	−0.760434	+	0.649416i	$0.775014\pi$
$62$	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	$-0.213359\pi$
$67$	859.529	1.56729	0.783643	+	0.621211i	$0.213359\pi$
$68$	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	$-0.400001\pi$
$83$	467.334	0.618031	0.309015	+	0.951057i	$0.400001\pi$
$84$	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	$-0.532286\pi$
$101$	−205.555	−0.202509	−0.101255	+	0.994861i	$0.532286\pi$
$102$	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	$-0.361309\pi$
$107$	934.274	0.844109	0.422055	+	0.906570i	$0.361309\pi$
$108$	0	0
$109$	584.123	0.513292	0.256646	−	0.966505i	$-0.417383\pi$
$109$	584.123	0.513292	0.256646	+	0.966505i	$0.417383\pi$
$110$	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	$-0.489550\pi$
$131$	98.4323	0.0656494	0.0328247	+	0.999461i	$0.489550\pi$
$132$	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	$-0.665090\pi$
$139$	−1624.70	−0.991407	−0.495703	+	0.868492i	$0.665090\pi$
$140$	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	$-0.444000\pi$
$149$	636.658	0.350047	0.175024	+	0.984564i	$0.444000\pi$
$150$	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	$-0.638331\pi$
$157$	−1656.50	−0.842059	−0.421029	+	0.907047i	$0.638331\pi$
$158$	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	$-0.320121\pi$
$163$	2228.82	1.07101	0.535505	+	0.844532i	$0.320121\pi$
$164$	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	$-0.685165\pi$
$173$	−2500.53	−1.09891	−0.549457	+	0.835522i	$0.685165\pi$
$174$	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	$-0.525196\pi$
$179$	−378.742	−0.158148	−0.0790740	+	0.996869i	$0.525196\pi$
$180$	0	0
$181$	3093.88	1.27053	0.635265	−	0.772294i	$-0.280891\pi$
$181$	3093.88	1.27053	0.635265	+	0.772294i	$0.280891\pi$
$182$	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	$-0.566548\pi$
$197$	−1147.74	−0.415091	−0.207546	+	0.978225i	$0.566548\pi$
$198$	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	$-0.323812\pi$
$211$	3222.35	1.05135	0.525677	+	0.850684i	$0.323812\pi$
$212$	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	$-0.677489\pi$
$227$	−3619.49	−1.05830	−0.529150	+	0.848529i	$0.677489\pi$
$228$	0	0
$229$	−305.759	−0.0882320	−0.0441160	−	0.999026i	$-0.514047\pi$
$229$	−305.759	−0.0882320	−0.0441160	+	0.999026i	$0.514047\pi$
$230$	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	$-0.552972\pi$
$251$	−1317.45	−0.331301	−0.165650	+	0.986185i	$0.552972\pi$
$252$	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	$-0.704761\pi$
$269$	−5292.72	−1.19964	−0.599820	+	0.800135i	$0.704761\pi$
$270$	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	$-0.288180\pi$
$277$	5692.81	1.23483	0.617415	+	0.786638i	$0.288180\pi$
$278$	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	$-0.508284\pi$
$283$	−247.761	−0.0520419	−0.0260210	+	0.999661i	$0.508284\pi$
$284$	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	$-0.198994\pi$
$293$	8133.61	1.62174	0.810871	+	0.585225i	$0.198994\pi$
$294$	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	$-0.589180\pi$
$307$	−2974.82	−0.553035	−0.276518	+	0.961009i	$0.589180\pi$
$308$	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	$-0.487899\pi$
$317$	429.036	0.0760160	0.0380080	+	0.999277i	$0.487899\pi$
$318$	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	$-0.738264\pi$
$331$	−8196.71	−1.36112	−0.680562	+	0.732691i	$0.738264\pi$
$332$	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	$-0.296679\pi$
$347$	7707.48	1.19239	0.596195	+	0.802840i	$0.296679\pi$
$348$	0	0
$349$	−9681.98	−1.48500	−0.742499	−	0.669847i	$-0.766360\pi$
$349$	−9681.98	−1.48500	−0.742499	+	0.669847i	$0.766360\pi$
$350$	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	$-0.660403\pi$
$373$	−6956.92	−0.965726	−0.482863	+	0.875696i	$0.660403\pi$
$374$	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	$-0.257039\pi$
$379$	10201.3	1.38260	0.691299	+	0.722569i	$0.257039\pi$
$380$	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	$-0.511338\pi$
$389$	−546.451	−0.0712240	−0.0356120	+	0.999366i	$0.511338\pi$
$390$	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	$-0.457936\pi$
$397$	2084.56	0.263529	0.131764	+	0.991281i	$0.457936\pi$
$398$	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	$-0.711487\pi$
$419$	−10576.7	−1.23318	−0.616592	+	0.787283i	$0.711487\pi$
$420$	0	0
$421$	−4871.09	−0.563901	−0.281951	−	0.959429i	$-0.590981\pi$
$421$	−4871.09	−0.563901	−0.281951	+	0.959429i	$0.590981\pi$
$422$	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	$-0.474844\pi$
$443$	1472.21	0.157893	0.0789465	+	0.996879i	$0.474844\pi$
$444$	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	$-0.852578\pi$
$461$	−17710.7	−1.78931	−0.894654	+	0.446759i	$0.852578\pi$
$462$	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	$-0.228372\pi$
$467$	15208.3	1.50697	0.753484	+	0.657466i	$0.228372\pi$
$468$	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	$-0.741028\pi$
$491$	−14946.7	−1.37380	−0.686898	+	0.726754i	$0.741028\pi$
$492$	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	$-0.462336\pi$
$499$	2631.77	0.236101	0.118051	+	0.993008i	$0.462336\pi$
$500$	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	$-0.324675\pi$
$509$	12020.3	1.04674	0.523368	+	0.852107i	$0.324675\pi$
$510$	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	$-0.378773\pi$
$523$	8891.64	0.743411	0.371706	+	0.928351i	$0.378773\pi$
$524$	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	$-0.329667\pi$
$541$	12833.5	1.01988	0.509940	+	0.860210i	$0.329667\pi$
$542$	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	$-0.719155\pi$
$547$	−16257.0	−1.27075	−0.635375	+	0.772204i	$0.719155\pi$
$548$	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	$-0.481122\pi$
$557$	1558.32	0.118542	0.0592712	+	0.998242i	$0.481122\pi$
$558$	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	$-0.380681\pi$
$563$	9782.16	0.732272	0.366136	+	0.930561i	$0.380681\pi$
$564$	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	$-0.205061\pi$
$571$	21819.3	1.59914	0.799570	+	0.600573i	$0.205061\pi$
$572$	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	$-0.345052\pi$
$587$	13305.6	0.935569	0.467785	+	0.883843i	$0.345052\pi$
$588$	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	$-0.234014\pi$
$613$	22514.2	1.48343	0.741713	+	0.670717i	$0.234014\pi$
$614$	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	$-0.502366\pi$
$619$	−228.949	−0.0148663	−0.00743315	+	0.999972i	$0.502366\pi$
$620$	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	$-0.497566\pi$
$643$	249.316	0.0152909	0.00764546	+	0.999971i	$0.497566\pi$
$644$	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	$-0.362116\pi$
$653$	14008.6	0.839511	0.419755	+	0.907637i	$0.362116\pi$
$654$	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	$-0.509521\pi$
$659$	−1011.91	−0.0598152	−0.0299076	+	0.999553i	$0.509521\pi$
$660$	0	0
$661$	23619.4	1.38985	0.694923	−	0.719084i	$-0.255439\pi$
$661$	23619.4	1.38985	0.694923	+	0.719084i	$0.255439\pi$
$662$	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	$-0.673387\pi$
$677$	−18255.2	−1.03634	−0.518172	+	0.855277i	$0.673387\pi$
$678$	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	$-0.690262\pi$
$683$	−20090.4	−1.12553	−0.562765	+	0.826617i	$0.690262\pi$
$684$	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	$-0.349717\pi$
$691$	16521.5	0.909563	0.454782	+	0.890603i	$0.349717\pi$
$692$	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	$-0.608702\pi$
$701$	−12431.4	−0.669795	−0.334897	+	0.942255i	$0.608702\pi$
$702$	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	$-0.491729\pi$
$709$	980.957	0.0519614	0.0259807	+	0.999662i	$0.491729\pi$
$710$	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	$-0.207956\pi$
$733$	31517.2	1.58815	0.794074	+	0.607821i	$0.207956\pi$
$734$	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	$-0.408303\pi$
$739$	11415.0	0.568213	0.284106	+	0.958793i	$0.408303\pi$
$740$	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	$-0.746463\pi$
$757$	−29125.9	−1.39841	−0.699206	+	0.714920i	$0.746463\pi$
$758$	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	$-0.610999\pi$
$773$	−14686.9	−0.683377	−0.341689	+	0.939813i	$0.610999\pi$
$774$	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	$-0.342228\pi$
$787$	21001.0	0.951214	0.475607	+	0.879658i	$0.342228\pi$
$788$	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	$-0.404032\pi$
$797$	13362.7	0.593892	0.296946	+	0.954894i	$0.404032\pi$
$798$	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	$-0.735654\pi$
$811$	−31157.5	−1.34906	−0.674531	+	0.738247i	$0.735654\pi$
$812$	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	$-0.649016\pi$
$821$	−21229.9	−0.902470	−0.451235	+	0.892405i	$0.649016\pi$
$822$	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	$-0.592776\pi$
$827$	−13668.1	−0.574710	−0.287355	+	0.957824i	$0.592776\pi$
$828$	0	0
$829$	27518.8	1.15291	0.576457	−	0.817127i	$-0.304435\pi$
$829$	27518.8	1.15291	0.576457	+	0.817127i	$0.304435\pi$
$830$	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	$-0.466882\pi$
$853$	5174.61	0.207708	0.103854	+	0.994593i	$0.466882\pi$
$854$	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	$-0.660690\pi$
$859$	−24353.0	−0.967304	−0.483652	+	0.875260i	$0.660690\pi$
$860$	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	$-0.909176\pi$
$877$	−49843.1	−1.91914	−0.959568	+	0.281476i	$0.909176\pi$
$878$	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	$-0.522425\pi$
$883$	−3693.99	−0.140784	−0.0703922	+	0.997519i	$0.522425\pi$
$884$	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	$-0.399173\pi$
$907$	17016.8	0.622971	0.311486	+	0.950251i	$0.399173\pi$
$908$	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	$-0.274406\pi$
$941$	37575.6	1.30173	0.650866	+	0.759193i	$0.274406\pi$
$942$	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	$-0.556489\pi$
$947$	−10289.3	−0.353070	−0.176535	+	0.984294i	$0.556489\pi$
$948$	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	$-0.581443\pi$
$971$	−15314.8	−0.506155	−0.253077	+	0.967446i	$0.581443\pi$
$972$	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	$-0.508682\pi$
$997$	−1717.01	−0.0545420	−0.0272710	+	0.999628i	$0.508682\pi$
$998$	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.v.1.4		4
3.2	odd	2		2304.4.a.by.1.1		4
4.3	odd	2		768.4.a.u.1.4		4
8.3	odd	2	inner	768.4.a.v.1.1		4
8.5	even	2		768.4.a.u.1.1		4
12.11	even	2		2304.4.a.cb.1.1		4
16.3	odd	4		384.4.d.f.193.1	✓	8
16.5	even	4		384.4.d.f.193.4	yes	8
16.11	odd	4		384.4.d.f.193.8	yes	8
16.13	even	4		384.4.d.f.193.5	yes	8
24.5	odd	2		2304.4.a.cb.1.4		4
24.11	even	2		2304.4.a.by.1.4		4
48.5	odd	4		1152.4.d.p.577.1		8
48.11	even	4		1152.4.d.p.577.2		8
48.29	odd	4		1152.4.d.p.577.7		8
48.35	even	4		1152.4.d.p.577.8		8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +17.4288 q^{5} +2.99032 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +17.4288 q^{5} +2.99032 q^{7} +9.00000 q^{9} -10.6274 q^{11} -43.3156 q^{13} +52.2865 q^{15} -37.8823 q^{17} +79.8823 q^{19} +8.97095 q^{21} +191.204 q^{23} +178.765 q^{25} +27.0000 q^{27} +138.918 q^{29} +212.136 q^{31} -31.8823 q^{33} +52.1177 q^{35} -270.404 q^{37} -129.947 q^{39} +441.411 q^{41} +64.1177 q^{43} +156.860 q^{45} -436.234 q^{47} -334.058 q^{49} -113.647 q^{51} +278.348 q^{53} -185.224 q^{55} +239.647 q^{57} +830.039 q^{59} -724.580 q^{61} +26.9128 q^{63} -754.940 q^{65} +859.529 q^{67} +573.613 q^{69} -681.264 q^{71} +785.058 q^{73} +536.294 q^{75} -31.7793 q^{77} -1018.82 q^{79} +81.0000 q^{81} +467.334 q^{83} -660.244 q^{85} +416.753 q^{87} -510.706 q^{89} -129.527 q^{91} +636.409 q^{93} +1392.26 q^{95} -234.235 q^{97} -95.6468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 36 * q^9	\( 4 q + 12 q^{3} + 36 q^{9} + 48 q^{11} + 120 q^{17} + 48 q^{19} + 172 q^{25} + 108 q^{27} + 144 q^{33} + 480 q^{35} + 408 q^{41} + 528 q^{43} + 836 q^{49} + 360 q^{51} + 144 q^{57} + 1872 q^{59} - 576 q^{65} + 2352 q^{67} + 968 q^{73} + 516 q^{75} + 324 q^{81} + 3408 q^{83} - 3672 q^{89} + 5184 q^{91} - 1480 q^{97} + 432 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more