LMFDB - Embedded newform 1728.4.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(1,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, 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("1728.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.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:	$101.955300490$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 216)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	1728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.2337 q^{5} +29.2337 q^{7} +O(q^{10})$	$q-21.2337 q^{5} +29.2337 q^{7} -1.00000 q^{11} +52.9348 q^{13} -96.9348 q^{17} -126.467 q^{19} -22.9348 q^{23} +325.870 q^{25} +133.533 q^{29} +101.832 q^{31} -620.739 q^{35} -105.065 q^{37} +16.3369 q^{41} +201.272 q^{43} -251.870 q^{47} +511.609 q^{49} +148.038 q^{53} +21.2337 q^{55} +73.6085 q^{59} +607.478 q^{61} -1124.00 q^{65} -761.945 q^{67} +701.196 q^{71} -287.000 q^{73} -29.2337 q^{77} -128.522 q^{79} -160.478 q^{83} +2058.28 q^{85} +430.206 q^{89} +1547.48 q^{91} +2685.37 q^{95} -31.1305 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{5} + 24 q^{7}+O(q^{10}) $ 2 * q - 8 * q^5 + 24 * q^7	$ 2 q - 8 q^{5} + 24 q^{7} - 2 q^{11} - 32 q^{13} - 56 q^{17} - 184 q^{19} + 92 q^{23} + 376 q^{25} + 336 q^{29} + 376 q^{31} - 690 q^{35} - 348 q^{37} - 312 q^{41} - 80 q^{43} - 228 q^{47} + 196 q^{49} - 152 q^{53} + 8 q^{55} - 680 q^{59} + 112 q^{61} - 2248 q^{65} - 352 q^{67} + 1816 q^{71} - 574 q^{73} - 24 q^{77} - 1360 q^{79} + 782 q^{83} + 2600 q^{85} + 240 q^{89} + 1992 q^{91} + 1924 q^{95} - 338 q^{97}+O(q^{100}) $ 2 * q - 8 * q^5 + 24 * q^7 - 2 * q^11 - 32 * q^13 - 56 * q^17 - 184 * q^19 + 92 * q^23 + 376 * q^25 + 336 * q^29 + 376 * q^31 - 690 * q^35 - 348 * q^37 - 312 * q^41 - 80 * q^43 - 228 * q^47 + 196 * q^49 - 152 * q^53 + 8 * q^55 - 680 * q^59 + 112 * q^61 - 2248 * q^65 - 352 * q^67 + 1816 * q^71 - 574 * q^73 - 24 * q^77 - 1360 * q^79 + 782 * q^83 + 2600 * q^85 + 240 * q^89 + 1992 * q^91 + 1924 * q^95 - 338 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−21.2337	−1.89920	−0.949599	−	0.313466i	$-0.898510\pi$
$5$	−21.2337	−1.89920	−0.949599	+	0.313466i	$0.898510\pi$
$6$	0	0
$7$	29.2337	1.57847	0.789235	−	0.614091i	$-0.210477\pi$
$7$	29.2337	1.57847	0.789235	+	0.614091i	$0.210477\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.0274101	−0.0137051	−	0.999906i	$-0.504363\pi$
$11$	−1.00000	−0.0274101	−0.0137051	+	0.999906i	$0.504363\pi$
$12$	0	0
$13$	52.9348	1.12934	0.564671	−	0.825316i	$-0.309003\pi$
$13$	52.9348	1.12934	0.564671	+	0.825316i	$0.309003\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−96.9348	−1.38295	−0.691474	−	0.722401i	$-0.743039\pi$
$17$	−96.9348	−1.38295	−0.691474	+	0.722401i	$0.743039\pi$
$18$	0	0
$19$	−126.467	−1.52703	−0.763516	−	0.645789i	$-0.776529\pi$
$19$	−126.467	−1.52703	−0.763516	+	0.645789i	$0.776529\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−22.9348	−0.207923	−0.103961	−	0.994581i	$-0.533152\pi$
$23$	−22.9348	−0.207923	−0.103961	+	0.994581i	$0.533152\pi$
$24$	0	0
$25$	325.870	2.60696
$26$	0	0
$27$	0	0
$28$	0	0
$29$	133.533	0.855048	0.427524	−	0.904004i	$-0.359386\pi$
$29$	133.533	0.855048	0.427524	+	0.904004i	$0.359386\pi$
$30$	0	0
$31$	101.832	0.589983	0.294992	−	0.955500i	$-0.404683\pi$
$31$	101.832	0.589983	0.294992	+	0.955500i	$0.404683\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−620.739	−2.99783
$36$	0	0
$37$	−105.065	−0.466828	−0.233414	−	0.972378i	$-0.574990\pi$
$37$	−105.065	−0.466828	−0.233414	+	0.972378i	$0.574990\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	16.3369	0.0622291	0.0311145	−	0.999516i	$-0.490094\pi$
$41$	16.3369	0.0622291	0.0311145	+	0.999516i	$0.490094\pi$
$42$	0	0
$43$	201.272	0.713805	0.356903	−	0.934142i	$-0.383833\pi$
$43$	201.272	0.713805	0.356903	+	0.934142i	$0.383833\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−251.870	−0.781680	−0.390840	−	0.920459i	$-0.627815\pi$
$47$	−251.870	−0.781680	−0.390840	+	0.920459i	$0.627815\pi$
$48$	0	0
$49$	511.609	1.49157
$50$	0	0
$51$	0	0
$52$	0	0
$53$	148.038	0.383671	0.191836	−	0.981427i	$-0.438556\pi$
$53$	148.038	0.383671	0.191836	+	0.981427i	$0.438556\pi$
$54$	0	0
$55$	21.2337	0.0520573
$56$	0	0
$57$	0	0
$58$	0	0
$59$	73.6085	0.162424	0.0812120	−	0.996697i	$-0.474121\pi$
$59$	73.6085	0.162424	0.0812120	+	0.996697i	$0.474121\pi$
$60$	0	0
$61$	607.478	1.27508	0.637538	−	0.770419i	$-0.279953\pi$
$61$	607.478	1.27508	0.637538	+	0.770419i	$0.279953\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1124.00	−2.14485
$66$	0	0
$67$	−761.945	−1.38935	−0.694675	−	0.719324i	$-0.744452\pi$
$67$	−761.945	−1.38935	−0.694675	+	0.719324i	$0.744452\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	701.196	1.17207	0.586033	−	0.810287i	$-0.300689\pi$
$71$	701.196	1.17207	0.586033	+	0.810287i	$0.300689\pi$
$72$	0	0
$73$	−287.000	−0.460148	−0.230074	−	0.973173i	$-0.573897\pi$
$73$	−287.000	−0.460148	−0.230074	+	0.973173i	$0.573897\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−29.2337	−0.0432661
$78$	0	0
$79$	−128.522	−0.183036	−0.0915181	−	0.995803i	$-0.529172\pi$
$79$	−128.522	−0.183036	−0.0915181	+	0.995803i	$0.529172\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−160.478	−0.212226	−0.106113	−	0.994354i	$-0.533841\pi$
$83$	−160.478	−0.212226	−0.106113	+	0.994354i	$0.533841\pi$
$84$	0	0
$85$	2058.28	2.62649
$86$	0	0
$87$	0	0
$88$	0	0
$89$	430.206	0.512380	0.256190	−	0.966626i	$-0.417533\pi$
$89$	430.206	0.512380	0.256190	+	0.966626i	$0.417533\pi$
$90$	0	0
$91$	1547.48	1.78263
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2685.37	2.90014
$96$	0	0
$97$	−31.1305	−0.0325858	−0.0162929	−	0.999867i	$-0.505186\pi$
$97$	−31.1305	−0.0325858	−0.0162929	+	0.999867i	$0.505186\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	983.103	0.968539	0.484269	−	0.874919i	$-0.339085\pi$
$101$	983.103	0.968539	0.484269	+	0.874919i	$0.339085\pi$
$102$	0	0
$103$	−952.152	−0.910857	−0.455429	−	0.890272i	$-0.650514\pi$
$103$	−952.152	−0.910857	−0.455429	+	0.890272i	$0.650514\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	272.087	0.245828	0.122914	−	0.992417i	$-0.460776\pi$
$107$	272.087	0.245828	0.122914	+	0.992417i	$0.460776\pi$
$108$	0	0
$109$	−1355.76	−1.19136	−0.595680	−	0.803222i	$-0.703117\pi$
$109$	−1355.76	−1.19136	−0.595680	+	0.803222i	$0.703117\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1938.36	−1.61368	−0.806838	−	0.590773i	$-0.798823\pi$
$113$	−1938.36	−1.61368	−0.806838	+	0.590773i	$0.798823\pi$
$114$	0	0
$115$	486.989	0.394887
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2833.76	−2.18294
$120$	0	0
$121$	−1330.00	−0.999249
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4265.20	−3.05193
$126$	0	0
$127$	232.375	0.162362	0.0811808	−	0.996699i	$-0.474131\pi$
$127$	232.375	0.162362	0.0811808	+	0.996699i	$0.474131\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2795.30	1.86433	0.932163	−	0.362038i	$-0.117919\pi$
$131$	2795.30	1.86433	0.932163	+	0.362038i	$0.117919\pi$
$132$	0	0
$133$	−3697.11	−2.41038
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1261.53	−0.786715	−0.393358	−	0.919386i	$-0.628687\pi$
$137$	−1261.53	−0.786715	−0.393358	+	0.919386i	$0.628687\pi$
$138$	0	0
$139$	307.369	0.187559	0.0937794	−	0.995593i	$-0.470105\pi$
$139$	307.369	0.187559	0.0937794	+	0.995593i	$0.470105\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−52.9348	−0.0309554
$144$	0	0
$145$	−2835.39	−1.62391
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1597.76	−0.878478	−0.439239	−	0.898370i	$-0.644752\pi$
$149$	−1597.76	−0.878478	−0.439239	+	0.898370i	$0.644752\pi$
$150$	0	0
$151$	−3415.41	−1.84067	−0.920337	−	0.391126i	$-0.872086\pi$
$151$	−3415.41	−1.84067	−0.920337	+	0.391126i	$0.872086\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2162.26	−1.12050
$156$	0	0
$157$	−476.826	−0.242387	−0.121194	−	0.992629i	$-0.538672\pi$
$157$	−476.826	−0.242387	−0.121194	+	0.992629i	$0.538672\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−670.467	−0.328200
$162$	0	0
$163$	−3304.04	−1.58768	−0.793842	−	0.608124i	$-0.791922\pi$
$163$	−3304.04	−1.58768	−0.793842	+	0.608124i	$0.791922\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2091.65	0.969203	0.484601	−	0.874735i	$-0.338965\pi$
$167$	2091.65	0.969203	0.484601	+	0.874735i	$0.338965\pi$
$168$	0	0
$169$	605.088	0.275416
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2924.56	−1.28526	−0.642631	−	0.766176i	$-0.722157\pi$
$173$	−2924.56	−1.28526	−0.642631	+	0.766176i	$0.722157\pi$
$174$	0	0
$175$	9526.37	4.11500
$176$	0	0
$177$	0	0
$178$	0	0
$179$	231.913	0.0968381	0.0484191	−	0.998827i	$-0.484582\pi$
$179$	231.913	0.0968381	0.0484191	+	0.998827i	$0.484582\pi$
$180$	0	0
$181$	−2291.74	−0.941125	−0.470562	−	0.882367i	$-0.655949\pi$
$181$	−2291.74	−0.941125	−0.470562	+	0.882367i	$0.655949\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2230.92	0.886598
$186$	0	0
$187$	96.9348	0.0379068
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4217.80	−1.59785	−0.798925	−	0.601430i	$-0.794598\pi$
$191$	−4217.80	−1.59785	−0.798925	+	0.601430i	$0.794598\pi$
$192$	0	0
$193$	−1188.35	−0.443208	−0.221604	−	0.975137i	$-0.571129\pi$
$193$	−1188.35	−0.443208	−0.221604	+	0.975137i	$0.571129\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3130.25	1.13209	0.566044	−	0.824375i	$-0.308473\pi$
$197$	3130.25	1.13209	0.566044	+	0.824375i	$0.308473\pi$
$198$	0	0
$199$	−659.146	−0.234802	−0.117401	−	0.993085i	$-0.537456\pi$
$199$	−659.146	−0.234802	−0.117401	+	0.993085i	$0.537456\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3903.65	1.34967
$204$	0	0
$205$	−346.892	−0.118185
$206$	0	0
$207$	0	0
$208$	0	0
$209$	126.467	0.0418561
$210$	0	0
$211$	−3613.95	−1.17912	−0.589560	−	0.807725i	$-0.700699\pi$
$211$	−3613.95	−1.17912	−0.589560	+	0.807725i	$0.700699\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4273.74	−1.35566
$216$	0	0
$217$	2976.91	0.931272
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5131.22	−1.56182
$222$	0	0
$223$	−2054.13	−0.616837	−0.308419	−	0.951251i	$-0.599800\pi$
$223$	−2054.13	−0.616837	−0.308419	+	0.951251i	$0.599800\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2153.61	−0.629692	−0.314846	−	0.949143i	$-0.601953\pi$
$227$	−2153.61	−0.629692	−0.314846	+	0.949143i	$0.601953\pi$
$228$	0	0
$229$	1819.87	0.525154	0.262577	−	0.964911i	$-0.415428\pi$
$229$	1819.87	0.525154	0.262577	+	0.964911i	$0.415428\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3989.68	−1.12177	−0.560886	−	0.827893i	$-0.689539\pi$
$233$	−3989.68	−1.12177	−0.560886	+	0.827893i	$0.689539\pi$
$234$	0	0
$235$	5348.12	1.48457
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5704.32	−1.54386	−0.771929	−	0.635709i	$-0.780708\pi$
$239$	−5704.32	−1.54386	−0.771929	+	0.635709i	$0.780708\pi$
$240$	0	0
$241$	6223.95	1.66357	0.831785	−	0.555099i	$-0.187319\pi$
$241$	6223.95	1.66357	0.831785	+	0.555099i	$0.187319\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10863.3	−2.83279
$246$	0	0
$247$	−6694.52	−1.72454
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4622.52	−1.16243	−0.581217	−	0.813749i	$-0.697423\pi$
$251$	−4622.52	−1.16243	−0.581217	+	0.813749i	$0.697423\pi$
$252$	0	0
$253$	22.9348	0.00569919
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4905.09	−1.19055	−0.595274	−	0.803523i	$-0.702957\pi$
$257$	−4905.09	−1.19055	−0.595274	+	0.803523i	$0.702957\pi$
$258$	0	0
$259$	−3071.44	−0.736874
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4432.85	1.03932	0.519660	−	0.854373i	$-0.326059\pi$
$263$	4432.85	1.03932	0.519660	+	0.854373i	$0.326059\pi$
$264$	0	0
$265$	−3143.39	−0.728668
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4454.81	−1.00972	−0.504860	−	0.863201i	$-0.668456\pi$
$269$	−4454.81	−1.00972	−0.504860	+	0.863201i	$0.668456\pi$
$270$	0	0
$271$	3256.23	0.729897	0.364948	−	0.931028i	$-0.381087\pi$
$271$	3256.23	0.729897	0.364948	+	0.931028i	$0.381087\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−325.870	−0.0714570
$276$	0	0
$277$	3421.61	0.742182	0.371091	−	0.928596i	$-0.378984\pi$
$277$	3421.61	0.742182	0.371091	+	0.928596i	$0.378984\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1715.11	−0.364109	−0.182055	−	0.983288i	$-0.558275\pi$
$281$	−1715.11	−0.364109	−0.182055	+	0.983288i	$0.558275\pi$
$282$	0	0
$283$	4487.60	0.942615	0.471307	−	0.881969i	$-0.343782\pi$
$283$	4487.60	0.942615	0.471307	+	0.881969i	$0.343782\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	477.587	0.0982268
$288$	0	0
$289$	4483.35	0.912548
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1685.31	−0.336031	−0.168016	−	0.985784i	$-0.553736\pi$
$293$	−1685.31	−0.336031	−0.168016	+	0.985784i	$0.553736\pi$
$294$	0	0
$295$	−1562.98	−0.308475
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1214.05	−0.234816
$300$	0	0
$301$	5883.91	1.12672
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12899.0	−2.42162
$306$	0	0
$307$	8079.07	1.50194	0.750972	−	0.660334i	$-0.229585\pi$
$307$	8079.07	1.50194	0.750972	+	0.660334i	$0.229585\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2557.63	0.466334	0.233167	−	0.972437i	$-0.425091\pi$
$311$	2557.63	0.466334	0.233167	+	0.972437i	$0.425091\pi$
$312$	0	0
$313$	−4081.43	−0.737049	−0.368524	−	0.929618i	$-0.620137\pi$
$313$	−4081.43	−0.737049	−0.368524	+	0.929618i	$0.620137\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7701.93	1.36462	0.682308	−	0.731065i	$-0.260976\pi$
$317$	7701.93	1.36462	0.682308	+	0.731065i	$0.260976\pi$
$318$	0	0
$319$	−133.533	−0.0234370
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12259.1	2.11181
$324$	0	0
$325$	17249.8	2.94415
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7363.07	−1.23386
$330$	0	0
$331$	357.314	0.0593346	0.0296673	−	0.999560i	$-0.490555\pi$
$331$	357.314	0.0593346	0.0296673	+	0.999560i	$0.490555\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16178.9	2.63865
$336$	0	0
$337$	−6659.09	−1.07639	−0.538195	−	0.842820i	$-0.680894\pi$
$337$	−6659.09	−1.07639	−0.538195	+	0.842820i	$0.680894\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−101.832	−0.0161715
$342$	0	0
$343$	4929.05	0.775929
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7894.04	−1.22125	−0.610626	−	0.791919i	$-0.709082\pi$
$347$	−7894.04	−1.22125	−0.610626	+	0.791919i	$0.709082\pi$
$348$	0	0
$349$	1254.46	0.192405	0.0962027	−	0.995362i	$-0.469330\pi$
$349$	1254.46	0.192405	0.0962027	+	0.995362i	$0.469330\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8604.92	1.29743	0.648716	−	0.761030i	$-0.275306\pi$
$353$	8604.92	1.29743	0.648716	+	0.761030i	$0.275306\pi$
$354$	0	0
$355$	−14889.0	−2.22598
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3875.61	−0.569768	−0.284884	−	0.958562i	$-0.591955\pi$
$359$	−3875.61	−0.569768	−0.284884	+	0.958562i	$0.591955\pi$
$360$	0	0
$361$	9135.00	1.33183
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6094.07	0.873913
$366$	0	0
$367$	−11890.1	−1.69117	−0.845583	−	0.533843i	$-0.820747\pi$
$367$	−11890.1	−1.69117	−0.845583	+	0.533843i	$0.820747\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4327.70	0.605614
$372$	0	0
$373$	−12384.5	−1.71916	−0.859578	−	0.511005i	$-0.829273\pi$
$373$	−12384.5	−1.71916	−0.859578	+	0.511005i	$0.829273\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7068.52	0.965642
$378$	0	0
$379$	−2062.80	−0.279576	−0.139788	−	0.990181i	$-0.544642\pi$
$379$	−2062.80	−0.279576	−0.139788	+	0.990181i	$0.544642\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11905.2	1.58833	0.794164	−	0.607704i	$-0.207909\pi$
$383$	11905.2	1.58833	0.794164	+	0.607704i	$0.207909\pi$
$384$	0	0
$385$	620.739	0.0821709
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12716.6	−1.65747	−0.828735	−	0.559641i	$-0.810939\pi$
$389$	−12716.6	−1.65747	−0.828735	+	0.559641i	$0.810939\pi$
$390$	0	0
$391$	2223.17	0.287547
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2729.00	0.347622
$396$	0	0
$397$	4531.59	0.572881	0.286441	−	0.958098i	$-0.407528\pi$
$397$	4531.59	0.572881	0.286441	+	0.958098i	$0.407528\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4829.15	−0.601388	−0.300694	−	0.953721i	$-0.597218\pi$
$401$	−4829.15	−0.601388	−0.300694	+	0.953721i	$0.597218\pi$
$402$	0	0
$403$	5390.43	0.666294
$404$	0	0
$405$	0	0
$406$	0	0
$407$	105.065	0.0127958
$408$	0	0
$409$	11484.9	1.38849	0.694245	−	0.719739i	$-0.255738\pi$
$409$	11484.9	1.38849	0.694245	+	0.719739i	$0.255738\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2151.85	0.256381
$414$	0	0
$415$	3407.54	0.403059
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8211.65	−0.957435	−0.478718	−	0.877969i	$-0.658898\pi$
$419$	−8211.65	−0.957435	−0.478718	+	0.877969i	$0.658898\pi$
$420$	0	0
$421$	−9788.13	−1.13312	−0.566561	−	0.824020i	$-0.691726\pi$
$421$	−9788.13	−1.13312	−0.566561	+	0.824020i	$0.691726\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−31588.1	−3.60529
$426$	0	0
$427$	17758.8	2.01267
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11206.3	1.25241	0.626207	−	0.779657i	$-0.284606\pi$
$431$	11206.3	1.25241	0.626207	+	0.779657i	$0.284606\pi$
$432$	0	0
$433$	719.306	0.0798329	0.0399165	−	0.999203i	$-0.487291\pi$
$433$	719.306	0.0798329	0.0399165	+	0.999203i	$0.487291\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2900.50	0.317505
$438$	0	0
$439$	8795.00	0.956179	0.478090	−	0.878311i	$-0.341329\pi$
$439$	8795.00	0.956179	0.478090	+	0.878311i	$0.341329\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3023.65	−0.324285	−0.162142	−	0.986767i	$-0.551840\pi$
$443$	−3023.65	−0.324285	−0.162142	+	0.986767i	$0.551840\pi$
$444$	0	0
$445$	−9134.87	−0.973111
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3137.67	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$449$	3137.67	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$450$	0	0
$451$	−16.3369	−0.00170571
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−32858.7	−3.38558
$456$	0	0
$457$	−9015.74	−0.922841	−0.461421	−	0.887182i	$-0.652660\pi$
$457$	−9015.74	−0.922841	−0.461421	+	0.887182i	$0.652660\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12262.6	1.23888	0.619442	−	0.785042i	$-0.287359\pi$
$461$	12262.6	1.23888	0.619442	+	0.785042i	$0.287359\pi$
$462$	0	0
$463$	3472.30	0.348534	0.174267	−	0.984698i	$-0.444244\pi$
$463$	3472.30	0.348534	0.174267	+	0.984698i	$0.444244\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11825.0	−1.17172	−0.585860	−	0.810412i	$-0.699243\pi$
$467$	−11825.0	−1.17172	−0.585860	+	0.810412i	$0.699243\pi$
$468$	0	0
$469$	−22274.5	−2.19305
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−201.272	−0.0195655
$474$	0	0
$475$	−41211.9	−3.98090
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1703.96	−0.162538	−0.0812692	−	0.996692i	$-0.525897\pi$
$479$	−1703.96	−0.162538	−0.0812692	+	0.996692i	$0.525897\pi$
$480$	0	0
$481$	−5561.60	−0.527208
$482$	0	0
$483$	0	0
$484$	0	0
$485$	661.015	0.0618869
$486$	0	0
$487$	7721.95	0.718512	0.359256	−	0.933239i	$-0.383031\pi$
$487$	7721.95	0.718512	0.359256	+	0.933239i	$0.383031\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20223.6	−1.85882	−0.929410	−	0.369049i	$-0.879683\pi$
$491$	−20223.6	−1.85882	−0.929410	+	0.369049i	$0.879683\pi$
$492$	0	0
$493$	−12944.0	−1.18249
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20498.5	1.85007
$498$	0	0
$499$	13702.4	1.22927	0.614633	−	0.788813i	$-0.289304\pi$
$499$	13702.4	1.22927	0.614633	+	0.788813i	$0.289304\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	13707.6	1.21509	0.607546	−	0.794285i	$-0.292154\pi$
$503$	13707.6	1.21509	0.607546	+	0.794285i	$0.292154\pi$
$504$	0	0
$505$	−20874.9	−1.83945
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12339.2	−1.07451	−0.537254	−	0.843421i	$-0.680538\pi$
$509$	−12339.2	−1.07451	−0.537254	+	0.843421i	$0.680538\pi$
$510$	0	0
$511$	−8390.07	−0.726330
$512$	0	0
$513$	0	0
$514$	0	0
$515$	20217.7	1.72990
$516$	0	0
$517$	251.870	0.0214259
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1416.70	0.119130	0.0595649	−	0.998224i	$-0.481029\pi$
$521$	1416.70	0.119130	0.0595649	+	0.998224i	$0.481029\pi$
$522$	0	0
$523$	−6696.15	−0.559851	−0.279925	−	0.960022i	$-0.590310\pi$
$523$	−6696.15	−0.559851	−0.279925	+	0.960022i	$0.590310\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9871.02	−0.815917
$528$	0	0
$529$	−11641.0	−0.956768
$530$	0	0
$531$	0	0
$532$	0	0
$533$	864.789	0.0702780
$534$	0	0
$535$	−5777.40	−0.466876
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−511.609	−0.0408841
$540$	0	0
$541$	−7105.69	−0.564691	−0.282345	−	0.959313i	$-0.591112\pi$
$541$	−7105.69	−0.564691	−0.282345	+	0.959313i	$0.591112\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	28787.8	2.26263
$546$	0	0
$547$	−12028.0	−0.940185	−0.470092	−	0.882617i	$-0.655779\pi$
$547$	−12028.0	−0.940185	−0.470092	+	0.882617i	$0.655779\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16887.5	−1.30569
$552$	0	0
$553$	−3757.17	−0.288917
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14529.5	−1.10527	−0.552634	−	0.833424i	$-0.686377\pi$
$557$	−14529.5	−1.10527	−0.552634	+	0.833424i	$0.686377\pi$
$558$	0	0
$559$	10654.3	0.806131
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10233.4	0.766053	0.383027	−	0.923737i	$-0.374882\pi$
$563$	10233.4	0.766053	0.383027	+	0.923737i	$0.374882\pi$
$564$	0	0
$565$	41158.5	3.06469
$566$	0	0
$567$	0	0
$568$	0	0
$569$	319.460	0.0235368	0.0117684	−	0.999931i	$-0.496254\pi$
$569$	319.460	0.0235368	0.0117684	+	0.999931i	$0.496254\pi$
$570$	0	0
$571$	6793.96	0.497931	0.248965	−	0.968512i	$-0.419909\pi$
$571$	6793.96	0.497931	0.248965	+	0.968512i	$0.419909\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7473.74	−0.542046
$576$	0	0
$577$	−11145.6	−0.804156	−0.402078	−	0.915605i	$-0.631712\pi$
$577$	−11145.6	−0.804156	−0.402078	+	0.915605i	$0.631712\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4691.36	−0.334992
$582$	0	0
$583$	−148.038	−0.0105165
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23817.6	1.67471	0.837356	−	0.546658i	$-0.184100\pi$
$587$	23817.6	1.67471	0.837356	+	0.546658i	$0.184100\pi$
$588$	0	0
$589$	−12878.4	−0.900924
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16416.2	−1.13682	−0.568408	−	0.822747i	$-0.692441\pi$
$593$	−16416.2	−1.13682	−0.568408	+	0.822747i	$0.692441\pi$
$594$	0	0
$595$	60171.2	4.14585
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4801.70	−0.327533	−0.163766	−	0.986499i	$-0.552364\pi$
$599$	−4801.70	−0.327533	−0.163766	+	0.986499i	$0.552364\pi$
$600$	0	0
$601$	4524.21	0.307066	0.153533	−	0.988144i	$-0.450935\pi$
$601$	4524.21	0.307066	0.153533	+	0.988144i	$0.450935\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	28240.8	1.89777
$606$	0	0
$607$	−14978.3	−1.00157	−0.500783	−	0.865573i	$-0.666955\pi$
$607$	−14978.3	−1.00157	−0.500783	+	0.865573i	$0.666955\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13332.6	−0.882784
$612$	0	0
$613$	−11651.6	−0.767709	−0.383854	−	0.923394i	$-0.625404\pi$
$613$	−11651.6	−0.767709	−0.383854	+	0.923394i	$0.625404\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−341.120	−0.0222576	−0.0111288	−	0.999938i	$-0.503542\pi$
$617$	−341.120	−0.0222576	−0.0111288	+	0.999938i	$0.503542\pi$
$618$	0	0
$619$	−18415.1	−1.19575	−0.597873	−	0.801591i	$-0.703987\pi$
$619$	−18415.1	−1.19575	−0.597873	+	0.801591i	$0.703987\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12576.5	0.808776
$624$	0	0
$625$	49832.2	3.18926
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10184.5	0.645599
$630$	0	0
$631$	13557.6	0.855341	0.427671	−	0.903935i	$-0.359334\pi$
$631$	13557.6	0.855341	0.427671	+	0.903935i	$0.359334\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4934.17	−0.308357
$636$	0	0
$637$	27081.9	1.68449
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9682.55	0.596627	0.298313	−	0.954468i	$-0.403576\pi$
$641$	9682.55	0.596627	0.298313	+	0.954468i	$0.403576\pi$
$642$	0	0
$643$	3259.50	0.199910	0.0999551	−	0.994992i	$-0.468130\pi$
$643$	3259.50	0.199910	0.0999551	+	0.994992i	$0.468130\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8115.35	−0.493118	−0.246559	−	0.969128i	$-0.579300\pi$
$647$	−8115.35	−0.493118	−0.246559	+	0.969128i	$0.579300\pi$
$648$	0	0
$649$	−73.6085	−0.00445206
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3427.74	0.205418	0.102709	−	0.994711i	$-0.467249\pi$
$653$	3427.74	0.205418	0.102709	+	0.994711i	$0.467249\pi$
$654$	0	0
$655$	−59354.6	−3.54073
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13820.2	0.816934	0.408467	−	0.912773i	$-0.366064\pi$
$659$	13820.2	0.816934	0.408467	+	0.912773i	$0.366064\pi$
$660$	0	0
$661$	22719.7	1.33690	0.668451	−	0.743756i	$-0.266957\pi$
$661$	22719.7	1.33690	0.668451	+	0.743756i	$0.266957\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	78503.2	4.57778
$666$	0	0
$667$	−3062.54	−0.177784
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−607.478	−0.0349500
$672$	0	0
$673$	−5133.17	−0.294011	−0.147005	−	0.989136i	$-0.546963\pi$
$673$	−5133.17	−0.294011	−0.147005	+	0.989136i	$0.546963\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9295.88	−0.527725	−0.263862	−	0.964560i	$-0.584996\pi$
$677$	−9295.88	−0.527725	−0.263862	+	0.964560i	$0.584996\pi$
$678$	0	0
$679$	−910.059	−0.0514357
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13824.3	−0.774486	−0.387243	−	0.921978i	$-0.626573\pi$
$683$	−13824.3	−0.774486	−0.387243	+	0.921978i	$0.626573\pi$
$684$	0	0
$685$	26787.0	1.49413
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7836.35	0.433296
$690$	0	0
$691$	11015.2	0.606424	0.303212	−	0.952923i	$-0.401941\pi$
$691$	11015.2	0.606424	0.303212	+	0.952923i	$0.401941\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6526.57	−0.356212
$696$	0	0
$697$	−1583.61	−0.0860596
$698$	0	0
$699$	0	0
$700$	0	0
$701$	460.523	0.0248127	0.0124064	−	0.999923i	$-0.496051\pi$
$701$	460.523	0.0248127	0.0124064	+	0.999923i	$0.496051\pi$
$702$	0	0
$703$	13287.3	0.712861
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28739.7	1.52881
$708$	0	0
$709$	−26445.0	−1.40080	−0.700398	−	0.713752i	$-0.746994\pi$
$709$	−26445.0	−1.40080	−0.700398	+	0.713752i	$0.746994\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2335.48	−0.122671
$714$	0	0
$715$	1124.00	0.0587905
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15524.4	0.805235	0.402617	−	0.915368i	$-0.368101\pi$
$719$	15524.4	0.805235	0.402617	+	0.915368i	$0.368101\pi$
$720$	0	0
$721$	−27834.9	−1.43776
$722$	0	0
$723$	0	0
$724$	0	0
$725$	43514.2	2.22907
$726$	0	0
$727$	18934.0	0.965922	0.482961	−	0.875642i	$-0.339561\pi$
$727$	18934.0	0.965922	0.482961	+	0.875642i	$0.339561\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19510.2	−0.987156
$732$	0	0
$733$	16779.9	0.845540	0.422770	−	0.906237i	$-0.361058\pi$
$733$	16779.9	0.845540	0.422770	+	0.906237i	$0.361058\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	761.945	0.0380823
$738$	0	0
$739$	−30309.0	−1.50871	−0.754353	−	0.656469i	$-0.772049\pi$
$739$	−30309.0	−1.50871	−0.754353	+	0.656469i	$0.772049\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32098.8	1.58492	0.792458	−	0.609927i	$-0.208801\pi$
$743$	32098.8	1.58492	0.792458	+	0.609927i	$0.208801\pi$
$744$	0	0
$745$	33926.2	1.66840
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7954.09	0.388032
$750$	0	0
$751$	9434.07	0.458394	0.229197	−	0.973380i	$-0.426390\pi$
$751$	9434.07	0.458394	0.229197	+	0.973380i	$0.426390\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	72521.7	3.49581
$756$	0	0
$757$	−1280.65	−0.0614876	−0.0307438	−	0.999527i	$-0.509788\pi$
$757$	−1280.65	−0.0614876	−0.0307438	+	0.999527i	$0.509788\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19624.7	0.934818	0.467409	−	0.884041i	$-0.345188\pi$
$761$	19624.7	0.934818	0.467409	+	0.884041i	$0.345188\pi$
$762$	0	0
$763$	−39633.9	−1.88053
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3896.45	0.183432
$768$	0	0
$769$	17644.9	0.827426	0.413713	−	0.910407i	$-0.364232\pi$
$769$	17644.9	0.827426	0.413713	+	0.910407i	$0.364232\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22541.7	1.04886	0.524430	−	0.851454i	$-0.324278\pi$
$773$	22541.7	1.04886	0.524430	+	0.851454i	$0.324278\pi$
$774$	0	0
$775$	33183.8	1.53806
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2066.08	−0.0950258
$780$	0	0
$781$	−701.196	−0.0321264
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10124.8	0.460342
$786$	0	0
$787$	39687.8	1.79761	0.898804	−	0.438350i	$-0.144437\pi$
$787$	39687.8	1.79761	0.898804	+	0.438350i	$0.144437\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−56665.4	−2.54714
$792$	0	0
$793$	32156.7	1.44000
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1991.97	0.0885309	0.0442654	−	0.999020i	$-0.485905\pi$
$797$	1991.97	0.0885309	0.0442654	+	0.999020i	$0.485905\pi$
$798$	0	0
$799$	24414.9	1.08102
$800$	0	0
$801$	0	0
$802$	0	0
$803$	287.000	0.0126127
$804$	0	0
$805$	14236.5	0.623317
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21679.4	−0.942160	−0.471080	−	0.882090i	$-0.656136\pi$
$809$	−21679.4	−0.942160	−0.471080	+	0.882090i	$0.656136\pi$
$810$	0	0
$811$	15099.7	0.653786	0.326893	−	0.945061i	$-0.393998\pi$
$811$	15099.7	0.653786	0.326893	+	0.945061i	$0.393998\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	70157.0	3.01533
$816$	0	0
$817$	−25454.3	−1.09000
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9902.77	−0.420961	−0.210480	−	0.977598i	$-0.567503\pi$
$821$	−9902.77	−0.420961	−0.210480	+	0.977598i	$0.567503\pi$
$822$	0	0
$823$	11488.7	0.486600	0.243300	−	0.969951i	$-0.421770\pi$
$823$	11488.7	0.486600	0.243300	+	0.969951i	$0.421770\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36938.8	−1.55319	−0.776594	−	0.630001i	$-0.783055\pi$
$827$	−36938.8	−1.55319	−0.776594	+	0.630001i	$0.783055\pi$
$828$	0	0
$829$	−3205.00	−0.134275	−0.0671376	−	0.997744i	$-0.521387\pi$
$829$	−3205.00	−0.134275	−0.0671376	+	0.997744i	$0.521387\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−49592.6	−2.06277
$834$	0	0
$835$	−44413.5	−1.84071
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1126.63	−0.0463594	−0.0231797	−	0.999731i	$-0.507379\pi$
$839$	−1126.63	−0.0463594	−0.0231797	+	0.999731i	$0.507379\pi$
$840$	0	0
$841$	−6558.04	−0.268893
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12848.2	−0.523069
$846$	0	0
$847$	−38880.8	−1.57728
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2409.65	0.0970641
$852$	0	0
$853$	−6254.66	−0.251061	−0.125531	−	0.992090i	$-0.540063\pi$
$853$	−6254.66	−0.251061	−0.125531	+	0.992090i	$0.540063\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19069.4	0.760090	0.380045	−	0.924968i	$-0.375908\pi$
$857$	19069.4	0.760090	0.380045	+	0.924968i	$0.375908\pi$
$858$	0	0
$859$	−26790.3	−1.06411	−0.532056	−	0.846709i	$-0.678580\pi$
$859$	−26790.3	−1.06411	−0.532056	+	0.846709i	$0.678580\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46056.3	1.81666	0.908329	−	0.418256i	$-0.137359\pi$
$863$	46056.3	1.81666	0.908329	+	0.418256i	$0.137359\pi$
$864$	0	0
$865$	62099.2	2.44097
$866$	0	0
$867$	0	0
$868$	0	0
$869$	128.522	0.00501704
$870$	0	0
$871$	−40333.4	−1.56905
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−124688.	−4.81738
$876$	0	0
$877$	−36490.2	−1.40500	−0.702500	−	0.711684i	$-0.747933\pi$
$877$	−36490.2	−1.40500	−0.702500	+	0.711684i	$0.747933\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−38136.3	−1.45839	−0.729197	−	0.684303i	$-0.760106\pi$
$881$	−38136.3	−1.45839	−0.729197	+	0.684303i	$0.760106\pi$
$882$	0	0
$883$	13336.1	0.508262	0.254131	−	0.967170i	$-0.418211\pi$
$883$	13336.1	0.508262	0.254131	+	0.967170i	$0.418211\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28027.2	−1.06095	−0.530474	−	0.847701i	$-0.677986\pi$
$887$	−28027.2	−1.06095	−0.530474	+	0.847701i	$0.677986\pi$
$888$	0	0
$889$	6793.17	0.256283
$890$	0	0
$891$	0	0
$892$	0	0
$893$	31853.3	1.19365
$894$	0	0
$895$	−4924.38	−0.183915
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13597.8	0.504464
$900$	0	0
$901$	−14350.0	−0.530598
$902$	0	0
$903$	0	0
$904$	0	0
$905$	48662.1	1.78738
$906$	0	0
$907$	45203.3	1.65485	0.827427	−	0.561573i	$-0.189804\pi$
$907$	45203.3	1.65485	0.827427	+	0.561573i	$0.189804\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−29822.0	−1.08457	−0.542287	−	0.840194i	$-0.682441\pi$
$911$	−29822.0	−1.08457	−0.542287	+	0.840194i	$0.682441\pi$
$912$	0	0
$913$	160.478	0.00581714
$914$	0	0
$915$	0	0
$916$	0	0
$917$	81717.0	2.94279
$918$	0	0
$919$	36967.7	1.32693	0.663467	−	0.748206i	$-0.269084\pi$
$919$	36967.7	1.32693	0.663467	+	0.748206i	$0.269084\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	37117.6	1.32366
$924$	0	0
$925$	−34237.6	−1.21700
$926$	0	0
$927$	0	0
$928$	0	0
$929$	50409.4	1.78028	0.890139	−	0.455690i	$-0.150607\pi$
$929$	50409.4	1.78028	0.890139	+	0.455690i	$0.150607\pi$
$930$	0	0
$931$	−64701.8	−2.27767
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2058.28	−0.0719925
$936$	0	0
$937$	−30156.5	−1.05141	−0.525705	−	0.850667i	$-0.676198\pi$
$937$	−30156.5	−1.05141	−0.525705	+	0.850667i	$0.676198\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42221.1	1.46267	0.731333	−	0.682020i	$-0.238898\pi$
$941$	42221.1	1.46267	0.731333	+	0.682020i	$0.238898\pi$
$942$	0	0
$943$	−374.682	−0.0129388
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40164.6	−1.37822	−0.689111	−	0.724656i	$-0.741999\pi$
$947$	−40164.6	−1.37822	−0.689111	+	0.724656i	$0.741999\pi$
$948$	0	0
$949$	−15192.3	−0.519665
$950$	0	0
$951$	0	0
$952$	0	0
$953$	7132.10	0.242425	0.121213	−	0.992627i	$-0.461322\pi$
$953$	7132.10	0.242425	0.121213	+	0.992627i	$0.461322\pi$
$954$	0	0
$955$	89559.5	3.03464
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36879.3	−1.24181
$960$	0	0
$961$	−19421.3	−0.651919
$962$	0	0
$963$	0	0
$964$	0	0
$965$	25233.0	0.841740
$966$	0	0
$967$	−549.206	−0.0182640	−0.00913199	−	0.999958i	$-0.502907\pi$
$967$	−549.206	−0.0182640	−0.00913199	+	0.999958i	$0.502907\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31024.8	−1.02537	−0.512685	−	0.858577i	$-0.671349\pi$
$971$	−31024.8	−1.02537	−0.512685	+	0.858577i	$0.671349\pi$
$972$	0	0
$973$	8985.52	0.296056
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7827.03	0.256304	0.128152	−	0.991755i	$-0.459096\pi$
$977$	7827.03	0.256304	0.128152	+	0.991755i	$0.459096\pi$
$978$	0	0
$979$	−430.206	−0.0140444
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−47421.5	−1.53867	−0.769334	−	0.638847i	$-0.779412\pi$
$983$	−47421.5	−1.53867	−0.769334	+	0.638847i	$0.779412\pi$
$984$	0	0
$985$	−66466.8	−2.15006
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4616.11	−0.148416
$990$	0	0
$991$	43260.0	1.38668	0.693339	−	0.720612i	$-0.256139\pi$
$991$	43260.0	1.38668	0.693339	+	0.720612i	$0.256139\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13996.1	0.445936
$996$	0	0
$997$	40209.1	1.27727	0.638634	−	0.769511i	$-0.279500\pi$
$997$	40209.1	1.27727	0.638634	+	0.769511i	$0.279500\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.a.bh.1.1		2
3.2	odd	2		1728.4.a.bt.1.2		2
4.3	odd	2		1728.4.a.bg.1.1		2
8.3	odd	2		432.4.a.s.1.2		2
8.5	even	2		216.4.a.h.1.2	yes	2
12.11	even	2		1728.4.a.bs.1.2		2
24.5	odd	2		216.4.a.e.1.1	✓	2
24.11	even	2		432.4.a.o.1.1		2
72.5	odd	6		648.4.i.s.217.2		4
72.13	even	6		648.4.i.m.217.1		4
72.29	odd	6		648.4.i.s.433.2		4
72.61	even	6		648.4.i.m.433.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.a.e.1.1	✓	2	24.5	odd	2
216.4.a.h.1.2	yes	2	8.5	even	2
432.4.a.o.1.1		2	24.11	even	2
432.4.a.s.1.2		2	8.3	odd	2
648.4.i.m.217.1		4	72.13	even	6
648.4.i.m.433.1		4	72.61	even	6
648.4.i.s.217.2		4	72.5	odd	6
648.4.i.s.433.2		4	72.29	odd	6
1728.4.a.bg.1.1		2	4.3	odd	2
1728.4.a.bh.1.1		2	1.1	even	1	trivial
1728.4.a.bs.1.2		2	12.11	even	2
1728.4.a.bt.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)

\(f(q)\)	\(=\)	\(q-21.2337 q^{5} +29.2337 q^{7} +O(q^{10})\)	\(q-21.2337 q^{5} +29.2337 q^{7} -1.00000 q^{11} +52.9348 q^{13} -96.9348 q^{17} -126.467 q^{19} -22.9348 q^{23} +325.870 q^{25} +133.533 q^{29} +101.832 q^{31} -620.739 q^{35} -105.065 q^{37} +16.3369 q^{41} +201.272 q^{43} -251.870 q^{47} +511.609 q^{49} +148.038 q^{53} +21.2337 q^{55} +73.6085 q^{59} +607.478 q^{61} -1124.00 q^{65} -761.945 q^{67} +701.196 q^{71} -287.000 q^{73} -29.2337 q^{77} -128.522 q^{79} -160.478 q^{83} +2058.28 q^{85} +430.206 q^{89} +1547.48 q^{91} +2685.37 q^{95} -31.1305 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{5} + 24 q^{7}+O(q^{10}) \) 2 * q - 8 * q^5 + 24 * q^7	\( 2 q - 8 q^{5} + 24 q^{7} - 2 q^{11} - 32 q^{13} - 56 q^{17} - 184 q^{19} + 92 q^{23} + 376 q^{25} + 336 q^{29} + 376 q^{31} - 690 q^{35} - 348 q^{37} - 312 q^{41} - 80 q^{43} - 228 q^{47} + 196 q^{49} - 152 q^{53} + 8 q^{55} - 680 q^{59} + 112 q^{61} - 2248 q^{65} - 352 q^{67} + 1816 q^{71} - 574 q^{73} - 24 q^{77} - 1360 q^{79} + 782 q^{83} + 2600 q^{85} + 240 q^{89} + 1992 q^{91} + 1924 q^{95} - 338 q^{97}+O(q^{100}) \) 2 * q - 8 * q^5 + 24 * q^7 - 2 * q^11 - 32 * q^13 - 56 * q^17 - 184 * q^19 + 92 * q^23 + 376 * q^25 + 336 * q^29 + 376 * q^31 - 690 * q^35 - 348 * q^37 - 312 * q^41 - 80 * q^43 - 228 * q^47 + 196 * q^49 - 152 * q^53 + 8 * q^55 - 680 * q^59 + 112 * q^61 - 2248 * q^65 - 352 * q^67 + 1816 * q^71 - 574 * q^73 - 24 * q^77 - 1360 * q^79 + 782 * q^83 + 2600 * q^85 + 240 * q^89 + 1992 * q^91 + 1924 * q^95 - 338 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more