LMFDB - Embedded newform 144.4.c.b.143.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,4,Mod(143,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("144.143");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	144.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$8.49627504083$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			143.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	144.143
Dual form			144.4.c.b.143.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264i q^{5} +13.8564i q^{7} +O(q^{10})$	$q-4.24264i q^{5} +13.8564i q^{7} +58.7878 q^{11} +28.0000 q^{13} -97.5807i q^{17} +110.851i q^{19} +176.363 q^{23} +107.000 q^{25} +4.24264i q^{29} +124.708i q^{31} +58.7878 q^{35} -118.000 q^{37} -190.919i q^{41} -249.415i q^{43} -293.939 q^{47} +151.000 q^{49} +700.036i q^{53} -249.415i q^{55} -587.878 q^{59} -802.000 q^{61} -118.794i q^{65} -138.564i q^{67} -646.665 q^{71} +496.000 q^{73} +814.587i q^{77} -1011.52i q^{79} -58.7878 q^{83} -414.000 q^{85} -1124.30i q^{89} +387.979i q^{91} +470.302 q^{95} -152.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 112 q^{13} + 428 q^{25} - 472 q^{37} + 604 q^{49} - 3208 q^{61} + 1984 q^{73} - 1656 q^{85} - 608 q^{97}+O(q^{100}) $ 4 * q + 112 * q^13 + 428 * q^25 - 472 * q^37 + 604 * q^49 - 3208 * q^61 + 1984 * q^73 - 1656 * q^85 - 608 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	− 4.24264i	− 0.379473i	−0.981835	−	0.189737i	$-0.939237\pi$
$5$	− 4.24264i	− 0.379473i	0.981835	−	0.189737i	$-0.0607634\pi$
$6$	0	0
$7$	13.8564i	0.748176i	0.927393	+	0.374088i	$0.122044\pi$
$7$	13.8564i	0.748176i	−0.927393	+	0.374088i	$0.877956\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	58.7878	1.61138	0.805690	−	0.592338i	$-0.201795\pi$
$11$	58.7878	1.61138	0.805690	+	0.592338i	$0.201795\pi$
$12$	0	0
$13$	28.0000	0.597369	0.298685	−	0.954352i	$-0.403452\pi$
$13$	28.0000	0.597369	0.298685	+	0.954352i	$0.403452\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 97.5807i	− 1.39216i	−0.717962	−	0.696082i	$-0.754925\pi$
$17$	− 97.5807i	− 1.39216i	0.717962	−	0.696082i	$-0.245075\pi$
$18$	0	0
$19$	110.851i	1.33847i	0.743049	+	0.669237i	$0.233379\pi$
$19$	110.851i	1.33847i	−0.743049	+	0.669237i	$0.766621\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	176.363	1.59888	0.799441	−	0.600745i	$-0.205129\pi$
$23$	176.363	1.59888	0.799441	+	0.600745i	$0.205129\pi$
$24$	0	0
$25$	107.000	0.856000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.24264i	0.0271668i	0.999908	+	0.0135834i	$0.00432387\pi$
$29$	4.24264i	0.0271668i	−0.999908	+	0.0135834i	$0.995676\pi$
$30$	0	0
$31$	124.708i	0.722521i	0.932465	+	0.361261i	$0.117654\pi$
$31$	124.708i	0.722521i	−0.932465	+	0.361261i	$0.882346\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	58.7878	0.283913
$36$	0	0
$37$	−118.000	−0.524299	−0.262150	−	0.965027i	$-0.584431\pi$
$37$	−118.000	−0.524299	−0.262150	+	0.965027i	$0.584431\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 190.919i	− 0.727232i	−0.931549	−	0.363616i	$-0.881542\pi$
$41$	− 190.919i	− 0.727232i	0.931549	−	0.363616i	$-0.118458\pi$
$42$	0	0
$43$	− 249.415i	− 0.884546i	−0.896880	−	0.442273i	$-0.854172\pi$
$43$	− 249.415i	− 0.884546i	0.896880	−	0.442273i	$-0.145828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−293.939	−0.912242	−0.456121	−	0.889918i	$-0.650762\pi$
$47$	−293.939	−0.912242	−0.456121	+	0.889918i	$0.650762\pi$
$48$	0	0
$49$	151.000	0.440233
$50$	0	0
$51$	0	0
$52$	0	0
$53$	700.036i	1.81429i	0.420820	+	0.907144i	$0.361743\pi$
$53$	700.036i	1.81429i	−0.420820	+	0.907144i	$0.638257\pi$
$54$	0	0
$55$	− 249.415i	− 0.611476i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−587.878	−1.29721	−0.648603	−	0.761127i	$-0.724646\pi$
$59$	−587.878	−1.29721	−0.648603	+	0.761127i	$0.724646\pi$
$60$	0	0
$61$	−802.000	−1.68337	−0.841685	−	0.539969i	$-0.818436\pi$
$61$	−802.000	−1.68337	−0.841685	+	0.539969i	$0.818436\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 118.794i	− 0.226686i
$66$	0	0
$67$	− 138.564i	− 0.252661i	−0.991988	−	0.126331i	$-0.959680\pi$
$67$	− 138.564i	− 0.252661i	0.991988	−	0.126331i	$-0.0403200\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−646.665	−1.08092	−0.540458	−	0.841371i	$-0.681749\pi$
$71$	−646.665	−1.08092	−0.540458	+	0.841371i	$0.681749\pi$
$72$	0	0
$73$	496.000	0.795238	0.397619	−	0.917551i	$-0.369837\pi$
$73$	496.000	0.795238	0.397619	+	0.917551i	$0.369837\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	814.587i	1.20559i
$78$	0	0
$79$	− 1011.52i	− 1.44056i	−0.693681	−	0.720282i	$-0.744012\pi$
$79$	− 1011.52i	− 1.44056i	0.693681	−	0.720282i	$-0.255988\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−58.7878	−0.0777445	−0.0388723	−	0.999244i	$-0.512377\pi$
$83$	−58.7878	−0.0777445	−0.0388723	+	0.999244i	$0.512377\pi$
$84$	0	0
$85$	−414.000	−0.528289
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1124.30i	− 1.33905i	−0.742789	−	0.669525i	$-0.766497\pi$
$89$	− 1124.30i	− 1.33905i	0.742789	−	0.669525i	$-0.233503\pi$
$90$	0	0
$91$	387.979i	0.446937i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	470.302	0.507915
$96$	0	0
$97$	−152.000	−0.159106	−0.0795529	−	0.996831i	$-0.525349\pi$
$97$	−152.000	−0.159106	−0.0795529	+	0.996831i	$0.525349\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	929.138i	0.915373i	0.889114	+	0.457687i	$0.151322\pi$
$101$	929.138i	0.915373i	−0.889114	+	0.457687i	$0.848678\pi$
$102$	0	0
$103$	1759.76i	1.68344i	0.539912	+	0.841722i	$0.318458\pi$
$103$	1759.76i	1.68344i	−0.539912	+	0.841722i	$0.681542\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	940.604	0.849828	0.424914	−	0.905234i	$-0.360304\pi$
$107$	940.604	0.849828	0.424914	+	0.905234i	$0.360304\pi$
$108$	0	0
$109$	628.000	0.551849	0.275924	−	0.961179i	$-0.411016\pi$
$109$	628.000	0.551849	0.275924	+	0.961179i	$0.411016\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1243.09i	− 1.03487i	−0.855722	−	0.517435i	$-0.826887\pi$
$113$	− 1243.09i	− 1.03487i	0.855722	−	0.517435i	$-0.173113\pi$
$114$	0	0
$115$	− 748.246i	− 0.606733i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1352.12	1.04158
$120$	0	0
$121$	2125.00	1.59654
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 984.293i	− 0.704302i
$126$	0	0
$127$	1621.20i	1.13274i	0.824151	+	0.566371i	$0.191653\pi$
$127$	1621.20i	1.13274i	−0.824151	+	0.566371i	$0.808347\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	823.029	0.548919	0.274459	−	0.961599i	$-0.411501\pi$
$131$	823.029	0.548919	0.274459	+	0.961599i	$0.411501\pi$
$132$	0	0
$133$	−1536.00	−1.00141
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1607.96i	− 1.00275i	−0.865229	−	0.501377i	$-0.832827\pi$
$137$	− 1607.96i	− 1.00275i	0.865229	−	0.501377i	$-0.167173\pi$
$138$	0	0
$139$	2133.89i	1.30211i	0.759029	+	0.651057i	$0.225674\pi$
$139$	2133.89i	1.30211i	−0.759029	+	0.651057i	$0.774326\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1646.06	0.962589
$144$	0	0
$145$	18.0000	0.0103091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	818.830i	0.450209i	0.974335	+	0.225104i	$0.0722723\pi$
$149$	818.830i	0.450209i	−0.974335	+	0.225104i	$0.927728\pi$
$150$	0	0
$151$	− 3256.26i	− 1.75490i	−0.479666	−	0.877451i	$-0.659242\pi$
$151$	− 3256.26i	− 1.75490i	0.479666	−	0.877451i	$-0.340758\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	529.090	0.274178
$156$	0	0
$157$	1730.00	0.879421	0.439710	−	0.898140i	$-0.355081\pi$
$157$	1730.00	0.879421	0.439710	+	0.898140i	$0.355081\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2443.76i	1.19624i
$162$	0	0
$163$	1635.06i	0.785690i	0.919605	+	0.392845i	$0.128509\pi$
$163$	1635.06i	0.785690i	−0.919605	+	0.392845i	$0.871491\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1998.78	−0.926171	−0.463085	−	0.886314i	$-0.653258\pi$
$167$	−1998.78	−0.926171	−0.463085	+	0.886314i	$0.653258\pi$
$168$	0	0
$169$	−1413.00	−0.643150
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2159.50i	− 0.949041i	−0.880245	−	0.474520i	$-0.842622\pi$
$173$	− 2159.50i	− 0.949041i	0.880245	−	0.474520i	$-0.157378\pi$
$174$	0	0
$175$	1482.64i	0.640438i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2351.51	−0.981900	−0.490950	−	0.871188i	$-0.663350\pi$
$179$	−2351.51	−0.981900	−0.490950	+	0.871188i	$0.663350\pi$
$180$	0	0
$181$	−620.000	−0.254609	−0.127305	−	0.991864i	$-0.540633\pi$
$181$	−620.000	−0.254609	−0.127305	+	0.991864i	$0.540633\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	500.632i	0.198958i
$186$	0	0
$187$	− 5736.55i	− 2.24331i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3644.84	−1.38079	−0.690396	−	0.723431i	$-0.742564\pi$
$191$	−3644.84	−1.38079	−0.690396	+	0.723431i	$0.742564\pi$
$192$	0	0
$193$	−946.000	−0.352822	−0.176411	−	0.984317i	$-0.556449\pi$
$193$	−946.000	−0.352822	−0.176411	+	0.984317i	$0.556449\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1913.43i	0.692012i	0.938232	+	0.346006i	$0.112462\pi$
$197$	1913.43i	0.692012i	−0.938232	+	0.346006i	$0.887538\pi$
$198$	0	0
$199$	− 734.390i	− 0.261605i	−0.991408	−	0.130803i	$-0.958245\pi$
$199$	− 734.390i	− 0.261605i	0.991408	−	0.130803i	$-0.0417555\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−58.7878	−0.0203256
$204$	0	0
$205$	−810.000	−0.275965
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6516.70i	2.15679i
$210$	0	0
$211$	− 360.267i	− 0.117544i	−0.998271	−	0.0587720i	$-0.981282\pi$
$211$	− 360.267i	− 0.117544i	0.998271	−	0.0587720i	$-0.0187185\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1058.18	−0.335662
$216$	0	0
$217$	−1728.00	−0.540573
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2732.26i	− 0.831637i
$222$	0	0
$223$	− 3865.94i	− 1.16091i	−0.814293	−	0.580454i	$-0.802875\pi$
$223$	− 3865.94i	− 1.16091i	0.814293	−	0.580454i	$-0.197125\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2763.02	−0.807878	−0.403939	−	0.914786i	$-0.632359\pi$
$227$	−2763.02	−0.807878	−0.403939	+	0.914786i	$0.632359\pi$
$228$	0	0
$229$	700.000	0.201997	0.100998	−	0.994887i	$-0.467796\pi$
$229$	700.000	0.201997	0.100998	+	0.994887i	$0.467796\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1964.34i	− 0.552311i	−0.961113	−	0.276155i	$-0.910940\pi$
$233$	− 1964.34i	− 0.552311i	0.961113	−	0.276155i	$-0.0890604\pi$
$234$	0	0
$235$	1247.08i	0.346172i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	235.151	0.0636429	0.0318215	−	0.999494i	$-0.489869\pi$
$239$	235.151	0.0636429	0.0318215	+	0.999494i	$0.489869\pi$
$240$	0	0
$241$	−6488.00	−1.73414	−0.867072	−	0.498182i	$-0.834001\pi$
$241$	−6488.00	−1.73414	−0.867072	+	0.498182i	$0.834001\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 640.639i	− 0.167057i
$246$	0	0
$247$	3103.84i	0.799564i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3586.05	−0.901791	−0.450896	−	0.892577i	$-0.648895\pi$
$251$	−3586.05	−0.901791	−0.450896	+	0.892577i	$0.648895\pi$
$252$	0	0
$253$	10368.0	2.57641
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5316.03i	1.29029i	0.764060	+	0.645145i	$0.223203\pi$
$257$	5316.03i	1.29029i	−0.764060	+	0.645145i	$0.776797\pi$
$258$	0	0
$259$	− 1635.06i	− 0.392268i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5996.35	1.40590	0.702948	−	0.711241i	$-0.251866\pi$
$263$	5996.35	1.40590	0.702948	+	0.711241i	$0.251866\pi$
$264$	0	0
$265$	2970.00	0.688474
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6520.94i	− 1.47802i	−0.673692	−	0.739012i	$-0.735292\pi$
$269$	− 6520.94i	− 1.47802i	0.673692	−	0.739012i	$-0.264708\pi$
$270$	0	0
$271$	734.390i	0.164616i	0.996607	+	0.0823081i	$0.0262292\pi$
$271$	734.390i	0.164616i	−0.996607	+	0.0823081i	$0.973771\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6290.29	1.37934
$276$	0	0
$277$	−4124.00	−0.894538	−0.447269	−	0.894399i	$-0.647603\pi$
$277$	−4124.00	−0.894538	−0.447269	+	0.894399i	$0.647603\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1294.01i	− 0.274711i	−0.990522	−	0.137356i	$-0.956140\pi$
$281$	− 1294.01i	− 0.274711i	0.990522	−	0.137356i	$-0.0438603\pi$
$282$	0	0
$283$	7482.46i	1.57168i	0.618428	+	0.785841i	$0.287770\pi$
$283$	7482.46i	1.57168i	−0.618428	+	0.785841i	$0.712230\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2645.45	0.544097
$288$	0	0
$289$	−4609.00	−0.938123
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3542.60i	− 0.706352i	−0.935557	−	0.353176i	$-0.885102\pi$
$293$	− 3542.60i	− 0.706352i	0.935557	−	0.353176i	$-0.114898\pi$
$294$	0	0
$295$	2494.15i	0.492255i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4938.17	0.955123
$300$	0	0
$301$	3456.00	0.661796
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3402.60i	0.638794i
$306$	0	0
$307$	249.415i	0.0463677i	0.999731	+	0.0231839i	$0.00738031\pi$
$307$	249.415i	0.0463677i	−0.999731	+	0.0231839i	$0.992620\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2233.93	−0.407315	−0.203657	−	0.979042i	$-0.565283\pi$
$311$	−2233.93	−0.407315	−0.203657	+	0.979042i	$0.565283\pi$
$312$	0	0
$313$	7946.00	1.43493	0.717467	−	0.696592i	$-0.245301\pi$
$313$	7946.00	1.43493	0.717467	+	0.696592i	$0.245301\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8964.70i	− 1.58835i	−0.607688	−	0.794176i	$-0.707903\pi$
$317$	− 8964.70i	− 1.58835i	0.607688	−	0.794176i	$-0.292097\pi$
$318$	0	0
$319$	249.415i	0.0437761i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10816.9	1.86338
$324$	0	0
$325$	2996.00	0.511348
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4072.94i	− 0.682517i
$330$	0	0
$331$	− 1856.76i	− 0.308328i	−0.988045	−	0.154164i	$-0.950732\pi$
$331$	− 1856.76i	− 0.308328i	0.988045	−	0.154164i	$-0.0492685\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−587.878	−0.0958782
$336$	0	0
$337$	−1952.00	−0.315526	−0.157763	−	0.987477i	$-0.550428\pi$
$337$	−1952.00	−0.315526	−0.157763	+	0.987477i	$0.550428\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7331.28i	1.16426i
$342$	0	0
$343$	6845.06i	1.07755i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5819.99	−0.900384	−0.450192	−	0.892932i	$-0.648644\pi$
$347$	−5819.99	−0.900384	−0.450192	+	0.892932i	$0.648644\pi$
$348$	0	0
$349$	−226.000	−0.0346633	−0.0173317	−	0.999850i	$-0.505517\pi$
$349$	−226.000	−0.0346633	−0.0173317	+	0.999850i	$0.505517\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1845.55i	0.278268i	0.990274	+	0.139134i	$0.0444319\pi$
$353$	1845.55i	0.278268i	−0.990274	+	0.139134i	$0.955568\pi$
$354$	0	0
$355$	2743.57i	0.410179i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−58.7878	−0.00864262	−0.00432131	−	0.999991i	$-0.501376\pi$
$359$	−58.7878	−0.00864262	−0.00432131	+	0.999991i	$0.501376\pi$
$360$	0	0
$361$	−5429.00	−0.791515
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2104.35i	− 0.301772i
$366$	0	0
$367$	− 2480.30i	− 0.352780i	−0.984320	−	0.176390i	$-0.943558\pi$
$367$	− 2480.30i	− 0.352780i	0.984320	−	0.176390i	$-0.0564421\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9699.98	−1.35741
$372$	0	0
$373$	−5338.00	−0.740995	−0.370498	−	0.928833i	$-0.620813\pi$
$373$	−5338.00	−0.740995	−0.370498	+	0.928833i	$0.620813\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	118.794i	0.0162286i
$378$	0	0
$379$	− 5376.29i	− 0.728658i	−0.931270	−	0.364329i	$-0.881298\pi$
$379$	− 5376.29i	− 0.728658i	0.931270	−	0.364329i	$-0.118702\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6349.08	0.847057	0.423528	−	0.905883i	$-0.360791\pi$
$383$	6349.08	0.847057	0.423528	+	0.905883i	$0.360791\pi$
$384$	0	0
$385$	3456.00	0.457491
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11238.8i	1.46485i	0.680847	+	0.732426i	$0.261612\pi$
$389$	11238.8i	1.46485i	−0.680847	+	0.732426i	$0.738388\pi$
$390$	0	0
$391$	− 17209.7i	− 2.22591i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4291.51	−0.546656
$396$	0	0
$397$	−466.000	−0.0589115	−0.0294558	−	0.999566i	$-0.509377\pi$
$397$	−466.000	−0.0589115	−0.0294558	+	0.999566i	$0.509377\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7403.41i	0.921966i	0.887409	+	0.460983i	$0.152503\pi$
$401$	7403.41i	0.921966i	−0.887409	+	0.460983i	$0.847497\pi$
$402$	0	0
$403$	3491.81i	0.431612i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6936.95	−0.844845
$408$	0	0
$409$	4024.00	0.486489	0.243244	−	0.969965i	$-0.421788\pi$
$409$	4024.00	0.486489	0.243244	+	0.969965i	$0.421788\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 8145.87i	− 0.970538i
$414$	0	0
$415$	249.415i	0.0295020i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15108.5	−1.76157	−0.880783	−	0.473520i	$-0.842983\pi$
$419$	−15108.5	−1.76157	−0.880783	+	0.473520i	$0.842983\pi$
$420$	0	0
$421$	−8540.00	−0.988632	−0.494316	−	0.869282i	$-0.664581\pi$
$421$	−8540.00	−0.988632	−0.494316	+	0.869282i	$0.664581\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 10441.1i	− 1.19169i
$426$	0	0
$427$	− 11112.8i	− 1.25946i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16049.1	−1.79363	−0.896817	−	0.442403i	$-0.854126\pi$
$431$	−16049.1	−1.79363	−0.896817	+	0.442403i	$0.854126\pi$
$432$	0	0
$433$	−11266.0	−1.25037	−0.625184	−	0.780477i	$-0.714976\pi$
$433$	−11266.0	−1.25037	−0.625184	+	0.780477i	$0.714976\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19550.1i	2.14006i
$438$	0	0
$439$	6110.68i	0.664343i	0.943219	+	0.332172i	$0.107781\pi$
$439$	6110.68i	0.664343i	−0.943219	+	0.332172i	$0.892219\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13580.0	1.45644	0.728221	−	0.685342i	$-0.240347\pi$
$443$	13580.0	1.45644	0.728221	+	0.685342i	$0.240347\pi$
$444$	0	0
$445$	−4770.00	−0.508134
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 547.301i	− 0.0575250i	−0.999586	−	0.0287625i	$-0.990843\pi$
$449$	− 547.301i	− 0.0575250i	0.999586	−	0.0287625i	$-0.00915665\pi$
$450$	0	0
$451$	− 11223.7i	− 1.17185i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1646.06	0.169601
$456$	0	0
$457$	15784.0	1.61563	0.807817	−	0.589434i	$-0.200649\pi$
$457$	15784.0	1.61563	0.807817	+	0.589434i	$0.200649\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18620.9i	1.88127i	0.339424	+	0.940634i	$0.389768\pi$
$461$	18620.9i	1.88127i	−0.339424	+	0.940634i	$0.610232\pi$
$462$	0	0
$463$	8244.56i	0.827554i	0.910378	+	0.413777i	$0.135791\pi$
$463$	8244.56i	0.827554i	−0.910378	+	0.413777i	$0.864209\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2057.57	−0.203882	−0.101941	−	0.994790i	$-0.532505\pi$
$467$	−2057.57	−0.203882	−0.101941	+	0.994790i	$0.532505\pi$
$468$	0	0
$469$	1920.00	0.189035
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 14662.6i	− 1.42534i
$474$	0	0
$475$	11861.1i	1.14573i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2175.15	−0.207484	−0.103742	−	0.994604i	$-0.533082\pi$
$479$	−2175.15	−0.207484	−0.103742	+	0.994604i	$0.533082\pi$
$480$	0	0
$481$	−3304.00	−0.313200
$482$	0	0
$483$	0	0
$484$	0	0
$485$	644.881i	0.0603764i
$486$	0	0
$487$	− 11708.7i	− 1.08947i	−0.838609	−	0.544733i	$-0.816631\pi$
$487$	− 11708.7i	− 1.08947i	0.838609	−	0.544733i	$-0.183369\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1646.06	0.151294	0.0756472	−	0.997135i	$-0.475898\pi$
$491$	1646.06	0.151294	0.0756472	+	0.997135i	$0.475898\pi$
$492$	0	0
$493$	414.000	0.0378207
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8960.46i	− 0.808715i
$498$	0	0
$499$	− 2992.98i	− 0.268506i	−0.990947	−	0.134253i	$-0.957137\pi$
$499$	− 2992.98i	− 0.268506i	0.990947	−	0.134253i	$-0.0428634\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9112.10	0.807731	0.403865	−	0.914818i	$-0.367666\pi$
$503$	9112.10	0.807731	0.403865	+	0.914818i	$0.367666\pi$
$504$	0	0
$505$	3942.00	0.347360
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 12384.3i	− 1.07843i	−0.842167	−	0.539217i	$-0.818720\pi$
$509$	− 12384.3i	− 1.07843i	0.842167	−	0.539217i	$-0.181280\pi$
$510$	0	0
$511$	6872.78i	0.594978i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7466.04	0.638822
$516$	0	0
$517$	−17280.0	−1.46997
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 3882.02i	− 0.326438i	−0.986590	−	0.163219i	$-0.947812\pi$
$521$	− 3882.02i	− 0.326438i	0.986590	−	0.163219i	$-0.0521877\pi$
$522$	0	0
$523$	15574.6i	1.30216i	0.759009	+	0.651080i	$0.225684\pi$
$523$	15574.6i	1.30216i	−0.759009	+	0.651080i	$0.774316\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12169.1	1.00587
$528$	0	0
$529$	18937.0	1.55642
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5345.73i	− 0.434426i
$534$	0	0
$535$	− 3990.65i	− 0.322487i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8876.95	0.709383
$540$	0	0
$541$	11188.0	0.889112	0.444556	−	0.895751i	$-0.353361\pi$
$541$	11188.0	0.889112	0.444556	+	0.895751i	$0.353361\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 2664.38i	− 0.209412i
$546$	0	0
$547$	21588.3i	1.68747i	0.536757	+	0.843737i	$0.319649\pi$
$547$	21588.3i	1.68747i	−0.536757	+	0.843737i	$0.680351\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−470.302	−0.0363621
$552$	0	0
$553$	14016.0	1.07780
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16346.9i	1.24352i	0.783208	+	0.621760i	$0.213582\pi$
$557$	16346.9i	1.24352i	−0.783208	+	0.621760i	$0.786418\pi$
$558$	0	0
$559$	− 6983.63i	− 0.528401i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21692.7	1.62387	0.811934	−	0.583749i	$-0.198415\pi$
$563$	21692.7	1.62387	0.811934	+	0.583749i	$0.198415\pi$
$564$	0	0
$565$	−5274.00	−0.392706
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9414.42i	− 0.693626i	−0.937934	−	0.346813i	$-0.887264\pi$
$569$	− 9414.42i	− 0.693626i	0.937934	−	0.346813i	$-0.112736\pi$
$570$	0	0
$571$	− 10364.6i	− 0.759623i	−0.925064	−	0.379811i	$-0.875989\pi$
$571$	− 10364.6i	− 0.759623i	0.925064	−	0.379811i	$-0.124011\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18870.9	1.36864
$576$	0	0
$577$	−9406.00	−0.678643	−0.339321	−	0.940670i	$-0.610197\pi$
$577$	−9406.00	−0.678643	−0.339321	+	0.940670i	$0.610197\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 814.587i	− 0.0581665i
$582$	0	0
$583$	41153.5i	2.92351i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21986.6	−1.54597	−0.772985	−	0.634424i	$-0.781237\pi$
$587$	−21986.6	−1.54597	−0.772985	+	0.634424i	$0.781237\pi$
$588$	0	0
$589$	−13824.0	−0.967076
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 13419.5i	− 0.929295i	−0.885496	−	0.464647i	$-0.846181\pi$
$593$	− 13419.5i	− 0.929295i	0.885496	−	0.464647i	$-0.153819\pi$
$594$	0	0
$595$	− 5736.55i	− 0.395253i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6878.17	−0.469172	−0.234586	−	0.972095i	$-0.575374\pi$
$599$	−6878.17	−0.469172	−0.234586	+	0.972095i	$0.575374\pi$
$600$	0	0
$601$	6086.00	0.413067	0.206533	−	0.978440i	$-0.433782\pi$
$601$	6086.00	0.413067	0.206533	+	0.978440i	$0.433782\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 9015.61i	− 0.605846i
$606$	0	0
$607$	− 7967.43i	− 0.532765i	−0.963867	−	0.266382i	$-0.914172\pi$
$607$	− 7967.43i	− 0.532765i	0.963867	−	0.266382i	$-0.0858284\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8230.29	−0.544946
$612$	0	0
$613$	20438.0	1.34663	0.673314	−	0.739357i	$-0.264870\pi$
$613$	20438.0	1.34663	0.673314	+	0.739357i	$0.264870\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3568.06i	− 0.232812i	−0.993202	−	0.116406i	$-0.962863\pi$
$617$	− 3568.06i	− 0.232812i	0.993202	−	0.116406i	$-0.0371373\pi$
$618$	0	0
$619$	− 1884.47i	− 0.122364i	−0.998127	−	0.0611820i	$-0.980513\pi$
$619$	− 1884.47i	− 0.122364i	0.998127	−	0.0611820i	$-0.0194870\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15578.8	1.00185
$624$	0	0
$625$	9199.00	0.588736
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11514.5i	0.729911i
$630$	0	0
$631$	− 9491.64i	− 0.598821i	−0.954124	−	0.299411i	$-0.903210\pi$
$631$	− 9491.64i	− 0.598821i	0.954124	−	0.299411i	$-0.0967900\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6878.17	0.429845
$636$	0	0
$637$	4228.00	0.262982
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16363.9i	1.00832i	0.863610	+	0.504161i	$0.168198\pi$
$641$	16363.9i	1.00832i	−0.863610	+	0.504161i	$0.831802\pi$
$642$	0	0
$643$	− 24193.3i	− 1.48381i	−0.670505	−	0.741905i	$-0.733922\pi$
$643$	− 24193.3i	− 1.48381i	0.670505	−	0.741905i	$-0.266078\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14520.6	−0.882323	−0.441161	−	0.897428i	$-0.645433\pi$
$647$	−14520.6	−0.882323	−0.441161	+	0.897428i	$0.645433\pi$
$648$	0	0
$649$	−34560.0	−2.09029
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4187.49i	0.250948i	0.992097	+	0.125474i	$0.0400452\pi$
$653$	4187.49i	0.250948i	−0.992097	+	0.125474i	$0.959955\pi$
$654$	0	0
$655$	− 3491.81i	− 0.208300i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4115.14	0.243252	0.121626	−	0.992576i	$-0.461189\pi$
$659$	4115.14	0.243252	0.121626	+	0.992576i	$0.461189\pi$
$660$	0	0
$661$	−22438.0	−1.32033	−0.660164	−	0.751121i	$-0.729513\pi$
$661$	−22438.0	−1.32033	−0.660164	+	0.751121i	$0.729513\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6516.70i	0.380010i
$666$	0	0
$667$	748.246i	0.0434366i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−47147.8	−2.71255
$672$	0	0
$673$	18962.0	1.08608	0.543040	−	0.839707i	$-0.317273\pi$
$673$	18962.0	1.08608	0.543040	+	0.839707i	$0.317273\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2278.30i	− 0.129338i	−0.997907	−	0.0646692i	$-0.979401\pi$
$677$	− 2278.30i	− 0.129338i	0.997907	−	0.0646692i	$-0.0205992\pi$
$678$	0	0
$679$	− 2106.17i	− 0.119039i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11581.2	−0.648817	−0.324408	−	0.945917i	$-0.605165\pi$
$683$	−11581.2	−0.648817	−0.324408	+	0.945917i	$0.605165\pi$
$684$	0	0
$685$	−6822.00	−0.380519
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19601.0i	1.08380i
$690$	0	0
$691$	11722.5i	0.645363i	0.946508	+	0.322681i	$0.104584\pi$
$691$	11722.5i	0.645363i	−0.946508	+	0.322681i	$0.895416\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9053.31	0.494118
$696$	0	0
$697$	−18630.0	−1.01243
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 9601.10i	− 0.517302i	−0.965971	−	0.258651i	$-0.916722\pi$
$701$	− 9601.10i	− 0.517302i	0.965971	−	0.258651i	$-0.0832779\pi$
$702$	0	0
$703$	− 13080.4i	− 0.701762i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12874.5	−0.684860
$708$	0	0
$709$	14572.0	0.771880	0.385940	−	0.922524i	$-0.373877\pi$
$709$	14572.0	0.771880	0.385940	+	0.922524i	$0.373877\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	21993.8i	1.15523i
$714$	0	0
$715$	− 6983.63i	− 0.365277i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9817.55	−0.509225	−0.254613	−	0.967043i	$-0.581948\pi$
$719$	−9817.55	−0.509225	−0.254613	+	0.967043i	$0.581948\pi$
$720$	0	0
$721$	−24384.0	−1.25951
$722$	0	0
$723$	0	0
$724$	0	0
$725$	453.963i	0.0232548i
$726$	0	0
$727$	− 33047.5i	− 1.68592i	−0.537975	−	0.842961i	$-0.680811\pi$
$727$	− 33047.5i	− 1.68592i	0.537975	−	0.842961i	$-0.319189\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24338.1	−1.23143
$732$	0	0
$733$	28636.0	1.44297	0.721483	−	0.692432i	$-0.243461\pi$
$733$	28636.0	1.44297	0.721483	+	0.692432i	$0.243461\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8145.87i	− 0.407133i
$738$	0	0
$739$	1496.49i	0.0744917i	0.999306	+	0.0372458i	$0.0118585\pi$
$739$	1496.49i	0.0744917i	−0.999306	+	0.0372458i	$0.988142\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13050.9	0.644402	0.322201	−	0.946671i	$-0.395577\pi$
$743$	13050.9	0.644402	0.322201	+	0.946671i	$0.395577\pi$
$744$	0	0
$745$	3474.00	0.170842
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13033.4i	0.635821i
$750$	0	0
$751$	− 8244.56i	− 0.400597i	−0.979735	−	0.200298i	$-0.935809\pi$
$751$	− 8244.56i	− 0.400597i	0.979735	−	0.200298i	$-0.0641912\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−13815.1	−0.665939
$756$	0	0
$757$	17740.0	0.851745	0.425873	−	0.904783i	$-0.359967\pi$
$757$	17740.0	0.851745	0.425873	+	0.904783i	$0.359967\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17254.8i	0.821927i	0.911652	+	0.410964i	$0.134808\pi$
$761$	17254.8i	0.821927i	−0.911652	+	0.410964i	$0.865192\pi$
$762$	0	0
$763$	8701.82i	0.412880i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16460.6	−0.774911
$768$	0	0
$769$	−17038.0	−0.798967	−0.399484	−	0.916740i	$-0.630811\pi$
$769$	−17038.0	−0.798967	−0.399484	+	0.916740i	$0.630811\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38739.6i	− 1.80254i	−0.433256	−	0.901271i	$-0.642635\pi$
$773$	− 38739.6i	− 1.80254i	0.433256	−	0.901271i	$-0.357365\pi$
$774$	0	0
$775$	13343.7i	0.618478i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21163.6	0.973382
$780$	0	0
$781$	−38016.0	−1.74177
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 7339.77i	− 0.333717i
$786$	0	0
$787$	3103.84i	0.140584i	0.997526	+	0.0702921i	$0.0223931\pi$
$787$	3103.84i	0.140584i	−0.997526	+	0.0702921i	$0.977607\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17224.8	0.774265
$792$	0	0
$793$	−22456.0	−1.00559
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1098.84i	0.0488370i	0.999702	+	0.0244185i	$0.00777341\pi$
$797$	1098.84i	0.0488370i	−0.999702	+	0.0244185i	$0.992227\pi$
$798$	0	0
$799$	28682.8i	1.26999i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	29158.7	1.28143
$804$	0	0
$805$	10368.0	0.453943
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38739.6i	1.68357i	0.539811	+	0.841786i	$0.318496\pi$
$809$	38739.6i	1.68357i	−0.539811	+	0.841786i	$0.681504\pi$
$810$	0	0
$811$	− 34419.3i	− 1.49029i	−0.666902	−	0.745145i	$-0.732380\pi$
$811$	− 34419.3i	− 1.49029i	0.666902	−	0.745145i	$-0.267620\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6936.95	0.298148
$816$	0	0
$817$	27648.0	1.18394
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 35387.9i	− 1.50432i	−0.658982	−	0.752159i	$-0.729013\pi$
$821$	− 35387.9i	− 1.50432i	0.658982	−	0.752159i	$-0.270987\pi$
$822$	0	0
$823$	11708.7i	0.495915i	0.968771	+	0.247958i	$0.0797594\pi$
$823$	11708.7i	0.495915i	−0.968771	+	0.247958i	$0.920241\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31392.7	1.31999	0.659994	−	0.751271i	$-0.270559\pi$
$827$	31392.7	1.31999	0.659994	+	0.751271i	$0.270559\pi$
$828$	0	0
$829$	43444.0	1.82011	0.910056	−	0.414486i	$-0.136039\pi$
$829$	43444.0	1.82011	0.910056	+	0.414486i	$0.136039\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 14734.7i	− 0.612877i
$834$	0	0
$835$	8480.12i	0.351457i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−15343.6	−0.631371	−0.315685	−	0.948864i	$-0.602234\pi$
$839$	−15343.6	−0.631371	−0.315685	+	0.948864i	$0.602234\pi$
$840$	0	0
$841$	24371.0	0.999262
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5994.85i	0.244058i
$846$	0	0
$847$	29444.9i	1.19450i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−20810.9	−0.838293
$852$	0	0
$853$	−4870.00	−0.195481	−0.0977407	−	0.995212i	$-0.531162\pi$
$853$	−4870.00	−0.195481	−0.0977407	+	0.995212i	$0.531162\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9414.42i	0.375251i	0.982241	+	0.187626i	$0.0600792\pi$
$857$	9414.42i	0.375251i	−0.982241	+	0.187626i	$0.939921\pi$
$858$	0	0
$859$	16600.0i	0.659353i	0.944094	+	0.329676i	$0.106940\pi$
$859$	16600.0i	0.659353i	−0.944094	+	0.329676i	$0.893060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36095.7	1.42377	0.711884	−	0.702297i	$-0.247842\pi$
$863$	36095.7	1.42377	0.711884	+	0.702297i	$0.247842\pi$
$864$	0	0
$865$	−9162.00	−0.360136
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 59464.9i	− 2.32130i
$870$	0	0
$871$	− 3879.79i	− 0.150932i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13638.8	0.526942
$876$	0	0
$877$	−19954.0	−0.768300	−0.384150	−	0.923271i	$-0.625505\pi$
$877$	−19954.0	−0.768300	−0.384150	+	0.923271i	$0.625505\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8744.08i	0.334388i	0.985924	+	0.167194i	$0.0534706\pi$
$881$	8744.08i	0.334388i	−0.985924	+	0.167194i	$0.946529\pi$
$882$	0	0
$883$	30068.4i	1.14596i	0.819570	+	0.572980i	$0.194213\pi$
$883$	30068.4i	1.14596i	−0.819570	+	0.572980i	$0.805787\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28100.5	1.06372	0.531862	−	0.846831i	$-0.321492\pi$
$887$	28100.5	1.06372	0.531862	+	0.846831i	$0.321492\pi$
$888$	0	0
$889$	−22464.0	−0.847490
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 32583.5i	− 1.22101i
$894$	0	0
$895$	9976.61i	0.372605i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−529.090	−0.0196286
$900$	0	0
$901$	68310.0	2.52579
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2630.44i	0.0966173i
$906$	0	0
$907$	− 1136.23i	− 0.0415962i	−0.999784	−	0.0207981i	$-0.993379\pi$
$907$	− 1136.23i	− 0.0415962i	0.999784	−	0.0207981i	$-0.00662072\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39387.8	1.43247	0.716233	−	0.697862i	$-0.245865\pi$
$911$	39387.8	1.43247	0.716233	+	0.697862i	$0.245865\pi$
$912$	0	0
$913$	−3456.00	−0.125276
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11404.2i	0.410688i
$918$	0	0
$919$	− 5861.26i	− 0.210386i	−0.994452	−	0.105193i	$-0.966454\pi$
$919$	− 5861.26i	− 0.210386i	0.994452	−	0.105193i	$-0.0335461\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18106.6	−0.645706
$924$	0	0
$925$	−12626.0	−0.448800
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 6614.28i	− 0.233592i	−0.993156	−	0.116796i	$-0.962738\pi$
$929$	− 6614.28i	− 0.233592i	0.993156	−	0.116796i	$-0.0372624\pi$
$930$	0	0
$931$	16738.5i	0.589241i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24338.1	−0.851275
$936$	0	0
$937$	−21610.0	−0.753434	−0.376717	−	0.926328i	$-0.622947\pi$
$937$	−21610.0	−0.753434	−0.376717	+	0.926328i	$0.622947\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	50865.0i	1.76212i	0.473007	+	0.881059i	$0.343168\pi$
$941$	50865.0i	1.76212i	−0.473007	+	0.881059i	$0.656832\pi$
$942$	0	0
$943$	− 33671.1i	− 1.16276i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22927.2	0.786731	0.393366	−	0.919382i	$-0.371311\pi$
$947$	22927.2	0.786731	0.393366	+	0.919382i	$0.371311\pi$
$948$	0	0
$949$	13888.0	0.475051
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8981.67i	0.305294i	0.988281	+	0.152647i	$0.0487797\pi$
$953$	8981.67i	0.305294i	−0.988281	+	0.152647i	$0.951220\pi$
$954$	0	0
$955$	15463.7i	0.523974i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	22280.6	0.750236
$960$	0	0
$961$	14239.0	0.477963
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4013.54i	0.133886i
$966$	0	0
$967$	983.805i	0.0327167i	0.999866	+	0.0163583i	$0.00520725\pi$
$967$	983.805i	0.0327167i	−0.999866	+	0.0163583i	$0.994793\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12639.4	0.417731	0.208865	−	0.977944i	$-0.433023\pi$
$971$	12639.4	0.417731	0.208865	+	0.977944i	$0.433023\pi$
$972$	0	0
$973$	−29568.0	−0.974210
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 31501.6i	− 1.03155i	−0.856724	−	0.515776i	$-0.827504\pi$
$977$	− 31501.6i	− 1.03155i	0.856724	−	0.515776i	$-0.172496\pi$
$978$	0	0
$979$	− 66095.1i	− 2.15772i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−56318.7	−1.82735	−0.913676	−	0.406444i	$-0.866769\pi$
$983$	−56318.7	−1.82735	−0.913676	+	0.406444i	$0.866769\pi$
$984$	0	0
$985$	8118.00	0.262600
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 43987.7i	− 1.41428i
$990$	0	0
$991$	− 5473.28i	− 0.175443i	−0.996145	−	0.0877217i	$-0.972041\pi$
$991$	− 5473.28i	− 0.175443i	0.996145	−	0.0877217i	$-0.0279586\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3115.75	−0.0992723
$996$	0	0
$997$	−22858.0	−0.726098	−0.363049	−	0.931770i	$-0.618264\pi$
$997$	−22858.0	−0.726098	−0.363049	+	0.931770i	$0.618264\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.c.b.143.2	yes	4
3.2	odd	2	inner	144.4.c.b.143.4	yes	4
4.3	odd	2	inner	144.4.c.b.143.1	✓	4
8.3	odd	2		576.4.c.c.575.3		4
8.5	even	2		576.4.c.c.575.4		4
12.11	even	2	inner	144.4.c.b.143.3	yes	4
16.3	odd	4		2304.4.f.f.1151.4		8
16.5	even	4		2304.4.f.f.1151.5		8
16.11	odd	4		2304.4.f.f.1151.7		8
16.13	even	4		2304.4.f.f.1151.2		8
24.5	odd	2		576.4.c.c.575.2		4
24.11	even	2		576.4.c.c.575.1		4
48.5	odd	4		2304.4.f.f.1151.1		8
48.11	even	4		2304.4.f.f.1151.3		8
48.29	odd	4		2304.4.f.f.1151.6		8
48.35	even	4		2304.4.f.f.1151.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.c.b.143.1	✓	4	4.3	odd	2	inner
144.4.c.b.143.2	yes	4	1.1	even	1	trivial
144.4.c.b.143.3	yes	4	12.11	even	2	inner
144.4.c.b.143.4	yes	4	3.2	odd	2	inner
576.4.c.c.575.1		4	24.11	even	2
576.4.c.c.575.2		4	24.5	odd	2
576.4.c.c.575.3		4	8.3	odd	2
576.4.c.c.575.4		4	8.5	even	2
2304.4.f.f.1151.1		8	48.5	odd	4
2304.4.f.f.1151.2		8	16.13	even	4
2304.4.f.f.1151.3		8	48.11	even	4
2304.4.f.f.1151.4		8	16.3	odd	4
2304.4.f.f.1151.5		8	16.5	even	4
2304.4.f.f.1151.6		8	48.29	odd	4
2304.4.f.f.1151.7		8	16.11	odd	4
2304.4.f.f.1151.8		8	48.35	even	4

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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	144.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.24264i q^{5} +13.8564i q^{7} +O(q^{10})\)	\(q-4.24264i q^{5} +13.8564i q^{7} +58.7878 q^{11} +28.0000 q^{13} -97.5807i q^{17} +110.851i q^{19} +176.363 q^{23} +107.000 q^{25} +4.24264i q^{29} +124.708i q^{31} +58.7878 q^{35} -118.000 q^{37} -190.919i q^{41} -249.415i q^{43} -293.939 q^{47} +151.000 q^{49} +700.036i q^{53} -249.415i q^{55} -587.878 q^{59} -802.000 q^{61} -118.794i q^{65} -138.564i q^{67} -646.665 q^{71} +496.000 q^{73} +814.587i q^{77} -1011.52i q^{79} -58.7878 q^{83} -414.000 q^{85} -1124.30i q^{89} +387.979i q^{91} +470.302 q^{95} -152.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 112 q^{13} + 428 q^{25} - 472 q^{37} + 604 q^{49} - 3208 q^{61} + 1984 q^{73} - 1656 q^{85} - 608 q^{97}+O(q^{100}) \) 4 * q + 112 * q^13 + 428 * q^25 - 472 * q^37 + 604 * q^49 - 3208 * q^61 + 1984 * q^73 - 1656 * q^85 - 608 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more