LMFDB - Embedded newform 48.3.g.a.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,3,Mod(31,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	48.g (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:	$1.30790526893$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.3.g.a.31.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +6.00000 q^{5} -6.92820i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +6.00000 q^{5} -6.92820i q^{7} -3.00000 q^{9} +20.7846i q^{11} -14.0000 q^{13} -10.3923i q^{15} -6.00000 q^{17} +6.92820i q^{19} -12.0000 q^{21} +11.0000 q^{25} +5.19615i q^{27} +30.0000 q^{29} -20.7846i q^{31} +36.0000 q^{33} -41.5692i q^{35} +26.0000 q^{37} +24.2487i q^{39} -54.0000 q^{41} +20.7846i q^{43} -18.0000 q^{45} -41.5692i q^{47} +1.00000 q^{49} +10.3923i q^{51} -18.0000 q^{53} +124.708i q^{55} +12.0000 q^{57} -20.7846i q^{59} -70.0000 q^{61} +20.7846i q^{63} -84.0000 q^{65} -117.779i q^{67} +83.1384i q^{71} +82.0000 q^{73} -19.0526i q^{75} +144.000 q^{77} -76.2102i q^{79} +9.00000 q^{81} -20.7846i q^{83} -36.0000 q^{85} -51.9615i q^{87} +114.000 q^{89} +96.9948i q^{91} -36.0000 q^{93} +41.5692i q^{95} +34.0000 q^{97} -62.3538i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{5} - 6 q^{9}+O(q^{10}) $ 2 * q + 12 * q^5 - 6 * q^9	$ 2 q + 12 q^{5} - 6 q^{9} - 28 q^{13} - 12 q^{17} - 24 q^{21} + 22 q^{25} + 60 q^{29} + 72 q^{33} + 52 q^{37} - 108 q^{41} - 36 q^{45} + 2 q^{49} - 36 q^{53} + 24 q^{57} - 140 q^{61} - 168 q^{65} + 164 q^{73} + 288 q^{77} + 18 q^{81} - 72 q^{85} + 228 q^{89} - 72 q^{93} + 68 q^{97}+O(q^{100}) $ 2 * q + 12 * q^5 - 6 * q^9 - 28 * q^13 - 12 * q^17 - 24 * q^21 + 22 * q^25 + 60 * q^29 + 72 * q^33 + 52 * q^37 - 108 * q^41 - 36 * q^45 + 2 * q^49 - 36 * q^53 + 24 * q^57 - 140 * q^61 - 168 * q^65 + 164 * q^73 + 288 * q^77 + 18 * q^81 - 72 * q^85 + 228 * q^89 - 72 * q^93 + 68 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	6.00000	1.20000	0.600000	−	0.800000i	$-0.295167\pi$
$5$	6.00000	1.20000	0.600000	+	0.800000i	$0.295167\pi$
$6$	0	0
$7$	− 6.92820i	− 0.989743i	−0.868966	−	0.494872i	$-0.835215\pi$
$7$	− 6.92820i	− 0.989743i	0.868966	−	0.494872i	$-0.164785\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	20.7846i	1.88951i	0.327777	+	0.944755i	$0.393700\pi$
$11$	20.7846i	1.88951i	−0.327777	+	0.944755i	$0.606300\pi$
$12$	0	0
$13$	−14.0000	−1.07692	−0.538462	−	0.842650i	$-0.680994\pi$
$13$	−14.0000	−1.07692	−0.538462	+	0.842650i	$0.680994\pi$
$14$	0	0
$15$	− 10.3923i	− 0.692820i
$16$	0	0
$17$	−6.00000	−0.352941	−0.176471	−	0.984306i	$-0.556468\pi$
$17$	−6.00000	−0.352941	−0.176471	+	0.984306i	$0.556468\pi$
$18$	0	0
$19$	6.92820i	0.364642i	0.983239	+	0.182321i	$0.0583610\pi$
$19$	6.92820i	0.364642i	−0.983239	+	0.182321i	$0.941639\pi$
$20$	0	0
$21$	−12.0000	−0.571429
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	11.0000	0.440000
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	30.0000	1.03448	0.517241	−	0.855840i	$-0.326959\pi$
$29$	30.0000	1.03448	0.517241	+	0.855840i	$0.326959\pi$
$30$	0	0
$31$	− 20.7846i	− 0.670471i	−0.942134	−	0.335236i	$-0.891184\pi$
$31$	− 20.7846i	− 0.670471i	0.942134	−	0.335236i	$-0.108816\pi$
$32$	0	0
$33$	36.0000	1.09091
$34$	0	0
$35$	− 41.5692i	− 1.18769i
$36$	0	0
$37$	26.0000	0.702703	0.351351	−	0.936244i	$-0.385722\pi$
$37$	26.0000	0.702703	0.351351	+	0.936244i	$0.385722\pi$
$38$	0	0
$39$	24.2487i	0.621762i
$40$	0	0
$41$	−54.0000	−1.31707	−0.658537	−	0.752549i	$-0.728824\pi$
$41$	−54.0000	−1.31707	−0.658537	+	0.752549i	$0.728824\pi$
$42$	0	0
$43$	20.7846i	0.483363i	0.970356	+	0.241682i	$0.0776989\pi$
$43$	20.7846i	0.483363i	−0.970356	+	0.241682i	$0.922301\pi$
$44$	0	0
$45$	−18.0000	−0.400000
$46$	0	0
$47$	− 41.5692i	− 0.884451i	−0.896904	−	0.442226i	$-0.854189\pi$
$47$	− 41.5692i	− 0.884451i	0.896904	−	0.442226i	$-0.145811\pi$
$48$	0	0
$49$	1.00000	0.0204082
$50$	0	0
$51$	10.3923i	0.203771i
$52$	0	0
$53$	−18.0000	−0.339623	−0.169811	−	0.985477i	$-0.554316\pi$
$53$	−18.0000	−0.339623	−0.169811	+	0.985477i	$0.554316\pi$
$54$	0	0
$55$	124.708i	2.26741i
$56$	0	0
$57$	12.0000	0.210526
$58$	0	0
$59$	− 20.7846i	− 0.352282i	−0.984365	−	0.176141i	$-0.943639\pi$
$59$	− 20.7846i	− 0.352282i	0.984365	−	0.176141i	$-0.0563614\pi$
$60$	0	0
$61$	−70.0000	−1.14754	−0.573770	−	0.819016i	$-0.694520\pi$
$61$	−70.0000	−1.14754	−0.573770	+	0.819016i	$0.694520\pi$
$62$	0	0
$63$	20.7846i	0.329914i
$64$	0	0
$65$	−84.0000	−1.29231
$66$	0	0
$67$	− 117.779i	− 1.75790i	−0.476912	−	0.878951i	$-0.658244\pi$
$67$	− 117.779i	− 1.75790i	0.476912	−	0.878951i	$-0.341756\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	83.1384i	1.17096i	0.810685	+	0.585482i	$0.199095\pi$
$71$	83.1384i	1.17096i	−0.810685	+	0.585482i	$0.800905\pi$
$72$	0	0
$73$	82.0000	1.12329	0.561644	−	0.827379i	$-0.310169\pi$
$73$	82.0000	1.12329	0.561644	+	0.827379i	$0.310169\pi$
$74$	0	0
$75$	− 19.0526i	− 0.254034i
$76$	0	0
$77$	144.000	1.87013
$78$	0	0
$79$	− 76.2102i	− 0.964687i	−0.875982	−	0.482343i	$-0.839786\pi$
$79$	− 76.2102i	− 0.964687i	0.875982	−	0.482343i	$-0.160214\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 20.7846i	− 0.250417i	−0.992130	−	0.125208i	$-0.960040\pi$
$83$	− 20.7846i	− 0.250417i	0.992130	−	0.125208i	$-0.0399600\pi$
$84$	0	0
$85$	−36.0000	−0.423529
$86$	0	0
$87$	− 51.9615i	− 0.597259i
$88$	0	0
$89$	114.000	1.28090	0.640449	−	0.768000i	$-0.278748\pi$
$89$	114.000	1.28090	0.640449	+	0.768000i	$0.278748\pi$
$90$	0	0
$91$	96.9948i	1.06588i
$92$	0	0
$93$	−36.0000	−0.387097
$94$	0	0
$95$	41.5692i	0.437571i
$96$	0	0
$97$	34.0000	0.350515	0.175258	−	0.984523i	$-0.443924\pi$
$97$	34.0000	0.350515	0.175258	+	0.984523i	$0.443924\pi$
$98$	0	0
$99$	− 62.3538i	− 0.629837i
$100$	0	0
$101$	−18.0000	−0.178218	−0.0891089	−	0.996022i	$-0.528402\pi$
$101$	−18.0000	−0.178218	−0.0891089	+	0.996022i	$0.528402\pi$
$102$	0	0
$103$	− 131.636i	− 1.27802i	−0.769199	−	0.639009i	$-0.779345\pi$
$103$	− 131.636i	− 1.27802i	0.769199	−	0.639009i	$-0.220655\pi$
$104$	0	0
$105$	−72.0000	−0.685714
$106$	0	0
$107$	145.492i	1.35974i	0.733332	+	0.679870i	$0.237964\pi$
$107$	145.492i	1.35974i	−0.733332	+	0.679870i	$0.762036\pi$
$108$	0	0
$109$	34.0000	0.311927	0.155963	−	0.987763i	$-0.450152\pi$
$109$	34.0000	0.311927	0.155963	+	0.987763i	$0.450152\pi$
$110$	0	0
$111$	− 45.0333i	− 0.405706i
$112$	0	0
$113$	−78.0000	−0.690265	−0.345133	−	0.938554i	$-0.612166\pi$
$113$	−78.0000	−0.690265	−0.345133	+	0.938554i	$0.612166\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	42.0000	0.358974
$118$	0	0
$119$	41.5692i	0.349321i
$120$	0	0
$121$	−311.000	−2.57025
$122$	0	0
$123$	93.5307i	0.760413i
$124$	0	0
$125$	−84.0000	−0.672000
$126$	0	0
$127$	103.923i	0.818292i	0.912469	+	0.409146i	$0.134173\pi$
$127$	103.923i	0.818292i	−0.912469	+	0.409146i	$0.865827\pi$
$128$	0	0
$129$	36.0000	0.279070
$130$	0	0
$131$	103.923i	0.793306i	0.917969	+	0.396653i	$0.129828\pi$
$131$	103.923i	0.793306i	−0.917969	+	0.396653i	$0.870172\pi$
$132$	0	0
$133$	48.0000	0.360902
$134$	0	0
$135$	31.1769i	0.230940i
$136$	0	0
$137$	186.000	1.35766	0.678832	−	0.734294i	$-0.262486\pi$
$137$	186.000	1.35766	0.678832	+	0.734294i	$0.262486\pi$
$138$	0	0
$139$	− 48.4974i	− 0.348902i	−0.984666	−	0.174451i	$-0.944185\pi$
$139$	− 48.4974i	− 0.348902i	0.984666	−	0.174451i	$-0.0558151\pi$
$140$	0	0
$141$	−72.0000	−0.510638
$142$	0	0
$143$	− 290.985i	− 2.03486i
$144$	0	0
$145$	180.000	1.24138
$146$	0	0
$147$	− 1.73205i	− 0.0117827i
$148$	0	0
$149$	−186.000	−1.24832	−0.624161	−	0.781296i	$-0.714559\pi$
$149$	−186.000	−1.24832	−0.624161	+	0.781296i	$0.714559\pi$
$150$	0	0
$151$	− 34.6410i	− 0.229411i	−0.993400	−	0.114705i	$-0.963408\pi$
$151$	− 34.6410i	− 0.229411i	0.993400	−	0.114705i	$-0.0365924\pi$
$152$	0	0
$153$	18.0000	0.117647
$154$	0	0
$155$	− 124.708i	− 0.804566i
$156$	0	0
$157$	170.000	1.08280	0.541401	−	0.840764i	$-0.317894\pi$
$157$	170.000	1.08280	0.541401	+	0.840764i	$0.317894\pi$
$158$	0	0
$159$	31.1769i	0.196081i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	284.056i	1.74268i	0.490683	+	0.871338i	$0.336747\pi$
$163$	284.056i	1.74268i	−0.490683	+	0.871338i	$0.663253\pi$
$164$	0	0
$165$	216.000	1.30909
$166$	0	0
$167$	− 207.846i	− 1.24459i	−0.782784	−	0.622294i	$-0.786201\pi$
$167$	− 207.846i	− 1.24459i	0.782784	−	0.622294i	$-0.213799\pi$
$168$	0	0
$169$	27.0000	0.159763
$170$	0	0
$171$	− 20.7846i	− 0.121547i
$172$	0	0
$173$	−42.0000	−0.242775	−0.121387	−	0.992605i	$-0.538734\pi$
$173$	−42.0000	−0.242775	−0.121387	+	0.992605i	$0.538734\pi$
$174$	0	0
$175$	− 76.2102i	− 0.435487i
$176$	0	0
$177$	−36.0000	−0.203390
$178$	0	0
$179$	− 145.492i	− 0.812806i	−0.913694	−	0.406403i	$-0.866783\pi$
$179$	− 145.492i	− 0.812806i	0.913694	−	0.406403i	$-0.133217\pi$
$180$	0	0
$181$	82.0000	0.453039	0.226519	−	0.974007i	$-0.427265\pi$
$181$	82.0000	0.453039	0.226519	+	0.974007i	$0.427265\pi$
$182$	0	0
$183$	121.244i	0.662533i
$184$	0	0
$185$	156.000	0.843243
$186$	0	0
$187$	− 124.708i	− 0.666886i
$188$	0	0
$189$	36.0000	0.190476
$190$	0	0
$191$	332.554i	1.74112i	0.492063	+	0.870560i	$0.336243\pi$
$191$	332.554i	1.74112i	−0.492063	+	0.870560i	$0.663757\pi$
$192$	0	0
$193$	−94.0000	−0.487047	−0.243523	−	0.969895i	$-0.578303\pi$
$193$	−94.0000	−0.487047	−0.243523	+	0.969895i	$0.578303\pi$
$194$	0	0
$195$	145.492i	0.746114i
$196$	0	0
$197$	−258.000	−1.30964	−0.654822	−	0.755783i	$-0.727257\pi$
$197$	−258.000	−1.30964	−0.654822	+	0.755783i	$0.727257\pi$
$198$	0	0
$199$	117.779i	0.591857i	0.955210	+	0.295928i	$0.0956289\pi$
$199$	117.779i	0.591857i	−0.955210	+	0.295928i	$0.904371\pi$
$200$	0	0
$201$	−204.000	−1.01493
$202$	0	0
$203$	− 207.846i	− 1.02387i
$204$	0	0
$205$	−324.000	−1.58049
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−144.000	−0.688995
$210$	0	0
$211$	− 90.0666i	− 0.426856i	−0.976959	−	0.213428i	$-0.931537\pi$
$211$	− 90.0666i	− 0.426856i	0.976959	−	0.213428i	$-0.0684629\pi$
$212$	0	0
$213$	144.000	0.676056
$214$	0	0
$215$	124.708i	0.580036i
$216$	0	0
$217$	−144.000	−0.663594
$218$	0	0
$219$	− 142.028i	− 0.648530i
$220$	0	0
$221$	84.0000	0.380090
$222$	0	0
$223$	353.338i	1.58448i	0.610212	+	0.792238i	$0.291084\pi$
$223$	353.338i	1.58448i	−0.610212	+	0.792238i	$0.708916\pi$
$224$	0	0
$225$	−33.0000	−0.146667
$226$	0	0
$227$	145.492i	0.640935i	0.947259	+	0.320468i	$0.103840\pi$
$227$	145.492i	0.640935i	−0.947259	+	0.320468i	$0.896160\pi$
$228$	0	0
$229$	226.000	0.986900	0.493450	−	0.869774i	$-0.335736\pi$
$229$	226.000	0.986900	0.493450	+	0.869774i	$0.335736\pi$
$230$	0	0
$231$	− 249.415i	− 1.07972i
$232$	0	0
$233$	114.000	0.489270	0.244635	−	0.969615i	$-0.421332\pi$
$233$	114.000	0.489270	0.244635	+	0.969615i	$0.421332\pi$
$234$	0	0
$235$	− 249.415i	− 1.06134i
$236$	0	0
$237$	−132.000	−0.556962
$238$	0	0
$239$	332.554i	1.39144i	0.718314	+	0.695719i	$0.244914\pi$
$239$	332.554i	1.39144i	−0.718314	+	0.695719i	$0.755086\pi$
$240$	0	0
$241$	178.000	0.738589	0.369295	−	0.929312i	$-0.379599\pi$
$241$	178.000	0.738589	0.369295	+	0.929312i	$0.379599\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	6.00000	0.0244898
$246$	0	0
$247$	− 96.9948i	− 0.392692i
$248$	0	0
$249$	−36.0000	−0.144578
$250$	0	0
$251$	103.923i	0.414036i	0.978337	+	0.207018i	$0.0663759\pi$
$251$	103.923i	0.414036i	−0.978337	+	0.207018i	$0.933624\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	62.3538i	0.244525i
$256$	0	0
$257$	258.000	1.00389	0.501946	−	0.864899i	$-0.332618\pi$
$257$	258.000	1.00389	0.501946	+	0.864899i	$0.332618\pi$
$258$	0	0
$259$	− 180.133i	− 0.695495i
$260$	0	0
$261$	−90.0000	−0.344828
$262$	0	0
$263$	− 374.123i	− 1.42252i	−0.702929	−	0.711260i	$-0.748125\pi$
$263$	− 374.123i	− 1.42252i	0.702929	−	0.711260i	$-0.251875\pi$
$264$	0	0
$265$	−108.000	−0.407547
$266$	0	0
$267$	− 197.454i	− 0.739527i
$268$	0	0
$269$	510.000	1.89591	0.947955	−	0.318403i	$-0.103147\pi$
$269$	510.000	1.89591	0.947955	+	0.318403i	$0.103147\pi$
$270$	0	0
$271$	− 450.333i	− 1.66175i	−0.556462	−	0.830873i	$-0.687842\pi$
$271$	− 450.333i	− 1.66175i	0.556462	−	0.830873i	$-0.312158\pi$
$272$	0	0
$273$	168.000	0.615385
$274$	0	0
$275$	228.631i	0.831384i
$276$	0	0
$277$	−14.0000	−0.0505415	−0.0252708	−	0.999681i	$-0.508045\pi$
$277$	−14.0000	−0.0505415	−0.0252708	+	0.999681i	$0.508045\pi$
$278$	0	0
$279$	62.3538i	0.223490i
$280$	0	0
$281$	354.000	1.25979	0.629893	−	0.776682i	$-0.283099\pi$
$281$	354.000	1.25979	0.629893	+	0.776682i	$0.283099\pi$
$282$	0	0
$283$	145.492i	0.514107i	0.966397	+	0.257053i	$0.0827517\pi$
$283$	145.492i	0.514107i	−0.966397	+	0.257053i	$0.917248\pi$
$284$	0	0
$285$	72.0000	0.252632
$286$	0	0
$287$	374.123i	1.30356i
$288$	0	0
$289$	−253.000	−0.875433
$290$	0	0
$291$	− 58.8897i	− 0.202370i
$292$	0	0
$293$	−498.000	−1.69966	−0.849829	−	0.527058i	$-0.823295\pi$
$293$	−498.000	−1.69966	−0.849829	+	0.527058i	$0.823295\pi$
$294$	0	0
$295$	− 124.708i	− 0.422738i
$296$	0	0
$297$	−108.000	−0.363636
$298$	0	0
$299$	0	0
$300$	0	0
$301$	144.000	0.478405
$302$	0	0
$303$	31.1769i	0.102894i
$304$	0	0
$305$	−420.000	−1.37705
$306$	0	0
$307$	187.061i	0.609321i	0.952461	+	0.304660i	$0.0985430\pi$
$307$	187.061i	0.609321i	−0.952461	+	0.304660i	$0.901457\pi$
$308$	0	0
$309$	−228.000	−0.737864
$310$	0	0
$311$	− 41.5692i	− 0.133663i	−0.997764	−	0.0668315i	$-0.978711\pi$
$311$	− 41.5692i	− 0.133663i	0.997764	−	0.0668315i	$-0.0212890\pi$
$312$	0	0
$313$	290.000	0.926518	0.463259	−	0.886223i	$-0.346680\pi$
$313$	290.000	0.926518	0.463259	+	0.886223i	$0.346680\pi$
$314$	0	0
$315$	124.708i	0.395897i
$316$	0	0
$317$	−210.000	−0.662461	−0.331230	−	0.943550i	$-0.607464\pi$
$317$	−210.000	−0.662461	−0.331230	+	0.943550i	$0.607464\pi$
$318$	0	0
$319$	623.538i	1.95467i
$320$	0	0
$321$	252.000	0.785047
$322$	0	0
$323$	− 41.5692i	− 0.128697i
$324$	0	0
$325$	−154.000	−0.473846
$326$	0	0
$327$	− 58.8897i	− 0.180091i
$328$	0	0
$329$	−288.000	−0.875380
$330$	0	0
$331$	200.918i	0.607003i	0.952831	+	0.303501i	$0.0981557\pi$
$331$	200.918i	0.607003i	−0.952831	+	0.303501i	$0.901844\pi$
$332$	0	0
$333$	−78.0000	−0.234234
$334$	0	0
$335$	− 706.677i	− 2.10948i
$336$	0	0
$337$	−302.000	−0.896142	−0.448071	−	0.893998i	$-0.647889\pi$
$337$	−302.000	−0.896142	−0.448071	+	0.893998i	$0.647889\pi$
$338$	0	0
$339$	135.100i	0.398525i
$340$	0	0
$341$	432.000	1.26686
$342$	0	0
$343$	− 346.410i	− 1.00994i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 62.3538i	− 0.179694i	−0.995956	−	0.0898470i	$-0.971362\pi$
$347$	− 62.3538i	− 0.179694i	0.995956	−	0.0898470i	$-0.0286378\pi$
$348$	0	0
$349$	−358.000	−1.02579	−0.512894	−	0.858452i	$-0.671427\pi$
$349$	−358.000	−1.02579	−0.512894	+	0.858452i	$0.671427\pi$
$350$	0	0
$351$	− 72.7461i	− 0.207254i
$352$	0	0
$353$	−558.000	−1.58074	−0.790368	−	0.612632i	$-0.790111\pi$
$353$	−558.000	−1.58074	−0.790368	+	0.612632i	$0.790111\pi$
$354$	0	0
$355$	498.831i	1.40516i
$356$	0	0
$357$	72.0000	0.201681
$358$	0	0
$359$	− 83.1384i	− 0.231583i	−0.993274	−	0.115792i	$-0.963059\pi$
$359$	− 83.1384i	− 0.231583i	0.993274	−	0.115792i	$-0.0369405\pi$
$360$	0	0
$361$	313.000	0.867036
$362$	0	0
$363$	538.668i	1.48393i
$364$	0	0
$365$	492.000	1.34795
$366$	0	0
$367$	− 214.774i	− 0.585216i	−0.956232	−	0.292608i	$-0.905477\pi$
$367$	− 214.774i	− 0.585216i	0.956232	−	0.292608i	$-0.0945231\pi$
$368$	0	0
$369$	162.000	0.439024
$370$	0	0
$371$	124.708i	0.336139i
$372$	0	0
$373$	554.000	1.48525	0.742627	−	0.669705i	$-0.233579\pi$
$373$	554.000	1.48525	0.742627	+	0.669705i	$0.233579\pi$
$374$	0	0
$375$	145.492i	0.387979i
$376$	0	0
$377$	−420.000	−1.11406
$378$	0	0
$379$	− 533.472i	− 1.40758i	−0.710410	−	0.703788i	$-0.751490\pi$
$379$	− 533.472i	− 1.40758i	0.710410	−	0.703788i	$-0.248510\pi$
$380$	0	0
$381$	180.000	0.472441
$382$	0	0
$383$	− 498.831i	− 1.30243i	−0.758893	−	0.651215i	$-0.774260\pi$
$383$	− 498.831i	− 1.30243i	0.758893	−	0.651215i	$-0.225740\pi$
$384$	0	0
$385$	864.000	2.24416
$386$	0	0
$387$	− 62.3538i	− 0.161121i
$388$	0	0
$389$	198.000	0.508997	0.254499	−	0.967073i	$-0.418090\pi$
$389$	198.000	0.508997	0.254499	+	0.967073i	$0.418090\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	180.000	0.458015
$394$	0	0
$395$	− 457.261i	− 1.15762i
$396$	0	0
$397$	−646.000	−1.62720	−0.813602	−	0.581422i	$-0.802496\pi$
$397$	−646.000	−1.62720	−0.813602	+	0.581422i	$0.802496\pi$
$398$	0	0
$399$	− 83.1384i	− 0.208367i
$400$	0	0
$401$	330.000	0.822943	0.411471	−	0.911423i	$-0.365015\pi$
$401$	330.000	0.822943	0.411471	+	0.911423i	$0.365015\pi$
$402$	0	0
$403$	290.985i	0.722046i
$404$	0	0
$405$	54.0000	0.133333
$406$	0	0
$407$	540.400i	1.32776i
$408$	0	0
$409$	130.000	0.317848	0.158924	−	0.987291i	$-0.449197\pi$
$409$	130.000	0.317848	0.158924	+	0.987291i	$0.449197\pi$
$410$	0	0
$411$	− 322.161i	− 0.783848i
$412$	0	0
$413$	−144.000	−0.348668
$414$	0	0
$415$	− 124.708i	− 0.300500i
$416$	0	0
$417$	−84.0000	−0.201439
$418$	0	0
$419$	− 353.338i	− 0.843290i	−0.906761	−	0.421645i	$-0.861453\pi$
$419$	− 353.338i	− 0.843290i	0.906761	−	0.421645i	$-0.138547\pi$
$420$	0	0
$421$	−398.000	−0.945368	−0.472684	−	0.881232i	$-0.656715\pi$
$421$	−398.000	−0.945368	−0.472684	+	0.881232i	$0.656715\pi$
$422$	0	0
$423$	124.708i	0.294817i
$424$	0	0
$425$	−66.0000	−0.155294
$426$	0	0
$427$	484.974i	1.13577i
$428$	0	0
$429$	−504.000	−1.17483
$430$	0	0
$431$	− 124.708i	− 0.289345i	−0.989480	−	0.144672i	$-0.953787\pi$
$431$	− 124.708i	− 0.289345i	0.989480	−	0.144672i	$-0.0462128\pi$
$432$	0	0
$433$	−142.000	−0.327945	−0.163972	−	0.986465i	$-0.552431\pi$
$433$	−142.000	−0.327945	−0.163972	+	0.986465i	$0.552431\pi$
$434$	0	0
$435$	− 311.769i	− 0.716711i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	561.184i	1.27832i	0.769072	+	0.639162i	$0.220719\pi$
$439$	561.184i	1.27832i	−0.769072	+	0.639162i	$0.779281\pi$
$440$	0	0
$441$	−3.00000	−0.00680272
$442$	0	0
$443$	436.477i	0.985275i	0.870235	+	0.492637i	$0.163967\pi$
$443$	436.477i	0.985275i	−0.870235	+	0.492637i	$0.836033\pi$
$444$	0	0
$445$	684.000	1.53708
$446$	0	0
$447$	322.161i	0.720719i
$448$	0	0
$449$	−198.000	−0.440980	−0.220490	−	0.975389i	$-0.570766\pi$
$449$	−198.000	−0.440980	−0.220490	+	0.975389i	$0.570766\pi$
$450$	0	0
$451$	− 1122.37i	− 2.48862i
$452$	0	0
$453$	−60.0000	−0.132450
$454$	0	0
$455$	581.969i	1.27905i
$456$	0	0
$457$	−446.000	−0.975930	−0.487965	−	0.872863i	$-0.662261\pi$
$457$	−446.000	−0.975930	−0.487965	+	0.872863i	$0.662261\pi$
$458$	0	0
$459$	− 31.1769i	− 0.0679236i
$460$	0	0
$461$	342.000	0.741866	0.370933	−	0.928660i	$-0.379038\pi$
$461$	342.000	0.741866	0.370933	+	0.928660i	$0.379038\pi$
$462$	0	0
$463$	159.349i	0.344166i	0.985082	+	0.172083i	$0.0550497\pi$
$463$	159.349i	0.344166i	−0.985082	+	0.172083i	$0.944950\pi$
$464$	0	0
$465$	−216.000	−0.464516
$466$	0	0
$467$	394.908i	0.845627i	0.906217	+	0.422813i	$0.138957\pi$
$467$	394.908i	0.845627i	−0.906217	+	0.422813i	$0.861043\pi$
$468$	0	0
$469$	−816.000	−1.73987
$470$	0	0
$471$	− 294.449i	− 0.625156i
$472$	0	0
$473$	−432.000	−0.913319
$474$	0	0
$475$	76.2102i	0.160443i
$476$	0	0
$477$	54.0000	0.113208
$478$	0	0
$479$	789.815i	1.64888i	0.565947	+	0.824442i	$0.308511\pi$
$479$	789.815i	1.64888i	−0.565947	+	0.824442i	$0.691489\pi$
$480$	0	0
$481$	−364.000	−0.756757
$482$	0	0
$483$	0	0
$484$	0	0
$485$	204.000	0.420619
$486$	0	0
$487$	− 6.92820i	− 0.0142263i	−0.999975	−	0.00711315i	$-0.997736\pi$
$487$	− 6.92820i	− 0.0142263i	0.999975	−	0.00711315i	$-0.00226420\pi$
$488$	0	0
$489$	492.000	1.00613
$490$	0	0
$491$	644.323i	1.31227i	0.754645	+	0.656133i	$0.227809\pi$
$491$	644.323i	1.31227i	−0.754645	+	0.656133i	$0.772191\pi$
$492$	0	0
$493$	−180.000	−0.365112
$494$	0	0
$495$	− 374.123i	− 0.755804i
$496$	0	0
$497$	576.000	1.15895
$498$	0	0
$499$	− 810.600i	− 1.62445i	−0.583345	−	0.812224i	$-0.698257\pi$
$499$	− 810.600i	− 1.62445i	0.583345	−	0.812224i	$-0.301743\pi$
$500$	0	0
$501$	−360.000	−0.718563
$502$	0	0
$503$	− 332.554i	− 0.661141i	−0.943781	−	0.330570i	$-0.892759\pi$
$503$	− 332.554i	− 0.661141i	0.943781	−	0.330570i	$-0.107241\pi$
$504$	0	0
$505$	−108.000	−0.213861
$506$	0	0
$507$	− 46.7654i	− 0.0922394i
$508$	0	0
$509$	−306.000	−0.601179	−0.300589	−	0.953754i	$-0.597183\pi$
$509$	−306.000	−0.601179	−0.300589	+	0.953754i	$0.597183\pi$
$510$	0	0
$511$	− 568.113i	− 1.11177i
$512$	0	0
$513$	−36.0000	−0.0701754
$514$	0	0
$515$	− 789.815i	− 1.53362i
$516$	0	0
$517$	864.000	1.67118
$518$	0	0
$519$	72.7461i	0.140166i
$520$	0	0
$521$	522.000	1.00192	0.500960	−	0.865471i	$-0.332980\pi$
$521$	522.000	1.00192	0.500960	+	0.865471i	$0.332980\pi$
$522$	0	0
$523$	48.4974i	0.0927293i	0.998925	+	0.0463646i	$0.0147636\pi$
$523$	48.4974i	0.0927293i	−0.998925	+	0.0463646i	$0.985236\pi$
$524$	0	0
$525$	−132.000	−0.251429
$526$	0	0
$527$	124.708i	0.236637i
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	62.3538i	0.117427i
$532$	0	0
$533$	756.000	1.41839
$534$	0	0
$535$	872.954i	1.63169i
$536$	0	0
$537$	−252.000	−0.469274
$538$	0	0
$539$	20.7846i	0.0385614i
$540$	0	0
$541$	802.000	1.48244	0.741220	−	0.671262i	$-0.234248\pi$
$541$	802.000	1.48244	0.741220	+	0.671262i	$0.234248\pi$
$542$	0	0
$543$	− 142.028i	− 0.261562i
$544$	0	0
$545$	204.000	0.374312
$546$	0	0
$547$	34.6410i	0.0633291i	0.999499	+	0.0316645i	$0.0100808\pi$
$547$	34.6410i	0.0633291i	−0.999499	+	0.0316645i	$0.989919\pi$
$548$	0	0
$549$	210.000	0.382514
$550$	0	0
$551$	207.846i	0.377216i
$552$	0	0
$553$	−528.000	−0.954792
$554$	0	0
$555$	− 270.200i	− 0.486847i
$556$	0	0
$557$	−474.000	−0.850987	−0.425494	−	0.904961i	$-0.639900\pi$
$557$	−474.000	−0.850987	−0.425494	+	0.904961i	$0.639900\pi$
$558$	0	0
$559$	− 290.985i	− 0.520545i
$560$	0	0
$561$	−216.000	−0.385027
$562$	0	0
$563$	− 685.892i	− 1.21828i	−0.793062	−	0.609140i	$-0.791515\pi$
$563$	− 685.892i	− 1.21828i	0.793062	−	0.609140i	$-0.208485\pi$
$564$	0	0
$565$	−468.000	−0.828319
$566$	0	0
$567$	− 62.3538i	− 0.109971i
$568$	0	0
$569$	−150.000	−0.263620	−0.131810	−	0.991275i	$-0.542079\pi$
$569$	−150.000	−0.263620	−0.131810	+	0.991275i	$0.542079\pi$
$570$	0	0
$571$	672.036i	1.17695i	0.808517	+	0.588473i	$0.200271\pi$
$571$	672.036i	1.17695i	−0.808517	+	0.588473i	$0.799729\pi$
$572$	0	0
$573$	576.000	1.00524
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−46.0000	−0.0797227	−0.0398614	−	0.999205i	$-0.512692\pi$
$577$	−46.0000	−0.0797227	−0.0398614	+	0.999205i	$0.512692\pi$
$578$	0	0
$579$	162.813i	0.281197i
$580$	0	0
$581$	−144.000	−0.247849
$582$	0	0
$583$	− 374.123i	− 0.641720i
$584$	0	0
$585$	252.000	0.430769
$586$	0	0
$587$	− 353.338i	− 0.601939i	−0.953634	−	0.300970i	$-0.902690\pi$
$587$	− 353.338i	− 0.601939i	0.953634	−	0.300970i	$-0.0973103\pi$
$588$	0	0
$589$	144.000	0.244482
$590$	0	0
$591$	446.869i	0.756124i
$592$	0	0
$593$	114.000	0.192243	0.0961214	−	0.995370i	$-0.469356\pi$
$593$	114.000	0.192243	0.0961214	+	0.995370i	$0.469356\pi$
$594$	0	0
$595$	249.415i	0.419185i
$596$	0	0
$597$	204.000	0.341709
$598$	0	0
$599$	− 249.415i	− 0.416386i	−0.978088	−	0.208193i	$-0.933242\pi$
$599$	− 249.415i	− 0.416386i	0.978088	−	0.208193i	$-0.0667582\pi$
$600$	0	0
$601$	626.000	1.04160	0.520799	−	0.853680i	$-0.325634\pi$
$601$	626.000	1.04160	0.520799	+	0.853680i	$0.325634\pi$
$602$	0	0
$603$	353.338i	0.585967i
$604$	0	0
$605$	−1866.00	−3.08430
$606$	0	0
$607$	− 672.036i	− 1.10714i	−0.832802	−	0.553571i	$-0.813265\pi$
$607$	− 672.036i	− 1.10714i	0.832802	−	0.553571i	$-0.186735\pi$
$608$	0	0
$609$	−360.000	−0.591133
$610$	0	0
$611$	581.969i	0.952486i
$612$	0	0
$613$	−694.000	−1.13214	−0.566069	−	0.824358i	$-0.691536\pi$
$613$	−694.000	−1.13214	−0.566069	+	0.824358i	$0.691536\pi$
$614$	0	0
$615$	561.184i	0.912495i
$616$	0	0
$617$	−30.0000	−0.0486224	−0.0243112	−	0.999704i	$-0.507739\pi$
$617$	−30.0000	−0.0486224	−0.0243112	+	0.999704i	$0.507739\pi$
$618$	0	0
$619$	339.482i	0.548436i	0.961668	+	0.274218i	$0.0884190\pi$
$619$	339.482i	0.548436i	−0.961668	+	0.274218i	$0.911581\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 789.815i	− 1.26776i
$624$	0	0
$625$	−779.000	−1.24640
$626$	0	0
$627$	249.415i	0.397792i
$628$	0	0
$629$	−156.000	−0.248013
$630$	0	0
$631$	464.190i	0.735641i	0.929897	+	0.367821i	$0.119896\pi$
$631$	464.190i	0.735641i	−0.929897	+	0.367821i	$0.880104\pi$
$632$	0	0
$633$	−156.000	−0.246445
$634$	0	0
$635$	623.538i	0.981950i
$636$	0	0
$637$	−14.0000	−0.0219780
$638$	0	0
$639$	− 249.415i	− 0.390321i
$640$	0	0
$641$	−390.000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$641$	−390.000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$642$	0	0
$643$	810.600i	1.26065i	0.776330	+	0.630326i	$0.217079\pi$
$643$	810.600i	1.26065i	−0.776330	+	0.630326i	$0.782921\pi$
$644$	0	0
$645$	216.000	0.334884
$646$	0	0
$647$	− 581.969i	− 0.899489i	−0.893157	−	0.449744i	$-0.851515\pi$
$647$	− 581.969i	− 0.899489i	0.893157	−	0.449744i	$-0.148485\pi$
$648$	0	0
$649$	432.000	0.665639
$650$	0	0
$651$	249.415i	0.383126i
$652$	0	0
$653$	774.000	1.18530	0.592649	−	0.805461i	$-0.298082\pi$
$653$	774.000	1.18530	0.592649	+	0.805461i	$0.298082\pi$
$654$	0	0
$655$	623.538i	0.951967i
$656$	0	0
$657$	−246.000	−0.374429
$658$	0	0
$659$	− 228.631i	− 0.346936i	−0.984840	−	0.173468i	$-0.944503\pi$
$659$	− 228.631i	− 0.346936i	0.984840	−	0.173468i	$-0.0554973\pi$
$660$	0	0
$661$	−454.000	−0.686838	−0.343419	−	0.939182i	$-0.611585\pi$
$661$	−454.000	−0.686838	−0.343419	+	0.939182i	$0.611585\pi$
$662$	0	0
$663$	− 145.492i	− 0.219445i
$664$	0	0
$665$	288.000	0.433083
$666$	0	0
$667$	0	0
$668$	0	0
$669$	612.000	0.914798
$670$	0	0
$671$	− 1454.92i	− 2.16829i
$672$	0	0
$673$	434.000	0.644874	0.322437	−	0.946591i	$-0.395498\pi$
$673$	434.000	0.644874	0.322437	+	0.946591i	$0.395498\pi$
$674$	0	0
$675$	57.1577i	0.0846780i
$676$	0	0
$677$	−234.000	−0.345643	−0.172821	−	0.984953i	$-0.555288\pi$
$677$	−234.000	−0.345643	−0.172821	+	0.984953i	$0.555288\pi$
$678$	0	0
$679$	− 235.559i	− 0.346920i
$680$	0	0
$681$	252.000	0.370044
$682$	0	0
$683$	270.200i	0.395608i	0.980242	+	0.197804i	$0.0633809\pi$
$683$	270.200i	0.395608i	−0.980242	+	0.197804i	$0.936619\pi$
$684$	0	0
$685$	1116.00	1.62920
$686$	0	0
$687$	− 391.443i	− 0.569787i
$688$	0	0
$689$	252.000	0.365747
$690$	0	0
$691$	− 20.7846i	− 0.0300790i	−0.999887	−	0.0150395i	$-0.995213\pi$
$691$	− 20.7846i	− 0.0300790i	0.999887	−	0.0150395i	$-0.00478741\pi$
$692$	0	0
$693$	−432.000	−0.623377
$694$	0	0
$695$	− 290.985i	− 0.418683i
$696$	0	0
$697$	324.000	0.464849
$698$	0	0
$699$	− 197.454i	− 0.282480i
$700$	0	0
$701$	−1074.00	−1.53210	−0.766049	−	0.642783i	$-0.777780\pi$
$701$	−1074.00	−1.53210	−0.766049	+	0.642783i	$0.777780\pi$
$702$	0	0
$703$	180.133i	0.256235i
$704$	0	0
$705$	−432.000	−0.612766
$706$	0	0
$707$	124.708i	0.176390i
$708$	0	0
$709$	898.000	1.26657	0.633286	−	0.773918i	$-0.281706\pi$
$709$	898.000	1.26657	0.633286	+	0.773918i	$0.281706\pi$
$710$	0	0
$711$	228.631i	0.321562i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 1745.91i	− 2.44183i
$716$	0	0
$717$	576.000	0.803347
$718$	0	0
$719$	956.092i	1.32975i	0.746954	+	0.664876i	$0.231516\pi$
$719$	956.092i	1.32975i	−0.746954	+	0.664876i	$0.768484\pi$
$720$	0	0
$721$	−912.000	−1.26491
$722$	0	0
$723$	− 308.305i	− 0.426425i
$724$	0	0
$725$	330.000	0.455172
$726$	0	0
$727$	− 810.600i	− 1.11499i	−0.830179	−	0.557496i	$-0.811762\pi$
$727$	− 810.600i	− 1.11499i	0.830179	−	0.557496i	$-0.188238\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 124.708i	− 0.170599i
$732$	0	0
$733$	370.000	0.504775	0.252387	−	0.967626i	$-0.418784\pi$
$733$	370.000	0.504775	0.252387	+	0.967626i	$0.418784\pi$
$734$	0	0
$735$	− 10.3923i	− 0.0141392i
$736$	0	0
$737$	2448.00	3.32157
$738$	0	0
$739$	852.169i	1.15314i	0.817048	+	0.576569i	$0.195609\pi$
$739$	852.169i	1.15314i	−0.817048	+	0.576569i	$0.804391\pi$
$740$	0	0
$741$	−168.000	−0.226721
$742$	0	0
$743$	1371.78i	1.84628i	0.384467	+	0.923139i	$0.374385\pi$
$743$	1371.78i	1.84628i	−0.384467	+	0.923139i	$0.625615\pi$
$744$	0	0
$745$	−1116.00	−1.49799
$746$	0	0
$747$	62.3538i	0.0834723i
$748$	0	0
$749$	1008.00	1.34579
$750$	0	0
$751$	− 76.2102i	− 0.101478i	−0.998712	−	0.0507392i	$-0.983842\pi$
$751$	− 76.2102i	− 0.101478i	0.998712	−	0.0507392i	$-0.0161577\pi$
$752$	0	0
$753$	180.000	0.239044
$754$	0	0
$755$	− 207.846i	− 0.275293i
$756$	0	0
$757$	514.000	0.678996	0.339498	−	0.940607i	$-0.389743\pi$
$757$	514.000	0.678996	0.339498	+	0.940607i	$0.389743\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−966.000	−1.26938	−0.634691	−	0.772766i	$-0.718873\pi$
$761$	−966.000	−1.26938	−0.634691	+	0.772766i	$0.718873\pi$
$762$	0	0
$763$	− 235.559i	− 0.308727i
$764$	0	0
$765$	108.000	0.141176
$766$	0	0
$767$	290.985i	0.379380i
$768$	0	0
$769$	−958.000	−1.24577	−0.622887	−	0.782312i	$-0.714040\pi$
$769$	−958.000	−1.24577	−0.622887	+	0.782312i	$0.714040\pi$
$770$	0	0
$771$	− 446.869i	− 0.579597i
$772$	0	0
$773$	−546.000	−0.706339	−0.353169	−	0.935559i	$-0.614896\pi$
$773$	−546.000	−0.706339	−0.353169	+	0.935559i	$0.614896\pi$
$774$	0	0
$775$	− 228.631i	− 0.295007i
$776$	0	0
$777$	−312.000	−0.401544
$778$	0	0
$779$	− 374.123i	− 0.480261i
$780$	0	0
$781$	−1728.00	−2.21255
$782$	0	0
$783$	155.885i	0.199086i
$784$	0	0
$785$	1020.00	1.29936
$786$	0	0
$787$	− 242.487i	− 0.308116i	−0.988062	−	0.154058i	$-0.950766\pi$
$787$	− 242.487i	− 0.308116i	0.988062	−	0.154058i	$-0.0492342\pi$
$788$	0	0
$789$	−648.000	−0.821293
$790$	0	0
$791$	540.400i	0.683186i
$792$	0	0
$793$	980.000	1.23581
$794$	0	0
$795$	187.061i	0.235297i
$796$	0	0
$797$	−1338.00	−1.67880	−0.839398	−	0.543518i	$-0.817092\pi$
$797$	−1338.00	−1.67880	−0.839398	+	0.543518i	$0.817092\pi$
$798$	0	0
$799$	249.415i	0.312159i
$800$	0	0
$801$	−342.000	−0.426966
$802$	0	0
$803$	1704.34i	2.12246i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 883.346i	− 1.09460i
$808$	0	0
$809$	−966.000	−1.19407	−0.597033	−	0.802216i	$-0.703654\pi$
$809$	−966.000	−1.19407	−0.597033	+	0.802216i	$0.703654\pi$
$810$	0	0
$811$	1517.28i	1.87087i	0.353497	+	0.935436i	$0.384992\pi$
$811$	1517.28i	1.87087i	−0.353497	+	0.935436i	$0.615008\pi$
$812$	0	0
$813$	−780.000	−0.959410
$814$	0	0
$815$	1704.34i	2.09121i
$816$	0	0
$817$	−144.000	−0.176255
$818$	0	0
$819$	− 290.985i	− 0.355292i
$820$	0	0
$821$	222.000	0.270402	0.135201	−	0.990818i	$-0.456832\pi$
$821$	222.000	0.270402	0.135201	+	0.990818i	$0.456832\pi$
$822$	0	0
$823$	− 1281.72i	− 1.55737i	−0.627413	−	0.778686i	$-0.715886\pi$
$823$	− 1281.72i	− 1.55737i	0.627413	−	0.778686i	$-0.284114\pi$
$824$	0	0
$825$	396.000	0.480000
$826$	0	0
$827$	− 1434.14i	− 1.73415i	−0.498182	−	0.867073i	$-0.665999\pi$
$827$	− 1434.14i	− 1.73415i	0.498182	−	0.867073i	$-0.334001\pi$
$828$	0	0
$829$	226.000	0.272618	0.136309	−	0.990666i	$-0.456476\pi$
$829$	226.000	0.272618	0.136309	+	0.990666i	$0.456476\pi$
$830$	0	0
$831$	24.2487i	0.0291802i
$832$	0	0
$833$	−6.00000	−0.00720288
$834$	0	0
$835$	− 1247.08i	− 1.49350i
$836$	0	0
$837$	108.000	0.129032
$838$	0	0
$839$	− 498.831i	− 0.594554i	−0.954791	−	0.297277i	$-0.903922\pi$
$839$	− 498.831i	− 0.594554i	0.954791	−	0.297277i	$-0.0960784\pi$
$840$	0	0
$841$	59.0000	0.0701546
$842$	0	0
$843$	− 613.146i	− 0.727338i
$844$	0	0
$845$	162.000	0.191716
$846$	0	0
$847$	2154.67i	2.54389i
$848$	0	0
$849$	252.000	0.296820
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−70.0000	−0.0820633	−0.0410317	−	0.999158i	$-0.513064\pi$
$853$	−70.0000	−0.0820633	−0.0410317	+	0.999158i	$0.513064\pi$
$854$	0	0
$855$	− 124.708i	− 0.145857i
$856$	0	0
$857$	42.0000	0.0490082	0.0245041	−	0.999700i	$-0.492199\pi$
$857$	42.0000	0.0490082	0.0245041	+	0.999700i	$0.492199\pi$
$858$	0	0
$859$	− 921.451i	− 1.07270i	−0.843995	−	0.536351i	$-0.819802\pi$
$859$	− 921.451i	− 1.07270i	0.843995	−	0.536351i	$-0.180198\pi$
$860$	0	0
$861$	648.000	0.752613
$862$	0	0
$863$	166.277i	0.192673i	0.995349	+	0.0963365i	$0.0307125\pi$
$863$	166.277i	0.192673i	−0.995349	+	0.0963365i	$0.969287\pi$
$864$	0	0
$865$	−252.000	−0.291329
$866$	0	0
$867$	438.209i	0.505431i
$868$	0	0
$869$	1584.00	1.82278
$870$	0	0
$871$	1648.91i	1.89313i
$872$	0	0
$873$	−102.000	−0.116838
$874$	0	0
$875$	581.969i	0.665108i
$876$	0	0
$877$	−166.000	−0.189282	−0.0946408	−	0.995511i	$-0.530170\pi$
$877$	−166.000	−0.189282	−0.0946408	+	0.995511i	$0.530170\pi$
$878$	0	0
$879$	862.561i	0.981298i
$880$	0	0
$881$	−702.000	−0.796822	−0.398411	−	0.917207i	$-0.630438\pi$
$881$	−702.000	−0.796822	−0.398411	+	0.917207i	$0.630438\pi$
$882$	0	0
$883$	− 630.466i	− 0.714005i	−0.934103	−	0.357003i	$-0.883799\pi$
$883$	− 630.466i	− 0.714005i	0.934103	−	0.357003i	$-0.116201\pi$
$884$	0	0
$885$	−216.000	−0.244068
$886$	0	0
$887$	− 1288.65i	− 1.45281i	−0.687265	−	0.726407i	$-0.741189\pi$
$887$	− 1288.65i	− 1.45281i	0.687265	−	0.726407i	$-0.258811\pi$
$888$	0	0
$889$	720.000	0.809899
$890$	0	0
$891$	187.061i	0.209946i
$892$	0	0
$893$	288.000	0.322508
$894$	0	0
$895$	− 872.954i	− 0.975367i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 623.538i	− 0.693591i
$900$	0	0
$901$	108.000	0.119867
$902$	0	0
$903$	− 249.415i	− 0.276207i
$904$	0	0
$905$	492.000	0.543646
$906$	0	0
$907$	− 1447.99i	− 1.59647i	−0.602349	−	0.798233i	$-0.705768\pi$
$907$	− 1447.99i	− 1.59647i	0.602349	−	0.798233i	$-0.294232\pi$
$908$	0	0
$909$	54.0000	0.0594059
$910$	0	0
$911$	332.554i	0.365043i	0.983202	+	0.182521i	$0.0584258\pi$
$911$	332.554i	0.365043i	−0.983202	+	0.182521i	$0.941574\pi$
$912$	0	0
$913$	432.000	0.473165
$914$	0	0
$915$	727.461i	0.795040i
$916$	0	0
$917$	720.000	0.785169
$918$	0	0
$919$	− 561.184i	− 0.610647i	−0.952249	−	0.305323i	$-0.901235\pi$
$919$	− 561.184i	− 0.610647i	0.952249	−	0.305323i	$-0.0987646\pi$
$920$	0	0
$921$	324.000	0.351792
$922$	0	0
$923$	− 1163.94i	− 1.26104i
$924$	0	0
$925$	286.000	0.309189
$926$	0	0
$927$	394.908i	0.426006i
$928$	0	0
$929$	−438.000	−0.471475	−0.235737	−	0.971817i	$-0.575751\pi$
$929$	−438.000	−0.471475	−0.235737	+	0.971817i	$0.575751\pi$
$930$	0	0
$931$	6.92820i	0.00744168i
$932$	0	0
$933$	−72.0000	−0.0771704
$934$	0	0
$935$	− 748.246i	− 0.800263i
$936$	0	0
$937$	1826.00	1.94877	0.974386	−	0.224881i	$-0.0721992\pi$
$937$	1826.00	1.94877	0.974386	+	0.224881i	$0.0721992\pi$
$938$	0	0
$939$	− 502.295i	− 0.534925i
$940$	0	0
$941$	−330.000	−0.350691	−0.175345	−	0.984507i	$-0.556104\pi$
$941$	−330.000	−0.350691	−0.175345	+	0.984507i	$0.556104\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	216.000	0.228571
$946$	0	0
$947$	187.061i	0.197531i	0.995111	+	0.0987653i	$0.0314893\pi$
$947$	187.061i	0.197531i	−0.995111	+	0.0987653i	$0.968511\pi$
$948$	0	0
$949$	−1148.00	−1.20969
$950$	0	0
$951$	363.731i	0.382472i
$952$	0	0
$953$	−1110.00	−1.16474	−0.582371	−	0.812923i	$-0.697875\pi$
$953$	−1110.00	−1.16474	−0.582371	+	0.812923i	$0.697875\pi$
$954$	0	0
$955$	1995.32i	2.08934i
$956$	0	0
$957$	1080.00	1.12853
$958$	0	0
$959$	− 1288.65i	− 1.34374i
$960$	0	0
$961$	529.000	0.550468
$962$	0	0
$963$	− 436.477i	− 0.453247i
$964$	0	0
$965$	−564.000	−0.584456
$966$	0	0
$967$	755.174i	0.780945i	0.920615	+	0.390473i	$0.127688\pi$
$967$	755.174i	0.780945i	−0.920615	+	0.390473i	$0.872312\pi$
$968$	0	0
$969$	−72.0000	−0.0743034
$970$	0	0
$971$	− 394.908i	− 0.406702i	−0.979106	−	0.203351i	$-0.934817\pi$
$971$	− 394.908i	− 0.406702i	0.979106	−	0.203351i	$-0.0651832\pi$
$972$	0	0
$973$	−336.000	−0.345324
$974$	0	0
$975$	266.736i	0.273575i
$976$	0	0
$977$	−918.000	−0.939611	−0.469806	−	0.882770i	$-0.655676\pi$
$977$	−918.000	−0.939611	−0.469806	+	0.882770i	$0.655676\pi$
$978$	0	0
$979$	2369.45i	2.42027i
$980$	0	0
$981$	−102.000	−0.103976
$982$	0	0
$983$	41.5692i	0.0422881i	0.999776	+	0.0211441i	$0.00673086\pi$
$983$	41.5692i	0.0422881i	−0.999776	+	0.0211441i	$0.993269\pi$
$984$	0	0
$985$	−1548.00	−1.57157
$986$	0	0
$987$	498.831i	0.505401i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 48.4974i	− 0.0489379i	−0.999701	−	0.0244689i	$-0.992211\pi$
$991$	− 48.4974i	− 0.0489379i	0.999701	−	0.0244689i	$-0.00778948\pi$
$992$	0	0
$993$	348.000	0.350453
$994$	0	0
$995$	706.677i	0.710228i
$996$	0	0
$997$	554.000	0.555667	0.277834	−	0.960629i	$-0.410384\pi$
$997$	554.000	0.555667	0.277834	+	0.960629i	$0.410384\pi$
$998$	0	0
$999$	135.100i	0.135235i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.3.g.a.31.1	✓	2
3.2	odd	2		144.3.g.b.127.1		2
4.3	odd	2	inner	48.3.g.a.31.2	yes	2
5.2	odd	4		1200.3.j.a.799.2		4
5.3	odd	4		1200.3.j.a.799.4		4
5.4	even	2		1200.3.e.h.751.2		2
7.6	odd	2		2352.3.m.a.1471.2		2
8.3	odd	2		192.3.g.a.127.1		2
8.5	even	2		192.3.g.a.127.2		2
9.2	odd	6		1296.3.o.n.271.1		2
9.4	even	3		1296.3.o.c.703.1		2
9.5	odd	6		1296.3.o.p.703.1		2
9.7	even	3		1296.3.o.a.271.1		2
12.11	even	2		144.3.g.b.127.2		2
15.2	even	4		3600.3.j.i.1999.3		4
15.8	even	4		3600.3.j.i.1999.1		4
15.14	odd	2		3600.3.e.t.3151.2		2
16.3	odd	4		768.3.b.b.127.1		4
16.5	even	4		768.3.b.b.127.2		4
16.11	odd	4		768.3.b.b.127.4		4
16.13	even	4		768.3.b.b.127.3		4
20.3	even	4		1200.3.j.a.799.1		4
20.7	even	4		1200.3.j.a.799.3		4
20.19	odd	2		1200.3.e.h.751.1		2
24.5	odd	2		576.3.g.i.127.1		2
24.11	even	2		576.3.g.i.127.2		2
28.27	even	2		2352.3.m.a.1471.1		2
36.7	odd	6		1296.3.o.c.271.1		2
36.11	even	6		1296.3.o.p.271.1		2
36.23	even	6		1296.3.o.n.703.1		2
36.31	odd	6		1296.3.o.a.703.1		2
48.5	odd	4		2304.3.b.n.127.2		4
48.11	even	4		2304.3.b.n.127.1		4
48.29	odd	4		2304.3.b.n.127.4		4
48.35	even	4		2304.3.b.n.127.3		4
60.23	odd	4		3600.3.j.i.1999.4		4
60.47	odd	4		3600.3.j.i.1999.2		4
60.59	even	2		3600.3.e.t.3151.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.g.a.31.1	✓	2	1.1	even	1	trivial
48.3.g.a.31.2	yes	2	4.3	odd	2	inner
144.3.g.b.127.1		2	3.2	odd	2
144.3.g.b.127.2		2	12.11	even	2
192.3.g.a.127.1		2	8.3	odd	2
192.3.g.a.127.2		2	8.5	even	2
576.3.g.i.127.1		2	24.5	odd	2
576.3.g.i.127.2		2	24.11	even	2
768.3.b.b.127.1		4	16.3	odd	4
768.3.b.b.127.2		4	16.5	even	4
768.3.b.b.127.3		4	16.13	even	4
768.3.b.b.127.4		4	16.11	odd	4
1200.3.e.h.751.1		2	20.19	odd	2
1200.3.e.h.751.2		2	5.4	even	2
1200.3.j.a.799.1		4	20.3	even	4
1200.3.j.a.799.2		4	5.2	odd	4
1200.3.j.a.799.3		4	20.7	even	4
1200.3.j.a.799.4		4	5.3	odd	4
1296.3.o.a.271.1		2	9.7	even	3
1296.3.o.a.703.1		2	36.31	odd	6
1296.3.o.c.271.1		2	36.7	odd	6
1296.3.o.c.703.1		2	9.4	even	3
1296.3.o.n.271.1		2	9.2	odd	6
1296.3.o.n.703.1		2	36.23	even	6
1296.3.o.p.271.1		2	36.11	even	6
1296.3.o.p.703.1		2	9.5	odd	6
2304.3.b.n.127.1		4	48.11	even	4
2304.3.b.n.127.2		4	48.5	odd	4
2304.3.b.n.127.3		4	48.35	even	4
2304.3.b.n.127.4		4	48.29	odd	4
2352.3.m.a.1471.1		2	28.27	even	2
2352.3.m.a.1471.2		2	7.6	odd	2
3600.3.e.t.3151.1		2	60.59	even	2
3600.3.e.t.3151.2		2	15.14	odd	2
3600.3.j.i.1999.1		4	15.8	even	4
3600.3.j.i.1999.2		4	60.47	odd	4
3600.3.j.i.1999.3		4	15.2	even	4
3600.3.j.i.1999.4		4	60.23	odd	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +6.00000 q^{5} -6.92820i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +6.00000 q^{5} -6.92820i q^{7} -3.00000 q^{9} +20.7846i q^{11} -14.0000 q^{13} -10.3923i q^{15} -6.00000 q^{17} +6.92820i q^{19} -12.0000 q^{21} +11.0000 q^{25} +5.19615i q^{27} +30.0000 q^{29} -20.7846i q^{31} +36.0000 q^{33} -41.5692i q^{35} +26.0000 q^{37} +24.2487i q^{39} -54.0000 q^{41} +20.7846i q^{43} -18.0000 q^{45} -41.5692i q^{47} +1.00000 q^{49} +10.3923i q^{51} -18.0000 q^{53} +124.708i q^{55} +12.0000 q^{57} -20.7846i q^{59} -70.0000 q^{61} +20.7846i q^{63} -84.0000 q^{65} -117.779i q^{67} +83.1384i q^{71} +82.0000 q^{73} -19.0526i q^{75} +144.000 q^{77} -76.2102i q^{79} +9.00000 q^{81} -20.7846i q^{83} -36.0000 q^{85} -51.9615i q^{87} +114.000 q^{89} +96.9948i q^{91} -36.0000 q^{93} +41.5692i q^{95} +34.0000 q^{97} -62.3538i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{5} - 6 q^{9}+O(q^{10}) \) 2 * q + 12 * q^5 - 6 * q^9	\( 2 q + 12 q^{5} - 6 q^{9} - 28 q^{13} - 12 q^{17} - 24 q^{21} + 22 q^{25} + 60 q^{29} + 72 q^{33} + 52 q^{37} - 108 q^{41} - 36 q^{45} + 2 q^{49} - 36 q^{53} + 24 q^{57} - 140 q^{61} - 168 q^{65} + 164 q^{73} + 288 q^{77} + 18 q^{81} - 72 q^{85} + 228 q^{89} - 72 q^{93} + 68 q^{97}+O(q^{100}) \) 2 * q + 12 * q^5 - 6 * q^9 - 28 * q^13 - 12 * q^17 - 24 * q^21 + 22 * q^25 + 60 * q^29 + 72 * q^33 + 52 * q^37 - 108 * q^41 - 36 * q^45 + 2 * q^49 - 36 * q^53 + 24 * q^57 - 140 * q^61 - 168 * q^65 + 164 * q^73 + 288 * q^77 + 18 * q^81 - 72 * q^85 + 228 * q^89 - 72 * q^93 + 68 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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