LMFDB - Embedded newform 2376.2.b.a.593.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2376,2,Mod(593,2376)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2376.593"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2376 = 2^{3} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2376.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$18.9724555203$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 35x^{10} + 432x^{8} + 2280x^{6} + 4784x^{4} + 2224x^{2} + 256 $ x^12 + 35x^10 + 432x^8 + 2280x^6 + 4784x^4 + 2224*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			593.1
Root			$-3.83827i$ of defining polynomial
Character	$\chi$	$=$	2376.593
Dual form			2376.2.b.a.593.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.83827i q^{5} +0.451228i q^{7} +(-1.33926 + 3.03420i) q^{11} -6.76223i q^{13} -4.29198 q^{17} -1.39167i q^{19} +6.04769i q^{23} -9.73229 q^{25} -7.95074 q^{29} +1.40571 q^{31} +1.73193 q^{35} -6.89638 q^{37} +8.17224 q^{41} -11.3754i q^{43} +10.5838i q^{47} +6.79639 q^{49} +5.06440i q^{53} +(11.6461 + 5.14042i) q^{55} -7.47678i q^{59} +4.85577i q^{61} -25.9553 q^{65} -7.54808 q^{67} +3.03902i q^{71} +6.54578i q^{73} +(-1.36912 - 0.604309i) q^{77} +0.242598i q^{79} -16.0549 q^{83} +16.4738i q^{85} -13.9496i q^{89} +3.05131 q^{91} -5.34160 q^{95} +6.53266 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 5 q^{11} - 10 q^{17} - 10 q^{25} + 2 q^{29} + 6 q^{31} - 10 q^{35} - 6 q^{37} + 2 q^{41} - 16 q^{49} - q^{55} - 32 q^{65} + 4 q^{67} + 7 q^{77} - 10 q^{83} + 24 q^{91} - 44 q^{95} + 8 q^{97}+O(q^{100}) $ 12 * q - 5 * q^11 - 10 * q^17 - 10 * q^25 + 2 * q^29 + 6 * q^31 - 10 * q^35 - 6 * q^37 + 2 * q^41 - 16 * q^49 - q^55 - 32 * q^65 + 4 * q^67 + 7 * q^77 - 10 * q^83 + 24 * q^91 - 44 * q^95 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2376\mathbb{Z}\right)^\times$.

$n$	$353$	$1189$	$1729$	$1783$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$		−	3.83827i		−	1.71652i	−0.513211	−	0.858262i	$-0.671544\pi$
$5$		−	3.83827i		−	1.71652i	0.513211	−	0.858262i	$-0.328456\pi$
$6$		0			0
$7$			0.451228i			0.170548i	0.996358	+	0.0852740i	$0.0271766\pi$
$7$			0.451228i			0.170548i	−0.996358	+	0.0852740i	$0.972823\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.33926	+	3.03420i	−0.403801	+	0.914847i
$11$	−1.33926	+	3.03420i	−0.403801	+	0.914847i
$12$		0			0
$13$		−	6.76223i		−	1.87551i	−0.347302	−	0.937753i	$-0.612902\pi$
$13$		−	6.76223i		−	1.87551i	0.347302	−	0.937753i	$-0.387098\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.29198			−1.04096			−0.520480	−	0.853874i	$-0.674247\pi$
$17$	−4.29198			−1.04096			−0.520480	+	0.853874i	$0.674247\pi$
$18$		0			0
$19$		−	1.39167i		−	0.319271i	−0.987176	−	0.159636i	$-0.948968\pi$
$19$		−	1.39167i		−	0.319271i	0.987176	−	0.159636i	$-0.0510319\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.04769i			1.26103i	0.776177	+	0.630515i	$0.217156\pi$
$23$			6.04769i			1.26103i	−0.776177	+	0.630515i	$0.782844\pi$
$24$		0			0
$25$	−9.73229			−1.94646
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.95074			−1.47642			−0.738208	−	0.674574i	$-0.764328\pi$
$29$	−7.95074			−1.47642			−0.738208	+	0.674574i	$0.764328\pi$
$30$		0			0
$31$	1.40571			0.252472			0.126236	−	0.992000i	$-0.459710\pi$
$31$	1.40571			0.252472			0.126236	+	0.992000i	$0.459710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.73193			0.292750
$36$		0			0
$37$	−6.89638			−1.13376			−0.566879	−	0.823801i	$-0.691849\pi$
$37$	−6.89638			−1.13376			−0.566879	+	0.823801i	$0.691849\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.17224			1.27629			0.638145	−	0.769917i	$-0.279702\pi$
$41$	8.17224			1.27629			0.638145	+	0.769917i	$0.279702\pi$
$42$		0			0
$43$		−	11.3754i		−	1.73473i	−0.497670	−	0.867367i	$-0.665811\pi$
$43$		−	11.3754i		−	1.73473i	0.497670	−	0.867367i	$-0.334189\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.5838i			1.54381i	0.635740	+	0.771903i	$0.280695\pi$
$47$			10.5838i			1.54381i	−0.635740	+	0.771903i	$0.719305\pi$
$48$		0			0
$49$	6.79639			0.970913
$50$		0			0
$51$		0			0
$52$		0			0
$53$			5.06440i			0.695649i	0.937560	+	0.347824i	$0.113079\pi$
$53$			5.06440i			0.695649i	−0.937560	+	0.347824i	$0.886921\pi$
$54$		0			0
$55$	11.6461	+	5.14042i	1.57036	+	0.693134i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	7.47678i		−	0.973394i	−0.873571	−	0.486697i	$-0.838202\pi$
$59$		−	7.47678i		−	0.973394i	0.873571	−	0.486697i	$-0.161798\pi$
$60$		0			0
$61$			4.85577i			0.621718i	0.950456	+	0.310859i	$0.100617\pi$
$61$			4.85577i			0.621718i	−0.950456	+	0.310859i	$0.899383\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−25.9553			−3.21935
$66$		0			0
$67$	−7.54808			−0.922145			−0.461072	−	0.887362i	$-0.652535\pi$
$67$	−7.54808			−0.922145			−0.461072	+	0.887362i	$0.652535\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.03902i			0.360665i	0.983606	+	0.180333i	$0.0577174\pi$
$71$			3.03902i			0.360665i	−0.983606	+	0.180333i	$0.942283\pi$
$72$		0			0
$73$			6.54578i			0.766126i	0.923722	+	0.383063i	$0.125131\pi$
$73$			6.54578i			0.766126i	−0.923722	+	0.383063i	$0.874869\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.36912	−	0.604309i	−0.156025	−	0.0688674i
$78$		0			0
$79$			0.242598i			0.0272945i	0.999907	+	0.0136472i	$0.00434418\pi$
$79$			0.242598i			0.0272945i	−0.999907	+	0.0136472i	$0.995656\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−16.0549			−1.76225			−0.881127	−	0.472880i	$-0.843214\pi$
$83$	−16.0549			−1.76225			−0.881127	+	0.472880i	$0.843214\pi$
$84$		0			0
$85$			16.4738i			1.78683i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	13.9496i		−	1.47865i	−0.673349	−	0.739325i	$-0.735145\pi$
$89$		−	13.9496i		−	1.47865i	0.673349	−	0.739325i	$-0.264855\pi$
$90$		0			0
$91$	3.05131			0.319864
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.34160			−0.548037
$96$		0			0
$97$	6.53266			0.663291			0.331646	−	0.943404i	$-0.392396\pi$
$97$	6.53266			0.663291			0.331646	+	0.943404i	$0.392396\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.3256			1.02744			0.513720	−	0.857958i	$-0.328267\pi$
$101$	10.3256			1.02744			0.513720	+	0.857958i	$0.328267\pi$
$102$		0			0
$103$	−1.99093			−0.196172			−0.0980860	−	0.995178i	$-0.531272\pi$
$103$	−1.99093			−0.196172			−0.0980860	+	0.995178i	$0.531272\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.7326			−1.52093			−0.760466	−	0.649378i	$-0.775029\pi$
$107$	−15.7326			−1.52093			−0.760466	+	0.649378i	$0.775029\pi$
$108$		0			0
$109$			3.30215i			0.316289i	0.987416	+	0.158144i	$0.0505511\pi$
$109$			3.30215i			0.316289i	−0.987416	+	0.158144i	$0.949449\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			5.69424i			0.535669i	0.963465	+	0.267835i	$0.0863081\pi$
$113$			5.69424i			0.535669i	−0.963465	+	0.267835i	$0.913692\pi$
$114$		0			0
$115$	23.2126			2.16459
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	1.93666i		−	0.177534i
$120$		0			0
$121$	−7.41279	−	8.12715i	−0.673890	−	0.738832i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			18.1638i			1.62462i
$126$		0			0
$127$			15.8174i			1.40356i	0.712391	+	0.701782i	$0.247612\pi$
$127$			15.8174i			1.40356i	−0.712391	+	0.701782i	$0.752388\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14.5048			−1.26729			−0.633647	−	0.773622i	$-0.718443\pi$
$131$	−14.5048			−1.26729			−0.633647	+	0.773622i	$0.718443\pi$
$132$		0			0
$133$	0.627960			0.0544511
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.16317i			0.355683i	0.984059	+	0.177842i	$0.0569115\pi$
$137$			4.16317i			0.355683i	−0.984059	+	0.177842i	$0.943088\pi$
$138$		0			0
$139$			0.0361845i			0.00306913i	0.999999	+	0.00153456i	$0.000488467\pi$
$139$			0.0361845i			0.00306913i	−0.999999	+	0.00153456i	$0.999512\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	20.5180	+	9.05636i	1.71580	+	0.757331i
$144$		0			0
$145$			30.5171i			2.53430i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.3697			−1.50490			−0.752451	−	0.658648i	$-0.771129\pi$
$149$	−18.3697			−1.50490			−0.752451	+	0.658648i	$0.771129\pi$
$150$		0			0
$151$		−	5.90244i		−	0.480334i	−0.970732	−	0.240167i	$-0.922798\pi$
$151$		−	5.90244i		−	0.480334i	0.970732	−	0.240167i	$-0.0772022\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	5.39547i		−	0.433375i
$156$		0			0
$157$	−2.48992			−0.198717			−0.0993587	−	0.995052i	$-0.531679\pi$
$157$	−2.48992			−0.198717			−0.0993587	+	0.995052i	$0.531679\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.72889			−0.215066
$162$		0			0
$163$	17.5792			1.37691			0.688456	−	0.725278i	$-0.258289\pi$
$163$	17.5792			1.37691			0.688456	+	0.725278i	$0.258289\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.85889			0.298610			0.149305	−	0.988791i	$-0.452296\pi$
$167$	3.85889			0.298610			0.149305	+	0.988791i	$0.452296\pi$
$168$		0			0
$169$	−32.7278			−2.51752
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.8395			0.900140			0.450070	−	0.892993i	$-0.351399\pi$
$173$	11.8395			0.900140			0.450070	+	0.892993i	$0.351399\pi$
$174$		0			0
$175$		−	4.39148i		−	0.331965i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	16.6600i		−	1.24523i	−0.782529	−	0.622615i	$-0.786070\pi$
$179$		−	16.6600i		−	1.24523i	0.782529	−	0.622615i	$-0.213930\pi$
$180$		0			0
$181$	−9.39614			−0.698409			−0.349205	−	0.937046i	$-0.613548\pi$
$181$	−9.39614			−0.698409			−0.349205	+	0.937046i	$0.613548\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			26.4702i			1.94612i
$186$		0			0
$187$	5.74806	−	13.0228i	0.420340	−	0.952318i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			13.6484i			0.987563i	0.869586	+	0.493782i	$0.164386\pi$
$191$			13.6484i			0.987563i	−0.869586	+	0.493782i	$0.835614\pi$
$192$		0			0
$193$		−	23.5615i		−	1.69599i	−0.530003	−	0.847996i	$-0.677809\pi$
$193$		−	23.5615i		−	1.69599i	0.530003	−	0.847996i	$-0.322191\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.44540			−0.174227			−0.0871136	−	0.996198i	$-0.527764\pi$
$197$	−2.44540			−0.174227			−0.0871136	+	0.996198i	$0.527764\pi$
$198$		0			0
$199$	−9.34483			−0.662437			−0.331219	−	0.943554i	$-0.607460\pi$
$199$	−9.34483			−0.662437			−0.331219	+	0.943554i	$0.607460\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	3.58759i		−	0.251800i
$204$		0			0
$205$		−	31.3672i		−	2.19078i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.22261	+	1.86380i	0.292084	+	0.128922i
$210$		0			0
$211$		−	14.6480i		−	1.00841i	−0.863585	−	0.504204i	$-0.831786\pi$
$211$		−	14.6480i		−	1.00841i	0.863585	−	0.504204i	$-0.168214\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−43.6618			−2.97771
$216$		0			0
$217$			0.634293i			0.0430586i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			29.0234i			1.95233i
$222$		0			0
$223$	−11.4199			−0.764731			−0.382366	−	0.924011i	$-0.624891\pi$
$223$	−11.4199			−0.764731			−0.382366	+	0.924011i	$0.624891\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.60498			0.239271			0.119635	−	0.992818i	$-0.461827\pi$
$227$	3.60498			0.239271			0.119635	+	0.992818i	$0.461827\pi$
$228$		0			0
$229$	−14.0319			−0.927255			−0.463627	−	0.886030i	$-0.653452\pi$
$229$	−14.0319			−0.927255			−0.463627	+	0.886030i	$0.653452\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.07429			0.266916			0.133458	−	0.991054i	$-0.457392\pi$
$233$	4.07429			0.266916			0.133458	+	0.991054i	$0.457392\pi$
$234$		0			0
$235$	40.6234			2.64998
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.707086			−0.0457376			−0.0228688	−	0.999738i	$-0.507280\pi$
$239$	−0.707086			−0.0457376			−0.0228688	+	0.999738i	$0.507280\pi$
$240$		0			0
$241$		−	2.33995i		−	0.150730i	−0.997156	−	0.0753648i	$-0.975988\pi$
$241$		−	2.33995i		−	0.150730i	0.997156	−	0.0753648i	$-0.0240121\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	26.0864i		−	1.66660i
$246$		0			0
$247$	−9.41080			−0.598795
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	17.1362i		−	1.08163i	−0.841143	−	0.540813i	$-0.818117\pi$
$251$		−	17.1362i		−	1.08163i	0.841143	−	0.540813i	$-0.181883\pi$
$252$		0			0
$253$	−18.3499	−	8.09940i	−1.15365	−	0.509205i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	20.6064i		−	1.28539i	−0.766122	−	0.642695i	$-0.777816\pi$
$257$		−	20.6064i		−	1.28539i	0.766122	−	0.642695i	$-0.222184\pi$
$258$		0			0
$259$		−	3.11184i		−	0.193360i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.67453			0.534895			0.267447	−	0.963572i	$-0.413820\pi$
$263$	8.67453			0.534895			0.267447	+	0.963572i	$0.413820\pi$
$264$		0			0
$265$	19.4385			1.19410
$266$		0			0
$267$		0			0
$268$		0			0
$269$			1.22282i			0.0745569i	0.999305	+	0.0372784i	$0.0118688\pi$
$269$			1.22282i			0.0745569i	−0.999305	+	0.0372784i	$0.988131\pi$
$270$		0			0
$271$		−	25.5118i		−	1.54973i	−0.632125	−	0.774866i	$-0.717817\pi$
$271$		−	25.5118i		−	1.54973i	0.632125	−	0.774866i	$-0.282183\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.0340	−	29.5298i	0.785981	−	1.78071i
$276$		0			0
$277$		−	2.00801i		−	0.120650i	−0.998179	−	0.0603249i	$-0.980786\pi$
$277$		−	2.00801i		−	0.120650i	0.998179	−	0.0603249i	$-0.0192137\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.26517			−0.194784			−0.0973918	−	0.995246i	$-0.531050\pi$
$281$	−3.26517			−0.194784			−0.0973918	+	0.995246i	$0.531050\pi$
$282$		0			0
$283$			25.2640i			1.50179i	0.660423	+	0.750894i	$0.270377\pi$
$283$			25.2640i			1.50179i	−0.660423	+	0.750894i	$0.729623\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			3.68754i			0.217669i
$288$		0			0
$289$	1.42113			0.0835957
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.8616			−0.985066			−0.492533	−	0.870294i	$-0.663929\pi$
$293$	−16.8616			−0.985066			−0.492533	+	0.870294i	$0.663929\pi$
$294$		0			0
$295$	−28.6979			−1.67085
$296$		0			0
$297$		0			0
$298$		0			0
$299$	40.8959			2.36507
$300$		0			0
$301$	5.13290			0.295855
$302$		0			0
$303$		0			0
$304$		0			0
$305$	18.6377			1.06719
$306$		0			0
$307$		−	2.84308i		−	0.162263i	−0.996703	−	0.0811317i	$-0.974147\pi$
$307$		−	2.84308i		−	0.162263i	0.996703	−	0.0811317i	$-0.0258534\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	3.65603i		−	0.207314i	−0.994613	−	0.103657i	$-0.966946\pi$
$311$		−	3.65603i		−	0.207314i	0.994613	−	0.103657i	$-0.0330545\pi$
$312$		0			0
$313$	−29.4606			−1.66521			−0.832606	−	0.553866i	$-0.813152\pi$
$313$	−29.4606			−1.66521			−0.832606	+	0.553866i	$0.813152\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	17.2944i		−	0.971348i	−0.874140	−	0.485674i	$-0.838574\pi$
$317$		−	17.2944i		−	0.971348i	0.874140	−	0.485674i	$-0.161426\pi$
$318$		0			0
$319$	10.6481	−	24.1242i	0.596178	−	1.35069i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			5.97303i			0.332348i
$324$		0			0
$325$			65.8120i			3.65059i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.77570			−0.263293
$330$		0			0
$331$	−3.37965			−0.185762			−0.0928811	−	0.995677i	$-0.529608\pi$
$331$	−3.37965			−0.185762			−0.0928811	+	0.995677i	$0.529608\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			28.9715i			1.58288i
$336$		0			0
$337$			16.9141i			0.921372i	0.887563	+	0.460686i	$0.152397\pi$
$337$			16.9141i			0.921372i	−0.887563	+	0.460686i	$0.847603\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.88260	+	4.26520i	−0.101948	+	0.230973i
$342$		0			0
$343$			6.22532i			0.336135i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.3183			−0.768645			−0.384322	−	0.923199i	$-0.625565\pi$
$347$	−14.3183			−0.768645			−0.384322	+	0.923199i	$0.625565\pi$
$348$		0			0
$349$			1.96342i			0.105100i	0.998618	+	0.0525499i	$0.0167348\pi$
$349$			1.96342i			0.105100i	−0.998618	+	0.0525499i	$0.983265\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	18.2149i		−	0.969479i	−0.874659	−	0.484739i	$-0.838914\pi$
$353$		−	18.2149i		−	0.969479i	0.874659	−	0.484739i	$-0.161086\pi$
$354$		0			0
$355$	11.6646			0.619091
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5.92794			0.312865			0.156432	−	0.987689i	$-0.450001\pi$
$359$	5.92794			0.312865			0.156432	+	0.987689i	$0.450001\pi$
$360$		0			0
$361$	17.0633			0.898066
$362$		0			0
$363$		0			0
$364$		0			0
$365$	25.1244			1.31507
$366$		0			0
$367$	24.6319			1.28577			0.642887	−	0.765961i	$-0.277736\pi$
$367$	24.6319			1.28577			0.642887	+	0.765961i	$0.277736\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.28520			−0.118642
$372$		0			0
$373$		−	8.89908i		−	0.460777i	−0.973099	−	0.230389i	$-0.926000\pi$
$373$		−	8.89908i		−	0.460777i	0.973099	−	0.230389i	$-0.0739997\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			53.7648i			2.76903i
$378$		0			0
$379$	12.4621			0.640135			0.320068	−	0.947395i	$-0.396294\pi$
$379$	12.4621			0.640135			0.320068	+	0.947395i	$0.396294\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	4.53654i		−	0.231807i	−0.993261	−	0.115903i	$-0.963024\pi$
$383$		−	4.53654i		−	0.231807i	0.993261	−	0.115903i	$-0.0369763\pi$
$384$		0			0
$385$	−2.31950	+	5.25504i	−0.118213	+	0.267821i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	16.3511i		−	0.829031i	−0.910042	−	0.414516i	$-0.863951\pi$
$389$		−	16.3511i		−	0.829031i	0.910042	−	0.414516i	$-0.136049\pi$
$390$		0			0
$391$		−	25.9566i		−	1.31268i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.931158			0.0468516
$396$		0			0
$397$	29.9493			1.50311			0.751556	−	0.659669i	$-0.229303\pi$
$397$	29.9493			1.50311			0.751556	+	0.659669i	$0.229303\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	8.54596i		−	0.426765i	−0.976969	−	0.213382i	$-0.931552\pi$
$401$		−	8.54596i		−	0.426765i	0.976969	−	0.213382i	$-0.0684480\pi$
$402$		0			0
$403$		−	9.50571i		−	0.473513i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.23602	−	20.9250i	0.457813	−	1.03722i
$408$		0			0
$409$		−	21.8721i		−	1.08151i	−0.841181	−	0.540754i	$-0.818139\pi$
$409$		−	21.8721i		−	1.08151i	0.841181	−	0.540754i	$-0.181861\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.37373			0.166010
$414$		0			0
$415$			61.6230i			3.02495i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			18.8013i			0.918504i	0.888306	+	0.459252i	$0.151882\pi$
$419$			18.8013i			0.918504i	−0.888306	+	0.459252i	$0.848118\pi$
$420$		0			0
$421$	23.2104			1.13121			0.565603	−	0.824677i	$-0.308643\pi$
$421$	23.2104			1.13121			0.565603	+	0.824677i	$0.308643\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	41.7708			2.02618
$426$		0			0
$427$	−2.19106			−0.106033
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.03871			0.435380			0.217690	−	0.976018i	$-0.430148\pi$
$431$	9.03871			0.435380			0.217690	+	0.976018i	$0.430148\pi$
$432$		0			0
$433$	−17.6604			−0.848703			−0.424352	−	0.905497i	$-0.639498\pi$
$433$	−17.6604			−0.848703			−0.424352	+	0.905497i	$0.639498\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.41639			0.402610
$438$		0			0
$439$		−	28.2453i		−	1.34808i	−0.738696	−	0.674038i	$-0.764558\pi$
$439$		−	28.2453i		−	1.34808i	0.738696	−	0.674038i	$-0.235442\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			13.8970i			0.660265i	0.943935	+	0.330133i	$0.107093\pi$
$443$			13.8970i			0.660265i	−0.943935	+	0.330133i	$0.892907\pi$
$444$		0			0
$445$	−53.5421			−2.53814
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	30.2241i		−	1.42636i	−0.700979	−	0.713182i	$-0.747253\pi$
$449$		−	30.2241i		−	1.42636i	0.700979	−	0.713182i	$-0.252747\pi$
$450$		0			0
$451$	−10.9447	+	24.7962i	−0.515366	+	1.16761i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	11.7117i		−	0.549054i
$456$		0			0
$457$			15.0513i			0.704070i	0.935987	+	0.352035i	$0.114510\pi$
$457$			15.0513i			0.704070i	−0.935987	+	0.352035i	$0.885490\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.51523			0.0705711			0.0352855	−	0.999377i	$-0.488766\pi$
$461$	1.51523			0.0705711			0.0352855	+	0.999377i	$0.488766\pi$
$462$		0			0
$463$	−12.2141			−0.567640			−0.283820	−	0.958878i	$-0.591602\pi$
$463$	−12.2141			−0.567640			−0.283820	+	0.958878i	$0.591602\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			5.35825i			0.247950i	0.992285	+	0.123975i	$0.0395643\pi$
$467$			5.35825i			0.247950i	−0.992285	+	0.123975i	$0.960436\pi$
$468$		0			0
$469$		−	3.40590i		−	0.157270i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	34.5153	+	15.2346i	1.58702	+	0.700487i
$474$		0			0
$475$			13.5441i			0.621448i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	36.3391			1.66038			0.830188	−	0.557484i	$-0.188233\pi$
$479$	36.3391			1.66038			0.830188	+	0.557484i	$0.188233\pi$
$480$		0			0
$481$			46.6350i			2.12637i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	25.0741i		−	1.13856i
$486$		0			0
$487$	27.2646			1.23548			0.617738	−	0.786384i	$-0.288049\pi$
$487$	27.2646			1.23548			0.617738	+	0.786384i	$0.288049\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.180583			0.00814961			0.00407481	−	0.999992i	$-0.498703\pi$
$491$	0.180583			0.00814961			0.00407481	+	0.999992i	$0.498703\pi$
$492$		0			0
$493$	34.1244			1.53689
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.37129			−0.0615107
$498$		0			0
$499$	41.7868			1.87063			0.935317	−	0.353812i	$-0.115115\pi$
$499$	41.7868			1.87063			0.935317	+	0.353812i	$0.115115\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	40.0842			1.78727			0.893633	−	0.448799i	$-0.148148\pi$
$503$	40.0842			1.78727			0.893633	+	0.448799i	$0.148148\pi$
$504$		0			0
$505$		−	39.6326i		−	1.76363i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	14.9296i		−	0.661745i	−0.943676	−	0.330872i	$-0.892657\pi$
$509$		−	14.9296i		−	0.661745i	0.943676	−	0.330872i	$-0.107343\pi$
$510$		0			0
$511$	−2.95364			−0.130661
$512$		0			0
$513$		0			0
$514$		0			0
$515$			7.64171i			0.336734i
$516$		0			0
$517$	−32.1134	−	14.1744i	−1.41235	−	0.623390i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	24.9031i		−	1.09102i	−0.838103	−	0.545512i	$-0.816335\pi$
$521$		−	24.9031i		−	1.09102i	0.838103	−	0.545512i	$-0.183665\pi$
$522$		0			0
$523$			29.0161i			1.26879i	0.773010	+	0.634394i	$0.218750\pi$
$523$			29.0161i			1.26879i	−0.773010	+	0.634394i	$0.781250\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.03327			−0.262813
$528$		0			0
$529$	−13.5745			−0.590198
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	55.2626i		−	2.39369i
$534$		0			0
$535$			60.3861i			2.61072i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.10211	+	20.6216i	−0.392056	+	0.888237i
$540$		0			0
$541$			23.4124i			1.00658i	0.864118	+	0.503289i	$0.167877\pi$
$541$			23.4124i			1.00658i	−0.864118	+	0.503289i	$0.832123\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.6745			0.542917
$546$		0			0
$547$		−	23.4098i		−	1.00093i	−0.865756	−	0.500466i	$-0.833162\pi$
$547$		−	23.4098i		−	1.00093i	0.865756	−	0.500466i	$-0.166838\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			11.0648i			0.471377i
$552$		0			0
$553$	−0.109467			−0.00465502
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.3854			−0.609528			−0.304764	−	0.952428i	$-0.598578\pi$
$557$	−14.3854			−0.609528			−0.304764	+	0.952428i	$0.598578\pi$
$558$		0			0
$559$	−76.9232			−3.25350
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.2840			−1.06559			−0.532797	−	0.846243i	$-0.678859\pi$
$563$	−25.2840			−1.06559			−0.532797	+	0.846243i	$0.678859\pi$
$564$		0			0
$565$	21.8560			0.919490
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−33.1918			−1.39147			−0.695737	−	0.718297i	$-0.744922\pi$
$569$	−33.1918			−1.39147			−0.695737	+	0.718297i	$0.744922\pi$
$570$		0			0
$571$		−	36.7701i		−	1.53878i	−0.638778	−	0.769391i	$-0.720560\pi$
$571$		−	36.7701i		−	1.53878i	0.638778	−	0.769391i	$-0.279440\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	58.8579i		−	2.45454i
$576$		0			0
$577$	31.3525			1.30522			0.652612	−	0.757692i	$-0.273673\pi$
$577$	31.3525			1.30522			0.652612	+	0.757692i	$0.273673\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	7.24441i		−	0.300549i
$582$		0			0
$583$	−15.3664	−	6.78253i	−0.636412	−	0.280904i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			23.2489i			0.959585i	0.877382	+	0.479793i	$0.159288\pi$
$587$			23.2489i			0.959585i	−0.877382	+	0.479793i	$0.840712\pi$
$588$		0			0
$589$		−	1.95628i		−	0.0806070i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−28.0473			−1.15176			−0.575881	−	0.817533i	$-0.695341\pi$
$593$	−28.0473			−1.15176			−0.575881	+	0.817533i	$0.695341\pi$
$594$		0			0
$595$	−7.43343			−0.304741
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1.51136i		−	0.0617523i	−0.999523	−	0.0308762i	$-0.990170\pi$
$599$		−	1.51136i		−	0.0617523i	0.999523	−	0.0308762i	$-0.00982975\pi$
$600$		0			0
$601$		−	21.1051i		−	0.860893i	−0.902616	−	0.430447i	$-0.858356\pi$
$601$		−	21.1051i		−	0.860893i	0.902616	−	0.430447i	$-0.141644\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−31.1942	+	28.4523i	−1.26822	+	1.15675i
$606$		0			0
$607$			2.65467i			0.107750i	0.998548	+	0.0538748i	$0.0171572\pi$
$607$			2.65467i			0.107750i	−0.998548	+	0.0538748i	$0.982843\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	71.5701			2.89542
$612$		0			0
$613$		−	28.8206i		−	1.16405i	−0.813170	−	0.582026i	$-0.802260\pi$
$613$		−	28.8206i		−	1.16405i	0.813170	−	0.582026i	$-0.197740\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	4.64880i		−	0.187154i	−0.995612	−	0.0935768i	$-0.970170\pi$
$617$		−	4.64880i		−	0.187154i	0.995612	−	0.0935768i	$-0.0298301\pi$
$618$		0			0
$619$	8.69227			0.349372			0.174686	−	0.984624i	$-0.444109\pi$
$619$	8.69227			0.349372			0.174686	+	0.984624i	$0.444109\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.29443			0.252181
$624$		0			0
$625$	21.0560			0.842240
$626$		0			0
$627$		0			0
$628$		0			0
$629$	29.5992			1.18020
$630$		0			0
$631$	−17.2387			−0.686262			−0.343131	−	0.939288i	$-0.611487\pi$
$631$	−17.2387			−0.686262			−0.343131	+	0.939288i	$0.611487\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	60.7113			2.40925
$636$		0			0
$637$		−	45.9588i		−	1.82095i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	12.7576i		−	0.503894i	−0.967741	−	0.251947i	$-0.918929\pi$
$641$		−	12.7576i		−	0.503894i	0.967741	−	0.251947i	$-0.0810709\pi$
$642$		0			0
$643$	30.6532			1.20884			0.604421	−	0.796665i	$-0.293404\pi$
$643$	30.6532			1.20884			0.604421	+	0.796665i	$0.293404\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9.84708i		−	0.387129i	−0.981088	−	0.193564i	$-0.937995\pi$
$647$		−	9.84708i		−	0.387129i	0.981088	−	0.193564i	$-0.0620048\pi$
$648$		0			0
$649$	22.6861	+	10.0133i	0.890506	+	0.393057i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			27.9502i			1.09377i	0.837206	+	0.546887i	$0.184187\pi$
$653$			27.9502i			1.09377i	−0.837206	+	0.546887i	$0.815813\pi$
$654$		0			0
$655$			55.6735i			2.17534i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.81712			0.265557			0.132779	−	0.991146i	$-0.457610\pi$
$659$	6.81712			0.265557			0.132779	+	0.991146i	$0.457610\pi$
$660$		0			0
$661$	0.290646			0.0113048			0.00565242	−	0.999984i	$-0.498201\pi$
$661$	0.290646			0.0113048			0.00565242	+	0.999984i	$0.498201\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	2.41028i		−	0.0934666i
$666$		0			0
$667$		−	48.0836i		−	1.86180i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−14.7334	−	6.50312i	−0.568777	−	0.251050i
$672$		0			0
$673$		−	40.7231i		−	1.56976i	−0.619649	−	0.784879i	$-0.712725\pi$
$673$		−	40.7231i		−	1.56976i	0.619649	−	0.784879i	$-0.287275\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−30.1353			−1.15819			−0.579097	−	0.815259i	$-0.696595\pi$
$677$	−30.1353			−1.15819			−0.579097	+	0.815259i	$0.696595\pi$
$678$		0			0
$679$			2.94772i			0.113123i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			5.68795i			0.217643i	0.994061	+	0.108822i	$0.0347078\pi$
$683$			5.68795i			0.217643i	−0.994061	+	0.108822i	$0.965292\pi$
$684$		0			0
$685$	15.9793			0.610539
$686$		0			0
$687$		0			0
$688$		0			0
$689$	34.2467			1.30469
$690$		0			0
$691$	−5.87194			−0.223379			−0.111690	−	0.993743i	$-0.535626\pi$
$691$	−5.87194			−0.223379			−0.111690	+	0.993743i	$0.535626\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.138886			0.00526823
$696$		0			0
$697$	−35.0751			−1.32856
$698$		0			0
$699$		0			0
$700$		0			0
$701$	37.0799			1.40049			0.700245	−	0.713903i	$-0.253074\pi$
$701$	37.0799			1.40049			0.700245	+	0.713903i	$0.253074\pi$
$702$		0			0
$703$			9.59749i			0.361976i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			4.65922i			0.175228i
$708$		0			0
$709$	−16.2107			−0.608805			−0.304402	−	0.952544i	$-0.598457\pi$
$709$	−16.2107			−0.608805			−0.304402	+	0.952544i	$0.598457\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			8.50127i			0.318375i
$714$		0			0
$715$	34.7607	−	78.7535i	1.29998	−	2.94522i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			3.88919i			0.145042i	0.997367	+	0.0725211i	$0.0231045\pi$
$719$			3.88919i			0.145042i	−0.997367	+	0.0725211i	$0.976896\pi$
$720$		0			0
$721$		−	0.898362i		−	0.0334567i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	77.3789			2.87378
$726$		0			0
$727$	−43.5652			−1.61574			−0.807872	−	0.589358i	$-0.799381\pi$
$727$	−43.5652			−1.61574			−0.807872	+	0.589358i	$0.799381\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			48.8231i			1.80579i
$732$		0			0
$733$			25.9513i			0.958534i	0.877669	+	0.479267i	$0.159097\pi$
$733$			25.9513i			0.958534i	−0.877669	+	0.479267i	$0.840903\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.1088	−	22.9024i	0.372363	−	0.843621i
$738$		0			0
$739$		−	10.0843i		−	0.370957i	−0.982648	−	0.185478i	$-0.940617\pi$
$739$		−	10.0843i		−	0.370957i	0.982648	−	0.185478i	$-0.0593835\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	14.6452			0.537280			0.268640	−	0.963241i	$-0.413426\pi$
$743$	14.6452			0.537280			0.268640	+	0.963241i	$0.413426\pi$
$744$		0			0
$745$			70.5077i			2.58320i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	7.09901i		−	0.259392i
$750$		0			0
$751$	−41.9234			−1.52981			−0.764904	−	0.644145i	$-0.777213\pi$
$751$	−41.9234			−1.52981			−0.764904	+	0.644145i	$0.777213\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−22.6551			−0.824505
$756$		0			0
$757$	−7.19615			−0.261549			−0.130774	−	0.991412i	$-0.541746\pi$
$757$	−7.19615			−0.261549			−0.130774	+	0.991412i	$0.541746\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−47.7652			−1.73149			−0.865743	−	0.500489i	$-0.833154\pi$
$761$	−47.7652			−1.73149			−0.865743	+	0.500489i	$0.833154\pi$
$762$		0			0
$763$	−1.49002			−0.0539424
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−50.5597			−1.82561
$768$		0			0
$769$		−	30.0122i		−	1.08227i	−0.840937	−	0.541133i	$-0.817995\pi$
$769$		−	30.0122i		−	1.08227i	0.840937	−	0.541133i	$-0.182005\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			27.1139i			0.975220i	0.873062	+	0.487610i	$0.162131\pi$
$773$			27.1139i			0.975220i	−0.873062	+	0.487610i	$0.837869\pi$
$774$		0			0
$775$	−13.6807			−0.491426
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	11.3731i		−	0.407482i
$780$		0			0
$781$	−9.22100	−	4.07002i	−0.329953	−	0.145637i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			9.55698i			0.341103i
$786$		0			0
$787$		−	6.96278i		−	0.248196i	−0.992270	−	0.124098i	$-0.960396\pi$
$787$		−	6.96278i		−	0.248196i	0.992270	−	0.124098i	$-0.0396038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.56940			−0.0913574
$792$		0			0
$793$	32.8359			1.16604
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	3.29030i		−	0.116548i	−0.998301	−	0.0582742i	$-0.981440\pi$
$797$		−	3.29030i		−	0.116548i	0.998301	−	0.0582742i	$-0.0185598\pi$
$798$		0			0
$799$		−	45.4255i		−	1.60704i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.8612	−	8.76647i	−0.700888	−	0.309362i
$804$		0			0
$805$			10.4742i			0.369167i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	20.4738			0.719822			0.359911	−	0.932987i	$-0.382807\pi$
$809$	20.4738			0.719822			0.359911	+	0.932987i	$0.382807\pi$
$810$		0			0
$811$			0.495125i			0.0173862i	0.999962	+	0.00869309i	$0.00276713\pi$
$811$			0.495125i			0.0173862i	−0.999962	+	0.00869309i	$0.997233\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	67.4738i		−	2.36350i
$816$		0			0
$817$	−15.8308			−0.553850
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.0895			−0.456827			−0.228413	−	0.973564i	$-0.573354\pi$
$821$	−13.0895			−0.456827			−0.228413	+	0.973564i	$0.573354\pi$
$822$		0			0
$823$	0.859785			0.0299702			0.0149851	−	0.999888i	$-0.495230\pi$
$823$	0.859785			0.0299702			0.0149851	+	0.999888i	$0.495230\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−44.0067			−1.53026			−0.765131	−	0.643875i	$-0.777326\pi$
$827$	−44.0067			−1.53026			−0.765131	+	0.643875i	$0.777326\pi$
$828$		0			0
$829$	19.1739			0.665937			0.332968	−	0.942938i	$-0.391950\pi$
$829$	19.1739			0.665937			0.332968	+	0.942938i	$0.391950\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−29.1700			−1.01068
$834$		0			0
$835$		−	14.8114i		−	0.512571i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			21.9087i			0.756373i	0.925729	+	0.378186i	$0.123452\pi$
$839$			21.9087i			0.756373i	−0.925729	+	0.378186i	$0.876548\pi$
$840$		0			0
$841$	34.2143			1.17980
$842$		0			0
$843$		0			0
$844$		0			0
$845$			125.618i			4.32139i
$846$		0			0
$847$	3.66720	−	3.34486i	0.126006	−	0.114931i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	41.7072i		−	1.42970i
$852$		0			0
$853$			4.49991i			0.154074i	0.997028	+	0.0770369i	$0.0245459\pi$
$853$			4.49991i			0.154074i	−0.997028	+	0.0770369i	$0.975454\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	23.0947			0.788901			0.394451	−	0.918917i	$-0.370935\pi$
$857$	23.0947			0.788901			0.394451	+	0.918917i	$0.370935\pi$
$858$		0			0
$859$	−10.1120			−0.345018			−0.172509	−	0.985008i	$-0.555187\pi$
$859$	−10.1120			−0.345018			−0.172509	+	0.985008i	$0.555187\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	50.1079i		−	1.70569i	−0.522161	−	0.852847i	$-0.674874\pi$
$863$		−	50.1079i		−	1.70569i	0.522161	−	0.852847i	$-0.325126\pi$
$864$		0			0
$865$		−	45.4431i		−	1.54511i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.736093	−	0.324901i	−0.0249703	−	0.0110215i
$870$		0			0
$871$			51.0419i			1.72949i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.19600			−0.277076
$876$		0			0
$877$			31.5725i			1.06613i	0.846075	+	0.533064i	$0.178960\pi$
$877$			31.5725i			1.06613i	−0.846075	+	0.533064i	$0.821040\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	28.6114i		−	0.963944i	−0.876187	−	0.481972i	$-0.839921\pi$
$881$		−	28.6114i		−	0.963944i	0.876187	−	0.481972i	$-0.160079\pi$
$882$		0			0
$883$	−53.3165			−1.79424			−0.897122	−	0.441783i	$-0.854346\pi$
$883$	−53.3165			−1.79424			−0.897122	+	0.441783i	$0.854346\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.3299			−0.816919			−0.408459	−	0.912777i	$-0.633934\pi$
$887$	−24.3299			−0.816919			−0.408459	+	0.912777i	$0.633934\pi$
$888$		0			0
$889$	−7.13724			−0.239375
$890$		0			0
$891$		0			0
$892$		0			0
$893$	14.7292			0.492892
$894$		0			0
$895$	−63.9456			−2.13747
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.1764			−0.372754
$900$		0			0
$901$		−	21.7363i		−	0.724142i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			36.0649i			1.19884i
$906$		0			0
$907$	−42.3878			−1.40746			−0.703732	−	0.710465i	$-0.748485\pi$
$907$	−42.3878			−1.40746			−0.703732	+	0.710465i	$0.748485\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	9.47079i		−	0.313781i	−0.987616	−	0.156891i	$-0.949853\pi$
$911$		−	9.47079i		−	0.313781i	0.987616	−	0.156891i	$-0.0501470\pi$
$912$		0			0
$913$	21.5016	−	48.7138i	0.711599	−	1.61219i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	6.54499i		−	0.216135i
$918$		0			0
$919$			42.4635i			1.40074i	0.713778	+	0.700372i	$0.246982\pi$
$919$			42.4635i			1.40074i	−0.713778	+	0.700372i	$0.753018\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	20.5506			0.676430
$924$		0			0
$925$	67.1176			2.20681
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	50.7844i		−	1.66618i	−0.553136	−	0.833091i	$-0.686569\pi$
$929$		−	50.7844i		−	1.66618i	0.553136	−	0.833091i	$-0.313431\pi$
$930$		0			0
$931$		−	9.45834i		−	0.309985i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−49.9848	−	22.0626i	−1.63468	−	0.721524i
$936$		0			0
$937$			11.2141i			0.366349i	0.983080	+	0.183174i	$0.0586373\pi$
$937$			11.2141i			0.366349i	−0.983080	+	0.183174i	$0.941363\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.53274			0.278159			0.139080	−	0.990281i	$-0.455586\pi$
$941$	8.53274			0.278159			0.139080	+	0.990281i	$0.455586\pi$
$942$		0			0
$943$			49.4232i			1.60944i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			34.7386i			1.12885i	0.825483	+	0.564427i	$0.190903\pi$
$947$			34.7386i			1.12885i	−0.825483	+	0.564427i	$0.809097\pi$
$948$		0			0
$949$	44.2641			1.43687
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.2936			0.365834			0.182917	−	0.983128i	$-0.441446\pi$
$953$	11.2936			0.365834			0.182917	+	0.983128i	$0.441446\pi$
$954$		0			0
$955$	52.3862			1.69518
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.87854			−0.0606611
$960$		0			0
$961$	−29.0240			−0.936258
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−90.4352			−2.91121
$966$		0			0
$967$			41.5835i			1.33724i	0.743606	+	0.668618i	$0.233114\pi$
$967$			41.5835i			1.33724i	−0.743606	+	0.668618i	$0.766886\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	26.4402i		−	0.848506i	−0.905544	−	0.424253i	$-0.860537\pi$
$971$		−	26.4402i		−	0.848506i	0.905544	−	0.424253i	$-0.139463\pi$
$972$		0			0
$973$	−0.0163274			−0.000523433
$974$		0			0
$975$		0			0
$976$		0			0
$977$			32.5068i			1.03999i	0.854171	+	0.519993i	$0.174065\pi$
$977$			32.5068i			1.03999i	−0.854171	+	0.519993i	$0.825935\pi$
$978$		0			0
$979$	42.3258	+	18.6820i	1.35274	+	0.597080i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	45.8661i		−	1.46290i	−0.681894	−	0.731451i	$-0.738844\pi$
$983$		−	45.8661i		−	1.46290i	0.681894	−	0.731451i	$-0.261156\pi$
$984$		0			0
$985$			9.38608i			0.299065i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	68.7949			2.18755
$990$		0			0
$991$	−17.4557			−0.554497			−0.277249	−	0.960798i	$-0.589423\pi$
$991$	−17.4557			−0.554497			−0.277249	+	0.960798i	$0.589423\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			35.8679i			1.13709i
$996$		0			0
$997$		−	13.2947i		−	0.421046i	−0.977589	−	0.210523i	$-0.932483\pi$
$997$		−	13.2947i		−	0.421046i	0.977589	−	0.210523i	$-0.0675167\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.b.a.593.1	✓	12
3.2	odd	2		2376.2.b.d.593.12	yes	12
4.3	odd	2		4752.2.b.l.593.1		12
11.10	odd	2		2376.2.b.d.593.1	yes	12
12.11	even	2		4752.2.b.i.593.12		12
33.32	even	2	inner	2376.2.b.a.593.12	yes	12
44.43	even	2		4752.2.b.i.593.1		12
132.131	odd	2		4752.2.b.l.593.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.b.a.593.1	✓	12	1.1	even	1	trivial
2376.2.b.a.593.12	yes	12	33.32	even	2	inner
2376.2.b.d.593.1	yes	12	11.10	odd	2
2376.2.b.d.593.12	yes	12	3.2	odd	2
4752.2.b.i.593.1		12	44.43	even	2
4752.2.b.i.593.12		12	12.11	even	2
4752.2.b.l.593.1		12	4.3	odd	2
4752.2.b.l.593.12		12	132.131	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2376.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.83827i q^{5} +0.451228i q^{7} +(-1.33926 + 3.03420i) q^{11} -6.76223i q^{13} -4.29198 q^{17} -1.39167i q^{19} +6.04769i q^{23} -9.73229 q^{25} -7.95074 q^{29} +1.40571 q^{31} +1.73193 q^{35} -6.89638 q^{37} +8.17224 q^{41} -11.3754i q^{43} +10.5838i q^{47} +6.79639 q^{49} +5.06440i q^{53} +(11.6461 + 5.14042i) q^{55} -7.47678i q^{59} +4.85577i q^{61} -25.9553 q^{65} -7.54808 q^{67} +3.03902i q^{71} +6.54578i q^{73} +(-1.36912 - 0.604309i) q^{77} +0.242598i q^{79} -16.0549 q^{83} +16.4738i q^{85} -13.9496i q^{89} +3.05131 q^{91} -5.34160 q^{95} +6.53266 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 5 q^{11} - 10 q^{17} - 10 q^{25} + 2 q^{29} + 6 q^{31} - 10 q^{35} - 6 q^{37} + 2 q^{41} - 16 q^{49} - q^{55} - 32 q^{65} + 4 q^{67} + 7 q^{77} - 10 q^{83} + 24 q^{91} - 44 q^{95} + 8 q^{97}+O(q^{100}) \) 12 * q - 5 * q^11 - 10 * q^17 - 10 * q^25 + 2 * q^29 + 6 * q^31 - 10 * q^35 - 6 * q^37 + 2 * q^41 - 16 * q^49 - q^55 - 32 * q^65 + 4 * q^67 + 7 * q^77 - 10 * q^83 + 24 * q^91 - 44 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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