LMFDB - Embedded newform 2394.2.b.e.1709.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1709,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2394.1709");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.b (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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1709.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	2394.1709
Dual form			2394.2.b.e.1709.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} +1.00000 q^{7} +1.00000 q^{8} -2.82843i q^{10} +5.65685i q^{11} +4.24264i q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.65685i q^{17} +(1.00000 + 4.24264i) q^{19} -2.82843i q^{20} +5.65685i q^{22} +5.65685i q^{23} -3.00000 q^{25} +4.24264i q^{26} +1.00000 q^{28} -6.00000 q^{29} -8.48528i q^{31} +1.00000 q^{32} +5.65685i q^{34} -2.82843i q^{35} +(1.00000 + 4.24264i) q^{38} -2.82843i q^{40} -4.00000 q^{43} +5.65685i q^{44} +5.65685i q^{46} +9.89949i q^{47} +1.00000 q^{49} -3.00000 q^{50} +4.24264i q^{52} -6.00000 q^{53} +16.0000 q^{55} +1.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} -8.48528i q^{62} +1.00000 q^{64} +12.0000 q^{65} -12.7279i q^{67} +5.65685i q^{68} -2.82843i q^{70} +6.00000 q^{71} +14.0000 q^{73} +(1.00000 + 4.24264i) q^{76} +5.65685i q^{77} -12.7279i q^{79} -2.82843i q^{80} -15.5563i q^{83} +16.0000 q^{85} -4.00000 q^{86} +5.65685i q^{88} -6.00000 q^{89} +4.24264i q^{91} +5.65685i q^{92} +9.89949i q^{94} +(12.0000 - 2.82843i) q^{95} +4.24264i q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 2 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{25} + 2 q^{28} - 12 q^{29} + 2 q^{32} + 2 q^{38} - 8 q^{43} + 2 q^{49} - 6 q^{50} - 12 q^{53} + 32 q^{55} + 2 q^{56} - 12 q^{58} + 24 q^{59} + 4 q^{61} + 2 q^{64} + 24 q^{65} + 12 q^{71} + 28 q^{73} + 2 q^{76} + 32 q^{85} - 8 q^{86} - 12 q^{89} + 24 q^{95} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 2 * q^14 + 2 * q^16 + 2 * q^19 - 6 * q^25 + 2 * q^28 - 12 * q^29 + 2 * q^32 + 2 * q^38 - 8 * q^43 + 2 * q^49 - 6 * q^50 - 12 * q^53 + 32 * q^55 + 2 * q^56 - 12 * q^58 + 24 * q^59 + 4 * q^61 + 2 * q^64 + 24 * q^65 + 12 * q^71 + 28 * q^73 + 2 * q^76 + 32 * q^85 - 8 * q^86 - 12 * q^89 + 24 * q^95 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		−	2.82843i		−	1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$		−	2.82843i		−	1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$	1.00000			0.353553
$9$		0			0
$10$		−	2.82843i		−	0.894427i
$11$			5.65685i			1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$			5.65685i			1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$			4.24264i			1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$			4.24264i			1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$			5.65685i			1.37199i	0.727607	+	0.685994i	$0.240633\pi$
$17$			5.65685i			1.37199i	−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	1.00000	+	4.24264i	0.229416	+	0.973329i
$19$	1.00000	+	4.24264i	0.229416	+	0.973329i
$20$		−	2.82843i		−	0.632456i
$21$		0			0
$22$			5.65685i			1.20605i
$23$			5.65685i			1.17954i	0.807573	+	0.589768i	$0.200781\pi$
$23$			5.65685i			1.17954i	−0.807573	+	0.589768i	$0.799219\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$			4.24264i			0.832050i
$27$		0			0
$28$	1.00000			0.188982
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		−	8.48528i		−	1.52400i	−0.647576	−	0.762001i	$-0.724217\pi$
$31$		−	8.48528i		−	1.52400i	0.647576	−	0.762001i	$-0.275783\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			5.65685i			0.970143i
$35$		−	2.82843i		−	0.478091i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	1.00000	+	4.24264i	0.162221	+	0.688247i
$39$		0			0
$40$		−	2.82843i		−	0.447214i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$			5.65685i			0.852803i
$45$		0			0
$46$			5.65685i			0.834058i
$47$			9.89949i			1.44399i	0.691898	+	0.721995i	$0.256775\pi$
$47$			9.89949i			1.44399i	−0.691898	+	0.721995i	$0.743225\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$	−3.00000			−0.424264
$51$		0			0
$52$			4.24264i			0.588348i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	16.0000			2.15744
$56$	1.00000			0.133631
$57$		0			0
$58$	−6.00000			−0.787839
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		−	8.48528i		−	1.07763i
$63$		0			0
$64$	1.00000			0.125000
$65$	12.0000			1.48842
$66$		0			0
$67$		−	12.7279i		−	1.55496i	−0.628906	−	0.777482i	$-0.716497\pi$
$67$		−	12.7279i		−	1.55496i	0.628906	−	0.777482i	$-0.283503\pi$
$68$			5.65685i			0.685994i
$69$		0			0
$70$		−	2.82843i		−	0.338062i
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	14.0000			1.63858			0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000			1.63858			0.819288	+	0.573382i	$0.194369\pi$
$74$		0			0
$75$		0			0
$76$	1.00000	+	4.24264i	0.114708	+	0.486664i
$77$			5.65685i			0.644658i
$78$		0			0
$79$		−	12.7279i		−	1.43200i	−0.698099	−	0.716002i	$-0.745970\pi$
$79$		−	12.7279i		−	1.43200i	0.698099	−	0.716002i	$-0.254030\pi$
$80$		−	2.82843i		−	0.316228i
$81$		0			0
$82$		0			0
$83$		−	15.5563i		−	1.70753i	−0.520658	−	0.853766i	$-0.674313\pi$
$83$		−	15.5563i		−	1.70753i	0.520658	−	0.853766i	$-0.325687\pi$
$84$		0			0
$85$	16.0000			1.73544
$86$	−4.00000			−0.431331
$87$		0			0
$88$			5.65685i			0.603023i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$			4.24264i			0.444750i
$92$			5.65685i			0.589768i
$93$		0			0
$94$			9.89949i			1.02105i
$95$	12.0000	−	2.82843i	1.23117	−	0.290191i
$96$		0			0
$97$			4.24264i			0.430775i	0.976529	+	0.215387i	$0.0691014\pi$
$97$			4.24264i			0.430775i	−0.976529	+	0.215387i	$0.930899\pi$
$98$	1.00000			0.101015
$99$		0			0
$100$	−3.00000			−0.300000
$101$		−	11.3137i		−	1.12576i	−0.826540	−	0.562878i	$-0.809694\pi$
$101$		−	11.3137i		−	1.12576i	0.826540	−	0.562878i	$-0.190306\pi$
$102$		0			0
$103$			16.9706i			1.67216i	0.548608	+	0.836080i	$0.315158\pi$
$103$			16.9706i			1.67216i	−0.548608	+	0.836080i	$0.684842\pi$
$104$			4.24264i			0.416025i
$105$		0			0
$106$	−6.00000			−0.582772
$107$	−18.0000			−1.74013			−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000			−1.74013			−0.870063	+	0.492941i	$0.835922\pi$
$108$		0			0
$109$			8.48528i			0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$			8.48528i			0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$	16.0000			1.52554
$111$		0			0
$112$	1.00000			0.0944911
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	16.0000			1.49201
$116$	−6.00000			−0.557086
$117$		0			0
$118$	12.0000			1.10469
$119$			5.65685i			0.518563i
$120$		0			0
$121$	−21.0000			−1.90909
$122$	2.00000			0.181071
$123$		0			0
$124$		−	8.48528i		−	0.762001i
$125$		−	5.65685i		−	0.505964i
$126$		0			0
$127$		−	12.7279i		−	1.12942i	−0.825289	−	0.564710i	$-0.808988\pi$
$127$		−	12.7279i		−	1.12942i	0.825289	−	0.564710i	$-0.191012\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	12.0000			1.05247
$131$			1.41421i			0.123560i	0.998090	+	0.0617802i	$0.0196778\pi$
$131$			1.41421i			0.123560i	−0.998090	+	0.0617802i	$0.980322\pi$
$132$		0			0
$133$	1.00000	+	4.24264i	0.0867110	+	0.367884i
$134$		−	12.7279i		−	1.09952i
$135$		0			0
$136$			5.65685i			0.485071i
$137$			9.89949i			0.845771i	0.906183	+	0.422885i	$0.138983\pi$
$137$			9.89949i			0.845771i	−0.906183	+	0.422885i	$0.861017\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$		−	2.82843i		−	0.239046i
$141$		0			0
$142$	6.00000			0.503509
$143$	−24.0000			−2.00698
$144$		0			0
$145$			16.9706i			1.40933i
$146$	14.0000			1.15865
$147$		0			0
$148$		0			0
$149$		−	7.07107i		−	0.579284i	−0.957135	−	0.289642i	$-0.906464\pi$
$149$		−	7.07107i		−	0.579284i	0.957135	−	0.289642i	$-0.0935363\pi$
$150$		0			0
$151$		−	12.7279i		−	1.03578i	−0.855446	−	0.517892i	$-0.826717\pi$
$151$		−	12.7279i		−	1.03578i	0.855446	−	0.517892i	$-0.173283\pi$
$152$	1.00000	+	4.24264i	0.0811107	+	0.344124i
$153$		0			0
$154$			5.65685i			0.455842i
$155$	−24.0000			−1.92773
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		−	12.7279i		−	1.01258i
$159$		0			0
$160$		−	2.82843i		−	0.223607i
$161$			5.65685i			0.445823i
$162$		0			0
$163$	8.00000			0.626608			0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000			0.626608			0.313304	+	0.949653i	$0.398564\pi$
$164$		0			0
$165$		0			0
$166$		−	15.5563i		−	1.20741i
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$	16.0000			1.22714
$171$		0			0
$172$	−4.00000			−0.304997
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$	−3.00000			−0.226779
$176$			5.65685i			0.426401i
$177$		0			0
$178$	−6.00000			−0.449719
$179$	18.0000			1.34538			0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000			1.34538			0.672692	+	0.739923i	$0.265138\pi$
$180$		0			0
$181$		−	4.24264i		−	0.315353i	−0.987491	−	0.157676i	$-0.949600\pi$
$181$		−	4.24264i		−	0.315353i	0.987491	−	0.157676i	$-0.0504003\pi$
$182$			4.24264i			0.314485i
$183$		0			0
$184$			5.65685i			0.417029i
$185$		0			0
$186$		0			0
$187$	−32.0000			−2.34007
$188$			9.89949i			0.721995i
$189$		0			0
$190$	12.0000	−	2.82843i	0.870572	−	0.205196i
$191$			14.1421i			1.02329i	0.859197	+	0.511645i	$0.170964\pi$
$191$			14.1421i			1.02329i	−0.859197	+	0.511645i	$0.829036\pi$
$192$		0			0
$193$			16.9706i			1.22157i	0.791797	+	0.610784i	$0.209146\pi$
$193$			16.9706i			1.22157i	−0.791797	+	0.610784i	$0.790854\pi$
$194$			4.24264i			0.304604i
$195$		0			0
$196$	1.00000			0.0714286
$197$			18.3848i			1.30986i	0.755689	+	0.654931i	$0.227302\pi$
$197$			18.3848i			1.30986i	−0.755689	+	0.654931i	$0.772698\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$	−3.00000			−0.212132
$201$		0			0
$202$		−	11.3137i		−	0.796030i
$203$	−6.00000			−0.421117
$204$		0			0
$205$		0			0
$206$			16.9706i			1.18240i
$207$		0			0
$208$			4.24264i			0.294174i
$209$	−24.0000	+	5.65685i	−1.66011	+	0.391293i
$210$		0			0
$211$		−	4.24264i		−	0.292075i	−0.989279	−	0.146038i	$-0.953348\pi$
$211$		−	4.24264i		−	0.292075i	0.989279	−	0.146038i	$-0.0466521\pi$
$212$	−6.00000			−0.412082
$213$		0			0
$214$	−18.0000			−1.23045
$215$			11.3137i			0.771589i
$216$		0			0
$217$		−	8.48528i		−	0.576018i
$218$			8.48528i			0.574696i
$219$		0			0
$220$	16.0000			1.07872
$221$	−24.0000			−1.61441
$222$		0			0
$223$		−	25.4558i		−	1.70465i	−0.523013	−	0.852325i	$-0.675192\pi$
$223$		−	25.4558i		−	1.70465i	0.523013	−	0.852325i	$-0.324808\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	−6.00000			−0.399114
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$	16.0000			1.05501
$231$		0			0
$232$	−6.00000			−0.393919
$233$		−	7.07107i		−	0.463241i	−0.972806	−	0.231621i	$-0.925597\pi$
$233$		−	7.07107i		−	0.463241i	0.972806	−	0.231621i	$-0.0744028\pi$
$234$		0			0
$235$	28.0000			1.82652
$236$	12.0000			0.781133
$237$		0			0
$238$			5.65685i			0.366679i
$239$			14.1421i			0.914779i	0.889267	+	0.457389i	$0.151215\pi$
$239$			14.1421i			0.914779i	−0.889267	+	0.457389i	$0.848785\pi$
$240$		0			0
$241$		−	12.7279i		−	0.819878i	−0.912113	−	0.409939i	$-0.865550\pi$
$241$		−	12.7279i		−	0.819878i	0.912113	−	0.409939i	$-0.134450\pi$
$242$	−21.0000			−1.34993
$243$		0			0
$244$	2.00000			0.128037
$245$		−	2.82843i		−	0.180702i
$246$		0			0
$247$	−18.0000	+	4.24264i	−1.14531	+	0.269953i
$248$		−	8.48528i		−	0.538816i
$249$		0			0
$250$		−	5.65685i		−	0.357771i
$251$			1.41421i			0.0892644i	0.999003	+	0.0446322i	$0.0142116\pi$
$251$			1.41421i			0.0892644i	−0.999003	+	0.0446322i	$0.985788\pi$
$252$		0			0
$253$	−32.0000			−2.01182
$254$		−	12.7279i		−	0.798621i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$		0			0
$259$		0			0
$260$	12.0000			0.744208
$261$		0			0
$262$			1.41421i			0.0873704i
$263$		−	28.2843i		−	1.74408i	−0.489432	−	0.872041i	$-0.662796\pi$
$263$		−	28.2843i		−	1.74408i	0.489432	−	0.872041i	$-0.337204\pi$
$264$		0			0
$265$			16.9706i			1.04249i
$266$	1.00000	+	4.24264i	0.0613139	+	0.260133i
$267$		0			0
$268$		−	12.7279i		−	0.777482i
$269$	12.0000			0.731653			0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000			0.731653			0.365826	+	0.930683i	$0.380786\pi$
$270$		0			0
$271$	26.0000			1.57939			0.789694	−	0.613501i	$-0.210239\pi$
$271$	26.0000			1.57939			0.789694	+	0.613501i	$0.210239\pi$
$272$			5.65685i			0.342997i
$273$		0			0
$274$			9.89949i			0.598050i
$275$		−	16.9706i		−	1.02336i
$276$		0			0
$277$	8.00000			0.480673			0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000			0.480673			0.240337	+	0.970690i	$0.422742\pi$
$278$	2.00000			0.119952
$279$		0			0
$280$		−	2.82843i		−	0.169031i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$	6.00000			0.356034
$285$		0			0
$286$	−24.0000			−1.41915
$287$		0			0
$288$		0			0
$289$	−15.0000			−0.882353
$290$			16.9706i			0.996546i
$291$		0			0
$292$	14.0000			0.819288
$293$	24.0000			1.40209			0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000			1.40209			0.701047	+	0.713115i	$0.252716\pi$
$294$		0			0
$295$		−	33.9411i		−	1.97613i
$296$		0			0
$297$		0			0
$298$		−	7.07107i		−	0.409616i
$299$	−24.0000			−1.38796
$300$		0			0
$301$	−4.00000			−0.230556
$302$		−	12.7279i		−	0.732410i
$303$		0			0
$304$	1.00000	+	4.24264i	0.0573539	+	0.243332i
$305$		−	5.65685i		−	0.323911i
$306$		0			0
$307$			25.4558i			1.45284i	0.687250	+	0.726421i	$0.258818\pi$
$307$			25.4558i			1.45284i	−0.687250	+	0.726421i	$0.741182\pi$
$308$			5.65685i			0.322329i
$309$		0			0
$310$	−24.0000			−1.36311
$311$			1.41421i			0.0801927i	0.999196	+	0.0400963i	$0.0127665\pi$
$311$			1.41421i			0.0801927i	−0.999196	+	0.0400963i	$0.987234\pi$
$312$		0			0
$313$	14.0000			0.791327			0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000			0.791327			0.395663	+	0.918396i	$0.370515\pi$
$314$	14.0000			0.790066
$315$		0			0
$316$		−	12.7279i		−	0.716002i
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$		0			0
$319$		−	33.9411i		−	1.90034i
$320$		−	2.82843i		−	0.158114i
$321$		0			0
$322$			5.65685i			0.315244i
$323$	−24.0000	+	5.65685i	−1.33540	+	0.314756i
$324$		0			0
$325$		−	12.7279i		−	0.706018i
$326$	8.00000			0.443079
$327$		0			0
$328$		0			0
$329$			9.89949i			0.545777i
$330$		0			0
$331$			29.6985i			1.63238i	0.577786	+	0.816188i	$0.303917\pi$
$331$			29.6985i			1.63238i	−0.577786	+	0.816188i	$0.696083\pi$
$332$		−	15.5563i		−	0.853766i
$333$		0			0
$334$	12.0000			0.656611
$335$	−36.0000			−1.96689
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−5.00000			−0.271964
$339$		0			0
$340$	16.0000			0.867722
$341$	48.0000			2.59935
$342$		0			0
$343$	1.00000			0.0539949
$344$	−4.00000			−0.215666
$345$		0			0
$346$	−6.00000			−0.322562
$347$		−	2.82843i		−	0.151838i	−0.997114	−	0.0759190i	$-0.975811\pi$
$347$		−	2.82843i		−	0.151838i	0.997114	−	0.0759190i	$-0.0241890\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$	−3.00000			−0.160357
$351$		0			0
$352$			5.65685i			0.301511i
$353$			22.6274i			1.20434i	0.798369	+	0.602168i	$0.205696\pi$
$353$			22.6274i			1.20434i	−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$		−	16.9706i		−	0.900704i
$356$	−6.00000			−0.317999
$357$		0			0
$358$	18.0000			0.951330
$359$		−	28.2843i		−	1.49279i	−0.665505	−	0.746393i	$-0.731784\pi$
$359$		−	28.2843i		−	1.49279i	0.665505	−	0.746393i	$-0.268216\pi$
$360$		0			0
$361$	−17.0000	+	8.48528i	−0.894737	+	0.446594i
$362$		−	4.24264i		−	0.222988i
$363$		0			0
$364$			4.24264i			0.222375i
$365$		−	39.5980i		−	2.07265i
$366$		0			0
$367$	−10.0000			−0.521996			−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000			−0.521996			−0.260998	+	0.965339i	$0.584052\pi$
$368$			5.65685i			0.294884i
$369$		0			0
$370$		0			0
$371$	−6.00000			−0.311504
$372$		0			0
$373$		−	25.4558i		−	1.31805i	−0.752119	−	0.659027i	$-0.770968\pi$
$373$		−	25.4558i		−	1.31805i	0.752119	−	0.659027i	$-0.229032\pi$
$374$	−32.0000			−1.65468
$375$		0			0
$376$			9.89949i			0.510527i
$377$		−	25.4558i		−	1.31104i
$378$		0			0
$379$			4.24264i			0.217930i	0.994046	+	0.108965i	$0.0347536\pi$
$379$			4.24264i			0.217930i	−0.994046	+	0.108965i	$0.965246\pi$
$380$	12.0000	−	2.82843i	0.615587	−	0.145095i
$381$		0			0
$382$			14.1421i			0.723575i
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$		0			0
$385$	16.0000			0.815436
$386$			16.9706i			0.863779i
$387$		0			0
$388$			4.24264i			0.215387i
$389$			9.89949i			0.501924i	0.967997	+	0.250962i	$0.0807470\pi$
$389$			9.89949i			0.501924i	−0.967997	+	0.250962i	$0.919253\pi$
$390$		0			0
$391$	−32.0000			−1.61831
$392$	1.00000			0.0505076
$393$		0			0
$394$			18.3848i			0.926212i
$395$	−36.0000			−1.81136
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$	8.00000			0.401004
$399$		0			0
$400$	−3.00000			−0.150000
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$	36.0000			1.79329
$404$		−	11.3137i		−	0.562878i
$405$		0			0
$406$	−6.00000			−0.297775
$407$		0			0
$408$		0			0
$409$			12.7279i			0.629355i	0.949199	+	0.314678i	$0.101896\pi$
$409$			12.7279i			0.629355i	−0.949199	+	0.314678i	$0.898104\pi$
$410$		0			0
$411$		0			0
$412$			16.9706i			0.836080i
$413$	12.0000			0.590481
$414$		0			0
$415$	−44.0000			−2.15988
$416$			4.24264i			0.208013i
$417$		0			0
$418$	−24.0000	+	5.65685i	−1.17388	+	0.276686i
$419$		−	15.5563i		−	0.759977i	−0.924991	−	0.379989i	$-0.875928\pi$
$419$		−	15.5563i		−	0.759977i	0.924991	−	0.379989i	$-0.124072\pi$
$420$		0			0
$421$		−	16.9706i		−	0.827095i	−0.910483	−	0.413547i	$-0.864290\pi$
$421$		−	16.9706i		−	0.827095i	0.910483	−	0.413547i	$-0.135710\pi$
$422$		−	4.24264i		−	0.206529i
$423$		0			0
$424$	−6.00000			−0.291386
$425$		−	16.9706i		−	0.823193i
$426$		0			0
$427$	2.00000			0.0967868
$428$	−18.0000			−0.870063
$429$		0			0
$430$			11.3137i			0.545595i
$431$	6.00000			0.289010			0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000			0.289010			0.144505	+	0.989504i	$0.453841\pi$
$432$		0			0
$433$		−	4.24264i		−	0.203888i	−0.994790	−	0.101944i	$-0.967494\pi$
$433$		−	4.24264i		−	0.203888i	0.994790	−	0.101944i	$-0.0325063\pi$
$434$		−	8.48528i		−	0.407307i
$435$		0			0
$436$			8.48528i			0.406371i
$437$	−24.0000	+	5.65685i	−1.14808	+	0.270604i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$	16.0000			0.762770
$441$		0			0
$442$	−24.0000			−1.14156
$443$		−	28.2843i		−	1.34383i	−0.740630	−	0.671913i	$-0.765473\pi$
$443$		−	28.2843i		−	1.34383i	0.740630	−	0.671913i	$-0.234527\pi$
$444$		0			0
$445$			16.9706i			0.804482i
$446$		−	25.4558i		−	1.20537i
$447$		0			0
$448$	1.00000			0.0472456
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$		0			0
$452$	−6.00000			−0.282216
$453$		0			0
$454$	−12.0000			−0.563188
$455$	12.0000			0.562569
$456$		0			0
$457$	32.0000			1.49690			0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000			1.49690			0.748448	+	0.663193i	$0.230799\pi$
$458$	14.0000			0.654177
$459$		0			0
$460$	16.0000			0.746004
$461$			22.6274i			1.05386i	0.849907	+	0.526932i	$0.176658\pi$
$461$			22.6274i			1.05386i	−0.849907	+	0.526932i	$0.823342\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$	−6.00000			−0.278543
$465$		0			0
$466$		−	7.07107i		−	0.327561i
$467$			18.3848i			0.850746i	0.905018	+	0.425373i	$0.139857\pi$
$467$			18.3848i			0.850746i	−0.905018	+	0.425373i	$0.860143\pi$
$468$		0			0
$469$		−	12.7279i		−	0.587721i
$470$	28.0000			1.29154
$471$		0			0
$472$	12.0000			0.552345
$473$		−	22.6274i		−	1.04041i
$474$		0			0
$475$	−3.00000	−	12.7279i	−0.137649	−	0.583997i
$476$			5.65685i			0.259281i
$477$		0			0
$478$			14.1421i			0.646846i
$479$			18.3848i			0.840022i	0.907519	+	0.420011i	$0.137974\pi$
$479$			18.3848i			0.840022i	−0.907519	+	0.420011i	$0.862026\pi$
$480$		0			0
$481$		0			0
$482$		−	12.7279i		−	0.579741i
$483$		0			0
$484$	−21.0000			−0.954545
$485$	12.0000			0.544892
$486$		0			0
$487$			12.7279i			0.576757i	0.957516	+	0.288379i	$0.0931162\pi$
$487$			12.7279i			0.576757i	−0.957516	+	0.288379i	$0.906884\pi$
$488$	2.00000			0.0905357
$489$		0			0
$490$		−	2.82843i		−	0.127775i
$491$		−	36.7696i		−	1.65939i	−0.558219	−	0.829693i	$-0.688515\pi$
$491$		−	36.7696i		−	1.65939i	0.558219	−	0.829693i	$-0.311485\pi$
$492$		0			0
$493$		−	33.9411i		−	1.52863i
$494$	−18.0000	+	4.24264i	−0.809858	+	0.190885i
$495$		0			0
$496$		−	8.48528i		−	0.381000i
$497$	6.00000			0.269137
$498$		0			0
$499$	−16.0000			−0.716258			−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000			−0.716258			−0.358129	+	0.933672i	$0.616585\pi$
$500$		−	5.65685i		−	0.252982i
$501$		0			0
$502$			1.41421i			0.0631194i
$503$			9.89949i			0.441397i	0.975342	+	0.220698i	$0.0708336\pi$
$503$			9.89949i			0.441397i	−0.975342	+	0.220698i	$0.929166\pi$
$504$		0			0
$505$	−32.0000			−1.42398
$506$	−32.0000			−1.42257
$507$		0			0
$508$		−	12.7279i		−	0.564710i
$509$	18.0000			0.797836			0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000			0.797836			0.398918	+	0.916987i	$0.369386\pi$
$510$		0			0
$511$	14.0000			0.619324
$512$	1.00000			0.0441942
$513$		0			0
$514$	−18.0000			−0.793946
$515$	48.0000			2.11513
$516$		0			0
$517$	−56.0000			−2.46288
$518$		0			0
$519$		0			0
$520$	12.0000			0.526235
$521$	−12.0000			−0.525730			−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000			−0.525730			−0.262865	+	0.964833i	$0.584667\pi$
$522$		0			0
$523$		−	16.9706i		−	0.742071i	−0.928619	−	0.371035i	$-0.879003\pi$
$523$		−	16.9706i		−	0.742071i	0.928619	−	0.371035i	$-0.120997\pi$
$524$			1.41421i			0.0617802i
$525$		0			0
$526$		−	28.2843i		−	1.23325i
$527$	48.0000			2.09091
$528$		0			0
$529$	−9.00000			−0.391304
$530$			16.9706i			0.737154i
$531$		0			0
$532$	1.00000	+	4.24264i	0.0433555	+	0.183942i
$533$		0			0
$534$		0			0
$535$			50.9117i			2.20110i
$536$		−	12.7279i		−	0.549762i
$537$		0			0
$538$	12.0000			0.517357
$539$			5.65685i			0.243658i
$540$		0			0
$541$	−40.0000			−1.71973			−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000			−1.71973			−0.859867	+	0.510518i	$0.829454\pi$
$542$	26.0000			1.11680
$543$		0			0
$544$			5.65685i			0.242536i
$545$	24.0000			1.02805
$546$		0			0
$547$		−	12.7279i		−	0.544207i	−0.962268	−	0.272103i	$-0.912281\pi$
$547$		−	12.7279i		−	0.544207i	0.962268	−	0.272103i	$-0.0877193\pi$
$548$			9.89949i			0.422885i
$549$		0			0
$550$		−	16.9706i		−	0.723627i
$551$	−6.00000	−	25.4558i	−0.255609	−	1.08446i
$552$		0			0
$553$		−	12.7279i		−	0.541246i
$554$	8.00000			0.339887
$555$		0			0
$556$	2.00000			0.0848189
$557$			35.3553i			1.49805i	0.662540	+	0.749027i	$0.269479\pi$
$557$			35.3553i			1.49805i	−0.662540	+	0.749027i	$0.730521\pi$
$558$		0			0
$559$		−	16.9706i		−	0.717778i
$560$		−	2.82843i		−	0.119523i
$561$		0			0
$562$	−18.0000			−0.759284
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$			16.9706i			0.713957i
$566$	−4.00000			−0.168133
$567$		0			0
$568$	6.00000			0.251754
$569$	42.0000			1.76073			0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000			1.76073			0.880366	+	0.474295i	$0.157297\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$	−24.0000			−1.00349
$573$		0			0
$574$		0			0
$575$		−	16.9706i		−	0.707721i
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$	−15.0000			−0.623918
$579$		0			0
$580$			16.9706i			0.704664i
$581$		−	15.5563i		−	0.645386i
$582$		0			0
$583$		−	33.9411i		−	1.40570i
$584$	14.0000			0.579324
$585$		0			0
$586$	24.0000			0.991431
$587$		−	15.5563i		−	0.642079i	−0.947066	−	0.321040i	$-0.895968\pi$
$587$		−	15.5563i		−	0.642079i	0.947066	−	0.321040i	$-0.104032\pi$
$588$		0			0
$589$	36.0000	−	8.48528i	1.48335	−	0.349630i
$590$		−	33.9411i		−	1.39733i
$591$		0			0
$592$		0			0
$593$			39.5980i			1.62609i	0.582198	+	0.813047i	$0.302193\pi$
$593$			39.5980i			1.62609i	−0.582198	+	0.813047i	$0.697807\pi$
$594$		0			0
$595$	16.0000			0.655936
$596$		−	7.07107i		−	0.289642i
$597$		0			0
$598$	−24.0000			−0.981433
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		−	4.24264i		−	0.173061i	−0.996249	−	0.0865305i	$-0.972422\pi$
$601$		−	4.24264i		−	0.173061i	0.996249	−	0.0865305i	$-0.0275780\pi$
$602$	−4.00000			−0.163028
$603$		0			0
$604$		−	12.7279i		−	0.517892i
$605$			59.3970i			2.41483i
$606$		0			0
$607$			33.9411i			1.37763i	0.724938	+	0.688814i	$0.241868\pi$
$607$			33.9411i			1.37763i	−0.724938	+	0.688814i	$0.758132\pi$
$608$	1.00000	+	4.24264i	0.0405554	+	0.172062i
$609$		0			0
$610$		−	5.65685i		−	0.229039i
$611$	−42.0000			−1.69914
$612$		0			0
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$			25.4558i			1.02731i
$615$		0			0
$616$			5.65685i			0.227921i
$617$		−	32.5269i		−	1.30948i	−0.755852	−	0.654742i	$-0.772777\pi$
$617$		−	32.5269i		−	1.30948i	0.755852	−	0.654742i	$-0.227223\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−24.0000			−0.963863
$621$		0			0
$622$			1.41421i			0.0567048i
$623$	−6.00000			−0.240385
$624$		0			0
$625$	−31.0000			−1.24000
$626$	14.0000			0.559553
$627$		0			0
$628$	14.0000			0.558661
$629$		0			0
$630$		0			0
$631$	−28.0000			−1.11466			−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000			−1.11466			−0.557331	+	0.830290i	$0.688175\pi$
$632$		−	12.7279i		−	0.506290i
$633$		0			0
$634$	18.0000			0.714871
$635$	−36.0000			−1.42862
$636$		0			0
$637$			4.24264i			0.168100i
$638$		−	33.9411i		−	1.34374i
$639$		0			0
$640$		−	2.82843i		−	0.111803i
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$	−22.0000			−0.867595			−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000			−0.867595			−0.433798	+	0.901010i	$0.642827\pi$
$644$			5.65685i			0.222911i
$645$		0			0
$646$	−24.0000	+	5.65685i	−0.944267	+	0.222566i
$647$		−	15.5563i		−	0.611583i	−0.952098	−	0.305792i	$-0.901079\pi$
$647$		−	15.5563i		−	0.611583i	0.952098	−	0.305792i	$-0.0989211\pi$
$648$		0			0
$649$			67.8823i			2.66461i
$650$		−	12.7279i		−	0.499230i
$651$		0			0
$652$	8.00000			0.313304
$653$		−	24.0416i		−	0.940822i	−0.882448	−	0.470411i	$-0.844106\pi$
$653$		−	24.0416i		−	0.940822i	0.882448	−	0.470411i	$-0.155894\pi$
$654$		0			0
$655$	4.00000			0.156293
$656$		0			0
$657$		0			0
$658$			9.89949i			0.385922i
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$		−	38.1838i		−	1.48518i	−0.669748	−	0.742588i	$-0.733598\pi$
$661$		−	38.1838i		−	1.48518i	0.669748	−	0.742588i	$-0.266402\pi$
$662$			29.6985i			1.15426i
$663$		0			0
$664$		−	15.5563i		−	0.603703i
$665$	12.0000	−	2.82843i	0.465340	−	0.109682i
$666$		0			0
$667$		−	33.9411i		−	1.31421i
$668$	12.0000			0.464294
$669$		0			0
$670$	−36.0000			−1.39080
$671$			11.3137i			0.436761i
$672$		0			0
$673$			42.4264i			1.63542i	0.575632	+	0.817709i	$0.304756\pi$
$673$			42.4264i			1.63542i	−0.575632	+	0.817709i	$0.695244\pi$
$674$		0			0
$675$		0			0
$676$	−5.00000			−0.192308
$677$	30.0000			1.15299			0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000			1.15299			0.576497	+	0.817099i	$0.304419\pi$
$678$		0			0
$679$			4.24264i			0.162818i
$680$	16.0000			0.613572
$681$		0			0
$682$	48.0000			1.83801
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	28.0000			1.06983
$686$	1.00000			0.0381802
$687$		0			0
$688$	−4.00000			−0.152499
$689$		−	25.4558i		−	0.969790i
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$		−	2.82843i		−	0.107366i
$695$		−	5.65685i		−	0.214577i
$696$		0			0
$697$		0			0
$698$	−10.0000			−0.378506
$699$		0			0
$700$	−3.00000			−0.113389
$701$		−	7.07107i		−	0.267071i	−0.991044	−	0.133535i	$-0.957367\pi$
$701$		−	7.07107i		−	0.267071i	0.991044	−	0.133535i	$-0.0426329\pi$
$702$		0			0
$703$		0			0
$704$			5.65685i			0.213201i
$705$		0			0
$706$			22.6274i			0.851594i
$707$		−	11.3137i		−	0.425496i
$708$		0			0
$709$	−4.00000			−0.150223			−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000			−0.150223			−0.0751116	+	0.997175i	$0.523931\pi$
$710$		−	16.9706i		−	0.636894i
$711$		0			0
$712$	−6.00000			−0.224860
$713$	48.0000			1.79761
$714$		0			0
$715$			67.8823i			2.53865i
$716$	18.0000			0.672692
$717$		0			0
$718$		−	28.2843i		−	1.05556i
$719$		−	7.07107i		−	0.263706i	−0.991269	−	0.131853i	$-0.957907\pi$
$719$		−	7.07107i		−	0.263706i	0.991269	−	0.131853i	$-0.0420927\pi$
$720$		0			0
$721$			16.9706i			0.632017i
$722$	−17.0000	+	8.48528i	−0.632674	+	0.315789i
$723$		0			0
$724$		−	4.24264i		−	0.157676i
$725$	18.0000			0.668503
$726$		0			0
$727$	2.00000			0.0741759			0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000			0.0741759			0.0370879	+	0.999312i	$0.488192\pi$
$728$			4.24264i			0.157243i
$729$		0			0
$730$		−	39.5980i		−	1.46559i
$731$		−	22.6274i		−	0.836905i
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$	−10.0000			−0.369107
$735$		0			0
$736$			5.65685i			0.208514i
$737$	72.0000			2.65215
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$	−6.00000			−0.220267
$743$	−48.0000			−1.76095			−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000			−1.76095			−0.880475	+	0.474093i	$0.842776\pi$
$744$		0			0
$745$	−20.0000			−0.732743
$746$		−	25.4558i		−	0.932005i
$747$		0			0
$748$	−32.0000			−1.17004
$749$	−18.0000			−0.657706
$750$		0			0
$751$			12.7279i			0.464448i	0.972662	+	0.232224i	$0.0746003\pi$
$751$			12.7279i			0.464448i	−0.972662	+	0.232224i	$0.925400\pi$
$752$			9.89949i			0.360997i
$753$		0			0
$754$		−	25.4558i		−	0.927047i
$755$	−36.0000			−1.31017
$756$		0			0
$757$	44.0000			1.59921			0.799604	−	0.600528i	$-0.205043\pi$
$757$	44.0000			1.59921			0.799604	+	0.600528i	$0.205043\pi$
$758$			4.24264i			0.154100i
$759$		0			0
$760$	12.0000	−	2.82843i	0.435286	−	0.102598i
$761$		−	2.82843i		−	0.102530i	−0.998685	−	0.0512652i	$-0.983675\pi$
$761$		−	2.82843i		−	0.102530i	0.998685	−	0.0512652i	$-0.0163254\pi$
$762$		0			0
$763$			8.48528i			0.307188i
$764$			14.1421i			0.511645i
$765$		0			0
$766$	24.0000			0.867155
$767$			50.9117i			1.83831i
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$	16.0000			0.576600
$771$		0			0
$772$			16.9706i			0.610784i
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$			25.4558i			0.914401i
$776$			4.24264i			0.152302i
$777$		0			0
$778$			9.89949i			0.354914i
$779$		0			0
$780$		0			0
$781$			33.9411i			1.21451i
$782$	−32.0000			−1.14432
$783$		0			0
$784$	1.00000			0.0357143
$785$		−	39.5980i		−	1.41331i
$786$		0			0
$787$		−	33.9411i		−	1.20987i	−0.796275	−	0.604935i	$-0.793199\pi$
$787$		−	33.9411i		−	1.20987i	0.796275	−	0.604935i	$-0.206801\pi$
$788$			18.3848i			0.654931i
$789$		0			0
$790$	−36.0000			−1.28082
$791$	−6.00000			−0.213335
$792$		0			0
$793$			8.48528i			0.301321i
$794$	2.00000			0.0709773
$795$		0			0
$796$	8.00000			0.283552
$797$	−48.0000			−1.70025			−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000			−1.70025			−0.850124	+	0.526583i	$0.823473\pi$
$798$		0			0
$799$	−56.0000			−1.98114
$800$	−3.00000			−0.106066
$801$		0			0
$802$	−6.00000			−0.211867
$803$			79.1960i			2.79476i
$804$		0			0
$805$	16.0000			0.563926
$806$	36.0000			1.26805
$807$		0			0
$808$		−	11.3137i		−	0.398015i
$809$			52.3259i			1.83968i	0.392293	+	0.919840i	$0.371682\pi$
$809$			52.3259i			1.83968i	−0.392293	+	0.919840i	$0.628318\pi$
$810$		0			0
$811$			25.4558i			0.893876i	0.894565	+	0.446938i	$0.147485\pi$
$811$			25.4558i			0.893876i	−0.894565	+	0.446938i	$0.852515\pi$
$812$	−6.00000			−0.210559
$813$		0			0
$814$		0			0
$815$		−	22.6274i		−	0.792604i
$816$		0			0
$817$	−4.00000	−	16.9706i	−0.139942	−	0.593725i
$818$			12.7279i			0.445021i
$819$		0			0
$820$		0			0
$821$			18.3848i			0.641633i	0.947141	+	0.320817i	$0.103957\pi$
$821$			18.3848i			0.641633i	−0.947141	+	0.320817i	$0.896043\pi$
$822$		0			0
$823$	20.0000			0.697156			0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000			0.697156			0.348578	+	0.937280i	$0.386665\pi$
$824$			16.9706i			0.591198i
$825$		0			0
$826$	12.0000			0.417533
$827$	−30.0000			−1.04320			−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000			−1.04320			−0.521601	+	0.853189i	$0.674665\pi$
$828$		0			0
$829$			46.6690i			1.62088i	0.585820	+	0.810442i	$0.300773\pi$
$829$			46.6690i			1.62088i	−0.585820	+	0.810442i	$0.699227\pi$
$830$	−44.0000			−1.52726
$831$		0			0
$832$			4.24264i			0.147087i
$833$			5.65685i			0.195998i
$834$		0			0
$835$		−	33.9411i		−	1.17458i
$836$	−24.0000	+	5.65685i	−0.830057	+	0.195646i
$837$		0			0
$838$		−	15.5563i		−	0.537385i
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		−	16.9706i		−	0.584844i
$843$		0			0
$844$		−	4.24264i		−	0.146038i
$845$			14.1421i			0.486504i
$846$		0			0
$847$	−21.0000			−0.721569
$848$	−6.00000			−0.206041
$849$		0			0
$850$		−	16.9706i		−	0.582086i
$851$		0			0
$852$		0			0
$853$	−34.0000			−1.16414			−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000			−1.16414			−0.582069	+	0.813139i	$0.697757\pi$
$854$	2.00000			0.0684386
$855$		0			0
$856$	−18.0000			−0.615227
$857$	6.00000			0.204956			0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000			0.204956			0.102478	+	0.994735i	$0.467323\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$			11.3137i			0.385794i
$861$		0			0
$862$	6.00000			0.204361
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$			16.9706i			0.577016i
$866$		−	4.24264i		−	0.144171i
$867$		0			0
$868$		−	8.48528i		−	0.288009i
$869$	72.0000			2.44243
$870$		0			0
$871$	54.0000			1.82972
$872$			8.48528i			0.287348i
$873$		0			0
$874$	−24.0000	+	5.65685i	−0.811812	+	0.191346i
$875$		−	5.65685i		−	0.191237i
$876$		0			0
$877$		−	16.9706i		−	0.573055i	−0.958072	−	0.286528i	$-0.907499\pi$
$877$		−	16.9706i		−	0.573055i	0.958072	−	0.286528i	$-0.0925010\pi$
$878$		0			0
$879$		0			0
$880$	16.0000			0.539360
$881$			5.65685i			0.190584i	0.995449	+	0.0952921i	$0.0303785\pi$
$881$			5.65685i			0.190584i	−0.995449	+	0.0952921i	$0.969621\pi$
$882$		0			0
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$	−24.0000			−0.807207
$885$		0			0
$886$		−	28.2843i		−	0.950229i
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$		0			0
$889$		−	12.7279i		−	0.426881i
$890$			16.9706i			0.568855i
$891$		0			0
$892$		−	25.4558i		−	0.852325i
$893$	−42.0000	+	9.89949i	−1.40548	+	0.331274i
$894$		0			0
$895$		−	50.9117i		−	1.70179i
$896$	1.00000			0.0334077
$897$		0			0
$898$	30.0000			1.00111
$899$			50.9117i			1.69800i
$900$		0			0
$901$		−	33.9411i		−	1.13074i
$902$		0			0
$903$		0			0
$904$	−6.00000			−0.199557
$905$	−12.0000			−0.398893
$906$		0			0
$907$		−	38.1838i		−	1.26787i	−0.773386	−	0.633936i	$-0.781438\pi$
$907$		−	38.1838i		−	1.26787i	0.773386	−	0.633936i	$-0.218562\pi$
$908$	−12.0000			−0.398234
$909$		0			0
$910$	12.0000			0.397796
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$	88.0000			2.91237
$914$	32.0000			1.05847
$915$		0			0
$916$	14.0000			0.462573
$917$			1.41421i			0.0467014i
$918$		0			0
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$	16.0000			0.527504
$921$		0			0
$922$			22.6274i			0.745194i
$923$			25.4558i			0.837889i
$924$		0			0
$925$		0			0
$926$	32.0000			1.05159
$927$		0			0
$928$	−6.00000			−0.196960
$929$		−	36.7696i		−	1.20637i	−0.797601	−	0.603185i	$-0.793898\pi$
$929$		−	36.7696i		−	1.20637i	0.797601	−	0.603185i	$-0.206102\pi$
$930$		0			0
$931$	1.00000	+	4.24264i	0.0327737	+	0.139047i
$932$		−	7.07107i		−	0.231621i
$933$		0			0
$934$			18.3848i			0.601568i
$935$			90.5097i			2.95998i
$936$		0			0
$937$	14.0000			0.457360			0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000			0.457360			0.228680	+	0.973502i	$0.426559\pi$
$938$		−	12.7279i		−	0.415581i
$939$		0			0
$940$	28.0000			0.913259
$941$	30.0000			0.977972			0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000			0.977972			0.488986	+	0.872292i	$0.337367\pi$
$942$		0			0
$943$		0			0
$944$	12.0000			0.390567
$945$		0			0
$946$		−	22.6274i		−	0.735681i
$947$		−	53.7401i		−	1.74632i	−0.487435	−	0.873160i	$-0.662067\pi$
$947$		−	53.7401i		−	1.74632i	0.487435	−	0.873160i	$-0.337933\pi$
$948$		0			0
$949$			59.3970i			1.92811i
$950$	−3.00000	−	12.7279i	−0.0973329	−	0.412948i
$951$		0			0
$952$			5.65685i			0.183340i
$953$	−18.0000			−0.583077			−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000			−0.583077			−0.291539	+	0.956559i	$0.594167\pi$
$954$		0			0
$955$	40.0000			1.29437
$956$			14.1421i			0.457389i
$957$		0			0
$958$			18.3848i			0.593985i
$959$			9.89949i			0.319671i
$960$		0			0
$961$	−41.0000			−1.32258
$962$		0			0
$963$		0			0
$964$		−	12.7279i		−	0.409939i
$965$	48.0000			1.54517
$966$		0			0
$967$	32.0000			1.02905			0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000			1.02905			0.514525	+	0.857475i	$0.327968\pi$
$968$	−21.0000			−0.674966
$969$		0			0
$970$	12.0000			0.385297
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		0			0
$973$	2.00000			0.0641171
$974$			12.7279i			0.407829i
$975$		0			0
$976$	2.00000			0.0640184
$977$	−6.00000			−0.191957			−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000			−0.191957			−0.0959785	+	0.995383i	$0.530598\pi$
$978$		0			0
$979$		−	33.9411i		−	1.08476i
$980$		−	2.82843i		−	0.0903508i
$981$		0			0
$982$		−	36.7696i		−	1.17336i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	52.0000			1.65686
$986$		−	33.9411i		−	1.08091i
$987$		0			0
$988$	−18.0000	+	4.24264i	−0.572656	+	0.134976i
$989$		−	22.6274i		−	0.719510i
$990$		0			0
$991$			38.1838i			1.21295i	0.795104	+	0.606474i	$0.207416\pi$
$991$			38.1838i			1.21295i	−0.795104	+	0.606474i	$0.792584\pi$
$992$		−	8.48528i		−	0.269408i
$993$		0			0
$994$	6.00000			0.190308
$995$		−	22.6274i		−	0.717337i
$996$		0			0
$997$	−10.0000			−0.316703			−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000			−0.316703			−0.158352	+	0.987383i	$0.550618\pi$
$998$	−16.0000			−0.506471
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.b.e.1709.1	yes	2
3.2	odd	2		2394.2.b.b.1709.2	yes	2
19.18	odd	2		2394.2.b.b.1709.1	✓	2
57.56	even	2	inner	2394.2.b.e.1709.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.b.b.1709.1	✓	2	19.18	odd	2
2394.2.b.b.1709.2	yes	2	3.2	odd	2
2394.2.b.e.1709.1	yes	2	1.1	even	1	trivial
2394.2.b.e.1709.2	yes	2	57.56	even	2	inner

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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} +1.00000 q^{7} +1.00000 q^{8} -2.82843i q^{10} +5.65685i q^{11} +4.24264i q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.65685i q^{17} +(1.00000 + 4.24264i) q^{19} -2.82843i q^{20} +5.65685i q^{22} +5.65685i q^{23} -3.00000 q^{25} +4.24264i q^{26} +1.00000 q^{28} -6.00000 q^{29} -8.48528i q^{31} +1.00000 q^{32} +5.65685i q^{34} -2.82843i q^{35} +(1.00000 + 4.24264i) q^{38} -2.82843i q^{40} -4.00000 q^{43} +5.65685i q^{44} +5.65685i q^{46} +9.89949i q^{47} +1.00000 q^{49} -3.00000 q^{50} +4.24264i q^{52} -6.00000 q^{53} +16.0000 q^{55} +1.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} -8.48528i q^{62} +1.00000 q^{64} +12.0000 q^{65} -12.7279i q^{67} +5.65685i q^{68} -2.82843i q^{70} +6.00000 q^{71} +14.0000 q^{73} +(1.00000 + 4.24264i) q^{76} +5.65685i q^{77} -12.7279i q^{79} -2.82843i q^{80} -15.5563i q^{83} +16.0000 q^{85} -4.00000 q^{86} +5.65685i q^{88} -6.00000 q^{89} +4.24264i q^{91} +5.65685i q^{92} +9.89949i q^{94} +(12.0000 - 2.82843i) q^{95} +4.24264i q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 2 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{25} + 2 q^{28} - 12 q^{29} + 2 q^{32} + 2 q^{38} - 8 q^{43} + 2 q^{49} - 6 q^{50} - 12 q^{53} + 32 q^{55} + 2 q^{56} - 12 q^{58} + 24 q^{59} + 4 q^{61} + 2 q^{64} + 24 q^{65} + 12 q^{71} + 28 q^{73} + 2 q^{76} + 32 q^{85} - 8 q^{86} - 12 q^{89} + 24 q^{95} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 2 * q^14 + 2 * q^16 + 2 * q^19 - 6 * q^25 + 2 * q^28 - 12 * q^29 + 2 * q^32 + 2 * q^38 - 8 * q^43 + 2 * q^49 - 6 * q^50 - 12 * q^53 + 32 * q^55 + 2 * q^56 - 12 * q^58 + 24 * q^59 + 4 * q^61 + 2 * q^64 + 24 * q^65 + 12 * q^71 + 28 * q^73 + 2 * q^76 + 32 * q^85 - 8 * q^86 - 12 * q^89 + 24 * q^95 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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