LMFDB - Embedded newform 444.2.e.a.73.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [444,2,Mod(73,444)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("444.73"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 444 = 2^{2} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	444.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$3.54535784974$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 4 $ x^4 + 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.4
Root			$2.28825i$ of defining polynomial
Character	$\chi$	$=$	444.73
Dual form			444.2.e.a.73.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.70246i q^{5} +3.23607 q^{7} +1.00000 q^{9} -6.47214 q^{11} -1.74806i q^{13} -3.70246i q^{15} +6.53089i q^{17} +4.57649i q^{19} -3.23607 q^{21} -7.19859i q^{23} -8.70820 q^{25} -1.00000 q^{27} +4.78282i q^{29} +6.32456i q^{31} +6.47214 q^{33} +11.9814i q^{35} +(2.23607 + 5.65685i) q^{37} +1.74806i q^{39} +2.00000 q^{41} -2.82843i q^{43} +3.70246i q^{45} +4.00000 q^{47} +3.47214 q^{49} -6.53089i q^{51} -6.94427 q^{53} -23.9628i q^{55} -4.57649i q^{57} -0.874032i q^{59} -5.65685i q^{61} +3.23607 q^{63} +6.47214 q^{65} +5.70820 q^{67} +7.19859i q^{69} +8.94427 q^{71} -1.23607 q^{73} +8.70820 q^{75} -20.9443 q^{77} -15.8902i q^{79} +1.00000 q^{81} +10.4721 q^{83} -24.1803 q^{85} -4.78282i q^{87} +7.19859i q^{89} -5.65685i q^{91} -6.32456i q^{93} -16.9443 q^{95} +5.24419i q^{97} -6.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - 8 q^{11} - 4 q^{21} - 8 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{41} + 16 q^{47} - 4 q^{49} + 8 q^{53} + 4 q^{63} + 8 q^{65} - 4 q^{67} + 4 q^{73} + 8 q^{75} - 48 q^{77} + 4 q^{81}+ \cdots - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - 8 * q^11 - 4 * q^21 - 8 * q^25 - 4 * q^27 + 8 * q^33 + 8 * q^41 + 16 * q^47 - 4 * q^49 + 8 * q^53 + 4 * q^63 + 8 * q^65 - 4 * q^67 + 4 * q^73 + 8 * q^75 - 48 * q^77 + 4 * q^81 + 24 * q^83 - 52 * q^85 - 32 * q^95 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/444\mathbb{Z}\right)^\times$.

$n$	$149$	$223$	$409$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			3.70246i			1.65579i	0.560883	+	0.827895i	$0.310462\pi$
$5$			3.70246i			1.65579i	−0.560883	+	0.827895i	$0.689538\pi$
$6$		0			0
$7$	3.23607			1.22312			0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607			1.22312			0.611559	+	0.791199i	$0.290543\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	−6.47214			−1.95142			−0.975711	−	0.219061i	$-0.929701\pi$
$11$	−6.47214			−1.95142			−0.975711	+	0.219061i	$0.929701\pi$
$12$		0			0
$13$		−	1.74806i		−	0.484826i	−0.970173	−	0.242413i	$-0.922061\pi$
$13$		−	1.74806i		−	0.484826i	0.970173	−	0.242413i	$-0.0779389\pi$
$14$		0			0
$15$		−	3.70246i		−	0.955971i
$16$		0			0
$17$			6.53089i			1.58397i	0.610539	+	0.791986i	$0.290953\pi$
$17$			6.53089i			1.58397i	−0.610539	+	0.791986i	$0.709047\pi$
$18$		0			0
$19$			4.57649i			1.04992i	0.851127	+	0.524960i	$0.175920\pi$
$19$			4.57649i			1.04992i	−0.851127	+	0.524960i	$0.824080\pi$
$20$		0			0
$21$	−3.23607			−0.706168
$22$		0			0
$23$		−	7.19859i		−	1.50101i	−0.660865	−	0.750505i	$-0.729811\pi$
$23$		−	7.19859i		−	1.50101i	0.660865	−	0.750505i	$-0.270189\pi$
$24$		0			0
$25$	−8.70820			−1.74164
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$			4.78282i			0.888148i	0.895990	+	0.444074i	$0.146467\pi$
$29$			4.78282i			0.888148i	−0.895990	+	0.444074i	$0.853533\pi$
$30$		0			0
$31$			6.32456i			1.13592i	0.823055	+	0.567962i	$0.192268\pi$
$31$			6.32456i			1.13592i	−0.823055	+	0.567962i	$0.807732\pi$
$32$		0			0
$33$	6.47214			1.12665
$34$		0			0
$35$			11.9814i			2.02523i
$36$		0			0
$37$	2.23607	+	5.65685i	0.367607	+	0.929981i
$37$	2.23607	+	5.65685i	0.367607	+	0.929981i
$38$		0			0
$39$			1.74806i			0.279914i
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$		−	2.82843i		−	0.431331i	−0.976467	−	0.215666i	$-0.930808\pi$
$43$		−	2.82843i		−	0.431331i	0.976467	−	0.215666i	$-0.0691921\pi$
$44$		0			0
$45$			3.70246i			0.551930i
$46$		0			0
$47$	4.00000			0.583460			0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000			0.583460			0.291730	+	0.956501i	$0.405769\pi$
$48$		0			0
$49$	3.47214			0.496019
$50$		0			0
$51$		−	6.53089i		−	0.914507i
$52$		0			0
$53$	−6.94427			−0.953869			−0.476935	−	0.878939i	$-0.658252\pi$
$53$	−6.94427			−0.953869			−0.476935	+	0.878939i	$0.658252\pi$
$54$		0			0
$55$		−	23.9628i		−	3.23115i
$56$		0			0
$57$		−	4.57649i		−	0.606171i
$58$		0			0
$59$		−	0.874032i		−	0.113789i	−0.998380	−	0.0568946i	$-0.981880\pi$
$59$		−	0.874032i		−	0.113789i	0.998380	−	0.0568946i	$-0.0181199\pi$
$60$		0			0
$61$		−	5.65685i		−	0.724286i	−0.932123	−	0.362143i	$-0.882045\pi$
$61$		−	5.65685i		−	0.724286i	0.932123	−	0.362143i	$-0.117955\pi$
$62$		0			0
$63$	3.23607			0.407706
$64$		0			0
$65$	6.47214			0.802770
$66$		0			0
$67$	5.70820			0.697368			0.348684	−	0.937240i	$-0.386629\pi$
$67$	5.70820			0.697368			0.348684	+	0.937240i	$0.386629\pi$
$68$		0			0
$69$			7.19859i			0.866608i
$70$		0			0
$71$	8.94427			1.06149			0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427			1.06149			0.530745	+	0.847532i	$0.321912\pi$
$72$		0			0
$73$	−1.23607			−0.144671			−0.0723354	−	0.997380i	$-0.523045\pi$
$73$	−1.23607			−0.144671			−0.0723354	+	0.997380i	$0.523045\pi$
$74$		0			0
$75$	8.70820			1.00554
$76$		0			0
$77$	−20.9443			−2.38682
$78$		0			0
$79$		−	15.8902i		−	1.78779i	−0.448279	−	0.893894i	$-0.647963\pi$
$79$		−	15.8902i		−	1.78779i	0.448279	−	0.893894i	$-0.352037\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	10.4721			1.14947			0.574733	−	0.818341i	$-0.305106\pi$
$83$	10.4721			1.14947			0.574733	+	0.818341i	$0.305106\pi$
$84$		0			0
$85$	−24.1803			−2.62273
$86$		0			0
$87$		−	4.78282i		−	0.512772i
$88$		0			0
$89$			7.19859i			0.763049i	0.924359	+	0.381524i	$0.124601\pi$
$89$			7.19859i			0.763049i	−0.924359	+	0.381524i	$0.875399\pi$
$90$		0			0
$91$		−	5.65685i		−	0.592999i
$92$		0			0
$93$		−	6.32456i		−	0.655826i
$94$		0			0
$95$	−16.9443			−1.73845
$96$		0			0
$97$			5.24419i			0.532467i	0.963909	+	0.266234i	$0.0857792\pi$
$97$			5.24419i			0.532467i	−0.963909	+	0.266234i	$0.914221\pi$
$98$		0			0
$99$	−6.47214			−0.650474
$100$		0			0
$101$	17.4164			1.73300			0.866499	−	0.499179i	$-0.166365\pi$
$101$	17.4164			1.73300			0.866499	+	0.499179i	$0.166365\pi$
$102$		0			0
$103$			8.07262i			0.795419i	0.917511	+	0.397709i	$0.130195\pi$
$103$			8.07262i			0.795419i	−0.917511	+	0.397709i	$0.869805\pi$
$104$		0			0
$105$		−	11.9814i		−	1.16927i
$106$		0			0
$107$	−6.47214			−0.625685			−0.312842	−	0.949805i	$-0.601281\pi$
$107$	−6.47214			−0.625685			−0.312842	+	0.949805i	$0.601281\pi$
$108$		0			0
$109$		−	5.24419i		−	0.502303i	−0.967948	−	0.251151i	$-0.919191\pi$
$109$		−	5.24419i		−	0.502303i	0.967948	−	0.251151i	$-0.0808092\pi$
$110$		0			0
$111$	−2.23607	−	5.65685i	−0.212238	−	0.536925i
$112$		0			0
$113$		−	11.1074i		−	1.04489i	−0.852672	−	0.522447i	$-0.825019\pi$
$113$		−	11.1074i		−	1.04489i	0.852672	−	0.522447i	$-0.174981\pi$
$114$		0			0
$115$	26.6525			2.48536
$116$		0			0
$117$		−	1.74806i		−	0.161609i
$118$		0			0
$119$			21.1344i			1.93739i
$120$		0			0
$121$	30.8885			2.80805
$122$		0			0
$123$	−2.00000			−0.180334
$124$		0			0
$125$		−	13.7295i		−	1.22800i
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$			2.82843i			0.249029i
$130$		0			0
$131$		−	15.6839i		−	1.37031i	−0.728399	−	0.685153i	$-0.759735\pi$
$131$		−	15.6839i		−	1.37031i	0.728399	−	0.685153i	$-0.240265\pi$
$132$		0			0
$133$			14.8098i			1.28418i
$134$		0			0
$135$		−	3.70246i		−	0.318657i
$136$		0			0
$137$	8.47214			0.723823			0.361912	−	0.932212i	$-0.382124\pi$
$137$	8.47214			0.723823			0.361912	+	0.932212i	$0.382124\pi$
$138$		0			0
$139$	−5.70820			−0.484164			−0.242082	−	0.970256i	$-0.577830\pi$
$139$	−5.70820			−0.484164			−0.242082	+	0.970256i	$0.577830\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$			11.3137i			0.946100i
$144$		0			0
$145$	−17.7082			−1.47059
$146$		0			0
$147$	−3.47214			−0.286377
$148$		0			0
$149$	−0.472136			−0.0386789			−0.0193394	−	0.999813i	$-0.506156\pi$
$149$	−0.472136			−0.0386789			−0.0193394	+	0.999813i	$0.506156\pi$
$150$		0			0
$151$	−3.05573			−0.248672			−0.124336	−	0.992240i	$-0.539680\pi$
$151$	−3.05573			−0.248672			−0.124336	+	0.992240i	$0.539680\pi$
$152$		0			0
$153$			6.53089i			0.527991i
$154$		0			0
$155$	−23.4164			−1.88085
$156$		0			0
$157$	1.81966			0.145225			0.0726123	−	0.997360i	$-0.476866\pi$
$157$	1.81966			0.145225			0.0726123	+	0.997360i	$0.476866\pi$
$158$		0			0
$159$	6.94427			0.550717
$160$		0			0
$161$		−	23.2951i		−	1.83591i
$162$		0			0
$163$			23.2951i			1.82461i	0.409506	+	0.912307i	$0.365701\pi$
$163$			23.2951i			1.82461i	−0.409506	+	0.912307i	$0.634299\pi$
$164$		0			0
$165$			23.9628i			1.86550i
$166$		0			0
$167$			10.0270i			0.775914i	0.921678	+	0.387957i	$0.126819\pi$
$167$			10.0270i			0.775914i	−0.921678	+	0.387957i	$0.873181\pi$
$168$		0			0
$169$	9.94427			0.764944
$170$		0			0
$171$			4.57649i			0.349973i
$172$		0			0
$173$	20.4721			1.55647			0.778234	−	0.627975i	$-0.216116\pi$
$173$	20.4721			1.55647			0.778234	+	0.627975i	$0.216116\pi$
$174$		0			0
$175$	−28.1803			−2.13023
$176$		0			0
$177$			0.874032i			0.0656963i
$178$		0			0
$179$		−	8.94665i		−	0.668704i	−0.942448	−	0.334352i	$-0.891483\pi$
$179$		−	8.94665i		−	0.668704i	0.942448	−	0.334352i	$-0.108517\pi$
$180$		0			0
$181$	2.76393			0.205441			0.102721	−	0.994710i	$-0.467245\pi$
$181$	2.76393			0.205441			0.102721	+	0.994710i	$0.467245\pi$
$182$		0			0
$183$			5.65685i			0.418167i
$184$		0			0
$185$	−20.9443	+	8.27895i	−1.53985	+	0.608681i
$186$		0			0
$187$		−	42.2688i		−	3.09100i
$188$		0			0
$189$	−3.23607			−0.235389
$190$		0			0
$191$			10.0270i			0.725529i	0.931881	+	0.362765i	$0.118167\pi$
$191$			10.0270i			0.725529i	−0.931881	+	0.362765i	$0.881833\pi$
$192$		0			0
$193$			11.3137i			0.814379i	0.913344	+	0.407189i	$0.133491\pi$
$193$			11.3137i			0.814379i	−0.913344	+	0.407189i	$0.866509\pi$
$194$		0			0
$195$	−6.47214			−0.463479
$196$		0			0
$197$	9.41641			0.670891			0.335446	−	0.942060i	$-0.391113\pi$
$197$	9.41641			0.670891			0.335446	+	0.942060i	$0.391113\pi$
$198$		0			0
$199$			2.82843i			0.200502i	0.994962	+	0.100251i	$0.0319646\pi$
$199$			2.82843i			0.200502i	−0.994962	+	0.100251i	$0.968035\pi$
$200$		0			0
$201$	−5.70820			−0.402626
$202$		0			0
$203$			15.4775i			1.08631i
$204$		0			0
$205$			7.40492i			0.517182i
$206$		0			0
$207$		−	7.19859i		−	0.500336i
$208$		0			0
$209$		−	29.6197i		−	2.04884i
$210$		0			0
$211$	−17.8885			−1.23150			−0.615749	−	0.787942i	$-0.711146\pi$
$211$	−17.8885			−1.23150			−0.615749	+	0.787942i	$0.711146\pi$
$212$		0			0
$213$	−8.94427			−0.612851
$214$		0			0
$215$	10.4721			0.714194
$216$		0			0
$217$			20.4667i			1.38937i
$218$		0			0
$219$	1.23607			0.0835257
$220$		0			0
$221$	11.4164			0.767951
$222$		0			0
$223$	−20.9443			−1.40253			−0.701266	−	0.712900i	$-0.747382\pi$
$223$	−20.9443			−1.40253			−0.701266	+	0.712900i	$0.747382\pi$
$224$		0			0
$225$	−8.70820			−0.580547
$226$		0			0
$227$		−	8.94665i		−	0.593810i	−0.954907	−	0.296905i	$-0.904046\pi$
$227$		−	8.94665i		−	0.593810i	0.954907	−	0.296905i	$-0.0959545\pi$
$228$		0			0
$229$	8.47214			0.559855			0.279927	−	0.960021i	$-0.409690\pi$
$229$	8.47214			0.559855			0.279927	+	0.960021i	$0.409690\pi$
$230$		0			0
$231$	20.9443			1.37803
$232$		0			0
$233$	−7.52786			−0.493167			−0.246583	−	0.969122i	$-0.579308\pi$
$233$	−7.52786			−0.493167			−0.246583	+	0.969122i	$0.579308\pi$
$234$		0			0
$235$			14.8098i			0.966087i
$236$		0			0
$237$			15.8902i			1.03218i
$238$		0			0
$239$			4.11512i			0.266185i	0.991104	+	0.133093i	$0.0424907\pi$
$239$			4.11512i			0.266185i	−0.991104	+	0.133093i	$0.957509\pi$
$240$		0			0
$241$		−	22.2148i		−	1.43098i	−0.698624	−	0.715489i	$-0.746204\pi$
$241$		−	22.2148i		−	1.43098i	0.698624	−	0.715489i	$-0.253796\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$			12.8554i			0.821304i
$246$		0			0
$247$	8.00000			0.509028
$248$		0			0
$249$	−10.4721			−0.663645
$250$		0			0
$251$		−	16.7642i		−	1.05815i	−0.848575	−	0.529074i	$-0.822539\pi$
$251$		−	16.7642i		−	1.05815i	0.848575	−	0.529074i	$-0.177461\pi$
$252$		0			0
$253$			46.5902i			2.92910i
$254$		0			0
$255$	24.1803			1.51423
$256$		0			0
$257$			6.78593i			0.423294i	0.977346	+	0.211647i	$0.0678828\pi$
$257$			6.78593i			0.423294i	−0.977346	+	0.211647i	$0.932117\pi$
$258$		0			0
$259$	7.23607	+	18.3060i	0.449627	+	1.13748i
$260$		0			0
$261$			4.78282i			0.296049i
$262$		0			0
$263$	0.583592			0.0359858			0.0179929	−	0.999838i	$-0.494272\pi$
$263$	0.583592			0.0359858			0.0179929	+	0.999838i	$0.494272\pi$
$264$		0			0
$265$		−	25.7109i		−	1.57941i
$266$		0			0
$267$		−	7.19859i		−	0.440546i
$268$		0			0
$269$	−3.88854			−0.237089			−0.118544	−	0.992949i	$-0.537823\pi$
$269$	−3.88854			−0.237089			−0.118544	+	0.992949i	$0.537823\pi$
$270$		0			0
$271$	−24.1803			−1.46885			−0.734426	−	0.678689i	$-0.762548\pi$
$271$	−24.1803			−1.46885			−0.734426	+	0.678689i	$0.762548\pi$
$272$		0			0
$273$			5.65685i			0.342368i
$274$		0			0
$275$	56.3607			3.39868
$276$		0			0
$277$			10.9010i			0.654980i	0.944855	+	0.327490i	$0.106203\pi$
$277$			10.9010i			0.654980i	−0.944855	+	0.327490i	$0.893797\pi$
$278$		0			0
$279$			6.32456i			0.378641i
$280$		0			0
$281$		−	11.5200i		−	0.687228i	−0.939111	−	0.343614i	$-0.888349\pi$
$281$		−	11.5200i		−	0.687228i	0.939111	−	0.343614i	$-0.111651\pi$
$282$		0			0
$283$			18.0509i			1.07302i	0.843895	+	0.536508i	$0.180257\pi$
$283$			18.0509i			1.07302i	−0.843895	+	0.536508i	$0.819743\pi$
$284$		0			0
$285$	16.9443			1.00369
$286$		0			0
$287$	6.47214			0.382038
$288$		0			0
$289$	−25.6525			−1.50897
$290$		0			0
$291$		−	5.24419i		−	0.307420i
$292$		0			0
$293$	−26.0000			−1.51894			−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000			−1.51894			−0.759468	+	0.650545i	$0.774541\pi$
$294$		0			0
$295$	3.23607			0.188411
$296$		0			0
$297$	6.47214			0.375551
$298$		0			0
$299$	−12.5836			−0.727728
$300$		0			0
$301$		−	9.15298i		−	0.527569i
$302$		0			0
$303$	−17.4164			−1.00055
$304$		0			0
$305$	20.9443			1.19927
$306$		0			0
$307$	−3.05573			−0.174400			−0.0871998	−	0.996191i	$-0.527792\pi$
$307$	−3.05573			−0.174400			−0.0871998	+	0.996191i	$0.527792\pi$
$308$		0			0
$309$		−	8.07262i		−	0.459235i
$310$		0			0
$311$			9.35931i			0.530718i	0.964150	+	0.265359i	$0.0854905\pi$
$311$			9.35931i			0.530718i	−0.964150	+	0.265359i	$0.914510\pi$
$312$		0			0
$313$		−	20.4667i		−	1.15685i	−0.815737	−	0.578423i	$-0.803668\pi$
$313$		−	20.4667i		−	1.15685i	0.815737	−	0.578423i	$-0.196332\pi$
$314$		0			0
$315$			11.9814i			0.675076i
$316$		0			0
$317$	−14.9443			−0.839354			−0.419677	−	0.907674i	$-0.637857\pi$
$317$	−14.9443			−0.839354			−0.419677	+	0.907674i	$0.637857\pi$
$318$		0			0
$319$		−	30.9551i		−	1.73315i
$320$		0			0
$321$	6.47214			0.361239
$322$		0			0
$323$	−29.8885			−1.66304
$324$		0			0
$325$			15.2225i			0.844392i
$326$		0			0
$327$			5.24419i			0.290004i
$328$		0			0
$329$	12.9443			0.713641
$330$		0			0
$331$			5.91189i			0.324947i	0.986713	+	0.162474i	$0.0519472\pi$
$331$			5.91189i			0.324947i	−0.986713	+	0.162474i	$0.948053\pi$
$332$		0			0
$333$	2.23607	+	5.65685i	0.122536	+	0.309994i
$334$		0			0
$335$			21.1344i			1.15470i
$336$		0			0
$337$	−4.47214			−0.243613			−0.121806	−	0.992554i	$-0.538869\pi$
$337$	−4.47214			−0.243613			−0.121806	+	0.992554i	$0.538869\pi$
$338$		0			0
$339$			11.1074i			0.603270i
$340$		0			0
$341$		−	40.9334i		−	2.21667i
$342$		0			0
$343$	−11.4164			−0.616428
$344$		0			0
$345$	−26.6525			−1.43492
$346$		0			0
$347$			2.20943i			0.118609i	0.998240	+	0.0593043i	$0.0188882\pi$
$347$			2.20943i			0.118609i	−0.998240	+	0.0593043i	$0.981112\pi$
$348$		0			0
$349$	30.1803			1.61552			0.807758	−	0.589514i	$-0.200681\pi$
$349$	30.1803			1.61552			0.807758	+	0.589514i	$0.200681\pi$
$350$		0			0
$351$			1.74806i			0.0933048i
$352$		0			0
$353$			2.20943i			0.117596i	0.998270	+	0.0587982i	$0.0187268\pi$
$353$			2.20943i			0.117596i	−0.998270	+	0.0587982i	$0.981273\pi$
$354$		0			0
$355$			33.1158i			1.75760i
$356$		0			0
$357$		−	21.1344i		−	1.11855i
$358$		0			0
$359$	−9.88854			−0.521897			−0.260949	−	0.965353i	$-0.584035\pi$
$359$	−9.88854			−0.521897			−0.260949	+	0.965353i	$0.584035\pi$
$360$		0			0
$361$	−1.94427			−0.102330
$362$		0			0
$363$	−30.8885			−1.62123
$364$		0			0
$365$		−	4.57649i		−	0.239544i
$366$		0			0
$367$	−12.0000			−0.626395			−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000			−0.626395			−0.313197	+	0.949688i	$0.601400\pi$
$368$		0			0
$369$	2.00000			0.104116
$370$		0			0
$371$	−22.4721			−1.16670
$372$		0			0
$373$	16.6525			0.862233			0.431116	−	0.902296i	$-0.358120\pi$
$373$	16.6525			0.862233			0.431116	+	0.902296i	$0.358120\pi$
$374$		0			0
$375$			13.7295i			0.708987i
$376$		0			0
$377$	8.36068			0.430597
$378$		0			0
$379$	36.1803			1.85846			0.929230	−	0.369503i	$-0.120472\pi$
$379$	36.1803			1.85846			0.929230	+	0.369503i	$0.120472\pi$
$380$		0			0
$381$	−12.0000			−0.614779
$382$		0			0
$383$		−	26.1723i		−	1.33734i	−0.743559	−	0.668670i	$-0.766864\pi$
$383$		−	26.1723i		−	1.33734i	0.743559	−	0.668670i	$-0.233136\pi$
$384$		0			0
$385$		−	77.5453i		−	3.95208i
$386$		0			0
$387$		−	2.82843i		−	0.143777i
$388$		0			0
$389$		−	18.5123i		−	0.938611i	−0.883036	−	0.469305i	$-0.844504\pi$
$389$		−	18.5123i		−	0.938611i	0.883036	−	0.469305i	$-0.155496\pi$
$390$		0			0
$391$	47.0132			2.37756
$392$		0			0
$393$			15.6839i			0.791147i
$394$		0			0
$395$	58.8328			2.96020
$396$		0			0
$397$	−14.7639			−0.740981			−0.370490	−	0.928836i	$-0.620810\pi$
$397$	−14.7639			−0.740981			−0.370490	+	0.928836i	$0.620810\pi$
$398$		0			0
$399$		−	14.8098i		−	0.741419i
$400$		0			0
$401$		−	29.1583i		−	1.45610i	−0.685526	−	0.728048i	$-0.740428\pi$
$401$		−	29.1583i		−	1.45610i	0.685526	−	0.728048i	$-0.259572\pi$
$402$		0			0
$403$	11.0557			0.550725
$404$		0			0
$405$			3.70246i			0.183977i
$406$		0			0
$407$	−14.4721	−	36.6119i	−0.717357	−	1.81479i
$408$		0			0
$409$			12.2364i			0.605053i	0.953141	+	0.302527i	$0.0978302\pi$
$409$			12.2364i			0.605053i	−0.953141	+	0.302527i	$0.902170\pi$
$410$		0			0
$411$	−8.47214			−0.417900
$412$		0			0
$413$		−	2.82843i		−	0.139178i
$414$		0			0
$415$			38.7727i			1.90327i
$416$		0			0
$417$	5.70820			0.279532
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$			14.3972i			0.701675i	0.936436	+	0.350838i	$0.114103\pi$
$421$			14.3972i			0.701675i	−0.936436	+	0.350838i	$0.885897\pi$
$422$		0			0
$423$	4.00000			0.194487
$424$		0			0
$425$		−	56.8723i		−	2.75871i
$426$		0			0
$427$		−	18.3060i		−	0.885888i
$428$		0			0
$429$		−	11.3137i		−	0.546231i
$430$		0			0
$431$			16.3516i			0.787627i	0.919190	+	0.393814i	$0.128844\pi$
$431$			16.3516i			0.787627i	−0.919190	+	0.393814i	$0.871156\pi$
$432$		0			0
$433$	26.1803			1.25815			0.629073	−	0.777346i	$-0.283434\pi$
$433$	26.1803			1.25815			0.629073	+	0.777346i	$0.283434\pi$
$434$		0			0
$435$	17.7082			0.849043
$436$		0			0
$437$	32.9443			1.57594
$438$		0			0
$439$		−	21.9597i		−	1.04808i	−0.851694	−	0.524040i	$-0.824424\pi$
$439$		−	21.9597i		−	1.04808i	0.851694	−	0.524040i	$-0.175576\pi$
$440$		0			0
$441$	3.47214			0.165340
$442$		0			0
$443$	−8.58359			−0.407819			−0.203909	−	0.978990i	$-0.565365\pi$
$443$	−8.58359			−0.407819			−0.203909	+	0.978990i	$0.565365\pi$
$444$		0			0
$445$	−26.6525			−1.26345
$446$		0			0
$447$	0.472136			0.0223313
$448$		0			0
$449$			11.7751i			0.555700i	0.960624	+	0.277850i	$0.0896219\pi$
$449$			11.7751i			0.555700i	−0.960624	+	0.277850i	$0.910378\pi$
$450$		0			0
$451$	−12.9443			−0.609522
$452$		0			0
$453$	3.05573			0.143571
$454$		0			0
$455$	20.9443			0.981883
$456$		0			0
$457$			12.2364i			0.572397i	0.958170	+	0.286198i	$0.0923917\pi$
$457$			12.2364i			0.572397i	−0.958170	+	0.286198i	$0.907608\pi$
$458$		0			0
$459$		−	6.53089i		−	0.304836i
$460$		0			0
$461$			21.5958i			1.00581i	0.864340	+	0.502907i	$0.167736\pi$
$461$			21.5958i			1.00581i	−0.864340	+	0.502907i	$0.832264\pi$
$462$		0			0
$463$			25.4558i			1.18303i	0.806293	+	0.591517i	$0.201471\pi$
$463$			25.4558i			1.18303i	−0.806293	+	0.591517i	$0.798529\pi$
$464$		0			0
$465$	23.4164			1.08591
$466$		0			0
$467$			13.1105i			0.606681i	0.952882	+	0.303340i	$0.0981019\pi$
$467$			13.1105i			0.606681i	−0.952882	+	0.303340i	$0.901898\pi$
$468$		0			0
$469$	18.4721			0.852964
$470$		0			0
$471$	−1.81966			−0.0838455
$472$		0			0
$473$			18.3060i			0.841709i
$474$		0			0
$475$		−	39.8530i		−	1.82858i
$476$		0			0
$477$	−6.94427			−0.317956
$478$		0			0
$479$			7.35621i			0.336114i	0.985777	+	0.168057i	$0.0537492\pi$
$479$			7.35621i			0.336114i	−0.985777	+	0.168057i	$0.946251\pi$
$480$		0			0
$481$	9.88854	−	3.90879i	0.450879	−	0.178225i
$482$		0			0
$483$			23.2951i			1.05996i
$484$		0			0
$485$	−19.4164			−0.881654
$486$		0			0
$487$		−	22.8825i		−	1.03690i	−0.855107	−	0.518452i	$-0.826509\pi$
$487$		−	22.8825i		−	1.03690i	0.855107	−	0.518452i	$-0.173491\pi$
$488$		0			0
$489$		−	23.2951i		−	1.05344i
$490$		0			0
$491$	8.58359			0.387372			0.193686	−	0.981064i	$-0.437956\pi$
$491$	8.58359			0.387372			0.193686	+	0.981064i	$0.437956\pi$
$492$		0			0
$493$	−31.2361			−1.40680
$494$		0			0
$495$		−	23.9628i		−	1.07705i
$496$		0			0
$497$	28.9443			1.29833
$498$		0			0
$499$			21.1344i			0.946105i	0.881034	+	0.473053i	$0.156848\pi$
$499$			21.1344i			0.946105i	−0.881034	+	0.473053i	$0.843152\pi$
$500$		0			0
$501$		−	10.0270i		−	0.447974i
$502$		0			0
$503$			13.9358i			0.621367i	0.950513	+	0.310683i	$0.100558\pi$
$503$			13.9358i			0.621367i	−0.950513	+	0.310683i	$0.899442\pi$
$504$		0			0
$505$			64.4835i			2.86948i
$506$		0			0
$507$	−9.94427			−0.441641
$508$		0			0
$509$	26.9443			1.19428			0.597142	−	0.802136i	$-0.296303\pi$
$509$	26.9443			1.19428			0.597142	+	0.802136i	$0.296303\pi$
$510$		0			0
$511$	−4.00000			−0.176950
$512$		0			0
$513$		−	4.57649i		−	0.202057i
$514$		0			0
$515$	−29.8885			−1.31705
$516$		0			0
$517$	−25.8885			−1.13858
$518$		0			0
$519$	−20.4721			−0.898627
$520$		0			0
$521$	−41.4164			−1.81449			−0.907243	−	0.420607i	$-0.861817\pi$
$521$	−41.4164			−1.81449			−0.907243	+	0.420607i	$0.861817\pi$
$522$		0			0
$523$		−	0.667701i		−	0.0291965i	−0.999893	−	0.0145983i	$-0.995353\pi$
$523$		−	0.667701i		−	0.0291965i	0.999893	−	0.0145983i	$-0.00464694\pi$
$524$		0			0
$525$	28.1803			1.22989
$526$		0			0
$527$	−41.3050			−1.79927
$528$		0			0
$529$	−28.8197			−1.25303
$530$		0			0
$531$		−	0.874032i		−	0.0379298i
$532$		0			0
$533$		−	3.49613i		−	0.151434i
$534$		0			0
$535$		−	23.9628i		−	1.03600i
$536$		0			0
$537$			8.94665i			0.386077i
$538$		0			0
$539$	−22.4721			−0.967943
$540$		0			0
$541$		−	20.0540i		−	0.862190i	−0.902307	−	0.431095i	$-0.858127\pi$
$541$		−	20.0540i		−	0.862190i	0.902307	−	0.431095i	$-0.141873\pi$
$542$		0			0
$543$	−2.76393			−0.118612
$544$		0			0
$545$	19.4164			0.831708
$546$		0			0
$547$		−	18.9737i		−	0.811255i	−0.914038	−	0.405628i	$-0.867053\pi$
$547$		−	18.9737i		−	0.811255i	0.914038	−	0.405628i	$-0.132947\pi$
$548$		0			0
$549$		−	5.65685i		−	0.241429i
$550$		0			0
$551$	−21.8885			−0.932483
$552$		0			0
$553$		−	51.4218i		−	2.18668i
$554$		0			0
$555$	20.9443	−	8.27895i	0.889035	−	0.351422i
$556$		0			0
$557$			29.1583i			1.23548i	0.786384	+	0.617738i	$0.211951\pi$
$557$			29.1583i			1.23548i	−0.786384	+	0.617738i	$0.788049\pi$
$558$		0			0
$559$	−4.94427			−0.209120
$560$		0			0
$561$			42.2688i			1.78459i
$562$		0			0
$563$		−	14.1908i		−	0.598073i	−0.954242	−	0.299036i	$-0.903335\pi$
$563$		−	14.1908i		−	0.598073i	0.954242	−	0.299036i	$-0.0966652\pi$
$564$		0			0
$565$	41.1246			1.73013
$566$		0			0
$567$	3.23607			0.135902
$568$		0			0
$569$			4.52778i			0.189815i	0.995486	+	0.0949073i	$0.0302555\pi$
$569$			4.52778i			0.189815i	−0.995486	+	0.0949073i	$0.969745\pi$
$570$		0			0
$571$	10.2918			0.430698			0.215349	−	0.976537i	$-0.430911\pi$
$571$	10.2918			0.430698			0.215349	+	0.976537i	$0.430911\pi$
$572$		0			0
$573$		−	10.0270i		−	0.418885i
$574$		0			0
$575$			62.6868i			2.61422i
$576$		0			0
$577$		−	34.4512i		−	1.43422i	−0.696959	−	0.717111i	$-0.745464\pi$
$577$		−	34.4512i		−	1.43422i	0.696959	−	0.717111i	$-0.254536\pi$
$578$		0			0
$579$		−	11.3137i		−	0.470182i
$580$		0			0
$581$	33.8885			1.40593
$582$		0			0
$583$	44.9443			1.86140
$584$		0			0
$585$	6.47214			0.267590
$586$		0			0
$587$		−	39.3916i		−	1.62587i	−0.582356	−	0.812934i	$-0.697869\pi$
$587$		−	39.3916i		−	1.62587i	0.582356	−	0.812934i	$-0.302131\pi$
$588$		0			0
$589$	−28.9443			−1.19263
$590$		0			0
$591$	−9.41641			−0.387339
$592$		0			0
$593$	35.8885			1.47377			0.736883	−	0.676020i	$-0.236297\pi$
$593$	35.8885			1.47377			0.736883	+	0.676020i	$0.236297\pi$
$594$		0			0
$595$	−78.2492			−3.20791
$596$		0			0
$597$		−	2.82843i		−	0.115760i
$598$		0			0
$599$	−5.52786			−0.225862			−0.112931	−	0.993603i	$-0.536024\pi$
$599$	−5.52786			−0.225862			−0.112931	+	0.993603i	$0.536024\pi$
$600$		0			0
$601$	26.7639			1.09172			0.545862	−	0.837875i	$-0.316202\pi$
$601$	26.7639			1.09172			0.545862	+	0.837875i	$0.316202\pi$
$602$		0			0
$603$	5.70820			0.232456
$604$		0			0
$605$			114.364i			4.64954i
$606$		0			0
$607$			18.5610i			0.753368i	0.926342	+	0.376684i	$0.122936\pi$
$607$			18.5610i			0.753368i	−0.926342	+	0.376684i	$0.877064\pi$
$608$		0			0
$609$		−	15.4775i		−	0.627181i
$610$		0			0
$611$		−	6.99226i		−	0.282876i
$612$		0			0
$613$	−46.7214			−1.88706			−0.943529	−	0.331290i	$-0.892516\pi$
$613$	−46.7214			−1.88706			−0.943529	+	0.331290i	$0.892516\pi$
$614$		0			0
$615$		−	7.40492i		−	0.298595i
$616$		0			0
$617$	0.111456			0.00448706			0.00224353	−	0.999997i	$-0.499286\pi$
$617$	0.111456			0.00448706			0.00224353	+	0.999997i	$0.499286\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$			7.19859i			0.288869i
$622$		0			0
$623$			23.2951i			0.933299i
$624$		0			0
$625$	7.29180			0.291672
$626$		0			0
$627$			29.6197i			1.18290i
$628$		0			0
$629$	−36.9443	+	14.6035i	−1.47306	+	0.582280i
$630$		0			0
$631$			19.3863i			0.771758i	0.922549	+	0.385879i	$0.126102\pi$
$631$			19.3863i			0.771758i	−0.922549	+	0.385879i	$0.873898\pi$
$632$		0			0
$633$	17.8885			0.711006
$634$		0			0
$635$			44.4295i			1.76313i
$636$		0			0
$637$		−	6.06952i		−	0.240483i
$638$		0			0
$639$	8.94427			0.353830
$640$		0			0
$641$	−33.4164			−1.31987			−0.659934	−	0.751323i	$-0.729416\pi$
$641$	−33.4164			−1.31987			−0.659934	+	0.751323i	$0.729416\pi$
$642$		0			0
$643$			38.5176i			1.51899i	0.650515	+	0.759493i	$0.274553\pi$
$643$			38.5176i			1.51899i	−0.650515	+	0.759493i	$0.725447\pi$
$644$		0			0
$645$	−10.4721			−0.412340
$646$		0			0
$647$			32.6544i			1.28378i	0.766797	+	0.641889i	$0.221849\pi$
$647$			32.6544i			1.28378i	−0.766797	+	0.641889i	$0.778151\pi$
$648$		0			0
$649$			5.65685i			0.222051i
$650$		0			0
$651$		−	20.4667i		−	0.802153i
$652$		0			0
$653$			14.3485i			0.561499i	0.959781	+	0.280749i	$0.0905830\pi$
$653$			14.3485i			0.561499i	−0.959781	+	0.280749i	$0.909417\pi$
$654$		0			0
$655$	58.0689			2.26894
$656$		0			0
$657$	−1.23607			−0.0482236
$658$		0			0
$659$	30.4721			1.18703			0.593513	−	0.804824i	$-0.297741\pi$
$659$	30.4721			1.18703			0.593513	+	0.804824i	$0.297741\pi$
$660$		0			0
$661$		−	36.6119i		−	1.42404i	−0.702160	−	0.712020i	$-0.747781\pi$
$661$		−	36.6119i		−	1.42404i	0.702160	−	0.712020i	$-0.252219\pi$
$662$		0			0
$663$	−11.4164			−0.443377
$664$		0			0
$665$	−54.8328			−2.12633
$666$		0			0
$667$	34.4296			1.33312
$668$		0			0
$669$	20.9443			0.809752
$670$		0			0
$671$			36.6119i			1.41339i
$672$		0			0
$673$	15.7082			0.605507			0.302753	−	0.953069i	$-0.402094\pi$
$673$	15.7082			0.605507			0.302753	+	0.953069i	$0.402094\pi$
$674$		0			0
$675$	8.70820			0.335179
$676$		0			0
$677$	−14.9443			−0.574355			−0.287178	−	0.957877i	$-0.592717\pi$
$677$	−14.9443			−0.574355			−0.287178	+	0.957877i	$0.592717\pi$
$678$		0			0
$679$			16.9706i			0.651270i
$680$		0			0
$681$			8.94665i			0.342836i
$682$		0			0
$683$		−	26.9976i		−	1.03303i	−0.856277	−	0.516517i	$-0.827228\pi$
$683$		−	26.9976i		−	1.03303i	0.856277	−	0.516517i	$-0.172772\pi$
$684$		0			0
$685$			31.3677i			1.19850i
$686$		0			0
$687$	−8.47214			−0.323232
$688$		0			0
$689$			12.1390i			0.462460i
$690$		0			0
$691$	10.6525			0.405239			0.202620	−	0.979258i	$-0.435054\pi$
$691$	10.6525			0.405239			0.202620	+	0.979258i	$0.435054\pi$
$692$		0			0
$693$	−20.9443			−0.795607
$694$		0			0
$695$		−	21.1344i		−	0.801673i
$696$		0			0
$697$			13.0618i			0.494750i
$698$		0			0
$699$	7.52786			0.284730
$700$		0			0
$701$		−	38.8214i		−	1.46626i	−0.680087	−	0.733131i	$-0.738058\pi$
$701$		−	38.8214i		−	1.46626i	0.680087	−	0.733131i	$-0.261942\pi$
$702$		0			0
$703$	−25.8885	+	10.2333i	−0.976405	+	0.385958i
$704$		0			0
$705$		−	14.8098i		−	0.557771i
$706$		0			0
$707$	56.3607			2.11966
$708$		0			0
$709$		−	23.0401i		−	0.865288i	−0.901565	−	0.432644i	$-0.857581\pi$
$709$		−	23.0401i		−	0.865288i	0.901565	−	0.432644i	$-0.142419\pi$
$710$		0			0
$711$		−	15.8902i		−	0.595929i
$712$		0			0
$713$	45.5279			1.70503
$714$		0			0
$715$	−41.8885			−1.56654
$716$		0			0
$717$		−	4.11512i		−	0.153682i
$718$		0			0
$719$	12.5836			0.469289			0.234644	−	0.972081i	$-0.424607\pi$
$719$	12.5836			0.469289			0.234644	+	0.972081i	$0.424607\pi$
$720$		0			0
$721$			26.1235i			0.972892i
$722$		0			0
$723$			22.2148i			0.826176i
$724$		0			0
$725$		−	41.6498i		−	1.54683i
$726$		0			0
$727$			7.65996i			0.284092i	0.989860	+	0.142046i	$0.0453681\pi$
$727$			7.65996i			0.284092i	−0.989860	+	0.142046i	$0.954632\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	18.4721			0.683217
$732$		0			0
$733$	14.9443			0.551979			0.275990	−	0.961161i	$-0.410994\pi$
$733$	14.9443			0.551979			0.275990	+	0.961161i	$0.410994\pi$
$734$		0			0
$735$		−	12.8554i		−	0.474180i
$736$		0			0
$737$	−36.9443			−1.36086
$738$		0			0
$739$	16.9443			0.623305			0.311653	−	0.950196i	$-0.399118\pi$
$739$	16.9443			0.623305			0.311653	+	0.950196i	$0.399118\pi$
$740$		0			0
$741$	−8.00000			−0.293887
$742$		0			0
$743$	−23.0557			−0.845833			−0.422916	−	0.906169i	$-0.638994\pi$
$743$	−23.0557			−0.845833			−0.422916	+	0.906169i	$0.638994\pi$
$744$		0			0
$745$		−	1.74806i		−	0.0640441i
$746$		0			0
$747$	10.4721			0.383155
$748$		0			0
$749$	−20.9443			−0.765287
$750$		0			0
$751$	9.70820			0.354257			0.177129	−	0.984188i	$-0.443319\pi$
$751$	9.70820			0.354257			0.177129	+	0.984188i	$0.443319\pi$
$752$		0			0
$753$			16.7642i			0.610923i
$754$		0			0
$755$		−	11.3137i		−	0.411748i
$756$		0			0
$757$			8.23024i			0.299133i	0.988752	+	0.149567i	$0.0477878\pi$
$757$			8.23024i			0.299133i	−0.988752	+	0.149567i	$0.952212\pi$
$758$		0			0
$759$		−	46.5902i		−	1.69112i
$760$		0			0
$761$	11.5279			0.417885			0.208942	−	0.977928i	$-0.432998\pi$
$761$	11.5279			0.417885			0.208942	+	0.977928i	$0.432998\pi$
$762$		0			0
$763$		−	16.9706i		−	0.614376i
$764$		0			0
$765$	−24.1803			−0.874242
$766$		0			0
$767$	−1.52786			−0.0551680
$768$		0			0
$769$			26.6336i			0.960433i	0.877150	+	0.480217i	$0.159442\pi$
$769$			26.6336i			0.960433i	−0.877150	+	0.480217i	$0.840558\pi$
$770$		0			0
$771$		−	6.78593i		−	0.244389i
$772$		0			0
$773$	−34.3607			−1.23587			−0.617934	−	0.786230i	$-0.712030\pi$
$773$	−34.3607			−1.23587			−0.617934	+	0.786230i	$0.712030\pi$
$774$		0			0
$775$		−	55.0755i		−	1.97837i
$776$		0			0
$777$	−7.23607	−	18.3060i	−0.259592	−	0.656723i
$778$		0			0
$779$			9.15298i			0.327940i
$780$		0			0
$781$	−57.8885			−2.07141
$782$		0			0
$783$		−	4.78282i		−	0.170924i
$784$		0			0
$785$			6.73722i			0.240462i
$786$		0			0
$787$	−30.8328			−1.09907			−0.549536	−	0.835470i	$-0.685195\pi$
$787$	−30.8328			−1.09907			−0.549536	+	0.835470i	$0.685195\pi$
$788$		0			0
$789$	−0.583592			−0.0207764
$790$		0			0
$791$		−	35.9442i		−	1.27803i
$792$		0			0
$793$	−9.88854			−0.351152
$794$		0			0
$795$			25.7109i			0.911871i
$796$		0			0
$797$		−	43.8105i		−	1.55185i	−0.630826	−	0.775924i	$-0.717284\pi$
$797$		−	43.8105i		−	1.55185i	0.630826	−	0.775924i	$-0.282716\pi$
$798$		0			0
$799$			26.1235i			0.924185i
$800$		0			0
$801$			7.19859i			0.254350i
$802$		0			0
$803$	8.00000			0.282314
$804$		0			0
$805$	86.2492			3.03989
$806$		0			0
$807$	3.88854			0.136883
$808$		0			0
$809$			52.0408i			1.82966i	0.403844	+	0.914828i	$0.367674\pi$
$809$			52.0408i			1.82966i	−0.403844	+	0.914828i	$0.632326\pi$
$810$		0			0
$811$	42.6525			1.49773			0.748865	−	0.662722i	$-0.230599\pi$
$811$	42.6525			1.49773			0.748865	+	0.662722i	$0.230599\pi$
$812$		0			0
$813$	24.1803			0.848042
$814$		0			0
$815$	−86.2492			−3.02118
$816$		0			0
$817$	12.9443			0.452863
$818$		0			0
$819$		−	5.65685i		−	0.197666i
$820$		0			0
$821$	−19.5279			−0.681527			−0.340764	−	0.940149i	$-0.610686\pi$
$821$	−19.5279			−0.681527			−0.340764	+	0.940149i	$0.610686\pi$
$822$		0			0
$823$	−17.8885			−0.623555			−0.311778	−	0.950155i	$-0.600924\pi$
$823$	−17.8885			−0.623555			−0.311778	+	0.950155i	$0.600924\pi$
$824$		0			0
$825$	−56.3607			−1.96223
$826$		0			0
$827$		−	19.3376i		−	0.672435i	−0.941784	−	0.336217i	$-0.890852\pi$
$827$		−	19.3376i		−	0.672435i	0.941784	−	0.336217i	$-0.109148\pi$
$828$		0			0
$829$		−	0.412662i		−	0.0143323i	−0.999974	−	0.00716617i	$-0.997719\pi$
$829$		−	0.412662i		−	0.0143323i	0.999974	−	0.00716617i	$-0.00228108\pi$
$830$		0			0
$831$		−	10.9010i		−	0.378153i
$832$		0			0
$833$			22.6761i			0.785681i
$834$		0			0
$835$	−37.1246			−1.28475
$836$		0			0
$837$		−	6.32456i		−	0.218609i
$838$		0			0
$839$	−33.3050			−1.14981			−0.574907	−	0.818219i	$-0.694962\pi$
$839$	−33.3050			−1.14981			−0.574907	+	0.818219i	$0.694962\pi$
$840$		0			0
$841$	6.12461			0.211194
$842$		0			0
$843$			11.5200i			0.396771i
$844$		0			0
$845$			36.8183i			1.26659i
$846$		0			0
$847$	99.9574			3.43458
$848$		0			0
$849$		−	18.0509i		−	0.619506i
$850$		0			0
$851$	40.7214	−	16.0965i	1.39591	−	0.551782i
$852$		0			0
$853$		−	40.1081i		−	1.37327i	−0.727001	−	0.686637i	$-0.759086\pi$
$853$		−	40.1081i		−	1.37327i	0.727001	−	0.686637i	$-0.240914\pi$
$854$		0			0
$855$	−16.9443			−0.579482
$856$		0			0
$857$		−	6.37326i		−	0.217707i	−0.994058	−	0.108853i	$-0.965282\pi$
$857$		−	6.37326i		−	0.217707i	0.994058	−	0.108853i	$-0.0347179\pi$
$858$		0			0
$859$		−	28.9520i		−	0.987829i	−0.869511	−	0.493914i	$-0.835566\pi$
$859$		−	28.9520i		−	0.987829i	0.869511	−	0.493914i	$-0.164434\pi$
$860$		0			0
$861$	−6.47214			−0.220570
$862$		0			0
$863$	8.36068			0.284601			0.142300	−	0.989824i	$-0.454550\pi$
$863$	8.36068			0.284601			0.142300	+	0.989824i	$0.454550\pi$
$864$		0			0
$865$			75.7972i			2.57718i
$866$		0			0
$867$	25.6525			0.871204
$868$		0			0
$869$			102.844i			3.48873i
$870$		0			0
$871$		−	9.97831i		−	0.338102i
$872$		0			0
$873$			5.24419i			0.177489i
$874$		0			0
$875$		−	44.4295i		−	1.50199i
$876$		0			0
$877$	14.7639			0.498543			0.249271	−	0.968434i	$-0.419809\pi$
$877$	14.7639			0.498543			0.249271	+	0.968434i	$0.419809\pi$
$878$		0			0
$879$	26.0000			0.876958
$880$		0			0
$881$	43.5279			1.46649			0.733246	−	0.679964i	$-0.238005\pi$
$881$	43.5279			1.46649			0.733246	+	0.679964i	$0.238005\pi$
$882$		0			0
$883$		−	15.0649i		−	0.506973i	−0.967339	−	0.253487i	$-0.918423\pi$
$883$		−	15.0649i		−	0.506973i	0.967339	−	0.253487i	$-0.0815774\pi$
$884$		0			0
$885$	−3.23607			−0.108779
$886$		0			0
$887$	−37.5279			−1.26006			−0.630031	−	0.776570i	$-0.716958\pi$
$887$	−37.5279			−1.26006			−0.630031	+	0.776570i	$0.716958\pi$
$888$		0			0
$889$	38.8328			1.30241
$890$		0			0
$891$	−6.47214			−0.216825
$892$		0			0
$893$			18.3060i			0.612586i
$894$		0			0
$895$	33.1246			1.10723
$896$		0			0
$897$	12.5836			0.420154
$898$		0			0
$899$	−30.2492			−1.00887
$900$		0			0
$901$		−	45.3523i		−	1.51090i
$902$		0			0
$903$			9.15298i			0.304592i
$904$		0			0
$905$			10.2333i			0.340168i
$906$		0			0
$907$			11.1561i			0.370432i	0.982698	+	0.185216i	$0.0592984\pi$
$907$			11.1561i			0.370432i	−0.982698	+	0.185216i	$0.940702\pi$
$908$		0			0
$909$	17.4164			0.577666
$910$		0			0
$911$			35.7379i			1.18405i	0.805920	+	0.592025i	$0.201671\pi$
$911$			35.7379i			1.18405i	−0.805920	+	0.592025i	$0.798329\pi$
$912$		0			0
$913$	−67.7771			−2.24309
$914$		0			0
$915$	−20.9443			−0.692396
$916$		0			0
$917$		−	50.7541i		−	1.67605i
$918$		0			0
$919$		−	42.0137i		−	1.38591i	−0.720983	−	0.692953i	$-0.756309\pi$
$919$		−	42.0137i		−	1.38591i	0.720983	−	0.692953i	$-0.243691\pi$
$920$		0			0
$921$	3.05573			0.100690
$922$		0			0
$923$		−	15.6352i		−	0.514638i
$924$		0			0
$925$	−19.4721	−	49.2610i	−0.640240	−	1.61969i
$926$		0			0
$927$			8.07262i			0.265140i
$928$		0			0
$929$	19.8885			0.652522			0.326261	−	0.945280i	$-0.394211\pi$
$929$	19.8885			0.652522			0.326261	+	0.945280i	$0.394211\pi$
$930$		0			0
$931$			15.8902i			0.520780i
$932$		0			0
$933$		−	9.35931i		−	0.306410i
$934$		0			0
$935$	156.498			5.11805
$936$		0			0
$937$	10.3607			0.338469			0.169234	−	0.985576i	$-0.445871\pi$
$937$	10.3607			0.338469			0.169234	+	0.985576i	$0.445871\pi$
$938$		0			0
$939$			20.4667i			0.667905i
$940$		0			0
$941$	−26.3607			−0.859334			−0.429667	−	0.902988i	$-0.641369\pi$
$941$	−26.3607			−0.859334			−0.429667	+	0.902988i	$0.641369\pi$
$942$		0			0
$943$		−	14.3972i		−	0.468837i
$944$		0			0
$945$		−	11.9814i		−	0.389755i
$946$		0			0
$947$		−	26.3299i		−	0.855606i	−0.903872	−	0.427803i	$-0.859288\pi$
$947$		−	26.3299i		−	0.855606i	0.903872	−	0.427803i	$-0.140712\pi$
$948$		0			0
$949$			2.16073i			0.0701401i
$950$		0			0
$951$	14.9443			0.484601
$952$		0			0
$953$	16.4721			0.533585			0.266792	−	0.963754i	$-0.414036\pi$
$953$	16.4721			0.533585			0.266792	+	0.963754i	$0.414036\pi$
$954$		0			0
$955$	−37.1246			−1.20132
$956$		0			0
$957$			30.9551i			1.00064i
$958$		0			0
$959$	27.4164			0.885322
$960$		0			0
$961$	−9.00000			−0.290323
$962$		0			0
$963$	−6.47214			−0.208562
$964$		0			0
$965$	−41.8885			−1.34844
$966$		0			0
$967$			0.667701i			0.0214718i	0.999942	+	0.0107359i	$0.00341741\pi$
$967$			0.667701i			0.0214718i	−0.999942	+	0.0107359i	$0.996583\pi$
$968$		0			0
$969$	29.8885			0.960158
$970$		0			0
$971$	22.2492			0.714012			0.357006	−	0.934102i	$-0.383798\pi$
$971$	22.2492			0.714012			0.357006	+	0.934102i	$0.383798\pi$
$972$		0			0
$973$	−18.4721			−0.592189
$974$		0			0
$975$		−	15.2225i		−	0.487510i
$976$		0			0
$977$		−	35.3252i		−	1.13015i	−0.825038	−	0.565077i	$-0.808846\pi$
$977$		−	35.3252i		−	1.13015i	0.825038	−	0.565077i	$-0.191154\pi$
$978$		0			0
$979$		−	46.5902i		−	1.48903i
$980$		0			0
$981$		−	5.24419i		−	0.167434i
$982$		0			0
$983$	31.7771			1.01353			0.506766	−	0.862084i	$-0.330841\pi$
$983$	31.7771			1.01353			0.506766	+	0.862084i	$0.330841\pi$
$984$		0			0
$985$			34.8639i			1.11086i
$986$		0			0
$987$	−12.9443			−0.412021
$988$		0			0
$989$	−20.3607			−0.647432
$990$		0			0
$991$		−	45.0972i		−	1.43256i	−0.697813	−	0.716280i	$-0.745843\pi$
$991$		−	45.0972i		−	1.43256i	0.697813	−	0.716280i	$-0.254157\pi$
$992$		0			0
$993$		−	5.91189i		−	0.187608i
$994$		0			0
$995$	−10.4721			−0.331989
$996$		0			0
$997$			8.32766i			0.263740i	0.991267	+	0.131870i	$0.0420981\pi$
$997$			8.32766i			0.263740i	−0.991267	+	0.131870i	$0.957902\pi$
$998$		0			0
$999$	−2.23607	−	5.65685i	−0.0707461	−	0.178975i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	444.2.e.a.73.4	yes	4
3.2	odd	2		1332.2.e.e.73.1		4
4.3	odd	2		1776.2.h.f.961.4		4
12.11	even	2		5328.2.h.g.2737.1		4
37.36	even	2	inner	444.2.e.a.73.1	✓	4
111.110	odd	2		1332.2.e.e.73.4		4
148.147	odd	2		1776.2.h.f.961.1		4
444.443	even	2		5328.2.h.g.2737.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
444.2.e.a.73.1	✓	4	37.36	even	2	inner
444.2.e.a.73.4	yes	4	1.1	even	1	trivial
1332.2.e.e.73.1		4	3.2	odd	2
1332.2.e.e.73.4		4	111.110	odd	2
1776.2.h.f.961.1		4	148.147	odd	2
1776.2.h.f.961.4		4	4.3	odd	2
5328.2.h.g.2737.1		4	12.11	even	2
5328.2.h.g.2737.4		4	444.443	even	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:	\( N \)	\(=\)	\( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	444.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.70246i q^{5} +3.23607 q^{7} +1.00000 q^{9} -6.47214 q^{11} -1.74806i q^{13} -3.70246i q^{15} +6.53089i q^{17} +4.57649i q^{19} -3.23607 q^{21} -7.19859i q^{23} -8.70820 q^{25} -1.00000 q^{27} +4.78282i q^{29} +6.32456i q^{31} +6.47214 q^{33} +11.9814i q^{35} +(2.23607 + 5.65685i) q^{37} +1.74806i q^{39} +2.00000 q^{41} -2.82843i q^{43} +3.70246i q^{45} +4.00000 q^{47} +3.47214 q^{49} -6.53089i q^{51} -6.94427 q^{53} -23.9628i q^{55} -4.57649i q^{57} -0.874032i q^{59} -5.65685i q^{61} +3.23607 q^{63} +6.47214 q^{65} +5.70820 q^{67} +7.19859i q^{69} +8.94427 q^{71} -1.23607 q^{73} +8.70820 q^{75} -20.9443 q^{77} -15.8902i q^{79} +1.00000 q^{81} +10.4721 q^{83} -24.1803 q^{85} -4.78282i q^{87} +7.19859i q^{89} -5.65685i q^{91} -6.32456i q^{93} -16.9443 q^{95} +5.24419i q^{97} -6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - 8 q^{11} - 4 q^{21} - 8 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{41} + 16 q^{47} - 4 q^{49} + 8 q^{53} + 4 q^{63} + 8 q^{65} - 4 q^{67} + 4 q^{73} + 8 q^{75} - 48 q^{77} + 4 q^{81}+ \cdots - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - 8 * q^11 - 4 * q^21 - 8 * q^25 - 4 * q^27 + 8 * q^33 + 8 * q^41 + 16 * q^47 - 4 * q^49 + 8 * q^53 + 4 * q^63 + 8 * q^65 - 4 * q^67 + 4 * q^73 + 8 * q^75 - 48 * q^77 + 4 * q^81 + 24 * q^83 - 52 * q^85 - 32 * q^95 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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