LMFDB - Embedded newform 198.2.b.a.197.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [198,2,Mod(197,198)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(198, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("198.197");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 198 = 2 \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	198.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:	$1.58103796002$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	198.197
Dual form			198.2.b.a.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} -1.00000 q^{8} +2.82843i q^{10} +(-3.00000 - 1.41421i) q^{11} -4.24264i q^{13} +1.00000 q^{16} +6.00000 q^{17} -4.24264i q^{19} -2.82843i q^{20} +(3.00000 + 1.41421i) q^{22} +1.41421i q^{23} -3.00000 q^{25} +4.24264i q^{26} +2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -10.0000 q^{37} +4.24264i q^{38} +2.82843i q^{40} +6.00000 q^{41} +12.7279i q^{43} +(-3.00000 - 1.41421i) q^{44} -1.41421i q^{46} +9.89949i q^{47} +7.00000 q^{49} +3.00000 q^{50} -4.24264i q^{52} +5.65685i q^{53} +(-4.00000 + 8.48528i) q^{55} +5.65685i q^{59} -12.7279i q^{61} -2.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +8.00000 q^{67} +6.00000 q^{68} -15.5563i q^{71} +8.48528i q^{73} +10.0000 q^{74} -4.24264i q^{76} +8.48528i q^{79} -2.82843i q^{80} -6.00000 q^{82} -6.00000 q^{83} -16.9706i q^{85} -12.7279i q^{86} +(3.00000 + 1.41421i) q^{88} -7.07107i q^{89} +1.41421i q^{92} -9.89949i q^{94} -12.0000 q^{95} -4.00000 q^{97} -7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{11} + 2 q^{16} + 12 q^{17} + 6 q^{22} - 6 q^{25} + 4 q^{31} - 2 q^{32} - 12 q^{34} - 20 q^{37} + 12 q^{41} - 6 q^{44} + 14 q^{49} + 6 q^{50} - 8 q^{55} - 4 q^{62} + 2 q^{64} - 24 q^{65} + 16 q^{67} + 12 q^{68} + 20 q^{74} - 12 q^{82} - 12 q^{83} + 6 q^{88} - 24 q^{95} - 8 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^11 + 2 * q^16 + 12 * q^17 + 6 * q^22 - 6 * q^25 + 4 * q^31 - 2 * q^32 - 12 * q^34 - 20 * q^37 + 12 * q^41 - 6 * q^44 + 14 * q^49 + 6 * q^50 - 8 * q^55 - 4 * q^62 + 2 * q^64 - 24 * q^65 + 16 * q^67 + 12 * q^68 + 20 * q^74 - 12 * q^82 - 12 * q^83 + 6 * q^88 - 24 * q^95 - 8 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/198\mathbb{Z}\right)^\times$.

$n$	$145$	$155$
$\chi(n)$	$-1$	$-1$

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
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$			2.82843i			0.894427i
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$12$		0			0
$13$		−	4.24264i		−	1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$		−	4.24264i		−	1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$		−	4.24264i		−	0.973329i	−0.873589	−	0.486664i	$-0.838214\pi$
$19$		−	4.24264i		−	0.973329i	0.873589	−	0.486664i	$-0.161786\pi$
$20$		−	2.82843i		−	0.632456i
$21$		0			0
$22$	3.00000	+	1.41421i	0.639602	+	0.301511i
$23$			1.41421i			0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$			1.41421i			0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$			4.24264i			0.832050i
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$	−6.00000			−1.02899
$35$		0			0
$36$		0			0
$37$	−10.0000			−1.64399			−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000			−1.64399			−0.821995	+	0.569495i	$0.807139\pi$
$38$			4.24264i			0.688247i
$39$		0			0
$40$			2.82843i			0.447214i
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$			12.7279i			1.94099i	0.241121	+	0.970495i	$0.422485\pi$
$43$			12.7279i			1.94099i	−0.241121	+	0.970495i	$0.577515\pi$
$44$	−3.00000	−	1.41421i	−0.452267	−	0.213201i
$45$		0			0
$46$		−	1.41421i		−	0.208514i
$47$			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
$49$	7.00000			1.00000
$50$	3.00000			0.424264
$51$		0			0
$52$		−	4.24264i		−	0.588348i
$53$			5.65685i			0.777029i	0.921443	+	0.388514i	$0.127012\pi$
$53$			5.65685i			0.777029i	−0.921443	+	0.388514i	$0.872988\pi$
$54$		0			0
$55$	−4.00000	+	8.48528i	−0.539360	+	1.14416i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			5.65685i			0.736460i	0.929735	+	0.368230i	$0.120036\pi$
$59$			5.65685i			0.736460i	−0.929735	+	0.368230i	$0.879964\pi$
$60$		0			0
$61$		−	12.7279i		−	1.62964i	−0.579712	−	0.814822i	$-0.696835\pi$
$61$		−	12.7279i		−	1.62964i	0.579712	−	0.814822i	$-0.303165\pi$
$62$	−2.00000			−0.254000
$63$		0			0
$64$	1.00000			0.125000
$65$	−12.0000			−1.48842
$66$		0			0
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$	6.00000			0.727607
$69$		0			0
$70$		0			0
$71$		−	15.5563i		−	1.84620i	−0.384561	−	0.923099i	$-0.625647\pi$
$71$		−	15.5563i		−	1.84620i	0.384561	−	0.923099i	$-0.374353\pi$
$72$		0			0
$73$			8.48528i			0.993127i	0.868000	+	0.496564i	$0.165405\pi$
$73$			8.48528i			0.993127i	−0.868000	+	0.496564i	$0.834595\pi$
$74$	10.0000			1.16248
$75$		0			0
$76$		−	4.24264i		−	0.486664i
$77$		0			0
$78$		0			0
$79$			8.48528i			0.954669i	0.878722	+	0.477334i	$0.158397\pi$
$79$			8.48528i			0.954669i	−0.878722	+	0.477334i	$0.841603\pi$
$80$		−	2.82843i		−	0.316228i
$81$		0			0
$82$	−6.00000			−0.662589
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$		−	16.9706i		−	1.84072i
$86$		−	12.7279i		−	1.37249i
$87$		0			0
$88$	3.00000	+	1.41421i	0.319801	+	0.150756i
$89$		−	7.07107i		−	0.749532i	−0.927119	−	0.374766i	$-0.877723\pi$
$89$		−	7.07107i		−	0.749532i	0.927119	−	0.374766i	$-0.122277\pi$
$90$		0			0
$91$		0			0
$92$			1.41421i			0.147442i
$93$		0			0
$94$		−	9.89949i		−	1.02105i
$95$	−12.0000			−1.23117
$96$		0			0
$97$	−4.00000			−0.406138			−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000			−0.406138			−0.203069	+	0.979164i	$0.565092\pi$
$98$	−7.00000			−0.707107
$99$		0			0
$100$	−3.00000			−0.300000
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$			4.24264i			0.416025i
$105$		0			0
$106$		−	5.65685i		−	0.549442i
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$			4.24264i			0.406371i	0.979140	+	0.203186i	$0.0651295\pi$
$109$			4.24264i			0.406371i	−0.979140	+	0.203186i	$0.934871\pi$
$110$	4.00000	−	8.48528i	0.381385	−	0.809040i
$111$		0			0
$112$		0			0
$113$		−	7.07107i		−	0.665190i	−0.943070	−	0.332595i	$-0.892076\pi$
$113$		−	7.07107i		−	0.665190i	0.943070	−	0.332595i	$-0.107924\pi$
$114$		0			0
$115$	4.00000			0.373002
$116$		0			0
$117$		0			0
$118$		−	5.65685i		−	0.520756i
$119$		0			0
$120$		0			0
$121$	7.00000	+	8.48528i	0.636364	+	0.771389i
$122$			12.7279i			1.15233i
$123$		0			0
$124$	2.00000			0.179605
$125$		−	5.65685i		−	0.505964i
$126$		0			0
$127$		−	8.48528i		−	0.752947i	−0.926427	−	0.376473i	$-0.877137\pi$
$127$		−	8.48528i		−	0.752947i	0.926427	−	0.376473i	$-0.122863\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	12.0000			1.05247
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$	−8.00000			−0.691095
$135$		0			0
$136$	−6.00000			−0.514496
$137$			1.41421i			0.120824i	0.998174	+	0.0604122i	$0.0192415\pi$
$137$			1.41421i			0.120824i	−0.998174	+	0.0604122i	$0.980758\pi$
$138$		0			0
$139$			12.7279i			1.07957i	0.841803	+	0.539784i	$0.181494\pi$
$139$			12.7279i			1.07957i	−0.841803	+	0.539784i	$0.818506\pi$
$140$		0			0
$141$		0			0
$142$			15.5563i			1.30546i
$143$	−6.00000	+	12.7279i	−0.501745	+	1.06436i
$144$		0			0
$145$		0			0
$146$		−	8.48528i		−	0.702247i
$147$		0			0
$148$	−10.0000			−0.821995
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$		−	8.48528i		−	0.690522i	−0.938507	−	0.345261i	$-0.887790\pi$
$151$		−	8.48528i		−	0.690522i	0.938507	−	0.345261i	$-0.112210\pi$
$152$			4.24264i			0.344124i
$153$		0			0
$154$		0			0
$155$		−	5.65685i		−	0.454369i
$156$		0			0
$157$	−10.0000			−0.798087			−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000			−0.798087			−0.399043	+	0.916932i	$0.630658\pi$
$158$		−	8.48528i		−	0.675053i
$159$		0			0
$160$			2.82843i			0.223607i
$161$		0			0
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$	6.00000			0.468521
$165$		0			0
$166$	6.00000			0.465690
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$			16.9706i			1.30158i
$171$		0			0
$172$			12.7279i			0.970495i
$173$	12.0000			0.912343			0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000			0.912343			0.456172	+	0.889892i	$0.349220\pi$
$174$		0			0
$175$		0			0
$176$	−3.00000	−	1.41421i	−0.226134	−	0.106600i
$177$		0			0
$178$			7.07107i			0.529999i
$179$			14.1421i			1.05703i	0.848923	+	0.528516i	$0.177252\pi$
$179$			14.1421i			1.05703i	−0.848923	+	0.528516i	$0.822748\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$		0			0
$184$		−	1.41421i		−	0.104257i
$185$			28.2843i			2.07950i
$186$		0			0
$187$	−18.0000	−	8.48528i	−1.31629	−	0.620505i
$188$			9.89949i			0.721995i
$189$		0			0
$190$	12.0000			0.870572
$191$		−	7.07107i		−	0.511645i	−0.966724	−	0.255822i	$-0.917654\pi$
$191$		−	7.07107i		−	0.511645i	0.966724	−	0.255822i	$-0.0823462\pi$
$192$		0			0
$193$			8.48528i			0.610784i	0.952227	+	0.305392i	$0.0987875\pi$
$193$			8.48528i			0.610784i	−0.952227	+	0.305392i	$0.901213\pi$
$194$	4.00000			0.287183
$195$		0			0
$196$	7.00000			0.500000
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$	3.00000			0.212132
$201$		0			0
$202$	−6.00000			−0.422159
$203$		0			0
$204$		0			0
$205$		−	16.9706i		−	1.18528i
$206$	−14.0000			−0.975426
$207$		0			0
$208$		−	4.24264i		−	0.294174i
$209$	−6.00000	+	12.7279i	−0.415029	+	0.880409i
$210$		0			0
$211$		−	21.2132i		−	1.46038i	−0.683246	−	0.730189i	$-0.739432\pi$
$211$		−	21.2132i		−	1.46038i	0.683246	−	0.730189i	$-0.260568\pi$
$212$			5.65685i			0.388514i
$213$		0			0
$214$	−12.0000			−0.820303
$215$	36.0000			2.45518
$216$		0			0
$217$		0			0
$218$		−	4.24264i		−	0.287348i
$219$		0			0
$220$	−4.00000	+	8.48528i	−0.269680	+	0.572078i
$221$		−	25.4558i		−	1.71235i
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$		0			0
$226$			7.07107i			0.470360i
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$	−4.00000			−0.263752
$231$		0			0
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	28.0000			1.82652
$236$			5.65685i			0.368230i
$237$		0			0
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$			16.9706i			1.09317i	0.837404	+	0.546585i	$0.184072\pi$
$241$			16.9706i			1.09317i	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−7.00000	−	8.48528i	−0.449977	−	0.545455i
$243$		0			0
$244$		−	12.7279i		−	0.814822i
$245$		−	19.7990i		−	1.26491i
$246$		0			0
$247$	−18.0000			−1.14531
$248$	−2.00000			−0.127000
$249$		0			0
$250$			5.65685i			0.357771i
$251$		−	19.7990i		−	1.24970i	−0.780744	−	0.624851i	$-0.785160\pi$
$251$		−	19.7990i		−	1.24970i	0.780744	−	0.624851i	$-0.214840\pi$
$252$		0			0
$253$	2.00000	−	4.24264i	0.125739	−	0.266733i
$254$			8.48528i			0.532414i
$255$		0			0
$256$	1.00000			0.0625000
$257$			1.41421i			0.0882162i	0.999027	+	0.0441081i	$0.0140446\pi$
$257$			1.41421i			0.0882162i	−0.999027	+	0.0441081i	$0.985955\pi$
$258$		0			0
$259$		0			0
$260$	−12.0000			−0.744208
$261$		0			0
$262$	12.0000			0.741362
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	16.0000			0.982872
$266$		0			0
$267$		0			0
$268$	8.00000			0.488678
$269$			5.65685i			0.344904i	0.985018	+	0.172452i	$0.0551690\pi$
$269$			5.65685i			0.344904i	−0.985018	+	0.172452i	$0.944831\pi$
$270$		0			0
$271$		−	8.48528i		−	0.515444i	−0.966219	−	0.257722i	$-0.917028\pi$
$271$		−	8.48528i		−	0.515444i	0.966219	−	0.257722i	$-0.0829719\pi$
$272$	6.00000			0.363803
$273$		0			0
$274$		−	1.41421i		−	0.0854358i
$275$	9.00000	+	4.24264i	0.542720	+	0.255841i
$276$		0			0
$277$			4.24264i			0.254916i	0.991844	+	0.127458i	$0.0406817\pi$
$277$			4.24264i			0.254916i	−0.991844	+	0.127458i	$0.959318\pi$
$278$		−	12.7279i		−	0.763370i
$279$		0			0
$280$		0			0
$281$	−30.0000			−1.78965			−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000			−1.78965			−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$			12.7279i			0.756596i	0.925684	+	0.378298i	$0.123491\pi$
$283$			12.7279i			0.756596i	−0.925684	+	0.378298i	$0.876509\pi$
$284$		−	15.5563i		−	0.923099i
$285$		0			0
$286$	6.00000	−	12.7279i	0.354787	−	0.752618i
$287$		0			0
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$			8.48528i			0.496564i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	16.0000			0.931556
$296$	10.0000			0.581238
$297$		0			0
$298$	−6.00000			−0.347571
$299$	6.00000			0.346989
$300$		0			0
$301$		0			0
$302$			8.48528i			0.488273i
$303$		0			0
$304$		−	4.24264i		−	0.243332i
$305$	−36.0000			−2.06135
$306$		0			0
$307$		−	21.2132i		−	1.21070i	−0.795959	−	0.605351i	$-0.793033\pi$
$307$		−	21.2132i		−	1.21070i	0.795959	−	0.605351i	$-0.206967\pi$
$308$		0			0
$309$		0			0
$310$			5.65685i			0.321288i
$311$			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
$313$	8.00000			0.452187			0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000			0.452187			0.226093	+	0.974106i	$0.427405\pi$
$314$	10.0000			0.564333
$315$		0			0
$316$			8.48528i			0.477334i
$317$			22.6274i			1.27088i	0.772149	+	0.635441i	$0.219182\pi$
$317$			22.6274i			1.27088i	−0.772149	+	0.635441i	$0.780818\pi$
$318$		0			0
$319$		0			0
$320$		−	2.82843i		−	0.158114i
$321$		0			0
$322$		0			0
$323$		−	25.4558i		−	1.41640i
$324$		0			0
$325$			12.7279i			0.706018i
$326$	16.0000			0.886158
$327$		0			0
$328$	−6.00000			−0.331295
$329$		0			0
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$	−6.00000			−0.329293
$333$		0			0
$334$		0			0
$335$		−	22.6274i		−	1.23627i
$336$		0			0
$337$			8.48528i			0.462223i	0.972927	+	0.231111i	$0.0742362\pi$
$337$			8.48528i			0.462223i	−0.972927	+	0.231111i	$0.925764\pi$
$338$	5.00000			0.271964
$339$		0			0
$340$		−	16.9706i		−	0.920358i
$341$	−6.00000	−	2.82843i	−0.324918	−	0.153168i
$342$		0			0
$343$		0			0
$344$		−	12.7279i		−	0.686244i
$345$		0			0
$346$	−12.0000			−0.645124
$347$	−30.0000			−1.61048			−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000			−1.61048			−0.805242	+	0.592946i	$0.797965\pi$
$348$		0			0
$349$			29.6985i			1.58972i	0.606791	+	0.794862i	$0.292457\pi$
$349$			29.6985i			1.58972i	−0.606791	+	0.794862i	$0.707543\pi$
$350$		0			0
$351$		0			0
$352$	3.00000	+	1.41421i	0.159901	+	0.0753778i
$353$			9.89949i			0.526897i	0.964673	+	0.263448i	$0.0848599\pi$
$353$			9.89949i			0.526897i	−0.964673	+	0.263448i	$0.915140\pi$
$354$		0			0
$355$	−44.0000			−2.33528
$356$		−	7.07107i		−	0.374766i
$357$		0			0
$358$		−	14.1421i		−	0.747435i
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$	−14.0000			−0.735824
$363$		0			0
$364$		0			0
$365$	24.0000			1.25622
$366$		0			0
$367$	−16.0000			−0.835193			−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000			−0.835193			−0.417597	+	0.908633i	$0.637127\pi$
$368$			1.41421i			0.0737210i
$369$		0			0
$370$		−	28.2843i		−	1.47043i
$371$		0			0
$372$		0			0
$373$		−	21.2132i		−	1.09838i	−0.835698	−	0.549189i	$-0.814937\pi$
$373$		−	21.2132i		−	1.09838i	0.835698	−	0.549189i	$-0.185063\pi$
$374$	18.0000	+	8.48528i	0.930758	+	0.438763i
$375$		0			0
$376$		−	9.89949i		−	0.510527i
$377$		0			0
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$	−12.0000			−0.615587
$381$		0			0
$382$			7.07107i			0.361787i
$383$			9.89949i			0.505841i	0.967487	+	0.252920i	$0.0813910\pi$
$383$			9.89949i			0.505841i	−0.967487	+	0.252920i	$0.918609\pi$
$384$		0			0
$385$		0			0
$386$		−	8.48528i		−	0.431889i
$387$		0			0
$388$	−4.00000			−0.203069
$389$			14.1421i			0.717035i	0.933523	+	0.358517i	$0.116718\pi$
$389$			14.1421i			0.717035i	−0.933523	+	0.358517i	$0.883282\pi$
$390$		0			0
$391$			8.48528i			0.429119i
$392$	−7.00000			−0.353553
$393$		0			0
$394$	−18.0000			−0.906827
$395$	24.0000			1.20757
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$	16.0000			0.802008
$399$		0			0
$400$	−3.00000			−0.150000
$401$			1.41421i			0.0706225i	0.999376	+	0.0353112i	$0.0112422\pi$
$401$			1.41421i			0.0706225i	−0.999376	+	0.0353112i	$0.988758\pi$
$402$		0			0
$403$		−	8.48528i		−	0.422682i
$404$	6.00000			0.298511
$405$		0			0
$406$		0			0
$407$	30.0000	+	14.1421i	1.48704	+	0.701000i
$408$		0			0
$409$			33.9411i			1.67828i	0.543915	+	0.839140i	$0.316941\pi$
$409$			33.9411i			1.67828i	−0.543915	+	0.839140i	$0.683059\pi$
$410$			16.9706i			0.838116i
$411$		0			0
$412$	14.0000			0.689730
$413$		0			0
$414$		0			0
$415$			16.9706i			0.833052i
$416$			4.24264i			0.208013i
$417$		0			0
$418$	6.00000	−	12.7279i	0.293470	−	0.622543i
$419$			14.1421i			0.690889i	0.938439	+	0.345444i	$0.112272\pi$
$419$			14.1421i			0.690889i	−0.938439	+	0.345444i	$0.887728\pi$
$420$		0			0
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$			21.2132i			1.03264i
$423$		0			0
$424$		−	5.65685i		−	0.274721i
$425$	−18.0000			−0.873128
$426$		0			0
$427$		0			0
$428$	12.0000			0.580042
$429$		0			0
$430$	−36.0000			−1.73607
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		0			0
$433$	−16.0000			−0.768911			−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000			−0.768911			−0.384455	+	0.923144i	$0.625611\pi$
$434$		0			0
$435$		0			0
$436$			4.24264i			0.203186i
$437$	6.00000			0.287019
$438$		0			0
$439$		−	33.9411i		−	1.61992i	−0.586484	−	0.809961i	$-0.699488\pi$
$439$		−	33.9411i		−	1.61992i	0.586484	−	0.809961i	$-0.300512\pi$
$440$	4.00000	−	8.48528i	0.190693	−	0.404520i
$441$		0			0
$442$			25.4558i			1.21081i
$443$			39.5980i			1.88136i	0.339300	+	0.940678i	$0.389810\pi$
$443$			39.5980i			1.88136i	−0.339300	+	0.940678i	$0.610190\pi$
$444$		0			0
$445$	−20.0000			−0.948091
$446$	−8.00000			−0.378811
$447$		0			0
$448$		0			0
$449$			9.89949i			0.467186i	0.972334	+	0.233593i	$0.0750483\pi$
$449$			9.89949i			0.467186i	−0.972334	+	0.233593i	$0.924952\pi$
$450$		0			0
$451$	−18.0000	−	8.48528i	−0.847587	−	0.399556i
$452$		−	7.07107i		−	0.332595i
$453$		0			0
$454$	−12.0000			−0.563188
$455$		0			0
$456$		0			0
$457$			16.9706i			0.793849i	0.917851	+	0.396925i	$0.129923\pi$
$457$			16.9706i			0.793849i	−0.917851	+	0.396925i	$0.870077\pi$
$458$	22.0000			1.02799
$459$		0			0
$460$	4.00000			0.186501
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$	−22.0000			−1.02243			−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000			−1.02243			−0.511213	+	0.859454i	$0.670804\pi$
$464$		0			0
$465$		0			0
$466$	18.0000			0.833834
$467$			14.1421i			0.654420i	0.944952	+	0.327210i	$0.106108\pi$
$467$			14.1421i			0.654420i	−0.944952	+	0.327210i	$0.893892\pi$
$468$		0			0
$469$		0			0
$470$	−28.0000			−1.29154
$471$		0			0
$472$		−	5.65685i		−	0.260378i
$473$	18.0000	−	38.1838i	0.827641	−	1.75569i
$474$		0			0
$475$			12.7279i			0.583997i
$476$		0			0
$477$		0			0
$478$	12.0000			0.548867
$479$	−36.0000			−1.64488			−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000			−1.64488			−0.822441	+	0.568850i	$0.807388\pi$
$480$		0			0
$481$			42.4264i			1.93448i
$482$		−	16.9706i		−	0.772988i
$483$		0			0
$484$	7.00000	+	8.48528i	0.318182	+	0.385695i
$485$			11.3137i			0.513729i
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$			12.7279i			0.576166i
$489$		0			0
$490$			19.7990i			0.894427i
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$		0			0
$494$	18.0000			0.809858
$495$		0			0
$496$	2.00000			0.0898027
$497$		0			0
$498$		0			0
$499$	−4.00000			−0.179065			−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000			−0.179065			−0.0895323	+	0.995984i	$0.528537\pi$
$500$		−	5.65685i		−	0.252982i
$501$		0			0
$502$			19.7990i			0.883672i
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$		−	16.9706i		−	0.755180i
$506$	−2.00000	+	4.24264i	−0.0889108	+	0.188608i
$507$		0			0
$508$		−	8.48528i		−	0.376473i
$509$		−	11.3137i		−	0.501471i	−0.968056	−	0.250736i	$-0.919328\pi$
$509$		−	11.3137i		−	0.501471i	0.968056	−	0.250736i	$-0.0806725\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		0			0
$514$		−	1.41421i		−	0.0623783i
$515$		−	39.5980i		−	1.74490i
$516$		0			0
$517$	14.0000	−	29.6985i	0.615719	−	1.30614i
$518$		0			0
$519$		0			0
$520$	12.0000			0.526235
$521$			35.3553i			1.54895i	0.632607	+	0.774473i	$0.281985\pi$
$521$			35.3553i			1.54895i	−0.632607	+	0.774473i	$0.718015\pi$
$522$		0			0
$523$			12.7279i			0.556553i	0.960501	+	0.278277i	$0.0897632\pi$
$523$			12.7279i			0.556553i	−0.960501	+	0.278277i	$0.910237\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$		0			0
$527$	12.0000			0.522728
$528$		0			0
$529$	21.0000			0.913043
$530$	−16.0000			−0.694996
$531$		0			0
$532$		0			0
$533$		−	25.4558i		−	1.10262i
$534$		0			0
$535$		−	33.9411i		−	1.46740i
$536$	−8.00000			−0.345547
$537$		0			0
$538$		−	5.65685i		−	0.243884i
$539$	−21.0000	−	9.89949i	−0.904534	−	0.426401i
$540$		0			0
$541$		−	21.2132i		−	0.912027i	−0.889973	−	0.456013i	$-0.849277\pi$
$541$		−	21.2132i		−	0.912027i	0.889973	−	0.456013i	$-0.150723\pi$
$542$			8.48528i			0.364474i
$543$		0			0
$544$	−6.00000			−0.257248
$545$	12.0000			0.514024
$546$		0			0
$547$		−	12.7279i		−	0.544207i	−0.962268	−	0.272103i	$-0.912281\pi$
$547$		−	12.7279i		−	0.544207i	0.962268	−	0.272103i	$-0.0877193\pi$
$548$			1.41421i			0.0604122i
$549$		0			0
$550$	−9.00000	−	4.24264i	−0.383761	−	0.180907i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	4.24264i		−	0.180253i
$555$		0			0
$556$			12.7279i			0.539784i
$557$	12.0000			0.508456			0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000			0.508456			0.254228	+	0.967144i	$0.418179\pi$
$558$		0			0
$559$	54.0000			2.28396
$560$		0			0
$561$		0			0
$562$	30.0000			1.26547
$563$	−6.00000			−0.252870			−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000			−0.252870			−0.126435	+	0.991975i	$0.540353\pi$
$564$		0			0
$565$	−20.0000			−0.841406
$566$		−	12.7279i		−	0.534994i
$567$		0			0
$568$			15.5563i			0.652730i
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$			4.24264i			0.177549i	0.996052	+	0.0887745i	$0.0282950\pi$
$571$			4.24264i			0.177549i	−0.996052	+	0.0887745i	$0.971705\pi$
$572$	−6.00000	+	12.7279i	−0.250873	+	0.532181i
$573$		0			0
$574$		0			0
$575$		−	4.24264i		−	0.176930i
$576$		0			0
$577$	−4.00000			−0.166522			−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000			−0.166522			−0.0832611	+	0.996528i	$0.526534\pi$
$578$	−19.0000			−0.790296
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	8.00000	−	16.9706i	0.331326	−	0.702849i
$584$		−	8.48528i		−	0.351123i
$585$		0			0
$586$		0			0
$587$		−	28.2843i		−	1.16742i	−0.811963	−	0.583708i	$-0.801601\pi$
$587$		−	28.2843i		−	1.16742i	0.811963	−	0.583708i	$-0.198399\pi$
$588$		0			0
$589$		−	8.48528i		−	0.349630i
$590$	−16.0000			−0.658710
$591$		0			0
$592$	−10.0000			−0.410997
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$		0			0
$595$		0			0
$596$	6.00000			0.245770
$597$		0			0
$598$	−6.00000			−0.245358
$599$		−	24.0416i		−	0.982314i	−0.871071	−	0.491157i	$-0.836574\pi$
$599$		−	24.0416i		−	0.982314i	0.871071	−	0.491157i	$-0.163426\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		−	8.48528i		−	0.345261i
$605$	24.0000	−	19.7990i	0.975739	−	0.804943i
$606$		0			0
$607$			8.48528i			0.344407i	0.985061	+	0.172203i	$0.0550887\pi$
$607$			8.48528i			0.344407i	−0.985061	+	0.172203i	$0.944911\pi$
$608$			4.24264i			0.172062i
$609$		0			0
$610$	36.0000			1.45760
$611$	42.0000			1.69914
$612$		0			0
$613$		−	12.7279i		−	0.514076i	−0.966401	−	0.257038i	$-0.917253\pi$
$613$		−	12.7279i		−	0.514076i	0.966401	−	0.257038i	$-0.0827465\pi$
$614$			21.2132i			0.856095i
$615$		0			0
$616$		0			0
$617$		−	24.0416i		−	0.967880i	−0.875101	−	0.483940i	$-0.839205\pi$
$617$		−	24.0416i		−	0.967880i	0.875101	−	0.483940i	$-0.160795\pi$
$618$		0			0
$619$	−40.0000			−1.60774			−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000			−1.60774			−0.803868	+	0.594808i	$0.797228\pi$
$620$		−	5.65685i		−	0.227185i
$621$		0			0
$622$		−	1.41421i		−	0.0567048i
$623$		0			0
$624$		0			0
$625$	−31.0000			−1.24000
$626$	−8.00000			−0.319744
$627$		0			0
$628$	−10.0000			−0.399043
$629$	−60.0000			−2.39236
$630$		0			0
$631$	−22.0000			−0.875806			−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000			−0.875806			−0.437903	+	0.899022i	$0.644279\pi$
$632$		−	8.48528i		−	0.337526i
$633$		0			0
$634$		−	22.6274i		−	0.898650i
$635$	−24.0000			−0.952411
$636$		0			0
$637$		−	29.6985i		−	1.17670i
$638$		0			0
$639$		0			0
$640$			2.82843i			0.111803i
$641$		−	15.5563i		−	0.614439i	−0.951639	−	0.307219i	$-0.900601\pi$
$641$		−	15.5563i		−	0.614439i	0.951639	−	0.307219i	$-0.0993986\pi$
$642$		0			0
$643$	32.0000			1.26196			0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000			1.26196			0.630978	+	0.775800i	$0.282654\pi$
$644$		0			0
$645$		0			0
$646$			25.4558i			1.00155i
$647$		−	32.5269i		−	1.27876i	−0.768889	−	0.639382i	$-0.779190\pi$
$647$		−	32.5269i		−	1.27876i	0.768889	−	0.639382i	$-0.220810\pi$
$648$		0			0
$649$	8.00000	−	16.9706i	0.314027	−	0.666153i
$650$		−	12.7279i		−	0.499230i
$651$		0			0
$652$	−16.0000			−0.626608
$653$			31.1127i			1.21753i	0.793349	+	0.608767i	$0.208336\pi$
$653$			31.1127i			1.21753i	−0.793349	+	0.608767i	$0.791664\pi$
$654$		0			0
$655$			33.9411i			1.32619i
$656$	6.00000			0.234261
$657$		0			0
$658$		0			0
$659$	18.0000			0.701180			0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000			0.701180			0.350590	+	0.936529i	$0.385981\pi$
$660$		0			0
$661$	2.00000			0.0777910			0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000			0.0777910			0.0388955	+	0.999243i	$0.487616\pi$
$662$	−20.0000			−0.777322
$663$		0			0
$664$	6.00000			0.232845
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$			22.6274i			0.874173i
$671$	−18.0000	+	38.1838i	−0.694882	+	1.47407i
$672$		0			0
$673$		−	42.4264i		−	1.63542i	−0.575632	−	0.817709i	$-0.695244\pi$
$673$		−	42.4264i		−	1.63542i	0.575632	−	0.817709i	$-0.304756\pi$
$674$		−	8.48528i		−	0.326841i
$675$		0			0
$676$	−5.00000			−0.192308
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$		0			0
$680$			16.9706i			0.650791i
$681$		0			0
$682$	6.00000	+	2.82843i	0.229752	+	0.108306i
$683$			31.1127i			1.19049i	0.803543	+	0.595247i	$0.202946\pi$
$683$			31.1127i			1.19049i	−0.803543	+	0.595247i	$0.797054\pi$
$684$		0			0
$685$	4.00000			0.152832
$686$		0			0
$687$		0			0
$688$			12.7279i			0.485247i
$689$	24.0000			0.914327
$690$		0			0
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$	12.0000			0.456172
$693$		0			0
$694$	30.0000			1.13878
$695$	36.0000			1.36556
$696$		0			0
$697$	36.0000			1.36360
$698$		−	29.6985i		−	1.12410i
$699$		0			0
$700$		0			0
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$		0			0
$703$			42.4264i			1.60014i
$704$	−3.00000	−	1.41421i	−0.113067	−	0.0533002i
$705$		0			0
$706$		−	9.89949i		−	0.372572i
$707$		0			0
$708$		0			0
$709$	−34.0000			−1.27690			−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000			−1.27690			−0.638448	+	0.769665i	$0.720423\pi$
$710$	44.0000			1.65129
$711$		0			0
$712$			7.07107i			0.264999i
$713$			2.82843i			0.105925i
$714$		0			0
$715$	36.0000	+	16.9706i	1.34632	+	0.634663i
$716$			14.1421i			0.528516i
$717$		0			0
$718$	−12.0000			−0.447836
$719$			1.41421i			0.0527413i	0.999652	+	0.0263706i	$0.00839501\pi$
$719$			1.41421i			0.0527413i	−0.999652	+	0.0263706i	$0.991605\pi$
$720$		0			0
$721$		0			0
$722$	−1.00000			−0.0372161
$723$		0			0
$724$	14.0000			0.520306
$725$		0			0
$726$		0			0
$727$	−40.0000			−1.48352			−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000			−1.48352			−0.741759	+	0.670667i	$0.766008\pi$
$728$		0			0
$729$		0			0
$730$	−24.0000			−0.888280
$731$			76.3675i			2.82456i
$732$		0			0
$733$			4.24264i			0.156706i	0.996926	+	0.0783528i	$0.0249660\pi$
$733$			4.24264i			0.156706i	−0.996926	+	0.0783528i	$0.975034\pi$
$734$	16.0000			0.590571
$735$		0			0
$736$		−	1.41421i		−	0.0521286i
$737$	−24.0000	−	11.3137i	−0.884051	−	0.416746i
$738$		0			0
$739$			21.2132i			0.780340i	0.920743	+	0.390170i	$0.127584\pi$
$739$			21.2132i			0.780340i	−0.920743	+	0.390170i	$0.872416\pi$
$740$			28.2843i			1.03975i
$741$		0			0
$742$		0			0
$743$	−36.0000			−1.32071			−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000			−1.32071			−0.660356	+	0.750953i	$0.729595\pi$
$744$		0			0
$745$		−	16.9706i		−	0.621753i
$746$			21.2132i			0.776671i
$747$		0			0
$748$	−18.0000	−	8.48528i	−0.658145	−	0.310253i
$749$		0			0
$750$		0			0
$751$	2.00000			0.0729810			0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000			0.0729810			0.0364905	+	0.999334i	$0.488382\pi$
$752$			9.89949i			0.360997i
$753$		0			0
$754$		0			0
$755$	−24.0000			−0.873449
$756$		0			0
$757$	−34.0000			−1.23575			−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000			−1.23575			−0.617876	+	0.786276i	$0.712006\pi$
$758$	−20.0000			−0.726433
$759$		0			0
$760$	12.0000			0.435286
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$		0			0
$764$		−	7.07107i		−	0.255822i
$765$		0			0
$766$		−	9.89949i		−	0.357683i
$767$	24.0000			0.866590
$768$		0			0
$769$			8.48528i			0.305987i	0.988227	+	0.152994i	$0.0488914\pi$
$769$			8.48528i			0.305987i	−0.988227	+	0.152994i	$0.951109\pi$
$770$		0			0
$771$		0			0
$772$			8.48528i			0.305392i
$773$			5.65685i			0.203463i	0.994812	+	0.101731i	$0.0324382\pi$
$773$			5.65685i			0.203463i	−0.994812	+	0.101731i	$0.967562\pi$
$774$		0			0
$775$	−6.00000			−0.215526
$776$	4.00000			0.143592
$777$		0			0
$778$		−	14.1421i		−	0.507020i
$779$		−	25.4558i		−	0.912050i
$780$		0			0
$781$	−22.0000	+	46.6690i	−0.787222	+	1.66995i
$782$		−	8.48528i		−	0.303433i
$783$		0			0
$784$	7.00000			0.250000
$785$			28.2843i			1.00951i
$786$		0			0
$787$		−	21.2132i		−	0.756169i	−0.925771	−	0.378085i	$-0.876583\pi$
$787$		−	21.2132i		−	0.756169i	0.925771	−	0.378085i	$-0.123417\pi$
$788$	18.0000			0.641223
$789$		0			0
$790$	−24.0000			−0.853882
$791$		0			0
$792$		0			0
$793$	−54.0000			−1.91760
$794$	−2.00000			−0.0709773
$795$		0			0
$796$	−16.0000			−0.567105
$797$			31.1127i			1.10207i	0.834483	+	0.551034i	$0.185767\pi$
$797$			31.1127i			1.10207i	−0.834483	+	0.551034i	$0.814233\pi$
$798$		0			0
$799$			59.3970i			2.10131i
$800$	3.00000			0.106066
$801$		0			0
$802$		−	1.41421i		−	0.0499376i
$803$	12.0000	−	25.4558i	0.423471	−	0.898317i
$804$		0			0
$805$		0			0
$806$			8.48528i			0.298881i
$807$		0			0
$808$	−6.00000			−0.211079
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$			46.6690i			1.63877i	0.573242	+	0.819386i	$0.305685\pi$
$811$			46.6690i			1.63877i	−0.573242	+	0.819386i	$0.694315\pi$
$812$		0			0
$813$		0			0
$814$	−30.0000	−	14.1421i	−1.05150	−	0.495682i
$815$			45.2548i			1.58521i
$816$		0			0
$817$	54.0000			1.88922
$818$		−	33.9411i		−	1.18672i
$819$		0			0
$820$		−	16.9706i		−	0.592638i
$821$	18.0000			0.628204			0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000			0.628204			0.314102	+	0.949389i	$0.398297\pi$
$822$		0			0
$823$	32.0000			1.11545			0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000			1.11545			0.557725	+	0.830026i	$0.311674\pi$
$824$	−14.0000			−0.487713
$825$		0			0
$826$		0			0
$827$	−42.0000			−1.46048			−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000			−1.46048			−0.730242	+	0.683189i	$0.760592\pi$
$828$		0			0
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		−	16.9706i		−	0.589057i
$831$		0			0
$832$		−	4.24264i		−	0.147087i
$833$	42.0000			1.45521
$834$		0			0
$835$		0			0
$836$	−6.00000	+	12.7279i	−0.207514	+	0.440204i
$837$		0			0
$838$		−	14.1421i		−	0.488532i
$839$			9.89949i			0.341769i	0.985291	+	0.170884i	$0.0546624\pi$
$839$			9.89949i			0.341769i	−0.985291	+	0.170884i	$0.945338\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	−26.0000			−0.896019
$843$		0			0
$844$		−	21.2132i		−	0.730189i
$845$			14.1421i			0.486504i
$846$		0			0
$847$		0			0
$848$			5.65685i			0.194257i
$849$		0			0
$850$	18.0000			0.617395
$851$		−	14.1421i		−	0.484786i
$852$		0			0
$853$		−	4.24264i		−	0.145265i	−0.997359	−	0.0726326i	$-0.976860\pi$
$853$		−	4.24264i		−	0.145265i	0.997359	−	0.0726326i	$-0.0231401\pi$
$854$		0			0
$855$		0			0
$856$	−12.0000			−0.410152
$857$	−30.0000			−1.02478			−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000			−1.02478			−0.512390	+	0.858753i	$0.671240\pi$
$858$		0			0
$859$	8.00000			0.272956			0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000			0.272956			0.136478	+	0.990643i	$0.456422\pi$
$860$	36.0000			1.22759
$861$		0			0
$862$	12.0000			0.408722
$863$		−	7.07107i		−	0.240702i	−0.992731	−	0.120351i	$-0.961598\pi$
$863$		−	7.07107i		−	0.240702i	0.992731	−	0.120351i	$-0.0384020\pi$
$864$		0			0
$865$		−	33.9411i		−	1.15403i
$866$	16.0000			0.543702
$867$		0			0
$868$		0			0
$869$	12.0000	−	25.4558i	0.407072	−	0.863530i
$870$		0			0
$871$		−	33.9411i		−	1.15005i
$872$		−	4.24264i		−	0.143674i
$873$		0			0
$874$	−6.00000			−0.202953
$875$		0			0
$876$		0			0
$877$		−	38.1838i		−	1.28937i	−0.764447	−	0.644687i	$-0.776988\pi$
$877$		−	38.1838i		−	1.28937i	0.764447	−	0.644687i	$-0.223012\pi$
$878$			33.9411i			1.14546i
$879$		0			0
$880$	−4.00000	+	8.48528i	−0.134840	+	0.286039i
$881$		−	24.0416i		−	0.809983i	−0.914320	−	0.404992i	$-0.867274\pi$
$881$		−	24.0416i		−	0.809983i	0.914320	−	0.404992i	$-0.132726\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		−	25.4558i		−	0.856173i
$885$		0			0
$886$		−	39.5980i		−	1.33032i
$887$	−36.0000			−1.20876			−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000			−1.20876			−0.604381	+	0.796696i	$0.706579\pi$
$888$		0			0
$889$		0			0
$890$	20.0000			0.670402
$891$		0			0
$892$	8.00000			0.267860
$893$	42.0000			1.40548
$894$		0			0
$895$	40.0000			1.33705
$896$		0			0
$897$		0			0
$898$		−	9.89949i		−	0.330350i
$899$		0			0
$900$		0			0
$901$			33.9411i			1.13074i
$902$	18.0000	+	8.48528i	0.599334	+	0.282529i
$903$		0			0
$904$			7.07107i			0.235180i
$905$		−	39.5980i		−	1.31628i
$906$		0			0
$907$	8.00000			0.265636			0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000			0.265636			0.132818	+	0.991140i	$0.457597\pi$
$908$	12.0000			0.398234
$909$		0			0
$910$		0			0
$911$		−	15.5563i		−	0.515405i	−0.966224	−	0.257702i	$-0.917035\pi$
$911$		−	15.5563i		−	0.515405i	0.966224	−	0.257702i	$-0.0829654\pi$
$912$		0			0
$913$	18.0000	+	8.48528i	0.595713	+	0.280822i
$914$		−	16.9706i		−	0.561336i
$915$		0			0
$916$	−22.0000			−0.726900
$917$		0			0
$918$		0			0
$919$		−	25.4558i		−	0.839711i	−0.907591	−	0.419855i	$-0.862081\pi$
$919$		−	25.4558i		−	0.839711i	0.907591	−	0.419855i	$-0.137919\pi$
$920$	−4.00000			−0.131876
$921$		0			0
$922$	12.0000			0.395199
$923$	−66.0000			−2.17242
$924$		0			0
$925$	30.0000			0.986394
$926$	22.0000			0.722965
$927$		0			0
$928$		0			0
$929$			9.89949i			0.324792i	0.986726	+	0.162396i	$0.0519222\pi$
$929$			9.89949i			0.324792i	−0.986726	+	0.162396i	$0.948078\pi$
$930$		0			0
$931$		−	29.6985i		−	0.973329i
$932$	−18.0000			−0.589610
$933$		0			0
$934$		−	14.1421i		−	0.462745i
$935$	−24.0000	+	50.9117i	−0.784884	+	1.66499i
$936$		0			0
$937$			33.9411i			1.10881i	0.832248	+	0.554404i	$0.187054\pi$
$937$			33.9411i			1.10881i	−0.832248	+	0.554404i	$0.812946\pi$
$938$		0			0
$939$		0			0
$940$	28.0000			0.913259
$941$	−30.0000			−0.977972			−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000			−0.977972			−0.488986	+	0.872292i	$0.662633\pi$
$942$		0			0
$943$			8.48528i			0.276319i
$944$			5.65685i			0.184115i
$945$		0			0
$946$	−18.0000	+	38.1838i	−0.585230	+	1.24146i
$947$		−	45.2548i		−	1.47058i	−0.677750	−	0.735292i	$-0.737045\pi$
$947$		−	45.2548i		−	1.47058i	0.677750	−	0.735292i	$-0.262955\pi$
$948$		0			0
$949$	36.0000			1.16861
$950$		−	12.7279i		−	0.412948i
$951$		0			0
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$	−20.0000			−0.647185
$956$	−12.0000			−0.388108
$957$		0			0
$958$	36.0000			1.16311
$959$		0			0
$960$		0			0
$961$	−27.0000			−0.870968
$962$		−	42.4264i		−	1.36788i
$963$		0			0
$964$			16.9706i			0.546585i
$965$	24.0000			0.772587
$966$		0			0
$967$			25.4558i			0.818605i	0.912399	+	0.409302i	$0.134228\pi$
$967$			25.4558i			0.818605i	−0.912399	+	0.409302i	$0.865772\pi$
$968$	−7.00000	−	8.48528i	−0.224989	−	0.272727i
$969$		0			0
$970$		−	11.3137i		−	0.363261i
$971$			5.65685i			0.181537i	0.995872	+	0.0907685i	$0.0289323\pi$
$971$			5.65685i			0.181537i	−0.995872	+	0.0907685i	$0.971068\pi$
$972$		0			0
$973$		0			0
$974$	−8.00000			−0.256337
$975$		0			0
$976$		−	12.7279i		−	0.407411i
$977$		−	24.0416i		−	0.769160i	−0.923092	−	0.384580i	$-0.874346\pi$
$977$		−	24.0416i		−	0.769160i	0.923092	−	0.384580i	$-0.125654\pi$
$978$		0			0
$979$	−10.0000	+	21.2132i	−0.319601	+	0.677977i
$980$		−	19.7990i		−	0.632456i
$981$		0			0
$982$	−12.0000			−0.382935
$983$		−	49.4975i		−	1.57872i	−0.613928	−	0.789362i	$-0.710411\pi$
$983$		−	49.4975i		−	1.57872i	0.613928	−	0.789362i	$-0.289589\pi$
$984$		0			0
$985$		−	50.9117i		−	1.62218i
$986$		0			0
$987$		0			0
$988$	−18.0000			−0.572656
$989$	−18.0000			−0.572367
$990$		0			0
$991$	−34.0000			−1.08005			−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000			−1.08005			−0.540023	+	0.841650i	$0.681584\pi$
$992$	−2.00000			−0.0635001
$993$		0			0
$994$		0			0
$995$			45.2548i			1.43467i
$996$		0			0
$997$			29.6985i			0.940560i	0.882517	+	0.470280i	$0.155847\pi$
$997$			29.6985i			0.940560i	−0.882517	+	0.470280i	$0.844153\pi$
$998$	4.00000			0.126618
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	198.2.b.a.197.1	✓	2
3.2	odd	2		198.2.b.b.197.2	yes	2
4.3	odd	2		1584.2.b.d.593.1		2
5.2	odd	4		4950.2.f.a.4949.1		4
5.3	odd	4		4950.2.f.a.4949.3		4
5.4	even	2		4950.2.d.e.4751.1		2
8.3	odd	2		6336.2.b.f.2177.2		2
8.5	even	2		6336.2.b.n.2177.2		2
9.2	odd	6		1782.2.i.b.1187.1		4
9.4	even	3		1782.2.i.g.593.1		4
9.5	odd	6		1782.2.i.b.593.2		4
9.7	even	3		1782.2.i.g.1187.2		4
11.10	odd	2		198.2.b.b.197.1	yes	2
12.11	even	2		1584.2.b.a.593.2		2
15.2	even	4		4950.2.f.b.4949.4		4
15.8	even	4		4950.2.f.b.4949.2		4
15.14	odd	2		4950.2.d.b.4751.2		2
24.5	odd	2		6336.2.b.a.2177.1		2
24.11	even	2		6336.2.b.i.2177.1		2
33.32	even	2	inner	198.2.b.a.197.2	yes	2
44.43	even	2		1584.2.b.a.593.1		2
55.32	even	4		4950.2.f.b.4949.3		4
55.43	even	4		4950.2.f.b.4949.1		4
55.54	odd	2		4950.2.d.b.4751.1		2
88.21	odd	2		6336.2.b.a.2177.2		2
88.43	even	2		6336.2.b.i.2177.2		2
99.32	even	6		1782.2.i.g.593.2		4
99.43	odd	6		1782.2.i.b.1187.2		4
99.65	even	6		1782.2.i.g.1187.1		4
99.76	odd	6		1782.2.i.b.593.1		4
132.131	odd	2		1584.2.b.d.593.2		2
165.32	odd	4		4950.2.f.a.4949.2		4
165.98	odd	4		4950.2.f.a.4949.4		4
165.164	even	2		4950.2.d.e.4751.2		2
264.131	odd	2		6336.2.b.f.2177.1		2
264.197	even	2		6336.2.b.n.2177.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.2.b.a.197.1	✓	2	1.1	even	1	trivial
198.2.b.a.197.2	yes	2	33.32	even	2	inner
198.2.b.b.197.1	yes	2	11.10	odd	2
198.2.b.b.197.2	yes	2	3.2	odd	2
1584.2.b.a.593.1		2	44.43	even	2
1584.2.b.a.593.2		2	12.11	even	2
1584.2.b.d.593.1		2	4.3	odd	2
1584.2.b.d.593.2		2	132.131	odd	2
1782.2.i.b.593.1		4	99.76	odd	6
1782.2.i.b.593.2		4	9.5	odd	6
1782.2.i.b.1187.1		4	9.2	odd	6
1782.2.i.b.1187.2		4	99.43	odd	6
1782.2.i.g.593.1		4	9.4	even	3
1782.2.i.g.593.2		4	99.32	even	6
1782.2.i.g.1187.1		4	99.65	even	6
1782.2.i.g.1187.2		4	9.7	even	3
4950.2.d.b.4751.1		2	55.54	odd	2
4950.2.d.b.4751.2		2	15.14	odd	2
4950.2.d.e.4751.1		2	5.4	even	2
4950.2.d.e.4751.2		2	165.164	even	2
4950.2.f.a.4949.1		4	5.2	odd	4
4950.2.f.a.4949.2		4	165.32	odd	4
4950.2.f.a.4949.3		4	5.3	odd	4
4950.2.f.a.4949.4		4	165.98	odd	4
4950.2.f.b.4949.1		4	55.43	even	4
4950.2.f.b.4949.2		4	15.8	even	4
4950.2.f.b.4949.3		4	55.32	even	4
4950.2.f.b.4949.4		4	15.2	even	4
6336.2.b.a.2177.1		2	24.5	odd	2
6336.2.b.a.2177.2		2	88.21	odd	2
6336.2.b.f.2177.1		2	264.131	odd	2
6336.2.b.f.2177.2		2	8.3	odd	2
6336.2.b.i.2177.1		2	24.11	even	2
6336.2.b.i.2177.2		2	88.43	even	2
6336.2.b.n.2177.1		2	264.197	even	2
6336.2.b.n.2177.2		2	8.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	198.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} -1.00000 q^{8} +2.82843i q^{10} +(-3.00000 - 1.41421i) q^{11} -4.24264i q^{13} +1.00000 q^{16} +6.00000 q^{17} -4.24264i q^{19} -2.82843i q^{20} +(3.00000 + 1.41421i) q^{22} +1.41421i q^{23} -3.00000 q^{25} +4.24264i q^{26} +2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -10.0000 q^{37} +4.24264i q^{38} +2.82843i q^{40} +6.00000 q^{41} +12.7279i q^{43} +(-3.00000 - 1.41421i) q^{44} -1.41421i q^{46} +9.89949i q^{47} +7.00000 q^{49} +3.00000 q^{50} -4.24264i q^{52} +5.65685i q^{53} +(-4.00000 + 8.48528i) q^{55} +5.65685i q^{59} -12.7279i q^{61} -2.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +8.00000 q^{67} +6.00000 q^{68} -15.5563i q^{71} +8.48528i q^{73} +10.0000 q^{74} -4.24264i q^{76} +8.48528i q^{79} -2.82843i q^{80} -6.00000 q^{82} -6.00000 q^{83} -16.9706i q^{85} -12.7279i q^{86} +(3.00000 + 1.41421i) q^{88} -7.07107i q^{89} +1.41421i q^{92} -9.89949i q^{94} -12.0000 q^{95} -4.00000 q^{97} -7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{11} + 2 q^{16} + 12 q^{17} + 6 q^{22} - 6 q^{25} + 4 q^{31} - 2 q^{32} - 12 q^{34} - 20 q^{37} + 12 q^{41} - 6 q^{44} + 14 q^{49} + 6 q^{50} - 8 q^{55} - 4 q^{62} + 2 q^{64} - 24 q^{65} + 16 q^{67} + 12 q^{68} + 20 q^{74} - 12 q^{82} - 12 q^{83} + 6 q^{88} - 24 q^{95} - 8 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^11 + 2 * q^16 + 12 * q^17 + 6 * q^22 - 6 * q^25 + 4 * q^31 - 2 * q^32 - 12 * q^34 - 20 * q^37 + 12 * q^41 - 6 * q^44 + 14 * q^49 + 6 * q^50 - 8 * q^55 - 4 * q^62 + 2 * q^64 - 24 * q^65 + 16 * q^67 + 12 * q^68 + 20 * q^74 - 12 * q^82 - 12 * q^83 + 6 * q^88 - 24 * q^95 - 8 * q^97 - 14 * q^98

Introduction

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