LMFDB - Embedded newform 666.2.c.b.73.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(73,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("666.73");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 666 = 2 \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	666.c (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:	$5.31803677462$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.1
Root			$-2.79129i$ of defining polynomial
Character	$\chi$	$=$	666.73
Dual form			666.2.c.b.73.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -3.79129i q^{5} -2.00000 q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -3.79129i q^{5} -2.00000 q^{7} +1.00000i q^{8} -3.79129 q^{10} -3.79129 q^{11} -0.791288i q^{13} +2.00000i q^{14} +1.00000 q^{16} -1.58258i q^{17} +7.58258i q^{19} +3.79129i q^{20} +3.79129i q^{22} +0.791288i q^{23} -9.37386 q^{25} -0.791288 q^{26} +2.00000 q^{28} -0.791288i q^{29} -5.37386i q^{31} -1.00000i q^{32} -1.58258 q^{34} +7.58258i q^{35} +(4.00000 - 4.58258i) q^{37} +7.58258 q^{38} +3.79129 q^{40} -5.20871 q^{41} +6.00000i q^{43} +3.79129 q^{44} +0.791288 q^{46} -1.58258 q^{47} -3.00000 q^{49} +9.37386i q^{50} +0.791288i q^{52} -7.58258 q^{53} +14.3739i q^{55} -2.00000i q^{56} -0.791288 q^{58} -7.58258i q^{59} -8.20871i q^{61} -5.37386 q^{62} -1.00000 q^{64} -3.00000 q^{65} -7.37386 q^{67} +1.58258i q^{68} +7.58258 q^{70} -9.16515 q^{71} +9.37386 q^{73} +(-4.58258 - 4.00000i) q^{74} -7.58258i q^{76} +7.58258 q^{77} -12.7913i q^{79} -3.79129i q^{80} +5.20871i q^{82} +3.16515 q^{83} -6.00000 q^{85} +6.00000 q^{86} -3.79129i q^{88} +6.00000i q^{89} +1.58258i q^{91} -0.791288i q^{92} +1.58258i q^{94} +28.7477 q^{95} +4.41742i q^{97} +3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 8 q^{7}+O(q^{10}) $ 4 * q - 4 * q^4 - 8 * q^7	$ 4 q - 4 q^{4} - 8 q^{7} - 6 q^{10} - 6 q^{11} + 4 q^{16} - 10 q^{25} + 6 q^{26} + 8 q^{28} + 12 q^{34} + 16 q^{37} + 12 q^{38} + 6 q^{40} - 30 q^{41} + 6 q^{44} - 6 q^{46} + 12 q^{47} - 12 q^{49} - 12 q^{53} + 6 q^{58} + 6 q^{62} - 4 q^{64} - 12 q^{65} - 2 q^{67} + 12 q^{70} + 10 q^{73} + 12 q^{77} - 24 q^{83} - 24 q^{85} + 24 q^{86} + 60 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^7 - 6 * q^10 - 6 * q^11 + 4 * q^16 - 10 * q^25 + 6 * q^26 + 8 * q^28 + 12 * q^34 + 16 * q^37 + 12 * q^38 + 6 * q^40 - 30 * q^41 + 6 * q^44 - 6 * q^46 + 12 * q^47 - 12 * q^49 - 12 * q^53 + 6 * q^58 + 6 * q^62 - 4 * q^64 - 12 * q^65 - 2 * q^67 + 12 * q^70 + 10 * q^73 + 12 * q^77 - 24 * q^83 - 24 * q^85 + 24 * q^86 + 60 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/666\mathbb{Z}\right)^\times$.

$n$	$371$	$631$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	3.79129i		−	1.69552i	−0.530384	−	0.847758i	$-0.677952\pi$
$5$		−	3.79129i		−	1.69552i	0.530384	−	0.847758i	$-0.322048\pi$
$6$		0			0
$7$	−2.00000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−3.79129			−1.19891
$11$	−3.79129			−1.14312			−0.571558	−	0.820562i	$-0.693661\pi$
$11$	−3.79129			−1.14312			−0.571558	+	0.820562i	$0.693661\pi$
$12$		0			0
$13$		−	0.791288i		−	0.219464i	−0.993961	−	0.109732i	$-0.965001\pi$
$13$		−	0.791288i		−	0.219464i	0.993961	−	0.109732i	$-0.0349992\pi$
$14$			2.00000i			0.534522i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	1.58258i		−	0.383831i	−0.981411	−	0.191915i	$-0.938530\pi$
$17$		−	1.58258i		−	0.383831i	0.981411	−	0.191915i	$-0.0614700\pi$
$18$		0			0
$19$			7.58258i			1.73956i	0.493438	+	0.869781i	$0.335740\pi$
$19$			7.58258i			1.73956i	−0.493438	+	0.869781i	$0.664260\pi$
$20$			3.79129i			0.847758i
$21$		0			0
$22$			3.79129i			0.808305i
$23$			0.791288i			0.164995i	0.996591	+	0.0824975i	$0.0262896\pi$
$23$			0.791288i			0.164995i	−0.996591	+	0.0824975i	$0.973710\pi$
$24$		0			0
$25$	−9.37386			−1.87477
$26$	−0.791288			−0.155184
$27$		0			0
$28$	2.00000			0.377964
$29$		−	0.791288i		−	0.146938i	−0.997297	−	0.0734692i	$-0.976593\pi$
$29$		−	0.791288i		−	0.146938i	0.997297	−	0.0734692i	$-0.0234071\pi$
$30$		0			0
$31$		−	5.37386i		−	0.965174i	−0.875848	−	0.482587i	$-0.839697\pi$
$31$		−	5.37386i		−	0.965174i	0.875848	−	0.482587i	$-0.160303\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−1.58258			−0.271409
$35$			7.58258i			1.28169i
$36$		0			0
$37$	4.00000	−	4.58258i	0.657596	−	0.753371i
$37$	4.00000	−	4.58258i	0.657596	−	0.753371i
$38$	7.58258			1.23006
$39$		0			0
$40$	3.79129			0.599455
$41$	−5.20871			−0.813464			−0.406732	−	0.913547i	$-0.633332\pi$
$41$	−5.20871			−0.813464			−0.406732	+	0.913547i	$0.633332\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	3.79129			0.571558
$45$		0			0
$46$	0.791288			0.116669
$47$	−1.58258			−0.230842			−0.115421	−	0.993317i	$-0.536822\pi$
$47$	−1.58258			−0.230842			−0.115421	+	0.993317i	$0.536822\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$			9.37386i			1.32566i
$51$		0			0
$52$			0.791288i			0.109732i
$53$	−7.58258			−1.04155			−0.520773	−	0.853695i	$-0.674356\pi$
$53$	−7.58258			−1.04155			−0.520773	+	0.853695i	$0.674356\pi$
$54$		0			0
$55$			14.3739i			1.93817i
$56$		−	2.00000i		−	0.267261i
$57$		0			0
$58$	−0.791288			−0.103901
$59$		−	7.58258i		−	0.987167i	−0.869698	−	0.493584i	$-0.835687\pi$
$59$		−	7.58258i		−	0.987167i	0.869698	−	0.493584i	$-0.164313\pi$
$60$		0			0
$61$		−	8.20871i		−	1.05102i	−0.850788	−	0.525509i	$-0.823875\pi$
$61$		−	8.20871i		−	1.05102i	0.850788	−	0.525509i	$-0.176125\pi$
$62$	−5.37386			−0.682481
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−3.00000			−0.372104
$66$		0			0
$67$	−7.37386			−0.900861			−0.450430	−	0.892812i	$-0.648729\pi$
$67$	−7.37386			−0.900861			−0.450430	+	0.892812i	$0.648729\pi$
$68$			1.58258i			0.191915i
$69$		0			0
$70$	7.58258			0.906291
$71$	−9.16515			−1.08770			−0.543852	−	0.839181i	$-0.683035\pi$
$71$	−9.16515			−1.08770			−0.543852	+	0.839181i	$0.683035\pi$
$72$		0			0
$73$	9.37386			1.09713			0.548564	−	0.836109i	$-0.315175\pi$
$73$	9.37386			1.09713			0.548564	+	0.836109i	$0.315175\pi$
$74$	−4.58258	−	4.00000i	−0.532714	−	0.464991i
$75$		0			0
$76$		−	7.58258i		−	0.869781i
$77$	7.58258			0.864115
$78$		0			0
$79$		−	12.7913i		−	1.43913i	−0.694424	−	0.719566i	$-0.744341\pi$
$79$		−	12.7913i		−	1.43913i	0.694424	−	0.719566i	$-0.255659\pi$
$80$		−	3.79129i		−	0.423879i
$81$		0			0
$82$			5.20871i			0.575206i
$83$	3.16515			0.347421			0.173710	−	0.984797i	$-0.444424\pi$
$83$	3.16515			0.347421			0.173710	+	0.984797i	$0.444424\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$	6.00000			0.646997
$87$		0			0
$88$		−	3.79129i		−	0.404153i
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$			1.58258i			0.165899i
$92$		−	0.791288i		−	0.0824975i
$93$		0			0
$94$			1.58258i			0.163230i
$95$	28.7477			2.94945
$96$		0			0
$97$			4.41742i			0.448521i	0.974529	+	0.224261i	$0.0719967\pi$
$97$			4.41742i			0.448521i	−0.974529	+	0.224261i	$0.928003\pi$
$98$			3.00000i			0.303046i
$99$		0			0
$100$	9.37386			0.937386
$101$	1.58258			0.157472			0.0787361	−	0.996895i	$-0.474912\pi$
$101$	1.58258			0.157472			0.0787361	+	0.996895i	$0.474912\pi$
$102$		0			0
$103$		−	2.20871i		−	0.217631i	−0.994062	−	0.108815i	$-0.965294\pi$
$103$		−	2.20871i		−	0.217631i	0.994062	−	0.108815i	$-0.0347058\pi$
$104$	0.791288			0.0775922
$105$		0			0
$106$			7.58258i			0.736485i
$107$	−8.37386			−0.809532			−0.404766	−	0.914420i	$-0.632647\pi$
$107$	−8.37386			−0.809532			−0.404766	+	0.914420i	$0.632647\pi$
$108$		0			0
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$	14.3739			1.37049
$111$		0			0
$112$	−2.00000			−0.188982
$113$		−	19.5826i		−	1.84217i	−0.389357	−	0.921087i	$-0.627303\pi$
$113$		−	19.5826i		−	1.84217i	0.389357	−	0.921087i	$-0.372697\pi$
$114$		0			0
$115$	3.00000			0.279751
$116$			0.791288i			0.0734692i
$117$		0			0
$118$	−7.58258			−0.698033
$119$			3.16515i			0.290149i
$120$		0			0
$121$	3.37386			0.306715
$122$	−8.20871			−0.743182
$123$		0			0
$124$			5.37386i			0.482587i
$125$			16.5826i			1.48319i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$			3.00000i			0.263117i
$131$		−	10.7477i		−	0.939033i	−0.882924	−	0.469517i	$-0.844428\pi$
$131$		−	10.7477i		−	0.939033i	0.882924	−	0.469517i	$-0.155572\pi$
$132$		0			0
$133$		−	15.1652i		−	1.31499i
$134$			7.37386i			0.637005i
$135$		0			0
$136$	1.58258			0.135705
$137$	17.3739			1.48435			0.742175	−	0.670207i	$-0.233794\pi$
$137$	17.3739			1.48435			0.742175	+	0.670207i	$0.233794\pi$
$138$		0			0
$139$	0.373864			0.0317107			0.0158553	−	0.999874i	$-0.494953\pi$
$139$	0.373864			0.0317107			0.0158553	+	0.999874i	$0.494953\pi$
$140$		−	7.58258i		−	0.640845i
$141$		0			0
$142$			9.16515i			0.769122i
$143$			3.00000i			0.250873i
$144$		0			0
$145$	−3.00000			−0.249136
$146$		−	9.37386i		−	0.775786i
$147$		0			0
$148$	−4.00000	+	4.58258i	−0.328798	+	0.376685i
$149$	13.5826			1.11273			0.556364	−	0.830939i	$-0.312196\pi$
$149$	13.5826			1.11273			0.556364	+	0.830939i	$0.312196\pi$
$150$		0			0
$151$	12.7477			1.03740			0.518698	−	0.854958i	$-0.326417\pi$
$151$	12.7477			1.03740			0.518698	+	0.854958i	$0.326417\pi$
$152$	−7.58258			−0.615028
$153$		0			0
$154$		−	7.58258i		−	0.611021i
$155$	−20.3739			−1.63647
$156$		0			0
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−12.7913			−1.01762
$159$		0			0
$160$	−3.79129			−0.299728
$161$		−	1.58258i		−	0.124724i
$162$		0			0
$163$		−	10.4174i		−	0.815956i	−0.912992	−	0.407978i	$-0.866234\pi$
$163$		−	10.4174i		−	0.815956i	0.912992	−	0.407978i	$-0.133766\pi$
$164$	5.20871			0.406732
$165$		0			0
$166$		−	3.16515i		−	0.245663i
$167$			0.956439i			0.0740115i	0.999315	+	0.0370057i	$0.0117820\pi$
$167$			0.956439i			0.0740115i	−0.999315	+	0.0370057i	$0.988218\pi$
$168$		0			0
$169$	12.3739			0.951836
$170$			6.00000i			0.460179i
$171$		0			0
$172$		−	6.00000i		−	0.457496i
$173$	3.16515			0.240642			0.120321	−	0.992735i	$-0.461608\pi$
$173$	3.16515			0.240642			0.120321	+	0.992735i	$0.461608\pi$
$174$		0			0
$175$	18.7477			1.41719
$176$	−3.79129			−0.285779
$177$		0			0
$178$	6.00000			0.449719
$179$		−	7.58258i		−	0.566748i	−0.959009	−	0.283374i	$-0.908546\pi$
$179$		−	7.58258i		−	0.566748i	0.959009	−	0.283374i	$-0.0914538\pi$
$180$		0			0
$181$	−18.7477			−1.39351			−0.696754	−	0.717310i	$-0.745373\pi$
$181$	−18.7477			−1.39351			−0.696754	+	0.717310i	$0.745373\pi$
$182$	1.58258			0.117308
$183$		0			0
$184$	−0.791288			−0.0583345
$185$	−17.3739	−	15.1652i	−1.27735	−	1.11496i
$186$		0			0
$187$			6.00000i			0.438763i
$188$	1.58258			0.115421
$189$		0			0
$190$		−	28.7477i		−	2.08558i
$191$		−	5.37386i		−	0.388839i	−0.980918	−	0.194420i	$-0.937718\pi$
$191$		−	5.37386i		−	0.388839i	0.980918	−	0.194420i	$-0.0622823\pi$
$192$		0			0
$193$		−	18.3303i		−	1.31944i	−0.751510	−	0.659722i	$-0.770674\pi$
$193$		−	18.3303i		−	1.31944i	0.751510	−	0.659722i	$-0.229326\pi$
$194$	4.41742			0.317153
$195$		0			0
$196$	3.00000			0.214286
$197$	−25.9129			−1.84622			−0.923108	−	0.384541i	$-0.874360\pi$
$197$	−25.9129			−1.84622			−0.923108	+	0.384541i	$0.874360\pi$
$198$		0			0
$199$		−	3.16515i		−	0.224372i	−0.993687	−	0.112186i	$-0.964215\pi$
$199$		−	3.16515i		−	0.224372i	0.993687	−	0.112186i	$-0.0357852\pi$
$200$		−	9.37386i		−	0.662832i
$201$		0			0
$202$		−	1.58258i		−	0.111350i
$203$			1.58258i			0.111075i
$204$		0			0
$205$			19.7477i			1.37924i
$206$	−2.20871			−0.153888
$207$		0			0
$208$		−	0.791288i		−	0.0548659i
$209$		−	28.7477i		−	1.98852i
$210$		0			0
$211$	−3.37386			−0.232266			−0.116133	−	0.993234i	$-0.537050\pi$
$211$	−3.37386			−0.232266			−0.116133	+	0.993234i	$0.537050\pi$
$212$	7.58258			0.520773
$213$		0			0
$214$			8.37386i			0.572426i
$215$	22.7477			1.55138
$216$		0			0
$217$			10.7477i			0.729603i
$218$	−6.00000			−0.406371
$219$		0			0
$220$		−	14.3739i		−	0.969086i
$221$	−1.25227			−0.0842370
$222$		0			0
$223$	14.0000			0.937509			0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000			0.937509			0.468755	+	0.883328i	$0.344703\pi$
$224$			2.00000i			0.133631i
$225$		0			0
$226$	−19.5826			−1.30261
$227$			19.9129i			1.32166i	0.750534	+	0.660832i	$0.229796\pi$
$227$			19.9129i			1.32166i	−0.750534	+	0.660832i	$0.770204\pi$
$228$		0			0
$229$	−20.7477			−1.37105			−0.685524	−	0.728050i	$-0.740427\pi$
$229$	−20.7477			−1.37105			−0.685524	+	0.728050i	$0.740427\pi$
$230$		−	3.00000i		−	0.197814i
$231$		0			0
$232$	0.791288			0.0519506
$233$	−25.1216			−1.64577			−0.822885	−	0.568208i	$-0.807637\pi$
$233$	−25.1216			−1.64577			−0.822885	+	0.568208i	$0.807637\pi$
$234$		0			0
$235$			6.00000i			0.391397i
$236$			7.58258i			0.493584i
$237$		0			0
$238$	3.16515			0.205166
$239$			24.9564i			1.61430i	0.590348	+	0.807149i	$0.298991\pi$
$239$			24.9564i			1.61430i	−0.590348	+	0.807149i	$0.701009\pi$
$240$		0			0
$241$		−	13.5826i		−	0.874931i	−0.899235	−	0.437465i	$-0.855876\pi$
$241$		−	13.5826i		−	0.874931i	0.899235	−	0.437465i	$-0.144124\pi$
$242$		−	3.37386i		−	0.216880i
$243$		0			0
$244$			8.20871i			0.525509i
$245$			11.3739i			0.726649i
$246$		0			0
$247$	6.00000			0.381771
$248$	5.37386			0.341241
$249$		0			0
$250$	16.5826			1.04877
$251$			13.5826i			0.857325i	0.903465	+	0.428662i	$0.141015\pi$
$251$			13.5826i			0.857325i	−0.903465	+	0.428662i	$0.858985\pi$
$252$		0			0
$253$		−	3.00000i		−	0.188608i
$254$		−	8.00000i		−	0.501965i
$255$		0			0
$256$	1.00000			0.0625000
$257$			22.7477i			1.41896i	0.704723	+	0.709482i	$0.251071\pi$
$257$			22.7477i			1.41896i	−0.704723	+	0.709482i	$0.748929\pi$
$258$		0			0
$259$	−8.00000	+	9.16515i	−0.497096	+	0.569495i
$260$	3.00000			0.186052
$261$		0			0
$262$	−10.7477			−0.663997
$263$	8.83485			0.544780			0.272390	−	0.962187i	$-0.412186\pi$
$263$	8.83485			0.544780			0.272390	+	0.962187i	$0.412186\pi$
$264$		0			0
$265$			28.7477i			1.76596i
$266$	−15.1652			−0.929835
$267$		0			0
$268$	7.37386			0.450430
$269$	10.7477			0.655300			0.327650	−	0.944799i	$-0.393743\pi$
$269$	10.7477			0.655300			0.327650	+	0.944799i	$0.393743\pi$
$270$		0			0
$271$	22.0000			1.33640			0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000			1.33640			0.668202	+	0.743980i	$0.267064\pi$
$272$		−	1.58258i		−	0.0959577i
$273$		0			0
$274$		−	17.3739i		−	1.04959i
$275$	35.5390			2.14308
$276$		0			0
$277$			23.3739i			1.40440i	0.711980	+	0.702200i	$0.247799\pi$
$277$			23.3739i			1.40440i	−0.711980	+	0.702200i	$0.752201\pi$
$278$		−	0.373864i		−	0.0224228i
$279$		0			0
$280$	−7.58258			−0.453146
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		−	24.0000i		−	1.42665i	−0.700832	−	0.713326i	$-0.747188\pi$
$283$		−	24.0000i		−	1.42665i	0.700832	−	0.713326i	$-0.252812\pi$
$284$	9.16515			0.543852
$285$		0			0
$286$	3.00000			0.177394
$287$	10.4174			0.614921
$288$		0			0
$289$	14.4955			0.852674
$290$			3.00000i			0.176166i
$291$		0			0
$292$	−9.37386			−0.548564
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$	−28.7477			−1.67376
$296$	4.58258	+	4.00000i	0.266357	+	0.232495i
$297$		0			0
$298$		−	13.5826i		−	0.786817i
$299$	0.626136			0.0362104
$300$		0			0
$301$		−	12.0000i		−	0.691669i
$302$		−	12.7477i		−	0.733549i
$303$		0			0
$304$			7.58258i			0.434891i
$305$	−31.1216			−1.78202
$306$		0			0
$307$	−12.3739			−0.706214			−0.353107	−	0.935583i	$-0.614875\pi$
$307$	−12.3739			−0.706214			−0.353107	+	0.935583i	$0.614875\pi$
$308$	−7.58258			−0.432057
$309$		0			0
$310$			20.3739i			1.15716i
$311$		−	9.62614i		−	0.545848i	−0.962036	−	0.272924i	$-0.912009\pi$
$311$		−	9.62614i		−	0.545848i	0.962036	−	0.272924i	$-0.0879908\pi$
$312$		0			0
$313$			33.1652i			1.87461i	0.348517	+	0.937303i	$0.386685\pi$
$313$			33.1652i			1.87461i	−0.348517	+	0.937303i	$0.613315\pi$
$314$			2.00000i			0.112867i
$315$		0			0
$316$			12.7913i			0.719566i
$317$	1.25227			0.0703347			0.0351673	−	0.999381i	$-0.488804\pi$
$317$	1.25227			0.0703347			0.0351673	+	0.999381i	$0.488804\pi$
$318$		0			0
$319$			3.00000i			0.167968i
$320$			3.79129i			0.211939i
$321$		0			0
$322$	−1.58258			−0.0881935
$323$	12.0000			0.667698
$324$		0			0
$325$			7.41742i			0.411445i
$326$	−10.4174			−0.576968
$327$		0			0
$328$		−	5.20871i		−	0.287603i
$329$	3.16515			0.174500
$330$		0			0
$331$			16.4174i			0.902383i	0.892427	+	0.451192i	$0.149001\pi$
$331$			16.4174i			0.902383i	−0.892427	+	0.451192i	$0.850999\pi$
$332$	−3.16515			−0.173710
$333$		0			0
$334$	0.956439			0.0523340
$335$			27.9564i			1.52742i
$336$		0			0
$337$	−8.12159			−0.442411			−0.221206	−	0.975227i	$-0.570999\pi$
$337$	−8.12159			−0.442411			−0.221206	+	0.975227i	$0.570999\pi$
$338$		−	12.3739i		−	0.673049i
$339$		0			0
$340$	6.00000			0.325396
$341$			20.3739i			1.10331i
$342$		0			0
$343$	20.0000			1.07990
$344$	−6.00000			−0.323498
$345$		0			0
$346$		−	3.16515i		−	0.170160i
$347$		−	1.58258i		−	0.0849571i	−0.999097	−	0.0424786i	$-0.986475\pi$
$347$		−	1.58258i		−	0.0849571i	0.999097	−	0.0424786i	$-0.0135254\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		−	18.7477i		−	1.00211i
$351$		0			0
$352$			3.79129i			0.202076i
$353$			7.58258i			0.403580i	0.979429	+	0.201790i	$0.0646758\pi$
$353$			7.58258i			0.403580i	−0.979429	+	0.201790i	$0.935324\pi$
$354$		0			0
$355$			34.7477i			1.84422i
$356$		−	6.00000i		−	0.317999i
$357$		0			0
$358$	−7.58258			−0.400752
$359$	27.1652			1.43372			0.716861	−	0.697216i	$-0.245578\pi$
$359$	27.1652			1.43372			0.716861	+	0.697216i	$0.245578\pi$
$360$		0			0
$361$	−38.4955			−2.02608
$362$			18.7477i			0.985359i
$363$		0			0
$364$		−	1.58258i		−	0.0829495i
$365$		−	35.5390i		−	1.86020i
$366$		0			0
$367$	−32.7477			−1.70942			−0.854709	−	0.519108i	$-0.826264\pi$
$367$	−32.7477			−1.70942			−0.854709	+	0.519108i	$0.826264\pi$
$368$			0.791288i			0.0412487i
$369$		0			0
$370$	−15.1652	+	17.3739i	−0.788399	+	0.903224i
$371$	15.1652			0.787335
$372$		0			0
$373$	14.7477			0.763608			0.381804	−	0.924243i	$-0.375303\pi$
$373$	14.7477			0.763608			0.381804	+	0.924243i	$0.375303\pi$
$374$	6.00000			0.310253
$375$		0			0
$376$		−	1.58258i		−	0.0816151i
$377$	−0.626136			−0.0322477
$378$		0			0
$379$	−21.1216			−1.08494			−0.542472	−	0.840074i	$-0.682511\pi$
$379$	−21.1216			−1.08494			−0.542472	+	0.840074i	$0.682511\pi$
$380$	−28.7477			−1.47473
$381$		0			0
$382$	−5.37386			−0.274951
$383$			18.3303i			0.936635i	0.883560	+	0.468317i	$0.155140\pi$
$383$			18.3303i			0.936635i	−0.883560	+	0.468317i	$0.844860\pi$
$384$		0			0
$385$		−	28.7477i		−	1.46512i
$386$	−18.3303			−0.932988
$387$		0			0
$388$		−	4.41742i		−	0.224261i
$389$		−	35.8693i		−	1.81865i	−0.416090	−	0.909323i	$-0.636600\pi$
$389$		−	35.8693i		−	1.81865i	0.416090	−	0.909323i	$-0.363400\pi$
$390$		0			0
$391$	1.25227			0.0633302
$392$		−	3.00000i		−	0.151523i
$393$		0			0
$394$			25.9129i			1.30547i
$395$	−48.4955			−2.44007
$396$		0			0
$397$	−12.7477			−0.639790			−0.319895	−	0.947453i	$-0.603648\pi$
$397$	−12.7477			−0.639790			−0.319895	+	0.947453i	$0.603648\pi$
$398$	−3.16515			−0.158655
$399$		0			0
$400$	−9.37386			−0.468693
$401$			10.7477i			0.536716i	0.963319	+	0.268358i	$0.0864810\pi$
$401$			10.7477i			0.536716i	−0.963319	+	0.268358i	$0.913519\pi$
$402$		0			0
$403$	−4.25227			−0.211821
$404$	−1.58258			−0.0787361
$405$		0			0
$406$	1.58258			0.0785419
$407$	−15.1652	+	17.3739i	−0.751709	+	0.861190i
$408$		0			0
$409$		−	8.83485i		−	0.436855i	−0.975853	−	0.218428i	$-0.929907\pi$
$409$		−	8.83485i		−	0.436855i	0.975853	−	0.218428i	$-0.0700927\pi$
$410$	19.7477			0.975271
$411$		0			0
$412$			2.20871i			0.108815i
$413$			15.1652i			0.746228i
$414$		0			0
$415$		−	12.0000i		−	0.589057i
$416$	−0.791288			−0.0387961
$417$		0			0
$418$	−28.7477			−1.40610
$419$	−6.79129			−0.331776			−0.165888	−	0.986145i	$-0.553049\pi$
$419$	−6.79129			−0.331776			−0.165888	+	0.986145i	$0.553049\pi$
$420$		0			0
$421$		−	3.95644i		−	0.192825i	−0.995341	−	0.0964125i	$-0.969263\pi$
$421$		−	3.95644i		−	0.192825i	0.995341	−	0.0964125i	$-0.0307368\pi$
$422$			3.37386i			0.164237i
$423$		0			0
$424$		−	7.58258i		−	0.368242i
$425$			14.8348i			0.719596i
$426$		0			0
$427$			16.4174i			0.794495i
$428$	8.37386			0.404766
$429$		0			0
$430$		−	22.7477i		−	1.09699i
$431$		−	27.1652i		−	1.30850i	−0.756279	−	0.654250i	$-0.772985\pi$
$431$		−	27.1652i		−	1.30850i	0.756279	−	0.654250i	$-0.227015\pi$
$432$		0			0
$433$	24.3739			1.17133			0.585667	−	0.810552i	$-0.300833\pi$
$433$	24.3739			1.17133			0.585667	+	0.810552i	$0.300833\pi$
$434$	10.7477			0.515907
$435$		0			0
$436$			6.00000i			0.287348i
$437$	−6.00000			−0.287019
$438$		0			0
$439$		−	26.3739i		−	1.25876i	−0.777099	−	0.629378i	$-0.783310\pi$
$439$		−	26.3739i		−	1.25876i	0.777099	−	0.629378i	$-0.216690\pi$
$440$	−14.3739			−0.685247
$441$		0			0
$442$			1.25227i			0.0595645i
$443$	−5.04356			−0.239627			−0.119813	−	0.992796i	$-0.538230\pi$
$443$	−5.04356			−0.239627			−0.119813	+	0.992796i	$0.538230\pi$
$444$		0			0
$445$	22.7477			1.07835
$446$		−	14.0000i		−	0.662919i
$447$		0			0
$448$	2.00000			0.0944911
$449$		−	21.1652i		−	0.998845i	−0.866358	−	0.499423i	$-0.833546\pi$
$449$		−	21.1652i		−	0.998845i	0.866358	−	0.499423i	$-0.166454\pi$
$450$		0			0
$451$	19.7477			0.929884
$452$			19.5826i			0.921087i
$453$		0			0
$454$	19.9129			0.934558
$455$	6.00000			0.281284
$456$		0			0
$457$		−	33.4955i		−	1.56685i	−0.621486	−	0.783426i	$-0.713471\pi$
$457$		−	33.4955i		−	1.56685i	0.621486	−	0.783426i	$-0.286529\pi$
$458$			20.7477i			0.969478i
$459$		0			0
$460$	−3.00000			−0.139876
$461$		−	24.3303i		−	1.13318i	−0.824002	−	0.566588i	$-0.808263\pi$
$461$		−	24.3303i		−	1.13318i	0.824002	−	0.566588i	$-0.191737\pi$
$462$		0			0
$463$			20.7042i			0.962204i	0.876665	+	0.481102i	$0.159763\pi$
$463$			20.7042i			0.962204i	−0.876665	+	0.481102i	$0.840237\pi$
$464$		−	0.791288i		−	0.0367346i
$465$		0			0
$466$			25.1216i			1.16374i
$467$			25.5826i			1.18382i	0.806004	+	0.591910i	$0.201626\pi$
$467$			25.5826i			1.18382i	−0.806004	+	0.591910i	$0.798374\pi$
$468$		0			0
$469$	14.7477			0.680987
$470$	6.00000			0.276759
$471$		0			0
$472$	7.58258			0.349016
$473$		−	22.7477i		−	1.04594i
$474$		0			0
$475$		−	71.0780i		−	3.26128i
$476$		−	3.16515i		−	0.145074i
$477$		0			0
$478$	24.9564			1.14148
$479$		−	0.791288i		−	0.0361549i	−0.999837	−	0.0180774i	$-0.994245\pi$
$479$		−	0.791288i		−	0.0361549i	0.999837	−	0.0180774i	$-0.00575454\pi$
$480$		0			0
$481$	−3.62614	−	3.16515i	−0.165338	−	0.144318i
$482$	−13.5826			−0.618669
$483$		0			0
$484$	−3.37386			−0.153357
$485$	16.7477			0.760475
$486$		0			0
$487$			36.6606i			1.66125i	0.556832	+	0.830625i	$0.312017\pi$
$487$			36.6606i			1.66125i	−0.556832	+	0.830625i	$0.687983\pi$
$488$	8.20871			0.371591
$489$		0			0
$490$	11.3739			0.513819
$491$	8.37386			0.377907			0.188954	−	0.981986i	$-0.439490\pi$
$491$	8.37386			0.377907			0.188954	+	0.981986i	$0.439490\pi$
$492$		0			0
$493$	−1.25227			−0.0563995
$494$		−	6.00000i		−	0.269953i
$495$		0			0
$496$		−	5.37386i		−	0.241294i
$497$	18.3303			0.822226
$498$		0			0
$499$		−	16.7477i		−	0.749731i	−0.927079	−	0.374866i	$-0.877689\pi$
$499$		−	16.7477i		−	0.749731i	0.927079	−	0.374866i	$-0.122311\pi$
$500$		−	16.5826i		−	0.741595i
$501$		0			0
$502$	13.5826			0.606220
$503$			29.3739i			1.30972i	0.755752	+	0.654858i	$0.227272\pi$
$503$			29.3739i			1.30972i	−0.755752	+	0.654858i	$0.772728\pi$
$504$		0			0
$505$		−	6.00000i		−	0.266996i
$506$	−3.00000			−0.133366
$507$		0			0
$508$	−8.00000			−0.354943
$509$	−13.5826			−0.602037			−0.301019	−	0.953618i	$-0.597327\pi$
$509$	−13.5826			−0.602037			−0.301019	+	0.953618i	$0.597327\pi$
$510$		0			0
$511$	−18.7477			−0.829351
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	22.7477			1.00336
$515$	−8.37386			−0.368997
$516$		0			0
$517$	6.00000			0.263880
$518$	9.16515	+	8.00000i	0.402694	+	0.351500i
$519$		0			0
$520$		−	3.00000i		−	0.131559i
$521$	−9.16515			−0.401533			−0.200766	−	0.979639i	$-0.564343\pi$
$521$	−9.16515			−0.401533			−0.200766	+	0.979639i	$0.564343\pi$
$522$		0			0
$523$			3.16515i			0.138402i	0.997603	+	0.0692012i	$0.0220450\pi$
$523$			3.16515i			0.138402i	−0.997603	+	0.0692012i	$0.977955\pi$
$524$			10.7477i			0.469517i
$525$		0			0
$526$		−	8.83485i		−	0.385218i
$527$	−8.50455			−0.370464
$528$		0			0
$529$	22.3739			0.972777
$530$	28.7477			1.24872
$531$		0			0
$532$			15.1652i			0.657493i
$533$			4.12159i			0.178526i
$534$		0			0
$535$			31.7477i			1.37257i
$536$		−	7.37386i		−	0.318502i
$537$		0			0
$538$		−	10.7477i		−	0.463367i
$539$	11.3739			0.489907
$540$		0			0
$541$			8.20871i			0.352920i	0.984308	+	0.176460i	$0.0564646\pi$
$541$			8.20871i			0.352920i	−0.984308	+	0.176460i	$0.943535\pi$
$542$		−	22.0000i		−	0.944981i
$543$		0			0
$544$	−1.58258			−0.0678524
$545$	−22.7477			−0.974406
$546$		0			0
$547$		−	19.9129i		−	0.851413i	−0.904861	−	0.425707i	$-0.860026\pi$
$547$		−	19.9129i		−	0.851413i	0.904861	−	0.425707i	$-0.139974\pi$
$548$	−17.3739			−0.742175
$549$		0			0
$550$		−	35.5390i		−	1.51539i
$551$	6.00000			0.255609
$552$		0			0
$553$			25.5826i			1.08788i
$554$	23.3739			0.993060
$555$		0			0
$556$	−0.373864			−0.0158553
$557$			41.7042i			1.76706i	0.468372	+	0.883531i	$0.344841\pi$
$557$			41.7042i			1.76706i	−0.468372	+	0.883531i	$0.655159\pi$
$558$		0			0
$559$	4.74773			0.200807
$560$			7.58258i			0.320422i
$561$		0			0
$562$		0			0
$563$		−	16.7477i		−	0.705833i	−0.935655	−	0.352916i	$-0.885190\pi$
$563$		−	16.7477i		−	0.705833i	0.935655	−	0.352916i	$-0.114810\pi$
$564$		0			0
$565$	−74.2432			−3.12343
$566$	−24.0000			−1.00880
$567$		0			0
$568$		−	9.16515i		−	0.384561i
$569$		−	26.8348i		−	1.12498i	−0.826806	−	0.562488i	$-0.809844\pi$
$569$		−	26.8348i		−	1.12498i	0.826806	−	0.562488i	$-0.190156\pi$
$570$		0			0
$571$	−28.3739			−1.18741			−0.593705	−	0.804683i	$-0.702335\pi$
$571$	−28.3739			−1.18741			−0.593705	+	0.804683i	$0.702335\pi$
$572$		−	3.00000i		−	0.125436i
$573$		0			0
$574$		−	10.4174i		−	0.434815i
$575$		−	7.41742i		−	0.309328i
$576$		0			0
$577$			4.41742i			0.183900i	0.995764	+	0.0919499i	$0.0293100\pi$
$577$			4.41742i			0.183900i	−0.995764	+	0.0919499i	$0.970690\pi$
$578$		−	14.4955i		−	0.602931i
$579$		0			0
$580$	3.00000			0.124568
$581$	−6.33030			−0.262625
$582$		0			0
$583$	28.7477			1.19061
$584$			9.37386i			0.387893i
$585$		0			0
$586$		−	6.00000i		−	0.247858i
$587$		−	25.9129i		−	1.06954i	−0.844998	−	0.534769i	$-0.820398\pi$
$587$		−	25.9129i		−	1.06954i	0.844998	−	0.534769i	$-0.179602\pi$
$588$		0			0
$589$	40.7477			1.67898
$590$			28.7477i			1.18353i
$591$		0			0
$592$	4.00000	−	4.58258i	0.164399	−	0.188343i
$593$	14.2087			0.583482			0.291741	−	0.956497i	$-0.405765\pi$
$593$	14.2087			0.583482			0.291741	+	0.956497i	$0.405765\pi$
$594$		0			0
$595$	12.0000			0.491952
$596$	−13.5826			−0.556364
$597$		0			0
$598$		−	0.626136i		−	0.0256046i
$599$	−27.1652			−1.10994			−0.554969	−	0.831871i	$-0.687270\pi$
$599$	−27.1652			−1.10994			−0.554969	+	0.831871i	$0.687270\pi$
$600$		0			0
$601$	8.12159			0.331287			0.165643	−	0.986186i	$-0.447030\pi$
$601$	8.12159			0.331287			0.165643	+	0.986186i	$0.447030\pi$
$602$	−12.0000			−0.489083
$603$		0			0
$604$	−12.7477			−0.518698
$605$		−	12.7913i		−	0.520040i
$606$		0			0
$607$			5.53901i			0.224822i	0.993662	+	0.112411i	$0.0358573\pi$
$607$			5.53901i			0.224822i	−0.993662	+	0.112411i	$0.964143\pi$
$608$	7.58258			0.307514
$609$		0			0
$610$			31.1216i			1.26008i
$611$			1.25227i			0.0506615i
$612$		0			0
$613$	5.49545			0.221959			0.110980	−	0.993823i	$-0.464601\pi$
$613$	5.49545			0.221959			0.110980	+	0.993823i	$0.464601\pi$
$614$			12.3739i			0.499368i
$615$		0			0
$616$			7.58258i			0.305511i
$617$	45.9564			1.85014			0.925068	−	0.379801i	$-0.124007\pi$
$617$	45.9564			1.85014			0.925068	+	0.379801i	$0.124007\pi$
$618$		0			0
$619$	6.12159			0.246048			0.123024	−	0.992404i	$-0.460741\pi$
$619$	6.12159			0.246048			0.123024	+	0.992404i	$0.460741\pi$
$620$	20.3739			0.818234
$621$		0			0
$622$	−9.62614			−0.385973
$623$		−	12.0000i		−	0.480770i
$624$		0			0
$625$	16.0000			0.640000
$626$	33.1652			1.32555
$627$		0			0
$628$	2.00000			0.0798087
$629$	−7.25227	−	6.33030i	−0.289167	−	0.252406i
$630$		0			0
$631$			3.95644i			0.157503i	0.996894	+	0.0787517i	$0.0250934\pi$
$631$			3.95644i			0.157503i	−0.996894	+	0.0787517i	$0.974907\pi$
$632$	12.7913			0.508810
$633$		0			0
$634$		−	1.25227i		−	0.0497341i
$635$		−	30.3303i		−	1.20362i
$636$		0			0
$637$			2.37386i			0.0940559i
$638$	3.00000			0.118771
$639$		0			0
$640$	3.79129			0.149864
$641$	35.5390			1.40371			0.701853	−	0.712321i	$-0.252356\pi$
$641$	35.5390			1.40371			0.701853	+	0.712321i	$0.252356\pi$
$642$		0			0
$643$		−	15.4955i		−	0.611081i	−0.952179	−	0.305541i	$-0.901163\pi$
$643$		−	15.4955i		−	0.611081i	0.952179	−	0.305541i	$-0.0988371\pi$
$644$			1.58258i			0.0623622i
$645$		0			0
$646$		−	12.0000i		−	0.472134i
$647$		−	32.7042i		−	1.28573i	−0.765978	−	0.642867i	$-0.777745\pi$
$647$		−	32.7042i		−	1.28573i	0.765978	−	0.642867i	$-0.222255\pi$
$648$		0			0
$649$			28.7477i			1.12845i
$650$	7.41742			0.290935
$651$		0			0
$652$			10.4174i			0.407978i
$653$			14.3739i			0.562493i	0.959636	+	0.281246i	$0.0907478\pi$
$653$			14.3739i			0.562493i	−0.959636	+	0.281246i	$0.909252\pi$
$654$		0			0
$655$	−40.7477			−1.59215
$656$	−5.20871			−0.203366
$657$		0			0
$658$		−	3.16515i		−	0.123390i
$659$	−33.9564			−1.32276			−0.661378	−	0.750053i	$-0.730028\pi$
$659$	−33.9564			−1.32276			−0.661378	+	0.750053i	$0.730028\pi$
$660$		0			0
$661$		−	36.7913i		−	1.43102i	−0.698605	−	0.715508i	$-0.746195\pi$
$661$		−	36.7913i		−	1.43102i	0.698605	−	0.715508i	$-0.253805\pi$
$662$	16.4174			0.638081
$663$		0			0
$664$			3.16515i			0.122832i
$665$	−57.4955			−2.22958
$666$		0			0
$667$	0.626136			0.0242441
$668$		−	0.956439i		−	0.0370057i
$669$		0			0
$670$	27.9564			1.08005
$671$			31.1216i			1.20144i
$672$		0			0
$673$	5.12159			0.197423			0.0987114	−	0.995116i	$-0.468528\pi$
$673$	5.12159			0.197423			0.0987114	+	0.995116i	$0.468528\pi$
$674$			8.12159i			0.312832i
$675$		0			0
$676$	−12.3739			−0.475918
$677$	9.16515			0.352245			0.176123	−	0.984368i	$-0.443644\pi$
$677$	9.16515			0.352245			0.176123	+	0.984368i	$0.443644\pi$
$678$		0			0
$679$		−	8.83485i		−	0.339050i
$680$		−	6.00000i		−	0.230089i
$681$		0			0
$682$	20.3739			0.780156
$683$		−	22.4174i		−	0.857779i	−0.903357	−	0.428889i	$-0.858905\pi$
$683$		−	22.4174i		−	0.857779i	0.903357	−	0.428889i	$-0.141095\pi$
$684$		0			0
$685$		−	65.8693i		−	2.51674i
$686$		−	20.0000i		−	0.763604i
$687$		0			0
$688$			6.00000i			0.228748i
$689$			6.00000i			0.228582i
$690$		0			0
$691$	−29.4955			−1.12206			−0.561030	−	0.827795i	$-0.689595\pi$
$691$	−29.4955			−1.12206			−0.561030	+	0.827795i	$0.689595\pi$
$692$	−3.16515			−0.120321
$693$		0			0
$694$	−1.58258			−0.0600738
$695$		−	1.41742i		−	0.0537660i
$696$		0			0
$697$			8.24318i			0.312233i
$698$		−	10.0000i		−	0.378506i
$699$		0			0
$700$	−18.7477			−0.708597
$701$		−	18.9564i		−	0.715975i	−0.933726	−	0.357987i	$-0.883463\pi$
$701$		−	18.9564i		−	0.715975i	0.933726	−	0.357987i	$-0.116537\pi$
$702$		0			0
$703$	34.7477	+	30.3303i	1.31054	+	1.14393i
$704$	3.79129			0.142890
$705$		0			0
$706$	7.58258			0.285374
$707$	−3.16515			−0.119038
$708$		0			0
$709$		−	27.7913i		−	1.04372i	−0.853030	−	0.521862i	$-0.825238\pi$
$709$		−	27.7913i		−	1.04372i	0.853030	−	0.521862i	$-0.174762\pi$
$710$	34.7477			1.30406
$711$		0			0
$712$	−6.00000			−0.224860
$713$	4.25227			0.159249
$714$		0			0
$715$	11.3739			0.425358
$716$			7.58258i			0.283374i
$717$		0			0
$718$		−	27.1652i		−	1.01379i
$719$	2.83485			0.105722			0.0528610	−	0.998602i	$-0.483166\pi$
$719$	2.83485			0.105722			0.0528610	+	0.998602i	$0.483166\pi$
$720$		0			0
$721$			4.41742i			0.164513i
$722$			38.4955i			1.43265i
$723$		0			0
$724$	18.7477			0.696754
$725$			7.41742i			0.275476i
$726$		0			0
$727$		−	28.1216i		−	1.04297i	−0.853260	−	0.521486i	$-0.825378\pi$
$727$		−	28.1216i		−	1.04297i	0.853260	−	0.521486i	$-0.174622\pi$
$728$	−1.58258			−0.0586542
$729$		0			0
$730$	−35.5390			−1.31536
$731$	9.49545			0.351202
$732$		0			0
$733$	−7.49545			−0.276851			−0.138425	−	0.990373i	$-0.544204\pi$
$733$	−7.49545			−0.276851			−0.138425	+	0.990373i	$0.544204\pi$
$734$			32.7477i			1.20874i
$735$		0			0
$736$	0.791288			0.0291673
$737$	27.9564			1.02979
$738$		0			0
$739$	6.12159			0.225186			0.112593	−	0.993641i	$-0.464084\pi$
$739$	6.12159			0.225186			0.112593	+	0.993641i	$0.464084\pi$
$740$	17.3739	+	15.1652i	0.638676	+	0.557482i
$741$		0			0
$742$		−	15.1652i		−	0.556730i
$743$	3.16515			0.116118			0.0580591	−	0.998313i	$-0.481509\pi$
$743$	3.16515			0.116118			0.0580591	+	0.998313i	$0.481509\pi$
$744$		0			0
$745$		−	51.4955i		−	1.88665i
$746$		−	14.7477i		−	0.539953i
$747$		0			0
$748$		−	6.00000i		−	0.219382i
$749$	16.7477			0.611949
$750$		0			0
$751$	13.4955			0.492456			0.246228	−	0.969212i	$-0.420809\pi$
$751$	13.4955			0.492456			0.246228	+	0.969212i	$0.420809\pi$
$752$	−1.58258			−0.0577106
$753$		0			0
$754$			0.626136i			0.0228025i
$755$		−	48.3303i		−	1.75892i
$756$		0			0
$757$			39.7913i			1.44624i	0.690723	+	0.723119i	$0.257292\pi$
$757$			39.7913i			1.44624i	−0.690723	+	0.723119i	$0.742708\pi$
$758$			21.1216i			0.767171i
$759$		0			0
$760$			28.7477i			1.04279i
$761$	−18.7913			−0.681184			−0.340592	−	0.940211i	$-0.610627\pi$
$761$	−18.7913			−0.681184			−0.340592	+	0.940211i	$0.610627\pi$
$762$		0			0
$763$			12.0000i			0.434429i
$764$			5.37386i			0.194420i
$765$		0			0
$766$	18.3303			0.662301
$767$	−6.00000			−0.216647
$768$		0			0
$769$			48.3303i			1.74284i	0.490542	+	0.871418i	$0.336799\pi$
$769$			48.3303i			1.74284i	−0.490542	+	0.871418i	$0.663201\pi$
$770$	−28.7477			−1.03600
$771$		0			0
$772$			18.3303i			0.659722i
$773$	13.9129			0.500411			0.250206	−	0.968193i	$-0.419502\pi$
$773$	13.9129			0.500411			0.250206	+	0.968193i	$0.419502\pi$
$774$		0			0
$775$			50.3739i			1.80948i
$776$	−4.41742			−0.158576
$777$		0			0
$778$	−35.8693			−1.28598
$779$		−	39.4955i		−	1.41507i
$780$		0			0
$781$	34.7477			1.24337
$782$		−	1.25227i		−	0.0447812i
$783$		0			0
$784$	−3.00000			−0.107143
$785$			7.58258i			0.270634i
$786$		0			0
$787$	8.00000			0.285169			0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000			0.285169			0.142585	+	0.989783i	$0.454459\pi$
$788$	25.9129			0.923108
$789$		0			0
$790$			48.4955i			1.72539i
$791$			39.1652i			1.39255i
$792$		0			0
$793$	−6.49545			−0.230660
$794$			12.7477i			0.452400i
$795$		0			0
$796$			3.16515i			0.112186i
$797$			21.6261i			0.766037i	0.923741	+	0.383019i	$0.125115\pi$
$797$			21.6261i			0.766037i	−0.923741	+	0.383019i	$0.874885\pi$
$798$		0			0
$799$			2.50455i			0.0886045i
$800$			9.37386i			0.331416i
$801$		0			0
$802$	10.7477			0.379515
$803$	−35.5390			−1.25414
$804$		0			0
$805$	−6.00000			−0.211472
$806$			4.25227i			0.149780i
$807$		0			0
$808$			1.58258i			0.0556748i
$809$			11.0780i			0.389483i	0.980855	+	0.194741i	$0.0623868\pi$
$809$			11.0780i			0.389483i	−0.980855	+	0.194741i	$0.937613\pi$
$810$		0			0
$811$	48.8693			1.71603			0.858017	−	0.513621i	$-0.171696\pi$
$811$	48.8693			1.71603			0.858017	+	0.513621i	$0.171696\pi$
$812$		−	1.58258i		−	0.0555375i
$813$		0			0
$814$	17.3739	+	15.1652i	0.608954	+	0.531538i
$815$	−39.4955			−1.38347
$816$		0			0
$817$	−45.4955			−1.59168
$818$	−8.83485			−0.308903
$819$		0			0
$820$		−	19.7477i		−	0.689621i
$821$	15.1652			0.529267			0.264634	−	0.964349i	$-0.414749\pi$
$821$	15.1652			0.529267			0.264634	+	0.964349i	$0.414749\pi$
$822$		0			0
$823$	−16.0000			−0.557725			−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000			−0.557725			−0.278862	+	0.960331i	$0.589957\pi$
$824$	2.20871			0.0769441
$825$		0			0
$826$	15.1652			0.527663
$827$			14.8348i			0.515858i	0.966164	+	0.257929i	$0.0830401\pi$
$827$			14.8348i			0.515858i	−0.966164	+	0.257929i	$0.916960\pi$
$828$		0			0
$829$		−	39.9564i		−	1.38774i	−0.720098	−	0.693872i	$-0.755903\pi$
$829$		−	39.9564i		−	1.38774i	0.720098	−	0.693872i	$-0.244097\pi$
$830$	−12.0000			−0.416526
$831$		0			0
$832$			0.791288i			0.0274330i
$833$			4.74773i			0.164499i
$834$		0			0
$835$	3.62614			0.125488
$836$			28.7477i			0.994261i
$837$		0			0
$838$			6.79129i			0.234601i
$839$	51.4955			1.77782			0.888910	−	0.458081i	$-0.151463\pi$
$839$	51.4955			1.77782			0.888910	+	0.458081i	$0.151463\pi$
$840$		0			0
$841$	28.3739			0.978409
$842$	−3.95644			−0.136348
$843$		0			0
$844$	3.37386			0.116133
$845$		−	46.9129i		−	1.61385i
$846$		0			0
$847$	−6.74773			−0.231855
$848$	−7.58258			−0.260387
$849$		0			0
$850$	14.8348			0.508831
$851$	3.62614	+	3.16515i	0.124302	+	0.108500i
$852$		0			0
$853$			5.70417i			0.195307i	0.995220	+	0.0976535i	$0.0311337\pi$
$853$			5.70417i			0.195307i	−0.995220	+	0.0976535i	$0.968866\pi$
$854$	16.4174			0.561793
$855$		0			0
$856$		−	8.37386i		−	0.286213i
$857$			14.8348i			0.506749i	0.967368	+	0.253374i	$0.0815405\pi$
$857$			14.8348i			0.506749i	−0.967368	+	0.253374i	$0.918460\pi$
$858$		0			0
$859$		−	54.6606i		−	1.86500i	−0.361175	−	0.932498i	$-0.617624\pi$
$859$		−	54.6606i		−	1.86500i	0.361175	−	0.932498i	$-0.382376\pi$
$860$	−22.7477			−0.775691
$861$		0			0
$862$	−27.1652			−0.925249
$863$	46.7477			1.59131			0.795656	−	0.605749i	$-0.207127\pi$
$863$	46.7477			1.59131			0.795656	+	0.605749i	$0.207127\pi$
$864$		0			0
$865$		−	12.0000i		−	0.408012i
$866$		−	24.3739i		−	0.828258i
$867$		0			0
$868$		−	10.7477i		−	0.364802i
$869$			48.4955i			1.64510i
$870$		0			0
$871$			5.83485i			0.197706i
$872$	6.00000			0.203186
$873$		0			0
$874$			6.00000i			0.202953i
$875$		−	33.1652i		−	1.12119i
$876$		0			0
$877$	38.7477			1.30842			0.654209	−	0.756314i	$-0.273002\pi$
$877$	38.7477			1.30842			0.654209	+	0.756314i	$0.273002\pi$
$878$	−26.3739			−0.890075
$879$		0			0
$880$			14.3739i			0.484543i
$881$	−28.1216			−0.947440			−0.473720	−	0.880675i	$-0.657089\pi$
$881$	−28.1216			−0.947440			−0.473720	+	0.880675i	$0.657089\pi$
$882$		0			0
$883$			13.9129i			0.468206i	0.972212	+	0.234103i	$0.0752152\pi$
$883$			13.9129i			0.468206i	−0.972212	+	0.234103i	$0.924785\pi$
$884$	1.25227			0.0421185
$885$		0			0
$886$			5.04356i			0.169442i
$887$	−33.8258			−1.13576			−0.567879	−	0.823112i	$-0.692236\pi$
$887$	−33.8258			−1.13576			−0.567879	+	0.823112i	$0.692236\pi$
$888$		0			0
$889$	−16.0000			−0.536623
$890$		−	22.7477i		−	0.762506i
$891$		0			0
$892$	−14.0000			−0.468755
$893$		−	12.0000i		−	0.401565i
$894$		0			0
$895$	−28.7477			−0.960931
$896$		−	2.00000i		−	0.0668153i
$897$		0			0
$898$	−21.1652			−0.706290
$899$	−4.25227			−0.141821
$900$		0			0
$901$			12.0000i			0.399778i
$902$		−	19.7477i		−	0.657527i
$903$		0			0
$904$	19.5826			0.651307
$905$			71.0780i			2.36271i
$906$		0			0
$907$		−	6.33030i		−	0.210194i	−0.994462	−	0.105097i	$-0.966485\pi$
$907$		−	6.33030i		−	0.210194i	0.994462	−	0.105097i	$-0.0335153\pi$
$908$		−	19.9129i		−	0.660832i
$909$		0			0
$910$		−	6.00000i		−	0.198898i
$911$			54.3303i			1.80004i	0.435845	+	0.900022i	$0.356449\pi$
$911$			54.3303i			1.80004i	−0.435845	+	0.900022i	$0.643551\pi$
$912$		0			0
$913$	−12.0000			−0.397142
$914$	−33.4955			−1.10793
$915$		0			0
$916$	20.7477			0.685524
$917$			21.4955i			0.709842i
$918$		0			0
$919$		−	36.0000i		−	1.18753i	−0.804638	−	0.593765i	$-0.797641\pi$
$919$		−	36.0000i		−	1.18753i	0.804638	−	0.593765i	$-0.202359\pi$
$920$			3.00000i			0.0989071i
$921$		0			0
$922$	−24.3303			−0.801276
$923$			7.25227i			0.238711i
$924$		0			0
$925$	−37.4955	+	42.9564i	−1.23284	+	1.41240i
$926$	20.7042			0.680381
$927$		0			0
$928$	−0.791288			−0.0259753
$929$	5.37386			0.176311			0.0881554	−	0.996107i	$-0.471903\pi$
$929$	5.37386			0.176311			0.0881554	+	0.996107i	$0.471903\pi$
$930$		0			0
$931$		−	22.7477i		−	0.745527i
$932$	25.1216			0.822885
$933$		0			0
$934$	25.5826			0.837087
$935$	22.7477			0.743930
$936$		0			0
$937$	−33.8693			−1.10646			−0.553231	−	0.833028i	$-0.686605\pi$
$937$	−33.8693			−1.10646			−0.553231	+	0.833028i	$0.686605\pi$
$938$		−	14.7477i		−	0.481530i
$939$		0			0
$940$		−	6.00000i		−	0.195698i
$941$	−33.4955			−1.09192			−0.545960	−	0.837811i	$-0.683835\pi$
$941$	−33.4955			−1.09192			−0.545960	+	0.837811i	$0.683835\pi$
$942$		0			0
$943$		−	4.12159i		−	0.134217i
$944$		−	7.58258i		−	0.246792i
$945$		0			0
$946$	−22.7477			−0.739592
$947$		−	28.7477i		−	0.934176i	−0.884211	−	0.467088i	$-0.845303\pi$
$947$		−	28.7477i		−	0.934176i	0.884211	−	0.467088i	$-0.154697\pi$
$948$		0			0
$949$		−	7.41742i		−	0.240780i
$950$	−71.0780			−2.30608
$951$		0			0
$952$	−3.16515			−0.102583
$953$	46.4519			1.50472			0.752362	−	0.658750i	$-0.228914\pi$
$953$	46.4519			1.50472			0.752362	+	0.658750i	$0.228914\pi$
$954$		0			0
$955$	−20.3739			−0.659283
$956$		−	24.9564i		−	0.807149i
$957$		0			0
$958$	−0.791288			−0.0255653
$959$	−34.7477			−1.12206
$960$		0			0
$961$	2.12159			0.0684384
$962$	−3.16515	+	3.62614i	−0.102049	+	0.116911i
$963$		0			0
$964$			13.5826i			0.437465i
$965$	−69.4955			−2.23714
$966$		0			0
$967$		−	0.956439i		−	0.0307570i	−0.999882	−	0.0153785i	$-0.995105\pi$
$967$		−	0.956439i		−	0.0307570i	0.999882	−	0.0153785i	$-0.00489532\pi$
$968$			3.37386i			0.108440i
$969$		0			0
$970$		−	16.7477i		−	0.537737i
$971$	−36.6261			−1.17539			−0.587694	−	0.809083i	$-0.699964\pi$
$971$	−36.6261			−1.17539			−0.587694	+	0.809083i	$0.699964\pi$
$972$		0			0
$973$	−0.747727			−0.0239710
$974$	36.6606			1.17468
$975$		0			0
$976$		−	8.20871i		−	0.262754i
$977$		−	23.0780i		−	0.738332i	−0.929364	−	0.369166i	$-0.879643\pi$
$977$		−	23.0780i		−	0.738332i	0.929364	−	0.369166i	$-0.120357\pi$
$978$		0			0
$979$		−	22.7477i		−	0.727021i
$980$		−	11.3739i		−	0.363325i
$981$		0			0
$982$		−	8.37386i		−	0.267221i
$983$	−18.3303			−0.584646			−0.292323	−	0.956320i	$-0.594428\pi$
$983$	−18.3303			−0.584646			−0.292323	+	0.956320i	$0.594428\pi$
$984$		0			0
$985$			98.2432i			3.13029i
$986$			1.25227i			0.0398805i
$987$		0			0
$988$	−6.00000			−0.190885
$989$	−4.74773			−0.150969
$990$		0			0
$991$			3.95644i			0.125680i	0.998024	+	0.0628402i	$0.0200159\pi$
$991$			3.95644i			0.125680i	−0.998024	+	0.0628402i	$0.979984\pi$
$992$	−5.37386			−0.170620
$993$		0			0
$994$		−	18.3303i		−	0.581402i
$995$	−12.0000			−0.380426
$996$		0			0
$997$			18.0000i			0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$			18.0000i			0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	−16.7477			−0.530140
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	666.2.c.b.73.1		4
3.2	odd	2		74.2.b.a.73.3	yes	4
4.3	odd	2		5328.2.h.m.2737.1		4
12.11	even	2		592.2.g.c.369.4		4
15.2	even	4		1850.2.c.g.1849.4		4
15.8	even	4		1850.2.c.h.1849.1		4
15.14	odd	2		1850.2.d.e.1701.2		4
24.5	odd	2		2368.2.g.j.961.3		4
24.11	even	2		2368.2.g.h.961.1		4
37.36	even	2	inner	666.2.c.b.73.4		4
111.68	even	4		2738.2.a.h.1.2		2
111.80	even	4		2738.2.a.k.1.2		2
111.110	odd	2		74.2.b.a.73.1	✓	4
148.147	odd	2		5328.2.h.m.2737.4		4
444.443	even	2		592.2.g.c.369.3		4
555.332	even	4		1850.2.c.h.1849.4		4
555.443	even	4		1850.2.c.g.1849.1		4
555.554	odd	2		1850.2.d.e.1701.4		4
888.221	odd	2		2368.2.g.j.961.4		4
888.443	even	2		2368.2.g.h.961.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.b.a.73.1	✓	4	111.110	odd	2
74.2.b.a.73.3	yes	4	3.2	odd	2
592.2.g.c.369.3		4	444.443	even	2
592.2.g.c.369.4		4	12.11	even	2
666.2.c.b.73.1		4	1.1	even	1	trivial
666.2.c.b.73.4		4	37.36	even	2	inner
1850.2.c.g.1849.1		4	555.443	even	4
1850.2.c.g.1849.4		4	15.2	even	4
1850.2.c.h.1849.1		4	15.8	even	4
1850.2.c.h.1849.4		4	555.332	even	4
1850.2.d.e.1701.2		4	15.14	odd	2
1850.2.d.e.1701.4		4	555.554	odd	2
2368.2.g.h.961.1		4	24.11	even	2
2368.2.g.h.961.2		4	888.443	even	2
2368.2.g.j.961.3		4	24.5	odd	2
2368.2.g.j.961.4		4	888.221	odd	2
2738.2.a.h.1.2		2	111.68	even	4
2738.2.a.k.1.2		2	111.80	even	4
5328.2.h.m.2737.1		4	4.3	odd	2
5328.2.h.m.2737.4		4	148.147	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.79129i q^{5} -2.00000 q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.79129i q^{5} -2.00000 q^{7} +1.00000i q^{8} -3.79129 q^{10} -3.79129 q^{11} -0.791288i q^{13} +2.00000i q^{14} +1.00000 q^{16} -1.58258i q^{17} +7.58258i q^{19} +3.79129i q^{20} +3.79129i q^{22} +0.791288i q^{23} -9.37386 q^{25} -0.791288 q^{26} +2.00000 q^{28} -0.791288i q^{29} -5.37386i q^{31} -1.00000i q^{32} -1.58258 q^{34} +7.58258i q^{35} +(4.00000 - 4.58258i) q^{37} +7.58258 q^{38} +3.79129 q^{40} -5.20871 q^{41} +6.00000i q^{43} +3.79129 q^{44} +0.791288 q^{46} -1.58258 q^{47} -3.00000 q^{49} +9.37386i q^{50} +0.791288i q^{52} -7.58258 q^{53} +14.3739i q^{55} -2.00000i q^{56} -0.791288 q^{58} -7.58258i q^{59} -8.20871i q^{61} -5.37386 q^{62} -1.00000 q^{64} -3.00000 q^{65} -7.37386 q^{67} +1.58258i q^{68} +7.58258 q^{70} -9.16515 q^{71} +9.37386 q^{73} +(-4.58258 - 4.00000i) q^{74} -7.58258i q^{76} +7.58258 q^{77} -12.7913i q^{79} -3.79129i q^{80} +5.20871i q^{82} +3.16515 q^{83} -6.00000 q^{85} +6.00000 q^{86} -3.79129i q^{88} +6.00000i q^{89} +1.58258i q^{91} -0.791288i q^{92} +1.58258i q^{94} +28.7477 q^{95} +4.41742i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 8 q^{7}+O(q^{10}) \) 4 * q - 4 * q^4 - 8 * q^7	\( 4 q - 4 q^{4} - 8 q^{7} - 6 q^{10} - 6 q^{11} + 4 q^{16} - 10 q^{25} + 6 q^{26} + 8 q^{28} + 12 q^{34} + 16 q^{37} + 12 q^{38} + 6 q^{40} - 30 q^{41} + 6 q^{44} - 6 q^{46} + 12 q^{47} - 12 q^{49} - 12 q^{53} + 6 q^{58} + 6 q^{62} - 4 q^{64} - 12 q^{65} - 2 q^{67} + 12 q^{70} + 10 q^{73} + 12 q^{77} - 24 q^{83} - 24 q^{85} + 24 q^{86} + 60 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^7 - 6 * q^10 - 6 * q^11 + 4 * q^16 - 10 * q^25 + 6 * q^26 + 8 * q^28 + 12 * q^34 + 16 * q^37 + 12 * q^38 + 6 * q^40 - 30 * q^41 + 6 * q^44 - 6 * q^46 + 12 * q^47 - 12 * q^49 - 12 * q^53 + 6 * q^58 + 6 * q^62 - 4 * q^64 - 12 * q^65 - 2 * q^67 + 12 * q^70 + 10 * q^73 + 12 * q^77 - 24 * q^83 - 24 * q^85 + 24 * q^86 + 60 * q^95

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