LMFDB - Embedded newform 666.2.c.a.73.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [666,2,Mod(73,666)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("666.73"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [2,0,0,-2,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$5.31803677462$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 222)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	666.73
Dual form			666.2.c.a.73.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{5} +3.00000 q^{7} -1.00000i q^{8} -2.00000 q^{10} +3.00000 q^{11} -1.00000i q^{13} +3.00000i q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000i q^{19} -2.00000i q^{20} +3.00000i q^{22} +1.00000i q^{23} +1.00000 q^{25} +1.00000 q^{26} -3.00000 q^{28} -6.00000i q^{29} +10.0000i q^{31} +1.00000i q^{32} -3.00000 q^{34} +6.00000i q^{35} +(-6.00000 + 1.00000i) q^{37} -1.00000 q^{38} +2.00000 q^{40} -2.00000 q^{41} +4.00000i q^{43} -3.00000 q^{44} -1.00000 q^{46} -8.00000 q^{47} +2.00000 q^{49} +1.00000i q^{50} +1.00000i q^{52} +11.0000 q^{53} +6.00000i q^{55} -3.00000i q^{56} +6.00000 q^{58} -6.00000i q^{59} +10.0000i q^{61} -10.0000 q^{62} -1.00000 q^{64} +2.00000 q^{65} +8.00000 q^{67} -3.00000i q^{68} -6.00000 q^{70} -2.00000 q^{71} -1.00000 q^{73} +(-1.00000 - 6.00000i) q^{74} -1.00000i q^{76} +9.00000 q^{77} -14.0000i q^{79} +2.00000i q^{80} -2.00000i q^{82} +1.00000 q^{83} -6.00000 q^{85} -4.00000 q^{86} -3.00000i q^{88} +9.00000i q^{89} -3.00000i q^{91} -1.00000i q^{92} -8.00000i q^{94} -2.00000 q^{95} -8.00000i q^{97} +2.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{7} - 4 q^{10} + 6 q^{11} + 2 q^{16} + 2 q^{25} + 2 q^{26} - 6 q^{28} - 6 q^{34} - 12 q^{37} - 2 q^{38} + 4 q^{40} - 4 q^{41} - 6 q^{44} - 2 q^{46} - 16 q^{47} + 4 q^{49} + 22 q^{53}+ \cdots - 4 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^7 - 4 * q^10 + 6 * q^11 + 2 * q^16 + 2 * q^25 + 2 * q^26 - 6 * q^28 - 6 * q^34 - 12 * q^37 - 2 * q^38 + 4 * q^40 - 4 * q^41 - 6 * q^44 - 2 * q^46 - 16 * q^47 + 4 * q^49 + 22 * q^53 + 12 * q^58 - 20 * q^62 - 2 * q^64 + 4 * q^65 + 16 * q^67 - 12 * q^70 - 4 * q^71 - 2 * q^73 - 2 * q^74 + 18 * q^77 + 2 * q^83 - 12 * q^85 - 8 * q^86 - 4 * q^95

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
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	3.00000			1.13389			0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000			1.13389			0.566947	+	0.823754i	$0.308125\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−2.00000			−0.632456
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$		−	1.00000i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$			3.00000i			0.801784i
$15$		0			0
$16$	1.00000			0.250000
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$		0			0
$19$			1.00000i			0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$			1.00000i			0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		−	2.00000i		−	0.447214i
$21$		0			0
$22$			3.00000i			0.639602i
$23$			1.00000i			0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$			1.00000i			0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$	1.00000			0.196116
$27$		0			0
$28$	−3.00000			−0.566947
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−3.00000			−0.514496
$35$			6.00000i			1.01419i
$36$		0			0
$37$	−6.00000	+	1.00000i	−0.986394	+	0.164399i
$37$	−6.00000	+	1.00000i	−0.986394	+	0.164399i
$38$	−1.00000			−0.162221
$39$		0			0
$40$	2.00000			0.316228
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−3.00000			−0.452267
$45$		0			0
$46$	−1.00000			−0.147442
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$			1.00000i			0.141421i
$51$		0			0
$52$			1.00000i			0.138675i
$53$	11.0000			1.51097			0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000			1.51097			0.755483	+	0.655168i	$0.227402\pi$
$54$		0			0
$55$			6.00000i			0.809040i
$56$		−	3.00000i		−	0.400892i
$57$		0			0
$58$	6.00000			0.787839
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$			10.0000i			1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$			10.0000i			1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$	−10.0000			−1.27000
$63$		0			0
$64$	−1.00000			−0.125000
$65$	2.00000			0.248069
$66$		0			0
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$		−	3.00000i		−	0.363803i
$69$		0			0
$70$	−6.00000			−0.717137
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	−1.00000			−0.117041			−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000			−0.117041			−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−1.00000	−	6.00000i	−0.116248	−	0.697486i
$75$		0			0
$76$		−	1.00000i		−	0.114708i
$77$	9.00000			1.02565
$78$		0			0
$79$		−	14.0000i		−	1.57512i	−0.616236	−	0.787562i	$-0.711343\pi$
$79$		−	14.0000i		−	1.57512i	0.616236	−	0.787562i	$-0.288657\pi$
$80$			2.00000i			0.223607i
$81$		0			0
$82$		−	2.00000i		−	0.220863i
$83$	1.00000			0.109764			0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000			0.109764			0.0548821	+	0.998493i	$0.482522\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$	−4.00000			−0.431331
$87$		0			0
$88$		−	3.00000i		−	0.319801i
$89$			9.00000i			0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$			9.00000i			0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$		0			0
$91$		−	3.00000i		−	0.314485i
$92$		−	1.00000i		−	0.104257i
$93$		0			0
$94$		−	8.00000i		−	0.825137i
$95$	−2.00000			−0.205196
$96$		0			0
$97$		−	8.00000i		−	0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$		−	8.00000i		−	0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$			2.00000i			0.202031i
$99$		0			0
$100$	−1.00000			−0.100000
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$		−	16.0000i		−	1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$		−	16.0000i		−	1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	−1.00000			−0.0980581
$105$		0			0
$106$			11.0000i			1.06841i
$107$	−13.0000			−1.25676			−0.628379	−	0.777908i	$-0.716281\pi$
$107$	−13.0000			−1.25676			−0.628379	+	0.777908i	$0.716281\pi$
$108$		0			0
$109$		−	19.0000i		−	1.81987i	−0.414751	−	0.909935i	$-0.636131\pi$
$109$		−	19.0000i		−	1.81987i	0.414751	−	0.909935i	$-0.363869\pi$
$110$	−6.00000			−0.572078
$111$		0			0
$112$	3.00000			0.283473
$113$		−	14.0000i		−	1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$		−	14.0000i		−	1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$		0			0
$115$	−2.00000			−0.186501
$116$			6.00000i			0.557086i
$117$		0			0
$118$	6.00000			0.552345
$119$			9.00000i			0.825029i
$120$		0			0
$121$	−2.00000			−0.181818
$122$	−10.0000			−0.905357
$123$		0			0
$124$		−	10.0000i		−	0.898027i
$125$			12.0000i			1.07331i
$126$		0			0
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$			2.00000i			0.175412i
$131$		−	10.0000i		−	0.873704i	−0.899533	−	0.436852i	$-0.856093\pi$
$131$		−	10.0000i		−	0.873704i	0.899533	−	0.436852i	$-0.143907\pi$
$132$		0			0
$133$			3.00000i			0.260133i
$134$			8.00000i			0.691095i
$135$		0			0
$136$	3.00000			0.257248
$137$	12.0000			1.02523			0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000			1.02523			0.512615	+	0.858619i	$0.328677\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		−	6.00000i		−	0.507093i
$141$		0			0
$142$		−	2.00000i		−	0.167836i
$143$		−	3.00000i		−	0.250873i
$144$		0			0
$145$	12.0000			0.996546
$146$		−	1.00000i		−	0.0827606i
$147$		0			0
$148$	6.00000	−	1.00000i	0.493197	−	0.0821995i
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$	−13.0000			−1.05792			−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000			−1.05792			−0.528962	+	0.848645i	$0.677419\pi$
$152$	1.00000			0.0811107
$153$		0			0
$154$			9.00000i			0.725241i
$155$	−20.0000			−1.60644
$156$		0			0
$157$	8.00000			0.638470			0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000			0.638470			0.319235	+	0.947676i	$0.396574\pi$
$158$	14.0000			1.11378
$159$		0			0
$160$	−2.00000			−0.158114
$161$			3.00000i			0.236433i
$162$		0			0
$163$			9.00000i			0.704934i	0.935824	+	0.352467i	$0.114657\pi$
$163$			9.00000i			0.704934i	−0.935824	+	0.352467i	$0.885343\pi$
$164$	2.00000			0.156174
$165$		0			0
$166$			1.00000i			0.0776151i
$167$		−	17.0000i		−	1.31550i	−0.753237	−	0.657750i	$-0.771508\pi$
$167$		−	17.0000i		−	1.31550i	0.753237	−	0.657750i	$-0.228492\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		−	6.00000i		−	0.460179i
$171$		0			0
$172$		−	4.00000i		−	0.304997i
$173$	−9.00000			−0.684257			−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000			−0.684257			−0.342129	+	0.939653i	$0.611148\pi$
$174$		0			0
$175$	3.00000			0.226779
$176$	3.00000			0.226134
$177$		0			0
$178$	−9.00000			−0.674579
$179$			24.0000i			1.79384i	0.442189	+	0.896922i	$0.354202\pi$
$179$			24.0000i			1.79384i	−0.442189	+	0.896922i	$0.645798\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$	3.00000			0.222375
$183$		0			0
$184$	1.00000			0.0737210
$185$	−2.00000	−	12.0000i	−0.147043	−	0.882258i
$186$		0			0
$187$			9.00000i			0.658145i
$188$	8.00000			0.583460
$189$		0			0
$190$		−	2.00000i		−	0.145095i
$191$		−	15.0000i		−	1.08536i	−0.839939	−	0.542681i	$-0.817409\pi$
$191$		−	15.0000i		−	1.08536i	0.839939	−	0.542681i	$-0.182591\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	8.00000			0.574367
$195$		0			0
$196$	−2.00000			−0.142857
$197$	17.0000			1.21120			0.605600	−	0.795769i	$-0.292933\pi$
$197$	17.0000			1.21120			0.605600	+	0.795769i	$0.292933\pi$
$198$		0			0
$199$		−	14.0000i		−	0.992434i	−0.868199	−	0.496217i	$-0.834722\pi$
$199$		−	14.0000i		−	0.992434i	0.868199	−	0.496217i	$-0.165278\pi$
$200$		−	1.00000i		−	0.0707107i
$201$		0			0
$202$			18.0000i			1.26648i
$203$		−	18.0000i		−	1.26335i
$204$		0			0
$205$		−	4.00000i		−	0.279372i
$206$	16.0000			1.11477
$207$		0			0
$208$		−	1.00000i		−	0.0693375i
$209$			3.00000i			0.207514i
$210$		0			0
$211$	22.0000			1.51454			0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000			1.51454			0.757271	+	0.653101i	$0.226532\pi$
$212$	−11.0000			−0.755483
$213$		0			0
$214$		−	13.0000i		−	0.888662i
$215$	−8.00000			−0.545595
$216$		0			0
$217$			30.0000i			2.03653i
$218$	19.0000			1.28684
$219$		0			0
$220$		−	6.00000i		−	0.404520i
$221$	3.00000			0.201802
$222$		0			0
$223$	4.00000			0.267860			0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000			0.267860			0.133930	+	0.990991i	$0.457240\pi$
$224$			3.00000i			0.200446i
$225$		0			0
$226$	14.0000			0.931266
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$	−20.0000			−1.32164			−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000			−1.32164			−0.660819	+	0.750546i	$0.729791\pi$
$230$		−	2.00000i		−	0.131876i
$231$		0			0
$232$	−6.00000			−0.393919
$233$	−24.0000			−1.57229			−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000			−1.57229			−0.786146	+	0.618041i	$0.787927\pi$
$234$		0			0
$235$		−	16.0000i		−	1.04372i
$236$			6.00000i			0.390567i
$237$		0			0
$238$	−9.00000			−0.583383
$239$		−	16.0000i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	2.00000i		−	0.128565i
$243$		0			0
$244$		−	10.0000i		−	0.640184i
$245$			4.00000i			0.255551i
$246$		0			0
$247$	1.00000			0.0636285
$248$	10.0000			0.635001
$249$		0			0
$250$	−12.0000			−0.758947
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$			3.00000i			0.188608i
$254$		−	7.00000i		−	0.439219i
$255$		0			0
$256$	1.00000			0.0625000
$257$			3.00000i			0.187135i	0.995613	+	0.0935674i	$0.0298271\pi$
$257$			3.00000i			0.187135i	−0.995613	+	0.0935674i	$0.970173\pi$
$258$		0			0
$259$	−18.0000	+	3.00000i	−1.11847	+	0.186411i
$260$	−2.00000			−0.124035
$261$		0			0
$262$	10.0000			0.617802
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$			22.0000i			1.35145i
$266$	−3.00000			−0.183942
$267$		0			0
$268$	−8.00000			−0.488678
$269$	5.00000			0.304855			0.152428	−	0.988315i	$-0.451291\pi$
$269$	5.00000			0.304855			0.152428	+	0.988315i	$0.451291\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$			3.00000i			0.181902i
$273$		0			0
$274$			12.0000i			0.724947i
$275$	3.00000			0.180907
$276$		0			0
$277$			7.00000i			0.420589i	0.977638	+	0.210295i	$0.0674423\pi$
$277$			7.00000i			0.420589i	−0.977638	+	0.210295i	$0.932558\pi$
$278$		0			0
$279$		0			0
$280$	6.00000			0.358569
$281$			15.0000i			0.894825i	0.894328	+	0.447412i	$0.147654\pi$
$281$			15.0000i			0.894825i	−0.894328	+	0.447412i	$0.852346\pi$
$282$		0			0
$283$			29.0000i			1.72387i	0.507018	+	0.861936i	$0.330748\pi$
$283$			29.0000i			1.72387i	−0.507018	+	0.861936i	$0.669252\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$	3.00000			0.177394
$287$	−6.00000			−0.354169
$288$		0			0
$289$	8.00000			0.470588
$290$			12.0000i			0.704664i
$291$		0			0
$292$	1.00000			0.0585206
$293$	−9.00000			−0.525786			−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000			−0.525786			−0.262893	+	0.964825i	$0.584677\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$	1.00000	+	6.00000i	0.0581238	+	0.348743i
$297$		0			0
$298$			10.0000i			0.579284i
$299$	1.00000			0.0578315
$300$		0			0
$301$			12.0000i			0.691669i
$302$		−	13.0000i		−	0.748066i
$303$		0			0
$304$			1.00000i			0.0573539i
$305$	−20.0000			−1.14520
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$	−9.00000			−0.512823
$309$		0			0
$310$		−	20.0000i		−	1.13592i
$311$			20.0000i			1.13410i	0.823685	+	0.567048i	$0.191915\pi$
$311$			20.0000i			1.13410i	−0.823685	+	0.567048i	$0.808085\pi$
$312$		0			0
$313$		−	6.00000i		−	0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$		−	6.00000i		−	0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$			8.00000i			0.451466i
$315$		0			0
$316$			14.0000i			0.787562i
$317$	2.00000			0.112331			0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000			0.112331			0.0561656	+	0.998421i	$0.482113\pi$
$318$		0			0
$319$		−	18.0000i		−	1.00781i
$320$		−	2.00000i		−	0.111803i
$321$		0			0
$322$	−3.00000			−0.167183
$323$	−3.00000			−0.166924
$324$		0			0
$325$		−	1.00000i		−	0.0554700i
$326$	−9.00000			−0.498464
$327$		0			0
$328$			2.00000i			0.110432i
$329$	−24.0000			−1.32316
$330$		0			0
$331$		−	20.0000i		−	1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$		−	20.0000i		−	1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$	−1.00000			−0.0548821
$333$		0			0
$334$	17.0000			0.930199
$335$			16.0000i			0.874173i
$336$		0			0
$337$	3.00000			0.163420			0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000			0.163420			0.0817102	+	0.996656i	$0.473962\pi$
$338$			12.0000i			0.652714i
$339$		0			0
$340$	6.00000			0.325396
$341$			30.0000i			1.62459i
$342$		0			0
$343$	−15.0000			−0.809924
$344$	4.00000			0.215666
$345$		0			0
$346$		−	9.00000i		−	0.483843i
$347$			18.0000i			0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$			18.0000i			0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$		0			0
$349$	30.0000			1.60586			0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000			1.60586			0.802932	+	0.596071i	$0.203272\pi$
$350$			3.00000i			0.160357i
$351$		0			0
$352$			3.00000i			0.159901i
$353$		−	34.0000i		−	1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$		−	34.0000i		−	1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$		0			0
$355$		−	4.00000i		−	0.212298i
$356$		−	9.00000i		−	0.476999i
$357$		0			0
$358$	−24.0000			−1.26844
$359$	−10.0000			−0.527780			−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000			−0.527780			−0.263890	+	0.964553i	$0.585006\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$			2.00000i			0.105118i
$363$		0			0
$364$			3.00000i			0.157243i
$365$		−	2.00000i		−	0.104685i
$366$		0			0
$367$	−17.0000			−0.887393			−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000			−0.887393			−0.443696	+	0.896177i	$0.646333\pi$
$368$			1.00000i			0.0521286i
$369$		0			0
$370$	12.0000	−	2.00000i	0.623850	−	0.103975i
$371$	33.0000			1.71327
$372$		0			0
$373$	4.00000			0.207112			0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000			0.207112			0.103556	+	0.994624i	$0.466978\pi$
$374$	−9.00000			−0.465379
$375$		0			0
$376$			8.00000i			0.412568i
$377$	−6.00000			−0.309016
$378$		0			0
$379$	−30.0000			−1.54100			−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000			−1.54100			−0.770498	+	0.637442i	$0.779993\pi$
$380$	2.00000			0.102598
$381$		0			0
$382$	15.0000			0.767467
$383$			11.0000i			0.562074i	0.959697	+	0.281037i	$0.0906783\pi$
$383$			11.0000i			0.562074i	−0.959697	+	0.281037i	$0.909322\pi$
$384$		0			0
$385$			18.0000i			0.917365i
$386$	16.0000			0.814379
$387$		0			0
$388$			8.00000i			0.406138i
$389$			4.00000i			0.202808i	0.994845	+	0.101404i	$0.0323335\pi$
$389$			4.00000i			0.202808i	−0.994845	+	0.101404i	$0.967667\pi$
$390$		0			0
$391$	−3.00000			−0.151717
$392$		−	2.00000i		−	0.101015i
$393$		0			0
$394$			17.0000i			0.856448i
$395$	28.0000			1.40883
$396$		0			0
$397$	−32.0000			−1.60603			−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000			−1.60603			−0.803017	+	0.595956i	$0.796773\pi$
$398$	14.0000			0.701757
$399$		0			0
$400$	1.00000			0.0500000
$401$		−	15.0000i		−	0.749064i	−0.927214	−	0.374532i	$-0.877803\pi$
$401$		−	15.0000i		−	0.749064i	0.927214	−	0.374532i	$-0.122197\pi$
$402$		0			0
$403$	10.0000			0.498135
$404$	−18.0000			−0.895533
$405$		0			0
$406$	18.0000			0.893325
$407$	−18.0000	+	3.00000i	−0.892227	+	0.148704i
$408$		0			0
$409$		−	24.0000i		−	1.18672i	−0.804936	−	0.593362i	$-0.797800\pi$
$409$		−	24.0000i		−	1.18672i	0.804936	−	0.593362i	$-0.202200\pi$
$410$	4.00000			0.197546
$411$		0			0
$412$			16.0000i			0.788263i
$413$		−	18.0000i		−	0.885722i
$414$		0			0
$415$			2.00000i			0.0981761i
$416$	1.00000			0.0490290
$417$		0			0
$418$	−3.00000			−0.146735
$419$	35.0000			1.70986			0.854931	−	0.518742i	$-0.173599\pi$
$419$	35.0000			1.70986			0.854931	+	0.518742i	$0.173599\pi$
$420$		0			0
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$			22.0000i			1.07094i
$423$		0			0
$424$		−	11.0000i		−	0.534207i
$425$			3.00000i			0.145521i
$426$		0			0
$427$			30.0000i			1.45180i
$428$	13.0000			0.628379
$429$		0			0
$430$		−	8.00000i		−	0.385794i
$431$		−	15.0000i		−	0.722525i	−0.932464	−	0.361262i	$-0.882346\pi$
$431$		−	15.0000i		−	0.722525i	0.932464	−	0.361262i	$-0.117654\pi$
$432$		0			0
$433$	−1.00000			−0.0480569			−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000			−0.0480569			−0.0240285	+	0.999711i	$0.507649\pi$
$434$	−30.0000			−1.44005
$435$		0			0
$436$			19.0000i			0.909935i
$437$	−1.00000			−0.0478365
$438$		0			0
$439$			26.0000i			1.24091i	0.784241	+	0.620456i	$0.213053\pi$
$439$			26.0000i			1.24091i	−0.784241	+	0.620456i	$0.786947\pi$
$440$	6.00000			0.286039
$441$		0			0
$442$			3.00000i			0.142695i
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$	−18.0000			−0.853282
$446$			4.00000i			0.189405i
$447$		0			0
$448$	−3.00000			−0.141737
$449$			34.0000i			1.60456i	0.596948	+	0.802280i	$0.296380\pi$
$449$			34.0000i			1.60456i	−0.596948	+	0.802280i	$0.703620\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$			14.0000i			0.658505i
$453$		0			0
$454$	−8.00000			−0.375459
$455$	6.00000			0.281284
$456$		0			0
$457$			2.00000i			0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$			2.00000i			0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$		−	20.0000i		−	0.934539i
$459$		0			0
$460$	2.00000			0.0932505
$461$		−	40.0000i		−	1.86299i	−0.363760	−	0.931493i	$-0.618507\pi$
$461$		−	40.0000i		−	1.86299i	0.363760	−	0.931493i	$-0.381493\pi$
$462$		0			0
$463$			34.0000i			1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$			34.0000i			1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$		−	6.00000i		−	0.278543i
$465$		0			0
$466$		−	24.0000i		−	1.11178i
$467$			8.00000i			0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$			8.00000i			0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$		0			0
$469$	24.0000			1.10822
$470$	16.0000			0.738025
$471$		0			0
$472$	−6.00000			−0.276172
$473$			12.0000i			0.551761i
$474$		0			0
$475$			1.00000i			0.0458831i
$476$		−	9.00000i		−	0.412514i
$477$		0			0
$478$	16.0000			0.731823
$479$		−	21.0000i		−	0.959514i	−0.877401	−	0.479757i	$-0.840725\pi$
$479$		−	21.0000i		−	0.959514i	0.877401	−	0.479757i	$-0.159275\pi$
$480$		0			0
$481$	1.00000	+	6.00000i	0.0455961	+	0.273576i
$482$		0			0
$483$		0			0
$484$	2.00000			0.0909091
$485$	16.0000			0.726523
$486$		0			0
$487$		−	8.00000i		−	0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$		−	8.00000i		−	0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	10.0000			0.452679
$489$		0			0
$490$	−4.00000			−0.180702
$491$	−27.0000			−1.21849			−0.609246	−	0.792981i	$-0.708528\pi$
$491$	−27.0000			−1.21849			−0.609246	+	0.792981i	$0.708528\pi$
$492$		0			0
$493$	18.0000			0.810679
$494$			1.00000i			0.0449921i
$495$		0			0
$496$			10.0000i			0.449013i
$497$	−6.00000			−0.269137
$498$		0			0
$499$		−	39.0000i		−	1.74588i	−0.487828	−	0.872940i	$-0.662211\pi$
$499$		−	39.0000i		−	1.74588i	0.487828	−	0.872940i	$-0.337789\pi$
$500$		−	12.0000i		−	0.536656i
$501$		0			0
$502$		0			0
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$		0			0
$505$			36.0000i			1.60198i
$506$	−3.00000			−0.133366
$507$		0			0
$508$	7.00000			0.310575
$509$	5.00000			0.221621			0.110811	−	0.993842i	$-0.464655\pi$
$509$	5.00000			0.221621			0.110811	+	0.993842i	$0.464655\pi$
$510$		0			0
$511$	−3.00000			−0.132712
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−3.00000			−0.132324
$515$	32.0000			1.41009
$516$		0			0
$517$	−24.0000			−1.05552
$518$	−3.00000	−	18.0000i	−0.131812	−	0.790875i
$519$		0			0
$520$		−	2.00000i		−	0.0877058i
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$			10.0000i			0.436852i
$525$		0			0
$526$		−	24.0000i		−	1.04645i
$527$	−30.0000			−1.30682
$528$		0			0
$529$	22.0000			0.956522
$530$	−22.0000			−0.955619
$531$		0			0
$532$		−	3.00000i		−	0.130066i
$533$			2.00000i			0.0866296i
$534$		0			0
$535$		−	26.0000i		−	1.12408i
$536$		−	8.00000i		−	0.345547i
$537$		0			0
$538$			5.00000i			0.215565i
$539$	6.00000			0.258438
$540$		0			0
$541$		−	45.0000i		−	1.93470i	−0.253442	−	0.967351i	$-0.581563\pi$
$541$		−	45.0000i		−	1.93470i	0.253442	−	0.967351i	$-0.418437\pi$
$542$		−	28.0000i		−	1.20270i
$543$		0			0
$544$	−3.00000			−0.128624
$545$	38.0000			1.62774
$546$		0			0
$547$			17.0000i			0.726868i	0.931620	+	0.363434i	$0.118396\pi$
$547$			17.0000i			0.726868i	−0.931620	+	0.363434i	$0.881604\pi$
$548$	−12.0000			−0.512615
$549$		0			0
$550$			3.00000i			0.127920i
$551$	6.00000			0.255609
$552$		0			0
$553$		−	42.0000i		−	1.78602i
$554$	−7.00000			−0.297402
$555$		0			0
$556$		0			0
$557$			38.0000i			1.61011i	0.593199	+	0.805056i	$0.297865\pi$
$557$			38.0000i			1.61011i	−0.593199	+	0.805056i	$0.702135\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$			6.00000i			0.253546i
$561$		0			0
$562$	−15.0000			−0.632737
$563$		−	14.0000i		−	0.590030i	−0.955493	−	0.295015i	$-0.904675\pi$
$563$		−	14.0000i		−	0.590030i	0.955493	−	0.295015i	$-0.0953246\pi$
$564$		0			0
$565$	28.0000			1.17797
$566$	−29.0000			−1.21896
$567$		0			0
$568$			2.00000i			0.0839181i
$569$		−	21.0000i		−	0.880366i	−0.897908	−	0.440183i	$-0.854914\pi$
$569$		−	21.0000i		−	0.880366i	0.897908	−	0.440183i	$-0.145086\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$			3.00000i			0.125436i
$573$		0			0
$574$		−	6.00000i		−	0.250435i
$575$			1.00000i			0.0417029i
$576$		0			0
$577$		−	8.00000i		−	0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$		−	8.00000i		−	0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$			8.00000i			0.332756i
$579$		0			0
$580$	−12.0000			−0.498273
$581$	3.00000			0.124461
$582$		0			0
$583$	33.0000			1.36672
$584$			1.00000i			0.0413803i
$585$		0			0
$586$		−	9.00000i		−	0.371787i
$587$		−	2.00000i		−	0.0825488i	−0.999148	−	0.0412744i	$-0.986858\pi$
$587$		−	2.00000i		−	0.0825488i	0.999148	−	0.0412744i	$-0.0131418\pi$
$588$		0			0
$589$	−10.0000			−0.412043
$590$			12.0000i			0.494032i
$591$		0			0
$592$	−6.00000	+	1.00000i	−0.246598	+	0.0410997i
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$	−18.0000			−0.737928
$596$	−10.0000			−0.409616
$597$		0			0
$598$			1.00000i			0.0408930i
$599$	−10.0000			−0.408589			−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000			−0.408589			−0.204294	+	0.978909i	$0.565490\pi$
$600$		0			0
$601$	7.00000			0.285536			0.142768	−	0.989756i	$-0.454400\pi$
$601$	7.00000			0.285536			0.142768	+	0.989756i	$0.454400\pi$
$602$	−12.0000			−0.489083
$603$		0			0
$604$	13.0000			0.528962
$605$		−	4.00000i		−	0.162623i
$606$		0			0
$607$			12.0000i			0.487065i	0.969893	+	0.243532i	$0.0783062\pi$
$607$			12.0000i			0.487065i	−0.969893	+	0.243532i	$0.921694\pi$
$608$	−1.00000			−0.0405554
$609$		0			0
$610$		−	20.0000i		−	0.809776i
$611$			8.00000i			0.323645i
$612$		0			0
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$		−	12.0000i		−	0.484281i
$615$		0			0
$616$		−	9.00000i		−	0.362620i
$617$	−18.0000			−0.724653			−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000			−0.724653			−0.362326	+	0.932051i	$0.618017\pi$
$618$		0			0
$619$	40.0000			1.60774			0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000			1.60774			0.803868	+	0.594808i	$0.202772\pi$
$620$	20.0000			0.803219
$621$		0			0
$622$	−20.0000			−0.801927
$623$			27.0000i			1.08173i
$624$		0			0
$625$	−19.0000			−0.760000
$626$	6.00000			0.239808
$627$		0			0
$628$	−8.00000			−0.319235
$629$	−3.00000	−	18.0000i	−0.119618	−	0.717707i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−14.0000			−0.556890
$633$		0			0
$634$			2.00000i			0.0794301i
$635$		−	14.0000i		−	0.555573i
$636$		0			0
$637$		−	2.00000i		−	0.0792429i
$638$	18.0000			0.712627
$639$		0			0
$640$	2.00000			0.0790569
$641$	8.00000			0.315981			0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000			0.315981			0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$		−	21.0000i		−	0.828159i	−0.910241	−	0.414080i	$-0.864104\pi$
$643$		−	21.0000i		−	0.828159i	0.910241	−	0.414080i	$-0.135896\pi$
$644$		−	3.00000i		−	0.118217i
$645$		0			0
$646$		−	3.00000i		−	0.118033i
$647$			3.00000i			0.117942i	0.998260	+	0.0589711i	$0.0187820\pi$
$647$			3.00000i			0.117942i	−0.998260	+	0.0589711i	$0.981218\pi$
$648$		0			0
$649$		−	18.0000i		−	0.706562i
$650$	1.00000			0.0392232
$651$		0			0
$652$		−	9.00000i		−	0.352467i
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$		0			0
$655$	20.0000			0.781465
$656$	−2.00000			−0.0780869
$657$		0			0
$658$		−	24.0000i		−	0.935617i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$			25.0000i			0.972387i	0.873851	+	0.486194i	$0.161615\pi$
$661$			25.0000i			0.972387i	−0.873851	+	0.486194i	$0.838385\pi$
$662$	20.0000			0.777322
$663$		0			0
$664$		−	1.00000i		−	0.0388075i
$665$	−6.00000			−0.232670
$666$		0			0
$667$	6.00000			0.232321
$668$			17.0000i			0.657750i
$669$		0			0
$670$	−16.0000			−0.618134
$671$			30.0000i			1.15814i
$672$		0			0
$673$	29.0000			1.11787			0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000			1.11787			0.558934	+	0.829212i	$0.311211\pi$
$674$			3.00000i			0.115556i
$675$		0			0
$676$	−12.0000			−0.461538
$677$	7.00000			0.269032			0.134516	−	0.990911i	$-0.457052\pi$
$677$	7.00000			0.269032			0.134516	+	0.990911i	$0.457052\pi$
$678$		0			0
$679$		−	24.0000i		−	0.921035i
$680$			6.00000i			0.230089i
$681$		0			0
$682$	−30.0000			−1.14876
$683$			16.0000i			0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$			16.0000i			0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$		0			0
$685$			24.0000i			0.916993i
$686$		−	15.0000i		−	0.572703i
$687$		0			0
$688$			4.00000i			0.152499i
$689$		−	11.0000i		−	0.419067i
$690$		0			0
$691$	2.00000			0.0760836			0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000			0.0760836			0.0380418	+	0.999276i	$0.487888\pi$
$692$	9.00000			0.342129
$693$		0			0
$694$	−18.0000			−0.683271
$695$		0			0
$696$		0			0
$697$		−	6.00000i		−	0.227266i
$698$			30.0000i			1.13552i
$699$		0			0
$700$	−3.00000			−0.113389
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−1.00000	−	6.00000i	−0.0377157	−	0.226294i
$704$	−3.00000			−0.113067
$705$		0			0
$706$	34.0000			1.27961
$707$	54.0000			2.03088
$708$		0			0
$709$		−	49.0000i		−	1.84023i	−0.391644	−	0.920117i	$-0.628094\pi$
$709$		−	49.0000i		−	1.84023i	0.391644	−	0.920117i	$-0.371906\pi$
$710$	4.00000			0.150117
$711$		0			0
$712$	9.00000			0.337289
$713$	−10.0000			−0.374503
$714$		0			0
$715$	6.00000			0.224387
$716$		−	24.0000i		−	0.896922i
$717$		0			0
$718$		−	10.0000i		−	0.373197i
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$		−	48.0000i		−	1.78761i
$722$			18.0000i			0.669891i
$723$		0			0
$724$	−2.00000			−0.0743294
$725$		−	6.00000i		−	0.222834i
$726$		0			0
$727$			42.0000i			1.55769i	0.627214	+	0.778847i	$0.284195\pi$
$727$			42.0000i			1.55769i	−0.627214	+	0.778847i	$0.715805\pi$
$728$	−3.00000			−0.111187
$729$		0			0
$730$	2.00000			0.0740233
$731$	−12.0000			−0.443836
$732$		0			0
$733$	−26.0000			−0.960332			−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000			−0.960332			−0.480166	+	0.877178i	$0.659424\pi$
$734$		−	17.0000i		−	0.627481i
$735$		0			0
$736$	−1.00000			−0.0368605
$737$	24.0000			0.884051
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$	2.00000	+	12.0000i	0.0735215	+	0.441129i
$741$		0			0
$742$			33.0000i			1.21147i
$743$	−34.0000			−1.24734			−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000			−1.24734			−0.623670	+	0.781688i	$0.714359\pi$
$744$		0			0
$745$			20.0000i			0.732743i
$746$			4.00000i			0.146450i
$747$		0			0
$748$		−	9.00000i		−	0.329073i
$749$	−39.0000			−1.42503
$750$		0			0
$751$	−48.0000			−1.75154			−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000			−1.75154			−0.875772	+	0.482724i	$0.839647\pi$
$752$	−8.00000			−0.291730
$753$		0			0
$754$		−	6.00000i		−	0.218507i
$755$		−	26.0000i		−	0.946237i
$756$		0			0
$757$			17.0000i			0.617876i	0.951082	+	0.308938i	$0.0999735\pi$
$757$			17.0000i			0.617876i	−0.951082	+	0.308938i	$0.900027\pi$
$758$		−	30.0000i		−	1.08965i
$759$		0			0
$760$			2.00000i			0.0725476i
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$		−	57.0000i		−	2.06354i
$764$			15.0000i			0.542681i
$765$		0			0
$766$	−11.0000			−0.397446
$767$	−6.00000			−0.216647
$768$		0			0
$769$			26.0000i			0.937584i	0.883309	+	0.468792i	$0.155311\pi$
$769$			26.0000i			0.937584i	−0.883309	+	0.468792i	$0.844689\pi$
$770$	−18.0000			−0.648675
$771$		0			0
$772$			16.0000i			0.575853i
$773$	−39.0000			−1.40273			−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000			−1.40273			−0.701366	+	0.712801i	$0.747426\pi$
$774$		0			0
$775$			10.0000i			0.359211i
$776$	−8.00000			−0.287183
$777$		0			0
$778$	−4.00000			−0.143407
$779$		−	2.00000i		−	0.0716574i
$780$		0			0
$781$	−6.00000			−0.214697
$782$		−	3.00000i		−	0.107280i
$783$		0			0
$784$	2.00000			0.0714286
$785$			16.0000i			0.571064i
$786$		0			0
$787$	38.0000			1.35455			0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000			1.35455			0.677277	+	0.735728i	$0.263160\pi$
$788$	−17.0000			−0.605600
$789$		0			0
$790$			28.0000i			0.996195i
$791$		−	42.0000i		−	1.49335i
$792$		0			0
$793$	10.0000			0.355110
$794$		−	32.0000i		−	1.13564i
$795$		0			0
$796$			14.0000i			0.496217i
$797$			8.00000i			0.283375i	0.989911	+	0.141687i	$0.0452527\pi$
$797$			8.00000i			0.283375i	−0.989911	+	0.141687i	$0.954747\pi$
$798$		0			0
$799$		−	24.0000i		−	0.849059i
$800$			1.00000i			0.0353553i
$801$		0			0
$802$	15.0000			0.529668
$803$	−3.00000			−0.105868
$804$		0			0
$805$	−6.00000			−0.211472
$806$			10.0000i			0.352235i
$807$		0			0
$808$		−	18.0000i		−	0.633238i
$809$		−	31.0000i		−	1.08990i	−0.838468	−	0.544951i	$-0.816548\pi$
$809$		−	31.0000i		−	1.08990i	0.838468	−	0.544951i	$-0.183452\pi$
$810$		0			0
$811$	22.0000			0.772524			0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000			0.772524			0.386262	+	0.922389i	$0.373766\pi$
$812$			18.0000i			0.631676i
$813$		0			0
$814$	−3.00000	−	18.0000i	−0.105150	−	0.630900i
$815$	−18.0000			−0.630512
$816$		0			0
$817$	−4.00000			−0.139942
$818$	24.0000			0.839140
$819$		0			0
$820$			4.00000i			0.139686i
$821$	3.00000			0.104701			0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000			0.104701			0.0523504	+	0.998629i	$0.483329\pi$
$822$		0			0
$823$	29.0000			1.01088			0.505438	−	0.862863i	$-0.331331\pi$
$823$	29.0000			1.01088			0.505438	+	0.862863i	$0.331331\pi$
$824$	−16.0000			−0.557386
$825$		0			0
$826$	18.0000			0.626300
$827$		−	2.00000i		−	0.0695468i	−0.999395	−	0.0347734i	$-0.988929\pi$
$827$		−	2.00000i		−	0.0695468i	0.999395	−	0.0347734i	$-0.0110710\pi$
$828$		0			0
$829$			11.0000i			0.382046i	0.981586	+	0.191023i	$0.0611805\pi$
$829$			11.0000i			0.382046i	−0.981586	+	0.191023i	$0.938820\pi$
$830$	−2.00000			−0.0694210
$831$		0			0
$832$			1.00000i			0.0346688i
$833$			6.00000i			0.207888i
$834$		0			0
$835$	34.0000			1.17662
$836$		−	3.00000i		−	0.103757i
$837$		0			0
$838$			35.0000i			1.20905i
$839$	20.0000			0.690477			0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000			0.690477			0.345238	+	0.938515i	$0.387798\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$	10.0000			0.344623
$843$		0			0
$844$	−22.0000			−0.757271
$845$			24.0000i			0.825625i
$846$		0			0
$847$	−6.00000			−0.206162
$848$	11.0000			0.377742
$849$		0			0
$850$	−3.00000			−0.102899
$851$	−1.00000	−	6.00000i	−0.0342796	−	0.205677i
$852$		0			0
$853$			49.0000i			1.67773i	0.544341	+	0.838864i	$0.316780\pi$
$853$			49.0000i			1.67773i	−0.544341	+	0.838864i	$0.683220\pi$
$854$	−30.0000			−1.02658
$855$		0			0
$856$			13.0000i			0.444331i
$857$			13.0000i			0.444072i	0.975039	+	0.222036i	$0.0712702\pi$
$857$			13.0000i			0.444072i	−0.975039	+	0.222036i	$0.928730\pi$
$858$		0			0
$859$			31.0000i			1.05771i	0.848713	+	0.528853i	$0.177378\pi$
$859$			31.0000i			1.05771i	−0.848713	+	0.528853i	$0.822622\pi$
$860$	8.00000			0.272798
$861$		0			0
$862$	15.0000			0.510902
$863$	36.0000			1.22545			0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000			1.22545			0.612727	+	0.790295i	$0.290072\pi$
$864$		0			0
$865$		−	18.0000i		−	0.612018i
$866$		−	1.00000i		−	0.0339814i
$867$		0			0
$868$		−	30.0000i		−	1.01827i
$869$		−	42.0000i		−	1.42475i
$870$		0			0
$871$		−	8.00000i		−	0.271070i
$872$	−19.0000			−0.643421
$873$		0			0
$874$		−	1.00000i		−	0.0338255i
$875$			36.0000i			1.21702i
$876$		0			0
$877$	−32.0000			−1.08056			−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000			−1.08056			−0.540282	+	0.841484i	$0.681682\pi$
$878$	−26.0000			−0.877457
$879$		0			0
$880$			6.00000i			0.202260i
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$		−	11.0000i		−	0.370179i	−0.982722	−	0.185090i	$-0.940742\pi$
$883$		−	11.0000i		−	0.370179i	0.982722	−	0.185090i	$-0.0592576\pi$
$884$	−3.00000			−0.100901
$885$		0			0
$886$		−	4.00000i		−	0.134383i
$887$	−28.0000			−0.940148			−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000			−0.940148			−0.470074	+	0.882627i	$0.655773\pi$
$888$		0			0
$889$	−21.0000			−0.704317
$890$		−	18.0000i		−	0.603361i
$891$		0			0
$892$	−4.00000			−0.133930
$893$		−	8.00000i		−	0.267710i
$894$		0			0
$895$	−48.0000			−1.60446
$896$		−	3.00000i		−	0.100223i
$897$		0			0
$898$	−34.0000			−1.13459
$899$	60.0000			2.00111
$900$		0			0
$901$			33.0000i			1.09939i
$902$		−	6.00000i		−	0.199778i
$903$		0			0
$904$	−14.0000			−0.465633
$905$			4.00000i			0.132964i
$906$		0			0
$907$			7.00000i			0.232431i	0.993224	+	0.116216i	$0.0370764\pi$
$907$			7.00000i			0.232431i	−0.993224	+	0.116216i	$0.962924\pi$
$908$		−	8.00000i		−	0.265489i
$909$		0			0
$910$			6.00000i			0.198898i
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	3.00000			0.0992855
$914$	−2.00000			−0.0661541
$915$		0			0
$916$	20.0000			0.660819
$917$		−	30.0000i		−	0.990687i
$918$		0			0
$919$			36.0000i			1.18753i	0.804638	+	0.593765i	$0.202359\pi$
$919$			36.0000i			1.18753i	−0.804638	+	0.593765i	$0.797641\pi$
$920$			2.00000i			0.0659380i
$921$		0			0
$922$	40.0000			1.31733
$923$			2.00000i			0.0658308i
$924$		0			0
$925$	−6.00000	+	1.00000i	−0.197279	+	0.0328798i
$926$	−34.0000			−1.11731
$927$		0			0
$928$	6.00000			0.196960
$929$	60.0000			1.96854			0.984268	−	0.176682i	$-0.0565363\pi$
$929$	60.0000			1.96854			0.984268	+	0.176682i	$0.0565363\pi$
$930$		0			0
$931$			2.00000i			0.0655474i
$932$	24.0000			0.786146
$933$		0			0
$934$	−8.00000			−0.261768
$935$	−18.0000			−0.588663
$936$		0			0
$937$	18.0000			0.588034			0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000			0.588034			0.294017	+	0.955800i	$0.405008\pi$
$938$			24.0000i			0.783628i
$939$		0			0
$940$			16.0000i			0.521862i
$941$	38.0000			1.23876			0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000			1.23876			0.619382	+	0.785090i	$0.287383\pi$
$942$		0			0
$943$		−	2.00000i		−	0.0651290i
$944$		−	6.00000i		−	0.195283i
$945$		0			0
$946$	−12.0000			−0.390154
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$			1.00000i			0.0324614i
$950$	−1.00000			−0.0324443
$951$		0			0
$952$	9.00000			0.291692
$953$	−44.0000			−1.42530			−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000			−1.42530			−0.712650	+	0.701520i	$0.752505\pi$
$954$		0			0
$955$	30.0000			0.970777
$956$			16.0000i			0.517477i
$957$		0			0
$958$	21.0000			0.678479
$959$	36.0000			1.16250
$960$		0			0
$961$	−69.0000			−2.22581
$962$	−6.00000	+	1.00000i	−0.193448	+	0.0322413i
$963$		0			0
$964$		0			0
$965$	32.0000			1.03012
$966$		0			0
$967$			52.0000i			1.67221i	0.548572	+	0.836104i	$0.315172\pi$
$967$			52.0000i			1.67221i	−0.548572	+	0.836104i	$0.684828\pi$
$968$			2.00000i			0.0642824i
$969$		0			0
$970$			16.0000i			0.513729i
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$		0			0
$974$	8.00000			0.256337
$975$		0			0
$976$			10.0000i			0.320092i
$977$			53.0000i			1.69562i	0.530300	+	0.847810i	$0.322079\pi$
$977$			53.0000i			1.69562i	−0.530300	+	0.847810i	$0.677921\pi$
$978$		0			0
$979$			27.0000i			0.862924i
$980$		−	4.00000i		−	0.127775i
$981$		0			0
$982$		−	27.0000i		−	0.861605i
$983$	16.0000			0.510321			0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000			0.510321			0.255160	+	0.966899i	$0.417872\pi$
$984$		0			0
$985$			34.0000i			1.08333i
$986$			18.0000i			0.573237i
$987$		0			0
$988$	−1.00000			−0.0318142
$989$	−4.00000			−0.127193
$990$		0			0
$991$			10.0000i			0.317660i	0.987306	+	0.158830i	$0.0507723\pi$
$991$			10.0000i			0.317660i	−0.987306	+	0.158830i	$0.949228\pi$
$992$	−10.0000			−0.317500
$993$		0			0
$994$		−	6.00000i		−	0.190308i
$995$	28.0000			0.887660
$996$		0			0
$997$			17.0000i			0.538395i	0.963085	+	0.269198i	$0.0867585\pi$
$997$			17.0000i			0.538395i	−0.963085	+	0.269198i	$0.913241\pi$
$998$	39.0000			1.23452
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	666.2.c.a.73.2		2
3.2	odd	2		222.2.c.a.73.1	✓	2
4.3	odd	2		5328.2.h.b.2737.2		2
12.11	even	2		1776.2.h.a.961.1		2
37.36	even	2	inner	666.2.c.a.73.1		2
111.68	even	4		8214.2.a.g.1.1		1
111.80	even	4		8214.2.a.c.1.1		1
111.110	odd	2		222.2.c.a.73.2	yes	2
148.147	odd	2		5328.2.h.b.2737.1		2
444.443	even	2		1776.2.h.a.961.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.c.a.73.1	✓	2	3.2	odd	2
222.2.c.a.73.2	yes	2	111.110	odd	2
666.2.c.a.73.1		2	37.36	even	2	inner
666.2.c.a.73.2		2	1.1	even	1	trivial
1776.2.h.a.961.1		2	12.11	even	2
1776.2.h.a.961.2		2	444.443	even	2
5328.2.h.b.2737.1		2	148.147	odd	2
5328.2.h.b.2737.2		2	4.3	odd	2
8214.2.a.c.1.1		1	111.80	even	4
8214.2.a.g.1.1		1	111.68	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{5} +3.00000 q^{7} -1.00000i q^{8} -2.00000 q^{10} +3.00000 q^{11} -1.00000i q^{13} +3.00000i q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000i q^{19} -2.00000i q^{20} +3.00000i q^{22} +1.00000i q^{23} +1.00000 q^{25} +1.00000 q^{26} -3.00000 q^{28} -6.00000i q^{29} +10.0000i q^{31} +1.00000i q^{32} -3.00000 q^{34} +6.00000i q^{35} +(-6.00000 + 1.00000i) q^{37} -1.00000 q^{38} +2.00000 q^{40} -2.00000 q^{41} +4.00000i q^{43} -3.00000 q^{44} -1.00000 q^{46} -8.00000 q^{47} +2.00000 q^{49} +1.00000i q^{50} +1.00000i q^{52} +11.0000 q^{53} +6.00000i q^{55} -3.00000i q^{56} +6.00000 q^{58} -6.00000i q^{59} +10.0000i q^{61} -10.0000 q^{62} -1.00000 q^{64} +2.00000 q^{65} +8.00000 q^{67} -3.00000i q^{68} -6.00000 q^{70} -2.00000 q^{71} -1.00000 q^{73} +(-1.00000 - 6.00000i) q^{74} -1.00000i q^{76} +9.00000 q^{77} -14.0000i q^{79} +2.00000i q^{80} -2.00000i q^{82} +1.00000 q^{83} -6.00000 q^{85} -4.00000 q^{86} -3.00000i q^{88} +9.00000i q^{89} -3.00000i q^{91} -1.00000i q^{92} -8.00000i q^{94} -2.00000 q^{95} -8.00000i q^{97} +2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{7} - 4 q^{10} + 6 q^{11} + 2 q^{16} + 2 q^{25} + 2 q^{26} - 6 q^{28} - 6 q^{34} - 12 q^{37} - 2 q^{38} + 4 q^{40} - 4 q^{41} - 6 q^{44} - 2 q^{46} - 16 q^{47} + 4 q^{49} + 22 q^{53}+ \cdots - 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^7 - 4 * q^10 + 6 * q^11 + 2 * q^16 + 2 * q^25 + 2 * q^26 - 6 * q^28 - 6 * q^34 - 12 * q^37 - 2 * q^38 + 4 * q^40 - 4 * q^41 - 6 * q^44 - 2 * q^46 - 16 * q^47 + 4 * q^49 + 22 * q^53 + 12 * q^58 - 20 * q^62 - 2 * q^64 + 4 * q^65 + 16 * q^67 - 12 * q^70 - 4 * q^71 - 2 * q^73 - 2 * q^74 + 18 * q^77 + 2 * q^83 - 12 * q^85 - 8 * q^86 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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