LMFDB - Embedded newform 222.2.c.a.73.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [222,2,Mod(73,222)] 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("222.73"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(222, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 222 = 2 \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	222.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.77267892487$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	222.73
Dual form			222.2.c.a.73.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{6} +3.00000 q^{7} +1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -1.00000i q^{13} -3.00000i q^{14} -2.00000i q^{15} +1.00000 q^{16} -3.00000i q^{17} -1.00000i q^{18} +1.00000i q^{19} +2.00000i q^{20} +3.00000 q^{21} +3.00000i q^{22} -1.00000i q^{23} +1.00000i q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} +6.00000i q^{29} -2.00000 q^{30} +10.0000i q^{31} -1.00000i q^{32} -3.00000 q^{33} -3.00000 q^{34} -6.00000i q^{35} -1.00000 q^{36} +(-6.00000 + 1.00000i) q^{37} +1.00000 q^{38} -1.00000i q^{39} +2.00000 q^{40} +2.00000 q^{41} -3.00000i q^{42} +4.00000i q^{43} +3.00000 q^{44} -2.00000i q^{45} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000i q^{50} -3.00000i q^{51} +1.00000i q^{52} -11.0000 q^{53} -1.00000i q^{54} +6.00000i q^{55} +3.00000i q^{56} +1.00000i q^{57} +6.00000 q^{58} +6.00000i q^{59} +2.00000i q^{60} +10.0000i q^{61} +10.0000 q^{62} +3.00000 q^{63} -1.00000 q^{64} -2.00000 q^{65} +3.00000i q^{66} +8.00000 q^{67} +3.00000i q^{68} -1.00000i q^{69} -6.00000 q^{70} +2.00000 q^{71} +1.00000i q^{72} -1.00000 q^{73} +(1.00000 + 6.00000i) q^{74} +1.00000 q^{75} -1.00000i q^{76} -9.00000 q^{77} -1.00000 q^{78} -14.0000i q^{79} -2.00000i q^{80} +1.00000 q^{81} -2.00000i q^{82} -1.00000 q^{83} -3.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} +6.00000i q^{87} -3.00000i q^{88} -9.00000i q^{89} -2.00000 q^{90} -3.00000i q^{91} +1.00000i q^{92} +10.0000i q^{93} -8.00000i q^{94} +2.00000 q^{95} -1.00000i q^{96} -8.00000i q^{97} -2.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} + 6 q^{7} + 2 q^{9} - 4 q^{10} - 6 q^{11} - 2 q^{12} + 2 q^{16} + 6 q^{21} + 2 q^{25} - 2 q^{26} + 2 q^{27} - 6 q^{28} - 4 q^{30} - 6 q^{33} - 6 q^{34} - 2 q^{36} - 12 q^{37} + 2 q^{38}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 + 6 * q^7 + 2 * q^9 - 4 * q^10 - 6 * q^11 - 2 * q^12 + 2 * q^16 + 6 * q^21 + 2 * q^25 - 2 * q^26 + 2 * q^27 - 6 * q^28 - 4 * q^30 - 6 * q^33 - 6 * q^34 - 2 * q^36 - 12 * q^37 + 2 * q^38 + 4 * q^40 + 4 * q^41 + 6 * q^44 - 2 * q^46 + 16 * q^47 + 2 * q^48 + 4 * q^49 - 22 * q^53 + 12 * q^58 + 20 * q^62 + 6 * q^63 - 2 * q^64 - 4 * q^65 + 16 * q^67 - 12 * q^70 + 4 * q^71 - 2 * q^73 + 2 * q^74 + 2 * q^75 - 18 * q^77 - 2 * q^78 + 2 * q^81 - 2 * q^83 - 6 * q^84 - 12 * q^85 + 8 * q^86 - 4 * q^90 + 4 * q^95 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/222\mathbb{Z}\right)^\times$.

$n$	$149$	$187$
$\chi(n)$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−1.00000			−0.500000
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		−	1.00000i		−	0.408248i
$7$	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
$9$	1.00000			0.333333
$10$	−2.00000			−0.632456
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$	−1.00000			−0.288675
$13$		−	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
$15$		−	2.00000i		−	0.516398i
$16$	1.00000			0.250000
$17$		−	3.00000i		−	0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$		−	3.00000i		−	0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$		−	1.00000i		−	0.235702i
$19$			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
$21$	3.00000			0.654654
$22$			3.00000i			0.639602i
$23$		−	1.00000i		−	0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$		−	1.00000i		−	0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$			1.00000i			0.204124i
$25$	1.00000			0.200000
$26$	−1.00000			−0.196116
$27$	1.00000			0.192450
$28$	−3.00000			−0.566947
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	−2.00000			−0.365148
$31$			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
$33$	−3.00000			−0.522233
$34$	−3.00000			−0.514496
$35$		−	6.00000i		−	1.01419i
$36$	−1.00000			−0.166667
$37$	−6.00000	+	1.00000i	−0.986394	+	0.164399i
$37$	−6.00000	+	1.00000i	−0.986394	+	0.164399i
$38$	1.00000			0.162221
$39$		−	1.00000i		−	0.160128i
$40$	2.00000			0.316228
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		−	3.00000i		−	0.462910i
$43$			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
$45$		−	2.00000i		−	0.298142i
$46$	−1.00000			−0.147442
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000			0.144338
$49$	2.00000			0.285714
$50$		−	1.00000i		−	0.141421i
$51$		−	3.00000i		−	0.420084i
$52$			1.00000i			0.138675i
$53$	−11.0000			−1.51097			−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000			−1.51097			−0.755483	+	0.655168i	$0.772598\pi$
$54$		−	1.00000i		−	0.136083i
$55$			6.00000i			0.809040i
$56$			3.00000i			0.400892i
$57$			1.00000i			0.132453i
$58$	6.00000			0.787839
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$			2.00000i			0.258199i
$61$			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
$63$	3.00000			0.377964
$64$	−1.00000			−0.125000
$65$	−2.00000			−0.248069
$66$			3.00000i			0.369274i
$67$	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
$69$		−	1.00000i		−	0.120386i
$70$	−6.00000			−0.717137
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$			1.00000i			0.117851i
$73$	−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
$75$	1.00000			0.115470
$76$		−	1.00000i		−	0.114708i
$77$	−9.00000			−1.02565
$78$	−1.00000			−0.113228
$79$		−	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
$81$	1.00000			0.111111
$82$		−	2.00000i		−	0.220863i
$83$	−1.00000			−0.109764			−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000			−0.109764			−0.0548821	+	0.998493i	$0.517478\pi$
$84$	−3.00000			−0.327327
$85$	−6.00000			−0.650791
$86$	4.00000			0.431331
$87$			6.00000i			0.643268i
$88$		−	3.00000i		−	0.319801i
$89$		−	9.00000i		−	0.953998i	−0.878904	−	0.476999i	$-0.841725\pi$
$89$		−	9.00000i		−	0.953998i	0.878904	−	0.476999i	$-0.158275\pi$
$90$	−2.00000			−0.210819
$91$		−	3.00000i		−	0.314485i
$92$			1.00000i			0.104257i
$93$			10.0000i			1.03695i
$94$		−	8.00000i		−	0.825137i
$95$	2.00000			0.205196
$96$		−	1.00000i		−	0.102062i
$97$		−	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
$99$	−3.00000			−0.301511
$100$	−1.00000			−0.100000
$101$	−18.0000			−1.79107			−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000			−1.79107			−0.895533	+	0.444994i	$0.853206\pi$
$102$	−3.00000			−0.297044
$103$		−	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
$105$		−	6.00000i		−	0.585540i
$106$			11.0000i			1.06841i
$107$	13.0000			1.25676			0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000			1.25676			0.628379	+	0.777908i	$0.283719\pi$
$108$	−1.00000			−0.0962250
$109$		−	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
$111$	−6.00000	+	1.00000i	−0.569495	+	0.0949158i
$112$	3.00000			0.283473
$113$			14.0000i			1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$			14.0000i			1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	1.00000			0.0936586
$115$	−2.00000			−0.186501
$116$		−	6.00000i		−	0.557086i
$117$		−	1.00000i		−	0.0924500i
$118$	6.00000			0.552345
$119$		−	9.00000i		−	0.825029i
$120$	2.00000			0.182574
$121$	−2.00000			−0.181818
$122$	10.0000			0.905357
$123$	2.00000			0.180334
$124$		−	10.0000i		−	0.898027i
$125$		−	12.0000i		−	1.07331i
$126$		−	3.00000i		−	0.267261i
$127$	−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
$129$			4.00000i			0.352180i
$130$			2.00000i			0.175412i
$131$			10.0000i			0.873704i	0.899533	+	0.436852i	$0.143907\pi$
$131$			10.0000i			0.873704i	−0.899533	+	0.436852i	$0.856093\pi$
$132$	3.00000			0.261116
$133$			3.00000i			0.260133i
$134$		−	8.00000i		−	0.691095i
$135$		−	2.00000i		−	0.172133i
$136$	3.00000			0.257248
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$	−1.00000			−0.0851257
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$			6.00000i			0.507093i
$141$	8.00000			0.673722
$142$		−	2.00000i		−	0.167836i
$143$			3.00000i			0.250873i
$144$	1.00000			0.0833333
$145$	12.0000			0.996546
$146$			1.00000i			0.0827606i
$147$	2.00000			0.164957
$148$	6.00000	−	1.00000i	0.493197	−	0.0821995i
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		−	1.00000i		−	0.0816497i
$151$	−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
$153$		−	3.00000i		−	0.242536i
$154$			9.00000i			0.725241i
$155$	20.0000			1.60644
$156$			1.00000i			0.0800641i
$157$	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
$159$	−11.0000			−0.872357
$160$	−2.00000			−0.158114
$161$		−	3.00000i		−	0.236433i
$162$		−	1.00000i		−	0.0785674i
$163$			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
$165$			6.00000i			0.467099i
$166$			1.00000i			0.0776151i
$167$			17.0000i			1.31550i	0.753237	+	0.657750i	$0.228492\pi$
$167$			17.0000i			1.31550i	−0.753237	+	0.657750i	$0.771508\pi$
$168$			3.00000i			0.231455i
$169$	12.0000			0.923077
$170$			6.00000i			0.460179i
$171$			1.00000i			0.0764719i
$172$		−	4.00000i		−	0.304997i
$173$	9.00000			0.684257			0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000			0.684257			0.342129	+	0.939653i	$0.388852\pi$
$174$	6.00000			0.454859
$175$	3.00000			0.226779
$176$	−3.00000			−0.226134
$177$			6.00000i			0.450988i
$178$	−9.00000			−0.674579
$179$		−	24.0000i		−	1.79384i	−0.442189	−	0.896922i	$-0.645798\pi$
$179$		−	24.0000i		−	1.79384i	0.442189	−	0.896922i	$-0.354202\pi$
$180$			2.00000i			0.149071i
$181$	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
$183$			10.0000i			0.739221i
$184$	1.00000			0.0737210
$185$	2.00000	+	12.0000i	0.147043	+	0.882258i
$186$	10.0000			0.733236
$187$			9.00000i			0.658145i
$188$	−8.00000			−0.583460
$189$	3.00000			0.218218
$190$		−	2.00000i		−	0.145095i
$191$			15.0000i			1.08536i	0.839939	+	0.542681i	$0.182591\pi$
$191$			15.0000i			1.08536i	−0.839939	+	0.542681i	$0.817409\pi$
$192$	−1.00000			−0.0721688
$193$		−	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
$195$	−2.00000			−0.143223
$196$	−2.00000			−0.142857
$197$	−17.0000			−1.21120			−0.605600	−	0.795769i	$-0.707067\pi$
$197$	−17.0000			−1.21120			−0.605600	+	0.795769i	$0.707067\pi$
$198$			3.00000i			0.213201i
$199$		−	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
$201$	8.00000			0.564276
$202$			18.0000i			1.26648i
$203$			18.0000i			1.26335i
$204$			3.00000i			0.210042i
$205$		−	4.00000i		−	0.279372i
$206$	−16.0000			−1.11477
$207$		−	1.00000i		−	0.0695048i
$208$		−	1.00000i		−	0.0693375i
$209$		−	3.00000i		−	0.207514i
$210$	−6.00000			−0.414039
$211$	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
$213$	2.00000			0.137038
$214$		−	13.0000i		−	0.888662i
$215$	8.00000			0.545595
$216$			1.00000i			0.0680414i
$217$			30.0000i			2.03653i
$218$	−19.0000			−1.28684
$219$	−1.00000			−0.0675737
$220$		−	6.00000i		−	0.404520i
$221$	−3.00000			−0.201802
$222$	1.00000	+	6.00000i	0.0671156	+	0.402694i
$223$	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
$225$	1.00000			0.0666667
$226$	14.0000			0.931266
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		−	1.00000i		−	0.0662266i
$229$	−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
$231$	−9.00000			−0.592157
$232$	−6.00000			−0.393919
$233$	24.0000			1.57229			0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000			1.57229			0.786146	+	0.618041i	$0.212073\pi$
$234$	−1.00000			−0.0653720
$235$		−	16.0000i		−	1.04372i
$236$		−	6.00000i		−	0.390567i
$237$		−	14.0000i		−	0.909398i
$238$	−9.00000			−0.583383
$239$			16.0000i			1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$			16.0000i			1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$		−	2.00000i		−	0.129099i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			2.00000i			0.128565i
$243$	1.00000			0.0641500
$244$		−	10.0000i		−	0.640184i
$245$		−	4.00000i		−	0.255551i
$246$		−	2.00000i		−	0.127515i
$247$	1.00000			0.0636285
$248$	−10.0000			−0.635001
$249$	−1.00000			−0.0633724
$250$	−12.0000			−0.758947
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−3.00000			−0.188982
$253$			3.00000i			0.188608i
$254$			7.00000i			0.439219i
$255$	−6.00000			−0.375735
$256$	1.00000			0.0625000
$257$		−	3.00000i		−	0.187135i	−0.995613	−	0.0935674i	$-0.970173\pi$
$257$		−	3.00000i		−	0.187135i	0.995613	−	0.0935674i	$-0.0298271\pi$
$258$	4.00000			0.249029
$259$	−18.0000	+	3.00000i	−1.11847	+	0.186411i
$260$	2.00000			0.124035
$261$			6.00000i			0.371391i
$262$	10.0000			0.617802
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		−	3.00000i		−	0.184637i
$265$			22.0000i			1.35145i
$266$	3.00000			0.183942
$267$		−	9.00000i		−	0.550791i
$268$	−8.00000			−0.488678
$269$	−5.00000			−0.304855			−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000			−0.304855			−0.152428	+	0.988315i	$0.548709\pi$
$270$	−2.00000			−0.121716
$271$	−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
$273$		−	3.00000i		−	0.181568i
$274$			12.0000i			0.724947i
$275$	−3.00000			−0.180907
$276$			1.00000i			0.0601929i
$277$			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
$279$			10.0000i			0.598684i
$280$	6.00000			0.358569
$281$		−	15.0000i		−	0.894825i	−0.894328	−	0.447412i	$-0.852346\pi$
$281$		−	15.0000i		−	0.894825i	0.894328	−	0.447412i	$-0.147654\pi$
$282$		−	8.00000i		−	0.476393i
$283$			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
$285$	2.00000			0.118470
$286$	3.00000			0.177394
$287$	6.00000			0.354169
$288$		−	1.00000i		−	0.0589256i
$289$	8.00000			0.470588
$290$		−	12.0000i		−	0.704664i
$291$		−	8.00000i		−	0.468968i
$292$	1.00000			0.0585206
$293$	9.00000			0.525786			0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000			0.525786			0.262893	+	0.964825i	$0.415323\pi$
$294$		−	2.00000i		−	0.116642i
$295$	12.0000			0.698667
$296$	−1.00000	−	6.00000i	−0.0581238	−	0.348743i
$297$	−3.00000			−0.174078
$298$			10.0000i			0.579284i
$299$	−1.00000			−0.0578315
$300$	−1.00000			−0.0577350
$301$			12.0000i			0.691669i
$302$			13.0000i			0.748066i
$303$	−18.0000			−1.03407
$304$			1.00000i			0.0573539i
$305$	20.0000			1.14520
$306$	−3.00000			−0.171499
$307$	−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
$309$		−	16.0000i		−	0.910208i
$310$		−	20.0000i		−	1.13592i
$311$		−	20.0000i		−	1.13410i	−0.823685	−	0.567048i	$-0.808085\pi$
$311$		−	20.0000i		−	1.13410i	0.823685	−	0.567048i	$-0.191915\pi$
$312$	1.00000			0.0566139
$313$		−	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
$315$		−	6.00000i		−	0.338062i
$316$			14.0000i			0.787562i
$317$	−2.00000			−0.112331			−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000			−0.112331			−0.0561656	+	0.998421i	$0.517887\pi$
$318$			11.0000i			0.616849i
$319$		−	18.0000i		−	1.00781i
$320$			2.00000i			0.111803i
$321$	13.0000			0.725589
$322$	−3.00000			−0.167183
$323$	3.00000			0.166924
$324$	−1.00000			−0.0555556
$325$		−	1.00000i		−	0.0554700i
$326$	9.00000			0.498464
$327$		−	19.0000i		−	1.05070i
$328$			2.00000i			0.110432i
$329$	24.0000			1.32316
$330$	6.00000			0.330289
$331$		−	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
$333$	−6.00000	+	1.00000i	−0.328798	+	0.0547997i
$334$	17.0000			0.930199
$335$		−	16.0000i		−	0.874173i
$336$	3.00000			0.163663
$337$	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
$339$			14.0000i			0.760376i
$340$	6.00000			0.325396
$341$		−	30.0000i		−	1.62459i
$342$	1.00000			0.0540738
$343$	−15.0000			−0.809924
$344$	−4.00000			−0.215666
$345$	−2.00000			−0.107676
$346$		−	9.00000i		−	0.483843i
$347$		−	18.0000i		−	0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$		−	18.0000i		−	0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$		−	6.00000i		−	0.321634i
$349$	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
$351$		−	1.00000i		−	0.0533761i
$352$			3.00000i			0.159901i
$353$			34.0000i			1.80964i	0.425797	+	0.904819i	$0.359994\pi$
$353$			34.0000i			1.80964i	−0.425797	+	0.904819i	$0.640006\pi$
$354$	6.00000			0.318896
$355$		−	4.00000i		−	0.212298i
$356$			9.00000i			0.476999i
$357$		−	9.00000i		−	0.476331i
$358$	−24.0000			−1.26844
$359$	10.0000			0.527780			0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000			0.527780			0.263890	+	0.964553i	$0.414994\pi$
$360$	2.00000			0.105409
$361$	18.0000			0.947368
$362$		−	2.00000i		−	0.105118i
$363$	−2.00000			−0.104973
$364$			3.00000i			0.157243i
$365$			2.00000i			0.104685i
$366$	10.0000			0.522708
$367$	−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
$369$	2.00000			0.104116
$370$	12.0000	−	2.00000i	0.623850	−	0.103975i
$371$	−33.0000			−1.71327
$372$		−	10.0000i		−	0.518476i
$373$	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
$375$		−	12.0000i		−	0.619677i
$376$			8.00000i			0.412568i
$377$	6.00000			0.309016
$378$		−	3.00000i		−	0.154303i
$379$	−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
$381$	−7.00000			−0.358621
$382$	15.0000			0.767467
$383$		−	11.0000i		−	0.562074i	−0.959697	−	0.281037i	$-0.909322\pi$
$383$		−	11.0000i		−	0.562074i	0.959697	−	0.281037i	$-0.0906783\pi$
$384$			1.00000i			0.0510310i
$385$			18.0000i			0.917365i
$386$	−16.0000			−0.814379
$387$			4.00000i			0.203331i
$388$			8.00000i			0.406138i
$389$		−	4.00000i		−	0.202808i	−0.994845	−	0.101404i	$-0.967667\pi$
$389$		−	4.00000i		−	0.202808i	0.994845	−	0.101404i	$-0.0323335\pi$
$390$			2.00000i			0.101274i
$391$	−3.00000			−0.151717
$392$			2.00000i			0.101015i
$393$			10.0000i			0.504433i
$394$			17.0000i			0.856448i
$395$	−28.0000			−1.40883
$396$	3.00000			0.150756
$397$	−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
$399$			3.00000i			0.150188i
$400$	1.00000			0.0500000
$401$			15.0000i			0.749064i	0.927214	+	0.374532i	$0.122197\pi$
$401$			15.0000i			0.749064i	−0.927214	+	0.374532i	$0.877803\pi$
$402$		−	8.00000i		−	0.399004i
$403$	10.0000			0.498135
$404$	18.0000			0.895533
$405$		−	2.00000i		−	0.0993808i
$406$	18.0000			0.893325
$407$	18.0000	−	3.00000i	0.892227	−	0.148704i
$408$	3.00000			0.148522
$409$		−	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
$411$	−12.0000			−0.591916
$412$			16.0000i			0.788263i
$413$			18.0000i			0.885722i
$414$	−1.00000			−0.0491473
$415$			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	$-0.826401\pi$
$419$	−35.0000			−1.70986			−0.854931	+	0.518742i	$0.826401\pi$
$420$			6.00000i			0.292770i
$421$		−	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
$423$	8.00000			0.388973
$424$		−	11.0000i		−	0.534207i
$425$		−	3.00000i		−	0.145521i
$426$		−	2.00000i		−	0.0969003i
$427$			30.0000i			1.45180i
$428$	−13.0000			−0.628379
$429$			3.00000i			0.144841i
$430$		−	8.00000i		−	0.385794i
$431$			15.0000i			0.722525i	0.932464	+	0.361262i	$0.117654\pi$
$431$			15.0000i			0.722525i	−0.932464	+	0.361262i	$0.882346\pi$
$432$	1.00000			0.0481125
$433$	−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
$435$	12.0000			0.575356
$436$			19.0000i			0.909935i
$437$	1.00000			0.0478365
$438$			1.00000i			0.0477818i
$439$			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
$441$	2.00000			0.0952381
$442$			3.00000i			0.142695i
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$	6.00000	−	1.00000i	0.284747	−	0.0474579i
$445$	−18.0000			−0.853282
$446$		−	4.00000i		−	0.189405i
$447$	−10.0000			−0.472984
$448$	−3.00000			−0.141737
$449$		−	34.0000i		−	1.60456i	−0.596948	−	0.802280i	$-0.703620\pi$
$449$		−	34.0000i		−	1.60456i	0.596948	−	0.802280i	$-0.296380\pi$
$450$		−	1.00000i		−	0.0471405i
$451$	−6.00000			−0.282529
$452$		−	14.0000i		−	0.658505i
$453$	−13.0000			−0.610793
$454$	−8.00000			−0.375459
$455$	−6.00000			−0.281284
$456$	−1.00000			−0.0468293
$457$			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
$459$		−	3.00000i		−	0.140028i
$460$	2.00000			0.0932505
$461$			40.0000i			1.86299i	0.363760	+	0.931493i	$0.381493\pi$
$461$			40.0000i			1.86299i	−0.363760	+	0.931493i	$0.618507\pi$
$462$			9.00000i			0.418718i
$463$			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
$465$	20.0000			0.927478
$466$		−	24.0000i		−	1.11178i
$467$		−	8.00000i		−	0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$		−	8.00000i		−	0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$			1.00000i			0.0462250i
$469$	24.0000			1.10822
$470$	−16.0000			−0.738025
$471$	8.00000			0.368621
$472$	−6.00000			−0.276172
$473$		−	12.0000i		−	0.551761i
$474$	−14.0000			−0.643041
$475$			1.00000i			0.0458831i
$476$			9.00000i			0.412514i
$477$	−11.0000			−0.503655
$478$	16.0000			0.731823
$479$			21.0000i			0.959514i	0.877401	+	0.479757i	$0.159275\pi$
$479$			21.0000i			0.959514i	−0.877401	+	0.479757i	$0.840725\pi$
$480$	−2.00000			−0.0912871
$481$	1.00000	+	6.00000i	0.0455961	+	0.273576i
$482$		0			0
$483$		−	3.00000i		−	0.136505i
$484$	2.00000			0.0909091
$485$	−16.0000			−0.726523
$486$		−	1.00000i		−	0.0453609i
$487$		−	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
$489$			9.00000i			0.406994i
$490$	−4.00000			−0.180702
$491$	27.0000			1.21849			0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000			1.21849			0.609246	+	0.792981i	$0.291472\pi$
$492$	−2.00000			−0.0901670
$493$	18.0000			0.810679
$494$		−	1.00000i		−	0.0449921i
$495$			6.00000i			0.269680i
$496$			10.0000i			0.449013i
$497$	6.00000			0.269137
$498$			1.00000i			0.0448111i
$499$		−	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
$501$			17.0000i			0.759504i
$502$		0			0
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$			3.00000i			0.133631i
$505$			36.0000i			1.60198i
$506$	3.00000			0.133366
$507$	12.0000			0.532939
$508$	7.00000			0.310575
$509$	−5.00000			−0.221621			−0.110811	−	0.993842i	$-0.535345\pi$
$509$	−5.00000			−0.221621			−0.110811	+	0.993842i	$0.535345\pi$
$510$			6.00000i			0.265684i
$511$	−3.00000			−0.132712
$512$		−	1.00000i		−	0.0441942i
$513$			1.00000i			0.0441511i
$514$	−3.00000			−0.132324
$515$	−32.0000			−1.41009
$516$		−	4.00000i		−	0.176090i
$517$	−24.0000			−1.05552
$518$	3.00000	+	18.0000i	0.131812	+	0.790875i
$519$	9.00000			0.395056
$520$		−	2.00000i		−	0.0877058i
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$	6.00000			0.262613
$523$			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
$525$	3.00000			0.130931
$526$		−	24.0000i		−	1.04645i
$527$	30.0000			1.30682
$528$	−3.00000			−0.130558
$529$	22.0000			0.956522
$530$	22.0000			0.955619
$531$			6.00000i			0.260378i
$532$		−	3.00000i		−	0.130066i
$533$		−	2.00000i		−	0.0866296i
$534$	−9.00000			−0.389468
$535$		−	26.0000i		−	1.12408i
$536$			8.00000i			0.345547i
$537$		−	24.0000i		−	1.03568i
$538$			5.00000i			0.215565i
$539$	−6.00000			−0.258438
$540$			2.00000i			0.0860663i
$541$		−	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
$543$	2.00000			0.0858282
$544$	−3.00000			−0.128624
$545$	−38.0000			−1.62774
$546$	−3.00000			−0.128388
$547$			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
$549$			10.0000i			0.426790i
$550$			3.00000i			0.127920i
$551$	−6.00000			−0.255609
$552$	1.00000			0.0425628
$553$		−	42.0000i		−	1.78602i
$554$	7.00000			0.297402
$555$	2.00000	+	12.0000i	0.0848953	+	0.509372i
$556$		0			0
$557$		−	38.0000i		−	1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$		−	38.0000i		−	1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$	10.0000			0.423334
$559$	4.00000			0.169182
$560$		−	6.00000i		−	0.253546i
$561$			9.00000i			0.379980i
$562$	−15.0000			−0.632737
$563$			14.0000i			0.590030i	0.955493	+	0.295015i	$0.0953246\pi$
$563$			14.0000i			0.590030i	−0.955493	+	0.295015i	$0.904675\pi$
$564$	−8.00000			−0.336861
$565$	28.0000			1.17797
$566$	29.0000			1.21896
$567$	3.00000			0.125988
$568$			2.00000i			0.0839181i
$569$			21.0000i			0.880366i	0.897908	+	0.440183i	$0.145086\pi$
$569$			21.0000i			0.880366i	−0.897908	+	0.440183i	$0.854914\pi$
$570$		−	2.00000i		−	0.0837708i
$571$	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
$573$			15.0000i			0.626634i
$574$		−	6.00000i		−	0.250435i
$575$		−	1.00000i		−	0.0417029i
$576$	−1.00000			−0.0416667
$577$		−	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
$579$		−	16.0000i		−	0.664937i
$580$	−12.0000			−0.498273
$581$	−3.00000			−0.124461
$582$	−8.00000			−0.331611
$583$	33.0000			1.36672
$584$		−	1.00000i		−	0.0413803i
$585$	−2.00000			−0.0826898
$586$		−	9.00000i		−	0.371787i
$587$			2.00000i			0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$			2.00000i			0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$	−2.00000			−0.0824786
$589$	−10.0000			−0.412043
$590$		−	12.0000i		−	0.494032i
$591$	−17.0000			−0.699287
$592$	−6.00000	+	1.00000i	−0.246598	+	0.0410997i
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$			3.00000i			0.123091i
$595$	−18.0000			−0.737928
$596$	10.0000			0.409616
$597$		−	14.0000i		−	0.572982i
$598$			1.00000i			0.0408930i
$599$	10.0000			0.408589			0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000			0.408589			0.204294	+	0.978909i	$0.434510\pi$
$600$			1.00000i			0.0408248i
$601$	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
$603$	8.00000			0.325785
$604$	13.0000			0.528962
$605$			4.00000i			0.162623i
$606$			18.0000i			0.731200i
$607$			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
$609$			18.0000i			0.729397i
$610$		−	20.0000i		−	0.809776i
$611$		−	8.00000i		−	0.323645i
$612$			3.00000i			0.121268i
$613$	−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
$615$		−	4.00000i		−	0.161296i
$616$		−	9.00000i		−	0.362620i
$617$	18.0000			0.724653			0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000			0.724653			0.362326	+	0.932051i	$0.381983\pi$
$618$	−16.0000			−0.643614
$619$	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
$621$		−	1.00000i		−	0.0401286i
$622$	−20.0000			−0.801927
$623$		−	27.0000i		−	1.08173i
$624$		−	1.00000i		−	0.0400320i
$625$	−19.0000			−0.760000
$626$	−6.00000			−0.239808
$627$		−	3.00000i		−	0.119808i
$628$	−8.00000			−0.319235
$629$	3.00000	+	18.0000i	0.119618	+	0.717707i
$630$	−6.00000			−0.239046
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	14.0000			0.556890
$633$	22.0000			0.874421
$634$			2.00000i			0.0794301i
$635$			14.0000i			0.555573i
$636$	11.0000			0.436178
$637$		−	2.00000i		−	0.0792429i
$638$	−18.0000			−0.712627
$639$	2.00000			0.0791188
$640$	2.00000			0.0790569
$641$	−8.00000			−0.315981			−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000			−0.315981			−0.157991	+	0.987441i	$0.550502\pi$
$642$		−	13.0000i		−	0.513069i
$643$		−	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
$645$	8.00000			0.315000
$646$		−	3.00000i		−	0.118033i
$647$		−	3.00000i		−	0.117942i	−0.998260	−	0.0589711i	$-0.981218\pi$
$647$		−	3.00000i		−	0.117942i	0.998260	−	0.0589711i	$-0.0187820\pi$
$648$			1.00000i			0.0392837i
$649$		−	18.0000i		−	0.706562i
$650$	−1.00000			−0.0392232
$651$			30.0000i			1.17579i
$652$		−	9.00000i		−	0.352467i
$653$		−	6.00000i		−	0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$		−	6.00000i		−	0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	−19.0000			−0.742959
$655$	20.0000			0.781465
$656$	2.00000			0.0780869
$657$	−1.00000			−0.0390137
$658$		−	24.0000i		−	0.935617i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		−	6.00000i		−	0.233550i
$661$			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
$663$	−3.00000			−0.116510
$664$		−	1.00000i		−	0.0388075i
$665$	6.00000			0.232670
$666$	1.00000	+	6.00000i	0.0387492	+	0.232495i
$667$	6.00000			0.232321
$668$		−	17.0000i		−	0.657750i
$669$	4.00000			0.154649
$670$	−16.0000			−0.618134
$671$		−	30.0000i		−	1.15814i
$672$		−	3.00000i		−	0.115728i
$673$	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
$675$	1.00000			0.0384900
$676$	−12.0000			−0.461538
$677$	−7.00000			−0.269032			−0.134516	−	0.990911i	$-0.542948\pi$
$677$	−7.00000			−0.269032			−0.134516	+	0.990911i	$0.542948\pi$
$678$	14.0000			0.537667
$679$		−	24.0000i		−	0.921035i
$680$		−	6.00000i		−	0.230089i
$681$		−	8.00000i		−	0.306561i
$682$	−30.0000			−1.14876
$683$		−	16.0000i		−	0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$		−	16.0000i		−	0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$		−	1.00000i		−	0.0382360i
$685$			24.0000i			0.916993i
$686$			15.0000i			0.572703i
$687$	−20.0000			−0.763048
$688$			4.00000i			0.152499i
$689$			11.0000i			0.419067i
$690$			2.00000i			0.0761387i
$691$	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
$693$	−9.00000			−0.341882
$694$	−18.0000			−0.683271
$695$		0			0
$696$	−6.00000			−0.227429
$697$		−	6.00000i		−	0.227266i
$698$		−	30.0000i		−	1.13552i
$699$	24.0000			0.907763
$700$	−3.00000			−0.113389
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−1.00000			−0.0377426
$703$	−1.00000	−	6.00000i	−0.0377157	−	0.226294i
$704$	3.00000			0.113067
$705$		−	16.0000i		−	0.602595i
$706$	34.0000			1.27961
$707$	−54.0000			−2.03088
$708$		−	6.00000i		−	0.225494i
$709$		−	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
$711$		−	14.0000i		−	0.525041i
$712$	9.00000			0.337289
$713$	10.0000			0.374503
$714$	−9.00000			−0.336817
$715$	6.00000			0.224387
$716$			24.0000i			0.896922i
$717$			16.0000i			0.597531i
$718$		−	10.0000i		−	0.373197i
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		−	2.00000i		−	0.0745356i
$721$		−	48.0000i		−	1.78761i
$722$		−	18.0000i		−	0.669891i
$723$		0			0
$724$	−2.00000			−0.0743294
$725$			6.00000i			0.222834i
$726$			2.00000i			0.0742270i
$727$			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
$729$	1.00000			0.0370370
$730$	2.00000			0.0740233
$731$	12.0000			0.443836
$732$		−	10.0000i		−	0.369611i
$733$	−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
$735$		−	4.00000i		−	0.147542i
$736$	−1.00000			−0.0368605
$737$	−24.0000			−0.884051
$738$		−	2.00000i		−	0.0736210i
$739$	−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
$741$	1.00000			0.0367359
$742$			33.0000i			1.21147i
$743$	34.0000			1.24734			0.623670	−	0.781688i	$-0.285641\pi$
$743$	34.0000			1.24734			0.623670	+	0.781688i	$0.285641\pi$
$744$	−10.0000			−0.366618
$745$			20.0000i			0.732743i
$746$		−	4.00000i		−	0.146450i
$747$	−1.00000			−0.0365881
$748$		−	9.00000i		−	0.329073i
$749$	39.0000			1.42503
$750$	−12.0000			−0.438178
$751$	−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
$756$	−3.00000			−0.109109
$757$			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
$759$			3.00000i			0.108893i
$760$			2.00000i			0.0725476i
$761$	−18.0000			−0.652499			−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000			−0.652499			−0.326250	+	0.945284i	$0.605785\pi$
$762$			7.00000i			0.253583i
$763$		−	57.0000i		−	2.06354i
$764$		−	15.0000i		−	0.542681i
$765$	−6.00000			−0.216930
$766$	−11.0000			−0.397446
$767$	6.00000			0.216647
$768$	1.00000			0.0360844
$769$			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
$771$		−	3.00000i		−	0.108042i
$772$			16.0000i			0.575853i
$773$	39.0000			1.40273			0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000			1.40273			0.701366	+	0.712801i	$0.252574\pi$
$774$	4.00000			0.143777
$775$			10.0000i			0.359211i
$776$	8.00000			0.287183
$777$	−18.0000	+	3.00000i	−0.645746	+	0.107624i
$778$	−4.00000			−0.143407
$779$			2.00000i			0.0716574i
$780$	2.00000			0.0716115
$781$	−6.00000			−0.214697
$782$			3.00000i			0.107280i
$783$			6.00000i			0.214423i
$784$	2.00000			0.0714286
$785$		−	16.0000i		−	0.571064i
$786$	10.0000			0.356688
$787$	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
$789$	24.0000			0.854423
$790$			28.0000i			0.996195i
$791$			42.0000i			1.49335i
$792$		−	3.00000i		−	0.106600i
$793$	10.0000			0.355110
$794$			32.0000i			1.13564i
$795$			22.0000i			0.780260i
$796$			14.0000i			0.496217i
$797$		−	8.00000i		−	0.283375i	−0.989911	−	0.141687i	$-0.954747\pi$
$797$		−	8.00000i		−	0.283375i	0.989911	−	0.141687i	$-0.0452527\pi$
$798$	3.00000			0.106199
$799$		−	24.0000i		−	0.849059i
$800$		−	1.00000i		−	0.0353553i
$801$		−	9.00000i		−	0.317999i
$802$	15.0000			0.529668
$803$	3.00000			0.105868
$804$	−8.00000			−0.282138
$805$	−6.00000			−0.211472
$806$		−	10.0000i		−	0.352235i
$807$	−5.00000			−0.176008
$808$		−	18.0000i		−	0.633238i
$809$			31.0000i			1.08990i	0.838468	+	0.544951i	$0.183452\pi$
$809$			31.0000i			1.08990i	−0.838468	+	0.544951i	$0.816548\pi$
$810$	−2.00000			−0.0702728
$811$	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
$813$	−28.0000			−0.982003
$814$	−3.00000	−	18.0000i	−0.105150	−	0.630900i
$815$	18.0000			0.630512
$816$		−	3.00000i		−	0.105021i
$817$	−4.00000			−0.139942
$818$	−24.0000			−0.839140
$819$		−	3.00000i		−	0.104828i
$820$			4.00000i			0.139686i
$821$	−3.00000			−0.104701			−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000			−0.104701			−0.0523504	+	0.998629i	$0.516671\pi$
$822$			12.0000i			0.418548i
$823$	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
$825$	−3.00000			−0.104447
$826$	18.0000			0.626300
$827$			2.00000i			0.0695468i	0.999395	+	0.0347734i	$0.0110710\pi$
$827$			2.00000i			0.0695468i	−0.999395	+	0.0347734i	$0.988929\pi$
$828$			1.00000i			0.0347524i
$829$			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
$831$			7.00000i			0.242827i
$832$			1.00000i			0.0346688i
$833$		−	6.00000i		−	0.207888i
$834$		0			0
$835$	34.0000			1.17662
$836$			3.00000i			0.103757i
$837$			10.0000i			0.345651i
$838$			35.0000i			1.20905i
$839$	−20.0000			−0.690477			−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000			−0.690477			−0.345238	+	0.938515i	$0.612202\pi$
$840$	6.00000			0.207020
$841$	−7.00000			−0.241379
$842$	−10.0000			−0.344623
$843$		−	15.0000i		−	0.516627i
$844$	−22.0000			−0.757271
$845$		−	24.0000i		−	0.825625i
$846$		−	8.00000i		−	0.275046i
$847$	−6.00000			−0.206162
$848$	−11.0000			−0.377742
$849$			29.0000i			0.995277i
$850$	−3.00000			−0.102899
$851$	1.00000	+	6.00000i	0.0342796	+	0.205677i
$852$	−2.00000			−0.0685189
$853$			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
$855$	2.00000			0.0683986
$856$			13.0000i			0.444331i
$857$		−	13.0000i		−	0.444072i	−0.975039	−	0.222036i	$-0.928730\pi$
$857$		−	13.0000i		−	0.444072i	0.975039	−	0.222036i	$-0.0712702\pi$
$858$	3.00000			0.102418
$859$			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
$861$	6.00000			0.204479
$862$	15.0000			0.510902
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$		−	1.00000i		−	0.0340207i
$865$		−	18.0000i		−	0.612018i
$866$			1.00000i			0.0339814i
$867$	8.00000			0.271694
$868$		−	30.0000i		−	1.01827i
$869$			42.0000i			1.42475i
$870$		−	12.0000i		−	0.406838i
$871$		−	8.00000i		−	0.271070i
$872$	19.0000			0.643421
$873$		−	8.00000i		−	0.270759i
$874$		−	1.00000i		−	0.0338255i
$875$		−	36.0000i		−	1.21702i
$876$	1.00000			0.0337869
$877$	−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
$879$	9.00000			0.303562
$880$			6.00000i			0.202260i
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		−	2.00000i		−	0.0673435i
$883$		−	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
$885$	12.0000			0.403376
$886$		−	4.00000i		−	0.134383i
$887$	28.0000			0.940148			0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000			0.940148			0.470074	+	0.882627i	$0.344227\pi$
$888$	−1.00000	−	6.00000i	−0.0335578	−	0.201347i
$889$	−21.0000			−0.704317
$890$			18.0000i			0.603361i
$891$	−3.00000			−0.100504
$892$	−4.00000			−0.133930
$893$			8.00000i			0.267710i
$894$			10.0000i			0.334450i
$895$	−48.0000			−1.60446
$896$			3.00000i			0.100223i
$897$	−1.00000			−0.0333890
$898$	−34.0000			−1.13459
$899$	−60.0000			−2.00111
$900$	−1.00000			−0.0333333
$901$			33.0000i			1.09939i
$902$			6.00000i			0.199778i
$903$			12.0000i			0.399335i
$904$	−14.0000			−0.465633
$905$		−	4.00000i		−	0.132964i
$906$			13.0000i			0.431896i
$907$			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
$909$	−18.0000			−0.597022
$910$			6.00000i			0.198898i
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$			1.00000i			0.0331133i
$913$	3.00000			0.0992855
$914$	2.00000			0.0661541
$915$	20.0000			0.661180
$916$	20.0000			0.660819
$917$			30.0000i			0.990687i
$918$	−3.00000			−0.0990148
$919$			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
$921$	−12.0000			−0.395413
$922$	40.0000			1.31733
$923$		−	2.00000i		−	0.0658308i
$924$	9.00000			0.296078
$925$	−6.00000	+	1.00000i	−0.197279	+	0.0328798i
$926$	34.0000			1.11731
$927$		−	16.0000i		−	0.525509i
$928$	6.00000			0.196960
$929$	−60.0000			−1.96854			−0.984268	−	0.176682i	$-0.943464\pi$
$929$	−60.0000			−1.96854			−0.984268	+	0.176682i	$0.943464\pi$
$930$		−	20.0000i		−	0.655826i
$931$			2.00000i			0.0655474i
$932$	−24.0000			−0.786146
$933$		−	20.0000i		−	0.654771i
$934$	−8.00000			−0.261768
$935$	18.0000			0.588663
$936$	1.00000			0.0326860
$937$	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
$939$		−	6.00000i		−	0.195803i
$940$			16.0000i			0.521862i
$941$	−38.0000			−1.23876			−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000			−1.23876			−0.619382	+	0.785090i	$0.712617\pi$
$942$		−	8.00000i		−	0.260654i
$943$		−	2.00000i		−	0.0651290i
$944$			6.00000i			0.195283i
$945$		−	6.00000i		−	0.195180i
$946$	−12.0000			−0.390154
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$			14.0000i			0.454699i
$949$			1.00000i			0.0324614i
$950$	1.00000			0.0324443
$951$	−2.00000			−0.0648544
$952$	9.00000			0.291692
$953$	44.0000			1.42530			0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000			1.42530			0.712650	+	0.701520i	$0.247495\pi$
$954$			11.0000i			0.356138i
$955$	30.0000			0.970777
$956$		−	16.0000i		−	0.517477i
$957$		−	18.0000i		−	0.581857i
$958$	21.0000			0.678479
$959$	−36.0000			−1.16250
$960$			2.00000i			0.0645497i
$961$	−69.0000			−2.22581
$962$	6.00000	−	1.00000i	0.193448	−	0.0322413i
$963$	13.0000			0.418919
$964$		0			0
$965$	−32.0000			−1.03012
$966$	−3.00000			−0.0965234
$967$			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
$969$	3.00000			0.0963739
$970$			16.0000i			0.513729i
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$	−1.00000			−0.0320750
$973$		0			0
$974$	−8.00000			−0.256337
$975$		−	1.00000i		−	0.0320256i
$976$			10.0000i			0.320092i
$977$		−	53.0000i		−	1.69562i	−0.530300	−	0.847810i	$-0.677921\pi$
$977$		−	53.0000i		−	1.69562i	0.530300	−	0.847810i	$-0.322079\pi$
$978$	9.00000			0.287788
$979$			27.0000i			0.862924i
$980$			4.00000i			0.127775i
$981$		−	19.0000i		−	0.606623i
$982$		−	27.0000i		−	0.861605i
$983$	−16.0000			−0.510321			−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000			−0.510321			−0.255160	+	0.966899i	$0.582128\pi$
$984$			2.00000i			0.0637577i
$985$			34.0000i			1.08333i
$986$		−	18.0000i		−	0.573237i
$987$	24.0000			0.763928
$988$	−1.00000			−0.0318142
$989$	4.00000			0.127193
$990$	6.00000			0.190693
$991$			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
$993$		−	20.0000i		−	0.634681i
$994$		−	6.00000i		−	0.190308i
$995$	−28.0000			−0.887660
$996$	1.00000			0.0316862
$997$			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
$999$	−6.00000	+	1.00000i	−0.189832	+	0.0316386i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.c.a.73.1	✓	2	1.1	even	1	trivial
222.2.c.a.73.2	yes	2	37.36	even	2	inner
666.2.c.a.73.1		2	111.110	odd	2
666.2.c.a.73.2		2	3.2	odd	2
1776.2.h.a.961.1		2	4.3	odd	2
1776.2.h.a.961.2		2	148.147	odd	2
5328.2.h.b.2737.1		2	444.443	even	2
5328.2.h.b.2737.2		2	12.11	even	2
8214.2.a.c.1.1		1	37.6	odd	4
8214.2.a.g.1.1		1	37.31	odd	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:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	222.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{6} +3.00000 q^{7} +1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -1.00000i q^{13} -3.00000i q^{14} -2.00000i q^{15} +1.00000 q^{16} -3.00000i q^{17} -1.00000i q^{18} +1.00000i q^{19} +2.00000i q^{20} +3.00000 q^{21} +3.00000i q^{22} -1.00000i q^{23} +1.00000i q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} +6.00000i q^{29} -2.00000 q^{30} +10.0000i q^{31} -1.00000i q^{32} -3.00000 q^{33} -3.00000 q^{34} -6.00000i q^{35} -1.00000 q^{36} +(-6.00000 + 1.00000i) q^{37} +1.00000 q^{38} -1.00000i q^{39} +2.00000 q^{40} +2.00000 q^{41} -3.00000i q^{42} +4.00000i q^{43} +3.00000 q^{44} -2.00000i q^{45} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000i q^{50} -3.00000i q^{51} +1.00000i q^{52} -11.0000 q^{53} -1.00000i q^{54} +6.00000i q^{55} +3.00000i q^{56} +1.00000i q^{57} +6.00000 q^{58} +6.00000i q^{59} +2.00000i q^{60} +10.0000i q^{61} +10.0000 q^{62} +3.00000 q^{63} -1.00000 q^{64} -2.00000 q^{65} +3.00000i q^{66} +8.00000 q^{67} +3.00000i q^{68} -1.00000i q^{69} -6.00000 q^{70} +2.00000 q^{71} +1.00000i q^{72} -1.00000 q^{73} +(1.00000 + 6.00000i) q^{74} +1.00000 q^{75} -1.00000i q^{76} -9.00000 q^{77} -1.00000 q^{78} -14.0000i q^{79} -2.00000i q^{80} +1.00000 q^{81} -2.00000i q^{82} -1.00000 q^{83} -3.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} +6.00000i q^{87} -3.00000i q^{88} -9.00000i q^{89} -2.00000 q^{90} -3.00000i q^{91} +1.00000i q^{92} +10.0000i q^{93} -8.00000i q^{94} +2.00000 q^{95} -1.00000i q^{96} -8.00000i q^{97} -2.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} + 6 q^{7} + 2 q^{9} - 4 q^{10} - 6 q^{11} - 2 q^{12} + 2 q^{16} + 6 q^{21} + 2 q^{25} - 2 q^{26} + 2 q^{27} - 6 q^{28} - 4 q^{30} - 6 q^{33} - 6 q^{34} - 2 q^{36} - 12 q^{37} + 2 q^{38}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 + 6 * q^7 + 2 * q^9 - 4 * q^10 - 6 * q^11 - 2 * q^12 + 2 * q^16 + 6 * q^21 + 2 * q^25 - 2 * q^26 + 2 * q^27 - 6 * q^28 - 4 * q^30 - 6 * q^33 - 6 * q^34 - 2 * q^36 - 12 * q^37 + 2 * q^38 + 4 * q^40 + 4 * q^41 + 6 * q^44 - 2 * q^46 + 16 * q^47 + 2 * q^48 + 4 * q^49 - 22 * q^53 + 12 * q^58 + 20 * q^62 + 6 * q^63 - 2 * q^64 - 4 * q^65 + 16 * q^67 - 12 * q^70 + 4 * q^71 - 2 * q^73 + 2 * q^74 + 2 * q^75 - 18 * q^77 - 2 * q^78 + 2 * q^81 - 2 * q^83 - 6 * q^84 - 12 * q^85 + 8 * q^86 - 4 * q^90 + 4 * q^95 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more