LMFDB - Embedded newform 552.2.m.b.137.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [552,2,Mod(137,552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(552, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("552.137");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	552.m (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.40774219157$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2x^{14} - 2x^{12} + 8x^{10} - 8x^{8} + 32x^{6} - 32x^{4} - 128x^{2} + 256 $ x^16 - 2x^14 - 2x^12 + 8x^10 - 8x^8 + 32x^6 - 32x^4 - 128*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			137.16
Root			$1.39495 + 0.232632i$ of defining polynomial
Character	$\chi$	$=$	552.137
Dual form			552.2.m.b.137.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.72779 + 0.121372i) q^{3} +2.32463 q^{5} +3.25516i q^{7} +(2.97054 + 0.419412i) q^{9} +O(q^{10})$	\(q+(1.72779 + 0.121372i) q^{3} +2.32463 q^{5} +3.25516i q^{7} +(2.97054 + 0.419412i) q^{9} -2.65123 q^{11} +1.62598 q^{13} +(4.01649 + 0.282146i) q^{15} -2.32463 q^{17} +2.69087i q^{19} +(-0.395086 + 5.62425i) q^{21} +(-2.65123 - 3.99637i) q^{23} +0.403920 q^{25} +(5.08157 + 1.08520i) q^{27} -4.83882i q^{29} +0.544414 q^{31} +(-4.58078 - 0.321786i) q^{33} +7.56706i q^{35} -1.29677i q^{37} +(2.80936 + 0.197349i) q^{39} -5.32431i q^{41} -12.6426i q^{43} +(6.90541 + 0.974980i) q^{45} +3.32431i q^{47} -3.59608 q^{49} +(-4.01649 - 0.282146i) q^{51} +12.2187 q^{53} -6.16314 q^{55} +(-0.326597 + 4.64927i) q^{57} +13.3967i q^{59} -8.82314i q^{61} +(-1.36525 + 9.66958i) q^{63} +3.77981 q^{65} +8.63690i q^{67} +(-4.09573 - 7.22669i) q^{69} +1.43490i q^{71} -8.85225 q^{73} +(0.697890 + 0.0490247i) q^{75} -8.63018i q^{77} -9.85439i q^{79} +(8.64819 + 2.49176i) q^{81} -15.6719 q^{83} -5.40392 q^{85} +(0.587299 - 8.36049i) q^{87} +9.14974 q^{89} +5.29284i q^{91} +(0.940635 + 0.0660767i) q^{93} +6.25529i q^{95} +10.1199i q^{97} +(-7.87558 - 1.11196i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 16 * q + 4 * q^3 + 4 * q^9	$ 16 q + 4 q^{3} + 4 q^{9} + 8 q^{13} + 32 q^{25} + 16 q^{27} + 56 q^{31} - 44 q^{39} - 32 q^{49} + 32 q^{55} + 4 q^{69} + 40 q^{73} - 20 q^{75} + 4 q^{81} - 112 q^{85} - 36 q^{87} - 36 q^{93}+O(q^{100}) $ 16 * q + 4 * q^3 + 4 * q^9 + 8 * q^13 + 32 * q^25 + 16 * q^27 + 56 * q^31 - 44 * q^39 - 32 * q^49 + 32 * q^55 + 4 * q^69 + 40 * q^73 - 20 * q^75 + 4 * q^81 - 112 * q^85 - 36 * q^87 - 36 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$.

$n$	$97$	$185$	$277$	$415$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.72779	+	0.121372i	0.997542	+	0.0700743i
$3$	1.72779	+	0.121372i	0.997542	+	0.0700743i
$4$		0			0
$5$	2.32463			1.03961			0.519804	−	0.854286i	$-0.326005\pi$
$5$	2.32463			1.03961			0.519804	+	0.854286i	$0.326005\pi$
$6$		0			0
$7$			3.25516i			1.23034i	0.788396	+	0.615168i	$0.210912\pi$
$7$			3.25516i			1.23034i	−0.788396	+	0.615168i	$0.789088\pi$
$8$		0			0
$9$	2.97054	+	0.419412i	0.990179	+	0.139804i
$10$		0			0
$11$	−2.65123			−0.799376			−0.399688	−	0.916651i	$-0.630882\pi$
$11$	−2.65123			−0.799376			−0.399688	+	0.916651i	$0.630882\pi$
$12$		0			0
$13$	1.62598			0.450967			0.225483	−	0.974247i	$-0.427604\pi$
$13$	1.62598			0.450967			0.225483	+	0.974247i	$0.427604\pi$
$14$		0			0
$15$	4.01649	+	0.282146i	1.03705	+	0.0728498i
$16$		0			0
$17$	−2.32463			−0.563806			−0.281903	−	0.959443i	$-0.590966\pi$
$17$	−2.32463			−0.563806			−0.281903	+	0.959443i	$0.590966\pi$
$18$		0			0
$19$			2.69087i			0.617328i	0.951171	+	0.308664i	$0.0998819\pi$
$19$			2.69087i			0.617328i	−0.951171	+	0.308664i	$0.900118\pi$
$20$		0			0
$21$	−0.395086	+	5.62425i	−0.0862149	+	1.22731i
$22$		0			0
$23$	−2.65123	−	3.99637i	−0.552820	−	0.833301i
$23$	−2.65123	−	3.99637i	−0.552820	−	0.833301i
$24$		0			0
$25$	0.403920			0.0807840
$26$		0			0
$27$	5.08157	+	1.08520i	0.977948	+	0.208847i
$28$		0			0
$29$		−	4.83882i		−	0.898547i	−0.893394	−	0.449274i	$-0.851683\pi$
$29$		−	4.83882i		−	0.898547i	0.893394	−	0.449274i	$-0.148317\pi$
$30$		0			0
$31$	0.544414			0.0977796			0.0488898	−	0.998804i	$-0.484432\pi$
$31$	0.544414			0.0977796			0.0488898	+	0.998804i	$0.484432\pi$
$32$		0			0
$33$	−4.58078	−	0.321786i	−0.797411	−	0.0560157i
$34$		0			0
$35$			7.56706i			1.27907i
$36$		0			0
$37$		−	1.29677i		−	0.213187i	−0.994303	−	0.106593i	$-0.966006\pi$
$37$		−	1.29677i		−	0.213187i	0.994303	−	0.106593i	$-0.0339943\pi$
$38$		0			0
$39$	2.80936	+	0.197349i	0.449858	+	0.0316012i
$40$		0			0
$41$		−	5.32431i		−	0.831518i	−0.909475	−	0.415759i	$-0.863516\pi$
$41$		−	5.32431i		−	0.831518i	0.909475	−	0.415759i	$-0.136484\pi$
$42$		0			0
$43$		−	12.6426i		−	1.92798i	−0.265942	−	0.963989i	$-0.585683\pi$
$43$		−	12.6426i		−	1.92798i	0.265942	−	0.963989i	$-0.414317\pi$
$44$		0			0
$45$	6.90541	+	0.974980i	1.02940	+	0.145341i
$46$		0			0
$47$			3.32431i			0.484901i	0.970164	+	0.242450i	$0.0779512\pi$
$47$			3.32431i			0.484901i	−0.970164	+	0.242450i	$0.922049\pi$
$48$		0			0
$49$	−3.59608			−0.513726
$50$		0			0
$51$	−4.01649	−	0.282146i	−0.562420	−	0.0395083i
$52$		0			0
$53$	12.2187			1.67836			0.839181	−	0.543852i	$-0.183035\pi$
$53$	12.2187			1.67836			0.839181	+	0.543852i	$0.183035\pi$
$54$		0			0
$55$	−6.16314			−0.831037
$56$		0			0
$57$	−0.326597	+	4.64927i	−0.0432588	+	0.615810i
$58$		0			0
$59$			13.3967i			1.74410i	0.489420	+	0.872048i	$0.337209\pi$
$59$			13.3967i			1.74410i	−0.489420	+	0.872048i	$0.662791\pi$
$60$		0			0
$61$		−	8.82314i		−	1.12969i	−0.825198	−	0.564844i	$-0.808936\pi$
$61$		−	8.82314i		−	1.12969i	0.825198	−	0.564844i	$-0.191064\pi$
$62$		0			0
$63$	−1.36525	+	9.66958i	−0.172006	+	1.21825i
$64$		0			0
$65$	3.77981			0.468828
$66$		0			0
$67$			8.63690i			1.05517i	0.849504	+	0.527583i	$0.176902\pi$
$67$			8.63690i			1.05517i	−0.849504	+	0.527583i	$0.823098\pi$
$68$		0			0
$69$	−4.09573	−	7.22669i	−0.493068	−	0.869991i
$70$		0			0
$71$			1.43490i			0.170292i	0.996368	+	0.0851459i	$0.0271356\pi$
$71$			1.43490i			0.170292i	−0.996368	+	0.0851459i	$0.972864\pi$
$72$		0			0
$73$	−8.85225			−1.03608			−0.518039	−	0.855357i	$-0.673338\pi$
$73$	−8.85225			−1.03608			−0.518039	+	0.855357i	$0.673338\pi$
$74$		0			0
$75$	0.697890	+	0.0490247i	0.0805854	+	0.00566088i
$76$		0			0
$77$		−	8.63018i		−	0.983501i
$78$		0			0
$79$		−	9.85439i		−	1.10871i	−0.832282	−	0.554353i	$-0.812966\pi$
$79$		−	9.85439i		−	1.10871i	0.832282	−	0.554353i	$-0.187034\pi$
$80$		0			0
$81$	8.64819	+	2.49176i	0.960910	+	0.276862i
$82$		0			0
$83$	−15.6719			−1.72021			−0.860106	−	0.510115i	$-0.829603\pi$
$83$	−15.6719			−1.72021			−0.860106	+	0.510115i	$0.829603\pi$
$84$		0			0
$85$	−5.40392			−0.586137
$86$		0			0
$87$	0.587299	−	8.36049i	0.0629651	−	0.896338i
$88$		0			0
$89$	9.14974			0.969871			0.484935	−	0.874550i	$-0.338843\pi$
$89$	9.14974			0.969871			0.484935	+	0.874550i	$0.338843\pi$
$90$		0			0
$91$			5.29284i			0.554840i
$92$		0			0
$93$	0.940635	+	0.0660767i	0.0975393	+	0.00685184i
$94$		0			0
$95$			6.25529i			0.641779i
$96$		0			0
$97$			10.1199i			1.02752i	0.857934	+	0.513761i	$0.171748\pi$
$97$			10.1199i			1.02752i	−0.857934	+	0.513761i	$0.828252\pi$
$98$		0			0
$99$	−7.87558	−	1.11196i	−0.791525	−	0.111756i
$100$		0			0
$101$		−	10.1631i		−	1.01127i	−0.862747	−	0.505635i	$-0.831258\pi$
$101$		−	10.1631i		−	1.01127i	0.862747	−	0.505635i	$-0.168742\pi$
$102$		0			0
$103$		−	2.60197i		−	0.256380i	−0.991750	−	0.128190i	$-0.959083\pi$
$103$		−	2.60197i		−	0.256380i	0.991750	−	0.128190i	$-0.0409167\pi$
$104$		0			0
$105$	−0.918431	+	13.0743i	−0.0896297	+	1.27592i
$106$		0			0
$107$	10.6842			1.03288			0.516440	−	0.856323i	$-0.327257\pi$
$107$	10.6842			1.03288			0.516440	+	0.856323i	$0.327257\pi$
$108$		0			0
$109$		−	11.0803i		−	1.06130i	−0.847591	−	0.530651i	$-0.821948\pi$
$109$		−	11.0803i		−	1.06130i	0.847591	−	0.530651i	$-0.178052\pi$
$110$		0			0
$111$	0.157391	−	2.24054i	0.0149389	−	0.212663i
$112$		0			0
$113$	−3.05711			−0.287588			−0.143794	−	0.989608i	$-0.545930\pi$
$113$	−3.05711			−0.287588			−0.143794	+	0.989608i	$0.545930\pi$
$114$		0			0
$115$	−6.16314	−	9.29010i	−0.574716	−	0.866306i
$116$		0			0
$117$	4.83004	+	0.681957i	0.446538	+	0.0630470i
$118$		0			0
$119$		−	7.56706i		−	0.693671i
$120$		0			0
$121$	−3.97098			−0.360998
$122$		0			0
$123$	0.646224	−	9.19931i	0.0582681	−	0.829474i
$124$		0			0
$125$	−10.6842			−0.955624
$126$		0			0
$127$	−9.17460			−0.814114			−0.407057	−	0.913403i	$-0.633445\pi$
$127$	−9.17460			−0.814114			−0.407057	+	0.913403i	$0.633445\pi$
$128$		0			0
$129$	1.53446	−	21.8438i	0.135102	−	1.92324i
$130$		0			0
$131$			15.0361i			1.31371i	0.754018	+	0.656854i	$0.228113\pi$
$131$			15.0361i			1.31371i	−0.754018	+	0.656854i	$0.771887\pi$
$132$		0			0
$133$	−8.75922			−0.759521
$134$		0			0
$135$	11.8128	+	2.52269i	1.01668	+	0.217118i
$136$		0			0
$137$	−6.18373			−0.528311			−0.264156	−	0.964480i	$-0.585093\pi$
$137$	−6.18373			−0.528311			−0.264156	+	0.964480i	$0.585093\pi$
$138$		0			0
$139$	−7.23440			−0.613614			−0.306807	−	0.951772i	$-0.599261\pi$
$139$	−7.23440			−0.613614			−0.306807	+	0.951772i	$0.599261\pi$
$140$		0			0
$141$	−0.403479	+	5.74373i	−0.0339791	+	0.483709i
$142$		0			0
$143$	−4.31086			−0.360492
$144$		0			0
$145$		−	11.2485i		−	0.934136i
$146$		0			0
$147$	−6.21328	−	0.436464i	−0.512463	−	0.0359990i
$148$		0			0
$149$	−11.5655			−0.947480			−0.473740	−	0.880665i	$-0.657096\pi$
$149$	−11.5655			−0.947480			−0.473740	+	0.880665i	$0.657096\pi$
$150$		0			0
$151$	0.426564			0.0347133			0.0173566	−	0.999849i	$-0.494475\pi$
$151$	0.426564			0.0347133			0.0173566	+	0.999849i	$0.494475\pi$
$152$		0			0
$153$	−6.90541	−	0.974980i	−0.558269	−	0.0788224i
$154$		0			0
$155$	1.26556			0.101652
$156$		0			0
$157$			9.95173i			0.794234i	0.917768	+	0.397117i	$0.129989\pi$
$157$			9.95173i			0.794234i	−0.917768	+	0.397117i	$0.870011\pi$
$158$		0			0
$159$	21.1113	+	1.48301i	1.67424	+	0.117610i
$160$		0			0
$161$	13.0088	−	8.63018i	1.02524	−	0.680154i
$162$		0			0
$163$	−15.5589			−1.21867			−0.609334	−	0.792913i	$-0.708563\pi$
$163$	−15.5589			−1.21867			−0.609334	+	0.792913i	$0.708563\pi$
$164$		0			0
$165$	−10.6486	−	0.748034i	−0.828995	−	0.0582344i
$166$		0			0
$167$		−	15.9927i		−	1.23756i	−0.785566	−	0.618778i	$-0.787628\pi$
$167$		−	15.9927i		−	1.23756i	0.785566	−	0.618778i	$-0.212372\pi$
$168$		0			0
$169$	−10.3562			−0.796629
$170$		0			0
$171$	−1.12858	+	7.99333i	−0.0863050	+	0.611265i
$172$		0			0
$173$			12.9710i			0.986165i	0.869982	+	0.493083i	$0.164130\pi$
$173$			12.9710i			0.986165i	−0.869982	+	0.493083i	$0.835870\pi$
$174$		0			0
$175$			1.31482i			0.0993914i
$176$		0			0
$177$	−1.62598	+	23.1467i	−0.122216	+	1.73981i
$178$		0			0
$179$			6.35334i			0.474871i	0.971403	+	0.237435i	$0.0763068\pi$
$179$			6.35334i			0.474871i	−0.971403	+	0.237435i	$0.923693\pi$
$180$		0			0
$181$			2.42535i			0.180275i	0.995929	+	0.0901374i	$0.0287306\pi$
$181$			2.42535i			0.180275i	−0.995929	+	0.0901374i	$0.971269\pi$
$182$		0			0
$183$	1.07088	−	15.2446i	0.0791621	−	1.12691i
$184$		0			0
$185$		−	3.01450i		−	0.221631i
$186$		0			0
$187$	6.16314			0.450693
$188$		0			0
$189$	−3.53250	+	16.5413i	−0.256951	+	1.20320i
$190$		0			0
$191$	1.26556			0.0915729			0.0457865	−	0.998951i	$-0.485421\pi$
$191$	1.26556			0.0915729			0.0457865	+	0.998951i	$0.485421\pi$
$192$		0			0
$193$	12.7601			0.918492			0.459246	−	0.888309i	$-0.348120\pi$
$193$	12.7601			0.918492			0.459246	+	0.888309i	$0.348120\pi$
$194$		0			0
$195$	6.53074	+	0.458765i	0.467676	+	0.0328528i
$196$		0			0
$197$			19.2914i			1.37445i	0.726444	+	0.687226i	$0.241172\pi$
$197$			19.2914i			1.37445i	−0.726444	+	0.687226i	$0.758828\pi$
$198$		0			0
$199$		−	19.0640i		−	1.35141i	−0.737171	−	0.675706i	$-0.763839\pi$
$199$		−	19.0640i		−	1.35141i	0.737171	−	0.675706i	$-0.236161\pi$
$200$		0			0
$201$	−1.04828	+	14.9228i	−0.0739400	+	1.05257i
$202$		0			0
$203$	15.7512			1.10551
$204$		0			0
$205$		−	12.3771i		−	0.864453i
$206$		0			0
$207$	−6.19945	−	12.9833i	−0.430892	−	0.902404i
$208$		0			0
$209$		−	7.13412i		−	0.493477i
$210$		0			0
$211$	26.4525			1.82107			0.910534	−	0.413435i	$-0.135671\pi$
$211$	26.4525			1.82107			0.910534	+	0.413435i	$0.135671\pi$
$212$		0			0
$213$	−0.174158	+	2.47922i	−0.0119331	+	0.169873i
$214$		0			0
$215$		−	29.3894i		−	2.00434i
$216$		0			0
$217$			1.77216i			0.120302i
$218$		0			0
$219$	−15.2949	−	1.07442i	−1.03353	−	0.0726024i
$220$		0			0
$221$	−3.77981			−0.254258
$222$		0			0
$223$	16.1051			1.07848			0.539238	−	0.842153i	$-0.318712\pi$
$223$	16.1051			1.07848			0.539238	+	0.842153i	$0.318712\pi$
$224$		0			0
$225$	1.19986	+	0.169409i	0.0799906	+	0.0112939i
$226$		0			0
$227$	−22.5547			−1.49701			−0.748503	−	0.663131i	$-0.769227\pi$
$227$	−22.5547			−1.49701			−0.748503	+	0.663131i	$0.769227\pi$
$228$		0			0
$229$			19.2503i			1.27209i	0.771651	+	0.636047i	$0.219431\pi$
$229$			19.2503i			1.27209i	−0.771651	+	0.636047i	$0.780569\pi$
$230$		0			0
$231$	1.04746	−	14.9112i	0.0689181	−	0.981083i
$232$		0			0
$233$			20.2768i			1.32838i	0.747564	+	0.664190i	$0.231224\pi$
$233$			20.2768i			1.32838i	−0.747564	+	0.664190i	$0.768776\pi$
$234$		0			0
$235$			7.72781i			0.504107i
$236$		0			0
$237$	1.19605	−	17.0263i	0.0776918	−	1.10598i
$238$		0			0
$239$		−	7.53607i		−	0.487468i	−0.969842	−	0.243734i	$-0.921628\pi$
$239$		−	7.53607i		−	0.487468i	0.969842	−	0.243734i	$-0.0783724\pi$
$240$		0			0
$241$			1.94996i			0.125608i	0.998026	+	0.0628040i	$0.0200043\pi$
$241$			1.94996i			0.125608i	−0.998026	+	0.0628040i	$0.979996\pi$
$242$		0			0
$243$	14.6398	+	5.35490i	0.939147	+	0.343517i
$244$		0			0
$245$	−8.35957			−0.534073
$246$		0			0
$247$			4.37531i			0.278394i
$248$		0			0
$249$	−27.0778	−	1.90213i	−1.71598	−	0.120543i
$250$		0			0
$251$	−24.9704			−1.57612			−0.788059	−	0.615599i	$-0.788914\pi$
$251$	−24.9704			−1.57612			−0.788059	+	0.615599i	$0.788914\pi$
$252$		0			0
$253$	7.02902	+	10.5953i	0.441911	+	0.666121i
$254$		0			0
$255$	−9.33686	−	0.655886i	−0.584697	−	0.0410732i
$256$		0			0
$257$		−	16.4980i		−	1.02912i	−0.857455	−	0.514559i	$-0.827955\pi$
$257$		−	16.4980i		−	1.02912i	0.857455	−	0.514559i	$-0.172045\pi$
$258$		0			0
$259$	4.22118			0.262291
$260$		0			0
$261$	2.02946	−	14.3739i	0.125621	−	0.889723i
$262$		0			0
$263$	29.0866			1.79356			0.896778	−	0.442481i	$-0.145902\pi$
$263$	29.0866			1.79356			0.896778	+	0.442481i	$0.145902\pi$
$264$		0			0
$265$	28.4039			1.74484
$266$		0			0
$267$	15.8089	+	1.11052i	0.967486	+	0.0679630i
$268$		0			0
$269$			27.4729i			1.67505i	0.546396	+	0.837527i	$0.315999\pi$
$269$			27.4729i			1.67505i	−0.546396	+	0.837527i	$0.684001\pi$
$270$		0			0
$271$	17.9420			1.08990			0.544948	−	0.838470i	$-0.316549\pi$
$271$	17.9420			1.08990			0.544948	+	0.838470i	$0.316549\pi$
$272$		0			0
$273$	−0.642404	+	9.14493i	−0.0388800	+	0.553476i
$274$		0			0
$275$	−1.07088			−0.0645768
$276$		0			0
$277$	21.2964			1.27958			0.639788	−	0.768552i	$-0.279022\pi$
$277$	21.2964			1.27958			0.639788	+	0.768552i	$0.279022\pi$
$278$		0			0
$279$	1.61720	+	0.228334i	0.0968194	+	0.0136700i
$280$		0			0
$281$	23.0398			1.37444			0.687221	−	0.726449i	$-0.258830\pi$
$281$	23.0398			1.37444			0.687221	+	0.726449i	$0.258830\pi$
$282$		0			0
$283$			16.6483i			0.989638i	0.868996	+	0.494819i	$0.164766\pi$
$283$			16.6483i			0.989638i	−0.868996	+	0.494819i	$0.835234\pi$
$284$		0			0
$285$	−0.759218	+	10.8078i	−0.0449722	+	0.640201i
$286$		0			0
$287$	17.3315			1.02305
$288$		0			0
$289$	−11.5961			−0.682122
$290$		0			0
$291$	−1.22828	+	17.4851i	−0.0720028	+	1.02500i
$292$		0			0
$293$	12.9511			0.756613			0.378307	−	0.925680i	$-0.376506\pi$
$293$	12.9511			0.756613			0.378307	+	0.925680i	$0.376506\pi$
$294$		0			0
$295$			31.1423i			1.81318i
$296$		0			0
$297$	−13.4724	−	2.87711i	−0.781749	−	0.166947i
$298$		0			0
$299$	−4.31086	−	6.49803i	−0.249303	−	0.375791i
$300$		0			0
$301$	41.1537			2.37206
$302$		0			0
$303$	1.23352	−	17.5598i	0.0708640	−	1.00878i
$304$		0			0
$305$		−	20.5106i		−	1.17443i
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$	0.315807	−	4.49566i	0.0179656	−	0.255749i
$310$		0			0
$311$			4.29529i			0.243564i	0.992557	+	0.121782i	$0.0388608\pi$
$311$			4.29529i			0.243564i	−0.992557	+	0.121782i	$0.961139\pi$
$312$		0			0
$313$			24.9127i			1.40815i	0.710126	+	0.704075i	$0.248638\pi$
$313$			24.9127i			1.40815i	−0.710126	+	0.704075i	$0.751362\pi$
$314$		0			0
$315$	−3.17372	+	22.4782i	−0.178819	+	1.26650i
$316$		0			0
$317$			23.3553i			1.31176i	0.754863	+	0.655882i	$0.227703\pi$
$317$			23.3553i			1.31176i	−0.754863	+	0.655882i	$0.772297\pi$
$318$		0			0
$319$			12.8288i			0.718277i
$320$		0			0
$321$	18.4601	+	1.29677i	1.03034	+	0.0723784i
$322$		0			0
$323$		−	6.25529i		−	0.348053i
$324$		0			0
$325$	0.656767			0.0364309
$326$		0			0
$327$	1.34484	−	19.1445i	0.0743700	−	1.05869i
$328$		0			0
$329$	−10.8212			−0.596591
$330$		0			0
$331$	2.48461			0.136566			0.0682832	−	0.997666i	$-0.478248\pi$
$331$	2.48461			0.136566			0.0682832	+	0.997666i	$0.478248\pi$
$332$		0			0
$333$	0.543879	−	3.85209i	0.0298044	−	0.211093i
$334$		0			0
$335$			20.0776i			1.09696i
$336$		0			0
$337$			10.6049i			0.577687i	0.957376	+	0.288843i	$0.0932707\pi$
$337$			10.6049i			0.577687i	−0.957376	+	0.288843i	$0.906729\pi$
$338$		0			0
$339$	−5.28205	−	0.371048i	−0.286881	−	0.0201526i
$340$		0			0
$341$	−1.44337			−0.0781627
$342$		0			0
$343$			11.0803i			0.598281i
$344$		0			0
$345$	−9.52107	−	16.7994i	−0.512597	−	0.904449i
$346$		0			0
$347$			11.2190i			0.602268i	0.953582	+	0.301134i	$0.0973651\pi$
$347$			11.2190i			0.602268i	−0.953582	+	0.301134i	$0.902635\pi$
$348$		0			0
$349$	−30.1383			−1.61327			−0.806634	−	0.591052i	$-0.798713\pi$
$349$	−30.1383			−1.61327			−0.806634	+	0.591052i	$0.798713\pi$
$350$		0			0
$351$	8.26254	+	1.76451i	0.441022	+	0.0941828i
$352$		0			0
$353$		−	34.9874i		−	1.86219i	−0.364772	−	0.931097i	$-0.618853\pi$
$353$		−	34.9874i		−	1.86219i	0.364772	−	0.931097i	$-0.381147\pi$
$354$		0			0
$355$			3.33563i			0.177037i
$356$		0			0
$357$	0.918431	−	13.0743i	0.0486085	−	0.691966i
$358$		0			0
$359$	23.6256			1.24691			0.623455	−	0.781859i	$-0.285728\pi$
$359$	23.6256			1.24691			0.623455	+	0.781859i	$0.285728\pi$
$360$		0			0
$361$	11.7592			0.618906
$362$		0			0
$363$	−6.86103	−	0.481967i	−0.360111	−	0.0252967i
$364$		0			0
$365$	−20.5782			−1.07711
$366$		0			0
$367$		−	5.19550i		−	0.271203i	−0.990763	−	0.135601i	$-0.956703\pi$
$367$		−	5.19550i		−	0.271203i	0.990763	−	0.135601i	$-0.0432967\pi$
$368$		0			0
$369$	2.23308	−	15.8161i	0.116250	−	0.823352i
$370$		0			0
$371$			39.7737i			2.06495i
$372$		0			0
$373$		−	31.7859i		−	1.64581i	−0.568178	−	0.822906i	$-0.692352\pi$
$373$		−	31.7859i		−	1.64581i	0.568178	−	0.822906i	$-0.307648\pi$
$374$		0			0
$375$	−18.4601	−	1.29677i	−0.953275	−	0.0669647i
$376$		0			0
$377$		−	7.86785i		−	0.405215i
$378$		0			0
$379$		−	23.2475i		−	1.19414i	−0.802187	−	0.597072i	$-0.796330\pi$
$379$		−	23.2475i		−	1.19414i	0.802187	−	0.597072i	$-0.203670\pi$
$380$		0			0
$381$	−15.8518	−	1.11354i	−0.812113	−	0.0570485i
$382$		0			0
$383$	−14.6010			−0.746076			−0.373038	−	0.927816i	$-0.621684\pi$
$383$	−14.6010			−0.746076			−0.373038	+	0.927816i	$0.621684\pi$
$384$		0			0
$385$		−	20.0620i		−	1.02245i
$386$		0			0
$387$	5.30246	−	37.5553i	0.269539	−	1.90904i
$388$		0			0
$389$	23.8300			1.20823			0.604115	−	0.796897i	$-0.293527\pi$
$389$	23.8300			1.20823			0.604115	+	0.796897i	$0.293527\pi$
$390$		0			0
$391$	6.16314	+	9.29010i	0.311683	+	0.469820i
$392$		0			0
$393$	−1.82496	+	25.9792i	−0.0920571	+	1.31048i
$394$		0			0
$395$		−	22.9078i		−	1.15262i
$396$		0			0
$397$	−5.82323			−0.292259			−0.146130	−	0.989265i	$-0.546682\pi$
$397$	−5.82323			−0.292259			−0.146130	+	0.989265i	$0.546682\pi$
$398$		0			0
$399$	−15.1341	−	1.06313i	−0.757653	−	0.0532229i
$400$		0			0
$401$	15.9985			0.798926			0.399463	−	0.916749i	$-0.369197\pi$
$401$	15.9985			0.798926			0.399463	+	0.916749i	$0.369197\pi$
$402$		0			0
$403$	0.885208			0.0440953
$404$		0			0
$405$	20.1039	+	5.79243i	0.998969	+	0.287828i
$406$		0			0
$407$			3.43802i			0.170417i
$408$		0			0
$409$	−4.21088			−0.208215			−0.104107	−	0.994566i	$-0.533199\pi$
$409$	−4.21088			−0.208215			−0.104107	+	0.994566i	$0.533199\pi$
$410$		0			0
$411$	−10.6842	−	0.750533i	−0.527013	−	0.0370211i
$412$		0			0
$413$	−43.6083			−2.14582
$414$		0			0
$415$	−36.4314			−1.78835
$416$		0			0
$417$	−12.4996	−	0.878056i	−0.612106	−	0.0429986i
$418$		0			0
$419$	−3.80139			−0.185710			−0.0928551	−	0.995680i	$-0.529599\pi$
$419$	−3.80139			−0.185710			−0.0928551	+	0.995680i	$0.529599\pi$
$420$		0			0
$421$			29.3702i			1.43141i	0.698400	+	0.715707i	$0.253895\pi$
$421$			29.3702i			1.43141i	−0.698400	+	0.715707i	$0.746105\pi$
$422$		0			0
$423$	−1.39426	+	9.87500i	−0.0677911	+	0.480139i
$424$		0			0
$425$	−0.938966			−0.0455465
$426$		0			0
$427$	28.7208			1.38990
$428$		0			0
$429$	−7.44827	−	0.523218i	−0.359606	−	0.0252612i
$430$		0			0
$431$	−22.3192			−1.07508			−0.537539	−	0.843239i	$-0.680646\pi$
$431$	−22.3192			−1.07508			−0.537539	+	0.843239i	$0.680646\pi$
$432$		0			0
$433$		−	13.8420i		−	0.665205i	−0.943067	−	0.332603i	$-0.892073\pi$
$433$		−	13.8420i		−	0.665205i	0.943067	−	0.332603i	$-0.107927\pi$
$434$		0			0
$435$	1.36525	−	19.4351i	0.0654590	−	0.931840i
$436$		0			0
$437$	10.7537	−	7.13412i	0.514420	−	0.341271i
$438$		0			0
$439$	−35.4411			−1.69151			−0.845755	−	0.533572i	$-0.820850\pi$
$439$	−35.4411			−1.69151			−0.845755	+	0.533572i	$0.820850\pi$
$440$		0			0
$441$	−10.6823	−	1.50824i	−0.508681	−	0.0718210i
$442$		0			0
$443$		−	25.0176i		−	1.18862i	−0.804235	−	0.594312i	$-0.797425\pi$
$443$		−	25.0176i		−	1.18862i	0.804235	−	0.594312i	$-0.202575\pi$
$444$		0			0
$445$	21.2698			1.00828
$446$		0			0
$447$	−19.9827	−	1.40373i	−0.945151	−	0.0663940i
$448$		0			0
$449$		−	17.0761i		−	0.805870i	−0.915229	−	0.402935i	$-0.867990\pi$
$449$		−	17.0761i		−	0.805870i	0.915229	−	0.402935i	$-0.132010\pi$
$450$		0			0
$451$			14.1160i			0.664696i
$452$		0			0
$453$	0.737015	+	0.0517731i	0.0346280	+	0.00243251i
$454$		0			0
$455$			12.3039i			0.576816i
$456$		0			0
$457$		−	2.90074i		−	0.135691i	−0.997696	−	0.0678454i	$-0.978388\pi$
$457$		−	2.90074i		−	0.135691i	0.997696	−	0.0678454i	$-0.0216125\pi$
$458$		0			0
$459$	−11.8128	−	2.52269i	−0.551374	−	0.117749i
$460$		0			0
$461$		−	14.1757i		−	0.660227i	−0.943941	−	0.330114i	$-0.892913\pi$
$461$		−	14.1757i		−	0.660227i	0.943941	−	0.330114i	$-0.107087\pi$
$462$		0			0
$463$	20.6157			0.958091			0.479046	−	0.877790i	$-0.340983\pi$
$463$	20.6157			0.958091			0.479046	+	0.877790i	$0.340983\pi$
$464$		0			0
$465$	2.18663	+	0.153604i	0.101403	+	0.00712322i
$466$		0			0
$467$	7.63891			0.353487			0.176743	−	0.984257i	$-0.443444\pi$
$467$	7.63891			0.353487			0.176743	+	0.984257i	$0.443444\pi$
$468$		0			0
$469$	−28.1145			−1.29821
$470$		0			0
$471$	−1.20786	+	17.1945i	−0.0556554	+	0.792282i
$472$		0			0
$473$			33.5184i			1.54118i
$474$		0			0
$475$			1.08690i			0.0498702i
$476$		0			0
$477$	36.2960	+	5.12466i	1.66188	+	0.234642i
$478$		0			0
$479$	−4.98768			−0.227893			−0.113946	−	0.993487i	$-0.536349\pi$
$479$	−4.98768			−0.227893			−0.113946	+	0.993487i	$0.536349\pi$
$480$		0			0
$481$		−	2.10852i		−	0.0961402i
$482$		0			0
$483$	23.5240	−	13.3323i	1.07038	−	0.606639i
$484$		0			0
$485$			23.5251i			1.06822i
$486$		0			0
$487$	18.5879			0.842300			0.421150	−	0.906991i	$-0.361627\pi$
$487$	18.5879			0.842300			0.421150	+	0.906991i	$0.361627\pi$
$488$		0			0
$489$	−26.8826	−	1.88842i	−1.21567	−	0.0853974i
$490$		0			0
$491$		−	11.3494i		−	0.512191i	−0.966651	−	0.256096i	$-0.917564\pi$
$491$		−	11.3494i		−	0.512191i	0.966651	−	0.256096i	$-0.0824362\pi$
$492$		0			0
$493$			11.2485i			0.506607i
$494$		0			0
$495$	−18.3078	−	2.58490i	−0.822876	−	0.116182i
$496$		0			0
$497$	−4.67085			−0.209516
$498$		0			0
$499$	−25.6187			−1.14685			−0.573426	−	0.819257i	$-0.694386\pi$
$499$	−25.6187			−1.14685			−0.573426	+	0.819257i	$0.694386\pi$
$500$		0			0
$501$	1.94108	−	27.6321i	0.0867208	−	1.23451i
$502$		0			0
$503$	0.115398			0.00514536			0.00257268	−	0.999997i	$-0.499181\pi$
$503$	0.115398			0.00514536			0.00257268	+	0.999997i	$0.499181\pi$
$504$		0			0
$505$		−	23.6256i		−	1.05132i
$506$		0			0
$507$	−17.8933	−	1.25695i	−0.794671	−	0.0558232i
$508$		0			0
$509$			33.4545i			1.48284i	0.671039	+	0.741422i	$0.265848\pi$
$509$			33.4545i			1.48284i	−0.671039	+	0.741422i	$0.734152\pi$
$510$		0			0
$511$		−	28.8155i		−	1.27472i
$512$		0			0
$513$	−2.92013	+	13.6738i	−0.128927	+	0.603715i
$514$		0			0
$515$		−	6.04862i		−	0.266534i
$516$		0			0
$517$		−	8.81352i		−	0.387618i
$518$		0			0
$519$	−1.57432	+	22.4112i	−0.0691048	+	0.983741i
$520$		0			0
$521$	4.79806			0.210207			0.105103	−	0.994461i	$-0.466483\pi$
$521$	4.79806			0.210207			0.105103	+	0.994461i	$0.466483\pi$
$522$		0			0
$523$		−	12.7315i		−	0.556710i	−0.960478	−	0.278355i	$-0.910211\pi$
$523$		−	12.7315i		−	0.556710i	0.960478	−	0.278355i	$-0.0897891\pi$
$524$		0			0
$525$	−0.159583	+	2.27175i	−0.00696478	+	0.0991471i
$526$		0			0
$527$	−1.26556			−0.0551288
$528$		0			0
$529$	−8.94196	+	21.1906i	−0.388781	+	0.921330i
$530$		0			0
$531$	−5.61872	+	39.7953i	−0.243832	+	1.72697i
$532$		0			0
$533$		−	8.65724i		−	0.374987i
$534$		0			0
$535$	24.8368			1.07379
$536$		0			0
$537$	−0.771119	+	10.9772i	−0.0332762	+	0.473703i
$538$		0			0
$539$	9.53404			0.410660
$540$		0			0
$541$	−25.2042			−1.08361			−0.541807	−	0.840503i	$-0.682260\pi$
$541$	−25.2042			−1.08361			−0.541807	+	0.840503i	$0.682260\pi$
$542$		0			0
$543$	−0.294370	+	4.19050i	−0.0126326	+	0.179832i
$544$		0			0
$545$		−	25.7577i		−	1.10334i
$546$		0			0
$547$	−5.51539			−0.235821			−0.117911	−	0.993024i	$-0.537620\pi$
$547$	−5.51539			−0.235821			−0.117911	+	0.993024i	$0.537620\pi$
$548$		0			0
$549$	3.70053	−	26.2095i	0.157935	−	1.11859i
$550$		0			0
$551$	13.0206			0.554698
$552$		0			0
$553$	32.0776			1.36408
$554$		0			0
$555$	0.365877	−	5.20844i	0.0155306	−	0.221086i
$556$		0			0
$557$	−20.4871			−0.868068			−0.434034	−	0.900897i	$-0.642910\pi$
$557$	−20.4871			−0.868068			−0.434034	+	0.900897i	$0.642910\pi$
$558$		0			0
$559$		−	20.5566i		−	0.869454i
$560$		0			0
$561$	10.6486	+	0.748034i	0.449585	+	0.0315820i
$562$		0			0
$563$	9.37781			0.395228			0.197614	−	0.980280i	$-0.436681\pi$
$563$	9.37781			0.395228			0.197614	+	0.980280i	$0.436681\pi$
$564$		0			0
$565$	−7.10665			−0.298979
$566$		0			0
$567$	−8.11108	+	28.1512i	−0.340633	+	1.18224i
$568$		0			0
$569$	−3.21567			−0.134808			−0.0674039	−	0.997726i	$-0.521472\pi$
$569$	−3.21567			−0.134808			−0.0674039	+	0.997726i	$0.521472\pi$
$570$		0			0
$571$		−	20.0536i		−	0.839216i	−0.907705	−	0.419608i	$-0.862168\pi$
$571$		−	20.0536i		−	0.839216i	0.907705	−	0.419608i	$-0.137832\pi$
$572$		0			0
$573$	2.18663	+	0.153604i	0.0913478	+	0.00641691i
$574$		0			0
$575$	−1.07088	−	1.61421i	−0.0446590	−	0.0673174i
$576$		0			0
$577$	16.1981			0.674337			0.337168	−	0.941444i	$-0.390531\pi$
$577$	16.1981			0.674337			0.337168	+	0.941444i	$0.390531\pi$
$578$		0			0
$579$	22.0468	+	1.54872i	0.916234	+	0.0643627i
$580$		0			0
$581$		−	51.0145i		−	2.11644i
$582$		0			0
$583$	−32.3945			−1.34164
$584$		0			0
$585$	11.2281	+	1.58530i	0.464224	+	0.0655441i
$586$		0			0
$587$		−	3.11255i		−	0.128469i	−0.997935	−	0.0642344i	$-0.979539\pi$
$587$		−	3.11255i		−	0.128469i	0.997935	−	0.0642344i	$-0.0204605\pi$
$588$		0			0
$589$			1.46495i			0.0603621i
$590$		0			0
$591$	−2.34144	+	33.3315i	−0.0963138	+	1.37107i
$592$		0			0
$593$			28.9565i			1.18910i	0.804059	+	0.594550i	$0.202670\pi$
$593$			28.9565i			1.18910i	−0.804059	+	0.594550i	$0.797330\pi$
$594$		0			0
$595$		−	17.5906i		−	0.721146i
$596$		0			0
$597$	2.31384	−	32.9387i	0.0946993	−	1.34809i
$598$		0			0
$599$		−	17.2531i		−	0.704943i	−0.935823	−	0.352471i	$-0.885341\pi$
$599$		−	17.2531i		−	0.704943i	0.935823	−	0.352471i	$-0.114659\pi$
$600$		0			0
$601$	−12.4175			−0.506522			−0.253261	−	0.967398i	$-0.581503\pi$
$601$	−12.4175			−0.506522			−0.253261	+	0.967398i	$0.581503\pi$
$602$		0			0
$603$	−3.62242	+	25.6562i	−0.147516	+	1.04480i
$604$		0			0
$605$	−9.23107			−0.375296
$606$		0			0
$607$	32.8368			1.33281			0.666403	−	0.745591i	$-0.267833\pi$
$607$	32.8368			1.33281			0.666403	+	0.745591i	$0.267833\pi$
$608$		0			0
$609$	27.2147	+	1.91175i	1.10280	+	0.0774682i
$610$		0			0
$611$			5.40528i			0.218674i
$612$		0			0
$613$			26.3873i			1.06577i	0.846187	+	0.532886i	$0.178893\pi$
$613$			26.3873i			1.06577i	−0.846187	+	0.532886i	$0.821107\pi$
$614$		0			0
$615$	1.50223	−	21.3850i	0.0605759	−	0.862328i
$616$		0			0
$617$	1.43155			0.0576321			0.0288161	−	0.999585i	$-0.490826\pi$
$617$	1.43155			0.0576321			0.0288161	+	0.999585i	$0.490826\pi$
$618$		0			0
$619$		−	20.9904i		−	0.843673i	−0.906672	−	0.421837i	$-0.861386\pi$
$619$		−	20.9904i		−	0.843673i	0.906672	−	0.421837i	$-0.138614\pi$
$620$		0			0
$621$	−9.13555	−	23.1849i	−0.366597	−	0.930380i
$622$		0			0
$623$			29.7839i			1.19327i
$624$		0			0
$625$	−26.8564			−1.07426
$626$		0			0
$627$	0.865884	−	12.3263i	0.0345801	−	0.492264i
$628$		0			0
$629$			3.01450i			0.120196i
$630$		0			0
$631$		−	5.12584i		−	0.204057i	−0.994781	−	0.102028i	$-0.967467\pi$
$631$		−	5.12584i		−	0.204057i	0.994781	−	0.102028i	$-0.0325332\pi$
$632$		0			0
$633$	45.7045	+	3.21060i	1.81659	+	0.127610i
$634$		0			0
$635$	−21.3276			−0.846359
$636$		0			0
$637$	−5.84717			−0.231673
$638$		0			0
$639$	−0.601817	+	4.26244i	−0.0238075	+	0.168619i
$640$		0			0
$641$	37.2299			1.47049			0.735246	−	0.677800i	$-0.237066\pi$
$641$	37.2299			1.47049			0.735246	+	0.677800i	$0.237066\pi$
$642$		0			0
$643$			46.1310i			1.81923i	0.415454	+	0.909614i	$0.363623\pi$
$643$			46.1310i			1.81923i	−0.415454	+	0.909614i	$0.636377\pi$
$644$		0			0
$645$	3.56706	−	50.7788i	0.140453	−	1.99941i
$646$		0			0
$647$			20.5310i			0.807157i	0.914945	+	0.403578i	$0.132234\pi$
$647$			20.5310i			0.807157i	−0.914945	+	0.403578i	$0.867766\pi$
$648$		0			0
$649$		−	35.5176i		−	1.39419i
$650$		0			0
$651$	−0.215091	+	3.06192i	−0.00843006	+	0.120006i
$652$		0			0
$653$			19.2663i			0.753947i	0.926224	+	0.376974i	$0.123035\pi$
$653$			19.2663i			0.753947i	−0.926224	+	0.376974i	$0.876965\pi$
$654$		0			0
$655$			34.9533i			1.36574i
$656$		0			0
$657$	−26.2959	−	3.71274i	−1.02590	−	0.144848i
$658$		0			0
$659$	−17.4108			−0.678228			−0.339114	−	0.940745i	$-0.610127\pi$
$659$	−17.4108			−0.678228			−0.339114	+	0.940745i	$0.610127\pi$
$660$		0			0
$661$		−	2.89112i		−	0.112451i	−0.998418	−	0.0562257i	$-0.982093\pi$
$661$		−	2.89112i		−	0.112451i	0.998418	−	0.0562257i	$-0.0179066\pi$
$662$		0			0
$663$	−6.53074	−	0.458765i	−0.253633	−	0.0178169i
$664$		0			0
$665$	−20.3620			−0.789603
$666$		0			0
$667$	−19.3377	+	12.8288i	−0.748760	+	0.496735i
$668$		0			0
$669$	27.8263	+	1.95471i	1.07583	+	0.0755735i
$670$		0			0
$671$			23.3922i			0.903045i
$672$		0			0
$673$	−18.6858			−0.720284			−0.360142	−	0.932898i	$-0.617272\pi$
$673$	−18.6858			−0.720284			−0.360142	+	0.932898i	$0.617272\pi$
$674$		0			0
$675$	2.05255	+	0.438333i	0.0790026	+	0.0168715i
$676$		0			0
$677$	−12.9873			−0.499141			−0.249570	−	0.968357i	$-0.580289\pi$
$677$	−12.9873			−0.499141			−0.249570	+	0.968357i	$0.580289\pi$
$678$		0			0
$679$	−32.9419			−1.26420
$680$		0			0
$681$	−38.9698	−	2.73751i	−1.49333	−	0.104902i
$682$		0			0
$683$		−	47.5216i		−	1.81836i	−0.416401	−	0.909181i	$-0.636709\pi$
$683$		−	47.5216i		−	1.81836i	0.416401	−	0.909181i	$-0.363291\pi$
$684$		0			0
$685$	−14.3749			−0.549237
$686$		0			0
$687$	−2.33645	+	33.2605i	−0.0891410	+	1.26897i
$688$		0			0
$689$	19.8673			0.756885
$690$		0			0
$691$	−2.45253			−0.0932986			−0.0466493	−	0.998911i	$-0.514854\pi$
$691$	−2.45253			−0.0932986			−0.0466493	+	0.998911i	$0.514854\pi$
$692$		0			0
$693$	3.61961	−	25.6363i	0.137497	−	0.973842i
$694$		0			0
$695$	−16.8173			−0.637918
$696$		0			0
$697$			12.3771i			0.468815i
$698$		0			0
$699$	−2.46105	+	35.0342i	−0.0930853	+	1.32512i
$700$		0			0
$701$	4.46640			0.168694			0.0843469	−	0.996436i	$-0.473120\pi$
$701$	4.46640			0.168694			0.0843469	+	0.996436i	$0.473120\pi$
$702$		0			0
$703$	3.48943			0.131606
$704$		0			0
$705$	−0.937942	+	13.3521i	−0.0353249	+	0.502868i
$706$		0			0
$707$	33.0827			1.24420
$708$		0			0
$709$		−	34.3987i		−	1.29187i	−0.763393	−	0.645935i	$-0.776468\pi$
$709$		−	34.3987i		−	1.29187i	0.763393	−	0.645935i	$-0.223532\pi$
$710$		0			0
$711$	4.13305	−	29.2728i	0.155002	−	1.09782i
$712$		0			0
$713$	−1.44337	−	2.17568i	−0.0540545	−	0.0814798i
$714$		0			0
$715$	−10.0212			−0.374770
$716$		0			0
$717$	0.914670	−	13.0208i	0.0341590	−	0.486270i
$718$		0			0
$719$		−	53.3676i		−	1.99028i	−0.0984873	−	0.995138i	$-0.531400\pi$
$719$		−	53.3676i		−	1.99028i	0.0984873	−	0.995138i	$-0.468600\pi$
$720$		0			0
$721$	8.46983			0.315433
$722$		0			0
$723$	−0.236671	+	3.36913i	−0.00880189	+	0.125299i
$724$		0			0
$725$		−	1.95450i		−	0.0725882i
$726$		0			0
$727$		−	37.9610i		−	1.40790i	−0.710251	−	0.703949i	$-0.751419\pi$
$727$		−	37.9610i		−	1.40790i	0.710251	−	0.703949i	$-0.248581\pi$
$728$		0			0
$729$	24.6447	+	11.0290i	0.912766	+	0.408482i
$730$		0			0
$731$			29.3894i			1.08701i
$732$		0			0
$733$			45.9351i			1.69665i	0.529474	+	0.848326i	$0.322389\pi$
$733$			45.9351i			1.69665i	−0.529474	+	0.848326i	$0.677611\pi$
$734$		0			0
$735$	−14.4436	−	1.01462i	−0.532760	−	0.0374248i
$736$		0			0
$737$		−	22.8984i		−	0.843474i
$738$		0			0
$739$	−20.3438			−0.748360			−0.374180	−	0.927356i	$-0.622076\pi$
$739$	−20.3438			−0.748360			−0.374180	+	0.927356i	$0.622076\pi$
$740$		0			0
$741$	−0.531041	+	7.55963i	−0.0195083	+	0.277710i
$742$		0			0
$743$	12.0290			0.441303			0.220651	−	0.975353i	$-0.429182\pi$
$743$	12.0290			0.441303			0.220651	+	0.975353i	$0.429182\pi$
$744$		0			0
$745$	−26.8855			−0.985007
$746$		0			0
$747$	−46.5539	−	6.57298i	−1.70332	−	0.240493i
$748$		0			0
$749$			34.7788i			1.27079i
$750$		0			0
$751$		−	3.99726i		−	0.145862i	−0.997337	−	0.0729310i	$-0.976765\pi$
$751$		−	3.99726i		−	0.145862i	0.997337	−	0.0729310i	$-0.0232353\pi$
$752$		0			0
$753$	−43.1437	−	3.03072i	−1.57224	−	0.110445i
$754$		0			0
$755$	0.991605			0.0360882
$756$		0			0
$757$			10.7731i			0.391555i	0.980648	+	0.195778i	$0.0627230\pi$
$757$			10.7731i			0.391555i	−0.980648	+	0.195778i	$0.937277\pi$
$758$		0			0
$759$	10.8587	+	19.1596i	0.394147	+	0.695450i
$760$		0			0
$761$			28.7478i			1.04211i	0.853524	+	0.521054i	$0.174461\pi$
$761$			28.7478i			1.04211i	−0.853524	+	0.521054i	$0.825539\pi$
$762$		0			0
$763$	36.0682			1.30576
$764$		0			0
$765$	−16.0525	−	2.26647i	−0.580381	−	0.0819444i
$766$		0			0
$767$			21.7827i			0.786529i
$768$		0			0
$769$			0.662816i			0.0239018i	0.999929	+	0.0119509i	$0.00380417\pi$
$769$			0.662816i			0.0239018i	−0.999929	+	0.0119509i	$0.996196\pi$
$770$		0			0
$771$	2.00240	−	28.5052i	0.0721148	−	1.02659i
$772$		0			0
$773$	27.9290			1.00454			0.502268	−	0.864712i	$-0.332499\pi$
$773$	27.9290			1.00454			0.502268	+	0.864712i	$0.332499\pi$
$774$		0			0
$775$	0.219900			0.00789903
$776$		0			0
$777$	7.29333	+	0.512334i	0.261647	+	0.0183799i
$778$		0			0
$779$	14.3270			0.513319
$780$		0			0
$781$		−	3.80426i		−	0.136127i
$782$		0			0
$783$	5.25108	−	24.5888i	0.187658	−	0.878733i
$784$		0			0
$785$			23.1341i			0.825692i
$786$		0			0
$787$			41.6499i			1.48466i	0.670035	+	0.742329i	$0.266279\pi$
$787$			41.6499i			1.48466i	−0.670035	+	0.742329i	$0.733721\pi$
$788$		0			0
$789$	50.2556	+	3.53030i	1.78915	+	0.125682i
$790$		0			0
$791$		−	9.95138i		−	0.353830i
$792$		0			0
$793$		−	14.3463i		−	0.509451i
$794$		0			0
$795$	49.0761	+	3.44745i	1.74055	+	0.122268i
$796$		0			0
$797$	52.1841			1.84846			0.924228	−	0.381840i	$-0.124710\pi$
$797$	52.1841			1.84846			0.924228	+	0.381840i	$0.124710\pi$
$798$		0			0
$799$		−	7.72781i		−	0.273390i
$800$		0			0
$801$	27.1796	+	3.83751i	0.960346	+	0.135592i
$802$		0			0
$803$	23.4693			0.828215
$804$		0			0
$805$	30.2408	−	20.0620i	1.06585	−	0.707093i
$806$		0			0
$807$	−3.33445	+	47.4675i	−0.117378	+	1.67094i
$808$		0			0
$809$			25.7063i			0.903784i	0.892073	+	0.451892i	$0.149251\pi$
$809$			25.7063i			0.903784i	−0.892073	+	0.451892i	$0.850749\pi$
$810$		0			0
$811$	−19.9015			−0.698835			−0.349418	−	0.936967i	$-0.613621\pi$
$811$	−19.9015			−0.698835			−0.349418	+	0.936967i	$0.613621\pi$
$812$		0			0
$813$	31.0000	+	2.17766i	1.08722	+	0.0763737i
$814$		0			0
$815$	−36.1688			−1.26694
$816$		0			0
$817$	34.0196			1.19019
$818$		0			0
$819$	−2.21988	+	15.7226i	−0.0775689	+	0.549391i
$820$		0			0
$821$		−	31.7788i		−	1.10909i	−0.832154	−	0.554544i	$-0.812893\pi$
$821$		−	31.7788i		−	1.10909i	0.832154	−	0.554544i	$-0.187107\pi$
$822$		0			0
$823$	47.1020			1.64187			0.820937	−	0.571019i	$-0.193452\pi$
$823$	47.1020			1.64187			0.820937	+	0.571019i	$0.193452\pi$
$824$		0			0
$825$	−1.85027	−	0.129976i	−0.0644180	−	0.00452517i
$826$		0			0
$827$	32.1680			1.11859			0.559295	−	0.828968i	$-0.311072\pi$
$827$	32.1680			1.11859			0.559295	+	0.828968i	$0.311072\pi$
$828$		0			0
$829$	−19.3749			−0.672918			−0.336459	−	0.941698i	$-0.609229\pi$
$829$	−19.3749			−0.672918			−0.336459	+	0.941698i	$0.609229\pi$
$830$		0			0
$831$	36.7957	+	2.58479i	1.27643	+	0.0896653i
$832$		0			0
$833$	8.35957			0.289642
$834$		0			0
$835$		−	37.1773i		−	1.28657i
$836$		0			0
$837$	2.76648	+	0.590797i	0.0956234	+	0.0204209i
$838$		0			0
$839$	−35.5392			−1.22695			−0.613475	−	0.789714i	$-0.710229\pi$
$839$	−35.5392			−1.22695			−0.613475	+	0.789714i	$0.710229\pi$
$840$		0			0
$841$	5.58578			0.192613
$842$		0			0
$843$	39.8081	+	2.79640i	1.37106	+	0.0963130i
$844$		0			0
$845$	−24.0743			−0.828182
$846$		0			0
$847$		−	12.9262i		−	0.444149i
$848$		0			0
$849$	−2.02064	+	28.7648i	−0.0693482	+	0.987205i
$850$		0			0
$851$	−5.18236	+	3.43802i	−0.177649	+	0.117854i
$852$		0			0
$853$	45.0000			1.54077			0.770385	−	0.637579i	$-0.220064\pi$
$853$	45.0000			1.54077			0.770385	+	0.637579i	$0.220064\pi$
$854$		0			0
$855$	−2.62354	+	18.5816i	−0.0897233	+	0.635476i
$856$		0			0
$857$		−	29.9796i		−	1.02408i	−0.858961	−	0.512041i	$-0.828889\pi$
$857$		−	29.9796i		−	1.02408i	0.858961	−	0.512041i	$-0.171111\pi$
$858$		0			0
$859$	11.4556			0.390860			0.195430	−	0.980718i	$-0.437390\pi$
$859$	11.4556			0.390860			0.195430	+	0.980718i	$0.437390\pi$
$860$		0			0
$861$	29.9453	+	2.10356i	1.02053	+	0.0716893i
$862$		0			0
$863$		−	19.4690i		−	0.662733i	−0.943502	−	0.331366i	$-0.892490\pi$
$863$		−	19.4690i		−	0.662733i	0.943502	−	0.331366i	$-0.107510\pi$
$864$		0			0
$865$			30.1528i			1.02522i
$866$		0			0
$867$	−20.0356	−	1.40744i	−0.680446	−	0.0477993i
$868$		0			0
$869$			26.1263i			0.886272i
$870$		0			0
$871$			14.0435i			0.475844i
$872$		0			0
$873$	−4.24441	+	30.0616i	−0.143652	+	1.01743i
$874$		0			0
$875$		−	34.7788i		−	1.17574i
$876$		0			0
$877$	−18.2894			−0.617589			−0.308794	−	0.951129i	$-0.599926\pi$
$877$	−18.2894			−0.617589			−0.308794	+	0.951129i	$0.599926\pi$
$878$		0			0
$879$	22.3769	+	1.57191i	0.754753	+	0.0530192i
$880$		0			0
$881$	−38.1786			−1.28627			−0.643135	−	0.765753i	$-0.722367\pi$
$881$	−38.1786			−1.28627			−0.643135	+	0.765753i	$0.722367\pi$
$882$		0			0
$883$	22.2000			0.747090			0.373545	−	0.927612i	$-0.378142\pi$
$883$	22.2000			0.747090			0.373545	+	0.927612i	$0.378142\pi$
$884$		0			0
$885$	−3.77981	+	53.8075i	−0.127057	+	1.80872i
$886$		0			0
$887$			20.2126i			0.678672i	0.940665	+	0.339336i	$0.110202\pi$
$887$			20.2126i			0.678672i	−0.940665	+	0.339336i	$0.889798\pi$
$888$		0			0
$889$		−	29.8648i		−	1.00163i
$890$		0			0
$891$	−22.9283	−	6.60623i	−0.768128	−	0.221317i
$892$		0			0
$893$	−8.94530			−0.299343
$894$		0			0
$895$			14.7692i			0.493679i
$896$		0			0
$897$	−6.65959	−	11.7505i	−0.222357	−	0.392337i
$898$		0			0
$899$		−	2.63432i		−	0.0878596i
$900$		0			0
$901$	−28.4039			−0.946271
$902$		0			0
$903$	71.1051	+	4.99492i	2.36623	+	0.166220i
$904$		0			0
$905$			5.63805i			0.187415i
$906$		0			0
$907$			39.9042i			1.32500i	0.749062	+	0.662499i	$0.230504\pi$
$907$			39.9042i			1.32500i	−0.749062	+	0.662499i	$0.769496\pi$
$908$		0			0
$909$	4.26254	−	30.1900i	0.141380	−	1.00134i
$910$		0			0
$911$	−17.2138			−0.570318			−0.285159	−	0.958480i	$-0.592046\pi$
$911$	−17.2138			−0.570318			−0.285159	+	0.958480i	$0.592046\pi$
$912$		0			0
$913$	41.5498			1.37510
$914$		0			0
$915$	2.48941	−	35.4380i	0.0822975	−	1.17154i
$916$		0			0
$917$	−48.9448			−1.61630
$918$		0			0
$919$		−	26.1246i		−	0.861772i	−0.902406	−	0.430886i	$-0.858201\pi$
$919$		−	26.1246i		−	0.861772i	0.902406	−	0.430886i	$-0.141799\pi$
$920$		0			0
$921$	6.91117	+	0.485489i	0.227731	+	0.0159974i
$922$		0			0
$923$			2.33313i			0.0767959i
$924$		0			0
$925$		−	0.523789i		−	0.0172221i
$926$		0			0
$927$	1.09130	−	7.72924i	0.0358429	−	0.253862i
$928$		0			0
$929$		−	27.3824i		−	0.898386i	−0.893435	−	0.449193i	$-0.851712\pi$
$929$		−	27.3824i		−	0.898386i	0.893435	−	0.449193i	$-0.148288\pi$
$930$		0			0
$931$		−	9.67658i		−	0.317137i
$932$		0			0
$933$	−0.521329	+	7.42137i	−0.0170676	+	0.242965i
$934$		0			0
$935$	14.3270			0.468544
$936$		0			0
$937$			35.8248i			1.17035i	0.810908	+	0.585173i	$0.198974\pi$
$937$			35.8248i			1.17035i	−0.810908	+	0.585173i	$0.801026\pi$
$938$		0			0
$939$	−3.02371	+	43.0440i	−0.0986751	+	1.40469i
$940$		0			0
$941$	11.3492			0.369974			0.184987	−	0.982741i	$-0.440776\pi$
$941$	11.3492			0.369974			0.184987	+	0.982741i	$0.440776\pi$
$942$		0			0
$943$	−21.2779	+	14.1160i	−0.692905	+	0.459680i
$944$		0			0
$945$	−8.21176	+	38.4525i	−0.267129	+	1.25086i
$946$		0			0
$947$		−	30.9439i		−	1.00554i	−0.864420	−	0.502771i	$-0.832314\pi$
$947$		−	30.9439i		−	1.00554i	0.864420	−	0.502771i	$-0.167686\pi$
$948$		0			0
$949$	−14.3936			−0.467236
$950$		0			0
$951$	−2.83469	+	40.3531i	−0.0919210	+	1.30854i
$952$		0			0
$953$	−53.8117			−1.74313			−0.871566	−	0.490277i	$-0.836895\pi$
$953$	−53.8117			−1.74313			−0.871566	+	0.490277i	$0.836895\pi$
$954$		0			0
$955$	2.94197			0.0951999
$956$		0			0
$957$	−1.55707	+	22.1656i	−0.0503328	+	0.716511i
$958$		0			0
$959$		−	20.1290i		−	0.650000i
$960$		0			0
$961$	−30.7036			−0.990439
$962$		0			0
$963$	31.7378	+	4.48108i	1.02274	+	0.144401i
$964$		0			0
$965$	29.6626			0.954871
$966$		0			0
$967$	−8.54441			−0.274770			−0.137385	−	0.990518i	$-0.543870\pi$
$967$	−8.54441			−0.274770			−0.137385	+	0.990518i	$0.543870\pi$
$968$		0			0
$969$	0.759218	−	10.8078i	0.0243896	−	0.347198i
$970$		0			0
$971$	59.9482			1.92383			0.961915	−	0.273350i	$-0.0881317\pi$
$971$	59.9482			1.92383			0.961915	+	0.273350i	$0.0881317\pi$
$972$		0			0
$973$		−	23.5492i		−	0.754951i
$974$		0			0
$975$	1.13476	+	0.0797133i	0.0363413	+	0.00255287i
$976$		0			0
$977$	−15.9192			−0.509300			−0.254650	−	0.967033i	$-0.581960\pi$
$977$	−15.9192			−0.509300			−0.254650	+	0.967033i	$0.581960\pi$
$978$		0			0
$979$	−24.2581			−0.775291
$980$		0			0
$981$	4.64722	−	32.9145i	0.148374	−	1.05088i
$982$		0			0
$983$	16.7428			0.534011			0.267006	−	0.963695i	$-0.413966\pi$
$983$	16.7428			0.534011			0.267006	+	0.963695i	$0.413966\pi$
$984$		0			0
$985$			44.8453i			1.42889i
$986$		0			0
$987$	−18.6968	−	1.31339i	−0.595124	−	0.0418057i
$988$		0			0
$989$	−50.5245	+	33.5184i	−1.60659	+	1.06582i
$990$		0			0
$991$	−38.4525			−1.22148			−0.610742	−	0.791829i	$-0.709129\pi$
$991$	−38.4525			−1.22148			−0.610742	+	0.791829i	$0.709129\pi$
$992$		0			0
$993$	4.29289	+	0.301562i	0.136231	+	0.00956980i
$994$		0			0
$995$		−	44.3169i		−	1.40494i
$996$		0			0
$997$	24.2408			0.767713			0.383856	−	0.923393i	$-0.374596\pi$
$997$	24.2408			0.767713			0.383856	+	0.923393i	$0.374596\pi$
$998$		0			0
$999$	1.40725	−	6.58960i	0.0445234	−	0.208486i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.m.b.137.16	yes	16
3.2	odd	2	inner	552.2.m.b.137.13	✓	16
4.3	odd	2		1104.2.m.e.689.2		16
12.11	even	2		1104.2.m.e.689.3		16
23.22	odd	2	inner	552.2.m.b.137.15	yes	16
69.68	even	2	inner	552.2.m.b.137.14	yes	16
92.91	even	2		1104.2.m.e.689.1		16
276.275	odd	2		1104.2.m.e.689.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.m.b.137.13	✓	16	3.2	odd	2	inner
552.2.m.b.137.14	yes	16	69.68	even	2	inner
552.2.m.b.137.15	yes	16	23.22	odd	2	inner
552.2.m.b.137.16	yes	16	1.1	even	1	trivial
1104.2.m.e.689.1		16	92.91	even	2
1104.2.m.e.689.2		16	4.3	odd	2
1104.2.m.e.689.3		16	12.11	even	2
1104.2.m.e.689.4		16	276.275	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.72779 + 0.121372i) q^{3} +2.32463 q^{5} +3.25516i q^{7} +(2.97054 + 0.419412i) q^{9} +O(q^{10})\)	\(q+(1.72779 + 0.121372i) q^{3} +2.32463 q^{5} +3.25516i q^{7} +(2.97054 + 0.419412i) q^{9} -2.65123 q^{11} +1.62598 q^{13} +(4.01649 + 0.282146i) q^{15} -2.32463 q^{17} +2.69087i q^{19} +(-0.395086 + 5.62425i) q^{21} +(-2.65123 - 3.99637i) q^{23} +0.403920 q^{25} +(5.08157 + 1.08520i) q^{27} -4.83882i q^{29} +0.544414 q^{31} +(-4.58078 - 0.321786i) q^{33} +7.56706i q^{35} -1.29677i q^{37} +(2.80936 + 0.197349i) q^{39} -5.32431i q^{41} -12.6426i q^{43} +(6.90541 + 0.974980i) q^{45} +3.32431i q^{47} -3.59608 q^{49} +(-4.01649 - 0.282146i) q^{51} +12.2187 q^{53} -6.16314 q^{55} +(-0.326597 + 4.64927i) q^{57} +13.3967i q^{59} -8.82314i q^{61} +(-1.36525 + 9.66958i) q^{63} +3.77981 q^{65} +8.63690i q^{67} +(-4.09573 - 7.22669i) q^{69} +1.43490i q^{71} -8.85225 q^{73} +(0.697890 + 0.0490247i) q^{75} -8.63018i q^{77} -9.85439i q^{79} +(8.64819 + 2.49176i) q^{81} -15.6719 q^{83} -5.40392 q^{85} +(0.587299 - 8.36049i) q^{87} +9.14974 q^{89} +5.29284i q^{91} +(0.940635 + 0.0660767i) q^{93} +6.25529i q^{95} +10.1199i q^{97} +(-7.87558 - 1.11196i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 16 * q + 4 * q^3 + 4 * q^9	\( 16 q + 4 q^{3} + 4 q^{9} + 8 q^{13} + 32 q^{25} + 16 q^{27} + 56 q^{31} - 44 q^{39} - 32 q^{49} + 32 q^{55} + 4 q^{69} + 40 q^{73} - 20 q^{75} + 4 q^{81} - 112 q^{85} - 36 q^{87} - 36 q^{93}+O(q^{100}) \) 16 * q + 4 * q^3 + 4 * q^9 + 8 * q^13 + 32 * q^25 + 16 * q^27 + 56 * q^31 - 44 * q^39 - 32 * q^49 + 32 * q^55 + 4 * q^69 + 40 * q^73 - 20 * q^75 + 4 * q^81 - 112 * q^85 - 36 * q^87 - 36 * q^93

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