LMFDB - Embedded newform 1503.2.c.a.1502.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1502,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1503.1502");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.0015154238$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1502.2
Character	$\chi$	$=$	1503.1502
Dual form			1503.2.c.a.1502.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18160i q^{2} +0.603817 q^{4} -4.40626 q^{5} -3.48779 q^{7} +3.07668i q^{8} +O(q^{10})$	\(q+1.18160i q^{2} +0.603817 q^{4} -4.40626 q^{5} -3.48779 q^{7} +3.07668i q^{8} -5.20644i q^{10} -5.26397i q^{11} +2.51240i q^{13} -4.12118i q^{14} -2.42777 q^{16} -4.09750 q^{17} +2.84113 q^{19} -2.66057 q^{20} +6.21992 q^{22} +3.67871 q^{23} +14.4151 q^{25} -2.96866 q^{26} -2.10599 q^{28} +1.94004i q^{29} +8.11309 q^{31} +3.28469i q^{32} -4.84162i q^{34} +15.3681 q^{35} -7.87574i q^{37} +3.35709i q^{38} -13.5566i q^{40} -5.05809 q^{41} +0.697951i q^{43} -3.17848i q^{44} +4.34677i q^{46} +0.734409i q^{47} +5.16471 q^{49} +17.0329i q^{50} +1.51703i q^{52} +8.65853 q^{53} +23.1944i q^{55} -10.7308i q^{56} -2.29236 q^{58} -0.555528 q^{59} +5.72436 q^{61} +9.58644i q^{62} -8.73674 q^{64} -11.0703i q^{65} +5.13152i q^{67} -2.47414 q^{68} +18.1590i q^{70} +6.37211 q^{71} -14.9516i q^{73} +9.30599 q^{74} +1.71552 q^{76} +18.3596i q^{77} +5.59514i q^{79} +10.6974 q^{80} -5.97665i q^{82} -0.595523 q^{83} +18.0546 q^{85} -0.824700 q^{86} +16.1955 q^{88} -13.6713i q^{89} -8.76274i q^{91} +2.22127 q^{92} -0.867779 q^{94} -12.5188 q^{95} -10.1399 q^{97} +6.10263i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 48 q^{4}+O(q^{10}) $ 56 * q - 48 * q^4	$ 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) $ 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.18160i			0.835519i	0.908558	+	0.417759i	$0.137184\pi$
$2$			1.18160i			0.835519i	−0.908558	+	0.417759i	$0.862816\pi$
$3$		0			0
$3$		0			0
$4$	0.603817			0.301908
$5$	−4.40626			−1.97054			−0.985269	−	0.171009i	$-0.945297\pi$
$5$	−4.40626			−1.97054			−0.985269	+	0.171009i	$0.945297\pi$
$6$		0			0
$7$	−3.48779			−1.31826			−0.659131	−	0.752028i	$-0.729076\pi$
$7$	−3.48779			−1.31826			−0.659131	+	0.752028i	$0.729076\pi$
$8$			3.07668i			1.08777i
$9$		0			0
$10$		−	5.20644i		−	1.64642i
$11$		−	5.26397i		−	1.58715i	−0.608474	−	0.793574i	$-0.708218\pi$
$11$		−	5.26397i		−	1.58715i	0.608474	−	0.793574i	$-0.291782\pi$
$12$		0			0
$13$			2.51240i			0.696815i	0.937343	+	0.348407i	$0.113277\pi$
$13$			2.51240i			0.696815i	−0.937343	+	0.348407i	$0.886723\pi$
$14$		−	4.12118i		−	1.10143i
$15$		0			0
$16$	−2.42777			−0.606943
$17$	−4.09750			−0.993790			−0.496895	−	0.867811i	$-0.665527\pi$
$17$	−4.09750			−0.993790			−0.496895	+	0.867811i	$0.665527\pi$
$18$		0			0
$19$	2.84113			0.651800			0.325900	−	0.945404i	$-0.394333\pi$
$19$	2.84113			0.651800			0.325900	+	0.945404i	$0.394333\pi$
$20$	−2.66057			−0.594922
$21$		0			0
$22$	6.21992			1.32609
$23$	3.67871			0.767065			0.383532	−	0.923527i	$-0.374707\pi$
$23$	3.67871			0.767065			0.383532	+	0.923527i	$0.374707\pi$
$24$		0			0
$25$	14.4151			2.88302
$26$	−2.96866			−0.582202
$27$		0			0
$28$	−2.10599			−0.397994
$29$			1.94004i			0.360257i	0.983643	+	0.180128i	$0.0576514\pi$
$29$			1.94004i			0.360257i	−0.983643	+	0.180128i	$0.942349\pi$
$30$		0			0
$31$	8.11309			1.45715			0.728577	−	0.684964i	$-0.240182\pi$
$31$	8.11309			1.45715			0.728577	+	0.684964i	$0.240182\pi$
$32$			3.28469i			0.580657i
$33$		0			0
$34$		−	4.84162i		−	0.830330i
$35$	15.3681			2.59769
$36$		0			0
$37$		−	7.87574i		−	1.29476i	−0.762166	−	0.647382i	$-0.775864\pi$
$37$		−	7.87574i		−	1.29476i	0.762166	−	0.647382i	$-0.224136\pi$
$38$			3.35709i			0.544591i
$39$		0			0
$40$		−	13.5566i		−	2.14349i
$41$	−5.05809			−0.789941			−0.394971	−	0.918694i	$-0.629245\pi$
$41$	−5.05809			−0.789941			−0.394971	+	0.918694i	$0.629245\pi$
$42$		0			0
$43$			0.697951i			0.106437i	0.998583	+	0.0532183i	$0.0169479\pi$
$43$			0.697951i			0.106437i	−0.998583	+	0.0532183i	$0.983052\pi$
$44$		−	3.17848i		−	0.479173i
$45$		0			0
$46$			4.34677i			0.640897i
$47$			0.734409i			0.107125i	0.998565	+	0.0535623i	$0.0170576\pi$
$47$			0.734409i			0.107125i	−0.998565	+	0.0535623i	$0.982942\pi$
$48$		0			0
$49$	5.16471			0.737815
$50$			17.0329i			2.40882i
$51$		0			0
$52$			1.51703i			0.210374i
$53$	8.65853			1.18934			0.594671	−	0.803969i	$-0.297282\pi$
$53$	8.65853			1.18934			0.594671	+	0.803969i	$0.297282\pi$
$54$		0			0
$55$			23.1944i			3.12754i
$56$		−	10.7308i		−	1.43396i
$57$		0			0
$58$	−2.29236			−0.301001
$59$	−0.555528			−0.0723236			−0.0361618	−	0.999346i	$-0.511513\pi$
$59$	−0.555528			−0.0723236			−0.0361618	+	0.999346i	$0.511513\pi$
$60$		0			0
$61$	5.72436			0.732929			0.366465	−	0.930432i	$-0.380568\pi$
$61$	5.72436			0.732929			0.366465	+	0.930432i	$0.380568\pi$
$62$			9.58644i			1.21748i
$63$		0			0
$64$	−8.73674			−1.09209
$65$		−	11.0703i		−	1.37310i
$66$		0			0
$67$			5.13152i			0.626915i	0.949602	+	0.313457i	$0.101487\pi$
$67$			5.13152i			0.626915i	−0.949602	+	0.313457i	$0.898513\pi$
$68$	−2.47414			−0.300034
$69$		0			0
$70$			18.1590i			2.17042i
$71$	6.37211			0.756231			0.378115	−	0.925759i	$-0.376572\pi$
$71$	6.37211			0.756231			0.378115	+	0.925759i	$0.376572\pi$
$72$		0			0
$73$		−	14.9516i		−	1.74995i	−0.484166	−	0.874976i	$-0.660877\pi$
$73$		−	14.9516i		−	1.74995i	0.484166	−	0.874976i	$-0.339123\pi$
$74$	9.30599			1.08180
$75$		0			0
$76$	1.71552			0.196784
$77$			18.3596i			2.09228i
$78$		0			0
$79$			5.59514i			0.629502i	0.949174	+	0.314751i	$0.101921\pi$
$79$			5.59514i			0.629502i	−0.949174	+	0.314751i	$0.898079\pi$
$80$	10.6974			1.19600
$81$		0			0
$82$		−	5.97665i		−	0.660011i
$83$	−0.595523			−0.0653671			−0.0326836	−	0.999466i	$-0.510405\pi$
$83$	−0.595523			−0.0653671			−0.0326836	+	0.999466i	$0.510405\pi$
$84$		0			0
$85$	18.0546			1.95830
$86$	−0.824700			−0.0889297
$87$		0			0
$88$	16.1955			1.72645
$89$		−	13.6713i		−	1.44916i	−0.689191	−	0.724580i	$-0.742034\pi$
$89$		−	13.6713i		−	1.44916i	0.689191	−	0.724580i	$-0.257966\pi$
$90$		0			0
$91$		−	8.76274i		−	0.918584i
$92$	2.22127			0.231583
$93$		0			0
$94$	−0.867779			−0.0895046
$95$	−12.5188			−1.28440
$96$		0			0
$97$	−10.1399			−1.02955			−0.514777	−	0.857324i	$-0.672125\pi$
$97$	−10.1399			−1.02955			−0.514777	+	0.857324i	$0.672125\pi$
$98$			6.10263i			0.616458i
$99$		0			0
$100$	8.70409			0.870409
$101$	4.02881			0.400882			0.200441	−	0.979706i	$-0.435763\pi$
$101$	4.02881			0.400882			0.200441	+	0.979706i	$0.435763\pi$
$102$		0			0
$103$			13.8776i			1.36740i	0.729763	+	0.683700i	$0.239630\pi$
$103$			13.8776i			1.36740i	−0.729763	+	0.683700i	$0.760370\pi$
$104$	−7.72984			−0.757973
$105$		0			0
$106$			10.2309i			0.993717i
$107$		−	9.43005i		−	0.911638i	−0.890073	−	0.455819i	$-0.849346\pi$
$107$		−	9.43005i		−	0.911638i	0.890073	−	0.455819i	$-0.150654\pi$
$108$		0			0
$109$		−	16.8519i		−	1.61412i	−0.590469	−	0.807060i	$-0.701057\pi$
$109$		−	16.8519i		−	1.61412i	0.590469	−	0.807060i	$-0.298943\pi$
$110$	−27.4066			−2.61311
$111$		0			0
$112$	8.46757			0.800110
$113$	16.2818			1.53167			0.765833	−	0.643040i	$-0.222327\pi$
$113$	16.2818			1.53167			0.765833	+	0.643040i	$0.222327\pi$
$114$		0			0
$115$	−16.2094			−1.51153
$116$			1.17143i			0.108765i
$117$		0			0
$118$		−	0.656413i		−	0.0604277i
$119$	14.2912			1.31008
$120$		0			0
$121$	−16.7094			−1.51904
$122$			6.76392i			0.612376i
$123$		0			0
$124$	4.89882			0.439927
$125$	−41.4854			−3.71057
$126$		0			0
$127$	−10.1841			−0.903697			−0.451848	−	0.892095i	$-0.649235\pi$
$127$	−10.1841			−0.903697			−0.451848	+	0.892095i	$0.649235\pi$
$128$		−	3.75397i		−	0.331807i
$129$		0			0
$130$	13.0807			1.14725
$131$	6.69558			0.584995			0.292498	−	0.956266i	$-0.405514\pi$
$131$	6.69558			0.584995			0.292498	+	0.956266i	$0.405514\pi$
$132$		0			0
$133$	−9.90928			−0.859244
$134$	−6.06341			−0.523799
$135$		0			0
$136$		−	12.6067i		−	1.08101i
$137$		−	12.3937i		−	1.05887i	−0.848351	−	0.529434i	$-0.822404\pi$
$137$		−	12.3937i		−	1.05887i	0.848351	−	0.529434i	$-0.177596\pi$
$138$		0			0
$139$		−	5.09173i		−	0.431875i	−0.976407	−	0.215937i	$-0.930719\pi$
$139$		−	5.09173i		−	0.431875i	0.976407	−	0.215937i	$-0.0692807\pi$
$140$	9.27953			0.784264
$141$		0			0
$142$			7.52930i			0.631845i
$143$	13.2252			1.10595
$144$		0			0
$145$		−	8.54833i		−	0.709900i
$146$	17.6668			1.46212
$147$		0			0
$148$		−	4.75551i		−	0.390900i
$149$	19.8634			1.62727			0.813636	−	0.581375i	$-0.197485\pi$
$149$	19.8634			1.62727			0.813636	+	0.581375i	$0.197485\pi$
$150$		0			0
$151$		−	17.2669i		−	1.40516i	−0.711604	−	0.702581i	$-0.752031\pi$
$151$		−	17.2669i		−	1.40516i	0.711604	−	0.702581i	$-0.247969\pi$
$152$			8.74124i			0.709008i
$153$		0			0
$154$	−21.6938			−1.74814
$155$	−35.7484			−2.87138
$156$		0			0
$157$	−3.94636			−0.314954			−0.157477	−	0.987523i	$-0.550336\pi$
$157$	−3.94636			−0.314954			−0.157477	+	0.987523i	$0.550336\pi$
$158$	−6.61123			−0.525961
$159$		0			0
$160$		−	14.4732i		−	1.14421i
$161$	−12.8306			−1.01119
$162$		0			0
$163$		−	12.2582i		−	0.960136i	−0.877231	−	0.480068i	$-0.840612\pi$
$163$		−	12.2582i		−	0.960136i	0.877231	−	0.480068i	$-0.159388\pi$
$164$	−3.05416			−0.238490
$165$		0			0
$166$		−	0.703671i		−	0.0546155i
$167$	−2.93836	+	12.5844i	−0.227377	+	0.973807i
$167$	−2.93836	+	12.5844i	−0.227377	+	0.973807i
$168$		0			0
$169$	6.68784			0.514449
$170$			21.3334i			1.63620i
$171$		0			0
$172$			0.421434i			0.0321341i
$173$			14.0140i			1.06547i	0.846283	+	0.532734i	$0.178835\pi$
$173$			14.0140i			1.06547i	−0.846283	+	0.532734i	$0.821165\pi$
$174$		0			0
$175$	−50.2770			−3.80058
$176$			12.7797i			0.963308i
$177$		0			0
$178$	16.1541			1.21080
$179$			8.33788i			0.623202i	0.950213	+	0.311601i	$0.100865\pi$
$179$			8.33788i			0.623202i	−0.950213	+	0.311601i	$0.899135\pi$
$180$		0			0
$181$	18.3077			1.36080			0.680402	−	0.732839i	$-0.261805\pi$
$181$	18.3077			1.36080			0.680402	+	0.732839i	$0.261805\pi$
$182$	10.3541			0.767494
$183$		0			0
$184$			11.3182i			0.834389i
$185$			34.7026i			2.55138i
$186$		0			0
$187$			21.5691i			1.57729i
$188$			0.443449i			0.0323418i
$189$		0			0
$190$		−	14.7922i		−	1.07314i
$191$		−	17.5191i		−	1.26764i	−0.773482	−	0.633819i	$-0.781487\pi$
$191$		−	17.5191i		−	1.26764i	0.773482	−	0.633819i	$-0.218513\pi$
$192$		0			0
$193$			24.6496i			1.77431i	0.461468	+	0.887157i	$0.347323\pi$
$193$			24.6496i			1.77431i	−0.461468	+	0.887157i	$0.652677\pi$
$194$		−	11.9814i		−	0.860212i
$195$		0			0
$196$	3.11854			0.222753
$197$	19.5011			1.38939			0.694697	−	0.719302i	$-0.255538\pi$
$197$	19.5011			1.38939			0.694697	+	0.719302i	$0.255538\pi$
$198$		0			0
$199$	4.17213			0.295754			0.147877	−	0.989006i	$-0.452756\pi$
$199$	4.17213			0.295754			0.147877	+	0.989006i	$0.452756\pi$
$200$			44.3506i			3.13606i
$201$		0			0
$202$			4.76045i			0.334944i
$203$		−	6.76647i		−	0.474913i
$204$		0			0
$205$	22.2873			1.55661
$206$	−16.3978			−1.14249
$207$		0			0
$208$		−	6.09953i		−	0.422927i
$209$		−	14.9556i		−	1.03450i
$210$		0			0
$211$	−14.2478			−0.980858			−0.490429	−	0.871481i	$-0.663160\pi$
$211$	−14.2478			−0.980858			−0.490429	+	0.871481i	$0.663160\pi$
$212$	5.22817			0.359072
$213$		0			0
$214$	11.1426			0.761690
$215$		−	3.07535i		−	0.209737i
$216$		0			0
$217$	−28.2968			−1.92091
$218$	19.9122			1.34863
$219$		0			0
$220$			14.0052i			0.944229i
$221$		−	10.2946i		−	0.692487i
$222$		0			0
$223$	17.4888			1.17114			0.585569	−	0.810622i	$-0.300871\pi$
$223$	17.4888			1.17114			0.585569	+	0.810622i	$0.300871\pi$
$224$		−	11.4563i		−	0.765458i
$225$		0			0
$226$			19.2386i			1.27974i
$227$	3.35896			0.222942			0.111471	−	0.993768i	$-0.464444\pi$
$227$	3.35896			0.222942			0.111471	+	0.993768i	$0.464444\pi$
$228$		0			0
$229$	6.38454			0.421902			0.210951	−	0.977497i	$-0.432344\pi$
$229$	6.38454			0.421902			0.210951	+	0.977497i	$0.432344\pi$
$230$		−	19.1530i		−	1.26291i
$231$		0			0
$232$	−5.96888			−0.391876
$233$			12.4748i			0.817252i	0.912702	+	0.408626i	$0.133992\pi$
$233$			12.4748i			0.817252i	−0.912702	+	0.408626i	$0.866008\pi$
$234$		0			0
$235$		−	3.23600i		−	0.211093i
$236$	−0.335437			−0.0218351
$237$		0			0
$238$			16.8866i			1.09459i
$239$			12.3230i			0.797107i	0.917145	+	0.398554i	$0.130488\pi$
$239$			12.3230i			0.797107i	−0.917145	+	0.398554i	$0.869512\pi$
$240$		0			0
$241$			15.0921i			0.972166i	0.873913	+	0.486083i	$0.161575\pi$
$241$			15.0921i			0.972166i	−0.873913	+	0.486083i	$0.838425\pi$
$242$		−	19.7439i		−	1.26918i
$243$		0			0
$244$	3.45647			0.221278
$245$	−22.7570			−1.45389
$246$		0			0
$247$			7.13806i			0.454184i
$248$			24.9613i			1.58505i
$249$		0			0
$250$		−	49.0193i		−	3.10025i
$251$		−	19.7564i		−	1.24702i	−0.781817	−	0.623508i	$-0.785707\pi$
$251$		−	19.7564i		−	1.24702i	0.781817	−	0.623508i	$-0.214293\pi$
$252$		0			0
$253$		−	19.3646i		−	1.21744i
$254$		−	12.0336i		−	0.755056i
$255$		0			0
$256$	−13.0378			−0.814862
$257$	−28.0851			−1.75190			−0.875951	−	0.482401i	$-0.839765\pi$
$257$	−28.0851			−1.75190			−0.875951	+	0.482401i	$0.839765\pi$
$258$		0			0
$259$			27.4690i			1.70684i
$260$		−	6.68443i		−	0.414551i
$261$		0			0
$262$			7.91151i			0.488775i
$263$			11.7708i			0.725817i	0.931825	+	0.362909i	$0.118216\pi$
$263$			11.7708i			0.725817i	−0.931825	+	0.362909i	$0.881784\pi$
$264$		0			0
$265$	−38.1517			−2.34364
$266$		−	11.7088i		−	0.717914i
$267$		0			0
$268$			3.09850i			0.189271i
$269$	−16.7667			−1.02228			−0.511142	−	0.859496i	$-0.670777\pi$
$269$	−16.7667			−1.02228			−0.511142	+	0.859496i	$0.670777\pi$
$270$		0			0
$271$		−	26.6298i		−	1.61764i	−0.588054	−	0.808822i	$-0.700106\pi$
$271$		−	26.6298i		−	1.61764i	0.588054	−	0.808822i	$-0.299894\pi$
$272$	9.94780			0.603174
$273$		0			0
$274$	14.6444			0.884704
$275$		−	75.8808i		−	4.57578i
$276$		0			0
$277$		−	7.40855i		−	0.445137i	−0.974917	−	0.222568i	$-0.928556\pi$
$277$		−	7.40855i		−	0.445137i	0.974917	−	0.222568i	$-0.0714441\pi$
$278$	6.01639			0.360839
$279$		0			0
$280$			47.2827i			2.82568i
$281$		−	22.5683i		−	1.34631i	−0.739500	−	0.673157i	$-0.764938\pi$
$281$		−	22.5683i		−	1.34631i	0.739500	−	0.673157i	$-0.235062\pi$
$282$		0			0
$283$	−15.9860			−0.950268			−0.475134	−	0.879913i	$-0.657600\pi$
$283$	−15.9860			−0.950268			−0.475134	+	0.879913i	$0.657600\pi$
$284$	3.84759			0.228312
$285$		0			0
$286$			15.6269i			0.924040i
$287$	17.6416			1.04135
$288$		0			0
$289$	−0.210486			−0.0123815
$290$	10.1007			0.593135
$291$		0			0
$292$		−	9.02803i		−	0.528325i
$293$		−	4.10312i		−	0.239707i	−0.992792	−	0.119853i	$-0.961758\pi$
$293$		−	4.10312i		−	0.239707i	0.992792	−	0.119853i	$-0.0382425\pi$
$294$		0			0
$295$	2.44780			0.142516
$296$	24.2311			1.40840
$297$		0			0
$298$			23.4706i			1.35962i
$299$			9.24240i			0.534502i
$300$		0			0
$301$		−	2.43431i		−	0.140311i
$302$	20.4026			1.17404
$303$		0			0
$304$	−6.89762			−0.395606
$305$	−25.2230			−1.44427
$306$		0			0
$307$			20.0994i			1.14714i	0.819158	+	0.573568i	$0.194441\pi$
$307$			20.0994i			1.14714i	−0.819158	+	0.573568i	$0.805559\pi$
$308$			11.0859i			0.631676i
$309$		0			0
$310$		−	42.2404i		−	2.39909i
$311$		−	22.8501i		−	1.29571i	−0.761762	−	0.647857i	$-0.775666\pi$
$311$		−	22.8501i		−	1.29571i	0.761762	−	0.647857i	$-0.224334\pi$
$312$		0			0
$313$		−	6.03741i		−	0.341255i	−0.985336	−	0.170627i	$-0.945421\pi$
$313$		−	6.03741i		−	0.341255i	0.985336	−	0.170627i	$-0.0545794\pi$
$314$		−	4.66302i		−	0.263150i
$315$		0			0
$316$			3.37844i			0.190052i
$317$		−	19.0557i		−	1.07028i	−0.844764	−	0.535138i	$-0.820259\pi$
$317$		−	19.0557i		−	1.07028i	0.844764	−	0.535138i	$-0.179741\pi$
$318$		0			0
$319$	10.2123			0.571781
$320$	38.4963			2.15201
$321$		0			0
$322$		−	15.1606i		−	0.844870i
$323$	−11.6415			−0.647753
$324$		0			0
$325$			36.2165i			2.00893i
$326$	14.4843			0.802212
$327$		0			0
$328$		−	15.5621i		−	0.859274i
$329$		−	2.56147i		−	0.141218i
$330$		0			0
$331$			2.01465i			0.110735i	0.998466	+	0.0553676i	$0.0176331\pi$
$331$			2.01465i			0.110735i	−0.998466	+	0.0553676i	$0.982367\pi$
$332$	−0.359587			−0.0197349
$333$		0			0
$334$	−14.8697	−	3.47198i	−0.813634	−	0.189978i
$335$		−	22.6108i		−	1.23536i
$336$		0			0
$337$	−4.09319			−0.222970			−0.111485	−	0.993766i	$-0.535561\pi$
$337$	−4.09319			−0.222970			−0.111485	+	0.993766i	$0.535561\pi$
$338$			7.90237i			0.429832i
$339$		0			0
$340$	10.9017			0.591228
$341$		−	42.7071i		−	2.31272i
$342$		0			0
$343$	6.40113			0.345628
$344$	−2.14737			−0.115778
$345$		0			0
$346$	−16.5590			−0.890218
$347$	36.2864			1.94796			0.973979	−	0.226639i	$-0.0727738\pi$
$347$	36.2864			1.94796			0.973979	+	0.226639i	$0.0727738\pi$
$348$		0			0
$349$			6.02652i			0.322592i	0.986906	+	0.161296i	$0.0515674\pi$
$349$			6.02652i			0.322592i	−0.986906	+	0.161296i	$0.948433\pi$
$350$		−	59.4073i		−	3.17546i
$351$		0			0
$352$	17.2905			0.921588
$353$			4.79288i			0.255099i	0.991832	+	0.127550i	$0.0407112\pi$
$353$			4.79288i			0.255099i	−0.991832	+	0.127550i	$0.959289\pi$
$354$		0			0
$355$	−28.0772			−1.49018
$356$		−	8.25499i		−	0.437514i
$357$		0			0
$358$	−9.85205			−0.520697
$359$			32.1257i			1.69553i	0.530374	+	0.847764i	$0.322051\pi$
$359$			32.1257i			1.69553i	−0.530374	+	0.847764i	$0.677949\pi$
$360$		0			0
$361$	−10.9280			−0.575156
$362$			21.6325i			1.13698i
$363$		0			0
$364$		−	5.29109i		−	0.277328i
$365$			65.8806i			3.44835i
$366$		0			0
$367$	−13.4552			−0.702356			−0.351178	−	0.936309i	$-0.614219\pi$
$367$	−13.4552			−0.702356			−0.351178	+	0.936309i	$0.614219\pi$
$368$	−8.93107			−0.465564
$369$		0			0
$370$	−41.0046			−2.13173
$371$	−30.1992			−1.56786
$372$		0			0
$373$			1.41387i			0.0732072i	0.999330	+	0.0366036i	$0.0116539\pi$
$373$			1.41387i			0.0732072i	−0.999330	+	0.0366036i	$0.988346\pi$
$374$	−25.4861			−1.31786
$375$		0			0
$376$	−2.25954			−0.116527
$377$	−4.87417			−0.251032
$378$		0			0
$379$		−	9.54187i		−	0.490133i	−0.969506	−	0.245067i	$-0.921190\pi$
$379$		−	9.54187i		−	0.490133i	0.969506	−	0.245067i	$-0.0788098\pi$
$380$	−7.55904			−0.387771
$381$		0			0
$382$	20.7006			1.05913
$383$		−	23.5423i		−	1.20296i	−0.798889	−	0.601478i	$-0.794579\pi$
$383$		−	23.5423i		−	1.20296i	0.798889	−	0.601478i	$-0.205421\pi$
$384$		0			0
$385$		−	80.8974i		−	4.12291i
$386$	−29.1260			−1.48247
$387$		0			0
$388$	−6.12266			−0.310831
$389$	21.0378			1.06666			0.533329	−	0.845908i	$-0.320941\pi$
$389$	21.0378			1.06666			0.533329	+	0.845908i	$0.320941\pi$
$390$		0			0
$391$	−15.0735			−0.762301
$392$			15.8901i			0.802572i
$393$		0			0
$394$			23.0425i			1.16086i
$395$		−	24.6536i		−	1.24046i
$396$		0			0
$397$	25.6386			1.28677			0.643383	−	0.765545i	$-0.277530\pi$
$397$	25.6386			1.28677			0.643383	+	0.765545i	$0.277530\pi$
$398$			4.92979i			0.247108i
$399$		0			0
$400$	−34.9966			−1.74983
$401$	15.1425			0.756178			0.378089	−	0.925769i	$-0.376581\pi$
$401$	15.1425			0.756178			0.378089	+	0.925769i	$0.376581\pi$
$402$		0			0
$403$			20.3833i			1.01537i
$404$	2.43267			0.121030
$405$		0			0
$406$	7.99527			0.396799
$407$	−41.4577			−2.05498
$408$		0			0
$409$	−5.81317			−0.287443			−0.143721	−	0.989618i	$-0.545907\pi$
$409$	−5.81317			−0.287443			−0.143721	+	0.989618i	$0.545907\pi$
$410$			26.3347i			1.30058i
$411$		0			0
$412$			8.37953i			0.412830i
$413$	1.93757			0.0953415
$414$		0			0
$415$	2.62403			0.128808
$416$	−8.25246			−0.404610
$417$		0			0
$418$	17.6716			0.864347
$419$		−	16.9719i		−	0.829133i	−0.910019	−	0.414567i	$-0.863933\pi$
$419$		−	16.9719i		−	0.829133i	0.910019	−	0.414567i	$-0.136067\pi$
$420$		0			0
$421$	5.31069			0.258827			0.129414	−	0.991591i	$-0.458691\pi$
$421$	5.31069			0.258827			0.129414	+	0.991591i	$0.458691\pi$
$422$		−	16.8352i		−	0.819525i
$423$		0			0
$424$			26.6395i			1.29373i
$425$	−59.0660			−2.86512
$426$		0			0
$427$	−19.9654			−0.966193
$428$		−	5.69403i		−	0.275231i
$429$		0			0
$430$	3.63384			0.175239
$431$		−	3.75488i		−	0.180866i	−0.995903	−	0.0904330i	$-0.971175\pi$
$431$		−	3.75488i		−	0.180866i	0.995903	−	0.0904330i	$-0.0288251\pi$
$432$		0			0
$433$	24.7305			1.18847			0.594236	−	0.804291i	$-0.297455\pi$
$433$	24.7305			1.18847			0.594236	+	0.804291i	$0.297455\pi$
$434$		−	33.4355i		−	1.60496i
$435$		0			0
$436$		−	10.1755i		−	0.487316i
$437$	10.4517			0.499973
$438$		0			0
$439$		−	19.4647i		−	0.929001i	−0.885573	−	0.464501i	$-0.846234\pi$
$439$		−	19.4647i		−	0.929001i	0.885573	−	0.464501i	$-0.153766\pi$
$440$	−71.3617			−3.40204
$441$		0			0
$442$	12.1641			0.578586
$443$	12.9593			0.615713			0.307856	−	0.951433i	$-0.400388\pi$
$443$	12.9593			0.615713			0.307856	+	0.951433i	$0.400388\pi$
$444$		0			0
$445$			60.2395i			2.85563i
$446$			20.6648i			0.978508i
$447$		0			0
$448$	30.4719			1.43966
$449$			16.7018i			0.788205i	0.919067	+	0.394102i	$0.128944\pi$
$449$			16.7018i			0.788205i	−0.919067	+	0.394102i	$0.871056\pi$
$450$		0			0
$451$			26.6257i			1.25375i
$452$	9.83124			0.462423
$453$		0			0
$454$			3.96896i			0.186272i
$455$			38.6109i			1.81011i
$456$		0			0
$457$		−	26.7008i		−	1.24901i	−0.781020	−	0.624506i	$-0.785300\pi$
$457$		−	26.7008i		−	1.24901i	0.781020	−	0.624506i	$-0.214700\pi$
$458$			7.54399i			0.352507i
$459$		0			0
$460$	−9.78748			−0.456344
$461$			5.74058i			0.267365i	0.991024	+	0.133683i	$0.0426803\pi$
$461$			5.74058i			0.267365i	−0.991024	+	0.133683i	$0.957320\pi$
$462$		0			0
$463$		−	32.0544i		−	1.48969i	−0.667236	−	0.744847i	$-0.732523\pi$
$463$		−	32.0544i		−	1.48969i	0.667236	−	0.744847i	$-0.267477\pi$
$464$		−	4.70998i		−	0.218655i
$465$		0			0
$466$	−14.7403			−0.682829
$467$			33.8769i			1.56763i	0.620992	+	0.783817i	$0.286730\pi$
$467$			33.8769i			1.56763i	−0.620992	+	0.783817i	$0.713270\pi$
$468$		0			0
$469$		−	17.8977i		−	0.826438i
$470$	3.82366			0.176372
$471$		0			0
$472$		−	1.70918i		−	0.0786714i
$473$	3.67399			0.168930
$474$		0			0
$475$	40.9552			1.87916
$476$	8.62929			0.395523
$477$		0			0
$478$	−14.5609			−0.665998
$479$	39.5033			1.80495			0.902477	−	0.430738i	$-0.141747\pi$
$479$	39.5033			1.80495			0.902477	+	0.430738i	$0.141747\pi$
$480$		0			0
$481$	19.7870			0.902210
$482$	−17.8328			−0.812263
$483$		0			0
$484$	−10.0894			−0.458610
$485$	44.6792			2.02878
$486$		0			0
$487$			0.0606595i			0.00274875i	0.999999	+	0.00137437i	$0.000437477\pi$
$487$			0.0606595i			0.00274875i	−0.999999	+	0.00137437i	$0.999563\pi$
$488$			17.6120i			0.797258i
$489$		0			0
$490$		−	26.8897i		−	1.21476i
$491$		−	13.7970i		−	0.622649i	−0.950304	−	0.311324i	$-0.899227\pi$
$491$		−	13.7970i		−	0.622649i	0.950304	−	0.311324i	$-0.100773\pi$
$492$		0			0
$493$		−	7.94933i		−	0.358020i
$494$	−8.43435			−0.379479
$495$		0			0
$496$	−19.6967			−0.884409
$497$	−22.2246			−0.996910
$498$		0			0
$499$		−	18.5004i		−	0.828192i	−0.910233	−	0.414096i	$-0.864098\pi$
$499$		−	18.5004i		−	0.828192i	0.910233	−	0.414096i	$-0.135902\pi$
$500$	−25.0496			−1.12025
$501$		0			0
$502$	23.3442			1.04190
$503$			5.17801i			0.230876i	0.993315	+	0.115438i	$0.0368272\pi$
$503$			5.17801i			0.230876i	−0.993315	+	0.115438i	$0.963173\pi$
$504$		0			0
$505$	−17.7520			−0.789953
$506$	22.8813			1.01720
$507$		0			0
$508$	−6.14936			−0.272834
$509$			24.2398i			1.07441i	0.843452	+	0.537205i	$0.180520\pi$
$509$			24.2398i			1.07441i	−0.843452	+	0.537205i	$0.819480\pi$
$510$		0			0
$511$			52.1481i			2.30690i
$512$		−	22.9134i		−	1.01264i
$513$		0			0
$514$		−	33.1854i		−	1.46375i
$515$		−	61.1483i		−	2.69452i
$516$		0			0
$517$	3.86591			0.170022
$518$	−32.4574			−1.42610
$519$		0			0
$520$	34.0597			1.49362
$521$	−3.04066			−0.133214			−0.0666070	−	0.997779i	$-0.521217\pi$
$521$	−3.04066			−0.133214			−0.0666070	+	0.997779i	$0.521217\pi$
$522$		0			0
$523$	28.9139			1.26432			0.632158	−	0.774839i	$-0.282169\pi$
$523$	28.9139			1.26432			0.632158	+	0.774839i	$0.282169\pi$
$524$	4.04290			0.176615
$525$		0			0
$526$	−13.9084			−0.606434
$527$	−33.2434			−1.44811
$528$		0			0
$529$	−9.46708			−0.411612
$530$		−	45.0802i		−	1.95816i
$531$		0			0
$532$	−5.98339			−0.259413
$533$		−	12.7080i		−	0.550443i
$534$		0			0
$535$			41.5513i			1.79642i
$536$	−15.7880			−0.681938
$537$		0			0
$538$		−	19.8116i		−	0.854137i
$539$		−	27.1869i		−	1.17102i
$540$		0			0
$541$			27.7001i			1.19092i	0.803384	+	0.595461i	$0.203030\pi$
$541$			27.7001i			1.19092i	−0.803384	+	0.595461i	$0.796970\pi$
$542$	31.4658			1.35157
$543$		0			0
$544$		−	13.4590i		−	0.577051i
$545$			74.2539i			3.18069i
$546$		0			0
$547$		−	28.2079i		−	1.20608i	−0.797710	−	0.603041i	$-0.793956\pi$
$547$		−	28.2079i		−	1.20608i	0.797710	−	0.603041i	$-0.206044\pi$
$548$		−	7.48354i		−	0.319681i
$549$		0			0
$550$	89.6609			3.82315
$551$			5.51192i			0.234816i
$552$		0			0
$553$		−	19.5147i		−	0.829849i
$554$	8.75396			0.371920
$555$		0			0
$556$		−	3.07447i		−	0.130387i
$557$		−	11.5198i		−	0.488111i	−0.969761	−	0.244055i	$-0.921522\pi$
$557$		−	11.5198i		−	0.488111i	0.969761	−	0.244055i	$-0.0784778\pi$
$558$		0			0
$559$	−1.75353			−0.0741665
$560$	−37.3103			−1.57665
$561$		0			0
$562$	26.6668			1.12487
$563$			10.5257i			0.443603i	0.975092	+	0.221802i	$0.0711938\pi$
$563$			10.5257i			0.443603i	−0.975092	+	0.221802i	$0.928806\pi$
$564$		0			0
$565$	−71.7419			−3.01821
$566$		−	18.8891i		−	0.793967i
$567$		0			0
$568$			19.6049i			0.822604i
$569$	4.53773			0.190232			0.0951158	−	0.995466i	$-0.469678\pi$
$569$	4.53773			0.190232			0.0951158	+	0.995466i	$0.469678\pi$
$570$		0			0
$571$		−	4.56537i		−	0.191055i	−0.995427	−	0.0955273i	$-0.969546\pi$
$571$		−	4.56537i		−	0.191055i	0.995427	−	0.0955273i	$-0.0304537\pi$
$572$	7.98560			0.333895
$573$		0			0
$574$			20.8453i			0.870067i
$575$	53.0291			2.21146
$576$		0			0
$577$	−1.32403			−0.0551201			−0.0275600	−	0.999620i	$-0.508774\pi$
$577$	−1.32403			−0.0551201			−0.0275600	+	0.999620i	$0.508774\pi$
$578$		−	0.248711i		−	0.0103450i
$579$		0			0
$580$		−	5.16163i		−	0.214325i
$581$	2.07706			0.0861710
$582$		0			0
$583$		−	45.5783i		−	1.88766i
$584$	46.0012			1.90354
$585$		0			0
$586$	4.84826			0.200280
$587$	35.0761			1.44775			0.723873	−	0.689934i	$-0.242360\pi$
$587$	35.0761			1.44775			0.723873	+	0.689934i	$0.242360\pi$
$588$		0			0
$589$	23.0504			0.949773
$590$			2.89233i			0.119075i
$591$		0			0
$592$			19.1205i			0.785848i
$593$	6.10461			0.250686			0.125343	−	0.992113i	$-0.459997\pi$
$593$	6.10461			0.250686			0.125343	+	0.992113i	$0.459997\pi$
$594$		0			0
$595$	−62.9709			−2.58155
$596$	11.9938			0.491287
$597$		0			0
$598$	−10.9208			−0.446586
$599$			8.87135i			0.362473i	0.983439	+	0.181237i	$0.0580100\pi$
$599$			8.87135i			0.362473i	−0.983439	+	0.181237i	$0.941990\pi$
$600$		0			0
$601$	6.37600			0.260082			0.130041	−	0.991509i	$-0.458489\pi$
$601$	6.37600			0.260082			0.130041	+	0.991509i	$0.458489\pi$
$602$	2.87638			0.117233
$603$		0			0
$604$		−	10.4261i		−	0.424230i
$605$	73.6260			2.99332
$606$		0			0
$607$			2.15408i			0.0874315i	0.999044	+	0.0437158i	$0.0139196\pi$
$607$			2.15408i			0.0874315i	−0.999044	+	0.0437158i	$0.986080\pi$
$608$			9.33224i			0.378472i
$609$		0			0
$610$		−	29.8036i		−	1.20671i
$611$	−1.84513			−0.0746460
$612$		0			0
$613$	−39.7249			−1.60447			−0.802237	−	0.597006i	$-0.796357\pi$
$613$	−39.7249			−1.60447			−0.802237	+	0.597006i	$0.796357\pi$
$614$	−23.7495			−0.958453
$615$		0			0
$616$	−56.4867			−2.27591
$617$		−	8.05681i		−	0.324355i	−0.986762	−	0.162177i	$-0.948148\pi$
$617$		−	8.05681i		−	0.324355i	0.986762	−	0.162177i	$-0.0518517\pi$
$618$		0			0
$619$		−	10.1789i		−	0.409126i	−0.978853	−	0.204563i	$-0.934423\pi$
$619$		−	10.1789i		−	0.409126i	0.978853	−	0.204563i	$-0.0655774\pi$
$620$	−21.5855			−0.866893
$621$		0			0
$622$	26.9998			1.08259
$623$			47.6828i			1.91037i
$624$		0			0
$625$	110.720			4.42880
$626$	7.13382			0.285125
$627$		0			0
$628$	−2.38288			−0.0950872
$629$			32.2709i			1.28672i
$630$		0			0
$631$	−37.5124			−1.49335			−0.746673	−	0.665191i	$-0.768350\pi$
$631$	−37.5124			−1.49335			−0.746673	+	0.665191i	$0.768350\pi$
$632$	−17.2144			−0.684753
$633$		0			0
$634$	22.5163			0.894236
$635$	44.8740			1.78077
$636$		0			0
$637$			12.9758i			0.514120i
$638$			12.0669i			0.477734i
$639$		0			0
$640$			16.5410i			0.653838i
$641$	12.0628			0.476453			0.238226	−	0.971210i	$-0.423434\pi$
$641$	12.0628			0.476453			0.238226	+	0.971210i	$0.423434\pi$
$642$		0			0
$643$			22.5546i			0.889466i	0.895663	+	0.444733i	$0.146701\pi$
$643$			22.5546i			0.889466i	−0.895663	+	0.444733i	$0.853299\pi$
$644$	−7.74733			−0.305287
$645$		0			0
$646$		−	13.7557i		−	0.541209i
$647$	12.3982			0.487424			0.243712	−	0.969848i	$-0.421635\pi$
$647$	12.3982			0.487424			0.243712	+	0.969848i	$0.421635\pi$
$648$		0			0
$649$			2.92428i			0.114788i
$650$	−42.7935			−1.67850
$651$		0			0
$652$		−	7.40171i		−	0.289873i
$653$		−	25.9530i		−	1.01562i	−0.861470	−	0.507809i	$-0.830456\pi$
$653$		−	25.9530i		−	1.01562i	0.861470	−	0.507809i	$-0.169544\pi$
$654$		0			0
$655$	−29.5025			−1.15276
$656$	12.2799			0.479449
$657$		0			0
$658$	3.02664			0.117991
$659$	−36.5225			−1.42271			−0.711356	−	0.702831i	$-0.751919\pi$
$659$	−36.5225			−1.42271			−0.711356	+	0.702831i	$0.751919\pi$
$660$		0			0
$661$			39.9017i			1.55199i	0.630736	+	0.775997i	$0.282753\pi$
$661$			39.9017i			1.55199i	−0.630736	+	0.775997i	$0.717247\pi$
$662$	−2.38052			−0.0925214
$663$		0			0
$664$		−	1.83223i		−	0.0711043i
$665$	43.6629			1.69317
$666$		0			0
$667$			7.13686i			0.276340i
$668$	−1.77423	+	7.59865i	−0.0686472	+	0.294000i
$669$		0			0
$670$	26.7170			1.03217
$671$		−	30.1329i		−	1.16327i
$672$		0			0
$673$			31.8824i			1.22897i	0.788927	+	0.614487i	$0.210637\pi$
$673$			31.8824i			1.22897i	−0.788927	+	0.614487i	$0.789363\pi$
$674$		−	4.83652i		−	0.186296i
$675$		0			0
$676$	4.03823			0.155317
$677$		−	6.38329i		−	0.245330i	−0.992448	−	0.122665i	$-0.960856\pi$
$677$		−	6.38329i		−	0.245330i	0.992448	−	0.122665i	$-0.0391440\pi$
$678$		0			0
$679$	35.3660			1.35722
$680$			55.5483i			2.13018i
$681$		0			0
$682$	50.4628			1.93232
$683$	−5.97499			−0.228627			−0.114313	−	0.993445i	$-0.536467\pi$
$683$	−5.97499			−0.228627			−0.114313	+	0.993445i	$0.536467\pi$
$684$		0			0
$685$			54.6100i			2.08654i
$686$			7.56359i			0.288779i
$687$		0			0
$688$		−	1.69447i		−	0.0646009i
$689$			21.7537i			0.828750i
$690$		0			0
$691$		−	21.1579i		−	0.804884i	−0.915446	−	0.402442i	$-0.868162\pi$
$691$		−	21.1579i		−	0.804884i	0.915446	−	0.402442i	$-0.131838\pi$
$692$			8.46191i			0.321674i
$693$		0			0
$694$			42.8761i			1.62756i
$695$			22.4355i			0.851026i
$696$		0			0
$697$	20.7255			0.785036
$698$	−7.12094			−0.269532
$699$		0			0
$700$	−30.3581			−1.14743
$701$		−	1.12606i		−	0.0425306i	−0.999774	−	0.0212653i	$-0.993231\pi$
$701$		−	1.12606i		−	0.0425306i	0.999774	−	0.0212653i	$-0.00676946\pi$
$702$		0			0
$703$		−	22.3760i		−	0.843928i
$704$			45.9900i			1.73331i
$705$		0			0
$706$	−5.66328			−0.213140
$707$	−14.0517			−0.528467
$708$		0			0
$709$			30.6936i			1.15272i	0.817194	+	0.576362i	$0.195528\pi$
$709$			30.6936i			1.15272i	−0.817194	+	0.576362i	$0.804472\pi$
$710$		−	33.1760i		−	1.24507i
$711$		0			0
$712$	42.0623			1.57635
$713$	29.8457			1.11773
$714$		0			0
$715$	−58.2737			−2.17931
$716$			5.03455i			0.188150i
$717$		0			0
$718$	−37.9597			−1.41664
$719$	−42.1938			−1.57356			−0.786782	−	0.617231i	$-0.788254\pi$
$719$	−42.1938			−1.57356			−0.786782	+	0.617231i	$0.788254\pi$
$720$		0			0
$721$		−	48.4022i		−	1.80259i
$722$		−	12.9125i		−	0.480554i
$723$		0			0
$724$	11.0545			0.410838
$725$			27.9659i			1.03863i
$726$		0			0
$727$		−	9.07942i		−	0.336737i	−0.985724	−	0.168368i	$-0.946150\pi$
$727$		−	9.07942i		−	0.336737i	0.985724	−	0.168368i	$-0.0538498\pi$
$728$	26.9601			0.999207
$729$		0			0
$730$	−77.8446			−2.88116
$731$		−	2.85985i		−	0.105776i
$732$		0			0
$733$	−37.4074			−1.38167			−0.690837	−	0.723011i	$-0.742758\pi$
$733$	−37.4074			−1.38167			−0.690837	+	0.723011i	$0.742758\pi$
$734$		−	15.8987i		−	0.586831i
$735$		0			0
$736$			12.0834i			0.445401i
$737$	27.0122			0.995006
$738$		0			0
$739$			30.3074i			1.11488i	0.830218	+	0.557438i	$0.188216\pi$
$739$			30.3074i			1.11488i	−0.830218	+	0.557438i	$0.811784\pi$
$740$			20.9540i			0.770284i
$741$		0			0
$742$		−	35.6834i		−	1.30998i
$743$		−	12.8422i		−	0.471134i	−0.971858	−	0.235567i	$-0.924305\pi$
$743$		−	12.8422i		−	0.471134i	0.971858	−	0.235567i	$-0.0756947\pi$
$744$		0			0
$745$	−87.5232			−3.20660
$746$	−1.67063			−0.0611660
$747$		0			0
$748$			13.0238i			0.476197i
$749$			32.8901i			1.20178i
$750$		0			0
$751$			39.1837i			1.42983i	0.699209	+	0.714917i	$0.253535\pi$
$751$			39.1837i			1.42983i	−0.699209	+	0.714917i	$0.746465\pi$
$752$		−	1.78298i		−	0.0650185i
$753$		0			0
$754$		−	5.75932i		−	0.209742i
$755$			76.0825i			2.76892i
$756$		0			0
$757$	23.8122			0.865471			0.432735	−	0.901521i	$-0.357548\pi$
$757$	23.8122			0.865471			0.432735	+	0.901521i	$0.357548\pi$
$758$	11.2747			0.409515
$759$		0			0
$760$		−	38.5162i		−	1.39713i
$761$			22.3933i			0.811756i	0.913927	+	0.405878i	$0.133034\pi$
$761$			22.3933i			0.811756i	−0.913927	+	0.405878i	$0.866966\pi$
$762$		0			0
$763$			58.7760i			2.12783i
$764$		−	10.5783i		−	0.382710i
$765$		0			0
$766$	27.8177			1.00509
$767$		−	1.39571i		−	0.0503961i
$768$		0			0
$769$			24.3971i			0.879784i	0.898051	+	0.439892i	$0.144983\pi$
$769$			24.3971i			0.879784i	−0.898051	+	0.439892i	$0.855017\pi$
$770$	95.5885			3.44477
$771$		0			0
$772$			14.8838i			0.535680i
$773$	7.83191			0.281694			0.140847	−	0.990031i	$-0.455017\pi$
$773$	7.83191			0.281694			0.140847	+	0.990031i	$0.455017\pi$
$774$		0			0
$775$	116.951			4.20101
$776$		−	31.1973i		−	1.11992i
$777$		0			0
$778$			24.8583i			0.891212i
$779$	−14.3707			−0.514884
$780$		0			0
$781$		−	33.5426i		−	1.20025i
$782$		−	17.8109i		−	0.636917i
$783$		0			0
$784$	−12.5387			−0.447812
$785$	17.3887			0.620628
$786$		0			0
$787$		−	49.0232i		−	1.74749i	−0.486386	−	0.873744i	$-0.661685\pi$
$787$		−	49.0232i		−	1.74749i	0.486386	−	0.873744i	$-0.338315\pi$
$788$	11.7751			0.419470
$789$		0			0
$790$	29.1308			1.03643
$791$	−56.7876			−2.01914
$792$		0			0
$793$			14.3819i			0.510716i
$794$			30.2946i			1.07512i
$795$		0			0
$796$	2.51920			0.0892907
$797$	5.26748			0.186584			0.0932919	−	0.995639i	$-0.470261\pi$
$797$	5.26748			0.186584			0.0932919	+	0.995639i	$0.470261\pi$
$798$		0			0
$799$		−	3.00924i		−	0.106459i
$800$			47.3492i			1.67405i
$801$		0			0
$802$			17.8924i			0.631801i
$803$	−78.7048			−2.77743
$804$		0			0
$805$	56.5349			1.99259
$806$	−24.0850			−0.848358
$807$		0			0
$808$			12.3954i			0.436067i
$809$		−	11.6037i		−	0.407963i	−0.978975	−	0.203982i	$-0.934612\pi$
$809$		−	11.6037i		−	0.407963i	0.978975	−	0.203982i	$-0.0653883\pi$
$810$		0			0
$811$		−	13.4554i		−	0.472484i	−0.971694	−	0.236242i	$-0.924084\pi$
$811$		−	13.4554i		−	0.472484i	0.971694	−	0.236242i	$-0.0759159\pi$
$812$		−	4.08571i		−	0.143380i
$813$		0			0
$814$		−	48.9865i		−	1.71698i
$815$			54.0128i			1.89199i
$816$		0			0
$817$			1.98297i			0.0693753i
$818$		−	6.86886i		−	0.240164i
$819$		0			0
$820$	13.4574			0.469954
$821$	−9.02522			−0.314983			−0.157491	−	0.987520i	$-0.550341\pi$
$821$	−9.02522			−0.314983			−0.157491	+	0.987520i	$0.550341\pi$
$822$		0			0
$823$		−	2.12684i		−	0.0741369i	−0.999313	−	0.0370684i	$-0.988198\pi$
$823$		−	2.12684i		−	0.0741369i	0.999313	−	0.0370684i	$-0.0118020\pi$
$824$	−42.6969			−1.48742
$825$		0			0
$826$			2.28943i			0.0796596i
$827$	−35.3167			−1.22808			−0.614041	−	0.789274i	$-0.710457\pi$
$827$	−35.3167			−1.22808			−0.614041	+	0.789274i	$0.710457\pi$
$828$		0			0
$829$			10.0905i			0.350458i	0.984528	+	0.175229i	$0.0560666\pi$
$829$			10.0905i			0.350458i	−0.984528	+	0.175229i	$0.943933\pi$
$830$			3.10056i			0.107622i
$831$		0			0
$832$		−	21.9502i		−	0.760986i
$833$	−21.1624			−0.733233
$834$		0			0
$835$	12.9472	−	55.4499i	0.448056	−	1.91892i
$836$		−	9.03047i		−	0.312325i
$837$		0			0
$838$	20.0541			0.692756
$839$			31.4208i			1.08477i	0.840131	+	0.542384i	$0.182478\pi$
$839$			31.4208i			1.08477i	−0.840131	+	0.542384i	$0.817522\pi$
$840$		0			0
$841$	25.2362			0.870215
$842$			6.27512i			0.216255i
$843$		0			0
$844$	−8.60305			−0.296129
$845$	−29.4684			−1.01374
$846$		0			0
$847$	58.2790			2.00249
$848$	−21.0209			−0.721862
$849$		0			0
$850$		−	69.7924i		−	2.39386i
$851$		−	28.9726i		−	0.993168i
$852$		0			0
$853$	25.8435			0.884865			0.442433	−	0.896802i	$-0.354116\pi$
$853$	25.8435			0.884865			0.442433	+	0.896802i	$0.354116\pi$
$854$		−	23.5911i		−	0.807272i
$855$		0			0
$856$	29.0132			0.991651
$857$		−	15.4308i		−	0.527105i	−0.964645	−	0.263552i	$-0.915106\pi$
$857$		−	15.4308i		−	0.527105i	0.964645	−	0.263552i	$-0.0848942\pi$
$858$		0			0
$859$	−12.8511			−0.438473			−0.219236	−	0.975672i	$-0.570357\pi$
$859$	−12.8511			−0.438473			−0.219236	+	0.975672i	$0.570357\pi$
$860$		−	1.85695i		−	0.0633214i
$861$		0			0
$862$	4.43677			0.151117
$863$			26.7454i			0.910423i	0.890383	+	0.455211i	$0.150436\pi$
$863$			26.7454i			0.910423i	−0.890383	+	0.455211i	$0.849564\pi$
$864$		0			0
$865$		−	61.7495i		−	2.09955i
$866$			29.2216i			0.992990i
$867$		0			0
$868$	−17.0861			−0.579939
$869$	29.4527			0.999113
$870$		0			0
$871$	−12.8924			−0.436843
$872$	51.8478			1.75579
$873$		0			0
$874$			12.3498i			0.417737i
$875$	144.693			4.89150
$876$		0			0
$877$	12.1905			0.411645			0.205823	−	0.978589i	$-0.434013\pi$
$877$	12.1905			0.411645			0.205823	+	0.978589i	$0.434013\pi$
$878$	22.9996			0.776198
$879$		0			0
$880$		−	56.3108i		−	1.89824i
$881$	21.1850			0.713742			0.356871	−	0.934154i	$-0.383844\pi$
$881$	21.1850			0.713742			0.356871	+	0.934154i	$0.383844\pi$
$882$		0			0
$883$	−16.9329			−0.569836			−0.284918	−	0.958552i	$-0.591966\pi$
$883$	−16.9329			−0.569836			−0.284918	+	0.958552i	$0.591966\pi$
$884$		−	6.21603i		−	0.209068i
$885$		0			0
$886$			15.3127i			0.514439i
$887$	−43.4435			−1.45869			−0.729345	−	0.684146i	$-0.760175\pi$
$887$	−43.4435			−1.45869			−0.729345	+	0.684146i	$0.760175\pi$
$888$		0			0
$889$	35.5202			1.19131
$890$	−71.1791			−2.38593
$891$		0			0
$892$	10.5600			0.353577
$893$			2.08655i			0.0698238i
$894$		0			0
$895$		−	36.7388i		−	1.22804i
$896$			13.0931i			0.437409i
$897$		0			0
$898$	−19.7348			−0.658560
$899$			15.7397i			0.524950i
$900$		0			0
$901$	−35.4784			−1.18196
$902$	−31.4609			−1.04753
$903$		0			0
$904$			50.0939i			1.66610i
$905$	−80.6687			−2.68152
$906$		0			0
$907$	55.5868			1.84573			0.922865	−	0.385125i	$-0.125842\pi$
$907$	55.5868			1.84573			0.922865	+	0.385125i	$0.125842\pi$
$908$	2.02820			0.0673081
$909$		0			0
$910$	−45.6227			−1.51238
$911$			15.4595i			0.512196i	0.966651	+	0.256098i	$0.0824370\pi$
$911$			15.4595i			0.512196i	−0.966651	+	0.256098i	$0.917563\pi$
$912$		0			0
$913$			3.13482i			0.103747i
$914$	31.5498			1.04357
$915$		0			0
$916$	3.85509			0.127376
$917$	−23.3528			−0.771177
$918$		0			0
$919$	2.76919			0.0913472			0.0456736	−	0.998956i	$-0.485457\pi$
$919$	2.76919			0.0913472			0.0456736	+	0.998956i	$0.485457\pi$
$920$		−	49.8709i		−	1.64420i
$921$		0			0
$922$	−6.78308			−0.223389
$923$			16.0093i			0.526952i
$924$		0			0
$925$		−	113.530i		−	3.73283i
$926$	37.8755			1.24467
$927$		0			0
$928$	−6.37244			−0.209186
$929$		−	46.8964i		−	1.53862i	−0.638875	−	0.769310i	$-0.720600\pi$
$929$		−	46.8964i		−	1.53862i	0.638875	−	0.769310i	$-0.279400\pi$
$930$		0			0
$931$	14.6736			0.480908
$932$			7.53250i			0.246735i
$933$		0			0
$934$	−40.0290			−1.30979
$935$		−	95.0392i		−	3.10811i
$936$		0			0
$937$		−	56.5656i		−	1.84792i	−0.382491	−	0.923959i	$-0.624934\pi$
$937$		−	56.5656i		−	1.84792i	0.382491	−	0.923959i	$-0.375066\pi$
$938$	21.1479			0.690504
$939$		0			0
$940$		−	1.95395i		−	0.0637308i
$941$	−25.5635			−0.833348			−0.416674	−	0.909056i	$-0.636804\pi$
$941$	−25.5635			−0.833348			−0.416674	+	0.909056i	$0.636804\pi$
$942$		0			0
$943$	−18.6073			−0.605936
$944$	1.34870			0.0438963
$945$		0			0
$946$			4.34120i			0.141145i
$947$		−	48.2481i		−	1.56785i	−0.620854	−	0.783926i	$-0.713214\pi$
$947$		−	48.2481i		−	1.56785i	0.620854	−	0.783926i	$-0.286786\pi$
$948$		0			0
$949$	37.5644			1.21939
$950$			48.3928i			1.57007i
$951$		0			0
$952$			43.9695i			1.42506i
$953$	4.60833			0.149279			0.0746393	−	0.997211i	$-0.476219\pi$
$953$	4.60833			0.149279			0.0746393	+	0.997211i	$0.476219\pi$
$954$		0			0
$955$			77.1937i			2.49793i
$956$			7.44082i			0.240653i
$957$		0			0
$958$			46.6772i			1.50807i
$959$			43.2268i			1.39586i
$960$		0			0
$961$	34.8222			1.12330
$962$			23.3804i			0.753814i
$963$		0			0
$964$			9.11285i			0.293505i
$965$		−	108.612i		−	3.49635i
$966$		0			0
$967$	15.3965			0.495119			0.247560	−	0.968873i	$-0.420371\pi$
$967$	15.3965			0.495119			0.247560	+	0.968873i	$0.420371\pi$
$968$		−	51.4094i		−	1.65236i
$969$		0			0
$970$			52.7930i			1.69508i
$971$	12.3162			0.395244			0.197622	−	0.980278i	$-0.436678\pi$
$971$	12.3162			0.395244			0.197622	+	0.980278i	$0.436678\pi$
$972$		0			0
$973$			17.7589i			0.569324i
$974$	−0.0716754			−0.00229663
$975$		0			0
$976$	−13.8974			−0.444846
$977$	18.6761			0.597502			0.298751	−	0.954331i	$-0.403430\pi$
$977$	18.6761			0.597502			0.298751	+	0.954331i	$0.403430\pi$
$978$		0			0
$979$	−71.9656			−2.30003
$980$	−13.7411			−0.438943
$981$		0			0
$982$	16.3025			0.520235
$983$	−26.4484			−0.843571			−0.421786	−	0.906696i	$-0.638597\pi$
$983$	−26.4484			−0.843571			−0.421786	+	0.906696i	$0.638597\pi$
$984$		0			0
$985$	−85.9268			−2.73786
$986$	9.39294			0.299132
$987$		0			0
$988$			4.31008i			0.137122i
$989$			2.56756i			0.0816437i
$990$		0			0
$991$		−	30.0360i		−	0.954126i	−0.878869	−	0.477063i	$-0.841701\pi$
$991$		−	30.0360i		−	0.954126i	0.878869	−	0.477063i	$-0.158299\pi$
$992$			26.6490i			0.846106i
$993$		0			0
$994$		−	26.2606i		−	0.832937i
$995$	−18.3835			−0.582795
$996$		0			0
$997$	35.7569			1.13243			0.566216	−	0.824257i	$-0.308407\pi$
$997$	35.7569			1.13243			0.566216	+	0.824257i	$0.308407\pi$
$998$	21.8601			0.691970
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.c.a.1502.2	yes	56
3.2	odd	2	inner	1503.2.c.a.1502.55	yes	56
167.166	odd	2	inner	1503.2.c.a.1502.56	yes	56
501.500	even	2	inner	1503.2.c.a.1502.1	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.2.c.a.1502.1	✓	56	501.500	even	2	inner
1503.2.c.a.1502.2	yes	56	1.1	even	1	trivial
1503.2.c.a.1502.55	yes	56	3.2	odd	2	inner
1503.2.c.a.1502.56	yes	56	167.166	odd	2	inner

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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.18160i q^{2} +0.603817 q^{4} -4.40626 q^{5} -3.48779 q^{7} +3.07668i q^{8} +O(q^{10})\)	\(q+1.18160i q^{2} +0.603817 q^{4} -4.40626 q^{5} -3.48779 q^{7} +3.07668i q^{8} -5.20644i q^{10} -5.26397i q^{11} +2.51240i q^{13} -4.12118i q^{14} -2.42777 q^{16} -4.09750 q^{17} +2.84113 q^{19} -2.66057 q^{20} +6.21992 q^{22} +3.67871 q^{23} +14.4151 q^{25} -2.96866 q^{26} -2.10599 q^{28} +1.94004i q^{29} +8.11309 q^{31} +3.28469i q^{32} -4.84162i q^{34} +15.3681 q^{35} -7.87574i q^{37} +3.35709i q^{38} -13.5566i q^{40} -5.05809 q^{41} +0.697951i q^{43} -3.17848i q^{44} +4.34677i q^{46} +0.734409i q^{47} +5.16471 q^{49} +17.0329i q^{50} +1.51703i q^{52} +8.65853 q^{53} +23.1944i q^{55} -10.7308i q^{56} -2.29236 q^{58} -0.555528 q^{59} +5.72436 q^{61} +9.58644i q^{62} -8.73674 q^{64} -11.0703i q^{65} +5.13152i q^{67} -2.47414 q^{68} +18.1590i q^{70} +6.37211 q^{71} -14.9516i q^{73} +9.30599 q^{74} +1.71552 q^{76} +18.3596i q^{77} +5.59514i q^{79} +10.6974 q^{80} -5.97665i q^{82} -0.595523 q^{83} +18.0546 q^{85} -0.824700 q^{86} +16.1955 q^{88} -13.6713i q^{89} -8.76274i q^{91} +2.22127 q^{92} -0.867779 q^{94} -12.5188 q^{95} -10.1399 q^{97} +6.10263i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 48 q^{4}+O(q^{10}) \) 56 * q - 48 * q^4	\( 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) \) 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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