LMFDB - Embedded newform 1881.2.h.g.208.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1881,2,Mod(208,1881)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1881, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1881.208"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1881 = 3^{2} \cdot 11 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1881.h (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,24,0,0,0,0,0,0,-10,0,0,0,0,-16,0,0,0,-104,0,0,20,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 627)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			208.3
Character	$\chi$	$=$	1881.208
Dual form			1881.2.h.g.208.4

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29042 q^{2} +3.24604 q^{4} -2.24604 q^{5} -2.78846i q^{7} -2.85397 q^{8} +5.14439 q^{10} +(3.31640 - 0.0382776i) q^{11} +6.74367 q^{13} +6.38676i q^{14} +0.0447120 q^{16} -6.10357i q^{17} +(-0.994366 - 4.24396i) q^{19} -7.29076 q^{20} +(-7.59597 + 0.0876720i) q^{22} +5.56652 q^{23} +0.0447120 q^{25} -15.4459 q^{26} -9.05147i q^{28} +3.42856 q^{29} +6.66253i q^{31} +5.60553 q^{32} +13.9798i q^{34} +6.26301i q^{35} -1.10091i q^{37} +(2.27752 + 9.72048i) q^{38} +6.41014 q^{40} -4.69431 q^{41} +7.61378i q^{43} +(10.7652 - 0.124251i) q^{44} -12.7497 q^{46} +5.34205 q^{47} -0.775527 q^{49} -0.102409 q^{50} +21.8902 q^{52} +11.4877i q^{53} +(-7.44879 + 0.0859732i) q^{55} +7.95819i q^{56} -7.85286 q^{58} -11.9423i q^{59} -9.96124i q^{61} -15.2600i q^{62} -12.9285 q^{64} -15.1466 q^{65} -6.39166i q^{67} -19.8124i q^{68} -14.3450i q^{70} +5.29076i q^{71} -2.27029i q^{73} +2.52154i q^{74} +(-3.22776 - 13.7761i) q^{76} +(-0.106736 - 9.24767i) q^{77} +7.84997 q^{79} -0.100425 q^{80} +10.7520 q^{82} +15.4616i q^{83} +13.7089i q^{85} -17.4388i q^{86} +(-9.46491 + 0.109243i) q^{88} +4.77843i q^{89} -18.8045i q^{91} +18.0692 q^{92} -12.2356 q^{94} +(2.23339 + 9.53213i) q^{95} +14.8414i q^{97} +1.77629 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q + 24 q^{4} - 10 q^{11} - 16 q^{16} - 104 q^{20} + 20 q^{23} - 16 q^{25} + 32 q^{26} + 4 q^{38} + 36 q^{44} + 20 q^{47} - 24 q^{49} - 46 q^{55} + 60 q^{58} - 68 q^{64} + 22 q^{77} - 12 q^{80} + 56 q^{82}+ \cdots + 120 q^{92}+O(q^{100})$ 24 * q + 24 * q^4 - 10 * q^11 - 16 * q^16 - 104 * q^20 + 20 * q^23 - 16 * q^25 + 32 * q^26 + 4 * q^38 + 36 * q^44 + 20 * q^47 - 24 * q^49 - 46 * q^55 + 60 * q^58 - 68 * q^64 + 22 * q^77 - 12 * q^80 + 56 * q^82 + 120 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1881\mathbb{Z}\right)^\times$ .

$n$	$343$	$496$	$1046$
$\chi(n)$	$-1$	$-1$	$1$

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

)

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

$2$	−2.29042			−1.61957			−0.809787	−	0.586724i	$-0.800418\pi$
$2$	−2.29042			−1.61957			−0.809787	+	0.586724i	$0.800418\pi$
$3$		0			0
$3$		0			0
$4$	3.24604			1.62302
$5$	−2.24604			−1.00446			−0.502231	−	0.864734i	$-0.667487\pi$
$5$	−2.24604			−1.00446			−0.502231	+	0.864734i	$0.667487\pi$
$6$		0			0
$7$		−	2.78846i		−	1.05394i	−0.849884	−	0.526970i	$-0.823328\pi$
$7$		−	2.78846i		−	1.05394i	0.849884	−	0.526970i	$-0.176672\pi$
$8$	−2.85397			−1.00903
$9$		0			0
$10$	5.14439			1.62680
$11$	3.31640	−	0.0382776i	0.999933	−	0.0115411i
$11$	3.31640	−	0.0382776i	0.999933	−	0.0115411i
$12$		0			0
$13$	6.74367			1.87036			0.935178	−	0.354177i	$-0.115239\pi$
$13$	6.74367			1.87036			0.935178	+	0.354177i	$0.115239\pi$
$14$			6.38676i			1.70693i
$15$		0			0
$16$	0.0447120			0.0111780
$17$		−	6.10357i		−	1.48033i	−0.672424	−	0.740166i	$-0.734747\pi$
$17$		−	6.10357i		−	1.48033i	0.672424	−	0.740166i	$-0.265253\pi$
$18$		0			0
$19$	−0.994366	−	4.24396i	−0.228123	−	0.973632i
$19$	−0.994366	−	4.24396i	−0.228123	−	0.973632i
$20$	−7.29076			−1.63026
$21$		0			0
$22$	−7.59597	+	0.0876720i	−1.61947	+	0.0186917i
$23$	5.56652			1.16070			0.580350	−	0.814367i	$-0.302916\pi$
$23$	5.56652			1.16070			0.580350	+	0.814367i	$0.302916\pi$
$24$		0			0
$25$	0.0447120			0.00894240
$26$	−15.4459			−3.02918
$27$		0			0
$28$		−	9.05147i		−	1.71057i
$29$	3.42856			0.636668			0.318334	−	0.947979i	$-0.396877\pi$
$29$	3.42856			0.636668			0.318334	+	0.947979i	$0.396877\pi$
$30$		0			0
$31$			6.66253i			1.19663i	0.801262	+	0.598313i	$0.204162\pi$
$31$			6.66253i			1.19663i	−0.801262	+	0.598313i	$0.795838\pi$
$32$	5.60553			0.990927
$33$		0			0
$34$			13.9798i			2.39751i
$35$			6.26301i			1.05864i
$36$		0			0
$37$		−	1.10091i		−	0.180988i	−0.995897	−	0.0904940i	$-0.971155\pi$
$37$		−	1.10091i		−	0.180988i	0.995897	−	0.0904940i	$-0.0288446\pi$
$38$	2.27752	+	9.72048i	0.369463	+	1.57687i
$39$		0			0
$40$	6.41014			1.01353
$41$	−4.69431			−0.733128			−0.366564	−	0.930393i	$-0.619466\pi$
$41$	−4.69431			−0.733128			−0.366564	+	0.930393i	$0.619466\pi$
$42$		0			0
$43$			7.61378i			1.16109i	0.814228	+	0.580545i	$0.197161\pi$
$43$			7.61378i			1.16109i	−0.814228	+	0.580545i	$0.802839\pi$
$44$	10.7652	−	0.124251i	1.62291	−	0.0187315i
$45$		0			0
$46$	−12.7497			−1.87984
$47$	5.34205			0.779218			0.389609	−	0.920980i	$-0.372610\pi$
$47$	5.34205			0.779218			0.389609	+	0.920980i	$0.372610\pi$
$48$		0			0
$49$	−0.775527			−0.110790
$50$	−0.102409			−0.0144829
$51$		0			0
$52$	21.8902			3.03563
$53$			11.4877i			1.57795i	0.614423	+	0.788976i	$0.289389\pi$
$53$			11.4877i			1.57795i	−0.614423	+	0.788976i	$0.710611\pi$
$54$		0			0
$55$	−7.44879	+	0.0859732i	−1.00439	+	0.0115926i
$56$			7.95819i			1.06346i
$57$		0			0
$58$	−7.85286			−1.03113
$59$		−	11.9423i		−	1.55475i	−0.629035	−	0.777377i	$-0.716550\pi$
$59$		−	11.9423i		−	1.55475i	0.629035	−	0.777377i	$-0.283450\pi$
$60$		0			0
$61$		−	9.96124i		−	1.27541i	−0.770282	−	0.637703i	$-0.779885\pi$
$61$		−	9.96124i		−	1.27541i	0.770282	−	0.637703i	$-0.220115\pi$
$62$		−	15.2600i		−	1.93803i
$63$		0			0
$64$	−12.9285			−1.61606
$65$	−15.1466			−1.87870
$66$		0			0
$67$		−	6.39166i		−	0.780866i	−0.920631	−	0.390433i	$-0.872325\pi$
$67$		−	6.39166i		−	0.780866i	0.920631	−	0.390433i	$-0.127675\pi$
$68$		−	19.8124i		−	2.40261i
$69$		0			0
$70$		−	14.3450i		−	1.71455i
$71$			5.29076i			0.627897i	0.949440	+	0.313949i	$0.101652\pi$
$71$			5.29076i			0.627897i	−0.949440	+	0.313949i	$0.898348\pi$
$72$		0			0
$73$		−	2.27029i		−	0.265718i	−0.991135	−	0.132859i	$-0.957584\pi$
$73$		−	2.27029i		−	0.265718i	0.991135	−	0.132859i	$-0.0424157\pi$
$74$			2.52154i			0.293123i
$75$		0			0
$76$	−3.22776	−	13.7761i	−0.370249	−	1.58023i
$77$	−0.106736	−	9.24767i	−0.0121637	−	1.05387i
$78$		0			0
$79$	7.84997			0.883190			0.441595	−	0.897214i	$-0.354413\pi$
$79$	7.84997			0.883190			0.441595	+	0.897214i	$0.354413\pi$
$80$	−0.100425			−0.0112279
$81$		0			0
$82$	10.7520			1.18736
$83$			15.4616i			1.69713i	0.529089	+	0.848566i	$0.322534\pi$
$83$			15.4616i			1.69713i	−0.529089	+	0.848566i	$0.677466\pi$
$84$		0			0
$85$			13.7089i			1.48694i
$86$		−	17.4388i		−	1.88047i
$87$		0			0
$88$	−9.46491	+	0.109243i	−1.00896	+	0.0116454i
$89$			4.77843i			0.506512i	0.967399	+	0.253256i	$0.0815015\pi$
$89$			4.77843i			0.506512i	−0.967399	+	0.253256i	$0.918498\pi$
$90$		0			0
$91$		−	18.8045i		−	1.97124i
$92$	18.0692			1.88384
$93$		0			0
$94$	−12.2356			−1.26200
$95$	2.23339	+	9.53213i	0.229141	+	0.977976i
$96$		0			0
$97$			14.8414i			1.50691i	0.657497	+	0.753457i	$0.271615\pi$
$97$			14.8414i			1.50691i	−0.657497	+	0.753457i	$0.728385\pi$
$98$	1.77629			0.179432
$99$		0			0
$100$	0.145137			0.0145137
$101$		−	1.53119i		−	0.152359i	−0.997094	−	0.0761795i	$-0.975728\pi$
$101$		−	1.53119i		−	0.152359i	0.997094	−	0.0761795i	$-0.0242722\pi$
$102$		0			0
$103$		−	7.21190i		−	0.710610i	−0.934750	−	0.355305i	$-0.884377\pi$
$103$		−	7.21190i		−	0.710610i	0.934750	−	0.355305i	$-0.115623\pi$
$104$	−19.2462			−1.88725
$105$		0			0
$106$		−	26.3116i		−	2.55561i
$107$	−3.70799			−0.358465			−0.179232	−	0.983807i	$-0.557361\pi$
$107$	−3.70799			−0.358465			−0.179232	+	0.983807i	$0.557361\pi$
$108$		0			0
$109$	9.13650			0.875118			0.437559	−	0.899190i	$-0.355843\pi$
$109$	9.13650			0.875118			0.437559	+	0.899190i	$0.355843\pi$
$110$	17.0609	−	0.196915i	1.62669	−	0.0187751i
$111$		0			0
$112$		−	0.124678i		−	0.0117809i
$113$		−	17.5728i		−	1.65311i	−0.562859	−	0.826553i	$-0.690299\pi$
$113$		−	17.5728i		−	1.65311i	0.562859	−	0.826553i	$-0.309701\pi$
$114$		0			0
$115$	−12.5027			−1.16588
$116$	11.1293			1.03333
$117$		0			0
$118$			27.3529i			2.51804i
$119$	−17.0196			−1.56018
$120$		0			0
$121$	10.9971	−	0.253888i	0.999734	−	0.0230807i
$122$			22.8155i			2.06562i
$123$		0			0
$124$			21.6269i			1.94215i
$125$	11.1298			0.995479
$126$		0			0
$127$	−11.1998			−0.993825			−0.496912	−	0.867801i	$-0.665533\pi$
$127$	−11.1998			−0.993825			−0.496912	+	0.867801i	$0.665533\pi$
$128$	18.4006			1.62640
$129$		0			0
$130$	34.6921			3.04270
$131$		−	13.4846i		−	1.17815i	−0.808077	−	0.589077i	$-0.799492\pi$
$131$		−	13.4846i		−	1.17815i	0.808077	−	0.589077i	$-0.200508\pi$
$132$		0			0
$133$	−11.8341	+	2.77275i	−1.02615	+	0.240428i
$134$			14.6396i			1.26467i
$135$		0			0
$136$			17.4194i			1.49370i
$137$	5.49582			0.469540			0.234770	−	0.972051i	$-0.424566\pi$
$137$	5.49582			0.469540			0.234770	+	0.972051i	$0.424566\pi$
$138$		0			0
$139$		−	5.91569i		−	0.501762i	−0.968018	−	0.250881i	$-0.919280\pi$
$139$		−	5.91569i		−	0.501762i	0.968018	−	0.250881i	$-0.0807203\pi$
$140$			20.3300i			1.71820i
$141$		0			0
$142$		−	12.1181i		−	1.01693i
$143$	22.3647	−	0.258131i	1.87023	−	0.0215860i
$144$		0			0
$145$	−7.70070			−0.639508
$146$			5.19994i			0.430350i
$147$		0			0
$148$		−	3.57359i		−	0.293747i
$149$			4.03938i			0.330919i	0.986217	+	0.165459i	$0.0529106\pi$
$149$			4.03938i			0.330919i	−0.986217	+	0.165459i	$0.947089\pi$
$150$		0			0
$151$	−11.3133			−0.920663			−0.460332	−	0.887747i	$-0.652269\pi$
$151$	−11.3133			−0.920663			−0.460332	+	0.887747i	$0.652269\pi$
$152$	2.83789	+	12.1121i	0.230183	+	0.982424i
$153$		0			0
$154$	0.244470	+	21.1811i	0.0197000	+	1.70682i
$155$		−	14.9643i		−	1.20196i
$156$		0			0
$157$	−19.3141			−1.54143			−0.770715	−	0.637180i	$-0.780101\pi$
$157$	−19.3141			−1.54143			−0.770715	+	0.637180i	$0.780101\pi$
$158$	−17.9798			−1.43039
$159$		0			0
$160$	−12.5903			−0.995347
$161$		−	15.5220i		−	1.22331i
$162$		0			0
$163$	20.5468			1.60935			0.804673	−	0.593718i	$-0.202340\pi$
$163$	20.5468			1.60935			0.804673	+	0.593718i	$0.202340\pi$
$164$	−15.2379			−1.18988
$165$		0			0
$166$		−	35.4137i		−	2.74863i
$167$	−7.67550			−0.593948			−0.296974	−	0.954886i	$-0.595978\pi$
$167$	−7.67550			−0.593948			−0.296974	+	0.954886i	$0.595978\pi$
$168$		0			0
$169$	32.4770			2.49823
$170$		−	31.3991i		−	2.40820i
$171$		0			0
$172$			24.7147i			1.88448i
$173$	−7.53723			−0.573045			−0.286522	−	0.958074i	$-0.592499\pi$
$173$	−7.53723			−0.573045			−0.286522	+	0.958074i	$0.592499\pi$
$174$		0			0
$175$		−	0.124678i		−	0.00942475i
$176$	0.148283	−	0.00171147i	0.0111773	−	0.000129007i
$177$		0			0
$178$		−	10.9446i		−	0.820334i
$179$		−	19.6996i		−	1.47242i	−0.676753	−	0.736210i	$-0.736613\pi$
$179$		−	19.6996i		−	1.47242i	0.676753	−	0.736210i	$-0.263387\pi$
$180$		0			0
$181$		−	12.1898i		−	0.906064i	−0.891494	−	0.453032i	$-0.850342\pi$
$181$		−	12.1898i		−	0.906064i	0.891494	−	0.453032i	$-0.149658\pi$
$182$			43.0702i			3.19258i
$183$		0			0
$184$	−15.8867			−1.17118
$185$			2.47268i			0.181795i
$186$		0			0
$187$	−0.233630	−	20.2419i	−0.0170847	−	1.48023i
$188$	17.3405			1.26469
$189$		0			0
$190$	−5.11541	−	21.8326i	−0.371111	−	1.58390i
$191$	2.09150			0.151335			0.0756677	−	0.997133i	$-0.475891\pi$
$191$	2.09150			0.151335			0.0756677	+	0.997133i	$0.475891\pi$
$192$		0			0
$193$	10.9658			0.789335			0.394667	−	0.918824i	$-0.370860\pi$
$193$	10.9658			0.789335			0.394667	+	0.918824i	$0.370860\pi$
$194$		−	33.9931i		−	2.44056i
$195$		0			0
$196$	−2.51740			−0.179814
$197$		−	7.66190i		−	0.545888i	−0.962030	−	0.272944i	$-0.912003\pi$
$197$		−	7.66190i		−	0.545888i	0.962030	−	0.272944i	$-0.0879974\pi$
$198$		0			0
$199$	14.7703			1.04704			0.523518	−	0.852014i	$-0.324619\pi$
$199$	14.7703			1.04704			0.523518	+	0.852014i	$0.324619\pi$
$200$	−0.127607			−0.00902315
$201$		0			0
$202$			3.50707i			0.246757i
$203$		−	9.56042i		−	0.671010i
$204$		0			0
$205$	10.5436			0.736398
$206$			16.5183i			1.15089i
$207$		0			0
$208$	0.301523			0.0209068
$209$	−3.46017	−	14.0366i	−0.239345	−	0.970935i
$210$		0			0
$211$	3.83576			0.264065			0.132032	−	0.991245i	$-0.457850\pi$
$211$	3.83576			0.264065			0.132032	+	0.991245i	$0.457850\pi$
$212$			37.2895i			2.56105i
$213$		0			0
$214$	8.49286			0.580560
$215$		−	17.1009i		−	1.16627i
$216$		0			0
$217$	18.5782			1.26117
$218$	−20.9265			−1.41732
$219$		0			0
$220$	−24.1791	+	0.279073i	−1.63015	+	0.0188151i
$221$		−	41.1604i		−	2.76875i
$222$		0			0
$223$			6.51940i			0.436571i	0.975885	+	0.218285i	$0.0700464\pi$
$223$			6.51940i			0.436571i	−0.975885	+	0.218285i	$0.929954\pi$
$224$		−	15.6308i		−	1.04438i
$225$		0			0
$226$			40.2491i			2.67733i
$227$	−27.5168			−1.82635			−0.913176	−	0.407566i	$-0.866378\pi$
$227$	−27.5168			−1.82635			−0.913176	+	0.407566i	$0.866378\pi$
$228$		0			0
$229$	−5.10277			−0.337200			−0.168600	−	0.985685i	$-0.553925\pi$
$229$	−5.10277			−0.337200			−0.168600	+	0.985685i	$0.553925\pi$
$230$	28.6364			1.88823
$231$		0			0
$232$	−9.78501			−0.642417
$233$		−	2.48713i		−	0.162937i	−0.996676	−	0.0814686i	$-0.974039\pi$
$233$		−	2.48713i		−	0.162937i	0.996676	−	0.0814686i	$-0.0259610\pi$
$234$		0			0
$235$	−11.9985			−0.782695
$236$		−	38.7652i		−	2.52340i
$237$		0			0
$238$	38.9820			2.52683
$239$			9.51458i			0.615447i	0.951476	+	0.307723i	$0.0995672\pi$
$239$			9.51458i			0.615447i	−0.951476	+	0.307723i	$0.900433\pi$
$240$		0			0
$241$	12.4989			0.805125			0.402562	−	0.915393i	$-0.368120\pi$
$241$	12.4989			0.805125			0.402562	+	0.915393i	$0.368120\pi$
$242$	−25.1880	+	0.581511i	−1.61914	+	0.0373810i
$243$		0			0
$244$		−	32.3346i		−	2.07001i
$245$	1.74187			0.111284
$246$		0			0
$247$	−6.70568	−	28.6199i	−0.426672	−	1.82104i
$248$		−	19.0147i		−	1.20743i
$249$		0			0
$250$	−25.4919			−1.61225
$251$	−11.3958			−0.719294			−0.359647	−	0.933088i	$-0.617103\pi$
$251$	−11.3958			−0.719294			−0.359647	+	0.933088i	$0.617103\pi$
$252$		0			0
$253$	18.4608	−	0.213073i	1.16062	−	0.0133958i
$254$	25.6524			1.60957
$255$		0			0
$256$	−16.2883			−1.01802
$257$		−	15.1013i		−	0.941992i	−0.882135	−	0.470996i	$-0.843895\pi$
$257$		−	15.1013i		−	0.941992i	0.882135	−	0.470996i	$-0.156105\pi$
$258$		0			0
$259$	−3.06984			−0.190750
$260$	−49.1664			−3.04917
$261$		0			0
$262$			30.8854i			1.90811i
$263$			16.7210i			1.03106i	0.856871	+	0.515530i	$0.172405\pi$
$263$			16.7210i			1.03106i	−0.856871	+	0.515530i	$0.827595\pi$
$264$		0			0
$265$		−	25.8018i		−	1.58499i
$266$	27.1052	−	6.35078i	1.66193	−	0.389392i
$267$		0			0
$268$		−	20.7476i		−	1.26736i
$269$		−	3.18586i		−	0.194246i	−0.995272	−	0.0971228i	$-0.969036\pi$
$269$		−	3.18586i		−	0.194246i	0.995272	−	0.0971228i	$-0.0309639\pi$
$270$		0			0
$271$		−	31.9237i		−	1.93922i	−0.244646	−	0.969612i	$-0.578672\pi$
$271$		−	31.9237i		−	1.93922i	0.244646	−	0.969612i	$-0.421328\pi$
$272$		−	0.272903i		−	0.0165472i
$273$		0			0
$274$	−12.5878			−0.760455
$275$	0.148283	−	0.00171147i	0.00894180	−	0.000103205i
$276$		0			0
$277$		−	9.26180i		−	0.556488i	−0.960510	−	0.278244i	$-0.910248\pi$
$277$		−	9.26180i		−	0.556488i	0.960510	−	0.278244i	$-0.0897523\pi$
$278$			13.5494i			0.812642i
$279$		0			0
$280$		−	17.8744i		−	1.06820i
$281$	6.24478			0.372533			0.186266	−	0.982499i	$-0.440361\pi$
$281$	6.24478			0.372533			0.186266	+	0.982499i	$0.440361\pi$
$282$		0			0
$283$			13.5842i			0.807497i	0.914870	+	0.403748i	$0.132293\pi$
$283$			13.5842i			0.807497i	−0.914870	+	0.403748i	$0.867707\pi$
$284$			17.1740i			1.01909i
$285$		0			0
$286$	−51.2247	+	0.591230i	−3.02898	+	0.0349602i
$287$			13.0899i			0.772673i
$288$		0			0
$289$	−20.2535			−1.19138
$290$	17.6379			1.03573
$291$		0			0
$292$		−	7.36947i		−	0.431266i
$293$	16.8099			0.982045			0.491022	−	0.871147i	$-0.336623\pi$
$293$	16.8099			0.982045			0.491022	+	0.871147i	$0.336623\pi$
$294$		0			0
$295$			26.8229i			1.56169i
$296$			3.14195i			0.182622i
$297$		0			0
$298$		−	9.25188i		−	0.535947i
$299$	37.5388			2.17092
$300$		0			0
$301$	21.2308			1.22372
$302$	25.9122			1.49108
$303$		0			0
$304$	−0.0444601	−	0.189756i	−0.00254996	−	0.0108833i
$305$			22.3734i			1.28110i
$306$		0			0
$307$	−3.49914			−0.199706			−0.0998531	−	0.995002i	$-0.531837\pi$
$307$	−3.49914			−0.199706			−0.0998531	+	0.995002i	$0.531837\pi$
$308$	−0.346469	−	30.0183i	−0.0197419	−	1.71045i
$309$		0			0
$310$			34.2747i			1.94667i
$311$	17.5166			0.993273			0.496636	−	0.867959i	$-0.334568\pi$
$311$	17.5166			0.993273			0.496636	+	0.867959i	$0.334568\pi$
$312$		0			0
$313$	9.66162			0.546107			0.273054	−	0.961999i	$-0.411966\pi$
$313$	9.66162			0.546107			0.273054	+	0.961999i	$0.411966\pi$
$314$	44.2374			2.49646
$315$		0			0
$316$	25.4813			1.43344
$317$			10.6406i			0.597633i	0.954311	+	0.298816i	$0.0965918\pi$
$317$			10.6406i			0.597633i	−0.954311	+	0.298816i	$0.903408\pi$
$318$		0			0
$319$	11.3705	−	0.131237i	0.636626	−	0.00734787i
$320$	29.0379			1.62327
$321$		0			0
$322$			35.5521i			1.98124i
$323$	−25.9033	+	6.06918i	−1.44130	+	0.337698i
$324$		0			0
$325$	0.301523			0.0167255
$326$	−47.0608			−2.60646
$327$		0			0
$328$	13.3974			0.739748
$329$		−	14.8961i		−	0.821249i
$330$		0			0
$331$			8.17528i			0.449354i	0.974433	+	0.224677i	$0.0721327\pi$
$331$			8.17528i			0.449354i	−0.974433	+	0.224677i	$0.927867\pi$
$332$			50.1891i			2.75448i
$333$		0			0
$334$	17.5802			0.961944
$335$			14.3560i			0.784349i
$336$		0			0
$337$	−8.84553			−0.481847			−0.240923	−	0.970544i	$-0.577450\pi$
$337$	−8.84553			−0.481847			−0.240923	+	0.970544i	$0.577450\pi$
$338$	−74.3862			−4.04608
$339$		0			0
$340$			44.4996i			2.41333i
$341$	0.255026	+	22.0957i	0.0138104	+	1.19655i
$342$		0			0
$343$		−	17.3567i		−	0.937174i
$344$		−	21.7295i		−	1.17158i
$345$		0			0
$346$	17.2634			0.928089
$347$		−	9.42824i		−	0.506134i	−0.967449	−	0.253067i	$-0.918561\pi$
$347$		−	9.42824i		−	0.506134i	0.967449	−	0.253067i	$-0.0814394\pi$
$348$		0			0
$349$			0.0334811i			0.00179220i	1.00000		0.000896102i	$0.000285238\pi$
$349$			0.0334811i			0.00179220i	−1.00000		0.000896102i	$0.999715\pi$
$350$			0.285565i			0.0152641i
$351$		0			0
$352$	18.5902	−	0.214566i	0.990861	−	0.0114364i
$353$	−24.6956			−1.31442			−0.657208	−	0.753709i	$-0.728263\pi$
$353$	−24.6956			−1.31442			−0.657208	+	0.753709i	$0.728263\pi$
$354$		0			0
$355$		−	11.8833i		−	0.630698i
$356$			15.5110i			0.822080i
$357$		0			0
$358$			45.1205i			2.38470i
$359$		−	34.3641i		−	1.81367i	−0.421488	−	0.906834i	$-0.638492\pi$
$359$		−	34.3641i		−	1.81367i	0.421488	−	0.906834i	$-0.361508\pi$
$360$		0			0
$361$	−17.0225	+	8.44011i	−0.895920	+	0.444216i
$362$			27.9199i			1.46744i
$363$		0			0
$364$		−	61.0401i		−	3.19937i
$365$			5.09918i			0.266903i
$366$		0			0
$367$	−10.6023			−0.553436			−0.276718	−	0.960951i	$-0.589247\pi$
$367$	−10.6023			−0.553436			−0.276718	+	0.960951i	$0.589247\pi$
$368$	0.248890			0.0129743
$369$		0			0
$370$		−	5.66350i		−	0.294431i
$371$	32.0329			1.66307
$372$		0			0
$373$	−15.7101			−0.813439			−0.406719	−	0.913553i	$-0.633327\pi$
$373$	−15.7101			−0.813439			−0.406719	+	0.913553i	$0.633327\pi$
$374$	0.535112	+	46.3625i	0.0276700	+	2.39735i
$375$		0			0
$376$	−15.2460			−0.786255
$377$	23.1211			1.19080
$378$		0			0
$379$			14.6023i			0.750070i	0.927011	+	0.375035i	$0.122369\pi$
$379$			14.6023i			0.750070i	−0.927011	+	0.375035i	$0.877631\pi$
$380$	7.24968	+	30.9417i	0.371901	+	1.58728i
$381$		0			0
$382$	−4.79042			−0.245099
$383$		−	5.90623i		−	0.301794i	−0.988549	−	0.150897i	$-0.951784\pi$
$383$		−	5.90623i		−	0.301794i	0.988549	−	0.150897i	$-0.0482162\pi$
$384$		0			0
$385$	0.239733	+	20.7707i	0.0122179	+	1.05857i
$386$	−25.1163			−1.27839
$387$		0			0
$388$			48.1758i			2.44575i
$389$	−10.5114			−0.532947			−0.266474	−	0.963842i	$-0.585859\pi$
$389$	−10.5114			−0.532947			−0.266474	+	0.963842i	$0.585859\pi$
$390$		0			0
$391$		−	33.9757i		−	1.71822i
$392$	2.21333			0.111790
$393$		0			0
$394$			17.5490i			0.884106i
$395$	−17.6314			−0.887131
$396$		0			0
$397$	3.57328			0.179338			0.0896690	−	0.995972i	$-0.471419\pi$
$397$	3.57328			0.179338			0.0896690	+	0.995972i	$0.471419\pi$
$398$	−33.8302			−1.69575
$399$		0			0
$400$	0.00199916			9.99582e−5
$401$			7.63438i			0.381243i	0.981664	+	0.190621i	$0.0610503\pi$
$401$			7.63438i			0.381243i	−0.981664	+	0.190621i	$0.938950\pi$
$402$		0			0
$403$			44.9299i			2.23812i
$404$		−	4.97031i		−	0.247282i
$405$		0			0
$406$			21.8974i			1.08675i
$407$	−0.0421401	−	3.65105i	−0.00208881	−	0.180976i
$408$		0			0
$409$	−6.26937			−0.310000			−0.155000	−	0.987914i	$-0.549538\pi$
$409$	−6.26937			−0.310000			−0.155000	+	0.987914i	$0.549538\pi$
$410$	−24.1494			−1.19265
$411$		0			0
$412$		−	23.4101i		−	1.15334i
$413$	−33.3006			−1.63862
$414$		0			0
$415$		−	34.7275i		−	1.70470i
$416$	37.8018			1.85339
$417$		0			0
$418$	7.92526	+	32.1499i	0.387637	+	1.57250i
$419$	−34.3573			−1.67846			−0.839232	−	0.543774i	$-0.816995\pi$
$419$	−34.3573			−1.67846			−0.839232	+	0.543774i	$0.816995\pi$
$420$		0			0
$421$			23.3758i			1.13927i	0.821898	+	0.569634i	$0.192915\pi$
$421$			23.3758i			1.13927i	−0.821898	+	0.569634i	$0.807085\pi$
$422$	−8.78552			−0.427672
$423$		0			0
$424$		−	32.7855i		−	1.59220i
$425$		−	0.272903i		−	0.0132377i
$426$		0			0
$427$	−27.7766			−1.34420
$428$	−12.0363			−0.581796
$429$		0			0
$430$			39.1683i			1.88886i
$431$	0.830910			0.0400235			0.0200118	−	0.999800i	$-0.493630\pi$
$431$	0.830910			0.0400235			0.0200118	+	0.999800i	$0.493630\pi$
$432$		0			0
$433$			16.4190i			0.789048i	0.918886	+	0.394524i	$0.129091\pi$
$433$			16.4190i			0.789048i	−0.918886	+	0.394524i	$0.870909\pi$
$434$	−42.5520			−2.04256
$435$		0			0
$436$	29.6575			1.42034
$437$	−5.53517	−	23.6241i	−0.264783	−	1.13010i
$438$		0			0
$439$	−11.0019			−0.525091			−0.262545	−	0.964920i	$-0.584562\pi$
$439$	−11.0019			−0.525091			−0.262545	+	0.964920i	$0.584562\pi$
$440$	21.2586	−	0.245365i	1.01346	−	0.0116973i
$441$		0			0
$442$			94.2748i			4.48420i
$443$	12.7663			0.606547			0.303274	−	0.952904i	$-0.401920\pi$
$443$	12.7663			0.606547			0.303274	+	0.952904i	$0.401920\pi$
$444$		0			0
$445$		−	10.7326i		−	0.508772i
$446$		−	14.9322i		−	0.707059i
$447$		0			0
$448$			36.0505i			1.70323i
$449$			14.1682i			0.668639i	0.942460	+	0.334320i	$0.108507\pi$
$449$			14.1682i			0.668639i	−0.942460	+	0.334320i	$0.891493\pi$
$450$		0			0
$451$	−15.5682	+	0.179687i	−0.733079	+	0.00846112i
$452$		−	57.0419i		−	2.68303i
$453$		0			0
$454$	63.0251			2.95791
$455$			42.2357i			1.98004i
$456$		0			0
$457$		−	2.00949i		−	0.0940000i	−0.998895	−	0.0470000i	$-0.985034\pi$
$457$		−	2.00949i		−	0.0940000i	0.998895	−	0.0470000i	$-0.0149661\pi$
$458$	11.6875			0.546121
$459$		0			0
$460$	−40.5842			−1.89225
$461$		−	12.8744i		−	0.599618i	−0.953999	−	0.299809i	$-0.903077\pi$
$461$		−	12.8744i		−	0.599618i	0.953999	−	0.299809i	$-0.0969230\pi$
$462$		0			0
$463$	40.1121			1.86417			0.932085	−	0.362241i	$-0.117988\pi$
$463$	40.1121			1.86417			0.932085	+	0.362241i	$0.117988\pi$
$464$	0.153298			0.00711668
$465$		0			0
$466$			5.69658i			0.263889i
$467$	37.6605			1.74272			0.871359	−	0.490645i	$-0.163239\pi$
$467$	37.6605			1.74272			0.871359	+	0.490645i	$0.163239\pi$
$468$		0			0
$469$	−17.8229			−0.822986
$470$	27.4816			1.26763
$471$		0			0
$472$			34.0829i			1.56879i
$473$	0.291437	+	25.2504i	0.0134003	+	1.16101i
$474$		0			0
$475$	−0.0444601	−	0.189756i	−0.00203997	−	0.00870661i
$476$	−55.2463			−2.53221
$477$		0			0
$478$		−	21.7924i		−	0.996762i
$479$		−	14.1961i		−	0.648638i	−0.945948	−	0.324319i	$-0.894865\pi$
$479$		−	14.1961i		−	0.648638i	0.945948	−	0.324319i	$-0.105135\pi$
$480$		0			0
$481$		−	7.42415i		−	0.338512i
$482$	−28.6278			−1.30396
$483$		0			0
$484$	35.6970	−	0.824132i	1.62259	−	0.0374605i
$485$		−	33.3344i		−	1.51364i
$486$		0			0
$487$		−	10.8842i		−	0.493209i	−0.969116	−	0.246605i	$-0.920685\pi$
$487$		−	10.8842i		−	0.493209i	0.969116	−	0.246605i	$-0.0793149\pi$
$488$			28.4291i			1.28692i
$489$		0			0
$490$	−3.98962			−0.180233
$491$			16.6282i			0.750418i	0.926940	+	0.375209i	$0.122429\pi$
$491$			16.6282i			0.750418i	−0.926940	+	0.375209i	$0.877571\pi$
$492$		0			0
$493$		−	20.9265i		−	0.942480i
$494$	15.3588	+	65.5517i	0.691027	+	2.94931i
$495$		0			0
$496$			0.297895i			0.0133759i
$497$	14.7531			0.661766
$498$		0			0
$499$	−6.85347			−0.306803			−0.153402	−	0.988164i	$-0.549023\pi$
$499$	−6.85347			−0.306803			−0.153402	+	0.988164i	$0.549023\pi$
$500$	36.1278			1.61568
$501$		0			0
$502$	26.1011			1.16495
$503$			21.7210i			0.968493i	0.874932	+	0.484247i	$0.160906\pi$
$503$			21.7210i			0.968493i	−0.874932	+	0.484247i	$0.839094\pi$
$504$		0			0
$505$			3.43912i			0.153039i
$506$	−42.2832	+	0.488028i	−1.87972	+	0.0216955i
$507$		0			0
$508$	−36.3552			−1.61300
$509$			11.6075i			0.514493i	0.966346	+	0.257247i	$0.0828153\pi$
$509$			11.6075i			0.514493i	−0.966346	+	0.257247i	$0.917185\pi$
$510$		0			0
$511$	−6.33063			−0.280051
$512$	0.505848			0.0223555
$513$		0			0
$514$			34.5883i			1.52563i
$515$			16.1982i			0.713780i
$516$		0			0
$517$	17.7164	−	0.204481i	0.779166	−	0.00899306i
$518$	7.03123			0.308935
$519$		0			0
$520$	43.2278			1.89567
$521$			3.15874i			0.138387i	0.997603	+	0.0691936i	$0.0220426\pi$
$521$			3.15874i			0.138387i	−0.997603	+	0.0691936i	$0.977957\pi$
$522$		0			0
$523$	1.71142			0.0748352			0.0374176	−	0.999300i	$-0.488087\pi$
$523$	1.71142			0.0748352			0.0374176	+	0.999300i	$0.488087\pi$
$524$		−	43.7716i		−	1.91217i
$525$		0			0
$526$		−	38.2982i		−	1.66988i
$527$	40.6652			1.77140
$528$		0			0
$529$	7.98620			0.347226
$530$			59.0971i			2.56701i
$531$		0			0
$532$	−38.4141	+	9.00048i	−1.66546	+	0.390220i
$533$	−31.6568			−1.37121
$534$		0			0
$535$	8.32830			0.360064
$536$			18.2416i			0.787917i
$537$		0			0
$538$			7.29698i			0.314595i
$539$	−2.57196	+	0.0296853i	−0.110782	+	0.00127864i
$540$		0			0
$541$			3.60070i			0.154806i	0.997000	+	0.0774030i	$0.0246628\pi$
$541$			3.60070i			0.154806i	−0.997000	+	0.0774030i	$0.975337\pi$
$542$			73.1188i			3.14072i
$543$		0			0
$544$		−	34.2137i		−	1.46690i
$545$	−20.5210			−0.879022
$546$		0			0
$547$	−16.1074			−0.688701			−0.344351	−	0.938841i	$-0.611901\pi$
$547$	−16.1074			−0.688701			−0.344351	+	0.938841i	$0.611901\pi$
$548$	17.8397			0.762074
$549$		0			0
$550$	−0.339631	+	0.00391999i	−0.0144819	+	0.000167149i
$551$	−3.40925	−	14.5507i	−0.145239	−	0.619881i
$552$		0			0
$553$		−	21.8893i		−	0.930830i
$554$			21.2135i			0.901273i
$555$		0			0
$556$		−	19.2026i		−	0.814371i
$557$			16.4237i			0.695897i	0.937514	+	0.347948i	$0.113122\pi$
$557$			16.4237i			0.695897i	−0.937514	+	0.347948i	$0.886878\pi$
$558$		0			0
$559$			51.3448i			2.17165i
$560$			0.280032i			0.0118335i
$561$		0			0
$562$	−14.3032			−0.603344
$563$	−22.1550			−0.933723			−0.466861	−	0.884331i	$-0.654615\pi$
$563$	−22.1550			−0.933723			−0.466861	+	0.884331i	$0.654615\pi$
$564$		0			0
$565$			39.4692i			1.66048i
$566$		−	31.1136i		−	1.30780i
$567$		0			0
$568$		−	15.0997i		−	0.633567i
$569$	16.1998			0.679131			0.339566	−	0.940582i	$-0.389720\pi$
$569$	16.1998			0.679131			0.339566	+	0.940582i	$0.389720\pi$
$570$		0			0
$571$		−	29.8989i		−	1.25123i	−0.780132	−	0.625615i	$-0.784848\pi$
$571$		−	29.8989i		−	1.25123i	0.780132	−	0.625615i	$-0.215152\pi$
$572$	72.5969	−	0.837906i	3.03543	−	0.0350346i
$573$		0			0
$574$		−	29.9814i		−	1.25140i
$575$	0.248890			0.0103794
$576$		0			0
$577$	−13.7060			−0.570590			−0.285295	−	0.958440i	$-0.592092\pi$
$577$	−13.7060			−0.570590			−0.285295	+	0.958440i	$0.592092\pi$
$578$	46.3892			1.92954
$579$		0			0
$580$	−24.9968			−1.03794
$581$	43.1141			1.78868
$582$		0			0
$583$	0.439721	+	38.0978i	0.0182114	+	1.57785i
$584$			6.47935i			0.268117i
$585$		0			0
$586$	−38.5018			−1.59049
$587$	−20.5883			−0.849769			−0.424884	−	0.905248i	$-0.639685\pi$
$587$	−20.5883			−0.849769			−0.424884	+	0.905248i	$0.639685\pi$
$588$		0			0
$589$	28.2756	−	6.62500i	1.16507	−	0.272978i
$590$		−	61.4358i		−	2.52927i
$591$		0			0
$592$		−	0.0492237i		−	0.00202308i
$593$			15.0049i			0.616176i	0.951358	+	0.308088i	$0.0996891\pi$
$593$			15.0049i			0.616176i	−0.951358	+	0.308088i	$0.900311\pi$
$594$		0			0
$595$	38.2267			1.56714
$596$			13.1120i			0.537088i
$597$		0			0
$598$	−85.9797			−3.51597
$599$			23.3878i			0.955600i	0.878469	+	0.477800i	$0.158566\pi$
$599$			23.3878i			0.955600i	−0.878469	+	0.477800i	$0.841434\pi$
$600$		0			0
$601$	−15.3685			−0.626895			−0.313447	−	0.949606i	$-0.601484\pi$
$601$	−15.3685			−0.626895			−0.313447	+	0.949606i	$0.601484\pi$
$602$	−48.6274			−1.98191
$603$		0			0
$604$	−36.7235			−1.49426
$605$	−24.6999	+	0.570244i	−1.00419	+	0.0231837i
$606$		0			0
$607$	−35.3215			−1.43366			−0.716829	−	0.697249i	$-0.754407\pi$
$607$	−35.3215			−1.43366			−0.716829	+	0.697249i	$0.754407\pi$
$608$	−5.57395	−	23.7897i	−0.226053	−	0.964798i
$609$		0			0
$610$		−	51.2445i		−	2.07483i
$611$	36.0250			1.45742
$612$		0			0
$613$		−	0.965128i		−	0.0389812i	−0.999810	−	0.0194906i	$-0.993796\pi$
$613$		−	0.965128i		−	0.0389812i	0.999810	−	0.0194906i	$-0.00620444\pi$
$614$	8.01451			0.323439
$615$		0			0
$616$	0.304620	+	26.3926i	0.0122735	+	1.06339i
$617$	2.95486			0.118958			0.0594791	−	0.998230i	$-0.481056\pi$
$617$	2.95486			0.118958			0.0594791	+	0.998230i	$0.481056\pi$
$618$		0			0
$619$	−30.5865			−1.22938			−0.614688	−	0.788770i	$-0.710718\pi$
$619$	−30.5865			−1.22938			−0.614688	+	0.788770i	$0.710718\pi$
$620$		−	48.5749i		−	1.95081i
$621$		0			0
$622$	−40.1203			−1.60868
$623$	13.3245			0.533833
$624$		0			0
$625$	−25.2216			−1.00886
$626$	−22.1292			−0.884461
$627$		0			0
$628$	−62.6943			−2.50177
$629$	−6.71946			−0.267922
$630$		0			0
$631$	−23.0028			−0.915728			−0.457864	−	0.889022i	$-0.651385\pi$
$631$	−23.0028			−0.915728			−0.457864	+	0.889022i	$0.651385\pi$
$632$	−22.4036			−0.891166
$633$		0			0
$634$		−	24.3714i		−	0.967911i
$635$	25.1553			0.998259
$636$		0			0
$637$	−5.22990			−0.207216
$638$	−26.0433	+	0.300589i	−1.03106	+	0.0119004i
$639$		0			0
$640$	−41.3286			−1.63366
$641$			30.2737i			1.19574i	0.801594	+	0.597869i	$0.203986\pi$
$641$			30.2737i			1.19574i	−0.801594	+	0.597869i	$0.796014\pi$
$642$		0			0
$643$	35.7411			1.40949			0.704746	−	0.709460i	$-0.251061\pi$
$643$	35.7411			1.40949			0.704746	+	0.709460i	$0.251061\pi$
$644$		−	50.3853i		−	1.98546i
$645$		0			0
$646$	59.3296	−	13.9010i	2.33429	−	0.546928i
$647$	20.8668			0.820357			0.410179	−	0.912005i	$-0.365466\pi$
$647$	20.8668			0.820357			0.410179	+	0.912005i	$0.365466\pi$
$648$		0			0
$649$	−0.457122	−	39.6054i	−0.0179436	−	1.55465i
$650$	−0.690615			−0.0270882
$651$		0			0
$652$	66.6957			2.61200
$653$	41.7945			1.63554			0.817772	−	0.575542i	$-0.195209\pi$
$653$	41.7945			1.63554			0.817772	+	0.575542i	$0.195209\pi$
$654$		0			0
$655$			30.2870i			1.18341i
$656$	−0.209892			−0.00819490
$657$		0			0
$658$			34.1184i			1.33007i
$659$	−7.91237			−0.308222			−0.154111	−	0.988054i	$-0.549251\pi$
$659$	−7.91237			−0.308222			−0.154111	+	0.988054i	$0.549251\pi$
$660$		0			0
$661$			7.10506i			0.276355i	0.990407	+	0.138177i	$0.0441244\pi$
$661$			7.10506i			0.276355i	−0.990407	+	0.138177i	$0.955876\pi$
$662$		−	18.7249i		−	0.727762i
$663$		0			0
$664$		−	44.1270i		−	1.71246i
$665$	26.5800	−	6.22773i	1.03073	−	0.241501i
$666$		0			0
$667$	19.0852			0.738981
$668$	−24.9150			−0.963991
$669$		0			0
$670$		−	32.8812i		−	1.27031i
$671$	−0.381292	−	33.0355i	−0.0147196	−	1.27532i
$672$		0			0
$673$	13.0813			0.504247			0.252124	−	0.967695i	$-0.418871\pi$
$673$	13.0813			0.504247			0.252124	+	0.967695i	$0.418871\pi$
$674$	20.2600			0.780387
$675$		0			0
$676$	105.422			4.05469
$677$	−36.6921			−1.41019			−0.705096	−	0.709112i	$-0.749096\pi$
$677$	−36.6921			−1.41019			−0.705096	+	0.709112i	$0.749096\pi$
$678$		0			0
$679$	41.3846			1.58820
$680$		−	39.1247i		−	1.50036i
$681$		0			0
$682$	−0.584117	−	50.6084i	−0.0223670	−	1.93790i
$683$			20.4401i			0.782117i	0.920366	+	0.391059i	$0.127891\pi$
$683$			20.4401i			0.782117i	−0.920366	+	0.391059i	$0.872109\pi$
$684$		0			0
$685$	−12.3439			−0.471635
$686$			39.7542i			1.51782i
$687$		0			0
$688$			0.340427i			0.0129787i
$689$			77.4690i			2.95133i
$690$		0			0
$691$	−39.1289			−1.48853			−0.744267	−	0.667882i	$-0.767201\pi$
$691$	−39.1289			−1.48853			−0.744267	+	0.667882i	$0.767201\pi$
$692$	−24.4662			−0.930064
$693$		0			0
$694$			21.5947i			0.819722i
$695$			13.2869i			0.504001i
$696$		0			0
$697$			28.6520i			1.08527i
$698$		−	0.0766860i		−	0.00290261i
$699$		0			0
$700$		−	0.404710i		−	0.0152966i
$701$			6.91306i			0.261103i	0.991442	+	0.130551i	$0.0416747\pi$
$701$			6.91306i			0.261103i	−0.991442	+	0.130551i	$0.958325\pi$
$702$		0			0
$703$	−4.67221	+	1.09470i	−0.176216	+	0.0412876i
$704$	−42.8760	+	0.494871i	−1.61595	+	0.0186511i
$705$		0			0
$706$	56.5635			2.12880
$707$	−4.26967			−0.160577
$708$		0			0
$709$	−16.4432			−0.617537			−0.308769	−	0.951137i	$-0.599917\pi$
$709$	−16.4432			−0.617537			−0.308769	+	0.951137i	$0.599917\pi$
$710$			27.2177i			1.02146i
$711$		0			0
$712$		−	13.6375i		−	0.511086i
$713$			37.0872i			1.38892i
$714$		0			0
$715$	−50.2321	+	0.579774i	−1.87858	+	0.0216823i
$716$		−	63.9459i		−	2.38977i
$717$		0			0
$718$			78.7084i			2.93737i
$719$	19.9186			0.742840			0.371420	−	0.928465i	$-0.378871\pi$
$719$	19.9186			0.742840			0.371420	+	0.928465i	$0.378871\pi$
$720$		0			0
$721$	−20.1101			−0.748940
$722$	38.9887	−	19.3314i	1.45101	−	0.719442i
$723$		0			0
$724$		−	39.5688i		−	1.47056i
$725$	0.153298			0.00569334
$726$		0			0
$727$	−14.7932			−0.548651			−0.274325	−	0.961637i	$-0.588455\pi$
$727$	−14.7932			−0.548651			−0.274325	+	0.961637i	$0.588455\pi$
$728$			53.6674i			1.98904i
$729$		0			0
$730$		−	11.6793i		−	0.432270i
$731$	46.4712			1.71880
$732$		0			0
$733$			30.0355i			1.10939i	0.832054	+	0.554694i	$0.187165\pi$
$733$			30.0355i			1.10939i	−0.832054	+	0.554694i	$0.812835\pi$
$734$	24.2838			0.896331
$735$		0			0
$736$	31.2033			1.15017
$737$	−0.244658	−	21.1973i	−0.00901208	−	0.780814i
$738$		0			0
$739$		−	15.0629i		−	0.554099i	−0.960856	−	0.277049i	$-0.910643\pi$
$739$		−	15.0629i		−	0.554099i	0.960856	−	0.277049i	$-0.0893566\pi$
$740$			8.02644i			0.295058i
$741$		0			0
$742$	−73.3690			−2.69346
$743$	−9.37768			−0.344034			−0.172017	−	0.985094i	$-0.555028\pi$
$743$	−9.37768			−0.344034			−0.172017	+	0.985094i	$0.555028\pi$
$744$		0			0
$745$		−	9.07261i		−	0.332395i
$746$	35.9828			1.31742
$747$		0			0
$748$	−0.758373	−	65.7061i	−0.0277289	−	2.40245i
$749$			10.3396i			0.377800i
$750$		0			0
$751$		−	29.9856i		−	1.09419i	−0.837071	−	0.547095i	$-0.815734\pi$
$751$		−	29.9856i		−	1.09419i	0.837071	−	0.547095i	$-0.184266\pi$
$752$	0.238854			0.00871010
$753$		0			0
$754$	−52.9571			−1.92858
$755$	25.4102			0.924770
$756$		0			0
$757$	−20.7974			−0.755894			−0.377947	−	0.925827i	$-0.623370\pi$
$757$	−20.7974			−0.755894			−0.377947	+	0.925827i	$0.623370\pi$
$758$		−	33.4455i		−	1.21479i
$759$		0			0
$760$	−6.37403	−	27.2044i	−0.231210	−	0.986807i
$761$		−	2.37920i		−	0.0862461i	−0.999070	−	0.0431230i	$-0.986269\pi$
$761$		−	2.37920i		−	0.0862461i	0.999070	−	0.0431230i	$-0.0137307\pi$
$762$		0			0
$763$		−	25.4768i		−	0.922322i
$764$	6.78909			0.245621
$765$		0			0
$766$			13.5278i			0.488779i
$767$		−	80.5348i		−	2.90794i
$768$		0			0
$769$			1.08952i			0.0392892i	0.999807	+	0.0196446i	$0.00625347\pi$
$769$			1.08952i			0.0392892i	−0.999807	+	0.0196446i	$0.993747\pi$
$770$	−0.549090	−	47.5737i	−0.0197878	−	1.71444i
$771$		0			0
$772$	35.5954			1.28111
$773$			11.3217i			0.407214i	0.979053	+	0.203607i	$0.0652664\pi$
$773$			11.3217i			0.407214i	−0.979053	+	0.203607i	$0.934734\pi$
$774$		0			0
$775$			0.297895i			0.0107007i
$776$		−	42.3568i		−	1.52052i
$777$		0			0
$778$	24.0755			0.863148
$779$	4.66786	+	19.9225i	0.167244	+	0.713797i
$780$		0			0
$781$	0.202517	+	17.5463i	0.00724664	+	0.627855i
$782$			77.8187i			2.78279i
$783$		0			0
$784$	−0.0346754			−0.00123841
$785$	43.3802			1.54831
$786$		0			0
$787$	17.1126			0.609998			0.304999	−	0.952353i	$-0.401344\pi$
$787$	17.1126			0.609998			0.304999	+	0.952353i	$0.401344\pi$
$788$		−	24.8709i		−	0.885988i
$789$		0			0
$790$	40.3833			1.43677
$791$	−49.0010			−1.74227
$792$		0			0
$793$		−	67.1753i		−	2.38546i
$794$	−8.18434			−0.290451
$795$		0			0
$796$	47.9450			1.69936
$797$			15.6088i			0.552892i	0.961029	+	0.276446i	$0.0891567\pi$
$797$			15.6088i			0.552892i	−0.961029	+	0.276446i	$0.910843\pi$
$798$		0			0
$799$		−	32.6056i		−	1.15350i
$800$	0.250634			0.00886126
$801$		0			0
$802$		−	17.4860i		−	0.617451i
$803$	−0.0869014	−	7.52921i	−0.00306669	−	0.265700i
$804$		0			0
$805$			34.8632i			1.22877i
$806$		−	102.909i		−	3.62480i
$807$		0			0
$808$			4.36997i			0.153735i
$809$			3.02548i			0.106370i	0.998585	+	0.0531850i	$0.0169373\pi$
$809$			3.02548i			0.106370i	−0.998585	+	0.0531850i	$0.983063\pi$
$810$		0			0
$811$	−4.78932			−0.168176			−0.0840879	−	0.996458i	$-0.526798\pi$
$811$	−4.78932			−0.168176			−0.0840879	+	0.996458i	$0.526798\pi$
$812$		−	31.0335i		−	1.08906i
$813$		0			0
$814$	0.0965186	+	8.36246i	0.00338298	+	0.293104i
$815$	−46.1489			−1.61653
$816$		0			0
$817$	32.3126	−	7.57089i	1.13048	−	0.264872i
$818$	14.3595			0.502069
$819$		0			0
$820$	34.2251			1.19519
$821$		−	19.7078i		−	0.687807i	−0.939005	−	0.343903i	$-0.888251\pi$
$821$		−	19.7078i		−	0.687807i	0.939005	−	0.343903i	$-0.111749\pi$
$822$		0			0
$823$	31.5377			1.09934			0.549668	−	0.835383i	$-0.314754\pi$
$823$	31.5377			1.09934			0.549668	+	0.835383i	$0.314754\pi$
$824$			20.5825i			0.717027i
$825$		0			0
$826$	76.2726			2.65386
$827$	10.1644			0.353452			0.176726	−	0.984260i	$-0.443449\pi$
$827$	10.1644			0.353452			0.176726	+	0.984260i	$0.443449\pi$
$828$		0			0
$829$		−	46.2995i		−	1.60805i	−0.594596	−	0.804025i	$-0.702688\pi$
$829$		−	46.2995i		−	1.60805i	0.594596	−	0.804025i	$-0.297312\pi$
$830$			79.5406i			2.76089i
$831$		0			0
$832$	−87.1852			−3.02260
$833$			4.73348i			0.164005i
$834$		0			0
$835$	17.2395			0.596598
$836$	−11.2319	−	45.5635i	−0.388462	−	1.57585i
$837$		0			0
$838$	78.6928			2.71840
$839$		−	42.1213i		−	1.45419i	−0.686538	−	0.727094i	$-0.740870\pi$
$839$		−	42.1213i		−	1.45419i	0.686538	−	0.727094i	$-0.259130\pi$
$840$		0			0
$841$	−17.2450			−0.594654
$842$		−	53.5406i		−	1.84513i
$843$		0			0
$844$	12.4510			0.428583
$845$	−72.9448			−2.50938
$846$		0			0
$847$	−0.707957	−	30.6649i	−0.0243257	−	1.05366i
$848$			0.513637i			0.0176384i
$849$		0			0
$850$			0.625063i			0.0214395i
$851$		−	6.12822i		−	0.210073i
$852$		0			0
$853$		−	1.18916i		−	0.0407161i	−0.999793	−	0.0203581i	$-0.993519\pi$
$853$		−	1.18916i		−	0.0407161i	0.999793	−	0.0203581i	$-0.00648062\pi$
$854$	63.6201			2.17703
$855$		0			0
$856$	10.5825			0.361702
$857$	24.0065			0.820047			0.410023	−	0.912075i	$-0.365521\pi$
$857$	24.0065			0.820047			0.410023	+	0.912075i	$0.365521\pi$
$858$		0			0
$859$	−18.1085			−0.617854			−0.308927	−	0.951086i	$-0.599970\pi$
$859$	−18.1085			−0.617854			−0.308927	+	0.951086i	$0.599970\pi$
$860$		−	55.5102i		−	1.89288i
$861$		0			0
$862$	−1.90314			−0.0648211
$863$		−	42.3524i		−	1.44169i	−0.693095	−	0.720846i	$-0.743754\pi$
$863$		−	42.3524i		−	1.44169i	0.693095	−	0.720846i	$-0.256246\pi$
$864$		0			0
$865$	16.9289			0.575601
$866$		−	37.6065i		−	1.27792i
$867$		0			0
$868$	60.3057			2.04691
$869$	26.0337	−	0.300478i	0.883132	−	0.0101930i
$870$		0			0
$871$		−	43.1032i		−	1.46050i
$872$	−26.0753			−0.883021
$873$		0			0
$874$	12.6779	+	54.1093i	0.428836	+	1.83027i
$875$		−	31.0350i		−	1.04918i
$876$		0			0
$877$	14.4810			0.488989			0.244495	−	0.969651i	$-0.421378\pi$
$877$	14.4810			0.488989			0.244495	+	0.969651i	$0.421378\pi$
$878$	25.1989			0.850423
$879$		0			0
$880$	−0.333050	+	0.00384403i	−0.0112271	+	0.000129582i
$881$	−24.9014			−0.838951			−0.419476	−	0.907767i	$-0.637786\pi$
$881$	−24.9014			−0.838951			−0.419476	+	0.907767i	$0.637786\pi$
$882$		0			0
$883$	49.4700			1.66480			0.832399	−	0.554178i	$-0.186967\pi$
$883$	49.4700			1.66480			0.832399	+	0.554178i	$0.186967\pi$
$884$		−	133.609i		−	4.49374i
$885$		0			0
$886$	−29.2403			−0.982348
$887$	−21.7252			−0.729460			−0.364730	−	0.931113i	$-0.618839\pi$
$887$	−21.7252			−0.729460			−0.364730	+	0.931113i	$0.618839\pi$
$888$		0			0
$889$			31.2303i			1.04743i
$890$			24.5821i			0.823994i
$891$		0			0
$892$			21.1622i			0.708564i
$893$	−5.31196	−	22.6715i	−0.177758	−	0.758672i
$894$		0			0
$895$			44.2463i			1.47899i
$896$		−	51.3094i		−	1.71413i
$897$		0			0
$898$		−	32.4512i		−	1.08291i
$899$			22.8429i			0.761854i
$900$		0			0
$901$	70.1158			2.33589
$902$	35.6578	−	0.411559i	1.18728	−	0.0137034i
$903$		0			0
$904$			50.1521i			1.66803i
$905$			27.3789i			0.910107i
$906$		0			0
$907$		−	6.01135i		−	0.199603i	−0.995007	−	0.0998017i	$-0.968179\pi$
$907$		−	6.01135i		−	0.199603i	0.995007	−	0.0998017i	$-0.0318209\pi$
$908$	−89.3206			−2.96421
$909$		0			0
$910$		−	96.7376i		−	3.20682i
$911$		−	24.0331i		−	0.796252i	−0.917331	−	0.398126i	$-0.869661\pi$
$911$		−	24.0331i		−	0.796252i	0.917331	−	0.398126i	$-0.130339\pi$
$912$		0			0
$913$	0.591833	+	51.2769i	0.0195868	+	1.69702i
$914$			4.60259i			0.152240i
$915$		0			0
$916$	−16.5638			−0.547284
$917$	−37.6013			−1.24170
$918$		0			0
$919$		−	12.9069i		−	0.425759i	−0.977078	−	0.212880i	$-0.931716\pi$
$919$		−	12.9069i		−	0.425759i	0.977078	−	0.212880i	$-0.0682842\pi$
$920$	35.6822			1.17641
$921$		0			0
$922$			29.4877i			0.971127i
$923$			35.6791i			1.17439i
$924$		0			0
$925$		−	0.0492237i		−	0.00161847i
$926$	−91.8738			−3.01916
$927$		0			0
$928$	19.2189			0.630891
$929$	−7.73250			−0.253695			−0.126848	−	0.991922i	$-0.540486\pi$
$929$	−7.73250			−0.253695			−0.126848	+	0.991922i	$0.540486\pi$
$930$		0			0
$931$	0.771159	+	3.29131i	0.0252737	+	0.107868i
$932$		−	8.07333i		−	0.264451i
$933$		0			0
$934$	−86.2585			−2.82246
$935$	0.524743	+	45.4642i	0.0171609	+	1.48684i
$936$		0			0
$937$		−	10.2538i		−	0.334977i	−0.985874	−	0.167489i	$-0.946434\pi$
$937$		−	10.2538i		−	0.334977i	0.985874	−	0.167489i	$-0.0535657\pi$
$938$	40.8220			1.33289
$939$		0			0
$940$	−38.9476			−1.27033
$941$	24.7981			0.808395			0.404197	−	0.914672i	$-0.367551\pi$
$941$	24.7981			0.808395			0.404197	+	0.914672i	$0.367551\pi$
$942$		0			0
$943$	−26.1310			−0.850942
$944$		−	0.533964i		−	0.0173790i
$945$		0			0
$946$	−0.667515	−	57.8341i	−0.0217028	−	1.88035i
$947$	4.32248			0.140462			0.0702309	−	0.997531i	$-0.477626\pi$
$947$	4.32248			0.140462			0.0702309	+	0.997531i	$0.477626\pi$
$948$		0			0
$949$		−	15.3101i		−	0.496987i
$950$	0.101833	+	0.434622i	0.00330388	+	0.0141010i
$951$		0			0
$952$	48.5733			1.57427
$953$	5.06287			0.164003			0.0820013	−	0.996632i	$-0.473869\pi$
$953$	5.06287			0.164003			0.0820013	+	0.996632i	$0.473869\pi$
$954$		0			0
$955$	−4.69759			−0.152011
$956$			30.8847i			0.998884i
$957$		0			0
$958$			32.5152i			1.05052i
$959$		−	15.3249i		−	0.494867i
$960$		0			0
$961$	−13.3893			−0.431914
$962$			17.0044i			0.548245i
$963$		0			0
$964$	40.5720			1.30674
$965$	−24.6296			−0.792856
$966$		0			0
$967$			11.2371i			0.361361i	0.983542	+	0.180680i	$0.0578300\pi$
$967$			11.2371i			0.361361i	−0.983542	+	0.180680i	$0.942170\pi$
$968$	−31.3853	+	0.724588i	−1.00876	+	0.0232892i
$969$		0			0
$970$			76.3499i			2.45145i
$971$		−	42.9353i		−	1.37786i	−0.724827	−	0.688930i	$-0.758081\pi$
$971$		−	42.9353i		−	1.37786i	0.724827	−	0.688930i	$-0.241919\pi$
$972$		0			0
$973$	−16.4957			−0.528827
$974$			24.9294i			0.798789i
$975$		0			0
$976$		−	0.445387i		−	0.0142565i
$977$			13.8317i			0.442514i	0.975216	+	0.221257i	$0.0710159\pi$
$977$			13.8317i			0.442514i	−0.975216	+	0.221257i	$0.928984\pi$
$978$		0			0
$979$	0.182907	+	15.8472i	0.00584572	+	0.506478i
$980$	5.65418			0.180616
$981$		0			0
$982$		−	38.0855i		−	1.21536i
$983$		−	6.87197i		−	0.219182i	−0.993977	−	0.109591i	$-0.965046\pi$
$983$		−	6.87197i		−	0.219182i	0.993977	−	0.109591i	$-0.0349541\pi$
$984$		0			0
$985$			17.2090i			0.548323i
$986$			47.9305i			1.52642i
$987$		0			0
$988$	−21.7669	−	92.9014i	−0.692498	−	2.95559i
$989$			42.3823i			1.34768i
$990$		0			0
$991$		−	37.2344i		−	1.18279i	−0.806382	−	0.591395i	$-0.798578\pi$
$991$		−	37.2344i		−	1.18279i	0.806382	−	0.591395i	$-0.201422\pi$
$992$			37.3470i			1.18577i
$993$		0			0
$994$	−33.7908			−1.07178
$995$	−33.1747			−1.05171
$996$		0			0
$997$			29.8335i			0.944837i	0.881374	+	0.472418i	$0.156619\pi$
$997$			29.8335i			0.944837i	−0.881374	+	0.472418i	$0.843381\pi$
$998$	15.6974			0.496891
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.h.g.208.3		24
3.2	odd	2		627.2.h.b.208.22	yes	24
11.10	odd	2	inner	1881.2.h.g.208.21		24
19.18	odd	2	inner	1881.2.h.g.208.22		24
33.32	even	2		627.2.h.b.208.4	yes	24
57.56	even	2		627.2.h.b.208.3	✓	24
209.208	even	2	inner	1881.2.h.g.208.4		24
627.626	odd	2		627.2.h.b.208.21	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
627.2.h.b.208.3	✓	24	57.56	even	2
627.2.h.b.208.4	yes	24	33.32	even	2
627.2.h.b.208.21	yes	24	627.626	odd	2
627.2.h.b.208.22	yes	24	3.2	odd	2
1881.2.h.g.208.3		24	1.1	even	1	trivial
1881.2.h.g.208.4		24	209.208	even	2	inner
1881.2.h.g.208.21		24	11.10	odd	2	inner
1881.2.h.g.208.22		24	19.18	odd	2	inner

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1881.2.h.g.208.3

Level	$1881$
Weight	$2$
Character	1881.208
Analytic conductor	$15.020$
Analytic rank	$0$
Dimension	$24$
Inner twists	$4$