LMFDB - Embedded newform 6039.2.a.o.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.a (trivial)

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:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.702784 q^{2} -1.50609 q^{4} -2.41068 q^{5} +0.380608 q^{7} -2.46403 q^{8} +O(q^{10})$	\(q+0.702784 q^{2} -1.50609 q^{4} -2.41068 q^{5} +0.380608 q^{7} -2.46403 q^{8} -1.69419 q^{10} +1.00000 q^{11} +2.87344 q^{13} +0.267485 q^{14} +1.28051 q^{16} +1.85240 q^{17} -2.38794 q^{19} +3.63071 q^{20} +0.702784 q^{22} -5.66757 q^{23} +0.811370 q^{25} +2.01941 q^{26} -0.573232 q^{28} +2.60447 q^{29} -4.67698 q^{31} +5.82798 q^{32} +1.30184 q^{34} -0.917524 q^{35} -10.7819 q^{37} -1.67821 q^{38} +5.93998 q^{40} -0.412567 q^{41} -2.52589 q^{43} -1.50609 q^{44} -3.98308 q^{46} +7.69707 q^{47} -6.85514 q^{49} +0.570218 q^{50} -4.32767 q^{52} +3.19883 q^{53} -2.41068 q^{55} -0.937829 q^{56} +1.83038 q^{58} +2.95404 q^{59} -1.00000 q^{61} -3.28691 q^{62} +1.53479 q^{64} -6.92694 q^{65} +8.09053 q^{67} -2.78990 q^{68} -0.644821 q^{70} -15.0776 q^{71} +12.1332 q^{73} -7.57734 q^{74} +3.59646 q^{76} +0.380608 q^{77} -11.3863 q^{79} -3.08690 q^{80} -0.289946 q^{82} +13.0055 q^{83} -4.46555 q^{85} -1.77516 q^{86} -2.46403 q^{88} +6.76539 q^{89} +1.09365 q^{91} +8.53590 q^{92} +5.40938 q^{94} +5.75655 q^{95} +6.74475 q^{97} -4.81768 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.702784	0.496943	0.248472	−	0.968639i	$-0.420072\pi$
$2$	0.702784	0.496943	0.248472	+	0.968639i	$0.420072\pi$
$3$	0	0
$3$	0	0
$4$	−1.50609	−0.753047
$5$	−2.41068	−1.07809	−0.539044	−	0.842278i	$-0.681214\pi$
$5$	−2.41068	−1.07809	−0.539044	+	0.842278i	$0.681214\pi$
$6$	0	0
$7$	0.380608	0.143856	0.0719282	−	0.997410i	$-0.477085\pi$
$7$	0.380608	0.143856	0.0719282	+	0.997410i	$0.477085\pi$
$8$	−2.46403	−0.871165
$9$	0	0
$10$	−1.69419	−0.535749
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.87344	0.796949	0.398474	−	0.917180i	$-0.369540\pi$
$13$	2.87344	0.796949	0.398474	+	0.917180i	$0.369540\pi$
$14$	0.267485	0.0714885
$15$	0	0
$16$	1.28051	0.320127
$17$	1.85240	0.449274	0.224637	−	0.974443i	$-0.427880\pi$
$17$	1.85240	0.449274	0.224637	+	0.974443i	$0.427880\pi$
$18$	0	0
$19$	−2.38794	−0.547831	−0.273915	−	0.961754i	$-0.588319\pi$
$19$	−2.38794	−0.547831	−0.273915	+	0.961754i	$0.588319\pi$
$20$	3.63071	0.811851
$21$	0	0
$22$	0.702784	0.149834
$23$	−5.66757	−1.18177	−0.590885	−	0.806756i	$-0.701221\pi$
$23$	−5.66757	−1.18177	−0.590885	+	0.806756i	$0.701221\pi$
$24$	0	0
$25$	0.811370	0.162274
$26$	2.01941	0.396038
$27$	0	0
$28$	−0.573232	−0.108331
$29$	2.60447	0.483639	0.241819	−	0.970321i	$-0.422256\pi$
$29$	2.60447	0.483639	0.241819	+	0.970321i	$0.422256\pi$
$30$	0	0
$31$	−4.67698	−0.840011	−0.420006	−	0.907522i	$-0.637972\pi$
$31$	−4.67698	−0.840011	−0.420006	+	0.907522i	$0.637972\pi$
$32$	5.82798	1.03025
$33$	0	0
$34$	1.30184	0.223264
$35$	−0.917524	−0.155090
$36$	0	0
$37$	−10.7819	−1.77253	−0.886265	−	0.463178i	$-0.846709\pi$
$37$	−10.7819	−1.77253	−0.886265	+	0.463178i	$0.846709\pi$
$38$	−1.67821	−0.272241
$39$	0	0
$40$	5.93998	0.939193
$41$	−0.412567	−0.0644321	−0.0322161	−	0.999481i	$-0.510256\pi$
$41$	−0.412567	−0.0644321	−0.0322161	+	0.999481i	$0.510256\pi$
$42$	0	0
$43$	−2.52589	−0.385195	−0.192598	−	0.981278i	$-0.561691\pi$
$43$	−2.52589	−0.385195	−0.192598	+	0.981278i	$0.561691\pi$
$44$	−1.50609	−0.227052
$45$	0	0
$46$	−3.98308	−0.587273
$47$	7.69707	1.12273	0.561367	−	0.827567i	$-0.310276\pi$
$47$	7.69707	1.12273	0.561367	+	0.827567i	$0.310276\pi$
$48$	0	0
$49$	−6.85514	−0.979305
$50$	0.570218	0.0806410
$51$	0	0
$52$	−4.32767	−0.600140
$53$	3.19883	0.439393	0.219697	−	0.975568i	$-0.429493\pi$
$53$	3.19883	0.439393	0.219697	+	0.975568i	$0.429493\pi$
$54$	0	0
$55$	−2.41068	−0.325056
$56$	−0.937829	−0.125323
$57$	0	0
$58$	1.83038	0.240341
$59$	2.95404	0.384584	0.192292	−	0.981338i	$-0.438408\pi$
$59$	2.95404	0.384584	0.192292	+	0.981338i	$0.438408\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−3.28691	−0.417438
$63$	0	0
$64$	1.53479	0.191849
$65$	−6.92694	−0.859181
$66$	0	0
$67$	8.09053	0.988416	0.494208	−	0.869344i	$-0.335458\pi$
$67$	8.09053	0.988416	0.494208	+	0.869344i	$0.335458\pi$
$68$	−2.78990	−0.338324
$69$	0	0
$70$	−0.644821	−0.0770709
$71$	−15.0776	−1.78938	−0.894692	−	0.446684i	$-0.852605\pi$
$71$	−15.0776	−1.78938	−0.894692	+	0.446684i	$0.852605\pi$
$72$	0	0
$73$	12.1332	1.42008	0.710041	−	0.704161i	$-0.248677\pi$
$73$	12.1332	1.42008	0.710041	+	0.704161i	$0.248677\pi$
$74$	−7.57734	−0.880847
$75$	0	0
$76$	3.59646	0.412542
$77$	0.380608	0.0433743
$78$	0	0
$79$	−11.3863	−1.28106	−0.640530	−	0.767933i	$-0.721285\pi$
$79$	−11.3863	−1.28106	−0.640530	+	0.767933i	$0.721285\pi$
$80$	−3.08690	−0.345125
$81$	0	0
$82$	−0.289946	−0.0320191
$83$	13.0055	1.42754	0.713770	−	0.700380i	$-0.246986\pi$
$83$	13.0055	1.42754	0.713770	+	0.700380i	$0.246986\pi$
$84$	0	0
$85$	−4.46555	−0.484357
$86$	−1.77516	−0.191420
$87$	0	0
$88$	−2.46403	−0.262666
$89$	6.76539	0.717130	0.358565	−	0.933505i	$-0.383266\pi$
$89$	6.76539	0.717130	0.358565	+	0.933505i	$0.383266\pi$
$90$	0	0
$91$	1.09365	0.114646
$92$	8.53590	0.889929
$93$	0	0
$94$	5.40938	0.557935
$95$	5.75655	0.590610
$96$	0	0
$97$	6.74475	0.684826	0.342413	−	0.939550i	$-0.388756\pi$
$97$	6.74475	0.684826	0.342413	+	0.939550i	$0.388756\pi$
$98$	−4.81768	−0.486659
$99$	0	0
$100$	−1.22200	−0.122200
$101$	−5.42967	−0.540272	−0.270136	−	0.962822i	$-0.587069\pi$
$101$	−5.42967	−0.540272	−0.270136	+	0.962822i	$0.587069\pi$
$102$	0	0
$103$	5.08542	0.501081	0.250541	−	0.968106i	$-0.419392\pi$
$103$	5.08542	0.501081	0.250541	+	0.968106i	$0.419392\pi$
$104$	−7.08023	−0.694274
$105$	0	0
$106$	2.24809	0.218354
$107$	6.74418	0.651984	0.325992	−	0.945373i	$-0.394302\pi$
$107$	6.74418	0.651984	0.325992	+	0.945373i	$0.394302\pi$
$108$	0	0
$109$	−14.1031	−1.35083	−0.675417	−	0.737436i	$-0.736036\pi$
$109$	−14.1031	−1.35083	−0.675417	+	0.737436i	$0.736036\pi$
$110$	−1.69419	−0.161534
$111$	0	0
$112$	0.487372	0.0460523
$113$	13.1051	1.23283	0.616413	−	0.787423i	$-0.288585\pi$
$113$	13.1051	1.23283	0.616413	+	0.787423i	$0.288585\pi$
$114$	0	0
$115$	13.6627	1.27405
$116$	−3.92258	−0.364203
$117$	0	0
$118$	2.07606	0.191116
$119$	0.705040	0.0646309
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.702784	−0.0636271
$123$	0	0
$124$	7.04398	0.632568
$125$	10.0974	0.903142
$126$	0	0
$127$	−3.54380	−0.314461	−0.157231	−	0.987562i	$-0.550257\pi$
$127$	−3.54380	−0.314461	−0.157231	+	0.987562i	$0.550257\pi$
$128$	−10.5773	−0.934912
$129$	0	0
$130$	−4.86814	−0.426964
$131$	2.56353	0.223976	0.111988	−	0.993710i	$-0.464278\pi$
$131$	2.56353	0.223976	0.111988	+	0.993710i	$0.464278\pi$
$132$	0	0
$133$	−0.908869	−0.0788090
$134$	5.68590	0.491187
$135$	0	0
$136$	−4.56437	−0.391392
$137$	10.1411	0.866413	0.433207	−	0.901295i	$-0.357382\pi$
$137$	10.1411	0.866413	0.433207	+	0.901295i	$0.357382\pi$
$138$	0	0
$139$	−10.9387	−0.927812	−0.463906	−	0.885885i	$-0.653552\pi$
$139$	−10.9387	−0.927812	−0.463906	+	0.885885i	$0.653552\pi$
$140$	1.38188	0.116790
$141$	0	0
$142$	−10.5963	−0.889222
$143$	2.87344	0.240289
$144$	0	0
$145$	−6.27855	−0.521405
$146$	8.52701	0.705700
$147$	0	0
$148$	16.2385	1.33480
$149$	−0.534433	−0.0437825	−0.0218912	−	0.999760i	$-0.506969\pi$
$149$	−0.534433	−0.0437825	−0.0218912	+	0.999760i	$0.506969\pi$
$150$	0	0
$151$	13.1663	1.07146	0.535729	−	0.844390i	$-0.320037\pi$
$151$	13.1663	1.07146	0.535729	+	0.844390i	$0.320037\pi$
$152$	5.88395	0.477251
$153$	0	0
$154$	0.267485	0.0215546
$155$	11.2747	0.905606
$156$	0	0
$157$	15.7696	1.25856	0.629278	−	0.777181i	$-0.283351\pi$
$157$	15.7696	1.25856	0.629278	+	0.777181i	$0.283351\pi$
$158$	−8.00212	−0.636615
$159$	0	0
$160$	−14.0494	−1.11070
$161$	−2.15712	−0.170005
$162$	0	0
$163$	10.1338	0.793742	0.396871	−	0.917874i	$-0.370096\pi$
$163$	10.1338	0.793742	0.396871	+	0.917874i	$0.370096\pi$
$164$	0.621365	0.0485204
$165$	0	0
$166$	9.14006	0.709407
$167$	1.44995	0.112201	0.0561003	−	0.998425i	$-0.482133\pi$
$167$	1.44995	0.112201	0.0561003	+	0.998425i	$0.482133\pi$
$168$	0	0
$169$	−4.74335	−0.364873
$170$	−3.13832	−0.240698
$171$	0	0
$172$	3.80423	0.290070
$173$	20.4074	1.55155	0.775774	−	0.631011i	$-0.217360\pi$
$173$	20.4074	1.55155	0.775774	+	0.631011i	$0.217360\pi$
$174$	0	0
$175$	0.308814	0.0233441
$176$	1.28051	0.0965220
$177$	0	0
$178$	4.75461	0.356373
$179$	2.18942	0.163645	0.0818224	−	0.996647i	$-0.473926\pi$
$179$	2.18942	0.163645	0.0818224	+	0.996647i	$0.473926\pi$
$180$	0	0
$181$	7.63386	0.567420	0.283710	−	0.958910i	$-0.408435\pi$
$181$	7.63386	0.567420	0.283710	+	0.958910i	$0.408435\pi$
$182$	0.768603	0.0569726
$183$	0	0
$184$	13.9651	1.02952
$185$	25.9916	1.91094
$186$	0	0
$187$	1.85240	0.135461
$188$	−11.5925	−0.845471
$189$	0	0
$190$	4.04561	0.293500
$191$	3.15980	0.228635	0.114318	−	0.993444i	$-0.463532\pi$
$191$	3.15980	0.228635	0.114318	+	0.993444i	$0.463532\pi$
$192$	0	0
$193$	21.0169	1.51283	0.756414	−	0.654093i	$-0.226949\pi$
$193$	21.0169	1.51283	0.756414	+	0.654093i	$0.226949\pi$
$194$	4.74011	0.340320
$195$	0	0
$196$	10.3245	0.737463
$197$	−26.6116	−1.89600	−0.948001	−	0.318269i	$-0.896899\pi$
$197$	−26.6116	−1.89600	−0.948001	+	0.318269i	$0.896899\pi$
$198$	0	0
$199$	−4.39584	−0.311613	−0.155807	−	0.987788i	$-0.549798\pi$
$199$	−4.39584	−0.311613	−0.155807	+	0.987788i	$0.549798\pi$
$200$	−1.99924	−0.141367
$201$	0	0
$202$	−3.81589	−0.268485
$203$	0.991284	0.0695745
$204$	0	0
$205$	0.994566	0.0694635
$206$	3.57395	0.249009
$207$	0	0
$208$	3.67946	0.255125
$209$	−2.38794	−0.165177
$210$	0	0
$211$	16.6321	1.14500	0.572500	−	0.819904i	$-0.305974\pi$
$211$	16.6321	1.14500	0.572500	+	0.819904i	$0.305974\pi$
$212$	−4.81774	−0.330884
$213$	0	0
$214$	4.73970	0.323999
$215$	6.08912	0.415274
$216$	0	0
$217$	−1.78010	−0.120841
$218$	−9.91145	−0.671288
$219$	0	0
$220$	3.63071	0.244782
$221$	5.32277	0.358048
$222$	0	0
$223$	−25.4938	−1.70719	−0.853595	−	0.520938i	$-0.825582\pi$
$223$	−25.4938	−1.70719	−0.853595	+	0.520938i	$0.825582\pi$
$224$	2.21818	0.148208
$225$	0	0
$226$	9.21007	0.612645
$227$	10.5081	0.697444	0.348722	−	0.937226i	$-0.386616\pi$
$227$	10.5081	0.697444	0.348722	+	0.937226i	$0.386616\pi$
$228$	0	0
$229$	17.1178	1.13117	0.565587	−	0.824689i	$-0.308650\pi$
$229$	17.1178	1.13117	0.565587	+	0.824689i	$0.308650\pi$
$230$	9.60192	0.633132
$231$	0	0
$232$	−6.41750	−0.421329
$233$	27.7248	1.81631	0.908156	−	0.418633i	$-0.137490\pi$
$233$	27.7248	1.81631	0.908156	+	0.418633i	$0.137490\pi$
$234$	0	0
$235$	−18.5552	−1.21041
$236$	−4.44907	−0.289610
$237$	0	0
$238$	0.495491	0.0321179
$239$	−6.31082	−0.408213	−0.204106	−	0.978949i	$-0.565429\pi$
$239$	−6.31082	−0.408213	−0.204106	+	0.978949i	$0.565429\pi$
$240$	0	0
$241$	13.6422	0.878774	0.439387	−	0.898298i	$-0.355196\pi$
$241$	13.6422	0.878774	0.439387	+	0.898298i	$0.355196\pi$
$242$	0.702784	0.0451767
$243$	0	0
$244$	1.50609	0.0964178
$245$	16.5255	1.05578
$246$	0	0
$247$	−6.86160	−0.436593
$248$	11.5242	0.731789
$249$	0	0
$250$	7.09632	0.448811
$251$	−14.8480	−0.937200	−0.468600	−	0.883410i	$-0.655241\pi$
$251$	−14.8480	−0.937200	−0.468600	+	0.883410i	$0.655241\pi$
$252$	0	0
$253$	−5.66757	−0.356317
$254$	−2.49053	−0.156270
$255$	0	0
$256$	−10.5032	−0.656448
$257$	28.5319	1.77977	0.889887	−	0.456181i	$-0.150783\pi$
$257$	28.5319	1.77977	0.889887	+	0.456181i	$0.150783\pi$
$258$	0	0
$259$	−4.10367	−0.254990
$260$	10.4326	0.647004
$261$	0	0
$262$	1.80161	0.111304
$263$	−7.41082	−0.456971	−0.228485	−	0.973547i	$-0.573377\pi$
$263$	−7.41082	−0.456971	−0.228485	+	0.973547i	$0.573377\pi$
$264$	0	0
$265$	−7.71136	−0.473705
$266$	−0.638739	−0.0391636
$267$	0	0
$268$	−12.1851	−0.744324
$269$	20.8269	1.26984	0.634919	−	0.772578i	$-0.281033\pi$
$269$	20.8269	1.26984	0.634919	+	0.772578i	$0.281033\pi$
$270$	0	0
$271$	17.8005	1.08131	0.540653	−	0.841246i	$-0.318177\pi$
$271$	17.8005	1.08131	0.540653	+	0.841246i	$0.318177\pi$
$272$	2.37202	0.143825
$273$	0	0
$274$	7.12701	0.430558
$275$	0.811370	0.0489274
$276$	0	0
$277$	11.6366	0.699176	0.349588	−	0.936904i	$-0.386322\pi$
$277$	11.6366	0.699176	0.349588	+	0.936904i	$0.386322\pi$
$278$	−7.68757	−0.461070
$279$	0	0
$280$	2.26080	0.135109
$281$	2.80094	0.167090	0.0835452	−	0.996504i	$-0.473376\pi$
$281$	2.80094	0.167090	0.0835452	+	0.996504i	$0.473376\pi$
$282$	0	0
$283$	7.27012	0.432164	0.216082	−	0.976375i	$-0.430672\pi$
$283$	7.27012	0.432164	0.216082	+	0.976375i	$0.430672\pi$
$284$	22.7083	1.34749
$285$	0	0
$286$	2.01941	0.119410
$287$	−0.157026	−0.00926897
$288$	0	0
$289$	−13.5686	−0.798153
$290$	−4.41247	−0.259109
$291$	0	0
$292$	−18.2737	−1.06939
$293$	−13.6581	−0.797914	−0.398957	−	0.916970i	$-0.630628\pi$
$293$	−13.6581	−0.797914	−0.398957	+	0.916970i	$0.630628\pi$
$294$	0	0
$295$	−7.12125	−0.414615
$296$	26.5669	1.54417
$297$	0	0
$298$	−0.375591	−0.0217574
$299$	−16.2854	−0.941810
$300$	0	0
$301$	−0.961375	−0.0554128
$302$	9.25307	0.532454
$303$	0	0
$304$	−3.05778	−0.175376
$305$	2.41068	0.138035
$306$	0	0
$307$	3.32393	0.189707	0.0948533	−	0.995491i	$-0.469762\pi$
$307$	3.32393	0.189707	0.0948533	+	0.995491i	$0.469762\pi$
$308$	−0.573232	−0.0326629
$309$	0	0
$310$	7.92368	0.450035
$311$	21.8525	1.23914	0.619572	−	0.784940i	$-0.287306\pi$
$311$	21.8525	1.23914	0.619572	+	0.784940i	$0.287306\pi$
$312$	0	0
$313$	8.51873	0.481507	0.240754	−	0.970586i	$-0.422605\pi$
$313$	8.51873	0.481507	0.240754	+	0.970586i	$0.422605\pi$
$314$	11.0827	0.625431
$315$	0	0
$316$	17.1489	0.964699
$317$	−4.34206	−0.243874	−0.121937	−	0.992538i	$-0.538911\pi$
$317$	−4.34206	−0.243874	−0.121937	+	0.992538i	$0.538911\pi$
$318$	0	0
$319$	2.60447	0.145823
$320$	−3.69989	−0.206830
$321$	0	0
$322$	−1.51599	−0.0844830
$323$	−4.42343	−0.246126
$324$	0	0
$325$	2.33142	0.129324
$326$	7.12189	0.394445
$327$	0	0
$328$	1.01658	0.0561311
$329$	2.92957	0.161512
$330$	0	0
$331$	14.9444	0.821419	0.410709	−	0.911766i	$-0.365281\pi$
$331$	14.9444	0.821419	0.410709	+	0.911766i	$0.365281\pi$
$332$	−19.5875	−1.07500
$333$	0	0
$334$	1.01900	0.0557573
$335$	−19.5037	−1.06560
$336$	0	0
$337$	−24.2822	−1.32273	−0.661367	−	0.750063i	$-0.730023\pi$
$337$	−24.2822	−1.32273	−0.661367	+	0.750063i	$0.730023\pi$
$338$	−3.33355	−0.181321
$339$	0	0
$340$	6.72554	0.364744
$341$	−4.67698	−0.253273
$342$	0	0
$343$	−5.27338	−0.284736
$344$	6.22387	0.335569
$345$	0	0
$346$	14.3420	0.771032
$347$	−2.45547	−0.131816	−0.0659082	−	0.997826i	$-0.520994\pi$
$347$	−2.45547	−0.131816	−0.0659082	+	0.997826i	$0.520994\pi$
$348$	0	0
$349$	7.87144	0.421349	0.210674	−	0.977556i	$-0.432434\pi$
$349$	7.87144	0.421349	0.210674	+	0.977556i	$0.432434\pi$
$350$	0.217030	0.0116007
$351$	0	0
$352$	5.82798	0.310632
$353$	−1.93527	−0.103004	−0.0515019	−	0.998673i	$-0.516401\pi$
$353$	−1.93527	−0.103004	−0.0515019	+	0.998673i	$0.516401\pi$
$354$	0	0
$355$	36.3473	1.92911
$356$	−10.1893	−0.540033
$357$	0	0
$358$	1.53869	0.0813223
$359$	27.2464	1.43801	0.719004	−	0.695006i	$-0.244598\pi$
$359$	27.2464	1.43801	0.719004	+	0.695006i	$0.244598\pi$
$360$	0	0
$361$	−13.2977	−0.699881
$362$	5.36496	0.281976
$363$	0	0
$364$	−1.64715	−0.0863339
$365$	−29.2492	−1.53097
$366$	0	0
$367$	13.1551	0.686691	0.343345	−	0.939209i	$-0.388440\pi$
$367$	13.1551	0.686691	0.343345	+	0.939209i	$0.388440\pi$
$368$	−7.25738	−0.378317
$369$	0	0
$370$	18.2665	0.949631
$371$	1.21750	0.0632095
$372$	0	0
$373$	−16.5086	−0.854785	−0.427392	−	0.904066i	$-0.640568\pi$
$373$	−16.5086	−0.854785	−0.427392	+	0.904066i	$0.640568\pi$
$374$	1.30184	0.0673165
$375$	0	0
$376$	−18.9658	−0.978086
$377$	7.48380	0.385435
$378$	0	0
$379$	−15.4445	−0.793332	−0.396666	−	0.917963i	$-0.629833\pi$
$379$	−15.4445	−0.793332	−0.396666	+	0.917963i	$0.629833\pi$
$380$	−8.66991	−0.444757
$381$	0	0
$382$	2.22066	0.113619
$383$	19.1481	0.978420	0.489210	−	0.872166i	$-0.337285\pi$
$383$	19.1481	0.978420	0.489210	+	0.872166i	$0.337285\pi$
$384$	0	0
$385$	−0.917524	−0.0467613
$386$	14.7703	0.751790
$387$	0	0
$388$	−10.1582	−0.515706
$389$	2.37394	0.120364	0.0601819	−	0.998187i	$-0.480832\pi$
$389$	2.37394	0.120364	0.0601819	+	0.998187i	$0.480832\pi$
$390$	0	0
$391$	−10.4986	−0.530939
$392$	16.8912	0.853137
$393$	0	0
$394$	−18.7022	−0.942205
$395$	27.4487	1.38110
$396$	0	0
$397$	−38.1843	−1.91641	−0.958207	−	0.286076i	$-0.907649\pi$
$397$	−38.1843	−1.91641	−0.958207	+	0.286076i	$0.907649\pi$
$398$	−3.08933	−0.154854
$399$	0	0
$400$	1.03897	0.0519483
$401$	−4.91341	−0.245364	−0.122682	−	0.992446i	$-0.539150\pi$
$401$	−4.91341	−0.245364	−0.122682	+	0.992446i	$0.539150\pi$
$402$	0	0
$403$	−13.4390	−0.669446
$404$	8.17760	0.406851
$405$	0	0
$406$	0.696659	0.0345746
$407$	−10.7819	−0.534438
$408$	0	0
$409$	−16.7395	−0.827713	−0.413857	−	0.910342i	$-0.635818\pi$
$409$	−16.7395	−0.827713	−0.413857	+	0.910342i	$0.635818\pi$
$410$	0.698966	0.0345194
$411$	0	0
$412$	−7.65912	−0.377338
$413$	1.12433	0.0553248
$414$	0	0
$415$	−31.3521	−1.53901
$416$	16.7463	0.821057
$417$	0	0
$418$	−1.67821	−0.0820837
$419$	−26.2978	−1.28473	−0.642365	−	0.766399i	$-0.722047\pi$
$419$	−26.2978	−1.28473	−0.642365	+	0.766399i	$0.722047\pi$
$420$	0	0
$421$	30.6961	1.49604	0.748019	−	0.663678i	$-0.231005\pi$
$421$	30.6961	1.49604	0.748019	+	0.663678i	$0.231005\pi$
$422$	11.6888	0.569001
$423$	0	0
$424$	−7.88201	−0.382784
$425$	1.50298	0.0729054
$426$	0	0
$427$	−0.380608	−0.0184189
$428$	−10.1574	−0.490975
$429$	0	0
$430$	4.27933	0.206368
$431$	5.76571	0.277724	0.138862	−	0.990312i	$-0.455655\pi$
$431$	5.76571	0.277724	0.138862	+	0.990312i	$0.455655\pi$
$432$	0	0
$433$	−33.7117	−1.62008	−0.810040	−	0.586375i	$-0.800554\pi$
$433$	−33.7117	−1.62008	−0.810040	+	0.586375i	$0.800554\pi$
$434$	−1.25103	−0.0600511
$435$	0	0
$436$	21.2406	1.01724
$437$	13.5338	0.647410
$438$	0	0
$439$	−26.5662	−1.26793	−0.633967	−	0.773360i	$-0.718574\pi$
$439$	−26.5662	−1.26793	−0.633967	+	0.773360i	$0.718574\pi$
$440$	5.93998	0.283177
$441$	0	0
$442$	3.74076	0.177930
$443$	29.8314	1.41733	0.708667	−	0.705543i	$-0.249297\pi$
$443$	29.8314	1.41733	0.708667	+	0.705543i	$0.249297\pi$
$444$	0	0
$445$	−16.3092	−0.773130
$446$	−17.9166	−0.848376
$447$	0	0
$448$	0.584154	0.0275987
$449$	22.1043	1.04316	0.521582	−	0.853201i	$-0.325342\pi$
$449$	22.1043	1.04316	0.521582	+	0.853201i	$0.325342\pi$
$450$	0	0
$451$	−0.412567	−0.0194270
$452$	−19.7376	−0.928376
$453$	0	0
$454$	7.38489	0.346590
$455$	−2.63645	−0.123599
$456$	0	0
$457$	−1.08398	−0.0507063	−0.0253531	−	0.999679i	$-0.508071\pi$
$457$	−1.08398	−0.0507063	−0.0253531	+	0.999679i	$0.508071\pi$
$458$	12.0301	0.562129
$459$	0	0
$460$	−20.5773	−0.959422
$461$	1.55423	0.0723878	0.0361939	−	0.999345i	$-0.488477\pi$
$461$	1.55423	0.0723878	0.0361939	+	0.999345i	$0.488477\pi$
$462$	0	0
$463$	−4.51024	−0.209609	−0.104804	−	0.994493i	$-0.533422\pi$
$463$	−4.51024	−0.209609	−0.104804	+	0.994493i	$0.533422\pi$
$464$	3.33505	0.154826
$465$	0	0
$466$	19.4845	0.902604
$467$	33.8119	1.56463	0.782315	−	0.622883i	$-0.214039\pi$
$467$	33.8119	1.56463	0.782315	+	0.622883i	$0.214039\pi$
$468$	0	0
$469$	3.07932	0.142190
$470$	−13.0403	−0.601503
$471$	0	0
$472$	−7.27885	−0.335036
$473$	−2.52589	−0.116141
$474$	0	0
$475$	−1.93750	−0.0888987
$476$	−1.06186	−0.0486701
$477$	0	0
$478$	−4.43514	−0.202859
$479$	−34.9203	−1.59555	−0.797775	−	0.602955i	$-0.793990\pi$
$479$	−34.9203	−1.59555	−0.797775	+	0.602955i	$0.793990\pi$
$480$	0	0
$481$	−30.9811	−1.41262
$482$	9.58755	0.436701
$483$	0	0
$484$	−1.50609	−0.0684588
$485$	−16.2594	−0.738303
$486$	0	0
$487$	5.76879	0.261409	0.130704	−	0.991421i	$-0.458276\pi$
$487$	5.76879	0.261409	0.130704	+	0.991421i	$0.458276\pi$
$488$	2.46403	0.111541
$489$	0	0
$490$	11.6139	0.524662
$491$	17.1848	0.775538	0.387769	−	0.921757i	$-0.373246\pi$
$491$	17.1848	0.775538	0.387769	+	0.921757i	$0.373246\pi$
$492$	0	0
$493$	4.82454	0.217286
$494$	−4.82222	−0.216962
$495$	0	0
$496$	−5.98892	−0.268910
$497$	−5.73866	−0.257414
$498$	0	0
$499$	5.81898	0.260494	0.130247	−	0.991482i	$-0.458423\pi$
$499$	5.81898	0.260494	0.130247	+	0.991482i	$0.458423\pi$
$500$	−15.2077	−0.680109
$501$	0	0
$502$	−10.4350	−0.465735
$503$	−8.74737	−0.390026	−0.195013	−	0.980801i	$-0.562475\pi$
$503$	−8.74737	−0.390026	−0.195013	+	0.980801i	$0.562475\pi$
$504$	0	0
$505$	13.0892	0.582461
$506$	−3.98308	−0.177069
$507$	0	0
$508$	5.33730	0.236804
$509$	38.5778	1.70993	0.854965	−	0.518685i	$-0.173578\pi$
$509$	38.5778	1.70993	0.854965	+	0.518685i	$0.173578\pi$
$510$	0	0
$511$	4.61799	0.204288
$512$	13.7732	0.608695
$513$	0	0
$514$	20.0518	0.884447
$515$	−12.2593	−0.540210
$516$	0	0
$517$	7.69707	0.338517
$518$	−2.88400	−0.126715
$519$	0	0
$520$	17.0682	0.748488
$521$	−10.0315	−0.439488	−0.219744	−	0.975558i	$-0.570522\pi$
$521$	−10.0315	−0.439488	−0.219744	+	0.975558i	$0.570522\pi$
$522$	0	0
$523$	−9.98871	−0.436776	−0.218388	−	0.975862i	$-0.570080\pi$
$523$	−9.98871	−0.436776	−0.218388	+	0.975862i	$0.570080\pi$
$524$	−3.86091	−0.168665
$525$	0	0
$526$	−5.20821	−0.227089
$527$	−8.66366	−0.377395
$528$	0	0
$529$	9.12136	0.396581
$530$	−5.41942	−0.235405
$531$	0	0
$532$	1.36884	0.0593469
$533$	−1.18549	−0.0513491
$534$	0	0
$535$	−16.2580	−0.702896
$536$	−19.9353	−0.861074
$537$	0	0
$538$	14.6368	0.631038
$539$	−6.85514	−0.295272
$540$	0	0
$541$	−19.2991	−0.829733	−0.414867	−	0.909882i	$-0.636172\pi$
$541$	−19.2991	−0.829733	−0.414867	+	0.909882i	$0.636172\pi$
$542$	12.5099	0.537348
$543$	0	0
$544$	10.7958	0.462865
$545$	33.9981	1.45632
$546$	0	0
$547$	−24.6238	−1.05284	−0.526420	−	0.850225i	$-0.676466\pi$
$547$	−24.6238	−1.05284	−0.526420	+	0.850225i	$0.676466\pi$
$548$	−15.2735	−0.652450
$549$	0	0
$550$	0.570218	0.0243142
$551$	−6.21933	−0.264952
$552$	0	0
$553$	−4.33373	−0.184289
$554$	8.17802	0.347451
$555$	0	0
$556$	16.4748	0.698686
$557$	36.0247	1.52642	0.763209	−	0.646152i	$-0.223623\pi$
$557$	36.0247	1.52642	0.763209	+	0.646152i	$0.223623\pi$
$558$	0	0
$559$	−7.25800	−0.306981
$560$	−1.17490	−0.0496485
$561$	0	0
$562$	1.96846	0.0830345
$563$	−9.03522	−0.380789	−0.190394	−	0.981708i	$-0.560977\pi$
$563$	−9.03522	−0.380789	−0.190394	+	0.981708i	$0.560977\pi$
$564$	0	0
$565$	−31.5922	−1.32910
$566$	5.10933	0.214761
$567$	0	0
$568$	37.1517	1.55885
$569$	−20.1685	−0.845506	−0.422753	−	0.906245i	$-0.638936\pi$
$569$	−20.1685	−0.845506	−0.422753	+	0.906245i	$0.638936\pi$
$570$	0	0
$571$	10.5977	0.443502	0.221751	−	0.975103i	$-0.428823\pi$
$571$	10.5977	0.443502	0.221751	+	0.975103i	$0.428823\pi$
$572$	−4.32767	−0.180949
$573$	0	0
$574$	−0.110356	−0.00460616
$575$	−4.59849	−0.191770
$576$	0	0
$577$	26.2551	1.09302	0.546508	−	0.837454i	$-0.315957\pi$
$577$	26.2551	1.09302	0.546508	+	0.837454i	$0.315957\pi$
$578$	−9.53580	−0.396637
$579$	0	0
$580$	9.45609	0.392643
$581$	4.95000	0.205361
$582$	0	0
$583$	3.19883	0.132482
$584$	−29.8965	−1.23713
$585$	0	0
$586$	−9.59869	−0.396518
$587$	−28.0666	−1.15843	−0.579216	−	0.815174i	$-0.696641\pi$
$587$	−28.0666	−1.15843	−0.579216	+	0.815174i	$0.696641\pi$
$588$	0	0
$589$	11.1684	0.460184
$590$	−5.00470	−0.206040
$591$	0	0
$592$	−13.8063	−0.567435
$593$	−1.25793	−0.0516569	−0.0258285	−	0.999666i	$-0.508222\pi$
$593$	−1.25793	−0.0516569	−0.0258285	+	0.999666i	$0.508222\pi$
$594$	0	0
$595$	−1.69962	−0.0696778
$596$	0.804907	0.0329703
$597$	0	0
$598$	−11.4451	−0.468026
$599$	1.05494	0.0431036	0.0215518	−	0.999768i	$-0.493139\pi$
$599$	1.05494	0.0431036	0.0215518	+	0.999768i	$0.493139\pi$
$600$	0	0
$601$	44.9488	1.83350	0.916749	−	0.399464i	$-0.130804\pi$
$601$	44.9488	1.83350	0.916749	+	0.399464i	$0.130804\pi$
$602$	−0.675639	−0.0275370
$603$	0	0
$604$	−19.8297	−0.806859
$605$	−2.41068	−0.0980080
$606$	0	0
$607$	14.9112	0.605229	0.302614	−	0.953113i	$-0.402141\pi$
$607$	14.9112	0.605229	0.302614	+	0.953113i	$0.402141\pi$
$608$	−13.9169	−0.564403
$609$	0	0
$610$	1.69419	0.0685956
$611$	22.1171	0.894761
$612$	0	0
$613$	−48.0971	−1.94263	−0.971313	−	0.237807i	$-0.923572\pi$
$613$	−48.0971	−1.94263	−0.971313	+	0.237807i	$0.923572\pi$
$614$	2.33600	0.0942734
$615$	0	0
$616$	−0.937829	−0.0377862
$617$	29.3356	1.18100	0.590502	−	0.807036i	$-0.298930\pi$
$617$	29.3356	1.18100	0.590502	+	0.807036i	$0.298930\pi$
$618$	0	0
$619$	−32.5047	−1.30647	−0.653236	−	0.757154i	$-0.726589\pi$
$619$	−32.5047	−1.30647	−0.653236	+	0.757154i	$0.726589\pi$
$620$	−16.9808	−0.681964
$621$	0	0
$622$	15.3576	0.615784
$623$	2.57496	0.103164
$624$	0	0
$625$	−28.3985	−1.13594
$626$	5.98683	0.239282
$627$	0	0
$628$	−23.7506	−0.947751
$629$	−19.9724	−0.796352
$630$	0	0
$631$	26.5348	1.05633	0.528167	−	0.849140i	$-0.322879\pi$
$631$	26.5348	1.05633	0.528167	+	0.849140i	$0.322879\pi$
$632$	28.0562	1.11602
$633$	0	0
$634$	−3.05153	−0.121192
$635$	8.54296	0.339017
$636$	0	0
$637$	−19.6978	−0.780456
$638$	1.83038	0.0724656
$639$	0	0
$640$	25.4985	1.00792
$641$	−6.11121	−0.241378	−0.120689	−	0.992690i	$-0.538510\pi$
$641$	−6.11121	−0.241378	−0.120689	+	0.992690i	$0.538510\pi$
$642$	0	0
$643$	25.1837	0.993146	0.496573	−	0.867995i	$-0.334591\pi$
$643$	25.1837	0.993146	0.496573	+	0.867995i	$0.334591\pi$
$644$	3.24883	0.128022
$645$	0	0
$646$	−3.10872	−0.122311
$647$	1.12978	0.0444163	0.0222081	−	0.999753i	$-0.492930\pi$
$647$	1.12978	0.0444163	0.0222081	+	0.999753i	$0.492930\pi$
$648$	0	0
$649$	2.95404	0.115956
$650$	1.63849	0.0642667
$651$	0	0
$652$	−15.2625	−0.597725
$653$	31.7849	1.24384	0.621919	−	0.783081i	$-0.286353\pi$
$653$	31.7849	1.24384	0.621919	+	0.783081i	$0.286353\pi$
$654$	0	0
$655$	−6.17984	−0.241466
$656$	−0.528296	−0.0206265
$657$	0	0
$658$	2.05885	0.0802625
$659$	−10.6011	−0.412962	−0.206481	−	0.978451i	$-0.566201\pi$
$659$	−10.6011	−0.412962	−0.206481	+	0.978451i	$0.566201\pi$
$660$	0	0
$661$	−9.22285	−0.358727	−0.179364	−	0.983783i	$-0.557404\pi$
$661$	−9.22285	−0.358727	−0.179364	+	0.983783i	$0.557404\pi$
$662$	10.5027	0.408199
$663$	0	0
$664$	−32.0459	−1.24362
$665$	2.19099	0.0849630
$666$	0	0
$667$	−14.7610	−0.571550
$668$	−2.18376	−0.0844923
$669$	0	0
$670$	−13.7069	−0.529543
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−40.0076	−1.54218	−0.771090	−	0.636726i	$-0.780288\pi$
$673$	−40.0076	−1.54218	−0.771090	+	0.636726i	$0.780288\pi$
$674$	−17.0651	−0.657324
$675$	0	0
$676$	7.14393	0.274767
$677$	29.5650	1.13628	0.568138	−	0.822933i	$-0.307664\pi$
$677$	29.5650	1.13628	0.568138	+	0.822933i	$0.307664\pi$
$678$	0	0
$679$	2.56711	0.0985166
$680$	11.0032	0.421955
$681$	0	0
$682$	−3.28691	−0.125862
$683$	−40.1267	−1.53541	−0.767703	−	0.640806i	$-0.778600\pi$
$683$	−40.1267	−1.53541	−0.767703	+	0.640806i	$0.778600\pi$
$684$	0	0
$685$	−24.4469	−0.934070
$686$	−3.70605	−0.141498
$687$	0	0
$688$	−3.23443	−0.123311
$689$	9.19165	0.350174
$690$	0	0
$691$	6.15715	0.234229	0.117114	−	0.993118i	$-0.462636\pi$
$691$	6.15715	0.234229	0.117114	+	0.993118i	$0.462636\pi$
$692$	−30.7355	−1.16839
$693$	0	0
$694$	−1.72566	−0.0655053
$695$	26.3698	1.00026
$696$	0	0
$697$	−0.764241	−0.0289477
$698$	5.53193	0.209387
$699$	0	0
$700$	−0.465103	−0.0175792
$701$	41.4955	1.56726	0.783631	−	0.621226i	$-0.213365\pi$
$701$	41.4955	1.56726	0.783631	+	0.621226i	$0.213365\pi$
$702$	0	0
$703$	25.7465	0.971047
$704$	1.53479	0.0578446
$705$	0	0
$706$	−1.36007	−0.0511871
$707$	−2.06658	−0.0777216
$708$	0	0
$709$	32.1998	1.20929	0.604645	−	0.796495i	$-0.293315\pi$
$709$	32.1998	1.20929	0.604645	+	0.796495i	$0.293315\pi$
$710$	25.5443	0.958660
$711$	0	0
$712$	−16.6701	−0.624739
$713$	26.5071	0.992700
$714$	0	0
$715$	−6.92694	−0.259053
$716$	−3.29747	−0.123232
$717$	0	0
$718$	19.1483	0.714609
$719$	9.41955	0.351290	0.175645	−	0.984454i	$-0.443799\pi$
$719$	9.41955	0.351290	0.175645	+	0.984454i	$0.443799\pi$
$720$	0	0
$721$	1.93555	0.0720837
$722$	−9.34545	−0.347801
$723$	0	0
$724$	−11.4973	−0.427294
$725$	2.11319	0.0784820
$726$	0	0
$727$	32.0232	1.18768	0.593838	−	0.804585i	$-0.297612\pi$
$727$	32.0232	1.18768	0.593838	+	0.804585i	$0.297612\pi$
$728$	−2.69479	−0.0998757
$729$	0	0
$730$	−20.5559	−0.760807
$731$	−4.67897	−0.173058
$732$	0	0
$733$	18.1215	0.669334	0.334667	−	0.942336i	$-0.391376\pi$
$733$	18.1215	0.669334	0.334667	+	0.942336i	$0.391376\pi$
$734$	9.24520	0.341246
$735$	0	0
$736$	−33.0305	−1.21752
$737$	8.09053	0.298019
$738$	0	0
$739$	37.9867	1.39736	0.698681	−	0.715433i	$-0.253771\pi$
$739$	37.9867	1.39736	0.698681	+	0.715433i	$0.253771\pi$
$740$	−39.1459	−1.43903
$741$	0	0
$742$	0.855641	0.0314116
$743$	−15.3100	−0.561671	−0.280835	−	0.959756i	$-0.590611\pi$
$743$	−15.3100	−0.561671	−0.280835	+	0.959756i	$0.590611\pi$
$744$	0	0
$745$	1.28835	0.0472014
$746$	−11.6020	−0.424780
$747$	0	0
$748$	−2.78990	−0.102009
$749$	2.56689	0.0937921
$750$	0	0
$751$	29.5527	1.07839	0.539197	−	0.842180i	$-0.318728\pi$
$751$	29.5527	1.07839	0.539197	+	0.842180i	$0.318728\pi$
$752$	9.85617	0.359418
$753$	0	0
$754$	5.25949	0.191539
$755$	−31.7397	−1.15513
$756$	0	0
$757$	−12.2744	−0.446122	−0.223061	−	0.974805i	$-0.571605\pi$
$757$	−12.2744	−0.446122	−0.223061	+	0.974805i	$0.571605\pi$
$758$	−10.8542	−0.394241
$759$	0	0
$760$	−14.1843	−0.514519
$761$	−43.1127	−1.56283	−0.781416	−	0.624010i	$-0.785502\pi$
$761$	−43.1127	−1.56283	−0.781416	+	0.624010i	$0.785502\pi$
$762$	0	0
$763$	−5.36777	−0.194326
$764$	−4.75896	−0.172173
$765$	0	0
$766$	13.4570	0.486220
$767$	8.48826	0.306493
$768$	0	0
$769$	−20.1046	−0.724992	−0.362496	−	0.931985i	$-0.618075\pi$
$769$	−20.1046	−0.724992	−0.362496	+	0.931985i	$0.618075\pi$
$770$	−0.644821	−0.0232377
$771$	0	0
$772$	−31.6534	−1.13923
$773$	−33.3091	−1.19804	−0.599022	−	0.800733i	$-0.704444\pi$
$773$	−33.3091	−1.19804	−0.599022	+	0.800733i	$0.704444\pi$
$774$	0	0
$775$	−3.79476	−0.136312
$776$	−16.6193	−0.596597
$777$	0	0
$778$	1.66837	0.0598140
$779$	0.985185	0.0352979
$780$	0	0
$781$	−15.0776	−0.539519
$782$	−7.37827	−0.263846
$783$	0	0
$784$	−8.77807	−0.313502
$785$	−38.0156	−1.35683
$786$	0	0
$787$	−23.3276	−0.831540	−0.415770	−	0.909470i	$-0.636488\pi$
$787$	−23.3276	−0.831540	−0.415770	+	0.909470i	$0.636488\pi$
$788$	40.0796	1.42778
$789$	0	0
$790$	19.2905	0.686327
$791$	4.98792	0.177350
$792$	0	0
$793$	−2.87344	−0.102039
$794$	−26.8353	−0.952349
$795$	0	0
$796$	6.62056	0.234659
$797$	1.30429	0.0462002	0.0231001	−	0.999733i	$-0.492646\pi$
$797$	1.30429	0.0462002	0.0231001	+	0.999733i	$0.492646\pi$
$798$	0	0
$799$	14.2581	0.504415
$800$	4.72864	0.167183
$801$	0	0
$802$	−3.45307	−0.121932
$803$	12.1332	0.428171
$804$	0	0
$805$	5.20013	0.183281
$806$	−9.44474	−0.332677
$807$	0	0
$808$	13.3789	0.470667
$809$	−40.5654	−1.42620	−0.713102	−	0.701060i	$-0.752710\pi$
$809$	−40.5654	−1.42620	−0.713102	+	0.701060i	$0.752710\pi$
$810$	0	0
$811$	−22.7975	−0.800530	−0.400265	−	0.916400i	$-0.631082\pi$
$811$	−22.7975	−0.800530	−0.400265	+	0.916400i	$0.631082\pi$
$812$	−1.49297	−0.0523929
$813$	0	0
$814$	−7.57734	−0.265585
$815$	−24.4294	−0.855724
$816$	0	0
$817$	6.03168	0.211022
$818$	−11.7642	−0.411327
$819$	0	0
$820$	−1.49791	−0.0523093
$821$	1.93050	0.0673750	0.0336875	−	0.999432i	$-0.489275\pi$
$821$	1.93050	0.0673750	0.0336875	+	0.999432i	$0.489275\pi$
$822$	0	0
$823$	−0.886777	−0.0309111	−0.0154556	−	0.999881i	$-0.504920\pi$
$823$	−0.886777	−0.0309111	−0.0154556	+	0.999881i	$0.504920\pi$
$824$	−12.5306	−0.436525
$825$	0	0
$826$	0.790164	0.0274933
$827$	−29.2678	−1.01774	−0.508871	−	0.860843i	$-0.669937\pi$
$827$	−29.2678	−1.01774	−0.508871	+	0.860843i	$0.669937\pi$
$828$	0	0
$829$	−45.6952	−1.58706	−0.793529	−	0.608532i	$-0.791759\pi$
$829$	−45.6952	−1.58706	−0.793529	+	0.608532i	$0.791759\pi$
$830$	−22.0338	−0.764803
$831$	0	0
$832$	4.41013	0.152894
$833$	−12.6985	−0.439976
$834$	0	0
$835$	−3.49536	−0.120962
$836$	3.59646	0.124386
$837$	0	0
$838$	−18.4817	−0.638438
$839$	−32.2318	−1.11277	−0.556383	−	0.830926i	$-0.687811\pi$
$839$	−32.2318	−1.11277	−0.556383	+	0.830926i	$0.687811\pi$
$840$	0	0
$841$	−22.2167	−0.766094
$842$	21.5727	0.743446
$843$	0	0
$844$	−25.0495	−0.862240
$845$	11.4347	0.393365
$846$	0	0
$847$	0.380608	0.0130779
$848$	4.09613	0.140662
$849$	0	0
$850$	1.05627	0.0362299
$851$	61.1071	2.09472
$852$	0	0
$853$	−2.70072	−0.0924710	−0.0462355	−	0.998931i	$-0.514722\pi$
$853$	−2.70072	−0.0924710	−0.0462355	+	0.998931i	$0.514722\pi$
$854$	−0.267485	−0.00915316
$855$	0	0
$856$	−16.6178	−0.567986
$857$	−14.4998	−0.495303	−0.247652	−	0.968849i	$-0.579659\pi$
$857$	−14.4998	−0.495303	−0.247652	+	0.968849i	$0.579659\pi$
$858$	0	0
$859$	27.3214	0.932195	0.466098	−	0.884733i	$-0.345660\pi$
$859$	27.3214	0.932195	0.466098	+	0.884733i	$0.345660\pi$
$860$	−9.17078	−0.312721
$861$	0	0
$862$	4.05205	0.138013
$863$	25.9733	0.884142	0.442071	−	0.896980i	$-0.354244\pi$
$863$	25.9733	0.884142	0.442071	+	0.896980i	$0.354244\pi$
$864$	0	0
$865$	−49.1957	−1.67271
$866$	−23.6920	−0.805088
$867$	0	0
$868$	2.68100	0.0909989
$869$	−11.3863	−0.386254
$870$	0	0
$871$	23.2477	0.787717
$872$	34.7505	1.17680
$873$	0	0
$874$	9.51135	0.321726
$875$	3.84317	0.129923
$876$	0	0
$877$	−2.74500	−0.0926922	−0.0463461	−	0.998925i	$-0.514758\pi$
$877$	−2.74500	−0.0926922	−0.0463461	+	0.998925i	$0.514758\pi$
$878$	−18.6703	−0.630091
$879$	0	0
$880$	−3.08690	−0.104059
$881$	51.4362	1.73293	0.866465	−	0.499239i	$-0.166387\pi$
$881$	51.4362	1.73293	0.866465	+	0.499239i	$0.166387\pi$
$882$	0	0
$883$	51.4088	1.73004	0.865021	−	0.501735i	$-0.167305\pi$
$883$	51.4088	1.73004	0.865021	+	0.501735i	$0.167305\pi$
$884$	−8.01659	−0.269627
$885$	0	0
$886$	20.9651	0.704335
$887$	32.0556	1.07632	0.538160	−	0.842843i	$-0.319120\pi$
$887$	32.0556	1.07632	0.538160	+	0.842843i	$0.319120\pi$
$888$	0	0
$889$	−1.34880	−0.0452373
$890$	−11.4618	−0.384202
$891$	0	0
$892$	38.3960	1.28559
$893$	−18.3801	−0.615068
$894$	0	0
$895$	−5.27799	−0.176424
$896$	−4.02582	−0.134493
$897$	0	0
$898$	15.5345	0.518394
$899$	−12.1811	−0.406262
$900$	0	0
$901$	5.92553	0.197408
$902$	−0.289946	−0.00965413
$903$	0	0
$904$	−32.2914	−1.07400
$905$	−18.4028	−0.611729
$906$	0	0
$907$	47.5518	1.57893	0.789466	−	0.613795i	$-0.210358\pi$
$907$	47.5518	1.57893	0.789466	+	0.613795i	$0.210358\pi$
$908$	−15.8261	−0.525208
$909$	0	0
$910$	−1.85285	−0.0614215
$911$	0.501609	0.0166191	0.00830953	−	0.999965i	$-0.497355\pi$
$911$	0.501609	0.0166191	0.00830953	+	0.999965i	$0.497355\pi$
$912$	0	0
$913$	13.0055	0.430419
$914$	−0.761802	−0.0251982
$915$	0	0
$916$	−25.7810	−0.851827
$917$	0.975699	0.0322204
$918$	0	0
$919$	−0.293282	−0.00967447	−0.00483724	−	0.999988i	$-0.501540\pi$
$919$	−0.293282	−0.00967447	−0.00483724	+	0.999988i	$0.501540\pi$
$920$	−33.6652	−1.10991
$921$	0	0
$922$	1.09229	0.0359726
$923$	−43.3246	−1.42605
$924$	0	0
$925$	−8.74809	−0.287635
$926$	−3.16973	−0.104164
$927$	0	0
$928$	15.1788	0.498269
$929$	−13.1097	−0.430114	−0.215057	−	0.976601i	$-0.568994\pi$
$929$	−13.1097	−0.430114	−0.215057	+	0.976601i	$0.568994\pi$
$930$	0	0
$931$	16.3697	0.536494
$932$	−41.7562	−1.36777
$933$	0	0
$934$	23.7625	0.777532
$935$	−4.46555	−0.146039
$936$	0	0
$937$	27.9120	0.911844	0.455922	−	0.890020i	$-0.349310\pi$
$937$	27.9120	0.911844	0.455922	+	0.890020i	$0.349310\pi$
$938$	2.16410	0.0706604
$939$	0	0
$940$	27.9458	0.911492
$941$	−40.5741	−1.32268	−0.661340	−	0.750087i	$-0.730012\pi$
$941$	−40.5741	−1.32268	−0.661340	+	0.750087i	$0.730012\pi$
$942$	0	0
$943$	2.33825	0.0761440
$944$	3.78268	0.123116
$945$	0	0
$946$	−1.77516	−0.0577154
$947$	−14.0369	−0.456138	−0.228069	−	0.973645i	$-0.573241\pi$
$947$	−14.0369	−0.456138	−0.228069	+	0.973645i	$0.573241\pi$
$948$	0	0
$949$	34.8639	1.13173
$950$	−1.36165	−0.0441776
$951$	0	0
$952$	−1.73724	−0.0563042
$953$	20.8014	0.673824	0.336912	−	0.941536i	$-0.390617\pi$
$953$	20.8014	0.673824	0.336912	+	0.941536i	$0.390617\pi$
$954$	0	0
$955$	−7.61726	−0.246489
$956$	9.50469	0.307404
$957$	0	0
$958$	−24.5415	−0.792899
$959$	3.85979	0.124639
$960$	0	0
$961$	−9.12582	−0.294381
$962$	−21.7730	−0.701990
$963$	0	0
$964$	−20.5465	−0.661758
$965$	−50.6649	−1.63096
$966$	0	0
$967$	29.9347	0.962636	0.481318	−	0.876546i	$-0.340158\pi$
$967$	29.9347	0.962636	0.481318	+	0.876546i	$0.340158\pi$
$968$	−2.46403	−0.0791968
$969$	0	0
$970$	−11.4269	−0.366895
$971$	−34.0269	−1.09198	−0.545988	−	0.837793i	$-0.683846\pi$
$971$	−34.0269	−1.09198	−0.545988	+	0.837793i	$0.683846\pi$
$972$	0	0
$973$	−4.16337	−0.133472
$974$	4.05421	0.129905
$975$	0	0
$976$	−1.28051	−0.0409881
$977$	8.40382	0.268862	0.134431	−	0.990923i	$-0.457079\pi$
$977$	8.40382	0.268862	0.134431	+	0.990923i	$0.457079\pi$
$978$	0	0
$979$	6.76539	0.216223
$980$	−24.8890	−0.795050
$981$	0	0
$982$	12.0772	0.385399
$983$	3.76815	0.120185	0.0600926	−	0.998193i	$-0.480860\pi$
$983$	3.76815	0.120185	0.0600926	+	0.998193i	$0.480860\pi$
$984$	0	0
$985$	64.1521	2.04406
$986$	3.39061	0.107979
$987$	0	0
$988$	10.3342	0.328775
$989$	14.3157	0.455212
$990$	0	0
$991$	−29.8258	−0.947448	−0.473724	−	0.880673i	$-0.657091\pi$
$991$	−29.8258	−0.947448	−0.473724	+	0.880673i	$0.657091\pi$
$992$	−27.2574	−0.865422
$993$	0	0
$994$	−4.03304	−0.127920
$995$	10.5970	0.335946
$996$	0	0
$997$	10.9003	0.345216	0.172608	−	0.984991i	$-0.444781\pi$
$997$	10.9003	0.345216	0.172608	+	0.984991i	$0.444781\pi$
$998$	4.08949	0.129451
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.15	yes	25
3.2	odd	2		6039.2.a.n.1.11	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.11	✓	25	3.2	odd	2
6039.2.a.o.1.15	yes	25	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+0.702784 q^{2} -1.50609 q^{4} -2.41068 q^{5} +0.380608 q^{7} -2.46403 q^{8} +O(q^{10})\)	\(q+0.702784 q^{2} -1.50609 q^{4} -2.41068 q^{5} +0.380608 q^{7} -2.46403 q^{8} -1.69419 q^{10} +1.00000 q^{11} +2.87344 q^{13} +0.267485 q^{14} +1.28051 q^{16} +1.85240 q^{17} -2.38794 q^{19} +3.63071 q^{20} +0.702784 q^{22} -5.66757 q^{23} +0.811370 q^{25} +2.01941 q^{26} -0.573232 q^{28} +2.60447 q^{29} -4.67698 q^{31} +5.82798 q^{32} +1.30184 q^{34} -0.917524 q^{35} -10.7819 q^{37} -1.67821 q^{38} +5.93998 q^{40} -0.412567 q^{41} -2.52589 q^{43} -1.50609 q^{44} -3.98308 q^{46} +7.69707 q^{47} -6.85514 q^{49} +0.570218 q^{50} -4.32767 q^{52} +3.19883 q^{53} -2.41068 q^{55} -0.937829 q^{56} +1.83038 q^{58} +2.95404 q^{59} -1.00000 q^{61} -3.28691 q^{62} +1.53479 q^{64} -6.92694 q^{65} +8.09053 q^{67} -2.78990 q^{68} -0.644821 q^{70} -15.0776 q^{71} +12.1332 q^{73} -7.57734 q^{74} +3.59646 q^{76} +0.380608 q^{77} -11.3863 q^{79} -3.08690 q^{80} -0.289946 q^{82} +13.0055 q^{83} -4.46555 q^{85} -1.77516 q^{86} -2.46403 q^{88} +6.76539 q^{89} +1.09365 q^{91} +8.53590 q^{92} +5.40938 q^{94} +5.75655 q^{95} +6.74475 q^{97} -4.81768 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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