LMFDB - Embedded newform 6039.2.a.o.1.4

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			1.4
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)

$f(q)$	$=$	$q-1.97018 q^{2} +1.88162 q^{4} +1.48404 q^{5} +2.73292 q^{7} +0.233227 q^{8} +O(q^{10})$	\(q-1.97018 q^{2} +1.88162 q^{4} +1.48404 q^{5} +2.73292 q^{7} +0.233227 q^{8} -2.92383 q^{10} +1.00000 q^{11} +2.15148 q^{13} -5.38436 q^{14} -4.22274 q^{16} -1.06202 q^{17} -6.24859 q^{19} +2.79240 q^{20} -1.97018 q^{22} +5.92666 q^{23} -2.79763 q^{25} -4.23881 q^{26} +5.14233 q^{28} -3.70060 q^{29} -1.98730 q^{31} +7.85312 q^{32} +2.09238 q^{34} +4.05577 q^{35} -11.5032 q^{37} +12.3109 q^{38} +0.346119 q^{40} -4.94074 q^{41} +8.44165 q^{43} +1.88162 q^{44} -11.6766 q^{46} +8.23891 q^{47} +0.468877 q^{49} +5.51184 q^{50} +4.04827 q^{52} -3.86573 q^{53} +1.48404 q^{55} +0.637393 q^{56} +7.29086 q^{58} +5.17477 q^{59} -1.00000 q^{61} +3.91535 q^{62} -7.02660 q^{64} +3.19288 q^{65} +10.5022 q^{67} -1.99832 q^{68} -7.99060 q^{70} +3.27381 q^{71} -0.437277 q^{73} +22.6635 q^{74} -11.7575 q^{76} +2.73292 q^{77} +0.577267 q^{79} -6.26672 q^{80} +9.73416 q^{82} +12.0375 q^{83} -1.57608 q^{85} -16.6316 q^{86} +0.233227 q^{88} +5.54239 q^{89} +5.87983 q^{91} +11.1517 q^{92} -16.2322 q^{94} -9.27316 q^{95} +0.0714936 q^{97} -0.923773 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.97018	−1.39313	−0.696565	−	0.717494i	$-0.745289\pi$
$2$	−1.97018	−1.39313	−0.696565	+	0.717494i	$0.745289\pi$
$3$	0	0
$3$	0	0
$4$	1.88162	0.940811
$5$	1.48404	0.663683	0.331841	−	0.943335i	$-0.392330\pi$
$5$	1.48404	0.663683	0.331841	+	0.943335i	$0.392330\pi$
$6$	0	0
$7$	2.73292	1.03295	0.516474	−	0.856303i	$-0.327244\pi$
$7$	2.73292	1.03295	0.516474	+	0.856303i	$0.327244\pi$
$8$	0.233227	0.0824583
$9$	0	0
$10$	−2.92383	−0.924596
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.15148	0.596713	0.298356	−	0.954455i	$-0.403562\pi$
$13$	2.15148	0.596713	0.298356	+	0.954455i	$0.403562\pi$
$14$	−5.38436	−1.43903
$15$	0	0
$16$	−4.22274	−1.05569
$17$	−1.06202	−0.257578	−0.128789	−	0.991672i	$-0.541109\pi$
$17$	−1.06202	−0.257578	−0.128789	+	0.991672i	$0.541109\pi$
$18$	0	0
$19$	−6.24859	−1.43353	−0.716763	−	0.697317i	$-0.754377\pi$
$19$	−6.24859	−1.43353	−0.716763	+	0.697317i	$0.754377\pi$
$20$	2.79240	0.624400
$21$	0	0
$22$	−1.97018	−0.420044
$23$	5.92666	1.23579	0.617897	−	0.786259i	$-0.287985\pi$
$23$	5.92666	1.23579	0.617897	+	0.786259i	$0.287985\pi$
$24$	0	0
$25$	−2.79763	−0.559525
$26$	−4.23881	−0.831299
$27$	0	0
$28$	5.14233	0.971809
$29$	−3.70060	−0.687184	−0.343592	−	0.939119i	$-0.611644\pi$
$29$	−3.70060	−0.687184	−0.343592	+	0.939119i	$0.611644\pi$
$30$	0	0
$31$	−1.98730	−0.356930	−0.178465	−	0.983946i	$-0.557113\pi$
$31$	−1.98730	−0.356930	−0.178465	+	0.983946i	$0.557113\pi$
$32$	7.85312	1.38825
$33$	0	0
$34$	2.09238	0.358840
$35$	4.05577	0.685550
$36$	0	0
$37$	−11.5032	−1.89112	−0.945560	−	0.325448i	$-0.894485\pi$
$37$	−11.5032	−1.89112	−0.945560	+	0.325448i	$0.894485\pi$
$38$	12.3109	1.99709
$39$	0	0
$40$	0.346119	0.0547262
$41$	−4.94074	−0.771613	−0.385807	−	0.922580i	$-0.626077\pi$
$41$	−4.94074	−0.771613	−0.385807	+	0.922580i	$0.626077\pi$
$42$	0	0
$43$	8.44165	1.28734	0.643670	−	0.765303i	$-0.277411\pi$
$43$	8.44165	1.28734	0.643670	+	0.765303i	$0.277411\pi$
$44$	1.88162	0.283665
$45$	0	0
$46$	−11.6766	−1.72162
$47$	8.23891	1.20177	0.600884	−	0.799336i	$-0.294815\pi$
$47$	8.23891	1.20177	0.600884	+	0.799336i	$0.294815\pi$
$48$	0	0
$49$	0.468877	0.0669824
$50$	5.51184	0.779492
$51$	0	0
$52$	4.04827	0.561394
$53$	−3.86573	−0.530999	−0.265500	−	0.964111i	$-0.585537\pi$
$53$	−3.86573	−0.530999	−0.265500	+	0.964111i	$0.585537\pi$
$54$	0	0
$55$	1.48404	0.200108
$56$	0.637393	0.0851752
$57$	0	0
$58$	7.29086	0.957337
$59$	5.17477	0.673697	0.336849	−	0.941559i	$-0.390639\pi$
$59$	5.17477	0.673697	0.336849	+	0.941559i	$0.390639\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	3.91535	0.497250
$63$	0	0
$64$	−7.02660	−0.878325
$65$	3.19288	0.396028
$66$	0	0
$67$	10.5022	1.28304	0.641521	−	0.767105i	$-0.278304\pi$
$67$	10.5022	1.28304	0.641521	+	0.767105i	$0.278304\pi$
$68$	−1.99832	−0.242332
$69$	0	0
$70$	−7.99060	−0.955060
$71$	3.27381	0.388530	0.194265	−	0.980949i	$-0.437768\pi$
$71$	3.27381	0.388530	0.194265	+	0.980949i	$0.437768\pi$
$72$	0	0
$73$	−0.437277	−0.0511794	−0.0255897	−	0.999673i	$-0.508146\pi$
$73$	−0.437277	−0.0511794	−0.0255897	+	0.999673i	$0.508146\pi$
$74$	22.6635	2.63458
$75$	0	0
$76$	−11.7575	−1.34868
$77$	2.73292	0.311446
$78$	0	0
$79$	0.577267	0.0649476	0.0324738	−	0.999473i	$-0.489661\pi$
$79$	0.577267	0.0649476	0.0324738	+	0.999473i	$0.489661\pi$
$80$	−6.26672	−0.700640
$81$	0	0
$82$	9.73416	1.07496
$83$	12.0375	1.32129	0.660645	−	0.750699i	$-0.270283\pi$
$83$	12.0375	1.32129	0.660645	+	0.750699i	$0.270283\pi$
$84$	0	0
$85$	−1.57608	−0.170950
$86$	−16.6316	−1.79343
$87$	0	0
$88$	0.233227	0.0248621
$89$	5.54239	0.587493	0.293746	−	0.955883i	$-0.405098\pi$
$89$	5.54239	0.587493	0.293746	+	0.955883i	$0.405098\pi$
$90$	0	0
$91$	5.87983	0.616374
$92$	11.1517	1.16265
$93$	0	0
$94$	−16.2322	−1.67422
$95$	−9.27316	−0.951406
$96$	0	0
$97$	0.0714936	0.00725908	0.00362954	−	0.999993i	$-0.498845\pi$
$97$	0.0714936	0.00725908	0.00362954	+	0.999993i	$0.498845\pi$
$98$	−0.923773	−0.0933152
$99$	0	0
$100$	−5.26408	−0.526408
$101$	11.8757	1.18167	0.590837	−	0.806791i	$-0.298798\pi$
$101$	11.8757	1.18167	0.590837	+	0.806791i	$0.298798\pi$
$102$	0	0
$103$	13.0250	1.28339	0.641695	−	0.766960i	$-0.278232\pi$
$103$	13.0250	1.28339	0.641695	+	0.766960i	$0.278232\pi$
$104$	0.501784	0.0492040
$105$	0	0
$106$	7.61620	0.739751
$107$	12.7242	1.23009	0.615047	−	0.788490i	$-0.289137\pi$
$107$	12.7242	1.23009	0.615047	+	0.788490i	$0.289137\pi$
$108$	0	0
$109$	9.98085	0.955992	0.477996	−	0.878362i	$-0.341363\pi$
$109$	9.98085	0.955992	0.477996	+	0.878362i	$0.341363\pi$
$110$	−2.92383	−0.278776
$111$	0	0
$112$	−11.5404	−1.09047
$113$	17.4671	1.64317	0.821583	−	0.570089i	$-0.193091\pi$
$113$	17.4671	1.64317	0.821583	+	0.570089i	$0.193091\pi$
$114$	0	0
$115$	8.79539	0.820174
$116$	−6.96313	−0.646510
$117$	0	0
$118$	−10.1952	−0.938548
$119$	−2.90242	−0.266065
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.97018	0.178372
$123$	0	0
$124$	−3.73935	−0.335803
$125$	−11.5720	−1.03503
$126$	0	0
$127$	0.625885	0.0555383	0.0277692	−	0.999614i	$-0.491160\pi$
$127$	0.625885	0.0555383	0.0277692	+	0.999614i	$0.491160\pi$
$128$	−1.86255	−0.164628
$129$	0	0
$130$	−6.29056	−0.551718
$131$	−2.40934	−0.210505	−0.105252	−	0.994446i	$-0.533565\pi$
$131$	−2.40934	−0.210505	−0.105252	+	0.994446i	$0.533565\pi$
$132$	0	0
$133$	−17.0769	−1.48076
$134$	−20.6912	−1.78745
$135$	0	0
$136$	−0.247693	−0.0212395
$137$	−1.14931	−0.0981923	−0.0490962	−	0.998794i	$-0.515634\pi$
$137$	−1.14931	−0.0981923	−0.0490962	+	0.998794i	$0.515634\pi$
$138$	0	0
$139$	0.490304	0.0415870	0.0207935	−	0.999784i	$-0.493381\pi$
$139$	0.490304	0.0415870	0.0207935	+	0.999784i	$0.493381\pi$
$140$	7.63142	0.644973
$141$	0	0
$142$	−6.45000	−0.541272
$143$	2.15148	0.179916
$144$	0	0
$145$	−5.49184	−0.456072
$146$	0.861516	0.0712996
$147$	0	0
$148$	−21.6447	−1.77919
$149$	11.9168	0.976261	0.488130	−	0.872771i	$-0.337679\pi$
$149$	11.9168	0.976261	0.488130	+	0.872771i	$0.337679\pi$
$150$	0	0
$151$	−13.5435	−1.10215	−0.551077	−	0.834454i	$-0.685783\pi$
$151$	−13.5435	−1.10215	−0.551077	+	0.834454i	$0.685783\pi$
$152$	−1.45734	−0.118206
$153$	0	0
$154$	−5.38436	−0.433884
$155$	−2.94923	−0.236888
$156$	0	0
$157$	7.81792	0.623938	0.311969	−	0.950092i	$-0.399011\pi$
$157$	7.81792	0.623938	0.311969	+	0.950092i	$0.399011\pi$
$158$	−1.13732	−0.0904804
$159$	0	0
$160$	11.6543	0.921357
$161$	16.1971	1.27651
$162$	0	0
$163$	22.9585	1.79825	0.899126	−	0.437690i	$-0.144203\pi$
$163$	22.9585	1.79825	0.899126	+	0.437690i	$0.144203\pi$
$164$	−9.29660	−0.725942
$165$	0	0
$166$	−23.7161	−1.84073
$167$	11.1457	0.862477	0.431238	−	0.902238i	$-0.358077\pi$
$167$	11.1457	0.862477	0.431238	+	0.902238i	$0.358077\pi$
$168$	0	0
$169$	−8.37114	−0.643934
$170$	3.10517	0.238156
$171$	0	0
$172$	15.8840	1.21114
$173$	14.4135	1.09584	0.547920	−	0.836531i	$-0.315420\pi$
$173$	14.4135	1.09584	0.547920	+	0.836531i	$0.315420\pi$
$174$	0	0
$175$	−7.64570	−0.577961
$176$	−4.22274	−0.318301
$177$	0	0
$178$	−10.9195	−0.818453
$179$	20.8579	1.55900	0.779498	−	0.626405i	$-0.215474\pi$
$179$	20.8579	1.55900	0.779498	+	0.626405i	$0.215474\pi$
$180$	0	0
$181$	−11.8764	−0.882769	−0.441384	−	0.897318i	$-0.645512\pi$
$181$	−11.8764	−0.882769	−0.441384	+	0.897318i	$0.645512\pi$
$182$	−11.5843	−0.858688
$183$	0	0
$184$	1.38226	0.101901
$185$	−17.0712	−1.25510
$186$	0	0
$187$	−1.06202	−0.0776627
$188$	15.5025	1.13064
$189$	0	0
$190$	18.2698	1.32543
$191$	−26.0776	−1.88691	−0.943456	−	0.331499i	$-0.892446\pi$
$191$	−26.0776	−1.88691	−0.943456	+	0.331499i	$0.892446\pi$
$192$	0	0
$193$	−4.45717	−0.320834	−0.160417	−	0.987049i	$-0.551284\pi$
$193$	−4.45717	−0.320834	−0.160417	+	0.987049i	$0.551284\pi$
$194$	−0.140856	−0.0101128
$195$	0	0
$196$	0.882249	0.0630178
$197$	5.45723	0.388812	0.194406	−	0.980921i	$-0.437722\pi$
$197$	5.45723	0.388812	0.194406	+	0.980921i	$0.437722\pi$
$198$	0	0
$199$	16.1212	1.14280	0.571400	−	0.820671i	$-0.306400\pi$
$199$	16.1212	1.14280	0.571400	+	0.820671i	$0.306400\pi$
$200$	−0.652483	−0.0461375
$201$	0	0
$202$	−23.3972	−1.64622
$203$	−10.1135	−0.709826
$204$	0	0
$205$	−7.33225	−0.512106
$206$	−25.6616	−1.78793
$207$	0	0
$208$	−9.08514	−0.629941
$209$	−6.24859	−0.432224
$210$	0	0
$211$	−9.88576	−0.680564	−0.340282	−	0.940323i	$-0.610523\pi$
$211$	−9.88576	−0.680564	−0.340282	+	0.940323i	$0.610523\pi$
$212$	−7.27384	−0.499570
$213$	0	0
$214$	−25.0690	−1.71368
$215$	12.5277	0.854385
$216$	0	0
$217$	−5.43114	−0.368690
$218$	−19.6641	−1.33182
$219$	0	0
$220$	2.79240	0.188264
$221$	−2.28492	−0.153700
$222$	0	0
$223$	16.8567	1.12881	0.564405	−	0.825498i	$-0.309106\pi$
$223$	16.8567	1.12881	0.564405	+	0.825498i	$0.309106\pi$
$224$	21.4620	1.43399
$225$	0	0
$226$	−34.4134	−2.28914
$227$	15.0773	1.00072	0.500358	−	0.865819i	$-0.333202\pi$
$227$	15.0773	1.00072	0.500358	+	0.865819i	$0.333202\pi$
$228$	0	0
$229$	−11.9161	−0.787438	−0.393719	−	0.919231i	$-0.628812\pi$
$229$	−11.9161	−0.787438	−0.393719	+	0.919231i	$0.628812\pi$
$230$	−17.3285	−1.14261
$231$	0	0
$232$	−0.863082	−0.0566641
$233$	8.05107	0.527443	0.263721	−	0.964599i	$-0.415050\pi$
$233$	8.05107	0.527443	0.263721	+	0.964599i	$0.415050\pi$
$234$	0	0
$235$	12.2269	0.797593
$236$	9.73695	0.633822
$237$	0	0
$238$	5.71831	0.370663
$239$	−17.2752	−1.11744	−0.558718	−	0.829358i	$-0.688707\pi$
$239$	−17.2752	−1.11744	−0.558718	+	0.829358i	$0.688707\pi$
$240$	0	0
$241$	13.8619	0.892925	0.446462	−	0.894802i	$-0.352684\pi$
$241$	13.8619	0.892925	0.446462	+	0.894802i	$0.352684\pi$
$242$	−1.97018	−0.126648
$243$	0	0
$244$	−1.88162	−0.120458
$245$	0.695832	0.0444550
$246$	0	0
$247$	−13.4437	−0.855403
$248$	−0.463493	−0.0294318
$249$	0	0
$250$	22.7989	1.44193
$251$	−24.4529	−1.54346	−0.771728	−	0.635953i	$-0.780607\pi$
$251$	−24.4529	−1.54346	−0.771728	+	0.635953i	$0.780607\pi$
$252$	0	0
$253$	5.92666	0.372606
$254$	−1.23311	−0.0773721
$255$	0	0
$256$	17.7228	1.10767
$257$	6.87515	0.428860	0.214430	−	0.976739i	$-0.431211\pi$
$257$	6.87515	0.428860	0.214430	+	0.976739i	$0.431211\pi$
$258$	0	0
$259$	−31.4375	−1.95343
$260$	6.00779	0.372587
$261$	0	0
$262$	4.74683	0.293260
$263$	21.8346	1.34638	0.673188	−	0.739471i	$-0.264924\pi$
$263$	21.8346	1.34638	0.673188	+	0.739471i	$0.264924\pi$
$264$	0	0
$265$	−5.73690	−0.352415
$266$	33.6447	2.06289
$267$	0	0
$268$	19.7611	1.20710
$269$	1.71049	0.104290	0.0521452	−	0.998640i	$-0.483394\pi$
$269$	1.71049	0.104290	0.0521452	+	0.998640i	$0.483394\pi$
$270$	0	0
$271$	−20.4720	−1.24359	−0.621793	−	0.783182i	$-0.713595\pi$
$271$	−20.4720	−1.24359	−0.621793	+	0.783182i	$0.713595\pi$
$272$	4.48464	0.271921
$273$	0	0
$274$	2.26436	0.136795
$275$	−2.79763	−0.168703
$276$	0	0
$277$	13.1619	0.790822	0.395411	−	0.918504i	$-0.370602\pi$
$277$	13.1619	0.790822	0.395411	+	0.918504i	$0.370602\pi$
$278$	−0.965988	−0.0579361
$279$	0	0
$280$	0.945916	0.0565293
$281$	11.8425	0.706465	0.353233	−	0.935536i	$-0.385082\pi$
$281$	11.8425	0.706465	0.353233	+	0.935536i	$0.385082\pi$
$282$	0	0
$283$	−0.546144	−0.0324649	−0.0162324	−	0.999868i	$-0.505167\pi$
$283$	−0.546144	−0.0324649	−0.0162324	+	0.999868i	$0.505167\pi$
$284$	6.16007	0.365533
$285$	0	0
$286$	−4.23881	−0.250646
$287$	−13.5027	−0.797037
$288$	0	0
$289$	−15.8721	−0.933654
$290$	10.8199	0.635368
$291$	0	0
$292$	−0.822790	−0.0481501
$293$	−16.2442	−0.948997	−0.474499	−	0.880256i	$-0.657371\pi$
$293$	−16.2442	−0.948997	−0.474499	+	0.880256i	$0.657371\pi$
$294$	0	0
$295$	7.67956	0.447121
$296$	−2.68287	−0.155939
$297$	0	0
$298$	−23.4782	−1.36006
$299$	12.7511	0.737414
$300$	0	0
$301$	23.0704	1.32976
$302$	26.6832	1.53544
$303$	0	0
$304$	26.3862	1.51335
$305$	−1.48404	−0.0849758
$306$	0	0
$307$	12.2548	0.699417	0.349709	−	0.936858i	$-0.386281\pi$
$307$	12.2548	0.699417	0.349709	+	0.936858i	$0.386281\pi$
$308$	5.14233	0.293011
$309$	0	0
$310$	5.81053	0.330016
$311$	0.895732	0.0507923	0.0253962	−	0.999677i	$-0.491915\pi$
$311$	0.895732	0.0507923	0.0253962	+	0.999677i	$0.491915\pi$
$312$	0	0
$313$	−30.9752	−1.75082	−0.875412	−	0.483377i	$-0.839410\pi$
$313$	−30.9752	−1.75082	−0.875412	+	0.483377i	$0.839410\pi$
$314$	−15.4027	−0.869227
$315$	0	0
$316$	1.08620	0.0611034
$317$	3.41938	0.192052	0.0960258	−	0.995379i	$-0.469387\pi$
$317$	3.41938	0.192052	0.0960258	+	0.995379i	$0.469387\pi$
$318$	0	0
$319$	−3.70060	−0.207194
$320$	−10.4278	−0.582929
$321$	0	0
$322$	−31.9113	−1.77834
$323$	6.63614	0.369245
$324$	0	0
$325$	−6.01904	−0.333876
$326$	−45.2325	−2.50520
$327$	0	0
$328$	−1.15232	−0.0636260
$329$	22.5163	1.24137
$330$	0	0
$331$	−12.9652	−0.712630	−0.356315	−	0.934366i	$-0.615967\pi$
$331$	−12.9652	−0.712630	−0.356315	+	0.934366i	$0.615967\pi$
$332$	22.6501	1.24308
$333$	0	0
$334$	−21.9590	−1.20154
$335$	15.5856	0.851533
$336$	0	0
$337$	28.8250	1.57020	0.785098	−	0.619372i	$-0.212613\pi$
$337$	28.8250	1.57020	0.785098	+	0.619372i	$0.212613\pi$
$338$	16.4927	0.897083
$339$	0	0
$340$	−2.96559	−0.160832
$341$	−1.98730	−0.107618
$342$	0	0
$343$	−17.8491	−0.963759
$344$	1.96882	0.106152
$345$	0	0
$346$	−28.3973	−1.52665
$347$	−21.2988	−1.14338	−0.571690	−	0.820470i	$-0.693712\pi$
$347$	−21.2988	−1.14338	−0.571690	+	0.820470i	$0.693712\pi$
$348$	0	0
$349$	19.2877	1.03245	0.516225	−	0.856453i	$-0.327337\pi$
$349$	19.2877	1.03245	0.516225	+	0.856453i	$0.327337\pi$
$350$	15.0634	0.805175
$351$	0	0
$352$	7.85312	0.418573
$353$	18.0954	0.963119	0.481560	−	0.876413i	$-0.340070\pi$
$353$	18.0954	0.963119	0.481560	+	0.876413i	$0.340070\pi$
$354$	0	0
$355$	4.85846	0.257860
$356$	10.4287	0.552719
$357$	0	0
$358$	−41.0940	−2.17188
$359$	26.9343	1.42154	0.710770	−	0.703425i	$-0.248347\pi$
$359$	26.9343	1.42154	0.710770	+	0.703425i	$0.248347\pi$
$360$	0	0
$361$	20.0449	1.05500
$362$	23.3988	1.22981
$363$	0	0
$364$	11.0636	0.579891
$365$	−0.648936	−0.0339669
$366$	0	0
$367$	−3.93047	−0.205169	−0.102584	−	0.994724i	$-0.532711\pi$
$367$	−3.93047	−0.205169	−0.102584	+	0.994724i	$0.532711\pi$
$368$	−25.0267	−1.30461
$369$	0	0
$370$	33.6335	1.74852
$371$	−10.5648	−0.548495
$372$	0	0
$373$	−6.87679	−0.356066	−0.178033	−	0.984024i	$-0.556973\pi$
$373$	−6.87679	−0.356066	−0.178033	+	0.984024i	$0.556973\pi$
$374$	2.09238	0.108194
$375$	0	0
$376$	1.92154	0.0990959
$377$	−7.96176	−0.410052
$378$	0	0
$379$	19.8839	1.02137	0.510684	−	0.859768i	$-0.329392\pi$
$379$	19.8839	1.02137	0.510684	+	0.859768i	$0.329392\pi$
$380$	−17.4486	−0.895093
$381$	0	0
$382$	51.3777	2.62871
$383$	−10.7375	−0.548658	−0.274329	−	0.961636i	$-0.588456\pi$
$383$	−10.7375	−0.548658	−0.274329	+	0.961636i	$0.588456\pi$
$384$	0	0
$385$	4.05577	0.206701
$386$	8.78144	0.446963
$387$	0	0
$388$	0.134524	0.00682942
$389$	−9.87375	−0.500619	−0.250310	−	0.968166i	$-0.580532\pi$
$389$	−9.87375	−0.500619	−0.250310	+	0.968166i	$0.580532\pi$
$390$	0	0
$391$	−6.29423	−0.318313
$392$	0.109355	0.00552326
$393$	0	0
$394$	−10.7517	−0.541665
$395$	0.856687	0.0431046
$396$	0	0
$397$	10.1898	0.511410	0.255705	−	0.966755i	$-0.417692\pi$
$397$	10.1898	0.511410	0.255705	+	0.966755i	$0.417692\pi$
$398$	−31.7617	−1.59207
$399$	0	0
$400$	11.8137	0.590683
$401$	−3.86852	−0.193185	−0.0965923	−	0.995324i	$-0.530794\pi$
$401$	−3.86852	−0.193185	−0.0965923	+	0.995324i	$0.530794\pi$
$402$	0	0
$403$	−4.27564	−0.212985
$404$	22.3455	1.11173
$405$	0	0
$406$	19.9254	0.988880
$407$	−11.5032	−0.570194
$408$	0	0
$409$	17.8473	0.882493	0.441246	−	0.897386i	$-0.354537\pi$
$409$	17.8473	0.882493	0.441246	+	0.897386i	$0.354537\pi$
$410$	14.4459	0.713431
$411$	0	0
$412$	24.5081	1.20743
$413$	14.1422	0.695895
$414$	0	0
$415$	17.8641	0.876917
$416$	16.8958	0.828386
$417$	0	0
$418$	12.3109	0.602144
$419$	−25.6592	−1.25353	−0.626767	−	0.779206i	$-0.715622\pi$
$419$	−25.6592	−1.25353	−0.626767	+	0.779206i	$0.715622\pi$
$420$	0	0
$421$	−31.1673	−1.51900	−0.759500	−	0.650507i	$-0.774557\pi$
$421$	−31.1673	−1.51900	−0.759500	+	0.650507i	$0.774557\pi$
$422$	19.4768	0.948114
$423$	0	0
$424$	−0.901595	−0.0437853
$425$	2.97114	0.144121
$426$	0	0
$427$	−2.73292	−0.132255
$428$	23.9421	1.15729
$429$	0	0
$430$	−24.6819	−1.19027
$431$	−13.3439	−0.642754	−0.321377	−	0.946951i	$-0.604146\pi$
$431$	−13.3439	−0.642754	−0.321377	+	0.946951i	$0.604146\pi$
$432$	0	0
$433$	26.0826	1.25345	0.626724	−	0.779241i	$-0.284395\pi$
$433$	26.0826	1.25345	0.626724	+	0.779241i	$0.284395\pi$
$434$	10.7003	0.513633
$435$	0	0
$436$	18.7802	0.899407
$437$	−37.0333	−1.77154
$438$	0	0
$439$	29.3693	1.40172	0.700860	−	0.713298i	$-0.252800\pi$
$439$	29.3693	1.40172	0.700860	+	0.713298i	$0.252800\pi$
$440$	0.346119	0.0165006
$441$	0	0
$442$	4.50170	0.214124
$443$	−25.2434	−1.19935	−0.599674	−	0.800244i	$-0.704703\pi$
$443$	−25.2434	−1.19935	−0.599674	+	0.800244i	$0.704703\pi$
$444$	0	0
$445$	8.22513	0.389909
$446$	−33.2108	−1.57258
$447$	0	0
$448$	−19.2032	−0.907265
$449$	−25.9684	−1.22552	−0.612762	−	0.790267i	$-0.709942\pi$
$449$	−25.9684	−1.22552	−0.612762	+	0.790267i	$0.709942\pi$
$450$	0	0
$451$	−4.94074	−0.232650
$452$	32.8665	1.54591
$453$	0	0
$454$	−29.7051	−1.39413
$455$	8.72590	0.409076
$456$	0	0
$457$	34.8817	1.63170	0.815849	−	0.578266i	$-0.196270\pi$
$457$	34.8817	1.63170	0.815849	+	0.578266i	$0.196270\pi$
$458$	23.4769	1.09700
$459$	0	0
$460$	16.5496	0.771629
$461$	28.2242	1.31453	0.657267	−	0.753658i	$-0.271712\pi$
$461$	28.2242	1.31453	0.657267	+	0.753658i	$0.271712\pi$
$462$	0	0
$463$	2.84727	0.132324	0.0661620	−	0.997809i	$-0.478925\pi$
$463$	2.84727	0.132324	0.0661620	+	0.997809i	$0.478925\pi$
$464$	15.6267	0.725451
$465$	0	0
$466$	−15.8621	−0.734796
$467$	−12.8337	−0.593872	−0.296936	−	0.954897i	$-0.595965\pi$
$467$	−12.8337	−0.593872	−0.296936	+	0.954897i	$0.595965\pi$
$468$	0	0
$469$	28.7016	1.32532
$470$	−24.0892	−1.11115
$471$	0	0
$472$	1.20690	0.0555520
$473$	8.44165	0.388147
$474$	0	0
$475$	17.4812	0.802094
$476$	−5.46126	−0.250317
$477$	0	0
$478$	34.0352	1.55673
$479$	−9.07271	−0.414543	−0.207271	−	0.978284i	$-0.566458\pi$
$479$	−9.07271	−0.414543	−0.207271	+	0.978284i	$0.566458\pi$
$480$	0	0
$481$	−24.7490	−1.12846
$482$	−27.3105	−1.24396
$483$	0	0
$484$	1.88162	0.0855282
$485$	0.106099	0.00481772
$486$	0	0
$487$	−7.96538	−0.360946	−0.180473	−	0.983580i	$-0.557763\pi$
$487$	−7.96538	−0.360946	−0.180473	+	0.983580i	$0.557763\pi$
$488$	−0.233227	−0.0105577
$489$	0	0
$490$	−1.37092	−0.0619317
$491$	−23.0482	−1.04015	−0.520074	−	0.854121i	$-0.674096\pi$
$491$	−23.0482	−1.04015	−0.520074	+	0.854121i	$0.674096\pi$
$492$	0	0
$493$	3.93012	0.177004
$494$	26.4866	1.19169
$495$	0	0
$496$	8.39186	0.376806
$497$	8.94707	0.401331
$498$	0	0
$499$	−28.9732	−1.29702	−0.648510	−	0.761206i	$-0.724607\pi$
$499$	−28.9732	−1.29702	−0.648510	+	0.761206i	$0.724607\pi$
$500$	−21.7741	−0.973767
$501$	0	0
$502$	48.1768	2.15023
$503$	−22.2722	−0.993068	−0.496534	−	0.868017i	$-0.665394\pi$
$503$	−22.2722	−0.993068	−0.496534	+	0.868017i	$0.665394\pi$
$504$	0	0
$505$	17.6240	0.784256
$506$	−11.6766	−0.519088
$507$	0	0
$508$	1.17768	0.0522510
$509$	4.13733	0.183384	0.0916920	−	0.995787i	$-0.470772\pi$
$509$	4.13733	0.183384	0.0916920	+	0.995787i	$0.470772\pi$
$510$	0	0
$511$	−1.19505	−0.0528657
$512$	−31.1920	−1.37850
$513$	0	0
$514$	−13.5453	−0.597457
$515$	19.3296	0.851763
$516$	0	0
$517$	8.23891	0.362347
$518$	61.9376	2.72138
$519$	0	0
$520$	0.744667	0.0326558
$521$	25.1690	1.10267	0.551337	−	0.834283i	$-0.314118\pi$
$521$	25.1690	1.10267	0.551337	+	0.834283i	$0.314118\pi$
$522$	0	0
$523$	−24.5867	−1.07510	−0.537551	−	0.843231i	$-0.680650\pi$
$523$	−24.5867	−1.07510	−0.537551	+	0.843231i	$0.680650\pi$
$524$	−4.53346	−0.198045
$525$	0	0
$526$	−43.0181	−1.87568
$527$	2.11056	0.0919373
$528$	0	0
$529$	12.1252	0.527184
$530$	11.3027	0.490960
$531$	0	0
$532$	−32.1323	−1.39311
$533$	−10.6299	−0.460432
$534$	0	0
$535$	18.8832	0.816392
$536$	2.44939	0.105798
$537$	0	0
$538$	−3.36998	−0.145290
$539$	0.468877	0.0201960
$540$	0	0
$541$	37.7216	1.62178	0.810889	−	0.585200i	$-0.198984\pi$
$541$	37.7216	1.62178	0.810889	+	0.585200i	$0.198984\pi$
$542$	40.3336	1.73248
$543$	0	0
$544$	−8.34019	−0.357582
$545$	14.8120	0.634475
$546$	0	0
$547$	−12.4600	−0.532751	−0.266376	−	0.963869i	$-0.585826\pi$
$547$	−12.4600	−0.532751	−0.266376	+	0.963869i	$0.585826\pi$
$548$	−2.16257	−0.0923804
$549$	0	0
$550$	5.51184	0.235026
$551$	23.1235	0.985096
$552$	0	0
$553$	1.57763	0.0670875
$554$	−25.9314	−1.10172
$555$	0	0
$556$	0.922566	0.0391255
$557$	3.92077	0.166128	0.0830641	−	0.996544i	$-0.473529\pi$
$557$	3.92077	0.166128	0.0830641	+	0.996544i	$0.473529\pi$
$558$	0	0
$559$	18.1620	0.768172
$560$	−17.1265	−0.723725
$561$	0	0
$562$	−23.3319	−0.984197
$563$	30.4188	1.28200	0.640999	−	0.767542i	$-0.278520\pi$
$563$	30.4188	1.28200	0.640999	+	0.767542i	$0.278520\pi$
$564$	0	0
$565$	25.9219	1.09054
$566$	1.07600	0.0452278
$567$	0	0
$568$	0.763542	0.0320375
$569$	21.1623	0.887169	0.443584	−	0.896233i	$-0.353707\pi$
$569$	21.1623	0.887169	0.443584	+	0.896233i	$0.353707\pi$
$570$	0	0
$571$	14.7107	0.615622	0.307811	−	0.951448i	$-0.400404\pi$
$571$	14.7107	0.615622	0.307811	+	0.951448i	$0.400404\pi$
$572$	4.04827	0.169267
$573$	0	0
$574$	26.6027	1.11038
$575$	−16.5806	−0.691458
$576$	0	0
$577$	17.6996	0.736843	0.368421	−	0.929659i	$-0.379898\pi$
$577$	17.6996	0.736843	0.368421	+	0.929659i	$0.379898\pi$
$578$	31.2710	1.30070
$579$	0	0
$580$	−10.3336	−0.429078
$581$	32.8976	1.36482
$582$	0	0
$583$	−3.86573	−0.160102
$584$	−0.101985	−0.00422017
$585$	0	0
$586$	32.0041	1.32208
$587$	−10.1097	−0.417272	−0.208636	−	0.977993i	$-0.566902\pi$
$587$	−10.1097	−0.417272	−0.208636	+	0.977993i	$0.566902\pi$
$588$	0	0
$589$	12.4178	0.511668
$590$	−15.1301	−0.622898
$591$	0	0
$592$	48.5752	1.99643
$593$	−14.8070	−0.608052	−0.304026	−	0.952664i	$-0.598331\pi$
$593$	−14.8070	−0.608052	−0.304026	+	0.952664i	$0.598331\pi$
$594$	0	0
$595$	−4.30731	−0.176583
$596$	22.4229	0.918476
$597$	0	0
$598$	−25.1219	−1.02731
$599$	9.79057	0.400032	0.200016	−	0.979793i	$-0.435901\pi$
$599$	9.79057	0.400032	0.200016	+	0.979793i	$0.435901\pi$
$600$	0	0
$601$	−7.03531	−0.286976	−0.143488	−	0.989652i	$-0.545832\pi$
$601$	−7.03531	−0.286976	−0.143488	+	0.989652i	$0.545832\pi$
$602$	−45.4529	−1.85252
$603$	0	0
$604$	−25.4837	−1.03692
$605$	1.48404	0.0603348
$606$	0	0
$607$	−22.1277	−0.898134	−0.449067	−	0.893498i	$-0.648244\pi$
$607$	−22.1277	−0.898134	−0.449067	+	0.893498i	$0.648244\pi$
$608$	−49.0710	−1.99009
$609$	0	0
$610$	2.92383	0.118382
$611$	17.7258	0.717111
$612$	0	0
$613$	−32.4063	−1.30888	−0.654438	−	0.756115i	$-0.727095\pi$
$613$	−32.4063	−1.30888	−0.654438	+	0.756115i	$0.727095\pi$
$614$	−24.1442	−0.974379
$615$	0	0
$616$	0.637393	0.0256813
$617$	−29.5002	−1.18763	−0.593816	−	0.804601i	$-0.702379\pi$
$617$	−29.5002	−1.18763	−0.593816	+	0.804601i	$0.702379\pi$
$618$	0	0
$619$	6.13979	0.246779	0.123389	−	0.992358i	$-0.460624\pi$
$619$	6.13979	0.246779	0.123389	+	0.992358i	$0.460624\pi$
$620$	−5.54934	−0.222867
$621$	0	0
$622$	−1.76476	−0.0707603
$623$	15.1469	0.606850
$624$	0	0
$625$	−3.18514	−0.127406
$626$	61.0269	2.43913
$627$	0	0
$628$	14.7104	0.587008
$629$	12.2167	0.487111
$630$	0	0
$631$	46.2770	1.84226	0.921130	−	0.389256i	$-0.127268\pi$
$631$	46.2770	1.84226	0.921130	+	0.389256i	$0.127268\pi$
$632$	0.134634	0.00535547
$633$	0	0
$634$	−6.73680	−0.267553
$635$	0.928838	0.0368598
$636$	0	0
$637$	1.00878	0.0399693
$638$	7.29086	0.288648
$639$	0	0
$640$	−2.76410	−0.109261
$641$	36.7404	1.45116	0.725580	−	0.688138i	$-0.241572\pi$
$641$	36.7404	1.45116	0.725580	+	0.688138i	$0.241572\pi$
$642$	0	0
$643$	−26.7744	−1.05588	−0.527939	−	0.849282i	$-0.677035\pi$
$643$	−26.7744	−1.05588	−0.527939	+	0.849282i	$0.677035\pi$
$644$	30.4768	1.20095
$645$	0	0
$646$	−13.0744	−0.514406
$647$	−16.7677	−0.659207	−0.329604	−	0.944119i	$-0.606915\pi$
$647$	−16.7677	−0.659207	−0.329604	+	0.944119i	$0.606915\pi$
$648$	0	0
$649$	5.17477	0.203127
$650$	11.8586	0.465133
$651$	0	0
$652$	43.1993	1.69181
$653$	−27.6734	−1.08294	−0.541471	−	0.840719i	$-0.682133\pi$
$653$	−27.6734	−1.08294	−0.541471	+	0.840719i	$0.682133\pi$
$654$	0	0
$655$	−3.57555	−0.139708
$656$	20.8635	0.814581
$657$	0	0
$658$	−44.3613	−1.72938
$659$	−21.4196	−0.834388	−0.417194	−	0.908817i	$-0.636986\pi$
$659$	−21.4196	−0.834388	−0.417194	+	0.908817i	$0.636986\pi$
$660$	0	0
$661$	2.24211	0.0872082	0.0436041	−	0.999049i	$-0.486116\pi$
$661$	2.24211	0.0872082	0.0436041	+	0.999049i	$0.486116\pi$
$662$	25.5438	0.992786
$663$	0	0
$664$	2.80748	0.108951
$665$	−25.3428	−0.982753
$666$	0	0
$667$	−21.9322	−0.849218
$668$	20.9719	0.811427
$669$	0	0
$670$	−30.7065	−1.18630
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	15.7390	0.606696	0.303348	−	0.952880i	$-0.401896\pi$
$673$	15.7390	0.606696	0.303348	+	0.952880i	$0.401896\pi$
$674$	−56.7905	−2.18749
$675$	0	0
$676$	−15.7513	−0.605820
$677$	12.1410	0.466617	0.233308	−	0.972403i	$-0.425045\pi$
$677$	12.1410	0.466617	0.233308	+	0.972403i	$0.425045\pi$
$678$	0	0
$679$	0.195387	0.00749825
$680$	−0.367585	−0.0140963
$681$	0	0
$682$	3.91535	0.149926
$683$	25.8861	0.990506	0.495253	−	0.868749i	$-0.335075\pi$
$683$	25.8861	0.990506	0.495253	+	0.868749i	$0.335075\pi$
$684$	0	0
$685$	−1.70562	−0.0651685
$686$	35.1659	1.34264
$687$	0	0
$688$	−35.6469	−1.35903
$689$	−8.31704	−0.316854
$690$	0	0
$691$	−42.5280	−1.61784	−0.808921	−	0.587917i	$-0.799948\pi$
$691$	−42.5280	−1.61784	−0.808921	+	0.587917i	$0.799948\pi$
$692$	27.1208	1.03098
$693$	0	0
$694$	41.9626	1.59288
$695$	0.727630	0.0276006
$696$	0	0
$697$	5.24717	0.198751
$698$	−38.0004	−1.43834
$699$	0	0
$700$	−14.3863	−0.543752
$701$	8.70577	0.328812	0.164406	−	0.986393i	$-0.447429\pi$
$701$	8.70577	0.328812	0.164406	+	0.986393i	$0.447429\pi$
$702$	0	0
$703$	71.8790	2.71097
$704$	−7.02660	−0.264825
$705$	0	0
$706$	−35.6512	−1.34175
$707$	32.4553	1.22061
$708$	0	0
$709$	17.7013	0.664786	0.332393	−	0.943141i	$-0.392144\pi$
$709$	17.7013	0.664786	0.332393	+	0.943141i	$0.392144\pi$
$710$	−9.57206	−0.359233
$711$	0	0
$712$	1.29264	0.0484437
$713$	−11.7780	−0.441091
$714$	0	0
$715$	3.19288	0.119407
$716$	39.2467	1.46672
$717$	0	0
$718$	−53.0656	−1.98039
$719$	18.2606	0.681005	0.340502	−	0.940244i	$-0.389403\pi$
$719$	18.2606	0.681005	0.340502	+	0.940244i	$0.389403\pi$
$720$	0	0
$721$	35.5963	1.32567
$722$	−39.4922	−1.46975
$723$	0	0
$724$	−22.3470	−0.830518
$725$	10.3529	0.384497
$726$	0	0
$727$	13.5400	0.502170	0.251085	−	0.967965i	$-0.419213\pi$
$727$	13.5400	0.502170	0.251085	+	0.967965i	$0.419213\pi$
$728$	1.37134	0.0508251
$729$	0	0
$730$	1.27852	0.0473203
$731$	−8.96521	−0.331590
$732$	0	0
$733$	−21.0598	−0.777861	−0.388930	−	0.921267i	$-0.627155\pi$
$733$	−21.0598	−0.777861	−0.388930	+	0.921267i	$0.627155\pi$
$734$	7.74374	0.285827
$735$	0	0
$736$	46.5428	1.71559
$737$	10.5022	0.386852
$738$	0	0
$739$	−25.0526	−0.921576	−0.460788	−	0.887510i	$-0.652433\pi$
$739$	−25.0526	−0.921576	−0.460788	+	0.887510i	$0.652433\pi$
$740$	−32.1216	−1.18081
$741$	0	0
$742$	20.8145	0.764124
$743$	24.0230	0.881319	0.440660	−	0.897674i	$-0.354745\pi$
$743$	24.0230	0.881319	0.440660	+	0.897674i	$0.354745\pi$
$744$	0	0
$745$	17.6850	0.647927
$746$	13.5485	0.496047
$747$	0	0
$748$	−1.99832	−0.0730659
$749$	34.7743	1.27062
$750$	0	0
$751$	4.13935	0.151047	0.0755235	−	0.997144i	$-0.475937\pi$
$751$	4.13935	0.151047	0.0755235	+	0.997144i	$0.475937\pi$
$752$	−34.7908	−1.26869
$753$	0	0
$754$	15.6861	0.571255
$755$	−20.0991	−0.731481
$756$	0	0
$757$	43.6351	1.58594	0.792972	−	0.609258i	$-0.208533\pi$
$757$	43.6351	1.58594	0.792972	+	0.609258i	$0.208533\pi$
$758$	−39.1750	−1.42290
$759$	0	0
$760$	−2.16275	−0.0784514
$761$	0.0336184	0.00121867	0.000609333	−	1.00000i	$-0.499806\pi$
$761$	0.0336184	0.00121867	0.000609333		1.00000i	$0.499806\pi$
$762$	0	0
$763$	27.2769	0.987490
$764$	−49.0682	−1.77523
$765$	0	0
$766$	21.1547	0.764352
$767$	11.1334	0.402004
$768$	0	0
$769$	6.30268	0.227280	0.113640	−	0.993522i	$-0.463749\pi$
$769$	6.30268	0.227280	0.113640	+	0.993522i	$0.463749\pi$
$770$	−7.99060	−0.287961
$771$	0	0
$772$	−8.38670	−0.301844
$773$	−0.807464	−0.0290425	−0.0145212	−	0.999895i	$-0.504622\pi$
$773$	−0.807464	−0.0290425	−0.0145212	+	0.999895i	$0.504622\pi$
$774$	0	0
$775$	5.55973	0.199711
$776$	0.0166743	0.000598572 0
$777$	0	0
$778$	19.4531	0.697427
$779$	30.8727	1.10613
$780$	0	0
$781$	3.27381	0.117146
$782$	12.4008	0.443452
$783$	0	0
$784$	−1.97995	−0.0707124
$785$	11.6021	0.414097
$786$	0	0
$787$	23.0904	0.823084	0.411542	−	0.911391i	$-0.364990\pi$
$787$	23.0904	0.823084	0.411542	+	0.911391i	$0.364990\pi$
$788$	10.2684	0.365798
$789$	0	0
$790$	−1.68783	−0.0600503
$791$	47.7362	1.69731
$792$	0	0
$793$	−2.15148	−0.0764013
$794$	−20.0757	−0.712461
$795$	0	0
$796$	30.3340	1.07516
$797$	−28.7063	−1.01683	−0.508415	−	0.861112i	$-0.669768\pi$
$797$	−28.7063	−1.01683	−0.508415	+	0.861112i	$0.669768\pi$
$798$	0	0
$799$	−8.74990	−0.309549
$800$	−21.9701	−0.776761
$801$	0	0
$802$	7.62169	0.269131
$803$	−0.437277	−0.0154312
$804$	0	0
$805$	24.0371	0.847198
$806$	8.42379	0.296715
$807$	0	0
$808$	2.76973	0.0974388
$809$	14.6813	0.516167	0.258084	−	0.966123i	$-0.416909\pi$
$809$	14.6813	0.516167	0.258084	+	0.966123i	$0.416909\pi$
$810$	0	0
$811$	41.6102	1.46113	0.730566	−	0.682842i	$-0.239256\pi$
$811$	41.6102	1.46113	0.730566	+	0.682842i	$0.239256\pi$
$812$	−19.0297	−0.667812
$813$	0	0
$814$	22.6635	0.794354
$815$	34.0714	1.19347
$816$	0	0
$817$	−52.7484	−1.84543
$818$	−35.1625	−1.22943
$819$	0	0
$820$	−13.7965	−0.481795
$821$	−7.21848	−0.251927	−0.125963	−	0.992035i	$-0.540202\pi$
$821$	−7.21848	−0.251927	−0.125963	+	0.992035i	$0.540202\pi$
$822$	0	0
$823$	−46.5143	−1.62139	−0.810694	−	0.585471i	$-0.800910\pi$
$823$	−46.5143	−1.62139	−0.810694	+	0.585471i	$0.800910\pi$
$824$	3.03778	0.105826
$825$	0	0
$826$	−27.8628	−0.969471
$827$	8.11571	0.282211	0.141105	−	0.989995i	$-0.454934\pi$
$827$	8.11571	0.282211	0.141105	+	0.989995i	$0.454934\pi$
$828$	0	0
$829$	25.8160	0.896626	0.448313	−	0.893877i	$-0.352025\pi$
$829$	25.8160	0.896626	0.448313	+	0.893877i	$0.352025\pi$
$830$	−35.1956	−1.22166
$831$	0	0
$832$	−15.1176	−0.524108
$833$	−0.497957	−0.0172532
$834$	0	0
$835$	16.5406	0.572411
$836$	−11.7575	−0.406641
$837$	0	0
$838$	50.5534	1.74634
$839$	−22.3395	−0.771246	−0.385623	−	0.922656i	$-0.626013\pi$
$839$	−22.3395	−0.771246	−0.385623	+	0.922656i	$0.626013\pi$
$840$	0	0
$841$	−15.3056	−0.527778
$842$	61.4053	2.11617
$843$	0	0
$844$	−18.6013	−0.640282
$845$	−12.4231	−0.427368
$846$	0	0
$847$	2.73292	0.0939044
$848$	16.3240	0.560568
$849$	0	0
$850$	−5.85369	−0.200780
$851$	−68.1757	−2.33703
$852$	0	0
$853$	13.9934	0.479125	0.239562	−	0.970881i	$-0.422996\pi$
$853$	13.9934	0.479125	0.239562	+	0.970881i	$0.422996\pi$
$854$	5.38436	0.184249
$855$	0	0
$856$	2.96763	0.101432
$857$	−13.5955	−0.464414	−0.232207	−	0.972666i	$-0.574595\pi$
$857$	−13.5955	−0.464414	−0.232207	+	0.972666i	$0.574595\pi$
$858$	0	0
$859$	35.0457	1.19574	0.597872	−	0.801592i	$-0.296013\pi$
$859$	35.0457	1.19574	0.597872	+	0.801592i	$0.296013\pi$
$860$	23.5725	0.803814
$861$	0	0
$862$	26.2900	0.895439
$863$	−3.86994	−0.131734	−0.0658672	−	0.997828i	$-0.520981\pi$
$863$	−3.86994	−0.131734	−0.0658672	+	0.997828i	$0.520981\pi$
$864$	0	0
$865$	21.3902	0.727290
$866$	−51.3875	−1.74622
$867$	0	0
$868$	−10.2194	−0.346868
$869$	0.577267	0.0195824
$870$	0	0
$871$	22.5952	0.765608
$872$	2.32781	0.0788295
$873$	0	0
$874$	72.9623	2.46799
$875$	−31.6254	−1.06913
$876$	0	0
$877$	−19.0090	−0.641887	−0.320943	−	0.947098i	$-0.604000\pi$
$877$	−19.0090	−0.641887	−0.320943	+	0.947098i	$0.604000\pi$
$878$	−57.8629	−1.95278
$879$	0	0
$880$	−6.26672	−0.211251
$881$	5.55898	0.187287	0.0936434	−	0.995606i	$-0.470149\pi$
$881$	5.55898	0.187287	0.0936434	+	0.995606i	$0.470149\pi$
$882$	0	0
$883$	−10.6502	−0.358408	−0.179204	−	0.983812i	$-0.557352\pi$
$883$	−10.6502	−0.358408	−0.179204	+	0.983812i	$0.557352\pi$
$884$	−4.29935	−0.144603
$885$	0	0
$886$	49.7340	1.67085
$887$	29.4528	0.988928	0.494464	−	0.869198i	$-0.335364\pi$
$887$	29.4528	0.988928	0.494464	+	0.869198i	$0.335364\pi$
$888$	0	0
$889$	1.71050	0.0573682
$890$	−16.2050	−0.543193
$891$	0	0
$892$	31.7180	1.06200
$893$	−51.4816	−1.72277
$894$	0	0
$895$	30.9540	1.03468
$896$	−5.09021	−0.170052
$897$	0	0
$898$	51.1625	1.70731
$899$	7.35421	0.245277
$900$	0	0
$901$	4.10549	0.136774
$902$	9.73416	0.324112
$903$	0	0
$904$	4.07381	0.135493
$905$	−17.6251	−0.585878
$906$	0	0
$907$	13.0974	0.434893	0.217447	−	0.976072i	$-0.430227\pi$
$907$	13.0974	0.434893	0.217447	+	0.976072i	$0.430227\pi$
$908$	28.3698	0.941484
$909$	0	0
$910$	−17.1916	−0.569897
$911$	0.0880556	0.00291741	0.00145871	−	0.999999i	$-0.499536\pi$
$911$	0.0880556	0.00291741	0.00145871	+	0.999999i	$0.499536\pi$
$912$	0	0
$913$	12.0375	0.398384
$914$	−68.7233	−2.27317
$915$	0	0
$916$	−22.4216	−0.740831
$917$	−6.58453	−0.217440
$918$	0	0
$919$	−13.5676	−0.447554	−0.223777	−	0.974640i	$-0.571839\pi$
$919$	−13.5676	−0.447554	−0.223777	+	0.974640i	$0.571839\pi$
$920$	2.05133	0.0676302
$921$	0	0
$922$	−55.6069	−1.83132
$923$	7.04353	0.231841
$924$	0	0
$925$	32.1818	1.05813
$926$	−5.60965	−0.184345
$927$	0	0
$928$	−29.0613	−0.953983
$929$	−39.4036	−1.29279	−0.646396	−	0.763002i	$-0.723725\pi$
$929$	−39.4036	−1.29279	−0.646396	+	0.763002i	$0.723725\pi$
$930$	0	0
$931$	−2.92982	−0.0960210
$932$	15.1491	0.496224
$933$	0	0
$934$	25.2847	0.827341
$935$	−1.57608	−0.0515434
$936$	0	0
$937$	−54.3177	−1.77448	−0.887241	−	0.461306i	$-0.847381\pi$
$937$	−54.3177	−1.77448	−0.887241	+	0.461306i	$0.847381\pi$
$938$	−56.5474	−1.84634
$939$	0	0
$940$	23.0063	0.750384
$941$	42.5623	1.38749	0.693746	−	0.720219i	$-0.255959\pi$
$941$	42.5623	1.38749	0.693746	+	0.720219i	$0.255959\pi$
$942$	0	0
$943$	−29.2820	−0.953554
$944$	−21.8517	−0.711213
$945$	0	0
$946$	−16.6316	−0.540740
$947$	−55.7071	−1.81024	−0.905119	−	0.425159i	$-0.860218\pi$
$947$	−55.7071	−1.81024	−0.905119	+	0.425159i	$0.860218\pi$
$948$	0	0
$949$	−0.940792	−0.0305394
$950$	−34.4412	−1.11742
$951$	0	0
$952$	−0.676925	−0.0219393
$953$	−10.4718	−0.339215	−0.169608	−	0.985512i	$-0.554250\pi$
$953$	−10.4718	−0.339215	−0.169608	+	0.985512i	$0.554250\pi$
$954$	0	0
$955$	−38.7002	−1.25231
$956$	−32.5053	−1.05130
$957$	0	0
$958$	17.8749	0.577512
$959$	−3.14098	−0.101428
$960$	0	0
$961$	−27.0506	−0.872601
$962$	48.7600	1.57208
$963$	0	0
$964$	26.0829	0.840073
$965$	−6.61461	−0.212932
$966$	0	0
$967$	−33.1759	−1.06687	−0.533433	−	0.845842i	$-0.679098\pi$
$967$	−33.1759	−1.06687	−0.533433	+	0.845842i	$0.679098\pi$
$968$	0.233227	0.00749621
$969$	0	0
$970$	−0.209035	−0.00671172
$971$	−22.4072	−0.719080	−0.359540	−	0.933130i	$-0.617066\pi$
$971$	−22.4072	−0.719080	−0.359540	+	0.933130i	$0.617066\pi$
$972$	0	0
$973$	1.33996	0.0429572
$974$	15.6933	0.502844
$975$	0	0
$976$	4.22274	0.135167
$977$	25.5190	0.816425	0.408212	−	0.912887i	$-0.366152\pi$
$977$	25.5190	0.816425	0.408212	+	0.912887i	$0.366152\pi$
$978$	0	0
$979$	5.54239	0.177136
$980$	1.30929	0.0418238
$981$	0	0
$982$	45.4091	1.44906
$983$	47.8848	1.52729	0.763643	−	0.645639i	$-0.223409\pi$
$983$	47.8848	1.52729	0.763643	+	0.645639i	$0.223409\pi$
$984$	0	0
$985$	8.09874	0.258047
$986$	−7.74305	−0.246589
$987$	0	0
$988$	−25.2960	−0.804772
$989$	50.0307	1.59088
$990$	0	0
$991$	3.76337	0.119547	0.0597737	−	0.998212i	$-0.480962\pi$
$991$	3.76337	0.119547	0.0597737	+	0.998212i	$0.480962\pi$
$992$	−15.6065	−0.495507
$993$	0	0
$994$	−17.6274	−0.559106
$995$	23.9245	0.758457
$996$	0	0
$997$	−1.25091	−0.0396167	−0.0198083	−	0.999804i	$-0.506306\pi$
$997$	−1.25091	−0.0396167	−0.0198083	+	0.999804i	$0.506306\pi$
$998$	57.0825	1.80692
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.4	yes	25
3.2	odd	2		6039.2.a.n.1.22	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.22	✓	25	3.2	odd	2
6039.2.a.o.1.4	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)

\(f(q)\)	\(=\)	\(q-1.97018 q^{2} +1.88162 q^{4} +1.48404 q^{5} +2.73292 q^{7} +0.233227 q^{8} +O(q^{10})\)	\(q-1.97018 q^{2} +1.88162 q^{4} +1.48404 q^{5} +2.73292 q^{7} +0.233227 q^{8} -2.92383 q^{10} +1.00000 q^{11} +2.15148 q^{13} -5.38436 q^{14} -4.22274 q^{16} -1.06202 q^{17} -6.24859 q^{19} +2.79240 q^{20} -1.97018 q^{22} +5.92666 q^{23} -2.79763 q^{25} -4.23881 q^{26} +5.14233 q^{28} -3.70060 q^{29} -1.98730 q^{31} +7.85312 q^{32} +2.09238 q^{34} +4.05577 q^{35} -11.5032 q^{37} +12.3109 q^{38} +0.346119 q^{40} -4.94074 q^{41} +8.44165 q^{43} +1.88162 q^{44} -11.6766 q^{46} +8.23891 q^{47} +0.468877 q^{49} +5.51184 q^{50} +4.04827 q^{52} -3.86573 q^{53} +1.48404 q^{55} +0.637393 q^{56} +7.29086 q^{58} +5.17477 q^{59} -1.00000 q^{61} +3.91535 q^{62} -7.02660 q^{64} +3.19288 q^{65} +10.5022 q^{67} -1.99832 q^{68} -7.99060 q^{70} +3.27381 q^{71} -0.437277 q^{73} +22.6635 q^{74} -11.7575 q^{76} +2.73292 q^{77} +0.577267 q^{79} -6.26672 q^{80} +9.73416 q^{82} +12.0375 q^{83} -1.57608 q^{85} -16.6316 q^{86} +0.233227 q^{88} +5.54239 q^{89} +5.87983 q^{91} +11.1517 q^{92} -16.2322 q^{94} -9.27316 q^{95} +0.0714936 q^{97} -0.923773 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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

Related objects

Downloads

Learn more