LMFDB - Embedded newform 6039.2.a.o.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.23548 q^{2} +2.99739 q^{4} -2.10645 q^{5} -1.54970 q^{7} -2.22964 q^{8} +O(q^{10})$	\(q-2.23548 q^{2} +2.99739 q^{4} -2.10645 q^{5} -1.54970 q^{7} -2.22964 q^{8} +4.70893 q^{10} +1.00000 q^{11} -7.11724 q^{13} +3.46434 q^{14} -1.01045 q^{16} -3.21614 q^{17} -3.37299 q^{19} -6.31384 q^{20} -2.23548 q^{22} +5.08722 q^{23} -0.562873 q^{25} +15.9105 q^{26} -4.64506 q^{28} +2.29976 q^{29} -2.78435 q^{31} +6.71812 q^{32} +7.18963 q^{34} +3.26437 q^{35} -9.34514 q^{37} +7.54027 q^{38} +4.69662 q^{40} -0.774810 q^{41} -3.87964 q^{43} +2.99739 q^{44} -11.3724 q^{46} -9.43926 q^{47} -4.59842 q^{49} +1.25829 q^{50} -21.3331 q^{52} -5.79127 q^{53} -2.10645 q^{55} +3.45528 q^{56} -5.14108 q^{58} -8.69383 q^{59} -1.00000 q^{61} +6.22436 q^{62} -12.9974 q^{64} +14.9921 q^{65} +0.877000 q^{67} -9.64002 q^{68} -7.29745 q^{70} -8.01889 q^{71} +11.3797 q^{73} +20.8909 q^{74} -10.1102 q^{76} -1.54970 q^{77} +3.88549 q^{79} +2.12846 q^{80} +1.73207 q^{82} +10.6148 q^{83} +6.77464 q^{85} +8.67288 q^{86} -2.22964 q^{88} +7.33188 q^{89} +11.0296 q^{91} +15.2484 q^{92} +21.1013 q^{94} +7.10504 q^{95} -17.1464 q^{97} +10.2797 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.23548	−1.58073	−0.790363	−	0.612639i	$-0.790108\pi$
$2$	−2.23548	−1.58073	−0.790363	+	0.612639i	$0.790108\pi$
$3$	0	0
$3$	0	0
$4$	2.99739	1.49869
$5$	−2.10645	−0.942033	−0.471016	−	0.882125i	$-0.656113\pi$
$5$	−2.10645	−0.942033	−0.471016	+	0.882125i	$0.656113\pi$
$6$	0	0
$7$	−1.54970	−0.585733	−0.292867	−	0.956153i	$-0.594609\pi$
$7$	−1.54970	−0.585733	−0.292867	+	0.956153i	$0.594609\pi$
$8$	−2.22964	−0.788296
$9$	0	0
$10$	4.70893	1.48909
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−7.11724	−1.97397	−0.986983	−	0.160824i	$-0.948585\pi$
$13$	−7.11724	−1.97397	−0.986983	+	0.160824i	$0.948585\pi$
$14$	3.46434	0.925884
$15$	0	0
$16$	−1.01045	−0.252613
$17$	−3.21614	−0.780029	−0.390015	−	0.920809i	$-0.627530\pi$
$17$	−3.21614	−0.780029	−0.390015	+	0.920809i	$0.627530\pi$
$18$	0	0
$19$	−3.37299	−0.773818	−0.386909	−	0.922118i	$-0.626457\pi$
$19$	−3.37299	−0.773818	−0.386909	+	0.922118i	$0.626457\pi$
$20$	−6.31384	−1.41182
$21$	0	0
$22$	−2.23548	−0.476607
$23$	5.08722	1.06076	0.530379	−	0.847760i	$-0.322049\pi$
$23$	5.08722	1.06076	0.530379	+	0.847760i	$0.322049\pi$
$24$	0	0
$25$	−0.562873	−0.112575
$26$	15.9105	3.12030
$27$	0	0
$28$	−4.64506	−0.877834
$29$	2.29976	0.427055	0.213527	−	0.976937i	$-0.431505\pi$
$29$	2.29976	0.427055	0.213527	+	0.976937i	$0.431505\pi$
$30$	0	0
$31$	−2.78435	−0.500083	−0.250042	−	0.968235i	$-0.580444\pi$
$31$	−2.78435	−0.500083	−0.250042	+	0.968235i	$0.580444\pi$
$32$	6.71812	1.18761
$33$	0	0
$34$	7.18963	1.23301
$35$	3.26437	0.551780
$36$	0	0
$37$	−9.34514	−1.53633	−0.768166	−	0.640251i	$-0.778830\pi$
$37$	−9.34514	−1.53633	−0.768166	+	0.640251i	$0.778830\pi$
$38$	7.54027	1.22319
$39$	0	0
$40$	4.69662	0.742601
$41$	−0.774810	−0.121005	−0.0605025	−	0.998168i	$-0.519270\pi$
$41$	−0.774810	−0.121005	−0.0605025	+	0.998168i	$0.519270\pi$
$42$	0	0
$43$	−3.87964	−0.591640	−0.295820	−	0.955244i	$-0.595593\pi$
$43$	−3.87964	−0.591640	−0.295820	+	0.955244i	$0.595593\pi$
$44$	2.99739	0.451873
$45$	0	0
$46$	−11.3724	−1.67677
$47$	−9.43926	−1.37686	−0.688429	−	0.725304i	$-0.741699\pi$
$47$	−9.43926	−1.37686	−0.688429	+	0.725304i	$0.741699\pi$
$48$	0	0
$49$	−4.59842	−0.656916
$50$	1.25829	0.177949
$51$	0	0
$52$	−21.3331	−2.95837
$53$	−5.79127	−0.795492	−0.397746	−	0.917496i	$-0.630207\pi$
$53$	−5.79127	−0.795492	−0.397746	+	0.917496i	$0.630207\pi$
$54$	0	0
$55$	−2.10645	−0.284034
$56$	3.45528	0.461731
$57$	0	0
$58$	−5.14108	−0.675057
$59$	−8.69383	−1.13184	−0.565920	−	0.824460i	$-0.691479\pi$
$59$	−8.69383	−1.13184	−0.565920	+	0.824460i	$0.691479\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	6.22436	0.790494
$63$	0	0
$64$	−12.9974	−1.62467
$65$	14.9921	1.85954
$66$	0	0
$67$	0.877000	0.107143	0.0535713	−	0.998564i	$-0.482940\pi$
$67$	0.877000	0.107143	0.0535713	+	0.998564i	$0.482940\pi$
$68$	−9.64002	−1.16902
$69$	0	0
$70$	−7.29745	−0.872212
$71$	−8.01889	−0.951668	−0.475834	−	0.879535i	$-0.657854\pi$
$71$	−8.01889	−0.951668	−0.475834	+	0.879535i	$0.657854\pi$
$72$	0	0
$73$	11.3797	1.33190	0.665948	−	0.745998i	$-0.268027\pi$
$73$	11.3797	1.33190	0.665948	+	0.745998i	$0.268027\pi$
$74$	20.8909	2.42852
$75$	0	0
$76$	−10.1102	−1.15971
$77$	−1.54970	−0.176605
$78$	0	0
$79$	3.88549	0.437151	0.218576	−	0.975820i	$-0.429859\pi$
$79$	3.88549	0.437151	0.218576	+	0.975820i	$0.429859\pi$
$80$	2.12846	0.237970
$81$	0	0
$82$	1.73207	0.191276
$83$	10.6148	1.16513	0.582563	−	0.812785i	$-0.302050\pi$
$83$	10.6148	1.16513	0.582563	+	0.812785i	$0.302050\pi$
$84$	0	0
$85$	6.77464	0.734813
$86$	8.67288	0.935221
$87$	0	0
$88$	−2.22964	−0.237680
$89$	7.33188	0.777177	0.388589	−	0.921411i	$-0.372963\pi$
$89$	7.33188	0.777177	0.388589	+	0.921411i	$0.372963\pi$
$90$	0	0
$91$	11.0296	1.15622
$92$	15.2484	1.58975
$93$	0	0
$94$	21.1013	2.17643
$95$	7.10504	0.728961
$96$	0	0
$97$	−17.1464	−1.74096	−0.870478	−	0.492207i	$-0.836190\pi$
$97$	−17.1464	−1.74096	−0.870478	+	0.492207i	$0.836190\pi$
$98$	10.2797	1.03840
$99$	0	0
$100$	−1.68715	−0.168715
$101$	−14.9192	−1.48452	−0.742258	−	0.670115i	$-0.766245\pi$
$101$	−14.9192	−1.48452	−0.742258	+	0.670115i	$0.766245\pi$
$102$	0	0
$103$	−12.1942	−1.20153	−0.600763	−	0.799427i	$-0.705137\pi$
$103$	−12.1942	−1.20153	−0.600763	+	0.799427i	$0.705137\pi$
$104$	15.8689	1.55607
$105$	0	0
$106$	12.9463	1.25745
$107$	−10.5505	−1.01995	−0.509977	−	0.860188i	$-0.670346\pi$
$107$	−10.5505	−1.01995	−0.509977	+	0.860188i	$0.670346\pi$
$108$	0	0
$109$	5.68537	0.544559	0.272280	−	0.962218i	$-0.412222\pi$
$109$	5.68537	0.544559	0.272280	+	0.962218i	$0.412222\pi$
$110$	4.70893	0.448979
$111$	0	0
$112$	1.56590	0.147964
$113$	−8.06689	−0.758869	−0.379435	−	0.925219i	$-0.623882\pi$
$113$	−8.06689	−0.758869	−0.379435	+	0.925219i	$0.623882\pi$
$114$	0	0
$115$	−10.7160	−0.999269
$116$	6.89327	0.640024
$117$	0	0
$118$	19.4349	1.78913
$119$	4.98407	0.456889
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.23548	0.202391
$123$	0	0
$124$	−8.34576	−0.749471
$125$	11.7179	1.04808
$126$	0	0
$127$	20.7131	1.83799	0.918995	−	0.394268i	$-0.129002\pi$
$127$	20.7131	1.83799	0.918995	+	0.394268i	$0.129002\pi$
$128$	15.6191	1.38055
$129$	0	0
$130$	−33.5146	−2.93942
$131$	−0.647790	−0.0565977	−0.0282989	−	0.999600i	$-0.509009\pi$
$131$	−0.647790	−0.0565977	−0.0282989	+	0.999600i	$0.509009\pi$
$132$	0	0
$133$	5.22714	0.453251
$134$	−1.96052	−0.169363
$135$	0	0
$136$	7.17084	0.614894
$137$	−12.2116	−1.04331	−0.521656	−	0.853156i	$-0.674685\pi$
$137$	−12.2116	−1.04331	−0.521656	+	0.853156i	$0.674685\pi$
$138$	0	0
$139$	−18.9723	−1.60921	−0.804603	−	0.593813i	$-0.797622\pi$
$139$	−18.9723	−1.60921	−0.804603	+	0.593813i	$0.797622\pi$
$140$	9.78459	0.826949
$141$	0	0
$142$	17.9261	1.50432
$143$	−7.11724	−0.595173
$144$	0	0
$145$	−4.84433	−0.402300
$146$	−25.4392	−2.10536
$147$	0	0
$148$	−28.0110	−2.30249
$149$	3.23280	0.264841	0.132421	−	0.991194i	$-0.457725\pi$
$149$	3.23280	0.264841	0.132421	+	0.991194i	$0.457725\pi$
$150$	0	0
$151$	8.86619	0.721521	0.360760	−	0.932659i	$-0.382517\pi$
$151$	8.86619	0.721521	0.360760	+	0.932659i	$0.382517\pi$
$152$	7.52055	0.609998
$153$	0	0
$154$	3.46434	0.279164
$155$	5.86508	0.471095
$156$	0	0
$157$	−17.3956	−1.38832	−0.694159	−	0.719822i	$-0.744224\pi$
$157$	−17.3956	−1.38832	−0.694159	+	0.719822i	$0.744224\pi$
$158$	−8.68594	−0.691016
$159$	0	0
$160$	−14.1514	−1.11877
$161$	−7.88369	−0.621322
$162$	0	0
$163$	6.29493	0.493057	0.246528	−	0.969136i	$-0.420710\pi$
$163$	6.29493	0.493057	0.246528	+	0.969136i	$0.420710\pi$
$164$	−2.32240	−0.181349
$165$	0	0
$166$	−23.7292	−1.84174
$167$	6.53551	0.505733	0.252867	−	0.967501i	$-0.418627\pi$
$167$	6.53551	0.505733	0.252867	+	0.967501i	$0.418627\pi$
$168$	0	0
$169$	37.6550	2.89654
$170$	−15.1446	−1.16154
$171$	0	0
$172$	−11.6288	−0.886687
$173$	−22.9563	−1.74534	−0.872668	−	0.488315i	$-0.837612\pi$
$173$	−22.9563	−1.74534	−0.872668	+	0.488315i	$0.837612\pi$
$174$	0	0
$175$	0.872287	0.0659387
$176$	−1.01045	−0.0761656
$177$	0	0
$178$	−16.3903	−1.22850
$179$	13.2157	0.987791	0.493895	−	0.869521i	$-0.335573\pi$
$179$	13.2157	0.987791	0.493895	+	0.869521i	$0.335573\pi$
$180$	0	0
$181$	−12.9759	−0.964493	−0.482247	−	0.876035i	$-0.660179\pi$
$181$	−12.9759	−0.964493	−0.482247	+	0.876035i	$0.660179\pi$
$182$	−24.6565	−1.82766
$183$	0	0
$184$	−11.3427	−0.836192
$185$	19.6851	1.44727
$186$	0	0
$187$	−3.21614	−0.235188
$188$	−28.2931	−2.06349
$189$	0	0
$190$	−15.8832	−1.15229
$191$	14.7375	1.06637	0.533185	−	0.845999i	$-0.320995\pi$
$191$	14.7375	1.06637	0.533185	+	0.845999i	$0.320995\pi$
$192$	0	0
$193$	−18.9417	−1.36345	−0.681726	−	0.731608i	$-0.738770\pi$
$193$	−18.9417	−1.36345	−0.681726	+	0.731608i	$0.738770\pi$
$194$	38.3306	2.75197
$195$	0	0
$196$	−13.7832	−0.984516
$197$	−14.6396	−1.04303	−0.521516	−	0.853242i	$-0.674633\pi$
$197$	−14.6396	−1.04303	−0.521516	+	0.853242i	$0.674633\pi$
$198$	0	0
$199$	−23.2753	−1.64994	−0.824972	−	0.565174i	$-0.808809\pi$
$199$	−23.2753	−1.64994	−0.824972	+	0.565174i	$0.808809\pi$
$200$	1.25500	0.0887421
$201$	0	0
$202$	33.3516	2.34661
$203$	−3.56395	−0.250140
$204$	0	0
$205$	1.63210	0.113991
$206$	27.2599	1.89928
$207$	0	0
$208$	7.19162	0.498649
$209$	−3.37299	−0.233315
$210$	0	0
$211$	23.3597	1.60815	0.804073	−	0.594531i	$-0.202662\pi$
$211$	23.3597	1.60815	0.804073	+	0.594531i	$0.202662\pi$
$212$	−17.3587	−1.19220
$213$	0	0
$214$	23.5854	1.61227
$215$	8.17227	0.557344
$216$	0	0
$217$	4.31491	0.292915
$218$	−12.7095	−0.860799
$219$	0	0
$220$	−6.31384	−0.425679
$221$	22.8900	1.53975
$222$	0	0
$223$	−4.09514	−0.274231	−0.137115	−	0.990555i	$-0.543783\pi$
$223$	−4.09514	−0.274231	−0.137115	+	0.990555i	$0.543783\pi$
$224$	−10.4111	−0.695621
$225$	0	0
$226$	18.0334	1.19956
$227$	−5.37273	−0.356600	−0.178300	−	0.983976i	$-0.557060\pi$
$227$	−5.37273	−0.356600	−0.178300	+	0.983976i	$0.557060\pi$
$228$	0	0
$229$	−12.6663	−0.837011	−0.418505	−	0.908214i	$-0.637446\pi$
$229$	−12.6663	−0.837011	−0.418505	+	0.908214i	$0.637446\pi$
$230$	23.9554	1.57957
$231$	0	0
$232$	−5.12764	−0.336646
$233$	19.1609	1.25527	0.627636	−	0.778507i	$-0.284022\pi$
$233$	19.1609	1.25527	0.627636	+	0.778507i	$0.284022\pi$
$234$	0	0
$235$	19.8833	1.29705
$236$	−26.0588	−1.69628
$237$	0	0
$238$	−11.1418	−0.722216
$239$	13.0438	0.843731	0.421865	−	0.906658i	$-0.361375\pi$
$239$	13.0438	0.843731	0.421865	+	0.906658i	$0.361375\pi$
$240$	0	0
$241$	2.51420	0.161954	0.0809769	−	0.996716i	$-0.474196\pi$
$241$	2.51420	0.161954	0.0809769	+	0.996716i	$0.474196\pi$
$242$	−2.23548	−0.143702
$243$	0	0
$244$	−2.99739	−0.191888
$245$	9.68633	0.618837
$246$	0	0
$247$	24.0064	1.52749
$248$	6.20808	0.394214
$249$	0	0
$250$	−26.1952	−1.65673
$251$	3.87919	0.244852	0.122426	−	0.992478i	$-0.460933\pi$
$251$	3.87919	0.244852	0.122426	+	0.992478i	$0.460933\pi$
$252$	0	0
$253$	5.08722	0.319831
$254$	−46.3038	−2.90536
$255$	0	0
$256$	−8.92156	−0.557598
$257$	−20.1362	−1.25606	−0.628030	−	0.778189i	$-0.716139\pi$
$257$	−20.1362	−1.25606	−0.628030	+	0.778189i	$0.716139\pi$
$258$	0	0
$259$	14.4822	0.899880
$260$	44.9371	2.78688
$261$	0	0
$262$	1.44812	0.0894654
$263$	9.01852	0.556106	0.278053	−	0.960566i	$-0.410311\pi$
$263$	9.01852	0.556106	0.278053	+	0.960566i	$0.410311\pi$
$264$	0	0
$265$	12.1990	0.749379
$266$	−11.6852	−0.716465
$267$	0	0
$268$	2.62871	0.160574
$269$	4.39179	0.267772	0.133886	−	0.990997i	$-0.457254\pi$
$269$	4.39179	0.267772	0.133886	+	0.990997i	$0.457254\pi$
$270$	0	0
$271$	−4.66984	−0.283673	−0.141836	−	0.989890i	$-0.545301\pi$
$271$	−4.66984	−0.283673	−0.141836	+	0.989890i	$0.545301\pi$
$272$	3.24976	0.197045
$273$	0	0
$274$	27.2989	1.64919
$275$	−0.562873	−0.0339425
$276$	0	0
$277$	−23.4584	−1.40948	−0.704740	−	0.709466i	$-0.748936\pi$
$277$	−23.4584	−1.40948	−0.704740	+	0.709466i	$0.748936\pi$
$278$	42.4122	2.54371
$279$	0	0
$280$	−7.27837	−0.434966
$281$	22.8244	1.36159	0.680793	−	0.732475i	$-0.261635\pi$
$281$	22.8244	1.36159	0.680793	+	0.732475i	$0.261635\pi$
$282$	0	0
$283$	−3.43936	−0.204449	−0.102224	−	0.994761i	$-0.532596\pi$
$283$	−3.43936	−0.204449	−0.102224	+	0.994761i	$0.532596\pi$
$284$	−24.0357	−1.42626
$285$	0	0
$286$	15.9105	0.940805
$287$	1.20073	0.0708766
$288$	0	0
$289$	−6.65642	−0.391554
$290$	10.8294	0.635925
$291$	0	0
$292$	34.1094	1.99610
$293$	−8.66879	−0.506436	−0.253218	−	0.967409i	$-0.581489\pi$
$293$	−8.66879	−0.506436	−0.253218	+	0.967409i	$0.581489\pi$
$294$	0	0
$295$	18.3131	1.06623
$296$	20.8363	1.21108
$297$	0	0
$298$	−7.22687	−0.418641
$299$	−36.2069	−2.09390
$300$	0	0
$301$	6.01230	0.346543
$302$	−19.8202	−1.14053
$303$	0	0
$304$	3.40824	0.195476
$305$	2.10645	0.120615
$306$	0	0
$307$	28.2928	1.61475	0.807376	−	0.590037i	$-0.200887\pi$
$307$	28.2928	1.61475	0.807376	+	0.590037i	$0.200887\pi$
$308$	−4.64506	−0.264677
$309$	0	0
$310$	−13.1113	−0.744671
$311$	−31.8058	−1.80354	−0.901771	−	0.432214i	$-0.857733\pi$
$311$	−31.8058	−1.80354	−0.901771	+	0.432214i	$0.857733\pi$
$312$	0	0
$313$	−20.6585	−1.16768	−0.583842	−	0.811867i	$-0.698451\pi$
$313$	−20.6585	−1.16768	−0.583842	+	0.811867i	$0.698451\pi$
$314$	38.8875	2.19455
$315$	0	0
$316$	11.6463	0.655156
$317$	−8.16217	−0.458433	−0.229217	−	0.973375i	$-0.573616\pi$
$317$	−8.16217	−0.458433	−0.229217	+	0.973375i	$0.573616\pi$
$318$	0	0
$319$	2.29976	0.128762
$320$	27.3783	1.53049
$321$	0	0
$322$	17.6239	0.982139
$323$	10.8480	0.603600
$324$	0	0
$325$	4.00610	0.222218
$326$	−14.0722	−0.779388
$327$	0	0
$328$	1.72755	0.0953877
$329$	14.6281	0.806472
$330$	0	0
$331$	−9.73896	−0.535301	−0.267651	−	0.963516i	$-0.586247\pi$
$331$	−9.73896	−0.535301	−0.267651	+	0.963516i	$0.586247\pi$
$332$	31.8167	1.74617
$333$	0	0
$334$	−14.6100	−0.799425
$335$	−1.84736	−0.100932
$336$	0	0
$337$	−17.8005	−0.969656	−0.484828	−	0.874609i	$-0.661118\pi$
$337$	−17.8005	−0.969656	−0.484828	+	0.874609i	$0.661118\pi$
$338$	−84.1772	−4.57864
$339$	0	0
$340$	20.3062	1.10126
$341$	−2.78435	−0.150781
$342$	0	0
$343$	17.9741	0.970511
$344$	8.65021	0.466388
$345$	0	0
$346$	51.3184	2.75890
$347$	−14.5629	−0.781780	−0.390890	−	0.920437i	$-0.627833\pi$
$347$	−14.5629	−0.781780	−0.390890	+	0.920437i	$0.627833\pi$
$348$	0	0
$349$	−2.54031	−0.135980	−0.0679900	−	0.997686i	$-0.521659\pi$
$349$	−2.54031	−0.135980	−0.0679900	+	0.997686i	$0.521659\pi$
$350$	−1.94998	−0.104231
$351$	0	0
$352$	6.71812	0.358077
$353$	−5.44682	−0.289905	−0.144953	−	0.989439i	$-0.546303\pi$
$353$	−5.44682	−0.289905	−0.144953	+	0.989439i	$0.546303\pi$
$354$	0	0
$355$	16.8914	0.896502
$356$	21.9765	1.16475
$357$	0	0
$358$	−29.5436	−1.56143
$359$	−28.6322	−1.51115	−0.755574	−	0.655063i	$-0.772642\pi$
$359$	−28.6322	−1.51115	−0.755574	+	0.655063i	$0.772642\pi$
$360$	0	0
$361$	−7.62292	−0.401206
$362$	29.0075	1.52460
$363$	0	0
$364$	33.0600	1.73282
$365$	−23.9708	−1.25469
$366$	0	0
$367$	27.2752	1.42375	0.711876	−	0.702305i	$-0.247846\pi$
$367$	27.2752	1.42375	0.711876	+	0.702305i	$0.247846\pi$
$368$	−5.14039	−0.267961
$369$	0	0
$370$	−44.0056	−2.28774
$371$	8.97475	0.465946
$372$	0	0
$373$	−31.8258	−1.64788	−0.823938	−	0.566681i	$-0.808227\pi$
$373$	−31.8258	−1.64788	−0.823938	+	0.566681i	$0.808227\pi$
$374$	7.18963	0.371767
$375$	0	0
$376$	21.0461	1.08537
$377$	−16.3679	−0.842992
$378$	0	0
$379$	5.68480	0.292009	0.146004	−	0.989284i	$-0.453359\pi$
$379$	5.68480	0.292009	0.146004	+	0.989284i	$0.453359\pi$
$380$	21.2965	1.09249
$381$	0	0
$382$	−32.9455	−1.68564
$383$	30.6669	1.56701	0.783503	−	0.621388i	$-0.213431\pi$
$383$	30.6669	1.56701	0.783503	+	0.621388i	$0.213431\pi$
$384$	0	0
$385$	3.26437	0.166368
$386$	42.3438	2.15524
$387$	0	0
$388$	−51.3945	−2.60916
$389$	18.9933	0.963001	0.481500	−	0.876446i	$-0.340092\pi$
$389$	18.9933	0.963001	0.481500	+	0.876446i	$0.340092\pi$
$390$	0	0
$391$	−16.3612	−0.827423
$392$	10.2528	0.517845
$393$	0	0
$394$	32.7267	1.64875
$395$	−8.18458	−0.411811
$396$	0	0
$397$	18.7287	0.939966	0.469983	−	0.882676i	$-0.344260\pi$
$397$	18.7287	0.939966	0.469983	+	0.882676i	$0.344260\pi$
$398$	52.0316	2.60811
$399$	0	0
$400$	0.568755	0.0284378
$401$	−5.05574	−0.252472	−0.126236	−	0.992000i	$-0.540290\pi$
$401$	−5.05574	−0.252472	−0.126236	+	0.992000i	$0.540290\pi$
$402$	0	0
$403$	19.8168	0.987147
$404$	−44.7186	−2.22483
$405$	0	0
$406$	7.96715	0.395403
$407$	−9.34514	−0.463221
$408$	0	0
$409$	−27.7592	−1.37260	−0.686301	−	0.727318i	$-0.740767\pi$
$409$	−27.7592	−1.37260	−0.686301	+	0.727318i	$0.740767\pi$
$410$	−3.64853	−0.180188
$411$	0	0
$412$	−36.5506	−1.80072
$413$	13.4729	0.662957
$414$	0	0
$415$	−22.3595	−1.09759
$416$	−47.8145	−2.34430
$417$	0	0
$418$	7.54027	0.368807
$419$	5.10399	0.249346	0.124673	−	0.992198i	$-0.460212\pi$
$419$	5.10399	0.249346	0.124673	+	0.992198i	$0.460212\pi$
$420$	0	0
$421$	18.1769	0.885890	0.442945	−	0.896549i	$-0.353934\pi$
$421$	18.1769	0.885890	0.442945	+	0.896549i	$0.353934\pi$
$422$	−52.2201	−2.54204
$423$	0	0
$424$	12.9124	0.627083
$425$	1.81028	0.0878114
$426$	0	0
$427$	1.54970	0.0749955
$428$	−31.6239	−1.52860
$429$	0	0
$430$	−18.2690	−0.881009
$431$	40.6305	1.95710	0.978552	−	0.205998i	$-0.0660441\pi$
$431$	40.6305	1.95710	0.978552	+	0.205998i	$0.0660441\pi$
$432$	0	0
$433$	−10.4178	−0.500650	−0.250325	−	0.968162i	$-0.580537\pi$
$433$	−10.4178	−0.500650	−0.250325	+	0.968162i	$0.580537\pi$
$434$	−9.64592	−0.463019
$435$	0	0
$436$	17.0412	0.816127
$437$	−17.1592	−0.820834
$438$	0	0
$439$	21.9407	1.04717	0.523585	−	0.851973i	$-0.324594\pi$
$439$	21.9407	1.04717	0.523585	+	0.851973i	$0.324594\pi$
$440$	4.69662	0.223903
$441$	0	0
$442$	−51.1703	−2.43392
$443$	32.4428	1.54140	0.770702	−	0.637196i	$-0.219906\pi$
$443$	32.4428	1.54140	0.770702	+	0.637196i	$0.219906\pi$
$444$	0	0
$445$	−15.4442	−0.732126
$446$	9.15462	0.433484
$447$	0	0
$448$	20.1421	0.951623
$449$	35.5000	1.67535	0.837675	−	0.546169i	$-0.183914\pi$
$449$	35.5000	1.67535	0.837675	+	0.546169i	$0.183914\pi$
$450$	0	0
$451$	−0.774810	−0.0364844
$452$	−24.1796	−1.13731
$453$	0	0
$454$	12.0106	0.563687
$455$	−23.2333	−1.08919
$456$	0	0
$457$	9.69562	0.453542	0.226771	−	0.973948i	$-0.427183\pi$
$457$	9.69562	0.453542	0.226771	+	0.973948i	$0.427183\pi$
$458$	28.3152	1.32308
$459$	0	0
$460$	−32.1199	−1.49760
$461$	−14.5967	−0.679836	−0.339918	−	0.940455i	$-0.610399\pi$
$461$	−14.5967	−0.679836	−0.339918	+	0.940455i	$0.610399\pi$
$462$	0	0
$463$	−17.8989	−0.831832	−0.415916	−	0.909403i	$-0.636539\pi$
$463$	−17.8989	−0.831832	−0.415916	+	0.909403i	$0.636539\pi$
$464$	−2.32380	−0.107880
$465$	0	0
$466$	−42.8339	−1.98424
$467$	23.4186	1.08368	0.541841	−	0.840481i	$-0.317727\pi$
$467$	23.4186	1.08368	0.541841	+	0.840481i	$0.317727\pi$
$468$	0	0
$469$	−1.35909	−0.0627570
$470$	−44.4488	−2.05027
$471$	0	0
$472$	19.3841	0.892226
$473$	−3.87964	−0.178386
$474$	0	0
$475$	1.89857	0.0871122
$476$	14.9392	0.684736
$477$	0	0
$478$	−29.1591	−1.33371
$479$	31.0334	1.41795	0.708976	−	0.705232i	$-0.249157\pi$
$479$	31.0334	1.41795	0.708976	+	0.705232i	$0.249157\pi$
$480$	0	0
$481$	66.5116	3.03267
$482$	−5.62045	−0.256004
$483$	0	0
$484$	2.99739	0.136245
$485$	36.1181	1.64004
$486$	0	0
$487$	−17.3844	−0.787764	−0.393882	−	0.919161i	$-0.628868\pi$
$487$	−17.3844	−0.787764	−0.393882	+	0.919161i	$0.628868\pi$
$488$	2.22964	0.100931
$489$	0	0
$490$	−21.6536	−0.978211
$491$	15.9176	0.718350	0.359175	−	0.933270i	$-0.383058\pi$
$491$	15.9176	0.718350	0.359175	+	0.933270i	$0.383058\pi$
$492$	0	0
$493$	−7.39636	−0.333115
$494$	−53.6659	−2.41454
$495$	0	0
$496$	2.81345	0.126327
$497$	12.4269	0.557423
$498$	0	0
$499$	15.7123	0.703380	0.351690	−	0.936116i	$-0.385607\pi$
$499$	15.7123	0.703380	0.351690	+	0.936116i	$0.385607\pi$
$500$	35.1231	1.57075
$501$	0	0
$502$	−8.67186	−0.387044
$503$	−9.04696	−0.403384	−0.201692	−	0.979449i	$-0.564644\pi$
$503$	−9.04696	−0.403384	−0.201692	+	0.979449i	$0.564644\pi$
$504$	0	0
$505$	31.4265	1.39846
$506$	−11.3724	−0.505565
$507$	0	0
$508$	62.0852	2.75458
$509$	35.8487	1.58896	0.794482	−	0.607287i	$-0.207742\pi$
$509$	35.8487	1.58896	0.794482	+	0.607287i	$0.207742\pi$
$510$	0	0
$511$	−17.6352	−0.780136
$512$	−11.2942	−0.499139
$513$	0	0
$514$	45.0141	1.98549
$515$	25.6864	1.13188
$516$	0	0
$517$	−9.43926	−0.415138
$518$	−32.3747	−1.42246
$519$	0	0
$520$	−33.4270	−1.46587
$521$	33.9162	1.48589	0.742947	−	0.669350i	$-0.233427\pi$
$521$	33.9162	1.48589	0.742947	+	0.669350i	$0.233427\pi$
$522$	0	0
$523$	0.886999	0.0387858	0.0193929	−	0.999812i	$-0.493827\pi$
$523$	0.886999	0.0387858	0.0193929	+	0.999812i	$0.493827\pi$
$524$	−1.94168	−0.0848226
$525$	0	0
$526$	−20.1608	−0.879051
$527$	8.95485	0.390080
$528$	0	0
$529$	2.87981	0.125209
$530$	−27.2707	−1.18456
$531$	0	0
$532$	15.6678	0.679284
$533$	5.51450	0.238860
$534$	0	0
$535$	22.2241	0.960830
$536$	−1.95539	−0.0844601
$537$	0	0
$538$	−9.81778	−0.423275
$539$	−4.59842	−0.198068
$540$	0	0
$541$	0.433662	0.0186446	0.00932230	−	0.999957i	$-0.497033\pi$
$541$	0.433662	0.0186446	0.00932230	+	0.999957i	$0.497033\pi$
$542$	10.4393	0.448408
$543$	0	0
$544$	−21.6064	−0.926369
$545$	−11.9759	−0.512993
$546$	0	0
$547$	24.5202	1.04841	0.524203	−	0.851593i	$-0.324363\pi$
$547$	24.5202	1.04841	0.524203	+	0.851593i	$0.324363\pi$
$548$	−36.6030	−1.56360
$549$	0	0
$550$	1.25829	0.0536538
$551$	−7.75708	−0.330463
$552$	0	0
$553$	−6.02136	−0.256054
$554$	52.4409	2.22800
$555$	0	0
$556$	−56.8672	−2.41171
$557$	−28.2399	−1.19656	−0.598281	−	0.801286i	$-0.704149\pi$
$557$	−28.2399	−1.19656	−0.598281	+	0.801286i	$0.704149\pi$
$558$	0	0
$559$	27.6123	1.16788
$560$	−3.29849	−0.139387
$561$	0	0
$562$	−51.0235	−2.15229
$563$	−25.1017	−1.05791	−0.528956	−	0.848649i	$-0.677416\pi$
$563$	−25.1017	−1.05791	−0.528956	+	0.848649i	$0.677416\pi$
$564$	0	0
$565$	16.9925	0.714880
$566$	7.68863	0.323177
$567$	0	0
$568$	17.8792	0.750196
$569$	14.3208	0.600357	0.300179	−	0.953883i	$-0.402954\pi$
$569$	14.3208	0.600357	0.300179	+	0.953883i	$0.402954\pi$
$570$	0	0
$571$	−17.2540	−0.722056	−0.361028	−	0.932555i	$-0.617574\pi$
$571$	−17.2540	−0.722056	−0.361028	+	0.932555i	$0.617574\pi$
$572$	−21.3331	−0.891982
$573$	0	0
$574$	−2.68420	−0.112036
$575$	−2.86346	−0.119414
$576$	0	0
$577$	−35.4179	−1.47447	−0.737234	−	0.675638i	$-0.763868\pi$
$577$	−35.4179	−1.47447	−0.737234	+	0.675638i	$0.763868\pi$
$578$	14.8803	0.618940
$579$	0	0
$580$	−14.5203	−0.602924
$581$	−16.4498	−0.682453
$582$	0	0
$583$	−5.79127	−0.239850
$584$	−25.3727	−1.04993
$585$	0	0
$586$	19.3789	0.800536
$587$	−29.9672	−1.23688	−0.618439	−	0.785833i	$-0.712235\pi$
$587$	−29.9672	−1.23688	−0.618439	+	0.785833i	$0.712235\pi$
$588$	0	0
$589$	9.39158	0.386973
$590$	−40.9387	−1.68542
$591$	0	0
$592$	9.44281	0.388097
$593$	20.2261	0.830585	0.415292	−	0.909688i	$-0.363679\pi$
$593$	20.2261	0.830585	0.415292	+	0.909688i	$0.363679\pi$
$594$	0	0
$595$	−10.4987	−0.430404
$596$	9.68995	0.396916
$597$	0	0
$598$	80.9400	3.30988
$599$	−32.1119	−1.31206	−0.656029	−	0.754735i	$-0.727765\pi$
$599$	−32.1119	−1.31206	−0.656029	+	0.754735i	$0.727765\pi$
$600$	0	0
$601$	9.92875	0.405002	0.202501	−	0.979282i	$-0.435093\pi$
$601$	9.92875	0.405002	0.202501	+	0.979282i	$0.435093\pi$
$602$	−13.4404	−0.547790
$603$	0	0
$604$	26.5754	1.08134
$605$	−2.10645	−0.0856393
$606$	0	0
$607$	−18.2430	−0.740460	−0.370230	−	0.928940i	$-0.620721\pi$
$607$	−18.2430	−0.740460	−0.370230	+	0.928940i	$0.620721\pi$
$608$	−22.6602	−0.918992
$609$	0	0
$610$	−4.70893	−0.190659
$611$	67.1815	2.71787
$612$	0	0
$613$	12.4164	0.501493	0.250746	−	0.968053i	$-0.419324\pi$
$613$	12.4164	0.501493	0.250746	+	0.968053i	$0.419324\pi$
$614$	−63.2480	−2.55248
$615$	0	0
$616$	3.45528	0.139217
$617$	15.0465	0.605748	0.302874	−	0.953031i	$-0.402054\pi$
$617$	15.0465	0.605748	0.302874	+	0.953031i	$0.402054\pi$
$618$	0	0
$619$	−18.2051	−0.731724	−0.365862	−	0.930669i	$-0.619226\pi$
$619$	−18.2051	−0.731724	−0.365862	+	0.930669i	$0.619226\pi$
$620$	17.5799	0.706026
$621$	0	0
$622$	71.1014	2.85091
$623$	−11.3622	−0.455219
$624$	0	0
$625$	−21.8688	−0.874752
$626$	46.1816	1.84579
$627$	0	0
$628$	−52.1412	−2.08066
$629$	30.0553	1.19838
$630$	0	0
$631$	24.4681	0.974058	0.487029	−	0.873386i	$-0.338081\pi$
$631$	24.4681	0.974058	0.487029	+	0.873386i	$0.338081\pi$
$632$	−8.66323	−0.344605
$633$	0	0
$634$	18.2464	0.724657
$635$	−43.6311	−1.73145
$636$	0	0
$637$	32.7280	1.29673
$638$	−5.14108	−0.203537
$639$	0	0
$640$	−32.9009	−1.30052
$641$	21.5755	0.852182	0.426091	−	0.904680i	$-0.359890\pi$
$641$	21.5755	0.852182	0.426091	+	0.904680i	$0.359890\pi$
$642$	0	0
$643$	25.6728	1.01244	0.506218	−	0.862406i	$-0.331043\pi$
$643$	25.6728	1.01244	0.506218	+	0.862406i	$0.331043\pi$
$644$	−23.6305	−0.931170
$645$	0	0
$646$	−24.2506	−0.954126
$647$	13.9609	0.548861	0.274430	−	0.961607i	$-0.411511\pi$
$647$	13.9609	0.548861	0.274430	+	0.961607i	$0.411511\pi$
$648$	0	0
$649$	−8.69383	−0.341263
$650$	−8.95557	−0.351266
$651$	0	0
$652$	18.8683	0.738941
$653$	−24.3856	−0.954281	−0.477141	−	0.878827i	$-0.658327\pi$
$653$	−24.3856	−0.954281	−0.477141	+	0.878827i	$0.658327\pi$
$654$	0	0
$655$	1.36454	0.0533169
$656$	0.782907	0.0305674
$657$	0	0
$658$	−32.7008	−1.27481
$659$	−7.00856	−0.273015	−0.136507	−	0.990639i	$-0.543588\pi$
$659$	−7.00856	−0.273015	−0.136507	+	0.990639i	$0.543588\pi$
$660$	0	0
$661$	−1.46887	−0.0571324	−0.0285662	−	0.999592i	$-0.509094\pi$
$661$	−1.46887	−0.0571324	−0.0285662	+	0.999592i	$0.509094\pi$
$662$	21.7713	0.846165
$663$	0	0
$664$	−23.6672	−0.918465
$665$	−11.0107	−0.426977
$666$	0	0
$667$	11.6994	0.453002
$668$	19.5895	0.757939
$669$	0	0
$670$	4.12973	0.159545
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	31.0943	1.19860	0.599298	−	0.800526i	$-0.295446\pi$
$673$	31.0943	1.19860	0.599298	+	0.800526i	$0.295446\pi$
$674$	39.7928	1.53276
$675$	0	0
$676$	112.867	4.34103
$677$	26.3752	1.01368	0.506840	−	0.862040i	$-0.330813\pi$
$677$	26.3752	1.01368	0.506840	+	0.862040i	$0.330813\pi$
$678$	0	0
$679$	26.5719	1.01974
$680$	−15.1050	−0.579250
$681$	0	0
$682$	6.22436	0.238343
$683$	39.5612	1.51377	0.756883	−	0.653550i	$-0.226721\pi$
$683$	39.5612	1.51377	0.756883	+	0.653550i	$0.226721\pi$
$684$	0	0
$685$	25.7232	0.982833
$686$	−40.1808	−1.53411
$687$	0	0
$688$	3.92019	0.149456
$689$	41.2178	1.57027
$690$	0	0
$691$	37.6958	1.43401	0.717007	−	0.697066i	$-0.245511\pi$
$691$	37.6958	1.43401	0.717007	+	0.697066i	$0.245511\pi$
$692$	−68.8089	−2.61572
$693$	0	0
$694$	32.5552	1.23578
$695$	39.9641	1.51593
$696$	0	0
$697$	2.49190	0.0943874
$698$	5.67883	0.214947
$699$	0	0
$700$	2.61458	0.0988218
$701$	−24.0001	−0.906473	−0.453236	−	0.891390i	$-0.649731\pi$
$701$	−24.0001	−0.906473	−0.453236	+	0.891390i	$0.649731\pi$
$702$	0	0
$703$	31.5211	1.18884
$704$	−12.9974	−0.489856
$705$	0	0
$706$	12.1763	0.458260
$707$	23.1203	0.869530
$708$	0	0
$709$	−23.9608	−0.899865	−0.449933	−	0.893062i	$-0.648552\pi$
$709$	−23.9608	−0.899865	−0.449933	+	0.893062i	$0.648552\pi$
$710$	−37.7604	−1.41712
$711$	0	0
$712$	−16.3474	−0.612646
$713$	−14.1646	−0.530468
$714$	0	0
$715$	14.9921	0.560673
$716$	39.6127	1.48039
$717$	0	0
$718$	64.0067	2.38871
$719$	−43.2374	−1.61248	−0.806241	−	0.591587i	$-0.798501\pi$
$719$	−43.2374	−1.61248	−0.806241	+	0.591587i	$0.798501\pi$
$720$	0	0
$721$	18.8974	0.703774
$722$	17.0409	0.634197
$723$	0	0
$724$	−38.8939	−1.44548
$725$	−1.29447	−0.0480755
$726$	0	0
$727$	49.7051	1.84346	0.921730	−	0.387832i	$-0.126776\pi$
$727$	49.7051	1.84346	0.921730	+	0.387832i	$0.126776\pi$
$728$	−24.5921	−0.911442
$729$	0	0
$730$	53.5863	1.98332
$731$	12.4775	0.461497
$732$	0	0
$733$	35.6594	1.31711	0.658555	−	0.752533i	$-0.271168\pi$
$733$	35.6594	1.31711	0.658555	+	0.752533i	$0.271168\pi$
$734$	−60.9732	−2.25056
$735$	0	0
$736$	34.1766	1.25977
$737$	0.877000	0.0323047
$738$	0	0
$739$	−29.9753	−1.10266	−0.551330	−	0.834288i	$-0.685879\pi$
$739$	−29.9753	−1.10266	−0.551330	+	0.834288i	$0.685879\pi$
$740$	59.0037	2.16902
$741$	0	0
$742$	−20.0629	−0.736533
$743$	−22.1438	−0.812378	−0.406189	−	0.913789i	$-0.633143\pi$
$743$	−22.1438	−0.812378	−0.406189	+	0.913789i	$0.633143\pi$
$744$	0	0
$745$	−6.80973	−0.249489
$746$	71.1459	2.60484
$747$	0	0
$748$	−9.64002	−0.352474
$749$	16.3501	0.597421
$750$	0	0
$751$	−23.9030	−0.872232	−0.436116	−	0.899890i	$-0.643646\pi$
$751$	−23.9030	−0.872232	−0.436116	+	0.899890i	$0.643646\pi$
$752$	9.53791	0.347812
$753$	0	0
$754$	36.5903	1.33254
$755$	−18.6762	−0.679696
$756$	0	0
$757$	−2.04527	−0.0743367	−0.0371683	−	0.999309i	$-0.511834\pi$
$757$	−2.04527	−0.0743367	−0.0371683	+	0.999309i	$0.511834\pi$
$758$	−12.7083	−0.461585
$759$	0	0
$760$	−15.8417	−0.574638
$761$	−4.94826	−0.179374	−0.0896872	−	0.995970i	$-0.528587\pi$
$761$	−4.94826	−0.179374	−0.0896872	+	0.995970i	$0.528587\pi$
$762$	0	0
$763$	−8.81064	−0.318966
$764$	44.1740	1.59816
$765$	0	0
$766$	−68.5553	−2.47701
$767$	61.8760	2.23421
$768$	0	0
$769$	28.1395	1.01474	0.507368	−	0.861729i	$-0.330618\pi$
$769$	28.1395	1.01474	0.507368	+	0.861729i	$0.330618\pi$
$770$	−7.29745	−0.262982
$771$	0	0
$772$	−56.7755	−2.04340
$773$	43.9205	1.57971	0.789856	−	0.613293i	$-0.210155\pi$
$773$	43.9205	1.57971	0.789856	+	0.613293i	$0.210155\pi$
$774$	0	0
$775$	1.56723	0.0562967
$776$	38.2304	1.37239
$777$	0	0
$778$	−42.4593	−1.52224
$779$	2.61343	0.0936357
$780$	0	0
$781$	−8.01889	−0.286939
$782$	36.5753	1.30793
$783$	0	0
$784$	4.64647	0.165946
$785$	36.6429	1.30784
$786$	0	0
$787$	−20.1447	−0.718080	−0.359040	−	0.933322i	$-0.616896\pi$
$787$	−20.1447	−0.718080	−0.359040	+	0.933322i	$0.616896\pi$
$788$	−43.8806	−1.56318
$789$	0	0
$790$	18.2965	0.650960
$791$	12.5013	0.444495
$792$	0	0
$793$	7.11724	0.252740
$794$	−41.8677	−1.48583
$795$	0	0
$796$	−69.7651	−2.47276
$797$	−29.0609	−1.02939	−0.514695	−	0.857373i	$-0.672095\pi$
$797$	−29.0609	−1.02939	−0.514695	+	0.857373i	$0.672095\pi$
$798$	0	0
$799$	30.3580	1.07399
$800$	−3.78145	−0.133694
$801$	0	0
$802$	11.3020	0.399088
$803$	11.3797	0.401582
$804$	0	0
$805$	16.6066	0.585305
$806$	−44.3002	−1.56041
$807$	0	0
$808$	33.2644	1.17024
$809$	32.8239	1.15403	0.577014	−	0.816734i	$-0.304218\pi$
$809$	32.8239	1.15403	0.577014	+	0.816734i	$0.304218\pi$
$810$	0	0
$811$	16.3460	0.573984	0.286992	−	0.957933i	$-0.407345\pi$
$811$	16.3460	0.573984	0.286992	+	0.957933i	$0.407345\pi$
$812$	−10.6825	−0.374884
$813$	0	0
$814$	20.8909	0.732226
$815$	−13.2599	−0.464476
$816$	0	0
$817$	13.0860	0.457822
$818$	62.0551	2.16971
$819$	0	0
$820$	4.89202	0.170837
$821$	−33.3538	−1.16406	−0.582029	−	0.813168i	$-0.697741\pi$
$821$	−33.3538	−1.16406	−0.582029	+	0.813168i	$0.697741\pi$
$822$	0	0
$823$	−14.2115	−0.495382	−0.247691	−	0.968839i	$-0.579672\pi$
$823$	−14.2115	−0.495382	−0.247691	+	0.968839i	$0.579672\pi$
$824$	27.1886	0.947159
$825$	0	0
$826$	−30.1184	−1.04795
$827$	−11.7177	−0.407464	−0.203732	−	0.979027i	$-0.565307\pi$
$827$	−11.7177	−0.407464	−0.203732	+	0.979027i	$0.565307\pi$
$828$	0	0
$829$	39.9499	1.38752	0.693759	−	0.720207i	$-0.255953\pi$
$829$	39.9499	1.38752	0.693759	+	0.720207i	$0.255953\pi$
$830$	49.9844	1.73498
$831$	0	0
$832$	92.5052	3.20704
$833$	14.7892	0.512414
$834$	0	0
$835$	−13.7667	−0.476417
$836$	−10.1102	−0.349667
$837$	0	0
$838$	−11.4099	−0.394148
$839$	11.9102	0.411185	0.205593	−	0.978638i	$-0.434088\pi$
$839$	11.9102	0.411185	0.205593	+	0.978638i	$0.434088\pi$
$840$	0	0
$841$	−23.7111	−0.817624
$842$	−40.6343	−1.40035
$843$	0	0
$844$	70.0179	2.41012
$845$	−79.3184	−2.72864
$846$	0	0
$847$	−1.54970	−0.0532485
$848$	5.85179	0.200951
$849$	0	0
$850$	−4.04685	−0.138806
$851$	−47.5408	−1.62968
$852$	0	0
$853$	−25.3527	−0.868059	−0.434029	−	0.900899i	$-0.642909\pi$
$853$	−25.3527	−0.868059	−0.434029	+	0.900899i	$0.642909\pi$
$854$	−3.46434	−0.118547
$855$	0	0
$856$	23.5238	0.804026
$857$	38.1116	1.30187	0.650933	−	0.759135i	$-0.274378\pi$
$857$	38.1116	1.30187	0.650933	+	0.759135i	$0.274378\pi$
$858$	0	0
$859$	−28.8693	−0.985006	−0.492503	−	0.870311i	$-0.663918\pi$
$859$	−28.8693	−0.985006	−0.492503	+	0.870311i	$0.663918\pi$
$860$	24.4955	0.835288
$861$	0	0
$862$	−90.8289	−3.09365
$863$	−26.1740	−0.890972	−0.445486	−	0.895289i	$-0.646969\pi$
$863$	−26.1740	−0.890972	−0.445486	+	0.895289i	$0.646969\pi$
$864$	0	0
$865$	48.3563	1.64416
$866$	23.2889	0.791390
$867$	0	0
$868$	12.9335	0.438990
$869$	3.88549	0.131806
$870$	0	0
$871$	−6.24181	−0.211496
$872$	−12.6763	−0.429274
$873$	0	0
$874$	38.3590	1.29751
$875$	−18.1593	−0.613896
$876$	0	0
$877$	−17.0105	−0.574405	−0.287202	−	0.957870i	$-0.592725\pi$
$877$	−17.0105	−0.574405	−0.287202	+	0.957870i	$0.592725\pi$
$878$	−49.0480	−1.65529
$879$	0	0
$880$	2.12846	0.0717505
$881$	−9.70384	−0.326931	−0.163465	−	0.986549i	$-0.552267\pi$
$881$	−9.70384	−0.326931	−0.163465	+	0.986549i	$0.552267\pi$
$882$	0	0
$883$	24.1772	0.813627	0.406814	−	0.913511i	$-0.366640\pi$
$883$	24.1772	0.813627	0.406814	+	0.913511i	$0.366640\pi$
$884$	68.6103	2.30761
$885$	0	0
$886$	−72.5253	−2.43653
$887$	−13.7330	−0.461109	−0.230555	−	0.973059i	$-0.574054\pi$
$887$	−13.7330	−0.461109	−0.230555	+	0.973059i	$0.574054\pi$
$888$	0	0
$889$	−32.0992	−1.07657
$890$	34.5253	1.15729
$891$	0	0
$892$	−12.2747	−0.410988
$893$	31.8386	1.06544
$894$	0	0
$895$	−27.8383	−0.930531
$896$	−24.2050	−0.808633
$897$	0	0
$898$	−79.3597	−2.64827
$899$	−6.40333	−0.213563
$900$	0	0
$901$	18.6255	0.620507
$902$	1.73207	0.0576718
$903$	0	0
$904$	17.9863	0.598214
$905$	27.3331	0.908584
$906$	0	0
$907$	−38.3704	−1.27407	−0.637034	−	0.770836i	$-0.719839\pi$
$907$	−38.3704	−1.27407	−0.637034	+	0.770836i	$0.719839\pi$
$908$	−16.1041	−0.534434
$909$	0	0
$910$	51.9377	1.72172
$911$	−0.616414	−0.0204227	−0.0102113	−	0.999948i	$-0.503250\pi$
$911$	−0.616414	−0.0204227	−0.0102113	+	0.999948i	$0.503250\pi$
$912$	0	0
$913$	10.6148	0.351299
$914$	−21.6744	−0.716926
$915$	0	0
$916$	−37.9657	−1.25442
$917$	1.00388	0.0331512
$918$	0	0
$919$	−27.8597	−0.919005	−0.459503	−	0.888176i	$-0.651972\pi$
$919$	−27.8597	−0.919005	−0.459503	+	0.888176i	$0.651972\pi$
$920$	23.8927	0.787720
$921$	0	0
$922$	32.6307	1.07463
$923$	57.0724	1.87856
$924$	0	0
$925$	5.26012	0.172952
$926$	40.0127	1.31490
$927$	0	0
$928$	15.4501	0.507174
$929$	12.1882	0.399882	0.199941	−	0.979808i	$-0.435925\pi$
$929$	12.1882	0.399882	0.199941	+	0.979808i	$0.435925\pi$
$930$	0	0
$931$	15.5104	0.508334
$932$	57.4326	1.88127
$933$	0	0
$934$	−52.3518	−1.71300
$935$	6.77464	0.221554
$936$	0	0
$937$	12.2225	0.399291	0.199645	−	0.979868i	$-0.436021\pi$
$937$	12.2225	0.399291	0.199645	+	0.979868i	$0.436021\pi$
$938$	3.03822	0.0992015
$939$	0	0
$940$	59.5980	1.94387
$941$	−10.9884	−0.358210	−0.179105	−	0.983830i	$-0.557320\pi$
$941$	−10.9884	−0.358210	−0.179105	+	0.983830i	$0.557320\pi$
$942$	0	0
$943$	−3.94163	−0.128357
$944$	8.78469	0.285917
$945$	0	0
$946$	8.67288	0.281980
$947$	1.56638	0.0509006	0.0254503	−	0.999676i	$-0.491898\pi$
$947$	1.56638	0.0509006	0.0254503	+	0.999676i	$0.491898\pi$
$948$	0	0
$949$	−80.9922	−2.62912
$950$	−4.24421	−0.137700
$951$	0	0
$952$	−11.1127	−0.360164
$953$	23.8850	0.773709	0.386855	−	0.922141i	$-0.373561\pi$
$953$	23.8850	0.773709	0.386855	+	0.922141i	$0.373561\pi$
$954$	0	0
$955$	−31.0438	−1.00455
$956$	39.0972	1.26449
$957$	0	0
$958$	−69.3747	−2.24139
$959$	18.9244	0.611102
$960$	0	0
$961$	−23.2474	−0.749917
$962$	−148.685	−4.79381
$963$	0	0
$964$	7.53602	0.242719
$965$	39.8997	1.28442
$966$	0	0
$967$	−8.73197	−0.280801	−0.140401	−	0.990095i	$-0.544839\pi$
$967$	−8.73197	−0.280801	−0.140401	+	0.990095i	$0.544839\pi$
$968$	−2.22964	−0.0716633
$969$	0	0
$970$	−80.7414	−2.59245
$971$	−32.3589	−1.03845	−0.519223	−	0.854639i	$-0.673779\pi$
$971$	−32.3589	−1.03845	−0.519223	+	0.854639i	$0.673779\pi$
$972$	0	0
$973$	29.4014	0.942566
$974$	38.8626	1.24524
$975$	0	0
$976$	1.01045	0.0323438
$977$	−24.1370	−0.772210	−0.386105	−	0.922455i	$-0.626180\pi$
$977$	−24.1370	−0.772210	−0.386105	+	0.922455i	$0.626180\pi$
$978$	0	0
$979$	7.33188	0.234328
$980$	29.0337	0.927446
$981$	0	0
$982$	−35.5835	−1.13551
$983$	−14.7630	−0.470866	−0.235433	−	0.971891i	$-0.575651\pi$
$983$	−14.7630	−0.470866	−0.235433	+	0.971891i	$0.575651\pi$
$984$	0	0
$985$	30.8377	0.982569
$986$	16.5344	0.526564
$987$	0	0
$988$	71.9564	2.28924
$989$	−19.7366	−0.627588
$990$	0	0
$991$	−5.70493	−0.181223	−0.0906116	−	0.995886i	$-0.528882\pi$
$991$	−5.70493	−0.181223	−0.0906116	+	0.995886i	$0.528882\pi$
$992$	−18.7056	−0.593903
$993$	0	0
$994$	−27.7802	−0.881133
$995$	49.0283	1.55430
$996$	0	0
$997$	55.3076	1.75161	0.875805	−	0.482664i	$-0.160331\pi$
$997$	55.3076	1.75161	0.875805	+	0.482664i	$0.160331\pi$
$998$	−35.1246	−1.11185
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.3	yes	25
3.2	odd	2		6039.2.a.n.1.23	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.23	✓	25	3.2	odd	2
6039.2.a.o.1.3	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)

\(f(q)\)	\(=\)	\(q-2.23548 q^{2} +2.99739 q^{4} -2.10645 q^{5} -1.54970 q^{7} -2.22964 q^{8} +O(q^{10})\)	\(q-2.23548 q^{2} +2.99739 q^{4} -2.10645 q^{5} -1.54970 q^{7} -2.22964 q^{8} +4.70893 q^{10} +1.00000 q^{11} -7.11724 q^{13} +3.46434 q^{14} -1.01045 q^{16} -3.21614 q^{17} -3.37299 q^{19} -6.31384 q^{20} -2.23548 q^{22} +5.08722 q^{23} -0.562873 q^{25} +15.9105 q^{26} -4.64506 q^{28} +2.29976 q^{29} -2.78435 q^{31} +6.71812 q^{32} +7.18963 q^{34} +3.26437 q^{35} -9.34514 q^{37} +7.54027 q^{38} +4.69662 q^{40} -0.774810 q^{41} -3.87964 q^{43} +2.99739 q^{44} -11.3724 q^{46} -9.43926 q^{47} -4.59842 q^{49} +1.25829 q^{50} -21.3331 q^{52} -5.79127 q^{53} -2.10645 q^{55} +3.45528 q^{56} -5.14108 q^{58} -8.69383 q^{59} -1.00000 q^{61} +6.22436 q^{62} -12.9974 q^{64} +14.9921 q^{65} +0.877000 q^{67} -9.64002 q^{68} -7.29745 q^{70} -8.01889 q^{71} +11.3797 q^{73} +20.8909 q^{74} -10.1102 q^{76} -1.54970 q^{77} +3.88549 q^{79} +2.12846 q^{80} +1.73207 q^{82} +10.6148 q^{83} +6.77464 q^{85} +8.67288 q^{86} -2.22964 q^{88} +7.33188 q^{89} +11.0296 q^{91} +15.2484 q^{92} +21.1013 q^{94} +7.10504 q^{95} -17.1464 q^{97} +10.2797 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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

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Groups

Database

Properties

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