LMFDB - Embedded newform 6039.2.a.l.1.20

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:	$21$
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+2.60374 q^{2} +4.77947 q^{4} -0.851432 q^{5} +3.04314 q^{7} +7.23702 q^{8} +O(q^{10})$	\(q+2.60374 q^{2} +4.77947 q^{4} -0.851432 q^{5} +3.04314 q^{7} +7.23702 q^{8} -2.21691 q^{10} +1.00000 q^{11} +1.25630 q^{13} +7.92356 q^{14} +9.28438 q^{16} +7.73575 q^{17} +6.82604 q^{19} -4.06939 q^{20} +2.60374 q^{22} -8.40462 q^{23} -4.27506 q^{25} +3.27109 q^{26} +14.5446 q^{28} -1.09847 q^{29} -2.03738 q^{31} +9.70010 q^{32} +20.1419 q^{34} -2.59103 q^{35} -6.32030 q^{37} +17.7732 q^{38} -6.16183 q^{40} +9.82455 q^{41} -8.23734 q^{43} +4.77947 q^{44} -21.8835 q^{46} -1.11751 q^{47} +2.26073 q^{49} -11.1312 q^{50} +6.00447 q^{52} +4.72415 q^{53} -0.851432 q^{55} +22.0233 q^{56} -2.86013 q^{58} -11.0384 q^{59} +1.00000 q^{61} -5.30480 q^{62} +6.68778 q^{64} -1.06966 q^{65} +6.91010 q^{67} +36.9728 q^{68} -6.74637 q^{70} +7.00240 q^{71} +1.27134 q^{73} -16.4564 q^{74} +32.6248 q^{76} +3.04314 q^{77} -3.71107 q^{79} -7.90502 q^{80} +25.5806 q^{82} -11.2940 q^{83} -6.58646 q^{85} -21.4479 q^{86} +7.23702 q^{88} +3.62680 q^{89} +3.82312 q^{91} -40.1696 q^{92} -2.90971 q^{94} -5.81191 q^{95} -0.113236 q^{97} +5.88635 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * 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 $

$2$	2.60374	1.84112	0.920562	−	0.390598i	$-0.127732\pi$
$2$	2.60374	1.84112	0.920562	+	0.390598i	$0.127732\pi$
$3$	0	0
$3$	0	0
$4$	4.77947	2.38973
$5$	−0.851432	−0.380772	−0.190386	−	0.981709i	$-0.560974\pi$
$5$	−0.851432	−0.380772	−0.190386	+	0.981709i	$0.560974\pi$
$6$	0	0
$7$	3.04314	1.15020	0.575100	−	0.818083i	$-0.304963\pi$
$7$	3.04314	1.15020	0.575100	+	0.818083i	$0.304963\pi$
$8$	7.23702	2.55867
$9$	0	0
$10$	−2.21691	−0.701048
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.25630	0.348436	0.174218	−	0.984707i	$-0.444260\pi$
$13$	1.25630	0.348436	0.174218	+	0.984707i	$0.444260\pi$
$14$	7.92356	2.11766
$15$	0	0
$16$	9.28438	2.32110
$17$	7.73575	1.87619	0.938097	−	0.346373i	$-0.112587\pi$
$17$	7.73575	1.87619	0.938097	+	0.346373i	$0.112587\pi$
$18$	0	0
$19$	6.82604	1.56600	0.783000	−	0.622021i	$-0.213688\pi$
$19$	6.82604	1.56600	0.783000	+	0.622021i	$0.213688\pi$
$20$	−4.06939	−0.909944
$21$	0	0
$22$	2.60374	0.555120
$23$	−8.40462	−1.75248	−0.876242	−	0.481871i	$-0.839958\pi$
$23$	−8.40462	−1.75248	−0.876242	+	0.481871i	$0.839958\pi$
$24$	0	0
$25$	−4.27506	−0.855013
$26$	3.27109	0.641514
$27$	0	0
$28$	14.5446	2.74867
$29$	−1.09847	−0.203980	−0.101990	−	0.994785i	$-0.532521\pi$
$29$	−1.09847	−0.203980	−0.101990	+	0.994785i	$0.532521\pi$
$30$	0	0
$31$	−2.03738	−0.365924	−0.182962	−	0.983120i	$-0.558568\pi$
$31$	−2.03738	−0.365924	−0.182962	+	0.983120i	$0.558568\pi$
$32$	9.70010	1.71475
$33$	0	0
$34$	20.1419	3.45430
$35$	−2.59103	−0.437964
$36$	0	0
$37$	−6.32030	−1.03905	−0.519525	−	0.854455i	$-0.673891\pi$
$37$	−6.32030	−1.03905	−0.519525	+	0.854455i	$0.673891\pi$
$38$	17.7732	2.88320
$39$	0	0
$40$	−6.16183	−0.974271
$41$	9.82455	1.53434	0.767168	−	0.641446i	$-0.221665\pi$
$41$	9.82455	1.53434	0.767168	+	0.641446i	$0.221665\pi$
$42$	0	0
$43$	−8.23734	−1.25618	−0.628091	−	0.778140i	$-0.716163\pi$
$43$	−8.23734	−1.25618	−0.628091	+	0.778140i	$0.716163\pi$
$44$	4.77947	0.720532
$45$	0	0
$46$	−21.8835	−3.22654
$47$	−1.11751	−0.163006	−0.0815029	−	0.996673i	$-0.525972\pi$
$47$	−1.11751	−0.163006	−0.0815029	+	0.996673i	$0.525972\pi$
$48$	0	0
$49$	2.26073	0.322961
$50$	−11.1312	−1.57418
$51$	0	0
$52$	6.00447	0.832670
$53$	4.72415	0.648912	0.324456	−	0.945901i	$-0.394819\pi$
$53$	4.72415	0.648912	0.324456	+	0.945901i	$0.394819\pi$
$54$	0	0
$55$	−0.851432	−0.114807
$56$	22.0233	2.94299
$57$	0	0
$58$	−2.86013	−0.375553
$59$	−11.0384	−1.43708	−0.718540	−	0.695485i	$-0.755189\pi$
$59$	−11.0384	−1.43708	−0.718540	+	0.695485i	$0.755189\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−5.30480	−0.673711
$63$	0	0
$64$	6.68778	0.835972
$65$	−1.06966	−0.132675
$66$	0	0
$67$	6.91010	0.844203	0.422102	−	0.906549i	$-0.361293\pi$
$67$	6.91010	0.844203	0.422102	+	0.906549i	$0.361293\pi$
$68$	36.9728	4.48361
$69$	0	0
$70$	−6.74637	−0.806346
$71$	7.00240	0.831032	0.415516	−	0.909586i	$-0.363601\pi$
$71$	7.00240	0.831032	0.415516	+	0.909586i	$0.363601\pi$
$72$	0	0
$73$	1.27134	0.148799	0.0743995	−	0.997229i	$-0.476296\pi$
$73$	1.27134	0.148799	0.0743995	+	0.997229i	$0.476296\pi$
$74$	−16.4564	−1.91302
$75$	0	0
$76$	32.6248	3.74233
$77$	3.04314	0.346798
$78$	0	0
$79$	−3.71107	−0.417529	−0.208764	−	0.977966i	$-0.566944\pi$
$79$	−3.71107	−0.417529	−0.208764	+	0.977966i	$0.566944\pi$
$80$	−7.90502	−0.883808
$81$	0	0
$82$	25.5806	2.82490
$83$	−11.2940	−1.23967	−0.619837	−	0.784731i	$-0.712801\pi$
$83$	−11.2940	−1.23967	−0.619837	+	0.784731i	$0.712801\pi$
$84$	0	0
$85$	−6.58646	−0.714402
$86$	−21.4479	−2.31279
$87$	0	0
$88$	7.23702	0.771469
$89$	3.62680	0.384440	0.192220	−	0.981352i	$-0.438431\pi$
$89$	3.62680	0.384440	0.192220	+	0.981352i	$0.438431\pi$
$90$	0	0
$91$	3.82312	0.400771
$92$	−40.1696	−4.18797
$93$	0	0
$94$	−2.90971	−0.300114
$95$	−5.81191	−0.596289
$96$	0	0
$97$	−0.113236	−0.0114974	−0.00574869	−	0.999983i	$-0.501830\pi$
$97$	−0.113236	−0.0114974	−0.00574869	+	0.999983i	$0.501830\pi$
$98$	5.88635	0.594611
$99$	0	0
$100$	−20.4325	−2.04325
$101$	3.13269	0.311714	0.155857	−	0.987780i	$-0.450186\pi$
$101$	3.13269	0.311714	0.155857	+	0.987780i	$0.450186\pi$
$102$	0	0
$103$	−0.612788	−0.0603798	−0.0301899	−	0.999544i	$-0.509611\pi$
$103$	−0.612788	−0.0603798	−0.0301899	+	0.999544i	$0.509611\pi$
$104$	9.09190	0.891534
$105$	0	0
$106$	12.3005	1.19473
$107$	−1.37535	−0.132960	−0.0664801	−	0.997788i	$-0.521177\pi$
$107$	−1.37535	−0.132960	−0.0664801	+	0.997788i	$0.521177\pi$
$108$	0	0
$109$	20.5325	1.96665	0.983327	−	0.181845i	$-0.0582068\pi$
$109$	20.5325	1.96665	0.983327	+	0.181845i	$0.0582068\pi$
$110$	−2.21691	−0.211374
$111$	0	0
$112$	28.2537	2.66973
$113$	0.112340	0.0105681	0.00528404	−	0.999986i	$-0.498318\pi$
$113$	0.112340	0.0105681	0.00528404	+	0.999986i	$0.498318\pi$
$114$	0	0
$115$	7.15596	0.667297
$116$	−5.25010	−0.487459
$117$	0	0
$118$	−28.7412	−2.64584
$119$	23.5410	2.15800
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.60374	0.235732
$123$	0	0
$124$	−9.73758	−0.874461
$125$	7.89709	0.706337
$126$	0	0
$127$	−12.5972	−1.11782	−0.558912	−	0.829227i	$-0.688781\pi$
$127$	−12.5972	−1.11782	−0.558912	+	0.829227i	$0.688781\pi$
$128$	−1.98695	−0.175624
$129$	0	0
$130$	−2.78511	−0.244271
$131$	−6.91322	−0.604011	−0.302006	−	0.953306i	$-0.597656\pi$
$131$	−6.91322	−0.604011	−0.302006	+	0.953306i	$0.597656\pi$
$132$	0	0
$133$	20.7726	1.80121
$134$	17.9921	1.55428
$135$	0	0
$136$	55.9837	4.80056
$137$	8.75852	0.748291	0.374146	−	0.927370i	$-0.377936\pi$
$137$	8.75852	0.748291	0.374146	+	0.927370i	$0.377936\pi$
$138$	0	0
$139$	13.2977	1.12790	0.563949	−	0.825809i	$-0.309281\pi$
$139$	13.2977	1.12790	0.563949	+	0.825809i	$0.309281\pi$
$140$	−12.3837	−1.04662
$141$	0	0
$142$	18.2324	1.53003
$143$	1.25630	0.105057
$144$	0	0
$145$	0.935271	0.0776700
$146$	3.31024	0.273957
$147$	0	0
$148$	−30.2077	−2.48305
$149$	−2.50057	−0.204855	−0.102428	−	0.994740i	$-0.532661\pi$
$149$	−2.50057	−0.204855	−0.102428	+	0.994740i	$0.532661\pi$
$150$	0	0
$151$	17.4446	1.41962	0.709811	−	0.704392i	$-0.248780\pi$
$151$	17.4446	1.41962	0.709811	+	0.704392i	$0.248780\pi$
$152$	49.4002	4.00688
$153$	0	0
$154$	7.92356	0.638499
$155$	1.73469	0.139334
$156$	0	0
$157$	−21.3392	−1.70305	−0.851526	−	0.524312i	$-0.824323\pi$
$157$	−21.3392	−1.70305	−0.851526	+	0.524312i	$0.824323\pi$
$158$	−9.66268	−0.768721
$159$	0	0
$160$	−8.25897	−0.652929
$161$	−25.5765	−2.01571
$162$	0	0
$163$	−13.4618	−1.05441	−0.527205	−	0.849738i	$-0.676760\pi$
$163$	−13.4618	−1.05441	−0.527205	+	0.849738i	$0.676760\pi$
$164$	46.9561	3.66666
$165$	0	0
$166$	−29.4065	−2.28239
$167$	−24.1963	−1.87237	−0.936184	−	0.351510i	$-0.885668\pi$
$167$	−24.1963	−1.87237	−0.936184	+	0.351510i	$0.885668\pi$
$168$	0	0
$169$	−11.4217	−0.878592
$170$	−17.1494	−1.31530
$171$	0	0
$172$	−39.3701	−3.00194
$173$	6.87500	0.522697	0.261348	−	0.965245i	$-0.415833\pi$
$173$	6.87500	0.522697	0.261348	+	0.965245i	$0.415833\pi$
$174$	0	0
$175$	−13.0096	−0.983436
$176$	9.28438	0.699837
$177$	0	0
$178$	9.44324	0.707801
$179$	7.18760	0.537227	0.268613	−	0.963248i	$-0.413435\pi$
$179$	7.18760	0.537227	0.268613	+	0.963248i	$0.413435\pi$
$180$	0	0
$181$	−3.28618	−0.244260	−0.122130	−	0.992514i	$-0.538972\pi$
$181$	−3.28618	−0.244260	−0.122130	+	0.992514i	$0.538972\pi$
$182$	9.95441	0.737870
$183$	0	0
$184$	−60.8244	−4.48403
$185$	5.38130	0.395641
$186$	0	0
$187$	7.73575	0.565694
$188$	−5.34111	−0.389541
$189$	0	0
$190$	−15.1327	−1.09784
$191$	15.9729	1.15576	0.577878	−	0.816123i	$-0.303881\pi$
$191$	15.9729	1.15576	0.577878	+	0.816123i	$0.303881\pi$
$192$	0	0
$193$	−5.16456	−0.371753	−0.185876	−	0.982573i	$-0.559512\pi$
$193$	−5.16456	−0.371753	−0.185876	+	0.982573i	$0.559512\pi$
$194$	−0.294837	−0.0211681
$195$	0	0
$196$	10.8051	0.771791
$197$	19.1277	1.36280	0.681398	−	0.731914i	$-0.261373\pi$
$197$	19.1277	1.36280	0.681398	+	0.731914i	$0.261373\pi$
$198$	0	0
$199$	0.0471165	0.00334000	0.00167000	−	0.999999i	$-0.499468\pi$
$199$	0.0471165	0.00334000	0.00167000	+	0.999999i	$0.499468\pi$
$200$	−30.9387	−2.18770
$201$	0	0
$202$	8.15672	0.573905
$203$	−3.34280	−0.234618
$204$	0	0
$205$	−8.36493	−0.584232
$206$	−1.59554	−0.111167
$207$	0	0
$208$	11.6640	0.808754
$209$	6.82604	0.472167
$210$	0	0
$211$	20.7627	1.42936	0.714681	−	0.699451i	$-0.246572\pi$
$211$	20.7627	1.42936	0.714681	+	0.699451i	$0.246572\pi$
$212$	22.5789	1.55073
$213$	0	0
$214$	−3.58106	−0.244796
$215$	7.01353	0.478319
$216$	0	0
$217$	−6.20003	−0.420886
$218$	53.4613	3.62085
$219$	0	0
$220$	−4.06939	−0.274358
$221$	9.71845	0.653734
$222$	0	0
$223$	10.9949	0.736273	0.368137	−	0.929772i	$-0.379996\pi$
$223$	10.9949	0.736273	0.368137	+	0.929772i	$0.379996\pi$
$224$	29.5188	1.97231
$225$	0	0
$226$	0.292505	0.0194572
$227$	−9.54696	−0.633654	−0.316827	−	0.948483i	$-0.602617\pi$
$227$	−9.54696	−0.633654	−0.316827	+	0.948483i	$0.602617\pi$
$228$	0	0
$229$	9.08724	0.600502	0.300251	−	0.953860i	$-0.402930\pi$
$229$	9.08724	0.600502	0.300251	+	0.953860i	$0.402930\pi$
$230$	18.6323	1.22858
$231$	0	0
$232$	−7.94963	−0.521919
$233$	9.22388	0.604277	0.302138	−	0.953264i	$-0.402300\pi$
$233$	9.22388	0.604277	0.302138	+	0.953264i	$0.402300\pi$
$234$	0	0
$235$	0.951486	0.0620681
$236$	−52.7578	−3.43424
$237$	0	0
$238$	61.2947	3.97314
$239$	14.5862	0.943501	0.471751	−	0.881732i	$-0.343622\pi$
$239$	14.5862	0.943501	0.471751	+	0.881732i	$0.343622\pi$
$240$	0	0
$241$	−10.7788	−0.694325	−0.347163	−	0.937805i	$-0.612855\pi$
$241$	−10.7788	−0.694325	−0.347163	+	0.937805i	$0.612855\pi$
$242$	2.60374	0.167375
$243$	0	0
$244$	4.77947	0.305974
$245$	−1.92486	−0.122975
$246$	0	0
$247$	8.57558	0.545651
$248$	−14.7445	−0.936279
$249$	0	0
$250$	20.5620	1.30045
$251$	−15.2761	−0.964220	−0.482110	−	0.876111i	$-0.660129\pi$
$251$	−15.2761	−0.964220	−0.482110	+	0.876111i	$0.660129\pi$
$252$	0	0
$253$	−8.40462	−0.528394
$254$	−32.7999	−2.05805
$255$	0	0
$256$	−18.5491	−1.15932
$257$	4.08000	0.254503	0.127252	−	0.991870i	$-0.459384\pi$
$257$	4.08000	0.254503	0.127252	+	0.991870i	$0.459384\pi$
$258$	0	0
$259$	−19.2336	−1.19512
$260$	−5.11240	−0.317057
$261$	0	0
$262$	−18.0002	−1.11206
$263$	−1.06475	−0.0656553	−0.0328276	−	0.999461i	$-0.510451\pi$
$263$	−1.06475	−0.0656553	−0.0328276	+	0.999461i	$0.510451\pi$
$264$	0	0
$265$	−4.02230	−0.247088
$266$	54.0865	3.31626
$267$	0	0
$268$	33.0266	2.01742
$269$	7.89164	0.481162	0.240581	−	0.970629i	$-0.422662\pi$
$269$	7.89164	0.481162	0.240581	+	0.970629i	$0.422662\pi$
$270$	0	0
$271$	−21.7155	−1.31912	−0.659560	−	0.751652i	$-0.729257\pi$
$271$	−21.7155	−1.31912	−0.659560	+	0.751652i	$0.729257\pi$
$272$	71.8216	4.35483
$273$	0	0
$274$	22.8049	1.37770
$275$	−4.27506	−0.257796
$276$	0	0
$277$	−17.6944	−1.06316	−0.531578	−	0.847009i	$-0.678401\pi$
$277$	−17.6944	−1.06316	−0.531578	+	0.847009i	$0.678401\pi$
$278$	34.6238	2.07660
$279$	0	0
$280$	−18.7513	−1.12061
$281$	−30.9826	−1.84827	−0.924133	−	0.382072i	$-0.875211\pi$
$281$	−30.9826	−1.84827	−0.924133	+	0.382072i	$0.875211\pi$
$282$	0	0
$283$	−9.38441	−0.557846	−0.278923	−	0.960314i	$-0.589977\pi$
$283$	−9.38441	−0.557846	−0.278923	+	0.960314i	$0.589977\pi$
$284$	33.4677	1.98594
$285$	0	0
$286$	3.27109	0.193424
$287$	29.8975	1.76479
$288$	0	0
$289$	42.8418	2.52010
$290$	2.43520	0.143000
$291$	0	0
$292$	6.07633	0.355590
$293$	1.25100	0.0730845	0.0365422	−	0.999332i	$-0.488366\pi$
$293$	1.25100	0.0730845	0.0365422	+	0.999332i	$0.488366\pi$
$294$	0	0
$295$	9.39847	0.547200
$296$	−45.7401	−2.65859
$297$	0	0
$298$	−6.51085	−0.377163
$299$	−10.5588	−0.610629
$300$	0	0
$301$	−25.0674	−1.44486
$302$	45.4212	2.61370
$303$	0	0
$304$	63.3756	3.63484
$305$	−0.851432	−0.0487529
$306$	0	0
$307$	−28.7361	−1.64005	−0.820027	−	0.572325i	$-0.806041\pi$
$307$	−28.7361	−1.64005	−0.820027	+	0.572325i	$0.806041\pi$
$308$	14.5446	0.828756
$309$	0	0
$310$	4.51668	0.256530
$311$	−15.3899	−0.872679	−0.436339	−	0.899782i	$-0.643725\pi$
$311$	−15.3899	−0.872679	−0.436339	+	0.899782i	$0.643725\pi$
$312$	0	0
$313$	14.0136	0.792096	0.396048	−	0.918230i	$-0.370381\pi$
$313$	14.0136	0.792096	0.396048	+	0.918230i	$0.370381\pi$
$314$	−55.5617	−3.13553
$315$	0	0
$316$	−17.7370	−0.997782
$317$	−18.4283	−1.03504	−0.517519	−	0.855672i	$-0.673144\pi$
$317$	−18.4283	−1.03504	−0.517519	+	0.855672i	$0.673144\pi$
$318$	0	0
$319$	−1.09847	−0.0615024
$320$	−5.69419	−0.318315
$321$	0	0
$322$	−66.5945	−3.71117
$323$	52.8045	2.93812
$324$	0	0
$325$	−5.37078	−0.297917
$326$	−35.0511	−1.94130
$327$	0	0
$328$	71.1004	3.92586
$329$	−3.40075	−0.187489
$330$	0	0
$331$	32.9871	1.81313	0.906567	−	0.422062i	$-0.138694\pi$
$331$	32.9871	1.81313	0.906567	+	0.422062i	$0.138694\pi$
$332$	−53.9791	−2.96249
$333$	0	0
$334$	−63.0010	−3.44726
$335$	−5.88348	−0.321449
$336$	0	0
$337$	−26.8517	−1.46271	−0.731354	−	0.681998i	$-0.761111\pi$
$337$	−26.8517	−1.46271	−0.731354	+	0.681998i	$0.761111\pi$
$338$	−29.7391	−1.61760
$339$	0	0
$340$	−31.4798	−1.70723
$341$	−2.03738	−0.110330
$342$	0	0
$343$	−14.4223	−0.778731
$344$	−59.6138	−3.21416
$345$	0	0
$346$	17.9007	0.962349
$347$	1.44142	0.0773795	0.0386897	−	0.999251i	$-0.487682\pi$
$347$	1.44142	0.0773795	0.0386897	+	0.999251i	$0.487682\pi$
$348$	0	0
$349$	−10.1255	−0.542004	−0.271002	−	0.962579i	$-0.587355\pi$
$349$	−10.1255	−0.542004	−0.271002	+	0.962579i	$0.587355\pi$
$350$	−33.8737	−1.81063
$351$	0	0
$352$	9.70010	0.517017
$353$	−6.94974	−0.369897	−0.184949	−	0.982748i	$-0.559212\pi$
$353$	−6.94974	−0.369897	−0.184949	+	0.982748i	$0.559212\pi$
$354$	0	0
$355$	−5.96206	−0.316434
$356$	17.3342	0.918708
$357$	0	0
$358$	18.7147	0.989100
$359$	−16.5539	−0.873681	−0.436840	−	0.899539i	$-0.643903\pi$
$359$	−16.5539	−0.873681	−0.436840	+	0.899539i	$0.643903\pi$
$360$	0	0
$361$	27.5948	1.45236
$362$	−8.55637	−0.449713
$363$	0	0
$364$	18.2725	0.957737
$365$	−1.08246	−0.0566585
$366$	0	0
$367$	−17.4739	−0.912132	−0.456066	−	0.889946i	$-0.650742\pi$
$367$	−17.4739	−0.912132	−0.456066	+	0.889946i	$0.650742\pi$
$368$	−78.0317	−4.06768
$369$	0	0
$370$	14.0115	0.728424
$371$	14.3763	0.746379
$372$	0	0
$373$	−32.0776	−1.66092	−0.830458	−	0.557082i	$-0.811921\pi$
$373$	−32.0776	−1.66092	−0.830458	+	0.557082i	$0.811921\pi$
$374$	20.1419	1.04151
$375$	0	0
$376$	−8.08745	−0.417079
$377$	−1.38001	−0.0710742
$378$	0	0
$379$	−5.75540	−0.295635	−0.147817	−	0.989015i	$-0.547225\pi$
$379$	−5.75540	−0.295635	−0.147817	+	0.989015i	$0.547225\pi$
$380$	−27.7778	−1.42497
$381$	0	0
$382$	41.5892	2.12789
$383$	25.7524	1.31589	0.657944	−	0.753067i	$-0.271426\pi$
$383$	25.7524	1.31589	0.657944	+	0.753067i	$0.271426\pi$
$384$	0	0
$385$	−2.59103	−0.132051
$386$	−13.4472	−0.684443
$387$	0	0
$388$	−0.541208	−0.0274757
$389$	−19.6307	−0.995314	−0.497657	−	0.867374i	$-0.665806\pi$
$389$	−19.6307	−0.995314	−0.497657	+	0.867374i	$0.665806\pi$
$390$	0	0
$391$	−65.0160	−3.28800
$392$	16.3609	0.826351
$393$	0	0
$394$	49.8037	2.50907
$395$	3.15973	0.158983
$396$	0	0
$397$	−16.9311	−0.849747	−0.424873	−	0.905253i	$-0.639681\pi$
$397$	−16.9311	−0.849747	−0.424873	+	0.905253i	$0.639681\pi$
$398$	0.122679	0.00614935
$399$	0	0
$400$	−39.6913	−1.98457
$401$	8.13681	0.406333	0.203167	−	0.979144i	$-0.434877\pi$
$401$	8.13681	0.406333	0.203167	+	0.979144i	$0.434877\pi$
$402$	0	0
$403$	−2.55957	−0.127501
$404$	14.9726	0.744914
$405$	0	0
$406$	−8.70378	−0.431961
$407$	−6.32030	−0.313286
$408$	0	0
$409$	16.5019	0.815966	0.407983	−	0.912990i	$-0.366232\pi$
$409$	16.5019	0.815966	0.407983	+	0.912990i	$0.366232\pi$
$410$	−21.7801	−1.07564
$411$	0	0
$412$	−2.92880	−0.144292
$413$	−33.5915	−1.65293
$414$	0	0
$415$	9.61604	0.472033
$416$	12.1863	0.597481
$417$	0	0
$418$	17.7732	0.869318
$419$	−4.06852	−0.198760	−0.0993802	−	0.995050i	$-0.531686\pi$
$419$	−4.06852	−0.198760	−0.0993802	+	0.995050i	$0.531686\pi$
$420$	0	0
$421$	−22.4972	−1.09645	−0.548223	−	0.836332i	$-0.684696\pi$
$421$	−22.4972	−1.09645	−0.548223	+	0.836332i	$0.684696\pi$
$422$	54.0606	2.63163
$423$	0	0
$424$	34.1888	1.66035
$425$	−33.0708	−1.60417
$426$	0	0
$427$	3.04314	0.147268
$428$	−6.57345	−0.317740
$429$	0	0
$430$	18.2614	0.880644
$431$	−14.8069	−0.713225	−0.356613	−	0.934252i	$-0.616068\pi$
$431$	−14.8069	−0.713225	−0.356613	+	0.934252i	$0.616068\pi$
$432$	0	0
$433$	13.8033	0.663343	0.331671	−	0.943395i	$-0.392387\pi$
$433$	13.8033	0.663343	0.331671	+	0.943395i	$0.392387\pi$
$434$	−16.1433	−0.774902
$435$	0	0
$436$	98.1343	4.69978
$437$	−57.3703	−2.74439
$438$	0	0
$439$	−19.4862	−0.930027	−0.465013	−	0.885304i	$-0.653950\pi$
$439$	−19.4862	−0.930027	−0.465013	+	0.885304i	$0.653950\pi$
$440$	−6.16183	−0.293754
$441$	0	0
$442$	25.3043	1.20360
$443$	−24.5693	−1.16732	−0.583661	−	0.811997i	$-0.698380\pi$
$443$	−24.5693	−1.16732	−0.583661	+	0.811997i	$0.698380\pi$
$444$	0	0
$445$	−3.08797	−0.146384
$446$	28.6279	1.35557
$447$	0	0
$448$	20.3519	0.961535
$449$	−20.1319	−0.950081	−0.475041	−	0.879964i	$-0.657567\pi$
$449$	−20.1319	−0.950081	−0.475041	+	0.879964i	$0.657567\pi$
$450$	0	0
$451$	9.82455	0.462620
$452$	0.536927	0.0252549
$453$	0	0
$454$	−24.8578	−1.16664
$455$	−3.25512	−0.152603
$456$	0	0
$457$	−14.6515	−0.685366	−0.342683	−	0.939451i	$-0.611336\pi$
$457$	−14.6515	−0.685366	−0.342683	+	0.939451i	$0.611336\pi$
$458$	23.6608	1.10560
$459$	0	0
$460$	34.2017	1.59466
$461$	8.90582	0.414785	0.207393	−	0.978258i	$-0.433502\pi$
$461$	8.90582	0.414785	0.207393	+	0.978258i	$0.433502\pi$
$462$	0	0
$463$	−32.9705	−1.53227	−0.766133	−	0.642681i	$-0.777822\pi$
$463$	−32.9705	−1.53227	−0.766133	+	0.642681i	$0.777822\pi$
$464$	−10.1986	−0.473458
$465$	0	0
$466$	24.0166	1.11255
$467$	−21.5869	−0.998921	−0.499461	−	0.866337i	$-0.666468\pi$
$467$	−21.5869	−0.998921	−0.499461	+	0.866337i	$0.666468\pi$
$468$	0	0
$469$	21.0284	0.971003
$470$	2.47742	0.114275
$471$	0	0
$472$	−79.8853	−3.67702
$473$	−8.23734	−0.378753
$474$	0	0
$475$	−29.1818	−1.33895
$476$	112.513	5.15704
$477$	0	0
$478$	37.9786	1.73710
$479$	13.8271	0.631774	0.315887	−	0.948797i	$-0.397698\pi$
$479$	13.8271	0.631774	0.315887	+	0.948797i	$0.397698\pi$
$480$	0	0
$481$	−7.94022	−0.362043
$482$	−28.0653	−1.27834
$483$	0	0
$484$	4.77947	0.217249
$485$	0.0964127	0.00437788
$486$	0	0
$487$	35.7656	1.62069	0.810347	−	0.585950i	$-0.199279\pi$
$487$	35.7656	1.62069	0.810347	+	0.585950i	$0.199279\pi$
$488$	7.23702	0.327604
$489$	0	0
$490$	−5.01183	−0.226411
$491$	30.4054	1.37217	0.686087	−	0.727519i	$-0.259327\pi$
$491$	30.4054	1.37217	0.686087	+	0.727519i	$0.259327\pi$
$492$	0	0
$493$	−8.49747	−0.382707
$494$	22.3286	1.00461
$495$	0	0
$496$	−18.9158	−0.849344
$497$	21.3093	0.955853
$498$	0	0
$499$	4.63293	0.207398	0.103699	−	0.994609i	$-0.466932\pi$
$499$	4.63293	0.207398	0.103699	+	0.994609i	$0.466932\pi$
$500$	37.7439	1.68796
$501$	0	0
$502$	−39.7751	−1.77525
$503$	−13.7105	−0.611319	−0.305660	−	0.952141i	$-0.598877\pi$
$503$	−13.7105	−0.611319	−0.305660	+	0.952141i	$0.598877\pi$
$504$	0	0
$505$	−2.66727	−0.118692
$506$	−21.8835	−0.972838
$507$	0	0
$508$	−60.2081	−2.67130
$509$	11.0272	0.488771	0.244386	−	0.969678i	$-0.421414\pi$
$509$	11.0272	0.488771	0.244386	+	0.969678i	$0.421414\pi$
$510$	0	0
$511$	3.86887	0.171149
$512$	−44.3231	−1.95882
$513$	0	0
$514$	10.6233	0.468572
$515$	0.521748	0.0229909
$516$	0	0
$517$	−1.11751	−0.0491481
$518$	−50.0793	−2.20036
$519$	0	0
$520$	−7.74113	−0.339471
$521$	−5.84854	−0.256229	−0.128115	−	0.991759i	$-0.540893\pi$
$521$	−5.84854	−0.256229	−0.128115	+	0.991759i	$0.540893\pi$
$522$	0	0
$523$	26.2499	1.14783	0.573914	−	0.818915i	$-0.305424\pi$
$523$	26.2499	1.14783	0.573914	+	0.818915i	$0.305424\pi$
$524$	−33.0415	−1.44343
$525$	0	0
$526$	−2.77233	−0.120879
$527$	−15.7606	−0.686544
$528$	0	0
$529$	47.6377	2.07120
$530$	−10.4730	−0.454919
$531$	0	0
$532$	99.2821	4.30442
$533$	12.3426	0.534618
$534$	0	0
$535$	1.17102	0.0506275
$536$	50.0085	2.16004
$537$	0	0
$538$	20.5478	0.885878
$539$	2.26073	0.0973764
$540$	0	0
$541$	21.5728	0.927487	0.463743	−	0.885970i	$-0.346506\pi$
$541$	21.5728	0.927487	0.463743	+	0.885970i	$0.346506\pi$
$542$	−56.5414	−2.42866
$543$	0	0
$544$	75.0375	3.21721
$545$	−17.4820	−0.748847
$546$	0	0
$547$	−13.2689	−0.567339	−0.283669	−	0.958922i	$-0.591552\pi$
$547$	−13.2689	−0.567339	−0.283669	+	0.958922i	$0.591552\pi$
$548$	41.8611	1.78822
$549$	0	0
$550$	−11.1312	−0.474634
$551$	−7.49819	−0.319434
$552$	0	0
$553$	−11.2933	−0.480241
$554$	−46.0717	−1.95740
$555$	0	0
$556$	63.5561	2.69538
$557$	19.7663	0.837525	0.418762	−	0.908096i	$-0.362464\pi$
$557$	19.7663	0.837525	0.418762	+	0.908096i	$0.362464\pi$
$558$	0	0
$559$	−10.3486	−0.437699
$560$	−24.0561	−1.01656
$561$	0	0
$562$	−80.6706	−3.40288
$563$	21.9105	0.923418	0.461709	−	0.887032i	$-0.347237\pi$
$563$	21.9105	0.923418	0.461709	+	0.887032i	$0.347237\pi$
$564$	0	0
$565$	−0.0956501	−0.00402403
$566$	−24.4346	−1.02706
$567$	0	0
$568$	50.6765	2.12634
$569$	33.2783	1.39510	0.697550	−	0.716536i	$-0.254273\pi$
$569$	33.2783	1.39510	0.697550	+	0.716536i	$0.254273\pi$
$570$	0	0
$571$	4.34024	0.181633	0.0908166	−	0.995868i	$-0.471052\pi$
$571$	4.34024	0.181633	0.0908166	+	0.995868i	$0.471052\pi$
$572$	6.00447	0.251059
$573$	0	0
$574$	77.8454	3.24920
$575$	35.9303	1.49840
$576$	0	0
$577$	25.8729	1.07710	0.538552	−	0.842592i	$-0.318971\pi$
$577$	25.8729	1.07710	0.538552	+	0.842592i	$0.318971\pi$
$578$	111.549	4.63982
$579$	0	0
$580$	4.47010	0.185611
$581$	−34.3691	−1.42587
$582$	0	0
$583$	4.72415	0.195654
$584$	9.20070	0.380728
$585$	0	0
$586$	3.25729	0.134558
$587$	13.9531	0.575908	0.287954	−	0.957644i	$-0.407025\pi$
$587$	13.9531	0.575908	0.287954	+	0.957644i	$0.407025\pi$
$588$	0	0
$589$	−13.9072	−0.573037
$590$	24.4712	1.00746
$591$	0	0
$592$	−58.6801	−2.41174
$593$	41.4876	1.70369	0.851846	−	0.523792i	$-0.175483\pi$
$593$	41.4876	1.70369	0.851846	+	0.523792i	$0.175483\pi$
$594$	0	0
$595$	−20.0436	−0.821706
$596$	−11.9514	−0.489549
$597$	0	0
$598$	−27.4923	−1.12424
$599$	−1.63508	−0.0668077	−0.0334038	−	0.999442i	$-0.510635\pi$
$599$	−1.63508	−0.0668077	−0.0334038	+	0.999442i	$0.510635\pi$
$600$	0	0
$601$	37.8743	1.54492	0.772462	−	0.635062i	$-0.219025\pi$
$601$	37.8743	1.54492	0.772462	+	0.635062i	$0.219025\pi$
$602$	−65.2690	−2.66017
$603$	0	0
$604$	83.3759	3.39252
$605$	−0.851432	−0.0346156
$606$	0	0
$607$	1.25664	0.0510055	0.0255027	−	0.999675i	$-0.491881\pi$
$607$	1.25664	0.0510055	0.0255027	+	0.999675i	$0.491881\pi$
$608$	66.2132	2.68530
$609$	0	0
$610$	−2.21691	−0.0897600
$611$	−1.40394	−0.0567972
$612$	0	0
$613$	22.7888	0.920430	0.460215	−	0.887807i	$-0.347772\pi$
$613$	22.7888	0.920430	0.460215	+	0.887807i	$0.347772\pi$
$614$	−74.8212	−3.01954
$615$	0	0
$616$	22.0233	0.887344
$617$	9.66482	0.389091	0.194545	−	0.980894i	$-0.437677\pi$
$617$	9.66482	0.389091	0.194545	+	0.980894i	$0.437677\pi$
$618$	0	0
$619$	12.2722	0.493263	0.246632	−	0.969109i	$-0.420676\pi$
$619$	12.2722	0.493263	0.246632	+	0.969109i	$0.420676\pi$
$620$	8.29089	0.332970
$621$	0	0
$622$	−40.0712	−1.60671
$623$	11.0369	0.442183
$624$	0	0
$625$	14.6515	0.586059
$626$	36.4878	1.45835
$627$	0	0
$628$	−101.990	−4.06984
$629$	−48.8922	−1.94946
$630$	0	0
$631$	32.5982	1.29772	0.648858	−	0.760910i	$-0.275247\pi$
$631$	32.5982	1.29772	0.648858	+	0.760910i	$0.275247\pi$
$632$	−26.8571	−1.06832
$633$	0	0
$634$	−47.9826	−1.90563
$635$	10.7257	0.425636
$636$	0	0
$637$	2.84016	0.112531
$638$	−2.86013	−0.113234
$639$	0	0
$640$	1.69176	0.0668725
$641$	−39.4363	−1.55764	−0.778820	−	0.627248i	$-0.784181\pi$
$641$	−39.4363	−1.55764	−0.778820	+	0.627248i	$0.784181\pi$
$642$	0	0
$643$	35.4044	1.39621	0.698107	−	0.715993i	$-0.254026\pi$
$643$	35.4044	1.39621	0.698107	+	0.715993i	$0.254026\pi$
$644$	−122.242	−4.81701
$645$	0	0
$646$	137.489	5.40944
$647$	25.6571	1.00869	0.504343	−	0.863504i	$-0.331735\pi$
$647$	25.6571	1.00869	0.504343	+	0.863504i	$0.331735\pi$
$648$	0	0
$649$	−11.0384	−0.433296
$650$	−13.9841	−0.548503
$651$	0	0
$652$	−64.3403	−2.51976
$653$	3.64441	0.142617	0.0713085	−	0.997454i	$-0.477283\pi$
$653$	3.64441	0.142617	0.0713085	+	0.997454i	$0.477283\pi$
$654$	0	0
$655$	5.88614	0.229991
$656$	91.2149	3.56134
$657$	0	0
$658$	−8.85467	−0.345191
$659$	3.49371	0.136095	0.0680477	−	0.997682i	$-0.478323\pi$
$659$	3.49371	0.136095	0.0680477	+	0.997682i	$0.478323\pi$
$660$	0	0
$661$	−49.5134	−1.92585	−0.962923	−	0.269775i	$-0.913051\pi$
$661$	−49.5134	−1.92585	−0.962923	+	0.269775i	$0.913051\pi$
$662$	85.8898	3.33820
$663$	0	0
$664$	−81.7346	−3.17192
$665$	−17.6865	−0.685852
$666$	0	0
$667$	9.23221	0.357473
$668$	−115.646	−4.47446
$669$	0	0
$670$	−15.3191	−0.591827
$671$	1.00000	0.0386046
$672$	0	0
$673$	−17.3434	−0.668538	−0.334269	−	0.942478i	$-0.608489\pi$
$673$	−17.3434	−0.668538	−0.334269	+	0.942478i	$0.608489\pi$
$674$	−69.9150	−2.69302
$675$	0	0
$676$	−54.5897	−2.09960
$677$	−23.0773	−0.886931	−0.443466	−	0.896291i	$-0.646251\pi$
$677$	−23.0773	−0.886931	−0.443466	+	0.896291i	$0.646251\pi$
$678$	0	0
$679$	−0.344593	−0.0132243
$680$	−47.6663	−1.82792
$681$	0	0
$682$	−5.30480	−0.203131
$683$	−41.1141	−1.57319	−0.786593	−	0.617471i	$-0.788157\pi$
$683$	−41.1141	−1.57319	−0.786593	+	0.617471i	$0.788157\pi$
$684$	0	0
$685$	−7.45729	−0.284928
$686$	−37.5519	−1.43374
$687$	0	0
$688$	−76.4786	−2.91572
$689$	5.93498	0.226105
$690$	0	0
$691$	0.00796003	0.000302814 0	0.000151407	−	1.00000i	$-0.499952\pi$
$691$	0.00796003	0.000302814 0	0.000151407		1.00000i	$0.499952\pi$
$692$	32.8589	1.24911
$693$	0	0
$694$	3.75308	0.142465
$695$	−11.3221	−0.429472
$696$	0	0
$697$	76.0002	2.87871
$698$	−26.3641	−0.997896
$699$	0	0
$700$	−62.1791	−2.35015
$701$	−15.6416	−0.590774	−0.295387	−	0.955378i	$-0.595449\pi$
$701$	−15.6416	−0.590774	−0.295387	+	0.955378i	$0.595449\pi$
$702$	0	0
$703$	−43.1426	−1.62715
$704$	6.68778	0.252055
$705$	0	0
$706$	−18.0953	−0.681026
$707$	9.53323	0.358534
$708$	0	0
$709$	16.5953	0.623250	0.311625	−	0.950205i	$-0.399127\pi$
$709$	16.5953	0.623250	0.311625	+	0.950205i	$0.399127\pi$
$710$	−15.5237	−0.582593
$711$	0	0
$712$	26.2472	0.983655
$713$	17.1234	0.641276
$714$	0	0
$715$	−1.06966	−0.0400029
$716$	34.3529	1.28383
$717$	0	0
$718$	−43.1020	−1.60855
$719$	25.8212	0.962967	0.481484	−	0.876455i	$-0.340098\pi$
$719$	25.8212	0.962967	0.481484	+	0.876455i	$0.340098\pi$
$720$	0	0
$721$	−1.86480	−0.0694489
$722$	71.8497	2.67397
$723$	0	0
$724$	−15.7062	−0.583716
$725$	4.69602	0.174406
$726$	0	0
$727$	−20.5085	−0.760619	−0.380310	−	0.924859i	$-0.624183\pi$
$727$	−20.5085	−0.760619	−0.380310	+	0.924859i	$0.624183\pi$
$728$	27.6680	1.02544
$729$	0	0
$730$	−2.81844	−0.104315
$731$	−63.7220	−2.35684
$732$	0	0
$733$	46.0269	1.70004	0.850021	−	0.526749i	$-0.176589\pi$
$733$	46.0269	1.70004	0.850021	+	0.526749i	$0.176589\pi$
$734$	−45.4976	−1.67935
$735$	0	0
$736$	−81.5256	−3.00508
$737$	6.91010	0.254537
$738$	0	0
$739$	33.7128	1.24015	0.620073	−	0.784544i	$-0.287103\pi$
$739$	33.7128	1.24015	0.620073	+	0.784544i	$0.287103\pi$
$740$	25.7198	0.945478
$741$	0	0
$742$	37.4321	1.37418
$743$	1.80937	0.0663793	0.0331896	−	0.999449i	$-0.489433\pi$
$743$	1.80937	0.0663793	0.0331896	+	0.999449i	$0.489433\pi$
$744$	0	0
$745$	2.12907	0.0780031
$746$	−83.5218	−3.05795
$747$	0	0
$748$	36.9728	1.35186
$749$	−4.18539	−0.152931
$750$	0	0
$751$	−17.2718	−0.630256	−0.315128	−	0.949049i	$-0.602047\pi$
$751$	−17.2718	−0.630256	−0.315128	+	0.949049i	$0.602047\pi$
$752$	−10.3754	−0.378352
$753$	0	0
$754$	−3.59319	−0.130856
$755$	−14.8529	−0.540552
$756$	0	0
$757$	−0.988737	−0.0359363	−0.0179681	−	0.999839i	$-0.505720\pi$
$757$	−0.988737	−0.0359363	−0.0179681	+	0.999839i	$0.505720\pi$
$758$	−14.9856	−0.544300
$759$	0	0
$760$	−42.0609	−1.52571
$761$	−47.0722	−1.70636	−0.853182	−	0.521613i	$-0.825330\pi$
$761$	−47.0722	−1.70636	−0.853182	+	0.521613i	$0.825330\pi$
$762$	0	0
$763$	62.4833	2.26205
$764$	76.3419	2.76195
$765$	0	0
$766$	67.0527	2.42271
$767$	−13.8676	−0.500731
$768$	0	0
$769$	26.8992	0.970009	0.485004	−	0.874512i	$-0.338818\pi$
$769$	26.8992	0.970009	0.485004	+	0.874512i	$0.338818\pi$
$770$	−6.74637	−0.243122
$771$	0	0
$772$	−24.6838	−0.888391
$773$	−12.9507	−0.465804	−0.232902	−	0.972500i	$-0.574822\pi$
$773$	−12.9507	−0.465804	−0.232902	+	0.972500i	$0.574822\pi$
$774$	0	0
$775$	8.70992	0.312869
$776$	−0.819491	−0.0294180
$777$	0	0
$778$	−51.1132	−1.83250
$779$	67.0627	2.40277
$780$	0	0
$781$	7.00240	0.250565
$782$	−169.285	−6.05362
$783$	0	0
$784$	20.9895	0.749623
$785$	18.1689	0.648475
$786$	0	0
$787$	−36.5969	−1.30454	−0.652270	−	0.757987i	$-0.726183\pi$
$787$	−36.5969	−1.30454	−0.652270	+	0.757987i	$0.726183\pi$
$788$	91.4204	3.25672
$789$	0	0
$790$	8.22711	0.292708
$791$	0.341868	0.0121554
$792$	0	0
$793$	1.25630	0.0446127
$794$	−44.0842	−1.56449
$795$	0	0
$796$	0.225192	0.00798172
$797$	−27.8407	−0.986167	−0.493083	−	0.869982i	$-0.664130\pi$
$797$	−27.8407	−0.986167	−0.493083	+	0.869982i	$0.664130\pi$
$798$	0	0
$799$	−8.64479	−0.305831
$800$	−41.4685	−1.46613
$801$	0	0
$802$	21.1862	0.748109
$803$	1.27134	0.0448646
$804$	0	0
$805$	21.7766	0.767525
$806$	−6.66445	−0.234745
$807$	0	0
$808$	22.6713	0.797575
$809$	1.23289	0.0433461	0.0216730	−	0.999765i	$-0.493101\pi$
$809$	1.23289	0.0433461	0.0216730	+	0.999765i	$0.493101\pi$
$810$	0	0
$811$	−30.2942	−1.06377	−0.531886	−	0.846816i	$-0.678516\pi$
$811$	−30.2942	−1.06377	−0.531886	+	0.846816i	$0.678516\pi$
$812$	−15.9768	−0.560676
$813$	0	0
$814$	−16.4564	−0.576797
$815$	11.4618	0.401490
$816$	0	0
$817$	−56.2284	−1.96718
$818$	42.9666	1.50229
$819$	0	0
$820$	−39.9799	−1.39616
$821$	−37.4922	−1.30849	−0.654244	−	0.756283i	$-0.727013\pi$
$821$	−37.4922	−1.30849	−0.654244	+	0.756283i	$0.727013\pi$
$822$	0	0
$823$	23.5323	0.820285	0.410142	−	0.912022i	$-0.365479\pi$
$823$	23.5323	0.820285	0.410142	+	0.912022i	$0.365479\pi$
$824$	−4.43476	−0.154492
$825$	0	0
$826$	−87.4636	−3.04325
$827$	−26.7937	−0.931707	−0.465854	−	0.884862i	$-0.654253\pi$
$827$	−26.7937	−0.931707	−0.465854	+	0.884862i	$0.654253\pi$
$828$	0	0
$829$	−9.76779	−0.339249	−0.169625	−	0.985509i	$-0.554256\pi$
$829$	−9.76779	−0.339249	−0.169625	+	0.985509i	$0.554256\pi$
$830$	25.0377	0.869070
$831$	0	0
$832$	8.40188	0.291283
$833$	17.4884	0.605938
$834$	0	0
$835$	20.6015	0.712945
$836$	32.6248	1.12835
$837$	0	0
$838$	−10.5934	−0.365942
$839$	2.24302	0.0774376	0.0387188	−	0.999250i	$-0.487672\pi$
$839$	2.24302	0.0774376	0.0387188	+	0.999250i	$0.487672\pi$
$840$	0	0
$841$	−27.7934	−0.958392
$842$	−58.5769	−2.01869
$843$	0	0
$844$	99.2345	3.41579
$845$	9.72480	0.334543
$846$	0	0
$847$	3.04314	0.104564
$848$	43.8609	1.50619
$849$	0	0
$850$	−86.1078	−2.95347
$851$	53.1197	1.82092
$852$	0	0
$853$	−44.3477	−1.51843	−0.759217	−	0.650837i	$-0.774418\pi$
$853$	−44.3477	−1.51843	−0.759217	+	0.650837i	$0.774418\pi$
$854$	7.92356	0.271139
$855$	0	0
$856$	−9.95345	−0.340202
$857$	−19.5439	−0.667608	−0.333804	−	0.942642i	$-0.608332\pi$
$857$	−19.5439	−0.667608	−0.333804	+	0.942642i	$0.608332\pi$
$858$	0	0
$859$	21.0704	0.718913	0.359456	−	0.933162i	$-0.382962\pi$
$859$	21.0704	0.718913	0.359456	+	0.933162i	$0.382962\pi$
$860$	33.5210	1.14306
$861$	0	0
$862$	−38.5534	−1.31314
$863$	55.0745	1.87476	0.937380	−	0.348309i	$-0.113244\pi$
$863$	55.0745	1.87476	0.937380	+	0.348309i	$0.113244\pi$
$864$	0	0
$865$	−5.85360	−0.199028
$866$	35.9401	1.22130
$867$	0	0
$868$	−29.6329	−1.00580
$869$	−3.71107	−0.125890
$870$	0	0
$871$	8.68119	0.294151
$872$	148.594	5.03202
$873$	0	0
$874$	−149.377	−5.05276
$875$	24.0320	0.812429
$876$	0	0
$877$	−14.0361	−0.473965	−0.236983	−	0.971514i	$-0.576158\pi$
$877$	−14.0361	−0.473965	−0.236983	+	0.971514i	$0.576158\pi$
$878$	−50.7371	−1.71229
$879$	0	0
$880$	−7.90502	−0.266478
$881$	23.7847	0.801326	0.400663	−	0.916226i	$-0.368780\pi$
$881$	23.7847	0.801326	0.400663	+	0.916226i	$0.368780\pi$
$882$	0	0
$883$	−13.7523	−0.462801	−0.231401	−	0.972859i	$-0.574331\pi$
$883$	−13.7523	−0.462801	−0.231401	+	0.972859i	$0.574331\pi$
$884$	46.4490	1.56225
$885$	0	0
$886$	−63.9721	−2.14918
$887$	−30.3096	−1.01770	−0.508849	−	0.860856i	$-0.669929\pi$
$887$	−30.3096	−1.01770	−0.508849	+	0.860856i	$0.669929\pi$
$888$	0	0
$889$	−38.3352	−1.28572
$890$	−8.04027	−0.269511
$891$	0	0
$892$	52.5498	1.75950
$893$	−7.62818	−0.255267
$894$	0	0
$895$	−6.11975	−0.204561
$896$	−6.04659	−0.202002
$897$	0	0
$898$	−52.4182	−1.74922
$899$	2.23799	0.0746413
$900$	0	0
$901$	36.5449	1.21749
$902$	25.5806	0.851740
$903$	0	0
$904$	0.813009	0.0270403
$905$	2.79796	0.0930073
$906$	0	0
$907$	−36.6102	−1.21562	−0.607812	−	0.794081i	$-0.707952\pi$
$907$	−36.6102	−1.21562	−0.607812	+	0.794081i	$0.707952\pi$
$908$	−45.6294	−1.51426
$909$	0	0
$910$	−8.47550	−0.280960
$911$	−34.9673	−1.15852	−0.579258	−	0.815144i	$-0.696658\pi$
$911$	−34.9673	−1.15852	−0.579258	+	0.815144i	$0.696658\pi$
$912$	0	0
$913$	−11.2940	−0.373776
$914$	−38.1486	−1.26184
$915$	0	0
$916$	43.4322	1.43504
$917$	−21.0379	−0.694734
$918$	0	0
$919$	6.04826	0.199514	0.0997568	−	0.995012i	$-0.468194\pi$
$919$	6.04826	0.199514	0.0997568	+	0.995012i	$0.468194\pi$
$920$	51.7878	1.70739
$921$	0	0
$922$	23.1884	0.763671
$923$	8.79714	0.289562
$924$	0	0
$925$	27.0197	0.888401
$926$	−85.8465	−2.82109
$927$	0	0
$928$	−10.6552	−0.349776
$929$	50.8532	1.66844	0.834219	−	0.551433i	$-0.185919\pi$
$929$	50.8532	1.66844	0.834219	+	0.551433i	$0.185919\pi$
$930$	0	0
$931$	15.4318	0.505757
$932$	44.0853	1.44406
$933$	0	0
$934$	−56.2066	−1.83914
$935$	−6.58646	−0.215400
$936$	0	0
$937$	27.2266	0.889453	0.444726	−	0.895666i	$-0.353301\pi$
$937$	27.2266	0.889453	0.444726	+	0.895666i	$0.353301\pi$
$938$	54.7526	1.78774
$939$	0	0
$940$	4.54760	0.148326
$941$	−18.7448	−0.611064	−0.305532	−	0.952182i	$-0.598834\pi$
$941$	−18.7448	−0.611064	−0.305532	+	0.952182i	$0.598834\pi$
$942$	0	0
$943$	−82.5716	−2.68890
$944$	−102.485	−3.33560
$945$	0	0
$946$	−21.4479	−0.697331
$947$	−51.2919	−1.66676	−0.833381	−	0.552698i	$-0.813598\pi$
$947$	−51.2919	−1.66676	−0.833381	+	0.552698i	$0.813598\pi$
$948$	0	0
$949$	1.59719	0.0518470
$950$	−75.9817	−2.46517
$951$	0	0
$952$	170.367	5.52161
$953$	−12.0797	−0.391301	−0.195651	−	0.980674i	$-0.562682\pi$
$953$	−12.0797	−0.391301	−0.195651	+	0.980674i	$0.562682\pi$
$954$	0	0
$955$	−13.5998	−0.440080
$956$	69.7142	2.25472
$957$	0	0
$958$	36.0021	1.16317
$959$	26.6535	0.860685
$960$	0	0
$961$	−26.8491	−0.866100
$962$	−20.6743	−0.666565
$963$	0	0
$964$	−51.5171	−1.65925
$965$	4.39727	0.141553
$966$	0	0
$967$	29.1060	0.935985	0.467992	−	0.883732i	$-0.344977\pi$
$967$	29.1060	0.935985	0.467992	+	0.883732i	$0.344977\pi$
$968$	7.23702	0.232607
$969$	0	0
$970$	0.251034	0.00806021
$971$	−6.38454	−0.204890	−0.102445	−	0.994739i	$-0.532666\pi$
$971$	−6.38454	−0.204890	−0.102445	+	0.994739i	$0.532666\pi$
$972$	0	0
$973$	40.4669	1.29731
$974$	93.1244	2.98390
$975$	0	0
$976$	9.28438	0.297186
$977$	33.0182	1.05634	0.528172	−	0.849137i	$-0.322878\pi$
$977$	33.0182	1.05634	0.528172	+	0.849137i	$0.322878\pi$
$978$	0	0
$979$	3.62680	0.115913
$980$	−9.19979	−0.293876
$981$	0	0
$982$	79.1677	2.52634
$983$	−48.1934	−1.53713	−0.768566	−	0.639770i	$-0.779029\pi$
$983$	−48.1934	−1.53713	−0.768566	+	0.639770i	$0.779029\pi$
$984$	0	0
$985$	−16.2860	−0.518914
$986$	−22.1252	−0.704611
$987$	0	0
$988$	40.9867	1.30396
$989$	69.2317	2.20144
$990$	0	0
$991$	42.8697	1.36180	0.680900	−	0.732376i	$-0.261589\pi$
$991$	42.8697	1.36180	0.680900	+	0.732376i	$0.261589\pi$
$992$	−19.7628	−0.627468
$993$	0	0
$994$	55.4839	1.75984
$995$	−0.0401165	−0.00127178
$996$	0	0
$997$	14.1040	0.446679	0.223340	−	0.974741i	$-0.428304\pi$
$997$	14.1040	0.446679	0.223340	+	0.974741i	$0.428304\pi$
$998$	12.0629	0.381846
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.l.1.20		21
3.2	odd	2		671.2.a.d.1.2	✓	21
33.32	even	2		7381.2.a.j.1.20		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.d.1.2	✓	21	3.2	odd	2
6039.2.a.l.1.20		21	1.1	even	1	trivial
7381.2.a.j.1.20		21	33.32	even	2

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)

\(f(q)\)	\(=\)	\(q+2.60374 q^{2} +4.77947 q^{4} -0.851432 q^{5} +3.04314 q^{7} +7.23702 q^{8} +O(q^{10})\)	\(q+2.60374 q^{2} +4.77947 q^{4} -0.851432 q^{5} +3.04314 q^{7} +7.23702 q^{8} -2.21691 q^{10} +1.00000 q^{11} +1.25630 q^{13} +7.92356 q^{14} +9.28438 q^{16} +7.73575 q^{17} +6.82604 q^{19} -4.06939 q^{20} +2.60374 q^{22} -8.40462 q^{23} -4.27506 q^{25} +3.27109 q^{26} +14.5446 q^{28} -1.09847 q^{29} -2.03738 q^{31} +9.70010 q^{32} +20.1419 q^{34} -2.59103 q^{35} -6.32030 q^{37} +17.7732 q^{38} -6.16183 q^{40} +9.82455 q^{41} -8.23734 q^{43} +4.77947 q^{44} -21.8835 q^{46} -1.11751 q^{47} +2.26073 q^{49} -11.1312 q^{50} +6.00447 q^{52} +4.72415 q^{53} -0.851432 q^{55} +22.0233 q^{56} -2.86013 q^{58} -11.0384 q^{59} +1.00000 q^{61} -5.30480 q^{62} +6.68778 q^{64} -1.06966 q^{65} +6.91010 q^{67} +36.9728 q^{68} -6.74637 q^{70} +7.00240 q^{71} +1.27134 q^{73} -16.4564 q^{74} +32.6248 q^{76} +3.04314 q^{77} -3.71107 q^{79} -7.90502 q^{80} +25.5806 q^{82} -11.2940 q^{83} -6.58646 q^{85} -21.4479 q^{86} +7.23702 q^{88} +3.62680 q^{89} +3.82312 q^{91} -40.1696 q^{92} -2.90971 q^{94} -5.81191 q^{95} -0.113236 q^{97} +5.88635 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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Database

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