LMFDB - Embedded newform 6039.2.a.n.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:	$1$
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)

$f(q)$	$=$	$q-2.28499 q^{2} +3.22117 q^{4} -2.86039 q^{5} +3.60040 q^{7} -2.79036 q^{8} +O(q^{10})$	\(q-2.28499 q^{2} +3.22117 q^{4} -2.86039 q^{5} +3.60040 q^{7} -2.79036 q^{8} +6.53597 q^{10} -1.00000 q^{11} -3.97298 q^{13} -8.22686 q^{14} -0.0663931 q^{16} -2.89791 q^{17} +3.91288 q^{19} -9.21383 q^{20} +2.28499 q^{22} +4.02846 q^{23} +3.18186 q^{25} +9.07821 q^{26} +11.5975 q^{28} -3.89190 q^{29} -0.470941 q^{31} +5.73244 q^{32} +6.62170 q^{34} -10.2986 q^{35} -1.32152 q^{37} -8.94089 q^{38} +7.98155 q^{40} +2.18046 q^{41} +11.4649 q^{43} -3.22117 q^{44} -9.20499 q^{46} -6.67167 q^{47} +5.96285 q^{49} -7.27051 q^{50} -12.7976 q^{52} +2.86420 q^{53} +2.86039 q^{55} -10.0464 q^{56} +8.89295 q^{58} +6.78636 q^{59} -1.00000 q^{61} +1.07609 q^{62} -12.9658 q^{64} +11.3643 q^{65} +6.63203 q^{67} -9.33468 q^{68} +23.5321 q^{70} +2.50557 q^{71} -16.0693 q^{73} +3.01965 q^{74} +12.6041 q^{76} -3.60040 q^{77} -1.93246 q^{79} +0.189911 q^{80} -4.98232 q^{82} -9.23389 q^{83} +8.28918 q^{85} -26.1971 q^{86} +2.79036 q^{88} -6.85941 q^{89} -14.3043 q^{91} +12.9764 q^{92} +15.2447 q^{94} -11.1924 q^{95} +11.7618 q^{97} -13.6250 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.28499	−1.61573	−0.807865	−	0.589367i	$-0.799377\pi$
$2$	−2.28499	−1.61573	−0.807865	+	0.589367i	$0.799377\pi$
$3$	0	0
$3$	0	0
$4$	3.22117	1.61059
$5$	−2.86039	−1.27921	−0.639604	−	0.768705i	$-0.720902\pi$
$5$	−2.86039	−1.27921	−0.639604	+	0.768705i	$0.720902\pi$
$6$	0	0
$7$	3.60040	1.36082	0.680411	−	0.732831i	$-0.261801\pi$
$7$	3.60040	1.36082	0.680411	+	0.732831i	$0.261801\pi$
$8$	−2.79036	−0.986543
$9$	0	0
$10$	6.53597	2.06686
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.97298	−1.10191	−0.550953	−	0.834536i	$-0.685736\pi$
$13$	−3.97298	−1.10191	−0.550953	+	0.834536i	$0.685736\pi$
$14$	−8.22686	−2.19872
$15$	0	0
$16$	−0.0663931	−0.0165983
$17$	−2.89791	−0.702847	−0.351424	−	0.936217i	$-0.614302\pi$
$17$	−2.89791	−0.702847	−0.351424	+	0.936217i	$0.614302\pi$
$18$	0	0
$19$	3.91288	0.897677	0.448839	−	0.893613i	$-0.351838\pi$
$19$	3.91288	0.897677	0.448839	+	0.893613i	$0.351838\pi$
$20$	−9.21383	−2.06027
$21$	0	0
$22$	2.28499	0.487161
$23$	4.02846	0.839993	0.419996	−	0.907526i	$-0.362031\pi$
$23$	4.02846	0.839993	0.419996	+	0.907526i	$0.362031\pi$
$24$	0	0
$25$	3.18186	0.636372
$26$	9.07821	1.78038
$27$	0	0
$28$	11.5975	2.19172
$29$	−3.89190	−0.722708	−0.361354	−	0.932429i	$-0.617685\pi$
$29$	−3.89190	−0.722708	−0.361354	+	0.932429i	$0.617685\pi$
$30$	0	0
$31$	−0.470941	−0.0845834	−0.0422917	−	0.999105i	$-0.513466\pi$
$31$	−0.470941	−0.0845834	−0.0422917	+	0.999105i	$0.513466\pi$
$32$	5.73244	1.01336
$33$	0	0
$34$	6.62170	1.13561
$35$	−10.2986	−1.74077
$36$	0	0
$37$	−1.32152	−0.217256	−0.108628	−	0.994082i	$-0.534646\pi$
$37$	−1.32152	−0.217256	−0.108628	+	0.994082i	$0.534646\pi$
$38$	−8.94089	−1.45040
$39$	0	0
$40$	7.98155	1.26199
$41$	2.18046	0.340530	0.170265	−	0.985398i	$-0.445538\pi$
$41$	2.18046	0.340530	0.170265	+	0.985398i	$0.445538\pi$
$42$	0	0
$43$	11.4649	1.74838	0.874190	−	0.485584i	$-0.161393\pi$
$43$	11.4649	1.74838	0.874190	+	0.485584i	$0.161393\pi$
$44$	−3.22117	−0.485610
$45$	0	0
$46$	−9.20499	−1.35720
$47$	−6.67167	−0.973164	−0.486582	−	0.873635i	$-0.661756\pi$
$47$	−6.67167	−0.973164	−0.486582	+	0.873635i	$0.661756\pi$
$48$	0	0
$49$	5.96285	0.851836
$50$	−7.27051	−1.02821
$51$	0	0
$52$	−12.7976	−1.77471
$53$	2.86420	0.393429	0.196714	−	0.980461i	$-0.436973\pi$
$53$	2.86420	0.393429	0.196714	+	0.980461i	$0.436973\pi$
$54$	0	0
$55$	2.86039	0.385696
$56$	−10.0464	−1.34251
$57$	0	0
$58$	8.89295	1.16770
$59$	6.78636	0.883509	0.441755	−	0.897136i	$-0.354356\pi$
$59$	6.78636	0.883509	0.441755	+	0.897136i	$0.354356\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	1.07609	0.136664
$63$	0	0
$64$	−12.9658	−1.62072
$65$	11.3643	1.40957
$66$	0	0
$67$	6.63203	0.810231	0.405116	−	0.914265i	$-0.367231\pi$
$67$	6.63203	0.810231	0.405116	+	0.914265i	$0.367231\pi$
$68$	−9.33468	−1.13200
$69$	0	0
$70$	23.5321	2.81262
$71$	2.50557	0.297356	0.148678	−	0.988886i	$-0.452498\pi$
$71$	2.50557	0.297356	0.148678	+	0.988886i	$0.452498\pi$
$72$	0	0
$73$	−16.0693	−1.88076	−0.940382	−	0.340119i	$-0.889533\pi$
$73$	−16.0693	−1.88076	−0.940382	+	0.340119i	$0.889533\pi$
$74$	3.01965	0.351027
$75$	0	0
$76$	12.6041	1.44579
$77$	−3.60040	−0.410303
$78$	0	0
$79$	−1.93246	−0.217419	−0.108710	−	0.994074i	$-0.534672\pi$
$79$	−1.93246	−0.217419	−0.108710	+	0.994074i	$0.534672\pi$
$80$	0.189911	0.0212326
$81$	0	0
$82$	−4.98232	−0.550205
$83$	−9.23389	−1.01355	−0.506776	−	0.862078i	$-0.669163\pi$
$83$	−9.23389	−1.01355	−0.506776	+	0.862078i	$0.669163\pi$
$84$	0	0
$85$	8.28918	0.899088
$86$	−26.1971	−2.82491
$87$	0	0
$88$	2.79036	0.297454
$89$	−6.85941	−0.727096	−0.363548	−	0.931576i	$-0.618435\pi$
$89$	−6.85941	−0.727096	−0.363548	+	0.931576i	$0.618435\pi$
$90$	0	0
$91$	−14.3043	−1.49950
$92$	12.9764	1.35288
$93$	0	0
$94$	15.2447	1.57237
$95$	−11.1924	−1.14832
$96$	0	0
$97$	11.7618	1.19423	0.597114	−	0.802157i	$-0.296314\pi$
$97$	11.7618	1.19423	0.597114	+	0.802157i	$0.296314\pi$
$98$	−13.6250	−1.37634
$99$	0	0
$100$	10.2493	1.02493
$101$	5.58800	0.556027	0.278014	−	0.960577i	$-0.410324\pi$
$101$	5.58800	0.556027	0.278014	+	0.960577i	$0.410324\pi$
$102$	0	0
$103$	−3.45640	−0.340570	−0.170285	−	0.985395i	$-0.554469\pi$
$103$	−3.45640	−0.340570	−0.170285	+	0.985395i	$0.554469\pi$
$104$	11.0861	1.08708
$105$	0	0
$106$	−6.54468	−0.635675
$107$	−10.2917	−0.994937	−0.497468	−	0.867482i	$-0.665737\pi$
$107$	−10.2917	−0.994937	−0.497468	+	0.867482i	$0.665737\pi$
$108$	0	0
$109$	−9.18170	−0.879447	−0.439724	−	0.898133i	$-0.644924\pi$
$109$	−9.18170	−0.879447	−0.439724	+	0.898133i	$0.644924\pi$
$110$	−6.53597	−0.623180
$111$	0	0
$112$	−0.239042	−0.0225873
$113$	−11.3407	−1.06684	−0.533421	−	0.845850i	$-0.679094\pi$
$113$	−11.3407	−1.06684	−0.533421	+	0.845850i	$0.679094\pi$
$114$	0	0
$115$	−11.5230	−1.07452
$116$	−12.5365	−1.16398
$117$	0	0
$118$	−15.5068	−1.42751
$119$	−10.4336	−0.956450
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.28499	0.206873
$123$	0	0
$124$	−1.51698	−0.136229
$125$	5.20060	0.465156
$126$	0	0
$127$	−0.801188	−0.0710940	−0.0355470	−	0.999368i	$-0.511317\pi$
$127$	−0.801188	−0.0710940	−0.0355470	+	0.999368i	$0.511317\pi$
$128$	18.1618	1.60529
$129$	0	0
$130$	−25.9673	−2.27748
$131$	13.2043	1.15366	0.576832	−	0.816863i	$-0.304289\pi$
$131$	13.2043	1.15366	0.576832	+	0.816863i	$0.304289\pi$
$132$	0	0
$133$	14.0879	1.22158
$134$	−15.1541	−1.30912
$135$	0	0
$136$	8.08624	0.693389
$137$	10.6488	0.909788	0.454894	−	0.890545i	$-0.349677\pi$
$137$	10.6488	0.909788	0.454894	+	0.890545i	$0.349677\pi$
$138$	0	0
$139$	−0.359916	−0.0305277	−0.0152638	−	0.999884i	$-0.504859\pi$
$139$	−0.359916	−0.0305277	−0.0152638	+	0.999884i	$0.504859\pi$
$140$	−33.1734	−2.80367
$141$	0	0
$142$	−5.72519	−0.480448
$143$	3.97298	0.332237
$144$	0	0
$145$	11.1324	0.924493
$146$	36.7181	3.03881
$147$	0	0
$148$	−4.25683	−0.349909
$149$	10.8472	0.888637	0.444319	−	0.895869i	$-0.353446\pi$
$149$	10.8472	0.888637	0.444319	+	0.895869i	$0.353446\pi$
$150$	0	0
$151$	−5.51221	−0.448577	−0.224289	−	0.974523i	$-0.572006\pi$
$151$	−5.51221	−0.448577	−0.224289	+	0.974523i	$0.572006\pi$
$152$	−10.9184	−0.885597
$153$	0	0
$154$	8.22686	0.662940
$155$	1.34708	0.108200
$156$	0	0
$157$	1.99760	0.159426	0.0797130	−	0.996818i	$-0.474600\pi$
$157$	1.99760	0.159426	0.0797130	+	0.996818i	$0.474600\pi$
$158$	4.41566	0.351291
$159$	0	0
$160$	−16.3970	−1.29630
$161$	14.5041	1.14308
$162$	0	0
$163$	−11.8657	−0.929396	−0.464698	−	0.885469i	$-0.653837\pi$
$163$	−11.8657	−0.929396	−0.464698	+	0.885469i	$0.653837\pi$
$164$	7.02363	0.548453
$165$	0	0
$166$	21.0993	1.63763
$167$	−2.08152	−0.161073	−0.0805365	−	0.996752i	$-0.525663\pi$
$167$	−2.08152	−0.161073	−0.0805365	+	0.996752i	$0.525663\pi$
$168$	0	0
$169$	2.78454	0.214195
$170$	−18.9407	−1.45268
$171$	0	0
$172$	36.9304	2.81592
$173$	23.4916	1.78603	0.893016	−	0.450026i	$-0.148585\pi$
$173$	23.4916	1.78603	0.893016	+	0.450026i	$0.148585\pi$
$174$	0	0
$175$	11.4560	0.865989
$176$	0.0663931	0.00500457
$177$	0	0
$178$	15.6737	1.17479
$179$	5.44522	0.406995	0.203497	−	0.979076i	$-0.434769\pi$
$179$	5.44522	0.406995	0.203497	+	0.979076i	$0.434769\pi$
$180$	0	0
$181$	−10.5035	−0.780718	−0.390359	−	0.920663i	$-0.627649\pi$
$181$	−10.5035	−0.780718	−0.390359	+	0.920663i	$0.627649\pi$
$182$	32.6851	2.42278
$183$	0	0
$184$	−11.2409	−0.828689
$185$	3.78006	0.277915
$186$	0	0
$187$	2.89791	0.211916
$188$	−21.4906	−1.56736
$189$	0	0
$190$	25.5745	1.85537
$191$	4.18007	0.302459	0.151230	−	0.988499i	$-0.451677\pi$
$191$	4.18007	0.302459	0.151230	+	0.988499i	$0.451677\pi$
$192$	0	0
$193$	1.86289	0.134093	0.0670467	−	0.997750i	$-0.478642\pi$
$193$	1.86289	0.134093	0.0670467	+	0.997750i	$0.478642\pi$
$194$	−26.8755	−1.92955
$195$	0	0
$196$	19.2074	1.37196
$197$	7.69073	0.547942	0.273971	−	0.961738i	$-0.411663\pi$
$197$	7.69073	0.547942	0.273971	+	0.961738i	$0.411663\pi$
$198$	0	0
$199$	−2.73005	−0.193528	−0.0967640	−	0.995307i	$-0.530849\pi$
$199$	−2.73005	−0.193528	−0.0967640	+	0.995307i	$0.530849\pi$
$200$	−8.87855	−0.627808
$201$	0	0
$202$	−12.7685	−0.898390
$203$	−14.0124	−0.983476
$204$	0	0
$205$	−6.23697	−0.435609
$206$	7.89784	0.550269
$207$	0	0
$208$	0.263778	0.0182897
$209$	−3.91288	−0.270660
$210$	0	0
$211$	−23.8354	−1.64090	−0.820448	−	0.571721i	$-0.806276\pi$
$211$	−23.8354	−1.64090	−0.820448	+	0.571721i	$0.806276\pi$
$212$	9.22610	0.633651
$213$	0	0
$214$	23.5164	1.60755
$215$	−32.7941	−2.23654
$216$	0	0
$217$	−1.69557	−0.115103
$218$	20.9801	1.42095
$219$	0	0
$220$	9.21383	0.621196
$221$	11.5133	0.774471
$222$	0	0
$223$	14.1106	0.944913	0.472457	−	0.881354i	$-0.343367\pi$
$223$	14.1106	0.944913	0.472457	+	0.881354i	$0.343367\pi$
$224$	20.6390	1.37900
$225$	0	0
$226$	25.9134	1.72373
$227$	4.64411	0.308240	0.154120	−	0.988052i	$-0.450746\pi$
$227$	4.64411	0.308240	0.154120	+	0.988052i	$0.450746\pi$
$228$	0	0
$229$	28.0873	1.85606	0.928031	−	0.372503i	$-0.121500\pi$
$229$	28.0873	1.85606	0.928031	+	0.372503i	$0.121500\pi$
$230$	26.3299	1.73614
$231$	0	0
$232$	10.8598	0.712982
$233$	3.59916	0.235789	0.117894	−	0.993026i	$-0.462386\pi$
$233$	3.59916	0.235789	0.117894	+	0.993026i	$0.462386\pi$
$234$	0	0
$235$	19.0836	1.24488
$236$	21.8600	1.42297
$237$	0	0
$238$	23.8407	1.54537
$239$	−6.67465	−0.431747	−0.215873	−	0.976421i	$-0.569260\pi$
$239$	−6.67465	−0.431747	−0.215873	+	0.976421i	$0.569260\pi$
$240$	0	0
$241$	−17.7520	−1.14351	−0.571754	−	0.820425i	$-0.693737\pi$
$241$	−17.7520	−1.14351	−0.571754	+	0.820425i	$0.693737\pi$
$242$	−2.28499	−0.146885
$243$	0	0
$244$	−3.22117	−0.206214
$245$	−17.0561	−1.08968
$246$	0	0
$247$	−15.5458	−0.989155
$248$	1.31410	0.0834452
$249$	0	0
$250$	−11.8833	−0.751567
$251$	−13.3388	−0.841940	−0.420970	−	0.907074i	$-0.638310\pi$
$251$	−13.3388	−0.841940	−0.420970	+	0.907074i	$0.638310\pi$
$252$	0	0
$253$	−4.02846	−0.253267
$254$	1.83071	0.114869
$255$	0	0
$256$	−15.5679	−0.972992
$257$	−25.6370	−1.59919	−0.799595	−	0.600540i	$-0.794952\pi$
$257$	−25.6370	−1.59919	−0.799595	+	0.600540i	$0.794952\pi$
$258$	0	0
$259$	−4.75798	−0.295646
$260$	36.6063	2.27023
$261$	0	0
$262$	−30.1716	−1.86401
$263$	−11.0083	−0.678802	−0.339401	−	0.940642i	$-0.610224\pi$
$263$	−11.0083	−0.678802	−0.339401	+	0.940642i	$0.610224\pi$
$264$	0	0
$265$	−8.19276	−0.503277
$266$	−32.1908	−1.97374
$267$	0	0
$268$	21.3629	1.30495
$269$	−12.1248	−0.739260	−0.369630	−	0.929179i	$-0.620516\pi$
$269$	−12.1248	−0.739260	−0.369630	+	0.929179i	$0.620516\pi$
$270$	0	0
$271$	−17.0028	−1.03284	−0.516422	−	0.856334i	$-0.672736\pi$
$271$	−17.0028	−1.03284	−0.516422	+	0.856334i	$0.672736\pi$
$272$	0.192402	0.0116661
$273$	0	0
$274$	−24.3324	−1.46997
$275$	−3.18186	−0.191873
$276$	0	0
$277$	−9.24926	−0.555734	−0.277867	−	0.960620i	$-0.589628\pi$
$277$	−9.24926	−0.555734	−0.277867	+	0.960620i	$0.589628\pi$
$278$	0.822404	0.0493245
$279$	0	0
$280$	28.7367	1.71735
$281$	−4.51310	−0.269229	−0.134614	−	0.990898i	$-0.542980\pi$
$281$	−4.51310	−0.269229	−0.134614	+	0.990898i	$0.542980\pi$
$282$	0	0
$283$	6.59603	0.392094	0.196047	−	0.980595i	$-0.437190\pi$
$283$	6.59603	0.392094	0.196047	+	0.980595i	$0.437190\pi$
$284$	8.07087	0.478918
$285$	0	0
$286$	−9.07821	−0.536806
$287$	7.85051	0.463401
$288$	0	0
$289$	−8.60210	−0.506006
$290$	−25.4373	−1.49373
$291$	0	0
$292$	−51.7619	−3.02913
$293$	4.98781	0.291391	0.145696	−	0.989329i	$-0.453458\pi$
$293$	4.98781	0.291391	0.145696	+	0.989329i	$0.453458\pi$
$294$	0	0
$295$	−19.4117	−1.13019
$296$	3.68751	0.214332
$297$	0	0
$298$	−24.7857	−1.43580
$299$	−16.0050	−0.925593
$300$	0	0
$301$	41.2782	2.37923
$302$	12.5953	0.724780
$303$	0	0
$304$	−0.259789	−0.0148999
$305$	2.86039	0.163786
$306$	0	0
$307$	19.2262	1.09730	0.548649	−	0.836053i	$-0.315142\pi$
$307$	19.2262	1.09730	0.548649	+	0.836053i	$0.315142\pi$
$308$	−11.5975	−0.660829
$309$	0	0
$310$	−3.07805	−0.174822
$311$	33.3358	1.89030	0.945151	−	0.326634i	$-0.105914\pi$
$311$	33.3358	1.89030	0.945151	+	0.326634i	$0.105914\pi$
$312$	0	0
$313$	24.5371	1.38692	0.693460	−	0.720495i	$-0.256085\pi$
$313$	24.5371	1.38692	0.693460	+	0.720495i	$0.256085\pi$
$314$	−4.56450	−0.257590
$315$	0	0
$316$	−6.22480	−0.350172
$317$	−14.7217	−0.826851	−0.413425	−	0.910538i	$-0.635668\pi$
$317$	−14.7217	−0.826851	−0.413425	+	0.910538i	$0.635668\pi$
$318$	0	0
$319$	3.89190	0.217905
$320$	37.0872	2.07324
$321$	0	0
$322$	−33.1416	−1.84691
$323$	−11.3392	−0.630930
$324$	0	0
$325$	−12.6415	−0.701222
$326$	27.1131	1.50165
$327$	0	0
$328$	−6.08427	−0.335948
$329$	−24.0207	−1.32430
$330$	0	0
$331$	2.22247	0.122158	0.0610790	−	0.998133i	$-0.480546\pi$
$331$	2.22247	0.122158	0.0610790	+	0.998133i	$0.480546\pi$
$332$	−29.7440	−1.63241
$333$	0	0
$334$	4.75625	0.260251
$335$	−18.9702	−1.03645
$336$	0	0
$337$	−15.0962	−0.822344	−0.411172	−	0.911558i	$-0.634880\pi$
$337$	−15.0962	−0.822344	−0.411172	+	0.911558i	$0.634880\pi$
$338$	−6.36264	−0.346082
$339$	0	0
$340$	26.7009	1.44806
$341$	0.470941	0.0255029
$342$	0	0
$343$	−3.73414	−0.201625
$344$	−31.9912	−1.72485
$345$	0	0
$346$	−53.6780	−2.88575
$347$	−4.99699	−0.268252	−0.134126	−	0.990964i	$-0.542823\pi$
$347$	−4.99699	−0.268252	−0.134126	+	0.990964i	$0.542823\pi$
$348$	0	0
$349$	25.6224	1.37154	0.685769	−	0.727820i	$-0.259466\pi$
$349$	25.6224	1.37154	0.685769	+	0.727820i	$0.259466\pi$
$350$	−26.1767	−1.39920
$351$	0	0
$352$	−5.73244	−0.305540
$353$	−3.60094	−0.191659	−0.0958294	−	0.995398i	$-0.530550\pi$
$353$	−3.60094	−0.191659	−0.0958294	+	0.995398i	$0.530550\pi$
$354$	0	0
$355$	−7.16691	−0.380380
$356$	−22.0953	−1.17105
$357$	0	0
$358$	−12.4423	−0.657594
$359$	6.52911	0.344593	0.172297	−	0.985045i	$-0.444881\pi$
$359$	6.52911	0.344593	0.172297	+	0.985045i	$0.444881\pi$
$360$	0	0
$361$	−3.68934	−0.194176
$362$	24.0003	1.26143
$363$	0	0
$364$	−46.0766	−2.41507
$365$	45.9644	2.40589
$366$	0	0
$367$	4.38807	0.229055	0.114528	−	0.993420i	$-0.463465\pi$
$367$	4.38807	0.229055	0.114528	+	0.993420i	$0.463465\pi$
$368$	−0.267462	−0.0139424
$369$	0	0
$370$	−8.63739	−0.449036
$371$	10.3123	0.535386
$372$	0	0
$373$	23.2972	1.20628	0.603142	−	0.797634i	$-0.293915\pi$
$373$	23.2972	1.20628	0.603142	+	0.797634i	$0.293915\pi$
$374$	−6.62170	−0.342400
$375$	0	0
$376$	18.6164	0.960068
$377$	15.4624	0.796355
$378$	0	0
$379$	−27.0154	−1.38769	−0.693844	−	0.720125i	$-0.744084\pi$
$379$	−27.0154	−1.38769	−0.693844	+	0.720125i	$0.744084\pi$
$380$	−36.0526	−1.84946
$381$	0	0
$382$	−9.55141	−0.488693
$383$	−5.32904	−0.272301	−0.136151	−	0.990688i	$-0.543473\pi$
$383$	−5.32904	−0.272301	−0.136151	+	0.990688i	$0.543473\pi$
$384$	0	0
$385$	10.2986	0.524863
$386$	−4.25667	−0.216659
$387$	0	0
$388$	37.8867	1.92341
$389$	−29.3245	−1.48681	−0.743405	−	0.668841i	$-0.766791\pi$
$389$	−29.3245	−1.48681	−0.743405	+	0.668841i	$0.766791\pi$
$390$	0	0
$391$	−11.6741	−0.590387
$392$	−16.6385	−0.840373
$393$	0	0
$394$	−17.5732	−0.885326
$395$	5.52761	0.278124
$396$	0	0
$397$	−33.8631	−1.69954	−0.849770	−	0.527153i	$-0.823260\pi$
$397$	−33.8631	−1.69954	−0.849770	+	0.527153i	$0.823260\pi$
$398$	6.23813	0.312689
$399$	0	0
$400$	−0.211254	−0.0105627
$401$	−24.3677	−1.21686	−0.608431	−	0.793607i	$-0.708201\pi$
$401$	−24.3677	−1.21686	−0.608431	+	0.793607i	$0.708201\pi$
$402$	0	0
$403$	1.87104	0.0932029
$404$	17.9999	0.895530
$405$	0	0
$406$	32.0181	1.58903
$407$	1.32152	0.0655051
$408$	0	0
$409$	−0.458378	−0.0226653	−0.0113327	−	0.999936i	$-0.503607\pi$
$409$	−0.458378	−0.0226653	−0.0113327	+	0.999936i	$0.503607\pi$
$410$	14.2514	0.703827
$411$	0	0
$412$	−11.1337	−0.548517
$413$	24.4336	1.20230
$414$	0	0
$415$	26.4126	1.29654
$416$	−22.7748	−1.11663
$417$	0	0
$418$	8.94089	0.437313
$419$	−2.26246	−0.110528	−0.0552642	−	0.998472i	$-0.517600\pi$
$419$	−2.26246	−0.110528	−0.0552642	+	0.998472i	$0.517600\pi$
$420$	0	0
$421$	−12.4635	−0.607434	−0.303717	−	0.952762i	$-0.598228\pi$
$421$	−12.4635	−0.607434	−0.303717	+	0.952762i	$0.598228\pi$
$422$	54.4636	2.65125
$423$	0	0
$424$	−7.99218	−0.388134
$425$	−9.22075	−0.447272
$426$	0	0
$427$	−3.60040	−0.174235
$428$	−33.1514	−1.60243
$429$	0	0
$430$	74.9342	3.61365
$431$	3.79155	0.182633	0.0913163	−	0.995822i	$-0.470893\pi$
$431$	3.79155	0.182633	0.0913163	+	0.995822i	$0.470893\pi$
$432$	0	0
$433$	8.14867	0.391600	0.195800	−	0.980644i	$-0.437270\pi$
$433$	8.14867	0.391600	0.195800	+	0.980644i	$0.437270\pi$
$434$	3.87436	0.185975
$435$	0	0
$436$	−29.5758	−1.41643
$437$	15.7629	0.754042
$438$	0	0
$439$	−9.96193	−0.475457	−0.237728	−	0.971332i	$-0.576403\pi$
$439$	−9.96193	−0.475457	−0.237728	+	0.971332i	$0.576403\pi$
$440$	−7.98155	−0.380505
$441$	0	0
$442$	−26.3079	−1.25134
$443$	−22.9958	−1.09256	−0.546282	−	0.837601i	$-0.683957\pi$
$443$	−22.9958	−1.09256	−0.546282	+	0.837601i	$0.683957\pi$
$444$	0	0
$445$	19.6206	0.930106
$446$	−32.2425	−1.52673
$447$	0	0
$448$	−46.6819	−2.20551
$449$	−6.99680	−0.330200	−0.165100	−	0.986277i	$-0.552795\pi$
$449$	−6.99680	−0.330200	−0.165100	+	0.986277i	$0.552795\pi$
$450$	0	0
$451$	−2.18046	−0.102674
$452$	−36.5303	−1.71824
$453$	0	0
$454$	−10.6117	−0.498033
$455$	40.9159	1.91817
$456$	0	0
$457$	12.7476	0.596307	0.298153	−	0.954518i	$-0.403629\pi$
$457$	12.7476	0.596307	0.298153	+	0.954518i	$0.403629\pi$
$458$	−64.1792	−2.99890
$459$	0	0
$460$	−37.1176	−1.73062
$461$	−23.8221	−1.10951	−0.554753	−	0.832015i	$-0.687187\pi$
$461$	−23.8221	−1.10951	−0.554753	+	0.832015i	$0.687187\pi$
$462$	0	0
$463$	−38.8683	−1.80637	−0.903183	−	0.429257i	$-0.858776\pi$
$463$	−38.8683	−1.80637	−0.903183	+	0.429257i	$0.858776\pi$
$464$	0.258395	0.0119957
$465$	0	0
$466$	−8.22403	−0.380971
$467$	−2.95532	−0.136756	−0.0683779	−	0.997659i	$-0.521782\pi$
$467$	−2.95532	−0.136756	−0.0683779	+	0.997659i	$0.521782\pi$
$468$	0	0
$469$	23.8779	1.10258
$470$	−43.6059	−2.01139
$471$	0	0
$472$	−18.9364	−0.871620
$473$	−11.4649	−0.527156
$474$	0	0
$475$	12.4502	0.571256
$476$	−33.6085	−1.54045
$477$	0	0
$478$	15.2515	0.697587
$479$	37.9012	1.73175	0.865875	−	0.500261i	$-0.166763\pi$
$479$	37.9012	1.73175	0.865875	+	0.500261i	$0.166763\pi$
$480$	0	0
$481$	5.25035	0.239395
$482$	40.5632	1.84760
$483$	0	0
$484$	3.22117	0.146417
$485$	−33.6433	−1.52766
$486$	0	0
$487$	−17.3128	−0.784520	−0.392260	−	0.919854i	$-0.628307\pi$
$487$	−17.3128	−0.784520	−0.392260	+	0.919854i	$0.628307\pi$
$488$	2.79036	0.126314
$489$	0	0
$490$	38.9730	1.76062
$491$	−25.1752	−1.13614	−0.568070	−	0.822980i	$-0.692310\pi$
$491$	−25.1752	−1.13614	−0.568070	+	0.822980i	$0.692310\pi$
$492$	0	0
$493$	11.2784	0.507953
$494$	35.5220	1.59821
$495$	0	0
$496$	0.0312672	0.00140394
$497$	9.02104	0.404649
$498$	0	0
$499$	31.4211	1.40660	0.703301	−	0.710892i	$-0.251709\pi$
$499$	31.4211	1.40660	0.703301	+	0.710892i	$0.251709\pi$
$500$	16.7520	0.749174
$501$	0	0
$502$	30.4791	1.36035
$503$	−13.3718	−0.596219	−0.298110	−	0.954532i	$-0.596356\pi$
$503$	−13.3718	−0.596219	−0.298110	+	0.954532i	$0.596356\pi$
$504$	0	0
$505$	−15.9839	−0.711274
$506$	9.20499	0.409212
$507$	0	0
$508$	−2.58077	−0.114503
$509$	39.9834	1.77223	0.886117	−	0.463461i	$-0.153393\pi$
$509$	39.9834	1.77223	0.886117	+	0.463461i	$0.153393\pi$
$510$	0	0
$511$	−57.8557	−2.55939
$512$	−0.751117	−0.0331950
$513$	0	0
$514$	58.5802	2.58386
$515$	9.88668	0.435659
$516$	0	0
$517$	6.67167	0.293420
$518$	10.8719	0.477685
$519$	0	0
$520$	−31.7105	−1.39060
$521$	−2.09718	−0.0918791	−0.0459396	−	0.998944i	$-0.514628\pi$
$521$	−2.09718	−0.0918791	−0.0459396	+	0.998944i	$0.514628\pi$
$522$	0	0
$523$	32.0727	1.40244	0.701220	−	0.712945i	$-0.252639\pi$
$523$	32.0727	1.40244	0.701220	+	0.712945i	$0.252639\pi$
$524$	42.5333	1.85808
$525$	0	0
$526$	25.1539	1.09676
$527$	1.36475	0.0594492
$528$	0	0
$529$	−6.77148	−0.294412
$530$	18.7204	0.813160
$531$	0	0
$532$	45.3797	1.96746
$533$	−8.66290	−0.375232
$534$	0	0
$535$	29.4383	1.27273
$536$	−18.5058	−0.799328
$537$	0	0
$538$	27.7049	1.19444
$539$	−5.96285	−0.256838
$540$	0	0
$541$	−2.24457	−0.0965016	−0.0482508	−	0.998835i	$-0.515365\pi$
$541$	−2.24457	−0.0965016	−0.0482508	+	0.998835i	$0.515365\pi$
$542$	38.8511	1.66880
$543$	0	0
$544$	−16.6121	−0.712238
$545$	26.2633	1.12500
$546$	0	0
$547$	−16.8367	−0.719887	−0.359944	−	0.932974i	$-0.617204\pi$
$547$	−16.8367	−0.719887	−0.359944	+	0.932974i	$0.617204\pi$
$548$	34.3016	1.46529
$549$	0	0
$550$	7.27051	0.310016
$551$	−15.2286	−0.648758
$552$	0	0
$553$	−6.95763	−0.295869
$554$	21.1345	0.897917
$555$	0	0
$556$	−1.15935	−0.0491675
$557$	−2.89212	−0.122543	−0.0612716	−	0.998121i	$-0.519516\pi$
$557$	−2.89212	−0.122543	−0.0612716	+	0.998121i	$0.519516\pi$
$558$	0	0
$559$	−45.5498	−1.92655
$560$	0.683753	0.0288939
$561$	0	0
$562$	10.3124	0.435001
$563$	5.29719	0.223250	0.111625	−	0.993750i	$-0.464394\pi$
$563$	5.29719	0.223250	0.111625	+	0.993750i	$0.464394\pi$
$564$	0	0
$565$	32.4389	1.36471
$566$	−15.0719	−0.633518
$567$	0	0
$568$	−6.99145	−0.293355
$569$	−32.0573	−1.34391	−0.671956	−	0.740591i	$-0.734546\pi$
$569$	−32.0573	−1.34391	−0.671956	+	0.740591i	$0.734546\pi$
$570$	0	0
$571$	−10.1557	−0.425004	−0.212502	−	0.977161i	$-0.568161\pi$
$571$	−10.1557	−0.425004	−0.212502	+	0.977161i	$0.568161\pi$
$572$	12.7976	0.535096
$573$	0	0
$574$	−17.9383	−0.748731
$575$	12.8180	0.534548
$576$	0	0
$577$	−18.3477	−0.763824	−0.381912	−	0.924199i	$-0.624734\pi$
$577$	−18.3477	−0.763824	−0.381912	+	0.924199i	$0.624734\pi$
$578$	19.6557	0.817569
$579$	0	0
$580$	35.8593	1.48898
$581$	−33.2457	−1.37926
$582$	0	0
$583$	−2.86420	−0.118623
$584$	44.8391	1.85546
$585$	0	0
$586$	−11.3971	−0.470810
$587$	−43.2491	−1.78508	−0.892540	−	0.450969i	$-0.851079\pi$
$587$	−43.2491	−1.78508	−0.892540	+	0.450969i	$0.851079\pi$
$588$	0	0
$589$	−1.84274	−0.0759286
$590$	44.3555	1.82609
$591$	0	0
$592$	0.0877396	0.00360607
$593$	−6.71395	−0.275709	−0.137854	−	0.990452i	$-0.544021\pi$
$593$	−6.71395	−0.275709	−0.137854	+	0.990452i	$0.544021\pi$
$594$	0	0
$595$	29.8443	1.22350
$596$	34.9407	1.43123
$597$	0	0
$598$	36.5712	1.49551
$599$	−35.7340	−1.46005	−0.730026	−	0.683419i	$-0.760492\pi$
$599$	−35.7340	−1.46005	−0.730026	+	0.683419i	$0.760492\pi$
$600$	0	0
$601$	40.8308	1.66552	0.832762	−	0.553632i	$-0.186759\pi$
$601$	40.8308	1.66552	0.832762	+	0.553632i	$0.186759\pi$
$602$	−94.3201	−3.84420
$603$	0	0
$604$	−17.7558	−0.722473
$605$	−2.86039	−0.116292
$606$	0	0
$607$	−40.6461	−1.64978	−0.824888	−	0.565296i	$-0.808762\pi$
$607$	−40.6461	−1.64978	−0.824888	+	0.565296i	$0.808762\pi$
$608$	22.4304	0.909671
$609$	0	0
$610$	−6.53597	−0.264634
$611$	26.5064	1.07233
$612$	0	0
$613$	−29.7392	−1.20116	−0.600578	−	0.799566i	$-0.705063\pi$
$613$	−29.7392	−1.20116	−0.600578	+	0.799566i	$0.705063\pi$
$614$	−43.9317	−1.77294
$615$	0	0
$616$	10.0464	0.404782
$617$	28.9410	1.16512	0.582561	−	0.812787i	$-0.302051\pi$
$617$	28.9410	1.16512	0.582561	+	0.812787i	$0.302051\pi$
$618$	0	0
$619$	−30.5570	−1.22819	−0.614095	−	0.789232i	$-0.710479\pi$
$619$	−30.5570	−1.22819	−0.614095	+	0.789232i	$0.710479\pi$
$620$	4.33916	0.174265
$621$	0	0
$622$	−76.1720	−3.05422
$623$	−24.6966	−0.989448
$624$	0	0
$625$	−30.7851	−1.23140
$626$	−56.0670	−2.24089
$627$	0	0
$628$	6.43462	0.256769
$629$	3.82964	0.152698
$630$	0	0
$631$	−26.6396	−1.06050	−0.530252	−	0.847840i	$-0.677903\pi$
$631$	−26.6396	−1.06050	−0.530252	+	0.847840i	$0.677903\pi$
$632$	5.39228	0.214493
$633$	0	0
$634$	33.6388	1.33597
$635$	2.29172	0.0909439
$636$	0	0
$637$	−23.6903	−0.938643
$638$	−8.89295	−0.352075
$639$	0	0
$640$	−51.9498	−2.05350
$641$	6.11961	0.241710	0.120855	−	0.992670i	$-0.461436\pi$
$641$	6.11961	0.241710	0.120855	+	0.992670i	$0.461436\pi$
$642$	0	0
$643$	−12.5702	−0.495719	−0.247859	−	0.968796i	$-0.579727\pi$
$643$	−12.5702	−0.495719	−0.247859	+	0.968796i	$0.579727\pi$
$644$	46.7201	1.84103
$645$	0	0
$646$	25.9099	1.01941
$647$	24.5827	0.966446	0.483223	−	0.875497i	$-0.339466\pi$
$647$	24.5827	0.966446	0.483223	+	0.875497i	$0.339466\pi$
$648$	0	0
$649$	−6.78636	−0.266388
$650$	28.8856	1.13299
$651$	0	0
$652$	−38.2216	−1.49687
$653$	44.7261	1.75027	0.875134	−	0.483881i	$-0.160773\pi$
$653$	44.7261	1.75027	0.875134	+	0.483881i	$0.160773\pi$
$654$	0	0
$655$	−37.7695	−1.47578
$656$	−0.144767	−0.00565222
$657$	0	0
$658$	54.8869	2.13972
$659$	−43.7516	−1.70432	−0.852160	−	0.523282i	$-0.824708\pi$
$659$	−43.7516	−1.70432	−0.852160	+	0.523282i	$0.824708\pi$
$660$	0	0
$661$	−12.9945	−0.505428	−0.252714	−	0.967541i	$-0.581323\pi$
$661$	−12.9945	−0.505428	−0.252714	+	0.967541i	$0.581323\pi$
$662$	−5.07832	−0.197375
$663$	0	0
$664$	25.7659	0.999912
$665$	−40.2970	−1.56265
$666$	0	0
$667$	−15.6784	−0.607069
$668$	−6.70494	−0.259422
$669$	0	0
$670$	43.3467	1.67463
$671$	1.00000	0.0386046
$672$	0	0
$673$	−32.7751	−1.26339	−0.631694	−	0.775218i	$-0.717640\pi$
$673$	−32.7751	−1.26339	−0.631694	+	0.775218i	$0.717640\pi$
$674$	34.4947	1.32869
$675$	0	0
$676$	8.96949	0.344980
$677$	50.4254	1.93801	0.969004	−	0.247047i	$-0.0794601\pi$
$677$	50.4254	1.93801	0.969004	+	0.247047i	$0.0794601\pi$
$678$	0	0
$679$	42.3470	1.62513
$680$	−23.1298	−0.886989
$681$	0	0
$682$	−1.07609	−0.0412058
$683$	−39.2346	−1.50127	−0.750634	−	0.660718i	$-0.770252\pi$
$683$	−39.2346	−1.50127	−0.750634	+	0.660718i	$0.770252\pi$
$684$	0	0
$685$	−30.4598	−1.16381
$686$	8.53247	0.325771
$687$	0	0
$688$	−0.761190	−0.0290201
$689$	−11.3794	−0.433521
$690$	0	0
$691$	−44.4727	−1.69182	−0.845912	−	0.533323i	$-0.820943\pi$
$691$	−44.4727	−1.69182	−0.845912	+	0.533323i	$0.820943\pi$
$692$	75.6704	2.87656
$693$	0	0
$694$	11.4181	0.433424
$695$	1.02950	0.0390512
$696$	0	0
$697$	−6.31878	−0.239341
$698$	−58.5470	−2.21604
$699$	0	0
$700$	36.9016	1.39475
$701$	−30.5556	−1.15407	−0.577034	−	0.816720i	$-0.695790\pi$
$701$	−30.5556	−1.15407	−0.577034	+	0.816720i	$0.695790\pi$
$702$	0	0
$703$	−5.17094	−0.195026
$704$	12.9658	0.488666
$705$	0	0
$706$	8.22811	0.309669
$707$	20.1190	0.756654
$708$	0	0
$709$	−28.9376	−1.08677	−0.543386	−	0.839483i	$-0.682858\pi$
$709$	−28.9376	−1.08677	−0.543386	+	0.839483i	$0.682858\pi$
$710$	16.3763	0.614592
$711$	0	0
$712$	19.1403	0.717311
$713$	−1.89717	−0.0710495
$714$	0	0
$715$	−11.3643	−0.425000
$716$	17.5400	0.655500
$717$	0	0
$718$	−14.9189	−0.556770
$719$	−27.1400	−1.01215	−0.506076	−	0.862489i	$-0.668904\pi$
$719$	−27.1400	−1.01215	−0.506076	+	0.862489i	$0.668904\pi$
$720$	0	0
$721$	−12.4444	−0.463455
$722$	8.43010	0.313736
$723$	0	0
$724$	−33.8335	−1.25741
$725$	−12.3835	−0.459911
$726$	0	0
$727$	5.34132	0.198099	0.0990494	−	0.995083i	$-0.468420\pi$
$727$	5.34132	0.198099	0.0990494	+	0.995083i	$0.468420\pi$
$728$	39.9142	1.47932
$729$	0	0
$730$	−105.028	−3.88727
$731$	−33.2243	−1.22884
$732$	0	0
$733$	41.6909	1.53989	0.769944	−	0.638111i	$-0.220284\pi$
$733$	41.6909	1.53989	0.769944	+	0.638111i	$0.220284\pi$
$734$	−10.0267	−0.370092
$735$	0	0
$736$	23.0929	0.851216
$737$	−6.63203	−0.244294
$738$	0	0
$739$	3.72294	0.136950	0.0684752	−	0.997653i	$-0.478187\pi$
$739$	3.72294	0.136950	0.0684752	+	0.997653i	$0.478187\pi$
$740$	12.1762	0.447607
$741$	0	0
$742$	−23.5634	−0.865040
$743$	−9.83128	−0.360675	−0.180337	−	0.983605i	$-0.557719\pi$
$743$	−9.83128	−0.360675	−0.180337	+	0.983605i	$0.557719\pi$
$744$	0	0
$745$	−31.0273	−1.13675
$746$	−53.2338	−1.94903
$747$	0	0
$748$	9.33468	0.341310
$749$	−37.0542	−1.35393
$750$	0	0
$751$	18.7431	0.683946	0.341973	−	0.939710i	$-0.388905\pi$
$751$	18.7431	0.683946	0.341973	+	0.939710i	$0.388905\pi$
$752$	0.442953	0.0161528
$753$	0	0
$754$	−35.3315	−1.28670
$755$	15.7671	0.573824
$756$	0	0
$757$	−16.2787	−0.591659	−0.295830	−	0.955241i	$-0.595596\pi$
$757$	−16.2787	−0.591659	−0.295830	+	0.955241i	$0.595596\pi$
$758$	61.7299	2.24213
$759$	0	0
$760$	31.2309	1.13286
$761$	10.4549	0.378989	0.189495	−	0.981882i	$-0.439315\pi$
$761$	10.4549	0.378989	0.189495	+	0.981882i	$0.439315\pi$
$762$	0	0
$763$	−33.0578	−1.19677
$764$	13.4647	0.487137
$765$	0	0
$766$	12.1768	0.439965
$767$	−26.9621	−0.973544
$768$	0	0
$769$	−29.6047	−1.06757	−0.533786	−	0.845620i	$-0.679231\pi$
$769$	−29.6047	−1.06757	−0.533786	+	0.845620i	$0.679231\pi$
$770$	−23.5321	−0.848037
$771$	0	0
$772$	6.00068	0.215969
$773$	2.70530	0.0973027	0.0486514	−	0.998816i	$-0.484508\pi$
$773$	2.70530	0.0973027	0.0486514	+	0.998816i	$0.484508\pi$
$774$	0	0
$775$	−1.49847	−0.0538265
$776$	−32.8196	−1.17816
$777$	0	0
$778$	67.0061	2.40229
$779$	8.53187	0.305686
$780$	0	0
$781$	−2.50557	−0.0896563
$782$	26.6753	0.953906
$783$	0	0
$784$	−0.395892	−0.0141390
$785$	−5.71393	−0.203939
$786$	0	0
$787$	40.4030	1.44021	0.720106	−	0.693864i	$-0.244093\pi$
$787$	40.4030	1.44021	0.720106	+	0.693864i	$0.244093\pi$
$788$	24.7732	0.882507
$789$	0	0
$790$	−12.6305	−0.449374
$791$	−40.8310	−1.45178
$792$	0	0
$793$	3.97298	0.141085
$794$	77.3768	2.74600
$795$	0	0
$796$	−8.79396	−0.311694
$797$	5.32615	0.188662	0.0943310	−	0.995541i	$-0.469929\pi$
$797$	5.32615	0.188662	0.0943310	+	0.995541i	$0.469929\pi$
$798$	0	0
$799$	19.3339	0.683985
$800$	18.2398	0.644875
$801$	0	0
$802$	55.6798	1.96612
$803$	16.0693	0.567072
$804$	0	0
$805$	−41.4874	−1.46224
$806$	−4.27530	−0.150591
$807$	0	0
$808$	−15.5926	−0.548545
$809$	25.7734	0.906143	0.453072	−	0.891474i	$-0.350328\pi$
$809$	25.7734	0.906143	0.453072	+	0.891474i	$0.350328\pi$
$810$	0	0
$811$	4.43894	0.155872	0.0779362	−	0.996958i	$-0.475167\pi$
$811$	4.43894	0.155872	0.0779362	+	0.996958i	$0.475167\pi$
$812$	−45.1363	−1.58397
$813$	0	0
$814$	−3.01965	−0.105839
$815$	33.9407	1.18889
$816$	0	0
$817$	44.8608	1.56948
$818$	1.04739	0.0366211
$819$	0	0
$820$	−20.0904	−0.701585
$821$	12.8609	0.448849	0.224424	−	0.974492i	$-0.427950\pi$
$821$	12.8609	0.448849	0.224424	+	0.974492i	$0.427950\pi$
$822$	0	0
$823$	−20.0539	−0.699034	−0.349517	−	0.936930i	$-0.613654\pi$
$823$	−20.0539	−0.699034	−0.349517	+	0.936930i	$0.613654\pi$
$824$	9.64463	0.335987
$825$	0	0
$826$	−55.8305	−1.94259
$827$	−5.37322	−0.186845	−0.0934225	−	0.995627i	$-0.529781\pi$
$827$	−5.37322	−0.186845	−0.0934225	+	0.995627i	$0.529781\pi$
$828$	0	0
$829$	34.0985	1.18429	0.592146	−	0.805831i	$-0.298281\pi$
$829$	34.0985	1.18429	0.592146	+	0.805831i	$0.298281\pi$
$830$	−60.3524	−2.09486
$831$	0	0
$832$	51.5127	1.78588
$833$	−17.2798	−0.598711
$834$	0	0
$835$	5.95398	0.206046
$836$	−12.6041	−0.435921
$837$	0	0
$838$	5.16969	0.178584
$839$	−41.5023	−1.43282	−0.716410	−	0.697680i	$-0.754216\pi$
$839$	−41.5023	−1.43282	−0.716410	+	0.697680i	$0.754216\pi$
$840$	0	0
$841$	−13.8531	−0.477694
$842$	28.4790	0.981449
$843$	0	0
$844$	−76.7779	−2.64281
$845$	−7.96489	−0.274000
$846$	0	0
$847$	3.60040	0.123711
$848$	−0.190164	−0.00653024
$849$	0	0
$850$	21.0693	0.722672
$851$	−5.32368	−0.182493
$852$	0	0
$853$	−28.1532	−0.963946	−0.481973	−	0.876186i	$-0.660080\pi$
$853$	−28.1532	−0.963946	−0.481973	+	0.876186i	$0.660080\pi$
$854$	8.22686	0.281517
$855$	0	0
$856$	28.7176	0.981548
$857$	2.25178	0.0769193	0.0384597	−	0.999260i	$-0.487755\pi$
$857$	2.25178	0.0769193	0.0384597	+	0.999260i	$0.487755\pi$
$858$	0	0
$859$	−38.2683	−1.30570	−0.652848	−	0.757489i	$-0.726426\pi$
$859$	−38.2683	−1.30570	−0.652848	+	0.757489i	$0.726426\pi$
$860$	−105.636	−3.60214
$861$	0	0
$862$	−8.66365	−0.295085
$863$	−16.0489	−0.546311	−0.273156	−	0.961970i	$-0.588067\pi$
$863$	−16.0489	−0.546311	−0.273156	+	0.961970i	$0.588067\pi$
$864$	0	0
$865$	−67.1952	−2.28470
$866$	−18.6196	−0.632720
$867$	0	0
$868$	−5.46173	−0.185383
$869$	1.93246	0.0655543
$870$	0	0
$871$	−26.3489	−0.892798
$872$	25.6203	0.867613
$873$	0	0
$874$	−36.0181	−1.21833
$875$	18.7242	0.632994
$876$	0	0
$877$	6.35401	0.214560	0.107280	−	0.994229i	$-0.465786\pi$
$877$	6.35401	0.214560	0.107280	+	0.994229i	$0.465786\pi$
$878$	22.7629	0.768211
$879$	0	0
$880$	−0.189911	−0.00640188
$881$	−56.1402	−1.89141	−0.945705	−	0.325026i	$-0.894627\pi$
$881$	−56.1402	−1.89141	−0.945705	+	0.325026i	$0.894627\pi$
$882$	0	0
$883$	−31.4041	−1.05683	−0.528416	−	0.848985i	$-0.677214\pi$
$883$	−31.4041	−1.05683	−0.528416	+	0.848985i	$0.677214\pi$
$884$	37.0865	1.24735
$885$	0	0
$886$	52.5452	1.76529
$887$	23.4516	0.787429	0.393714	−	0.919233i	$-0.371190\pi$
$887$	23.4516	0.787429	0.393714	+	0.919233i	$0.371190\pi$
$888$	0	0
$889$	−2.88460	−0.0967462
$890$	−44.8329	−1.50280
$891$	0	0
$892$	45.4526	1.52186
$893$	−26.1055	−0.873587
$894$	0	0
$895$	−15.5755	−0.520630
$896$	65.3895	2.18451
$897$	0	0
$898$	15.9876	0.533514
$899$	1.83285	0.0611291
$900$	0	0
$901$	−8.30022	−0.276520
$902$	4.98232	0.165893
$903$	0	0
$904$	31.6447	1.05249
$905$	30.0441	0.998700
$906$	0	0
$907$	5.58667	0.185502	0.0927512	−	0.995689i	$-0.470434\pi$
$907$	5.58667	0.185502	0.0927512	+	0.995689i	$0.470434\pi$
$908$	14.9595	0.496448
$909$	0	0
$910$	−93.4924	−3.09924
$911$	−37.8502	−1.25403	−0.627017	−	0.779006i	$-0.715724\pi$
$911$	−37.8502	−1.25403	−0.627017	+	0.779006i	$0.715724\pi$
$912$	0	0
$913$	9.23389	0.305597
$914$	−29.1281	−0.963471
$915$	0	0
$916$	90.4741	2.98935
$917$	47.5407	1.56993
$918$	0	0
$919$	4.39329	0.144921	0.0724606	−	0.997371i	$-0.476915\pi$
$919$	4.39329	0.144921	0.0724606	+	0.997371i	$0.476915\pi$
$920$	32.1534	1.06007
$921$	0	0
$922$	54.4333	1.79266
$923$	−9.95456	−0.327658
$924$	0	0
$925$	−4.20488	−0.138255
$926$	88.8137	2.91860
$927$	0	0
$928$	−22.3101	−0.732364
$929$	20.1988	0.662701	0.331351	−	0.943508i	$-0.392496\pi$
$929$	20.1988	0.662701	0.331351	+	0.943508i	$0.392496\pi$
$930$	0	0
$931$	23.3319	0.764674
$932$	11.5935	0.379758
$933$	0	0
$934$	6.75287	0.220961
$935$	−8.28918	−0.271085
$936$	0	0
$937$	−4.49533	−0.146856	−0.0734280	−	0.997301i	$-0.523394\pi$
$937$	−4.49533	−0.146856	−0.0734280	+	0.997301i	$0.523394\pi$
$938$	−54.5608	−1.78147
$939$	0	0
$940$	61.4716	2.00498
$941$	−1.89630	−0.0618177	−0.0309088	−	0.999522i	$-0.509840\pi$
$941$	−1.89630	−0.0618177	−0.0309088	+	0.999522i	$0.509840\pi$
$942$	0	0
$943$	8.78389	0.286043
$944$	−0.450568	−0.0146647
$945$	0	0
$946$	26.1971	0.851743
$947$	2.31483	0.0752218	0.0376109	−	0.999292i	$-0.488025\pi$
$947$	2.31483	0.0752218	0.0376109	+	0.999292i	$0.488025\pi$
$948$	0	0
$949$	63.8428	2.07243
$950$	−28.4487	−0.922997
$951$	0	0
$952$	29.1137	0.943579
$953$	−44.6867	−1.44754	−0.723772	−	0.690039i	$-0.757593\pi$
$953$	−44.6867	−1.44754	−0.723772	+	0.690039i	$0.757593\pi$
$954$	0	0
$955$	−11.9567	−0.386908
$956$	−21.5002	−0.695366
$957$	0	0
$958$	−86.6038	−2.79804
$959$	38.3399	1.23806
$960$	0	0
$961$	−30.7782	−0.992846
$962$	−11.9970	−0.386798
$963$	0	0
$964$	−57.1823	−1.84172
$965$	−5.32859	−0.171533
$966$	0	0
$967$	−49.3989	−1.58856	−0.794280	−	0.607551i	$-0.792152\pi$
$967$	−49.3989	−1.58856	−0.794280	+	0.607551i	$0.792152\pi$
$968$	−2.79036	−0.0896857
$969$	0	0
$970$	76.8746	2.46829
$971$	10.6647	0.342245	0.171122	−	0.985250i	$-0.445261\pi$
$971$	10.6647	0.342245	0.171122	+	0.985250i	$0.445261\pi$
$972$	0	0
$973$	−1.29584	−0.0415427
$974$	39.5596	1.26757
$975$	0	0
$976$	0.0663931	0.00212519
$977$	−24.9408	−0.797928	−0.398964	−	0.916967i	$-0.630630\pi$
$977$	−24.9408	−0.797928	−0.398964	+	0.916967i	$0.630630\pi$
$978$	0	0
$979$	6.85941	0.219228
$980$	−54.9407	−1.75502
$981$	0	0
$982$	57.5250	1.83570
$983$	20.1772	0.643554	0.321777	−	0.946815i	$-0.395720\pi$
$983$	20.1772	0.643554	0.321777	+	0.946815i	$0.395720\pi$
$984$	0	0
$985$	−21.9985	−0.700931
$986$	−25.7710	−0.820716
$987$	0	0
$988$	−50.0757	−1.59312
$989$	46.1859	1.46863
$990$	0	0
$991$	−4.07447	−0.129430	−0.0647148	−	0.997904i	$-0.520614\pi$
$991$	−4.07447	−0.129430	−0.0647148	+	0.997904i	$0.520614\pi$
$992$	−2.69964	−0.0857136
$993$	0	0
$994$	−20.6130	−0.653804
$995$	7.80902	0.247562
$996$	0	0
$997$	22.2645	0.705125	0.352562	−	0.935788i	$-0.385310\pi$
$997$	22.2645	0.705125	0.352562	+	0.935788i	$0.385310\pi$
$998$	−71.7969	−2.27269
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.28499 q^{2} +3.22117 q^{4} -2.86039 q^{5} +3.60040 q^{7} -2.79036 q^{8} +O(q^{10})\)	\(q-2.28499 q^{2} +3.22117 q^{4} -2.86039 q^{5} +3.60040 q^{7} -2.79036 q^{8} +6.53597 q^{10} -1.00000 q^{11} -3.97298 q^{13} -8.22686 q^{14} -0.0663931 q^{16} -2.89791 q^{17} +3.91288 q^{19} -9.21383 q^{20} +2.28499 q^{22} +4.02846 q^{23} +3.18186 q^{25} +9.07821 q^{26} +11.5975 q^{28} -3.89190 q^{29} -0.470941 q^{31} +5.73244 q^{32} +6.62170 q^{34} -10.2986 q^{35} -1.32152 q^{37} -8.94089 q^{38} +7.98155 q^{40} +2.18046 q^{41} +11.4649 q^{43} -3.22117 q^{44} -9.20499 q^{46} -6.67167 q^{47} +5.96285 q^{49} -7.27051 q^{50} -12.7976 q^{52} +2.86420 q^{53} +2.86039 q^{55} -10.0464 q^{56} +8.89295 q^{58} +6.78636 q^{59} -1.00000 q^{61} +1.07609 q^{62} -12.9658 q^{64} +11.3643 q^{65} +6.63203 q^{67} -9.33468 q^{68} +23.5321 q^{70} +2.50557 q^{71} -16.0693 q^{73} +3.01965 q^{74} +12.6041 q^{76} -3.60040 q^{77} -1.93246 q^{79} +0.189911 q^{80} -4.98232 q^{82} -9.23389 q^{83} +8.28918 q^{85} -26.1971 q^{86} +2.79036 q^{88} -6.85941 q^{89} -14.3043 q^{91} +12.9764 q^{92} +15.2447 q^{94} -11.1924 q^{95} +11.7618 q^{97} -13.6250 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 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

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Database

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