LMFDB - Embedded newform 6039.2.a.o.1.2

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			1.2
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)

$f(q)$	$=$	$q-2.44801 q^{2} +3.99278 q^{4} +1.89686 q^{5} -0.969789 q^{7} -4.87834 q^{8} +O(q^{10})$	\(q-2.44801 q^{2} +3.99278 q^{4} +1.89686 q^{5} -0.969789 q^{7} -4.87834 q^{8} -4.64354 q^{10} +1.00000 q^{11} -1.38040 q^{13} +2.37406 q^{14} +3.95671 q^{16} +0.855355 q^{17} +5.83285 q^{19} +7.57373 q^{20} -2.44801 q^{22} +3.24050 q^{23} -1.40193 q^{25} +3.37923 q^{26} -3.87215 q^{28} +7.17008 q^{29} -3.46194 q^{31} +0.0706103 q^{32} -2.09392 q^{34} -1.83955 q^{35} +5.28798 q^{37} -14.2789 q^{38} -9.25353 q^{40} +6.09420 q^{41} +1.07616 q^{43} +3.99278 q^{44} -7.93279 q^{46} +6.02763 q^{47} -6.05951 q^{49} +3.43194 q^{50} -5.51162 q^{52} -2.06446 q^{53} +1.89686 q^{55} +4.73096 q^{56} -17.5525 q^{58} +5.72346 q^{59} -1.00000 q^{61} +8.47487 q^{62} -8.08627 q^{64} -2.61842 q^{65} -11.3644 q^{67} +3.41524 q^{68} +4.50325 q^{70} -7.88328 q^{71} +0.611985 q^{73} -12.9451 q^{74} +23.2893 q^{76} -0.969789 q^{77} +10.1792 q^{79} +7.50532 q^{80} -14.9187 q^{82} -4.28509 q^{83} +1.62249 q^{85} -2.63446 q^{86} -4.87834 q^{88} -14.0115 q^{89} +1.33869 q^{91} +12.9386 q^{92} -14.7557 q^{94} +11.0641 q^{95} -2.51129 q^{97} +14.8338 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.44801	−1.73101	−0.865504	−	0.500902i	$-0.833002\pi$
$2$	−2.44801	−1.73101	−0.865504	+	0.500902i	$0.833002\pi$
$3$	0	0
$3$	0	0
$4$	3.99278	1.99639
$5$	1.89686	0.848301	0.424151	−	0.905592i	$-0.360573\pi$
$5$	1.89686	0.848301	0.424151	+	0.905592i	$0.360573\pi$
$6$	0	0
$7$	−0.969789	−0.366546	−0.183273	−	0.983062i	$-0.558669\pi$
$7$	−0.969789	−0.366546	−0.183273	+	0.983062i	$0.558669\pi$
$8$	−4.87834	−1.72476
$9$	0	0
$10$	−4.64354	−1.46842
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.38040	−0.382853	−0.191427	−	0.981507i	$-0.561311\pi$
$13$	−1.38040	−0.382853	−0.191427	+	0.981507i	$0.561311\pi$
$14$	2.37406	0.634493
$15$	0	0
$16$	3.95671	0.989177
$17$	0.855355	0.207454	0.103727	−	0.994606i	$-0.466923\pi$
$17$	0.855355	0.207454	0.103727	+	0.994606i	$0.466923\pi$
$18$	0	0
$19$	5.83285	1.33815	0.669073	−	0.743196i	$-0.266691\pi$
$19$	5.83285	1.33815	0.669073	+	0.743196i	$0.266691\pi$
$20$	7.57373	1.69354
$21$	0	0
$22$	−2.44801	−0.521918
$23$	3.24050	0.675691	0.337846	−	0.941202i	$-0.390302\pi$
$23$	3.24050	0.675691	0.337846	+	0.941202i	$0.390302\pi$
$24$	0	0
$25$	−1.40193	−0.280385
$26$	3.37923	0.662722
$27$	0	0
$28$	−3.87215	−0.731767
$29$	7.17008	1.33145	0.665725	−	0.746197i	$-0.268122\pi$
$29$	7.17008	1.33145	0.665725	+	0.746197i	$0.268122\pi$
$30$	0	0
$31$	−3.46194	−0.621782	−0.310891	−	0.950446i	$-0.600627\pi$
$31$	−3.46194	−0.621782	−0.310891	+	0.950446i	$0.600627\pi$
$32$	0.0706103	0.0124823
$33$	0	0
$34$	−2.09392	−0.359105
$35$	−1.83955	−0.310941
$36$	0	0
$37$	5.28798	0.869339	0.434670	−	0.900590i	$-0.356865\pi$
$37$	5.28798	0.869339	0.434670	+	0.900590i	$0.356865\pi$
$38$	−14.2789	−2.31634
$39$	0	0
$40$	−9.25353	−1.46311
$41$	6.09420	0.951754	0.475877	−	0.879512i	$-0.342131\pi$
$41$	6.09420	0.951754	0.475877	+	0.879512i	$0.342131\pi$
$42$	0	0
$43$	1.07616	0.164113	0.0820567	−	0.996628i	$-0.473851\pi$
$43$	1.07616	0.164113	0.0820567	+	0.996628i	$0.473851\pi$
$44$	3.99278	0.601934
$45$	0	0
$46$	−7.93279	−1.16963
$47$	6.02763	0.879221	0.439610	−	0.898189i	$-0.355117\pi$
$47$	6.02763	0.879221	0.439610	+	0.898189i	$0.355117\pi$
$48$	0	0
$49$	−6.05951	−0.865644
$50$	3.43194	0.485349
$51$	0	0
$52$	−5.51162	−0.764324
$53$	−2.06446	−0.283576	−0.141788	−	0.989897i	$-0.545285\pi$
$53$	−2.06446	−0.283576	−0.141788	+	0.989897i	$0.545285\pi$
$54$	0	0
$55$	1.89686	0.255772
$56$	4.73096	0.632202
$57$	0	0
$58$	−17.5525	−2.30475
$59$	5.72346	0.745131	0.372566	−	0.928006i	$-0.378478\pi$
$59$	5.72346	0.745131	0.372566	+	0.928006i	$0.378478\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	8.47487	1.07631
$63$	0	0
$64$	−8.08627	−1.01078
$65$	−2.61842	−0.324775
$66$	0	0
$67$	−11.3644	−1.38839	−0.694193	−	0.719789i	$-0.744239\pi$
$67$	−11.3644	−1.38839	−0.694193	+	0.719789i	$0.744239\pi$
$68$	3.41524	0.414159
$69$	0	0
$70$	4.50325	0.538241
$71$	−7.88328	−0.935574	−0.467787	−	0.883841i	$-0.654949\pi$
$71$	−7.88328	−0.935574	−0.467787	+	0.883841i	$0.654949\pi$
$72$	0	0
$73$	0.611985	0.0716274	0.0358137	−	0.999358i	$-0.488598\pi$
$73$	0.611985	0.0716274	0.0358137	+	0.999358i	$0.488598\pi$
$74$	−12.9451	−1.50483
$75$	0	0
$76$	23.2893	2.67146
$77$	−0.969789	−0.110518
$78$	0	0
$79$	10.1792	1.14525	0.572627	−	0.819816i	$-0.305924\pi$
$79$	10.1792	1.14525	0.572627	+	0.819816i	$0.305924\pi$
$80$	7.50532	0.839120
$81$	0	0
$82$	−14.9187	−1.64749
$83$	−4.28509	−0.470350	−0.235175	−	0.971953i	$-0.575566\pi$
$83$	−4.28509	−0.470350	−0.235175	+	0.971953i	$0.575566\pi$
$84$	0	0
$85$	1.62249	0.175983
$86$	−2.63446	−0.284082
$87$	0	0
$88$	−4.87834	−0.520033
$89$	−14.0115	−1.48521	−0.742606	−	0.669728i	$-0.766411\pi$
$89$	−14.0115	−1.48521	−0.742606	+	0.669728i	$0.766411\pi$
$90$	0	0
$91$	1.33869	0.140333
$92$	12.9386	1.34894
$93$	0	0
$94$	−14.7557	−1.52194
$95$	11.0641	1.13515
$96$	0	0
$97$	−2.51129	−0.254983	−0.127492	−	0.991840i	$-0.540693\pi$
$97$	−2.51129	−0.254983	−0.127492	+	0.991840i	$0.540693\pi$
$98$	14.8338	1.49844
$99$	0	0
$100$	−5.59758	−0.559758
$101$	14.2789	1.42080	0.710401	−	0.703798i	$-0.248514\pi$
$101$	14.2789	1.42080	0.710401	+	0.703798i	$0.248514\pi$
$102$	0	0
$103$	−7.93576	−0.781934	−0.390967	−	0.920405i	$-0.627859\pi$
$103$	−7.93576	−0.781934	−0.390967	+	0.920405i	$0.627859\pi$
$104$	6.73405	0.660328
$105$	0	0
$106$	5.05384	0.490872
$107$	12.2576	1.18499	0.592494	−	0.805575i	$-0.298143\pi$
$107$	12.2576	1.18499	0.592494	+	0.805575i	$0.298143\pi$
$108$	0	0
$109$	7.54910	0.723073	0.361536	−	0.932358i	$-0.382252\pi$
$109$	7.54910	0.723073	0.361536	+	0.932358i	$0.382252\pi$
$110$	−4.64354	−0.442744
$111$	0	0
$112$	−3.83717	−0.362579
$113$	6.19099	0.582400	0.291200	−	0.956662i	$-0.405946\pi$
$113$	6.19099	0.582400	0.291200	+	0.956662i	$0.405946\pi$
$114$	0	0
$115$	6.14677	0.573189
$116$	28.6285	2.65809
$117$	0	0
$118$	−14.0111	−1.28983
$119$	−0.829513	−0.0760414
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.44801	0.221633
$123$	0	0
$124$	−13.8227	−1.24132
$125$	−12.1436	−1.08615
$126$	0	0
$127$	−17.7585	−1.57581	−0.787907	−	0.615794i	$-0.788835\pi$
$127$	−17.7585	−1.57581	−0.787907	+	0.615794i	$0.788835\pi$
$128$	19.6541	1.73719
$129$	0	0
$130$	6.40993	0.562188
$131$	8.09154	0.706961	0.353480	−	0.935442i	$-0.384998\pi$
$131$	8.09154	0.706961	0.353480	+	0.935442i	$0.384998\pi$
$132$	0	0
$133$	−5.65663	−0.490492
$134$	27.8203	2.40331
$135$	0	0
$136$	−4.17272	−0.357807
$137$	−2.99237	−0.255656	−0.127828	−	0.991796i	$-0.540800\pi$
$137$	−2.99237	−0.255656	−0.127828	+	0.991796i	$0.540800\pi$
$138$	0	0
$139$	−11.5771	−0.981953	−0.490976	−	0.871173i	$-0.663360\pi$
$139$	−11.5771	−0.981953	−0.490976	+	0.871173i	$0.663360\pi$
$140$	−7.34492	−0.620759
$141$	0	0
$142$	19.2984	1.61949
$143$	−1.38040	−0.115435
$144$	0	0
$145$	13.6006	1.12947
$146$	−1.49815	−0.123988
$147$	0	0
$148$	21.1137	1.73554
$149$	13.7136	1.12346	0.561731	−	0.827320i	$-0.310135\pi$
$149$	13.7136	1.12346	0.561731	+	0.827320i	$0.310135\pi$
$150$	0	0
$151$	17.8159	1.44984	0.724921	−	0.688833i	$-0.241876\pi$
$151$	17.8159	1.44984	0.724921	+	0.688833i	$0.241876\pi$
$152$	−28.4546	−2.30798
$153$	0	0
$154$	2.37406	0.191307
$155$	−6.56681	−0.527459
$156$	0	0
$157$	13.5321	1.07998	0.539989	−	0.841672i	$-0.318428\pi$
$157$	13.5321	1.07998	0.539989	+	0.841672i	$0.318428\pi$
$158$	−24.9189	−1.98244
$159$	0	0
$160$	0.133938	0.0105887
$161$	−3.14260	−0.247672
$162$	0	0
$163$	14.8613	1.16402	0.582012	−	0.813180i	$-0.302266\pi$
$163$	14.8613	1.16402	0.582012	+	0.813180i	$0.302266\pi$
$164$	24.3328	1.90007
$165$	0	0
$166$	10.4900	0.814179
$167$	−13.7794	−1.06628	−0.533141	−	0.846026i	$-0.678989\pi$
$167$	−13.7794	−1.06628	−0.533141	+	0.846026i	$0.678989\pi$
$168$	0	0
$169$	−11.0945	−0.853423
$170$	−3.97187	−0.304629
$171$	0	0
$172$	4.29688	0.327634
$173$	12.4898	0.949582	0.474791	−	0.880098i	$-0.342524\pi$
$173$	12.4898	0.949582	0.474791	+	0.880098i	$0.342524\pi$
$174$	0	0
$175$	1.35957	0.102774
$176$	3.95671	0.298248
$177$	0	0
$178$	34.3003	2.57091
$179$	19.0115	1.42098	0.710492	−	0.703705i	$-0.248472\pi$
$179$	19.0115	1.42098	0.710492	+	0.703705i	$0.248472\pi$
$180$	0	0
$181$	0.660327	0.0490818	0.0245409	−	0.999699i	$-0.492188\pi$
$181$	0.660327	0.0490818	0.0245409	+	0.999699i	$0.492188\pi$
$182$	−3.27714	−0.242918
$183$	0	0
$184$	−15.8083	−1.16540
$185$	10.0306	0.737461
$186$	0	0
$187$	0.855355	0.0625497
$188$	24.0670	1.75527
$189$	0	0
$190$	−27.0850	−1.96496
$191$	23.1136	1.67244	0.836222	−	0.548391i	$-0.184760\pi$
$191$	23.1136	1.67244	0.836222	+	0.548391i	$0.184760\pi$
$192$	0	0
$193$	1.20067	0.0864258	0.0432129	−	0.999066i	$-0.486241\pi$
$193$	1.20067	0.0864258	0.0432129	+	0.999066i	$0.486241\pi$
$194$	6.14769	0.441378
$195$	0	0
$196$	−24.1943	−1.72816
$197$	−5.69586	−0.405813	−0.202907	−	0.979198i	$-0.565039\pi$
$197$	−5.69586	−0.405813	−0.202907	+	0.979198i	$0.565039\pi$
$198$	0	0
$199$	−5.20131	−0.368711	−0.184356	−	0.982860i	$-0.559020\pi$
$199$	−5.20131	−0.368711	−0.184356	+	0.982860i	$0.559020\pi$
$200$	6.83908	0.483596
$201$	0	0
$202$	−34.9549	−2.45942
$203$	−6.95346	−0.488038
$204$	0	0
$205$	11.5598	0.807374
$206$	19.4269	1.35353
$207$	0	0
$208$	−5.46183	−0.378710
$209$	5.83285	0.403466
$210$	0	0
$211$	−7.38195	−0.508195	−0.254097	−	0.967179i	$-0.581778\pi$
$211$	−7.38195	−0.508195	−0.254097	+	0.967179i	$0.581778\pi$
$212$	−8.24294	−0.566128
$213$	0	0
$214$	−30.0068	−2.05122
$215$	2.04133	0.139218
$216$	0	0
$217$	3.35735	0.227912
$218$	−18.4803	−1.25164
$219$	0	0
$220$	7.57373	0.510621
$221$	−1.18073	−0.0794245
$222$	0	0
$223$	7.03118	0.470843	0.235421	−	0.971893i	$-0.424353\pi$
$223$	7.03118	0.470843	0.235421	+	0.971893i	$0.424353\pi$
$224$	−0.0684771	−0.00457532
$225$	0	0
$226$	−15.1556	−1.00814
$227$	−19.4013	−1.28771	−0.643855	−	0.765148i	$-0.722666\pi$
$227$	−19.4013	−1.28771	−0.643855	+	0.765148i	$0.722666\pi$
$228$	0	0
$229$	9.22075	0.609324	0.304662	−	0.952461i	$-0.401456\pi$
$229$	9.22075	0.609324	0.304662	+	0.952461i	$0.401456\pi$
$230$	−15.0474	−0.992195
$231$	0	0
$232$	−34.9781	−2.29643
$233$	−20.1940	−1.32295	−0.661476	−	0.749966i	$-0.730070\pi$
$233$	−20.1940	−1.32295	−0.661476	+	0.749966i	$0.730070\pi$
$234$	0	0
$235$	11.4336	0.745844
$236$	22.8525	1.48757
$237$	0	0
$238$	2.03066	0.131628
$239$	−5.63737	−0.364651	−0.182325	−	0.983238i	$-0.558362\pi$
$239$	−5.63737	−0.364651	−0.182325	+	0.983238i	$0.558362\pi$
$240$	0	0
$241$	17.1402	1.10410	0.552050	−	0.833811i	$-0.313846\pi$
$241$	17.1402	1.10410	0.552050	+	0.833811i	$0.313846\pi$
$242$	−2.44801	−0.157364
$243$	0	0
$244$	−3.99278	−0.255611
$245$	−11.4940	−0.734327
$246$	0	0
$247$	−8.05165	−0.512314
$248$	16.8885	1.07242
$249$	0	0
$250$	29.7276	1.88014
$251$	−1.16912	−0.0737943	−0.0368971	−	0.999319i	$-0.511747\pi$
$251$	−1.16912	−0.0737943	−0.0368971	+	0.999319i	$0.511747\pi$
$252$	0	0
$253$	3.24050	0.203729
$254$	43.4731	2.72775
$255$	0	0
$256$	−31.9410	−1.99631
$257$	−14.1284	−0.881307	−0.440654	−	0.897677i	$-0.645253\pi$
$257$	−14.1284	−0.881307	−0.440654	+	0.897677i	$0.645253\pi$
$258$	0	0
$259$	−5.12823	−0.318653
$260$	−10.4548	−0.648377
$261$	0	0
$262$	−19.8082	−1.22375
$263$	5.39904	0.332919	0.166460	−	0.986048i	$-0.446766\pi$
$263$	5.39904	0.332919	0.166460	+	0.986048i	$0.446766\pi$
$264$	0	0
$265$	−3.91600	−0.240558
$266$	13.8475	0.849045
$267$	0	0
$268$	−45.3756	−2.77176
$269$	−23.2326	−1.41652	−0.708258	−	0.705954i	$-0.750519\pi$
$269$	−23.2326	−1.41652	−0.708258	+	0.705954i	$0.750519\pi$
$270$	0	0
$271$	0.968446	0.0588289	0.0294144	−	0.999567i	$-0.490636\pi$
$271$	0.968446	0.0588289	0.0294144	+	0.999567i	$0.490636\pi$
$272$	3.38439	0.205209
$273$	0	0
$274$	7.32537	0.442542
$275$	−1.40193	−0.0845394
$276$	0	0
$277$	−0.925052	−0.0555810	−0.0277905	−	0.999614i	$-0.508847\pi$
$277$	−0.925052	−0.0555810	−0.0277905	+	0.999614i	$0.508847\pi$
$278$	28.3408	1.69977
$279$	0	0
$280$	8.97397	0.536297
$281$	22.4103	1.33688	0.668442	−	0.743764i	$-0.266961\pi$
$281$	22.4103	1.33688	0.668442	+	0.743764i	$0.266961\pi$
$282$	0	0
$283$	−26.1312	−1.55334	−0.776669	−	0.629909i	$-0.783092\pi$
$283$	−26.1312	−1.55334	−0.776669	+	0.629909i	$0.783092\pi$
$284$	−31.4762	−1.86777
$285$	0	0
$286$	3.37923	0.199818
$287$	−5.91009	−0.348861
$288$	0	0
$289$	−16.2684	−0.956963
$290$	−33.2946	−1.95512
$291$	0	0
$292$	2.44352	0.142996
$293$	−11.5708	−0.675974	−0.337987	−	0.941151i	$-0.609746\pi$
$293$	−11.5708	−0.675974	−0.337987	+	0.941151i	$0.609746\pi$
$294$	0	0
$295$	10.8566	0.632096
$296$	−25.7966	−1.49940
$297$	0	0
$298$	−33.5711	−1.94472
$299$	−4.47318	−0.258691
$300$	0	0
$301$	−1.04365	−0.0601551
$302$	−43.6137	−2.50969
$303$	0	0
$304$	23.0789	1.32366
$305$	−1.89686	−0.108614
$306$	0	0
$307$	−17.6298	−1.00619	−0.503093	−	0.864232i	$-0.667805\pi$
$307$	−17.6298	−1.00619	−0.503093	+	0.864232i	$0.667805\pi$
$308$	−3.87215	−0.220636
$309$	0	0
$310$	16.0756	0.913035
$311$	−4.59243	−0.260413	−0.130206	−	0.991487i	$-0.541564\pi$
$311$	−4.59243	−0.260413	−0.130206	+	0.991487i	$0.541564\pi$
$312$	0	0
$313$	14.0068	0.791709	0.395854	−	0.918313i	$-0.370448\pi$
$313$	14.0068	0.791709	0.395854	+	0.918313i	$0.370448\pi$
$314$	−33.1268	−1.86945
$315$	0	0
$316$	40.6434	2.28637
$317$	27.7798	1.56027	0.780136	−	0.625610i	$-0.215150\pi$
$317$	27.7798	1.56027	0.780136	+	0.625610i	$0.215150\pi$
$318$	0	0
$319$	7.17008	0.401448
$320$	−15.3385	−0.857449
$321$	0	0
$322$	7.69313	0.428722
$323$	4.98915	0.277604
$324$	0	0
$325$	1.93522	0.107346
$326$	−36.3806	−2.01494
$327$	0	0
$328$	−29.7296	−1.64154
$329$	−5.84553	−0.322275
$330$	0	0
$331$	34.0813	1.87328	0.936640	−	0.350294i	$-0.113918\pi$
$331$	34.0813	1.87328	0.936640	+	0.350294i	$0.113918\pi$
$332$	−17.1094	−0.939001
$333$	0	0
$334$	33.7322	1.84574
$335$	−21.5567	−1.17777
$336$	0	0
$337$	1.84351	0.100422	0.0502112	−	0.998739i	$-0.484011\pi$
$337$	1.84351	0.100422	0.0502112	+	0.998739i	$0.484011\pi$
$338$	27.1595	1.47728
$339$	0	0
$340$	6.47823	0.351331
$341$	−3.46194	−0.187474
$342$	0	0
$343$	12.6650	0.683844
$344$	−5.24990	−0.283056
$345$	0	0
$346$	−30.5752	−1.64373
$347$	3.94391	0.211720	0.105860	−	0.994381i	$-0.466240\pi$
$347$	3.94391	0.211720	0.105860	+	0.994381i	$0.466240\pi$
$348$	0	0
$349$	10.1806	0.544956	0.272478	−	0.962162i	$-0.412157\pi$
$349$	10.1806	0.544956	0.272478	+	0.962162i	$0.412157\pi$
$350$	−3.32825	−0.177903
$351$	0	0
$352$	0.0706103	0.00376354
$353$	13.6707	0.727616	0.363808	−	0.931474i	$-0.381477\pi$
$353$	13.6707	0.727616	0.363808	+	0.931474i	$0.381477\pi$
$354$	0	0
$355$	−14.9535	−0.793648
$356$	−55.9446	−2.96506
$357$	0	0
$358$	−46.5404	−2.45974
$359$	3.82844	0.202057	0.101029	−	0.994884i	$-0.467787\pi$
$359$	3.82844	0.202057	0.101029	+	0.994884i	$0.467787\pi$
$360$	0	0
$361$	15.0221	0.790637
$362$	−1.61649	−0.0849609
$363$	0	0
$364$	5.34510	0.280160
$365$	1.16085	0.0607616
$366$	0	0
$367$	24.5024	1.27901	0.639506	−	0.768786i	$-0.279139\pi$
$367$	24.5024	1.27901	0.639506	+	0.768786i	$0.279139\pi$
$368$	12.8217	0.668378
$369$	0	0
$370$	−24.5550	−1.27655
$371$	2.00209	0.103944
$372$	0	0
$373$	22.3103	1.15518	0.577591	−	0.816326i	$-0.303993\pi$
$373$	22.3103	1.15518	0.577591	+	0.816326i	$0.303993\pi$
$374$	−2.09392	−0.108274
$375$	0	0
$376$	−29.4049	−1.51644
$377$	−9.89756	−0.509750
$378$	0	0
$379$	12.0926	0.621156	0.310578	−	0.950548i	$-0.399477\pi$
$379$	12.0926	0.621156	0.310578	+	0.950548i	$0.399477\pi$
$380$	44.1764	2.26620
$381$	0	0
$382$	−56.5825	−2.89501
$383$	−36.3134	−1.85553	−0.927764	−	0.373168i	$-0.878271\pi$
$383$	−36.3134	−1.85553	−0.927764	+	0.373168i	$0.878271\pi$
$384$	0	0
$385$	−1.83955	−0.0937523
$386$	−2.93925	−0.149604
$387$	0	0
$388$	−10.0270	−0.509046
$389$	−1.16648	−0.0591427	−0.0295713	−	0.999563i	$-0.509414\pi$
$389$	−1.16648	−0.0591427	−0.0295713	+	0.999563i	$0.509414\pi$
$390$	0	0
$391$	2.77178	0.140175
$392$	29.5604	1.49302
$393$	0	0
$394$	13.9436	0.702466
$395$	19.3086	0.971520
$396$	0	0
$397$	8.54740	0.428982	0.214491	−	0.976726i	$-0.431191\pi$
$397$	8.54740	0.428982	0.214491	+	0.976726i	$0.431191\pi$
$398$	12.7329	0.638242
$399$	0	0
$400$	−5.54701	−0.277351
$401$	−34.2861	−1.71217	−0.856084	−	0.516837i	$-0.827109\pi$
$401$	−34.2861	−1.71217	−0.856084	+	0.516837i	$0.827109\pi$
$402$	0	0
$403$	4.77885	0.238051
$404$	57.0124	2.83647
$405$	0	0
$406$	17.0222	0.844797
$407$	5.28798	0.262116
$408$	0	0
$409$	18.5074	0.915134	0.457567	−	0.889175i	$-0.348721\pi$
$409$	18.5074	0.915134	0.457567	+	0.889175i	$0.348721\pi$
$410$	−28.2987	−1.39757
$411$	0	0
$412$	−31.6857	−1.56104
$413$	−5.55055	−0.273125
$414$	0	0
$415$	−8.12821	−0.398998
$416$	−0.0974703	−0.00477887
$417$	0	0
$418$	−14.2789	−0.698404
$419$	20.0178	0.977932	0.488966	−	0.872303i	$-0.337374\pi$
$419$	20.0178	0.977932	0.488966	+	0.872303i	$0.337374\pi$
$420$	0	0
$421$	33.2596	1.62097	0.810486	−	0.585758i	$-0.199203\pi$
$421$	33.2596	1.62097	0.810486	+	0.585758i	$0.199203\pi$
$422$	18.0711	0.879689
$423$	0	0
$424$	10.0712	0.489099
$425$	−1.19914	−0.0581671
$426$	0	0
$427$	0.969789	0.0469314
$428$	48.9419	2.36570
$429$	0	0
$430$	−4.99721	−0.240987
$431$	−29.9785	−1.44401	−0.722007	−	0.691886i	$-0.756780\pi$
$431$	−29.9785	−1.44401	−0.722007	+	0.691886i	$0.756780\pi$
$432$	0	0
$433$	−11.4446	−0.549991	−0.274995	−	0.961446i	$-0.588676\pi$
$433$	−11.4446	−0.549991	−0.274995	+	0.961446i	$0.588676\pi$
$434$	−8.21883	−0.394517
$435$	0	0
$436$	30.1419	1.44353
$437$	18.9013	0.904174
$438$	0	0
$439$	7.54994	0.360339	0.180170	−	0.983636i	$-0.442335\pi$
$439$	7.54994	0.360339	0.180170	+	0.983636i	$0.442335\pi$
$440$	−9.25353	−0.441145
$441$	0	0
$442$	2.89044	0.137484
$443$	−6.08354	−0.289038	−0.144519	−	0.989502i	$-0.546163\pi$
$443$	−6.08354	−0.289038	−0.144519	+	0.989502i	$0.546163\pi$
$444$	0	0
$445$	−26.5778	−1.25991
$446$	−17.2124	−0.815033
$447$	0	0
$448$	7.84197	0.370498
$449$	1.43753	0.0678413	0.0339207	−	0.999425i	$-0.489201\pi$
$449$	1.43753	0.0678413	0.0339207	+	0.999425i	$0.489201\pi$
$450$	0	0
$451$	6.09420	0.286965
$452$	24.7192	1.16270
$453$	0	0
$454$	47.4947	2.22904
$455$	2.53931	0.119045
$456$	0	0
$457$	−30.9601	−1.44825	−0.724127	−	0.689667i	$-0.757757\pi$
$457$	−30.9601	−1.44825	−0.724127	+	0.689667i	$0.757757\pi$
$458$	−22.5725	−1.05474
$459$	0	0
$460$	24.5427	1.14431
$461$	24.2494	1.12941	0.564704	−	0.825294i	$-0.308991\pi$
$461$	24.2494	1.12941	0.564704	+	0.825294i	$0.308991\pi$
$462$	0	0
$463$	−28.0262	−1.30249	−0.651245	−	0.758868i	$-0.725753\pi$
$463$	−28.0262	−1.30249	−0.651245	+	0.758868i	$0.725753\pi$
$464$	28.3699	1.31704
$465$	0	0
$466$	49.4352	2.29004
$467$	5.16196	0.238867	0.119434	−	0.992842i	$-0.461892\pi$
$467$	5.16196	0.238867	0.119434	+	0.992842i	$0.461892\pi$
$468$	0	0
$469$	11.0211	0.508907
$470$	−27.9895	−1.29106
$471$	0	0
$472$	−27.9210	−1.28517
$473$	1.07616	0.0494821
$474$	0	0
$475$	−8.17722	−0.375197
$476$	−3.31206	−0.151808
$477$	0	0
$478$	13.8004	0.631214
$479$	0.552200	0.0252307	0.0126153	−	0.999920i	$-0.495984\pi$
$479$	0.552200	0.0252307	0.0126153	+	0.999920i	$0.495984\pi$
$480$	0	0
$481$	−7.29952	−0.332829
$482$	−41.9596	−1.91121
$483$	0	0
$484$	3.99278	0.181490
$485$	−4.76357	−0.216303
$486$	0	0
$487$	−28.7046	−1.30073	−0.650366	−	0.759621i	$-0.725384\pi$
$487$	−28.7046	−1.30073	−0.650366	+	0.759621i	$0.725384\pi$
$488$	4.87834	0.220832
$489$	0	0
$490$	28.1376	1.27113
$491$	19.7847	0.892870	0.446435	−	0.894816i	$-0.352693\pi$
$491$	19.7847	0.892870	0.446435	+	0.894816i	$0.352693\pi$
$492$	0	0
$493$	6.13296	0.276215
$494$	19.7105	0.886820
$495$	0	0
$496$	−13.6979	−0.615053
$497$	7.64512	0.342930
$498$	0	0
$499$	18.8129	0.842182	0.421091	−	0.907018i	$-0.361647\pi$
$499$	18.8129	0.842182	0.421091	+	0.907018i	$0.361647\pi$
$500$	−48.4865	−2.16838
$501$	0	0
$502$	2.86203	0.127738
$503$	−8.44822	−0.376687	−0.188344	−	0.982103i	$-0.560312\pi$
$503$	−8.44822	−0.376687	−0.188344	+	0.982103i	$0.560312\pi$
$504$	0	0
$505$	27.0850	1.20527
$506$	−7.93279	−0.352656
$507$	0	0
$508$	−70.9058	−3.14594
$509$	−3.24496	−0.143830	−0.0719152	−	0.997411i	$-0.522911\pi$
$509$	−3.24496	−0.143830	−0.0719152	+	0.997411i	$0.522911\pi$
$510$	0	0
$511$	−0.593496	−0.0262547
$512$	38.8838	1.71844
$513$	0	0
$514$	34.5866	1.52555
$515$	−15.0530	−0.663315
$516$	0	0
$517$	6.02763	0.265095
$518$	12.5540	0.551590
$519$	0	0
$520$	12.7736	0.560157
$521$	11.3259	0.496196	0.248098	−	0.968735i	$-0.420195\pi$
$521$	11.3259	0.496196	0.248098	+	0.968735i	$0.420195\pi$
$522$	0	0
$523$	34.8169	1.52244	0.761218	−	0.648496i	$-0.224602\pi$
$523$	34.8169	1.52244	0.761218	+	0.648496i	$0.224602\pi$
$524$	32.3077	1.41137
$525$	0	0
$526$	−13.2169	−0.576286
$527$	−2.96118	−0.128991
$528$	0	0
$529$	−12.4992	−0.543442
$530$	9.58642	0.416407
$531$	0	0
$532$	−22.5857	−0.979212
$533$	−8.41242	−0.364382
$534$	0	0
$535$	23.2510	1.00523
$536$	55.4396	2.39463
$537$	0	0
$538$	56.8737	2.45200
$539$	−6.05951	−0.261002
$540$	0	0
$541$	−41.2014	−1.77139	−0.885693	−	0.464271i	$-0.846316\pi$
$541$	−41.2014	−1.77139	−0.885693	+	0.464271i	$0.846316\pi$
$542$	−2.37077	−0.101833
$543$	0	0
$544$	0.0603969	0.00258949
$545$	14.3196	0.613383
$546$	0	0
$547$	9.19226	0.393033	0.196516	−	0.980501i	$-0.437037\pi$
$547$	9.19226	0.393033	0.196516	+	0.980501i	$0.437037\pi$
$548$	−11.9479	−0.510388
$549$	0	0
$550$	3.43194	0.146338
$551$	41.8220	1.78168
$552$	0	0
$553$	−9.87171	−0.419788
$554$	2.26454	0.0962111
$555$	0	0
$556$	−46.2246	−1.96036
$557$	5.68316	0.240803	0.120402	−	0.992725i	$-0.461582\pi$
$557$	5.68316	0.240803	0.120402	+	0.992725i	$0.461582\pi$
$558$	0	0
$559$	−1.48553	−0.0628314
$560$	−7.27857	−0.307576
$561$	0	0
$562$	−54.8607	−2.31416
$563$	2.01368	0.0848663	0.0424332	−	0.999099i	$-0.486489\pi$
$563$	2.01368	0.0848663	0.0424332	+	0.999099i	$0.486489\pi$
$564$	0	0
$565$	11.7434	0.494050
$566$	63.9695	2.68884
$567$	0	0
$568$	38.4574	1.61364
$569$	−14.5668	−0.610670	−0.305335	−	0.952245i	$-0.598768\pi$
$569$	−14.5668	−0.610670	−0.305335	+	0.952245i	$0.598768\pi$
$570$	0	0
$571$	−2.23602	−0.0935744	−0.0467872	−	0.998905i	$-0.514898\pi$
$571$	−2.23602	−0.0935744	−0.0467872	+	0.998905i	$0.514898\pi$
$572$	−5.51162	−0.230452
$573$	0	0
$574$	14.4680	0.603882
$575$	−4.54294	−0.189454
$576$	0	0
$577$	19.6123	0.816472	0.408236	−	0.912876i	$-0.366144\pi$
$577$	19.6123	0.816472	0.408236	+	0.912876i	$0.366144\pi$
$578$	39.8252	1.65651
$579$	0	0
$580$	54.3043	2.25486
$581$	4.15563	0.172405
$582$	0	0
$583$	−2.06446	−0.0855014
$584$	−2.98547	−0.123540
$585$	0	0
$586$	28.3255	1.17012
$587$	−10.0921	−0.416546	−0.208273	−	0.978071i	$-0.566784\pi$
$587$	−10.0921	−0.416546	−0.208273	+	0.978071i	$0.566784\pi$
$588$	0	0
$589$	−20.1929	−0.832036
$590$	−26.5771	−1.09416
$591$	0	0
$592$	20.9230	0.859930
$593$	47.5052	1.95081	0.975403	−	0.220431i	$-0.0707464\pi$
$593$	47.5052	1.95081	0.975403	+	0.220431i	$0.0707464\pi$
$594$	0	0
$595$	−1.57347	−0.0645060
$596$	54.7554	2.24287
$597$	0	0
$598$	10.9504	0.447795
$599$	39.8977	1.63017	0.815087	−	0.579339i	$-0.196689\pi$
$599$	39.8977	1.63017	0.815087	+	0.579339i	$0.196689\pi$
$600$	0	0
$601$	25.2220	1.02883	0.514413	−	0.857543i	$-0.328010\pi$
$601$	25.2220	1.02883	0.514413	+	0.857543i	$0.328010\pi$
$602$	2.55487	0.104129
$603$	0	0
$604$	71.1351	2.89445
$605$	1.89686	0.0771183
$606$	0	0
$607$	38.3949	1.55840	0.779201	−	0.626774i	$-0.215625\pi$
$607$	38.3949	1.55840	0.779201	+	0.626774i	$0.215625\pi$
$608$	0.411859	0.0167031
$609$	0	0
$610$	4.64354	0.188011
$611$	−8.32053	−0.336613
$612$	0	0
$613$	1.87322	0.0756586	0.0378293	−	0.999284i	$-0.487956\pi$
$613$	1.87322	0.0756586	0.0378293	+	0.999284i	$0.487956\pi$
$614$	43.1580	1.74172
$615$	0	0
$616$	4.73096	0.190616
$617$	20.1184	0.809935	0.404967	−	0.914331i	$-0.367283\pi$
$617$	20.1184	0.809935	0.404967	+	0.914331i	$0.367283\pi$
$618$	0	0
$619$	−13.2869	−0.534046	−0.267023	−	0.963690i	$-0.586040\pi$
$619$	−13.2869	−0.534046	−0.267023	+	0.963690i	$0.586040\pi$
$620$	−26.2198	−1.05301
$621$	0	0
$622$	11.2423	0.450776
$623$	13.5882	0.544398
$624$	0	0
$625$	−16.0250	−0.640999
$626$	−34.2887	−1.37045
$627$	0	0
$628$	54.0306	2.15606
$629$	4.52310	0.180348
$630$	0	0
$631$	8.52495	0.339373	0.169686	−	0.985498i	$-0.445725\pi$
$631$	8.52495	0.339373	0.169686	+	0.985498i	$0.445725\pi$
$632$	−49.6578	−1.97528
$633$	0	0
$634$	−68.0055	−2.70084
$635$	−33.6854	−1.33676
$636$	0	0
$637$	8.36453	0.331415
$638$	−17.5525	−0.694909
$639$	0	0
$640$	37.2810	1.47366
$641$	−1.32404	−0.0522966	−0.0261483	−	0.999658i	$-0.508324\pi$
$641$	−1.32404	−0.0522966	−0.0261483	+	0.999658i	$0.508324\pi$
$642$	0	0
$643$	−0.703946	−0.0277609	−0.0138805	−	0.999904i	$-0.504418\pi$
$643$	−0.703946	−0.0277609	−0.0138805	+	0.999904i	$0.504418\pi$
$644$	−12.5477	−0.494449
$645$	0	0
$646$	−12.2135	−0.480535
$647$	6.76756	0.266060	0.133030	−	0.991112i	$-0.457529\pi$
$647$	6.76756	0.266060	0.133030	+	0.991112i	$0.457529\pi$
$648$	0	0
$649$	5.72346	0.224665
$650$	−4.73744	−0.185818
$651$	0	0
$652$	59.3377	2.32384
$653$	24.1785	0.946176	0.473088	−	0.881015i	$-0.343139\pi$
$653$	24.1785	0.946176	0.473088	+	0.881015i	$0.343139\pi$
$654$	0	0
$655$	15.3485	0.599716
$656$	24.1130	0.941453
$657$	0	0
$658$	14.3099	0.557860
$659$	31.2395	1.21692	0.608458	−	0.793586i	$-0.291788\pi$
$659$	31.2395	1.21692	0.608458	+	0.793586i	$0.291788\pi$
$660$	0	0
$661$	24.3254	0.946150	0.473075	−	0.881022i	$-0.343144\pi$
$661$	24.3254	0.946150	0.473075	+	0.881022i	$0.343144\pi$
$662$	−83.4316	−3.24266
$663$	0	0
$664$	20.9041	0.811238
$665$	−10.7298	−0.416085
$666$	0	0
$667$	23.2347	0.899650
$668$	−55.0181	−2.12871
$669$	0	0
$670$	52.7711	2.03873
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−6.12355	−0.236045	−0.118023	−	0.993011i	$-0.537656\pi$
$673$	−6.12355	−0.236045	−0.118023	+	0.993011i	$0.537656\pi$
$674$	−4.51294	−0.173832
$675$	0	0
$676$	−44.2979	−1.70376
$677$	12.4601	0.478879	0.239440	−	0.970911i	$-0.423036\pi$
$677$	12.4601	0.478879	0.239440	+	0.970911i	$0.423036\pi$
$678$	0	0
$679$	2.43543	0.0934630
$680$	−7.91505	−0.303528
$681$	0	0
$682$	8.47487	0.324520
$683$	0.225362	0.00862325	0.00431162	−	0.999991i	$-0.498628\pi$
$683$	0.225362	0.00862325	0.00431162	+	0.999991i	$0.498628\pi$
$684$	0	0
$685$	−5.67611	−0.216873
$686$	−31.0040	−1.18374
$687$	0	0
$688$	4.25807	0.162337
$689$	2.84978	0.108568
$690$	0	0
$691$	−17.8903	−0.680578	−0.340289	−	0.940321i	$-0.610525\pi$
$691$	−17.8903	−0.680578	−0.340289	+	0.940321i	$0.610525\pi$
$692$	49.8690	1.89573
$693$	0	0
$694$	−9.65475	−0.366489
$695$	−21.9600	−0.832992
$696$	0	0
$697$	5.21270	0.197445
$698$	−24.9223	−0.943323
$699$	0	0
$700$	5.42847	0.205177
$701$	33.9483	1.28221	0.641104	−	0.767454i	$-0.278477\pi$
$701$	33.9483	1.28221	0.641104	+	0.767454i	$0.278477\pi$
$702$	0	0
$703$	30.8440	1.16330
$704$	−8.08627	−0.304763
$705$	0	0
$706$	−33.4660	−1.25951
$707$	−13.8475	−0.520789
$708$	0	0
$709$	37.8681	1.42217	0.711083	−	0.703108i	$-0.248205\pi$
$709$	37.8681	1.42217	0.711083	+	0.703108i	$0.248205\pi$
$710$	36.6063	1.37381
$711$	0	0
$712$	68.3528	2.56163
$713$	−11.2184	−0.420133
$714$	0	0
$715$	−2.61842	−0.0979233
$716$	75.9086	2.83684
$717$	0	0
$718$	−9.37208	−0.349763
$719$	−12.9954	−0.484647	−0.242324	−	0.970195i	$-0.577910\pi$
$719$	−12.9954	−0.484647	−0.242324	+	0.970195i	$0.577910\pi$
$720$	0	0
$721$	7.69601	0.286614
$722$	−36.7743	−1.36860
$723$	0	0
$724$	2.63654	0.0979862
$725$	−10.0519	−0.373319
$726$	0	0
$727$	−41.5701	−1.54175	−0.770875	−	0.636987i	$-0.780181\pi$
$727$	−41.5701	−1.54175	−0.770875	+	0.636987i	$0.780181\pi$
$728$	−6.53061	−0.242040
$729$	0	0
$730$	−2.84177	−0.105179
$731$	0.920502	0.0340460
$732$	0	0
$733$	−32.3844	−1.19614	−0.598072	−	0.801442i	$-0.704066\pi$
$733$	−32.3844	−1.19614	−0.598072	+	0.801442i	$0.704066\pi$
$734$	−59.9821	−2.21398
$735$	0	0
$736$	0.228813	0.00843415
$737$	−11.3644	−0.418614
$738$	0	0
$739$	30.9538	1.13865	0.569326	−	0.822112i	$-0.307204\pi$
$739$	30.9538	1.13865	0.569326	+	0.822112i	$0.307204\pi$
$740$	40.0498	1.47226
$741$	0	0
$742$	−4.90115	−0.179927
$743$	4.25715	0.156180	0.0780899	−	0.996946i	$-0.475118\pi$
$743$	4.25715	0.156180	0.0780899	+	0.996946i	$0.475118\pi$
$744$	0	0
$745$	26.0128	0.953035
$746$	−54.6158	−1.99963
$747$	0	0
$748$	3.41524	0.124874
$749$	−11.8873	−0.434352
$750$	0	0
$751$	8.95075	0.326618	0.163309	−	0.986575i	$-0.447783\pi$
$751$	8.95075	0.326618	0.163309	+	0.986575i	$0.447783\pi$
$752$	23.8496	0.869705
$753$	0	0
$754$	24.2294	0.882382
$755$	33.7943	1.22990
$756$	0	0
$757$	18.4763	0.671533	0.335767	−	0.941945i	$-0.391005\pi$
$757$	18.4763	0.671533	0.335767	+	0.941945i	$0.391005\pi$
$758$	−29.6029	−1.07523
$759$	0	0
$760$	−53.9744	−1.95786
$761$	19.7949	0.717566	0.358783	−	0.933421i	$-0.383192\pi$
$761$	19.7949	0.717566	0.358783	+	0.933421i	$0.383192\pi$
$762$	0	0
$763$	−7.32103	−0.265039
$764$	92.2876	3.33885
$765$	0	0
$766$	88.8957	3.21193
$767$	−7.90065	−0.285276
$768$	0	0
$769$	−1.56813	−0.0565482	−0.0282741	−	0.999600i	$-0.509001\pi$
$769$	−1.56813	−0.0565482	−0.0282741	+	0.999600i	$0.509001\pi$
$770$	4.50325	0.162286
$771$	0	0
$772$	4.79399	0.172539
$773$	20.2661	0.728921	0.364460	−	0.931219i	$-0.381254\pi$
$773$	20.2661	0.728921	0.364460	+	0.931219i	$0.381254\pi$
$774$	0	0
$775$	4.85338	0.174339
$776$	12.2510	0.439784
$777$	0	0
$778$	2.85555	0.102376
$779$	35.5465	1.27359
$780$	0	0
$781$	−7.88328	−0.282086
$782$	−6.78535	−0.242644
$783$	0	0
$784$	−23.9757	−0.856275
$785$	25.6685	0.916147
$786$	0	0
$787$	27.6985	0.987346	0.493673	−	0.869648i	$-0.335654\pi$
$787$	27.6985	0.987346	0.493673	+	0.869648i	$0.335654\pi$
$788$	−22.7423	−0.810161
$789$	0	0
$790$	−47.2677	−1.68171
$791$	−6.00395	−0.213476
$792$	0	0
$793$	1.38040	0.0490193
$794$	−20.9242	−0.742571
$795$	0	0
$796$	−20.7677	−0.736091
$797$	11.7709	0.416945	0.208473	−	0.978028i	$-0.433151\pi$
$797$	11.7709	0.416945	0.208473	+	0.978028i	$0.433151\pi$
$798$	0	0
$799$	5.15577	0.182398
$800$	−0.0989905	−0.00349984
$801$	0	0
$802$	83.9329	2.96378
$803$	0.611985	0.0215965
$804$	0	0
$805$	−5.96107	−0.210100
$806$	−11.6987	−0.412069
$807$	0	0
$808$	−69.6573	−2.45053
$809$	−0.583252	−0.0205061	−0.0102530	−	0.999947i	$-0.503264\pi$
$809$	−0.583252	−0.0205061	−0.0102530	+	0.999947i	$0.503264\pi$
$810$	0	0
$811$	10.2823	0.361062	0.180531	−	0.983569i	$-0.442218\pi$
$811$	10.2823	0.361062	0.180531	+	0.983569i	$0.442218\pi$
$812$	−27.7636	−0.974312
$813$	0	0
$814$	−12.9451	−0.453724
$815$	28.1897	0.987443
$816$	0	0
$817$	6.27710	0.219608
$818$	−45.3065	−1.58410
$819$	0	0
$820$	46.1558	1.61183
$821$	−22.6306	−0.789812	−0.394906	−	0.918722i	$-0.629223\pi$
$821$	−22.6306	−0.789812	−0.394906	+	0.918722i	$0.629223\pi$
$822$	0	0
$823$	14.1404	0.492905	0.246453	−	0.969155i	$-0.420735\pi$
$823$	14.1404	0.492905	0.246453	+	0.969155i	$0.420735\pi$
$824$	38.7134	1.34864
$825$	0	0
$826$	13.5878	0.472781
$827$	14.3417	0.498710	0.249355	−	0.968412i	$-0.419781\pi$
$827$	14.3417	0.498710	0.249355	+	0.968412i	$0.419781\pi$
$828$	0	0
$829$	37.1526	1.29036	0.645181	−	0.764030i	$-0.276782\pi$
$829$	37.1526	1.29036	0.645181	+	0.764030i	$0.276782\pi$
$830$	19.8980	0.690669
$831$	0	0
$832$	11.1623	0.386982
$833$	−5.18303	−0.179581
$834$	0	0
$835$	−26.1376	−0.904529
$836$	23.2893	0.805476
$837$	0	0
$838$	−49.0038	−1.69281
$839$	19.6392	0.678021	0.339011	−	0.940783i	$-0.389908\pi$
$839$	19.6392	0.678021	0.339011	+	0.940783i	$0.389908\pi$
$840$	0	0
$841$	22.4101	0.772762
$842$	−81.4199	−2.80592
$843$	0	0
$844$	−29.4745	−1.01455
$845$	−21.0447	−0.723960
$846$	0	0
$847$	−0.969789	−0.0333223
$848$	−8.16848	−0.280507
$849$	0	0
$850$	2.93552	0.100688
$851$	17.1357	0.587405
$852$	0	0
$853$	−24.3177	−0.832622	−0.416311	−	0.909222i	$-0.636677\pi$
$853$	−24.3177	−0.832622	−0.416311	+	0.909222i	$0.636677\pi$
$854$	−2.37406	−0.0812386
$855$	0	0
$856$	−59.7969	−2.04382
$857$	39.1805	1.33838	0.669190	−	0.743092i	$-0.266641\pi$
$857$	39.1805	1.33838	0.669190	+	0.743092i	$0.266641\pi$
$858$	0	0
$859$	−42.7150	−1.45742	−0.728709	−	0.684824i	$-0.759879\pi$
$859$	−42.7150	−1.45742	−0.728709	+	0.684824i	$0.759879\pi$
$860$	8.15058	0.277932
$861$	0	0
$862$	73.3878	2.49960
$863$	−3.67478	−0.125091	−0.0625454	−	0.998042i	$-0.519922\pi$
$863$	−3.67478	−0.125091	−0.0625454	+	0.998042i	$0.519922\pi$
$864$	0	0
$865$	23.6914	0.805532
$866$	28.0165	0.952039
$867$	0	0
$868$	13.4051	0.455000
$869$	10.1792	0.345307
$870$	0	0
$871$	15.6874	0.531548
$872$	−36.8271	−1.24712
$873$	0	0
$874$	−46.2708	−1.56513
$875$	11.7767	0.398124
$876$	0	0
$877$	−6.17385	−0.208476	−0.104238	−	0.994552i	$-0.533240\pi$
$877$	−6.17385	−0.208476	−0.104238	+	0.994552i	$0.533240\pi$
$878$	−18.4824	−0.623750
$879$	0	0
$880$	7.50532	0.253004
$881$	23.7564	0.800372	0.400186	−	0.916434i	$-0.368945\pi$
$881$	23.7564	0.800372	0.400186	+	0.916434i	$0.368945\pi$
$882$	0	0
$883$	−48.0631	−1.61745	−0.808727	−	0.588185i	$-0.799843\pi$
$883$	−48.0631	−1.61745	−0.808727	+	0.588185i	$0.799843\pi$
$884$	−4.71439	−0.158562
$885$	0	0
$886$	14.8926	0.500326
$887$	−57.4994	−1.93064	−0.965320	−	0.261070i	$-0.915925\pi$
$887$	−57.4994	−1.93064	−0.965320	+	0.261070i	$0.915925\pi$
$888$	0	0
$889$	17.2220	0.577608
$890$	65.0628	2.18091
$891$	0	0
$892$	28.0739	0.939985
$893$	35.1583	1.17653
$894$	0	0
$895$	36.0621	1.20542
$896$	−19.0603	−0.636760
$897$	0	0
$898$	−3.51910	−0.117434
$899$	−24.8224	−0.827873
$900$	0	0
$901$	−1.76585	−0.0588290
$902$	−14.9187	−0.496738
$903$	0	0
$904$	−30.2018	−1.00450
$905$	1.25255	0.0416361
$906$	0	0
$907$	−32.7430	−1.08721	−0.543607	−	0.839340i	$-0.682942\pi$
$907$	−32.7430	−1.08721	−0.543607	+	0.839340i	$0.682942\pi$
$908$	−77.4651	−2.57077
$909$	0	0
$910$	−6.21628	−0.206068
$911$	10.1712	0.336988	0.168494	−	0.985703i	$-0.446110\pi$
$911$	10.1712	0.336988	0.168494	+	0.985703i	$0.446110\pi$
$912$	0	0
$913$	−4.28509	−0.141816
$914$	75.7908	2.50694
$915$	0	0
$916$	36.8164	1.21645
$917$	−7.84708	−0.259133
$918$	0	0
$919$	−26.9136	−0.887799	−0.443900	−	0.896077i	$-0.646405\pi$
$919$	−26.9136	−0.887799	−0.443900	+	0.896077i	$0.646405\pi$
$920$	−29.9861	−0.988612
$921$	0	0
$922$	−59.3629	−1.95501
$923$	10.8821	0.358187
$924$	0	0
$925$	−7.41337	−0.243750
$926$	68.6086	2.25462
$927$	0	0
$928$	0.506282	0.0166195
$929$	−29.6568	−0.973007	−0.486504	−	0.873679i	$-0.661728\pi$
$929$	−29.6568	−0.973007	−0.486504	+	0.873679i	$0.661728\pi$
$930$	0	0
$931$	−35.3442	−1.15836
$932$	−80.6301	−2.64113
$933$	0	0
$934$	−12.6366	−0.413481
$935$	1.62249	0.0530610
$936$	0	0
$937$	−4.53343	−0.148101	−0.0740503	−	0.997255i	$-0.523593\pi$
$937$	−4.53343	−0.148101	−0.0740503	+	0.997255i	$0.523593\pi$
$938$	−26.9798	−0.880921
$939$	0	0
$940$	45.6517	1.48899
$941$	−12.7808	−0.416641	−0.208320	−	0.978061i	$-0.566800\pi$
$941$	−12.7808	−0.416641	−0.208320	+	0.978061i	$0.566800\pi$
$942$	0	0
$943$	19.7483	0.643092
$944$	22.6461	0.737067
$945$	0	0
$946$	−2.63446	−0.0856538
$947$	16.7702	0.544959	0.272480	−	0.962162i	$-0.412156\pi$
$947$	16.7702	0.544959	0.272480	+	0.962162i	$0.412156\pi$
$948$	0	0
$949$	−0.844782	−0.0274228
$950$	20.0180	0.649468
$951$	0	0
$952$	4.04665	0.131153
$953$	46.2301	1.49754	0.748769	−	0.662831i	$-0.230645\pi$
$953$	46.2301	1.49754	0.748769	+	0.662831i	$0.230645\pi$
$954$	0	0
$955$	43.8433	1.41874
$956$	−22.5087	−0.727985
$957$	0	0
$958$	−1.35179	−0.0436745
$959$	2.90197	0.0937095
$960$	0	0
$961$	−19.0150	−0.613387
$962$	17.8693	0.576130
$963$	0	0
$964$	68.4371	2.20421
$965$	2.27749	0.0733151
$966$	0	0
$967$	36.8990	1.18659	0.593296	−	0.804985i	$-0.297827\pi$
$967$	36.8990	1.18659	0.593296	+	0.804985i	$0.297827\pi$
$968$	−4.87834	−0.156796
$969$	0	0
$970$	11.6613	0.374422
$971$	4.65238	0.149302	0.0746510	−	0.997210i	$-0.476216\pi$
$971$	4.65238	0.149302	0.0746510	+	0.997210i	$0.476216\pi$
$972$	0	0
$973$	11.2273	0.359931
$974$	70.2694	2.25158
$975$	0	0
$976$	−3.95671	−0.126651
$977$	−25.0717	−0.802113	−0.401057	−	0.916053i	$-0.631357\pi$
$977$	−25.0717	−0.802113	−0.401057	+	0.916053i	$0.631357\pi$
$978$	0	0
$979$	−14.0115	−0.447808
$980$	−45.8931	−1.46600
$981$	0	0
$982$	−48.4332	−1.54556
$983$	−30.2899	−0.966098	−0.483049	−	0.875593i	$-0.660471\pi$
$983$	−30.2899	−0.966098	−0.483049	+	0.875593i	$0.660471\pi$
$984$	0	0
$985$	−10.8042	−0.344252
$986$	−15.0136	−0.478130
$987$	0	0
$988$	−32.1484	−1.02278
$989$	3.48731	0.110890
$990$	0	0
$991$	−29.9628	−0.951800	−0.475900	−	0.879499i	$-0.657878\pi$
$991$	−29.9628	−0.951800	−0.475900	+	0.879499i	$0.657878\pi$
$992$	−0.244448	−0.00776125
$993$	0	0
$994$	−18.7154	−0.593615
$995$	−9.86615	−0.312778
$996$	0	0
$997$	30.0514	0.951739	0.475869	−	0.879516i	$-0.342133\pi$
$997$	30.0514	0.951739	0.475869	+	0.879516i	$0.342133\pi$
$998$	−46.0543	−1.45782
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.2	yes	25
3.2	odd	2		6039.2.a.n.1.24	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.24	✓	25	3.2	odd	2
6039.2.a.o.1.2	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)

\(f(q)\)	\(=\)	\(q-2.44801 q^{2} +3.99278 q^{4} +1.89686 q^{5} -0.969789 q^{7} -4.87834 q^{8} +O(q^{10})\)	\(q-2.44801 q^{2} +3.99278 q^{4} +1.89686 q^{5} -0.969789 q^{7} -4.87834 q^{8} -4.64354 q^{10} +1.00000 q^{11} -1.38040 q^{13} +2.37406 q^{14} +3.95671 q^{16} +0.855355 q^{17} +5.83285 q^{19} +7.57373 q^{20} -2.44801 q^{22} +3.24050 q^{23} -1.40193 q^{25} +3.37923 q^{26} -3.87215 q^{28} +7.17008 q^{29} -3.46194 q^{31} +0.0706103 q^{32} -2.09392 q^{34} -1.83955 q^{35} +5.28798 q^{37} -14.2789 q^{38} -9.25353 q^{40} +6.09420 q^{41} +1.07616 q^{43} +3.99278 q^{44} -7.93279 q^{46} +6.02763 q^{47} -6.05951 q^{49} +3.43194 q^{50} -5.51162 q^{52} -2.06446 q^{53} +1.89686 q^{55} +4.73096 q^{56} -17.5525 q^{58} +5.72346 q^{59} -1.00000 q^{61} +8.47487 q^{62} -8.08627 q^{64} -2.61842 q^{65} -11.3644 q^{67} +3.41524 q^{68} +4.50325 q^{70} -7.88328 q^{71} +0.611985 q^{73} -12.9451 q^{74} +23.2893 q^{76} -0.969789 q^{77} +10.1792 q^{79} +7.50532 q^{80} -14.9187 q^{82} -4.28509 q^{83} +1.62249 q^{85} -2.63446 q^{86} -4.87834 q^{88} -14.0115 q^{89} +1.33869 q^{91} +12.9386 q^{92} -14.7557 q^{94} +11.0641 q^{95} -2.51129 q^{97} +14.8338 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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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