LMFDB - Embedded newform 6039.2.a.m.1.13

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			1.13
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)

$f(q)$	$=$	$q-0.536820 q^{2} -1.71182 q^{4} -1.82707 q^{5} +2.96010 q^{7} +1.99258 q^{8} +O(q^{10})$	\(q-0.536820 q^{2} -1.71182 q^{4} -1.82707 q^{5} +2.96010 q^{7} +1.99258 q^{8} +0.980806 q^{10} +1.00000 q^{11} +4.10754 q^{13} -1.58904 q^{14} +2.35399 q^{16} +0.658907 q^{17} -4.67764 q^{19} +3.12762 q^{20} -0.536820 q^{22} -1.60420 q^{23} -1.66183 q^{25} -2.20501 q^{26} -5.06718 q^{28} -6.71713 q^{29} -3.25036 q^{31} -5.24883 q^{32} -0.353714 q^{34} -5.40831 q^{35} +3.16378 q^{37} +2.51105 q^{38} -3.64058 q^{40} -9.51933 q^{41} +7.90466 q^{43} -1.71182 q^{44} +0.861168 q^{46} -3.05519 q^{47} +1.76221 q^{49} +0.892104 q^{50} -7.03138 q^{52} +6.17173 q^{53} -1.82707 q^{55} +5.89825 q^{56} +3.60589 q^{58} +4.74234 q^{59} +1.00000 q^{61} +1.74486 q^{62} -1.89030 q^{64} -7.50474 q^{65} -10.1663 q^{67} -1.12793 q^{68} +2.90329 q^{70} +3.32298 q^{71} +4.59025 q^{73} -1.69838 q^{74} +8.00729 q^{76} +2.96010 q^{77} +7.97046 q^{79} -4.30089 q^{80} +5.11017 q^{82} -10.7338 q^{83} -1.20387 q^{85} -4.24338 q^{86} +1.99258 q^{88} +6.38437 q^{89} +12.1587 q^{91} +2.74611 q^{92} +1.64009 q^{94} +8.54635 q^{95} +16.3258 q^{97} -0.945992 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 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 $

$2$	−0.536820	−0.379589	−0.189795	−	0.981824i	$-0.560782\pi$
$2$	−0.536820	−0.379589	−0.189795	+	0.981824i	$0.560782\pi$
$3$	0	0
$3$	0	0
$4$	−1.71182	−0.855912
$5$	−1.82707	−0.817089	−0.408544	−	0.912738i	$-0.633963\pi$
$5$	−1.82707	−0.817089	−0.408544	+	0.912738i	$0.633963\pi$
$6$	0	0
$7$	2.96010	1.11881	0.559407	−	0.828893i	$-0.311029\pi$
$7$	2.96010	1.11881	0.559407	+	0.828893i	$0.311029\pi$
$8$	1.99258	0.704484
$9$	0	0
$10$	0.980806	0.310158
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.10754	1.13923	0.569613	−	0.821913i	$-0.307093\pi$
$13$	4.10754	1.13923	0.569613	+	0.821913i	$0.307093\pi$
$14$	−1.58904	−0.424690
$15$	0	0
$16$	2.35399	0.588497
$17$	0.658907	0.159808	0.0799042	−	0.996803i	$-0.474539\pi$
$17$	0.658907	0.159808	0.0799042	+	0.996803i	$0.474539\pi$
$18$	0	0
$19$	−4.67764	−1.07312	−0.536562	−	0.843861i	$-0.680277\pi$
$19$	−4.67764	−1.07312	−0.536562	+	0.843861i	$0.680277\pi$
$20$	3.12762	0.699356
$21$	0	0
$22$	−0.536820	−0.114450
$23$	−1.60420	−0.334499	−0.167250	−	0.985915i	$-0.553489\pi$
$23$	−1.60420	−0.334499	−0.167250	+	0.985915i	$0.553489\pi$
$24$	0	0
$25$	−1.66183	−0.332366
$26$	−2.20501	−0.432438
$27$	0	0
$28$	−5.06718	−0.957606
$29$	−6.71713	−1.24734	−0.623670	−	0.781688i	$-0.714359\pi$
$29$	−6.71713	−1.24734	−0.623670	+	0.781688i	$0.714359\pi$
$30$	0	0
$31$	−3.25036	−0.583781	−0.291891	−	0.956452i	$-0.594284\pi$
$31$	−3.25036	−0.583781	−0.291891	+	0.956452i	$0.594284\pi$
$32$	−5.24883	−0.927872
$33$	0	0
$34$	−0.353714	−0.0606615
$35$	−5.40831	−0.914170
$36$	0	0
$37$	3.16378	0.520123	0.260062	−	0.965592i	$-0.416257\pi$
$37$	3.16378	0.520123	0.260062	+	0.965592i	$0.416257\pi$
$38$	2.51105	0.407346
$39$	0	0
$40$	−3.64058	−0.575626
$41$	−9.51933	−1.48667	−0.743335	−	0.668920i	$-0.766757\pi$
$41$	−9.51933	−1.48667	−0.743335	+	0.668920i	$0.766757\pi$
$42$	0	0
$43$	7.90466	1.20545	0.602725	−	0.797949i	$-0.294082\pi$
$43$	7.90466	1.20545	0.602725	+	0.797949i	$0.294082\pi$
$44$	−1.71182	−0.258067
$45$	0	0
$46$	0.861168	0.126972
$47$	−3.05519	−0.445645	−0.222823	−	0.974859i	$-0.571527\pi$
$47$	−3.05519	−0.445645	−0.222823	+	0.974859i	$0.571527\pi$
$48$	0	0
$49$	1.76221	0.251745
$50$	0.892104	0.126163
$51$	0	0
$52$	−7.03138	−0.975077
$53$	6.17173	0.847752	0.423876	−	0.905720i	$-0.360669\pi$
$53$	6.17173	0.847752	0.423876	+	0.905720i	$0.360669\pi$
$54$	0	0
$55$	−1.82707	−0.246362
$56$	5.89825	0.788187
$57$	0	0
$58$	3.60589	0.473477
$59$	4.74234	0.617400	0.308700	−	0.951159i	$-0.400106\pi$
$59$	4.74234	0.617400	0.308700	+	0.951159i	$0.400106\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	1.74486	0.221597
$63$	0	0
$64$	−1.89030	−0.236287
$65$	−7.50474	−0.930848
$66$	0	0
$67$	−10.1663	−1.24201	−0.621006	−	0.783805i	$-0.713276\pi$
$67$	−10.1663	−1.24201	−0.621006	+	0.783805i	$0.713276\pi$
$68$	−1.12793	−0.136782
$69$	0	0
$70$	2.90329	0.347009
$71$	3.32298	0.394365	0.197183	−	0.980367i	$-0.436821\pi$
$71$	3.32298	0.394365	0.197183	+	0.980367i	$0.436821\pi$
$72$	0	0
$73$	4.59025	0.537248	0.268624	−	0.963245i	$-0.413431\pi$
$73$	4.59025	0.537248	0.268624	+	0.963245i	$0.413431\pi$
$74$	−1.69838	−0.197433
$75$	0	0
$76$	8.00729	0.918499
$77$	2.96010	0.337335
$78$	0	0
$79$	7.97046	0.896747	0.448373	−	0.893846i	$-0.352003\pi$
$79$	7.97046	0.896747	0.448373	+	0.893846i	$0.352003\pi$
$80$	−4.30089	−0.480855
$81$	0	0
$82$	5.11017	0.564324
$83$	−10.7338	−1.17819	−0.589093	−	0.808065i	$-0.700515\pi$
$83$	−10.7338	−1.17819	−0.589093	+	0.808065i	$0.700515\pi$
$84$	0	0
$85$	−1.20387	−0.130578
$86$	−4.24338	−0.457576
$87$	0	0
$88$	1.99258	0.212410
$89$	6.38437	0.676742	0.338371	−	0.941013i	$-0.390124\pi$
$89$	6.38437	0.676742	0.338371	+	0.941013i	$0.390124\pi$
$90$	0	0
$91$	12.1587	1.27458
$92$	2.74611	0.286302
$93$	0	0
$94$	1.64009	0.169162
$95$	8.54635	0.876837
$96$	0	0
$97$	16.3258	1.65763	0.828816	−	0.559522i	$-0.189015\pi$
$97$	16.3258	1.65763	0.828816	+	0.559522i	$0.189015\pi$
$98$	−0.945992	−0.0955596
$99$	0	0
$100$	2.84476	0.284476
$101$	−9.93237	−0.988308	−0.494154	−	0.869374i	$-0.664522\pi$
$101$	−9.93237	−0.988308	−0.494154	+	0.869374i	$0.664522\pi$
$102$	0	0
$103$	−6.27839	−0.618628	−0.309314	−	0.950960i	$-0.600099\pi$
$103$	−6.27839	−0.618628	−0.309314	+	0.950960i	$0.600099\pi$
$104$	8.18460	0.802566
$105$	0	0
$106$	−3.31311	−0.321797
$107$	0.939378	0.0908131	0.0454066	−	0.998969i	$-0.485542\pi$
$107$	0.939378	0.0908131	0.0454066	+	0.998969i	$0.485542\pi$
$108$	0	0
$109$	5.15498	0.493757	0.246879	−	0.969046i	$-0.420595\pi$
$109$	5.15498	0.493757	0.246879	+	0.969046i	$0.420595\pi$
$110$	0.980806	0.0935162
$111$	0	0
$112$	6.96805	0.658419
$113$	−6.15127	−0.578662	−0.289331	−	0.957229i	$-0.593433\pi$
$113$	−6.15127	−0.578662	−0.289331	+	0.957229i	$0.593433\pi$
$114$	0	0
$115$	2.93098	0.273316
$116$	11.4985	1.06761
$117$	0	0
$118$	−2.54578	−0.234358
$119$	1.95043	0.178796
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.536820	−0.0486014
$123$	0	0
$124$	5.56404	0.499665
$125$	12.1716	1.08866
$126$	0	0
$127$	10.9037	0.967547	0.483774	−	0.875193i	$-0.339266\pi$
$127$	10.9037	0.967547	0.483774	+	0.875193i	$0.339266\pi$
$128$	11.5124	1.01756
$129$	0	0
$130$	4.02870	0.353340
$131$	6.64522	0.580596	0.290298	−	0.956936i	$-0.406246\pi$
$131$	6.64522	0.580596	0.290298	+	0.956936i	$0.406246\pi$
$132$	0	0
$133$	−13.8463	−1.20063
$134$	5.45748	0.471455
$135$	0	0
$136$	1.31293	0.112582
$137$	−16.2075	−1.38470	−0.692350	−	0.721562i	$-0.743424\pi$
$137$	−16.2075	−1.38470	−0.692350	+	0.721562i	$0.743424\pi$
$138$	0	0
$139$	−20.4228	−1.73224	−0.866120	−	0.499836i	$-0.833394\pi$
$139$	−20.4228	−1.73224	−0.866120	+	0.499836i	$0.833394\pi$
$140$	9.25807	0.782449
$141$	0	0
$142$	−1.78384	−0.149697
$143$	4.10754	0.343489
$144$	0	0
$145$	12.2726	1.01919
$146$	−2.46414	−0.203934
$147$	0	0
$148$	−5.41584	−0.445180
$149$	−13.6619	−1.11923	−0.559613	−	0.828754i	$-0.689050\pi$
$149$	−13.6619	−1.11923	−0.559613	+	0.828754i	$0.689050\pi$
$150$	0	0
$151$	0.844381	0.0687147	0.0343574	−	0.999410i	$-0.489062\pi$
$151$	0.844381	0.0687147	0.0343574	+	0.999410i	$0.489062\pi$
$152$	−9.32057	−0.755998
$153$	0	0
$154$	−1.58904	−0.128049
$155$	5.93861	0.477001
$156$	0	0
$157$	−24.1223	−1.92517	−0.962585	−	0.270979i	$-0.912653\pi$
$157$	−24.1223	−1.92517	−0.962585	+	0.270979i	$0.912653\pi$
$158$	−4.27870	−0.340395
$159$	0	0
$160$	9.58997	0.758153
$161$	−4.74861	−0.374243
$162$	0	0
$163$	−7.32068	−0.573400	−0.286700	−	0.958020i	$-0.592558\pi$
$163$	−7.32068	−0.573400	−0.286700	+	0.958020i	$0.592558\pi$
$164$	16.2954	1.27246
$165$	0	0
$166$	5.76211	0.447227
$167$	−7.32205	−0.566597	−0.283299	−	0.959032i	$-0.591429\pi$
$167$	−7.32205	−0.566597	−0.283299	+	0.959032i	$0.591429\pi$
$168$	0	0
$169$	3.87184	0.297834
$170$	0.646260	0.0495659
$171$	0	0
$172$	−13.5314	−1.03176
$173$	13.2768	1.00942	0.504709	−	0.863289i	$-0.331600\pi$
$173$	13.2768	1.00942	0.504709	+	0.863289i	$0.331600\pi$
$174$	0	0
$175$	−4.91919	−0.371856
$176$	2.35399	0.177439
$177$	0	0
$178$	−3.42726	−0.256884
$179$	4.61966	0.345290	0.172645	−	0.984984i	$-0.444769\pi$
$179$	4.61966	0.345290	0.172645	+	0.984984i	$0.444769\pi$
$180$	0	0
$181$	−12.1579	−0.903693	−0.451847	−	0.892096i	$-0.649235\pi$
$181$	−12.1579	−0.903693	−0.451847	+	0.892096i	$0.649235\pi$
$182$	−6.52705	−0.483817
$183$	0	0
$184$	−3.19651	−0.235650
$185$	−5.78044	−0.424987
$186$	0	0
$187$	0.658907	0.0481840
$188$	5.22995	0.381433
$189$	0	0
$190$	−4.58785	−0.332838
$191$	−24.0576	−1.74075	−0.870374	−	0.492392i	$-0.836123\pi$
$191$	−24.0576	−1.74075	−0.870374	+	0.492392i	$0.836123\pi$
$192$	0	0
$193$	−1.60630	−0.115624	−0.0578121	−	0.998327i	$-0.518412\pi$
$193$	−1.60630	−0.115624	−0.0578121	+	0.998327i	$0.518412\pi$
$194$	−8.76401	−0.629219
$195$	0	0
$196$	−3.01660	−0.215471
$197$	−17.1109	−1.21910	−0.609551	−	0.792746i	$-0.708650\pi$
$197$	−17.1109	−1.21910	−0.609551	+	0.792746i	$0.708650\pi$
$198$	0	0
$199$	13.9889	0.991645	0.495822	−	0.868424i	$-0.334867\pi$
$199$	13.9889	0.991645	0.495822	+	0.868424i	$0.334867\pi$
$200$	−3.31133	−0.234147
$201$	0	0
$202$	5.33190	0.375151
$203$	−19.8834	−1.39554
$204$	0	0
$205$	17.3924	1.21474
$206$	3.37036	0.234824
$207$	0	0
$208$	9.66909	0.670431
$209$	−4.67764	−0.323559
$210$	0	0
$211$	2.66352	0.183364	0.0916821	−	0.995788i	$-0.470776\pi$
$211$	2.66352	0.183364	0.0916821	+	0.995788i	$0.470776\pi$
$212$	−10.5649	−0.725601
$213$	0	0
$214$	−0.504277	−0.0344717
$215$	−14.4423	−0.984959
$216$	0	0
$217$	−9.62139	−0.653142
$218$	−2.76730	−0.187425
$219$	0	0
$220$	3.12762	0.210864
$221$	2.70648	0.182058
$222$	0	0
$223$	−1.81616	−0.121619	−0.0608096	−	0.998149i	$-0.519368\pi$
$223$	−1.81616	−0.121619	−0.0608096	+	0.998149i	$0.519368\pi$
$224$	−15.5371	−1.03812
$225$	0	0
$226$	3.30212	0.219654
$227$	2.64065	0.175266	0.0876331	−	0.996153i	$-0.472070\pi$
$227$	2.64065	0.175266	0.0876331	+	0.996153i	$0.472070\pi$
$228$	0	0
$229$	18.0352	1.19180	0.595899	−	0.803059i	$-0.296796\pi$
$229$	18.0352	1.19180	0.595899	+	0.803059i	$0.296796\pi$
$230$	−1.57341	−0.103748
$231$	0	0
$232$	−13.3844	−0.878731
$233$	−14.7825	−0.968433	−0.484217	−	0.874948i	$-0.660895\pi$
$233$	−14.7825	−0.968433	−0.484217	+	0.874948i	$0.660895\pi$
$234$	0	0
$235$	5.58204	0.364132
$236$	−8.11805	−0.528440
$237$	0	0
$238$	−1.04703	−0.0678690
$239$	4.75396	0.307508	0.153754	−	0.988109i	$-0.450864\pi$
$239$	4.75396	0.307508	0.153754	+	0.988109i	$0.450864\pi$
$240$	0	0
$241$	13.2189	0.851505	0.425753	−	0.904840i	$-0.360009\pi$
$241$	13.2189	0.851505	0.425753	+	0.904840i	$0.360009\pi$
$242$	−0.536820	−0.0345081
$243$	0	0
$244$	−1.71182	−0.109588
$245$	−3.21968	−0.205698
$246$	0	0
$247$	−19.2136	−1.22253
$248$	−6.47660	−0.411265
$249$	0	0
$250$	−6.53396	−0.413244
$251$	4.35690	0.275005	0.137503	−	0.990501i	$-0.456092\pi$
$251$	4.35690	0.275005	0.137503	+	0.990501i	$0.456092\pi$
$252$	0	0
$253$	−1.60420	−0.100855
$254$	−5.85333	−0.367270
$255$	0	0
$256$	−2.39950	−0.149969
$257$	−24.7380	−1.54311	−0.771557	−	0.636160i	$-0.780522\pi$
$257$	−24.7380	−1.54311	−0.771557	+	0.636160i	$0.780522\pi$
$258$	0	0
$259$	9.36513	0.581921
$260$	12.8468	0.796724
$261$	0	0
$262$	−3.56729	−0.220388
$263$	−10.8043	−0.666223	−0.333112	−	0.942887i	$-0.608099\pi$
$263$	−10.8043	−0.666223	−0.333112	+	0.942887i	$0.608099\pi$
$264$	0	0
$265$	−11.2762	−0.692688
$266$	7.43297	0.455744
$267$	0	0
$268$	17.4029	1.06305
$269$	−16.9256	−1.03197	−0.515986	−	0.856597i	$-0.672574\pi$
$269$	−16.9256	−1.03197	−0.515986	+	0.856597i	$0.672574\pi$
$270$	0	0
$271$	−11.7489	−0.713695	−0.356847	−	0.934163i	$-0.616148\pi$
$271$	−11.7489	−0.713695	−0.356847	+	0.934163i	$0.616148\pi$
$272$	1.55106	0.0940468
$273$	0	0
$274$	8.70051	0.525617
$275$	−1.66183	−0.100212
$276$	0	0
$277$	1.40038	0.0841406	0.0420703	−	0.999115i	$-0.486605\pi$
$277$	1.40038	0.0841406	0.0420703	+	0.999115i	$0.486605\pi$
$278$	10.9634	0.657540
$279$	0	0
$280$	−10.7765	−0.644019
$281$	32.3460	1.92960	0.964802	−	0.262979i	$-0.0847050\pi$
$281$	32.3460	1.92960	0.964802	+	0.262979i	$0.0847050\pi$
$282$	0	0
$283$	31.2262	1.85620	0.928102	−	0.372327i	$-0.121440\pi$
$283$	31.2262	1.85620	0.928102	+	0.372327i	$0.121440\pi$
$284$	−5.68835	−0.337542
$285$	0	0
$286$	−2.20501	−0.130385
$287$	−28.1782	−1.66331
$288$	0	0
$289$	−16.5658	−0.974461
$290$	−6.58820	−0.386872
$291$	0	0
$292$	−7.85770	−0.459837
$293$	−2.69227	−0.157284	−0.0786422	−	0.996903i	$-0.525058\pi$
$293$	−2.69227	−0.157284	−0.0786422	+	0.996903i	$0.525058\pi$
$294$	0	0
$295$	−8.66457	−0.504471
$296$	6.30410	0.366418
$297$	0	0
$298$	7.33398	0.424846
$299$	−6.58932	−0.381070
$300$	0	0
$301$	23.3986	1.34867
$302$	−0.453281	−0.0260834
$303$	0	0
$304$	−11.0111	−0.631530
$305$	−1.82707	−0.104618
$306$	0	0
$307$	0.697397	0.0398025	0.0199013	−	0.999802i	$-0.493665\pi$
$307$	0.697397	0.0398025	0.0199013	+	0.999802i	$0.493665\pi$
$308$	−5.06718	−0.288729
$309$	0	0
$310$	−3.18797	−0.181064
$311$	−16.7415	−0.949324	−0.474662	−	0.880168i	$-0.657430\pi$
$311$	−16.7415	−0.949324	−0.474662	+	0.880168i	$0.657430\pi$
$312$	0	0
$313$	11.8295	0.668643	0.334321	−	0.942459i	$-0.391493\pi$
$313$	11.8295	0.668643	0.334321	+	0.942459i	$0.391493\pi$
$314$	12.9493	0.730774
$315$	0	0
$316$	−13.6440	−0.767536
$317$	−32.2617	−1.81200	−0.905999	−	0.423279i	$-0.860879\pi$
$317$	−32.2617	−1.81200	−0.905999	+	0.423279i	$0.860879\pi$
$318$	0	0
$319$	−6.71713	−0.376087
$320$	3.45370	0.193068
$321$	0	0
$322$	2.54915	0.142058
$323$	−3.08213	−0.171494
$324$	0	0
$325$	−6.82602	−0.378640
$326$	3.92989	0.217656
$327$	0	0
$328$	−18.9680	−1.04733
$329$	−9.04368	−0.498594
$330$	0	0
$331$	7.54175	0.414532	0.207266	−	0.978285i	$-0.433543\pi$
$331$	7.54175	0.414532	0.207266	+	0.978285i	$0.433543\pi$
$332$	18.3744	1.00842
$333$	0	0
$334$	3.93062	0.215074
$335$	18.5745	1.01483
$336$	0	0
$337$	−1.21029	−0.0659287	−0.0329644	−	0.999457i	$-0.510495\pi$
$337$	−1.21029	−0.0659287	−0.0329644	+	0.999457i	$0.510495\pi$
$338$	−2.07848	−0.113055
$339$	0	0
$340$	2.06081	0.111763
$341$	−3.25036	−0.176017
$342$	0	0
$343$	−15.5044	−0.837158
$344$	15.7507	0.849220
$345$	0	0
$346$	−7.12727	−0.383164
$347$	−8.00583	−0.429775	−0.214888	−	0.976639i	$-0.568939\pi$
$347$	−8.00583	−0.429775	−0.214888	+	0.976639i	$0.568939\pi$
$348$	0	0
$349$	−8.03771	−0.430249	−0.215124	−	0.976587i	$-0.569016\pi$
$349$	−8.03771	−0.430249	−0.215124	+	0.976587i	$0.569016\pi$
$350$	2.64072	0.141152
$351$	0	0
$352$	−5.24883	−0.279764
$353$	−12.1828	−0.648423	−0.324211	−	0.945985i	$-0.605099\pi$
$353$	−12.1828	−0.648423	−0.324211	+	0.945985i	$0.605099\pi$
$354$	0	0
$355$	−6.07130	−0.322231
$356$	−10.9289	−0.579231
$357$	0	0
$358$	−2.47993	−0.131068
$359$	−34.8027	−1.83681	−0.918407	−	0.395636i	$-0.870524\pi$
$359$	−34.8027	−1.83681	−0.918407	+	0.395636i	$0.870524\pi$
$360$	0	0
$361$	2.88028	0.151593
$362$	6.52663	0.343032
$363$	0	0
$364$	−20.8136	−1.09093
$365$	−8.38669	−0.438980
$366$	0	0
$367$	−19.7187	−1.02931	−0.514653	−	0.857399i	$-0.672079\pi$
$367$	−19.7187	−1.02931	−0.514653	+	0.857399i	$0.672079\pi$
$368$	−3.77628	−0.196852
$369$	0	0
$370$	3.10306	0.161320
$371$	18.2689	0.948477
$372$	0	0
$373$	32.1156	1.66289	0.831443	−	0.555611i	$-0.187515\pi$
$373$	32.1156	1.66289	0.831443	+	0.555611i	$0.187515\pi$
$374$	−0.353714	−0.0182901
$375$	0	0
$376$	−6.08772	−0.313950
$377$	−27.5908	−1.42100
$378$	0	0
$379$	0.494227	0.0253868	0.0126934	−	0.999919i	$-0.495959\pi$
$379$	0.494227	0.0253868	0.0126934	+	0.999919i	$0.495959\pi$
$380$	−14.6298	−0.750495
$381$	0	0
$382$	12.9146	0.660769
$383$	12.0656	0.616525	0.308262	−	0.951301i	$-0.400253\pi$
$383$	12.0656	0.616525	0.308262	+	0.951301i	$0.400253\pi$
$384$	0	0
$385$	−5.40831	−0.275633
$386$	0.862296	0.0438897
$387$	0	0
$388$	−27.9469	−1.41879
$389$	−9.69134	−0.491370	−0.245685	−	0.969350i	$-0.579013\pi$
$389$	−9.69134	−0.491370	−0.245685	+	0.969350i	$0.579013\pi$
$390$	0	0
$391$	−1.05702	−0.0534558
$392$	3.51136	0.177350
$393$	0	0
$394$	9.18549	0.462758
$395$	−14.5626	−0.732722
$396$	0	0
$397$	−22.1784	−1.11310	−0.556550	−	0.830814i	$-0.687875\pi$
$397$	−22.1784	−1.11310	−0.556550	+	0.830814i	$0.687875\pi$
$398$	−7.50951	−0.376418
$399$	0	0
$400$	−3.91193	−0.195596
$401$	−18.4895	−0.923320	−0.461660	−	0.887057i	$-0.652746\pi$
$401$	−18.4895	−0.923320	−0.461660	+	0.887057i	$0.652746\pi$
$402$	0	0
$403$	−13.3509	−0.665058
$404$	17.0025	0.845905
$405$	0	0
$406$	10.6738	0.529732
$407$	3.16378	0.156823
$408$	0	0
$409$	1.87986	0.0929531	0.0464765	−	0.998919i	$-0.485201\pi$
$409$	1.87986	0.0929531	0.0464765	+	0.998919i	$0.485201\pi$
$410$	−9.33661	−0.461103
$411$	0	0
$412$	10.7475	0.529491
$413$	14.0378	0.690756
$414$	0	0
$415$	19.6113	0.962683
$416$	−21.5598	−1.05705
$417$	0	0
$418$	2.51105	0.122819
$419$	9.60282	0.469128	0.234564	−	0.972101i	$-0.424634\pi$
$419$	9.60282	0.469128	0.234564	+	0.972101i	$0.424634\pi$
$420$	0	0
$421$	17.3356	0.844884	0.422442	−	0.906390i	$-0.361173\pi$
$421$	17.3356	0.844884	0.422442	+	0.906390i	$0.361173\pi$
$422$	−1.42983	−0.0696031
$423$	0	0
$424$	12.2977	0.597228
$425$	−1.09499	−0.0531148
$426$	0	0
$427$	2.96010	0.143249
$428$	−1.60805	−0.0777280
$429$	0	0
$430$	7.75294	0.373880
$431$	−21.8464	−1.05231	−0.526153	−	0.850390i	$-0.676366\pi$
$431$	−21.8464	−1.05231	−0.526153	+	0.850390i	$0.676366\pi$
$432$	0	0
$433$	25.7062	1.23536	0.617680	−	0.786429i	$-0.288073\pi$
$433$	25.7062	1.23536	0.617680	+	0.786429i	$0.288073\pi$
$434$	5.16496	0.247926
$435$	0	0
$436$	−8.82442	−0.422613
$437$	7.50388	0.358959
$438$	0	0
$439$	0.504684	0.0240872	0.0120436	−	0.999927i	$-0.496166\pi$
$439$	0.504684	0.0240872	0.0120436	+	0.999927i	$0.496166\pi$
$440$	−3.64058	−0.173558
$441$	0	0
$442$	−1.45289	−0.0691072
$443$	5.62560	0.267280	0.133640	−	0.991030i	$-0.457333\pi$
$443$	5.62560	0.267280	0.133640	+	0.991030i	$0.457333\pi$
$444$	0	0
$445$	−11.6647	−0.552958
$446$	0.974953	0.0461654
$447$	0	0
$448$	−5.59548	−0.264362
$449$	29.3841	1.38672	0.693361	−	0.720591i	$-0.256129\pi$
$449$	29.3841	1.38672	0.693361	+	0.720591i	$0.256129\pi$
$450$	0	0
$451$	−9.51933	−0.448248
$452$	10.5299	0.495284
$453$	0	0
$454$	−1.41756	−0.0665292
$455$	−22.2148	−1.04145
$456$	0	0
$457$	−21.5675	−1.00889	−0.504443	−	0.863445i	$-0.668302\pi$
$457$	−21.5675	−1.00889	−0.504443	+	0.863445i	$0.668302\pi$
$458$	−9.68165	−0.452394
$459$	0	0
$460$	−5.01733	−0.233934
$461$	13.0759	0.609007	0.304503	−	0.952511i	$-0.401509\pi$
$461$	13.0759	0.609007	0.304503	+	0.952511i	$0.401509\pi$
$462$	0	0
$463$	−17.4764	−0.812199	−0.406100	−	0.913829i	$-0.633111\pi$
$463$	−17.4764	−0.812199	−0.406100	+	0.913829i	$0.633111\pi$
$464$	−15.8120	−0.734056
$465$	0	0
$466$	7.93554	0.367607
$467$	1.67808	0.0776524	0.0388262	−	0.999246i	$-0.487638\pi$
$467$	1.67808	0.0776524	0.0388262	+	0.999246i	$0.487638\pi$
$468$	0	0
$469$	−30.0933	−1.38958
$470$	−2.99655	−0.138221
$471$	0	0
$472$	9.44950	0.434949
$473$	7.90466	0.363457
$474$	0	0
$475$	7.77343	0.356670
$476$	−3.33880	−0.153034
$477$	0	0
$478$	−2.55202	−0.116727
$479$	30.7304	1.40411	0.702054	−	0.712124i	$-0.252267\pi$
$479$	30.7304	1.40411	0.702054	+	0.712124i	$0.252267\pi$
$480$	0	0
$481$	12.9954	0.592537
$482$	−7.09618	−0.323222
$483$	0	0
$484$	−1.71182	−0.0778102
$485$	−29.8283	−1.35443
$486$	0	0
$487$	−38.0671	−1.72498	−0.862492	−	0.506071i	$-0.831097\pi$
$487$	−38.0671	−1.72498	−0.862492	+	0.506071i	$0.831097\pi$
$488$	1.99258	0.0902000
$489$	0	0
$490$	1.72839	0.0780807
$491$	−29.7719	−1.34359	−0.671793	−	0.740739i	$-0.734476\pi$
$491$	−29.7719	−1.34359	−0.671793	+	0.740739i	$0.734476\pi$
$492$	0	0
$493$	−4.42596	−0.199335
$494$	10.3142	0.464059
$495$	0	0
$496$	−7.65130	−0.343554
$497$	9.83636	0.441221
$498$	0	0
$499$	−27.7321	−1.24146	−0.620729	−	0.784025i	$-0.713163\pi$
$499$	−27.7321	−1.24146	−0.620729	+	0.784025i	$0.713163\pi$
$500$	−20.8356	−0.931798
$501$	0	0
$502$	−2.33887	−0.104389
$503$	6.89167	0.307284	0.153642	−	0.988127i	$-0.450900\pi$
$503$	6.89167	0.307284	0.153642	+	0.988127i	$0.450900\pi$
$504$	0	0
$505$	18.1471	0.807535
$506$	0.861168	0.0382836
$507$	0	0
$508$	−18.6652	−0.828135
$509$	25.8868	1.14741	0.573706	−	0.819061i	$-0.305505\pi$
$509$	25.8868	1.14741	0.573706	+	0.819061i	$0.305505\pi$
$510$	0	0
$511$	13.5876	0.601081
$512$	−21.7367	−0.960637
$513$	0	0
$514$	13.2799	0.585749
$515$	11.4710	0.505474
$516$	0	0
$517$	−3.05519	−0.134367
$518$	−5.02739	−0.220891
$519$	0	0
$520$	−14.9538	−0.655768
$521$	15.7926	0.691886	0.345943	−	0.938255i	$-0.387559\pi$
$521$	15.7926	0.691886	0.345943	+	0.938255i	$0.387559\pi$
$522$	0	0
$523$	−16.7661	−0.733131	−0.366565	−	0.930392i	$-0.619466\pi$
$523$	−16.7661	−0.733131	−0.366565	+	0.930392i	$0.619466\pi$
$524$	−11.3755	−0.496939
$525$	0	0
$526$	5.79998	0.252891
$527$	−2.14168	−0.0932931
$528$	0	0
$529$	−20.4265	−0.888110
$530$	6.05327	0.262937
$531$	0	0
$532$	23.7024	1.02763
$533$	−39.1010	−1.69365
$534$	0	0
$535$	−1.71631	−0.0742024
$536$	−20.2572	−0.874979
$537$	0	0
$538$	9.08600	0.391725
$539$	1.76221	0.0759039
$540$	0	0
$541$	23.8101	1.02367	0.511837	−	0.859083i	$-0.328965\pi$
$541$	23.8101	1.02367	0.511837	+	0.859083i	$0.328965\pi$
$542$	6.30705	0.270911
$543$	0	0
$544$	−3.45849	−0.148282
$545$	−9.41849	−0.403444
$546$	0	0
$547$	39.1878	1.67555	0.837774	−	0.546018i	$-0.183857\pi$
$547$	39.1878	1.67555	0.837774	+	0.546018i	$0.183857\pi$
$548$	27.7444	1.18518
$549$	0	0
$550$	0.892104	0.0380394
$551$	31.4203	1.33855
$552$	0	0
$553$	23.5934	1.00329
$554$	−0.751751	−0.0319389
$555$	0	0
$556$	34.9603	1.48265
$557$	−14.1224	−0.598387	−0.299193	−	0.954193i	$-0.596718\pi$
$557$	−14.1224	−0.598387	−0.299193	+	0.954193i	$0.596718\pi$
$558$	0	0
$559$	32.4687	1.37328
$560$	−12.7311	−0.537987
$561$	0	0
$562$	−17.3640	−0.732457
$563$	3.46110	0.145868	0.0729341	−	0.997337i	$-0.476764\pi$
$563$	3.46110	0.145868	0.0729341	+	0.997337i	$0.476764\pi$
$564$	0	0
$565$	11.2388	0.472819
$566$	−16.7628	−0.704595
$567$	0	0
$568$	6.62131	0.277824
$569$	41.8059	1.75259	0.876297	−	0.481771i	$-0.160006\pi$
$569$	41.8059	1.75259	0.876297	+	0.481771i	$0.160006\pi$
$570$	0	0
$571$	−35.5434	−1.48745	−0.743723	−	0.668488i	$-0.766942\pi$
$571$	−35.5434	−1.48745	−0.743723	+	0.668488i	$0.766942\pi$
$572$	−7.03138	−0.293997
$573$	0	0
$574$	15.1266	0.631373
$575$	2.66591	0.111176
$576$	0	0
$577$	−35.5075	−1.47820	−0.739098	−	0.673597i	$-0.764748\pi$
$577$	−35.5075	−1.47820	−0.739098	+	0.673597i	$0.764748\pi$
$578$	8.89288	0.369895
$579$	0	0
$580$	−21.0086	−0.872334
$581$	−31.7731	−1.31817
$582$	0	0
$583$	6.17173	0.255607
$584$	9.14645	0.378483
$585$	0	0
$586$	1.44527	0.0597035
$587$	−3.20023	−0.132087	−0.0660437	−	0.997817i	$-0.521038\pi$
$587$	−3.20023	−0.132087	−0.0660437	+	0.997817i	$0.521038\pi$
$588$	0	0
$589$	15.2040	0.626469
$590$	4.65132	0.191492
$591$	0	0
$592$	7.44752	0.306091
$593$	−5.07358	−0.208347	−0.104174	−	0.994559i	$-0.533220\pi$
$593$	−5.07358	−0.208347	−0.104174	+	0.994559i	$0.533220\pi$
$594$	0	0
$595$	−3.56357	−0.146092
$596$	23.3868	0.957959
$597$	0	0
$598$	3.53728	0.144650
$599$	37.0798	1.51504	0.757519	−	0.652813i	$-0.226411\pi$
$599$	37.0798	1.51504	0.757519	+	0.652813i	$0.226411\pi$
$600$	0	0
$601$	−16.4700	−0.671825	−0.335912	−	0.941893i	$-0.609045\pi$
$601$	−16.4700	−0.671825	−0.335912	+	0.941893i	$0.609045\pi$
$602$	−12.5608	−0.511942
$603$	0	0
$604$	−1.44543	−0.0588138
$605$	−1.82707	−0.0742808
$606$	0	0
$607$	39.2240	1.59205	0.796027	−	0.605261i	$-0.206931\pi$
$607$	39.2240	1.59205	0.796027	+	0.605261i	$0.206931\pi$
$608$	24.5521	0.995720
$609$	0	0
$610$	0.980806	0.0397117
$611$	−12.5493	−0.507690
$612$	0	0
$613$	−10.9678	−0.442987	−0.221494	−	0.975162i	$-0.571093\pi$
$613$	−10.9678	−0.442987	−0.221494	+	0.975162i	$0.571093\pi$
$614$	−0.374377	−0.0151086
$615$	0	0
$616$	5.89825	0.237647
$617$	31.2353	1.25748	0.628742	−	0.777614i	$-0.283570\pi$
$617$	31.2353	1.25748	0.628742	+	0.777614i	$0.283570\pi$
$618$	0	0
$619$	−20.3446	−0.817718	−0.408859	−	0.912597i	$-0.634073\pi$
$619$	−20.3446	−0.817718	−0.408859	+	0.912597i	$0.634073\pi$
$620$	−10.1659	−0.408271
$621$	0	0
$622$	8.98718	0.360353
$623$	18.8984	0.757148
$624$	0	0
$625$	−13.9292	−0.557167
$626$	−6.35031	−0.253810
$627$	0	0
$628$	41.2932	1.64778
$629$	2.08464	0.0831200
$630$	0	0
$631$	19.7537	0.786381	0.393190	−	0.919457i	$-0.371371\pi$
$631$	19.7537	0.786381	0.393190	+	0.919457i	$0.371371\pi$
$632$	15.8818	0.631744
$633$	0	0
$634$	17.3187	0.687815
$635$	−19.9218	−0.790572
$636$	0	0
$637$	7.23835	0.286794
$638$	3.60589	0.142759
$639$	0	0
$640$	−21.0339	−0.831440
$641$	−23.4802	−0.927412	−0.463706	−	0.885989i	$-0.653481\pi$
$641$	−23.4802	−0.927412	−0.463706	+	0.885989i	$0.653481\pi$
$642$	0	0
$643$	1.39790	0.0551278	0.0275639	−	0.999620i	$-0.491225\pi$
$643$	1.39790	0.0551278	0.0275639	+	0.999620i	$0.491225\pi$
$644$	8.12878	0.320319
$645$	0	0
$646$	1.65455	0.0650973
$647$	−22.2494	−0.874715	−0.437358	−	0.899288i	$-0.644086\pi$
$647$	−22.2494	−0.874715	−0.437358	+	0.899288i	$0.644086\pi$
$648$	0	0
$649$	4.74234	0.186153
$650$	3.66435	0.143728
$651$	0	0
$652$	12.5317	0.490780
$653$	−45.9766	−1.79920	−0.899602	−	0.436711i	$-0.856143\pi$
$653$	−45.9766	−1.79920	−0.899602	+	0.436711i	$0.856143\pi$
$654$	0	0
$655$	−12.1413	−0.474398
$656$	−22.4084	−0.874901
$657$	0	0
$658$	4.85483	0.189261
$659$	19.9901	0.778705	0.389353	−	0.921089i	$-0.372699\pi$
$659$	19.9901	0.778705	0.389353	+	0.921089i	$0.372699\pi$
$660$	0	0
$661$	2.35196	0.0914808	0.0457404	−	0.998953i	$-0.485435\pi$
$661$	2.35196	0.0914808	0.0457404	+	0.998953i	$0.485435\pi$
$662$	−4.04856	−0.157352
$663$	0	0
$664$	−21.3880	−0.830014
$665$	25.2981	0.981018
$666$	0	0
$667$	10.7756	0.417234
$668$	12.5341	0.484957
$669$	0	0
$670$	−9.97118	−0.385220
$671$	1.00000	0.0386046
$672$	0	0
$673$	−1.24577	−0.0480210	−0.0240105	−	0.999712i	$-0.507644\pi$
$673$	−1.24577	−0.0480210	−0.0240105	+	0.999712i	$0.507644\pi$
$674$	0.649708	0.0250258
$675$	0	0
$676$	−6.62792	−0.254920
$677$	4.04512	0.155467	0.0777333	−	0.996974i	$-0.475232\pi$
$677$	4.04512	0.155467	0.0777333	+	0.996974i	$0.475232\pi$
$678$	0	0
$679$	48.3260	1.85458
$680$	−2.39880	−0.0919899
$681$	0	0
$682$	1.74486	0.0668140
$683$	−37.1501	−1.42151	−0.710755	−	0.703440i	$-0.751646\pi$
$683$	−37.1501	−1.42151	−0.710755	+	0.703440i	$0.751646\pi$
$684$	0	0
$685$	29.6122	1.13142
$686$	8.32307	0.317776
$687$	0	0
$688$	18.6075	0.709404
$689$	25.3506	0.965780
$690$	0	0
$691$	−0.675687	−0.0257044	−0.0128522	−	0.999917i	$-0.504091\pi$
$691$	−0.675687	−0.0257044	−0.0128522	+	0.999917i	$0.504091\pi$
$692$	−22.7276	−0.863973
$693$	0	0
$694$	4.29769	0.163138
$695$	37.3138	1.41539
$696$	0	0
$697$	−6.27235	−0.237582
$698$	4.31481	0.163318
$699$	0	0
$700$	8.42078	0.318276
$701$	39.7941	1.50300	0.751502	−	0.659731i	$-0.229330\pi$
$701$	39.7941	1.50300	0.751502	+	0.659731i	$0.229330\pi$
$702$	0	0
$703$	−14.7990	−0.558156
$704$	−1.89030	−0.0712433
$705$	0	0
$706$	6.53995	0.246134
$707$	−29.4009	−1.10573
$708$	0	0
$709$	−37.3479	−1.40263	−0.701315	−	0.712851i	$-0.747404\pi$
$709$	−37.3479	−1.40263	−0.701315	+	0.712851i	$0.747404\pi$
$710$	3.25920	0.122316
$711$	0	0
$712$	12.7214	0.476754
$713$	5.21423	0.195274
$714$	0	0
$715$	−7.50474	−0.280661
$716$	−7.90805	−0.295538
$717$	0	0
$718$	18.6828	0.697235
$719$	4.88561	0.182202	0.0911012	−	0.995842i	$-0.470961\pi$
$719$	4.88561	0.182202	0.0911012	+	0.995842i	$0.470961\pi$
$720$	0	0
$721$	−18.5847	−0.692129
$722$	−1.54619	−0.0575433
$723$	0	0
$724$	20.8123	0.773482
$725$	11.1627	0.414573
$726$	0	0
$727$	−1.91715	−0.0711033	−0.0355516	−	0.999368i	$-0.511319\pi$
$727$	−1.91715	−0.0711033	−0.0355516	+	0.999368i	$0.511319\pi$
$728$	24.2273	0.897922
$729$	0	0
$730$	4.50215	0.166632
$731$	5.20843	0.192641
$732$	0	0
$733$	18.0332	0.666071	0.333036	−	0.942914i	$-0.391927\pi$
$733$	18.0332	0.666071	0.333036	+	0.942914i	$0.391927\pi$
$734$	10.5854	0.390713
$735$	0	0
$736$	8.42019	0.310372
$737$	−10.1663	−0.374481
$738$	0	0
$739$	4.76304	0.175211	0.0876057	−	0.996155i	$-0.472078\pi$
$739$	4.76304	0.175211	0.0876057	+	0.996155i	$0.472078\pi$
$740$	9.89510	0.363751
$741$	0	0
$742$	−9.80714	−0.360031
$743$	−36.9064	−1.35396	−0.676982	−	0.736000i	$-0.736713\pi$
$743$	−36.9064	−1.35396	−0.676982	+	0.736000i	$0.736713\pi$
$744$	0	0
$745$	24.9612	0.914507
$746$	−17.2403	−0.631213
$747$	0	0
$748$	−1.12793	−0.0412413
$749$	2.78066	0.101603
$750$	0	0
$751$	−28.2591	−1.03119	−0.515594	−	0.856833i	$-0.672429\pi$
$751$	−28.2591	−1.03119	−0.515594	+	0.856833i	$0.672429\pi$
$752$	−7.19189	−0.262261
$753$	0	0
$754$	14.8113	0.539396
$755$	−1.54274	−0.0561460
$756$	0	0
$757$	−2.90282	−0.105505	−0.0527523	−	0.998608i	$-0.516799\pi$
$757$	−2.90282	−0.105505	−0.0527523	+	0.998608i	$0.516799\pi$
$758$	−0.265311	−0.00963654
$759$	0	0
$760$	17.0293	0.617718
$761$	−24.0628	−0.872276	−0.436138	−	0.899880i	$-0.643654\pi$
$761$	−24.0628	−0.872276	−0.436138	+	0.899880i	$0.643654\pi$
$762$	0	0
$763$	15.2593	0.552423
$764$	41.1824	1.48993
$765$	0	0
$766$	−6.47707	−0.234026
$767$	19.4793	0.703358
$768$	0	0
$769$	−31.6169	−1.14013	−0.570067	−	0.821598i	$-0.693083\pi$
$769$	−31.6169	−1.14013	−0.570067	+	0.821598i	$0.693083\pi$
$770$	2.90329	0.104627
$771$	0	0
$772$	2.74971	0.0989642
$773$	−13.9588	−0.502064	−0.251032	−	0.967979i	$-0.580770\pi$
$773$	−13.9588	−0.502064	−0.251032	+	0.967979i	$0.580770\pi$
$774$	0	0
$775$	5.40154	0.194029
$776$	32.5305	1.16778
$777$	0	0
$778$	5.20251	0.186519
$779$	44.5279	1.59538
$780$	0	0
$781$	3.32298	0.118906
$782$	0.567430	0.0202912
$783$	0	0
$784$	4.14823	0.148151
$785$	44.0731	1.57304
$786$	0	0
$787$	−40.5357	−1.44494	−0.722470	−	0.691402i	$-0.756993\pi$
$787$	−40.5357	−1.44494	−0.722470	+	0.691402i	$0.756993\pi$
$788$	29.2909	1.04344
$789$	0	0
$790$	7.81748	0.278133
$791$	−18.2084	−0.647416
$792$	0	0
$793$	4.10754	0.145863
$794$	11.9058	0.422521
$795$	0	0
$796$	−23.9465	−0.848761
$797$	−21.5848	−0.764571	−0.382286	−	0.924044i	$-0.624863\pi$
$797$	−21.5848	−0.764571	−0.382286	+	0.924044i	$0.624863\pi$
$798$	0	0
$799$	−2.01309	−0.0712179
$800$	8.72267	0.308393
$801$	0	0
$802$	9.92553	0.350483
$803$	4.59025	0.161986
$804$	0	0
$805$	8.67602	0.305789
$806$	7.16706	0.252449
$807$	0	0
$808$	−19.7911	−0.696247
$809$	2.71633	0.0955010	0.0477505	−	0.998859i	$-0.484795\pi$
$809$	2.71633	0.0955010	0.0477505	+	0.998859i	$0.484795\pi$
$810$	0	0
$811$	−40.3769	−1.41782	−0.708912	−	0.705297i	$-0.750814\pi$
$811$	−40.3769	−1.41782	−0.708912	+	0.705297i	$0.750814\pi$
$812$	34.0369	1.19446
$813$	0	0
$814$	−1.69838	−0.0595283
$815$	13.3754	0.468519
$816$	0	0
$817$	−36.9751	−1.29360
$818$	−1.00915	−0.0352840
$819$	0	0
$820$	−29.7728	−1.03971
$821$	−23.0630	−0.804904	−0.402452	−	0.915441i	$-0.631842\pi$
$821$	−23.0630	−0.804904	−0.402452	+	0.915441i	$0.631842\pi$
$822$	0	0
$823$	−11.4394	−0.398754	−0.199377	−	0.979923i	$-0.563892\pi$
$823$	−11.4394	−0.398754	−0.199377	+	0.979923i	$0.563892\pi$
$824$	−12.5102	−0.435814
$825$	0	0
$826$	−7.53579	−0.262203
$827$	−25.5440	−0.888251	−0.444125	−	0.895965i	$-0.646486\pi$
$827$	−25.5440	−0.888251	−0.444125	+	0.895965i	$0.646486\pi$
$828$	0	0
$829$	3.83211	0.133095	0.0665474	−	0.997783i	$-0.478802\pi$
$829$	3.83211	0.133095	0.0665474	+	0.997783i	$0.478802\pi$
$830$	−10.5278	−0.365424
$831$	0	0
$832$	−7.76447	−0.269184
$833$	1.16113	0.0402309
$834$	0	0
$835$	13.3779	0.462960
$836$	8.00729	0.276938
$837$	0	0
$838$	−5.15499	−0.178076
$839$	−35.8072	−1.23620	−0.618101	−	0.786098i	$-0.712098\pi$
$839$	−35.8072	−1.23620	−0.618101	+	0.786098i	$0.712098\pi$
$840$	0	0
$841$	16.1198	0.555855
$842$	−9.30609	−0.320709
$843$	0	0
$844$	−4.55947	−0.156944
$845$	−7.07412	−0.243357
$846$	0	0
$847$	2.96010	0.101710
$848$	14.5282	0.498900
$849$	0	0
$850$	0.587813	0.0201618
$851$	−5.07535	−0.173981
$852$	0	0
$853$	−22.2174	−0.760708	−0.380354	−	0.924841i	$-0.624198\pi$
$853$	−22.2174	−0.760708	−0.380354	+	0.924841i	$0.624198\pi$
$854$	−1.58904	−0.0543760
$855$	0	0
$856$	1.87179	0.0639764
$857$	10.9067	0.372564	0.186282	−	0.982496i	$-0.440356\pi$
$857$	10.9067	0.372564	0.186282	+	0.982496i	$0.440356\pi$
$858$	0	0
$859$	−16.5394	−0.564317	−0.282158	−	0.959368i	$-0.591050\pi$
$859$	−16.5394	−0.564317	−0.282158	+	0.959368i	$0.591050\pi$
$860$	24.7227	0.843038
$861$	0	0
$862$	11.7276	0.399444
$863$	43.9687	1.49671	0.748356	−	0.663297i	$-0.230843\pi$
$863$	43.9687	1.49671	0.748356	+	0.663297i	$0.230843\pi$
$864$	0	0
$865$	−24.2576	−0.824785
$866$	−13.7996	−0.468930
$867$	0	0
$868$	16.4701	0.559032
$869$	7.97046	0.270379
$870$	0	0
$871$	−41.7585	−1.41493
$872$	10.2717	0.347844
$873$	0	0
$874$	−4.02823	−0.136257
$875$	36.0292	1.21801
$876$	0	0
$877$	44.9662	1.51840	0.759200	−	0.650857i	$-0.225590\pi$
$877$	44.9662	1.51840	0.759200	+	0.650857i	$0.225590\pi$
$878$	−0.270924	−0.00914326
$879$	0	0
$880$	−4.30089	−0.144983
$881$	25.7353	0.867044	0.433522	−	0.901143i	$-0.357271\pi$
$881$	25.7353	0.867044	0.433522	+	0.901143i	$0.357271\pi$
$882$	0	0
$883$	−8.61567	−0.289940	−0.144970	−	0.989436i	$-0.546309\pi$
$883$	−8.61567	−0.289940	−0.144970	+	0.989436i	$0.546309\pi$
$884$	−4.63302	−0.155825
$885$	0	0
$886$	−3.01994	−0.101457
$887$	−33.1107	−1.11175	−0.555874	−	0.831266i	$-0.687616\pi$
$887$	−33.1107	−1.11175	−0.555874	+	0.831266i	$0.687616\pi$
$888$	0	0
$889$	32.2761	1.08251
$890$	6.26183	0.209897
$891$	0	0
$892$	3.10895	0.104095
$893$	14.2911	0.478232
$894$	0	0
$895$	−8.44043	−0.282132
$896$	34.0780	1.13846
$897$	0	0
$898$	−15.7740	−0.526384
$899$	21.8331	0.728173
$900$	0	0
$901$	4.06659	0.135478
$902$	5.11017	0.170150
$903$	0	0
$904$	−12.2569	−0.407659
$905$	22.2134	0.738397
$906$	0	0
$907$	1.90022	0.0630959	0.0315480	−	0.999502i	$-0.489956\pi$
$907$	1.90022	0.0630959	0.0315480	+	0.999502i	$0.489956\pi$
$908$	−4.52033	−0.150013
$909$	0	0
$910$	11.9254	0.395322
$911$	19.9002	0.659323	0.329661	−	0.944099i	$-0.393065\pi$
$911$	19.9002	0.659323	0.329661	+	0.944099i	$0.393065\pi$
$912$	0	0
$913$	−10.7338	−0.355236
$914$	11.5779	0.382963
$915$	0	0
$916$	−30.8731	−1.02007
$917$	19.6706	0.649579
$918$	0	0
$919$	−40.5758	−1.33847	−0.669236	−	0.743049i	$-0.733379\pi$
$919$	−40.5758	−1.33847	−0.669236	+	0.743049i	$0.733379\pi$
$920$	5.84023	0.192547
$921$	0	0
$922$	−7.01942	−0.231172
$923$	13.6493	0.449271
$924$	0	0
$925$	−5.25767	−0.172871
$926$	9.38171	0.308302
$927$	0	0
$928$	35.2571	1.15737
$929$	11.4805	0.376662	0.188331	−	0.982106i	$-0.439692\pi$
$929$	11.4805	0.376662	0.188331	+	0.982106i	$0.439692\pi$
$930$	0	0
$931$	−8.24299	−0.270153
$932$	25.3050	0.828894
$933$	0	0
$934$	−0.900828	−0.0294760
$935$	−1.20387	−0.0393706
$936$	0	0
$937$	1.18099	0.0385811	0.0192906	−	0.999814i	$-0.493859\pi$
$937$	1.18099	0.0385811	0.0192906	+	0.999814i	$0.493859\pi$
$938$	16.1547	0.527470
$939$	0	0
$940$	−9.55546	−0.311665
$941$	−30.1737	−0.983634	−0.491817	−	0.870699i	$-0.663667\pi$
$941$	−30.1737	−0.983634	−0.491817	+	0.870699i	$0.663667\pi$
$942$	0	0
$943$	15.2709	0.497290
$944$	11.1634	0.363338
$945$	0	0
$946$	−4.24338	−0.137964
$947$	−7.72440	−0.251009	−0.125505	−	0.992093i	$-0.540055\pi$
$947$	−7.72440	−0.251009	−0.125505	+	0.992093i	$0.540055\pi$
$948$	0	0
$949$	18.8546	0.612047
$950$	−4.17294	−0.135388
$951$	0	0
$952$	3.88640	0.125959
$953$	20.4499	0.662439	0.331219	−	0.943554i	$-0.392540\pi$
$953$	20.4499	0.662439	0.331219	+	0.943554i	$0.392540\pi$
$954$	0	0
$955$	43.9548	1.42235
$956$	−8.13794	−0.263200
$957$	0	0
$958$	−16.4967	−0.532984
$959$	−47.9758	−1.54922
$960$	0	0
$961$	−20.4352	−0.659200
$962$	−6.97617	−0.224921
$963$	0	0
$964$	−22.6285	−0.728813
$965$	2.93482	0.0944753
$966$	0	0
$967$	6.76713	0.217616	0.108808	−	0.994063i	$-0.465297\pi$
$967$	6.76713	0.217616	0.108808	+	0.994063i	$0.465297\pi$
$968$	1.99258	0.0640440
$969$	0	0
$970$	16.0124	0.514128
$971$	52.4375	1.68280	0.841400	−	0.540413i	$-0.181732\pi$
$971$	52.4375	1.68280	0.841400	+	0.540413i	$0.181732\pi$
$972$	0	0
$973$	−60.4536	−1.93805
$974$	20.4352	0.654785
$975$	0	0
$976$	2.35399	0.0753494
$977$	26.4722	0.846921	0.423461	−	0.905915i	$-0.360815\pi$
$977$	26.4722	0.846921	0.423461	+	0.905915i	$0.360815\pi$
$978$	0	0
$979$	6.38437	0.204045
$980$	5.51153	0.176059
$981$	0	0
$982$	15.9822	0.510011
$983$	38.6605	1.23308	0.616538	−	0.787325i	$-0.288534\pi$
$983$	38.6605	1.23308	0.616538	+	0.787325i	$0.288534\pi$
$984$	0	0
$985$	31.2628	0.996115
$986$	2.37595	0.0756655
$987$	0	0
$988$	32.8902	1.04638
$989$	−12.6807	−0.403222
$990$	0	0
$991$	−2.47031	−0.0784719	−0.0392360	−	0.999230i	$-0.512492\pi$
$991$	−2.47031	−0.0784719	−0.0392360	+	0.999230i	$0.512492\pi$
$992$	17.0606	0.541674
$993$	0	0
$994$	−5.28036	−0.167483
$995$	−25.5586	−0.810262
$996$	0	0
$997$	−52.1001	−1.65003	−0.825014	−	0.565113i	$-0.808833\pi$
$997$	−52.1001	−1.65003	−0.825014	+	0.565113i	$0.808833\pi$
$998$	14.8871	0.471244
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.13	✓	25
3.2	odd	2		6039.2.a.p.1.13	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.13	✓	25	1.1	even	1	trivial
6039.2.a.p.1.13	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)

\(f(q)\)	\(=\)	\(q-0.536820 q^{2} -1.71182 q^{4} -1.82707 q^{5} +2.96010 q^{7} +1.99258 q^{8} +O(q^{10})\)	\(q-0.536820 q^{2} -1.71182 q^{4} -1.82707 q^{5} +2.96010 q^{7} +1.99258 q^{8} +0.980806 q^{10} +1.00000 q^{11} +4.10754 q^{13} -1.58904 q^{14} +2.35399 q^{16} +0.658907 q^{17} -4.67764 q^{19} +3.12762 q^{20} -0.536820 q^{22} -1.60420 q^{23} -1.66183 q^{25} -2.20501 q^{26} -5.06718 q^{28} -6.71713 q^{29} -3.25036 q^{31} -5.24883 q^{32} -0.353714 q^{34} -5.40831 q^{35} +3.16378 q^{37} +2.51105 q^{38} -3.64058 q^{40} -9.51933 q^{41} +7.90466 q^{43} -1.71182 q^{44} +0.861168 q^{46} -3.05519 q^{47} +1.76221 q^{49} +0.892104 q^{50} -7.03138 q^{52} +6.17173 q^{53} -1.82707 q^{55} +5.89825 q^{56} +3.60589 q^{58} +4.74234 q^{59} +1.00000 q^{61} +1.74486 q^{62} -1.89030 q^{64} -7.50474 q^{65} -10.1663 q^{67} -1.12793 q^{68} +2.90329 q^{70} +3.32298 q^{71} +4.59025 q^{73} -1.69838 q^{74} +8.00729 q^{76} +2.96010 q^{77} +7.97046 q^{79} -4.30089 q^{80} +5.11017 q^{82} -10.7338 q^{83} -1.20387 q^{85} -4.24338 q^{86} +1.99258 q^{88} +6.38437 q^{89} +12.1587 q^{91} +2.74611 q^{92} +1.64009 q^{94} +8.54635 q^{95} +16.3258 q^{97} -0.945992 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Introduction

L-functions

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