LMFDB - Embedded newform 6039.2.a.m.1.12

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			1.12
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)

$f(q)$	$=$	$q-0.649968 q^{2} -1.57754 q^{4} +3.31582 q^{5} +1.03899 q^{7} +2.32529 q^{8} +O(q^{10})$	\(q-0.649968 q^{2} -1.57754 q^{4} +3.31582 q^{5} +1.03899 q^{7} +2.32529 q^{8} -2.15517 q^{10} +1.00000 q^{11} -6.28797 q^{13} -0.675310 q^{14} +1.64372 q^{16} +6.50322 q^{17} -5.89638 q^{19} -5.23084 q^{20} -0.649968 q^{22} +1.29669 q^{23} +5.99465 q^{25} +4.08698 q^{26} -1.63905 q^{28} +1.33708 q^{29} -8.72432 q^{31} -5.71894 q^{32} -4.22689 q^{34} +3.44510 q^{35} -4.00745 q^{37} +3.83246 q^{38} +7.71023 q^{40} -7.11281 q^{41} -1.18723 q^{43} -1.57754 q^{44} -0.842807 q^{46} -4.79883 q^{47} -5.92050 q^{49} -3.89633 q^{50} +9.91953 q^{52} +3.64627 q^{53} +3.31582 q^{55} +2.41595 q^{56} -0.869057 q^{58} +8.28781 q^{59} +1.00000 q^{61} +5.67053 q^{62} +0.429682 q^{64} -20.8498 q^{65} -14.3363 q^{67} -10.2591 q^{68} -2.23921 q^{70} -5.20372 q^{71} -16.2402 q^{73} +2.60471 q^{74} +9.30179 q^{76} +1.03899 q^{77} +6.50865 q^{79} +5.45028 q^{80} +4.62310 q^{82} +14.0862 q^{83} +21.5635 q^{85} +0.771664 q^{86} +2.32529 q^{88} -8.87822 q^{89} -6.53314 q^{91} -2.04558 q^{92} +3.11908 q^{94} -19.5513 q^{95} -3.27774 q^{97} +3.84813 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.649968	−0.459597	−0.229798	−	0.973238i	$-0.573807\pi$
$2$	−0.649968	−0.459597	−0.229798	+	0.973238i	$0.573807\pi$
$3$	0	0
$3$	0	0
$4$	−1.57754	−0.788771
$5$	3.31582	1.48288	0.741439	−	0.671020i	$-0.234143\pi$
$5$	3.31582	1.48288	0.741439	+	0.671020i	$0.234143\pi$
$6$	0	0
$7$	1.03899	0.392702	0.196351	−	0.980534i	$-0.437091\pi$
$7$	1.03899	0.392702	0.196351	+	0.980534i	$0.437091\pi$
$8$	2.32529	0.822113
$9$	0	0
$10$	−2.15517	−0.681526
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.28797	−1.74397	−0.871984	−	0.489534i	$-0.837167\pi$
$13$	−6.28797	−1.74397	−0.871984	+	0.489534i	$0.837167\pi$
$14$	−0.675310	−0.180484
$15$	0	0
$16$	1.64372	0.410931
$17$	6.50322	1.57726	0.788632	−	0.614866i	$-0.210790\pi$
$17$	6.50322	1.57726	0.788632	+	0.614866i	$0.210790\pi$
$18$	0	0
$19$	−5.89638	−1.35272	−0.676362	−	0.736570i	$-0.736444\pi$
$19$	−5.89638	−1.35272	−0.676362	+	0.736570i	$0.736444\pi$
$20$	−5.23084	−1.16965
$21$	0	0
$22$	−0.649968	−0.138574
$23$	1.29669	0.270379	0.135189	−	0.990820i	$-0.456836\pi$
$23$	1.29669	0.270379	0.135189	+	0.990820i	$0.456836\pi$
$24$	0	0
$25$	5.99465	1.19893
$26$	4.08698	0.801522
$27$	0	0
$28$	−1.63905	−0.309752
$29$	1.33708	0.248289	0.124145	−	0.992264i	$-0.460381\pi$
$29$	1.33708	0.248289	0.124145	+	0.992264i	$0.460381\pi$
$30$	0	0
$31$	−8.72432	−1.56693	−0.783467	−	0.621433i	$-0.786551\pi$
$31$	−8.72432	−1.56693	−0.783467	+	0.621433i	$0.786551\pi$
$32$	−5.71894	−1.01098
$33$	0	0
$34$	−4.22689	−0.724905
$35$	3.44510	0.582329
$36$	0	0
$37$	−4.00745	−0.658821	−0.329410	−	0.944187i	$-0.606850\pi$
$37$	−4.00745	−0.658821	−0.329410	+	0.944187i	$0.606850\pi$
$38$	3.83246	0.621707
$39$	0	0
$40$	7.71023	1.21909
$41$	−7.11281	−1.11083	−0.555417	−	0.831572i	$-0.687441\pi$
$41$	−7.11281	−1.11083	−0.555417	+	0.831572i	$0.687441\pi$
$42$	0	0
$43$	−1.18723	−0.181052	−0.0905258	−	0.995894i	$-0.528855\pi$
$43$	−1.18723	−0.181052	−0.0905258	+	0.995894i	$0.528855\pi$
$44$	−1.57754	−0.237823
$45$	0	0
$46$	−0.842807	−0.124265
$47$	−4.79883	−0.699981	−0.349991	−	0.936753i	$-0.613815\pi$
$47$	−4.79883	−0.699981	−0.349991	+	0.936753i	$0.613815\pi$
$48$	0	0
$49$	−5.92050	−0.845785
$50$	−3.89633	−0.551024
$51$	0	0
$52$	9.91953	1.37559
$53$	3.64627	0.500854	0.250427	−	0.968136i	$-0.419429\pi$
$53$	3.64627	0.500854	0.250427	+	0.968136i	$0.419429\pi$
$54$	0	0
$55$	3.31582	0.447105
$56$	2.41595	0.322845
$57$	0	0
$58$	−0.869057	−0.114113
$59$	8.28781	1.07898	0.539490	−	0.841992i	$-0.318617\pi$
$59$	8.28781	1.07898	0.539490	+	0.841992i	$0.318617\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	5.67053	0.720158
$63$	0	0
$64$	0.429682	0.0537102
$65$	−20.8498	−2.58609
$66$	0	0
$67$	−14.3363	−1.75146	−0.875729	−	0.482803i	$-0.839619\pi$
$67$	−14.3363	−1.75146	−0.875729	+	0.482803i	$0.839619\pi$
$68$	−10.2591	−1.24410
$69$	0	0
$70$	−2.23921	−0.267636
$71$	−5.20372	−0.617568	−0.308784	−	0.951132i	$-0.599922\pi$
$71$	−5.20372	−0.617568	−0.308784	+	0.951132i	$0.599922\pi$
$72$	0	0
$73$	−16.2402	−1.90077	−0.950385	−	0.311076i	$-0.899311\pi$
$73$	−16.2402	−1.90077	−0.950385	+	0.311076i	$0.899311\pi$
$74$	2.60471	0.302792
$75$	0	0
$76$	9.30179	1.06699
$77$	1.03899	0.118404
$78$	0	0
$79$	6.50865	0.732281	0.366140	−	0.930560i	$-0.380679\pi$
$79$	6.50865	0.732281	0.366140	+	0.930560i	$0.380679\pi$
$80$	5.45028	0.609360
$81$	0	0
$82$	4.62310	0.510536
$83$	14.0862	1.54616	0.773080	−	0.634308i	$-0.218715\pi$
$83$	14.0862	1.54616	0.773080	+	0.634308i	$0.218715\pi$
$84$	0	0
$85$	21.5635	2.33889
$86$	0.771664	0.0832107
$87$	0	0
$88$	2.32529	0.247876
$89$	−8.87822	−0.941090	−0.470545	−	0.882376i	$-0.655943\pi$
$89$	−8.87822	−0.941090	−0.470545	+	0.882376i	$0.655943\pi$
$90$	0	0
$91$	−6.53314	−0.684859
$92$	−2.04558	−0.213267
$93$	0	0
$94$	3.11908	0.321709
$95$	−19.5513	−2.00592
$96$	0	0
$97$	−3.27774	−0.332804	−0.166402	−	0.986058i	$-0.553215\pi$
$97$	−3.27774	−0.332804	−0.166402	+	0.986058i	$0.553215\pi$
$98$	3.84813	0.388720
$99$	0	0
$100$	−9.45681	−0.945681
$101$	0.0574927	0.00572074	0.00286037	−	0.999996i	$-0.499090\pi$
$101$	0.0574927	0.00572074	0.00286037	+	0.999996i	$0.499090\pi$
$102$	0	0
$103$	18.1900	1.79231	0.896155	−	0.443740i	$-0.146349\pi$
$103$	18.1900	1.79231	0.896155	+	0.443740i	$0.146349\pi$
$104$	−14.6213	−1.43374
$105$	0	0
$106$	−2.36996	−0.230191
$107$	−0.879973	−0.0850702	−0.0425351	−	0.999095i	$-0.513543\pi$
$107$	−0.879973	−0.0850702	−0.0425351	+	0.999095i	$0.513543\pi$
$108$	0	0
$109$	−14.4500	−1.38406	−0.692028	−	0.721870i	$-0.743283\pi$
$109$	−14.4500	−1.38406	−0.692028	+	0.721870i	$0.743283\pi$
$110$	−2.15517	−0.205488
$111$	0	0
$112$	1.70781	0.161373
$113$	−11.5185	−1.08357	−0.541784	−	0.840518i	$-0.682251\pi$
$113$	−11.5185	−1.08357	−0.541784	+	0.840518i	$0.682251\pi$
$114$	0	0
$115$	4.29959	0.400939
$116$	−2.10930	−0.195843
$117$	0	0
$118$	−5.38681	−0.495896
$119$	6.75679	0.619394
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.649968	−0.0588453
$123$	0	0
$124$	13.7630	1.23595
$125$	3.29807	0.294988
$126$	0	0
$127$	−15.3909	−1.36572	−0.682862	−	0.730547i	$-0.739265\pi$
$127$	−15.3909	−1.36572	−0.682862	+	0.730547i	$0.739265\pi$
$128$	11.1586	0.986290
$129$	0	0
$130$	13.5517	1.18856
$131$	−11.2617	−0.983944	−0.491972	−	0.870611i	$-0.663724\pi$
$131$	−11.2617	−0.983944	−0.491972	+	0.870611i	$0.663724\pi$
$132$	0	0
$133$	−6.12629	−0.531217
$134$	9.31813	0.804964
$135$	0	0
$136$	15.1219	1.29669
$137$	11.7867	1.00700	0.503502	−	0.863994i	$-0.332045\pi$
$137$	11.7867	1.00700	0.503502	+	0.863994i	$0.332045\pi$
$138$	0	0
$139$	−9.50504	−0.806207	−0.403103	−	0.915154i	$-0.632069\pi$
$139$	−9.50504	−0.806207	−0.403103	+	0.915154i	$0.632069\pi$
$140$	−5.43480	−0.459324
$141$	0	0
$142$	3.38225	0.283832
$143$	−6.28797	−0.525826
$144$	0	0
$145$	4.43350	0.368183
$146$	10.5556	0.873588
$147$	0	0
$148$	6.32192	0.519659
$149$	10.6395	0.871624	0.435812	−	0.900038i	$-0.356461\pi$
$149$	10.6395	0.871624	0.435812	+	0.900038i	$0.356461\pi$
$150$	0	0
$151$	3.51632	0.286154	0.143077	−	0.989712i	$-0.454300\pi$
$151$	3.51632	0.286154	0.143077	+	0.989712i	$0.454300\pi$
$152$	−13.7108	−1.11209
$153$	0	0
$154$	−0.675310	−0.0544181
$155$	−28.9283	−2.32357
$156$	0	0
$157$	−18.8937	−1.50788	−0.753941	−	0.656942i	$-0.771850\pi$
$157$	−18.8937	−1.50788	−0.753941	+	0.656942i	$0.771850\pi$
$158$	−4.23041	−0.336554
$159$	0	0
$160$	−18.9630	−1.49915
$161$	1.34725	0.106178
$162$	0	0
$163$	20.2437	1.58561	0.792806	−	0.609474i	$-0.208619\pi$
$163$	20.2437	1.58561	0.792806	+	0.609474i	$0.208619\pi$
$164$	11.2208	0.876194
$165$	0	0
$166$	−9.15557	−0.710610
$167$	−16.3606	−1.26602	−0.633011	−	0.774143i	$-0.718181\pi$
$167$	−16.3606	−1.26602	−0.633011	+	0.774143i	$0.718181\pi$
$168$	0	0
$169$	26.5385	2.04143
$170$	−14.0156	−1.07495
$171$	0	0
$172$	1.87291	0.142808
$173$	−22.4222	−1.70473	−0.852364	−	0.522948i	$-0.824832\pi$
$173$	−22.4222	−1.70473	−0.852364	+	0.522948i	$0.824832\pi$
$174$	0	0
$175$	6.22838	0.470821
$176$	1.64372	0.123900
$177$	0	0
$178$	5.77056	0.432522
$179$	17.1196	1.27958	0.639789	−	0.768551i	$-0.279022\pi$
$179$	17.1196	1.27958	0.639789	+	0.768551i	$0.279022\pi$
$180$	0	0
$181$	13.9749	1.03874	0.519372	−	0.854548i	$-0.326166\pi$
$181$	13.9749	1.03874	0.519372	+	0.854548i	$0.326166\pi$
$182$	4.24633	0.314759
$183$	0	0
$184$	3.01518	0.222282
$185$	−13.2880	−0.976951
$186$	0	0
$187$	6.50322	0.475563
$188$	7.57035	0.552125
$189$	0	0
$190$	12.7077	0.921916
$191$	9.35123	0.676631	0.338316	−	0.941033i	$-0.390143\pi$
$191$	9.35123	0.676631	0.338316	+	0.941033i	$0.390143\pi$
$192$	0	0
$193$	14.0457	1.01103	0.505515	−	0.862818i	$-0.331303\pi$
$193$	14.0457	1.01103	0.505515	+	0.862818i	$0.331303\pi$
$194$	2.13042	0.152956
$195$	0	0
$196$	9.33983	0.667131
$197$	−10.5967	−0.754985	−0.377492	−	0.926013i	$-0.623214\pi$
$197$	−10.5967	−0.754985	−0.377492	+	0.926013i	$0.623214\pi$
$198$	0	0
$199$	3.02639	0.214535	0.107268	−	0.994230i	$-0.465790\pi$
$199$	3.02639	0.214535	0.107268	+	0.994230i	$0.465790\pi$
$200$	13.9393	0.985656
$201$	0	0
$202$	−0.0373684	−0.00262923
$203$	1.38921	0.0975035
$204$	0	0
$205$	−23.5848	−1.64723
$206$	−11.8229	−0.823740
$207$	0	0
$208$	−10.3357	−0.716650
$209$	−5.89638	−0.407861
$210$	0	0
$211$	−0.775671	−0.0533994	−0.0266997	−	0.999643i	$-0.508500\pi$
$211$	−0.775671	−0.0533994	−0.0266997	+	0.999643i	$0.508500\pi$
$212$	−5.75214	−0.395059
$213$	0	0
$214$	0.571954	0.0390980
$215$	−3.93665	−0.268478
$216$	0	0
$217$	−9.06449	−0.615338
$218$	9.39202	0.636108
$219$	0	0
$220$	−5.23084	−0.352663
$221$	−40.8921	−2.75070
$222$	0	0
$223$	−1.96530	−0.131607	−0.0658033	−	0.997833i	$-0.520961\pi$
$223$	−1.96530	−0.131607	−0.0658033	+	0.997833i	$0.520961\pi$
$224$	−5.94193	−0.397012
$225$	0	0
$226$	7.48664	0.498004
$227$	12.6172	0.837432	0.418716	−	0.908117i	$-0.362480\pi$
$227$	12.6172	0.837432	0.418716	+	0.908117i	$0.362480\pi$
$228$	0	0
$229$	4.54784	0.300529	0.150265	−	0.988646i	$-0.451987\pi$
$229$	4.54784	0.300529	0.150265	+	0.988646i	$0.451987\pi$
$230$	−2.79460	−0.184270
$231$	0	0
$232$	3.10909	0.204122
$233$	20.7431	1.35892	0.679462	−	0.733711i	$-0.262213\pi$
$233$	20.7431	1.35892	0.679462	+	0.733711i	$0.262213\pi$
$234$	0	0
$235$	−15.9120	−1.03799
$236$	−13.0744	−0.851069
$237$	0	0
$238$	−4.39169	−0.284671
$239$	29.5762	1.91313	0.956563	−	0.291524i	$-0.0941624\pi$
$239$	29.5762	1.91313	0.956563	+	0.291524i	$0.0941624\pi$
$240$	0	0
$241$	11.0280	0.710377	0.355189	−	0.934795i	$-0.384417\pi$
$241$	11.0280	0.710377	0.355189	+	0.934795i	$0.384417\pi$
$242$	−0.649968	−0.0417815
$243$	0	0
$244$	−1.57754	−0.100992
$245$	−19.6313	−1.25420
$246$	0	0
$247$	37.0763	2.35911
$248$	−20.2866	−1.28820
$249$	0	0
$250$	−2.14364	−0.135576
$251$	2.67622	0.168921	0.0844607	−	0.996427i	$-0.473083\pi$
$251$	2.67622	0.168921	0.0844607	+	0.996427i	$0.473083\pi$
$252$	0	0
$253$	1.29669	0.0815223
$254$	10.0036	0.627682
$255$	0	0
$256$	−8.11209	−0.507006
$257$	19.9491	1.24439	0.622194	−	0.782863i	$-0.286241\pi$
$257$	19.9491	1.24439	0.622194	+	0.782863i	$0.286241\pi$
$258$	0	0
$259$	−4.16370	−0.258720
$260$	32.8914	2.03984
$261$	0	0
$262$	7.31977	0.452217
$263$	−14.7400	−0.908910	−0.454455	−	0.890770i	$-0.650166\pi$
$263$	−14.7400	−0.908910	−0.454455	+	0.890770i	$0.650166\pi$
$264$	0	0
$265$	12.0904	0.742705
$266$	3.98189	0.244145
$267$	0	0
$268$	22.6161	1.38150
$269$	4.30202	0.262299	0.131149	−	0.991363i	$-0.458133\pi$
$269$	4.30202	0.262299	0.131149	+	0.991363i	$0.458133\pi$
$270$	0	0
$271$	−7.15233	−0.434473	−0.217236	−	0.976119i	$-0.569704\pi$
$271$	−7.15233	−0.434473	−0.217236	+	0.976119i	$0.569704\pi$
$272$	10.6895	0.648146
$273$	0	0
$274$	−7.66096	−0.462815
$275$	5.99465	0.361491
$276$	0	0
$277$	−24.8344	−1.49215	−0.746076	−	0.665861i	$-0.768064\pi$
$277$	−24.8344	−1.49215	−0.746076	+	0.665861i	$0.768064\pi$
$278$	6.17797	0.370530
$279$	0	0
$280$	8.01085	0.478740
$281$	−26.3744	−1.57336	−0.786681	−	0.617360i	$-0.788202\pi$
$281$	−26.3744	−1.57336	−0.786681	+	0.617360i	$0.788202\pi$
$282$	0	0
$283$	11.0509	0.656908	0.328454	−	0.944520i	$-0.393472\pi$
$283$	11.0509	0.656908	0.328454	+	0.944520i	$0.393472\pi$
$284$	8.20908	0.487119
$285$	0	0
$286$	4.08698	0.241668
$287$	−7.39014	−0.436226
$288$	0	0
$289$	25.2919	1.48776
$290$	−2.88164	−0.169215
$291$	0	0
$292$	25.6196	1.49927
$293$	29.7007	1.73513	0.867566	−	0.497322i	$-0.165683\pi$
$293$	29.7007	1.73513	0.867566	+	0.497322i	$0.165683\pi$
$294$	0	0
$295$	27.4809	1.60000
$296$	−9.31847	−0.541625
$297$	0	0
$298$	−6.91535	−0.400595
$299$	−8.15355	−0.471532
$300$	0	0
$301$	−1.23353	−0.0710992
$302$	−2.28550	−0.131516
$303$	0	0
$304$	−9.69202	−0.555875
$305$	3.31582	0.189863
$306$	0	0
$307$	−4.77361	−0.272445	−0.136222	−	0.990678i	$-0.543496\pi$
$307$	−4.77361	−0.272445	−0.136222	+	0.990678i	$0.543496\pi$
$308$	−1.63905	−0.0933936
$309$	0	0
$310$	18.8024	1.06791
$311$	19.9841	1.13320	0.566598	−	0.823994i	$-0.308259\pi$
$311$	19.9841	1.13320	0.566598	+	0.823994i	$0.308259\pi$
$312$	0	0
$313$	−34.3848	−1.94354	−0.971772	−	0.235920i	$-0.924190\pi$
$313$	−34.3848	−1.94354	−0.971772	+	0.235920i	$0.924190\pi$
$314$	12.2803	0.693018
$315$	0	0
$316$	−10.2677	−0.577602
$317$	−6.08883	−0.341983	−0.170991	−	0.985273i	$-0.554697\pi$
$317$	−6.08883	−0.341983	−0.170991	+	0.985273i	$0.554697\pi$
$318$	0	0
$319$	1.33708	0.0748620
$320$	1.42475	0.0796457
$321$	0	0
$322$	−0.875669	−0.0487991
$323$	−38.3455	−2.13360
$324$	0	0
$325$	−37.6941	−2.09090
$326$	−13.1578	−0.728742
$327$	0	0
$328$	−16.5393	−0.913231
$329$	−4.98594	−0.274884
$330$	0	0
$331$	16.7845	0.922561	0.461280	−	0.887254i	$-0.347390\pi$
$331$	16.7845	0.922561	0.461280	+	0.887254i	$0.347390\pi$
$332$	−22.2216	−1.21957
$333$	0	0
$334$	10.6339	0.581859
$335$	−47.5366	−2.59720
$336$	0	0
$337$	−13.6093	−0.741344	−0.370672	−	0.928764i	$-0.620873\pi$
$337$	−13.6093	−0.741344	−0.370672	+	0.928764i	$0.620873\pi$
$338$	−17.2492	−0.938232
$339$	0	0
$340$	−34.0173	−1.84485
$341$	−8.72432	−0.472449
$342$	0	0
$343$	−13.4243	−0.724843
$344$	−2.76066	−0.148845
$345$	0	0
$346$	14.5737	0.783488
$347$	−23.5563	−1.26457	−0.632283	−	0.774737i	$-0.717882\pi$
$347$	−23.5563	−1.26457	−0.632283	+	0.774737i	$0.717882\pi$
$348$	0	0
$349$	−16.6256	−0.889949	−0.444975	−	0.895543i	$-0.646787\pi$
$349$	−16.6256	−0.889949	−0.444975	+	0.895543i	$0.646787\pi$
$350$	−4.04825	−0.216388
$351$	0	0
$352$	−5.71894	−0.304821
$353$	−2.52322	−0.134298	−0.0671489	−	0.997743i	$-0.521390\pi$
$353$	−2.52322	−0.134298	−0.0671489	+	0.997743i	$0.521390\pi$
$354$	0	0
$355$	−17.2546	−0.915778
$356$	14.0058	0.742304
$357$	0	0
$358$	−11.1272	−0.588090
$359$	−18.4997	−0.976376	−0.488188	−	0.872738i	$-0.662342\pi$
$359$	−18.4997	−0.976376	−0.488188	+	0.872738i	$0.662342\pi$
$360$	0	0
$361$	15.7673	0.829860
$362$	−9.08322	−0.477404
$363$	0	0
$364$	10.3063	0.540197
$365$	−53.8495	−2.81861
$366$	0	0
$367$	28.7976	1.50322	0.751610	−	0.659608i	$-0.229277\pi$
$367$	28.7976	1.50322	0.751610	+	0.659608i	$0.229277\pi$
$368$	2.13140	0.111107
$369$	0	0
$370$	8.63676	0.449004
$371$	3.78844	0.196686
$372$	0	0
$373$	4.89364	0.253383	0.126691	−	0.991942i	$-0.459564\pi$
$373$	4.89364	0.253383	0.126691	+	0.991942i	$0.459564\pi$
$374$	−4.22689	−0.218567
$375$	0	0
$376$	−11.1587	−0.575464
$377$	−8.40750	−0.433008
$378$	0	0
$379$	−13.9331	−0.715693	−0.357847	−	0.933780i	$-0.616489\pi$
$379$	−13.9331	−0.715693	−0.357847	+	0.933780i	$0.616489\pi$
$380$	30.8431	1.58222
$381$	0	0
$382$	−6.07800	−0.310977
$383$	−8.51127	−0.434906	−0.217453	−	0.976071i	$-0.569775\pi$
$383$	−8.51127	−0.434906	−0.217453	+	0.976071i	$0.569775\pi$
$384$	0	0
$385$	3.44510	0.175579
$386$	−9.12923	−0.464665
$387$	0	0
$388$	5.17077	0.262506
$389$	9.37811	0.475489	0.237744	−	0.971328i	$-0.423592\pi$
$389$	9.37811	0.475489	0.237744	+	0.971328i	$0.423592\pi$
$390$	0	0
$391$	8.43267	0.426459
$392$	−13.7669	−0.695331
$393$	0	0
$394$	6.88752	0.346988
$395$	21.5815	1.08588
$396$	0	0
$397$	−21.2989	−1.06896	−0.534480	−	0.845181i	$-0.679492\pi$
$397$	−21.2989	−1.06896	−0.534480	+	0.845181i	$0.679492\pi$
$398$	−1.96706	−0.0985997
$399$	0	0
$400$	9.85354	0.492677
$401$	−33.1701	−1.65644	−0.828218	−	0.560406i	$-0.810645\pi$
$401$	−33.1701	−1.65644	−0.828218	+	0.560406i	$0.810645\pi$
$402$	0	0
$403$	54.8583	2.73268
$404$	−0.0906972	−0.00451235
$405$	0	0
$406$	−0.902942	−0.0448123
$407$	−4.00745	−0.198642
$408$	0	0
$409$	−7.73865	−0.382652	−0.191326	−	0.981527i	$-0.561279\pi$
$409$	−7.73865	−0.382652	−0.191326	+	0.981527i	$0.561279\pi$
$410$	15.3293	0.757063
$411$	0	0
$412$	−28.6954	−1.41372
$413$	8.61096	0.423717
$414$	0	0
$415$	46.7073	2.29277
$416$	35.9605	1.76311
$417$	0	0
$418$	3.83246	0.187452
$419$	−18.6441	−0.910825	−0.455413	−	0.890280i	$-0.650508\pi$
$419$	−18.6441	−0.910825	−0.455413	+	0.890280i	$0.650508\pi$
$420$	0	0
$421$	−27.7930	−1.35455	−0.677273	−	0.735731i	$-0.736839\pi$
$421$	−27.7930	−1.35455	−0.677273	+	0.735731i	$0.736839\pi$
$422$	0.504161	0.0245422
$423$	0	0
$424$	8.47862	0.411758
$425$	38.9845	1.89103
$426$	0	0
$427$	1.03899	0.0502803
$428$	1.38819	0.0671009
$429$	0	0
$430$	2.55870	0.123391
$431$	−29.7851	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$431$	−29.7851	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$432$	0	0
$433$	−19.6885	−0.946168	−0.473084	−	0.881017i	$-0.656859\pi$
$433$	−19.6885	−0.946168	−0.473084	+	0.881017i	$0.656859\pi$
$434$	5.89163	0.282807
$435$	0	0
$436$	22.7954	1.09170
$437$	−7.64579	−0.365748
$438$	0	0
$439$	2.01586	0.0962118	0.0481059	−	0.998842i	$-0.484681\pi$
$439$	2.01586	0.0962118	0.0481059	+	0.998842i	$0.484681\pi$
$440$	7.71023	0.367571
$441$	0	0
$442$	26.5785	1.26421
$443$	−35.1907	−1.67196	−0.835980	−	0.548759i	$-0.815100\pi$
$443$	−35.1907	−1.67196	−0.835980	+	0.548759i	$0.815100\pi$
$444$	0	0
$445$	−29.4386	−1.39552
$446$	1.27738	0.0604859
$447$	0	0
$448$	0.446435	0.0210921
$449$	−31.1965	−1.47226	−0.736128	−	0.676843i	$-0.763348\pi$
$449$	−31.1965	−1.47226	−0.736128	+	0.676843i	$0.763348\pi$
$450$	0	0
$451$	−7.11281	−0.334929
$452$	18.1709	0.854687
$453$	0	0
$454$	−8.20076	−0.384881
$455$	−21.6627	−1.01556
$456$	0	0
$457$	−7.65961	−0.358301	−0.179151	−	0.983822i	$-0.557335\pi$
$457$	−7.65961	−0.358301	−0.179151	+	0.983822i	$0.557335\pi$
$458$	−2.95595	−0.138122
$459$	0	0
$460$	−6.78279	−0.316249
$461$	−9.43077	−0.439235	−0.219617	−	0.975586i	$-0.570481\pi$
$461$	−9.43077	−0.439235	−0.219617	+	0.975586i	$0.570481\pi$
$462$	0	0
$463$	14.7708	0.686455	0.343228	−	0.939252i	$-0.388480\pi$
$463$	14.7708	0.686455	0.343228	+	0.939252i	$0.388480\pi$
$464$	2.19778	0.102030
$465$	0	0
$466$	−13.4823	−0.624557
$467$	4.02696	0.186346	0.0931728	−	0.995650i	$-0.470299\pi$
$467$	4.02696	0.186346	0.0931728	+	0.995650i	$0.470299\pi$
$468$	0	0
$469$	−14.8953	−0.687800
$470$	10.3423	0.477055
$471$	0	0
$472$	19.2715	0.887044
$473$	−1.18723	−0.0545891
$474$	0	0
$475$	−35.3467	−1.62182
$476$	−10.6591	−0.488560
$477$	0	0
$478$	−19.2236	−0.879267
$479$	−7.92580	−0.362139	−0.181070	−	0.983470i	$-0.557956\pi$
$479$	−7.92580	−0.362139	−0.181070	+	0.983470i	$0.557956\pi$
$480$	0	0
$481$	25.1987	1.14896
$482$	−7.16786	−0.326487
$483$	0	0
$484$	−1.57754	−0.0717065
$485$	−10.8684	−0.493508
$486$	0	0
$487$	8.54032	0.386999	0.193499	−	0.981100i	$-0.438016\pi$
$487$	8.54032	0.386999	0.193499	+	0.981100i	$0.438016\pi$
$488$	2.32529	0.105261
$489$	0	0
$490$	12.7597	0.576425
$491$	16.0656	0.725030	0.362515	−	0.931978i	$-0.381918\pi$
$491$	16.0656	0.725030	0.362515	+	0.931978i	$0.381918\pi$
$492$	0	0
$493$	8.69531	0.391617
$494$	−24.0984	−1.08424
$495$	0	0
$496$	−14.3404	−0.643901
$497$	−5.40661	−0.242520
$498$	0	0
$499$	8.61901	0.385840	0.192920	−	0.981215i	$-0.438204\pi$
$499$	8.61901	0.385840	0.192920	+	0.981215i	$0.438204\pi$
$500$	−5.20284	−0.232678
$501$	0	0
$502$	−1.73946	−0.0776357
$503$	23.7779	1.06020	0.530102	−	0.847934i	$-0.322154\pi$
$503$	23.7779	1.06020	0.530102	+	0.847934i	$0.322154\pi$
$504$	0	0
$505$	0.190635	0.00848316
$506$	−0.842807	−0.0374674
$507$	0	0
$508$	24.2798	1.07724
$509$	42.7061	1.89291	0.946457	−	0.322829i	$-0.104634\pi$
$509$	42.7061	1.89291	0.946457	+	0.322829i	$0.104634\pi$
$510$	0	0
$511$	−16.8734	−0.746436
$512$	−17.0446	−0.753272
$513$	0	0
$514$	−12.9663	−0.571917
$515$	60.3146	2.65778
$516$	0	0
$517$	−4.79883	−0.211052
$518$	2.70627	0.118907
$519$	0	0
$520$	−48.4817	−2.12606
$521$	−6.40857	−0.280765	−0.140382	−	0.990097i	$-0.544833\pi$
$521$	−6.40857	−0.280765	−0.140382	+	0.990097i	$0.544833\pi$
$522$	0	0
$523$	−43.5989	−1.90645	−0.953223	−	0.302268i	$-0.902256\pi$
$523$	−43.5989	−1.90645	−0.953223	+	0.302268i	$0.902256\pi$
$524$	17.7659	0.776106
$525$	0	0
$526$	9.58055	0.417732
$527$	−56.7362	−2.47147
$528$	0	0
$529$	−21.3186	−0.926895
$530$	−7.85835	−0.341345
$531$	0	0
$532$	9.66448	0.419008
$533$	44.7251	1.93726
$534$	0	0
$535$	−2.91783	−0.126149
$536$	−33.3360	−1.43990
$537$	0	0
$538$	−2.79617	−0.120552
$539$	−5.92050	−0.255014
$540$	0	0
$541$	−36.2204	−1.55724	−0.778618	−	0.627499i	$-0.784079\pi$
$541$	−36.2204	−1.55724	−0.778618	+	0.627499i	$0.784079\pi$
$542$	4.64878	0.199682
$543$	0	0
$544$	−37.1915	−1.59457
$545$	−47.9135	−2.05239
$546$	0	0
$547$	−38.0309	−1.62609	−0.813043	−	0.582204i	$-0.802190\pi$
$547$	−38.0309	−1.62609	−0.813043	+	0.582204i	$0.802190\pi$
$548$	−18.5940	−0.794295
$549$	0	0
$550$	−3.89633	−0.166140
$551$	−7.88392	−0.335866
$552$	0	0
$553$	6.76243	0.287568
$554$	16.1415	0.685788
$555$	0	0
$556$	14.9946	0.635913
$557$	−19.5930	−0.830182	−0.415091	−	0.909780i	$-0.636250\pi$
$557$	−19.5930	−0.830182	−0.415091	+	0.909780i	$0.636250\pi$
$558$	0	0
$559$	7.46529	0.315748
$560$	5.66279	0.239297
$561$	0	0
$562$	17.1425	0.723112
$563$	5.58308	0.235299	0.117649	−	0.993055i	$-0.462464\pi$
$563$	5.58308	0.235299	0.117649	+	0.993055i	$0.462464\pi$
$564$	0	0
$565$	−38.1932	−1.60680
$566$	−7.18273	−0.301912
$567$	0	0
$568$	−12.1001	−0.507710
$569$	18.0752	0.757753	0.378877	−	0.925447i	$-0.376310\pi$
$569$	18.0752	0.757753	0.378877	+	0.925447i	$0.376310\pi$
$570$	0	0
$571$	−18.7850	−0.786126	−0.393063	−	0.919511i	$-0.628585\pi$
$571$	−18.7850	−0.786126	−0.393063	+	0.919511i	$0.628585\pi$
$572$	9.91953	0.414756
$573$	0	0
$574$	4.80336	0.200488
$575$	7.77321	0.324165
$576$	0	0
$577$	−4.42505	−0.184217	−0.0921087	−	0.995749i	$-0.529361\pi$
$577$	−4.42505	−0.184217	−0.0921087	+	0.995749i	$0.529361\pi$
$578$	−16.4389	−0.683769
$579$	0	0
$580$	−6.99404	−0.290412
$581$	14.6354	0.607180
$582$	0	0
$583$	3.64627	0.151013
$584$	−37.7631	−1.56265
$585$	0	0
$586$	−19.3045	−0.797461
$587$	8.03413	0.331604	0.165802	−	0.986159i	$-0.446979\pi$
$587$	8.03413	0.331604	0.165802	+	0.986159i	$0.446979\pi$
$588$	0	0
$589$	51.4420	2.11963
$590$	−17.8617	−0.735353
$591$	0	0
$592$	−6.58714	−0.270730
$593$	39.8339	1.63578	0.817891	−	0.575373i	$-0.195143\pi$
$593$	39.8339	1.63578	0.817891	+	0.575373i	$0.195143\pi$
$594$	0	0
$595$	22.4043	0.918486
$596$	−16.7843	−0.687512
$597$	0	0
$598$	5.29954	0.216715
$599$	30.3276	1.23915	0.619576	−	0.784937i	$-0.287305\pi$
$599$	30.3276	1.23915	0.619576	+	0.784937i	$0.287305\pi$
$600$	0	0
$601$	−21.3363	−0.870324	−0.435162	−	0.900352i	$-0.643309\pi$
$601$	−21.3363	−0.870324	−0.435162	+	0.900352i	$0.643309\pi$
$602$	0.801752	0.0326770
$603$	0	0
$604$	−5.54715	−0.225710
$605$	3.31582	0.134807
$606$	0	0
$607$	−37.3615	−1.51646	−0.758228	−	0.651989i	$-0.773935\pi$
$607$	−37.3615	−1.51646	−0.758228	+	0.651989i	$0.773935\pi$
$608$	33.7211	1.36757
$609$	0	0
$610$	−2.15517	−0.0872605
$611$	30.1749	1.22074
$612$	0	0
$613$	34.2083	1.38166	0.690830	−	0.723017i	$-0.257245\pi$
$613$	34.2083	1.38166	0.690830	+	0.723017i	$0.257245\pi$
$614$	3.10269	0.125215
$615$	0	0
$616$	2.41595	0.0973415
$617$	4.37034	0.175943	0.0879716	−	0.996123i	$-0.471962\pi$
$617$	4.37034	0.175943	0.0879716	+	0.996123i	$0.471962\pi$
$618$	0	0
$619$	−4.05118	−0.162831	−0.0814154	−	0.996680i	$-0.525944\pi$
$619$	−4.05118	−0.162831	−0.0814154	+	0.996680i	$0.525944\pi$
$620$	45.6355	1.83277
$621$	0	0
$622$	−12.9890	−0.520813
$623$	−9.22439	−0.369568
$624$	0	0
$625$	−19.0374	−0.761498
$626$	22.3490	0.893247
$627$	0	0
$628$	29.8056	1.18937
$629$	−26.0613	−1.03913
$630$	0	0
$631$	−30.0729	−1.19718	−0.598591	−	0.801055i	$-0.704272\pi$
$631$	−30.0729	−1.19718	−0.598591	+	0.801055i	$0.704272\pi$
$632$	15.1345	0.602017
$633$	0	0
$634$	3.95754	0.157174
$635$	−51.0335	−2.02520
$636$	0	0
$637$	37.2279	1.47502
$638$	−0.869057	−0.0344063
$639$	0	0
$640$	36.9999	1.46255
$641$	0.747701	0.0295324	0.0147662	−	0.999891i	$-0.495300\pi$
$641$	0.747701	0.0295324	0.0147662	+	0.999891i	$0.495300\pi$
$642$	0	0
$643$	15.5274	0.612343	0.306171	−	0.951976i	$-0.400952\pi$
$643$	15.5274	0.612343	0.306171	+	0.951976i	$0.400952\pi$
$644$	−2.12534	−0.0837503
$645$	0	0
$646$	24.9233	0.980596
$647$	28.6653	1.12695	0.563475	−	0.826133i	$-0.309464\pi$
$647$	28.6653	1.12695	0.563475	+	0.826133i	$0.309464\pi$
$648$	0	0
$649$	8.28781	0.325325
$650$	24.5000	0.960968
$651$	0	0
$652$	−31.9354	−1.25069
$653$	38.3160	1.49942	0.749711	−	0.661766i	$-0.230193\pi$
$653$	38.3160	1.49942	0.749711	+	0.661766i	$0.230193\pi$
$654$	0	0
$655$	−37.3419	−1.45907
$656$	−11.6915	−0.456476
$657$	0	0
$658$	3.24070	0.126336
$659$	−1.38946	−0.0541258	−0.0270629	−	0.999634i	$-0.508615\pi$
$659$	−1.38946	−0.0541258	−0.0270629	+	0.999634i	$0.508615\pi$
$660$	0	0
$661$	10.2420	0.398369	0.199185	−	0.979962i	$-0.436171\pi$
$661$	10.2420	0.398369	0.199185	+	0.979962i	$0.436171\pi$
$662$	−10.9094	−0.424006
$663$	0	0
$664$	32.7544	1.27112
$665$	−20.3137	−0.787730
$666$	0	0
$667$	1.73378	0.0671321
$668$	25.8095	0.998601
$669$	0	0
$670$	30.8972	1.19366
$671$	1.00000	0.0386046
$672$	0	0
$673$	−17.9716	−0.692752	−0.346376	−	0.938096i	$-0.612588\pi$
$673$	−17.9716	−0.692752	−0.346376	+	0.938096i	$0.612588\pi$
$674$	8.84559	0.340719
$675$	0	0
$676$	−41.8657	−1.61022
$677$	35.8373	1.37734	0.688669	−	0.725076i	$-0.258195\pi$
$677$	35.8373	1.37734	0.688669	+	0.725076i	$0.258195\pi$
$678$	0	0
$679$	−3.40554	−0.130693
$680$	50.1413	1.92283
$681$	0	0
$682$	5.67053	0.217136
$683$	−48.2546	−1.84641	−0.923205	−	0.384309i	$-0.874440\pi$
$683$	−48.2546	−1.84641	−0.923205	+	0.384309i	$0.874440\pi$
$684$	0	0
$685$	39.0825	1.49326
$686$	8.72535	0.333135
$687$	0	0
$688$	−1.95148	−0.0743996
$689$	−22.9276	−0.873473
$690$	0	0
$691$	43.3072	1.64748	0.823741	−	0.566966i	$-0.191883\pi$
$691$	43.3072	1.64748	0.823741	+	0.566966i	$0.191883\pi$
$692$	35.3720	1.34464
$693$	0	0
$694$	15.3108	0.581190
$695$	−31.5170	−1.19551
$696$	0	0
$697$	−46.2562	−1.75208
$698$	10.8061	0.409018
$699$	0	0
$700$	−9.82554	−0.371370
$701$	0.890920	0.0336496	0.0168248	−	0.999858i	$-0.494644\pi$
$701$	0.890920	0.0336496	0.0168248	+	0.999858i	$0.494644\pi$
$702$	0	0
$703$	23.6295	0.891202
$704$	0.429682	0.0161942
$705$	0	0
$706$	1.64001	0.0617228
$707$	0.0597344	0.00224654
$708$	0	0
$709$	6.15437	0.231132	0.115566	−	0.993300i	$-0.463132\pi$
$709$	6.15437	0.231132	0.115566	+	0.993300i	$0.463132\pi$
$710$	11.2149	0.420888
$711$	0	0
$712$	−20.6444	−0.773682
$713$	−11.3127	−0.423666
$714$	0	0
$715$	−20.8498	−0.779737
$716$	−27.0069	−1.00929
$717$	0	0
$718$	12.0242	0.448739
$719$	−13.6746	−0.509978	−0.254989	−	0.966944i	$-0.582072\pi$
$719$	−13.6746	−0.509978	−0.254989	+	0.966944i	$0.582072\pi$
$720$	0	0
$721$	18.8992	0.703843
$722$	−10.2483	−0.381401
$723$	0	0
$724$	−22.0460	−0.819332
$725$	8.01531	0.297681
$726$	0	0
$727$	−43.0844	−1.59791	−0.798956	−	0.601389i	$-0.794614\pi$
$727$	−43.0844	−1.59791	−0.798956	+	0.601389i	$0.794614\pi$
$728$	−15.1914	−0.563032
$729$	0	0
$730$	35.0004	1.29542
$731$	−7.72085	−0.285566
$732$	0	0
$733$	−52.7317	−1.94769	−0.973844	−	0.227216i	$-0.927038\pi$
$733$	−52.7317	−1.94769	−0.973844	+	0.227216i	$0.927038\pi$
$734$	−18.7175	−0.690875
$735$	0	0
$736$	−7.41570	−0.273346
$737$	−14.3363	−0.528084
$738$	0	0
$739$	−2.12724	−0.0782518	−0.0391259	−	0.999234i	$-0.512457\pi$
$739$	−2.12724	−0.0782518	−0.0391259	+	0.999234i	$0.512457\pi$
$740$	20.9623	0.770591
$741$	0	0
$742$	−2.46236	−0.0903962
$743$	40.0136	1.46796	0.733978	−	0.679173i	$-0.237661\pi$
$743$	40.0136	1.46796	0.733978	+	0.679173i	$0.237661\pi$
$744$	0	0
$745$	35.2787	1.29251
$746$	−3.18071	−0.116454
$747$	0	0
$748$	−10.2591	−0.375110
$749$	−0.914284	−0.0334072
$750$	0	0
$751$	−34.1592	−1.24649	−0.623244	−	0.782028i	$-0.714186\pi$
$751$	−34.1592	−1.24649	−0.623244	+	0.782028i	$0.714186\pi$
$752$	−7.88794	−0.287644
$753$	0	0
$754$	5.46460	0.199009
$755$	11.6595	0.424332
$756$	0	0
$757$	9.15298	0.332671	0.166335	−	0.986069i	$-0.446807\pi$
$757$	9.15298	0.332671	0.166335	+	0.986069i	$0.446807\pi$
$758$	9.05604	0.328930
$759$	0	0
$760$	−45.4625	−1.64910
$761$	−1.83381	−0.0664755	−0.0332378	−	0.999447i	$-0.510582\pi$
$761$	−1.83381	−0.0664755	−0.0332378	+	0.999447i	$0.510582\pi$
$762$	0	0
$763$	−15.0134	−0.543521
$764$	−14.7520	−0.533707
$765$	0	0
$766$	5.53205	0.199881
$767$	−52.1135	−1.88171
$768$	0	0
$769$	17.6934	0.638039	0.319019	−	0.947748i	$-0.396646\pi$
$769$	17.6934	0.638039	0.319019	+	0.947748i	$0.396646\pi$
$770$	−2.23921	−0.0806954
$771$	0	0
$772$	−22.1576	−0.797470
$773$	8.76718	0.315334	0.157667	−	0.987492i	$-0.449603\pi$
$773$	8.76718	0.315334	0.157667	+	0.987492i	$0.449603\pi$
$774$	0	0
$775$	−52.2992	−1.87864
$776$	−7.62168	−0.273603
$777$	0	0
$778$	−6.09547	−0.218533
$779$	41.9399	1.50265
$780$	0	0
$781$	−5.20372	−0.186204
$782$	−5.48096	−0.195999
$783$	0	0
$784$	−9.73166	−0.347559
$785$	−62.6481	−2.23601
$786$	0	0
$787$	24.0259	0.856429	0.428215	−	0.903677i	$-0.359143\pi$
$787$	24.0259	0.856429	0.428215	+	0.903677i	$0.359143\pi$
$788$	16.7168	0.595510
$789$	0	0
$790$	−14.0273	−0.499068
$791$	−11.9676	−0.425519
$792$	0	0
$793$	−6.28797	−0.223292
$794$	13.8436	0.491290
$795$	0	0
$796$	−4.77426	−0.169219
$797$	53.8145	1.90621	0.953104	−	0.302644i	$-0.0978693\pi$
$797$	53.8145	1.90621	0.953104	+	0.302644i	$0.0978693\pi$
$798$	0	0
$799$	−31.2079	−1.10405
$800$	−34.2830	−1.21209
$801$	0	0
$802$	21.5595	0.761292
$803$	−16.2402	−0.573104
$804$	0	0
$805$	4.46724	0.157449
$806$	−35.6561	−1.25593
$807$	0	0
$808$	0.133687	0.00470310
$809$	30.4182	1.06945	0.534723	−	0.845028i	$-0.320416\pi$
$809$	30.4182	1.06945	0.534723	+	0.845028i	$0.320416\pi$
$810$	0	0
$811$	43.4739	1.52657	0.763287	−	0.646059i	$-0.223584\pi$
$811$	43.4739	1.52657	0.763287	+	0.646059i	$0.223584\pi$
$812$	−2.19154	−0.0769079
$813$	0	0
$814$	2.60471	0.0912952
$815$	67.1246	2.35127
$816$	0	0
$817$	7.00039	0.244913
$818$	5.02987	0.175865
$819$	0	0
$820$	37.2060	1.29929
$821$	1.41291	0.0493109	0.0246554	−	0.999696i	$-0.492151\pi$
$821$	1.41291	0.0493109	0.0246554	+	0.999696i	$0.492151\pi$
$822$	0	0
$823$	3.84780	0.134126	0.0670630	−	0.997749i	$-0.478637\pi$
$823$	3.84780	0.134126	0.0670630	+	0.997749i	$0.478637\pi$
$824$	42.2969	1.47348
$825$	0	0
$826$	−5.59684	−0.194739
$827$	−29.0483	−1.01011	−0.505054	−	0.863087i	$-0.668528\pi$
$827$	−29.0483	−1.01011	−0.505054	+	0.863087i	$0.668528\pi$
$828$	0	0
$829$	38.4992	1.33713	0.668566	−	0.743653i	$-0.266908\pi$
$829$	38.4992	1.33713	0.668566	+	0.743653i	$0.266908\pi$
$830$	−30.3582	−1.05375
$831$	0	0
$832$	−2.70182	−0.0936689
$833$	−38.5023	−1.33403
$834$	0	0
$835$	−54.2488	−1.87736
$836$	9.30179	0.321709
$837$	0	0
$838$	12.1181	0.418612
$839$	51.0036	1.76084	0.880420	−	0.474196i	$-0.157261\pi$
$839$	51.0036	1.76084	0.880420	+	0.474196i	$0.157261\pi$
$840$	0	0
$841$	−27.2122	−0.938353
$842$	18.0645	0.622545
$843$	0	0
$844$	1.22365	0.0421199
$845$	87.9969	3.02719
$846$	0	0
$847$	1.03899	0.0357001
$848$	5.99346	0.205816
$849$	0	0
$850$	−25.3387	−0.869110
$851$	−5.19643	−0.178131
$852$	0	0
$853$	−27.0403	−0.925842	−0.462921	−	0.886399i	$-0.653199\pi$
$853$	−27.0403	−0.925842	−0.462921	+	0.886399i	$0.653199\pi$
$854$	−0.675310	−0.0231086
$855$	0	0
$856$	−2.04619	−0.0699374
$857$	50.6974	1.73179	0.865895	−	0.500225i	$-0.166749\pi$
$857$	50.6974	1.73179	0.865895	+	0.500225i	$0.166749\pi$
$858$	0	0
$859$	36.4380	1.24325	0.621625	−	0.783315i	$-0.286473\pi$
$859$	36.4380	1.24325	0.621625	+	0.783315i	$0.286473\pi$
$860$	6.21024	0.211767
$861$	0	0
$862$	19.3594	0.659383
$863$	2.40141	0.0817449	0.0408724	−	0.999164i	$-0.486986\pi$
$863$	2.40141	0.0817449	0.0408724	+	0.999164i	$0.486986\pi$
$864$	0	0
$865$	−74.3480	−2.52791
$866$	12.7969	0.434856
$867$	0	0
$868$	14.2996	0.485360
$869$	6.50865	0.220791
$870$	0	0
$871$	90.1462	3.05449
$872$	−33.6003	−1.13785
$873$	0	0
$874$	4.96952	0.168096
$875$	3.42666	0.115842
$876$	0	0
$877$	−0.646214	−0.0218211	−0.0109105	−	0.999940i	$-0.503473\pi$
$877$	−0.646214	−0.0218211	−0.0109105	+	0.999940i	$0.503473\pi$
$878$	−1.31025	−0.0442186
$879$	0	0
$880$	5.45028	0.183729
$881$	12.8910	0.434309	0.217154	−	0.976137i	$-0.430323\pi$
$881$	12.8910	0.434309	0.217154	+	0.976137i	$0.430323\pi$
$882$	0	0
$883$	−20.1858	−0.679305	−0.339653	−	0.940551i	$-0.610310\pi$
$883$	−20.1858	−0.679305	−0.339653	+	0.940551i	$0.610310\pi$
$884$	64.5089	2.16967
$885$	0	0
$886$	22.8728	0.768427
$887$	52.0114	1.74637	0.873187	−	0.487386i	$-0.162049\pi$
$887$	52.0114	1.74637	0.873187	+	0.487386i	$0.162049\pi$
$888$	0	0
$889$	−15.9910	−0.536322
$890$	19.1341	0.641377
$891$	0	0
$892$	3.10035	0.103807
$893$	28.2957	0.946881
$894$	0	0
$895$	56.7654	1.89746
$896$	11.5937	0.387318
$897$	0	0
$898$	20.2767	0.676644
$899$	−11.6651	−0.389053
$900$	0	0
$901$	23.7125	0.789978
$902$	4.62310	0.153932
$903$	0	0
$904$	−26.7838	−0.890815
$905$	46.3382	1.54033
$906$	0	0
$907$	16.6601	0.553189	0.276595	−	0.960987i	$-0.410794\pi$
$907$	16.6601	0.553189	0.276595	+	0.960987i	$0.410794\pi$
$908$	−19.9041	−0.660542
$909$	0	0
$910$	14.0801	0.466749
$911$	−0.0930752	−0.00308372	−0.00154186	−	0.999999i	$-0.500491\pi$
$911$	−0.0930752	−0.00308372	−0.00154186	+	0.999999i	$0.500491\pi$
$912$	0	0
$913$	14.0862	0.466185
$914$	4.97850	0.164674
$915$	0	0
$916$	−7.17440	−0.237049
$917$	−11.7009	−0.386396
$918$	0	0
$919$	20.9935	0.692512	0.346256	−	0.938140i	$-0.387453\pi$
$919$	20.9935	0.692512	0.346256	+	0.938140i	$0.387453\pi$
$920$	9.99778	0.329617
$921$	0	0
$922$	6.12970	0.201871
$923$	32.7208	1.07702
$924$	0	0
$925$	−24.0233	−0.789880
$926$	−9.60051	−0.315492
$927$	0	0
$928$	−7.64667	−0.251014
$929$	18.8811	0.619470	0.309735	−	0.950823i	$-0.399760\pi$
$929$	18.8811	0.619470	0.309735	+	0.950823i	$0.399760\pi$
$930$	0	0
$931$	34.9095	1.14411
$932$	−32.7231	−1.07188
$933$	0	0
$934$	−2.61739	−0.0856438
$935$	21.5635	0.705202
$936$	0	0
$937$	5.36028	0.175113	0.0875563	−	0.996160i	$-0.472094\pi$
$937$	5.36028	0.175113	0.0875563	+	0.996160i	$0.472094\pi$
$938$	9.68145	0.316111
$939$	0	0
$940$	25.1019	0.818734
$941$	−20.0174	−0.652550	−0.326275	−	0.945275i	$-0.605794\pi$
$941$	−20.0174	−0.652550	−0.326275	+	0.945275i	$0.605794\pi$
$942$	0	0
$943$	−9.22312	−0.300346
$944$	13.6229	0.443386
$945$	0	0
$946$	0.771664	0.0250890
$947$	24.4734	0.795280	0.397640	−	0.917542i	$-0.369829\pi$
$947$	24.4734	0.795280	0.397640	+	0.917542i	$0.369829\pi$
$948$	0	0
$949$	102.118	3.31488
$950$	22.9742	0.745383
$951$	0	0
$952$	15.7115	0.509212
$953$	13.3783	0.433365	0.216682	−	0.976242i	$-0.430476\pi$
$953$	13.3783	0.433365	0.216682	+	0.976242i	$0.430476\pi$
$954$	0	0
$955$	31.0070	1.00336
$956$	−46.6577	−1.50902
$957$	0	0
$958$	5.15151	0.166438
$959$	12.2462	0.395452
$960$	0	0
$961$	45.1138	1.45528
$962$	−16.3784	−0.528059
$963$	0	0
$964$	−17.3972	−0.560325
$965$	46.5729	1.49923
$966$	0	0
$967$	21.1259	0.679363	0.339681	−	0.940541i	$-0.389681\pi$
$967$	21.1259	0.679363	0.339681	+	0.940541i	$0.389681\pi$
$968$	2.32529	0.0747375
$969$	0	0
$970$	7.06410	0.226815
$971$	21.6575	0.695021	0.347511	−	0.937676i	$-0.387027\pi$
$971$	21.6575	0.695021	0.347511	+	0.937676i	$0.387027\pi$
$972$	0	0
$973$	−9.87565	−0.316599
$974$	−5.55093	−0.177863
$975$	0	0
$976$	1.64372	0.0526143
$977$	−42.7909	−1.36900	−0.684501	−	0.729012i	$-0.739980\pi$
$977$	−42.7909	−1.36900	−0.684501	+	0.729012i	$0.739980\pi$
$978$	0	0
$979$	−8.87822	−0.283749
$980$	30.9692	0.989274
$981$	0	0
$982$	−10.4421	−0.333221
$983$	41.4675	1.32261	0.661304	−	0.750118i	$-0.270003\pi$
$983$	41.4675	1.32261	0.661304	+	0.750118i	$0.270003\pi$
$984$	0	0
$985$	−35.1368	−1.11955
$986$	−5.65167	−0.179986
$987$	0	0
$988$	−58.4894	−1.86079
$989$	−1.53948	−0.0489525
$990$	0	0
$991$	−7.94757	−0.252463	−0.126231	−	0.992001i	$-0.540288\pi$
$991$	−7.94757	−0.252463	−0.126231	+	0.992001i	$0.540288\pi$
$992$	49.8939	1.58413
$993$	0	0
$994$	3.51412	0.111461
$995$	10.0350	0.318130
$996$	0	0
$997$	−24.0237	−0.760840	−0.380420	−	0.924814i	$-0.624221\pi$
$997$	−24.0237	−0.760840	−0.380420	+	0.924814i	$0.624221\pi$
$998$	−5.60208	−0.177331
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.12	✓	25
3.2	odd	2		6039.2.a.p.1.14	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.12	✓	25	1.1	even	1	trivial
6039.2.a.p.1.14	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)

\(f(q)\)	\(=\)	\(q-0.649968 q^{2} -1.57754 q^{4} +3.31582 q^{5} +1.03899 q^{7} +2.32529 q^{8} +O(q^{10})\)	\(q-0.649968 q^{2} -1.57754 q^{4} +3.31582 q^{5} +1.03899 q^{7} +2.32529 q^{8} -2.15517 q^{10} +1.00000 q^{11} -6.28797 q^{13} -0.675310 q^{14} +1.64372 q^{16} +6.50322 q^{17} -5.89638 q^{19} -5.23084 q^{20} -0.649968 q^{22} +1.29669 q^{23} +5.99465 q^{25} +4.08698 q^{26} -1.63905 q^{28} +1.33708 q^{29} -8.72432 q^{31} -5.71894 q^{32} -4.22689 q^{34} +3.44510 q^{35} -4.00745 q^{37} +3.83246 q^{38} +7.71023 q^{40} -7.11281 q^{41} -1.18723 q^{43} -1.57754 q^{44} -0.842807 q^{46} -4.79883 q^{47} -5.92050 q^{49} -3.89633 q^{50} +9.91953 q^{52} +3.64627 q^{53} +3.31582 q^{55} +2.41595 q^{56} -0.869057 q^{58} +8.28781 q^{59} +1.00000 q^{61} +5.67053 q^{62} +0.429682 q^{64} -20.8498 q^{65} -14.3363 q^{67} -10.2591 q^{68} -2.23921 q^{70} -5.20372 q^{71} -16.2402 q^{73} +2.60471 q^{74} +9.30179 q^{76} +1.03899 q^{77} +6.50865 q^{79} +5.45028 q^{80} +4.62310 q^{82} +14.0862 q^{83} +21.5635 q^{85} +0.771664 q^{86} +2.32529 q^{88} -8.87822 q^{89} -6.53314 q^{91} -2.04558 q^{92} +3.11908 q^{94} -19.5513 q^{95} -3.27774 q^{97} +3.84813 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 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

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